JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS VOLUME 27, 2019

Table of contents :
BLOCK-2019-V27-1
FACE-2019-V27-1
JCAAA-2019-V27-front-1
BACK-JOCAAA-2019
SCOPE--JOCAAA--2019
EB--JOCAAA--2019
Instructions--JOCAAA--2019
Binder-27-1
172-2019-Dong Yun Shin-jocaaa-8-4-2017
173-2019-REV-ELAIW-JOCAAA-12-25-2017
174-2019-REV-ALSHAREEF-JOCAAA-12-26-2017
175-2019-FNL-YIGIT-EFE-JOCAAA-12-27-2017
177-2019-Zhengping Zhang-Gaowen Xi-jocaaa-8-11-2017
178-2019-REV-MURSALEEN-12-19-2017
179-2019-Feilong Cao-jocaaa-8-11-2017
Introduction
Preliminaries
Strictly positive definite kernel
The hybrid interpolation
Native space and Sobolev space
Pointwise error estimates
181-2019-REV-Taher Hassan-JOCAAA-1-31-2018
1. Introduction
2. Stability analysis of the equilibrium points
3. Analytical expressions of ( xn) n-1
4. Main Results
4.1. The forbidden set
4.2. Convergence
4.3. Oscillation about the equilibrium point x1=0
4.4. Periodicity
5. Numerical simulation
to 0ptsection*.1namesection.5XYZ-8.39996pt plus -12.0pt8.39996pt plus 12.0ptAcknowledgement
to 0ptsection*.2namesection.5XYZ-8.39996pt plus -12.0pt8.39996pt plus 12.0ptReferences
182-2019-Feilong Cao-JOCAAA-8-12-2017
Introduction
Preliminaries
Global error estimates for Lp norm
Inf-sup condition and improved global error estimates
Hybrid interpolation for rough native space
183-2019-Mansour-BADR-JOCAAA-8-14-2017
186-2019-REV-Jung Yoog Kang-JOCAAA-12-20-2017
187-2019-Ding-Zhang-Cao-JOCAAA--8-17-2017
Introduction
Preliminaries
Spherical harmonics, sphere function spaces, and sphere point sets
Laplace-Beltrami operator
Moving least squares
Regularized moving least squares with Laplace-Beltrami operator
Error estimates
Numerical experiments
189-2019-Almohammadi-JOCAAA-8-19-2017
190-2019-Jongkyum Kwon-JOCAAA-8-21-2017
BLANK-JoCAAA-2019-1
BLANK-JoCAAA-2019-2
BLOCK-2019-V27-2
FACE-2019-V27-2
JCAAA-2019-V27-front-2
BACK-JOCAAA-2019
SCOPE--JOCAAA--2019
EB--JOCAAA--2019
Instructions--JOCAAA--2019
Binder-27-2
191-2019-REV-Lingling Lv-JOCAAA-12-21-2017
193-2019-Dong Yun Shin-JOCAAA-8-25-2017
194-2019-Nazeer-Munir-Naqvi-Jung-Kang-jocaaa-8-26-2017
195-2019-Hoewoon Kim-jocaaa-8-26-2017
196-2019-REV-Cui-Ahn-JOCAAA-12-25-2017
198-2019-Sungsik Yun-jocaaa-8-27-2017
200-2019-Nazim Mahmudov-JOCAAA-8-29-2017
201-2019-Shexiang Hai-JOCAAA-8-30-2017
202-2019-FNL-El-Dessoky-JOCAAA-1-29-2018
203-2019-Zhihua Zhang-JOCAAA-8-31-2017
204-2019-REV-Pattrawut Chansangiam-JOCAAA-12-26-2017
206-2019-WANG-ZHU-WU-JOCAAA-9-4-2017
Introduction
preliminaries
Coupled coincidence point results in partially ordered complete Menger probabilistic G-metric spaces
Coupled common fixed point results in partially ordered complete Menger probabilistic G-metric spaces
An example
208-2019-Jongkyum Kwon-jocaaa-9-5-2017
209-2019-RYOOCS-JOCAAA-9-4-2017
210-2019-Dong Yun Shin-PARK-ANASTASSIOU-JOCAAA-9-6-2017
BLANK-JoCAAA-2019-1
BLANK-JoCAAA-2019-2
BLANK-JoCAAA-2019-3
BLOCK-2019-V27-3
FACE-2019-V27-3
JCAAA-2019-V27-front-3
BACK-JOCAAA-2019
SCOPE--JOCAAA--2019
EB--JOCAAA--2019
Instructions--JOCAAA--2019
Binder-27-3
211-2019-Kangtunyakarn-jocaaa-9-7-2017
212-2019-Kyung Soo Kim-JOCAAA-9-8-2017
213-2019-Sungsik Yun-PARK-ANASTASSIOU-jocaaa-9-11-2017
214-2019-Zhong-Qi Xiang-jocaaa-9-11-2017
1. Introduction
2. Preliminaries
3. Main results and their proofs
Acknowledgements
References
215-2019-Jung Rye Lee-park-anastassiou-jocaaa-9-12-2017
216-2019-anastassiou-park-jocaaa-9-13-2017
218-2019-Hwan-Yong Shin -JOCAAA-9-14-2017
219-2019-Won-Gil Park-Jae-Hyeong Bae-JOCAAA-9-15-2017
221-2019-kulenovic-jocaaa-9-18-2017
222-2019-Qing-Bo Cai -JOCAAA-9-19-2017
223-2019-alinaLUPAS-JOCAAA-9-20-2017
224-2019-alinaLUPAS-JOCAAA-9-21-2017
225-2019-Dawei Meng-JOCAAA-9-23-2017
226-2019-Afrah Abdou-JOCAAA-9-24-2017
Introduction and Preliminaries
Main result
Application to metric space
Application to system of integral equations
228-2019-ChangIl Kim-jocaaa-9-24-2017
BLANK-JoCAAA-2019-1
BLANK-JoCAAA-2019-2
BLANK-JoCAAA-2019-3
BLOCK-2019-V27-4
FACE-2019-V27-4
JCAAA-2019-V27-front-4
BACK-JOCAAA-2019
SCOPE--JOCAAA--2019
EB--JOCAAA--2019
Instructions--JOCAAA--2019
Binder-27-4
230-2019-OBIEDAT-JOCAAA-9-26-2017
231-2019-REV-Kamal Abodayeh-JOCAAA-1-14-2018
233-2019-REV-Mi Zhou-JOCAAA-12-26-2017
234-2019-REV-Tongxing LI-JOCAAA-1-6-2018
235-2019-XU-JOCAAA--9-29-2017
236-2019-REV- Asim Asiri-JOCAAA-12-27-2017
237-2019-REV-YING-HE-JOCAAA-1-2-2018
239-2019-REV-Jin Tu-JOCAAA-12-21-2017
240-2019-REV-Tingsong Du-JOCAAA-12-26-2017
Introduction
New definitions and a lemma
Main results
Applications to special means
241-2019-REV-Chaojun Yang-JOCAAA-2-1-2018
242-2019-Muawya Elsheikh Hamid-jocaaa-10-8-2017
243-2019-REV-Chatthai Thaiprayoon-JOCAAA-12-27-2017
244-2019-MansourYassen-JOCAAA-10-10-2017
BLANK-JoCAAA-2019-1
BLANK-JoCAAA-2019-2
BLOCK-2019-V27-5
FACE-2019-V27-5
JCAAA-2019-V27-front-5
BACK-JOCAAA-2019
SCOPE--JOCAAA--2019
EB--JOCAAA--2019
Instructions--JOCAAA--2019
Binder-27-5
246-2019-Jung Rye Lee-JOCAAA-10-15-2017
247-2019-REV-Mohiuddine-JOCAAA-1-8-2018
249-2019-REV-Ozkan Ocalan-JOCAAA-12-26-2017
251-2019-REV-ZALIK-JOCAAA-12-24-2017
550-2019-REV-Ko-Ahn-JOCAAA-1-18-2018
601-2019-fnl-TAMER-NABIL-JOCAAA-2-22-2018
602-2019-RYOOCS-JOCAAA-10-20-2017
604-2019-Hongwei Zhang-JOCAAA-10-24-2017
606-2019-Cheng-fu Yang-JOCAAA-10-31-2017
609-2019-Gafel-JOCAAA-11-6-2017
611-2019-Pattrawut Chansangiam-JOCAAA-1-6-2018
612-2019-Keum Sook So-JOCAAA-11-10-2017
1. Introduction and Preliminaries
2. Fibonacci frequency
3. Radical functions
4. Powers of primes
5. Comments
6. Appendix
References
614-2019-ZENGTAIGONG-WENJINLEI-JOCAAA-11-15-2017
618-2019-REV-Lingling Lv-JOCAAA-2-1-2018
619-2019-FNL-Nak Eun Cho-JOCAAA-1-25-2019
1. Introduction
2. Main results
Acknowledgements
References
620-2019-Choonkil Park -JOCAAA-11-24-2017
621-2019-FNL-Kang-Qadri-Nazeer-Haq-JOCAAA-2-20-2018
BLANK-JoCAAA-2019-1
BLANK-JoCAAA-2019-2
BLOCK-2019-V27-6
FACE-2019-V27-6
JCAAA-2019-V27-front-6
BACK-JOCAAA-2019
SCOPE--JOCAAA--2019
EB--JOCAAA--2019
Instructions--JOCAAA--2019
Binder-27-6
623-2019-FNL-Changyou WANG-JOCAAA-2-2-2018
626-2019-REV-Kang-Farid-Nazeer-Naqvi-JOCAAA-2-1-2018
627-2019-Haidong Zhang-jocaaa-12-5-2017
628-2019-Giljun Han-JOCAAA-12-5-2017
629-2019-SHU-LIAO-JOCAAA-12-6-2017
638-2019-Khalil Salem Al-Basyouni-JOCAAA-12-19-2017
641-2019-Kyung Soo Kim-JOCAAA-12-21-2017
642-2019-REV-Kang-Haq-Nazeer-Ahmed-Ahmad-JOCAAA-2-1-2018
643-2019-REV-Kang-Haq-Nazeer-Ahmad-JOCAAA-2-1-2018
645-2019-Mensah Folly-Gbetoula-JOCAAA-12-28-2017
BLANK-JoCAAA-2019-1
BLANK-JoCAAA-2019-2
BLANK-JoCAAA-2019-3
BLOCK-2019-V27-7
FACE-2019-V27-7
JCAAA-2019-V27-front-7
BACK-JOCAAA-2019
SCOPE--JOCAAA--2019
EB--JOCAAA--2019
Instructions--JOCAAA--2019
Binder-27-7
646-2019-Youqing Shen-jocaaa-12-29-2017
Preliminaries
Common fixed point theorems in Gb-metric space
An example
647-2019-Guang Zeng-JOCAAA-12-30-2017
648-2019-Park_Cho_Kumar_JOCAAA-1-5-2018
1. Introduction
2. Coefficient Estimates
Acknowledgement
References
649-2019-Ming Zhao-JOCAAA-1-2-2018
650-2019- Manli Liu-JOCAAA-1-11-2018
651-2019-REV-Muawya Elsheikh Hamid-JOCAAA-2-6-2018
652-2019 -REV-Pattrawut Chansangiam-JOCAAA-1-23-2018
653-2019-dessoky elabbasy asim-JOCAAA-1-16-2018
700-2019-ANASTASSIOU-JOCAAA-3-28-2018
BLANK-JoCAAA-2019-1
BLANK-JoCAAA-2019-2

Citation preview

Volume 27, Number 1 ISSN:1521-1398 PRINT,1572-9206 ONLINE

Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC

July 15, 2019

Journal of Computational Analysis and Applications ISSNno.’s:1521-1398 PRINT,1572-9206 ONLINE SCOPE OF THE JOURNAL An international publication of Eudoxus Press, LLC (fifteen times annually) Editor in Chief: George Anastassiou Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152-3240, U.S.A [email protected] http://www.msci.memphis.edu/~ganastss/jocaaa The main purpose of "J.Computational Analysis and Applications" is to publish high quality research articles from all subareas of Computational Mathematical Analysis and its many potential applications and connections to other areas of Mathematical Sciences. Any paper whose approach and proofs are computational,using methods from Mathematical Analysis in the broadest sense is suitable and welcome for consideration in our journal, except from Applied Numerical Analysis articles. Also plain word articles without formulas and proofs are excluded. The list of possibly connected mathematical areas with this publication includes, but is not restricted to: Applied Analysis, Applied Functional Analysis, Approximation Theory, Asymptotic Analysis, Difference Equations, Differential Equations, Partial Differential Equations, Fourier Analysis, Fractals, Fuzzy Sets, Harmonic Analysis, Inequalities, Integral Equations, Measure Theory, Moment Theory, Neural Networks, Numerical Functional Analysis, Potential Theory, Probability Theory, Real and Complex Analysis, Signal Analysis, Special Functions, Splines, Stochastic Analysis, Stochastic Processes, Summability, Tomography, Wavelets, any combination of the above, e.t.c. "J.Computational Analysis and Applications" is a peer-reviewed Journal. See the instructions for preparation and submission of articles to JoCAAA. Assistant to the Editor: Dr.Razvan Mezei,[email protected], Madison,WI,USA.

Journal of Computational Analysis and Applications(JoCAAA) is published by EUDOXUS PRESS,LLC,1424 Beaver Trail Drive,Cordova,TN38016,USA,[email protected] http://www.eudoxuspress.com. Annual Subscription Prices:For USA and Canada,Institutional:Print $800, Electronic OPEN ACCESS. Individual:Print $400. For any other part of the world add $160 more(handling and postages) to the above prices for Print. No credit card payments. Copyright©2019 by Eudoxus Press,LLC,all rights reserved.JoCAAA is printed in USA. JoCAAA is reviewed and abstracted by AMS Mathematical Reviews,MATHSCI,and Zentralblaat MATH. It is strictly prohibited the reproduction and transmission of any part of JoCAAA and in any form and by any means without the written permission of the publisher.It is only allowed to educators to Xerox articles for educational purposes.The publisher assumes no responsibility for the content of published papers.

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Editorial Board Associate Editors of Journal of Computational Analysis and Applications Francesco Altomare Dipartimento di Matematica Universita' di Bari Via E.Orabona, 4 70125 Bari, ITALY Tel+39-080-5442690 office +39-080-3944046 home +39-080-5963612 Fax [email protected] Approximation Theory, Functional Analysis, Semigroups and Partial Differential Equations, Positive Operators.

Fractional Differential Equations Nonlinear Analysis, Fractional Dynamics Carlo Bardaro Dipartimento di Matematica e Informatica Universita di Perugia Via Vanvitelli 1 06123 Perugia, ITALY TEL+390755853822 +390755855034 FAX+390755855024 E-mail [email protected] Web site: http://www.unipg.it/~bardaro/ Functional Analysis and Approximation Theory, Signal Analysis, Measure Theory, Real Analysis.

Ravi P. Agarwal Department of Mathematics Texas A&M University - Kingsville 700 University Blvd. Kingsville, TX 78363-8202 tel: 361-593-2600 [email protected] Differential Equations, Difference Equations, Inequalities

Martin Bohner Department of Mathematics and Statistics, Missouri S&T Rolla, MO 65409-0020, USA [email protected] web.mst.edu/~bohner Difference equations, differential equations, dynamic equations on time scale, applications in economics, finance, biology.

George A. Anastassiou Department of Mathematical Sciences The University of Memphis Memphis, TN 38152,U.S.A Tel.901-678-3144 e-mail: [email protected] Approximation Theory, Real Analysis, Wavelets, Neural Networks, Probability, Inequalities.

Jerry L. Bona Department of Mathematics The University of Illinois at Chicago 851 S. Morgan St. CS 249 Chicago, IL 60601 e-mail:[email protected] Partial Differential Equations, Fluid Dynamics

J. Marshall Ash Department of Mathematics De Paul University 2219 North Kenmore Ave. Chicago, IL 60614-3504 773-325-4216 e-mail: [email protected] Real and Harmonic Analysis

Luis A. Caffarelli Department of Mathematics The University of Texas at Austin Austin, Texas 78712-1082 512-471-3160 e-mail: [email protected] Partial Differential Equations George Cybenko Thayer School of Engineering

Dumitru Baleanu Department of Mathematics and Computer Sciences, Cankaya University, Faculty of Art and Sciences, 06530 Balgat, Ankara, Turkey, [email protected]

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Dartmouth College 8000 Cummings Hall, Hanover, NH 03755-8000 603-646-3843 (X 3546 Secr.) e-mail:[email protected] Approximation Theory and Neural Networks

011-49-203-379-3542 e-mail: [email protected] Approximation Theory, Computer Aided Geometric Design John R. Graef Department of Mathematics University of Tennessee at Chattanooga Chattanooga, TN 37304 USA [email protected] Ordinary and functional differential equations, difference equations, impulsive systems, differential inclusions, dynamic equations on time scales, control theory and their applications

Sever S. Dragomir School of Computer Science and Mathematics, Victoria University, PO Box 14428, Melbourne City, MC 8001, AUSTRALIA Tel. +61 3 9688 4437 Fax +61 3 9688 4050 [email protected] Inequalities, Functional Analysis, Numerical Analysis, Approximations, Information Theory, Stochastics.

Weimin Han Department of Mathematics University of Iowa Iowa City, IA 52242-1419 319-335-0770 e-mail: [email protected] Numerical analysis, Finite element method, Numerical PDE, Variational inequalities, Computational mechanics

Oktay Duman TOBB University of Economics and Technology, Department of Mathematics, TR06530, Ankara, Turkey, [email protected] Classical Approximation Theory, Summability Theory, Statistical Convergence and its Applications

Tian-Xiao He Department of Mathematics and Computer Science P.O. Box 2900, Illinois Wesleyan University Bloomington, IL 61702-2900, USA Tel (309)556-3089 Fax (309)556-3864 [email protected] Approximations, Wavelet, Integration Theory, Numerical Analysis, Analytic Combinatorics

Saber N. Elaydi Department Of Mathematics Trinity University 715 Stadium Dr. San Antonio, TX 78212-7200 210-736-8246 e-mail: [email protected] Ordinary Differential Equations, Difference Equations

Margareta Heilmann Faculty of Mathematics and Natural Sciences, University of Wuppertal Gaußstraße 20 D-42119 Wuppertal, Germany, [email protected] Approximation Theory (Positive Linear Operators)

J .A. Goldstein Department of Mathematical Sciences The University of Memphis Memphis, TN 38152 901-678-3130 [email protected] Partial Differential Equations, Semigroups of Operators

Xing-Biao Hu Institute of Computational Mathematics AMSS, Chinese Academy of Sciences Beijing, 100190, CHINA [email protected]

H. H. Gonska Department of Mathematics University of Duisburg Duisburg, D-47048 Germany

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Computational Mathematics Ram N. Mohapatra Department of Mathematics University of Central Florida Orlando, FL 32816-1364 tel.407-823-5080 [email protected] Real and Complex Analysis, Approximation Th., Fourier Analysis, Fuzzy Sets and Systems

Jong Kyu Kim Department of Mathematics Kyungnam University Masan Kyungnam,631-701,Korea Tel 82-(55)-249-2211 Fax 82-(55)-243-8609 [email protected] Nonlinear Functional Analysis, Variational Inequalities, Nonlinear Ergodic Theory, ODE, PDE, Functional Equations.

Gaston M. N'Guerekata Department of Mathematics Morgan State University Baltimore, MD 21251, USA tel: 1-443-885-4373 Fax 1-443-885-8216 Gaston.N'[email protected] [email protected] Nonlinear Evolution Equations, Abstract Harmonic Analysis, Fractional Differential Equations, Almost Periodicity & Almost Automorphy

Robert Kozma Department of Mathematical Sciences The University of Memphis Memphis, TN 38152, USA [email protected] Neural Networks, Reproducing Kernel Hilbert Spaces, Neural Percolation Theory Mustafa Kulenovic Department of Mathematics University of Rhode Island Kingston, RI 02881,USA [email protected] Differential and Difference Equations

M.Zuhair Nashed Department Of Mathematics University of Central Florida PO Box 161364 Orlando, FL 32816-1364 e-mail: [email protected] Inverse and Ill-Posed problems, Numerical Functional Analysis, Integral Equations, Optimization, Signal Analysis

Irena Lasiecka Department of Mathematical Sciences University of Memphis Memphis, TN 38152 PDE, Control Theory, Functional Analysis, [email protected]

Mubenga N. Nkashama Department OF Mathematics University of Alabama at Birmingham Birmingham, AL 35294-1170 205-934-2154 e-mail: [email protected] Ordinary Differential Equations, Partial Differential Equations

Burkhard Lenze Fachbereich Informatik Fachhochschule Dortmund University of Applied Sciences Postfach 105018 D-44047 Dortmund, Germany e-mail: [email protected] Real Networks, Fourier Analysis, Approximation Theory

Vassilis Papanicolaou Department of Mathematics National Technical University of Athens Zografou campus, 157 80 Athens, Greece tel:: +30(210) 772 1722 Fax +30(210) 772 1775 [email protected] Partial Differential Equations, Probability

Hrushikesh N. Mhaskar Department Of Mathematics California State University Los Angeles, CA 90032 626-914-7002 e-mail: [email protected] Orthogonal Polynomials, Approximation Theory, Splines, Wavelets, Neural Networks

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Choonkil Park Department of Mathematics Hanyang University Seoul 133-791 S. Korea, [email protected] Functional Equations

Anfithea - Paleon Faliron GR-175 64 Athens, Greece [email protected] Numerical Analysis H. M. Srivastava Department of Mathematics and Statistics University of Victoria Victoria, British Columbia V8W 3R4 Canada tel.250-472-5313; office,250-4776960 home, fax 250-721-8962 [email protected] Real and Complex Analysis, Fractional Calculus and Appl., Integral Equations and Transforms, Higher Transcendental Functions and Appl.,q-Series and q-Polynomials, Analytic Number Th.

Svetlozar (Zari) Rachev, Professor of Finance, College of Business, and Director of Quantitative Finance Program, Department of Applied Mathematics & Statistics Stonybrook University 312 Harriman Hall, Stony Brook, NY 11794-3775 tel: +1-631-632-1998, [email protected] Alexander G. Ramm Mathematics Department Kansas State University Manhattan, KS 66506-2602 e-mail: [email protected] Inverse and Ill-posed Problems, Scattering Theory, Operator Theory, Theoretical Numerical Analysis, Wave Propagation, Signal Processing and Tomography

I. P. Stavroulakis Department of Mathematics University of Ioannina 451-10 Ioannina, Greece [email protected] Differential Equations Phone +3-065-109-8283

Tomasz Rychlik Polish Academy of Sciences Instytut Matematyczny PAN 00-956 Warszawa, skr. poczt. 21 ul. Śniadeckich 8 Poland [email protected] Mathematical Statistics, Probabilistic Inequalities

Manfred Tasche Department of Mathematics University of Rostock D-18051 Rostock, Germany [email protected] Numerical Fourier Analysis, Fourier Analysis, Harmonic Analysis, Signal Analysis, Spectral Methods, Wavelets, Splines, Approximation Theory

Boris Shekhtman Department of Mathematics University of South Florida Tampa, FL 33620, USA Tel 813-974-9710 [email protected] Approximation Theory, Banach spaces, Classical Analysis

Roberto Triggiani Department of Mathematical Sciences University of Memphis Memphis, TN 38152 PDE, Control Theory, Functional Analysis, [email protected]

T. E. Simos Department of Computer Science and Technology Faculty of Sciences and Technology University of Peloponnese GR-221 00 Tripolis, Greece Postal Address: 26 Menelaou St.

Juan J. Trujillo University of La Laguna Departamento de Analisis Matematico C/Astr.Fco.Sanchez s/n 38271. LaLaguna. Tenerife. SPAIN Tel/Fax 34-922-318209 [email protected]

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Fractional: Differential EquationsOperators-Fourier Transforms, Special functions, Approximations, and Applications

Ahmed I. Zayed Department of Mathematical Sciences DePaul University 2320 N. Kenmore Ave. Chicago, IL 60614-3250 773-325-7808 e-mail: [email protected] Shannon sampling theory, Harmonic analysis and wavelets, Special functions and orthogonal polynomials, Integral transforms

Ram Verma International Publications 1200 Dallas Drive #824 Denton, TX 76205, USA [email protected] Applied Nonlinear Analysis, Numerical Analysis, Variational Inequalities, Optimization Theory, Computational Mathematics, Operator Theory

Ding-Xuan Zhou Department Of Mathematics City University of Hong Kong 83 Tat Chee Avenue Kowloon, Hong Kong 852-2788 9708,Fax:852-2788 8561 e-mail: [email protected] Approximation Theory, Spline functions, Wavelets

Xiang Ming Yu Department of Mathematical Sciences Southwest Missouri State University Springfield, MO 65804-0094 417-836-5931 [email protected] Classical Approximation Theory, Wavelets

Xin-long Zhou Fachbereich Mathematik, Fachgebiet Informatik Gerhard-Mercator-Universitat Duisburg Lotharstr.65, D-47048 Duisburg, Germany e-mail:[email protected] Fourier Analysis, Computer-Aided Geometric Design, Computational Complexity, Multivariate Approximation Theory, Approximation and Interpolation Theory

Xiao-Jun Yang State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining and Technology, Xuzhou 221116, China Local Fractional Calculus and Applications, Fractional Calculus and Applications, General Fractional Calculus and Applications, Variable-order Calculus and Applications, Viscoelasticity and Computational methods for Mathematical [email protected]

Jessada Tariboon Department of Mathematics, King Mongkut's University of Technology N. Bangkok 1518 Pracharat 1 Rd., Wongsawang, Bangsue, Bangkok, Thailand 10800 [email protected], Time scales, Differential/Difference Equations, Fractional Differential Equations

Richard A. Zalik Department of Mathematics Auburn University Auburn University, AL 36849-5310 USA. Tel 334-844-6557 office 678-642-8703 home Fax 334-844-6555 [email protected] Approximation Theory, Chebychev Systems, Wavelet Theory

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Instructions to Contributors Journal of Computational Analysis and Applications An international publication of Eudoxus Press, LLC, of TN.

Editor in Chief: George Anastassiou Department of Mathematical Sciences University of Memphis Memphis, TN 38152-3240, U.S.A.

1. Manuscripts files in Latex and PDF and in English, should be submitted via email to the Editor-in-Chief: Prof.George A. Anastassiou Department of Mathematical Sciences The University of Memphis Memphis,TN 38152, USA. Tel. 901.678.3144 e-mail: [email protected] Authors may want to recommend an associate editor the most related to the submission to possibly handle it. Also authors may want to submit a list of six possible referees, to be used in case we cannot find related referees by ourselves.

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4. The paper starts with the title of the article, author's name(s) (no titles or degrees), author's affiliation(s) and e-mail addresses. The affiliation should comprise the department, institution (usually university or company), city, state (and/or nation) and mail code. The following items, 5 and 6, should be on page no. 1 of the paper. 5. An abstract is to be provided, preferably no longer than 150 words. 6. A list of 5 key words is to be provided directly below the abstract. Key words should express the precise content of the manuscript, as they are used for indexing purposes. The main body of the paper should begin on page no. 1, if possible. 7. All sections should be numbered with Arabic numerals (such as: 1. INTRODUCTION) . Subsections should be identified with section and subsection numbers (such as 6.1. Second-Value Subheading). If applicable, an independent single-number system (one for each category) should be used to label all theorems, lemmas, propositions, corollaries, definitions, remarks, examples, etc. The label (such as Lemma 7) should be typed with paragraph indentation, followed by a period and the lemma itself. 8. Mathematical notation must be typeset. Equations should be numbered consecutively with Arabic numerals in parentheses placed flush right, and should be thusly referred to in the text [such as Eqs.(2) and (5)]. The running title must be placed at the top of even numbered pages and the first author's name, et al., must be placed at the top of the odd numbed pages. 9. Illustrations (photographs, drawings, diagrams, and charts) are to be numbered in one consecutive series of Arabic numerals. The captions for illustrations should be typed double space. All illustrations, charts, tables, etc., must be embedded in the body of the manuscript in proper, final, print position. In particular, manuscript, source, and PDF file version must be at camera ready stage for publication or they cannot be considered. Tables are to be numbered (with Roman numerals) and referred to by number in the text. Center the title above the table, and type explanatory footnotes (indicated by superscript lowercase letters) below the table. 10. List references alphabetically at the end of the paper and number them consecutively. Each must be cited in the text by the appropriate Arabic numeral in square brackets on the baseline. References should include (in the following order): initials of first and middle name, last name of author(s) title of article,

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Book 2. G.G.Lorentz, (title of book in italics) Bernstein Polynomials (2nd ed.), Chelsea,New York,1986.

Contribution to a Book 3. M.K.Khan, Approximation properties of beta operators,in(title of book in italics) Progress in Approximation Theory (P.Nevai and A.Pinkus,eds.), Academic Press, New York,1991,pp.483-495.

11. All acknowledgements (including those for a grant and financial support) should occur in one paragraph that directly precedes the References section. 12. Footnotes should be avoided. When their use is absolutely necessary, footnotes should be numbered consecutively using Arabic numerals and should be typed at the bottom of the page to which they refer. Place a line above the footnote, so that it is set off from the text. Use the appropriate superscript numeral for citation in the text. 13. After each revision is made please again submit via email Latex and PDF files of the revised manuscript, including the final one. 14. Effective 1 Nov. 2009 for current journal page charges, contact the Editor in Chief. Upon acceptance of the paper an invoice will be sent to the contact author. The fee payment will be due one month from the invoice date. The article will proceed to publication only after the fee is paid. The charges are to be sent, by money order or certified check, in US dollars, payable to Eudoxus Press, LLC, to the address shown on the Eudoxus homepage. No galleys will be sent and the contact author will receive one (1) electronic copy of the journal issue in which the article appears.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.1, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

On common fixed point theorems of weakly compatible mappings in fuzzy metric spaces Afshan Batool1 , Tayyab Kamran2 , Dong Yun Shin3 and Choonkil Park4 1,2

Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan Department of Mathematics, University of Seoul, Seoul 02504, Korea 4 Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea [email protected], [email protected], [email protected], [email protected] 3

Abstract: The purpose of this paper is to obtain common fixed point theorem involving two pair of weakly compatible mappings in complete fuzzy metric spaces. Some related results and illustrative examples are also discussed. Keywords: common fixed point; weakly compatible mapping; complete fuzzy metric space; coincidence point; point of coincidence 2010 MSC: 47H10, 54E50, 54E40, 46S50.

1. Introduction and preliminaries Let (X, d) be a metric space. A mapping T : X → X is said to be contraction if there exists α ∈ (0, 1) such that for all x, y ∈ X, d(T x, T y) ≤ αd(x, y).

(1)

If the metric space (X, d) is complete, then the mapping satisfying (1) has a unique fixed point. Rhoades [11] assumed a weakly contractive mapping f : X → X which satisfies the condition d(f x, f y) ≤ d(x, y) − ϕ(d(x, y)), (2) where x, y ∈ X and ϕ : [0, ∞) → [0, ∞) is a continuous and nondecreasing function such that ϕ(t) = 0 if and only if t = 0. Rhoades [11] obtained the following extension. Theorem 1.1. ([11]) Let T : X → X be a weakly contractive mapping, where (X, d) is a complete metric space. Then T has a unique fixed point. Dutta and Choudhury [7] introduced a new generalization of contraction principle in the following theorem. Theorem 1.2. ([7]) Let (X, d) be a complete metric space and let T : X → X be a self-mapping satisfying the inequality ψ(d(f x, f y)) ≤ ψ(d(x, y)) − ϕ(d(x, y))

(3)

for all x, y ∈ X, where φ, ϕ : [0, ∞) → [0, ∞) are both continuous and monotone nondecreasing functions with ψ(t) = ϕ(t) = 0 if and only if t = 0. Then T has a unique fixed point. 0

Corresponding authors: [email protected] (Dong Yun Shin), [email protected] (Choonkil Park)

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Several researchers have studied the existence of fixed points and common fixed points of mappings (see [1, 2, 3, 4, 5, 6, 8, 9, 10, 12]). In this article, we give a fixed point theorem for contraction maps in complete fuzzy metric space, which improves and generalizes the above-mentioned result of Dutta and Choudhury. We recall some definitions before giving the main result of this article. Definition 1.3. A binary operation ∗ : [0, 1]2 → [0, 1] is called a continuous t-norm if ([0, 1], ∗) is an Abelian topological monoid, i.e., (1) ∗ is associative and commutative; (2) ∗ is continuous; (3) a ∗ 1 = a for all a ∈ [0, 1]; (4) a ∗ b ≤ c ∗ d whenever a ≤ c and b ≤ d for all a, b, c, d ∈ [0, 1]. Definition 1.4. A 3-tuple (X, M, ∗) is called a fuzzy metric space if X is an arbitrary set, ∗ is a continuous t-norm and M is a fuzzy set on X 2 ×(0, ∞) satisfying the following conditions: (1) M (x, y, t) > 0, (2) M (x, y, t) = 1 if and only if x = y, (3) M (x, y, t) = M (y, x, t), (4) M (x, y, t) ∗ M (y, z, s) ≤ M (x, z, t + s), (5) M (x, y, .) : (0, ∞) → [0, 1] is continuous, for all x, y, z ∈ X and t, s > 0. Definition 1.5. Let f and g be self-maps on a set X. If w = f x = gx for some x ∈ X, then x is called coincidence point of f and g, and w is called a point of coincidence of f and g. Definition 1.6. Let f and g be two self-maps on a set X. Then f and g are said to be weakly compatible if they commute at every coincidence point. 2. Main results Theorem 2.1. Let (X, M, t) be a complete fuzzy metric space, and let E be a nonempty closed subset of X. Let S, T : E → E and I, J : E → X be mappings satisfying T (E) ⊂ I(E) and S(E) ⊂ J(E) and for every x, y ∈ X, ψ(M (Sx, T y, t)) ≤ ψ(MI,J (x, y)) − ϕ(MI,J (x, y)),

(4)

where ψ : [0, ∞) → [0, ∞) is a continuous and nondecreasing function such that ψ(t) = 0 if and only if t = 0. ϕ : [0, ∞) → [0, ∞) is a lower semi-continuous function such that ϕ(t) = 0 if and only if t = 0, and n MI,J (x, y) = max M (Ix, Jy, t), M (Ix, Sx, t), M (Jy, T y, t), (5)  o 1 M (Ix, T y, t) + M (Jy, Sx, t) . 2

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If one of S(E), T (E), I(E), JE is a closed subset of X, then {S, I} and {T, J} have a unique point of coincidence in X. Moreover, if {S, I} and {T, J} are weakly compatible, then S, T, I and J have a unique common fixed point in X. Proof. Let x0 be an arbitrary point in X. Since T (E) ⊂ I(E) and S(E) ⊂ J(E), we can define the sequences {xn } and {yn } in X by y2n−1 = Sx2n−2 = Jx2n−1 , y2n = T x2n−1 = Ix2n ,

n = 1, 2, · · · .

Suppose that yn0 = yn0 +1 for some n0 . Then the sequence {yn } is constant for n ≥ n0 . Indeed, let n0 = 2k. Then y2k = y2k+1 and it follows from (4) that ψ(M (y2k+1 , y2k+2 , t)) = ψ(M (Sx2k , T x2k+1 , t)) ≤ ψ(MI,J (x2k , x2k+1 )) − ϕ(MI,J (x2k , x2k+1 )),

(6)

where MI,J (x2k , x2k+1 ) n = max M (y2k , y2k+1 , t), M (y2k , Sx2k , t), M (y2k+1 , T x2k+1 , t), o 1 M (y2k , T x2k+1 , t) + M (y2k+1 , Sx2k , t) 2 n o 1 = max 0, 0, M (y2k+1 , y2k+2 , t), M (y2k , y2k+2 , t) + 0 2 n o 1 = max M (y2k+1 , y2k+2 , t), M (y2k , y2k+2 , t) 2 = M (y2k+1 , y2k+2 , t). By (6), we get ψ(M (y2k+1 , y2k+2 , t)) ≤ ψ(M (y2k+1 , y2k+2 , t)) − ϕ(M (y2k+1 , y2k+2 , t)), and so ϕ(M (y2k+1 , y2k+2 , t)) ≤ 0 and y2k+1 = y2k+2 . Similarly, if n0 = 2k + 1, then one easily obtains that y2k+2 = y2k+3 and the sequence {yn } is constant (starting from some n0 ). Therefore, {S, I} and {T, J} have a point of coincidence in X. Now, suppose that M (yn , yn+1 , t) > 0 for each n. We shall show that for each n = 0, 1, · · · , M (yn+1 , yn+2 , t) ≤ MI,J (xn , xn+1 ) = M (yn , yn+1 , t). (7) Using (4), we obtain that ψ(M (y2n+1 , y2n+2 , t)) = ψ(M (Sx2n , T x2n+1 , t)) ≤ ψ(MI,J (x2n , x2n+1 )) − ϕ(MI,J (x2n , x2n+1 ))

(8)

< ψ(MI,J (x2n , x2n+1 )). On the other hand, the control function ψ is nondecreasing. Then M (y2n+1 , y2n+2 , t) ≤ MI,J (x2n , x2n+1 ).

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(9)

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Common fixed point theorems of weakly compatible mappings

Moreover, we have MI,J (x2n , x2n+1 ) n = max M (y2n , y2n+1 , t), M (y2n , Sx2n , t), M (y2n+1 , T x2n+1 , t), o 1 M (y2n , T x2n+1 , t) + M (y2n+1 , Sx2n , t) n2 = max M (y2n , y2n+1 , t), M (y2n , y2n+1 , t), o 1 M (y2n+1 , y2n+2 , t), M (y2n , y2n+2 , t) 2 n ≤ max M (y2n , y2n+1 , t), M (y2n+1 , y2n+2 , t), o 1 M (y2n , y2n+1 , t) + M (y2n+1 , y2n+2 ) n2 o ≤ max M (y2n , y2n+1 , t), M (y2n+1 , y2n+2 , t) . If M (y2n+1 , y2n+2 , t) ≥ M (y2n , y2n+1 , t), then by using the last inequality and (9), we have MI,J (x2n , x2n+1 ) = M (y2n+1 , y2n+2 , t) and (8) implies that ψ(M (y2n+1 , y2n+2 , t)) = ψ(M (Sx2n , T x2n+1 , t)) ≤ ψ(M (y2n+1 , y2n+2 , t)) − ϕ(M (y2n+1 , y2n+2 , t)), which is only possible when M (y2n+1 , y2n+2 , t) = 0. It is a contradiction. Hence M (y2n+1 , y2n+2 , t) ≤ M (y2n , y2n+1 , t) and MI,J (x2n , x2n+1 ) ≤ M (y2n , y2n+1 , t). By definition, MI,J (x2n , x2n+1 ) ≥ M (y2n , y2n+1 , t), and so (7) is proved for {M (y2n+1 , y2n+2 , t)}. In a similar way, one can obtain that M (y2n+3 , y2n+2 , t) ≤ MI,J (x2n+2 , x2n+1 ) = M (y2n+2 , y2n+1 , t). So (7) holds for each n ∈ N. It follows that the sequence {M (yn , yn+1 , t)} is nondecreasing and the limit lim M (yn , yn+1 , t) = lim MI,J (xn , xn+1 )

n→∞

n→∞

∗

exists. We denote this limit by d . We have d∗ ≥ 0. Suppose that d∗ > 0. Then ψ(M (yn+1 , yn+2 , t)) ≤ ψ(MI,J (xn , xn+1 )) − ϕ(MI,J (xn , xn+1 )). Passing to the (upper) limit when n → ∞, we get ψ(d∗ ) ≤ ψ(d∗ ) − lim inf ϕ(MI,J (xn , xn+1 )) ≤ ψ(d∗ ) − ϕ(d∗ ), n→∞

∗

i.e., ϕ(d ) ≤ 0. Using the properties of control functions, we get that d∗ = 0, which is a contradiction. Hence we have limn→∞ M (yn , yn+1 , t) = 0. Now we show that {yn } is a Cauchy sequence in X. It is enough to prove that {y2n } is a Cauchy sequence. Suppose the contrary. Then,

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for some  > 0, there exist subsequences {y2n(k) } and {y2m(k) } of {y2n } such that n(k) is the smallest index satisfying n(k) > m(k) and M (yn(k) , ym(k) , t) ≥ . In particular, M (yn(k)−2 , ym(k) , t) < . Using the triangle inequality and the known relation |d(x, z) − d(x, y)| ≤ d(x, z), we obtain that lim M (y2n(k) , y2m(k) , t) = lim M (y2n(k) , y2m(k)−1 , t) = lim M (y2n(k)+1 , y2m(k) , t)

k→∞

k→∞

k→∞

= lim M (y2n(k)+1 , y2m(k)−1 , t) = . k→∞

By the definition of M (x, y, t) and by using the previous limits, we get that lim MI,J (x2n(k) , x2m(k)−1 ) = .

k→∞

Indeed, MI,J (x2n(k) , x2m(k)−1 ) n = max M (y2n(k) , y2m(k)−1 , t), M (y2n(k) , y2n(k)+1 , t), M (y2m(k)−1 , y2m(k) , t), o 1 M (y2n(k) , y2m(k) , t) + M (y2n(k)+1 , y2m(k)−1 , t) 2 o n 1 → max , 0, 0, ( + ) = . 2 Applying (4), we obtain ψ(M (y2n(k)+1 , y2m(k) , t)) = ψ(M (Sx2n(k) , T x2m(k)−1 , t)) ≤ ψ(MI,J (x2n(k) , x2m(k)−1 )) − ϕ(MI,J (x2n(k) , x2m(k)−1 )). Passing to the limit k → ∞, we obtain that ψ() ≤ ψ()−ϕ(), which is a contradiction. Therefore, {yn } is a Cauchy sequence in the complete metric (X, d). So there exists u ∈ X such that limn→∞ yn = u. On the other hand, E is closed and {yn } ⊂ E. Then u ∈ E. Suppose that I(E) is closed. Then there exists v ∈ E such that u = Iv.

(10)

We claim that Sv = u. Using (4) and (10), we have ψ(M (Sv, y2n , t)) = ψ(M (Sv, T x2n−1 , t)) ≤ ψ(MI,J (v, x2n−1 ))−ϕ(MI,J (v, x2n−1 )), (11) where n MI,J (v, x2n−1 ) = max M (y2n−1 , u, t), M (u, Sv, t), M (y2n−1 , T x2n−1 , t), o 1 M (y2n−1 , Sv, t) + M (u, T x2n−1 , t) 2  1 → max 0, M (u, Sv, t), 0, M (u, Sv, t) = M (u, Sv, t). 2

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Passing to the limit when n → ∞ in (11), we get ψ(M (u, Sv, t)) ≤ ψ(M (u, Sv, t)) − ϕ(M (u, Sv, t)). It follows that u = Sv. Since u = Sv ∈ SE ⊂ JE, there exists w ∈ E such that

(12)

u = Jw.

(13)

We claim that T w = u. By (4), we get ψ(M (u, T w, t)) = ψ(M (Sv, T w, t)) ≤ ψ(MI,J (v, w)) − ϕ(MI,J (v, w)), where n MI,J (v, w) = max M (u, u, t), M (Iv, Sv, t), M (Jw, T w, t), o 1 M (Jw, Sv, t) + M (Iv, T w, t) 2 n o 1 = max 0, 0, M (u, T w, t), M (u, T w, t) = M (u, T w, t). 2 Hence (2) implies that ψ(M (u, T w, t)) ≤ ψ(M (u, T w, t)) − ϕ(M (u, T w, t)). It follows that u = T w.

(14)

Combining (10) and (12) yields u = Iv = Sv, that is, u is a point of coincidence of I and S. Combining (13) and (14) yields u = Jw = T w,

(15) (16)

that is, u is a point of coincidence of J and T . 0 To prove the uniqueness property of u, suppose that u is another point of coincidence of I and S, that is, 0 0 0 u = Iv = Sv 0 for some v ∈ E. By (4), we have 0

0

0

0

ψ(M (u , u, t)) = ψ(M (Sv , T w, t)) ≤ ψ(MI,J (v , w)) − ϕ(MI,J (v , w)), where o 1 0 0 MI,J (v , w) = max M (u , u, t), 0, 0, M (u , u, t) + M (u , u, t) 2 0 = M (u , u, t). n

0

0

0

It follows from (2) that u = u. Now, suppose that u is another point of coincidence of J and T , that is, 0

u = jw = T w

16

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for some w ∈ E. Using (4), we obtain 0

0

0

ψ(M (u, u, t)) = ψ(M (Sv, T w, t )) ≤ ψ(MI,J (v, w )) − ϕ(MI,J (v, w )), where n o 1 MI,J (v, w ) = max M (u, u, t), 0, 0, M (u, u, t) + M (u, u, t) 2 = M (u, u, t). 0

It follows from (2) that u = u. Therefore, u is the unique point of coincidence of {S, I} and {T, J}. Now, if {S, I} and {T, J} are weakly compatible, then by (15) and (16), we have Su = S(Iv) = I(Sv) = Iu = w1 and T u = T (Jw) = J(T w) = Ju = w2 . By (4), we have ψ(M (w1 , w2 , t)) = ψ(M (Su, T u, t)) ≤ ψ(MI,J (u, u)) − ϕ(MI,J (u, u)), where o 1 MI,J (u, u) = max M (w1 , w2 , t), 0, 0, M (w1 , w2 , t) + M (w1 , w2 , t) 2 = M (w1 , w2 , t). n

It follows that w1 = w2 , that is, Su = Iu = T u = Ju.

(17)

By (4) and (17), we have ψ(M (Sv, T u, t)) ≤ ψ(MI,J (v, u) − ϕ(MI,J (v, u)), where n MI,J (v, u) = max M (Iv, Ju, t), M (Iv, Sv, t), M (Ju, T u, t), o 1 M (Iv, T u, t) + M (Sv, T u, t) 2 o n 1 = max M (Sv, T u, t), 0, 0, M (Sv, T u, t) + M (Sv, T u, t) 2 = M (Sv, T u, t). Therefore, we deduce that Sv = T u, that is, u = T u. It follows from (17) that u = Su = Iu = T u = Ju. Then u is the unique common fixed point of S, I, J and T . The rest of the proof is similar to the above case and so the rest will be omitted.  Example 2.2. Let X = [0, 1] be equipped with the natural metric d(x, y) = |x − y|. Now for t ∈ [0, ∞) define  0 if t = 0 and x, y ∈ X M (x, y, t) = t if t 6= 0 and x, y ∈ X. t+|x−y|

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Clearly, (X, M, ∗) is a fuzzy metric on X, where ∗ is defined as a ∗ b = ab. This fuzzy metric space is complete. Let E = {0, 12 , 1} and we define T, S : E → E as T 0 = T 1 = 0 and T 21 = 1, Sx = 0. We also define I, J : E → X as I0 = I1 = 0 and I 21 = 1, J0 = J1 = 0 and J 21 = 1. The functions ψ : ϕ : [0, ∞) → [0, ∞) are defined as ψ(t) = t and ϕ(t) = 4t . Then ψ(M (Sx, T y, t)) ≤ ψ(MI,J (x, y)) − ϕ(MI,J (x, y)). References [1] I. Beg, M. Abbas, Coincidence point and invariant approximation for mappings satisfying generalized weak contractive condition, Fixed Point Theory Appl. 2006, Article ID 74503 (2006). [2] G.A. Anastassiou, I.K. Argyros, Approximating fixed points with applications in fractional calculus, J. Comput. Anal. Appl. 21 (2016), 1225–1242. [3] S. Banach, Sur les oprations dans les ensembles abstraits et leur application aux quations intgrales, Fund. Math. 3 (1922), 133–181. [4] A. Batool, T. Kamran, S. Jang, C. Park, Generalized ϕ-weak contractive fuzzy mappings and related fixed point results on complete metric space, J. Comput. Anal. Appl. 21 (2016), 729–737. [5] V. Berinde, Approximating fixed points of weak ϕ-contractions, Fixed Point Theory 4 (2003), 131–142. [6] C. E. Chidume, H. Zegeye, S. J. Aneke, Approximation of fixed points of weakly contractive nonself maps in Banach spaces, J. Math. Anal. Appl. 270 (2002), 189–199. [7] P. N. Dutta, B. S. Choudhury, A generalization of contraction principle in metric spaces, Fixed Point Theory Appl. 2008, Article ID 406368 (2008). [8] G. Jungck, B. E. Rhoades, Fixed points for set valued functions without continuity, Indian J. Pure Appl. Math. 29 (1998), 227–238. [9] J. H. Mai, X. H. Liu, Fixed points of weakly contractive maps and boundedness of orbits, Fixed Point Theory Appl. 2007, Article ID 20962 (2007). [10] S. Moradi, Z. Fathi, E. Analouee, The common fixed point of single-valued generalized ϕf -weakly contractive mappings, Appl. Math. Lett. 24 (2011), 771–776. [11] B.E. Rhoades, Some theorems on weakly contractive maps, Nonlinear Anal. 47 (2001), 2683–2693. [12] Q. Zhang, Y. Song, Y Fixed point theory for generalized ϕ-weak contractions, Appl. Math. Lett. 22 (2009), 75–78.

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Analysis of latent CHIKV dynamics model with time delays Ahmed. M. Elaiw, Taofeek O. Alade and Saud M. Alsulami Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia. Email: a m [email protected] (A. Elaiw) Abstract This paper proposes a latent Chikungunya viral infection model with saturated incidence rate. To take into account the time lag between the initial viral contacts uninfected monocytes and the production of new active CHIKV particles the model is incorporated by intracellular discrete or distributed time delays. We study the qualitative behavior of the model. Using the method of Lyapunov function, we established the global stability of the steady states of the model. The effect of the time delay on the stability of the steady states has also been shown by numerical simulations. Keywords: Chikungunya virus infection; Latency; Time delay; Global stability; Lyapunov function.

1

Introduction

Mathematical analysis of viral infection models plays a substantial role in understanding the dynamics of human viruses (such as HIV, HCV, HBV, HTLV and Chikungunya virus). The models have been developed to mainly describe the relation among virus particles, uninfected target cells and infected cells [1]-[15]. The effect of Cytotoxic T Lymphocytes (CTL) immune response or humoral immune response has also been modeled (see e.g. [10]-[15]. Two main classes of mathematical models of viral infection have been proposed in the literature. The first class of models are given by ordinary differential equations. The second class of models is given by delay differential equations which incorporate the time lag between the initial viral contacts a target cell and the production of new active viruses. Modeling the virus dynamics with two types of infected cells, latently infected cells and actively infected cells has been studied by several researchers (see e.g. [2] and [14]). The latent viral infection model has been formulated as [2]: ˙ S(t) = µ − aS(t) − bV (t)S(t), ˙ L(t) = (1 − ρ)bV (t)S(t) − (θ + λ)L(t),

(1)

˙ = ρbV (t)S(t) + λL(t) − I(t), I(t) V˙ (t) = mI(t) − rV (t),

(3)

(2)

(4)

where, S, L, I and V are the concentrations of uninfected cells, latently infected cells, actively infected cells and free virus particles. Parameters a and µ represent the death rate and birth rate constants of the uninfected cells, respectively. The uninfected cells become infected at rate bSV , where b is a constant. The parameters θ,  and r denote the death rate constants of the latently infected cells, actively infected cells and free virus particles, respectively. An actively infected cell produces an average number m of virus particles. The parameter λ is the latent to active transmission rate constant. A fraction (1 − ρ) of infected cells is assumed to be latently infected cells and the remaining ρ becomes actively infected cells, where 0 < ρ < 1.

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Chikungunya virus (CHIKV) is an alphavirus and is transmitted to humans by Aedes aegypti and Aedes albopictus mosquitos. In the CHIKV literature, most of the mathematical models have been presented to describe the disease transmission in mosquito and human populations (see e.g. [17]-[22]). However, only few works have devoted for mathematical modeling of the dynamics of the CHIKV within host. In 2017, Wang and Liu [16] have presented a mathematical model for in host CHIKV infection model without considering the latent infection. The objective of this paper is to propose a CHIKV infection model which improves the model presented in [16] by taking into account (i) two types of infected monocytes, latently infected monocytes and actively infected monocytes, (ii) two types of discrete or distributed time delays (iii) saturated incidence rate which is suitable to model the nonlinear dynamics of the CHIKV especially when its concentration is high. We investigate the nonnegativity and boundedness of the solutions of the CHIKV dynamics model. We show that the CHIKV dynamics is governed by one bifurcation parameter (the basic reproduction numbers R0 ). We use Lyapunov direct method to establish the global stability of the model’s equilibria.

2

CHIKV model with discrete time delays

We consider a within-host CHIKV dynamics model with latently infected monocytes taking into account two discrete time delays. bV (t)S(t) ˙ S(t) = µ − aS(t) − , 1 + πV (t) (1 − ρ)e−δ1 τ1 bV (t − τ1 )S(t − τ1 ) ˙ L(t) = − (θ + λ)L(t), 1 + πV (t − τ1 ) −δ2 τ2 bV (t − τ2 )S(t − τ2 ) ˙ = ρe I(t) + λL(t) − I(t), 1 + πV (t − τ2 ) V˙ (t) = mI(t) − rV (t) − qB(t)V (t),

(5) (6) (7) (8)

˙ B(t) = η + cB(t)V (t) − δB(t),

(9)

where, S, L, I, V , and B are the concentrations of uninfected monocytes, latently infected monocytes, actively infected monocytes CHIKV particles and B cells, respectively. The CHIKV particles are attacked by the B cells at rate qV B. The B cells are produced at constant rate η, proliferated at rate cBV and die at rate δB. τ1 denotes the time between the CHIKV contacts the uninfected monocytes and latent infection, while τ2 denotes the time between monocytes infection and the production of active CHIKV particles. The probability of latently and actively infected monocytes surviving to the age of τ1 and τ2 are represented by e−δ1 τ1 and e−δ2 τ2 , respectively, where δ1 and δ2 are. We consider the following initial conditions: S(ϑ) = ϕ1 (ϑ), L(ϑ) = ϕ2 (ϑ), I(ϑ) = ϕ3 (ϑ), V (ϑ) = ϕ4 (ϑ), B(ϑ) = ϕ5 (ϑ), ϕi (ϑ) ≥ 0, ϑ ∈ [−τ, 0] and ϕi ∈ C ([−τ, 0] , R≥0 ) , i = 1, 2, ..., 5,

(10)

where τ = max {τ1 , τ2 } and C is the Banach space of continuous functions mapping the interval [−τ, 0] into R≥0 with norm kϕj k = sup |ϕj (ϑ)| . Then the uniqueness of the solution for t > 0 is guaranteed [23]. −τ ≤ϑ≤0

2.1

Preliminaries

In this subsection we show the nonnegativity and boundedness of solutions as well as the existence of the steady states of system (5)-(9).

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Lemma 1. The solutions of system (5)-(9) with the initial states (10) are nonnegative and ultimately bounded. = η > 0. Thus, S(t) > 0 and B(t) > 0 = µ > 0 and B˙ Proof. From Eqs. (5) and (9) we have S˙ B=0 S=0 for all t ≥ 0. Moreover, for t ∈ [0, τ ] we have  Z t (1 − ρ)e−δ1 τ1 bS(ω − τ1 )V (ω − τ1 ) −(θ+λ)(t−ω) −(θ+λ)t L(t) = ϕ2 (0)e + e dω ≥ 0, 1 + πV (ω − τ1 ) 0  Z t  −δ2 τ2 ρe bS(ω − τ2 )V (ω − τ2 ) I(t) = ϕ3 (0)e−t + + λL(ω) e−(t−ω) dω ≥ 0, 1 + πV (ω − τ2 ) 0 Rt Rt Z t − (c+qB(u))u − (c+qB(u))du dω ≥ 0. mI(ω)e ω + V (t) = ϕ4 (0)e 0 0

By recursive argument, we get L(t) ≥ 0, I(t) ≥ 0 and V (t) ≥ 0 for all t ≥ 0. Next, we establish the boundedness of the model’s solutions. The nonnegativity of the model’s solution µ implies that dS(t) dt ≤ µ − aS(t), which yields lim sup S(t) ≤ a . Let us define t→∞

X1 (t) = (1 − ρ)e−δ1 τ1 S(t − τ1 ) + L(t), then X˙ 1 (t) = (1 − ρ)e−δ1 τ1

 bV (t − τ1 )S(t − τ1 ) bV (t − τ1 )S(t − τ1 ) + (1 − ρ)e−δ1 τ1 − (θ + λ)L(t) µ − aS(t − τ1 ) − 1 + πV (t − τ1 ) 1 + πV (t − τ1 )
− σ1 (1 − ρ)e−δ1 τ1 S(t − τ1 ) + L(t) ≤ µ(1 − ρ) − σ1 X1 (t),



≤ µ(1 − ρ)e−δ1 τ1

where σ1 = min{a, θ + λ}. Then, lim sup X1 (t) ≤ M1 , and lim sup L(t) ≤ M1 , where M1 = t→∞

t→∞

X2 (t) = ρe−δ2 τ2 S(t − τ2 ) + I(t) +

µ(1−ρ) σ1 .

Let

 q V (t) + B(t), 2m 2mc

then we get  bV (t − τ2 )S(t − τ2 ) bV (t − τ2 )S(t − τ2 ) + ρe−δ2 τ2 + λL(t) − I(t) 1 + πV (t − τ2 ) 1 + πV (t − τ2 )  q + (mI(t) − rV (t) − qV (t)B(t)) + (η + cB(t)V (t) − δB(t)) 2m 2mc  qη r qδ = ρµe−δ2 τ2 − ρe−δ2 τ2 aS(t − τ2 ) + λL(t) − I(t) + − V (t) − B(t) 2 2mc 2m 2mc   qη  q ≤ ρµ + λM1 + − σ2 ρe−δ2 τ2 S(t − τ2 ) + I(t) + V (t) + B(t) 2mc 2m 2mc qη = ρµ + λM1 + − σ2 X2 (t), 2mc

X˙ 2 (t) = ρe−δ2 τ2



µ − aS(t − τ2 ) −

where σ2 = min{a, 2 , r, δ}. It follows that lim sup I(t) ≤ M2 , lim sup V (t) ≤ M3 and lim sup B(t) ≤ t→∞

t→∞

t→∞

qη 2mc 1 M4 , where M2 = ρµ+λM + 2mcσ , M3 = 2m σ2  and M4 = q . This shows the ultimate boundedness of 2 S(t), L(t), I(t), V (t) and B(t).  Lemma 2. For system (5)-(9) there exists a threshold parameter R0 > 0, such that (i) if R0 ≤ 1, then there exists only one positive steady state, virus-free steady state Q0 . (i) if R0 > 1, then in addition to Q0 , there exists an endemic steady state Q1 Proof.

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To calculate the steady states we let the R.H.S of system (5)-(9) be equal zero bV S , 1 + πV e−δ1 τ1 bV S − (θ + λ)L, 0 = (1 − ρ) 1 + πV ρe−δ2 τ2 bV S 0= + λL − I, 1 + πV 0 = mI − rV − qV B, 0 = µ − aS −

(11) (12) (13) (14)

0 = η + cBV − δB.

(15)

From Eqs. (11)-(15) we obtain S=

(1 − ρ)e−δ1 τ1 bV S bβV S η µ (1 + πV ) , L= , I= , B= . bV + a (1 + πV ) (1 + πV ) (θ + λ)  (1 + πV ) (θ + λ) δ − cV

where β = λ(1 − ρ)e−δ1 τ1 + ρe−δ2 τ2 (θ + λ). Substituting Eq. (16) into Eq. (14) we have   mµbβ qη −r− V = 0.  (θ + λ) (bV + a (1 + πV )) δ − cV Equation (17) has two possibilities: (i) V = 0 which gives the virus-free steady state Q0 = (S0 , L0 , I0 , V0 , B0 ) = ( µa , 0, 0, 0, ηδ ), (ii) V 6= 0 which gives mµbβ qη −r− = 0.  (θ + λ) (bV + a (1 + πV )) δ − cV

(16)

(17)

(18)

Equation (18) takes the form P1 V 2 − P2 V + P3 = 0, where P1 = rc(θ + λ)(b + πa), P2 = −rca(θ + λ) + mµbc(β) +  (rδ + qη) (θ + λ)(b + πa), P3 = mρbµδ(θ + λ)e−δ2 τ2 + mµbλδ(1 − ρ)e−δ1 τ1 − a (rδ + qη) (θ + λ). The constants P1 , P2 and P3 can be rewritten as P1 = rc(θ + λ)(b + πa), caqη(θ + λ) ca (rδ + qη) (θ + λ) (R0 − 1) +  (rδ + qη) (θ + λ)(b + πa) + , δ δ P3 = a (rδ + qη) (θ + λ)(R0 − 1),

P2 =

where R0 =

mµbδβ . a(rδ + qη)(θ + λ)

Let Θ1 (V ) = P1 V 2 − P2 V + P3 = 0.

(19)

If R0 > 1, then P2 > 0 and P3 > 0. We have Θ1 (0) = P3 > 0, Θ1 c = − qη(θ+λ)(ca+δ(b+πa)) < 0, and c 0 Θ1 (0) = −P2 < 0. Then, Eq. (19) has two positive roots p p P2 + P22 − 4P1 P3 P2 − P22 − 4P1 P3 δ δ V1 = < and V2 = > . 2P1 c 2P1 c
δ

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If V = V2 , then from Eq. (16) we get B2 = Q1 = (S1 , L1 , I1 , V1 , B1 ) will appear, where

η δ−cV2

< 0. Thus, when R0 > 1, a positive endemic steady state

µ (1 + πV1 ) bµV1 (1 − ρ)e−δ1 τ1 bµβV1 , L1 = , I1 = , bV1 + a (1 + πV1 ) (θ + λ)(bV1 + a (1 + πV1 )) (θ + λ)(bV1 + a (1 + πV1 )) p ρ2 − ρ22 − 4ρ1 ρ3 η V1 = , B1 = . 2ρ1 δ − cV1

S1 =

The parameter R0 represents the basic reproduction number. 

2.2

Global stability

We define H(x) = x − ln x − 1. Clearly, H(1) = 0 and H(u) ≥ 0 for u > 0. Denote (S, L, I, V, B) = (S(t), L(t), I(t), V (t), B(t)). Theorem 1. Suppose that R0 ≤ 1, then Q0 is globally asymptotically stable (GAS). Proof. We define a Lyapunov functional Y0 as:       β S  q B λ Y0 (S, L, I, V, B) = S0 H L+I + V + B0 H + θ+λ S0 θ+λ m mc B0 Z Z τ2 bV (t − ϑ)S(t − ϑ) λ(1 − ρ)e−δ1 τ1 τ1 bV (t − ϑ)S(t − ϑ) dϑ + ρe−δ2 τ2 dϑ. (20) + θ+λ 1 + πV (t − ϑ) 1 + πV (t − ϑ) 0 0 dY0 Note that, Y0 (S, L, I, V, B) > 0 for all S, L, I, V, B > 0 and Y0 (S0 , 0, 0, 0, B0 ) = 0. Calculating along the dt trajectories of (5)-(9) we get    β S0 bV S dY0 = 1− µ − aS − dt θ+λ S 1 + πV   −δ1 τ1 (1 − ρ)e bV (t − τ1 )S(t − τ1 ) ρe−δ2 τ2 bV (t − τ2 )S(t − τ2 ) λ − (θ + λ)L + + θ+λ 1 + πV (t − τ1 ) 1 + πV (t − τ2 )    q B0 + λL − I + (mI − rV − qV B) + 1− (η + cBV − δB) m mc B     λ(1 − ρ)e−δ1 τ1 bV S bV (t − τ1 )S(t − τ1 ) bV S bV (t − τ2 )S(t − τ2 ) −δ2 τ2 + − + ρe − θ+λ 1 + πV 1 + πV (t − τ1 ) 1 + πV 1 + πV (t − τ2 )   2 aβ (S − S0 ) β bS0 V rV qB0 V q B0 =− + − − + 1− (δB0 − δB) θ+λ S θ + λ 1 + πV m m mc B   aβ (S − S0 )2 qδ (B − B0 )2 (rδ + qη) mµbδβ =− − + −1 V θ+λ S mc B mδ a(rδ + qη)(θ + λ)(1 + πV ) aβ (S − S0 )2 qδ (B − B0 )2 (rδ + qη) (rδ + qη)R0 πV 2 =− − + (R0 − 1)V − . (21) θ+λ S mc B mδ mδ(1 + πV ) dY0 dY0 Therefore, ≤ 0 holds if R0 ≤ 1. Further, = 0 if and only if S = S0 , B = B0 and V = 0. By LaSalle’s dt dt invariance principle, Q0 is GAS.  In the next theorem we show the global stability of Q1 . Theorem 2. Suppose that R0 > 1, then Q1 is GAS.

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Proof. Consider         λ L I  V q B L1 H + I1 H + V1 H + B1 H θ+λ L1 I1 m V1 mc B1  Z τ1  −δ1 τ1 V (t − ϑ)S(t − ϑ)(1 + πV1 ) λ(1 − ρ)e bS1 V1 H + dϑ θ+λ 1 + πV1 0 S1 V1 (1 + πV (t − ϑ))  Z τ2  bS1 V1 V (t − ϑ)S(t − ϑ)(1 + πV1 ) H + ρe−δ2 τ2 dϑ. 1 + πV1 0 S1 V1 (1 + πV (t − ϑ))

Y1 (S, L, I, V, B) =

β S1 H θ+λ



S S1



+

dY1 We have Y1 (S, L, I, V, B) > 0 for all S, L, I, V, B > 0 and Y1 (S1 , L1 , I1 , V1 , B1 ) = 0. Calculating along the dt trajectories of (5)-(9) we get    dY1 β S1 bV S = 1− µ − aS − dt θ+λ S 1 + πV    −δ1 τ1 λ L1 (1 − ρ)e bV (t − τ1 )S(t − τ1 ) + 1− − (θ + λ)L θ+λ L 1 + πV (t − τ1 )    −δ2 τ2    I1 ρe bV (t − τ2 )S(t − τ2 )  V1 + 1− + λL − I + 1− (mI − rV − qV B) I 1 + πV (t − τ2 ) m V     B1 λ(1 − ρ)e−δ1 τ1 bV S bV (t − τ1 )S(t − τ1 ) q 1− (η + cBV − δB) + − + mc B θ+λ 1 + πV 1 + πV (t − τ1 )     λ(1 − ρ)e−δ1 τ1 bS1 V1 bV (t − τ2 )S(t − τ2 ) V (t − τ1 )S(t − τ1 )(1 + πV ) bV S −δ2 τ2 + + ρe − ln θ+λ 1 + πV1 V S (1 + πV (t − τ1 )) 1 + πV 1 + πV (t − τ2 )   bS V V (t − τ )S(t − τ )(1 + πV ) 1 1 2 2 + ρe−δ2 τ2 ln . (22) 1 + πV1 V S (1 + πV (t − τ2 )) Applying µ = aS1 +

bS1 V1 , η = δB1 − cB1 V1 , 1 + πV1

we obtain β dY1 = dt θ+λ

    S1 β bS1 V1 S1 β bS1 V 1− (aS1 − aS) + 1− + S θ + λ 1 + πV1 S θ + λ 1 + πV

λ(1 − ρ)e−δ1 τ1 bV (t − τ1 )S(t − τ1 )L1 bV (t − τ2 )S(t − τ2 )I1 λLI1 IV1 + λL1 − ρe−δ2 τ2 − + I1 − θ+λ 1 + πV (t − τ1 )L 1 + πV (t − τ2 )I I V     rV1 qBV1 q B1 qB1 V qB1 V1 qB1 V1 B1 rV + + + 1− (δB1 − δB) − − + − m m m mc B m m m B      bS1 V1 λ(1 − ρ)e−δ1 τ1 V (t − τ1 )S(t − τ1 )(1 + πV ) V (t − τ )S(t − τ2 )(1 + πV ) 2 + ln + ρe−δ2 τ2 ln . 1 + πV1 θ+λ V S (1 + πV (t − τ1 )) V S (1 + πV (t − τ2 )) −

Using the equilibrium conditions for Q1 : (1 − ρ)e−δ1 τ1

bS1 V1 bS1 V1 = (θ + λ)L1 , ρe−δ1 τ1 + λL1 = I1 , mI1 = rV1 + qB1 V1 , 1 + πV1 1 + πV1

we get I1 =

β bS1 V1 , θ + λ (1 + πV1 )

rV1 β bS1 V1 qB1 V1 = − , m θ + λ (1 + πV1 ) m

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and   dY1 aβ (S − S1 )2 λ(1 − ρ)e−δ1 τ1 bS1 V1 S1 =− + 1− dt θ+λ S θ+λ (1 + πV1 ) S       bS1 V1 S1 β bS1 V1 (1 + πV1 )V V + ρe−δ2 τ2 1− + − 1 + πV1 S θ + λ 1 + πV1 (1 + πV )V1 V1 λ(1 − ρ)e−δ1 τ1 bS1 V1 V (t − τ1 )S(t − τ1 )(1 + πV1 )L1 λ(1 − ρ)e−δ1 τ1 bS1 V1 + θ+λ 1 + πV1 (1 + πV (t − τ1 ))S1 V1 L θ+λ (1 + πV1 ) −δ1 τ1 bS1 V1 V (t − τ2 )S(t − τ2 )(1 + πV1 )I1 λ(1 − ρ)e bS1 V1 I1 L − ρe−δ2 τ2 − 1 + πV1 (1 + πV (t − τ2 ))S1 V1 I θ+λ 1 + πV1 L1 I λ(1 − ρ) −δ1 τ1 bS1 V1 bS1 V1 λ(1 − ρ)e−δ1 τ1 bS1 V1 IV1 + e + ρe−δ2 τ2 − θ+λ (1 + πV1 ) (1 + πV1 ) θ+λ 1 + πV1 I1 V −δ1 τ1 bS1 V1 IV1 λ(1 − ρ)e bS1 V1 bS1 V1 − ρe−δ2 τ2 + + ρe−δ2 τ2 1 + πV1 I1 V θ+λ (1 + πV1 ) (1 + πV1 )   qBV1 qB1 V1 B1 qδ (B − B1 )2 2qB1 V1 + + − − m m m B mc B      bS1 V1 λ(1 − ρ)e−δ1 τ1 V (t − τ1 )S(t − τ1 )(1 + πV ) V (t − τ2 )S(t − τ2 )(1 + πV ) −δ2 τ2 + ln + ρe ln . 1 + πV1 θ+λ V S (1 + πV (t − τ1 )) V S (1 + πV (t − τ2 )) −

Using the following equalities:           V (t − τ1 )S(t − τ1 )(1 + πV ) S1 IV1 LI1 1 + πV ln = ln + ln + ln + ln V S (1 + πV (t − τ1 )) S I1 V L1 I 1 + πV1   V (t − τ1 )S(t − τ1 )(1 + πV1 )L1 + ln , (1 + πV (t − τ1 ))S1 V1 L         V (t − τ2 )S(t − τ2 )(1 + πV ) S1 IV1 1 + πV ln = ln + ln + ln V S (1 + πV (t − τ2 )) S I1 V 1 + πV1   V (t − τ2 )S(t − τ2 )(1 + πV1 )I1 + ln , (1 + πV (t − τ2 ))S1 V1 I we get   dY1 aβ (S − S1 )2 β bS1 V1 (1 + πV1 )V V 1 + πV =− + −1 + − + dt θ+λ S θ + λ 1 + πV1 (1 + πV )V1 V1 1 + πV1       −δ1 τ1 −δ1 τ1 λ(1 − ρ)e bS1 V1 S1 S1 λ(1 − ρ)e bS1 V1 IV1 IV1 + 1− + ln + 1− + ln θ+λ (1 + πV1 ) S S θ+λ (1 + πV1 ) I1 V I1 V    −δ1 τ1 λ(1 − ρ)e bS1 V1 V (t − τ1 )S(t − τ1 )(1 + πV1 )L1 V (t − τ1 )S(t − τ1 )(1 + πV1 )L1 + 1− + ln θ+λ (1 + πV1 ) (1 + πV (t − τ1 ))S1 V1 L (1 + πV (t − τ1 ))S1 V1 L    −δ1 τ1 λ(1 − ρ)e bS1 V1 1 + πV 1 + πV + 1− + ln θ+λ (1 + πV1 ) 1 + πV1 1 + πV1    λ(1 − ρ)e−δ1 τ1 bS1 V1 LI1 LI1 + 1− + ln θ+λ (1 + πV1 ) L1 I L1 I       bS V S S IV1 IV1 1 1 1 1 −δ2 τ2 −δ2 τ2 bS1 V1 + ρe 1− + ln + ρe 1− + ln 1 + πV1 S S (1 + πV1 ) I1 V I1 V    bS V V (t − τ )S(t − τ V (t − τ )S(t − τ )(1 + πV )I 1 1 2 2 1 1 2 2 )(1 + πV1 )I1 −δ2 τ2 + ρe 1− + ln 1 + πV1 (1 + πV (t − τ2 ))S1 V1 I (1 + πV (t − τ2 ))S1 V1 I      1 + πV 1 + πV qδ (B − B1 )2 qB1 V1 B B1 −δ2 τ2 bS1 V1 1− − − 2− + ρe + ln − 1 + πV1 1 + πV1 1 + πV1 mc B m B1 B

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   aβ(S − S1 )2 λ(1 − ρ)e−δ1 τ1 bS1 V1 qη (B − B1 )2 β πbS1 (V − V1 )2 S1 − − − H (θ + λ)S mcB1 B θ + λ (1 + πV )(1 + πV1 )2 θ+λ (1 + πV1 ) S         IV1 1 + πV V (t − τ1 )S(t − τ1 )(1 + πV1 )L1 LI1 +H +H +H +H I1 V 1 + πV1 (1 + πV (t − τ1 ))S1 V1 L L1 I          bS1 V1 S1 IV1 1 + πV V (t − τ2 )S(t − τ2 )(1 + πV1 )I1 − ρe−δ2 τ2 H +H +H +H . (23) (1 + πV1 ) S I1 V 1 + πV1 (1 + πV (t − τ2 ))S1 V1 I =−

dY1 dY1 It can be seen that if R0 > 1, then ≤ 0 for all S, L, I, V, B > 0 and = 0 if and only if S = S1 , dt dt L = L1 , I = I1 , V = V1 , and B = B1 . It follows from LaSalle’s invariance principle that, Q1 is GAS. 

3

CHIKV model with delay-distributed

We suggest a dynamical model for within-host CHIKV infection with latently infected monocytes taking into account the distributed delays. bV (t)S(t) ˙ S(t) = µ − aS(t) − , 1 + πV (t) Z κ1 V (t − τ )S(t − τ ) ˙ L(t) = (1 − ρ)b dτ − (θ + λ)L(t), ξ1 (τ )e−δ1 τ 1 + πV (t − τ ) Z κ2 0 V (t − τ )S(t − τ ) ˙ = ρb I(t) ξ2 (τ )e−δ2 τ dτ + λL(t) − I(t), 1 + πV (t − τ ) 0 V˙ (t) = mI(t) − rV (t) − qV (t)B(t), ˙ B(t) = η + cB(t)V (t) − δB(t).

0

(25) (26) (27) (28)

where, ξ1 (τ ) and ξ2 (τ ) are probability distribution functions which satisfy ξ1 (τ ) > 0 and ξ2 (τ ) > 0, and Z κ1 Z κ2 Z κ1 Z κ2 nu ξ1 (τ )dτ = ξ2 (τ )dτ = 1, ξ1 (u)e du < ∞, ξ2 (u)enu du < ∞, 0

(24)

0

(29)

0

where n is a positive number. Let Z E=

κ1

ξ1 (τ )e−δ1 τ dτ and K =

0

Z

κ2

ξ2 (τ )e−δ2 τ dτ

0

Then 0 < E ≤ 1, 0 < K ≤ 1. The initial conditions for model (24)-(28) take the form S(ϕ) = ψ1 (ϕ), L(ϕ) = ψ2 (ϕ), I(ϕ) = ψ3 (ϕ), V (ϕ) = ψ4 (ϕ), B(ϕ) = ψ5 (ϕ), ψj (ϕ) ≥ 0, ϕ ∈ [−`, 0], j = 1, ..., 5,

(30)

where ` = max{κ1 , κ2 }, ψj ∈ C([−`, 0], R≥0 ). This guarantees the uniqueness of solution of the system [23].

3.1

Preliminaries

Lemma 3. The solutions of system (24)-(28) with the initial states (30) are nonnegative and ultimately bounded.

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Proof. From Lemma 1 we have S(t) > 0 and B(t) > 0 for all t ≥ 0. Moreover, one can show that for t ≥ 0 Z κ1 S(u − τ )V (u − τ ) e−δ1 τ ξ1 (τ ) dτ du ≥ 0, e−(θ+λ)(t−u) 1 + πV (u − τ ) 0 0 Z κ2  Z t S(u − τ )V (u − τ ) e−δ2 τ ξ2 (τ ) e−(t−u) I(t) = e−t ψ3 (0) + ρb dτ + λL(u) du ≥ 0, 1 + πV (u − τ ) 0 0 Rt Rt Z t − (c+qB(u))u − (c+qB(u))du dω ≥ 0. V (t) = e 0 ψ4 (0) + mI(ω)e ω L(t) = e−(θ+λ)t ψ2 (0) + (1 − ρ)b

Z

t

0

From (24), we have lim sup S(t) ≤ t→∞

µ a.

Let T1 (t) = (1 − ρ)

κ1

R κ1 0

ξ1 (τ )e−δ1 τ S(t − τ )dτ + L(t), then

  bV (t − τ )S(t − τ ) ξ1 (τ )e µ − aS(t − τ ) − dτ 1 + πV (t − τ ) 0 Z κ1 V (t − τ )S(t − τ ) dτ − (θ + λ)L(t) ξ1 (τ )e−δ1 τ + (1 − ρ)b 1 + πV (t − τ ) 0   Z κ1 −δ1 τ ≤ µ(1 − ρ)E − σ1 (1 − ρ) ξ1 (τ )e S(t − τ )dτ + L(t)

T˙1 (t) = (1 − ρ)

Z

−δ1 τ

0

≤ µ(1 − ρ) − σ1 T1 (t). It follows that, lim sup T1 (t) ≤ M1 . Since t→∞

Z T2 (t) = ρ

κ2

R κ1 0

ξ1 (τ )e−δ1 τ S(t − τ )dτ > 0, then lim sup L(t) ≤ M1 . Let t→∞

ξ2 (τ )e−δ2 τ S(t − τ )dτ + I(t) +

0

q  V (t) + B(t), 2m 2mc

then we have κ2



 bV (t − τ )S(t − τ ) ξ2 (τ )e µ − aS(t − τ ) − dτ 1 + πV (t − τ ) 0 Z κ2 V (t − τ )S(t − τ ) dτ + λL(t) − I(t) + ρb ξ2 (τ )e−δ2 τ 1 + πV (t − τ ) 0  q + (mI(t) − rV (t) − qV (t)B(t)) + (η + cB(t)V (t) − δB(t)) 2m 2mc  Z κ2   q −δ2 τ ≤ µρK + λL1 − σ2 ρ ξ2 (τ )e S(t − τ )dτ + I(t) + V (t) + B(t) 2m 2mc 0

T˙2 (t) = ρ

Z

−δ2 τ

≤ µρ + λL1 − σ2 T2 (t). Then lim sup T2 (t) ≤ M2 . It follows that lim sup I(t) ≤ M2 , lim sup V (t) ≤ M3 and lim sup B(t) ≤ M4 . t→∞

t→∞

t→∞

t→∞

Therefore S(t), L(t), I(t), V (t), and B(t) are ultimately bounded.  Lemma 4. For system (24)-(28) there exists a threshold parameter RD 0 > 0, such that D (i) if R0 ≤ 1, then there exists only one positive steady state, virus-free steady state Q0 . (i) if RD 0 > 1, then in addition to Q0 , there exists an endemic steady state Q1 Proof. Similar to the proof of Lemma 2 we can show that if RD 0 ≤ 1 then there exists Q0 = (S0 , 0, 0, 0, B0 ), µ η D where S0 = a and B0 = δ , and if R0 > 1 then there exists Q1 = (S1 , L1 , I1 , V1 , B1 ), with µ (1 + πV1 ) E(1 − ρ)bµV1 bµV1 γ , L1 = , I1 = , bV1 + a (1 + πV1 ) (θ + λ)(bV1 + a (1 + πV1 )) (θ + λ)(bV1 + a (1 + πV1 )) q
2 P2D − P2D − 4P1D P3D δ η V1 = < , B1 = , c δ − cV1 2P1D

S1 =

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where γ = Eλ(1 − ρ) + Kρ(θ + λ),

P1D = rc(θ + λ)(b + πa),

ca (rδ + qη) (θ + λ) D caqη(θ + λ) (R0 − 1) +  (rδ + qη) (θ + λ)(b + πa) + , δ δ = a (rδ + qη) (θ + λ)(RD 0 − 1).

P2D = P3D

The basic reproduction number for system (24)-(28) is defined as RD 0 =

3.2

mµbδγ . a(rδ + qη)(θ + λ)

Global stability

In this section we construct suitable Lyapunov functions to prove that the steady states Q0 and Q1 of system (24)-(28) are GAS. Theorem 3. Suppose that RD 0 ≤ 1, then Q0 is GAS. D Proof. Let us define Y0 (S, L, I, V, B) as:     γ S  q B λ Y0D = S0 H L+I + V + B0 H + θ+λ S0 θ+λ m mc B0 Z Z κ2 Z τ Z τ λ(1 − ρ)b κ1 V (t − ϑ)S(t − ϑ) V (t − ϑ)S(t − ϑ) dϑdτ + ρb dϑdτ. (31) ξ1 (τ )e−δ1 τ ξ2 (τ )e−δ2 τ + θ+λ 1 + πV (t − ϑ) 1 + πV (t − ϑ) 0 0 0 0 dY0D Note that, Y0D (S, L, I, V, B) > 0 for all S, L, I, V, B > 0 and Y0D (S0 , 0, 0, 0, B0 ) = 0. Calculating along dt the trajectories of (24)-(28) we get    dY0D γ S0 bV S = 1− µ − aS − dt θ+λ S 1 + πV   Z κ1 λ V (t − τ )S(t − τ ) + (1 − ρ)b ξ1 (τ )e−δ1 τ dτ − (θ + λ)L θ+λ 1 + πV (t − τ ) 0 Z κ2  V (t − τ )S(t − τ ) + ρb dτ + λL − I + (mI − rV − qV B) ξ2 (τ )e−δ2 τ 1 + πV (t − τ ) m 0     Z q B0 λ(1 − ρ) κ1 bV S bV (t − τ )S(t − τ ) + 1− (η + cBV − δB) + ξ1 (τ )e−δ1 τ − dτ mc B θ+λ 0 1 + πV 1 + πV (t − τ )   Z κ2 bV (t − τ )S(t − τ ) bV S + ρb − dτ ξ2 (τ )e−δ2 τ 1 + πV 1 + πV (t − τ ) 0     aγ (S − S0 )2 γ bS0 V rV qB0 V q B0 =− + − − + 1− (δB0 − δB) θ+λ S (θ + λ) 1 + πV m m mc B =−

2 aγ (S − S0 )2 qδ (B − B0 )2 (rδ + qη) D (rδ + qη)RD 0 πV − + (R0 − 1)V − . θ+λ S mc B mδ mδ(1 + πV )

(32)

dY0D dY0D Therefore, ≤ 0 holds if RD = 0 if and only if S = S0 , B = B0 , V = 0. Applying 0 ≤ 1. Further, dt dt LaSalle’s invariance principle, we get that Q0 is GAS .  Theorem 4. Suppose that RD 0 > 1, then Q1 is GAS. Proof. Consider         γ S λ L I  V Y1D (S, L, I, V, B) = S1 H + L1 H + I1 H + V1 H θ+λ S1 θ+λ L1 I1 m V1    Z τ  Z κ1 q B λ(1 − ρ) bS1 V1 V (t − ϑ)S(t − ϑ)(1 + πV1 ) −δ1 τ + B1 H + ξ1 (τ )e H dϑdτ mc B1 θ + λ 1 + πV1 0 S1 V1 (1 + πV (t − ϑ)) 0   Z Z τ ρbS1 V1 κ2 V (t − ϑ)S(t − ϑ)(1 + πV1 ) + ξ2 (τ )e−δ2 τ H dϑdτ. 1 + πV1 0 S1 V1 (1 + πV (t − ϑ)) 0 28

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dY1D We have Y1D (S, L, I, V, B) > 0 for all S, L, I, V, B > 0 and Y1D (S1 , L1 , I1 , V1 , B1 ) = 0. Calculating along dt the trajectories of (24)-(28) we get    γ S1 bV S dY1D = 1− µ − aS − dt θ+λ S 1 + πV    Z κ1 λ L1 −δ1 τ V (t − τ )S(t − τ ) ξ1 (τ )e + 1− (1 − ρ)b dτ − (θ + λ)L θ+λ L 1 + πV (t − τ ) 0    Z κ2    I1  V1 −δ2 τ V (t − τ )S(t − τ ) + 1− ρb ξ2 (τ )e dτ + λL − I + 1− (mI − rV − qV B) I 1 + πV (t − τ ) m V    0  Z κ1 q bV S B1 λ(1 − ρ) bV (t − τ )S(t − τ ) −δ1 τ + ξ1 (τ )e 1− (η + cBV − δB) + − dτ mc B (θ + λ) 0 1 + πV 1 + πV (t − τ )   Z κ1 λ(1 − ρ) bS1 V1 V (t − τ )S(t − τ )(1 + πV ) + ξ1 (τ )e−δ1 τ ln dτ (θ + λ) 1 + πV1 0 V S (1 + πV (t − τ ))   Z κ2 bV S bV (t − τ )S(t − τ ) +ρ ξ2 (τ )e−δ2 τ − dτ 1 + πV 1 + πV (t − τ ) 0   Z κ2 V (t − τ )S(t − τ )(1 + πV ) ρbS1 V1 −δ2 τ dτ. (33) ξ2 (τ )e ln + 1 + πV1 0 V S (1 + πV (t − τ )) Applying µ = aS1 +

bS1 V1 , η = δB1 − cB1 V1 , 1 + πV1

we obtain       γ dY1D S1 γ bS1 V1 S1 = 1− (aS1 − aS) + 1− dt θ+λ S θ + λ 1 + πV1 S Z κ1 bS1 V λ(1 − ρ)b γ V (t − τ )S(t − τ )L1 − dτ + λL1 ξ1 (τ )e−δ1 τ + θ + λ 1 + πV θ+λ (1 + πV (t − τ )) L 0 Z κ2 V (t − τ )S(t − τ )I1 λLI1 IV1 rV rV1 qBV1 − ρb ξ2 (τ )e−δ2 τ dτ − + I1 − − + + (1 + πV (t − τ )) I I V m m m 0     q qB1 V1 qB1 V1 B1 B1 qB1 V + − + 1− (δB1 − δB) − mc B m m m B   Z κ1 λ(1 − ρ) bS1 V1 V (t − τ )S(t − τ )(1 + πV ) + ξ1 (τ )e−δ1 τ ln dτ θ + λ 1 + πV1 0 V S (1 + πV (t − τ ))   Z V (t − τ )S(t − τ )(1 + πV ) ρbS1 V1 κ2 ξ2 (τ )e−δ2 τ ln dτ. + 1 + πV1 0 V S (1 + πV (t − τ )) The components of the steady state Q1 satisfy E(1 − ρ)

bS1 V1 bS1 V1 = (θ + λ)L1 , Kρ + λL1 = I1 , mI1 = rV1 + qB1 V1 , 1 + πV1 1 + πV1

then I1 =

γ bS1 V1 , θ + λ (1 + πV1 )

rV1 γ bS1 V1 qB1 V1 = − , m θ + λ (1 + πV1 ) m

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and   dY1D aγ (S − S1 )2 Eλ(1 − ρ) bS1 V1 S1 =− + 1− dt θ+λ S θ + λ 1 + πV1 S       S1 γ bS1 V1 (1 + πV1 )V V bS1 V1 1− + − + Kρ (1 + πV1 ) S θ + λ 1 + πV1 (1 + πV )V1 V1 Z κ1 λ(1 − ρ) bS1 V1 V (t − τ )S(t − τ )(1 + πV )L Eλ(1 − ρ) bS1 V1 1 1 − ξ1 (τ )e−δ1 τ dτ + θ + λ 1 + πV1 0 (1 + πV (t − τ ))S1 V1 L θ + λ (1 + πV1 ) Z ρbS1 V1 κ2 V (t − τ )S(t − τ )(1 + πV )I Eλ(1 − ρ) bS 1 1 1 V 1 I1 L ξ2 (τ )e−δ2 τ − dτ − 1 + πV1 0 (1 + πV (t − τ ))S1 V1 I θ + λ 1 + πV1 L1 I Eλ(1 − ρ) bS1 V1 bS1 V1 Eλ(1 − ρ) bS1 V1 IV1 bS1 V1 IV1 + + Kρ − − Kρ θ + λ (1 + πV1 ) (1 + πV1 ) θ + λ 1 + πV1 I1 V 1 + πV1 I1 V   Eλ(1 − ρ) bS1 V1 bS1 V1 2qB1 V1 qBV1 qB1 V1 B1 + + Kρ − + + θ + λ (1 + πV1 ) (1 + πV1 ) m m m B   Z κ1 2 qδ (B − B1 ) λ(1 − ρ) bS1 V1 V (t − τ )S(t − τ )(1 + πV ) −δ1 τ ξ1 (τ )e − + ln dτ mc B (θ + λ) 1 + πV1 0 V S (1 + πV (t − τ ))   Z κ2 V (t − τ )S(t − τ )(1 + πV ) ρbS1 V1 ξ2 (τ )e−δ2 τ ln dτ. + 1 + πV1 0 V S (1 + πV (t − τ )) Utilizing the following equalities           V (t − τ )S(t − τ )(1 + πV ) S1 IV1 1 + πV LI1 ln = ln + ln + ln + ln V S (1 + πV (t − τ )) S I1 V 1 + πV1 L1 I   V (t − τ )S(t − τ )(1 + πV1 )L1 + ln , (1 + πV (t − τ ))S1 V1 L         V (t − τ )S(t − τ )(1 + πV ) S1 IV1 1 + πV ln = ln + ln + ln V S (1 + πV (t − τ )) S I1 V 1 + πV1   V (t − τ )S(t − τ )(1 + πV1 )I1 , + ln (1 + πV (t − τ ))S1 V1 I we have   dY1D aγ (S − S1 )2 γ bS1 V1 (1 + πV1 )V V 1 + πV =− + −1 + − + dt θ+λ S θ + λ 1 + πV1 (1 + πV )V1 V1 1 + πV1       S1 S1 Eλ(1 − ρ) bS1 V1 IV1 IV1 Eλ(1 − ρ) bS1 V1 1− + ln + 1− + ln + (θ + λ) (1 + πV1 ) S S (θ + λ) (1 + πV1 ) I1 V I1 V  Z Eλ(1 − ρ) bS1 V1 1 κ1 V (t − τ )S(t − τ )(1 + πV )L 1 1 + ξ1 (τ )e−δ1 τ 1 − θ + λ 1 + πV1 E 0 (1 + πV (t − τ ))S1 V1 L   V (t − τ )S(t − τ )(1 + πV1 )L1 + ln dτ (1 + πV (t − τ ))S1 V1 L       Eλ(1 − ρ) bS1 V1 1 + πV 1 + πV Eλ(1 − ρ) bS1 V1 LI1 LI1 + 1− + ln + 1− + ln θ + λ 1 + πV1 1 + πV1 1 + πV1 θ + λ (1 + πV1 ) L1 I L1 I       bS1 V1 S1 S1 bS1 V1 IV1 IV1 + Kρ 1− + ln + Kρ 1− + ln (1 + πV1 ) S S 1 + πV1 I1 V I1 V  Z κ2 bS1 V1 1 V (t − τ )S(t − τ )(1 + πV )I 1 1 + Kρ ξ2 (τ )e−δ2 τ 1 − 1 + πV1 K 0 (1 + πV (t − τ ))S1 V1 I   V (t − τ )S(t − τ )(1 + πV1 )I1 + ln dτ (1 + πV (t − τ ))S1 V1 I      bS1 V1 1 + πV 1 + πV qδ (B − B1 )2 qB1 V1 B B1 + Kρ 1− + ln − − 2− − (1 + πV1 ) 1 + πV1 1 + πV1 mc B m B1 B

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   γ (S − S1 )2 γ πbS1 (V − V1 )2 = −a − θ+λ S θ + λ (1 + πV )(1 + πV1 )2          2 qη (B − B1 ) Eλ(1 − ρ) bS1 V1 S1 IV1 1 + πV LI1 − − H +H +H +H mcB1 B θ + λ 1 + πV1 S I1 V 1 + πV1 L1 I    Z 1 κ1 V (t − τ )S(t − τ )(1 + πV )L 1 1 ξ1 (τ )e−δ1 τ H + dτ E 0 (1 + πV (t − τ ))S1 V1 L        bS1 V1 S1 IV1 1 + πV − Kρ H +H +H (1 + πV1 ) S I1 V 1 + πV1    Z V (t − τ )S(t − τ )(1 + πV 1 κ2 1 )I1 −δ2 τ ξ2 (τ )e H + dτ . K 0 (1 + πV (t − τ ))S1 V1 I 

It can be seen that if RD 0 > 1, then S1 , L1 , I1 , V1 , B1 > 0 and

dY1D ≤ 0 for all S, L, I, V, B > 0. We have dt

dY1D = 0 if and only if S = S1 , L = L1 , I = I1 , V = V1 , B = B1 and H = 0. Then using from LaSalle’s dt invariance principle, we show that Q1 is GAS. 

4

Numerical simulations

Next we conduct numerical simulations for system (5)-(9). The values of the parameters are listed in Table 1. We let τi = τ1 = τ2 . The following initial conditions are used: ϕ1 (ϑ) = 1.7, ϕ2 (ϑ) = 0.4, ϕ3 (ϑ) = 0.6, ϕ4 (ϑ) = 0.6, ϕ5 (ϑ) = 1.6, ϑ ∈ [−τi , 0] In Figures 1-5, we show the evolution of the five states of the system S, L, I, V and B with respect to the time. The effect of τi on the stability of Q0 and Q1 is also shown. We can see that, for smaller values of τi e.g. τi = 0.0, 0.5, 1.0 and 2.0, the corresponding values of R0 satisfy R0 > 1, and the trajectory of the system converges to the steady states Q1 . This confirm the results of Theorem 2 that Q1 is GAS. On the the other hand, when τi become larger e.g. τi = 3.0 and 5.0, then R0 < 1, and the system has one steady state Q0. and according to Theorem 1 it is GAS. For this case, the concentrations of the uninfected monocytes and B cells return to their values S0 = µa = 2.2885 and B0 = ηδ = 1.1207, respectively, while the CHIKV particles are cleared from the body. Let τ cr be the critical value of the parameter τi , such that cr

R0 =

cr

bmδµ(λ(1 − ρ)e−δ1 τ + ρ(θ + λ)e−δ1 τ ) = 1. a(rδ + qη)(θ + λ)

Using the data given in Table 1 we obtain τ cr = 2.01206. The value of R0 for different values of τi are listed in Table 2. We can observed that as τi is increased then R0 is decreased. Moreover, we have the following cases: (i) if 0 ≤ τi < 2.01206, then Q1 exists and it is GAS, (ii) if τi ≥ 2.01206, then Q0 is GAS. It is clearly seen that, an increasing in time delay will stabilize the system around Q0 . Biologicaly, the time delay has a similar effect as the antiviral treatment which can be used to eliminate the CHIKV. We observe that, when the delay period is sufficiently long the CHIKV replication will be cleared.

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Table 1: The values of the parameters of model (5)-(9). Parameter

Value

Parameter

Value

µ

1.826

m

2.02

π

0.1

q

0.5964

c

1.2129

r

0.4418

a

0.7979

η

1.402

θ

0.5

δ1

0.5

λ

0.1

τ1

varied



0.4441

τ2

varied

δ

1.251

b

0.5

ρ

0.5

Table 2: The values of steady states, R0 for model (5)-(9) with different values of τi . τi 0.0 0.5 1.0 1.5 2.0 2.01206 2.5 3.0 3.5 4.0 4.5 5.0

Q1 Q1 Q1 Q1 Q1

Steady states = (1.6788, 0.4054, 0.6390, 0.6152, 2.7772) = (1.7636, 0.2718, 0.4284, 0.4986, 2.1694) = (1.8827, 0.1637, 0.2580, 0.3562, 1.7120) = (2.0497, 0.0750, 0.1182, 0.1895, 1.3729) = (2.2819, 0.0016, 0.0025, 0.0046, 1.1257) Q0 = (2.2885, 0, 0, 0, 1.1207) Q0 = (2.2885, 0, 0, 0, 1.1207) Q0 = (2.2885, 0, 0, 0, 1.1207) Q0 = (2.2885, 0, 0, 0, 1.1207) Q0 = (2.2885, 0, 0, 0, 1.1207) Q0 = (2.2885, 0, 0, 0, 1.1207) Q0 = (2.2885, 0, 0, 0, 1.1207)

32

R0 2.7347 2.1298 1.6587 1.2918 1.0060 1.0000 0.7835 0.6102 0.4752 0.3701 0.2882 0.2245

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2.3 2.2 Uninfected monocytes

τ = 3.0

τ i= 5.0

i

τ i= 2.0

2.1 2 1.9

τ i= 1.0

1.8

τ = 0.5

1.7

τ i= 0.0

i

1.6 0

10

20

30

40

50

60

Time

Figure 1: The evolution of uninfected monocytes.

0.45 τ i= 0.0

Latently infected monocytes

0.4 0.35 0.3

τ i= 0.5

0.25 0.2

τ i= 1.0

0.15 0.1 0.05 τ i= 5.0

τ i= 2.0

τ i= 3.0

0 0

10

20

30

40

50

60

Time

Figure 2: The evolution of latently infected monocytes.

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0.7 τ i= 0.0

Actively infected monocytes

0.6 0.5 τ i= 0.5

0.4 τ i= 1.0

0.3 0.2 0.1

τ i= 2.0

τ i= 3.0

τ i= 5.0

0 0

10

20

30

40

50

60

Time

Figure 3: The evolution of actively infected monocytes.

0.8 0.7 Free CHIKV particles

τ i= 0.0

0.6 τ i= 0.5

0.5 0.4

τ i= 1.0

0.3 0.2 τ i= 2.0

0.1

τ i= 5.0

τ i= 3.0

0 0

10

20

30

40

50

60

Time

Figure 4: The evolution of free CHIKV particles.

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3 τ i= 0.0

2.5

B cells

τ i= 0.5

2 τ i= 1.0

1.5 τ i= 5.0

τ = 2.0

τ i= 3.0

i

1 0

10

20

30

40

50

60

Time

Figure 5: The evolution of B cells.

5

Acknowledgment

This article was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR for technical and financial support.

References [1] M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996) 74-79. [2] D.S. Callaway, and A.S. Perelson, HIV-1 infection and low steady state viral loads, Bull. Math. Biol., 64 (2002), 29-64. [3] C. Connell McCluskey, Y. Yang, Global stability of a diffusive virus dynamics model with general incidence function and time delay, Nonlinear Analysis: Real World Applications, 25 (2015), 64-78. [4] A. M. Elaiw and S.A. Azoz, Global properties of a class of HIV infection models with Beddington-DeAngelis functional response, Mathematical Methods in the Applied Sciences, 36 (2013), 383-394. [5] A. M. Elaiw, Global properties of a class of HIV models, Nonlinear Analysis: Real World Applications, 11 (2010), 2253-2263. [6] G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for delay differential equations model of viral infections, SIAM J. Appl. Math., 70(7) (2010), 2693-2708. [7] D. Huang, X. Zhang, Y. Guo, and H. Wang, Analysis of an HIV infection model with treatments and delayed immune response, Applied Mathematical Modelling, 40(4) (2016), 3081-3089.

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[8] K. Wang, A. Fan, and A. Torres, Global properties of an improved hepatitis B virus model, Nonlinear Analysis: Real World Applications, 11 (2010), 3131-3138. [9] A. U. Neumann, N. P. Lam, H. Dahari, D. R. Gretch, T. E. Wiley, T. J, Layden, and A. S. Perelson, Hepatitis C viral dynamics in vivo and the antiviral efficacy of interferon-alpha therapy, Science, 282 (1998), 103-107. [10] A. M. Elaiw, A. A. Raezah and A. S. Alofi, Stability of delay-distributed virus dynamics model with cell-tocell transmission and CTL immune response, Journal of Computational Analysis and Applications, 25(8) (2018), 1518-1531. [11] X. Shi, X. Zhou, and X. Son, Dynamical behavior of a delay virus dynamics model with CTL immune response, Nonlinear Analysis: Real World Applications, 11 (2010), 1795-1809. [12] H. Shu, L. Wang and J. Watmough, Global stability of a nonlinear viral infection model with infinitely distributed intracellular delays and CTL imune responses, SIAM Journal of Applied Mathematics, 73(3) (2013), 1280-1302. [13] A. M. Elaiw, A. M. Althiabi, M. A. Alghamdi and N. Bellomo, Dynamical behavior of a general HIV-1 infection model with HAART and cellular reservoirs, Journal of Computational Analysis and Applications, 24(4) (2018), 728-743. [14] A. M. Elaiw and N. H. AlShamrani, Global stability of humoral immunity virus dynamics models with nonlinear infection rate and removal, Nonlinear Analysis: Real World Applications, 26, (2015), 161-190. [15] A. M. Elaiw and N. H. AlShamrani, Stability of a general delay-distributed virus dynamics model with multi-staged infected progression and immune response, Mathematical Methods in the Applied Sciences, 40(3) (2017), 699-719. [16] Y. Wang, X. Liu, Stability and Hopf bifurcation of a within-host chikungunya virus infection model with two delays, Mathematics and Computers in Simulation, 138 (2017), 31-48. [17] Y. Dumont, F. Chiroleu, Vector control for the chikungunya disease, Mathematical Biosciences and Engineering, 7 (2010), 313-345. [18] Y. Dumont, J. M. Tchuenche, Mathematical studies on the sterile insect technique for the chikungunya disease and aedes albopictus, Journal of Mathematical Biology 65(5) (2012), 809-854. [19] D. Moulay, M. Aziz-Alaoui, M.Cadivel, The chikungunya disease: modeling, vector and transmission global dynamics, Mathematical Biosciences, 229 (2011) 50-63. [20] C. A. Manore, K. S. Hickmann, S. Xu, H. J. Wearing, J. M. Hyman, Comparing dengue and chikungunya emergence and endemic transmission in A. aegypti and A. albopictus, Journal of Theoretical Biology 356 (2014), 174-191. [21] L. Yakob, A.C. Clements, A mathematical model of chikungunya dynamics and control: the major epidemic on Reunion Island, PLoS One, 8 (2013), e57448. [22] X. Liu, and P. Stechlinski, Application of control strategies to a seasonal model of chikungunya disease, Applied Mathematical Modelling, 39 (2015), 3194-3220. [23] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer Verlag, New York, 1993.

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Dynamical behavior of MERS-CoV model with discrete delays H. Batarfi, A. Elaiw and A. Alshareef Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia Emails: [email protected] (H. Batarfi), a m [email protected] (A. Elaiw), [email protected] (A. Alshareef) Abstract A nonlinear mathematical model for Middle East Respiratory Syndrome Corona Virus (MERS-CoV) with two discrete time delays is proposed and analyzed. We show that the solutions of the model are nonnegative and bounded. We derive the basic reproduction number for the MERS-CoV model, R0 . we prove that if R0 ≤ 1 then there exists a disease-free equilibrium P0 and R0 > 1 then in addition to P0 the model has an endemic equilibrium P ∗ . Utilizing Lyapunov method, the global asymptotic stability of disease-free equilibrium of the proposed model is obtained. The dynamical behaviour of the model is also shown by numerical simulations.

Keywords: Infectious diseases; global stability; Lyapunov functional.

1

Introduction

Mathematical of infectious diseases have received the attention of several researchers during the past decides. Some of the models are given by a set of ODEs (see e.g. [1]-[12]). For some disease such as influenza, on adequate contact with an infective, a susceptible individual becomes exposed, that is, infected but not infective. This individual stays in exposed class for a certain latent period before becoming infective. This period can been described as delays on the spread of infectious diseases, and thus, delays should be incorporated into infection term in the system. As a result, the models are given by DDEs (see e.g. [13]-[19]). There are two types of time delays: (i) discrete delay, where the time delay is assumed to be constant (see e.g. [13]-[15]), (ii) distributed delays, where the time delay is assumed to be random parameter taken from probability distributed function (see e.g. [16]-[19]). Recently, Chowell et al. [20] have studied the spread of a Middle East Respiratory Syndrome

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Corona virus (MERS-CoV) by using a SEIR-type compartmental transmission model as: dS dt dEi dt dEs dt dIi dt dAi dt dIs dt dAs dt dH dt dR dt

=−

β S (Ii + Is + ` H) − α, N

(1)

= α − k Ei , =

(2)

β S (Ii + Is + ` H) − k Es , N

(3)

= k ρc,i Ei − γa Ii − γI,i Ii

(4)

= k (1 − ρc,i ) Ei ,

(5)

= k ρc,s Es − γa Is − γI,s Is ,

(6)

= k (1 − ρc,s ) Es ,

(7)

= γa (Ii + Is ) − γr H,

(8)

= γr H + γI,i Ii + γI,s Is .

(9)

In model (1)-(9), the populations divided into 9 compartment: susceptible individuals S, individuals exposed to the zoonotic reservoir Ei or to infectious humans Es , infectious and symptomatic individuals arising from reservoir Ii , or from human-to-human transmission Is , asymptomatic and non-infectious individuals arising from environmental/animal exposure Ai or arising from human-to-human transmission As , hospitalized individuals H, and removed individuals after recovery or disease-induced death R [20]. Susceptible individuals are infected uniformly at random from the zoonotic reservoir at rate α. The parameter β is the mean human-to-human transmission rate per day, ` is relative transmissibility of hospitalized cases, k1 mean latent period (days), ρc,i is proportion of symptomatic and infectious cases among index cases, ρs,i denote to proportion of symptomatic and infectious cases among secondary cases, ρh,i proportion of hospitalized individuals among symptomatic and infectious index cases, ρh,s is proportion of hospitalized individuals among symptomatic and infectious secondary 1 1 represent the mean infectious period among primary cases (days), γI,s is the mean infectious period cases, γI,i 1 among secondary cases (days), γa is the mean time from symptom onset to hospital admission (days) and γ1r denote to mean length of hospital stay (days). Chowell et al., assume that the asymptomatic individuals do not contribute to the transmission process. Moreover, the basic properties of model (1)-(9) are not well studied. Therefore, the aim of this paper is to study the effect of asymptomatic individuals on the transmission of MERSCoV. Our proposed model is a modification of model (1)-(9) by incorporate the asymptomatic individuals as a carrier individuals. We assume that the first scenario describes only the carrier cases and the second one describes the infected cases which demonstrate symptoms. We introduce two types of discrete time delays into the MERS-CoV model. We study the basic properties of the model such as nonnegativity and boundedness of the solutions, stability analysis of the equilibria. At the end we perform some numerical simulations.

2

The MERS-CoV model

In this section, we propose a MERS-CoV model with two discrete delays . Let us define Υ(t) = S(t)(βIc (t) + γIm (t) + `H(t)).

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Then we propose the following model: ˙ S(t) = b − Υ(t) − d1 S(t), ˙ Ec (t) = p e−µ1 τ1 Υ(t − τ1 ) − (kρ1 + d2 )Ec (t),

(10) (11)

E˙ m (t) = (1 − p)e Υ(t − τ2 ) − (kρ2 + d3 )Em (t), I˙c (t) = k ρ1 Ec (t) − γa Ic (t) − qIc (t) − γ1 Ic (t) − d4 Ic (t),

(12)

I˙m (t) = k ρ2 Em (t) − γa Im (t) − γ2 Im (t) + qIc (t) − d5 Im (t), ˙ H(t) = γa (Ic (t) + Im (t)) − γr H(t) − d6 H(t),

(14)

−µ2 τ2

(13)

(15)

˙ R(t) = γ1 Ic (t) + γ2 Im (t) + γr H(t) − d7 R(t),

(16)

where S is susceptible individuals, Ec exposed individuals to carrier, Em exposed individuals to infected, Ic carrier individuals, Im infected individuals, H hospitalized infected and R recovered individuals. The parameters τ1 ≥ 0 and τ2 ≥ 0 represents for the time between contact the susceptible individuals with exposed to carrier Ec and exposed to infected Em , respectively. The factors e−µ1 τ1 and e−µ2 τ2 are the probability that an individuals survives during the delay periods [0, τ1 ] and [0, τ2 ], respectively. The other parameters are defined in section 6. The initial conditions of system (10)-(16) are given by

S(θ) = ϕ1 (θ), Ec (θ) = ϕ2 (θ), Em (θ) = ϕ3 (θ), (17)

Ic (θ) = ϕ4 (θ), Im (θ) = ϕ5 (θ), H(θ) = ϕ6 (θ), R(θ) = ϕ7 (θ), ϕi (θ) ≥ 0, θ ∈ [−%, 0], i = 1, ..., 7,

where, % = max {τ1 , τ2 ] and (ϕ1 (θ), ϕ2 (θ), ..., ϕ7 (θ)) ∈ C ([−%, 0], R7≥0 ) where C is the Banach space of continuous functions mapping the interval [−%, 0] into R7≥ 0 . By the fundamental theory of functional differential equations [21], system (10)-(16) has a unique solution satisfying the initial conditions.

3

Nonnegativity and boundedness

In this section, we will study the nonnegativity and boundedness of the model’s solutions. Theorem 1. The solutions of system (10)-(16) are nonnegative and there exist a positive number Q such that the compact set: Γ = {(S, Ec , Em , Ic , Im , H, R) ∈ R7≥ 0 : 0 ≤ S, Ec , Em , Ic , Im , H, R ≤ Q} is positively invariant. Proof First, we show the nonnegativity solutions, we will write the system in the matrix form as Y˙ = φ(Y ), where Y = (S, Ec , Em , Ic , Im , H, R)T and φ = (φ1 , φ2 , φ3 , φ4 , φ5 , φ6 , φ7 )T . Then, 

   φ1 (Y ) b − Υ(t) − d1 S(t) φ (Y )   p e−µ1 τ1 Υ(t − τ1 ) − k ρ1 Ec (t) − d2 Ec (t)  2        φ3 (Y )  (1 − p) e−µ2 τ2 Υ(t − τ2 ) − k ρ2 Em (t) − d3 Em (t)      φ4 (Y ) =  k ρ1 Ec (t) − γa Ic (t) − q Ic (t) − γ1 Ic (t) − d4 Ic (t)  .         φ5 (Y ) k ρ2 Em (t) − γa Im (t) − γ2 Im (t) + q Ic (t) − d5 Im (t)     φ6 (Y )   γa (Ic (t) + Im (t)) − γr H(t) − d6 H(t) φ7 (Y )

γ1 Ic (t) + γ2 Im (t) + γr H(t) − d7 R(t)

It is easy to see that functions φi satisfies the following condition φi (Y (t))|Yi (t)=0,Y (t)∈R7 ≥ 0 . >0

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Due to lemma 2 in [22], any solution of (10 )-(16) with initial (17) is such that Y (t) ∈ R7≥0 for all t ≥ 0. Next, we prove the ultimate bound of the solutions of system (10)-(16). Let us define L(t) = p e−µ1 τ1 S(t − τ1 ) + (1 − p) e−µ2 τ2 S(t − τ2 ) + Ec (t) + Em (t) + Ic (t) + Im (t) + H(t) + R(t). Then, ˙ L(t) = p e−µ1 τ1 (b − Υ(t − τ1 ) − d1 S(t − τ1 )) + (1 − p) e−µ2 τ2 (b − Υ(t − τ2 ) − d1 S(t − τ2 )) + p e−µ1 τ1 Υ(t − τ1 ) − k ρ1 Ec (t) − d2 Ec (t) + (1 − p)e−µ2 τ2 Υ(t − τ2 ) − kρ2 Em (t) − d3 Em (t) + k ρ1 Ec (t) − γa Ic (t) − q Ic (t) − γ1 Ic (t) − d4 Ic (t) + k ρ2 Em (t) − γa Im (t) − γ2 Im (t) + q Ic (t) − d5 Im (t) + γa (Ic (t) + Im (t)) − γr H(t) − d6 H(t) + γ1 Ic (t) + γ2 Im (t) + γr H(t) − d7 R(t), = (p e−µ1 τ1 + (1 − p) e−µ2 τ2 ) b − p e−µ1 τ1 d1 S(t − τ1 ) − (1 − p) e−µ2 τ2 d1 S(t − τ2 ) − d2 Ec (t) − d3 Em (t) − d4 Ic (t) − d5 Im (t) − d6 H(t) − d7 R(t) ≤ b − d L(t), where d = min{di }, i = 1, .., 7. It follows that, lim supt→∞ L(t) ≤ Q, where Q = db . Then, lim supt→∞ S(t) ≤ Q, lim supt→∞ Ec (t) ≤ Q, lim supt→∞ Em (t) ≤ Q, lim supt→∞ Ic (t) Q, lim supt→∞ Im (t) ≤ Q, lim supt→∞ H(t) ≤ Q, and lim supt→∞ R(t) ≤ Q. 

4

≤

Equilibria and biological thresholds

To calculated the equilibria of model (10)-(16), we put the R.H.S of Eqs. (10)-(16) equals zero, we get b − S (d1 + β Ic + γ Im + ` H) = 0, −µ1 τ1

(18)

S (β Ic + γ Im + ` H) − a1 Ec = 0,

(19)

(1 − p) e−µ2 τ2 S( (β Ic + γ Im + ` H) − a2 Em , = 0,

(20)

λ1 Ec − a3 Ic = 0,

(21)

λ2 Em − a4 Im + q Ic = 0,

(22)

γa (Ic + Im ) − a5 H = 0,

(23)

γ1 Ic + γ2 Im + γr H − d7 R = 0,

(24)

pe

where a1 = k ρ1 + d2 ,

a 2 = k ρ 2 + d3

a3 = γa + γ1 + q + d4 ,

a4 = γa + γ2 + d5 ,

a5 = γr + d6 ,

λ1 = kρ1 , λ2 = kρ2 .

Solving system (18)-(24), we find that the system has two equilibria • The disease-free equilibrium  P0 = (S0 , 0, 0, 0, 0, 0, 0) =

40

 b , 0, 0, 0, 0, 0, 0 . d1

(25)

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• The endemic equilibrium ∗ ∗ P ∗ = (S ∗ , Ec∗ , Em , Ic∗ , Im , H ∗ , R∗ ),

(26)

where a0 p(A2 − a0 d1 eµ2 τ2 ) (1 − p) (A4 − a0 d1 eµ1 τ1 ) ∗ , Ec∗ = , Em , = A1 a1 A3 a2 A3 λ1 p (A2 − a0 d1 eµ2 τ2 ) A5 + a2 (A6 − 2 a1 a3 A7 ) ∗ Ic∗ = , Im = , a1 a3 A3 a1 a2 a3 a4 A3 γa (2 a1 a3 (a2 A10 + A9 ) − A8 ) A11 − a1 a3 (A12 + a2 A13 ) H∗ = , R∗ = , a0 A3 d1 a0 A3 S∗ =

and a0 = a1 a2 a3 a4 a5 , A1 = ((a4 β + γ q)a5 + γa `(q + a4 ))a2 λ1 p e(−µ1 τ1 ) + (1 − p)λ2 a3 a1 (a5 γ + `γa ) e(−µ2 τ2 ) , A2 = ((a4 β + γ q)a5 + γa `(q + a4 ))a2 λ1 p b e(−µ1 τ1 +µ2 τ2 ) + (1 − p)λ2 b a3 a1 (a5 γ + `γa ), A3 = ((a4 β + γ q)a5 + γa `(q + a4 ))a2 λ1 p e(µ2 τ2 ) + (1 − p)λ2 a3 a1 (a5 γ + `γa ) e(µ1 τ1 ) , A4 = ((a4 β + γ q)a5 + γa `(q + a4 ))a2 λ1 p b + (1 − p)λ2 a3 a1 b (a5 γ + `γa ) e(µ1 τ1 −µ2 τ2 ) , A5 = b λ22 a21 a23 (p − 1)2 (a5 γ + `γa ) eµ1 τ1 −µ2 τ2 , A6 = q b λ21 ((a4 β + γ q) a5 + ` γa (q + a4 )) p2 a2 e−µ1 τ1 +µ2 τ2 ,  1 1 µ1 τ1 + λ1 d1 eµ2 τ2 q a2 a4 a5 A7 = − d1 λ2 a1 a3 a4 a5 (p − 1) e 2 2      1 1 +b a4 β + γ q a5 + `γa q + a4 (p − 1)λ2 p, 2 2 A8 = b λ21 ((a4 β + γ q) a5 + ` γa (q + a4 )) (q + a4 ) p2 a22 e−µ1 τ1 +µ2 τ2 , 1 A9 = − (bλ22 a1 a3 (p − 1)2 (a5 γ + `γa ) eµ1 τ1 −µ2 τ2 ), 2 1 1 A10 = − d1 λ2 a1 a3 a4 a5 (p − 1) eµ1 τ1 + λ1 d1 a2 a4 a5 (q + a4 ) eµ2 τ2 2   2   1 1 β + γ a4 + γ a5 + `γa (q + a4 ) λ2 p, + b (p − 1) 2 2 A11 = b λ21 ((a4 γ1 + γ2 q) a5 + γr γa (q + a4 )) p2 a22 ((a4 β + γ q) a5 + `γa (q + a4 )) e−µ1 τ1 +µ2 τ2 , A12 = −bλ22 a1 a3 (p − 1)2 (a5 γ2 + γa γr ) (a5 γ + ` γa ) eµ1 τ1 −µ2 τ2 , A13 = −d1 λ2 a1 a3 a4 a5 (p − 1) (a5 γ2 + γa γr ) eµ1 τ1 + λ1 (((a4 γ1 + γ2 q) a5 + γr γa (q + a4 )) a4 a2 a5 d1 eµ2 τ2 + λ2 b (((βγ2 + γ γ1 ) a4 + 2γ γ2 q) a25 + γa (((β + γ)γr + `(γ1 + γ2 )) a4 + 2 q (γ γr + γ2 `)) a5 + 2`γa2 γr (q + a4 )) (p − 1)) p.

4.1

Calculating the basic reproduction number

We will apply the next generation method [23] to determine the basic reproduction number R0 for system (10)-(16). We follow the following steps (i) We evaluated the matrix F at P0 as:   p e−µ1 τ1 γ db1 p e−µ1 τ1 ` db1 0 0 p e−µ1 τ1 β db1   0 0 (1 − p) e−µ2 τ2 β db1 (1 − p) e−µ2 τ2 γ db1 (1 − p) e−µ2 τ2 ` db1    . F = 0 0 0  0 0   0 0 0  0 0 0 0 0 0 0

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(ii) We get the matrix V at P0 as: 

a1   0  V = −λ1   0 0

0 a2 0 −λ2 0

0 0 a3 −q −γa

0 0 0 a4 −γa

 0  0  0 .  0 a5

(iii) Finally, the basic reproduction number is given by R0 = ρ(F V −1 ) =

4.2

A1 S0 a0 .

Existence of equilibria

Theorem 2. For system (10)-(16), we have (i) If R0 ≤ 1, then there exists only one positive equilibria P0 . (ii) If R0 > 1, then there exist two positive equilibria P0 and P ∗ . Proof We have a0 S0 = , A1 R0   p (A2 − a0 d1 eµ2 τ2 ) p A2 p p Ec∗ = − a d = = (b A1 − a0 d1 ) = (R0 − 1), 0 1 µ τ 2 2 a1 A3 a1 A3 e a1 A3 a1 A3   (1 − p) A4 (1 − p)(A4 − a0 d1 eµ1 τ1 ) ∗ = − a d Em = , 0 1 a2 A3 a2 A3 eµ1 τ1 (1 − p) (1 − p) = (b A1 − a0 d1 ) = (R0 − 1). a2 A3 a2 A3 S∗ =

From Eq. (13)-(16), we have λ1 p λ1 p λ1 ∗ E = (R0 − 1) = (R0 − 1), a3 c a3 a1 A3 a1 a3 A3 1 ∗ ∗ Im = (λ2 Em + q Ic∗ ) = C1 (R0 − 1), a4 1 ∗ H∗ = (γa (Ic∗ + Im )) = C2 (R0 − 1)), a5 1 ∗ + γr H ∗ ) = C3 (R0 − 1), R∗ = (γ1 Ic∗ + γ2 Im d7 Ic∗ =

(27) (28) (29) (30)

where, 1 (1 − p) λ1 p (λ2 +q ), a4 a2 A3 a1 a3 A3 λ1 p 1 (1 − p) λ1 p γa + (λ2 +q )), C2 = ( a5 a1 a3 A3 a4 a2 A3 a1 a3 A3 1 λ1 p C3 = (γ1 + γ2 C1 + γr C2 ). d7 a1 a3 A3 C1 =

(31)



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5

Global stability analysis of P0

In this section, we use Lyapunov function and LaSalle’s invariance principle to establish the global stability of P0 . Theorem 3. For system (10)-(16), if R0 ≤ 1, then P0 is GAS. Proof We define the following Lyapunov functional   S S W0 = S0 − 1 − ln + ε1 Ec + ε2 Em + ε3 Ic + ε4 Im + ε5 H S0 S0 Z τ1 + ε6 S(t − s)(βIc (t − s) + γIm (t − s) + `H(t − s)) ds 0 Z τ2 + ε7 S(t − s)(βIc (t − s) + γIm (t − s) + `H(t − s)) ds. 0

The time derivative of W0 along the trajectory of system (10)-(16) is given by   S0 dW0 = 1− (b − S(t) (β Ic (t) + γ Im (t) + ` H(t)) − d1 S(t)) dt S(t) + ε1 (p e−µ1 τ1 S(t − τ1 ) (β Ic (t − τ1 ) + γ Im (t − τ1 ) + ` H(t − τ1 )) − a1 Ec (t)) + ε2 ((1 − p) e−µ2 τ2 S(t − τ2 ) (β Ic (t − τ2 ) + γ Im (t − τ2 ) + ` H(t − τ2 )) − a2 Em (t))

(32)

+ ε3 (λ1 Ec (t) − a3 Ic (t)) + ε4 (λ2 Em (t) − a4 Im (t) + q Ic (t)) + ε5 (γa (Ic (t) + Im ) − a5 H(t)) + ε6 {(S(t) (β Ic (t) + γ Im (t) + ` H(t))) − (S(t − τ1 ) (β Ic (t − τ1 ) + γ Im (t − τ1 ) + `H(t − τ1 )))} + ε7 {(S(t) (β Ic (t) + γ Im (t) + ` H(t))) − (S(t − τ2 ) (β Ic (t − τ2 ) + γ Im (t − τ2 ) + `H(t − τ2 )))}, The parameters εi , i = 1, ..., 7 are chosen such that ε6 + ε7 = 1,

(33)

p ε1 e

−µ1 τ1

− ε6 = 0,

(34)

(1 − p) ε2 e

−µ2 τ2

− ε7 = 0,

(35)

−ε1 a1 + λ1 ε3 = 0,

(36)

−ε2 a2 + λ2 ε4 = 0,

(37)

−a3 ε3 + q ε4 + γa ε5 + βS0 = 0,

(38)

−a4 ε4 + γa ε5 + γS0 = 0.

(39)

Solving Eqs. (33)-(39), we get ε5 = where G=

G(1 − R0 ) + `S0 , a5

a1 a2 a3 a4 a5 . γa (λ1 pa2 (a4 + q) e−µ1 τ1 + λ2 e−µ2 τ2 a1 a3 (1 − p))

We can see that ε5 > 0 if R0 ≤ 1. From Eqs. (34)-(38) we get

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1 (γa ε5 + γS0 ) > 0. a4 1 ε3 = (qε4 + γa ε5 + βS0 ) > 0. a3 λ2 ε4 ε2 = > 0. a2 λ1 ε3 ε1 = > 0. a1 ε4 =

ε7 = (1 − p) ε2 e−µ2 τ2 > 0. ε6 = p ε1 e−µ1 τ1 > 0, Thus, Eq. (32) becomes dW0 (S − S0 )2 = −b + (`S0 − a5 ε5 ) H, dt S

(40)

we have `S0 − a5 ε5 = G(R0 − 1). . Then

(S − S0 )2 G dW0 = −b + (R0 − 1) H, dt S a5

(41)

0 0 From Eq (41), dW ≤ 0 if R0 ≤ 1. Then, dW equal to zero if S = S0 and H = 0. Let Ω = dt dt {(S, Ec , Em , Ic , Im , H, R) : S = S0 , H = 0}. From system (10)-(16), if H = 0, then H˙ = 0 and 0 = γa (Ic + Im ). ˙ = 0. From system (10)-(16), we have 0 = I˙c = λ1 Ec ⇒ Since, Ic ≥ 0, Im ≥ 0 then Ic = 0, Im = 0, ⇒ I˙c = Im ˙ = λ2 Em ⇒ Em = 0. Finally, R(t) ˙ Ec = 0. Similarly, we have 0 = Im = −d7 R it follows that R → 0 as t → ∞. From LaSalle’s invariance principle, P0 is GAS in Γ. 

6

Numerical simulations and discussions

In this section, we introduce the numerical results of system (10)-(16). We consider the following initial conditions IC : S(θ) = 600, Ec (θ) = 30, Em (θ) = 80, Ic (θ) = 3, Im (θ) = 12, H(θ) = 8, R(θ) = 40, θ ∈ [−max{τ1 , τ2 }, 0]. we use the values of the parameters in Table 1. In addition we choose µ1 = µ2 = 1. We study the following cases:

6.1

Effect of parameters β, γ and ` on the stability of equilibria:

In this case, we fix the values τ1 = τ2 = 0.01. Figure 1 shows the evaluation of system states for two scenarios: i) R0 ≤ 1. We choose β = 0.002, γ = 0.0001, and ` = 0.0001 then we compute R0 = 0.23. We can see from the figure that the states of the system approach P0 = (1000, 0, 0, 0, 0, 0, 0). This means that according to Theorem 3 P0 is GAS. ii) R0 > 1. We choose β = 0.02, γ = 0.001, and ` = 0.001 then we compute R0 = 2.37 and P ∗ = (421.6, 55.4, 129.2, 5.08, 20.9, 14.5, 72.4). Then P ∗ exists and this confirm the results of Theorem 2. Figure 1 shows that the states of the system converge to P ∗ .

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Table 1: The parameters values of MERS-CoV model

Symbol b β γ ` γa d1 d2 d3 d4 d5 d6 d7 k ρ1 = ρ2 γ1 = γ2 γr p q

6.2

Parameter Rate of generation of new susceptible individuals Rate constant of transmission for carriers Rate constant of symptomatically infected individuals Relative transmissibility of hospitalized cases Mean time from carrier and infected to hospital admission (days) Death rate of susceptible individuals Death rate of exposed to carrier Death rate of exposed to infected individuals Death rate of carrier individuals Death rate of infected individuals Death rate of hospitalized individuals Death rate of recovered individuals Mean latent period Proportion of carrier and infected cases Mean infectious period Mean length of hospital stay Rate of infected individual who becomes carrier Rate of carrier individual who becomes infected

Value 100 Varied Varied Varied 0.3 0.1 0.2 0.2 0.2 0.3 0.4 0.1 0.19 0.58 0.2 0.14 0.3 0.5

Effect of the time delays on the asymptotic behaviour of the equilibria:

In this case, we take the values β = 0.02, γ = 0.001, and ` = 0.001. Let us consider the case τ1 = τ2 = τ . In Table 2, we present the values of R0 and the equilibria of system (10)-(16) with different values of τ .From Table 2: Values of R0 and steady states of system (10)-(16) with different values of τ τ 0.067 0.082 0.67 0.8735928143 1.2 1.5 2.5 3.1 3.5

R0 2.24 2.21 1.23 1.00 0.72 0.5 0.19 0.11 0.07

Steady states P = (446.37, 50.07, 116.83, 4.6, 18.97, 13.09, 65.46) P ∗ = (453.12, 48.73, 113.69, 4.47, 18.97, 12.74, 63.70) P ∗ = (815.79, 9.12, 21.27, 0.84, 3.45, 2.38, 11.92) P0 = (1000, 0, 0, 0, 0, 0, 0) P0 = (1000, 0, 0, 0, 0, 0, 0) P0 = (1000, 0, 0, 0, 0, 0, 0) P0 = (1000, 0, 0, 0, 0, 0, 0) P0 = (1000, 0, 0, 0, 0, 0, 0) P0 = (1000, 0, 0, 0, 0, 0, 0) ∗

Table 2, we can observe that the value of R0 is decreased as τ is increased. Moreover, for small values of τ , P ∗ exists and for large values of τ the system moved from P ∗ to P0 with is GAS. Figures 2 shows the effect of the parameter τ on the evaluation of the states of the system. We can see that as the time delay parameter is increased, the number of susceptible individuals are increased and tend to its normal number, while the number

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of individuals in other groups is reduced and tends to zero. It means that, the time delay play the role of controlling the disease transmission.

(a) Evaluation of S(t).

(b) Evaluation of Ec (t).

(c) Evaluation of Em (t).

(d) Evaluation of Ic (t).

(e) Evaluation of carrier Im (t).

(f) Evaluation of H(t).

(g) Evaluation of R(t).

Figure 1: The evaluations of the system states (10)-(16) with two delays τ1 = τ2 = 0.01.

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. 1000

Exposed to carrier group

Susecptible group

60

τ = 0.06 τ=0.8 τ=1.5

900 800 700 600 500 400 0

10

20

30

40 Time

50

60

70

τ = 0.06 τ=0.8 τ=1.5

50 40 30 20 10 0 0

80

10

20

40 Time

50

60

70

80

(b) Evaluation of Ec (t).

(a) Evaluation of S(t). 6

140 τ = 0.06 τ=0.8 τ=1.5

120 100

τ = 0.06 τ=0.8 τ=1.5

5 Carrier group

Exposed group

30

80 60

4 3 2

40 1

20 0 0

10

20

30

40

50

60

70

0 0

80

10

20

30

Time

(c) Evaluation of Em (t).

50

60

70

80

(d) Evaluation of Ic (t).

25

15

Hospitlaized group

τ = 0.06 τ=0.8 τ=1.5

20

15

10

τ = 0.06 τ=0.8 τ=1.5 10

5

5

0 0

10

20

30

40 Time

50

60

70

0 0

80

10

(e) Evaluation of carrier Im (t).

20

30

40 Time

50

60

70

80

(f) Evaluation of H(t).

70 τ = 0.06 τ=0.8 τ=1.5

60 Recovered group

Infected group

40 Time

50 40 30 20 10 0 0

10

20

30

40 Time

50

60

70

80

(g) Evaluation of R(t).

Figure 2: The evaluations of system (10) -(16) with different values of τ .

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7

Conclusion

We have proposed a MERS-CoV model with two times delay. We have obtained the biological threshold, the basic reproduction number R0 . The existence of the model’s equilibria has been proven. The global asymptotic stability of the disease free equilibria P0 has been investigated by constructing Lyapunov functional and using LaSalle’s invariance principle. To support our theoretical results, we have presented the numerical simulations.

References [1] Rachsh, A., and Torres, D., (2015), Mathematical modelling, simulation, and optimal control of 2014 Ebola outbreak in West Africa, Discrete Dynamics in Nature and Society, Vol. 2015, 1-9. [2] Elaiw, A., (2012), Global properties of a class of virus infection models with multi target cells, Nonlinear Dynamics , Vol.69, 423-435. [3] Wester, T., (2015), Analysis and simulation of a mathematical model of Ebola virus dynamics in vivo, Society for Industrial and Applied Mathematics, Vol. 8, 236-256. [4] Ullah, R., Zaman, G., and Islam, S., (2013), Stability analysis of a general SIR epidemic model, VFAST Transactions on Mathematics, Vol. 1, 16-20. [5] Ma, X., Zhou, Y., and Cao, H., (2013), Global stability of the endemic equilibrium of a discrete SIR epidemic model, Advances in Difference Equations, Vol. 2013, 1-19. [6] Korobeinikov, A., (2004), Lyapunov functions and global properties for SEIR and SEIS epidemic models, Mathematical Medicine and Biology, Vol. 21, 75-83. [7] Grigorieva, E., Khailov, E., and Korobeinikov A., (2016), Optimal control for a SIR epidemic model with nonlinear incident rate, Mathematical Modelling of Natural Phenomena, Vol. 11, 89-104. [8] Ledzewicz, U., and Schattler, H., (2011), On optimal Singular control for a general SIR-model with vaccination and treatment, Discrete and Continuous Dynamical System, Vol. 2011, 981-990. [9] Bakare1, E., Nwagwo, A., and Danso-Addo, E., (2014), Optimal control analysis of an SIR epidemic model with constant recruitment, International Journal of Applied Mathematical Research, Vol. 3, 273-285. [10] Kara, T., and Batabyalb, A., (2011), Stability analysis and optimal control of an SIR epidemic model with vaccination, BioSystems, Vol. 104, 127-135. [11] Grigorieva, E., Khailov, E., and Korobeinikov, A., (2015), Optimal control for an epidemic in populations of varying size, American Institute of Mathematical Sciences, Vol. 2015, 549-561. [12] Pinho, M., and Nogueira, F., (2017), On application of optimal control to SEIR normalized models: pros and cons, Mathematical Biosciences and Engineering, Vol. 14, 111-126. [13] Rui Xu, (2013), Global dynamics of a delayed epidemic model with latency and relapse, Nonlinear Analysis: Modelling and Control, Vol. 18, 250-263. [14] Gau, S., Chen, L., and Teng, Z., (2008), Pulse vaccination of an SEIR epidemic model with time delay, Nonlinear Analysis: Real world Applications, Vol. 9, 599-607.

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[15] Gao, S., Teng, Z., and Xie, D., (2008), The effects of pulse vaccination on SEIR model with two time delays, Applied Mathematics and Computation, Vol. 201, 282-292. [16] Connell McCluskey, C., (2010), Complete global stability for an SIR epidemic model with delay distributed or discrete, Nonlinear Analysis: Real World Applications, Vol. 11, 55-59. [17] Enatsu, Y., and Nakata, Y., (2010), Global stability for a class of discrete SIR epidemic models, Mathematical Bioscience and Engineering, Vol. 7, 347-361. [18] Beretta, E., and Takeuchi, Y., (1997), Convergence results in SIR epidemic models with varying population size, Nonlinear Analysis: Theory, Methods and Application, Vol. 28, 1909-1921. [19] Enatsu, Y., Yukihiko and Muroya, Y., (2012), Global stability of SIRS epidemic models with a class of nonlinear incidence rate and distributed delayes, Acta Mathematical Scientia, Vol. 32, 851-865. [20] Chowell, G., Blumberg, S., Simonsen, L., Miller, M., and Viboud, C., (2014), Synthesizing data and models for the spread of MERS-CoV, 2013:Key role of index cases and hospital transmission, Epidemics, Vol. 9, 40-51. [21] Hale, J., and Lunel, S., (1993), Introduction to functional differential equations, Science and Business Media. [22] X. Yang, L.S. Chen and J.F. Chen, (1996), Permanence and positive periodic solution for the single-species nonautonomous delay diffusive models, Computers and Mathematics with Applications, Vol. 32, 109-116. [23] Heffernan, J., smith, R., and Wahl, L., (2005), Perspectives on the basic reproduction ratio, Journal of the Royal Society Interface, Vol. 2, 281-293.

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CONVEXITY AND HYPERCONVEXITY IN FUZZY METRIC SPACE ˘ EBRU YIGIT AND HAKAN EFE

Abstract. In this paper, firstly we give the definion of fuzzy convex metric, in a different way. Then we introduce the concept of hyperconvexity in fuzzy metric space and prove that every fuzzy hyperconvex metric space is complete. Also it is proved that for m−seperable fuzzy metric spaces, fuzzy m−hyperconvexity is equivalent to fuzzy hyperconvexity.

1. INTRODUCTION The concept of convex metric space has been studied by many authors, in some different ways [7, 9, 11, 14, 15]. After that, some authors examined this concept for fuzzy metric space by using the definition of fuzzy metric which is introduced by George and Veeremani [1], for example; Thanithamil [4] introduced the convex structure in fuzzy metric spaces and Vosoughi and Hosseni [8] gave the definion of metrically convex fuzzy metric space (X, M, ∗). The other common concept for metric space is hyperconvexity which was introduced by Aronszajn and Panitchpakdi [10] in 1956. Since then many interesting works have been appeared for hyperconvex spaces [5, 11, 13]. In this paper, we give the notion of fuzzy convex metric space by using the closed balls, in a different way. Also, we introduce a new notion for fuzzy metric space which is called fuzzy hyperconvex metric space. One of the main result of this paper is that every fuzzy hyperconvex metric space is complete. Also, the fuzzy m−hyperconvexity is introduced for any cardinal m ≥ 3, which is a weaker property than fuzzy hyperconvexity. The definition m−seperability for fuzzy metric space is used, so the other result for this paper is that for any m−seperable fuzzy metric spaces, fuzzy m−hyperconvexity is equivalent to fuzzy hyperconvexity. 2. PRELIMINARIES Definition 1. [6] A binary operation ∗ : [0, 1] × [0, 1] → [0, 1] is continuous t-norm if ∗ satisfies the following conditions: (i) ∗ is commutative and associative; (ii) ∗ is continuous; (iii) a ∗ 1 = a for all a ∈ [0, 1] ; (iv) a ∗ b ≤ c ∗ d whenever a ≤ c and b ≤ d, for a, b, c, d ∈ [0, 1] . Remark 1. [1] (i) For any r1 ∈ (0, 1) with r1 > r2 , there exist r3 ∈ (0, 1) such that r1 ∗ r3 ≥ r2 . Date: July 8, 2017. 1991 Mathematics Subject Classification. 46S40, 54A40 . Key words and phrases. convexity, hyperconvexity, fuzzy metric space. 1

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2

(ii) For any r4 ∈ (0, 1) , there exist r5 ∈ (0, 1) such that r5 ∗ r5 ≥ r4 . Definition 2. [1] The 3-tuple (X, M, ∗) is said to be a fuzzy metric space if X is an arbitrary set, ∗ is a continuous t-norm and M is a fuzzy set on X 2 × (0, ∞) satisfying the following conditions, for all x, y, z ∈ X and s, t > 0: (FM-1) (FM-2) (FM-3) (FM-4) (FM-5)

M (x, y, t) > 0, M (x, y, t) = 1 if and only if x = y, M (x, y, t) = M (y, x, t), M (x, z, t + s) ≥ M (x, y, t) ∗ M (y, z, s), M (x, y, .) : (0, ∞) → [0, 1] is continuous.

Example 1. [1] (Induced fuzzy metric). Let (X, d) be a metric space. Define a ∗ b = min {a, b} for all ∀a, b ∈ [0, 1] and let M be fuzzy set on X × X × (0, ∞) as follows: ktn , k, m, n ∈ R+ . M (x, y, t) = n kt + md(x, y) Then (X, M, ∗) is a fuzzy metric space. In this example by taking k = m = n = 1, we get t M (x, y, t) = . t + d(x, y) We call this fuzzy metric induced by a metric d the standard fuzzy metric. Definition 3. [1] Let (X, M, ∗) be a fuzzy metric space and let r ∈ (0, 1), t > 0 and x ∈ X. The open ball and the closed ball with center x and radius r with respect to t are defined as follows, respectively BM (x, r, t) = {y ∈ X : M (x, y, t) > 1 − r} ¯M (x, r, t) = {y ∈ X : M (x, y, t) ≥ 1 − r} . B Remark 2. [1] Every open ball is an open set and every closed ball is a closed set in a fuzzy metric space (X, M, ∗). Theorem 1. [1] Let (X, M, ∗) be a fuzzy metric space. Define τM = {A ⊂ X : ∀ x ∈ A, ∃r ∈ (0, 1) and t > 0 3 BM (x, r, t) ⊂ A } . Then τM is a topology on X. Definition 4. [1] Let (X, M, ∗) be a fuzzy metric space. Then (a) A sequence {xn } in X is said to be Cauchy sequence if for each ε > 0 and each t > 0, there exists n0 ∈ N such that M (xn , xm , t) > 1 − ε, for all n, m ≥ n0 . (b) (X, M, ∗) is called complete if every Cauchy sequence is convergent with respect to τM . Definition 5. [3] Let (X, M, ∗) be a fuzzy metric space. A collection of sets {Fn }n∈N is said to have fuzzy diameter zero if and only if for each pair r, t > 0, (r ∈ (0, 1) and t > 0), there exists n ∈ N such that M (x, y, t) > 1 − r for all x, y ∈ Fn . Remark 3. [3] A non-empty subset F of a fuzzy metric space X has fuzzy diameter zero if and only if F is a singleton set.

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CONVEXITY AND HYPERCONVEXITY IN FUZZY METRIC SPACE

3

Theorem 2. [3] A necessary and sufficient condition that a fuzzy metric space ∞ (X, M, ∗) be complete is that every nested sequence of non-empty closed sets {FT n }n=1 with fuzzy diameter zero have non-empty intersection. And the element x ∈ Fn n∈N

is unique. Definition 6. [12] Let (X, M, ∗) be a fuzzy metric space. Let the mappings δA (t) : (0, ∞) −→ [0, 1] be defined as δA (t) = inf supM (x, y, ε). x,y∈A ε0

set A will be called F-strongly bounded. Definition 7. [11] Let (X, d) be a metric space. We say that X is metrically convex if for any points x1 , x2 ∈ X and positive numbers α and β such that d(x1 , x2 ) ≤ α + β, there exists z ∈ X such that d(x1 , z) ≤ α and d(x2 , z) ≤ β, or equivalently ¯ 1 , α) ∩ B(x ¯ 2 , β). z ∈ B(x Definition 8. [11] Let (X, d) be a metric space and Γ be an index set. The T ¯metric space X is said to has the ball intersection property (BIP in short) if Bα 6= ∅ α∈Γ T ¯α )α∈Γ such that ¯α 6= ∅, for any finite for any collection of closed balls (B B α∈Γf

subset Γf ⊂ Γ. Definition 9. [11] Let (X, d) be a metric T ¯ space and Γ be an index set. The metric space X is said to be hyperconvex if B(xα , rα ) 6= ∅ for any collection of points α∈Γ

{xα }α∈Γ in X and positive numbers {rα }α∈Γ such that d(xα , xβ ) ≤ rα + rβ for any α and β in Γ. Example 2. [11] The real line R is hyperconvex with the usual metric d. Example 3. [11] The infinite dimensional Banach space l∞ is hyperconvex. Definition 10. [10] A metric space (X, d) is called m−seperable if it contains a dense subset of cardinal < m. 3. MAIN RESULTS Before we give the definition of fuzzy metrically convexity, we give the following Lemma for the definition to be clear. Lemma 1. Let (X, M, ∗) be a fuzzy metric space, x1 , x2 ∈ X, r1 , r2 ∈ (0, 1) and ¯M (x1 , r1 , t1 ) ∩ B ¯M (x2 , r2 , t2 ) 6= ∅ then M (x1 , x2 , t1 + t2 ) ≥ t1 , t2 ∈ (0, ∞). If B (1 − r1 ) ∗ (1 − r2 ) for any x1 , x2 ∈ X and each pair of r1 , t1 > 0 and r2 , t2 > 0. ¯M (x1 , r1 , t1 ) ∩ B ¯M (x2 , r2 , t2 ) 6= ∅. Then there exists z ∈ X such that Proof. Let B ¯M (x1 , r1 , t1 ) ∩ B ¯M (x2 , r2 , t2 ) z ∈ B ¯ ¯M (x2 , r2 , t2 ) =⇒ z ∈ BM (x1 , r1 , t1 ) and z ∈ B =⇒

M (x1 , z, t1 ) ≥ (1 − r1 ) and M (x2 , z, t2 ) ≥ (1 − r2 ).

By the Definition 1-(vi) we have M (x1 , z, t1 )∗M (x2 , z, t2 ) ≥ (1−r1 )∗(1−r2 ) and by the condition (FM-4) of fuzzy metric we get M (x1 , x2 , t1 +t2 ) ≥ (1−r1 )∗(1−r2 ). 

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4

The converse of Lemma 1 may not be true. Example 4 explain this situation. Example 4. Let X = N. Define a ∗ b = a.b for all ∀a, b ∈ [0, 1] and let M be fuzzy set on N × N × (0, ∞) as follows: M (x, y, t) =

min {x, y} + t . max {x, y} + t

In this case we know that M is a fuzzy metric on N. If we choose t1 = 1, t2 = 1, r1 = 0.3, r2 = 0.5, x1 = 3 and x2 = 10 then the inequality M (x1 , x2 , t1 + t2 ) ≥ ¯M (3, 0.3, 1)∩ B ¯M (10, 0.5, 1) = ∅ and so we can not (1−r1 )∗(1−r2 ) is satisfied but B find any point z ∈ X such that M (x1 , z, t1 ) ≥ (1 − r1 ) and M (x2 , z, t2 ) ≥ (1 − r2 ). Consequently, when the converse of Lemma 1 also be true, we give Definition 11. Definition 11. Let (X, M, ∗) be a fuzzy metric space. We say that X is fuzzy metrically convex if for any points x1 , x2 ∈ X and for each pair r1 , t1 > 0 and r2 , t2 > 0 (r1 , r2 ∈ (0, 1) and t1 , t2 ∈ (0, ∞)) such that M (x1 , x2 , t1 + t2 ) ≥ (1 − r1 ) ∗ (1 − r2 ), there exists z ∈ X such that M (x1 , z, t1 ) ≥ (1 − r1 ) and M (x2 , z, t2 ) ≥ ¯M (x1 , r1 , t1 ) ∩ B ¯M (x2 , r2 , t2 ). (1 − r2 ) or equivalently z ∈ B Example 5. Let the metric space (X, d) be metrically convex. Define continuous t-norm as a ∗ b = a.b for all ∀a, b ∈ [0, 1] and let M be fuzzy set on X × X × (0, ∞) as follows: M (x, y, t) = e

−d(x,y) t

.

Then the 3-tuple (X, M, ∗) is a fuzzy metric space and under these conditions (X, M, ∗) is fuzzy metrically convex. Indeed, let (X, d) be metrically convex then for any points x1 , x2 ∈ X and positive numbers α and β such that d(x1 , x2 ) ≤ α + β, there exists z ∈ X such that d(x1 , z) ≤ α and d(x2 , z) ≤ β, or equivalently z ∈ B(x1 , α) ∩ B(x2 , β). Take α = −t1 ln(1 − r1 ) and β = −t2 ln(1 − r2 ). By the choices of α, β, the inequality M (x1 , x2 , t1 + t2 ) ≥ (1 − r1 ) ∗ (1 − r2 ) is satisfied and also r1 , r2 ∈ (0, 1). By using the metrically convexity of (X, d); d(x1 , z)

≤ =⇒

−t1 ln(1 − r1 ) and d(x2 , z) ≤ −t2 ln(1 − r2 ) −d(x1 , z) ≥ t1 ln(1 − r1 ) and − d(x2 , z) ≥ t2 ln(1 − r2 )

=⇒

e−d(x1 ,z) ≥ et1 ln(1−r1 ) and e−d(x2 ,z) ≥ et2 ln(1−r2 )

=⇒ =⇒

e t1 ≥ (1 − r1 ) and e t2 ≥ (1 − r2 ) M (x1 , z, t1 ) ≥ (1 − r1 ) and M (x2 , z, t2 ) ≥ (1 − r2 ).

−d(x1 ,z)

−d(x2 ,z)

¯M (x1 , r1 , t1 ) ∩ B ¯M (x2 , r2 , t2 ), then the fuzzy metric space This implies that z ∈ B (X, M, ∗) is fuzzy metrically convex. Definition 12. Let (X, M, ∗) be a metric space, Γ be an index set, ri ∈ (0, 1) and ti ∈ (0, ∞) for all i ∈ Γ. The fuzzy T ¯metric space X is said to has the ball intersection property (BIP in short) if BM (xi , ri , ti ) 6= ∅ for any collection of i∈Γ T ¯M (xi , ri , ti ))i∈Γ such that ¯M (xi , ri , ti ) 6= ∅ for any finite subset closed balls (B B i∈Γf

Γf ⊂ Γ. Definition 13. Let (X, M, ∗) be a metric space, Γ be an index set, ri ∈ (0, 1) and ti ∈ (0, ∞) for all i ∈ Γ. The fuzzy metric space X is said to be fuzzy hyperconvex

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¯M (xi , ri , ti ) in X, satifying the condition if for any indexed class of closed balls B that M (xi , xj , ti + tj ) ≥ (1 − ri ) ∗ (1 − rj ) T ¯ for all i, j ∈ Γ, the intersection BM (xi , ri , ti ) 6= ∅. i∈Γ

Theorem 3. Let (X = R, d) be the usual metric space. Consider the standard fuzzy t metric M where M (x, y, t) = t+d(x,y) with a ∗ b = min {a, b} for all a, b ∈ (0, 1). Then (X, M, ∗) is fuzzy metrically hyperconvex (or fuzzy hyperconvex). ¯ α , rα ) Proof. Since (R, d) is hyperconvex, for any collection of closed balls B(x satisfying the that d(xα , xβ ) ≤ rα + rβ for any α and β in Γ, the inT condition ¯ α , rα ) 6= ∅. Now choose Rα = rα ve Rβ = rβ . It tersection B(x tα +rα tβ +rβ α∈Γ

is clear that Rα , Rβ ∈ (0, 1) and by these choices and the minimum t-norm, M (xα , xβ , tα + tβ ) ≥ (1 − Rα ) ∗ (1 − Rβ ) = min {(1 − Rα ), (1 − Rβ )} = (1 − Rα ) (without lost generality we can take Rα ≥ Rβ ) is satisfied. By the hyperconvexity of (R, d); \ ¯ α , rα ) B(x 6= ∅, for all α ∈ Γ , then there exsists z ∈ X such that α∈Γ

z

∈

\

¯ α , rα ) B(x

α∈Γ

=⇒ =⇒ =⇒ =⇒ =⇒ =⇒

d(xα , z) ≤ rα tα + d(xα , z) ≤ tα + rα tα tα ≥ tα + d(xα , z) tα + rα M (xα , z, tα ) ≥ 1 − Rα ¯M (xα , rα , tα ), for all α ∈ Γ z∈B \ ¯M (xα , rα , tα ). z∈ B i∈Γ

So (R, M, ∗) is fuzzy hyperconvex.



Example 6. In particular if we take t = 1 in the Theorem 3 M becomes M (x, y) = 1 1+d(x,y) . M is stationary fuzzy metric on R with the continuous minimum t-norm and (R, M, ∗) is fuzzy hyperconvex. Proposition 1. If the space (X, M, ∗) is fuzzy hyperconvex then it has the ball intersection property. T ¯ Proof. Let (X, M, ∗) be fuzzy hyperconvex and be BM (xi , ri , ti ) 6= ∅ for any i∈Γf

finite subset Γf ⊂ Γ. Then it follows that \ ¯M (xi , ri , ti ) B 6= ∅, then there exists z ∈ X such that i∈Γf

z

∈

\

¯M (xi , ri , ti ) for i = {1, 2, ..., n} B

i∈Γf

=⇒ =⇒

¯M (x1 , r1 , t1 ) ∩ B ¯M (x2 , r2 , t2 ) ∩ ... ∩ B ¯M (xn , rn , tn ) z∈B M (x1 , z, t1 ) ≥ (1 − r1 ),M (x2 , z, t2 ) ≥ (1 − r2 ), ...,M (xn , z, tn ) ≥ (1 − rn )

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so M (xi , z, ti ) ≥ (1 − ri ) and M (xj , z, tj ) ≥ (1 − rj ) for arbitrary i, j ∈ Γf . By the condition (FM-4) (3.1)

M (xi , xj , ti + tj ) ≥ M (xi , z, ti ) ∗ M (xj , z, tj ) ≥ (1 − ri ) ∗ (1 − rj ).

Since (X, M, T ∗) is fuzzy hyperconvex and the inequality (3.1) is satisfied for all ¯M (xi , ri , ti ) 6= ∅ for all i ∈ Γ and so (X, M, ∗) has the ball i, j ∈ Γ, then B i∈Γ

intersection property.



Theorem 4. Any fuzzy metric space (X, M, ∗) which has the ball intersection property is complete. In particular any fuzzy hyperconvex metric space is complete. Proof. Let (X, M, ∗) be a fuzzy metric space which has ball intersection property and let {xn } be a Cauchy sequence in X. For any n ≥ 1, take the set    rn = sup inf sup {M (xn , xm , s)} . tn >0

m≥n

s 0 there exists n1 ∈ N such that M (xn , z, tn ) > 1 − rn for all n ≥ n1 . Therefore, M (xn , z, tn ) converges to 1 when n −→ ∞, for each tn > 0 and (X, M, ∗) is complete.  Proposition 2. Fuzzy hyperconvexity is equivalent to the ball intersection property and fuzzy metrically convexity. Proof. If (X, M, ∗) is fuzzy hyperconvex, by Proposition 1 X satisfies the ball intersection property and it is easy to see that X is fuzzy convex metric space. ¯M (xi , ri , ti ) and B ¯M (xj , rj , tj ) satisfy the relation Conversely, if two closed balls B M (xi , xj , ti +tj ) ≥ (1−ri )∗(1−rj ), they must intersect since X has ball intersection property. 

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7

Now we give the definition of fuzzy m−hyperconvexity. Note that fuzzy mhyperconvexity is a weaker property than fuzzy hyperconvexity. The definitions of fuzzy hyperconvexity and fuzzy m−hyperconvexity can be considered structurally similar. Definition 14. Let (X, M, ∗) be a metric space, Γ be an index set such that card(Γ)< m, ri ∈ (0, 1) and ti ∈ (0, ∞) for all i ∈ Γ. The fuzzy metric space X is ¯M (xi , ri , ti ) said to be fuzzy m−hyperconvex if for any indexed class of closed balls B in X, satifying the condition that M (xi , xj , ti + tj ) ≥ (1 − ri ) ∗ (1 − rj ) T ¯ for all i, j ∈ Γ, the intersection BM (xi , ri , ti ) 6= ∅. i∈Γ

Proposition 3. (i) It is clear that fuzzy hyperconvexity is stronger than fuzzy m−hyperconvexity, which is stronger than fuzzy n−hyperconvexity if n < m. (ii) It is easy to see that every fuzzy metric space (X, M, ∗) is fuzzy 1−hyperconvex. Theorem 5. For m = 3, fuzzy 3−hyperconvexity is equivalent to fuzzy metrically convexity. Proof. Let (X, M, ∗) be fuzzy 3−hyperconvex. Since card(Γ)< m = 3, the index set Γ is Γ = {1, 2}. It follows that for any points x1 , x2 ∈ X and for each pair r1 , t1 > 0 and r2 , t2 > 0 (r1 , r2 ∈ (0, 1) and t1 , t2 ∈ (0, ∞)) such that M (x1 , x2 , t1 + t2 ) ≥ (1 − r1 ) ∗ (1 − r2 ), there exists z ∈ X such that M (x1 , z, t1 ) ≥ (1 − r1 ) and M (x2 , z, t2 ) ≥ (1 − r2 ). This means that (X, M, ∗) is fuzzy metrically convex. Conversely, Let (X, M, ∗) be fuzzy metrically convex. Then for Γ = {1, 2}, we ¯M (x1 , r1 , t1 ) ∩ B ¯M (x2 , r2 , t2 ) 6= ∅. So for any indexed class of closed balls have B ¯ BM (xi , ri , ti ) in X, satifying the condition that M (xi , xj , ti + tj ) ≥ (1 − ri ) ∗ (1 − rj ) for all i, j ∈ Γ = {1, 2}, the intersection

2 T ¯M (xi , ri , ti ) 6= ∅. So (X, M, ∗) is fuzzy B i=1

3−hyperconvex.



Definition 15. A fuzzy metric space (X, M, ∗) is called m−seperable if it contains a dense subset of cardinal (K) < m where K ⊂ Γ, Γ is index set. (This definition is the same with Definition 10 except for the spaces.) Note that when n < m, m−seperability is weaker than n−seperability for any fuzzy metric space (X, M, ∗). m−seperability for a finite cardinal m means that the fuzzy metric space (X, M, ∗) is a finite set, and at the same time it contains at most m − 1 points. Theorem 6. If the fuzzy metric space (X, M, ∗) is fuzzy m−hyperconvex and at the same time m−seperable, then it is fuzzy hyperconvex. ¯M (xi , ri , ti ) satisfying Proof. Consider an arbitrary indexed family of closed balls B the condition that M (xi , xj , ti + tj ) ≥ (1 − ri ) ∗ (1 − rj ), for all i, j ∈ Γ. Let X be fuzzy m-hyperconvex and let {pk }, k ∈ K with card(K)< m, be an indexed set of 0 0 points, which is dense in X. Take the pair of rk , tk > 0 as follows, respectively 0

0

(3.2) rk , tk = 00 the infimum of all r ∈ (0, 1) and the infimum of all t > 0 ¯M (xi , ri , ti ) ⊂ B ¯M (pk , r, t)00 . such that ∃ i ∈ Γ with B

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¯M (pk , r0 , t0 ), k ∈ K, satisfies the Now we claim that the class of closed balls B k k requirement of fuzzy m−hyperconvexity. So indeed, take any indices k, l ∈ K and 0 arbitrary ε ∈ (0, 1) and arbitrary ε > 0. By (3.2) there exist i, j ∈ Γ such that ¯M (xi , ri , ti ) ⊂ B ¯M (pk , rk0 + ε, t0k + ε0 ) B

(3.3) and

¯M (xj , rj , tj ) ⊂ B ¯M (pl , rl0 + ε, t0l + ε0 ). B ¯M (xi , ri , ti )∩B ¯M (xj , rj , tj ), Since X is fuzzy m−hyperconvex, there exist a point q in B 0 0 0 0 0 ¯ ¯ at the same time by (3.3), (3.4) q is in BM (pk , rk + ε, tk + ε ) ∩ BM (pl , rl + ε, tl + ε0 ). Then,

(3.4)

q

∈ =⇒

(3.5) =⇒

¯M (pk , rk0 + ε, t0k + ε0 ) ∩ B ¯M (pl , rl0 + ε, t0l + ε0 ) B 0

0

0

0

0

0

M (pk , q, tk + ε ) ≥ 1 − (rk + ε) and M (pl , q, tl + ε ) ≥ 1 − (rl + ε) h i h i 0 0 0 0 0 M (pk , pl , tk + tl + 2ε ) ≥ 1 − (rk + ε) ∗ 1 − (rl + ε) .

Since ε ∈ (0, 1) and ε0 > 0 are arbitrary, by (3.5) we find the requirement for ¯M (pk , r0 , t0 ), k ∈ K. So, m−hyperconvexity for the collection of closed balls B k k T ¯ 0 0 there is a point x in BM (pk , rk , tk ). k∈K

¯M (xi , ri , ti ), for all i ∈ Γ, i.e. M (x, xi , ti ) ≥ 1−ri Now we need to show that x ∈ B to see the fuzzy hyperconvexity of X. For this, take an arbitrary ε ∈ (0, 1) and arbitrary ε0 > 0. Since the set {pk }, k ∈ K is dense in X, there exists a point pk for each xi ∈ X such that M (xi , pk , ε0 ) > 1 − ε.

(3.6) Therefore

¯M (xi , ri , ti ) ⊂ B ¯M (pk , ri + ε, ti + ε0 ). B 0 ¯M (pk , ri +ε, ti +ε0 ) ¯M (pk , r0 , t0 ) ⊂ B Due to the choices of r and t , it follows that B 0

0

k

k

k

0

k

and so, we get that rk ≤ ri + ε and tk ≤ ti + ε0 .Therefore, by the triangle inequality for fuzzy metric (i.e. the condition (FM-4)), 0

0

M (x, xi , ti + 2ε ) ≥ M (x, pk , ti + ε ) ∗ M (pk , xi , ε0 ) 0

≥ M (x, pk , tk ) ∗ M (pk , xi , ε0 ) 0

> (1 − rk ) ∗ (1 − ε) ≥ [1 − (ri + ε)] ∗ (1 − ε). Since ε and ε0 is arbitrary, M (x, xi , ti ) ≥ 1−ri . This means that x ∈

T ¯ BM (xi , ri , ti ) i∈Γ

and so X is fuzzy hyperconvex.



Remark 4. It is clear that if the fuzzy metric space (X, M, ∗) is m−seperable and the space X has finite number of points, then we can not mention fuzzy m−hyperconvexity. So indeed, since fuzzy m−hyperconvexity (m ≥ 3) implies the fuzzy metrically convexity (Proposition 2), X can not be a finite set except when the set is reduced to a single point. Consequently, Teorem 6 indicate this situation i.e. if the fuzzy metric space (X, M, ∗) is fuzzy m− hyperconvex and m−seperable then (X, M, ∗) is fuzzy hyperconvex.

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9

References [1] A.George and P.Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems, 64(3), 395-399 (1994). [2] A.George and P.Veeramani, Some theorems in fuzzy metric spaces, Journal of Fuzzy Mathematics, 3, 933-940 (1995). [3] A.George and P.Veeramani, On some results of analysis for fuzzy metric spaces, Fuzzy Sets and Systems, 90(3), 365-368 (1997). [4] A.Thanithamil and P.Thirunavukarasu, Some Results in Fuzzy Metric Spaces with Convex Structure, International Journal of Mathematical Analysis, 8(57), 2827-2836 (2014). [5] B.Miesch, M.Pav´ on, Ball intersection properties in metric Spaces, arXiv preprint arXiv:1610.03307 (2016). [6] B.Schweizer and A.Sklar, Statistical metric spaces, Pacific J. Math, 10(3), 313-334 (1960). [7] B.K.Sharma and C.L.Dewangan, Fixed point theorem in convex metric space, Novi Sad Journal of Mathematics, 25(1), 9-18 (1995). [8] H.Vosoughi and S.J.Hosseini Ghoncheh, Extension of fuzzy contraction mappings, Iranian Journal of Fuzzy Systems, 9(5), 1-6 (2014). [9] K.Menger, Untersuchungen u ¨ber allegemeine Metrik, Math. Ann., 100, 75–63 (1928). [10] N.Aronszajn and P.Panitchpakdi, Extension of uniformly continuous transformations and hyperconvex metric spaces, Pacific J. Math, 6, 405–439. MR 18:917c (1956). [11] R.Esp´ınola and M.A.Khamsi, Introduction to hyperconvex spaces, in Handbook of Metric Fixed Point Theory (W. A. Kirk and B. Sims, eds.), Kluwer Acad. Publ., Dordrecht,2001,pp.391-435. [12] S.N.Jeˇsi´ c, N.A.Babaˇ cev and R.M.Nikoli´ c, A common fixed point theorem in fuzzy metric spaces with nonlinear contractive type condition defined using Φ-function, In Abstract and Applied Analysis (Vol. 2013). Hindawi Publishing Corporation (2013). [13] S.Park, Fixed point theorems in hyperconvex metric spaces, Nonlinear Analysis: Theory, Methods & Applications, 37(4), 467-472 (1999). [14] T.Shimizu, Fixed Points of Multivalued Nonexpansive Mappings in Certain Convex Metric Spaces, Nonlinear Analysıs and Convex Analysıs, (1998). [15] W.Takahashi, A convexity in metric space and nonexpansive mappings, I. In Kodai Mathematical Seminar Reports (Vol. 22, No. 2, pp. 142-149). Department of Mathematics, Tokyo Institute of Technology (1970). Graduate School of Natural and Applied Science Gazi University Teknikokullar, Ankara, 06500, TURKEY E-mail address: [email protected] Department of Mathematics Faculty of Science Gazi University Teknikokullar, Ankara, 06500, TURKEY E-mail address: [email protected]

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ON GENERALIZATIONS OF A REVERSE HARDY-HILBERT’S TYPE INEQUALITY ZHENGPING ZHANG, GAOWEN XI Abstract. By introducing a parameter α and using the expression of the β function establishing the inequality of the weight coefficient, a generalizations of the reverse Hardy-Hilbert’s type inequality is proved. As applications, some equivalent form and a number of particular cases are obtained.

Let p > 1, then

1 p

+

1 q

1. Introduction P∞ q P p = 1, an ≥ 0, bn ≥ 0, and 0 < ∞ n=0 bn < ∞, n=0 an < ∞, 0
1, 1r + 1s = 1, t ∈ [0, 1], r s 2 − min{r, s}t < λ ≤ 2 − min{r, s}t + min{r, s}). In [5] and [6], Xi gave a generalizations and reinforcements of inequalities (1.2: ∞ X ∞ X

am bn < max(mλ , nλ ) n=1 m=1

(

∞ X

" κ (λ ) −

( ×

∞ X n=1

(∞  X

∞ X ∞ X n=1 m=1

max{mλ

3qn

n=1

n1−λ apn

q+λ−2 q

" 3pn

p+λ−2 p



) 1q

#

1

κ (λ ) −

) p1

#

1

n1−λ bqn

,

(1.8)



) p1

am bn 1 1 B < κ(λ) − q+λ−2 − n1−λ apn λ + A, n + B} 3q 1 + B n q n=1 (∞  ) 1q   X 1 1 A × κ(λ) − p+λ−2 − n1−λ bqn , (1.9) 3p 1 + A p n n=1

p qλ 1 1 where κ(λ) = (p+λ−2)(q+λ−2) > 0, 2−min{p, q} < λ ≤ 2, 0 ≤ A ≤ B ≤ min{ 3p−1 , 3q−1 }. For the reverse Hardy-Hilbert’s inequality, recently, Yang [12] gave a reverse form of inequalities (1.5), (1.6) and (1.7) for λ = 2. In [4], Xi gave an extension of the above Yang’s work for 1.5 ≤ λ < 3:

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ON GENERALIZATIONS OF A REVERSE HARDY-HILBERT’S TYPE INEQUALITY 3

∞ X ∞ X

  ∞ X (n + 1)2−λ (λ − 1)2 2 1− apn > 2 λ − 1 n=0 2n + 3 − λ 4(n + 1) )1/q (∞ X (n + 1)2−λ bqn × , 2n + 3 − λ n=0 (

am bn (m + n + 1)λ n=0 m=0

)1/p

(1.10)

where 0 < p < 1, p1 + 1q = 1, 1.5 ≤ λ < 3 and an ≥ 0, bn > 0, such that 0 < P∞ (n+1)2−λ apn P∞ (n+1)2−λ bqn n=0 2n+3−λ < ∞, 0 < n=0 2n+3−λ < ∞. In this paper, by introducing a parameter α and using the expression of the β function establishing the inequality of the weight coefficient. The purpose of this paper is to give a generalization of inequality (1.10). For this, we need the following expression of the β function B(p, q) (see [3]) Z ∞ 1 B(p, q) = B(q, p) = up−1 du (p, q > 0), (1.11) p+q (1 + u) 0 and the following inequality [8]: Z ∞ Z ∞ ∞ X 1 1 1 f (x)dx + f (0) − f 0 (0) f (m) < (1.12) f (x)dx + f (0) < 2 2 12 0 0 m=0 R∞ where f (x) ∈ C 3 [0, ∞), and 0 f (x)dx < ∞, (−1)n f (n) (x) > 0, f (n) (∞) = 0(n = 0, 1, 2, 3). 2. Main Results Lemma 2.1. Let N0 be the set of non-negative integers, N be the set of positive integers and R be the set of real numbers. The weight coefficient ωλ (n, α) is defined by ωλ (n, α) =

∞ X

1 , λ (m + n + α) m=0

n ∈ N0 , 1.5 ≤ λ < 3, α ≥ 1.

Then we have   2(n + α)2−λ (λ − 1)2 1− < ωλ (n, α) (λ − 1)(2n + 2α − λ + 1) 4(n + α)2 2(n + α)2−λ < . (λ − 1)(2n + 2α − λ + 1)

(2.1)

1 Proof If n ∈ N0 , let f (x) = (x+n+α) x ∈ [0, ∞). By (1.12), we obtain λ, Z ∞ dx 1 1 1 ωλ (n, α) > + = + . λ λ λ−1 (x + n + α) 2(n + α) (λ − 1)(n + α) 2(n + α)λ 0

Z

∞

1 λ 1 dx + + λ λ (x + n + α) 2(n + α) 12(n + α)λ+1 0 1 1 λ = + + . λ−1 λ (λ − 1)(n + α) 2(n + α) 12(n + α)λ+1

ωλ (n, α)
0, (2λ−3)(λ−1) ≥ 0, 12(n+α)2 have (2.1). The lemma is proved.

λ(λ−1)2 24(n+α)3

> 0. Then we

Theorem 2.2. Let 0 < p < 1, p1 + 1q = 1, 1.5 ≤ λ < 3, α ≥ 1, and an ≥ 0, bn > 0, such i h P P (n+α)2−λ q (n+α)2−λ (λ−1)2 apn < ∞, 0 < ∞ that 0 < ∞ 1 − n=0 2n+2α−λ+1 bn < ∞. Then we n=0 2n+2α−λ+1 4(n+α)2 have (∞  )1/p ∞ X ∞ X X (n + α)2−λ  am bn 2 (λ − 1)2 > 1− apn λ 2 (m + n + α) λ − 1 2n + 2α − λ + 1 4(n + α) n=0 m=0 n=0 (∞ )1/q X (n + α)2−λ q × b . (2.2) 2n + 2α − λ + 1 n n=0 Proof By the reverse H¨older ’s inequality [2], we have ∞ X ∞ X

∞ X ∞ X am bn am bn = λ · λ λ (m + n + α) p (m + n + α) q n=0 m=0 n=0 m=0 (m + n + α) ( ∞ ∞ ) p1 ( ∞ ∞ ) 1q q p XX X X am bn · ≥ λ (m + n + α) (m + n + α)λ m=0 n=0 n=0 m=0 ( ∞ ) p1 ( ∞ ) 1q X X = ωλ (m, α)apm · ωλ (n, α)bqn · m=0

n=0

Since 0 < p < 1 and q < 0, then by (2.1), we obtain (2.2). The theorem is proved. In Theorem 2.2, for α = 1 we have

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ON GENERALIZATIONS OF A REVERSE HARDY-HILBERT’S TYPE INEQUALITY 5

Corollary 2.3. Let 0 < p < 1, p1 + 1q = 1, 1.5 ≤ λ < 3 and an ≥ 0, bn > 0, such that P P∞ (n+1)2−λ bqn (n+1)2−λ apn 0< ∞ n=0 2n+3−λ < ∞, 0 < n=0 2n+3−λ < ∞. Then we have (∞  )1/p ∞ X ∞ X (n + 1)2−λ  X (λ − 1)2 am bn 2 1− apn > λ 2 (m + n + 1) λ − 1 2n + 3 − λ 4(n + 1) n=0 n=0 m=0 (∞ )1/q X (n + 1)2−λ q × b . (2.3) 2n + 3 − λ n n=0 Remark. Inequality (2.3) is inequality (1.10). Hence, inequality (2.2) is an extension inequality (1.10) Theorem 2.4. Let 0 < p < 1, p1 + 1q = 1, 1.5 ≤ λ < 3, α ≥ 1, and an ≥ 0, such that P (n+α)2−λ apn 0< ∞ n=0 2n+2α−λ+1 < ∞. Then we have #p 1−p " X ∞  ∞ X (n + α)2−λ am 2n + 2α − λ + 1 (m + n + α)λ n=0 m=0  p X   ∞ 2 (n + α)2−λ (λ − 1)2 > 1− ap . (2.4) λ − 1 n=0 2n + 2α − λ + 1 4(n + α)2 n Inequalities (2.4) and (2.2) are equivalent. Proof

Let (n + 1)2−λ bn = 2n + 2α − λ + 1 

1−p " X ∞

am (m + n + α)λ m=0

#p−1 ,

n ∈ N0 .

By (2.2), we have (∞ )p ( ∞  #p )p 1−p " X ∞ X (n + 1)2−λ bq X (n + 1)2−λ am n = 2n + 2α − λ + 1 2n + 2α − λ + 1 (m + n + α)λ n=0 n=0 m=0 (∞ ∞ )p XX am bn = (m + n + α)λ n=0 m=0   p X  ∞ (n + α)2−λ 2 (λ − 1)2 ≥ 1− apn 2 λ − 1 n=0 2n + 2α − λ + 1 4(n + α) (∞ )p−1 X (n + α)2−λ bq n × . 2n + 2α − λ + 1 n=0 Then we obtain #p 1−p " X ∞ ∞ ∞  X X (n + α)2−λ bqn am (n + α)2−λ = 2n + 2α − λ + 1 n=0 2n + 2α − λ + 1 (m + n + α)λ n=0 m=0  p X   ∞ 2 (n + α)2−λ (λ − 1)2 ≥ 1− ap . (2.5) λ − 1 n=0 2n + 2α − λ + 1 4(n + α)2 n

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6

ZHENGPING ZHANG, GAOWEN XI

If

(n+α)2−λ bqn n=0 2n+2α−λ+1

P∞

= ∞, then in view of

  X ∞ ∞ X (n + α)2−λ apn (λ − 1)2 (n + α)2−λ apn 0< 1− ≤

(m + n + α)λ λ−1 m=0   ∞ X (λ − 1)2 (n + α)λ−2 × 1− apn ; 2 2n + 2α − λ + 1 4(n + α) n=0

(n + α)2−λ 2n + 2α − λ + 1

(n+α)λ−2 bqn n=0 2n+2α−λ+1

P∞

∞  X n=0

1−p " X ∞

< ∞, then by (2.2), we find #p  p am 2 > (m + n + α)λ λ−1 m=0   ∞ X (n + α)2−λ (λ − 1)2 × 1− apn . 2 2n + 2α − λ + 1 4(n + α) n=0

(n + α)2−λ 2n + 2α − λ + 1

1−p " X ∞

Hence we obtain (2.4). On the other-hand, by the reverse H¨older ’s inequality [2], we have "∞ # ∞ X ∞ ∞ X X λ−2 1 X am bn a m = (n + α) q (2n + 2α − λ + 1) q λ (m + n + α) (m + n + α)λ n=0 m=0 n=0 m=0 " # bn × λ−2 1 (n + α) q (2n + 2α − λ + 1) q (∞  #p ) p1 1−p " X ∞ 2−λ X (n + α) am ≥ 2n + 2α − λ + 1 (m + n + α)λ n=0 m=0 ) 1q (∞ X (n + α)2−λ bq n . × 2n + 2α − λ + 1 n=0 Hence by (2.4), it follows (∞ ) p1 2 X (n + α)2−λ ap  am bn 2 (λ − 1) n > 1− λ (m + n + α) λ − 1 2n + 2α − λ + 1 4(n + α)2 n=0 m=0 n=0 (∞ ) 1q X (n + α)2−λ bq n × . 2n + 2α − λ + 1 n=0

∞ X ∞ X

Then, (2.4) and (2.2) are equivalent. The theorem is proved. In (2.4), for α = 1, we have

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ON GENERALIZATIONS OF A REVERSE HARDY-HILBERT’S TYPE INEQUALITY 7

P Corollary 2.5. Let 1.5 ≤ λ < 3, 0 < p < 1, p1 + 1q = 1, an ≥ 0, 0 < ∞ n=0 ∞, Then we have #p 1−p " X ∞  ∞ X (n + 1)2−λ am 2n − λ + 3 (m + n + 1)λ n=0 m=0 p X    ∞ (n + 1)2−λ (λ − 1)2 2 1− ap . > λ − 1 n=0 2n − λ + 3 4(n + 1)2 n

(n+1)2−λ apn 2n−λ+3


k. 0 q r q There are two q-analogue of the exponential function ex ∞ X 1 1 xk eq (x) = = , |x| < , |q| < 1, ∞ [k] (1 − (1 − q)x) 1 − q q! q k=0 and Eq (x) =

∞ X

q

k(k−1) 2

k=0

xk = (1 + (1 − q)x)∞ q , |q| < 1, [k]q !

where (1 −

x)∞ q

∞ Y = (1 − q j x). j=0

Our investigation is to construct a linear positive operators generated by generalization of exponential function for defined by [15] ∞ X xn . eµ (x) = γ (n) n=0 µ

Here

and

22k k!Γ k + µ +
γµ (2k) = Γ µ + 21

1 2




22k+1 k!Γ k + µ +
γµ (2k + 1) = Γ µ + 12 The recursion formula for γµ is given by

, 3 2


.

γµ (k + 1) = (k + 1 + 2µθk+1 )γµ (k), k = 0, 1, 2, · · · ,

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3

where µ > − 12 and

( θk =

if k ∈ 2N if k ∈ 2N + 1.

0 1

Sucu [16] defined a Dunkl analogue of Sz´asz operators via a generalization of the exponential function [15] as follows: ∞

Sn∗ (f ; x)

1 X (nx)k := f eµ (nx) k=0 γµ (k)



k + 2µθk n

 ,

(1.3)

where x ≥ 0, f ∈ C[0, ∞), µ ≥ 0, n ∈ N. Cheikh et al., [2] stated the q-Dunkl classical q-Hermite type polynomials and gave definitions of q-Dunkl analogues of exponential functions and recursion relations for µ > − 12 and 0 < q < 1. eµ,q (x) =

∞ X n=0

Eµ,q (x) =

xn , x ∈ [0, ∞) γµ,q (n)

n(n−1) ∞ X q 2 xn

n=0

γµ,q (n)

, x ∈ [0, ∞)

 1 − q 2µθn+1 +n+1 γµ,q (n + 1) = γµ,q (n), n ∈ N, 1−q ( 0 if n ∈ 2N, θn = 1 if n ∈ 2N + 1.

(1.4)

(1.5)



(1.6)

An explicit formula for γµ,q (n) is γµ,q (n) =

(q 2µ+1 , q 2 )[ n+1 ] (q 2 , q 2 )[ n2 ] 2

(1 − q)n

γµ,q (n), n ∈ N.

And some of the special cases of γµ,q (n) are defined as:    1 − q 2µ+1 1 − q2 1 − q 2µ+1 , γµ,q (2) = , γµ,q (0) = 1, γµ,q (1) = 1−q 1−q 1−q     1 − q2 1 − q 2µ+3 1 − q 2µ+1 γµ,q (3) = , 1−q 1−q 1−q      1 − q 2µ+1 1 − q2 1 − q 2µ+3 1 − q4 γµ,q (4) = . 1−q 1−q 1−q 1−q In [4], G¨ urhan I¸c¨oz gave the Dunkl generalization of Sz´asz operators via q-calculus as:   ∞ X 1 ([n]q x)k 1 − q 2µθk +k , (1.7) Dn,q (f ; x) = f eµ,q ([n]q x) k=0 γµ,q (k) 1 − qn for µ > 21 , x ≥ 0, 0 < q < 1 and f ∈ C[0, ∞).

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Previous studies demonstrate that providing a better error estimation for positive linear operators plays an important role in approximation theory, which allows us to approximate much faster to the function being approximated. Motivated essentially by by I¸co¨z [4] the recent investigation of Dunkl generalization of Sz´asz-Mirakjan operators via q-calculus we have showed that our modified operators have better error estimation than [4]. We have proved several approximation results. We have successfully extended these results and modifying the results of papers [4].

2. Construction of operators and moments estimation We modify the q Dunkl analogue of Sz´asz-operators [4]. Let {r[n]q (x)} be a sequence of real-valued continuous functions defined on [0, ∞) with 0 ≤ r[n]q < ∞. Then we define ∞

? Dn,q (f ; x)

X ([n]q r[n]q (x))k 1 = f eµ,q ([n]q r[n]q (x)) k=0 γµ,q (k)



1 − q 2µθk +k 1 − qn

 .

(2.1)

Now, if we replace r[n]q (x) as r[n]q (x) = x − Then for any ∗ Dn,q (f ; x)

1 2n

1 1 1 , where ≤x< and n ∈ N. 2[n]q 2n 1 − qn

≤x


1 2n

and n ∈ N we have

  ∞ X 1 − q 2µθk +k (2[n]q x − 1)k   f . kγ n 2[n]q x−1 2 1 − q µ,q (k) k=0 1

= eµ,q

(2.2)

(2.3)

2

where eµ,q (x), γµ,q are defined in (1.4),(1.6) by [16] and f ∈ Cζ [0, ∞) with ζ ≥ 0 and Cζ [0, ∞) = {f ∈ C[0, ∞) :| f (t) |≤ M (1 + t)ζ , for some M > 0, ζ > 0}. (2.4) ∗ Lemma 2.1. Let Dn,q (. ; .) be the operators given by (2.3). Then for each 1 1 ≤ x < 1−qn , n ∈ N, we have we have the following identities: 2n ∗ (1) Dn,q (1; x) = 1, 1 ∗ (2) Dn,q (t; x) = r[n]q (x) = x − 2[n] , q     e (q[n] r (x)) eµ,q (q[n]q r[n]q (x)) µ,q q [n]q x 1 2µ − 1 − 1 ≤ (3) x2 + q 2µ [1 − 2µ]q eµ,q ([n]q r[n] − 2q [1 − 2µ] q eµ,q ([n]q r (x)) [n]q 4[n]2 [n] (x)) q

q

∗ Dn,q (t2 ; x) ≤ x2 + ([1 + 2µ]q − 1) [n]x q −

Proof.

∗ (1) Dn,q (1; x) =

1 eµ,q ([n]q r[n]q (x))

P∞

k=0

69

1 4[n]2q

q

(2[1 + 2µ]q − 1).

([n]q r[n]q (x))k γµ (k)

= 1.

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(2) ∞

∗ Dn,q (t; x)

X ([n]q r[n]q (x))k 1 = eµ,q ([n]q r[n]q (x)) k=0 γµ (k)



1 − q 2µθk +k 1 − qn



∞

X ([n]q r[n]q (x))k 1 = [n]q eµ,q ([n]q r[n]q (x)) k=1 γµ (k − 1) = x−

1 2[n]q

(3) ∞

∗ Dn,q (t2 ; x)

2 1 − q 2µθk +k 1 − qn   ∞ X ([n]q r[n]q (x))k 1 − q 2µθk +k 1 = [n]2q eµ,q ([n]q r[n]q (x)) k=0 γµ (k − 1) 1−q   ∞ X ([n]q r[n]q (x))k+1 1 − q 2µθk+1 +k+1 1 = . [n]2q eµ,q ([n]q r[n]q (x)) k=0 γµ (k) 1−q

X ([n]q r[n]q (x))k 1 = eµ,q ([n]q r[n]q (x)) k=0 γµ (k)



From [4] we know that [2µθk+1 + k + 1]q = [2µθk + k]q + q 2µθk +k [2µ(−1)k + 1]q ,

(2.5)

Now by separating to the even and odd terms and using (2.5), we get   ∞ X ([n]q r[n]q (x))k+1 1 − q 2µθk+1 +k+1 1 ∗ 2 Dn,q (t ; x) = [n]2q eµ,q ([n]q r[n]q (x)) k=0 γµ (k) 1−q ∞

X ([n]q r[n]q (x))2k+1 [1 + 2µ]q + q 2µθ2k +2k [n]2q eµ,q ([n]q r[n]q (x)) k=0 γµ (2k) ∞

X ([n]q r[n]q (x))2k+2 [1 − 2µ]q + q 2µθ2k+1 +2k+1 . 2 [n]q eµ,q ([n]q r[n]q (x)) k=0 γµ (2k) We know the inequality [1 − 2µ]q ≤ [1 + 2µ]q .

(2.6)

Therefore by using (2.6) we have ∞

∗ Dn,q (t2 ; x)

r[n]q (x)[1 − 2µ]q X (q[n]q r[n]q (x))2k ≥ (r[n]q (x)) + [n]q eµ,q ([n]q r[n]q (x)) k=0 γµ (2k) 2

∞

+

q 2µ r[n]q (x)[1 − 2µ]q X (q[n]q r[n]q (x))2k+1 [n]q eµ,q ([n]q r[n]q (x)) k=0 γµ (2k + 1)

≥ (r[n]q (x))2 + q 2µ [1 − 2µ]q

70

eµ,q (q[n]q r[n]q (x)) r[n]q (x) . eµ,q ([n]q r[n]q (x)) [n]q

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Similarly on the other hand we have ∗ Dn,q (t2 ; x) ≤ (r[n]q (x))2 + [1 + 2µ]q

r[n]q (x) . [n]q

Which completes the proof.  ∗ (. ; .) be given by (2.3). Then for each Lemma 2.2. Let the operators Dn,q 1 x ≥ 2n , n ∈ N, we have 1 ∗ , (t − x; x) = − 2[n] (1) Dn,q q 2 ∗ (2) Dn,q ((t − x) ; x) ≤ [1 + 2µ]q [n]x q −

1 4[n]2q

(2[1 + 2µ]q − 1).

3. Main results We obtain the Korovkin’s type approximation properties for our operators defined by (2.3). Let CB (R+ ) be the set of all bounded and continuous functions on R+ = [0, ∞), which is linear normed space with k f kCB = sup | f (x) | . x≥0

Let H := {f : x ∈ [0, ∞),

f (x) is convergent as x → ∞}. 1 + x2

∗ Remark 3.1. By lemma 2.1, it is clear that the positive liner operators Dn,q given by (2.3) preserve a linear functions, that is for φ(y) = cy + d, c, d ∈ 1 ∗ R(Real numbers), Dn,q (φ; x) = φ(x) for all x ≥ 2n , n ∈ N. 1 Now, fix b > 2 and consider the lattice homomorphism Hb : C[0, ∞] → C[0, b] defined by Hb (f ) = f [0,b] for every f ∈ C[0, ∞], where f [0,b] denotes the restriction of the domain of f to the interval [0, b]. In this case for each j = 0, 1, 2, we have
1  ∗ lim Hb Dn,q (ej ) = Hb (ej ) unif ormly on ,b . (3.1) 2

Thus, by using (3.1) and with the universal Korovkin-type property with respect to the monotone operators. And hence we have the following Korovkintype approximation result. ∗ Theorem 3.2. Let Dn,q (. ; .) be the operators defined by (2.3). Then for any function f ∈ Cζ [0, ∞) ∩ H, ζ ≥ 2, ∗ lim Dn,q (f ; x) = f (x)

n→∞

is uniformly on each compact subset of [0, ∞), where x ∈

71

1 2

, b],

1 2

< b < ∞.

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Proof. The proof is based on Lemma 2.1 and well known Korovkin’s theorem regarding the convergence of a sequence of linear and positive operators, so it is enough to prove the conditions ∗ lim Dn,q ((tj ; x) = xj , j = 0, 1, 2, {as n → ∞}

n→∞

uniformly on [0, 1]. Clearly [n]1 q → 0 (n → ∞) we have ∗ ∗ lim Dn,q (t; x) = x, lim Dn,q (t2 ; x) = x2 .

n→∞

n→∞

Which complete the proof.



We recall the weighted spaces of the functions on R+ , which are defined as follows: Pρ (R+ ) = {f :| f (x) |≤ Mf ρ(x)} ,  Qρ (R+ ) = f : f ∈ Pρ (R+ ) ∩ C[0, ∞) ,   f (x) k + + Qρ (R ) = f : f ∈ Qρ (R ) and lim = k(k is a constant) , x→∞ ρ(x) where ρ(x) = 1 + x2 is a weight function and Mf is a constant depending only (x)| on f . Note that Qρ (R+ ) is a normed space with the norm k f kρ = supx≥0 |fρ(x) . Lemma 3.3. ([3]) The linear positive operators Ln , n ≥ 1 act from Qρ (R+ ) → Pρ (R+ ) if and only if k Ln (ϕ; x) k≤ Kϕ(x), 2 + where ϕ(x) = 1 + x , x ∈ R and K is a positive constant. Theorem 3.4. ([3]) Let {Ln }n≥1 be a sequence of positive linear operators acting from Qρ (R+ ) → Pρ (R+ ) and satisfying the condition lim k Ln (ρτ ) − ρτ kϕ = 0, τ = 0, 1, 2.

n→∞

Then for any function f ∈ Qkρ (R+ ), we have lim k Ln (f ; x) − f kϕ = 0.

n→∞

∗ Theorem 3.5. Let Dn,q (. ; .) be the operators defined by (2.3). Then for each k + function f ∈ Qρ (R ) we have ∗ lim k Dn,q (f ; x) − f kρ = 0.

n→∞

Proof. From Lemma 2.1 and Theorem 3.4 for τ = 0, the first condition is fulfilled. Therefore ∗ lim k Dn,q (1; x) − 1 kρ = 0. n→∞

Similarly From Lemma 2.1 and Theorem 3.4 for τ = 1, 2 we have that ∗ | Dn,q (t; x) − x | 1 1 sup ≤ sup 2 1+x 2[n]q x∈[0,∞) 1 + x2 x∈[0,∞) 1 = , 2[n]q

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which imply that ∗ (t; x) − x kρ = 0. lim k Dn,q

n→∞

∗ | Dn,q (t2 ; x) − x2 | x | [1 + 2µ]q − 1 | sup ≤ 2 1 + x2 [n]q x∈[0,∞) x∈[0,∞) 1 + x 1 1 + | [1 + 2µ]q − 1 | sup 2 2 4[n]q x∈[0,∞) 1 + x

sup

which imply that ∗ lim k Dn,q (t2 ; x) − x2 kρ = 0.

n→∞

This complete the proof.

 4. Rate of Convergence

Here we calculate the rate of convergence of operators (2.3) by means of modulus of continuity and Lipschitz type maximal functions. Let f ∈ CB [0, ∞], the space of all bounded and continuous functions on 1 , n ∈ N. Then for δ > 0, the modulus of continuity of f [0, ∞) and x ≥ 2n denoted by ω(f, δ) gives the maximum oscillation of f in any interval of length not exceeding δ > 0 and it is given by ω(f, δ) = sup | f (t) − f (x) |, t ∈ [0, ∞).

(4.1)

|t−x|≤δ

It is known that limδ→0+ ω(f, δ) = 0 for f ∈ CB [0, ∞) and for any δ > 0 one has   |t−x| | f (t) − f (x) |≤ + 1 ω(f, δ). (4.2) δ ∗ (. ; .) be the operators defined by (2.3). Then for f ∈ Theorem 4.1. Let Dn,q 1 CB [0, ∞), x ≥ 2n and n ∈ N we have ∗ | Dn,q (f ; x) − f (x) |≤ 2ω (f ; δn,x ) ,

where CB [0, ∞) is the space of uniformly continuous bounded functions on R+ , ω(f, δ) is the modulus of continuity of the function f ∈ CB [0, ∞) defined in (4.1) and s x 1 δn,x = [1 + 2µ]q − (2[1 + 2µ]q − 1). (4.3) [n]q 4[n]2q Proof. We prove it by using (4.1), (4.2) and Cauchy-Schwarz inequality we can easily get  
12 1 ∗ ∗ 2 | Dn,q (f ; x) − f (x) | ≤ 1+ Dn,q (t − x) ; x ω(f ; δ) δ if we choose δ = δn,x and by applying the result (2) of Lemma 2.2 complete the proof. 

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Remark 4.2. For the operators Dn,q (. ; .) defined by (1.7) we may write that, for every f ∈ CB [0, ∞), x ≥ 0 and n ∈ N | Dn,q (f ; x) − f (x) |≤ 2ω (f ; λn,x ) ,

(4.4)

where by [4] we have λn,x =

q

Dn,q ((t −

x)2 ; x)

≤

r [1 + 2µ]q

x . [n]q

(4.5)

Now we claim that the error estimation in Theorem 4.1 is better than that 1 1 1 of (4.4) provided f ∈ CB [0, ∞) and x ≥ 2n , n ∈ N. Indeed, for x ≥ 2n , µ ≥ 2n and n ∈ N, it is guarantees that

[1 + 2µ]q

∗ Dn,q ((t − x)2 ; x) ≤ Dn,q ((t − x)2 ; x),

(4.6)

x 1 x − (2[1 + 2µ]q − 1) ≤ [1 + 2µ]q . 2 [n]q 4[n]q [n]q

(4.7)

Which imply that s

x 1 − (2[1 + 2µ]q − 1) ≤ [1 + 2µ]q [n]q 4[n]2q

r x . [1 + 2µ]q [n]q

(4.8)

∗ Now we give the rate of convergence of the operators Dn,q (f ; x) defined in (2.3) in terms of the elements of the usual Lipschitz class LipM (ν). Let f ∈ CB [0, ∞), M > 0 and 0 < ν ≤ 1. The class LipM (ν) is defined as

LipM (ν) = {f :| f (ζ1 ) − f (ζ2 ) |≤ M | ζ1 − ζ2 |ν (ζ1 , ζ2 ∈ [0, ∞))}

(4.9)

∗ Theorem 4.3. Let Dn,q (. ; .) be the operator defined in (2.3).Then for each f ∈ LipM (ν), (M > 0, 0 < ν ≤ 1) satisfying (4.9) we have ν

∗ | Dn,q (f ; x) − f (x) |≤ M (δn,x ) 2

where δn,x is given in Theorem 4.1. Proof. We prove it by using (4.9) and H¨older inequality. ∗ ∗ | Dn,q (f ; x) − f (x) | ≤ | Dn,q (f (t) − f (x); x) | ∗ ≤ Dn,q (| f (t) − f (x) |; x) ∗ ≤ | M Dn,q (| t − x |ν ; x) .

Therefore

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10 ∗ | Dn,q (f ; x) − f (x) |

≤

≤ × ≤

× =

ν ∞ X ([n]q r[n]q (x))k 1 − q 2µθk +k [n]q M − x dt n eµ,q ([n]q r[n]q (x)) k=0 γµ,q (k) 1−q  2−ν ∞  X ([n]q r[n]q (x))k 2 [n]q M eµ,q ([n]q r[n]q (x)) k=0 γµ,q (k) ν ν   ([n]q r[n]q (x))k 2 1 − q 2µθk +k dt − x 1 − qn γµ,q (k) ! 2−ν ∞ 2 X ([n]q r[n]q (x))k n M dt eµ,q ([n]q r[n]q (x)) k=0 γµ,q (k) 2 ! ν2 ∞ X ([n]q r[n]q (x))k 1 − q 2µθk +k [n]q − x dt n eµ,q ([n]q r[n]q (x)) k=0 γµ,q (k) 1−q
ν ∗ M Dn,q (t − x)2 ; x 2 .

Which complete the proof.



Let CB [0, ∞) denote the space of all bounded and continuous functions on R+ = [0, ∞) and CB2 (R+ ) = {g ∈ CB (R+ ) : g 0 , g 00 ∈ CB (R+ )},

(4.10)

k g kCB2 (R+ ) =k g kCB (R+ ) + k g 0 kCB (R+ ) + k g 00 kCB (R+ ) ,

(4.11)

k g kCB (R+ ) = sup | g(x) | .

(4.12)

with the norm

also x∈R+

∗ (. ; .) be the operator defined in (2.3). Then for any Theorem 4.4. Let Dn,q 2 + g ∈ CB (R ) we have    1 δn,x ∗ | Dn,q (f ; x) − f (x) |≤ − + k g kCB2 (R+ ) , 2[n]q 2

where δn,x is given in Theorem 4.1. Proof. Let g ∈ CB2 (R+ ), then by using the generalized mean value theorem in the Taylor series expansion we have (t − x)2 , ψ ∈ (x, t). 2 ∗ By applying linearity property on Dn,q , we have g(t) = g(x) + g 0 (x)(t − x) + g 00 (ψ)

∗ ∗ Dn,q (g, x) − g(x) = g 0 (x)Dn,q ((t − x); x) +

75


g 00 (ψ) ∗ Dn,q (t − x)2 ; x , 2

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which imply that    1 ∗ 0 + | Dn,q (g; x)−g(x) |≤ − 2[n] k g k + [1 + 2µ]q [n]x q − CB (R ) q From (4.11) we have k g 0 kCB [0,∞) ≤k g kCB2 [0,∞) .    1 ∗ 2 | Dn,q (g; x)−g(x) |≤ − 2[n] k g k + [1 + 2µ]q [n]x q − + CB (R ) q This completes the proof from 2 of Lemma 2.2.

1 4[n]2q

1 4[n]2q

 kg00 k CB (R+ ) (2[1 + 2µ]q − 1) . 2  kgk 2 + C (R ) B (2[1 + 2µ]q − 1) . 2 

Acknowledgement This project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, under grant no. G-312-130-38. The authors, therefore, acknowledge with thanks DSR for technical and financial support. References [1] S.N. Bernstein, D´emonstration du th´eor´eme de Weierstrass fond´ee sur le calcul des probabilit´es, Commun. Soc. Math. Kharkow, 2(13) (2012), 1–2. [2] B. Cheikh, Y. Gaied, M. Zaghouani, A q-Dunkl-classical q-Hermite type polynomials, Georgian Math. J., 21(2) (2014), 125–137. [3] A. D. Gadzhiev, The convergence problem for a sequence of positive linear operators on unbounded sets and theorems analogues to that of PP Korovkin. Soviet Mathematics Doklady, 15(5) (1974), 1453-1436. ˙ c¯ [4] G. I¸ oz, B. C ¸ ekim, Dunkl generalization of Sz´ asz operators via q-calculus, Jour. Ineq. Appl., 284:(2015), 2015. [5] A. Lupa¸s, A q-analogue of the Bernstein operator, In Seminar on Numerical and Statistical Calculus, University of Cluj-Napoca, Cluj-Napoca 9 (1987), 85–92. [6] N.I. Mahmudov, V. Gupta, On certain q-analogue of Sz´asz Kantorovich operators, J. Appl. Math. Comput., 37 (2011), 407–419. [7] M. Mursaleen, K.J. Ansari, Approximation of q-Stancu-Beta operators which preserve x2 , Bull. Malaysian Math. Sci. Soc., DOI: 10.1007/s40840-015-0146-9. [8] M. Mursaleen, A. Khan, Statistical approximation properties of modified q- Stancu-Beta operators, Bull. Malays. Math. Sci. Soc. (2), 36(3) (2013), 683–690. [9] M. Mursaleen, A. Khan, Generalized q-Bernstein-Schurer operators and some approximation theorems, Jour. Function Spaces Appl., Volume (2013), Article ID 719834, 7 pages. [10] M. Mursaleen, Faisal Khan, Asif Khan, Approximation properties for modified q-bernsteinkantorovich operators, Numerical Functional Analysis and Optimization, 36(9) (2015) 1178–1197. [11] M. Mursaleen, Faisal Khan, Asif Khan, Approximation properties for King’s type modified q-Bernstein-Kantorovich operators, Math. Meth. Appl. Sci., 38 (2015) 5242–5252. ¨ u, O. Do˘ [12] M. Orkc¨ gru, Weighted statistical approximation by Kantorovich type q-Sz´asz Mirakjan operators, Appl. Math. Comput., 217 (2011), 7913–7919. ¨ u, O. Do˘ [13] M. Orkc¨ gru, q-Sz´ asz-Mirakyan-Kantorovich type operators preserving some test functions, Applied Mathematics Letters 24 (2011), 1588-1593. [14] G.M. Phillips, Bernstein polynomials based on the q- integers, Ann. Numer. Math., 4 (1997), 511–518. [15] M. Rosenblum, Generalized Hermite polynomials and the Bose-like oscillator calculus, Oper. Theory, Adv. Appl., 73 (1994), 369–396. [16] S. Sucu, Dunkl analogue of Sz´ asz operators, Appl. Math. Comput., 244 (2014), 42–48. [17] O. Sz´ asz, Generalization of S. Bernstein’s polynomials to the infinite interval, J. Res. Natl. Bur. Stand., 45 (1950), 239–245.

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Pointwise error estimates for spherical hybrid interpolation Chunmei Ding

Ming Li

∗

Feilong Cao

Department of Applied Mathematics, College of Sciences, China Jiliang University, Hangzhou 310018, Zhejiang Province, P R China. E-mail: [email protected]

Abstract This paper studies pointwise error estimates for spherical hybrid interpolation, which combines spherical polynomials together with spherical radial basis functions constructed by a strictly positive definite zonal kernel. The study is first carried out in the native space associated with the kernel, and then refined error estimates for a target function in a still smaller space are established. MSC(2000): 41A17, 41A30 Keywords: Sphere; Interpolation; Approximation; Pointwise Error

1

Introduction

In recent years, fitting a surface to scattered data arising from sampling an unknown function defined on an underlying manifold comes up frequently in applied problems. If the underlying manifold is S2 , the unit sphere embedded in the Euclidean space R3 , then there are applications to astrophysics, meteorology, geodesy, geophysics and other areas (see [5, 6, 27]). Amongst approaches for scattered data interpolation and approximation on S2 , many authors have used spherical polynomials or spherical radial basis functions (see [5, 6, 9, 12, 18, 20, 25, 26, 27, 13, 2]). Motivated by the fact that the spherical radial basis functions are helpful to handle scattered data and rapid changes, at the same time, the spherical polynomials contribute to handle the slowly varying large-scale features, a hybrid interpolation scheme was given in [23]. The hybrid interpolation scheme combines spherical radial basis functions together with spherical polynomials, that is a little different from interpolation by radial basis functions constructed from conditionally positive definite kernels (in which case a polynomial part is needed to make the theory work, see [8]). Sloan and Sommariva [23] restricted their attention to the case of strictly positive definite kernels, so that the polynomial component is voluntary rather than forced. This paper studies the hybrid interpolation problem in an appropriate native space Nφ of continuous functions on S2 , which is defined by an underlying strictly positive definite kernel φ. We use the method in [23] to get the pointwise error estimate for the hybrid interpolation. It is known that if φ is smooth, the native space Nφ is small in the sense that it is composed of very smooth functions. That is so called “native space barrier” problem and there are several literatures focus on it. We refer the readers to [10, 11, 15, 16, 17] for more details. In this paper, we combine the approach which was used by Levesley and Sun in [10] with the techniques in [24], and embed the smooth radial basis functions in a larger native space generated by a less smooth kernel ψ and still use the hybrid interpolation associated with the smooth kernel φ to interpolate the target function from the larger native space. In the process of obtaining the corresponding error estimates, we will use the “norming set” method developed by Jetter in [9] and a special case of the general Bernstein-type inequality established by Ditzian [4]. This paper is organized as follows. In Section 2, we give some notations and preliminary results. The hybrid interpolation is introduced and the crucial condition for the scheme to be well defined ∗ Supported

by the National Natural Science Foundation of China (No. 61672477)

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and discussed in Subsection 2.2, and native space and Sobolev space are introduced in Subsection 2.3. Finally, the pointwise errors are estimated in Section 3. In the following, we adopt the following convention regarding symbols. Let C be a positive constant, whose value will be different at different occurrence even within the same formula. The symbol A ∼ B means that there exist positive constant C1 and C2 such that C1 B ≤ A ≤ C2 B.

2

Preliminaries

Let S2 be the unit sphere embedded in the Euclidean space R3 , i.e.,  S2 := x := (x1 , x2 , x3 ) ∈ R3 : x21 + x22 + x23 = 1 . For integer l ≥ 0, the restriction to S2 of a homogeneous harmonic polynomial with degree l is called a spherical harmonic of degree l. The class of all spherical harmonics with degree l is denoted by Hl , and it is well know that spherical harmonics of different degrees are orthogonal with respect to the L2 (S2 ) inner product Z hf, gi :=

f (x)g(x)dω(x), S2

where dω denotes surface measure on S2 . Hence, if we choose an orthogonal basis {Yl,k : k = 1, . . . , 2l + 1} for each Hl , then the set {Yl,k : l = 0, 1, . . . , k = 1, . . . , 2l + 1} is an orthogonal basis for L2 (S2 ). The class of all spherical harmonics with total degree l ≤ L is denoted by PL . LL Of course, PL = l=0 Hl , and the dimension of Hl is 2l + 1 and that of PL is (L + 1)2 . We denote by Lp (S2 ) the space of p-integrable functions on S2 endowed with the norms kf k∞ := kf kL∞ (S2 ) := ess sup |f (x)|,

p = ∞,

x∈S2

and

1/p |f (x)| dω(x) < ∞,

Z kf kp := kf kLp (S2 ) :=

p

1 ≤ p < ∞.

S2

The well known addition formula is given by (see [14]) 2l+1 X

Yl,k (x)Yl,k (y) =

k=1

2l + 1 Pl (x · y), 4π

where Pl is the Legendre polynomial with degree l and dimension three, which is normalized such that Pl (1) = 1, and satisfies the orthogonality relation (see [14]) Z 1 2 Pk (t)Pj (t)dt = δk,j , 2l +1 −1 where the symbol δk,j denotes the usual Kronecker symbol. The addition formula also yields the following useful relation 2l+1 X

|Yl,k (x)Yl,k (y)| ≤

2l+1 X

k=1

2.1

2 Yl,k (x) =

k=1

2l + 1 , 4π

x, y ∈ S2 .

(2.1)

Strictly positive definite kernel

Definition 2.1 (see [27]). A continuous and symmetric function φ : S2 × S2 −→ R is called positive definite kernel, if, for any N ∈ N+ , α = (αi )i=1,...,N ∈ RN and {x1 , . . . , xN } ⊂ S2 , we have N X N X αi αj φ(xi , xj ) ≥ 0. i=1 j=1

When for any N distinct points {x1 , . . . , xN }, the above quadratic form is positive for all α = (αi )i=1,...,N ∈ RN /{0}, then φ is called strictly positive definite kernel.

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A kernel φ is called rotational invariant, if φ(ρx, ρy) = φ(x, y) for all x, y ∈ S2 and for all rotations ρ. It can be shown that a continuous rotational invariant kernel depends only on the distance between x and y, that is, there is a function ϕ : [−1, 1] → R , such that ϕ(xy) = φ(x, y) for all x, y ∈ S2 (see [22]). Therefore, a rotational invariant kernel is also called a zonal kernel in the literature. Schoenberg characterized the positive definite zonal kernels in [21] and the notation of strictly positive definiteness on spheres was first introduced by Xu and Cheney [28]. It is important to characterize all the strictly positive definite functions on spheres and such an endeavor has been taken by Ron and Sun in [19]. In [3], Chen et al. established a necessary and sufficient condition for strictly positive definite zonal kernels: the kernel φ is strictly positive definite and zonal if and only if φ(x, y) =

∞ X l=0

with al ≥ 0 for all l, many odd values of l.

2.2

P∞

l=0

al

2l+1 X

Yl,k (x)Yl,k (y) =

k=1

∞ X (2l + 1)al l=0

4π

Pl (x · y),

lal < ∞ and al > 0 for infinitely many even values of l and infinitely

The hybrid interpolation

Assume that we are given a strictly positive definite kernel φ(·, ·) and a set of distinct points X = {x1 , . . . , xN } ⊂ S2 . Then for a target function f ∈ C(S2 ) we can take the hybrid interpolation for f in the form L 2l+1 N X X X βl,k Yl,k , αj φ(·, xj ) + IX,L f = j=1

l=0 k=1

where we fix L ≥ 0 as the desired degree of the polynomial component of the hybrid interpolation and the coefficients {αj }N j=1 , {βl,k }k=1,...,2l+1, l=0,...,L are determined by the interpolation conditions IX,L f (xi ) = f (xi ), i = 1, . . . , N, (2.2) and also (in order to give a square linear system) the side conditions N X

αj p(xj ) = 0, ∀p ∈ PL .

j=1

In order to give the conditions which will make sure that the interpolation is exist and unique, we shall impose a condition on the point set X. Definition 2.2 (see [23, Definition 3.1]). The set X = {x1 , . . . , xN } ⊂ S2 is said to be PL -unisolvent if p ∈ PL , p(xj ) = 0 for j = 1, . . . , N ⇒ p = 0. For the analysis of the interpolation error in the later sections it is convenient to define a finite-dimensional space VX,L within the interpolation IX,L f lies. VX,L :=

X N

αj φ(·, xj ) + q : q ∈ PL , αj ∈ R for j = 1, . . . , N, and

j=1

N X

 αj p(xj ) = 0, ∀p ∈ PL .

j=1

The following Theorem 2.1 gives a crucial condition for the interpolation to be well defined, whose proof can be find in [23]. Theorem 2.1 Let φ(·, ·) be a strictly positive definite kernel, let L ≥ 0 and X = {x1 , . . . , xN } ⊂ S2 be a set of distinct points which is PL -unisolvent. Then for each f ∈ C(S2 ) there exists a unique IX,L f ∈ VX,L that satisfies the interpolation conditions in (2.2).

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2.3

Native space and Sobolev space

Here and in the other sections we assume that the strictly positive definite kernel φ is zonal and has the expansion φ(x, y) =

∞ X

al

l=0

2l+1 X

Yl,k (x)Yl,k (y)

(2.3)

k=1

P∞ with al > 0 for all l, l=0 lal < ∞, in which case the series of the right side in (2.3) converges uniformly for x, y ∈ S2 . For f, g ∈ L2 (S2 ), they can be represented by their Fourier series f=

∞ 2l+1 X X

fˆl,k Yl,k ,

g=

l=0 k=1

∞ 2l+1 X X

gˆl,k Yl,k ,

l=0 k=1

respectively. With respect to the inner product expressed as (see [27]) (f, g)Nφ =

∞ 2l+1 X X fˆl,k gˆl,k l=0 k=1

al

,

the native space Nφ , which is the subspace of L2 (S2 ), can be defined by ( ) ∞ 2l+1 X X |fˆl,k |2 2 2 Nφ := f ∈ L2 (S ) : kf kNφ = 1, then the space Hs is continuously embedded in C(S2 ), so that Hs is a reproducing kernel Hilbert space.

3

Pointwise error estimates

As we can see that the uniqueness result in Theorem 2.1 ensures the existence and uniqueness of the lagrangians lj := lj,X,L : S2 → R, which is defined by lj ∈ VX,L , lj (xi ) = δi,j , i, j = 1, . . . , N. The following Theorem 3.1 is a little different from the obtained result in [23] and it is the difference that helps us to extend the error estimates for hybrid interpolation to Lp norm in the next section. Theorem 3.1 Let φ ∈ C(S2 × S2 ) be a strictly positive definite kernel defined in (2.3), and let X = {x1 , . . . , xN } ⊂ S2 be a PL -unisolvent set of distinct points on S2 . For f ∈ Nφ , let IX,L f ∈ VX,L be the hybrid interpolation defined in Section 2.2. Then for a fixed x ∈ S2 , we have |f (x) − IX,L f (x)| ≤ kf − IX,L f kNφ Pφ,X,L (x),

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where Pφ,X,L is the power function defined by  Pφ,X,L (x) = φ(x, x) − 2

N X

lj (x)φ(x, xj ) +

j=1

N X N X

1/2 li (x)lj (x)φ(xi , xj )

.

i=1 j=1

Proof. With the help of the reproducing property of φ, we can rewrite the form of IX,L f as   N N N X X X IX,L f (x) = f (xj )lj (x) = (f, φ(·, xj ))Nφ lj (x) = f, φ(·, xj )lj (x) , x ∈ S2 . j=1

j=1

j=1

Nφ

Since we also have f (x) = (f, φ(·, x))Nφ , x ∈ S2 , from the reproducing property of φ, for IX,L f ∈ VX,L , we have, since VX,L ⊂ Nφ ,   N N X X IX,L f, φ(·, x) −  φ(·, xj )lj (x) = IX,L f (x) − lj (x)IX,L f (xj ) = 0, j=1

j=1

Nφ

here the Lagrange representation of IX,L f ∈ VX,L ensures that N X

lj (x)IX,L f (xj ) = IX,L f (x),

∀x ∈ S2 .

j=1

So the pointwise error turns into  f (x) − IX,L f (x) = f − IX,L f, φ(·, x) −

N X

 φ(·, xj )lj (x)

j=1

,

(3.4)

Nφ

and by the Cauchy-Schwarz inequality, we have |f (x) − IX,L f (x)| ≤ kf − IX,L f kNφ Pφ,X,L (x), where Pφ,X,L is the power function defined by



N X

φ(·, x )l (x) Pφ,X,L (x) = φ(·, x) − j j



j=1

,

x ∈ S2 ,

x ∈ S2 .

Nφ

1/2

On using the definition k · k = (·, ·)Nφ and the reproducing property of φ, the power function turns into  1/2 N N X N X X Pφ,X,L (x) = φ(x, x) − 2 lj (x)φ(x, xj ) + li (x)lj (x)φ(xi , xj ) , j=1

i=1 j=1

completing the proof of Theorem 3.1. The following Lemma 3.1 is taken from [27] and it is also established by Sloan and Sommariva in [23]. Lemma 3.1 (see [23, Lemma 5.3]). Let φ ∈ C(S2 × S2 ) be a strictly positive definite kernel on S2 , and assume that X = {x1 , . . . , xN } ⊂ S2 is a PL - unisolvent set of distinct points on S2 . For a fixed x ∈ S2 , we define the quadratic functional Lx : RN → R by Lx (α) := φ(x, x) − 2

N X j=1

αj φ(x, xj ) +

N X N X

αi αj φ(xi , xj ), α = (α1 , . . . , αN ).

i=1 j=1

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Then the minimum of Lx (α) on the set   N   X Mx,L := α ∈ RN : αj p(xj ) = p(x), ∀p ∈ PL ,   j=1

is achieved by the vector (l1 (x), . . . , lN (x)), that is, Lx (l1 (x), . . . , lN (x)) ≤ Lx (α), for all α ∈ Mx,L . Follows from Theorem 3.1 and 3.1, we can easily obtain the next Theorem 3.2. Theorem 3.2 Under the conditions of Theorem 3.1, for a fixed x ∈ S2 , we have 1/2

|f (x) − IX,L f (x)| ≤ kf − IX,L f kNφ (Lx (α))

,

PN for any real number αj := αj (x), j = 1, . . . , N , such that j=1 αj p(xj ) = p(x), for all p ∈ PL , and N N X N X X Lx (α) := φ(x, x) − 2 αj φ(x, xj ) + αi αj φ(xi , xj ). j=1

i=1 j=1

The error estimates are general expressed in terms of the mesh norm of X = {x1 , . . . , xN } ⊂ S2 , which is defined by hX := sup inf d(x, xj ), x∈S2 xj ∈X

where d(x, xj ) = arccos(x · xj ) is the geodesic distance between xj and x. Next, we state the following Lemma 3.2, whose proof can be found in [27, Corollary 17.12]. Lemma 3.2 Suppose that X = {x1 , . . . , xN } ⊂ S2 has mesh norm hX ≤ L ≥ 1. Then there exist functions αj : S2 → R for j = 1, . . . , N such that PN (i) j=1 αj (x)p(xj ) = p(x), ∀p ∈ PL , ∀x ∈ S2 , PN (ii) j=1 |αj (x)| ≤ 2, ∀x ∈ S2 .

1 2L

for some integer

With the above obtained results we can provide the following crucial result about the pointwise error estimate for the hybrid interpolation. Theorem 3.3 Let φ ∈ C(S2 × S2 ) be a strictly positive definite kernel defined by (2.3) and al ∼ (l + 1)−2s , s > 1. Assume that integer L ≥ 1 and that X = {x1 , . . . , xN } ⊂ S2 is a set of distinct points on S2 with mesh norm 1/(2L + 2) < hX ≤ 1/(2L). For f ∈ Nφ , let IX,L f ∈ VX,L be the hybrid interpolation defined in Section 2.2. Then for a fixed x ∈ S2 , we have |f (x) − IX,L f (x)| ≤ Chs−1 X kf − IX,L f kNφ . 1 Proof. Because hX ≤ 2L , it follows that for each x ∈ S2 there exists α = α(x) ∈ RN satisfying (i) and (ii) in Lemma 3.2. For (i), it means that a polynomial p ∈ PL that vanishes at x1 , . . . , xN must vanish identically, which verify that X = {x1 , . . . , xN } ⊂ S2 is a PL -unisolvent set of distinct
1/2 points on S2 . By using Theorem 3.2, we only have to give the estimate of the factor Lx (α) ,

Lx (α) := φ(x, x) − 2

N X j=1

=

αj φ(x, xj ) +

N X N X

αi αj φ(xi , xj )

i=1 j=1

∞ N N N h X X X
i 1 X (2l + 1)al Pl (x · x) − αj Pl (x · xj ) − αj Pl (x · xj ) − αi Pl (xi · xj ) , 4π j=1 j=1 i=1 l=0

in which the terms with l ≤ L vanish by property (i) of Lemma 3.2. Hence Lx (α) :=

∞ N N X N   X X 1 X (2l + 1)al Pl (x · x) − 2 αj Pl (x · xj ) + αi αj Pl (xi · xj ) , 4π j=1 i=1 j=1 l=L+1

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and since |Pl (z)| ≤ 1,

PN

|Lx (α)|

|αj | ≤ 2 and al ∼ (l + 1)−2s , we have   ∞ N N X N X X X 1 ≤ (2l + 1)al 1 + 2 |αj | + |αi ||αj | 4π j=1 i=1 j=1

j=1

l=L+1

≤ C

∞ X

(2l + 1)al ≤ C

l=L+1 Z ∞

≤ C L

∞ X

(l + 1)−2s+1

l=L+1

(x + 1)−2s+1 dx = C(L + 1)−2s+2 ≤ Ch2s−2 . X

With the help of Theorem 3.2, we see that |f (x) − IX,L f (x)| ≤ Chs−1 X kf − IX,L f kNφ . This completes the proof of Theorem 3.3.

References [1] S. C. Brenner, R. L. Scott, The Mathematical Theory of Finite Element Methods, Springer, New York, 1994. [2] F. Cao, M. Li, Spherical data fitting by multiscale moving least squares, Applied Math. Model., 39 (2015) 3448-3458. [3] D. Chen, V. A. Menegatto, X. Sun, A necessary and sufficient condition for strictly positive definite functions on spheres, Proc. Amer. Math. Soc., 131 (2003) 2733-2740. [4] Z. Ditzian, Fractional derivatives and best approximation, Acta. Math. Hungar., 81 (1998) 323-348. [5] G. E. Fasshauer, L. L. Schumaker, Scattered data fitting on the sphere, in Mathematical Methods for Curves and Surfaces II (M. Dælen, T. Lyche, and L. L. Schumaker, eds. ), Vanderbilt University Press, Nashville, TN, (1998) 117-166. [6] W. Freeden, T. Gervens, M. Schreiner, Constructive Approximation on the Sphere, Oxford University Press Inc., New York, 1998. [7] P. B. Gilkey, The Index Theorem and the Heat Equation, Publish or Perish, Boston, MA, 1974. [8] M. v. Golitschek, W. A. Light, Interpolation by polynomials and radial basis functions on spheres, Constr. Approx., 17 (2001) 1-18. [9] K. Jetter, J. St¨ ockler, J. D. Ward, Error estimates for scattered data interpolation on spheres, Math. Comp., 68 (1999) 733-747. [10] J. Levesley, X. Sun, Approximation in rough native spaces by shifts of smooth kernels on spheres, J. Approx. Theory, 133 (2005) 269-283. [11] J. Levesley, X. Sun, Corrigendum to and two open questions arising from the article “Approximation in rough native spaces by shifts of smooth kernels on spheres” [J. Approx. Theory, 133 (2005) 269-283], J. Approx. Theory, 138 (2006) 124-127. [12] Q. T. Le Gia, F. J. Narcowich, J. D. Ward, H. Wendland, Continuous and discrete leastsquares approximation by radial basis functions on spheres, J. Approx. Theory, 143 (2006) 124-133. [13] M. Li, F. L. Cao, Local uniform error estimates for spherical basis functions interpolation, Math. Meth. Applied Sci., 37 (2014) 1364-1376.

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[14] C. M¨ uller, Spherical Harmonics, Lecture Notes in Mathematics, Vol. 17, Springer-Verlag, Berlin, 1966. [15] F. J. Narcowich, R. Schaback, J. D. Ward, Approximation in Sobolev spaces by kernel expansions, J. Approx. Theory, 114 (2002) 70-83. [16] F. J. Narcowich, J. D. Ward, Scattered data interpolation on spheres: Error estimates and locally supported basis functions, SIAM J. Math. Anal., 33 (2002) 1393-1410. [17] F. J. Narcowich, X. Sun, J. D. Ward, H. Wendland, Direct and inverse sobolev error estimates for scattered data interpolation via spherical basis functions, Found. Comput. Math., (2007) 369-390. [18] F. J. Narcowich, X. Sun, J. D. Ward, Approximation power of RBFs and their associated SBFs: A connection, Adv. Comput. Math., 27 (2007) 107-124. [19] A. Ron, X. Sun, Strictly positive definite functions on spheres in Enclidean spaces, Math. Comp., 65 (1996) 1513-1530. [20] R. Schaback, Improved error bounds for scattered data interpolation by radial basis functions, Math. Comp., 68 (1999) 201-216. [21] I. J. Schoenberg, Positive definite functions on spheres, Duke Math. J., 9 (1942) 96-108. [22] E. M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princetion, NJ, 1971. [23] I. H. Sloan, A. Sommariva, Approximation on the sphere using radial basis function plus polynomials, Adv. Comput. Math., 29 (2008) 147-177. [24] I. H. Sloan, H. Wendland, Inf-sup condition for spherical polynomials and radial basis functions on spheres, Math. Comp., 78 (2009) 1319-1331. [25] I. H. Sloan, Polynomial interpolation and hyperinterpolation over general regions, J. Approx. Theory, 83 (1995) 238-254. [26] I. H. Sloan, R. S. Womersley, Constructive polynomial approximation on the sphere, J. Approx. Theory, 103 (2000) 91-118. [27] H. Wendland, Scattered Data Approximation, Cambridge University Press, Cambridge, Uk, 2005. [28] Y. Xu, E. W. Cheney, Strictly positive definite functions on spheres, Proc. Amer. Math. Soc., 116 (1992) 977-981.

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INVESTIGATING DYNAMICS OF THE RATIONAL DIFFERENCE EQUATION xn+1 =

xn−1 A + Bxn xn−1

MALEK GHAZEL, TAHER S. HASSAN, AND AHMED M. MOSALLEM Abstract.

This paper is devoted to investigate the dynamics of the rational dierence equation xn+1 =

xn−1 A + Bxn xn−1

with arbitrary initial conditions A and B as nonzero real numbers. The solution is obtained and analytical study and asymptotic behavior are investigated. The forbidden set is determined. The existence of periodic and oscillatory solutions are discussed. Our results are illustrated with numerical simulations.

1.

Introduction

The study of dierence equation has been of great interest and many spectacular developments have been witnessed in the last decade. They are also used to present many numerical schemes in an easiest manner [116]. This is largely due to the fact that it appears as direct mathematical models describing real life situations in physics and engineering [5, 6], biology [8], game theory [7, 9, 10, 12, 13, 19] and economy [14, 15]. Therefore, the study of behavior and global stability of nonlinear dierence equations is of paramount importance and rational dierence equations are one of the most practical classes of equations. Immense literature is available on the second order dierence equations of the form α + βxn + γxn−1 xn+1 = , A + Bxn + Cxn−1 where α, β , γ , A, B and C and the initial conditions x−1 , x0 are real numbers. In a particular case when γ = C = 0, this equation is known as the rst order Riccati dierence equation which can also be written b in the form xn+1 = a + . The results such as Agarwal et al [17], investigated the global stability, xn periodic nature and solved some particular cases of the dierence equation dxn−l xn−k xn+1 = a + . b − cxn−s Elsayed [18] studied the dynamical behavior and gave the solution of the dierence equation xn−5 . xn+1 = ±1 ± xn−2 xn−5 Aloqeili [11] found the solution of the dierence equation xn−1 xn+1 = . a − xn xn−1 Cinar [20] determined the global stability and obtained the positive solutions of the following dierence equation axn−1 xn+1 = . 1 + bxn xn−1 1991 Mathematics Subject Classication. 34K13, 34K05, 34K20, 39A10. Key words and phrases. Rational dierence equations, Stability, Innite products, Forbidden set, Asymptotic behavior, Periodicity, Oscillation, Numerical Simulation. 1

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2

M. GHAZEL, T. S. HASSAN, AND A. M. MOSALLEM

Elabbasy et al [21, 22] obtained the solution in some particular cases and studied the global stability, periodicity of the following dierence equations

xn+1 = axn −

bxn αxn−k . and xn+1 = cxn − dxn−1 β + γΠkp=0 xn−p

The problem of existence of solutions for a given dierence equation is of great importance. The primary aim is to nd the set F of all initial values at which the solution of the given equation is not dened for all natural number n. The set of this nature is called the forbidden set of the equation. In order to avoid the appearance of the forbidden set, the common assumption used by some authors, while studying rational dierence equations, is to choose positive initial values and coecients. The interest of this problem has increased in the literature recently [2325]. Azizi [26] found the forbidden set of the second order rational Riccati dierence equation. Also, Balibrea et al [27] gave sucient conditions for a rational dierence equation of order two to be not uniformly eventually positive outside a bounded set. Camouzis et al [28] described the forbidden set of the dierence equation xn−1 . xn+1 = p + xn In [29] Sedaghat studied the existence of solutions of certain singular dierence equations. Stevic` [30] studied the domains for which the solutions of some equations and systems of dierence equations are not well-dened. The study of existence of oscillatory solutions (periodic or aperiodic) of dierence equations is in a great concern and it is extremely useful in the behavior of mathematical models describing real live situations, for some results in this area. Ladas [31] studied the oscillation of positive solutions about the positive steady state N in the delay logistic dierence equation   m X Nn+1 = Nn exp r − r pj Nn−j  , j=0

which describes that the population growth is not continuous but seasonal. Matti [32] studied the oscillations in some nonlinear economic relationships modeled by a dierence equations. Sedaghat [33] studied the oscillations and chaos in a discrete model of combat. See also related results [3437]. Motivated by above, in this paper, we will present complete analytical study and asymptotic behavior of the solutions of the more general second order dierence equation xn−1 (1.1) xn+1 = , x0 = c and x−1 = d, A + Bxn xn−1 with arbitrary parameters A and B . To the best of our knowledge, the analysis for convergence, oscillation and periodicity of equation (1.1) have not been considered till now and other results extend and improve existing results in the literature, especially those established in [11, 20]. Throughout the paper we use the convention that N = {0, 1, 2, . . .} ,

m Y

ap = 1 and

p=n

m X

ap = 0, where

p=n

(ap )p is a sequence of real numbers and m < n for m, n ∈ Z and the cases when AB = 0 and A + B 6= 0 are trivial, therefore we will assume that A 6= 0 and B 6= 0. 2.

Stability analysis of the equilibrium points

Before stating stability analysis of the equilibrium points, we begin with the following theorem which will given equilibrium points of Eq. (1.1).

Theorem 1. Let (xn )n≥−1 be a solution of Eq.

(1.1).

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(1) (2)

3

If B(1 − A) ≤ 0, then the Eq. (1.1) has a unique equilibrium point x¯1 = 0. If B(1 − A) > 0, then the Eq. (1.1) has exactly three equilibrium points r

x ¯1 = 0

and x¯2,3 = ±

1−A . B

Proof. Let x¯ be a equilibrium point of Eq. (1.1). It is easy to see that x ¯ = 0 or x ¯2 =

1−A . B

This completes the proof. Now, we will prove the following stability analysis of the equilibrium points for equation (1.1).

Theorem 2. Let (xn )n≥−1 be a solution of Eq. (1.1). Then: (1) For A < 0, the characteristic equation about the equilibrium point x ¯1 has no real roots. (2) For 0 < A < 1, the equilibrium point x ¯1 is a repeller. (3) For A = 1, the equilibrium point x ¯1 is nonhyperbolic. (4) For A > 1, the equilibrium point x ¯1 is locally asymptotically stable. Moreover, for B(1 − A) > 0, (i) The equilibrium points x ¯2,3 are nonhyperbolic. (ii) If 0 < |A| < 1, then the equilibrium points x ¯2,3 are unstable. Proof. Denote by U := (u0 , u1 ) an arbitrary point in the good set of Eq. (1.1) and x¯ be an equilibrium point of Eq. (1.1), recall that the characteristic equation about the equilibrium point x ¯ is dened as (2.2) where qk =

F are (2.3)

λ2 − q0 λ − q1 = 0, ∂F u1 (¯ x, x ¯), k = 0, 1 with F (u0 , u1 ) = . Since the partial derivative of the function ∂uk A + Bu0 u1 −Bu1 ∂F = ∂u0 (A + Bu0 u1 )2

and

∂F A = , ∂u1 (A + Bu0 u1 )2

so, for the equilibrium point x ¯1 = 0, the coecients of the characteristic equation are q0 =

∂F (0, 0) = 0 ∂u0

∂F 1 ¯1 is (0, 0) = . Hence the characteristic equation about the equilibrium point x ∂u1 A 1 (2.4) λ2 − = 0. A Thus, we have the following cases: (1) If A < 0, then the Eq. (2.4) has no real roots. r 1 (2) If 0 < A < 1, then the real roots of Eq. (2.4) are ± , their absolute values are greater than A one which implies that the equilibrium point x ¯1 is a repeller. (3) If A = 1, then the real roots of Eq. (2.4) are ±1, so x ¯1 is nonhyperbolic. (4) If A > 1, then all real roots of Eq. (2.4) have absolute value less than one, so x ¯1 is locally asymptotically stable. In the case when B(1 − A) > 0, two new equilibrium points appear x ¯2 and x ¯3 . According to the Eq. (2.3), the coecients q0 and q1 of their characteristic equations are the same and they are given as q0 = A − 1 and q1 = A, so the characteristic equation about x ¯k , k = 2, 3 is and q1 =

λ2 − (A − 1)λ − A = 0,

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which has −1 and A as real roots, then x ¯2 and x ¯3 are nonhyperbolic. Furthermore, if |A| < 1, then x ¯2 and x ¯3 are unstable. This completes the proof.

3.

Analytical expressions of

(xn )n≥−1

In this section, we give some analytical expressions of the sequence (xn )n≥−1 , where (xn )n≥−1 is a solution of Eq. (1.1).

Theorem 3. Let (xn )n≥−1 be a solution of Eq.

(1.1).

 (3.5)

x2n−1

Then for all integer n ∈ N, 2p−1 X

 k

 A2p + Bcd A     k=0 =d ,  2p   X p=0  2p+1 k A + Bcd A n−1 Y

k=0

and  (3.6)

x2n

2p X

 k

 A2p+1 + Bcd A     k=0 =c .  2p+1   X p=0  2p+2 k A + Bcd A n−1 Y

k=0

Proof. We show it by induction. First we have  x−1

2p−1 X



 A2p + Bcd Ak  −1   Y   k=0 =d  =d 2p   X p=0  2p+1 k A + Bcd A k=0

and



2p X



 A2p+1 + Bcd Ak  −1   Y   k=0 x0 = c  =c 2p+1   X p=0  2p+2 k A + Bcd A k=0

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5

This shows that (3.5) and (3.6) hold for n = 0. Assume (3.5) and (3.6) hold with n replaced by some k ∈ N. From Eq. (1.1) we get

x2(k+1)−1

= x2k+1 =

x2k−1 A + Bx2k x2k−1

2p−1 2p+1  n−1  n−1  Y X Y X k 2p+2 2p d A + Bcd A A + Bcd Ak

=

p=0

p=0

k=0

,

k=0

2p  n−1  Y X A2p+1 + Bcd Ak p=0

h

A

k=0

n−1 Y

A2p+2 + Bcd

p=0

= d

k Y

2p−1 n−1  i Y X Ak + Bcd A2p + Bcd Ak p=0

k=0 2p

A

P2p−1

+ Bcd

A2p+1

p=0

2p+1 X

+ Bcd

k=0 P 2p

k

A

k=0

k=0

!

Ak

.

and

x2(k+1)

= x2k+1+1 = =

x2k A + Bx2k+1 x2k

2p 2p k   k−1 Y  Y X X 2p+1 k 2p+1 c A + Bcd A A + Bcd Ak p=0

p=0

k=0

k  Y

A2p + Bcd

p=1

2p−1 X

Ak

,

k=0



k=0

2p 2p k  k−1 h Y  i X Y X A A2p+1 + Bcd Ak + Bcd A2p+1 + Bcd Ak p=0

= c

k  Y p=0

p=0

k=0

A2p+1 + Bcd A2p+2 + Bcd

2p X

k=0

Ak

k=0 2p+1 X



.

k

A

k=0

This shows that (3.5) and (3.6) hold for k + 1. Therefore, (3.5) and (3.6) hold for n ∈ N. This completes the proof.

Corollary 4. Let (xn )n≥−1 be a solution of Eq. (1) for A 6= 1, (3.7)

x2n−1 = d

n−1 Y p=0

(1.1).

Then:

(A − 1 + Bcd)A2p − Bcd , (A − 1 + Bcd)A2p+1 − Bcd

and x2n = c

n−1 Y p=0

(A − 1 + Bcd)A2p+1 − Bcd . (A − 1 + Bcd)A2p+2 − Bcd

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(2)

for A = 1,

(3.8)

x2n−1 = d

n−1 Y p=0

and x2n = c

n−1 Y p=0

1 + 2pBcd  , 1 + (2p + 1)Bcd

1 + (2p + 1)Bcd  . 1 + (2p + 2)Bcd

Proof. It is sucient to use in the (3.5) and (3.6), the identity p X

xk =

k=0

1 − xp+1 , 1−x

where p is a nonnegative integer and x is a real numbers dierent of one, and the proof is directly obtained.

4.

Main Results

4.1. The forbidden set. The determination of the set of all initial conditions through which the solution of a given dierence equation is dened for all n ∈ N is in general a problem of great diculty. This problem leads to introduce the notion of forbidden set.

Denition 1. (4.9)

Consider a dierence equation of order k in N

xn+1 = F (xn , xn−1 , ..., xn−(k−1) ) for n ∈ N,

where F = F (u0 , u1 , ..., uk−1 ) is a function that maps on some subset Ω in Rk , and let (x0 , x−1 , . . . , x−k+1 ) ∈ Ω be the vector of initial conditions of the Eq. (4.9). The forbidden set of Eq. (4.9) is the set denoted F dened as the set of all vectors of initial conditions (x0 , x−1 , ..., x−k+1 ) through which the solution of Eq. (4.9) is not dened for all positive integer n. The good set G is the complementary in Ω of the forbidden set, consequently, the solution (xn )n of Eq. (4.9) is well dened for all n ∈ N if and only if (x0 , x−1 , ..., x−k+1 ) ∈ G. When we obtain the analytic expression of the solution for a given dierence equation, the determination of the forbidden set becomes more easy to obtain. However it can be gotten in some particular cases by the mean of substitution, in the following Theorem, we give the forbidden set in the case when A = 1.

Theorem 5. Let (xn )n≥−1 be a solution of the Eq. (xn )n≥−1 . If A = 1, then n F = (c, d) ∈ R2

(1.1)

such that cd ∈

and F be the forbidden set of the sequence n −1 nB

oo , n∈N .

Proof. The sequence (xn )n≥−1 satises the equation Bxn+1 xn =

Bxn xn−1 , A + Bxn xn−1

x−1 = d, x0 = c,

Hence, (4.10)

A + Bxn+1 xn = A + 1 −

A . A + Bxn xn−1

Let (yn )n≥0 be the sequence dened as (4.11)

yn := A + Bxn xn−1 ,

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7

So the Eq. (4.10) can be written as

yn+1 = A + 1 −

A , yn

which is a rst order Ricatti dierence equation. If A = 1, then

yn+1 = 2 −

1 yn

for n ∈ N.

Let n ∈ N, for yn to exist a necessary and sucient condition are that for all integer 0 ≤ k ≤ n − 1, yk 6= 0, −1 y0 6= 0 is equivalent to cd 6= , B

y1 6= 0 is equivalent to y0 6= 0 and y0 6=

1 , 2

and

y2 6= 0

1 2o . 2 3

i y0 ∈/ 0, , n

By induction, we can easily prove that for all n ∈ N,

yn 6= 0

i for all k ≤ n + 1, y0 6=

k−1 . k

n−1 n−1 So the forbidden set of the sequence Y = (yn )n≥0 is FY = { , n ∈ N}. Now, let n ∈ N, y0 = n n n−1 −1 is equivalent to 1 + Bcd = which is equivalent to cd = . Thus, the forbidden set of the sequence n nB (xn )n≥−1 is given by n n −1 oo F = (c, d) ∈ R2 such that cd ∈ , n∈N . nB The proof is complete. This results can be immediately found by using Corollary 4. Also, in the case when A 6= 1, the forbidden set F of Eq. (1.1) can be easily obtained by using Corollary 4 as in the following theorem.

Theorem 6. Let (xn )n≥−1 be a solution of the Eq. the sequence (xn )n≥−1 is      F = (c, d) ∈ R2    

such that

(1.1).

Suppose that A 6= 1, then the forbidden set of

 1   A = −1 and cd =   B or n o  (1 − A)An    A 6= −1 and cd ∈ , n ∈ N , B(An − 1)

    

.

   

4.2. Convergence. In this section, we study the asymptotic behavior of a solution of the dierence Eq. (1.1). 4.2.1.

The case when 0 < |A| < 1.

Theorem 7. Let (xn )n≥−1 be a solution of the Eq. (x2n−1 ) and (x2n ) converge.

(1.1).

91

Assume that |A| < 1, then the subsequences

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Proof. Using Corollary 4, we obtain x2n−1

= d

n−1 Y p=0

= d

n−1 Y p=0

= d

n−1 Y

(A − 1 + Bcd)A2p − Bcd (A − 1 + Bcd)A2p+1 − Bcd A−1+Bcd 2p A Bcd A−1+Bcd 2p+1 A Bcd

1− 1− Up ,

p=0

where

A − 1 + Bcd 1 − αA2p with α := . 1 − αA2p+1 Bcd One of the following cases holds: For p big enough, Up is always is in (0, 1) or lies greater than one, this allows us to apply of the Taylor expansion to the sequence (Up )p≥0 which gives that Up :=

Up is asymptotically equivalent to 1 − α(A − 1)A2p , which is the general term of convergent innite product, thus (x2n−1 ) converges. Again by using Corollary 4, we get

x2n

= c

n−1 Y p=0

= c

= c

(A − 1 + Bcd)A2p+1 − Bcd (A − 1 + Bcd)A2p+2 − Bcd

n−1 Y

1−

p=0

1−

n−1 Y

A−1+Bcd 2p+1 A Bcd A−1+Bcd 2p+2 A Bcd

Tp ,

p=0

where

Tp :=

1 − αA2p+1 A − 1 + Bcd with α = . 2p+2 1 − αA Bcd

Hence,

Tp is asymptotically equivalent to 1 − α(1 − A)A2p+1 , the last term is the general term of convergent innite product, then (x2n )n converges. This completed the proof. 4.2.2.

The case when A = −1.

Lemma 8. Let (xn )n≥−1 be a solution of the Eq. (1.1). Assume that A = −1, then (1) The subsequence (x2n−1 )n converges i Bcd ∈ (−∞, 0) ∪ [2, ∞). (2) The subsequence (x2n )n converges i Bcd ∈ (0, 2]. Proof.

1. Replacing A by −1 in Corollary (4), for the subsequence (x2n−1 )n , we obtain

x2n−1

−2 )n 2 − 2Bcd d , (Bcd − 1)n

= d( =

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then

(x2n−1 )n

converges i

9

  |Bcd − 1| > 1, or  Bcd − 1 = 1,

the last system is equivalent to Bcd ∈ (−∞, 0) ∪ [2, ∞). 2. To prove the second part of the Theorem, we replace A by (−1) in Corollary (4) for the subsequence (x2n )n , we get  2Bcd − 2 n x2n = c −2 = c(1 − Bcd)n , then

converges i

(x2n )n

  |Bcd − 1| < 1, or  Bcd − 1 = 1,

the last system holds i Bcd ∈ (0, 2]. As a result, the proof is completed.

Remark 1. Using the computation in the proof of Lemma (8), we can easily deduce that when A = −1, we have (1) If Bcd ∈ (−∞, 0) ∪ (2, ∞), then (x2n−1 ) converges to zero and (|x2n |) goes to innity. (2) If Bcd ∈ (0, 2), then (|x2n−1 |) goes to innity and (x2n ) converges to zero. (3) If Bcd = 2, then the subsequences (x2n−1 ) and (x2n ) are constant, x2n−1 = d and x2n = c. The following theorem is now proved.

Theorem 9. Let (xn )n≥−1 be a solution of the Eq. (1.1). Assume that A = −1, then r

The whole sequence (xn )n≥−1 converges i B > 0 and c = d = ± r

In this case, (xn )n≥−1 is constant and equal ± 4.2.3.

2 . B

2 . B

The case when A = 1.

Theorem 10. Let (xn )n≥−1 be a solution of the Eq. (1.1). Assume that A = 1, then (xn )n≥−1 converges to zero. Proof. Replacing A by 1, then by Eq. (3.8), x2n−1

= d = d

n−1 Y p=0 n−1 Y

1 + 2pBcd  1 + (2p + 1)Bcd

Vp ,

p=0

where (Vp )p≥1 is the sequence dened as (4.12)

Vp = 1 −

Bcd . 1 + (2p + 1)Bcd

It can be easily veried that there exists a positive integer r0 such that for all p ≥ r0 , we have Vp ∈ (0, 1). Therefore, if p is big enough, the x2n−1 is then written in innite series form as

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(4.13)

x2n−1

0 −1  rY   n−1  X =d Vp exp ln Vp .

p=r0

p=0

−1 which is a general term divergence innite series, since for all p ≥ r0 2p X ln Vp goes to −∞, consequently (x2n−1 )n converges to zero. Vp ∈ (0, 1), then the innite series We have ln Vp is equivalent to

p≥r0

Although the proof of the convergence of the subsequence (x2n )n to zero can be done similarly, we describe in order to use its notations in the sequel, from Eq. (3.7), we can see that

x2n

= c = c

n−1 Y p=0 n−1 Y

1 + (2p + 1)Bcd  1 + (2p + 2)Bcd

Wp ,

p=0

where (Wp )p≥0 is the sequence dened as (4.14)

Wp = 1 −

Bcd . 1 + (2p + 2)Bcd

Similarly, it can be easily checked that there exist a positive integer s0 such that for all p ≥ s0 , we have Wp ∈ (0, 1). Hence if p is big enough, the subsequence x2n is then written as (4.15)

x2n = c

0 −1  sY

  n−1  X Wp exp ln Wp . p=s0

p=0

−1 which is a general term divergence innite series, since for all p ≥ s0 2p n−1 X Wp ∈ (0, 1), then the innite series ln Wp goes to −∞, consequently (x2n )n converges to zero. This We have ln Wp is equivalent to

complete the proof of Theorem.

4.2.4.

p=s0

The case when |A| > 1.

Theorem 11. Let (xn )n≥−1 be a solution of the Eq. (1.1). Assume that |A| > 1, then (1) The subsequences (x2n−1 )n and (x2n )n converges. (2) The whole sequence (xn )n≥−1 converge i  A − 1 + Bcd 6= 0    or r  1−A   (1 − A)B > 0 and c = d = ± . B

Proof. We distinguish two cases:

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(1) (I) If A − 1 + Bcd 6= 0, then using Corollary (4),

x2n−1

n−1 Y

(A − 1 + Bcd)A2p − Bcd  (A − 1 + Bcd)A2p+1 − Bcd p=0 Bcd n−1  Y  1 − (A − 1 + Bcd)A2p = d Bcd p=0 A(1 − ) (A − 1 + Bcd)A2p+1 n−1 d Y = Yp , An p=0

= d

where (Yp )p≥0 is the sequence dened as

β 2p Bcd A Yp = and β = . β A − 1 + Bcd 1 − 2p+1 A 1−

It can be easily veried that for p big enough, always Yp is in the interval (0, 1) or lies in the interval (1, ∞). The Taylor expansion applied to the sequence (Yp )p≥0 gives

(Yp )p≥0

1 A

is equivalent to 1 + β( − 1)

1 , A2p

the last term is a general term of convergent innite product so (x2n−1 )n converges to zero. An easy calculus gives that n−1 c Y Zp , x2n = n A p=0 where (Zp )p≥0 is the sequence dened as

1− Zp = 1−

β A2p+1 β

,

A2p+2

we have

1 1 , A A2p+1 the last term is a general term of convergent innite product, so (x2n )n converges to zero. (II) If A−1+Bcd = 0, then the subsequences (x2n−1 )n and (x2n )n are constant x2n−1 = d and x2n = c, so they converge. By the calculus in the preview part of the proof, if A − 1 + Bcd 6= 0, then the whole sequence (xn )n≥−1 converges to zero. When A − 1 + Bcd = 0 that is (Zp )p≥0

(4.16)

is asymptotically equivalent to 1 + β( − 1)

cd =

1−A , B

the subsequences (x2n−1 )n and (x2n )n are constant equal d and c respectively, then the whole sequence (xn )n≥−1 converges if and only if c = d, using Eq. (4.16) the last proposition is r 1−A , for this to can hold it is necessary and sucient that (1−A)B > equivalent to c = d = ± B 0. Hence, the proof is achieved.

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4.3. Oscillation about the equilibrium point x1 = 0. In this section, we study the oscillation the solution of dierence Eq. (1.1) about the equilibrium point x1 = 0.

Theorem 12. Let (xn )n≥−1 be a solution of the Eq. (1.1). Assume that |A| < 1, then the subsequences (x2n−1 )n and (x2n )n converge, then (1) For |A| < 1 (xn )n≥−1

0 −1 0 −1  nY  mY  cd Up Tp < 0,

is oscillatory about zero i

p=0

p=0

where (Up )p , (Tp )p are the sequences dened in the proof of Theorem (7) and n0 , m0 , are integers such that, for all p ≥ n0 , Up is positive and for all p ≥ m0 , Tp is positive. (2) For A = −1, (xn )n≥−1 is oscillatory about zero. (3) For A = 1, (xn )n≥−1

is oscillatory about zero i cd

rY 0 −1

Vp

p=0

(4)

sY 0 −1

Wp < 0,

p=0

where (Vp )p≥1 , (Wp )p≥0 , r0 and s0 are dened in the proof of Theorem (10). For |A| > 1, (xn )n≥−1 is oscillatory about zero i    A − 1 + Bcd = 0 and cd < 0,   or    

  A − 1 + Bcd 6= 0      

 A < −1,     or

and

    A>1

and cd

pY 0 −1 p=0

Yp

qY 0 −1

Zp < 0,

p=0

where (Yp )p≥1 , (Zp )p≥0 , p0 and q0 are dened in the proof of Theorem (11). Proof.

1. For |A| < 1 The sequences (x2n−1 )n and (x2n )n have a constant signs which are these of

d

nY 0 −1

Up and c

p=0

mY 0 −1

Tp ,

p=0

respectively, so we can immediately obtain the aimed result. d 2. For A = −1, in this case x2n−1 = and x2n−1 = c(1 − Bcd)n . Hence, if Bcd − 1 < 0, (Bcd − 1)n then (x2n−1 )n and therefore (xn )n is oscillatory about zero. If Bcd − 1 > 0, then (x2n )n and therefore (xn )n is oscillatory about zero. 3. For A = 1, the Eq. (4.13) and (4.15) give rY 0 −1  n−1  X x2n−1 = d Vp exp ln Vp , p=r0

p=0

and

x2n = c

qY 0 −1

Wp exp

 n−1 X

 ln Wp ,

p=q0

p=0

in this case the sequences (x2n−1 )n and (x2n )n have a constant signs which are these of

d

rY 0 −1

Vp and c

p=0

qY 0 −1

Wp ,

p=0

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respectively, we nd that (xn )n is oscillatory about zero i

cd

rY 0 −1 p=0

Vp

qY 0 −1

Wp < 0.

p=0

4. For |A| > 1, if A − 1 + Bcd = 0, then the subsequences are constant (x2n−1 )n and (x2n )n equal d and c respectively, so (xn )n is oscillatory about zero i cd < 0. If A − 1 + Bcd 6= 0, the the sequence (xn )n≥−1 converges to zero and we have

x2n−1 =

n−1 n−1 d Y c Y Y and x = Zp , p 2n An p=0 An p=0

where (Yp )p and (Zp )p are the sequences dened in the proof of Theorem (11). It has been seen that there exists integers p0 and q0 such that for all p ≥ p0 , Yp is positive and for all p ≥ q0 , Zp n−1 n−1 Y Y is positive, then for n big enough, the sign of d Yp and c Zp are constant. Then, we have p=0

p=0

the following cases: (a) When A < −1, the sequence (x2n−1 )n and consequently (xn )n≥−1 are oscillatory about zero. pY qY 0 −1 0 −1 (b) When A > 1, the sign of x2n−1 is that of d Yp and the sign of x2n is that of c Zp . p=0

Thus, we can immediately have the target result and the proof is complete.

p=0

4.4. Periodicity. Firstly, we recall the following Lemma, which describes sucient conditions for Eq. (1.1) to have a periodic solution.

Lemma 13. Let (xn )n≥−k+1 be a solution of Eq. (1.1). Suppose that there are real numbers lr , 0, 1, ..., p − 1 such that lim xpn+r = lr for all r = 0, 1, ..., p − 1. n→∞ Finally, let (yn )n≥−k+1 be the periodic-p sequence such that yr = lr for all r = 0, 1, ..., p − 1. Then (yn )n≥−k+1 is a periodic-p solution of Eq. (1.1).

r =

Note that the zero sequence is a solution of Eq. (1.1) corresponding to the initial conditions x−1 = 0 and x0 = 0, this solution is called trivial solution of of Eq. (1.1). The periodicity results are given by the following Theorem

Theorem 14. Let (xn )n≥−1 be a solution of the Eq. (1.1). (1) For |A| < 1, Eq. (1.1) has a nontrivial periodic-2 solution. 2 (2) For A = −1, Eq. (1.1) has a nontrivial periodic-2 solution if and only if cd = . B (3) For A = 1, Eq. (1.1) has no nontrivial periodic-2 solution. 1−A (4) For |A| > 1, Eq. (1.1) has a nontrivial periodic-2 solution if and only if cd = . Proof.

B 1. If |A| < 1, then by Theorem (7), the subsequences (x2n−1 )n and (x2n )n converge, let l1 and l0 be their limits respectively. Applying Lemma (13), it follow that the sequence l1 , l0 , l1 , l0 , ...

is a periodic-2 solution of Eq. (1.1). 2. Suppose that A = −1, we distinguish two cases:

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(a) If Bcd 6= 2, then using Lemma (8), every solution of Eq. (1.1) is unbounded, so Eq. (1.1) has no periodic solutions. (b) If Bcd = 2, then using Lemma (8), the subsequences (x2n−1 )n and (x2n )n are constant x2n−1 = d and x2n = c, therefore (xn )n≥−1 is the periodic-2 solution

d, c, d, c, ... . 3. If A = 1, then by using the proof of Theorem (10), every solution of of Eq. (1.1) converges to zero, so Eq. (1.1) has no nontrivial solution. 4. If |A| > 1, we distinguish two cases: (a) If A − 1 + Bcd 6= 0, then by using the proof of Theorem (11), every solution of Eq. (1.1) converges to zero, so Eq. (1.1) has no nontrivial solution. (b) If A−1+Bcd = 0, then by Theorem (11), the subsequences (x2n−1 )n and (x2n )n are constant x2n−1 = d and x2n = c, consequently (xn )n≥−1 is the periodic-2 solution d, c, d, c,. . . This achieves the proof. 5.

Numerical simulation

1 , B = 4, c = 3 and d = 2. 2 The subsequences (x2n−1 )n and (x2n )n converge. This is coherent with Theorem (7). 1 In Fig. (2) (case A = −1 and Bcd ∈ (−∞, 0) ∪ (2, ∞), we choose A = −1, B = , c = 1 2 and d = −2. The subsequence (x2n−1 )n converges to zero and the subsequence (|x2n |)n goes to innity and oscillates about zero which matches Lemma (8), Remark (1) and Theorem (12). 1 The case A = −1 and Bcd ∈ (0, 2) is studied using the parameters values A = −1, B = , c = 3 2 and d = 1. The subsequence (|x2n−1 |)n goes to innity and the subsequence (x2n )n converges to zero as depicted in Fig. (3) which is coherent to Lemma (8), Remark (1) and Theorem (12). 1 In order to illustrate the case A = −1 and Bcd = 2, we choose A = −1, B = , c = 1 and 2 d = 4. In Fig. (4), it is shown that the subsequences (x2n−1 )n and (x2n )n are constant x2n−1 = d and x2n = c which agrees Lemma (8) and Remark (1), consequently (xn )n≥−1 is the periodic-2 solution d, c, d, c, ... . This is in harmony with Theorem (14). The case A = 1 is investigated using the parameters values A = 1, B = 3, c = 0.5 and d = 3. In Fig. (5), the simulation results show that the whole sequence (xn )n≥−1 converges to zero which matches Theorem (10). The case |A| > 1 and A − 1 + Bcd 6= 0 can be taken by choosing A = 5, B = 1, c = 3 and d = 0.5. The whole sequence (xn )n≥−1 converges to zero as depicted in Fig. (6) which is coherent to Theorem (11). Fig. (7) illustrates the case |A| > 1 and A − 1 + Bcd = 0, we choose A = 5, B = 1, c = 2 and d = −2, the subsequences (x2n−1 )n and (x2n )n are constant: x2n−1 = d and x2n = c, we obtain a periodic-2 solution. This case is justied analytically in the proofs of Theorems (11), (12) and (14).

(1) The case |A| < 1 is illustrated in Fig. (1), in which we set A = (2)

(3)

(4)

(5) (6) (7)

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3 4

3 2

2 X 2 n -1

1

X n

X n

X 2 n

1

0 0

b )

a ) -1

-1 0

1 0

2 0

3 0

0

4 0

1 0

2 0

3 0

Figure 1.

4 0

5 0

n n

|A| < 1, A − 1 + Bcd 6= 0: (x2n−1 )n and (x2n )n converge.

2 .0

1 0 0 8 0

1 .5

6 0

a )

b )

4 0

1 .0

2 0 0 .5

X n

X n

0

0 .0

-2 0 -4 0

X 2 n -1

-6 0

-0 .5

X 2 n

-8 0 -1 .0

-1 0 0 0

2 0

4 0

6 0

8 0

1 0 0

0

n

1 0

2 0

3 0

n

Figure 2. A = −1 and Bcd ∈ (−∞, 0)∪(2, ∞): (x2n−1 )n converges to zero and (|x2n |)n goes to innity, the solution is unbounded.

Acknowledgement

The authors thank the Deanship of Research at the University of Hail, Saudi Arabia, for funding this work under Grant no. 0150287.

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1 0

1 0

9 8 8 7 6

X 2 n -1

6

X 2 n

5

X n

X n

4

4 3

2 2

b ) 1

0

a ) 0 -1

-2 0

2 0

4 0

6 0

8 0

0

1 0 0

5

1 0

1 5

2 0

2 5

3 0

3 5

4 0

4 5

5 0

n n

Figure 3. A = −1 and Bcd ∈ (0, 2): (|x2n−1 |)n goes to innity and (x2n )n converges to zero, the solution is unbounded.

5 5

4 4 3 3 X 2 n -1

2

X n

X n

X 2 n

2 1 1 0

b )

a ) 0

-1 0

1 0

2 0

3 0

4 0

5 0 0

1 0

2 0

3 0

n

5 0

A = −1 and Bcd = 2: (x2n−1 )n and (x2n )n are constants, (xn )n is periodic-2 solution. 3

3

2 2

X 2 n -1 X 2 n

X n

1

X n

Figure 4.

4 0

n

1

0 0

a )

b )

-1 0

1 0

2 0

3 0

4 0

5 0

n

Figure 5.

0

5

1 0

1 5

2 0

2 5

3 0

3 5

4 0

4 5

5 0

n

A = 1: the solution converges to zero.

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3 3

2 2

X 2 n -1 X 2 n

b )

X n

X n

a ) 1

1

0 0

-1 0

1 0

2 0

3 0

4 0

0

5 0

5

1 0

1 5

2 0

2 5

n

Figure 6.

3 0

3 5

4 0

4 5

5 0

n

|A| > 1 and A − 1 + Bcd 6= 0: the solution converges to zero.

3 3

2 2

1 1

0 0

X 2 n -1

X n

X n

X 2 n

-1

-1

-2

-2

a )

b )

-3 0

-3 1 0

2 0

3 0

4 0

5 0 0

n

5

1 0

1 5

2 0

2 5

3 0

3 5

4 0

4 5

5 0

n

Figure 7. |A| > 1 and A − 1 + Bcd = 0: (x2n−1 )n and (x2n )n are constants, (xn )n is periodic-2 solution.

References

[1] R. Karatas, C. Cinar, D. Simsek, On positive solutions of the dierence equation xn+1 = xn−5 /(1 + xn−2 xn−5 ), Int. Journal of Contemp. Math. Sciences 1(10) (2006) 494500. ˘anin, On a max-type and a min-type dierence equation, Appl. Math. Comput. 215 [2] E. M. Elsayed, Bratislav D. Iric (2009) 608614. [3] H. A. El-Morshedy, E. Liz, Globally attracting xed points in higher order discrete population models, J. Math. Biology 53 (2006) 365384. [4] H. El-Metwally, E. M. Elsayed, H. El-Morshedy, Dynamics of some rational dierence equations, J. Comput. Anal. Applic. 18 (2015) 9931003. [5] E. M. Elabbasy, A. A. Elsadany, Y. Zhang, Bifurcation analysis and chaos in a discrete reduced Lorenz system, Appl. Math. Comput. 228(1) (2014) 184194. [6] R. P. Agarwal, A. M. A. El-Sayed, S. M. Salman, Fractional-order Chua's system discretization, bifurcation and chaos, Advances in Dierence Equations 320 (2013) 13 pages. [7] S. S. Askar, A. M. Alshamrani, K. Alnowibet, The arising of cooperation in Cournot duopoly games, Appl. Math. Comput. 273 (2016) 535542. [8] A. E. Matouk, A. A. Elsadany, E. Ahmed, H. N. Agiza, Dynamical behavior of fractional-order Hastings-Powell food chain model and its discretization, Communications in Nonlinear Science and Numerical Simulation 27(1-3) (2015) 153167. [9] E. Ahmed, A. S. Hegazi, On dynamical multi-team and signaling games, Appl. Math. Comput. 172(1) (2006) 524530.

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[10] S. S. Askar, The impact of cost uncertainty on Cournot oligopoly game with concave demand function. Appl. Math. Comput. 232 (2014) 144149. [11] A. A. Elsadany, A. E. Matouk, Dynamic Cournot duopoly game with delay, Journal of Complex Systems 2014Article ID 384843 (2014) 7 pages. [12] A. A. Elsadany, H. N. Agiza, E. M. Elabbasy, Complex dynamics and chaos control of heterogeneous quadropoly game, Appl. Math. Comput. 219(24) (2013) 1111011118. [13] E. Ahmed, A. A. Elsadany, Tonu Puu, On Bertrand duopoly game with dierentiated goods, Appl. Math. Comput. 251(15) (2015) 169179. [14] V. L. Kocic, G. Ladas, I.W. Rodrigues, On the rational recursive sequences, J. Math. Anal. Appl. 173 (1993) 127157, Chapman and Hall/CRC Boca Raton (2002). [15] L. A. Moye, A.S. Kapadia, Dierence equations with public health applications, (2000)Marcel Dekker, Inc. [16] A. A. Elsadany, A dynamic Cournot duopoly model with dierent strategies, Journal of the Egyptian Mathematical Society 23(1) (2015) 5661. [17] R. P. Agarwal, E. M. Elsayed, Periodicity and stability of solutions of higher order rational dierence equation, Advanced Studies in Contemporary Mathematics 17(2) (2008) 181201. [18] E. M. Elsayed, Dynamics of a rational recursive sequences, International Journal of Dierence Equations 4(2) (2009) 185200. [19] M. Aloqeili, Dynamics of a rational dierence equation, Appl. Math. Comput. 176(2) (2006) 768774. axn−1 [20] C. Cinar, On the positive solutions of the dierence equation xn+1 = , Appl. Math. Comput. 156 (2004) 1 + bxn xn−1

587590. [21] E. M. Elabbasy, H. El-Metwalli, E. M. Elsayed, On the dierence equation xn+1 = axn − Equ. 2006 Article ID 82579 (2006) 10 pages. [22] E. M. Elabbasy, H. El-Metwalli, E. M. Elsayed, On the dierence equation xn+1 =

bxn , Adv. Dier. cxn − dxn−1

αxn−k Qk

β+γ

i=0

xn−i

, J. Conc. Appl.

Math. 5(2) (2007) 101113. [23] E. A. Grove, G. Ladas, Periodicities in Nonlinear Dierence Equations, Chapman and Hall/CRC Press, London/Boca Raton (2005). [24] M. R. S. Kulenovic, G. Ladas, Dynamics of Second Order Rational Dierence Equations with Open Problems and Conjectures, Chapman and Hall/CRC Press, London/Boca Raton (2002). [25] J. Rubi-Masseg, Global periodicity and openness of the set of solutions for discrete dynamical systems, J. Dier. Equ. Appl. 15 (2009) 569578. [26] R. Azizi, Global behaviour of the rational Riccati dierence equation of order two: the general case, J. Dier. Equ. Appl. 18 (2012) 947961. [27] F Balibrea and A Cascales, Eventually positive solutions in rational dierence equations, Comp and Math with Appl 64(7) (2012) 22752281. xn−1 , Special Session of the American Mathematical [28] E. Camouzis, R. Devault, The forbidden set of xn+1 = p + xn

[29] [30] [31] [32] [33] [34] [35] [36] [37]

Society Meeying, Part II, San Diego (2002). H Sedaghat, Existence of solutions of certain singular dierence equations, J. Dier. Equ. Appl., 6 535561 (2000). ˘, Domains of undenable solutions of some equations and systems of dierence equations, Appl Math Comput, S Stevic 219 1120611213 (2013). G. Ladas, Recent developments in the oscillation of delay dierence equations, In: Int Conf. on Dierential Equations, Theory Appl. Stab. Control, pp (1989) 710. L. Matti, Oscillations in some nonlinear economic relationships, Chaos, Solitons and Fractals, 7 (1996) 22352245. H. Sedaghat, Converges, oscillations, and chaos in a discrete model of combat, SIAM 44 (2002) 7492. L. Erbe, T. S. Hassan, A. Peterson, S. H. Saker, Interval oscillation criteria for forced second-order nonlinear delay dynamic equations with oscillatory potential. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 17 (2010) no. 4 533542. T. S. Hassan, Interval oscillation for second order nonlinear dierential equations with a damping term. Serdica Math. J. 34 (2008) no. 4 715732. E. M. Elabbasy, T. S. Hassan, Interval oscillation for second order sublinear dierential equations with a damping term. Int. J. Dyn. Syst. Dier. Equ. 1 (2008) no. 4 291299. T. S. Hassan, Oscillation criteria for second-order nonlinear dynamic equations. Adv. Dierence Equ. 2012, 2012:171 13 pp.

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19

Department of Mathematics, Faculty of Science,University of Hail, Hail 2440, Saudi Arabia.

E-mail address : [email protected]

Department of Mathematics, Faculty of Science, Mansoura University Mansoura, 35516, Egypt.

E-mail address : [email protected]

Department of Mathematics, Faculty of Science,University of Hail, Hail 2440, Saudi Arabia.

E-mail address : [email protected]

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Lp approximation errors for hybrid interpolation on the unit sphere ∗ Chunmei Ding

Ming Li

Feilong Cao

Department of Applied Mathematics, College of Sciences, China Jiliang University, Hangzhou 310018, Zhejiang Province, P R China. E-mail: [email protected]

Abstract This paper discusses Lp approximation error estimates for hybrid interpolation on the unit sphere. This interpolation scheme is integrated by spherical polynomials and radial basis functions. The smooth radial basis functions generated by a strictly positive definite zonal kernel are embedded in a larger native space generated by a less smooth kernel, and the error estimates for hybrid interpolation to a target function from the larger native space are given. In a sense, the results of this paper show that the hybrid interpolation associated with the smooth kernel enjoys the same order of error estimate as hybrid interpolation associated with the less smooth kernel for a target function from the rough native space. MSC(2000): 41A17, 41A30 Keywords: Sphere; Interpolation; Approximation; Error

1

Introduction

Recently, fitting spherical scattered data comes up in many application areas, such as astrophysics, meteorology, geodesy, geophysics, and so on [5, 6, 29]. As interpolation or approximation tools, spherical polynomials or spherical radial basis functions were used to handle spherical scattered data in more studies [5, 6, 11, 14, 20, 22, 27, 28, 29, 15, 2]. Since spherical polynomials can handle the slowly varying large-scale features, and spherical radial basis functions are helpful to handle scattered and rapidly changed data, Sloan and Sommariv [25] introduced a hybrid interpolation scheme, which combines spherical radial basis functions together with spherical polynomials, and restricts the radial basis functions to the case of strictly positive definite kernels, so that the polynomial component is voluntary rather than forced. This paper studies the hybrid interpolation in an appropriate native space Nφ of continuous functions on the unit sphere, which is defined by a underlying strictly positive definite kernel φ. We apply the approach used by Hubbert and Morton [9, 10] to obtain error estimates in Lp norm. However, if the target function is from a subspace of the native space Nφ , we then adopt the inf-sup condition [26] and the method of constructing a convolution kernel to improve the error estimates. So called “native space barrier” problem means that if φ is smooth, then the native space Nφ is small. There have been much literature to focus on it, for example, [12, 13, 17, 18, 19]. In this paper, we employ the approach in [12] and the techniques in [26], and embed the smooth radial basis functions in a larger native space generated by a less smooth kernel ψ. At same time, we utilize the hybrid interpolation associated with the smooth kernel φ to interpolate the target function from the larger native space. In the process of error estimates, the “norming set” method developed by Jetter [11] and a special case of the general Bernstein-type inequality in [4] are used. This paper is organized as follows. Section 2 is preliminary, which is related to introducing notations, hybrid interpolation and its crucial condition, native space, and Sobolev space. The Lp approximation error estimates are established in Section 3. In Section 4, for a target function f ∗ Supported

by the National Natural Science Foundation of China (No. 61672477)

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C. Ding et al.: Lp approximation error for hybrid interpolation on the unit sphere

in a subspace of the original native space, we improve the global Lp -error estimates. In Section 5, we still use the hybrid interpolation defined in Section 2 to interpolate and approximate a target function f from a larger native space generated by a less smooth kernel.

2

Preliminaries

This paper uses C to denote a positive constant, whose value may be different at different occurrence even within the same formula. The symbol A ∼ B means that there exist positive constant C1 and C2 such that  C1 B ≤ A ≤ C2 B. We use S2 := x := (x1 , x2 , x3 ) ∈ R3 : x21 + x22 + x23 = 1 to denote the unit sphere embedded in the Euclidean space R3 , and denote by Lp (S2 ) the space of p-integrable functions on S2 endowed with the norms kf k∞ := kf kL∞ (S2 ) := esssupx∈S2 |f (x)|(p = ∞), and kf kp := kf kLp (S2 ) := R 1/p |f (x)|p dω(x) < ∞(1 ≤ p < ∞). The so called spherical harmonic with degree l is the S2 restriction to S2 of a homogeneous harmonic polynomial with degree l ≥ 0. The class of all spherical harmonics with degree l is denoted by Hl , and the class of all spherical harmonics with total degree l ≤ L is denoted by PL . Clearly, spherical harmonics with different degrees are R orthogonal with respect to the L2 (S2 ) inner product: hf, gi := S2 f (x)g(x)dω(x), where dω is surface measure on S2 . P2l+1 The famous addition formula k=1 Yl,k (x)Yl,k (y) = 2l+1 4π Pl (x · y) yields the following useful relation [16]: 2l+1 2l+1 X X 2l + 1 2 Yl,k (x) = |Yl,k (x)Yl,k (y)| ≤ , x, y ∈ S2 . (2.1) 4π k=1

k=1

Here Pl is the Legendre polynomial with degree l and dimension three, which is normalized such R1 2 that Pl (1) = 1, and satisfies the orthogonality relation: −1 Pk (t)Pj (t)dt = 2l+1 δk,j , where the symbol δk,j denotes the usual Kronecker symbol. The definition of strictly positive definite kernel is given by Definition 2.1 (see [29]). A continuous and symmetric function φ : S2 × S2 −→ R is called positive definite kernel, if, for any N ∈ N+ , α = (αi )i=1,...,N ∈ RN and {x1 , . . . , xN } ⊂ S2 , we have N N X X αi αj φ(xi , xj ) ≥ 0. i=1 j=1

When for any N distinct points {x1 , . . . , xN }, the above quadratic form is positive for all α = (αi )i=1,...,N ∈ RN /{0}, then φ is called strictly positive definite kernel. We say that a kernel φ is called rotational invariant if φ(ρx, ρy) = φ(x, y) for all x, y ∈ S2 and for all rotations ρ. So a continuous rotational invariant kernel depends only on the distance between x and y [24], that is, there is a function ϕ : [−1, 1] → R , such that ϕ(xy) = φ(x, y) for all x, y ∈ S2 . Therefore, a rotational invariant kernel is also called a zonal kernel. In [23], Schoenberg characterized the positive definite zonal kernels. In [30], Xu and Cheney introduced the notation of strictly positive definiteness on the sphere. Clearly, it is important to characterize all the strictly positive definite functions on the sphere, and such an endeavor has been taken by Ron and Sun in [21]. In [3], Chen et al. established a necessary and sufficient condition for strictly positive definite zonal kernels: the kernel φ is strictly positive definite and zonal if and only if φ(x, y) =

∞ X l=0

with al ≥ 0 for all l, many odd values of l.

P∞

l=0

al

2l+1 X

Yl,k (x)Yl,k (y) =

k=1

∞ X (2l + 1)al l=0

4π

Pl (x · y),

lal < ∞ and al > 0 for infinitely many even values of l and infinitely

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C. Ding et al.: Lp approximation error for hybrid interpolation on the unit sphere

For given strictly positive definite kernel φ(·, ·), a set of distinct points X = {x1 , . . . , xN } ⊂ S2 , and target function f ∈ C(S2 ), we take the hybrid interpolation for f in the form IX,L f =

N X

αj φ(·, xj ) +

j=1

L 2l+1 X X

βl,k Yl,k ,

l=0 k=1

where we fix L ≥ 0 as the desired degree of the polynomial component of the hybrid interpolation and the coefficients {αj }N j=1 , {βl,k }k=1,...,2l+1, l=0,...,L are determined by the interpolation conditions IX,L f (xi ) = f (xi ), i = 1, . . . , N, (2.2) PN and also (in order to give a square linear system) the side conditions j=1 αj p(xj ) = 0, ∀p ∈ PL . Now we give a condition on the point set X, which makes sure that the interpolation is exist and unique. Definition 2.2 (see [25, Definition 3.1]). The set X = {x1 , . . . , xN } ⊂ S2 is said to be PL -unisolvent if p ∈ PL , p(xj ) = 0 for j = 1, . . . , N ⇒ p = 0. In order to analyze the interpolation error in the later sections it is convenient to define a finite-dimensional space VX,L within the interpolation IX,L f lies. X  N N X αj φ(·, xj ) + q : q ∈ PL , αj ∈ R for j = 1, . . . , N, and αj p(xj ) = 0, ∀p ∈ PL . VX,L := j=1

j=1

The following Theorem 2.1 gives a crucial condition for the interpolation to be well defined, whose proof can be find in [25]. Theorem 2.1 Let φ(·, ·) be a strictly positive definite kernel, and X = {x1 , . . . , xN } ⊂ S2 be a set of distinct points which is PL -unisolvent for L ≥ 0. Then for each f ∈ C(S2 ) there exists a unique IX,L f ∈ VX,L that satisfies the interpolation conditions in (2.2). In this paper, we assume that the strictly positive definite kernel φ is zonal and has the expansion φ(x, y) =

∞ X

al

l=0

2l+1 X

Yl,k (x)Yl,k (y)

(2.3)

k=1

P∞ with al > 0 for all l, l=0 lal < ∞, in which case the series of the right side in (2.3) converges uniformly for x, y ∈ S2 . P∞ P2l+1 ˆ For f, g ∈ L2 (S2 ), they can be represented by their Fourier series f = l=0 k=1 fl,k Yl,k P∞ P2l+1 and g = l=0 k=1 gˆl,k Yl,k , respectively. With respect to the inner product expressed as (see P∞ P2l+1 fˆ gˆ [29]) (f, g)Nφ = l=0 k=1 l,kal l,k , the native space Nφ , which is the subspace of L2 (S2 ), can be defined by ( ) ∞ 2l+1 X X |fˆl,k |2 2 2 Nφ := f ∈ L2 (S ) : kf kNφ = 1, then the space Hs is continuously embedded in C(S2 ), so that Hs is a reproducing kernel Hilbert space. The error estimates are general expressed in terms of the mesh norm of X = {x1 , . . . , xN } ⊂ S2 , which is defined by hX := supx∈S2 inf xj ∈X d(x, xj ), where d(x, xj ) = arccos(x · xj ) is the geodesic distance between xj and x.

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C. Ding et al.: Lp approximation error for hybrid interpolation on the unit sphere

3

Global error estimates for Lp norm

We first give the following three lemmas, which can be found in [9] and [10]. √ 1 Lemma 3.1 Let d ≥ 1 be an integer and set M := 2 d and δd := 4(d+1) 3/2 . Let M1 be an arbitrary positive number, θ ∈ (0, π3 ) and set h0 := M +Mθ 1 +δd . Then for any h ∈ (0, h0 ), there exists a set of S points Zh ⊂ Sd such that Sd = z∈Zh D(z, M h). Here we denote by D(x0 , γ) the spherical cap with  center x0 and angle γ, i.e., D(x0 , γ) := x ∈ Sd : x · x0 > cos γ , and then denote by A(x0 , γ) the Rγ volume of D(x0 , γ), i.e., A(x0 , γ) := Ωd 0 sind−1 θdθ, where Ωd denotes the volume of Sd . Let FA denote the characteristic function of a set A ⊂ Sd . There exists a positive integer Q independent of h such that X 0 FD(z,M 0 h) ≤ Q, where M = M + M1 . z∈Zh

Further, the cardinality of Zh is bounded above by CQ h−d , where CQ is independent of h. d Lemma 3.2 Let z ∈ Sd and X = {xi }N i=1 denote a set of distinct points on S . Let s ∈ [k, k + 1], d where k > 2 is a positive integer. There exist positive numbers C1 and C2 such that if we let M1 > max{C1 − 2d1/2 , 0} be a fixed positive number and let

h0 =

C2 , where M2 = 2d1/2 + M1 , 3M2

then, assuming that X has mesh norm h := hX ∈ (0, h0 ), there exists an extension operator ED(z,M2 h) : Hs (D(z, M2 h)) −→ Hs (Sd ) satisfying (1) (ED(z,M2 h) f )|D(z,M2 h) = f , for all f ∈ Hs (D(z, M2 h)), (2) there exists a positive constant C, independent of h and z such that kED(z,M2 h) f kHs (Sd ) ≤ Ckf kHs (D(z,M2 h)) , T for all f ∈ Hs (D(z, M2 h)) such that f (ξ) = 0 for ξ ∈ X D(z, M2 h). Lemma 3.3 Let s > 0 and let M1 be any positive number. Let h ∈ (0, h0 ) and let Zh denote the corresponding quasi-uniform mesh for Sd from Lemma 3.1. Then, for any f ∈ Hs (Sd ), we have X kf k2Hs (D(z,M2 h)) ≤ Qkf k2Hs (Sd ) , z∈Zh

where Q is the constant (independent of h) from Lemma 3.1. We are now ready to state the main results for the error estimates of the hybrid interpolation in Lp norm. Theorem 3.1 Let φ ∈ C(S2 × S2 ) be a strictly positive definite kernel on S2 , having the representation in (2.3) and al ∼ (l + 1)−2s . Assume that integer L ≥ 1 and X = {x1 , . . . , xN } ⊂ S2 is a set of distinct points on S2 with mesh norm 1/(2L + 2) < hX ≤ 1/(2L). For f ∈ Nφ , let IX,L f ∈ VX,L be the hybrid interpolation defined in Section 2. Then we have 2

+s−1

p kf − IX,L f kLp (S2 ) ≤ ChX

kf − IX,L f kNφ ,

p ∈ [2, +∞),

and kf − IX,L f kLp (S2 ) ≤ ChsX kf − IX,L f kNφ ,

p ∈ [1, 2),

where the constant C is independent of f and hX .

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C. Ding et al.: Lp approximation error for hybrid interpolation on the unit sphere

Proof. For the case S2 , we can take d = 2 in Lemma 3.1, Lemma 3.2 and Lemma 3.3. By using Lemma 3.1, for arbitrary 1 ≤ p < ∞, we have Z X Z kf − IX,L f kpLp (S2 ) = |(f − IX,L f )(ξ)|p dω(ξ) ≤ |(f − IX,L f )(ξ)|p dω(ξ), (3.4) S2

D(z,M h)

z∈Zh

where M = 23/2 . This step motivates us to consider the error estimates locally. In particular, f − IX,L f is continuous on D(z, M h), which is a compact subset of S2 , so there exists a point ξz ∈ D(z, M h) where f − IX,L f attains its maximum. Now we can write Z X X p p kf − IX,L f kLp (S2 ) ≤ |(f − IX,L f )(ξ)| dω(ξ) ≤ Ch2X |(f − IX,L f )(ξ)|p , (3.5) D(z,M h)

z∈Zh

z∈Zh

where the constant C satisfies A(z, M h) ≤ Ch2X . We know that f − IX,L f ∈ Nφ and Nφ is norm equivalent to the Sobolev space H s . Now, rather than consider f − IX,L f , we choose instead to consider the restriction f − IX,L f D(z,M2 h) , where M2 = 23/2 + M1 . We should choose a suitable M1 to fit the conditions of Lemma 3.2, because we can find constant C1 , C2 such that 2C2 L 2C2 L > C1 , > 23/2 . 3 3 1 , M1 = 2C3 2 L − 23/2 , and M2 = 2C32 L , then it is easy to prove that Lemma 3.2 holds. So set h0 = 2L If we let vz := f − IX,L f D(z,M2 h) and use Lemma 3.2, we have (E1) ED(z,M2 h) vz ∈ Hs (S2 ), T (E2) ED(z,M2 h) vz (ξ) = 0 for all ξ ∈ X D(z, M2 h), (E3) there exists a positive constant C, independent of hX and z such that

ED(z,M h) vz ≤ Ckvz kHs (D(z,M2 h)) . 2 H (S2 ) s

Hence, with the help of Theorem and (E3) we can obtain



ED(z,M h) vz |(f − IX,L f )(ξz )| = ED(z,M2 h) vz (ξz ) ≤ Chs−1 2 X Nφ

ED(z,M h) vz ≤ Chs−1 kvz k ≤ Chs−1 . 2 X X Hs (D(z,M2 h)) H s

Substituting this into (3.5) gives 2+p(s−1)

kf − IX,L f kpLp (S2 ) ≤ ChX

X

kvz kpHs (D(z,M2 h)) .

(3.6)

z∈Zh

For p ∈ [2, ∞) we use Jensen’s inequality [1] give kf −

IX,L f kpLp (S2 )

≤

PN

i=1

api ≤

P

N i=1

followed by Lemma 3.3 to !p/2

Hs (D(z,M2 h))

z∈Zh 2+p(s−1)

 p2

2 X

f − IX,L f D(z,M2 h)

2+p(s−1) ChX

≤ ChX

a2i



kf − IX,L f k2Hs (S2 )

p/2

2+p(s−1)

≤ ChX

kf − IX,L f kpNφ .

Finally, taking the p-th root gives 2

+s−1

p kf − IX,L f kLp (S2 ) ≤ ChX

kf − IX,L f kNφ ,

p ∈ [2, +∞),

(3.7)

where the constant C is independent of f and hX . For p ∈ [1, 2) we conduct the same steps as above, however we replace Jensen’s inequality with N X

api ≤ N

1− p 2

i=1

N X

! p2 a2i

.

i=1

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C. Ding et al.: Lp approximation error for hybrid interpolation on the unit sphere

Moreover, we use the fact that the cardinality of Zh is bounded by CQ h−2 (see Lemma 3.1), and we obtain !p/2

2 X

p ps kf − IX,L f k ≤ Ch

f − IX,L f 2 X

Lp (S )

D(z,M2 h) H (D(z,M h)) s 2

z∈Zh

 p/2 p 2 ≤ Chps ≤ Chps X kf − IX,L f kHs (S2 ) X kf − IX,L f kNφ . Finally, taking the p-th root gives kf − IX,L f kLp (S2 ) ≤ ChsX kf − IX,L f kNφ ,

p ∈ [1, 2),

(3.8)

where the constant C is independent of f and hX . Combining (3.7) and (3.8) yields Theorem 3.1.

4

Inf-sup condition and improved global error estimates

As we can see that the factor kf − IX,L f kNφ in Theorem 3.1 may be harder to estimate than factor kf kNφ . Considering the fact that the hybrid interpolation defined in Section 2 is different from the interpolation scheme only by radial basis functions constructed from strictly positive definite kernels or conditionally positive definite kernels (see [10, ]), we should find the other method to characterize the relationship between kf − IX,L f kNφ and kf kNφ . The following Inf-sup condition is quoted from [26], whose method is helpful to “tidy up” the error results in Theorem 3.1. Theorem 4.1 (see [26, Theorem 6.1]). Let φ ∈ C(S2 × S2 ) be a strictly positive definite kernel on S2 , having the representation in (2.3) and al ∼ (l + 1)−2s . Then there exist constants γ > 0 and τ > 0 depending only on s such that for all L ≥ 1 and all X = {x1 , . . . , xN } ⊂ S2 satisfying hX ≤ τ /L, the following inequality holds: (p, v)Nφ ≥ γkpkNφ , p ∈ PL , v∈RX \{0} kvkNφ sup

(4.9)

where RX = span{φ(·, x1 ), . . . , φ(·, xN )}. In order to use the same method in [26], we simply denote that IX,L f = uX,L + pX,L , where PL P2l+1 PN pX,L = l=0 k=1 βl,k Yl,k , and uX,L = j=1 φ(·, xj ). For a given f ∈ Nφ , the interpolation conditions IX,L f (xi ) = f (xi ), i = 1, . . . , N, and the side PN conditions j=1 αj q(xj ) = 0, ∀q ∈ PL , are equivalent to (uX,L , vX )Nφ + (pX,L , vX )Nφ = (f, vX )Nφ , vX ∈ RX ,

(4.10)

(q, uX,L )Nφ = 0, ∀q ∈ PL .

(4.11)

and Now we can write the target function f ∈ Nφ in an analogous way to IX,L f as f := u + p, where p ∈ PL and u ∈ Nφ are defined by (p, q)Nφ = (f, q)Nφ , q ∈ PL , which means that p is the Nφ -orthogonal project of f onto PL . Similar to (4.10) and (4.11), we have (u, v)Nφ + (p, v)Nφ = (f, v)Nφ , vX ∈ RX ,

(4.12)

(q, u)Nφ = 0, q ∈ PL .

(4.13)

and By subtracting (4.10) from (4.12) (with v replaced by vX ) and (4.11) from (4.13), we can obtain (u − uX,L , vX )Nφ + (p − pX,L , vX )Nφ = 0, vX ∈ RX ,

109

(4.14)

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C. Ding et al.: Lp approximation error for hybrid interpolation on the unit sphere

and (q, u − uX,L )Nφ = 0, q ∈ PL .

(4.15)

Now we define u ˜X ∈ RX to be the Nφ -orthogonal project of u onto RX , that is, (˜ uX , vX )Nφ = (u, vX )Nφ , vX ∈ RX .

(4.16)

From (4.14), (4.15) and (4.16), we clearly have (˜ uX − uX,L , vX )Nφ + (p − pX,L , vX )Nφ = 0, vX ∈ RX ,

(4.17)

(q, u ˜X − uX,L )Nφ = (q, u ˜X − u)Nφ , q ∈ PL .

(4.18)

and With the help of Theorem 4.1, we have kp − pX,L kNφ

≤

1 γ

(p − pX,L , vX )Nφ kvX kNφ vX ∈RX \{0}

=

1 γ

(uX,L − u ˜X , vX )Nφ 1 ≤ kuX,L − u ˜X kNφ . kv k γ X Nφ vX ∈RX \{0}

sup sup

By using (4.17) with vX = u ˜X − uX,L and (4.18), we also have k˜ uX − uX,L k2Nφ

= −(p − pX,L , u ˜X − uX,L )Nφ = −(p − pX,L , u ˜X − u)Nφ 1 ˜X kNφ k˜ uX − ukNφ . ≤ kp − pX,L kNφ k˜ uX − ukNφ ≤ kuX,L − u γ

So we obtain that k˜ uX − uX,L kNφ ≤

1 k˜ uX − ukNφ ≤ Ck˜ uX − ukNφ , γ

(4.19)

and

1 k˜ uX − ukNφ ≤ Ck˜ uX − ukNφ . (4.20) γ2 With the above inequalities (4.19) and (4.20), we can establish the following Theorem 4.2, which indicates the relationship between kf − IX,L f kNφ and kf kNφ . kp − pX,L kNφ ≤

Theorem 4.2 Let φ ∈ C(S2 × S2 ) be a strictly positive definite kernel on S2 , having the representation in (2.3) and al ∼ (l + 1)−2s , s > 1. Assume that integer L ≥ 1 and X = {x1 , . . . , xN } ⊂ S2 is a set of distinct points on S2 with mesh norm hX ≤ τ /L, where τ is as in Theorem 4.1. For f ∈ Nφ , let IX,L f ∈ VX,L be the hybrid interpolation defined in Section 2. Then we have kf − IX,L f kNφ ≤ C inf kf − qkNφ ≤ Ckf kNφ . q∈PL

Proof. Using the representation IX,L f = uX,L + pX,L , f = u + p and (4.19), (4.20) we have kf − IX,L f kNφ

≤ ku − uX,L kNφ + kp − pX,L kNφ ≤ k˜ uX − ukNφ + k˜ uX − uX,L kNφ + kp − pX,L kNφ ≤ Ck˜ uX − ukNφ ,

and also we have k˜ uX − ukNφ ≤ kukNφ = kf − pkNφ = inf q∈PL kf − qkNφ , which yields kf − IX,L f kNφ ≤ C inf q∈PL kf − qkNφ ≤ Ckf kNφ , and the proof of Theorem 4.2 is completed. Combining Theorem 4.2 with Theorem 3.1, we can easily verify the following Corollary 4.1. Corollary 4.1 Under the conditions of Theorem 3.1 apart from the mesh norm 1/(2L + 2) < hX ≤ min{1/(2L), τ /L}, where τ is as in Theorem 4.1. For f ∈ Nφ , let IX,L f ∈ VX,L be the hybrid interpolation defined in Section 2. Then we have 2

+s−1

p kf − IX,L f kLp (S2 ) ≤ ChX

kf kNφ , p ∈ [2, +∞),

and kf − IX,L f kLp (S2 ) ≤ ChsX kf kNφ , p ∈ [1, 2), where the constant C is independent of f and hX .

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In the rest part of this section, unlike the above arguments we used to perform the “cleaner” error estimates in Corollary 4.1, we will show that improved global error estimates are available, provided that the target function f belongs to a certain subspace of Nφ , which defined by Nφ∗φ . This procedure is the same as in [10] and the following Definition 4.1 is about the convolution kernel of φ, which generates the corresponding native space Nφ∗φ . Definition 4.1 Let φ be a strictly positive definite zonal kernel that defined in (2.3) We define the convolution kernel of φ by Z (φ ∗ φ)(x, y) := φ(x, z)φ(y, z)dω(z), x, y ∈ S2 . S2

P∞ P2l+1 Working in terms of Fourier expansions, we have (φ ∗ φ)(x, y) := l=0 a2l k=1 Yl,k (x)Yl,k (y). Executing the same arguments as in Section 2, we know that the native space Nφ∗φ associated with kernel (φ ∗ φ)(·, ·) can be defined by ) ( ∞ 2l+1 X X |fˆl,k |2 2 2 1, we know that the native space Nφ is norm equivalent to the Sobolev space Hs . So Nφ∗φ ∼ = denotes norm = Nφ , where ∼ = H2s ⊂ Hs ∼ equivalence. Obviously, we see Nφ∗φ ⊂ Nφ . The following Lemma 4.1 gives a crucial inequality, which is helpful to improve the global error estimates of the hybrid interpolation for a target function f ∈ Nφ∗φ . Lemma 4.1 Let u ∈ Nφ∗φ and u ˜X ∈ RX be the Nφ -orthogonal project of u onto RX , which has the property as in (4.16), then we have k˜ uX − uk2Nφ ≤ kukNφ∗φ · k˜ uX − ukL2 (S2 ) ,

(4.21)

where RX is the same as in Theorem 4.1. Proof. By using (4.16), the definition of (·, ·)Nφ∗φ , and Cauchy-Schwarz inequality respectively, we have 
 ∞ 2l+1 ˆl,k · u ˆl,k − u ˜c X Xu X l,k k˜ uX − uk2Nφ = (u, u ˜X − u)Nφ = al l=0 k=1 ! !1/2 1/2 ∞ 2l+1 ∞ 2l+1 X X |ˆ X X
2 ul,k |2 ≤ u ˆl,k − u ˜c ≤ kukNφ∗φ · k˜ uX − ukL2 (S2 ) X l,k a2l l=0 k=1

l=0 k=1

With this in place we can provide the following improved global error estimates. Theorem 4.3 Under the conditions of Corollary 4.1 and assume further that f ∈ Nφ∗φ , we have 2

+2s−1

p kf − IX,L f kLp (S2 ) ≤ ChX

kf kNφ∗φ , p ∈ [2, +∞),

(4.22)

and kf − IX,L f kLp (S2 ) ≤ Ch2s X kf kNφ∗φ , p ∈ [1, 2),

(4.23)

where the constant C is independent of f and hX . Proof. First we have, from Theorem 3.1, with p = 2, that k˜ uX − ukL2 (S2 ) ≤ ChsX k˜ uX − ukNφ .

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C. Ding et al.: Lp approximation error for hybrid interpolation on the unit sphere

Substituting this into (4.21) gives k˜ uX − uk2Nφ ≤ ChsX kukNφ∗φ · k˜ uX − ukNφ .

(4.24)

k˜ uX − ukNφ ≤ ChsX kukNφ∗φ .

(4.25)

So, Using the same procedure as in the proof of Theorem 4.2, we see that kf − IX,L f kNφ ≤ ku − uX,L kNφ + kp − pX,L kNφ ≤ Ck˜ uX − ukNφ ≤ ChsX kukNφ∗φ . Clearly, kukNφ∗φ = kf − pkNφ∗φ = inf kf − qkNφ∗φ ≤ kf kNφ∗φ ,

(4.26)

q∈PL

which implies kf − IX,L f kNφ ≤ ChsX kf kNφ∗φ .

(4.27)

With the help of Theorem 3.1 we see 2

+2s−1

p kf − IX,L f kLp (S2 ) ≤ ChX

kf kNφ∗φ , p ∈ [2, +∞),

and kf − IX,L f kLp (S2 ) ≤ Ch2s X kf kNφ∗φ , p ∈ [1, 2), where the constant C is independent of f and hX .

5

Hybrid interpolation for rough native space

In order to generate a larger native space than Nφ , we should give a new kernel defined in the form ψ(x, y) =

∞ X l=0

with bl > 0 for all l, and

P∞

l=0

bl

2l+1 X

Yl,k (x)Yl,k (y),

(5.28)

k=1

lbl < ∞.

P∞ P2l+1 With respect to the inner product expressed as (f, g)Nψ = l=0 k=1 Nψ may alternatively be characterized as the following set ( ) ∞ 2l+1 X X |fˆl,k |2 2 2 Nψ := f ∈ L2 (S ) : kf kNψ = al , for all l = 0, 1, . . . , we then see that Nφ ⊂ Nψ . Next, we will consider the error estimates for the hybrid interpolation of a target function f ∈ Nψ ⊃ Nφ . Obviously, if we take the hybrid interpolation associated with the less smooth PN PL P2l+1 kernel ψ in the form IX,L,ψ f = j=1 αj ψ(·, xj ) + l=0 k=1 βl,k Yl,k , then Theorem 3.1 above still holds for f ∈ Nψ . However, motivated by the idea in [12], we still take the initial hybrid interpolation IX,L,φ f in the form IX,L,φ f =

N X

αj φ(·, xj ) +

j=1

L 2l+1 X X

βl,k Yl,k ,

(5.29)

l=0 k=1

and consider the error estimate kf − IX,L,φ f kLp (S2 ) . Lemma 5.1 Let α be a nonnegative real number, P and let M be the multiplier PL PL operator P2l+1 defined 2l+1 on PL (embedded in C(S2 )) by M (p) = l=0 (λl )α k=1 cl,k Yl,k , where p = l=0 k=1 cl,k Yl,k . Then we have kM (p)k ≤ C(λL )α kpk, where C is a constant independent of p and L.

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C. Ding et al.: Lp approximation error for hybrid interpolation on the unit sphere

Lemma 5.1 is a special case of Theorem 3.2 in [4]. Lemma 5.2 Let X = {x1 , . . . , xN } ⊂ S2 satisfying hX ≤ 1/(2L), then for any linear functional σ on PL (embedded in C(S2 )), such that kσk∗ = 1, there exist N real numbers αj := αj (x) (x is PN PN fixed) with j=1 |αj | ≤ 2, so that σ(f ) = j=1 αj δj (f ) for all f ∈ PL , where δj denotes the point evaluation functional at the point xj in X. The proof of the following Lemma 5.3 can be found in [29, Corollary 17.12]. Lemma 5.3 Suppose that X = {x1 , . . . , xN } ⊂ S2 has mesh norm hX ≤ L ≥ 1. Then there exist functions αj : S2 → R for j = 1, . . . , N such that PN (i) j=1 αj (x)p(xj ) = p(x), ∀p ∈ PL , ∀x ∈ S2 , PN (ii) j=1 |αj (x)| ≤ 2, ∀x ∈ S2 .

1 2L

for some integer

The following Theorem 5.1 is about the pointwise error estimate |f (x) − IX,L,φ f (x)|, by which we can obtain the global error estimate kf − IX,L,φ f kLp (S2 ) . Theorem 5.1 Let φ ∈ C(S2 × S2 ) be a strictly positive definite kernel defined by (2.3), let ψ ∈ C(S2 ×S2 ) be a strictly positive definite kernel on S2 , having the representation in (5.28), bl /al = λl for l ≥ 1 and bl ∼ (l + 1)−2s , s > 1, l ≥ 0. Assume that integer L ≥ 1 and X = {x1 , . . . , xN } ⊂ S2 is a set of distinct points on S2 with mesh norm 1/(2L + 2) < hX ≤ 1/(2L). For f ∈ Nψ , let IX,L,φ f ∈ VX,L be the hybrid interpolation defined in (5.29). Then for a fixed x ∈ S2 , we have |f (x) − IX,L,φ f (x)| ≤ Chs−1 X kf − IX,L,φ f kNψ . Proof. For ∀f ∈ Nψ , we simply take the hybrid interpolation associated with the smooth kernel PN φ by IX,L,φ f (x) = uX,L,φ + pX,L , where uX,L,φ = j=1 αj φ(·, xj ), xj ∈ X = {x1 , x2 , . . . , xN }, PL P2l+1 and pX,L = l=0 k=1 βl,k Yl,k , such that IX,L,φ f (xj ) = f (xj )(j = 1, 2, . . . , N ). However, if we just use the hybrid interpolation associated with the less smooth kernel ψ, we PN have IX,L,ψ f (x) = uX,L,ψ + p0X,L , where uX,L,ψ = j=1 γj ψ(·, xj ), xj ∈ X = {x1 , x2 , . . . , xN }, and PL P2l+1 0 p0X,L = l=0 k=1 βl,k Yl,k . First, we consider the estimate of kψ(·, x) − uX,L,ψ kNψ . Using the same method as that in [12], we have kψ(·, x) − uX,L,ψ kNψ =

sup v∈Nψ kvkNψ =1

=

sup

∞ X

v∈Nψ l=0 kvkNψ =1

=

sup

b−1 l

∞ 2l+1 X X

v∈Nψ l=0 k=1 kvkNψ =1

2l+1 X

 vˆl,k bl

(ψ(·, x) − uX,L,ψ , v)Nψ

N X

 γj Yl,k (xj ) − bl Yl,k (x)

j=1

k=1

  N X vˆl,k  γj Yl,k (xj ) − Yl,k (x) . j=1

By using Lemma 5.3, for a fixed x, we see that there exist N real numbers γj with such that N X γj Yl,k (xj ) = Yl,k (x), l = 0, 1, . . . , L,

PN

j=1

|γj | ≤ 2 (5.30)

j=1

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C. Ding et al.: Lp approximation error for hybrid interpolation on the unit sphere

which yields kψ(·, x) − uX,L,ψ kNψ =

∞ 2l+1 X X

sup

v∈Nψ l=L+1 k=1 kvkNψ =1

=

sup v∈Nψ kvkNψ =1

≤

sup v∈Nψ kvkNψ =1

 vˆl,k 

N X

 γj Yl,k (xj ) − Yl,k (x)

j=1

  N ∞ 2l+1 ∞ 2l+1 X X X X X  γj vˆl,k Yl,k (xj ) − vˆl,k Yl,k (x) j=1

l=L+1 k=1

l=L+1 k=1

 ∞ 2l+1  N X X X  |γj | vˆl,k Yl,k (xj )  + j=1

l=L+1 k=1

∞ 2l+1 X X vˆl,k Yl,k (x) .

sup

v∈Nψ l=L+1 k=1 kvkNψ =1

By using the Cauchy-Schwarz inequality, (5.30) and the relation in (2.1), we see that kψ(·, x) − uX,L,ψ kNψ ≤

N X

|γj | max

xj ∈X

j=1 ∞ X

+

2l+1 X

bl

l=L+1

≤2

∞ X l=L+1

≤ C1

t

2l + 1 bl 4π

sup

+

∞ X l=L+1

2l+1 X

! 12 2 Yl,k (xj )

l=L+1 k=1

2l + 1 bl 4π

∞ 2l+1 2 X X vˆl,k

sup v∈Nψ kvkNψ =1

k=1

l=L+1 k=1

! 12

bl

! 12

∞ 2l+1 2 X vˆl,k X

v∈Nψ kvkNψ =1

! 21

bl

l=L+1

! 21 2 Yl,k (x)

k=1

∞ X

∞ X

bl

! 12 ≤ C1

∞ X

! 21 (2l + 1)bl

l=L+1

! 21 (l + 1)

−2s+1

≤ C1 (L + 1)−s+1 ≤ C1 hs−1 X .

(5.31)

l=L+1

Next we consider the estimate of kψ(·, x) − uX,L,φ kNψ , in which we will use Lemma 5.1 and Lemma 5.2. kψ(·, x) − uX,L,φ kNψ

=

sup v∈Nψ kvkNψ =1

=

sup

(ψ(·, x) − uX,L,φ , v)Nψ ∞ X

v∈Nψ l=0 kvkNψ =1

=

sup

b−1 l

2l+1 X

 vˆl,k al

k=1

N X

 αj Yl,k (xj ) − bl Yl,k (x)

j=1

  ∞ 2l+1 N X X X bl b−1 vˆl,k  αj Yl,k (xj ) − Yl,k (x) . l al a l j=1

v∈Nψ l=0 kvkNψ =1

k=1

Let TL be the multiplier operator defined on PL (embedded in C(S2 )) by TL (Yl,k ) = abll Yl,k , for each l = 0, 1, . . . , L and all k = 1, 2, . . . , 2l + 1, and extended linearly throughout PL . Let σ be the linear functional on PL defined by σ = δx ◦ TL . That is σ(p) = (TL (p))(x) for each p ∈ PL . By Lemma 5.1 with α = 1 and the assumption that bl /al = λl , l ≥ 1, we have |σ(p)| = |(TL (p))(x)| ≤ kTL (p)k ≤ CλL kpk = C

bL kpk, aL

in whichP C is a constant independent of p and L. Then by Lemma 5.2, there exist N real numbers N αj with j=1 |αj | ≤ 2C abLL such that N X j=1

αj Yl,k (xj ) =

bl Yl,k (x), l = 0, 1, . . . , L. al

114

(5.32)

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With the help of (5.32), we see that kψ(·, x) − uX,L,φ kNψ =

∞ X

sup

v∈Nψ l=L+1 kvkNψ =1

=

sup v∈Nψ kvkNψ =1

≤

sup v∈Nψ kvkNψ =1

b−1 l al

2l+1 X

 vˆl,k 

N X

 bl Yl,k (x) al

αj Yl,k (xj ) −

j=1

k=1

  N ∞ 2l+1 ∞ 2l+1 X X X X X  αj b−1 vˆl,k Yl,k (xj ) − vˆl,k Yl,k (x) l al j=1

l=L+1

k=1

l=L+1 k=1

  ∞ N X a 2l+1 X X l  vˆl,k Yl,k (xj )  + |αj | b l j=1 k=1

l=L+1

∞ 2l+1 X X vˆl,k Yl,k (x) .

sup

v∈Nψ l=L+1 k=1 kvkNψ =1

Using the Cauchy-Schwarz inequality, we see that kψ(·, x) − uX,L,φ kNψ ≤

N X

|αj | max

xj ∈X

j=1

+

∞ X

2l+1 X

bl

l=L+1

With the help of

! 21 2 Yl,k (x)

k=1

PN

j=1

! 12 ∞ 2l+1 X a2l X 2 Yl,k (xj ) bl

l=L+1

k=1

∞ 2l+1 2 X X vˆl,k

sup v∈Nψ kvkNψ =1

l=L+1 k=1

∞ 2l+1 2 X X vˆl,k

sup v∈Nψ kvkNψ =1

l=L+1 k=1

! 12

bl

! 12

bl

.

|αj | ≤ 2C abLL , bl > al and the relation in (2.1), we have

kψ(·, x) − uX,L,φ kNψ

! 12 ∞ X a2l 2l + 1 + bl 4π l=L+1 ! 21 ∞ X bL 2l + 1 + ≤ 2C bl aL 4π l=L+1 ! 12 ∞ X ≤ C2 (2l + 1)bl ≤ C2 bL ≤ 2C aL

l=L+1 −s+1

≤ C2 (L + 1)

! 12 2l + 1 bl 4π l=L+1 ! 21 ∞ X 2l + 1 bl 4π l=L+1 ! 21 ∞ X −2s+1 (l + 1) ∞ X

l=L+1

≤

C2 hs−1 X .

(5.33)

With the above obtained results, we can provide the following pointwise error estimate: |f (x) − IX,L,φ f (x)| = (f − IX,L,φ f, ψ(·, x))Nψ = (f − IX,L,φ f, ψ(·, x) − uX,L,φ )Nψ + (f − IX,L,φ f, uX,L,ψ )Nψ + (f − IX,L,φ f, uX,L,φ − uX,L,ψ )Nψ := |I1 + I2 + I3 | . It is easy to verify that  I2 :

=

(f − IX,L,φ f, uX,L,ψ )Nψ = f − IX,L,φ f,

N X j=1

=

N X

 γj ψ(·, xj ) Nψ

γj (f (xj ) − IX,L,φ f (xj )) = 0.

j=1

With the help of (5.33), we have |I1 | := (f − IX,L,φ f, ψ(·, x) − uX,L,φ )Nψ ≤ kf − IX,L,φ f kNψ kψ(·, x) − uX,L,φ kNψ ≤ C2 hs−1 X kf − IX,L,φ f kNψ .

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C. Ding et al.: Lp approximation error for hybrid interpolation on the unit sphere

We denote I4 := (f − IX,L,φ f, ψ(·, x) − uX,L,ψ )Nψ so that we have I3 = I4 − I1 . With the help of (5.31) we can see that |I4 | = (f − IX,L,φ f, ψ(·, x) − uX,L,ψ )Nψ ≤ kf − IX,L,φ f kNψ kψ(·, x) − uX,L,ψ kNψ ≤ C1 hs−1 X kf − IX,L,φ f kNψ , which yields |I3 | ≤ (C1 + C2 )hs−1 X kf − IX,L,φ f kNψ . Then |f (x) − IX,L,φ f (x)| ≤ |I1 | + |I2 | + |I3 | ≤ (C1 + 2C2 )hs−1 X kf − IX,L,φ f kNψ ≤ Chs−1 X kf − IX,L,φ f kNψ . This completes the proof of Theorem 5.1. Having the pointwise error estimate in Theorem 5.1, we can perform the same steps in Theorem 3.1, where the local-global strategy is the key to establish the error estimates. So we are now ready to state the error estimates of the hybrid interpolation for a target function f ∈ Nψ for Lp norm. Theorem 5.2 Under the conditions of Theorem 5.1, we have 2

+s−1

p kf − IX,L,φ f kLp (S2 ) ≤ ChX

kf − IX,L,φ f kNψ ,

p ∈ [2, +∞),

(5.34)

p ∈ [1, 2),

(5.35)

and k f − IX,L,φ f kLp (S2 ) ≤ ChsX k f − IX,L,φ f kNψ , where the constant C is independent of f and hX .

References [1] S. C. Brenner, R. L. Scott, The Mathematical Theory of Finite Element Methods, Springer, New York, 1994. [2] F. Cao, M. Li, Spherical data fitting by multiscale moving least squares, Applied Math. Model., 39 (2015) 3448-3458. [3] D. Chen, V. A. Menegatto, X. Sun, A necessary and sufficient condition for strictly positive definite functions on spheres, Proc. Amer. Math. Soc., 131 (2003) 2733-2740. [4] Z. Ditzian, Fractional derivatives and best approximation, Acta. Math. Hungar., 81 (1998) 323-348. [5] G. E. Fasshauer, L. L. Schumaker, Scattered data fitting on the sphere, in Mathematical Methods for Curves and Surfaces II (M. Dælen, T. Lyche, and L. L. Schumaker, eds. ), Vanderbilt University Press, Nashville, TN, (1998) 117-166. [6] W. Freeden, T. Gervens, M. Schreiner, Constructive Approximation on the Sphere, Oxford University Press Inc., New York, 1998. [7] P. B. Gilkey, The Index Theorem and the Heat Equation, Publish or Perish, Boston, MA, 1974. [8] M. v. Golitschek, W. A. Light, Interpolation by polynomials and radial basis functions on spheres, Constr. Approx., 17 (2001) 1-18. [9] S. Hubbert, T. M. Morton, A Duchon framework for the sphere, J. Approx. Theory, 129 (2004) 28-57. [10] S. Hubbert, T. M. Morton, Lp -error estimates for radial basis function interpolation on the sphere, J. Approx. Theory, 129 (2004) 58-77.

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C. Ding et al.: Lp approximation error for hybrid interpolation on the unit sphere

[11] K. Jetter, J. St¨ ockler, J. D. Ward, Error estimates for scattered data interpolation on spheres, Math. Comp., 68 (1999) 733-747. [12] J. Levesley, X. Sun, Approximation in rough native spaces by shifts of smooth kernels on spheres, J. Approx. Theory, 133 (2005) 269-283. [13] J. Levesley, X. Sun, Corrigendum to and two open questions arising from the article “Approximation in rough native spaces by shifts of smooth kernels on spheres” [J. Approx. Theory, 133 (2005) 269-283], J. Approx. Theory, 138 (2006) 124-127. [14] Q. T. Le Gia, F. J. Narcowich, J. D. Ward, H. Wendland, Continuous and discrete leastsquares approximation by radial basis functions on spheres, J. Approx. Theory, 143 (2006) 124-133. [15] M. Li, F. Cao, Local uniform error estimates for spherical basis functions interpolation, Math. Meth. Applied Sci., 37 (2014) 1364-1376. [16] C. M¨ uller, Spherical Harmonics, Lecture Notes in Mathematics, Vol. 17, Springer-Verlag, Berlin, 1966. [17] F. J. Narcowich, R. Schaback, J. D. Ward, Approximation in Sobolev spaces by kernel expansions, J. Approx. Theory, 114 (2002) 70-83. [18] F. J. Narcowich, J. D. Ward, Scattered data interpolation on spheres: Error estimates and locally supported basis functions, SIAM J. Math. Anal., 33 (2002) 1393-1410. [19] F. J. Narcowich, X. Sun, J. D. Ward, H. Wendland, Direct and inverse sobolev error estimates for scattered data interpolation via spherical basis functions, Found. Comput. Math., (2007) 369-390. [20] F. J. Narcowich, X. Sun, J. D. Ward, Approximation power of RBFs and their associated SBFs: A connection, Adv. Comput. Math., 27 (2007) 107-124. [21] A. Ron, X. Sun, Strictly positive definite functions on spheres in Enclidean spaces, Math. Comp., 65 (1996) 1513-1530. [22] R. Schaback, Improved error bounds for scattered data interpolation by radial basis functions, Math. Comp., 68 (1999) 201-216. [23] I. J. Schoenberg, Positive definite functions on spheres, Duke Math. J., 9 (1942) 96-108. [24] E. M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princetion, NJ, 1971. [25] I. H. Sloan, A. Sommariva, Approximation on the sphere using radial basis function plus polynomials, Adv. Comput. Math., 29 (2008) 147-177. [26] I. H. Sloan, H. Wendland, Inf-sup condition for spherical polynomials and radial basis functions on spheres, Math. Comp., 78 (2009) 1319-1331. [27] I. H. Sloan, Polynomial interpolation and hyperinterpolation over general regions, J. Approx. Theory, 83 (1995) 238-254. [28] I. H. Sloan, R. S. Womersley, Constructive polynomial approximation on the sphere, J. Approx. Theory, 103 (2000) 91-118. [29] H. Wendland, Scattered Data Approximation, Cambridge University Press, Cambridge, Uk, 2005. [30] Y. Xu, E. W. Cheney, Strictly positive definite functions on spheres, Proc. Amer. Math. Soc., 116 (1992) 977-981.

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Some best approximation formulas and inequalities for the Bateman’s G−function Ahmed Hegazi1 , Mansour Mahmoud2 , Ahmed Talat3 and Hesham Moustafa4 1,2,4 Mansoura 3 Port

University, Faculty of Science, Mathematics Department, Mansoura 35516, Egypt.

Said University, Faculty of Science, Mathematics and Computer Sciences Department, Port Said, Egypt. 1 3a

2 [email protected],

[email protected],

− t− [email protected],

4 [email protected]

. . Abstract In the paper, the authors established two best approximation formulas for the Bateman’s G−function. Also, they studied the completely monotonicity of some functions involving G(x). Some new inequalities are deduced for the function and its derivative such as # # " " 1 2x + a 1 2x + b ln 1 + 2 < G(x + 2) < ln 1 + 2 , x>0 2 2 x + 2x + 34 x + 2x + 43 4

−16 are the best possible constants. Our results improve some recent where a = 3 and b = e 12 inequalities about the function G(x).

2010 Mathematics Subject Classification: 33B15, 26D15, 41A25, 26A48. Key Words: Digamma function, Bateman G−function , best approximation, completely monotonic, monotonicity, bounds, rate of convergence, best possible constant.

1

Introduction.

In 2010, Mortici [21] presented the following Lemma which is considered as a powerful tool to constructing asymptotic expansions and to measure the rate of convergence: Lemma 1.1. If {τs }s∈N is convergent to zero and there exist h in R and k > 1 such that lim sk (τs − τs+1 ) = h,

(1)

s→∞

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then we get lim sk−1 τs =

s→∞

h . k−1

It is clear from lemma (1.1) that, the sequence {τs }s∈N will converge more quickly when the value of k is large in the relation (1). This Lemma has been applied successfully to produce several approximations and inequalities in several papers such as [6], [7], [11], [15], [16], [22], [24], [28]. In this paper, Lemma (1.1) will be an effectively tool in producing best approximations of the Bateman’s G−function defined by [9]   x x+1 G(x) = ψ −ψ , x 6= 0, −1, −2, ... (2) 2 2 where ψ(x) is the digamma or Psi function which is defined by ψ(x) =

d ln Γ(x) dx

and Γ(x) is the classical Euler gamma given for x > 0 by Z ∞ Γ(x) = e−w wx−1 dw. 0

The hypergeometric representation of the function G(x) is given by 1 2 F1 (1, 1; 1 + x; 1/2) x

G(x) =

(3)

and it satisfies the following relations [9]: G(x + 1) + G(x) = and

Z G(x) = 2 0

∞

2 x

e−xw dw, 1 + e−w

(4)

x > 0.

(5)

Qiu and Vuorinen [30] established the inequality x + (6 − 4 ln 4) x + 1/2 < G(x) < , 2 x x2

x > 1/2

(6)

x ≥ 1; j ∈ (0, 1)

(7)

and Mortici [23] presented the general inequality 0 < ψ(x + j) − ψ(x) ≤ ψ(j) + γ − j + j −1 ,

where γ is Euler−Mascheroni constant (also called Euler’s constant) defined by ! m X 1 . γ = lim − ln m + m→∞ w w=1

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Mahmoud and Agarwal [17] deduced the following asymptotic formula for x → ∞ ∞

1 X (22w − 1)B2w −2w G(x) − ∼ x , x w=1 w

(8)

where Bw 0 s are the Bernoulli numbers [1] defined by the generating series ∞ X

Bw

w=0

vw v = v . w! e −1

They also presented the following double inequality 1 1 + 2 x 2x +

3 2

< G(x)
and Almuashi proved the following inequality 2r X (22w − 1) B2w w=1

w

x2w

< G(x) − x

−1


0

(9)


9−12 ln 2 1/2 . 16 ln 2−11

(22w − 1) B2w , w x2w

In [18] Mahmoud

r∈N

(10)

2w

where (2 w−1) B2w are the best possible constants. In [19], Mahmoud, Talat and Moustafa presented the following approximations of the Bateman’s G−function   1 2 , c ∈ [1, 2], x>0 (11) G(x) ≈ ln 1 + + x+c x(x + 1) and they deduced the following double inequality     1 1 2 2 ln 1 + < G(x) < ln 1 + , + + x + α2 x(x + 1) x + α1 x(x + 1) where the constants α1 = 1 and α2 =

4 e2 −4

x>0

(12)

are the best possible constants.

Recently, Mahmoud, Talat, Moustafa and Agarwal [20] improved the double inequality (9) by 1 1 1 1 + 2 < G(x) < + 2 , x 2x + a x 2x + b where a = 1 and b = 0 are the best possible constants.

x>0

(13)

A function T defined on an interval I is said to be completely monotonic if it possesses derivatives T (s) (x) for all s = 0, 1, 2, ... such that (−1)s T (s) (x) ≥ 0

x ∈ I; s = 0, 1, 2, ... .

(14)

Such functions occur in several areas such as numerical analysis, elasticity and probability theory, for more details see [2], [5], [12]-[14], [26], [27], [29]. According to Bernstein theorem [31],

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the necessary and sufficient condition for the function T (x) to be completely monotonic for 0 < x < ∞ is that Z ∞ e−xt dλ(t), (15) T (x) = 0

where λ(t) is non-decreasing and the integral converges for 0 < x < ∞. In this paper, we presented two best approximation formulas of the Bateman’s G−function and some completely monotonic functions involving it. Some new inequalities of G(x) and its derivative will be deduced, which improve some pervious results.

2

Auxiliary Results

We can easily prove the following simple modification of Lemma (1.1): Lemma 2.1. If {τs }s∈N is convergent to zero and there exist h ∈ R, m ∈ N and k > 1 such that . lims→∞ sk (τs − τs+m ) = h, then we get lims→∞ sk−1 τs = h/m k−1 Proof. Using the relation k

lim s (τs − τs+m ) =

s→∞

= =

lim s

s→∞ m−1 X i=0 m−1 X i=0

then lim v k (τv − τv+1 ) = v→∞

h . m

k

m−1 X

(τs+i − τs+i+1 ) = lim

s→∞

i=0

m−1 X

(

i=0

s k ) (s + i)k (τs+i − τs+i+1 ) s+i

m−1 X s k k lim ( ) (s + i) (τs+i − τs+i+1 ) = lim (s + i)k (τs+i − τs+i+1 ) s→∞ s + i s→∞ i=0

lim v k (τv − τv+1 ) = m lim v k (τv − τv+1 ),

v→∞

v→∞

Applying Lemma (1.1) to get lims→∞ sk−1 τs =

h/m . k−1

Lemma 2.2.

1. For x > x0 ≈ 4.02361, we have N (x) = ln



 √ (x+1)(3− 6+3x) √ (x+2)(− 6+3x)

 2. For x > xλ ≈ 2.02059, we have M (x) = ln 3. For x > 0, we have H(x) = ln





−

√

4 )(3+ 6+3x) e2 −4 √ (x+1+ 24 )( 6+3x) e −4

(x+

 √ ( 6+3x)2 (13+12x+3x2 ) √ (3+ 6+3x)2 (4+6x+3x2 )



+2

√2

1+

3

x(x+1)

 −

< 0.

 √  1− 23 x(x+1)

> 0.

√2

1−

3

x(x+1)

> 0.

Proof. q

3

√

√

2

√

−(9+14 6)x −(6+11 6)x−2 6 √ √ N 0 (x) = 9x , nn12 (x) , where the polynomial n2 (x) (x) x2 (x+1)2 (x+2)(3x− 6)(3x+3− 6) q is positive for x > 23 and n1 (x) is a polynomial of degree 3 has only one positive real root x1 ≈ 5.49455 and n1 (x) > ( ( 0 for x > x1 with

1. For x >

2 , 3

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q  2 limx→∞ N (x) = 0 and hence N (x) < 0 for x > x1 . Also, N (x) is decreasing on , x 1 3 with N (4.023) ≈ 0.0000005 >q 0 and  N (4.024) ≈ −0.0000003 < 0. Then N (x) has only

one real root x0 ≈ 4.02361 ∈ for x > x0 . 2. For x > 0, M 0 (x) =

2 , x1 3

and N (x) < 0 for x0 < x < x1 . Hence, N (x) < 0

m(x) √ √ , x2 (1+x)2 ( 6+3x)(3+ 6+3x)(4+(e2 −4)x)(e2 +(e2 −4)x)

where

 √ √ √ √ 
m(x) = 4 6e2 + −16 6 + (−12 + 20 6)e2 + 6e4 x + 576 − 216e2 + 18e4 x5  √ √ 2 √ 4 2 + 144 − 16 6 − (72 + 20 6)e + (3 + 9 6)e x   √ √ √ + 384 + 224 6 − (144 + 160 6)e2 + (12 + 20 6)e4 x3  √ √ 2 √ 4 4 + 432 + 384 6 − (252 + 144 6)e + (27 + 12 6)e x and  √ √ √ 
m0 (x) = 5 576 − 216e2 + 18e4 x4 + −16 6 + (−12 + 20 6)e2 + 6e4  √ 2 √ 4 3 √ + 4 432 + 384 6 + (−252 − 144 6)e + (27 + 12 6)e x   √ √ √ + 3 384 + 224 6 + (−144 − 160 6)e2 + (12 + 20 6)e4 x2   √ √ √ + 2 144 − 16 6 + (−72 − 20 6)e2 + (3 + 9 6)e4 x. The polynomial m0 (x) of fourth degree has only one positive real root xα ≈ 2.57862 also m0 (x) < 0 for x > xα and m0 (x) > 0 for 0 < x < xα . Hence m(x) is increasing on (0, xα ) and is decreasing on (xα , ∞) with m(0) > 0, m(3.453) ≈ 22.157 > 0 and m(3.455) ≈ −6.01919 < 0. Then m(x) has only one positive real root xβ ≈ 3.45457 with m(x) < 0 for x > xβ and m(x) > 0 for 0 < x < xβ . Now M (x) is decreasing on (xβ , ∞) and lim M (x) = 0, then M (x) > 0 for x > xβ . Also, M (x) is increasing on (0, xβ ) with x→∞

M (2.0205) ≈ −0.0000006 < 0 and M (2.0206) ≈ 0.0000001 > 0, then M (x) has only one positive real root xλ ≈ 2.02059. Hence, M (x) > 0 for x > xλ . 3. H 0 (x) =

x2 (1

+

√

x)2 (

−4h(x) √ , 6 + 3x)(3 + 6 + 3x)(4 + 6x + 3x2 )(13 + 12x + 3x2 )

where √ √ √ √ h(x) = 26 6 + (−78 + 193 6)x + (−300 + 477 6)x2 + (−324 + 498 6)x3 √ √ +(−126 + 234 6)x4 + 36 6x5 > 0, x > 0. Hence H 0 (x) < 0 for all x > 0 with lim H(x) = 0 , then H(x) > 0 for x > 0 . x→∞

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The following result is considered as a method presented by Elbert and Laforgia in [8] (see also, [4], [25] and [32]): Corollary 2.3. Let K be a real-valued function defined on x > a, a ∈ R with limx→∞ K(x) = 0. Then K(x) > 0, if K(x) > K(x + 1) for all x > a and K(x) < 0, if K(x) < K(x + 1) for all x > a. This result has the following simple modification [20]: Corollary 2.4. Let K be a real-valued function defined on x > a, a ∈ R with limx→∞ K(x) = 0. Then for m ∈ N, K(x) > 0, if K(x) > K(x+m) for all x > a and K(x) < 0, if K(x) < K(x+m) for all x > a.

3

First formula of the best approximations and some its related inequalities

With the help of Mortici’s technique in Lemma(1.1), we provide the first best approximation formula of the Bateman’s G−function. Lemma 3.1. The best approximation G(n) ≈ ln(1 +

b 1 )+ n+a n(n + c)

(16)

holds for θ1 + θ2 − 5 θ2 + θ22 + 2θ1 + 2θ2 − 21 , b = a + 1 and c = 1 , (17) 9 54 p √ 3 1 b where θ1 , θ2 = 91 ± 63 2 and the sequence G(n) − ln(1 + n+a ) − n(n+c) converges to zero with −5 speed estimated by n .
1 b Proof. Define the error sequence by υn = G(n) − ln 1 + n+a − n(n+c) . Using the functional equation (4), we get ∞ X (−1)r−1  r−1 υn − υn+2 = b[c + 2r−1 − (2 + c)r−1 ]/c + [(a + 3)r − (a + 2)r − (a + 1)r r n r=3 a=

+ ar ]/r − 2} 4(a − b + 1) 2(7 + 3a(a + 3) − 3b(c + 2)) − = n3 n4 4(a(16 + a(9 + 2a)) − 2(−5 + b(4 + c(3 + c)))) + n5 326/3 + 10a(3 + a)(7 + a(3 + a)) − 10b(2 + c)(4 + c(2 + c)) − + O(n−7 ). 6 n According to Lemma (2.1), the three parameters a, b and c which produce the fastest convergence of the sequence υn satisfy the system a−b+1=0 3a + 9a − 3b(c + 2) + 7 = 0 2

9 a3 + a2 + 8a + 5 − b(c2 + 3c + 4) = 0. 2

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.1, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

Now, the values of a, b and c determined in (17) form solution of this system and the sequence υn converges to zero with speed estimated by n−5 . Now we will prove the complete monotonicity of some functions involving the function G(x) depending on the approximation formula (16). Lemma 3.2.
a+1 1 + x(x+c) − G(x) is 1. For the values of a and c in (17), the function L1 (x) = ln 1 + x+a completely monotonic on (0, ∞).   √ q  (1− 23 ) 1 2 √ 2. The function L2 (x) = ln 1 + + x(x+1) −G(x) is completely monotonic on ,∞ . 2 3 x−

3

 3. The function L3 (x) = G(x) − ln 1 + Proof.

√



1 √ x+ 23

−

(1+ 23 ) x(x+1)

is completely monotonic on (0, ∞).

1. Using the formula [1] 1 1 = xk (k − 1)!

∞

Z

tk−1 e−xt dt,

k∈N

(18)

0

and the integral representation (5) of G0 (x), we get L01 (x)

∞

Z = 0

e−(x+a+1)t ν1 (t)dt, 1 + et

where ν1 (t) =

h ∞ (2 − X

(a+1) )(a c

+ 2)k +

(a+1) [(a c

+ 1 − c)k + (a + 2 − c)k − (a + 1)k ] − k!

k=0 5

6

7

= −0.0316t − 0.0381t − 0.243t +

∞ X (a + 2)k [C1 (k)] − k=7

with

2k+1 (k+1)

(a+1) (a c

+ 1)k −

i tk+1

2k+1 (k+1) k+1

k!

t

"   k  k # a+1 a+1 a+1−c a+2−c C1 (k) = 2 − + + . c c a+2 a+2

The sequences ( a+1−c )k and ( a+2−c )k are decreasing for k ≥ 7, hence a+2 a+2 "   7  7 # a+1 a+1 a+1−c a+2−c + + ≈ −0.05248 < 0 C1 (k) < 2 − c c a+2 a+2 and consequently ν1 (t) < 0. Then −L01 (x) is completely monotonic. The function L1 (x) is decreasing on (0, ∞) and lim L1 (x) = 0, then L1 (x) > 0 and hence L1 (x) is completely x→∞

monotonic on (0, ∞).

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2. Z

L02 (x)

∞

e−(x+1)t ν2 (t)dt, 1 + et



k+1

= 0

where k+1

ν2 (t) =

∞ 2 X



√1 6

k+1

−

√1 6

+1

+ (k + 1)



√1 6

+

1 2

 tk+1 .

(k + 1)!

k=3

Now, consider the following sequence for k = 3, 4, 5, ... k+1   k+1   1 1 1 1 √ − √ +1 C2 (k) = + (k + 1) √ + 6 6 6 2     k X
1 r 1 1 k+1 √ = − + (k + 1) √ + r 6 6 2 r=0     2 X
1 r 1 1 1 k+1 √ < − + (k + 1) √ + < − (k − 2)(k − 3) < 0. r 12 6 6 2 r=0 0 Hence q ν2(t) < 0 and −L2 (x) is completely monotonic. The function L2 (x) is decreasing on 2 , ∞ with lim L2 (x) = 0 and then L2 (x) > 0. Hence L2 (x) is completely monotonic 3 q  x→∞ 2 on ,∞ . 3

3. L03 (x) =

∞

Z

 √  − x+ 23 +1 t

e

0

1 + et

ν3 (t)dt,

where k+1

ν3 (t) =

∞ 2 X



−1 2

+

√1 6

 1+

√1 6

k

+

1 (k+1)

−



1 2

+

√1 6



√1 6

k 

k!

k=3

The sequence   k   k −1 1 1 1 −1 1 X C3 (k) = +√ 1+ √ + = +√ 2 k+1 2 6 6 6 r=0 X    2
1 r 1 −1 1 k √ < +√ + r 2 k+1 6 r=0 6 √ √ √ (k − 3)(( 6 − 3)k 2 + (3 − 3 6)k − (12 + 4 6)) < < 0, 72(k + 1)

k r






1 √ 6

tk+1 .

r +

1 k+1

k = 3, 4, 5, ... .

Then ν3 (t) < 0 and hence −L03 (x) is completely monotonic . The function L3 (x) is decreasing for all x > 0 with lim L3 (x) = 0. and hence L3 (x) > 0 for all x > 0. Then L3 (x) x→∞

is completely monotonic on (0, ∞).

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From the complete monotonicity of the functions L1 (x), L2 (x) and L3 (x), we deduce the following result: Lemma 3.3. 1. 





1+

q 





2 3



1−

q  2

3 1 1 q + q + ln 1 + < G(x) < ln 1 − , x(x + 1) x(x + 1) 2 2 x+ 3 x− 3 q where the upper bound holds for x > 23 and the lower bound holds for x > 0.

2.



1 G(x) < ln 1 + x+a where the values of a and c are in (17)

 +

1+a , x(x + c)

x>0

(19)

(20)

Remark 1. From Lemma (2.2), we can conclude that the inequality (19) improves the lower bound of the inequality (12) for x > xλ ' 2.02059 and improves its upper bound for x > x0 ' 4.02361. Lemma 3.4. The following inequality holds q  q        2 2 1 + 1 − 3 3 1 1 1 1 q + q + + √ − √ ln 1 + < G(x) < ln 1 + , x(x + 1) x(x + 1) 6 6x4 6 6x4 x + 23 x − 23 q where the upper bound holds for x > 23 and the lower bound holds for x ≥ 2. Proof. Consider the function 





1−

q  2

3 1 1 q + T (x) = ln 1 + − √ − G(x), 4 x(x + 1) 2 6 6x x− 3

r x>

2 3

and use the functional equation (4) to obtain 2l(x)

T 0 (x + 2) − T 0 (x) = 81x5 (1

+

x)2 (2

+

x)5 (3

+

x)2

P3

i=0



x+i−

q , 2 3

where l(x) =

√   √  25920 − 10080 6 + 197856 − 75408 6 x  √  2  √  3 + 677952 − 257008 6 x + 1367472 − 520800 6 x   √  √  + 1777284 − 666478 6 x4 + 1535268 − 502094 6 x5   √  √  + 879720 − 127639 6 x6 + 321960 + 145938 6 x7   √  √  + 66960 + 180403 6 x8 + 4596 + 95742 6 x9   √  √  √ + −864 + 28431 6 x10 + −144 + 4590 6 x11 + 315 6x12 > 0, x > 0. 

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q Then T 0 (x + 2) − T 0 (x) > 0 for x > 23 and also lim T 0 (x) = 0. Using Corollary (2.4), we get x→∞ q q  2 2 0 that T (x) < 0 for all x > 3 . Hence T (x) is decreasing on , ∞ with lim T (x) = 0 , thus 3 x→∞ q  2 T (x) > 0 for all x ∈ , ∞ . Now consider the function 3 q

(1 + 23 ) 1 1 q )− − √ Q(x) = G(x) − ln(1 + , x(x + 1) 6 6x4 x + 23 Then Q0 (x + 2) − Q0 (x) =

x > 0.

2u(x − 2) q , P3  2 5 2 5 2 81x (1 + x) (2 + x) (3 + x) i=0 x + i + 3

where u(x) =

√   √  −207466560 + 113432160 6 + −585268704 + 582357840 6 x   √  √  + −729011328 + 1250421968 6 x2 + −523396080 + 1539421184 6 x3   √  √  + −235893516 + 1231511026 6 x4 + −67175076 + 680979590 6 x5  √  6  √  7 + −10943256 + 268473813 6 x + −465384 + 76331554 6 x   √  √  + 195912 + 15574939 6 x8 + 44364 + 2228562 6 x9   √  √  √ + 4032 + 212571 6 x10 + 144 + 12150 6 x11 + 315 6x12 > 0, x ≥ 0. 

Thus Q0 (x + 2) − Q0 (x) > 0 for x ≥ 2 and also lim Q0 (x) = 0. Using Corollary (2.4), we obtain x→∞

that Q0 (x) < 0 for all x ≥ 2. and then Q(x) is decreasing on [2, ∞) with lim Q(x) = 0. Then x→∞

Q(x) > 0 for all x ≥ 2.

4

Second formula of the best approximations and some of its related inequalities

In this section, we will present the best constants of the approximation of formula   P1 (n) 2 1 + , n∈N G(n) ≈ ln 1 + 2 P2 (n) n(n + 1) where P1 (n) and P2 (n) are two polynomials of degrees one and two (resp.). Also, some inequalities of the function G(x) will provided, which improve some results of the previous section. Lemma 4.1. The best approximation of the formula   1 αn + β 2 G(n) ≈ ln 1 + 2 + , 2 n + ρx + σ n(n + 1)

127

n∈N

Ahmed Hegazi ET AL 118-135

(21)

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.1, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

h holds for α = 2, β = 3, ρ = 2 and σ = 4/3 and the sequence G(n) − 21 ln 1 +

αn+β n2 +ρx+σ

i

−

2 n(n+1)

converges to zero with speed estimated by n−5 . h Proof. Consider the error sequence χn = G(n) − 21 ln 1 + χn − χn+2 = + + + + + + +

αn+β n2 +ρx+σ

i

−

2 , n(n+1)

then we have

1 1 (2 − α) + 3 (−2(5 + β) + α(2 + α + 2ρ)) 2 n n 1 (38 − α3 − 3α2 (1 + ρ) + 3β(2 + ρ) + α(−4 + 3β − 3ρ(2 + ρ) + 3σ)) n4 1 4 (α + 4α3 (1 + ρ) + 2(−65 + β 2 − 2β(4 + ρ(3 + ρ) − σ)) n5 α2 (8 − 4β + 6ρ(2 + ρ) − 4σ) + 4α(2 − β(3 + 2ρ) + ρ(4 + ρ(3 + ρ) − 2σ) − 3σ)) 1 (422 − α5 − 5α4 (1 + ρ) + 5β(2 + ρ)(4 − β + ρ(2 + ρ) − 2σ) n6 5/3α3 (−8 + 3β − 6ρ(2 + ρ) + 3σ) + 5α2 (β(4 + 3ρ) − 2(1 + ρ)(2 + ρ(2 + ρ)) (4 + 3ρ)σ) − α(16 + 5β 2 − 5β(8 + ρ(8 + 3ρ) − 2σ) − 40σ + 5(ρ(2 + ρ)(4 ρ(2 + ρ)) − ρ(8 + 3ρ)σ + σ 2 ))) + O(n−7 ).

According to Lemma (2.1), the fastest convergence of the sequence χn satisfies if α = 2, β = 3, ρ = 2 and σ = 4/3 with speed estimated by n−5 . Lemma 4.2. For x > −1, the function 4 R(x) = (e2G(x+2) − 1)(x2 + 2x + ) − 2x 3

(22)

is strictly decreasing and convex. As consequence, we have " #   4 −16 2x + e 12 1 2x + 3 1 ln 1 + 2 , < G(x + 2) < ln 1 + 2 2 2 x + 2x + 43 x + 2x + 34 where the constants 3 and

e4 −16 12

x>0

(23)

are the best possible.

Proof. 1 0 4 R (x) = −x − 2 + [(x2 + 2x + )G0 (x + 2) + (x + 1)]e2G(x+2) , 2 3 1 00 4 R (x) = −1 + 2[(x2 + 2x + )G0 (x + 2) + (x + 1)]G0 (x + 2)e2G(x+2) 2 3 4 +[2(x + 1)G0 (x + 2) + (x2 + 2x + )G00 (x + 2) + 1]e2G(x+2) 3 and 1 2e2G(x+2)

4 R000 (x) = 4(x2 + 2x + )(G0 (x + 2))3 + 12(x + 1)(G0 (x + 2))2 3 4 + 6(x2 + 2x + )G0 (x + 2)G00 (x + 2) 3 4 + 6(x + 1)G00 (x + 2) + 6G0 (x + 2) + (x2 + 2x + )G000 (x + 2) , U (x). 3

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Now, U (x + 2) − U (x) = 16(x + 2)(G0 (x + 2))3 + 4(x + 2)G000 (x + 2) + 24(x + 2)G0 (x + 2)G00 (x + 2) 8(4 + x)(62 + 66x + 24x2 + 3x3 ) 0 + (G (x + 2))2 (x + 2)2 (x + 3)2 8(1448 + 2564x + 2008x2 + 915x3 + 258x4 + 42x5 + 3x6 ) 0 + G (x + 2) (x + 2)4 (x + 3)4 4(4 + x)(62 + 66x + 24x2 + 3x3 ) 00 + G (x + 2) (x + 2)2 (x + 3)2 1 + 4(−33056 − 88128x − 92112x2 − 46976x3 − 10548x4 3(2 + x)6 (3 + x)6 + 360x5 + 705x6 + 138x7 + 9x8 ). Also, let V (x) =

U (x+2)−U (x) , 4(x+2)

V (x + 2) − V (x) =

− + + + − + + + + + + + +

then

−4 [34576 + 91136x + 105392x2 + 68520x3 (2 + + x)2 (4 + x)3 (5 + x)2 +27152x4 + 6698x5 + 1005x6 + 84x7 + 3x8 ](G0 (x + 2))2 4 [376528768 + 1642942016x + 3297590048x2 5 4 (2 + x) (3 + x) (4 + x)5 (5 + x)4 4031614688x3 + 3354474592x4 + 2010658592x5 + 896184192x6 302070808x7 + 77457190x8 + 15051780x9 2183975x10 + 229624x11 + 16548x12 + 732x13 + 15x14 ]G0 (x + 2) 2 (34576 + 91136x + 105392x2 3 2 (2 + x) (3 + x) (4 + x)3 (5 + x)2 68520x3 + 27152x4 + 6698x5 + 1005x6 + 84x7 + 3x8 )G00 (x + 2) 2 [(331346962432 + 5157549202432x 7 3(2 + x) (3 + x)6 (4 + x)7 (5 + x)6 24078469545984x2 + 60544326323200x3 + 99175110059776x4 116067402353280x5 + 102360341211232x6 + 70356164081536x7 38541166023024x8 + 17080773307136x9 + 6184124004420x10 1839407553792x11 + 450403876283x12 + 90669453918x13 + 14930598072x14 1992453932x15 + 212255598x16 + 17635104x17 + 1101756x18 + 48708x19 1359x20 + 18x21 )] x)3 (3

Using the completely monotonicity of the functions X1 (x) =

1 x

− G(x) +

2m−1 P k=1

(22k −1)B2k kx2k

and

, X2 (x) = G(x) − x1 − 2x21+1 for x > 0 (see [20]), we get the following inequalities: G0 (x) > − x+1 x3 h 4 3 2 i2 6 5 4 3 2 −2x +6x +1) +4x +1 (G0 (x))2 > 4x x+4x and G00 (x) > 2(8x +12xx3+12x for x > 0. Hence, 2 (2x2 +1)2 (2x2 +1)3 V (x + 2) − V (x) < F (x),

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where F (x) =

3(2 +

x)8 (3

+

−2A(x + 1) + x)7 (5 + x)6 (9 + 8x + 2x2 )4

x)6 (4

with A(x) = + + + + + + +

74586749184 + 1263034988928x + 9398610597600x2 + 42943724513952x3 138441396784472x4 + 339577282357568x5 + 663837528239296x6 1065966556249164x7 + 1434284269631783x8 + 1638094930661455x9 1600662870950288x10 + 1344078197917456x11 + 971670315225407x12 604754235543331x13 + 323563802759956x14 + 148404632743888x15 58109372496201x16 + 19315095938361x17 + 5408999070416x18 1263403407224x19 + 242838160053x20 + 37707313393x21 + 4608156812x22 426325500x23 + 28044100x24 + 1167972x25 + 23136x26 .

Using A(x) > 0 for all x > 0, then we obtain F (x) < 0 for all x > −1 and hence V (x+2)−V (x) < 0 for all x > −1. Using the asymptotic expansion (8) and its derivatives, we have

and

G0 (x) =

−1 1 1 3 17 − 3 + 5 − 7 + 9 + O(x−11 ) 2 x x x x x

(24)

G00 (x) =

2 3 5 21 153 + 4 − 6 + 8 − 10 + O(x−12 ), 3 x x x x x

(25)

−6 12 30 168 1530 − 5 + 7 − 9 + 11 + O(x−13 ). x4 x x x x   64x25 −9 + O(x ) = 0 lim V (x) = lim x→∞ x→∞ (2 + x)27 (3 + x)6 G000 (x) =

Then

(26)

and hence V (x) > 0 for all x > −1. Now, U (x + 2) − U (x) > 0 with   −64x21 −7 lim U (x) = lim + O(x ) = 0 x→∞ x→∞ 3(x + 2)27 and so U (x) < 0 for all x > −1. Thus, R000 (x) < 0 and   128 448 2368 00 −8 lim R (x) = lim − + + O(x ) = 0. x→∞ x→∞ 15x5 9x6 21x7 Then R00 (x) > 0 for all x > −1 and so the function R(x) is convex for x ∈ (−1, ∞). Also,   −32 448 1184 688 11104 0 −9 lim R (x) = lim + − − + + O(x ) = 0 x→∞ x→∞ 15x4 45x5 63x6 63x7 81x8 and thus R0 (x) < 0 for all x > −1. Hence we conclude that R(x) is decreasing on (−1, ∞) with 4 −16 R(0) = e 12 and   32 112 1184 −6 lim R(x) = lim 3 + − + + O(x ) = 3. x→∞ x→∞ 45x3 45x4 315x5

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Then

e4 − 16 4 , 3 < (e2G(x+2) − 1)(x2 + 2x + ) − 2x < 3 12

where the constants 3 and

e4 −16 12

are the best possible.

Lemma 4.3. For every x ≥ 0, we have 1 ln[ 2 (x2 + 6x + < 7e

where a =

10 3 −e4

12

4x + a 28 )e 3

4 − (x+2)(x+3)

1 ln[ 2 (x2 + 6x +

− (x2 + 2x + 34 )

] ≤ G(x + 2)

4x + b 28 )e 3

4 − (x+2)(x+3)

− (x2 + 2x + 43 )

]

(27)

and b = 12 are the best possible constants.

Proof. For x ≥ 0, consider f (x) = R(x + 2) − R(x), where R(x) defined in (22). Then f 0 (x) = R0 (x + 2) − R0 (x) and R(x) is convex function for x ∈ (−1, ∞). Hence f (x) is increasing with f (0) =

7e

10 3 −e4

− 12 and lim f (x) = 0, where lim R(x) = 3. Then

12

x→∞

7e

10 3 −e4

12

x→∞

− 12 ≤ f (x) < 0 or

10

4 28 4 7e 3 − e4 ≤ e2G(x+2) [(x2 + 6x + )e− (x+2)(x+3) − (x2 + 2x + )] − 4x < 12, 12 3 3

where

7e

10 3 −e4

12

and 12 are the best possible constants.

Lemma 4.4. For every x > 0, then we have (x + β)e−2G(x+2) − (x + 1) (x + α)e−2G(x+2) − (x + 1) 0 < G (x + 2) < , (x2 + 2x + 43 ) (x2 + 2x + 34 ) where α =

(2π 2 −15)e4 144

(28)

and β = 2 are the best possible constants.

Proof. The function R(x) defined in (22) is convex for x ∈ (−1, ∞) and hence R0 (x) is increasing. 2 4 Then R0 (0) < R0 (x) < lim R0 (x) with R0 (0) = (2π −15)e − 4 and lim R0 (x) = 0. Hence 72 x→∞

x→∞

(2π 2 − 15)e4 4 < −x + [(x2 + 2x + )G0 (x + 2) + (x + 1)]e2G(x+2) < 2, 144 3 where

(2π 2 −15)e4 144

and 2 are the best possible constants.

Lemma 4.5. The following inequality holds " #   1 2x + 3 2 1 2x + 3 2 ln 1 + 2 < G(x) < ln 1 + 2 + , 4 + 48 2 x(x + 1) 2 x(x + 1) x + 2x + 3 x + 2x + e4 −16 where the upper bound holds for x > xδ ≈ 0.575833 and the lower bound holds for x > 0.

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h Proof. Consider the function F (x) = G(x + 2) − 12 ln 1 +

2x+3 x2 +2x+ 43

i , then

F 0 (x + 2) − F 0 (x) = G0 (x + 4) − G0 (x + 2) 54(5 + 2x)(56 + 140x + 103x2 + 30x3 + 3x4 ) − . (4 + 6x + 3x2 )(13 + 12x + 3x2 )(28 + 18x + 3x2 )(49 + 24x + 3x2 ) Using the functional equation (4) and its derivative, we get F 0 (x + 2) − F 0 (x) = 32(5 + 2x)(1057 + 1680x + 1011x2 + 270x3 + 27x4 ) . (2 + x)2 (3 + x)2 (4 + 6x + 3x2 )(13 + 12x + 3x2 )(28 + 18x + 3x2 )(49 + 24x + 3x2 ) Thus F 0 (x + 2) − F 0 (x) > 0, for x > 0 and also lim F 0 (x) = 0. Using Corollary (2.4), we get x→∞

that F 0 (x) < 0 for all x > 0. Then F (x) is decreasing function on (0, ∞) with lim F (x) = 0, x→∞

thus F (x) > 0 for x > 0. Now, let # " 2x + 3 1 S(x) = G(x + 2) − ln 1 + 2 2 x + 2x + e448 −16 and then S 0 (x + 2) − S 0 (x) =

−8(5 + 2x)W (x) )(x2 + 4x + (e4 − 16)4 (2 + x)2 (3 + x)2 (x2 + 2x + e448 −16

3e4 e4 −16

)D(x)

,

where  D(x) = (x + 2)2 + 2(x + 2) +

48 4 e − 16

 (x + 2)2 + 4(x + 2) +

3e4 e4 − 16

 > 0,

x>0

and W (x) =


21233664 − 6856704e4 + 720576e8 − 28944e12 + 324e16
+ 100270080 − 26173440e4 + 2361600e8 − 84240e12 + 900e16 x
+ 152764416 − 35570688e4 + 2920320e8 − 96948e12 + 1005e16 x2
+ 106332160 − 23142400e4 + 1795200e8 − 57040e12 + 580e16 x3
+ 37257216 − 7818240e4 + 587520e8 − 18204e12 + 183e16 x4
+ 6389760 − 1320960e4 + 97920e8 − 3000e12 + 30e16 x5
+ 425984 − 88064e4 + 6528e8 − 200e12 + 2e16 x6 .

W 0 (x) =


100270080 − 26173440e4 + 2361600e8 − 84240e12 + 900e16
+2 152764416 − 35570688e4 + 2920320e8 − 96948e12 + 1005e16 x
+3 106332160 − 23142400e4 + 1795200e8 − 57040e12 + 580e16 x2
+4 37257216 − 7818240e4 + 587520e8 − 18204e12 + 183e16 x3
+5 6389760 − 1320960e4 + 97920e8 − 3000e12 + 30e16 x4
+6 425984 − 88064e4 + 6528e8 − 200e12 + 2e16 x5

Then

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and
W 00 (x) = 2 152764416 − 35570688e4 + 2920320e8 − 96948e12 + 1005e16
+6 106332160 − 23142400e4 + 1795200e8 − 57040e12 + 580e16 x
+12 37257216 − 7818240e4 + 587520e8 − 18204e12 + 183e16 x2
+20 6389760 − 1320960e4 + 97920e8 − 3000e12 + 30e16 x3
+30 425984 − 88064e4 + 6528e8 − 200e12 + 2e16 x4 > 0, x > 0. Thus W 0 (x) is increasing on (0, ∞) which implies that W 0 (x) > W 0 (0.1) > 0. Then W (x) is increasing on (0.1, ∞) with W (0.57583) ≈ −475.425 < 0 and W (0.57584) ≈ 1147.33 > 0. Hence W (x) has only one positive root on (0.57583, ∞) say xδ ≈ 0.575833 and then W (x) > 0 for x > 0.575833. Now, S 0 (x + 2) − S 0 (x) < 0 for x > 0.575833 and also lim S 0 (x) = 0. Using x→∞

Corollary (2.4), then S 0 (x) > 0 for all x > 0.575833 or S(x) is increasing on (0.575833, ∞) with lim S(x) = 0. Thus S(x) < 0 for all x > 0.575833.

x→∞

Remark 2. Using the inequalities 1 + (2x + 3)/(x2 + 2x + 48/(e4 − 16)) < (1 + 1/(x + 1))2 ,

x>0

1 + (2x + 3)/(x2 + 2x + 4/3) > (1 + 1/(x + 4/(e2 − 4)))2 ,

x > xµ

and √

−112+68e2 −7e4 −

2

4

6

8

52480−30208e +6672e −664e +25e ' 0.465586, we can conclude that the where xµ = 6(32−12e2 +e4 ) inequality (29) improves the lower bound of the inequality (12) for x > xµ and improves its upper bound for x > 0.

Remark 3. The inequality (29) improves the lower bound of the inequality (19) for x > 0.

References [1] M. Abramowitz and I. A. Stegun (Eds.), Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Dover, New York, 1965. [2] H. Alzer and C. Berg, Some classes of completely monotonic functions, Annales Acad. Sci. Fenn. Math. 27(2), 445-460, 2002. [3] G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge Univ. Press, 1999. [4] N. Batir, Sharp bounds for the psi function and harmonic numbers, Math. Inequal.Appl, Vol. 14, No. 4, 917-925, 2011. [5] T. Buri´ c, N. Elezovi´ c, Some completely monotonic functions related to the psi function, Math. Inequal. Appl., 14(3), 679-691, 2011. [6] C.-P. Chen and H. M. Srivastava, New representations for the Lugo and Euler-Mascheroni constants, II, Appl. Math. Lett. 25, no. 3, 333-338, 2012.

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[7] C.-P. Chen and C. Mortici, Sharpness of Muqattash-Yahdi problem, Comput. Appl. Math., Vol. 31, No.1 , 85-93, 2012. ´ Elbert and A. Laforgia, On some properties of the gamma function, Proc. Am. Math. [8] A. Soc. 128(9), 2667-2673, 2000. [9] A. Erd´ elyi et al., Higher Transcendental Functions Vol. I-III, California Institute of Technology - Bateman Manuscript Project, 1953-1955 McGraw-Hill Inc., reprinted by Krieger Inc. 1981. [10] W. Feller, An Introduction to probability theory and its applications, Vol. 2, 3rd ed. New York, Wiley, 1971. [11] O. Furdui, A class of fractional part integrals and zeta function values, Integral Transforms Spec. Funct. 24, no. 6, 485-490, 2013. [12] A. Z. Grinshpan and M. E. H. Ismail, Completely monotonic functions involving the gamma and q−gamma functions, Proc. Amer. Math. Soc., 134, 1153-1160, 2006. [13] H. Van Haeringen, Completely monotonic and related functions, J. Math. Anal. Appl., 204, 389-408, 1996. [14] M. E. H. Ismail, L. Lorch, and M. E. Muldon, Completely monotonic functions associated with the gamma function and its q−analogues, J. Math. Anal. Appl. 116, 1-9, 1986. [15] A. Laforgia and P. Natalini, On the asymptotic expansion of a ratio of gamma functions, J. Math. Anal. Appl. 389 , no. 2, 833837, 2012. [16] M. Mahmoud, M. A. Alghamdi and R. P. Agarwal, New upper bounds of n!, J. Inequal. Appl. 2012;2012 doi: 10.1186/1029-242X-2012-27. [17] M. Mahmoud and R. P. Agarwal, Bounds for Bateman’s G-function and its applications, Georgian Mathematical Journal, Vol. 23, Issue 4, 579-586, 2016. [18] M. Mahmoud and H. Almuashi, On some inequalities of the Bateman’s G−function, J. Comput. Anal. Appl., Vol. 22, No.4, , 672-683, 2017. [19] M. Mahmoud, A. Talat and H. Moustafa, Some approximations of the Bateman’s G−function, J. Comput. Anal. Appl., Vol. 23, No. 6, 1165-1178, 2017. [20] M. Mahmoud, A. Talat, H. Moustafa and R. P. Agarwal, Completely monotonic functions involving Bateman’s G−function, Submitted for publication. [21] C. Mortici, New approximations of the gamma function in terms of the digamma function, Appl. Math. Lett., Vol. 23, Issue 1, 97-100, 2010. [22] C. Mortici, The proof of Muqattash-Yahdi conjecture, Math. Comput. Mod., Vol. 51, Issue 9, 1154-1159, 2010. [23] C. Mortici, A sharp inequality involving the psi function, Acta Universitatis Apulensis, Vol. 22, 41-45, 2010.

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[24] C. Mortici, A new Stirling series as continued fraction, Numer. Algorithms 56, no. 1, 1726, 2011. [25] F. Qi, The best bounds in Kershaw’s inequality and two completely monotonic functions, RGMIA Res. Rep. Coll. 9 (2006), no. 4, Art. 2. [26] F. Qi, S. Guo, B.-N. Guo, Complete monotonicity of some functions involving polygamma functions, J. Comput. Appl. Math., 233(9), 2149-2160, 2010. [27] F. Qi and S.-H. Wang, Complete monotonicity, completely monotonic degree, integral representations, and an inequality related to the exponential, trigamma, and modified Bessel functions, Glob. J. Math. Anal. 2, no. 3, 91-97, 2014. [28] F. Qi and C. Mortici, Some best approximation formulas and inequalities for the Wallis ratio, Applied Mathematics and Computation, Vol. 253 (15), 363-368, 2015. [29] F. Qi and W.-H. Lic, Integral representations and properties of some functions involving the logarithmic function, Filomat 30:7, 1659-1674, 2016. [30] S.-L. Qiu and M. Vuorinen, Some properties of the gamma and psi functions with applications, Math. Comp., Vol.74, No. 250, 723-742, 2004. [31] D. V. Widder, The Laplace Transform, Princeton University Press, Princeton, 1946. [32] T.-H. Zhao, Z.-H. Yang and Y.-M. Chu, Monotonicity properties of a function involving the Psi function with applications, Journal of Inequalities and Applications (2015) 2015:193.

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A new q-extension of Euler polynomial of the second kind and some related polynomials R. P. Agarwal 1 , J. Y. Kang2* , C. S. Ryoo3

Abstract : We define q-Euler polynomials of the second kind using q-analogue within exponential function. We have some basic properties of this polynomials such as addition, alternative finite sum, and symmetry property. We also investigate some relations of q-Euler, q-Bernoulli, and q-tangent polynomials using q-Euler polynomials of the second kind including two parameters. Key words : q-Euler polynomials of the second kind, q-Euler polynomials, q-Bernoulli polynomials, q-tangent polynomials 2000 Mathematics Subject Classification : 11B68, 11B75, 12D10 1. Introduction The main aim of this paper is to extend Euler numbers and polynomials of the second kind, and study some of their properties. Our paper is organised as follows: in Section 2, we define q-Euler numbers and polynomials of the second kind. From this definition we investigate some interesting properties of these numbers and polynomials using q-analogue of exponential function. In Section 3, we consider q-Euler polynomials of the second kind in two parameters and make some relations between q-Euler polynomials of the second kind and q-Euler , q-Bernoulli, q-tangent polynomials. Furthermore, we derive a symmetric relation, multiple q-derivative, and multiple q-integral. For any n ∈ C, the q-number is defined by [n]q =

X 1 − qn = q i = 1 + q + q 2 + · · · + q n−1 . 1−q 0≤i≤n

An intensive and somewhat surprising interest in q-numbers appeared in many areas of mathematics and applications including q-difference equations, special functions, q-combinatorics, q-integrable systems, variational q-calculus, q-series, and so on. In this paper, we introduce some basic definitions and theorems(see [1-18]). Definition 1.1.[1,3-5,10-13] The Gaussian binomial coefficients are defined by " # (   0 if r > m m m = = , (1−q m )(1−q m−1 )···(1−q m−r+1 ) r q r if r ≤ m (1−q)(1−q 2 )···(1−q r ) q

where m and r are non-negative integers. For r = 0 the value is 1 since the numerator and the denominator are both empty products. Like the classical binomial coefficients, the Gaussian binomial coefficients are center-symmetric. There are analogues of the binomial formula, and this definition has a number of properties. 1 2 3

Department of Mathematics, Texas A & M University, Kingsville, USA Department of Information and Statistics, Anyang University, Anyang, KOREA Department of Mathematics, Hanman University, Daejeon, KOREA

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Theorem 1.2.[5] Let n, k be non-negative integers. Then we get " # n−1 n Y X k n k k ( ) (i) (1 + q t) = t , q 2 k q k=0 k=0 " # n−1 ∞ Y X n+k−1 k 1 (ii) = t . (1 − q k t) k k=0 k=0 q

Definition 1.3.[1,4,12-13] Let z be any complex numbers with |z| < 1. Two forms of qexponential functions are defined by eq (z) =

∞ X zn , [n]q ! n=0

eq−1 (z) =

∞ X

∞ X n zn zn = . q( 2 ) [n]q−1 ! n=0 [n]q ! n=0

Definition 1.4.[4,10-11,13] The q-derivative operator of any function f is defined by Dq f (x) =

f (x) − f (qx) , (1 − q)x

x 6= 0,

and Dq f (0) = f 0 (0). We can prove that f is differentiable at 0, and it is clear that Dq xn = [n]q xn−1 . Definition 1.5.[4,10-11,13] We define the q-integral as Z

b

f (x)dq x = (1 − q)b 0

∞ X

q j f (q j b).

j=0

If this function, f (x), is differentiable on the point x, the q-derivative in Definition 1.4 goes to the ordinary derivative in the classical analysis when q → 1. In 1961, L.Calitz introduced several properties of the Bernoulli and Euler polynomials of the second kind(see [6]). Euler numbers of the second kind was expanded, and C. S. Ryoo have studied these numbers and polynomials of the second kind in [17]. He also developed several properties of these numbers and polynomials. en , and the classical Euler polynomiDefinition 1.6.[7-8, 6, 17] The classical Euler numbers, E en (x), of the second kind are defined by means of the generating functions als, E ∞ X

n 2 en t = E , t n! e + e−t n=0

∞ X

n 2 en (x) t = E etx . t −t n! e + e n=0

Theorem 1.7.[17] For any positive integer n, we have (i)

(ii) (iii)

For any positive integer m(=odd),   m−1 X en (x) = mn en 2i + x + 1 − m for n ≥ 0, E (−1)i E m i=0 l   X l e el (x + y) = E En (x)y l−n , n n=0 en (x) = (−1)n E en (−x). E

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2. Some basic properties of the q-Euler polynomials of the second kind In this section, we define the q-Euler numbers and polynomials of the second kind, and investigate basic properties of these numbers and polynomials. Furthermore, we find the alternative finite sum which is related to the q-Euler numbers and polynomials of second kind. Definition 2.1. Let n be any non-negative integer. For |q| < 1, x ∈ C, we define q-Euler numbers and polynomials of the second kind as ∞ X

tn 2 Een,q = , [n]q ! eq (t) + eq (−t) n=0 ∞ X

2 tn = eq (tx). Een,q (x) [n]q ! eq (t) + eq (−t) n=0 Substituting x = 0 in the q-Euler polynomials of the second kind, they can be simplified as follows: ∞ X tn tn 2 1 Een,q Een,q (0) = = = , [n] ! [n] ! e (t) + e (−t) cosh q q q q q (t) n=0 n=0 ∞ X

where Een,q is q-Euler numbers of the second kind. If q → 1, then we can find the classical Euler polynomials of the second kind in Een,q (x)(see [6,17]). Theorem 2.2. Let |q| < 1, x be any complex numbers. Then, we have " # n X n e Ek,q xn−k . Een,q (x) = k k=0 q

Proof. From the generating function of the q-Euler polynomials of second kind, Een,q (x), we can find ∞ X

∞ ∞ X tn 2 tn X n tn Een,q (x) = eq (tx) = Een,q x [n]q ! eq (t) + eq (−t) [n]q ! n=0 [n]q ! n=0 n=0   " # n ∞ X X n tn  Eek,q (x)xn−k  = , [n]q ! k n=0 k=0 q

which gives the required result. Theorem 2.3. For |q| < 1, the following holds: Dq Een,q (x) = [n]q Een−1,q (x). Proof. Considering q-derivative of xn−k in Theorem 2.2, we get " # n−1 X n Dq Een,q (x) = [n − k]q Eek,q xn−k−1 . k k=0 q

Transforming a binomial operation of q and using Theorem 2.2 again, we obtain " # n X n e Dq Een+1,q (x) = [n + 1]q Ek,q xn−k = [n + 1]q Een,q (x). k k=0 q The required relation now follows at once. 138

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Theorem 2.4. Let n be any non-negative integer. Then, the following holds: x

Z 0

Een+1,q (x) − Een+1,q Een,q (x)dq x = , [n + 1]q

where Een,q (0) = Een,q is q-Euler numbers of the second kind. Proof. Using q-integral in Theorem 2.2, we have " # " # Z xX Z x n n x X n n e 1 n−k Eek,q x dq x = Ek,q Een,q (x)dq x = xn−k+1 [n − k + 1] 0 k k q 0 k=0 0 k=0 q q " # n+1 x   X n+1 1 1 = Eek,q xn−k+1 = Een+1,q (x) − Een+1,q , [n + 1]q [n + 1]q 0 k k=0 q

and we obtain the required relation at once. Corollary 2.5. In Theorem 2.4, we get Z

b

a

Een+1,q (b) − Een+1,q (a) Een,q (x)dq x = . [n + 1]q

Now we find some properties of q-exponential function to obtain the next theorem. From Definition 1.3 and Theorem 1.2, we find that (i) (ii)

(iii)

n

[n]q−1 ! = q −( 2 ) [n]q !,   " # ∞ ∞ ∞ n n X n X X X k n t t tn  eq (t)eq−1 (t) = = q (2)  [n]q ! n=0 [n]q−1 ! n=0 [n]q ! k n=0 k=0 q ! ∞ n−1 X Y tn k = (1 + q ) , [n]q ! n=0 k=0   " # ∞ ∞ ∞ n X X X k n tn tn X (−t)n  eq (t)eq−1 (−t) = = (−1)k q (2)  [n]q ! n=0 [n]q−1 ! n=0 [n]q ! k n=0 k=0 q ! ∞ n−1 X Y tn = (1 − q k ) , [n]q ! n=0 k=0

(iv)

(v)

  " # ∞ ∞ ∞ n X X X k n (−t)n X tn tn (−1)n eq (−t)eq−1 (t) = = q (2) (−1)k  [n]q ! n=0 [n]q−1 ! n=0 [n]q ! k n=0 k=0 q ! ∞ n−1 Y X tn n k = (−1) (1 − q ) , [n]q ! n=0 k=0   " # ∞ ∞ ∞ n n X n X X X k n (−t) (−t) tn (−1)n eq (−t)eq−1 (−t) = = q (2)  [n]q ! n=0 [n]q−1 ! n=0 [n]q ! k n=0 k=0 q ! ∞ n−1 X Y tn = (−1)n (1 + q k ) . [n]q ! n=0 k=0

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Theorem 2.6. For |q| < 1, we find  " #  n−l " # " # n n−l X X n X n − l n − l (i) (−1)k + (−1)n−l  Eel,q = 2(−1)n , l k k l=0 k=0 k=0 q q q   " # " # " # n n−l n−l X X n X n − l n − l (ii) (−1)k + (−1)n−l  Eel,q (x) l k k l=0 k=0 k=0 q q q " # n X n =2 (−1)n−l xl . l l=0 q

Proof. (i) Loading eq (t)eq (−t) + eq (−t)eq (−t) 6= 0 for the generating function of q-Euler numbers of the second kind, one obtains ∞ X

tn (eq (t)eq (−t) + eq (−t)eq (−t)) = 2eq (−t), Een,q [n]q ! n=0 and we can transform such as ∞ X

tn Een,q (eq (t)eq (−t) + eq (−t)eq (−t)) [n]q ! n=0   " # " # ∞ n ∞ n n X X X X n n tn t  = Een,q (−1)k + (−1)n  [n]q ! n=0 [n]q ! k k n=0 k=0 k=0 q q     " # " # " # ∞ X n n−l n−l  tn X X n X n − l n − l = (−1)k + (−1)n−l  Eel,q   [n]q ! l k k n=0 l=0 k=0 k=0 q

=2

∞ X

q

q

n

(−1)n

n=0

t . [n]q !

The required relation now follows at once. (ii) We omit a proof of the q-Euler polynomials of the second kind due to its similarity to (i). Corollary 2.7. For q → 1, in Theorem 2.6, one holds !   n   X n−l  n−l  X X n n−l n−l k n−l el = 2(−1)n , (i) (−1) + (−1) E l k k l=0 k=0 k=0 !   n   X n−l  n−l  n   X X X n n−l n − l n k n−l e (ii) (−1) + (−1) El (x) = 2 (−1)n−l xl , l k k k l=0

k=0

k=0

l=0

en (x) is the classical Euler polynomials of the second kind and E en is the classical Euler where E numbers of the second kind(see [16]). Theorem 2.8. Let |q| < 1. Then we have " # n−l−1 ! n n−l−1 X Y Y n n k n−l k (i) (1 + q ) + (−1) (1 − q ) Eel,q = 2q ( 2 ) , l l=0 k=0 k=0 q " # n−l−1 ! " # n n−l−1 n X n Y Y X n−l n k n−l k e (ii) (1 + q ) + (−1) (1 − q ) El,q (x) = 2 q ( 2 ) xl . l l l=0 k=0 k=0 l=0 q

q

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Proof. (i) For eq (t)eq−1 (t) + eq (−t)eq−1 (t) 6= 0, we have ∞ X

tn Een,q (eq (t)eq−1 (t) + eq (−t)eq−1 (t)) = 2eq−1 (t). [n]q ! n=0 To obtain the result, we can calculate the above equation as ∞ X

tn (eq (t)eq−1 (t) + eq (−t)eq−1 (t)) Een,q [n]q ! n=0 ∞ X

! ∞ n−1 n−1 n X Y Y t tn = Een,q (1 + q k ) + (−1)n (1 − q k ) [n]q ! n=0 [n]q ! n=0 k=0 k=0 " # ! ∞ X n n−l−1 n−l−1 X Y Y n tn k n−l = (1 + q ) + (−1) (1 − q k ) Eel,q [n]q ! l q n=0 l=0 k=0 k=0 =2

∞ X n=0

n

q( 2 )

tn . [n]q !

The required relation now follows on comparing the coefficients of tn on both sides. (ii) Using the same method as (i) we can find the required result, so we omit the proof. Corollary 2.9. In Theorem 2.8, we can see " # n−l−1 ! n n−l−1 X Y Y n n 1 k n−l k ( ) (1 + q ) + (−1) (1 − q ) Eel,q , (i) q 2 = 2 l l=0 k=0 k=0 q " # " # n−l−1 ! n n n−l−1 X n X n Y Y n−l 1 k n−l k l ( ) (1 + q ) + (−1) (1 − q ) Eel,q (x). (ii) q 2 x = 2 l q l q l=0 l=0 k=0 k=0 Theorem 2.10. For |q| < 1, k ∈ N, one holds " # n−l−1 ! n n−l−1 X Y Y n n k n−l k (i) (1 − q ) + (−1) (1 + q ) Eel,q = 2(−1)n q ( 2 ) , l l=0 k=0 k=0 q " " # # n−l−1 ! n n n−l−1 X n X Y Y n−l n k n−l k (ii) (1 − q ) + (−1) (1 + q ) Eel,q (x) = 2 (−1)n−l q ( 2 ) xl . l l l=0 k=0 k=0 l=0 q

q

Proof. (i) Let eq (t)eq−1 (−t) + eq (−t)eq−1 (−t) 6= 0. From the generating function of q-Euler numbers of the second kind, we can find ∞ X

tn Een,q (eq (t)eq−1 (−t) + eq (−t)eq−1 (−t)) = 2eq−1 (−t), [n]q ! n=0 or, equivalently, ∞ X

tn Een,q (eq (t)eq−1 (−t) + eq (−t)eq−1 (−t)) [n]q ! n=0 ∞ X

! ∞ n−1 n−1 n X Y Y t tn = Een,q (1 − q k ) + (−1)n (1 + q k ) [n]q ! n=0 [n]q ! n=0 k=0 k=0   ! " # n−l−1 n ∞ ∞ X n−l−1  tn X Y Y X n n tn = (1 − q k ) + (−1)n−l (1 + q k ) Eel,q =2 (−1)n q ( 2 ) .   [n]q ! [n]q ! l q n=0 n=0 k=0 k=0 l=0 141

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n

t Comparing the coefficients of [n] , the proof is complete. q! (ii) We omit the proof of the q-Euler polynomials because we can derive it in the same method as (i).

Corollary 2.11. In Theorem 2.10, we get " # n−l−1 ! n n−l−1 Y Y 1X n n (n k n−l k ) 2 (i) (−1) q = (1 − q ) + (−1) (1 + q ) Eel,q , 2 l l=0 k=0 k=0 q " # n−l−1 ! " # n n−l−1 n X n Y Y X n 1 n−l (n−l l k n−l k ) (ii) (−1) q 2 x = (1 − q ) + (−1) (1 + q ) Eel,q (x). 2 l l l=0 k=0 k=0 l=0 q q

Theorem 2.12. For x ∈ C, we hold " # ( n X n 2 if n = 0 (i) (1 + (−1)k )Een−k,q = , k 0 if n 6= 0 k=0 q " # n X n (1 + (−1)k )Een−k,q (x) = 2xn . (ii) k q k=0 Proof. (i) From Definition 2.1, we can represent q-Euler numbers, Een,q , as ∞ tn tn X (1 + (−1)n ) = 2. Een,q [n]q ! n=0 [n]q ! n=0 ∞ X

Now using the Cauchy’s product, we find the relation,   " # n ∞ X X n tn  (1 + (−1)k )Een−k,q  = 2, [n]q ! k n=0 k=0 q

and the proof is done. (ii) We omit a proof of (ii) since we can obtain (ii) using Cauchy’s product and the method of coefficient comparison for Definition 2.1 using the same method (i). Theorem 2.13. Let x ∈ C and |q| < 1. Then, the following holds: (i)

# " # [ n2 ] " n X X n n k e (1 + (−1) )En−k,q = 2 Een−2k,q , k q n − 2k q k=0 k=0

(ii)

" # # [ n2 ] " n X X n n k e (1 + (−1) )En−k,q (x) = 2 Een−2k,q (x), k n − 2k k=0 k=0 q

q

where [x] is the greatest integer not exceeding x. Proof. (i) In Theorem 2.12. (i), the left-side is changed as: ∞ X

∞ ∞ ∞ X tn X tn tn X t2n Een,q (1 + (−1)n ) =2 Een,q [n]q ! n=0 [n]q ! [n]q ! n=0 [2n]q ! n=0 n=0     " # # [ n2 ] " ∞ n ∞ 2n−k X X X X 2n − k e  t n   tn  =2 Ek,q =2 Een−2k,q  .  [2n − k]q ! [n]q ! k n − 2k n=0 n=0 k=0

k=0

q

142

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The required relation now follows on comparing the coefficients of tn on both sides. (ii) Now following the same procedure as (i), we find (ii). Corollary 2.14. From the Theorem 2.12 and Theorem 2.13, the following relations hold: (i)

[ n2 ] " X k=0

[ ]" X n 2

(ii)

n n − 2k

k=0

#

n n − 2k

( Een−2k,q = q

1 0

if n = 0 , if n = 6 0

# Een−2k,q (x) = xn , q

where [x] is the greatest integer not exceeding x. Theorem 2.15. For x ∈ C, the following relation holds Een,q (x) = (−1)n Een,q (−x). Proof. Replacing t, x with −t, −x, respectively, we get ∞ X tn 2 (−t)n Een,q (x) = eq (tx) = , Een,q (−x) [n]q ! eq (−t) + eq (t) [n]q ! n=0 n=0 ∞ X

which on comparing the coefficients immediately gives the required relation. Corollary 2.16. Putting x = 1 in Theorem 2.15, we see Een,q (1) = (−1)n Een,q (−1). 3. Some special properties of the q-Euler polynomials of the second kind In this section, we define the q-Euler polynomials of the second kind in two parameters. From these polynomials, we can find some relations between these polynomials and other polynomials. We can also observe a symmetric property of the q-Euler polynomials of the second kind. Definition 3.1. Let x, y ∈ C. We then define the q-Euler polynomials of the second kind in two parameters as: ∞ X tn 2 Een,q (x, y) = eq (tx)eq (ty). [n] ! e (t) + eq (−t) q q n=0 For y = 0, we can see that Een,q (x, 0) = Een,q (x). Theorem 3.2. Let x be any complex numbers. Then we hold " # n X n e e (i) En,q (x, y) = Ek,q (x)y n−k , k k=0 q " # " # n l X n X l e e xl−k y k . (ii) En,q (x, y) = En−l l k l=0 k=0 q q Proof. From Definition 3.1, we find ∞ X

tn 2 Een,q (x, y) = eq (tx)eq (ty) [n]q ! eq (t) + eq (−t) n=0   " # ∞ ∞ n n X n X X n t t tn  = Een,q (x) yn = Eek,q (x)y n−k  . [n]q ! n=0 [n]q ! n=0 [n]q ! k n=0 ∞ X

k=0

143

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The required relation now follows immediately. Theorem 3.3. For x ∈ C, we hold Een,q (x, 1) + Een,q (x, −1) = 2xn . Proof. Setting y = 1 and −1, we can get ∞  X n=0

=

 tn 2 2 = eq (tx)eq (t) + eq (tx)eq (−t) Een,q (x, 1) + Een,q (x, −1) [n]q ! eq (t) + eq (−t) eq (t) + eq (−t)

∞ X 2 tn eq (tx) (eq (t) + eq (−t)) = 2eq (tx) = 2 , xn eq (t) + eq (−t) [n]q ! n=0

and the proof is complete on comparing the coefficient of both sides. Corollary 3.4. From Theorem 3.3, we see  1 e En,q (x, 1) + Een,q (x, −1) . xn = 2 To investigate some relations of other polynomials, we define q-Euler, q-Bernoulli, and q-tangent polynomials. These polynomials have a lot of properties, applications, and identities. Definition 3.5. We define q-tangent polynomials, T (x); q-Euler polynomials, E(x); and qBernoulli polynomials, B(x) as ∞ X n=0 ∞ X n=0 ∞ X n=0

En,q (x)

tn [2]q = eq (tx), [n]q ! eq (t) + 1

|t| < π,

Tn,q (x)

[2]q tn = eq (tx), [n]q ! eq (2t) + 1

|t|
1, such that qX,S2 ≤ hX,S2 ≤ c1 qX,S2 ,

(2.3)

then X is called quasi-uniform. In addition, the set X is said to be PL -unisolvent (see [32]), if p ∈ PL , p(xi ) = 0 for i = 1, 2, . . . , N ⇒ p = 0.

2.2

Laplace-Beltrami operator

In this subsection, we introduce Laplace-Beltrami operator on S2 (see [23, 33]). The LaplaceBeltrami operator is defined by   3 X x ∂ 2 g(x) , g(x) := f ∆f := . ∂x2i kxk2 i=1

kxk2 =1

In fact, the Laplace-Beltrami operator is the angular part of the Laplace operator in three dimensions ∂2 ∂2 ∂2 + + . 2 2 ∂x1 ∂x2 ∂x23 Giving point x := (x1 , x2 , x3 ) on S2 , then the related spherical polar coordinate system is (θ, ϕ), 0 ≤ θ ≤ π, 0 ≤ ϕ < 2π, in terms of polar coordinate transformation x1 = sin θ cos ϕ, x2 = sin θ sin ϕ, x3 = cos θ, the Laplace-Beltrami operator acting as a differential operator can be written by ∆ :=

1 ∂ ∂ 1 ∂2 (sin θ ) + . sin θ ∂θ ∂θ sin2 θ ∂ϕ2

The literature has pointed an intrinsic characterization of spherical harmonics, which is every element of Hl is an eigenfunction corresponding to the eigenvalue −l(l + 1) of the Laplace-Beltrami operator ∆, namely that ∆Yl,k (x) = −l(l + 1)Yl,k (x). In fact, ∆ is a semi-positive operator, and for any s > 0 we can define (−∆)s as s

(−∆)s Yl,k = (l(l + 1)) Yl,k (x) = βl Yl,k (x). So, for p(x) ∈ PL , (−∆)s p(x) can be represented by (−∆)s p(x) =

L X l=0

βl

2l+1 X

Yl,k (x)hYl,k , pi =

k=1

L X l=0

Z βl S2

2l + 1 Pl (x, y)p(y)dω(y), 4π

where βµ = (µ(µ + 1))s , µ = 0, 1, . . . , L, and Pl is the Legendre polynomial with degree l.

3

Moving least squares

Moving least squares (MLS) approximation has been frequently applied to potential energy surfaces [20], surface reconstruction [15], and partial differential equations [7]. In order to propose regularized moving least squares (RMLS) in the next section, we should first review some details about MLS approximation on the sphere [35]. The issue of MLS approximation on the sphere has been given some detailed discussions by Wendland in [34, 35, 36]. Suppose an unknown continuous function f ∈ C(S2 ) and x ∈ S2 , we can construct an approximation of f (x) from values {f (xi )}N i=1 of f on a given point set

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X = {x1 , . . . , xN } ⊆ S2 . Then the approximate value p∗ (x) of f (x) can be obtained by the solution of following minimization problem (N ) X 2 min (f (xi ) − p(xi )) w(x, xi ) : p ∈ P , (3.4) i=1 2

where P ⊆ C(S ) is a finite dimensional subspace, usually spanned by spherical harmonics, and w : S2 × S2 → [0, ∞) is a continuous function. Since we consider a local process, we choose w(x, y) as a radial and compactly supported function, even if it is not really necessary. So Wendland [34, 35, 36] chose continuous function φ : [0, ∞) → [0, ∞) with • φ(r) > 0, 0 ≤ r < 1, • φ(r) = 0, r ≥ 1, and define 1  , x, y ∈ S2 , (3.5) θδ (x, y) :=  d(x,y) φ δ where δ > 0 is a scale. Then above weight function w(x, xi ) has the following form   1 d(x, y) w(x, xi ) = =φ . θδ (x, xi ) δ For X = {x1 , x2 , . . . , xN }, we further define the index set I(x) as I(x) := I(x, δ, X) = {i ∈ {1, 2, . . . , N } : d(x, xi ) < δ},

(3.6)

which contains the subscripts of points within the spherical cap of radius δ centered at x. And we choose P = PL . Then the MLS approximation (3.4) takes the form (see [18, 34, 35, 36]) X sf,X (x) = a∗i (x)f (xi ), i∈I(x)

where the coefficients

a∗i (x)

are determined by minimizing 1 X 2 ai (x)θδ (x, xi ) 2

(3.7)

i∈I(x)

under the constraints X

ai (x)p(xi ) = p(x), p ∈ PL .

(3.8)

i∈I(x)

If X satisfies certain conditions, then we have the following theorem [35]. Theorem 3.1 Assume that Z = {xi ∈ X : i ∈ I(x, δ, X)} is PL -unisolvent. Then the minimization problem (3.7) with constraint (3.8) has an unique solution a∗i (x):   L 2µ+1 d(x, xi ) X X ∗ ai (x) = φ λµ,ν Yµ,ν (xi ), δ µ=0 ν=1 where i ∈ I(x), xi ∈ Z, and the Lagrange multipliers λl,k have unique solution by solving the following system of equations: L 2µ+1 X X X  d(x, xi )  λµ,ν φ Yµ,ν (xi )Yl,k (xi ) = Yl,k (x) δ µ=0 ν=1 i∈I(x)

with 0 ≤ l ≤ L, 1 ≤ k ≤ 2µ + 1. Since Z = {xi1 , xi2 , . . . , xiM } = {xi , i ∈ I(x)} ⊆ X involves the choice of scale δ, so it is also an interesting research direction. From [36] we know that if x lies in a region with a high data density, then the δ should be chosen small. However, we should choose a bigger δ, since our method is local. Therefore, we often choose δ = δX = C1 hX , (3.9) where C1 is a constant.

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4

Regularized moving least squares approximation on the sphere

Regularized moving least squares with Laplace-Beltrami operator

In this section, we propose a category of local polynomial approximation on the unit sphere S2 in terms of an improvement of MLS, and give the model of the RMLS. For an unknown continuous function f ∈ C(S2 ), X = {x1 , x2 , · · · , xN } ⊆ S2 , and x ∈ S2 , we can get an approximate value p(x) of f (x) from values {f (xi )}N i=1 by the solution p of following minimization problem (N )   d(x, x )  X i 2 2 : p ∈ PL , min (f (xi ) − p(xi )) + λ ((−∆)s p(xi )) φ (4.10) δ i=1 where (−∆)s and φ is defined as above, λ > 0 is a regularization parameter. Similar to [18, 34, 35, 36], we want to use polynomial local reconstruction to estimate approximation order. So, the new approximation form can be constructed and it is the same as the solution of (4.10). We construct the new approximation form: X sf,X (x) = a∗i (x)f (xi ), (4.11) i∈I(x)

where the coefficients are determined by minimizing 1 X 2 ai (x)θδ (x, xi ) 2

(4.12)

i∈I(x)

under the constraints X

ai (x)p(xi ) = q(x), p ∈ PL ,

(4.13)

i∈I(x)

where

L 2µ+1 X X q(x) = (1 + λβµ2 )−1 pˆµ,ν Yµ,ν (x), µ=0 ν=1

pˆµ,ν is the Fourier coefficient of p, and βµ = (µ(µ + 1))s . The following (2) of Theorem 4.1 shows that the constructed approximation form (4.11) and constrained optimization problems (4.12)(4.13) are valid. In the following, we focus on how to solve the new constrained optimization problem, where Z = {xi1 , xi2 , . . . , xiM } = {xi , i ∈ I(x)} ⊆ X. We need the following notations: f = (f (x1 ), f (x2 ), . . . , f (xM ))T ; a = (a∗1 (x), a∗2 (x), . . . , a∗M (x))T ; α = (α0,1 , . . . , αL,2L+1 )T ; ϕ = (Y0,1 (x), . . . , YL,2L+1 (x))T ;        d(x, x1 ) d(x, x2 ) d(x, xM ) W = diag φ ,φ ,...,φ ; δ δ δ B = diag{β0 , β1 , β1 , β1 , . . . , βµ , . . . , βµ , . . . , β2L+1 , . . . , β2L+1 }; | {z } 2µ+1



Y0,1 (x1 )  Y0,1 (x2 )  Y = ..  .

Y1,1 (x1 ) Y1,1 (x2 ) .. .

Y1,2 (x1 ) Y1,2 (x2 ) .. .

··· ··· .. .

Y0,1 (xM ) Y1,1 (xM ) Y1,2 (xM ) · · ·

 YL,2L+1 (x1 ) YL,2L+1 (x2 )   . ..  . YL,2L+1 (xM )

The following Theorem 4.1 will give the concrete form of the solution of RMLS approximation, and proves that the solution of (4.10) is equivalent to the solution of the minimization problem (4.12) with constraint (4.13). Namely, the constructions (4.11)-(4.13) are valid.

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Theorem 4.1 The following statements hold. (1). For a given point set X = {x1 , . . . , xN } ⊂ S2 , if Z = {xi ∈ X : i ∈ I(x, δ, X)} is PL unisolvent, then the minimization problem (4.12) with constraint (4.13) has unique solution a∗i (x): a∗i (x)

 =φ

d(x, xi ) δ

X L 2µ+1 X

zµ,ν Yµ,ν (xi ), i ∈ I(x),

(4.14)

µ=0 ν=1

where zl,k can be obtained by solving the following system of equations: L 2µ+1 X X

zµ,ν

µ=0 ν=1



X

φ

i∈I(x)

d(x, xi ) δ



Yµ,ν (xi )Yl,k (xi ) = (1 + λβl2 )−1 Yl,k (x)

(4.15)

with 0 ≤ l ≤ L, 1 ≤ k ≤ 2l + 1, λ > 0; (2). The solution of (4.10) is equivalent to the solution of the minimization problem (4.12) with constraint (4.13). Proof. We first prove (1). Similar to [34], [35] and [36], we introduce Lagrange multiplies z = (ˆ z0,1 , . . . , zˆL,2L+1 ) to solve the optimal problem (4.12) with constraint (4.13). Let   L 2µ+1 X X X 1 X 2 ai (x)θδ (x, xi ) − zµ,ν  ai (x)Yµ,ν (xi ) − (1 + λβµ2 )−1 Yµ,ν (x) , J= 2 µ=0 ν=1 i∈I(x)

i∈I(x)

where zµ,ν = zˆµ,ν pˆµ,ν . We solve partial derivatives about ai (x) and zl,k for J, respectively, L 2µ+1 X X ∂J = ai (x)θδ (x, xi ) − zµ,ν Yµ,ν (xi ) = 0, i ∈ I(x), ∂ai (x) µ=0 ν=1

X ∂J =− ai (x)Yl,k (xi ) + (1 + λβl2 )−1 Yl,k (x) = 0, 0 ≤ l ≤ L, 1 ≤ k ≤ 2l + 1, ∂zl,k i∈I(x)

then, solving the above equations, we can get (4.14) and (4.15). In order to prove equivalent conditions, the solution (4.14) of the optimal problem (4.12) under constraint (4.13) can be written as matrix form a = W Y (Y T W Y + λB T Y T W Y B)−1 ϕ. Next, we prove the uniqueness of the solution. In fact, we only need to prove that Y T W Y + λB Y T W Y B is a positive definite matrix. For any vector T

2

r = (r0,1 , . . . , rL,2L+1 )T ∈ R(L+1) , and for i ∈ I(x), w(x, xi ) > 0, rT (Y T W Y + λB T Y T W Y B)r = (Y r)T W (Y r) + rT λB T Y T W Y Br !2 !2 L 2l+1 L 2l+1 X  d(x, xi )  X X X  d(x, xi )  X X = φ Yl,k + φ rl,k βl Yl,k δ δ i∈I(x)

l=0 k=1

l=0 k=1

i∈I(x)

≥ 0. Since the entries of diagonal line of matrix B are not all 0, and Z is PL -unisolvent, we see that X i∈I(x)

 φ

d(x, xi ) δ

 X L 2l+1 X l=0 k=1

!2 Yl,k

+

X i∈I(x)

 φ

d(x, xi ) δ

 X L 2l+1 X

!2 rl,k βl Yl,k

= 0,

l=0 k=1

implies r = 0. So Y T W Y + λB T Y T W Y B is a positive definite matrix.

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We now prove property (2). Let {Y0,1 , . . . , YL,2L+1 } be a set of the spherical harmonics in PL . Then the minimizer of (4.10) can be written as p∗ (x) =

L 2µ+1 X X

αµ,ν Yµ,ν (x).

µ=0 ν=1

Thus αT = f T W Y (Y T W Y + λB T Y T W Y B)−1 , hence ∗

p (x)

=

L 2µ+1 X X

αµ,ν Yµ,ν (x) = αT ϕ

µ=0 ν=1

= f T W Y (Y T W Y + λB T Y T W Y B)−1 ϕ = f T a = sf,X (x). The proof of Theorem 4.1 is complete.  The following Theorem 4.2 devotes that p(x) of the solution of the minimization problem (4.12) with constraint (4.13) uniformly converges to “s-smoothed” solution fs : fs (x) :=

∞ X l=0

Z 2l+1 ∞ X X 1 1 1 ˆl,k Yl,k (x) = f Pl (x · y)f (y)dω(y), 1 + λ(l(l + 1))2s 1 + λ(l(l + 1))2s 4π S2 k=1

l=0

where the last equation uses the addition theorem (2.1). Theorem 4.2 Assume that the order of Laplace-Beltrami operator s > 1/2, p(x) is the solution of the minimization problem (4.12) with constraints (4.13), and L is the order of p(x), then we have limL→∞ kp − fs kC(S2 ) = 0. This theorem has been proved in [1].

5

Error estimates

In this section, we will give an error estimate for RMLS approximation, which ensures the fact that RMLS approximation scheme is reasonable (see Theorem 6 below). But before starting the error analysis, we need to collect a few auxiliary results. The following Lemma 4 indicates the local polynomial reproduction on the sphere, which is quoted from [35]. We also refer the reader to [18] and [34] for a general form of the local polynomial reproduction property, and it plays an important role in the error estimates for RMLS approximation. Lemma 5.1 There exist constants h0 , C2 , C3 > 0 such that for every point set X = {x1 , x2 , . . . , xN } ⊆ X X S2 with hX,S2 ≤ h0 and every x ∈ S2 , there exist aX 1 (x), a2 (x), . . . , aN (x) satisfying that PN X (1) i=1 ai (x)p(xi ) = q(x), for any p ∈ PL ; (2) aX i (x) = 0, if d(x, xi ) > C2 hX,S2 ; PN X (3) i=1 |ai (x)| ≤ C3 , where L 2µ+1 X X q(x) = (1 + λβµ2 )−1 pˆµ,ν Yµ,ν (x). µ=0 ν=1

The following Lemma 5.2 is quoted from [35], which shows that |I(x)| is uniformly bounded in terms of packing argument from [27], and it plays an important role in the error estimates for RMLS and MLS approximation.

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Lemma 5.2 Assume that X = {x1 , x2 , . . . , xN } ⊂ S2 is quasi-uniform with hX,S2 ≤ h0 , I(x) := {i ∈ {1, 2, . . . , N } : d(x, xi ) < δ}, and δ = C1 hX,S2 . Then |I(x)| ≤

qX + δ ≤ (1 + c1 C1 ), qX

where the c1 and C1 are constants which associate with (2.3) and (3.9), respectively. From Lemma 5.1 and Lemma 5.2, we use the similar techniques of [35], and obtain the following Theorem 5.1, which is an error estimate for RMLS approximation. Theorem 5.1 Let C1 , C3 , δ be given in (3.9), Lemma 5.1, and Lemma 5.2. Suppose that X = {x1 , x2 , . . . , xN } ⊆ S2 is quasi-uniform, and sX,f is the RMLS approximation of f ∈ C(S2 ) by minimization (4.12) under the constraint (4.13). Then there exist constants h0 and C which are independent of f and X, such that for every X with hX,S2 ≤ h0 and every x ∈ S2 , the error between f and sX,f can be bounded by |f (x) − sX,f (x)| ≤ Ccf C1l+1 hl+1 X,S2 . Proof. Let q ∈ PL and B(x, δ) = {y ∈ S2 ; d(x, y) ≤ δ}. We adopt the standard arguments to estimate the error of RMLS approximation: |f (x) − sX,f (x)| = |f (x) − q(x) + q(x) − sX,f (x)| ≤ |f (x) − q(x)| + |q(x) − sX,f (x)| X ≤ kf (x) − q(x)k∞,B(x,δ) + |a∗i (x)|kf (x) − p(x)k∞,B(x,δ) i∈I(x)

PL P2µ+1 where the relationship between p(x) and q(x) is q(x) = µ=0 ν=1 (1 + λβµ2 )−1 pˆµ,ν Yµ,ν (x), pˆµ,ν is the Fourier coefficient of p, and βµ = (µ(µ + 1))s . So we can write  max kf (x) − q(x)k∞,B(x,δ) , kf (x) − p(x)k∞,B(x,δ) := kf (x) − G(x)k∞,B(x,δ) , then |f (x) − sX,f (x)| ≤ (1 +

X

|a∗i (x)|)kf (x) − G(x)k∞,B(x,δ) .

i∈I(x)

For

∗ i∈I(x) |ai (x)|,

P

using Cauchy inequality we have 1/2 

 X i∈I(x)

X

|a∗i (x)| ≤ 

X

|a∗i (x)|2 θδ (x, xi )



i∈I(x)

i∈I(x)

1/2 d(x, xi )  ) . φ( δ

(5.16)

Now we prove that the first term of the right of (5.16) is bounded. According to hX,S2 ≤ h0 , g = we can get ai (x) that reproduces spherical harmonics and vanishes if d(x, xi ) > 2δ . We set I(x) δ {i : d(x, xi ) ≤ 2 }, then, by the minimization condition it is not difficult for us to obtain that X

|a∗i (x)|2 θδ (x, xi ) ≤

i∈I(x)

X

|ai (x)|2 θδ (x, xi ) ≤

g i∈I(x)

≤

N X

!2 |ai (x)|

i=1

and X

φ(

1

X

d(x,xi ) mini∈I(x) ) g g φ( δ i∈I(x)

1 miny∈Z φ( d(x,y) δ )

≤ C32

|ai (x)|2

1 miny∈Z φ( d(x,y) δ )

,

d(x, xi ) ) ≤ |I(x)|kφk∞ . δ

g i∈I(x)

From Lemma 5.2, we see that |I(x)| is uniformly which implies that (5.16) is bounded.  bounded,  P Therefore, there exists a constant C, such that 1 + i∈I(x) |a∗i (x)| < C.

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Next, we prove that kf (x)−G(x)k∞,Z is bounded. According to [35], without loss of generality, we suppose that x = (0, 0, 1)T . Then B(x, δ) = {y ∈ S2 : d(x, y) < δ} = {y ∈ S2 : y3 > cos δ}. p y ∈ R2 : k˜ y k22 < We define the bijective map T : U → B(x, δ) by y˜ → (˜ y , 1 − k˜ y k22 )T , where U = {˜ 2 −1 T 1 − cos δ}. Obviously, the inverse of T is T (y) = y˜ = (y1 , y2 ) . Then, the Taylor expansion of g around x ˜ = 0 is X g (α) (0) X g (α) (ξ) g(˜ y) = y˜α + y˜α . α! α! |α|≤l

|α|=l+1

So f (y) = g ◦ T −1 (y) =

X

X

cα y α +

|α|≤l

|α|=l+1

g (α) (ξ) α y˜ , α!

and G(y) =

X

cα y α .

|α|≤l

Hence |f (y) − G(y)|

≤ cf k˜ y kl+1 = cf (1 − y32 )(l+1)/2 ≤ cf (1 − cos2 δ)(l+1)/2 = cf (sin δ)l+1 2 l+1 ≤ cf δ = cf C1 hl+1 X,S2 .

Therefore, |f (x) − sX,f (x)| ≤ Ckf − Gk∞,B(x,δ) ≤ Ccf C1l+1 hl+1 X,S2 . The proof of Theorem 5.1 is complete.

6



Numerical experiments

In order to further validate our theoretical results derived in the previous sections, this section presents some numerical experiments handling data set with high level noise. In our experiments, we choose two test functions, where the Franke function f (x, y, z) is chosen as the first test function which has been frequently used in the other literature (for example, [28, 35]),   (9x − 2)2 (9y − 2)2 (9z − 2)2 3 exp − − − f1 (x, y, z) = 4 4 4 4   (9x + 1)2 (9y + 1)2 (9z + 1)2 3 − − + exp − 4 49 10 10   1 (9x − 7)2 (9y − 3)2 (9z − 5)2 + exp − − − 2 4 4 4
1 − exp −(9x − 4)2 − (9y − 7)2 − (9z − 5)2 , (x, y, z) ∈ S2 . 5 This function is shown in the Figure 2 (a), and it is C ∞ (S2 ). The second test function is spherical cap function which is a sum of the Franke function f1 and an other function fcap (see [38]), which is defined by f2 := f1 + fcap ,where      ρ cos π arccos(hxc , xi) , x ∈ C(x , r); c 2r fcap :=   0, otherwise, and ρ is a positive number. We set xc = (− 21 , − 12 , function is shown in the Figure 3 (a).

157

q

1 2 ), ρ

= 2, and r =

1 2

in the experiment. This

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In the RMLS approximation, the weight function plays an important role. We choose a famous radial basis function φ(r) as weight function in our numerical experiments, that is φ(r) = (1 − r)4+ (4r + 1), which is called Wendland function (see [37]). The uniform error of the approximation is estimated by kf − pkC(S2 ) ≈ max |f (xi ) − p(xi )|. xi ∈X

In our numerical experiments, we choose X to be a set of 1024 points generated from the equal area algorithm [30], which is shown in Figure 1.

1

0.5

0

−0.5

−1 1 1

0.5 0.5

0 0 −0.5

−0.5 −1

−1

Figure 1: A set of 1024 points generated from the equal area algorithm Next, we consider two groups of numerical experiments reconstructing the test function f1 and f2 in terms of RMLS and MLS, where the data set X has been contaminated by high levels of noise. In the experiment 1 and 2, X = {x1 , x2 , . . . , xN }, and N = 1024, meanwhile, 30% noise have been used in X, where the noise is a sample of a normal random variable with mean 0 and standard deviation σ = 0.1. In order to achieve uniform standard of comparison, we take polynomial degree L = 2 and scale δ = 0.25. Experiment 1. We want to reconstruct the Frank function f1 from contaminated data and compare approximation results of RMLS (λ = 0.2) and MLS (λ = 0), meanwhile, s is set as 2. Figure 2 illustrates that RMLS exists more obvious advantages than MLS when we reconstruct test function f1 from data set with high level noise. The Figure 2 (a) shows original function f1 , the Figure 2 (b) reports f1 with high level noise, and the Figure 2 (c) reveals approximation result of RMLS for reconstructing f1 , and the uniform error of RMLS approximation is 0.0868. At last, the Figure 2 (d) shows approximation result of MLS for reconstructing f1 , and the uniform error of MLS approximation is 0.1363. As we known, the test function f1 called Franke function is C ∞ (S2 ), however, test function f2 is continuous on the unit sphere S2 but not differentiable on the boundary of spherical cap C(xc , r). In order to show the effect of RMLS approximation for reconstructing function, we reconstruct f2 from data set with high level noise in the following experiments. Experiment 2. Test function f2 is reconstructed from data set with high level noise, and its designing approach is similar with Experiment 1. First of all, we fix the order of Laplace-Beltrami

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Figure 2: A result of test function f1 in experiment 1

Figure 3: A result of test function f2 in experiment 2

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Table 1: The uniform error of MLS and RMLS for f1 and f2 when λ and s were changed λ 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

uniform error for f1 s=2 s=4 0.1363 0.1363 0.0868 0.0805 0.0909 0.0881 0.0936 0.0839 0.0936 0.0832 0.0900 0.0933 0.0983 0.0929 0.0847 0.0902 0.0834 0.0845 0.1033 0.0854 0.0900 0.0844

uniform error for f2 s=2 s=4 0.1303 0.1303 0.0991 0.0958 0.0990 0.0818 0.0851 0.0755 0.0836 0.0801 0.0905 0.0829 0.0932 0.0932 0.0934 0.0758 0.0899 0.0791 0.0982 0.0863 0.0833 0.0830

operator s = 4. Secondly, in order to compare RMLS with MLS, we let λ = 0 (MLS) and λ = 1.4 (RMLS). At last, we show the superiority in terms of uniform error. Figure 3 illustrates that RMLS exists more obvious advantages than MLS when we reconstruct test function f2 from data set with high level noise. The Figure 3 (a) shows original function f2 , the Figure 3 (b) reports f2 with high level noise, and the Figure 3 (c) reveals approximation result of RMLS for reconstructing f2 , and the uniform error of RMLS approximation 0.0758. Finally, the Figure 3 (d) shows approximation result of MLS for reconstructing f2 , and the uniform error of MLS approximation 0.1330. Table 1 gives the values of uniform error for Experiment 1 and 2, when we choose different regularized parameter λ and order of Lplace-Beltrami operator s for (4.10). The results indicate that the choosing method of λ and s are uncertain, and the optimal combination of λ and s is λ = 0.2, s = 4 for approximation f1 . However, the optimal combination of λ and s is λ = 1.4, s = 4 for approximation f2 . The different choices for the order of Lplace-Beltrami operator and regularlized parameter can provide different pointwise approximation results. From what has been discussed above, the RMLS is better than the MLS for recoving a function from data set with high level noise. However, the choice of λ and s is critical. How to automatically choose the proper λ and s is a challenging problem.

References [1] An C, Chen X, Sloan I H, Womersley R S. Regularized least squares approximations on the sphere using spherical designs. SIAM J. Num. Anal., 2012, 50(3): 1513-1534. [2] Armentano M G. Error estimates in Sobolev spaces for moving least square approximations. SIAM J. Num. Anal., 2001, 39(1): 38-51. [3] Armentano M G, Dur´ an R G. Error estimates for moving least square approximations. Appl. Num. Math., 2001, 37(3): 397-416. [4] Backus G, Gilbert F. Uniqueness in the inversion of inaccurate gross earth data. Phil. Trans. Roy. Soc. London, Ser. A, Math. & Phys. Sci., 1970: 123-192. [5] Backus G, Gilbert F. Numerical applications of a formalism for geophysical inverse problems. Geophy. J. Int., 1967, 13(1-3): 247-276. [6] Backus G, Gilbert F. The resolving power of gross earth data. Geophy. J. Roy. Astr. Soc., 2007, 16(2): 169-205. [7] Belytschko T, Krongauz Y, Organ D, Fleming M, Krysl P. Meshless methods: an overview and recent developments. Comp. Meth. Appl. Mech. Eng., 1996, 139(1): 3-47. [8] Bos L P, Salkauskas K. Moving least-squares are Backus-Gilbert optimal. J. Approx. Theory, 1989, 59(3): 267-275. [9] Fasshauer G E, Schumaker L L. Scattered data fitting on the sphere. Math. Meth. Curv. Surf. II, 1998: 117-166.

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[10] Franke R, Nielson G. Smooth interpolation of large sets of scattered data. Int. J. Num. Meth. Eng., 1980, 15(11): 1691-1704. [11] Freeden W, Gervens T, Schreiner M. Constructive approximation on the sphere: with applications to geomathematics. Oxford: Clarendon Press, 1998. [12] Golitschek M, Light W A. Interpolation by polynomials and radial basis functions on spheres. Constr. Approx., 2001, 17(1): 1-18. [13] Hubbert S, Morton T M. Lp -error estimates for radial basis function interpolation on the sphere. J. Approx. Theory, 2004, 129(1): 58-77. [14] Jetter K, St¨ ockler J, Ward J. Error estimates for scattered data interpolation on spheres. Math. Comp., 1999, 68(226): 733-747. [15] Lancaster P, Salkauskas K. Surfaces generated by moving least squares methods. Math. Comp., 1981, 37(155): 141-158. [16] Le Gia Q T, Narcowich F J, Ward J D, Wendland, H. Continuous and discrete least-squares approximation by radial basis functions on spheres. J. Approx. Theory, 2006, 143(1): 124-133. [17] Levesley J, Sun X. Approximation in rough native spaces by shifts of smooth kernels on spheres. J. Approx. Theory, 2005, 133(2): 269-283. [18] Levin D. The approximation power of moving least-squares. Math. Comp., 1998, 67(224): 1517-1531. [19] Li L Q. Regularized least square regression with spherical polynomial kernels. Int. J. Wav. Multires. Inf. Proc., 2009, 7(06): 781-801. [20] Maisuradze G G, Thompson D L, Wagner A F, Minkoff, M. Interpolating moving least-squares methods for fitting potential energy surfaces: Detailed analysis of one-dimensional applications. The J. Chem. Phys., 2003, 119(19): 10002-10014. [21] McLain D H. Drawing contours from arbitrary data points. The Comp. J., 1974, 17(4): 318-324. [22] McLain D H. Two dimensional interpolation from random data. The Comp. J., 1976, 19(2): 178-181. [23] M¨ uller C. Spherical harmonics. Springer, 1966. [24] Narcowich F J, Ward J D. Scattered data interpolation on spheres: error estimates and locally supported basis functions. SIAM J. Math. Anal., 2002, 33(6): 1393-1410. [25] Narcowich F J, Sun X, Ward J D, Wendland H. Direct and inverse Sobolev error estimates for scattered data interpolation via spherical basis functions. Found. Comp. Math., 2007, 7(3): 369-390. [26] Narcowich F J, Sun X, Ward J D. Approximation power of RBFs and their associated SBFs: a connection. Adv. Comp. Math., 2007, 27(1): 107-124. [27] Narcowich F J, Sivakumar N, Ward J D. Stability results for scattered-data interpolation on Euclidean spheres. Adv. Comp. Math., 1998, 8(3): 137-163. [28] Renka R J. Multivariate interpolation of large sets of scattered data. ACM Trans. Math. Softw., 1988, 14(2): 139-148. [29] Shepard D. A two-dimensional interpolation function for irregularly-spaced data. In: Proc. the 1968 23rd ACM Nat. Conf. ACM, 1968: 517-524. [30] Sloan I H. Polynomial interpolation and hyperinterpolation over general regions. J. Approx. Theory, 1995, 83(2): 238-254. [31] Sloan I H, Womersley R S. Constructive polynomial approximation on the sphere. J. Approx. Theory, 2000, 103(1): 91-118. [32] Sloan I H, Sommariva A. Approximation on the sphere using radial basis functions plus polynomials. Adv. Comp. Math., 2008, 29(2): 147-177. [33] Wang K Y, Li L Q. Harmonic Analysis and Approximation on the Unit Sphere. Sci. Press, Beijing, 2000. [34] Wendland H. Local polynomial reproduction and moving least squares approximation. IMA J. Num. Anal., 2001, 21(1): 285-300. [35] Wendland H. Moving least squares approximation on the sphere. Mathematical Methods for Curves and Surfaces, Vanderbilt Univ. Press, Nashville, TN, 2001: 517-526. [36] Wendland H. Scattered data approximation. Cambridge: Cambr. Univ. Press, 2005. [37] Wendland H. Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Adv. Comp. Math., 1995, 4(1): 389-396. [38] Williamson D L, Drake J B, Hack J J, Jakob R, Swarztrauber P N. A standard test set for numerical approximations to the shallow water equations in spherical geometry. J. Comp. Phys., 1992, 102(1): 211-224.

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Chaos Control and Function Projective Synchronization of Noval Chaotic Dynamical System M. M. El-Dessoky1;2 , E. O. Alzahrani1 and N.A. Almohammadi3 1

Mathematics Department, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia. 2 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt. 3 Mathematics Department, Faculty of science - AL Salmania Campus, King Abdulaziz University, Jeddah, Saudi Arabia. E-mail: [email protected]; [email protected]; [email protected] ABSTRACT

In this paper, a Noval chaotic dynamical system is proposed and the basic properties of the system are investigated. Linear feedback control technique is used to suppress chaos. The controlled system is stable under some conditions on the parameters of the system determined by Lyapunov direct method. In addition, a function projective synchronization of two identical Noval system is presented. Lyapunov method of stability is used to prove the asymptotic stability of solutions for the error dynamical system. Numerical simulations results are included to show the e¤ectiveness of the proposed schemes.

1. INTRODUCTION Chaos has been developed and thoroughly studied over the past two decades. A chaotic system is a nonlinear deterministic system that displays complex and unpredictable behavior. The sensitive dependence on the initial conditions and on the system’s parameter variation is a prominent characteristic of chaotic behavior. Research e¤orts have investigated chaos control and chaos synchronization problems in many physical chaotic systems. Controlling chaos has become a challenging topic in nonlinear dynamics. Feedback control methods are used to control chaos by stabilizing a desired unstable periodic solution which is embedded in a chaotic attractor [1-12]. Generalized synchronization is another interesting chaos synchronization technique. Li considered a new type of projective synchronization method, called a modi…ed projective synchronization (MPS). Chen et al. introduced another new projective synchronization which is called a function projective synchronization (FPS), where the response of the synchronized dynamical states synchronizes up to scaling function factor [11-29]. The object of this paper is to study the function project synchronization (FPS) of two identical Noval chaotic system with known parameters. The paper is organized as follows. In Section 2, presented the model of Noval chaotic system. In Section 3, the dissipation, symmetry, equilibrium points and lyapunov exponents. In Section 4, the feedback control method is applied to Noval system and numerical simulations are presented to show the e¤ectiveness of the proposed method. In Section 5, the proposed scheme is applied to function projective synchronize two identical Noval chaotic systems. Also numerical simulations are presented in order to validate the proposed synchronization approach. Finally, in Section 6 the conclusion of the paper is given.

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2. THE MODEL OF NOVAL CHAOTIC SYSTEM The Noval chaotic system [30] is described by the following system of di¤erential equations: :

x :

y : z

1 = ( ¡ a + )x + xy + z b = ¡by ¡ x2 = ¡x ¡ cz

(1)

Where the parameters a; b; c are positive real constants. A new chaotic attractor for the parameters a = 2; b = 0:1; c = 1 is shown in Fig. 1.

Figure 1: Noval Chaotic System at a = 2; b = 0:1; c = 1:

3. DYNAMICAL BEHAVIOR OF THE NOVAL CHAOTIC SYSTEM 3.1. The dissipation The divergence of Noval system is given by; rV =

@ x_ @ y_ @ z_ 1 + + = ¡a + + y ¡ b ¡ c: @x @y @z b

When y < a + b + c ¡ 1b , then Noval system is dissipative.

3.2. Symmetry The relation of (x; y; z) ! ( ¡ x; y; ¡z) is transformed, the system remains unchanged. The system trajectory in the x,z plane symmetry of y axis.

3.3. Equilibrium points and stability By putting the right side of equation of system (1) equal to zero, that is; 1 ( ¡ a + )x + xy + z b ¡by ¡ x2 ¡x ¡ cz

= 0 = 0 = 0

This system has three equilibrium points: p p P1 = (0; 0; 0); P2;3 = ( § 1 ¡ ab ¡ b=c; a ¡ 1=b + 1=c; ¨ 1c 1 ¡ ab ¡ b=c)

The eigenvalues at each equilibrium point can be obtained as shown in Table 1. And all the equilibrium points are unstable, since at least one eigenvalue has positive real part for each equilibrium point.

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Table 1. Eigenvalues and stability of equilibrium points

Equilibrium points P1 P2 P3

Eigenvalues ¸1 = ¡0:1 , ¸2 = ¡0:887 , ¸3 = 7:887 ¸1 = ¡0:7446 , ¸2 = 0:32 + 1:33i , ¸3 = 0:32 ¡ 1:33i ¸1 = ¡0:7446 , ¸2 = 0:32 + 1:33i , ¸3 = 0:32 ¡ 1:33i

Stable/Unstable Unstable Unstable Unstable

3.4. Lyapunov exponents and its dimension By using singular value decomposition method and we may get three Lyapunov exponents of system,¸1 = 0:13; ¸2 = 0; ¸3 = ¡0:52: and the Lyapunov dimension of the new chaotic system is as follows: j

1 X ¸1 + ¸2 0:13 + 0 DL = j + ¸i = 2 + =2+ = 2:254 j¸j+1 j i=1 j¸3 j j ¡ 0:52j Thus, the Lyapunov dimension is the fractal dimension, shows that the system is a chaotic system

4. CONTROLLING NOVAL SYSTEM In order to control the Noval system to the unstable …xed point (xi ; yi ; zi ), we introduce the feedback control to guide the chaotic trajectory (x(t); y(t); z(t)) to the unstable …xed point (xi ; yi ; zi ). let system (1) be controlled by the following form: 1 : x = ( ¡ a + )x + xy + z ¡ ki1 (x ¡ xi ) b : y = ¡by ¡ x2 ¡ ki2 (y ¡ yi ) : z = ¡x ¡ cz ¡ ki3 (z ¡ zi )

(2)

where i = 1; 2; 3.

4.1. First For i = 1, the controlled system (2) has one equilibrium point (x1 ; y1 ; z1 ) = (0; 0; 0). Let system (2) be controlled by a linear feedback control of the form: 1 : x = ( ¡ a + )x + xy + z ¡ k11 (x ¡ x1 ) b : y = ¡by ¡ x2 ¡ k12 (y ¡ y1 ) : z = ¡x ¡ cz ¡ k13 (z ¡ z1 )

(3)

The controlled system (3) has one equilibrium point (x1 ; y1 ; z1 ). We linearize (3) about this equilibrium point. Then the linearized system is given by: 8 1 > _ > < X = ( ¡ a + b ¡ k11 + y1 )X + x1 Y + Z (4) Y_ = ¡(b + k12 )Y ¡ 2x1 X > > : _ Z = ¡X ¡ (c + k13 )Z where (x1 ; y1 ; z1 ) = (0; 0; 0), that is;

8 1 > _ > < X = ( ¡ a + b ¡ k11 )X + Z Y_ = ¡(b + k12 )Y > > : _ Z = ¡X ¡ (c + k13 )Z

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(5)

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To prove the asymptotic stability we use the direct method of Lyapunov. De…ne the Lyapunov function for system (5) by: 1 (6) V (X; Y; Z) = (X 2 + Y 2 + Z 2 ) 2 The function V satis…ed: i V (0; 0; 0) = 0 ii V (X; Y; Z) > 0 for X; Y and Z in the neighbourhood of the origin. So, V (X; Y; Z) is positive de…nite. Also, we have: 1 dV = ¡f(a ¡ + k11 )X 2 + (b + k12 )Y 2 + (c + k13 )Z 2 g dt b therefore, the derivative

dV dt

(7)

· 0 if, k11 ¸

1 ¡ a; b

k12 ¸ ¡b;

k13 ¸ ¡c

(8)

i.e. dV =dt is negative de…nite under condition (8). We deduce the following lemma, Lemma 4.1. The equilibrium solution (x1 ; y1 ; z1 ) of the controlled system (3) is asymptotically stable such that the feedback control gain K satisfy: k11 ¸ 1b ¡ a and k12 = k13 = 0:

4.2. Second we introduce the conventional feedback control p (x(t); y(t); z(t)) to the second p to guide the chaotic trajectory unstable equilibrium point (x2 ; y2 ; z2 ) = ( 1 ¡ ab ¡ b=c; a ¡ 1b + 1c ; ¡ 1c 1 ¡ ab ¡ b=c) 8 1 > > < x_ = ( ¡ a + b )x + xy + z ¡ k21 (x ¡ x2 ) y_ = ¡by ¡ x2 ¡ k22 (y ¡ y2 ) > > : z_ = ¡x ¡ cz ¡ k23 (z ¡ z2 )

(9)

The controlled system (9) has one equilibrium point (x2 ; y2 ; z2 ). We linearize (9) about this equilibrium point. Then the linearized system is given by: 8 1 _ > > < X = ( ¡ a + b ¡ k21 + y2 )X + x2 Y + Z (10) Y_ = ¡(b + k22 )Y ¡ 2x2 X > > : _ Z = ¡X ¡ (c + k23 )Z p where (x2 ; y2 ; z2 ) = ( 1 ¡ ab ¡ b=c; a ¡

1 b

+ 1c ; ¡ 1c

p 1 ¡ ab ¡ b=c), that is;

8 p 1 _ > > < X = ( c ¡ k21 )X + ( 1 ¡ ab ¡ b=c)Y + Z p Y_ = ¡(b + k22 )Y ¡ 2( 1 ¡ ab ¡ b=c)X > > : _ Z = ¡X ¡ (c + k23 )Z

(11)

To prove the asymptotic stability we use the direct method of Lyapunov. De…ne the Lyapunov function for system(10) by: 1 V (X; Y; Z) = (X 2 + Y 2 + Z 2 ) (12) 2 The function V satis…ed:

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i V (0; 0; 0) = 0 ii V (X; Y; Z) > 0 for X; Y and Z in the neighbourhood of the origin. So, V (X; Y; Z) is positive de…nite. Also, we have: 1 dV = ¡f2(k21 ¡ )X 2 + (b + k22 )Y 2 + 2(c + k23 )Z 2 g dt c therefore, the derivative

i.e.

dV dt

dV dt

· 0 if,

1 k21 ¸ ; c

k22 ¸ ¡b;

k23 ¸ ¡c

(13)

(14)

is negative de…nite under condition (14). We deduce the following lemma,

Lemma 4.2. The equilibrium solution (x2 ; y2 ; z2 ) of the controlled system (9) is asymptotically stable such that the feedback control gain K has the simple choice k21 ¸ 1c and k22 = k23 = 0:

4.3. Third we introduce the conventional feedback trajectory (x(t),y(t),z(t)) to the third unstable p control to guide the chaoticp equilibrium point(x3 ; y3 ; z3 ) = ( ¡ 1 ¡ ab ¡ b=c; a ¡ 1b + 1c ; 1c 1 ¡ ab ¡ b=c) 8 1 > > < x_ = ( ¡ a + b )x + xy + z ¡ k31 (x ¡ x3 ) y_ = ¡by ¡ x2 ¡ k32 (y ¡ y3 ) > > : z_ = ¡x ¡ cz ¡ k33 (z ¡ z3 )

(15)

The controlled system (14) has one equilibrium point (x3 ; y3 ; z3 ). We linearize (14) about this equilibrium point. Then the linearized system is given by: 8 1 > _ > < X = ( ¡ a + b ¡ k31 + y3 )X + x3 Y + Z (16) Y_ = ¡(b + k32 )Y ¡ 2x3 X > > : _ Z = ¡X ¡ (c + k33 )Z where (x3 ; y3 ; z3 ) = ( ¡

p

1 ¡ ab ¡ b=c; a ¡

1 b

+ 1c ; 1c

p

1 ¡ ab ¡ b=c), that is;

8 p 1 _ > > < X = ( c ¡ k31 )X ¡ ( 1 ¡ ab ¡ b=c)Y + Z p Y_ = ¡(b + k32 )Y + 2( 1 ¡ ab ¡ b=c)X > > : _ Z = ¡X ¡ (c + k33 )Z

(17)

To prove the asymptotic stability we use the direct method of Lyapunov. De…ne the Lyapunov function for system(16) by: 1 V (X; Y; Z) = (X 2 + Y 2 + Z 2 ) (18) 2 The function V satis…ed: i V (0; 0; 0) = 0 ii V (X; Y; Z) > 0 for X; Y and Z in the neighbourhood of the origin.

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So, V (X; Y; Z) is positive de…nite. Also, we have: 1 dV = ¡f2(k31 ¡ )X 2 + (b + k32 )Y 2 + 2(c + k33 )Z 2 g dt c therefore, the derivative

i.e.

dV dt

dV dt

· 0 if,

1 k31 ¸ ; c

k32 ¸ ¡b;

(19)

k33 ¸ ¡c

(20)

is negative de…nite under condition (20). We deduce the following lemma,

Lemma 4.3. The equilibrium solution (x3 ; y3 ; z3 ) of the controlled system (15) is asymptotically stable such that the feedback control gain K has the simple choice k31 ¸ 1c and k32 = k33 = 0:

5. THE SCHEME OF GENERALIZED FUNCTION PROJECTIVE SYNCHRONIZATION OF CHAOTIC SYSTEMS The chaotic (master and slave) systems can be given in the following form: X_ = F (X)

(21)

Y_ = G(Y ) + U (X; Y; t)

(22)

T T Where X = (x1 ; x2 ; : : : ; xn ) ; Y = (y1 ; y2 ; : : : ; yn ) 2 Rn are state vectors of the system (20) and (21), n n respectively; F; G : R ! R are two continuous vector functions and U : (Rn ; Rn ; Rn ) ! Rn is a controller which will be designed later.

Definition 5.1. For the master system (20) and the slave system (21), there is said to be generalized function projective synchronization (GFPS) if there exists a vector function U (X; Y; t)such that; limt!+1 kY ¡ ¤(X)Xk = 0 where ¤(X) = diagfh1 (X); h2 (X); : : : ; hn (X)g where hi (X) are continuous functions, k:k represents a vector norm induced by the matrix norm. Remark 1. We de…ne e = Y ¡¤(X)X which is called the error vector between systems (20) and (21) for GFPS, where e = (e1 ; e2 ; : : : ; en )T ,and ei = Yi ¡ hi (X)Xi ; (i = 1; 2; : : : ; n) Remark 2. If ¤ = ¾I; ¾ 2 R, the GFPS problem will be reduced to projective synchronization, where I is an n £ n identity matrix. In particular if ¾ = 1and ¡ 1 the problem is further simpli…ed to complete synchronization and antiphase synchronization, respectively. And if ¤ = diagfa1 ; a2 ; : : : ; an g,the modi…ed projective synchronization will appear. We will study the FPS of novel system with known parameters and determine controller function for the FPS of the derive and response systems. Our aim is to design a controller and make the response system trace the drive system and become ultimately. The Noval system as a drive system is given as below; 8 1 >

: z_1 = ¡x1 ¡ cz1

the Noval system as the response system is also given by; 8 1 >

: z_2 = ¡x2 ¡ cz2 + u3

167

(24)

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According to the FPS scheme presented in the previous section, without loss of generality,we choose the scaling function matrix ¤(X) = diagfd11 x1 + d12 ; d21 y1 + d22 ; d31 z1 + d32 g where dij (i = 1; 2; 3; j = 1; 2) are constant numbers. The error vector can be de…ned as 8 >

: ez = z2 ¡ (d31 z1 + d32 )z1 The error dynamical system between (23) and (24) is; 8 e_ x = ( ¡ a + 1b )ex + x2 y2 + z2 ¡ d11 ( ¡ a + 1b )x21 ¡ 2d11 x1 z1 > > > < ¡2d11 x21 y1 ¡ d12 x1 y1 ¡ d12 z1 + u1 > e_ y = ¡bey ¡ x22 + d21 by12 + 2d21 y1 x21 + d22 x21 + u2 > > : e_ z = ¡cez ¡ x2 + d31 cz12 + 2d31 z1 x1 + d32 x1 + u3

(26)

we can get the controller 8 ¡2 1 2 2 >

: u3 = x2 ¡ d31 cz12 ¡ 2d31 z1 x1 ¡ d32 x1 then the error dynamical system is described by 8 1 >

: e_ z = ¡cez

(27)

(28)

for this choice, the closed loop system (28) has three negative eigenvalues ¡(a + 1b ); ¡b; ¡c which implies that the error state ex ; ey and ez converge to zero as time t tends to in…nity. Hence the FPS between the identical Noval chaotic system is achieved.

5.1. Numerical Results In this section, some numerical simulation results are presented to verify the previous analytical results where a = 2; b = 0:1; c = 1. Figure 2: shows the convergence of the trajectory of the controlled system to the unstable equilibrium point (x1 ; y1 ; z1 ) = (0; 0; 0) of the uncontrolled system (1). Figure 3: shows p the convergence of the trajectory of the controlled system to the unstable equilibrium point (x2 ; y2 ; z2 ) = ( 1 ¡ ab ¡ b=c; a ¡ 1b + p 1 1 1 ¡ ab ¡ b=c) of the uncontrolled system (1). Figure 4: shows the convergence of the trajectory of thec c; ¡ c p p ontrolled system to the unstable equilibrium point (x3 ; y3 ; z3 ) = ( ¡ 1 ¡ ab ¡ b=c; a ¡ 1b + 1c ; 1c 1 ¡ ab ¡ b=c) of the uncontrolled system (1).

Figure 2: The time responses for the states of the controlled Noval system to a …xed point (x1 ; y1 ; z1 ).

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Figure 3: The time responses for the states of the controlled Noval system to a …xed point (x2 ; y2 ; z2 ).

Figure 4: The time responses for the states of the controlled Noval system to a …xed point (x3 ; y3 ; z3 ): The initial values of the drive system and response system are taken as: (x1 (0); y1 (0); z1 (0))T = (1; ¡6; 0:1)T ; (x2 (0); y2 (0); z2 (0))T = (10; 12; ¡3)T : We choose the scaling function factors as: h1 = x1 + 2; h2 = ¡2y1 ¡ 2 and h3 = z1 ¡ 2: Figure 5: show the FPS between two identical Noval systems. When the scaling factors are simpli…ed as hi = 1 (i = 1; 2; 3), the complete synchronization between two identical Noval systems are shown in Figure 6. Furthermore, when the scaling factors are simpli…ed as hi = ¡1 (i = 1; 2; 3), the anti synchronization between two identical Noval systems are shown in Figure 7. Finally, when the scaling factors are simpli…ed as h1 = 1:5; h2 = 2 and h3 = 2:5, the modi…ed projective synchronization (MPS) between two identical Noval

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systems are shown in Figure 8.

Figure 5: The behaviour of the trajectories ex ; ey and ez of the error system tends to zero for FPS.

Figure 6: The behaviour of the trajectories ex; ey and ez of the error system tends to zero for complete synchronization

Figure 7: The behaviour of the trajectories ex; ey and ez of the error system tends to zero for anti synchronization

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Figure 8: The behaviour of the trajectories ex; ey and ez of the error system tends to zero for MPS.

6. CONCLUSION The paper has studied the noval chaotic dynamical system, including some basic dynamical properties, Lyapunov exponents, Lyapunov dimension. A feedback control has been proposed to the noval chaotic dynamical system. The controlling conditions are derived from the Lyapunov direct method. The function projective synchronization has been used to synchronize two identical chaotic systems with known parameters. By the Lyapunov stability theory, the su¢cient condition of the function projective synchronization has been obtained. Finally, numerical simulations are provided to verify the e¤ectiveness of the results obtained. Acknowledgments This article was funded by the Deanship of Scienti…c Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR technical and …nancial support.

References [1] E.Ott, C. Grebogi, J A. Yorke, Controlling Chaos, Phys Rev Lett (64), (1999), 1179-1184. [2] G. Chen, Chaos on Some Controllability Conditions for Chaotic Dynamics Control, Chaos Solitons Fractals, 8(9), (1997), 1461-1470. [3] C. Hwang , J. Hsheh , and R. Lin, A Linear Continuous Feedback Control of Chua’s Circuit, Chaos Solitons Fractals 8(9), 1997, 1507-1515. [4] G. Chen and X. Dong, On feedback control of chaotic dynamic systems, Int. J. Bifurcation and Chaos 2, (1995), 407-411. [5] K. Pyragas, Continuous control of chaos by self-controlling feedback, Physics letters A 170, pp. 421-428, (1992). [6] A. Hegazi, H. N. Agiza, and M. M. El-Dessoky. Controlling chaotic behaviour for spin generator and Rossler dynamical systems with feedback control, Chaos Solitons Fractals, 12, (2001), 631-658. [7] Sara Dadras, and Hamid Momeni, Control of a fractional-order economical system via sliding mode, Physica A (389), (2010), 2434-2442. [8] H. N. Agiza, On The Analysis of Stability, Bifurcation, Chaos and Chaos Control of Kopel map, Chaos Solitons Fractals 10(11), (1999), 1909-1916. [9] C. Hwang, J. Hsheh and R. Lin, A linear continuous feedback control of Chua’s circuit, Chaos Solitons Fractals, 8(9), (1997), 1507-1515. [10] Seyed Mehdi Abedi Pahnehkolaei, Alireza Al…, J. A. Tenreiro Machado, Chaos suppression in fractional systems using adaptive fractional state feedback control, Chaos Solitons Fractals, 103, (2017), 488-503.

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[11] V. Bala Shunmuga Jothi, S. Selvaraj, V. Chinnathambi, S. Rajasekar, Bifurcations and chaos in two-coupled periodically driven four-well Du¢ng-van der Pol oscillators, Chinese J. Phys., 55(5), (2017), 1849-1856. [12] Anuraj Singh, Sunita Gakkhar, Controlling chaos in a food chain model, Math. Comput. Simulation, 115, (2015), 24-36. [13] L. M. Pecora, T. L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett.64 (8), (1990), 821-824. [14] T. L. Carroll, L. M. Perora, Synchronizing a chaotic systems, IEEE Trans, Circuits Systems 38, (1991), 453-456. [15] G. Chen, Control and Synchronization of Chaos, a Bibliography, Dept. of Elect. Eng., Univ. Houston, TX, (1997). [16] Yongguang Yua, and Han-Xiong Li. Adaptive generalized function projective synchronization of uncertain chaotic systems, Nonlinear Analysis: Real World Applications, Vol.11, (2010), 2456-2464. [17] E. M. Elabbasy,and M. M. El-Dessoky, Adaptive Coupled Synchronization of Coupled Chaotic Dynamical Systems, Applied Sciences Research , (2), (2007), 88-102. [18] Na Cai, Yuanwei Jing, and Siying Zhang, Modi. . . ed projective synchronization of chaotic systems with disturbances via active sliding mode control, Commun Nonlinear Sci Numer Simulat., (15), (2010), 1613-1620. [19] Guo-Hui Li, Generalized Projective Synchronization between Lorenz System and Chen’s System, Chaos,Solitons and Fractals (32), 2007, 1454-1458. [20] Guo-Hui Li. Modi. . . ed Projective Synchronization of Chaotic System, Chaos Solitons Fractals (32), (2007), 1786-1790. [21] Johannes Petereit, Arkady Pikovsky, Chaos synchronization by nonlinear coupling, Commun. Nonlinear Sci. Numer. Simul., 44, (2017), 344-351. [22] K. Vishal, Saurabh K. Agrawal, On the dynamics, existence of chaos, control and synchronization of a novel complex chaotic system, Chinese J. Phys., 55(2), (2017), 519-532. [23] M. M. El-Dessoky, M. T. Yassen and E. Salah, Adaptive Modi. . . ed Function Projective Synchronization between two dix oerent Hyperchaotic Dynamical Systems, Math. Probl. Eng. , Vol., 2012, (2012), Article ID 810626, 16 pages, doi:10.1155/2012/810626. [24] N. F. Rulkov, M. M. Sushchik, L. S. Tsimring, and Henry D. I. Abarbanel, Generalized Synchronization of Chaos in Directionally Coupled Chaotic Systems, Phys. Rev. E, (51),1995, 980- 994. [25] Yong Chen, X. Li. Function Projective Synchronization between Two Identical Chaotic Systems, Int. J. Mod. Phys. C(18), 2007, 883-888. [26] M. M. El-Dessoky, and M. T. Yassen, Adaptive feedback control for chaos control and synchronization for new chaotic dynamical system, Math. Probl. Eng. , Vol. 2012, (2012), Article ID 347210, 12 pages, doi:10.1155/2012/347210. [27] Guo-Hui Li, Generalized Synchronization of Chaos Based on Suitable Separation, Chaos Solitons Fractals, (39), (2009), 2056-2062. [28] M. M. El-Dessoky, E. O. Alzahrany, and N. A. Almohammadi. Function Projective Synchronization for Four Scroll Attractor by Nonlinear Control, Appl. Math. Sci., Vol.11(26), (2017), 1247-1259. [29] Er-Wei Bai and Karl E. Lonngren, Sequential synchronization of two Lorenz system using active control, Chaos Solitons Fractals, 11(1), (2000), 1041-1044. [30] Shao FuWang, and Da-zhuan Xu, The dynamic analysis of a chaotic system, Adv. Mech. Eng. , Vol. 9(3), 2017, 1-6.

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UMBRAL CALCULUS APPROACH TO r-STIRLING NUMBERS OF THE SECOND KIND AND r-BELL POLYNOMIALS TAEKYUN KIM1 , DAE SAN KIM2 , HYUCK-IN KWON3 , AND JONGKYUM KWON4,∗

Abstract. In this paper, we would like to use umbral calculus in order to derive some properties, recurrence relations and identities related to r-Stirling numbers of second kind and r-Bell polynomials. In particular, we will express the r-Bell polynomials as linear combinations of many well-known families of special polynomials.

1. Introduction The Stirling numbers S2 (n, k) of the second kind counts the number of partitions of the set [n] = {1, 2, · · · , n} into k nonempty disjoint subsets. The S2 (n, k), (n, k ≥ 0) are given by the recurrence relation S2 (n, k) = kS2 (n − 1, k) + S2 (n − 1, k − 1), (n, k ≥ 1),

(1.1)

with the initial conditions S2 (n, 0) = δ0n , S2 (0, k) = δ0k .

(1.2)

They are also given by xn =

n X

S2 (n, k)(x)k ,

(1.3)

k=0

with (x)0 = 1, (x)k = x(x − 1) · · · (x − k + 1), for k ≥ 1, and by ∞ X 1 t tn (e − 1)k = S2 (n, k) . k! n!

(1.4)

n=k

More explicitly, they are given by S2 (n, k) =

k   1 X k (−1)k−j j n k! j=0 j

(1.5) 1 k n = 4 0 , (n ≥ k), k! where 4k 0n = 4k xn |x=0 , and 4f (x) = f (x + 1) − f (x) is the forward difference operator. For these well known facts, one may refer to [3,4].

2010 Mathematics Subject Classification. 05A19, 05A40, 11B73, 11B83. Key words and phrases. r-Stirling numbers of the second kind, r-Bell polynomials, umbral calculus. ∗ corresponding author. 1

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2

Umbral calculus approach to r-Stirling numbers of the second kind and r-Bell polynomials

Let r be any positive integer. These ’classical’ Stirling numbers S2 (n, k) of the second kind were generalized to the r-Stirling numbers S2,r (n, k) of the second kind (see, [1,2,7]). The S2,r (n, k) enumerates the number of partitions of the set [n] = {1, 2, · · · , n} into k nonempty disjoint subsets in such a way that 1, 2, · · ·, r are in distinct subsets. They are given by the recurrence relation S2,r (n, k) = kS2,r (n − 1, k) + S2,r (n − 1, k − 1), (n > r),

(1.6)

with the initial conditions S2,r (n, k) = 0, (n < r); S2,r (n, k) = δkr , (n = r).

(1.7)

The S2,r (n, k) are also given by (x + r)n =

n X

S2,r (n + r, k + r)(x)k ,

(1.8)

k=0

and by ∞ X tn 1 rt t k S2,r (n + r, k + r) . e (e − 1) = k! n!

(1.9)

n=k

Analogously to the classical case, they are explicitly given by k   1 X k (−1)k−j (r + j)n S2,r (n + r, k + r) = k! j=0 j

(1.10)

1 = 4k rn , (n ≥ k), k! where 4k rn = 4k xn |x=r . For more details about r-Stirling numbers of the second kind, one may refer to [1,2,7]. The Bell polynomials Beln (x) (also called exponential or Touchard polynomials) are defined by ∞ X tn x(et −1) Beln (x) , (see [3, 4, 8, 9]). e = (1.11) n! n=0 Then it is immediate to see that Beln (x) =

n X

S2 (n, k)xk .

(1.12)

k=0

For x = 1, Beln = Beln (1) =

Pn

ee

k=0

t

−1

=

S2 (n, k) are called Bell numbers so that ∞ X

Beln

n=0

tn . n!

(1.13)

Further, the Bell polynomial is given by Dobinski’s formula Beln (x) = e−x

∞ X kn k=0

174

k!

xk .

(1.14)

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On the other hand, the r-Bell polynomials Beln,r (x) are defined by ert ex(e

t

−1)

=

∞ X

Beln,r (x)

n=0

tn , (see [5]). n!

(1.15)

Then it is easy to see that Beln,r (x) =

n X

S2,r (n + r, k + r)xk .

(1.16)

k=0

Moreover, they satisfy the generalized Dobinski’s formula Beln,r (x) = e−x

∞ X (k + r)n k=0

When x = 1, Beln,r = Beln,r (1) = numbers so that e

et −1+rt

=

Pn

k=0

∞ X n=0

k!

xk .

(1.17)

S2,r (n + r, k + r) are called r-Bell

Beln,r

tn . n!

(1.18)

We note here, in passing, that r-Bell numbers were called in another name, namely extended Bell numbers,(see [6]). In this paper, we would like to use umbral calculus in order to derive some properties, recurrence relations and identities related to r-Stirling numbers of the second kind and r-Bell polynomials. In particular, we will express the r-Bell polynomials as linear combinations of many well-known families of special polynomials. 2. Review on umbral calculus Here we will go over some of the basic facts about umbral calculus. For a complete treatment, the reader may refer to [4]. Let F be the algebra of all formal power series in the single variable t with the coefficients in the field C of complex numbers: ( ) ∞ X tk F = f (t) = ak ak ∈ C . (2.1) k! k=0

Let P = C[x] denote the ring of polynomials in x with the coefficients in C, and let P∗ be the vector space of all linear functionals on P. For L ∈ P∗ , p(x) ∈ P, < L | p(x) > denotes the action of the linear functional L on p(x). For f (t) = P∞ tk k=0 ak k! ∈ F, the linear functional < f (t) | · > on P is defined by < f (t) | xn >= an , (n ≥ 0). (2.2) D E k D E P ∞ For L ∈ P∗ , let fL (t) = k=0 L|xk tk! ∈ F. Then we evidently have fL (t)|xn = D E L|xn , and the map L → fL (t) is a vector space isomorphism from P∗ to F. Thus F may be viewed as the vector space of all linear functionals on P as well as the algebra of formal power series in t. So an element f (t) ∈ F will be thought of as both a formal power series and a linear functional on P. F is called the umbral algebra, the study of which is the umbral calculus.

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Umbral calculus approach to r-Stirling numbers of the second kind and r-Bell polynomials

The order o(f (t)) of 0 6= f (t) ∈ F is the smallest integer k such that the coefficients of tk does not vanish. In particular, for 0 6= f (t) ∈ F, it is called an invertible series if o(f (t)) = 0 and a delta series if o(f (t)) = 1. Let f (t), g(t) ∈ F, with o(g(t)) = 0, o(f (t)) = 1. Then there exists a unique D E sequence of polynomials Sn (x) (deg Sn (x) = n) such that g(t)f (t)k |Sn (x) = n!δn,k , for n, k ≥ 0. Such a sequence is called the Sheffer sequence for the Sheffer pair (g(t), f (t)), which is concisely denoted by Sn (x) ∼ (g(t), f (t)). It is known that Sn (x) ∼ (g(t), f (t)) if and only if ∞ X 1 tn xf¯(t) e , = S (x) n n! g(f¯(t)) n=0

(2.3)

where f¯(t) is the compositional inverse of f (t) satisfying f (f¯(t)) = f¯(f (t)) = t. Let pn (x) ∼ (1, f (t)), qn (x) ∼ (1, l(t)). Then the transfer formula says that  n f (t) (2.4) qn (x) = x x−1 pn (x), (n ≥ 1). l(t) Let Sn (x) ∼ (g(t)), f (t)). Then we have the Sheffer identity: Sn (x + y) =

n   X n k=0

k

Sk (x)pn−k (y),

(2.5)

where pn (x) = g(t)Sn (x) ∼ (1, f (t)). The derivative of Sn (x) is given by n−1 E X nD d Sn (x) = f¯(t)|xn−k Sk (x), (n ≥ 1). dx k

(2.6)

k=0

Also, we have the recurrence formula:   g 0 (t) 1 Sn+1 (x) = x − Sn (x). g(t) f 0 (t)

(2.7)

Assume that Sn (x) ∼ (g(t), f (t)), rn (x) ∼ (h(t), l(t)). Then Sn (x) =

n X

Cn,k rk (x),

(2.8)

k=0

where Cn,k =

1 D h(f¯(t)) ¯ k n E l(f (t)) |x . k! g(f¯(t))

Finally, we also need the following: for any h(t) ∈ F, p(x) ∈ P, D E D E h(t)|xp(x) = ∂t h(t)|p(x) .

176

(2.9)

(2.10)

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3. Main Results As we can see from (1.15) and (2.3), we see that   1 Beln,r (x) ∼ , log((1 + t) = (g(t), f (t)) . (1 + t)r

(3.1)

Let n ≥ 1. Then, using (2.10), we have Beln,r (y) =

∞ DX

tm n E |x m! E

Belm,r (y)

m=0

D t = ert ey(e −1) |xn E D t = ∂t (ert ey(e −1) )|xn−1 E D t t = rert ey(e −1) + ert ey(e −1) yet |xn−1 E E D D t t = r ert ey(e −1) |xn−1 + y e(r+1)t ey(e −1) |xn−1

(3.2)

= rBeln−1,r (y) + yBeln−1,r+1 (y). Thus we obtain the following recurrence relation for r-Bell polynomials. Theorem 3.1. For all integers n ≥ 1, we have the recurrence relation. Beln,r (x) = rBeln−1,r (x) + xBeln−1,r+1 (x), (n ≥ 1). From (2.6), we have n−1 E X nD d Beln,r (x) = et − 1|xn−k Belk,r (x) dx k k=0  n−1 X n = (1 − δn,k )Belk,r (x) k k=0 n−1 X n = Belk,r (x), (n ≥ 1). k

(3.3)

k=0

Using (2.7), we obtain 1 )(1 + t)Beln,r (x) 1+t = x(1 + t)Beln,r (x) + rBeln,r (x)

Beln+1,r (x) = (x + r

= xBeln,r (x) + x

(3.4)

d Beln,r (x) + rBeln,r (x), dx

from which it follows that d Beln,r (x) dx Beln+1,r (x) rBeln,r (x) = − − Beln,r (x). x x

(3.5)

This agrees with the result in [2].

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Noting that pn (x) = g(t)Beln,r (x) ∼ (1, log(1 + t)), we have pn (x) = Beln (x). Hence from (2.5), we get the following Sheffer identity

Beln,r (x + y) =

n   X n k=0

k

Belk,r (x)Beln−k (y).

(3.6)

E D t Beln,r (y) = ert ey(e −1) |xn E D t = ert |ey(e −1) xn ∞ D X tm E = ert | Belm (y) xn m! m=0   n E D X n = Belm (y) ert |xn−m m m=0  n X n Belm (y)rn−m . = m m=0

(3.7)

Hence we get n   X n n−m Beln,r (x) = r Belm (x). m m=0

Here we apply the transfer formula in (2.4) to xn ∼ (1, t), (1, log((1 + t)). For n ≥ 1, we have n t x−1 xn log(1 + t) ∞ k X (n) t xn−1 =x bk k! k=0 n−1 X n − 1 (n) = bk xn−k . k

1 Beln,r (x) = x (1 + t)r

(3.8)

1 (1+t)r Beln,r (x)

∼



(3.9)

k=0

(n)

Here bk

are the Bernoulli numbers of the second kind of order n defined by



t log(1 + t)

n

178

=

∞ X k=0

k (n) t

bk

k!

.

(3.10)

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(k−n+1)

7

(n)

Here, as is well known, bk = Bk (1), with Bk (x) denoting the Bernoulli polynomials of order n. Thus we obtain n−1 X n − 1 (n) Beln,r (x) = bk (1 + t)r xn−k k k=0 n−1 r   X n − 1 (n) X r l n−k bk tx = (3.11) k l k=0 l=0   n−1 r  XX n−1 r (n) = (n − k)l bk xn−k−l . k l k=0 l=0 Pn 1 j As (1+t) r Beln,r (x) = Beln (x) = j=0 S2 (n, j)x , we can proceed as follows. Beln,r (x) = (1 + t)r Beln (x) ∞   X r k = t Beln (x) k k=0 n   X n X r k = t S2 (n, j)xj k j=0 k=0 n n  X X r S2 (n, j)(j)k xj−k = k j=k

k=0

=

n  X k=0

=

(3.12)

n X

r k

 n−k X l=0

n−l  X

l=0

k=0

S2 (n, k + l)(k + l)k xl

!  r (k + l)k S2 (n, k + l) xl . k

r

Also, from Beln,r (x) = (1 + t) Beln (x), (1 + t)s Beln,r (x) = Beln,r+s (x), (s ≥ 0).

(3.13)

In particular, for s = 1, we have d Beln,r (x). (3.14) dx Hence in addition to (3.3) and (3.4) we obtain another expression for the derivative of Beln,r (x), namely Beln,r+1 (x) = Beln,r (x) +

d Beln,r (x) = Beln,r+1 (x) − Beln,r (x). dx Combining this with (3.3), we get n   X n Beln,r+1 (x) = Belk,r (x). k

(3.15)

(3.16)

k=0

We are now going to summarize the results obtained so far as the following three theorems. Theorem 2 follows from (3.3), (3.5) and (3.15), Theorem 3 from (3.6), (3.8) and (3.16), and Theorem 4 from (3.11) and (3.12).

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Theorem 3.2. For all integers n ≥ 1, the derivative of r-Bell polynomials can be given as follows: n−1 X n d Beln,r (x) = Belk,r (x) dx k k=0

Beln+1,r (x) rBeln,r (x) − − Beln,r (x) x x = Beln,r+1 (x) − Beln,r (x). =

Theorem 3.3. For all integers n ≥ 0, the following identities hold true. Beln,r (x + y) =

n   X n k=0

k

Belk,r (x)Beln−k (y),

n   X n n−m Beln,r (x) = r Belm (x), m m=0 n   X n Beln,r+1 (x) = Belk,r (x). k k=0

Theorem 3.4. For all integers n ≥ 0, we have the following expressions of r-Bell polynomials. n−1 r  XX

  n−1 r (n) (n − k)l bk xn−k−l k l k=0 l=0 ! n n−l   X X r = (k + l)k S2 (n, k + l) xl , k

Beln,r (x) =

l=0

(n)

where bk

k=0

are the Bernoulli numbers of the second kind of order n given by (3.10).

From now on, we will apply the formula (2.9) in order to express Beln,r (x) as linear combinations of well-known families of special polynomials. For this, let us remind you of the fact in (3.1), namely   1 , log(1 + t) . (3.17) Beln,r (x) ∼ (1 + t)r  t  Noting that the Bernoulli polynomial Bn (x) is Sheffer for e −1 , t , we write t Pn Beln,r (x) = k=0 Cn,k Bk (x).Then D eet −1 − 1 1 E | ert (et − 1)k xn t e − 1 k! ∞ D eet −1 − 1 X tl E = | S2,r (l + r, k + r) xn t e −1 l! l=k   t n D ee −1 − 1 E X n n−l = S2,r (l + r, k + r) |x . l et − 1

Cn,k =

(3.18)

l=k

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Here we observe that D eet −1 − 1 et

|xn−l

−1 D eet −1 − 1

9

E

E t xn−l t −1 ∞ D eet −1 − 1 X tm = | Bm xn−l t m! m=0  n−l  E D eet −1 − 1 X n−l |xn−l−m = Bm t m m=0   n−l E D t X 1 n−l = Bm ee −1 − 1|xn−l−m+1 n−l−m+1 m m=0   n−l X 1 n−l = Bm Beln−l−m+1 . n−l−m+1 m m=0 =

|

et

(3.19)

Thus we see that Cn,k

  n n−l  1 X X n+1 n−l+1 S2,r (l + r, k + r) = l m n+1 m=0

(3.20)

l=k

× Bm Beln−l−m+1 . Finally, we obtain   n n−l  n 1 X X X n+1 n−l+1 S2,r (l + r, k + r) m l n+1 (3.21) k=0 l=k m=0
× Bm Beln−l−m+1 Bk (x). Pn Let Beln,r (x) = k=0 Cn,k Ek (x). Here En (x) are the Euler polynomials with t En (x) ∼ ( e 2+1 , t). Then Beln,r (x) =

E 1 D et −1 1 e + 1| ert (et − 1)k xn 2 k! n   D t E 1X n S2,r (l + r, k + r) ee −1 + 1|xn−l = 2 l l=k   n 1X n = S2,r (l + r, k + r)(Beln−l + δn,l ). 2 l

Cn,k =

(3.22)

l=k

Hence we get ! n n   1X X n Beln,r (x) = S2,r (l + r, k + r)(Beln−l + δn, l) Ek (x). 2 l k=0

(3.23)

l=k

We summarize the expressions of Beln,r (x) in (3.21) and (3.23) as a theorem.

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Umbral calculus approach to r-Stirling numbers of the second kind and r-Bell polynomials

Theorem 3.5. For all integers n ≥ 0, we have the following expressions.   n n n−l  1 X X X n+1 n−l+1 Beln,r (x) = S2,r (l + r, k + r) n+1 l m k=0 l=k m=0
× Bm Beln−l−m+1 Bk (x) ! n n   1X X n = S2,r (l + r, k + r)(Beln−l + δn, l) Ek (x). 2 l k=0 l=k Pn Write Beln,r (x) = k=0 Cn,k (x)k , where (x)n are the falling factorials with (x)n ∼ (1, et − 1). Then E D t 1 Cn,k = ert | (ee −1 − 1)k xn k! ∞ D E X 1 = ert | S2 (l, k) (et − 1)l xn l! l=k

∞ D X tm E S2 (l, k) ert | S2 (m, l) xn m! l=k m=l   n n D E X n X S2 (m, l) ert |xn−m S2 (l, k) = m m=l l=k   n X n X n = S2 (l, k)S2 (m, l)rn−m . m

=

n X

(3.24)

l=k m=l

Thus we have Beln,r (x) =

n n X n   X X n k=0

l=k m=l

m Pn

! S2 (l, k)S2 (m, l)r

n−m

(x)k .

(3.25)

As in (3.24), we let Beln,r (x) = k=0 Cn,k (x)k . But here we compute the coefficients Cn,k in a way different from (3.24). Then E 1 D rt et −1 Cn,k = e |(e − 1)k xn k! k   E t 1 D rt X k e | (−1)k−l el(e −1) xn = k! l l=0 k   ∞ D X 1 X k tm E = (−1)k−l ert | Belm (l) xn (3.26) l k! m! m=0 l=0 k   n   X n 1 X k (−1)k−l Belm (l)rn−m = k! l m m=0 l=0    k n 1 XX k n n−m = (−1)k−l r Belm (l). k! l m m=0 l=0

Hence we obtain !    n k X n X X (−1)k−l k n n−m Beln,r (x) = r Belm (l) (x)k . k! l m m=0 k=0

(3.27)

l=0

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11

Combining (3.25) and (3.27), we get the following theorem. Theorem 3.6. For all integers n ≥ 0, we have the following expressions. ! n n X n   X X n n−m Beln,r (x) = S2 (l, k)S2 (m, l)r (x)k m k=0 l=k m=l !    n k X n X X (−1)k−l k n n−m = r Belm (l) (x)k . k! l m m=0 k=0

l=0

We recall here that the Abel polynomial An (x; a)(a 6= 0) isPthe associated sen quence for teat , namely An (x; a) ∼ (1, teat ). Let Beln,r (x) = k=0 Cn,k Ak (x; a). Then E D t 1 Cn,k = ert eak(e −1) | (et − 1)k xn k! ∞ D X t tl E = ert eak(e −1) | S2 (l, k) xn l! l=k   n D E X n t = S2 (l, k) ert |eak(e −1) xn−l l l=k   ∞ n D E X X tm n Belm (ak) xn−l S2 (l, k) ert | = l m! m=0 l=k     n n−l X n X n−l Belm (ak)rn−l−m = S2 (l, k) m l m=0 l=k  n X n−l   X n n−l S2 (l, k)rn−l−m Belm (ak). = m l m=0

(3.28)

l=k

Thus we have the following result. Theorem 3.7. For all integers n ≥ 0, we have the following expression. !  n n X n−l   X X n n−l S2 (l, k)rn−l−m Belm (ak) Ak (x; a), Beln,r (x) = l m m=0 k=0

l=k

where An (x; a) are the Abel polynomials. The ordered Bell polynomials PnObn (x) are the Appell polynomial with Obn (x) ∼ (2 − et , t). Write Beln,r (x) = k=0 Cn,k Obk (x). Then D E t 1 Cn,k = 2 − ee −1 | ert (et − 1)k xn k! n   D E X t n = S2,r (l + r, k + r) 2 − ee −1 |xn−l (3.29) l l=k n   X
n = S2,r (l + r, k + r) 2δn,l − Beln−l . l l=k

Hence we obtain the following theorem.

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Umbral calculus approach to r-Stirling numbers of the second kind and r-Bell polynomials

Theorem 3.8. For all integers n ≥ 0, we have the following expression. ! n n   X X
n Beln,r (x) = S2,r (l + r, k + r) 2δn,l − Beln−l Obk (x), l k=0

l=k

where Obn (x) are the ordered Bell polynomials. In (3.29), we saw that the ordered Bell polynomials Obm (x) are given by generating function ∞ X tm 1 xt e = Ob (x) . m 2 − et m! m=0

(3.30)

(α)

More generally, the ordered Bell polynomials Obm (x) of order α are defined by α  ∞ X 1 tm xt (α) e = Ob . (x) (3.31) m 2 − et m! m=0 (α)

(α)

For x = 0, Obm = Obm (0) are called the ordered Bell numbers of order α and given by  α ∞ m X 1 (α) t . = Ob (3.32) m 2 − et m! m=0 Pn (α) (α) Let Beln,r (x) = k=0 Cn,k Lk (x). Here Ln (x) are the Laguerre polynomials
(α) t of order α with Ln (x) ∼ (1 − t)−α−1 , t−1 . Then Cn,k

 t k E
e −1 1D n t −α−1 rt 2−e e = x k! et − 2 D
−(k+α+1) 1 rt t
k E = (−1)k 2 − et e e − 1 xn k! n   D X
−(k+α+1) n−l E n = (−1)k S2,r (l + r, k + r) 2 − et x l l=k n   X n (k+α+1) k = (−1) S2,r (l + r, k + r)Obn−l . l

(3.33)

l=k

Then we have the following theorem. Theorem 3.9. For all integers n ≥ 0, we have the following expression.   n n X X n Beln,r (x) = (−1)k S2,r (l + r, k + r) l k=0 l=k ! (k+α+1)

× Obn−l (α)

(α)

Lk (x),

(α)

where Obn (x) and Ln (x) are the higher-order ordered Bell polynomials and the Laguerre polynomials of order α, respectively.

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13

Pn Let Beln,r (x) = k=0 Cn,k Dk (x), where Dn (x) are the Daehee polynomials with
t t Dn (x) ∼ e −1 t , e − 1 . Then et −1
k n E 1D t − 1 et −1 rt e Cn,k = e e − 1 x k! et − 1 t D
k n+1 E 1 1 t rt et −1 et −1 = e (e − 1) e − 1 x k! n + 1 et − 1
k+1 n+1 E t k + 1D t 1 = ee −1 − 1 ert x t n+1 e −1 (k + 1)! ∞ X
l n+1 E 1 t k + 1D t rt S (l, k + 1) = e e − 1 x 2 n + 1 et − 1 l! l=k+1 (3.34) ∞ n+1 X E D t m k+1 X t = S2 (m, l) xn+1 S2 (l, k + 1) t ert n+1 e −1 m! m=l l=k+1   n+1 n+1 E D t X n+1 k+1 X = S2 (l, k + 1) ert xn+1−m S2 (m, l) t n+1 e −1 m l=k+1 m=l n+1 X n+1 X k + 1 n + 1  = S2 (l, k + 1)S2 (m, l)Bn+1−m (r). n+1 m l=k+1 m=l

Thus we have the following theorem. Theorem 3.10. For all integers n ≥ 0, we have the following expression. ! n n+1 X X n+1 X k + 1 n + 1  Beln,r (x) = S2 (l, k + 1)S2 (m, l)Bn+1−m (r) Dk (x), n+1 m k=0

l=k+1 m=l

where Dn (x) are the Daehee polynomials. Pn (ν) (ν) Write Beln,r (x) = k=0 Cn,k Hk (x). Here Hn (x) are the Hermite polynomi(ν)

als with Hn (x) ∼ (e

νt2 2

, t). Then

D ν(et −1)2 1
k E Cn,k = e 2 ert et − 1 xn k! n   D ν(et −1)2 E X n = S2,r (l + r, k + r) e 2 xn−l l l=k   n ∞ DX X n ν m (et − 1)2m n−l E S2,r (l + r, k + r) = x l m! 2m m=0 l=k

=

n   X n l=k

=

l

n   X n l=k

l

[ n−l 2 ]

S2,r (l + r, k + r)

(3.35)

E X (2m)!ν m D 1 (et − 1)2m xn−l m m!2 (2m)! m=0

[ n−l 2 ]

S2,r (l + r, k + r)

∞ E X (2m)! ν D X ti ( )m S2 (i, 2m) xn−l m! 2 i! m=0 i=2m

[ n−l 2 ]

n X   X n (2m)! ν m = ( ) S2,r (l + r, k + r)S2 (n − l, 2m). l m! 2 m=0 l=k

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Umbral calculus approach to r-Stirling numbers of the second kind and r-Bell polynomials

Hence we obtain the following result. Theorem 3.11. For all integers n ≥ 0, we have the following expression.

Beln,r (x) =

n X k=0

n−l

n [X 2 ]  X n (2m)! ν m ( ) S2,r (l + r, k + r) l m! 2 l=0 m=0 ! (ν)

× S2 (n − l, 2m) Hk (x), (ν)

where Hn (x) are the Hermite polynomials. Pn Let Beln,r (x) = k=0 Cn,k pk (x). Here pn (x) = xn yn−1 ( x1 ) ∼ (1, t − 21 t2 ), where
Pn (n+k)! x k yn (x) = k=0 (n−k)!k! are called Bessel polynomials and satisfy the differential 2 equation x2 y 00 + (2x + 2)y 0 + n(n + 1)y = 0.

(3.36)


k 1
k E et − 3 ert et − 1 xn k! k X  n   E D
k n 1 S2,r (l + r, k + r) et − 3 xn−l = − 2 l l=k  k X k   n   DX E k 1 n (−3)k−m emt xn−l = − S2,r (l + r, k + r) m l 2 m=0 l=k  k X n   k   X 1 k n = − (−3)k−m mn−l S2,r (l + r, k + r) m 2 l m=0 l=k  k X n X k    3 n k 1 = (− )m mn−l S2,r (l + r, k + r). m 2 l 3 m=0

(3.37)



Cn,k =

−

1 2

k D

l=k

Hence we have the following result. Theorem 3.12. For all integers n ≥ 0, we have the following expression.

Beln,r (x) =

n X k=0

n X k     k X n k 3 1 (− )m mn−l l m 2 3 l=k m=0 !

× S2,r (l + r, k + r) pk (x), where pn (x) = xn yn−1 ( x1 ), with yn (x) the Bessel polynomials.

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Let Beln,r (x) =

15

Pn

Cn,kbk (x), wherebn (x) are the Bernoulli polynomials of t the second kind with bn (x) ∼ et −1 , et − 1 . k=0

E D et − 1 rt 1 et −1 k n e (e − 1) x t k! ee −1 − 1 ∞ X D et − 1 E 1 = et −1 S2 (l, k) (et − 1)l xn ert l! e −1

Cn,k =

l=k

∞ X D et − 1 tm E = S2 (m, l) xn S2 (l, k) et −1 ert m! e −1 m=l l=k   n n E D et − 1 X X n = xn−m S2 (l, k) S2 (m, l) ert et −1 m e −1 l=k m=l   n ∞ n E D X X X 1 n = S2 (l, k) Bi (et − 1)i xn−m S2 (m, l) ert i! m i=0 l=k m=l   ∞ n n n−m E XX n X D X tj rt S2 (j, i) xn−m = S2 (l, k)S2 (m, l) Bi e m j! j=i i=0 n X

(3.38)

l=k m=l

n X n   n−m n−m X X X n − m n = S2 (l, k)S2 (m, l) Bi S2 (j, i)rn−m−j m j i=0 j=i l=k m=l

n X n n−m X X n−m X  n n − m = S2 (l, k)S2 (m, l)S2 (j, i)rn−m−j Bi . m j i=0 j=i l=k m=l

Thus we get the final result of this paper. Theorem 3.13. For all integers n ≥ 0, we have the following expression. n n X n n−m X  n n − m X X X n−m Beln,r (x) = m j k=0 l=k m=l i=0 j=i ! × S2 (l, k)S2 (m, l)S2 (j, i)rn−m−j Bi bk (x), where bn (x) are the Bernoulli polynomials of the second kind. References 1. A.Z. Broder, The r-Stirling numbers, Discrete Math., 49 (1984), 241-259. 2. I. Mez¨ o, On the maximum of r-Stirmilg numbers, Adv. Appl. Math., 41 (2008), 293-306. 3. J. Quaintance, H. W. Gould, Combinatorial identities for Stirling numbers. The unpublished notes of H. W. Gould. With a foreword by George E. Andrews. World Scientific Publishing Co. Pte. Ltd., Singapore, 2016. xv+260 pp. ISBN: 978-981-4725-26-2 4. S. Roman,The umbral calculus, Pure and Applied Mathematics, Vol.111, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1984. 5. M. Mihoubi, H. Belbachir, Linear recurrences for r-Bell polynomials, J. Integer Seq., 17 (2014), Article 14.10.6. 6. T. Kim, D. S. Kim, Extended Stirling polynomials of the second kind and Bell polynomials, Preprint. 7. D. S. Kim, T. Kim, Identities involving r-Stirling numbers, J. Comput. Anal. Appl., 17 (2014), no. 4, 674-680.

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Umbral calculus approach to r-Stirling numbers of the second kind and r-Bell polynomials

8. D. S. Kim, T. Kim, Some identities of Bell polynomials, Sci. China Math., 58 (2015), no. 10, 2095-2104. 9. T. Kim, D. S. Kim, On λ-Bell polynomials associated with umbral calculus, Russ. J. Math. Phys., 24 (2017), no. 1, 69-78. 1

Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin 300160, China, Department of Mathematics, Kwangwoon University, Seoul, 139701, Republic of Korea E-mail address: [email protected] 2

Department of Mathematics, Sogang University, Seoul, 121-742, Republic of Korea E-mail address: [email protected] 3 Department of Mathematics, Kwangwoon University, Seoul, 139-701, Republic of Korea E-mail address: [email protected] 4,∗ Department of Mathematics Education and RINS, Gyeongsang National University, Jinju, Gyeongsangnamdo, 52828, Republic of Korea E-mail address: [email protected]

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO. 1, 2019

On common fixed point theorems of weakly compatible mappings in fuzzy metric spaces, Afshan Batool, Tayyab Kamran, Dong Yun Shin, and Choonkil Park,……………………11 Analysis of latent CHIKV dynamics model with time delays, Ahmed. M. Elaiw, Taofeek O. Alade, and Saud M. Alsulami,………………………………………………………………19 Dynamical behavior of MERS-CoV model with discrete delays, H. Batarfi, A. Elaiw, and A. Alshareef,…………………………………………………………………………………….37 Convexity and hyperconvexity in fuzzy metric space, Ebru Yiğit and Hakan Efe,…………50 On generalizations of a reverse Hardy-Hilbert's type inequality, Zhengping Zhang and Gaowen Xi,……………………………………………………………………………………………59 Dunkl generalization of q-Szász-Mirakjan-Kantrovich type operators and approximation, Abdullah Alotaibi and M. Mursaleen,……………………………………………………….66 Pointwise error estimates for spherical hybrid interpolation, Chunmei Ding, Ming Li, and Feilong Cao,…………………………………………………………………………………77 𝑥𝑥

Investigating dynamics of the rational difference equation 𝑥𝑥𝑛𝑛+1 = 𝐴𝐴+𝐵𝐵𝑥𝑥𝑛𝑛−1𝑥𝑥

𝑛𝑛 𝑛𝑛−1

, Malek Ghazel,

Taher S. Hassan, and Ahmed M. Mosallem,…………………………………………………85 𝐿𝐿𝑝𝑝 approximation errors for hybrid interpolation on the unit sphere, Chunmei Ding, Ming Li, and Feilong Cao,…………………………………………………………………………………104 Some best approximation formulas and inequalities for the Bateman's G-function, Ahmed Hegazi, Mansour Mahmoud, Ahmed Talat, and Hesham Moustafa,…………………………118 A new q-extension of Euler polynomial of the second kind and some related polynomials, R. P. Agarwal, J. Y. Kang, and C. S. Ryoo,……………………………………………………….136 Regularized moving least squares approximation with Laplace-Beltrami operator on the sphere, Chunmei Ding, Yongli Zhang, and Feilong Cao,…………………………………………… 149 Chaos Control and Function Projective Synchronization of Noval Chaotic Dynamical System, M. M. El-Dessoky, E. O. Alzahrani, and N.A. Almohammadi,……………………………..162

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO. 1, 2019 (continued)

Umbral calculus approach to r-Stirling numbers of the second kind and r-Bell polynomials, Taekyun Kim, Dae San Kim, Hyuck-In Kwon, and Jongkyum Kwon,………………………173

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.2, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

An iterative algorithm of poles assignment for LDP systems Lingling Lv †, Zhe Zhang ‡, Lei Zhang §, Xianxing Liu

∗

¶

Abstract The problem of poles assignment and robust poles assignment in linear discrete-time periodic (LDP) system via periodic state feedback is discussed in this paper. Based on a numerical solution to the periodic Sylvester matrix equation, an iterative algorithm of computing the periodic feedback gain can be obtained. By optimizing the free parameter matrix in the proposed algorithm, according to robustness principle, an algorithm on the minimum norm and robust poles assignment for the LDP systems is presented. Two numerical examples are worked out to illustrate the effect of the proposed approaches. Keywords: Linear discrete-time periodic (LDP) systems; poles assignment; robustness.

1

Introduction

Linear discrete-time periodic (LDP) systems are important bridges connecting time-varying systems and time-invariant systems. In fact, Many natural and engineering phenomena can be reduced to a composite of periodic systems thus applications of periodic systems would be found in different field, where periodic controllers could be used to dealing with the problem in which time-invariant controllers is helpless(for example, [1–3]). Moreover, another major role of the periodic controller is to improve the performance of the closed-loop system, which has also been extensively studied(one can see [4, 5] and references therein). Therefore, researches on LDP systems have attracted more and more attentions. Since poles assignment techniques to modify the dynamic response of linear systems are the most studied problems among modern control theory, the above mentioned advantages of periodic systems and periodic controllers provide sufficient impetus for the researchers to carry out the study of poles assignment for periodic systems (see [6–9] and literatures therein). Due to the constraints of the constant controller in the periodic system, it is advocated in [6] that linear periodic output feedback is adequate to assign poles of a linear periodic discrete-time system. By utilizing a computational method on Sylvester equation, [7] proposes a complete parametric approach for pole assignment via periodic output feedback, in which parameter existed in the feedback gain could be used to accomplish some properties of plant system, robustness for instance. Using gradient search methods on the defined cost function, a computational approach is proposed in [8] to solve the minimum norm and robust pole assignment problem for linear periodic discrete-time system. Based on the proposed algorithm for parametric pole assignment problem, [9] considers the robust and minimum norm pole assignment problem and an explicit algorithm is proposed. In this paper, the problem of poles assignment and robust poles assignment in LDP systems via state feedback is considered. Based on an iterative algorithm proposed in [13] for periodic Sylvester matrix equation, an algorithm on the problem of poles assignment in periodic linear discrete-time system with periodic state ∗ This work is supported by the Programs of National Natural Science Foundation of China (Nos. 11501200, U1604148, 61402149), Innovative Talents of Higher Learning Institutions of Henan (No. 17HASTIT023), China Postdoctoral Science Foundation (No. 2016M592285). † 1. College of Environment and Planning, Henan University, Kaifeng, 475004, P. R. China. 2. Institute of Electric power, North China University of Water Resources and Electric Power, Zhengzhou 450011, P. R. China. Email: lingling [email protected] (Lingling Lv). ‡ Institute of electric power, North China University of Water Resources and Electric Power, Zhengzhou 450011, P. R. China. Email: zhe [email protected] (Zhe Zhang) § Computer and Information Engineering College, Henan University, Kaifeng 475004, P. R. China. Email: [email protected] (Lei Zhang). ¶ Computer and Information Engineering College, Henan University, Kaifeng 475004, P. R. China. Email: [email protected] (Xianxing Liu). Corresponding author.

1

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feedback is presented. The algorithm can realize accurate configuration of the closed-loop poles and obtain the numerical solution of the control gain. After solving the basic poles assignment problem, it is tempting to think: can we improve this algorithm to achieve the robustness of the system? The answer is positive. By optimizing the free parameter matrix in the proposed algorithm, this paper presents an algorithm on the minimum norm and robust poles assignment for the periodic linear discrete-time system. This algorithm can significantly improve the robust performance of closed-loop system. Two numerical examples are worked out to illustrate the effect of the proposed approaches. Here, we give descriptions of some symbols which will be encountered in the rest of this paper. tr(A) means the trace of matrix A. Norm ∥A∥ is a Frobenius norm of matrix A. Λ(A) means the eigenvalue set of matrix A and ΦAk denotes the monodromy matrix AK−1 AK−2 · · · A0 .

2

Main Discussions

2.1

Poles Assignment with Periodic State Feedback

Consider the completely reachable LDP systems as: qk+1 = Ak qk + Bk uk ,

(1)

where state matrix Ak ∈ Rn×n and input matrix Bk ∈ Rn×r are K-periodic. Based on the periodic feedback law in the form of u k = Fk qk , (2) where Fk is the K-periodic control gain, the closed-loop system can be obtained as qk+1 = Ac,k qk ,

(3)

where Ac,k denotes (Ak + Bk Fk ). Then the problem of poles assignment for periodic discrete-time linear system by control law (2) can be represented as Problem 1 Consider the completely reachable periodic discrete-time linear system (1), seek the periodic state feedback gain Fk ∈ Rm×n , k ∈ 0, K − 1, such that the poles of corresponding periodic closed-loop system (3) are set to the predetermined position Γ = {λ1 , · · · , λn }, where Γ should be symmetrical about the real axis. In the following, we will first present a new poles assignment algorithm via periodic state feedback, then give strict mathematical argument to show the correctness of the proposed algorithm. Algorithm 1 (Poles assignment with periodic state feedback) ek ∈ Rn×n , k ∈ 0, K − 1, satisfying Λ(Φ e ) = Γ. Further, 1. Choose the appropriate K-periodic matrices A Ak ek , Gk ) are completely observable and choose Gk ∈ Rr×n , k ∈ 0, K − 1 such that periodic matrix pairs (A Λ(ΦAek ) ∩ Λ(ΦAk ) = 0; 2. Set tolerance ε, for arbitrary initial matrix Xk (0) ∈ Rn×n , k ∈ 0, K − 1, calculate ek ; Qk (0) = Bk Gk + Ak Xk (0) − Xk+1 (0)A eT Rk (0) = −AT k Qk (0) + Qk−1 (0)Ak−1 ; Pk (0) = −Rk (0); j := 0;

2

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3. While ∥Rk (j)∥ ≤ ε, k ∈ 0, K − 1, calculate [ ] tr PkT (j)Rk (j)

∑K−1 k=0

α(j) = ∑

2 ;

K−1 e P (j) + P (j) A k+1 k k=0 −Ak k Xk (j + 1) = Xk (j) + α(j)Pk (j) ∈ Rn×n ; ek ∈ Rn×n ; Qk (j + 1) = Bk Gk + Ak Xk (j + 1) − Xk+1 (j + 1)A eT Rk (j + 1) = −AT k Qk (j + 1) + Qk−1 (j + 1)Ak−1 ; ∑K−1 2 k=0 ∥Rk (j + 1)∥ Pk (j + 1) = −Rk (j + 1) + ∑ Pk (j) ∈ Rn×n ; K−1 2 ∥R (j)∥ k k=0 j = j + 1; 4. Let Xk∗ = Xk (j), calculate the periodic state feedback gain Fk by Fk = Gk (Xk∗ )−1 , k ∈ 0, K − 1. To verify the validity of the above algorithm, we would provide several necessary lemmas for the problem under discussion, whose correctness can be easily checked by detail computation or derivation, and their proof is omitted due to space limitations. Lemma 1 For k ≥ 0, the following equation holds: T −1 ∑

[ ] tr RkT (j + 1)Pk (j) = 0

k=0

for all {Rk (j)} and {Pk (j)} derived from Algorithm 1. Lemma 2 For k ≥ 0, the following equation holds: T −1 ∑

T −1 ∑ ] [ 2 ∥Rk (j)∥ tr RkT (j)Pk (j) = − j=0

k=0

for all {Rk (j)} and {Pk (j)} generated by Algorithm 1. Lemma 3 For k ≥ 0, the following relation holds: ∑ j≥0

(∑

T −1 k=0

∥Rk (j)∥

∑K−1 k=0

2

∥Pk (j)∥

)2

2

< ∞.

for all {Rk (j)} and {Pk (j)} generated by Algorithm 1. Based on these lemmas, we can further draw the following conclusion. Theorem 1 The matrices Xk∗ , k ∈ 0, T − 1 generated by Algorithm 1 satisfy periodic Sylvester matrix equation ek + Bk Gk = 0, k ∈ 0, K − 1. (4) Ak Xk − Xk+1 A Proof. To explain matrices Xk , k ∈ 0, K − 1 generated by Algorithm 1 are solutions to equation (10), we first illustrate that this problem is related to the convergence of matrix sequence {Rk (j)}, k ∈ 0, T − 1 generated by Algorithm 1.

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According to equation (10), construct the following index function: J=

K−1 ∑ k=0

2 1

ek

Bk Gk + Ak Xk − Xk+1 A

. 2

(5)

It is easily obtained that for k ∈ 0, K − 1, ) ( ) ( ∂J ek + Bk−1 Gk−1 + Ak−1 Xk−1 − Xk A ek−1 A eTk−1 = −ATk Bk Gk + Ak Xk − Xk+1 A ∂Xk So far, if we can find matrices Xk∗ , k ∈ 0, K − 1 such that ∂J = 0, ∂Xk ∗ Xk =Xk

then matrices Xk∗ , k ∈ 0, K − 1 must be the solution to equation (10) in the meaning of least squares. From the formulation of sequence {Rk (j)}, k ∈ 0, T − 1 in Algorithm 1, we can see ∂J Rk (j) = . ∂Xk Xk =Xk (j) That is to say, if matrix sequence {Rk (j)}, k ∈ 0, T − 1 can converge to zero, matrices Xk∗ , k ∈ 0, K − 1 generated by Algorithm 1 must satisfy periodic matrix equation (10). In the remaining, we only need proof that, for k ∈ 0, K − 1 lim ∥Rk (j)∥ = 0.

j→∞

By Lemma 1 and the expressions of Pk (j + 1) in Algorithm 1, we have

2

∑K−1 K−1 K−1 2

∑ ∑ ∥R (j + 1)∥

k 2 k=0 P (j) ∥Pk (j + 1)∥ =

−Rk (j + 1) + ∑ k K−1 2

k=0 ∥Rk (j)∥ k=0 k=0 (∑ ) K−1 K−1 2 2 K−1 ∑ ∑ 2 2 k=0 ∥Rk (j + 1)∥ ∥P (j)∥ + ∥Rk (j + 1)∥ . = ∑K−1 k 2 ∥R (j)∥ k k=0 k=0 k=0 Let

∑K−1 tj = (∑ k=0 K−1 k=0

∥Pk (j)∥

∥Rk (j)∥

2

2

)2 .

Then the preceding relation can be written as tj+1 = tj + ∑K−1 k=0

1 ∥Rk (j + 1)∥

2

.

(6)

equivalently. We now proceed by contradiction and assume that lim

j→∞

K−1 ∑

2

∥Rk (j)∥ ̸= 0.

(7)

k=0

This relation implies that there exists a constant δ > 0 such that K−1 ∑

2

∥Rk (j)∥ ≥ δ

k=0

for all j ≥ 0. It follows from (6) and (7) that tj+1 ≤ t0 +

j+1 . δ

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And it shows that

So we have

δ 1 ≥ . tj+1 δt0 + j + 1 ∞ ∞ ∑ ∑ 1 δ ≥ = ∞. t δt + j+1 0 j=1 j j=1

However, it follows from Lemma 3 that

∞ ∑ 1 < ∞. t j=1 j

This gives a contradiction. Thus, the correctness of the theorem has been proved. As for the effectiveness of Algorithm 1, we have the following conclusion: Theorem 2 Consider completely reachable periodic discrete-time linear system (1), the K-periodic matrix Fk generated from Algorithm 1 is a solution of the problem of poles assignment with periodic state feedback. Proof. Notice that the poles of LDP system (1) are the poles of the monodromy matrix ΦAk . According to algorithm 1, ΦAfk possesses the desired pole set Γ. To assign the poles of the closed-loop system (3) to set Γ, we just need find n-order invertible matrices Xk , k ∈ 0, K − 1, such that

namely

−1 ek , Xk+1 Ack Xk = A

(8)

−1 ek , Xk+1 (Ak + Bk Fk )Xk = A

(9)

Pre-multiplying the above equation by matrix Xk+1 gives ek + Bk Fk Xk = 0, k ∈ 0, K − 1, Ak Xk − Xk+1 A Let Gk = Fk Xk , then Problem 1 is converted to the problem of solving the periodic Sylvester matrix equation in the form of ek + Bk Gk = 0, k ∈ 0, K − 1. Ak Xk − Xk+1 A

(10)

The step 2-3 in Algorithm 1 involve the solution of this matrix equation, and its correctness has been proved in [13]. By solving the solution matrix Xk , the periodic feedback gain can be obtained as Fk = Gk Xk−1 , k ∈ 0, K − 1.

(11)

That is, the periodic feedback gain Fk derived from (11) is a solution to Problem 1. ek , it should satisfy Λ(Φ e) = Γ. This requirement can be achieved by Remark 1 For the periodic matrix A A letting F0 be the real Jordan canonical form of the desired pole set and Fk , k ∈ 1, K − 1 be unit matrices of corresponding dimension. Remark 2 If system (1) is completely reachable and Λ(ΦAe)∩Λ(ΦA ) = 0, then Xk will be invertible naturally. That’s why the algorithm requires condition Λ(ΦAe) ∩ Λ(ΦA ) = 0.

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2.2

Robust Consideration

In the previous subsection, the iterative algorithm can provide infinite numerical solutions for the pole assignment problem via periodic state feedback by choose different parameter matrix Gk . Therefore, by adding some additional conditions to the feedback gain matrix Fk , k ∈ 0, K − 1 and transforming matrix Xk , k ∈ 0, K − 1, the free parameter matrix Gk can be used to achieve the robustness of the system. In general, the small feedback gain is robust. Because small gain means small control signals, that is beneficial to reduce noise amplification. At the same time, in the sense of poles assignment, the closed-loop poles to be configured should be not as sensitive as possible to disturbances in the system matrix. Thus, the following robust and minimum norm pole assignment problem via periodic state feedback is proposed. Problem 2 Consider the completely reachable linear periodic discrete-time system (1), seek the K-periodic state feedback gain Fk ∈ Rm×n , such that 1. the poles of corresponding periodic closed-loop system are set to the predetermined position Γ = {λ1 , · · · , λn }; 2. The periodic feedback gain is as small as possible and the closed-loop poles are not as sensitive as possible to disturbances in the system matrix. In order to solve Problem 2, the index function in [8] is introduced as follows: J(Gk ) = γ

K−1 K−1 1 ∑ 1 ∑ 2 2 κF (Xk ) + (1 − γ) ∥Fk ∥ , 2 2 k=0

(12)

k=0

where 0 ≤ γ ≤ 1 is a weighting factor. It is noted that when γ = 0, J(Gk ) degenerates into the index function of the minimum norm problem; when γ = 1, J(Gk ) becomes a purely objective function to solve the robust problem. Obviously, the weight γ leads to the combination of these two problems. [8] gives explicit analytical expressions for the index function J and its gradient. So it’s easy to minimize J(Gk ) by using any gradient-based search method. Therefore, we can present an algorithm for the problem of periodic robust and minimum norm poles assignment. Algorithm 2 (Robust and minimum norm poles assignment) ek ∈ Rn×n satisfying Λ(Φ e ) = Γ, and initialize Gk ∈ 1. Choose the appropriate K-periodic matrices A Ak r×n ek , Gk ) are completely observable and Λ(Φ e ) ∩ Λ(ΦA ) = 0; R such that periodic matrix pairs (A Ak

k

2. Set tolerance ε, for arbitrary initial matrix Xk (0) ∈ Rn×n , k ∈ 0, K − 1, calculate ek ; Qk (0) = Bk Gk + Ak Xk (0) − Xk+1 (0)A eT ; Rk (0) = −AT Qk (0) + Qk−1 (0)A k

k−1

Pk (0) = −Rk (0); j := 0; 3. While ∥Rk (j)∥ ≤ ε, k ∈ 0, K − 1, calculate ] ∑K−1 [ T k=0 tr Pk (j)Rk (j) α(j) = ∑

2 ; K−1 ek P (j) + P (j) A

−A

k k k+1 k=0 Xk (j + 1) = Xk (j) + α(j)Pk (j) ∈ Rn×n ; ek ∈ Rn×n ; Qk (j + 1) = Bk Gk + Ak Xk (j + 1) − Xk+1 (j + 1)A eT Rk (j + 1) = −AT k Qk (j + 1) + Qk−1 (j + 1)Ak−1 ; ∑K−1 2 k=0 ∥Rk (j + 1)∥ Pk (j + 1) = −Rk (j + 1) + ∑ Pk (j) ∈ Rn×n ; K−1 2 ∥R (j)∥ k k=0 j = j + 1; 6

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4. Based on gradient-based search methods and the index (12), choosing the appropriate weighting factor γ, solve the optimization problem Minimize J (Gk ). Denote the optimal decision matrix by Gopt,k ; 5. Substituting Gopt,k into step 2-3 gives optimization solution Xopt,k (j); 6. Let Xopt,k = Xopt,k (j), calculate the robust and minimum norm periodic state feedback gain Fopt,k by −1 Fopt,k = Gopt,k Xopt,k , k ∈ 0, K − 1.

3

Numerical examples

Example 1 Consider the completely reachable system described by q(t + 1) = A(t)q(t) + B(t)u(t) with

   A0 =    

0 1 0 0 0

e 0 0 0 0

0 0 e 0 0

0 0 0

e−1 0

1 0  0 1  e − 1 0 B0 =    0 1 − e−1 1 0





0 0   0 1    , A1 =  1 0     0 1 − e−1 0 0   1 0   0 1      , B1 =  e − 1 0  .      0 1  1 0 0 0 0 0 1 

1 0 e 0 0

0 0 0

e−1 0

0 0 0 0 1

   ,  

Find 2-periodic control law u(t) = F (t)q(t) such that the poles of the periodic close-loop system are assigned at Γ = {0.5 ± 0.5i, 0.7 ± 0.7i, −0.6}. Specially, let [ ] e 0 2 0 1 G(t) = , t = 0, 1 0.5 −e−1 0 1 2    0.5 0.5 0 0 0      −0.5 0.5 0 0 0         0 0 0.7 0.7 0     ,t = 0    0 0 −0.7 0.7 0      0 0 0 0 0.6  e = A(t)     1 0 0 0 0      0 1 0 0 0         0 0 1 0 0 ,t = 1        0 0 0 1 0     0 0 0 0 1 The proposed Algorithm 1 applied to the example gives the following 2-periodic feedback gain: [ ] 2.8249 −0.4278 −2.6334 2.3210 0.4035 F (0) = , 1.1033 0.2796 −0.8349 1.4695 0.2045 [ ] −0.2648 −1.0196 −0.7015 −0.2593 −0.0573 F (1) = . 1.0698 −1.7859 1.4382 −0.7656 −0.2827 What can be verified is that the poles assignment is valid.

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Example 2 This example is borrowed from [12]. The desired close-loop eigenvalues set is Γ = {0.5, 0.6, 0.7, −0.6, −0.7}. Arbitrarily assigning the parameter matrix Gk as [ ] 0.3 0.5 2.1 0 1.1 G(t) = , t = 0, 1 0.6 1.1 0.7 1.2 0.2 gives a group of feedback gains as [ Frand (0) = [ Frand (1) =

follows: 1.0000 −0.0000 36.9007 −19.7886 −0.0045 0.0419 −0.8356 0.1582

0.0000 93.1374

0.0000 19.1142

−1.3397 −0.0351 1.9971 0.4532

Applying Algorithm 2 with γ = 0.5 gives the following [ 1.0000 0.0000 Frobu (0) = −0.0289 −2.6601 [ −0.0332 0.0005 Frobu (1) = 0.0042 −0.8145

0.0000 −9.4571 0.0476 −0.5408

] , ] .

robust feedback gains: 0.0000 −0.0603 −1.2358 −0.0068

] −0.0000 , 0.0054 ] −0.0004 0.0200 . 1.0742 0.0029

−0.0000 2.9199

Let the close-loop system matrices be perturbed by ∆k ∈ Rn×n , k = 0, 1, which are random perturbations satisfying ∥∆k ∥ = 1, k = 0, 1. Then the close-loop system with perturbations can be represented as: Ack + µ∆k , k = 0, 1, where µ > 0 is a factor representing the disturbance level. According to [14], the following index can be adopted to measure the robustness of the corresponding close-loop system: dµ (∆k ) = max {|λi {(Ac1 + µ∆1 )(Ac0 + µ∆0 )}|}, 1≤i≤5

where λi {A} denotes the i-th eigenvalue of matrix A. 3,000 randomized trials were performed at µ equal to 0.002, 0.003 and 0.005, respectively. The worst and the average value of dµ (∆k ) corresponding to Frobu and Frand respectively are listed in Table 1. Polar plots of the trials are depicted in Fig.1, where the left hand side refers to Frobu and the right hand side refers to Frand . As we can see, in the presence of disturbances, the robust periodic feedback gain Frobu always performs better than Frand .

µ dµ Worst Mean

Table 1: Comparison between Krobu µ=0.002 µ=0.003 Frobu Frand Frobu Frand 1.0237 3.3798 1.0197 4.7927 0.7262 1.3667 0.7244 1.5881

and Krand µ=0.005 Frobu Frand 1.1561 10.9309 0.9022 2.5102

In terms of minimum norm, we compute the robust periodic feedback gains by minimize the √ index J(Gk ) at 2 2 γ equal to 0,0.5 and 1 respectively and the feedback norm ∥F0 ∥, ∥F1 ∥ together with ∥F ∥ = ∥F0 ∥ + ∥F1 ∥ . The results can be see in Table 2. Table 2: Comparison between Krobu and Krand Factor ∥F0 ∥ ∥F1 ∥ ∥F ∥ γ=0 2.2230 2.2549 3.1665 γ = 0.5 4.0751 1.8292 4.4668 γ=1 4.0727 1.8289 4.4645

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2

90

1

150

120 30

180

210

60 1

150

30

180

0

0

330

210

330 240

2

90 60

120

300

240

300

270

270

(a) Perturbed eigenvalues of the close-loop system with µ = 0.002 90

2

120

90 60

1

150

30

180

1

30

180

0

330

210

330

240

300

240

60

150

0

210

2

120

300 270

270

(b) Perturbed eigenvalues of the close-loop system with µ = 0.003

90

2

120

90 60

1

150

30

180

30

180

0

210

330 240

60 1

150

0

210

2

120

330 300

240

300

270

270

(c) Perturbed eigenvalues of the close-loop system with µ = 0.005

Figure 1: Perturbed eigenvalues of the close-loop system with different disturbance levels

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4

Conclusions

Poles assignment with periodic state feedback and periodic robust and minimum norm poles assignment are discussed in this paper. Through mathematical derivation, the poles assignment problem is transformed into the solution to the periodic Sylvester matrix equation. Based on the recent method of solving the equation, an algorithm for solving the problem of poles assignment is presented. In this algorithm, the parameter matrix Gk can be used for further discussion on robustness. By analyzing the theory of robustness and the minimum norm, an index function of matrix Gk is adopted. Based on the gradient search algorithm, the optimization decision matrix is finally given, and the robust and minimum norm gain is thus obtained. Two examples demonstrate the effectiveness of the proposed approaches.

References [1] P Khargonekar, K Poolla, A Tannenbaum. Robust control of linear time-invariant plants using periodic compensation. IEEE Transactions on Automatic Control, 1985, 30(11):1088-1096. [2] E Carlos. De Souza, A. Trofino. An LMI approach to stabilization of linear discrete-time periodic systems. International Journal of Control, 2000, 73(8):696-703. [3] C Farges, D Peaucelle, D Arzelier, et al. Robust H2 performance analysis and synthesis of linear polytopic discrete-time periodic systems via LMIs. Systems & Control Letters, 2007, 56(2):159-166. [4] S Longhi, R Zulli. A robust periodic pole assignment algorithm. Automatic Control IEEE Transactions on, 1995, 40(5):890-894. [5] J Lavaei, S Sojoudi, Aghdam A G. Pole Assignment With Improved Control Performance by Means of Periodic Feedback. IEEE Transactions on Automatic Control, 2007, 55(1):248-252. [6] D Aeyels, J L Willems. Pole assignment for linear periodic systems by memoryless output feedback. IEEE Transactions on Automatic Control, 1995, 40(4):735-739. [7] L L Lv, G R Duan, B Zhou. Parametric Pole Assignment for Discrete-time Linear Periodic Systems via Output Feedback. Acta Automatica Sinica, 2010, 36(36):113-120. [8] A Varga. Robust and minimum norm pole assignment with periodic state feedback. Automatic Control IEEE Transactions on, 2000, 45(5):1017-1022. [9] L L Lv, G Duan, B Zhou. Parametric pole assignment and robust pole assignment for discrete-time linear periodic systems. SIAM Journal on Control and Optimization, 2010, 48(6): 3975-3996. [10] L H Keel, J A Fleming, S P Bhattacharyya. Minimum Norm Pole Assignment via Sylvester’s Equation. Linear Algebra & Its Role in Systems Theory, 1985, 47:265-272. [11] G R Duan. Solutions of the equation AV+ BW= VF and their application to eigenstructure assignment in linear systems. IEEE Transactions on Automatic Control, 1993, 38(2): 276-280. [12] S. Longhi, R. Zulli. A note on robust pole assignment for periodic systems. IEEE Transactions on Automatic Control, 1996, 41(10):1493-1497. [13] L. Lv, Z. Zhang. Finite Iterative Solutions to Periodic Sylvester Matrix Equations. Journal of the Franklin Institute, 2017, 354(5):2358-2370. [14] L. James, H. K. TSO, N. K. Tsing. Robust deadbeat regulation. International Journal of Control, 1997, 67(4):587-602.

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C ∗ -ALGEBRA-VALUED MODULAR METRIC SPACES AND RELATED FIXED POINT RESULTS BAHMAN MOEINI1 , ARSLAN HOJAT ANSARI2 , CHOONKIL PARK3 AND DONG YUN SHIN4 Abstract. In this paper, a concept of C ∗ -algebra-valued modular metric space is introduced which is a generalization of a modular metric space of Chistyakov (Folia Math. 14 (2008), 3-25). Next, some common fixed point theorems are proved for generalized contraction type mappings on such spaces. Also, to support of our results an application is provided for existence and uniqueness of solution for a system of integral equations.

1. Introduction One of the main directions in obtaining possible generalizations of fixed point results is introduced in new types of spaces. The notion of modular spaces, as a generalization of metric spaces, was introduced by Nakano [18] and was intensively developed by Koshi and Shimogaki [12], Yamamuro [23] and others. Also, the theory of fixed points in the content of modular spaces was investigated by Khamsi et al. [11] and many authors generalized these results [1, 2, 9, 10, 15, 22]. In 2008, Chistyakov [3] introduced the notion of modular metric spaces generated by F -modular and developed the theory of this space. In 2010, Chistyakov [4] defined the notion of modular on an arbitrary set and developed the theory of metric spaces generated by modular which are t called the modular metric spaces. Recently, Mongkolkeha et al. [16, 17] have introduced some notions and established some fixed point results in modular metric spaces. In [14], Ma et al. introduced the concept of C ∗ -algebra-valued metric spaces. The main idea consists in using the set of all positive elements of a unital C ∗ -algebra instead of the set of real numbers. They showed that if (X, A, d) is a complete C ∗ -algebra-valued metric space and T : X → X is a contractive mapping, i.e., there exists an a ∈ A with kak < 1 such that d(T x, T y)  a∗ d(x, y)a, (∀x, y ∈ X). Then T has a unique fixed point in X. This line of research was continued in [7, 8, 13, 21, 24], where several other fixed point results were obtained in the framework of C ∗ -algebra valued metric, as well as (more general) C ∗ -algebra-valued b-metric spaces. Recently, Shateri [20] introduced the concept of C ∗ -algebra-valued modular space which is a generalization of a modular space and next proved some fixed point theorems for self-mappings with contractive or expansive conditions on such spaces. In this paper, new type of modular metric space is introduced and by using some ideas of [19] some common fixed point results are proved for self-mappings with contractive 0

Corresponding authors: [email protected] (Choonkil Park), [email protected] (Dong Yun Shin) 2010 Mathematics Subject Classification. Primary 47H10; 54H25; 46L05. Key words and phrases. modular metric space, C ∗ -algebra-valued modular metric space, common fixed point, occasionally weakly compatible, integral equation.

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B. MOEINI, A.H. ANSARI, C. PARK, D. SHIN

conditions on such spaces. Also, some examples to elaborate and illustrate our results are constructed. Finally, as application, existence and uniqueness of solution for a type of system of nonlinear integral equations is established. 2. Basic notions Let X be a nonempty set, λ ∈ (0, ∞) and due to the disparity of the arguments, function ω : (0, ∞) × X × X → [0, ∞] will be written as ωλ (x, y) = ω(λ, x, y) for all λ > 0 and x, y ∈ X. Definition 2.1. [3] Let X be a nonempty set. A function ω : (0, ∞) × X × X → [0, ∞] is said to be a modular metric on X if it satisfies the following three axioms: (i) given x, y ∈ X, ωλ (x, y) = 0 for all λ > 0 if and only if x = y; (ii) ωλ (x, y) = ωλ (y, x) for all λ > 0 and x, y ∈ X; (iii) ωλ+µ (x, y) ≤ ωλ (x, z) + ωµ (z, y) for all λ > 0 and x, y, z ∈ X. Then (X, ω) is called a modular metric space. Recall that a Banach algebra A (over the field C of complex numbers) is said to be a C ∗ -algebra if there is an involution ∗ in A (i.e., a mapping ∗ : A → A satisfying a∗∗ = a for each a ∈ A) such that, for all a, b ∈ A and λ, µ ∈ C, the following holds: ¯ ∗+µ (i) (λa + µb)∗ = λa ¯ b∗ ; (ii) (ab)∗ = b∗ a∗ ; (iii) ka∗ ak = kak2 . Note that, from (iii), it follows that kak = ka∗ k for each a ∈ A. Moreover, the pair (A, ∗) is called a unital ∗-algebra if A contains the unit element 1A . A positive element of A is an element a ∈ A such that a∗ = a and its spectrum σ(a) ⊂ R+ , where σ(a) = {λ ∈ R : λ1A − a is noninvertible}. The set of all positive elements will be denoted by A+ . Such elements allow us to define a partial ordering ‘’ on the elements of A. That is, b  a if and only if b − a ∈ A+ . If a ∈ A is positive, then we write a  θ, where θ is the zero element of A. Each positive element a of a C ∗ -algebra A has a unique positive square root. From now on, by A we mean a unital C ∗ -algebra with unit element 1A . Further, a+ = {a ∈ A : a  θ} and 1 (a∗ a) 2 = |a|. Lemma 2.2. [5] Suppose that A is a unital C ∗ -algebra with a unit 1A . (1) For any x ∈ A+ , we have x  1A ⇔ kxk ≤ 1. (2) If a ∈ A+ with kak < 12 , then 1A − a is invertible and ka(1A − a)−1 k < 1. (3) Suppose that a, b ∈ A with a, b  θ and ab = ba. Then ab  θ. (4) By A0 we denote the set {a ∈ A : ab = ba, ∀b ∈ A}. Let a ∈ A0 . If b, c ∈ A with b  c  θ and 1A − a ∈ A0 is an invertible operator, then (1A − a)−1 b  (1A − a)−1 c. Notice that in a C ∗ -algebra, if θ  a, b, one cannot that  conclude   θ  ab.For ex3 2 1 −2 ∗ ample, consider the C -algebra M2 (C) and set a = ,b= . Then 2 3 −2 4   −1 2 ab = . Clearly a, b ∈ M2 (C)+ , while ab is not. −4 8 In the following we begin to introduce and study a new type of modular metric space that is called a C ∗ -algebra-valued modular metric space.

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Definition 2.3. Let X be a nonempty set. A function ω : (0, ∞) × X × X → A is said to be a C ∗ -algebra-valued modular metric (briefly, C ∗ .m.m) on X if it satisfies the following three axioms: (i) given x, y ∈ X, ωλ (x, y) = θ for all λ > 0 if and only if x = y; (ii) ωλ (x, y) = ωλ (y, x) for all λ > 0 and x, y ∈ X; (iii) ωλ+µ (x, y)  ωλ (x, z) + ωµ (z, y) for all λ, µ > 0 and x, y, z ∈ X. The truple (X, A, ω) is called a C ∗ .m.m space. If instead of (i), we have the condition ωλ (x, x) = θ for all λ > 0 and x ∈ X, then ω is said to be a C ∗ -algebra-valued pseudo modular metric (briefly, C ∗ .p.m.m) on X and if ω satisfies (i0 ), (iii) and (i00 ) given x, y ∈ X, if there exists a number λ > 0, possibly depending on x and y, such that ωλ (x, y) = θ, then x = y, then ω is called a C ∗ -algebra-valued strict modular metric (briefly, C ∗ .s.m.m) on X. (i0 )

A C ∗ .m.m (or C ∗ .p.m.m, C ∗ .s.m.m) ω on X is said to be convex if, instead of (iii), we replace the following condition: µ λ ωλ (x, z) + λ+µ ωµ (z, y) for all λ, µ > 0 and x, y, z ∈ X. (iv) ωλ+µ (x, y)  λ+µ Clearly, if ω is a C ∗ .s.m.m, then ω is a C ∗ .m.m, which in turn implies that ω is a C ∗ .p.m.m on X, and similar implications hold for convex ω. The essential property of a C ∗ .m.m ω on a set X is as follows: given x, y ∈ X, the function 0 < λ → ωλ (x, y) ∈ A is non increasing on (0, ∞). In fact, if 0 < µ < λ, then we have ωλ (x, y)  ωλ−µ (x, x) + ωµ (x, y) = ωµ (x, y).

(2.1)

It follows that at each point λ > 0 the right limit ωλ+0 (x, y) := limε→+0 ωλ+ε (x, y) and the left limit ωλ−0 (x, y) := limε→+0 ωλ−ε (x, y) exist in A and the following two inequalities hold: ωλ+0 (x, y)  ωλ (x, y)  ωλ−0 (x, y). It can be checked that if x0 ∈ X, then the set Xω = {x ∈ X : lim ωλ (x, x0 ) = θ} λ→∞

is a C ∗ -algebra-valued metric space, called a C ∗ -algebra-valued modular space, where d0ω : Xω × Xω → A is given by d0ω = inf{λ > 0 : kωλ (x, y)k ≤ λ} for all x, y ∈ Xω . Moreover, if ω is convex, then the set Xω is equal to Xω∗ = {x ∈ X : ∃ λ = λ(x) > 0 such that kωλ (x, x0 )k < ∞} and d∗ω : Xω∗ × Xω∗ → A is given by d∗ω = inf{λ > 0 : kωλ (x, y)k ≤ 1} for all x, y ∈ Xω∗ . It is easy to see that if X is a real linear space, ρ : X → A and x−y ) for all λ > 0 and x, y ∈ X, (2.2) ωλ (x, y) = ρ( λ then ρ is a C ∗ -algebra valued modular (convex C ∗ -algebra-valued modular) on X if and only if ω is C ∗ .m.m (convex C ∗ .m.m, respectively) on X. On the other hand, assume that ω satisfies the following two conditions: (i) ωλ (µx, 0) = ω λ (x, 0) for all λ, µ > 0 and x ∈ X; µ

(ii) ωλ (x + z, y + z) = ωλ (x, y) for all λ > 0 and x, y, z ∈ X.

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If we set ρ(x) = ω1 (x, 0) with (2.2), where x ∈ X, then Xρ = Xω is a linear subspace of X and the functional kxkρ = d0ω (x, 0), x ∈ Xρ is an F -norm on Xρ . If ω is convex, then Xρ∗ ≡ Xω∗ = Xρ is a linear subspace of X and the functional kxkρ = d∗ω (x, 0), x ∈ Xρ∗ , is a norm on Xρ∗ . Similar assertions hold if we replace C ∗ .m.m by C ∗ .p.m.m. If ω is C ∗ .m.m in X, then the set Xω is a C ∗ .m.m space. By the idea of property in C ∗ -algebra-valued metric spaces and C ∗ -algebra-valued modular spaces, we define the following: Definition 2.4. Let Xω be a C ∗ .m.m space. (1) The sequence (xn )n∈N in Xω is said to be ω-convergent to x ∈ Xω with respect to A if ωλ (xn , x) → θ as n → ∞ for all λ > 0. (2) The sequence (xn )n∈N in Xω is said to be ω-Cauchy with respect to A if ωλ (xm , xn ) → θ as m, n → ∞ for all λ > 0. (3) A subset C of Xω is said to be ω-closed with respect to A if the limit of the ω-convergent sequence of C always belongs to C. (4) Xω is said to be ω-complete if any ω-Cauchy sequence with respect to A is ω-convergent. (5) A subset C of Xω is said to be ω-bounded with respect to A if for all λ > 0 δω (C) = sup{kωλ (x, y)k; x, y ∈ C} < ∞. Definition 2.5. Let Xω be a C ∗ .m.m space. Let f, g be self-mappings of Xω . A point x in Xω is called a coincidence point of f and g if and only if f x = gx. We shall call w = f x = gx a point of coincidence of f and g. Definition 2.6. Let Xω be a C ∗ .m.m space. Two self-mappings f and g of Xω are said to be weakly compatible if they commute at coincidence points. Definition 2.7. Let Xω be a C ∗ .m.m space. Two self-mappings f and g of Xω are occasionally weakly compatible (owc) if and only if there is a point x in Xω which is a coincidence point of f and g at which f and g commute. Lemma 2.8. [6] Let Xω be a C ∗ .m.m space and f, g owc self-mappings of Xω . If f and g have a unique point of coincidence, w = f x = gx, then w is a unique common fixed point of f and g. 3. Main results Theorem 3.1. Let Xω be a C ∗ .m.m space and I, J, R, S, T, U : Xω → Xω be selfmappings of Xω such that the pairs (SR, I) and (T U, J) are occasionally weakly compatible. Suppose there exist a, b, c ∈ A with 0 < kak2 + kbk2 + kck2 < 1 such that the following assertion for all x, y ∈ Xω and λ > 0 hold: (3.1.1) ωλ (SRx, T U y)  a∗ ωλ (Ix, Jy)a + b∗ ωλ (SRx, Jy)b + c∗ ω2λ (T U y, Ix)c; (3.1.2) kωλ (SRx, T U y)k < ∞. Then SR, T U, I and J have a common fixed point in Xω . Furthermore, if the pairs (S, R), (S, I), (R, I), (T, J), (T, U ), (U, J) are commuting pairs of mappings, then I, J, R, S, T and U have a unique common fixed point in Xω . Proof. Since the pair (SR, I) and (T U, J) are occasionally weakly compatible, there exist u, v ∈ Xω : SRu = Iu and T U v = Jv. Moreover, SR(Iu) = I(SRu) and

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C ∗ -ALGEBRA-VALUED MODULAR METRIC SPACES AND RELATED FIXED POINT RESULTS

T U (Jv) = J(T U v). Now we can assert that SRu = T U v. If not then by (3.1.1) ωλ (SRu, T U v)  a∗ ωλ (Iu, Jv)a + b∗ ωλ (SRu, Jv)b + c∗ ω2λ (T U v, Iu)c = a∗ ωλ (Iu, Jv)a + b∗ ωλ (Iu, Jv)b + c∗ ω2λ (Jv, Iu)c = a∗ ωλ (Iu, Jv)a + b∗ ωλ (Iu, Jv)b + c∗ ω2λ (Iu, Jv)c.

(3.1)

By definition of C ∗ .m.m space and (2.1) and (3.1), we have ωλ (SRu, T U v)  a∗ ωλ (Iu, Jv)a + b∗ ωλ (Iu, Jv)b + c∗ (ωλ (Iu, Iu) + ωλ (Iu, Jv))c = a∗ ωλ (Iu, Jv)a + b∗ ωλ (Iu, Jv)b + c∗ ωλ (Iu, Jv)c 1 1 1 1 = a∗ (ωλ (Iu, Jv)) 2 (ωλ (Iu, Jv)) 2 a + b∗ (ωλ (Iu, Jv)) 2 (ωλ (Iu, Jv)) 2 b 1 1 +c∗ (ωλ (Iu, Jv)) 2 (ωλ (Iu, Jv)) 2 c 1 1 = (a(ωλ (Iu, Jv)) 2 )∗ (a(ωλ (Iu, Jv)) 2 ) 1 1 +(b(ωλ (Iu, Jv)) 2 )∗ (b(ωλ (Iu, Jv)) 2 ) 1 1 +(c(ωλ (Iu, Jv)) 2 )∗ (c(ωλ (Iu, Jv)) 2 ) 1 1 1 = |a(ωλ (Iu, Jv)) 2 |2 + |b(ωλ (Iu, Jv)) 2 |2 + |c(ωλ (Iu, Jv)) 2 |2 1 1 1  ka(ωλ (Iu, Jv)) 2 k2 1A + kb(ωλ (Iu, Jv)) 2 k2 1A + kc(ωλ (Iu, Jv)) 2 k2 1A . Thus

kωλ (SRu, T U v)k ≤ kωλ (Iu, Jv)k(kak2 + kbk2 + kck2 ) < kωλ (Iu, Jv)k. So kωλ (Iu, Jv)k < kωλ (Iu, Jv)k, which is a contradiction. Hence SRu = T U v and thus SRu = Iu = T U v = Jv. Moreover, assume that there is another point z such that SRz = Iz. Using (3.1.1), ωλ (SRz, T U v)  a∗ ωλ (Iz, Jv)a + b∗ ωλ (SRz, Jv)b + c∗ ω2λ (T U v, Iz)c = a∗ ωλ (SRz, T U v)a + b∗ ωλ (SRz, T U v)b + c∗ ω2λ (SRz, T U v)c. (3.2) By a similar way, kωλ (SRz, T U v)k < kωλ (SRz, T U v)k(kak2 + kbk2 + kck2 ), which is a contradiction. Hence we get SRu = Iu = T U v = Jv.

(3.3)

Thus from (3.2) and (3.3), it follows that SRu = SRz. Hence w = SRu = Iu, for some w ∈ Xω , is the unique point of coincidence of SR and I. Then by Lemma 2.8, w is a unique common fixed point of SR and I. So SRw = Iw = w. Similarly, there is another common fixed point w0 ∈ Xω : T U w0 = Jw0 = w0 . For the uniqueness, suppose w 6= w0 . Then by (3.1.1), we have ωλ (SRw, T U w0 ) = ωλ (w, w0 )  a∗ ωλ (Iw, Jw0∗ ωλ (SRw, Jw0∗ ω2λ (T U w, Iw0 )c = a∗ ωλ (w, w0∗ ωλ (w, w0∗ ω2λ (w, w0 )c. Thus kωλ (w, w0 )k < kωλ (w, w02 + kbk2 + kck2 ), which is a contradiction. Hence w = w0 . So w is a unique common fixed point of SR, T U, I and J. Furthermore, if (S, R), (S, I), (R, I), (T, J), (T, U ), (U, J) are commuting pairs, then Sw = S(SRw) = S(RS)w = SR(Sw) Sw = S(Iw) = S(RS)w = I(Sw) Rw = R(SRw) = RS(Rw) = SR(Rw) Rw = R(Iw) = (Rw), which show that Sw and Rw is a common fixed point of (SR, I), which gives SRw = Sw = Rw = Iw = w.

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Similarly, we have T U w = T w = U w = Jw = w. Hence w is a unique common fixed point of S, R, I, J, T, U .  Corollary 3.2. Let Xω be a C ∗ .m.m space and I, J, S, T : Xω → Xω be self-mappings of Xω such that the pairs (S, I) and (T, J) are occasionally weakly compatible. Suppose there exist a, b, c ∈ A with 0 < kak2 + kbk2 + kck2 < 1 such that the following assertions for all x, y ∈ Xω and λ > 0 hold: (3.2.1) ωλ (Sx, T y)  a∗ ωλ (Ix, Jy)a + b∗ ωλ (Sx, Jy)b + c∗ ω2λ (T y, Ix)c; (3.2.2) kωλ (Sx, T y)k < ∞. Then S, T, I and J have a unique common fixed point in Xω . Proof. If we put R = U := IXω where IXω is an identity mapping on Xω , then the result follows from Theorem 3.1.  Corollary 3.3. Let Xω be a C ∗ .m.m space and S, T : Xω → Xω be self-mappings of Xω such that S and T are occasionally weakly compatible. Suppose there exist a, b, c ∈ A with 0 < kak2 + kbk2 + kck2 < 1 such that the following assertions for all x, y ∈ Xω and λ > 0 hold: (3.3.1) ωλ (T x, T y)  a∗ ωλ (Sx, Sy)a + b∗ ωλ (T x, Sy)b + c∗ ω2λ (T y, Sx)c; (3.3.2) kωλ (T x, T y)k < ∞. Then S and T have a unique common fixed point in Xω . Proof. If we put I = J := S and S := T in (3.2.1) and (3.2.2), then the result follows from Theorem 3.1.  Corollary 3.4. Let Xω be a C ∗ .m.m space and S, T : Xω → Xω be self-mappings of Xω such that S and T are occasionally weakly compatible. Suppose there exists a ∈ A with 0 < kak < 1 such that the following assertions for all x, y ∈ Xω and λ > 0 hold: (3.4.1) ωλ (T x, T y)  a∗ ωλ (Sx, Sy)a; (3.4.2) kωλ (T x, T y)k < ∞. Then S and T have a unique common fixed point in Xω . Proof. If we put b = c := 0A in (3.3.1), then the result follows from Corollary 3.3.



4. Examples In this section we provide some nontrivial examples in favour of our results. Example 4.1. Let X = R and consider A = M2 (R) of all 2 × 2 matrices with the usual operation of addition, scalar multiplication and matrix multiplication. Define a norm P 1 2 2 2 and ∗ : A → A, given by A∗ = A for all A ∈ A, defines |a | on A by kAk = i,j=1 ij an involution on A. Thus A becomes a C ∗ -algebra. For     a11 a12 b11 b12 A= ,B = ∈ A = M2 (R), a21 a22 b21 b22 we denote A  B if and only if (aij − bij ) ≤ 0 for all i, j = 1, 2. Define ω : (0, ∞) × X × X → A by  x−y  | λ | 0 ωλ (x, y) = 0 | x−y λ | for all x, y ∈ X and λ > 0. It is easy to check that ω satisfies all the conditions of Definition 2.3. So (X, A, ω) is a C ∗ .m.m space.

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Example 4.2. Let X = { c1n : n = 1, 2, · · · } where 0 < c < 1 and A = M2 (R). Define ω : (0, ∞) × X × X → A by  x−y  k λ k 0 ωλ (x, y) = 0 αk x−y λ k for all x, y ∈ X, α ≥ 0 and λ > 0. Then it is easy to check that ω is a C ∗ .m.m. space. Example 4.3. Let X = L∞ (E) and H = L2 (E), where E is a Lebesgue measurable set. By B(H) we denote the set of bounded linear operators on the Hilbert space H. Clearly, B(H) is a C ∗ -algebra with the usual operator norm. Define ω : (0, ∞) × X × X → B(H) by ωλ (f, g) = π| f −g | ,

(∀f, g ∈ X).

λ

Here πh : H → H is the multiplication operator defined by πh (φ) = h · φ for φ ∈ H. Then ω is a C ∗ .m.m and (Xω , B(H), ω) is an ω-complete C ∗ .m.m space. It suffices to verify the completeness of Xω . For this, let {fn } be an ω-Cauchy sequence with respect to B(H), that is, for an arbitrary ε > 0, there is N ∈ N such that for all m, n ≥ N , fm − fn k∞ ≤ ε. λ So {fn } is a Cauchy sequence in Banach space X. Hence there are a function f ∈ X and N1 ∈ N such that kωλ (fm , fn )k = kπ| fm −fn | k = k λ

k

fn − f k∞ ≤ ε (n ≥ N1 ), λ

which implies that fn − f k∞ ≤ ε, (n ≥ N1 ). λ Consequently, the sequence {fn } is an ω-convergent sequence in Xω and so Xω is an ω-complete C ∗ .m.m space. kωλ (fn , f )k = kπ| fn −f | k = k λ

Example 4.4. Let (X, A, ω) be C ∗ .m.m space defined as in Example 4.1. Define S, T, I, J : Xω → Xω by  x if x ∈ (−∞, 1),  2 1 if x = 1, Sx = T x = 1, Jx = 2 − x, Ix =  0 if x ∈ (1, ∞) for all x,y ∈ Xω =  R and λ > 0 . Then we have

0

0 0=

= kωλ (Sx, T y)k < ∞. 0 0 For all a, b,c ∈ A with 0 < kak2 + kbk2 + kck2 < 1, we get  0 0 = ωλ (Sx, T y)  a∗ ωλ (Ix, Jy)a + b∗ ωλ (Sx, Jy)b + c∗ ω2λ (T y, Ix)c for all 0 0 x, y ∈ Xω and λ > 0. Also clearly, the pairs (S, I) and (T, J) are occasionally weakly compatible. So all the conditions of Corollary 3.2 are satisfied and x = 1 is a unique common fixed point of S, T, I and J.

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5. Application Remind that if for λ > 0 and x, y ∈ L∞ (E), we define ω : (0, ∞)×L∞ (E)×L∞ (E) → B(H) by ωλ (x, y) = π| x−y | , λ

where πh : H → H is defined as in Example 4.3, then (L∞ (E)ω , B(H), ω) is an ωcomplete C ∗ .m.m space. Let E be a Lebesgue measurable set, X = L∞ (E) and H = L2 (E) be the Hilbert space. Consider the following system of nonlinear integral equations: Z n(t, s)hj (s, x(s))ds (5.1) x(t) = w(t) + ki (t, x(t)) + µ E

L∞ (E)

for all t ∈ E, where w ∈ ω is known, ki (t, x(t)), n(t, s), hj (s, x(s)), i, j = 1, 2 and i 6= j are real or complex valued functions that are measurable both in t and s on E and µ is a real or complex number, and assume the following conditions: R (a) sups∈E E |n(t, s)|dt = M1 < +∞, (b) ki (s, x(s)) ∈ L∞ (E)ω for all x ∈ L∞ (E)ω , and there exists L1 > 1 such that for all s ∈ E, |k1 (s, x(s)) − k2 (s, y(s))| ≥ L1 |x(s) − y(s)| for all x, y ∈ L∞ (E)ω , (c) hi (s, x(s)) ∈ L∞ (E)ω for all x ∈ L∞ (E)ω , and there exists L2 > 0 such that for all s ∈ E, |h1 (s, x(s)) − h2 (s, y(s))| ≤ L2 |x(s) − y(s)| for all x, y ∈ L∞ (E)ω , (d) there exists x(t) ∈ L∞ (E)ω such that Z x(t) − w(t) − µ n(t, s)h1 (s, x(s))ds = k1 (t, x(t)), E

which implies R k1 (t, x(t)) − w(t) − µ ER n(t, s)h1 (s, k1 (s, x(s)))ds = k1 (t, x(t) − w(t) − µ E n(t, s)h1 (s, x(s))ds), (e) there exists y(t) ∈ L∞ (E)ω such that Z y(t) − w(t) − µ n(t, s)h2 (s, y(s))ds = k2 (t, y(t)), E

which implies R k2 (t, y(t)) − w(t) − µ ER n(t, s)hi (s, k2 (s, y(s)))ds = k2 (t, y(t) − w(t) − µ E n(t, s)h2 (s, y(s))ds). Theorem 5.1. With the assumptions (a)-(e), the system of nonlinear integral equations (5.1) has a unique solution x∗ in L∞ (E)ω for each real or complex number µ with 1+|µ|L2 M1 < 1. L1 Proof. Define Z Sx(t) = x(t) − w(t) − µ

n(t, s)h1 (s, x(s))ds, E

Z T x(t) = x(t) − w(t) − µ

n(t, s)h2 (s, x(s))ds, E

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Ix(t) = k1 (t, x(t)), Jx(t) = k2 (t, x(t)). 1+|µ|M1 L2 · 1B(H) , L1 1+|µ|M1 L2 < 1. L1

Set a = kck2 =

q

b = c = 0B(H) . Then a ∈ B(H)+ and 0 < kak2 + kbk2 +

For any h ∈ H, we have kωλ (Sx, T y)k = supkhk=1 (π| Sx−T y | h, h) λ

i R R h 1 = supkhk=1 E λ (x − y) + µ E n(t, s)(h2 (s, y(s) − h1 (s, x(s))ds h(t)h(t)dt ≤ supkhk=1 ≤

1 λ

i R h 1 R (x − y) + µ n(t, s)(h (s, y(s) − h (s, x(s))ds |h(t)|2 dt 2 1 E λ E

supkhk=1

h i 2 dt kx − yk + |µ|M L kx − yk |h(t)| ∞ 1 2 ∞ E

R

1 L2 ≤ ( 1+|µ|M )kx − yk∞ λ 1 L2 2 (t,y(t)) ≤ ( 1+|µ|M )k k1 (t,x(t))−k k∞ L1 λ 1 L2 = ( 1+|µ|M )kωλ (Ix, Jy)k L1

= kak2 kωλ (Ix, Jy)k. Then kωλ (Sx, T y)k ≤ kak2 kωλ (Ix, Jy)k + kbk2 kωλ (Sx, Jy)k + kck2 kω2λ (T y, Ix)k for all x, y ∈ L∞ (E)ω and λ > 0. Also by conditions (d) and (e) the pairs (S, I) and (T, J) are occasionally weakly compatible. Therefore, by Corollary 3.2, there exists a unique common fixed point x∗ ∈ L∞ (E)ω such that x∗ = Sx∗ = T x∗ = Ix∗ = Jx∗ , which proves the existence of unique solution of (5.1) in L∞ (E)ω . This completes the proof.  References [1] G.A. Anastassiou, I.K. Argyros, Approximating fixed points with applications in fractional calculus, J. Comput. Anal. Appl. 21 (2016), 1225–1242. [2] A. Batool, T. Kamran, S. Jang, C. Park, Generalized ϕ-weak contractive fuzzy mappings and related fixed point results on complete metric space, J. Comput. Anal. Appl. 21 (2016), 729–737. [3] V.V. Chistyakov, Modular metric spaces generated by F -modulars, Folia Math. 14 (2008), 3–25. [4] V.V. Chistyakov, Modular metric spaces I: basic concepts, Nonlinear Anal. 72 (2010), 1–14. [5] R. Douglas, Banach Algebra Techniques in Operator Theory, Springer, Berlin, 1998. [6] G. Jungck and B.E. Rhoades, Fixed point theorems for occasionally weakly compatible mappings, Fixed Point Theory, 7 (2006), 287–296. [7] Z. Kadelburg and S. Radenovi´ c, Fixed point results in C ∗ -algebra-valued metric spaces are direct consequences of their standard metric counterparts, Fixed Point Theory Appl. 2016, 2016:53. [8] T. Kamran, M. Postolache, A. Ghiura, S. Batul and R. Ali, The Banach contraction principle in C ∗ -algebra-valued b-metric spaces with application, Fixed Point Theory Appl. 2016, 2016:10. [9] M.A. Khamsi, A convexity property in modular function spaces, Math. Jpn. 44 (1996), 269–279. [10] M.A. Khamsi, Quasicontraction mapping in modular spaces without ∆2 -condition, Fixed Point Theory Appl. 2008, Article ID 916187 (2008). [11] M.A. Khamsi, W.M. Kozlowski and S. Reich, Fixed point theory in modular function spaces, Nonlinear Anal. 14 (1990), 935–953.

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B. MOEINI, A.H. ANSARI, C. PARK, D. SHIN

[12] S. Koshi and T. Shimogaki, On F -norms of quasi-modular spaces, J. Fac. Sci. Hokkaido Univ. Ser. I, 15 (1961), 202–218. [13] Z. Ma and L. Jiang, C ∗ -Algebra-valued b-metric spaces and related fixed point theorems, Fixed Point Theory Appl. 2015, 2015:222. [14] Z. Ma, L. Jiang and H. Sun, C ∗ -Algebra-valued metric spaces and related fixed point theorems, Fixed Point Theory Appl. 2014, 2014:206. [15] C. Mongkolkeha and P. Kumam, Common fixed points for generalized weak contraction mappings in modular spaces, Sci. Math. Jpn. 75 (2012), 69–79. [16] C. Mongkolkeha, W. Sintunavarat and P. Kumam, Fixed point theorems for contraction mappings in modular metric spaces, Fixed Point Theory Appl. 2011, 2011:93. [17] C. Mongkolkeha, W. Sintunavarat and P. Kumam, Fixed point theorems for contraction mappings in modular metric spaces, Fixed Point Theory Appl. 2012, 2012:103. [18] H. Nakano, Modulared: Semi-Ordered Linear Spaces, In Tokyo Math. Book Ser. Vol. 1, Maruzen Co., Tokyo, 1950. [19] A. Parya, P. Pathak, V.H. Badshah and N. Gupta, Common fixed point theorems for generalized contraction mappings in modular metric spaces, Adv. Inequal. Appl. 2017, 2017:9. [20] T.L. Shateri, C ∗ -algebra-valued modular spaces and fixed point theorems, J. Fixed Point Theory Appl. 19 (2017), 1551–1560. [21] D. Shehwar and T. Kamran, C ∗ -Valued G-contraction and fixed points, J. Inequal. Appl. 2015, 2015:304. [22] X. Wang and Y. Chen, Fixed points of asymptotic pointwise nonexpansive mappings in modular spaces, Appl. Math. 2012, Article ID 319394 (2012). [23] S. Yamamuro, On conjugate spaces of Nakano spaces, Trans. Amer. Math. Soc. 90 (1959), 291–311. [24] A. Zada, S. Saifullah and Z. Ma, Common fixed point theorems for G-contraction in C ∗ -algebravalued metric spaces, Int. J. Anal. Appl. 11 (2016), 23–27. 1

Department of Mathematics, Hidaj Branch, Islamic Azad University, Hidaj, Iran E-mail address: [email protected] 2

Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran E-mail address: [email protected] 3

Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea E-mail address: [email protected] 4

Department of Mathematics, University of Seoul, Seoul 02504, Korea E-mail address: [email protected]

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.2, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

Strong Convergence Theorems and Applications of a New Viscosity Rule for Nonexpansive Mappings Waqas Nazeer1, Mobeen Munir1, Sayed Fakhar Abbas Naqvi2, Chahn Yong Jung3,∗ and Shin Min Kang4,5,∗

1

Division of Science and Technology, University of Education, Lahore 54000, Pakistan e-mails: [email protected] (W.N); [email protected] (M.M) 2

3

Department of Mathematics, Lahore Leads University, Lahore 54810, Pakistan e-mail: [email protected]

Department of Business Administration, Gyeongsang National University, Jinju 52828, Korea e-mail: [email protected] 4

5

Center for General Education, China Medical University, Taichung 40402, Taiwan

Department of Mathematics and RINS, Gyeongsang National University, Jinju 52828, Korea e-mail: [email protected] Abstract We introduced new viscosity rule for nonexpansive mappings in Hilbert Spaces. The strong convergence theorem of the new rule is proved under certain assumptions imposed on the sequence of parameters. Moreover, applications of proposed viscosity rule are also given. 2010 Mathematics Subject Classification: 47H09 Key words and phrases: viscosity rule, Hilbert space, nonexpansive mapping, variational inequality

1

Introduction

In this paper, we shall take H as a real Hilbert space, h·, ·i as inner product, k · k as the induced norm, and C as a nonempty closed subset of H. Definition 1.1. Let T : H → H be a mapping. T is called nonexpansive if kT x − T yk ≤ kx − yk,

∀x, y ∈ H.

Definition 1.2. A mapping f : H → H is called a contraction if for all x, y ∈ H and θ ∈ [0, 1) kf x − f yk ≤ θkx − yk. ∗

Corresponding authors

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Definition 1.3. Pc : H → C is called a metric projection if for every x ∈ H there exists a unique nearest point in C, denoted by Pc x, such that kx − Pc xk ≤ kx − yk,

∀y ∈ C.

The following theorem gives the condition for a projection mapping to be nonexpansive. Theorem 1.4. Let C be a nonempty closed convex subset of the real Hilbert space H and Pc : H → H a metric projection. Then (a) kPc x − Pc yk2 ≤ hx − y, Pcx − Pc yi for all x, y ∈ H. (b) Pc is a nonexpansive mapping, that is, kx − Pc xk ≤ kx − yk for all y ∈ C. (c) hx − Pc x, y − Pc xi ≤ 0 for all x ∈ H and y ∈ C. In order to verify the weak convergence of an algorithm to a fixed point of a nonexpansive mapping we need the demiclosedness principle: Theorem 1.5. (The demiclosedness principle) ([2]) Let C be a nonempty closed convex subset of the real Hilbert space H and T : C → C such that xn * x∗ ∈ C and (I − T )xn → 0. Then x∗ = T x∗ . (Here → and * denote strong and weak convergence, respectively). Moreover, the following result gives the conditions for the convergence of a nonnegative real sequence. Theorem 1.6. ([9]) Assume that {an } is a sequence of nonnegative real numbers such that an+1 ≤ (1 − γn )an + δn , ∀n ≥ 0, where {γn} is a sequence in (0, 1) and {δn } is a sequence with P (1) ∞ n=0 γn = ∞, P (2) limn→∞ sup γδnn ≤ 0 or ∞ n=0 |δn | < ∞. Then an → 0 as n → ∞. The following strong convergence theorem, which is also called the viscosity approximation method, for nonexpansive mappings in real Hilbert spaces is given by Moudafi [8] in 2000. Theorem 1.7. Let C be a noneempty closed convex subset of the real Hilbert space H. Let T be a nonexpansive mapping of C into itself such that F (T ) := {x ∈ H : T (x) = x} is nonempty. Let f be a contraction of C into itself. Consider the sequence xn+1 =

n 1 f (xn ) + T (xn ), 1 + n 1 + n

n ≥ 0,

where the sequence {n } ∈ (0, 1) satisfies (1) limn→∞ n = 0, P (2) ∞ n=0 n = ∞, 1 (3) limn→∞ | n+1 − 1n | = 0. Then {xn } converges strongly to a fixed point x∗ of the nonexpansive mapping T, which is also the unique solution of the variational inequality h(I − f )x, y − xi ≥ 0,

222

∀ ∈ F (T ).

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.2, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

In 2015, Xu et al. [9] applied the viscosity method on the midpoint rule for nonexpansive mappings and give the generalized viscosity implicit rule:   xn + xn+1 xn+1 = αn f (xn ) + (1 − αn )T , ∀n ≥ 0. 2 This, using contraction, regularizes the implicit midpoint rule for nonexpansive mappings. They also proved that the sequence generated by the generalized viscosity implicit rule converges strongly to a fixed point of T . Ke and Ma [6], motivated and inspired by the idea of Xu et al. [9], proposed two generalized viscosity implicit rules: xn+1 = αn f (xn ) + (1 − αn )T (sn xn + (1 − sn )xn+1 ) and xn+1 = αn xn + βf (xn ) + γn T (sn xn + (1 − sn )xn+1 ). In [3], Jung et al. presented the following viscosity rule   xn+1 = T (yn ), yn = αn (wn ) + βn f (wn ) + γn T (wn ),   n+1 wn = xn +x . 2

In [7], Kwun et al. proved the strong convergence of the following viscosity rule ( xn+1 = T (yn ),
yn = αn (xn ) + βn f (xn ) + γn T xn +x2 n+1 .

Our contribution in this direction is the following new viscosity rule       xn + xn+1 xn + xn+1 xn + xn+1 xn+1 = αn + βn f + γn T . 2 2 2

2

(1.1)

New viscosity rule

Theorem 2.1. Let C be a nonempty closed convex subset of the real Hilbert space H. Let T : C → C be a nonexpansive mapping with F (T ) 6= ∅ and f : C → C a contraction with coefficient θ ∈ [0, 1). Pick any x0 ∈ C, let {xn } be a sequence generated by the new viscosity rule (1.1), where {αn }, {βn } and {γn} are sequences in (0, 1) satisfying the following conditions: (i) αn + βn + γn = 1, (ii) limn→∞ αn = 0 = limn→∞ βn and limn→∞ γn → 1, P (iii) ∞ n=0 |αn+1 − αn | < ∞, P∞ (iv) n=0 |βn+1 − βn | < ∞. Then {xn } converges strongly to a fixed point x∗ of the nonexpansive mapping T, which is also the unique solution of the variational inequality h(I − f )x, y − xi ≥ 0, ∀y ∈ F (T ). In other words, x∗ is the unique fixed point of the contraction PF (T )f, that is, PF (T ) f (x∗ ) = x∗ .

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Proof. This proof is divided into five steps. Step 1. ({xn } is bounded) Taking an arbitrary point p of F (T ), we have kxn+1 − pk



    

xn + xn+1 xn + xn+1 xn + xn+1

= αn + βn f + γn T − p

2 2 2

   

xn + xn+1 xn + xn+1 − αn p + βn f − βn p =

αn 2 2

 

xn + xn+1 + γn T + (αn + βn − 1)p

2







  

xn + xn+1

xn + xn+1

x + x n n+1 ≤ αn − p + βn f − p + γn T − p



2 2 2





xn + xn+1

αn αn kxn − pk + kxn+1 − pk + βn − f (p) f ≤

2 2 2

xn + xn+1

+ βn kf (p) − pk + γn − p

2

xn + xn+1

αn αn

kxn − pk + kxn+1 − pk + θβ − p ≤

+ βkf (p) − pk 2 2 2   1 1 + γn kxn − pk + kxn+1 − pk 2 2     αn + γn + θβn αn + γn + θβn = kxn − pk + kxn+1 − pk 2 2 γn + kxn+1 − pk + βn kf (p) − pk  2    1 − βn + θβn 1 − βn + θβn = kxn − pk + kxn+1 − pk 2 2 γn + kxn+1 − pk + βn kf (p) − pk. 2 It follows that     1 − βn + θβn 1 − βn + θβn 1− kxn+1 − pk ≤ kxn − pk + βn kf (p) − pk 2 2 implies (1 + βn (1 − θ))kxn+1 − pk ≤ (1 − βn (1 − θ))kxn − pk + 2βn kf (p) − pk.

(2.1)

Since βn , θ ∈ (0, 1), 1 − βn (1 − θ) ≥ 0. Moreover, by (2.1) and αn + βn + γn = 1 we get kxn+1 − pk 1 − βn (1 − θ) 2βn ≤ kxn − pk + kf (p) − pk 1 + βn (1 − θ) 1 + βn (1 − θ)     2βn (1 − θ) 2βn (1 − θ) 1 ≤ 1− kxn − pk + kf (p) − pk . 1 + βn (1 − θ) 1 + βn (1 − θ) 1 − θ

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Thus, we have  kxn+1 − pk ≤ max kxn − pk,

 1 kf (p) − pk . 1−θ

 kxn+1 − pk ≤ max kx0 − pk,

 1 kf (p) − pk . 1−θ

By induction we obtain

n+1 n+1 Hence, we concluded that {xn } is bounded. Consequently, {f ( xn +x )} and {T ( xn+x )} 2 2 are bounded. Step 2. (limn→∞ kxn+1 − xn k = 0)

kxn+1 − xn k

     

xn + xn+1 xn + xn+1 xn + xn+1

= αn + βn f + γn T 2 2 2        xn + xn−1 xn + xn−1 xn−1 + xn

− αn−1 + βn−1 f + γn−1 T

2 2 2

αn αn 1 1 =

2 (xn+1 − xn ) + 2 (xn − xn−1 ) + 2 (αn − αn−1 )xn + 2 (αn − αn−1 )xn−1        xn + xn−1 xn + xn+1 xn + xn−1 + βn f −f + (βn − βn−1 )f 2 2 2        xn−1 + xn xn+1 + xn xn−1 + xn

+ γn T −T + (γn − γn−1 )T

2 2 2

αn αn 1 =

2 (xn+1 − xn ) + 2 (xn − xn−1 ) + 2 (αn − αn−1 )(xn + xn−1 )        xn + xn−1 xn + xn+1 xn + xn−1 + βn f −f + (βn − βn−1 )f 2 2 2      xn+1 + xn xn−1 + xn + γn T −T 2 2     xn−1 + xn

− (αn − αn−1 ) + (βn − βn−1 ) T

2 αn αn ≤ kxn+1 − xn k + kxn − xn−1 k 2

2  

xn−1 + xn 1

+ |αn − αn−1 | xn−1 + xn − 2T

2 2

   

xn + xn+1 xn + xn−1

+ βn −f

f

2 2

   

xn + xn−1 xn + xn−1

+ |βn − βn−1 | f −T

2 2

   

xn+1 + xn xn−1 + xn

+ γn −T

T

2 2   αn αn 1 ≤ kxn+1 − xn k + kxn − xn−1 k + |αn − αn−1 | + |βn − βn−1 | M 2 2 2



xn+1 + xn xn − xn−1

xn+1 + xn xn − xn−1



+ θβn − −

+ γn

2 2 2 2 225

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  αn αn 1 = kxn+1 − xn k + kxn − xn−1 k + |αn − αn−1 | + |βn − βn−1 | M 2 2 2 θβn γn θβn γn + kxn+1 − xn k + kxn − xn−1 k + kxn+1 − xn k + kxn − xn− k 2 2 2 2 αn + θβn + γn αn + θβn + γn kxn+1 − xn k + kxn − xn−1 k = 2  2  1 + |αn − αn−1 | + |βn − βn−1 | M, 2 where M > 0 is a constant such that

  

xn−1 + xn

, M ≥ max sup xn + xn−1 − 2T

2 n≥0

    

xn + xn−1 xn + xn−1

. sup f −T

2 2 n≥0

It gives

  αn + θβn + γn 1− kxn+1 − xn k 2   αn + θβn + γn 1 ≤ kxn − xn−1 k + |αn − αn−1 | + |βn − βn−1 | M 2 2 implies   1 − βn + θβn 1− kxn+1 − xn k 2   1 − βn + θβn 1 ≤ kxn − xn−1 k + |αn − αn−1 | + |βn − βn−1 | M 2 2 implies (1 + βn (1 − θ))kxn+1 − xn k ≤ (1 − βn (1 − θ))kxn − xn−1 k + (|αn − αn−1 | + 2|βn − βn−1 |)M. Thus we have  1 − βn (1 − θ) kxn+1 − xn k ≤ kxn − xn−1 k 1 + βn (1 − θ) M (|αn − αn−1 | − 2|βn − βn−1 |). + 1 + βn (1 − θ) 

Since θ, βn ∈ (0, 1), 1 + βn (1 − θ) ≥ 1 and hence 1 − βn (1 − θ) ≤ 1 − βn (1 − θ). 1 + βn (1 − θ) Thus





kxn+1 − xn k ≤ 1 − βn (1 − θ) kxn − xn−1 k +

M (|αn − αn−1 | − 2|βn − βn−1 |). 1 + βn (1 − θ)

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Since

∞ X

βn = ∞,

n=0

∞ X

|αn+1 − αn | < ∞,

and

n=0

∞ X

|βn+1 − βn | < ∞,

n=0

by Theorem 1.6, we have kxn+1 − xn k → 0 as n → ∞. Step 3. (kxn − T xn k → 0 as → ∞) Consider kxn − T xn k

   

x + x x + x n n+1 n n+1 +T − T xn =

xn − xn+1 + xn+1 − T

2 2

   

xn + xn+1 xn + xn+1

≤ kxn − xn+1 k + xn+1 − T + T − T xn

2 2

   

xn + xn+1 xn + xn+1 + βn f ≤ kxn − xn+1 k +

αn 2 2

   

xn + xn+1

xn + xn+1 xn + xn+1

+ γn T −T − xn +

2 2 2

 

αn xn + xn+1 = kxn − xn+1 k +

2 (xn + xn+1 ) + βn f 2   xn + xn+1

+ 1 kxn+1 − xn k − (1 − γn )T

2 2

 

αn xn + xn+1 3

≤ kxn − xn+1 k + (xn + xn+1 ) + βn f 2 2 2   xn + xn+1

− (αn + βn )T

2

  3 αn xn + xn+1

≤ kxn − xn+1 k + xn + xn+1 − 2T

2 2 2

   

xn + xn+1 xn + xn+1

+ βn −T

f 2 2   αn 3 + βn M. ≤ kxn+1 − xn k + 2 2 Then by limn→∞ kxn+1 − xn k = 0 and limn→∞ γn = 1, we get kxn − T xn k → 0. Step 4. (limn→∞ suphx∗ − f (x∗ ), x∗ − xn i ≤ 0, where x∗ = PF (T ) f (x∗ )) Indeed, we take a subsequence {xni } of {xn } which converges weakly to a fixed point p of T . Without loss of generality, we may assume that {xni } * p. From limn→∞ kxn − T xn k = 0 and Theorem 1.5 we have p = T p. This, together with the property of the metric projection, implies that lim suphx∗ − f (x∗ ), x∗ − xn i = lim suphx∗ − f (x∗ ), x∗ − xni i

n→∞

n→∞ ∗

= hx − f (x∗ ), x∗ − pi ≤ 0.

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Step 5. (xn → x∗ as n → ∞) Now we again take x∗ ∈ F (T ) as the unique fixed point of the contraction PF (T )f . Consider kxn+1 − x∗ k2

2

     

x + x x + x x + x n n+1 n n+1 n n+1 ∗ + βn f + γn T −x =

αn 2 2 2

   

xn + xn+1 xn + xn+1 − αn x∗ + βn f − β n x∗ =

αn 2 2

2  

xn + xn+1 ∗ + γn T + (αn + βn − 1)x 2



2

2

  



xn + xn+1 2 xn + xn+1 ∗ ∗ 2 = αn − x + βn f −x 2 2

2

 

xn + xn+1 − x∗ + γn2

T 2     xn + xn+1 xn + xn+1 ∗ ∗ −x ,f −x + 2αn βn 2 2     xn + xn+1 xn + xn+1 ∗ ∗ + 2αn γn −x ,T −x 2 2       xn + xn+1 xn + xn+1 ∗ ∗ + 2βn γn f −x ,T −x 2 2



2



2

2  

xn + xn+1

2 xn + xn+1 ∗ ∗ ∗ 2 2 xn+1 + xn ≤ αn − x + βn f − x + γn −x 2 2 2      xn + xn+1 x + x n n+1 − x∗ , f − x∗ + 2αn βn 2 2





xn + xn+1 ∗ ∗ + 2αn γn kxn − x k T −x 2       xn + xn+1 xn + xn+1 − f (x∗ ), T − x∗ + 2βn γn f 2 2     xn + xn+1 + 2βn γn f (x∗ ) − x∗ , T − x∗ 2

2

2

xn+1 + xn

2 2 xn + xn+1 ∗ ∗

≤ (αn + γn ) − x + 2αn γn −x 2 2





xn + xn+1

∗

xn+1 + xn ∗ + 2βn γn f − f (x ) − x



+ Kn 2 2

2

2

xn+1 + xn

2 xn+1 + xn ∗ ∗

γ ≤ (αn + γn ) − x + 2θβn n − x + Kn 2 2

2  

xn+1 + xn

≤ (αn + γn )2 + 2θβn γn − x∗

+ Kn 2

2  

xn+1 + xn

2 ∗

≤ (1 − βn ) + 2θβn γn − x + Kn , 2 228

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where



2 

xn+1 + xn

∗ Kn = −x 2      xn+1 + xn xn+1 + xn ∗ ∗ −x ,f −x + 2αn βn 2 2     xn+1 + xn + 2βn γn f (x∗ ) − x∗ , T − x∗ . 2 βn2

f

It follows that  implies

p

implies

implies

2 

xn+1 + xn

∗

(1 − βn ) + 2θβn γn − x ≥ kxn+1 − xn k2 − Kn 2 2

)2

(1 − βn



xn+1 + xn

p ∗

+ 2θβn γn − x ≥ kxn+1 − xn k2 − Kn 2

p 1p (1 − βn )2 + 2θβn γn (kxn+1 − x∗ k + kxn − x∗ k) ≥ kxn+1 − xn k2 − Kn 2 1 ((1 − βn )2 + 2θβn γn )(kxn+1 − x∗ k2 + kxn − x∗ k2 4 + 2kxn+1 − x∗ kkxn − x∗ k) ≥ kxn+1 − xn k2 − Kn

implies

1 ((1 − βn )2 + 2θβn γn )(kxn+1 − x∗ k2 + kxn − x∗ k2 4 + (kxn+1 − x∗ k2 + kxn − x∗ k2 )) ≥ kxn+1 − xn k2 − Kn

implies 

 1 2 1 − ((1 − βn ) + 2θβn γn ) kxn+1 − x∗ k2 2   1 2 ≤ ((1 − βn ) + 2θβn γn ) kxn − x∗ k2 + Kn . 2

Thus we have kxn+1 − x∗ k2 ≤ =

1 2 Kn 2 ((1 − βn ) + 2θβn γn ) kxn − x∗ k2 + 1 1 2 1 − 2 ((1 − βn ) + 2θβn γn ) 1 − 2 ((1 − βn )2 + 2θβn γn ) 1 − 21 ((1 − βn )2 + 2θβn γn ) − 1 + ((1 − βn )2 + 2θβn γn ) kxn − x∗ k2 1 − 12 ((1 − βn )2 + 2θβn γn )

+

1−

1 2 ((1 −

Kn βn )2 + 2θβn γn )

 1 − ((1 − βn )2 + 2θβn γn ) Kn = 1− kxn − x∗ k2 + . 1 1 2 1 − 2 ((1 − βn ) + 2θβn γn ) 1 − 2 ((1 − βn )2 + 2θβn γn ) 

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Note that

1 0 < 1 − ((1 − βn )2 + 2θβn γn ) < 1 2

implies 1 − ((1 − βn )2 + 2θβn γn ) ≥ 1 − ((1 − βn )2 + 2θβn γn ). 1 − 12 ((1 − βn )2 + 2θβn γn ) Thus we have kxn+1 − x∗ k2 ≤ [1 − (1 − ((1 − βn )2 + 2θβn γn ))]kxn − x∗ k2 +

1 2 ((1 −

Kn βn )2 + 2θβn γn )

1− Kn = [(1 − βn )2 − 2θβn γn ]kxn − x∗ k2 + 1 1 − 2 ((1 − βn )2 + 2θβn γn ) Kn ≤ (1 − βn )2 kxn − x∗ k2 + . 1 − 21 ((1 − βn )2 + 2θβn γn )

Since 0 < 1 − βn < 1, this give (1 − βn )2 < (1 − βn ) and kxn+1 − x∗ k2 ≤ (1 − βn )kxn − x∗ k2 +

1−

Kn 1 2 2 ((1 − βn )

+ 2θβn γn )

(2.2)

.

By limn→∞ αn = limn→∞ βn = 0 and limn→∞ γn = 1 we have lim

n→∞

1

Kn 1 − 2 ((1 − βn )2 + 2θβn γn )
 2 βn kf xn+12+xn − x∗ k2


+ 2αn βn xn+12+xn − x∗ , f = lim n→∞ 1 − 12 ((1 − βn )2 + 2θβn γn )
 2βn γn hf (x∗ ) − x∗ , T xn+12+xn − x∗ i + 1 − 12 ((1 − βn )2 + 2θβn γn )

xn+1 +xn
2

− x∗



(2.3)

≤ 0. From (2.2), (2.3), and Theorem 1.6 we have limn→∞ kxn+1 − x∗ k2 = 0, which implies that xn → x∗ as n → ∞. This completes the proof.

3

Applications

The scheme can be used to solve problems of system of variational inequalities and constrained convex minimization. Moreover, it can be applied to find a fixed point in Kmappings.

3.1

A more general system of variational inequalities

Let C be a nonempty closed convex subset of the real Hilbert Space H and {Ai}N i=1 : C → H be a family of mappings. In [1] Cai and Bu considered the problem of finding

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(x∗1 , x∗2 , . . . , x∗N ) ∈ C × C × · · · × C such that  ∗ ∗ ∗ ∗   hλN AN xN + x1 − xN , x − x1 i ≥ 0,   ∗ ∗ ∗ ∗   hλN −1 AN −1 xN −1 + xN − xN −1 , x − xN i ≥ 0, .. .     hλ2 A2 x∗2 + x∗3 − x∗2 , x − x∗3 i ≥ 0,    hλ A x∗ + x∗ − x∗ , x − x∗ i ≥ 0, ∀x ∈ C. 1 1 1 2 1 2

(3.1)

The equation (3.1) can be written as   hx∗1 − (I − λN AN )x∗N , x − x∗1 i ≥ 0,     ∗ ∗ ∗   hxN − (I − λN −1 AN −1 )xN −1 , x − xN i ≥ 0, .. .    ∗  hx3 − (I − λ2 A2 )x∗2 , x − x∗3 i ≥ 0,    hx∗ − (I − λ A )x∗ , x − x∗ i ≥ 0, 1 1 1 2 2

which is a more general system of variational inequalities in Hilbert spaces with λi > 0 for all i ∈ {1, 2, 3, . . ., N }. Moreover, we have some useful results: Lemma 3.1. ([1]) Let C be a nonempty closed convex subset of the real Hilbert spaces H. For i ∈ {1, 2, 3, · · · , N }, let Ai : C → H be δi -inverse strongly monotone for some positive real number δi , namely, hAi x − Ai y, x − yi ≥ δi kAix − Ai yk2 , ∀x, y ∈ C Let G : C → C be a mapping defined by G(x) = PC (I − λN AN )PC (I − λN −1 AN −1 ) · · · PC (I − λ2 A2 )PC (I − λ1 A1 )x,

∀x ∈ C.

(3.2)

If 0 < λi ≤ 2δi for all i ∈ {1, 2, 3, · · · , N }, then G is nonexpansive. Lemma 3.2. ([5]) Let C be a nonempty closed convex subject of the real Hilbert Spaces H. Let Ai : C → H be a nonlinear mapping,where i ∈ {1, 2, 3, ..., N}. For given x∗i ∈ C, i ∈ {1, 2, 3, ..., N}, (x∗1 , x∗2 , x∗3 , ..., x∗N ) is a solution of the problem (3.1) if and only if x∗1 = PC (I − λN AN )x∗N , x∗i = PC (I − λi−1 Ai−1 )x∗i−1 ,

i = 2, 3, 4, ..., N,

that is, x∗1 = PC (I − λN AN )PC (I − λN −1 AN −1 ) · · · PC (I − λ2 A2 )PC (I − λ1 A1 )x∗1 ,

∀x ∈ C.

From Lemma 3.2, we know that x∗1 = G(x∗1 ), that is, x∗1 is a fixed point of the mapping G, where G is defined by (3.2). Moreover, if we find the fixed point x∗1 , it is easy to get the other points by (3.3). Applying Theorem 2.1 we get the result.

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Theorem 3.3. Let C be a nonempty closed convex subject of the real Hilbert spaces H. For i ∈ {1, 2, 3, ..., N}, let Ai : C → H be δi -inverse-strongly monotone for some positive real number δi with F (G) 6= ∅, where G : C → C is defined by G(x) = PC (I − λN AN )PC (I − λN −1 AN −1 ) · · · PC (I − λ2 A2 )PC (I − λ1 A1 )x,

∀x ∈ C.

Let f : C → C be a contraction with coefficient θ ∈ [0, 1). Pick any x0 ∈ C, let {xn } be a sequence generated by       xn + xn+1 xn + xn+1 xn + xn+1 xn+1 = αn + βn f + γn G , 2 2 2 where {αn }, {βn } and {γn } are sequences in (0, 1) satisfying the conditions (i)-(iv). Then {xn } converges strongly to a fixed point x∗ of the nonexpansive mapping G, which is also the unique solution of the variational inequality h(I − f )x, y − xi ≥ 0, ∀y ∈ F (T ). In other words, x∗ is the unique fixed point of the contraction PF (G) f, that is, PF (G) f (x∗ ) = x∗ .

3.2

The constrained convex minimization problem

Now, we consider the following constrained convex minimization problem; min φ(x), x∈C

(3.4)

where φ : C → R is a real-valued convex function and assumes that the problem (3.4)is consistent. Let Ω denote its solution set. For the minimization problem (3.4), if φ is (Fr´echet)differentiable, then we have the following lemma. Lemma 3.4. (Optimality Condition) ([5]) A necessary condition of optimality for a point x∗ ∈ C to be a solution of the minimization problem (3.4) is that x∗ solves the variational inequality h∇φ(x∗ ), x − x∗ i ≥ 0, ∀x ∈ C. (3.5) Equivalently, x∗ ∈ C solves the fixed point equation   x∗ = PC x∗ − λ∇φ(x∗ ) for every constant λ > 0. If, in a addition φ is convex, then the optimality condition (3.5) is also sufficient. It is well known that the mapping PC (I − λA) is nonexpansive when the mapping A is δ-inverse-strongly monotone and 0 < λ < 2δ. We therefore have the following result. Theorem 3.5. Let C be a nonempty closed convex subset of the real Hilbert Space H. For the minimization problem (3.4), assume that φ is (Fr´echet) differentiable and the gradient ∇φ is a δ-inverse-strongly monotone mapping for some positive real number δ. Let f : C → C be a contraction with coefficient θ ∈ [0, 1). Pick any x0 ∈ C. Let {xn } be a sequence generated by       xn + xn+1 xn + xn+1 xn + xn+1 xn+1 = αn + βn f + γn PC (I − λ∇φ) , 2 2 2

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where {αn }, {βn } and {γn } are sequences in (0, 1) satisfying the conditions (i)-(iv). Then {xn } converges strongly to a solution x∗ of the minimization problem (3.4), which is also the unique solution of the variational inequality h(I − f )x, y − xi ≥ 0, ∀y ∈ Ω. In other words, x∗ is the unique fixed point of the contraction PΩ f, that is, PΩ f (x∗ ) = ∗ x .

3.3

K-mapping

Kangtunyakarn and Suantai [4] in 2009 gave K-mapping generated by T1 , T2 , T3, ..., TN and λ1 , λ2 , λ3, ..., λN as follows. Definition 3.6. ([4]) Let C be a nonempty convex subset of real Banach Space. Let {Ti}N i=1 be a family of mappings of C into itself and let λ1 , λ2, λ3, ..., λN be real numbers such that 0 ≤ λi ≤ 1 for every i = 1, 2, 3, ..., N . We define a mapping K : C → C as follows;   U1 = λ1 T1 + (1 − λ1 )I,       U2 = λ2 T2 U1 + (1 − λ2 )U1 , .. .     UN −1 = λN −1 TN −1 UN −2 + (1 − λN −1 )UN −2 ,     UN = λN TN UN −1 + (1 − λN )UN −1 .

Such a mapping is called a K-mapping generated by T1 , T2 , T3, ..., TN and λ1 , λ2, λ3 , ..., λN . In 2014, Kangtunyakarn and Suwannaut [10] established the following result for Kmapping generated by T1 , T2 , T3, ..., TN and λ1 , λ2 , λ3, ..., λN . Lemma 3.7. ([10]) Let C be a nonempty closed convex subset of the real Hilbert space H. For i = 1, 2, 3, ..., N, let {Ti}N i=1 be a finite TN family of Ki-strictly pseudo-contractive mapping of C into itself with Ki ≤ ωi and i=1 F (Ti ) 6= ∅, namely, there exist constants Ki ∈ [0, 1) such that kTix − Tiyk2 ≤ kx − yk2 + Ki k(I − Ti)x − (I − Ti )yk2 ,

∀x, y ∈ C.

Let λ1 , λ2 , λ3, ..., λN be real numbers with 0 < λi < ω2 , ∀i = 1, 2, 3, ..., N and ω1 + ω2 < 1. Let K be the K-mapping generated by T1 , T2, T3 , ..., TN and λ1 , λ2, λ3 , ..., λN . Then the following properties hold: T (a) F (K) = N i=1 F (Ti ). (b) K is a nonexpansive mapping. On the bases of above lemma, we have the following results. Theorem 3.8. Let C be a nonempty closed convex subset of the real Hilbert space H. For i = 1, 2, 3, ..., N, let {Ti}N i=1 be a finite TN family of Ki -strictly pseudo-contractive mapping of C into itself with Ki ≤ ωi and i=1 F (Ti ) 6= ∅. Let λ1 , λ2 , λ3, ..., λN be real numbers with 0 < λi < ω2 , ∀i = 1, 2, 3, ..., N and ω1 + ω2 < 1. Let K be the K-mapping generated by T1 , T2, T3 , ..., TN and λ1 , λ2, λ3 , ..., λN . Let f : C → C be a contraction with coefficient θ ∈ [0, 1). Pick any x0 ∈ C, let {xn } be sequence generated by       xn + xn+1 xn + xn+1 xn + xn+1 xn+1 = αn + βn f + γn K , 2 2 2

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where {αn }, {βn } and {γn } are sequences in (0, 1) satisfying the conditions (i)-(iv). Then {xn } converges strongly to a fixed point x∗ of the mappings {Ti }N i=1 , which T is also the unique solution of the variational inequality h(I −f )x, y −xi, ∀y ∈ F (K) = N i=1 F (Ti ). ∗ T In other words, x is the unique fixed point of the contraction P N F (Ti ) f, that is, i=1 T P N F (Ti )f (x∗ ) = x∗ . i=1

References [1] G. Cai and S. Q. Bu, Hybrid algorithm for generalized mixed equilibrium problems and variational inequality problems and fixed point problems, Comput. Math. Appl., 62 (2011), 4772–4782. [2] K. Goebel and W. Kirk, Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics, vol. 28. Cambridge University Press, Cambridge, 1990. [3] C. Y. Jung, W. Nazeer, S. F. A. Naqvi and S. M. Kang, An implicit viscosity technique of nonexpansive mappings in Hilbert spaces, Int. J. Pure Appl. Math., 108 (2016), 635–650. [4] A. Kangtunyakarn and S. Suantai, A new mapping for finding common solutions of equilibrium problems and fixed point problems of finite family of nonexpansive mappings, Nonlinear Anal., 71 (2009), 4448–4460. [5] Y. F. Ke and C. F. Ma, A new relaxed extragradient-like algorithm for approaching common solutions of generalized mixed equilibrium problems, a more general system of variational inequalities and a fixed point problem, Fixed point Theory Appl., 126 (2013), 21 pages. [6] Y. F. Ke and C. F. Ma, The generalized viscosity implicit rules of nonexpansive mappings in Hilbert spaces, Fixed point Theory Appl., 190 (2015), 21 pages. [7] Y. C. Kwun, W. Nazeer, S. F. A. Naqvi and S. M. Kang, Viscosity approximation methods of nonexpansive mappings in Hilbert spaces and applications, Int. J. Pure Appl. Math., 108 (2016), 929–944. [8] A. Moudafi, Viscosity approximation methods for fixed points problems, J. Math. Anal. Appl., 241 (2000), 46–55. [9] H. K. Xu, M. A. Alghamdi and N. Shahzad., The viscosity technique for the implicit mid point rule of nonexpansive mappings in Hilbert spaces, Fixed point Theory Appl., 41 (2015), 12 pages. [10] S. Suwannaut and A. Kangtunyakarn, Strong convergence theorem for the modified generalized equilibrium problem and fixed point problem of strictly pseudocontractive mappings, Fixed Point Theory Appl., 86 (2014), 31 pages.

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GENERALIZED STABILITY OF CUBIC FUNCTIONAL EQUATIONS WITH AN AUTOMORPHISM ON A QUASI-β NORMED SPACE DONGSEUNG KANG1 AND HOEWOON B. KIM2

1 Mathematics

Education, Dankook University, 152, Jukjeon, Suji, Yongin, Gyeonggi,

16890, Korea E-mail address: [email protected] 2 Department

of Mathematics, Oregon State University, Corvallis, Oregon 97331

E-mail address: [email protected]

Abstract. We introduce a generalized cubic functional equation with an automorphism and investigate the generalized stability of the cubic functions as solutions to the generalized cubic functional equation on a quasi-β Banach space by the fixed point of the alternative method.

Keywords: Hyers-Ulam Stability, Cubic functional equations, Quasi-β normed space, Fixed Point, Functional equations 1. Introduction In a talk before the Mathematics Club of the University of Wisconsin in 1940, a Polish-American mathematician, S. M. Ulam [25] proposed the stability problem of the linear functional equation f (x + y) = f (x) + f (y) where any solution f (x) of this equation is called a linear function. To make the statement of the problem precise, let G1 be a group and G2 a metric group with the metric d(·, ·). Then given  > 0, does there exist a δ > 0 such that if a function f : G1 −→ G2 satisfies the inequality d(f (xy), f (x)f (y)) < δ for all x, y ∈ G1 , then there is a homomorphism F : G1 −→ G2 with d(f (x), F (x)) <  for all x ∈ G1 ?. In other words, the question would be generalized as “Under what conditions a mathematical object satisfying a certain property approximately must 2000 Mathematics Subject Classification. 39B52. Correspondence: Hoewoon B. Kim, [email protected]. 1

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GENERALIZED STABILITY OF CUBIC FUNCTIONAL EQUATIONS

be close to an object satisfying the property exactly?”. In 1941, the first, affirmative, and partial solution to Ulam’s question was provided by D. H. Hyers [10]. In his celebrated theorem Hyers explicitly constructed the linear function (or additive function) in Banach spaces directly from a given approximate function satisfying the well-known weak Hyers inequality with a positive constant. The Hyers stability result was first generalized in the stability of additive mappings by Aoki [1] allowing the Cauchy difference to become unbounded. In 1978 Th. M. Rassias [16] also provided a generalization of Hyers’ theorem with the possibly unbounded Cauchy difference for linear mappings. For the last decades, stability problems of various functional equations, not only linear case, have been extensively investigated and generalized by many mathematicians (see [4, 7, 9, 11, 17, 20, 21]). The functional equation f (x + y) + f (x − y) = 2f (x) + 2f (y)

(1.1)

is called a quadratic functional equation and every solution of this functional equation is said to be a quadratic function or mapping (e.g. f (x) = cx2 ). The HyersUlam stability problem for the quadratic functional equation was first studied by Skof [23] in a normed space as the domain of a quadracitc mapping of the equation. Cholewa [6] noticed that the results of Skof still hold in abelian groups. In [7] Czerwik obtained the Hyers-Ulam-Rassias stability (or generalized Hyers-Ulam stability) of the quadratic functional equation. See [2, 15, 27] for more results on the equation (1.1). Also the quadratic equation (1.1) was generalized by Stetkær in [24] introducing an involution σ of an abelian group G, i.e., an automorphism σ : G → G with σ 2 = I (I denotes the identity) and considering the following functional equation (1.2)

f (x + y) + f (x + σ(y)) = 2f (x) + 2f (y)

for all x, y ∈ G. As we already notice the equation (1.1) corresponds to the equation (1.2) with σ = −I. Jun and Kim [11] considered the following cubic functional equation (1.3)

f (2x + y) + f (2x − y) = 2f (x + y) + 2f (x − y) + 12f (x)

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since it should be easy to see that a function f (x) = cx3 is a solution of the equation (1.3) as the quadratic equation case. In a year they [12] proved the generalized Hyers-Ulam stability of a different version of a cubic functional equation (1.4)

f (x + 2y) + f (x − 2y) + 6f (x) = f (x + y) + 4f (x − y).

Since then the stability of cubic functional equations has been investigated by a number of authors (see [5, 14] for details). In particular, Najati [14] investigated the following generalized cubic functional equation (1.5)

f (sx + y) + f (sx − y) = sf (x + y) + sf (x − y) + 2(s3 − s)f (x)

for a positive integer s ≥ 2. As we might notice there are various definitions for the stability of the cubic functional equations and here we consider the following definition of a generalized cubic functional equation f (ax + y) − f (x + ay) + a(a − 1)f (x − y) (1.6) = (a − 1)(a + 1)2 f (x) − (a − 1)(a + 1)2 f (y) for all a ∈ Z (a 6= 0, ±1) and generalized the equation (1.6) with the involution σ of a linear space X when a = 2; (1.7)

f (2x + y) − f (x + 2y) + 2f (x + σ(y)) − 9f (x) + 9f (y) = 0.

In this paper we will study the generalized Hyers-Ulam stability problem of the equation (1.7). In order to give our results in Section 3 it is convenient to state the definition of a generalized metric on a set X and a result on a fixed point theorem of the alternative by Diaz and Margolis [8]. Let X be a set. A function d : X × X −→ [0, ∞] is called a generalized metric on X if d satisfies (1) d(x, y) = 0 if and only if x = y; (2) d(x, y) = d(y, x) for all x, y ∈ X; (3) d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ X. Theorem 1.1. Let (X, d) be a complete generalized metric space and let J : X −→ X be a strictly contractive mapping with Lipschitz constant 0 < L < 1. Then for

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each element x ∈ X, either d(J n x, J n+1 x) = ∞ for all nonnegative integers n or there exists a positive n0 such that (1) d(J n x, J n+1 x) < ∞ for all n ≥ n0 ; (2) the sequence {J n x} converges to a fixed point y ∗ of J; (3) y ∗ is the unique fixed point of J in the set Y = {y ∈ X|d(J n0 x, y) < ∞}; (4) d(y, y ∗ ) ≤ (1/(1 − L))d(y, Jy) for all y ∈ Y . In 2009, Rassias and Kim [18] investigated the Hyers-Ulam stability of Cauchy and Jensen type additive mappings in quasi-β-normed spaces with the following definition of a quasi-β-norm: Definition 1.2. Let β be a real number with 0 < β ≤ 1 and K be either R or C. Let X be a linear space over a field K. A quasi-β-norm || · || is a real-valued function on X satisfying the following properties: (1) ||x|| ≥ 0 for all x ∈ X and ||x|| = 0 if and only if x = 0 (2) ||λx|| = |λ|β ||x|| for all λ ∈ K and all x ∈ X (3) There is a constant K ≥ 1 such that ||x + y|| ≤ K(||x|| + ||y||) for all x, y ∈ X. The pair (X, || · ||) is called a quasi-β-normed space if || · || is a quasi-β-norm on X. A smallest possible constant K is called the modulus of concavity of || · ||. A quasi-β-Banach space is a complete quasi-β-normed space. A quasi-β-norm || · || is called a (β, p)-norm (0 < p ≤ 1) if the property (3) of the Definition 1.2 takes the form ||x + y||p ≤ ||x||p + ||y||p for all x, y ∈ X. In this case, a quasi-β-Banach space is referred to as a (β, p)-Banach space; see [3, 18, 19] for datails. In this paper, using the Fixed Point method we prove the generalized Hyers-Ulam stability of the generalized cubic functional equation (1.7) in a quasi-β-normed linear space we just defined above (Definition 1.2). In Section 2 we establish the general solution of the cubic functional equation (1.7) applying the symmetric nadditive mappings for the cubic functional equation (1.7) that will be explained in detail in the Section. Finally, we obtain, in Section 3, the generalized Hyers-Ulam stability of the generalized cubic functional equation (1.7) with the Fixed Point theorem of the Alternative.

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2. The General Solution with σ = −I In this section we study the general solution of the cubic functional equation (1.7) with σ = −I by introducing and applying n-additive symmetric mappings and their properties that are found in [22, 26]. Before we proceed, let us give some basic backgrounds of n-additive symmetric mappings. Let X and Y be real vector spaces and n a positive integer. A function An : X n −→ Y is called nadditive if it is additive in each of its variables. A function An : X n −→ Y is said to be symmetric if An (x1 , x2 , · · · , xn ) = An (xσ(1) , xσ(2) , · · · , xσ(n) ) for every permutation {σ(1), σ(2), · · · , σ(n)} of {1, 2, · · · , n}. If An (x1 , x2 , · · · , xn ) is an nadditive symmetric map, then An (x) will denote the diagonal An (x, x, · · · , x) and An (rx) = rn An (x) for all x ∈ X and r ∈ Q. Such a function An (x) will be called a monomial function of degree n assuming An (x) 6≡ 0. Moreover, the resulting function after substituting x1 = x2 = · · · = xs = x and xs+1 , xs+2 , · · · = xn = y in An (x1 , x2 , · · · , xn ) will be denoted by As,n−s (x, y). Theorem 2.1. A function f : X −→ Y is a solution of the functional equation (1.7) with σ = −I if and only if f is of the form f (x) = A3 (x) for all x ∈ X, where A3 (x) is the diagonal of the 3-additive symmetric mapping A3 : X 3 −→ Y . Proof. Assume that f satisfies the functional equation (1.7). Taking x = y = 0 in the equation (1.7) it’s not hard to have f (0) = 0 since σ(0) = 0. Substituting y = 0 in (1.7) also gives f (2x) − f (x) + 2f (x) − 9f (x) = 0, that is, (2.1)

f (2x) = 23 f (x)

for all x ∈ X. Similarly, when x = 0 in the equation (1.7) we have 2f (y) + 2f (σ(y)) = 0, i.e., (2.2)

f (y) + f (−y)) = 0

for all y ∈ X since σ(y) = −y. This observation leads us to f (−y) = −f (y) for all y ∈ X and hence f is an odd function. Rewriting the equation (1.7) as

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1 1 2 f (x) − f (2x + y) + f (x + 2y) − f (x − y) − f (y) = 0 9 9 9

(2.3)

and applying Theorems 3.5 and 3.6 in [26] we express f as f (x) = A3 (x) + A2 (x) + A1 (x) + A0

(2.4)

where A0 is an arbitrary element in Y and Ai (x) is the diagonal of the i-additive symmetric mapping Ai : X i −→ Y for i = 1, 2, 3. Since f is odd and f (0) = 0 it follows that f (x) = A3 (x) + A1 (x) for all x ∈ X. By the property (2.1) of f and An (rx) = rn An (x) for all x ∈ X and r ∈ Q we should obtain A1 (x) = 0 for all x ∈ X. Therefore we conclude that f (x) = A3 (x) for all x ∈ X. Conversely, let us assume that f (x) = A3 (x) for all x ∈ X, where A3 (x) is the diagonal of a 3-additive symmetric mapping A3 : X 3 −→ Y . Noting that A3 (qx + ry) = q 3 A3 (x) + 3q 2 rA2,1 (x, y) + 3qr2 A1,2 (x, y) + r3 A3 (y) and calculating simple computation for the equation (1.7) with σ = −I in term of A3 (x), we show that the function f satisfies the cubic equation (1.7) with σ = −I, which completes the proof.



3. General Hyers-Ulam Stability in a Quasi-β Banach Space: A Fixed Point Theorem of the Alternative Approach In this section we will investigate the generalized Hyers-Ulam stability of the cubic functional equation (1.7) which is introduced earlier in previous sections f (2x + y) − f (x + 2y) + 2f (x + σ(y)) − 9f (x) + 9f (y) = 0. for all x, y ∈ X by the approach of the fixed point of the alternative. As we used the notations in the previous sections we assume that X is a vector space and (Y, || · ||) is a quasi-β-Banach space in this section. A set R+ denotes the set of all nonnegative real numbers.

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Theorem 3.1. Suppose that a function φ : X 2 −→ R+ is given and there exists a constant L with 0 < L < 1 such that φ(2x, 2y) ≤ 2Lφ(x, y)

(3.1)

and

φ(x + σ(x), y + σ(y)) ≤ 2Lφ(x, y)

for all x, y ∈ X. Furthermore, let f : X −→ Y be a mapping such that f (0) = 0 and ||f (2x + y) − f (x + 2y) + 2f (x + σ(y)) − 9f (x) + 9f (y)|| ≤ φ(x, y)

(3.2)

for all x, y ∈ X where σ is an automorphism on X with σ 2 = I where I is the identity. Then there existsthe unique generalized cubic function C : X −→ Y defined by  1 C(x) := limn→∞ (f (2n x) + (2n − 1)f (2n−1 x + 2n−1 σ(x))) such that 23n   1+L Φ(x) (3.3) ||f (x) − C(x)|| ≤ 23 (1 − L) for all x ∈ X where Φ(x) = max{φ(x, 0), φ(0, x)} for all x ∈ X. Proof. First, we put y = 0 in the inequality (3.2) to obtain ||f (2x) − 23 f (x)|| ≤ φ(x, 0)

(3.4)

for x ∈ X since σ(0) = 0. Similarly we substitute x = 0 into the inequality (3.2) again to have (3.5)

||10f (y) − f (2y) + 2f (σ(y))|| ≤ φ(0, y)

for all y ∈ X. Combining the two inequalities (3.4) and (3.5) we note that ||2f (x) + 2f (σ(x))|| = ||10f (x) − f (2x) + 2f (σ(x)) + f (2x) − 23 f (x)|| ≤ φ(x, 0) + φ(0, x) and hence we conclude that (3.6)

||f (x) + f (σ(x))|| ≤

1 (φ(x, 0) + φ(0, x)) 2

Then we let x = x + σ(x) in the above inequality (3.6) and we are able to get (3.7) ||f (x + σ(x))|| ≤

1 L (φ(x + σ(x), 0) + φ(0, x + σ(x))) ≤ (φ(x, 0) + φ(0, x)) 4 2

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GENERALIZED STABILITY OF CUBIC FUNCTIONAL EQUATIONS

We also define a function T (f ) from X to Y by T (f )(x) =

1 (f (2x) + f (x + σ(x))) 23

and we then consider the following estimation

(3.8)



1

||T (f )(x) − f (x)|| = 3 (f (2x) + f (x + σ(x))) − f (x)

2

1

1 3

= (f (2x) − 2 f (x)) + f (x + σ(x))

23

3 2   1 1 L ≤ 3 φ(x, 0) + 3 (φ(x, 0) + φ(0, x)) 2 2 2 1 ≤ 3 (1 + L)Φ(x) 2

This idea enables us to define a sequence {T n (f )} in Y for each x ∈ X by T n (f )(x) =

1 (f (2n x) + (2n − 1)f (2n−1 x + 2n−1 σ(x))) 23n

for a nonnegative integer n with T 0 (f ) = f and we claim that it should be a Cauchy sequence in Y . In order to show this we use the inequalities (3.4), (3.7), and (3.8) to compute the following estimations; (3.9) ||T n (f )(x) − T n−1 (f )(x)|| = ||

1 (f (2n x) + (2n − 1)f (2n−1 x + 2n−1 σ(x))) 23n

1 (f (2n−1 x) + (2n−1 − 1)f (2n−2 x + 2n−2 σ(x)))|| 23(n−1) 1 = || 3n (f (2n x) + f (2n−1 x + 2n−1 σ(x)) + (2n − 2)f (2n−1 x + 2n−1 σ(x))) 2 1 − 3(n−1) (f (2n−1 x) + (2n−1 − 1)f (2n−2 x + 2n−2 σ(x)))|| 2 1 = || 3n (f (2n x) + f (2n−1 x + 2n−1 σ(x)) − 23 f (2n−1 x)) 2 1 + 3n ((2n − 2)f (2n−1 x + 2n−1 σ(x)) − 22 (2n − 2)f (2n−2 x + 2n−2 σ(x)))|| 2 1 = || 3n (f (2n x) + f (2n−1 x + 2n−1 σ(x)) − 23 f (2n−1 x)) 2   1 2n − 2 (2f (2n−1 x + 2n−1 σ(x)) − 23 f (2n−2 x + 2n−2 σ(x)))|| + 2 23n 1 L ≤ 3n (φ(2n−1 x, 0) + (φ(2n−1 x, 0) + φ(0, 2n−1 x))) 2 2   n    1 2 −2 L n−2 n−2 n−2 n−2 n−2 n−2 + φ(2 x+2 σ(x), 0) + (φ(2 x+2 σ(x), 0) + φ(0, 2 x+2 σ(x))) 2 23n 2  n−1 (2L)n−1 2n−1 − 1 1 L n−1 ≤ (1 + L)Φ(x) + (2L) (1 + L)Φ(x) = 3 (1 + L) Φ(x) 23n 23n 2 2 −

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9

for all x ∈ X and all nonnegative integer n. Hence we note that n−1  j 1+L X L kT (f )(x) − T (f )(x)k ≤ Φ(x) 23 j=m 2 n

(3.10)

m

for all x ∈ X and n > m ∈ N. With this result in mind we consider the set Ω = {g|g : X −→ Y, g(0) = 0} and then define a generalized metric d on Ω as follows: d(g, h) = inf {λ ∈ [0, ∞] : kg(x) − h(x)k ≤ λΦ(x) for all x ∈ X} with inf ∅ = ∞. Then (S, d) is a complete generalized metric space; see Lemma 2.1 in [13]. Now we define a mapping T : Ω −→ Ω by (3.11)

T (g)(x) =

1 (g(2x) + g(x + σ(x))) 23

for all x ∈ X. We, then, will show that T is strictly contractive on Ω. Given g, h ∈ Ω, let λ ∈ [0, ∞] be a constant with d(g, h) ≤ λ. Then we have kg(x) − h(x)k ≤ λΦ(x) for all x ∈ X. By the equation (3.1) we have 1 kg(2x) − h(2x) + g(x + σ(x)) − h(x + σ(x))k 23 1 1 ≤ 3 kg(2x) − h(2x)k + 3 kg(x + σ(x)) − h(x + σ(x))k 2 2 λ λ 1 ≤ 3 Φ(2x) + 3 Φ(x + σ(x)) ≤ Lλ ≤ Lλ 2 2 2

kT (g)(x) − T (h)(x)k =

for all x ∈ G, which implies d(T (g), T (h)) ≤ Lλ. Therefore we may conclude that d(T (g), T (h)) ≤ Ld(g, h) for any g, h ∈ Ω. Since L is a constant with 0 < L < 1, T is strictly contractive as claimed. Also the inequality (3.8) implies that (3.12)

d(T (f ), f ) ≤

1 (1 + L) < ∞. 23

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By the Alternative of Fixed Point as we introduced in the Introduction Section, there exists a mapping C : X −→ Y which is a fixed point of T such that d(T n (f ), C) → 0 as n → ∞, that is, C(x) = lim T n (f )(x) n→∞

for all x ∈ X. Then we will show that C is cubic and it would not be hard if we recall the approximation inequality (3.2) for f where we let x = 2n x, y = 2n y and x = 2n−1 (x + σ(x)), y = 2n−1 (y + σ(y)), respectably, as follows; kC(2x + y) − C(x + 2y) + 2C(x + σ(y)) − 9C(x) + 9C(y)k 1 2n − 1 n n φ(2 x, 2 y) + lim φ(2n−1 (x + σ(x)), 2n−1 (y + σ(y))) n→∞ 23n n→∞ 23n (2L)n (2n − 1)(2L)n ≤ lim φ(x, y) + lim φ(x, y) 3n n→∞ 2 n→∞ 23n  n L φ(x, y) = 0 = lim n→∞ 2

≤ lim

for all x, y ∈ X, which implies that C is cubic. By the Alternative of Fixed Point theorem and the inequality (3.12) we get d(f, C) ≤

1+L 1 d(f, T (f )) ≤ 3 . 1−L 2 (1 − L)

Hence the inequality (3.3) is true for all x ∈ X. By the uniqueness of the fixed point of T , the cubic function C should be unique, which completes the proof.



Let us give the classical Cauchy difference type stability of the generalized cubic functional equation (1.7) when σ = −I from Theorem 3.1 as we see the following Corollary. Corollary 3.2. Let  ≥ 0, 0 < p
µ(a) = µ(b) > µ(c). Then µ is a fuzzy subalgebra of X. Proposition 3.3. Let µ be a fuzzy subalgebra of a BI-algebra X. Then µ(0) ≥ µ(x) for all x ∈ X. Proof. By (B1), we have x ∗ x = 0 for all x ∈ X. Using (F0), µ(0) = µ(x ∗ x) ≥ min{µ(x), µ(x)} = µ(x) for all x ∈ X. We denote a notation

□ ∏n

x ∗ x by

∏n

x ∗ x := x ∗ (x ∗ (x ∗ (· · · ∗ (x ∗ x)) · · · ) for any natural number n. | {z } n

Proposition 3.4. Let µ be a fuzzy subalgebra of a BI-algebra X and let n ∈ N. Then ∏n (i) µ( x ∗ x) ≥ µ(x) whenever n is odd, ∏n (ii) µ( x ∗ x) = µ(x) whenever n is even. Proof. Let x ∈ X and n be an odd natural number. Then n = 2k − 1 for some positive integer k. Then ∏2k−1 ∏2k−1 ∏2k+1 ∏2(k+1)−1 x ∗ x) ≥ µ(x) which proves (i). x ∗ (x ∗ (x ∗ x))) = µ( x ∗ x) = µ( x ∗ x) = µ( µ( □

Similarly we can prove the second part, but we omit it. Definition 3.5. A fuzzy set µ in a BI-algebra X is said to be fuzzy normal if it satisfies the inequality µ((x ∗ a) ∗ (y ∗ b)) ≥ min{µ(x ∗ y), µ(a ∗ b)} for all a, b, x, y ∈X. Example 3.6. Let X := {0, 1, 2, 3} be a BI-algebra [1] set with the following table: ∗ 0 1 2 3

0 0 1 2 3

1 0 0 2 3

2 0 1 0 3

3 0 1 2 0

Define a fuzzy set µ : X → [0, 1] by µ(0) > µ(1) > µ(2) = µ(3). Then it easy to see that µ is fuzzy normal of X. Theorem 3.7 Every fuzzy normal set µ in a BI-algebra X is a fuzzy subalgebra of X. Proof. Let x, y ∈ X. Since µ is fuzzy normal, we have µ(x ∗ y) = µ((x ∗ y) ∗ (0 ∗ 0)) ≥ min{µ(x ∗ 0), µ(y ∗ 0)} = □

min{µ(x), µ(y)}, which shows that µ is a fuzzy subalgebra of X. The converse of Theorem 3.7 may not be true in general.

Example 3.8. Consider a BI-algebra X = {0, a, b, c} and a fuzzy set µ as in Example 3.2. Then µ is a fuzzy subalgebra of X, but not fuzzy normal, since µ((c ∗ b) ∗ (c ∗ c)) = µ(c) ≱ µ(b) = min{µ(c ∗ c), µ(b ∗ c)}.

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Y. Cui and S. S. Ahn Definition 3.9. A fuzzy set µ in a BI-algebra X is called a fuzzy normal subalgebra of X if it is both a fuzzy subalgebra and a fuzzy normal subset of X. Example 3.10. Consider a BI-algebra X = {0, 1, 2, 3} as in Example 3.6. Define a fuzzy set ν : X → [0, 1] by { 0.7 if x ∈ {0, 1}, ν(x) := 0.3 if x ∈ {2, 3}. It is easy to show that ν is a fuzzy normal subalgebra of X. Proposition 3.11. If a fuzzy set µ in a BI-algebra X is fuzzy normal, then µ(x ∗ y) = µ(y ∗ x) for all x, y ∈ X. Proof. Let x, y ∈ X. Using Proposition 3.3, we have µ(x∗y) = µ((x∗y)∗(x∗x)) ≥ min{µ(x∗x), µ(y ∗x)} = µ(y ∗x). Interchanging x with y, we obtain µ(y ∗ x) ≥ µ(x ∗ y), which proves the proposition.

□

Theorem 3.12. Let µ be a fuzzy normal BI-algebra X. Then the set Xµ := {x ∈ X|µ(x) = µ(0)} is a normal subalgebra of X. Proof. Let a, b, x, y ∈ X be such that x ∗ y ∈ Xµ and a ∗ b ∈ Xµ . Then µ(x ∗ y) = µ(0) = µ(a ∗ b). Since µ is fuzzy normal, we have µ((x ∗ a) ∗ (y ∗ b)) ≥ min{µ(x ∗ y), µ(a ∗ b)} = µ(0). It follows from Proposition 3.3 that µ((x ∗ a) ∗ (y ∗ b)) = µ(0). Hence (x ∗ a) ∗ (y ∗ b) ∈ Xµ . This completes the proof.

□

Theorem 3.13. The intersection of a family of fuzzy normal subalgebras of a BI-algebra X is also a fuzzy normal subalgebra of X. Proof. Let {µα |α ∈ Λ} be a family of fuzzy normal subalgebras and let a, b, x, y ∈ X. Then ∩α∈Λ µα ((x ∗ a) ∗ (y ∗ b)) = inf µα ((x ∗ a) ∗ (y ∗ b)) α∈Λ

≥ inf {min{µα (x ∗ y), µα (a ∗ b)}} α∈Λ

= min{ inf µα (x ∗ y), inf µα (a ∗ b)} α∈Λ

α∈Λ

= min{∩α∈Λ µα (x ∗ y), ∩α∈Λ µα (a ∗ b)} which shows that ∩α∈Λ µα is fuzzy normal of X. By Proposition 3.7, we know that ∩α∈Λ µα is a fuzzy normal □

subalgebra of X.

Suppose that µ is a fuzzy normal subalgebra of a BI-algebra X. Define a binary relation “ ∼µ ” on X by putting x ∼µ y if and only if µ(x ∗ y) = µ(0) for any x, y ∈ X. Lemma 3.14. The relation ∼µ is an equivalence relation on a BI-algebra X. Proof. Using (B1), µ(x ∗ x) = µ(0) and so x ∼µ x, which means ∼µ is reflexive. Suppose that x ∼µ y for any x, y ∈ X. Then µ(0) = µ(x ∗ y). By Proposition 3.11, µ(y ∗ x) = µ(0). So y ∼µ x, which means ∼µ is symmetric.

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Two quotient BI-algebras Suppose that x ∼µ y and y ∼µ z for any x, y, z ∈ X. Then µ(x ∗ y) = µ(0), µ(y ∗ z) = µ(0) = µ(z ∗ y) and µ(x ∗ z) =µ((x ∗ z) ∗ 0) = µ((x ∗ z) ∗ (y ∗ y)) ≥ min{µ(x ∗ y), µ(z ∗ y)} = min{µ(0), µ(0)} = µ(0). Also since µ(0) ≥ µ(x) for all x ∈ X, µ(0) ≥ µ(x ∗ z) and so µ(x ∗ z) = µ(0). Hence x ∼µ z. Therefore ∼µ is an □

equivalence relation on a BI-algebra X. Lemma 3.15. For all x, y, z in a BI-algebra X, x ∼µ y implies x ∗ z ∼µ y ∗ z and z ∗ x ∼µ z ∗ y. Proof. Let x ∼µ y. Then µ(x ∗ y) = µ(0). Since x ∗ x = 0 and µ(0) ≥ µ(x) for all x ∈ X, we have µ((x ∗ z) ∗ (y ∗ z)) ≥ min{µ(x ∗ y), µ(z ∗ z)} = min{µ(0), µ(0)} = µ(0).

Since µ(0) ≥ µ(x) for all x ∈ X, µ(0) ≥ µ((x ∗ z) ∗ (y ∗ z)). Therefore µ(0) = µ((x ∗ z) ∗ (y ∗ z)), so x ∗ z ∼µ y ∗ z. By a similar way, we can prove that z ∗ x ∼µ z ∗ y. The proof is complete.

□

Lemma 3.16. Let X be a BI-algebra. For any x, y, z, w ∈ X, x ∼µ y and z ∼µ w imply x ∗ z ∼µ y ∗ w. Proof. Let x ∼µ y and z ∼µ w for any x, y, z, w ∈ X. Then µ(x ∗ y) = µ(0) and µ(z ∗ w) = µ(0). Hence µ((x ∗ z) ∗ (y ∗ w)) ≥ min{µ(x ∗ y), µ(z ∗ w)} = min{µ(0), µ(0)} = µ(0). Since µ(0) ≥ µ(x) for all x ∈ X, µ(0) ≥ µ((x ∗ z) ∗ (y ∗ w)). Thus µ(0) = µ((x ∗ z) ∗ (y ∗ w)), so x ∗ z ∼µ y ∗ w. The proof is complete.

□

The above Lemmas 3.14, 3.15 and 3.16 give the following theorem. Theorem 3.17. The relation “ ∼µ ” is a congruence relation on a BI-algebra X. Denote by µx the equivalence class containing x, and let X/µ be the set of all equivalence classes with respect to ∼µ , that is, µx = {y ∈ X|y ∼µ x} and X/µ = {µx |x ∈ X}. Now we define a binary operation “ ∗ ” in X/µ by putting µx ∗ µy := µx∗y . Theorem 3.17 guarantees that this operation is well defined. Theorem 3.18. Let µ be a fuzzy normal subalgebra in a BI1 -algebra X. Then (X/µ, ∗, µ0 ) is a BI1 -algebra. Proof. Let µx , µy , µz ∈ X/µ. Then µx ∗ µx = µx∗x = µ0 and µx = µx∗(y∗x) = µx ∗ µy∗x = µx ∗ (µy ∗ µx ). If µx ∗ µy = µ0 and µy ∗ µx = µ0 , then µx∗y = µ0 = µy∗x and so x ∗ y = 0 = y ∗ x. Hence x = y and therefore µx = µy . Thus (X/µ, ∗, µ0 ) is a BI1 -algebra.

□

Corollary 3.19. Let µ be a fuzzy normal subalgebra in a BI-algebra. Then (X/µ; ∗, µ0 ) is a BI-algebra. This algebra X/µ is called the quotient BI-algebra of a BI-algebra X induced by a fuzzy normal subalgebra µ. If µ is a fuzzy normal subalgebra in a BI-algebra X, then the set Xµ := {x ∈ X|µ(x) = µ(0)} is a normal subalgebra of X. Theorem 3.20. Let µ be a fuzzy normal subalgebra of a BI-algebra X. The mapping γ : X → X/µ, given by γ(x) = µx , is a surjective homomorphism, and kerγ = {x ∈ X|γ(x) = µ0 } = Xµ .

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Y. Cui and S. S. Ahn Proof. Let µx ∈ X/µ. Then there exists an element x0 ∈ µx , so x0 ∈ X such that γ(x0 ) = µx . Hence γ is surjective. For any x, y ∈ X, γ(x ∗ y) = µx∗y = µx ∗ µy = γ(x) ∗ γ(y). Thus γ is a homomorphism. And kerγ = {x ∈ X|γ(x) = µ0 } = {x ∈ X|x ∼µ 0} = {x ∈ X|µ(x) = µ(0)} = Xµ .

□

Let X, Y be BI-algebras. If we define (x1 , y1 ) ∗ (x2 , y2 ) := (x1 ∗ x2 , y1 ∗ y2 ) in X × Y , then (X × Y, ∗, (0, 0)) becomes a BI-algebra, and we call it a product BI-algebra. Theorem 3.21. Let µ (resp., ν) be a fuzzy normal subalgebra in a BI-algebra X (resp., Y ). If we define (µ × ν)(x, y) := min{µ(x), ν(x)} in X × Y for x ∈ X, y ∈ Y , then µ × ν is also a fuzzy normal subalgebra in X × Y . Proof. Let µ (resp., ν) be a fuzzy normal subalgebra in X (resp., Y ). Then (µ × ν)((x1 , y1 )∗(x2 , y2 )) = (µ × ν)(x1 ∗ x2 , y1 ∗ y2 ) = min{µ(x1 ∗ x2 ), ν(y1 ∗ y2 )} ≥ min{min{µ(x1 ), µ(x2 )}, min{ν(y1 ), ν(y2 )}} = min{min{µ(x1 ), ν(y1 )}, min{µ(x2 ), ν(y2 )}} = min{(µ × ν)(x1 , y1 ), (µ × ν)(x2 , y2 )} for any (x1 , y1 ), (x2 , y2 ) ∈ X × Y . Hence µ × ν is a fuzzy subalgebra of X × Y . And (µ × ν)(((x1 , y1 ) ∗ (a1 , b1 )) ∗ ((x2 , y2 ) ∗ (a2 , b2 ))) =(µ × ν)((x1 ∗ a1 , y1 ∗ b1 ) ∗ (x2 ∗ a2 , y2 ∗ b2 )) =(µ × ν)((x1 ∗ a1 ) ∗ (x2 ∗ a2 ), (y1 ∗ b1 ) ∗ (y2 ∗ b2 )) = min{µ((x1 ∗ a1 ) ∗ (x2 ∗ a2 )), ν((y1 ∗ b1 ) ∗ (y2 ∗ b2 ))} ≥ min{min{µ(x1 ∗ x2 ), µ(a1 ∗ a2 )}, min{ν(y1 ∗ y2 ), ν(b1 ∗ b2 )}} = min{min{µ(x1 ∗ x2 ), ν(y1 ∗ y2 )}, min{µ(a1 ∗ a2 ), ν(b1 ∗ b2 )}} = min{(µ × ν)((x1 ∗ x2 ), (y1 ∗ y2 )), (µ × ν)((a1 ∗ a2 ), (b1 ∗ b2 ))} = min{(µ × ν)((x1 , y1 ) ∗ (x2 , y2 )), (µ × ν)((a1 , b1 ) ∗ (a2 , b2 ))}. So µ × ν is fuzzy normal. Therefore µ × ν is also a fuzzy normal subalgebra of X × Y .

□

Proposition 3.22. Let µ be a fuzzy normal subalgebra of a BI-algebra X. If J is a normal subalgebra of X, then J/µ is a normal subalgebra of X/µ. Proof. Let µ be a fuzzy normal subalgebra of X and J be a normal subalgebra of X. Then for any x, y ∈ J, x ∗ y ∈ J. Let µx , µy ∈ J/µ. Then µx ∗ µy = µx∗y ∈ J/µ. So J/µ = {µx |x ∈ J} is a subalgebra of X/µ. For any x ∗ y, a ∗ b ∈ J, (x ∗ a) ∗ (y ∗ b) ∈ J. For any µx ∗ µy , µa ∗ µb ∈ J/µ, we have (µx ∗ µa ) ∗ (µy ∗ µb ) =µx∗a ∗ µy∗b =µ(x∗a)∗(y∗b) ∈ J/µ. □

Hence J/µ is a normal subalgebra of X/µ.

Theorem 3.23. If J ∗ is a normal subalgebra of X/µ, then there exists a normal subalgebra J = ∪{x ∈ X|µx ∈ J ∗ } in X such that J/µ = J ∗ .

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Two quotient BI-algebras Proof. Since J ∗ is a normal subalgebra of X/µ, we have µx ∗ µy = µx∗y ∈ J ∗ for any µx , µy ∈ J ∗ . Hence x ∗ y ∈ J for any x, y ∈ J. And µx∗a ∗ µy∗b = µ(x∗a)∗(y∗b) ∈ J ∗ for any µx∗y , µa∗b ∈ J ∗ . Therefore (x ∗ a) ∗ (y ∗ b) ∈ J for any x ∗ y, a ∗ b ∈ J. Thus J is a normal subalgebra of X. By Theorem 3.20, J/µ ={µj |j ∈ J} ={µj |∃µx ∈ J ∗ such that j ∼µ x} ={µj |∃µx ∈ J ∗ such that µx = µj } ={µj |µj ∈ J ∗ } = J ∗ . □

This completes the proof.

4. Quotient BI-algebras induced by fuzzy congruence relations Definition 4.1. [10] A binary operation θ from X × X → [0, 1] is a fuzzy equivalence relation on X if for all x, y, z, u ∈ X (FC1) θ(x, x) = sup{θ(y, z)|y, z ∈ X} = θ(0, 0), (FC2) θ(x, y) = θ(y, x), (FC3) θ(x, z) ≥ min{θ(x, y), θ(y, z)}. Moreover, if it satisfies (FC4) θ(x ∗ u, y ∗ u) ≥ θ(x, y), θ(u ∗ x, u ∗ y) ≥ θ(x, y) for all x, y, u ∈ X, we say that θ is a fuzzy congruence relation on (X, ∗, 0). Let F Co(X) be the set of all fuzzy congruence relations on a BI-algebra X. Lemma 4.2. If θ satisfies the condition (FC2) ∼ (FC4) above, then (FC1) is equivalent to θ(0, 0) ≥ θ(x, y) for all x, y ∈ X. Proof. Suppose that θ(0, 0) = θ(x, x). By (FC2) and (FC3), we have θ(0, 0) = θ(x, x) ≥ min{θ(x, y), θ(y, x)} = θ(x, y) for all x, y ∈ X. Conversely, assume that θ(0, 0) ≥ θ(x, y) for all x, y ∈ X. It follows from (FC4) that θ(0, 0) ≤ θ(x ∗ 0, x ∗ 0) = □

θ(x, x) By assumption, we have θ(0, 0) = θ(x, x). Hence (FC1) holds.

Proposition 4.3. Let θ be a fuzzy congruence relation on a BI-algebra X. Then θ(x, y) = θ(x ∗ y, 0) for all x, y ∈ X. Proof. By (FC4) and Lemma 4.2, we have min{θ(x, y), θ(y, y)} = min{θ(x, y), θ(0, 0)} = θ(x, y) ≤ θ(x ∗ y, y ∗ y) = θ(x ∗ y, 0) for all x, y ∈ X. On the other hand, θ(x ∗ y, 0) = θ(x ∗ y, x ∗ x) ≥ θ(y, x). Hence θ(x, y) = θ(x ∗ y, 0)

□

For every element x ∈ X, we define θx := {y ∈ X|θ(x, y) = θ(0, 0)} of X and X/θ := {θx |x ∈ X}. Define an operation “ • ” on the set X/θ by θx • θy := θx∗y .

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Y. Cui and S. S. Ahn This operation is well defined. In fact, if θx = θx′ and θy = θy′ , then we have θ(x, x′ ) = θ(y, y ′ ) = θ(0, 0). Since θ(0, 0) = min{θ(x, x′ ), θ(y, y ′ )} ≤ θ(x ∗ y, x′ ∗ y ′ ) ≤ θ(0, 0), we have θ(x ∗ y, x′ ∗ y ′ ) = θ(0, 0) and so θx∗y = θx′ ∗y′ . Hence • is well defined. Theorem 4.4. If θ ∈ F Co(X), where X is a BI-algebra, then (X/θ, •, θ0 ) is a BI-algebra. □

Proof. Straightforward.

Proposition 4.5. Let f : X → Y be a homomorphism of BI-algebras. If θ is a fuzzy congruence relation of Y , ¯ y) := θ(f (x), f (y)) is a fuzzy congruence relation of X. then θ(x, Proof. It is obvious that θ¯ is well-defined. Let x, y, z, u ∈ X. Then ¯ x) = θ(f (x), f (x)) = θ(0, 0). (i) θ(x, ¯ y) = θ(f (x), f (y)) = θ(f (y), f (x)) = θ(y, ¯ x). (ii) θ(x, ¯ z) = θ(f (x), f (z)) ≥ min{θ(f (x), f (y)), θ(f (y), f (z))} = min{θ(x, ¯ y), θ(y, ¯ z)}. (iii) It can be shown that θ(x, ¯ ∗ u, y ∗ u) = θ(f (x ∗ u), f (y ∗ u)) = θ(f (x) ∗ f (u), f (y)∗ f (u)) ≥ θ(f (x), f (y)) = θ(x, ¯ y). (iv) It can be shown that θ(x ¯ ∗ x, u ∗ y) ≥ θ(x, ¯ y). Thus θ¯ is a fuzzy congruence relation. By a similar way, we have θ(u □ Proposition 4.6. Let θ be a fuzzy congruence relation of a BI-algebra X. Then the mapping γ : X → X/θ, given by γ(x) := θx , is a surjective homomorphism. Proof. Let θx ∈ X/θ. Then there exists an element x0 ∈ θx such that γ(x0 ) = θx . Hence γ is surjective. For any x, y ∈ X, γ(x ∗ y) = θx∗y = θx • θy = γ(x) • γ(y). Thus γ is a homomorphism.

□

Theorem 4.7. Let f : (X, ∗, 0X ) → (Y, ∗, 0Y ) be an epimorphism of BI1 -algebras and let θ be a fuzzy congruence relation of Y . If θ¯ = θ ◦ f , then the quotient algebra X/θ¯ := (X/(θ ◦ f ), •X , θ¯0 ) is isomorphic to the quotient X

algebra Y /θ := (Y /θ, •Y , θ0Y ). Proof. By Theorem 4.4 and Proposition 4.5, X/(θ ◦ f ) := (X/(θ ◦ f ), •X , θ¯0X ) is a BI-algebra and Y /θ := (Y /θ, •Y , θ0Y ) is a BI-algebra. Define a map η : X/(θ ◦ f ) → Y /θ, (θ ◦ f )x 7→ θf (x) for all x ∈ X. Then the function η is well-defined. In fact, assume that (θ ◦ f )x = (θ ◦ f )y for all x, y ∈ X. Then we have θ(f (x) ∗Y f (y)) = θ(f (x ∗X y)) = (θ ◦ f )(x ∗X y) = (θ ◦ f )(0X ) = θ(f (0X )) = θ(0Y ) and θ(f (y) ∗Y f (x)) = θ(f (y ∗X x)) = (θ ◦ f )(y ∗X x) = (θ ◦ f )(0X ) = θ(f (0X )) = θ(0Y ). Hence θf (x) = θf (y) . For any (θ ◦ f )x , (θ ◦ f )y ∈ X/(θ ◦ f ), we have η((θ ◦ f )x •X (θ ◦ f )y ) = η((θ ◦ f )x∗y ) = θf (x∗X y) = θf (x)∗Y f (y) = θf (x) • θf (y) = η((θ ◦ fx )) •Y η((θ ◦ f )y ). Therefore η is a homomorphism. Let θa ∈ Y /θ. Then there exists x ∈ X such that f (x) = a, since f is surjective. Hence η((θ ◦ f )x ) = θf (x) = θa and so η is surjective. Let x, y ∈ X be such that θf (x) = θf (y) . Then we have (θ◦f )(x∗X y) = θ(f (x∗X y)) = θ(f (x)∗Y f (y)) = θ(0Y ) = θ(f (0X )) = (θ ◦ f )(0X ) and (θ ◦ f )(y ∗X x) = θ(f (y ∗X x)) = θ(f (y) ∗Y f (x)) = θ(0Y ) = θ(f (0X )) = (θ ◦ f )(0X ). It follows that (θ ◦ f )x = (θ ◦ f )y . Thus η is injective. This completes the proof.

254

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Two quotient BI-algebras The homomorphism π : X → X/θ, x → θx , is called the natural homomorphism of X onto X/θ. In Theorem 4.7, if we define natural homomorphisms πX : X → X/θ ◦ f and πY : Y → Y /θ, then it is easy to show that η ◦ πX = πY ◦ f , i.e., the following diagram commutes. X   πX y

f

−−−−→

Y   πY y

η

X/(θ ◦ f ) −−−−→ Y /θ. The fuzzy subset θx of a BI-algebra X, which is defined by θx (y) := θ(x, y), is called the fuzzy congruence class containing x and X/θ is the set of all fuzzy congruences classes θx . Proposition 4.8. Let θ be a fuzzy congruence relation in a BI-algebra X. Then θ0 is a fuzzy ideal of X. Proof. Let x, y ∈ X. Then θ0 (0) = θ(0, 0) ≥ θ(x, y) by Lemma 4.2. Put y := 0 in above inequality. Then θ0 (0) ≥ θ(x, 0) = θ0 (x). By (FC3), (FC2) and Proposition 5.3, we have θ0 (y) = θ(0, y) ≥ min{θ(0, x), θ(x, y)} = min{θ(x, 0), θ(x ∗ y, 0)} = min{θ0 (x), θ0 (x ∗ y)}. Thus θ0 is a fuzzy ideal of X.

□

References [1] S. S. Ahn, J. M. Ko and A. B. Saeid, Normal subalgebras of BI-algebras and its analytic constructions, (submitted). [2] G. Dymek and A. Walendziak, (Fuzzy) ideals of BN -algebras, The Scientific World Journal 2015 (2015), Article ID 925049. [3] M. Khan, F. Feng and M. N. A. Khan, On minimal fuzzy ideals of semigroups, Journal of Mathematics 2013 (2013), Article ID 475190. [4] H. S. Kim, C. B. Kim and K. S. So, Radical structures of fuzzy polynomial ideals in a ring, Discrete Dynamics in Nature and Society 2016 (2016), Article ID: 782178. [5] H. V. Kumbhojar, Proper fuzzification of prime ideals of a hemiring, Advances in fuzzy systems 2012 (2012), Article ID: 801650. [6] T. Kuraoka and N. Kuroki, On fuzzy quotient rings induced by fuzzy ideals, Fuzzy Sets and Systems 47 (1992), 381-386. [7] W. J. Liu, Fuzzy invariant subgroups and fuzzy ideals, Fuzzy sets and Systems 8 (1982), 133-139. [8] B. L. Meng and X. L. Xin, On fuzzy ideals of BL-algebras, The Scientific World Journal 2014 (2014), Article ID 757382. [9] T. K. Mukherjee and M. K. Sen, On fuzzy ideals of a ring 1, Fuzzy Sets and Systems 21 (1987), 99-104. [10] V. Murali, Fuzzy congruences relations, Fuzzy Sets and Systems 30 (1989), 155-163. [11] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971), 512-517. [12] A. B. Saeid, H. S. Kim and A. Rezaei, On BI-algebras, An. S¸t. Univ. Ovidius Constant¸a 25 (2017), 177-194. [13] S. Z. Song, Y. B. Jun and H. S. Kim, Characterizations of positive implicative superior ideals induced by superior mappings, J. Computational Anal. and Appl. 25 (2018), 634-643. [14] L. A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965), 338-353.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.2, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

General quadratic functional equations in quasi-β-normed spaces: solution, superstability and stability Shahrokh Farhadabadi1∗ , Choonkil Park2∗ and Sungsik Yun3∗ 1

Young Researchers and Elite Club, Parand Brunch, Islamic Azad University, Parand, Iran 2 Research Institute for Natural Sciences Hanyang University, Seoul 04763, Korea 3 Department of Financial Mathematics, Hanshin University, Gyeonggi-do 18101, Korea e-mail: shahrokh [email protected]; [email protected]; [email protected]

Abstract. Let f : X → Y be a mapping where X is a quasi-α-normed space and Y is a quasi-β-normed space. The following quadratic functional equation n n X X

f

i=1

j=1 j6=i

xj +

2−n xi 2



=

n n2 X f (xi ), 4

(n ≥ 3)

(0.1)

i=1

is introduced and solved by giving its general solution. Moreover, we prove the Hyers-Ulam stability of the functional equation (0.1) by using a direct method.

1. Introduction and preliminaries Studying functional equations by focusing on their approximate and exact solutions conduces to one of the most substantial significant study brunches in functional equations, what we call “the theory of stability of functional equations”. This theory specifically analyzes relationships between approximate and exact solutions of functional equations. Actually a functional equation is considered to be stable if one can find an exact solution for any approximate solution of that certain functional equation. Another related and close term in this area is superstability, which has a similar nature and concept to the stability problem. As a matter of fact, superstability for a given functional equation occurs when any approximate solution is an exact solution too. In such this situation the functional equation is called superstable. In 1940, the most preliminary form of stability problems was proposed by Ulam [35]. He gave a talk and asked the following: “when and under what conditions does an exact solution of a functional equation near an approximately solution of that exist?” In 1941, this question that today is considered as the source of the stability theory, was formulated and solved by Hyers [13] for the Cauchy’s functional equation in Banach spaces. Then the result of Hyers was generalized by Aoki [1] for additive mappings and by Rassias [24] for linear mappings by considering an unbounded Cauchy difference. In 1994, G˘ avrut¸a [12] provided a further generalization of Rassias’ theorem in which he replaced the unbounded Cauchy difference by a general control function for the existence of a unique linear mapping. For more epochal information and various aspects about the stability of functional equations theory, we refer the reader to the monographs [10, 11, 14, 15, 18, 20, 25, 27, 29, 30, 31, 32, 33], which also include many interesting results concerning the stability of different functional equations in many various spaces. Now we present some brief explanations about the functional equation (0.1) and also generally about quadratic functional equations. Consider the functional equation f (x + y) + f (x − y) = 2f (x) + 2f (y)

(1.1)

0

2010 Mathematics Subject Classification: 39B52, 39B72, 46Bxx, 39Bxx. Keywords: Hyers-Ulam stability; functional equation; quadratic functional equation; superstability; direct method. ∗ Corresponding authors. 0

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S. Farhadabadi, C. Park, S. Yun which is called the classic quadratic functional equation. Obviously, the function f (x) = cx2 is its solution and so it is called quadratic. There are some other different types of quadratic functional equations. For examples, the following n-dimensional quadratic functional equations n h k k+1 X X X k=2

n X

···

i1 =2 i2 =i1 +1

f

n X



n−k+1

X

xi −

i=1 i6=i1 ,··· ,in−k+1

in−k+1 =in−k +1

xir

i

r=1

+f

n X 

xi

= 2n−1

n X 

X

xi +

f (xi − xj ) = n

1≤i 0 there exist if x 2 F ; k 2 N ; s1 ; s2 2 (tk Lemma 5 For

is given by

1 ; tk ], and js1

s2 j < ; x(s1)

2 P C(J; R), the solution of the following ISFDEs 8 c q < ( D + c Dq 1 )x(t) = (t); xjt=tk = 'k (x(tk )); x0 jt=tk = 'k (x(tk )); 0 0 : 1 x(0) + 1 x (0) = 1 ; 2 x(T ) + 2 x (T ) = 2 ;

x (t) =

Z

t (t s) q 1

e

I

(s)ds + v1 (t)

0

+

(2) k = 1; :::; p;

T (T

e

s) q 1

I

(s)ds

(3)

0

+ v2 (t)I q p X

Z

> 0 such that

x(s2 ) < ":

1

(T ) + v3 (t)

z1j (t) 'j (x(tj )) +

j=1

p X

'j (x(tj )) + v4 (t)

j=1 p X

p X

'j (x(tj )

k=1 p X

z2j (t) 'j (x(tj ))

j=k+1

'j (x(tj )) + z3 (t) ;

j=k+1

t 2 [tk ; tk+1 ) ; k = 0; 1; :::; p; where =( v1 (t) = v2 (t) =

1)

1 1e

t

1e

t

T

2e

2 1

+

1

(

1

+

1

2

t

1

2e

T

e

1 2

e

t

1

2e

T

v4 (t) =

tj

e

t

t

z2;j (t) = e z3 (t) =

e

1 t

e

tj

2

2

( 1

1)

1

1

6= 0;

;

;

1 2

e

2)

2

v3 (t) =

z1;j (t) =

T

2e

+e

2e

T

2e

T

tj

(

; ; 1)

1

2e

T

2e

T

;

; 2e

T

T

2e

!

1

e

t

1

(

1

1)

2:

3

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Proof. Assume that x is a solution of (c Dq +

c

Dq

1

)x(t) = (t);

on (tk ; tk+1 ]; (k = 1; 2 : : : ; p). Applying the operator Iq get Iq

1 c

( Dq +

c

Dq

1

1

operator to both sides of the above equation, we

)x(t) = Iq

1

(t);

(D + ) x(t) = c0 + Iq

1

(t):

This can be expressed as e t ((D + ) x (t)) = e

t

c0 + Iq

1

(t) ;

Solving the above equation, we see that the general solution of (1) on each interval (tk ; tk+1 ]; (k = 1; 2 : : : p); can be written as Z t x(t) = e t Ak + Bk + e (t s) Iq 1 (s)ds; t 2 J: 0

Next, solving the obtained linear equation on J0 ; we get Z t x(t) = e t A0 + B0 + e (t

s) q 1

I

0

(s)ds; t 2 J0 ;

(4)

where A0 and B0 are arbitrary constants. Taking the derivative to (4), we get Z t 0 t x (t) = e A0 e (t s) Iq 1 (s)ds + I q 1 (t); t 2 J0 :

(5)

0

Now, applying the boundary condition, we have ( In general, for t 2 [tk ; tk+1 ), we …nd x(t) = e 0

x (t) =

1 ) A0

1

+

Z t Ak + Bk + e 0 Z t t e Ak e t

1 B0

=

(t s) q 1

I

(t s) q 1

I

1:

(6)

(s)ds;

(7)

(s)ds + Iq

1

(t):

0

Now, applying the boundary condition at tk+1 = T , we have 2e

T

2e

T

Ap +

2 Bp =

2

(

2)

2

Z

T

e

(T

s) q 1

I

(s)ds

2I

q 1

(T ):

(8)

0

From

x0 (tk ) = 'k (x(tk )), we have

ek tk Ak + ek 1tk Ak 1 ; 1 tk e 'k (x(tk )); k = 1; :::; p:

'k (x(tk )) = Ak Similarly, from

Ak

1

=

(9)

x(tk ) = 'k (x(tk )), we get 'k (x(tk )) = e Bk

Bk

1

tk

Ak

tk

e

= 'k (x(tk )) +

1

Ak

1

+ Bk

Bk

1;

'k (x(tk )); k = 1; :::; p:

(10)

4

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Next, it follows from (9) and (10) that Ap

p 1 X

Ak =

tj

e

'j (x(tj ));

(11)

j=k+1

Bp

Bk =

p X

p 1 X

'j (x(tj )) +

It follows that for k = 0 from ( (

1 ) Ap

1

+

1 Bp

1 ) A0

1

1

=

1

'j (x(tj ));

k = 0; 1; :::; p

1:

(12)

j=k+1

j=k+1

(

+

1 B0

1)

1

p X

e

= tj

that

1

'j (x(tj )) +

1

j=1

p X

1

'j (x(tj )) +

1

j=1

p X

'j (x(tj )):

j=1

Solving the last equation together(8), for Ap and Bp ; we get 1

Ap =

(

Z

2)

2

T (T

e

s) q 1

1 2

(s)ds +

I

Iq

1

(T )

0

p X

1 2

+

p X

1 2

'j (x(tj )) +

j=1

+

2

1

2

'j (x(tj ))

(

p X

1)

1

j=1

e

tj

'j (x(tj ))

j=1

2;

1

and (

Bp =

2e

1

+ where that

(

1) ( 2

1

T

2e

1)

1

=(

1)

T

T

2e

1

Z

2)

!

p X

s) q 1

I

(

(s)ds

1

'j (x(tj ))

j=1

!

T

T

2e

Ak = Ap +

(T

e

1)

1

0

2e

2

T

p X

e T

e

tj

1

!

T

p X

1

'j (x(tj ))

j=1

2e

(T )

!

T

1

+

(

1

1)

6= 0. Now, from the equations (11) and (12) it follows

'j (x(tj ));

j=k+1 p X

Bk = Bp

2e

'j (x(tj ))

T

2e

Iq

j=1

2e

p 1 X

tj

T

2e

2

p 1 X

'j (x(tj ))

'j (x(tj )); k = 1; :::; p

1:

j=k+1

j=k+1

So

Ak =

1

(

2)

2

Z

T (T

e

s) q 1

(s)ds +

I

1 2

Iq

1

(T )

0

+

p X

1 2

1 2

'j (x(tj )) +

j=1

+

2

'j (x(tj ))

j=1

1 1

p X

2

+

1

p X

e

tj

2

(

1

1)

p X

e

tj

'j (x(tj ))

j=1

'j (x(tj )):

j=k+1

5

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2;

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.2, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

t

Multiplying the above equation by e e

t

Ak =

e

t

1

(

Z

2)

2

, we get

T (T

e

s) q 1

t

e

(s)ds +

I

1 2

Iq

1

(T )

0

+

e

t

p X

1 2

j (x(tj ))

t

e

+

p X

1 2

j=1

+

e

t

t

e

2

1

1

t

e

+

2

j=1 p X

t

e

j (x(tj ))

2

(

p X

1)

1

e

tj

j

(x(tj ))

j=1

e

tj

j (x(tj )):

j=k+1

and (

Bk =

1

+

+

(

(

1

1

1) ( 2

1

2e

T

1)

2)

T

2e

2e

1) 2

T

p X

!

Z

T

0 p X

s) q 1

I

T

!

p X

e

(s)ds 1

'j (x(tj ))

j=1

2e

(T

e

tj

2e

( T

'j (x(tj ))

j=1

1

'j (x(tj ))

p X

1)

1

T

2e

2e

2

T

Iq !

1

p X

(T )

'j (x(tj ))

j=1

2e

T

!

1

'j (x(tj )):

j=k+1

j=k+1

Combining the last two equations, we get

6

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e

t

Ak + Bk =

t

e

1

(

Z

2)

2

T (T

e

s) q 1

t

e

(s)ds +

I

1 2

Iq

1

(T )

0

+

t

e

p X

1 2

j (x(tj ))

t

e

+

p X

1 2

j=1

t

e

2

(

j (x(tj ))

j=1

p X

1)

1

tj

e

j

(x(tj ))

j=1

+

t

e

t

e

2

1

1

(

1) ( 2

1

+

2 2)

Z

p X

t

e

e

tj

j (x(tj ))

j=k+1 T (T

e

s) q 1

I

(s)ds

0

(

1

+ +

1)

1

( (

2e

2

I

T

2e

1)

1

q 1

!

T

2e

p X

!

p X

'j (x(tj ))

j=1

'j (x(tj ))

j=1

!

T

2e

T

2e

p X

e

tj

'j (x(tj ))

2e

T

2e

j=1

T

!

1

2 p X

1

'j (x(tj ))

j=k+1

t

T

1)

1

p X

e

(T )

T

2e

1

'j (x(tj )):

j=k+1

Ak + Bk = v1 (t)

Z

T

e

(T

s) q 1

I

(s)ds + v2 (t)Iq

1

(T ) + v3 (t)

0

+ v4 (t)

p X

'j (x(tj ))

(13)

j=1

'j (x(tj ) +

j=1

p X

p X

p X

z1;j (t) 'j (x(tj )) +

j=1

p X

z2;j (t) 'j (x(tj ))

j=k+1

'j (x(tj )) + z3 (t):

j=k+1

Inserting (13) into (7), thus we obtain the desired formula (3). The converse of the lemma follows by direct computation. This completes the proof.

3

Main results

This section deals with the existence and uniqueness of solutions for the problem (1). Before stating and proving the main results, we introduce the following hypotheses. (H1 ) the function f : J

R ! R is jointly continuous .

(H2 ) there exists a constant Lf > 0 such that jf (t; x)

f (t; y)j

Lf jx

yj ;

t 2 J; x; y 2 R:

7

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(H3 ) There exist a positive constants L' ; L' ; M' ; M' such that j'k (x)

'k (y)j

L' jx

yj ; j'k (x)

'k (y)j

L' jx

yj ; j'k (x)j

M' ; j'k (x)j

M' :

From (G1 )-(G3 ) it follows that jf (t; x)j j'k (x)j

Lf jxj + Mf ; t 2 J; x 2 R; Mf := sup fjf (t; 0)j : 0 < t L' jxj + M' ; j'k (x)j L' jxj + M' :

Tg;

Theorem 6 Suppose that (H1 ), (H2 ) and (H3 ) hold. If Tq

LT :=

1

(q)

1

T

e

(1 + k 1 k) +

Tq 1 k 2 k Lf (q)

(14)

+ (1 + k 3 k) pL' + (kv4 k + kz1j k + kz2j k) pL' < 1; then the equation (1) has a unique solution on J. Proof. In view of Lemma 5, we can transform problem (1) into a …xed point problem. Consider the operator T : P C (J; R) ! P C (J; R) de…ned by Z t Z T (Tx)(t) := e (t s) Iq 1 f (s; x (s))ds + v1 (t) e (T s) Iq 1 f (s; x (s))ds (15) 0

0

+ v2 (t)I

q 1

f (T; x (T )) + v3 (t)

p X

'j (x(tj )) + v4 (t)

j=1

+

p X

z1j (t) 'j (x(tj )) +

j=1

p X

z2j (t) 'j (x(tj ))

j=k+1

; t 2 Jk ; k = 0; 1; :::; p:

p X

j=1 p X

'j (x(tj ) 'j (x(tj )) + z3 (t)

j=k+1

It is obvious that T is well de…ned due to (H1 ) and sends P C (J; R) into itself. Step 1. T maps Br = fx 2 P C ([0; T ] ; R) ; kxk rg into itself for some r > 0: Let r > (1

LT )

1

Tq

1

(q)

1

e

T

Tq 1 k 2 k Ef (q)

(1 + k 1 k) +

+ (1 + k 3 k) p (L' r + M' ) + (k 4 k + kz1j k + kz2j k) p (L' r + M' ) + kz3 k : For t 2 Jk ; k = 0; 1; :::; p; x 2 Br ; we have Z t 1 e j(Tx)(t)j (q 1) 0 Z T jv1 (t)j + e (q 1) 0 Z T jv2 (t)j + (T (q 1) 0 + jv4 (t)j +

p X

j=k+1

p X j=1

Z

(t s)

s

(s

q 2

)

0

(T

Z

s)

s

(s

0

q 2

s)

'j (x(tj ) +

jf ( ; x( ))j d

q 2

)

jf ( ; x( ))j d

jf (s; x(s))j ds + jv3 (t)j

p X j=1

p X j=1

ds

j'j (x(tj ))j

jz1j (t)j 'j (x(tj ))

p X

jz2j (t)j 'j (x(tj ))

ds

j=k+1

j'j (x(tj ))j + jz3 (t)j ;

8

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Thus tq

j(Tx)(t)j

1

1

(q)

t

e

(Lf r + Mf ) + jv1 (t)j

Tq

1

(q)

1

T

e

(Lf r + Mf )

Tq 1 (Lf r + Mf ) + jv3 (t)j p (L' r + M' ) + jv4 (t)j p (Lr + M' ) (q) + jz1j (t)j p (L' r + M' ) + jz2j (t)j p (L' r + M' ) + p (L' r + M' ) + jz3 (t)j + jv2 (t)j Tq

1

(q)

1

T

e

Tq 1 k 2 k (Lf r + Mf ) + (1 + k 3 k) p (L' r + M' ) (q)

(1 + k 1 k) +

+ (k 4 k + kz1j k + kz2j k) p (L' r + M' ) + kz3 k We use the following estimation in what follows Z

1

Z

t (t s)

e

(q 1) 0 Tq 1 1 e = (q)

s

q 2

(s

)

( )d

tq

ds

1

(q)

0

T

1

t

e

k kP C

(16)

k kP C ; 2 P C (J; R)

We obtain that Tq

j(Tx)(t)j

1

(q)

1

T

e

Tq 1 k 2 k (Lf r + Mf ) + (1 + k 3 k) p (L' r + M' ) (q)

(1 + k 1 k) +

+ (k 4 k + kz1j k + kz2j k) p (L' r + M' ) + kz3 k < r: This implies that Tx 2 Br : Thus TBr Br : Step 2. T is a contraction operator on P C (J; R). Let x; y 2 Br . Then For each t 2 J , we have j(Tx)(t)

(Ty)(t)j :=

Z

t

e

(t s) q 1

I

f (s; x (s))ds + v1 (t)

0

Z

T

e

(T

s) q 1

I

+ v2 (t)I q

1

f (T; x (T )) + v3 (t)

p X

'j (x(tj )) + v4 (t)

j=1

+

p X

z1j (t) 'j (x(tj )) +

j=1

Z

p X

p X

j=k+1 (t s) q 1

I

f (s; y (s))ds + v1 (t)

Z

T

e

(T

q 1

f (T; y (T )) + v3 (t)

p X

s) q 1

I

'j (y(tj )) + v4 (t)

j=1

j=1

'j (x(tj )) + z3 (t)

f (s; y (s))ds

0

+ v2 (t)I p X

'j (x(tj )

j=k+1

t

e

p X j=1

z2j (t) 'j (x(tj ))

0

+

f (s; x (s))ds

0

z1j (t) 'j (y(tj )) +

p X

j=k+1

z2j (t) 'j (y(tj )) +

p X

'j (y(tj )

j=1

p X

'j (y(tj )) + z3 (t) ;

j=k+1

9

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j(Tx)(t)

(Ty)(t)j :=

Z

t (t s) q 1

e

I

0

+ jv1 (t)j

Z

p X

f (s; y (s))j ds

T (T

e

s) q 1

I

0

+ jv2 (t)j I q + v4 (t)

jf (s; x (s))

1

jf (s; x (s))

jf (T; x (T ))

'j (x(tj )

f (T; y (T ))j + jv3 (t)j

'j (y(tj )) +

j=1

+

p X

j=k+1

f (s; y (s))j ds

p X j=1

jz2j j (t) 'j (x(tj ))

p X j=1

j'j (x(tj ))

jz1j j (t) 'j (x(tj ))

'j (y(tj )) +

p X

j=k+1

'j (y(tj ))j

'j (y(tj ))

j'j (x(tj ))j :

Therefore, j(Tx)(t)

Tq

(Ty)(t)j

1

(q)

1

T

e

(1 + k 1 k) +

Tq 1 k 2 k Lf (q)

+ (1 + k 3 k) pL' + (k 4 k + kz1j k + kz2j k) pL' ) kx = LT kx ykP C :

ykP C

Thus, T is a contraction mapping on P C(J; R) due to condition (14). By applying the well-known Banach’s contraction mapping we see that the operator T has a unique …xed point on P C(J; R ). Therefore, the problem (1) has a unique solution. This completes the proof. The second result is based on a known result due to Krasnoselskii. We state the Krasnoselskii theorem which is needed to prove the existence of at least one solution of (1). Theorem 7 . Let M be a closed convex and nonempty subset of a Banach space X. Let T1 , T2 be the operators such that: 1. T1 x + T2 y 2 M whenever x; y 2 M ; 2. T1 is compact and continuous; 3. T2 is a contraction mapping. Then there exists z 2 M such that z = T1 z + T2 z. Now, we replace (H2 ) into the following condition: (H4 ) jf (t; x)j

(t) for (t; x) 2 J

R where

1

2 L (J) ;

2 (0; q

1) :

Theorem 8 Suppose that (H1 ),(H3 ) and (H4 ) hold. If (1 + k 3 k) pL' + (kv4 k + kz1j k + kz2j k) pL' < 1. Then (1) has at least one solution on J. Proof. Let Br = fx 2 P C(J; R); kxkP C rg. We choose 0 1 k k 1 BTq 1 e T L r (1 + kv1 k) + @ 1 (q) q 1

T q

1

+ (1 + k 3 k) pL' + (kv4 k + kz1j k + kz2j k) pL' :

q

1 1

1

1

1

C kv2 kA

The operators T1 and T2 on Br are de…ned as:

10

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(T1 x)(t) =

Z

Z

t (t s) q 1

e

I

f (s; x (s))ds + v1 (t)

0

T (T

e

s) q 1

I

f (s; x (s))ds + v2 (t)Iq

1

f (T; x (T ));

0

and

(T2 x)(t) := v3 (t)

p X

'j (x(tj )) + v4 (t)

j=1

p X

+

p X

'j (x(tj ) +

j=1

p X

z2j (t) 'j (x(tj ))

j=k+1

p X

z1j (t) 'j (x(tj ))

j=1

'j (x(tj )); t 2 Jk ; k = 0; 1; :::; p:

j=k+1

Step 1. T1 x + T2 y 2 Br whenever x; y 2 Br : For any x; y 2 Br and t 2 Jk , using the assumption (H4 ) with the Holder inequality we get 1 (q

1)

Z

(t s)

e

0

1 (q

t

1)

1 tq (q)

Z

s

q 2

(s

)

jf ( ; x( ))j d

0

Z

t (t s)

e

0

1

1

q

e

t

0 @

q 1 )

s

(s

L

11

2

0 Z @

d A

0

k k

1

1

Z

ds

0

1

1

d A ds

jf ( ; x( ))j

;

1

1

Z

Z

T

e

(T

s)

0

s

q 2

(s

) (q

0

1)

f ( ; x( ))d

!

1 Tq (q)

ds

1 q

1

T

e

k k

1

1

L

1

:

1

and v2 (t) (q 1)

Z

T

(T

s)q

2

1 (q)

f (s; x(s))ds

0

Tq q

1

k k

1

1

L

1

:

1

Therefore,

kT1 x + T2 ykP C

k k

L

1

0

1 BT @ (q)

q

1

1

q

e 1

1

T

(1 + k 1 k) +

Tq q

1

1 1

+ ((1 + k 3 k) pM' + (k 4 k + kz1j k + kz2j k) pM'

1

r:

1

C k 2 kA

Thus, kT1 x + T2 yk r; so T1 x + T2 y 2 Br . Step 2. T1 is compact and continuous. The continuity of f implies T1 is continuous, also T1 is uniformly bounded on Br as 1 0 kT1 xkP C

k k

L

1

1 BTq @ (q)

1

q

1

e

1

1

1

T

(1 + k 1 k) +

Tq

q

1

1

1

1

1

C k 2 kA

r:

11

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For equicontinuity on [0; t1 ] ; let x 2 Br and for any s1 ; s2 2 [0; t1 ] ; s1 < s2 ; we have j(T1 x)(s2 )

(T1 x)(s1 )j =

Z

s2

Z

(s2 s)

e

s

+ v1 (s2 )

Z

Z

T (T

e

) (q

0

0

q 2

(s

s)

s

ds

q 2

(s

) (q

0

0

f ( ; x( ))d

1)

!

f ( ; x( ))d

1)

!

ds

Z T v2 (s2 ) (T s)q 2 f (s; x(s))ds + (q 1) 0 ! Z s Z s1 q 1 (s ) (s1 s) e f ( ; x( ))d ds (q 1) 0 0 ! Z s Z T q 1 (s ) v1 (s1 ) e (T s) f ( ; x( ))d ds (q 1) 0 0 Z T v2 (s1 ) (T s)q 2 f (s; x(s))ds ; + (q 1) 0

j(T1 x)(s2 )

(T1 x)(s1 )j

e

(s2 )

e

Z

(s1 )

s1

Z

s

e

0

+

Z

s2

e

(s2 s)

s1

Z

s

Z

!

q 1

) (q

1)

d Z

T

e

(T

+ jv1 (s2 )

v1 (s1 )j

+ jv2 (s2 )

v2 (s1 ) v2 (s1 )j (q 1)

s)

0

q 1

(s

) (q

0

(s

0

s

s

d

ds

ds q 1

(s

) (q

0

Z

1)

!

1)

d

!

ds

T

(T

s)q

2

ds:

0

It tends to zero as s1 ! s2 . This implies that T1 is equicontinuous on the interval [0; t1 ]. In general, for the time interval (tk ; tk+1 ], we similarly obtain the same inequality, which yields that T1 is equicontinuous on interval (tk ; tk+1 ]. Together with the P C-type Arzela-Ascoli (Lemma 4) theorem, we can conclude that T1 : Br ! Br is continuous and compact. Step 3. It is clearly that T2 is contraction mapping. Thus all the assumptions of the Krasnoselskii theorem are satis…ed. In consequence, the the Krasnoselskii theorem is applied and hence the problem (1) has at least one solution on J. Our second existence result is based on the nonlinear alternative of Leray-Schauder type. Assume that (H5 ) There exist #f 2 P C (J; R) and : R+ ! R+ continuous and nondecreasing such that jf (t; x)j

#f (t) (kxk); for all (t; x) 2 J

R;

(H6 ) There exist an number N > 0 such that N > 1: LT k#k (N ) Theorem 9 Suppose that (H1 ), (H2 ), (H5 ),(H6 ) are hold. Then our BVP in (1) has at least one solution on J: Proof. Consider the operator T : P C (J; R) ! P C (J; R) de…ned by (15). It can be easily shown that T is continuous and compact. maps bounded sets into bounded sets in P C (J; R). Repeating the same process 12

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in Step 2 of Theorem 8, we get Z t e (t s) Iq j(Tx)(t)j

1

0

+ jv2 (t)j Iq +

p X j=1

Theorem 10 Proof. Z t e

1

jf (s; x (s))j ds + jv1 (t)j

j (T )j + jv3 (t)j

jz1j (t)j 'j (x(tj )) +

(t s) q 1

I

#f (s)

+

p X j=1

jx(t)j

1

j (T )j #f (s)

jz1j (t)j 'j (x(tj )) + 0

1 BT @ (q)

j(Tx)(t)j

j=1 p X

j=k+1

j=k+1

q

1

Z

I

p X

e

(T

'j (x(tj ) p X

j=k+1

j'j (x(tj ))j + jz3 (t)j ;

s) q 1

I

#f (s)

(kxk) ds

0

p X j=1

j'j (x(tj ))j + jv4 (t)j

(1 + k 1 k) +

1

jf (s; x (s))j ds

j=1

p X

j=k+1

T

e 1

s) q 1

T

jz2j (t)j 'j (x(tj )) +

1

q

(T

e

0

jz2j (t)j 'j (x(tj )) +

(kxk) + jv3 (t)j p X

T

j'j (x(tj ))j + jv4 (t)j

(kxk) ds + jv1 (t)j

0

+ jv2 (t)j Iq

p X

Z

T q

1

p X

'j (x(tj )

j=1

j'j (x(tj ))j + jz3 (t)j ;

q

1 1

1

1

+ (1 + k 3 k) pM' + (k 4 k + kz1j k + kz2j k) pM' + kz3 k :

1

C k 2 kA k#k (kxk)

Now, construct the set = fx 2 P C (J; R) : kxk < N g :The operator T : ! P C (J; R) is continuous and completely continuous. From the choice of , there is no x 2 @ such that x = Tx, 0 1: As a consequence of the nonlinear alternative of Leray–Schauder type, we deduce that T has a …xed point x 2 @ , which implies that the problem (1) has at least one solution. This completes the proof.

4

Example

In this section we give some examples to illustrate the usefulness of our main results. Example 1. Consider the following ISFDE: 3

1

(c D 2 + 2 c D 2 )x (t) = 0:01 t2 + sin t + 1 + tan 0

x(0) + x (0) = 1 x( ) = 4

2 ; x(1)

3

LT :=

+ x (1) =

x(tk ) = 0:01

Here t 2 [0; 1]; let 1 = 1; 2 = 1; f (t; x)) = L t2 + sin t + 1 + tan 1 x : A simple calculations show that T2 1 (1 2 ( 32 )

0

1

= 1;

2

1

x(t) ; t 2 [0; 1] ;

2;

kxk 1 kxk ; x0 ( ) = 0:01 ; k = 1; 2; :::; p: 1 + kxk 4 1 + kxk = 1;

=

3 2;

= 2; T = 1;

1; 2

(17)

= 0; L' ; L' ; = 0:01;

! 3 2 1 1 e 2 ) (1 + 2:312) + 2:312 0:01 + (1 + 1:312) 0:01 + (0:656 + 1:152 + 0:002) 0:01 < 1; ( 32 ) 13

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where we used the inequality 0:88 < ( 32 ) < 0:89. To apply Theorem 6 we need to show conditions (H1 ) (H3 ) are satis…ed. Indeed, f is jointly continuous and (H1 ) jf (t; x) f (t; y)j = 0:01 tan 1 x tan 1 y 0:01jx yj: (H2 ) LT = 0:042 + 0:248 < 1: Therefore, by (6), ISFDE (17) has a unique solution on [0; 1].

References [1] LakshmikanthamV, Leela S, Vasundhara Devi J: Theory of Fractional Dynamic Systems. Cambridge Scienti…c Publishers, Cambridge; 2009. [2] Kilbas A. A, Srivastava H.M, Trujillo J., North-Holland Mathematics Studies 204. In Theory and Applications of Fractional Di¤erential Equations. Elsevier, Amsterdam; 2006. [3] Liang S, Zhang J: Existence of multiple positive solutions for m-point fractional boundary value problems on an in…nite interval. Math. Comput. Model. 54, 1334-1346 (2011). [4] Su X., Solutions to boundary value problem of fractional order on unbounded domains in a Banach space. Nonlinear Anal. 74, 2844-2852 (2011). [5] Bai Z. B, Sun W., Existence and multiplicity of positive solutions for singular fractional boundary value problems. Comput. Math. Appl. 63, 1369-1381 (2012). [6] Agarwal R.P, O’Regan D, Stanek S., Positive solutions for mixed problems of singular fractional di¤erential equations. Math. Nachr. 285, 27-41 (2012). [7] Podlubny I. Fractional Di¤erential Equations. Academic Press, San Diego (1999). [8] Kilbas A.A, Srivastava„H.M, Trujillo„J.J.,: Theory and Applications of Fractional Di¤erential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006). [9] Bainov D, Simeonov P., Impulsive Di¤erential Equations: Periodic Solutions and Applications. Pitman Monographs and Surveys in Pure and Applied Mathematics (1993). [10] Zhang L, Wang G, Ahmad B, Agarwal R.P., Nonlinear fractional integro-di¤erential equations on unbounded domains in a Banach space. J. Comput. Appl. Math. 249, 51-56 (2013). [11] Yang, X-J: Fractional derivatives of constant and variable orders applied to anomalous relaxation models in heat-transfer problems. Therm. Sci. (2016). [12] Yang X-J, Srivastava H.M, Machado J. A., A new fractional derivative without singular kernel: application to the modelling of the steady heat ‡ow. Therm. Sci. 20, 753-756 (2016). [13] Miller K.S, Ross B., An Introduction to the Fractional Calculus and Fractional Di¤erential Equations. Wiley, New York (1993). [14] Delbosco D, Rodino L., Existence and uniqueness for a nonlinear fractional di¤erential equation, J. Math. Anal. Appl. 204 (1996), 609-625. [15] TianY , Bai Z., :Existence results for the three-point impulsive boundary value problem involving fractional di¤erential equations. 59, Issue 8, Pages 2601-2609, April (2010). [16] Wang X., Impulsive boundary Value Problem for nonlinear di¤erential Equations with Fractional order. Computers and Mathematics with Applications. Vol 62. September 2011. Pages 2383-2391.

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[17] Mahmudov, N. I, Unul, S: On existence of BVP’s for impulsive fractional di¤erential equations. Advances in Di¤erence Equations. 2017, 2017: Article ID 15. [18] Mahmudov, N. I.; Mahmoud, H. Four-point impulsive multi-orders fractional boundary value problems, Journal of Computational Analysis and Applications, 2017, Volume: 22 Issue: 7 Pages: 1249-1260 [19] Ahmad B., Ntouyas S.K., Existence results for a coupled system of Caputo type sequential fractional di¤erential equations with nonlocal integral boundary conditions, Appl. Math. Comput., 2015, 266, 615-622 [20] Ahmad B, Ntouyas S. K., On higher-order sequential fractional di¤erential inclusions with nonlocal three-point boundary conditions. Abstr. Appl. Anal. 2014, Article ID 659405 (2014). [21] Alsaedi A, Ahmad, B, Aqlan, M: Sequential fractional di¤erential equations and uni…cation of antiperiodic and multi-point boundary conditions. J. Nonlinear Sci. Appl. 10, 71–83, (2017).

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The Differentiability and Gradient for Fuzzy Mappings Based on The Generalized Difference of Fuzzy Numbers

∗

Shexiang Hai†, Fangdi Kong a

School of Science, Lanzhou University of Technology, Lanzhou, 730050, P.R. China

Abstract In this paper, the concepts of differentiability and gradient for fuzzy mappings are presented and discussed using the characteristic theorem for generalized difference of n dimensional fuzzy numbers. The relationships of gradient, support-f unction-wise gradient and level-wise gradient are characterized. Keywords: Fuzzy numbers, Fuzzy mappings, Differentiability, Gradient. 1. Introduction Since the concept and operations of fuzzy set were introduced by Zadeh [1], many studies have focused on the theoretical aspects and applications of fuzzy sets. Soon after, Zadeh proposed the notion of fuzzy numbers in [2, 3, 4]. Since then, fuzzy numbers have been extensively investigated by many authors. Since then, fuzzy numbers have been extensively investigated by many authors. Fuzzy numbers are a powerful tool for modeling uncertainty and for processing vague or subjective information in mathematical models. As part of the development of theories about fuzzy numbers and its applications, researchers began to study the differentiability and integrability of fuzzy mappings. Initially, the derivative for fuzzy mappings from an open subset of a normed space into the n dimension fuzzy number space E n was developed by Puri and Ralescu [5], which generalized and extended the concept of Hukuhara differentiability for setvalued mappings. In 1987, Kaleva [6] discussed the G-derivative, and obtained a sufficient condition for the H-differentiability of the fuzzy mappings from [a, b] into E n as well as a necessary condition for the Hdifferentiability of fuzzy mapping from [a, b] into E 1 . In 2003, Wang and Wu [7] put forward the concepts of directional derivative, differential and sub-differential of fuzzy mappings from Rn into E 1 by using Hukuhara difference. However, the Hukuhara difference between two fuzzy numbers exists only under very restrictive conditions [6] and the H-difference of two fuzzy numbers does not always exist [8]. The g-difference proposed in [8, 9] overcomes these shortcomings of the above discussed concepts and the g-difference of two fuzzy numbers always exists. Based on the novel generalizations of the Hukuhara difference for fuzzy sets, Bede [10] introduced and studied new generalized differentiability concepts for fuzzy valued functions in 2013. The purpose of the present paper is to use the fuzzy g-difference introduced in [10] to define and study differentiability and gradient for fuzzy mappings. First of all, we give the preliminary terminology used in the present paper. And then, in Section 3, the differentiability and gradient were presented and the relations among gradient, support-f unction-wise gradient and level-wise gradient for fuzzy mappings are examined. 2. Preliminaries In this section, basic definitions and operations for fuzzy numbers are presented [11, 12, 13, 14]. Throughout this paper, F (Rn ) denote the set of all fuzzy subsets on n dimensional Euclidean space Rn . A fuzzy subset u e (in short, a fuzzy set) on Rn is a function u e : Rn → [0, 1]. For each fuzzy sets u e, we ∗ This

work is supported by National Natural Science Fund of China (11761047). author. Tel.: +86 931 2973590. E-mail address: [email protected].

† Corresponding

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Shexiang Hai and Fangdi Kong: The Differentiability and Gradient for Fuzzy Mappings Based on ...

denote its r-level set as [e u]r = {x ∈ Rn : u e(x) ≥ r} for any r ∈ (0, 1]. The support of u e is denoted by n 0 suppe u = {x ∈ R : u e(x) > 0}. The closure of suppe u defines the 0-level of u e, i.e. [e u] = cl(suppe u). Here cl(M ) denotes the closure of set M. Fuzzy set u e ∈ F (Rn ) is called a fuzzy number if (1) u e is a normal fuzzy set, i.e., there exists an x0 ∈ Rn such that u e(x0 ) = 1, (2) u e is a convex fuzzy set, i.e., u e(λx + (1 − λ)y) ≥ min{e u(x), u e(y)} for any x, y ∈ Rn and λ ∈ [0, 1], (3) u e is upper semicontinuous , S (4) [e u]0 = cl(suppe u) = cl( r∈(0,1] [e u]r ) is compact. We will denote E n the set of fuzzy numbers [11, 12, 13]. It is clear that any u ∈ Rn can be regarded as a fuzzy number u e defined by ( 1, x = u, u e(x) = 0, otherwise. In particular, the fuzzy number e 0 is defined as e 0(x) = 1 if x = 0, and e 0(x) = 0 otherwise. Theorem 2.1.[6, 13] If u e ∈ E n , then (1) [e u]r is a nonempty compact convex subset of Rn for any r ∈ (0, 1], (2) [e u]r1 ⊆ [e u]r2 , whenever 0 ≤ r2 ≤ r1 ≤ 1, T∞ (3) if rk > 0 and rk is a nondecreasing sequence converging to r ∈ (0, 1], then k=1 [e u]rn = [e u]r . r n Conversely, if {[A] ⊆ R : r ∈ [0, 1]} satisfies the conditions (1)-(3), then there exists a unique u e ∈ En S such that [e u]r = [A]r for each r ∈ (0, 1] and [e u]0 = cl( r∈(0,1] [e u]r ) ⊆ A0 . Let u e, ve ∈ E n and k ∈ R. For any x ∈ Rn , the addition u e + ve and scalar multiplication ke u can be defined, respectively, as: (e u + ve)(x) = sup min{e u(s), ve(t)}, s+t=x

x (ke u)(x) = u e( ), k 6= 0, k ( 0, x 6= 0, (0e u)(x) = 1, x = 0. It is well known that for any u e, ve ∈ E n and k ∈ R, the addition u e + ve and the scalar multiplication ke u have the level sets [e u + ve]r = [e u]r + [e v ]r = {x + y : x ∈ [e u]r , y ∈ [e v ]r }, [ke u]r = k[e u]r = {kx : x ∈ [e u]r }, for any r ∈ [0, 1]. The Hausdorff distance D : E n × E n → [0, +∞) on E n is defined by D(e u, ve) = sup d([e u]r , [e v ]r ), r∈[0,1]

where d is the Hausdorff metric given by d([e u]r , [e v ]r )

=

inf{ε : [e u]r ⊂ N ([e v ]r , ε), [e v ]r ⊂ N ([e u]r , ε)}

=

max{supa∈[eu]r inf b∈[ev]r ka − bk, supb∈[ev]r inf a∈[eu]r ka − bk}.

N ([e u]r , ε) = {x ∈ Rn : d(x, [e u]r ) = inf y∈[eu]r d(x, y) ≤ ε} is the ε-neighborhood of [e u]r . Then (E n , D) is a complete metric space, and satisfies D(e u + w, e ve + w) e = D(e u, ve), D(ke u, ke v ) = |k|D(e u, ve) for any u e, ve, w e ∈ En and k ∈ R.

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Shexiang Hai and Fangdi Kong: The Differentiability and Gradient for Fuzzy Mappings Based on ...

Let S n−1 = {x ∈ Rn : kxk = 1} be the unit sphere of Rn and h·, ·i be the inner product in Rn , i.e. Pn hx, yi = i=1 xi yi , where x = (x1 , x2 , · · ·, xn ) ∈ Rn , y = (y1 , y2 , · · ·, yn ) ∈ Rn . Suppose u e ∈ E n , r ∈ [0, 1] n−1 and x ∈ S , the support function of u e is defined by u e∗ (r, x) = sup ha, xi. a∈[e u]r

Theorem 2.2.[14] Suppose u e ∈ E n , r ∈ [0, 1], then [e u]r = {y ∈ Rn : hy, xi ≤ u e∗ (r, x), x ∈ S n−1 }. The theorem below will give some basic properties of the support function. Theorem 2.3.[14, 15] Suppose u e ∈ E n , then (1) u e∗ (r, x + y) ≤ u e∗ (r, x) + u e∗ (r, y), ∗ (2) u e (r, x) ≤ supa∈[eu]r k a k, i.e. u e∗ (r, x) is bounded on S n−1 for each fixed r ∈ [0, 1], (3) u e∗ (r, x) is nonincreasing and left continuous in r ∈ [0, 1], right continuous at r = 0, for each fixed x ∈ S n−1 , (4) u e∗ (r, x) is Lipschitz continuous in x, i.e. |e u∗ (r, x) − u e∗ (r, y)| ≤ ( sup kak)kx − yk, a∈[e u]r

(5) if u e, ve ∈ E n , r ∈ [0, 1], then d([e u]r , [e v ]r ) = sup |e u∗ (r, x) − ve∗ (r, x)|, x∈S n−1

(6) (7) (8) (9)

(e u + ve)∗ (r, x) = u e∗ (r, x) + ve∗ (r, x), (ke u)∗ (r, x) = ke u∗ (r, x), for any k ≥ 0, ∗ −e u (r, −x) ≤ u e∗ (r, x), (−e u)∗ (r, x) = u e∗ (r, −x).

Definition 2.1. [10] The generalized difference (g-difference for short) of two fuzzy numbers u e, ve ∈ E n is given by its level sets as [ [e u g ve]r = cl( ([e u]β gH [e v ]β )), r ∈ [0, 1], β≥r

where the gH-difference gH is with interval operands [e u]β and [e v ]β . Remark 2.1. A necessary condition for u e g ve to exist is that either [e u]r contains a translate of [e v ]r or [e v ]r contains a translate of [e u]r for any r ∈ [0, 1]. Theorem 2.4. [15] Let u e, ve ∈ E n . If the g-difference u e g ve of u e and ve exists, then for any r ∈ [0, 1] and n−1 x∈S , we have ( (1) supβ≥r (e u∗ (β, x) − ve∗ (β, x)), ∗ (e u g ve) (r, x) = or (2) supβ≥r ((−e v )∗ (β, x) − (−e u)∗ (β, x)), ( (1) supβ≥r (e u∗ (β, x) − ve∗ (β, x)), = or (2) supβ≥r (e v ∗ (β, −x) − u e∗ (β, −x)). Theorem 2.5.[15] Let u e, ve ∈ E n . Then (1) if the g-difference exists, it is unique, (2) u e g u e = 0, (3) (e u + ve) g ve = u e, (e u + ve) g u e = ve, (4) u e g ve = −(e v g u e).

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3. The differentiability and gradient for fuzzy mappings In [5], Puri and Ralescu defined the g-derivative of fuzzy mappings from an open subset of a normed space into n-dimension fuzzy number space E n by using Hukuhara difference. In [7], Wang and Wu defined the directional g-derivative of fuzzy mappings from Rn into E 1 . Based on the generalizations of the Hukuhara difference for fuzzy sets, Bede [10] introduced and studied new generalized differentiability concepts for fuzzy valued functions from R into E 1 . The new generalized differentiability concept is a useful and applicable tool dealing with fuzzy differential equations and fuzzy optimization problems. In the following, using the characteristic theorem for generalized difference of n dimensional fuzzy numbers introduced in [15], we define and study differentiability and gradient for fuzzy mappings. Definition 3.1. Let Fe : M → E n , t0 = (t01 , t02 , · · · , t0m ) ∈ intM and t = (t1 , t2 , · · · , tm ) ∈ intM. If g-difference Fe(t) g Fe(t0 ) exists and there exist u ej ∈ E n (j = 1, 2, · · · , m), such that Pm D(Fe(t) g Fe(t0 ), j=1 u ej (tj − t0j )) lim = 0, t→t0 d(t, t0 ) then we say that Fe is differentiable at t0 and the fuzzy vector (e u1 , u e2 , · · · , u em ) is the gradient of Fe at t0 , denoted by ∇Fe(t0 ), i.e., ∇Fe(t0 ) = (e u1 , u e2 , · · · , u em ). Remark 3.1. Let Fe : M → E n , t0 = (t01 , · · · , t0j , · · · , t0m ) ∈ intM and h ∈ R with t = (t01 , · · · , t0j + h, · · · , t0m ) ∈ intM. Then the gradient ∇Fe(t0 ) exists at t0 if and only if Fe(t) g Fe(t0 ) exists and there are u ej ∈ E n (j = 1, 2, · · · , m), such that Fe(t01 , · · · , t0j + h, · · · , t0m ) g Fe(t01 , · · · , t0j , · · · , t0m ) . h→0 h

u ej = lim

Here the limit is taken in the metric space (E n , D). Theorem 3.1. The gradient ∇Fe(t) of fuzzy mapping Fe : M → E n is unique if it exists. Proof. Suppose we have two gradients (e u1 , u e2 , · · · , u em ) and (e v1 , ve2 , · · · , vem ) for fuzzy mapping Fe at t0 . For any ε > 0, according to Remark 3.1, there exist two positive real numbers δ1 and δ2 , when |h| < δ1 , we have |h| D(Fe(t01 , · · · , t0j + h, · · · , t0m ) g Fe(t01 , · · · , t0j , · · · , t0m ), he ε (j = 1, 2, · · · , m), uj ) < 2 when |h| < δ2 , we have |h| D(Fe(t01 , · · · , t0j + h, · · · , t0m ) g Fe(t01 , · · · , t0j , · · · , t0m ), he vj ) < ε (j = 1, 2, · · · , m). 2 Setting |h| < min(δ1 , δ2 ), we obtain, D(e uj , vej ) =

1 uj , he vj ) |h| D(he

≤

1 e 0 |h| D(F (t1 , · · ·

, t0j + h, · · · , t0m ) g Fe(t01 , · · · , t0j , · · · , t0m ), he uj )

+

1 e 0 |h| D(F (t1 , · · ·

, t0j + h, · · · , t0m ) g Fe(t01 , · · · , t0j , · · · , t0m ), he vj )


0, it follows from Theorem 2.3 that, (

S

u] r∈(0,1] [e

r

⊆ A0 (j = 1, 2, · · · , m) for any

Fe(t) g Fe(t0 ) ∗ 1 ) (r, x) = (Fe(t) g Fe(t0 ))∗ (r, x), h h

for any r ∈ [0, 1] and x ∈ S n−1 . For any r ∈ [0, 1] and x ∈ S n−1 , if taking (Fe(t) g Fe(t0 ))∗ (r, x) = sup(Fe(t)∗ (β, x) − Fe(t0 )∗ (β, x)), β≥r

then u e∗j (r, x)

e(t) g F e(t0 ) ∗ F ) (r, x) h

=

limh→0 (

=

limh→0 supβ≥r

=

supβ≥r vej∗ (β, x).

289

e(t)∗ (β,x)−F e(t0 )∗ (β,x) F h

Shexiang Hai ET AL 284-293

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Shexiang Hai and Fangdi Kong: The Differentiability and Gradient for Fuzzy Mappings Based on ...

According to Theorem 2.3, for any ε > 0, there is δ > 0, when h < δ, we have D(

e(t) g F e(t0 ) F ,u ej ) h e(t) g F e(t0 ) ∗ F ) (r, x) h

=

supr∈[0,1] supx∈S n−1 |(

=

supr∈[0,1] supx∈S n−1 | supβ≥r


0, there is δ > 0, when h < δ, we have D(

e(t0 ) e(t) g F F ,u ej ) h e(t) g F e(t0 ) ∗ F ) (r, x) h

=

supr∈[0,1] supx∈S n−1 |(

=

supr∈[0,1] supx∈S n−1 | supβ≥r


0, there is δ > 0, when −h < δ, we have D(

e(t) g F e(t0 ) F ,u ej ) h e(t) g F e(t0 ) ∗ F ) (r, x) h

=

supr∈[0,1] supx∈S n−1 |(

=

supr∈[0,1] supx∈S n−1 | supβ≥r


0, there is δ > 0, when −h < δ, we have D(

e(t) g F e(t0 ) F ,u ej ) h e(t) g F e(t0 ) ∗ F ) (r, x) h

=

supr∈[0,1] supx∈S n−1 |(

=

supr∈[0,1] supx∈S n−1 | supβ≥r


0, there is δ > 0, when |h| < δ, we have D( =

e(t0 ,··· , t0 ,··· , t0 ) e(t0 ,··· ,t0 +h,··· ,t0 ) g F F 1 j m 1 j m , vej ) h

S supr∈[0,1] d(cl( β≥r

≤ supr∈[0,1] supβ≥r d(
0;

k; l; s; t

even.

(7)

Proof: First suppose that there exists a prime period two solution :::; p; q; p; q; :::; of Eq. (1). We will prove that Condition (7) holds. We see from Eq. (1) ( when k; l; s; t even ) that p = aq +

b + cq ; dq + eq

q = ap +

b + cp dp + ep

p = aq +

b + cq ; (d + e)q

q = ap +

b + cp (d + e)p

Then

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(d + e)pq = a(d + e)q 2 + b + cq;

(8)

(d + e)pq = a(d + e)p2 + b + cp:

(9)

and Subtracting (9) from (8) gives 0 = a(d + e)(p2 Since p 6= q; it follows that

q 2 ) + c(p

q).

c : a(d + e)

p+q =

(10)

Again; adding (8) and (9) yields 2(d + e)pq = a(d + e)(p + q)2 It follows by (10); (11) and the relation p2 + q 2 = (p + q)2 pq =

2a(d + e)pq + 2b + c(p + q)

(11)

2pq for all p; q 2 R that

b : (a + 1)(d + e)

(12)

Now it is clear from Eq. (10) and Eq. (12) that p and q are the two positive distinct roots of the quadratic equation b c t + (a+1)(d+e) = 0; (13) t2 + a(d+e) a(d + e)(a + 1)t2 + c(a + 1)t + ab = 0; and so

2

(c(a + 1))

4a2 b(d + e)(a + 1) > 0

thus c2 (a + 1)

4a2 b(d + e) > 0

Therefore Inequality (7) holds. Second suppose that Inequality (7) is true. We will show that Eq. (1) has a prime period two solution. Assume that p p c(a + 1) + cA + p= = ; 2a(a + 1)(d + e) 2aAB and q= where

= c2 (a + 1)2

p cA ; where A = (a + 1); B = (d + e) 2aAB

4a2 b(a + 1)(d + e):

We see from Inequality (7) that 2

(c(a + 1))

4a2 b(d + e)(a + 1) > 0

then after dividing by (a + 1)we see that )

c2 > 4a2 b(d + e)

Therefore p and q are distinct real numbers. Set x x

l s

= p; x = p; x

= q; ; x k = p; x k+1 = q; s+1 = q; x t = p; x t+1 = q and x0 = p:

l+1

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We wish to show that x1 = x

1

=q

and

x2 = x0 = p:

It follows from Eq. (1) that x1 = ax

l

+

b + cx k dx s + ex

b + cp b + cp = ap + dp + ep (d + e)p

= ap + t

b + c(

= ap +

(d + e)

p cA+ 2aAB ) p cA+ 2aAB

:

Multiplying the denominator and numerator of the right side by 2aAB gives x1 = ap +

p ) 2abAB+c( cA+ p ; (d+e)( cA+ )

p

Multiplying the denominator and numerator of the right side by ( cA 2

and by Replacing A = (a + 1) , B = (d + e) and numerator of above equation gives

x1

= ap + = ap +

2

)

2

= c (a + 1)

4a b(a + 1)(d + e)in denominator and

p 2abAB( cA )+c(c2 A2 ) ; (d+e)(c2 A2 ) p 2 2ab(a+1)(d+e)( cA )+c(c (a+1)2 c2 (a+1)2 +4a2 b(a+1)(d+e)) ; (d+e)(c2 (a+1)2 c2 (a+1)2 +4a2 b(a+1)(d+e))

= ap +

p )+4a2 bc(a+1)(d+e) 2ab(a+1)(d+e)( cA ; 4a2 b(a+1)(d+e)2

Dividing numerator and denominator by (2ab(a + 1)(d + e)) we get = ap +

=

p +2ac cA 2a(d+e)

2a2 (d+e)p cA 2a(d+e)

p

+2ac

Now inserting the value of p we get x1

p 1 ca(a+1)+a 2a(d + e) p 1 2a(a + 1)(d + e) p c(a + 1) 2a(a + 1)(d + e)

= = =

c(a+1)2 (a+1) (a+1)

p

+2ac(a+1)

c(a + 1)2 + ca(a + 1)

But (a + 1) = A and (d + e) = B we get x1 =

p cA 2aAB

=q

Similarly as before one can easily show that x2 = p: Then it follows by induction that x2n = p

and

x2n+1 = q

299

for all

n

1:

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Thus Eq. (1) has the positive prime period two solution ...; p; q; p; q; ..., where p and q are the distinct roots of the quadratic equation (13) and the proof is completed. The following Theorems can be proved similarly. Theorem 3.2. Eq. (1) has a prime period two solutions if and only if c2 + 4b(d + e)(1

a) > 0

(l; k; s; t

odd).

Theorem 3.3. Eq. (1) has a prime period two solutions if and only if c2 (d

4(b(ad + e)2

e)(1 + a)

ec2 ) > 0

(l; k; s

even and t

odd).

even and s

odd).

Theorem 3.4. Eq. (1) has a prime period two solutions if and only if c2 (e

4(b(ae + d)2

d)(1 + a)

c2 d) > 0

(l; k; t

Theorem 3.5. Eq. (1) has a prime period two solutions if and only if c2 (1 + a)

4a(ab(d + e) + c2 ) > 0

(l; s; t

even and k

odd).

odd and t

even).

Theorem 3.6. Eq. (1) has a prime period two solutions if and only if c2 (e

d)

4bd2 (1

a) > 0

(l; k; s

Theorem 3.7. Eq. (1) has a prime period two solutions if and only if c2 (d

e)

4be2 (1

a) > 0

(l; k; t

odd and s

even).

Theorem 3.8. Eq. (1) has a prime period two solutions if and only if c2

4(b(d + e)(a

1) + c2 ) > 0

(l; s; t

odd and k

even).

odd and l

even).

Theorem 3.9. Eq. (1) has a prime period two solutions if and only if c2 (1 + a) + 4b(d + e) > 0

(k; s; t

Theorem 3.10. Eq. (1) has a prime period two solutions if and only if c2 (1 + a)

4(c2

b(d + e)) > 0

(l; k

even and s; t

odd).

even and k; t

odd).

Theorem 3.11. Eq. (1) has a prime period two solutions if and only if c2 (1 + a)(d

e)

4(b(ad + e)2 + ac2 d) > 0

300

(l; s

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Theorem 3.12. Eq. (1) has a prime period two solutions if and only if c2 (e

d)

4e(be(a

1) + c2 ) > 0

(s; k

even and l; t

odd).

odd and k; t

even).

Theorem 3.13. Eq. (1) has a prime period two solutions if and only if c2 (d

e)

4d(bd(a

1) + c2 ) > 0

(l; s

Theorem 3.14. Eq. (1) has a prime period two solutions if and only if c2 (a + 1)(e

4(b(ae + d)2 + ac2 e) > 0

d)

(s; k

odd and l; t

even).

Theorem 3.15. Eq. (1) has no prime period two solutions if one of the following statements holds (i) c 6= 0

(k; s; t

(ii) c 6= 0

(s; t

even and l

odd),

even and l; k

odd).

4. GLOBAL ATTRACTIVITY OF THE EQUILIBRIUM POINT OF EQ. (1) In this section we investigate the global attractivity character of solutions of Eq. (1). Theorem 4.1. The equilibrium point x of Eq. (1) is global attractor: Proof: Let p; q are a real numbers and assume that f : [p; q]4 ! [p; q] be a function de…ned by f (u0 ; u1 ; u2 ; u3 ) = au0 +

b + cu1 : du2 + eu3

We can easily see that the function f (u0 ; u1 ; u2 ; u3 ) increasing in u0 ; u1 and decreasing in u2 ; u3 . Suppose that (m; M ) is a solution of the system m = f (m; m; M; M )

and

M = f (M; M; m; m):

Then from Eq. (1), we see that m = am +

b + cm ; (d + e)M

M = aM +

b + cM ; (d + e)m

(1

b + cm ; (d + e)M

(1

b + cM ; (d + e)m

That is a)m =

a)M =

or, b + cm = b + cM Thus m = M: It follows by the Theorem B that x is a global attractor of Eq. (1) and then the proof is complete.

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5.

NUMERICAL EXAMPLES

For con…rming the results of this paper, we consider numerical examples which represent di¤erent types of solutions to Eq. (1). Example 1. We assume l = 5; k = 4; s = 3; t = 5; x 5 = 6; x 4; a = 0:1; b = 0:2; c = 0:9; d = 0:7 e = 0:8. [See Fig. 1]

plot of x(n+1)= a.X(n-l)+((b+c.X(n-k))/((d.X(n-s)+e.X(n-t)))) 18

= 9; x

= 8; x

3

2

= 9; x

1

= 12; x

1

=

10 of x(n+1)= a.X(n-l)+((b+c.X(n-k))/((d.X(n-s)+e.X(n-t)))) plot 10

14

16

4

12

14 10

8

10

x(n)

x(n)

12

8

6

6 4 4 2

2 0

0 0

50

100

150

200

250

300

350

400

450

500

0

10

20

30

40

n

50

60

70

80

90

100

n

Figure 1.

Figure 2.

Example 2. See Fig. 2, since l = 1; k = 2; s = 1; t = 3; x 1:6; b = 0:2; c = 0:9; d = 0:09, e = 0:01: Example 3. See Fig. 3, since l = 1; k = 2; s = 1; t = 1 x 0:2; c = 0:5; d = 0:6; e = 0:2.

3

3

= 1:2; x

= 12; x

2

2

= 0:7; x

= 7; x

1

1

= 8:5; x0 = 5; a =

= 8; x0 = 3; a = 0:1; b =

Example 4. Fig. 4. shows the solutions when a = 0:1; b = 0:2; c = 0:5; d = 0:6; e = 0:9; l = 4; k = 2; s = 4; t = 2; x 4 = p; x 3 = q; x 2 = p; x 1 = q; x0 = p: p2 c(a+1) c (a+1)2 4a2 b(a+1)(d+e) Since p; q = 2a(a+1)(d+e)

plot of x(n+1)= a.X(n-l)+((b+c.X(n-k))/((d.X(n-s)+e.X(n-t))))

plot of x(n+1)= a.X(n-l)+((b+c.X(n-k))/((d.X(n-s)+e.X(n-t))))

25

0

-0.5 20 -1 15

x(n)

x(n)

-1.5

-2

10

-2.5 5 -3

0

-3.5 0

10

20

30

40

50

60

70

80

90

100

0

n

5

10

15

20

25

30

35

40

45

50

n

Figure 3.

Figure 4.

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Acknowledgements This article was funded by the Deanship of Scienti…c Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR technical and …nancial support.

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Asymptotic Representations for Fourier Approximation of Functions on the Unit Square ∗ Zhihua Zhang College of Global Change and Earth System Science, Beijing Normal University, Beijing, China, 100875 E-mail: [email protected]

Abstract.

In this paper, for any smooth function on [0, 1]2 , we give an asymptotic representation

of hyperbolic cross approximations of its Fourier series whose principal part is determined by the values of the function at vertexes of [0, 1]2 and present a novel approach to estimates of the upper bounds of approximation errors. At the same time, we also give an asymptotic formula of partial sum approximations whose principal part is determined by not only partial derivatives at vertexes of [0, 1]2 , but also mean values on each side. Comparing asymptotic representations of these two kinds of approximation, we find that although in general the hyperbolic cross approximation is better than the partial sum approximation, the partial sum approximation possibly work better under some cases, and we also give the corresponding necessary and sufficient condition to characterize these cases. 1. Introduction For a function f on [0, 1]2 , regardless of how smooth it is, by the Riemann-Lebesgue lemma, we only know that its Fourier coefficients cmn (f ) = o(1). In this paper, we first obtain a precise asymptotic formula of the Fourier coefficients (see Theorem 2.2) by using our novel decomposition formula of f :   q(x, y) + τ (x, y) (x, y) ∈ [0, 1]2 , f (x, y) =  q(x, y) (x, y) ∈ ∂([0, 1]2 ), where q(x, y) is a combination of the boundary function and four simple polynomial factors x, 1 − x, y, and 1 − y. After that, we will discuss further two kinds of Fourier approximations of functions on the unit square. The sparse approximation has received much attention in recent years [1,6,7,8]. As an approximation tool, hyperbolic cross truncations of Fourier series has obvious advantages over partial sums of Fourier ∗ Zhihua Zhang is a full professor at Beijing Normal University, China. He has published more than 50 first-authored papers in applied mathematics, signal processing and climate change. His research is supported by National Key Science Program No.2013CB956604; Fundamental Research Funds for the Central Universities (Key Program) No.105565GK; Beijing Young Talent fund and Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry. Zhihua Zhang is an associate editor of “EURASIP Journal on Advances in Signal Processing” (Springer, SCIindexed), an editorial board member in applied mathematics of “SpringerPlus” (Springer, SCI-indexed) and an editorial board member of “Journal of Applied Mathematics” (Hindawi).

1

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series since the hyperbolic cross truncations [8]: (h) sN (f ; x, y)

=

N X

2πimx

cm0 (f ) e

N X

+

|m|=0

c0n (f ) e2πiny +

|n|=1

X

cmn (f ) e2πi(mx+ny)

(1.1)

1≤|mn|≤N

can make full use of the decay of Fourier coefficients to reconstruct the target function f . Throughout this paper, we always assume that f ∈ C (3,3) ([0, 1]2 ) which means that

∂f i+j ∂xi ∂y j (0

≤ i, j ≤

2

3) are continuous on [0, 1] . We will show that, for the hyperbolic cross truncations of its Fourier series, the following asymptotic representation holds (see Theorem 3.1): kf−

(h) sN (f )

2 1 2 log Nd (f (0, 0) + f (1, 1) − f (0, +O 1) − f (1, 0)) 4π 4 Nd

k22 =

(h)

where Nd is the number of Fourier coefficients in sN (f ) and k F k22 =

R1R1 0

0

µ

log Nd Nd

¶ ,

(1.2)

|F (x, y)|2 dxdy.

For the partial sum approximation of the Fourier series of f on [0, 1]2 , we will give another asymptotic representation. The corresponding principal part will become more complicated. It depends on not only values of function f and its partial derivatives

∂f ∂x

and

∂f ∂y

at vertexes of [0, 1]2 , but also the mean values

of f on each side of the boundary ∂([0, 1]2 ) (in detail, see Theorem 4.1). Comparing asymptotic representations of two kinds of Fourier approximations, we find that for hyperbolic cross approximation, the approximation order is in general the approximation order is

√1 , Nd

log2 Nd Nd ,

while for the partial sum approximation,

and under some cases the approximation order is

1 Nd .

More-

over, we further give a corresponding necessary and sufficient condition for these cases (see Corollary 4.2). 2. Asymptotic representation of Fourier coefficients Let f ∈ C (3,3) ([0, 1]2 ). Expand f into Fourier series: f (x, y) =

P m,n

Z

1

Z

cmn (f ) = 0

and

P m,n

means

∞ P

∞ P

1

cmn e2πi(mx+ny) , where

f (x, y) e−2πi(mx+ny) dxdy

0

. We extend f from [0, 1]2 to R2 . Then f is a function on the whole plane

m=−∞ n=−∞

R2 with period 1 and f is discontinuous at the integral points {m, n}m,n∈Z . By the Riemann-Lebesgue lemma, we only know that cmn (f ) = o(1) as m → 0 or n → ∞, where “o” means high-order infinitesimal. To obtain the precise asymptotic formula of Fourier coefficients, we construct a combination q(x, y) of the boundary functions f (x, 0), f (x, 1), f (0, y), f (1, y) and factors x, (1 − x), y, (1 − y) such that the difference f (x, y) − q(x, y) vanishes on the boundary ∂([0, 1]2 ). Now we define three functions as follows. q1 (x, y) = (f (x, 0) − f (0, 0)(1 − x) − f (1, 0)x)(1 − y) + (f (x, 1) − f (0, 1)(1 − x) − f (1, 1)x)y, q2 (x, y) = (f (0, y) − f (0, 0)(1 − y) − f (0, 1)y)(1 − x) + (f (1, y) − f (1, 0)(1 − y) − f (1, 1)y)x,

(2.1)

q3 (x, y) = f (0, 0)(1 − x)(1 − y) + f (0, 1)(1 − x)y + f (1, 0)x(1 − y) + f (1, 1)xy. Then q(x, y) = q1 (x, y) + q2 (x, y) + q3 (x, y) is the desired function, i.e., we have the following theorem. 2

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Theorem 2.1. Let f be defined on [0, 1]2 and q(x, y) be stated as above. Then τ (x, y) = f (x, y) − q(x, y) vanished on the boundary ∂([0, 1]2 ). From this, we deduce that if f ∈ C (3,3) ([0, 1]2 ), then τ (x, y) ∈ C (3,3) ([0, 1]2 ) and satisfies that for i = 1, 2, 3, ∂iτ ∂xi (x, 0)

=

∂iτ ∂xi (x, 1)

=0

(0 ≤ x ≤ 1),

∂iτ ∂y i (0, y)

=

∂iτ ∂y i (1, y)

=0

(0 ≤ y ≤ 1).

(2.2)

Now we further explain the relationship between q(x, y) and f (x, y). By (2.1), it follows that ∂q ∂x (x, y)

∂f ∂x (x, 0)(1

=

− y) +

∂f ∂x (x, 1)y

− f (0, y) + f (1, y)

+(f (0, 0) − f (1, 0))(1 − y) + (f (0, 1) − f (1, 1))y, ∂q ∂y (x, y)

∂f ∂y (0, y)(1

=

− x) +

∂f ∂y (1, y)x

− f (x, 0) + f (x, 1)

+(f (0, 0) − f (0, 1))(1 − x) + (f (1, 0) − f (1, 1))x, ∂2q ∂x∂y (x, y)

= − ∂f ∂x (x, 0) +

∂f ∂x (x, 1)

∂f ∂y (0, y)

−

+

∂f ∂y (1, y)

−f (0, 0) + f (1, 0) + f (0, 1) − f (1, 1), ∂3q ∂x2 ∂y (x, y)

= − ∂∂xf2 (x, 0) +

∂3q ∂x∂y 2 (x, y)

= − ∂∂yf2 (0, y) +

∂4q ∂x2 ∂y 2 (x, y)

2

∂2f ∂x2 (x, 1),

2

∂2f ∂y 2 (1, y),

= 0.

From this, we get ∂2q ∂x∂y (1, 1)

−

∂2q ∂x∂y (1, 0)

−

³

∂q ∂x (1, y)

−

∂q ∂x (0, y)

=

∂q ∂y (x, 1)

−

∂q ∂y (x, 0)

=

³

∂2q ∂x∂y (0, 1)

+

∂2q ∂x∂y (0, 0)

= 0,

´

∂f ∂x (1, 0)

−

∂f ∂x (0, 0)

∂f ∂y (0, 1)

−

∂f ∂y (0, 0)

³ (1 − y) +

´

³ (1 − x) +

´

∂f ∂x (1, 1)

−

∂f ∂x (0, 1)

∂f ∂y (1, 1)

−

∂f ∂y (1, 0)

y,

(2.3)

´ x.

Since cmn (f ) = cmn (q) + cmn (τ ) and cmn (q) = cmn (q1 ) + cmn (q2 ) + cmn (q3 ), by (2.1), cmn (q1 ) = cm (R(x, 0))cn (1 − y) + cm (R(x, 1))cn (y),

(2.4)

where R(x, ν) = f (x, ν) − f (0, ν)(1 − x) − f (1, ν)x

(ν = 0, 1).

(2.5)

Since R(0, ν) = R(1, ν), cm (R(x, ν)) = = c0 (R(x, ν)) =

R1 0

R(x, ν) e−2πimx dx =

1 4π 2 m2

R1 0

³

∂R ∂x (1, ν)

−

1 2πim

∂R ∂x (0, ν)

R1

−

∂R (x, ν) e−2πimx dx 0 ∂x

R1

´

∂2R (x, ν) e−2πimx dx 0 ∂x2

(m 6= 0),

f (x, ν)dx − 12 (f (0, ν) + f (1, ν)). 3

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Noticing that

∂R ∂x (x, ν)

=

∂f ∂x (x, ν)

+ f (0, 0) − f (1, ν), we get

∂R ∂x (1, ν)

∂R ∂x (0, ν)

−

∂2R ∂x2 (x, ν)

=

∂f ∂x (1, ν)

=

∂2f ∂x2 (x, ν)

−

∂f ∂x (0, ν),

(ν = 0, 1).

1 1 and − 2πim (m 6= 0), respectively, we get by (2.5) Since mth Fourier coefficients of (1 − x) and x are 2πim µ ¶ Z 1 2 ∂f 1 ∂f ∂ f −2πimx (x, ν) e cm (R(x, ν)) = (1, ν) − (0, ν) − dx 2 4π 2 m2 ∂x ∂x 0 ∂x

while Z 0

1

∂2f 1 (x, ν) e−2πimx dx = − ∂x2 2πim

So cm (R(x, ν)) = c0 (R(x, ν)) =

1

R1 0

³

1

8π 2 m2

³R

i c0n (q1 ) = − 2πn

∂f ∂x (1, 0)

−

∂2f ∂2f (1, ν) − (0, ν) − 2 ∂x ∂x2

∂f ∂x (1, ν)

´

−

∂f ∂x (0, ν)

+O

¡

Z

1

0

¶ ∂3f 2πimx (x, ν) e dx . ∂x3

¢

1 m3

(m 6= 0),

f (x, ν)dx − 12 (f (0, 0) + f (1, ν)).

From this and (2.4), it follows that ³ ∂f i cmn (q1 ) = − 8π3 m 2n ∂x (1, 0) + cm0 (q1 ) =

³

4π 2 m2

µ

∂f ∂x (0, 1)

∂f ∂x (0, 1)

−

−

∂f ∂x (0, 0)

∂f ∂x (0, 0)

´

−

∂f ∂x (1, 1)

´

+

∂f ∂x (1, 1)

¡

+O

+O

¡

1 m3

1 m3 n

¢

¢

(m 6= 0, n 6= 0), (m 6= 0),

´ − f (x, 1))dx − 12 (f (0, 1) + f (1, 0) − f (0, 1) − f (1, 1))

1 (f (x, 0) 0

(n 6= 0).

Similarly, we have i cmn (q2 ) = − 8π3 mn 2

c0n (q2 ) =

1 8π 2 n2

³

i cm0 (q2 ) = − 2πm

and

³

∂f ∂y (1, 0)

∂f ∂y (0, 1)

³R

−

∂f ∂y (0, 1)

∂f ∂y (1, 0)

1 (f (0, y) 0

cmn (q3 ) =

+

−

−

∂f ∂y (0, 0)

∂f ∂y (0, 0)

´

−

∂f ∂y (1, 1)

+O

´

+

∂f ∂y (1, 1)

+O

¡

1 n3

¡

¢

1 mn3

¢

(m 6= 0, n 6= 0),

(n 6= 0),

´ − f (1, y))dy − 12 (f (0, 0) + f (0, 1) − f (1, 0) − f (1, 1))

1 4π 2 mn (f (1, 0)

+ f (0, 1) − f (0, 0) − f (1, 1))

i cm0 (q3 ) = − 4πm (f (0, 0) − f (0, 1) − f (1, 0) + f (1, 1)) i c0n (q3 ) = − 4πn (f (0, 0) − f (0, 1) + f (1, 0) − f (1, 1))

(m 6= 0).

(m 6= 0, n 6= 0), (m 6= 0), (n 6= 0).

From this, we get an asymptotic representation of cmn (q) by q(x, y) = q1 (x, y)+q2 (x, y)+q3 (x, y). Finally, we write out the asymptotic representation of cmn (τ ). Using the integration by parts, it follows by Theorem 2.1, (2,2) and (2.4) that (i) For m 6= 0, n 6= 0, ¶ µ µ 2 ¶µ ¶ ∂2f ∂2f ∂2f 1 1 ∂ f 1 1 (1, 1) − (1, 0) − (0, 1) + (0, 0) + O + ; cmn (τ ) = 16π 4 m2 n2 ∂x∂y ∂x∂y ∂x∂y ∂x∂y m2 n 2 m n 4

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(ii) For m 6= 0, µZ 1 µ ¶ µ ¶¶ µ ¶ ∂f 1 ∂f 1 ∂f ∂f ∂f ∂f 1 cm0 (τ ) = 2 2 (1, y)− (0, y) dy+ (0, 0)− (1, 0)+ (0, 1)− (1, 1) +O 3 4π m ∂x ∂x 2 ∂x ∂x ∂x ∂x m 0 (iii) For n 6= 0, µZ 1 µ µ ¶ ¶ µ ¶¶ ∂f 1 1 ∂f ∂f 1 ∂f ∂f ∂f c0n (τ ) = 2 2 (x, 1)− (x, 0) dx + (0, 0)− (0, 1)+ (1, 0)− (1, 1) +O 3 . 4π n ∂y ∂y 2 ∂y ∂y ∂y ∂y n 0 From this and cmn (f ) = cmn (q) + cmn (τ ), we get the following asymptotic representation of Fourier coefficients of f (x, y). Theorem 2.2. Let f ∈ C (3,3) ([0, 1]2 ). Then Fourier coefficients of f (x, y) satisfy (i) for m 6= 0, n 6= 0, cmn (f ) =

1 β γ δ (−α + i +i + )+O 4π 2 mn 2πm 2πn 4π 2 mn

where

µ

1 m2 n

¶µ

1 1 + m n

¶ ,

α = f (0, 0) − f (0, 1) − f (1, 0) + f (1, 1), β=

∂f ∂x (0, 0)

−

∂f ∂x (0, 1)

−

∂f ∂x (1, 0)

+

∂f ∂x (1, 1),

γ=

∂f ∂y (0, 0)

−

∂f ∂y (0, 1)

−

∂f ∂y (1, 0)

+

∂f ∂y (1, 1),

δ=

∂2f ∂x∂y (0, 0)

−

∂2f ∂x∂y (0, 1)

−

∂2f ∂x∂y (1, 0)

(ii) for m 6= 0, b a + +O cm0 (f ) = i 2πm 4π 2 m2 where a = f (0, 1) − f (1, 0) − b=

0

∂x (1, y)

−

∂f ∂x (0, y)

,

dy;

d c + 2 2 +O c0n (f ) = i 2πn 4π n where c = f (1, 0) − f (0, 1) − R 1 ³ ∂f

R1 0

µ

1 n3

¶ ,

(f (x, 0) − f (x, 1))dx, ´

∂y (x, 1) −

0

¶

(f (0, y) − f (1, y))dy,

(iii) for n 6= 0,

d=

1 m3

´

R 1 ³ ∂f 0

R1

µ

∂2f ∂x∂y (1, 1);

+

∂f ∂y (x, 0)

dx + O

¡

1 n3

¢

.

Now we compute |cmn (f )|2 . Since f is a real-valued function, it is clear that α, β, γ, δ and a, b, c, d in Theorem 2.2 are all real numbers. So we get the following corollary. Corollary 2.3. Let f ∈ C (3,3) ([0, 1]2 ). Then ³ |cmn (f )|2 = 16π41m2 n2 α2 + βγ−αδ 2π 2 mn + |cm0 (f )|2 = |c0n (f )|2 =

a2 4π 2 m2 c2 4π 2 n2

+O

+O

¡

¡

1 m4

1 n4

¢

¢

β2 4π 2 m2

+

γ2 4π 2 n2

´ +O

¡

1 m3 n3

¢¡ 1 m

+

1 n

¢

,

,

, 5

309

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.2, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

where α, β, γ, δ and a, b, c, d are stated as above. 3. Asymptotic representation of hyperbolic cross approximation Let f ∈ C (3,3) ([0, 1]2 ). We expand it into a Fourier series. Consider the hyperbolic cross truncations of its Fourier series: N P

(h)

sN (f ; x, y) =

|m|=0

N P

+

N P

cm0 (f ) e2πimx +

c0n (f ) e2πiny

|n|=1

P

cmn (f ) e2πi(mx+ny) ,

N |n|=1 |m|≤ |n|

where cmn (f ) =

R1R1 0

0

f (x, y) e−2πi(mx+ny) dxdy. So P

(h)

f (x, y) − sN (f ; x, y) =

|m|≥N +1

c0n (f ) e2πiny

|n|=N +1

P

+

∞ P

cm0 (f ) e2πimx + ∞ P

N P

P

|n|=1

N |m|> |n|

cmn (f ) e2πi(mx+ny) +

|n|≥N +1 |m|=1

cmn (f ) e2πi(mx+ny) .

Using the Parseval identity [4,5,9] of bivariate Fourier series, P (h) k f − sN k22 = (|c0n (f )|2 + |cn0 (f )|2 ) |n|≥N +1

P

+

∞ P

|cmn (f )|2 +

|n|≥N +1 |m|=1

N P

P

|n|=1

N |m|> |n|

(3.1)

|cmn (f )|2

=: PN + QN + RN . By Corollary 2.3, |cmn (f )|2 =

α2 +O 4 16π m2 n2

µ

1 m3

¶

1 +O n2

µ

1 n3

¶

1 . m2

We first compute RN : RN

=

N P

P

|cmn (f )|2

N |n|=1 |m|> |n|

=

α2 16π 4

N P |n|=1

(1)

1 n2

P N |m|> |n|

(2)

1 m2

N P

+ O(1)

1 n4

|n|=1

P N |m|> |n|

1 m3

+ O(1)

N P |n|=1

1 n3

P N |m|> |n|

1 m4

(3.2)

(3)

= : RN + RN + RN . Note that (1)

RN = P N |m|≥ |n|

α2 16π 4

1 m2

N P |n|=1

=2

1 n2

R∞ N |n|

P N |m|≥ |n|

1 t2 dt

1 m2 ,

³ +O

n2 N2

´ =

2|n| N

³ +O

n2 N2

´ .

6

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This implies that (1)

RN = (2)

Similarly, RN = O

¡1¢ N

µ ¶ µ ¶ N α2 X 1 1 α2 log N 1 + O = + O . 4 4 4π n=1 nN N 4π N N

(3)

and RN = O

¡1¢ N

α2 log N +O = 4π 4 N

RN By |cmn |2 = O

¡

1 m2 n2

|n|≥N +1

From |c0n (f )|2 = O

µ

1 N

¶ .

¢

, it follows that  X QN = O(1) 

¡

. So

1 n2

¢

  µ ¶ µ ¶ 1  X 1  1 1 +O =O . n2 m2 N2 N

and |cm0 (f )|2 = O X

PN =

|n|≥N +1

|m|=1

¡

1 m2

¢

, it is easy to deduce that µ ¶ X 1 2 |c0n (f )| + |cm0 (f )| = O . N |m|≥N +1

Therefore, by (3,1), (h)

k f − sN (f ) k22 =

α2 log N +O 4π 4 N

µ

1 N

¶ . (h)

The number Nd of Fourier coefficients in the hyperbolic cross truncation sN (f ) is equal to Nd = 2N + 1 +

¸ N · X N = 2N log N + O(N ). |n1 | n =1 1

Theorem 3.1. Let f ∈ C (3,3) ([0, 1]2 ). Then the asymptotic representation of the hyperbolic cross approximation of Fourier series of f is kf−

(h) sN (f )

k22 =

α2 log2 Nd 4π 4 Nd

µ

µ 1+O

1 log Nd

¶¶ ,

(3.3) (h)

where Nd is the number of Fourier coefficients in hyperbolic cross truncation sN (f ) and α = f (0, 0) − f (0, 1) − f (1, 0) + f (1, 1). (2,2) 2 Corollary 3.2. Let f ∈ ´([0, 1] ). Then ³C (h) d (i) k f − sN (f ) k22 = O logNN if and only if f (0, 0) + f (1, 1) = f (0, 1) + f (1, 0). d

(ii) when F (x, y) = f (x, y) + (f (0, 1) + f (1, 0) − f (0, 0) − f (1, 1))xy, ¶ µ log Nd (h) . k F − sN (F ) k22 = O Nd Now we show an approach to estimates of the bound of the term “ O” in Theorem 3.1 using the Sobolev norm. For

∂6f ∂x3 ∂y 3

∈ C([0, 1]2 ), its Sobolev norm is defined as M (f ) =

¯ i+j ¯ ¯∂ f ¯ ¯ ¯ ¯ i j¯. x,y∈∂([0,1]2 ) ∂x ∂y max

i,j=0,1,2,3

7

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By Theorem 2.1 and (2.2), and (2.4), we get Z 1Z 1 cmn (τ ) = τ (x, y) e−2πi(mx+ny) dxdy = 0

where δ=

0

+ Jmn ,

∂2f ∂2f ∂2f ∂2f (0, 0) − (0, 1) − (1, 0) + (1, 1) ∂x∂y ∂x∂y ∂x∂y ∂x∂y

and Jmn =

δ 16π 4 m2 n2

³

1

32π 5 m2 n3

− 32π51m2 n3

∂3f ∂x∂y 2 (1, 1)

R1³ 0

³

+ 32iπ51m3 n2

R1³ 0

−

∂3f ∂x∂y 2 (1, 0)

´

−

∂4f ∂x∂y 2 (0, y)

´

+

∂3f ∂x∂y 2 (0, 0)

e−2πiny dy

´

−

∂3f ∂x2 ∂y (0, 0)

´

∂4f ∂x2 ∂y 2 (1, y) ∂4f ∂x3 ∂y (x, 1)

−

∂4f ∂x2 ∂y 2 (0, y)

´

∂4f ∂x3 ∂y (x, 0)

−

e−2πiny dy

e−2πimx dx

R1R1

− 32iπ51m3 n2 |Jmn | ≤

∂3f ∂x2 ∂y (1, 0)

0

+ 32iπ51m3 n2

So

∂4f ∂x∂y 3 (1, y)

R1³

− 32iπ51m3 n2

∂3f ∂x∂y 2 (0, 1)

−

∂5f (x, y) e−2πi(mx+ny) dxdy 0 ∂x3 ∂y 2

0

6M (f ) 7M (f ) 13M (f ) + ≤ 32π 5 m2 n3 32π 5 m3 n2 32π 5 m2 n2

µ

1 1 + m n

¶ .

For cm0 and c0n , we have cm0 (τ ) = c0n (τ ) = where

(1)

1 (2πm)2 1 (2πn)2

1 (2πm)3 i

Tm =

1 (2πn)3 i

1 0

∂f ∂y (x, 1)

³

0

∂2f ∂x2 (1, 0)

0

´

−

∂f ∂x (0, y)

´

−

∂f ∂y (x, 0)

´ (1) dy + 21 β + Tm ,

´ (2) dx + 12 γ + Tn ,

´

−

−

∂2f ∂x2 (0, y) ∂2f ∂x2 (0, 0)

R 1 R 1 ³ ∂3f

∂x3 (x, y)

−

−

dy ∂2f ∂x2 (1, 1)

3 1∂ f 2 ∂x3 (x, 0)

´

+

∂2f ∂x2 (0, 1)

´

−

3 1∂ f 2 ∂x3 (x, 1)

dxdy.

R 1 ³ ∂2f 0

1 − 2(2πn) 3i 1 − (2πn) 3i

So

³R ³

∂f ∂x (1, y)

∂x2 (1, y)

0

1 − (2πm) 3i (2)

1 0

R 1 ³ ∂2f

1 − 2(2πm) 3i

Tn =

³R ³

³

´ ∂2f (x, 1) − (x, 0) dx 2 2 ∂y ∂y

∂2f ∂y 2 (0, 1)

−

R 1 R 1 ³ ∂3f 0

0

∂2f ∂y 2 (0, 0)

∂y 3 (x, y)

−

−

3 1∂ f 2 ∂y 3 (0, y)

|Tm | ≤

(1)

6M (f ) (2πm)3 ,

(2)

6M (f ) (2πn)3 .

|Tn | ≤

∂2f ∂y 2 (1, 1)

´

+

∂2f ∂y 2 (1, 0)

´

−

3 1∂ f 2 ∂y 3 (1, y)

dxdy.

8

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Now we estimate cmn (q). Note that 1 cm (R(x, ν)) = 4π 2 m2 where L(ν) m (ν)

Then |Lm | ≤

µ

1 = 8π 3 m3 i

5M (f ) 8π 3 m3 .

µ

¶ ∂f ∂f (1, ν) − (0, ν) + L(ν) m ∂x ∂x Z

∂2f ∂2f (1, (0, ν) − ν) − ∂x2 ∂x2

1

0

(ν = 0, 1),

¶ ∂3f −2πimx (x, ν) e dx ∂x3

(ν = 0, 1).

This implies that (1)

β cmn (q1 ) = − 8π3 m 2 ni + Hmn , (2)

γ cmn (q2 ) = − 8π3 mn 2 i + Hm ,

where

(1)

5M (f ) 8π 4 m3 n ,

|H,mn | ≤ (2)

5M (f ) 8π 4 mn3 .

|Hmn | ≤ From this and cmn (q3 ) =

α 4π 2 mn ,

we get

cmn (q) = where |Hmn | ≤

5M (f ) 8π 4 mn

¡

1 m2

+

1 n2

¢

1 4π 2 mn

µ

β γ − 2πmi 2πni

α−

¶ + Hmn ,

.

Similarly, we may estimate cm0 (q) and c0n (q). Using cmn (f ) = cmn (q) + cmn (τ ) and the above estimates, we easily obtain the estimates of upper bounds of |cmn (f )|2 . Again, using the method of argument in Theorem 3.1, we finally can give the bound of the term “ O ” in (3.3). 4. Asymptotic representation of square errors of partial sums Let f ∈ C (3,3) ([0, 1]2 ). Consider the partial sums of its Fourier series: X

sN (f ; x, y) =

X

cmn (f ) e2πi(mx+ny) .

|m|≤N |n|≤N

Then the square errors are equal to k f − sN (f ) k22 =

P |n|≥N +1

+

P

|c0n (f )|2 + P

|cm0 (f )|2

|m|≥N +1 ∞ P

|cmn (f )|2 +

|n|≥N +1 |m|=1

N P

P

|cmn (f )|2

(4.1)

|n|=1 |m|≥N +1

=: KN + LN + IN + JN . By Corollary 2,3 (ii) and (iii), KN =

c2 2π 2 N

+O

LN =

a2 2π 2 N

+O

¡ ¡

1 N3

1 N3

¢ ¢

,

.

9

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By Corollary 2.3 (i), µ

1 |cmn (f )| = 4 16π m2 n2 and so

à IN =

α2 48π 2

=

1 8π 2

α2 16π 4

α2 3

2

∞ P n=1

1 ns

!Ã

N P

|n|=1

γ + 64π 6

à +

¶

β2 64π 6

+O

!

P

1 n2

|n|>N

µ

à ζ(4) +

γ2 192π 4

1 3 m n3

¶ ,

!

P

|n|>N

1 n4

+O

¡

1 N2

¢

´ ¡ 1 ¢ β2 1 + 2π ζ(4) 4 N + O N2 ,

Ã

where ζ(s) =

1 n2

|n|>N

³

à JN =

!

P

β2 γ2 α + 2 2+ 2 2 4π m 4π n 2

2

1 n2

N P

|n|=1

!

P

|m|>N

!Ã 1 n4

Ã

1 m2

P

|m|>N

+

β2 64

|n|=1

! 1 m2

!Ã

N P

+O

¡

1 N2

¢

1 n2

!

P

|m|>N

1 m4

,

is the Riemann-Zeta function. Note that N P |n|=1 N P |n|=1

1 n2

=

1 n4

=

Then JN =

∞ P |n|=1 ∞ P |n|=1

1 8π 2

µ

1 n2

−

1 n4

−

P |n|>N

P |n|>N

π2 3

¡1¢

1 n2

=

1 n4

= ζ(4) + O

+O

N

¡

,

1 N3

¢

.

¶ µ ¶ α2 γ2 1 1 + 4 ζ(4) +O . 3 π N N2

Finally, by (4.1), we get the following theorem. Theorem 4.1. Let f ∈ C (3,3) ([0, 1]2 ). Then the partial sums sN (f ) of its Fourier series satisfy µ 2 ¶ µ ¶ a + c2 1 1 α2 β2 + γ2 k f − sN (f ) k22 = + + ζ(4) + O , 2π 2 24π 2 8π 6 N N2 where a, c, α, β, γ are stated in Theorem 2.2 and ζ(4) is the Riemann-Zeta function. Note that the number Nd of Fourier coefficients in the sum sN (f ) is (2N + 1)2 . From Theorem 4.1, it follows that

1 k f − sN (f ) k22 ∼ √ . Nd

Again, by Theorem 4.1, we get the following corollary. Corollary 4.2. Let f ∈ C (3,3) ([0, 1]2 ). Then the partial sums sN (f ) of its Fourier series satisfy ¶ µ 1 2 k f − sN (f ) k2 = O N2

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if and only if a = c = α = β = γ = 0, i.e., f (1, 0) − f (0, 1) = f (0, 1) − f (1, 0) =

R1 0

R1 0

(f (x, 0) − f (x, 1))dx, (f (0, y) − f (1, y))dy, (4.2)

f (0, 1) + f (1, 0) = f (0, 0) + f (1, 1), ∂f ∂x (0, 1)

+

∂f ∂x (1, 0)

=

∂f ∂x (0, 0)

+

∂f ∂x (1, 1),

∂f ∂y (0, 1)

+

∂f ∂y (1, 0)

=

∂f ∂y (0, 0)

+

∂f ∂y (1, 1).

Since the number of Fourier coefficients is 2N + 1 in sN (f ), it is clear that when (4.2) holds, µ ¶ 1 k f − sN (f ) k22 = O . Nd Comparing it with Theorem 3.1, we see that in this case the partial sum approximation is better than the hyperbolic cross approximation.

References [1] V. Barthelmann, E. Novak, and K. Ritter, High dimensional polynomial interpolation on sparse grids, Advances in Computation Mathematics, 12(4) (2000), 273-288. [2] B. Boashash, Time-frequency signal analysis and processing, Second edition, Academic press, 2016. [3] W. Cheney and W. Light, A course in approximation theory, Thomson Learning, 2000. [4] A. DoVore and G. G. Lorentz, Constructive approximation, Vol. 303 of Grundlehren, Springer, Hcidelberg, 1993. [5] G.G. Lorentz, M.von Golitschck, and Ju. Makovoz, Constructive approximation, Advanced Problems, Springer, Borlin, 1996. [6] M. Griebil and J. Hamaekers, Sparse grids for the Schr¨ odinger equation, ESAIM Math. Model, Numer. Anal. 41 (2007). [7] J. Shen and H. Yu, Efficient spectral sparse grid methods and applications to high-dimensional elliptic problems, SIAM Journal on Scientific Computing, 32(6) (2010), 3228-3250. [8] J. Shen and L. Wang, Sparse spectral approximation of high-dimensional problems based on the hyperbolic cross, SIAM, J. Num. Anal., 48 (2010). [9] E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press, 1971. [10] P. Stoica and R. Moses, Spectral analysis of signals, Prentice Hall, 2005. [11] A. F. Timan, Theory at approximation of Functions of a real variable, Pergamon, 1963. [12] Z. Zhang and John C. Moore, Mathematical and physical fundamentals of climate change, Elsevier, 2015. [13] Z. Zhang, Approximation of bivariate functions via smooth extensions, The Scientific World Journal, vol. 2014, Article ID 102062, 2014. doi:10.1155/2014/102062. 119-136. [14] Z. Zhang, Environmental Data Analysis, DeGruyter, December 2016. [15] Z. Zhang, P. Jorgensen, Modulated Haar wavelet analysis of climatic background noise, Acta Appl Math, 140, 71-93, 2015

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Khatri-Rao Products and Selection Operators Arnon Ploymukda and Pattrawut Chansangiam∗ Department of Mathematics, Faculty of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand.

Abstract We develop further theory for Khatri-Rao products of Hilbert space operators in connections with selection operators. We provide two constructions related to selection operators. Then we establish certain identities and inequalities involving Khatri-Rao and Tracy-Singh products. As consequences, we obtain some characterizations for the mixed product property concerning the Khatri-Rao product of operators.

Keywords: tensor product, Khatri-Rao product, Tracy-Singh product, operator matrix Mathematics Subject Classifications 2010: 47A80, 15A69, 47A05.

1

Introduction

This paper concerns operator extensions of certain matrix products, namely, the Kronecker (tensor) product, the Tracy-Singh product, and the Khatri-Rao product. Fundamental theory for these matrix products are collected, for instance, in [1, 2, 4, 5, 10, 11, 12] and references therein. Denote by Mm,n (C) the algebra of m-by-n complex matrices. Recall that the Kronecker product of A = [aij ] ∈ Mm,n (C) and B ∈ Mp,q (C) is given by ˆ B = [aij B]ij . A⊗ Consider partitioned matrices A and B such that the (i, j)th block of A is Aij and the (k, l)th block of B is Bkl . The Tracy-Singh product [9] of A and B is defined by [[ ] ] ˆ B = Aij ⊗B ˆ kl . (1) A kl ij The Khatri-Rao product [3] is defined for two partitioned matrices A = [Aij ] and B = [Bij ] as follows [ ] ˆ ij . (2) A ˆ B = Aij ⊗B ij ∗ Corresponding

author. Email: [email protected]

1

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.2, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

Khatri-Rao Products and Selection Operators

The tensor product of Hilbert space operators can be viewed as an extension of the Kronecker product of complex matrices. Recall that the tensor product of A ∈ B(H, H′ ) and B ∈ B(K, K′ ) is the unique bounded linear operator from H ⊗ K into H′ ⊗ K′ such that (A ⊗ B)(x ⊗ y) = Ax ⊗ By for all x ∈ H and y ∈ K. Recently, the Tracy-Singh product and the Khatri-Rao product for matrices were generalized to those for operators acting on the direct sum of Hilbert spaces, see [6, 7, 8]. Fundamental algebraic and order properties of operator Khatri-Rao products are investigated in [8]. That paper also provides a construction of a unital positive linear map taking the Tracy-Singh product of two operators to their Khatri-Rao product. Such a linear map appears in the form X 7→ Z ∗ AZ where Z is an isometry, called a selection operator. See details in Section 2. The present paper contains further development on operator Khatri-Rao products in relations with Tracy-Singh products and selection operators. First, we provide two constructions related to selection operators (see Section 3). Consequently, we establish some operator identities and inequalities involving Khatri-Rao and Tracy-Singh products (see Section 4). Finally, we obtain some characterizations for the mixed product property concerning the Khatri-Rao product of operators (see Section 5).

2

Tracy-Singh products and Khatri-Rao products for operators

Throughout this paper, let H, H′ , K and K′ be complex separable Hilbert spaces. When X and Y are Hilbert spaces, let us denote by B(X , Y) the space of all bounded linear operators from X into Y and abbreviate B(X , X ) to B(X ). Capital letters always denote a Hilbert space operator. In particular, I and O stand for the identity and the zero operator, respectively. In order to define Tracy-Singh products of operators, we fix the following decompositions H =

n ⊕ j=1

Hj ,

H′ =

m ⊕ i=1

Hi′ ,

K =

q ⊕ j=1

Kj ,

K′ =

p ⊕

Ki′ .

(3)

i=1

where all of Hj , Hi′ , Kl , Kk′ are Hilbert spaces. For each j and l, let Mj : Hj → H and Nl : Kl → K be the canonical injections. For each i and k, let Pi : H′ → Hi′ and Qk : K′ → Kk′ be the canonical projections. Given A ∈ B(H, H′ ), put Aij = Pi AMj ∈ B(Hj , Hi′ ) for each i, j. Thus we can write A in the operatorm,n matrix form A = [Aij ]i,j=1 . Similarly, given B ∈ B(K, K′ ), let Bkl = Qk BNl ∈ B(Kl , Kk′ ) for each k = 1, . . . , p and l = 1, . . . , q. We can identify B with the p,q operator matrix B = [Bkl ]k,l=1 . Definition 1. The Tracy-Singh product of⊕A and B is defined to be the bounded ⊕n,q m,p ′ ′ linear operator from j,l=1 Hj ⊗ Kl to i,k=1 Hi ⊗ Kk represented by [ ] A  B = [Aij ⊗ Bkl ]kl ij . (4)

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A. Ploymukda and P. Chansangiam

If both factor A and B consist of only one block, then A  B = A ⊗ B. Lemma 2 ([6]). The following properties of the Tracy-Singh product for operators hold (provided that each term is well-defined): 1. Compatibility with adjoints: (A  B)∗ = A∗  B ∗ . 2. Mixed-product property: (A  B)(C  D) = AC  BD. 3. Monotonicity: if A > B > 0 and C > D > 0, then A  B > C  D > 0. From now on, we fix the decomposition (3), and assume n = q and m = p. Definition 3. The Khatri-Rao product of A = [Aij ]m,n and B = [Bij ]m,n is ⊕n i,j=1 ⊕m i,j=1 ′ ′ defined to be a bounded linear operator from j=1 Hj ⊗ Kj to i=1 Hi ⊗ Ki represented by the operator matrix A

m,n

B = [Aij ⊗ Bij ]i,j=1 .

(5)

Lemma 4 ([8]). For A ∈ B(H, H′ ) and B ∈ B(K, K′ ), we have (A A∗ B ∗ .

B)∗ =

Fix an ordered tuple (H, H′ , K, K′ ) of Hilbert spaces. Define the ordered pair (Z1 , Z2 ) of selection operators associated with (H, H′ , K, K′ ) by [8]:     E1 F1  ..   ..  Z1 =  .  and Z2 =  .  . (6) Em Here, for each r = 1, ..., m [ ]m,m (r) Er = Egh

Fn

:

g,h=1

m ⊕

Hk′ ⊗ Kk′ →

k=1

m ⊕

Hr′ ⊗ Kl′

l=1

(r)

with Egh is an identity operator if g = h = r and the others are zero operators. For each s = 1, ..., n, the operator Fs is defined by n n [ ]n,n ⊕ ⊕ (s) Fs = Fgh : Hi ⊗ Ki → Hs ⊗ Kj g,h=1

i=1

j=1

(s)

with Fgh is an identity operator if g = h = s and the others are zero operators. From the construction, the operator Zi is an isometry and Zi Zi∗ 6 I for i = 1, 2. When H = H′ and K = K′ , we have Z1 = Z2 . Lemma 5 ([8]). Let (Z1 , Z2 ) be the ordered pair of selection operators associated with the ordered tuple (H, H′ , K, K′ ). For any operator matrices A ∈ B(H, H′ ) and B ∈ B(K, K′ ), we have A ′

B = Z1∗ (A  B)Z2 .

(7)

′

For the case H = H and K = K , we have Z1 = Z2 := Z and hence for any A ∈ B(H) and B ∈ B(K), A

B = Z ∗ (A  B)Z.

318

(8)

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Khatri-Rao Products and Selection Operators

3

Two constructions related to selection operators

In this section, we construct certain operators related to selection operators. Theorem 6. Let (Z1 , Z2 ) be the ordered pair of selection operators associated with an ordered tuple (H, H′ , K, K′ ). Then there exist operators V :

m−1 m ⊕⊕

m ⊕ m ⊕

Hi′ ⊗ Kj′ →

i=1 j=1

W :

n−1 n ⊕⊕

Hi′ ⊗ Kj′ ,

i=1 j=1 n ⊕ n ⊕

Hi ⊗ Kj →

i=1 j=1

Hi ⊗ Kj

i=1 j=1

such that Z1∗ V = 0, Z2∗ W = 0, Z1 Z1∗ + V V ∗ = I and Z2 Z2∗ + W W ∗ = I. If, in addition, H = H′ and K = K′ , we have V = W . Proof. Let 

 V1   V =  ... 

(9)

Vm where V

(r)

m m m [ ]m,m2 −1 ⊕ ⊕ ⊕ (r) ′ ′ Hr′ ⊗ Ki′ = Vkl : Hi ⊗ Ki → k,l=1

i=1

i=1

j=1 i+j 0. Definition 2.1 ([9]). A function △ : [0, 1] × [0, 1] → [0, 1] is called a triangular norm (for short, t − norm) if the following conditions are satisfied for any a, b, c, d ∈ [0, 1] : (△ − 1) △(a, 1) = a; (△ − 2) △(a, b) = △(b, a); (△ − 3) △(a, b) ≥ △(c, d), for a ≥ c, b ≥ d; (△ − 4) △(△(a, b), c) = △(a, △(b, c)). Definition 2.2 ([2]). Let Φ denote the class of all functions ϕ : R+ → R+ satisfies the following conditions: (i) ϕ(t) = 0 if and only if t = 0; (ii) ϕ(t) is strictly increasing and ϕ(t) → ∞ as t → ∞; (iii) ϕ is left continuous in (0, +∞); (iv) ϕ is continuous at 0. Definition 2.3 ([8]). Let Ψ denote the class of all functions ψ : R+ → R+ satisfies the following conditions: 2 327

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(1)ψ is non-decreasing; (2) ψ(t + s) ≤ ψ(t) + ψ(s) for all t, s ∈ [0, 1). Remark 2.1 ([8]). Ψ also satisfies that Ψ is continuous and Ψ(t) = 0 if and only if t = 0. It is easy to see that the notion of Ψ is stronger than Definition 2.3 in [8]. And it is obvious that the following condition holds: (3) ψ(p + q + t + s) ≤ ψ(p) + ψ(q) + ψ(t) + ψ(s) for all p, q, t, s ∈ [0, 1). Definition 2.4 ([18]). A Menger probabilistic G-metric space (briefly, a PGM-space) is a triple (X, G∗ , △), where X is a nonempty set, △ is a continuous t-norm, and G∗ is a mapping from X ×X ×X into D + (G∗x,y,z denotes the value of G∗ at the point (x,y,z)) satisfying the following conditions: (PGM-1) G∗x,y,z (t) = 1 for x, y, z ∈ X and t > 0 if and only if x = y = z; (PGM-2) G∗x,x,y (t) ≥ G∗x,y,z (t) for x, y, z ∈ X with z ̸= y and t > 0; (PGM-3) G∗x,y,z (t) = G∗x,z,y (t) = G∗y,x,z (t) = · · · (symmetry in all three variables); (PGM-4) G∗x,y,z (t + s) ≥ △(G∗x,a,a (t), G∗a,y,z (s)) for x, y, z, a ∈ X and s, t > 0. Definition 2.5 ([1]). Let (X, G∗ , △) be a PGM-space, and {xn } is a sequence in X. (1) {xn } is said to be convergent to x ∈ X (write xn → x), if for any ε > 0 and 0 < δ < 1, there exists a positive integer Mε,λ such that xn ∈ Nx0 (ε, λ) whenever n > Mε,λ ; (2) {xn } is said to be Cauchy sequence, if for any ε > 0 and 0 < δ < 1, there exists a positive integer Mε,λ such that G∗xn ,xm ,xl > 1 − δ whenever n, m, l > Mε,λ ; (3) (X, G∗ , △) is said to be complete, if every Cauchy sequence in X converges to a point in X. Definition 2.6 ([7]). Let X be a non-empty set and F : X × X → X and g : X → X. We say F and g are commutative if g(F (x, y)) = F (g(x), g(y)) for all x, y ∈ X. Definition 2.7 ([7]). Let (X, ≤) be a partially ordered set and F : X × X → X is said to possess the mixed monotone property if F is monotone non-decreasing in its first argument and is monotone non-increasing in its second argument, that is, for any x, y ∈ X, x1 , x2 ∈ X, x1 ≤ x2 ⇒ F (x1 , y) ≤ F (x2 , y) and y1 , y2 ∈ X, y1 ≤ y2 ⇒ F (x, y2 ) ≤ F (x, y1 ) 3 328

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Definition 2.8 ([11]). Let (X, ≤) be a partially ordered set and F : X × X → X is said to have the mixed g−monotone property if F is monotone g−non-decreasing in its first argument and is monotone g−non-decreasing in its second argument, that is, for any x, y ∈ X. x1 , x2 ∈ X, g(x1 ) ≤ g(x2 ) ⇒ F (x1 , y) ≤ F (x2 , y) and y1 , y2 ∈ X, g(y1 ) ≤ g(y2 ) ⇒ F (x, y2 ) ≤ F (x, y1 ).

3

Coupled coincidence point results in partially ordered complete Menger probabilistic G-metric spaces In this section, We begin with the following definition which is useful to prove some new coupled

coincidence point theorems and coupled fixed point theorems in partially ordered complete Menger probabilistic G-metric spaces. Definition 3.1 Let (X, G∗ , △) be a Menger PGM-space with △ (a continuous t − norm), T : X 4 → X and g : X → X be two mappings satisfying the following condition: ψ(

1 G∗T (x,y,z,w),T (u,v,p,q),T (a,b,c,d) (ϕ(λt))

1 1 1 − 1) ≤ ψ( ∗ −1+ ∗ −1 4 Gg(x),g(u),g(a) (ϕ(t)) Gg(y),g(v),g(b) (ϕ(t)) +

1 G∗g(z),g(p),g(c) (ϕ(t))

−1+

1 G∗g(w),g(q),g(d) (ϕ(t))

− 1). (3.1)

for all t > 0, and x, y, z, w, u, v, p, q, a, b, c, d ∈ X, g(x) ≤ g(u) ≤ g(a), g(y) ≥ g(v) ≥ g(b), g(z) ≤ g(p) ≤ g(c) and g(w) ≥ g(q) ≥ g(d), where λ ∈ (0, 1), ψ ∈ Ψ and ϕ ∈ Φ. Then mappings T and g are said to satisfy ψ-contractive condition. Theorem 3.1 Let(X, ≤) be a partially ordered set and (X, G∗ , △) be a complete PGM-space with a continuous t − norm. suppose that T : X 4 → X and g : X → X are the mappings with mixed g−monotone property and satisfy ψ-contractive condition, such that G∗g(x),g(u),g(a) > 0, G∗g(y),g(v),g(b) > 0, G∗g(z),g(p),g(c) > 0, G∗g(w),g(q),g(d) > 0. Suppose T (X 4 ) ⊆ g(X), g is continuous and commutes with T . Assuming that either (a) T is continuous, or (b) X has the following properties: (I) If a non-decreasing sequence xn → x, zn → z, then xn ≤ x, zn ≤ z for all n; 4 329

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(II) If a non-increasing sequence yn → y, wn → w, then yn ≤ y, wn ≤ w for all n. If there exist x0 , y0 , z0 , w0 ∈ X, such that g(x0 ) ≤ T (x0 , y0 , z0 , w0 ), g(z0 ) ≤ T (z0 , w0 , x0 , y0 ), g(y0 ) ≥ T (y0 , z0 , w0 , x0 ) and g(w0 ) ≥ T (w0 , x0 , y0 , z0 ), then there exist x, y, z, w ∈ X, such that g(x) = T (x, y, z, w), g(y) = T (y, z, w, x), g(z) = T (z, w, x, y), g(w) = T (w, x, y, z), that is, T and g have a coupled coincidence point.

Proof

Let x0 , y0 , z0 , w0 ∈ X, such that g(x0 ) ≤ T (x0 , y0 , z0 , w0 ), g(z0 ) ≤ T (z0 , w0 , x0 , y0 ) and

g(y0 ) ≥ T (y0 , z0 , w0 , x0 ), g(w0 ) ≥ T (w0 , x0 , y0 , z0 ), since T (X 4 ) ⊆ g(X), we can choose x1 , y1 , z1 , w1 ∈ X, such that g(x1 ) = T (x0 , y0 , z0 , w0 ), g(y1 ) = T (y0 , z0 , w0 , x0 ),

(3.2)

g(z1 ) = T (z0 , w0 , x0 , y0 ), g(w1 ) = T (w0 , x0 , y0 , z0 ).

(3.3)

Continuing this process we can construct sequences {xn }, {yn }, {zn } and {wn } in X, such that g(xn+1 ) = T (xn , yn , zn , wn ), g(yn+1 ) = T (yn , zn , wn , xn ) for all n ≥ 0, g(zn+1 ) = T (zn , wn , xn , yn ), g(wn+1 ) = T (wn , xn , yn , zn ) for all n ≥ 0, we shall show that g(xn ) ≤ g(xn+1 ), g(yn ) ≥ g(yn+1 ), g(zn ) ≤ g(zn+1 ), g(wn ) ≥ g(wn+1 ). We shall use the mathematical induction to show that (3.4) holds. Let n = 0, since g(x0 ) ≤ T (x0 , y0 , z0 , w0 ), g(y0 ) ≥ T (y0 , z0 , w0 , x0 ), g(z0 ) ≤ T (z0 , w0 , x0 , y0 ), g(w0 ) ≥ T (w0 , x0 , y0 , z0 ), by (3.2) and (3.3), we have g(x0 ) ≤ g(x1 ), g(y0 ) ≥ g(y1 ), g(z0 ) ≤ g(z1 ), g(w0 ) ≥ g(w1 ). Thus (3.4) holds for n = 0. Now we suppose that (3.4) holds for some n = i, i ∈ Z+ , we get g(xi ) ≤ g(xi+1 ), g(yi ) ≥ g(yi+1 ), g(zi ) ≤ g(zi+1 ), g(wi ) ≥ g(wi+1 ).

5 330

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(3.4)

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.2, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

Let n = i + 1, owing to the property of mixed g-monotone, we have g(xi+2 ) = T (xi+1 , yi+1 , zi+1 , wi+1 ) ≥ T (xi , yi+1 , zi , wi+1 ) ≥ T (xi , yi , zi , wi ) = g(xi+1 ), g(yi+2 ) = T (yi+1 , zi+1 , wi+1 , xi+1 ) ≤ T (yi , zi+1 , wi , xi+1 ) ≤ T (yi , zn , wi , xi ) = g(yi+1 ). Similarly, we obtain g(zi+2 ) ≥ g(zi+1 ), g(wi+2 ) ≤ g(wi+1 ). By the mathematical induction, we conclude that (3.4) holds for all n > 0. Therefore g(x0 ) ≤ g(x1 ) ≤ g(x2 ) ≤ ... ≤ g(xn ) ≤ g(xn+1 ) ≤ · · ·; g(y0 ) ≥ g(y1 ) ≥ g(y2 ) ≥ ... ≥ g(yn ) ≥ g(yn+1 ) ≤ · · ·; g(z0 ) ≤ g(z1 ) ≤ g(z2 ) ≤ ... ≤ g(zn ) ≤ g(zn+1 ) ≤ · · ·; g(w0 ) ≥ g(w1 ) ≥ g(w2 ) ≥ ... ≥ g(wn ) ≥ g(wn+1 ) ≤ · · ·. In view of the fact, we have sup G∗g(x2 ),g(x1 ),g(x0 ) (t) = 1, sup G∗g(y2 ),g(y1 ),g(y0 ) (t) = 1, t∈R

t∈R

sup G∗g(z2 ),g(z1 ),g(z0 ) (t) t∈R

= 1, sup G∗g(w2 ),g(w1 ),g(w0 ) (t) = 1, t∈R

and by (ii) of Definition 2.2, we can find some t > 0, such that G∗g(x2 ),g(x1 ),g(x0 ) (ϕ(t)) > 0, G∗g(y2 ),g(y1 ),g(y0 ) (ϕ(t)) > 0, G∗g(z2 ),g(z1 ),g(z0 ) (ϕ(t)) > 0, G∗g(w2 ),g(w1 ),g(w0 ) (ϕ(t)) > 0, for g(x0 ) ≤ g(x1 ) ≤ g(x2 ), g(y0 ) ≥ g(y1 ) ≥ g(y2 ), g(z0 ) ≤ g(z1 ) ≤ g(z2 , g(w0 ) ≥ g(w1 ) ≥ g(w2 ), which implies that t t G∗g(x2 ),g(x1 ),g(x0 ) (ϕ( )) > 0, G∗g(y2 ),g(y1 ),g(y0 ) (ϕ( )) > 0, λ λ t t ∗ ∗ Gg(z2 ),g(z1 ),g(z0 ) (ϕ( )) > 0, Gg(w2 ),g(w1 ),g(w0 ) (ϕ( )) > 0. λ λ Then by (3.1), we get ψ(

1 G∗g(x3 ),g(x2 ),g(x1 ) (ϕ(t))

− 1) = ψ(

1 G∗T (x2 ,y2 ,z2 ,w2 ),T (x1 ,y1 ,z1 ,w1 ),T (x0 ,y0 ,z0 ,w0 ) (ϕ(t))

− 1)

1 t t ≤ ψ(G∗g(x2 ),g(x1 ),g(x0 ) (ϕ( )) − 1 + G∗g(y2 ),g(y1 ),g(y0 ) (ϕ( )) − 1 4 λ λ t t ∗ ∗ + Gg(z2 ),g(z1 ),g(z0 ) (ϕ( )) − 1 + G(w2 ),g(w1 ),g(w0 ) (ϕ( )) − 1). λ λ 6 331

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Similarly, ψ(

1 t t 1 − 1) ≤ ψ(G∗g(y2 ),g(y1 ),g(y0 ) (ϕ( )) − 1 + G∗g(z2 ),g(z1 ),g(z0 ) (ϕ( )) − 1 G∗g(y3 ),g(y2 ),g(y1 ) (ϕ(t)) 4 λ λ +

ψ(

G∗g(w2 ),g(w1 ),g(w0 ) (ϕ(

t t )) − 1 + G∗(x2 ),g(x1 ),g(x0 ) (ϕ( )) − 1), λ λ

1 1 t t − 1) ≤ ψ(G∗g(z2 ),g(z1 ),g(z0 ) (ϕ( )) − 1 + G∗g(w2 ),g(w1 ),g(w0 ) (ϕ( )) − 1 G∗g(z3 ),g(z2 ),g(z1 ) (ϕ(t)) 4 λ λ

t t + )) − 1 + G∗(y2 ),g(y1 ),g(y0 ) (ϕ( )) − 1), λ λ 1 1 t t ψ( ∗ − 1) ≤ ψ(G∗g(w2 ),g(w1 ),g(w0 ) (ϕ( )) − 1 + G∗g(x2 ),g(x1 ),g(x0 ) (ϕ( )) − 1 Gg(w3 ),g(w2 ),g(w1 ) (ϕ(t)) 4 λ λ

(3.6)

(3.7)

G∗g(x2 ),g(x1 ),g(x0 ) (ϕ(

+

G∗g(y2 ),g(y1 ),g(y0 ) (ϕ(

t t )) − 1 + G∗(z2 ),g(z1 ),g(z0 ) (ϕ( )) − 1). λ λ

(3.8)

From (3.5)-(3.8), we have ψ(

1 G∗g(x3 ),g(x2 ),g(x1 ) (ϕ(t)) + ψ(

≤ ψ( +

− 1) + ψ(

1 G∗g(w3 ),g(w2 ),g(w1 ) (ϕ(t)) 1

G∗g(x2 ),g(x1 ),g(x0 ) (ϕ( λt ))

1 G∗g(y3 ),g(y2 ),g(y1 ) (ϕ(t))

1

1

− 1)

G∗g(z3 ),g(z2 ),g(z1 ) (ϕ(t))

− 1)

−1+

G∗g(w2 ),g(w1 ),g(w0 ) (ϕ( λt ))

− 1) + ψ(

1 G∗g(y2 ),g(y1 ),g(y0 ) (ϕ( λt ))

−1+

1 G∗g(z2 ),g(z1 ),g(z0 ) (ϕ( λt ))

−1

− 1).

By (3) of Remark 2.1, we have ψ(

1 1 1 −1+ ∗ −1+ ∗ −1 G∗g(x3 ),g(x2 ),g(x1 ) (ϕ(t)) Gg(y3 ),g(y2 ),g(y1 ) (ϕ(t)) Gg(z3 ),g(z2 ),g(z1 ) (ϕ(t)) 1 − 1) G∗g(w3 ),g(w2 ),g(w1 ) (ϕ(t))

+ ≤ ψ(

1 1 1 − 1) + ψ( ∗ − 1) + ψ( ∗ − 1) G∗g(x3 ),g(x2 ),g(x1 ) (ϕ(t)) Gg(y3 ),g(y2 ),g(y1 ) (ϕ(t)) Gg(z3 ),g(z2 ),g(z1 ) (ϕ(t))

+ ψ(

1 − 1), G∗g(w3 ),g(w2 ),g(w1 ) (ϕ(t))

which implies that ψ(

1 1 1 −1+ ∗ −1+ ∗ −1 G∗g(x3 ),g(x2 ),g(x1 ) (ϕ(t)) Gg(y3 ),g(y2 ),g(y1 ) (ϕ((t))) Gg(z3 ),g(z2 ),g(z1 ) (ϕ(t)) +

≤ ψ( +

1 − 1) G∗g(w3 ),g(w2 ),g(w1 ) (ϕ(t)) 1 G∗g(x2 ),g(x1 ),g(x0 ) (ϕ( λt ))

−1+

1 G∗g(w2 ),g(w1 ),g(w0 ) (ϕ( λt ))

1 G∗g(y2 ),g(y1 ),g(y0 ) (ϕ( λt ))

−1+

1 G∗g(z2 ),g(z1 ),g(z0 ) (ϕ( λt ))

−1

− 1). 7 332

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Using the fact that ψ is non-decreasing, we get 1 1 1 −1+ ∗ −1+ ∗ −1 G∗g(x3 ),g(x2 ),g(x1 ) (ϕ(t)) Gg(y3 ),g(y2 ),g(y1 ) (ϕ(t)) Gg(z3 ),g(z2 ),g(z1 ) (ϕ(t)) + ≤

1 −1 G∗g(w3 ),g(w2 ),g(w1 ) (ϕ(t))

1 1 1 −1+ ∗ −1+ ∗ −1 G∗g(x2 ),g(x1 ),g(x0 ) (ϕ( λt )) Gg(y2 ),g(y1 ),g(y0 ) (ϕ( λt )) Gg(z2 ),g(z1 ),g(z0 ) (ϕ( λt )) +

1 G∗g(w2 ),g(w1 ),g(w0 ) (ϕ( λt ))

− 1.

From the above inequalities we deduce that G∗g(x3 ),g(x2 ),g(x1 ) (ϕ(t)) > 0, G∗g(y3 ),g(y2 ),g(y1 ) (ϕ(t)) > 0, G∗g(z3 ),g(z2 ),g(z1 ) (ϕ(t)) > 0, G∗g(w3 ),g(w2 ),g(w1 ) (ϕ(t)) > 0, and t t G∗g(x3 ),g(x2 ),g(x1 ) (ϕ( )) > 0, G∗g(y3 ),g(y2 ),g(y1 ) (ϕ( )) > 0, λ λ t t ∗ ∗ Gg(z3 ),g(z2 ),g(z1 ) (ϕ( )) > 0, Gg(w3 ),g(w2 ),g(w1 ) (ϕ( )) > 0. λ λ Again, by using (3.1), we have 1

+ ≤

1

+ ≤

1

−1+

1 G∗g(w3 ),g(w2 ),g(w1 ) (ϕ( λt )) 1

G∗g(x2 ),g(x1 ),g(x0 ) (ϕ( λt2 )) +

G∗g(y4 ),g(y3 ),g(y2 ) (ϕ(t))

−1+

1 G∗g(z4 ),g(z3 ),g(z2 ) (ϕ(t))

−1

−1

G∗g(w4 ),g(w3 ),g(w2 ) (ϕ(t))

G∗g(x3 ),g(x2 ),g(x1 ) (ϕ( λt ))

1

−1+

G∗g(x4 ),g(x3 ),g(x2 ) (ϕ(t))

1 G∗g(y3 ),g(y2 ),g(y1 ) (ϕ( λt ))

−1+

1 G∗g(z3 ),g(z2 ),g(z1 ) (ϕ( λt ))

−1

−1

−1+

1 G∗g(w2 ),g(w1 ),g(w0 ) (ϕ( λt2 ))

1 G∗g(y2 ),g(y1 ),g(y0 ) (ϕ( λt2 ))

−1+

1 G∗g(z2 ),g(z1 ),g(z0 ) (ϕ( λt2 ))

−1

− 1.

Repeating the above procedure successively, we obtain 1 1 1 −1+ ∗ −1+ ∗ −1 G∗g(xn+2 ),g(xn+1 ),g(xn ) (ϕ(t)) Gg(yn+2 ),g(yn+1 ),g(yn ) (ϕ(t)) Gg(zn+2 ),g(zn+1 ),g(zn ) (ϕ(t)) + ≤

1 −1 G∗g(wn+2 ),g(wn+1 ),g(wn ) (ϕ(t)) 1

G∗g(x2 ),g(x1 ),g(x0 ) (ϕ( λtn )) +

−1+

1 G∗g(w2 ),g(w1 ),g(w0 ) (ϕ( λtn ))

1 G∗g(y2 ),g(y1 ),g(y0 ) (ϕ( λtn ))

−1+

1 G∗g(z2 ),g(z1 ),g(z0 ) (ϕ( λtn ))

−1

− 1. 8 333

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If we replace x0 with xk in the previous inequalities, then for all n > k, we get 1 1 −1+ ∗ −1 ∗ k Gg(xn+2 ),g(xn+1 ),g(xn ) (ϕ(λ t)) Gg(yn+2 ),g(yn+1 ),g(yn ) (ϕ(λk t)) + ≤

1 G∗g(zn+2 ),g(zn+1 ),g(zn ) (ϕ(λk t)) 1

kt )) G∗g(xk+2 ),g(xk+1 ),g(xk ) (ϕ( λλn−k

+

−1+

−1+

1

1 G∗g(wn+2 ),g(wn+1 ),g(wn ) (ϕ(λk t)) 1

t )) G∗g(yk+2 ),g(yk+1 ),g(yk ) (ϕ( λλn−k k

−1+

kt )) G∗g(zk+2 ),g(zk+1 ),g(zk ) (ϕ( λλn−k

−1

−1

1 kt )) G∗g(wk+2 ),g(wk+1 ),g(wk ) (ϕ( λλn−k

− 1.

k

t Since ϕ( λλn−k ) → ∞ as n → ∞ for all 0 < k < n, we have

λk t λk t ∗ )) = 1, lim G (ϕ( )) = 1, n→∞ n→∞ g(yk+2 ),g(yk+1 ),g(yk ) λn−k λn−k λk t λk t lim G∗g(zk+2 ),g(zk+1 ),g(zk ) (ϕ( n−k )) = 1, lim G∗g(wk+2 ),g(wk+1 ),g(wk ) (ϕ( n−k )) = 1. n→∞ n→∞ λ λ lim G∗g(xk+2 ),g(xk+1 ),g(xk ) (ϕ(

Thus, lim (

n→∞

1 − 1) G∗g(xn+2 ),g(xn+1 ),g(xn ) (ϕ(λk t)) 1

≤ lim (

k n→∞ G∗ g(xn+2 ),g(xn+1 ),g(xn ) (ϕ(λ t))

+ lim (

n→∞

1 G∗g(zn+2 ),g(zn+1 ),g(zn ) (ϕ(λk t))

1 G∗g(yn+2 ),g(yn+1 ),g(yn ) (ϕ(λk t)) 1

−1+

G∗g(wn+2 ),g(wn+1 ),g(wn ) (ϕ(λk t))

−1

− 1) ≤ 0,

1 − 1) G∗g(yn+2 ),g(yn+1 ),g(yn ) (ϕ(λk t))

≤ lim ( n→∞

+

−1+

1 1 −1+ ∗ −1 G∗g(yn+2 ),g(yn+1 ),g(yn ) (ϕ(λk t)) Gg(zn+2 ),g(zn+1 ),g(zn ) (ϕ(λk t))

1 1 −1+ ∗ − 1) ≤ 0, G∗g(wn+2 ),g(wn+1 ),g(wn ) (ϕ(λk t)) Gg(xn+2 ),g(xn+1 ),g(xn ) (ϕ(λk t))

similarly lim (

n→∞

1 G∗g(zn+2 ),g(zn+1 ),g(zn ) (ϕ(λk t))

− 1) ≤ 0,

lim (

n→∞

1 G∗g(wn+2 ),g(wn+1 ),g(wn ) (ϕ(λk t))

− 1) ≤ 0,

which implies that lim (G∗g(xn+2 ),g(xn+1 ),g(xn ) (ϕ(λk t)) = 1, lim (G∗g(yn+2 ),g(yn+1 ),g(yn ) (ϕ(λk t)) = 1,

(3.9)

lim (G∗g(zn+2 ),g(zn+1 ),g(zn ) (ϕ(λk t)) = 1, lim (G∗g(wn+2 ),g(wn+1 ),g(wn ) (ϕ(λk t)) = 1.

(3.10)

n→∞

n→∞

n→∞

n→∞

Now, let ϵ > 0 be given, by (i) and (iv) of Definition 2.2, we can find k ∈ Z+ such that ϕ(λk t) < ϵ, it follows from (3.9) and (3.10) that lim (G∗g(xn+2 ),g(xn+1 ),g(xn ) (ϵ)) ≥ lim (G∗g(xn+2 ),g(xn+1 ),g(xn ) (ϕ(λk t)) = 1,

n→∞

lim (G∗g(yn+2 ),g(yn+1 ),g(yn ) (ϵ)) n→∞

n→∞

≥ lim (G∗g(yn+2 ),g(yn+1 ),g(yn ) (ϕ(λk t)) = 1, n→∞

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similarly, lim (G∗g(zn+2 ),g(zn+1 ),g(zn ) (ϵ)) ≥ 1.

lim (G∗g(wn+2 ),g(wn+1 ),g(wn ) (ϵ)) ≥ 1,

n→∞

n→∞

By using Menger triangle inequality, we obtain ϵ ϵ G∗g(xn+p ),g(xn+1 ),g(xn ) (ϵ) ≥ △(G∗g(xn+p ),g(xn+p−1 ),g(xn+p−1 ) ( ), △(G∗g(xn+p−1 ),g(xn+p−2 ),g(xn+p−2 ) ( ) p p ϵ · ··, G∗g(xn+2 ),g(xn+1 ),g(xn ) ( )). p Thus, letting n → ∞ and making use of (3.9) and (3.10), for any integer, we get lim G∗g(xn+p ),g(xn+1 ),g(xn ) (ϵ) = 1 for every ϵ > 0.

n→∞

Hence g(xn ) is a Cauchy sequence. Similarly, we can prove that g(yn ), g(zn ), g(wn ) are also Cauchy sequences. Since (X, G∗ , △) is complete, there exist x, y, z, w ∈ X such that lim g(xn ) = x,

n→∞

lim g(yn ) = y,

n→∞

lim g(zn ) = z,

n→∞

(3.11)

lim g(wn ) = w.

n→∞

From (3.11) and the continuity of g, we have lim g(g(xn )) = g(x),

n→∞

lim g(g(yn )) = g(y),

n→∞

lim g(g(zn )) = g(z),

n→∞

lim g(g(wn )) = g(w).

n→∞

From (3.2), (3.3) and the commutativity of T and g, we have g(g(xn+1 )) = g(T (xn , yn , zn , wn )) = T (g(xn ), g(yn ), g(zn ), g(wn )),

(3.12)

g(g(yn+1 )) = g(T (yn , zn , wn , xn )) = T (g(yn ), g(zn ), g(wn ), g(xn )),

(3.13)

g(g(zn+1 )) = g(T (zn , wn , xn , yn )) = T (g(zn ), g(wn ), g(xn ), g(yn )),

(3.14)

g(g(wn+1 )) = g(T (wn , xn , yn , zn )) = T (g(wn ), g(xn ), g(yn ), g(zn )).

(3.15)

Now,we show that g(x) = T (x, y, z, w), g(y) = T (y, z, w, x), g(z) = T (z, w, x, y), g(w) = T (w, x, y, z). Suppose that the assumption (a) holds. Taking the limit of (3.11) as n → ∞, by (3.12) ∼ (3.15) and the continuity of T , we get g(x) = lim g(g(xn+1 )) = lim T (g(xn , yn , zn , wn )) = T ( lim g(xn ), lim g(yn ), lim g(zn ), lim g(wn )) n→∞

n→∞

n→∞

n→∞

n→∞

n→∞

= T (x, y, z, w),

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g(y) = lim g(g(yn+1 )) = lim T (g(yn , zn , wn , xn )) = T ( lim g(yn ), lim g(zn ), lim g(wn ), lim g(xn )) n→∞

n→∞

n→∞

n→∞

n→∞

n→∞

= T (y, z, w, x). Similarly, g(z) = T (z, w, x, y), g(w) = T (w, x, y, z). Thus we prove that g(x) = T (x, y, z, w), g(y) = T (y, z, w, x), g(z) = T (z, w, x, y), g(w) = T (w, x, y, z). Suppose now that (b) holds, since ϵ ϵ G∗g(x),T (x,y,z,w),T (x,y,z,w) (ϵ) ≥ △(G∗g(x),g(g(xn+1 )),g(g(xn+1 )) ( ), G∗g(g(xn+1 )),T (x,y,z,w),T (x,y,z,w) ( )). (3.16) 2 2 and using (i) of Definition 2.2, we find some s > 0 such that ϕ(s) < 2ϵ , since lim g(g(xn )) = g(x), lim g(g(yn )) = g(y), lim g(g(zn )) = g(z), lim g(g(wn )) = g(w).

n→∞

n→∞

n→∞

n→∞

then there exists n0 ∈ Z+ , such that G∗g(g(xn )),g(x),g(x) (ϕ(s)) > 0, G∗g(g(yn )),g(y),g(y) (ϕ(s)) > 0, G∗g(g(zn )),g(z),g(z) (ϕ(s)) > 0, G∗g(g(wn )),g(w),g(w) (ϕ(s)) > 0. for all n > n0 . Since {g(xn )}, {g(zn )} is non-decreasing and as {g(yn )}, {g(wn )} is non-increasing and g(xn ) → x, g(yn ) → y, g(zn ) → z, g(wn ) → w. By (3.1) and (3.12)-(3,15), we get ψ(

1 1 − 1) = ψ( ∗ − 1) G∗g(g(xn+1 )),T (x,y,z,w),T (x,y,z,w) (ϕ(s)) GT (g(xn ),g(yn ),g(zn ),g(wn )),T (x,y,z,w),T (x,y,z,w) (ϕ(s))

1 1 1 1 ≤ ψ( ∗ −1+ ∗ −1+ ∗ −1 4 Gg(g(xn )),g(x),g(x) (ϕ( λs )) Gg(g(yn )),g(y),g(y) (ϕ( λs )) Gg(g(zn )),g(z),g(z) (ϕ( λs )) +

1 − 1). ∗ Gg(g(w (ϕ( λs )) n )),g(w),g(w)

By the same way, we obtain ψ(

1 G∗g(g(yn+1 )),T (y,z,w,x),T (y,z,w,x) (ϕ(s))

− 1)

1 1 1 1 ≤ ψ( ∗ −1 s −1+ s −1+ ∗ ∗ 4 Gg(g(xn )),g(x),g(x) ϕ( λ ) Gg(g(yn )),g(y),g(y) ϕ( λ ) Gg(g(zn )),g(z),g(z) ϕ( λs ) +

1 G∗g(g(wn )),g(w),g(w) ϕ( λs )

− 1), 11 336

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ψ(

1 − 1) G∗g(g(zn+1 )),T (z,w,x,y),T (z,w,x,y) (ϕ(s))

1 1 1 1 ≤ ψ( ∗ −1+ ∗ −1+ ∗ −1 4 Gg(g(xn )),g(x),g(x) ϕ( λs ) Gg(g(yn )),g(y),g(y) ϕ( λs ) Gg(g(zn )),g(z),g(z) ϕ( λs ) + ψ(

1 − 1), G∗g(g(wn )),g(w),g(w) ϕ( λs ) 1

− 1)

G∗g(g(wn+1 )),T (w,x,y,z),T (w,x,y,z) (ϕ(s))

1 1 1 1 ≤ ψ( ∗ −1 s −1+ s −1+ ∗ ∗ 4 Gg(g(xn )),g(x),g(x) ϕ( λ ) Gg(g(yn )),g(y),g(y) ϕ( λ ) Gg(g(zn )),g(z),g(z) ϕ( λs ) +

1 G∗g(g(wn )),g(w),g(w) ϕ( λs )

− 1).

By the above inequalities and (3) of Remark 2.1, we have 1 G∗g(g(xn+1 )),T (x,y,z,w),T (x,y,z,w) (ϕ( 2ϵ )) ≤

1 G∗g(g(xn+1 )),T (x,y,z,w),T (x,y,z,w) (ϕ(s)) +

≤

−1≤

1 1

+

−1+

G∗g(g(zn+1 )),T (z,w,x,y),T (z,w,x,y) (ϕ(s))

G∗g(g(xn )),g(x),g(x) (ϕ( λs ))

−1+

1 G∗g(g(wn )),g(w),g(w) (ϕ( λs ))

1 G∗g(g(xn+1 )),T (x,y,z,w),T (x,y,z,w) (ϕ(s))

−1

1 G∗g(g(yn+1 )),T (y,z,w,x),T (y,z,w,x) (ϕ(s))

−1+

−1

1 G∗g(g(wn+1 )),T (w,x,y,z),T (w,x,y,z) (ϕ(s))

1 G∗g(g(yn )),g(y),g(y) (ϕ( λs ))

−1+

−1

1 G∗g(g(zn )),g(z),g(z) (ϕ( λs ))

(3.17)

−1

− 1.

Letting n → ∞ in above inequalities (3.17), we obtain ϵ lim G∗g(g(xn+1 )),T (x,y,z,w),T (x,y,z,w) ( ) = 1. 2

(3.18)

n→∞

From (3.16) and (3.18), we get G∗g(x),T (x,y,z,w),T (x,y,z,w) (ϵ) = 1 for every ϵ > 0, which implies that g(x) = T (x, y, z, w). Similarly, we show that g(y) = T (y, z, w, x), g(z) = T (z, w, x, y), g(w) = T (w, x, y, z). Thus we prove that g and T have a coupled coincidence point. Corollary 3.1 Let (X, ≤) be a partially ordered set and (X, G∗ , △) be a complete PGM-space with a continuous t − norm. Assume that T : X 4 → X has the mixed monotone property, and satisfying the following: 1 G∗T (x,y,z,w),T (u,v,p,q),T (a,b,c,d) ( 2t )

1 1 1 1 1 −1≤ ( ∗ −1+ ∗ −1+ ∗ −1+ ∗ − 1) 4 Gx,u,a (t) Gy,v,b (t) Gz,p,c (t) Gw,q,d (t)

for t > 0, G∗x,u,a (t) > 0, G∗y,v,b (t) > 0, G∗z,p,c (t) > 0, G∗w,q,d (t) > 0 and x, y, z, w, u, v, p, q, a, b, c, d ∈ X satisfying x ≤ u ≤ a, z ≤ p ≤ c , y ≥ v ≥ b and w ≥ q ≥ d. Suppose that either (a) T is continuous, or (b) X has the following properties: 12 337

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(I) if non-decreasing sequences {xn } → x, {zn } → z, then xn ≤ x, zn ≤ z for all n, (II) if non-increasing sequences {yn } → y, {wn } → w, then yn ≤ y, wn ≤ w for all n. If there exist x0 , y0 , z0 , w0 ∈ X, such that x0 ≤ T (x0 , y0 , z0 , w0 ), z0 ≤ T (z0 , w0 , x0 , y0 ) and y0 ≥ T (y0 , z0 , w0 , x0 ), w0 ≥ T (w0 , x0 , y0 , z0 ), then their exist x, y, z, w ∈ X, such that x = T (x, y, z, w), y = T (y, z, w, x), z = T (z, w, x, y), w = T (w, x, y, z), that is, T has a coupled coincidence point.

Proof Taking g = IX (the identity mapping on X), λ =

1 2

and ϕ(t) = φ(t) = t for all t ≥ 0 in

Theorem 3.1, we can easily obtain the above corollary.

4

Coupled common fixed point results in partially ordered complete Menger probabilistic G-metric spaces In the section, we prove the existence and uniqueness theorem of a coupled fixed point in partially

ordered complete Menger probabilistic G-metric spaces. Theorem 3.2 In addition to the hypotheses of Theorem 3.1, suppose that for every (x, y, z, w), (x∗ , y ∗ , z ∗ , w∗ ) ∈ X 4 there exists a (u, v, p, q) ∈ X 4 , such that (T (u, v, p, q), T (v, p, q, u), T (p, q, u, v), T (q, u, v, p)) are comparable to (T (x, y, z, w), T (y, z, w, x), T (z, w, x, y), T (w, x, y, z)) and (T (x∗ , y ∗ , z ∗ , w∗ ), T (y ∗ , z ∗ , w∗ , x∗ ), T (z ∗ , w∗ , x∗ , y ∗ ), T (w∗ , x∗ , y ∗ , z ∗ )). Then T and g have a unique coupled common fixed point, that is, there exists a unique (x, y, z, w) ∈ X 4 , such that x = g(x) = T (x, y, z, w), y = g(y) = T (y, z, w, x), z = g(z) = T (z, w, x, y), w = g(w) = T (w, x, y, z). Proof From Theorem 3.1, the set of coupled coincidences is non-empty, we shall first show that if (x, y, z, w) and (x∗ , y ∗ , z ∗ , w∗ ) are coupled coincidence points, that is, if g(x) = T (x, y, z, w), g(y) = T (y, z, w, x), g(z) = T (z, w, x, y), g(w) = T (w, x, y, z) and g(x∗ ) = T (x∗ , y ∗ , z ∗ , w∗ ), g(y ∗ ) = T (y ∗ , z ∗ , w∗ , x∗ ), g(z ∗ ) = T (z ∗ , w∗ , x∗ , y ∗ ), g(w∗ ) = T (w∗ , x∗ , y ∗ , z ∗ ), then g(x) = g(x∗ ), g(y) = g(y ∗ ), g(z) = g(z ∗ ), g(w) = g(w∗ ). 13 338

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(4.1)

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By assumption, there exists a (u, v, p, q) ∈ X 4 , such that (T (u, v, p, q), T (v, p, q, u), T (p, q, u, v), T (q, u, v, p)) is comparable to (T (x, y, z, w), T (y, z, w, x), T (z, w, x, y), T (w, x, y, z)) and (T (x∗ , y ∗ , z ∗ , w∗ ), T (y ∗ , z ∗ , w∗ , x∗ ), T (z ∗ , w∗ , x∗ , y ∗ ), T (w∗ , x∗ , y ∗ , z ∗ )). Putting u0 = u, v0 = v, p0 = p, q0 = q and u1 , v1 , p1 , q1 ∈ X, such that g(u1 ) = T (u0 , v0 , p0 , q0 ), g(v1 ) = T (v0 , p0 , q0 , u0 ), g(p1 ) = T (p0 , q0 , u0 , v0 ), g(q1 ) = T (q0 , u0 , v0 , p0 ). The proof of Theorems is similar to Theorem 3.1. We inductively define sequences {g(un )}, {g(vn )}, {g(pn )}, {g(qn )}, such that g(un+1 ) = T (un , vn , pn , qn ), g(vn+1 ) = T (vn , pn , qn , un ), g(pn+1 ) = T (pn , qn , un , vn ), g(qn+1 ) = T (qn , un , vn , pn ). Similarly, setting x0 = x, y0 = y, z0 = z, w0 = w, and x∗0 = x∗ , y0∗ = y ∗ , z0∗ = z ∗ , w0∗ = w∗ . We also define sequences {g(xn )}, {g(yn )}, {g(zn )}, {g(wn )} and {g(x∗n )}, {g(yn∗ )}, {g(zn∗ )}, {g(wn∗ )}, then it is easy to show that g(xn ) = T (x, y, z, w), g(yn ) = T (y, z, w, x), g(zn ) = T (z, w, x, y), g(wn ) = T (w, x, y, z) and g(x∗n ) = T (x∗ , y ∗ , z ∗ , w∗ ), g(yn∗ ) = T (y ∗ , z ∗ , w∗ , x∗ ), g(zn∗ ) = T (z ∗ , w∗ , x∗ , y ∗ ), g(wn∗ ) = T (w∗ , x∗ , y ∗ , z ∗ ). Since (T (x, y, z, w), T (y, z, w, x), T (z, w, x, y), T (w, x, y, z)) = (g(x1 ), g(y1 ), g(z1 ), g(w1 )) = (g(x), g(y), g(z), g(w)) and (T (u, v, p, q), T (v, p, q, u), T (p, q, u, v), T (q, u, v, p)) = (g(u1 ), g(v1 ), g(p1 ), g(q1 )) are comparable, then we have g(x) ≤ g(u1 ), g(z) ≤ g(p1 ), g(y) ≥ g(v1 ) and g(w) ≥ g(q1 ). It is easy to show that (g(x), g(y), g(w), g(z)) and (g(un ), g(vn ), g(pn ), g(qn )) are comparable, that is, g(x) ≤ g(xn ), g(z) ≤ g(zn ), g(y) ≥ g(yn ) and g(w) ≥ g(wn ), for all n ≥ 1. Following the proof of Theorem 3.1, we can find some t > 0 such that t t G∗g(x),g(un ,g(un ) (ϕ( )) > 0, G∗g(y),g(vn ,g(vn ) (ϕ( )) > 0 for all n ≥ 0, λ λ t t G∗g(z),g(pn ,g(pn ) (ϕ( )) > 0, G∗g(z),g(qn ,g(qn ) (ϕ( )) > 0 for all n ≥ 0. λ λ Thus from (3.1) ψ(

1 G∗g(x),g(un+1 ),g(un+1 ) (ϕ(t))

− 1) = ψ(

1 G∗T (x,y,z,w),T (un ,vn ,pn ,qn ),T (un ,vn ,pn ,qn ) (ϕ(t))

− 1)

1 1 1 1 1 ≤ ψ( ∗ −1+ ∗ −1+ ∗ −1+ ∗ − 1). t t t 4 Gx,un ,un (ϕ( λ )) Gy,vn ,vn (ϕ( λ )) Gz,pn ,pn (ϕ( λ )) Gw,qn ,qn (ϕ( λt ))

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By Remark 2.4, we get 1 1 1 −1+ ∗ −1+ ∗ −1 G∗g(x),g(un+1 ),g(un+1 ) (ϕ(t)) Gg(y),g(vn+1 ),g(vn+1 ) (ϕ(t)) Gg(z),g(pn+1 ),g(pn+1 ) (ϕ(t)) + ≤

1 −1 G∗g(w),g(qn+1 ),g(qn+1 ) (ϕ(t))

1 1 1 −1+ ∗ −1+ ∗ −1 G∗g(x),g(un ),g(un ) (ϕ( λt )) Gg(y),g(vn ),g(vn ) (ϕ( λt )) Gg(z),g(pn ),g(pn ) (ϕ( λt )) +

1 −1 G∗g(w),g(qn ),g(qn ) (ϕ( λt ))

(4.2)

.. . ≤

1 G∗g(x),g(u0 ),g(u0 ) (ϕ( λtn )) +

1

−1+

1 G∗g(w),g(q0 ),g(q0 ) (ϕ( λtn ))

G∗g(y),g(v0 ),g(v0 ) (ϕ( λtn ))

−1+

1 G∗g(z),g(p0 ),g(p0 ) (ϕ( λtn ))

−1

− 1.

We replace uk with u0 in (4.2), we get 1 G∗g(x),g(un+1 ),g(un+1 ) ϕ(λk t) + ≤

−1+

1 G∗g(w),g(qn+1 ),g(qn+1 ) ϕ(λk t) 1

G∗g(x),g(uk ),g(uk ) (ϕ( +

λk t λn−k

))

1 G∗g(y),g(vn+1 ),g(vn+1 ) ϕ(λk t)

1

1 G∗g(z),g(pn+1 ),g(pn+1 ) ϕ(λk t)

−1

−1

−1+

kt G∗g(w),g(qk ),g(qk ) (ϕ( λλn−k ))

−1+

1 G∗g(y),g(vk ),g(vk ) (ϕ(

λk t λn−k

))

−1+

1 t G∗g(z),g(pk ),g(pk ) (ϕ( λλn−k )) k

−1

− 1,

for all n > k. Letting n → ∞, we obtain lim G∗g(x),g(un+1 ,g(un+1 )(ϕ(λk t)) = 1, lim G∗g(y),g(vn+1 ,g(vn+1 )(ϕ(λk t)) = 1.

n→∞

lim G∗ k n→∞ g(z),g(pn+1 ,g(pn+1 )(ϕ(λ t))

n→∞

= 1, lim G∗g(w),g(qn+1 ,g(qn+1 )(ϕ(λk t)) = 1. n→∞

Let ϵ > 0 be given. By (i) and (iv) of Definition 2.2, there exists k ∈ Z+ , such that ϕ(λk t) < 2ϵ . Then we have ϵ lim G∗g(x),g(un+1 ),g(un+1 ) ( ) ≥ lim G∗g(x),g(un+1 ),g(un+1 ) (ϕ(λk t)) = 1, n→∞ 2 ϵ ∗ lim G ( ) ≥ lim G∗g(y),g(vn+1 ),g(vn+1 ) (ϕ(λk t)) = 1. n→∞ g(y),g(vn+1 ),g(vn+1 ) 2 n→∞

n→∞

(4.3) (4.4)

Similarly, we prove that ϵ ϵ lim G∗g(x∗ ),g(un+1 ),,g(un+1 ) ( ) = 1, lim G∗g(y∗ ),g(vn+1 ),,g(vn+1 ) ( ) = 1. n→∞ n→∞ 2 2 ϵ ϵ ∗ ∗ lim Gg(z ∗ ),g(pn+1 ),,g(pn+1 ) ( ) = 1, lim Gg(w∗ ),g(qn+1 ),,g(qn+1 ) ( ) = 1. n→∞ n→∞ 2 2 15 340

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(4.5) (4.6)

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By using Menger triangle inequality, and (4.3)-(4.6), we get ϵ ϵ G∗g(x),g(un+1 ),g(x∗ ) (ϵ) ≥ △(G∗g(x),g(un+1 ),g(un+1 ) ( ), G∗g(un+1 ),g(un+1 ),g(x∗ ) ( )) → 1 2 2 ϵ ϵ G∗g(y),g(vn+1 ),g(y∗ ) (ϵ) ≥ △(G∗g(y),g(vn+1 ),g(vn+1 ) ( ), G∗g(vn+1 ),g(vn+1 ),g(y∗ ) ( )) → 1 2 2 ϵ ϵ ∗ ∗ ∗ Gg(z),g(pn+1 ),g(z ∗ ) (ϵ) ≥ △(Gg(z),g(pn+1 ),g(pn+1 ) ( ), Gg(pn+1 ),g(pn+1 ),g(z ∗ ) ( )) → 1 2 2 ϵ ϵ ∗ ∗ ∗ Gg(w),g(qn+1 ),g(w∗ ) (ϵ) ≥ △(Gg(w),g(qn+1 ),g(qn+1 ) ( ), Gg(qn+1 ),g(qn+1 ),g(w∗ ) ( )) → 1 2 2

as n → ∞, as n → ∞, as n → ∞, as n → ∞.

Hence g(x) = g(x∗ ), g(y) = g(y ∗ ), g(z) = g(z ∗ ), g(w) = g(w∗ ), thus (4.1) holds. Since g(x) = T (x, y, z, w), g(y) = T (y, z, w, x), g(z) = T (z, w, x, y), g(w) = T (w, x, y, z), by commutativity of T and g, we have g(g(x)) = g(T (x, y, z, w)) = T (g(x), g(y), g(z), g(w)),

(4.7)

g(g(y)) = g(T (y, z, w, x)) = T (g(y), g(z), g(w), g(x)),

(4.8)

g(g(z)) = g(T (z, w, x, y)) = T (g(z), g(w), g(x), g(y)),

(4.9)

g(g(w)) = g(T (w, x, y, z)) = T (g(w), g(x), g(y), g(z)).

(4.10)

Denote g(x) = α, g(y) = β, g(z) = γ, g(w) = σ. From (4.7)-(4.10), we obtain g(α) = T (α, β, γ, σ), g(β) = T (β, γ, σ, α), g(γ) = T (γ, σ, α, β), g(σ) = T (σ, α, β, γ),

(4.11)

thus (α, β, γ, σ) is a coupled coincidence point. Owing to (4.1) with x∗ = α, y ∗ = β, z ∗ = γ, and w∗ = σ, it follows g(α) = g(x), g(β) = g(y), g(γ) = g(z), g(σ) = g(w), that is g(α) = α, g(β) = β, g(γ) = γ, g(σ) = σ.

(4.12)

From (4.11) and (4.12), we have α = g(α) = T (α, β, γ, σ), β = g(β) = T (β, γ, σ, α), γ = g(γ) = T (γ, σ, α, β), σ = g(σ) = T (σ, α, β, γ). Therefore, (α, β, γ, σ) is a coupled common fixed point of T and g. Suppose that (α∗ , β ∗ , γ ∗ , σ ∗ ) is another coupled common fixed point. By (4.1), we have α∗ = g(α∗ ) = g(x) = x, β ∗ = g(β ∗ ) = g(y) = y, γ ∗ = g(γ ∗ ) = g(z) = z, σ ∗ = g(σ ∗ ) = g(w) = w, which implies that T and g has a unique coupled common fixed point. This completes the proof. 16 341

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5

An example In this section, an example are presented to verify the effectiveness and applicability of Theorem

3.1. Example 5.1 Let X = [0, 1] be given. Define G(x, y, z) = |x − y| + |y − z| + |z − x|. A mapping T : X 4 → X define by T (x1 , x2 , x3 , x4 ) =

x1 +x2 +x3 +x4 . 16

G∗x,y,z (t)

  

=

And g : X → X define by g(x) = x2 . Define

t t+G(x,y,z) ,

 0,

if t > 0, if t < 0.

for x1 , x2 , x3 , x4 , x, y, z ∈ X, where T (X 4 ) ⊂ g(X). Then (X, G∗ , △m ) is a complete Menger PGMspace with a continuous t-norm △m . Let λ = 21 , φ(t) =

9t 10

and ϕ(t) =

t 2

be given for all t > 0. Then

we have ψ( =

1 G∗T (x,y,z,w),T (u,v,p,q),T (a,b,c,d) (ϕ(λt))

− 1) = ψ(

1 (G(T (x, y, z, w), T (u, v, p, q), T (a, b, c, d)))) ϕ(λt)

9 (|x + y + z + w − u − v − p − q| + |u + v + p + q − a − b − c − d| 40t + |a + b + c + d − x − y − z − w|), (5.1)

1 1 1 1 1 ψ( ∗ −1+ ∗ −1+ ∗ −1+ ∗ − 1) 4 Gg(x),g(u),g(a) ϕ(t) Gg(y),g(v),g(b) ϕ(t) Gg(z),g(p),g(c) ϕ(t) Gg(w),g(q),g(d) ϕ(t) =

9 (|x − u| + |u − a| + |a − x| + |y − v| + |v − b| + |b − y| + |z − p| + |p − c| + |c − z| 40t + |w − q| + |q − d| + |d − w|). (5.2)

By (5.1) and (5.2), we obtain ψ(

1 G∗T (x,y,z,w),T (u,v,p,q),T (a,b,c,d) (ϕ(λt))

1 1 1 − 1) ≤ ψ( ∗ −1+ ∗ −1 4 Gg(x),g(u),g(a) ϕ(t) Gg(y),g(v),g(b) ϕ(t) +

1 G∗g(z),g(p),g(c) ϕ(t)

−1+

1 G∗g(w),g(q),g(d) ϕ(t)

− 1),

which implies that T and g satisfy ψ-contractive condition. Thus, all the conditions of Theorem 3.1 are satisfied. And (0,0,0,0) is the coupled coincidence point of T and g.

Acknowledgement The authors would like to thank the editor and the referees for their constructive comments and suggestions. The research was supported by the Natural Science Foundation of China (11361042,11071108, 11461045,71363043), the Natural Science Foundation of Jiangxi Province of China (2010GZS0147,2011 17 342

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4BAB 201003, 20142BAB211016, 20132BAB201001), and partly supported by the NSF of Education Department of Jiangxi Province of China (GJJ150008).

References [1] T.G. Bhaskar and V. Lakshmikantham, Fixed point theory in partially ordered metric spaces and applications, Nonlinear Anal. 65, 1379-1393 (2006). [2] B.S. Choudhury and K. Das, A new contraction principle in Menger spaces, Acta Math. Sin. 24, 1379-1386 (2008). [3] E. Karapinar, Coupled fixed point theorems for nonlinear contractions in cone metric spaces, Comput. Math. Appl. 59, 3656-3668 (2010). [4] S.S. Chang, Y.J. Cho and S.M. Kang, Nonlinear Operator Theory in Probabilistic Metric Space, Nova Science, Huntington, NY, USA. (2001). [5] B.S Choudhury and A. Kundu, A coupled coincidence point result in partially ordered metric spaces for compatible mappings, Nonlinear Anal. 73, 2524-2531 (2010). [6] M.A. Kutbi, D. Gopal, C. Vetro and W. Sintunavarat, Further generalization of fixed point theorems in Menger PM-spaces, Fixed Point Theory Appl. 2015, 32 (2015). ´ c, Coupled fxed point theorems for nonlinear contractions in [7] V. Lakshmikantham and L.Ciri´ partially ordered metric spaces, Nonlinear Anal. 70, 4341-4349 (2009). [8] V. Luong and N. Thuan, Coupled fixed points in partially ordered metric spaces and application, Nonlinear Anal. 74, 983-992 (2011). [9] J. Jachymski, On probabilistic φ-contractions on Menger spaces, Nonlinear Anal. 73, 2199-2203 (2010). [10] J.M. Jin, C.X. Zhu and Z.Q. Wu, New fixed point theorem for phi-contractions in KM-fuzzy metric spaces, J. of Nonlinear Sci. Appl. 9, 6204-6209 (2016). [11] J.Z. Xiao, X.H. Zhu and Y.F. Cao, Common coupled fixed point results for probabilistic ϕcontractions in Menger spaces, Nonlinear Anal. 74, 4589-4600 (2011). [12] B.S. Choudhury and P. Maity, coupled fixed point results in generalized metric spaces, Math. Comput. Modelling. 54, 73-79 (2011). [13] J.H. Cheng and X.J. Huang, coupled fixed point theorems for compatible mappings in partially ordered G-metric spaces, J.of Nonlinear Sci. Appl. 8, 130-141 (2015). 18 343

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[14] Q.Tu, C. X. Zhu and Z. Q. Wu, Some new coupled fixed point theorems in partially ordered complete probabilistic metric spaces, J. of Nonlinear Sci. Appl. 9 1116-1128 (2016). [15] N. Wairojjana, T. Do˘ senovi´ c, D. Raki´ c, D. Gopal and P. Kuman, An altering distance function in fuzzy metric fixed point theorems, Fixed Point Theory Appl. 2015, 69 (2015). [16] C.X. Zhu, Several nonlinear operator problems in the Menger PN space, Nonlinear Anal. 65, 1281-1284 (2006). [17] C.X. Zhu, Research on some problems for nonlinear operators, Nonlinear Anal. 71, 4568-4571 (2009). ´ c and S.M. Alsulami, Generalized probabilistic metric spaces and [18] C.L. Zhou, S.H. Wang, L. Ciri´ fixed point theorems, Fixed point Theory Appl. 2014, 91 (2014).

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FOURIER SERIES OF SUMS OF PRODUCTS OF HIGHER-ORDER EULER FUNCTIONS TAEKYUN KIM1 , DAE SAN KIM2 , GWAN-WOO JANG3 , AND JONGKYUM KWON4,∗

Abstract. In this paper, we consider three types of functions given by sums of products of higher-order Euler functions and derive their Fourier series expansions. Moreover, we express each of them in terms of Bernoulli functions.

1. Introduction

(r)

Let r be a nonnegative integer. The Euler polynomials Em (x) of order r are defined by the generating function (see [2, 9–12, 17, 19])  r ∞ X 2 tm (r) xt Em (x) . e = (1.1) t e +1 m! m=0 (r)

(r)

When x = 0, Em = Em (0) are called the Euler numbers of order r. For r = 1, (1) (1) Em (x) = Em (x), and Em = Em are called Euler polynomials and numbers, respectively. From (1.1), it is immediate to see that d (r) (r) (r) (r) (r−1) E (x) = mEm−1 (x), m ≥ 1, Em (x+1)+Em (x) = 2Em (x), m ≥ 0. (1.2) dx m These in turn imply that (r) (r−1) (r) Em (1) = 2Em − Em , (m ≥ 0),

(1.3)

and Z 0

1 (r) Em (x)dx =

 2  (r−1) (r) Em+1 − Em+1 , (m ≥ 0). m+1

(1.4)

For any real number x, the fractional part of x is denoted by < x >= x − [x] ∈ [0, 1).

(1.5)

We will need the following facts about the Fourier series expansion of the Bernoulli function Bm (< x >): 2010 Mathematics Subject Classification. 11B68, 42A16. Key words and phrases. Fourier series, sums of products of higher-order Euler functions. ∗ corresponding author. 1

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2

Fourier series of sums of products of higher-order Euler functions

(a) for m ≥ 2, Bm (< x >) = −m!

∞ X n=−∞,n6=0

e2πinx , (2πin)m

(1.6)

(b) for m = 1, ∞ X

−

n=−∞,n6=0

e2πinx = 2πin

( B1 (< x >), 0,

for x ∈ / Z, for x ∈ Z.

(1.7)

In the present paper, we will study the following three types of sums of products of higher-order Euler functions and find Fourier series expansions for them. Furthermore, we will express them in terms of Bernoulli functions. In the following, we let r, s be positive integers. Pm (r) (s) (1) αm (< x >) = k=0 Ek (< x >)Em−k (< x >), (m ≥ 1); Pm (r) (s) 1 Ek (< x >)Em−k (< x >), (m ≥ 1); (2) βm (< x >) = k=0 k!(m−k)! Pm−1 (r) (s) 1 Ek (< x >)Em−k (< x >), (m ≥ 2). (3) γm (< x >) = k=1 k(m−k) For elementary facts about Fourier analysis, the reader may refer to any book (for example, see [1, 20]). As to γm (< x >), we note that the polynomial identity (1.8) follows immediately from the Fourier series expansion of γm (< x >) in Theorems 4.1 and 4.2: m−1 X

1 (r) (s) E (x)Em−k (x) k(m − k) k k=1 m   2(Hm−1 − Hm−k ) 1 X m n Λm−k+1 + = m m−k+1 k

(1.8)

k=0

×

(r−1) (Em−k+1

o (s−1) (r) (s) + Em−k+1 − Em−k+1 − Em−k+1 ) Bk (x),

where, for each integer l ≥ 2, Λl =

l−1 X k=1

2 (r−1) (s−1) (r) (s−1) (r−1) (s)
2Ek El−k − Ek El−k − Ek El−k , k(l − k)

(1.9)

Pm and Hm = j=1 1j are the harmonic numbers. The obvious polynomial identities can be derived also for αm (< x >) and βm (< x >) from Theorems 2.1 and 2.2, and Theorems 3.1 and 3.2, respectively. It is noteworthy that from the Fourier series expansion of the function m−1 X k=1

1 Bk (hxi)Bm−k (hxi) k(m − k)

(1.10)

we can derive the famous Faber-Pandharipande-Zagier identity (see [4, 7, 8]) and the Miki’s identity (see [3, 5, 7, 8, 18]). Hence our problem here is a natural extension of the previous works which lead to a simple proof for the important FaberPandharipande-Zagier and Miki’s identities (see [15]). Some related recent works can be found in [6, 13–16].

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3

2. The function αm (< x >) (r)

Pm

Let αm (x) = tion

(s)

Ek (x)Em−k (x), (m ≥ 1). Then we will consider the func-

k=0

αm (< x >) =

m X

(r)

(s)

Ek (< x >)Em−k (< x >), (m ≥ 1),

(2.1)

k=0

defined on R, which is periodic with period 1. The Fourier series of αm (< x >) is ∞ X

2πinx A(m) , n e

(2.2)

n=−∞

where A(m) n

1

Z

αm (< x >)e

=

−2πinx

1

Z

αm (x)e−2πinx dx.

dx =

(2.3)

0

0

To proceed further, we need to observe the following. m   X (r) (s) (r) (s) 0 αm (x) = kEk−1 (x)Em−k (x) + (m − k)Ek (x)Em−k−1 (x) k=0

=

m X

(r)

(s)

kEk−1 (x)Em−k (x) +

k=1

=

m−1 X

(r)

(s)

(m − k)Ek (x)Em−k−1 (x)

k=0

m−1 X

(r)

(s)

(k + 1)Ek (x)Em−1−k (x) +

k=0

m−1 X

(r)

(s)

(m − k)Ek (x)Em−1−k (x)

(2.4)

k=0 m−1 X

= (m + 1)

(r)

(s)

Ek (x)Em−1−k (x)

k=0

= (m + 1)αm−1 (x). From this, we have 

αm+1 (x) m+2

0 = αm (x),

(2.5)

and Z

1

αm (x)dx = 0

1 (αm+1 (1) − αm+1 (0)) . m+2

(2.6)

For m ≥ 1, we put ∆m = αm (1) − αm (0) m   X (r) (s) (r) (s) = Ek (1)Em−k (1) − Ek Em−k =

k=0 m  X

k=0 m  X

=2

(r−1)

(2Ek

(r−1)

2Ek

(r)

(s−1)

(s)

(r)

(s)

− Ek )(2Em−k − Em−k ) − Ek Em−k (s−1)

(r)

(s−1)

(r−1)

Em−k − Ek Em−k − Ek



(2.7)

 (s) Em−k .

k=0

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Fourier series of sums of products of higher-order Euler functions

We now see that αm (0) = αm (1) ⇐⇒ ∆m = 0,

(2.8)

and 1

Z

αm (x)dx = 0

1 ∆m+1 . m+2

(2.9)

(m)

Next, we want to determine the Fourier coefficients An . Case 1 : n 6= 0. Z 1 (m) αm (x)e−2πinx dx An = 0

1 1 [αm (x)e−2πinx ]10 + =− 2πin 2πin

Z

1 0 αm (x)e−2πinx dx

0

Z 1 m+1 1 αm−1 (x)e−2πinx dx (αm (1) − αm (0)) + =− 2πin 2πin 0 m + 1 (m−1) 1 = A − ∆m , 2πin n 2πin from which by induction on m, we can easily derive that

(2.10)

m

A(m) =− n

1 X (m + 2)j ∆m−j+1 . m + 2 j=1 (2πin)j

Case 2 : n = 0. Z

(2.11)

1

1 ∆m+1 . (2.12) m +2 0 αm (< x >), (m ≥ 1) is piecewise C ∞ . In addition, αm (< x >) is continuous for those positive integers m with ∆m = 0, and discontinuous with jump discontinuities at integers for those positive integers with ∆m 6= 0. (m) A0

=

αm (x)dx =

Assume first that ∆m = 0, for a positive integer m. Then αm (0) = αm (1). Hence αm (< x >) is piecewise C ∞ , and continuous. Thus the Fourier series of αm (< x >) converges uniformly to αm (< x >) , and  m X 1 (m + 2) j − ∆m−j+1  e2πinx m + 2 j=1 (2πin)j n=−∞,n6=0    m  ∞ 2πinx X 1 1 X m+2 e  = ∆m+1 + ∆m−j+1 −j! m+2 m + 2 j=1 (2πin)j j

1 αm (< x >) = ∆m+1 + m+2

∞ X



n=−∞,n6=0

 m  1 1 X m+2 = ∆m+1 + ∆m−j+1 Bj (< x >) m+2 m + 2 j=2 j ( B1 (< x >), for x ∈ / Z, + ∆m × 0, for x ∈ Z. (2.13)

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5

We are now going to state our first result. Theorem 2.1. For each positive integer l, we let ∆l = 2

l  X

(r−1)

2Ek

(s−1)

El−k

(r)

(s−1)

− Ek El−k

(r−1)

− Ek

 (s) El−k .

k=0

Assume that ∆m = 0, for a positive integer m. Then we have the following. Pm (r) (s) (a) k=0 Ek (< x >)Em−k (< x >) has the Fourier series expansion m X

(r)

(s)

Ek (< x >)Em−k (< x >)

k=0

1 ∆m+1 + = m+2

 m X 1 (m + 2) j − ∆m−j+1  e2πinx , m + 2 j=1 (2πin)j 

∞ X n=−∞,n6=0

(2.14)

for all x ∈ R, where the convergence is uniform.

(b)

m X

(r)

(s)

Ek (< x >)Em−k (< x >) =

k=0

1 m+2

  m X m+2 ∆m−j+1 Bj (< x >), j

j=0,j6=1

(2.15) for all x in R. Assume next that ∆m 6= 0, for a positive integer m. Then αm (0) 6= αm (1). Hence αm (< x >) is piecewise C ∞ , and discontinuous with jump discontinuities at integers. Then the Fourier series of αm (< x >) converges pointwise to αm (< x >) , for x∈ / Z, and converges to 1 1 (αm (0) + αm (1)) = αm (0) + ∆m , (2.16) 2 2 for x ∈ Z. Now, we are going to state our second result. Theorem 2.2. For each positive integer l, we let ∆l = 2

l  X

(r−1)

2Ek

(s−1)

El−k

(r)

(s−1)

− Ek El−k

(r−1)

− Ek

 (s) El−k .

k=0

Assume that ∆m 6= 0, for a positive integer m. Then we have the following.   ∞ m X X 1 1 (m + 2) j − (a) ∆m+1 + ∆m−j+1  e2πinx m+2 m + 2 j=1 (2πin)j n=−∞,n6=0 (2.17) (P (r) (s) m E (< x >)E (< x >), for x ∈ / Z, k m−k = Pk=0 (r) (s) m 1 E E + for x ∈ Z. k=0 k m−k 2 ∆m ,

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Fourier series of sums of products of higher-order Euler functions

 m  m X 1 X m+2 (r) (s) (b) ∆m−j+1 Bj (< x >) = Ek (< x >)Em−k (< x >), f or x ∈ / Z; m + 2 j=0 j k=0

(2.18) 1 m+2

  m m X X m+2 1 (r) (s) ∆m−j+1 Bj (< x >) = Ek Em−k + ∆m , f or x ∈ Z. j 2

j=0,j6=1

k=0

(2.19)

3. The function βm (< x >) Let βm (x) = the function

(r) (s) 1 k=0 k!(m−k)! Ek (x)Em−k (x),

Pm

m X

βm (< x >) =

k=0

(m ≥ 1). Then we will consider

1 (r) (s) E (< x >)Em−k (< x >), (m ≥ 1), k!(m − k)! k

defined on R, which is periodic with period 1. The Fourier series of βm (< x >) is ∞ X

Bn(m) e2πinx ,

(3.1)

n=−∞

where Bn(m) =

Z

1

βm (< x >)e−2πinx dx =

Z

0

1

βm (x)e−2πinx dx.

(3.2)

0

Before continuing further, we need to note the following.  m  X k (m − k) (r) (r) (s) (s) 0 βm (x) = E (x)Em−k (x) + E (x)Em−k−1 (x) k!(m − k)! k−1 k!(m − k)! k =

k=0 m X

k=1

+

m−1 X k=0

=

m−1 X k=0

+

m−1 X k=0

1 (r) (s) E (x)Em−k (x) (k − 1)!(m − k)! k−1 1 (r) (s) E (x)Em−k−1 (x) k!(m − k − 1)! k 1 (r) (s) E (x)Em−1−k (x) k!(m − 1 − k)! k 1 (r) (s) E (x)Em−1−k (x) k!(m − 1 − k)! k

= 2βm−1 (x). (3.3) From this, we have 

βm+1 (x) 2

0

350

= βm (x),

(3.4)

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7

and Z

1

 1 βm+1 (1) − βm+1 (0) . 2

βm (x)dx = 0

(3.5)

For m ≥ 1, we set Ωm = βm (1) − βm (0) = = =

m X k=0 m X k=0 m X k=0

1 (r) (s) (r) (s)
Ek (1)Em−k (1) − Ek Em−k k!(m − k)!   1 (r−1) (r) (s−1) (s) (r) (s) (2Ek − Ek )(2Em−k − Em−k ) − Ek Em−k k!(m − k)!

(3.6)

2 (r−1) (s−1) (r) (s−1) (r−1) (s)
2Ek Em−k − Ek Em−k − Ek Em−k . k!(m − k)!

Now, it is immediate to see that βm (0) = βm (1) ⇐⇒ Ωm = 0,

(3.7)

and 1

Z

βm (x)dx = 0

1 Ωm+1 . 2

(3.8) (m)

We now would like to determine the Fourier coefficients Bn . Case 1:n 6= 0 Z 1 Bn(m) = βm (x)e−2πinx dx 0

Z 1 i1 1 h 1 β 0 (x)e−2πinx dx βm (x)e−2πinx + 2πin 2πin 0 m 0 Z 1  1  2 =− βm (1) − βm (0) + βm−1 (x)e−2πinx dx 2πin 2πin 0 2 1 = Bn(m−1) − Ωm , 2πin 2πin from which by induction on m gives =−

Bn(m) = −

m X 2j−1 Ωm−j+1 . (2πin)j j=1

(3.9)

(3.10)

Case 2: n = 0 (m) B0

Z =

1

βm (x)dx = 0

1 Ωm+1 . 2

(3.11)

βm (< x >), (m ≥ 1) is piecewise C ∞ . Further, βm (< x >) is continuous for those positive integers m with Ωm = 0, and discontinuous with jump discontinuities at integers for those positive integers m with Ωm 6= 0 .

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Fourier series of sums of products of higher-order Euler functions

Assume first that Ωm = 0, for a positive integer m. Then βm (0) = βm (1). Hence βm (< x >) is piecewise C ∞ , and continuous. Thus the Fourier series of βm (< x >) converges uniformly to βm (< x >), and βm (< x >) =

=

1 Ωm+1 + 2

∞ X

m   X 2j−1 Ω e2πinx − m−j+1 j (2πin) j=1

n=−∞,n6=0 m j−1 X

 1 2 Ωm+1 + Ωm−j+1 −j! 2 j! j=1

∞ X n=−∞,n6=0

e2πnx  (2πin)j

(3.12)

m X

1 2j−1 Ωm+1 + Ωm−j+1 Bj (< x >) 2 j! j=2 ( B1 (< x >), for x ∈ / Z, + Ωm × 0, for x ∈ Z. =

Now, we are going to state our first result. Theorem 3.1. For each positive integer l, we let Ωl =

l X k=0

2 (r−1) (s−1) (r) (s−1) (r−1) (s)
2Ek El−k − Ek El−k − Ek El−k . k!(l − k)!

(3.13)

Assume that Ωm = 0, for a positive integer m. Then we have the following. (a)

(r) 1 k=0 k!(m−k)! Ek (
)Em−k (< x >) has the Fourier series expansion

1 (r) (s) E (< x >)Em−k (< x >) k!(m − k)! k

1 = Ωm+1 + 2

∞ X

m  X  2j−1 − Ω e2πinx , m−j+1 j (2πin) j=1

(3.14)

n=−∞,n6=0

for all x ∈ R, where the convergence is uniform. (b)

m X k=0

=

1 (r) (s) E (< x >)Em−k (< x >) k!(m − k)! k m X

j=0,j6=1

2j−1 Ωm−j+1 Bj (< x >), j!

(3.15)

for all x ∈ R. Assume next that Ωm 6= 0, for a positive integer m. Then, βm (0) 6= βm (1). Hence βm (< x >) is piecewise C ∞ , and discontinuous with jump discontinuities at integers. Then the Fourier series of βm (< x >) converges pointwise to βm (< x >), for x ∈ / Z, and converges to 1 1 (βm (0) + βm (1)) = βm (0) + Ωm , (3.16) 2 2 for x ∈ Z. Next, we are going to state our second result.

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Theorem 3.2. For each positive integer l, we let Ωl =

l X k=0

2 (r−1) (s−1) (r) (s−1) (r−1) (s)
2Ek El−k − Ek El−k − Ek El−k . k!(l − k)!

(3.17)

Assume that Ωm 6= 0, for a positive integer m. Then we have the following.

(a)

(b)

∞ m   X X 1 2j−1 Ω e2πinx Ωm+1 + − m−j+1 j 2 (2πin) j=1 n=−∞,n6=0 (Pm (r) (s) 1 for x ∈ / Z, k!(m−k)! Ek (< x >)Em−k (< x >), = Pk=0 (r) (s) m 1 1 for x ∈ Z. k=0 k!(m−k)! Ek Em−k + 2 Ωm ,

m X 2j−1 j=0

=

j!

m X

k=0 m X

=

k=0

Ωm−j+1 Bj (< x >)

1 (r) (s) E (< x >)Em−k (< x >), k!(m − k)! k

j=0,j6=1 m X

(3.18)

for x ∈ / Z; (3.19)

2j−1 Ωm−j+1 Bj (< x >) j!

1 1 (r) (s) E E + Ωm , k!(m − k)! k m−k 2

for x ∈ Z.

4. The function γm (< x >) Let γm (x) = the function

(r) (s) 1 k=1 k(m−k) Ek (x)Em−k (x),

Pm−1

γm (< x >) =

m−1 X k=1

(m ≥ 2). Then we will consider

1 (r) (s) E (< x >)Em−k (< x >), k(m − k) k

defined on R, which is periodic with period 1. The Fourier series of γm (< x >) is ∞ X

Cn(m) e2πinx ,

(4.1)

n=−∞

where Cn(m)

Z =

1 −2πinx

γm (< x >)e 0

Z dx =

1

γm (x)e−2πinx dx.

(4.2)

0

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Fourier series of sums of products of higher-order Euler functions

To proceed further, we need to observe the following. 0 γm (x) =

m−1 X

m−1 X 1 (r) 1 (r) (s) (s) Ek−1 (x)Em−k (x) + E (x)Em−k−1 (x) m−k k k

k=1

=

m−2 X k=0

k=1

1 (r) (s) E (x)Em−1−k (x) + m−1−k k

= (m − 1)

m−2 X k=1

m−1 X k=1

1 (r) (s) E (x)Em−1−k (x) k k

1 1 1 (r) (s) (s) (r) E (x)Em−1−k (x) + E (x) + E (x) k(m − 1 − k) k m − 1 m−1 m − 1 m−1

= (m − 1)γm−1 (x) +

1 1 (s) (r) Em−1 (x) + E (x). m−1 m − 1 m−1 (4.3)

From this, we easily see that  
0 1 1 1 (r) (s) γm+1 (x) − E (x) − E (x) = γm (x), m m(m + 1) m+1 m(m + 1) m+1

(4.4)

and Z

1

γm (x)dx 0

i1 1h 1 1 (r) (s) γm+1 (x) − Em+1 (x) − Em+1 (x) m m(m + 1) m(m + 1) 0  1 1 (r) (r) = (E (1) − Em+1 (0)) γm+1 (1) − γm+1 (0) − m m(m + 1) m+1  1 (s) (s) − (Em+1 (1) − Em+1 (0)) m(m + 1) 2 1 (r−1) (r) γm+1 (1) − γm+1 (0) − (E − Em+1 ) = m m(m + 1) m+1  2 (s−1) (s) − (Em+1 − Em+1 ) . m(m + 1) =

(4.5)

Let Λ1 = 0, and for m ≥ 2, we let Λm = γm (1) − γm (0) =

m−1 X k=1

=

m−1 X k=1

=

m−1 X k=1

  1 (r) (s) (r) (s) Ek (1)Em−k (1) − Ek Em−k k(m − k)   1 (r−1) (r) (s−1) (s) (r) (s) (2Ek − Ek )(2Em−k − Em−k ) − Ek Em−k k(m − k)

(4.6)

2 (r−1) (s−1) (r) (s−1) (r−1) (s)
2Ek Em−k − Ek Em−k − Ek Em−k . k(m − k)

Then we have γm (0) = γm (1) ⇔ Λm = 0, (m ≥ 2),

(4.7)

and

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1

11

    2 2 (r−1) (r) (s−1) (s) Em+1 − Em+1 − Em+1 − Em+1 . m(m + 1) m(m + 1) 0 (4.8) (m) We now want to determine the Fourier coefficients Cn . Case 1: n 6= 0

Z

γm (x)dx =

Cn(m) =

1 m

 Λm+1 −

1

Z

γm (x)e−2πinx dx

0

Z 1 i1 1 1 h −2πinx + =− γm (x)e γ 0 (x)e−2πinx dx 2πin 2πin 0 m 0  1  γm (1) − γm (0) =− 2πin Z 1 1 1 1 (r) (s) {(m − 1)γm−1 (x) + + E (x) + E (x)}e−2πinx dx 2πin 0 m − 1 m−1 m − 1 m−1 Z 1 1 1 m − 1 (m−1) (r) C − Λm + E (x)e−2πinx dx = 2πin n 2πin 2πin(m − 1) 0 m−1 Z 1 1 (s) + E (x)e−2πinx dx 2πin(m − 1) 0 m−1   1 1 m − 1 (m−1) (s) Cn − Λm − Φ(r) = m + Φm , 2πin 2πin 2πin(m − 1) (4.9) where Φ(r) m =

m−1 X k=1

Z

1

 2(m − 1)k−1  (r−1) (r) E − E m−k m−k , (2πin)k

(r)

El (x)e−2πinx dx 0  P   (r−1) (r) 2(l)k−1 − l k=1 (2πin)k El−k+1 − El−k+1 , for n 6= 0,   =  2 E (r−1) − E (r) , for n = 0. l+1 l+1 l+1

(4.10)

Thus we have shown that Cn(m) =

  m − 1 (m−1) 1 1 (s) Cn − Λm − Φ(r) + Φ m . 2πin 2πin 2πin(m − 1) m

(4.11)

An easy induction on m now gives

Cn(m) = −

m−1 X j=1

m−1 X (m − 1)j−1 (m − 1)j−1 (r) (s) Λ − (Φm−j+1 + Φm−j+1 ). m−j+1 j j (m − j) (2πin) (2πin) j=1

(4.12)

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Fourier series of sums of products of higher-order Euler functions (m)

To find a more explicit expression for Cn , we need to observe the following.

m−1 X j=1

=

m−1 X j=1

=2

(m − 1)j−1 (r) Φ (2πin)j (m − j) m−j+1 m−j (m − 1)j−1 X 2(m − j)k−1 (r−1) (r) (Em−j−k+1 − Em−j−k+1 ) (2πin)j (m − j) (2πin)k k=1

m−1 X j=1

=2

m−1 X j=1

=2

=2

1 m−j

m−j X k=1

m X 1 (m − 1)k−2 (r−1) (r) (Em−k+1 − Em−k+1 ) m−j (2πin)k

k=1

(4.13)

k=j+1

m X (m − 1)k−2 k=2 m X

(m − 1)j+k−2 (r−1) (r) (Em−j−k+1 − Em−j−k+1 ) (2πin)j+k

(2πin)k

(r−1)

(r)

(Em−k+1 − Em−k+1 )

k−1 X j=1

1 m−j

(m − 1)k−2 (r−1) (r) (Em−k+1 − Em−k+1 ) (Hm−1 − Hm−k ) (2πin)k (r−1)

(r)

m 2 X (m)k Em−k+1 − Em−k+1 = (Hm−1 − Hm−k ) . m (2πin)k m−k+1 k=1

(m)

Recalling that Λ1 = 0, we get the following expression of Cn : for n 6= 0,

m

1 X (m)k  2(Hm−1 − Hm−k ) Λm−k+1 + m (2πin)k m−k+1 k=1  (r−1) (s−1) (r) (s) × (Em−k+1 + Em−k+1 − Em−k+1 − Em−k+1 ) .

Cn(m) = −

(4.14)

Case 2: n = 0

1

 2 (r−1) (s−1) (r) (s) + Em+1 − Em+1 − Em+1 ) . (E = m(m + 1) m+1 0 (4.15) γm (< x >), (m ≥ 2) is piecewise C ∞ . Furthermore, γm (< x >) is continuous for those integers m ≥ 2 with Λm = 0, and discontinuous with jump discontinuities at integers for those integer m ≥ 2 with Λm 6= 0. Assume first that Λm = 0, for an integer m ≥ 2. Then γm (0) = γm (1). Hence γm (< x >) is piecewise C ∞ , and continuous. Thus the Fourier series of γm (< x >) (m) C0

Z

1 γm (x)dx = m

 Λm+1 −

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13

converges uniformly to γm (< x >), and γm (< x >)   1 2 (r−1) (s−1) (r) (s) = Λm+1 − (E + Em+1 − Em+1 − Em+1 ) m m(m + 1) m+1 ∞ m nX X 1 2(Hm−1 − Hm−k ) (m)k  − Λm−k+1 + k m (2πin) m−k+1 n=−∞,n6=0 k=1
o (r−1) (s−1) (r) (s) × (Em−k+1 + Em−k+1 − Em−k+1 − Em−k+1 ) e2πinx   1 2 (r−1) (s−1) (r) (s) Λm+1 − (E + Em+1 − Em+1 − Em+1 ) = m m(m + 1) m+1 m   1 X m n 2(Hm−1 − Hm−k ) + Λm−k+1 + m k m−k+1 k=1

×

(r−1) (Em−k+1

+

(s−1) Em−k+1

−

(r) Em−k+1

−

o −k!

(s) Em−k+1 )

∞ X n=−∞,n6=0

e2πinx  (2πin)k (4.16)



 2 (r−1) (s−1) (r) (s) Λm+1 − (E + Em+1 − Em+1 − Em+1 ) m(m + 1) m+1 m   2(Hm−1 − Hm−k ) 1 X m n Λm−k+1 + + m k m−k+1 k=2 o (r−1) (s−1) (r) (s) × (Em−k+1 + Em−k+1 − Em−k+1 − Em−k+1 ) Bk (< x >) ( B1 (< x >), for x ∈ / Z, + Λm × 0, for x ∈ Z  n m 1 X m 2(Hm−1 − Hm−k ) = Λm−k+1 + m k m−k+1 k=0,k6=1 o (r−1) (s−1) (r) (s) × (Em−k+1 + Em−k+1 − Em−k+1 − Em−k+1 ) Bk (< x >) ( B1 (< x >), for x ∈ / Z, + Λm × 0, for x ∈ Z. 1 = m

Now, we can state our first result. Theorem 4.1. For each integer l ≥ 2, we let

Λl =

l−1 X k=1

2 (r−1) (s−1) (r) (s−1) (r−1) (s)
2Ek El−k − Ek El−k − Ek El−k , k(l − k)

(4.17)

with Λ1 = 0. Assume that Λm = 0, for an integer m ≥ 2. Then we have the following. (a)

(r) 1 k=1 k(m−k) Ek (
)Em−k (< x >) has the Fourier series expansion

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Fourier series of sums of products of higher-order Euler functions

m−1 X

1 (r) (s) E (< x >)Em−k (< x >) k(m − k) k k=1   1 2 (r−1) (s−1) (r) (s) = Λm+1 − (E + Em+1 − Em+1 − Em+1 ) m m(m + 1) m+1 ∞ m nX X 1 (m)k 2(Hm−1 − Hm−k ) − Λm−k+1 + m (2πin)k m−k+1 n=−∞,n6=0 k=1
o (r−1) (s−1) (r) (s) × (Em−k+1 + Em−k+1 − Em−k+1 − Em−k+1 ) e2πinx ,

(4.18)

for all x ∈ R, where the convergence is uniform.

(b)

m−1 X

1 (r) (s) E (< x >)Em−k (< x >) k(m − k) k k=1  n m m 2(Hm−1 − Hm−k ) 1 X Λm−k+1 + = m m−k+1 k k=0,k6=1 o (r−1) (s−1) (r) (s) × (Em−k+1 + Em−k+1 − Em−k+1 − Em−k+1 ) Bk (< x >)

(4.19)

for all x ∈ R. Assume next that Λm 6= 0, for an integers m ≥ 2. Then γm (0) 6= γm (1). Hence γm (< x >) is piecewise C ∞ , and discontinuous with jump discontinuities at integers. Thus the Fourier series of γm (< x >) converges pointwise to γm (< x >), for x ∈ / Z, and converges to 1 1 (γm (0) + γm (1)) = γm (0) + Λm , 2 2

(4.20)

for x ∈ Z. We can now state our second result. Theorem 4.2. For each integer l ≥ 2, let Λl =

l−1 X k=1

2 (r−1) (s−1) (r) (s−1) (r−1) (s)
2Ek El−k − Ek El−k − Ek El−k , k(l − k)

with Λ1 = 0. Assume that Λm 6= 0, for an integer m ≥ 2. Then we have the following.   2 1 (r−1) (s−1) (r) (s) (a) Λm+1 − (E + Em+1 − Em+1 − Em+1 ) m m(m + 1) m+1 ∞ m nX X 1 2(Hm−1 − Hm−k ) (m)k − Λm−k+1 + k m (2πin) m−k+1 n=−∞,n6=0 k=1
o (r−1) (s−1) (r) (s) × (Em−k+1 + Em−k+1 − Em−k+1 − Em−k+1 ) e2πinx (Pm−1 (r) (s) 1 for x ∈ / Z, k=1 k(m−k) Ek (< x >)Em−k (< x >), = Pm−1 (r) (s) 1 1 for x ∈ Z. k=1 k(m−k) Ek Em−k + 2 Λm ,

358

(4.21)

(4.22)

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T. Kim, D. S. Kim, G.-W. Jang, J. Kwon

(b)

15

m   1 X m n 2(Hm−1 − Hm−k ) Λm−k+1 + m k m−k+1 k=0

o (r−1) (s−1) (r) (s) × (Em−k+1 + Em−k+1 − Em−k+1 − Em−k+1 ) Bk (< x >) m−1 X

1 (r) (s) E (< x >)Em−k (< x >), for x ∈ / Z; k(m − k) k k=1  n m m 2(Hm−1 − Hm−k ) 1 X Λm−k+1 + m k m−k+1 k=0,k6=1 o (r−1) (s−1) (r) (s) × (Em−k+1 + Em−k+1 − Em−k+1 − Em−k+1 ) Bk (< x >) =

=

m−1 X k=1

(4.23)

1 1 (r) (s) Ek Em−k + Λm , for x ∈ Z. k(m − k) 2

References [1] M. Abramowitz, IA. Stegun, Handbook of Mathematical Functions, Dover, New York, 1970. [2] A. Bayad, T. Kim, Identities involving values of Bernstein, q-Bernoulli, and q-Euler polynomials, Russ. J. Math. Phys., 18(2011), no. 2, 133-143. [3] G. V. Dunne, C. Schubert, Bernoulli number identities from quantum field theory and topological string theory, Commun. Number Theory Phys., 7(2)(2013), 225-249. [4] C. Faber, R. Pandharipande, Hodge integrals and Gromov-Witten theory, Invent. Math. 139(1)(2000), 173-199. [5] I. M. Gessel, On Miki’s identities for Bernoulli numbers, J. Number Theory, 110(1)(2005), 75-82. [6] G.-W. Jang, T. Kim, D.S. Kim, T. Mansour, Fourier series of functions related to Bernoulli polynomials, Adv. Stud. Contemp. Math., 27(2017), no.1, 49-62. [7] D.S. Kim, T. Kim, Some identities of higher order Euler polynomials arising from Euler basis, Integral Transforms Spec. Funct., 24(9) (2013), 734-738. [8] D.S. Kim, T. Kim, Euler basis, identities, and their applications, Int. J. Math. Math. Sci. 2012, Art. ID 343981. [9] D.S. Kim, T. Kim, Y.H. Lee, Some arithemetic properties of Bernoulli and Euler nembers, Adv. Stud. Contemp. Math., 22(2010), no.4, 467-480. [10] T. Kim, Euler numbers and polynomials associated with zeta functions, Abstr. Appl. Anal., 2008, Art. ID 581582, 11pp. [11] T. Kim, Some identities for the Bernoulli, the Euler and Genocchi numbers and polynomials, Adv. Stud. Contemp. Math., 20(2015), no.1, 23-28. [12] T. Kim, On the Multiple q-Genocchi and Euler Numbers, Russ. J. Math. Phys., 15(2008), 481-486. [13] T. Kim, D.S. Kim, D.Dolgy, and J.-W. Park, Fourier series of sums of products of polyBernoulli functions and their applications, J. Nonlinear Sci. Appl., 10(2017), no.4, 23842401. [14] T. Kim, D.S. Kim, D.Dolgy, and J.-W. Park, Fourier series of sums of products of ordered Bell and poly-Bernoulli functions, J. Inequal. Appl., 2017 Article ID 13660, 17pages,(2017). [15] T. Kim, D.S. Kim, G.-W. Jang, and J. Kwon, Fourier series of sums of products of Genocchi functions and their applications, J. Nonlinear Sci. Appl., 10(2017), no.4, 1683-1694. [16] T. Kim, D.S. Kim, S.-H. Rim, and D.Dolgy, Fourier series of higher-order Bernoulli functions and their applications, J. Inequal. Appl., 2017 Article ID 71452, 8pages,(2017). [17] H. Liu, and W. Wang, Some identities on the Bernoulli, Euler and Genocchi poloynomials via power sums and alternate power sums, Disc. Math., 309(2009), 3346-3363.

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Fourier series of sums of products of higher-order Euler functions

[18] K. Shiratani, S. Yokoyama, An application of p-adic convolutions, Mem. Fac. Sci. Kyushu Univ. Ser. A 36(1)(1982), 7383. [19] H. M. Srivastava, Some generalizations and basic extensions of the Bernoulli, Euler and Genocchi polynomials, Appl. Math. and Inf. Sci., 5(2011), no. 3, 390-414. [20] D. G. Zill, M. R. Cullen, Advanced Engineering Mathematics, Jones and Bartlett Publishers 2006. 1

Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin 300160, China, Department of Mathematics, Kwangwoon University, Seoul, 139701, Republic of Korea E-mail address: [email protected] 2

Department of Mathematics, Sogang University, Seoul, 121-742, Republic of Korea E-mail address: [email protected] 3 Department of Mathematics, Kwangwoon University, Seoul, 139-701, Republic of Korea E-mail address: [email protected] 4,∗ Department of Mathematics Education and ERI, Gyeongsang National University, Jinju, Gyeongsangnamdo, 52828, Republic of Korea E-mail address: [email protected]

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Some symmetric identities for (p, q)-Euler zeta function Cheon Seoung Ryoo Department of Mathematics, Hannam University, Daejeon 306-791, Korea

Abstract : In this paper we obtain several symmetric identities of the (p, q)-Euler zeta function. We also give some new interesting properties, explicit formulas, a connection with (p, q)-Euler numbers and polynomials. Key words : Euler numbers and polynomials, q-Euler numbers and polynomials, (p, q)-Euler numbers and polynomials, (p, q)-analogue of Euler zeta function. 2000 Mathematics Subject Classification : 11B68, 11S40, 11S80. 1. Introduction Many mathematicians have studied in the area of the Bernoulli numbers and polynomials, Euler numbers and polynomials, Genocchi numbers and polynomials, tangent numbers and polynomials(see [1-10]). The Euler numbers and the Euler polynomials have been extensively worked in many different contexts in such branches of mathematics as, for instance, complex analytic number theory, elementary number theory, differential topology, q-adic analytic number theory and quantum physics. In this paper, we obtain symmetric properties of the (p, q)-Euler zeta function. As applications of these properties, we study some interesting identities for the (p, q)-Euler polynomials and numbers. Throughout this paper, we always make use of the following notations: N denotes the set of natural numbers, Z+ = N ∪ {0} denotes the set of nonnegative integers, Z− 0 = {0, −1, −2, −2, . . .} denotes the set of nonpositive integers, Z denotes the set of integers, R denotes the set of real numbers, and C denotes the set of complex numbers. We remember that the classical Euler numbers En and Euler polynomials En (x) are defined by the following generating functions ∞ ∑ 2 tn = En , t e + 1 n=0 n!

and

(

2 t e +1

) ext =

∞ ∑

En (x)

n=0

(|t| < π).

tn , n!

(|t| < π).

(1.1)

(1.2)

respectively. Some interesting properties of the (p, q)-Euler numbers and polynomials were first investigated by Ryoo[6]. The (p, q)-number is defined by [n]p,q =

pn − q n . p−q

It is clear that (p, q)-number contains symmetric property, and this number is q-number when p = 1. In particular, we can see limq→1 [n]p,q = n with p = 1. By using (p, q)-number, we introduced the (p, q)-Euler polynomials and numbers, which generalized the previously known numbers and polynomials, including the Carlitz’s type q-Euler numbers

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and polynomials. We begin by recalling here the Carlitz’s type (p, q)-Euler numbers and polynomials(see [2]). Definition 1. For 0 < q < p ≤ 1, the Carlitz’s type (p, q)-Euler numbers En,p,q and polynomials En,p,q (x) are defined by means of the generating functions Fp,q (t) =

∞ ∑

En,p,q

n=0

and

∞ ∑ tn = [2]q (−1)m q m e[m]p,q t . n! m=0

(1.1)

∞ ∑ tn Fp,q (t, x) = En,p,q (x) = [2]q (−1)m q m e[m+x]p,q t , n! m=0 n=0 ∞ ∑

(1.2)

respectively. The following elementary properties of Carlitz’s type (p, q)-Euler numbers En,p,q and polynomials En,p,q (x) are readily derived from (1.1) and (1.2). We, therefore, choose to omit the details involved. More studies and results in this subject we may see reference [6]. Theorem 2. For n ∈ Z+ , we have ( (h) En,p,q (x) = [2]q

1 p−q

)n ∑ n ( ) n 1 (−1)l q xl p(n−l)x . l+1 l 1 + q pn−l+h l=0

Theorem 3 (Distribution relation). For any positive integer m(=odd), we have En,p,q (x) =

( ) m−1 ∑ [2]q a+x [m]np,q (−1)a q a En,pm ,qm , [2]qm m a=0

n ∈ N0 .

(h)

Next, we introduce Carlitz’s type (h, p, q)-Euler polynomials En,p,q (x). The Carlitz’s type (h) (h, p, q)-Euler polynomials En,p,q (x) are defined by (h) En,p,q (x) = [2]q

∞ ∑

(−1)m q m phm [m + x]np,q .

m=0

By (p, q)-number, we have the following theorem. Theorem 4. For n ∈ Z+ , we have En,p,q (x) =

n ( ) ∑ n l=0

l

(n−l)

xl [x]n−l p,q q El,p,q .

By using Carlitz’s type (p, q)-Euler numbers and polynomials, (p, q)-Euler zeta function and Hurwitz (p, q)-Euler zeta functions are defined. These functions interpolate the Carlitz’s type (p, q)Euler numbers En,p,q , and polynomials En,p,q (x), respectively. From (1.1), we note that ∞ ∑ dk = [2] F (t) (−1)n q m [m]kp,q q p,q dtk t=0 m=0 = Ek,p,q , (k ∈ N). By using the above equation, we are now ready to define (p, q)-Euler zeta function.

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Definition 5. Let s ∈ C with Re(s) > 0. ζp,q (s) = [2]q

∞ ∑ (−1)n q n . [n]sp,q n=1

(1.3)

Note that ζp,q (s) is a meromorphic function on C. Note that, if p = 1, q → 1, then ζp,q (s) = ζE (s) which is the Euler zeta function(see [3, 4]). Relation between ζp,q (s) and Ek,p,q is given by the following theorem. Theorem 6. For k ∈ N, we have ζp,q (−k) = Ek,p,q . Observe that ζp,q (s) function interpolates Ek,p,q numbers at non-negative integers. By using (1.2), we note that ∞ ∑ dk F (t, x) = [2] (−1)m q m [m + x]kp,q (1.4) p,q q dtk t=0 m=0 and

(

d dt

)k ( ∑ ∞

tn En,p,q (x) n! n=0

)

= Ek,p,q (x), for k ∈ N.

(1.5)

t=0

By (1.4) and (1.5), we are now ready to define the Hurwitz (p, q)-Euler zeta function. Definition 7. Let s ∈ C with Re(s) > 0 and x ∈ / Z− 0. ζp,q (s, x) = [2]q

∞ ∑ (−1)n q n . [n + x]sp,q n=0

(1.6)

Note that ζp,q (s, x) is a meromorphic function on C. Obverse that, if p = 1 and q → 1, then ζp,q (s, x) = ζE (s, x) which is the Hurwitz Euler zeta function(see [3, 4]). Relation between ζp,q (s, x) and Ek,p,q (x) is given by the following theorem. Theorem 8. For k ∈ N, we have ζp,q (−k, x) = Ek,p,q (x). Observe that ζp,q (−k, x) function interpolates Ek,p,q (x) numbers at non-negative integers. 2. Symmetric properties about (p, q)-analogue of Euler zeta functions In this section, we are going to obtain the main results of (p, q)-Euler zeta function. We also establish some interesting symmetric identities for (p, q)-Euler polynomials by using (p, q)-Euler zeta function. Observe that [xy]p,q = [x]py ,qy [y]p,q for any x, y ∈ C. By substitute w1 x + we derive

w1 i w2

for x in Definition 7, replace p by pw2 and replace q by q w2 , respectively, ) w1 i s, w1 x + w2 ∞ ∑ (−1)n q w2 n

( ζ

pw2 ,q w2

= [2]qw2

n=0

[w1 x +

= [2]qw2 [w2 ]sp,q

w1 i w2

+ n]spw2 ,qw2

∞ ∑

(−1)n q w2 n . [w1 w2 x + w1 i + w2 n]sp,q n=0

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Since for any non-negative integer m and odd positive integer w1 , there exist unique non-negative integer r such that m = w1 r + j with 0 ≤ j ≤ w1 − 1. Hence, this can be written as ( ) w1 i ζpw2 ,qw2 s, w1 x + w2 ∞ ∑ (−1)w1 r+j q w2 (w1 r+j) = [2]qw2 [w2 ]sp,q [w2 (w1 r + j) + w1 w2 x + w1 i]sp,q w r+j=0 1

0≤j≤w1 −1

= [2]qw2 [w2 ]sp,q

w∑ ∞ 1 −1 ∑ j=0 r=0

(−1)w1 r+j q w2 (w1 r+j) . [w1 w2 (r + x) + w1 i + w2 j]sp,q

It follows from the above equation that [2]qw1 [w1 ]sp,q

w∑ 2 −1

( i w1 i

(−1) q

ζpw2 ,qw2

i=0

=

w1 i s, w1 x + w2

w∑ ∞ 2 −1 w 1 −1 ∑ ∑

[2]qw1 [2]qw2 [w1 ]sp,q [w2 ]sp,q

i=0

j=0 r=0

) (2.1)

(−1)r+i+j q (w1 w2 r+w1 i+w2 j) . [w1 w2 (r + x) + w1 i + w2 j]sq

From the similar method, we can have that ( ) ∞ ∑ w2 j (−1)n q w1 n ζpw1 ,qw1 s, w2 x + = [2]qw1 w1 [w2 x + ww21j + n]spw1 ,qw1 n=0 = [2]qw1 [w1 ]sp,q

∞ ∑

(−1)n q w1 n . [w1 w2 x + w2 j + w1 n]sp,q n=0

After some calculations in the above, we have [2]

q w2

[w2 ]sp,q

w∑ 1 −1

(h) (−1)j q w2 j ζpw1 ,qw1

j=0

= [2]qw1 [2]qw2 [w1 ]sp,q [w2 ]sp,q

( ) w2 j s, w2 x + w1

w∑ ∞ 2 −1 w 1 −1 ∑ ∑ j=0 r=0

i=0

(2.2) (−1)r+i+j q (w1 w2 r+w1 i+w2 j) . [w1 w2 (r + x) + w1 i + w2 j]sp,q

Thus, we have the following theorem from (2.1) and (2.2). Theorem 9. Let s ∈ C with Re(s) > 0 and w1 , w2 : odd positive integers. Then one has ( ) w∑ 2 −1 w1 i [2]qw1 [w1 ]sp,q (−1)i q w1 i ζpw2 ,qw2 s, w1 x + w2 i=0 ( ) w∑ 1 −1 w2 j j w j s 2 = [2]qw2 [w2 ]p,q (−1) q ζpw1 ,qw1 s, w2 x + . w1 j=0 In Theorem 9, we get the following formulas for the (p, q)-tangent zeta function. Corollary 10. Let w2 = 1 in Theorem 9. Then we get ζp,q (s, x) =

[w1 ]−s p,q

w∑ 1 −1

( j j

(−1) q ζpw1 ,qw1

j=0

x+j s, w1

) .

Corollary 11. Let w1 = 2, w2 = 1 in Theorem 9. Then we have ( ) ( x) x+1 s − qζp2 ,q2 s, = [2]q2 [2]−1 ζp2 ,q2 s, q [2]p,q ζp,q (s, x). 2 2

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For n ∈ N, we have ζp,q (−n, x) = En,p,q (x), (see Theorem 8). By substituting En,p,q (x) for ζp,q (s, x) in Theorem 9, we can derive that ( ) w1 i −n, w1 x + [2] (−1) q ζ w2 i=0 ) ( w∑ 2 −1 w1 i i w1 i w w , (−1) q E w x + = [2]qw1 [w1 ]−n n,p 2 ,q 2 1 p,q w2 i=0 q w1

[w1 ]−n p,q

and [2]qw2 [w2 ]−n p,q

w∑ 2 −1

w∑ 1 −1

i w1 i

j w2 j

(−1) q

pw2 ,q w2

ζpw1 ,qw1

j=0

=

[2]qw2 [w2 ]−n p,q

w∑ 1 −1

j w2 j

(−1) q

( ) w2 j −n, w2 x + w1

En,pw1 ,qw1

j=0

( ) w2 j w2 x + . w1

Thus, we obtain the following theorem from Theorem 9. Theorem 12. Let w1 , w2 be any odd positive integer. Then for non-negative integers n, one has

( ) w1 i [2] (−1) q E w1 x + w2 i=0 ) ( w∑ 1 −1 w2 j . = [2]qw2 [w1 ]np,q (−1)j q w2 j En,pw1 ,qw1 w2 x + w1 j=0 q w1

[w2 ]np,q

w∑ 2 −1

i w1 i

n,pw2 ,q w2

Considering w1 = 1 in the Theorem 12, we obtain as below equation(see Theorem 3). ) ( w∑ 2 −1 [2]q x+j n j j En,p,q (x) = [w2 ]p,q . (−1) q En,pw2 ,qw2 [2]qw2 w2 j=1 We obtain another result by applying the addition theorem for the Carlitz’s type (h, p, q)(h) tangent polynomials En,p,q (x). From the Theorem 12, we have [2]qw1 [w2 ]np,q

w∑ 2 −1

( (−1)i q w1 i En,pw2 ,qw2

i=0

w1 x +

w1 i w2

)

( )l n ( ) ∑ n w1 (n−l)i w1 w2 xl (l) [w1 ]p,q q p En−l,pw2 ,qw2 (w1 x) [i]lpw1 ,qw1 [w ] l 2 p,q i=0 l=0 ( ) ( )l w∑ n 2 −1 ∑ n [w1 ]p,q (l) n = [2]qw1 [w2 ]p,q pw1 w2 xl En−l,pw2 ,qw2 (w1 x) (−1)i q w1 i q (n−l)w1 i [i]lpw1 ,qw1 . l [w2 ]p,q i=0 = [2]qw1 [w2 ]np,q

w∑ 2 −1

(−1)i q w1 i

l=0

Therefore, we obtain that [2]qw1 [w2 ]np,q

w∑ 2 −1

(−1)i q w1 i En,pw2 ,qw2

i=0

= [2]qw1

) ( w1 i w1 x + w2

n ( ) ∑ n w1 w2 xl (l) [w1 ]lp,q [w2 ]n−l En−l,pw2 ,qw2 (w1 x)En,l,pw1 ,qw1 (w2 ), p,q p l

(2.3)

l=0

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and [2]qw2 [w1 ]np,q

w∑ 1 −1

( j w2 j

(−1) q

En,pw1 ,qw1

j=0

= [2]qw2

w2 j w2 x + w1

)

n ( ) ∑ n w1 w2 xl (l) [w2 ]lp,q [w1 ]n−l En−l,pw1 ,qw1 (w2 x)En,l,pw2 ,qw2 (w1 ). p,q p l

(2.4)

l=0

where En,l,p,q (k) =

∑k−1 i=0

(−1)i q (1+n−l)i [i]lp,q is called as the sums of powers.

Hence, from (2.3) and (2.4), we have the following theorem. Theorem 13. Let w1 , w2 be any odd positive integer. Then we have n ( ) ∑ n w1 w2 xl (l) [2]qw2 [w2 ]lp,q [w1 ]n−l En−l,pw1 ,qw1 (w2 x)En,l,pw2 ,qw2 (w1 ) p,q p l l=0 n ( ) ∑ n w1 w2 xl (l) w = [2]q 1 [w1 ]lp,q [w2 ]n−l En−l,pw2 ,qw2 (w1 x)En,l,pw1 ,qw1 (w2 ). p,q p l l=0

Acknowledgement: This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MEST) (No. 2017R1A2B4006092). REFERENCES 1. R. P. Agarwal, J. Y. Kang, C. S. Ryoo, Some properties of (p, q)-tangent polynomials, J. Computational Analysis and Applications, 24 (2018), 1439-1454. 2. N. S. Jung, C. S. Ryoo, A research on a new approach to Euler polynomials and Bernstein polynomials with variable [x]q , J. Appl. Math. & Informatics, 35 (2017), 205-215. 3. A. M. Robert, A Course in p-adic Analysis, Graduate Text in Mathematics, Vol. 198, Springer, 2000. 4. H. Ozden, Y. Simsek, A new extension of q-Euler numbers and polynomials related to their interpolation functions, Appl. Math. Letters, 21 (2008), 934-938. 5. C. S. Ryoo, A numerical investigation on the zeros of the tangent polynomials, J. Appl. Math. & Informatics, 32 (2014), 315-322. 6. C. S. Ryoo, On the (p, q)-analogue of Euler zeta function, J. Appl. Math. & Informatics, 35 (2017), 303-311. 7. C. S. Ryoo, On degenerate q-tangent polynomials of higher order, J. Appl. Math. & Informatics 35 (2017), 113-120. 8. C. S. Ryoo, R .P. Agarwal, Some identities involving q-poly-tangent numbers and polynomials and distribution of their zeros, Advances in Difference Equations 2017:213 (2017), 1-14. 9. H. Shin, J. Zeng, The q-tangent and q-secant numbers via continued fractions, European J. Combin., 31 (2010), 1689-1705. 10. P. T. Young, Degenerate Bernoulli polynomials, generalized factorial sums, and their applications, Journal of Number Theorey, 128 (2008), 738-758.

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ADDITIVE (ρ1 , ρ2 )-FUNCTIONAL INEQUALITIES IN COMPLEX BANACH SPACES CHOONKIL PARK, DONG YUN SHIN∗ , AND GEORGE A. ANASTASSIOU Abstract. In this paper, we introduce and solve the following additive (ρ1 , ρ2 )-functional inequalities kf (x + y + z) − f (x) − f (y) − f (z)k ≥ kρ1 (f (x + y − z) − f (x) − f (y) + f (z))k + kρ2 (f (x − y + z) − f (x) + f (y) − f (z))k ,

(0.1)

where ρ1 and ρ2 are fixed complex numbers with |ρ1 | · |ρ2 | > 1, and kf (x + y − z) − f (x) − f (y) + f (z)k ≥ kρ1 (f (x + y + z) − f (x) − f (y) − f (z))k + kρ2 (f (x − y + z) − f (x) + f (y) − f (z))k

(0.2)

where ρ1 and ρ2 are fixed complex numbers with |ρ1 | > 1. Using the fixed point method and the direct method, we prove the Hyers-Ulam stability of the additive (ρ1 , ρ2 )-functional inequalities (0.1) and (0.2) in complex Banach spaces.

1. Introduction and preliminaries The stability problem of functional equations originated from a question of Ulam [29] concerning the stability of group homomorphisms. The functional equation f (x + y) = f (x) + f (y) is called the Cauchy equation. In particular, every solution of the Cauchy equation is said to be an additive mapping. Hyers [13] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [2] for additive mappings and by Rassias [23] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Rassias theorem was obtained by G˘ avruta [12] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias’ approach. The stability of quadratic functional equation was proved by Skof [28] for mappings f : E1 → E2 , where E1 is a normed space and E2 is a Banach space. Cholewa [8] noticed that the theorem of Skof is still true if the relevant domain E1 is replaced by an Abelian group. Park [18, 19] defined additive ρ-functional inequalities and proved the Hyers-Ulam stability of the additive ρ-functional inequalities in Banach spaces and non-Archimedean Banach spaces. The stability problems of various functional equations have been extensively investigated by a number of authors (see [1, 3, 7, 10, 11, 15, 17, 20, 21, 24, 25, 26, 27, 30, 31, 32]). We recall a fundamental result in fixed point theory. Theorem 1.1. [4, 9] Let (X, d) be a complete generalized metric space and let J : X → X be a strictly contractive mapping with Lipschitz constant α < 1. Then for each given element x ∈ X, either d(J n x, J n+1 x) = ∞ for all nonnegative integers n or there exists a positive integer n0 such that (1) d(J n x, J n+1 x) < ∞, ∀n ≥ n0 ; (2) the sequence {J n x} converges to a fixed point y ∗ of J; 2010 Mathematics Subject Classification. Primary 39B62, 47H10, 39B52. Key words and phrases. Hyers-Ulam stability; additive (ρ1 , ρ2 )-functional inequality; fixed point method; direct method; Banach space. ∗ Corresponding author (Dong Yun Shin).

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(3) y ∗ is the unique fixed point of J in the set Y = {y ∈ X | d(J n0 x, y) < ∞}; 1 (4) d(y, y ∗ ) ≤ 1−α d(y, Jy) for all y ∈ Y . In 1996, G. Isac and Th.M. Rassias [14] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [5, 6, 22]). In Section 2, we solve the additive (ρ1 , ρ2 )-functional inequality (0.1) and prove the HyersUlam stability of the additive (ρ1 , ρ2 )-functional inequality (0.1) in Banach spaces by using the fixed point method. In Section 3, we prove the Hyers-Ulam stability of the additive (ρ1 , ρ2 )-functional inequality (0.1) in Banach spaces by using the direct method. In Section 4, we solve the additive (ρ1 , ρ2 )-functional inequality (0.1) and prove the HyersUlam stability of the additive (ρ1 , ρ2 )-functional inequality (0.1) in Banach spaces by using the fixed point method. In Section 5, we prove the Hyers-Ulam stability of the additive (ρ1 , ρ2 )-functional inequality (0.1) in Banach spaces by using the direct method. Throughout this paper, let X be a real or complex normed space with norm k · k and Y a complex Banach space with norm k · k. Assume that ρ1 and ρ2 are fixed complex numbers with |ρ1 | · |ρ2 | > 1. 2. Additive (ρ1 , ρ2 )-functional inequality (0.1): a fixed point method In this section, we solve and investigate the additive (ρ1 , ρ2 )-functional inequality (0.1) in complex Banach spaces. Lemma 2.1. If a mapping f : X → Y satisfies f (0) = 0 and kf (x + y + z) − f (x) − f (y) − f (z)k ≥ kρ1 (f (x + y − z) − f (x) − f (y) + f (z))k + kρ2 (f (x − y + z) − f (x) + f (y) − f (z))k (2.1) for all x, y, z ∈ X, then f : X → Y is additive. Proof. Assume that f : X → Y satisfies (2.1). Since |ρ1 | · |ρ2 | > 1, |ρ1 | > 1 or |ρ2 | > 1. (i) Assume that |ρ1 | > 1. Letting z = 0 in (4.1), we get (1 − |ρ1 |)kf (x + y) − f (x) − f (y)k ≥ |ρ2 |kf (x − y) − f (x) + f (y)k for all x, y ∈ X. So f (x + y) = f (x) + f (y) for all x, y ∈ X, since |ρ1 | > 1. So f is additive. (ii) Assume that |ρ2 | > 1. Letting y = 0 in (4.1), we get (1 − |ρ2 |)kf (x + z) − f (x) − f (z)k ≥ |ρ1 |kf (x − z) − f (x) + f (z)k for all x, z ∈ X. So f (x + z) = f (x) + f (z) for all x, z ∈ X, since |ρ2 | > 1. So f is additive.



Using the fixed point method, we prove the Hyers-Ulam stability of the additive (ρ1 , ρ2 )functional inequality (2.1) in complex Banach spaces. Since |ρ1 | · |ρ2 | > 1, |ρ1 | > 1 or |ρ2 | > 1. One can exchange y and z and from now on, one can assume that |ρ1 | > 1. Theorem 2.2. Let ϕ : X 3 → [0, ∞) be a function such that there exists an L < 1 with x y z L ϕ , , ≤ ϕ (x, y, z) 2 2 2 2 for all x, y, z ∈ X. Let f : X → Y be a mapping satisfying f (0) = 0 and 



(2.2)

kρ1 (f (x + y − z) − f (x) − f (y) + f (z))k + kρ2 (f (x − y + z) − f (x) + f (y) − f (z))k ≤ kf (x + y + z) − f (x) − f (y) − f (z)k + ϕ(x, y, z) (2.3)

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ADDITIVE (ρ1 , ρ2 )-FUNCTIONAL INEQUALITY

for all x, y, z ∈ X. Then there exists a unique additive mapping A : X → Y such that L kf (x) − A(x)k ≤ ϕ (x, x, 0) 2(1 − L)(|ρ1 | − 1) for all x ∈ X. Proof. Letting z = 0 and y = x in (2.3), we get kf (2x) − 2f (x)k ≤

1 ϕ(x, x, 0) |ρ1 | − 1

(2.4)

for all x ∈ X. Consider the set S := {h : X → Y, h(0) = 0} and introduce the generalized metric on S: d(g, h) = inf {µ ∈ R+ : kg(x) − h(x)k ≤ µϕ (x, x, 0) , ∀x ∈ X} , where, as usual, inf φ = +∞. It is easy to show that (S, d) is complete (see [16]). Now we consider the linear mapping J : S → S such that   x Jg(x) := 2g 2 for all x ∈ X. Let g, h ∈ S be given such that d(g, h) = ε. Then kg(x) − h(x)k ≤ εϕ (x, x, 0) for all x ∈ X. Hence

kJg(x) − Jh(x)k =

2g

x 2

 

− 2h

    x

≤ 2εϕ x , x , 0 2 2 2

L ≤ 2ε ϕ (x, x, 0) = Lεϕ (x, x, 0) 2 for all x ∈ X. So d(g, h) = ε implies that d(Jg, Jh) ≤ Lε. This means that d(Jg, Jh) ≤ Ld(g, h) for all g, h ∈ S. It follows from (2.4) that

   

1 x x L

f (x) − 2f x ≤ ϕ , ,0 ≤ ϕ(x, x, 0)

2 |ρ1 | − 1 2 2 2(|ρ1 | − 1)

for all x ∈ X So d(f, Jf ) ≤ 2(|ρ1L|−1) . By Theorem 1.1, there exists a mapping A : X → Y satisfying the following: (1) A is a fixed point of J, i.e.,   x A (x) = 2A 2 for all x ∈ X. The mapping A is a unique fixed point of J in the set

(2.5)

M = {g ∈ S : d(f, g) < ∞}. This implies that A is a unique mapping satisfying (2.5) such that there exists a µ ∈ (0, ∞) satisfying kf (x) − A(x)k ≤ µϕ (x, x, 0) for all x ∈ X; (2) d(J l f, A) → 0 as l → ∞. This implies the equality

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n



lim 2 f

l→∞

for all x ∈ X; (3) d(f, A) ≤

1 1−L d(f, Jf ),

x 2n



= A(x)

which implies

kf (x) − A(x)k ≤

L ϕ (x, x, 0) 2(1 − L)(|ρ1 | − 1)

for all x ∈ X. It follows from (2.2) and (2.3) that kA (x + y + z) − A(x) − A(y) − A(z)k

         

x y z x+y+z x y z n n

+ lim 2 ϕ n , n , n −f −f −f = lim 2 f n→∞ 2n 2n 2n 2n n→∞ 2 2 2

       

x+y−z x y z ≥ lim 2n |ρ1 | −f −f +f

f n n n n→∞ 2 2 2 2n

       

x−y+z x y z

+ lim 2n |ρ2 | −f +f −f

f n n n n→∞ 2 2 2 2n = kρ1 (A(x + y − z) − A(x) − A(y) + A(z))k + kρ2 (A(x − y + z) − A(x) + A(y) − A(z))k for all x, y, z ∈ X. So kA (x + y + z) − A(x) − A(y) − A(z)k ≥ kρ1 (A(x + y − z) − A(x) − A(y) + A(z))k + kρ2 (A(x − y + z) − A(x) + A(y) − A(z))k for all x, y, z ∈ X. By Lemma 2.1, the mapping A : X → Y is additive.



Corollary 2.3. Let r > 1 and θ be nonnegative real numbers, and let f : X → Y be a mapping satisfying f (0) = 0 and kρ1 (f (x + y − z) − f (x) − f (y) + f (z))k + kρ2 (f (x − y + z) − f (x) + f (y) − f (z))k ≤ kf (x + y + z) − f (x) − f (y) − f (z)k + θ(kxkr + kykr + kzkr ) (2.6) for all x, y, z ∈ X. Then there exists a unique additive mapping A : X → Y such that kf (x) − A(x)k ≤

(2r

2θ kxkr − 2)(|ρ1 | − 1)

for all x ∈ X. Proof. The proof follows from Theorem 2.2 by taking ϕ(x, y, z) = θ(kxkr + kykr + kzkr ) for all x, y, z ∈ X. Choosing L = 21−r , we obtain the desired result.  Theorem 2.4. Let ϕ : X 3 → [0, ∞) be a function such that there exists an L < 1 with x y z ϕ (x, y, z) ≤ 2Lϕ , , 2 2 2 



(2.7)

for all x, y, z ∈ X. Let f : X → Y be a mapping satisfying f (0) = 0 and (2.3). Then there exists a unique additive mapping A : X → Y such that kf (x) − A(x)k ≤

1 ϕ (x, x, 0) 2(1 − L)(|ρ1 | − 1)

for all x ∈ X.

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CHOONKIL PARK ET AL 367-379

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.2, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

ADDITIVE (ρ1 , ρ2 )-FUNCTIONAL INEQUALITY

Proof. Let (S, d) be the generalized metric space defined in the proof of Theorem 2.2. Now we consider the linear mapping J : S → S such that 1 Jg(x) := g (2x) 2 for all x ∈ X. It follows from (2.4) that



1

f (x) − 1 f (2x) ≤ ϕ(x, x, 0)

2 2(|ρ1 | − 1) for all x ∈ X. The rest of the proof is similar to the proof of Theorem 2.2.



Corollary 2.5. Let r < 1 and θ be positive real numbers, and let f : X → Y be a mapping satisfying f (0) = 0 and (2.6). Then there exists a unique additive mapping A : X → Y such that 2θ kf (x) − A(x)k ≤ kxkr (2 − 2r )(|ρ1 | − 1) for all x ∈ X. Proof. The proof follows from Theorem 2.4 by taking ϕ(x, y, z) = θ(kxkr + kykr + kzkr ) for all x, y, z ∈ X. Choosing L = 2r−1 , we obtain the desired result.  Remark 2.6. If ρ1 and ρ2 are real numbers such that |ρ1 | · |ρ2 | > 1 and Y is a real Banach space, then all the assertions in this section remain valid. 3. Additive (ρ1 , ρ2 )-functional inequality (0.1): a direct method In this section, we prove the Hyers-Ulam stability of the additive (ρ1 , ρ2 )-functional inequality (2.1) in complex Banach spaces by using the direct method. Theorem 3.1. Let ϕ : X 3 → [0, ∞) be a function such that ∞ X

x y z 2 ϕ j, j, j Ψ(x, y, z) := 2 2 2 j=1 

j



l and all x ∈ X. It follows from (3.4) that the sequence {2k f ( 2xk )} is Cauchy for all x ∈ X. Since Y is a Banach space, the sequence {2k f ( 2xk )} converges. So one can define the mapping A : X → Y by   x k A(x) := lim 2 f k→∞ 2k for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (3.4), we get (3.2). It follows from (2.3) and (3.1) that kA (x + y + z) − A(x) − A(y) − A(z)k

         

x+y+z x y z x y z n n

= lim 2 f + lim 2 ϕ n , n , n −f −f −f n→∞ 2n 2n 2n 2n n→∞ 2 2 2

       

x + y − z x y z

−f −f +f ≥ lim 2n |ρ1 |

f n→∞ 2n 2n 2n 2n

       

x−y+z x y z n

+ lim 2 |ρ2 | f −f +f −f n n n n→∞ 2 2 2 2n = kρ1 (A(x + y − z) − A(x) − A(y) + A(z))k + kρ2 (A(x − y + z) − A(x) + A(y) − A(z))k for all x, y, z ∈ X. So kA (x + y + z) − A(x) − A(y) − A(z)k ≥ kρ1 (A(x + y − z) − A(x) − A(y) + A(z))k + kρ2 (A(x − y + z) − A(x) + A(y) − A(z))k for all x, y, z ∈ X. By Lemma 2.1, the mapping A : X → Y is additive. Now, let T : X → Y be another additive mapping satisfying (3.2). Then we have

   

q x x q

kA(x) − T (x)k = 2 A q − 2 T 2 2q

       

q

x x q

+ 2q T x − 2q f x ≤ 2 A q − 2 f

q q q 2 2 2 2   q

≤

2 x x Ψ q, q,0 , |ρ1 | − 1 2 2

which tends to zero as q → ∞ for all x ∈ X. So we can conclude that A(x) = T (x) for all x ∈ X. This proves the uniqueness of A.  Corollary 3.2. Let r > 1 and θ be nonnegative real numbers, and let f : X → Y be a mapping satisfying f (0) = 0 and (2.6). Then there exists a unique additive mapping A : X → Y such that kf (x) − A(x)k ≤

(2r

2θ kxkr − 2)(|ρ1 | − 1)

for all x ∈ X. Theorem 3.3. Let ϕ : X 3 → [0, ∞) be a function and let f : X → Y be a mapping satisfying f (0) = 0, (2.3) and Ψ(x, y, z) :=

∞ X 1

2j j=0

ϕ(2j x, 2j y, 2j z) < ∞

(3.5)

for all x, y, z ∈ X. Then there exists a unique additive mapping A : X → Y such that 1 kf (x) − A(x)k ≤ Ψ(x, x, 0) 2(|ρ1 | − 1) for all x ∈ X.

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CHOONKIL PARK ET AL 367-379

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.2, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

ADDITIVE (ρ1 , ρ2 )-FUNCTIONAL INEQUALITY

Proof. It follows from (3.3) that



1

f (x) − 1 f (2x) ≤ ϕ(x, x)

2 2(|ρ1 | − 1)

for all x ∈ X. Hence

1

f (2l x) − 1 f (2m x) ≤

2l

2m

≤

m−1 X j=l m−1 X j=l

   

1 f 2j x − 1 f 2j+1 x

2j

2j+1

1 2j+1 (|ρ

1|

− 1)

ϕ(2j x, 2j x, 0)

(3.6)

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (3.6) that the sequence { 21n f (2n x)} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence { 21n f (2n x)} converges. So one can define the mapping A : X → Y by 1 f (2n x) n→∞ 2n for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (3.6), we get (3.6). The rest of the proof is similar to the proof of Theorem 3.1. A(x) := lim



Corollary 3.4. Let r < 1 and θ be positive real numbers, and let f : X → Y be a mapping satisfying f (0) = 0 and (2.6). Then there exists a unique additive mapping A : X → Y such that 2θ kxkr kf (x) − A(x)k ≤ r (2 − 2 )(|ρ1 | − 1) for all x ∈ X. 4. Additive (ρ1 , ρ2 )-functional inequality (0.2): a fixed point method In this section, we solve and investigate the additive (ρ1 , ρ2 )-functional inequality (0.2) in complex Banach spaces. From now on, assume that ρ1 | > 1. Lemma 4.1. If a mapping f : X → Y satisfies f (0) = 0 and kf (x + y − z) − f (x) − f (y) + f (z)k ≥ kρ1 (f (x + y + z) − f (x) − f (y) − f (z))k + kρ2 (f (x − y + z) − f (x) + f (y) − f (z))k (4.1) for all x, y, z ∈ X, then f : X → Y is additive. Proof. Assume that f : X → Y satisfies (4.1). Letting z = 0 in (4.1), we get (1 − |ρ1 |)kf (x + y) − f (x) − f (y)k ≥ |ρ2 |kf (x − y) − f (x) + f (y)k for all x, y ∈ X. So f (x + y) = f (x) + f (y) for all x, y ∈ X, since |ρ1 | > 1. So f is additive.



Using the fixed point method, we prove the Hyers-Ulam stability of the additive (ρ1 , ρ2 )functional inequality (4.1) in complex Banach spaces. Theorem 4.2. Let ϕ : X 3 → [0, ∞) be a function such that there exists an L < 1 with   x y z L ϕ , , ≤ ϕ (x, y, z) 2 2 2 2 for all x, y, z ∈ X. Let f : X → Y be a mapping satisfying f (0) = 0 and

(4.2)

kρ1 (f (x + y + z) − f (x) − f (y) − f (z))k + kρ2 (f (x − y + z) − f (x) + f (y) − f (z))k ≤ kf (x + y − z) − f (x) − f (y) + f (z)k + ϕ(x, y, z) (4.3)

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.2, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

C. PARK, D.Y. SHIN, AND G.A. ANASTASSIOU

for all x, y, z ∈ X. Then there exists a unique additive mapping A : X → Y such that L kf (x) − A(x)k ≤ ϕ (x, x, 0) 2(1 − L)(|ρ1 | − 1) for all x ∈ X. Proof. Letting y = x and z = 0 in (4.3), we get kf (2x) − 2f (x)k ≤

1 ϕ(x, x, 0) |ρ1 | − 1

(4.4)

for all x ∈ X. Consider the set S := {h : X → Y, h(0) = 0} and introduce the generalized metric on S: d(g, h) = inf {µ ∈ R+ : kg(x) − h(x)k ≤ µϕ (x, x, 0) , ∀x ∈ X} , where, as usual, inf φ = +∞. It is easy to show that (S, d) is complete (see [16]). Now we consider the linear mapping J : S → S such that   x Jg(x) := 2g 2 for all x ∈ X. Let g, h ∈ S be given such that d(g, h) = ε. Then kg(x) − h(x)k ≤ εϕ (x, x, 0) for all x ∈ X. Hence

kJg(x) − Jh(x)k =

2g

x 2

 

− 2h

    x

≤ 2εϕ x , x , 0 2 2 2

L ≤ 2ε ϕ (x, x, 0) = Lεϕ (x, x, 0) 2 for all x ∈ X. So d(g, h) = ε implies that d(Jg, Jh) ≤ Lε. This means that d(Jg, Jh) ≤ Ld(g, h) for all g, h ∈ S. It follows from (4.4) that

   

1 x x L

f (x) − 2f x ≤ ϕ , ,0 ≤ ϕ(x, x, 0)

2 |ρ1 | − 1 2 2 2(|ρ1 | − 1)

for all x ∈ X So d(f, Jf ) ≤ 2(|ρ1L|−1) . By Theorem 1.1, there exists a mapping A : X → Y satisfying the following: (1) A is a fixed point of J, i.e.,   x A (x) = 2A 2 for all x ∈ X. The mapping A is a unique fixed point of J in the set

(4.5)

M = {g ∈ S : d(f, g) < ∞}. This implies that A is a unique mapping satisfying (4.5) such that there exists a µ ∈ (0, ∞) satisfying kf (x) − A(x)k ≤ µϕ (x, x, 0) for all x ∈ X; (2) d(J l f, A) → 0 as l → ∞. This implies the equality

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.2, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

ADDITIVE (ρ1 , ρ2 )-FUNCTIONAL INEQUALITY

n



lim 2 f

l→∞

for all x ∈ X; (3) d(f, A) ≤

1 1−L d(f, Jf ),

x 2n



= A(x)

which implies

kf (x) − A(x)k ≤

L ϕ (x, x, 0) 2(1 − L)(|ρ1 | − 1)

for all x ∈ X. It follows from (4.2) and (4.3) that kA (x + y − z) − A(x) − A(y) + A(z)k

         

x y z x+y−z x y z n n

+ lim 2 ϕ n , n , n −f −f +f = lim 2 f n→∞ 2n 2n 2n 2n n→∞ 2 2 2

       

x+y+z x y z ≥ lim 2n |ρ1 | −f −f −f

f n n n n→∞ 2 2 2 2n

       

x−y+z x y z

+ lim 2n |ρ2 | −f +f −f

f n n n n→∞ 2 2 2 2n = kρ1 (A(x + y + z) − A(x) − A(y) − A(z))k + kρ2 (A(x − y + z) − A(x) + A(y) − A(z))k for all x, y, z ∈ X. So kA (x + y − z) − A(x) − A(y) + A(z)k ≥ kρ1 (A(x + y + z) − A(x) − A(y) − A(z))k + kρ2 (A(x − y + z) − A(x) + A(y) − A(z))k for all x, y, z ∈ X. By Lemma 4.1, the mapping A : X → Y is additive.



Corollary 4.3. Let r > 1 and θ be nonnegative real numbers, and let f : X → Y be a mapping satisfying f (0) = 0 and kρ1 (f (x + y + z) − f (x) − f (y) − f (z))k + kρ2 (f (x − y + z) − f (x) + f (y) − f (z))k ≤ kf (x + y − z) − f (x) − f (y) + f (z)k + θ(kxkr + kykr + kzkr ) (4.6) for all x, y, z ∈ X. Then there exists a unique additive mapping A : X → Y such that kf (x) − A(x)k ≤

(2r

2θ kxkr − 2)(|ρ1 | − 1)

for all x ∈ X. Proof. The proof follows from Theorem 4.2 by taking ϕ(x, y, z) = θ(kxkr + kykr + kzkr ) for all x, y, z ∈ X. Choosing L = 21−r , we obtain the desired result.  Theorem 4.4. Let ϕ : X 3 → [0, ∞) be a function such that there exists an L < 1 with x y z ϕ (x, y, z) ≤ 2Lϕ , , 2 2 2 



(4.7)

for all x, y, z ∈ X. Let f : X → Y be a mapping satisfying f (0) = 0 and (4.3). Then there exists a unique additive mapping A : X → Y such that kf (x) − A(x)k ≤

1 ϕ (x, x, 0) 2(1 − L)(|ρ1 | − 1)

for all x ∈ X.

375

CHOONKIL PARK ET AL 367-379

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.2, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

C. PARK, D.Y. SHIN, AND G.A. ANASTASSIOU

Proof. Let (S, d) be the generalized metric space defined in the proof of Theorem 4.2. Now we consider the linear mapping J : S → S such that 1 Jg(x) := g (2x) 2 for all x ∈ X. It follows from (4.4) that



1

f (x) − 1 f (2x) ≤ ϕ(x, x, 0)

2 2(|ρ1 | − 1) for all x ∈ X. The rest of the proof is similar to the proof of Theorem 4.2.



Corollary 4.5. Let r < 1 and θ be positive real numbers, and let f : X → Y be a mapping satisfying f (0) = 0 and (4.6). Then there exists a unique additive mapping A : X → Y such that 2θ kf (x) − A(x)k ≤ kxkr (2 − 2r )(|ρ1 | − 1) for all x ∈ X. Proof. The proof follows from Theorem 4.4 by taking ϕ(x, y, z) = θ(kxkr + kykr + kzkr ) for all x, y, z ∈ X. Choosing L = 2r−1 , we obtain the desired result.  Remark 4.6. If ρ1 and ρ2 are real numbers such that |ρ1 | > 1 and Y is a real Banach space, then all the assertions in this section remain valid. 5. Additive (ρ1 , ρ2 )-functional inequality (0.2): a direct method In this section, we prove the Hyers-Ulam stability of the additive (ρ1 , ρ2 )-functional inequality (4.1) in complex Banach spaces by using the direct method. Theorem 5.1. Let ϕ : X 3 → [0, ∞) be a function such that ∞ X

x y z 2 ϕ j, j, j Ψ(x, y, z) := 2 2 2 j=1 

j



l and all x ∈ X. It follows from (5.4) that the sequence {2k f ( 2xk )} is Cauchy for all x ∈ X. Since Y is a Banach space, the sequence {2k f ( 2xk )} converges. So one can define the mapping A : X → Y by   x k A(x) := lim 2 f k→∞ 2k for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (5.4), we get (5.2). It follows from (5.4) and (5.1) that kA (x + y − z) − A(x) − A(y) + A(z)k

         

x+y−z x y z x y z n n

= lim 2 f + lim 2 ϕ n , n , n −f −f +f n→∞ 2n 2n 2n 2n n→∞ 2 2 2

       

x + y + z x y z

−f −f −f ≥ lim 2n |ρ1 |

f n→∞ 2n 2n 2n 2n

       

x−y+z x y z n

+ lim 2 |ρ2 | f −f +f −f n n n n→∞ 2 2 2 2n = kρ1 (A(x + y + z) − A(x) − A(y) − A(z))k + kρ2 (A(x − y + z) − A(x) + A(y) − A(z))k for all x, y, z ∈ X. So kA (x + y − z) − A(x) − A(y) + A(z)k ≥ kρ1 (A(x + y + z) − A(x) − A(y) − A(z))k + kρ2 (A(x − y + z) − A(x) + A(y) − A(z))k for all x, y, z ∈ X. By Lemma 4.1, the mapping A : X → Y is additive. Now, let T : X → Y be another additive mapping satisfying (5.2). Then we have

   

q x x q

kA(x) − T (x)k = 2 A q − 2 T 2 2q

       

q

x x q

+ 2q T x − 2q f x ≤ 2 A q − 2 f

q q q 2 2 2 2   q

≤

2 x x Ψ q, q,0 , |ρ1 | − 1 2 2

which tends to zero as q → ∞ for all x ∈ X. So we can conclude that A(x) = T (x) for all x ∈ X. This proves the uniqueness of A.  Corollary 5.2. Let r > 1 and θ be nonnegative real numbers, and let f : X → Y be a mapping satisfying f (0) = 0 and (4.6). Then there exists a unique additive mapping A : X → Y such that kf (x) − A(x)k ≤

(2r

2θ kxkr − 2)(|ρ1 | − 1)

for all x ∈ X. Theorem 5.3. Let ϕ : X 3 → [0, ∞) be a function and let f : X → Y be a mapping satisfying f (0) = 0, (4.3) and Ψ(x, y, z) :=

∞ X 1

2j j=0

ϕ(2j x, 2j y, 2j z) < ∞

(5.5)

for all x, y, z ∈ X. Then there exists a unique additive mapping A : X → Y such that 1 kf (x) − A(x)k ≤ Ψ(x, x, 0) 2(|ρ1 | − 1) for all x ∈ X.

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CHOONKIL PARK ET AL 367-379

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.2, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

C. PARK, D.Y. SHIN, AND G.A. ANASTASSIOU

Proof. It follows from (5.3) that



1

f (x) − 1 f (2x) ≤ ϕ(x, x, 0)

2 2(|ρ1 | − 1)

for all x ∈ X. Hence



1

f (2l x) − 1 f (2m x) ≤

2l m 2

≤

m−1 X

j=l

   

1 f 2j x − 1 f 2j+1 x

2j

j+1 2

m−1 X j=l

1 2j+1 (|ρ

1 | − 1)

ϕ(2j x, 2j x, 0)

(5.6)

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (5.6) that the sequence { 21n f (2n x)} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence { 21n f (2n x)} converges. So one can define the mapping A : X → Y by 1 f (2n x) 2n for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (5.6), we get (5.6). The rest of the proof is similar to the proof of Theorem 5.1. A(x) := lim

n→∞



Corollary 5.4. Let r < 1 and θ be positive real numbers, and let f : X → Y be a mapping satisfying f (0) = 0 and (4.6). Then there exists a unique additive mapping A : X → Y such that kf (x) − A(x)k ≤

2θ (2 −

2r )(|ρ1 |

− 1)

kxkr

for all x ∈ X. References [1] M. Adam, On the stability of some quadratic functional equation, J. Nonlinear Sci. Appl. 4 (2011), 50–59. [2] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. [3] L. C˘ adariu, L. G˘ avruta, P. G˘ avruta, On the stability of an affine functional equation, J. Nonlinear Sci. Appl. 6 (2013), 60–67. [4] L. C˘ adariu, V. Radu, Fixed points and the stability of Jensen’s functional equation, J. Inequal. Pure Appl. Math. 4, no. 1, Art. ID 4 (2003). [5] L. C˘ adariu, V. Radu, On the stability of the Cauchy functional equation: a fixed point approach, Grazer Math. Ber. 346 (2004), 43–52. [6] L. C˘ adariu, V. Radu, Fixed point methods for the generalized stability of functional equations in a single variable, Fixed Point Theory Appl. 2008, Art. ID 749392 (2008). [7] A. Chahbi, N. Bounader, On the generalized stability of d’Alembert functional equation, J. Nonlinear Sci. Appl. 6 (2013), 198–204. [8] P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76–86. [9] J. Diaz, B. Margolis, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 74 (1968), 305–309. [10] N. Eghbali, J. M. Rassias, M. Taheri, On the stability of a k-cubic functional equation in intuitionistic fuzzy n-normed spaces, Results Math. 70 (2016), 233–248. [11] G. Z. Eskandani, P. Gˇ avruta, Hyers-Ulam-Rassias stability of pexiderized Cauchy functional equation in 2Banach spaces, J. Nonlinear Sci. Appl. 5 (2012), 459–465. [12] P. Gˇ avruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436. [13] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A. 27 (1941), 222–224. [14] G. Isac, Th. M. Rassias, Stability of ψ-additive mappings: Appications to nonlinear analysis, Internat. J. Math. Math. Sci. 19 (1996), 219–228. [15] H. Khodaei, On the stability of additive, quadratic, cubic and quartic set-valued functional equations, Results Math. 68 (2015), 1–10.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.2, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

ADDITIVE (ρ1 , ρ2 )-FUNCTIONAL INEQUALITY

[16] D. Mihet¸, V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl. 343 (2008), 567–572. [17] C. Park, Orthogonal stability of a cubic-quartic functional equation, J. Nonlinear Sci. Appl. 5 (2012), 28–36. [18] C. Park, Additive ρ-functional inequalities and equations, J. Math. Inequal. 9 (2015), 17–26. [19] C. Park, Additive ρ-functional inequalities in non-Archimedean normed spaces, J. Math. Inequal. 9 (2015), 397–407. [20] C. Park, K. Ghasemi, S. G. Ghaleh, S. Jang, Approximate n-Jordan ∗-homomorphisms in C ∗ -algebras, J. Comput. Anal. Appl. 15 (2013), 365–368. [21] C. Park, A. Najati, S. Jang, Fixed points and fuzzy stability of an additive-quadratic functional equation, J. Comput. Anal. Appl. 15 (2013), 452–462. [22] V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), 91–96. [23] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [24] K. Ravi, E. Thandapani, B. V. Senthil Kumar, Solution and stability of a reciprocal type functional equation in several variables, J. Nonlinear Sci. Appl. 7 (2014), 18–27. [25] S. Shagholi, M. Bavand Savadkouhi, M. Eshaghi Gordji, Nearly ternary cubic homomorphism in ternary Fr´echet algebras, J. Comput. Anal. Appl. 13 (2011), 1106–1114. [26] D. Shin, C. Park, Sh. Farhadabadi, On the superstability of ternary Jordan C ∗ -homomorphisms, J. Comput. Anal. Appl. 16 (2014), 964–973. [27] D. Shin, C. Park, Sh. Farhadabadi, Stability and superstability of J ∗ -homomorphisms and J ∗ -derivations for a generalized Cauchy-Jensen equation, J. Comput. Anal. Appl. 17 (2014), 125–134. [28] F. Skof, Propriet locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53 (1983), 113–129. [29] S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960. [30] Z. Wang, Stability of two types of cubic fuzzy set-valued functional equations, Results Math. 70 (2016), 1–14. [31] C. Zaharia, On the probabilistic stability of the monomial functional equation, J. Nonlinear Sci. Appl. 6 (2013), 51–59. [32] S. Zolfaghari, Approximation of mixed type functional equations in p-Banach spaces, J. Nonlinear Sci. Appl. 3 (2010), 110–122. Choonkil Park Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Republic of Korea E-mail address: [email protected] Dong Yun Shin Department of Mathematics, University of Seoul, Seoul 02504, Republic of Korea E-mail address: [email protected] George A. Anastassiou Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA E-mail address: [email protected]

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO. 2, 2019

An iterative algorithm of poles assignment for LDP systems, Lingling Lv, Zhe Zhang, Lei Zhang, and Xianxing Liu,…………………………………………………………………201 C*-algebra-valued modular metric spaces and related fixed point results, Bahman Moeini, Arslan Hojat Ansari, Choonkil Park, and Dong Yun Shin,……………………………………….211 Strong Convergence Theorems and Applications of a New Viscosity Rule for Nonexpansive Mappings, Waqas Nazeer, Mobeen Munir, Sayed Fakhar Abbas Naqvi, Chahn Yong Jung, and Shin Min Kang,……………………………………………………………………………221 Generalized stability of cubic functional equations with an automorphism on a quasi-𝛽𝛽 normed space, Dongseung Kang and Hoewoon B. Kim,…………………………………………..235 Two quotient BI-algebras induced by fuzzy normal subalgebras and fuzzy congruence relations, Yinhua Cui and Sun Shin Ahn,……………………………………………………………247 General quadratic functional equations in quasi-𝛽𝛽-normed spaces: solution, superstability and stability, Shahrokh Farhadabadi, Choonkil Park, and Sungsik Yun,………………………256 On Impulsive Sequential Fractional Differential Equations, N. I. Mahmudov and B. Sami,269 The Differentiability and Gradient for Fuzzy Mappings Based on the Generalized Difference of Fuzzy Numbers, Shexiang Hai and Fangdi Kong,…………………………………………284 Global Attractivity and Periodic Nature of a Higher order Difference Equation, M. M. ElDessoky, Abdul Khaliq, Asim Asiri, and Ansar Abbas,……………………………………294 Asymptotic Representations for Fourier Approximation of Functions on the Unit Square, Zhihua Zhang,………………………………………………………………………………………305 Khatri-Rao Products and Selection Operators, Arnon Ploymukda, Pattrawut Chansangiam,316 Some new coupled fixed point theorems in partially ordered complete Menger probabilistic Gmetric spaces, Gang Wang, Chuanxi Zhu, and Zhaoqi Wu,…………………………………326 Fourier series of sums of products of higher-order Euler functions, Taekyun Kim, Dae San Kim, Gwan-Woo Jang, and Jongkyum Kwon,……………………………………………………345 Some symmetric identities for (p, q)-Euler zeta function, Cheon Seoung Ryoo,…………...361

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO. 2, 2019 (continued)

Additive (𝜌𝜌1 , 𝜌𝜌2 )-functional inequalities in complex Banach spaces, Choonkil Park, Dong Yun Shin, and George A. Anastassiou,…………………………………………………………367

Volume 27, Number 3 ISSN:1521-1398 PRINT,1572-9206 ONLINE

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Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC

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Modified Halpern’s iteration without assumptions on fixed point set in metric space Kanyarat Cheawchan, Atid Kangtunyakarn∗ Department of Mathematics, Faculty of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand E-mail addresses: [email protected]; [email protected]

Abstract By improving Halpern’s iteration and studing convergence theorem of [1] and [2] in a complete uniformly convex metric space, we prove convergence theorem of a finite family of nonexpansive mappings without the assumption that 88 the set of common fixed points of nonexpansive mappings is nonempty00 . We also introduce a mapping in metric space using a concept of the S-mapping defined by [3] for proving our main results. Keywords: Convex metric space; Nonexpansive mapping; S-mapping. Mathematics Subject Classification (2000): 31E05, 54E40, 54E50, 47H09.

1

Introduction

Many researchers have theorized for finding a solution of fixed point problems by taking advantage of iteration process, see for instance [4], [5], [6]. Halpern’s iteration is a method which has been very popular for finding a solution to fixed point problem. It was introduced for the first time by Halpern [7] and defined by the vector u, x0 belonging to a closed convex C subset of Hilbert (Banach) space and xn+1 = αn u + (1 − αn ) T xn , for all n ≥ 1, where T : C → C is a mapping and parameter {αn } ⊆ [0, 1]. It has been developed and improved to fixed point theorem to increase efficiency by several researchers, see example [4], [5], [6]. Although the proof of the theorem has been well developed, but the proof is still under critical conditions below; ∗ Corresponding

author

1

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i)∗ F (T ) 6= ∅; ii)∗ limn→∞ αn = 0 and

P∞

n=1

αn = ∞.

Can we prove a convergence theorem by developing Halpern iteration and without conditions i)∗ and ii)∗ in space which is more general than Hilbert and Banach spaces? Throughout this paper, we assume that (X, d) is a complete metric space and C is a nonempty closed convex subset of (X, d). A point x is called a fixed point of T if T x = x. We use F (T ) to denote the set of fixed point of T . Recall the following definitions; Definition 1.1. The mapping T : C → C is said to be nonexpansive if d(T x, T y) ≤ d(x, y), ∀x, y ∈ C. In 1970, Takahashi [8] introduced the following definition: Definition 1.2. Let (X, d) be a metric space. A mapping W : X × X × [0, 1] → X is said to be a convex structure on X if for each (x, y, λ) ∈ X × X × [0, 1] and for all u ∈ X,
d u, W (x, y, λ) ≤ λd(u, x) + (1 − λ)d(u, y). If the mapping W is defined by W (x, y, λ) = λx + (1 − λ)y, then it is a convex structure on a normed linear space. A metric space (X, d) together with a convex structure W is called a convex metric space denoted by (X, d, W ). A nonempty subset C of X is said to be convex if W (x, y, λ) ∈ C for all x, y ∈ C and λ ∈ [0, 1]. Definition 1.3. (See [9]) A convex metric space (X, d, W ) is said to be uniformly convex if for any  > 0, there exists δ = δ() > 0 such that for all r > 0 and x, y, z ∈ X with d(z, x) < r, d(z, y) < r and d(x, y) ≥ r, 1
d z, W (x, y, ) ≤ (1 − δ)r. 2 It is well known that Hilbert space is uniformly convex metric space. Very recently, Hafiz Fukhar-ud-din [1] proved convergence theorem in uniformly convex metric spaces (X, d, W ) with convex structure but he still assumed the fixed point set is nonempty as follows; Theorem 1.1. Let C be a nonempty, closed and convex subset of a uniformly convex complete metric space X with continuous convex structure W and S, T : C → C be nonexpansive mappings with the sequence   F (S) ∩ F (T ) 6=  ∅. Then  βn {xn }, defined by xn+1 = W T xn , W Sxn , xn , , αn , ∆-converges to 1 − αn an element of F (S) ∩ F (T ), where 0 < a ≤ αn , βn ≤ b < 1 with αn + βn < 1.

2

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.3, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

In 2013, Phuengrattana and Suantai [2] proved convergence theorem in uniformly convex metric space for infinite family of nonexpansive mapping T∞ by leveraging the map Kn , see [2] for more details, but still assume that i=1 F (Ti ) 6= ∅ as follows; Theorem 1.2. Let C be a nonempty compact convex subset of a complete uniformly convex metric space (X, d, W ) with the property (H).T Let {Ti } be a ∞ family of nonexpansive mappings of C into itself such that i=1 F (Ti ) 6= ∅ and let λ1 , λ2 , . . . be real numbers such that 0 < λi < 1 for every i ∈ N with P ∞ i=1 λi < ∞. Let Kn be K-mapping generated by T1 , T2 , . . . and λ1 , λ2 , . . .. Assume that x1 ∈ C and the sequence {xn } is generated by xn+1 = W (xn , Kn xn , αn ) , P∞ for all n ≥ 1 where {αn } is a sequence in [0, 1]Twith n=1 αn (1 − αn ) = ∞. ∞ Then sequence {xn } converges to an element of i=1 F (Ti ) 6= ∅. Inspired by Theorem 1.1 and 1.2 and improved process of Halpern’s iteration, we prove convergence theorem in uniformly convex metric space for a finite family of nonexpansive mappings without using the conditions i)∗ and ii)∗ .

2

Preliminaries

In this section, in order to prove our main theorem, we provide definitions, lemma and also prove the importance lemma to be used as a tool to prove the main theorem: Lemma 2.1. (See [8], [10]) Let (X, d, W ) be a convex metric space. For each x, y ∈ X and λ, λ1 , λ2 ∈ [0, 1], we have the following. (i) W (x, x, λ) = x,
W (x, y, 0) = y and W (x, y, 1) = x.
(ii) d x, W (x, y, λ) = (1 − λ)d(x, y) and d y, W
(x, y, λ) = λd(x, y).
(iii) d(x, y) = d x, W (x, y, λ) + d W (x, y, λ), y
. (iv) |λ1 − λ2 |d(x, y) ≤ d W (x, y, λ1 ), W (x, y, λ2 ) . We say that a convex metric space (X, d, W ) has the following properties: (C) if W (x, y, λ) = W (y, x, 1 − λ)
for all x, y ∈ X and λ ∈ [0, 1], (I) if d W (x, y, λ1 ), W (x, y, λ2 ) ≤ |λ1 − λ2 |d(x, y) for all x, y ∈ X and λ1 , λ2 ∈ [0, 1],
(H) if d W (x, y, λ), W (x, z, λ) ≤ (1 − λ)d(y, z) for all x, y, z ∈ X and λ ∈ [0, 1],
(S) if d W (x, y, λ), W (z, w, λ) ≤ λd(x, z) + (1 − λ)d(y, w) for all x, y, z, w ∈ X and λ ∈ [0, 1]. Remark 2.2. It is easy to see that the property (C) and (H) imply continuity of a convex structure W : X × X × [0, 1] → X and the property (S) implies the property (H). In 2005, Aoyama et al. [10] proved that a convex metric space with property (C) and (H) has the property (S).

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In 2011, Phuengrattana and Suantai [2] proved the following lemma as follows; Lemma 2.3. (See [2]) Property (C) holds in uniformly convex metric space. Remark 2.4. (See [2]) From Lemma 2.3, a uniformly convex metric space (X, d, W ) with the property (H) has the property S and the convex structure W is also continuous. Lemma 2.5. (See [11]) Let (X, d, W ) be a uniformly convex metric space with continuous convex structure. Then for arbitrary positive number , there exists η = η() > 0 such that
d z, W (x, y, λ) ≤ (1 − 2 min{λ, 1 − λ}η)r, for all r > 0 and x, y, z ∈ X, d(z, x) ≤ r, d(z, y) ≤ r, d(x, y) ≥ rε and λ ∈ [0, 1]. We introduce the following definition to use in the next section. Definition 2.1. Let (X, d, W ) be a complete convex metric space and C be a nonempty closed convex subset of (X, d, W ). Let {Ti }N i=1 be a finite family of mappings of C into C. For each j = 1, 2, · · · , N , let αj = (α1j , α2j , α3j ) where α1j , α2j , α3j ∈ [0, 1] and α1j +α2j +α3j = 1. For every x ∈ C, we define the mapping S : C × C × [0, 1] → C as follows; U0 x = x,
α21 ), α11 , 1 1 − α1
α22 U2 x = W T2 U1 x, W (U1 x, x, ), α12 , 2 1 − α1 .. .
α2N −1 ), α1N −1 , UN −1 x = W TN −1 UN −2 x, W (UN −2 x, x, N −1 1 − α1
α2N Sx = UN x = W TN UN −1 x, W (UN −1 x, x, ), α1N . N 1 − α1 U1 x = W T1 U0 x, W (U0 x, x,

This mapping is called S−mapping generated by T1 , T2 , . . . , TN and α1 , α2 , . . . , αN . Lemma 2.6. Let C be a nonempty closed convex subset of a complete uniformly convex metric space (X, d, W ) with property (H). Let {Ti }N i=1 be a finite family TN of nonexpansive mappings of C into itself with i=1 F (Ti ) 6= ∅ and let αj = (α1j , α2j , α3j ) ∈ I × I × I, j = 1, 2, . . . , N , where I = [0, 1] , α1j + α2j + α3j = 1, α1j ∈ (0, 1) for all j = 1, 2, ..., N − 1, α1N ∈ (0, 1] α2j , α3j ∈ [0, 1) for all j = 1, 2, ..., N. Let T S be the mapping generated by T1 , T2 , ...., TN and α1 , α2 , ..., αN . N Then F (S) = i=1 F (Ti ).

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Proof. From Lemma 2.1 and definition of S−mapping, it is easy to see that TN TN F (T ) ⊆ F (S). Next, we show that F (S) ⊆ F (T i i ). To show this let i=1 i=1 TN x0 ∈ F (S) and q ∈ i=1 F (Ti ), we have  
α2N N d(q, Sx0 ) = d q, W TN UN −1 x0 , W (UN −1 x0 , x0 , ), α1 1 − α1N   α2N N N ) ≤ α1 d(q, TN UN −1 x0 ) + (1 − α1 )d q, W (UN −1 x0 , x0 , 1 − α1N  α2N ≤ α1N d(q, TN UN −1 x0 ) + (1 − α1N ) d(q, UN −1 x0 ) 1 − α1N  α2N )d(q, x ) + (1 − 0 1 − α1N = α1N d(q, TN UN −1 x0 ) + α2N d(q, UN −1 x0 ) + α3N d(q, x0 ) ≤ (1 − α3N )d(q, UN −1 x0 ) + α3N d(q, x0 )   N −1 N −1 N ≤ (1 − α3 ) (1 − α3 )d(q, UN −2 x0 ) + α3 d(q, x0 ) +α3N d(q, x0 ) = (1 − α3N )(1 − α3N −1 )d(q, UN −2 x0 ) + α3N −1 (1 − α3N )d(q, x0 ) +α3N d(q, x0 ) j j
N = ΠN j=N −1 (1 − α3 )d(q, UN −2 x0 ) + 1 − Πj=N −1 (1 − α3 ) d(q, x0 ) .. . j j
N ≤ ΠN j=3 (1 − α3 )d(q, U2 x0 ) + 1 − Πj=3 (1 − α3 ) d(q, x0 )

α22 j ), α12 = ΠN j=3 (1 − α3 )d q, W T2 U1 x0 , W (U1 x0 , x0 , 2 1 − α1 j
N + 1 − Πj=3 (1 − α3 ) d(q, x0 )   α22
j 2 2 ) ≤ ΠN (1 − α ) α d(q, T U x ) + (1 − α )d q, W (U x , x , 2 1 0 1 0 0 j=3 1 1 3 1 − α12
j + 1 − ΠN j=3 (1 − α3 ) d(q, x0 )  α22 j 2 2 ≤ ΠN d(q, U1 x0 ) j=3 (1 − α3 ) α1 d(q, T2 U1 x0 ) + (1 − α1 ) 1 − α12 
α22 j
+(1 − )d(q, x0 ) + 1 − ΠN j=3 (1 − α3 ) d(q, x0 ) 1 − α12   j N 2 2 2 = Πj=3 (1 − α3 ) α1 d(q, T2 U1 x0 ) + α2 d(q, U1 x0 ) + α3 d(q, x0 ) j
+ 1 − ΠN j=3 (1 − α3 ) d(q, x0 )  
j N 2 2 ≤ Πj=3 (1 − α3 ) (1 − α3 )d(q, U1 x0 ) + α3 d(q, x0 ) j
+ 1 − ΠN j=3 (1 − α3 ) d(q, x0 ) 5

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.3, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

= =

= ≤ ≤ =

j j
N ΠN j=2 (1 − α3 )d(q, U1 x0 ) + 1 − Πj=2 (1 − α3 ) d(q, x0 )



α21 j ΠN ), α11 j=2 (1 − α3 )d q, W T1 U0 x0 , W (U0 x0 , x0 , 1 − α11 j
+ 1 − ΠN j=2 (1 − α3 ) d(q, x0 )

j j
1 ΠN + 1 − ΠN j=2 (1 − α3 )d q, W T1 x0 , x0 , α1 j=2 (1 − α3 ) d(q, x0 )
j j
1 1 N ΠN j=2 (1 − α3 ) α1 d(q, T1 x0 ) + (1 − α1 )d(q, x0 ) + 1 − Πj=2 (1 − α3 ) d(q, x0 ) j j
N ΠN j=2 (1 − α3 )d(q, x0 ) + 1 − Πj=2 (1 − α3 ) d(q, x0 ) d(q, x0 ). (2.1)

From (2.1), we have
d(q, U1 x0 ) = d q, W (T1 x0 , x0 , α11 ) = d(q, x0 ) and d(q, T1 x0 ) = d(q, x0 ). Suppose x0 6= T1 x0 , we have d(x0 , T1 x0 ) > 0. Choose r = d(q, x0 ) > 0 and  = d(x0 , T1 x0 ) , we have d(q, T1 x0 ) ≤ d(q, x0 ) = r, d(q, x0 ) ≤ r and d(x0 , T1 x0 ) ≥ r r. From Lemma 2.5, we have
d q, W (T1 x0 , x0 , α11 ) < d(q, x0 ) for α11 ∈ (0, 1). This is a contradiction, we have x0 = T1 x0 that is x0 ∈ F (T1 ). Since x0 = T1 x0 definition of U1 and Lemma 2.1, we have U1 x0 = x0 that is x0 ∈ F (U1 ). From (2.1) and x0 = U1 x0 , we have

d(q, U2 x0 ) = d q, W T2 x0 , x0 , α12 = d(q, x0 ) and d(q, T2 x0 ) = d(q, x0 ). Suppose x0 6= T2 x0 , we have d(x0 , T2 x0 ) > 0. Choose r1 = d(q, x0 ) > 0 and  = d(x0 , T2 x0 ) , we have d(q, T2 x0 ) ≤ d(q, x0 ) = r1 , d(q, x0 ) ≤ r1 and d(x0 , T2 x0 ) ≥ r1 r1 . From Lemma 2.5, we have
d q, W (T2 x0 , x0 , α12 ) < d(q, x0 ) for α12 ∈ (0, 1). This is a contradiction, we have x0 = T2 x0 that is x0 ∈ F (T2 ). Since x0 = T2 x0 definition of U2 and Lemma 2.1, we have U2 x0 = x0 that is x0 ∈ F (U2 ). By continuing on this way, we can conclude that x0 ∈ F (Ti ) and x0 ∈ F (Ui ) for all i = 1, 2, . . . , N − 1. Finally, we show that x0 ∈ F (TN ). From definition of S and Lemma 2.1, we have Sx0 = W TN UN −1 x0 , W (UN −1 x0 , x0 ,


α2N ), α1N = W (TN x0 , x0 , α1N ). 1 − α1N

Since 0 = d(x0 , Sx0 ) = d(x0 , W (TN x0 , x0 , α1N )) = α1N d(TN x0 , x0 ), TN we have x0 = TN x0 , that is, x0 ∈ F (TN ). Hence F (S) ⊆ i=1 F (Ti ). 6

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Remark 2.7. From Theorem 2.6, we have the mapping S is nonexpansive. To show this, let x, y ∈ C. By Remark 2.4, we have 
α2N d(Sx, Sy) = d W TN UN −1 x, W (UN −1 x, x, ), α1N , N 1 − α1 
α2N N ), α W TN UN −1 y, W (UN −1 y, y, 1 1 − α1N ≤ α1N d(TN UN −1 x, TN UN −1 y) +(1 − α1N )d W (UN −1 x, x,

α2N α2N
), W (UN −1 y, y, ) N 1 − α1 1 − α1N

≤ α1N d(TN UN −1 x, TN UN −1 y)   α2N α2N
N +(1 − α1 ) d(UN −1 x, UN −1 y) + 1 − d(x, y) 1 − α1N 1 − α1N ≤ α1N d(UN −1 x, UN −1 y) + α2N d(UN −1 x, UN −1 y) + α3N d(x, y) = (1 − α3N )d(UN −1 x, UN −1 y) + α3N d(x, y)
≤ (1 − α3N ) (1 − α3N −1 )d(UN −2 x, UN −2 y) + α3N −1 d(x, y) + α3N d(x, y) j j
N = ΠN j=N −1 (1 − α3 )d(UN −2 x, UN −2 y) + 1 − Πj=N −1 (1 − α3 ) d(x, y) ≤ .. . j j
N = ΠN j=1 (1 − α3 )d(U0 x, U0 y) + 1 − Πj=1 (1 − α3 ) d(x, y) = d(x, y). Example 2.8. Let the metric d : R2 × R2 → R be defined by d (x, y) = max {|x1 − y1 | , |x2 − y2 |} , for all x = (x1 , x2 ) , y = (y1 , y2 ) ∈ R2 . Let the mapping W : R2 × R2 × [0, 1] → R2 be defined by W (x, y, λ) = λx + (1 − λ) y = (λx1 + (1 − λ) y1 , λx2 + (1 − λ) y2 ) , for all x = (x1 , x2 ) , y = (y1 , y2 ) ∈ R2 . For every i = 1, 2, . . . , N, let the mapping Ti : R2 → R2 be defined by Ti x =

ix , i+1

for all x = (x1 , x2 ) ∈ R2 . Let S be the mapping generated by T1 , T2 , ...., T N and  1  2j − 1 2j − 1 j+1 j j j α1 , α2 , ..., αN , where αj = α1, α2, α3 = , , · for 2j 2i (2 + j) 2j j+2 TN all j = 1, 2, . . . , N . Then F (S) = i=1 F (Ti ) .
Solution. From the properties of d, W, R2 , R2 , d, W is a complete uniformly

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convex metric space. Next, we show that R2 , d, W has a property (H). Let x = (x1 , x2 ) , y = (y1 , y2 ) , z = (z1, z2 ) ∈ R2 and a ∈ [0, 1], then d (W (x, y, a) , W (x, z, a)) = max {(1 − a) |y1 − z1 | , (1 − a) |y2 − z2 |} . Since d (y, z) = max {|y1 − z1 | , |y2 − z2 |}, we get d (W (x, y, a) , W (x, z, a)) ≤ (1 − a) d (y, z) .


Then R2 , d, W has a property (H). TN It is clear that Ti is a nonexpansive mapping for all i = 1, 2, . . . , N and ) = {0}, due to the properties of Ti . From Lemma 2.6, we have i=1 F (T TiN F (S) = i=1 F (Ti ) . Remark 2.9. Lemma 2.8 in [3] is a spacial case of Lemma 2.6.

3

Main Results

Theorem 3.1. Let C be a nonempty closed convex subset of a complete uniformly convex metric space (X, d, W ) with property (H). Let {Ti }N i=1 be a finite family of nonexpansive mappings of C into itself and let αj = (α1j , α2j , α3j ) ∈ I × I × I, j = 1, 2, 3, ..., N , where I = [0, 1] , α1j + α2j + α3j = 1, α1j ∈ (0, 1) for all j = 1, 2, ..., N − 1, α1N ∈ (0, 1] α2j , α3j ∈ [0, 1) for all j = 1, 2, ..., N. Let S be the mapping generated by T1 , T2 , ...., TN and α1 , α2 , ..., αN . Let {xn } be a sequence generated by x1 , u ∈ C and xn+1 = W (u, Sxn , α)

(3.1)

for all n ≥ 1 and α ∈ [0, 1]. Then the following statements are equivalent: TN i) The sequence {xn } converges to z ∈ i=1 F (Ti ), ii) limn→∞ d (xn , Ti xn ) = 0 for all i = 1, 2, . . . , N. TN Proof. i) ⇒ ii). Since {xn } converges to z ∈ i=1 F (Ti ) and d (xn , Ti xn ) ≤ d (xn , z) + d (Ti xn , z) ≤ 2d (xn , z) for all i = 1, 2, . . . , N , so we can prove that ii) is true. For the next result, we prove ii) ⇒ i). For every n ∈ N and remark (S property), we have d (xn+1 , xn ) ≤ d (W (u, Sxn , α) , W (u, Sxn−1 , α)) ≤ (1 − α) d (xn , xn−1 ) 2

≤ (1 − α) d (xn−1 , xn−2 ) .. . n ≤ (1 − α) d (x1 , x0 ) .

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Using the benefits from the inequality above, we have d (xn+k , xn ) ≤

n+k−1 X

d (xj+1 , xj )

j=n

≤

n+k−1 X

j

(1 − α) d (x1 , x0 )

j=n n

≤

(1 − α) · d (x1 , x0 ) , α n

for all k ∈ N. Since limn→∞ (1 − α) = 0, we get the sequence {xn } is Cauchy. Then there exists z ∈ C such that limn→∞ xn = z. From the condition ii) and d (z, Ti z) ≤ d (xn , z) + d (xn , Ti xn ) + d (Ti xn , Ti z) ≤ 2d (xn , z) + d (xn , Ti xn ) , for all i = 1, 2, . . . , N , we have d (z, Ti z) = 0. We can conclude that z ∈ TN TN i=1 F (Ti ) . Hence the sequence {xn } converges to z ∈ i=1 F (Ti ) . Theorem 3.2. Let C be a nonempty closed convex subset of a complete uniformly convex metric space (X, d, W ) with property (H). Let {Ti }N i=1 be a finite family of nonexpansive mappings of C into itself with limn→∞ d (xn , Ti xn ) = 0 for all i = 1, 2, . . . , N and let αj = (α1j , α2j , α3j ) ∈ I × I × I, j = 1, 2, 3, ..., N , where I = [0, 1] , α1j + α2j + α3j = 1, α1j ∈ (0, 1) for all j = 1, 2, ..., N − 1, α1N ∈ (0, 1], α2j , α3j ∈ [0, 1) for all j = 1, 2, ..., N. Let S be the mapping generated by T1 , T2 , ...., TN and α1 , α2 , ..., αN . Let {xn } be a sequence generated by x1 , u ∈ C and xn+1 = W (u, Sxn , α) (3.2) for all n ≥ 1 and α ∈ [0, 1]. Then the sequence {xn } converges to z ∈ F (S). Proof. The sequence {xn } is a Cauchy by using the same method of Theorem 3.1. Then there exists z ∈ C such that limn→∞ xn = z. Since limn→∞ d (xn , Ti xn ) = 0 for all i = 1, 2, . . . , N and d (z, Ti z) ≤ 2d (xn , z) + d (xn , Ti xn ) , TN for all i = 1, 2, . . . , N , we have z ∈ i=1 F (Ti ). From Lemma 2.6, we have z ∈ F (S). Hence the sequence {xn } converges to z ∈ F (S). If the condition ii) in Theorem are replaced by    3.1 and 3.2  TN TN 00 lim inf n→∞ d xn , i=1 F (Ti ) = 0 where d xn , i=1 F (Ti ) = inf v∈TN F (Ti ) d (xn , v)”. Then, the following theorems are still true.

88

i=1

Theorem 3.3. Let C be a nonempty closed convex subset of a complete uniformly convex metric space (X, d, W ) with property (H). Let {Ti }N i=1 be a finite family of nonexpansive mappings of C into itself and let αj = (α1j , α2j , α3j ) ∈ 9

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I × I × I, j = 1, 2, 3, ..., N , where I = [0, 1] , α1j + α2j + α3j = 1, α1j ∈ (0, 1) for all j = 1, 2, ..., N −1, α1N ∈ (0, 1], α2j , α3j ∈ [0, 1) for all j = 1, 2, ..., N. Let S be the mapping generated by T1 , T2 , ...., TN and α1 , α2 , ..., αN . Let {xn } be a sequence generated by x1 , u ∈ C and xn+1 = W (u, Sxn , α)

(3.3)

for all n ≥ 1 and α ∈ [0, 1]. Then the following statements are equivalent: TN i) The sequence  {xn } converges to  z ∈ i=1 F (T  i ). T  TN N ii) lim inf n→∞ d xn , i=1 F (Ti ) = 0 where d xn , i=1 F (Ti ) = inf v∈TN F (Ti ) d (xn , v). i=1

Proof. It is very clear that case i) ⇒ ii). Next, we show that case ii) ⇒ i). Using the same method in Theorem 3.1, we obtain that the sequence {xn } is a Cauchy sequence. Then, there exists z ∈ C such that limn→∞ xn = z. For every ε > 0, there exists N0 ∈ N such that
ε d xn , ∩N j=1 F (Ti ) < 2 and d (xn , z)
12 d(x, y), we have 

1 d(T x, T y) ≤ d(x, y) − ϕ(d(T x, T y)) ≤ d(x, y) − ϕ d(x, y) 2



on account of monotonocity of ϕ and finally d(T x, T y) ≤ d(x, y) − ϕ(d(x, ˜ y)). On the other hand, if d(T x, T y) < 12 d(x, y), we get 1 d(T x, T y) < d(x, y) − d(x, y) ≤ d(x, y) − ϕ(d(x, ˜ y)). 2 So T is just thr ϕ-weak ˜ contractive mapping. The continuity and monotonocity of ϕ˜ follows directly from properties of min function, ϕ and the metric.

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4

K. S. Kim

One of the most interesting aspects of metric fixed point theory is to extend a linear version of known result to the nonlinear case in metric spaces. To achieve this, Takahashi [16] introduced a convex structure in a metric space (X, d). A mapping W : X × X × [0, 1] → X is a convex structure in X if d(u, W (x, y, λ)) ≤ λd(u, x) + (1 − λ)d(u, y) for all x, y ∈ X and λ ∈ [0, 1]. A metric space with a convex structure W is known as a convex metric space which denoted by (X, d, W ). A nonempty subset K of a convex metric space is said to be convex if W (x, y, λ) ∈ K for all x, y ∈ K and λ ∈ [0, 1]. In fact, every normed linear space and its convex subsets are convex metric spaces but the converse is not true, in general (see, [16]). Example 1.3. ([9]) Let X = {(x1 , x2 ) ∈ R2 : x1 > 0, x2 > 0}. For all x = (x1 , x2 ), y = (y1 , y2 ) ∈ X and λ ∈ [0, 1]. We define a mapping W : X × X × [0, 1] → X by   λx1 x2 + (1 − λ)y1 y2 W (x, y, λ) = λx1 + (1 − λ)y1 , λx1 + (1 − λ)y1 and define a metric d : X × X → [0, ∞) by d(x, y) = |x1 − y1 | + |x1 x2 − y1 y2 |. Then we can show that (X, d, W ) is a convex metric space but not a normed linear space. A metric space X is a CAT (0) space. This term is due to M. Gromov [6] and it is an acronym for E. Cartan, A.D. Aleksandrov and V.A. Toponogov. If X is geodesically connected, and if every geodesic triangle in X is at least as ‘thin’ as its comparison triangle in the Euclidean plane(see, e.g., [2, p.159]). It is well known that any complete, simply connected Riemannian manifold nonpositive sectional curvature is a CAT (0) space. The precise definition is given below. For a thorough discussion of these spaces and of the fundamental role they play in various branches of mathematics, see Bridson and Haefliger [2] or Burago et al. [4]. Let (X, d) be a metric space. A geodesic path joining x ∈ X to y ∈ X (or, more briefly, a geodesic from x to y) is a mapping c from a closed interval [0, l] ⊂ R to X such that c(0) = x, c(l) = y, and d(c(t), c(t0 )) = |t − t0 | for all t, t0 ∈ [0, l]. In particular, c is an isometry and d(x, y) = l. The image α of c is called a geodesic (or, metric) segment joining x and y. When it is unique, this geodesic is denoted by [x, y]. The space (X, d) is said to be a geodesic space if every two points of X are joined by a geodesic, and X is said to be uniquely

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Convergence of double acting iterative scheme in CAT (0) spaces

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geodesic if there is exactly one geodesic joining x and y for each x, y ∈ X. A subset Y ⊆ X is said to be convex if Y includes every geodesic segment joining any two of its points. A geodesic triangle 4(x1 , x2 , x3 ) is a geodesic metric space (X, d) consists of three points x1 , x2 , x3 ∈ X (the vertices of 4) and a geodesic segment between each pair of vertices (the edges of 4). A comparison triangle for the geodesic ¯ 1 , x2 , x3 ) = 4(x¯1 , x¯2 , x¯3 ) in triangle 4(x1 , x2 , x3 ) in (X, d) is a triangle 4(x 2 R such that dR2 (x¯i , x¯j ) = d(xi , xj ) for i, j ∈ {1, 2, 3}. Such a triangle always exists(see, [2]). A geodesic metric space is said to be a CAT (0) space if all geodesic triangles of appropriate size satisfy the following CAT (0) comparison axiom. ¯ ⊂ R2 be a comparison Let 4 be a geodesic triangle in X and let 4 triangle for 4. Then 4 is said to satisfy the CAT (0) inequality if for ¯ all x, y ∈ 4 and all comparison points x ¯, y¯ ∈ 4, d(x, y) ≤ d(¯ x, y¯). Complete CAT (0) spaces are often called Hadamard spaces(see, [11]). If x, y1 , y2 are points of a CAT (0) space and if y0 is the midpoint of the segment 2 [y1 , y2 ], which we will denote by y1 ⊕y 2 , then the CAT (0) inequality implies   y1 ⊕ y2 1 1 1 2 d x, ≤ d2 (x, y1 ) + d2 (x, y2 ) − d2 (y1 , y2 ). 2 2 2 4 This inequality is the (CN) inequality of Bruhat and Tits [3]. In fact, a geodesic space is a CAT (0) space if and only if satisfies the (CN) inequality (cf. [2, p.163]). The above inequality has been extended by [5] as d2 (z, αx ⊕ (1 − α)y) ≤ αd2 (z, x) + (1 − α)d2 (z, y) − α(1 − α)d2 (x, y),

(CN∗ )

for any α ∈ [0, 1] and x, y, z ∈ X. Let us recall that a geodesic metric space is a CAT (0) space if and only if it satisfies the (CN) inequality(see, [2, p.163]). Moreover, if X is a CAT (0) metric space and x, y ∈ X, then for any α ∈ [0, 1], there exists a unique point αx ⊕ (1 − α)y ∈ [x, y] such that d(z, αx ⊕ (1 − α)y) ≤ αd(z, x) + (1 − α)d(z, y)

(1.3)

for any z ∈ X and [x, y] = {αx ⊕ (1 − α)y : α ∈ [0, 1]}. In view of the above inequality, CAT (0) space have Takahashi’s convex structure W (x, y, α) = αx ⊕ (1 − α)y.

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It is easy to see that for any x, y ∈ X and λ ∈ [0, 1], d(x, (1 − λ)x ⊕ λy) = λd(x, y), d(y, (1 − λ)x ⊕ λy) = (1 − λ)d(x, y). As a consequence, 1 · x ⊕ 0 · y = x, (1 − λ)x ⊕ λx = λx ⊕ (1 − λ)x = x. Moreover, a subset K of CAT (0) space X is convex if for any x, y ∈ K, we have [x, y] ⊂ K(see, [1, 10, 13]). The purpose of this paper, we discuss the convergence theorems for the double acting iterative scheme to a common fixed point for a family of generalized ϕ-weak contraction mappings in CAT (0) spaces. 2. Convergence theorems of double acting iterative schemes Xue [18] proved the following very intersting fixed point theorem in complete metric space. Theorem 2.1. ([18]) Let (X, d) be a complete metric space and let T : X → X be a generalized ϕ-weak contraction. Then the Picard iterative scheme ([14]) xn+1 = T xn converges to the unique fixed point. Theorem 2.2. Let T be a generalized ϕ-weak contractive self mapping of a closed convex subset K of a Banach space X. Then the Picard iterative scheme xn+1 = T xn converges strongly to the fixed point p with the following error estimate: kxn+1 − pk ≤ Φ−1 (Φ(kx1 − pk − n)), where Φ is defined by the antiderivative Z 1 dt, Φ(t) = ϕ(t)

Φ(0) = 0

and Φ−1 is the inverse of Φ. Proof. The proof is similar as Rhoades ([15], Theorem 2). However, for completeness, we give a sketch of the proof. We can obtain convergence follows from Theorem 2.1. To establish the error estimete, from (1.2) with λn = kxn − pk,

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λn+1 = kxn+1 − pk = kT xn − pk ≤ kxn − pk − ϕ(kxn+1 − pk) = λn − ϕ(λn+1 ), so, we have ϕ(λn+1 ) ≤ λn − λn+1 .

(2.1)

Thus Z

λn

Φ(λn ) − Φ(λn+1 ) = λn+1

1 λn − λn+1 dt = , ϕ(t) ϕ(µn )

for some λn+1 < µn < λn . Since ϕ is nondecreasing, from (2.1), Φ(λn ) − Φ(λn+1 ) =

λn − λn+1 λn − λn+1 ≥ ≥ 1. ϕ(µn ) ϕ(λn )

Thus Φ(λn+1 ) ≤ Φ(λn ) − 1 ≤ · · · ≤ Φ(λ1 ) − n. This completes the proof of Theorem 2.2.



In this section, we will use I = {1, 2, · · · , r} , where r ≥ 1. Let {Ti : i ∈ I} be a family of generalized ϕ-weak contraction self mappings on K. The scheme introduced in [8] is x1 ∈ K,

xn+1 = Un(r) xn ,

n ≥ 1,

(2.2)

where Un(0) = Id (: the identity mapping), Un(1) x = αn(1) x ⊕ (1 − αn(1) )T1n Un(0) x, Un(2) x = αn(2) x ⊕ (1 − αn(2) )T2n Un(1) x, .. . n Un(r−1) x = αn(r−1) x ⊕ (1 − αn(r−1) )Tr−1 Un(r−2) x,

Un(r) x = αn(r) x ⊕ (1 − αn(r) )Trn Un(r−1) x, where αn(i) ∈ [0, 1] for each i ∈ I. After this, the we called the iterative scheme (2.2) is double acting iterative scheme. The existence of fixed (or common fixed) points of one mapping (or two mappings or a family of mappigs) is not known in many situations. So the

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approximation of fixed (or common fixed) points of one or more mappings by various iterations have been extensively studied in many other spaces. In the sequel, it is assumed that F=

r \

F (Ti ) 6= ∅,

i=1

where F (Ti ) = {x ∈ K : Ti x = x, i ∈ I} . Now, we shall investigate the convergence of double acting iterative scheme applied to {Ti : i ∈ I} . Theorem 2.3. Let (X, d) be a complete CAT (0) space, K be a closed convex subset of X, {Ti : i ∈ I} be a family of generalized ϕ-weak contraction self mappings of K. Then the double acting iterative scheme (2.2) satisfies (i) 0 ≤ αn(i) ≤ 1, i ∈ I, P∞ (ii) n=1 (1 − αn(1) )(1 − αn(2) ) · · · (1 − αn(r) ) = ∞ converges to commom fixed point p ∈ F. Proof. For p ∈ F, using (2.2) and (1.3), d(Un(1) xn , p) = d(αn(1) xn ⊕ (1 − αn(1) )T1n Un(0) xn , p) ≤ αn(1) d(xn , p) + (1 − αn(1) )d(T1n xn , p) ≤ αn(1) d(xn , p) + (1 − αn(1) )[d(xn , p) − ϕ(d(T1n xn , p))] ≤ d(xn , p) − (1 − αn(1) )ϕ(d(T1n xn , p)).

(2.3)

Using (2.3), we get d(Un(2) xn , p) = d(αn(2) xn ⊕ (1 − αn(2) )T2n Un(1) xn , p) ≤ αn(2) d(xn , p) + (1 − αn(2) )d(T2n Un(1) xn , p) ≤ αn(2) d(xn , p) + (1 − αn(2) )[d(Un(1) xn , p) − ϕ(d(T2n Un(1) xn , p))] ≤ αn(2) d(xn , p) + (1 − αn(2) )[d(xn , p) − (1 − αn(1) )ϕ(d(T1n xn , p))] − (1 − αn(2) )ϕ(d(T2n Un(1) xn , p)) ≤ d(xn , p) − (1 − αn(1) )(1 − αn(2) )ϕ(d(T1n xn , p)) − (1 − αn(1) )(1 − αn(2) )ϕ(d(T2n Un(1) xn , p))

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and d(Un(3) xn , p) = d(αn(3) xn ⊕ (1 − αn(3) )T2n Un(2) xn , p) ≤ d(xn , p) − (1 − αn(1) )(1 − αn(2) )(1 − αn(3) )ϕ(d(T1n xn , p)) − (1 − αn(1) )(1 − αn(2) )(1 − αn(3) )ϕ(d(T2n Un(1) xn , p)) − (1 − αn(1) )(1 − αn(2) )(1 − αn(3) )ϕ(d(T3n Un(2) xn , p)). Continue this processing, we obtain d(Un(r) xn , p) = d(αn(r) xn ⊕ (1 − αn(r) )Trn Un(r−1) xn , p) ≤ d(xn , p) − (1 − αn(1) )(1 − αn(2) ) · · · (1 − αn(r) )ϕ(d(T1n xn , p)) − (1 − αn(1) )(1 − αn(2) ) · · · (1 − αn(r) )ϕ(d(T2n Un(1) xn , p)) .. . − (1 − αn(1) )(1 − αn(2) ) · · · (1 − αn(r) )ϕ(d(Trn Un(r−1) xn , p)) ≤ d(xn , p) − (1 − αn(1) )(1 − αn(2) ) · · · (1 − αn(r) )ϕ(d(Tin Un(i−1) xn , p)), (2.4) for each i ∈ I. From property of ϕ, we conclude d(Un(r) xn , p) ≤ d(xn , p), that is d(xn+1 , p) ≤ d(xn , p). Therefore, {d(xn , p)} is a nonnegative nonincreasing sequence, which converges to a limit L ≥ 0. (I) Most of all, we want to show that d(Tin Un(i−1) xn , p) ≥ L,

∀ n ≥ 1, i ∈ I.

(2.5)

To show (2.5), it is sufficient to show that there exists k ∈ N such that d(xk , p) ≤ d(Tin Un(i−1) xn , p),

n ≥ 1, i ∈ I.

To verify (2.5), suppose that d(Tin Un(i−1) xn , p) < L. Then d(xk , p) > d(Tin Un(i−1) xn , p),

∀ k ∈ N,

(2.6)

for n ≥ 1, i ∈ I. Since {d(xn , p)} is a nonincreasing sequence, we have d(xn , p) ≥ d(xn+1 , p) ≥ · · · ≥ L,

∀ n ≥ 1.

(2.7)

Let ε = L − d(Tin Un(i−1) xn , p) > 0. 2n

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Since limn→∞ d(xn , p) = L and (2.6), there exists N ∈ N with ε d(xN , p) < d(Tin Un(i−1) xn , p) + 4n such that

(2.9)

|d(xn , p) − L| ≤ |L − d(Tin Un(i−1) xn , p)| + |d(Tin Un(i−1) xn , p) − d(xn , p)| = L − d(Tin Un(i−1) xn , p) + d(xn , p) − d(Tin Un(i−1) xn , p) ε ≤ + d(xN , p) − d(Tin Un(i−1) xn , p) (from (2.7)) 2n ε ε < + < ε, ∀ n ≥ N. 2n 4n On the other hand, from (2.9), (2.8) and (2.6), we obtain ε d(xN , p) < d(Tin Un(i−1) xn , p) + 4n 1 n = d(Ti Un(i−1) xn , p) + (L − d(Tin Un(i−1) xn , p)) 2 1 = (L + d(Tin Un(i−1) xn , p)) 2 1 < (L + d(xN , p)), 2 i.e., d(xN , p) < L. This is a contradiction to (2.7). Therefore, (2.5) holds. That is d(Tin Un(i−1) xn , p) ≥ L,

∀ n ≥ 1, i ∈ I.

(II) We claim that L = 0. Suppose that L > 0. It follows that, from (2.4) and (2.5), for any fixed integer N ∈ N and i ∈ I ∞ X (1 − αn(1) )(1 − αn(2) ) · · · (1 − αn(r) )ϕ(L) n=N ∞ X

≤

≤

n=N ∞ X

(1 − αn(1) )(1 − αn(2) ) · · · (1 − αn(r) )ϕ(d(Tin Un(i−1) xn , p)) (d(xn , p) − d(xn+1 , p))

n=N

≤ d(xN , p). This is a contradiction to the condition (ii). Therefore, L ≤ 0. Thus lim d(xn , p) = L = 0.

n→∞

This completes the proof of Theorem 2.3.

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Remark 2.1. The author does not apply the real CAT (0) properties of a space such as for example (CN∗ ) inequality, d2 (αx ⊕ (1 − α)y, z) ≤ αd2 (x, z) + (1 − α)d2 (y, z) − α(1 − α)d2 (x, y),

(CN∗ )

but only the fact that d(αx ⊕ (1 − α)y, z) ≤ αd(x, z) + (1 − α)d(y, z), i.e., the convexity of the metric. Corollary 2.1. Let (X, d) be a complete CAT (0) space, K be a closed convex subset of X, T be a generalized ϕ-weak contraction self mapping of K. Then the Noor iterative scheme ([17]) xn+1 = αn xn ⊕ (1 − αn )T yn , yn = βn xn ⊕ (1 − βn )T zn , zn = γn xn ⊕ (1 − γn )T xn satisfies (i) P 0 ≤ αn , βn , γn ≤ 1, ∞ (ii) n=1 (1 − αn )(1 − βn )(1 − γn ) = ∞ converges to fixed point p ∈ F (T ). Proof. In the double acting iterative scheme (2.2), if r = 3 and T1 = T2 = T3 = T , then it reduces to the Noor iterative scheme. So the proof is similar to that of Theorem 2.3, and will be omitted.  Corollary 2.2. Let (X, d) be a complete CAT (0) space, K be a closed convex subset of X, T be a generalized ϕ-weak contraction self mapping of K. Then the Ishikawa iterative scheme ([7]) xn+1 = αn xn ⊕ (1 − αn )T yn , yn = βn xn ⊕ (1 − βn )T xn satisfies (i) P 0 ≤ αn , βn ≤ 1, ∞ (ii) n=1 (1 − αn )(1 − βn ) = ∞ converges to fixed point p ∈ F (T ). Proof. In the double acting iterative scheme (2.2), if r = 2 and T1 = T2 = T , then it reduces to the Ishikawa iterative scheme. So the proof is similar to that of Theorem 2.3, and will be omitted. 

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Corollary 2.3. Let (X, d) be a complete CAT (0) space, K be a closed convex subset of X, T be a generalized ϕ-weak contraction self mapping of K. Then the Mann iterative scheme ([12]) xn+1 = αn xn ⊕ (1 − αn )T xn , satisfies (i) P 0 ≤ αn ≤ 1, ∞ (ii) n=1 (1 − αn ) = ∞ converges to fixed point p ∈ F (T ). Proof. In the double acting iterative scheme (2.2), if r = and T1 = T , then it reduces to the Mann iterative scheme. So the proof is similar to that of Theorem 2.3, and will be omitted.  Competing interests The authors declares that there is no conflict of interest regarding the publication of this paper. Acknowledgments This work was supported by Kyungnam University Research Fund, 2017. References [1] Y.I. Alber and S. Guerre-Delabriere, Principle of weakly contractive maps in Hilbert spaces, in: I. Gohberg, Yu. Lyubich(Eds.), New Results in Operator Theory, in: Advances and Appl., vol. 98, Birkh¨ auser, Basel, 1997, 7–22. [2] M. Bridson and A. Haefliger, Metric spaces of Non-Positive Curvature, Springer-Verlag, Berlin, Heidelberg, 1999. [3] F. Bruhat and J. Tits, Groups r´ eductifss sur un corps local. I. Donn´ ees radicielles ´ valu´ ees, Publ. Math. Inst. Hautes Etudes Sci., 41 (1972), 5–251. [4] D. Burago, Y. Burago and S. Ivanov, A course in metric Geometry, in:Graduate studies in Math., 33, Amer. Math. Soc., Providence, Rhode Island, 2001. [5] S. Dhompongsa and B. Panyanak, On triangle-convergence theorems in CAT (0) spaces, Comput. Math. Anal., 56 (2008), 2572–2579. [6] M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ. 8. Springer, New York, 1987. [7] S. Ishikawa, Fixed point by a new iteration, Proc. Amer. Math. Soc., 44 (1974), 147–150. [8] A.R. Khan, M.A. Khamsi and H. Fukhar-ud-din, Strong convergence of a general iteration scheme in CAT (0) spaces, Nonlinear Anal., 74(3) (2011), 783-791. [9] J.K. Kim, K.S. Kim and S.M. Kim, Convergence theorems of implicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings in convex metric spaces, Proc. of RIMS Kokyuroku, Kyoto Univ., 1484 (2006), 40–51. [10] K.S. Kim, Some convergence theorems for contractive type mappings in CAT (0) spaces, Abstract and Applied Analysis, 2013, Article ID 381715, 9 pages, http://dx.doi.org/10.1155/2013/381715

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[11] W.A. Kirk, A fixed point theorem in CAT (0) spaces and R-trees, Fixed Point Theory Appl., 2004(4) (2004), 309–316. [12] W.R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4 (1953), 506– 510. [13] A. Nicolae, Asymptotic behavior of averaged and firmly nonexpansive mappings in geodesic spaces, Nonlinear Analysis, 2013, 87, 102–115 [14] E. Picard, Sur les groupes de transformation des e´quations diff´ erentielles lin´ eaires, Comptes Rendus Acad. Sci. Paris, 96 (1883), 1131–1134. [15] B.E. Rhoades, Some theorems on weakly contractive maps, Nonlinear Anal., 47 (2001), 2683–2693. [16] W. Takahashi, A convexity in metric spaces and nonexpansive mappings, Kodai Math. Sem. Rep., 22 (1970), 142–149. [17] B.L. Xu and M.A. Noor, Fixed-point iterations for asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 267 (2002), 444–453. [18] Z. Xue, The convergence of fixed point for a kind of weak contraction, Nonlinear Func. Anal. Appl., 21(3) (2016), 497–500.

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On solution of a system of differential equations via fixed point theorem ¨ Muhammad Nazam1 , Muhammad Arshad1 , Choonkil Park2∗ , Ozlem Acar3 , Sungsik Yun4∗ , George A. Anastassiou5 1 Department of Mathematics and Statistics, International Islamic University, H-10, Islamabad, Pakistan

e-mail: [email protected]; [email protected] 2 Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea

e-mail: [email protected] 3 Department of Mathematics, Faculty of Science and Arts, Mersin University, 33343, Yeni¸sehir, Mersin, Turkey

e-mail: [email protected] 4 Department of Financial Mathematics, Hanshin University, Gyeonggi-do 18101, Korea

e-mail: [email protected] 5 Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA

e-mail: ganastss@@memphis.edu Abstract. The purpose of the present paper is to study the existence of solution of a system of differential equations using fixed point technique. In this regard, in the first part of this article, along with some properties of partial b-metric topology, we prove a common fixed point theorem for generalized Geraghty type contraction mappings in a complete partial b-metric spaces. Then in second part we apply this result to show the existence of the solution of a system of ordinary differential equations.

1. Introduction and preliminaries One of the most important results in fixed point theory is the Banach contraction principle introduced by Banach [4]. There were many authors who have studied and proved the results for fixed point theory by generalizing the Banach contraction principle in several directions (see [1, 5–7, 18, 22, 24]). Czerwik [9] introduced the notion of b-metric to generalize the concept of a distance. The analog of the famous Banach fixed point theorem was proved by Czerwik in the frame of complete b-metric spaces. Following these initial papers, the existence and the uniqueness of (common) fixed points for the classes of both singlevalued and multivalued operators in the setting of (generalized) b-metric spaces have been investigated extensively (see [2, 3, 10, 13, 15, 16, 20, 23, 26–28] and related references therein). Shukla [29] introduced the concept of partial b-metric space and established some fixed point theorems. Shukla, in fact, generalized Matthews partial metric to partial b-metric. Recently, Mustafa et al. [20], Latif et al. [19] and Piri et al. [21] have established some fixed point results in complete partial b-metric spaces. In this paper, we introduce the notion of generalized Geraghty type contraction mappings and develop new common fixed point theorems for such mappings in complete partial b-metric spaces and properties of partial b-metric topology. Examples are given to support the usability of our results. In the last section of this paper, we utilize our results to present an application on existence of a solution of a pair of ordinary differential equations. We also study well-posedness of common fixed point problem for generalized Geraghty type contraction mappings. First of all, we recall some definitions and properties of partial b-metric spaces. Definition 1. [29] Let X be a nonempty set and s ≥ 1 be a real number. A function pb : X × X → [0, ∞) is said to be a partial b-metric if for all x, y, z ∈ X, we have 0

2010 Mathematics Subject Classification: 47H10; 54H25 Keywords: complete partial b-metric space; generalized Geraghty type contraction mapping; differential equation; well posed. ∗ Corresponding authors. 0

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(pb 1) (pb 2) (pb 3) (pb 4)

x = y if and only if pb (x, y) = pb (x, x) = pb (y, y), pb (x, x) ≤ pb (x, y), pb (x, y) = pb (y, x), pb (x, y) ≤ s [pb (x, z) + pb (z, y)] − pb (z, z).

In this case, the pair (X, pb ) is called a partial b-metric space (with constant s). It is clear that every partial metric space is a partial b-metric space with coefficient s = 1 and every b-metric space is a partial b-metric space with the same coefficient and zero self-distance. However, the converse of this fact need not to hold. The self distance pb (x, x), referred to as the size or weight of x, is a feature used to describe the amount of information contained in x. Definition 2. Let (X, pb ) be a partial b-metric space. The distance function dpb : X × X → R+ 0 , defined by dpb (x, y) = 2pb (x, y) − pb (x, x) − pb (y, y), for all x, y ∈ X, defines a metric on X called an induced metric. Example 1. [29] Let X = R+ and l > 1. Then the functional pb : X × X → R+ , defined by n o pb (x, y) = (max{x, y})l + |x − y|l , for all x, y ∈ X, is a partial b-metric. Example 2. [29] Let X be a nonempty set such that p is a partial metric and d is a b-metric with coefficient s > 1 on X. Then the function pb : X × X → R+ , defined by pb (x, y) = p(x, y) + d(x, y) for all x, y ∈ X, is a partial b-metric on X and (X, pb ) is a partial b-metric space. Example 3. [29] Let X be a nonempty set and p be a partial metric defined on X. The functional pb : X × X → R+ , defined by pb (x, y) = [p(x, y)]q for all x, y ∈ X and q > 1, defines a partial b-metric. For a partial b-metric space (X, pb ), we immediately have a natural definition for the open balls: B (x; pb ) = {y ∈ X|pb (x, y) < pb (x, x) + } for each x ∈ X and  > 0. Proposition 1. The set {B (x; pb )|x ∈ X,  > 0} of open balls forms the basis for partial b-metric topology denoted by T [pb ]. Proof. It is obvious that X = ∪x∈X B (x; pb ) and for any two open balls B (x; pb ), Bδ (y; pb ) we note that B (x; pb ) ∩ Bδ (y; pb ) = ∪ {Bκ (c; pb )| c ∈ B (x; pb ) ∩ Bδ (y; pb )} where, κ = pb (c, c) + min { − pb (x, c), δ − pb (y, c)} , as desired.



Proposition 2. Each partial b-metric topology is T0 topology but not T1 . Proof. Suppose pb : X × X → R+ 0 is a partial b-metric and x 6= y. Then without loss of generality, we have pb (x, x) < pb (x, y) for all x, y ∈ X. Choose  = pb (x, y) − pb (x, x). Since pb (x, x) < pb (x, x) +  = pb (x, y) , x ∈ B (x; pb ) and y ∈ / B (x; pb ). Otherwise we obtain an absurdity (pb (x, y) < pb (x, y)). It is obvious that for x 6= v, x ∈ Bδ (x; pb ) ⊆ B (v; pb ),

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which contradicts T1 axiom.



The following definition and lemma describe the convergence criteria established by Shukla in [29]. Definition 3. [29] Let (X, pb ) be a partial b-metric space. (1) A sequence {xn }n∈N in (X, pb ) is called a Cauchy sequence if limn,m→∞ pb (xn , xm ) exists and is finite. (2) A partial b-metric space (X, pb ) is said to be complete if every Cauchy sequence {xn }n∈N in X converges, with respect to T [pb ], to a point υ ∈ X such that pb (x, x) =

lim

n,m→∞

pb (xn , xm ).

Lemma 1. [29] Let (X, pb ) be a partial b-metric space. (1) Every Cauchy sequence in (X, dpb ) is also a Cauchy sequence in (X, pb ). (2) A partial b-metric (X, pb ) is complete if and only if the metric space (X, dpb ) is complete. (3) A sequence {xn }n∈N in X converges to a point υ ∈ X with respect to T [(dpb )] if and only if lim pb (υ, xn ) = pb (υ, υ) = lim pb (xn , xm ).

n→∞

n→∞

(4) If limn→∞ xn = υ such that pb (υ, υ) = 0, then limn→∞ pb (xn , k) = pb (υ, k) for every k ∈ X. The following important lemma is useful in the sequel. Lemma 2. [20] Let (X, pb ) be a partial b-metric space with coefficient s > 1. Suppose that the sequences {xn },{yn } converge to x, y, respectively. Then we have 1 1 pb (x, y) − pb (x, x) − pb (y, y) ≤ lim inf pb (xn , yn ) ≤ lim sup pb (xn , yn ) n→∞ n→∞ s2 s 2 ≤ spb (x, x) + s pb (y, y) + s2 pb (x, y). If pb (x, y) = 0 then we have limn→∞ pb (xn , yn ) = 0. Moreover, for each x∗ ∈ X we obtain 1 pb (x, x∗ ) − pb (x, x) ≤ lim inf pb (xn , x∗ ) ≤ lim sup pb (xn , x∗ ) n→∞ n→∞ s ≤ spb (x, x∗ ) + spb (x, x). If pb (x, x) = 0, then we have 1 pb (x, x∗ ) ≤ lim inf pb (xn , x∗ ) ≤ lim sup pb (xn , x∗ ) ≤ spb (x, x∗ ). n→∞ n→∞ s Let Ω denote to the class of all functions β : [0, +∞) → [0, 1) such that for any bounded sequence {tn } of positive reals, β (tn ) → 1 implies tn → 0. Geraghty [11] presented a very important generalization of Banach Contraction Principle as follows: Theorem 1. [11] Let (X, d) be a metric space. Let S : X → X be a self-mapping. Suppose that there exists β ∈ Ω such that for all x, y ∈ X, d (Sx, Sy) ≤ β (d (x, y)) d (x, y) . Then S has a unique fixed point x∗ ∈ X and {S n x} converges to x∗ for each x ∈ X. Following [8], we let Ψ denote to the class of functions ψ : [0, ∞) → [0, ∞) satisfying the following conditions: (1) ψ is nondecreasing, (2) ψ is continuous, (3) ψ (t) = 0 if and only if t = 0. Definition 4. Let S, T : X → X be two self-mappings and F (S) and F (T ) denote the set of fixed points of S and T , respectively. Then a fixed point problem for S and T is well posed if for any sequence {xn } in X and x∗ ∈ F (S) ∩ F (T ), limn→∞ pb (xn , S(xn )) = 0 or limn→∞ pb (xn , T (xn )) = 0 implies limn→∞ pb (xn , x∗ ) = pb (x∗ , x∗ ).

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2. Fixed point results We begin with the introduction of the concept of generalized Geraghty type contraction mappings as follows: Definition 5. Let (X, pb ) be a partial b-metric space. The pair S, T : X → X of self-mappings is called a generalized Geraghty type contraction mapping if there exist β ∈ Ω and ψ ∈ Ψ such that for x, y ∈ X, the pair (S, T ) satisfies the following inequality:
ψ s3 pb (Sx, T y) ≤ β (ψ (M (x, y))) · ψ (M (x, y)) (2.1) where



pb (x, T y) + pb (y, Sx) M (x, y) = max pb (x, y) , pb (x, Sx) , pb (y, T y) , 2s

 .

The main result of this section is the following. Theorem 2. Let (X, pb ) be a complete partial b-metric space and S, T : X → X be two self-mappings satisfying the following conditions: (1) (S, T ) is a pair of generalized Geraghty type contraction mappings; (2) S or T is a continuous mapping. Then S and T have a common fixed point x∗ ∈ X. Proof. First, we suppose that s > 1. Let x0 ∈ X and choose x1 = S(x0 ), x2 = T (x1 ). Continuing in the same way we construct a sequence {xn } in X such that x2i+1 = S(x2i ) and x2i+2 = T (x2i+1 ), i = 0, 1, 2, .... Without loss of generality, we can assume that M(x, y) > 0 for x 6= y. Now, for i ∈ N, we have
0 < ψ (pb (x2i+1 , x2i+2 )) ≤ ψ s3 pb (Sx2i , T x2i+1 ) ≤

β (ψ (M (x2i , x2i+1 ))) .ψ (M (x2i , x2i+1 )) ,

(2.2)

where ( M (x2i , x2i+1 )

=

max

pb (x2i , x2i+1 ) , pb (x2i , Sx2i ) , pb (x2i+1 , T x2i+1 ) , pb (x2i ,T x2i+1 )+pb (x2i+1 ,Sx2i )

)

2s

(

=

≤ =

) pb (x2i , x2i+1 ) , pb (x2i , x2i+1 ) , pb (x2i+1 , x2i+2 ) , max pb (x2i , x2i+2 ) + pb (x2i+1 , x2i+1 ) 2s ) ( pb (x2i , x2i+1 ) , pb (x2i , x2i+1 ) , pb (x2i+1 , x2i+2 ) , max pb (x2i , x2i+1 ) + pb (x2i+1 , x2i+2 ) 2s max {pb (x2i , x2i+1 ) , pb (x2i+1 , x2i+2 )} .

If max {pb (x2i , x2i+1 ) , pb (x2i+1 , x2i+2 )} = pb (x2i+1 , x2i+2 ) , then from (2.2) we have ψ (pb (x2i+1 , x2i+2 ))

≤

β (ψ (pb (x2i+1 , x2i+2 ))) .ψ (pb (x2i+1 , x2i+2 ))


0. From (2.1), we have
ψ (pb (xn+1 , xn+2 )) ≤ ψ s3 pb (Sxn , T xn+1 ) ≤

β (ψ (M (xn , xn+1 ))) .ψ (M (xn , xn+1 )) ,

which implies ψ (pb (xn+1 , xn+2 )) ≤ β (ψ (pb (xn , xn+1 ))) .ψ (pb (xn , xn+1 )) . Hence ψ (pb (xn+1 , xn+2 )) ≤ β (ψ (pb (xn , xn+1 ))) < 1. ψ (pb (xn , xn+1 )) This implies that lim β (ψ (pb (xn , xn+1 ))) = 1. Since β ∈ Ω, we have n→∞

lim ψ (pb (xn , xn+1 )) = 0,

n→∞

which yields r = lim pb (xn , xn+1 ) = 0,

(2.3)

n→∞

which is a contradiction. Now we will show that {xn } is a Cauchy sequence. For this purpose, we use Lemma 1. Suppose that there
exists ε > 0 such that for all k ∈ N, there exists m (k) > n (k) > k with dpb xn(k) , xm(k) ≥ ε. Let m (k) be the
smallest number satisfying the condition above. Then we have dpb xn(k) , xm(k)−1 < ε. Therefore,


 (2.4) ε ≤ dpb xn(k) , xm(k) ≤ s dpb xn(k) , xm(k)−1 + dpb xm(k)−1 , xm(k)
  < s ε + dpb xm(k)−1 , xm(k) . By taking the upper limit as k → ∞ in (2.4) and using (2.3) , we get
ε ≤ lim sup dpb xn(k) , xm(k) < sε.

(2.5)

k→∞

From the triangular inequality, we have


dpb xn(k) , xm(k) ≤ s[dpb xn(k) , xn(k)+1 + dpb xn(k)+1 , xm(k) ]

(2.6)




dpb xn(k)+1 , xm(k) ≤ s[dpb xn(k)+1 , xn(k) + dpb xn(k) , xm(k) ].

(2.7)

and By taking upper limit as k → ∞ in (2.6) and applying (2.3) and (2.5) , 

 ε ≤ lim sup dpb xn(k) , xm(k) ≤ s lim sup dpb xn(k)+1 , xm(k) . k→∞

k→∞

Again, by taking the upper limit as k → ∞ in (2.7), we get 

 lim sup dpb xn(k)+1 , xm(k) ≤ s lim sup dpb xn(k) , xm(k) ≤ s.sε = s2 ε. k→∞

k→∞

Thus
ε ≤ lim sup dpb xn(k)+1 , xm(k) ≤ s2 ε. k→∞ s

(2.8)

Similarly

ε ≤ lim sup dpb xn(k) , xm(k)+1 = lim sup dpb xn(k)+1 , xm(k)+2 ≤ s2 ε. k→∞ k→∞ s By the triangular inequality, we have


dpb xn(k)+1 , xm(k) ≤ s[dpb xn(k)+1 , xm(k)+1 + dpb xm(k)+1 , xm(k) ]. Letting k → ∞ in (2.10) and using (2.3) and (2.8), we get
ε ≤ lim sup dpb xn(k)+1 , xm(k)+1 . k→∞ s2 Following the above process, we find
lim sup dpb xn(k)+1 , xm(k)+1 ≤ s3 ε. k→∞

421

(2.9)

(2.10)

(2.11)

(2.12)

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From (2.11) and (2.12) , we get
ε ≤ lim sup dpb xn(k)+1 , xm(k)+1 ≤ s3 ε. 2 k→∞ s Since xn(k) 6= xm(k)+1 , we get ψ dpb xn(k)+1 , xm(k)+2 ≤ ≤ ≤







ψ s3 dpb Sxn(k) , T xm(k)+1




β ψ M xn(k) , xm(k)+1 · ψ M xn(k) , xm(k)+1




β ψ M xn(k) , xm(k)+1 · ψ M xn(k) , xm(k)+1 ,

where

M xn(k) , xm(k)+1




=



 dpb xn(k) , xm(k)+1 , dpb xn(k) , Sxn(k) ,    dpb xm(k)+1 , T xm(k)+1 , max      dpb xn(k) ,T xm(k)+1 +dpb (xm(k)+1 ,Sxn(k) ) 2s

=



 dpb xn(k) , xm(k)−1 ,  dpb xn(k) , xn(k)+1 ,    dpb xm(k)+1 , xm(k)+2 , max      dpb xn(k) ,xm(k)+2 +dpb (xm(k)+1 ,xn(k)+1 )

          

.

  

2s

Taking the limit as k → ∞ and using (2.3) , (2.5) , (2.8) and (2.9), we get   n ε sε o
ε s2 ε = max , ≤ lim sup M xn(k) , xm(k)+1 ≤ max s2 ε, = s2 ε. k→∞ s s 4 4 Similarly, we can show that   n ε sε o
ε s2 ε = max , ≤ lim inf M xn(k) , xm(k)+1 ≤ max s2 ε, = s2 ε. k→∞ s s 4 4 From (2.9) , we have
ψ s2 ε = ≤ ≤
0 and β (0) = 0. Then β ∈ Ω. Let ψ be a function on [0, +∞) defined by ψ (t) = t. Then ψ ∈ Ψ. Define the mappings S, T : X → X by  2 
 245 x, if x ∈ 0, 12 T (x) =  and S(x) = 0.   1, if x ∈ 12 , 1 


If {xn } is a Cauchy sequence such that {xn } ⊆ 0, 12 . Since 0, 12 , pb is a complete partial b-metric space,  1
the sequence {xn } converges in 0, 2 ⊆ X. Thus (X, pb ) is a complete partial b-metric space. We note that

 x, y, Sy, T y ∈ 0, 21 and S is continuous. It is easy to check that for all x, y ∈ 0, 12 , the following inequality is true
ψ s3 pb (Sx, T y) ≤ β (ψ (M (x, y))) · ψ (M (x, y)) , Thus all the conditions of Theorem 2 are satisfied. Hence S and T have a common fixed point (x = 0) . 3. Derived results In Theorem 2, if we set S = T and   pb (x, Sy) + pb (y, Sx) , M (x, y) = max pb (x, y) , pb (x, Sx) , pb (y, Sy) , 2s then we obtain the following result. Corollary 1. Let (X, pb ) be a complete partial b-metric space. Suppose that S : X → X is a self-mapping satisfying the following conditions: (1) S is a generalized Geraghty type contraction mapping; (2) S is a continuous mapping. Then S has a fixed point x∗ ∈ X. In Theorem 2, if ψ (t) = t, then we obtain the following corollary. Corollary 2. Let (X, pb ) be a complete partial b-metric space. Suppose that S, T : X → X are two self-mappings such that (1) there exists β ∈ Ω such that for x, y ∈ X, the pair (S, T ) satisfies the following inequality s3 pb (Sx, T y) ≤ β ((M (x, y))) . (M (x, y)) , where   pb (x, T y) + pb (y, Sx) M (x, y) = max pb (x, y) , pb (x, Sx) , pb (y, T y) , . 2s (2) S or T is a continuous mapping Then S and T have a common fixed point x∗ ∈ X.

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In particular, if pb (x, x) = 0 for all x ∈ X, then the following result can easily be obtained from Theorem 2. Corollary 3. Let (X, d) be a b-metric space. Suppose that S, T : X → X are two self-mappings satisfying the following conditions: (1) (S, T ) is a pair of Geraghty type contraction mappings; (2) S or T is a continuous mapping. Then S and T have a common fixed point x∗ ∈ X. In the following, we see that the problem stated in Theorem 2 is well posed. Theorem 3. Let (X, pb ) be a complete partial b-metric space. Let S, T : X → X be two self-mappings as in Theorem 2 with ψ(t) = t. Then the fixed point problem for S and T is well posed. Proof. Let {xn } be a sequence in X and x∗ ∈ F (S) ∩ F (T ). Suppose that limn→∞ pb (xn , S(xn )) = 0. If limn→∞ pb (xn , x∗ ) = 0, then we are done. Assume that limn→∞ pb (xn , x∗ ) = r > 0. Using (pb 3), we have s3 pb (xn , x∗ )

≤

s4 [pb (xn , S(xn )) + pb (S(xn ), x∗ ) − pb (S(xn ), S(xn ))],

s2 pb (xn , x∗ )

≤

s3 pb (xn , S(xn )) + s3 pb (S(xn ), T (x∗ ))

≤

s3 pb (xn , S(xn )) + β (M (xn , x∗ )) · M (xn , x∗ ) ,

1 lim pb (xn , x∗ ) ≤ s3 lim pb (xn , S(xn )) + lim β(pb (xn , x∗ )) · pb (xn , x∗ ), n→∞ n→∞ s n→∞ r r ≤ 0 + 3 β(r), a contradiction due to the definition of β. s s ∗ Similarly, we obtain limn→∞ xn = x if we assume limn→∞ d(xn , T (xn )) = 0.



4. Application In this section, we present an application on existence of a solution of a pair of ordinary differential equations. In particular, inspired from [17] and using Theorem 2, we consider the following pair of differential equations: (  d2 x 2 − dt2 = f (t, x (t)) , t ∈ [0, 1] − ddt2y = K (t, y (t)) , t ∈ [0, 1] and (4.1) x(0) = x (1) = 0 y(0) = y (1) = 0 where f, K : [0, 1] × R → R are continuous functions. The Green function associated to (4.1) is defined by  t (1 − s) , 0 ≤ t ≤ s ≤ 1 G (t, s) = s (1 − t) , 0 ≤ t ≤ s ≤ 1. Let C (I) be the space of all continuous functions defined on I, where I = [0, 1]. Suppose that  2 pb (x, y) = sup |x(t) − y(t)| + (max{x(t), y(t)})2 . t∈I

It is known that (C (I) , pb ) is a complete partial b-metric space with constant s = 2. Now, define the operators S, T : C (I) → C (I) by Z 1 Z 1 Sx(t) = G(t, s)f (s, x(s))ds and T x(t) = G(t, s)K(s, y(s))ds 0

0

for all t ∈ I. Note that (4.1) has a solution if and only if the operators S and T have a common fixed point. The main result is the following. Theorem 4. Assume that (1) there exist continuous functions f, K : [0, 1] × R → R such that for all a, b, ρ ∈ R, we have   M(a, b) + 1 for all t ∈ I, |f (t, a) − K(t, b)|2 ≤ 64 ln ρ

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where   pb (a, S(b)) + pb (b, T (a)) M(a, b) = max pb (a, b), pb (a, S(a)), pb (b, T (b)), > ρ; 2s (2) the operators S, T are such that 

2

(max{Sx(t), T y(t)}) ≤ ln(ρ) sup t∈I

2

1

Z

G(t, s) ds

.

0

Then the system of ordinary differential equations (4.1) has a solution. Proof. It is well known that x∗ ∈ C 2 (I) is a solution of (4.1) if and only if x∗ ∈ C (I) is a solution of the integral equation (see [17]). Define the mappings S, T : C (I) → C (I) by Z 1 Z 1 Sx(t) = G(t, s)f (s, x(s))ds and T x(t) = G(t, s)K(s, y(s))ds. 0

0

Hence the solution of (4.1) is equivalent to find x∗ ∈ C (I), that is, a fixed point of T . By (1), we get  2 sup |Sx(t) − T y(t)| + (max{Sx(t), T y(t)})2 pb (Sx, T y) = t∈I

≤

Z 1 2   2 Z 1 sup G(t, s) [f (s, x(s)) − K(s, y(s))] ds + ln(ρ) sup G(t, s) ds t∈I 0 t∈I 0 " #  2 2 Z Z 1

≤

sup t∈I

"

1

G(t, s) ds

|f (s, x(s)) − K(s, y(s))|2 + ln(ρ) sup t∈I

0

G(t, s) ds 0

#  2 Z 1 M(a, b) + 1 ≤ 64 sup G(t, s) ds ln + ln(ρ) sup G(t, s) ds ρ t∈I 0 t∈I 0   Z 1 2 !   M(a, b) + 1 + ln(ρ) sup G(t, s)ds . = 82 ln ρ t∈I 0  hR i2  R 2 1 Since G(t, s)ds = − t2 + 2t for all t ∈ I, we have sup 0 G(t, s)ds = 812 . Therefore, 

Z

2

1



t∈I

pb (Sx, T y) ≤ ln (M(a, b) + 1) , which implies that ln (pb (Sx, T y) + 1)

≤ =

ln (ln (M(x, y) + 1) + 1) ln (ln (M(x, y) + 1) + 1) ln (M(x, y) + 1) . ln (M(x, y) + 1)

Define the functions ψ : [0, ∞) → [0, ∞) and β : [0, ∞) → [0, 1) by  ψ(x) , x ψ (x) = ln (x + 1) and β (x) = 0,

if x 6= 0 otherwise.

Note that ψ : [0, ∞) → [0, ∞) is continuous, nondecreasing, positive in (0, ∞), ψ (0) = 0 and ψ (x) < x. Hence β ∈ Ω, ψ ∈ Ψ and
ψ s3 pb (Sx, T y) ≤ β (ψ (M (x, y))) .ψ (M (x, y)) for all x, y ∈ C(I). Therefore, all the assumptions of Theorem 2 are satisfied. Hence S and T have a common fixed point x∗ ∈ C (I), that is, Sx∗ = T x∗ = x∗ , which is a solution of (4.1).  References [1] G. A. Anastassiou, I. K. Argyros, Approximating fixed points with applications in fractional calculus, J. Comput. Anal. Appl. 21 (2016), 1225–1242. [2] H. Aydi, M. F. Bota, E. Karapinar, S. Moradi, A common fixed point for weak φ-contractions on b-metric spaces, Fixed Point Theory 13 (2012), 337–346. [3] H. Aydi, A. Felhi, S. Sahmim, Common fixed points in rectangular b-metric spaces using (E : A) property, J. Adv. Math. Studies 8 (2015), 159–169.

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[4] S. Banach, Sur les op´erations dans les ensembles abstraits et leur application aux equations itegrales, Fund. Math. 3 (1922), 133–181. [5] A. Batool, T. Kamran, S. Jang, C. Park, Generalized ϕ-weak contractive fuzzy mappings and related fixed point results on complete metric space, J. Comput. Anal. Appl. 21 (2016), 729–737. [6] M. Berzig, E. Karapınar, On modified α-ψ-contractive mappings with application, Thai J. Math. 13 (2015), 147–152. [7] S. H. Cho, J. S. Bae, E. Karapinar, Fixed point theorems for α-Geraghty contraction type maps in metric spaces, Fixed Point Theory Appl. 2013, 2013:329. [8] P. Chuadchawna, A. Kaewcharoen, S. Plubtieng, Fixed point theorems for generalized α-η-ψ-Geraghty contraction type mappings in α-η-complete metric spaces, J. Nonlinear Sci. App. 9 (2016), 471–485. [9] S. Czerwik, Contraction mappings in b-metric spaces. Acta Math. Inf. Univ. Ostrav. 1 (1993), 5–11. [10] A. Felhi, S. Sahmim, H. Aydi, Ulam-Hyers stability and well-posedness of fixed point problems for α-λcontractions on quasi b-metric spaces, Fixed Point Theory Appl. 2016, 2016:1. [11] M. Geraghty, On contractive mappings, Proc. Amer. Math. Soc. 40 (1973), 604–608. [12] V. Gupta, W. Shatanawi, N. Mani, Fixed point theorems for (Ψ, β)-Geraghty contraction type maps in ordered metric spaces and some applications to integral and ordinary differential equations, J. Fixed Point Theory Appl. 19 (2017). 1251–1267. [13] H. Huang, S. Xu, Fixed point theorems of contractive mappings in cone b-metric spaces and applications, Fixed Point Theory Appl. 2013, 2013:112. [14] N. Hussain, M. A. Kutbi, P. Salimi, Fixed point theory in α-complete metric space with applications, Abstr. Appl. Anal. 2014, Art. ID 280817 (2014). [15] N. Hussain, M. H. Shah, KKM mappings in cone b-metric spaces, Comput. Math. Appl. 62 (2011), 1677–1684. [16] M. Jovanovic, Z. Kadelburg, S. Radenovic, Common fixed point results in metric type spaces, Fixed Point Theory Appl. 2010 , Art. ID 978121 (2010). [17] E. Karapinar, α-ψ-Geraghty contraction type mappings and some related fixed point results, Filomat 28 (2014), 37–48. [18] E. Karapınar, P. Kumam, P. Salimi, On α-ψ-Meir-Keeler contractive mappings, Fixed Point Theory Appl. 2013, 2013:94. [19] A. Latif, J. R. Roshan, V. Parvaneh, N. Hussain, Fixed point results via α-admissible mappings and cyclic contractive mappings in partial b-metric spaces, J. Inequal. Appl. 2014, 2014:345. [20] Z. Mustafa, J. R. Roshan, V. Parvaneh, Z. Kadelburg, Some common fixed point results in ordered partial b-metric spaces, J. Inequal. Appl. 2013, 2013:562. [21] H. Piri, H. Afshari, Some fixed point theorems in complete partial b-metric spaces, Adv. Fixed Point Theory 4 (2014), 444–461. [22] O. Popescu, Some new fixed point theorems for α-Geraghty contraction type maps in metric spaces, Fixed Point Theory Appl. 2014, 2014:190. [23] J. R. Roshan, V. Parvaneh, Sh. Sedghi, N. Shobkolaei, W. Shatanawi, Common fixed points of almost generalized (ψ-ϕ)s -contraction mappings in ordered b-metric spaces. Fixed Point Theory Appl. 2013, 2013:159. [24] P. Salimi, A. Latif, N. Hussain, Modified α-ψ-contractive mappings with applications, Fixed Point Theory Appl. 2013, 2013:151. [25] B. Samet, C. Vetro, P. Vetro, Fixed point theorems for α-ψ-contractive type mappings, Nonlinear Anal. 75 (2012), 2154–2165. [26] R. J. Shahkoohi, A. Razani, Some fixed point theorems for rational Geraghty contractive mappings in ordered b-metric spaces, J. Inequal. Appl. 2014, 2014:373. [27] W. Shatanawi, M. B. Hani, A fixed point theorem in b-metric spaces with nonlinear contractive condition, Far East J. Math. Sci. 100 (2016), 1901–1908. [28] L. Shi, S. Xu, Common fixed point theorems for two weakly compatible self-mappings in cone b-metric spaces, Fixed Point Theory Appl. 2013, 2013:120. [29] S. Shukla, Partial b-metric spaces and fixed point theorems, Mediterr. J. Math. 11 (2014), 703–711.

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SOME EQUALITIES AND INEQUALITIES FOR K-G-FRAMES ZHONG-QI XIANG† AND YIN-SUO JIA

Abstract. In this paper we establish some equalities and inequalities for K-g-frames. Our results generalize the remarkable results obtained by Balan et al. and G˘avrut¸a. We also give several new inequalities for K-g-frames by using operator theory methods, which differ in structure from those for frames.

1. Introduction Throughout this paper, H and K are separable Hilbert spaces, {K j } j∈J is a sequence of closed subspaces of K , where J is a finite or countable index set. For any I ⊂ J, we denote Ic = J\I. The notation B(H , K ) is reserved for the set of all linear bounded operators from H to K , and B(H , H ) is abbreviated to B(H ); K ∈ B(H ). Frames for Hilbert spaces, appeared first in the early 1950’s, have now been applied in a variety of fields because of their redundancy and flexibility. For more information on frame theory and its applications, the interested reader can consult [4–8,16,19]. G-frames, proposed by Sun in [17], generalize the concept of frames extensively and possess some distinct properties though they share many similar properties with frames, see [15, 18]. A K-frame is a generalization of a frame, which was put forward by G˘avrut¸a in [10] to investigate the atomic systems associated with a linear bounded operator K. When K is an orthogonal projection, a K-frame is just an atom system for subspace which was introduced by Feichtinger and Werther in [9]. It should be remarked that the properties of K-frames are quite different from those of frames as shown in [1, 12, 20, 22], though the definition of a K-frame is similar to a frame in form. Recently, Xiao et al. [23] applied G˘avrut¸a’s idea to the case of g-frames, thereby leading to the notion of K-g-frames, which have attracted much attention, see [2, 13]. Balan et al. [3] found a surprising identity for Parseval frames when they devoted to the study of efficient algorithms for signal reconstruction, given below. Theorem 1.1. Let { f j } j∈J be a Parseval frame for H , then for every I ⊂ J and every f ∈ H , we have

2 X

X

2

X X (1.1) |h f, f j i|2 −

h f, f j i f j

= |h f, f j i|2 −

h f, f j i f j

. j∈I

j∈Ic

j∈I

j∈Ic

In [3], the following inequality was also obtained. Theorem 1.2. Let { f j } j∈J be a Parseval frame for H , then for every I ⊂ J and every f ∈ H , we have

X

2 X 3 (1.2) |h f, f j i|2 +

h f, f j i f j

≥ k f k2 . 4 j∈I j∈Ic † Corresponding author. 2010 Mathematics Subject Classification. Primary 42C15; Secondary 42C40. Key words and phrases. Parseval K-g-frame; K-dual g-frame; operator; pseudo-inverse. 1

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Later on, G˘avrut¸a [11] extended Theorems 1.1 and 1.2 to alternate dual frames: Theorem 1.3. Let { f j } j∈J be a frame for H and {g j } j∈J be an alternate dual frames of { f j } j∈J . Then for all I ⊂ J and all f ∈ H , we have

2

X X Re h f, g j ih f, f j i +

h f, g j i f j

j∈Ic

j∈I

2

X X 3

= Re h f, g j ih f, f j i + h f, g j i f j

≥ k f k2 . 4 j∈Ic j∈I

(1.3)

In fact, Theorem 1.3 is a particular case of the following result, given in [11]. Theorem 1.4. Let { f j } j∈J be a frame for H and {g j } j∈J be an alternate dual frame of { f j } j∈J . Then for all bounded sequence {ω j } j∈J and all f ∈ H , we have

X

2 X Re ω j h f, g j ih f, f j i +

(1 − ω j )h f, g j i f j

j∈J

j∈J

(1.4)

X

2 X 3 = Re (1 − ω j )h f, g j ih f, f j i +

ω j h f, g j i f j

≥ k f k2 . 4 j∈J j∈J

In this paper we generalize the equalities and inequalities (1.1), (1.2) and (1.4) to K-gframes. Since g-frames can be considered as a class of K-g-frames, Theorem 2.2 in [21] and Theorem 4.1 in [24] which are a generalization of Theorems 1.1 and 1.2, and Theorem 1.4 respectively, can be obtained as a special case of the results we establish on K-g-frames. We also present some new inequalities for K-g-frames by using operator theory methods, which are different in structure from those in (1.2)–(1.4). 2. Preliminaries In the following we briefly recall some definitions and basic properties of operators. Definition 2.1. We call a sequence {Λ j ∈ B(H , K j )} j∈J a K-g-frame for H with respect to {K j } j∈J , if there exist 0 < C ≤ D < ∞ such that X (2.1) CkK ∗ f k2 ≤ kΛ j f k2 ≤ Dk f k2 , ∀ f ∈ H . j∈J

If we only require the right-hand inequality of (2.1), then {Λ j } j∈J is said to be a g-Bessel sequence for H with respect to {K j } j∈J with g-Bessel bound D. Remark 2.2. If K = IH , the identity operator on H , then a K-g-frame is just a g-frame. A K-g-frame {Λ j ∈ B(H , K j )} j∈J for H is said to be Parseval if X (2.2) kK ∗ f k2 = kΛ j f k2 , ∀ f ∈ H . j∈J

Let {Λ j } j∈J be a Parseval K-g-frame for H with respect to {K j } j∈J . Then it is easy to check that X (2.3) KK ∗ f = Λ∗j Λ j f, ∀ f ∈ H . j∈J

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SOME EQUALITIES AND INEQUALITIES FOR K-G-FRAMES

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Definition 2.3. Let {Λ j } j∈J be a K-g-frame for H with respect to {K j } j∈J . A g-Bessel sequence {Γ j } j∈J for H with respect to {K j } j∈J is called a K-dual g-frame of {Λ j } j∈J , if X (2.4) Kf = Λ∗j Γ j f, ∀ f ∈ H . j∈J

To prove the main results, we need the following lemmas. Lemma 2.4. (see [6]) Suppose that T ∈ B(H ) has closed range, then there exists a unique operator T † ∈ B(H ), called the pseudo-inverse of T , satisfying T T †T = T ,

T †T T † = T †,

(T † T )∗ = T † T ,

(T † )∗ = (T ∗ )† .

In the following, the notation Θ† is reserved to denote the pseudo-inverse of the linear bounded operator Θ (if it exists). Lemma 2.5. (see [14]) Suppose that U, V, T ∈ B(H ), that U + V = T , and that T has closed range. Then we have T ∗ T † U + V ∗ T † V = V ∗ T † T + U ∗ T † U. Lemma 2.6. If U, V, K ∈ B(H ) satisfy U + V = K, then 3 1 U ∗ U + (V ∗ K + K ∗ V) ≥ K ∗ K. 2 4 Proof. We have 1 1 U ∗ U + (V ∗ K + K ∗ V) = (K ∗ − V ∗ )(K − V) + (V ∗ K + K ∗ V) 2 2 1 = V ∗ V − (K ∗ V + V ∗ K) + K ∗ K + (V ∗ K + K ∗ V) 2 1 = V ∗ V − (V ∗ K + K ∗ V) + K ∗ K 2  1 ∗  1  3 = V − K V − K + K∗ K 2 2 4 3 ∗ ≥ K K. 4  3. Main results and their proofs We begin with several equalities and inequalities for Parseval K-g-frames and K-dual g-frames. Theorem 3.1. Let {Λ j } j∈J be a Parseval K-g-frame for H with respect to {K j } j∈J . Then for every I ⊂ J and every f ∈ H we have

X

2 X

2

X X (3.1) hΛ j f, Λ j KK ∗ f i −

Λ∗j Λ j f

= hΛ j f, Λ j KK ∗ f i −

Λ∗j Λ j f

. j∈I

j∈Ic

j∈Ic

j∈I

Proof. For every I ⊂ J, one can easily check that the operators S I and S Ic defined by X X SI f = Λ∗j Λ j f, S Ic f = Λ∗j Λ j f j∈Ic

j∈I

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are positive, linear bounded and self-adjoint. Moreover, the definition of a Parseval K-gframe gives S I + S Ic = KK ∗ . Hence for each f ∈ H ,

2

X X hΛ j f, Λ j KK ∗ f i −

Λ∗j Λ j f

= hS I f, KK ∗ f i − kS I f k2 j∈I

j∈I

= h(KK ∗ − S I )S I f, f i = hS Ic (KK ∗ − S Ic ) f, f i = hS Ic KK ∗ f, f i − hS I2c f, f i = hS Ic f, KK ∗ f i − kS Ic f k2

2

X X = hΛ j f, Λ j KK ∗ f i −

Λ∗j Λ j f

. j∈Ic

j∈Ic

 A version of the equality obtained in Theorem 3.1 for overlapping divisions is derived in the following result. Theorem 3.2. Let {Λ j } j∈J be a Parseval K-g-frame for H with respect to {K j } j∈J . Then for every I ⊂ J, every E ⊂ Ic , and every f ∈ H , we have

X

2

X

2



∗ ∗

Λ j Λ j f − Λ j Λ j f

j∈Ic \E

j∈I∪E

X

2

X

2 X =

Λ∗j Λ j f

−

Λ∗j Λ j f

+2Re hΛ j f, Λ j KK ∗ f i. j∈Ic

j∈I

j∈E

Proof. Applying Theorem 3.1 twice yields

X

2

X

2



∗ ∗

Λ j Λ j f − Λ j Λ j f

j∈Ic \E

j∈I∪E

=

X

hΛ j f, Λ j KK ∗ f i −

hΛ j f, Λ j KK ∗ f i

j∈Ic \E

j∈I∪E

=

X

X X X hΛ j f, Λ j KK ∗ f i − hΛ j f, Λ j KK ∗ f i + 2Re hΛ j f, Λ j KK ∗ f i j∈Ic

j∈I

j∈E

2

X

2

X X =

Λ∗j Λ j f

−

Λ∗j Λ j f

+2Re hΛ j f, Λ j KK ∗ f i. j∈Ic

j∈I

j∈E

 Theorem 3.3. Let {Λ j } j∈J be a K-g-frame for H with respect to {K j } j∈J and {Γ j } j∈J be a K-dual g-frame of {Λ j } j∈J . Then for every {α j } j∈J ∈ `∞ (J) and every f ∈ H we have

2

X X (1 − α j )hΓ j f, Λ j K f i +

α j Λ∗j Γ j f

j∈J

j∈J

(3.2)

=

X j∈J

X

2

∗

α j hΓ j f, Λ j K f i + (1 − α j )Λ j Γ j f

. j∈J

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SOME EQUALITIES AND INEQUALITIES FOR K-G-FRAMES

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Proof. For any {α j } j∈J ∈ `∞ (J) and any f ∈ H , we let X X Sα f = α j Λ∗j Γ j f, S 1−α f = (1 − α j )Λ∗j Γ j f. j∈J

j∈J

Denote by D1 and D2 the g-Bessel bounds of {Λ j } j∈J and {Γ j } j∈J respectively. Then X

2

X  2

α Λ∗ Γ f

= sup ∗ α Λ Γ f, g j j j j j j

g∈H ,kgk=1

j∈J

=

sup

j∈J

2 X α hΓ f, Λ gi j j j

g∈H ,kgk=1 j∈J

≤

sup

2 |α j | |hΓ j f, Λ j gi|

X

g∈H ,kgk=1 j∈J

≤

sup

g∈H ,kgk=1

k{α j } j∈J k2

X j∈J

kΓ j f k2 ·

X

kΛ j gk2

j∈J

≤ D1 D2 k{α j } j∈J k k f k . 2

2

It follows that S α is well defined and bounded. By the same way we can show that S 1−α is also well defined and bounded. Since {Γ j } j∈J is a K-dual g-frame of {Λ j } j∈J , we have X X X S α f + S 1−α f = α j Λ∗j Γ j f + (1 − α j )Λ∗j Γ j f = Λ∗j Γ j f = K f j∈J

j∈J

j∈J

for each f ∈ H . It follows that

X

2 X (1 − α j )hΓ j f, Λ j K f i +

α j Λ∗j Γ j f

= hS 1−α f, K f i + hS α f, S α f i j∈J

j∈J

= h(K − S α ) f, K f i + hS α f, S α f i = hK f, K f i − hS α f, K f i + hS α f, S α f i,

(3.3) and X j∈J

X

2 α j hΓ j f, Λ j K f i +

(1 − α j )Λ∗j Γ j f

j∈J

= hS α f, K f i + hS 1−α f, S 1−α f i = hK f, S α f i + h(K − S α ) f, (K − S α ) f i = hK f, S α f i + hK f, K f i + hS α f, S α f i − hS α f, K f i − hK f, S α f i = hK f, K f i − hS α f, K f i + hS α f, S α f i.

(3.4)



Combination of (3.3) and (3.4) yields (3.2).

Corollary 3.4. Let {Λ j } j∈J be a K-g-frame for H with respect to {K j } j∈J and {Γ j } j∈J be a K-dual g-frame of {Λ j } j∈J . Then for every I ⊂ J and every f ∈ H we have

2 X

X

2

X X hΓ j f, Λ j K f i +

Λ∗j Γ j f

= hΓ j f, Λ j K f i +

Λ∗j Γ j f

. j∈Ic

j∈I

j∈I

j∈Ic

Proof. The result follows directly from Theorem 3.3 if we take I ⊂ J and ( 1, j ∈ I, αj = 0, j ∈ Ic . 

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Theorem 3.5. Let {Λ j } j∈J be a K-g-frame for H with respect to {K j } j∈J and {Γ j } j∈J be a K-dual g-frame of {Λ j } j∈J . Then for every {α j } j∈J ∈ `∞ (J) and every f ∈ H we have

2

X X 

α Λ∗ Γ f

+Re (1 − α )hΓ f, Λ K f i j j j j j j

j∈J

j∈J

2

X  3 X α j hΓ j f, Λ j K f i ≥ kK f k2 . =

(1 − α j )Λ∗j Γ j f

+Re 4 j∈J j∈J

(3.5)

Proof. The equality is obtained immediately if we take the real part on both sides of (3.2). For the inequality, taking X X Uf = α j Λ∗j Γ j f and V f = (1 − α j )Λ∗j Γ j f j∈J

j∈J

for each f ∈ H in Lemma 2.6, then we have

X

2 X 

α Λ∗ Γ f

+Re (1 − α j )hΓ j f, Λ j K f i j j j

j∈J

j∈J


1 = kU f k2 + RehV f, K f i = hU f, U f i + hV f, K f i + hK f, V f i 2    1 = U ∗ U + (V ∗ K + K ∗ V) f, f 2 3 ∗ 3 ≥ hK K f, f i = kK f k2 . 4 4  Theorem 3.6. Let {Λ j } j∈J be a Parseval K-g-frame for H with respect to {K j } j∈J . Then for every I ⊂ J and every f ∈ H we have

X

2 X

Λ∗ Λ f

+Re hΛ f, Λ KK ∗ f i j j j j

j∈I

j∈Ic

X

2 X 3 =

Λ∗j Λ j f

+Re hΛ j f, Λ j KK ∗ f i ≥ kKK ∗ f k2 . 4 j∈I j∈Ic

(3.6)

Proof. The equality follows if we take the real part on both sides of (3.1). It remains to prove the inequality. Since S I + S Ic = KK ∗ , it follows that S I2 + S I2c = S I2 + (KK ∗ − S I )2 = 2S I2 + (KK ∗ )2 − KK ∗ S I − S I KK ∗ 2 (KK ∗ )2  KK ∗ =2 − SI + . 2 2

(3.7) Therefore,

KK ∗ S Ic + S Ic KK ∗ + 2S I2 = KK ∗ (S I + S Ic ) − KK ∗ S I + S Ic KK ∗ + 2S I2 = (KK ∗ )2 − (S I + S Ic )S I + S Ic (S I + S Ic ) + 2S I2 = (KK ∗ )2 − S I2 − S Ic S I + S Ic S I + S I2c + 2S I2 3 = (KK ∗ )2 + S I2 + S I2c ≥ (KK ∗ )2 . 2

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SOME EQUALITIES AND INEQUALITIES FOR K-G-FRAMES

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Thus for every f ∈ H we have

2

X X

Λ∗ Λ f

+Re hΛ f, Λ KK ∗ f i j j j j

j∈Ic

j∈I

= kS I f k2 +


1 hS Ic f, KK ∗ f i + hKK ∗ f, S Ic f i 2


1 2hS I2 f, f i + hS Ic f, KK ∗ f i + hKK ∗ f, S Ic f i 2 3 3 1 = h(KK ∗ S Ic + S Ic KK ∗ + 2S I2 ) f, f i ≥ h(KK ∗ )2 f, f i = kKK ∗ f k2 . 2 4 4 =

 We give an upper bound condition for the left-hand-side of the equality in (3.6) under the condition that K has closed range. Theorem 3.7. Suppose that K has closed range and that {Λ j } j∈J is a Parseval K-g-frame for H with respect to {K j } j∈J . Then for every I ⊂ J and every f ∈ H we obtain

X

2 X

Λ∗ Λ f

+Re hΛ f, Λ KK ∗ f i ≤ kKk kK † k(1 + kKk kK † k)kKK ∗ f k2 . j j j j

j∈Ic

j∈I

Proof. For each f ∈ H , by Lemma 2.4 we have

X

2 X X

Λ∗ Λ f

≤ kS k 2 kΛ f k ≤ kS k kΛ j f k2 I j I j j

j∈I

j∈I

j∈J

≤ kKk kK f k = kKk kK (K ∗ )† K ∗ f k2 2

∗

2

2

∗

= kKk2 kK † KK ∗ f k2 ≤ kKk2 kK † k2 kKK ∗ f k2 ,

(3.8) and

X  1 X 1 X ∗ 2 2 ∗ 2 2 Re hΛ j f, Λ j KK f i ≤ kΛ j f k kΛ j KK f k j∈Ic

j∈J ∗

j∈J

= kK f k kK KK f k ∗

∗

= kK † KK ∗ f k kK ∗ KK ∗ f k ≤ kKk kK † k kKK ∗ f k2 .

(3.9)



Now, the result follows by combining (3.8) and (3.9).

In the following we give some new inequalities for K-g-frames, which possess different structure comparing with the inequalities for frames shown in Theorems 1.2, 1.3 and 1.4. Theorem 3.8. Let {Λ j } j∈J be a K-g-frame for H with respect to {K j } j∈J and {Γ j } j∈J be a K-dual g-frame of {Λ j } j∈J . Then for every I ⊂ J and every f ∈ H we have

X

2 X 3kKk2 + kFI − FIc k2 3

2 ∗

(3.10) kK f k ≤ Λ j Γ j f

+Re hΓ j f, Λ j K f i ≤ k f k2 , 4 4 j∈I

j∈Ic

where the operators FI is defined by FI f =

P

j∈I

Λ∗j Γ j f .

Proof. The left-hand inequality follows from Theorem 3.5 if we consider I ⊂ J and ( 1, j ∈ I, αj = 0, j ∈ Ic .

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We now prove the right-hand inequality of (3.10). For any f ∈ H we get

2

X X

Λ∗ Γ f

+Re hΓ f, Λ K f i j j j j

j∈Ic

j∈I

= hFI f, FI f i + RehK f, FIc f i = hFI f, FI f i + RehK f, (K − FI ) f i = hFI f, FI f i + hK f, K f i − RehK f, FI f i
= hK f, K f i − Re h(K − FI ) f, FI f i = hK f, K f i − RehFIc f, FI f i 1 1 = hK f, K f i − hFI f, FIc f i − hFIc f, FI f i 2 2 3 1 1 1 2 = kK f k + hFI f + FIc f, FI f + FIc f i − hFI f, FIc f i − hFIc f, FI f i 4 4 2 2 1 3 2 = kK f k + h(FI − FIc ) f, (FI − FIc ) f i 4 4 1 3 2 2 ≤ kKk k f k + kFI − FIc k2 k f k2 4 4 3kKk2 + kFI − FIc k2 k f k2 . = 4 This completes the proof.



Theorem 3.9. Suppose that K is positive and that it has closed range. Let {Λ j } j∈J be a K-g-frame for H with respect to {K j } j∈J and {Γ j } j∈J be a K-dual g-frame of {Λ j } j∈J . Then for every I ⊂ J and every f ∈ H we have X  X X Re hΓ j f, Λ j K † K f i + K † Λ∗j Γ j f, Λ∗j Γ j f j∈Ic

j∈I

j∈Ic

X  3 1 X X K † Λ∗j Γ j f, Λ∗j Γ j f ≥ kK 2 f k2 . = Re hΓ j f, Λ j K † K f i + 4 j∈I j∈I j∈Ic Proof. Since K is positive, it is self-adjoint and thus by Lemma 2.4, (K † )∗ = (K ∗ )† = K † . Hence, hK † FI f, FI f i and hK † FIc f, FIc f i are real numbers for every f ∈ H . From Lemma 2.5 it follows that X  X X Re hΓ j f, Λ j K † K f i + K † Λ∗j Γ j f, Λ∗j Γ j f j∈Ic

j∈I

j∈Ic

= RehFI f, K K f i + hK FIc f, FIc f i †

†

= RehKK † FI f, f i + hFI∗c K † FIc f, f i = Reh(KK † FI + FI∗c K † FIc ) f, f i = Reh(FI∗c K † K + FI∗ K † FI ) f, f i
= Re hFI∗c K † K f, f i + hFI∗ K † FI f, f i
= Re hK † K f, FIc f i + hK † FI f, FI f i = RehFIc f, K † K f i + hK † FI f, FI f i X  X X = Re hΓ j f, Λ j K † K f i + K † Λ∗j Γ j f, Λ∗j Γ j f . j∈Ic

j∈I

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SOME EQUALITIES AND INEQUALITIES FOR K-G-FRAMES

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Again by Lemma 2.4 we have  X X X Λ∗j Γ j f K † Λ∗j Γ j f, Re hΓ j f, Λ j K † K f i + j∈Ic

j∈I

j∈I

= Reh(KK † FI + FI∗c K † FIc ) f, f i = Reh(KK † (K − FIc ) + FI∗c K † FIc ) f, f i = hK f, f i − RehKK † FIc f, f i + hFI∗c K † FIc f, f i 1

1

1

1

1

1

= hK 2 f, K 2 f i − RehK 2 K 2 K † FIc f, f i + h(K 2 K † FIc )∗ (K 2 K † FIc ) f, f i 1 1  3 1 1 1 1 1 3 1 = kK 2 f k2 + K 2 f − K 2 K † FIc f, K 2 f − K 2 K † FIc f ≥ kK 2 f k2 4 2 2 4 for each f ∈ H and the proof is finished.



Theorem 3.10. Let {Λ j } j∈J be a Parseval K-g-frame for H with respect to {K j } j∈J . If S I commutes with S Ic for every I ⊂ J, then for every f ∈ H we have

X

2

X

2 1 (3.11) kKK ∗ f k2 ≤

Λ∗j Λ j f

+

Λ∗j Λ j f

≤ kKK ∗ f k2 . 2 j∈Ic

j∈I

0≤

(3.12)

X

2 X 1 hΛ j KK ∗ f, Λ j f i −

Λ∗j Λ j f

≤ kKK ∗ f k2 . 4 j∈I j∈I

Proof. From (3.7) it follows that

2

X

2 X

Λ∗ Λ f

+

Λ∗ Λ f

= kS f k2 + kS c f k2 = hS f, S f i + hS c f, S c f i I I I I I I j j j j

j∈J

j∈Ic

1 1 h(KK ∗ )2 f, f i = kKK ∗ f k2 2 2 for every f ∈ H . Since S I commutes with S Ic , S Ic S I ≥ 0 and (3.13)

= h(S I2 + S I2c ) f, f i ≥

(3.14)

0 ≤ S I S Ic = S I (KK ∗ − S I ) = S I KK ∗ − S I2 .

It follows that S I2 + S I2c = S I2 + (KK ∗ )2 − KK ∗ S I − S I KK ∗ + S I2 = (KK ∗ )2 + (S I2 − S I KK ∗ ) + (S I2 − KK ∗ S I ) = (KK ∗ )2 − (S I KK ∗ − S I2 ) − S Ic S I ≤ (KK ∗ )2 . Hence for every f ∈ H we have

2

2 X

X

Λ∗ Λ f

+

Λ∗ Λ f

= h(S 2 + S 2 ) f, f i ≤ h(KK ∗ )2 f, f i = kKK ∗ f k2 . j j j j I Ic

j∈J

j∈Ic

This together with (3.13) gives (3.11). We next prove (3.12). Using formula (3.14) we get

X

2 X

∗ ∗ hΛ j KK f, Λ j f i −

Λ j Λ j f

= hS I KK ∗ f, f i − kS I f k2 = h(S I KK ∗ − S I2 ) f, f i j∈I

j∈I

= hS I (KK ∗ − S I ) f, f i = hS I S Ic f, f i ≥ 0

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ZHONG-QI XIANG† AND YIN-SUO JIA

10

for every f ∈ H . On the other hand we obtain

2

X X hΛ j KK ∗ f, Λ j f i −

Λ∗j Λ j f

= h(S I KK ∗ − S I2 ) f, f i j∈I

j∈I

   KK ∗ 2 (KK ∗ )2 = − SI − f+ f, f 2 4  (KK ∗ )2  ≤ f, f 4 1 = kKK ∗ f k2 . 4 

This completes the proof. Acknowledgements

The research was partially supported by the National Natural Science Foundation of China (Grant Nos. 11761057 and 11561057), the Natural Science Foundation of Jiangxi Province (Grant No. 20151BAB201007), and the Science Foundation of Jiangxi Education Department (Grant No. GJJ151061). References [1] F. Arabyani Neyshaburi and A. Arefijamaal, Some constructions of K-frames and their duals, to appear in Rocky Mountain J. Math. [2] M.S. Asgari and H. Rahimi, Generalized frames for operators in Hilbert spaces, Infin. Dimens. Anal. Quantum. Probab. Relat. Top. 17, 1450013, 20 pp (2014). [3] R. Balan, P.G. Casazza, D. Edidin, and G. Kutyniok, A new identity for Parseval frames, Proc. Amer. Math. Soc. 135, 1007–1015 (2007). [4] J.J. Benedetto, A.M. Powell, and O. Yilmaz, Sigma-Delta (Σ∆) quantization and finite frames, IEEE Trans. Inform. Theory 52, 1990–2005 (2006). [5] P.G. Casazza, The art of frame theory, Taiwanese J. Math. 4, 129–201 (2000). [6] O. Christensen, An Introduction to Frames and Riesz Bases, Birkh¨auser, Boston, 2003. [7] I. Daubechies, A. Grossmann, and Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys. 27, 1271–1283 (1986). [8] R.J. Duffin and A.C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72, 341– 366 (1952). [9] H.G. Feichtinger and T. Werther, Atomic systems for subspaces, in Proceedings SampTA 2001 (L. Zayed, eds.), Orlando, FL, 2001, pp. 163–165. [10] L. G˘avrut¸a, Frames for operators, Appl. Comput. Harmon. Anal. 32, 139–144 (2012). [11] P. G˘avrut¸a, On some identities and inequalities for frames in Hilbert spaces, J. Math. Anal. Appl. 321, 469–478 (2006). [12] X.X. Guo, Canonical dual K-Bessel sequences and dual K-Bessel generators for unitary systems of Hilbert spaces, J. Math. Anal. Appl. 444, 598–609 (2016). [13] D.L. Hua and Y.D. Huang, K-g-frames and stability of K-g-frames in Hilbert spaces, J. Korean Math. Soc. 53, 1331-1345 (2016). [14] J.Z. Li and Y.C. Zhu, Some equalities and inequalities for g-Bessel sequences in Hilbert spaces, Appl. Math. Lett. 25, 1601–1607 (2012). [15] J.Z. Li and Y.C. Zhu, Exact g-frames in Hilbert spaces, J. Math. Anal. Appl. 374, 201–209 (2011). [16] T. Strohmer and R. Heath, Grassmannian frames with applications to coding and communication, Appl. Comput. Harmon. Anal. 14, 257–275 (2003). [17] W. Sun, G-frames and g-Riesz bases, J. Math. Anal. Appl. 322, 437–452 (2006). [18] W. Sun, Stability of g-frames, J. Math. Anal. Appl. 326, 858–868 (2007). [19] W. Sun, Asymptotic properties of Gabor frame operators as sampling density tends to infinity, J. Funct. Anal. 258, 913–932 (2010). [20] Z.Q. Xiang and Y.M. Li, Frame sequences and dual frames for operators, ScienceAsia 42, 222–230 (2016).

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SOME EQUALITIES AND INEQUALITIES FOR K-G-FRAMES

11

[21] X.C. Xiao, Y.C. Zhu, and X.M. Zeng, Some properties of g-Parseval frames in Hilbert spaces, Acta Math. Sin. (Chin. Ser.) 51, 1143–1150 (2008). [22] X.C. Xiao, Y.C. Zhu, and L. G˘avrut¸a, Some properties of K-frames in Hilbert spaces, Results Math. 63, 1243–1255 (2013). [23] X.C. Xiao, Y.C. Zhu, Z.B. Shu, and M.L. Ding, G-frames with bounded linear operators, Rocky Mountain J. Math. 45, 675–693 (2015). [24] X.H. Yang and D.F. Li, Some new equalities and inequalities for g-frames and their dual frames, Acta Math. Sin. (Chin. Ser.) 52, 1033–1040 (2009). College of Mathematics and Computer Science, Shangrao Normal University, Shangrao, Jiangxi 334001, P.R. China E-mail address: [email protected]; [email protected].

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.3, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

AQ-functional equation in matrix non-Archimedean fuzzy normed spaces Jung-Rye Lee1 , George A. Anastassiou2 , Choonkil Park3∗ , Murali Ramdoss4∗ and Vithya Veeramani5 1

Department of Mathematics, Daejin University, Kyunggi 11159, Korea Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA 3 Research Institute for Natural Sciences Hanyang University, Seoul 04763, Korea 4,5 Department of Mathematics, Sacred Heart College, Tirupattur - 635 601, TamilNadu, India e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected] 2

Abstract. Using the fixed point method, we establish some stability results concerning the following new mixed type AQ-functional equation f (−x + 2y) + 2[f (3x − 2y) + f (2x + y) − f (y) − f (y − x)] = 3[f (x + y) + f (x − y) + f (−x)] + 4f (2x − y) in matrix non-Archimedean fuzzy normed spaces.

1. Introduction and preliminaries A basic question in the theory of functional equations is as follows: “When is it true that a function, which approximately satisfies a functional equation must be close to an act solution of the equation? If the problem accepts a solution, we say the equation is stable. The first stability problem concerning group homomorphisms was raised by Ulam [29] in 1940 and affirmatively solved by Hyers [7]. The result of Hyers was generalized by Aoki [1] for approximate additive mappings and by Rassias [24] for approximate linear mappings by allowing the difference Cauchy equation kf (x+y)−f (x)−f (y)k to be controlled by (kxkp +kykp ). In 1994, a generalization of the Rassias theorem was obtained by Gavruta [6] who replaced (kxkp +kykp ) by a general control function χ(x, y). In addition, Rassias [20]–[23] generalized the Hyers-Ulam stability result by introducing two weaker conditions controlled by a product of different powers of norms and a mixed product sum of powers of norms, respectively. applied to the cases of other functional equations in various spaces [2, 5, 13, 15, 16, 26, 27]. In particular Mirmostafafe and Moslehian [14] introduced a notation of non-Archimedean fuzzy normed spaces. They presented the interdisciplinary relation between the theory of fuzzy spaces, the theory of non-Archimedean spaces and the theory of functional equations. Many authors [8, 11, 12, 14, 19, 25, 32] investigated the Hyers-Ulam stability in non-Archimedean fuzzy normed spaces. Definition 1. [8, 32] Let X be a linear space over a non-Archimedean field K. A function N : X × R → [0, 1] is said to be a non-Archimedean fuzzy norm on X if for all x, y ∈ X and all s, t ∈ R (N1) N (x, c) = 0 for c ≤ 0; (N2) x = 0 ⇔ N (x, c) = 1 for all c > 0; t (N3) N (cx, t)= N (x, |c| ) if c 6= 0; (N4) N (x + y, max {s + t}) ≥ min {N (x, s), N (y, t)}; (N5) lim N (x, t) = 1. t→∞

The pair (X, N ) is called a non-Archimedean fuzzy normed space. Clearly, if (N 4) holds then so does (N6) N (x + y, s + t) ≥ min {N (x, s), N (y, t)}. A classical vector space over a complex or real field satisfying (N 1) and (N 5) is called fuzzy normed space. It is easy to see that (N 4) is equivalent to the following condition (N7) N (x + y, t) ≥ min {N (x, t), N (y, t)} (x, y ∈ X; t ∈ R). 0

2010 Mathematics Subject Classification: 46S40, 46S50, 47L25, 47H10, 54C30, 54E70. Keywords: Hyers-Ulam stability, fixed point, mixed type additive-quadratic functional equation, matrix nonArchemedian fuzzy normed space. ∗ Corresponding authors. 0

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J. Lee, G.A. Anastassiou, C. Park, M. Ramdoss, V. Veeramani Definition 2. Let (X, N ) be a non-Archimedean fuzzy normed space. A sequence {xn } in X is said to be convergent or converge if there exists an x ∈ X such that lim N (xn − x, t) = 1 for all t > 0. In this case, x is n→∞

called the limit of the sequence {xn } and we denote it by N − lim xn = x n→∞

Definition 3. Let (X, N ) be a non-Archimedean fuzzy normed space. A sequence {xn } in X is said to be Cauchy if for each  > 0 and each t > 0 there exists an n0 ∈ N such that for all n ≥ n0 and all p > 0, we have N (xn+p − xn , t) > 1 − . Due to N (xn+p − xn , t) ≥ min {N (xn+p − xn+p−1 , t), ..., N (xn+1 − xn , t)} the sequence {xn } is Cauchy if for each  > 0 and each t > 0 there exists an n0 ∈ N such that for all n ≥ n0 we have N (xn+1 − xn , t) > 1 − . It is well known that every convergent sequence in a (non-Archimedean) fuzzy normed space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the (non-Archimedean) fuzzy normed space is called a (non-Archimedean) fuzzy Banach space. The abstract characterization given for linear spaces of bounded Hilbert space operators in terms of matricially normed spaces [4] implies that quotients, mapping spaces and various tensor products of operator spaces may again be regarded as operator spaces. Owing in part to this result, the theory of operator spaces is having an increasingly significant effect on operator algebra theory [18]. Recently, Lee et al. [9] researched the Hyers-Ulam stability of the Cauchy functional equation and the quadratic functional equation in matrix normed spaces. This terminology may also be applied to the cases of other functional equations [3, 10, 28, 30, 31]. We will use the following notations: Mn (X) is the set of all n × n-matrices in X; ej ∈ M1,n (C) is that jth component is 1 and the other components are zero; Eij ∈ Mn (C) is that (i,j)-component is 1 and the other components are zero; Eij ⊗ x ∈ Mn (X) is that (i,j)-component is x and the other components are zero. For x ∈ Mn (X), y ∈ Mk (X),

 x ⊕ y :=



x 0



0 . y



Note that (X, k.kn ) is a matrix normed space if and only if (Mn (X), k.kn ) is a normed space for each positive integer  n and kAxBkk ≤ kAk kBk kxkn holds for A ∈ Mk,n (C), B ∈ Mn,k (C) and  x = (xij ) ∈ Mn (X), and that (X, k.kn ) is a matrix Banach space if and only if X is a Banach space and (X, k.kn ) is a matrix normed space.   A matrix normed space (X, k.kn ) is called an L∞ -matrix normed space if kx ⊗ ykn+k = max kxkn , kykk holds for all x ∈ Mn (X) and all y ∈ Mk (X). Let E, F be vector spaces. For a given mapping h : E → F and a given positive integer n, define hn : Mn (E) → Mn (F ) by hn ([xij ]) = [h(xij )] for all [xij ] ∈ Mn (E). We introduce the concept of matrix non-Archimedean fuzzy normed space. Definition 4. Let (X, N ) be a non-Archimedean fuzzy normed space. (i) (X, {Nn }) is called a matrix non-Archimedean fuzzy normed  space if for  each positive integer n, (Mn (X), Nn ) is a

non-Archimedean fuzzy normed space and Nk (AxB, t) ≥ Nn x,

t kAk·kBk

for all t > 0, A ∈ Mk,n (R), B ∈ Mn,k (R)

and x = [xij ] ∈ Mn (X) with kAk · kBk 6= 0. (ii) (X, {Nn }) is called a complete matrix non-Archimedean fuzzy normed space if (X, N ) is a non-Archimedean fuzzy Banach space and (X, {Nn }) is a matrix non-Archimedean fuzzy normed space.





Example 5. Let (X, k.kn ) is a matrix normed space. Let Nn (x, t) = Then Nk (AxB, t) =

t t+kAxBkk

≥

t t+kAk·kxkn ·kBk

t t+kxkn

=

for all t > 0 and x = [xij ] ∈ Mn (X).

t kAk·kBk t kAkkBk+kxkn

for all t > 0, A ∈ Mk,n (R), B ∈ Mn,k (R) and x = [xij ] ∈ Mn (X) with kAk · kBk 6= 0. So (X, {Nn }) is a matrix non-Archimeaden fuzzy normed space. Definition 6. Let X be a set. A function d : X × X → [0, ∞] is called a generalized metric on X if d satisfies (1) d(x, y) = 0 if and only if x = y; (2) d(x, y) = d(y, x) forall x, y ∈ X; (3) d(x, z) ≤ d(x, y) + d(y, z) forall x, y, z ∈ X.

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AQ-functional equation in matrix non-Archimedean fuzzy spaces Theorem 7. [17] Let (x, d) be a complete generalized metric space and let J : X → Y be a strictly contractive mapping with a Lipschitz constant α < 1.Then, for each given element x ∈ X, either d(J n x, J n+1 x) = ∞. for all nonnegative integers n or there exists a positive integer n0 such that (1) (2) (3) (4)

d(J n x, J n+1 x) < ∞ for all n ≥ n0 ; the sequence {J n x}converges to a fixed point y∗ of J; y∗ is the unique fixed pointof J; 1 d(y, y∗) ≤ 1−α d(y, Jy) forall y ∈ Y .

In this paper, we establish some stability results concerning the following new mixed type AQ-functional equation f (−x + 2y) + 2[f (3x − 2y) + f (2x + y) − f (y) − f (y − x)]

(1.1)

= 3[f (x + y) + f (x − y) + f (−x)] + 4f (2x − y) in matrix non-Archimedean fuzzy normed spaces by using the fixed point method. Theorem 8. Let A and B be real vector spaces. If an odd mapping f : A → B satisfies (1.1), then f is additive. Proof. Suppose that f is an odd mapping. Then (1.1) is equivalent to − f (x − 2y) + 2[f (3x − 2y) + f (2x + y) − f (y)] = 3[f (x + y) − f (x)] + f (x − y) + 4f (2x − y)

(1.2)

for all x, y ∈ A. Replacing x by x + y in (1.2), we obtain − f (x − y) + 2[f (3x + y) + f (2x + 3y) − f (y)] = 3[f (x + 2y) − f (x + y)] + f (x) + 4f (2x + y)

(1.3)

for all x, y ∈ A. Replacing (x, y) by (x + y, −y) in (1.3), we obtain − f (x + 2y) + 2[f (3x + 2y) + f (2x − y) + f (y)] = 3[f (x − y) − f (x)] + f (x + y) + 4f (2x + y)

(1.4)

for all x, y ∈ A. Subtracting (1.3) from (1.4) and then dividing the resulting equation by 2, we get −f (x + 2y) + f (x − y) + f (2x + 3y) − f (3x + 2y) + f (3x + y) − f (2x − y) = −2f (x + y) + 2f (x) + 2f (y)

(1.5)

for all x, y ∈ A. Interchanging x and y in (1.5) and then adding the resulting equation to (1.5), we get −f (x + 2y) − f (2x + y) + f (3x + y) + f (x + 3y) − f (2x − y) + f (x − 2y) = −4f (x + y) + 4f (x) + 4f (y)

(1.6)

for all x, y ∈ A. Replacing x by x − y in (1.6), we obtain −f (x + y) − f (2x − y) + f (3x − 2y) + f (x + 2y) − f (2x − 3y) + f (x − 3y) = −4f (x) + 4f (x − y) + 4f (y)

(1.7)

for all x, y ∈ A. Replacing y by −y in (1.7), we obtain −f (x − y) − f (2x + y) + f (3x + 2y) + f (x − 2y) − f (2x + 3y) + f (x + 3y) = −4f (x) + 4f (x + y) − 4f (y)

(1.8)

for all x, y ∈ A. Adding (1.7) to (1.8), we get −f (x + 2y) − f (2x + y) + f (3x + y) + f (x + 3y) − f (2x − y) + f (x − 2y) = −2f (x) + 2f (x + y) − 2f (y)

(1.9)

for all x, y ∈ A. Subtracting (1.9) from (1.6) and then dividing the resulting equation by 6, we get f (x + y) = f (x) + f (y) for all x, y ∈ A, as desired.



Theorem 9. Let A and B be real vector spaces. If an even mapping f : A → B satisfies (1.1), then f is quadratic.

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J. Lee, G.A. Anastassiou, C. Park, M. Ramdoss, V. Veeramani Proof. Suppose that f is an even mapping. Then (1.1) is equivalent to f (x − 2y) + 2[f (3x − 2y) + f (2x + y) − f (y)]

(1.10)

= 3[f (x + y) + f (x)] + 5f (x − y) + 4f (2x − y) for all x, y ∈ A. Replacing x by x + y and y by x + y in (1.10) and then comparing the two resulting equations, we get 2f (x) + 4f (x + 2y) − 2f (2x + 3y)

(1.11)

= −f (x + y) + 5f (x − y) + 3f (y) − f (2x + y) − 2f (x − 2y) for all x, y ∈ A. Interchanging x and y in (1.11), we obtain 2f (y) + 4f (2x + y) − 2f (3x + 2y)

(1.12)

= −f (x + y) + 5f (x − y) + 3f (x) − f (x + 2y) − 2f (2x − y) for all x, y ∈ A. Replacing y by −y in (1.12), we get 2f (y) + 4f (2x − y) − 2f (3x − 2y)

(1.13)

= −f (x − y) + 5f (x + y) + 3f (x) − f (x − 2y) − 2f (2x + y) for all x, y ∈ A. Subtracting (1.13) from (1.10) and then dividing the resulting equation by 2, we get 2f (y) + 4f (2x − y) − 2f (3x − 2y) = −3f (x − y) + f (x + y)

(1.14)

for all x, y ∈ A. Replacing x by x + y in (1.14), we get f (x + 2y) + 2[f (3x + y) − f (y)] = 3f (x) + 4f (2x + y)

(1.15)

for all x, y ∈ A. Replacing y by y − x in (1.15), we get f (−x + 2y) + 2[f (2x + y) − f (y − x)] = 3f (x) + 4f (x + y)

(1.16)

for all x, y ∈ A. Replacing y by −y in (1.16), we obtain f (x + 2y) + 2[f (2x − y) − f (x + y)] = 3f (x) + 4f (x − y)

(1.17)

for all x, y ∈ A. Replacing x by y and y by x in (1.16), we obtain that f (2x − y) + 2[f (x + 2y) − f (x − y)] = 3f (y) + 4f (x + y)

(1.18)

for all x, y ∈ A. Adding (1.17) to (1.18) and then dividing the resulting equation by 3, we get f (x + 2y) + f (2x − y) = f (x) + f (y) + 2f (x + y) + 2f (x − y)

(1.19)

for all x, y ∈ A. Subtracting (1.17) from (1.18) and then adding the resulting equation to (1.19), we get f (x + 2y) + f (x) = 2f (y) + 2f (x + y)

(1.20)

for all x, y ∈ A. Replacing x by x − y in (1.20), we obtain f (x + y) + f (x − y) = 2f (x) + 2f (y) for all x, y ∈ A. This completes the proof.  2. Hyers-Ulam stability of the additive-quadratic functional equation (1.1) Throughout this paper, we assume that K is a non-Archimedean field, X is a vector space over K and (Y, Nn ) is a complete matrix non-Archimedean fuzzy normed space over K, and (Z, N 0 ) is (an Archimedean or a nonArchimedean fuzzy) normed space. For a mapping f : X → Y , define G f : X 2 → Y and G fn : Mn (X 2 ) → Mn (Y ) by Gf (a, b) = f (−a + 2b) + 2[f (3a − 2b) + f (2a + b) − f (b) − f (b − a)] − 3[f (a + b) + f (a − b) + f (−a)] − 4f (2a − b), Gfn ([xij ], [yij ]) = fn ([−xij + 2yij ]) + 2[fn ([3xij − 2yij ]) + fn ([2xij + yij ]) − fn ([yij ]) − fn ([yij − xij ])] − 3[fn ([xij + yij ]) + fn ([xij − yij ]) + fn ([−xij ])] − 4fn ([2xij − yij ]) for all a, b ∈ X and all x = [xij ], y = [yij ] ∈ Mn (X). In this section, we investigate the Hyers-Ulam stability for the functional equation (1.1) in matrix non-Archimedean fuzzy normed spaces by using the fixed point method.

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AQ-functional equation in matrix non-Archimedean fuzzy spaces Theorem 10. Let q = ±1 be fixed and let ψ : X × X → Z be a mapping such that for some η 6= 2 with 0

q

q

N (ψ(2 a, 2 b) ≥ N

0

ψ(a, b), η

−q

t




η q 2




0, and lim N (2−kq Gf (2kq a, 2kq b), t) = 1 k→∞

for all a, b ∈ X and t > 0. Suppose that an odd mapping f : X → Y satisfies the inequality N (Gfn ([xij ], [yij ]), t) ≥ N

n X

0

! ψ(xij , yij ), t

∀ x = [xij ], y = [yij ] ∈ Mn (X), and t > 0.

(2.2)

i,j=1

Then there exists a unique additive mapping A : X → Y such that Nn (fn ([xij ]) − An ([xij ]), t) ≥ min N 0 (ψ(xij , 0), |η − 2| n−2 t) : i, j = 1, 2, · · · , n





(2.3)

for all x = [xij ] ∈ Mn (X) and t > 0. Proof. For the cases q = 1 and q = −1, we consider η < 2 and η > 2, respectively. Letting n = 1 in (2.2), we obtain N (Gf (a, b), t) ≥ N 0 (ψ(a, b), t)

(2.4)

for all a, b ∈ X and t > 0. Replacing (a, b) by (0, a) in (2.4), we get N (f (2a) − 2f (a), t) ≥ N 0 (ψ(0, a), t) for all a ∈ X and t > 0. Thus q−1

N

η 2 1 f (a) − q f (2q a), 1+q t 2 |2|( 2 )

! ≥ N 0 (ψ(0, a), t)

∀ a ∈ X and t > 0.

(2.5)

Consider the set M = {f : X → Y } and introduce the generalized metric ρ on M as follows: ρ(f, g) =∈ ρ(f, g) = f µ ∈ R+ : N (f (a) − g(a), µt) ≥ N 0 (ψ(0, a), t) , ∀a ∈ X, t > 0





We will prove that (M, ρ) is a complete generalized metric, First we will prove that ρ is a generalized metric on M. Let ρ(f, g) = µ1 and ρ(g, h) = µ2 . Then N (f (a) − g(a), µ1 t) ≥ N 0 (ψ(0, a), t) and N (g(a) − h(a), µ2 t) ≥ N 0 (ψ(0, a), t) for all a ∈ X and t > 0. Therefore, N (f (a) − h(a), (µ1 + µ2 )t) ≥ N 0 (ψ(0, a), t). By definition of ρ, ρ(f, h) ≤ µ1 + µ2 = ρ(f, g) + ρ(g, h). which means that ρ satisfies the triangle inequality. One can show that other properties are satisfied. So ρ is a generalized metric on M. Next we will prove that (M, ρ) is a complete generalized metric. Suppose that {fn } is ρ-Cauchy, i.e., for any τ > 0, there exist n0 , n > m ≥ n0 , such that ρ(fn , fm ) < τ . By definition of ρ, there exists 0 < µ0 < τ , which satisfies N (fn (a) − fm (a), τ t) ≥ N 0 (ψ(0, a), t) for all a ∈ X and t > 0, n > m ≥ n0 , i.e., {fn (a)} is a Cauchy sequence in Y . Since Y is complete, there exists {f0 (a)} ⊆ Y and {fn (a)} → {f0 (a)}. Taking the limit as m → ∞, we obtain N (fn (a) − f0 (a), τ t) ≥ N 0 (ψ(0, a), t) for all a ∈ X and t > 0, n ≥ n0 . Therefore, ρ(fn , f0 ) = inf µ ∈ R+ : N (fn (a) − f0 (a), µt) ≥ N 0 (ψ(0, a), t) < τ.





for all n ≥ n0 , so that {fn } is ρ-convergent, i.e., (M, ρ) is a complete generalized metric. Now consider the mapping P : M → M by 1 Pf (a) = q f (2q a) ∀ f ∈ M and a ∈ X. 2 Let f, g ∈ M and ν be an arbitrary constant with ρ(f, g) ≤ ν. Then N (f (a) − g(a), νt) ≥ N 0 (ψ(0, a), t) for all a ∈ X and t > 0. Therefore, using (2.1), we get N Pf (a) − Pg(a), 2−q νt = N (f (2q a) − g(2q a), νt) , ≥ N 0 ψ(0, a), η −q t




442




Jung-Rye Lee ET AL 438-446

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.3, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

J. Lee, G.A. Anastassiou, C. Park, M. Ramdoss, V. Veeramani

 q

η ν, that is, ρ(Pf, Pg) ≤ Lρ(f, g) for all f, g ∈ M. 2  q η This means that P is a contractive mapping with Lipschitz constant L = < 1. 2 q−1 η( 2 ) It follows from (2.5) that ρ(f, Pf ) ≤ 1+q . Therefore according to Theorem 7, there exists a mapping |2|( 2 ) A : X → Y which satisfies for all a ∈ X and t > 0. Hence by definition ρ(Pf, Pg) ≤

(1) A is a unique fixed point of P in the set S = {g ∈ M : ρ(f, g) < ∞}, which satisfies A(2q a) = 2q A(a)

∀ a ∈ X.

In other words, there exists a µ > 0 satisfying N (f (a) − g(a), µt) ≥ N 0 (ψ(0, a), t)

∀ a ∈ X and t > 0.

(2) ρ(P k f, VU ) → 0 as k → ∞. This implies the equality lim k→∞

(3) ρ(f, A) ≤

1 f (2kq a) = A(a) 2kq

∀ a ∈ X.

1 1 ρ(f, Pf ), which implies the inequality ρ(f, A) ≤ . So 1−η |2 − η|

 N

1 f (a) − A(a), t |2 − η|



≥ N 0 (ψ(0, a), t)

∀ a ∈ X and t > 0.

(2.6)

By (2.4), N (GA(a, b), t) = lim N (2−kq Gf (2kq a, 2kq b), t) ≥ lim N 0 (2−kq ψ(2kq a, 2kq b), t) = 1. k→∞

k→∞

Hence by (N2), GA(a, b) = 0. Thus A is additive. We note that ej ∈ M1,n (R) means that the j-th component is 1 and the others are zero, Eij ∈ Mn (X) means that (i, j)-component is 1 and the others are zero, and Eij ⊗ x ∈ Mn (X) means that (i, j)-component is x and the others are zero. Since N (Ekl ⊗ x, t) = N (x, t), we have

!

n X

Nn ([xij ], t) = Nn

Eij ⊗ xij , t

≥ min {Nn (Eij ⊗ xij , tij ) : i, j = 1, 2, ..., n}

i,j=1

= min {N (xij , tij ) : i, j = 1, 2, ..., n} , where t =

n P

tij . So Nn ([xij ], t) ≥ T



N (xij ,

t ) n2



: i, j = 1, 2, · · · , n .

i,j=1

By (2.6),

n



t : i, j = 1, 2, · · · , n. n2
ψ(0, xij ), |2 − η| n−2 t : i, j = 1, 2, ..., n

N (fn ([xij ]) − An ([xij ]), t) ≥ min N f (xij ) − A(xij ), ≥ min N 0





o

for all x = [xij ] ∈ Mn (X) and t > 0. Thus A : X → Y is a unique additive mapping satisfying (2.3).



Corollary 1. Let q = ±1 be fixed and let p be a nonnegative real number with p 6= 1 and Υ ∈ Z. Let f : X → Y be an odd mapping such that Nn (Gfn ([xij ], [yij ]), t) ≥

n X

N 0 (Υ(kxij kp + kyij kp ), t)

i,j=1

for all x = [xij ], y = [yij ] ∈ Mn (X) and t > 0. Then there exists a unique additive mapping A : X → Y such that N (fn ([xij ]) − An ([xij ]), t) ≥ min N 0 (kxkp Υ, |2 − 2p | n−2 t) : i, j = 1, 2, · · · , n





(2.7)

for all x = [xij ] ∈ Mn (X) and t > 0. Proof. The proof follows from Theorem 10 by taking ψ(a, b) = Υ(kakp + kbkp ) for all a, b ∈ X. Then we can choose η = 2q(p−1) , and we can obtain the required result.  The following corollary gives the Hyers-Ulam stability for the additive functional equation (1.1).

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Jung-Rye Lee ET AL 438-446

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.3, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

AQ-functional equation in matrix non-Archimedean fuzzy spaces Corollary 2. Let q = ±1 be fixed and let p be a nonnegative real number with p = v + w 6= 1 and Υ ∈ Z. Let f : X → Y be an odd mapping such that Nn (Gfn ([xij ], [yij ]), t) ≥

n X

N 0 (Υ(kxij kv · kyij kw + kxij kv+w + kyij kv+w ), t)

i,j=1

for all x = [xij ], y = [yij ] ∈ Mn (X) and t > 0. Then there exists a unique additive mapping A : X → Y satisfying (2.7). Proof. The proof follows from Theorem 10 by taking ψ(a, b) = Υ(kakv · kbkw + kakp + kbkp ) for all a, b ∈ X. Then we can choose η = 2q(p−1) , and we can obtain the required result.  Theorem 11. Let q = ±1 be fixed and let ψ : X × X → Z be a mapping such that for some η 6= 4 with 0

q

q

N (ψ(2 a, 2 b) ≥ N

0

ψ(a, b), η

−q

t




η q 4




0, and lim N (4−kq Gf (2kq a, 2kq b), t) = 1 for all a, b ∈ X and t > 0. Suppose that an even k→∞

mapping f : X → Y with f (0) = 0 satisfies the inequality N (Gfn ([xij ], [yij ]), t) ≥ N

0

n X

! ψ(xij , yij ), t

∀ x = [xij ], y = [yij ] ∈ Mn (X), and t > 0.

(2.9)

i,j=1

Then there exists a unique quadratic mapping Q : X → Y such that Nn (fn ([xij ]) − Qn ([xij ]), t) ≥ min N 0 (ψ(0, xij ), |η − 4| n−2 t) : i, j = 1, 2, · · · , n





(2.10)

for all x = [xij ] ∈ Mn (X) and t > 0. Proof. For the cases q = 1 and q = −1, we consider η < 4 and η > 4, respectively. Letting n = 1 in (2.9), we obtain N (Gf (a, b), t) ≥ N 0 (ψ(a, b), t)

(2.11)

for all a, b ∈ X and t > 0. Replacing (a, b) by (0, a) in (2.11), we get N (f (2a) − 4f (a), t) ≥ N 0 (ψ(0, a), t)

(2.12)

for all a ∈ X and t > 0. Thus q−1

N

η 2 1 f (a) − q f (2q a), 1+q t 4 |4|( 2 )

! ≥ N 0 (ψ(0, a), t)

∀ a ∈ X and t > 0.

(2.13)

We consider the set M = {f : X → Y } and introduce the generalized metric ρ on M as follows: ρ(f, g) = inf µ ∈ R+ : N (f (a) − g(a), µt) ≥ N 0 (ψ(0, a), t) , ∀a ∈ X, t > 0 .





It is easy to check that (M, ρ) is a complete generalized metric (see also Theorem 10). 1 Define the mapping P : M → M by Pf (a) = q f (2q a) for all f ∈ M and a ∈ X. 4 Let f, g ∈ M and ν be an arbitrary constant with ρ(f, g) ≤ ν. Then N (f (a) − g(a), νt) ≥ N 0 (ψ(0, a), t) for all a ∈ X and t > 0. Therefore, using (2.8), we get N Pf (a) − Pg(a), 4−q νt = N (f (2q a) − g(2q a), νt) , ≥ N 0 ψ(0, a), η −q t







 q

η ν, that is, ρ(Pf, Pg) ≤ Lρ(f, g) for all f, g ∈ M. 4  q η This means that P is a contractive mapping with Lipschitz constant L = < 1. 4 q−1 η( 2 ) It follows from (2.13) that ρ(f, Pf ) ≤ 1+q . Therefore according to Theorem 7, there exists a mapping |4|( 2 ) Q : X → Y which satisfies

for all a ∈ X and t > 0. Hence by definition ρ(Pf, Pg) ≤

(1) Q is a unique fixed point of P, which satisfies Q(2q a) = 4q Q(a) for all a ∈ X.

444

Jung-Rye Lee ET AL 438-446

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.3, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

J. Lee, G.A. Anastassiou, C. Park, M. Ramdoss, V. Veeramani (2) ρ(f, Q) ≤

1 1 ρ(f, Pf ), which implies the inequality ρ(f, Q) ≤ . So 1−η |4 − η|

 N

f (a) − Q(a),

1 t |4 − η|



≥ N 0 (ψ(0, a), t)

∀ a ∈ X and t > 0.

(2.14)

By (2.11), N (GQ(a, b), t) = lim N (4−kq Gf (2kq a, 2kq b), t) ≥ lim N 0 (4−kq ψ(2kq a, 2kq b), t) = 1. k→∞

k→∞

Hence by (N2), GQ(a, b) = 0. Thus Q is quadratic. Since N (Ekl ⊗ x, t) = N (x, t), we have n X

Nn ([xij ], t) = Nn

! Eij ⊗ xij , t

≥ min {Nn (Eij ⊗ xij , tij ) : i, j = 1, 2, ..., n}

i,j=1

= min {N (xij , tij ) : i, j = 1, 2, ..., n} , where t =

n P



tij . So Nn ([xij ], t) ≥ min N (xij ,

t ) n2



: i, j = 1, 2, ..., n .

i,j=1

By (2.14),

n



t : i, j = 1, 2, · · · , n. n2
ψ(0, xij ), |4 − η| n−2 t : i, j = 1, 2, · · · , n

N (fn ([xij ]) − Qn ([xij ]), t) ≥ min N f (xij ) − Q(xij ), ≥ min N 0





o

for all x = [xij ] ∈ Mn (X) and t > 0. Thus Q : X → Y is a unique quadratic mapping satisfying (2.10).



Corollary 3. Let q = ±1 be fixed and let p be a nonnegative real number with p 6= 2 and Υ ∈ Z . Let f : X → Y be an even mapping satisfying f (0) = 0 and Nn (Gfn ([xij ], [yij ]), t) ≥ N 0 (

n X

Υ(kxij kp + kyij kp ), t)

i,j=1

for all x = [xij ], y = [yij ] ∈ Mn (X) and t > 0. Then there exists a unique quadratic mapping Q : X → Y such that N (fn ([xij ]) − Qn ([xij ]), t) ≥ min N 0 (kxkp Υ, |4 − 2p | n−2 t) : i, j = 1, 2, · · · , n





(2.15)

for all x = [xij ] ∈ Mn (X) and t > 0. Proof. The proof follows from Theorem 11 by taking ψ(a, b) = Υ(kakp + kbkp ) for all a, b ∈ X. Then we can choose η = 2q(p−2) , and we can obtain the required result.  The following corollary gives the Hyers-Ulam stability for the quadratic functional equation (1.1). Corollary 4. Let q = ±1 be fixed and let p be a nonnegative real number with p = v + w 6= 2 and Υ ∈ Z. Let f : X → Y be an even mapping satisfying f (0) = 0 and Nn (Gfn ([xij ], [yij ]), t) ≥ N 0 (

n X

Υ(kxij kv · kyij kw + kxij kv+w + kyij kv+w ), t)

i,j=1

for all x = [xij ], y = [yij ] ∈ Mn (X) and t > 0. Then there exists a unique quadratic mapping Q : X → Y satisfying (2.15). Proof. The proof follows from Theorem 11 by taking ψ(a, b) = Υ(kakv · kbkw + kakp + kbkp ) for all a, b ∈ X. Then we can choose η = 2q(p−2) , and we can obtain the required result.  References [1] T. Aoki , On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. [2] A. Bodaghi, C. Park, J. M. Rassias, Fundamental stabilities of the nonic functional equation In intuitionistic fuzzy normed spaces , Commun. Korean Math. Soc. 31 (2016), 729–743. [3] A. Ebadian, S. Zolfaghari, S. Ostadbashi, C. Park, Approximation on the reciprocal functional equation in several variables in matrix non-Archimedean random normed spaces, Adv. Difference Equ. 2015, 2015:314. [4] E. Effros, Z. J. Ruan, On approximation properties for operator spaces, Int. J. Math. 1 (1990), 163–187. [5] Iz. EL-Fassi, S. Kabbaj, Non-Archimedean random stability of σ-quadratic functional equation, Thai J. Math. 14) (2016), 151–165.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.3, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

AQ-functional equation in matrix non-Archimedean fuzzy spaces [6] P. G˘ avruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436. [7] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA, 27 (1941), 222–224. [8] A. Kumar, S. Kumar, On stability of cubic functional equation in non-Archimedean fuzzy normed spaces , Int. J. Math. Archive 7 (2016), No. 10, 167–174. [9] J. Lee, C. Park, D. Shin, Functional equations in matrix normed spaces, Proc. Indian Acad. Sci. 125 (2015), 399–412. [10] J. Lee, D. Shin, C. Park, Fuzzy stabilty of functional inequalities in matrix fuzzy normed spaces, J. Inequal. Appl. 2013, 2013:224. [11] D. Mihet, Fuzzy ϕ-contractive mapping in non-Archimedean fuzzy metric spaces, Fuzzy Sets Syst. 159 (2008), 739–744. [12] D. Mihet, The stability of the additive Cauchy functional equation in non-Archimedean fuzzy normed spaces, Fuzzy Sets Syst. 161 (2010), 2206–2212. [13] D. Mihet, V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl. 343 (2008), 567–572. [14] A. K. Mirmostafaee, M. S. Moslehian, Stability of additive mappings in non- Archimedean fuzzy normed spaces, Fuzzy Sets Syst. 160 (2009), 1643–1652. [15] C. Park, K. Ghasemi, S. G. Ghaleh, S. Jang, Approximate n-Jordan ∗-homomorphisms in C ∗ -algebras, J. Comput. Anal. Appl. 15 (2013), 365–368. [16] C. Park, A. Najati, S. Jang, Fixed points and fuzzy stability of an additive-quadratic functional equation, J. Comput. Anal. Appl. 15 (2013), 452–462. [17] V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), 91–96. [18] Z. J. Ruan, Subspaces of C∗-algebras, J. Funct. Anal. 76 (1988), 217–230. [19] C. Renu, Sushma, A fixed point approach to Ulam stability problem for cubic and quartic mapping in nonArchimedean fuzzy normed spaces, Proceeding of the World Congress on Engineering 2010 vol VIII WCE, (2010). [20] J. M. Rassias, On approximation of approximately linear mappings by linear mappings, J. Funct. Anal. 46 (1982), 126–130. [21] J. M. Rassias, On approximation of approximately linear mappings by linear mappings, Bull. Sci. Math. 108 (1984), 445–446. [22] J. M. Rassias, On a new approximation of approximately linear mappings by linear mappings, Discuss. Math. 7 (1985), 193–196. [23] J. M. Rassias, Solution of a problem of Ulam, J. Approx. Theory 57 (1989), 268–273. [24] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Am. Math. Soc. 72 (1978), 297–300. [25] B. V. Senthil Kumar, A. Kumar, P. Narasimman, Estimation of approximate nonic functional equation in non-Archimedean fuzzy normed spaces, Int. J. Pure Appl. Math. Tech. 1 (2016), No. 2, 18–29. [26] D. Shin, C. Park, Sh. Farhadabadi, On the superstability of ternary Jordan C ∗ -homomorphisms, J. Comput. Anal. Appl. 16 (2014), 964–973. [27] D. Shin, C. Park, Sh. Farhadabadi, Stability and superstability of J ∗ -homomorphisms and J ∗ -derivations for a generalized Cauchy-Jensen equation, J. Comput. Anal. Appl. 17 (2014), 125–134. [28] A. Song, The Ulam stability of matrix intuitionistic fuzzy normed spaces, J. Intelligent Fuzzy Syst. 32 (2017), 629–641. [29] S. M. Ulam, Problems in Modern Mathematics, Science Editions, Wiley, NewYork, 1964. [30] Z. Wang, P. K. Sahoo, Stability of an ACQ-functional equation in various matrix normed spaces, J. Nonlinear Sci. Appl. 8 (2015), 64–85. [31] Z. Wang, P. K. Sahoo, Stability of the generalized quadratic and quartic type functional equation in nonArchimedean fuzzy normed spaces, J. Appl. Anal. Comput. 6 (2016), 917–938. [32] T. Z. Xu, J. M. Rassias, W. X. Xu, Stability of a general mixed additive-cubic functional equation in nonArchimedean fuzzy normed spaces, J. Math. Phys. 51 (2010), 1–19.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.3, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

Continuous selection for multivalued mappings

Existence of continuous selection for some special kind of multivalued mappings G. Poonguzalia , Muthiah Marudaib , George A. Anastassiouc and Choonkil Parkd∗ a

Department of Mathematics, Bharathidasan University, Tiruchirappalli 620 024, Tamil Nadu, India

b

Department of Mathematics, Bharathidasan University, Tiruchirappalli 620 024, Tamil Nadu, India c

Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA d

Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea

E-mail: [email protected]; [email protected]; [email protected]; [email protected]

Abstract This paper deals with the existence of continuous selection of a multivalued mapping in product space. Many authors provided existence of continuous map for lower semicontinuous. We provide continuous selection for weakly lower semicontinuous. Rhybinski [9] proved the existence for contraction type mapping. We prove the existence for some general type of mapping different from contraction mapping.

1

Introduction and preliminaries

Let X be a normed linear space. Then B = {x ∈ X : kxk ≤ 1} represents the closed unit ball and B 0 = {x ∈ X : kxk < 1} represents the open unit ball in X. First we quote some notations and basic facts that are used in the sequal P(X) = {A ⊂ X : A 6= ∅}, Pcl (X) = {A ∈ P(X) : A is closed}, Pcv (X) = {A ∈ P(X) : A is convex}, Pcl,cv (X) = {A ∈ P(X) : A is closed, convex}. For x ∈ X, A, B ∈ P(X), δ(A, B) = sup{d(x, B) : x ∈ A}. H(A, B) = max{δ(A, B), δ(B, A)}. Let us consider the mapping T : X × Y → Pcl,cv (Y ). Then the fixed point set is defined as PT (x) := {y ∈ Y : y ∈ T (x, y)}. See [1, 2] for more information on fixed point theory. Definition 1.1. A multivalued mapping F : X → P(Y ) is called lower semicontinuous (l.s.c.) at x0 ∈ X if and only if for every  > 0 and z ∈ F (x0 ) there exists a neighborhood Uz containing x0 with the property that z ∈ ∩{F (x) + B 0 : x ∈ Uz }. 2010 Mathematics Subject Classification: 47H04, 47H10. Keywords: weakly lower semicontinuous map, continuous selection, paracompact space, perfectly normal space. ∗ Corresponding author: Choonkil Park (email: [email protected], office: +82-2-2220-0892).

447

G. Poonguzali ET AL 447-452

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.3, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

G. Poonguzali, M. Marudai, G. A. Anastassiou, C. Park

Definition 1.2. A multivalued mapping F is said to be weakly lower semicontinuous (w.l.s.c.) at x0 ∈ X if ond only if for every  > 0 and for every neighborhood V containing x0 , there exists a point x1 ∈ V so that for every z ∈ F (x1 ) there is a neighborhood Uz containing x0 satifying the condition that z ∈ ∩{F (x) + B 0 : x ∈ Uz }. It is well known that F is l.s.c. (w.l.s.c.) if and only if F is l.s.c. (w.l.s.c.) at every x ∈ X. Also, it is easy to see that f is l.s.c. implies that F is w.l.s.c., but the converse is not true [8]. A topological space X is said to be paracompact if every open cover of X has a locally finite refinement. A cover {Uβ }β∈J is called a refinement of {Wα }α∈I if for all β ∈ J, there exists α ∈ I such that Uβ ⊂ Wα . Also, a collection {Ai : i ∈ I} of subsets of X is locally finite if and only if for each x ∈ X there is an open U 3 x with |{i ∈ I : Ai ∩ U 6= ∅}| < ∞. A topological space X is said to be perfectly normal if it is normal and every closed subset is a Gδ subset. A multivalued mapping T : X × Y → Pcl,cv (Y ) is said to satisfy condition C if there exists K < 1 such that H(T (x, y1 ), T (x, y2 )) ≤ Kky1 − y2 k f or x ∈ X, y1 , y2 ∈ Y. In a similar way, a multivalued mapping H : X × Y → Pcl,cv (Y ) is said to satisfy condition N if it satisfies H(T (x, y1 ), T (x, y2 )) ≤ ky1 − y2 k f or x ∈ X, y1 , y2 ∈ Y. In 1956, Michael [6] was the first person to study about continuous selection for a given multivalued mapping under some suitable conditions. The following theorem is due to Michael. Theorem 1.3. [6] In a paracompact space X, the lower semi-continuous multivalued mapping F : X → Pcl,cv (Y ) has a continuous selection, where Y is a Banach space. The importance of the above theorem was first noticed by Browder [4], who used the theorem to prove Fan Browder theorem. Later, many researchers established results on continuous selections with applications (see [3, 5, 7, 10, 11]). Further, in [8], Przeslawski and Rybinski has generalized Michael selection theorem for weakly lower semicontinuous mapping. They proved the existence of continuous selection for w.l.s.c. which is weaker than l.s.c. Rybinski [9] proved the following theorem. Theorem 1.4. Let X be a paracompact and perfectly normal topological space and Y be a closed subset of a Banach space (Z, k · k). Assume that T : X × Y → Pcl,cv (Y ) satisfies condition C and also, satisfies the condition that for every y ∈ Y the multivalued mapping T (·, y) is w.l.s.c. Then there exists a continuous mapping h : X × Y → Y such that h(x, y) ∈ PT (x) for every (x, y) ∈ X × Y . In this direction, we study the existence of continuous selection for multivalued mapping with certain conditions. For that, we need the following lemma and theorem. Lemma 1.5. [8] Let X and Y be any topological spaces. If T : X → Pcl,cv (Y ) is a w.l.s.c. multivalued mapping and f : X → Y is a continuous and open mapping, then T ◦ f is w.l.s.c. Theorem 1.6. [8] If X is a paracompact topological space, Y is a normed linear space and F : X → Pcl,cv (Y ) is w.l.s.c., then F has a continuous selection.

2

Existence of continuous selections

In this section, we provide continuous selection for some general type of mapping. Theorem 2.1. Let F : X1 × X2 → X2 be any multivalued mapping with the property that

448

G. Poonguzali ET AL 447-452

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.3, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

Continuous selection for multivalued mappings

1. F (., x2 ) is w.l.s.c for every x2 ∈ X2 , 2. F satisfies property (N ). Then, for a given continuous mapping α : X1 × X2 → X2 , the new mapping (x1 , x2 ) → F (x1 , α(x1 , x2 )) is w.l.s.c. Proof. Let us define S := X1 ×X2 and define G : S×X2 → Pcl,cv (X2 ) by G(s, u) = F (PX (s), u) = F (x1 , u) for s ∈ S and u ∈ X2 . Now our aim is to show that the mapping s → G(s, α(s)) is w.l.s.c. By Lemma 1.5, it is clear that G(., u) is w.l.s.c. for every u ∈ X2 . Step 1: For an s0 ∈ S and an  > 0 and a neighborhood O 3 s0 , by continuity of α, we can choose a neighborhood V ⊂ O of s0 with the property that kα(s) − α(s0 )k
0. Fix  > 0 and any neighborhood V of x0 . Then choose δ > 0 such that δ < min{d(x0 ), }. Now, choose a neighborhood W of x0 , W ⊆ V, such that for x1 , x2 ∈ W, |d(x1 ) − d(x2 )| < kf (x1 ) − f (x2 )k
0, does there exist a δ > 0 such that if a function h : G1 → G2 satisfies the inequality d(h(xy), h(x)h(y)) ≤ δ for all x, y ∈ G1 , then there exists a homomorphism H : G1 → G2 with d(h(x), H(x)) < ε for all x ∈ G1 . Ulam’s question was partially solved by D. H. Hyers [6] in the case of approximately additive functions and when the groups in the question are Banach spaces. In fact, Hyers proved that each solution of the inequality kf (x + y) − f (x) − f (y)k ≤ ε for all x and y can be approximated by an exact solution, say an additive function. In this case, it is said that the Cauchy additive functional equation f (x + y) = f (x) + f (y) satisfies Hyers–Ulam stability or that the equation is stable in the sense of Hyers–Ulam. Many mathematicians attempted to moderate the condition for the bound of the norm of the Cauchy difference. First, T. Aoki [1] proved the stability of Cauchy functional equations by changing the bound of Cauchy difference as follows kf (x + y) − f (x) − f (y)k ≤ ε(kxkp + kykp ), where p ∈ (0, 1), and Rassias [14] obtained additional linear properties of this results. Furthermore, the control function of Cauchy difference with some regularity conditions has been employed by Gˇavrutˇa [5] as follows kf (x + y) − f (x) − f (y)k ≤ ϕ(x, y). 1991 Mathematics Subject Classification. 39B52, 39B82, 54C65. Key words and phrases. set-valued functional equation; generalized Hyers–Ulam stability; Cantor intersection theorem; cone subset in Banach spaces. † Corresponding author. [email protected]. 1

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HONG-MEI LIANG, HARK-MAHN KIM, AND HWAN-YONG SHIN

Recently, as the development of non-convex analysis, cone sets were investigated by many authors and were applied for various regions of optimization theory and mathematical physics [10, 11]. Let X be a real Banach space and P a subset of X. P is called a cone [15] if (i) P is closed, non-empty and P 6= {0}, (ii) ax + by ∈ P for all x, y ∈ P and non-negative real numbers a, b, (iii) P ∩ (−P ) = {0}. Set-valued functions in Banach spaces also have received a lot of attention in the literature [2]. Functional inclusion is a tool for defining many notions of set-valued analysis, e.g., linear, affine, convex, concave, subadditive, superadditive set-valued maps. Finding a selection of such set-valued maps, with some special properties, is one of the main problems of set-valued analysis (see [2]). The stability theory of functional equations leads in some cases to such problems and solving them provides Hyers–Ulam stability results [3, 4, 7]. In setting domain of set-valued functions as a cone, some stability results of set-valued functional equations were obtained by several authors [9, 13]. In this sequel, we introduce a result concerning with stability of set-valued functional equations under cone domain. Let Y be a Banach space and P be a cone. We define the following families of sets : P0 (Y ) := {A ⊆ Y : A is nonempty set} cl(Y ) := {A ∈ P0 (A) : A is closed set} cz(Y ) := {A ∈ P0 (A) : A is closed set containing zero}. Theorem 1.1. (C. Park, D. O’Regan, R. Saadati, [13]) If F : P → cz(Y ) is a set-valued mapping satisfying F (0) = {0}, x + y
F (x) + F (y) ⊆ 2F (1.1) 2 and sup{diam(F (x)) : x ∈ P } < +∞ for all x, y ∈ P , then there exists a unique additive mapping g : P → Y such that g(x) ∈ F (x) for all x ∈ P . In view of Theorem 1.1, if diam(F (x)) = ε, then sup{diam(F (x)) : x ∈ P } < +∞. So if, in addition, F satisfies (1.1), we may understand Theorem 1.1 works good in the sense of Hyers–Ulam. On the other hand, if diam(F (x)) = kxkp , p 6= 0, we confirm that sup{diam(F (x)) : x ∈ P } = ∞, and so Theorem 1.1 cannot be favorably applied in this case. Thus, in this paper, we are devoted to investigate refined stability results of Theorem 1.1, and also we present alternative new stability theorems and examples to provide refined stability theorems of Theorem 1.1.

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3

Let A, B be nonempty subsets of a real vector space X and λ a real number. We define A + B = {x ∈ X : x = a + b, a ∈ A, b ∈ B} λA = {x ∈ X : x = λa, a ∈ A}. Lemma 1.2. [12] Let λ and µ be real numbers. If A and B are empty subsets of a real vector space X, then λ(A + B) = λA + λB (λ + µ)A ⊆ λA + µA. Moreover, if A is convex in X and λµ ≥ 0, then we have (λ + µ)A = λA + µA. Lemma 1.3. If An and Bn are non-empty subsets of a real vector space X for all nonnegative positive integer n, then l \

An +

n=0

l \ n=0

Bn =

l \

(An + Bn )

n=0

for any given l ∈ N. The following famous theorem is a crucial tool to prove our main theorems. Theorem 1.4. (Cantor Intersection Theorem, [8]) Suppose (X, d) is a non-empty complete metric space, and {Cn }n≥0 closed subsets of X which satisfies C1 ⊇ C2 ⊇ · · · ⊇ Cn ⊇ Cn+1 ⊇ · · · . If limn→∞ diam(Cn ) = 0, where diam(Cn ) is defined by diam(Cn ) = sup{d(x, y)|x, y ∈ Cn }, T then ∞ n=1 Cn consists of a single point. From now on, let P be a cone for a Banach space Y . We present a main theorem, which is an extended Hyers–Ulam stability of a set-valued functional equations on the domain of cones. Theorem 1.5. If F : P → cl(Y ) is a set-valued mapping satisfying Pm m X j=1 xj
F (xi ) ⊆ mF (1.2) m i=1

and diam(F (mn x)) =0 n→∞ mn for all x1 , · · · , xm , x ∈ P , where m > 1 is a positive integer, then there exists a unique additive mapping g : P → cl(Y ) such that g(x) ⊆ F (x) + (−1)F (0) for all x ∈ P .

(1.3)

lim

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Proof. Since F (0) ∈ cl(Y ), F (0) has at least an element, say p ∈ F (0). Letting x1 = x and xk = 0 for all k 6= 1, in (1.2), we have F (x) + (m − 1){p} ⊆ F (x) + (m − 1)F (0) ⊆ mF (

(1.4)

x ) m

and so F (x) + (−1){p} ⊆ m F (

(1.5)


x ) + (−1){p} m

for all x ∈ P. Replacing x by mn+1 x in (1.5), then we obtain F (mn+1 x) + (−1){p} ⊆ m F (mn x) + (−1){p}




and hence F (mn x) + (−1){p} F (mn+1 x) + (−1){p} ⊆ mn+1 mn F (mn x) + (−1){p} for all x ∈ P and mn is a decreasing sequence of closed subsets of the

for all x ∈ P and all n ∈ N ∪ {0}. Denoting Fn (x) := all n ∈ N ∪ {0}, it results that {Fn (x)}≥0 Banach space Y . We have also diam(Fn (x)) =

1 1 diam(F (mn x) + (−1){p}) = n diam(F (mn x)). n m m

By (1.3), we get limn→∞ diam(Fn (x)) = 0 for all x ∈ P. Using the Cantor Intersection T Theorem for the sequence {Fn (x)}n≥0 , the intersection n≥0 Fn (x) is a singleton and we denote this intersection by g(x) for all x ∈ P . Thus we obtain a mapping g : P → cl(Y ), T defined as g(x) := n≥0 Fn (x), which is a singleton from F because g(x) ⊆ F0 (x) = F (x) + (−1){p} ⊆ F (x) + (−1)F (0) for all x ∈ P . Now, we show that g is additive. It follows from the definition of g and Lemma 1.3 that m m \ l l X m l   Pm xj  X X \ \ j=1 g(xi ) ⊆ Fn (xi ) = Fn (xi ) ⊆ mFn m i=1

i=1 n=0

n=0 i=1

n=0

for any l ∈ N ∪ {0}, thus m X i=1

g(xi ) ⊆

∞  \

mFn

 Pm xj  j=1

n=0

m

for all x1 , · · · , xm ∈ P . On the other hand, one obtains vacuously ∞   Pm xj   Pm xj  \ j=1 j=1 mg ⊆ mFn m m n=0

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STABILITY OF SET-VALUED FUNCTIONAL EQUATIONS

for all x1 , · · · , xm ∈ P . Thus, since

T∞ 

 Pm xj  j=1

mFn m m  Pm xj  X j=1 g(xi ) = mg m n=0

5

is a singleton, we arrive at

i=1

for all x1 , · · · , xm ∈ P . Thus g is additive since g(0) = {0}. Therefore, we conclude that there exists an additive mapping g : P → cl(Y ) such that g(x) ⊆ F0 (x) ⊆ F (x) + (−1)F (0) for all x ∈ P . Next, we will finalize the proof by proving the uniqueness of g. Suppose that g 0 : P → cl(Y ) is another additive mapping such that g 0 (x) ⊆ F (x) + (−1)F (0) for all x ∈ P . Then we have mn g(x) = g(mn x) ⊆ F (mn x) + (−1)F (0) mn g 0 (x) = g 0 (mn x) ⊆ F (mn x) + (−1)F (0) for all n ∈ N ∪ {0} and all x ∈ P . Thus, we get mn diam(g(x) − g 0 (x)) = diam(mn g(x) − mn g 0 (x)) = diam(g(mn x) − g 0 (mn x)) ≤ diam(F (mn x) + (−1)F (0)) = diam(F (mn x)) + diam((−1)F (0)) which implies diam(g(x) − g 0 (x)) ≤

 1  diam(F (mn x)) + diam((−1)F (0)) n m

for all n ∈ N ∪ {0} and all x ∈ P . Therefore, it follows from limn→∞ g(x) = g 0 (x) for all x ∈ P , as desired.

diam(F (mn x)) mn

= 0 that 

The following corollary is a refined stability result of Theorem 1.1, if we take m = 2. Corollary 1.6. If F : P → cl(Y ) is a set-valued mapping satisfying Pm m X j=1 xj
F (xi ) ⊆ mF m i=1

and sup{diam(F (x)) : x ∈ X} < +∞ for all x1 , · · · , xm , x ∈ P , then there exists a unique additive mapping g : P → cl(Y ) such that g(x) ⊆ F (x) + (−1)F (0) for all x ∈ P . Proof. Since sup{diam(F (x)) : x ∈ P } < +∞, limn→∞ Applying Theorem 1.5, we complete the proof.

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diamF (mn x) = 0 for all x ∈ P . mn 

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Now, let us consider the following example with nontrivial set-valued function at zero. Example 1.7. Let F : [0, ∞) → cl(R) be defined by  [ax, ax + bxp ], if x 6= 0, F (x) = [0, c], if x = 0, where a, b are positive real numbers, c ≥ 0 and p ∈ (−∞, 0) ∪ (0, 1). It is easy to see that F (x) + F (y) ⊆ 2F

x + y
, 2

diam(F (2n x)) =0 n→∞ 2n lim

for all x, y ∈ [0, ∞). Also, we can check that ∞ \ F (2n x) + (−1)F (0) = {ax} 2n

n=0

for all x ∈ [0, ∞). Therefore, there exists additive mapping g : [0, ∞) → cl(R) defined by g(x) = {ax} such that g(x) ⊆ F (x) + (−1)F (0) = [ax − c, ax + bxp ] for all x ∈ [0, ∞). This result can be found by applying Theorem 1.5. However, it is noted that we cannot apply Theorem 1.1 to this example because sup{diam(F (x)) : x ∈ [0, ∞)} = +∞. Next, we provide an alternative main theorem of Theorem 1.5. Theorem 1.8. If F : P → cl(Y ) is a set-valued mapping satisfying Pm m X j=1 xj
mF (1.6) ⊆ F (xi ) m i=1

and (1.7)

lim mn diam(F (

n→∞

x )) = 0 mn

for all x1 , · · · , xm , x ∈ P , then there exists a unique additive mapping g : P → cl(Y ) such that g(x) ⊆ F (x) + (−1)F (0) for all x ∈ P . Proof. By assumption (1.7), one has lim mn diam(F (0)) = 0

n→∞

and so F (0) is a singleton, say F (0) = {p}. Taking x1 = x and xk = 0 for all k 6= 0 in (1.6), we obtain
x
(1.8) m F + (−1){p} ⊆ F (x) + (−1){p}. m

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7

x in (1.8), then we obtain mn
x x
m F ( n+1 ) + (−1){p} ⊆ F + (−1){p} m mn

for all x ∈ P . And if we replace x by

and so mn+1 F

x




mn+1


+ (−1){p} ⊆ mn F


x
+ (−1){p} n m


for all x ∈ P and all n ∈ N ∪ {0}. Defining Fn (x) = mn F ( mxn ) + (−1){p} for all x ∈ P and all n ∈ N ∪ {0}, we obtain that {Fn (x)}n≥0 is a decreasing sequence of closed subsets of the Banach space Y . It is noted that

x

x , diam(Fn (x)) = diam mn F ( n ) + (−1){p} = mn diam F m mn which implies limn→∞ diam(Fn (x)) = 0 for all x ∈ P by (1.7). T Employing the Cantor Intersection Theorem to the sequence {Fn (x)}n≥0 , n≥0 Fn (x) is a T singleton set and so we may define a mapping g : P → cl(Y ) by g(x) := n≥0 Fn (x), x ∈ P , which satisfies g(x) ⊆ F0 (x) = F (x) + (−1){p} ⊆ F (x) + (−1)F (0) for all x ∈ P . Now, we show that g is additive. It follows from Lemma 1.2 that m  Pm xj   X  xj
j=1 n mFn =m·m F + (−1){p} m mn · m j=1

⊆ mn

m X i=1

m

F(


X xi ) + (−1){p} = Fn (xi ) mn i=1

for all x1 , · · · , xm ∈ P . By the definition of g, we can get l l X m  Pm xj   Pm xj  \ \ j=1 j=1 ⊆ mFn ⊆ Fn (xi ) mg m m n=0

n=0 i=1

for any l ∈ N ∪ {0} and all x1 , · · · , xm ∈ P , which yields ∞ X m  Pm xj  \ j=1 (1.9) mg ⊆ Fn (xi ). m n=0 i=1

Moreover, it is easy to show that, for all x1 , · · · , xm ∈ P , m X

g(xi ) ⊆

i=1

m X

Fn (xi ), ∀n ∈ N ∪ {0}

i=1

and so (1.10)

m X i=1

g(xi ) ⊆

∞ X m \

Fn (xi ).

n=0 i=1

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HONG-MEI LIANG, HARK-MAHN KIM, AND HWAN-YONG SHIN

Therefore, it follows from (1.9) and (1.10) that m  Pm xj  X j=1 mg = g(xi ), m i=1

that is, g is additive because g(0) = {0}. Finally, let us prove the uniqueness of g. Suppose that g 0 : P → cl(Y ) is an additive mapping such that g 0 (x) ⊆ F (x) + (−1)F (0) for all x ∈ P . Then we have x
x
1 g(x) = g n ⊆ F + (−1)F (0), n m m mn
1 0 x
0 x g (x) = g ⊆ F + (−1)F (0) mn mn mn for all n ∈ N ∪ {0} and all x ∈ P . Thus, noting singleton F (0), we get 1 x x diam(g(x) − g 0 (x)) = diam(g( n ) − g 0 ( n )) mn m m    x  x ≤ diam F ( n ) + (−1)F (0) = diam F ( n ) m m 0 for all x ∈ P and all n ∈ N ∪ {0}. It follows from (1.7) that g(x) = g (x) for all x ∈ P , as desired.  Corollary 1.9. If F : P → cl(Y ) is a set-valued mapping satisfying F (0) = {0}, Pm m X j=1 xj
mF ⊆ F (xi ) m i=1

and x )) = 0 n→∞ mn for all x1 , · · · , xm , x ∈ P , then there exists a unique additive mapping g : P → cl(Y ) such that g(x) ⊆ F (x) for all x ∈ P . lim mn diam(F (

Example 1.10. Let F : [0, ∞) → cl(R) be defined by F (x) = [ax, ax + bxp ], where a, b are positive real numbers and p > 1. Then, since the function xp is convex, it is easily checked
x that 2F x+y ⊆ F (x) + F (y) and limn→∞ 2n diam(F ( n )) = 0 for all x, y ∈ [0, ∞). Thus, 2 2 there exists an additive mapping g : [0, ∞) → cl(R) such that g(x) = {ax} ⊆ F (x) for all x ∈ [0, ∞) by Corollary 1.9. Example 1.11. Finally, let H : [0, ∞) → cl(R) be defined by H(x) = [ax, ax + bx], where
= H(x) + H(y) for all a, b are positive real numbers. Then, it follows easily that 2H x+y 2 x, y ∈ [0, ∞). However, there are two different additive mappings g1 (x) := {ax}, g2 (x) := {(a + b)x} such that g1 (x), g2 (x) ⊆ H(x) for all x ∈ [0, ∞). In fact, one notes that either

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(1.3) or (1.7) is not satisfied for the function H, and so one cannot apply Theorems 1.5 and 1.8 to this example. Thus, we remark that the set-valued function F (x) = [ax, ax + bx] has no Hyers–Ulam stability property for the set-valued Cauchy–Jensen additive functional equation. Acknowledgements This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(2016R1D1A3B03930971). Correspondence should be addressed to Hark-Mahn Kim([email protected]) and Hwan-Yong Shin([email protected]). References [1] T. Aoki : On the stability of the linear transformation in Banach spaces. J. Math. Soc. Japan 2, 64–66 (1950) [2] J.P. Aubin, H. Frankowska, : Set-valued analysis, in Modern Birkh¨ auser Classics. Birkh¨ auser, Boston (2008) [3] H.-Y. Chu, S. K. Yoo, : On the Stability of the Generalized Quadratic Set-Valued Functional Equation. Journal of Computational Analysis and Applicartions. 20, 1007-1020 (2016) [4] J. Brzd¸ek, D. Popa, B. Xu,: Selections of set-valued maps satisfying a linear inclusion in a single variable. Nonlinear Analysis: Theory, Methods and Applications 74, 324–330 (2011) [5] P. Gˇ avruta,: A generalization of the Hyers–Ulam–Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 184, 431–436 (1994) [6] D.H. Hyers,: On the stability of the linear functional equation. Proc. Nat. Acad. Sci. 27, 222–224 (1941) [7] D. Kang,: Stabiltiy of generalized cubic set-valued functional equations Journal of Computational Analysis and Applicartions. 20, 296-306 (2016) [8] J. Lewin,: An Interactive Introduction to Mathematical Analysis. Cambridge University Press (2003) [9] G. Lu, C. Park,: Hyers–Ulam stability of additive set-valued functional equations. Appl. Math. Lett. 24, 1312–1316 (2011) [10] H. Mohebi,: Topical functions and their properties in a class of ordered Banach spaces, in Continuous Optimization. Applied Optimization, Springer 99, 343–361 (2005) [11] H. Mohebi, H. Sadeghi, A.M. Rubinov, : Best approximation in a class of normed spaces with star-shaped cone. Current Numer. Funct. Anal. Optim. 27, 411–436 (2006) [12] K. Nikodem,: K-convex and K-concave set-valued functions. Z. K. Nr. 559 (1989) [13] C. Park, D. O’Regon, R. Saadati,: Stability of some set-valued functional equations. Appl. Math. Lett. 24, 1910–1914 (2011) [14] T.M. Rassias,: On the stability of the linear mapping in Banach spaces. Proc. Amer. Math. Soc. 72, 297–300 (1978) [15] S. Rezapour, R. Hamlbarani,: Some notes on the paper “Cone metric spaces and fixed point theorems of contractive mappings”. J. Math. Anal. Appl. 345, 719–724 (2008) [16] S.M. Ulam,: Problems in Modern Mathematics. Chapter 6 Wiley Interscience, New York (1964)

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HONG-MEI LIANG, HARK-MAHN KIM, AND HWAN-YONG SHIN

Hong-Mei Liang, Department of Mathematics, Qiqihar University, Qiqihar, 161006, China; Department of Mathematics, Chungnam National University,99 Daehangno, Yuseong-gu, Daejeon 34134, Korea E-mail address: [email protected] Hark-Mahn Kim, Department of Mathematics, Chungnam National University,99 Daehangno, Yuseong-gu, Daejeon 34134, Korea E-mail address: [email protected] Department of Mathematics, Chungnam National University,99 Daehangno, Yuseong-gu, Daejeon 34134, Korea E-mail address: [email protected]

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APPROXIMATE CAUCHY-JENSEN AND BI-QUADRATIC MAPPINGS IN 2-BANACH SPACES WON-GIL PARK AND JAE-HYEONG BAE Abstract. In this paper, we obtain the stability of the Cauchy-Jensen and bi-quadratic functional equation ) ( z+w = f (x, z) + f (x, w) + f (y, z) + f (y, w), 2f x + y, 2 f (x + y, z + w) + f (x + y,z − w) + f (x − y, z + w) + f (x − y, z − w) = 4[f (x, z) + f (x, w) + f (y, z) + f (y, w)], respectively, in 2-Banach spaces.

1. Introduction In 1940, Ulam [7] suggested the stability problem of functional equations concerning the stability of group homomorphisms: Let a group G and a metric group H with the metric ρ be given. For each ( ε > 0, the )question is whether or not there is a δ > 0 such that if f : G → H satisfies ρ f (xy), f (x)f (y) < δ for all x, y ∈ G, then there exists a group homomorphism h : G → H satisfying ρ(f (x), h(x)) < ε for all x ∈ G. We introduce some definitions on 2-Banach spaces [2], [3]. Definition 1. Let X be a real linear space with dim X ≥ 2 and ∥·, ·∥ : X 2 → R be a function. Then (X, ∥·, ·∥) is called a linear 2-normed space if the following conditions hold: (a) ∥x, y∥ = 0 if and only if x and y are linearly dependent, (b) ∥x, y∥ = ∥y, x∥, (c) ∥αx, y∥ = |α|∥x, y∥, (d) ∥x, y + z∥ ≤ ∥x, y∥ + ∥x, z∥ for all α ∈ R and x, y, z ∈ X. In this case, the function ∥·, ·∥ is called a 2-norm on X. Definition 2. Let {xn } be a sequence in a linear 2-normed space X. The sequence {xn } is said to convergent in X if there exits an element x ∈ X such that lim ∥xn − x, y∥ = 0

n→∞

1991 Mathematics Subject Classification. 39B52, 39B72. Key words and phrases. linear 2-normed space, Cauchy-Jensen mapping, bi-quadratic mapping. This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology(grant number 2017028238). Competing interests. The authors declare that they have no competing interests. 1

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for all y ∈ X. In this case, we say that a sequence {xn } converges to the limit x, simply dented by limn→∞ xn = x. Definition 3. A sequence {xn } in a linear 2-normed space X is called a Cauchy sequence if for any ε > 0, there exists N ∈ N such that for all m, n ≥ N , ∥xm − xn , y∥ < ε for all y ∈ X. For convenience, we will write limm,n→∞ ∥xn − xm , y∥ = 0 for a Cauchy sequence {xn }. A 2-Banach space is defined to be a linear 2-normed space in which every Cauchy sequence is convergent. In the following lemma, we obtain some basic properties in a linear 2-normed space which will be used to prove the stability results. Lemma 4. ([1]) Let (X, ∥·, ·∥) be a linear 2-normed space and x ∈ X. (a) If ∥x, y∥ = 0 for all y ∈ X, then x = 0. (b) ∥x, z∥ − ∥y, z∥ ≤ ∥x − y, z∥ for all x, y, z ∈ X. (c) If a sequence {xn } is convergent in X, then limn→∞ ∥xn , y∥ = ∥ limn→∞ xn , y∥ for all y ∈ X. Throughout this paper, let X be a normed space and Y a 2-Banach space. We introduce the definitions of Cauchy-Jensen and bi-quadratic mappings. Definition 5. A mapping f : X × X → Y is called a Cauchy-Jensen mapping if f satisfies the system of equations (1)

f (x( + y, z)) = f (x, z) + f (y, z), 2f x, y+z = f (x, y) + f (x, z). 2

Definition 6. A mapping f : X × X → Y is called bi-quadratic if f satisfies the system of equations (2)

f (x + y, z) + f (x − y, z) = 2f (x, z) + 2f (y, z), f (x, y + z) + f (x, y − z) = 2f (x, y) + 2f (x, z).

For a mapping f : X × X → Y , consider the functional equations: ) ( z+w (3) 2f x + y, = f (x, z) + f (x, w) + f (y, z) + f (y, w) 2 and f (x + y, z + w) + f (x + y,z − w) + f (x − y, z + w) + f (x − y, z − w) (4)

= 4[f (x, z) + f (x, w) + f (y, z) + f (y, w)].

When X = Y = R, the function f : R × R → R given by f (x, y) := axy + bx and f (x, y) := ax2 y 2 are solutions of (3) and (4), respectively. In 2011, W.-G. Park [4] investigate approximate additive, Jensen and quadratic mappings in 2Banach spaces. In this papaer, we also investigate Cauchy-Jensen and bi-quadratic mappings in 2-Banach spaces with different assumptions from [4]. 2. Approximate Cauchy-Jensen mappings

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3

Let φ : X 5 → [0, ∞) be a function satisfying [ ∞ ∑ 1 (5) φ(x, ˜ y, z, w, s) := φ(2j x, 2j y, 3j z, 3j w, s) + 2φ(2j x, 2j y, −3j z, 3j w, s) 6j+1 j=0

1 + φ(2j x, 2j y, −3j z, 3j+1 w, s) + φ(2j x, 2j y, 3j+1 z, 3j+1 w, s) 2 ] + 3φ(2j x, 2j y, 3j z, −3j w, s) + 2∥f (2j+1 x, 0), t∥ + 5∥f (x, 0), t∥ < ∞ for all x, y, z, w, s ∈ X, where t = f (s). Theorem 7. Suppose that f : X × X → Y is a surjective mapping such that

(

)

z+w

(6) − f (x, z) − f (x, w) − f (y, z) − f (y, w), t

2f x + y, 2

≤ φ(x, y, z, w, s) for all x, y, z, w, s ∈ X, where t = f (s). Then there exists a unique Cauchy-Jensen mapping F : X × X → Y such that (7)

∥f (x, y) − f (x, 0) − F (x, y), t∥ ≤ φ(x, ˜ x, y, y, s)

for all x, y, s ∈ X, where t = f (s). Proof. Let t = f (s). Letting y = x in (6), we gain

(

)

z + w

2f 2x, − 2f (x, z) − 2f (x, w), t (8)

≤ φ(x, x, z, w, s) 2 for all x, z, w, s ∈ X. Putting w = −z in (8), we get (9)

∥ − 2f (2x, 0) + 2f (x, z) + 2f (x, −z), t∥ ≤ φ(x, x, z, −z, s)

for all x, z, s ∈ X. Replacing z by −z and w by −z in (8), we have 1 ∥f (2x, −z) − 2f (x, −z), t∥ ≤ φ(x, x, −z, −z, s) 2 for all x, z, s ∈ X. By (9) and (10), (10)

1 ∥f (2x, −z) + 2f (x, z) − 2f (2x, 0), t∥ ≤ φ(x, x, −z, −z, s) + φ(x, x, z, −z, s) 2 for all x, z, s ∈ X. Setting w = −3z in (8), (11)

∥2f (2x, −z) − 2f (x, z) − 2f (x, −3z), t∥ ≤ φ(x, x, z, −3z, s) for all x, z, s ∈ X. By (11) and the above inequality, (12) ∥6f (x, z) + 2f (x, −3z) − 4f (2x, 0), t∥ ≤ φ(x, x, −z, −z, s) + 2φ(x, x, z, −z, s) + φ(x, x, z, −3z, s) for all x, z, s ∈ X. Replacing z by 3z in (10), 1 ∥f (2x, −3z) − 2f (x, −3z), t∥ ≤ φ(x, x, −3z, −3z, s) 2

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for all x, z, s ∈ X. By (12) and the above inequality, ∥6f (x, z) + f (2x, −3z) − 4f (2x, 0), t∥ 1 ≤ φ(x, x, −z, −z, s) + 2φ(x, x, z, −z, s) + φ(x, x, z, −3z, s) + φ(x, x, −3z, −3z, s) 2 for all x, z, s ∈ X. Replacing z by −z in the above inequality, ∥6f (x, −z) + f (2x, 3z) − 4f (2x, 0), t∥ 1 ≤ φ(x, x, z, z, s) + 2φ(x, x, −z, z, s) + φ(x, x, −z, 3z, s) + φ(x, x, 3z, 3z, s) 2 for all x, z, s ∈ X. By (9) and the above inequality, ∥6f (x, z) − f (2x, 3z) − 2f (2x, 0), t∥ 1 ≤ φ(x, x, z, z, s) + 2φ(x, x, −z, z, s) + φ(x, x, −z, 3z, s) + φ(x, x, 3z, 3z, s) + 3φ(x, x, z, −z, s) 2 for all x, z, s ∈ X. Replacing x by 2j x and z by 3j y in the above inequality and dividing 6j+1 ,

1

f (2j x, 3j y) − 1 f (2j+1 x, 3j+1 y) − 2 f (2j+1 x, 0), t

6j

j+1 j+1 6 6 [ 1 ≤ j+1 φ(2j x, 2j x, 3j y, 3j y, s) + 2φ(2j x, 2j x, −3j y, 3j y, s) 6 ] 1 j j j j+1 j j j j j j j+1 j+1 +φ(2 x, 2 x, −3 y, 3 y, s) + φ(2 x, 2 x, 3 y, 3 y, s) + 3φ(2 x, 2 x, 3 y, −3 y, s) 2 for all x, y, s ∈ X. For given integers l, m(0 ≤ l < m),



m−1 ∑ 2

1

1 l l m m j+1

f (2 x, 3 y) − f (2 x, 3 y) − f (2 x, 0), t (13)

m j+1 l 6 6

6

j=l m−1 ∑ 1 [ φ(2j x, 2j x, 3j y, 3j y, s) + 2φ(2j x, 2j x, −3j y, 3j y, s) ≤ 6j+1 j=l

1 +φ(2j x, 2j x, −3j y, 3j+1 y, s) + φ(2j x, 2j x, 3j+1 y, 3j+1 y, s) + 3φ(2j x, 2j x, 3j y, −3j y, s) 2

]

for all x, y, s ∈ X. By (14) and (13), the sequence { 61j f (2j x, 3j y)} is a Cauchy sequence for all x, y ∈ X. Since Y is complete, the sequence { 61j f (2j x, 3j y)} converges for all x, y ∈ X. Define F : X × X → Y by 1 F (x, y) := lim j f (2j x, 3j y) j→∞ 6 for all x, y ∈ X.

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APPROXIMATE CAUCHY-JENSEN AND BI-QUADRATIC MAPPINGS IN 2-BANACH SPACES

By (6),

5

( ) j (z + w)

1 3 1 1 j

f 2 (x + y), − j f (2j x, 3j z) − j f (2j x, 3j w)

6j 2 6 6

1 1 1 − j f (2j y, 3j z) − j f (2j y, 3j w), t ≤ j φ(2j x, 2j y, 3j z, 3j w, s)

6 6 6

for all x, y, z, w, s ∈ X. Letting j → ∞ and using (14), F satisfies (3). By Theorem 4 in [6], F is a Cauchy-Jensen mapping. Setting l = 0 and taking m → ∞ in (13), one can obtain the inequality (7). If G : X × X → Y is another Cauchy-Jensen mapping satisfying (7), 1 ∥F (2n x, 3n y) − G(2n x, 3n y), t∥ ∥F (x, y) − G(x, y), t∥ = 6n 1 ≤ ∥F (2n x, 3n y) − f (2n x, 0) − f (2n x, 3n y), t∥ 6n 1 + n ∥f (2n x, 2n y) + f (2n x, 0) − G(2n x, 3n y), t∥ 6 2 ≤ φ(2 ˜ n x, 2n x, 3n y, 3n y, s) → 0 as n → ∞ 6n for all x, y, s ∈ X. Hence the mapping F is the unique Cauchy-Jensen mapping, as desired. □ Corollary 8. Let ε > 0. Suppose that f : X × X → Y is a surjective mapping satisfying

(

)

2f x + y, z + w − f (x, z) − f (x, w) − f (y, z) − f (y, w), t ≤ ε,

2 ∑ 2 j+1 x, 0), t∥ < ∞ for all x, y, z, w, s ∈ X, where t = f (s) and φε (x, s) := 32 ε+∥f (x, 0), t∥+ ∞ j=0 6j+1 ∥f (2 for all x, s ∈ X, where t = f (s). Then there exists a unique Cauchy-Jensen mapping F : X × X → Y such that ∥f (x, y) − f (x, 0) − F (x, y), t∥ ≤ φε (x, s) for all x, y, s ∈ X, where t = f (s). Proof. Taking φ(x, y, z, w, s) := ε in Theorem 7, we have ∞

∑ 2 3 φ(x, ˜ x, y, y, s) = ε + ∥f (x, 0), t∥ + ∥f (2j+1 x, 0), t∥ = φε (x, s) 2 6j+1 j=0

for all x, y, z, w, s ∈ X, where t = f (s). □ 3. Approximate bi-quadratic mappings From now on, let φ : X 5 → [0, ∞) be a function satisfying (14)

φ(x, ˜ y, z, w, s) :=

∞ ∑ j=0

1 φ(2j x, 2j y, 2j z, 2j w, s) < ∞ 16j+1

for all x, y, z, w, s ∈ X.

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Theorem 9. Let f : X × X → Y be a surjective mapping such that (15)

∥f (x + y, z + w) + f (x + y, z − w) + f (x − y, z + w) + f (x − y, z − w) −4[f (x, z) − f (x, w) − f (y, z) − f (y, w)], t∥ ≤ φ(x, y, z, w, s)

and let f (x, 0) = 0 and f (0, y) = 0 for all x, y, z, w, s ∈ X, where t = f (s). Then there exists a unique bi-quadratic mapping F : X × X → Y such that (16)

∥f (x, y) − F (x, y), t∥ ≤ φ(x, ˜ x, y, y, s)

for all x, y, s ∈ X, where t = f (s). Proof. Let t = f (s). Putting y = x and w = z in (15), we have



f (x, z) − 1 f (2x, 2z), t ≤ 1 φ(x, x, z, z, s)

16 16 for all x, z, s ∈ X. Thus we obtain

1

1 1 j j j+1 j+1 j j j j

16j f (2 x, 2 z) − 16j+1 f (2 x, 2 z), t ≤ 16j+1 φ(2 x, 2 x, 2 z, 2 z, s) for all x, z, s ∈ X and all j. Replacing z by y in the above inequality, we see that

1

1 j j j+1 j+1

≤ 1 φ(2j x, 2j x, 2j y, 2j y, s) f (2 x, 2 y) − f (2 x, 2 y), t

16j

16j+1 16j+1 for all x, y, s ∈ X and all j. For given integers l, m(0 ≤ l < m), we get

m−1

1

∑ 1 1 l l m m

≤ f (2 x, 2 y) − f (2 x, 2 y), t φ(2j x, 2j x, 2j y, 2j y, s) (17)

16l

16m 16j+1 j=l

for all x, y, s ∈ X. By (17), the sequence { 161j f (2j x, 2j y)} is a Cauchy sequence for all x, y ∈ X. Since Y is complete, the sequence { 161j f (2j x, 2j y)} converges for all x, y ∈ X. Define F : X × X → Y by 1 f (2j x, 2j y) j→∞ 16j

F (x, y) := lim

for all x, y ∈ X. By (15), we have

1 ( j ) ) 1 ( j j j

16j f 2 (x + y), 2 (z + w) + 16j f 2 (x + y), 2 (z − w) ) ) 1 ( 1 ( + j f 2j (x − y), 2j (z + w) + j f 2j (x − y), 2j (z − w) 16 16

4 4 4 4 j j j j j j j j − j f (2 x, 2 z) − j f (2 x, 2 w) − j f (2 y, 2 z) − j f (2 y, 2 w), t

16 16 16 16 1 ≤ j φ(2j x, 2j y, 2j z, 2j w, s) 16 for all x, y, z, w, s ∈ X and all j. Letting j → ∞ and using (14), we see that F satisfies (4). By Theorem 4 in [5], we obtain that F is bi-quadratic. Setting l = 0 and taking m → ∞ in (17), one

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APPROXIMATE CAUCHY-JENSEN AND BI-QUADRATIC MAPPINGS IN 2-BANACH SPACES

7

can obtain the inequality (16). If G : X × X → Y is another bi-quadratic mapping satisfying (16), we obtain ∥F (x, y) − G(x, y), t∥ 1 = n ∥F (2n x, 2n y) − G(2n x, 2n y), t∥ 16 1 1 ≤ n ∥F (2n x, 2n y) − f (2n x, 2n y), t∥ + n ∥f (2n x, 2n y) − G(2n x, 2n y), t∥ 16 16 2 ˜ n x, 2n x, 2n y, 2n y, s) ≤ n φ(2 16 → 0 as n → ∞ for all x, y, s ∈ X. Hence the mapping F is the unique bi-quadratic mapping, as desired. □ Corollary 10. Let ε > 0. Suppose that f : X × X → Y is a surjective mapping satisfying ∥f (x + y, z + w) + f (x + y, z − w) + f (x − y, z + w) + f (x − y, z − w) −4[f (x, z) − f (x, w) − f (y, z) − f (y, w)], t∥ ≤ ε, for all x, y, z, w, s ∈ X, where t = f (s). Then there exists a unique bi-quadratic mapping F : X × X → Y such that 1 ∥f (x, y) − F (x, y), t∥ ≤ ε 15 for all x, y, s ∈ X, where t = f (s). Proof. Taking φ(x, y, z, w, s) := ε in Theorem 9, we have φ(x, ˜ x, y, y, s) = X, where t = f (s). □

1 15 ε

for all x, y, z, w, s ∈

References [1] H.-Y. Chu, A. Kim and J. Park, On the Hyers-Ulam stabilities of functional equations on n-Banach spaces, Math. Nachr. 289 (2016), 1177–1188. [2] S. Gähler, 2-metrische Räume und ihre topologische Struktur, Math. Nachr. 26 (1963), 115–148. [3] S. Gähler, Lineare 2-normierte Räumen, Math. Nachr. 28 (1964), 1–43. [4] W.-G. Park, Approximate additive mappings in 2-Banach spaces and related topics, J. Math. Anal. Appl. 376 (2011), 193–202. [5] W.-G. Park and J.-H. Bae, On a bi-quadratic functional equation and its stability, Nonlinear Anal. 62 (2005), 643–654. [6] W.-G. Park and J.-H. Bae, On a Cauchy-Jensen functional equation and its stability, J. Math. Anal. Appl. 323 (2006), 634–643. [7] S. M. Ulam, A Collection of Mathematical Problems, Interscience Publishers, New York, 1960. Won-Gil Park, Department of Mathematics Education, College of Education, Mokwon University, Daejeon 35349, Republic of Korea E-mail address: [email protected] Jae-Hyeong Bae, Humanitas College, Kyung Hee University, Yongin 17104, Republic of Korea E-mail address: [email protected]

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Birkhoff Normal Forms, KAM theory and continua of periodic points for certain planar system M. R. S. Kulenovi´c†12 E. Pilav‡ and N. Muji´c‡ †

Department of Mathematics University of Rhode Island, Kingston, Rhode Island 02881-0816, USA ‡

Department of Mathematics University of Sarajevo, Sarajevo, Bosnia and Herzegovina

Abstract. By using the KAM theory and time reversal symmetries we investigate the stability of the equilibrium solutions of the system:  a  xn+1 = xn +y n , n = 0, 1, 2, . . . ,  yn+1 = xn yn where the parameter a > 0, and initial conditions x0 and y0 are positive numbers. We obtain the Birkhoff normal form for this system and prove the existence of periodic points with arbitrarily large periods in every neighborhood of the unique positive equilibrium. We also use the time reversal symmetry method to find effectively some feasible periods and the corresponding periodic orbits. Finally, we give computational procedure for finding an infinite number of periodic solutions with the given period. The second order difference equation obtained by eliminating xn from this system is an equation of the type yn+1 = f (yn , yn−1 ), where f is decreasing in both variables. Such equation can be embedded into fifth order difference equation which is increasing in all its arguments and it exhibits chaotic behavior. Keywords. area preserving map, Birkhoff normal form, difference equation, KAM theory, periodic solutions, symmetry, time reversal, competitive map, global stable manifold, monotonicity, period-two solution.

AMS 2010 Mathematics Subject Classification: 37E40, 37J40, 37N25, 39A28, 39A30

1

Introduction

In this paper we consider the following rational system of difference equations  a  xn+1 = xn +y n , n = 0, 1, 2, . . . ,  yn+1 = xn yn and the corresponding equation yn+1 =

a , yn yn−1 (1 + yn )

(1)

n = 0, 1, 2, . . . ,

(2)

where the parameter a > 0, and initial conditions x0 and y0 are positive numbers. System (1) was first considered in [6], where boundedness of all its solutions was proved using the invariant. We will use this invariant in Section 3 to prove the stability of the unique equilibrium. Equation (2) gives an example of second order difference equation where transition function decreases in both variables and yet equation exhibits complicated dynamics. First such example was given in [5]. We will use similar techniques as in [5] with the addition of the new computational procedure from [7], which uses an invariant of the system to find effectively continua of periodic solutions of certain feasible periods. We will show that the corresponding map can be transformed into an area preserving map and using Birkhoff Normal form we will apply the KAM theorem to prove stability of the unique positive equilibrium and the existence of periodic points with arbitrarily large period in every neighborhood of the unique positive equilibrium. In addition, we will prove that the corresponding map is conjugate to its inverse map through the involution map and then use this conjugacy to find some feasible periods of this map. The method of invariants for proving stability of the equilibrium solution for all values of parameter a will be used along with Morse’s lemma to prove that the level sets of the invariants are diffeomorphic with circles. This method was used successfully in [11, 12] and the KAM theory was used for the same objective in [8, 10, 13, 14]. Let T be the map associated to the system (1), i.e., T 1 2

   a  x . = x+y x y y

(3)

Corresponding author, e-mail: [email protected] Partially supported by Maitland P. Simmons Foundation

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The map (3) has the unique fixed point (¯ y 2 , y¯) in the positive quadrant, where y¯3 (¯ y + 1) = a. An invertible mapping T is area preserving if the area of T (A) coincides with the area of A for all measurable subsets A [9, 12, 18]. We claim that in logarithmic coordinates, i.e., u = ln (x/¯ y 2 ), v = ln (y/¯ y ) the map (3) is area preserving. Lemma 1 The map (3) is area preserving in the logarithmic coordinates. Proof. The Jacobian matrix of the corresponding transformation T is JT (x, y) =

a − (x+y) 2 1 y

with detJT (x, y) =

a − (x+y) 2 x − y2

! (4)

a . y 2 (x + y)

We substitute u = ln (x/¯ y 2 ), v = ln (y/¯ y ) and rewrite the map in (u, v) coordinates to obtain the transformation     ln a − 3 ln y¯ − ln(¯ y eu + ev ) u (5) → u−v v The Jacobian of this transformation is  J(u, v) =

u

y ¯ − euey¯+e v 1

v

− eu y¯e+ev −1

 (6) 2

It is easy to see that detJ(u, v) = 1.

A point (¯ x, y¯) is a fixed point of T if T (¯ x, y¯) = (¯ x, y¯). A fixed point is elliptic if the eigenvalues of JT (¯ x, y¯) form a ¯ on the unit circle and is hyperbolic if the the modulus of the eigenvalues is different from 1, complex conjugate pair λ, λ see [9, 12]. Lemma 2 The map T in the (x, y) coordinates has elliptic fixed point (¯ y 2 , y¯). In the logarithmic coordinates, the corresponding fixed points is (0, 0). Proof. For the fixed points in (x, y) coordinates, solving a/(¯ x + y¯) = x and x ¯/¯ y = y¯ yields the fixed points (¯ y 2 , y¯) where 3 2 y¯ (¯ y + 1) = a. Evaluating the Jacobian matrix (4) of T at (¯ y , y¯) gives ! − 2a 2 − 2a 2 2 y ¯ +¯ y) y ¯ +¯ y) ( ( JT (¯ y , y¯) = (7) 1 −1 y ¯ ¯ where By using a = y¯3 (1 + y¯) we obtain that the eigenvalues of JT (¯ y 2 , y¯) are λ and λ √ −1 − 2¯ y + i 4¯ y+3 λ= . 2¯ y+2

(8)

It is easy to see that |λ| = 1 and so (¯ y 2 , y¯) is an elliptic fixed point. Under the logarithmic coordinate change (x, y) → (u, v), the fixed point (¯ y 2 , y¯) becomes (0, 0). Evaluating the Jacobian matrix (6) of T at (0, 0) gives   y ¯ 1 − y¯+1 − y¯+1 J(0, 0) = (9) 1 −1 2

with eigenvalues which are given by (8).

This paper is organized as follows. In section 2 the KAM theorem is explained in some detail and Birkhoff normal form for map T is derived. By using the KAM theory stability of the unique equilibrium and existence of infinite number of periodic solution is proven except for a single value of the parameter a. Section 3 uses the invariant of the equation (2) in proving stability for all values of a. In section 4 by using symmetries it is shown that the map T is conjugate to its inverse through an involution. Then by using time reversal symmetry method some feasible periods and corresponding orbits of the map T are found. Finally in Section 5 we use the recent method of Gasull and al. [7] to find continua of p-periodic points lying on the level sets of the invariant I. The method is based on use of resultants and is implemented by Mathematica. The special attention is given to period-seven solution.

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2

The KAM theory and Birkhoff normal form

The KAM Theorem asserts that in any sufficiently small neighborhood of a non degenerate elliptic fixed point of a smooth area-preserving map there exists many invariant closed curves.We explain this theorem in some detail. Consider a smooth, area-preserving mapping (x, y) → T (x, y) of the plane that has (0, 0) as an elliptic fixed point. After a linear transformation one can put the map in the form z → λz + g(z, z¯) where λ is the eigenvalue of the elliptic fixed point, z = x + iy and z¯ = x − iy are complex variables, and g vanishes with its derivative at z = 0. Assume that the eigenvalue λ of the elliptic fixed point satisfies the non-resonance condition ¯ λk 6= 1 for k = 1, . . . , q, for some q ≥ 4. Then Birkhoff showed that there exist new, canonical complex coordinates (ζ, ζ) relative to which the mapping takes the normal form ¯ ¯ ζ → λζeiτ (ζ ζ) + h(ζ, ζ) ¯ = τ1 |ζ|2 +. . .+τs |ζ|2s is a real polynomial, s = [(q −2)/2], and h in a neighborhood of the elliptic fixed point, where τ (ζ ζ) vanishes with its derivatives up to order q − 1. The numbers τ1 , . . . , τs are called twist coefficients. Consider an invariant annulus  < |ζ| < 2 in a neighborhood of the elliptic fixed point, for  a very small positive number. Note that under ¯ the neglect of the remainder h, the normal form approximation ζ → λζeiτ (ζ ζ) leaves invariant all circles |ζ|2 = const. The motion restricted to each of these circles is a rotation by some angle. Also note that if at least one of the twist coefficients τj is nonzero, the angle of rotation will vary from circle to circle. A radial line through the fixed point will undergo twisting under the mapping. The KAM theorem (Moser’s twist theorem) says that, under the addition of the remainder term, most of these invariant circles will survive as invariant closed curves under the full map.

¯ is not identically zero and  is sufficiently small, then the map T has a set of invariant Theorem 1 Assuming that τ (ζ ζ) closed curves of positive Lebesque measure close to the original invariant circles. Moreover the relative measure of the set of surviving invariant curves approaches full measure as  approaches 0. The surviving invariant closed curves are filled with dense irrational orbits. The KAM theorem requires that the elliptic fixed point be non-resonant and non degenerate. Note that for q = 4 the non-resonance condition λk 6= 1 requires that λ 6= ±1 or λ 6= ±i. The above normal form yields the approximation ζ → λζ + c1 ζ 2 ζ¯ + O(|ζ|4 ) with c1 = iλτ1 and τ1 being the first twist coefficient. We will call an elliptic fixed point non-degenerate if τ1 6= 0. ¯ satisfying |λ| = 1 Consider a general map T that has a fixed point at the origin with complex eigenvalues λ and λ and Im(λ) 6= 0. By putting the linear part of such a map into Jordan Canonical form, we may assume T to have the following form near the origin        x1 Re(λ) −Im(λ) x1 g1 (x1 , x2 ) T = + (10) x2 Im(λ) Re(λ) x2 g2 (x1 , x2 ) One can now switch to the complex coordinates z = x1 + ix2 to obtain the complex form of the system z → λz + ξ20 z 2 + ξ11 z z¯ + ξ02 z¯2 + ξ30 z 3 + ξ21 z 2 z¯ + ξ12 z z¯2 + ξ03 z¯3 + O(|z|4 ) The coefficient c1 can be computed directly using the formula below derived by Wan in the context of Hopf bifurcation theory [19]. In [16] it is shown that when one uses area-preserving coordinate changes this formula by Wan yields the twist coefficient τ1 that is used to verify the non-degeneracy condition necessary to apply the KAM theorem. We use the formula: c1 = where

¯ + 2λ − 3) ξ20 ξ11 (λ |ξ11 |2 2|ξ02 |2 + + 2 ¯ + ξ21 2 ¯ ¯ (λ − λ)(λ − 1) 1−λ λ −λ

(11)

1 {(g1 )x1 x1 − (g1 )x2 x2 + 2(g2 )x1 x2 + i [(g2 )x1 x1 − (g2 )x2 x2 − 2(g1 )x1 x2 ]} , 8 1 ξ11 = {(g1 )x1 x1 + (g1 )x2 x2 + i [(g2 )x1 x1 + (g2 )x2 x2 ]} , 4 1 = {(g1 )x1 x1 − (g1 )x2 x2 − 2(g2 )x1 x2 + i [(g2 )x1 x1 − (g2 )x2 x2 + 2(g1 )x1 x2 ]} , 8

ξ20 =

ξ02 ξ21 =

1 {(g1 )x1 x1 x1 + (g1 )x1 x2 x2 + (g2 )x1 x1 x2 + (g2 )x2 x2 x2 + i [(g2 )x1 x1 x1 + (g2 )x1 x2 x2 − (g1 )x1 x1 x2 − (g1 )x2 x2 x2 ]} . 16

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Theorem 2 The elliptic fixed point (0, 0), in the (u, v) coordinates, is non-degenerate for a 6= a > 0.

3 16

and non-resonant for

Proof. Let F be the function defined by F

    ln a − 3 ln y¯ − ln(¯ y eu + ev ) u . = u−v v

(12)

Then F has the unique elliptic fixed point (0, 0). The Jacobian matrix of F is given by   u v y ¯ − euey¯+e − eu y¯e+ev v JF (u, v) = . 1 −1

(13)

At (0, 0), JF (u, v) has the form  J0 = JF (0, 0) =

y ¯ − y¯+1 1

1 − y¯+1 −1

 .

(14)

¯ where The eigenvalues of (14) are λ and λ λ=

√ −1 − 2y + i 4y + 3 . 2y + 2

(15)

One can prove that |λ| = 1, √ i (2¯ y + 1) 4¯ y+3 2¯ y −1 − , λ2 = 2 (¯ y + 1)2 2 (¯ y + 1)2 √ y + 3 (3¯ y + 2) y¯ ((3 − 2¯ y ) y¯ + 6) + 2 i¯ y 4¯ λ3 = + , 2 (¯ y + 1)3 2 (¯ y + 1)3
√ i (2¯ y + 1) 4¯ y + 3 2¯ y2 − 1 2¯ y (¯ y ((¯ y − 4) y¯ − 8) − 4) − 1 λ4 = − , 2 (¯ y + 1)4 2 (¯ y + 1)4 2

from which follows that λk 6= 1 for k = 1, 2, 3, 4 and a > 0. Then we have that        y ¯ 1 − y¯+1 − y¯+1 u u f1 (δ, u, v) F = + , v v f2 (δ, u, v) 1 −1

(16)

(17)

where f1 (δ, u, v) = − ln (eu y¯ + ev ) +

u¯ y v + − 3 ln y¯ + ln a y¯ + 1 y¯ + 1

(18)

f2 (δ, u, v) =0. The system (un+1 , vn+1 ) = F (un , vn ) takes the form        y ¯ 1 − y¯+1 − y¯+1 un+1 un f1 (un , vn ) = + , vn f2 (un , vn ) vn+1 1 −1 Let



un vn



 =P

 u ˜n , v˜n

where 1 2¯ y +2

1 P = √ D and P −1 =

√ D

−

√ 4¯ y +3 2¯ y +2

1 

1

y +2 − √2¯ 4¯ y +3

√ 1 4¯ y +3

D=

!

0 0

with

(19)

 ,

√ 4¯ y+3 . 2¯ y+2

Then the system (un+1 , vn+1 ) = F (un , vn ) becomes 

u ˜n+1 v˜n+1

 =

−2¯ y −1 2¯ y +2 √ 4¯ y +3 2¯ y +2

√

4¯ y +3 2¯ y +2 −2¯ y −1 2¯ y +2

−

473

!

u ˜n v˜n



+ P −1 H

   u ˜n P , v˜n

(20)

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where

    f1 (u, v) u . := H f2 (u, v) v

Let G

       u g1 (u, v) u . = P −1 H P = v g2 (u, v) v

The straightforward calculation yields g1 (u, v) =0 1 g2 (u, v) = − √ ln D





a y¯3

−

 u−2Dv(y+1)  ¯ √ y¯ (u − 2Dv (¯ y + 1)) 1 u √u ¯ + √ ln y¯e 2 D(y+1) +e D − . 2 D (¯ y + 1) 2D (¯ y + 1) D

(21)

By straightforward calculation we obtain that
√
√ 4¯ y + 3 + i¯ y + 4¯ y+3−i p , 8 (¯ y + 1)3 D (4¯ y + 3) i¯ y p , = 2 (¯ y + 1) D (4¯ y + 3)

√ √ i¯ y 2¯ y y¯ + i 4¯ y + 3 + i 4¯ y+3−1 p = , 8 (¯ y + 1)3 D (4¯ y + 3)
√ (¯ y − 1) y¯ 2i¯ y + 4¯ y+3+i = . 3√ 16D (¯ y + 1) 4¯ y+3

ξ20 |u=v=0 = ξ11 |u=v=0 ξ02 |u=v=0 ξ21 |u=v=0

y¯ 2¯ y

(22)

Since

ξ21 ξ11 =

y i¯ y 2 2¯

√


√
4¯ y + 3 + i¯ y + 4¯ y+3−i

16D (¯ y + 1)4 (4¯ y + 3)

ξ11 ξ11 =

y¯2 , 4D (¯ y + 1)2 (4¯ y + 3)

ξ02 ξ02 =

y¯2 . 16D (¯ y + 1)2 (4¯ y + 3)

, (23)

the simplification of the expression for c1 yields

c1 = =

¯ + 2λ − 3) |ξ11 |2 2|ξ02 |2 ξ20 ξ11 (λ + + ¯ − 1) ¯ ¯ + ξ21 (λ2 − λ)(λ 1−λ λ2 − λ
√ y+3+i y¯ (2¯ y − 1) (2¯ y + 2) 2i¯ y + 4¯

(24)

8 (¯ y + 1)2 (4¯ y + 3)2

One can prove that ¯ 1=− τ1 = −iλc which implies that τ1 6= 0 for a 6=

3 16

y¯ (2¯ y − 1) , 2 (4¯ y + 3)2

since y¯2 (1 + y¯) = a. 2

The following result is a consequence of Moser’s twist map theorem [8, 15, 17, 18]. 3 Theorem 3 Let T be a map (3) associated to the system (1), and (¯ x, y¯) a non-degenerate elliptic fixed point. If a 6= 16 then there exist periodic points with arbitrarily large period in every neighbourhood of (¯ x, y¯). In adition, (¯ x, y¯) is a stable fixed point.

3

Invariant

3 In this section we prove that the restriction a 6= 16 is not necessary for stability of the equilibrium solution. The system (1) possesses the invariant given by

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(a)

(b)

Figure 1: Some orbits of the map T for (a) a = 0.5 and (b) a = 10.0

a xn + . (25) xn yn Indeed, it is easy to see that I is continuous and that I(xn+1 , yn+1 ) = I(xn , yn ). In this section we use the invariant I to find a Lyapunov function and prove stability of the equilibrium point for all values of parameter a > 0, see[11, 12]. The partial derivatives of the function I(x, y) are given with I(xn , yn ) = xn + yn +

∂I ∂x ∂I ∂y

a 1 + + 1, x2 y x 1− 2. y −

= =

(26)

The unique positive equilibrium of (1) satisfies that x ¯ = y¯2 and y¯3 (¯ y + 1) = a. Equation (26) implies that any critical point (x, y) of (25) satisfies the system x

=

y2

y4 + y3

=

a.

2

Hance, (¯ y , y¯) of (1) is the unique positive solution of this system and (¯ y 2 , y¯) is critical point of the invariant (25). Thus the unique equilibrium (¯ y 2 , y¯) is critical point of the invariant (25). Lemma 3 The graph of the function I(x, y) associated with (25) is a simple closed curve in a neighborhood of the equilibrium point of (1). The equilibrium point (¯ y 2 , y¯) is stabile. Proof. The Hessian matrix assosiated with I(x, y) is H(x, y) =

2a x3 − y12

with determinant det(H(x, y)) =

− y12

!

2x y3

4ay − x2 . x2 y 4

For the equilibrium (¯ y 2 , y¯) we have det(H(¯ y 2 , y¯)) =

4¯ y (¯ y 4 + y¯3 ) − y¯4 4a¯ y−x ¯2 4¯ y 5 + 3¯ y4 4¯ y+3 = = = >0 x ¯2 y¯4 y¯8 y¯8 y¯4

Thus, in view of Morse’s lema, [9], the level sets of the function I(x, y) are diffeomorphic to circles in the neighborhood of (¯ x, y¯). In adition, the function V (x, y) = I(x, y) − I(¯ x, y¯) 2

is Lyapunov function, and so the equilibrium point (¯ x, y¯) is stable, see [11].

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4

Symmetries

In this section we will show that mat T is conjugate to its inverse map and use this conjugacy to find some feasible periods of T and corresponding periodic orbits. A transformation R of the plane is said to be a time reversal symmetry for T if R−1 ◦ T ◦ R = T −1 . If the time reversal symmetry R is an involution, i.e. R2 = I, where I is identity map then the time reversal symmetry condition is equivalent to R ◦ T ◦ R = T −1 , and T can be written as the composition of two involutions T = I1 ◦ I0 where I0 = R and I1 = T ◦ R. Let us note here that if I0 = R is reversor then so is I1 = T ◦ R. Also, the j th involution defined as Ij = T j ◦ R is also a reversor. The invariant sets of the involution maps S0,1 = {(x, y)|I0,1 (x, y) = (x, y)} are one-dimensional sets called the symmetry lines of the map. When the sets S0,1 are known the search for periodic orbits can be reduced to one-dimensional root finding problem using the following result, see [2, 8] Theorem 4 If (x, y) ∈ S0,1 then T n (x, y) = (x, y) if and only if  T n/2 (x, y) ∈ S0,1 , (n±1)/2 T (x, y) ∈ S1,0 ,

(a)

for n even for n odd.

(b)

(c)

Figure 2: a) The first fourteen iterations of symmetry line S0 of the map T for a = 0.02 (b) The first twelve iterations of symmetry line S1 of the map T for a = 0.02 (c) The periodic orbits of period 14 (blue) and 17 (red) The inverse map of the map T is T −1 (x, y) = 

The involution R = x,

x y



ay a , x(y + 1) x(y + 1)

 .



is reversor for T . Indeed,       x ay ay a (R ◦ T ◦ R)(x, y) = (R ◦ T ) x, =R ,y = , = T −1 (x, y). y x(y + 1) x(y + 1) x(y + 1)

Thus T = I1 ◦ I0 where I0 (x, y) = R(x, y) and  I1 (x, y) = T ◦ R =

 ay ,y . x(y + 1)

The symmetry lines corresponding to I0 and I1 are S0 = {(x, y) : x = y 2 },

S1 = {(x, y) : ay = x2 (y + 1)}.

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Periodic orbits of different orders can be found at the intersection of the symmetry lines Sj , j = 1, 2, ... associated to the j th involution. So if (x, y) ∈ Sj ∩ Sk then T j−k (x, y) = (x, y). The symmetry lines are also related to each other by the relation S2j+i = T j (Si ), S2j−i = Ij (Si ), ∀i, j. √ Now we start with the point (x0 , x0 ) ∈ S0 in search for periodic orbits on the symmetry line S0 with even period n and impose that (xn/2 , yn/2 ) ∈ S0 , where (xn/2 , yn/2 ) = T n/2 (x0 ,

√

x0 ).

2 This reduces to one-dimensional root finding for the equation xn/2 = yn/2 , where the unknown is x0 . Periodic orbits on S0 with odd period n are obtained by solving for x0 the equation

ay(n+1)/2 = x2(n+1)/2 (y(n+1)/2 + 1), where (x(n+1)/2 , y(n+1)/2 ) = T (n+1)/2 (x0 ,

√

x0 ).

For example, for a = 0.02 in Figure 2 we have an intersection between the symmetry lines S0 and S14 = T 7 (S0 ), S4 = T 2 (S0 ) and S18 = T 9 (S0 ), S5 = T 2 (S1 ) and S19 = T 9 (S1 ), and S11 = T 5 (S1 ) and S25 = T 12 (S1 ) of the map T . The intersection points of these lines correspond to the periodic orbits of period 14.

5

Continua of periodic points for map T

In this section we use resultants and technique from [7] for finding continua of p-periodic points lying on the level sets of the invariant I. Let T p (x, y) = (T1p (x, y), T2p (x, y)) . The idea is to find the values of h for which the system T1p (x, y) = x

(27)

I(x, y) = h has continua of solutions. Let F (y, h) := Res(numerator (T1p (x, y) − x), numerator (I(x, y) − h)),

(28)

where Res denote the resultant of corresponding expressions. The values of h have to be such that F (y, h) vanishes identically. We need to collect the factors of the above resultant that only depend on h. Denote by Dp (a, h) the product of these factors. We introduce the functions dp (a, h) as those factors of Dp (a, h) that remain after removing from this polynomial all the factors that already appear in some Dk (a, h) where k is either 1 or a proper divisor of p. We call the conditions dp (a, h) = 0 the resultant p-periodicity conditions associated to the invariant I (RPC from now on), see [7]. The main fact is that the energy levels filled with periodic points must satisfy the RPC what gives us the necessary condition for periodic point because the resultant (28) can contain some spurious factors. We will prove in our examples that the RPC we obtain actually give continua of p-periodic points. Theorem 5 The RPC of the map T associated to the invariant I for p ≤ 10 are given by dp (a, h) = 0, where: d2 (a, h) = a d3 (a, h) = 1 d4 (a, h) = 1 + h d5 (a, h) = a − h − 1 d6 (a, h) = a − h2 − 3h − 2 d7 (a, h) = a2 − ah − a − h3 − 3h2 − 3h − 1 d8 (a, h) = 2a2 − ah2 − 5ah − 4a + h2 + 2h + 1 d9 (a, h) = −3 + 4a − 3a2 + a3 − 12h + 9ah − 3a2 h − 19h2 + 6ah2 − 15h3 + ah3 − 6h4 − h5 d10 (a, h) = 1 + 5a − 5a2 + a3 + 5h + 15ah − 8a2 h + 10h2 + 16ah2 − 3a2 h2 + 10h3 + 7ah3 + 5h4 + ah4 + h5

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Proof. For p = 2 we obtain numerator(T12 (x, y) − x) = −x3 − x2 y + ay 2 , and Res(numerator(T12 (x, y) − x), numerator(I(x, y) − h), x) = ay 3 (a2 + 4ay + 4ahy + 4ay 2 + 3ahy 2 − h2 y 2 − h3 y 2 + 2ay 3 + 2hy 3 + 2h2 y 3 − y 4 + ay 4 − hy 4 ). So there is no factor of resultant without dependence on the variable y that can be equal to zero since a > 0 so we can ensure there are no energy levels formed by continua of period-two points. We come to the same conclusion for p = 3, so we continue with p = 4 where we have numerator(T14 (x, y) − x) = −x7 + ax4 y − ax5 y − 4x6 y − x7 y + 2a2 x2 y 2 + 2ax3 y 2 − 2ax4 y 2 − 6x5 y 2 − 4x6 y 2 + a3 y 3 + 2a2 xy 3 + ax2 y 3 + a2 x2 y 3 − ax3 y 3 − 4x4 y 3 − 6x5 y 3 + a2 xy 4 − x3 y 4 − 4x4 y 4 − x3 y 5 , and Res(numerator(T14 (x, y) − x), numerator(I(x, y) − h), x) = a3 (1 + h)2 y 8 (1 + y) (a − 2a2 + a3 + 2ah − 2a2 h + ah2 + 8ay − 4a2 y + 20ahy − 4a2 hy + 16ah2 y + 4ah3 y + 8ay 2 − 4a2 y 2 − hy 2 + 22ahy 2 − 5a2 hy 2 − 4h2 y 2 + 19ah2 y 2 − 6h3 y 2 + 5ah3 y 2 − 4h4 y 2 − h5 y 2 + 2y 3 − 4ay 3 + 2a2 y 3 + 8hy 3 − 6ahy 3 + 12h2 y 3 − 2ah2 y 3 + 8h3 y 3 + 2h4 y 3 − y 4 − ay 4 + a2 y 4 − 3hy 4 − ahy 4 − 3h2 y 4 − h3 y 4 ). The only factor independent of y is 1 + h which gives d4 (a, h) = 1 + h. In an analogous way we compute d5 (a, h) and d6 (a, h). For p = 7 we consider the equation T 4 (x, y) = T −3 (x, y) so we obtain Res(numerator(T14 (x, y) − T1−3 (x, y)), numerator(I(x, y) − h), x) = a4 (−1 − a + a2 − 3h − ah − 3h2 − h3 )2 y 11 (1 + y)3 (a + 4ay + 4ay 2 − hy 2 − h2 y 2 + 2y 3 + 2hy 3 ), and d7 (a, h) = −1 − a + a2 − 3h − ah − 3h2 − h3 . The computation of d8 (a, h), d9 (a, h) and d10 (a, h) is analogous to the previous computation. 2 Let us now determine the feasibility region R of a map T , that is those pairs (a, h) ∈ R2 that satisfy the condition {I(x, y) = h} ∩ R2 = {x2 + ay − hxy + x2 y + xy 2 = 0} ∩ R2 6= 0. From the property of the invariant I in Lemma 3 we obtain that the equilibrium point (¯ y 2 , y¯) is the absolute minimum of the invariant (25). Let us denote the value of I in the absolute minimum with hc (¯ y ) = I(¯ y 2 , y¯) = y¯(2¯ y + 3) and therefore the region R = {(a, h), a > 0 and h ≥ hc (¯ y ) and a = y¯3 (¯ y + 1)} is a feasibility region for the map T.

5.1

Analysis of the 7-periodic RPC

In this section we will determine the number of the level curves associated to the 7-periodic RPC. We will use the following Lemma from [7], Lemma 4 Let Ga (h) = gn (a)hn + gn−1 (a)hn−1 + ... + g1 (a)h + g0 (a), be a family of real polynomials depending also polynomially on a real parameter a. Set Ia = (φ(a), +∞) where φ(a) is a continuous function. Suppose that there exists an open interval Λ ⊂ R such that i) There exists a0 ∈ Λ such that Ga0 (h) has exactly r ≥ 0 simple roots in Ia0 . ii) For all a ∈ Λ, Ga (φ(a)) · gn (a) 6= 0. iii) For all a ∈ Λ, ∆h (Ga ) 6= 0, where ∆h (Ga ) is disriminant of the polynomial Ga (h). Then for all a ∈ Λ, Ga (h) has exactly r ≥ 0 simple roots in Ia .

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The disriminant ∆h (Ga ) of the polynomial Ga (h) is given as ∆h (Ga ) = (−1)

n(n−1) 2

1 Res(Ga (h), G0a (h), h). an

Let us now for the sake of the convenience rewrite d7 (a, h) as one-parametric family of polynomials in h depending on the parameter y¯ where a = y¯3 (1 + y¯): Gy¯ (h) := d7 (a, h) = g3 (¯ y )h3 + g2 (¯ y )h2 + g1 (¯ y )h + g0 (¯ y ),
where g3 (¯ y ) = −1, g2 (¯ y ) = −3, g1 (¯ y ) = −3 − y¯3 (1 + y¯) and g0 (¯ y ) = y 3 (y + 1) y 4 + y 3 − 1 − 1. Since (a, h) ∈ R if and 3 only if h ∈ [hc (¯ y ), +∞) = [¯ y (2¯ y + 3), +∞) and a = y¯ (1 + y¯), we have to study the number of real roots of Gy¯ (h) = 0 in the feasibility region. Let us note that
∆h (Gy¯ (h)) = y¯9 (¯ y + 1)3 27¯ y 4 + 27¯ y 3 + 4 > 0, so hypotheses (iii) of the Lemma 4 is satisfied. Further, according to Lemma 4, since g3 (¯ y ) 6= 0, the number of real simple roots in Gy¯ (h) is constant on any open interval where Ga (hc (¯ y )) 6= 0. We have
Gy¯ (¯ y (2¯ y + 3)) = (¯ y + 1)5 y¯3 − 3¯ y 2 − 4¯ y−1 , and it vanishes in y¯0 ≈ 4.04892, which is the only positive root of y¯3 − 3¯ y 2 − 4¯ y − 1 = 0. Hence, the map T has a constant number of real roots in Iy¯ = [hc (¯ y ), +∞) = [¯ y (2¯ y + 3), +∞) for y¯ in each of the intervals (0, y¯0 ) and (¯ y0 , ∞). So we have to determine the number of roots of Gy¯ in Iy¯ = [¯ y (2¯ y + 3), +∞) for the intervals (0, y¯0 ) and (¯ y0 , +∞). We can reduce the problem to study one concrete value of y¯ in each of the intervals mentioned above. Let us consider the value y¯ = 2 ∈ (0, y¯0 ). We can compute G2 (h) = −h3 − 3h2 − 27h + 551, and it is easy to see that G2 (h) has no simple roots in Iy¯ . By Lemma 4 we have that Gy¯ (h) has no simple roots in Iy¯ = [¯ y (2¯ y + 3), +∞) for y¯ ∈ (0, y¯0 ), i.e. for a ∈ (0, y¯03 (¯ y0 + 1)). Similarly, one can see that Gy¯ (h) has one simple root in Iy¯ = [¯ y (2¯ y + 3), +∞) for y¯ ∈ (¯ y0 , +∞), i.e. a ∈ (¯ y03 (¯ y0 + 1), +∞). From the previous discussion we obtain the following theorem: Theorem 6 Consider the map T given by (3) with positive parameter a and value a0 = y¯03 (¯ y0 + 1) ≈ 335.13213. The set of real 7-periodic points is empty set for a ∈ (0, a0 ) and it is given by smooth non-empty level sets Ia (x, y) = h for the values of h satisfying d7 (a, h) = 0 for a > a0 , with d7 given in Theorem 5 and it is formed by one closed curve diffeomorphic to S1 .

References [1] A. M. Amleh, E. Camouzis, and G. Ladas, On the Dynamics of a Rational Difference Equation, Part I, Int. J. Difference Equ. 3(2008), 1–35. [2] D. del-Castillo-Negrete, J. M. Greene, E. J. Morrison, Area preserving nontwist maps: periodic orbits and transition to chaos, Physica D, 91(1996), 1–23. [3] A. Cima, A. Gasull, V. Ma˜ nosa, Studying discrete dynamical systems through differential equations, J. Differential Equations 244 (2008), 630-648. [4] A. Cima, A. Gasull, V. Ma˜ nosa, Non-autonomus 2-periodic Gumovski-Mira difference equations, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 22, (2012), 1250264, 14 pp. [5] E. Denette, M. R. S. Kulenovi´c and E. Pilav, Birkhoff normal forms, KAM theory and time reversal symmetry for certain rational map, Mathematics, MDPI, 2016; 4(1):20. [6] E. Drymmonis, E. Camouzis, G. Ladas, G. and W. Tikjha, Patterns of boundedness of the rational system xn+1 = +β2 xn +γ2 yn α1 and yn+1 = Aα22+B . J. Difference Equ. Appl. 18 (2012), 89–110. A1 +B1 xn +C1 yn 2 xn +C2 yn [7] A. Gasull, M. Liorens, V. Ma˜ nosa, Continua of periodic points for planar integrable rational maps, Int. J. Difference Equ., 11(2016), 37–63. [8] M. Gidea, J. D. Meiss, I. Ugarcovici, H. Weiss, Applications of KAM Theory to Population Dynamics, J. Biological Dynamics 5:1,44-63 (2011). [9] J. K. Hale and H. Kocak, Dynamics and Bifurcation, Springer-Verlag, New York, (1991). [10] V. L. Kocic, G. Ladas, G. Tzanetopoulos, and E. Thomas, On the stability of Lyness’ equation, Dynam. Contin. Discrete Impuls. Systems, 1(1995), 245–254.

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[11] M. R. S. Kulenovi´c, Invariants and related Liapunov functions for difference equations, Appl. Math. Lett. 13(2000), 1-8. [12] M. R. S. Kulenovi´c and O. Merino, Discrete Dynamical Systems and Difference Equations with Mathematica, Chapman and Hall/CRC, Boca Raton, London, 2002. [13] M. R. S. Kulenovi´c and Z. Nurkanovi´c, Stability of Lyness’ Equation with Period-Two Coefficient via KAM Theory, J. Concr. Appl. Math., 6(2008), 229-245. [14] G. Ladas, G. Tzanetopoulos, and A. Tovbis, On May’s host parasitoid model, J. Difference Equ. Appl. 2 (1996), 195–204. [15] R. S. MacKay, Renormalization in Area-Preserving Maps, World Scientific, River Edge, NJ, 1993. [16] R. Moeckel, Generic bifurcations of the twist coefficient, Ergodic Theory Dyn. Syst. 10(1) (1990), pp. 185–195. [17] C. Siegel and J. Moser, Lectures on Celestial Mechanics, Springer-Varlag, New York, 1971. [18] M. Tabor, Chaos and integrability in nonlinear dynamics. An introduction. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1989. [19] Y. H. Wan, Computation of the stability condition for the Hopf bifurcation of diffeomorphisms on R2 , SIAM J. Appl. Math. 34(1) (1978), pp. 167–175.

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Durrmeyer type (p, q)-Baskakov operators for functions of one and two variables Qing-Bo Caia and Guorong Zhoub,∗ a

School of Mathematics and Computer Science, Quanzhou Normal University, Quanzhou 362000, China b

School of Applied Mathematics, Xiamen University of Technology, Xiamen 361024, China E-mail: [email protected], [email protected].

Abstract. In this paper, we construct a generalization of Durrmeyer type Baskakov operators based on the concept of (p, q)-integers and bivariate tensor product form. For the univariate case, we obtain the estimates of moments and central moments of these operators, establish a local approximation theorem, obtain the estimates on the rate of convergence and weighted approximation of those operators. For the bivariate case, we give the rate of convergence by using the weighted modulus of continuity, give some graphs and numerical examples to illustrate the convergent properties of these operators to certain functions. We also compare these operators Dn,p,q with another forms. 2000 Mathematics Subject Classification: 41A10, 41A25, 41A36. Key words and phrases: (p, q)-integers, Baskakov operators, modulus of continuity, rate of convergence, bivariate tensor product.

1

Introduction

In recent years, (p, q)-integers have been introduced to linear positive operators to construct new approximation processes. A sequence of (p, q)-analogue of Bernstein operators was first introduced by Mursaleen [1, 2]. Besides, (p, q)-analogues of Sz´asz-Mirakyan operators [3] , (p, q)-Baskakov Kantorovich operators [4, 5], (p, q)-Baskakov-Beta operators [6] and Kantorovich-type Bernstein-Stancu-Schurer operators [7] were also considered. For further developments, one can also refer to [8, 9, 28]. These operators are double parameters corresponding to p and q versus single parameter q-type operators [11, 12, 13]. The aim of these generalizations is to provide appropriate and powerful tools to these application areas such as numerical analysis, CAGD and solutions of differential equations (see, e. g., [14]). ∗

Corresponding author.

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In 2010, Aral and Gupta [15], Gupta [16] introduced certain Durrmeyer type qBaskakov operators and got some important approximation properties, motivated by them, in 2012, Cai and Zeng [17] introduced a new modification of Durrmeyer type one. Recently, Acar et al. [18] introduced a generalization of Durrmeyer type (p, q)-Baskakov operators which having Baskakov and Sz´asz basis functions defined by  k+n−1  Z ∞ ∞ k X k(k−1) ([n]p,q t) Ep,q (−q[n]p,q t) p p,q 2 Bn (f ; x) = [n]p,q bn,k (p, q; x) p t dp,q t, (1) f [k] ! q k−1 p,q 0 k=0

where " bn,k (p, q; x) =

n+k−1 k

# pk+

n(n−1) 2

p,q

q

k(k−1) 2

xk . (1 + x)n+k p,q

(2)

P From [5], we know ∞ k=0 bn,k (p, q; x) = 1. In 2016, Mishra and Pandey [19] introduced the Stancu type base on operators (1). Inspired by these results, in this paper, we introduce a generalization of Durrmeyer type (p, q)-Baskakov operators Dn,p,q (f ; x) as Z ∞ ∞ X k Dn,p,q (f ; x) = [n − 1]p,q bg (p, q; µ(x)) bg (3) n,k n,k (p, q; pu)f (p u)dp,q u, 0

k=0 pn−2 (p2 q[n−2]p,q x−1) , [n]p,q

 h 1 , ∞ , 0 < q < p ≤ 1 and x ∈ p2 q[n−2] p,q " # n(n−1)+(k+1)(k+2) k2 −1 xk n + k − 1 4 2 p bg (p, q; x) = q . n,k k (1 + x)n+k p,q

where µ(x) =

(4)

p,q

The paper is organized as follows: In section 2, we give some basic definitions regarding (p, q)-integers and (p, q)-calculus. In section 3, we estimate the moments and central moments of these operators (3). In section 4, we establish a local approximation theorem, obtain the estimates on the rate of convergence and weighted approximation. In section 5, we give some graphs and numerical examples to illustrate the convergent properties for one variable functions. In section 6-7, we propose the bivariate case, give the rate of convergence by using the weighted modulus of continuity and give some graphs and numerical analysis for two variables functions. In the last section, we compare the operators Dn,p,q ^ with D n,p,q , and show the former operators give better approximation to f than the latter ones by graphs.

2

Some notations

We mention some definitions based on (p, q)-integers, details can be found in [20, 21, 22, 23, 24]. For any fixed real number 0 < q < p ≤ 1 and each nonnegative integer k, we denote (p, q)-integers by [k]p,q , where [k]p,q =

pk − q k . p−q

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Also (p, q)-factorial and (p, q)-binomial coefficients are defined as follows: ( [k]p,q [k − 1]p,q ...[1]p,q , k = 1, 2, ..., [k]p,q ! = 1, k = 0, " # [n]p,q ! n = , (n ≥ k ≥ 0). [k]p,q ![n − k]p,q ! k p,q

Let n be a non-negative integer, the (p, q)-Gamma function is defined as Γp,q (n + 1) =

(p − q)np,q = [n]p,q !, (p − q)n

where (p − q)np,q = (p − q)(p2 − q 2 )...(pn − q n ). For m, n ∈ N, the (p, q)-Beta function of second kind is given by Z ∞ tm−1 Bp,q (m, n) = dp,q t, (1 + pt)m+n p,q 0 where the (p, q)-power basis is given by (1 + pt)m+n = (1 + pt)(p + pqt)(p2 + pq 2 t)...(pm+n−1 + pq m+n−1 t). p,q The relationship by the (p, q)-Beta and Gamma functions is shown as follows Bp,q (m, n) =

qΓp,q (m)Γp,q (n) m+1 (p q m−1 )m/2 Γp,q (m

, + n)

if p = 1, q → 1− , it reduces to the classic type B(m, n) = Γ(m)Γ(n) Γ(m+n) . The improper (p, q)-integral of f (x) on [0, ∞) is defined to be Z

∞

f (x)dp,q x = 0

∞ Z X j=−∞

qj pj q j+1 pj+1

f (x)dp,q x = (p − q)

∞ X j=−∞

qj

p



f j+1

qj pj+1

 .

When p = 1, all the definitions of (p, q)-calculus above are reduced to q-calculus.

3

Auxiliary results

Lemma 3.1. For x ∈ [0, ∞) and sufficiently large n, the following equalities hold Dn,p,q (1; x) = 1,

(5)

Dn,p,q (t; x) = x, Dn,p,q (t2 ; x) =

(6) p2

q2




2pn−2

+ [n − 2]p,q [n + 1]p,q 2 x + 3 3 x− 3 x 2 q [n − 3]p,q [n]p,q p q [n − 3]p,q q [n − 3]p,q [n]p,q pn−4 1 + 4 − 3 4 , q [n − 2]p,q [n − 3]p,q [n]p,q p q [n − 2]p,q [n − 3]p,q

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Dn,p,q (t3 ; x)
[n + 1]p,q [n + 2]p,q [n − 2]2p,q 3 [5]p,q + [2]2p,q pq pn−2 [2]p,q [n + 1]p,q [n + 2]p,q 2 x = x + [n − 3]p,q [n − 4]p,q [n]2p,q p4 q 7 [n]2p,q [n − 3]p,q [n − 4]p,q
[5]p,q q 2 + [2]2p,q pq 3 − 3p2 [n + 1]p,q [n + 2]p,q [n − 2]p,q 2 q 5 + 3p3 q 2 − p5 + x + x p4 q 7 [n]2p,q [n − 3]p,q [n − 4]p,q p7 q 7 [n − 3]p,q [n − 4]p,q
pn−6 2p4 + 2q 4 + 4pq 3 + p3 q 3p2n−4 [2]p,q + 8 2 x− x q [n]p,q [n − 3]p,q [n − 4]p,q q 8 [n]p,q [n − 3]p,q [n − 4]p,q
[2]2p,q pn−8 [5]p,q + [2]p,q pq 2 + 9 − 7 7 p q [n − 2]p,q [n − 3]p,q [n − 4]p,q q [n]p,q [n − 2]p,q [n − 3]p,q [n − 4]p,q 2n+3 p [2]p,q − 9 9 2 , (8) p q [n]p,q [n − 2]p,q [n − 3]p,q [n − 4]p,q   [n + 1]p,q [n + 2]p,q [n + 3]p,q [n − 2]3p,q 4 1 4 Dn,p,q (t ; x) = 12 x +O φ(x), (9) q [n − 3]p,q [n − 4]p,q [n − 5]p,q [n]3p,q [n]p,q where φ(x) is depend on x. Proof. Since Z ∞ 0

qΓp,q (k + 1)Γp,q (n − 1) uk d u = Bp,q (k + 1, n − 1) = k+1 n+k p,q (1 + pu)p,q (pk+2 q k ) 2 Γp,q (n + k) q[k]p,q ![n − 2]p,q ! = , (k+1)(k+2) k(k+1) 2 p q 2 [n + k − 1]p,q !

we have Dn,p,q (1; x) = [n − 1]p,q

∞ X

bg n,k (p, q; pu)dp,q u 0

k=0

= [n − 1]p,q × =

p ∞ X

∞ X

[n + k − 1]p,q ! n(n−1)+(k+1)(k+2) k2 −1 4 p q 2 bg n,k (p, q; µ(x)) [k]p,q ![n − 1]p,q !

k=0 k p q[k]p,q ![n (k+1)(k+2) 2

q

∞

Z bg n,k (p, q; µ(x))

k(k+1) 2

− 2]p,q !

[n + k − 1]p,q !

bn,k (p, q; µ(x)) = 1.

k=0

Similarly, we get Z ∞ 0

q[k + 1]p,q ![n − 3]p,q ! uk+1 d u = (k+2)(k+3) (k+1)(k+2) , n+k p,q (1 + pu)p,q 2 2 p q [n + k − 1]p,q !

thus, Dn,p,q (t; x) = [n − 1]p,q

∞ X k=0

Z bg n,k (p, q; µ(x))

∞ k bg n,k (p, q; pu)p udp,q u

0

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= [n − 1]p,q

∞ X k=0

p2k q[k + 1]p,q ![n − 3]p,q !

× p =

[n + k − 1]p,q ! n(n−1)+(k+1)(k+2) k2 −1 4 bg q 2 p n,k (p, q; µ(x)) [k]p,q ![n − 1]p,q !

∞ X

(k+2)(k+3) 2

q

(k+1)(k+2) 2

bg n,k (p, q; µ(x))

[n + k − 1]p,q !

p2k q[k + 1]p,q p p

k=0

(k+2)(k+3) 2

q

n(n−1)+(k+1)(k+2) 4

(k+1)(k+2) 2

q

k2 −1 2

.

[n − 2]p,q

Since [k + 1]p,q = q k + p[k]p,q , by simple computations, we have " # n(n+1) k(k−1) ∞ [n]p,q µ(x) X n + k pk+ 2 q 2 (µ(x))k Dn,p,q (t; x) = pn q 2 [n − 2]p,q k (1 + µ(x))n+k+1 p,q + =

p2 q[n

1 − 2]p,q

k=0 ∞ X

p,q

bn,k (p, q; µ(x))

k=0

[n]p,q µ(x) 1 + 2 = x. n 2 p q [n − 2]p,q p q[n − 2]p,q

Next, Z

∞

0

q[k + 2]p,q ![n − 4]p,q ! uk+2 d u = (k+3)(k+4) (k+2)(k+3) , n+k p,q (1 + pu)p,q 2 2 p q [n + k − 1]p,q !

we get Dn,p,q (t2 ; x) Z ∞ X = [n − 1]p,q bg (p, q; µ(x)) n,k = [n − 1]p,q ∞

Z ×

k=0 ∞ X

∞ 2k 2 bg n,k (p, q; pu)p u dp,q u

0

" bg n,k (p, q; µ(x))

k=0 p3k uk+2

n+k−1 k

#

dp,q u (1 + pu)n+k p,q " # ∞ X n + k − 1 = [n − 1]p,q bg n,k (p, q; µ(x)) k

p

n(n−1)+(k+1)(k+2) 4

q

k2 −1 2

p

n(n−1)+(k+1)(k+2) 4

q

k2 −1 2

p,q

0

k=0 3k p q[k

× p =

∞ X k=0

(k+3)(k+4) 2

"

q

p,q

+ 2]p,q ![n − 4]p,q ! (k+2)(k+3) 2

n+k−1 k

# p,q

[n + k − 1]p,q ! n(n−1)+(k+1)(k+2)

2

2 p p3k q k [k + 1]p,q [k + 2]p,q (µ(x))k (. 10) (k+3)(k+4) (k+2)(k+3) [n − 2]p,q [n − 3]p,q (1 + µ(x))n+k p,q 2 2 p q

Using [k + 1]p,q = q k + p[k]p,q and some computations, we obtain [k + 1]p,q [k + 2]p,q = [2]p,q q 2k + p[2]2p,q q k−1 [k]p,q + p4 [k]p,q [k − 1]p,q .

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Since ∞ X

"

n(n−1)+(k+1)(k+2)

#

2

2 p (µ(x))k p3k q k p4 [k]p,q [k − 1]p,q (k+3)(k+4) (k+2)(k+3) [n − 2]p,q [n − 3]p,q (1 + µ(x))n+k p,q 2 2 p q k=0 p,q " # ∞ X (n+1)(n+2) k(k−1) [n]p,q [n + 1]p,q x2 (µ(x))k n+k+1 2 = pk+ q 2 2n 6 n+k+2 p q [n − 2]p,q [n − 3]p,q k (1 + µ(x)) p,q k=0 p,q

n+k−1 k

[n]p,q [n + 1]p,q (µ(x))2 , p2n q 6 [n − 2]p,q [n − 3]p,q

=

(12)

and ∞ X

"

n(n−1)+(k+1)(k+2)

#

2

2 q k−1 [k]p,q (µ(x))k p p3k q k (k+3)(k+4) (k+2)(k+3) [n − 2]p,q [n − 3]p,q (1 + µ(x))n+k p,q 2 2 p q k=0 p,q " # ∞ X n(n+1) k(k−1) [n]p,q µ(x) (µ(x))k n+k pk+ 2 q 2 = n+4 5 n+k+1 p q [n − 2]p,q [n − 3]p,q k (1 + µ(x))p,q k=0 p,q

=

n+k−1 k

[n]p,q µ(x) , − 2]p,q [n − 3]p,q

(13)

pn+4 q 5 [n

and ∞ X k=0

"

n+k−1 k

# p,q

n(n−1)+(k+1)(k+2)

2

2 [2]p,q q 2k p3k q k (µ(x))k p (k+3)(k+4) (k+2)(k+3) [n − 2]p,q [n − 3]p,q (1 + µ(x))n+k p,q 2 2 q p

=

∞ X [2]p,q bn,k (p, q; µ(x)) p5 q 3 [n − 2]p,q [n − 3]p,q

=

[2]p,q , 5 3 p q [n − 2]p,q [n − 3]p,q

k=0

(14)

combining (10), (11), (12), (13) and (14), we have Dn,p,q (t2 ; x) [2]2p,q [n]p,q µ(x) [n]p,q [n + 1]p,q (µ(x))2 [2]p,q + + 5 3 p2n q 6 [n − 2]p,q [n − 3]p,q pn+3 q 5 [n − 2]p,q [n − 3]p,q p q [n − 2]p,q [n − 3]p,q
p2 + q 2 [n − 2]p,q [n + 1]p,q 2 2pn−2 = x + x − x q 2 [n − 3]p,q [n]p,q p3 q 3 [n − 3]p,q q 3 [n − 3]p,q [n]p,q pn−4 1 + 4 − 3 4 . q [n − 2]p,q [n − 3]p,q [n]p,q p q [n − 2]p,q [n − 3]p,q

=

Using the same methods, we have Dn,p,q (t3 ; x) Z ∞ X g = [n − 1]p,q bn,k (p, q; µ(x)) k=0

∞ 3k 3 bg n,k (p, q; pu)p u dp,q u

0

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∞ X

=

"

n+k−1 k

k=0

#

n(n−1)+2k−18 k2 −7k−12 [k + 1]p,q [k + 2]p,q [k + 3]p,q (µ(x))k 2 2 p q , [n − 2]p,q [n − 3]p,q [n − 4]p,q (1 + µ(x))n+k p,q

p,q

since [k + 1]p,q [k + 2]p,q [k + 3]p,q
= p9 [k]p,q [k − 1]p,q [k − 2]p,q + p4 q k−2 [5]p,q + [2]2p,q pq [k]p,q [k − 1]p,q
+pq 2k−2 [2]p,q [5]p,q + [2]2p,q pq [k]p,q + q 3k [2]p,q [3]p,q , by some computations, we have Dn,p,q (t3 ; x)
[5]p,q + [2]2p,q pq [n]p,q [n + 1]p,q (µ(x))2 [n]p,q [n + 1]p,q [n + 2]p,q (µ(x))3 = + p3n q 12 [n − 2]p,q [n − 3]p,q [n − 4]p,q p2n+4 q 11 [n − 2]p,q [n − 3]p,q [n − 4]p,q
2 [2]p,q [5]p,q + [2]p,q pq [n]p,q µ(x) [2]p,q [3]p,q + n+7 9 + 9 6 p q [n − 2]p,q [n − 3]p,q [n − 4]p,q p q [n − 2]p,q [n − 3]p,q [n − 4]p,q
[5]p,q + [2]2p,q pq pn−2 [2]p,q [n + 1]p,q [n + 2]p,q 2 [n + 1]p,q [n + 2]p,q [n − 2]2p,q 3 x + x = [n − 3]p,q [n − 4]p,q [n]2p,q p4 q 7 [n]2p,q [n − 3]p,q [n − 4]p,q
[5]p,q q 2 + [2]2p,q pq 3 − 3p2 [n + 1]p,q [n + 2]p,q [n − 2]p,q 2 q 5 + 3p3 q 2 − p5 + x + x p4 q 7 [n]2p,q [n − 3]p,q [n − 4]p,q p7 q 7 [n − 3]p,q [n − 4]p,q
pn−6 2p4 + 2q 4 + 4pq 3 + p3 q 3p2n−4 [2]p,q x− x + 8 2 q [n]p,q [n − 3]p,q [n − 4]p,q q 8 [n]p,q [n − 3]p,q [n − 4]p,q
[2]2p,q pn−8 [5]p,q + [2]p,q pq 2 − 7 7 + 9 p q [n − 2]p,q [n − 3]p,q [n − 4]p,q q [n]p,q [n − 2]p,q [n − 3]p,q [n − 4]p,q p2n+3 [2]p,q − 9 9 2 . p q [n]p,q [n − 2]p,q [n − 3]p,q [n − 4]p,q Finally, Dn,p,q (t4 ; x) Z ∞ X g = [n − 1]p,q bn,k (p, q; µ(x)) =

k=0

"

n+k−1 k

4k 4 bg n,k (p, q; pu)p u dp,q u

0

k=0 ∞ X

∞

# p

n(n−1)+2k−28 2

q

k2 −9k−20 2

p,q

[k + 1]p,q [k + 2]p,q [k + 3]p,q [k + 4]p,q [n − 2]p,q [n − 3]p,q [n − 4]p,q [n − 5]p,q

k

×

(µ(x)) , (1 + µ(x))n+k p,q

since [k + 1]p,q [k + 2]p,q [k + 3]p,q [k + 4]p,q
= p16 [k]p,q [k − 1]p,q [k − 2]p,q [k − 3]p,q + [7]p,q + pq[5]p,q + [2]2p,q p2 q 2 p9 q k−3 [k]p,q

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×[k − 1]p,q [k − 2]p,q + p4 q 2k−4 [5]p,q + [2]2p,q pq [6]p,q + p2 q 2 [2]p,q [k]p,q [k − 1]p,q
+pq 3k−3 [2]p,q [5]2p,q + [2]2p,q [5]p,q pq + p3 q 3 [3]p,q [k]p,q + q 4k [2]p,q [3]p,q [4]p,q , we have Dn,p,q (t4 ; x) [n]p,q [n + 1]p,q [n + 2]p,q [n + 3]p,q = (µ(x))4 p4n q 20 [n − 2]p,q [n − 3]p,q [n − 4]p,q [n − 5]p,q

[7]p,q + pq[5]p,q + [2]2p,q p2 q 2 [n]p,q [n + 1]p,q [n + 2]p,q 3 + µ(x) p3n+5 q 19 [n − 2]p,q [n − 3]p,q [n − 4]p,q [n − 5]p,q


[5]p,q + [2]2p,q pq [6]p,q + p2 q 2 [2]p,q + 2n+9 17 µ(x)2 p q [n − 2]p,q [n − 3]p,q [n − 4]p,q [n − 5]p,q
[2]p,q [5]2p,q + [2]2p,q [5]p,q pq + p3 q 3 [3]p,q [n]p,q + n+12 14 µ(x) p q [n − 2]p,q [n − 3]p,q [n − 4]p,q [n − 5]p,q [2]p,q [3]p,q [4]p,q + 14 10 p q [n − 2]p,q [n − 3]p,q [n − 4]p,q [n − 5]p,q   [n + 1]p,q [n + 2]p,q [n + 3]p,q [n − 2]3p,q 4 1 = x + O φ(x). q 12 [n − 3]p,q [n − 4]p,q [n − 5]p,q [n]3p,q [n]p,q Lemma 3.1 is proved. Lemma 3.2. For sufficiently large n, we have Dn,p,q (t − x; x) = 0,

(15)

Dn,p,q ((t − x)2 ; x)
p2 + q 2 x pn x2 pn−3 x2 p2n−3 x2 2pn−2 x = + + 2 + 3 3 − 3 q[n]p,q q[n − 3]p,q q [n − 3]p,q [n]p,q p q [n − 3]p,q q [n − 3]p,q [n]p,q n−4 1 p − 3 4 = Bn,p,q (x) (16) + 4 q [n − 2]p,q [n − 3]p,q [n]p,q p q [n − 2]p,q [n − 3]p,q  
1 = O x2 + x + 1 , (17) [n]p,q  
1 4 Dn,p,q ((t − x) ; x) = O x4 + x3 + x2 + x + 1 . (18) [n]p,q Proof. (15) is obtained by (5) and (6). Since [n − 2]p,q [n + 1]p,q q 2 [n − 3]p,q [n]p,q


pn−3 + q[n − 3]p,q (pn + q[n]p,q ) = q 2 [n − 3]p,q [n]p,q pn pn−3 p2n−3 = 1+ + + 2 , q[n]p,q q[n − 3]p,q q [n − 3]p,q [n]p,q

using lemma 3.1, we have Dn,p,q ((t − x)2 ; x)

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= Dn,p,q (t2 ; x) − 2xMn,p,q (t; x) + x2 = Dn,p,q (t2 ; x) − x2
  p2 + q 2 [n − 2]p,q [n + 1]p,q 2pn−2 2 = − 1 x + x − x q 2 [n − 3]p,q [n]p,q p3 q 3 [n − 3]p,q q 3 [n − 3]p,q [n]p,q pn−4 1 + 4 − 3 4 q [n − 2]p,q [n − 3]p,q [n]p,q p q [n − 2]p,q [n − 3]p,q
n 2 n−3 2 2n−3 p2 + q 2 x p x p x p x2 2pn−2 x = + + 2 + 3 3 − 3 q[n]p,q q[n − 3]p,q q [n − 3]p,q [n]p,q p q [n − 3]p,q q [n − 3]p,q [n]p,q n−4 p 1 + 4 − 3 4 . q [n − 2]p,q [n − 3]p,q [n]p,q p q [n − 2]p,q [n − 3]p,q Similarly, by some computations, we can obtain (18). Lemma 3.3. (See theorem 2.1 of [25]). For 0 < qn < pn ≤ 1, set qn := 1−αn , pn := 1−βn such that 0 ≤ βn < αn < 1, αn → 0, βn → 0 as n → ∞. The following statements are true (A) If lim en(βn −αn ) = 1 and enβn /n → 0, then [n]pn ,qn → ∞. n→∞

(B) If lim en(βn −αn ) < 1 and enβn (αn − βn ) → 0, then [n]pn ,qn → ∞. n→∞

(C) If limn→∞ en(βn −αn ) < 1, lim en(βn −αn ) = 1 and max{enβn /n, enβn (αn − βn )} n→∞

→ 0, then [n]pn ,qn → ∞.

4

Approximation properties

In this section, we establish a local approximation theorem. We give the following definitions at first, the space of all real valued continuous bounded functions f defined on the interval [0, ∞) is denoted by CB [0, ∞). The norm on CB [0, ∞) is defined by ||f || = sup{|f (x)| : x ∈ [0, ∞)}. The Peetre’s K-functional is given by K2 (f ; δ) = inf {||f − g|| + δ||g 00 ||}, g∈W 2

(19)

where δ > 0, W 2 = {g ∈ CB [0, ∞) : g 0 , g 00 ∈ CB [0, ∞)}. For f ∈ CB [0, ∞), the second order modulus of smoothness is defined as √ ω2 (f ; δ) = sup sup |f (x + 2h) − 2f (x + h) + f (x)|. (20) 0 0, such that K2 (f ; δ) ≤ Cω2 (f ;

√ δ).

(21)

In order to obtain the convergence of operators defined in (3), in the sequel, let p = {pn } and q = {qn } be sequences satisfying pnn → 1(n → ∞) and the conditions of lemma 3.3 (A), (B) or (C).

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Theorem 4.1. For f ∈ CB [0, ∞) and n ≥ 6, we have  q  |Dn,p,q (f ; x) − f (x)| ≤ Cω2 f ; Bn,p,q (x)/2 ,

(22)

where C is a positive constant, Bn,p,q (x) is defined in (16). Proof. Let g ∈ W 2 , by Taylor’s expansion, we have Z

0

t

g(t) = g(x) + g (x)(t − x) +

(t − u)g 00 (u)du,

(23)

x

applying Dn,p,q to (23), using (15), we get t

Z Dn,p,q (g; x) − g(x) = Dn,p,q

 (t − u)g 00 (u)du; x .

x

Thus, we have,  Z t 00 (t − u)g (u)du; x |Dn,p,q (g; x) − g(x)| = Dn,p,q x   Z t 00 ≤ Dn,p,q |t − u||g (u)|du ; x x
≤ Dn,p,q (t − x)2 ; x ||g 00 || = Bn,p,q (x)||g 00 ||.

(24)

On the other hand, by (3) and (5), we have Dn,p,q (f ; x) = [n − 1]p,q

∞ X k=0

Z bg n,k (p, q; x)

∞ k bg n,k (p, q; pu)|f (p u)|dp,q u ≤ ||f ||.

(25)

0

Now (24) and (25) imply |Dn,p,q (f ; x) − f (x)| ≤ |Dn,p,q (f − g; x) − (f − g)(x)| + |Dn,p,q (g; x) − g(x)| ≤ 2||f − g|| + Bn,p,q (x)||g 00 ||, from (19), taking infimum on the right hand side over all g ∈ W 2 , we obtain |Dn,p,q (f ; x) − f (x)| ≤ 2K2 (f ; Bn,p,q (x)/2) . Finally, using (21), we get  q  |Dn,p,q (f ; x) − f (x)| ≤ Cω2 f ; Bn,p,q (x)/2 . Theorem 4.1 is proved.

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Let Bx2 [0, ∞) be the set of all functions f defined on [0, ∞) satisfying the condition |f (x)| ≤ Mf (1 + x2 ), where Mf is the constant depending only on f . We denote the subspace of all continuous functions belonging to Bx2 [0, ∞) by Cx2 [0, ∞). Let Cx∗2 [0, ∞) f (x) be the subspace of all functions f ∈ Cx2 [0, ∞), for which limx→∞ 1+x 2 is finite. The norm ∗ on Cx2 [0, ∞) is |f (x)| . 2 x∈[0,∞) 1 + x

||f ||x2 = sup

We denote the usual modulus of continuity of f on the closed interval [0, a] (a > 0) by sup |f (t) − f (x)|.

ωa (f ; δ) = sup

|t−x|≤δ x,t∈[0,a]

Obviously, for a function f ∈ Cx2 [0, ∞), the modulus of continuity ωa (f, δ) tends to zero. Theorem 4.2. Let f ∈ Cx2 [0, ∞), ωa+1 (f ; δ) be the modulus of continuity on the finite interval [0, a + 1] ⊂ [0, ∞), where a > 0. Then for n ≥ 6, we have  q  2 ||Dn,p,q (f ) − f ||Cx2 [0,a] ≤ 4Mf (1 + a )Bn,p,q (a) + 2ωa+1 f ; Bn,p,q (a) , where Bn,p,q (a) is defined in (16). Proof. For x ∈ [0, a] and t > a + 1, we have

|f (t) − f (x)| ≤ Mf 2 + x2 + t2 ≤ Mf 2 + 3x2 + 2(t − x)2 . Since t − x ≥ t − a > 1, then (t − x)2 > 1. Thus 2 + 3x2 + 2(t − x)2 ≤ (2 + 3x2 )(t − x)2 + 2(t − x)2 = (4 + 3x2 )(t − x)2 ≤ (4 + 3a2 )(t − x)2 ≤ 4(1 + a2 )(t − x)2 . Thus, we obtain |f (t) − f (x)| ≤ 4Mf (1 + a2 )(t − x)2 .

(26)

For x ∈ [0, a] and t ≤ a + 1, we have   |t − x| ωa+1 (f ; δ), (δ > 0) |f (t) − f (x)| ≤ ωa+1 (f ; |t − x|) ≤ 1 + δ

(27)

From (26) and (27), we get   |t − x| |f (t) − f (x)| ≤ 4Mf (1 + a )(t − x) + 1 + ωa+1 (f ; δ). δ 2

2

(28)

For x ∈ [0, a] and t ≥ 0, by Schwarz’s inequality and lemma 3.2, we have |Dn,p,q (f ; x) − f (x)| ≤ Dn,p,q (|f (t) − f (x)|; x)   q 1 2 ≤ 4Mf (1 + a )Dn,p,q (t − x) ; x + ωa+1 (f ; δ) 1 + Dn,p,q ((t − x) ; x) δ ! p Bn,p,q (x) 2 ≤ 4Mf (1 + a )Bn,p,q (x) + ωa+1 (f ; δ) 1 + . δ p By taking δ = Bn,p,q (x), we get the assertion of theorem 4.2. 2

2




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Now we discuss the weighted approximation theorem. Theorem 4.3. For f ∈ Cx∗2 [0, ∞) and n ≥ 6, we have lim ||Dn,pn ,qn (f ) − f ||x2 = 0.

n→∞

(29)

Proof. By using the Korovkin theorem, we see that it is sufficient to verify the following three conditions lim ||Dn,pn ,qn (ti ; x) − xi ||x2 = 0, i = 0, 1, 2.

n→∞

(30)

Since Dn,pn ,qn (1; x) = 1 and Dn,pn ,qn (t; x) = x, equality (30) holds true for i = 0 and i = 1. Finally, for i = 2, from lemma 3.2, we have ||Dn,pn ,qn (t2 ; x) − x2 ||x2 |Dn,pn ,qn (t2 ; x) − x2 | = sup 1 + x2 x∈[0,∞)   pnn pn−3 p2n−3 x2 n n ≤ + + 2 sup qn [n]pn ,qn qn [n − 3]pn ,qn qn [n − 3]pn ,qn [n]pn ,qn x∈[0,∞) 1 + x2   x p2n + qn2 2pn−2 n + + 3 sup 3 3 pn qn [n − 3]pn ,qn qn [n − 3]pn ,qn [n]pn ,qn x∈[0,∞) 1 + x2   1 1 pn−4 n + 3 4 sup + 4 qn [n − 2]pn ,qn [n − 3]pn ,qn [n]pn ,qn pn qn [n − 2]pn ,qn [n − 3]pn ,qn x∈[0,∞) 1 + x2 ≤

pnn p2 + q 2 + pnn qn2 p2n−3 qn + 2pn−2 1 n + 3n 3 n + 3 n + 3 4 qn [n]pn ,qn pn qn [n − 3]pn ,qn qn [n − 3]pn ,qn [n]pn ,qn pn qn [n − 2]pn ,qn [n − 3]pn ,qn n−4 pn . + 4 qn [n − 2]pn ,qn [n − 3]pn ,qn [n]pn ,qn

We can obtain limn→∞ ||Dn,pn ,qn (t2 ; x) − x2 || = 0 by using lemma 3.3 and limn→∞ pnn = 1, theorem 4.3 is proved.

Table 1: The errors of the approximation of Dn,pn ,qn (t2 ; x) with pn = 0.999999 and different values of qn and n .

qn 0.95 0.99 0.999 0.9999 0.99999

kf (x) − Dn,pn ,qn (f ; x)k∞ n = 10 n = 20 n = 30 n = 50 0.756459 0.471385 0.396404 0.348978 0.545694 0.264869 0.185654 0.126539 0.502874 0.224749 0.146152 0.087113 0.498679 0.220856 0.142352 0.083372 0.498261 0.220468 0.141973 0.083000

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Table 2: The errors of the approximation of Dn,pn ,qn (t2 ; x) with qn = 0.99 and different values of pn and n .

pn = 1 − 1/10m m=3 m=4 m=5 m=6 m=7

n = 10 0.545703746 0.545694253 0.545693781 0.545693738 0.545693733

kf (x) − Dn,pn ,qn (f ; x)k∞ n = 20 n = 30 0.264908767 0.185736472 0.264872541 0.185660670 0.264869729 0.185654271 0.264869456 0.185653644 0.264869429 0.185653581

1.4

1.2

n = 50 0.126723749 0.126555374 0.126540622 0.126539167 0.126539022

1.5 y = x2 q = 0.95 q = 0.99 q = 0.999

y = x2 n = 10 n = 20 n = 30 n = 50

1

(t 2 , x)

(t 2 ,x)

1

n

D n,p

D n,p

n

,q

,q

n

n

0.8

0.6

0.5 0.4

0.2

0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.1

1

x (n=50, p=0.99999)

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x (p = 0.99999, q=0.9999)

Figure 1: The figures of Dn,pn ,qn (t2 ; x) for n =

Figure 2: The figures of Dn,pn ,qn (t2 ; x) for

50, pn = 0.99999 and different values of qn .

pn = 0.99999, qn = 0.9999 and different values of n.

5

Graphical and numerical analysis for one variable functions

In this section, we give several graphs and numerical examples to show the convergence of Dn,pn ,qn (f ; x) to f (x) with different values of parameters which satisfy the conclusions of lemma 3.3. Let f (x) = x2 , the graphs of Dn,pn ,qn (f ; x) with n = 50, pn = 0.99999 and different values of qn is shown in Figure 1. The graphs of Dn,pn ,qn (f ; x) with pn = 0.99999, qn = 0.9999 and different values of n is shown in Figure 2. The graphs of Dn,pn ,qn (f ; x) with n = 50, qn = 0.95 and different values of pn is shown in Figure 3. Moreover, we give the errors of the approximation of Dn,pn ,qn (f ; x) to f (x) with different parameters in Table 1 and Table 2.

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1.4 2

1.2

y=x p = 0.99 p = 0.9999

(t 2 , x)

1

D n,p

n

,q

n

0.8

0.6

0.4

0.2

0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x (n = 50, q = 0.95)

Figure 3: The figures of Dn,pn ,qn (t2 ; x) for n = 50, qn = 0.95 and different values of pn .

6

Construction of bivariate operators and approximation properties

We introduce the bivariate tensor product (p, q)-analogue of Durrmeyer type Baskakov operators as follows Dpnn1 ,n,q2n

1 ,pn2 ,qn2

1

(f ; x, y)

= [n1 − 1]pn1 ,qn1 [n2 − 1]pn2 ,qn2

∞ X ∞ X

Z

∞Z ∞

] b] n1 ,k1 (pn1 , qn1 ; µ(x))bn2 ,k2 (pn2 , qn2 ; ν(y)) 0

k1 =0 k2 =0





k1 k2 ] b] n1 ,k1 (pn1 , qn1 ; pn1 u)bn2 ,k2 (pn2 , qn2 ; pn2 v)f pn1 u, pn2 v dpn1 ,qn1 udpn2 ,qn2 v,

0

(31)

where µ(x) = ν(y) =

!
pnn11 −2 qn1 p2n1 qn1 [n1 − 2]pn1 ,qn1 x − 1 1 , x≥ 2 , [n1 ]pn1 ,qn1 pn1 qn1 [n1 − 2]pn1 ,qn1 !
pnn22 −2 qn2 p2n2 qn2 [n2 − 2]pn2 ,qn2 y − 1 1 , y≥ 2 , [n2 ]pn2 ,qn2 pn2 qn2 [n2 − 2]pn2 ,qn2

0 < qn1 , qn2 < pn1 , pn2 ≤ 1 and bg n,k (p, q; x) is defined in (4). Lemma 6.1. Let ei,j (x, y) = xi y j , i, j ∈ N, (x, y) ∈ ([0, ∞) × [0, ∞)) be the two dimensional test functions and n1 , n2 ≥ 6, using lemma 3.1, we easily obtain the following equalities Dpnn1 ,n,q2n

(e0,0 ; x, y) = 1,

(32)

Dpnn1 ,n,q2n

(e1,0 ; x, y) = x,

(33)

1

1

1 ,pn2 ,qn2 1 ,pn2 ,qn2

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Dpnn1 ,n,q2n

(e0,1 ; x, y) 1 1 ,pn2 ,qn2 n1 ,n2 Dpn ,qn ,pn ,qn (e1,1 ; x, y) 1 1 2 2

= y,

(34)

= xy,

(35)

[n1 − 2]pn1 ,qn1 [n1 + 1]pn1 ,qn1 2 p2n1 + qn2 1 x + x qn2 1 [n1 − 3]pn1 ,qn1 [n1 ]pn1 ,qn1 p3n1 qn3 1 [n1 − 3]pn1 ,qn1


Dpnn1 ,n,q2n 1

(e2,0 ; x, y) 1 ,pn2 ,qn2

=

2pnn11 −2 pnn11 −4 x + qn3 1 [n1 − 3]pn1 ,qn1 [n1 ]pn1 ,qn1 qn4 1 [n1 − 2]pn1 ,qn1 [n1 − 3]pn1 ,qn1 [n1 ]pn1 ,qn1 1 − 3 4 , pn1 qn1 [n1 − 2]pn1 ,qn1 [n1 − 3]pn1 ,qn1

−

Dpnn1 ,n,q2n ,pn ,qn (e0,2 ; x, y) 1 1 2 2

(36)


[n2 − 2]pn2 ,qn2 [n2 + 1]pn2 ,qn2 2 p2n2 + qn2 2 = 2 y + 3 3 y qn2 [n2 − 3]pn2 ,qn2 [n2 ]pn2 ,qn2 pn2 qn2 [n2 − 3]pn2 ,qn2

2pnn22 −2 pnn22 −4 y + qn3 2 [n2 − 3]pn2 ,qn2 [n2 ]pn2 ,qn2 qn4 2 [n2 − 2]pn2 ,qn2 [n2 − 3]pn2 ,qn2 [n2 ]pn2 ,qn2 1 . − 3 4 pn2 qn2 [n2 − 2]pn2 ,qn2 [n2 − 3]pn2 ,qn2 −

(37)

Lemma 6.2. For sufficiently large n1 and n2 , using lemma 6.1 and lemma 3.2, we have the following statements Dpnn1 ,n,q2n

(t − x; x, y) 1 1 ,pn2 ,qn2 n1 ,n2 Dpn ,qn ,pn ,qn (s − y; x, y) 1 1 2 2 Dpnn1 ,n,q2n ,pn ,qn ((t 2 2 1 1 Dpnn1 ,n,q2n 1

1 ,pn2 ,qn2

Dpnn1 ,n,q2n 1

1 ,pn2 ,qn2

= 0, = 0,

2

− x) ; x, y) = O

((s − y)2 ; x, y) = O ((t − x)4 ; x, y) = O = O

Dpnn1 ,n,q2n ,pn ,qn ((s 1 1 2 2

4

− y) ; x, y) = O = O

!

1

2

x +x+1 =O

[n1 ]pn1 ,qn1

y2 + y + 1 = O

!

1




[n2 ]pn2 ,qn2

(x + 1)2 ,

[n1 ]pn1 ,qn1

!

1

!

1




[n2 ]pn2 ,qn2

(y + 1)2 ,

!

1

x4 + x3 + x2 + x + 1

[n1 ]pn1 ,qn1




!

1

(x + 1)4 ,

[n1 ]pn1 ,qn1 !

1

y4 + y3 + y2 + y + 1

[n2 ]pn2 ,qn2




!

1

(y + 1)4 .

[n2 ]pn2 ,qn2

Let Bρ be the space of all functions f defined on [0, ∞)×[0, ∞) satisfying the condition |f (x)| ≤ Mf ρ(x, y), where Mf is a positive constant depending only on f and ρ(x, y) = 1+x2 +y 2 is a weighted function. We denote the subspace of all continuous functions belong (x,y) to Bρ by Cρ . Let Cρ∗ be the subspace of all functions f ∈ Cρ , for which lim√x2 +y2 →∞ fρ(x,y) is finite. The norm on Cρ∗ is ||f ||ρ = supx,y∈[0,∞)

|f (x,y)| ρ(x,y) .

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˙ f ∈ Cρ∗ and δ1 , δ2 > 0, Ispir and Atakut [28] introduced the weighted modulus of continuity as Ωρ (f ; δ1 , δ2 ) =

sup

sup

x,y∈[0,∞) 0≤|k1 |≤δ1 ,0≤|k2 |≤δ2

|f (x + k1 , y + k2 ) − f (x, y)| , ρ(x, y)ρ(k1 , k2 )

which satisfy the following inequality Ωρ (f ; d1 δ1 , d2 δ2 ) ≤ 4(1 + d1 )(1 + d2 )(1 + δ12 )(1 + δ22 )Ωρ (f ; δ1 , δ2 ), d1 , d2 > 0.

(38)

From the definition of Ωρ , we have |f (t, s) − f (x, y)| ≤ ρ(x, y)ρ(|t − x|, |s − y|)Ωρ (f ; |t − x|, |s − y|)


≤ 1 + x2 + y 2 1 + (t − x)2 1 + (s − y)2 Ωρ (f ; |t − x|, |s − y|)

(39)

Now, we establish the degree approximation of operators Dpnn11,n,q2n1 ,pn2 ,qn2 in the weighted space Cρ∗ by the weighted modulus of continuity Ωρ . Theorem 6.3. For f ∈ Cρ∗ , then for sufficiently large n1 , n2 , we have the following inequality   Dpnn11,n,q2n1 ,pn2 ,qn2 (f ; x, y) − f (x, y) 1 1 , ≤ CΩρ f ; q ,q sup (ρ(x, y))3 x,y∈[0,∞) [n1 ]pn1 ,qn1 [n2 ]pn2 ,qn2 where C is a positive constant. Proof. From (38) and (39), for δn1 , δn2 > 0, we get |f (t, s) − f (x, y)|   


|s − y| |t − x| 2 2 2 2 1+ 1 + δn2 1 = 4 1+x +y 1 + (t − x) (1 + (s − y) ) 1 + δn1 δn 2
2 × 1 + δn2 Ωρ (f ; δn1 , δn2 )  


|t − x| |t − x| 2 2 2 2 2 2 + (t − x) + (t − x) = 4 1+x +y 1 + δn1 1 + δn2 1 + δn1 δn 1   |s − y| |s − y| 2 2 × 1+ + (s − y) + (s − y) Ωρ (f ; δn1 , δn2 ), δn2 δn2 applying the operators Dpnn11,n,q2n1 ,pn2 ,qn2 on the above inequality, we have |Dpnn1 ,n,q2n

1 ,pn2 ,qn2

1

≤ Dpnn1 ,n,q2n 1

1 ,pn2 ,qn2

2

2

(f ; x, y) − f (x, y)|

(|f (t, s) − f (x, y)|; x, y)

≤ 4(1 + x + y ) 1 +

δn2 1




1+

δn2 2




 |t − x| + (t − x)2 1+ δn1   |s − y| |s − y| 2 2 1+ + (s − y) + (s − y) ; x, y δn 2 δn 2

Dpnn1 ,n,q2n ,pn ,qn 2 1 1 2

 |t − x| 2 + (t − x) ; x, y Dpnn1 ,n,q2n ,pn ,qn 1 1 2 2 δn1

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×Ωρ (f ; δn1 , δn2 ) 


Dpnn11,n,q2n1 ,pn2 ,qn2 (|t − x|; x, y) 2 2 2 2 = 4 1+x +y 1 + δn1 1 + δn2 1 + δn1 + Dpnn1 ,n,q2n

1 ,pn2 ,qn2

1

Dpnn11,n,q2n1 ,pn2 ,qn2 |t − x|(t − x)2 ; x, y (t − x)2 ; x, y + δn1


!





Dpnn11,n,q2n1 ,pn2 ,qn2 (|s − y|; x, y) + 1+ + Dpnn1 ,n,q2n ,pn ,qn (s − y)2 ; x, y 1 1 2 2 δn 2
! n1 ,n2 Dpn1 ,qn1 ,pn2 ,qn2 |s − y|(s − y)2 ; x, y + Ωρ (f ; δn1 , δn2 ). δn 2 Using Cauchy-Schwarz inequality, we get |Dpnn1 ,n,q2n

(f ; x, y) − f (x, y)| q  Dpnn11,n,q2n1 ,pn2 ,qn2 ((t − x)2 ; x, y)
2 2  + Dpnn1 ,n,q2n ,pn ,qn (t − x)2 ; x, y ≤ 4(1 + x + y ) 1 + 1 1 2 2 δn1 q q  Dpnn11,n,q2n1 ,pn2 ,qn2 ((t − x)2 ; x, y) Dpnn11,n,q2n1 ,pn2 ,qn2 ((t − x)4 ; x, y)  + δn1 q  Dpnn11,n,q2n1 ,pn2 ,qn2 ((s − y)2 ; x, y)

2 2  × 1 + δn 1 1 + δn 2 1+ δn2
q + Dpnn1 ,n,q2n ,pn ,qn (s − y)2 ; x, y + Dpnn11,n,q2n1 ,pn2 ,qn2 ((s − y)2 ; x, y) 2 2 1 1 q  n1 ,n2 Dpn1 ,qn1 ,pn2 ,qn2 ((s − y)4 ; x, y)  Ωρ (f ; δn1 , δn2 ). × δn2 1

1 ,pn2 ,qn2

Using lemma 6.2, we have |Dpnn1 ,n,q2n 1

1 ,pn2 ,qn2

(f ; x, y) − f (x, y)| 

v ! u 1 1 u 2 2 2 2  t O (x + 1)2 ≤ 4(1 + x + y )(1 + δn1 )(1 + δn2 ) 1 + δn1 [n1 ]pn1 ,qn1 v v  ! ! ! u u u u 1 1 1 1 t O +O (x + 1)2 + (x + 1)2 tO (x + 1)4  [n1 ]pn1 ,qn1 δn1 [n1 ]pn1 ,qn1 [n1 ]pn1 ,qn1 v  ! ! u u 1 1 1 tO (y + 1)2 + O (y + 1)2 × 1 + δn 2 [n2 ]pn2 ,qn2 [n2 ]pn2 ,qn2 v v  ! ! u u u u 1 t 1 1 + O (y + 1)2 tO (y + 1)4  Ωρ (f ; δn1 , δn2 ). δn2 [n2 ]pn2 ,qn2 [n2 ]pn2 ,qn2

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Then, we have |Dpnn1 ,n,q2n 1

1 ,pn2 ,qn2

(f ; x, y) − f (x, y)|

s 1 C1 C1 ≤ 4(1 + x2 + y 2 )(1 + δn2 1 )(1 + δn2 2 ) 1 + (x + 1) + (x + 1)2 δn1 [n1 ]pn1 ,qn1 [n1 ]pn1 ,qn1 ! s s 1 C2 C12 C2 3 + (x + 1) 1 + δn2 (y + 1) + (y + 1)2 2 δn1 [n1 ]pn ,qn [n2 ]pn2 ,qn2 [n2 ]pn2 ,qn2 1 1 ! s 1 C22 3 + (y + 1) Ωρ (f ; δn1 , δn2 ). δn2 [n2 ]2pn ,qn 2

Let δn1 = √[n

2

1

and δn2 = √[n

1 ]pn1 ,qn1

1

2 ]pn2 ,qn2

, we have

|Dpnn1 ,n,q2n

(f ; x, y) − f (x, y)| ! ! 1 1 ≤ 4(1 + x2 + y 2 ) 1 + 1+ C(1 + x2 + y 2 )2 Ωρ (f ; δn1 , δn2 ), [n1 ]pn1 ,qn1 [n2 ]pn2 ,qn2 1

1 ,pn2 ,qn2

where C is a positive constant. Theorem 6.3 is proved.

7

Graphical and numerical analysis for two variables functions

In this section, we give several graphs and numerical examples to show the convergence of Dpnn11,n,q2n1 ,pn2 ,qn2 (f ; x, y) to f (x, y) with different values of parameters which satisfy the conclusions of lemma 3.3. Let f (x, y) = x2 y 2 , the graphs of Dpnn11,n,q2n1 ,pn2 ,qn2 (f ; x, y) with n1 = n2 = 30, pn1 = pn2 = 0.9999, qn1 = qn2 = 0.999 and f (x, y) = x2 y 2 are shown in Figure 4. The graphs of Dpnn11,n,q2n1 ,pn2 ,qn2 (f ; x, y) with n1 = n2 = 50, pn1 = pn2 = 0.99999, qn1 = qn2 = 0.9999 and f (x, y) = x2 y 2 are shown in Figure 5. Moreover, we give the errors of the approximation of Dpnn11,n,q2n1 ,pn2 ,qn2 (f ; x, y) to f (x, y) with different parameters in Table 3 and Table 4. Table 3: The errors of the approximation of Dpnn11,n,q2n1 ,pn2 ,qn2 (f ; x, y) with pn1 = pn2 = 0.99999 and different values of qn1 = qn2 = q and n1 = n2 = n .

q 0.95 0.99 0.999 0.9999

kf (x) − Dpnn11,n,q2n1 ,pn2 ,qn2 (f ; x, y)k∞ n = 10 n = 20 n = 30 n = 50 2.085144 1.164978 0.949957 0.819780 1.389169 0.599895 0.405776 0.269094 1.258631 0.500011 0.313666 0.181816 1.246041 0.490490 0.3049678 0.173697

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Figure 4:

The figures of (the upper one) Dpnn11,n,q2n1 ,pn2 ,qn2 (f ; x, y) for n1 = n2 = 30, pn1 = pn2 = 0.9999, qn1 = qn2 = 0.999, and (the below one) f (x, y) = x2 y 2 .

Figure 5:

The figures of (the upper one) Dpnn11,n,q2n1 ,pn2 ,qn2 (f ; x, y) for n1 = n2 = 50, pn1 = pn2 = 0.99999, qn1 = qn2 = 0.9999, and (the below one) f (x, y) = x2 y 2 .

Table 4: The errors of the approximation of Dpnn11,n,q2n1 ,pn2 ,qn2 (f ; x, y) with qn1 = qn2 = 0.999 and different values of pn1 = pn2 = p and n1 = n2 = n .

p = 1 − 1/10m m=4 m=5 m=6 m=7

8

kf (x) − Dpnn11,n,q2n1 ,pn2 ,qn2 (f ; x, y)k∞ n = 10 n = 20 n = 30 n = 50 1.25863727 0.50001245 0.31366763 0.18181861 1.25863086 0.50001052 0.31366566 0.18181572 1.25863023 0.50001034 0.31366548 0.18181546 1.25863016 0.50001033 0.31366546 0.18181544

Further discussion ^ If we consider the following modified forms D n,p,q ,

^ D n,p,q (f ; x) = [n − 1]p,q

∞ X k=0

Z

∞

bg n,k (p, q; x)

k bg n,k (p, q; pu)f (p u)dp,q u,

(40)

0

where x ∈ [0, ∞), bg n,k (p, q; x) is defined in (4). Here we omit the bivariate forms of operators (40). By similar computations in section 3, we know these operators (40) reproduce only constant functions, but not linear functions. We also provide two graphs to show that ^ the operators Dn,p,q give a better approximation to f than D n,p,q and so is the bivariate case (See Figure 6 and Figure 7), hence it is more appropriate to consider the operators Dn,p,q and the bivariate ones defined in (31).

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1.4

1.2

1

y

0.8

0.6

0.4

0.2

0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x (n=50, p=0.99999, q=0.999)

Figure 6: The figures of Dn,pn ,qn (f ; x) (the red one), D^ n,pn ,qn (f ; x) (the yellow one)for n = 50, pn = 0.99999, qn = 0.999, and f (x) = x2 (the blue one).

Figure 7: dle one) 2 Dpn1 ,n^ ,q ,p

The figures of (the midDpnn11,n,q2n1 ,pn2 ,qn2 (f ; x, y) and

(f ; x, y) (the upper one) for n1 = n2 = 50, pn1 = pn2 = 0.99999, qn1 = qn2 = 0.9999, and f (x, y) = x2 y 2 (the below one). n1

n1

n2 ,qn2

Acknowledgement This work is supported by the National Natural Science Foundation of China (Grant Nos. 11601266 and 11626201), the Natural Science Foundation of Fujian Province of China (Grant No. 2016J05017) and the Program for New Century Excellent Talents in Fujian Province University. We also thank Fujian Provincial Key Laboratory of Data Intensive Computing and Key Laboratory of Intelligent Computing and Information Processing of Fujian Province University.

References [1] M. Mursaleen, K. J. Ansari, A. Khan, On (p, q)-analogue of Bernstein operators. Appl. Math. Comput., 266 (2015), 874-882. [2] M. Mursaleen, K. J. Ansari, A. Khan, Erratum to ”On (p, q)-analogue of Bernstein operators [Appl. Math. Comput. 266 (2015) 874-882]”, Appl. Math. Comput., 278 (2016), 70-71. [3] T. Acar, (p, q)-Generalization of Sz´ asz-Mirakyan operators, Math. Methods Appl. Sci., 39(10) (2016), 2685-2695. [4] T. Acar, A. Aral, S. A. Mohiuddine, On Kantorovich modification of (p, q)-Baskakov operators, J. Ineq. App., (2016), Doi: 10.1186/s13660-016-1045-9. [5] V. Gupta, (p, q)-Baskakov-Kantorovich operators, Appl. Math. Inf. Sci., 10(4) (2016), 15511556. [6] N. Malik, V. Gupta, Approximation by (p, q)-Baskakov-Beta operators, Appl. Math. Comput., 293 (2017), 49-56.

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[7] Q. -B. Cai, G. Zhou, On (p, q)-analogue of Kantorovich type Bernstein-Stancu-Schurer operators, Appl. Math. Comput., 276 (2016), 12-20. [8] T. Acar, On pointwise convergence of q-Bernstein operators and their q-derivatives, Nurmer. Funct. Anal. Optim., 36(3) (2015), 287-304. [9] T. Acar, P. Agrawal, A. Kumar, On a modification of (p, q)-Sz´asz-Mirakyan operators, Complex Anal. Oper. Theory, (2016), Doi: 10.1007/s11785-016-0613-9. [10] H. Ilarslan, T. Acar, Approximation by bivariate (p, q)-Baskakov-Kantorovich operators, Georgian Math. J., (2016), Doi: 10.1515/gmj-2016-0057. [11] G. M. Phillips, Bernstein polynomials based on the q-integers, Ann. Number. Math., 4 (1997), 511-518. [12] V. Gupta, T. Kim, On the rate of approximation by q modified Beta operators, J. Math. Anal. Appl., 377 (2011), 471-480. [13] V. Gupta, A. Aral, Convergence of the q analogue of Sz´asz-Beta operators, Appl. Math. Comput., 216 (2010), 374-380. [14] K. Khan, D. K. Lobiyal, B´ezier curves based on Lupas (p, q)-analogue of Bernstein functions in CAGD, J. Comput. Appl. Math., 317 (2017), 458-477. [15] A. Aral, V. Gupta, On the Durrmeyer type modification of the q-Baskakov type operators, Nonlinear Anal., 72 (2010), 1171-1180. [16] V. Gupta, On certain Durrmeyer type q Baskakov operators, Ann. Univ. Ferrara, 56 (2010), 295-303. [17] Q. -B. Cai, X. -M. Zeng, Convergence of modification of the Durrmeyer type q-Baskakov operators, Georgian Math. J., 19 (2012), 49-61. [18] T. Acar, A. Aral, M. Mursaleen, Approximation by Baskakov-Durrmeyer operators based on (p, q)-integers, arXiv: submit/1450876 [math. CA]. [19] V. N. Mishra, S. Pandey, On (p, q)-Baskakov-Durrmeyer-Stancu operators, (2016), arXiv: 1602. 06719. [20] M. N. Hounkonnou, J. D´esir´e, B. Kyemba, R(p, q)-calculus: differentiation and integration, SUT Journal of Mathematics, 49 (2013), 145-167. [21] R. Jagannathan, K. S. Rao, Two-parameter quantum algebras, twin-basic numbers, and associated generalized hypergeometric series, Proceedings of the International Conference on Number Theory and Mathematical Physics, (2005) 20-21. [22] J. Katriel, M. Kibler, Normal ordering for deformed boson operators and operator-valued deformed Stirling numbers, J. Phys. A: Math. Gen. (1992) 24, 2683-2691, printed in the UK. [23] P. N. Sadjang, On the fundamental theorem of (p, q)-calculus and some (p, q)-Taylor formulas, (2015) arXiv: 1309.3934v1. [24] V. Sahai, S. Yadav, Representations of two parameter quantum algebras and p, q-special functions, J. Math. Anal. Appl., 335(2007), 268-279. [25] Q. -B. Cai, X. -W. Xu, A basic problem of (p, q)-Bernstein operators, J. Ineq. Appl., 140 (2017), Doi: 10. 1186/s13660-017-1413-0. [26] G. A. Anastassiou, S. G. Gal, Approximation theory: moduli of continuity and global smoothness preservation, Birkhauser, Boston, 2000. [27] R. A. DeVore, G. G. Lorentz, Constructive Approximation, Springer, Berlin, 1993. ˙ [28] N. Ispir, C. Atakut, Approximation by modified Sz´asz-Mirakjan operators on weighted spaces, Proc. Indian Acad. Sci. Math. Sci., 112(4) (2002), 571-578.

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A subclass of analytic functions defined by a fractional integral operator Alb Lupa¸s Alina Department of Mathematics and Computer Science University of Oradea str. Universitatii nr. 1, 410087 Oradea, Romania [email protected], [email protected] Abstract Making use the fractional integral associated with the convolution product of S˘ al˘ agean operator and Ruscheweyh derivative, we introduce a new class of analytic functions D(µ, λ, α, β) defined on the open unit disc, and investigate its various characteristics. Further we obtain distortion bounds, extreme points and radii of close-to-convexity, starlikeness and convexity for functions belonging to the class D(µ, λ, α, β).

Keywords: Analytic functions, univalent functions, radii of starlikeness and convexity, neighborhood property, Salagean operator, Ruscheweyh operator. 2000 Mathematical Subject Classification: 30C45, 30A20, 34A40.

1

Introduction

Denote by U the unit disc of the complex plane, U = {z ∈ C : |z| < 1} and H(U ) the space of holomorphic functions in U . P∞ Let A (p, t) = {f ∈ H(U ) : f (z) = z p + j=p+t aj z j , z ∈ U }, with A (1, t) = At and H[a, t] = {f ∈ H(U ) : f (z) = a + at z t + at+1 z t+1 + . . . , z ∈ U }, where p, t ∈ N, a ∈ C. Definition 1.1 (S˘ al˘ agean [4]) For f ∈ At , and n ∈ N, the operator S n is defined by S n : At → At , 0

S 0 f (z) = f (z) , S 1 f (z) = zf 0 (z), ..., S n+1 f (z) = z (S n f (z)) , z ∈ U. P∞ P∞ Remark 1.1 If f ∈ At , f (z) = z + j=t+1 aj z j , then S n f (z) = z + j=t+1 j n aj z j , z ∈ U . Definition 1.2 (Ruscheweyh [3]) For f ∈ At and n ∈ N, the operator Rn is defined by Rn : At → At , 0

R0 f (z) = f (z) , R1 f (z) = zf 0 (z) , ..., (n + 1) Rn+1 f (z) = z (Rn f (z)) + nRn f (z) , z ∈ U. P∞ P∞ Γ(n+j) Remark 1.2 If f ∈ At , f (z) = z + j=t+1 aj z j , then Rn f (z) = z + j=t+1 Γ(n+1)Γ(j) aj z j for z ∈ U . Definition 1.3 Let n, m ∈ N. Denote by SRλm,n : At → At the operator given by the Hadamard product of the S˘ al˘ agean operator S m and the Ruscheweyh derivative Rn , SRm,n f (z) = (S m ∗ Rn ) f (z) , for any z ∈ U and each nonnegative integers m, n. P∞ P∞ Γ(n+j) Remark 1.3 If f ∈ At and f (z) = z + j=t+1 aj z j , then SRm,n f (z) = z + j=t+1 j m Γ(n+1)Γ(j) a2j z j , z ∈ U . Definition 1.4 ([2]) The fractional integral of order λ (λ > 0) is defined for a function f by Dz−λ f (z) = R z f (t) 1 Γ(λ) 0 (z−t)1−λ dt, where f is an analytic function in a simply-connected region of the z-plane containing the λ−1

origin, and the multiplicity of (z − t)

is removed by requiring log (z − t) to be real, when (z − t) > 0.

From Definition 1.3 and Definition 1.4, we get the fractional integral associated with the convolution product of S˘al˘agean operator and Ruscheweyh derivative, which has the following form Dz−λ SRm,n f (z) =

∞ X 1 j m+1 Γ (n + j) z λ+1 + a2j z j+λ , Γ (λ + 2) Γ (n + 1) Γ (j + λ + 1) j=t+1

P∞ for a function f (z) = z + j=t+1 aj z j ∈ At . Following the work from [1] we can define the class D(µ, λ, α, β) as follows.

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Definition 1.5 For µ ≥ 0, λ ∈ N, α ∈ C − {0} and 0 < β ≤ 1, let D(µ, λ, α, β) be the subclass of At consisting of functions that satisfying the inequality λ (1 − µ) Dz−λ SRm,n f (z) + µ D−λ SRm,n f (z)
0 z z (1.1) 0) is defined for a function f by Dz−λ f (z) = R z f (t) 1 Γ(λ) 0 (z−t)1−λ dt, where f is an analytic function in a simply-connected region of the z-plane containing the λ−1

origin, and the multiplicity of (z − t)

is removed by requiring log (z − t) to be real, when (z − t) > 0.

From Definition 1.3 and Definition 1.4, we get the fractional integral associated with the convolution product 1 of S˘al˘agean operator and Ruscheweyh derivative, which has the following form Dz−λ SRm,n f (z) = Γ(λ+2) z λ+1 + m+1 P∞ P ∞ j Γ(n+j) 2 j+λ , for a function f (z) = z + j=2 aj z j ∈ A. j=t+1 Γ(n+1)Γ(j+λ+1) aj z We follow the works from [1].

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Definition 1.5 Let the function f ∈ A. Then f (z) is said to be in the class L(λ, d, α, β) if it satisfies the following criterion:   1 z(Dz−λ SRm,n f (z))0 + αz 2 (Dz−λ SRm,n f (z))00 < β, (1.1) − 1 d (1 − α)D−λ SRm,n f (z) + αz(D−λ SRm,n f (z))0 z z where λ > 0, ∈ C − {0}, 0 ≤ α ≤ 1, 0 < β ≤ 1, m, n ∈ N, z ∈ U . In this paper we shall first deduce a necessary and sufficient condition for a function f (z) to be in the class L(λ, d, α, β). Then obtain the distortion and growth theorems, closure theorems, neighborhood and radii of univalent starlikeness, convexity and close-to-convexity of order δ, 0 ≤ δ < 1, for these functions.

2

Coefficient Inequality

Theorem 2.1 Let the function f ∈ A. Then f (z) is said to be in the class L(λ, d, α, β) if and only if ∞ X j m+1 Γ (n + j)  2 αj + [α (2λ − 2 + β |d|) + 1] j + [α (λ − 1) + 1] (λ − 1 + β |d|) a2j ≤ Γ (j + λ + 1) j=2

(αλ + 1) (β |d| − λ)

Γ (n + 1) , Γ (λ + 2)

(2.1)

where λ > 0, ∈ C − {0}, 0 ≤ α ≤ 1, 0 < β ≤ 1, m, n ∈ N, z ∈ U . Let f (z) ∈ L(λ, d, α, β). Assume that inequality (2.1) holds true. Then we find that Proof. z(Dz−λ SRm,n f (z))0 +αz2 (Dz−λ SRm,n f (z))00 − 1 = (1−α)D−λ SRm,n f (z)+αz(D−λ SRm,n f (z))0 z λ(αλ+1)z λ+1 P∞ jm+1 Γ(n+j) Γ(λ+2) z + j=2 Γ(n+1)Γ(j+λ+1) {αj 2 +[2α(λ−1)+1]j+(λ−1)[α(λ−1)+1]}a2j zj+λ ≤ P j m+1 Γ(n+j) αλ+1 z λ+1 + ∞ [αj+α(λ−1)+1]a2 z j+λ j=2 Γ(n+1)Γ(j+λ+1) j Γ(λ+2) λ(αλ+1) P∞ j m+1 Γ(n+j) 2 αj +[2α(λ−1)+1]j+(λ−1)[α(λ−1)+1] + j=2 Γ(n+1)Γ(j+λ+1) Γ(λ+2) P j m+1 Γ(n+j) αλ+1 2 j−1 | − ∞ j=2 Γ(n+1)Γ(j+λ+1) [αj+α(λ−1)+1]aj |z Γ(λ+2)

{

}a2j |zj−1 |

≤ β|d|.

Choosing values of z on real axis and letting z → 1− , we have P∞ j m+1 Γ(n+j)  2 αj + [α (2λ − 2 + β |d|) + 1] j + [α (λ − 1) + 1] (λ − 1 + β |d|) a2j ≤ j=2 Γ(j+λ+1) (αλ + 1) (β |d| − λ) Γ(n+1) Γ(λ+2) . Conversely, assume that f (z) ∈ L(λ, d, α, β), then o we get the following inequality n z(Dz−λ SRm,n f (z))0 +αz 2 (Dz−λ SRm,n f (z))00 Re (1−α)D−λ SRm,n f (z)+αz(D−λ SRm,n f (z))0 − 1 > −β|d| z z  (λ+1)(αλ+1)  P j m+1 Γ(n+j) 2 2 j+λ z λ+1 + ∞ j=2 Γ(n+1)Γ(j+λ+1) [αj +(2aλ−α+1)j+λ(αλ−α+1)]aj z Γ(λ+2) Re − 1 + β|d| >0 P∞ j m+1 Γ(n+j) αλ+1 2 j+λ λ+1 Re

z + j=2 Γ(n+1)Γ(j+λ+1) [αj+α(λ−1)+1]aj z Γ(λ+2) (αλ+1)(β|d|−λ) λ+1 P∞ j m+1 Γ(n+j) 2 z + j=2 Γ(n+1)Γ(j+λ+1) αj +[α(2λ−2+β|d|)+1]j+[α(λ−1)+1](λ−1+β|d|) Γ(λ+2) P∞ j m+1 Γ(n+j) αλ+1 λ+1 z + j=2 Γ(n+1)Γ(j+λ+1) [αj+α(λ−1)+1]a2j z j+λ Γ(λ+2)

{

}a2j zj+λ

Since Re(−eiθ ) ≥ −|eiθ | = −1, the above inequality reduces to (αλ+1)(β|d|−λ) λ+1 P∞ j m+1 Γ(n+j) r − j=2 Γ(n+1)Γ(j+λ+1) {αj 2 +[α(2λ−2+β|d|)+1]j+[α(λ−1)+1](λ−1+β|d|)}a2j r j+λj Γ(λ+2) P j m+1 Γ(n+j) αλ+1 2 j+λ r λ+1 − ∞ j=2 Γ(n+1)Γ(j+λ+1) [αj+α(λ−1)+1]aj r Γ(λ+2)

> 0.

> 0.

Letting r → 1− and by the mean value theorem we have desired inequality (2.1). This completes the proof of Theorem 2.1 Corollary 2.2sLet the function f ∈ A be in the class L(λ, d, α, β). Then aj ≤

3

Γ(n+1) (αλ+1)(β|d|−λ) Γ(λ+2) j m+1 Γ(n+j) 2 {αj +[α(2λ−2+β|d|)+1]j+[α(λ−1)+1](λ−1+β|d|)} Γ(j+λ+1)

, j ≥ 2.

Distortion Theorems

Theorem q 3.1 Let the function f ∈ A be in the class L(λ, d, α,qβ). Then for |z| = r < 1, we have (αλ+1)(λ+2)(β|d|−λ) (αλ+1)(λ+2)(β|d|−λ) r − 2m+1 (n+1){(λ+1)[α(λ+1+β|d|)+1]+β|d|} r2 ≤ |f (z)| ≤ r + 2m+1 (n+1){(λ+1)[α(λ+1+β|d|)+1]+β|d|} r2 . q (αλ+1)(λ+2)(β|d|−λ) The result is sharp for the function f (z) given by f (z) = z + 2m+1 (n+1){(λ+1)[α(λ+1+β|d|)+1]+β|d|} z2.

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Proof. Given that f (z) ∈ L(λ, d, α, β), from the equation (2.1) and since 2m+1 and positive for j ≥ 2, then we have p(n + 1) {(λ + 1) [α (λ + 1 + β |d|) + 1] + β |d|} is nonPdecreasing ∞ 2m+1 (n + 1) {(λ + 1) [α (λ + 1 + β |d|) + 1] + β |d|} j=2 aj ≤ P∞ q j m+1 Γ(n+j) 2 Γ(j+λ+1) {αj + [α (2λ − 2 + β |d|) + 1] j + [α (λ − 1) + 1] (λ − 1 + β |d|)}aj ≤ q j=2 q P∞ (αλ+1)(λ+2)(β|d|−λ) (αλ + 1) (β |d| − λ) Γ(n+1) , which is equivalent to, a ≤ j=2 j Γ(λ+2) 2m+1 (n+1){(λ+1)[α(λ+1+β|d|)+1]+β|d|} . P∞ We obtain for f (z) = z + j=2 aj z j , q P∞ P∞ P∞ (αλ+1)(λ+2)(β|d|−λ) |f (z)| ≤ |z| + j=2 aj |z|j ≤ r + j=2 aj rj ≤ r + r2 j=2 aj ≤ r + 2m+1 (n+1){(λ+1)[α(λ+1+β|d|)+1]+β|d|} r2 . q (αλ+1)(λ+2)(β|d|−λ) Similarly, |f (z)| ≥ r − 2m+1 (n+1){(λ+1)[α(λ+1+β|d|)+1]+β|d|} r2 . This completes the proof of Theorem 3.1. Theorem 3.2 Let the function f ∈ A be in the class L(λ, q q d, α, β). Then for |z| = r < 1, we have (αλ+1)(λ+2)(β|d|−λ) (αλ+1)(λ+2)(β|d|−λ) − 2m−1 (n+1){(λ+1)[α(λ+1+β|d|)+1]+β|d|} r ≤ |f 0 (z)| ≤ 2m−1 (n+1){(λ+1)[α(λ+1+β|d|)+1]+β|d|} r. q (αλ+1)(λ+2)(β|d|−λ) z2. The result is sharp for the function f (z) given by f (z) = z + 2m+1 (n+1){(λ+1)[α(λ+1+β|d|)+1]+β|d|} P∞ Proof. We have f 0 (z) = 1 + j=2 jaj z j−1 and q P∞ P∞ (αλ+1)(λ+2)(β|d|−λ) |f 0 (z)| ≤ 1 − j=2 jaj |z|j−1 ≤ 1 + j=2 jaj rj−1 ≤ 1 + 2m−1 (n+1){(λ+1)[α(λ+1+β|d|)+1]+β|d|} r. q (αλ+1)(λ+2)(β|d|−λ) r. This completes the proof of Theorem 3.2. Similarly, |f 0 (z)| ≥ 1 − 2m−1 (n+1){(λ+1)[α(λ+1+β|d|)+1]+β|d|}

4

Closure Theorems

P∞ Theorem 4.1 Let the functions fk , k = 1, 2, ..., l, defined by fk (z) = z + j=2 aj,k z j , aj,k ≥ 0, be in the class Pl L(λ, d, α, β). Then the function h(z) defined by h(z) = k=1 µk fk (z), µk ≥ 0, is also in the class L(λ, d, α, β), Pl where k=1 µk = 1. Pl Pl P∞ P∞ Pl Proof. We can write h(z) = k=1 µk z + k=1 j=2 µk aj,k z j = z + j=2 k=1 µk aj,k z j . Furthermore, since the functions fk (z), k = 1, 2, ..., l, are in the class L(λ, d, α, β), then from Corollary 2.2 we have P∞ q j m+1 Γ(n+j) 2 Γ(j+λ+1) {αj + [α (2λ − 2 + β |d|) + 1] j + [α (λ − 1) + 1] (λ − 1 + β |d|)}aj ≤ q j=2 (αλ + 1) (β |d| − λ) Γ(n+1) Γ(λ+2) . Thus it is enough to prove that P∞ q j m+1 Γ(n+j) Pm 2 j=2 k=1 µk aj,k ) = Γ(j+λ+1) {αj + [α (2λ − 2 + β |d|) + 1] j + [α (λ − 1) + 1] (λ − 1 + β |d|)} ( Pm P∞ q j m+1 Γ(n+j) 2 k=1 µk j=2 Γ(j+λ+1) {αj + [α (2λ − 2 + β |d|) + 1] j + [α (λ − 1) + 1] (λ − 1 + β |d|)}aj,k ≤ q q Pm (αλ + 1) (β |d| − λ) Γ(n+1) (αλ + 1) (β |d| − λ) Γ(n+1) k=1 µk Γ(λ+2) = Γ(λ+2) . Hence the proof is complete. P∞ Corollary 4.2 Let the functions fk , k = 1, 2, defined by fk (z) = z + j=2 aj,k z j , aj,k ≥ 0 be in the class L(λ, d, α, β). Then the function h(z) defined by h(z) = (1 − ζ)f1 (z) + ζf2 (z), 0 ≤ ζ ≤ 1, is also in the class L(λ, d, α, β). s Theorem 4.3 Let f1 (z) = z, and fj (z) = z +

Γ(n+1) (αλ+1)(β|d|−λ) Γ(λ+2) j m+1 Γ(n+j) {αj 2 +[α(2λ−2+β|d|)+1]j+[α(λ−1)+1](λ−1+β|d|)} Γ(j+λ+1)

z j , j ≥ 2.

Then the P function f (z) is in the class L(λ, d, α, β) if and only P∞if it can be expressed in the form f (z) = ∞ µ1 f1 (z) + j=2 µj fj (z), where µ1 ≥ 0, µj ≥ 0, j ≥ 2 and µ1 + j=2 µj = 1. Proof. Assume that f (z) can be expressed s in the form Γ(n+1) P∞ P∞ (αλ+1)(β|d|−λ) Γ(λ+2) f (z) = µ1 f1 (z) + j=2 µj fj (z) = z + j=2 jm+1 Γ(n+j) 2 µj z j . Thus {αj +[α(2λ−2+β|d|)+1]j+[α(λ−1)+1](λ−1+β|d|)} Γ(j+λ+1) s j m+1 Γ(n+j) P∞ {αj 2 +[α(2λ−2+β|d|)+1]j+[α(λ−1)+1](λ−1+β|d|)} Γ(j+λ+1) · Γ(n+1) j=2 (αλ+1)(β|d|−λ) Γ(λ+2) s Γ(n+1) P∞ (αλ+1)(β|d|−λ) Γ(λ+2) µj = j=2 µj = 1 − µ1 ≤ 1. Hence f (z) ∈ L(λ, d, α, β). j m+1 Γ(n+j) 2 Γ(j+λ+1)

{αj +[α(2λ−2+β|d|)+1]j+[α(λ−1)+1](λ−1+β|d|)}

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s

j m+1 Γ(n+j)

{αj 2 +[α(2λ−2+β|d|)+1]j+[α(λ−1)+1](λ−1+β|d|)}

Γ(j+λ+1) Conversely, assume that f (z) ∈ L(λ, d, α, β). Setting µj = Γ(n+1) (αλ+1)(β|d|−λ) Γ(λ+2) P∞ P∞ since µ1 = 1 − j=2 µj . Thus f (z) = µ1 f1 (z) + j=2 µj fj (z). Hence the proof is complete.

Corollary 4.4 The extreme points of the class L(d, α, β) are the functions f1 (z) = z, and fj (z) = z + s Γ(n+1) (αλ+1)(β|d|−λ) Γ(λ+2) j m+1 Γ(n+j) 2 {αj +[α(2λ−2+β|d|)+1]j+[α(λ−1)+1](λ−1+β|d|)} Γ(j+λ+1)

5

z j , j ≥ 2.

Inclusion and Neighborhood Results

P∞ We define the δ- neighborhood of a function f (z) ∈ A by Nδ (f ) = {g ∈ A : g(z) = z + j=2 bj z j and P∞ j=2 j|aj − bj | ≤ δ}. P∞ P∞ In particular, for e(z) = z, Nδ (e) = {g ∈ A : g(z) = z + j=2 bj z j and j=2 j|bj | ≤ δ}. Furthermore, a function f ∈ A is said to be in the class Lξ (λ, d, α, β) if there exists a function h(z) ∈ (z) − 1 < 1 − ξ, z ∈ U, 0 ≤ ξ < 1. L(λ, d, α, β) such that fh(z) Theorem 5.1 If δ =

q

(αλ+1)(λ+2)(β|d|−λ) 2m−1 (n+1){(λ+1)[α(λ+1+β|d|)+1]+β|d|} ,

then L(λ, d, α, β) ⊂ Nδ (e).

Proof. Let f ∈ L(λ, d, α, β). Then in view of assertion of Corollary 2.2 and since β |d|) ≥

j m+1 Γ(n+j)  αj 2 + [α (2λ − 2 + β |d|) + 1] j + [α (λ − 1) + 1] (λ − 1 + Γ(j+λ+1) m−1 2 Γ(n+2) {(λ + 1) [α (λ + 1 + β |d|) + 1] + β |d|} for j ≥ 2, we get Γ(λ+3) q m−1 P∞ 2 Γ(n+2) {(λ + 1) [α (λ + 1 + β |d|) + 1] + β |d|} j=2 aj ≤ Γ(λ+3)

P∞ q j m+1 Γ(n+j)

2 Γ(j+λ+1) {αj + [α (2λ − 2 + β |d|) + 1] j + [α (λ − 1) + 1] (λ − 1 + β |d|)}aj ≤ q j=2 q P∞ (αλ+1)(λ+2)(β|d|−λ) (αλ + 1) (β |d| − λ) Γ(n+1) , which implise a ≤ j=2 j Γ(λ+2) 2m+1 (n+1){(λ+1)[α(λ+1+β|d|)+1]+β|d|} . q P∞ (αλ+1)(λ+2)(β|d|−λ) Applying assertion of Corollary 2.2, we obtain j=2 jaj ≤ 2m−1 (n+1){(λ+1)[α(λ+1+β|d|)+1]+β|d|} = δ, so we have f ∈ Nδ (e). This completes the proof of the Theorem 5.1.

Theorem 5.2 If h ∈ L(λ, d, α, β) and s (αλ + 1) (λ + 2) (β |d| − λ) δ , ξ =1+ m+1 2 2 (n + 1) {(λ + 1) [α (λ + 1 + β |d|) + 1] + β |d|}

(5.1)

then Nδ (h) ⊂ Lξ (d, α, β). P∞ Proof. Suppose that f ∈ Nδ (h), we then find that j=2 j|aj − bj | ≤ δ, which readily implies the following P∞ coefficient inequality j=2 |aj − bj | ≤ 2δ . q P∞ (z) (αλ+1)(λ+2)(β|d|−λ) and we get fh(z) − 1 ≤ Next, since h ∈ L(d, α, β), we have j=2 bj ≤ 2m+1 (n+1){(λ+1)[α(λ+1+β|d|)+1]+β|d|} P∞ |aj −bj | j=2 P 1− ∞ j=2 bj

≤

δ  r 2 1−

(αλ+1)(λ+2)(β|d|−λ) 2m+1 (n+1){(λ+1)[α(λ+1+β|d|)+1]+β|d|}



= 1 − ξ, provided that ξ is given by (5.1), thus f ∈

Lξ (λ, d, α, β), where ξ is given by (5.1).

6

Radii of Starlikeness, Convexity and Close-to-Convexity

Theorem 6.1 Let the function f ∈ A be in the class L(λ, d, α, β). Then f is univalent starlike of order δ, 1   2(j−1) j m+1 Γ(n+j) (1−δ)2 Γ(j+λ+1) {αj 2 +[α(2λ−2+β|d|)+1]j+[α(λ−1)+1](λ−1+β|d|)} 0 ≤ δ < 1, in |z| < r1 , where r1 = inf j . Γ(n+1) 2 (αλ+1)(β|d|−λ) Γ(λ+2) (j−δ)

The result is sharp for the function f (z) given by v u u (αλ + 1) (β |d| − λ) Γ(n+1) Γ(λ+2) t fj (z) = z + z j , j ≥ 2. j m+1 Γ(n+j) 2 {αj + [α (2λ − 2 + β |d|) + 1] j + [α (λ − 1) + 1] (λ − 1 + β |d|)} Γ(j+λ+1)

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(6.1)

aj ,

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P 0 0 ∞ (j−1)aj z j−1 (z) (z) j=2 ≤ P∞ | − 1 ≤ 1 − δ, |z| < r1 . Since zff (z) − 1 = 1+ Proof. It suffices to show that zff (z) k−1 j=2 aj z P∞ (j−1)aj |z|j−1 j=2 P j−1 . 1− ∞ j=2 aj |z|

P∞

(j−1)aj |z|j−1

j=2 P∞ To prove the theorem, we must show that 1− ≤ 1 − δ. j−1 j=2 aj |z| P∞ j−1 It is equivalent to j=2 (j − δ)aj |z| ≤ 1 − δ, using Theorem 2.1, we obtain 1   2(j−1) m+1 j Γ(n+j) (1−δ)2 Γ(j+λ+1) {αj 2 +[α(2λ−2+β|d|)+1]j+[α(λ−1)+1](λ−1+β|d|)} |z| ≤ . Hence the proof is complete. Γ(n+1) 2

(αλ+1)(β|d|−λ) Γ(λ+2) (j−δ)

Theorem 6.2 Let the function f ∈ A be in the class L(λ, d, α, β). Then f is univalent convex of order δ, 1   2(j−1) j m−1 Γ(n+j) (1−δ)2 Γ(j+λ+1) {αj 2 +[α(2λ−2+β|d|)+1]j+[α(λ−1)+1](λ−1+β|d|)} 0 ≤ δ ≤ 1, in |z| < r2 , where r2 = inf j . Γ(n+1) 2 (αλ+1)(β|d|−λ) Γ(λ+2) (j−δ)

The result is sharp for the function f (z) given by (6.1). 00 P∞ 00 j(j−1)aj z j−1 zf (z) j=2 (z) P ≤ 1−δ, |z| < r . Since = Proof. It suffices to show that zf ∞ 2 f 0 (z)) f 0 (z) jaj z j−1 ≤ 1+ j=2

P∞

To prove the theorem, we must show that s Theorem 2.1, we obtain |z|j−1 ≤  |z| ≤

(1−δ)2

j m−1 Γ(n+j) Γ(j+λ+1)

(1−δ) j(j−δ)

j(j−1)aj |z|j−1 j=2 P j−1 1− ∞ j=2 jaj |z|

≤ 1 − δ, i.e.

P∞

j=2



j(j−1)aj |z|j−1 j=2 P j−1 . 1− ∞ j=2 jaj |z|

j(j − δ)aj |z|j−1 ≤ 1 − δ, using

j m+1 Γ(n+j) {αj 2 +[α(2λ−2+β|d|)+1]j+[α(λ−1)+1](λ−1+β|d|)} Γ(j+λ+1) Γ(n+1) (αλ+1)(β|d|−λ) Γ(λ+2)

{αj 2 +[α(2λ−2+β|d|)+1]j+[α(λ−1)+1](λ−1+β|d|)}

P∞

, or

1 2(j−1)

. Hence the proof is complete.

Γ(n+1)

(αλ+1)(β|d|−λ) Γ(λ+2) (j−δ)2

Theorem 6.3 Let the function f ∈ A be in the class L(λ, d, α, β). Then f is univalent close-to-convex of order 1   2(j−1) j m−1 Γ(n+j) (1−δ)2 Γ(j+λ+1) {αj 2 +[α(2λ−2+β|d|)+1]j+[α(λ−1)+1](λ−1+β|d|)} δ, 0 ≤ δ < 1, in |z| < r3 , where r3 = inf j . Γ(n+1) (αλ+1)(β|d|−λ) Γ(λ+2)

The result is sharp for the function f (z) given by (6.1). P ∞ Proof. It suffices to show that |f 0 (z) − 1| ≤ 1 − δ, |z| < r3 . Then |f 0 (z) − 1| = j=2 jaj z j−1 ≤ P∞ P∞ ja j−1 . Thus |f 0 (z) − 1| ≤ 1 − δ if j=2 1−δj |z|j−1 ≤ 1. Using Theorem 2.1, the above inequality holds j=2 jaj |z| s true if |z|j−1 ≤  |z| ≤

(1−δ)2

(1−δ) j

j m−1 Γ(n+j) Γ(j+λ+1)

j m+1 Γ(n+j) {αj 2 +[α(2λ−2+β|d|)+1]j+[α(λ−1)+1](λ−1+β|d|)} Γ(j+λ+1) Γ(n+1) (αλ+1)(β|d|−λ) Γ(λ+2)

{αj 2 +[α(2λ−2+β|d|)+1]j+[α(λ−1)+1](λ−1+β|d|)} Γ(n+1)

(αλ+1)(β|d|−λ) Γ(λ+2)

or

1  2(j−1)

. Hence the proof is complete.

References [1] A. Alb Lupa¸s, Properties on a subclass of univalent functions defined by using S˘ al˘ agean operator and Ruscheweyh derivative, J. of Comput. Anal. Appl., 21 (2016), No.7, 1213-1217. [2] N.E.Cho, A.M.K. Aouf, Some applications of fractional calculus operators to a certain subclass of analytic functions with negative coefficients, Tr. J. of Mathematics, Vol. 20, 1996, 553-562. [3] St. Ruscheweyh, New criteria for univalent functions, Proc. Amet. Math. Soc., 49(1975), 109-115. [4] G. St. S˘al˘agean, Subclasses of univalent functions, Lecture Notes in Math., Springer Verlag, Berlin, 1013 (1983), 362-372.

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Normal criteria of meromorphic functions concerning holomorphic functions∗ Da-Wei Meng1 , San-Yang Liu1 , and Hong-Yan Xu1,2

1

School of Mathematics and statistics, Xidian University, Xi’an 710071, Shaanxi, P. R. China 2 Department of Informatics and Engineering, Jingdezhen Ceramic Institute, Jingdezhen, Jiangxi, 333403, China E-mail: [email protected](Da-Wei Meng); [email protected](San-Yang Liu); [email protected]

Abstract In this paper, we mainly investigate the problem of normal families of meromorphic functions concerning shared functions, and obtain some normality criteria of meromorphic functions sharing a holomorphic function. Our results generalize or extend the previous theorems given by Ding J. J., Ding L. W. and Yuan W. J..

1

Introduction and main results

Let F be a family of meromorphic functions defined in a domain D. In the sense of Montel, F is said to be normal in D, if for any sequence {fn } ⊂ F there exists a

2010 Mathematics Subject Classification. Primary 30D35, 30D45. This work is supported by Natural Science Foundation of China (Grant No.11271227, No.11201360, No.61373174 and No.11561033), and the Fundamental Research Funds for the Central Universities of China (Grant No.800272125771). ∗

Key words: meromorphic functions; normal family; sharing a holomorphic function.

1

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subsequence {fnj } which converges spherically locally uniformly in D, to a meromorphic function or ∞. For simplicity, we take → to stand for convergence, ⇒ for convergence spherically locally uniformly, and M(D) (resp. A(D)) for the set of meromorphic (resp. holomorphic) functions on D. Let F and G two non-constant meromorphic functions defined in D. Then we say that f and g share a IM if F − a and G − a assume the same zeros ignoring multiplicity. The zeros of F − a mean the poles of F when a = ∞ . In 1959, Hayman [9] proposed a conjecture: if F ∈ M(C) is transcendental, then F n F ′ assumes every finite non-zero complex number infinitely often for any positive integer n. The conjecture is showed to be true by many authors, such as Hayman [10], Mues [17], Clunie [6], Bergweiler and Eremenko [2], Chen and Fang [4]. Accordingly, Hayman [10] conjectured that if F is the family of M(D) such that each f ∈ F satisfies f n f ′ ̸= a for a positive integer n and a non-zero complex number a, then F is normal. This conjecture has been confirmed by some authors, such as Yang and Zhang [26] , Gu [8], Pang [20], Oshkin [18] and Pang [20]. In 2008, from the point of shared values, Zhang [29] concluded that if F is the family of M(D) such that each pair (f, g) of F satisfies that f n f ′ and g n g ′ share a finite non-zero complex number a IM for n ≥ 2, then F is normal. Recently, Jiang and Gao [12] generalized Zhang’s result based on the ideas of shared functions. For other generations, we can refer to [3, 15, 24]. For the case of F n F (k) , Zhang and Li [31] proved that if F ∈ M(C) is transcendental, then F n L[F ] − a has infinitely many zeros for n ≥ 2 and a ̸= 0, ∞, where L[F ] = ak F (k) + ak−1 F (k−1) + · · · + a0 F in which ai (i = 0, 1, 2, · · · , k) are small functions of F . Pang and Zalcman [22] further obtained the corresponding normality criterion as follows: If F is the family of A(D) such that zeros of each f ∈ F have multiplicities at least k and such that each f ∈ F satisfies f n f (k) ̸= a for a non-zero complex number a, then F is normal. Recently, Meng and Hu [16] extended Pang’s result, by replacing f n f (k) ̸= a into the condition that f n f k and g n g k share a IM. Similarly, we also have analogues related to ( )l some conditions of f f (k) for a positive integer l (refer to [1, 11, 13, 30]). In 2013, considering the general case of F n (F (k) )l from the view of shared values, Ding, Ding and Yuan [7] proved a normality criterion as follows: Let a be a non-zero value, if F is the family of M(D) such that each pair (f, g) of F satisfies that f n (f (k) )l and g n (g (k) )l share a non-zero value a, where each f ∈ F has only zeros of multiplicity at least max(k, 2), then F is normal. Naturally we ask: whether there exists normality theorem when a is a function? Take four integers k ≥ 1, m ≥ 0, n ≥ 1 and l ≥ 2. Let a (̸≡ 0) be a holomorphic function in a domain D such that multiplicities of zeros of a are at most m and divisible by n + l. In this paper, we prove the following normality criterion:

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Theorem 1.1. Let F be the family of M(D) such that multiplicities of zeros of each f ∈ F are at least k + m + 1 and such that multiplicities of poles of f are at least m + 1 whenever f have zeros and poles. If each pair (f, g) of F satisfies that f n (f (k) )l and g n (g (k) )l share a IM, then F is normal in D. In special, when k = 1, we may modify Theorem 1.1 as follows: Theorem 1.2. Suppose a = a(z) as in Theorem 1.1, if F is the family of M(D) such that each f ∈ F satisfies that f n (f ′ )l ̸= a, then F is normal in D. Similar to the proof of Theorem 1.2, we conclude the following result: Theorem 1.3. Suppose a = a(z) as in Theorem 1.1, if F is the family of M(D) such that each f ∈ F satisfies that f n (f ′ (z))l = a implies |f (z)| > A for a positive number A, then F is normal in D. As a matter of fact, Theorem 1.3 is inspired by the ideas of papers [11, 13] initially.

2

Preliminary lemmas First of all, we introduce the following Zalcman’s lemma [28]:

Lemma 2.1. Take a positive integer k. Let F be a family of meromorphic functions in the unit disc △ with the property that zeros of each f ∈ F are of multiplicity at least k. If F is not normal at a point z0 ∈ ∆, then for 0 ≤ α < k, there exist a sequence {zn } ⊂ ∆ of complex numbers with zn → z0 ; a sequence {fn } of F ; and a sequence {ρn } of positive numbers with ρn → 0 such that gn (ξ) = ρ−α n fn (zn + ρn ξ) locally uniformly (with respect to the spherical metric) to a nonconstant meromorphic function g(ξ) on C. Moreover, the zeros of g(ξ) are of multiplicity at least k, and the function g(ξ) may be taken to satisfy the normalization g ♯ (ξ) ≤ g ♯ (0) = 1 for any ξ ∈ C. In particular, g(ξ) has at most order 2. This Lemma is Pang’s generalization [19, 21, 25] to the Main Lemma in [27] (where α is taken to be 0), with improvements due to Schwick [23], Chen and Gu [5]. Next, by applying the results from [1, 14, 31, 30] we can deduce the following lemma: Lemma 2.2. Let f be a transcendental meromorphic function in the complex plane. Let n, l, k be three positive integers and a = a(z) ̸≡ 0 be a polynomial. Then for l ≥ 2, f n (f (k) )l − a has infinitely many zeros. Finally, we investigate the zeros of f n (f (k) )l − a if f is rational, and thus give Lemma 2.3 and 2.4: 3

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Lemma 2.3. Let p ≥ 0, n, k ≥ 1 and l ≥ 2 be four integers, and let a be a non-zero polynomial of degree p. If f is a non-constant rational function which has only zeros of multiplicity at least k+p+1 and has only poles of multiplicity at least p+1, then f n (f (k) )l −a has at least two distinct zeros. Proof. Firstly, we assume that f is a non-constant polynomial. It follows that f (k) ̸≡ 0 from f has only zeros of multiplicity at least k + p + 1. Hence we have ( ) deg f n (f (k) )l ≥ n(k + p + 1) + l(p + 1) > p = deg(a). Therefore, it follows that f n (f (k) )l −a is also a non-constant polynomial, and hence f n (f (k) )l − a has at least one zero. Further, we claim that f n (f (k) )l − a has at least two distinct zeros if f is a non-constant polynomial. To the contrary, suppose that f n (f (k) )l − a has only one zero z0 , which means f n (z)(f (k) )l (z) − a(z) = A′ (z − z0 )d , where A′ is a non-zero constant and d is a positive integer. Since f is a non-constant polynomial which has only zeros of multiplicity at least k + p + 1, we find f (k) ̸≡ 0, and hence d = deg(f n (f (k) )l − a) > deg(f n ) ≥ n(k + p + 1) ≥ p + 2. By computing we find { }(p+1) f n (f (k) )l (z) = A′ d(d − 1)...(d − p)(z − z0 )d−p−1 , {

f n (f (k))

l

}(p+1)

has a unique zero z0 . Take a zero ξ0 of f , then it is a zero of f n { }(p) with multiplicity at least n(k + p + 1). It follows that ξ0 is a zero of f n (f (k) )l and { n (k) l }(p+1) { n (k) l }(p) f (f ) , which further implies that ξ0 = z0 . Therefore, we obtain f (f ) (z0 ) = { n (k) l }(p) ′ d−p 0. On the other hand, we get f (f ) (z) = b + A d(d − 1)...(d − p + 1)(z − z0 ) , in { }(p) (z0 ) = which b is a non-zero constant such that b = a(p) (z). This yields that f n (f (k) )l { n (k) l }(p) (z0 ) = 0. The claim is proved. b ̸= 0, which is contradictory to f (f ) Secondly, we assume that f has poles, and then set hence

f (z) =

A(z − α1 )m1 (z − α2 )m2 · · · (z − αs )ms , (z − β1 )n1 (z − β2 )n2 · · · (z − βt )nt

(2.1)

where A is a non-zero constant, αi distinct zeroes of f with s ≥ 0, and βj distinct poles of f with t ≥ 1. For simplicity, we put m1 + m2 + · · · + ms = M ≥ (k + p + 1)s,

(2.2)

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n1 + n2 + · · · + nt = N ≥ (p + 1)t.

(2.3)

From (2.1), we obtain f (k) (z) =

(z − α1 )m1 −k (z − α2 )m2 −k · · · (z − αs )ms −k g(z) , (z − β1 )n1 +k (z − β2 )n2 +k · · · (z − βt )nt +k

(2.4)

where g is a polynomial of degree ≤ kl(s + t − 1). From (2.1) and (2.4), we get f n (z)(f (k) )l (z) =

An (z − α1 )M1 (z − α2 )M2 · · · (z − αs )Ms g l (z) , (z − β1 )N1 (z − β2 )N2 · · · (z − βt )Nt

(2.5)

in which Mi = (n + l)mi − kl, i = 1, 2, · · · , s, Nj = (n + l)nj + kl, j = 1, 2, · · · , t. Differentiating (2.5) yields { }(p+1) (z − α1 )M1 −p−1 (z − α2 )M2 −p−1 · · · (z − αs )Ms −p−1 g0 (z) f n (f (k) )l (z) = , (z − β1 )N1 +p+1 · · · (z − βt )Nt +p+1

(2.6)

where g0 (z) is a polynomial of degree ≤ (p + kl + 1)(s + t − 1). We claim that f n (f (k) )l − a has at least one zero if f is a non-polynomial rational function. In order to prove this claim, suppose the contrary holds, thus we set f n (z)(f (k) )l (z) = a(z) +

(z − β1

)N1 (z

C , − β2 )N2 · · · (z − βt )Nt

(2.7)

where C is a non-zero constant. Subsequently, (2.7) yields {

f n (f (k) )l

}(p+1) (z) =

g1 (z) , (z − β1 )N1 +p+1 · · · (z − βt )Nt +p+1

(2.8)

where g1 (z) is a polynomial of degree ≤ (p + 1)(t − 1). Comparing (2.6) with (2.8), we get (p + 1)(t − 1) ≥ deg(g1 ) ≥ (n + l)M − kls − (p + 1)s, and hence M
N . It follows that p + kl + 1 1 + p. The expression (2.12) yields {

n

f (f

(k) l

}(p+1)

)

(z) =

(z − z0 )d−p−1 g3 (z) , (z − β1 )N1 +p+1 · · · (z − βt )Nt +p+1

(2.14)

where g3 (z) is a polynomial with deg(g3 ) ≤ (p + 1)t. We claim that z0 ̸= αi for each i. Otherwise, if z0 = αi for some i, then (2.12) yields { }(p) { }(p) a(p) (z0 ) = f n (f (k) )l (z0 ) = f n (f (k) )l (αi ) = 0 because each αi is a zero of f n (f (k) )l of multiplicity > n(k+p+1) ≥ p+2. This is impossible since deg(a) = p. Hence (z − z0 )d−p−1 is a factor of the polynomial g0 in (2.6). By (2.6) and (2.14), we conclude that (p + 1)t ≥ deg(g3 ) ≥ (n + l)M − kls − (p + 1)s, which is equivalent to M≤

p+1 p + kl + 1 s+ t. n+l n+l

(2.15)

If d ̸= (n + l)N + klt + p, then (2.5) together with (2.12) implies (n + l)N + klt + p ≤ (n + l)M − kls + deg(g l ), so we get N < M from deg(g l ) ≤ kl(s + t − 1). Therefore, by using the facts M ≥ (k + p + 1)s, N ≥ (p + 1)t, (2.15) implies a contradiction { } p + kl + 1 1 M< + M ≤ M. (n + l)(k + p + 1) n + l Hence d = (n + l)N + klt + p. Now we must have N ≥ M , otherwise, when N < M , we can deduce the contradiction M < M from (2.15). Comparing (2.6) with (2.14), we find (p + kl + 1)(s + t − 1) ≥ deg(g0 ) ≥ d − p − 1 since (z − z0 )l−p−1 |g0 , and hence (n + l)N + klt + p = d ≤ (p + kl + 1)s + (p + kl + 1)t − kl, which further yields p+1 p+k+1 s+ t. n+1 n+1 Since M ≥ (k + p + 1)s and N ≥ (p + 1)t, it follows from that N
0 such that if a mapping f : G1 −→ G2 satisfies d(f (xy), f (x)f (y)) < c for all x, y ∈ G1 , then there exists a unique homomorphism h : G1 −→ G2 with d(f (x), h(x)) < δ for all x ∈ G1 ?” In the next year, Hyers [12] gave a partial solution of Ulam, s problem for the case of approximate additive mappings. Subsequently, his result was generalized by Aoki ([1]) for additive mappings and by Rassias [24] for linear mappings to consider the stability problem with unbounded Cauchy differences. During the last decades, the stability problem of functional equations have been extensively investigated by a number of mathematicians (see [4], [5], [6], [9], and [19]). In 2001, Rassias [23] introduced the following cubic functional equation (1.1)

f (x + 2y) − 3f (x + y) + 3f (x) − f (x − y) − 6f (y) = 0

and every solution of the cubic functional equation is called a cubic mapping and in ([14]), the following cubic functional equation was investigated (1.2)

f (2x + y) + f (2x − y) = 2f (x + y) + 2f (x − y) + 12f (x).

Katsaras [15] defined a fuzzy norm on a vector space to construct a fuzzy vector topological structure on the space. Later, some mathematicians have defined fuzzy norms on a vector space in different points of view. In particular, Bag and Samanta [2], following Cheng and Mordeson [3], gave an idea of fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michalek type [16]. In [10], Gl´ anyi showed that if a mapping f : X −→ Y satisfies the following functional inequality (1.3)

k2f (x) + 2f (y) − f (xy −1 )k ≤ kf (xy)k,

2010 Mathematics Subject Classification. 39B62, 39B72, 54A40, 47H10. Key words and phrases. Hyers-Ulam stability, fuzzy normed space, fixed point theorem. * Corresponding author. The second author was supported by the research fund of Dankook University in 2018. 1

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then f satisfies the Jordan-Von Neumann functional equation 2f (x) + 2f (y) − f (xy −1 ) = f (xy). Gl´ anyi [11] and Fechner [8] proved the Hyers-Ulam stability of (1.3). Park, Cho, and Han [22] proved the Hyers-Ulam stability of the following functional inequality: (1.4)

kf (x) + f (y) + f (z)k ≤ kf (x + y + z)k.

Further, Park [21] proved the generalized Hyers-Ulam stability of the Cauchy additive functional inequality (1.4) in fuzzy Banach spaces using the fixed point method if f is an odd mapping. In this paper, we investigate the following functional inequality related by (1.1) and (1.2) (1.5)

N (f (x + 2y) − 3f (x + y) + 3f (x) − f (x − y) − 6f (y), t) ≥ N (f (2x + y) + f (2x − y) − 2f (x + y) − 2f (x − y) − 12f (x), kt)

for some fixed nonzero real number k and prove the generalized Hyers-Ulam stability for (1.5) in fuzzy Banach spaces by fixed point methods. 2. preliminaries In this paper, we use the definition of fuzzy normed spaces given in [2], [17], and [18]. Definition 2.1. Let X be a real vector space. A function N : X × R −→ [0, 1] is called a fuzzy norm on X if for any x, y ∈ X and any s, t ∈ R, (N1) N (x, t) = 0 for t ≤ 0; (N2) x = 0 if and only if N (x, t) = 1 for all t > 0; t (N3) N (cx, t) = N (x, |c| ) if c 6= 0; (N4) N (x + y, s + t) ≥ min{N (x, s), N (y, t)}; (N5) N (x, ·) is a nondecreasing function of R and limt→∞ N (x, t) = 1; (N6) for any x 6= 0, N (x, ·) is continuous on R. In this case, the pair (X, N ) is called a fuzzy normed space. Let (X, N ) be a fuzzy normed space and {xn } a sequence in X. Then (i) {xn } is said to be Cauchy in (X, N ) if for any  > 0, there exists an m ∈ N such that N (xn+p − xn , t) > 1 −  for all n ≥ m, all positive integer p, and any t > 0 and (ii) {xn } is said to be convergent in (X, N ) if there exists an x ∈ X such that limn→∞ N (xn − x, t) = 1 for all t > 0. In this case, x is called the limit of the sequence {xn } in X and one denotes it by N − limn→∞ xn = x. Example 2.2. For example, it is well known that for any normed space (X, || · ||) and any nonnegative real number , the mapping NX : X × R −→ [0, 1], defined by ( 0, if t ≤ 0 NX (x, t) = t , if t>0, t+||x|| is a fuzzy norm on X([17], [18], and [19]). It is well known that every convergent sequence in a fuzzy normed space is Cauchy. A fuzzy normed space is said to be complete if each Cauchy sequence in it is convergent and a complete fuzzy normed space is called a fuzzy Banach space.

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In 1996, Isac and Rassias [13] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. Theorem 2.3. [7] Let (X, d) be a complete generalized metric space and let J : X −→ X be a strictly contractive mapping with some Lipschitz constant L with 0 < L < 1. Then for each given element x ∈ X, either d(J n x, J n+1 x) = ∞ for all nonnegative integer n or there exists a positive integer n0 such that (1) d(J n x, J n+1 x) < ∞ for all n ≥ n0 ; (2) the sequence {J n x} converges to a fixed point y ∗ of J ; (3) y ∗ is the unique fixed point of J in the set Y = {y ∈ X | d(J n0 x, y) < ∞}; 1 (4) d(y, y ∗ ) ≤ d(y, Jy) for all y ∈ Y . 1−L Throughout this paper, we assume that X is a linear space, (Y, N ) is a fuzzy Banach space, and (Z, N 0 ) is a fuzzy normed space. 3. Solutions of (1.5) In this section, we investigate the solution and prove the generalized Hyers-Ulam stability of the functional inequality (1.5) in fuzzy Banach spaces. For any mapping f : X −→ Y , let Af (x, y) = f (2x + y) + f (2x − y) − 2f (x + y) − 2f (x − y) − 12f (x) and Bf (x, y) = f (x + 2y) − 3f (x + y) + 3f (x) − f (x − y) − 6f (y). By (N5), we can easily shown the following lemma. Lemma 3.1. Let αi : [0, ∞) −→ [0, ∞)(i = 1, 2, · · ·, n) be a mapping and r a positive real numbers with r > 1 and y, z, z1 , z2 , ·, zn ∈ Y . Suppose that N (y, t) ≥ min{N (z, rn t), N (z1 , α1 (t)), N (z2 , α2 (t)), · · ·, N (zn , αn (t))} for all t > 0 and all n ∈ N. Then N (y, t) ≥ min{N (z1 , α1 (t)), N (z2 , α2 (t)), · · ·, N (zn , αn (t))} for all t > 0. By Lemma 3.1, we have the following corollary. Corollary 3.2. Let r be a real number with r > 1 and y ∈ Y . Suppose that N (y, t) ≥ N (y, rt) for all t > 0. Then y = 0. Using Lemma 3.1 and Corollary 3.2, we will prove the following theorem : Theorem 3.3. Let f : X −→ Y be a mapping. Suppose that k is a real number with k > 4. Then f is cubic if and only if f is a solution of (1.5). Proof. Letting x = 0 and y = 0 in (1.5), we have  3  N (f (0), t) ≥ N f (0), kt 7 3 for all t > 0 and sicne 7 k > 1, by Corollary 3.2, we get f (0) = 0.

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Suppose that f is a solution of (1.5). Letting x = 0 in (1.5), we have (3.1)

N (f (2y) − 9f (y) − f (−y), t) ≥ N (f (y) + f (−y), kt)

for all y ∈ X and all t > 0 and letting y = −x in (1.5), we have (3.2)

N (f (2x) − 3f (x) + 5f (−x), t) ≥ N (f (3x) − 2f (2x) − 11f (x), kt)

for all x ∈ X and all t > 0. Letting y = x in (1.5), we have (3.3)

N (f (3x) − 3f (2x) − 3f (x), t) ≥ N (f (3x) − 2f (2x) − 11f (x), kt)

for all x ∈ X and all t > 0. By (3.1) and (3.2), we get N (6f (x) + 6f (−x), t) n  t  t o ≥ min N f (2x) − 9f (x) − f (−x), , N f (2x) − 3f (x) + 5f (−x), 2 2 n  kt   kt o ≥ min N f (x) + f (−x), , N f (3x) − 2f (2x) − 11f (x), 2 2 for all x ∈ X and all t > 0 and so we obtain N (f (x) + f (−x), t) (3.4) ≥ min{N (f (x) + f (−x), 3kt), N (f (3x) − 2f (2x) − 11f (x), 3kt)} for all x ∈ X and all t > 0. For any x ∈ X, let G(x) = f (3x) − 2f (2x) − 11f (x), H(x) = f (x) + f (−x) for all x ∈ X. By (3.1), (3.4), and (N5), we have n  t  t o N (f (2x) − 8f (x), t) ≥ min N f (2x) − 9f (x) − f (−x), , N H(x), 2 2 n  3kt   3kt o kt   (3.5) , N H(x), , N G(x), ≥ min N H(x), 2 2 2 n    o kt 3kt ≥ min N H(x), , N G(x), 2 2 for all x ∈ X and all t > 0. Further, by (3.3), (3.5), and (N5), we have n  t  t o N (G(x), t) ≥ min N f (3x) − 3f (2x) − 3f (x), , N f (2x) − 8f (x), 2 2 n  kt   kt   3kt o (3.6) ≥ min N G(x), , N H(x), , N G(x), 2 4 4 n    o kt kt ≥ min N G(x), , N H(x), 2 4 for all x ∈ X and all t > 0 and since k > 4, by (3.6) and (N5), we have n  kt   kt o N (G(x), t) ≥ min N G(x), , N H(x), 2 4 n  k2 t   k2 t   kt o ≥ min N G(x), 2 , N H(x), 3 , N H(x), 2 2 4 n  k2 t   kt o ≥ min N G(x), 2 , N H(x), 2 4 for all x ∈ X and all t > 0. Hence by induction, we get n  kn t   kt o (3.7) N (G(x), t) ≥ min N G(x), n , N H(x), 2 4

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for all x ∈ X, all t > 0 and all n ∈ N. By Lemma 3.1 and (3.7), we obtain  kt  N (G(x), t) ≥ N H(x), 4 for all x ∈ X and all t > 0. By (3.4) and (N5), we have  kt  N (G(x), t) ≥ N H(x), 4 n  (3k)2 t   (3k)2 t o ≥ min N H(x), , N G(x), 12 12 (3.8) n  (3k)3 t   (3k)3 t   (3k)2 t o ≥ min N H(x), , N G(x), , N G(x), 12 12 12 n  (3k)2 t o (3k)3 t   , N G(x), ≥ min N H(x), 12 12 for all x ∈ X and all t > 0. By induction and (3.8), we get  (3k)2 t  (3.9) N (G(x), t) ≥ N G(x), 12 for all x ∈ X and all t > 0. By (3.9) and Corollary 3.2, we get (3.10)

G(x) = f (3x) − 2f (2x) − 11f (x) = 0

for all x ∈ X. By (3.4) and (3.10), we get (3.11)

N (H(x), t) ≥ N (H(x), 3kt)

for all x ∈ X and by Corollary 3.2, we have (3.12)

H(x) = f (x) + f (−x) = 0

for all x ∈ X. Hence f is an odd mapping. Further, by (3.5), (3.10), and (3.12), we get (3.13)

f (2x) = 8f (x)

for all x ∈ X. Now, letting x = 2y in (1.5), by (3.13), we have (3.14)

N (8f (x + y) − 3f (2x + y) + 24f (x) − f (2x − y) − 6f (y), t) ≥ N (f (4x + y) + f (4x − y) − 2f (2x + y) − 2f (2x − y) − 96f (x), kt)

for all x, y ∈ X and all t > 0 and letting y = −y in (3.14), by (3.12), we have (3.15)

N (8f (x − y) − 3f (2x − y) + 24f (x) − f (2x + y) + 6f (y), t) ≥ N (f (4x + y) + f (4x − y) − 2f (2x + y) − 2f (2x − y) − 96f (x), kt)

for all x, y ∈ X and all t > 0. By (3.14) and (3.15), we have  kt  N (4Af (x, y), t) ≥ N Af (2x, y), 2 for all x, y ∈ X and all t > 0 and so we have (3.16)

N (Af (x, y), t) ≥ N (Af (2x, y), 2kt)

for all x, y ∈ X and all t > 0. Letting y = 2y in (1.5), we have (3.17)

N (f (x + 4y) − 3f (x + 2y) + 3f (x) − f (x − 2y) − 48f (y), t) ≥ N (8f (x + y) + 8f (x − y) − 2f (x + 2y) − 2f (x − 2y) − 12f (x), kt)

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for all x, y ∈ X and all t > 0 and interchang x and y in (3.17), by (3.12) and (1.5), we get N (f (4x + y) − 3f (2x + y) + 3f (y) + f (2x − y) − 48f (x), t) ≥ N (8f (x + y) − 8f (x − y) − 2f (2x + y) + 2f (2x − y) − 12f (y), kt) = N (−2Af (x, y) − 4Bf (y, −x), kt) n  kt   kt o ≥ min N A (x, y), , N B (y, −x), (3.18) f f 6 6   n  k 2 t o kt , N Af (y, −x), ≥ min N Af (x, y), 6 6 n    kt k 2 t o ≥ min N Af (x, y), , N Af (y, x), 6 6 for all x, y ∈ X and all t > 0. Letting y = −y in (3.18), we get N (f (4x − y) − 3f (2x − y) − 3f (y) + f (2x + y) − 48f (x), t) n  kt   k 2 t o ≥ min N A (x, −y), , N A (−y, −x), f f (3.19) 6 6 n  kt   k 2 t o ≥ min N Af (x, y), , N Af (y, x), 6 6 for all x, y ∈ X. By (3.16), (3.18), (3.19), and (N5), we have n  k2 t   k 3 t o N (Af (x, y), t) ≥ N (Af (2x, y), 2kt) ≥ min N Af (x, y), , N Af (y, x), 6 6 for all x, y ∈ X and all t > 0. By Lemma 3.1 and induction, we get Af (x, y) = f (2x + y) + f (2x − y) − 2f (x + y) − 2f (x − y) − 12f (x) = 0 for all x, y ∈ X. Thus f is a cubic mapping.



By Theorem 3.3, we have the following corollaries : Corollary 3.4. Let f : X −→ Y be a mapping. Suppose that a, b are real numbers with |a| > 4|b| > 0. Then f is cubic if and only if f satisfies the following inequlaity (3.20)

N (aBf (x, y), t) ≥ N (bAf (x, y), t)

for all x, y ∈ X and all t > 0. Corollary 3.5. Let f : X −→ Y be a mapping. Suppose that a is a real number with |a| > 8. Then f is cubic if and only if f satisfies the following inequlaity (3.21)

N (aBf (x, y) + Af (x, y), t) ≥ N (Af (x, y), t)

for all x, y ∈ X and all t > 0. Proof. By (3.21) and (N5), we have N (Bf (x, y), t) = N (aBf (x, y), |a|t) n  |a|   |a| o ≥ min N aBf (x, y) + Af (x, y), t , N Af (x, y), t 2 2  |a|  = N Af (x, y), t 2 for all x, y ∈ X and all t > 0. Hence by Theorem 3.3, we have the result.

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Using the fuzzy norm NX : X ×R −→ [0, 1] in Exmaple 2.2, we have the following corollary : Corollary 3.6. Let (X, || · ||) be a normed space and f : X −→ Y a mapping. Suppose that a is a real number with |a| > 8. Then f is cubic if and only if f satisfies the following inequlaity kaBf (x, y) + Af (x, y)k ≤ kAf (x, y)k

(3.22) for all x, y ∈ X.

4. The generalized Hyers-Ulam stability for (1.5) Now, we will prove the generalized Hyers-Ulam stability for (1.5) in fuzzy normed spaces. Theorem 4.1. Assume that φ : X 3 −→ [0, ∞) is a function such that N 0 (φ(2x, 2y), t) ≥ N 0 (8Lφ(x, y), t)

(4.1)

for all x, y ∈ X, t > 0 and some real number L with 0 < L < 1. Let f : X −→ Y be a mapping such that f (0) = 0 and (4.2)

N (Bf (x, y), t) ≥ min{N (Af (x, y), kt), N 0 (φ(x, y), t)}

for all x, y ∈ X and t > 0. Then there exists a unique cubic mapping C : X −→ Y such that N (f (x) − C(x),

(4.3)

1 t) ≥ Ψ(x, t) 8(1 − L)

for all x ∈ X, t > 0,nandsome real number   o  k with k > 4,  3kt t 0 0 0 where Ψ(x, t) = min N φ(x.x), 4 , N φ(x. − x), 3t , N . φ(0, x), 2 2 Proof. Letting x = 0 in (4.2), by (N2), we have (4.4)

N (f (2y) − 9f (y) − f (−y), t) ≥ min{N (H(y), kt), N 0 (φ(0, y), t)}

for all y ∈ X and t > 0 and letting y = −x in (4.2), by (N2), we have (4.5)

N (f (2x) + 5f (−x) − 3f (x), t) ≥ min{N (G(x), kt), N 0 (φ(x. − x), t)}

for all x ∈ X and t > 0. Letting y = x in (4.2), we have (4.6)

N (f (3x) − 3f (2x) − 3f (x), t) ≥ min{N (G(x), kt), N 0 (φ(x.x), t)}

for all x ∈ X and all t > 0. By (4.4) and (4.5), we get (4.7)

N (H(x), t) ≥ min{N (H(x), 3kt), N (G(x), 3kt), N 0 (φ(0, x), 3t), N 0 (φ(x. − x), 3t)}

for all x ∈ X and all t > 0. Similar to the proof of Theorem 3.3, by (4.7), we have (4.8)

N (H(x), t) ≥ min{N (G(x), 3kt), N 0 (φ(0, x), 3t), N 0 (φ(x. − x), 3t)}

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for all x ∈ X and all t > 0. By (4.4), (4.8), and (N5), we get (4.9) n  t  t o N (f (2x) − 8f (x), t) ≥ min N H(x), , N f (2x) − 9f (x) − f (−x), 2 2 n  t o kt  0  t  , N H(x), , N φ(0, x), ≥ min N H(x), 2 2 2 n  3t  0  t o 3kt  0  ≥ min N G(x), , N φ(x. − x), , N φ(0, x), 2 2 2 for all x ∈ X and all t > 0 and by (4.6), (4.9), and (N5), we get (4.10) n  t  t o N (G(x), t) ≥ min N f (3x) − 3f (2x) + 5f (−x), , N f (2x) − 8f (x), 2 2 n  t  0 3t  0  t o kt  0  , N φ(x.x), , N φ(x. − x), , N φ(0, x), ≥ min N G(x), 2 2 4 4 for all x ∈ X and all t > 0. Since k > 4, by (4.10) and (N5), we obtain n  3t  0  t o t  0 (4.11) N (G(x), t) ≥ min N 0 φ(x.x), , N φ(x. − x), , N φ(0, x), 2 4 4 for all x ∈ X and all t > 0. By (4.9), (4.11), and (N5), we get N (f (2x) − 8f (x), t) n  3kt  0  9kt  0  3kt  ≥ min N 0 φ(x.x), , N φ(x. − x), , N φ(0, x), , 4 8 8  o   (4.12) 3t t , N 0 φ(0, x), N 0 φ(x. − x), 2 2 n  3kt  0  3t  0  t o 0 ≥ min N φ(x.x), , N φ(x. − x), , N φ(0, x), 4 2 2 for all x ∈ X and all t > 0. Consider the set S = {g | g : X −→ Y } and the generalized metric d on S defined by d(g, h) = inf{c ∈ [0, ∞) | N (g(x) − h(x), ct) ≥ Ψ(x, t), ∀x ∈ X, ∀t > 0}. Then (S, d) is a complete metric space(See [20]). Define a mapping J : S −→ S by Jg(x) = 81 g(2x) for all x ∈ X and all g ∈ S. Let g, h ∈ S and d(g, h) ≤ c for some c ∈ [0, ∞). Then by (4.1), we have t N (Jg(x) − Jh(x), ct) = N (g(2x) − h(2x), 8ct) ≥ Ψ(2x, 8t) ≥ Ψ(x, ) L for all x ∈ X and t > 0. Hence N (Jg(x) − Jh(x), cLt) ≥ Ψ(x, t) for all x ∈ X and t > 0 and thus d(Jg, Jh) ≤ Ld(g, h) for any g, h ∈ S. Moreover, by (4.12), we have d(Jf, f ) ≤ 18 < ∞. By Theorem 2.3, there exists a mapping C : X −→ Y which is a fixed point of J such that d(J n f, A) → 0 as n → ∞. That is, f (2n x) n→∞ 23n for all x ∈ X. Replacing x, and y by 2n x and 2n y in (4.2), respectively, by (4.1), we have N (Bf (2n x, 2n ), 23n t) n  (4.14) 1 o ≥ min N (Af (2n x, 2n y), 23n t), N 0 φ(x, y), n t L (4.13)

C(x) = N − lim

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for all x, y ∈ X and all n ∈ N. Letting n → ∞ in (4.14), C is a solution of (1.5) and so by Theorem 3.3, C is a cubic mapping. Since d(f, Jf ) ≤ 18 , by Theorem 2.3, we have (4.17). Now, we show the uniqueness of C. Let C0 be another cubic mapping with (4.17). Then for any positive integer n, C0 (2n x) C(2n x) , C (x) = 0 23n 23n for all x ∈ X. Hence by (4.17), (N3) and (N4), we have C(x) =

N (C(x) − C0 (x), t) = N (C(2n x) − C0 (2n x), 23n t) ≥ Ψ(2n x, 23n 8(1 − L)t)  8(1 − L)t  ≥ Ψ x, Ln for all x ∈ X, t > 0, and all n ∈ N. Hence, letting n → ∞ in the above inequality, we have C(x) = C0 (x) for all x ∈ X.  By Corollary 3.5 and Theorem 4.1, we can show that the following corollaries: Corollary 4.2. Let  and p be real numbers with  ≥ 0 and 0 < p < 23 . Let f : X −→ Y be a mapping such that n o t (4.15) N (Bf (x, y), t) ≥ min N (Af (x, y), kt), 2p 2p p p t + (kxk + kyk + kxk kyk ) for all x, y ∈ X, all t > 0 and some real number k with k > 4. Then there exists a unique cubic mapping C : X −→ Y such that N (f (x) − C(x), t) ≥

(8 − 22p )t (8 − 22p )t + 2kxk2p

for all x ∈ X and all t > 0. Corollary 4.3. Assume that φ : X 3 −→ [0, ∞) is a function with (4.1) Let f : X −→ Y be a mapping such that f (0) = 0 and (4.16)

N (aBf (x, y) + Af (x, y), t) ≥ min{N (Af (x, y), t), N 0 (φ(x, y), t)}

for all x, y ∈ X, all t > 0 and some real numbers a with |a| > 8. Then there exists a unique cubic mapping C : X −→ Y such that (4.17)

N (f (x) − C(x),

1 t) ≥ Ψ(x, t) 8(1 − L)

for all x ∈ X, t > 0,nandsome fixed real k with k > 4,  number  o 3|a|t t 0 0 0 where Ψ(x, t) = min N φ(x.x), 8 , N φ(x. − x), 3t , N φ(0, x), . 2 2 Proof. By (N5) and (4.16), we have n  |a|   |a| o N (Bf (x, y), t) ≥ min N aBf (x, y) + Af (x, y), t , N Af (x, y), t 2 2 n  |a|  0  |a| o ≥ min N Af (x, y), t , N φ(x, y), t 2 2 n  o |a|  0  ≥ min N Af (x, y), t , N φ(x, y), t 2 for all x, y ∈ X, all t > 0. Hence we have the results.

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Corollary 4.4. Let  and p be real numbers with  ≥ 0 and 0 < p < f : X −→ Y be a mapping such that (4.18)

3 2.

Let

N (aBf (x, y) + Af (x, y), t) n o t ≥ min N (Af (x, y), t), t + (kxk2p + kyk2p + kxkp kykp )

for all x, y ∈ X, all t > 0 and some real number a with |a| > 8. Then there exists a unique cubic mapping C : X −→ Y such that N (f (x) − C(x), t) ≥

(8 − 22p )t (8 − 22p )t + 2kxk2p

for all x ∈ X and all t > 0. Related with Theorem 4.1, we can also have the following theorem. And the proof is similar to that of Theorem 4.1. Theorem 4.5. Assume that φ : X 3 −→ [0, ∞) is a function such that L   x y  , t ≥ N0 φ(x, y), t (4.19) N0 φ , 2 2 8 for all x, y ∈ X, t > 0 and some L with 0 ≤ L < 1. Let f : X −→ Y be a mapping with f (0) = 0 and (4.2). Then there exists a unique cubic mapping C : X −→ Y such that  L  (4.20) N f (x) − C(x), t ≥ Ψ0 (x, t) 1−L for all x ∈ X, t > 0, and fixed real k with k> 4,  n some   number  o where Ψ0 (x, t) = min N 0 φ(x.x), 6kt , N 0 φ(x. − x), 12t , N 0 φ(0, x), 4t . Proof. By (4.12) in Theorem 4.1, we get  x  N f (x) − 8f ,t 2 (4.21) n  6kt  0  12t  0  4t o ≥ min N 0 φ(x.x), , N φ(x. − x), , N φ(0, x), L L L for all x ∈ X and all t > 0. Consider the set S = {g | g : X −→ Y } and the generalized metric d on S defined by d(g, h) = inf{c ∈ [0, ∞) | N (g(x) − h(x), ct) ≥ Ψ0 (x, t), ∀x ∈ X, ∀t > 0}. Then (S, d)is acomplete metric space(See [20]). Define a mapping J : S −→ S by Jg(x) = 8g 12 x for all x ∈ X and all g ∈ S. Let g, h ∈ S and d(g, h) ≤ c for some c ∈ [0, ∞). Then by (4.19), we have  1  1   1 t   t N (Jg(x) − Jh(x), ct) = N 8g x − 8h x , ct ≥ Ψ0 x, ≥ Ψ0 x, 2 2 2 8 L for all x ∈ X and t > 0. Hence N (Jg(x) − Jh(x), cLt) ≥ Ψ0 (x, t) for all x ∈ X and t > 0 and thus d(Jg, Jh) ≤ Ld(g, h) for any g, h ∈ S. Moreover, by (4.21) we have d(f, Jf ) ≤ L < ∞. The rest of the proof is similar to Theorem 4.1.  By Corollary 3.6 and Theorem 4.5, we can show that the following corollaries:

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Corollary 4.6. Let  and p be real numbers with  ≥ 0 and p > 32 . Let f : X −→ Y be a mapping such that o n t (4.22) N (Bf (x, y), t) ≥ min N (Af (x, y), kt), t + (kxk2p + kyk2p + kxkp kykp ) for all x, y ∈ X, all t > 0 and some real number k with k > 4. Then there exists a unique cubic mapping C : X −→ Y such that N (f (x) − C(x), t) ≥

(22p − 8)t (22p − 8)t + 2kxk2p

for all x ∈ X and all t > 0. Corollary 4.7. Assume that φ : X 3 −→ [0, ∞) is a function with (4.1) Let f : X −→ Y be a mapping such that f (0) = 0 and (4.23)

N (aBf (x, y) + Af (x, y), t) ≥ min{N (Af (x, y), t), N 0 (φ(x, y), t)}

for all x, y ∈ X, all t > 0 and some real numbers a, b with |a| > 8. Then there exists a unique cubic mapping C : X −→ Y such that L t) ≥ Ψ0 (x, t) (4.24) N (f (x) − C(x), 1−L for all x ∈ X, t > 0, and fixed real k with k > n some   number   4,  o 0 0 where Ψ0 (x, t) = min N φ(x.x), 3|a|t , N φ(x. − x), 12t , N 0 φ(0, x), 4t . Corollary 4.8. Let  and p be real numbers with  ≥ 0 and p > 32 . Let f : X −→ Y be a mapping such that N (aBf (x, y) + Af (x, y), t) n o (4.25) t ≥ min N (Af (x, y), t), t + (kxk2p + kyk2p + kxkp kykp ) for all x, y ∈ X, all t > 0 and some real number a with |a| > 8. Then there exists a unique cubic mapping C : X −→ Y such that N (f (x) − C(x), t) ≥

(22p − 8)t (22p − 8)t + 2kxk2p

for all x ∈ X and all t > 0. References [1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2(1950), 64-66. [2] T. Bag and S. K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math. 11(2003), 687-705. [3] S. C. Cheng and J. N. Mordeson, Fuzzy linear operator and fuzzy normed linear spaces, Bull. Calcutta Math. Soc. 86(1994), 429-436. [4] P.W.Cholewa, Remarkes on the stability of functional equations, Aequationes Math. 27(1984), 76-86. [5] K Cieplinski, Applications of fixed point theorems to the hyers-ulam stability of functional equation-A survey, Ann. Funct. Anal. 3(2012), no. 1, 151-164. [6] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Bull. Abh. Math. Sem. Univ. Hamburg 62(1992), 59-64. [7] J. B. Diaz and B. Margolis, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 74 (1968), 305-309. [8] W. Fechner, Stability of a functional inequalty associated with the Jordan-Von Neumann functional equation, Aequationes Math. 71(2006), 149-161.

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[9] P. Gˇ avruta, A generalization of the Hyer-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184(1994), 431-436. . [10] A. Gil´ anyi, Eine zur Parallelogrammgleichung a ¨quivalente Ungleichung, Aequationes Mathematicae, 62(2001), 303-309. [11] A. Gil´ anyi, On a problem by K. Nikoden, Mathematical Inequalities and Applications, 5(2002), 701-710. [12] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. 27(1941), 222-224. [13] G. Isac and Th. M. Rassias, Stability of ψ-additive mappings, Appications to nonlinear analysis, Internat. J. Math. and Math. Sci. 19(1996), 219-228. [14] K. W. Jun and H. M. Kim, The generalized Hyers-Ulam-Rassias stability of a cubic functional equation, J. Math. Anal. Appl. 274(2002), 867-878. [15] A. K. Katsaras, Fuzzy topological vector spaces II, Fuzzy Sets Syst 12(1984), 143-154. [16] I. Kramosil and J. Mich´ alek, Fuzzy metric and statistical metric spaces, Kybernetica 11(1975), 336-344. [17] A. K. Mirmostafaee and M. S. Moslehian, Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Sets Syst. 159(2008), 720-729. [18] A. K. Mirmostafaee, M. Mirzavaziri, and M. S. Moslehian, Fuzzy stability of the Jensen functional equation, Fuzzy Sets Syst. 159(2008), 730-738. [19] M. Mirzavaziri and M. S. Moslehian, A fixed point approach to stability of a quadratic equation, Bulletin of the Brazilian Mathematical Society 37(2006), no. 3, 361-376 [20] M. S. Moslehian and T. H. Rassias, Stability of functional equations in non-Archimedean spaces, Applicable Anal. Discrete Math. 1(2007), 325-334. [21] C. Park, Fuzzy Stability of Additive Functional Inequalities with the Fixed Point Alternative, J. Inequal. Appl. 2009(2009), 1-17. [22] C. Park, Y. S. Cho, and M. H. Han, Functional inequalities associated with Jordan-von Neumann type additive functional equations, J. Inequal. Appl. 2007(2007), 1-13. [23] J. M. Rassias, Solution of the Ulam stability problem for cubic mappings, Glasnik Matematiˇ cki, 36(2001), 63-72. [24] Th. M. Rassias, On the stability of the linear mapping in Banach sapces, Proc. Amer. Math. Sco. 72(1978), 297-300. [25] S. M. Ulam, Problems in modern mathematics, Science Editions John Wiley and Sons, Inc., New York, 1964. Department of Mathematics Education, Dankook University, 152, Jukjeon-ro, Sujigu, Yongin-si, Gyeonggi-do, 448-701, Korea E-mail address: [email protected] Department of Mathematics Education, Dankook University, 126, Jukjeon, Yongin, Gyeonggi, South Korea 448-701, KOREA E-mail address: [email protected]

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO. 3, 2019

Modified Halpern's iteration without assumptions on fixed point set in metric space, Kanyarat Cheawchan and Atid Kangtunyakarn,………………………………………………………393 Convergence of double acting iterative scheme for a family of generalized 𝜑𝜑-weak contraction mappings in CAT(0) spaces, Kyung Soo Kim,…………………………………………….404 On solution of a system of differential equations via fixed point theorem, Muhammad Nazam, Muhammad Arshad, Choonkil Park, Ozlem Acar, Sungsik Yun, and George A. Anastassiou,417 Some equalities and inequalities for K-g-frames, Zhong-Qi Xiang and Yin-Suo Jia,………427 AQ-functional equation in matrix non-Archimedean fuzzy normed spaces, Jung-Rye Lee, George A. Anastassiou, Choonkil Park, Murali Ramdoss, and Vithya Veeramani,…………438 Existence of continuous selection for some special kind of multivalued mappings, G. Poonguzali, Muthiah Marudai, George A. Anastassiou, and Choonkil Park,…………………………….447 Refined stability of set-valued functional equations, Hong-Mei Liang, Hark-Mahn Kim, and Hwan-Yong Shin,……………………………………………………………………………453 Approximate Cauchy-Jensen and bi-quadratic mappings in 2-Banach spaces, Won-Gil Park and Jae-Hyeong Bae,…………………………………………………………………………….463 Birkhoff Normal Forms, KAM theory and continua of periodic points for certain planar system, M. R. S. Kulenović, E. Pilav, and N. Mujić,…………………………………………………470 Durrmeyer type (p, q)-Baskakov operators for functions of one and two variables, Qing-Bo Cai and Guorong Zhou,…………………………………………………………………………..481 A subclass of analytic functions defined by a fractional integral operator, Alb Lupaș Alina,502 Properties on a subclass of analytic functions defined by a fractional integral operator, Alb Lupaș Alina,…………………………………………………………………………………………506 Normal criteria of meromorphic functions concerning holomorphic functions, Da-Wei Meng, San-Yang Liu, and Hong-Yan Xu,……………………………………………………………511

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO. 3, 2019 (continued) Mixed Weakly Monotone Mappings and its Application to System of Integral Equations via Fixed Point Theorems, Deepak Singh, Om Prakash Chauhan, Afrah A N Abdou, and Garima Singh,………………………………………………………………………………………527 Functional inequalities in fuzzy normed spaces and its stability, Giljun Han, Chang Il Kim,544

Volume 27, Number 4 ISSN:1521-1398 PRINT,1572-9206 ONLINE

October 15, 2019

Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC

Journal of Computational Analysis and Applications ISSNno.’s:1521-1398 PRINT,1572-9206 ONLINE SCOPE OF THE JOURNAL An international publication of Eudoxus Press, LLC (fifteen times annually) Editor in Chief: George Anastassiou Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152-3240, U.S.A [email protected] http://www.msci.memphis.edu/~ganastss/jocaaa The main purpose of "J.Computational Analysis and Applications" is to publish high quality research articles from all subareas of Computational Mathematical Analysis and its many potential applications and connections to other areas of Mathematical Sciences. Any paper whose approach and proofs are computational,using methods from Mathematical Analysis in the broadest sense is suitable and welcome for consideration in our journal, except from Applied Numerical Analysis articles. Also plain word articles without formulas and proofs are excluded. The list of possibly connected mathematical areas with this publication includes, but is not restricted to: Applied Analysis, Applied Functional Analysis, Approximation Theory, Asymptotic Analysis, Difference Equations, Differential Equations, Partial Differential Equations, Fourier Analysis, Fractals, Fuzzy Sets, Harmonic Analysis, Inequalities, Integral Equations, Measure Theory, Moment Theory, Neural Networks, Numerical Functional Analysis, Potential Theory, Probability Theory, Real and Complex Analysis, Signal Analysis, Special Functions, Splines, Stochastic Analysis, Stochastic Processes, Summability, Tomography, Wavelets, any combination of the above, e.t.c. "J.Computational Analysis and Applications" is a peer-reviewed Journal. See the instructions for preparation and submission of articles to JoCAAA. Assistant to the Editor: Dr.Razvan Mezei,[email protected], Madison,WI,USA.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.4, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

Some Fixed Point Results of Caristi Type in G Metric Spaces Hamed M. Obiedat1 and Ameer A. Jaber2 1;2 Department of Mathematics Hashemite University P.O.Box150459 Zarqa13115-Jordan 1 email : [email protected], email2 : [email protected], September 4, 2017 Abstract In this paper, we prove several …xed point results for mappings of Caristi type in the setting of G metric spaces.

1

Introduction

The class of G metric spaces introduced by Z. Mustafa and B. Sims (See [7]) was to provide a new class of generalized metric spaces and to extend the …xed point theory for a variety of mappings. Moreover, many theorems were proved in this new setting with most of them recognizable as counterparts of well-known metric space theorems (See [6], [8], [9]). Caristi’s …xed point theorem provides a generalization of Banach’s contraction mapping principle (See [2]). Due to the importance of Caristi’s …xed point theorem, it has been improved, generalized, extended and used in many application ( See [1], [3], [4], [5]). In this paper, we prove several …xed point results for mappings of Caristi type in the setting of G metric spaces. 0

2000 Mathematics Subject Classi…cation. 47H10, 54E50. Key words and phrases. Caristi’s Fixed Point Theorem; G-Metric Spaces; Lower semiContinuous Functions.

1

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De…nition 1 ([7]) G-metric space is a pair (X; G), where X is a nonempty set, and G is a nonnegative real-valued function de…ned on X X X such that for all x; y; z; a 2 X; we have: (G1) G(x; y; z) = 0 if x = y = z; (G2) 0 < G(x; x; y); for all x; y 2 X, with x 6= y; (G3) G(x; x; y)

G(x; y; z); for all x; y; z 2 X; with z 6= y;

(G4) G(x; y; z) = G(pfx; z; yg) (symmetry in all three variables); (G5) G(x; y; z)

G(x; a; a) + G(a; y; z); (rectangle inequality).

The function G is called a G metric on X . De…nition 2 ([7])A sequence (xn ) in a G metric space X is said to converge if there exists x 2 X such that lim G(x; xn ; xm ) = 0; and one say n;m!1

that the sequence (xn ) is G convergent to x.

Proposition 3 ([7])Let X be G metric space. Then the following statements are equivalent. 1. (xn ) is G convergent to x. 2. G(xn ; xn ; x) ! 0; as n ! 1: 3. G(xn ; x; x) ! 0, as n ! 1: 4. G(xm ; xn ; x) ! 0, as m; n ! 1: In a G metric space X, a sequence (xn ) is said to be G Cauchy if given " > 0, there is N" 2 N such that G(xn ; xm ; xl ) < ", for all n; m; l N" . Proposition 4 ([7])In a G metric space X, the following statements are equivalent. 1. The sequence (xn ) is G Cauchy. 2. For every " > 0, there exists N" 2 N such that G(xn ; xm ; xm ) < ", for all n; m N: 2

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De…nition 5 ([7])Let (X; G) and (X 0 ; G0 ) be two G metric spaces, and let f : (X; G) ! (X 0 ; G0 ) be a function, then f is said to be G continuous at a point a 2 X if and only if, given " > 0, there exists > 0 such that x; y 2 X; and G(a; x; y) < implies G0 (f (a); f (x); f (y)) < ". A function f is G continuous on X if it is G continuous at all a 2 X. Proposition 6 ([7])Let (X; G) and (X 0 ; G0 ) be two G-metric spaces. Then a function f : (X; G) ! (X 0 ; G0 ) is G continuous at a point x 2 X if and only if it is G sequentially continuous at x; that is, whenever (xn ) is G convergent to x we have (f (xn )) is G convergent to f (x). A G metric space (X; G) is called symmetric G metric space if G(x; y; y) = G(y; x; x) for all x; y 2 X, and called nonsymmetric if it is not symmetric. Proposition 7 ([7])Let X be a G metric space, then the function G(x; y; z) is jointly continuous in all three of its variables. A G metric space X is said to be complete if every G Cauchy sequence in X is G convergent in X. De…nition 8 With M we indicate the space of functions , where 1.

: [0; 1) ! [0; 1) is strictly increasing, continuous and concave,

2.

(0) = 0.

Lemma 9 Let (X; G) be a complete G metric space and let (X; G) is a complete G metric space.

2 M. Then

Proof. First let us prove that is subadditive. To do so, let x; y 2 [0; 1) (x) . Then since is increasing, we have and set k = (x) + (y) k 1 (x + y) = ( x + k 1 Since

k y) k

x y maxf ( ); ( )g: k 1 k

is concave and (0) = 0; we have

k k (x) = ( x) = ( x + (1 k k

k):0)

1 k ( x) + (1 k

1 k) (0) = k ( x) k

3

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which implies that fore, (x + y)

1 (x) k

( k1 x): Similarly

x y maxf ( ); ( )g k 1 k

maxf

1 1

k

(y)

(

1 1

1 1 (x); (y)g k 1 k

k

x): There-

(x) + (y):

This completes the proof that is subadditive. Now to prove that G de…nes G metric on X, we let x; y; z; a 2 X. Then G1) Since is strictly increasing and (0) = 0 then G(x; y; z) = 0 implies G(x; y; z) = 0 which means x = y = z; G2) Since 0 < G(x; x; y); with x 6= y and is strictly increasing with (0) = 0; then 0 < G(x; x; y); with x 6= y; G3) Since G(x; x; y) G(x; y; z); with z 6= y and is strictly increasing then G(x; x; y) G(x; y; z); with z 6= y; G4) Since G(x; y; z) = G(pfx; z; yg) and is strictly increasing(injective) then G(x; y; z) = G(pfx; z; yg) (symmetry in all three variables); G5) Since G(x; y; z) G(x; a; a) + G(a; y; z) and is strictly increasing and subadditive then G(x; y; z)

(G(x; a; a) + G(a; y; z))

G(x; a; a) +

G(a; y; z);

which proves that G de…nes G metric on X. We still need to prove that (X; G) is complete, so let fxn g be a Cauchy sequence in (X; G). Then lim G(xn ; xm ; xm ) = 0. Since is continuous and strictly inn;m!1

creasing with (0) = 0; we have ( lim G(xn ; xm ; xm )) = 0: This implies n;m!1

lim G(xn ; xm ; xm ) = 0, which means that fxn g is Cauchy sequence in the

n;m!1

complete G metric space (X; G): Therefore, there exists x 2 X such that fxn g G-converges to x 2 X: Hence lim G(xn ; x ; x ) = 0, which implies n!1

( lim G(xn ; x ; x )) = (0) = 0: By continuity of ; we have lim (G(xn ; x ; x )) = n!1

0, which implies fxn g is proof of Lemma 9.

n!1

G convergent in (X;

G): This completes the

De…nition 10 With L(X) we indicate the space of functions , where X ! R+ is lower semi-continuous. Remark 11 Let (X; G) be a G metric space and 2 L(X). De…ne X by x y () G(x; y; y) (x) (y) 8x; y 2 X;

: on

4

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then (X; G; ) is partially ordered G metric space. In fact, 8x; y; z 2 X the following conditions are satis…ed i) since 0 = G(x; x; x) ii) if x

y and y G(y; x; x) x = y:

iii) if x

(x) = 0; we have that x

x

x;then 0 G(x; y; y) (x) (y) = ( (y) (x)) 0: This implies that G(x; y; y) = G(y; x; x) = 0: Hence,

y and y

z;then

G(x; z; z)

G(x; y; y) + G(y; z; z) by rectangle inequality (x) (y) + (y) (z) = (x) (z);

which implies x

2

(x)

z:

Main Results

In this section, we introduce several …xed point results for mappings of Caristi type in the setting of G metric spaces. We use the existence of a maximal element to prove Caristi’s …xed point theorem in the setting of G metric spaces. Theorem 12 Let (X; G; ) be a partially ordered G metric space with as de…ned in Remark 11. Then the following statements are equivalent: 1 Any selfmapping T on X satis…es G(x; T x; T x) point.

(x)

(T x) has a …xed

2 X has a maximal element. Proof. 1 =) 2) Suppose that T : X ! X has a …xed point, say x ; and x1 x2 ::: be a chain in X: Fix xj ; then G(xj ; x ; x ) = G(xj ; T x ; T x ) (xj ) (T x ) = (xj ) (x ) which implies that xj x : Hence X has x as the maximal element. 2 =) 1) Suppose X has x as a maximal element, then T x x : Since T satis…es G(x ; T x ; T x ) (x ) (T x ) which implies that x Tx . Therefore, T x = x : 5

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Theorem 13 Let (X; G; ) be a partially ordered complete G metric space, : X ! R+ be a lower semi-continuous and T : X ! X be selfmapping satisfying the inequality; G(x; T x; T x) (x) (T x): Then T has a …xed point. Proof. Let C = fxt : t 2 4g X be any chain in X and let ftn g be any increasing sequence of elements of 4: We prove …rst that (C) is a decreasing net. To do so, let ct and cs be any pair of elements in C with xt xs for t; s 2 4. Then G(xt ; xs ; xs ) (xt ) (xs ), which implies that (xs ) (xt ) G(xt ; xs ; xs ): Therefore, f (xt )gt24 is a decreasing net of positive real numbers. Thus inff (xt ) : t 2 4g exists by completeness property of R. Now choose ftn gn2N to be an increasing sequence of 4 such that lim (xtn ) = inff (xt ) : t 2 4g: Then fxtn g is G Cauchy since for n!1 n; m 2 N, we have G(xtn ; xtm ; xtm )

(xtn )

(xtm ):

(1)

Thus passing to the limit in the inequality (1) implies G(xtn ; xtm ; xtm ) = 0 as n; m ! 1: Since (X; G; ) is G complete then there exists x 2 X such that fxtn g converges to x : To prove that x is an upper bound of the set C, let m; n 2 N since fxtn g converges to x and fxtn g is increasing imply xtn x 8n 1. Therefore, G(xtn ; x ; x ) =

lim G(xtn ; xtm ; xtm )

m!1

(xtn ) (xtn ) (xtn )

lim (xtm )

m!1

lim (xtm )

m!1

(x ):

(xtn ) 8n 1 which implies that (x ) inff (xt ) : t 2 4g. Then (x ) Hence xt x 8t 2 4 since is decreasing which means that x is an upper bound of the chain C. Therefore Zorn’s lemma implies that (X; ) has a maximal element. By Theorem 12 any selfmapping T : X ! X satis…es the inequality G(x; T x; T x) (x) (T x) has a …xed point. Corollary 14 Let (X; G) be a partially ordered G metric space. Suppose f : X ! X is any function and T : X ! X be G continuous. If there exists a real number r < 0 such that for all x 2 X G(f (x); T f (x); T f (x)

G(x; T x; T x) + rG(x; f (x); f (x)); 6

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then f has a …xed point. G(x; T x; T x) . Then the lower semi Proof. De…ne : X ! R+ by (x) = r continuity of follows from the G continuity of T: Now G(f (x); T f (x); T f (x) =

r (f (x))

r (x) + rG(x; f (x); f (x)):

Then (f (x))

(x)

G(x; f (x); f (x));

which implies G(x; f (x); f (x)) De…ne

(x)

(f (x)):

on X by x

y () G(x; y; y)

(y) 8x; y 2 X.

(x)

Then by Theorem 13, there exists x 2 X such that f (x ) = x : Corollary 15 Let (X; G) be a complete G metric space and let 2 M. Then (X; G) is a complete G metric space. Then any selfmapping T on X satis…es G(x; T x; T x) (x) (T x) has a …xed point. Corollary 16 Let (X; G) be a complete G metric space and let 2 M. Suppose f : X ! X is any function and T : X ! X is G continuous. If for all x 2 X G(f (x); T f (x); T f (x)

G(x; T x; T x)

G(x; f (x); f (x)):

Then f has a …xed point. Proof. De…ne (x) = 1 G(x; T x; T x): Then lower semi continuity of follows from the G continuity of T and continuity of 1 : Now G(f (x); T f (x); T f (x) = ( (f (x)))

( (x))

G(x; f (x); f (x)):

Then ( (f (x)))

( (x))

G(x; f (x); f (x)):

By the subaddivity of ; the above inequality becomes ( (f (x)) + G(x; f (x); f (x)))

( (f (x))) + ( (x)):

G(x; f (x); f (x))

7

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Now since

is increasing, we obtain (f (x)) + G(x; f (x); f (x))

(x):

Hence G(x; f (x); f (x)) De…ne

(x)

(f (x)):

on X by y () G(x; y; y)

x

(x)

(y) 8x; y 2 X:

Then by Theorem 13 there exists x 2 X such that f (x ) = x : Corollary 17 Let (X; G) be a complete G metric space and Suppose f : X ! X is any function and T : X ! X is G continuous. If there exist a real number r < 0 and n 2 N such that for all x; y 2 X G(f (x); T f (x); T n f (x)

G(x; T x; T n x) + rG(x; f (x); f (x))

then f has a …xed point. G(x; T x; T n x) . Then lower semi Proof. De…ne : X ! R+ by (x) = r continuity of follows from the G continuity of T: Then r (f (x)) = G(x; T f (x); T n f (x) r (x) + rG(x; f (x); f (x)): Then we obtain G(x; f (x); f (x)) De…ne

(x)

(f (x)):

on X by x

y () G(x; y; y)

(x)

(y) 8x; y 2 X:

Then by Theorem 13 there exists x 2 X such that f (x ) = x . Corollary 18 Let (X; G) be a complete G metric space and let 2 M. Suppose f : X ! X is any function and T : X ! X is G continuous. If there exist 2 M and n 2 N such that for all x 2 X G(f (x); T f (x); T n f (x)

G(x; T x; T n x)

G(x; f (x); f (x));

then f has a …xed point. 8

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The following theorem gives a natural generalization of Caristi type mapping in the setting of G metric spaces. Theorem 19 Let (X; G) be a complete G metric space. Suppose T : X ! X is G continuous. If there exists y 2 L(X) for all y 2 X such that for all x 2 X G(T x; T 2 x; T y) y (x) y (T x), then T has a …xed point. Proof. Fix x0 2 X and let xn = T n x0 n = 1; 2; 3; :::: Then G(xn ; xn+1 ; T y) = G(T xn 1 ; T xn ; T y) = G(T xn 1 ; T 2 xn 1 ; T y) y (xn 1 ) y (T xn 1 ) = y (xn 1 ) y (xn ): Then for each y 2 X; n X

G(xn ; xn+1 ; T y) = G(x1 ; x2 ; T y) + G(x2 ; x3 ; T y) + ::: + G(xn ; xn+1 ; T y)

j=1

y (x0 )

y (x1 )

y (x0 )

y (xn )

y (x0 )

+

y (x1 )

y (x2 )

+ :::

y (xn 1 )

+

y (xn 1 )

+ C;

where C > 0; which implies that 1 X

G(xn ; xn+1 ; T y)

y (x0 )

j=1

Then

1 P

+ C < 1:

G(xn ; xn+1 ; T y) is a convergent series. Hence lim G(xn ; xn+1 ; T y) = n!1

j=1

0: Therefore, G(xn ; xm ; xm )

G(xn ; T y; T y) + G(T y; xm ; xm ) G(xn ; xm ; T y) + G(xn ; xm ; T y) m X 2 G(xj ; xj+1 ; T y) ! 0 as m; n ! 1 j=n

9

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y (xn )

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which implies that fxn g is Cauchy in complete G metric space. That is, there exists x 2 X such that fxn g converges to x : Now G(T x ; T 2 x ; x ) = lim G(T xn ; T 2 xn ; xn ) = lim G(xn+1 ; xn+2 ; xn ) = 0; n!1

n!1

which implies that x is a …xed point of T: Corollary 20 Let (X; G) be a complete G metric space. Suppose T : X ! X is G continuous and for all x 2 X G(T f (x); T 2 f (x); T y)

G(x; T x; T y)

G(f (x); f 2 (x); T y):

Then f has a …xed point. Proof. For each y 2 X, choose by applying Theorem 19

y (x)

= G(x; T x; T y) then the result follows

References [1] R. P. Agarwal, M. A. Khamsi, Extension of Caristi’s …xed point theorem to vector valued metric space. Nonlinear Anal. 74, (2011), 141–145. [2] J. Caristi, Fixed point theorems for mappings satisfying inwardness conditions, Trans. Amer. Math. Soc. 215 (1976), 241-251. [3] W.A. Kirk, Caristi’s …xed-point theorem and metric convexity, Colloq.Math.36 (1976), 81-86. [4] D. Downing, W. A. Kirk, A generalization of Caristi’s theorem with applications to nonlinear mapping theory. Paci…c J. Math. 69, (1977), 339– 345. [5] M.A. Khamsi, Remarks on Caristi’s …xed point theorem, Nonlinear Anal. TMA70 (2009), 4341-4349. [6] Mustafa, Z, A new structure for generalized metric spaces-with applications to …xed point theory. PhD thesis, the University of Newcastle, Australia (2005). [7] Z. Mustafa, B. Sims, A new approach to generalized metric spaces. J. Nonlinear Convex Anal. 7(2), (2006), 289–297. 10

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[8] Z. Mustafa, H. Obiedat , A …xed point theorem of Reich in G-metric spaces. CUBO. 12(1), (2010), 83–93. Publisher Full Text OpenURL. [9] Z. Mustafa, H. Obiedat, F. Awawdeh, Some …xed point theorem for mapping on complete G-metric spaces. Fixed Point Theory Appl. 2008, Article ID 189870 (2008).

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Meir-Keeler contraction mappings in Mb-metric Spaces N. Mlaiki1 , N. Souayah2 , K. Abodayeh1 , T. Abdeljawad1 Department of Math & General Sciences, Prince Sultan University1 Department of Natural Sciences, Community College, King Saud University2 Riyadh 11586, Saudi Arabia E-mail: [email protected] [email protected] [email protected] [email protected] Abstract In this paper, we generalize the notion of Meir-Keeler contraction condition in Mb metric spaces. We prove some fixed point theorems for this class of contractions which enables us to extend and generalize the recent results of Gholmian and Khanehgir [2].

1

Introduction and preliminaries

First of all, we would like to mention that this work is inspired by the work of Gholmian and Khanehgir [2]. In 1922 Banach established one of the most important theorem in fixed point theory known as the ”Banach contraction principle”. Subsequently, many authors have extended this theorem in many different ways. For example, in 1969, Meir and Keeler [3] generalize the Banach’s theorem using the weakly uniformly strict contraction and proved the following theorem: Theorem 1. Let (X, d) be a complete metric space and f a mapping of X into itself satisfying the following condition: given  > 0, there exists δ > 0 such that  ≤ d(x, y) <  + δ implies d(f (x), f (y)) < . Then f has a unique fixed point ξ. Moreover, For any x ∈ X, lim f n (x) = ξ. n→∞

The Theorem 1 has been extended in many different metric spaces under several contractive definitions, see [2], [5]. On the other hand, several types of generalized metric spaces are proposed and a series of fixed point theorems for various classes of mapping are obtained, see [4], [6], [8], [9], [10], [11], [12]. M -metric spaces was introduced by Asadi see [1], which is an extension of partial metric spaces. So, first we remind the reader of the definition of an M -metric spaces along with some other notations. 1

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Notation 1. [1] 1. mx,y := min{m(x, x), m(y, y)} 2. Mx,y := max{m(x, x), m(y, y)} Definition 1. [1] Let X be a nonempty set, if the function m : X 2 → R+ satisfies the following conditions: for all x, y, z ∈ X (1) m(x, x) = m(y, y) = m(x, y) if and only if x = y, (2) mx,y ≤ m(x, y), (3) m(x, y) = m(y, x), (4) (m(x, y) − mx,y ) ≤ (m(x, z) − mx,z ) + (m(z, y) − mz,y ). Then the pair (X, m) is called an M -metric space. Recently, Mlaiki et al. [7] developed the concept of Mb -metric spaces which extends the M -metric spaces and some fixed point theorems are established. Motivated by the properties of this original metric space, we introduce the notion of generalized Meir-Keeler contraction mappings in the Mb -metric spaces. Now, let’s recall some definitions and notations of Mb -metric spaces. Notation 2. [7] 1. mbx,y := min{mb (x, x), mb (y, y)} 2. Mbx,y := max{mb (x, x), mb (y, y)} Definition 2. [7] An Mb -metric space on a nonempty set X is a function mb : X 2 → R+ that satisfies the following conditions, for all x, y, z ∈ X we have (1) mb (x, x) = mb (y, y) = mb (x, y) if and only if x = y, (2) mbx,y ≤ mb (x, y), (3) mb (x, y) = mb (y, x), (4) There exists a real number s ≥ 1 such that for all x, y, z ∈ X we have (mb (x, y) − mbx,y ) ≤ s[(mb (x, z) − mbx,z ) + (mb (z, y) − mbz,y )] − mb (z, z). The number s is called the coefficient of the Mb -metric space (X, mb ). Now, we give an example of an Mb -metric which is not an M -metric space. Example 1. [7] Let X = [0, ∞) and p > 1 be constant and mb : X 2 → [0, ∞) defined by for all x, y ∈ X we have mb (x, y) = max{x, y}p + |x − y|P . Note that (X, mb ) is an Mb -metric with coefficient s = 2p . Now, we show that (X, mb ) is not an M -metric space. Take x = 5, y = 1 and z = 4, we get mb (x, y) − mbx,y = 5p + 4p − 1 and (mb (x, z) − mbx,z ) + (mb (z, y) − mbz,y ) = 5p + 1 − 4p + 4p + 3p − 1 = 5p + 3p . Therefore, mb (x, y) − mbx,y > (mb (x, z) − mbx,z ) + (mb (z, y) − mbz,y ), as required.

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Definition 3. [7] Let (X, mb ) be a Mb -metric space. Then: 1) A sequence {xn } in X converges to a point x if and only if lim (mb (xn , x) − mbxn ,x ) = 0.

n→∞

2) A sequence {xn } in X is said to be mb -Cauchy sequence if and only if lim (mb (xn , xm ) − mbxn ,xm ), and lim (Mbxn ,xm − mbxn ,xm )

n,m→∞

n→∞

exist and finite. 3) An Mb -metric space is said to be complete if every mb -Cauchy sequence {xn } converges to a point x such that lim (mb (xn , x) − mbxn ,x ) = 0 and lim (Mbxn ,x − mbxn ,x ) = 0.

n→∞

n→∞

Definition 4. Each mb -metric generates a topology τmb on X whose base is the family of open mb -balls {Bmb (x, ) | x ∈ X,  > 0} where Bmb (x, ) = {y ∈ X | mb (x, y) − mbx,y < }. Definition 5. Let X be a nonempty set, T : X −→ X be a mapping and α : X×X −→ [0, ∞) be a function. Then, T is said to be α-admissible if for all x, y ∈ X we have α(x, y) ≥ 1 =⇒ α(T x, T y) ≥ 1.

(1)

Definition 6. A mapping T : X → X is called triangular α-admissible if it is α-admissible and it satisfies the following condition: α(x, y) ≥ 1 and α(y, z) ≥ 1, then α(x, z) ≥ 1 where x, y, z ∈ X. Definition 7. Let (X, mb ) be an mb -metric space with coefficient s, an α-admissible mapping T : X −→ X is said to be generalized Meir-Keeler contraction of type (I) if for every  > 0 there exists δ > 0 such that  ≤ β(mb (x, y))M (x, y) <  + δ implies α(x, y)mb (T x, T y) < 

(2)

M (x, y) = max{mb (x, y), mb (T x, x), mb (T y, y)}, for all x, y ∈ IN

(3)

where and β : [0, ∞) −→ (0,

1 ) s

is a given function.

Definition 8. Let (X, mb ) be an mb -metric space with coefficient s. A triangular α-admissible mapping T : X −→ X is said to be generalized Meir-Keeler contraction of type (II) if for every  > 0 there exists δ > 0 such that  ≤ β(mb (x, y))N (x, y) <  + δ implies α(x, y)mb (T x, T y) < 

(4)

where

1 N (x, y) = max{mb (x, y), [mb (T x, x) + mb (T y, y)]}, forall x, y ∈ IN 2 1 and β : [0, ∞) −→ (0, s ) is a given function.

(5)

Remark 1. 1. Suppose that T : X −→ X is a generalized Meir-Keeler contraction of type (I). Then α(x, y)mb (T x, T y) < β(mb (x, y))M (x, y) (6) for all x, y ∈ X when M (x, y) > 0. 2. Note that for all x, y ∈ X, we have N (x, y) ≤ M (x, y). 3

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2

Main Results

Theorem 2. Let (X, mb ) be a complete Mb metric space and T : X → X be a triangular α-admisible mapping. Suppose that the following conditions hold: (a) There exists x0 ∈ X such that α(x0 , T x0 ) ≥ 1, α(T x0 , x0 ) ≥ 1. (b) If {xn } is a sequence in X that converges to z as n → ∞, and α(xn , xm ) ≥ 1 for all n, m ∈ IN, then α(xn , z) ≥ 1 for all n ∈ IN. (c) If for each  > 0 there exists δ > 0 such that 2s ≤ mb (y, T y)

1 + mb (x, T x) + N (x, y) < s(2 + δ), 1 + M (x, y)

then we have α(x, y)mb (T x, T y) < . Then, T has a fixed point in X. Proof. Note that condition (c) implies that α(x, y)mb (T x, T y)
k. Suppose that 2s ≤ mb (xl , xk ), so that 2s ≤ mb (xl , xk ) − mbxl ,xk < η. Note mb (xl , xk ) ≤ N (xl , xk ). Hence, 2s ≤ mb (xl , xk ) and this implies that  ≤ Thus, 1 1 + mb (xl , xl+1 ) 1  ≤ mb (xk , xk+1 ) + N (xl , xk ). 2s 1 + M (xl , xk ) 2s

1 m (x , xk ). 2s b l

5

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Therefore,

1 1 + mb (xl , xl+1 ) 1 δ0 mb (xk , xk+1 ) + N (xl , xk ) <  + , 2s 1 + M (xl , xk ) 2s 2 and this implies that 2s ≤ mb (xk , T xk )

1 + mb (xl , T xl ) + N (xl , xk ) < s(2 + δ 0 ). 1 + M (xl , xk )

Thus, by part (c) of the theorem, we have mb (T xl , T xk ) ≤ α(xl , xk )mb (T xl , T xk ) < . Therefore, mb (T xl , xk ) − mbT xl ,xk ≤ mb (T xl , xk ) ≤ s [(mb (T xl , T xk ) − mbT xl ,T xk ) + (mb (T xk , xk ) − mbT xk ,xk )] ≤ s [mb (T xl , T xk ) + mb (T xk , xk )] δ0 < s[ + ] 4 δ0 < s[2 + ] 2 which implies that xl+1 ∈ B[xk , η] as desired. Now assume that mb (xl , xk ) < 2s. Then we have mb (T xl , xk ) − mbT xl ,xk ≤ ≤ ≤ ≤

mb (T xl , xk ) s [(mb (T xl , T xk ) − mbT xl ,T xk ) + (mb (T xl , xk ) − mbT xl ,xk )] s [mb (T xl , T xk ) + mb (T xk , xk )] sα(xl , xk )mb (T xl , T xk ) + smb (T xk , xk )   1 + mb (xl , T xl+1 ) 1 1 mb (xk , xk+1 ) + N (xl , xk ) + smb (xk+1 , xk ) < s 2s 1 + M (xl , xk ) 2s 1 mb (xk , xk+1 )mb (xl , xl+1 ) 1 ≤ mb (xk , xk+1 ) + + N (xl , xk ) + smb (xk+1 , xk ) 2 2(1 + mb (xl , xk )) 2 0 mb (xk , xk+1 )mb (xl , xl+1 ) 1 δ0 δ + + N (xl , xk ) + s . ≤ 8 2(1 + mb (xl , xk )) 2 4

On the other hand, note that mb (xk , xk+1 ) δ0 ≤ mb (xk , xk+1 ) < < 1. 1 + mb (xl , xk ) 4 Hence, mb (T xl , xl ) − mbT xl xl ≤ mb (T xl , xk ) δ0 1 1 δ0 = + mb (xl , xl+1 ) + N (T xl , xk ) + s 2 4 8 0 2 0  δ δ δ0 < + + s + s 8 8 4 0 δ ≤ s( + 2). 2 6

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Therefore, for all m > k, we have  mb (xm , xk ) − mbxm ,xk < s

 δ0 + 2 . 2

Now, for every m, n ∈ IN such that m > n > k, we have mb (xm , xn ) − mbxm ,xn ≤ s [(mb (xm , xk ) − mbxm ,xk ) + (mb (xk , xn ) − mbxk ,xn )] ≤ smb (xm , xk ) + smb (xk , xn ) δ0 δ0 < s.s( + 2) + s.s( + 2) 2 2 = s2 (4 + δ 0 ) ≤ 5s2  which implies that limn,m→∞ mb (xm , xn )−mbxm ,xn exists and finite. Using the same argument it is not difficult to show that limn,m→∞ Mb (xm , xn ) − mbxm ,xn exists and finite. Therefore, the sequence {xn } is an mb -Cauchy sequence and since X is complete, there exists u ∈ X such that limn→∞ Mbxn ,u − mbxn ,u = 0 . Finally, we show that is a fixed point for T ; that is T u = u. lim (Mbxn ,u − mbxn,u ) = 0

n→∞

lim (Mbxn+1 ,u − mbxn+1,u ) = 0

n→∞

lim (MbT xn ,u − mbT xn,u ) = 0

n→∞

Mb (T u, u) − mbT u,u = 0

Then, MbT u,u = mbT u,u , and similarly by the convergence of xn we obtain that mb (T u, u) = mbT u,u , which implies that T u = u as required. Definition 9. Let (X, mb ) be an mb -metric space and let T be a self mapping on X. T is called mb -orbitally continuous if whenever lim mb (Txn , z) = mb (z, z) ⇒ lim mb (T Txn , Tz ) = mb (Tz , Tz )∀x, z ∈ X.

n→+∞

n→+∞

(7)

Remark 2. Note that, continuous mappings are mb -orbitally continuous. But the converse is not necessary true, for example, consider the mb -metric space defined by mb (x, y) = [max(x, y)]q (q ≥ 1) for all x, y ∈ X where X = [0, 1] and the map T : X −→ X defined by  x if 0 ≤ x < 1 2 T = 0 if x=1 It is not difficult to see that T is not continuous, but T is mb -orbitally continuous. Theorem 3. Let (X, mb ) be a complete mb -metric space with coefficient s and T : X −→ T be a mapping. Suppose that the following conditions hold: 7

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a) T is an mb -orbitally continuous generalized Meir-Keeler contraction of type (I), b) there exists x0 ∈ X such that α(x0 , T x0 ) ≥ 1, α(T x0 , x0 ) ≥ 1, c) if {xn } is a sequence in X such that xn −→ z as n −→ ∞ and α(xn , xm ) ≥ 1 forall n, m ∈ IN, then α(z, z) ≥ 1, d) s > 1 or β is a continuous function. then, T has a fixed point in X. Proof. Let x0 ∈ X be such that condition b) holds and define {xn } in X so that x1 = T x0 , xn+1 = T x0 ∀n ∈ IN. Without lose of generality, we may suppose that xn+1 6= xn ∀n ∈ IN∪0. Since T is α-admissible, then α(xn , xn+1 ) ≥ 1 ∀n ∈ IN. As T is a generalized Meir-Keeler contraction of type (I), then by replacing x by xn and y by yn in (4), we observe that for every  > 0 there exists δ > 0 such that  ≤ β(mb (xn , xn+1 ))M (xn , xn+1 ) <  + δ =⇒ α(xn , xn+1 )mb (T xn , T xn+1 ) < 

(8)

M (xn , xn+1 ) = max[mb (xn , xn+1 ), mb (xn+2 , xn+1 )].

(9)

where Next, we distinguish two following cases: Case 1. Assume that M (xn , xn+1 ) = mb (xn+2 , xn+1 ). In this case, equation (8) becomes  ≤ β(mb (xn , xn+1 ))mb (xn+2 , xn+1 ) <  + δ =⇒ α(xn , xn+1 )mb (T xn , T xn+1 ) <  and using that α(xn , xn+1 ) ≥ 1 ∀n ∈ IN, we have mb (T xn , T xn+1 )mb (xn+1 , xn+2 ) <  ≤ β(mb (xn , xn+1 ))mb (xn+2 , xn+1 ). Then mb (xn+1 , xn+2 ) < mb (xn+2 , xn+1 ) ∀n ∈ IN which gives a contradiction. Case 2. Assume that M (xn , xn+1 ) = mb (xn , xn+1 ). Since M (xn , xn+1 ) > 0 ∀n ∈ IN due to Remark 1, we get mb (xn+1 , xn+2 ) ≤ α(xn , xn+1 )mb (T xn , T xn+1 ) <  ≤ β(mb (xn , xn+1 ))mb (xn , xn+1 ) 1 mb (xn , xn+1 ) ≤ mb (xn , xn+1 ). < s

(10)

That is {mb (xn , xn+1 )} is a strictly decreasing positive sequence in R+ and it converges to some r ≥ 0. Let prove that r = 0. Let be untrue, the we have r > 0. We assert that 0 < r ≤ mb (xn , xn+1 ) ∀n ∈ IN. 1 First, suppose that s > 1. Applying equation (10), we have mb (xn+1 , xn+2 ) < mb (xn , xn+1 ). s 1 By taking the limit as n tends to infinity, we get r ≤ r < r which is a contradiction and so s r = 0. Next, suppose that β is a continuous function. We prove in the following claim that {β(mb (xn , xn+1 ))mb (xn , xn+1 )} is a strictly decreasing positive sequence in R+ . 8

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1 Claim 1. Let β : [0, ∞[−→ [0, ) a continuous function. Then, {β(mb (xn , xn+1 ))mb (xn , xn+1 )} s is strictly decreasing positive sequence in R+ . First, note that β(mb (xn+1 , xn+2 ))mb (xn+1 , xn+2 ) < mb (xn+1 , xn+2 ) ≤ α(xn , xn+1 )mb (T xn , T xn+1 ) < β(mb (xn , xn+1 ))M (xn , xn+1 ). If M (xn , xn+1 ) = mb (xn , xn+1 ), we obtain β(mb (xn+1 , xn+2 ))mb (xn+1 , xn+2 ) < β(mb (xn , xn+1 ))mb (xn , xn+1 ). If M (xn , xn+1 ) = mb (xn+2 , xn+1 ), we have mb (xn+2 , xn+1 ) < mb (xn , xn+1 ) (as mb (xn , xn+1 ) is a strictly decerasing). Then, β(mb (xn+1 , xn+2 ))mb (xn+1 , xn+2 ) < β(mb (xn , xn+1 ))mb (xn , xn+1 ). Thus, {β(mb (xn , xn+1 ))mb (xn , xn+1 )} is strictly decreasing positive sequence in R+ which prove our claim as desired. From Claim 1, we have {β(mb (xn , xn+1 ))mb (xn , xn+1 )} converges to some r0 ≥ 0. We consider the two follwing cases: Case 1. r0 = 0 Since lim mb (xn , xn+1 ) 6= 0 so we have n→∞

∃ > 0, ∀k ∈ IN, ∃nk ≥ k, mb (xnk , xnk+1 ) ≥ . Now, let 0 > 0 be given. Since lim β(mb (xnk , xnk+1 ))mb (xnk , xnk+1 ) = 0. n→∞

Therefore, using (4), we derive ∃k 0 ∈ IN, ∀k ≥ k 0 , β(mb (xnk , xnk+1 )) ≤ β(mb (xnk , xnk+1 ))mb (xnk , xnk+1 ) < 0 . It enforces that lim β(mb (xnk , xnk+1 )) = 0. n→∞

By continuity of β, we obtain β(r) = 0 =⇒ r = 0 which is a contradiction. Case 2. r0 > 0 we can distinguish two subcases: r < r0 and r > r0 . 1 If r < r0 , then β(mb (xn , xn+1 ))mb (xn , xn+1 ) < mb (xn , xn+1 ) and by taking the limit as n s r 0 tends to infinity we get r ≤ ≤ r which is a contradiction with r0 > 0. s If r > r0 , let δ > 0 be such that satisfying (4) whenever  = r0 . We know there exists N0 ∈ IN such that r0 ≤ β(mb (xN0 , xN0 +1 ))mb (xN0 , xN0 +1 ) < r0 + δ. Thus r < mb (xN0 +1 , xN0 +2 ) ≤ α(xN0 , xN0 +1 )mb (T xN0 , T xN0 +1 ) < r0 ≤ r which leads to contradiction with 0 < r ≤ mb (xn , xn+1 ) ∀n ∈ IN.

9

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Thus, r = 0 and so lim mb (xn , xn+1 ). n→∞

Next, we intend to show that the sequence {xn } is an mb - Cauch sequence. For this purpose, we will prove that for every  > 0 there exists N ∈ IN such that lim mb (xn , xm ) − n,m→∞

mbxn,m < ∞. We will prove that for every  > 0, there exists N ∈ IN such that mb (xl , xl+k ) − mbxl,l+k < .

(11)

Since the sequence {mb (xn , xn+1 )} −→ 0, n −→ ∞, for every δ > 0 there exists N ∈ IN such that mb (xn , xn+1 ) < δ for all n ≥ N . Choose δ < . We will prove equation (11) by using induction on k. • for k = 1, we have mb (xl , xl+1 ) <  ⇒ mb (xl , xl+1 ) − mbxl,l+1 <  so, (11) it clearly holds for all l ≥ N (due to the choice of δ). • Assume that the inequality (11) holds for some k = m, that is mb (xl , xl+m ) − mbxl,l+m <  ∀l ≥ N . For k = m + 1, we have to show that mb (xl , xl+m+1 ) − mbxl,l+m+1 <  ∀ l ≥ N

(12)

Employing condition (4) of the definition fo the Mb -metric space, we get mb (xl−1 , xl+m ) − mbxl,l+m < s[mb (xl−1 , xl ) − mbxl−1,l + mb (xl , xl+m ) − mbxl,l+m − mb (xl , xl )] ≤ s[mb (xl−1 , xl ) + mb (xl , xl+m )] ≤ s[δ + ] ∀l ≥ N. If β(mb (xl−1 , xl+m ))mb (xl−1 , xl+m ) ≥ , then we deduce  ≤ ≤ = <
0, otherwise mb (xl , xl−1 ) = 0 and hence xl = xl−1 , which is contradiction. Thus, mb (xl , xl+k ) <  ∀l ≥ N and k ≥ 1, it means mb (xn , xm ) <  ∀ m ≥ n ≥ N.

(13)

Hence, it is easy to deduce that {xn } is an mb −Cauchy sequence. Since X is a complete mb -metric space, there exists u ∈ X such that lim (Mbxn ,u − mbxn,u ) = 0. n→∞ Now, we will show that T u = u for any n ∈ IN. We have lim (Mbxn ,u − mbxn,u ) = 0

n→∞

lim (Mbxn+1 ,u − mbxn+1,u ) = 0

n→∞

lim (MbT xn ,u − mbT xn,u ) = 0

n→∞

Mb (T u, u) − mbT u,u = 0 Then, MbT u,u = mbT u,u , and similarly by the convergence we obtain that mb (T u, u) = mbT u,u , which implies that T u = u as desired. Next, we prove the same result for a self mapping T on X which is an mb -orbitally continuous generalized Meir-Keeler contraction of type (II). Theorem 4. Let (X, mb ) be a complete mb -metric space, T : X −→ X be a mapping. Assume that the following conditions are satisfied: a) T is an mb -orbitally continuous generalized Meir-Keeler contraction of type (II), b) there exists x0 ∈ X such that α(x0 , Tx0 ) ≥ 1, α(Tx0 , x0 ), c) If {xn } is a sequence in X such that xn −→ z as n −→ ∞ and α(xn , xm ) ≥ 1 for all n, m ∈ N, then α(z, z) ≥ 1, d) s > 1 or β is a continuous function, then T has a unique fixed point in X. Proof. By remark 1 , we have N (x, y) ≤ M (x, y). Hence, similarly to the proof of theorem 3, the result of our theorem will follow as desired. Theorem 5. Let (X, mb ) be a complete mb -metric space with coefficient s and satisfies the following conditions: 11

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a) if {xn } is a sequence in X which converges to z with respect to τmb and satisfies α(xn+1 , xn ) ≥ 1 and α(xn , xn+1 ) ≥ 1 for all n, then there exists a subsequence {xnk } of {xn } such that α(xz , xnk ) ≥ 1 and α(xnk , xz ) ≥ 1 for all k, b) T : X −→ X is a generalized Meir-Keeler contraction of type (II), c) there exists x0 ∈ X such that α(x0 , Tx0 ) ≥ 1, α(Tx0 , x0 ) ≥ 1 d) s > 1 or β is a continuous function then, T has a fixed point in X. Proof. By the proof of theorem 2, one can easly deduce that {xn } defined by x1 = Tx0 and xn+1 = Txn (n ∈ N) converges to some z ∈ X with mb (z, z) = 0, by condition a), there exist a subsequence {xnk } of xn such that α(z, xnk ) ≥ 1 and α(xnk , z) ≥ 1 for all k. Note that, if N (z, xnk ) = 0, then Tz = z and we are done. Now, by remark 1 for all k ∈ N we have mb (Tz , xn+1 ) = mb (Tz , T xn ) ≤ α(z, xnk )mb (Tz , Txnk ) < β(mb (z, xnk ))N (z, xnk ). 1 1 Taking the limit k −→ ∞ we obtain lim N (z, xnk ) = max{0, mb (Tz , z)} = mb (Tz , z). k→∞ 2 2 1 Thus, lim mb (Tz , xnk+1 ) ≤ mb (Tz , z). Hence, n→∞ 2s mb (Tz , z) ≤ smb (Tz , xnk+1 ) + smb (xnk+1 , z). Taking the limit k −→ ∞ we obtain 1 mb (Tz , z) ≤ mb (Tz , z). 2 which implies mb (Tz , z) = 0, similarly we can show that MbTz ,z = 0 and therefore, Tz = z as desired.

3

Acknowledgement

The authors would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RG-DES-2017-01-17.

References [1] Asadi et al: “New extention of p-metric spaces with some fixed point results on M metric spaces.” Journal of Inequalities and Applications. 2014, 2014:18 [2] N. Gholamian, M. Khanehgir, Fixed points of generalized Meir-Keeler contraction mappings in b-metric-like spaces, Fixed Point Theory and Applications, (2016) 2016:34 DOI: 10.1186/s13663-016-0507-6. 12

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[3] A. Meir, E. Keeler, A Theorem on Contraction Mappings, Journal of Mathematical Analysis and Applications, 28 (1969), 326-369. [4] W. Shatanawi, MB. Hani, A Coupled Fixed Point Theorem in b-metric spaces, International Journal of Pure and Applied Mathematics, 4 109, 889-897. [5] T. Abdeljawad, Meir-Keeler α-contractive fixed and common fixed point theorems, Fixed Point Theorem and Apllications, 2013, 2013:19, DOI: 10.1186/1687-1812-2013-19. [6] T. Abeljawad, K. Abodayeh, N. Mlaiki,On Fixed Point Generalizations to Partial bmetric Spaces, Journal of Computational Analysis & Applications, 19 (2015), 883-891. [7] N. Mlaiki, A. Zarrad, N. Souayah, A. Mukheimer, T. Abdeljawed, Fixed Point Theorems in Mb -metric spaces, Journal of Mathematical Analysis, 7 (2016),1-9. [8] N. Souayah and N. Mlaiki, A coincident point principle for two weakly compatible mappings in partial S-metric spaces, Journal of Nonlinear science and applications, 9 (2016), 2217-2223. [9] N. Mlaiki, A. Zarrad, N. Souayah, A. Mukheimer, T. Abdeljawed, Fixed point theorems in Mb -metric spaces, Journal of Mathematical Analysis, 7 (5)(2016), 1-9. [10] N. Souayah, A fixed point in partial Sb -metric spaces, An. St. Univ. Ovidius Constanta, 24(3) (2016), 351-362. [11] N. Mlaiki, M. Souayah, K. Abodayeh, T. Abdeljawad, Contraction principles in Ms metric spaces, Journal of Nonlinear Sciences and Applications, 10 (2017), 575-582 [12] N. Souayah and N. Mlaiki, A fixed point theorem in Sb -metric spaces, J. Math. Computer Sci. 16 (2016), 131-139.

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Generalized Ulam-Hyers Stability for Generalized types of (γ − ψ)−Meir-Keeler Mappings via Fixed Point Theory in S−metric spaces Mi Zhou1 , Xiao-lan Liu2,3∗, Arslan Hojat Ansari 4 , Yeol Je Cho 5 , Stojan Radenovi´c6

Abstract: In this paper, we introduce several extensions of Meir-Keeler contractive mappings in the structure of S−metric spaces. Then we investigate some existence, uniqueness, and generalized Ulam-Hyers stability results for the classes of MKC mappings via fixed point theory. Besides the theoretical results, we also present some illustrative examples to verify the effectiveness and applicability of our main results. MSC: 47H10;54H25 Keywords: Generalized Ulam-Hyers stability; (γ − ψ)−Meir-Keeler contraction mappings; S−metric space; fixed point theory.

1. 1.1.

Introduction S−metric spaces

Very recently, Sedghi et al.[1] have introduced the notion of an S−metric space and proved that this notion is a generalization of a G−metric space and D∗ −metric space. Also, they have proved some properties of an S−metric and some fixed point results for a self-map on S−metric spaces. After that, many interesting results were obtained by transporting certain results in metric spaces and known generalizes metric spaces to S−metric spaces, see ([2]-[10]). First, we recall the definition of an S−metric space and some useful notions and lemmas for the following discussion. In the sequel, the letters N, R+ and R will denote the sets of positive integers, nonnegative real numbers and real numbers, respectively. Definition 1.1. [1] Let X be a nonempty set. An S−metric on X is a function S : X 3 7→ [0, ∞) that satisfies the following conditions for ∀x, y, z, a ∈ X: (S1) S(x, y, z) = 0 if and only if x = y = z; (S2) S(x, y, z) ≤ S(x, x, a) + S(y, y, a) + S(z, z, a). ∗ Correspondence:

[email protected]

2

College of Science, Sichuan University of Science and Engineering, Zigong, Sichuan 643000, China

3

Sichuan Province University Key Laboratory of Bridge Non-destruction Detecting and Engineering Computing, Zigong,

Sichuan 643000, China Full list of author information is available at the end of the article

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The pair (X, S) is called an S−metric space. Immediate examples of such S−metric spaces are: (1) Let X = R+ and k · k be a norm on X, then S(x, y, z) = k2x + y − 3zk + kx − zk is an S−metric on X, for ∀x, y, z ∈ X. (2) Let X be a nonempty set, d is ordinary metric on X, the Sd (x, y, z) = d(x, z) + d(y, z) is an S−metric on X, for ∀x, y, z ∈ X. Lemma 1.1. [1] Let (X, S) be an S−metric space. Then S(x, x, z) ≤ 2S(x, x, y) + S(y, y, z), and S(x, x, z) ≤ 2S(x, x, y) + S(z, z, y), for ∀x, y, z ∈ X. Lemma 1.2. [1] Let (X, S) be an S−metric space. Then S(x, x, y) = S(y, y, x), for ∀x, y ∈ X. Lemma 1.3. Let (X, S) be an S−metric space. Then, for ∀x, y, z ∈ X, it follows that (1) S(x, y, y) ≤ S(x, x, y). (2) S(x, y, x) ≤ S(x, x, y). (3) S(x, y, z) ≤ S(x, x, z) + S(y, y, z). (4) S(x, y, z) ≤ S(y, y, z) + S(x, x, z). (5) S(x, y, z) ≤ S(y, y, x) + S(z, z, x). (6) S(x, x, z) ≤ 23 [S(y, y, z) + S(y, y, x)]. (7) S(x, y, z) ≤ 23 [S(x, x, y) + S(y, y, z) + S(z, z, x)]. Proof. It follows from (S2) and Lemma 1.2, one can easily obtain (1) − (5). Now, we prove (6) and (7) also hold true. By Lemma 1.1 and Lemma 1.2, we have 2S(x, x, z) = S(x, x, z) + S(z, z, x) ≤ [2S(x, x, y) + S(y, y, z)] + [2S(z, z, y) + S(x, x, y)] = 3[S(y, y, z) + S(y, y, x)]. Hence, S(x, x, z) ≤ 32 [S(y, y, z) + S(y, y, x)]. Then (6) holds true. By virtue of (3) − (5) and Lemma 1.2, we have 3S(x, y, z) = 2[S(x, x, y) + S(y, y, z) + S(z, z, x)], which implies (7) holds true. Definition 1.2. [1] Let (X, S) be an S−metric space. (1) A sequence {xn } ⊂ X is said to convergent to x ∈ X if S(xn , xn , x) → 0 as n → ∞. That is, for each  > 0, there exists n0 ∈ N such that for ∀n ≥ n0 , we have S(xn , xn , x) < . (2) A sequence {xn } ⊂ X is said to be a Cauchy sequence if S(xn , xn , xm ) → 0 as n, m → ∞. That is, for each  > 0, there exists n0 ∈ N such that for ∀n, m ≥ n0 , we have S(xn , xn , xm ) < , or for each  > 0, there exists n0 ∈ N such that for ∀l, m, n ≥ n0 , we have S(xl , xm , xn ) <  . 594

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(3) The S−metric space (X, S) is said to be complete if every Cauchy sequence is a convergent sequence. (4) A mapping T : X 7→ X is said to be S−continuous if {T xn } is S−convergent to T x, where {xn } is an S−convergent sequence converging to x. Lemma 1.4. [1] Let (X, S) be an S−metric space. If there exist sequences {xn } and {yn } such that xn → x and yn → y as n → ∞, then S(xn , xn , yn ) → S(x, x, y). Lemma 1.5. [1] Let (X, S) be an S−metric space. If the sequences {xn } in X such that xn → x, then x is unique.

1.2.

The generalized Ulam-Hyers Stability

The stability problem of functional equations, originated from a question of Ulam [11], in 1940, concerns the stability of group homomorphism which stated as follows: Let G1 be a group and G2 be a metric group with a metric d(·, ·). Given  > 0, does there exist δ > 0 such that if a function h : G1 7→ G2 satisfies the inequality d(h(xy), h(x)h(y)) < δ, ∀x, y ∈ G1 , then there is a homomorphism H : G1 7→ G2 with d(h(x), H(x)) < , ∀x ∈ G1 ? If the answer is affirmative, then we say that the equation of homomorphism H(xy) = H(x)H(y) is stable. The first affirmative partial answer to the equation of Ulam for Banach spaces was given by Hyers [12] in 1941. Thereafter, this type of stability is called the Ulam-Hyers stability and has attracted attentions of many mathematicians. In particular, Ulam-Hyers stability results in fixed point theory and remarkable results on the stability of certain classes of functional equation via fixed point approach have been studied densely, see ([13]-[16]).

Definition 1.3. Let (X, S) be an S−metric space and T : X 7→ X be a mapping. By definition, the fixed point equation x = T x, x ∈ X

(1)

is said to be generalized Ulam-Hyers stable in the framework of an S−metric space if there exists an increasing operator ϕ : [0, ∞) 7→ [0, ∞), continuous at 0 and ϕ(0) = 0, such that for each  > 0 and an −solution w∗ ∈ X, that is S(w∗ , T w∗ , T w∗ ) ≤ ,

(2)

there is a solution x∗ ∈ X of the fixed point equation (1) such that S(w∗ , x∗ , x∗ ) ≤ ϕ().

(3)

If ϕ(t) = ct, ∀t ≥ 0, where c > 0, then (1) is said to be Ulam-Hyers stable in the framework of an S−metric space.

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1.3.

The generalized (γ − ψ)−Meir-Keeler contractive mappings

In 1969, Meir and Keeler [17] established a fixed point theorem in a metric space (X, d) for mappings satisfying the condition that for each  > 0 there exists δ() > 0 such that  ≤ d(x, y) <  + δ

implies

d(T x, T y) < ,

(4)

∀x, y ∈ X. This condition is called the Meri-Keeler contractive (M KC, for short) type condition. Since then, many authors extended and improved this condition and established fixed point results for new generalized conditions, see Maiti and Pal [18], Park and Rhoades [19], Mongkolkeha and Kuman [20] and so on. On the other hand, Samet et al.[21] introduced the concepts of α − ψ−contractive mapping and α−admissible mapping in metric spaces. Also they proved a fixed point theorem for α − ψ contractive mappings in complete metric spaces using the concept of α−addmissible mappings. Motivated by Samet’s work, Latif et al.[22] introduced a new type of a generalized (α − ψ)−MeirKeeler contractive mapping and established some interesting theorems on the existence of fixed points for such mappings via admissible mappings. Admissible mappings in the setting of S−metric spaces can be defined as follows. Definition 1.4. A mapping T : X 7→ X is called γ−admissible if for ∀x, y, z ∈ X, we have γ(x, y, z) ≥ 1 ⇒ γ(T x, T y, T z) ≥ 1, where, γ : X 3 7→ [0, ∞) is a given function. If in addition,   γ(x, y, y) ≥ 1 implies γ(x, z, z) ≥ 1, ∀x, y, z ∈ X. Then T is called triangular γ−admissible.  γ(y, z, z) ≥ 1   2, Example 1.1. Let X = [1, ∞) and T : X → 7 X. Define T x = x2 and γ(x, y, z)=  0, Then T is γ−admissible.

if

x ≥ y ≥ z;

otherwise.

Definition 1.5. We say that: (1) A sequence {xn } in X is (T, γ)−orbital if xn = T n x0 and γ(xn , xn+1 , xn+1 ) ≥ 1, ∀n ∈ {0} ∪ N. (2) T is γ−orbital continuous if, for every (T, γ)−orbital sequence {xn } in X such that xn → x as n → ∞, there exists a subsequence {xnk } of {xn } such that T xnk → T x as k → ∞. (3) X is (T, γ)−regular if, for every (T, γ)−orbital sequence {xn } in X such that xn → x as n → ∞, there exists a subsequence {xnk } of {xn } such that γ(xnk , x, x) ≥ 1, ∀k ∈ N. (4) X is γ−regular if, for every sequence {xn } in X such that xn → x as n → ∞ and γ(xn , xn+1 , xn+1 ) ≥ 1, ∀n ∈ {0} ∪ N, there exists a subsequence {xnk } of {xn } such that γ(xnk , x, x) ≥ 1, ∀k ∈ N. (5) X is (T, γ)−limit if, for every sequence {xn } in X such that xn → x as n → ∞ and γ(xn , xn+1 , xn+1 ) ≥ 1, ∀n ∈ {0} ∪ N, then γ(x, T x, T x) ≥ 1. Remark 1.1. (1) If T is continuous, then T is γ−orbital continuous (for any γ). (2) If X is γ−regular, then X is also (T, γ)−regular (for any γ).

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Lemma 1.6. Let γ : X 3 7→ [0, ∞) and T : X 7→ X be γ−admissible with triangular admissibility. Assume that there exists x0 ∈ X such that γ(x0 , T x0 , T x0 ) ≥ 1. Define a sequence {xn } by xn = T n x0 . Then γ(xm , xn , xn ) ≥ 1, for ∀m, n ∈ N with m < n. Proof. Since there exists x0 ∈ X such that γ(x0 , T x0 , T x0 ) ≥ 1, then from the definition of γ− admissibility, we deduce that γ(x1 , x2 , x2 ) = γ(T x0 , T x1 , T x1 ) ≥ 1. By continuing this process, we  get γ(xn , xn+1 , xn+1 ) ≥ 1, ∀n ∈ 0 ∪ N.  γ(xm , xm+1 , xm+1 ) ≥ 1 Suppose that m < n. Since  γ(x m+1 , xm+2 , xm+2 ) ≥ 1, by the definition of triangular γ− admissibility, we deduce that γ(xm , xm+2 , xm+2 ) ≥ 1. By continuing this process, we get γ(xm , xn , xn ) ≥ 1, ∀m, n ∈ N with m < n. Let Ψ stand for the family of nondecreasing functions ψ : [0, ∞) 7→ [0, ∞) satisfying conditions: n n th (Ψ1) Σ∞ iterate of ψ; n=1 ψ (t) < ∞, ∀t > 0, where ψ is the n

(Ψ2) ψ(0) = 0. Remark 1.2. For every function ψ : [0, ∞) 7→ [0, ∞) the following holds: if ψ is nondecreasing, then for each t > 0, lim ψ n (t) = 0 ⇒ ψ(t) < t ⇒ ψ(0) = 0.

n→∞

Therefore, if ψ ∈ Ψ, then for every t > 0, ψ(t) < t and ψ is continuous at 0. Definition 1.6. Let (X, S) be an S−metric space and T : X 7→ X. The mapping T is called a (γ − ψ)−Meir-Keeler contractive mapping if there exist two functions ψ ∈ Ψ and γ : X 3 7→ [0, ∞) satisfying the following condition: for each  > 0 there exists δ() > 0 such that  ≤ ψ(S(x, y, y)) <  + δ() implies γ(x, y, y)S(T x, T y, T y) < , ∀x, y ∈ X. Remark 1.3. It is easily shown that if T : X 7→ X is a (γ − ψ)−Meir-Keeler contractive mapping, then γ(x, y, y)S(T x, T y, T y) < ψ(S(x, y, y)), ∀x, y ∈ X, when x 6= y. Definition 1.7. Let (X, S) be an S−metric space and T : X 7→ X. The mapping T is called a (γ − ψ)−Meir-Keeler contractive mapping of dim3 if there exist two functions ψ ∈ Ψ and γ : X 3 7→ [0, ∞) satisfying the following condition: for each  > 0 there exists δ() > 0 such that  ≤ ψ(S(x, y, z)) <  + δ() implies γ(x, T x, T x)γ(y, T y, T y)γ(z, T z, T z)S(T x, T y, T z) < . Remark 1.4. It is easily shown that if T : X 7→ X is a (γ − ψ)−Meir-Keeler contractive mapping of dim3, then γ(x, T x, T x)γ(y, T y, T y)γ(z, T z, T z)S(T x, T y, T z) < ψ(S(x, y, z)), ∀x, y, z ∈ X when x 6= y 6= z.

597

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Definition 1.8. Let (X, S) be an S−metric space and T : X 7→ X. The mapping T is called a generalized (γ − ψ)−Meir-Keeler contractive mapping of type A if there exist two functions ψ ∈ Ψ and γ : X 3 7→ [0, ∞) satisfying the following condition: for each  > 0 there exists δ() > 0 such that  ≤ ψ(M1 (x, y)) <  + δ() implies γ(x, y, y)S(T x, T y, T y) < , where M1 (x, y) = max{S(x, y, y), S(x, T x, T x), S(y, T y, T y)}, ∀x, y ∈ X. Definition 1.9. Let (X, S) be an S−metric space and T : X 7→ X. The mapping T is called a generalized (γ − ψ)−Meir-Keeler contractive mapping of type B if there exist two functions ψ ∈ Ψ and γ : X 3 7→ [0, ∞) satisfying the following condition: for each  > 0 there exists δ() > 0 such that  ≤ ψ(M2 (x, y)) <  + δ() implies γ(x, y, y)S(T x, T y, T y) < , where M2 (x, y) = max{S(x, y, y), 12 [S(x, T x, T x) + S(y, T y, T y)]}, ∀x, y ∈ X. Remark 1.5. (1) It is obviously that M2 (x, y) ≤ M1 (x, y), ∀x, y ∈ X, where M1 (x, y), M2 (x, y) are defined in Definition 1.8 and Definition 1.9, respectively. (2) Let T : X 7→ X be a generalized (γ − ψ)−Meir-Keeler contractive mapping of type A (resp., type B). Then γ(x, y, y)S(T x, T y, T y) < ψ(M1 (x, y)), (resp., ψ(M2 (x, y))), ∀x, y ∈ X. Definition 1.10. Let (X, S) be an S−metric space and T : X 7→ X. The mapping T is called a generalized (γ − ψ)−Meir-Keeler contractive mapping of dim3 of type A if there exist two functions ψ ∈ Ψ and γ : X 3 7→ [0, ∞) satisfying the following condition: for each  > 0 there exists δ() > 0 such that 0

 ≤ ψ(M1 (x, y, z)) <  + δ() implies γ(x, T x, T x)γ(y, T y, T y)γ(z, T z, T z)S(T x, T y, T z) < , where 0

M1 (x, y, z) = max{S(x, y, y), S(y, z, z), S(z, x, x), S(x, T x, T x), S(y, T y, T y)S(z, T z, T z)}, ∀x, y, z ∈ X. Definition 1.11. Let (X, S) be an S−metric space and T : X 7→ X. The mapping T is called a generalized (γ − ψ)−Meir-Keeler contractive mapping of dim3 of type B if there exist two functions ψ ∈ Ψ and γ : X 3 7→ [0, ∞) satisfying the following condition: for each  > 0 there exists δ() > 0 such that 0

 ≤ ψ(M2 (x, y, z)) <  + δ() implies γ(x, T x, T x)γ(y, T y, T y)γ(z, T z, T z)S(T x, T y, T y) < , where 0 1 M2 (x, y, z) = max{S(x, y, y), S(y, z, z), S(z, x, x), [S(x, T x, T x) + S(y, T y, T y)], 2 1 1 [S(y, T y, T y) + S(z, T z, T z)], [S(z, T z, T z) + S(x, T x, T x)]}, 2 2

∀x, y, z ∈ X. 0

0

0

0

Remark 1.6. (1) It is obviously that M2 (x, y, z) ≤ M1 (x, y, z), ∀x, y, z ∈ X, where M1 (x, y, z), M2 (x, y, z) are defined in Definition 1.10 and Definition 1.11, respectively. (2) Let T : X 7→ X be a generalized (γ − ψ)−Meir-Keeler contractive mapping of dim3 of type A (re0

0

sp., type B). Then γ(x, T x, T x)γ(y, T y, T y)γ(z, T z, T z)S(T x, T y, T z) < ψ(M1 (x, y, z)), (resp., ψ(M2 (x, y, z))), ∀x, y, z ∈ X. 598

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2.

Fixed point theorems for several types of (γ − ψ)−MeirKeeler contractive mappings in S−metric spaces

In this section, by introducing the class of (γ − ψ)−Meir-Keeler contractive mapping and the classes of generalized (γ − ψ)−Meir-Keeler contractive mappings, we study the existence and uniqueness of fixed points for these contractive mappings via γ−admissible mappings. Proposition 2.1. Assume that T is γ−admissible and (γ −ψ)−Meir-Keeler contractive. Let x, y ∈ X such that γ(x, y, y) ≥ 1. Then γ(T n x, T n y, T n y) ≥ 1, ∀n ∈ N,

(5)

the sequence {S(T n x, T n y, T n y)} is non-increasing, bounded and S(T n x, T n y, T n y) → 0 as n → ∞. Proof. Since T is γ−admissible and γ(x, y, y) ≥ 1, then (5) follows directly by induction on n. Next, let n ∈ N. If T n x 6= T n y, by (5) and Remark 1.3, it follows that S(T n+1 x, T n+1 y, T n+1 y) ≤γ(T n x, T n y, T n y)S(T n+1 x, T n+1 y, T n+1 y) = γ(T n x, T n y, T n y)S(T (T n x), T (T n y), T (T n y)) < ψ(S(T n x, T n y, T n y)) < S(T n x, T n y, T n y). Else, if T n x = T n y, then S(T n x, T n y, T n y) = S(T n+1 x, T n+1 y, T n+1 y). Eventually, we conclude that {S(T n x, T n y, T n y)} is a non-increasing and bounded sequence. Hence, there exists r ∈ [0, ∞) such that lim S(T n x, T n y, T n y) = r. n→∞

In what follows, we will prove that r = 0. Suppose, on the contrary, that r > 0. Since T is a (γ − ψ)−Meir-Keeler contractive mapping, for  = ψ(r) > 0, there exists δ > 0 and a p ∈ N such that  ≤ ψ(S(T p x, T p y, T p y)) <  + δ implies γ(T p x, T p y, T p y)S(T p+1 x, T p+1 y, T p+1 y) < . By taking (5) into account, we get that S(T p+1 x, T p+1 y, T p+1 y) <  = ψ(r) < r, which is a contradiction, since r = inf{S(T n x, T n y, T n y)}∞ n=1 . Consequently, we have lim S(T n x, T n y, T n y) = 0. n→∞

Proposition 2.2. Assume that T is γ−admissible and (γ − ψ)−Meir-Keeler contractive of dim3. Let x, y, z ∈ X such that γ(x, T x, T x) ≥ 1, γ(y, T y, T y) ≥ 1, γ(z, T z, T z) ≥ 1. Then γ(T n x, T n y, T n z) ≥ 1, ∀n ∈ N,

(6)

the sequence {S(T n x, T n y, T n z)} is non-increasing, bounded and S(T n x, T n y, T n z) → 0 as n → ∞. Proof. Using similar process to the proof of Proposition 2.1, one can safely draw the conclusion. Theorem 2.1. Let (X, S) be a complete S−metric space and T : X 7→ X be a (γ − ψ)−MKC mapping. Assume that (A1) T is γ−admissible; 599

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(A2) there exists x0 ∈ X such that γ(x0 , T x0 , T x0 ) ≥ 1; (A3) T is γ−orbital continuous. Then, there exists x∗ ∈ X such that T x∗ = x∗ . Proof. Due to assumption (A2), there exists x0 ∈ X such that γ(x0 , T x0 , T x0 ) ≥ 1. Define an iterative sequence {xn } in X by xn+1 = T xn , ∀n ∈ {0}∪N. Note that if xn0 = xn0 +1 for some n0 , then x∗ = xn0 is a fixed point of T . So we suppose that xn 6= xn+1 for ∀n ∈ {0} ∪ N. Since T is γ−admissible, we have that γ(x0 , x1 , x1 ) = γ(x0 , T x0 , T x0 ) ≥ 1 ⇒ γ(T x0 , T x1 , T x1 ) = γ(x1 , x2 , x2 ) ≥ 1. By induction, we get that γ(xn , xn+1 , xn+1 ) ≥ 1,

∀n ∈ {0} ∪ N.

(7)

From (7) together with the assumption of the theorem that T is a (γ − ψ)−MKC mapping, it follows that for ∀n ∈ N, we have that S(xn , xn+1 , xn+1 )

= S(T xn−1 , T xn , T xn ) ≤ γ(xn−1 , xn , xn )S(T xn−1 , T xn , T xn ) ≤ ψ(S(xn−1 , xn , xn )).

Since ψ ∈ Ψ, by induction, we have that S(xn , xn+1 , xn+1 ) < ψ n (S(x0 , x1 , x1 )),

∀n ∈ N.

(8)

Using (S2) and (8), for ∀m, n ∈ N with m < n, we have that S(xm , xn , xn ) ≤ 2

n−2 X

S(xk , xk+1 , xk+1 ) + S(xn−1 , xn , xn )

k=m

≤ 2

n−2 X

ψ k (S(x0 , x1 , x1 )) + ψ n−1 (S(x0 , x1 , x1 )).

k=m

Since ψ ∈ Ψ and S(x0 , x1 , x1 ) > 0, by Remark 1.2, we get that lim S(xm , xn , xn ) = 0.

n,m→∞

This implies that {xn } is a Cauchy sequence in the S−metric space (X, S). As (X, S) is complete, then there exists x∗ ∈ X such that lim S(xn , xn , x∗ ) = 0.

n→∞

(9)

Since T is γ−orbital continuous, then there exists a subsequence {xnk } of {xn } such that T xnk converges to T x∗ as k → ∞. By the uniqueness of this limit, we get x∗ = T x∗ , that is x∗ is a fixed point of T . Theorem 2.2. Let (X, S) be a complete S−metric space and T : X 7→ X be a (γ − ψ)−MKC mapping of dim3. Assume that 600

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(A1) T is γ−admissible; (A2) there exists x0 ∈ X such that γ(x0 , T x0 , T x0 ) ≥ 1; (A3) T is γ−orbital continuous. Then, there exists x∗ ∈ X such that T x∗ = x∗ . Proof. Due to assumption (A2), there exists x0 ∈ X such that γ(x0 , T x0 , T x0 ) ≥ 1. Define an iterative sequence {xn } in X by xn+1 = T xn for all n ∈ {0} ∪ N. Note that if xn0 = xn0 +1 for some n0 , then x∗ = xn0 is a fixed point of T . So we suppose that xn 6= xn+1 for all n ∈ {0} ∪ N. Since T is γ−admissible, we have that γ(x0 , x1 , x1 ) = γ(x0 , T x0 , T x0 ) ≥ 1 ⇒ γ(T x0 , T x1 , T x1 ) = γ(x1 , x2 , x2 ) ≥ 1. By induction, we get that γ(xn , xn+1 , xn+1 ) ≥ 1, ∀n ∈ {0} ∪ N.

(10)

From (10) together with the assumption of the theorem that T is a (γ − ψ)−MKC mapping of dim3, it follows that for ∀n ∈ N, we have that S(xn , xn+1 , xn+1 ) = S(T xn−1 , T xn , T xn ) ≤ γ(xn−1 , xn , xn )γ(xn , xn+1 , xn+1 )γ(xn , xn+1 , xn+1 )S(T xn−1 , T xn , T xn ) ≤ ψ(S(xn−1 , xn , xn )). Since ψ ∈ Ψ, by induction, we have that S(xn , xn+1 , xn+1 ) < ψ n (S(x0 , x1 , x1 )),

∀n ∈ N.

Using Lemma 1.3 and (10), for l, m, n ∈ N with l < m < n, we have that S(xl , xm , xn ) ≤ S(xl , xl , xm ) + S(xm , xm , xn ) ≤2

m−2 X k=l

≤2

m−2 X

n−2 X

S(xk , xk+1 , xk+1 ) + S(xm−1 , xm , xm ) + 2

S(xk , xk+1 , xk+1 ) + S(xn−1 , xn , xn )

k=m

ψ k (S(x0 , x1 , x1 )) + ψ m−1 (S(x0 , x1 , x1 )) + 2

k=l

n−2 X

ψ k (S(x0 , x1 , x1 )) + ψ n−1 (S(x0 , x1 , x1 )).

k=m

Since ψ ∈ Ψ and S(x0 , x1 , x1 ) > 0, by Remark 1.2, we get that lim

S(xl , xm , xn ) = 0.

l,m,n→∞

This implies that {xn } is a Cauchy sequence in the S−metric space (X, S). As (X, S) is complete, then there exists x∗ ∈ X such that lim S(xn , xn , x∗ ) = 0.

n→∞

Since T is γ−orbital continuous, then there exists a subsequence {xnk } of {xn } such that T xnk converges to T x∗ as k → ∞. By the uniqueness of this limit, we get x∗ = T x∗ , that is x∗ is a fixed point of T . 601

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In the next theorems, we replace the γ−orbital continuity of T by a regularity condition or (T − γ)−limit condition over the S−metric spaces (X, S). Theorem 2.3. Let (X, S) be a complete S−metric space and T : X 7→ X be a (γ − ψ)−MKC mapping. Assume that (A1) T is γ−admissible; (A2) there exists x0 ∈ X such that γ(x0 , T x0 , T x0 ) ≥ 1; (A3) (X, S) is (T, γ)− regular. Then, there exists x∗ ∈ X such that T x∗ = x∗ . Proof. Following the line of the proof of Theorem 2.1, it follows that the sequence {xn } defined by xn+1 = T xn , ∀n ∈ {0} ∪ N is a Cauchy sequence in the complete S−metric space (X, S), that is convergent to x∗ ∈ X. Since {xn } is a (T, γ)−orbital sequence, by (A3), there exists a subsequence {xnk } of {xn } such that γ(xnk , x∗ , x∗ ) ≥ 1, ∀k ∈ N.

(11)

Using Remark 1.3 and (11), we have that S(xnk +1 , T x∗ , T x∗ )

= S(T xnk , T x∗ , T x∗ ) ≤ γ(xnk , x∗ , x∗ )S(T xnk , T x∗ , T x∗ ) ≤ ψ(S(xnk , x∗ , x∗ )).

Letting k → ∞, since ψ is continuous at t = 0, it follows that S(x∗ , T x∗ , T x∗ ) = 0, then x∗ = T x∗ . Theorem 2.4. Let (X, S) be a complete S−metric space and T : X 7→ X be a (γ − ψ)−MKC mapping of dim3. Assume that (A1) T is γ−admissible; (A2) there exists x0 ∈ X such that γ(x0 , T x0 , T x0 ) ≥ 1; (A3) (X, S) is (T, γ)−limit. Then, there exists x∗ ∈ X such that T x∗ = x∗ . Proof. Following the line of the proof of Theorem 2.1, it follows that the sequence {xn } defined by xn+1 = T xn , for all n ∈ {0} ∪ N is a Cauchy sequence in the complete S−metric space (X, S), that is convergent to x∗ ∈ X. By (A3), we have γ(x∗ , T x∗ , T x∗ ) ≥ 1.

(12)

Using Remark 1.4 and (12), we have that S(xn+ 1 , T x∗ , T x∗ ) =S(T xn , T x∗ , T x∗ ) ≤ γ(xn , xn+1 , xn+1 )γ(x∗ , T x∗ , T x∗ )γ(x∗ , T x∗ , T x∗ )S(T xn , T x∗ , T x∗ ) ≤ ψ(S(xn , x∗ , x∗ )). Letting n → ∞, since ψ is continuous at t = 0, it follows that S(x∗ , T x∗ , T x∗ ) = 0, then x∗ = T x∗ . 602

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Example 2.1. Let X = [0, ∞) be an S−metric space with the S−metric defined by S(x, y, z) = |x − z|  + |y − z|, ∀x, y, z ∈ X. For ∀k > 1, consider the self-mapping T : X 7→ X given by  ex−1 , x ≥ 1, T x=  x2 , 0 ≤ x < 1. 4 Also, defineγ : X 3 7→ [0, 1) as  1, x, y, z ∈ [0, 1), γ(x, y, z)=  0, otherwise. Let ψ(t) = 2t for t ≥ 0. Clearly, T is not continuous at x = 1. Then we will claim that T is a (γ − ψ)−MKC. Let  > 0 be given. Take δ =  and suppose that  ≤

1 2 |x

− y| <  + δ, we want to show that

γ(x, y, y)S(T x, T y, T y) < . Suppose that γ(x, y, y) = 1, then x, y ∈ [0, ∞) and |x + y| < 2. So T x = Hence, S(T x, T y, T y) =

2 | x4

−

y2 4 |

|x2 −y 2 | 4

=

=

|x+y||x−y| 4


0 be given. Take δ =  and suppose that  ≤ 12 (|x − y| + |y − z|) <  + δ, we want to show that γ(x, T x, T x)γ(y, T y, T y)γ(z, T z, T z)S(T x, T y, T z) < . Suppose that γ(x, T x, T x) = γ(y, T y, T y) = γ(z, T z, T z) = 1, then x, y, z, T x, T y, T z ∈ [0, 1) and |x + y| < 2, |y + z| < 2. So T x =

x2 4

∈ [0, 1), T y =

603

y2 4

∈ [0, 1), T z =

z2 4

∈ [0, 1).

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Hence, x2 y2 y2 z2 − |+| − | 4 4 4 4 |x2 − y 2 | |y 2 − z 2 | + = 4 4 |x + y||x − y| |y + z||y − z| = + 4 4 |x − y| |y − z| < + 2 2 +δ < 2 = .

S(T x, T y, T z) = |

Also, T is γ−admissible. To see that, let x, y, z ∈ X such that γ(x, y, z) ≥ 1, which implies that x, y, z ∈ [0, 1). Due to the definitions of γ and T , we have that

Tx =

x2 ∈ [0, 1), 4

Ty =

y2 ∈ [0, 1), 4

Tz =

z2 ∈ [0, 1). 4

Hence, γ(T x, T y, T z) ≥ 1. Moreover, there exists x0 ∈ X such that γ(x0 , T x0 , T x0 ) ≥ 1. Indeed, for any x0 ∈ [0, 1), we have γ(x0 ,

x20 x20 4 , 4 )

≥ 1.

Finally, let {xn } be a sequence such that xn → x as n → ∞ with γ(xn , xn+1 , xn+1 ) ≥ 1. By the definition of γ, we have that x, T x ∈ [0, 1). Then γ(x, T x, T x) ≥ 1. So we conclude that all the hypotheses of Theorem 2.4 are fulfilled. In fact, 0 and 1 are two fixed points of T . Now, we propose the following conditions for the uniqueness of a fixed point of a (γ − ψ)−MKC mapping and a (γ − ψ)−MKC mapping of dim3. Let F ix(T ) denote the set of fixed points of the mapping T . (U 1) For ∀x, y ∈ F ix(T ), there exists z ∈ X such that γ(x, z, z) ≥ 1 and γ(y, z, z) ≥ 1. Theorem 2.5. Adding the condition (U 1) to the hypotheses of Theorem 2.1(resp.Theorem 2.3), we obtain the uniqueness of a fixed point T .

Proof. Let u, v ∈ X be two fixed points of T . By (U 1), there exists z ∈ X such that γ(u, z, z) ≥ 1 and γ(v, z, z) ≥ 1. Since T is γ−admissible, we get by induction that γ(u, u, T n z) ≥ 1 and

γ(v, v, T n z) ≥ 1,

∀n ∈ N.

(13)

From (13), we have that S(u, u, T n z)

= S(T u, T u, T (T n−1 z)) ≤ γ(u, u, T n−1 z)S(T u, T u, T (T n−1 z))
0. Due to the property of ψ, we get that ψ(S(u, v, v)) > 0. Let  = ψ(S(u, v, v)) > 0; then, for any δ > 0, we find that  = ψ(S(u, v, v)) <  + δ. Considering (U 2) and the assumption of theorem that T is a (γ − ψ)−MKC mapping, we obtain that S(u, v, v) ≤ γ(u, v, v)S(T u, T v, T v) < ψ(S(u, v, v)) < S(u, v, v), which is a contradiction. Then u = v. As a uniqueness condition for fixed points of (γ − ψ)−MKC mappings of dim3, we suggest the following hypothesis: (U 3) For ∀x ∈ F ix(T ), then γ(x, x, x) ≥ 1. Theorem 2.7. Adding the condition (U 3) to the hypotheses of Theorem 2.2(resp.Theorem 2.4), we obtain the uniqueness of a fixed point T . Proof. Let u, v be two distinct fixed points of T . Due to the property of ψ, we get that ψ(S(u, v, v)) > 0. Let  = ψ(S(u, v, v)) > 0; then, for any δ > 0, we find that  = ψ(S(u, v, v)) <  + δ. Considering (U 3) and the assumption of theorem that T is a (γ − ψ)−MKC mapping of dim3, we obtain that S(u, v, v) ≤ γ(u, T u, T u)γ(v, T v, T v)γ(v, T v, T v)S(T u, T v, T v) < ψ(S(u, v, v)) < S(u, v, v), which is a contradiction. Then u = v. 605

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Theorem 2.8. Let (X, S) be a complete S−metric space and T : X 7→ X be a generalized (γ − ψ)−MKC mapping of type A. Assume also that: (A1) T is triangular γ−admissible; (A2) there exists x0 ∈ X such that γ(x0 , T x0 , T x0 ) ≥ 1; (A3) (X, S) is (T, γ)−regular. Then, there exists x∗ ∈ X such that T x∗ = x∗ . Proof. In view of assumption (A2), let x0 ∈ X be such that γ(x0 , T x0 , T x0 ) ≥ 1. Define the sequence {xn } in X by xn+1 = T xn , ∀n ∈ {0} ∪ N. Without loss of generality, we assume that xn 6= xn+1 , for ∀n ∈ {0} ∪ N, then S(xn , xn+1 , xn+1 ) > 0, ∀n{0} ∪ N.

(16)

Indeed, if there exists some n0 ∈ N such that xn0 = xn0 +1 , then the proof is complete, since x∗ = xn0 +1 = T xn0 = T x∗ . Since T is triangular γ−admissible, by Lemma 1.6, we have that γ(xn , xm , xm ) ≥ 1, ∀n, m ∈ N with n < m.

(17)

Step1. We will prove that lim S(xn , xn+1 , xn+1 ) = 0.

(18)

n→∞

Taking (16) and (17) into account together with the fact that T is generalized (γ − ψ)−MKC mapping of type A, for each n ∈ {0} ∪ N, we get S(xn , xn+1 , xn+1 ) = S(T xn−1 , T xn , T xn ) ≤ γ(xn−1 , xn , xn )S(T xn−1 , T xn , T xn ) ≤ ψ(M1 (xn−1 , xn )) < ψ(M1 (xn−1 , xn )), where M1 (xn−1 , xn ) = max{S(xn−1 , xn , xn ), S(xn−1 , T xn−1 , T xn−1 ), S(xn , T xn , T xn )} = max{S(xn−1 , xn , xn ), S(xn , xn , xn+1 )}. If M1 (xn−1 , xn ) = S(xn , xn+1 , xn+1 ). Since ψ is nondecreasing, from the inequality above, we have that S(xn , xn+1 , xn+1 ) ≤ ψ(S(xn , xn+1 , xn+1 )) < S(xn , xn+1 , xn+1 ), ∀n ∈ N, which is a contradiction. Thus, M1 (xn−1 , xn ) = S(xn−1 , xn , xn ) and we also have that S(xn , xn+1 , xn+1 ) ≤ ψ(S(xn−1 , xn , xn )) < S(xn−1 , xn , xn ), ∀n ∈ N.

(19)

So, we deduce that the sequence {S(xn , xn+1 , xn+1 )} is non-increasing and bounded below by zero. Hence, there exists t ∈ [0, ∞) such that lim S(xn , xn+1 , xn+1 ) = t.

n→∞

606

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Iteratively, we derive from (19) that S(xn , xn+1 , xn+1 ) ≤ ψ n (S(x0 , x1 , x1 )), ∀n ∈ N.

(21)

On account of (21) and Remark 1.2, we obtain lim S(xn , xn+1 , xn+1 ) = 0.

(22)

n→∞

Step2. We will show that {xn } is a Cauchy sequence. Suppose, on the contrary, that there exist  > 0 and a subsequence {xn(i) } of {xn } such that S(xn(i) , xn(i+1) , xn(i+1) ) > 2.

(23)

First, we will show that the existence of k ∈ N such that n(i) < k ≤ n(i + 1). Later, we will prove that for given  > 0 above, there exists δ > 0 such that +δ  < ψ(M1 (xn(i) , xk )) < , 2 2 but γ(xn(i) , xk , xk )S(T xn(i) , T xk , T xk ) ≥ , which contradicts (23), where M1 (xn(i) , xk ) = max{S(xn(i) , xk , xk ), S(xn(i) , xn(i)+1 , xn(i)+1 ), S(xk , xk+1 , xk+1 )} Let r = min{, 2δ }. Taking Step1 into account, we will choose n0 ∈ N such that S(xn , xn+1 , xn+1 )
n0 . Let n(i) > n0 . According to our construction, we have n(i) ≤ n(i + 1) − 1. If S(xn(i) , xn(i+1)−1 , xn(i+1)−1 )


+r 2 .

Indeed, if S(xn(i) , xn(i)+1 , xn(i)+1 ) ≥

+r 2 ,

then we have S(xn(i) , xn(i)+1 , xn(i)+1 ) ≥ 8r , which contra-

dicts (24). Hence, we can choose the smallest integer k > n(i) such that S(xn(i) , xk , xk ) ≥ So, necessarily, we also have S(xn(i) , xk−1 , xk−1 )
0, we can choose n0 ∈ N such that S(xn , xn+1 , xn+1 )
0. From assumption (A3), we have that γ(x∗ , T x∗ , T x∗ ) ≥ 1,

∀k ∈ N.

By using Lemma 1.1 and above inequality together with the assumption of the theorem that T is a generalized (γ − ψ)−MKC mapping of dim3 of type A, we get that S(x∗ , T x∗ , T x∗ ) ≤ 2S(T xnk , T x∗ , T x∗ ) + S(T xnk , x∗ , x∗ ) ≤ 2γ(xnk , T xnk , T xnk )γ(x∗ , T x∗ , T x∗ )γ(x∗ , T x∗ , T x∗ )S(T xnk , T x∗ , T x∗ ) + S(xnk +1 , x∗ , x∗ ) 0

≤ 2ψ(M1 (xnk , x∗ , x∗ )) + S(xnk +1 , x∗ , x∗ ), 0

where, M1 (xnk , x∗ , x∗ ) = max{S(xnk , x∗ , x∗ ), S(xnk , xnk +1 , xnk +1 ), S(x∗ , T x∗ , T x∗ )}. Suppose that 0

M1 (xnk , x∗ , x∗ ) = S(xnk , x∗ , x∗ ), then from the above inequality, we get that S(x∗ , T x∗ , T x∗ ) ≤ 2ψ(S(xnk , x∗ , x∗ )) + S(xnk +1 , x∗ , x∗ ) 1, x ≥ 1; and γ(x, y, z)= T x=  0,  x , x ∈ [0, 1). 4 Let ψ(t) = 2t , t ≥ 0.

if

x, y, z ∈ [0, 1);

otherwise.

We first show that T is a triangular γ−admissible mapping. Let x, y, z ∈ X, if γ(x, y, z) ≥ 1, the x, y, z ∈ [0, 1). On the other hand, for ∀x, y, z ∈ [0, 1), we have T x = Tz =

z 4

x 4

∈ [0, 1), T y =

y 4

∈ [0, 1),

∈ [0, 1). It follows that γ(T x, T y, T z) ≥ 1. Also, if γ(x, y, y) ≥ 1 and γ(y, y, z) ≥ 1, then

x, y, z ∈ [0, 1) and hence γ(x, z, z) ≥ 1. Thus, the first assertion holds. Notice that γ(0, 0, 0) = 1. Next, if {xn } is a (T, γ)−orbital sequence such that xn → x as n → ∞. By the definition of γ, we have that xn ∈ [0, 1) and x ∈ [0, 1). Then there exists a subsequence {xnk } of {xn } such that γ(xnk , x, x) ≥ 1, ∀k ∈ N. Finally, we will show that T is generalized (γ − ψ)−MKC mapping of type A. If γ(x, y, y) = 0, it is obviously to verify the assertion. If γ(x, y, y) 6= 0, it follows that x, y ∈ [0, 1) and γ(x, y, y) = 1. For  > 0, Case 1. If M1 (x, y) = 2|x − y|, taking δ = , then  ≤ ψ(M1 (x, y)) = |x − y| < 2 implies that |x−y| < ε. 2 3|x| Case 2. If M1 (x, y) = 2 , taking δ = 3 , then  ≤ ψ(M1 (x, y)) = 3|x| 4 γ(x, y, y)S(T x, T y, T y) = 12 |x − y| ≤ 21 (|x| + |y|) < 12 (|x| + |x|) = |x| < . 3|y|  Case 3. If M1 (x, y) = 3|y| 2 , taking δ = 3 , then  ≤ ψ(M1 (x, y)) = 4 γ(x, y, y)S(T x, T y, T y) = 12 |x − y| ≤ 21 (|x| + |y|) < 12 (|y| + |y|) = |y| < .

γ(x, y, y)S(T x, T y, T y) =

< +

 3

implies that

< +

 3

implies that

Therefore, conditions of Theorem 2.8 hold and T has a fixed point. Indeed, x∗ = 0 and x∗ = 1 are two fixed points. In what follows, we present an existence theorem for fixed point of a generalized (γ − ψ)−MKC mapping of type B and a generalized (γ − ψ)−MKC mapping of dims3 of type B. Taking Remark 1.5 and Remark 1.6 into account, we observe that the proof of this theorem is similar to the proof of Theorem 2.8 and Theorem 2.9. Theorem 2.10. Let (X, S) be a complete S−metric space and T : X 7→ X be a generalized (γ − ψ)−MKC mapping of type B. Assume also that: 612

Mi Zhou ET AL 593-628

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(A1) T is triangular γ−admissible; (A2) there exists x0 ∈ X such that γ(x0 , T x0 , T x0 ) ≥ 1; (A3) (X, S) is (T, γ)−regular. Then, there exists x∗ ∈ X such that T x∗ = x∗ . Theorem 2.11. Let (X, S) be a complete S−metric space and T : X 7→ X be a generalized (γ − ψ)−MKC mapping of dim3 of type B. Assume also that: (A1) T is triangular γ−admissible; (A2) there exists x0 ∈ X such that γ(x0 , T x0 , T x0 ) ≥ 1; (A3) (X, S) is (T, γ)−limit. Then, there exists x∗ ∈ X such that T x∗ = x∗ . Definition 2.1. Let (X, S) be an S−metric space and T : X 7→ X. The mapping T is called a generalized (γ − ψ)−Meir-Keeler contractive mapping of type C if there exist two functions ψ ∈ Ψ and γ : X 3 7→ [0, ∞) satisfying the following condition: for each  > 0 there exists δ() > 0 such that  ≤ ψ(M3 (x, y)) <  + δ() implies γ(x, y, y)S(T x, T y, T y) < ,

(36)

where M3 (x, y) = max{S(x, y, y), S(x, T x, T x), S(y, T y, T y), 18 [S(x, T y, T y) + S(y, T x, T x)]}, ∀x, y ∈ X. Theorem 2.12. Let (X, S) be a complete S−metric space and T : X 7→ X be a generalized (γ − ψ)−MKC mapping of type C. Assume also that: (A1) T is triangular γ−admissible; (A2) there exists x0 ∈ X such that γ(x0 , T x0 , T x0 ) ≥ 1; (A3) (X, S) is (T, γ)−regular. Then, there exists x∗ ∈ X such that T x∗ = x∗ . Proof. In view of assumption (A2), let x0 ∈ X be such that γ(x0 , T x0 , T x0 ) ≥ 1. Define the sequence {xn } in X by xn+1 = T xn , ∀n ∈ {0} ∪ N. Since T is triangular γ−admissible, by Lemma 1.6, we have that γ(xn , xm , xm ) ≥ 1, ∀n, m ∈ N with n < m. If there exists some n0 ∈ N such that xn0 = xn0 +1 , then the proof is complete, since x∗ = xn0 +1 = T x0 = T x∗ . For this, we assume that xn 6= xn+1 , ∀n ∈ {0} ∪ N, then S(xn , xn+1 , xn+1 ) > 0, ∀n ∈ {0} ∪ N.

(37)

lim S(xn , xn+1 , xn+1 ) = 0.

(38)

Step1. We will prove that n→∞

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Taking (36) and (38) into account together with the fact that T is generalized (γ − ψ)−MKC mapping of type C, for each n ∈ {0} ∪ N, we get S(xn , xn+1 , xn+1 ) = S(T xn−1 , T xn , T xn ) ≤ γ(xn−1 , xn , xn )S(T xn−1 , T xn , T xn ) ≤ ψ(M3 (xn−1 , xn )) < ψ(M3 (xn−1 , xn )), where M3 (xn−1 , xn ) = max{S(xn−1 , xn , xn ), S(xn−1 , T xn−1 , T xn−1 ), S(xn , T xn , T xn ), 1 [S(xn−1 , T xn , T xn ) + S(xn , T xn−1 , T xn−1 )]} 8 1 = max{S(xn−1 , xn , xn ), S(xn , xn , xn+1 ), [S(xn−1 , xn+1 , xn+1 ) + S(xn , xn , xn )]}. 8 Regarding Lemma 1.1, we estimate the last term in the expression of M3 (xn−1 , xn ) as follows: 1 [S(xn−1 , xn+1 , xn+1 ) + S(xn , xn , xn )] 8 1 = S(xn−1 , xn+1 , xn+1 ) 8 1 ≤ [2S(xn−1 , xn , xn ) + S(xn , xn+1 , xn+1 ) 8 1 1 = S(xn−1 , xn , xn ) + S(xn , xn+1 , xn+1 ) 4 8 ≤ max{S(xn−1 , xn+1 , xn+1 ), S(xn , xn , xn )}. Consequently, we get that M3 (xn−1 , xn ) = max{S(xn−1 , xn , xn ), S(xn , xn+1 , xn+1 )}.

(39)

Let us consider the two cases. If M3 (xn−1 , xn ) = S(xn , xn+1 , xn+1 ). Since ψ is nondecreasing, then we have that S(xn , xn+1 , xn+1 ) ≤ ψ(S(xn , xn+1 , xn+1 )) < S(xn , xn+1 , xn+1 ),

(40)

which is a contradiction. Thus, M3 (xn−1 , xn ) = S(xn−1 , xn , xn ) and we also have that S(xn , xn+1 , xn+1 ) ≤ ψ(S(xn−1 , xn , xn )) < S(xn−1 , xn , xn ), ∀n ∈ N.

(41)

So, we derive that the sequence {S(xn , xn+1 , xn+1 )} is non-increasing and bounded below by zero. Hence, there exists t ∈ [0, ∞) such that lim S(xn , xn+1 , xn+1 ) = t.

(42)

n→∞

Recursively, we deduce from (41) that S(xn , xn+1 , xn+1 ) ≤ ψ n (S(x0 , x1 , x1 )),

∀n ∈ N.

(43)

On account of (43) and Remark 1.2, we obtain lim S(xn , xn+1 , xn+1 ) = 0.

n→∞

614

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Step2. We will show that {xn } is a Cauchy sequence. Suppose, on the contrary, that there exist  > 0 and a subsequence {xn(i) } of {xn } such that S(xn(i) , xn(i+1) , xn(i+1) ) > 2.

(45)

First, we will show that the existence of k ∈ N such that n(i) < k ≤ n(i + 1). Later, we will prove that for given  > 0 above, there exists δ > 0 such that +δ  < ψ(M3 (xn(i) , xk )) < , 2 2 but γ(xn(i) , xk , xk )S(T xn(i) , T xk , T xk ) ≥ , which contradicts (45), where M3 (xn(i) , xk ) = max{S(xn(i) , xk , xk ), S(xn(i) , T xn(i) , T xn(i)1 ), S(xk , T xk , T xk ), 1 [S(xn(i) , T xk , T xk ) + S(xk , T xn(i) , T xn(i) )]}. 8 Let r = min{, 2δ }. Taking Step1 into account, we will choose n0 ∈ N such that S(xn , xn+1 , xn+1 )
n0 . Let n(i) > n0 . According to our construction, we have n(i) ≤ n(i + 1) − 1. If S(xn(i) , xn(i+1)−1 , xn(i+1)−1 )


+r 2 .

Indeed, if S(xn(i) , xn(i)+1 , xn(i)+1 ) ≥

+r 2 ,

then we have S(xn(i) , xn(i)+1 , xn(i)+1 ) ≥ 8r , which contra-

dicts (46). Hence, we can choose the smallest integer k > n(i) such that S(xn(i) , xk , xk ) ≥ So, necessarily, we also have S(xn(i) , xk−1 , xk−1 )
0. From Lemma 1.1, it follows that 1 [S(xnk , T x∗ , T x∗ ) + S(x∗ , xnk +1 , xnk +1 )] 8 1 ≤ [2S(xnk , x∗ , x∗ ) + S(x∗ , T x∗ , T x∗ ) + 2S(x∗ , xnk , xnk ) + S(xnk , xnk +1 , xnk +1 )] 8 1 1 1 = S(xnk , x∗ , x∗ ) + S(x∗ , T x∗ , T x∗ ) + S(xnk , xnk +1 , xnk +1 ) 2 8 8 ≤ max{S(xnk , x∗ , x∗ ), S(x∗ , T x∗ , T x∗ ), S(xnk , xnk +1 , xnk +1 )}. By the above inequality, we have that M3 (xnk , x∗ ) =

max{S(xnk , x∗ , x∗ ), S(x∗ , T x∗ , T x∗ ), S(xnk , xnk +1 , xnk +1 )}.

Suppose that M3 (xnk , x∗ ) = S(xnk , x∗ , x∗ ), then, we get that S(x∗ , T x∗ , T x∗ ) ≤ ψ(S(xnk , x∗ , x∗ )) + S(xnk +1 , x∗ , x∗ ) 0 there exists δ() > 0 such that 0

 ≤ ψ(M3 (x, y, z)) <  + δ() implies γ(x, T x, T x)γ(y, T y, T y)γ(z, T z, T z)S(T x, T y, T z) < , where 0

M3 (x, y, z) = max{S(x, y, y), S(y, z, z), S(z, x, x), S(x, T x, T x), S(y, T y, T y), S(z, T z, T z), 1 1 [S(x, T y, T y) + S(y, T x, T x)], [S(y, T z, T z) + S(z, T y, T y)], 8 8 1 [S(z, T x, T x) + S(x, T z, T z)]}, 8 ∀x, y, z ∈ X. Theorem 2.13. Let (X, S) be a complete S−metric space and T : X 7→ X be a generalized (γ − ψ)−MKC mapping of dim3 of type C. Assume also that: (A1) T is γ−admissible; (A2) there exists x0 ∈ X such that γ(x0 , T x0 , T x0 ) ≥ 1; (A3) (X, S) is (T, γ)−limit. Then, there exists x∗ ∈ X such that T x∗ = x∗ . Proof. In view of assumption (A2), let x0 ∈ X be such that γ(x0 , T x0 , T x0 ) ≥ 1. Define the sequence {xn } in X by xn+1 = T xn , for all n ∈ {0} ∪ N. Since T is triangular γ−admissible, by Lemma 1.6, we have that γ(xn , xn+1 , xn+1 ) ≥ 1, ∀n ∈ N .

(50)

If there exists some n0 ∈ N such that xn0 = xn0 +1 , then the proof is complete, since x∗ = xn0 +1 = T xn0 = T x∗ . For this, we assume that xn 6= xn+1 , for all n ∈ N, then S(xn , xn+1 , xn+1 ) > 0, ∀n ∈ {0} ∪ N.

(51)

Step1. We will prove that lim S(xn , xn+1 , xn+1 ) = 0.

n→∞

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Taking (50) and (51) into account together with the fact that T is generalized (γ − ψ)−MKC mapping of dim3 of type C, for each n ∈ N, we get S(xn , xn+1 , xn+1 ) = S(T xn−1 , T xn , T xn ) ≤ γ(xn−1 , xn , xn )γ(xn , xn+1 , xn+1 )γ(xn , xn+1 , xn+1 )S(T xn−1 , T xn , T xn ) 0

≤ ψ(M3 (xn−1 , xn , , xn )) 0

< M3 (xn−1 , xn , xn ), where 0

M3 (xn−1 , xn , xn ) = max{S(xn−1 , xn , xn ), S(xn , xn , xn ), S(xn , xn−1 , xn−1 ), S(xn−1 , T xn−1 , T xn−1 ), 1 S(xn , T xn , T xn ), S(xn , T xn , T xn ), [S(xn−1 , T xn , T xn ) + S(xn , T xn−1 , T xn−1 )], 8 1 1 [S(xn , T xn , T xn ) + S(xn , T xn , T xn )], [S(xn , T xn−1 , T xn−1 ) + S(xn−1 , T xn , T xn )]} 8 8 1 = max{S(xn−1 , xn , xn ), S(xn , xn , xn+1 ), [S(xn−1 , xn+1 , xn+1 ) + S(xn , xn , xn )], 8 1 1 [2S(xn , xn+1 , xn+1 )], [S(xn−1 , xn+1 , xn+1 ) + S(xn , xn , xn ]}. 8 8 0

Regarding Lemma 1.1, we estimate the last term in the expression of M3 (xn−1 , xn , xn ) as follows: 1 [S(xn−1 , xn+1 , xn+1 ) + S(xn , xn , xn )] 8 1 = S(xn−1 , xn+1 , xn+1 ) 8 1 ≤ [2S(xn−1 , xn , xn ) + S(xn , xn+1 , xn+1 ) 8 1 1 = S(xn−1 , xn , xn ) + S(xn , xn+1 , xn+1 ) 4 8 ≤ max{S(xn−1 , xn+1 , xn+1 ), S(xn , xn , xn )}. Consequently, we get that M3 (xn−1 , xn , xn ) = max{S(xn−1 , xn , xn ), S(xn , xn+1 , xn+1 )}. 0

Let us consider the two cases. If M3 (xn−1 , xn , xn ) = S(xn , xn+1 , xn+1 ). Since ψ is nondecreasing, then we have that S(xn , xn+1 , xn+1 ) ≤ ψ(S(xn , xn+1 , xn+1 )) < S(xn , xn+1 , xn+1 ), 0

which is a contradiction. Thus, M3 (xn−1 , xn , xn ) = S(xn−1 , xn , xn ) and we also have that S(xn , xn+1 , xn+1 ) ≤ ψ(S(xn−1 , xn , xn )) < S(xn−1 , xn , xn ), ∀n ∈ N.

(52)

So, we derive that the sequence {S(xn , xn+1 , xn+1 )} is non-increasing and bounded below by zero. Hence, there exists t ∈ [0, ∞) such that lim S(xn , xn+1 , xn+1 ) = t.

n→∞

Recursively, we deduce from (52) that S(xn , xn+1 , xn+1 ) ≤ ψ n (S(x0 , x1 , x1 )), 619

∀n ∈ N.

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On account of (53) and Remark 1.2, we obtain lim S(xn , xn+1 , xn+1 ) = 0.

n→∞

Step2. We will show that {xn } is a Cauchy sequence. We will prove that for each  > 0, there exists n0 ∈ N such that for ∀m, n ≥ n0 , S(xm , xm , xn ) < .

(54)

Taking Step1 into account, for each  > 0, we can choose n0 ∈ N such that S(xn , xn+1 , xn+1 )
0. From assumption (A3), we have that γ(x∗ , T x∗ , T x∗ ) ≥ 1,

∀k ∈ N.

By using Lemma 1.1 and above inequality together with the assumption of the theorem that T is a generalized (γ − ψ)−MKC mapping of dim3 of type C, S(x∗ , T x∗ , T x∗ ) ≤ 2S(T xnk , T x∗ , T x∗ ) + S(T xnk , x∗ , x∗ ) ≤ 2γ(xnk , T xnk , T xnk )γ(x∗ , T x∗ , T x∗ )γ(x∗ , T x∗ , T x∗ )S(T xnk , T x∗ , T x∗ ) + S(xnk +1 , x∗ , x∗ ) 0

≤ ψ(M3 (xnk , x∗ , x∗ )) + S(xnk +1 , x∗ , x∗ ), where, 0

M3 (xnk , x∗ , x∗ ) = max{S(xnk , x∗ , x∗ ), S(x∗ , x∗ , x∗ ), S(x∗ , xnk , xnk ), S(xnk , xnk +1 , xnk +1 ), S(x∗ , T x∗ , T x∗ ), S(x∗ , T x∗ , T x∗ ), S(x∗ , T x∗ , T x∗ ), 1 [S(xnk , T x∗ , T x∗ ) + S(x∗ , xnk +1 , xnk +1 )]}. 8

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0

Notice that as S(x∗ , T x∗ , T x∗ ) > 0, then we have that M3 (xnk , x∗ , x∗ ) > 0. From Lemma 1.1, it follows that 1 [S(xnk , T x∗ , T x∗ ) + S(x∗ , xnk +1 , xnk +1 )] 8 1 ≤ [2S(xnk , x∗ , x∗ ) + S(x∗ , T x∗ , T x∗ ) + 2S(x∗ , xnk , xnk ) + S(xnk , xnk +1 , xnk +1 )] 8 1 1 1 = S(xnk , x∗ , x∗ ) + S(x∗ , T x∗ , T x∗ ) + S(xnk , xnk +1 , xnk +1 ) 2 8 8 ≤ max{S(xnk , x∗ , x∗ ), S(x∗ , T x∗ , T x∗ ), S(xnk , xnk +1 , xnk +1 )}. By the above inequality, we have that 0

M3 (xnk , x∗ , x∗ ) = max{S(xnk , x∗ , x∗ ), S(x∗ , T x∗ , T x∗ ), S(xnk , xnk +1 , xnk +1 )}. 0

Suppose that M3 (xnk , x∗ , x∗ ) = S(xnk , x∗ , x∗ ), then, we get that S(x∗ , T x∗ , T x∗ ) ≤ ψ(S(xnk , x∗ , x∗ )) + S(xnk +1 , x∗ , x∗ ) 0, then M3 (x∗ , zn ) > 0. Since T is a generalized (γ − ψ)−MKC mapping of type C, we get S(x∗ , zn+1 , zn+1 ) ≤ γ(x∗ , zn , zn )S(T x∗ , T zn , T zn ) ≤ ψ(M3 (x∗ , zn )) < M3 (x∗ , zn ), 622

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.4, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

where M3 (x∗ , zn ) = max{S(x∗ , zn , zn ), S(x∗ , T x∗ , T x∗ ), S(zn , T zn , T zn ), 81 [S(x∗ , T zn , T zn )+S(zn , T x∗ , T x∗ )]}. Taking (U 20 ) and Lemma 1.1 into account, we have M3 (x∗ , zn ) = S(x∗ , zn , zn ). Thus, S(x∗ , zn+1 , zn+1 ) < S(x∗ , zn , zn ). Letting n → ∞ in the inequality above, we obtain lim S(x∗ , zn+1 , zn+1 ) < lim S(x∗ , zn , zn ),

n→∞

n→∞

which is a contradiction. Then, M3 (x∗ , zn ) = S(x∗ , zn , zn ) = 0. Hence, we get that lim S(x∗ , zn , zn ) = 0.

n→∞

Step2. We will prove that lim S(y ∗ , zn , zn ) = 0. n→∞

In a analogous way of Step1., we can complete the proof of lim S(y ∗ , zn , zn ) = 0. n→∞

By Lemma 1.1, S(x∗ , y ∗ , y ∗ ) ≤ 2S(x∗ , zn , zn ) + S(y ∗ , zn , zn ). Letting n → ∞ in the above inequality, we get S(x∗ , y ∗ , y ∗ ) = 0, therefore, we have x∗ = y ∗ . As a uniqueness condition for fixed points of (γ − ψ)−MKC mappings of dim3 of type C, we suggest the following hypothesis: (U 30 ) For ∀x∗ , y ∗ ∈ F ix(T ), γ(x∗ , x∗ , x∗ ) ≥ 1, γ(y ∗ , y ∗ , y ∗ ) ≥ 1. Theorem 2.15. Adding condition (U 30 ) to the statements of Theorem 2.13, one has that T has the unique fixed point. Proof. Let x∗ , y ∗ be two distinct fixed points of T . Form condition (U 30 ) γ(x∗ , x∗ , x∗ ) ≥ 1, γ(y ∗ , y ∗ , y ∗ ) ≥ 1.

(58)

By (58) and the statement of the theorem that T is generalized (γ − ψ)−MKC mapping of dim3 of type C. we have S(x∗ , y ∗ , y ∗ ) ≤ γ(x∗ , T x∗ , T x∗ )γ(y ∗ , T y ∗ , T y ∗ )γ(y ∗ , T y ∗ , T y ∗ )S(T x∗ , T y ∗ , T y ∗ ) 0

0

≤ ψ(M3 (x∗ , y ∗ , y ∗ )) < M3 (x∗ , y ∗ , y ∗ ). but 0

M3 (x∗ , y ∗ , y ∗ ) = max{S(x∗ , y ∗ , y ∗ ), S(y ∗ , y ∗ , y ∗ ), S(y ∗ , x∗ , x∗ ), S(x∗ , T x∗ , T x∗ ), S(y ∗ , T y ∗ , T y ∗ ), 1 S(y ∗ , T y ∗ , T y ∗ ), [S(x∗ , T y ∗ , T y ∗ ) + S(y ∗ , T x∗ , T x∗ )], 8 1 1 [S(y ∗ , T y ∗ , T y ∗ ) + S(y ∗ , T y ∗ , T y ∗ )], [S(y ∗ , T x∗ , T x∗ ) + S(x∗ , T y ∗ , T y ∗ )]} 8 8 ∗ ∗ ∗ = S(x , y , y ). 623

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so, S(x∗ , y ∗ , y ∗ ) < S(x∗ , y ∗ , y ∗ ) which is again a contradiction.therefore, we have x∗ = y ∗ .

3.

Generalized Ulam-Hyers Stability for MKC mappings

In the following section, by introducing the generalized Ulam-Hyers stability in the framework of S−metric spaces, we study the stability for MKC mappings. Theorem 3.1. Let (X, S) be a complete S−metric and T : X → X be a self-mapping. Suppose that all the hypotheses of Theorem 2.12 hold. In addition, assume that (A1) the function β : [0, ∞) → [0, ∞), β(r) = r − ψ(r) is strictly increasing and onto. (A2) for any −solution w∗ ∈ X of (2), one has γ(w∗ , x∗ , x∗ ) ≥ 1, where x∗ ∈ F ix(T ). Then, the fixed point problem (1) is generalized Ulam-Hyers stable. Proof. From the conclusion of Theorem 2.12, it follows that there exists x∗ ∈ F ix(T ) such that S(x∗ , T x∗ , T x∗ ) = 0. Let  > 0 and w∗ be a −solution of (2). From (A2), we have γ(x∗ , w∗ , w∗ ) ≥ 1. Since T is triangular γ−admissible, we can obtain that γ(T x∗ , T w∗ , T w∗ ) = γ(x∗ , T w∗ , T w∗ ) ≥ 1. Thus, we also get that S(x∗ , w∗ , w∗ ) = S(T x∗ , w∗ , w∗ ) ≤ S(T x∗ , T w∗ , T w∗ ) + 2S(w∗ , T w∗ , T w∗ ) ≤ γ(T x∗ , T w∗ , T w∗ )S(T x∗ , T w∗ , T w∗ ) + 2S(w∗ , T w∗ , T w∗ ) < ψ(M3 (x∗ , w∗ )) + 2, where M3 (x∗ , w∗ ) = max{S(x∗ , w∗ , w∗ ), S(x∗ , T x∗ , T x∗ ), S(w∗ , T w∗ , T w∗ ), 18 [S(x∗ , T w∗ , T w∗ )+S(w∗ , T x∗ , T x∗ )]}. We also get 1 [S(x∗ , T w∗ , T w∗ ) + S(w∗ , T x∗ , T x∗ )] 8 1 ≤ [2S(x∗ , w∗ , w∗ ) + S(w∗ , T w∗ , T w∗ ) + 2S(w∗ , x∗ , x∗ ) + S(x∗ , T x∗ , T x∗ )] 8 1 = [4S(x∗ , w∗ , w∗ ) + S(w∗ , T w∗ , T w∗ )] 8 1 1 = S(x∗ , w∗ , w∗ ) + S(w∗ , T w∗ , T w∗ ) 2 8 1 1 ≤ S(x∗ , w∗ , w∗ ) +  2 8 < max{S(x∗ , w∗ , w∗ ), }. From the inequality above, we have that M3 (x∗ , w∗ ) < max{S(x∗ , w∗ , w∗ ), }. It is obviously that if S(x∗ , w∗ , w∗ ) < , then the proof is complete. Suppose that max{S(x∗ , w∗ , w∗ ), } = S(x∗ , w∗ , w∗ ). Then, we have M3 (x∗ , w∗ ) < S(x∗ , w∗ , w∗ ). 624

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So, we can deduce that S(x∗ , w∗ , w∗ ) ≤ ψ(S(x∗ , w∗ , w∗ )) + 2, S(x∗ , w∗ , w∗ ) − ψ(S(x∗ , w∗ , w∗ )) ≤ 2. From assumption (A1), we get that β(S(x∗ , w∗ , w∗ )) ≤ 2. Hence, S(x∗ , w∗ , w∗ ) ≤ β −1 (2). Therefore, (1) is generalized Ulam-Hyers stable. Theorem 3.2. Let (X, S) be a complete S−metric and T : X → X be a self-mapping. Suppose that all the hypotheses of Theorem 2.13 hold. In addition, assume that (A1) the function β : [0, ∞) → [0, ∞), β(r) = r − ψ(r) is strictly increasing and onto. (A2) for any −solution w∗ ∈ X of (2), one has γ(w∗ , x∗ , x∗ ) ≥ 1, where x∗ ∈ F ix(T ). Then, the fixed point problem (1) is generalized Ulam-Hyers stable. Proof. From the conclusion of Theorem 2.13, it follows that there exists x∗ ∈ F ix(T ) such that S(x∗ , T x∗ , T x∗ ) = 0. Let  > 0 and w∗ be a −solution of (2). From (A2), we have γ(x∗ , w∗ , w∗ ) ≥ 1. Since T is triangular γ−admissible, we can obtain that γ(T x∗ , T w∗ , T w∗ ) = γ(x∗ , T w∗ , T w∗ ) ≥ 1. Thus, we also get that S(x∗ , w∗ , w∗ ) = S(T x∗ , w∗ , w∗ ) ≤ S(T x∗ , T w∗ , T w∗ ) + 2S(w∗ , T w∗ , T w∗ ) ≤ γ(T x∗ , T w∗ , T w∗ )S(T x∗ , T w∗ , T w∗ ) + 2S(w∗ , T w∗ , T w∗ ) 0

< ψ(M3 (x∗ , w∗ , w∗ )) + 2, where 0

M3 (x∗ , w∗ ) = max{S(x∗ , w∗ , w∗ ), S(w∗ , w∗ , w∗ ), S(w∗ , x∗ , x∗ ), 1 S(x∗ , T x∗ , T x∗ ), S(w∗ , T w∗ , T w∗ ), [S(x∗ , T w∗ , T w∗ ) + S(w∗ , T x∗ , T x∗ )] 8 1 1 ∗ ∗ ∗ ∗ ∗ [S(w , T w , T w ) + S(w , T w , T w∗ )], [S(w∗ , T x∗ , T x∗ ) + S(x∗ , T w∗ , T w∗ )]} 8 8 ∗ ∗ ∗ ∗ ∗ ∗ 1 = max{S(x , w , w ), S(w , T w , T w ), [S(x∗ , T w∗ , T w∗ ) + S(w∗ , T x∗ , T x∗ )]} 8 We also get 1 [S(x∗ , T w∗ , T w∗ ) + S(w∗ , T x∗ , T x∗ )] 8 1 ≤ [2S(x∗ , w∗ , w∗ ) + S(w∗ , T w∗ , T w∗ ) + 2S(w∗ , x∗ , x∗ ) + S(x∗ , T x∗ , T x∗ )] 8 1 = [4S(x∗ , w∗ , w∗ ) + S(w∗ , T w∗ , T w∗ )] 8 1 1 = S(x∗ , w∗ , w∗ ) + S(w∗ , T w∗ , T w∗ ) 2 8 1 1 ≤ S(x∗ , w∗ , w∗ ) +  2 8 < max{S(x∗ , w∗ , w∗ ), }. 625

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From the inequality above, we have that 0

M3 (x∗ , w∗ , w∗ ) < max{S(x∗ , w∗ , w∗ ), }. It is obviously that if S(x∗ , w∗ , w∗ ) < , then the proof is complete. Suppose that max{S(x∗ , w∗ , w∗ ), } = S(x∗ , w∗ , w∗ ). Then, we have 0

M3 (x∗ , w∗ , w∗ ) < S(x∗ , w∗ , w∗ ). So, we can deduce that S(x∗ , w∗ , w∗ ) ≤ ψ(S(x∗ , w∗ , w∗ )) + 2, S(x∗ , w∗ , w∗ ) − ψ(S(x∗ , w∗ , w∗ )) ≤ 2. From assumption (A1), we get that β(S(x∗ , w∗ , w∗ )) ≤ 2. Hence, S(x∗ , w∗ , w∗ ) ≤ β −1 (2). Therefore, (1) is generalized Ulam-Hyers stable. Corollary 3.1. Let (X, S) be a complete S−metric and T : X → X be a self-mapping. Suppose that all the hypotheses of Theorem 2.8(resp., Theorem 2.10) hold. In addition, assume that (A1) the function β : [0, ∞) → [0, ∞), β(r) = r − ψ(r) is strictly increasing and onto. (A2) for any −solution w∗ ∈ X of (2), one has γ(w∗ , x∗ , x∗ ) ≥ 1, where x∗ ∈ F ix(T ). Then, the fixed point problem (1) is generalized Ulam-Hyers stable. Proof. The proof is an analog of the proof of Theorem 3.1. Corollary 3.2. Let (X, S) be a complete S−metric and T : X → X be a self-mapping. Suppose that all the hypotheses of Theorem 2.9(resp., Theorem 2.11) hold. In addition, assume that (A1) the function β : [0, ∞) → [0, ∞), β(r) = r − ψ(r) is strictly increasing and onto. (A2) for any −solution w∗ ∈ X of (2), one has γ(w∗ , x∗ , x∗ ) ≥ 1, where x∗ ∈ F ix(T ). Then, the fixed point problem (1) is generalized Ulam-Hyers stable. Proof. The proof is an analog of the proof of Theorem 3.2. Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript. Author details 1

School of Science and Technology, Sanya University, Sanya, Hainan 572000, China.

2

College of Science, Sichuan University of Science and Engineering, Zigong, Sichuan 643000, China.

3

Sichuan Province University Key Laboratory of Bridge Non-destruction Detecting and Engineering Com-

puting, Zigong, Sichuan 643000, China. 626

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4

Department of Mathematics, Karaj Branch Islamic Azad University, Karaj, Iran.

5

Department of Mathematics Education and RINS, Gyeongsang National University, Gajwa-dong,900, 52828

Jinju, Korea. 6

Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11120 Belgrade, Serbia. Acknowledgements

Mi Zhou was supported by Scientific Research Fund of Hainan Province Education Department (Grant No.Hnjg2016ZD-20). Xiao-lan Liu was partially supported by National Natural Science Foundation of China (Grant No.61573010), Artificial Intelligence of Key Laboratory of Sichuan Province(No.2015RZJ01), Science Research Fund of Science and Technology Department of Sichuan Province(N0.2017JY0125), Scientific Research Fund of Sichuan Provincial Education Department(No.16ZA0256), Scientific Research Fund of Sichuan University of Science and Engineering (No.2014RC01, No.2014RC03, No.2017RCL54). Yeol Je Cho was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and future Planning (2014R1A2A2A01002100). Stojan Radenovi´c was supported by the Ministry of Education, Science and Technological Development of Serbia. The authors thank the editor and the referees for their useful comments and suggestions.

References [1] S. Sedghi, N. Shobe, A. Aliouche, A generalized of fixed point theorems in S−metric spaces, Mat. Vesn., 64(2012), 258–266. [2] J. M. Afra, Fixed point type theorem for weak contraction in S−metric spaces, Int. J. Res. Rer. Appl. Sci., 22(2015),11–14. [3] P. Chouhan, N. Malviya, A common unique fixed point theorem for expansive type mappings in S−metric spaces, Int. Math. Forum, 8(2013), 1287–1293. ´ c quasi-contractions for maps on S−metric [4] N. T. Hieu, N. T. Thanhly, N. V. Dung, A generalization of Ciri´ spaces, Thai. J. Math., 13 (2015), 369–380. [5] S. Sedghi, N. V. Dung, Fixed point theorems in S−metric spaces, Mat. Vesn., 66(2014), 113–124. [6] N. V. Dung, N. T. Hieu, S. Radojevi´c, Fixed point theorems for g−monotone maps on partially orderded S−metric spaces, Filomat, 28(2014),1685–1898. [7] M. Zhou, X. L. Liu, On coupled common fixed point theorems for nonlinear contractions with the mixed weakly monotone property in partially ordered S−metric spaces, Journal of Function Spaces, 2016(2016), Article ID 7529523, 9 pages. [8] M. Zhou, X. L. Liu, D. D. Diana, B. Damjanovi´c, Coupled coincidence point results for Geraghty-type contraction by using monotone property in partially ordered S−metric spaces, J. Nonlinear Sci. Appl., 9(2016), 5950-5969. [9] S. Sedghi, A. Gholidahneh, T. Doˇsenovi´c, J. Esfahani, S. Radenovi´c, Common fixed point of four maps in Sb −metric spaces, to appear in Journal of Linear and Topol. Algebra. [10] A. Gholidahneh, S. Sedghi, T. Doˇsenovi´c, S. Radenovi´c, Ordered S−metric spaces and coupled common fixed point theorems of integral type contraction, to appear in Mathematics Interdisciplinary Research. [11] S. M. Ulam, Problems in Modern Mathematics, John Wiley & Sons, New York, NY, USA, 1964. [12] D. H. Hyers, On the stability of linear functional equation, Proceedings of the National Academy of Sciences of the United States of America, 27(1941), 222–224.

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[13] M. F. Bota-Boriceanu, A. Petursel, Ulam-Hyers stability for operatorial equations, Analele Stiintifice ale Universitatii, 57(1) (2011), 65–74. [14] V. L. Lazˇ ar, Ulam-Hyers stability for partial differential inclusions, Electronic Journal of Qualitative Theory of Differential Equations, 21(2012), 1–19. [15] J. Brzdek, K. Ciepli´ nski, A fixed point theorems and the Hyers-Ulam stability in non-Archimedean spaces, J. Math. Anal. Appl., 400(1)(2013), 68–75. [16] L. Cadariu, L. Gavruta, and P. Gavruta, Fixed points and generalized Hyers-Ulam stability, Abstract and Applied Analysis, 2012(2012), Article ID 712743, 10 pages. [17] A. Meir, E. Keeler, A theorem on contraction mappings, J.Math.Anal.Appl., 28(1969), 326–329. [18] M. Maiti, T. K. Pal, Generalizations of two fixed points theorems, Bull. Calcutta Math. Soc., 70(1978), 57–61. [19] S. Park, B. E. Rhoades, Meir-Keeler type contractive conditions, Math. Japn., 26(1)(1981), 13–20. [20] C. Mongkolkeha, P. Kumam, Best proximity points for asympototic proximal pointwise weaker MeirKeeler-type ψ−contraction mappings, J. Egypt. Math. Soc., 26(1)(1981), 13–20. [21] B. Samet, C. Vetro, P. Vetro, Fixed point theorem for α − ψ contractive type mappings, Nonlinear Anal., 75(2012), 2154–2165. [22] A. Latif, M. E. Gordji, E. Karapinar, W. Sintunavarat, Fixed point results for generalized (α − ψ)−MeirKeeler contractive mappings and applications, Journal of Inequalities and Applications, 68(2014), 11 pages.

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New oscillation criteria for second-order nonlinear delay dynamic equations with nonpositive neutral coefficients on time scales Ming Zhanga,b∗, Wei Chena†, M.M.A. El-Sheikhc‡, R.A. Sallamc§, A.M. Hassand¶, and Tongxing Lib∥

a School

of Information Engineering, Wuhan University of Technology, Wuhan, Hubei 430070, P. R. China of Information Science and Engineering, Linyi University, Linyi, Shandong 276005, P. R. China c Department of Mathematics, Faculty of Science, Menoufia University, Shebin El-Koom 32511, Egypt d Department of Mathematics, Faculty of Science, Benha University, Benha-Kalubia 13518, Egypt

b School

Abstract We analyze the oscillatory behavior of solutions to a nonlinear second-order neutral delay dynamic equation with a nonpositive neutral coefficient under the assumptions that allow applications to Emden–Fowler type dynamic equations. New theorems complement and improve related contributions to the subject. An example is included. Keywords: Oscillation, second-order delay dynamic equation, neutral type equation, Emden–Fowler type equation. Mathematics Subject Classification 2010: 34K11, 34N05.

1

Introduction

In this paper, we study the oscillation of a class of second-order neutral dynamic equations [ ]∆ r(t)(z ∆ (t))α + q(t)f (x(δ(t))) = 0.

(1.1)

Here t ∈ [t0 , ∞)T , α ≥ 1 is a quotient of odd natural numbers, and z(t) = x(t) − p(t)x(τ (t)). The increasing interest in oscillation of solutions to various classes of equations is motivated by their applications in natural sciences, engineering, and control; see, for instance, [1–30] and the references cited therein. Analysis of qualitative properties of (1.1) is important not only for the sake of further development of the oscillation theory, but for practical reasons too. As a matter of fact, a particular case of (1.1), an Emden–Fowler dynamic equation [ ]∆ r(t)(x∆ (t))α + q(t)xβ (δ(t)) = 0, has applications in mathematical, theoretical, and chemical physics; see Li and Rogovchenko [15–18]. Throughout the paper, we assume that the following assumptions are satisfied: ∗

e–mail: e–mail: ‡ e–mail: § e–mail: ¶ e–mail: ∥ e–mail: †

[email protected] [email protected] msheikh [email protected] [email protected] [email protected] [email protected] (Corresponding author)

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(H1 ) r ∈ Crd ([t0 , ∞)T , (0, ∞)),

∫∞ t0

r− α (t)∆t = ∞, R(t) = 1

∫t t1

r− α (s)∆s, where t1 ∈ [t0 , ∞)T is sufficiently 1

large;

(H2 ) p, q ∈ Crd ([t0 , ∞)T , R), 0 ≤ p(t) ≤ p0 < 1, q(t) ≥ 0, and q(t) is not identically zero for large t; (H3 ) τ, δ ∈ Crd ([t0 , ∞)T , T), τ (t) ≤ t, δ(t) ≤ t, and limt→∞ τ (t) = limt→∞ δ(t) = ∞; (H4 ) f ∈ C(R, R), uf (u) > 0 for all u ̸= 0, and there exists a positive constant k such that f (u)/uβ ≥ k for all u ̸= 0, where β ≥ α is a quotient of odd natural numbers. By a solution to (1.1) we mean a function x ∈ Crd [Tx , ∞)T , Tx ∈ [t0 , ∞)T , such that r(z ∆ )α ∈ C1rd [Tx , ∞)T and x satisfies (1.1) on [Tx , ∞)T . We consider only those solutions x of (1.1) which satisfy sup{|x(t)| : t ∈ [T, ∞)T } > 0 for all T ∈ [Tx , ∞)T and assume that (1.1) possesses such solutions. As usual, a solution of (1.1) is said to be oscillatory if it is not of the same sign eventually; otherwise, it is called nonoscillatory. Recently, a great deal of interest in oscillatory properties of solutions to various classes of equations with nonnegative neutral coefficients has been shown; see, for instance, [2,4,5,14–17,19,20,22,27,28] and the references cited therein. However, there are relatively fewer results for equations with nonpositive neutral coefficients; see [3, 4, 7, 13, 21, 23–25, 29]. In the papers by Arul and Shobha [3] and Li et al. [21], a particular case of (1.1), a neutral differential equation ′ [r(t)(z ′ (t))α ] + q(t)f (x(δ(t))) = 0 was studied. Seghar et al. [23] investigated the neutral difference equation ∆(an ∆(xn − pn xn−k )) + qn f (xn−l ) = 0. Bohner and Li [7] and Karpuz [13] established oscillation results for neutral dynamic equations (

r(t)|z ∆ (t)|p−2 z ∆ (t)

)∆

+ q(t)|x(δ(t))|p−2 x(δ(t)) = 0,

z(t) = x(t) − p(t)x(τ (t))

and (x(t) − p(t)x(τ (t)))∆∆ + q(t)x(δ(t)) = 0, whereas Zhang et al. [29] explored (1.1) assuming that α = β. It should be noted that research in this paper was strongly motivated by the paper [29]. Our principal goal is to analyze the oscillatory behavior of solutions to (1.1) in the case where β ≥ α. As customary for papers on oscillation, all functional inequalities are supposed to hold eventually. Without loss of generality, we can deal only with positive solutions of (1.1).

2

Main results

For the proofs of our oscillation criteria we need the following lemmas. The first lemma is extracted from the monograph by Bohner and Peterson [9, Theorem 1.93], and the latter lemmas can be obtained by similar techniques to those used in [3, 21].

Lemma 2.1. Assume that v : T → R is strictly increasing and T˜ := v(T) is a time scale. Let y : T˜ → R. If ˜

v ∆ (t) and y ∆ (v(t)) exist for t ∈ Tκ , then ˜

∆

(y(v(t))) = y ∆ (v(t))v ∆ (t).

Lemma 2.2. Let x be a positive solution of (1.1). Then z has the following two possible cases: (I) z(t) > 0,

z ∆ (t) > 0,

(r(t)(z ∆ (t))α )∆ ≤ 0;

(II) z(t) < 0,

z ∆ (t) > 0,

(r(t)(z ∆ (t))α )∆ ≤ 0

for t ∈ [t1 , ∞)T , where t1 ∈ [t0 , ∞)T is sufficiently large.

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Lemma 2.3. Let x be a positive solution of (1.1) and assume that the corresponding z has property (II) of Lemma 2.2. Then lim x(t) = 0. t→∞

Lemma 2.4. If x is a positive solution of (1.1) such that case (I) of Lemma 2.2 is satisfied, then x(t) ≥ z(t) and z(t)/R(t) is strictly decreasing for large t.

Theorem 2.1. Assume that δ([t0 , ∞)T ) = [δ(t0 ), ∞)T and δ ∆ (t) > 0. If for any M > 0, [ ∫ lim sup Q(t) + α t→∞

where Q(t) = kM β−α

∞

δ ∆ (s)r− α (δ(s))Q 1

]( ∫

α+1 α

t

∫∞ t

δ(t)

(σ(s))∆s

r− α (s)∆s

)α

1

> 1,

(2.1)

t0

q(u)∆u, then solutions of (1.1) are either oscillatory or converge to zero as t → ∞.

Proof. Let x be a nonoscillatory solution of (1.1) such that x(t) > 0, x(τ (t)) > 0, and x(δ(t)) > 0 for t ∈ [t1 , ∞)T . It follows from Lemma 2.2 that z satisfies either (I) or (II) for t ∈ [t1 , ∞)T . Case 1. Suppose first that z satisfies case (I). By virtue of the definition of z, x(t) = z(t) + p(t)x(τ (t)) ≥ z(t) and so we can write (1.1) in the form [

r(t)(z ∆ (t))α

]∆

≤ −kq(t)z β (δ(t)).

Defining the Riccati transformation ν(t) =

r(t)(z ∆ (t))α , z α (δ(t))

(2.2)

then ν(t) > 0 and there exists a constant M > 0 such that ∆

ν (t)

=

[ ]∆ [ ]∆ [ ] r(t)(z ∆ (t))α 1 ∆ α σ + r(t)(z (t)) z α (δ(t)) z α (δ(t))

≤ −kM β−α q(t) − αδ ∆ (t)ν(σ(t)) 1

z ∆ (δ(t)) . z(δ(σ(t)))

(2.3)

1

Taking into account that ν α (σ(t)) = r α (σ(t))z ∆ (σ(t))/z(δ(σ(t))), r(t)(z ∆ (t))α ≤ 0, and δ(t) ≤ t ≤ σ(t), we conclude that 1 z ∆ (δ(t)) ν α (σ(t)) ≥ 1 . (2.4) z(δ(σ(t))) r α (δ(t)) Combining (2.3) and (2.4), we arrive at ν ∆ (t) ≤ −kM β−α q(t) − αδ ∆ (t)r− α (δ(t))ν 1

Integrating (2.5) from t to s, we deduce that ∫ s ∫ q(u)∆u − α ν(s) − ν(t) ≤ −kM β−α t

which yields

∫ ν(t) ≥ kM β−α

s

α+1 α

(σ(t)).

δ ∆ (u)r− α (δ(u))ν 1

α+1 α

(2.5)

(σ(u))∆u,

t

∫

s

s

q(u)∆u + α t

δ ∆ (u)r− α (δ(u))ν 1

α+1 α

(σ(u))∆u.

t

Passing to the limit as s → ∞, we have ∫ ν(t) ≥ Q(t) + α

∞

δ ∆ (u)r− α (δ(u))ν

α+1 α

δ ∆ (u)r− α (δ(u))Q

α+1 α

1

(σ(u))∆u.

(2.6)

(σ(u))∆u.

(2.7)

t

An application of (2.6) implies that ∫ ν(t) ≥ Q(t) + α

∞

1

t

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By virtue of (2.2), we conclude that 1 ν(t)

= = ≥

( )α 1 z(δ(t)) r(t) z ∆ (t) ∫ δ(t) 1 1 ( ) z(t2 ) + t2 r α (s)z ∆ (s)r− α (s)∆s α 1 z ∆ (t) ∫ 1 ( ) δ(t) r (t)z ∆ (t) t2 r− α (s)∆s α 1 , r(t) z ∆ (t) r(t)

1 α

that is,

(∫

δ(t)

ν(t)

r

1 −α

)α (s)∆s ≤ 1.

(2.8)

t2

Using (2.7) and (2.8), we deduce that [ ∫ lim sup Q(t) + α t→∞

∞

δ ∆ (s)r− α (δ(s))Q 1

α+1 α

]( ∫

δ(t)

(σ(s))∆s

t

r− α (s)∆s 1

)α ≤ 1,

t2

which contradicts (2.1). Case 2. Suppose now that z satisfies case (II). It follows from Lemma 2.3 that limt→∞ x(t) = 0. This completes the proof. Theorem 2.2. If there exists a positive function β ∈ C1rd ([t0 , ∞)T , R) such that for all sufficiently large t1 ∈ [t0 , ∞)T , for some t2 ∈ [t1 , ∞)T , and for any M > 0, )α ] ( ∫ ∞[ 1 (β ∆ (s))α+1 r(s) R(δ(s)) − ∆s = ∞, (2.9) kM β−α q(s)β(s) R(s) (α + 1)α+1 β α (s) t2 then conclusion of Theorem 2.1 remains intact. Proof. Assume that x is a nonoscillatory solution of (1.1) on [t0 , ∞)T that satisfies x(t) > 0, x(τ (t)) > 0, and x(δ(t)) > 0 for t ∈ [t1 , ∞)T . By virtue of Lemma 2.2, z satisfies either (I) or (II) for t ∈ [t1 , ∞)T . Case 1. Suppose that z satisfies case (I). Define the Riccati transformation ω(t) = β(t)

r(t)(z ∆ (t))α . z α (t)

Then ω(t) > 0 and there exists a constant M > 0 such that ω ∆ (t) =

[ ]∆ [ ]∆ β(t) [ ] β(t) ∆ α σ r(t)(z ∆ (t))α + r(t)(z (t)) z α (t) z α (t)

z α (δ(t)) β ∆ (t) β(t) z ∆ (t) + ω(σ(t)) − α σ ω(σ(t)) α z (t) β(σ(t)) β (t) z(t) α+1 z α (δ(t)) β ∆ (t) β(t) ≤ −kM β−α q(t)β(t) α + ω(σ(t)) − α α+1 ω α (σ(t)). 1 z (t) β(σ(t)) α α β (σ(t))r (t)

≤ −kM β−α q(t)β(t)

In view of Lemma 2.4, we obtain (

ω (t) ≤ −kM ∆

β−α

R(δ(t)) q(t)β(t) R(t)

)α +

α+1 β ∆ (t) β(t) ω(σ(t)) − α α+1 ω α (σ(t)). 1 β(σ(t)) β α (σ(t))r α (t)

(2.10)

Applying the inequality Bω − Aω with B=

α+1 α

β ∆ (t) β(σ(t))

≤

αα B α+1 , α+1 (α + 1) Aα

and

A=α

A>0

β(t) β

α+1 α

1

(σ(t))r α (t)

,

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and using (2.10), we deduce that (

ω (t) ≤ −kM ∆

β−α

R(δ(t)) q(t)β(t) R(t)

)α +

(β ∆ (t))α+1 r(t) 1 . (α + 1)α+1 β α (t)

(2.11)

Integrating (2.11) from t2 (t2 ∈ [t1 , ∞)T ) to t, we arrive at ( )α ] ∫ t[ (β ∆ (s))α+1 r(s) R(δ(s)) 1 β−α kM q(s)β(s) ∆s ≤ ω(t2 ), − R(s) (α + 1)α+1 β α (s) t2 which contradicts (2.9). Case 2. If z satisfies case (II), then limt→∞ x(t) = 0 due to Lemma 2.3. The proof is complete. Remark 2.1. On the basis of Theorem 2.2, one can obtain Philos-type oscillation criteria for equation (1.1). The details are left to the reader. Example 2.1. For t ∈ [1, ∞)T , consider the second-order superlinear Emden–Fowler neutral delay dynamic equation ( ( ))∆∆ ( ) t γ β t 1 + x = 0, β > 1, γ > 0. x(t) − x (2.12) 3 2 t 4 Let β(t) = 1. It follows from Theorem 2.2 that every solution x of equation (2.12) is either oscillatory or satisfies limt→∞ x(t) = 0. Remark 2.2. For a class of second-order neutral delay dynamic equations (1.1), we derived two new oscillation results which complement and improve those obtained by Zhang et al. [29]. A distinguishing feature of our criteria is that we do not impose specific restriction α = β. Since the sign of the derivative z ∆ is not known, it is difficult to establish sufficient conditions which ensure that every solution x of (1.1) is just oscillatory and does not satisfy limt→∞ x(t) = 0. Neither is it possible to use the technique exploited in this paper for proving that all solutions of (1.1) approach zero at infinity. ∫ ∞ − 1 As mentioned in the paper by Zhang et al. [29], it would be of interest to study (1.1) in the case where t0 r α (t)∆t < ∞.

Acknowledgements This research is supported by Project of Shandong Province Independent Innovation and Achievement Transformation (Grant No. 2014ZZCX02702), Shandong Province Key Research and Development Project (Grant No. 2016GGX109001), Shandong Provincial Natural Science Foundation (Grant Nos. ZR2017MF050, ZR2014FL008, and ZR2015FL014), and Project of Shandong Province Higher Educational Science and Technology Program (Grant No. J17KA049).

References [1] R. P. Agarwal, M. Bohner, T. Li, and C. Zhang. Oscillation criteria for second-order dynamic equations on time scales. Applied Mathematics Letters, 31 (2014) 34–40. [2] R. P. Agarwal, D. O’Regan, and S. H. Saker. Oscillation criteria for second-order nonlinear neutral delay dynamic equations. Journal of Mathematical Analysis and Applications, 300 (2004) 203–217. [3] R. Arul and V. S. Shobha. Improvement results for oscillatory behavior of second order neutral differential equations with nonpositive neutral term. British Journal of Mathematics & Computer Science, 12 (2016) 1–7. [4] B. Bacul´ıkov´ a and J. Dˇzurina. Oscillation of third-order neutral differential equations. Mathematical and Computer Modelling, 52 (2010) 215–226. [5] M. Bohner, S. R. Grace, and I. Jadlovsk´a. Oscillation criteria for second-order neutral delay differential equations. Electronic Journal of Qualitative Theory of Differential Equations, 2017 (2017) 1–12. [6] M. Bohner, T. S. Hassan, and T. Li. Fite–Hille–Wintner-type oscillation criteria for second-order half-linear dynamic equations with deviating arguments. Indagationes Mathematicae, (2018) in press.

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[7] M. Bohner and T. Li. Oscillation of second-order p-Laplace dynamic equations with a nonpositive neutral coefficient. Applied Mathematics Letters, 37 (2014) 72–76. [8] M. Bohner and T. Li. Kamenev-type criteria for nonlinear damped dynamic equations. Science China Mathematics, 58 (2015) 1445–1452. [9] M. Bohner and A. Peterson. Dynamic Equations on Time Scales: An Introduction with Applications. Birkh¨auser, Boston, 2001. [10] J. Dˇzurina and I. Jadlovsk´ a. A note on oscillation of second-order delay differential equations. Applied Mathematics Letters, 69 (2017) 126–132. [11] L. Erbe, A. Peterson, and S. H. Saker. Oscillation criteria for second-order nonlinear delay dynamic equations. Journal of Mathematical Analysis and Applications, 333 (2007) 505–522. [12] S. Hilger. Analysis on measure chain–a unified approach to continuous and discrete calculus. Results in Mathematics, 18 (1990) 18–56. [13] B. Karpuz. Sufficient conditions for the oscillation and asymptotic behaviour of higher-order dynamic equations of neutral type. Applied Mathematics and Computation, 221 (2013) 453–462. [14] T. Li, R. P. Agarwal, and M. Bohner. Some oscillation results for second-order neutral dynamic equations. Hacettepe Journal of Mathematics and Statistics, 41 (2012) 715–721. [15] T. Li and Yu. V. Rogovchenko. Asymptotic behavior of higher-order quasilinear neutral differential equations. Abstract and Applied Analysis, 2014 (2014) 1–11. [16] T. Li and Yu. V. Rogovchenko. Oscillation of second-order neutral differential equations. Mathematische Nachrichten, 288 (2015) 1150–1162. [17] T. Li and Yu. V. Rogovchenko. Oscillation criteria for second-order superlinear Emden–Fowler neutral differential equations. Monatshefte f¨ ur Mathematik, 184 (2017) 489–500. [18] T. Li and Yu. V. Rogovchenko. On asymptotic behavior of solutions to higher-order sublinear Emden– Fowler delay differential equations. Applied Mathematics Letters, 67 (2017) 53–59. [19] T. Li, Yu. V. Rogovchenko, and C. Zhang. Oscillation of second-order neutral differential equations. Funkcialaj Ekvacioj, 56 (2013) 111–120. [20] T. Li and S. H. Saker. A note on oscillation criteria for second-order neutral dynamic equations on isolated time scales. Communications in Nonlinear Science and Numerical Simulation, 19 (2014) 4185–4188. [21] Q. Li, R. Wang, F. Chen, and T. Li. Oscillation of second-order nonlinear delay differential equations with nonpositive neutral coefficients. Advances in Difference Equations, 2015 (2015) 1–7. [22] T. Li, C. Zhang, and E. Thandapani. Asymptotic behavior of fourth-order neutral dynamic equations with noncanonical operators. Taiwanese Journal of Mathematics, 18 (2014) 1003–1019. [23] D. Seghar, E. Thandapani, and S. Pinelas. Oscillation theorems for second order difference equations with negative neutral term. Tamkang Journal of Mathematics, 46 (2015) 441–451. [24] E. Thandapani, V. Balasubramanian, and J. R. Graef. Oscillation criteria for second order neutral difference equations with negative neutral term. International Journal of Pure and Applied Mathematics, 87 (2013) 283–292. [25] E. Thandapani and K. Mahalingam. Necessary and sufficient conditions for oscillation of second order neutral difference equations. Tamkang Journal of Mathematics, 34 (2003) 137–145. [26] J. Wang, M. M. A. El-Sheikh, R. A. Sallam, D. I. Elimy, and T. Li. Oscillation results for nonlinear secondorder damped dynamic equations. Journal of Nonlinear Sciences and Applications, 8 (2015) 877–883. [27] C. Zhang, R. P. Agarwal, M. Bohner, and T. Li. New oscillation results for second-order neutral delay dynamic equations. Advances in Difference Equations, 2012 (2012) 1–14. [28] C. Zhang, R. P. Agarwal, M. Bohner, and T. Li. Oscillation of second-order nonlinear neutral dynamic equations with noncanonical operators. Bulletin of the Malaysian Mathematical Sciences Society, 38 (2015) 761–778.

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[29] M. Zhang, W. Chen, M. M. A. El-Sheikh, R. A. Sallam, A. M. Hassan, and T. Li. Oscillation criteria for second-order nonlinear delay dynamic equations of neutral type. Advances in Difference Equations, (2018) in press. [30] C. Zhang and T. Li. Some oscillation results for second-order nonlinear delay dynamic equations. Applied Mathematics Letters, 26 (2013) 1114–1119.

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A Consistency Reaching Approach for Probability-interval Valued Hesitant Fuzzy Preference Relations Jiuping Xu1,∗ ,

Kang Xu1,2 ,

Zhibin Wu1

1. Business School, Sichuan University, Chengdu 610065, P R China 2. School of Economics and Management, Hubei University of Automotive Technology, Shiyan, Hubei, 442002, P R China

Abstract In a group decision making (GDM) situation with qualitative settings and complex environments, experts may require intervals with corresponding possibility values, rather than only intervalvalued hesitant fuzzy sets (IVHFSs) or probability-hesitant fuzzy sets (P-HFSs), to express their preferences. In this paper, in line with such situations, probability-interval valued hesitant fuzzy sets (P-IVHFSs) are presented to address GDM problems with hesitant fuzzy intervals and the corresponding possibility values. A P-IVHFS can serve as an extension of both a P-HFS and an IVHFS. As important tools in GDM, P-IVHFSs can describe the actual preferences of decisionmakers and better reflect their uncertainty, hesitancy, and inconsistency, thus enhancing the modeling abilities of HFSs. Firstly, the concept of P-IVHFSs is defined, and then some properties of P-IVHFSs are presented. Furthermore, probability-interval valued hesitant fuzzy preference relations (P-IVHFPRs) are defined and the consistency of P-IVHFPRs is discussed. Then, based on related research, a decomposition method is developed to deal with the consistency of P-IVHFPRs. Finally an example is provided to illustrate the proposed approach. Keywords: Decision making, Fuzzy sets, P-IVHFS, Preference relation, Consistency

1. Introduction Torra initiated the notable concept of HFSs, which represented a new generalization for fuzzy sets, as this method permits an element to have not just one but a set of several possible membership values. Consequently, HFSs can describe the hesitancy experienced by decision makers (DMs) in the decision-making process. As a result of this innovation, the HFS has attracted an increasing amount of attention in academia since its introduction. In recent years, there have been a number of developments regarding the theory of HFSs. For example, Xu and Xia defined the concept of the hesitant fuzzy element (HFE), which can be considered to be the basic unit of a HFS. Moreover, Rodr´ıguez et al. proposed the hesitant fuzzy linguistic term set to deal with linguistic decision making. Chen et al. extended HFSs to IVHFSs, which represent the membership degrees of an element to a set with several possible interval values. Farhadinia proposed a series of score functions for HFSs and Wei, Zhang, Yu, and Ai et al. studied their aggregation operators. Farhadinia, Xu and Xia, Peng et al., and Chen et al. discussed the information measures of HFSs. Wang et al. studied the interval-valued hesitant fuzzy linguistic set, which can serve as an extension of both a linguistic term set and an interval-valued hesitant fuzzy set. Finally, Wu and Xu presented the concept of possibility distribution for a hesitant fuzzy linguistic term set and Zhu and Xu extended HFSs to P-HFSs. * Corresponding author. Tel: +86 28 85418191; Fax: +86 28 85415143. E-mail address: [email protected](J. Xu)

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.4, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

In group decision making (GDM) problems with fuzzy preference relations, some of the experts’ preference properties are often assumed and it is desirable to avoid contradictions or, in other words, inconsistent opinions. One of these properties is associated with the pairwise comparison transitivity between any three alternatives. For fuzzy preference relations, the transitivity has been modeled in many different ways depending on the role of the preference intensities. The purpose of consistency control is to measure the level of consistency of each individual preference relation so as to identify the expert, alternative and preference values that are the most inconsistent within the GDM problem. This inconsistency identification is also used to suggest possible new consistent preference valuee. In the process of GDM, preference relations are very popular tools for expressing the DM’s preferences when they compare a set of alternatives. Various types of preference relations have been suggested for different environments. For example, Orlovsky proposed the concept of fuzzy preference relations, and Xu introduced the concept of interval fuzzy preference relations to express uncertainty and vagueness. In many practical decision making problems, due to a lack of available information, it may be difficult for DMs to quantify their opinions precisely with a crisp number; however they can be represented by an interval number within [0, 1]. This means that it is vital to introduce the concept of IVHFSs, which permit the membership degrees of an element to a given set to have some different interval values. Chen et al. introduced interval-valued hesitant preference relations and their applications to GDM. Moreover, Farhadinia discussed the information measures of IVHFSs and Wang et al. developed interval-valued hesitant fuzzy linguistic sets, and discussed their applications in multi-criteria decision-making problems. However, in a GDM situation with qualitative settings and in complex environments, experts may require intervals with corresponding possibility values rather than only IVHFSs or P-HFSs, to express their preferences. Consider the following case for example: the DMs of a large organization discuss the membership of x into a set A; forty percent of them want to assign values between 0.3 and 0.4, while the remaining sixty percent wish to assign values between 0.5 and 0.6. At this point, interval numbers with probability values can be used, i.e., {[0.3, 0.4](40%), [0.5, 0.6](60%)}, or {[0.3, 0.4](0.4), [0.5, 0.6](0.6)}, to represent the preferences of the large organization. In accordance with such cases, in this paper, P-IVHFSs are presented to address GDM problems with hesitant fuzzy intervals and the corresponding possibility values. A P-IVHFS can serve as an extension of both a P-HFS and an IVHFS. Furthermore, as a powerful tool in GDM, P-IVHFSs can describe the actual preferences of decision-makers flexibly and better reflect their uncertainty, hesitancy, and inconsistency , and thus enhance the modeling abilities of HFSs. The consistency of preference relations has become a research topic of great interest in recent years. For example, Liao et al. defined the concept of the multiplicative consistent hesitant fuzzy preference relation. Furthermore, Wu and Xu developed separate consistency and consensus processes to deal with the hesitant fuzzy linguistic preference relations of individual rationality and group rationality. Zhu and Xu proposed the concept of the probability-hesitant fuzzy preference relation. As mentioned earlier, to date there has been a great deal of research into preference relations and interval preference relations. Nevertheless, in a probability-interval valued hesitant fuzzy environment, it is still not known how to calculate or improve the consistency of preference relations. Therefore, this study focuses on resolving this problem. In this paper, based on the P-HFS and IVHFS, a definition of P-IVHFS is provided, and the relationship between the P-HFS, IVHFS and P-IVHFS is illustrated. Furthermore, motivated by the comparison method of HFEs, the comparison method of P-IVHFEs is defined. Additionally, inspired by the operations of IVHFEs, the complement, union and intersection and operational laws of P-IVHFEs are provided. Moreover, based on related studies, the definition of P-IVHFPRs is also provided. Subsequently, the consistency of P-IVHFPRs is discussed, using the multiplicative transitivity to verify the consistency of a P-IVHFPR. Finally, based on the method in a hesitant fuzzy environment, some definitions related to multiplicative consistent P-

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.4, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

IVHFPRs are provided, and a decomposition method to repair the consistency of P-IVHFPRs is proposed. The rest of this paper is organized as follows. In Section 2, some concepts and properties associated with the topic are briefly reviewed. In Section 3, P-IVHFSs are proposed and some of their properties are discussed. In Section 4, P-IVHFPRs are proposed and in Section 5, the consistency of P-IVHFPRs is discussed. In Section 6, based on the multiplicative consistency of hesitant fuzzy preference relations, a decomposition method to deal with the consistency of P-IVHFPRs is proposed. Finally an example is provided to illustrate the algorithm. 2. Preliminaries In this section, some concepts and properties associated with the topic are briefly reviewed. Definition 1. Let a ˜ = [aL , aU ] = {x|aL ≤ x ≤ aU }, and then a ˜ is called an interval number. For convenience, interval numbers are sometimes also called interval values. In particular, if aL = aU , a ˜ is a real number. If aL ≥ 0, then a ˜ is called a positive interval number. For any two positive interval numbers a ˜ = [aL , aU ], ˜b = [bL , bU ] and λ ≥ 0, δ > 0, then L L U U ˜ (1) a ˜ = b if a = b and a = b ; (2) a ˜ + ˜b = [aL + bL , aU + bU ]; (3) a ˜ · ˜b = [aL · bL , aU · bU ]; (4) λ˜ a = [λaL , λaU ]; (5) a ˜δ = [aL , aU ]δ = [(aL )δ , (aU )δ ]; ( U L [δ a , δ a ], if 0 < δ < 1; L ,aU ] L U L U a ˜ [a a a a a (6) δ = δ = [min{δ , δ }, max{δ , δ }] = L U [δ a , δ a ], if δ ≥ 1. U U Definition 2. [29] Let a ˜1 = [aL ˜2 = [aL a1 ) = 1 , a1 ] and a 2 , a2 ] be two interval numbers, and len(˜ U L U L a1 − a1 , len(˜ a2 ) = a2 − a2 , then the degree of possibility of a ˜1 ≥ a ˜2 is defined as follows:

p(˜ a1 ≥ a ˜2 ) = max{1 − max{

L aU 2 − a1 , 0}, 0} len(˜ a1 ) + len(˜ a2 )

(1)

Similarly, the degree of possibility of a ˜2 ≥ a ˜1 is defined as follows: p(˜ a2 ≥ a ˜1 ) = max{1 − max{

L aU 1 − a2 , 0}, 0} len(˜ a1 ) + len(˜ a2 )

(2)

Based on Definition 2, the following results hold: (1) 0 ≤ p(˜ a1 ≥ a ˜2 ) ≤ 1, 0 ≤ p(˜ a2 ≥ a ˜1 ) ≤ 1. (2) p(˜ a1 ≥ a ˜2 ) + p(˜ a2 ≥ a ˜1 ) = 1. Especially, p(˜ a1 ≥ a ˜1 ) = p(˜ a2 ≥ a ˜2 ) = 1. Definition 3. [15, 16] Let X be a universal set, a hesitant fuzzy set (HFS) on X is in terms of a function that when applied to X returns a subset of [0, 1]. To be easily understood, the HFS can be expressed by a mathematical symbol [21]: nD E o ˜ ˜ (x) |x ∈ X A˜ = x, h A ˜ ˜ (x) is a set of some values in [0, 1], denoting the possible membership degrees of the where h A ˜ ˜ (x) is called a hesitant fuzzy element (HFE) and Θ the set of all ˜ h element x ∈ X to the set A. A HFEs [22]. For three HFEs h, h1 and h2 , Torra and Narukawa [15, 16] defined the corresponding complement, union and intersection, namely (1) hc = ∪γ∈h {1 − γ}; (2) h1 ∪ h2 = ∪γ1 ∈h1 ,γ2 ∈h2 max{γ1 , γ2 }; (3) h1 ∩ h2 = ∩γ1 ∈h1 ,γ2 ∈h2 min{γ1 , γ2 }.

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Operational laws on the HFEs h, h1 and h2 have been given as follows [22]: (1) hλ = ∪γ∈h {γ λ }, λ > 0; (2) λh = ∪γ∈h {1 − (1 − γ)λ }, λ > 0; (3) h1 ⊕ h2 = ∪γ1 ∈h1 ,γ2 ∈h2 {γ1 + γ2 − γ1 γ2 }; (4) h1 ⊗ h2 = ∪γ1 ∈h1 ,γ2 ∈h2 {γ1 γ2 }. Definition 4. [2] Let X be a universal set, and D[0, 1] be the set of all closed subintervals of [0, 1]. An interval-valued hesitant fuzzy set (IVHFS) on X is nD E o ˜ ˜ (xi ) |xi ∈ X, i = 1, 2, . . . , n A˜ = xi , h A ˜ ˜ (xi ) : X → D[0, 1] denotes all possible interval-valued membership degrees of the elwhere h A ˜ ˜ (xi ) an interval-valued hesitant fuzzy ˜ For convenience, we call h ement xi ∈ X to the set A. A element (IVHFE), which is denoted by n o ˜ ˜ ˜ (xi ) hA˜ (xi ) = γ˜ γ˜ ∈ h A Here γ˜ = [˜ γ L , γ˜ U ] is an interval number. γ˜ L = inf γ˜ and γ˜ U = sup γ˜ represent the lower and upper limits of γ˜ , respectively. When the lower and upper limits of the interval numbers are identical, IVHFS reduces to HFS [15]. Namely HFS is a special case of IVHFS. ˜ ˜ (x1 ) = {[0.1, 0.3], [0.4, 0.5]} Example 1. Let X = {x1 , x2 } be a universal set, and the two IVHFEs h A ˜ ˜ (x2 ) = {[0.1, 0.2], [0.4, 0.6], [0.7, 0.8]} denote the membership degrees of xi (i = 1, 2) to the and h A ˜ A˜ is an IVHFS, where set A. A˜ = {hx1 , {[0.1, 0.3], [0.4, 0.5]}i , hx2 , {[0.1, 0.2], [0.4, 0.6], [0.7, 0.8]}i} ˜ s(h) ˜ = 1 P ˜ γ˜ is called the score function of h ˜ where l˜ Definition 5. [2] For an IVHFE h, γ ˜ ∈h h lh ˜ ˜ and s(h) ˜ is an interval value belonging to [0, 1]. For is the number of the interval values in h, ˜ 1 and h ˜ 2 , if s(h ˜ 1 ) ≥ s(h ˜ 2 ), then h ˜1 ≥ h ˜ 2. two IVHFEs h Definition 6. [2] For three IVHFEs h, h1 and h2 , the corresponding complement, union and intersection and operational laws have been given as follows. If γ˜ L = γ˜ U , then the following operations reduce to those of HFEs: c U L ˜ ˜ }; (1) h = {[1 − γ˜ , 1 − γ˜ ] γ˜ ∈ h L L U U ˜ ˜ ˜ 1 , γ˜2 ∈ h ˜ 2 }; (2) h1 ∪ h2 = {[max(˜ γ1 , γ˜2 ), max(˜ γ1 , γ˜2 )] γ˜1 ∈ h ˜1 ∩ h ˜ 2 = {[min(˜ ˜ 1 , γ˜2 ∈ h ˜ 2 }; (3) h γ1L , γ˜2L ), min(˜ γ1U , γ˜2U )] γ˜1 ∈ h ˜ λ = {[(˜ ˜ λ > 0; (4) h γ L )λ , (˜ γ U )λ ] γ˜ ∈ h}, ˜ = {[1 − (1 − γ˜ L )λ , 1 − (1 − γ˜ U )λ ] γ˜ ∈ h}, ˜ λ > 0; (5) λh ˜1 ⊕ h ˜ 2 = {[˜ ˜ 1 , γ˜2 ∈ h ˜ 2 }; (6) h γ1L + γ˜2L − γ˜1L · γ˜2L , γ˜1U + γ˜2U − γ˜1U · γ˜2U ] γ˜1 ∈ h ˜1 ⊗ h ˜ 2 = {[˜ ˜ 1 , γ˜2 ∈ h ˜ 2 }. (7) h γ1L · γ˜2L , γ˜1U · γ˜2U ] γ˜1 ∈ h 3. P-IVHFS Inspired by the P-HFS [19, 37] and IVHFS [2], the definition of a P-IVHFS is provided.

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Jiuping Xu ET AL 636-655

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.4, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

Definition 7. Let X be a universal set, and D[0, 1] be the set of all closed subintervals of [0, 1]. A P-IVHFS on X is nD E o ˜ ˜ (xi , pij ) |xi ∈ X, i = 1, 2, . . . , n, j = 1, 2, . . . , mi A˜ = xi , h A P i ˜ ˜ (xi , pij ), pij denotes the where m j=1 pij = 1, mi denotes the number of the interval values in hA ˜ ˜ (xi , pij ), and h ˜ ˜ (xi , pij ) : X → D[0, 1] corresponding probability of the jth interval value in h A A ˜ For denotes all possible interval-valued membership degrees of the element xi ∈ X to the set A. ˜ convenience, hA˜ (xi , pij ) is called a probability-interval valued hesitant fuzzy element(P-IVHFE), which is denoted by n o ˜ ˜ (xi , pij ) = γ˜ γ˜ ∈ h ˜ ˜ (xi , pij ) h A A Here γ˜ is an interval number with a corresponding possibility. For simplicity, P-IVHFE can be ˜ i , i = 1, 2, · · · . A P-IVHFE is the basic unit of a P-IVHFS, and the former can be denoted by h considered as a special case of the latter. The relationship between a P-IVHFE and a P-IVHFS is similar to that between an IVHFE and an IVHFS [2]. Suppose that γ˜ L = inf γ˜ and γ˜ U = sup γ˜ represent the lower and upper limits of γ˜ , respectively. When the lower and upper limits of the interval numbers are identical, the interval numbers are reduced to crisp numbers, and a P-IVHFS is reduced to a P-HFS. Thus, a P-HFS is a special case of a P-IVHFS. Meanwhile, it is clear that without the probability description pij , that is the probability values pij (j = 1, 2, · · · ) are identical, a P-IVHFE is reduced to an IVHFE, and a P-IVHFS is reduced to an IVHFS. Thus, an IVHFE is a special case of a P-IVHFE, and an IVHFS is a special case of a P-IVHFS. Example 2. Let X = {x1 , x2 } be a universal set, and the two P-IVHFEs ˜ ˜ (x1 , p1j ) = {[0.2, 0.3](p11 = 0.4), [0.5, 0.6](p12 = 0.6)} h A ˜ ˜ (x2 , p2j ) = {[0.1, 0.2](p21 = 0.3), [0.3, 0.5](p22 = 0.5), [0.6, 0.7](p23 = 0.2)} h A ˜ A˜ is a P-IVHFS, where denote the membership degrees of xi (i = 1, 2) to the set A.   hx1 , {[0.2, 0.3](p11 = 0.4), [0.5, 0.6](p12 = 0.6)}i , ˜ A= hx2 , {[0.1, 0.2](p21 = 0.3), [0.3, 0.5](p22 = 0.5), [0.6, 0.7](p23 = 0.2)}i Based on the comparison method of HFEs [21], the following comparison method of PIVHFEs is defined: ˜ = P ˜ γ˜ pγ˜ is called the score of h, ˜ where pγ˜ is the Definition 8. For a P-IVHFE, s(h) γ ˜ ∈h ˜ is also an interval number. corresponding probability of γ˜ . It is clear that s(h) ˜ 1 ) ≥ s(h ˜ 2 ) and s(h ˜ 2 ) ≥ s(h ˜ 1 ), Then by Eqs. (1) and (2), we can get the possibilities of s(h ˜ ˜ ˜ ˜ namely p(h1 ≥ h2 ) and p(h2 ≥ h1 ). ˜ 1 ) ≥ s(h ˜ 2 )) > 0.5, then s(h ˜ 1 ) is superior to s(h ˜ 2 ), and thus h ˜ 1 is superior to h ˜ 2, If p(s(h ˜ ˜ ˜ ˜ denoted by h1 > h2 or h2 < h1 . ˜ 1 ) ≥ s(h ˜ 2 )) < 0.5, then s(h ˜ 2 ) is superior to s(h ˜ 1 ), and thus h ˜ 2 is superior to h ˜ 1, If p(s(h ˜ ˜ ˜ ˜ denoted by h2 > h1 or h1 < h2 . ˜ 1 ) ≥ s(h ˜ 2 )) = 0.5, then h ˜ 1 is indifferent to h ˜ 2 , denoted by h ˜1 ∼ h ˜ 2. In particular, if p(s(h Example 3. In Example 2, for the two P-IVHFEs ˜ 1 = {[0.2, 0.3](p11 = 0.4), [0.5, 0.6](p12 = 0.6)} h ˜ 2 = {[0.1, 0.2](p21 = 0.3), [0.3, 0.5](p22 = 0.5), [0.6, 0.7](p23 = 0.2)} h according to Definition 1, we have ˜ 1 ) = [0.2, 0.3] × 0.4 + [0.5, 0.6] × 0.6 = [0.38, 0.48] s(h

640

Jiuping Xu ET AL 636-655

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.4, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

˜ 2 ) = [0.1, 0.2] × 0.3 + [0.3, 0.5] × 0.5 + [0.6, 0.7] × 0.2 = [0.3, 0.45] s(h Using Definition 2, we obtain ˜ 1 ) ≥ s(h ˜ 2 )) = max{1 − max{ 0.45 − 0.38 , 0}, 0} = 0.72 p(s(h 0.15 + 0.1 ˜1 > h ˜ 2. which indicates that h ˜ }, ˜ = {[˜ To be easily formulated, a P-IVHFE can be denoted by h γiL , γ˜iU ](p[˜γ L ,˜γ U ] ) γ˜i ∈ h i i ˜ = {[˜ ˜ }, where p L U (or pγ˜ ) denotes the corfor simplicity, denoted by h γiL , γ˜iU ](pγ˜i ) γ˜i ∈ h i [˜ γ ,˜ γ ] i

i

responding probability value of [˜ γiL , γ˜iU ] (i.e., γ˜i ). Based on the operations of IVHFEs [2], the complement, union and intersection and operational laws of P-IVHFEs can be provided as follows: ˜ h ˜ 1 and h ˜ 2 be three P-IVHFEs, then Definition 9. Let h, ˜ c = {[1 − γ˜ U , 1 − γ˜ L ](pγ˜ ) γ˜i ∈ h ˜ }; (1) h i i i

˜ 1 , γ˜2 ∈ h ˜ 2 }; ˜1 ∪ h ˜ 2 = {[max(˜ (2) h γ1L , γ˜2L ), max(˜ γ1U , γ˜2U )](pγ˜1 · pγ˜2 ) γ˜1 ∈ h ˜1 ∩ h ˜ 2 = {[min(˜ ˜ 1 , γ˜2 ∈ h ˜ 2 }; γ1U , γ˜2U )](pγ˜1 · pγ˜2 ) γ˜1 ∈ h (3) h γ1L , γ˜2L ), min(˜ ˜ λ = {[(˜ ˜ }; (4) h γiL )λ , (˜ γiU )λ ](pγ˜i ) γ˜i ∈ h ˜ = {[1 − (1 − γ˜ L )λ , 1 − (1 − γ˜ U )λ ](pγ˜ ) γ˜i ∈ h ˜ }, λ > 0; (5) λh i i i ˜1 ⊕ h ˜ 2 = {[˜ ˜ 1 , γ˜2 ∈ h ˜ 2 }; (6) h γ1L + γ˜2L − γ˜1L · γ˜2L , γ˜1U + γ˜2U − γ˜1U · γ˜2U ](pγ˜1 · pγ˜2 ) γ˜1 ∈ h ˜ 1 , γ˜2 ∈ h ˜ 2 }. ˜1 ⊗ h ˜ 2 = {[˜ (7) h γ1L · γ˜2L , γ˜1U · γ˜2U ](pγ˜1 · pγ˜2 ) γ˜1 ∈ h

It is clear that without the probability description pij , that is the probability values pij (j = 1, 2, · · · ) are identical, then the operational laws of P-IVHFEs are reduced to those of the IVHFEs. Theorem 1. When IVHFEs are extended to P-IVHFEs, the following operational laws [2] still ˜ h ˜ 1 and h ˜ 2 be three P-IVHFEs, then are true in the P-IVHFS environment. Let h, ˜1 ⊕ h ˜2 = h ˜2 ⊕ h ˜ 1; (1) h ˜ ˜ ˜ ˜ 1; (2) h1 ⊗ h2 = h2 ⊗ h ˜1 ⊕ h ˜ 2 ) = λh ˜ 1 ⊕ λh ˜ 2 , λ > 0; (3) λ(h λ λ λ ˜ ˜ ˜ (4) (h1 ⊗ h2 ) = h1 ⊗ h2 , λ > 0; ˜ ⊕ λ2 h ˜ = (λ1 + λ2 )h, ˜ λ1 , λ2 > 0; (5) λ1 h λ λ (λ +λ ) ˜ 1 ⊕h ˜ 2 =h ˜ 1 2 , λ1 , λ2 > 0. (6) h c c ˜ ˜ ˜ ˜ 2 )c ; (7) h1 ∪ h2 = (h1 ∩ h c c ˜ ∩h ˜ = (h ˜1 ∪ h ˜ 2 )c ; (8) h 1 2 ˜ c )λ = (λh) ˜ c; (9) (h c ˜ ) = (h ˜ λ )c ; (10) λ(h ˜c ⊕ h ˜ c = (h ˜1 ⊗ h ˜ 2 )c ; (11) h 1 2 ˜c ⊗ h ˜ c = (h ˜1 ⊕ h ˜ 2 )c . (12) h 1 2 Since they can be proven analogously, like those in an IVHFS environment, they are just listed without any proof. Meanwhile, according to Definition 9, the following operational laws also hold:

641

Jiuping Xu ET AL 636-655

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.4, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

˜ h ˜ 1 and h ˜ 2 be three P-IVHFEs, then Theorem 2. Let h, ˜∪h ˜ 1) ∪ h ˜2 = h ˜ ∪ (h ˜1 ∪ h ˜ 2 ); (1) (h ˜∩h ˜ 1) ∩ h ˜2 = h ˜ ∩ (h ˜1 ∩ h ˜ 2 ); (2) (h ˜ ˜ ˜ ˜ ˜ ˜ 2 ); (3) (h ⊕ h1 ) ⊕ h2 = h ⊕ (h1 ⊕ h ˜⊗h ˜ 1) ⊗ h ˜2 = h ˜ ⊗ (h ˜1 ⊗ h ˜ 2 ). (4) (h Proof . In the following, only (3) is proven; others can be obtained directly by Definition 9. Suppose that ˜ 2 }, ˜ 1 }; h ˜ 2 = {[˜ ˜ = {[˜ ˜ }; h ˜ 1 = {[˜ γ2L , γ˜2U ](pγ˜2 ) γ˜2 ∈ h h γ L , γ˜ U ](pγ˜ ) γ˜ ∈ h γ1L , γ˜1U ](pγ˜1 ) γ˜1 ∈ h then according to Definition 9, ˜ , γ˜1 ∈ h ˜ 1} ˜⊕h ˜ 1 = {[˜ h γ L + γ˜1L − γ˜ L · γ˜1L , γ˜ U + γ˜1U − γ˜ U · γ˜1U ](pγ˜ · pγ˜1 ) γ˜ ∈ h is obtained, therefore, ˜⊕h ˜ 1) ⊕ h ˜2 (h L L L γU + γ = {[(˜ γ + γ˜1L − γ˜ L · γ˜1L ) + γ˜2L − (˜ γ L + γ˜1L − γ˜ L ˜1U − γ˜ U · γ˜1U ) + γ˜2U · γ˜1 ) · γ˜2 , (˜ ˜ , γ˜1 ∈ h ˜ 1 , γ˜2 ∈ h ˜ 2} −(˜ γ U + γ˜1U − γ˜ U · γ˜1U ) · γ˜2U ]((pγ˜ · pγ˜1 ) · pγ˜2 ) γ˜ ∈ h

= {[˜ γ L + γ˜1L + γ˜2L − γ˜ L · γ˜1L − γ˜ L · γ˜2L − γ˜1L · γ˜2L + γ˜ L · γ˜1L · γ˜2L , γ˜ U + γ˜1U + γ˜2U − γ˜ U · γ˜1U ˜ , γ˜1 ∈ h ˜ 1 , γ˜2 ∈ h ˜ 2} −˜ γ U · γ˜ U − γ˜ U · γ˜ U + γ˜ U · γ˜ U · γ˜ U ](pγ˜ · pγ˜ · pγ˜ ) γ˜ ∈ h 2

1

2

1

2

1

2

Likewise, ˜ ⊕ (h ˜1 ⊕ h ˜ 2) h L = {[˜ γ + (˜ γ1L + γ˜2L − γ˜1L · γ˜2L ) − γ˜ L · (˜ γ1L + γ˜2L − γ˜1L · γ˜2L ), γ˜ U + (˜ γ1U + γ˜2U − γ˜1U · γ˜2U ) ˜ , γ˜1 ∈ h ˜ 1 , γ˜2 ∈ h ˜ 2} −˜ γ U · (˜ γ1U + γ˜2U − γ˜1U · γ˜2U )](pγ˜ · (pγ˜1 · pγ˜2 )) γ˜ ∈ h

= {[˜ γ L + γ˜1L + γ˜2L − γ˜ L · γ˜1L − γ˜ L · γ˜2L − γ˜1L · γ˜2L + γ˜ L · γ˜1L · γ˜2L , γ˜ U + γ˜1U + γ˜2U − γ˜ U · γ˜1U ˜ , γ˜1 ∈ h ˜ 1 , γ˜2 ∈ h ˜ 2} −˜ γ U · γ˜2U − γ˜1U · γ˜2U + γ˜ U · γ˜1U · γ˜2U ](pγ˜ · pγ˜1 · pγ˜2 ) γ˜ ∈ h

can be obtained. Therefore, we have ˜⊕h ˜ 1) ⊕ h ˜2 = h ˜ ⊕ (h ˜1 ⊕ h ˜ 2) (h which completes the proof. 4. P-IVHFPRs and Consistency In this section, we present P-IVHFPRs and discuss their consistency. 4.1. P-IVHFPRs In the GDM process, preference relations are very popular tools for expressing the DM’s preferences when they compare a set of alternatives. Various types of preference relations have been given for different environments [2]. In order to represent preference relations more objectively, suppose that DMs are allowed to provide several possible interval fuzzy preference values and the associated probability values when they compare two alternatives, then we get the following P-IVHFPR:

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Jiuping Xu ET AL 636-655

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.4, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

Definition 10. Let X = {x1 , x2 , . . . xn } be a universal set. A P-IVHFPR on X is denoted ˜ ij )n×n ⊂ X × X, where h ˜ ij = {h ˜ t (pt ), t = 1, 2, · · · , mij } is a P-IVHFE, ˜ = (h by a matrix R ij ij indicating all possible degrees to which xi is preferred to xj and the corresponding probability ˜ t should values with mij representing the number of intervals in the P-IVHFE. In addition, h ij satisfy σ(t)

˜ inf h ij σ(t) pij

σ(mij +1−t)

˜ + sup h ji

σ(t)

˜ = sup h ij

σ(mij +1−t)

˜ + inf h ji

= 1,

σ(m +1−t) pji ij ,

= ˜ ii = {[0.5, 0.5](p = 1)}, h

i, j = 1, 2, · · · , n

(3)

˜ ij in an increasing order, and let h ˜ σ(t) be the tth smallest where we arrange the intervals in h ij ˜ ij . inf h ˜ σ(t) and sup h ˜ σ(t) denote the lower and upper limits of h ˜ σ(t) respectively, interval in h ij ij ij σ(mij +1−t) σ(mij +1−t) σ(t) σ(t) ˜ ˜ p and p denote the corresponding values of h and h respectively, p = 1 ij

ji

ij

ji

denotes the corresponding value is equal to 1. Example 4. The following matrix in which every element is a P-IVHFE can represent a probability-interval valued hesitant fuzzy preference relation: 0

˜ ij )3×3 ˜ e = (h R

1 {[0.5, 0.5](1)} {[0.4, 0.5](0.6), [0.7, 0.8](0.4)} {[0.5, 0.6](1)} = @ {[0.2, 0.3](0.4), [0.5, 0.6](0.6)} {[0.5, 0.5](1)} {[0.3, 0.4](0.2), [0.5, 0.7](0.5), [0.8, 0.9](0.3)} A {[0.4, 0.5](1)} {[0.1, 0.2](0.3), [0.3, 0.5](0.5), [0.6, 0.7](0.2)} {[0.5, 0.5](1)}

˜ ij denotes the group preference degree that the alternative xi is superior to the alternative where h xj . Motivated by [2, 35], it can be explained how the elements in the matrix are obtained. ˜ 23 as an example. Since h ˜ 23 represents all possible probability-interval valued preference Take h ˜ 1 = [0.1, 0.2], h ˜ 2 = [0.3, 0.5], degrees to which x3 is preferred to x2 , its values come from h 23 23 3 ˜ h23 = [0.6, 0.7] which is provided by a DM. The DM is sure that the preference value is the interval [0.1, 0.2] with a probability of 30%, and the interval [0.3, 0.5] with a probability of ˜ 23 can be denoted 50%, and the interval [0.6, 0.7] with a probability of 20%. Therefore, the h ˜ 23 , i.e., by {[0.1, 0.2](0.3), [0.3, 0.5](0.5), [0.6, 0.7](0.2)}. Similarly the symmetric element of h ˜ h32 can be denoted by {[0.3, 0.4](0.2), [0.5, 0.7](0.5), [0.8, 0.9](0.3)}. Other symmetric elements ˜ ij and h ˜ ji in R ˜ e are obtained in an analogous way, and satisfy the complementary properties h ˜ ij represents the preference degree to which xi defined in Eq.(3). In addition, when i = j, h ˜ ii = {[0.5, 0.5](1)}(i = 1, 2, 3). is preferred to itself; namely, it is preferred equally , therefore h ˜ Through the above procedure, the aforementioned matrix Re is obtained. 4.2. The Consistency of P-IVHFPRs Cardinal consistency is a stronger concept than ordinal consistency. In the analytic hierarchy process, Saaty [13] first addressed the issue of consistency, and developed the notions of perfect consistency and acceptable consistency. Ordinal consistency is based on the notion of transitivity, meaning that if A is preferred to B and B is preferred to C, it perceives A to be preferred to C, which is normally referred to as weak transitivity [4, 26]. The weak transitivity is the minimum requirement condition to ensure that the hesitant fuzzy preference relation is consistent. There are further two conditions, named additive transitivity and multiplicative transitivity [14] which are more restrictive than weak transitivity and can imply reciprocity. Even though both additive transitivity and multiplicative transitivity can be used to measure consistency, the additive consistency may produce infeasible results [27]. Thus, the multiplicative transitivity is also used to verify the consistency of a P-IVHFPR. Let U = (uij )n×n , where uij denotes a ratio of preference intensity for the alternative Ai to that for Aj . Then the condition of multiplicative transitivity can be rewritten as follows: uij ujk uki = uik ukj uji

(4)

643

Jiuping Xu ET AL 636-655

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.4, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

Under the assumption of reciprocity, and in the case where (uik , ukj ) ∈ / {(0, 1), (1, 0)}, Eq.(4) can be expressed as follows [4]: uij =

uik ukj uik ukj + (1 − uik )(1 − ukj )

(5)

in the case where (uik , ukj ) ∈ {(0, 1), (1, 0)}, stipulating uij = 0. Based on the multiplicative consistency of hesitant fuzzy preference relations, and using a decomposition method, the following definition is obtained: ˜ ij )n×n be a P-IVHFPR on a fixed set X = {x1 , x2 , · · · , xn } and ˜ = (h Definition 11. Let R t t ˜ ˜ ˜ t = [#h ˜ t (x), †h ˜ t (x)], where hij = {hij (pij ), t = 1, 2, · · · , mij } be a P-IVHFE; suppose that h ij ij ij ˜ t (x) is the left endpoint of h ˜ t , and †h ˜ t (x) is the right endpoint of h ˜ t , let #h ij ij ij ij ˜ t (pt )} {0.5(1)} {#h 12 12 t t ˜ (p )}  {†h {0.5(1)} 21 21  = ··· ··· ˜ t (pt )} {†h ˜ t (pt )} {†h n1 n1 n2 n2 

˜ A = [R ˜ t (pt )]n×n R ij ij

 ˜t  #hij , t ˜ 0.5, namely, Rij =  ˜t †hij , if i > j, taking the right

if i < j, ˜ t , while if i = j, which means if i < j, taking the left endpoint of h ij if i > j ˜ t . And let endpoint of h ij ˜ t (pt )} {0.5(1)} {†h 12 12 ˜ t (pt )}  {#h {0.5(1)} 21 21 =  ··· ··· t t t ˜ ˜ {#hn1 (pn1 )} {#hn2 (ptn2 )} 

t ˜ B = [˜ R rij (ptij )]n×n

˜ t (pt )}  {#h 1n 1n ˜ t (pt )}  {#h 2n 2n   ··· {0.5(1)}

··· ··· ··· ···

··· ··· ··· ···

˜ t (pt )}  {†h 1n 1n ˜ t (pt )}  {†h 2n 2n   ··· {0.5(1)}

˜ t , while if i > j, taking the left endpoint of which means if i < j, taking the right endpoint of h ij ˜ t , namely, h ij  ˜t  †hij , if i < j, t 0.5, if i = j, r˜ij =  ˜t #hij , if i > j ˜ A and R ˜ B are called the decomposition of R, ˜ while R ˜ is the composition of for convenience, R A B ˜ ij )n×n is multiplicative consistent if and only if R ˜ and R ˜ . Then R ˜ = (h ˜ A and R ˜ B are both R multiplicative consistent, i.e., the following two conditions are satisfied simultaneously:  ˜ kj ) ∈ {({0}, {1}), ({1}, {0})} ˜ ik , R if (R  0, σ(s) σ(s) σ(s) σ(s) σ(s) ˜ ˜ ˜ R (1) R = ik pik Rkj pkj ij , otherwise,  ˜ σ(s) σ(s) ˜ σ(s) σ(s) ˜ σ(s) σ(s) ˜ σ(s) σ(s) Rik pik Rkj pkj +(1−Rik )pik (1−Rkj )pkj

f or all i ≤ k  ≤j 0,  ˜ σ(s) = ˜ σ(s) R ˜ σ(s) R i.e., R ij  ˜ σ(s) ˜ σ(s) ik ˜ kj σ(s)

˜ ik , R ˜ kj ) ∈ {({0}, {1}), ({1}, {0})} if (R σ(s)

˜ Rik Rkj +(1−Rik )(1−R kj )

f or all i ≤ k ≤ j σ(s) σ(s) σ(s) pij = pik pkj   σ(s) (2) r˜ij =  σ(s) r˜ik

otherwise,

if (˜ rik , r˜kj ) ∈ {({0}, {1}), ({1}, {0})}

0, σ(s) σ(s) r˜ik r˜kj σ(s) σ(s) σ(s) r˜kj +(1−˜ rik )(1−˜ rkj )

,

,

otherwise,

f or all i ≤ k ≤ j σ(s) σ(s) σ(s) pij = pik pkj

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Jiuping Xu ET AL 636-655

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.4, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

˜ σ(s) , R ˜ σ(s) and R ˜ σ(s) are the sth smallest values in R ˜ ij , R ˜ ik and R ˜ kj respectively; pσ(s) , where R ij ij ik kj σ(s)

pik

σ(s)

and pkj

σ(s)

σ(s)

are their corresponding probability values, respectively; r˜ij , r˜ik

the sth smallest values in r˜ij , r˜ik and r˜kj respectively; and sponding probability values, respectively.

σ(s) pij ,

σ(s) pik

and

σ(s) pkj

σ(s)

and r˜kj

are

are their corre-

If without the probability description and #htik = †htik , #htkj = †htkj , then Definition 11 is reduced to that of a hesitant fuzzy preference relation. ˜ ij )2×2 is multiplicative consistent. ˜ = (h It can be proven that any P-IVHFPR R By extending the definitions in a hesitant fuzzy environment, the following definitions in a probability-interval valued hesitant fuzzy environment are obtained: ˜ ij )n×n be a P-IVHFPR on a fixed set X = {x1 , x2 , · · · , xn } and ˜ = (h Definition 12. Let R t t ˜ ˜ hij = {hij (pij ), t = 1, 2, · · · , mij } be a P-IVHFE in which mij represents the number of intervals ˜ t = [#h ˜ t (x), †h ˜ t (x)], let suppose that h ij ij ij

˜ A = [R ˜ t (pt )]n×n R ij ij

˜ t (pt )} {0.5(1)} {#h 12 12 ˜ t (pt )}  {†h {0.5(1)} 21 21 =  ··· ··· t t t ˜ ˜ {†hn1 (pn1 )} {†hn2 (ptn2 )}

··· ··· ··· ···

˜ t (pt )}  {#h 1n 1n ˜ t (pt )}  {#h 2n 2n   ··· {0.5(1)}

t ˜ B = [˜ R rij (ptij )]n×n

˜ t (pt )} {0.5(1)} {†h 12 12 ˜ t (pt )}  {#h {0.5(1)} 21 21 =  ··· ··· t t t ˜ ˜ {#hn1 (pn1 )} {#hn2 (ptn2 )}

··· ··· ··· ···

˜ t (pt )}  {†h 1n 1n ˜ t (pt )}  {†h 2n 2n   ··· {0.5(1)}





¯ a prefect multiplicative consistent P-IVHFPR, if R ¯ is the composition of R ¯ A and then we call R B A t B t ¯ ¯ ¯ ¯ R , R = (Rij (x))n×n (pij )), R = (¯ rij (x))n×n (pij )), and

¯ σ(s) (x) = R ij

σ(s)

r¯ij (x) =

      

1 j−i−1

      

1 j−i−1

j−1 P

˜ σ(s) (x)R ˜ σ(s) (x) R ik kj

˜ σ(s) ˜ σ(s) ˜ σ(s) ˜ σ(s) k=i+1 Rik (x)Rkj (x)+(1−Rik (x))(1−Rkj (x))

,

i+1 j 1−R ji j−1 P k=i+1

(6)

σ(s) σ(s) r˜ik (x)˜ rkj (x) σ(s) σ(s) σ(s) σ(s) r˜ik (x)˜ rkj (x)+(1−˜ rik (x))(1−˜ rkj (x))

σ(s)

,

i+1 j

¯ σ(s) (x), R ˜ σ(s) (x), R ˜ σ(s) (x), r¯σ(s) (x), r˜σ(s) (x), r˜σ(s) (x) denote the sth smallest values in where R ij ij ik kj ik kj ¯ ij (x), R ˜ ik (x), R ˜ kj (x), r¯ij (x), r˜ik (x), r˜kj (x) respectively, and s = 1, 2, · · · , l , l = max{mik , mkj }, R ˜ ik and h ˜ kj respectively. in which mik , mkj represent the number of intervals in h If the two endpoints of the intervals are considered, the two corresponding probabilityhesitant fuzzy preference relations are multiplicative consistent, thus it is believed that the P-IVHFPR is multiplicative consistent. ˜ ij )n×n , R ˜ ij )n×n an ˜ = (h ˜A, R ˜B , R ¯A, R ¯ B be as before, then we call R ˜ = (h Definition 13. Let R acceptable multiplicative consistent P-IVHFPR if  ˜A, R ¯ A ) < θ0 d(R ˜B , R ¯ B ) < θ0 d(R

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.4, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

˜A, R ¯ A ) is the distance measure between R ˜ A and R ¯ A , d(R ˜B , R ¯ B ) is the distance measure where d(R B B A A B B ˜ and R ¯ . d(R ˜ ,R ¯ ) and d(R ˜ ,R ¯ ) can be calculated by the following Eqs.(8) and between R (9). θ0 is the consistency level. We usually take θ0 = 0.1 in practice. 4.3. An Iterative Algorithm for Improving the Consistency of P-IVHFPR In general, the P-IVHFPR constructed by the decision maker often has an unacceptable ˜A, R ¯ A ) ≥ θ0 or d(R ˜B , R ¯ B ) ≥ θ0 . At this time, it is multiplicative consistency which means d(R necessary to adjust the elements in the P-IVHFPR in order to improve the consistency. Based on the algorithm in a hesitant fuzzy environment [9], An iterative algorithm is proposed as follows to repair the consistency of the P-IVHFPR. An Iterative Algorithm for Improving the Consistency of P-IVHFPR ˜ ij )n×n ; k, the number of iterations; δ, the step size, 0 ≤ λ = kδ ≤ 1; ˜ = (h Input: P-IVHFPR R θ0 , the consistency level. Hereby we take θ0 = 0.1. ˜ (k) , with satisfactory consistency. Output: P-IVHFPR R ¯ where R ¯ Step 1. Let k = 1, and construct a perfect multiplicative consistent P-IVHFPR R, A B A B A B ¯ and R ¯ are defined ¯ and R ¯ ,R ¯ = (R ¯ ij (x))n×n , R ¯ = (¯ is the composition of R rij (x))n×n . R in Definition 12. ˜ (k)A , R ¯ A ) and d(R ˜ (k)B , R ¯ B ). Eqs.(8) and (9) are given Step 2. Calculate the deviations d(R as follows:    mij n P n P P  (k)σ(s) σ(s) σ(s) 1 (k)A A  ˜ ¯ )= ¯  ,R −R Rij  dHam min g (R ij pij (n−1)(n−2) i=1 j=1 s=1  (8) m n P n Pij (k)σ(s) P  σ(s) σ(s) 1 (k)B , R B) =  ˜ ¯ d ( R r − r ¯ p  ij ij ij  Ham min g (n−1)(n−2) i=1 j=1 s=1

or  #1 "  mij   2  2  n P n  P P  (k)σ(s) σ(s) σ(s) 1 (k)A A ¯ ˜ ¯ )=  −R pij ,R  Rij ij  dEuclidean (R (n−1)(n−2) i=1 j=1 s=1 #1 "  mij   2  2  n P n  P P  (k)σ(s) σ(s) σ(s) 1  dEuclidean (R ˜ (k)B , R ¯B ) =  − r¯ij pij rij  (n−1)(n−2) i=1 j=1

(9)

s=1

(k)σ(s) ¯ σ(s) (k)σ(s) σ(s) ˜ (k)A , R ¯A, R ˜ (k)B , R ¯ B respecwhere Rij , Rij , rij , r¯ij are the sth smallest values in R ˜ (k)A and R ˜ (k)B are the resolution of R ˜ (k) . R ¯ A and R ¯ B are the resolution of R, ¯ which is the tively. R ˜ (k)B , R ¯ B ) < θ0 , ˜ If d(R ˜ (k)A , R ¯ A ) < θ0 and d(R corresponding perfect multiplicative relation of R. then go to Step 4; Otherwise, go to Step 3. _ (k)A ˜ (k)A to R Step 3. Repair the inconsistent multiplicative P-IVHFPR, transforming R and _ (k)B

˜ (k)B to R R _ (k)σ(s)

Rij

=

by using the following equations. We give Eqs.(10) and (11).

” “ ” “ (k)σ(s) 1−λ ¯ σ(s) λ Rij Rij “ ” “ ” “ ” “ ”λ (k)σ(s) 1−λ ¯ σ(s) λ (k)σ(s) 1−λ ¯ σ(s) Rij Rij + 1−Rij 1−R ij

i, j = 1, 2, · · · , n _(k)σ(s)

r ij

=

(10)

“ ” “ ” (k)σ(s) 1−λ σ(s) λ rij r¯ij ” “ ” “ ” “ ” “ σ(s) λ (k)σ(s) 1−λ σ(s) λ (k)σ(s) 1−λ 1−¯ rij rij r¯ij + 1−rij

i, j = 1, 2, · · · , n _ (k)σ(s)

where Rij (k)σ(s)

, rij

σ(s)

, r¯ij

(k)σ(s)

, Rij

_ (k)

(k)σ(s) ¯ σ(s) are the sth smallest values in R , R(k) , R ¯ ij respectively, _ ,R r ij ij ij ij _(k)

(k)

_ (k)A

are the sth smallest values in r ij , rij , r¯ij respectively. Let R(k+1)A = R

_ (k)B

R(k+1)B = R

(11)

,

and k = k + 1, then go to Step 2.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.4, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

˜ (k) . Step 4. Output R Step 5. End. From the calculation process, it can be seen that the iterative process is convergent; for example when we take λ = 1. Therefore, only the steps are listed without providing any proof. 5. An Illustrative Example and Discussion In this section, an example is used to illustrate the algorithm. 5.1. Illustrative Example A large project of Jiudianxia reservoir operation [2, 25] is employed to demonstrate the validity of our approach. The reservoir is designed for many purposes, such as power generation, irrigation, total water supply for industry, agriculture, residents and environment. Because of different requirements for the partition of the amount of water, four reservoir operation schemes x1 , x2 , x3 and x4 are suggested. x1 : maximum plant output, enough supply of water used in the Tao River basin, higher and lower supply for society and economy; x2 : maximum plant output, enough supply of water used in the Tao River basin, higher and lower supply for society and economy, lower supply for ecosystem; x3 : maximum plant output, enough supply of water used in the Tao River basin, higher and lower supply for society and economy, total supply for ecosystem and environment, whose 90% is used for flushing sands at low water period; x4 : maximum plant output, enough supply of water used in the Tao River basin, higher and lower supply for society and economy, total supply for ecosystem and environment, whose 50% is used for flushing sands at low water period. To select the best scheme, the government assigns a large consultancy organization to evaluate four competing schemes. Due to uncertainties, the DMs give their preference information regarding alternatives in the form of interval values with probabilities. Take schemes x1 and x2 as an example; the DMs evaluate the degrees to which x1 is preferred to x2 , where 40% give a rating of [0.2,0.3] and the remaining 60% give [0.5,0.6]. Assume that these DMs in the consultancy firm cannot be persuaded each other to change their minds, the preference information that x1 is preferred to x2 provided by the organization can be considered as a P-IVHFE, i.e., {[0.2, 0.3](0.4), [0.5, 0.6](0.6)}. The preference information of the organization is listed as a ˜ P-IVHFPR R. ˜ ˜ = (h R  ij )4×4

{[0.5, 0.5](1)} {[0.4, 0.5](0.6), [0.7, 0.8](0.4)}  {[0.2, 0.3](0.4), [0.5, 0.6](0.6)} {[0.5, 0.5](1)} =  {[0.4, 0.5](1)} {[0.3, 0.4](1)} {[0.5, 0.6](1)} {[0.3, 0.5](0.6), [0.5, 0.6](0.4)} {[0.5, 0.6](1)} {[0.4, 0.5](1)} {[0.6, 0.7](1)} {[0.4, 0.5](0.4), [0.5, 0.7](0.6)} {[0.5, 0.5](1)} {[0.1, 0.2](0.3), [0.3, 0.5](0.5), [0.6, 0.7](0.2) {[0.3, 0.4](0.2), [0.5, 0.7](0.5), [0.8, 0.9](0.3)} {[0.5, 0.5](1)}

   

To get the optimal alternative, the following steps are adopted. ¯ Step 1. First of all, let k = 1 and construct the perfect multiplicative consistent P-IVHFPR R. By Definition 12, we get ˜A R 2

{0.5(1)} 6 {0.3(0.4), 0.6(0.6)} 6 =4 {0.5(1)} {0.6(1)}

{0.4(0.6), 0.7(0.4)} {0.5(1)} {0.4(1)} {0.5(0.6), 0.6(0.4)}

{0.5(1)} {0.6(1)} {0.5(1)} {0.4(0.2), 0.7(0.5), 0.9(0.3)}

647

3 {0.4(1)} 7 {0.4(0.4), 0.5(0.6)} 7 {0.1(0.3), 0.3(0.5), 0.6(0.2)} 5 {0.5(1)}

Jiuping Xu ET AL 636-655

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.4, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

˜B R 2

{0.5(1)} 6 {0.2(0.4), 0.5(0.6)} =6 4 {0.4(1)} {0.5(1)}

{0.6(1)} {0.7(1)} {0.5(1)} {0.3(0.2), 0.5(0.5), 0.8(0.3)}

{0.5(0.6), 0.8(0.4)} {0.5(1)} {0.3(1)} {0.3(0.6), 0.5(0.4)}

3 {0.5(1)} 7 {0.5(0.4), 0.7(0.6)} 7 {0.2(0.3), 0.5(0.5), 0.7(0.2)} 5 {0.5(1)}

Therefore, according to Eq.(6), we have ¯ σ(1) = R 13 σ(1)

p13

12

σ(2)

23

12

=

0.4×0.6 0.4×0.6+(1−0.4)×(1−0.6)

= 0.5

=

0.7×0.6 0.7×0.6+(1−0.7)×(1−0.6)

= 0.778

23

= 0.6

¯ σ(2) = R 13 p13

˜ σ(1) ˜ σ(1) R R 23 12 ˜ σ(1) R ˜ σ(1) +(1−R ˜ σ(1) )(1−R ˜ σ(1) ) R

σ(2)

σ(2)

˜ ˜ R 12 R23 σ(2) σ(2) ˜ σ(2) ˜ σ(2) ) ˜ ˜ R12 R23 +(1−R12 )(1−R 23

= 0.4

˜ 23 , i.e., {0.6(1)} can be regarded as {0.6(0.6), 0.6(0.4)}. So, where R ¯ 13 = {0.5(0, 6), 0.778(0.4)} R hence, ¯ 31 = {0.222(0.4), 0.5(0, 6)} R Analogously, by Eq.(6), we have  ˜ σ(s) ˜ σ(s) R R σ(s) 1 ¯ R14 = 2 ˜ σ(s) ˜ σ(s) 12 ˜ 24 σ(s)

σ(s)

˜ R12 R24 +(1−R12 )(1−R 24 )

+

˜ σ(s) R ˜ σ(s) R 13 34 σ(s) σ(s) ˜ ˜ ˜ σ(s) )(1−R ˜ σ(s) ) R R +(1−R 13

34

13



34

s = 1, 2, · · · Similar to the previous method to deal with P-IVHFE {0.6(1)}, in order to facilitate observing ˜ 34 = {0.1(0.3), 0.3(0.5), 0.6(0.2)} can be regarded as, or in other words, the probability values, R ˜ 34 = {0.1(0.3), 0.3(0.5), 0.6(0.2)} R = {0.1(0.3), 0.3(0.1), 0.3(0.2), 0.3(0.2), 0.6(0.2)} Similarly, ˜ 12 = {0.4(0.6), 0.7(0.4)} R = {0.4(0.3), 0.4(0.1), 0.4(0.2), 0.70.2), 0.7(0.2)} ˜ 24 = {0.4(0.4), 0.5(0.6)} R = {0.4(0.3), 0.4(0.1), 0.5(0.2), 0.5(0.2), 0.5(0.2)} ˜ 13 = {0.5(1)} R = {0.5(0.3), 0.5(0.1), 0.5(0.2), 0.5(0.2), 0.5(0.2)} therefore, ¯ σ(1) = R 14 = σ(1) p14

1 2 1 2



˜ σ(1) R ˜ σ(1) R 12 24 σ(1) σ(1) ˜ ˜ ˜ σ(1) )(1−R ˜ σ(1) ) R R +(1−R 12



24

12

0.4×0.4 0.4×0.4+(1−0.4)×(1−0.4)

24

+

+

˜ σ(1) R ˜ σ(1) R 13 34 σ(1) σ(1) ˜ ˜ ˜ σ(1) )(1−R ˜ σ(1) ) R R +(1−R 13

34

0.5×0.1 0.5×0.1+(1−0.5)×(1−0.1)

13



34

= 0.204 = 0.3

Similarly, we have ¯ σ(2) R 14 ¯ σ(3) R 14 σ(4) ¯ R14 ¯ σ(5) R 14

σ(2)

= 0.304, p14 = 0.1, σ(3) = 0.35, p14 = 0.2, σ(4) = 0.5, p14 = 0.2, σ(5) = 0.65, p14 = 0.2

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.4, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

thus, ¯ 14 = {0.204(0.3), 0.304(0.1), 0.35(0.2), 0.5(0.2), 0.65(0.2)} R ¯ R41 = {0.35(0.2), 0.5(0.2), 0.65(0.2), 0.696(0.1), 0.796(0.3)} Analogously, we get ¯ 24 = {0.143(0.3), 0.391(0.5), 0.692(0.2)} R ¯ 42 = {0.308(0.2), 0.609(0.5), 0.857(0.3)} R hence, 

{0.5(1)} {0.4(0.6), 0.7(0.4)} {0.3(0.4), 0.6(0.6)} {0.5(1)} {0.222(0.4), 0.5(0, 6)} {0.4(1)} {0.35(0.2), 0.5(0.2), 0.65(0.2), 0.696(0.1), 0.796(0.3)} {0.308(0.2), 0.609(0.5), 0.857(0.3)}  {0.5(0, 6), 0.778(0.4)} {0.204(0.3), 0.304(0.1), 0.35(0.2), 0.5(0.2), 0.65(0.2)}  {0.4(0.4), 0.5(0.6)} {0.143(0.3), 0.391(0.5), 0.692(0.2)}   {0.5(1)} {0.1(0.3), 0.3(0.5), 0.6(0.2) {0.4(0.2), 0.7(0.5), 0.9(0.3)} {0.5(1)}

 ¯A =  R 

In the similar way, according to Eq.(7), we can obtain  {0.5(1)} {0.5(0.6), 0.8(0.4)}  {0.2(0.4), 0.5(0.6)} {0.5(1)} ¯B =  R  {0.097(0.4), 0.3(0.6)} {0.3(1)} {0.159(0.2), 0.248(0.2), 0.35(0.2), 0.45(0.1), 0.614(0.3)} {0.155(0.2), 0.3(0.5), 0.632(0.3)}  {0.7(0.6), 0.903(0.4)} {0.386(0.3), 0.55(0.1), 0.65(0.2), 0.752(0.2), 0.841(0.2)}  {0.7(1)} {0.368(0.3), 0.7(0.5), 0.845(0.2)}   {0.5(1)} {0.2(0.3), 0.5(0.5), 0.7(0.2)} {0.3(0.2), 0.5(0.5), 0.8(0.3)} {0.5(1)} ˜ (k)A , R ¯ A ) and d(R ˜ (k)B , R ¯ B ). Step 2. Calculate the deviations d(R Using Eq.(8), we get  mij  n P n P (k)σ(s) ¯ σ(s) σ(s) P 1 A A ˜ ¯ dHam min g (R , R ) = (n−1)(n−2) − Rij pij Rij s=1 i=1 j=1  mij  4 P 4 P P σ(s) ¯ σ(s) σ(s) = 16 R − R p ij ij ij i=1 j=1 s=1

= 16 [(|0.5 − 0.5| × 0.6 + |0.5 − 0.778| × 0.4) + (|0.204 − 0.4| × 0.3 + |0.304 − 0.4| × 0.1 + |0.35 − 0.4| × 0.2 + |0.5 − 0.4| × 0.2 + |0.65 − 0.4| × 0.2) + (|0.143 − 0.4| × 0.3 + |0.391 − 0.4| × 0.1 + |0.391 − 0.5| × 0.4 + |0.692 − 0.5| × 0.2) + (|0.5 − 0.222| × 0.4 + |0.5 − 0.5| × 0.6) + (|0.35 − 0.6| × 0.2 + |0.5 − 0.6| × 0.1 + |0.65 − 0.6| × 0.2 + |0.696 − 0.6| × 0.1 + |0.796 − 0.6| × 0.3) + (|0.308 − 0.5| × 0.2 + |0.609 − 0.5| × 0.4 + |0.609 − 0.6| × 0.1 + |0.857 − 0.6| × 0.3)] = 0.8092 = 0.135 > θ0 = 0.1 6 Analogously, by Eq.(8), we can obtain ˜B , R ¯B ) = dHam min g (R

0.9152 = 0.153 > θ0 = 0.1 6

˜ A and R ˜ B are both not multiplicative consistent P-IVHFPR. R ˜ A and R ˜ B need to Therefore, R be repaired by Eqs.(10) and (11). Step 3. Repair the inconsistent multiplicative P-IVHFPR.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.4, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

Hereby we try to assign a value, such as let λ = 0.7, then 2 _(1)A

R

6 =6 4

2 _(1)B

R

6 =6 4

{0.5(1)} {0.4(0.6), 0.7(0.4)} {0.3(0.4), 0.6(0.6)} {0.5(1)} {0.294(0.4), 0.5(0, 6)} {0.4(1)} {0.423(0.2), 0.53(0.2), 0.635(0.2), 0.669(0.1), 0.745(0.3)} {0.362(0.2), 0.577(0.4), 0.606(0.1), 0.798(0.3)} 3 {0.5(0.6), 0.706(0.4)} {0.255(0.3), 0.331(0.1), 0.365(0.2), 0.47(0.2), 0.577(0.2)} 7 {0.4(0.4), 0.5(0.6)} {0.202(0.3), 0.394(0.1), 0.423(0.4), 0.638(0.2)} 7 5 {0.5(1)} {0.1(0.3), 0.3(0.5), 0.6(0.2) {0.4(0.2), 0.7(0.5), 0.9(0.3)} {0.5(1)} {0.5(1)} {0.5(0.6), 0.8(0.4)} {0.2(0.4), 0.5(0.6)} {0.5(1)} {0.157(0.4), 0.329(0.6)} {0.3(1)} {0.238(0.2), 0.315(0.2), 0.393(0.2), 0.465(0.1), 0.581(0.3)} {0.191(0.2), 0.3(0.4), 0.356(0.1), 0.594(0.3)} 3 {0.671(0.6), 0.843(0.4)} {0.419(0.3), 0.535(0.1), 0.607(0.2), 0.685(0.2), 0.762(0.2)} 7 {0.7(1)} {0.406(0.3), 0.644(0.1), 0.7(0.4), 0.809(0.2)} 7 5 {0.5(1)} {0.2(0.3), 0.5(0.5), 0.7(0.2)} {0.3(0.2), 0.5(0.5), 0.8(0.3)} {0.5(1)}

It follows that the normalized Hamming distance _ (1)A

dHam min g (R

_ (1)B

dHam min g (R _ (1)A

Let R(2)A = R

¯ A ) = 0.037 < θ0 = 0.1 ,R ¯ B ) = 0.038 < θ0 = 0.1 ,R _ (1)B

, R(2)B = R

, then we have

¯ A ) = 0.037 < 0.1 dHam min g (R(2)A , R (2)B ¯ B ) = 0.038 < 0.1 dHam min g (R ,R the normalized Hamming distances are less than the consistency level 0.1, so R(2)A and R(2)B ˜ A and R ˜ B respectively. are the repaired R ˜ (k) . Step 4. Output R The composition of R(2)A and R(2)B , i.e., 2

{[0.5, 0.5](1)} {[0.2, 0.3](0.4), [0.5, 0.6](0.6)} {[0.157, 0.294](0.4), [0.329, 0.5](0.6)} {[0.238, 0.423](0.2), [0.315, 0.53](0.2), [0.393, 0.635](0.2), [0.465, 0.669](0.1), [0.581, 0.745](0.3)} {[0.4, 0.5](0.6), [0.7, 0.8](0.4)} {[0.5, 0.671](0.6), [0.706, 0.843](0.4)} {[0.5, 0.5](1)} {[0.6, 0.7](1)} {[0.3, 0.4](1)} {[0.5, 0.5](1)} {[0.191, 0.362](0.2), [0.3, 0.577](0.4), [0.356, 0.606](0.1), [0.594, 0.798](0.3)} {[0.3, 0.4](0.2), [0.5, 0.7](0.5), [0.8, 0.9](0.3)}3 {[0.255, 0.419](0.3), [0.33, 0.535](0.1), [0.365, 0.607](0.2), [0.47, 0.685](0.2), [0.577, 0.762](0.2)} 7 {[0.202, 0.406](0.3), [0.394, 0.644](0.1), [0.423, 0.7](0.4), [0.638, 0.809](0.2)} 7 5 {[0.1, 0.2](0.3), [0.3, 0.5](0.5), [0.6, 0.7](0.2) {[0.5, 0.5](1)}

6 ˜ (2) = 6 R 4

˜ is the repaired multiplicative P-IVHFPR of R. Step 5. The last step is to sort the four schemes (alternatives). ˜ (2) ≥ R ˜ (2) ), then we get the following complementary matrix: Using Definition 8, let pij = p(R ij ji   0.5 1 1 0.454  0 0.5 1 0.554   P =  0 0 0.5 0  0.546 0.446 1 0.5 If critical value λ is allowed to be an appropriate value, such as a value between the largest and the second largest value of pij , i, j = 1, 2, 3, 4 (not including 1), e.g., λ = 0.55, and  1, if pij ≥ λ, p˜ij = 0, if pij < λ

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then further we  0  0 P˜ =   0 0

get 1 0 0 0

1 1 0 1

 0 1   0  0

According to p˜, we have x1  x2 , x1  x3 , x2  x3 , x2  x4 , x4  x3 namely, x1  x2  x4  x3 which indicates that the first scheme is the most desirable according to the opinion of the large consultancy firm. 5.2. Discussion and Comparison Having carefully analyzed the calculation process and results, the following conclusions can be drawn: (1) Since there are probability values in a probability-interval valued hesitant fuzzy environment, after the multiplications, there are decimals which are not the integer multiples of 0.1 in the calculated results, such as 0.696, 0.857, 0.391 · · · . If one searches the relevant documents on interval-valued preference relations, it can be found that this inevitably happens in the calculation process. Therefore, future research could try and explain this phenomenon. (2) It can be seen that there are overlapping intervals in the calculated results. Such as a P-IVHFE, ˜ (2) = {[0.238, 0.423](0.2), [0.315, 0.53](0.2), [0.393, 0.635](0.2), [0.465, 0.669](0.1), [0.581, 0.745](0.3)} R 12

where between the intervals [0.238, 0.423] and [0.315, 0.53], there is an overlapping interval [0.315, 0.423]. To deal with this problem, without a loss of generality, it is assumed that all the interval values have a uniform distribution, then they can be changed into an equivalent ˜ (2) , we have expression in which the intervals are not overlapping. For example, as for R 12 [0.238, 0.423](0.2) 0.315−0.238 ={[0.238, 0.315](0.2 × 0.423−0.238 ), [0.315, 0.393](0.2 × 0.393−0.315 0.423−0.238 ), [0.393, 0.423](0.2 × = {[0.238, 0.315](0.083), [0.315, 0.393](0.084), [0.393, 0.423](0.033)}

0.423−0.393 0.423−0.238 )}

In a similar way, we get [0.315, 0.53](0.2) = {[0.315, 0.393](0.073), [0.393, 0.465](0.067), [0.465, 0.53](0.060)} [0.393, 0.635](0.2) = {[0.393, 0.465](0.059), [0.465, 0.581](0.096), [0.581, 0.635](0.045)} [0.465, 0.669](0.1) = {[0.465, 0.581](0.057), [0.581, 0.669](0.043)} therefore, ˜ (2) = {[0.238, 0.315](0.083), [0.315, 0.393](0.084 + 0.073), [0.393, 0.465](0.033 + 0.067 + 0.059), R 12 [0.465, 0.581](0.060 + 0.096 + 0.057), [0.581, 0.745](0.045 + 0.043 + 0.3)} = {[0.238, 0.315](0.083), [0.315, 0.393](0.157), [0.393, 0.465](0.159), [0.465, 0.581](0.213), [0.581, 0.745](0.388)} There are not any overlapping intervals in this new expression.

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It can be seen from the above example that P-IVHFPRs are useful in resolving large GDM problems, because they express intuitively the uncertain and hesitant preference information provided by each DM in a decision-making organization. This differs from the approach of interval-valued fuzzy sets for GDM, where the opinions of the DMs based on a pairwise comparison of alternatives, are first aggregated and, correspondingly, only the average interval-valued preference information is obtained. However, the use of, P-IVHFPRs does not need to perform such an aggregation and, hence, provides a more comprehensive description of the opinions of these DMs [2]. In the above example, if the probability-interval valued preference information is firstly aggregated at the beginning of the calculation, with regard to the probability values as the corresponding weights, using Definition 8, e.g., ˜ 12 ) = ({[0.4, 0.5](0.6), [0.7, 0.8](0.4)}) s(h = [0.4 × 0.6 + 0.7 × 0.4, 0.5 × 0.6 + 0.8 × 0.4] = [0.52, 0.62] then we get ˜ ˜ = (h R  ij )4×4

{[0.5, 0.5](1)} {[0.4, 0.5](0.6), [0.7, 0.8](0.4)}  {[0.2, 0.3](0.4), [0.5, 0.6](0.6)} {[0.5, 0.5](1)} =  {[0.4, 0.5](1)} {[0.3, 0.4](1)} {[0.5, 0.6](1)} {[0.3, 0.5](0.6), [0.5, 0.6](0.4)} {[0.5, 0.6](1)} {[0.4, 0.5](1)} {[0.6, 0.7](1)} {[0.4, 0.5](0.4), [0.5, 0.7](0.6)} {[0.5, 0.5](1)} {[0.1, 0.2](0.3), [0.3, 0.5](0.5), [0.6, 0.7](0.2) {[0.3, 0.4](0.2), [0.5, 0.7](0.5), [0.8, 0.9](0.3)}  {[0.5, 0.5](1)} [0.5, 0.5] [0.52, 0.62] [0.5, 0.6] [0.4, 0.5]  [0.38, 0.48] [0.5, 0.5] [0.6, 0.7] [0.46, 0.62]  ˜ ij )]4×4 =   → [s(h  [0.4, 0.5] [0.3, 0.4] [0.5, 0.5] [0.3, 0.45]  [0.5, 0.6] [0.38, 0.54] [0.55, 0.7] [0.5, 0.5] Further, in the same way mentary matrix is obtained:  0.5 1 1  0 0.5 1 P0 =   0 0 0.5 1 0.25 1

˜ ij ) ≥ s(h ˜ ji )), then the following compleas before, let pij = (s(h  0 0.75   0  0.5

indicating that x1  x2 , x1  x3 , x2  x3 , x2  x4 , x4  x1 , x4  x3 which is heavily inconsistent. Let  1, if p0 ij ≥ λ, 0 p˜ij = 0, if p0 ij < λ Only when we let critical value λ > 0.75, can a consistent result be obtained. At this time,   0 1 1 0  0 0 1 0   P˜ 0 =   0 0 0 0  1 0 1 0 which indicates that x1  x2 , x1  x3 , x2  x3 , x4  x1 , x4  x3

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namely, x4  x1  x2  x3 From the results of the calculations, one can find a difference in the ranking results derived in these two approaches. The reason is that for each decision-making organization composed of multiple DMs, a group’s preference value is obtained by aggregating (namely, averaging) individual preference values. Such an aggregation actually amounts to implementing a transformation of P-IVHFEs into an interval-valued fuzzy number. As a result, it leads to the loss of information, which affects the final ranking results. Thus, the comparison clearly shows the benefits of the proposed GDM approach based on P-IVHFPRs [2]. Compared with that in a hesitant fuzzy environment, this method’s implementation could be far more sophisticated in a probability-interval valued hesitant fuzzy environment, but has led to some new problems. For example, in order to get the equivalent expression in which the intervals are not overlapping, it is assumed that all the interval values have a uniform distribution. If they are not have a uniform distribution, but some other type, e.g., a normal distribution, it is not known what would happen. Therefore this should be a topic for future research. In spite of what has been mentioned above, compared with P-HFSs, IVHFSs and a possibilityhesitant fuzzy linguistic term set, P-IVHFSs can describe the actual preferences of decisionmakers and better reflect their uncertainty, hesitancy, and inconsistency, and thus enhance the modeling abilities of HFSs. The proposed method using P-IVHFSs has the following advantages. First, compared with P-HFSs, P-IVHFSs can better depict uncertainty. Second, compared with IVHFSs, P-IVHFSs can depict hesitancy more accurately and differentiate intervals according to their possibilities. Third, compared with a possibility-hesitant fuzzy linguistic term set, P-IVHFSs can express the evaluation information more flexibly. Possibility-hesitant fuzzy linguistic term sets can therefore be regarded as a special case of P-IVHFSs. Although the representation of P-IVHFSs looks complex, they can depict fuzzy information clearly and retain the completeness of original data or the inherent thoughts of decision-makers, which is a prerequisite of guaranteeing the accuracy of final outcomes. Additionally, as far as the applicability of P-IVHFSs is concerned, decision-makers can make a trade-off between the features of P-IVHFSs and the relative computational cost. Moreover, the complexity and amount of computation can be clearly reduced with the assistance of programming software [17]. 6. Conclusion In this paper, P-HFSs and IVHFSs have been extended to P-IVHFSs. As an important tool in GDM, P-IVHFSs can describe the actual preferences of decision-makers and better reflect their uncertainty, hesitancy, and inconsistency, and thus enhance the modeling abilities of HFSs. Based on related research, a decomposition method has been proposed to deal with the consistency of P-IVHFPRs. A simulated example has also been provided to illustrate the use of the proposed approach. The main contributions of this paper are summarized as follows. (1) The concept of P-IVHFSs has been defined and some desirable properties of P-IVHFSs have been discussed. P-IVHFSs are a natural development to manage the possible preferences in decision making following the introduction of P-HFSs and IVHFSs. (2) P-IVHFPRs have been proposed and the consistency of P-IVHFPRs has been discussed, using the multiplicative transitivity to verify the consistency of a P-IVHFPR. Moreover, a decomposition method has been proposed to deal with the consistency of P-IVHFPRs. (3) Based on the multiplicative consistency of hesitant fuzzy preference relations, an iterative algorithm has been proposed for improving the consistency of P-IVHFPR. In future research, the developed theoretical structure could be extended to the probability distributions of preferences on the intervals. Another potential area of research would be to analyze the hesitant fuzzy information in P-IVHFPRs.

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References [1] Ai, F.Y., Yang, J.Y., Zhang, P.D.: An approach to multiple attribute decision making problems based on hesitant fuzzy set. J. Intell. Fuzzy Syst. 27(6), 2749-2755 (2014) [2] Chen, N., Xu, Z.S., Xia, M.M.: Interval-valued hesitant preference relations and their applications to group decision making. Knowledge-Based Syst. 37, 528-540 (2013). [3] Chen, N., Xu, Z.S., Xia, M.M.: Correlation coefficients of hesitant fuzzy sets and their applications to clustering analysis. Appl. Math. Model. 37, 2197-2211 (2013) [4] Chiclana, F., Herrera-Viedma, E., Alonso, S., Herrera, F.: Cardinal consistency of reciprocal preference relations: a characterization of multiplicative transitivity. IEEE Trans. Fuzzy Syst. 17, 14-23 (2009) [5] Chiclana, F., Mata, F., Alonso, S., Herrera-viedma, E., Mart´ınez, L.: Group decision making: From consistency to consensus. Lect. Note. Comput. Sci. 2(2), 80-91 (2007) [6] Farhadinia, B.: A series of score functions for hesitant fuzzy sets. Inf. Sci. 277, 102-110 (2014) [7] Farhadinia, B.: Information measures for hesitant fuzzy sets and interval-valued hesitant fuzzy sets. Inf. Sci. 240, 129-144 (2013) [8] Liao, H.C., Xu, Z.S.: A VIKOR-based method for hesitant fuzzy multi-criteria decision making. Fuzzy Optim. Decis. Mak. 12(4), 373-392 (2013) [9] Liao, H.C., Xu, Z.S., Xia, M.M.: Multiplicative consistency on hesitant fuzzy preference relation and its application in group decision making. Int. J. Inf. Tech. Decis. 13 (1), 47-76 (2014) [10] Orlovsky, S.A.: Decision-making with a fuzzy preference relation. Fuzzy Sets Syst. 1, 155-167 (1978) [11] Peng, D.H., Gao, C.Y., Gao, Z.F.: Generalized hesitant fuzzy synergetic weighted distance measures and their application to multiple criteria decision-making. Appl. Math. Model. 37, 5837-5850 (2013) [12] Rodr´ıguez, Mart´ınez, L., Herrera, F.: Hesitant fuzzy linguistic term sets for decision making. IEEE Trans. Fuzzy Syst. 20(1), 109-119 (2012) [13] Saaty, T.L.: The Analytic Hierarchy Process. McGraw-Hill, New York, 1980. [14] Tanino, T.: Fuzzy preference relation in group decision making. Springer Berlin Heidelberg. 301, 54-71 (1988) [15] Torra, V.: Hesitant fuzzy sets. Int. J. Intell. Syst. 25(6), 529-539 (2010) [16] Torra, V., Narukawa, Y.: On hesitant fuzzy sets and decision. In: IEEE International Conference on Fuzzy Systems 1378-1382 (2009) [17] Wang, J.Q., Wu, J.T., Wang, J., Zhang, H.Y., Chen, X.H.: Interval-valued hesitant fuzzy linguistic sets and their applications in multi-criteria decision-making problems. Inf. Sci. 288, 55-72 (2014) [18] Wei, G.W. Hesitant fuzzy prioritized operators and their application to multiple attribute decision making. Knowledge-Based Syst. 31, 176-182 (2012) [19] Wu, Z.B., Xu, J.P.: Possibility distribution-based approach for MAGDM with hesitant fuzzy linguistic information. IEEE Trans. Cybernetics 46 (3), 694-705 (2016) [20] Wu, Z.B., Xu, J.P.: Managing consistency and consensus in group decision making with hesitant fuzzy linguistic preference relations. Omega 65, 28-40 (2016) [21] Xia, M.M., Xu, Z.S.: Studies on the aggregation of intuitionistic fuzzy and hesitant fuzzy information. Technical Report (2011) [22] Xia, M.M., Xu, Z.S.: Hesitant fuzzy information aggregation in decision making, Int. J. Approx. Reason. 52, 395-407 (2011) [23] Xia, M.M., Xu, Z.S.: On distance and correlation measures of hesitant fuzzy information. Int. J. Intell. Syst. 26(5), 410-425 (2011) [24] Xia, M.M., Xu, Z.S., Chen, N.: Some hesitant fuzzy aggregation operators with their application in group decision making. Group Decis. Negotiation 22(2), 259-279 (2013) [25] Xu, K., Zhou, J.Z., Gu, R., Qin, H.: Approach for aggregating interval-valued intuitionistic fuzzy information and its application to reservoir operation. Expert Syst. Appl. 38, 9032-9035 (2011) [26] Xu, Y.J., Herrera, F., Wang, H.M.: A distance-based framework to deal with ordinal and additive inconsistencies for fuzzy reciprocal preference relations. Inf. Sci. 328, 189-205 (2016) [27] Xu, Z.S.: Hesitant Fuzzy Sets Theory, Studies in Fuzziness and Soft Computing. Springer International Publishing Switzerland. 314, (2014) [28] Xu, Z.S.: On compatibility of interval fuzzy preference matrices. Fuzzy Optim. Decis. Mak. 3, 217-225 (2004) [29] Xu,Z.S., Da, Q.L.: The uncertain OWA operator. Int. J. Intell. Syst. 17, 569-575 (2002) [30] Xu, Z.S., Xia, M.M.: Hesitant fuzzy entropy and cross-entropy and their use in multiattribute decision making. Int. J. Intell. Syst. 27, 799-822 (2012)

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[31] Xu, Z.S., Xia, M.M.: Distance and similarity measures for hesitant fuzzy sets. Inf. Sci. 181, 2128-2138 (2011) [32] Yu, D.J.: Some hesitant fuzzy information aggregation operators based on Einstein operational laws. IEEE Trans. Fuzzy Syst. 29, 320-340 (2014) [33] Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning-I. Inf. Sci. 8, 199-249 (1975) [34] Zadeh, L.A.: Fuzzy sets. Inf. Control 8(3), 338-353 (1965) [35] Zhang, Y.X., Xu, Z.S., Wang, H., Liao, H.C.: Consistency-based risk assessment with probabilistic linguistic preference relation. Appl. Soft Comput. 49 (2016) 817-833. [36] Zhang, Z.M.: Hesitant fuzzy power aggregation operators and their application to multiple attribute group decision making. Inf. Sci. 234, 150-181 (2013) [37] Zhu, B., Xu, Z.S.: Probability-hesitant fuzzy sets and the presentation of preference relations. Technological and Economic Development of Economy. In press. [38] Zhu, B., Xu, Z.S.: Consistency measures for hesitant fuzzy linguistic preference relations. IEEE Trans. Fuzzy Syst. 24(1), 72-85 (2014) [39] Zhu, B., Xu, Z.S., Xu, J.P.: Deriving a ranking from hesitant fuzzy preference relations under group decision making. IEEE Trans. Cybernetics 44(8), 1328-1337 (2014)

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Dynamics and Solutions of Some Recursive Sequences of Higher Order Asim Asiri1 and E. M. Elsayed1,2 1

King Abdulaziz University, Faculty of Science, Mathematics Department, P. O. Box 80203, Jeddah 21589, Saudi Arabia. 2 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt. E-mail: [email protected], [email protected]. ABSTRACT In this article we study the existence of solutions and some of their qualitative behavior of the following rational nonlinear difference equation axn−(2k+1) , n = 0, 1, ..., xn+1 = b + cxn−k xn−(2k+1) where a, b and c are real numbers, k is a non-negative integer number and the initial conditions x−2k−1 , x−2k , ..., x−1 , x0 are arbitrary non-negative real numbers. Also, the solutions of some special cases of the equation under consideration will be obtained. Keywords: recursive sequence, periodicity, solutions of difference equations. Mathematics Subject Classification: 39A10 ––––––––––––––––––––––

1. INTRODUCTION During the last decade, the research on difference equations has been increasing. The fact that difference equations demonstrate themselves as mathematical models representing some real life phenomena is a significant reason of this concern. For example, the are used in probability theory,economics, genetics in biology, geometry, electrical network, quanta in radiation, psychology, sociology, etc. Actually, no doubt that the difference equations play and will play a remarkable role in applicable analysis and in mathematics generally. Recently, many authors’ attention was on studying the global attractivity, boundedness character, periodicity and the solution form of nonlinear difference equations. Now, we write some results in this area: Cinar [3—4] obtained the solutions of the following difference equations xn+1 =

xn−1 , 1 + xn xn−1

xn+1 =

xn−1 . −1 + xn xn−1

Cinar et al. [5] discussed the solutions and attractivity of the difference equation xn+1 =

xn−3 −1+xn xn−1 xn−2 xn−3 .

Elabbasy et al. [8—9] looked at the global stability, periodicity character and derive the solution of some special cases of the following difference equations xn+1 = axn −

bxn αxn−k , xn+1 = . Qk cxn − dxn−1 β + γ i=0 xn−i

Elsayed [13] examined the behavior and found the form of solution of the nonlinear difference equation xn+1 = axn−1 +

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In [2], Belhannache et al. investigated the global behavior of the solutions of the difference equation xn+1 =

A + Bxn−2k−1 . Q i C + D ki=1 xm n−2i

Karatas et al. [29] achieved the solution of the following difference equation xn+1 =

axn−(2k+2) . Q −a + 2k+2 i=0 xn−i

In [35] Simsek and Abdullayev found the solution of the recursive sequence xn+1 =

1+

xn−(4k+3) 2 t=0 xn−(k+1)t−k

.

Other related results on rational difference equations can be found in the references. [1-52]. Our aim in this paper is to investigate the dynamics of the solution of the following nonlinear difference equation of higher order axn−(2k+1) , n = 0, 1, ..., (1) xn+1 = b + cxn−k xn−(2k+1) where a, b and c are real numbers, k a is non negative integer number and the initial conditions x−2k−1 , x−2k , ..., x−1 , x0 are arbitrary non-negative real numbers. Also, we obtain the solutions of some special cases of Eq.(1). Suppose that I is an interval of real numbers and let f : I k+1 → I, be a continuously differentiable function. Then for every set of initial conditions x−k , x−k+1 , ..., x0 ∈ I, the difference equation xn+1 = f (xn , xn−1 , ..., xn−k ), n = 0, 1, ...,

(2)

has a unique solution {xn }∞ n=−k .

Definition 1. (Equilibrium Point) A point x ∈ I is called an equilibrium point of Eq.(2) if x = f (x, x, ..., x). That is, xn = x for n ≥ 0, is a solution of Eq.(2), or equivalently, x is a fixed point of f . Definition 2. (Periodicity) A sequence {xn }∞ n=−k is said to be periodic with period p if xn+p = xn for all n ≥ −k.

Definition 3. (Stability)

(i) The equilibrium point x of Eq.(2) is locally stable if for every x−k , x−k+1 , ..., x−1 , x0 ∈ I with

> 0, there exists δ > 0 such that for all

|x−k − x| + |x−k+1 − x| + ... + |x0 − x| < δ, we have |xn − x|
0, such that for all x−k , x−k+1 , ..., x−1 , x0 ∈ I with |x−k − x| + |x−k+1 − x| + ... + |x0 − x| < γ, we have limn→∞ xn = x. (iii) The equilibrium point x of Eq.(2) is a global attractor if for all x−k , x−k+1 , ..., x−1 , x0 ∈ I, we have limn→∞ xn = x. (iv) The equilibrium point x of Eq.(2) is globally asymptotically stable if x is locally stable, and x is also a global attractor of Eq.(2).

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(v) The equilibrium point x of Eq.(2) is unstable if x is not locally stable. The linearized equation of Eq.(2) about the equilibrium x is the linear difference equation yn+1 =

k X ∂f (x, x, ..., x)

∂xn−i

i=0

yn−i .

(3)

Theorem A [32]: Assume that pi ∈ R, i = 1, 2, ..., k and k ∈ {0, 1, 2, ...}. Then k X i=1

|pi | < 1,

is a sufficient condition for the asymptotic stability of the difference equation xn+k + p1 xn+k−1 + ... + pk xn = 0, n = 0, 1, ... .

2. DYNAMICS OF SOLUTIONS OF EQ.(1) In this section we look at some qualitative behavior of Eq.(1) such as local stability, periodicity and boundedness character of solutions of Eq.(1) when the constants a, b and c are positive real numbers.

2.1. Local Stability of the Equilibrium Points We now investigate the local stability character of the solutions of Eq.(1). The equilibrium points of Eq.(1) are given by the relation x = x=0

or

x=

r

ax , b+cx2

which gives

a−b . c

Note that if a > b, then Eq.(1) has a unique positive equilibrium point. Let f : (0, ∞)2 −→ (0, ∞) be a function defined by f (u, v) = Therefore it follows that

ab ∂f (u, v) = , ∂u (b + cuv)2

au . b + cuv

(4)

−acu2 ∂f (u, v) = . ∂v (b + cuv)2

Theorem 2.1. The following statements are true: (1) If a ≤ b, then x = 0 is the only equilibrium point of Eq.(1) and it is locally stable. r a−b of Eq.(1) are unstable. (2) If a > b, then the equilibrium points x = 0 and x = c Proof. (1) If a ≤ b, then we see from Eq.(4) that a ∂f (0, 0) = , ∂u b

∂f (0, 0) = 0. ∂v

Then the linearized equation associated with Eq.(1) about x = 0 is a yn+1 − yn−2k−1 = 0, b and whose characteristic equation is λ2k+2 −

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(5)

a = 0. b

(6)

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It follows by Theorem A that, Eq.(5) is asymptotically stable. Then the equilibrium point x = 0 of Eq.(1) is locally stable. (2) Assume r that a > b. (i) At x = 0 it follows again from Eq.(6) and Theorem A that x = 0 is unstable. a−b we see from Eq.(4) that (ii) At x = c b ∂f (x, x) = , ∂u a Then the linearized equation of Eq.(1) about x = yn+1 +

r

∂f (x, x) −(a − b) = . ∂v a a−b is c

a−b b yn−k − yn−2k−1 = 0, a a

(7)

and whose characteristic equation is λ2k+2 +

a − b k+1 b λ − = 0. a a

(8)

b . Then it follows by Theorem A that the equilibrium point x = a Eq.(1) is unstable. The proof is complete.

Therefore λk+1 = −1 or λk+1 =

r

a−b of c

2.2. Existence of Period (2k+2) Solutions In this section we look at the existence of period (2k + 2) solutions of Eq.(1). Remark: The initial values {x−2k−1 , x−2k , x−2k+1 , ..., x−1 , x0 } of Eq.(1) have not to be equal zero at the same time, otherwise Eq.(1) will have only the zero solution. In the sequel we assume that any element of the set {x−2k−1 , x−2k , x−2k+1 , ..., x−1 , x0 } doesn’t equal zero.

Theorem 2.2. Eq.(1) has positive prime period (2k + 2) solutions if and only if (b + cAi − a) = 0,

(9)

where Ai = x−k+i x−2k−1+i (f or i = 0, 1, 2, ..., k) and Ak+1+i = Ai . Proof. Firstly, we suppose that there exists a prime period (2k + 2) solution of Eq.(1) of the form ..., x−2k−1 , x−2k , x−2k+1 , ..., x−1 , x0 , x−2k−1 , x−2k , x−2k+1 , ..., x−1 , x0 , ... . That is xN +1 = xN −2k−1 f or N ≥ 0. We now will show that (9) holds. We see from Eq.(1) that ax−2k−1 ax−2k ax−2k+1 , x−2k = x2 = , x−2k+1 = x3 = , ..., x−2k−1 = x1 = b + cA0 b + cA1 b + cA2 ax−k−2 ax−k−1 x−k−2 = xk = , x−k−1 = xk+1 = , b + cAk−1 b + cAk ax−k ax−k ax−2 ax−2 x−k = xk+2 = = , ..., x−2 = x2k = = , b + cAk+1 b + cA0 b + cA2k−1 b + cAk−2 ax−1 ax−1 ax0 ax0 x−1 = x2k+1 = = , x0 = x2k+2 = = . b + cA2k b + cAk−1 b + cA2k+1 b + cAk Then it is easy to see that x−2k−1 (b + cA0 ) x−2k (b + cA1 ) x−2k+1 (b + cA2 ) x−1 (b + cAk−1 ) x0 (b + cAk )

= = = = =

ax−2k−1 ⇒ x−2k−1 (b + cA0 − a) = 0, ax−2k ⇒ x−2k (b + cA1 − a) = 0, ax−2k+1 ⇒ x−2k+1 (b + cA2 − a) = 0, ..., ax−1 ⇒ x−1 (b + cAk−1 − a) = 0, ax0 ⇒ x0 (b + cAk − a) = 0.

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Since xj 6= 0 for all −(2k + 1) ≤ j ≤ 0, then Condition (9) is satisfied. Secondly, we suppose that (9) is true. We will prove that Eq.(1) has a prime period (2k + 2) solution. It follows from Eq.(1) and Eq.(9) that ax−2k−1 ax−2k ax−2k+1 = x−2k−1 , x2 = = x−2k , x3 = = x−2k+1 , ..., x1 = b + cA0 b + cA1 b + cA2 ax−k−2 ax−k−1 ax−k xk = = x−k−2 , xk+1 = = x−k−1 , xk+2 = = x−k , ..., b + cAk−1 b + cAk b + cAk+1 ax−2 ax−1 ax0 x2k = = x−2 , x2k+1 = = x−1 , x2k+2 = = x0 , b + cA2k−1 b + cA2k b + cA2k+1 which completes the proof.

2.3. Boundedness and Global Stability of Solutions Here we examine the boundedness nature of the solutions of Eq.(1). In addition, we deal with the global stability of the equilibrium point x = 0. Theorem 2.3. Every solution of Eq.(1) is bounded. Proof. Let {xn }∞ n=−2k−1 be a solution of Eq.(1), we have to look at the following two cases

(1) If a ≤ b. It follows from Eq.(1) that axn−(2k+1) axn−(2k+1) ≤ xn−(2k+1) . ≤ xn+1 = b + cxn−k xn−(2k+1) b

∞ }∞ {x(2k+2)n }∞ Then the subsequences {x(2k+2)n−2k−1 }∞ n=0 , {x(2k+2)n−2k n=0 , ..., {x(2k+2)n−1 }n=0 , r n=0 are decreas¾ ½ a . ing and so are bounded from above by M = max x−2k−1 , x−2k , x−2k+1 , ..., x−1 , x0 , c (2) If a > b. For the sake of contradiction, we suppose that there exists a subsequence {x(2k+2)n−2k−1 }∞ n=0 and it is not bounded from above. Then we obtain from Eq.(1), for sufficiently large n, that ax(2k+2)n−(2k+1) ∞ = lim x(2k+2)n+1 = lim n→∞ n→∞ b + cx(2k+2)n−k x(2k+2)n−(2k+1) ax(2k+2)n−(2k+1) a < lim = lim . (10) n→∞ cx(2k+2)n−k x(2k+2)n−(2k+1) n→∞ cx(2k+2)n−k

It follows that the limit of the right hand side of (10) is bounded which is a contradiction, and so the proof of the theorem is complete. Theorem 2.4. If a ≤ b, then every solution of Eq.(1) converges to the equilibrium point x = 0. Proof. It was shown in Theorem 2.1 that x = 0 is local stable and then it suffices to show that x = 0 is global attractor of the solutions of Eq.(1). ∞ ∞ ∞ We claim that each one of the subsequences {x(2k+2)n−2k−1 }∞ n=0 , {x(2k+2)n−2k }n=0 , ..., {x(2k+2)n−1 }n=0 , {x(2k+2)n }n=0 ∞ has limit equal to zero. For the sake of contradiction, suppose that there exists a subsequence {x(2k+2)n−2k−1 }n=0 with limit doesn’t zero. Now we see from Eq.(1) that bx(2k+2)n+1 + cx(2k+2)n−k x(2k+2)n−(2k+1) = ax(2k+2)n−(2k+1) , or x(2k+2)n−(2k+1) =

bx(2k+2)n+1 . a − cx(2k+2)n+1 x(2k+2)n−k

Now it follows from the boundedness of the solution that bx(2k+2)n+1 bM < < 0, lim x(2k+2)n−(2k+1) = lim n→∞ n→∞ a − cx(2k+2)n+1 x(2k+2)n−k a − cM 2 r a where M ≥ which is a contradiction and this completes the proof of the theorem. c

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Numerical Examples For confirming the results of this section, we present some numerical examples which show the behavior of solutions of Eq.(1). See Figures 1, 2 and 3 below.

plot of x(n+1)= (ax(n−(2k+1))/(b+cx(n−k)*x(n−(2k+1))) 14

12

10

x(n)

8

6

4

2

0

0

10

20

30

40

50 n

60

70

80

90

100

Figure 1: a = 3, b = 2, c = 5, k = 2, x−5 = 0.4, x−4 = 0.2, x−3 = 13, x−2 = 9, x−1 = 7, x0 = 5. plot of x(n+1)= (ax(n−(2k+1))/(b+cx(n−k)*x(n−(2k+1))) 10

8

6

x(n)

4

2

0

−2

−4

−6

0

5

10

15

20

25 n

30

35

40

45

50

Figure 2: a = 10, b = 6, c = 2, k = 3, x−7 = 4, x−6 = 7, x−5 = 2/9, x−4 = −6, x−3 = 0.5, x−2 = 2/7, x−1 = 9, x0 = −2/6.

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plot of x(n+1)= (ax(n−(2k+1))/(b+cx(n−k)*x(n−(2k+1))) 9

8

7

6

x(n)

5

4

3

2

1

0

0

5

10

15

20

25 n

30

35

40

45

50

Figure 3: a = 3, b = 7, c = 9, k = 2, x−5 = 4, x−4 = 1.7, x−3 = 3, x−2 = 1.9, x−1 = 9, x0 = 3.

3. THE SOLUTIONS FORM OF SOME SPECIAL CASES OF EQ.(1) Our goal in this section is to find a specific form of the solutions of some special cases of Eq.(1) and give numerical examples in each case when the constants a, b and c are integer numbers.

3.1. On the Difference Equation xn+1 =

xn−(2k+1) −1 + xn−k xn−(2k+1)

In this section we obtain the solution of the following equation xn+1 =

xn−(2k+1) , n = 0, 1, ..., −1 + xn−k xn−(2k+1)

(11)

where the initial values are arbitrary non zero real numbers with x−k+i x−2k−1+i 6= 1 (f or i = 0, 1, 2, ..., k).

Theorem 3.1. Let {xn }∞ n=−2k−1 be a solution of Eq.(11). Then for n = 1, 2, ... x(2k+2)n−2k−1

=

x(2k+2)n−2k+1

=

x(2k+2)n−k−1

=

x(2k+2)n−k+1 x(2k+2)n−1

= =

x−2k−1 x−2k x(2k+2)n−2k = , n, (−1 + x−k x−2k−1 ) (−1 + x−k+1 x−2k )n x−2k+1 , ..., (−1 + x−k+2 x−2k+1 )n x−k−1 x(2k+2)n−k = x−k (−1 + x−k x−2k−1 )n , n, (−1 + x0 x−k−1 ) n x−k+1 (−1 + x−k+1 x−2k ) , ..., x−1 (−1 + x−1 x−k−2 )n , x(2k+2)n = x0 (−1 + x0 x−k−1 )n .

Proof: For n = 1 the result holds. Now suppose that n > 1 and that our assumption holds for n − 1. That is; x−2k−1

x(2k+2)n−4k−3

=

n−1 ,

x(2k+2)n−4k−2 =

x(2k+2)n−4k−1

=

(−1 + x−k x−2k−1 ) x−2k+1

x(2k+2)n−3k−3

=

x−k−1

x(2k+2)n−3k−1

= x−k+1 (−1 + x−k+1 x−2k )n−1 , ...,

(−1 + x−k+2 x−2k+1 )n−1 (−1 + x0 x−k−1 )

n−1 ,

x−2k

(−1 + x−k+1 x−2k )

n−1 ,

, ...,

x(2k+2)n−3k−2 = x−k (−1 + x−k x−2k−1 )

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x(2k+2)n−2k−3 = x−1 (−1 + x−1 x−k−2 )n−1 ,

x(2k+2)n−2k−2 = x0 (−1 + x0 x−k−1 )n−1 .

Now, it follows from Eq.(11) that x(2k+2)n−2k−1

=

=

x(2k+2)n−(4k+3) −1 + x(2k+2)n−3k−2 x(2k+2)n−(4k+3) x−2k−1 (−1 + x−k x−2k−1 )

x−2k−1

n−1

= x−2k−1 −1+x−k (−1+x−k x−2k−1 )n−1 n (−1 + x−k x−2k−1 ) − 1

Hence, we have x(2k+2)n−2k−1 =

n−1

(−1 + x−k x−2k−1 ) −1 + x−k x−2k−1

.

x−2k−1 n. (−1 + x−k x−2k−1 )

Also, we see from Eq.(11) that x(2k+2)n−k−1

=

x(2k+2)n−(3k+3) −1 + x(2k+2)n−2k−2 x(2k+2)n−(3k+3) x−k−1 (−1 + x0 x−k−1 )

=

−1+x0 (−1+x0 x−k−1 )n−1

Thus x(2k+2)n−k−1 =

x−k−1

n−1

x−k−1

(−1 + x0 x−k−1 )

= n−1

(−1 + x0 x−k−1 ) 1 + x0 x−k−1

n−1

.

x−k−1 . (−1 + x0 x−k−1 )n

Similarly x(2k+2)n−1

= =

x(2k+2)n−(2k+3) = −1 + x(2k+2)n−k−2 x(2k+2)n−(2k+3) x−1 (−1+x−1 x−k−2 )n−1 x−1 x−k−2 −1+ (−1 + x−1 x−k−2 )

x−1 (−1+x−1 x−k−2 )n−1

x−k−2 x−1 (−1+x−1 x−k−2 )n−1 −1+ (−1 + x−1 x−k−2 )n ³ ´ −1+x−1 x−k−2 n−1 (−1 + x−1 x−k−2 ) . −1+x−1 x−k−2 = x−1 (−1 + x−1 x−k−2 )

Then, we get

n

x(2k+2)n−1 = x−1 (−1 + x−1 x−k−2 ) . Similarly, one can obtain the other relations. Thus, the proof is completed. x 2 Note that the equilibrium points of Eq.(11) are given by the equation x = 2 . Then we have x(x −2) = −1 + x √ √ 0. Thus Eq.(11) has the equilibrium points 0, 2, − 2.

Theorem 3.2. The following statements are true:

(a) If x−k+i x−2k−1+i 6= 2 (f or i = 0, 1, 2, ..., k), then all the solutions of Eq.(11) are unbounded.

(b) Eq.(11) has a periodic solutions of period (2k + 2) iff x−k+i x−2k−1+i = 2 (f or i = 0, 1, 2, ..., k) and will be take the form {x−2k−1 , x−2k , ..., x−1 , x0 , x−2k−1 , x−2k , ..., x−1 , x0 , ...}. Proof: (a) The proof in this case follows directly from the form of the solution as given in Theorem 3.1. (b) First suppose that there exists a prime period (2k + 2) solution of Eq.(11) of the form x−2k−1 , x−2k , ..., x−1 , x0 , x−2k−1 , x−2k , ..., x−1 , x0 , ... .

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Then we see from the form of solution of Eq.(11) that x−2k−1

=

x−2k+1

=

x−k−1

=

x−k+1 x−1

= =

x−2k−1 x−2k x−2k = , n, (−1 + x−k x−2k−1 ) (−1 + x−k+1 x−2k )n x−2k+1 , ..., (−1 + x−k+2 x−2k+1 )n x−k−1 , x−k = x−k (−1 + x−k x−2k−1 )n , (−1 + x0 x−k−1 )n n x−k+1 (−1 + x−k+1 x−2k ) , ..., n n x−1 (−1 + x−1 x−k−2 ) , x0 = x0 (−1 + x0 x−k−1 ) ,

Then x−k x−2k−1 x−k x−2k−1

= x−k+1 x−2k = x−k+2 x−2k+1 = ... = −1 + x−k x−2k−1 = = x−k+1 x−2k = ... = x0 x−k−1 = 2,

or x−k+i x−2k−1+i = 2. (f or i = 0, 1, 2, ..., k). Second suppose that x−k x−2k−1 x−k x−2k−1

= x−k+1 x−2k = x−k+2 x−2k+1 = ... = −1 + x−k x−2k−1 = = x−k+1 x−2k = ... = x0 x−k−1 = 2.

Then we see from Eq.(11) that x(2k+2)n−2k−1 x(2k+2)n−k−1 x(2k+2)n−1

= x−2k−1 , x(2k+2)n−2k = x−2k , x(2k+2)n−2k+1 = x−2k+1 , ..., = x−k−1 , x(2k+2)n−k = x−k , x(2k+2)n−k+1 = x−k+1 , ..., = x−1 , x(2k+2)n = x0 .

Thus we have a period (2k + 2) solution and the proof is complete. In the following we give some numerical examples to confirm the obtained results for Eq.(11). See Figures 4 and 5 below. plot of x(n+1)= (x(n−(2k+1))/(−1+x(n−k)*x(n−(2k+1))) 1000

500

0

−500

x(n)

−1000

−1500

−2000

−2500

−3000

−3500

0

5

10

15

20 n

25

30

35

40

Figure 4: k = 2, x−5 = 2.4, x−4 = −6.2, x−3 = 4, x−2 = 0.9, x−1 = 0.7, x0 = 0.5.

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plot of x(n+1)= (x(n−(2k+1))/(−1+x(n−k)*x(n−(2k+1))) 8

6

4

x(n)

2

0

−2

−4

−6

−8

0

5

10

15

20

25 n

30

35

40

45

50

Figure 5: k = 3, x−7 = 6, x−6 = −7, x−5 = 4, x−4 = 1/4, x−3 = 2/6, x−2 = −2/7, x−1 = 2/4, x0 = 8.

3.2. On the Difference Equation xn+1 =

xn−(2k+1) 1 − xn−k xn−(2k+1)

In this section we get the solution form of the difference equation xn+1 =

xn−(2k+1) , n = 0, 1, ..., 1 − xn−k xn−(2k+1)

(12)

where the initial values are arbitrary non zero real numbers. Theorem 3.3. Let {xn }∞ n=−2k−1 be a solution of Eq.(12). Then for n = 1, 2, ... x(2k+2)n−2k−1 x(2k+2)n−k−1 x(2k+2)n−1

¶ ¶ n−1 Y µ 1 − 2ix−k+1 x−2k 1 − 2ix−k x−2k−1 , x(2k+2)n−2k = x−2k , ..., 1 − (2i + 1) x−k x−2k−1 1 − (2i + 1) x−k+1 x−2k i=0 i=0 n−1 n−1 Y µ 1 − 2ix0 x−k−1 ¶ Y µ 1 − (2i + 1)x−k x−2k−1 ¶ , x(2k+2)n−k = x−k , ..., = x−k−1 1 − (2i + 1) x0 x−k−1 1 − (2i + 2) x−k x−2k−1 i=0 i=0 n−1 n−1 Y µ 1 − (2i + 1)x−1 x−k−2 ¶ Y µ 1 − (2i + 1)x0 x−k−1 ¶ , x(2k+2)n = x0 . = x−1 1 − (2i + 2) x−1 x−k−2 1 − (2i + 2) x0 x−k−1 i=0 i=0 = x−2k−1

n−1 Yµ

Proof: For n = 1 the result holds. Now suppose that n > 1 and that our assumption holds for n − 1. That is; x(2k+2)n−4k−3 = x−2k−1

n−2 Yµ i=0

x(2k+2)n−3k−3 = x−k−1

x(2k+2)n−3k−1 x(2k+2)n−2k−2

1 − 2ix−k x−2k−1 1 − (2i + 1) x−k x−2k−1

n−2 Yµ i=0

1 − 2ix0 x−k−1 1 − (2i + 1) x0 x−k−1

¶

¶

, x(2k+2)n−3k−2 = x−k

n−2 Yµ

1 − (2i + 1)x−k+1 x−2k 1 − (2i + 2) x−k+1 x−2k i=0 n−2 Y µ 1 − (2i + 1)x0 x−k−1 ¶ . = x0 1 − (2i + 2) x0 x−k−1 i=0 = x−k+1

, x(2k+2)n−4k−2 = x−2k

665

¶

n−2 Yµ i=0

n−2 Yµ i=0

, ..., x(2k+2)n−2k−3 = x−1

1 − 2ix−k+1 x−2k 1 − (2i + 1) x−k+1 x−2k

1 − (2i + 1)x−k x−2k−1 1 − (2i + 2) x−k x−2k−1

n−2 Yµ i=0

¶

¶

, ...,

,

1 − (2i + 1)x−1 x−k−2 1 − (2i + 2) x−1 x−k−2

Asim Asiri ET AL 656-670

¶

,

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.4, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

Now, it follows from Eq.(12) that x(2k+2)n−2k−1

=

x(2k+2)n−(4k+3) 1 − x(2k+2)n−3k−2 x(2k+2)n−(4k+3) n−2 Y 1 − 2ix−k x−2k−1 x−2k−1

=

1 − (2i + 1) x−k x−2k−1 i=0 n−2 Y 1 − (2i + 1)x−k x−2k−1 ⎞ Y 1 − 2ix−k x−2k−1 ⎠x−2k−1 ⎝ 1−x−k 1 − (2i + 2) x−k x−2k−1 1 − (2i + 1) x−k x−2k−1 i=0 i=0 n−2⎛

Y

n−2

x−2k−1

= 1−

1 − 2ix−k x−2k−1 µ ¶ 1 − (2i + 1) x−k x−2k−1 1 − (2n − 2) x−k x−2k−1 i=0 x−k x−2k−1 1 − (2n − 2) x−k x−2k−1 1 − (2n − 2) x−k x−2k−1 n−2 Y µ 1 − 2ix−k x−2k−1 ¶ µ 1 − (2n − 2) x−k x−2k−1 ¶

= x−2k−1

i=0

1 − (2i + 1) x−k x−2k−1

1 − (2n − 1) x−k x−2k−1

n−1 Yµ

¶

Hence, we have x(2k+2)n−2k−1 = x−2k−1 Similarly x(2k+2)n−k−1

=

i=0

1 − 2ix−k x−2k−1 1 − (2i + 1) x−k x−2k−1

.

.

x(2k+2)n−(3k+3) 1 − x(2k+2)n−2k−2 x(2k+2)n−(3k+3) n−2 Y 1 − 2ix0 x−k−1 x−k−1

1 − (2i + 1) x0 x−k−1 i=0 n−2 Y 1 − (2i + 1)x0 x−k−1 ⎞ Y 1 − 2ix0 x−k−1 ⎠x−k−1 ⎝ 1−x0 1 − (2i + 2) x x 1 − (2i + 1) x0 x−k−1 0 −k−1 i=0 i=0 µ ¶ n−2 Y 1 − 2ix0 x−k−1 x−k−1 1 − (2i + 1) x0 x−k−1 i=0 = µ ¶. n−2 Y 1 − 2ix0 x−k−1 1 − x0 x−k−1 1 − (2i + 2) x0 x−k−1 i=0 µ ¶µ ¶ n−2 Y 1 − 2ix0 x−k−1 1 − (2n − 2) x0 x−k−1 . = x−k−1 1 − (2i + 1) x0 x−k−1 1 − (2n − 1) x0 x−k−1 i=0 =

n−2⎛

Hence, we have x(2k+2)n−k−1 = x−k−1

n−1 Yµ i=0

1 − 2ix0 x−k−1 1 − (2i + 1) x0 x−k−1

¶

.

Similarly, we can easily get the other relations. Thus, the proof is completed. Theorem 3.4. Eq.(12) has the unique equilibrium point x = 0. Proof: For the equilibrium points of Eq.(12), we can write x = equilibrium point of Eq.(12) is x = 0.

x . Then we have x3 = 0. Thus the 1 − x2

The following figures show the behavior of the solutions of Eq.(12) with a fixed order and some numerical values of the initial values.

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plot of x(n+1)= (x(n−(2k+1))/(1−x(n−k)*x(n−(2k+1))) 15

10

x(n)

5

0

−5

−10

−15

0

10

20

30

40

50 n

60

70

80

90

100

Figure 6: k = 4, x−9 = 8, x−8 = 11, x−7 = 6, x−6 = 7, x−5 = 4, x−4 = 0.2, x−3 = 1.1, x−2 = 0.6, x−1 = 2, x0 = 4. plot of x(n+1)= (x(n−(2k+1))/(1−x(n−k)*x(n−(2k+1))) 14

12

10

x(n)

8

6

4

2

0

−2

0

10

20

30

40

50 n

60

70

80

90

100

Figure 7: k = 3, x−7 = 0.8, x−6 = 0.7, x−5 = 0.4, x−4 = 2, x−3 = 13, x−2 = 6, x−1 = 0.2, x0 = 4. Notice: The proofs of the theorems in the following section are similar to that are presented in the previous sections and so they will be omitted.

3.3. On the Difference Equation xn+1 =

xn−(2k+1) −1 − xn−k xn−(2k+1)

Here we obtain the form of the solutions of the following equation xn+1 =

xn−(2k+1) , n = 0, 1, ..., −1 − xn−k xn−(2k+1)

(13)

where the initial values are arbitrary non zero real numbers with x−k+i x−2k−1+i 6= −1 (f or i = 0, 1, 2, ..., k).

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Theorem 3.5. Let {xn }∞ n=−2k−1 be a solution of Eq.(13). Then for n = 1, 2, ... x(2k+2)n−2k−1

=

x(2k+2)n−2k+1

=

x(2k+2)n−k−1

=

x(2k+2)n−k+1 x(2k+2)n−1

= =

(−1)n x−2k−1 (−1)n x−2k , x = , (2k+2)n−2k (1 + x−k x−2k−1 )n (1 + x−k+1 x−2k )n n (−1) x−2k+1 , ..., (1 + x−k+2 x−2k+1 )n n (−1) x−k−1 n n , x(2k+2)n−k = (−1) x−k (1 + x−k x−2k−1 ) , (1 + x0 x−k−1 )n n n (−1) x−k+1 (1 + x−k+1 x−2k ) , ..., n n n n (−1) x−1 (1 + x−1 x−k−2 ) , x(2k+2)n = (−1) x0 (1 + x0 x−k−1 ) .

Theorem 3.6. Eq.(13) has a unique equilibrium point which is zero. Theorem 3.7. Let {xn }∞ n=−2k−1 be a solution of Eq.(13). Then the following statements are true: (1) If x−k+i x−2k−1+i 6= −2 (f or i = 0, 1, 2, ..., k), Then {xn }∞ n=−2k−1 is unbounded.

(2) Eq.(13) has a periodic solutions of period (2k + 2) iff x−k+i x−2k−1+i = −2 (f or i = 0, 1, 2, ..., k) and will be take the form {x−2k−1 , x−2k , ..., x−1 , x0 , x−2k−1 , x−2k , ..., x−1 , x0 , ...}.

Acknowledgements This Project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, Saudi Arabia under grant no. G—214—130—38. The authors, therefore, acknowledge with thanks DSR for technical and financial support. Last, but not least, sincere appreciations are dedicated to all our colleagues in the Faculty of Science for their nice wishes.

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Extremal solutions for a coupled system of nonlinear fractional differential equations with p-Laplacian operator ∗ Ying He† School of Mathematics and Statistics, Northeast Petroleum University, Daqing163318, P.R.China.

Abstract. This paper studies the existence of extremal solutions for nonlinear fractional differential coupled systems with p-Laplacian operator. The monotone iterative method combined with lower and upper solutions is applied. As an application, an example is presented to illustrate the main result.

Key words. Fractional differential system; p-Laplacian operator; Extremal solution; Monotone iterative technique; MR(2000) Subject Classifications: 34B15. 1. Introduction In recent years, fractional differential equations have been of great interest due to the intensive development of the theory of fractional calculus itself and its applications. The study of coupled systems involving fractional-order differential equation is also very significant as such systems appear in a variety of problems of applied nature, especially in bioscience. For details and examples the reader is referred to the papers [1 − 4] and the reference therein. In addition, much effort has been made towards the study of the existence of solutions for fractional differential equations involving the p-Laplacian operator based on different fractional derivatives[5 − 9]. In [10], Li and Lin considered a Hadamard fractional boundary value problem with p-Laplacian operator as below: (

Dβ (ϕp (Dα x(t))) = f (t, x(t)), 0 < t < e, x(1) = x0 (1) = x0 (e) = 0, Dα x(1) = Dα x(e) = 0

where 2 < α ≤ 3, 1 < β ≤ 2, ϕp (s) = |s|p−2 s, p > 1, and f : [1, e] × [0, +∞) −→ [0, +∞) is a positive continuous function. By using the Leray-Schauder type alternative and the Guo∗

This work is supported by the Guiding Innovation Foundation of Northeast Petroleum University (No.2016YDL-02) and Fostering Foundation of Northeast Petroleum University (No.2017PYYL-08). † Corresponding author. E-mail adress:[email protected];

1

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Krasnoselskii fixed point theorem, the existence and the uniqueness of the positive solutions were established. To best of our knowledge, only few papers considered the method of upper and lower solutions for a coupled system of fractional p-Laplacian equation. Motivated by [11 − 12], in this paper, we use the monotone iterative technique, combined with the method of upper and lower solution to study the coupled system of fractional differential equations with p-Laplacian operator, which is given by   Dβ (φp (Dα x(t))) = f (t, x(t), y(t), Dα x(t), Dα y(t)),    β α α α

t ∈ [0, 1], t ∈ [0, 1],

D (φp (D y(t))) = g(t, y(t), x(t), D y(t), D x(t)), t1−α x(t)|t=0 = r1 , D y(t)|t=0 = 0, t1−α y(t)|t=0 = r2 ,

 Dα x(t)|t=0 = 0,    α

(1.1)

where J = [0, 1], f, g ∈ C(J ×R4 , R), r1 , r2 ∈ R and r1 ≤ r2 , Dα , Dβ are the standard RiemannLiouville fractional derivatives, satisfying 0 < α, β < 1, 1 < α + β < 2, φp (t) = |t|p−2 t, p > 1, is the p-Laplacian operator and (φp )−1 = φq , p1 + 1q = 1. The rest of this paper is organized as follows. In section 2, we give some necessary definitions and lemmas. In section 3, the main result and proof are given. Finally, an example is presented to illustrate the main result. 2. Preliminaries In this section, we establish some preliminary results that will be used in the next section to attain existence results for the nonlinear system (1.1) Let C[0, 1] denote the Banach space of continuous functions from [0, 1] into R with the norm kukC = maxt∈[0,1] |u(t)|. Denote C1−α [0, 1] by C1−α [0, 1] = {x ∈ C(0, 1] : t1−α x ∈ C[0, 1]}. Then, C1−α [0, 1] is a Banach spaces with the norm kxk1−α = kt1−α x(t)kC . It is clear that C[0, 1] := C0 [0, 1] ⊂ C1−α [0, 1] with kxkC1−α ≤ kxkC for 0 < α ≤ 1 and C1−α [0, 1] ⊂ L[0, 1] (note L[0, 1] is the space of Lebesgue integrable functions defined on [0, 1]). Denote C α [0, 1] by C α [0, 1] = {x(t) ⊂ C[0, 1] : (Dα x)(t) ⊂ C[0, 1] and Dα x(t)|t = 0 = 0} . Lemma 2.1: Let 0 < β < 1, σ ∈ C[0, 1] ,M ≥ 0 and M Γ(1 − β) < 1, then the problem (

Dβ u(t) + M u(t) = σ(t), u(0) = 0,

0 ≤ t ≤ 1,

(2.1)

has a unique solution. Proof. Equation (2.1) is equivalent to the following integral equation 1 u(t) = Γ(β)

Z

t

(t − s)β−1 (σ(s) − M u(s))ds,

∀t∈J

0

2

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.4, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

Let

t 1 (t − s)β−1 (σ(s) − M u(s))ds, Au(t) = Γ(β) 0 By M ≥ 0 and M Γ(1 − β) < 1, for any u, v ∈ C[0, 1], we have

Z

kAu(t) − Av(t)kC

∀t∈J

t M (t − s)β−1 dsku − vkC Γ(β) 0 M ku − vkC Γ(β)β 1 1 1 · · ku − vkC Γ(β) β Γ(1 − β) sin Πβ ku − vkC Πβ ku − vkC

Z

≤ ≤ < =
1. r→∞

r→∞

q

q

Proposition 1.7. ([17]) If φ(r) satisfies the above two conditions (i) − (ii) in Proposition 1.6: (i) then for any entire function f (z), we have logp T (r, f ) logp+1 M (r, f ) = lim ; r→∞ logq φ(r) r→∞ logq φ(r) logp T (r, f ) logp+1 M (r, f ) µ[p,q] (f, φ) = lim = lim . r→∞ logq φ(r) r→∞ logq φ(r) σ[p,q] (f, φ) = lim

(ii) then for any meromorphic function f (z), we have λ[p,q] (f, φ) = lim

r→∞

λ[p,q] (f, φ) = lim

r→∞

logp n(r, f1 ) logq φ(r) logp n(r, f1 ) logq φ(r)

= lim

r→∞

= lim

r→∞

logp N (r, f1 ) logq φ(r) logp N (r, f1 ) logq φ(r)

; .

In this paper, we investigate the growth and zeros of solutions of (1.1) and (1.2) with entire coefficients of [p, q] − φ(r) order and obtain the following results.

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Theorem 1.8. Let Aj (z) (j = 0, 1, · · · , k − 1) be entire functions satisfying max{σ[p,q] (Aj , φ)|j = 1, 2, · · · , k −1} < σ[p,q] (A0 , φ) < ∞. Then every solution f (z) ̸≡ 0 of (1.1) satisfies σ[p+1,q] (f, φ) = σ[p,q] (A0 , φ). Theorem 1.9. Let F (z) ̸≡ 0, Aj (z)(j = 0, · · · , k − 1) be entire functions, and let f (z) be a solution of (1.2) satisfying max{σ[p,q] (Aj , φ), σ[p,q] (F, φ)|j = 0, 1, · · · , k − 1} < σ[p,q] (f, φ). Then λ[p,q] (f, φ) = λ[p,q] (f, φ) = σ[p,q] (f, φ). Theorem 1.10. Let F (z) ̸≡ 0, Aj (z) (j = 0, 1, · · · , k − 1) be entire functions satisfying max{σ[p,q] (Aj , φ), σ[p+1,q] (F, φ)|j = 1, · · · , k − 1} < σ[p,q] (A0 , φ). Then every solution f (z) of (1.2) satisfies λ[p+1,q] (f, φ) = λ[p+1,q] (f, φ) = σ[p+1,q] (f, φ) = σ[p,q] (A0 , φ), with at most one exceptional solution f0 satisfying σ[p+1,q] (f0 , φ) < σ[p,q] (A0 , φ). Remark 1.11. The above Theorems 1.8-1.10 generalize and extend Theorems A-C and some previous results. 2. Preliminary Lemmas Lemma 2.1. ([10, 14]) Let f (z) be a transcendental entire function, and let z be a point with |z| = r at which |f (z)| = M (r, f ). Then for all |z| = r outside a set E1 of r of finite logarithmic measure, we have ( ) vf (r) j f (j) (z) = (1 + o(1)) (j ∈ N), f (z) z where vf (r) is the central index of f (z), E1 ⊂ (1, +∞) is a set of r of finite logarithmic measure or finite linear measure in this paper, not necessarily the same at each occurrence. Lemma 2.2. ([7, 14]) Let g : [0, +∞) −→ R and h : [0, +∞) −→ R be monotone increasing functions such that g(r) ≤ h(r) outside of an exceptional set E1 ⊂ [1, +∞) of finite logarithmic measure or finite linear measure. Then for any d > 1, there exists r0 > 0 such that g(r) ≤ h(dr) for all r > r0 . Lemma 2.3.([17]) Let f (z) be an entire function satisfying σ[p,q] (f, φ) = σ1 and µ[p,q] (f, φ) = µ1 . Then logp vf (r) logp vf (r) lim = σ1 , lim = µ1 . r→∞ logq φ(r) r→∞ logq φ(r) Lemma 2.4. Let f (z) be an entire function of [p, q] − φ(r) order satisfying σ[p,q] (f, φ) = σ2 , log φ(αr)

where φ(r) only satisfies lim logq φ(r) = 1 for some α > 1. Then there exists a set E2 ⊂ (1, +∞) q r→∞ having infinite logarithmic measure such that for all r ∈ E2 , we have logp T (r, f ) = σ2 r→∞ logq φ(r) lim

684

(r ∈ E2 ).

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.4, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

Growth and Zeros of Solutions of Linear Differential Equations with Entire Coefficients of [p, q] − φ(r) Order

5

Proof. By Definition 1.3, there exists an increasing sequence {rn }∞ n=1 tending to ∞ satisfying (1 + n1 )rn < rn+1 and logp T (rn , f ) = σ[p,q] (f, φ) = σ2 , lim n→∞ logq φ(rn ) 1 there exists an n1 (∈ N) such that for n ≥ n1 and for any r ∈ E2 = ∪∞ n=n1 [rn , (1 + n )rn ], we have

logp T (rn , f ) logq φ((1 +

1 n )rn )

≤

logp T (r, f ) . logq φ(r)

(2.1)

By (2.1), for all r ∈ E2 , we have logp T (rn , f ) logq φ(rn ) logp T (r, f ) · lim . ≤ lim 1 n→∞ logq φ(rn ) n→∞ logq φ((1 + )rn ) r→∞ logq φ(r) n lim

logq φ(αr) r→∞ logq φ(r)

By (2.12) and lim

(2.2)

= 1 (α > 1), for all r ∈ E2 , we have logp T (r, f ) ≥ σ2 . r→∞ logq φ(r) lim

(2.3)

On the other hand, by Definition 1.3, for all r ∈ E2 , we have logp T (r, f ) ≤ σ2 . r→∞ logq φ(r) lim

(2.4)

By (2.3) and (2.4), for any r ∈ E2 , we have logp T (r, f ) = σ2 . r→∞ logq φ(r) lim

where ml E2 =

∫ (1+ 1 )rn dt n n=n1 rn t

∑∞

=

∑∞

n=n1

log(1 + n1 ) = ∞.

By Lemma 2.4, it is easy to obtain the following Lemma 2.5. Lemma 2.5. Let f1 (z), f2 (z) be entire functions of [p, q] − φ(r) order satisfying σ[p,q] (f1 , φ) > σ[p,q] (f2 ). Then there exists a set E3 ⊂ (1, +∞) having infinite logarithmic measure such that for all r ∈ E3 , we have T (r, f2 ) = 0 (r ∈ E3 ). lim r→∞ T (r, f1 ) Lemma 2.6. ([8]) Let f (z) be a transcendental meromorphic function, and let β > 1 be a given constant, for any given ε > 0, there exist a set E1 ⊂ (1, +∞) that has finite logarithmic measure

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and a constant B > 0 that depends only on β and (i, j) (i, j are integers with 0 ≤ i < j) such that for all |z| = r ∈ / [0, 1] ∪ E1 , we have [ ]j−i f (j) (z) T (βr, f ) β (log r) log T (βr, f ) . (i) ≤ B f (z) r 3. Proofs of Theorems 1.8 - 1.10 Proof of Theorem 1.8. We divide the proof into two parts. (i) Set σ[p,q] (A0 , φ) = σ3 , first, we prove that every solution of (1.1) satisfies σ[p+1,q] (f, φ) ≤ σ3 . It is easy to know that equation (1.1) has no polynomial solutions under the assumptions. If f (z) is a transcendental solution of (1.1), by (1.1), we get f (k−1) (z) f (k) (z) f (s) (z) (3.1) ≤ |Ak−1 | + · · · + |As | + · · · + |A0 |. f (z) f (z) f (z) Since max{σ[p,q] (Aj , φ)|j = 0, 1, · · · , k − 1} ≤ σ3 , for any given ε > 0 and for sufficiently large r, we have |Aj (z)| ≤ expp+1 {(σ3 + ε) logq φ(r)} (j = 0, 1, · · · , k − 1). (3.2) By Lemma 2.1, there exists a set E1 ⊂ (1, +∞) having finite logarithmic measure such that for all z satisfying |z| = r ̸∈ [0, 1] ∪ E1 and |f (z)| = M (r, f ), we have ( ) vf (r) j f (j) (z) = (1 + o(1)) (j = 1, · · · , k − 1). (3.3) f (z) z By (3.1)-(3.3), for all z satisfying |z| = r ̸∈ [0, 1] ∪ E1 and |f (z)| = M (r, f ), we get ( ) ( ) vf (r) k vf (r) k−1 (1 + o(1)) ≤ k expp+1 {(σ3 + ε) logq φ(r)} (1 + o(1)), r r

(3.4)

by (3.4) and Lemma 2.2, there exists some α1 (1 < α1 < α) and r ≥ r0 , we have vf (r) ≤ kα1 r expp+1 {(σ3 + ε) logq φ(α1 r)}.

(3.5)

By Lemma 2.3 and the Proposition 1.6, we have σ[p+1,q] (f, φ) ≤ σ3 . (ii) On the other hand, if f ̸≡ 0, (1.1) can be written −A0 =

f (k) (z) f (j) (z) f ′ (z) + · · · + Aj + · · · + A1 . f (z) f (z) f (z)

By (3.6), we get m(r, A0 ) ≤

k−1 ∑

m(r, Aj ) +

i=1

k ∑ j=1

686

(

f (k) m r, f

(3.6)

) + log k.

(3.7)

Sheng Gui Liu ET AL 681-689

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.4, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

Growth and Zeros of Solutions of Linear Differential Equations with Entire Coefficients of [p, q] − φ(r) Order

7

Since max{σ[p,q] (Aj , φ)|j = 1, 2, · · · , k − 1} < σ3 and by Lemma 2.5, there exists a set E2 ⊂ (1, +∞) with infinite logarithmic measure such that for all z satisfying |z| = r ∈ E2 , we have logp m(r, A0 ) = σ3 , r→∞ logq φ(r) lim

m(r, Aj ) −→ 0 (r ∈ E2 , j = 1, · · · , k − 1). m(r, A0 )

By the lemma of logarithmic derivative, we have ( ) f (j) m r, = O{log rT (r, f )} (r ∈ / E1 ). f

(3.8)

(3.9)

By (3.7)-(3.9), for all sufficiently large r ∈ E2 \ E1 , we have 1 m(r, A0 ) ≤ O{log rT (r, f )}. 2 Hence by Proposition 1.6, we have σ[p+1,q] (f, φ) ≥ σ3 . Therefore, every solution f (z) ̸≡ 0 of (1.1) satisfies σ[p+1,q] (f, φ) = σ[p,q] (A0 , φ). Proof of Theorem 1.9. Proof. If f (z) ̸≡ 0, by (1.2), we get 1 1 = f F

(

f (k−1) f (k) + Ak−1 + · · · + A0 f f

) ,

(3.10)

it is easy to see that if f (z) has a zero at z0 of order α (α > k), and A0 , · · · , Ak−1 are analytic at z0 , then F must have a zero at z0 of order α − k, hence ( ) ( ) ( ) 1 1 1 n r, ≤ kn r, + n r, , (3.11) f f F and

( ) ( ) ( ) 1 1 1 N r, ≤ kN r, + N r, . f f F

(3.12)

By the lemma of logarithmic derivative and (3.10), we have ( ) ( ) ∑ k−1 1 1 m r, ≤ m r, + m(r, Aj ) + O (log T (r, f ) + log r) (r ̸∈ E1 ). f F

(3.13)

j=0

By (3.12), (3.13), we get ( ) k−1 ∑ 1 T (r, Aj ) + O {log (rT (r, f ))} (r ̸∈ E1 ). T (r, f ) ≤ kN r, + T (r, F ) + f

(3.14)

j=0

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8

G. S. Liu, J. Tu, H. Zhang

Since max{σ[p,q] (F, φ), σ[p,q] (Aj , φ)|j = 0, 1, · · · , k − 1} < σ[p,q] (f, φ), by Lemma 2.5, there exists a set E3 ⊂ (1, +∞) having infinite logarithmic measure such that { } T (r, F ) T (r, Aj ) max , −→ 0 (r ∈ E3 , j = 0, · · · , k − 1). (3.15) T (r, f ) T (r, f ) Since for all sufficiently large r, we have log T (r, f ) = o{T (r, f )}.

(3.16)

By (3.14)-(3.16), for all |z| = r ∈ E3 \E1 , we have { ( )} 1 (1 − o(1)) T (r, f ) ≤ O N r, + O{log r}. f

(3.17)

By Definition 1.4 and Proposition 1.7 and (3.17), we get σ[p,q] (f, φ) ≤ λ[p,q] (f, φ).

(3.18)

Since σ[p,q] (f, φ) ≥ λ[p,q] (f, φ) ≥ λ[p,q] (f, φ), and by (3.18), we have λ[p,q] (f, φ) = λ[p,q] (f, φ) = σ[p,q] (f, φ). Proof of Theorem 1.10. We assume that f is a solution of (1.2). By the elementary theory of differential equations, all the solutions of (1.2) are entire functions and have the form f = f ∗ + C1 f1 + C2 f2 + · · · + Ck fk , where C1 , · · · , Ck are complex constants, {f1 , · · · , fk } is a solution base of (1.1), f ∗ is a solution of (1.2) and has the form f ∗ = D1 f1 + D2 f2 + · · · + Dk fk , (3.19) where D1 , · · · , Dk are certain entire functions satisfying ′

Dj = F · Gj (f1 , · · · , fk ) · W (f1 , · · · , fk )−1

(j = 1, · · · , k),

(3.20)

where Gj (f1 , · · · , fk ) are differential polynomials in f1 , · · · , fk and their derivatives with constant coefficients, and W (f1 , · · · , fk ) is the Wronskian of f1 , · · · , fk . By Theorem 1.8, we have σ[p+1,q] (fj , φ) = σ[p,q] (A0 , φ)(j = 1, 2, · · · , k) , then by (3.19) and (3.20), we get σ[p+1,q] (f, φ) ≤ max{σ[p+1,q] (fj , φ), σ[p+1,q] (F, φ)|j = 1, · · · , k} ≤ σ[p,q] (A0 , φ). We affirm that (1.2) can only possess at most one exceptional solution f0 satisfying σ[p+1,q] (f0 , φ) < σ[p,q] (A0 , φ). In fact, if f∗ is another solution satisfying σ[p+1,q] (f∗ , φ) < σ[p,q] (A0 , φ), then σ[p+1,q] (f0 − f∗ , φ) < σ[p,q] (A0 , φ). But f0 − f∗ is a solution of (1.1), this contradicts Theorem 1.8. Then σ[p+1,q] (f, φ) = σ[p,q] (A0 , φ) holds for all solutions of (1.2) with at most one exceptional solution f0 satisfying σ[p+1,q] (f0 , φ) < σ[p,q] (A0 , φ). By Theorem 1.9, we get that λ[p+1,q] (f, φ) = λ[p+1,q] (f, φ) = σ[p+1,q] (f, φ) holds for all solutions satisfying σ[p+1,q] (f, φ) = σ[p,q] (A0 , φ) with at most one exceptional solution f0 satisfying σ[p+1,q] (f0 , φ) < σ[p,q] (A0 , φ).

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.4, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

Growth and Zeros of Solutions of Linear Differential Equations with Entire Coefficients of [p, q] − φ(r) Order

9

References [1] S. Bank and I. Laine, On the oscillation theory of f ′′ + Af = 0 where A is entire, Trans. Amer. Math. Soc. 273 (1982), 352-363. [2] B. Bela¨ıdi, On the iterated order and the fixed points of entire solutions of some complex linear differential equations, Electron. J. Qual. Theory Differ. Equ. No. 9, (2006), 1-11. [3] B. Bela¨ıdi, On the [p, q]-order of meromorphic solutions of linear differential equations. Acta Univ. M. Belii Ser. Math. 2015, 37-49 [4] L. G. Bernal, On growth k-order of solutions of a complex homogeneous linear differential equations, Proc. Amer. Math. Soc. 101 (1987), 317-322. [5] Z. X. Chen and S. A. Gao, The complex oscillation theory of certain non-homogeneous linear differential equations with transcendental entire coefficients, J. Math. Anal. Appl. 179 (1993), 403-416. [6] G. Frank and S. Hellerstein, On the meromorphic solutions of non-homogeneous linear differential equations with polynomial coefficients, Proc. London Math. Soc. 53 (3) (1986), 407-428. [7] S. A. Gao, Z. X. Chen and T. W. Chen, The Complex Oscillation Theory of Linear Differential Equations, HuaZhong Univ. Sci. Tech. Press 1998 (in Chinese). [8] G. Gundersen, Estimates for the logarithmic derivate of a meromorphic function, plus similar estimates, J. London Math. Soc. 37 (2) (1988), 88-104. [9] W. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964. [10] W. Hayman, The local growth of power series: A survey of the Wiman-Valiron method, Canad. Math. Bull. 17 (1974), 317-358 . [11] H. Hu and X. M. Zheng, Growth of solutions of linear differential equations with meromorphic coefficients of [p, q]-order, Math. Commun. 19 (1) (2014), 29-42. [12] O. P. Juneja, G. P. Kapoor and S. K. Bajpai, On the (p, q)-order and lower (p,q)-order of an entire function, J. Reine Angew. Math. 282 (1976), 53-67. [13] L. Kinnunen, Linear differential equations with solutions of finite iterated order, Southeast Asian Bull. Math. (4) 22 (1998), 385-405. [14] I. Laine, Nevanlinna Theory and Complex Differential Equations, Walter de Gruyter, Berlin, 1993. [15] J. Liu, J. Tu and L. Z. Shi, Linear differential equantials with entire cofficients of [p, q]-order in the complex plane, J. Math. Anal. Appl. 372 (2010), 55-67. [16] L. M. Li and T. B. Cao, Solutions for linear differential equations with meromorphic coefficients of [p,q]-order in the plane, Electron. J. Diff. Equ. No. 195, (2012), 1-15. [17] X. Shen, J. Tu and H. Y. Xu, Complex oscillation of a second-order linear differential equation with entire coefficients of [p, q] − φ(r) order, Advances in Difference Equations No. 200, (2014), 1-14. [18] J. Tu, C. Y. Liu and H, Y. Xu, Meromorphic Functions of Relative [p, q] Order to φ(r), J. Jiangxi Norm. Univ., Nat. Sci. 36 (1), (2012), 47-50. [19] J. Tu, J. S. Wei, H. Y. Xu, The order and type of meromorphic functions and analytic functions of [p, q] − φ(r) order in the unit disc, J. Jiangxi Norm. Univ., Nat. Sci. 39 (2), (2015), 207-211.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.4, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

Some k-fractional integrals inequalities through generalized λφm-MT-preinvexity Chunyan Luo1 Tingsong Du1,2∗ Muhammad Adil Khan3 Artion Kashuri4 Yanjun Shen5 1

Department of Mathematics, College of Science, China Three Gorges University, Yichang 443002, China 2 Three Gorges Mathematical Research Center, China Three Gorges University, Yichang 443002, China E-mail: [email protected] [email protected] 3 Department of Mathematics, University of Peshawar, Pakistan E-mail: [email protected] 4 Department of Mathematics, Faculty of Technical Science, University “Ismail Qemali”, Vlora, Albania E-mail: [email protected] 5 Hubei Provincial Collaborative Innovation Center for New Energy Microgrid, China Three Gorges University, Yichang 443002, China E-mail: [email protected] ∗ Corresponding author Tingsong Du

Abstract The authors introduce the concept of the generalized λφm -MT-preinvex functions and discover a new k-fractional integral identity concerning twice differentiable preinvex mappings defined on (φ, m)-invex set. By using this identity, we establish the right-sided new HermiteHadamard type inequalities for the generalized λφm -MT-preinvex mappings via k-fractional integrals. The new k-fractional integral inequalities are also applied to some special means. 2010 Mathematics Subject Classification: Primary 26A33; Secondary 26D07, 26D20, 41A55. Key words and phrases: Hermite-Hadamard’s inequality; λφm -MT-preinvex functions; kRiemann-Liouville fractional integrals.

1

Introduction

Let f : I ⊆ R → R be a convex mapping on the interval I of real numbers and u, v ∈ I with u < v. Then the following well-know Hermite-Hadamard inequality holds  f

u+v 2

 ≤

1 v−u

Z

v

f (x)dx ≤ u

f (u) + f (v) . 2

(1.1)

This inequality is one of the famous results for convex functions. Many researchers generalized and extended the inequalities (1.1) involving a variety of convex functions one can see [8, 9, 12, 15, 20–22, 40, 41] and the references mentioned in these papers.

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In 2013, Sarikaya et al. [32] considered the following Hermite-Hadamard type inequalities via Riemann-Liouville fractional integrals. Theorem 1.1 Let f : [u, v] → R be a positive function along with 0 ≤ u < v and let f ∈ L1 [u, v]. Suppose f is a convex function on [u, v], then the subsequent inequalities for fractional integrals hold: u + v  f (u) + f (v) Γ(α + 1)  α ≤ Ju+ f (v) + Jvα− f (u) ≤ , f (1.2) α 2 2(v − u) 2 where the symbols Juα+ f and Jvα− f denote respectively the left-sided and right-sided RiemannLiouville fractional integrals of the order α ∈ R+ defined by Z x 1 (x − t)α−1 f (t)dt, u < x Juα+ f (x) = Γ(α) u and Jvα− f (x) =

1 Γ(α)

Z

v

(t − x)α−1 f (t)dt, x < v.

x

Here, Γ(α) is the gamma function and its definition is Γ(α) =

R∞ 0

e−µ µα−1 dµ.

Due to the wide applications of Riemann-Liouville fractional Hermite-Hadamard type inequalities in mathematical analysis, many researchers extended Hermite-Hadamard inequality for different classes of convex functions. For example, see for convex mappings [7, 10, 16, 17, 29], for m-convex mappings [37] and (s, m)-convex mappings [3], for h-preinvex mappings[13], for harmonically convex mappings [18], for preinvex mappings [25, 31] and the references mentioned in these papers. Also in [4], Anastassiou presented a complete theory with respect to fractional differentiation inequalities. In 2012, Mubeen and Habibullah [24] introduced a new fractional integral that generalizes the Riemann-Liouville fractional integrals. Definition 1.1 ([24]) Let f ∈ L1 [a, b], then k-Riemann-Liouville fractional integrals k Jaµ+ f (x) and k Jbµ− f (x) of order µ > 0 are defined by Z x µ 1 µ J f (x) = (x − t) k −1 f (t)dt, (0 ≤ a < x < b) k a+ kΓk (µ) a and µ k Jb− f (x)

=

1 kΓk (µ)

Z

b

µ

(t − x) k −1 f (t)dt, (0 ≤ a < x < b),

x

respectively, where k > 0 and Γk (µ) is the k-gamma function given as Γk (µ) = Note that Γk (µ + k) = µΓk (µ) and k Ja0+ f (x) = k Jb0− f (x) = f (x).

R∞ 0

tk

tµ−1 e− k dt.

The notion of k-Riemann-Liouville fractional integral is an significant extension of RiemannLiouville fractional integrals. It is stressed that for k 6= 1 the properties of k-Riemann-Liouville fractional integrals are quite dissimilar from those of general Riemann-Liouville fractional integrals. For this, the k-Riemann-Liouville fractional integrals have aroused the interest of many researchers. Properties and integral inequalities concerning this operator can refer to [1, 2, 6, 33, 34, 38] and the references mentioned in these papers. Let us evoke some basic definitions as follows. 2 691

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Definition 1.2 ([5]) A set K ⊆ Rn is said to be invex set respecting the mapping η : K × K → Rn if x + tη(y, x) ∈ K for any x, y ∈ K and t ∈ [0, 1]. Definition 1.3 ([39]) A function f defined on the invex set K ⊆ Rn is said to be preinvex with respect to η, if
f x + tη(y, x) ≤ (1 − t)f (x) + tf (y),

∀ x, y ∈ K, t ∈ [0, 1].

Definition 1.4 ([27]) Let x ∈ K ⊆ Rn and let φ : K → R be a continuous function. Then the set K is said to be φ-convex at x respecting φ, if x + λeiφ (y − x) ∈ K,

∀ x, y ∈ K, λ ∈ [0, 1].

Definition 1.5 ([26]) A set K ⊆ Rn is called φ-invex at x with respect to φ(·), if there a continuous function φ(·) : K → R and a bifunction η(·, ·) : K × K → Rn , such that x + teiφ η(y, x) ∈ K,

∀ x, y ∈ K, t ∈ [0, 1].

Definition 1.6 ([11]) A set K ⊆ Rn is said to be m-invex with respect to the mapping η : K × K × (0, 1] → Rn for some fixed m ∈ (0, 1], if mx + tη(y, x, m) ∈ K holds for each x, y ∈ K and any t ∈ [0, 1]. Definition 1.7 ([27]) The function f on the φ-convex set K is said to be φ-convex with respect to φ, if
f x + λeiφ (y − x) ≤ (1 − λ)f (x) + λf (y), ∀ x, y ∈ K, λ ∈ [0, 1]. (1.3) Definition 1.8 ([42]) The function f defined on the φ-invex set K ⊆ Rn is said to be φ-MTpreinvex, if it is nonnegative and for ∀ x, y ∈ K and t ∈ (0, 1) satisfies the following inequality √ √
1−t t iφ √ f (x) + √ f (y). (1.4) f x + te η(y, x) ≤ 2 1−t 2 t Definition 1.9 ([28]) A function: I ⊆ R → R is said to be m-MT-convex, if f is positive and for ∀ x, y∈I, and t ∈ (0, 1), with m ∈ [0, 1], satisfies the following inequality √ √
m 1−t t √ f (x) + f tx + m(1 − t)y ≤ √ f (y). (1.5) 2 1−t 2 t Definition 1.10 ([14]) A function f: I ⊆ R → R is said to be λ-MT-convex function, if f is positve and ∀ x, y∈I, λ ∈ (0, 21 ] and t ∈ (0, 1), satisfies the following inequality √ √
t (1 − λ) 1 − t √ f tx + (1 − t)y ≤ √ f (x) + f (y). 2 1−t 2λ t

(1.6)

Clearly, when choosing m = 1 and λ = 12 in Definition 1.9 and Definition 1.10, respectively, the function f reduces to MT-convex function in [35]. For some significant integral inequalities in association with MT-convex functions, one can see [19, 23, 30, 36] and the references therein. The main purpose of this paper is to introduce the class of generalized λφm -MT-preinvex functions on (φ, m)-invex and to prove a k-fractional integral identity. By using this identity, we establish the right-sided new Hadamard-type inequalities for the generalized λφm -MT-preinvex functions via k-Riemann-Liouville fractional integrals. These inequalities can be viewed as generalization of recent results that appeared in Refs. [30] and [42].

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2

New definitions and a lemma

As one can see, the definitions of the φ-invex and m-invex have similar configurations. This observation leads us to generalized these concepts. Firstly, the so-called ‘(φ, m)-invex’ may be introduced as follows. Definition 2.1 A set Kφm ⊆ Rn is said to be (φ, m)-invex with respect to a continuous function φ(·) : Kφm → R and the mapping η : Kφm × Kφm × (0, 1] → Rn , for some fixed m ∈ (0, 1], if mx + teiφ η(y, x, m) ∈ Kφm holds for any x, y ∈ Kφm and t ∈ (0, 1). Let us note that: - if φ = 0, then we get the definition of an m-invex set, - if the mapping η(y, x, m) with m = 1 reduces to η(y, x), then we obtain the definition of a φ-invex set, - if φ = 0 and η(y, x, m) = y − mx with m = 1, then we obtain the definition of a convex set. Now we define the concept of generalized λφm -MT-preinvex functions. Definition 2.2 Let Kφm ⊆ R is a (φ, m)-invex set with respect to η and φ. A function f : Kφm → R0 is said to be generalized λφm -MT-preinvex, according to η and φ, and ∀ x, y ∈ Kφm , t ∈ (0, 1) and λ ∈ (0, 12 ], along with some fixed m ∈ (0, 1] satisfies the coming inequality √ √
m(1 − λ) 1 − t t (2.1) √ f mx + teiφ η(y, x, m) ≤ f (x) + √ f (y). 2 1−t 2λ t Let us note that: - if the mapping η(y, x, m) with m = 1 degenerates into η(y, x), then we obtain the definition of λφ -MT-preinvex function, - if the mapping η(y, x, m) with m = 1 degenerates into η(y, x) and λ = 21 , then we obtain the definition of φ-MT-preinvex function, - if φ = 0, the mapping η(y, x, m) = y − mx, and λ = 12 , then we obtain the definition of m-MT-convex function, - if φ = 0 and the mapping η(y, x, m) = y − mx with m = 1, then we obtain the definition of λ-MT-convex function, - if φ = 0, the mapping η(y, x, m) = y − mx with m = 1, and λ = 12 , then we obtain the definition of MT-convex function. Before presenting our main results, we prove the following lemma. Lemma 2.1 Let Kφm ⊆ R be a (φ, m)-invex subset respecting φ(·) and η : Kφm ×Kφm ×(0, 1] → R, a, b ∈ Kφm with η(b, a, m) > 0 and some fixed m ∈ (0, 1]. Suppose that f : Kφm → R is a twice differentiable mapping such that f 00 ∈ L[ma, ma + eiφ η(b, a, m)], we have the following identity via k-fractional integral with k, α > 0 holds:
2 Z 1 α α
eiφ η(b, a, m) 1 − t k +1 − (1 − t) k +1 00 Rf (α, k; φ, η, m, a, b) = f ma + teiφ η(b, a, m) dt, α 2 0 k +1 (2.2) where
f (ma) + f ma + eiφ η(b, a, m) Γk (α + k) Rf (α, k; φ, η, m, a, b) := −
α 2 2k eiφ η(b, a, m) k h i
α iφ α × k Jma η(b, a, m) + k J(ma+e + f ma + e iφ η(b,a,m))− f (ma) .

4 693

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Proof. Set eiφ η(b, a, m) I = 2 ∗


2 Z 0

1

α

α


1 − t k +1 − (1 − t) k +1 00 f ma + teiφ η(b, a, m) dt. α + 1 k

Since a, b ∈ Kφm and Kφm is a (φ, m)-invex subset respecting φ and η, for ∀ t ∈ (0, 1), we have ma + teiφ η(b, a, m) ∈ Kφm . Integrating by part gives, we have
2  α α
1 eiφ η(b, a, m) 1 − t k +1 − (1 − t) k +1 0 ∗ iφ I = f ma + te η(b, a, m) α 2 ( k + 1)eiφ η(b, a, m) 0  α α Z 1 α α
−( k + 1)t k + ( k + 1)(1 − t) k 0 − f ma + teiφ η(b, a, m) dt α iφ ( k + 1)e η(b, a, m) 0
2  α α iφ
1 e η(b, a, m) t k − (1 − t) k iφ = f ma + te η(b, a, m)
2 2 0 eiφ η(b, a, m)  α Z 1 α α −1 α k
+ k (1 − t) k −1 kt − f ma + teiφ η(b, a, m) dt
2 0 eiφ η(b, a, m)
Z 1  
f (ma) + f ma + teiφ η(b, a, m) α α α −1 −1 iφ k k = − t + (1 − t) f ma + te η(b, a, m) dt . 2 2k 0 Using the reduction formula Γk (α + k) = αΓk (α) (α > 0), we have Z
α 1 α −1 Γk (α + k) α t k f ma + teiφ η(b, a, m) dt =
α k J(ma+eiφ η(b,a,m))− f (ma) 2k 0 2k eiφ η(b, a, m) k and α 2k

Z

1

Γk (α + k)


α (1 − t) k −1 f ma + teiφ η(b, a, m) dt =

0

2k eiφ η(b, a, m)


iφ α
αk k Jma+ f ma + e η(b, a, m) .

Thus, we obtain conclusion (2.2). Remark 2.1 If we put k = 1 in Lemma 2.1, then we have: (a) for the mapping η(b, a, m) with m = 1 reduces to η(b, a), we obtain Lemma 3.1 in [14], (b) for α = 1 = m with the mapping η(b, a, m) reduces to η(b, a), we obtain Lemma 2.3 in [42], (c) for φ = 0, α = 1 = m with the mapping η(b, a, m) = b − ma, we obtain Lemma 1.3 in [37].

3

Main results

Using Lemma 2.1, we now state the following theorem. Theorem 3.1 Let Aφm ⊆ R0 be an open (φ, m)-invex subset respecting φ(·) and η : Aφm × Aφm × (0, 1] → R0 , a, b ∈ Aφm with η(b, a, m) > 0, λ ∈ (0, 12 ] and some fixed m ∈ (0, 1]. If f : Aφm → R is a twice differentiable mapping such that f 00 ∈ L[ma, ma + eiφ η(b, a, m)] and |f 00 |q for q ≥ 1 is generalized λφm -M T -preinvex on Aφm and x ∈ [ma, ma + eiφ η(b, a, m)], then we have the following inequality for k-fractional integrals with k, α > 0 Rf (α, k; φ, η, m, a, b) )1
# q1 (
2 " √ (3.1) 00 q q k eiφ η(b, a, m) πΓ q( αk + 1) + 12 π m(1 − λ) 00 q f (b)
− f (a)| + . ≤ 2(α + k) 4 λ 2Γ q( αk + 1) + 1 5 694

Chunyan Luo ET AL 690-705

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Proof. Using Lemma 2.1 and the power-mean integral inequality, we obtain Rf (α, k; φ, η, m, a, b)
2 Z 1 1 − t αk +1 − (1 − t) αk +1 00
eiφ η(b, a, m) f ma + teiφ η(b, a, m) dt ≤ α 2 0 k +1
1   Z 2 1− q 1 eiφ η(b, a, m) ≤ 1dt 2( αk + 1) 0 ) q1 (Z q 1 q
α α 1 − t k +1 − (1 − t) k +1 f 00 ma + teiφ η(b, a, m) dt × 0


2  Z 1   q1 
q k e η(b, a, m) α α ≤ 1 − tq( k +1) − (1 − t)q( k +1) f 00 ma + teiφ η(b, a, m) dt . 2(α + k) 0 iφ

To prove the third inequality above, we use the following inequality  q α α α α 1 − (1 − t) k +1 − t k +1 ≤ 1 − (1 − t)q( k +1) − tq( k +1) ,

(3.2)

for any t ∈ (0, 1), which follows from (A − B)q ≤ Aq − B q , for any A > B ≥ 0 and q ≥ 1. Since |f 00 |q is generalized λφm -M T -preinvex on Aφm , it follows that Z

1


q α α 1 − tq( k +1) − (1 − t)q( k +1) f 00 ma + teiφ η(b, a, m) dt 0 ( ) √ Z 1  m(1 − λ)√1 − t α t +1) q( +1) 00 q q( α 00 q √ ≤ − (1 − t) k |f (a)| + √ 1−t k |f (b)| dt 2 1−t 2λ t 0 ( )    1 3 1 1 α  3 α m(1 − λ) π 1 f 00 (a) q − β q +1 + , − β ,q +1 + = λ 4 2 k 2 2 2 2 k 2 ( )       3 1  1 00 q π 1 α 1 3 α f (b) + − β q +1 + , − β ,q +1 + 4 2 k 2 2 2 2 k 2 " )
#( √ 00 q πΓ q( αk + 1) + 12 π m(1 − λ) 00 q
− f (a)| + f (b) . = 4 λ 2Γ q( αk + 1) + 1 

Here, we utilize the following fact that Z 1√ Z 1 √ 1−t t 1 1 3 π √ dt = √ dt = β , = , 2 2 2 4 2 t 0 0 2 1−t  Z 1 α  3 1 1 1 q( α +1)+ − 2 (1 − t) 2 dt = β t k q +1 + , k 2 2 0 and Z

1

α

1

1

(1 − t)q( k +1)− 2 t 2 dt = β

0



 1 3 α ,q +1 + 2 k 2



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where the beta function Z β(x, y) =

1

tx−1 (1 − t)y−1 dt =

0

Γ(x)Γ(y) , Γ(x + y)

∀ x, y > 0.

Hence, the proof is completed. We now discuss some special cases of Theorem 3.1. Corollary 3.1 In Theorem 3.1, if q = 1, then we have " #( ) k eiφ η(b, a, m)
2 π √πΓ α + 3
00 m(1 − λ) 00 k 2
f (a)| + f (b) . − Rf (α, k; φ, η, m, a, b) ≤ 2(α + k) 4 λ 2Γ αk + 2 Corollary 3.2 In Theorem 3.1, if we take λ = 21 , q = 1 and the mapping η(b, a, m) with m = 1 degenerates into η(b, a), then we have the following inequality for φ-MT-preinvex functions   f (a) + f a + eiφ η(b, a)

Γk (α + k) iφ α α −
αk k Ja+ f a + e η(b, a) + k J(a+eiφ η(b,a))− f (a) 2 iφ 2k e η(b, a) ) "

#( √ 2 00 00 k eiφ η(b, a) πΓ αk + 32 π f (a)| + f (b) .
≤ − 2(α + k) 4 2Γ αk + 2 Remark 3.1 In Corollary 3.2, if we put φ = 0 and η(b, a) = b − a, then we have the succeeding inequality for MT-preinvex functions   f (a) + f (b) Γk (α + k) α α J + f (b) + k Jb− f (a) − α 2 2k(b − a) k k a #( ) "
√ 00 00 πΓ αk + 23 k(b − a)2 π f (a)| + f (b) .
≤ − 2(α + k) 4 2Γ αk + 2 Especially if we take k = 1 and α = 1, we have  Z b f (a) + f (b) π(b − a)2  00 1 00 − f (a) + f (b) . f (x)dx ≤ 2 b−a a 64 Corollary 3.3 In Theorem 3.1, if |f 00 (x)| ≤ M , λ = 21 and η(b, a, m) with m = 1 degenerates into η(b, a), then we have the forthcoming inequality for φ-MT-preinvex functions   f (a) + f a + eiφ η(b, a)

Γk (α + k) α iφ α −
α k Ja+ f a + e η(b, a) + k J(a+eiφ η(b,a))− f (a) 2 2k eiφ η(b, a) k
# q1
2 " √ kM eiφ η(b, a) πΓ q( αk + 1) + 12 π
− . ≤ 2(α + k) 2 Γ q( αk + 1) + 1 Especially if we take α = 1, q = 1 and k = 1, we get Z a+eiφ η(b,a) M π eiφ η(b, a)
2 f (a) + f (a + eiφ η(b, a)) 1 − iφ f (x)dx ≤ , 2 e η(b, a) a 32

(3.3)

which is the result given in [42],Theorem 2.5. Obviously, if we choose φ = 0 and η(b, a) = b − a in (3.3), then we obtain the result given in [30],Theorem 2.1. 7 696

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Now, we are ready to prove our second theorem. Theorem 3.2 Suppose that all the assumptions of Theorem 3.1 are satisfied, then we have the following inequality Rf (α, k; φ, η, m, a, b) # q1 " # q1 "
2 √ k eiφ η(b, a, m)  α 1− q1 π πΓ( αk + 32 ) m(1 − λ) 00 q
≤ f (a)|q + f 00 (b) − . 2(α + k) α + 2k 4 λ 2Γ αk + 2 (3.4) Proof. Using Lemma 2.1 and the H¨ older’s integral inequality for q ≥ 1, we get Rf (α, k; φ, η, m, a, b)
2 Z 1 1 − t αk +1 − (1 − t) αk +1 00
eiφ η(b, a, m) f ma + teiφ η(b, a, m) dt ≤ α 2 0 k +1 )1− q1
2 ( Z 1   eiφ η(b, a, m) α α +1 +1 ≤ dt 1 − t k − (1 − t) k 2( αk + 1) 0 (Z ) q1 1 
q α α +1 +1 00 iφ × 1 − t k − (1 − t) k f ma + te η(b, a, m) dt 0

k eiφ η(b, a, m) = 2(α + k)


2

α α + 2k

!1− q1 ( Z

1



1−t

α k +1

− (1 − t)

0

α k +1

) q1 q 
00 f ma + teiφ η(b, a, m) dt .

By the generalized λφm -M T -preinvexity of |f 00 |q on Aφm for q ≥ 1, we have Z 1 
q α α 1 − t k +1 − (1 − t) k +1 f 00 ma + teiφ η(b, a, m) dt 0 √  Z 1  m(1 − λ)√1 − t α α t 00 q 00 q +1 +1 k k √ ≤ 1−t |f (a)| + √ |f (b)| dt − (1 − t) 2 1−t 2λ t 0 " # m(1 − λ) π 1  α 3 3  1  1 α 5  00 q = − β + , − β , + f (a) λ 4 2 k 2 2 2 2 k 2 " # π 1  α 5 1  1  3 α 3  00 q + − β + , − β , + f (b) 4 2 k 2 2 2 2 k 2 " #" # √ 00 q πΓ( αk + 32 ) m(1 − λ) 00 π q
= − f (a)| + f (b) . 4 λ 2Γ αk + 2 Hence, the proof is completed. Let us discuss some special cases of Theorem 3.2. Corollary 3.4 In Theorem 3.2, if the mapping η(b, a, m) with m = 1 degenerates into η(b, a), then we obtain the following inequality for λφ -MT-preinvex functions   f (a) + f a + eiφ η(b, a)

Γk (α + k) α iφ α −
α k Ja+ f a + e η(b, a) + k J(a+eiφ η(b,a))− f (a) 2 2k eiφ η(b, a) k # q1 " #1
2  √ 1− q1 " 00 q q k eiφ η(b, a) πΓ( αk + 32 ) α π (1 − λ) 00 q
− f (a)| + f (b) . ≤ 2(α + k) α + 2k 4 λ 2Γ αk + 2 8 697

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Corollary 3.5 In Theorem 3.2, if φ = 0, λ = 12 and η(b, a, m) = b − ma with m = 1, then we have the following inequality for MT-convex functions   f (a) + f (b) Γk (α + k) α α − J + f (b) + k Jb− f (a) α 2 2k(b − a) k k a 1" " # # q1 √ q 00 πΓ( αk + 32 ) k(b − a)2  α 1− q1 π q f (a)|q + f 00 (b)
≤ − . 2(α + k) α + 2k 4 2Γ αk + 2 Corollary 3.6 In Theorem 3.2, if |f 00 (x)| ≤ M , λ = 21 and η(b, a, m) with m = 1 degenerates into η(b, a), then we have the following inequality for φ-MT-preinvex functions   f (a) + f a + eiφ η(b, a)

Γk (α + k) iφ α α −
αk k Ja+ f a + e η(b, a) + k J(a+eiφ η(b,a))− f (a) 2 iφ 2k e η(b, a) " # q1
√ 2 kM eiφ η(b, a)  α 1− q1 π πΓ( αk + 23 )
≤ − . 2(α + k) α + 2k 2 Γ αk + 2 Especially if we take α = 1 = k, we get Z a+eiφ η(b,a) f (a) + f (a + eiφ η(b, a)) M (eiφ η(b, a))2  1 1− q1  π  q1 1 − iφ f (x)dx ≤ , 2 e η(b, a)) a 2 6 16 (3.5) which is the result given in [42],Theorem 2.15. Clearly, if we put φ = 0 and η(b, a) = b − a in (3.5), we obtain the result given in [30], Theorem 2.4. A different approach leads to the following results. Theorem 3.3 Let Aφm ⊆ R0 be an open (φ, m)-invex subset respecting φ(·) and η : Aφm × Aφm × (0, 1] → R0 , a, b ∈ Aφm with η(b, a, m) > 0, and let f : Aφm → R be a twice differentiable mapping such that f 00 ∈ L[ma, ma + eiφ η(b, a, m)]. If |f 00 |q is generalized λφm -M T -preinvex on p Aφm , λ ∈ (0, 21 ], m ∈ (0, 1], q = p−1 , q 6= p > 1 and x ∈ [ma, ma + eiφ η(b, a, m)], then we have the following inequality for k-fractional integrals with k, α > 0 " # q1 k eiφ η(b, a, m)
2  p(α + k) − k  p1 π  m(1 − λ) q q f 00 (a) + f 00 (b) . Rf (α, k; φ, η, m, a, b) ≤ 2(α + k) p(α + k) + k 4 λ (3.6) Proof. Using Lemma 2.1 and H¨ older’s integral inequality leads to Rf (α, k; φ, η, m, a, b)
2 Z 1 1 − t αk +1 − (1 − t) αk +1 00
eiφ η(b, a, m) f ma + teiφ η(b, a, m) dt ≤ α 2 0 k +1
  q1 Z 2 1 p  p1  Z 1
q k eiφ η(b, a, m) α α 00 +1 +1 iφ k k 1−t − (1 − t) dt ≤ f ma + te η(b, a, m) dt 2(α + k) 0 0

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2  Z 1    p1 k eiφ η(b, a, m) p( α +1) p( α +1) k k 1−t ≤ dt − (1 − t) 2(α + k) 0 (Z  √ √  ) q1 1 m(1 − λ) 1 − t 00 t q 00 q √ |f (a)| + √ × |f (b)| dt 2 1−t 2λ t 0
2  # q1 1 "  k eiφ η(b, a, m) p(α + k) − k p π m(1 − λ) 00 q 00 q = f (a) + f (b) . 2(α + k) p(α + k) + k 4 λ To prove the third inequality above, we use the same inequality (3.2) as Theorem 3.1, the generalized λφm -M T -preinvexity of |f 00 |q on Aφm for q > 1, and the following fact Z 1  α α p(α + k) − k 1 − (1 − t)p( k +1) − tp( k +1) dt = . p(α + k) + k 0 This ends the proof of Theorem 3.3. Let us point out some special cases of Theorem 3.3. Corollary 3.7 In Theorem 3.3, if the mapping η(b, a, m) with m = 1 degenerates into η(b, a) and λ = 12 , then we have the following inequality for φ-MT-preinvex functions   f (a) + f a + eiφ η(b, a)

Γk (α + k) iφ α α −
αk k Ja+ f a + e η(b, a) + k J(a+eiφ η(b,a))− f (a) 2 iφ 2k e η(b, a)
 1 "  # q1 2 k eiφ η(b, a) p(α + k) − k p π ≤ |f 00 (a)|q + |f 00 (b)|q . 2(α + k) p(α + k) + k 4 Corollary 3.8 In Theorem 3.3, if we put φ = 0 and η(b, a, m) = b − ma with m = 1, then we have the following inequality for λ-MT-convex functions   f (a) + f (b) Γk (α + k) α α − J + f (b) + k Jb− f (a) α 2 2k(b − a) k k a "  1  # q1 k(b − a)2 p(α + k) − k p π (1 − λ) 00 q 00 q ≤ f (a) + f (b) . 2(α + k) p(α + k) + k 4 λ Especially if we take k = 1 and λ = 12 , we have   f (a) + f (b) Γ(α + 1) α − Ja+ f (b) + Jbα− f (a) α 2 2(b − a)  1  1 (b − a)2 p(α + 1) − 1 p π  00 q 00 q  q f (a) + f (b) . ≤ 2(α + 1) p(α + 1) + 1 4 Corollary 3.9 In Theorem 3.3, if |f 00 (x)| ≤ M , φ = 0, λ = 12 and the mapping η(b, a, m) = b − ma with m = 1 , then we have the following inequality for MT-convex functions    1  1 f (a) + f (b) Γk (α + k) kM (b − a)2 π q p(α + k) − k p α α − J + f (b) + k Jb− f (a) ≤ . α 2 2(α + k) 2 p(α + k) + k 2k(b − a) k k a 10 699

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Especially if we take k = 1 and α = 1, then we get  1  1 Z b M (b − a)2 π q 2p − 1 p f (a) + f (b) 1 ≤ f (x)dx − . 2 b−a a 4 2 2p + 1 Finally, we are in a position to present the following result. Theorem 3.4 Suppose that the assumptions of Theorem 3.3 are satisfied, then we have the following inequality Rf (α, k; φ, η, m, a, b)
2   q−1 q k eiφ η(b, a, m) (q − p)α − pk + k ≤ (3.7) 2(α + k) (q − p)α + 2qk − pk − k # q1 # q1 " "
√ q πΓ p( αk + 1) + 21 m(1 − λ) 00 π
f (a)|q + f 00 (b) − . × α 4 λ 2Γ p( k + 1) + 1 Proof. Using Lemma 2.1 and H¨ older’s inequality, we have Rf (α, k; φ, η, m, a, b)
2 Z 1 1 − t αk +1 − (1 − t) αk +1 00
eiφ η(b, a, m) f ma + teiφ η(b, a, m) dt ≤ α 2 0 k +1 # q−1 "
2 Z 1  q  q−p eiφ η(b, a, m) α α q−1 +1 +1 k k dt − (1 − t) ≤ 1 − t 2( αk + 1) 0 "Z # q1 q 1 p
α α × 1 − t k +1 − (1 − t) k +1 f 00 ma + teiφ η(b, a, m) dt

(3.8)

0


2  Z 1   q−1 q ( α +1)(q−p) ( α +1)(q−p)  k eiφ η(b, a, m) k k q−1 q−1 ≤ 1−t − (1 − t) dt 2(α + k) 0  q1 Z 1  
q α α × 1 − tp( k +1) − (1 − t)p( k +1) f 00 ma + teiφ η(b, a, m) dt . 0

By the generalize λφm -M T -preinvexity of |f 00 |q on Aφm for q > 1, we have Z

1


q α α 1 − tp( k +1) − (1 − t)p( k +1) f 00 ma + teiφ η(b, a, m) dt 0 √  Z 1  m(1 − λ)√1 − t α t p( α +1) p( +1) 00 q 00 q √ − (1 − t) k |f (a)| + √ |f (b)| dt ≤ 1−t k 2 1−t 2λ t 0 " #    1 3 1 1 α  3 m(1 − λ) π 1 α f 00 (a) q = − β p +1 + , − β ,p +1 + λ 4 2 k 2 2 2 2 k 2 # "    3 1 1 3 α  1 π 1 α f 00 (b) q + − β p +1 + , − β ,p +1 + 4 2 k 2 2 2 2 k 2 # " #"
√ πΓ p( αk + 1) + 12 m(1 − λ) 00 π q
− f (a)|q + f 00 (b) . = 4 λ 2Γ p( αk + 1) + 1 

(3.9)

11 700

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Also Z

1

 1−t

( α +1)(q−p) k q−1

− (1 − t)

( α +1)(q−p) k q−1

 dt =

0

(q − p)α − pk + k . (q − p)α + 2qk − pk − k

(3.10)

Utilizing (3.9) and (3.10) in (3.8), we deduce the inequality (3.7). This completes the proof of Theorem 3.4 as well. We next discuss some special cases of Theorem 3.4. Corollary 3.10 In Theorem 3.4, if the mapping η(b, a, m) with m = 1 degenerates into η(b, a), then we obtain the following inequality for λφ -MT-preinvex functions   f (a) + f a + eiφ η(b, a)

Γk (α + k) α iφ α −
αk k Ja+ f a + e η(b, a) + k J(a+eiφ η(b,a))− f (a) 2 iφ 2k e η(b, a) " # q−1
2 q k eiφ η(b, a) (q − p)α − pk + k ≤ 2(α + k) (q − p)α + 2qk − pk − k " #1
# q1 " √ 00 q q πΓ p( αk + 1) + 21 π (1 − λ) 00 q
× − f (a)| + f (b) . 4 λ 2Γ p( αk + 1) + 1 Corollary 3.11 In Theorem 3.4, if we put φ = 0 and η(b, a, m) = b − ma with m = 1, then we obtain the following inequality for λ-MT-convex functions   f (a) + f (b) Γk (α + k) α α − J + f (b) + k Jb− f (a) α 2 2k(b − a) k k a q−1   q k(b − a)2 (q − p)α − pk + k ≤ 2(α + k) (q − p)α + 2qk − pk − k # q1 "
# q1 " √ πΓ p( αk + 1) + 12 π (1 − λ) 00 q
f (a)|q + f 00 (b) × − . 4 λ 2Γ p( αk + 1) + 1 Especially if we take k = 1 and λ = 12 , then we have the following inequality for MT-convex functions   f (a) + f (b) Γ(α + 1) α − J + f (b) + Jbα− f (a) 2 2(b − a)α a " #1
# q1 " √   q−1 q 00 00 q q πΓ p(α + 1) + 12 (q − p)α − p + 1 π (b − a)2 q
− f (a)| + f (b) . ≤ 2(α + 1) (q − p)α + 2q − p − 1 4 2Γ p(α + 1) + 1 Corollary 3.12 In Theorem 3.4, if |f 00 (x)| ≤ M , φ = 0, λ = 12 and the mapping η(b, a, m) = b − ma with m = 1 , then we have the following inequality for MT-convex functions   f (a) + f (b) Γk (α + k) α α − J + f (b) + k Jb− f (a) α 2 2k(b − a) k k a " # q−1 "
# q1 √ q πΓ p( αk + 1) + 12 (q − p)α − pk + k π kM (b − a)2
− . ≤ 2(α + k) (q − p)α + 2qk − pk − k 2 Γ p( αk + 1) + 1 12 701

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4

Applications to special means

We begin this section by considering some particular means for two positive real numbers a, b and for this purpose we recall the following well-known means: Arithmetic mean: A := A(a, b) = Geometric mean: G := G(a, b) =

a+b 2 ,

√ ab,

2ab Harmonic mean: H := H(a, b) = a+b ,  r r  r1 Power mean: Pr := Pr (a, b) = a +b , r ≥ 1, 2  1  1  bb  b−a , a 6= b, e aa Identric mean: = I(a, b) =  a, a = b,  b−a ln b−ln a , a 6= b, Logarthmic mean: L(a, b) = a, a = b,

and  h i1  bp+1 −ap+1 p    (p+1)(b−a) , p 6= 0, −1, and a 6= b, L(a, b), p = −1 and a 6= b, Generalized mean: Lp := Lp (a, b) =   I(a, b), p = 0 and a 6= b,   a, a = b. Clearly, Lp is monotonic nondecreasing over p ∈ R, with L−1 := L and L0 := I. In particular, we have H ≤ G ≤ L ≤ I ≤ A. Let 0 < a < b, λ ∈ (0, 12 ] and let M := M (a, b) : [a + η(b, a)] × [a, a + η(b, a)] → R+ , which is one of the above mentioned means, one can obtain various inequalities for these means. Now, if η(b, a, m) with m=1 degenerates into η(b, a) and η(b, a) := M (b, a), for φ = 0 in (3.1), (3.4), (3.6) and (3.7), we have the following interesting inequalities concerning the above means " )1
# q1 ( √ 00 q q πΓ q( αk + 1) + 21 kM 2 π (1 − λ) 00 q
− f (a)| + f (b) , Rf (α, k; 0, η, 1, a, b) ≤ 2(α + k) 4 λ 2Γ q( αk + 1) + 1 (4.1)

Rf (α, k; 0, η, 1, a, b) ≤

" # q1 ( )1 √ 00 q q πΓ( αk + 23 ) kM 2  α 1− q1 π (1 − λ) 00 q
− f (a)| + f (b) , 2(α + k) α + 2k 4 λ 2Γ αk + 2 (4.2)

Rf (α, k; 0, η, 1, a, b) ≤

 1 (  ) q1 kM 2 p(α + k) − k p π (1 − λ) 00 q 00 q f (a) + f (b) 2(α + k) p(α + k) + k 4 λ (4.3) 13 702

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and   q−1 q kM 2 (q − p)α − pk + k 2(α + k) (q − p)α + 2qk − pk − k " ) q1
# q1 ( √ πΓ p( αk + 1) + 12 π (1 − λ) 00 q
f (a)|q + f 00 (b) × − , 4 λ 2Γ p( αk + 1) + 1

Rf (α, k; 0, η, 1, a, b) ≤

(4.4)

where f (a) + f a + M (a, b)
Γk (α + k) − Rf (α, k; 0, η, 1, a, b) = α 2 2kM k (a, b) i h
α × k Jaα+ f a + M (b, a) + k J(a+M f (a) . − (b,a)) Letting M = A, G, H, Pr , I, L, Lp in (4.1), (4.2), (4.3) and (4.4), we also get the required inequalities, and the more details are left to the reader to explore.

Acknowledgments This work was partially supported by the National Natural Science Foundation of China under Grant No. 61374028.

References [1] P. Agarwal, J. Tariboon, S. K. Ntouyas, Some generalized Riemann-Liouville k-fractional integral inequalities , J. Inequal. Appl., 2016 (2016), Article ID 122, 13 pages. [2] A. Ali, G. Gulshan, R. Hussain, A. Latif, M. Muddassar, Generalized inequalities of the type of Hermite-Hadamard-Fejer with quasi-convex functions by way of k-fractional derivatives, J. Comput. Anal. Appl., 22 (7) (2017), 1208-1219. [3] G. A. Anastassiou, Generalised fractional Hermite-Hadamard inequalities involving mconvexity and (s, m)-convexity, Facta Univ. Ser. Math. Inform., 28 (2) (2013), 107-126. [4] G. A. Anastassiou, Fractional Differentiation Inequalities, Research Monograph, Springer, New York, 2009. [5] T. Antczak, Mean value in invexity analysis, Nonlinear Anal., 60 (2005), 1473-1484. [6] M. U. Awan, M. A. Noor, M. V. Mihai, K. I. Noor, On bounds involving k-Appell’s hypergeometric functions, J. Inequal. Appl., 2017 (2017), Article ID 118, 15 pages. [7] F. X. Chen, Extensions of the Hermite-Hadamard inequality for convex functions via fractional integrals, J. Math. Inequal., 10 (2016), 75–81. [8] Y.-M. Chu, M. A. Khan, T. Ali, S. S. Dragomir, Inequalities for α-fractional differentiable functions, J. Inequal. Appl., 2017 (2017), Article ID 93, 12 pages. [9] Y.-M. Chu, M. A. Khan, T. U. Khan, T. Ali, Generalizations of Hermite-Hadamard type inequalities for MT-convex functions, J. Nonlinear Sci. Appl., 9 (2016), 4305–4316.

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[10] S. S. Dragomir, M. I. Bhatti, M. Iqbal, M. Muddassar, Some new Hermite-Hadamard’s type fractional integral inequalities, J. Comput. Anal. Appl., 18 (4) (2015), 655-661. [11] T. S. Du, J. G. Liao, Y. J. Li, Properties and integral inequalities of Hadamard-Simpson type for the generalized (s, m)-preinvex functions, J. Nonlinear Sci. Appl., 9 (5) (2016), 3112-3126. [12] T. S. Du, Y. J. Li, Z. Q. Yang, A generalization of Simpson’s inequality via differentiable mapping using extended (s, m)-convex functions, Appl. Math. Comput., 293 (2017), 358369. [13] T. S. Du, S. H. Wu, S. J. Zhao, M. U. Awan, Riemann-Liouville fractional HermiteHadamard inequalities for h-preinvex functions, J. Comput. Anal. Appl., 25 (2) (2018), 364-384. [14] S. Ermeydan, H. Yildirim, Riemann-Liouville Fractional Hermite-Hadamard Inequalities for differentiable λφ-preinvex functions, Malaya J. Mat., 4 (3) (2016), 430-437. [15] S. Hussain, S. Qaisar, More results on Hermite-Hadamard type inequality through (α, m)preinvexity, J. Appl. Anal. Comput., 6 (2016), 293-305. [16] S. R. Hwang, K. L. Tseng, K. C. Hsu, New inequalities for fractional integrals and their applications, Turkish J. Math., 40 (2016), 471-486. [17] M. Iqbal, M.I. Bhatti, K. Nazeer, Generalization of inequalities analogous to HermiteHadamard inequality via fractional integrals, Bull Korean Math. Soc., 52 (3) (2015), 707716. ˙ I¸ ˙ scan, S. H. Wu, Hermite-Hadamard type inequalities for harmonically convex functions [18] I. via fractional integrals, Appl. Math. Comput., 238 (2014), 237-244. [19] A. Kashuri, R. Liko, Generalizations of Hermite-Hadamard and Ostrowski type inequalities for MTm -preinvex functions, Proyecciones (Antofagasta), 36 (1) (2017), 45-80. [20] M. A. Khan, T. Ali, S. S. Dragomir, Hermite-Hadamard type inequalities for conformable fractional integrals, Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Math., (2017), DOI 10.1007/s13398-017-0408-5. [21] M. A. Latif, S. S. Dragomir, Generalization of Hermite-Hadamard type inequalities for n-times differentiable functions which are s-preinvex in the second sense with applications, Hacet. J. Math. Stat., 44 (2015), 839-853. [22] Y. J. Li, T. S. Du, A generalization of Simpson type inequality via differentiable functions using extended (s, m)φ -preinvex functions, J. Comput. Anal. Appl., 22 (4) (2017), 613-632. [23] W. J. Liu, Ostrowski type fractional integral inequalities for MT-convex functions, Miskolc Math. Notes, 16 (2015), 249-256. [24] S. Mubeen, G. M. Habibullah, k-fractional integrals and applications, Int. J. Contemp. Math. Sciences, 7 (2) (2012), 89-94. [25] M. A. Noor, K. I. Noor, M. V. Mihai, M. U. Awana, Fractional Hermite-Hadamard inequalities for some classes of differentiable preinvex functions, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., 78 (3) (2016), 163-174. 15 704

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[26] M. A. Noor, K. I. Noor, M. U. Awan, S. Khan, Hermite-Hadamard type inequalities for differentiable hϕ -preinvex functions, Arab. J. Math., 4 (2015), 63-76. [27] M. A. Noor, Some new classes of nonconvex functions, Nonlinear Funct. Anal. Appl., 12 (2006), 165-171. [28] O. Omotoyinbo, A. Mogbodemu, Some new Hermite-Hadamard integral inequalities for convex functions, Int. J. Sci. Innovation Tech., 1 (1) (2014), 001-012. ¨ [29] M. E. Ozdemir, S. S. Dragomir, C ¸ . Yildiz, The Hadamard inequality for convex function via fractional integrals, Acta. Math. Sci. Ser. B Engl. Ed., 33B (5) (2013), 1293-1299. [30] J. Park, Hermite-Hadamard-like type inequalities for twice differentiable M T -convex functions, Appl. Math. Sci., 9 (2015), 5235-5250. [31] S. Qaisar, M. Iqbal, M. Muddassar, New Hermite-Hadamard’s inequalities for preinvex functions via fractional integrals, J. Comput. Anal. Appl., 20 (7) (2016), 1318-1328. [32] M. Z. Sarikaya, E. Set, H. Yaldiz, N. Ba¸sak, Hermite-Hadamard’s inequalities for fractional integrals and related fractional inequalities, Math. Comput. Modelling, 57 (2013), 24032407. [33] M. Z. Sarikaya, A. Karaca, On the k-Riemann-Liouville fractional integral and applications, Int. J. Stat. Math., 1 (3) (2014), 33-43. [34] E. Set, M. Tomar, M. Z. Sarikaya, On generalized Gr¨ uss type inequalities for k-fractional integrals, Appl. Math. Comput., 269 (2015), 29-34. [35] M. Tun¸c, Y. S ¸ uba¸s, I. Karabayir, On some Hadamard type inequalities for M T -convex functions, Int. J. Open Problems Compt. Math., 6 (2) (2013), 102-113. [36] M. Tun¸c, Ostrowski type inequalities for functions whose derivatives are M T -convex, J. Comput. Anal. Appl., 17 (2014), 691-696. [37] J. R. Wang, X. Z. Li, M. Feˇckan, Y. Zhou, Hermite-Hadamard-type inequalities for Riemann-Liouville fractional integrals via two kinds of convexity, Appl. Anal., 92 (11) (2013), 2241-2253. [38] H. Wang, T. S. Du, Y. Zhang, k-fractional integral trapezium-like inequalities through (h, m)-convex and (α, m)-convex mappings, J. Inequal. Appl., 2017 (2017), Article ID 311, 20 pages. [39] T. Weir, B. Mond, Pre-invex functions in multiple objective optimization, J. Math. Anal. Appl., 136 (1988), 29-38. [40] S. H. Wu, B. Sroysang, J. S. Xie, Y. M. Chu, Parametrized inequality of Hermite-Hadamard type for functions whose third derivative absolute values are quasi-convex, SpringerPlus, 4 (2015), Article ID 831, 9 pages. [41] Y. C. Zhang, T. S. Du, J. Pan, On new inequalities of Fej´er-Hermite-Hadamard type for differentiable (α, m)-preinvex mappings, ScienceAsia, 43 (4) (2017), 258-266. [42] S. Zheng, T. S. Du, S. S. Zhao, L. Z. Chen, New Hermite-Hadamard inequalities for twice differentiable φ-MT-preinvex functions, J. Nonlinear Sci. Appl., 9 (10) (2016), 5648-5660. 16 705

Chunyan Luo ET AL 690-705

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.4, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

Some generalizations of operator inequalities for positive linear map Chaojun Yang and Fangyan Lu∗

Abstract: We generalize some inequalities for positive unital linear map as follows: Let A, B be positive operators on a Hilbert space with 0 < m ≤ A ≤ m0 < M 0 ≤ B ≤ M . Then for every positive unital linear map Φ, µ ∈ [0, 1] and p > 0, Φp (A∇µ B + 2rM m(A−1 ∇B −1 − A−1 ]B −1 )) ≤ αp Φp (A]µ B) and Φp (A∇µ B + 2rM m(A−1 ∇B −1 − A−1 ]B −1 )) ≤ αp (Φ(A)]µ Φ(B))p  2 (M +m)2 M0 0 √ where r = min {µ, 1 − µ}, h = m0 and α = max 4M mK( , 2 (M +m) √ h0 ,2)R

4 p M mK( h0 ,2)R

 . Fur-

thermore, we give a squaring reversed Karcher mean inequality involving positive linear map.

1. Introduction Through this paper, let m, m0 , M, M 0 be scalars. Other capital letters denote general elements of the C ∗ -algebra B(H) of all bounded linear operators on a complex separable Hilbert space (H, h·, ·i). 2 The Kantorovich constant is defined by K(h, 2) = (h+1) for h > 0. We write A ≥ 0(A > 0) to 4h mean the self-adjoint operator A is positive( strictly positive). The partial order A ≤ B is defined as B − A ≥ 0. For each µ ∈ [0, 1], the weighted arithmetic mean ∇µ and weight geometric mean ]µ for strictly positive operator A and B are defined below: A∇µ B = (1 − µ)A + µB

and

1

A]µ B = A 2 (A

−1 2

BA

−1 2

1

)µ A 2 .

When µ = 12 we write A∇B and A]B for brevity, respectively. A linear map Φ : B(H) → B(H) is called positive (strictly positive) if Φ(A) ≥ 0 (Φ(A) > 0) whenever A ≥ 0 (A > 0), and Φ is said to be unital if Φ(I) = I. The arithmetic-geometric mean for positive operator A and B states A+B 2

≥ A]B.

In [8], Lin give a reversed arithmetic-geometric mean inequality involving a positive linear map 2

(M +m) Φ( A+B 2 ) ≤ 4M m Φ(A]B) where 0 < m ≤ A, B ≤ M and Φ is a positive unital linear map.

(1)

∗

Corresponding author. 2010 Mathematics Subject Classification. Primary 47A63 ; Secondary 47B20 The research was supported by NNSFC (No. 11571247). Keywords and phrases : positive linear map; operator inequality; Kantorovich constant; Karcher mean. 1

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Chaojun Yang ET AL 706-713

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.4, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

2

It is well known that tα is operator monotone function on [0, ∞) if and only if α ∈ [0, 1]. Since t2 is not an operator monotone function, we can not obtain A2 ≥ B 2 directly by A ≥ B ≥ 0. However Lin [8] gave a p-th powering(0 < p ≤ 2) of inequality (1), namely the inequality p  (M +m)2 A+B p (2) Φ ( 2 )≤ Φp (A]B) 2 4 p Mm

and Φp ( A+B 2 )

 ≤

(M +m)2

p

2

(Φ(A)]Φ(B))p

(3)

4 p Mm

where 0 < m ≤ A, B ≤ M and Φ is a positive unital linear map. In [6], the authors extend (2) and (3) to p > 2, which states p  (M +m)2 A+B p Φ ( 2 )≤ Φp (A]B) 2

(4)

4 p Mm

and Φp ( A+B 2 )

 ≤

(M +m)2 2

p

(Φ(A)]Φ(B))p

(5)

4 p Mm

where 0 < m ≤ A, B ≤ M and Φ is a positive unital linear map. Recently the author in [4] gives inequalities that generalize the inequalities (2) to (5) and state as follows Φp (A∇µ B + 2rM m(A−1 ∇B −1 − A−1 ]B −1 )) ≤ αp Φp (A]µ B) (6) and Φp (A∇µ B + 2rM m(A−1 ∇B −1 − A−1 ]B −1 )) ≤ αp (Φ(A)]µ Φ(B))p (7) 0 0 where 0 < m ≤ A ≤ m < M ≤ B ≤  M , Φ be a positive  unital linear map on B(H), µ ∈ [0, 1], p > 0, r = min {µ, 1 − µ} and α = max

(M +m)2 (M +m)2 4M m , p2 4 Mm

.

The ω−weighted Karcher mean Λ(ω; A1 , . . . , An )(orΛ(ω; A)) of A1 , · · · , An > 0 is defined to be the unique positive definite solution of equation n X 1 1 ωi log(X − 2 Ai X − 2 ) = 0, i=1

where ω = (w1 , . . . , wn ) is a probability vector. Next we cite some basic properties of the Karcher mean as follows, for more details about Karcher mean, see[7]. Proposition 1.1. [7] The Karcher mean satisfies the following properties: (i) (consistency with scalars) Λ(ω; A) = Aω1 1 · · · Aωnn if the Ai is commute. −1 −1 = Λ(ω; A , · · · , A ). (ii) (self duality) Λ(ω; A−1 1 n 1 , · · · , An ) P Pn −1 −1 (iii) (AGH weighted mean inequalities) ( i=1 ωi Ai ) ≤ Λ(ω; A1 , · · · , An ) ≤ ni=1 ωi Ai .

(iv) Φ(Λ(ω; A)) ≤ Λ(ω; Φ(A)) f or any positive unital linear map Φ. (v) (monotonicity) If Bi ≤ Ai f or all 1 ≤ i ≤ n, then Λ(ω; B) ≤ Λ(ω; A). As mentioned in the abstract, we shall give refinements of inequalities (6) and (7), along with presenting a reversed Karcher mean inequality related to (iv) in Proposition 1.1 and a squaring version thereafter. 2. Main Results

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.4, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

3

Lemma 2.1.( Choi inequality.) [5] Let Φ be a unital positive linear map, then (i) when A > 0 and −1 ≤ p ≤ 0, then Φ(A)p ≤ Φ(Ap ), in particular,Φ(A)−1 ≤ Φ(A−1 ); (ii) when A ≥ 0 and 0 ≤ p ≤ 1, then Φ(A)p ≥ Φ(Ap ); (iii) when A ≥ 0 and 1 ≤ p ≤ 2, then Φ(A)p ≤ Φ(Ap ). Lemma 2.2. [1] Let Φ be a unital positive linear map and A, B be positive operators. Then for α ∈ [0, 1] Φ(A]α B) ≤ Φ(A)]α Φ(B). Lemma 2.3. [3] Let A, B ≥ 0. Then the following norm inequality holds: kABk≤ 41 kA + Bk2 . Lemma 2.4. [2] Let A, B ≥ 0. Then for 1 ≤ r < +∞, kAr + B r k≤ k(A + B)r k. Lemma 2.5. [5] (L-H inequality) If 0 ≤ α ≤ 1, A, B ≥ 0 and A ≥ B, then Aα ≥ B α . 1

1

1

1

Lemma 2.6. [9] For two operators A, B ≥ 0 and 1 < h ≤ A− 2 BA− 2 ≤ h0 or 0 < h0 ≤ A− 2 BA− 2 ≤ h < 1, we have √ A∇µ B − 2r(A∇B − A]B) ≥ K( h, 2)R A]µ B for all µ ∈ [0, 1], where r = min {µ, 1 − µ} and R = min {2r, 1 − 2r}. Lemma 2.7. Let 0 < m ≤ A ≤ m0 < M 0 ≤ B ≤ M , then √ A−1 ∇µ B −1 − 2r(A−1 ∇B −1 − A−1 ]B −1 ) ≥ K( h0 , 2)R A−1 ]µ B −1 for all µ ∈ [0, 1], where r = min {µ, 1 − µ}, h0 =

M0 m0

and R = min {2r, 1 − 2r}. 1

1

m ≤ (A−1 )− 2 (B −1 )(A−1 )− 2 ≤ Proof. Since 0 < m ≤ A ≤ m0 < M 0 ≤ B ≤ M , we have 0 < M m0 M 0 < 1. Thus by Lemma 2.6 we have √ A−1 ∇µ B −1 − 2r(A−1 ∇B −1 − A−1 ]B −1 ) ≥ K( h0 , 2)R A−1 ]µ B −1

q √ where K( h0 , 2) = K( h10 , 2). Theorem 2.8. Let 0 < m ≤ A ≤ m0 < M 0 ≤ B ≤ M , Φ be a positive unital linear map on B(H), µ ∈ [0, 1] and p > 0, we have Φp (A∇µ B + 2rM m(A−1 ∇B −1 − A−1 ]B −1 )) ≤ αp Φp (A]µ B)

(8)

Φp (A∇µ B + 2rM m(A−1 ∇B −1 − A−1 ]B −1 )) ≤ αp (Φ(A)]µ Φ(B))p

(9)

and

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Chaojun Yang ET AL 706-713

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.4, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

4

where r = min {µ, 1 − µ},

h0

=

M0 m0

 and α = max

(M +m)2 (M +m)2 √ , 4M mK( h0 ,2)R 4 p2 M mK(√h0 ,2)R

 .

Proof. By < m ≤ A ≤ m0 < M 0 ≤ B ≤ M we have A + M mA−1 ≤ M + m

and

B + M mB −1 ≤ M + m.

Thus we have (1 − µ)A + (1 − µ)M mA−1 ≤ (1 − µ)(M + m)

and

µB + µM mB −1 ≤ µ(M + m).

(10)

By (10) we obtain A∇µ B + M mA−1 ∇µ B −1 ≤ M + m.

(11)

√ A−1 ∇µ B −1 − 2r(A−1 ∇B −1 − A−1 ]B −1 ) ≥ K( h0 , 2)R A−1 ]µ B −1

(12)

By Lemma 2.7 we have

Compute √ ||Φ(A∇µ B + 2rM m(A−1 ∇B −1 − A−1 ]B −1 ))K( h0 , 2)R M mΦ−1 (A]µ B)|| √ ≤ 41 ||Φ(A∇µ B + 2rM m(A−1 ∇B −1 − A−1 ]B −1 )) + K( h0 , 2)R M mΦ−1 (A]µ B)||2 √ ≤ 41 ||Φ(A∇µ B + 2rM m(A−1 ∇B −1 − A−1 ]B −1 )) + K( h0 , 2)R Φ((A]µ B)−1 )||2 √ = 41 ||Φ(A∇µ B + 2rM m(A−1 ∇B −1 − A−1 ]B −1 )) + K( h0 , 2)R Φ(A−1 ]µ B −1 )||2 √ = 41 ||Φ(A∇µ B) + M mΦ(2r(A−1 ∇B −1 − A−1 ]B −1 ) + K( h0 , 2)R (A−1 ]µ B −1 ))||2 ≤ 41 ||Φ(A∇µ B) + M mΦ(A−1 ∇µ B −1 )||2 , ≤ 41 (M + m)2 where the first inequality is derived by Lemma 2.3, the second one is derived by Lemma 2.1, the third one is derived by (12) and the last one is derived by (11). Therefore ||Φ(A∇µ B + 2rM m(A−1 ∇B −1 − A−1 ]B −1 ))Φ−1 (A]µ B)|| ≤

(M +m)2 √ . 4M mK( h0 ,2)R

Hence Φ2 (A∇µ B + 2rM m(A−1 ∇B −1 − A−1 ]B −1 )) ≤ ( If 0 < p ≤ 2, then 0
2. We can obtain √ p p p (K( h0 , 2)R M m) 2 ||Φ 2 (A∇µ B + 2rM m(A−1 ∇B −1 − A−1 ]B −1 ))Φ− 2 (A]µ B)|| √ p p p = ||Φ 2 (A∇µ B + 2rM m(A−1 ∇B −1 − A−1 ]B −1 ))(K( h0 , 2)R M m) 2 Φ− 2 (A]µ B)|| √ p p p ≤ 41 ||Φ 2 (A∇µ B + 2rM m(A−1 ∇B −1 − A−1 ]B −1 )) + (K( h0 , 2)R M m) 2 Φ− 2 (A]µ B)||2 √ ≤ 14 ||Φ(A∇µ B + 2rM m(A−1 ∇B −1 − A−1 ]B −1 )) + K( h0 , 2)R M mΦ−1 (A]µ B)||p √ ≤ 41 ||Φ(A∇µ B + 2rM m(A−1 ∇B −1 − A−1 ]B −1 )) + K( h0 , 2)R M mΦ(A−1 ]µ B −1 )||p √ = 41 ||Φ(A∇µ B) + M mΦ(2r(A−1 ∇B −1 − A−1 ]B −1 ) + K( h0 , 2)R (A−1 ]µ B −1 ))||p ≤ 41 ||Φ(A∇µ B) + M mΦ(A−1 ∇µ B −1 )||p , ≤ 41 (M + m)p where the second inequality is obtained by Lemma 2.4. Therefore, we get inequality (8) for p > 2. Likewise, we have √ p p p (K( h0 , 2)R M m) 2 ||Φ 2 (A∇µ B + 2rM m(A−1 ∇B −1 − A−1 ]B −1 ))(Φ(A)]µ Φ(B))− 2 || √ p p p = ||Φ 2 (A∇µ B + 2rM m(A−1 ∇B −1 − A−1 ]B −1 ))(K( h0 , 2)R M m) 2 (Φ(A)]µ Φ(B))− 2 || √ p p p ≤ 14 ||Φ 2 (A∇µ B + 2rM m(A−1 ∇B −1 − A−1 ]B −1 )) + (K( h0 , 2)R M m) 2 (Φ(A)]µ Φ(B))− 2 ||2 √ ≤ 41 ||Φ(A∇µ B + 2rM m(A−1 ∇B −1 − A−1 ]B −1 )) + K( h0 , 2)R M m(Φ(A)]µ Φ(B))−1 ||p √ ≤ 41 ||Φ(A∇µ B + 2rM m(A−1 ∇B −1 − A−1 ]B −1 )) + K( h0 , 2)R M mΦ(A]µ B)−1 ||p √ ≤ 41 ||Φ(A∇µ B + 2rM m(A−1 ∇B −1 − A−1 ]B −1 )) + K( h0 , 2)R M mΦ((A]µ B)−1 )||p √ = 41 ||Φ(A∇µ B + 2rM m(A−1 ∇B −1 − A−1 ]B −1 )) + K( h0 , 2)R Φ(A−1 ]µ B −1 )||p √ = 41 ||Φ(A∇µ B) + M mΦ(2r(A−1 ∇B −1 − A−1 ]B −1 ) + K( h0 , 2)R (A−1 ]µ B −1 ))||p ≤ 41 ||Φ(A∇µ B) + M mΦ(A−1 ∇µ B −1 )||p , ≤ 41 (M + m)p . Remark 2.9. Since

(M +m)2 √ 4M mK( h0 ,2)R

≤

(M +m)2 4M m

and

(M +m)2

2 √ 4 p M mK( h0 ,2)R

≤

(M +m)2 2

, so under a stronger

4 p Mm

condition as Theorem 2.8, we see (8) and (9) are refinements of (6) and (7), respectively. Corollary 2.10. Let 0 < m ≤ A ≤ m0 < M 0 ≤ B ≤ M , µ ∈ [0, 1] and p > 0, we have

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Chaojun Yang ET AL 706-713

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.4, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

6

(A∇µ B + 2rM m(A−1 ∇B −1 − A−1 ]B −1 ))p ≤ αp (A]µ B)p where r = min {µ, 1 − µ},

h0

M0 m0

=

 and α = max

(M +m)2 (M +m)2 √ , 4M mK( h0 ,2)R 4 p2 M mK(√h0 ,2)R

 .

Proof. Put Φ(A) = A for all A ∈B(H) in Theorem 2.3, we then get the desired result. Theorem 2.11. Let Φ be a strictly unital positive linear map, 0 < m ≤ Ai ≤ M for i = 1, · · · n, ω = (ω1 , · · · , ωn ) be a probability vector, t ∈ [−1, 0). Then we have Λ(ω; Φ(A)) ≤

(m+M )2 4mM Φ(Λ(ω; A)).

(13)

Proof. By Proposition 1.1 and 0 < m ≤ Ai ≤ M we have n n P P ωi Ai + M m( ωi A−1 i ) ≤ M + m. i=1

i=1

First we show (

n P i=1

2

) 2 ωi Ai )2 ≤ ( (m+M 4mM ) (

n P i=1

−2 ωi A−1 i ) .

This inequality equals to k

n X

ωi Ai

i=1

n X

ωi A−1 i k≤

i=1

(M + m)2 . 4M m

Note that k(

n P

ωi Ai )M m(

i=1

≤ 41 k ≤

n P i=1

n P

ωi A−1 i )k

ωi Ai + M m(

i=1 1 4 (M +

n P i=1

2 ωi A−1 i )k

m)2 .

Use Lemma 2.5 we get n P

ωi A i ≤

i=1

n (m+M )2 P −1 ( ωi A−1 i ) . 4mM i=1

(14)

Thus by Proposition 1.1 and (14) we get Λ(ω; Φ(A)) ≤

n X i=1

n n X X (m + M )2 (m + M )2 −1 ωi Φ(Ai ) = Φ( ωi Ai ) ≤ Φ(( ωi A−1 ) ) ≤ Φ(Λ(ω; A)). i 4mM 4mM i=1

i=1

Remark 2.12. Since Φ(Λ(ω; A)) ≤ Λ(ω; Φ(A)) for any positive unital linear map, we get a reversed version of this inequality by Theorem 2.11. Next we give a squaring version of inequality (13). Theorem 2.13. Suppose all the assumptions of Theorem 2.11 be satisfied. Then (Λ(ω; Φ(A)))2 ≤ ψΦ2 (Λ(ω; A))

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Chaojun Yang ET AL 706-713

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.4, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

7

where ψ =

 M  K( m ,2)2 (M +m)2 4M m



K( M ,2)(M +m)−M m m

f or m ≤ t0 f or m ≥ t0

, t0 =

2M m K( M ,2)(M +m) m

and K( M m , 2) =

Proof. According to the assumption one can see that m ≤ Φ(Λ(ω; A)) ≤ M

(M +m)2 4M m .

(15)

and m ≤ Λ(ω; Φ(A)) ≤ M

(16)

inequality (15) implies Φ2 (Λ(ω; A)) ≤ (M + m)Φ(Λ(ω; A)) − M m, and inequality (16) give us Λ2 (ω; Φ(A)) ≤ (M + m)Λ(ω; Φ(A)) − M m. Hence Φ−1 (Λ(ω; A))Λ2 (ω; Φ(A))Φ−1 (Λ(ω; A)) ≤ Φ−1 (Λ(ω; A))((M + m)Λ(ω; Φ(A)) − M m)Φ−1 (Λ(ω; A)) ≤

(K( M m , 2)(M

+ m)Φ(Λ(ω; A)) −

(17)

M m)Φ−2 (Λ(ω; A))

where the second inequality is derived by Theorem 2.11. Consider the real function f (t) on (0, ∞) defined as f (t) =

K( M ,2)(M +m)t−M m m . t2

As a matter of fact, the inequality (17) implies that Φ−1 (Λ(ω; A))Λ2 (ω; Φ(A))Φ−1 (Λ(ω; A)) ≤ max f (t). m≤t≤M

Notice that f (m) ≥ f (M ) and f 0 (t) =

2M m−K( M ,2)(M +m)t m . t3

The function has an maximum point on t0 =

2M m K( M ,2)(M +m) m

with the maximum value f (t0 ) =

K( M ,2)2 (M +m)2 m . 4M m

Whence ( f (t0 ) f or m ≤ t0 max f (t) = m≤t≤M f (m) f or m ≥ t0 .

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Chaojun Yang ET AL 706-713

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.4, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

8

Notice that f (m) =

K( M ,2)(M +m)−M m . m

This completes the proof.

References [1] F. Kubo, T. Ando, Means of positive linear operators, Math. Ann., 246 (1980), 205-224. [2] R. Bhatia, Positive definite matrices, Princeton (NJ): Princeton University Press, (2007). [3] R. Bhatia, F. Kittaneh, Notes on matrix arithmetic-geometric mean inequalities, Linear Algebra Appl., 308 (2000), 203–211. [4] M. Bakherad, Refinements of a reversed AM-GM operator inequality, Linear Multilinear Algebra, 64 (2016), 1687-1695. [5] T. Furuta, J. Mi´ ci´ c Hot, J. Peˇ cari´ c and Y. Seo, Mond-Peˇ cari´ c method in operator inequalities, Monographs in inequalities 1, Element, Z´ agreb, (2005). [6] X. Fu, C. He, Some operator inequalities for positive linear maps, Linear Multilinear Algebra, 63 (2015), 571-577. [7] Y. Lim , M. P´ alfia, Matrix power means and the Karcher mean, J.Funct.Anal., 262 (2012), 1498-1514. [8] M. Lin, Squaring a reverse AM-GM inequality, Studia Math., 215 (2013), 187-194. [9] X. Zhao, L. Li and H Zuo Operator iteration on the Young inequality, Journal of Inequalities and Applications, Doi: 10.1186/s13660-016-1249-z. Chaojun Yang Department of Mathematics, Soochow University, Suzhou 215006, P. R. China E-mail address: [email protected] Fangyan Lu Department of Mathematics, Soochow University, Suzhou 215006, P. R. China E-mail address: [email protected]

713

Chaojun Yang ET AL 706-713

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.4, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

Locally and globally small Riemann sums and Henstock integral of fuzzy-number-valued functions in En a

Muawya Elsheikh Hamida,b ∗, Luoshan Xu a School of Mathematical Science, Yangzhou University, Yangzhou 225002, P.R. China b School of Management, Ahfad University for Women, Omdurman, Sudan

Abstract: In this paper, the notions of locally and globally small Riemann sums modifications with respect to a fuzzy-number-valued functions in E n are introduced and studied. The basic properties and characterizations are presented. In particular, it is proved that a fuzzy-number-valued functions in E n is Henstock (H) integrable on [a, b] if and only if it has (LSRS), and also it is proved that a fuzzy-number-valued functions in E n is Henstock (H) integrable on [a, b] if and only if it has (GSRS). Keywords: Fuzzy-number-valued functions in E n ; Henstock integral (H); Locally small Riemann sums (LSRS); Globally small Riemann sums (GSRS).

1

Introduction

Since the concept of fuzzy sets was firstly introduced by Zadeh in 1965 [12], it has been studied extensively from many different aspects of the theory and applications, such as fuzzy topology, fuzzy analysis, fuzzy decision making and fuzzy logic, information science and so on. The locally and globally small Riemann sums have been introduced by many authors from different points of views including [2, 3, 5, 6, 8, 9]. In 1986, Schurle characterized the Lebesgue integral in (LSRS) (locally small Riemann sums) property [8]. The (LSRS) property has been used to characterized the Perron (P ) integral on [a, b] [9]. By considering the equivalency between the (P ) integral and the Henstock-Kurzweil (HK) integral, the (LSRS) property has been used to characterized the (HK) integral on [a, b] [6]. The (LSRS) property brought a research to have global characterization on the Riemann sums of an (HK) integrable function on [a, b]. This research has been done by considering the following fact: Every (HK) integrable function on [a, b] is measurable, however, there is no guarantee the boundedness of the function. A measurable function f is (HK) integrable on [a, b] depends on it behaves on the set of x in which |f (x)| is large, i.e. |f (x)| ≥ N for some N . This fact has been characterized in (GSRS) (globally small Riemann sums) property [6]. The (GSRS) property involves one characteristic of the primitive of an (HK) integrable function. That is the primitive of the (HK) integral on [a, b] is ACG∗ (generalized strongly absolutely continuous) on [a, b]. This is not a simple concept. In 2015, Indrati [5] introduced a countably Lipschitz condition of a function which is simpler than the ACG∗ , and proved that the (HK) integrable function or it, s primitive could be characterized in countably Lipschitz condition. Also, by considering the characterization of the (HK) integral in the (GSRS) property, it showed that the relationship between (GSRS) property and countably Lipschitz condition of an (HK) integrable function on [a, b]. In 2018, Hamid et al. [2] investigated locally and globally small Riemann sums for fuzzy-number-valued functions and proved two main theorems: (1) A fuzzy-number-valued functions f˜(x) is Henstock integrable on [a, b] if and only if f˜(x) has (LSRS). (2) A fuzzy-number-valued functions f˜(x) is Henstock integrable on [a, b] if and only if f˜(x) has (GSRS). ∗ Corresponding author. Tel.: +8613218977118. E-mail address: [email protected], [email protected] (M.E. Hamid). 714 Hamid ET AL 714-722

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M.E. Hamid, L.S. Xu: Locally and globally small Riemann sums and Henstock integral...

In this paper, we generalize locally and globally small Riemann sums from fuzzy-valued functions to n-dimensional fuzzy-numbers by means of support function. The notions of locally small Riemann sums for n-dimensional fuzzynumber-valued functions are presented and discussed. Finally, we provide a characterizations of globally small Riemann sums in n-dimensional fuzzy-number-valued functions. The rest of this paper is organized as follows, in Section 2 we shall review the relevant concepts and properties of fuzzy-number-valued functions in E n and the definition of Henstock integrals for fuzzy-number-valued functions in E n . Section 3 is devoted to discussing the support function characterizations of locally small Riemann sums and Henstock integral for fuzzy-number-valued functions in E n . In section 4 we shall investigate the support function characterizations of globally small Riemann sums and Henstock integral for fuzzy-number-valued functions in E n . The last section provides the Conclusions.

2

Preliminaries

 In this paper the close interval [a, b] denotes a compact interval on R. The set of intervals-point ([a1 , b1 ], ξ1 ), ([a2 , b2 ], ξ2 ), · · · , ([ak , bk ], ξk ) is called a division of [a, b] that is ξ1 , ξ2 , · · · , ξk ∈ [a, b], intervals [a1 , b1 ], [a2 , b2 ], · · · , [ak , bk ] k  S are non-intersect and [ai , bi ] = [a, b]. Marking the division of [a, b] as P = ([a1 , b1 ], ξ1 ), ([a2 , b2 ], ξ2 ), · · · , ([ak , bk ], ξk ) ,  i=1 shortening as P = [u, v]; ξ [7]. Definition 2.1 [4, 6] Let δ : [a, b] → R+ be a positive real-valued function. P = {[xi−1 , xi ]; ξi } is said to be a δ-fine division, if the following conditions are satisfied: (1) a = x0 < x1 < x2 < ... < xn = b; (2) ξi ∈ [xi−1 , xi ] ⊂ (ξi − δ(ξi ), ξi + δ(ξi ))(i = 1, 2, · · · , n). For brevity, we write P = {[u, v]; ξ}, where [u, v] denotes a typical interval in P and ξ is the associated point of [u, v]. Definition 2.2 [11] E n is said to be a fuzzy number space if E n = {u : Rn → [0, 1] : u satisfies (1)-(4) below}: (1) u is normal, i.e., there exists a x0 ∈ Rn such that u(x0 ) = 1; (2) u is a convex fuzzy set, i.e., u(rx + (1 − r)y) > min(u(x), u(y)), x, y ∈ Rn , r ∈ [0, 1]; (3) u is upper semi-continuous; (4) [u]0 = {x ∈ Rn : u(x) > 0} is compact, for 0 < r ≤ 1, denote [u]r = {x : x ∈ Rn and u(x) > r}, [u]0 = r r∈(0,1] [u] .

S

Form (1)-(4), it follows that for any u ∈ E n and r ∈ [0, 1] the r−level set [u]r is a compact convex set. For any u, v ∈ E n D(u, v) = sup d([u]r , [v]r ),

(2.1)

r∈[0,1]

where d is Hausdorff metric. It is well known that (E n , d) is an metric space [11]. The norm of fuzzy number u ∈ E n is defined by ˜ = sup |α|, (2.2) kuk = D(u, 0) α∈[u]0

where the k · k is norm on E n , ˜ 0 is fuzzy number on E n and ˜ 0 = χ{0} . Definition 2.3 [11] For A ∈ Pk (Rn ), x ∈ S n−1 , define the support function of A as σ(x, A) = sup hy, xi, whereS n−1 y∈A

is the unit sphere of Rn , i.e., S n−1 = {x ∈ Rn : kxk = 1}, h·, ·i is the inner product in Rn . Definition 2.4 [10] A fuzzy-number-valued function f˜ : [a, b] → E n is said to be Henstock integrable to A˜ ∈ E n if for every ε > 0, there is a function δ(t) > 0 such that for any δ-fine division P = {[u, v]; ξ} of [a, b], we have D where the sum

P

X


f˜(ξ)(v − u), A˜ < ε,

is understood to be over P and we write (F H) 715

Rb a

(2.3)

f˜(t)dt = A˜ , and f˜(t) ∈ F H[a, b]. Hamid ET AL 714-722

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.4, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

M.E. Hamid, L.S. Xu: Locally and globally small Riemann sums and Henstock integral...

Lemma 2.1 [11] If u, v ∈ E n , k ∈ R, for any r ∈ [0, 1], we have [u + v]r = [u]r + [v]r , [ku]r = k[u]r . Lemma 2.2 [11] Suppose u ∈ E n , then (1) u∗ (r, x + y) ≤ u∗ (r, x) + u∗ (r, y), (2) if u, v ∈ E n , r ∈ [0, 1], then d([u]r , [v]r ) =

(2.4)

sup |u∗ (r, x) − v ∗ (r, x)|,

(2.5)

x∈S n−1

(3) (u + v)∗ (r, x) = u∗ (r, x) + v ∗ (r, x), (4) (ku)∗ (r, x) = ku∗ (r, x), k ≥ 0. Lemma 2.3 [1, 11] Given u, v ∈ E n the distance D : E n × E n → [0, +∞) between u and v is defined by the equation D(u, v) = sup d([u]r , [v]r ), then r∈[0,1]

(1) (2) (3) (4) (5) (6) Where

(E n , D) is a complete metric space, D(u + w, v + w) = D(u, v), D(u + v, w + e) 6 D(u, w) + D(v, e), D(ku, kv) = |k|D(u, v), k ∈ R, D(u + v, ˜ 0) 6 D(u, ˜ 0) + D(v, ˜ 0), D(u + v, w) 6 D(u, w) + D(v, ˜ 0). n e e u, v, w, e, 0 ∈ E , 0 = X({0}) .

Lemma 2.4 [1] If f˜ : [a, b] → E n , then the following statements are equivalent: (1) f˜ is (F H) integrable. (2) f ∗ (ξ)(r, x) is (RH) integrable for any r ∈ [0, 1] uniformly, i.e., for every ε > 0 there is a δ(ξ) > 0 which is independent of r ∈ [0, 1], such that for any δ-fine division P = {[u, v]; ξ} and r ∈ [0, 1] we have |

3

X

f ∗ (ξ)(r, x)(v − u) − A∗ (r, x)| < ε.

(2.6)

Support function characterizations of locally small Riemann sums and Henstock integral for fuzzy-number-valued functions in E n

In this section, we define the locally small Riemann sums for fuzzy-number-valued functions in n-dimensional and investigate their properties. We star with the following definition. Definition 3.1 A fuzzy-number-valued function f˜ : [a, b] → E n is said to be have locally small Riemann sums or (LSRS) if for every ε > 0 there is a δ(ξ) > 0 such that for every t ∈ [a, b], we have ||

X

f˜(ξ)(v − u)||E n < ε,

(3.1)

whenever P = {[u, v]; ξ} is a δ-fine division of an interval C ⊂ (t − δ(t), t + δ(t)), t ∈ C and Σ sums over P . (Where C = [y, z]). The following Theorem 3.1 shows that f˜ has (LSRS) is equal to the type of it, s support functions. Theorem 3.1 Let f˜ : [a, b] → E n be a fuzzy-number-valued function, the support-function-wise f ∗ (ξ)(r, x) of f˜ has locally small Riemann sums or (LSRS) if and only if for every ε > 0, there is a δ(ξ) > 0 such that for every t ∈ [a, b], we have X ∗ | f (ξ)(r, x)(v − u)| < ε, (3.2) uniformly for any r ∈ [0, 1] and x ∈ S n−1 , whenever P = {[u, v]; ξ} is a δ-fine division of an interval C ⊂ (t − δ(t), t + δ(t)), t ∈ C and Σ sums over P. 716

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M.E. Hamid, L.S. Xu: Locally and globally small Riemann sums and Henstock integral...

Proof Let ˜ 0 ∈ E n denote the (F H) integral of f˜ on [a, b]. Given ε > 0 there is a δ(ξ) > 0 such that for any δ-fine division P = {[u, v]; ξ} of [a, b], we have X D( f˜(ξ)(v − u), ˜ 0) < ε. (3.3) That is sup d([

X

˜ r ) < ε. f˜(ξ)(v − u)]r , [0]

(3.4)

r∈[0,1]

By Lemma 2.2 we have sup |(

sup

X

f˜(ξ)(v − u))∗ (r, x) − σ(x, 0)| < ε.

(3.5)

f ∗ (ξ)(r, x)(v − u) − σ(x, 0)| < ε.

(3.6)

r∈[0,1] x∈S n−1

Furthermore, by σ(x, A) = sup hy, xi, we have y∈A

sup |

sup

X

r∈[0,1] x∈S n−1

Hence, for any r ∈ [0, 1], x ∈ S n−1 and for any δ-fine division P we have X ∗ | f (ξ)(r, x)(v − u)| < ε. Where σ(x, 0) = 0. This completes the proof.

(3.7)



Lemma 3.1 (Henstock Lemma). Let f˜ : [a, b] → E n be a fuzzy-number-valued function and Henstock integrable ˜ Then, the support-function-wise f ∗ (ξ)(r, x) of f˜ on [a, b] is Henstock integrable to A∗ (r, x) uniformly for any to A. r ∈ [0, 1], x ∈ S n−1 and A˜ ∈ E n , i.e., for every ε > 0 there is a positive function δ(ξ) > 0, for δ-fine division P = {[u, v]; ξ} of [a, b] and for any x ∈ S n−1 , we have X ∗ | f (ξ)(r, x)(v − u) − A∗ (r, x)| < ε. (3.8) Furthermore, for any sum of parts

P

from

P

we have

1

|

X

f ∗ (ξ)(r, x)(v − u) − A∗ (r, x)| < ε.

(3.9)

1

Proof Let A˜ ∈ E n denote the (F H) integral of f˜ on [a, b]. Given ε > 0 there is a δ(ξ) > 0 such that for any δ-fine division P = {[u, v]; ξ} of [a, b], we have X ˜ < ε. D( f˜(ξ)(v − u), A) (3.10) That is sup d([

X

˜ r ) < ε. f˜(ξ)(v − u)]r , [A]

(3.11)

r∈[0,1]

By Lemma 2.2 we have sup

X sup |( f˜(ξ)(v − u))∗ (r, x) − A∗ (r, x)| < ε.

(3.12)

r∈[0,1] x∈S n−1

Furthermore, by A∗ (r, x) = sup hy, xi, we have y∈[A]r

sup |

sup

X

f ∗ (ξ)(r, x)(v − u) − A∗ (r, x)| < ε.

(3.13)

r∈[0,1] x∈S n−1

Hence, for any r ∈ [0, 1], x ∈ S n−1 and for any δ-fine division P we have X ∗ | f (ξ)(r, x)(v − u) − A∗ (r, x)| < ε. For proof |

X

f ∗ (ξ)(r, x)(v − u) − A∗ (r, x)| < ε,

(3.14)

1

the proof is similar to the Theorem 3.7 in [6]. This completes the proof.  ˜ ˜ Hamid et al. [2] showed that if a fuzzy-number-valued functions f (x) is Henstock integrable on [a, b] then f (x) has LSRS. In next Theorem, we prove the above result to n-dimensional fuzzy-number-valued functions, which is an extension of the above result of Muawya et al. [2]. 717

Hamid ET AL 714-722

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M.E. Hamid, L.S. Xu: Locally and globally small Riemann sums and Henstock integral...

Theorem 3.2 Let f˜ : [a, b] → E n be a fuzzy-number-valued function. If f˜ is Henstock integrable to F˜ ([a, b]), then f˜ has LSRS. Proof Since f˜ is Henstock integrable to F˜ ([a, b]), by Theorem 3.1 the support-function-wise f ∗ (ξ)(r, x) of f˜ on [a, b] is Henstock integrable to F ∗ ([a, b])(r, x) uniformly for any r ∈ [0, 1], x ∈ S n−1 , i.e., for every ε > 0 there is a positive function δ(ξ) > 0, for δ-fine division P = {[u, v]; ξ} of [a, b] and for any x ∈ S n−1 , we have X ∗ ε | f (ξ)(r, x)(v − u) − F ∗ ([a, b])(r, x)| < . (3.15) 2 For each t ∈ [a, b], there is a closed interval C = [y, z] ⊂ (t − δ(t), t + δ(t)) such that ε |F ∗ ([y, z])(r, x)| < . (3.16) 2 According to Henstock lemma, for each t ∈ [a, b] and δ-fine division P = {[u, v]; ξ} of C ⊂ (t − δ(t), t + δ(t)), we have X ∗ X ∗ | f (ξ)(r, x)(v − u)| ≤ | f (ξ)(r, x)(v − u) − F ∗ ([a, b])(r, x)| + |F ∗ ([y, z])(r, x)|
0 there is a positive function δ(ξ) > 0, such that for any [u, v] ⊂ [a, b] with v − u < δ(ξ), we have Z kF˜ ([u, v])kE n = k(F H) f˜dxkE n < ε. (3.17) [u,v]

Proof The continuity follows from Lemma 3.1 and the following inequality: kF˜ (t) − F˜ (ξ)kE n

≤

kF˜ (t) − F˜ (ξ) − f˜(ξ)(t − ξ)kE n + kf˜(ξ)(t − ξ)kE n


0 and P = {([a, b], ξ)} = {([a1 , b1 ], ξ1 ), ([a2 , b2 ], ξ2 ), · · · , ([an , bn ], ξn )} is a δ-fine partition of [a, b]. For each i(i = 1, 2, · · · , n) there is a positive function δi with Pi = {([ui , vi ], ξi )} is a δi -fine partition of [ai , bi ]. Since f˜ has LSRS on [ai , bi ], then we have X ε k f˜(ξ)(v − u)kE n < . (3.18) 2n P i

Taken η = max{δ(ξ), ξ ∈ [a, b]}, according to the Lemma 3.2 we have Z ε f˜dxkE n < kF˜ ([ai , bi ])kE n = k(F H) . 2n [ai ,bi ]

(3.19)

Therefore, for any δi -fine partition Pi = {([ui , vi ], ξi )} of [ai , bi ], we have X X D( f˜(ξ)(v − u), F˜ ([ai , bi ])) ≤ k f˜(ξ)(v − u)kE n + kF˜ ([ai , bi ])kE n Pi

Pi


0 there exists a positive integer N such that for every n > N there is a δn (ξ) > 0 and for every δn -fine division P = {[u, v]; ξ} of [a, b], we have X k f˜(ξ)(v − u)kE n < ε, (4.1) kf˜(ξ)kE n >n

where the

P

is taken over P and for which kf˜(ξ)kE n >n .

The following Theorem 4.1 shows that f˜ has (GSRS) is equal to the type of it, s support functions. Theorem 4.1 Let f˜ : [a, b] → E n be a fuzzy-number-valued function, the support-function-wise f ∗ (ξ)(r, x) of f˜ has globally small Riemann sums or (GSRS) if and only if for every ε > 0, there exists a positive integer N such that for every n > N there is a δn (ξ) > 0 and for every δn -fine division P = {[u, v]; ξ} of [a, b], we have X f ∗ (ξ)(r, x)(v − u) < ε, (4.2) |f ∗ (ξ)(r,x)|>n

uniformly for any r ∈ [0, 1] and x ∈ S n−1 , where the

P

is taken over P and for which |f ∗ (ξ)(r, x)| > n.

Proof First, we can prove the following statements are equivalent: (1) kf˜(ξ)kE n > n. (2) |f ∗ (ξ)(r, x)| > n. In fact kf˜(ξ)kE n > n

sup d([f˜(ξ)]r , [˜ 0]r )

=

r∈[0,1]

=

sup

sup |f ∗ (ξ)(r, x)|.

r∈[0,1] x∈S n−1

Second, let ˜ 0 ∈ E n denote the (F H) integral of f˜ on [a, b]. Given ε > 0 there exists a positive integer N such that for every n > N there is a δn (ξ) > 0 and for every δn -fine division P = {[u, v]; ξ} of [a, b], we have X D( f˜(ξ)(v − u), ˜ 0) < ε. (4.3) kf˜(ξ)kE n >n

That is sup d([ r∈[0,1]

X

f˜(ξ)(v − u)]r , [˜ 0]r ) < ε.

(4.4)

kf˜r (ξ)kE n >n

By Lemma 2.2 we have sup

sup |(

r∈[0,1] x∈S n−1

X

f (ξ)(v − u))∗ (r, x) − σ(x, 0)| < ε.

(4.5)

f ∗ (ξ)(r, x)(v − u)) − σ(x, 0)| < ε.

(4.6)

|f ∗ (ξ)(r,x)|>n

Furthermore, by σ(x, A) = sup hy, xi, we have y∈A

sup

sup |

r∈[0,1] x∈S n−1

X |f ∗ (ξ)(r,x)|>n

Hence, for any r ∈ [0, 1], x ∈ S n−1 and for any δ-fine division P we have X | f ∗ (ξ)(r, x)(v − u)| < ε. |f ∗ (ξ)(r,x)|>n

719

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M.E. Hamid, L.S. Xu: Locally and globally small Riemann sums and Henstock integral...

Where σ(x, 0) = 0. This completes the proof.  Hamid et al. [2] investigated that a fuzzy-number-valued functions f˜(x) is Henstock integrable on [a, b] if and only if f˜(x) has GSRS. In next Theorem 4.3, we extend this result to n-dimensional fuzzy-number-valued functions. To prove this result, we need to prove the following Theorem. Theorem 4.2 Let f˜ : [a, b] → E n be a fuzzy-number-valued function. If f˜ has GSRS then f˜ is Henstock integrable on [a, b]. Proof Because f˜ has GSRS, then by Theorem 4.1 for every ε > 0, there exists a positive integer N such that for every n > N there is a δn (ξ) > 0 and for every δn -fine division P = {[u, v]; ξ} of [a, b], we have X

|

f ∗ (ξ)(r, x)(v − u)| < ε.

(4.8)

|f ∗ (ξ)(r,x)|>n

P uniformly for any r ∈ [0, 1] and x ∈ S n−1 , where the is taken over P and for which |f ∗ (ξ)(r, x)| > n. For each two δ-fine divisions P1 = {[u1 , v1 ]; ξ1 }, P2 = {[u2 , v2 ]; ξ2 } of [a, b], we have

≤ ≤

X

f ∗ (ξ1 )(r, x)(v1 − u1 ) −

X

f ∗ (ξ2 )(r, x)(v2 − u2 )| X ∗ X ∗ | f (ξ1 )(r, x)(v1 − u1 )| + | f (ξ2 )(r, x)(v2 − u2 )| X X ∗ | f (ξ1 )(r, x)(v1 − u1 )| + | f ∗ (ξ1 )(r, x)(v1 − u1 )| |

|f ∗ (ξ1 )(r,x)|>n

+

X

|

|f ∗ (ξ1 )(r,x)|≤n ∗

f (ξ2 )(r, x)(v2 − u2 )| + |

|f ∗ (ξ2 )(r,x)|>n


0 there is a positive function δ ∗ , for δ ∗ -fine division P = {[u, v]; ξ} of [a, b], we have |

X

f ∗ (ξ)(r, x)(v − u) − F ∗ ([a, b])(r, x)|
N there is a δn (ξ) > 0 and for every δn -fine division P = {[u, v]; ξ} of [a, b], we have X f ∗ (ξ)(r, x)(v − u) < ε, (4.13) |f ∗ (ξ)(r,x)|>n

P uniformly for any r ∈ [0, 1] and x ∈ S , where the is taken over P and for which |f ∗ (ξ)(r, x)| > n. Note that f˜n , is Henstock integrable on [a, b] for all n. Choose N so that whenever n, m > N we have n−1

∗ |Fn∗ ([a, b])(r, x) − Fm ([a, b])(r, x)| < ε.

(4.14)

Then for n, m > N and a suitably chosen δ-fine division P = {[u, v]; ξ}, we have

≤

∗ |Fn∗ ([a, b])(r, x) − Fm ([a, b])(r, x)| X ∗ |Fn ([a, b])(r, x) − f ∗ (ξ)(r, x)(v − u)| + | |f ∗ (ξ)(r,x)|≤n

X

+

|


n

∗ f ∗ (ξ)(r, x)(v − u) − Fm ([a, b])(r, x)| + |

|f ∗ (ξ)(r,x)|≤m

X

f ∗ (ξ)(r, x)(v − u)|

|f ∗ (ξ)(r,x)|>m

That is, {Fn∗ ([a, b])(r, x)} converge to F ∗ ([a, b])(r, x), as n → ∞. Again, for suitably chosen N and δ(ξ) and for every δ-fine division P = {[u, v]; ξ}, we have |

X

f ∗ (ξ)(r, x)(v − u) − F ∗ ([a, b])(r, x)|

≤

|

X

≤

|

∗ ∗ f ∗ (ξ)(r, x)(v − u) − FN ([a, b])(r, x)| + |FN ([a, b])(r, x) − F ∗ ([a, b])(r, x)| X X ∗ f ∗ (ξ)(r, x)(v − u) − FN ([a, b])(r, x)| + | f ∗ (ξ)(r, x)(v − u)|

+

∗ |FN ([a, b])(r, x) − F ∗ ([a, b])(r, x)|


N

That is, f˜ is Henstock integrable on [a, b]. This completes the proof.

5



conclusions

This paper introduces, first of all, the generalization of locally and globally small Riemann sums from fuzzy-valued functions to n-dimensional fuzzy-numbers by means of support function. In addition, the concept of locally small Riemann sums for n-dimensional fuzzy-number-valued functions is presented and discussed. Finally, an important result of this paper is a characterizations of globally small Riemann sums for n-dimensional fuzzy-number-valued functions. 721

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M.E. Hamid, L.S. Xu: Locally and globally small Riemann sums and Henstock integral...

References [1] S.X. Hai, Z.T. Gong, On Henstock integral of fuzzy-number-valued functions in Rn , International Journal of Pure and Applied Mathematics, 7(1), 111-121(2003). [2] M.E. Hamid, L.S. Xu, Z.T. Gong, Locally and globally small Riemann sums and Henstock integral of fuzzynumber-valued functions, Journal of Computational analysis and applications, 25(1), 11-18(2018). [3] M.E. Hamid, L.S. Xu, Z.T. Gong, Locally and globally small Riemann sums and Henstock-Stieltjes integral of fuzzy- number-valued functions, Journal of Computational analysis and applications, 25(6), 1107-1115(2018). [4] R. Henstock, Theory of Integration, Butterworth, London (1963). [5] C.R. Indrati, Some Characteristics of the Henstock-Kurzweil in Countably Lipschitz Condition, The 7th SEAMSUGM Conference (2015). [6] P.Y. Lee, Lanzhou Lectures on Henstock Integration, World Scientific, Singapore (1989). [7] P.Y. Lee, R. Vyborny, The Integral: An Easy Approach after Kurzweil and Henstock, Cambridge University Press (2000). [8] A.W. Schurle, A new property equivalent to Lebesgue integrability, Proceedings of the American Mathematical Society, 96(1), 103-106(1986). [9] A.W. Schurle, A function is Perron integrable if it has locally small Riemann sums, Journal of the Australian Mathematical Society (Series A), 41(2), 224-232(1986). [10] C.X. Wu, Zengtai Gong, On Henstock integral of fuzzy-number-valued functions (I), Fuzzy Sets and Systems, 120, 523-532(2001). [11] C.X. Wu, M. Ma, J.X. Fang, Structure Theory of Fuzzy Analysis, Guizhou Scientific Publication (1994), In Chinese. [12] L.A. Zadeh, Fuzzy sets, Information Control, 8, 338-353(1965).

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On systems of fractional Langevin equations of Riemann-Liouville type with generalized nonlocal fractional integral boundary conditions Chatthai Thaiprayoona,∗, Sotiris K. Ntouyasb,c and Jessada Tariboond,e a

Department of Mathematics, Faculty of Science, Burapha University, Chonburi, 20131, Thailand E-mail: [email protected] b

Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece E-mail: [email protected] c

Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia d

Nonlinear Dynamic Analysis Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand e

Centre of Excellence in Mathematics, CHE, Sri Ayutthaya Rd., Bangkok 10400, Thailand E-mail: [email protected] Abstract By applying Krasnoselskii’s and O’Regan’s fixed point theorems, in this paper, we study the existence of solutions for a coupled system consisting from Langevin fractional differential equations of Riemann-Liouville type subject to the generalized nonlocal integral boundary conditions. Examples illustrating our results are also presented.

Key words and phrases: Fractional differential equations, Krasnoselskii’s fixed point theorem, O’Regan’s fixed point theorem, generalized fractional integral. AMS (MOS) Subject Classifications: 26A33; 34A08.

1

Introduction

In this paper we concentrate on the study of existence of solutions for a coupled system of Langevin fractional differential equations of Riemann-Liouville type subject to the generalized nonlocal integral boundary conditions of the form  p D 1 (Dp2 + λ1 ) x(t) = f (t, x(t), y(t)), 0 < t < T,     Dq1 (Dq2 + λ2 ) y(t) = g(t, x(t), y(t)), 0 < t < T,    n  ∑  x(0) = 0, x(η) = αi µi I γi x(ξi ), (1)  i=1   m  ∑    y(0) = 0, y(κ) = βj δj I φj y(ζj ),   j=1

where Dχ is the Riemann-Liouville fractional derivative of order χ ∈ {p1 , p2 , q1 , q2 }, µi I γi , δj I φj are the Katugampola fractional integrals of orders γi , φj > 0, respectively, ξi , ζj ∈ (0, T ) and αi , βj ∈ R for ∗ Corresponding

author

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C. THAIPRAYOON, S. K. NTOUYAS AND J. TARIBOON all i = 1, 2, . . . , n, j = 1, 2, . . . , m, f, g : [0, T ] × R2 → R are continuous functions and λ1 , λ2 are given constants. Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physics, chemistry, control theory, biology, economics, etc. A comprehensive study of fractional calculus and its applications is introduced in several books (see [1]-[3]). Initial and boundary value problems of nonlinear fractional differential equations and inclusions have been addressed by several researchers. For some recent results on fractional differential equations we refer in a series of papers ([4]-[12]). In fractional calculus, the fractional derivatives are defined via fractional integrals. There are several known forms of the fractional integrals which have been studied extensively for their applications. Two of the most known fractional integrals are the Riemann-Liouville and the Hadamard fractional integral. A new fractional integral, called generalized Riemann-Liouville fractional integral, which generalizes the Riemann-Liouville and the Hadamard integrals into a single form, was introduced in [13]. The corresponding fractional derivatives were introduced in [14]. This integral is now known as ”Katugampola fractional integral” see for example [15, pp 15, 123]. For some recent work with this new operator, for example, see [16]-[17] and the references cited therein. The Langevin equation (first formulated by Langevin in 1908) is found to be an effective tool to describe the evolution of physical phenomena in fluctuating environments [18]. For some new developments on the fractional Langevin equation in physics, see, for example, [19]-[23]. For recent results on Langevin equations with different kinds of boundary conditions we refer to [24]-[28] and the references therein. Recently in [16], we have studied the existence and the uniqueness of solutions of a class of boundary value problems for fractional Langevin equations of Riemann-Liouville type with generalized nonlocal integral boundary conditions. Here we extend the results of [16], to a coupled system of Langevin fractional differential equations of Riemann-Liouville type subject to the generalized nonlocal integral boundary conditions. Usually in the literature the Banach’s contraction mapping principle is used to prove he existence and the uniqueness of solutions, and he existence of solutions is proved via LeraySchauder alternative. Here we apply Krasnoselskii’s and O’Regan’s fixed point theorems. To the best of our knowledge this is the first paper using Krasnoselskii’s and O’Regan’s fixed point theorems to prove the existence of solutions for coupled systems. The paper is organized as follows: In Section 2 we will present some useful preliminaries and some auxiliary lemmas. In Section 3, we establish the main existence results by using Krasnoselskii’s and O’Regan’s fixed point theorems. Examples illustrating our results are presented in the final Section 4.

2

Preliminaries

In this section, we introduce some notations and definitions of fractional calculus [1, 2] and present preliminary results needed in our proofs later. Definition 2.1 [2] The Riemann-Liouville fractional integral of order p > 0 of a continuous function f : (0, ∞) → R is defined by ∫ t 1 p J f (t) = (t − s)p−1 f (s)ds, Γ(p) 0 provided∫ the right-hand side is point-wise defined on (0, ∞), where Γ is the gamma function defined by ∞ Γ(p) = 0 e−s sp−1 ds. Definition 2.2 [2] The Riemann-Liouville fractional derivative of order p > 0 of a continuous function f : (0, ∞) → R is defined by ( )n ∫ t d 1 (t − s)n−p−1 f (s)ds, n − 1 ≤ p < n, Dp f (t) = Γ(n − p) dt 0 where n = [p]+1, [p] denotes the integer part of a real number p, provided the right-hand side is point-wise defined on (0, ∞).

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ON SYSTEMS OF FRACTIONAL LANGEVIN EQUATIONS . . . Lemma 2.3 [2] Let p > 0 and x ∈∑C(0, T )∩L(0, T ). Then the fractional differential equation Dp x(t) = n p−i 0∑has a unique solution x(t) = , and the following formula holds: J p Dp x(t) = x(t) + i=1 ci t n p−i , where ci ∈ R, i = 1, 2, . . . , n, and n − 1 ≤ p < n. i=1 ci t Lemma 2.4 ([2], page 71) Let α > 0, β > 0 and a ≥ 0. Then the following properties hold: J α (x − a)β−1 (t)

=

Γ(β) (t − a)β+α−1 Γ(β + α)

Definition 2.5 [14] The generalized (Katugampola) fractional integral of order q > 0 and ρ > 0, of a function f, for all 0 < t < ∞, is defined as ∫ ρ1−q t sρ−1 f (s) ρ q ds, I f (t) = Γ(q) 0 (tρ − sρ )1−q provided the right-hand side is point-wise defined on (0, ∞). Lemma 2.6 [16] Let constants ρ, q > 0 and p > 0. Then the following formula holds ) ( Γ p+ρ ρ tp+ρq ρ q p ) q . I t = ( ρ Γ p+ρq+ρ ρ

(2)

For convenience to prove our results, we set constants Γ(p1 ) η p1 +p2 −1 , Γ(p1 + p2 ) ( ) i −1 n Γ p1 +p2µ+µ ∑ ξ p1 +p2 +µi γi −1 αi Γ(p1 ) i ) i ( , Ω2 = Γ(p1 + p2 ) Γ p1 +p2 +µi γi +µi −1 µγi i i=1

Ω1 =

(3)

(4)

µi

Ω = Ω2 − Ω1 6= 0,

(5)

and Γ(q1 ) κq1 +q2 −1 , Γ(q1 + q2 ) ( ) q1 +q2 +δj −1 q +q +δ φ −1 m Γ ∑ ζ 1 2 j j δj βj Γ(q1 ) ( ) j Ψ2 = , φ Γ(q1 + q2 ) Γ q1 +q2 +δj φj +δj −1 δ j j=1 Ψ1 =

(6)

(7)

j

δj

Ψ = Ψ2 − Ψ1 6= 0.

(8)

Lemma 2.7 Let Ω, Ψ 6= 0, 0 < p1 , p2 , q1 , q2 ≤ 1, µi , γi > 0, δj , φj > 0, η, κ, ξi , ζj ∈ (0, T ), αi , βj ∈ R for all i = 1, 2, . . . , n, j = 1, 2, . . . , m and h, g ∈ C([0, T ], R). Then the problem Dp1 (Dp2 + λ1 )x(t) = h(t), Dq1 (Dq2 + λ2 )y(t) = g(t), n ∑ x(0) = 0, x(η) = αi y(0) = 0,

y(κ) =

i=1 m ∑

βj

0 < t < T, 0 < t < T,

(9) (10)

µi γi

(11)

δj φj

(12)

I x(ξi ),

I y(ζj ),

j=1

has a unique solution given by x(t)

=

[ Γ(p1 ) tp1 +p2 −1 p1 +p2 J h(η) − λ1 J p2 x(η) Γ(p1 + p2 ) Ω

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C. THAIPRAYOON, S. K. NTOUYAS AND J. TARIBOON

−

n ∑

αi

µi γ i

I

(

J

p1 +p2

] ) h(s) − λ1 J x(s) (ξi ) + J p1 +p2 h(t) − λ1 J p2 x(t), p2

i=1

and

[ Γ(q1 ) tq1 +q2 −1 q1 +q2 J g(κ) − λ2 J q2 y(κ) Γ(q1 + q2 ) Ψ ] m ∑ ( q1 +q2 ) δj φj q2 − βj I J g(s) − λ2 J y(s) (ζj ) + J q1 +q2 g(t) − λ2 J q2 y(t).

y(t) =

j=1

Proof. Applying Lemma 2.3 to the equations (9) and (10), we obtain (Dp2 + λ1 )x(t) = J p1 h(t) + c1 tp1 −1 ,

and

(Dq2 + λ2 )y(t) = J q1 g(t) + d1 tq1 −1 ,

which give Γ(p1 ) tp1 +p2 −1 + c2 tp2 −1 , Γ(p1 + p2 ) Γ(q1 ) y(t) = J q1 +q2 g(t) − λ2 J q2 y(t) + d1 tq1 +q2 −1 + d2 tq2 −1 , Γ(q1 + q2 ) for c1 , c2 , d1 , d2 ∈ R. It is easy to see that the conditions x(0) = 0, y(0) = 0 imply that c2 = 0, d2 = 0. Thus Γ(p1 ) x(t) = J p1 +p2 h(t) − λ1 J p2 x(t) + c1 tp1 +p2 −1 , (13) Γ(p1 + p2 ) Γ(q1 ) y(t) = J q1 +q2 g(t) − λ2 J q2 y(t) + d1 tq1 +q2 −1 . (14) Γ(q1 + q2 ) Taking the generalized fractional integral of order µi > 0, γi > 0, to (13) and φj > 0, δj > 0 to (14) , we have x(t) = J p1 +p2 h(t) − λ1 J p2 x(t) + c1

( ) I x(t) = µi I γi J p1 +p2 h(s) − λ1 J p2 x(s) (t) + c1

µi γ i

i −1 Γ( p1 +p2µ+µ ) tp1 +p2 +µi γi −1 Γ(p1 ) i , (15) p +p +µ γ +µ −1 1 2 i i i Γ(p1 + p2 ) Γ( µγi i ) µ i

and δj φj

I y(t) =

δj φj

I

(

J

q1 +q2

) g(s) − λ2 J y(s) (t) + d1 q2

q +q +δ −1

Γ( 1 2δj j ) tq1 +q2 +δj φj −1 Γ(q1 ) . (16) φ Γ(q1 + q2 ) Γ( q1 +q2 +δj φj +δj −1 ) δ j j

δj

Using the second condition of (11), (12) to (15), (16) respectively, we get J p1 +p2 h(η) − λ1 J p2 x(η) + c1 Ω1 =

n ∑

αi

µi γ i

I

(

) J p1 +p2 h(s) − λ1 J p2 x(s) (ξi ) + c1 Ω2 ,

i=1

and J q1 +q2 g(κ) − λ2 J p2 y(κ) + d1 Ψ1 m ∑ ( ) = βj δj I φj J q1 +q2 g(s) − λ2 J q2 y(s) (ζj ) + d1 Ψ2 . j=1

Solving the above equations for finding constants c1 , d1 , we obtain [ ] n ∑ ( p1 +p2 ) 1 p1 +p2 p2 µi γ i p2 c1 = J h(η) − λ1 J x(η) − αi I J h(s) − λ1 J x(s) (ξi ) , Ω i=1 and

 d1 =

m ∑

1  q1 +q2 β j δj I J g(κ) − λ2 J q2 y(κ) − Ψ j=1

( φj

)



J q1 +q2 g(s) − λ2 J q2 y(s) (ζj ) .

Substituting the constants c1 , d1 into (13), (14), we obtain (13) and (13). The proof is completed.

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ON SYSTEMS OF FRACTIONAL LANGEVIN EQUATIONS . . .

3

Main results

Let C = C([0, T ], R) denotes the Banach space of all continuous functions from [0, T ] to R. Let us introduce the space X = {x(t)|x(t) ∈ C([0, T ])} endowed with the norm kxk = sup{|x(t)|, t ∈ [0, T ]}. Obviously (X, k · k) is a Banach space. Also let Y = {y(t)|y(t) ∈ C([0, T ])} be endowed with the norm kyk = sup{|y(t)|, t ∈ [0, T ]}. Obviously the product space (X × Y, k(x, y)k) is a Banach space with norm k(x, y)k = kxk + kyk. Throughout this paper, for convenience, we use the following expressions ∫ τ 1 (τ − s)z−1 f (s, x(s), y(s))ds, J z h(s, x(s), y(s))(τ ) = Γ(z) 0 and ρ1−z I h(s, x(s), y(s))(τ ) = Γ(z)

∫

τ

ρ z

0

sρ−1 f (s, x(s), y(s)) ds, (τ ρ − sρ )1−z

where ρ, z > 0 and τ ∈ [0, T ]. In view of Lemma 2.7, we define an operator F : X × Y → X × Y by ( ) P(x, y)(t) F(x, y)(t) = , Q(x, y)(t) where P(x, y)(t) =

(17)

[ Γ(p1 ) tp1 +p2 −1 p1 +p2 J f (s, x(s), y(s))(η) − λ1 J p2 x(s)(η) Γ(p1 + p2 ) Ω ] n ( ) ∑ µi γ i p1 +p2 p2 − αi I J f (s, x(s), y(s))(τ ) − λ1 J x(s)(τ ) (ξi )

(18)

i=1

+J p1 +p2 f (s, x(s), y(s))(t) − λ1 J p2 x(s)(t), and Q(x, y)(t) =

[ Γ(q1 ) tq1 +q2 −1 q1 +q2 J g(s, x(s), y(s))(κ) − λ2 J q2 y(s)(κ) Γ(q1 + q2 ) Ψ ] m ∑ ( q1 +q2 ) δj φj q2 − βj I J g(s, x(s), y(s))(s) − λ2 J x(s) (ζj )

(19)

j=1

+J q1 +q2 g(t) − λ2 J q2 y(t). To simplify the notations, we use in the following constants Φ(a) =

Γ(p1 ) T p1 +p2 −1 T a+p2 + Γ(1 + a + p2 ) Γ(p1 + p2 ) |Ω| +

n ∑

[ |αi |

i=1

1 ξia+p2 +µi γi Γ(1 + a + p2 ) µγi i Γ

and Λ(b) =

Γ(q1 ) T q1 +q2 −1 T b+q2 + Γ(1 + b + q2 ) Γ(q1 + q2 ) |Ψ| +

m ∑ j=1

[ |βj |

η a+p2 Γ(1 + a + p2 ) ) ( ]) Γ a+pµ2i+µi ( ) , a+p2 +µi γi +µi µi

(20)

(

b+q2 +δj φj

ζj 1 φ Γ(1 + b + q2 ) δj j

(

Γ

κb+q2 Γ(1 + b + q2 ) ( ) b+q2 +δj ]) Γ δj ( ) , b+q2 +δj φj +δj δj

(21)

where a ∈ {p1 , 0} and b ∈ {q1 , 0}.

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3.1

Existence result via Krasnoselskii’s fixed point theorem

The next result is based on the following fixed point theorem. Lemma 3.1 (Krasnoselskii’s fixed point theorem) [29]. Let M be a closed, bounded, convex and nonempty subset of a Banach space X. Let A, B be the operators such that (a) Ax + By ∈ M whenever x, y ∈ M ; (b) A is compact and continuous; (c) B is a contraction mapping. Then there exists z ∈ M such that z = Az + Bz. Theorem 3.2 Suppose that the folloeing conditions hold: (H1 ) |f (t, u, v)| ≤ ψ(t),

∀(t, u, v) ∈ [0, T ] × R2 ,

and ψ ∈ C([0, T ], R+ );

(H2 ) |g(t, u, v)| ≤ ω(t),

∀(t, u, v) ∈ [0, T ] × R2 ,

and ω ∈ C([0, T ], R+ );

If Υ = max{|λ1 |Φ(0), |λ2 |Λ(0)} < 1,

(22)

where Φ(0) and Λ(0) are defined by (20) and (21) with a = b = 0, respectively. Then the problem (1) has at least one solution on [0, T ]. Proof. To prove our result, we set supt∈[0,T ] |ψ(t)| = kψk, supt∈[0,T ] |ω(t)| = kωk and choose R≥

kψkΦ(p1 ) + kωkΛ(q1 ) , 1−Υ

(23)

where Φ(p1 ) and Λ(q1 ) are defined by (20) and (21) with a = p1 and b = q1 , respectively. Let BR = {(x, y) ∈ X × Y : k(x, y)k ≤ R}. We define four operators by [ p1 +p2 −1 Γ(p ) t 1 P1 (x, y)(t) = J p1 +p2 f (s, x(s), y(s))(t) + J p1 +p2 f (s, x(s), y(s))(η) Γ(p1 + p2 ) Ω ] n ( ) ∑ µi γi p1 +p2 − αi I J f (s, x(s), y(s))(τ ) (ξi ) , i=1

[ Γ(p1 ) tp1 +p2 −1 p2 J x(s)(η) P2 (x)(t) = −λ1 J x(s)(t) − λ1 Γ(p1 + p2 ) Ω ] n ( ) ∑ µi γ i p2 − αi I J x(s)(τ ) (ξi ) , p2

i=1

and Q1 (x, y)(t) =

J

q1 +q2

−

m ∑

[ Γ(q1 ) tq1 +q2 −1 q1 +q2 g(s, x(s), y(s))(t) + J g(s, x(s), y(s))(κ) Γ(q1 + q2 ) Ψ ]

βj

δj φj

I (J q1 +q2 g(s, x(s), y(s))(s))(ζj ) ,

j=1

and Q2 (y)(t)

=

[ Γ(q1 ) tq1 +q2 −1 q2 −λ2 J y(s)(t) − λ2 J y(s)(κ) Γ(q1 + q2 ) Ψ ] m ∑ δj φj q2 − βj I (J x(s)(τ ))(ζj ) , q2

j=1

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ON SYSTEMS OF FRACTIONAL LANGEVIN EQUATIONS . . . (

and F1 (x, y)(t) =

P1 (x, y)(t) Q1 (x, y)(t)

)

( ,

F2 (x, y)(t) =

P2 (x)(t) Q2 (y)(t)

) .

(24)

Observe that P = P1 + P2 , Q = Q1 + Q2 and F = F1 + F2 . For any (x1 , y1 ), (x2 , y2 ) ∈ BR we have |P1 (x1 , y1 )(t) + P2 (x2 )(t)| ¯ [ ¯ Γ(p1 ) tp1 +p2 −1 p1 +p2 ¯ p1 +p2 f (s, x1 (s), y1 (s))(t) + J f (s, x1 (s), y1 (s))(η) = ¯J ¯ Γ(p1 + p2 ) Ω ] n ( ) ∑ µi γ i p1 +p2 − αi I J f (s, x1 (s), y1 (s))(τ ) (ξi ) − λ1 J p2 x2 (s)(t) i=1

]¯ [ n ¯ ( ) ∑ Γ(p1 ) tp1 +p2 −1 p2 ¯ p2 µi γ i J x2 (s)(η) − αi I J x2 (s)(τ ) (ξi ) ¯ −λ1 ¯ Γ(p1 + p2 ) Ω i=1 [ ( ]) n ( ) ∑ Γ(p1 ) T p1 +p2 −1 p1 +p2 p1 +p2 µi γi p1 +p2 J (η) + |αi | I J ≤ kψk J (T ) + (τ ) (ξi ) Γ(p1 + p2 ) |Ω| i=1 ]) [ ( n ∑ Γ(p1 ) T p1 +p2 −1 p2 µi γi p2 p2 J (η) + |αi | I (J (τ ))(ξi ) +|λ1 |kx2 k J (T ) + Γ(p1 + p2 ) |Ω| i=1 ≤

kψkΦ(p1 ) + |λ1 |kx2 kΦ(0).

In a similar way, we get |Q (x , y )(t) + Q2 (y2 )(t)| ¯ 1 1 1 [ ¯ Γ(q1 ) tq1 +q2 −1 q1 +q2 ¯ q1 +q2 = ¯J g(s, x1 (s), y1 (s))(t) + J g(s, x1 (s), y1 (s))(κ) ¯ Γ(q1 + q2 ) Ψ ] m ∑ − βj δj I φj (J q1 +q2 g(s, x1 (s), y1 (s))(s))(ζj ) − λ2 J q2 y2 (s)(t) j=1

[ ]¯ m ¯ ∑ Γ(q1 ) tq1 +q2 −1 q2 ¯ −λ2 J y2 (s)(κ) − βj δj I φj (J q2 y2 (s)(τ ))(ζj ) ¯ ¯ Γ(q1 + q2 ) Ψ j=1 [ ]) ( m ∑ Γ(q1 ) T q1 +q2 −1 q1 +q2 δj φj q1 +q2 q1 +q2 J (κ) + |βj | I (J (τ ))(ζj ) ≤ kωk J (T ) + Γ(q1 + q2 ) |Ψ| j=1 ( [ ]) m ∑ Γ(q1 ) T q1 +q2 −1 q2 q2 δj φj q2 +|λ2 |ky2 k J (T ) + J (κ) + |βj | I (J (τ ))(ζj ) Γ(q1 + q2 ) |Ψ| j=1 ≤ kωkΛ(q1 ) + |λ2 |ky2 kΛ(0), which imply that kF1 (x, y) + F2 (x, y)k ≤ R. This shows that F1 (x, y) + F2 (x, y) ∈ BR . For (x1 , y1 ), (x2 , y2 ) ∈ X × Y and for each t ∈ [0, T ] we have kP2 (x1 ) − P2 (x2 )k ≤ |λ1 |Φ(0)kx1 − x2 k, and kQ2 (y1 ) − Q2 (y2 )k ≤ |λ2 |Λ(0)ky1 − y2 k. Thus kF2 (x1 , y1 ) − F2 (x2 , y2 )k ≤ Υkx1 − x2 k + Υky1 − y2 k = Υk(x1 − x2 , y1 − y2 )k, which implies that F2 is a contraction mapping by (22). The continuity of f implies that the operator F1 is continuous. Also, F1 is uniformly bounded on BR as kP1 (x, y)k ≤ kψkΦ(p1 ),

and kQ1 (x, y)k ≤ kωkΛ(q1 ).

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C. THAIPRAYOON, S. K. NTOUYAS AND J. TARIBOON

Thus kF1 (x, y)k ≤ kψkΦ(p1 ) + kωkΛ(q1 ). Next we will prove the compactness of the operator F1 . Let t1 , t2 ∈ [0, T ] with t1 < t2 . Then we have |P1 (x, y)(t2 ) − P1 (x, y)(t1 )| ¯ ¯ ≤ ¯J p1 +p2 f (s, x(s), y(s))(t2 ) − J p1 +p2 f (s, x(s), y(s))(t1 ) [ Γ(p1 )(tp21 +p2 −1 − tp11 +p2 −1 ) p1 +p2 + J f (s, x(s), y(s))(η) ΩΓ(p1 + p2 ) ]¯ n ¯ ( ) ∑ ¯ µi γ i p1 +p2 − αi I J f (s, x(s), y(s))(τ ) (ξi ) ¯ ¯ i=1

Γ(p1 )(tp21 +p2 −1 − tp11 +p2 −1 ) p1 +p2 (tp21 +p2 − t1p1 +p2 ) + 2(t2 − t1 )p1 +p2 + kψk J (η) Γ(p1 + p2 + 1) |Ω|Γ(p1 + p2 ) n ( ) ∑ + |αi | µi I γi J p1 +p2 (τ ) (ξi )

≤ kψk

i=1

and

≤

|Q (x, y)(t2 ) − Q1 (x, y)(t1 )| ¯ 1 ¯ q1 +q2 g(s, x(s), y(s))(t2 ) − J q1 +q2 g(s, x(s), y(s))(t1 ) ¯J [ Γ(q1 )(t2q1 +q2 −1 − tq11 +q2 −1 ) q1 +q2 J g(s, x(s), y(s))(κ) + ΨΓ(q1 + q2 ) ]¯ m ¯ ( ) ∑ ¯ δj φj q1 +q2 − βj I J g(s, x(s), y(s))(τ ) (ζj ) ¯ ¯ j=1

Γ(q1 )(tq21 +q2 −1 − t1q1 +q2 −1 ) q1 +q2 (tq21 +q2 − tq11 +q2 ) + 2(t2 − t1 )q1 +q2 + kωk J (κ) Γ(q1 + q2 + 1) |Ψ|Γ(q1 + q2 ) m ( ) ∑ + |βj | δj I φj J q1 +q2 (τ ) (ζj ),

≤ kωk

j=1

which is independent of (x, y) and tends to zero as t2 − t1 → 0. Thus, F1 is equicontinuous. So F1 is relatively compact on BR . Hence, by the Arzel´a-Ascoli theorem, F1 is compact on BR . Thus all the assumptions of Lemma 3.1 are satisfied. So the conclusion of Lemma 3.1 implies that the problem (1) has at least one solution on [0, T ]. This completes the proof. ¤

3.2

Existence result via O’Regan’s fixed point theorem

Our next existence result relies on a fixed point theorem due to O’Regan in [30]. Lemma 3.3 Denote by U an open set in a closed, convex set C of a Banach space E. Assume 0 ∈ U. ¯ ) is bounded and that F : U ¯ → C is given by F = F1 + F2 , in which F1 : U ¯ →E Also assume that F (U ¯ → E is a nonlinear contraction (i.e., there exists is continuous and completely continuous and F2 : U a nonnegative nondecreasing function φ : [0, ∞) → [0, ∞) satisfying φ(z) < z for z > 0, such that ¯ ). Then, either kF2 (x) − F2 (y)k ≤ φ(kx − yk) for all x, y ∈ U ¯ ; or (C1) F has a fixed point u ∈ U ¯ and ∂U, respectively, represent (C2) there exist a point u ∈ ∂U and λ ∈ (0, 1) with u = λF (u), where U the closure and boundary of U.

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ON SYSTEMS OF FRACTIONAL LANGEVIN EQUATIONS . . .

In the sequel, we will use Lemma 3.3 by taking C to be E. For more details of such fixed point theorems, we refer a paper [31] by Petryshyn. Let Kr = {(x, y) ∈ X × Y : k(x, y)k ≤ R}. Theorem 3.4 Let f, g : [0, T ] × R → R be continuous functions. Suppose that (22) holds. In addition we assume that: (H3 ) there exist a nonnegative function z1 ∈ C([0, T ], R) and nondecreasing functions ψ1 , ψ2 : [0, ∞) → [0, ∞) such that |f (t, u, v)| ≤ z1 (t)[ψ1 (kuk) + ψ2 (kvk)] for all (t, u, v) ∈ [0, T ] × R2 ; (H4 ) there exist a nonnegative function z2 ∈ C([0, T ], R) and nondecreasing functions ω1 , ω2 : [0, ∞) → [0, ∞) such that |g(t, u, v)| ≤ z2 (t)[ω1 (kuk) + ω2 (kvk)] for all (t, u, v) ∈ [0, T ] × R2 ; (H5 )

r 1 > , where Φ(p1 ), Λ(q1 ) and Υ kz1 k[ψ1 (r) + ψ2 (r)]Φ(p1 ) + kz2 k[ω1 (r) + ω2 (r)]Λ(q1 ) 1−Υ are defined in (20), (21)and (22) respectively. sup

r∈(0,∞)

Then the he problem (1) has at least one solution on [0, T ]. Proof. Consider the operator F : X × Y → X × Y as that defined in (18). We decompose F into a sum of two operators F(x, y)(t) = F1 (x, y)(t) + F2 (x, y)(t) where F1 (x, y), F2 (x, y) defined in (24). From (H5 ) there exists a number r0 > 0 such that r0 1 . > kz1 k[ψ1 (r0 ) + ψ2 (r0 )]Φ(p1 ) + kz2 k[ω1 (r0 ) + ω2 (r0 )]Λ(q1 ) 1−Υ

(25)

We shall prove that the operators F1 and F2 satisfy all the conditions of Lemma 3.3. ¯r Step 1. The set F(Kr0 ) is bounded. We first show that F1 (Kr0 ) is bounded. For any (x, y) ∈ K 0 we have kP1 (x, y)k ≤ kz1 k[ψ1 (r0 ) + ψ2 (r0 )]Φ(p1 ), and kQ1 (x, y)k ≤ kz2 k[ω1 (r0 ) + ω2 (r0 )]Λ(q1 ). Thus F1 (x, y)k ≤ kz1 k[ψ1 (r0 ) + ψ2 (r0 )]Φ(p1 ) + kz2 k[ω1 (r0 ) + ω2 (r0 )]Λ(q1 ). ¯ r ) is uniformly bounded. In a similar way we have This proves that F1 (K 0 kP2 (x)k ≤ |λ1 |Φ(0)kxk, and kQ2 (y)k ≤ |λ2 |Λ(0)kyk, and thus kF2 (x, y)k ≤ Υr0 . Step 2. The operator F1 is continuous and completely continuous. ¯ r ) is uniformly bounded. In addition for any t1 , t2 ∈ [0, T ], we have: By Step 1, F1 (K 0 |P1 (x, y)(t2 ) − P1 (x, y)(t1 )| [

( ) 1 tp21 +p2 − tp11 +p2 + 2(t2 − t1 )p1 +p2 Γ(p1 + p2 + 1) ] n ( ) ∑ Γ(p1 )(tp21 +p2 −1 − t1p1 +p2 −1 ) p1 +p2 + J (η) + |αi | µi I γi J p1 +p2 (τ ) (ξi ) , |Ω|Γ(p1 + p2 ) i=1

≤ kz1 k[ψ1 (r0 ) + ψ2 (r0 )]

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and |Q1 (x, y)(t2 ) − Q1 (x, y)(t1 )| [

( ) 1 t2q1 +q2 − t1q1 +q2 + 2(t2 − t1 )q1 +q2 Γ(q1 + q2 + 1) ] m ( ) ∑ Γ(q1 )(tq21 +q2 −1 − tq11 +q2 −1 ) q1 +q2 q1 +q2 δj φ j J (τ ) (ζj ) , J (κ) + |βj | I + |Ψ|Γ(q1 + q2 ) j=1

≤

kz2 k[ω1 (r0 ) + ω2 (r0 )]

which are independent of (x, y) and tends to zero as t2 − t1 → 0. Thus, F1 is equicontinuous. Hence, ¯ r ) is a relatively compact set. Now, let (xn , yn ) ⊂ K ¯ r with by the Arzel´ a-Ascoli Theorem, F1 (K 0 0 k(xn , yn ) − (x, y)k → 0. Then the limit k(xn , yn )(t) − (x, y)(t)k → 0 is uniformly valid on [0, T ]. From the uniform continuity of f (t, x, y) and g(t, x, y) on the compact set [0, T ] × [−r0 , r0 ] × [−r0 , r0 ], it follows that kf (t, xn (t), yn (t)) − f (t, x(t), y(t))k → 0 and kg(t, xn (t), yn (t)) − g(t, x(t), y(t))k → 0 are uniformly valid on [0, T ]. Hence kF1 (xn , yn ) − F1 (x, y)k → 0 as n → ∞ which proves the continuity of F1 . Therefore the operator F1 is continuous and completely continuous Step 3. The operator F2 is contractive. This was proved in Theorem 3.2. Step 4. Finally, it will be shown that the case (C2) in Lemma 3.3 does not hold. On the contrary, ¯ r such that we suppose that (C2) holds. Then, we have that there exist θ ∈ (0, 1) and (x, y) ∈ ∂ K 0 (x, y) = θF(x, y). So, we have k(x, y)k = r0 and kxk ≤ kz1 k[ψ1 (r0 ) + ψ2 (r0 )]Φ(p1 ) + |λ1 |Φ(0)kxk, and kyk ≤ kz2 k[ω1 (r0 ) + ω2 (r0 )]Λ(q1 ) + |λ2 |Λ(0)kyk, from which we get kxk + kyk ≤ kz1 k[ψ1 (r0 ) + ψ2 (r0 )]Φ(p1 ) + kz2 k[ω1 (r0 ) + ω2 (r0 )]Λ(q1 ) + Υr0 , or

r0 1 , ≤ kz1 k[ψ1 (r0 ) + ψ2 (r0 )]Φ(p1 ) + kz2 k[ω1 (r0 ) + ω2 (r0 )]Λ(q1 ) 1−Υ

which contradicts to (25). Consequently, we have proved that the operators F1 and F2 satisfy all the ¯ r , which is the conditions in Lemma 3.3. Hence, the operator F has at least one fixed point (x, y) ∈ K 0 solution of the he problem (1). The proof is completed. ¤ Theorem 3.5 Let f, g : [0, T ] × R → R be continuous functions. Suppose that (22) holds. In addition we assume that: (H6 ) there exist a nonnegative function z1 ∈ C([0, T ], R) and a nondecreasing function ψ : [0, ∞) → [0, ∞) such that |f (t, u, v)| ≤ z1 (t)ψ(kuk + kvk) for all (t, u, v) ∈ [0, T ] × R2 ; (H7 ) there exist a nonnegative function z2 ∈ C([0, T ], R) and a nondecreasing function ω : [0, ∞) → [0, ∞) such that |g(t, u, v)| ≤ z2 (t)ω(kuk + kvk) for all (t, u, v) ∈ [0, T ] × R2 ; (H8 )

1 r > , where Φ(p1 ), Λ(q1 ) and Υ are defined in (20), kz kψ(r)Φ(q ) + kz kω(r)Λ(q 1 − Υ ) 1 1 2 1 r∈(0,∞) (21)and (22) respectively. sup

Then the he problem (1) has at least one solution on [0, T ].

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ON SYSTEMS OF FRACTIONAL LANGEVIN EQUATIONS . . . ¤

Proof. The proof is similar to that of Theorem 3.4 and it is omitted. To establish some special cases, we set constants R1 =

n ∑ αi Γ(p1 ) Γ (p1 + p2 ) ξip1 +p2 +γi −1 , R = R1 − Ω1 6= 0, Γ(p + p ) Γ (p + p + γ ) 1 2 1 2 i i=1

L1 =

m ∑ Γ (q1 + q2 ) βj Γ(q1 ) q +q +φ −1 ζj 1 2 j , L = L1 − Ψ1 6= 0, Γ(q + q ) Γ (q + q + φ ) 1 2 1 2 j j=1

and

χ(a) =

and Θ(b) =

( Γ(p1 ) T p1 +p2 −1 η a+p2 T a+p2 + Γ(1 + a + p2 ) Γ(p1 + p2 ) |R| Γ(1 + a + p2 ) [ ]) n ∑ ξia+p2 +γi Γ (a + p2 + 1) + |αi | , Γ(1 + a + p2 ) Γ (a + p2 + γi + 1) i=1 ( Γ(q1 ) T q1 +q2 −1 κb+q2 T b+q2 + Γ(1 + b + q2 ) Γ(q1 + q2 ) |L| Γ(1 + b + q2 ) [ ]) b+q2 +φj m ∑ ζj Γ (b + q2 + 1) + |βj | , Γ(1 + b + q2 ) Γ (b + q2 + φj + 1) j=1

(26)

(27)

where a = {p1 , 0} and b = {q1 , 0} By setting µi = 1 and δj = 1, we have a boundary value problem with nonlocal Riemann-Liouville fractional integral conditions  p D 1 (Dp2 + λ1 ) x(t) = f (t, x(t), y(t)), 0 < t < T,    q1  D (Dq2 + λ2 ) y(t) = g(t, x(t), y(t)), 0 < t < T,    n  ∑  x(0) = 0, x(η) = αi J γi x(ξi ), (28)  i=1   m  ∑    y(0) = 0, y(κ) = βj J φj y(ζj ).   j=1

Using the above constants, we have the following corollaries. Corollary 3.6 Suppose that (H1) and (H2) holds. If M = max{|λ1 |χ(0), |λ2 |Θ(0)} < 1,

(29)

then the problem (28) has at least one solution on [0, T ]. Corollary 3.7 Let f, g : [0, T ] × R → R be continuous functions. Suppose that (29), (H3 ) and (H4 ) holds. In addition we assume that: 1 r > , where χ(p1 ), Θ(q1 ) and M (H9 ) sup kz k[ψ (r) + ψ (r)]χ(p 1 − M ) + kz k[ω (r) + ω (r)]Θ(q ) 1 1 2 1 2 1 2 1 r∈(0,∞) are defined in (26), (27)and (29) respectively. Then the he problem (28) has at least one solution on [0, T ]. Corollary 3.8 Let f, g : [0, T ] × R → R be continuous functions. Suppose that (29), (H6 ) and (H7 ) holds. In addition we assume that: 1 r > , where χ(p1 ), Θ(q1 ) and M are defined in (26), (H10 ) sup 1−M r∈(0,∞) kz1 kψ(r)χ(q1 ) + kz2 kω(r)Θ(q1 ) (27)and (29) respectively. Then the he problem (28) has at least one solution on [0, T ].

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4

Examples

In this section we present examples to illustrate our results. Example 4.1 Consider the following system of fractional Langevin equation subject to the nonlocal Katugampola fractional integral conditions  ( ) 2 cos t sin y(t) t sin 3t arctan x(t)  1/2 3/5  + 2 , 0 < t < 1, D D + 0.2 x(t) =   t + 1 3|x(t)| + 2 3t + 2 2|y(t)| + 3   ( )  2 3t 3x(t) 2y(t) + 3 D2/5 D4/5 + 0.25 y(t) = + , 0 < t < 1, (30) 4t + 3 5|x(t)| + 1 3|y(t)| +4    1/2 7/10 2/5 3/5  x(0) = 0, x(0.6) = 0.2 I x(0.3) + 0.3 I x(0.6),    3/10 4/5 3/5 2/5 y(0) = 0, y(0.2) = 0.2 I y(0.3) + 0.3 I y(0.7) + 0.4 2/5 I 9/10 y(0.9), Here p1 = 1/2, p2 = 3/5, q1 = 2/5, q2 = 4/5, λ1 = 0.2, λ2 = 0.25, η = 0.6, κ = 0.2, α1 = 0.2, α2 = 0.3, β1 = 0.2, β2 = 0.3, β3 = 0.4, µ1 = 1/2, µ2 = 2/5, γ1 = 7/10, γ2 = 3/5, δ1 = 3/10, δ2 = 3/5, δ3 = 2/5, φ1 = 4/5, φ2 = 2/5, φ3 = 9/10, ξ1 = 0.3, ξ2 = 0.6, ζ1 = 0.3, ζ2 = 0.7, ζ3 = 0.9, T = 1, f (t, x, y) = (t sin 3t arctan x(t))/((t + 1)(3|x(t)| + 2)) + (2 cos t sin y(t))/((3t2 + 2)(2|y(t)| + 3)) and g(t, x, y) = (9t2 x(t))/((4t + 3)(5|x(t)| + 1)) + (2y(t) + 3)/(3|y(t)| + 4). Since f (t, x, y) ≤ (t sin 3t)/(3t + 3) + (2 cos t)/(6t2 + 4), g(t, x, y ≤ (9t2 )/(20t + 15) + (2/3) and by using the Maple program, we can find Φ(0) =

T p2 Γ(1 + p2 ) p1 +p2 −1

Γ(p1 ) T + Γ(p1 + p2 ) |Ω|

(

2 ∑

p2

[

η 1 + |αi | Γ(1 + p2 ) i=1 Γ(1 +

( ξip2 +µi γi p2 ) µγi i

Γ ( Γ

p2 +µi µi

)

p2 +µi γi +µi µi

]) )

≈ 4.318646369, and Λ(0)

=

T q2 Γ(1 + q2 ) Γ(q1 ) T q1 +q2 −1 + Γ(q1 + q2 ) |Ψ|

(

) ( q2 +δj ]) [ 3 q +δ φ ∑ δj κq2 ζi 2 j j Γ 1 ( ) + |βj | Γ(1 + q2 ) j=1 Γ(1 + q2 ) δ φj Γ q2 +δj φj +δj j δj

≈ 3.234126953. Thus Υ ≈ 0.8637292738 < 1. Hence, by Theorem 3.2, the system (30) has at least one solution on [0, 1]. Example 4.2 Consider the following system of fractional Langevin equation subject to the nonlocal Katugampola fractional integral conditions ( )  ( ) t |x|2 + 2|x| |y|2 + 2|y| + 2  4/5 3/10  + , 0 < t < 1, D + 0.25 x(t) = D   15 |x| + 4 3|y| +)4  (  ( )  t |x|2 + |x| + 1 |y|2 + 1 + , 0 < t < 1, D2/5 D9/10 + 0.2 y(t) = (31) 5 2|x| + 5 |y| + 5    7/10 1/2 3/10 1/5 3/5 3/10  x(0) = 0, x(0.1) = 1.5 I x(0.6) + 2 I x(0.8) + 2.5 I x(0.9),    y(0) = 0, y(0.8) = 3 7/10 I 4/5 y(0.7) + 2.5 3/10 I 9/10 y(0.8), Here p1 = 3/10, p2 = 4/5, q1 = 2/5, q2 = 9/10, λ1 = 0.25, λ2 = 0.2, η = 0.1, κ = 0.8, α1 = 1.5, α2 = 2, α3 = 2.5, β1 = 3, β2 = 2.5, µ1 = 7/10, µ2 = 3/10, µ3 = 3/5, γ1 = 1/2, γ2 = 1/5, γ3 = 3/10, δ1 = 7/10, δ2 = 3/10, φ1 = 4/5, φ2 = 9/10, ξ1 = 0.6, ξ2 = 0.8, ξ3 = 0.9, ζ1 = 0.7, ζ2 = 0.8, T = 1, f (t, x, y) = (t/15)[((|x|2 + 2|x|)/(|x| + 4)) + ((|y|2 + 2|y| + 2)/(3|y| + 4))] and g(t, x, y) = (t/5)[((|x|2 + |x| + 1)/(2|x| + 5)) + ((|y|2 + 1)/(|y| + 5))]. By using the Maple program, we can find Φ(0) =

T p2 Γ(1 + p2 )

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ON SYSTEMS OF FRACTIONAL LANGEVIN EQUATIONS . . .

Γ(p1 ) T p1 +p2 −1 + Γ(p1 + p2 ) |Ω|

(

) ( p2 +µi [ ]) 3 ∑ µi η p2 ξip2 +µi γi Γ 1 ) ( + |αi | Γ(1 + p2 ) i=1 Γ(1 + p2 ) µγi i Γ p2 +µi γi +µi µi

≈ 1.892763483, and Λ(0)

=

T q2 Γ(1 + q2 ) q1 +q2 −1

Γ(q1 ) T + Γ(q1 + q2 ) |Ψ|

(

(

[

2 ∑

q2

κ 1 + |βj | Γ(1 + q2 ) j=1 Γ(1 +

q +δ φ ζi 2 j j q2 ) δ φj j

Γ ( Γ

q2 +δj δj

)

q2 +δj φj +δj δj

]) )

≈ 1.824427804. Thus Υ ≈ 0.4731908708 < 1. Since |f (t, x, y)| ≤ (t/15)[(|x|2 + 2|x|)/4 + (|y|2 + 2|y| + 2)/4], |g(t, x, y)| ≤ (t/5)[(|x|2 + |x| + 1)/5 + (|y|2 + 1)/5], we choose z1 (t) = t/15, ψ1 (x) = (|x|2 + 2|x|)/4, ψ2 (y) = (|y|2 + 2|y| + 2)/4, z2 (t) = t/5, ω1 (x) = (|x|2 + |x| + 1)/5, ω2 (y) = (|y|2 + 1)/5. We can show that r kz k[ψ (r) + ψ (r)]Φ(p ) + kz2 k[ω1 (r) + ω2 (r)]Λ(q1 ) 1 1 2 1 r∈(0,∞) sup

≈ 2.080080186 > 1.898220711 ≈

1 . 1−Υ

Hence, by Theorem 3.4, the system (31) has at least one solution on [0, 1]. Example 4.3 Consider the following system of fractional Langevin equation subject to the nonlocal Katugampola fractional integral conditions ( )  ( ) 2 t 2(|x + y|)3 + 2|x| + |y|  4/5 9/10  , 0 0). (1.4)

Some interesting properties and general of Mittag-Leffler function can be found e.g. in [2], [3], [4], [5], [6], [9], [13], [14], [15], [16], [18], [21], [22] and [23]. The function η,k Eα,β (z)(z ∈ C) introduced by Srivastava and Tomovski [20] in the form: η,k Eα,β (z)

=

∞ X

(η)nk z n , Γ(αn + β)n! n=0

(α, β, η ∈ C; Re(α) > max{0, Re(k)−1}; Re(k) > 0), (1.5)

where Γ(η + n) (η)n = = Γ(η)



1, η(η + 1)(η + 2) . . . (η + n − 1),

n = 0, n ∈ N.

(1.6)

2010 Mathematics Subject Classification. 30C45. Key words and phrases. Analytic functions, Hadamard product,starlike functions, prestar-like functions, Differential subordination, Mittag-Leffler function. 1

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MANSOUR F. YASSEN

η,k The function Eα,β (z) proved by Srivastava and Tomovski [20] is an entire function

in the complex z-plane. Attiya [1] defined the function Qη,k α,β (z) by   Γ(α + β) 1 η,k Qη,k Eα,β (z) − , (z ∈ U), α,β (z) = (η)k Γ(β)

(1.7)

very recently, Attiya [1] introduce the operator η,k Hα,β (f (z)) : A(1) → A(1),

defined, in terms of convolution by η,k Hα,β (f (z))

= Qη,k α,β (z) ∗ f (z) = z+

∞ X Γ(η + nk)Γ(α + β) an z n Γ(η + k)Γ(nα + β) n=2

(z ∈ U).

(1.8)

η,k η,k Analogous to Hα,β (f (z)), we introduce the operator Hα,β,p (f (z)) as follows η,k Hα,β,p (f (z)) : A(p) → A(p),

(1.9)

where η,k Hα,β,p (f (z)) = Qη,k α,β,p (z) ∗ f (z),

(z ∈ U).

(1.10)

and Qη,k α,β,p (z)

z p−1 Γ(α + β) = (η)k



η,k Eα,β (z)

1 − Γ(β)

 ,

(z ∈ U),

(1.11)

from equations (1.9), (1.10) and (1.11) we not that η,k Hα,β,p (f (z))

= Qη,k α,β,p (z) ∗ f (z) = zp +

∞ X Γ(η + nk)Γ(α + β) an z n+p−1 Γ(η + k)Γ(nα + β) n=2

(z ∈ U), (1.12)

η,k η,k when p = 1, the operator Hα,β,1 (f (z)) is the Attiya operator Hα,β (f (z)) [1]. A function f (z) ∈ A(1) is said to be in the class S ∗ (σ) [7] and [19] or (star-like of order σ in U) if:    0  zf (z) ∗ S (σ) := f (z) : Re > σ, 0 ≤ σ < 1, z ∈ U . (1.13) f (z)

A function f (z) ∈ A(1) is said to be in the class 0, ρ > 0, and f (z) ∈ Tη,k α,β,p (δ; ρw + 1 − ρ). If ρ ≤ ρ0 , where ρ0 =

1 2



(p − 1) δ

1−

1 ((p−1)/δ)−1

Z

t

1+t

0

−1 dt .

(3.6)

Then f (z) ∈ Tη,k α,β,p (0; w). The bound ρ0 is sharp in the case w(z) = 1/(1 − z). Proof. Let f (z) ∈ Tη,k α,β,p (δ; ρw + 1 − ρ) with δ > 0, ρ > 0. Suppose that 0 z −p+1  η,k Hα,β,p (f (z)) . p

φ(z) =

(3.7)

Then we have φ(z) +

δzφ0 (z) (p − 1)

0 00 (1 − δ)z −p+1  η,k δz −p+2  η,k Hα,β,p (f (z)) + Hα,β,p (f (z)) p p(p − 1) ≺ ρw(z) + 1 − ρ. (3.8) =

Using Lemma 1.1, we have φ(z) ≺

ρ(p − 1)z (−(p−1)/δ) δ

Z

z

u((p−1)/δ) w(u)du + 1 − ρ = (w ∗ ϕ)(z),

(3.9)

0

where ρ(p − 1)z (−(p−1)/δ) ϕ(z) = δ

Z 0

z

u((p−1)/δ)−1 du + 1 − ρ. 1−u

(3.10)

If 0 < ρ ≤ ρ0 where ρ0 (> 1) is given by (3.6), then it follows from (3.10) that   Z ρ(p − 1) 1 ((p−1)/δ)−1 1 Re {ϕ(z)} = t Re dt + 1 − ρ δ 1 − tz 0 Z ρ(p − 1) 1 t((p−1)/δ)−1 > dt + 1 − ρ δ 1+t 0 1 ≥ (z ∈ U). (3.11) 2 Using the Herglotz representation for ϕ(z). Also, from Equations (3.7) and (3.9) we obtain 0 z −p+1  η,k Hα,β,p (f (z)) ≺ (w ∗ ϕ)(z) ≺ w(z), (3.12) p

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since w(z) is convex univalent in U. Therefore f (z) ∈ Tη,k α,β,p (0; w). If w(z) = 1/(1 − z) and f (z) ∈ A(p) defined by: Z 0 ρ(p − 1)z −(p−1)/δ z u((p−1)/δ)−1 z −p+1  η,k Hα,β,p (f (z)) = du + 1 − ρ, (3.13) p δ 1−u 0 we can see that 0 δz −p+2  00 (1 − δ)z −p+1  η,k η,k Hα,β,p (f (z)) + Hα,β,p (f (z)) = ρw(z) + 1 − ρ. (3.14) p p(p − 1) Thus f (z) ∈ Tη,k α,β,p (δ; ρh + 1 − ρ). Also, for ρ > ρ0 , we have (at z → −1)  −p+1  Z 0  z ρ(p − 1) 1 t((p−1)/δ)−1 1 η,k Re Hα,β,p (f (z)) −→ dt + 1 − ρ < , (3.15) p δ 1 + t 2 0 which obtains f (z) 6∈ Tη,k α,β,p (0; w). Therefore, the value ρ0 cannot be increased when w(z) = 1/(1 − z). Theorem 3.3. Let f (z) ∈ Tη,k α,β,p (δ; w), φ(z) ∈ A(p), and  1 Re z −p φ(z) > 2

(z ∈ U).

(3.16)

Then (f ∗ φ)(z) ∈ Tη,k α,β,p (δ; w). Proof. For f (z) ∈ Tη,k α,β,p (δ; w) and φ(z) ∈ A(p), we have 0 00 (1 − δ)z −p+1  η,k δz −p+2  η,k Hα,β,p ((f ∗ φ)(z)) + Hα,β,p ((f ∗ φ)(z)) p p(p − 1)   0 
(1 − δ) −p η,k −p+1 = z φ(z) ∗ z Hα,β,p (f )(z)) p   00 
δ η,k + z −p φ(z) ∗ z −p+2 Hα,β,p (f )(z)) p(p − 1) =

(z −p φ(z)) ∗ ϕ(z),

(3.17)

where ϕ(z) =

0 00 (1 − δ)z −p+1  η,k δz −p+2  η,k Hα,β,p (f (z)) + Hα,β,p (f (z)) . p p(p − 1)

Using (3.16), the function z −p φ(z) has the Herglotz Representation Z dµ(y) z −p φ(z) = (z ∈ U), |y|=1 (1 − yz)

(3.18)

(3.19)

where µ(y) is a probability measure defined on the circle |y| = 1 and Z dµ(y) = 1. |y|=1

Since w(z) is convex univalent in U, it follows from (3.17) to (3.19) that 0 00 (1 − δ)z −p+1  η,k δz −p+2  η,k Hα,β,p ((f ∗ φ)(z)) + Hα,β,p ((f ∗ φ)(z)) p p(p − 1) Z = ϕ(yz)dµ(y) ≺ w(z). (3.20) |y|=1

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MANSOUR F. YASSEN

This shows that (f ∗ φ)(z) ∈ Tη,k α,β,p (δ; w) and the theorem is proved. Theorem 3.4. Let f (z) ∈ Tη,k α,β,p (δ; w), φ(z) ∈ A(p), and z −p+1 φ(z) ∈ 0. by putting 0 00 (1 − δ)z −p+1  η,k δz −p+2  η,k Gi (z) = Hα,β,p (fi (z)) + Hα,β,p (fi (z)) p p(p − 1)

(i = 1, 2), (3.28)

for fi (z), (i = 1, 2) given by (3.23), we find that ∞ X 1+z Gi (z) = 1 + bn,i z n−1 ≺ γi + (1 − γi ) 1 −z n=2

(i = 1, 2),

and Z  0 p(p − 1)z −(p−1)(1−δ)/δ z ((p−1)/δ)−1 η,k Hα,β,p (fi (z)) = u Gi (u)du (i = 1, 2). δ 0

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Now, if f (z) ∈ A(p) is defined by (3.25), we find from (3.30) that  0  0  0 η,k η,k η,k Hα,β,p (f (z)) = Hα,β,p (f1 (z)) ∗ Hα,β,p (f2 (z))   Z p(p − 1)z p−1 1 ((p−1)/δ)−1 = t G1 (tz)dt δ 0   Z p(p − 1)z p−1 z ((p−1)/δ)−1 ∗ t G2 (tz)dt δ 1   Z p(p − 1)z p−1 z ((p−1)/δ)−1 = t G(tz)dt δ 1

(3.29)

where G(z) =

p(p − 1) δ

Z

1

u((p−1)/δ)−1 (G1 ∗ G2 )(tz)dt.

(3.30)

0

Also, by using (3.29) and the Herglotz theorem,we see that     G1 (z) − γ1 1 G2 (z) − γ2 Re + >0 (z ∈ U), ∗ 1 − γ1 2 2(1 − γ2 )

(3.31)

which gives Re {(G1 ∗ G2 ) (z)} > γ0 = 1 − 2(1 − γ1 )(1 − γ2 )

(z ∈ U).

(3.32)

According to Lemma 1.3, we have  Re {(G1 ∗ G2 ) (z)} ≥ γ0 + (1 − γ0 )

1 − |z| 1 + |z|

 (z ∈ U).

(3.33)

Now it follows from (3.31) to (3.35) that  0 00  (1 − δ)z −p+1  η,k δz −p+2  η,k Re Hα,β,p (f (z)) + Hα,β,p (f (z)) = Re{G(z)} p p(p − 1) Z 1 p(p − 1) = t((p−1)/δ)−1 Re{(G1 ∗ G2 )(tz)}dt δ 0   Z p(p − 1) 1 ((p−1)/δ)−1 1 − t|z| ≥ t β0 + (1 − β0 ) dt δ 1 + t|z| 0 Z p(p − 1)(1 − γ0 ) 1 ((p−1)/δ)−1 1 − t > pγ0 + t dt δ 1+t 0   Z p − 1 1 t((p−1)/δ)−1 = p − 4p(1 − γ1 )(1 − γ2 ) 1 − dt δ 1+t 0 = γ (z ∈ U). (3.34) which proves that f (z) ∈ Tη,k α,β,p (δ; w) for the function w(z) given by (3.26).In order to show that the bound γ is Sharp, we take the functions fi (z) ∈ A(p) (i = 1, 2) defined by  0 p(p − 1)z −(p−1)(1−δ)/δ η,k Hα,β,p (fi (z)) = δ   Z z 1+u ((p−1)/δ)−1 u γi + (1 − γi ) × du, (3.35) 1−u 0

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MANSOUR F. YASSEN

for i = 1, 2 and, we have Gi (z)

0 00 (1 − δ)z −p+1  η,k δz −p+2  η,k Hα,β,p (fi (z)) + Hα,β,p (fi (z)) p p(p − 1) 1+z = γi + (1 − γi ) (i = 1, 2), (3.36) 1−z

=

and (G1 ∗ G2 ) (z) = 1 + 4(1 − γ1 )(1 − γ2 )

z . 1−z

(3.37)

Hence, for f (z) ∈ A(p) given by (3.25), we obtain 0 00 (1 − δ)z −p+1  η,k δz −p+2  η,k Hα,β,p (f (z)) + Hα,β,p (f (z)) p p(p − 1)   Z 1 p(p − 1) tz t((p−1)/δ)−1 1 + 4(1 − γ1 )(1 − γ2 ) = dt δ 1 − tz 0 −→ γ (as z −→ −1).

(3.38)

The proof is simple in the case of δ = 0, therefore, we omit the details involved. Conclusions we introduced the class Tη,k α,β,p (δ; w) of analytic functions associated with MittagLeffler function. Conclusion property of the class Tη,k α,β,p (δ; w) is obtained, sufficient condition of the class Tη,k α,β,p (δ; w) is also derived. Furthermore, several properties of functions belonging to this class are derived. References [1] A. A. Attiya, Some Applications of Mittag-Leffler Function in the Unit Disk, Filomat,30,(7)(2016), 2075-2081. [2] M. Garg, P. Manoha and S.L. Kalla, A Mittag-Leffler-type function of two variables, Integral Transforms Spec. Funct.24,(11)(2013), 934-944. [3] A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam,(2006). [4] V. Kiryakova, Generalized fractional calculus and applications, Pitman Research Notes in Mathematics Series,301. Longman Scientific Technical, Harlow; copublished in the United States with John Wiley Sons, Inc., New York,1994. [5] V. S. Kiryakova, Multiple (multiindex) Mittag-Leffler functions and relations to generalized fractional calculus.Higher transcendental functions and their applications, J. Comput. Appl. Math., 118,(1-2)(2000), 241-259. [6] V. Kiryakova, The multi-index Mittag-Leffler functions as an important class of special functions of fractional calculus, Comput. Math. Appl. 59,(5)(2010), 1885-1895. [7] J. Liu, subordinations for certain multivalent analytic funcations associated with the generalized Srivastava-Attiya operator, Integral Trans. and Special Functions, 19,(12)(2008), 893-901. [8] T. H. MacGregor, Functions whose derivative has a postitive real part, Trans. Am. Math. Soc., 104,(1962), 532-537. [9] F. Mainardia and R. Gorenflo, On Mittag-Leffler-type functions in fractional evolution processes. Higher transcendental functions and their applications, J. Comput. Appl. Math. 118,(1-2) (2000), 283299. [10] S. S. Miller and P. T. Mocanu, Differential subordinations and univalent functions, Michigan Math. J., 28,(1981), 157-171. [11] G. M. Mittag-leffler, Sur la nouvelle function, C.R. Acad. Sci.,Paris, 137,(1903), 554-558.

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[12] G. M. Mittag-leffler, Sur la representation analytique d’une function monogene (cinquieme note), Acta Math., 29,(1905), 101-181. [13] M. A. Ozarslan and B. Ylmaz, The extended Mittag-Leffler function and its properties, J. Inequal. Appl.No.1, Vol. 2014,(2014). [14] I. Podlubny, Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Mathematics in Science and Engineering, 198. Academic Press, Inc.,San Diego, CA,(1999). [15] T. R. Prabhakar, A singular integral equation with a generalized Mittag-Leffeer function in the Kernal, Yokohoma Math. J., 19,(1971), 7-15. [16] J. C. Prajapati, R.K.Jana, R. K. Saxena and A. K. Shukla, Some results on the generalized Mittag-Leffler function operator, J. Inequal.Appl., 6,(2013),1-6. [17] S. Ruscheweyh, Convolutions in geometric function theory, Gaetan Morin Editeur Ltee,83,(1982). [18] A. K. Shukla and J.C. Prajapati, On a generalization of MittagLeffler function and its properties, J. Math. Anal. Appl.,336, (2007),797-811. [19] R. Singh, On a class of star-like functions, Compositio Mathematica,19,(1)(1968), 78-82. [20] H. M. Srivastava and Z. Tomovski, Fractional calculus with an itegral operator containing a generalized Mittag-Leffler function in the kernal, Appl. Math. Comp., 211,(2009), 198-210. [21] Z. Tomovski, R. Hilfer and H.M. Srivastava, Fractional and operational calculus with generalized fractional derivative operators and Mittag-Leffler type functions, Integral Transforms Spec. Funct., 21,(11) (2010), 797-814. [22] Z. Tomovski, Generalized Cauchy type problems for nonlinear fractional differential equations with composite fractional derivative operator, Nonlinear Anal., 75,(7) (2012), 3364-3384. [23] A. Wiman, Uber den Fundamental Salz in der Theorie der Funktionen, Acta. Math.,29, (1905),191-201. Department of Mathematics, Faculty of Science, Damietta University, New Damietta 34517, Egypt. Department of Mathematics, College of Science and Humanities in Al-Aflaj, Prince Sattam Bin Abdulaziz University, Kingdom of Saudi Arabia E-mail address: [email protected]

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO. 4, 2019

Some Fixed Point Results of Caristi Type in G-Metric Spaces, Hamed M. Obiedat and Ameer A. Jaber,………………………………………………………………………………………569 Meir-Keeler contraction mappings in 𝑀𝑀𝑏𝑏 -metric Spaces, N. Mlaiki, N. Souayah, K. Abodayeh, and T. Abdeljawad,…………………………………………………………………………580 Generalized Ulam-Hyers Stability for Generalized types of (𝛾𝛾 − 𝜓𝜓)-Meir-Keeler Mappings via Fixed Point Theory in S-metric spaces, Mi Zhou, Xiao-lan Liu, Arslan Hojat Ansari, Yeol Je Cho, Stojan Radenović,………………………………………………………………………593 New oscillation criteria for second-order nonlinear delay dynamic equations with nonpositive neutral coefficients on time scales, Ming Zhang, Wei Chen, M.M.A. El-Sheikh, R.A. Sallam, A.M. Hassan, and Tongxing Li,………………………………………………………………629 A Consistency Reaching Approach for Probability-interval Valued Hesitant Fuzzy Preference Relations, Jiuping Xu, Kang Xu, and Zhibin Wu,……………………………………………636 Dynamics and Solutions of Some Recursive Sequences of Higher Order, Asim Asiri and E. M. Elsayed,……………………………………………………………………………………….656 Extremal solutions for a coupled system of nonlinear fractional differential equations with pLaplacian operator, Ying He,…………………………………………………………………671 The Growth and Zeros of Linear Differential Equations with Entire Coefficients of [p, q] − φ(r) Order, Sheng Gui Liu, Jin Tu, and Hong Zhang,……………………………………………..681 Some k-fractional integrals inequalities through generalized 𝜆𝜆𝜙𝜙𝜙𝜙 -MT-preinvexity, Chunyan Luo, Tingsong Du, Muhammad Adil Khan, Artion Kashuri, and Yanjun Shen,……………………690 Some generalizations of operator inequalities for positive linear map, Chaojun Yang and Fangyan Lu,……………………………………………………………………………………706 Locally and globally small Riemann sums and Henstock integral of fuzzy-number-valued functions in 𝐸𝐸 𝑛𝑛 , Muawya Elsheikh Hamid and Luoshan Xu,…………………………………714 On systems of fractional Langevin equations of Riemann-Liouville type with generalized nonlocal fractional integral boundary conditions, Chatthai Thaiprayoon, Sotiris K. Ntouyas, and Jessada Tariboon,………………………………………………………………………………723

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO. 4, 2019 (continued)

Subordination results for certain class of analytic functions associated with Mittag-Leffler function, Mansour F. Yassen,……………………………………………………………738

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.5, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

Fixed point theorems for F-contractions on closed ball in partial metric spaces Muhammad Nazam1 , Choonkil Park2 , Aftab Hussain3 , Muhammad Arshad1 and Jung-Rye Lee4∗ 1 Department of Mathematics and Statistics, International Islamic University, H-10, Islamabad, Pakistan

e-mail: [email protected], [email protected] 2 Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea

e-mail: [email protected] 3 Depatment of Mathematics, Khawaja Farid University, Rahim Yar Khan, Punjab, Pakistan

e-mail: [email protected] 4 Depatment of Mathematics, Daejin University,Kyunggi 11159

e-mail: [email protected] Abstract. In this paper, we present new fixed point theorems for Kannan type Fp -contraction and Kannan type (α, η, GFp )-contraction on a closed ball contained in a complete partial metric space. Some comparative examples are constructed to illustrate the significance of these results. Our results provide substantial generalizations and improvements of several well known results existing in the comparable literature.

1. Introduction and preliminaries The recent study in Fixed Point Theory is due to a Polish mathematician Stefan Banach who, in 1922, presented a revolutionary contraction principle known as Banach’s Contraction Principle. He proved that every contraction T in a complete metric space X has a unique fixed point (T (x) = x; x ∈ X). After the appearance of this remarkable result many generalizations of this result have appeared in literature (see for example [1–3,6–11,13,14,16,19,20,22,24,25,29]). One of these generalizations is known as F-contraction presented by Wardowski [30]. Wardowski [30] evinced that every F-contraction defined on a complete metric space has a unique fixed point. The concept of F-contraction proved another milestone in fixed point theory and numerous research papers on F-contraction have been published (see [21,23,28,31]). Hussain et al. [12] introduced an α-GF-contraction with respect to a general family of functions G and established Wardowski type fixed point results in ordered metric spaces. Batra et al. [4, 5] extended the concept of F-contraction on graphs and altered distances and proved some fixed point and coincidence point results. Motivated by Kannan [15], Wardowski [30], Matthews [18] and Kryeyszig [17], in this paper, we introduce Kannan type F-contraction and Kannan type (α, η, GF )-contraction on a closed ball contained in a complete partial metric space and present related fixed point theorems. We construct examples to illustrate these results. F-contraction on partial metric spaces is more general than F-contraction defined on metric spaces. The notion of a partial metric space (PMS) was introduced in 1992 by Matthews [18] to model computation over a metric space. The PMS is a generalization of the usual metric space in which the self-distance is no longer necessarily zero. Definition 1. [18] Let X be a nonempty set and p : X × X → R+ 0 satisfy the following properties: for all x, y, z ∈ X, (p1 ) (p2 ) (p3 ) (p4 )

x = y ⇔ p (x, x) = p (x, y) = p (y, y) , p (x, x) ≤ p (x, y) , p (x, y) = p (y, x) , p (x, z) + p (y, y) ≤ p (x, y) + p (y, z) .

0

2010 Mathematics Subject Classification: 47H09; 47H10; 54H25 Keywords: partial metric space; fixed point; F-contraction; closed ball. ∗ Corresponding author. 0

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Then (X, p) is called PMS. We present some new nontrivial examples of PMS. Example 1. Let the set of rational numbers be Q = {x1 , x2 , · · · }. We define p : R × R → R+ by  1 if x = y ∈ R − Q;        3  if x = 6 y ∈ R − Q;  2     1 if x = y ∈ Q; p(x, y) = 3      1 1    1 + m + n if x = xm , y = xn and m 6= n;      1 + n1 if {x, y} ∩ Q = {xn } and {x, y} − Q 6= φ. Clearly p satisfies (p1 ) − (p3 ). To prove p4 , let x, y, z ∈ R − Q and m 6= n. Then p (x, y) + p (z, z)

≤

p (xn , y) + p (z, z)

=

p (xn , xn ) + p (z, z)

=

p (xm , xn ) + p (z, z)

=

p (x, y) + p (xk , xk )


0 ⇒ τ + F (d(T (r1 ), T (r2 )) ≤ F (d(r1 , r2 ))) for all r1 , r2 ∈ M and some τ > 0. Then T has a unique fixed point υ ∈ M and for every r0 ∈ M the sequence {T n (r0 )} for all n ∈ N is convergent to υ. Remark 2. [30, Remark 2.1] In metric spaces a mapping giving fulfillment to F-contraction, is always a Banach contraction and hence a continuous map. Example 6 explains that Fp -contraction is more general than Fd -contraction.

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Example 6. Let M = [0, 1] and define partial metric by p(r1 , r2 ) = max {r1 , r2 } for all r1 , r2 ∈ M . The metric d induced by partial metric p is given by d(r1 , r2 ) = |r1 − r2 | for all r1 , r2 ∈ M . Define F : R+ → R by F (r) = ln(r) and T by  r  if r ∈ [0, 1),  5 T (r) =   0 if r = 1. Note that for all r1 , r2 ∈ M with r1 ≤ r2 or r2 ≤ r1 τ + F (p(T (r1 ), T (r2 ))) r  1 τ +F 5

≤ ≤

F (p(r1 , r2 )) implies r  2 F (r1 ) or τ + F ≤ F (r2 ) . 5

But T is neither continuous and nor satisfies F -contraction in a metric space (M, d). Indeed, for r1 = 1 and r2 = d(T (r1 ), T (r2 )) > 0 and we have τ + F (d(T (r1 ), T (r2 )))   5 τ + F d(T (1), T ( )) 6   1 τ + F d(0, ) 6 1 6 which is a contradiction for all possible values of τ .

≤ ≤ ≤
0 and x0 be an arbitrary point in X. Suppose there exists k ∈ [0, 1) with d(T (x), T (y)) ≤ kd(x, y), for all x, y ∈ Y = B(x0 , r) and d(x0 , T (x0 )) < (1 − k)r. Then there exists a unique point x∗ in B(x0 , r) such that x∗ = T (x∗ ). Definition 3. [15] Let (X, p) be a partial metric space. A mapping T : X → X is said to be a Kannan contraction if it satisfies the following condition: p (T (x) , T (y)) ≤

k [p (x, T (x)) + p (y, T (y))] 2

for all x, y ∈ X and some k ∈ [0, 1[. 2. Kannan type Fp -contraction on closed ball Definition 4. Let (X, p) be a partial metric space, r > 0 and x0 be an arbitrary point in X. The mapping T : X → X is called Kannan type Fp -contraction on closed ball if for all x, y ∈ Bp (x0 , r) ⊆ X we have   k τ + F (p(T (x), T (y))) ≤ F [p(x, T (x)) + p(y, T (y))] , (2.1) 2 where 0 ≤ k < 1, F ∈ ∆F and τ > 0. Remark 3. (1) Fp -contraction and Kannan type Fp -contraction are independent. (2) Let F be a Kannan type Fp -contraction. From (2.1), for all x, y ∈ Bp (x0 , r) with T (x) 6= T (y), we have   k F (p(T (x), T (y))) ≤ τ + F (p(T (x), T (y))) ≤ F [p(x, T (x)) + p(y, T (y))] . 2

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Due to (F1 ), we obtain p(T (x), T (y))
0 and x0 be an arbitrary point in X. Assume that T : X → X is a Kannan type Fp -contraction on closed ball Bp (x0 , r) ⊆ X with p(x0 , T (x0 )) ≤ (1 − λ)[r + p(x0 , x0 )], λ =

k . 2−k

(2.2)

If T or F is continuous, then there exists a point x∗ in Bp (x0 , r) such that T (x∗ ) = x∗ with p(x∗ , x∗ ) = 0. Proof. Let x0 be an initial point in X such that x1 = T (x0 ), x2 = T (x1 ) = T 2 (x0 ). Continuing in this way we can construct an iterative sequence {xn } such that xn+1 = T (xn ) = T n (x0 ), for all n ≥ 0. We show that xn ∈ Bp (x0 , r) for all n ∈ N. From (2.2), we have p(x0 , x1 ) = p(x0 , T (x0 )) ≤ (1 − λ)[r + p(x0 , x0 )] < r + p(x0 , x0 ), which shows that x1 ∈ Bp (x0 , r). From (2.1) and (F1 ), we get   k F (p(x1 , x2 )) = F (p(T (x0 ), T (x1 ))) ≤ F [p(x0 , x1 ) + p(x1 , x2 )] − τ, 2 which implies k [p(x0 , x1 ) + p(x1 , x2 )] < λp(x0 , x1 ) ≤ λ[r + p(x0 , x0 )] 2 p(x0 , x2 ) ≤ p(x0 , x1 ) + p(x1 , x2 ) − p(x1 , x1 ) < (1 − λ)[r + p(x0 , x0 )] + λ[r + p(x0 , x0 )] = r + p(x0 , x0 ). p(x1 , x2 )
n ≥ n1 , p(xn , xm )

≤

p(xn , xn+1 ) + p(xn+1 , xn+2 ) + p(xn+2 , xn+3 ) + · · · + p(xm−1 , xm ) −

(2.9)

m−1 X

p(xj , xj )

j=n+1

≤ =

p(xn , xn+1 ) + p(xn+1 , xn+2 ) + p(xn+2 , xn+3 ) + · · · + p(xm−1 , xm ) m−1 X

p(xi , xi+1 ) ≤

i=n

∞ X

p(xi , xi+1 ) ≤

i=n

∞ X 1 1

i=n

.

ik

P 1 The convergence of the series ∞ i=n 1 entails that limn,m→∞ p(xn , xm ) = 0. Hence {xn } is a Cauchy sequence in κ i       Bp (x0 , r), p . By Lemma 1, {xn } is a Cauchy sequence in B(x0 , r), dp . Moreover, since Bp (x0 , r), p is a complete   partial metric space, by Lemma 1, B(x0 , r), dp is also a complete metric space. Thus there exists x∗ ∈ (B(x0 , r), dp ) such that xn → x∗ as n → ∞ and using Lemma 1, we have lim p(x∗ , xn ) = p(x∗ , x∗ ) =

n→∞

lim

n,m→∞

p(xn , xm ).

(2.10)

Due to limn,m→∞ p(xn , xm ) = 0, we infer from (2.10) that p(x∗ , x∗ ) = 0 and {xn } converges to x∗ with respect to Tp . In order to show that x∗ is a fixed point of T , we have two cases. Case (1). T is continuous. We have x∗ = lim xn = lim T n (x0 ) = lim T n+1 (x0 ) = T ( lim T n (x0 )) = T (x∗ ). n→∞

n→∞

n→∞

n→∞

Hence x∗ = T (x∗ ), that is, x∗ is a fixed point of T . Case (2). F is continuous. We complete this case in two steps. First, if for each n ∈ N there exists bn ∈ N such that xbn +1 = T (x∗ ) and bn > bn−1 with b0 = 1. Then we have x∗ = lim xbn +1 = lim T (x∗ ) = T (x∗ ). n→∞

n→∞

This shows that x∗ is a fixed point of T. Second, there exists n0 ∈ N such that xn+1 6= T (x∗ ) for all n ≥ n0 . Using contractive condition (2.1), we obtain   k F (p(T (xn ), T (x∗ ))) ≤ F [p (xn , xn+1 ) + p (x∗ , T (x∗ ))] − τ. 2 On taking limit as n → ∞ and using the continuity of F and the fact that limn→∞ p(xn , xn+1 ) = 0, we have k F (p(x∗ , T (x∗ ))) < F ( p (x∗ , T (x∗ ))). 2 Since F is strictly increasing, the above inequality leads us to conclude that p(x∗ , T (x∗ )) = 0. Thus, by using the properties (p1 ) and (p2 ), we obtain x∗ = T (x∗ ), which completes the proof. To prove the uniqueness of x∗ , assume on contrary, that y ∗ ∈ Bp (x0 , r) is another fixed point of T , that is, y ∗ = T (y ∗ ). From (2.1), we have     k k τ + F (p(T (x∗ ), T (y ∗ ))) ≤ F [p(x∗ , T (x∗ )) + p(y ∗ , T (y ∗ ))] ≤ F × 2p(x∗ , y ∗ ) . (2.11) 2 2 The inequality (2.11) leads to a contradiction. Hence p(x∗ , y ∗ ) = 0. Thus, due to (p1 ) and (p2 ), we obtain x∗ = y ∗ .



The following example explains the significance of Theorem 3. Example 7. Let X = R+ . Define p : X 2 → [0, ∞) by p (x, y) = max {x, y} for all (x, y) ∈ X 2 . Then (X, p) is a complete partial metric space. Define the mapping T : X → X by  x if x ∈ [0, 1], 14 T (x) = x − 21 if x ∈ (1, ∞).

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Set k = 52 , x0 = 12 , r =

1 2

and p(x0 , x0 ) = 12 . Then Bp (x0 , r) = [0, 1] and   1 1 1 p(x0 , T (x0 )) = max , = < (1 − λ)[r + p(x0 , x0 )]. 2 28 2

For all x, y ∈ Bp (x0 , r), we note that nx y o 1 , = max {x, y} 14 14 14 h n n y oi 1 1 xo < [x + y] = max x, + max y, 5 5 14 14 k = [p(x, T (x)) + p(y, T (y)] 2 Thus   k τ + ln (p(T (x), T (y))) ≤ ln [p(x, T (x)) + p(y, T (y)] . 2 If F (α) = ln(α), α > 0 and τ > 0, then   k [p(x, T (x)) + p(y, T (y)] . τ + F (p(T (x), T (y))) ≤ F 2 p(T (x), T (y))

=

max

However, for x = 100, y = 10 ∈ (1, ∞) ,   1 1 max x − , y − 2 2 1 k ≥ [x + y] = [p(x, T (x)) + p(y, T (y)] . 5 2 Consequently, the contractive condition (2.1) does not hold on X. Hence, all the hypotheses of Theorem 3 are satisfied on closed ball and so x = 0 is a fixed point of T . p(T (x), T (y))

=

3. Kannan type (α, η, GFp )-contraction on closed ball Definition 5. [27]. Let T : X → X and α : X × X → [0, +∞) be two functions. We say that T is an α-admissible if for all x, y ∈ X, α(x, y) ≥ 1 implies α(T (x), T (y)) ≥ 1. Example 8. Let X = R. Define α : X × X → [0, ∞) and f : X → X by  x+y  x2 e if x, y ∈ [0, 1], 7 α (x, y) = f (x) = 0 otherwise . ln(x)

if x ∈ [0, 1], if x ∈ (1, ∞).

Apparently, α(x, y) ≥ 1 implies α(f x, f y) ≥ 1. Definition 6. [26]. Let T : X → X and α, η : X × X → [0, +∞) be two functions. We say that T is an α-admissible mapping with respect to η if for all x, y ∈ X, α(x, y) ≥ η(x, y) implies α(T (x), T (y)) ≥ η(T (x), T (y)). Example 9. Let X = R. Define α, η : X × X → [0, ∞) and f : X → X by  x+y  x+y π if x, y ∈ [0, 1], e α (x, y) = η (x, y) = 0 otherwise , 0  x2 if x ∈ [0, 1], 7 f (x) = ln(x) if x ∈ (1, ∞). Apparently, α(x, y) ≥ η(x, y) implies α(f x, f y) ≥ η(f x, f y).

if x, y ∈ [0, 1], otherwise .

If η(x, y) = 1, then the above definition reduces to Definition 5. We begin by introducing the following family of new functions. Let ∆G denote the set of all functions G : (R+ )4 → R+ which satisfy the property (G): for all p1 , p2 , p3 , p4 ∈ R+ , if p1 + p2 + p3 + p4 pi + pi+1 ≤ , i = 1, 2, 3, 4, 4 2 then there exists τ > 0 such that G(p1 , p2 , p3 , p4 ) = τ .

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Definition 7. Let (X, p) be a partial metric space and T be a self mapping on X. Suppose that α, η : X ×X → [0, +∞) are two functions. The mapping T is said to be an (α, η, GFp )-contraction if for all x, y ∈ X, with η(x, y) ≤ α(x, y) and d(T (x), T (y)) > 0, we have G(p(x, T (x)), p(y, T (y)), p(x, T (y)), p(y, T (x))) + F (p(T (x), T (y))) ≤ F (p(x, y)) , where G ∈ ∆G and F ∈ ∆F . Definition 8. Let (X, p) be a partial metric space and T : X → X and α, η : X × X → [0, +∞) be two functions. T is said to be an (α, η)-continuous mapping on (X, p) if for a given x ∈ X, and the sequence {xn }n∈N converging to x α(xn , xn+1 ) ≥ η(xn , xn+1 ) implies T (xn ) → T (x). Example 10. Let X = [0, ∞) and p : X × X → [0, ∞) be defined by p(r1 , r2 ) = max{r1 , r2 } for all r1 , r2 ∈ X. Define   3 sin(πr) if r ∈ [0, 1], r1 + r23 + 1 if r1 , r2 ∈ [0, 1], T (r) = α(r1 , r2 ) = cos(πr) + 2 if r ∈ (1, ∞), 0 otherwise,  3 3 r1 + r2 if r1 , r2 ∈ [0, 1], η(r1 , r2 ) = 0 otherwise. Then apparently, T is not continuous on X, however T is an (α, η)-continuous. Definition 9. Let (X, p) be a partial metric space and α, η : X × X → [0, +∞) are two functions, r > 0 and x0 be an arbitrary point in X. The mapping T : X → X is said to be a Kannan type (α, η, GFp )-contraction on closed ball if for all x, y ∈ Bp (x0 , r) ⊆ X with η(x, y) ≤ α(x, y) and p(T (x), T (y)) > 0, we have   k τ (G) + F (p(T (x), T (y))) ≤ F [p(x, T (x)) + p(y, T (y)] , (3.1) 2 where τ (G) = G(p(x, T (x)), p(y, T (y)), p(x, T (y)), p(y, T (x))), 0 ≤ k < 1, G ∈ ∆G and F ∈ ∆F . Theorem 4. Let (X, p) be a complete metric space and T : X → X be a Kannan type (α, η, GFp )-contraction mapping on a closed ball Bp (x0 , r) satisfying the following assertions (1) T is an α-admissible mapping with respect to η, (2) there exists x0 ∈ X such that α(x0 , T (x0 )) ≥ η(x0 , T (x0 )), (3) there exist r > 0 and x0 ∈ X such that p(x0 , T (x0 )) ≤ (1 − λ)[r + p(x0 , x0 )], where λ =

k . 2−k

Then there exists a point x∗ in Bp (x0 , r) such that T (x∗ ) = x∗ with p(x∗ , x∗ ) = 0. Proof. Suppose that x0 is an initial point of X, we can construct a sequence {xn }∞ n=1 such that xn+1 = T (xn ) = T n+1 (x0 ) for all n ∈ N. By assumption (2) there exists x0 ∈ X such that α(x0 , T (x0 )) ≥ η(x0 , T (x0 )). Since T is an α-admissible mapping with respect to η, α(x0 , T (x0 )) ≥ η(x0 , T (x0 )) implies α(x1 , x2 ) ≥ η(x1 , x2 ), which implies α(x2 , x3 ) ≥ η(x2 , x3 ). In general, we have η(xn−1 , xn ) ≤ α(xn−1 , xn ), for all n ∈ N. If there exists n0 ∈ N such that p(xn0 , T (xn0 )) = 0, then xn0 is a fixed point of T . We assume that p(xn , T (xn )) > 0 for all n ∈ N. We show that xn ∈ Bp (x0 , r) for all n ∈ N . Assumption (3) implies p(x0 , x1 ) = p(x0 , T (x0 )) ≤ (1 − λ)[r + p(x0 , x0 )] < [r + p(x0 , x0 )] and thus x1 ∈ Bp (x0 , r). Note that τ (G) = τ . Indeed, τ (G) = G(p(x0 , x1 ), p(x1 , x2 ), p(x0 , x2 ), p(x1 , x1 )) satisfies p(x0 , x1 ) + p(x1 , x2 ) + p(x0 , x2 ) + p(x1 , x1 ) p(x0 , x1 ) + p(x1 , x2 ) ≤ . 4 2 By the property (G), there exists τ > 0 such that G(p(x0 , x1 ), p(x1 , x2 ), p(x0 , x2 ), p(x1 , x1 )) = τ.

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Due to (3.1) and (F1 ), we have F (p(x1 , x2 ))

= ≤

This implies p(x1 , x2 ) p(x0 , x2 )


bn−1 with b0 = 1, then we have x∗ = lim xbn +1 = lim T (x∗ ) = T (x∗ ). n→∞

n→∞

This shows that x∗ is a fixed point of T. Second, there exists n0 ∈ N such that xn+1 6= T (x∗ ) for all n ≥ n0 . Using the contractive condition (3.1), we obtain   k [p (xn−1 , xn ) + p (x∗ , T (x∗ ))] − τ (G), F (p(xn , T (x∗ ))) ≤ F 2

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where τ (G) = G(p(xn−1 , xn ), p(x∗ , T (x∗ )), p(xn−1 , T (x∗ )), p(x∗ , xn )). Using the continuity of F and the property (G), we have  h   i k F lim p(xn , T (x∗ )) ≤ F lim p (xn−1 , xn ) + lim p (x∗ , T (x∗ )) − τ. n→∞ n→∞ 2 n→∞ Since F is strictly increasing, the above inequality leads us to conclude that p(x∗ , T (x∗ )) = 0. Thus, by using properties (p1 ) and (p2 ), we obtain x∗ = T (x∗ ), which completes the proof.  Example 11. Let X = R+ . Define p : X 2 → [0, ∞) by p (x, y) = max {x, y} . space. Define T : X → X, α : X × X → [0, +∞), η : X × X → R+ , G : (R+ )4  5x  x+y if x ∈ [0, 1], e 19 T (x) = α(x, y) = 1 1 x − 3 if x ∈ (1, ∞), 3

Then (X, p) is a complete partial metric → R+ and F : R+ → R by if x ∈ [0, 1], otherwise,

η(x, y) = 12 for all x, y ∈ X, G(t1 , t2 , t3 , t4 ) = τ > 0 and F (t) = ln(t) with t > 0. Set k = p(x0 , x0 ) = 21 . Then B(x0 , r) = [0, 1], α(0, T (0)) ≥ η(0, T (0)) and      1 1 1 5 p ,T = max , < (1 − λ)[r + p(x0 , x0 )]. 2 2 2 38

4 5

x0 =

1 , 2

r =

1 2

and

For if x, y ∈ B(x0 , r), then α(x, y) = ex+y ≥ 12 = η(x, y). On the other hand, T (x) ∈ [0, 1] for all x ∈ [0, 1] and so  α(T (x), T (y)) ≥ η(T (x), T (y)) for x 6= y, p(T (x), T (y)) = 5x , 5y > 0. For all x, y ∈ Bp (x0 , r), we have 19 19   5x 5y 5 p(T (x), T (y)) = , = max {x, y} , 19 19 19      5 k 5x 5y 14k max {x, y} < max x, + max y, = [x + y]. 19 2 19 19 38 Thus p(T (x), T (y))
0 be any

real number. The! Euler transform (E, q) of the sequence S = (sn ) is defined by Enq (S) = P Pn n 1 q n−v sv . A series ∞ n=0 an is said to be summable (E, q) to the number s if v=0 (1+q)n v ! Pn n q 1 En (S) = (1+q)n v=0 q n−v sv → s as n → ∞, and is said to be absolutely summable v P q (E, q) or summable |E, q|, if k |Ekq(S) − Ek−1 (S)| < ∞. Let x = (xn ) be a sequence of scalars, q for k ≥ 1 we will denote by Nn (x) the difference Enq (x) − En−1 (x), where Enq is defined as above.

Using Abel’s transform we have n−2 X 1 sn−1 An−1 sn q n−1 Nn (x) = − A + x + − s0 , k k+1 (1 + q)n−1 (1 + q)n−1 (1 + q)n (1 + q)n k=0

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where

k h X q Ak = 1+q i=0

n i

!

n−1

−

i

!

i

q n−i−1 .

Clearly, for any sequence x = (xn ), y = (yn ) and scalar λ, we have: Nn (x + y) = Nn (x) + Nn (y) and Nn (λx) = λNn (x). Let



(1)

p1

w1

(2)

w1

...



 (−1)  (1)  w  p w . . . 2 1 2   M = [mij ] =  (−2) , (−1)  w1 w2 p3 . . .    .. .. .. .. . . . .

where p = (pi) and w (t) = (wi )(t) are some fixed numerical ( sequences t ∈ Z\{0}. For a fixed n+1 2 , n is odd . We construct kf ∈ N, we define a finite sequence tn with kf terms as tn = −n n is even 2 , ti t a matrix M(p,w ,kf ) = M, w = 0 ∀ i > kf and for i = 1, 2, . . ., kf we have some fixed sequences w ti and p. Example 1.1. For kf = 2 we have t1 = 1, t2 = −1, we define pi = −1 ∀ i and ( 1, for t = 1, −1 (t) , wi = 0, ∀ t ∈ Z\{0, 1, −1} then we have M(p,wt ,2)x =

*

∞ X j=1

mij ξj

+

=< −ξ1 + ξ2 , ξ1 − ξ2 + ξ3 , ξ2 − ξ3 + ξ4 , ξ3 − ξ4 + ξ5 · · · > . n

An Orlicz function is a function M : [0, ∞) → [0, ∞) which is continuous, non-decreasing and convex with M (0) = 0, M (x) > 0 as x > 0 and M (x) → ∞ (x → ∞). Clerly, if M is a convex function and M (0) = 0, then M (λx) ≤ λM (x) for all λ ∈ (0, 1). Using the idea of Orlicz function, Lindenstrauss and Tzafriri [15] constructed the sequence space ( )   ∞ X |xk | `M = (xk ) ∈ w : M < ∞, for some ρ > 0 , ρ k=1

is called Orlicz sequence space and showed that `M is a Banach space with the following norm: ( )   ∞ X |xk | ||x|| = inf ρ > 0 : M ≤1 . ρ k=1

The space `M is closely related to the space `p which is an Orlicz sequence space with M (t) = |t|p for 1 ≤ p < ∞. A sequence M = (Mk ) of Orlicz functions is said to be a Musielak-Orlicz function [22]. A sequence V = (Vk ) is defined by Vk (v) = sup{|v|u − (Mk ) : u ≥ 0} (k = 1, 2, · · · ) is said to

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complementary function of M. For a given M, the Musielak-Orlicz sequence space tM and its subspace hM are defined by n o tM = x ∈ w : IM (cx) < ∞ for some c > 0 , n o hM = x ∈ w : IM (cx) < ∞ for all c > 0 ,

where IM denotes the convex modular and is defined by IM (x) =

∞ X

(Mk )(xk )

(x = (xk ) ∈ tM ).

k=1

It is noted that tM equipped with the Luxemburg norm or equipped with the Orlicz norm, where Luxemburg and Orlicz norms are given by n x n1  o o ||x|| = inf k > 0 : IM ≤ 1 and ||x||0 = inf 1 + IM (kx) : k > 0 , k k respectively. Kızmaz [14] was the first who introduced the idea of difference sequence spaces and studied Z(∆) = {x = (xk ) ∈ w : ∆x ∈ Z} (Z = l∞ , c, c0), where ∆x = xk − xk+1 for all k ∈ N (N and w denote the set of natural numbers and the set of all real and complex sequences) and the standard notations l∞ , c and c0 denote bounded, convergent and null sequences respectively. Et and C ¸ olak [7] presented a generalization of these difference sequence spaces and introduced the space Z(∆n ) (n ∈ N), in this case, ∆n x is given by ∆n x = ∆(∆n−1 x) = ∆n−1 xk − ∆n−1 xk+1 for n ≥ 2, which is equivalent to the following binomial representation   n X n v n ∆ xk = (−1) xk+v . v v=0

We remark that if we take n = 1, then difference sequence space Z(∆n ) is reduced to Z(∆). G¨ahler [12] extended the usual notion of normed spaces into 2-normed spaces while the notion was again extended to n-normed spaces by Misiak [16]. Assume that X is a linear space over the field K of real or complex numbers of dimension d ≥ n ≥ 2, n ∈ N (N denotes the set of natural numbers). A real valued function ||·, · · · , ·|| on X n satisfying the conditions: (N1) ||x1 , x2 , · · · , xn || = 0 if and only if x1 , · · · , xn are linearly dependent in X; (N2) ||x1 , x2 , · · · , xn || is invariant under permutation; (N3) ||αx1 , x2 , · · · , xn|| = |α| ||x1 , x2 , · · · , xn || for any α ∈ K; (N4) ||x1 + x01 , x2 , · · · , xn|| ≤ ||x1 , x2 , · · · , xn || + ||x01 , x2 , · · · , xn || is called a n-norm on X, and the pair (X, ||·, · · · , ·||) is called a n-normed space over K. For more details about these notions we refer to [3–5,13,18,19,21,23] and references therein. We used the standard notation θ = (kr ) to denotes the lacunary sequence, where θ is a sequence of positive integers such that k0 = 0, 0 < kr < kr+1 and hr = kr − kr−1 → ∞ (r → ∞). The intervals determined by θ will be denoted by Ir = (kr−1 , kr] and the ratio

772

kr kr −1

by qr (see [9]).

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2

Main results

Let M = (Mk ) be a Musielak-Orlicz function, p = (pk ) be a bounded sequence of positive real numbers and u = (uk ) a sequence of strictly positive numbers. We define the following sequence space in the present paper: Enq (M, u, p, s, M(p,wt,kf ) , ∆r , k·, · · ·

 ∞ 1 X −s , ·k) = x = (xk ) : lim k r hr k=1

   r

pk

uk Nk (M(p,wt,kf )∆ x) < ∞, s ≥ 0, for some ρ > 0 . , z1 , · · · , zn−1 × Mk ρ

We will use the following inequality to prove our results. If 0 ≤ pk ≤ sup pk = H, K = max(1, 2H−1) then

|ak + bk |pk ≤ K{|ak |pk + |bk |pk } for all k and ak , bk ∈ C. Also |a|pk ≤ max(1, |a|H ) for all a ∈ C. Theorem 2.1. Let M = (Mk ) be a Musielak-Orlicz function, p = (pk ) be a bounded sequence of positive real numbers and u = (uk ) be a sequence of strictly positive real numbers. Then the q

space En (M, u, p, s, M(p,wt,kf ), ∆r , k·, · · · , ·k) is linear over the field R of real numbers. Proof. Let x = (xk ), y = (yk ) ∈ Enq (M, u, p, s, M(p,wt,kf ) , ∆r , k·, · · · , ·k) and α, β ∈ R. Then there exist positive integers ρ1 and ρ2 such that   ∞ r

pk 1 X −s

uk Nk (M(p,wt,kf ) ∆ x) lim 0, let n0 be a positive

integer such that 1 lim r hr

∞ X

k

−s

k=n0 +1



 r

pk 

uk Nk (M(p,wt,kf ) ∆ x) < Mk , z1 , · · · , zn−1 ρ 2

for some ρ > 0. This implies that  1 lim r hr

∞ X

k=n0 +1

k

−s



 r

pk  H1 

uk Nk (M(p,wt,kf ) ∆ x) Mk , z1 , · · · , zn−1 ≤ . ρ 2

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Let 0 < |λ| < 1, then using convexity of (Mk ), we get 1 lim r hr

∞ X

k

−s

k=n0 +1

1 < |λ| lim r hr



 r

pk

λuk Nk (M(p,wt,kf )∆ x)

Mk , z1 , · · · , zn−1 ρ

∞ X

k

−s

k=n0 +1



 r

pk   H

uk Nk (M(p,wt,kf ) ∆ x)

Mk < , z1 , · · · , zn−1 . ρ 2

Since (Mk ) is continuous everywhere in [0, ∞), so

  n0 r

pk 1 X

tuk Nk (M(p,wt,kf ) ∆ x)

−s k Mk h(t) = lim , z1 , · · · , zn−1 r hr ρ k=1

is continuous at 0. Hence, there is 0 < δ < 1 such that |h(t)| < /2 for 0 < t < δ. Let K be such that |λn | < δ for n > K we have 

  n0 r

pk  H1  1 X

λn uk Nk (M(p,wt,kf )∆ x) −s lim k Mk , z1 , · · · , zn−1 < . r hr ρ 2 k=1

Thus, for n > K,    ∞ r

pk  1 1 X −s H

λn uk Nk (M(p,wt,kf )∆ x) lim k Mk , z1 , · · · , zn−1 < . r hr ρ k=1

Hence, g(λx) → 0 as λ → 0. This completes the proof of the theorem.

Theorem 2.3. If M0 = (Mk0 ) and M00 = (Mk00) are two Musielak-Orlicz functions and s, s1 , s2 be non-negative real numbers, then we have (i) Enq (M0 , u, p, s, M(p,wt,kf ) , ∆r , k·, · · · , ·k) ∩ Enq (M00, u, p, s, M(p,wt,kf ) , ∆r , k·, · · · , ·k) Enq (M0

⊆

+ M00, u, p, s, M(p,wt,kf ), ∆r , k·, · · · , ·k).

(ii) If the inequality s1

≤

s2 holds, then Enq (M0 , u, p, s1, M(p,wt,kf ), ∆r , k·, · · · , ·k) ⊆

Enq (M0 , u, p, s2, M(p,wt,kf ) , ∆r , k·, · · · , ·k). (iii) If M0 and M00 are equivalent, Enq (M00 , u, p, s, M(p,wt,kf ) , ∆r , k·, · · ·

q

then En (M0 , u, p, s, M(p,wt,kf ) , ∆r , k·, · · · , ·k)

=

, ·k).

Proof. It is obvious so we omit the details. Theorem 2.4. Suppose that 0 < rk ≤ pk < ∞ for each k. Then Enq (M, u, r, s, M(p,wt,kf ) , ∆r , k·, · · · , ·k) ⊆ Enq (M, u, p, s, M(p,wt,kf ) , ∆r , k·, · · · , ·k). Proof. Let x ∈ Enq (M, u, r, s, M(p,wt,kf ) , ∆r , k·, · · · , ·k). Then there exists some ρ > 0 such that   ∞ r

rk 1 X −s

uk Nk (M(p,wt,kf )∆ x) lim < ∞. k Mk , z1 , · · · , zn−1 r hr ρ k=1

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This implies that

 r



uk Nk (M(p,wt,kf ) ∆ x)

Mk , z1 , · · · , zn−1 ≤ 1 ρ

for sufficiently large value of k, say k ≥ k0 for some fixed k0 ∈ N. Since (Mk ) is nondecreasing, we have   ∞ r pk 1 X −s

uk Nk (M(p,wt,kf ) ∆ x) lim k Mk , z1 , · · · , zn−1 r hr ρ k≥k0

  ∞ r

rk 1 X −s

uk Nk (M(p,wt,kf ) ∆ x)

, z1 , · · · , zn−1 k Mk ≤ lim < ∞. r hr ρ k≥k0

Hence, x ∈

Enq (M, u, p, s, M(p,wt,kf ) , ∆r , k·, · · ·

, ·k).

Theorem 2.5. (i) If 0 < pk ≤ 1 for each k, then Enq (M, u, p, s, M(p,wt,kf ), ∆r , k·, · · · , ·k) ⊆ Enq (M, u, s, M(p,wt,kf ) , ∆r , k·, · · · , ·k). (ii)

If

pk

≥

1

for

all

Enq (M, u, p, s, M(p,wt,kf ) , ∆r , k·, · · ·

k,

then

Enq (M, u, s, M(p,wt,kf ), ∆r , k·, · · · , ·k)

⊆

, ·k).

Proof. It is easy to prove by using above so we omit the details.

3

Applications to statistical convergence

Fast [8] extended the notion of usual convergence of a sequence of real or complex numbers and called it statistical convergence. This notion turned out to be one of the most active areas of ˘ at [24]. Fridy and Orhan [11] research in summability theory after the works of Fridy [10] and Sal´ defined and studied the notion of lacunary statistical convergence. Some recent related work and applications we refer to [1, 2, 6, 17, 20]. We are now ready to define following notions: Definition 3.1. Let θ = (kr ) be a lacunary sequence. Then, the sequence x = (xk ) is Nk (u)lacunary statistically convergent to the number l provided that for every  > 0,   r

uk Nk (M 1 (p,wt ,kf ) ∆ x) − l

lim k ∈ Ir : , z1 · · · , zn−1 ≥  = 0. r hr ρ In

symbols,

we

shall

write

[Nk , u, M(p,wt,kf ), S, ∆r ]θ - lim x

=

l

or

xk

→

l([Nk , u, M(p,wt,kf ) , S, ∆r]θ ). If we take θ = (2r ), then we shall write [Nk , u, M(p,wt,kf ), S, ∆r ] instead of [Nk , u, M(p,wt,kf ), S, ∆r ]θ . Definition 3.2. Let θ = (kr ) be a lacunary sequence, M = (Mk ) be a Musielak-Orlicz function, u = (uk ) be a sequence of strictly positive real numbers and p = (pk ) be a bounded sequence of positive real numbers. We say that x = (xk ) is strongly Nk (u, M(p,wt,kf ) , ∆r )-lacunary convergent to l with respect to M provided that   r

pk 1 X

uk Nk (M(p,wt,kf )∆ x) − l lim = 0. Mk , z1 , · · · , zn−1 r hr ρ k∈Ir

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The set of all strongly Nk (u, M(p,wt,kf ) , ∆r )-lacunary convergent sequences to l with respect to M is denoted by [Nk , u, M, p, M(p,wt,kf ) , ∆r , k·, · · · , ·k]θ . In symbols, we shall write xk → l([Nk , u, M, p, M(p,wt,kf ), ∆r , k·, · · · , ·k]θ ). Note that, in the special case, M(x) = x, pk = p0 for all k ∈ N, we shall write [Nk , u, M(p,wt,kf ), ∆r , k·, · · · , ·k]θ instead of [Nk , u, M, p, M(p,wt,kf ) , ∆r , k·, · · · , ·k]θ. Theorem 3.3. Let θ = (kr ) be a lacunary sequence. (i) If a sequence x = (xk ) is strongly Nk (u, M(p,wt,kf ), ∆r )-lacunary convergent to l, then it is Nk (u, M(p,wt,kf ) , ∆r )-lacunary statistically convergent to l. (ii) If a bounded sequence x = (xk ) is Nk (u, M(p,wt,kf ), ∆r )-lacunary statistically convergent to l, then it is strongly Nk (u, M(p,wt,kf ) , ∆r )-lacunary convergent to l. Proof. (i) Let  > 0 and xk → l([Nk, u, M(p,wt,kf ) , ∆r , k·, · · · , ·k]θ ). Then, we have r

p0 X

uk Nk (M(p,wt,kf ) ∆ x) − l

, z1 , · · · , zn−1

ρ

k∈Ir

X

≥

k∈I

uk Nk (M (p,wt ,k ) ∆rr x)−l

f

,z1 ,··· ,zn−1 ≥ ρ

r

uk Nk (M

p0 (p,wt ,kf ) ∆ x) − l

, z1 , · · · , zn−1

ρ

  r

uk Nk (M

(p,wt ,kf ) ∆ x) − l

p0 ≥  k ∈ Ir : , z1 , · · · , zn−1 ≥  . ρ

Hence, xk → l([Nk , u, M(p,wt,kf ) , S, ∆r ]θ ).

(ii) Suppose that xk → l([Nk , u, M(p,wt,kf ) , S, ∆r]θ ) and let x ∈ l∞ . Let  > 0 be given and take N such that  r

   1 

uk Nk (M 1 (p,wt ,kf ) ∆ x) − l p0

≤  lim k ∈ Ir : , z1 , · · · , zn−1 ≥ 2K p0 r hr ρ 2

for all r > N and set  r

uk Nk (M

 1  (p,wt ,kf ) ∆ x) − l p0

T r = k ∈ Ir : , z1 , · · · , zn−1 ≥ , ρ 2 where K = supk |xk | < ∞. Now for all r > N we have lim r

r

p0 1 X

uk Nk (M(p,wt,kf ) ∆ x) − l , z1 , · · · , zn−1

hr ρ k∈Ir

= lim r

r

p0 1 X

uk Nk (M(p,wt,kf ) ∆ x) − l , z1 , · · · , zn−1

hr ρ k∈Ir

k∈Tr

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.5, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

+ lim r

r

p0 1 X

uk Nk (M(p,wt,kf ) ∆ x) − l

, z1 , · · · , zn−1

hr ρ k∈Ir

k∈T / r

≤ lim r

 1  hr   p0 hr = . K + p 0 hr 2K 2hr

Thus, (xk ) ∈ [Nk , u, M(p,wt,kf ), ∆r , k·, · · · , ·k]θ .

Theorem 3.4. For any lacunary sequence θ, if limr→∞ inf qr > 1 then [Nk , u, M(p,wt,kf ) , S, ∆r] ⊂ [Nk , u, M(p,wt,kf ) , S, ∆r]θ . Proof. If limr→∞ inf qr > 1, then there exists a δ > 0 such that 1 + δ ≤ qr for sufficiently large r. Since hr = kr − kr−1 , we have  > 0, we have  1 k ≤ kr kr

kr hr

≤

1+δ δ .

Let xk → l([Nk, u, M(p,wt,kf ) , S, ∆r]). Then for every

 r

uk Nk (M (p,wt ,kf ) ∆ x) − l

, z1 , · · · , zn−1 ≥  : ρ   r

uk Nk (M 1 (p,wt ,kf ) ∆ x) − l

≥ k ∈ Ir : , z1 , · · · , zn−1 ≥  kr ρ   r

uk Nk (M  δ 1 (p,wt ,kf ) ∆ x) − l

≥ , z1 , · · · , zn−1 ≥  . k ∈ Ir : 1 + δ hr ρ

Hence, [Nk , u, M(p,wt,kf ) , S, ∆r] ⊂ [Nk , u, M(p,wt,kf ) , S, ∆r]θ . In the next results we denote the quantity

uk Nk (M(p,wt ,k

f)

∆r x)−l

ρ

by xl,ρ k .

Theorem 3.5. Let θ = (kr ) be a lacunary sequence, M = (Mk ) be a Musielak-Orlicz function and 0 < h = inf k pk ≤ pk ≤ supk pk = H. Then [Nk , M, u, M(p,wt,kf ) , ∆r , k·, · · · , ·k]θ ⊂ [Nk , u, M(p,wt,kf ), S, ∆r ]θ . Proof. Let x ∈ [Nk , M, u, M(p,wt,kf ), ∆r , k·, · · · , ·k]θ . Then there exists a number ρ > 0 such ipk h  P that limr h1r k∈Ir Mk kxl,ρ , z , · · · , z k → 0, as r → ∞. Then given  > 0, we have 1 n−1 k  

pk 1 X

l,ρ lim Mk xk , z1 , · · · , zn−1 r hr k∈Ir

1 ≥ lim r hr



X

[Mk (1 )]pk , where /ρ = 1

k∈Ir

kxl,ρ k ,z1 ,··· ,zn−1 k≥

≥ lim r

1 hr l,ρ



pk

l,ρ

Mk xk , z1 , · · · , zn−1

X

k∈Ir

kxk ,z1 ,··· ,zn−1 k≥

≥ lim r

n o 1 X min [Mk (1 )]h, [Mk (1 )]H hr k∈Ir

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Alotaibi ET AL 770-780

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.5, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

 

n o 1

l,ρ

≥ lim k ∈ Ir : xk , z1 , · · · , zn−1 ≥  . min [Mk (1 )]h , [Mk (1 )]H . r hr

Hence, x ∈ [Nk , u, M(p,wt,kf ) , S, ∆r]θ . This completes the proof of the theorem.

Theorem 3.6. Let θ = (kr ) be a lacunary sequence, M = (Mk ) be a MusielakOrlicz function and p = (pk ) be a bounded sequence, then [Nk , u, M(p,wt,kf ) , S, ∆r]θ ⊂ [Nk , M, u, M(p,wt,kf ) , ∆r , k·, · · · , ·k]θ . Proof. Let x ∈ l∞ and xk → l([Nk, u, M(p,wt,kf ) , S, ∆r]θ ). Since x ∈ l∞ , there is a constant T > 0 such that kxl,ρ k , z1 , · · · , zn−1 k ≤ T and given  > 0 we have lim r

ipk 1 X h  l,ρ

, z , · · · , z Mk xk 1 n−1 hr k∈Ir

= lim r

1 hr

X

k∈Ir

kxl,ρ k ,z1 ,··· ,zn−1 k≥

+ lim r

1 hr l,ρ

h 

ipk

Mk xl,ρ , z , · · · , z 1 n−1 k

X

k∈Ir

kxk ,z1 ,··· ,zn−1 k n!1

and (1.6)

n X

lim sup n!1

p(j) > 1

j=n l

is su¢ cient for all solutions of (1:4) to be oscillatory. In the same year, 1989, Ladas, Philos and S…cas [11] established that all solutions of (1:4) are oscillatory if (1.7)

lim inf [ n!1

n 1 1 X ll : p(j)] > l (l + 1)l+1 j=n l

Clearly, condition (1:6) improves to (1:4). In 1991, Philos [14] extended the oscillation criterion (1:7) to the general case of the Eq.(1:3), by establishing that, if the sequence ( (n))n 0 is increasing, then the condition 2 3 n X1 1 (n (n))n (n) (1.8) lim inf 4 p(j)5 > lim sup n (n)+1 n!1 n (n) n!1 (n (n) + 1) j= (n)

su¢ ces for the oscillation of all solutions of Eq.(1:3). In 1998, Zhang and Tian [19] obtained that if ( (n)) is non-decreasing, (1.9)

lim (n

(n)) = 1

n!1

and (1.10)

lim inf n!1

n X1

p(j) >

j= (n)

1 ; e

then all solutions of Eq.(1:3) are oscillatory. Later, in 1998, Zhang and Tian [20] obtained that if ( (n)) is non-decreasing or non-monotone, (1.11)

lim sup p(n) > 0 n!1

and (1:10) holds, then all solutions of Eq.(1:3) are oscillatory. In 2008, Chatzarakis, Koplatadze and Stavroulakis [3] proved that if ( (n)) is non-decreasing or non-monotone, h(n) = max0 s n (s); (1.12)

lim sup n!1

n X

p(j) > 1;

j= (n)

then all solutions of Eq.(1:3) are oscillatory. In the same year, Chatzarakis, Koplatadze and Stavroulakis [4] proved that if ( (n)) is non-decreasing or non-monotone, h(n) = max0 s n (s); (1.13)

lim sup n!1

n X

j= (n)

p(j) < 1

and (1:10) holds, then all solutions of Eq.(1:3) are oscillatory.

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OSCILLATION ANALYSIS FOR HIGHER ORDER DIFFERENCE EQUATION

3

In 2006, Yan, Meng and Yan [17] obtained that if ( (n)) is non-decreasing, (1.14)

n X1

lim inf n!1

p(j) > 0

j= (n)

and (1.15)

lim inf n!1

n X1

p(j)

j

j= (n)

j

(j) + 1 j (j)

(j)+1

> 1;

then all solutions of Eq.(1:3) are oscillatory. Finally, in 2016, Öcalan [16] proved that if ( (n)) is non-decreasing or nonmonotone, h(n) = max0 s n (s) and (1:15) holds, then all solutions of Eq.(1:3) are oscillatory. Set (1.16)

k(n) =

n

(n) + 1 n (n)

e

k(n)

n

(n)+1

; n

1:

Clearly (1.17)

4; n

1:

Observe that, it is easy to see that n X1

p(j)k(j)

e

j= (n)

n X1

p(j)

j= (n)

and therefore condition (1:15) is better than condition (1:10). In 2006, Zhou [22] studied the following delay di¤erence equation with constant delays m

(1.18)

x(n) +

l X

pi (n)x(n

ki ) = 0; n = 0; 1;

;

i=1

where (pi (n))n 0 are sequences of nonnegative real numbers and ki is a positive integer for i = 1; 2; ; l: He obtained some new criteria for all solutions of Eq.(1.18) to be oscillatory. 2. Main Results In this section we investigated the oscillatory behavior of all solutions of Eq.(1:1). Further, we need the following lemmas proved in [1; 2]: Lemma 2.1. (Discrete Kneser’s Theorem) Let x(n) be de…ned for n n0 , and x(n) > 0 with m x(n) of constant sing for n n0 and not identically zero. Then, there exists an integer j, 0 j m with (m + j) odd for m x(n) 0 or (m + j) even for m x(n) 0 and such that j

m

1 implies ( 1)j+i

i

x(n) > 0,

for all n

n0 , j

i

m

1 implies i x(n) > 0; for all large n n0 ; 1 i j m x(n) 0 for n n0 ; and (x(n)) is bounded, then

1:

1,

and j Specially, if

( 1)i+1

m i

x(n)

0; for all n

783

n0 ; 1

i

m

1,

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.5, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

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ÖZKAN ÖCALAN AND UM UT M UTLU ÖZKAN

and i

lim

n!1

x(n) = 0; 1

i

m

1:

Lemma 2.2. Let x(n) be de…ned for n n0 , and x(n) > 0 with m x(n) 0 for n n0 and not identically zero. Then, there exists a large n1 n0 such that (n n1 )m (m 1)!

x(n)

1

m 1

x(2m

j 1

n) ; n

n1 ;

where j is de…ned in Lemma 2:1. Further, if x(n) is increasing, then m 1

n

1

x(n)

(m

m 1

2m 1

1)!

2m

x(n) ; n

1

n1 :

Set (2.1)

h(n) = max

(s)

0 s n

Clearly, h(n) is nondecreasing, and (n) h(n) for all n is nondecreasing, then we have (n) = h(n) for all n 0: Theorem 2.3. Assume that (1:2) holds. monotone, (2.2)

n X1

lim inf n!1

0: We note that if (n)

If ( (n)) is non-decreasing or non-

p(j)k(j) > (m

1)!;

j= (n)

where k(n) is de…ned by (1:16), then every solution of Eq.(1:1) either oscillates or limn!1 x(n) = 0: Proof. Assume, for the sake of contradiction, that (x(n)) is an eventually positive solution of (1:1) and limn!1 x(n) > 0. Then there exists n1 n0 such that x(n); x ( (n)) ; x (h(n)) > 0; for all n n1 : Thus, from Eq.(1:1) we have m

(2.3)

x(n) =

p(n)x( (n))

0;

for all n

n1 :

By Lemma 2:1, i x(n) are eventually of one sign for every i 2 f1; 2; : : : ; m 1g; and m 1 x(n) > 0 holds for large n; and there exist two cases to consider: (A) x(n) > 0 and (B) x(n) < 0: Case A: This says that (x(n)) is increasing. By Lemma 2:2, there exists an integer n2 n1 such that x(n)

(m

m 1

n

1 1)!

2m 1

m 1

x(n) ; n

n2

and (2.4)

x( (n))

Letting y(n) =

m 1

1 (m

m 1

(n) 1)!

2m

m 1

1

x( (n)) ; n

n2

x(n): So, we have y(n) > 0; y ( (n)) > 0 for n

n2 ;

y(n) + p(n)x( (n)) = 0; n

n2 :

which implies that (2.5)

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OSCILLATION ANALYSIS FOR HIGHER ORDER DIFFERENCE EQUATION

5

On the other hand, by (2:4) and since limn!1 (n) = 1; there exists an integer n3 n2 such that 1

x( (n))

m 1

(n)

(m

2m

1)! 1

(m

1)!

y ( (n))

1

y ( (n)) ; n

(2.6)

n3 :

In view of (2:6), Eq.(2:5) gives (2.7)

y(n) +

1 (m

1)!

p(n)y ( (n))

0; n

n3 :

Taking into account that y(n) is nonincreasing and h(n) is nondecreasing, (n) h(n) for all n 0; from (2:7) we get (2.8)

y(n) +

1 (m

1)!

p(n)y (h(n))

0; n

n3 :

It follows that (2.9)

y(n) + p(n)y (h(n))

0; n

n3 ;

where p(n) = (mp(n)1)! ; which means that inequality (2:9) has an eventually positive solution. On the other hand, we know from Lemma 2:3 in [16] that (2.10)

n X1

lim inf n!1

p(j)k(j) = lim inf n!1

j= (n)

n X1

p(j)k(j);

j=h(n)

where h(n) is de…ned by (2:1). Therefore, condition (2:2) and (2:10) imply that (2.11)

lim inf n!1

n X1

p(j)k(j) =

j=h(n)

1 (m

1)!

n X1

lim inf n!1

p(j)k(j) > 1

j=h(n)

Thus, by Theorem 1 in [16], Eq.(2:9) has no eventually positive solution. This is a contradiction. Case B: Note that, by Lemma 2:1, it is impossible that the case that m is even. In what follows, we only consider the case that m is odd. Case B says that x(n) is decreasing and bounded, and so, (x(n)) converges a constant a: By Lemma 2:1, we get (2.12)

( 1)i+1

m i

x(n) > 0; for all large n

n1 ; 1

i

m

1,

and (2.13)

lim

n!1

By (2:13), there exists an integer n4 (2.14)

0

m 1

x(n)

m 1

x(n) = 0:

n1 such that "; for any " > 0; n

It is obvious that a > 0: So, there exists an integer n5 (2.15)

x(n) >

1 1 a; x( (n)) > a; n 2 2

785

n4 : n4 such that

n5 :

OCALAN ET AL 781-789

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.5, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

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ÖZKAN ÖCALAN AND UM UT M UTLU ÖZKAN

Thus, Eq.(1:1) implies that a x(n) + p(n) 0; n n5 : 2 Summing both sides of (2:16) from n5 to n, we obtain m

(2.16)

(2.17)

m 1

x(n + 1)

m 1

n a X p(s) 2 s=n

x(n5 ) +

0; n

n5 :

5

Letting n ! 1; we have

n a X p(s) 2 s=n

(2.18)

"; for large n:

5

On the other hand, condition (2:2) says that there exist an integer n6 that n X1 (m 1)! ; n n6 : p(s)k(s) > (2.19) 2

n5 such

s= (n)

Since k(n)

4 for n

1; by (2:19) we get n 1 a X p(s) 2

(2.20)

a(m

1)! 8

s= (n)

; for large n;

which contradicts (2:18) and (2:20). The proof is completed. Theorem 2.4. Assume that m is even and (1:2) holds. If ( (n)) is non-decreasing or non-monotone, (2.21)

lim inf n!1

n X1

m 1

(j)p(j)k(j) > 2(m

1)2

(m

1)!;

j= (n)

where k(n) is de…ned by (1:16), then every solution of Eq.(1:1) oscillates. Proof. Assume, for the sake of contradiction, that (x(n)) is an eventually positive solution of (1:1). Then there exists n1 n0 such that x(n); x ( (n)) ; x (h(n)) > 0; for all n n1 : According to the proof of Theorem 2:3, there exists a positive integer n1 such that (2:3) holds. By Lemma 2:1, we have x(n) > 0 which implies x(n) is increasing. In view of proof of Theorem 2:3, we have (2.22) where y(n) = (2.23)

x( (n)) m 1

1 (m

(n) 1)!

2m

1

m 1

y ( (n)) ;

x(n): Therefore, from Eq.(2:5) and (2:22), we obtain

y(n) +

1 (m

(n) 1)!

2m

m 1

p(n)y ( (n))

1

0; n

n2 :

Taking into account that y(n) is nonincreasing and h(n) is nondecreasing, (n) h(n) for all n 0; from (2:23) we get, (2.24)

y(n) +

1 (m

(n) 1)!

2m

m 1

p(n)y (h(n))

1

786

0; n

n3 :

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OSCILLATION ANALYSIS FOR HIGHER ORDER DIFFERENCE EQUATION

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It follows that (2.25)

y(n) + p(n)y (h(n))

0; n

n3 ;

m 1

p(n) where p(n) = 2m(n)1 (m 1)! ; which means that inequality (2:25) has an eventually positive solution. On the other hand, we know from Lemma 2.3 in [16] that

(2.26)

lim inf n!1

n X1

p(j)k(j) = lim inf n!1

j= (n)

n X1

p(j)k(j);

j=h(n)

where h(n) is de…ned by (2:1). Therefore, condition (2:21) and (2:26) imply that (2.27) n n X1 X1 1 1 lim inf p(j)k(j) = lim inf 2 n!1 (m 1)! 2(m 1) n!1

m 1

(j)p(j)k(j) > 1

j=h(n)

j=h(n)

Thus, by Theorem 1 in [16], Eq.(2:25) has no eventually positive solution. This contradiction completes the proof. Now, using (1.16), (1.17), Theorem 2.3 and Theorem 1 in [16], we have the following results immediately. Corollary 2.5. Assume that (1:2) holds. If ( (n)) is non-decreasing or nonmonotone, (2.28)

n X1

lim inf n!1

p(j) >

j= (n)

1 (m e

1)!;

then every solution of Eq.(1:1) either oscillates or limn!1 x(n) = 0: Corollary 2.6. Assume that m is even and (1:2) holds. If ( (n)) is non-decreasing or non-monotone, (2.29)

lim inf n!1

n X1

2

m 1

(j)p(j) >

j= (n)

2(m 1) (m e

1)!;

then every solution of Eq.(1:1) oscillates. Finally, using the proofs of Theorem 2.3 and Theorem 2.4, and from the Theorem 2.1 in [3], we obtain the following results by removing the proofs. Theorem 2.7. Assume that (1:2) and (1:14) hold. If ( (n)) is non-decreasing or non-monotone, (2.30)

lim sup n!1

n X

p(j) > (m

1)!;

j=h(n)

where h(n) is de…ned by (2:1), then every solution of Eq.(1:1) either oscillates or limn!1 x(n) = 0:

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ÖZKAN ÖCALAN AND UM UT M UTLU ÖZKAN

Theorem 2.8. Assume that m is even and (1:2) holds. If ( (n)) is non-decreasing or non-monotone, n X 2 m 1 (2.31) lim sup (j)p(j) > 2(m 1) (m 1)!; n!1

j=h(n)

where h(n) is de…ned by (2:1), then every solution of Eq.(1:1) oscillates. we present an example to show the signi…cance of our new result. Example 2.1. Consider the retarded di¤ erence equation 3 3 (2.32) x(n) + x( (n)) = 0; n 0, e with n 3; if n is even (n) = : n 1; if n is odd Here, it is clear that (1:2) is satis…ed. By (2:1), we see that h(n) = max

0 s n

n n

(s) =

2; if n is even : 1; if n is odd

Computing, we get n X1

p(j) =

j= (n)

Thus lim inf n!1

n X1

j= (n)

6=e; if n is even : 3=e; if n is odd

p(j) =

3 1 > (m e e

1)! =

2 ; e

that is, condition (2:28) of Corollary 2:5 is satis…ed and therefore every solution of Eq.(2:32) either oscillates or limn!1 x(n) = 0: References [1] R. P. Agarwal, Di¤erence Equations and Inequalities, Marcel Dekker, New York, 2000. [2] R. P. Agarwal, S.R. Grace and D. O’Regan, Oscillation Theory for Di¤erence and Functional Di¤erential Equations, Kluwer Academic Publishers, The Netherlands, 2000. [3] G. E. Chatzarakis, R. Koplatadze, and I. P. Stavroulakis, Oscillation criteria of …rst order linear di¤erence equations with delay argument, Nonlinear Anal., 68 (2008), 994–1005. [4] G. E. Chatzarakis, R. Koplatadze, and I. P. Stavroulakis, Optimal oscillation criteria for …rst order di¤erence equations with delay argument, Paci…c J. Math., 235 (2008), 15–33. [5] G. E. Chatzarakis, Ch. G. Philos, and I. P. Stavroulakis, On the oscillation of the solutions to linear di¤erence equations with variable delay, Electron. J. Di¤erential Equations, No. 50 (2008), 15 pp. [6] G. E. Chatzarakis, Ch. G. Philos, and I. P. Stavroulakis, An oscillation criterion for linear di¤erence equations with general delay argument, Portugal. Math., (N.S.), 66 (4) (2009), 513–533. [7] M.-P. Chen and J. S. Yu, Oscillations of delay di¤erence equations with variable coe¢ cients, In Proceedings of the First International Conference on Di¤erence Equations, Gordon and Breach, London 1994, 105–114. [8] L. H. Erbe and B. G. Zhang, Oscillation of discrete analogues of delay equations, Di¤erential Integral Equations, 2 (1989), 300–309. [9] I. Györi and G. Ladas, Linearized oscillations for equations with piecewise constant arguments, Di¤erential Integral Equations, 2 (1989), 123–131. [10] I. Györi and G. Ladas, Oscillation Theory of Delay Di¤erential Equations with Applications, Clarendon Press, Oxford, 1991.

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OSCILLATION ANALYSIS FOR HIGHER ORDER DIFFERENCE EQUATION

9

[11] G. Ladas, Ch. G. Philos, and Y. G. S…cas, Sharp conditions for the oscillation of delay di¤erence equations, J. Appl. Math. Simulation, 2 (1989), 101–111. [12] G. Ladas, Explicit conditions for the oscillation of di¤erence equations, J. Math. Anal. Appl., 153 (1990), 276–287. [13] G. S. Ladde, V. Lakshmikantham and B. G. Zhang, Oscillation theory of di¤erential equations with deviating arguments, Marcel Dekker, New York, (1987). [14] Ch. G. Philos, On oscillations of some di¤erence equations, Funkcial. Ekvac., 34 (1991), 157–172. [15] Ö. Öcalan and S. S ¸ . Öztürk, An Oscillation Criterion for First Order Di¤erence Equations, Results Math., 68 (2015), no. 1-2, 105–116. [16] Ö. Öcalan, An improved oscillation criterion for …rst order di¤erence equations, Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 59(107) (2016), no. 1, 65–73. [17] W. Yan, Q. Meng and J. Yan, Oscillation criteria for di¤erence equation of variable delays, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 13A (2006), Part 2, suppl., 641–647. [18] J. S. Yu and Z. C. Wang, Some further results on oscillation of neutral di¤erential equations, Bull. Austral. Math. Soc., 46 (1992) 149–157. [19] B.G. Zhang and C. J. Tian, Oscillation criteria for di¤erence equations with unbounded delay, Comput. Math. Appl., 35 (4), (1998), 19–26. [20] B.G. Zhang and C. J. Tian, Nonexistence and existence of positive solutions for di¤erence equations with unbounded delay. Comput. Math. Appl., 36, (1998), 1–8. [21] B. G. Zhang, X. Z. Yan and X. Y. Liu, Oscillation criteria of certain delay dynamic equations on time scales, J. Di¤erence Equ. Appl., 11(10) (2005), 933–946. [22] Y. Zhou, Oscillation of higher-order delay di¤erence equations, Adv. Di¤erence Equ., Art. ID 65789 (2006), 7 pp. Akdeniz University, Faculty of Science, Department of Mathematics, 07058, Antalya, Turkey E-mail address : [email protected] Afyon Kocatepe University, Faculty of Science and Arts, Department of Mathematics, ANS Campus, 03200, Afyon, Turkey E-mail address : [email protected]

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OCALAN ET AL 781-789

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.5, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

On Orthonormal Wavelet Bases Richard A. Zalik

∗

Abstract Given a multiresolution analysis with one generator in L2 (Rd ), we give a characterization in closed form and in the frequency domain, of all orthonormal wavelets associated to this MRA. Examples are given. This theorem corrects a previous result of the author.

1

Introduction

In what follows Z will denote the set of integers, and R the set of real numbers. We will always assume that A is a dilation matrix preserving the lattice Zd ; that is, AZd ⊂ Zd and all its eigenvalues have modulus greater than 1; A∗ will denote the transpose of A and B := (A∗ )−1 . The underlying space will be L2 (Rd ), where d ≥ 1 is an integer and I will stand for the identity matrix. Boldface lowcase letters will denote elements of Rd , which will be represented as column vectors; x · y will stand for the standard dot product of the vectors x and y; ||x||2 := x · x. Let A ∈ Rd×d and a := | det A|. For every j ∈ Z and k ∈ Zd the dilation operator A D and the translation operator Tk are defined on L2 (Rd ) by DA f (t) := a1/2 f (At)

and

Tk f (t) := f (t + k)

respectively. Let u = {u1 , . . . , um } ⊂ L2 (Rd ); then T (u1 , . . . , um ) = T (u), S(u1 , . . . , um ) = S(u) and S(A; u1 , . . . , um ) = S(A; u) are respectively defined by T (u) := {Tk u; u ∈ u, k ∈ Zd },

S(u) := span T (u),

and S(A, u) := span {DA Tk u; u ∈ u, k ∈ Zd }. In [5] we formulated a representation theorem for multiresolution analyses having an arbitrary set u1 , . . . , un of scaling functions, i.e., the set of translates of all these functions constitutes an orthonormal basis of V0 . However the proof was based on the implicit (and incorrect) assumption that any such function u` is contained in S(A, u` ), and it is therefore not valid. The purpose of this paper is to apply the method of proof ∗

Department of Mathematics and Statistics, Auburn University, AL 36849-5310, [email protected] 790

Zalik 790-797

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.5, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

employed in [5] to prove a representation theorem for MRA’s having a single scaling function, and to provide some examples. A function f will be called Zd –periodic if it is defined on Rd and Tk f = f for every k ∈ Zd . The Fourier transform of a function f will be denoted by fb or F(f ). If f ∈ L(Rd ), Z b e−i2πx·t f (t) dt. f (x) := Rd

The Fourier transform is extended to L2 (Rd ) in the usual way. Our starting point and motivation is the following well known characterization in Fourier space of affine MRA orthonormal wavelets in L2 (R) (see e.g. Hern´ andez and Weiss [2], Wojtaszczyk [4]) which, with the definition of Fourier transform we have adopted, may be stated as follows. Theorem A. Let ϕ be a scaling function for a multiresolution analysis M with associated low pass filter p. The following propositions are equivalent: (a) ψ is an MRA orthonormal wavelet associated with M . (b) There is a measurable unimodular Z–periodic function µ(x) such that b ψ(2x) = ei2πx µ(2x)p(x + 1/2)ϕ(x) b

a.e.

Recall that a multiresolution analysis (MRA) in L2 (Rd ) (generated by A) is a sequence {Vj ; j ∈ Z} of closed linear subspaces of L2 (Rd ) such that: (i) Vj ⊂ Vj+1 for every j ∈ Z. (ii) For every j ∈ Z, f (t) ∈ Vj if and only if f (At) ∈ Vj+1 . j∈Z Vj

is dense in L2 (Rd ).

j∈Z Vj

= ∅.

(iii)

S

(iv)

T

(v) There is a function u (called the scaling function of the MRA) such that T (u) is an orthonormal basis of V0 . A finite set of functions ψ = {ψ1 , · · · , ψm } ∈ L2 (Rd ) is called an orthonormal wavelet system if the affine sequence {DjA Tk ψ` ; j ∈ Z, k ∈ Zd , ` = 1, · · · , m} is an orthonormal basis of L2 (Rd ). Let ψ := {ψ1 , · · · , ψm } be an orthonormal wavelet system in L2 (Rd ) generated by a matrix A; for j ∈ Z we define X Vj = S(Ar ; ψ). r TA (at ∗ bt ), max{IA (ai ), IA (bi )} < IA (ai ∗ bi ), and max{FA (af ), FA (bf )} < FA (af ∗ bf ). Then min{TA (at ), TA (bt )} ≥ tα1 > TA (at ∗ bt ), max{IA (ai ), IA (bi )} ≤ tα2 < IA (ai ∗ bi ), and max{FA (af ), FA (bf )} ≤ tα3 < FA (af ∗ bf ) for some tα1 ∈ (0, 1], and tα2 , tα3 ∈ [0, 1). Hence at , bt , ai , bi , af , bf ∈ A(tα1 ,tα2 ,tα3 ) , but at ∗ bt , ai ∗ bi , af ∗ bf ∈ / A(tα1 ,tα2 ,tα3 ) , which is a contradiction. Hence min{TA (x), TA (y)} ≤ TA (x ∗ y), max{IA (x), IA (y)} ≥ IA (x ∗ y), and max{FA (x), FA (y)} ≥ FA (x ∗ y) for any x, y ∈ X. Therefore A is a neutrosophic subalgebra of X.

□

Since [0, 1] is a completely distributive lattice with respect to the usual ordering, we have the following theorem. Theorem 3.7. If {Ai |i ∈ N} is a family of neutrosopic subalgebras of a BE-algebra X, then ({Ai |i ∈ N}, ⊆) forms a complete distributive lattice. Proposition 3.8. If A is a neutrosopic subalgebra of a BE-algebra X, then TA (x) ≤ TA (1), IA (x) ≥ IA (1), and FA (x) ≥ FA (1) for all x ∈ X. □

Proof. Straightforward.

Theorem 3.9. Let A be a neutrosophic subalgebra of a BE-algebra X. If there exists a sequence {an } in X such that limn→∞ TA (an ) = 1, limn→∞ IA (an ) = 0, and limn→∞ FA (an ) = 0, then TA (1) = 1, IA (1) = 0, and FA (1) = 0. Proof. By Proposition 3.8, we have TA (x) ≤ TA (1), IA (x) ≥ IA (1), and FA (x) ≥ FA (1) for all x ∈ X. Hence we have TA (an ) ≤ TA (1), IA (an ) ≥ IA (1), and FA (an ) ≥ FA (1) for every positive integer n. Therefore 1 = limn→∞ TA (an ) ≤ TA (1) ≤ 1, 0 = limn→∞ IA (an ) ≥ IA (1) ≥ 0, and 0 = limn→∞ FA (an ) ≥ FA (1) ≥ 0. Thus we □

have TA (1) = 1, TA (1) = 0, and FA (1) = 0. Proposition 3.10. If every neutrosophic subalgebra A of a BE-algebra X satisfies the condition (3.1) TA (x ∗ y) ≥ TA (x), IA (x ∗ y) ≤ IA (x), FA (x ∗ y) ≤ FA (x), for any x, y ∈ X, then TA , IA , and FA are constant functions.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.5, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

Neutrosophic sets applied to mighty filters in BE-algebras Proof. It follows from (3.1) that TA (x) = TA (1 ∗ x) ≥ TA (1), IA (x) = IA (1 ∗ x) ≤ IA (1), and FA (x) = FA (1 ∗ x) ≤ FA (1) for any x ∈ X. By Proposition 3.8, we have TA (x) = TA (1), IA (x) = IA (1), and FA (x) = FA (1) for any x ∈ X. Hence TA , IA , and FA are constant functions.

□

Proposition 3.11. Let A be a neutrosophic filter of a BE-algebra X. Then (i) min{TA (x ∗ (y ∗ z)), TA (y)} ≤ TA (x ∗ z), max{IA (x ∗ (y ∗ z)), IA (y)} ≥ IA (x ∗ z), and max{FA (x ∗ (y ∗ z)), FA (y)} ≥ FA (x ∗ z) for any x, y ∈ X. (ii) TA (a) ≤ TA ((a ∗ x) ∗ x), IA (a) ≥ IA ((a ∗ x) ∗ x), and FA (a) ≥ FA ((a ∗ x) ∗ x) for any a, x ∈ X. Proof. (i) Using (BE4) and (NSF2), we have TA (x ∗ z) ≥ min{TA (y ∗ (x ∗ z)), TA (y)} = min{TA (x ∗ (y ∗ z)), TA (y)}, IA (x ∗ z) ≤ max{IA (y ∗ (x ∗ z)), IA (y)} = max{IA (x ∗ (y ∗ z)), IA (y)}, and FA (x ∗ z) ≤ max{FA (y ∗ (x ∗ z)), FA (y)} = max{FA (x ∗ (y ∗ z)), FA (y)} for any x, y ∈ X. (ii) Taking y := (a ∗ x) ∗ x and x := a in (NSF2), we have TA ((a ∗ x) ∗ x) ≥ min{TA (a ∗ ((a ∗ x) ∗ x)), TA (a)} = min{TA ((a ∗ x) ∗ (a ∗ x)), TA (a)} = min{TA (1), TA (a)} = TA (a), IA ((a ∗ x) ∗ x) ≤ max{IA (a ∗ ((a ∗ x) ∗ x)), IA (a)} = max{IA ((a ∗ x) ∗ (a ∗ x)), IA (a)} = max{IA (1), IA (a)} = IA (a), and FA ((a ∗ x) ∗ x) ≤ max{FA (a ∗ ((a ∗ x) ∗ x)), FA (a)} = max{FA ((a ∗ x) ∗ (a ∗ x)), FA (a)} = max{FA (1), FA (a)} = FA (a) for any a, x ∈ X. □ Theorem 3.12. ([12]) Let A be a neutrosophic set in a BE-algebra. Then A is a neutrosophic filter of X if and only if it satisfies (NSF1) and (3.2) if x ≤ y ∗ z for any x, y ∈ X, then min{TA (x), TA (y)} ≤ TA (z), max{IA (x), IA (y)} ≥ IA (z), and max{FA (x), FA (y)} ≥ FA (z). Theorem 3.13. If every neutrosophic set of a BE-algebra X satisfies (NSF1) and Proposition 3.11(i), then it is a neutrosophic filter of X. Proof. Taking x := 1 in Proposition 3.11(i) and using (BE3), we get TA (z) = TA (1 ∗ z) ≥ min{TA (1 ∗ (y ∗ z)), TA (y)} = min{TA (y ∗ z), TA (y)}, IA (z) = IA (1 ∗ z) ≤ max{IA (1 ∗ (y ∗ z)), TA (y)} = max{IA (y ∗ z), IA (y)}, and FA (z) = FA (1 ∗ z) ≤ max{FA (1 ∗ (y ∗ z)), FA (y)} = max{FA (y ∗ z), FA (y)} for any y, z ∈ X. Hence A is a □

neutrosophic filter of X.

Corollary 3.14. Let A be a neutrosophic set of a BE-algebra X. Then A is a neutrosophic filter of X if and only if it satisfies (NSF1) and Proposition 3.11(i). Theorem 3.15. Let A be a neutrosophic set of a BE-algebra X. Then A is a neutrosophic filter of X if and only if it satisfies the following conditions: (i) TA (y ∗ x) ≥ TA (x), IA (y ∗ x) ≤ IA (x), and FA (y ∗ x) ≤ FA (x); (ii) TA ((a∗(b∗x))∗x) ≥ min{TA (a), TA (b)}, IA ((a∗(b∗x))∗x) ≤ max{IA (a), IA (b)}, and FA ((a∗(b∗x))∗x) ≤ max{FA (a), FA (b)} for any a, b, x ∈ X. Proof. Assume that A is a neutrosophic filter of X. Using (NSF2), we have TA (y ∗ x) ≥ min{TA (x ∗ (y ∗ x)), TA (x)} = min{TA (1), TA (x)} = TA (x), IA (y ∗ x) ≤ max{IA (x ∗ (y ∗ x)), IA (x)} = max{IA (1), IA (x)} = IA (x), and FA (y ∗ x) ≤ max{FA (x ∗ (y ∗ x)), FA (x)} = max{FA (1), FA (x)} = FA (x), for any x, y ∈ X. It follows

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Jung Mi Ko ET AL 798-806

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.5, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

Jung Mi Ko and S. S. Ahn from Proposition 3.11 that TA ((a ∗ (b ∗ x)) ∗ x) ≥ min{TA ((a ∗ (b ∗ x)) ∗ (b ∗ x)), TA (b)} ≥ min{TA (a), TA (b)}, IA ((a ∗ (b ∗ x)) ∗ x) ≤ max{IA ((a ∗ (b ∗ x)) ∗ (b ∗ x)), IA (b)} ≤ max{IA (a), IA (b)}, and FA ((a ∗ (b ∗ x)) ∗ x) ≤ max{FA ((a ∗ (b ∗ x)) ∗ (b ∗ x)), FA (b)} ≤ max{FA (a), FA (b)} for any x, a, b ∈ X. Conversely, assume that A is a neutrosophic set of X satisfying conditions (i) and (ii). Taking y := x in (i), we have TA (1) = TA (x ∗ x) ≥ TA (x), IA (1) = IA (x ∗ x) ≤ IA (x) and FA (1) = FA (x ∗ x) ≤ FA (x) for any x ∈ X. Using (ii), we get TA (y) = TA (1 ∗ y) = TA (((x ∗ y) ∗ (x ∗ y)) ∗ y) ≥ min{TA (x ∗ y), TA (x)}, IA (y) = IA (1 ∗ y) = IA (((x∗y)∗(x∗y))∗y) ≤ max{IA (x∗y), IA (x)}, FA (y) = FA (1∗y) = FA (((x∗y)∗(x∗y))∗y) ≤ max{FA (x∗y), FA (x)} for any x, y ∈ X. Hence A is a neutrosophic filter of X.

□

4. Neutrosophic mighty filters in BE-algebras Definition 4.1. A neutrosophic set A in a BE-algebra X is called a neutrosophic mighty filter of X if it satisfies (NSF1) and (NSF3) min{TA (z ∗ (y ∗ x)), TA (z)} ≤ TA (((x ∗ y) ∗ y) ∗ x)), max{IA (z ∗ (y ∗ x)), IA (z)} ≥ IA (((x ∗ y) ∗ y) ∗ x), and max{FA (z ∗ (y ∗ x)), FA (z)} ≥ FA (((x ∗ y) ∗ y) ∗ x) for any x, y, z ∈ X. Example 4.2. Let X := {1, a, b, c, d, 0} be a BE-algebra ([8]) with the following table: ∗ 1 a b c d 0 Define a neutrosophic set A in X as follows: TA (x) =

IA (x) =

FA (x) =

1 1 1 1 1 1 1

a b c d 0 a b c d 0 1 b c b c a 1 b a d a 1 1 a a 1 1 b 1 b 1 1 1 1 1

{ 0.83, if x ∈ {1, b, c} 0.12, otherwise, { 0.14, if x ∈ {1, b, c} 0.81, otherwise, { 0.14, if x ∈ {1, b, c} 0.81, otherwise.

It is easy to check that A is a neutrosophic mighty filter of X. Proposition 4.3. Every neutrosophic mighty filter of a BE-algebra X is a neutrosophic filter of X. Proof. Let A be a neutrosophic mighty filter of X. Putting y := 1 in (NSF3), we obtain min{TA (z∗(1∗x)), TA (z)} = min{TA (z ∗ x), TA (z)} ≤ TA (((x ∗ 1) ∗ 1) ∗ x) = TA (x), max{IA (z ∗ (1 ∗ x)), IA (z)} = max{IA (z ∗ x), IA (z)} ≥ IA (((x∗1)∗1)∗x) = IA (x), and max{FA (z ∗(1∗x)), FA (z)} = max{FA (z ∗x), FA (z)} ≥ FA (((x∗1)∗1)∗x) = FA (x) for any x, y, z ∈ X. Hence A is a neutrosophic filter of X.

□

The converse of Proposition 4.3 may be not true in general (see Example 4.4).

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Jung Mi Ko ET AL 798-806

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.5, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

Neutrosophic sets applied to mighty filters in BE-algebras Example 4.4. Let X := {1, a, b, c, d} be a BE-algebra ([5]) with the following table: ∗ 1 a b c d

1 a b c d 1 a b c d 1 1 b c d 1 a 1 c c 1 1 b 1 b 1 1 1 1 1

Define a neutrosophic set A in X as follows: TA (x) =

IA (x) =

FA (x) =

{ 0.84, if x = 1 0.11, otherwise, { 0.13, if x = 1 0.81, otherwise, { 0.13, if x = 1 0.81, otherwise.

Then A is a neutrosophic filter of X, but not a neutrosophic mighty filter of X, since min{TA (1 ∗ (c ∗ a)), TA (1)} = TA (1) = 0.84 ≰ TA (((a ∗ c) ∗ c) ∗ a) = TA (a) = 0.11. Theorem 4.5. Any neutrosophic filter A of a BE-algebra X is mighty if and only if it satisfies the following conditions: (4.1) TA (y ∗ x) ≤ TA (((x ∗ y) ∗ y) ∗ x), IA (y ∗ x) ≥ IA (((x ∗ y) ∗ y) ∗ x), and FA (y ∗ x) ≥ FA (((x ∗ y) ∗ y) ∗ x) for any x, y ∈ X. Proof. Suppose that a neutrosophic filter A of a BE-algebra X satisfies the condition (4.1). Using (NSF2) and (4.1), we have min{TA (z ∗ (y ∗ x)), TA (z)} ≤ TA (y ∗ x) ≤ TA (((x ∗ y) ∗ y) ∗ x), max{IA (z ∗ (y ∗ x)), IA (z)} ≥ IA (y ∗ x) ≥ IA (((x ∗ y) ∗ y) ∗ x), and max{FA (z ∗ (y ∗ x)), FA (z)} ≥ FA (y ∗ x) ≥ FA (((x ∗ y) ∗ y) ∗ x) for any x, y ∈ X. Hence A is a neutrosophic mighty filter of X. Conversely, assume that the neutrosophic filter A of X is mighty. Setting z := 1 in (NSF3), we have min{TA (1 ∗ (y ∗ x)), TA (1)} = TA (y ∗ x) ≤ TA (((x ∗ y) ∗ y) ∗ x), max{IA (1 ∗ (y ∗ x)), IA (1)} = IA (y ∗ x) ≥ IA (((x ∗ y) ∗ y) ∗ x), and max{FA (1 ∗ (y ∗ x)), FA (1)} = FA (y ∗ x) ≥ FA (((x ∗ y) ∗ y) ∗ x) for any x, y ∈ X. Hence (4.1) holds.

□

Proposition 4.6. Let A be a neutrosophic mighty filter of a BE-algebra X. Denote that XT := {x ∈ X|TA (x) = TA (1)}, XI := {x ∈ X|IA (x) = IA (1)}, and XF := {x ∈ X|FA (x) = FA (1)}. Then XT , XI , and XF are mighty filters of X. Proof. Clearly, 1 ∈ XT , XI , XF . Let z ∗ (y ∗ x), z ∈ XT . Then TA (z ∗ (y ∗ x)) = TA (1), TA (z) = TA (1). Hence min{TA (z ∗ (y ∗ x)), TA (z)} = TA (1) ≤ TA (((x ∗ y) ∗ y) ∗ x) and so TA ((x ∗ y) ∗ y) ∗ x) = TA (1). Therefore ((x ∗ y) ∗ y) ∗ x ∈ XT . Thus XT is a mighty filter of X. Similarly, XI , XF are mighty filters of X.

□

Theorem 4.7. Let A, B be neutrosophic filters of a transitive BE-algebra X such that A ⊆ B and TA (1) = TB (1), IA (1) = IB (1), FA (1) = FB (1). If A is mighty, then B is mighty.

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Jung Mi Ko ET AL 798-806

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.5, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

Jung Mi Ko and S. S. Ahn Proof. Let x, y ∈ X. Since A is a neutrosophic mighty filter of a BE-algebra X, by (4.1) and A⊆B we have TA (1) = TA (y ∗ ((y ∗ x) ∗ x)) ≤ TA (((((y ∗ x) ∗ x) ∗ y) ∗ y) ∗ ((y ∗ x) ∗ x)) ≤ TB (((((y ∗ x) ∗ x) ∗ y) ∗ y) ∗ ((y ∗ x) ∗ x)). Since TA (1) = TB (1), we get TB ((y ∗ x) ∗ ((((y ∗ x) ∗ x) ∗ y) ∗ y) ∗ x)) = TB (((((y ∗ x) ∗ x) ∗ y) ∗ y) ∗ ((y ∗ x) ∗ x)) = TB (1). It follows from (NSF1) and (NSF2) that TB (y ∗ x) = min{TB (1), TB (y ∗ x)} = min{TB ((y ∗ x) ∗ (((((y ∗ x) ∗ x) ∗ y) ∗ y) ∗ x)), TB (y ∗ x)}

(4.2)

≤TB (((((y ∗ x) ∗ x) ∗ y) ∗ y) ∗ x). Since X is transitive, we get [((((y ∗ x) ∗ x) ∗ y) ∗ y) ∗ x]∗[((x ∗ y) ∗ y) ∗ x] ≥ ((x ∗ y) ∗ y) ∗ ((((y ∗ x) ∗ x) ∗ y) ∗ y) ≥ (((y ∗ x) ∗ x) ∗ y) ∗ (x ∗ y) ≥ x ∗ ((y ∗ x) ∗ x) = (y ∗ x) ∗ (x ∗ x) = (y ∗ x) ∗ 1 = 1. It follows from Theorem 3.12 that min{TB (((((y ∗ x) ∗ x) ∗ y) ∗ y) ∗ x), TB (1)} = TB (((((y ∗ x) ∗ x) ∗ y) ∗ y) ∗ x) ≤ TB (((x ∗ y) ∗ y) ∗ x). Using (4.2), we have TB (y ∗ x) ≤ TB (((((y ∗ x) ∗ x) ∗ y) ∗ y) ∗ x) ≤ TB (((x ∗ y) ∗ y) ∗ x). Therefore TB (y∗x) ≤ TB (((x∗y)∗y)∗x). Similarly, we have IB (y∗x) ≥ TB (((x∗y)∗y)∗x) and FB (y∗x) ≥ FB (((x∗y)∗y)∗x). □

By Theorem 4.5, B is a neutrosophic mighty filter of X.

Theorem 4.8. Let A be a neutrosophic set in a BE-algebra X and let α, β, γ ∈ [0, 1] with 0 ≤ α + β + γ ≤ 3. Then A is a neutrosophic mighty filter of X if and only if all of (α, β, γ)-level set A(α,β,γ) are mighty filters of X when A(α,β,γ) ̸= ∅. Proof. Assume that A is a neutrosophic mighty filter of X. Let α, β, γ ∈ [0, 1] be such that 0 ≤ α + β + γ ≤ 3 and A(α,β,γ) ̸= ∅. Let z ∗ (y ∗ x), z ∈ A(α,β,γ) . Then TA (z ∗ (y ∗ x)) ≥ α, TA (z) ≥ α, IA (z ∗ (y ∗ x)) ≤ β, IA (z) ≤ β, and FA (z ∗ (y ∗ x)) ≤ γ, FA (z) ≤ γ. By Definition 4.1, we have TA (1) ≥ TA (((x ∗ y) ∗ y) ∗ x) ≥ min{TA (z ∗ (y ∗ x)), TA (z)} ≥ α, IA (1) ≤ IA (((x ∗ y) ∗ y) ∗ x) ≤ max{IA (z ∗ (y ∗ x)), IA (z)} ≤ β, and FA (1) ≤ FA (((x ∗ y) ∗ y) ∗ x) ≤ max{FA (z ∗ (y ∗ x)), FA (z)} ≤ γ. Hence 1, ((x ∗ y) ∗ y) ∗ x ∈ A(α,β,γ) . Therefore A(α,β,γ) are mighty filters of X. Conversely, suppose that there exist a, b, c ∈ X such that TA (a) > TA (1), IA (b) < IA (1), and FA (c) < FA (1). Then there exist at ∈ (0, 1] and bt , ct ∈ [0, 1) such that TA (a) ≥ at > TA (1), IA (b) ≤ bt < IA (1) and FA (c) ≤ ct < FA (1). Hence 1 ∈ / A(at ,bt ,ct ) , which is a contradiction. Therefore TA (x) ≤ TA (1), IA (x) ≥ IA (1) and FA (x) ≥ FA (1) for all x ∈ X. Assume that there exist at , bt , ct , ai , bi , ci ∈ X and af , bf , cf ∈ X such that TA (((at ∗ bt ) ∗ bt ) ∗ at ) < min{TA (ct ∗ (bt ∗ at )), TA (ct )}, IA (((ai ∗ bi ) ∗ bi ) ∗ ai ) > max{IA (ci ∗ (bi ∗ ai )), IA (ci )}, and FA (((af ∗ bf ) ∗ bf ) ∗ af ) > max{FA (cf ∗ (bf ∗ af )), FA (cf )}. Then there exist st ∈ (0, 1] and si , sf ∈ [0, 1) such that TA (((at ∗bt )∗bt )∗at ) < st ≤ min{TA (ct ∗(bt ∗at )), TA (ct )}, IA (((ai ∗bi )∗bi )∗ai ) > si ≥ max{IA (ci ∗(bi ∗ai )), IA (ci )}, and FA (((af ∗bf )∗bf )∗af ) > sf ≥ max{FA (cf ∗(bf ∗af )), FA (cf )}. Hence ct ∗(bt ∗at ), ct , ci ∗(bi ∗ai ), ci ∈ A(st ,si ,sf ) and cf ∗(bf ∗af ), cf ∈ A(st ,si ,sf ) but ((at ∗bt )∗bt )∗at , ((ai ∗bi )∗bi )∗ai ∈ / A(st ,si ,sf ) , and ((af ∗bf )∗bf )∗af ∈ / A(st ,si ,sf ) ,

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Neutrosophic sets applied to mighty filters in BE-algebras which is a contradiction. Therefore min{TA (z ∗ (y ∗ x)), TA (z)} ≤ TA (((x ∗ y) ∗ y) ∗ x)), max{IA (z ∗ (y ∗ x)), IA (z)} ≥ IA (((x ∗ y) ∗ y) ∗ x)), and max{FA (z ∗ (y ∗ x)), FA (z)} ≥ FA (((x ∗ y) ∗ y) ∗ x)) for any x, y, z ∈ X. Thus A is a □

neutrosophic mighty filter of X

References [1] S. S. Ahn and K. S. So, On ideals and upper sets in BE-algerbas, Sci. Math. Jpn. 68 (2008), 279–285 . [2] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy sets and Systems 20 (1986), 87–96. [3] J. S. Han and S. S. Ahn, Hesitant fuzzy implicative filters in BE-algebras, J. Comput. Anal. Appl. 23 (2017), 530–543. [4] Y. B. Jun and S. S. Ahn, On hesitant fuzzy filters in BE-algebras, J. Comput. Anal. Appl. 22 (2017), 346-358. [5] Y. B. Jun and S. S. Ahn, On hesitant fuzzy mighty filters in BE-algebras, J. Comput. Anal. Appl. 23 (2017), 1112–1119. [6] H. S. Kim and Y. H. Kim, On BE-algerbas, Sci. Math. Jpn. 66 (2007), no. 1, 113–116. [7] M. Khan, S. Anis, F. Smarandache and Y. B. Jun, Neutrosophic N -structures and their applications in semigroups, Ann. Fuzzy Math. Inform., (to appear). [8] H. R. Lee and S. S. Ahn, Mighty filters in BE-algebras, Honam Mathematical J. 37(2) (2015), 221-233. [9] A. Rezei, A. B. Saeid, and F. Smarnadache, Neutrosophic filters in BE-algebras, Ration Mathematica 29 (2015), 65–79. [10] F. Smarandache, Neutrosophy, Neutrosophic Probablity, Sets, and Logic, Amer. Res. Press, Rehoboth, USA, 1998. [11] L. A. Zadeh, Fuzzy sets, Information and Control 8 (1965), 338-353. [12] X. Zhang, P. Yu, F. Smarandache, and C. Park, Redefined Neutrosophic filters in BE-algebras, Information, (to submit).

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Coupled fixed point of firmly nonexpansive mappings by Mann’s iterative processes in Hilbert spaces

TAMER NABILa,b1 a

b

King Khalid University, College of Science, Department of Mathematics, 61413, Abha, Saudi Arabia

Suez Canal University, Faculty of Computers and Informatics, Department of Basic Science, Ismailia, Egypt

Abstract We study the weak convergence of Mann’s explicit iteration processes to common coupled fixed point of firmly nonexpansive coupled mappings in Hilbert spaces.Our results extend and generalized the results due to Nabil and Soliman for coupled fixed point approach (T. Nabil and A. H. Soliman, weak convergence theorems of explicit iteration process with errors and applications in optimization, J. Ana. Num. Theor., 5(2017) 81: 89).

Key words and phrases. explicit iteration process, firmly coupled nonexpansive mapping; coupled fixed point; Hilbert space.

AMS Mathematics subject Classification. 47H09,47H10,47H20.

1

Introduction The study of finding the fixed point of iterative processes has attracted the interest of many researchers due

to its applications in physics, optimization, image processing and economics can be recast in terms of a fixed point problem of nonlinear mappings in Hilbert space [[1],[2], [3], [4], [5], [6]]. A lot of this studies consider this mappings as nonexpansive which is defined as: let H be a real Hilbert space and K be a nonempty closed convex subset of H. Then, a mapping R of K into H is called nonexpansive if kRx − Ryk ≤ kx − yk for all x, y ∈ K. R is called firmly nonexpansive if ||Rx − Ry||2 + ||(Id − R)x − (Id − R)y||2 ≤ ||x − y||2

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1 t [email protected] −

1

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for all x, y ∈ K, where Id : K → K denote the identity operator. We have known that every firmly nonexpansive mapping is a nonexpansive mapping. The finding of common fixed point for iteration process have been investigated since the early 1953 by Mann [7] which consider the following iteration scheme     x1 ∈ C is chosen arbitrarily    xn+1 = αn xn + (1 − αn )Rxn , ∀ n ∈ N where {αn } is a sequence in [0,1]. Several authors studied another types of iteration process such as: Halpern [8] , Bauschke [9] and Xu and Ori [10] . In 2005, Kimura et al. [11], studied the convergence of an iterative scheme to a common fixed point of a finite family of nonexpansive mappings in Banach space. The problem of finding a common fixed point of families of nonlinear mappings has been investigated by many researchers; see, for instance, ([12]-[17]). Recently, Chuang and Takahashi [18] defined the new Mann’s type iteration process by metric projection from H to K and gave weak convergence theorems for finding a common fixed point of a sequence of firmly nonexpansive mappings in a Hilbert space. More recently, in 2017 Nabil and Soliman [19] studied the weak convergen theorem f a new Mann iterative proesses with errors. The idea of coupled fixed point was started in 1987 by Guo and Lakshmikantham [20]. Several authors studied the coupled fixed point Theorem See [[21],[22], [23], [24], [25]]

In this work, we prove the weak convergence theorem for finding the coupled fixed points of iteration processes for the families of nonlinear coupled mappings in Hilbert spaces.

2

Firmly nonexpansive coupled mappings Throughout this paper we denote by N the set of positive integers and strongly (respectively weak) conver-

gence of {xn } to x ∈ H by xn → x (respectively xn * x). Let H be a Hilbert space . The inner product and the induced norm on H are denoted by < ., . > and k . k respectively. Consider F (T ) be the set of fixed points of T (i.e., F (T ) = {x ∈ C : T x = x}). Let C 6= ∅ be a closed and convex subset of a real Hilbert space H, and consider the coupled mapping 2

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T : C × C → H. Then (w1 , w2 ) ∈ C × C is said to be coupled fixed point of T if T (w1 , w2 ) = w1 and T (w2 , w1 ) = w2 , thus we can define the set of all coupled fixed points of T ( denoted by CF (T ) ) as : CF (T ) = {(x, y) ∈ C × C : T (x, y) = x, T (y, x) = y}. T : C × C → C is said to be nonexpansive coupled mapping ( denoted by N CM ) if for every (x, y) and (u, v) ∈ C × C, 1 [kx − uk + ky − vk] 2

kT (x, y) − T (u, v)k ≤

;T is said to be firmly nonexpansive coupled mapping ( denoted ny F N CM ) if, kT (x, y) − T (u, v)k2 ≤

1 [hx − u, T (x, y) − T (u, v)i + hy − v, T (x, y) − T (u, v)i], 2

equivalent; kT (x, y) − T (u, v)k2 ≤

1 hx − u + y − v, T (x, y) − T (u, v)i, 2

for all (x, y), (u, v) ∈ C × C. The following lemma give the relation between N CM and F N CM. Lemma 2.1 Let C 6= ∅ be subset of real Hilbert space H. If T : C × C → H be F N CM. Then T is N CM Proof. Since T is F N CM, for all (x, u), (u, v) ∈ C × C we get that, kT (x, y) − T (u, v)k2 ≤ ≤

1 [hx − u, T (x, y) − T (u, v)i + hy − v, T (x, y) − T (u, v)i] 2

1 [kx − ukkT (x, y) − T (u, v)k + ky − vkkT (x, y) − T (u, v)k]. 2

Therefore, we get that; kT (x, y) − T (u, v)k ≤

1 [kx − uk + ky − vk]. 2

Thus , T is N CM. The following example show that the converse of lemma 2.1 is may not be true. Example 2.1 Let H = 0, then (xn , yn ) * (x, y) where (x, y) ∈ CF (T ). Lemma 3.2.Let C 6= φ be closed and convex subset of a Hilbert space H. Consider {Tn } : C × C → H be a sequence of F N CM and be even mappings in the second variable (i.e. Tn (x, −y) = Tn (x, y), for all (x, y) ∈ C × C). Suppose that :

Pn=∞ n=1

sup{kTn+1 (x, y) − Tn (x, y)k < ∞ : (x, y) ∈ C × C}. Then {Tn (x, y)} converges

strongly to some point of C × C. In the other hand, if T : C × C → C defined by: T (x, y) = limn→∞ Tn (x, y), for all (x, y) ∈ C × C. Then limn→∞ sup{kT (x, y) − Tn (x, y)k : (x, y) ∈ C × C} = 0. Proof. First, we will prove that {Tn (x, y)} is Cauchy sequence for all (x, y) ∈ C × C. Let i, j ∈ N and i > j. we get that: kTi (x, y) − Tj (x, y)k ≤ k sup{kTi (x, y) − Tj (x, y)k : (x, y) ∈ C × C} ≤ sup{kTi (x, y) − Ti−1 (x, y)k : (x, y) ∈ C × C} + sup{kTi−1 (x, y) − Tj (x, y)k : (x, y) ∈ C × C} ≤ .... ≤

∞ X

sup{kTn+i (x, y) − Tn (x, y)k : (x, y) ∈ C × C}

i=1

Let i → ∞, Then we get that {Tn (x, y)} is a Cauchy sequence. Thus {Tn (x, y)} converges strongly to some point of C × C. Also, we have : kT (x, y) − Tj (x, y)k ≤

∞ X

sup{kTn+i (x, y) − Tn (x, y)k : (x, y) ∈ C × C}

i=1

Thus, we have that: limj→∞ sup{kT (x, y) − Tj (x, y)k : (x, y) ∈ C × C} = 0. Theorem 3.2. Let C 6= φ be closed and convex subset of a Hilbert space H. Consider {Tn } : C × C → H be a sequence of F N CM and be even mappings in the second variable (i.e. Tn (x, −y) = Tn (x, y), for all 10

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(x, y) ∈ C × C) with S :=

T∞

n=1

CF (Tn ) 6= φ. Let {αn } be a sequence of real numbers in (0,2). Let {(xn , yn )}

be a sequence in C × C defined by:     (x1 , y1 ) ∈ C is chosen arbitrarily     xn+1 := Pc ((1 − αn )xn + αn Tn (xn , yn )), ∀ n ∈ N,        yn+1 := Pc ((1 − αn )yn + αn Tn (yn , xn )), ∀ n ∈ N. If {Tn } satisfies the property:

Pn=∞ n−1

sup{kTn+1 (x, y) − Tn (x, y)k < ∞ : (x, y) ∈ C × C} and lim inf n→∞ αn (2 −

αn ) > 0, then (xn , yn ) * (x, y) where (x, y) ∈

T∞

n=1

CF (Tn ).

Proof. First, we will apply lemma 3.2. Define T : C × C → H by T (x, y) = limn→∞ Tn (x, y) kT (x, y) − T (u, v)k = k lim Tn (x, y) − lim Tn (u, v)k n→∞

n→∞

= lim kTn (x, y) − Tn (u, v)k ≤ lim n→∞

n→∞

1 (kx − uk + ky − vk). 2

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For all (x, y), (u, v) ∈ C × C. Hence T is a N CM . Therefore, we get that: lim sup{kT (x, y) − Tn (x, y)k : (x, y) ∈ B} = 0,

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n→∞

for each bounded subset B of C × C. Then by doing the same steps as in Theorem 3.1, we get that kxn − w1 k2 ≤ kxn − w1 k2 − αn (2 − αn )kxn − Tn (xn , yn )k2 .

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therefore, kyn − w2 k2 ≤ kyn − w2 k2 − αn (2 − αn )kyn − Tn (yn , xn )k2 Thus , we have that: lim kT (xn , yn ) − Tn (xn , yn )k = 0.

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n→∞

Then, we get the following: kxn − T (xn , yn )k ≤ kxn − Tn (xn , yn )k + kT (xn , yn ) − Tn (xn , yn )k. therefore, we get that: lim kxn − T (xn , yn )k = 0.

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n→∞

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By doing the same step we can prove that:

lim kyn − T (yn , xn )k = 0.

n→∞

Agian , by doing the same steps as the proof of Theorem 3.1, we get the proof of Theorem 3.2. Let C be a nonempty closed convex subset of a Hilbert space H and let {Tn } and Γ be two families of N CM mappings of C × C into C and even in the second variable, such that: ∅ 6= CF (Γ) =

T∞

n=1

CF (Tn ), where

CF (Tn ) is the set of all coupled fixed points of {Tn } and CF (Γ) is the set of all common coupled fixed points of Γ. We gave the following Condition. Condition 3.1. For each bounded sequence {(xn , yn )} of C ×C, if we have that: limn→∞ kxn −Tn (xn , yn )k = 0 and limn→∞ kyn − Tn (yn , xn )k = 0, then limn→∞ kxn − T (xn , yn )k = 0 and limn→∞ kyn − T (yn , xn )k = 0 for all T ∈ Γ. Theorem 3.3. Let H be a Hilbert space, C be a nonempty, closed and convex subset of H. Consider {Tn } : C × C → C be a sequence of F N CM mappings. Let Γ be a family of N CM of C × C into C , which satisfies ∅ 6= CF (Γ) ⊆

T∞

n=1

CF (Tn ) and condition (3.1). Let {αn } be a sequence of real numbers in (0,2), and {(xn , yn )}

be a sequence in C × C defined by:     (x1 , y1 ) ∈ C is chosen arbitrarily     xn+1 := Pc ((1 − αn )xn + αn Tn (xn , yn )), ∀ n ∈ N,        yn+1 := Pc ((1 − αn )yn + αn Tn (yn , xn )), ∀ n ∈ N. If lim inf n→∞ αn (2 − αn ) > 0, then (xn , yn ) * (x, y) where (x, y) ∈

T∞

n=1

CF (Tn ).

Proof. By doing the same steps as in the proof of Theorem 3.1, we get {(xn , yn )} is bounded and lim kxn − Tn (xn , yn )k = 0,

n→∞

also, lim kyn − Tn (yn , xn )k = 0.

n→∞

By condition (3.1), lim kxn − T (xn , yn )k = 0, lim kyn − T (yn , xn )k = 0,

n→∞

n→∞

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for all T ∈ Γ. Since {(xn , yn )} is bounded , there exist a subsequence {(xnk , ynk )} of {(xn , yn )} and (u1 , u2 ) ∈ C × C such that: xnk → u1 and ynk → u2 . By lemma 2.6, we have that (u1 , u2 ) ∈ CF (T ) for all T ∈ Γ. Thus we have that: (u1 , u2 ) ∈ CF (Γ) ⊆ (xn , yn ) * (x, y), where (x, y) ∈

T∞

n=1

T∞

n=1

CF (Tn ). Then the same steps as in the proof of Theorem 3.1 lead to

CF (Tn ).

Acknowledgments The author would like to express their gratitude to King Khalid University, Saudi Arabia, for providing administrative and technical support.

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Dynamics of the zeros of analytic continued the second kind q-Euler polynomial Cheon Seoung Ryoo Department of Mathematics, Hannam University, Daejeon 306-791, Korea

Abstract : In this paper we study that the second kind q-Euler numbers En,q and q-Euler Euler polynomials En,q (x) are analytic continued to Eq (s) and Eq (s, w). We investigate the new concept of dynamics of the zeros of analytic continued polynomials. Finally, we observe an interesting phenomenon of ‘scattering’ of the zeros of Eq (s, w). Key words : Second kind Euler polynomial, Euler Zeta function, Analytic Continuation, complex zeros, dynamics. 2000 Mathematics Subject Classification : 11B68, 11S40, 11S80. 1. Introduction Several mathematicians have studied the Bernoulli numbers and polynomials, Euler numbers and polynomials, q-Bernoulli numbers and polynomials, q-Euler numbers and polynomials, the second kind Euler numbers and polynomials(see [1-11]). These numbers and polynomials posses many interesting properties and arising in many areas of mathematics and physics. Throughout this paper, we always make use of the following notations:N = {1, 2, 3, · · · } denotes the set of natural numbers, N0 = {0, 1, 2, 3, · · · } denotes the set of nonnegative integers, Z denotes the set of integers, R denotes the set of real numbers, C denotes the set of complex numbers. We introduced the second kind q-Euler numbers En,q and polynomials En,q (x) and investigate their properties(see [6]). Let q be a complex number with |q| < 1. We define the second kind q-Euler numbers En,q and polynomials En,q (x) as follows: ∞ ∑ 2et tn Fq (t) = 2t = (1) En,q , qe + 1 n=0 n! ( ) ∞ ∑ 2et tn xt Fq (x, t) = e = En,q (x) . (2) 2t qe + 1 n! n=0 By the above definition (2) and Cauchy product, we have ∞ ∑ l=0

El,q (x)

) ∞ ∞ ∑ 2et tn ∑ m tm xt e = E x n,q e2t + 1 n! m=0 m! n=0 ( ) ( l ( ) ) ∞ ∞ l ∑ ∑ l ∑ ∑ tl tn l−n tl−n l−n = En,q x . = En,q x n! (l − n)! n l! n=0 n=0

tl = l!

(

l=0

l=0

tl , we have the following theorem. l! Theorem 1. For n ∈ N0 , one has n ( ) ∑ n En,q (x) = Ek,q xn−k . k

By using comparing coefficients

k=0

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By Theorem 1 and some calculations, we have ∫

b

En,q (x)dx =

n ( ) ∑ n

a

l

l=0

∫

b

El,q

x

n−l

dx =

a

n ( ) ∑ n l=0

l

El,q

b xn−l+1 n − l + 1 a

) n+1 ( b 1 ∑ n+1 = El,q xn−l+1 a . n+1 l l=0

By Theorem 1, we get

∫

b

En,q (x)dx = a

En+1,q (b) − En+1,q (a) . n+1

(3)

Since En,q (0) = En,q , by (3), we have the following theorem. Theorem 2. For n ∈ N, one has ∫ En,q (x) = En,q + n

x

En−1,q (t)dt. 0

By using computer, the second kind q-Euler polynomials En,q (x) can be determined explicitly. A few of them are 2 , 1+q 2 2q 2x − + , E1,q (x) = (1 + q)2 (1 + q)2 (1 + q) 4 8q 4q 2 2 2q 4x 4qx 2x2 E2,q (x) = − + − − + − + . (1 + q)3 (1 + q)3 (1 + q)3 (1 + q)2 (1 + q)2 (1 + q)2 (1 + q)2 (1 + q) E0,q (x) =

2. Analytic Continuation of the second kind q-Euler numbers and the q-Euler Zeta function By using the second kind q-Euler numbers and polynomials, the second kind q-Euler zeta function and Hurwitz q-Euler zeta functions are defined. From (1), we note that ∞ ∑ dk F (t) = 2 (−1)n q n (2n + 1)k = Ek,q , (k ∈ N). q dtk t=0 n=0 By using the above equation, we are now ready to define the second kind q-Euler zeta functions. Definition 3. For s ∈ C with Re(s) > 0, define the second kind q-Euler zeta function by ζE (s) = 2

∞ ∑ (−1)n q n . (2n + 1)s n=0

Notice that the Euler zeta function can be analytically continued to the whole complex plane, and these q-zeta function have the values of the q-Euler numbers at negative integers. That is, the second kind q-Euler numbers are related to the second kind q-Euler zeta function as ζE,q (−k) = Ek,q .

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By using (2), we note that ∞ ∑ dk F (x, t) = 2 (−1)n q n (2n + x + 1)k , (k ∈ N), q dtk t=0 n=0 and

(

d dt

)k ( ∑ ∞

tn En,q (x) n! n=0

)

= Ek,q (x), for k ∈ N.

(4)

(5)

t=0

By (4) and (5), we are now ready to define the Hurwitz q-Euler zeta functions. Definition 4. We define the Hurwitz q-zeta function ζE,q (s, x) for s ∈ C with Re(s) > 0 by ζE,q (s, x) = 2

∞ ∑

(−1)n q n . (2n + x + 1)s n=0

Note that ζE,q (s, x) is a meromorphic function on C. Relation between ζE,q (s, x) and Ek,q (x) is given by the following theorem. Theorem 5. For k ∈ N, we have ζE,q (−k, x) = Ek,q (x).

(6)

We now consider the function Eq (s) as the analytic continuation of the second kind q-Euler numbers. From the above analytic continuation of the second kind q-Euler numbers, we consider En,q 7→ Eq (s),

(7)

ζE,q (−n) = En,q 7→ ζE,q (−s) = Eq (s).

All the second kind q-Euler number En,q agree with Eq (n), the analytic continuation of the second

40

20

0

Eq HsL -20

-40

-60 0

1

2

4

3

5

6

7

s

Figure 1: The curve Eq (s) runs through the points of all En,q

kind q-Euler numbers evaluated at n(see Figure 1). Consider En,q = Eq (n) for n ≥ 0

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′ (s), the derivative of In Figure 1, we choose q = 1/3. In fact, we can express Eq′ (s) in terms of ζE,q ζE,q (s). Consider Eq (s) = ζE,q (−s), ′ Eq′ (s) = −ζE,q (−s)

(9)

′ Eq′ (2n + 1) = −ζE,q (−2n − 1) for, n ∈ N0 .

From the relation (9), we can define the other analytic continued half of the second kind q-Euler numbers Eq (s) = ζE,q (−s), Eq (−s) = ζE,q (s) (10) ⇒ Eq (−n) = ζE,q (n), n ∈ N. By (10), we have lim Eq (−n) = ζE,q (n) = 2.

n→∞

The curve Eq (s) runs through the points E−n,q = Eq (−n) and grows ∼ 2 asymptotically as −n → ∞(see Figure 2). 2

1.9

Eq HsL 1.8

1.7

1.6 -12

-10

-8

-6

-4

-2

0

s

Figure 2: The curve Eq (s) runs through the points E−n,q for q =

1 3

3. Dynamics of the zeros of analytic continued polynomials Our main purpose in this section is to investigate the new concept of dynamics of the zeros of analytic continued polynomials. Let Γ(s) be the gamma function. The analytic continuation can be then obtained as n 7→ s ∈ R, x 7→ w ∈ C, Ek,q 7→ Eq (k + s − [s]) = ζE,q (−(k + (s − [s]))), ( ) n Γ(1 + s) 7→ k Γ(1 + k + (s − [s]))Γ(1 + [s] − k) ⇒ En,q (w) 7→ Eq (s, w) =

[s] ∑ k=−1

Γ(1 + s)Eq (k + s − [s])w[s]−k Γ(1 + k + (s − [s]))Γ(1 + [s] − k)

(11)

∑ Γ(1 + s)Eq ((k − 1) + s − [s])w[s]+1−k , Γ(k + (s − [s]))Γ(2 + [s] − k)

[s]+1

=

k=0

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where [s] gives the integer part of s, and so s − [s] gives the fractional part. By (11), we obtain analytic continuation of the second kind q-Euler polynomials for q = 1/3. Consider E0,q (w) ≈ 1.5, Eq (1, w) ≈ 0.75 + 1.5w, Eq (2, w) ≈ −0.75 + 1.5w + 1.5w2 , Eq (2.2, w) ≈ −1.14137 + 1.13863w + 1.84171w2 + 0.15595w3 ,

(12)

Eq (2.4, w) ≈ −1.54674 + 0.60395w + 2.13491w2 + 0.38568w3 , Eq (2.6, w) ≈ −1.94844 − 0.12719w + 2.33741w2 + 0.69096w3 , Eq (2.8, w) ≈ −2.32024 − 1.07449w + 2.39690w2 + 1.06697w3 , Eq (3, w) ≈ −2.625 − 2.25w + 2.25w2 + 1.5w3 .

By using (12), we plot the deformation of the curve Eq (2, w) into the curve of Eq (3, w) via the real analytic continuation Eq (s, w), 2 ≤ s ≤ 3, w ∈ R(see Figure 3). In [6], we observe that Eq (n, w), w ∈ 2

Eq H2,wL

1

0

Eq Hs,wL -1

-2

Eq H3,wL

-3 -0.2

0

0.2

0.4

0.6

0.8

1

w

Figure 3: The curve of Eq (s, w), 2 ≤ s ≤ 3, −0.3 ≤ w ≤ 1 C, has Im(w) = 0 reflection symmetry analytic complex functions(see Figure 4). The zeros of Eq (n, w) will also inherit these symmetries. If Eq (n, w0 ) = 0, then Eq (n, w0∗ ) = 0, where ∗ denotes complex conjugation. For n ∈ N0 , it is easy to deduce that the second kind q-Euler polynomials En,q (x) satisfy ∞ ∑ n=0

En,q−1 (−x)

∞ ∑ 2e−t 2qet xt tn (−t)n = −1 −2t e(−x)(−t) = 2t e =q En,q (x) . n! q e +1 e +1 n! n=0

tn in the above equation, we have the following theorem. n! Theorem 6 (Theorem of complement). For any positive integer n, we have

By using comparing coefficients

En,q (x) = (−1)n q −1 En,q−1 (−x).

(13)

The question is as follows: what happens with the reflexive symmetry (13), when one considers the second kind q-Euler polynomials? Prove that Eq (n, w), w ∈ C, has not Re(w) = 0 reflection

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symmetry analytic complex functions(see Figure 4). Next, we investigate the beautiful zeros of the Eq (s, w) by using a computer. We plot the zeros of Eq (s, w) for s = 9, 9.3, 9.7, 10, q = 1/3, and w ∈ C(Figure 4). In Figure 4(top-left), we choose s = 9. In Figure 4(top-right), we choose s = 9.3.

ImHwL

2

2

1

1

ImHwL

0

-1

-1

-2

-2 -10

ImHwL

0

-7.5

-5

-2.5 ReHwL

0

2.5

5

2

2

1

1

ImHwL

0

-1

-2

-2 -7.5

-5

-2.5 ReHwL

0

2.5

5

-7.5

-5

-2.5 ReHwL

0

2.5

5

-10

-7.5

-5

-2.5 ReHwL

0

2.5

5

0

-1

-10

-10

Figure 4: Zeros of Eq (s, w) for s = 9, 9.3, 9.7, 10

In Figure 4(bottom-left), we choose s = 9.7. In Figure 4(bottom-right), we choose s = 10. Stacks of zeros of Eq (s, w) for s = n + 1/3, 1 ≤ n ≤ 50, forming a 3D structure are presented(Figure 5). In Figure 5(top-right), we draw y and z axes but no x axis in three dimensions. In Figure 5(bottom-left), we draw x and y axes but no z axis in three dimensions. In Figure 5(bottom-right), we draw x and z axes but no y axis in three dimensions. However, we observe that Eq (n, w), w ∈ C, has Im(w) = 0 reflection symmetry analytic complex functions(see Figure 4 and Figure 5). Our numerical results for approximate solutions of real zeros of Eq (s, w), q = 1/3, are displayed. We observe a remarkably regular structure of the complex roots of the second kind q-Euler polynomials. We hope to verify a remarkably regular structure of the complex roots of the second kind q-Euler polynomials(Table 1).

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5

ImHwL

2.5

0 -2.5

40

-5

40 s

n

20 20

0 -5

0

5

2.5

0

-2.5

-5

0

ReHwL ImHwL

5

ReHwL -5

HL ReHwL 0

5

-5

0

5

5 40 2.5 s

ImHwLL 0

20 -2.5 -5

0

Figure 5: Stacks of zeros of Eq (s, w) for 1 ≤ n ≤ 50

Table 1. Numbers of real and complex zeros of Eq (s, w) s

real zeros

complex zeros

1.5

2

0

2.5

3

0

3.5

4

0

4.5

3

2

5.5

4

2

6.5

5

2

7.5

6

2

8.5

3

6

9

3

6

9.3

4

6

9.5

4

6

9.8

4

6

10

4

6

Next, we calculated an approximate solution satisfying Eq (s, w), q = 1/3, w ∈ R. The results

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are given in Table 2. Table 2. Approximate solutions of Eq (s, w) = 0, w ∈ R s

w −2.25291,

6

−8.19021,

6.5

7.5

−1.97235,

−2.65744,

7

−9.25827,

−0.499167,

1.90361,

3.03711

0.286584,

2.31062,

3.2536

−1.32105, −0.927418,

8

2.89899

−0.106447,

−1.71446,

−2.51685,

1.50121,

0.679634,

2.83991,

1.07258

8.5

−10.3265,

−0.534533,

1.46541

9

−2.1399,

−0.141641,

1.85831

−35.7141,

9.2

−1.98173,

3.19538

0.0155236,

2.01523

9.5

−11.3949,

−1.74785,

0.251276,

2.2499

9.7

−6.68645,

−1.59132,

0.408446,

2.40587

10

−3.09896,

−1.3558,

0.644202,

2.64146

In Figure 6, we plot the real zeros of the the second kind q-Euler polynomials Eq (s, w) for 1 s = n + , 1 ≤ n ≤ 30, q = 1/3, and w ∈ C (Figure 7). In Figure 6(right), we choose Eq (s, w) for 3 30

30

20

20

n

n

10

10

0 -4

-2

0

-2

2

ReHxL

0 -1

0

1

2

ReHxL

Figure 6: Real zeros of Eq (s, w) 1 s = n + , 1 ≤ n ≤ 30. In Figure 6(left), we choose Eq (n, w) for 1 ≤ n ≤ 30. 3 The second kind q-Euler polynomials En,q (w) is a polynomials of degree n. Thus, En,q (w) has n zeros and En+1,q (w) has n + 1 zeros. When discrete n is analytic continued to continuous parameter s, it naturally leads to the question: How does Eq (s, w), the analytic continuation of En,q (w), pick up an additional zero as s increases continuously by one? This introduces the exciting concept of the dynamics of the zeros of analytic continued polynomials-the idea of looking at how the zeros

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move about in the w complex plane as we vary the parameter s. To have a physical picture of the motion of the zeros in the complex w plane, imagine that each time, as s increases gradually and continuously by one, an additional real zero flies in from positive infinity along the real positive axis, gradually slowing down as if ” it is flying through a viscous medium ”. More studies and results in this subject we may see references [5], [6], [7], [10]. Acknowledgement: This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MEST) (No. 2017R1A2B4006092). REFERENCES 1. R. P. Agarwal, J. Y. Kang, C. S. Ryoo, Some properties of (p, q)-tangent polynomials, J. Computational Analysis and Applications, 24 (2018), 1439-1454. 2. I. N. Cangul, H. Ozden, and Y. Simsek, A new approach to q-Genocchi numbers and their interpolation functions, Nonlinear Analysis, 71 (2009), 793-799. 3. M. S. Kim, S. Hu, On p-adic Hurwitz-type Euler Zeta functions, J. Number Theory (132) (2012), 2977-3015. 4. T. Kim, C.S. Ryoo, L.C. Jang, S.H. Rim, Exploring the q-Riemann Zeta function and qBernoulli polynomials, Discrete Dynamics in Nature and Society, 2005, 171-181. 5. C. S. Ryoo, A numerical computation of the roots of q-Euler polynomials, J. Comput. Anal. Appl., 12 (2010), 148-156. 6. C. S. Ryoo, A numerical computation of the structure of the roots of the second kind q-Euler polynomials, J. Comput. Anal. Appl., 14 (2012), 321-327. 7. C. S. Ryoo, Analytic Continuation of Euler Polynomials and the Euler Zeta Function Discrete Dynamics in Nature and Society, Volume 2014 (2014), Article ID 568129, 6 pages. 8. C. S. Ryoo, On the (p, q)-analogue of Euler zeta function, J. Appl. Math. & Informatics, 35 (2017), 303-311. 9. C. S. Ryoo, On degenerate q-tangent polynomials of higher order, J. Appl. Math. & Informatics 35 (2017), 113-120. 10. C. S. Ryoo, R .P. Agarwal, Some identities involving q-poly-tangent numbers and polynomials and distribution of their zeros, Advances in Difference Equations 2017:213 (2017), 1-14. 11. Y. Simesk, V. Kurt and D. Kim, New approach to the complete sum of products of the twisted (h, q)-Bernoulli numbers and polynomials, J. Nonlinear Math. Phys., 14 (2007), 44-56.

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Remarks on the blow-up for damped Klein-Gordon equations with a gradient nonlinearity ∗ Hongwei Zhang, Jian Dang, Qingying Hu (Department of Mathematics, Henan University of Technology, Zhengzhou 450001, China)

Abstract We consider initial boundary value problem for a class of damped Klein-Gordon type wave equations with a gradient nonlinearity and derive sufficient conditions for finite time blow-up of its solutions. To prove blow-up of the solution, we use eigenfunction method combining with a modification of Glassey’s inequality. This extend the early results. Keywords Klein-Gordon equations; blow-up; initial-boundary value problem; gradient nonlinearity AMS Classification (2010): 35L20,35B44.

1

Introduction

The aim of this paper is to give some sufficient conditions for blow-up of solutions to the following damped Klein-Gordon type wave equations with a gradient nonlinearity utt − ∆u + cut = f (u, ∇u), in Ω × (0, T ),

(1.1)

u(x, t) = 0, x ∈ ∂Ω, t ∈ (0, T ),

(1.2)

u(x, 0) = u0 (x),

ut (x, 0) = u1 (x), x ∈ Ω;

(1.3)

where Ω is a bounded domain in Rn with sufficiently smooth boundary ∂Ω, f (u, ∇u) = a|u|p−1 u+ b|∇u|q ,p, q > 1, a, b ∈ R, ab < 0, and c > 0. Nonlinear wave equations of the form (1.1) arise in differential geometry, controllability theory of partial differential equations, and in various areas of physics(see [1] and its references). The derivative Klein-Gordon type wave problem (1.1)-(1.3) can be viewed as a simplification of the Boussinesq equation [2, 3, 4, 1] with higher order spatial derivative terms appearing neither in the linear part nor in the nonlinearity. It belongs to the family of nonlinear wave equations of the form utt +Au = ρ(u)∇u+g(u). This family of wave equations have as an important subclass the Yang-Mills-type equations with ρ(u) = u and g(u) = u3 . Yang-Mills-type wave equations have the same scaling as the cubic nonlinear wave equation, but are more difficult technically because of the derivative term u∇u. Other important examples of the type equations include the Maxwell-Klein-Gordon and Yang-Mills-Higgs equations in the Lorenz gauge at least, as well as the simplified model equations of these (see [1]). If b = 0 and a 6= 0, then equation (1.1) is the standard Klein-Gordon wave problem. The standard Klein-Gordon wave problem in the critical exponent has been studied by many authors. In this case, the blowup behavior of solutions is by ∗

Corresponding author:Zhang H.W., Email: [email protected]

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now fairly well understood, and various sufficient conditions for blowup have been provided and qualitative properties have also been investigated (see for example [5, 6, 7, 8, 9, 10, 11, 12, 13, 14], to cite just a few). However, very little is known in literature concerning the asymptotic dynamics exhibited by the derivative Klein-Gordon type wave equations of the form (1.1) and in space dimensions greater than one or two (see [1]). Recently, D’Abbicco [15] proved global existence of small data solutions to the following Cauchy problem for the doubly dissipative wave equation with power nonlinearity |∇u|p : utt − ∆u + ut − ∆ut = |∇u|p , 1 , in any space dimension n ≥ 1, and he also derive optimal energy estimates for p > 1 + n+1 1 1 and L − L estimates for the solution to the semilinear problems. Willie[1] studied a nonlinear wave problem of the form

utt − ∆u + dut = −ρ|∇u|2 + γ|u|p−1 u, ρ ≥ 0, γ > 0, its linear problem well-posedness, behaviour of the spectrum of the wave differential operator in varied damping and diffusion constants, as well as the asymptotic dynamics defined by the derivative Klein-Gordon type wave problem. We mention also some related mathematical work involving the derivative nonlinearity term in the literature. Ebihara [16, 17, 18] established global existence of classical solutions and asymptotic behavior of solutions of the following nonlinear wave equation utt − ∆u = f (u, ut , ∇u),

(1.4)

where f (u, ut , ∇u) = −up − |∇u|2r − uqt (or f (u, ut , ∇u) = −up |∇u|q urt , here p, q, r > 0). When f (u, ut , ∇u) = −a(x)β(ut , ∇u) in (1.4), where β(λ1 , λ2 , ..., λn )λ1 ≥ 0, Slemrod [19], Vancostenoble [20] and Haraux [21] proved the weak asymptotic stabilization of solutions. Quite recently, Nakao [22, 23, 24, 25, 26] considered the nonlinear wave equations of the form utt − ∆u + ρ(x, ut ) = f (u, ut , ∇u),

(1.5)

and he proved the global existence and decay of solutions. On the other hand, relatively little is known on the blowup for nonlinearities with a dependence on spatial derivatives of u. As far as we know, the previous studies of blow-up of solutions of (1.1) were performed in [27, 28, 29, 30]. In [27], Sideris gives blow-up of small data solutions in finite time for the Cauchy problem in three dimensions when the nonlinear gradient term a|u|p−1 u + b|∇u|q in (1.1) is replaced with term f (u, ut , ∇u) = a2 |∇u|2 + b2 |∆u|2 . To our knowledge, this is the first blow-up result for nonlinear wave equation when the nonlinear perturbation term depends on the derivatives of u. Then the result was extended by Schaeffer [28] and Rammaha [29, 30]. However, very little is known in the literature concerning the blow-up of solutions for initial boundary problem of equation (1.1) and such a method in [27, 28, 29, 30] cannot applied this case. Levine [7] has pointed that the eigenfunction method can easily be modified to include nonlinear terms of the form f (u, ∇u) provided that for all s ∈ R1 , p ∈ Rn ,f (s, p) ≥ G(s), where G(s) is a convex function and the function G(s) satisfy the 2 832

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following conditions:(1)G(s) − (λ + 1)s is nonnegative and nondecreasing on (s0 , ∞) for some Rs Rs −1 s0 > 0; (2) 0 G(ρ)dρ − 21 s2 is nondecreasing on (s0 , ∞);(3)[ 0 G(ρ)dρ − 21 s2 ] 2 is integrable at +∞ for s. However, when f (u, ∇u) = a|u|p−1 u + b|∇u|q , a, b ∈ R, we can’t find any function G(s) such that f (s, p) ≥ G(s). Motivated by the eigenfunction method in [5, 7], the main purpose of this paper is to give sufficient conditions for finite time blow-up of solutions for the initial boundary value problem of equation (1.1) under certain conditions. We will generalize Glassey’s inequality (Lemma 1.1 in [5],and see also [7]), and get sufficient conditions for blow-up of solutions to problem (1.1)-(1.3) for various a, b ∈ R and ab < 0 by eigenfunction method. In this sense, we extend the result [5, 7]. This method applies also to the case of the equation (1.1) with Neumann boundary condition and it remains valid for more general equation utt − ∆u + cut = |u|p−1 u + f (u, |∇u|),

(1.6)

where f is locally Lipschitz continuous and satisfies certain growth condition (see remark 2.4).

2

Main results

Throughout this paper we assume all function spaces are considered over real field and their notations and definitions are same as those [31]. By the usual Galerkin method and similar to the proof in [16], we can obtain regular solution in the local sense. Now we extend Lemma 1.1 in [5](see also [7]) to the following lemma, which play an essential role in this paper. Lemma 1 Let φ(t) ∈ C 2 satisfy φtt + k1 φt ≥ h(φ),

t≥0

(2.1)

with φ(0) = α > 0, φt (0) = β > 0, where k1 > 0. Suppose that h(s) ≥ 0 for all s ≥ α. If Rs R +∞ 1 δ0 = k1 α [β 2 + 2 α h(ρ)dρ]− 2 ds < 1, then φt (t) > 0 where φt (t) exists and lim φ(t) = +∞ t→T −

T∗

− k11 ln(1

where T ≤ = − δ0 ). Proof Because φ(0) = α > 0 and φt (0) = β > 0 then there exist an interval [0, T0 ) such that φt (t) > 0 and φ(t) > α for t ∈ [0, T0 ). If it is false, let t1 = inf{t : φ(t) = α}, t2 = inf{t : φt (t) = 0}. If t2 < t1 , taking into account the condition (2.1) and the fact that h(s) ≥ 0 for all s ≥ α, we have d k1 t (e φt ) = ek1 t (φtt + k1 φt ) ≥ ek1 t h(φ) > 0. dt Thus φt (t2 ) > e−k1 t2 φt (0) > 0, which contradicts φt (t2 ) = 0, and so we have t2 ≥ t1 . FurtherRt more, we have φt (t) > 0 for t ∈ [0, t1 ). In this case, we get that φ(t1 ) = φ(0) + 0 1 φt (s)ds > φ(0) = α > 0, this is a contradiction of the fact φ(t1 ) = α. Thus, there exist an interval [0, T0 ) such that φt (t) > 0 and φ(t) > α for t ∈ [0, T0 ). A multiplication of (2.1) by 2e2k1 t φt (t) gives 2e2k1 t φt φtt + 2k1 e2k1 t (φt )2 ≥ 2e2k1 t h(φ)φt , 3 833

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that is, d 2k1 t (φt )2 ] dt [e

d ≥ 2e2k1 t h(φ)φt ≥ 2h(φ)φt = 2 dt

Rφ α

h(s)ds.

(2.2)

Integrating (2.2) from 0 to t yields e2k1 t (φt )2 − (φt (0))2 ≥ 2

Rφ α

h(s)ds,

since φt > 0, hence −k1 t

φt ≥ e

Z

2

φ

(β + 2

1

h(s)ds)− 2 .

(2.3)

α

We may separate variables and integrate over (0, t) to obtain Z +∞ Z y 1 (β 2 + 2 h(s)ds)− 2 dy = δ0 . 1 − e−k1 t ≤ k1 α

α

Therefore we get the result. We consider the following spectral problem ∆w + λw = 0 in Ω,

(2.4)

w = 0, on ∂Ω.

(2.5)

It is well known that problem (2.4)-(2.5) has the smallest eigenvalue λ1 > 0 and the correspondR ing normalized eigenfunction w1 > 0 in Ω, Ω w1 (x)dx = 1. Then we denote q Z q−1 |∇w1 | q−1 q . k0 = ( dx) 1/(q−1) Ω w1

(2.6)

Theorem 2 Suppose q > 1, a = 0 and b > 0. Let u(x, t) be a regular solution of problem (1.1)-(1.3). Suppose that the following conditions are satisfied: Z Z u0 (x)w1 (x)dx = α, u1 (x)ψ1 (x)dx = β, Ω

Ω

q/(q−1) k0 λ1

> 0, β > 0, and that ( λk01 )q sq − λ1 s is a nongeative, nondecreasing function R +∞ Rs 1 for s ≥ α. If δ1 = c α [β 2 + 2 α [( λk01 )q ρq − λ1 ρ]dρ]− 2 ds < 1, then the solution of problem (1.1)-(1.3) blows up in a finite time. Proof Let Z u(x, t)w1 (x)dx. U (t) = where α >

Ω

Then U (0) = α > 0, Ut (0) = β > 0 and as it follows from (1.1)-(1.3), U (t) satisfies Z Utt + cUt + λ1 U = |∇u|q w1 dx.

(2.7)

Ω

By (2.4) and Holder inequality, we get R R λ1 U ≤ | Ω u(x, t)λ1 w1 (x)dx| = | Ω u(x, t)∆w1 (x)dx| R R R 1/q 1| = | Ω ∇u∇w1 dx| ≤ Ω |∇u||∇w1 |dx = Ω (|∇u|w1 ) |∇w 1/q dx w1

R ≤( Ω

|∇w1

q | q−1

1/(q−1)

w1

dx)

q−1 q

1 q

R 1 ( Ω |∇u|q w1 dx) = k0 ( Ω |∇u|q w1 dx) q , R

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that is to say Z

|∇u|q w1 dx ≥ (

Ω

λ1 q q ) U . k0

(2.8)

Therefore, from (2.7) and inequality(2.2), we obtain the ordinary differential inequality Utt + cUt ≥ (

λ1 q q ) U − λ1 U, k0

(2.9)

with U (0) = α > 0, Ut (0) = β > 0. Denote h(s) = ( λk01 )q sq − λ1 s, since h(s) > 0 for s ≥ α, it follows from Lemma 6 that lim U (t) = ∞, for some T0 ≤ T ∗ = − 1c ln(1 − δ1 ). Furthermore, t→T0− R since U (t) > 0, we have U (t) = |U (t)| ≤ supΩ |u(x, t)| Ω w1 dx ≤ supΩ |u(x, t)|, and we get lim ||u||pp = ∞, ∀1 ≤ p ≤ ∞, for some T0 ≤ T ∗ = − 1c ln(1 − δ1 ), which proves the theorem. t→T0−

Theorem 3 Suppose q ≥ 2, 0 < p < 2, a < 0 and b > 0. Let u(x, t) be a regular solution of problem (1.1)-(1.3). Suppose that the following conditions are satisfied: Z Z u0 (x)w1 (x)dx = α0 , u1 (x)ψ1 (x)dx = β0 , Ω

Ω

where β0 > 0 and α0 is the positive root of the equation b( λk01 )q sq − |a|sp − λ1 s = 0. If δ2 = Rs R +∞ 1 c α [β 2 + 2 α [( λk01 )q ρq − |a|ρp − λ1 ρ]dρ]− 2 ds < 1, then the solution of problem (1.1)-(1.3) blows up in a finite time. Proof Let Z U (t) = u(x, t)w1 (x)dx. Ω

Then U (0) = α0 > 0, Ut (0) = β0 > 0 and as it follows from (1.1)-(1.3), U (t) satisfies Z Z p Utt + cUt + λ1 U = a |u| w1 dx + b |∇u|q w1 dx. Ω

Then (2.8) and the inequality

p Ω |u| w1 dx

R

Utt + cUt ≥ b(

(2.10)

Ω

≥ U p yield the ordinary differential inequality

λ1 q q ) U − |a|U p − λ1 U = h2 (U ), k0

(2.11)

with U (0) = α0 > 0, Ut (0) = β0 > 0. Since h2 (s) > 0 for s ≥ α0 , then the rest of the proof is similar to the proof of Theorem 2 and the proof is complete. Theorem 4 Suppose p ≥ 2, 0 < q < 2, b < 0 and a > 0. Let u(x, t) be a regular solution of problem (1.1)-(1.3). Suppose that the following conditions are satisfied: Z Z u0 (x)w1 (x)dx = α1 , u1 (x)ψ1 (x)dx = β1 , Ω

Ω

where β1 > 0 and α1 is the positive root of the equation asp − |b|( λk01 )q sq − λ1 s = 0. If δ3 = R +∞ Rs 1 c α [β 2 + 2 α [aρp − |b|( λk01 )q ρq − λ1 ρ]dρ]− 2 ds < 1, then the solution of problem (1.1)-(1.3) blows up in a finite time. 5 835

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Proof Similar to the proof Theorem 3, U (t) satisfies Z Z p Utt + cUt + λ1 U = a |u| w1 dx + b |∇u|q w1 dx, Ω

(2.12)

Ω

with U (0) = α1 > 0, Ut (0) = β1 > 0, and then we have Utt + cUt ≥ aU p − |b|(

λ1 q q ) U − λ1 U = h3 (U ), k0

(2.13)

with U (0) = α1 > 0, Ut (0) = β1 > 0. Since h3 (s) > 0 for s ≥ α0 , then the rest of the proof is similar to the proof of Theorem 3 and the proof is complete. Remark 1 By Theorem 2-Theorem 4, we can also prove that the blowup result holds under the similar initial conditions for the case a > 0, p > 2, p > q or b > 0, q > 2, q > p. ∂u Remark 2 The same results hold if the boundary condition is of the form a ∂n + bu = 0. Remark 3 The results remain true when 4u is replaced by p-Laplace operator div(|∇u|p ∇u). Remark 4 The method remains valid for more general equation (1.6), where f is locally Lipschitz continuous and satisfies the growth condition f (u, |∇u|) ≤ C(1 + |u|k + |∇u|q ). ACKNOWLEDGEMENTS This work is supported by the National Natural Science Foundation of China (No.11601122).

References [1] R. Willie. Spectral analysis, an integral mild solution formula and asymptotic dynamics of the derivative Klein-Gordon type wave equation. Journal of Abstract Differential Equations and Applications, 2011, 2(1):54-83. [2] D. Henry. Geometric theory of semilinear parabolic problems. Lecture notes in mathematics 840. Springer Verlag. New York 1981. [3] G. Sell, and Y. You. Dynamics of evolutionary equations. Applied Mathematical Sciences 143. Springer-Verlag 2002. [4] Y. You. Global dynamics of 2D Boussinesq equations. Nonlinear Analysis. T.M.A., 1997, 30: 46434654. [5] R. T. Glassey. Blow-up theorems for nonlinear wave equations. Mathematische Zeitschrift, 1973, 132(3): 183-203. [6] H.A.Levine. Instability and nonexistence of global solutions to nonlinear wave equations of the form P utt = Au + F (u). Transactions of the American Mathematical Society, 1974, 192: 1-21. [7] H.A.Levine. Nonexistence of global solutions to some properly and improperly posed problem of mathematical physics: The method of unbounded fourier coefficients. Mathematische Annalen, 1975, 214(3): 205-220. [8] L.E.Payne, D.Sattinger. Saddle points and instability of nonlinear hyperbolic equations. Israel Journal of Mathematics, 1975, 22(3-4): 273-303. [9] V.K.Kalantarov, O.A.Ladyzhenskaya. The occurrence of collapse for quasilinear equations of parabolic and hyperbolic types. Journal of Soviet Mathematics, 1978, 10(1): 77-102. [10] V. Georgiev, G.Todorova. Existence of a solution of the wave equation with nonlinear damping and source terms. Journal of Differential Equations, 1994, 109: 295-308. [11] S. A. Messaoudi. Blow-up in a nonlinear damped wave equation. Mathematische Nachrichten, 2001, 231(1): 105-111.

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[12] V.Barbu, I.Lasiecka, M.A.Rammaha. On nonlinear wave equations with degenerate damping and source terms. Transactions of the American Mathematical Society, 2005, 357(7): 2571-2611. [13] E.Vitillaro. Global nonexistence theorems for a class of evolution equationwith dissipation. Archive for Rational Mechanics and Analysis, 1999, 149(2): 155-182. [14] M.O.Korpusov, A.A.Panin. Blow-up of solutions of an abstract Cauchy problem for a formally hyperbolic equation with double non-linearity. Izvestiya: Mathematics, 2014, 78(5): 937-985. [15] Marcello D’Abbicco. L1 −L1 estimates for a doubly dissipative semilinear wave equation. Nonlinear Differential Equations and Applications, 2017, 24(1):article5,1-23. [16] Y. Ebihara. On the local classical solution of the mixed problem for nonlinear wave equation. Mathematical reports of College of General Education, Kyushu University, 1974, 9(2): 43-65. [17] Y. Ebihara. On the equations with variable coefficients utt − uxx = F (x, t, u, ux , ut ). Funkcialaj Ekvacioj, 1977, 20(1): 77-95. [18] Y. Ebihara. Nonlinear wave equations with variable coefficients. Funkcialaj Ekvacioj, 1979, 22(2): 143-159. [19] M. Slemrod. Weak asymptotic decay via a ”relaxed invariance principle” for a wave equation with nonlinear, nonmonotone damping. Proceedings of the Royal Society of Edinburgh Sect. A, 1989, 113(1): 87-97. [20] J. Vancostenoble. Weak asymptotic decay for a wave equation with gradient dependent damping. Asymptotic Analysis,2001, 26: 1-20. [21] A. Haraux. Remarks on weak stabilization of semilinear wave equations. ESAIM: Control, Optimisation and Calculus of Variations, 2001,6: 553-560. [22] M. Nakao. Existence of global decaying solution to the Cauchy problem for a nonlinear dissipative wave equations of Klein-Gordon type with a derivative nonlinearity. Funkcialaj Ekvacioj, 2012, 55: 457-477. [23] M. Nakao. Global existence and decay for nonlinear dissipative wave equations with a derivative nonlinearity. Nonlinear Analysis, 2012, 75: 2236-2248. [24] M.Nakao. Existence of global decaying solutions to the exterior problem for the Klein-Gordon equation with a nonlinear localized dissipation and a derivative nonlinearity. Journal of Differential Equations, 2013, 255: 3940-3970. [25] M. Nakao. Global solutions to the initial-boundary value problem for the quasilinear viscoelastic equation with a derivative nonlinearity. Opuscula Mathatics, 2014, 34(3): 569-590. [26] M. Nakao. Existence of global decaying solutions to the initial boundary value problem for the quasilinear wave equation of p-Laplacian type with Kelvin-Voigt dissipation and a derivative nonlinearity. Kyushu Journal of Mathematics, 2016,70: 63-82. [27] T. Sideris. Global behavior of solutions to nonloinear wave equations in three dimensions. Communications in Partial Differential Equations, 1983, 8(12):1291-1323. [28] J. Schaeffer. Finite-time blow-up for utt −∆u = H(ur , ut ) in two space dimensions. Communications in Partial Differential Equations, 1986, 11(5):513-543. [29] M.A. Rammaha. Finite-time blow-up for nonloinear wave equations in high dimensions. Communications in Partial Differential Equations, 1987, 12(6):677-700. [30] M.A. Rammaha. On the blowing up solutions to nonlinear wave equations in two space dimensions. Journal fur Die Reine Und Angewandte Mathematik, 1988, 391: 55-64. [31] J.L.Lions. Quelques methodes de resolution des problemes aur limites non lineaires. Dunod Gauthier-villars, Paris,1969.

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The γ-fuzzy topological semigroups and γ-fuzzy topological ideals Cheng-Fu Yang (School of Mathematics and Statistics of Hexi University, Zhangye Gansu,734000, P. R. China) Abstract: Based on the concepts of semigroup and Chang's fuzzy topological space, this paper gives the defines of the γ-fuzzy topological semigroups, γ-fuzzy topological left ideals (γ-fuzzy topological right ideals, γ-fuzzy topological intrinsic ideals and γ-fuzzy topological double ideals) and discusses the fuzzy continuous homomorphic image and the fuzzy continuous homomorphic inverse image of them. Keywords: Fuzzy topological space; γ-fuzzy topological semigroup; γ-fuzzy topological ideal; F-continuous; homomorphic image and homomorphic inverse image

1.Introduction Since Zadeh [15] introduced fuzzy sets and fuzzy set operations in 1965. The concept of fuzzy sets has been widely used in various fields. For example, in 1968, Chang [2] applied the fuzzy set to topological space to give fuzzy topological space. After that, Pu and Liu [9,10] introduced neighborhood structure of a fuzzy point, moore-smith convergence and product and quotient spaces in fuzzy topological space. Afterwards Rosenfeld [12] formulated the elements of the theory of fuzzy groups and Foster [4] introduced the fuzzy topological groups. In 2011, Tanay et al. [13] gave the notion of fuzzy soft topological spaces and studied neighborhood and interior of a fuzzy soft set and then used these to characterize fuzzy soft open sets. Then Nazmul and Samanta [8] introduced the fuzzy soft topological groups. Subsequently, Coker [3] used the notion of intuitionistic fuzzy sets gave by Atanassov in [1] to introduce the notion of intuitionistic fuzzy topological spaces and obtained several preservation properties and some characterizations concerning fuzzy compactness and fuzzy connectedness. After that, Kul [6] introduced the intuitionistic fuzzy topological groups. Recently, Rajesh gave the notion of γ-fuzzy topological group in [11] and discussed the connection between fuzzy topological group and γ-fuzzy topological group. Based on this idea, in this paper, we give the concepts of the γ-fuzzy topological semigroups, γ-fuzzy topological left ideals (right ideals, intrinsic ideals and double ideals) and then discuss the homomorphic image and inverse image of them.

2.Preliminary Definition 2.1.[15] A fuzzy set A in X is a set of ordered pairs: A  {( x, A ( x)) : x  X }

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Where A ( x) : X  I  [0,1] is a mapping and A ( x) states the grade of belongness of x in A . The family of all fuzzy sets in X is denoted by IX . Particularly, the fuzzy set  , if x  y , y  X x ( y )   0, otherwise is called a fuzzy point in X, denoted by x . Deflnition 2.2.[15] Let A , B be two fuzzy sets of

IX

1) A is contained in B if and only if A ( x)  B ( x) for every x  X . 2)The union of A and B is a fuzzy set C , denoted by A  B  C , whose membership function C ( x)  A ( x)  B ( x) for every x  X . 3)The intersection of A and B is a fuzzy set C , denoted by A  B  C , whose membership function C ( x)  A ( x)  B ( x) for every x  X . 4)The complement of A is a fuzzy set, denoted by A c , whose membership function A c ( x)  1  A ( x) for every x  X . Definition 2.3.[2] Let X, Y be two nonempty sets, f a function from X to Y and B a fuzzy set in Y with membership function B (y). Then the inverse of B , written as f -1  ( B ), is a fuzzy set in X whose membership function is defined by f -1( B )(x) = B (f (x)) for all x in X. Conversely, let A be a fuzzy set in X with membership function A (x). The image of A , written as f ( A ), is a fuzzy set in Y whose membership function is given by 1   x f 1 ( y ) A( x), if f ( y )    for y  Y f ( A)( y )   otherwise, 0,

where

f 1 ( y )   x | f ( x)  y .

Proposition 2.4.[2] Let f be a function from X to Y. Then: (1) f 1[ B c ]  [ f 1 B ]c for any fuzzy set B in Y. (2) f [ Ac ]  [ f ( A)]c for any fuzzy set A in X. (3) B1  B 2  f 1[ B1 ]  f 1[ B 2 ] , where B1 , B 2 are fuzzy sets in Y. (4) A1  A 2  f [ A1 ]  f [ A 2 ] , where A1 , A 2 are fuzzy sets in X.

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(5) B  f [ f 1[ B ]] for any fuzzy set B in Y. (6) A  f [ f 1[ A ]] for any fuzzy set A in X. Proposition 2.5. Let f be a function from X to Y. Then: (1) f ( A  B )  f ( A )  f ( B ) and f ( A  B )  f ( A )  f ( B ) for any A , B  I X . (2)

f 1 ( A  B )  f 1 ( A )  f 1 ( B )

and

f 1 ( A  B )  f 1 ( A )  f 1 ( B )

for any

A , B  I X . Proof. This proposition can be directly verified by the Definition 2.3. Next example shows that f ( A  B )  f ( A )  f ( B ) for any A , B  I X has not hold. Example.

Let

X

=

a, b, c, d 

,

Y =

 x, y

,

0.3 0.2 0.6 0.1 , A     a b c d

0.5 0.8 0.1 0.3 . Define f : X  Y as f (a) = f (b) = f (c) = x, f (d) = y. Then B     a b c d

A  B 

0.3 0.2 0.1 0.1 ,    a b c d

0.3 0.1 f ( A  B )   ， x y

f ( A ) 

0.6 0.1  ， x y

0.8 0.3 0.6 0.1 f ( B )   ，f ( A )  f ( B )   . Thus f ( A  B )  f ( A )  f ( B ) for any x y x y A , B  I X has not hold.

3.Fuzzy topological space Definition 3.1.[2] A fuzzy topology is a family  of fuzzy sets in X which satisfies the following conditions:    ; (1) 0,1 (2)If A , B  ,then A  B  ; (3) If Ai  , i   ,then  i Ai  ;

 is called a fuzzy topology for X, and the pair ( X , ) is called a fuzzy topological space, or fts for short. Every member of  is called a  -open fuzzy set. A fuzzy set is  -closed if and only if its complement is  -open. In the sequel, when no confusion is likely to arise, we shall call a  -open (  -closed) fuzzy set simply an open (closed) set. Proposition 3.2. Let X be a nonempty set. If  and J are two fuzzy topologicals for X, then   J is a fuzzy topology for X, where   J   A  B | A  , B  J  .

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Proof. Straightforward. Proposition 3.3. Let X, Y be two nonempty sets, f be an one-to-one mapping from X to Y. If  is a fuzzy topological for X, then f ( ) is a fuzzy topology for Y, where

f ( )   f ( A) | A   .   0 , f (1)   1 ; Proof. (1) Obviously, f (0) (2) If A , B  f ( ) , then there exist A1 , B1  such that A  f ( A1 ) and B  f ( B1 ) respectively. y  Y , ( A  B )( y )  ( f ( A1 )  f ( B1 ))( y )  f ( A1 )( y )  f ( B1 )( y )  ( x f 1 ( y ) A1 ( x))  ( x f 1 ( y ) B1 ( x))

=  x f 1 ( y ) ( A1 ( x)  B1 ( x))   x f 1 ( y ) ( A1  B1 )( x)  f ( A1  B1 )( y ). This means A  B  f ( A1  B1 ) . Since A1  B1  , thus f ( A1  B1 )  f ( ) (3) If Ai  f ( ), i   , then for any i   there exists a A1 

such that

Ai  f ( Ai). And then ( i Ai )( y )   i ( Ai ( y ))   i ( f ( Ai)( y ))   i ( x f 1 ( y ) A1( x))

=  x f 1 ( y ) ( i A1( x))   x f 1 ( y ) (( i A1)( x))  f ( i A )( y ). This means  i Ai  f ( i A ) . Since  i Ai  , thus f ( i A )  f ( ) . This completes the proof. Proposition 3.4. Let X, Y be two nonempty sets and f a mapping from X to Y. If  is a fuzzy topological for Y, then



f 1 ( ) is a fuzzy topology for X, where



f 1 ( )  f 1 ( A ) | A  .

  0 , f 1 (1)   1 ; Proof. (1) Obviously, f 1 (0) (2)If A , B  f 1 ( ) ,then there exist

A1 , B1  , such that

A  f 1 ( A1 )

and

B  f 1 ( B1 ) respectively. x  X , ( A  B )( x)  ( f 1 ( A1 )  f 1 ( B1 ))( x)  f 1 ( A1 )( x)  f 1 ( B1 )( x)  A1 ( f ( x))  B1 ( f ( x))

= ( A1  B1 )( f ( x))  f 1 ( A1  B1 )( x).

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This

means

A  B  f 1 ( A1  B1 )

.

Since

A1  B1 

,

thus

A  B  f 1 ( A1  B1 )  f 1 ( ) .

(3) If Ai  f 1 ( ), i   , then for any i   there exists a A1  such that Ai  f 1 ( Ai). And then ( i Ai )( x)   i ( Ai ( x))   i ( f 1 ( Ai)( x))   i ( A1( f ( x)))   i ( A1( f ( x)))  ( i A1)( f ( x))  f 1 ( i A1)( x).

This

means

 i Ai  f 1 ( i A1)

.

Since

 i A1 

,

thus

 i Ai  f 1 ( i A1)  f 1 ( ) .

This complete the proof. Definition 3.5.[2] A function f from a fts ( X , ) to a fts ( Y , U ) is F-continuous iff the inverse of each open set in Y is open set in X. Definition 3.6.[9] Let A be a fuzzy set in (X, τ) and the union of all the open sets contained in A is called the interior of A , denoted by A O . Evidently A O is the largest open set contained in A and A OO = A O . Proposition 3.7.[2] Let A be a fuzzy set in a fts (X, τ). Then A is open iff A = A O . Definition 3.8.[9] The intersection of all the closed sets containing A is called the closure of A , denoted by A . Obviously A is the smallest closed set containing A and A = A . By the definitions of the interior and closure, obviously A O ⊂ A ⊂ A . Proposition 3.9.[9] Let A be a fuzzy set in a fts (X, τ). Then A is closed iff A = A . Proposition 3.10. Let A be a fuzzy set in a fts (X, τ). (1) If A ⊂ B , then A O ⊂ B O . (2) If A ⊂ B , then A ⊂ B . Proof. According to the definition can be directly proved. Proposition 3.11.[10] Let f : (X, τ)  (Y, U) be a function, then the ollowing are equivalent: (1) f is F-continuous.

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(2) For every closed set A in Y, f -1( A ) is closed set in X. (3) For any fuzzy set A in X, f( A ) ⊂ f ( A ) . (4) For any fuzzy set B in Y, f 1 ( B )  f 1 ( B ) . Proposition 3.12. Let f : (X, τ)  (Y, U) be a function; then the following are equivalent: (1) f is F-continuous. (2) For any fuzzy set B in Y,

f 1 ( B O )  ( f 1 ( B ))o .

Proof. (1)  (2). For any fuzzy set B in Y, by the definition of the interior and f is F-continuous, this means f 1 ( B O ) is an open set in X. On the other hand, since B O  B , by (3) of Proposition 2.4, f 1 ( B O )  f 1 ( B ) . Considering ( f 1 ( B ))o

is

the union of all the open sets contained in f 1 ( B ) , thus f 1 ( B O )  ( f 1 ( B ))o . (2)  (1). Let B be any open fuzzy set in Y, then B O  B . By condition, f 1 ( B )  f 1 ( B O )  ( f 1 ( B ))o . On the other hand, since f 1 ( B )  ( f 1 ( B ))O , thus f 1 ( B )  ( f 1 ( B ))o . This means f 1 ( B ) is an open set in X, thus f is F-continuous. Proposition 3.13. Let (X, τ) be a fts and f : X  Y be an one-to-one F-continuous mapping, then the following are hold: (1) For any fuzzy set A in X,

f ( A O )  ( f ( A ))o .

(2) For any fuzzy set A in X, f (( A )O )  ( f ( A )) o Proof. According to the previous conclusion, (Y, f(τ)) is a fts.





(1) For any fuzzy set A in X, since A O   B | B  A , B  , by (1) of Proposition 2.5,





f ( A O )   f ( B ) | f ( B )  f ( A ), f ( B )  f ( ) .On the other hand, by the





definition of the interior, ( f ( A ))O   f ( B ) | f ( B )  f ( A ), f ( B )  f ( ) . Thus f ( A O )  ( f ( A ))o .



 =   f ( B ) | f ( B )  f ( A ), f ( B )  f ( ) .Considering f

(2) For any fuzzy set A in X, since A O   B | B  A , B  , by (1) of Proposition 2.5, thus f (( A )O )

is a F-continuous, thus for any

f ( B )  f ( A ) and

843

f ( B )  f ( ) , by (3) of

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Proposition 3.11, f ( B )  f ( A )  f ( A )

holds. By the definition of the interior of

( f ( A )) o and (1) of Proposition 3.10, and then f ( B )  ( f ( B ))O  ( f ( A )) o , thus f (( A )O )  ( f ( A )) o . Definition 3.14.[5, 11] A fuzzy set A in fts (X, τ) is said to be fuzzy γ-open if A ⊂ ( A )o  ( A )O . The complement of a fuzzy γ-open set is called a fuzzy γ-closed set. The family of all fuzzy γ-open sets of X is denoted by γO(X). Proposition 3.15. Let (X, τ) be a fts and f : X  Y be an one-to-one F-continuous mapping. If A is a γ-open set in X, then f( A ) is a γ-open set in Y. Proof. Since A is a γ-open set in X, then A ⊂ ( A )o  ( A )O . And then f ( A ) ⊂ f (( A )o  ( A )O )  f (( A )o )  f (( A )O ) .By Proposition 3.11 and Proposition 3.13, f (( A )O )  f ( A O )  ( f ( A ))O

and f (( A )o )  ( f ( A ))O . This means

f ( A )

⊂

f (( A )o  ( A )O )  f (( A )o )  f (( A )O )  ( f ( A )) o  ( f ( A ))o . By the Definition of the fuzzy γ-open set, f ( A ) is a γ-open set in Y. Proposition 3.16. Let (Y，J) be a fts and f -1 : Y  X be an one-to-one F-continuous. If B is a γ-open set in Y, then f -1( B ) is a γ-open set in X. Proof. Let f -1 as f in proposition 3.15, the proof is similar to proposition 3.15. Definition 3.17. A fuzzy set A in a fts (X, τ) is called a γ-neighborhood of fuzzy point x , if there exists a γ-open set B  such that x  B  A . The family consisting of all γ-neighborhoods of x is called the system of γ-neighborhoods of x . Definition 3.18. A fuzzy point x is said to be quasi-coincident with a fuzzy set A , denoted by x q A , if

λ + A ( x) > 1. A fuzzy set A is said to be a Qγ-neighborhood

x of if there exists a γ-open set B  such that x q B  A . The family consisting of all the Qγ-neighborhoods of x is called the system of Qγ-neighborhoods of x . Proposition 3.19. Let ( X , τ) be a fts and f an one-to-one F-continuous mapping from X to Y . If U is a Qγ-neighborhood of a in X , then f (U ) is a Qγ-neighborhood

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of f (a )  [ f (a)] in fts ( Y , f ( ) ). Proof. In order to avoid confusion, here record y  f (a ) . Without losing generality, let U   O( X ) , Since f (U )( f (a))    f (U )( y )    this

means

[ f (a ) qf (U )] .Considering

 U ( x)    U (a)  1,

x f 1 ( y )

f (U )   O(Y ) ,

thus

f (U )

is

a

Qγ-neighborhood of [ f (a)] . Proposition 3.20. Let (Y, J) be a fts and f -1 an one-to-one F-continuous mapping from Y to X. If V is a Qγ-neighborhood of (f(a))λ in Y, then f -1(V) is a Qγ-neighborhood of aλ in fts (X, f -1(J)). Proof. Without losing generality, let V   O(Y ) , Since f -1(V)(a) + λ = V( f(a)) + λ > 1, this means aλq f -1(V). Considering f -1(V)  γO(X), thus f -1(V) is a Qγ-neighborhood of aλ.

4  -Fuzzy topological semigroup Definition 4.1.[7] Let X be a semigroup and A , B two fuzzy sets in X. A B is defined as a fuzzy set in X, which membership function is as follows:

  ( x)     AB x1 x2  x ( A( x1 )  B ( x2 ))

for x  X .

Proposition 4.2. Let X, Y be two semigroups and f an epimorphism from X to Y. If A ,   )  f ( A ) f ( B ) . B are any two the fuzzy sets in X, then f ( AB Proof. For any y  Y , since   )( y )  f ( AB

 )   ( AB (x)

x f 1 ( y )

=  (



x1 x2  x x1 x2  f 1 ( y )

=



 (  ( A ( x1 )  B ( x2 )))   ( 1 ( A ( x1 )  B ( x2 )))

x f 1 ( y ) x1 x2  x

( A ( x1 )  B ( x2 ))) 

(



f ( x1 ) f ( x2 )  f ( x )  y x1 x2  f 1 ( y )

=  (



y1 y2  y x1 x2  f 1 ( y1 y2 )

=  (( y1 y2  y

x1 x2  x x f





(

f ( x1 x2 )  f ( x )  y x1 x2  f 1 ( y )

( A ( x1 )  B ( x2 )))   (

( y)

( A ( x1 )  B ( x2 )))



y1 y2  y x1 x2  f 1 ( y )

( A ( x1 )  B ( x2 )))

( A ( x1 )  B ( x2 )))   (



( A ( x1 )  B ( x2 )))

( A ( x1 ))  (



B ( x2 )))

y1 y2  y x1 x2  f 1 ( y1 ) f 1 ( y2 )



x1 x2  f 1 ( y1 ) f 1 ( y2 )

=  (( y1 y2  y



x1 f 1 ( y1 )

( A ( x1 ))  (

845

x1 x2  f 1 ( y1 ) f 1 ( y2 )



x2  f 1 ( y2 )

B ( x2 )))

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.5, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

  )( y ) ， =  ( f ( A )( y1 )  f ( B )( y2 ))  f ( AB y1 y2  y

  )  f ( A ) f ( B ) . thus f ( AB Proposition 4.3. Let X, Y be two semigroups and f a monomorphism from X to Y.   )  f 1 (C ) f 1 ( D ) . If C , D are any two the fuzzy sets in Y, then f 1 (CD Proof. For any x  X , since   )( x)  CD   ( f ( x))  f 1 (CD



y1 y2  f ( x )

(C ( y1 )  D ( y2 )) 



f 1 ( y1 ) f 1 ( y2 )  x

(C ( y1 )  D ( y2 ))

=  (C ( f ( x1 ))  D ( f ( x2 )))   ( f 1 (C ( x1 ))  f 1 ( D ( x2 )))  ( f 1 (C ) f 1 ( D ))( x), x1 x2  x

x1 x2  x

  )  f 1 (C ) f 1 ( D ) . thus f 1 (CD Definition 4.4. Let X be a semigroup and ( X , ) a fts. Then ( X , ) is called a

 -fuzzy topological semigroup, or  -ftsg for short, if for all a, b  X and any Qγ-neighborhood W of fuzzy point (ab) there exist Qγ-neighborhoods U of a and V of b such that UV  W . Proposition 4.5. Let X, Y be two semigroups and ( X , ) a  -ftsg. If f is an one-to-one F-continuous homomorphic mapping from X to Y, then ( Y , f ( ) ) is a

 -ftsg. Proof. By Proposition 3.3, ( Y , f ( ) ) is a fts. For any Qγ-neighborhood W of fuzzy point (ab) in Y, according to Proposition 3.20, f 1 (W ) is a Qγ-neighborhood of fuzzy point

f 1 ((ab) ) in X. Since ( X , ) is a  -ftsg, thus there exist

Qγ-neighborhoods U of f 1 (a ) and V of f 1 (b ) such that UV  f 1 (W ) , and then f (UV )  W . By proposition 3.19, f (U ) and f (V ) is the Qγ-neighborhoods of a and b respectively. Again by Proposition 4.2, f (U ) f (V )  f (UV )  W . Thus ( Y , f ( ) ) is a  -ftsg. Proposition 4.6. Let X, Y be two semigroups and (Y, J) a  -ftsg. If f 1 is an

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one-to-one F-continuous homomorphic mapping from X to Y, then ( X , f 1 ( J ) ) is a

 -ftsg. Proof. By Proposition 3.4, f 1 ( J ) is a fts. For any Qγ-neighborhood W of fuzzy point (ab) in X, according to Proposition 3.19, f (W ) is a Qγ-neighborhood W of fuzzy point

f ((ab) )

in Y. Since (Y, J) is a  -ftsg, thus there exist

Qγ-neighborhoods U of f (a ) and V of

f (b ) such that UV  f (W ) , and then

f 1 (UV )  W . By proposition 3.20, f 1 (U ) and f 1 (V ) is the Qγ-neighborhoods of a and b respectively.Again by Proposition 4.3, f 1 (U ) f 1 (V )  f 1 (UV )  W . Thus ( X , f 1 ( J ) ) is a  -ftsg.

5.  -Fuzzy topological ideal Definition 5.1. Let X be a semigroup and ( X , ) a fts. Then ( X , ) is called a -fuzzy topological left ideal (right ideal), if for all a, b  X and any Qγ-neighborhood W of fuzzy point (ab) there exists Qγ-neighborhood U of b ( Qγ-neighborhood V of a ) such that U  W ( V  W ). Definition 5.2. Let X be a semigroup and ( X , ) a fts. Then ( X , ) is called a

 -fuzzy topological intrinsic ideal (double ideal), if for all a, b, c  X and any Qγ-neighborhood W of fuzzy point (abc) there exists Qγ-neighborhood U of b (Qγ-neighborhood U of a and Q -neighborhood V of c ) such that U  W (such that UV  W ). Proposition 5.3. Let X, Y be two semigroups and ( X , ) a -fuzzy topological left ieal (right ieal, intrinsic ideal, double ideal). If

f is an one-to-one F-continuous

homomorphic mapping from X to Y, then ( Y , f ( ) ) is a  -fuzzy topological left ieal (right ieal, intrinsic ideal, double ideal). Proof. Similar to the proof of Proposition 4.5.

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Proposition 5.4. Let X, Y be two semigroups and (Y, J) a  -fuzzy topological left ieal (right ieal, intrinsic ideal, double ideal). If f 1 is an one-to-one F-continuous homomorphic mapping from X to Y, then ( X , f 1 ( J ) ) is a  -fuzzy topological left ieal (right ieal, intrinsic ideal, double ideal). Proof. Similar to the proof of Proposition 4.6. References [1]K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Syst. 20(1) (1986) 87-96. [2]C. L. Chang, Fuzzy Topological Spaces, J. Math. Anal. Appl 24(1968)182-190. [3]D. Coker, An introduction to intuitionistic fuzzy topological space, Fuzzy sets and systems, 1997, 81-89. [4]D. H. Foster, fuzzy topological groups, J. Math. Anal. Appl. 67(1979)549-564. [5]I. M. Hanafy, Fuzzy -open sets and fuzzy -continuity, J. Fuzzy Math., 7(2)(1999)419-430. [6]Kul Hur, Young Bae Jun, Jang Hyun Ryou, Intuitionistic fuzzy topological groups. Honam Math. J. 26(2004)163-192. [7]J. L. Ma, C. H. Yu, Fuzzy topological groups. Fuzzy Sets and Systems 12(1984)289-299. [8]S. Nazmul, S. K. Samanta, Fuzzy Soft Topological Groups, Fuzzy Inf. Eng. 6(2014)71-92. [9]Pu Pao-ming and Liu Ying-ming, Fuzy topology I, J. Math. Anal. Appl. 76(1980)571-599. [10]Pu Pao ming and Liu Ying ming, Fuzy topology II, J. Math. Anal. Appl. 77(1980)20-37. [11]N. Rajesh, A New Type of Fuzzy Topological Groups, The journal of fuzzy mathematics, 21(1)(2013)99-103. [12]A. Rosenfeld, fuzzy groups, J. Math. Anal. Appl. 35(1971)512-517. [13]B. Tanay and M. Burc Kandemir, Topological structure of fuzzy soft sets, Comp. Math. Appl. 61(2011)2952-2957. [14]C. H. Yu, On fuzzy topological groups, Fuzzy Sets and Systems 23(1987)281-287. [15]L. A. Zadeh, Fuzzy Sets, Inform. Control 8 (1965) 338-353.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.5, 2019, 849-863 COPYRIGHT 2019 EUDOXUS PRESS, LLC The Behavior and Closed Form of the Solutions of Some Difference Equations 1 E. M. Elsayed1,2, and Hanan S. Gafel1 2 4

¹Mathematics Department, Faculty of Science, King Abdulaziz University, 3 P. O. Box 80203, Jeddah 21589, Saudi Arabia. ²Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt. 5 E-mail: [email protected], [email protected].

ABSTRACT: In this paper, we will investigate the local stability of the equilibrium points, global attractor, boundedness and the form of the solutions for the following equations x

=

Ax x Bx + Cx

and x

=

Ax x Bx − Cx

,

where the coefficients A, B and C are real positive numbers, and the initial conditions x x are arbitrary non zero real numbers.

,x

,x

,

Keywords: difference equations, global attractor, local stability, equilibrium point, boundedness. Mathematics Subject Classification: 39A10.

1. Introduction The study of difference equations has been growing continuously for the last decade. This is largely due to the fact that difference equations manifest themselves as mathematical models describing real life situations in probability theory ,quelling theory, statistical problems, stochastic time series. combinatorial analysis number theory, geometry, electrical network, quanta in radiation, genetics in biology ,economics, psychology. sociology, etc. In fact, now it occupies a central position in applicable analysis and will no doubt continue to play an important role in mathematics as a whole see [1]-[20]. The purpose of this article is to investigate the global attractivity of the equilibrium point, and the asymptotic behavior of the solutions of the following difference equations x

=

Ax x Bx + Cx

.

x

=

Ax x Bx − Cx

.

( 1)

(2)

Where the initial conditions x₋₃, x₋₂, x₋₁, x₀ are arbitrary positive real numbers, and A, B, C are positive constants. We obtain the form of the solution of some special cases of equation (1) and (2) and some numerical simulation to the equation are given to illustrate our results. Lemma 1.1. Let I be some interval of real numbers and let : → , be a continuously differentiable function. Then for every initial conditions x , … , x , x ∈ , ∈ {1, 2, 3, … }, the difference equation = ( has a unique solution {

}

,

,…,

), = 0, 1, 2, …, (3)

.

Difinition 1.1. A point ̅ ∈ is called an equilibrium point of equation (3) if ̅ = ( ̅ , ̅ , … , ̅ ). That is, = ̅ ≥ 0, is a solution of equation (3), or equivalently, ̅ is a fixed point of . Difinition 1.2. The equilibrium point ̅ of equation (3) is called locally stable if for every > 0 such that for all x , x

,…,x ,x ∈

with │x

− ̅ │ + │x

> 0 there exists

− ̅ │ + ⋯ + │x − ̅│ < ,

we have │x − ̅ │ < , ∀ ≥ − . Difinition 1.3. The equilibrium point ̅ of equation (3) is called locally asymptotically stable if it is locally stable and if there exists > 0 such that for all x , x , … , x , x ∈ with │x

− ̅ │ + │x

− ̅ │ + ⋯ + │x − ̅│ < ,

we have lim x = ̅ . →∞

Difinition 1.4. The equilibrium point ̅ of equation (3) is called global attractor if for all x , x , … , x , x ∈ , we have lim x = ̅. →∞

Difinition 1.5. The equilibrium point ̅ of equation (3) is called global asymptotically stable if it is locally stable and a global attractor of equation (3). The equilibrium point ̅ of equation (3) is unstable if it is not locally stable. The linearized equation of equation (3) about the equilibrium point ̅ is the linear difference equation

=

( ̅, ̅, … ,

̅)

(4)

Now suppose that the characteristic equation associated with equation (4) is ( )= where

=

( ̅ , ̅ ,… , ̅ )

.

+

+⋯+

+

= 0,

Theorem A. [32] Suppose that ∈ , = 1,2,3, … and k ∈ {0,1,2, ⋯ }. Then ∑ sufficient condition for the asymptotic stability of the difference equation +

+ ⋯+

= 0, = 0,1, 2, … .

Theorem B. [33] Let [ , ] be an interval of real numbers and assume that continuous function satisfying the following properties: (a) ( , , ) is non-decreasing in [ , ] for each x and z in [ , ]; (b) If ( ,

and

in [ , ] for each

,

,

: [ , ] → [ , ] is a

∈ [ , ], and is non-increasing in

) ∈ [ , ] × [ , ] is a solution of the system = ( , , ) = ( ,

Then = . Then equation = ( solution of this equation converges to ̅ .

| | < 1, is a

,

∈

).

), has a unique equilibrium

̅ ∈ [ , ] and every

Theorem C. [33] Let [ , ] be an interval of real numbers and assume that : [ , ] → [ , ] is a continuous function satisfying the following properties: (a) ( , , ) is non-decreasing in [ , ] for each y and z in [ , ]; (b) If ( ,

and

in [ , ] for each

∈ [ , ], and is non-increasing in

∈

) ∈ [ , ] × [ , ] is a solution of the system = ( ,

Then = . Then equation = ( solution of this equation converges to ̅ .

, ,

) ,



= ( ,

,

).

), has a unique equilibrium

̅ ∈ [ , ] and every

Theorem D. [33] Let [ , ] be an interval of real numbers and assume that : [ , ] → [ , ] is a continuous function satisfying the following properties: (a) ( , , ) is non-decreasing in ∈ [ , ] for each y in [ , ]; (b) If ( ,

in [ , ] for each

∈ [ , ], and is non-increasing in and

) ∈ [ , ] × [ , ] is a solution of the system = ( , , ) = ( ,

Then = . Then equation = ( solution of this equation converges to ̅ .

,

,

,

).

), has a unique equilibrium

̅ ∈ [ , ] and every

Many researchers have investigated the behavior of the solution of difference equation for example: Cinar [5-7] has got the solutions of the following difference =

1+

,

=

,

−1 +

=

1+

.

Elabbasy et al. [10] studied the behavior of the difference equation =

−

.

−

El-Metwally et al. [12] investigated the asymptotic behavior of the population model =

+

.

Karatas et al. [28] got the form of the solution of the difference equation =

1+

.

Zayed and El-Moneam [45] deal with the behavior of the following rational recursive sequence x

=

α + βx + γx A + Bx + Cx

.

Wang and Zhang [38] considered the sufficient and necessary condition for the existence and uniqueness of the initial value problem of difference equations of higher order. In addition, they investigated the local stability, asymptotic behavior, periodicity and oscillation of solutions for the same equation. See also [21][47].

2. The Behavior of Equation (1) This section will examine the behavior of solutions of equation (1). The constants A, B and C within the equation are real positive numbers. 2.1. Local Stability of Equation (1) In this subsection, we explore the local stability character of the solution of equation (1). Equation (1) make sure has a unique equilibrium point is set as follows: x=

Axx ⇒ x = 0. Bx + Cx

Then the unique equilibrium point is x = 0 if B + C ≠ A. Theorem 2.1. Assume that A(B + 3C) < (B + C) , then the equilibrium point of equation (1) is locally asymptotically stable. Proof: Let f: (0, ∞) → (0, ∞) be a function define by f(u, v, w) =

Auw . (5) Bu + Cv

Thus, it follows that ∂f ACvw ∂f −ACuw = , = , ∂u (Bu + Cv) ∂v (Bu + Cv)

∂f Au = . ∂w Bu + Cv

As it can be seen ∂f AC ∂f −AC ∂f A │ = , │ = , │ = . ∂u (B + C) ∂v (B + C) ∂w B+C Then the linearized equation associated with equation (1) about x = 0 is AC AC A y − y + y − y = 0, (B + C) (B + C) B+C and it associated characteristic equation is AC AC A λ − λ + λ− = 0. (B + C) ( B + C) B+C It follows by theorem A that equation (1) is asymptotically stable if AC AC A │ │+│ │+│ │ < 1, (B + C) (B + C) B+C thus, A(B + 3C) < (B + C) . Therefore, the proof is complete.

2.2. Global Attractivity of the Equilibrium Point of Equation (1) The global attractivity character of solutions of equation (1) will be investigated in this section. Theorem 2.2. The equilibrium point of equation (1) is global attractor if B ≠ A. Proof. Let p, q are real numbers and suppose that f: [p, q] → [p, q] be a function define by equation f(u, v, w) =

, then we can easily see that the function increasing in u, w and decreasing in v. Assume

that (m, M) is a solution of the system M = f(M, m, M) and m = f(m, M, m). Then from equation (1), we see that AM Am , m = BM + Cm Bm + CM ⇒ BM + CMm = AM , Bm + CMm = Am . M=

Formerly (B − A)(M − m)(M + m) = 0. Then M = m if B ≠ A. Therefore, it can be concluded from Theorem B that x¯ is a global attractor. 2.3. Boundedness of the Solutions of Equation (1) The boundedness of the solutions of Equation (1) will be discussed in this section. Theorem 2.3. Every solution of equation (1) is bounded if ∞

Proof. Let {x }

< 1.

be a solution of equation (1). It follows from equation (1) that Ax x Ax x A x = ≤ = x , for all n ≥ 1. Bx + Cx Bx B

By using a comparison, we can write the right hand side as follows y So y =

= y

.

K, K is constant, and this equation is locally asymptotically stable if

< 1, and converges

to the equilibrium point y = 0. Thus the solution of equation (1) is bounded. 2.4. Special Case of Equation (1) In this subsection, the solution of the fourth order difference equation will be presented here x

=

x x x +x

. (6)

Such that the initial conditions x , x , x , x are arbitrary non zero real numbers. Theorem 2.4. The solution of equation (6) is given by the following formulas for n = 0,1,2, … x

=

(abc) d , (a + c) (b + d) (ab + ad + bc)

x

=

(abd) c , (a + c) (b + d) (ab + ad + bc)

x

=

x

=

x

(a + c)

=

x

=

(acd) b , (a + c) (b + d) (ab + ad + bc)

(a + c)

(a + c)

x

(ad) (bc) , (b + d) (ab + ad + bc) (cbd) (b + d) (abd) (b + d)

=

x

=

a ,x (ab + ad + bc) c ,x (ab + ad + bc)

(bcd) a , (a + c) (b + d) (ab + ad + bc)

=

=

(acd) b (a + c) (b + d) (ab + ad + bc)

(bca) d (a + c) (b + d) (ab + ad + bc)

(a + c)

(cd) b a (b + d) (ab + ad + bc)

,

,

,

where x = d, x = c, x = b, x = a . Proof. By using mathematical induction, we can prove as follow. For n = 0 the result holds. Assume that the result holds for n − 1, as follows x

=

x

(cbd) a (a + c) (b + d) (ab + ad + bc)

=

(abd) c (a + c) (b + d) (ab + ad + bc)

,x

=

,x

(a + c)

=

(bca) d (b + d) (ab + ad + bc)

(cd) b a . (a + c) (b + d) (ab + ad + bc)

We see from equation (6) that x

=

=

= x

x x

x +x

(cd) b a(acbd) a (a + c) (b + d) (ab + ad + bc)

(acbd) b a(a + c) + (bcad) d ] (a + c) (b + d) (ab + ad + bc)

÷[

(cd) b a ÷ [b d (a + c) (da + b(a + c)] (acb) d = . (a + c) (b + d) (ab + ad + bc) (a + c) (b + d) (ab + ad + bc) =

=

x x

(a + c)

=

x +x (acbd) (b + d) (ab + ad + bc)

(acbd) d(ab + ad + bc) + (abcd) c (a + c) (b + d) (ab + ad + bc)

÷[

]

(acbd) ÷ (ab + ad + bc) c [dc + (ab + ad + bc)] (abd) c = . (a + c) (b + d) (ab + ad + bc) (a + c) (b + d) (ab + ad + bc)

Also, the other relations can be proved similarly. The proof is completed. 2.5. Numerical Examples In this subsection, numerical examples which represent different types of solutions to equation (1). Are considered to confirm the results. Example 5.1. We assume the initial condition as follows: x = 14, x = 2, x = 7, x = 5 and the constants = 2, = 4, =1. See Fig. 1.

Example 5.2. See Fig. 2 since we put x 4, = 3.

Figure 1. = 4, x = 2, x = 7, x = 5 and the constants = 12,

=

Figure 2.

3. The Behavior of Equation (2) This section will examine the behavior of solutions of equation (2). The constants A, B and C within the equation are real positive numbers.

3.1. Local Stability of Equation (2) In this subsection, we explore the local stability character of the solution of equation (2). Equation (2) make sure a unique equilibrium point is set as follows: x=

Axx ⇒ x = 0. Bx − Cx

Then the unique equilibrium point is x = 0 if A + C ≠ B. Theorem 3.1. Assume that A(B − 3C) < (B − C) , then the equilibrium point of equation (2) is locally asymptotically stable. Proof: Let f: (0, ∞) → (0, ∞) be a function define by f(u, v, w) =

Auw . (7) Bu − Cv

Thus, it follows that ∂f −ACvw = , ∂u (Bu − Cv)

∂f ACuw = , ∂v (Bu − Cv)

∂f Au = . ∂w Bu − Cv

As it can be seen ∂f −AC ∂f AC ∂f A │ = , │ = , │ = . ∂u (B − C) ∂v (B − C) ∂w B−C Then the linearized equation associated with equation (2) about x = 0 is AC AC A y + y − y − y = 0, ( B − C) (B − C) B−C and it associated characteristic equation is AC AC A λ + λ − λ− = 0. (B − C) (B − C) B−C

It follows by theorem A that equation (2) is asymptotically stable if AC AC A │ │+│ │+│ │ < 1, (B − C) (B − C) B−C or A(B − 3C) < (B − C) . Therefore, the proof is complete.

3.2. Global Attractivity of the Equilibrium Point of Equation (2) The global attractivity character of solutions of equation (2) will be investigated in this section. Theorem 3.2. The equilibrium point of equation (2) is global attractor if C ≠ A. Proof. Let p, q are real numbers and suppose that f: [p, q] → [p, q] be a function define by f(u, v, w) = . Then we can easily see that the function f(u, v, w) decreasing in u and increasing in v. So we have two cases we prove case (1) and case (2) is similar and so will be omitted. Case (1):- If Bu − Cv > 0, then we can easily see that the function f(u, v, w) increasing in w. Assume that (m, M) is a solution of the system M = f(m, M, M) and m = f(M, m, m). Then from equation (2), we see that A , m = Bm − CM BM − Cm ⇒ BMm − CM = AMm, BMm − Cm = AMm. Formerly C(M − m)(M + m) = 0. Then M = m. Therefore, it can be concluded from Theorem C that ̅ is a global attractor. M=

3.3. Special Case of Equation (2) In this section, we study the following special cases of equation (2) where the constants A, B and C are integers numbers. The solution of the fourth order difference equation will be presented here x

=

x x x −x

. (8)

Such that the initial conditions x , x , x , x are arbitrary nonzero real numbers. Theorem 3.3. The solution of equation (8) is given by the following formulas for n = 0,1,2, … x

=

(abc) d , (a − c) (b − d) (bc − ab + ad)

x

=

(abd) c , (a − c) (b − d) (bc − ab + ad)

x

x

x

=

=

(acd) b , (a − c) (b − d) (bc − ab + ad)

(a − c)

=

(a − c)

(ad) (bc) , (b − d) (bc − ab + ad) (cbd) (b − d)

=

=

x

a ,x (bc − ab + ad)

(abd) c ,x (a − c) (b − d) (bc − ab + ad) Proof. As the proof of Theorem 2.4 and will be omitted. x

x

(bcd) a , (a − c) (b − d) (bc − ab + ad)

=

=

=

(acd) b (a − c) (b − d) (bc − ab + ad)

(bca) d (a − c) (b − d) (bc − ab + ad)

(a − c)

3.4. Numerical Examples Example 3.4. We suppose that initial condition are taken as follows: x and the constants = 12, = 3, = 8. See Fig. 3.

(cd) b a (b − d) (bc − ab + ad)

= 14, x

= 32, x

Figure 3. Example 3.5. The Figure 4 shows the behavior of the solutions of equation (2) when x 2.20, x = 5.45, x = 7 and the constants = 4, = 2, = 3. See Fig. 4.

,

,

.

= −7, x = 5

= 1.55, x

=

Figure 4.

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=

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,

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.5, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

Convexity and Monotonicity of Certain Maps Involving Hadamard Products and Bochner Integrals for Continuous Fields of Operators Pattrawut Chansangiam∗ Department of Mathematics, Faculty of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand.

Abstract We investigate the convexity and the monotonicity of certain maps involving Hadamard products and Bochner integrals for continuous fields of Hilbert space operators. Their special cases and consequences are then discussed. In particular, we obtain certain arithmetic mean-harmonic mean, Jensen, and Fiedler type inequalities.

Keywords: Hadamard product, tensor product, continuous field of operators, Bochner integral Mathematics Subject Classifications 2010: 26D15, 474A63, 46G10, 47A80.

1

Introduction

Throughout, let B(H) be the algebra of bounded linear operators on a complex separable Hilbert space H. The positive cone B(H)+ of B(H) consists of all positive operators on H. The identity operator is denoted by I, where the underlying space should be clear from contexts. The spectrum of A ∈ B(H) is written as sp(A). For self-adjoint operators A and B, the situation A > B means that A − B ∈ B(H)+ . If A is an invertible positive operator, we write A > 0. The operator norm of A ∈ B(H) is denoted by ∥A∥. The notation ∥·∥∞,X is used for the supremum norm on the set X. The Hadamard product of A and B in B(H) is defined to be the bounded linear operator A ◦ B satisfying ⟨(A ◦ B)e, e⟩ = ⟨Ae, e⟩⟨Be, e⟩ ∗ Corresponding

for all e ∈ E.

(1.1)

author. Email: [email protected]

1

864

Chansangiam 864-873

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.5, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

Here, E is a fixed countable orthonormal basis for H. This definition is independent on a choice of the orthonormal basis. In [7], it was shown that there is a positive linear map Φ taking the tensor product A⊗B to the Hadamard product A ◦ B for any A, B ∈ B(H). Indeed, the map Φ is given by Φ(X) = Z ∗ XZ where Z : H → H ⊗ H is the isometry defined on the basis E by Ze = e ⊗ e for all e ∈ E.

(1.2)

From the condition (1.1), the Hadamard product is commutative, bilinear, and positivity preserving. When H is the finite-dimensional space Cn , the Hadamard product for square complex matrices is just a principal submatrix of their Kronecker product, and it can be computed easily as the entrywise product. In the literature, there are many results concerning Hadamard products for matrices/operators, see e.g. [5, 8, 10]. A well known result is Fiedler’s inequality: Theorem 1.1 ([6]). For any positive definite matrix A, we have A ◦ A−1 > I. Theorem 1.1 can be extended in the following way: Theorem 1.2 ([11]). For each i = 1, 2, . . . , n, let Ai be a positive definite matrix and Xi a positive semidefinite matrix of the same size. Then the map α 7→

n ∑

1/2

Xi

1/2

Aα i Xi

i=1

◦

n ∑

1/2

Xi

A−α i Xi

1/2

i=1

is increasing on [0, ∞). In this paper, we shall investigate the convexity and the monotonicity of an integral map ∫ ∫ (1.3) X dµ(t) ◦ Xt∗ A−α α 7→ Xt∗ Aα t t Xt dµ(t) t Ω

Ω

where α is a real constant. Here, (At )t∈Ω and (Xt )t∈Ω are two operator-valued maps parametrized by a locally compact Hausdorff space Ω. Some interesting special cases of this map are discussed. Moreover, we obtain certain arithmetic mean - harmonic mean (AM-HM), Jensen, and Fiedler type inequalities as consequences. When we set Ω to be a finite space endowed with the counting measure, our results are reduced to the corresponding discrete inequalities. In particular, these include Theorems 1.1 and 1.2. The paper is structured as follows. In Section 2, we set up basic notations, and discuss Bochner integrability of continuous field of operators on a locally compact Hausdorff space. The main part of the paper, Section 3, discusses convexity and monotonicity of the map (1.3) and its interesting special cases. As consequences, we obtain certain AM-GM, Jensen, and Fielder type inequalities in the last section. 2

865

Chansangiam 864-873

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.5, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

2

Continuous field of operators on a locally compact Hausdorff space

In this section, we provide fundamental background on continuous fields of operators and their integrability. See, e.g., [1, 3, 12] for more information. Let Ω be a locally compact Hausdorff space endowed with a Radon measure µ. A family (At )t∈Ω of operators in B(H) is said to be a continuous field of operators if the parametrization t 7→ At is norm-continuous on Ω. If, in addition, the norm function∫ t 7→ ∥At ∥ is Lebesgue integrable on Ω, then we can form the Bochner integral Ω At dµ(t) which is the unique operator in B(H) such that (∫

) At dµ(t)

ϕ

∫ =

Ω

ϕ(At ) dµ(t) Ω

for every bounded linear functional ϕ on B(H) (see e.g. [14, pp. 75-78]). In what follows, suppose further that the measure µ on Ω is finite. Next, we shall prove the Bochner integrability of an operator-valued map involving a continuous field of operators (Proposition 2.3). To do this we need some auxiliary results about functional calculus and vector-valued integration. Lemma 2.1. Let ∆ be a nonempty compact subset of C and f : ∆ → C a continuous function. Let A be the subset of B(H) consisting of all normal operators whose spectra are contained in ∆. Then the map Ψ : A → B(H), A 7→ f (A) is continuous. Here, f (A) is the continuous functional calculus of f on the spectrum of A. Proof. See [4, Lemma 2.1]. Lemma 2.2. Let (X, ∥·∥X ) be a Banach space, and let (Γ, ν) be a finite measure space. Suppose that f : Γ → X is a measurable function. Then f is Bochner integrable if and only if its norm function ∥f ∥ is Lebesgue integrable, i.e., ∫ ∥f ∥ dν < ∞. Γ

Here, ∥f ∥ is defined by ∥f ∥(x) = ∥f (x)∥X for any x ∈ X. Proof. See e.g. [1, Theorem 11.44]. Now we are in a position to prove the Bochner integrability of a map related to the map (1.3). Proposition 2.3. Let (At )t∈Ω be a continuous field of normal operators in B(H) such that sp(At ) ⊆ [m, M ] for all t ∈ Ω. Let (Xt )t∈Ω be a bounded continuous field of operators in B(H). For any continuous function f : [m, M ] → C, we can form the Bochner integral ∫ Xt∗ f (At )Xt dµ(t). Ω

In addition, if f ([m, M ]) ⊆ [0, ∞), then this operator is positive. 3

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Proof. Let K > 0 be such that ∥Xt ∥ 6 K for all t ∈ Ω. By Lemma 2.2, it suffices to prove the Lebesgue integrability of the norm function t 7→ ∥Xt∗ f (At )Xt ∥. We shall show that the map t 7→ Xt∗ f (At )Xt is continuous and bounded. Since t 7→ At is continuous, the map t 7→ f (At ) is continuous on Ω by Lemma 2.1, and hence so is the map t 7→ Xt∗ f (At )Xt . For boundedness, we have that for each t ∈ Ω, ∥Xt∗ f (At )Xt ∥ 6 ∥Xt∗ ∥ · ∥f (At )∥ · ∥Xt ∥ 6 ∥Xt ∥2 · ∥f ∥∞,[m,M ] 6 K 2 ∥f ∥∞,[m,M ] . Now, suppose that f is positive on [m, M ]. Then f (At ) is a positive operator for all t ∈ Ω. Hence the resulting integral is positive since the integrand is positive. Remark 2.4. For convenience to all results in this paper, we may assume that Ω is a compact Hausdorff space. In this case, any Radon measure on Ω is always finite. It follows that every continuous field of operators on Ω is automatically bounded, and hence is Bochner integrable. Lemma 2.5. Let X and Y be Banach spaces and let (Γ, ν) be a measure space. Suppose that a function f : Γ → X is Bochner integrable. If T : X → Y be a bounded linear operator, then the composition T ◦ f is Bochner integrable and (∫ ) ∫ (T ◦ f ) dν = T f dν . Γ

Γ

Proof. See e.g. [1, Lemma 11.45]. The next property will be used to prove the main result of the paper. Proposition 2.6. Let (At )t∈Ω be a bounded continuous field of operators in B(H). For any X ∈ B(H), we have ∫ ∫ At dµ(t) ◦ X = (At ◦ X) dµ(t). (2.1) Ω

Ω

Proof. By Lemma 2.2, the map t 7→ At is Bochner integrable on Ω since it is continuous and bounded. Note that the map T 7→ T ◦ X is a bounded linear operator from B(H) to itself. It follows from Lemma 2.5 that the map t 7→ At ◦X is Bochner integrable on Ω and the property (2.1) holds.

3

Convexity and Monotonicity of certain maps for Hadamard products of operators

In this section, we consider convexity and monotonicity of the map ∫ ∫ α 7→ Xt∗ Aα X dµ(t) ◦ Xt∗ A−α t t Xt dµ(t) t Ω

Ω

where α is a real constant. We start with an auxiliary result. 4

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Lemma 3.1. For each A > 0, the map α 7→ Aα +A−α is convex on R, increasing on [0, ∞), decreasing on (−∞, 0] and attaining its minimum at α = 0. Proof. The convexity of the map F (α) = Aα +A−α means that for each α, β ∈ R and ω ∈ (0, 1), we have F ((1−ω)α+ωβ) 6 (1−ω)F (α)+ωF (β) or equivalently, A(1−ω)α+ωβ + A−((1−ω)α+ωβ) 6 (1 − ω)(Aα + A−α ) + ω(Aβ + A−β ).

(3.1)

Indeed, for each fixed x > 0, consider the function f (α) = xα + x−α in a real variable α. Then f ′′ (α) = (ln x)2 (xα + x−α ) > 0,

α ∈ R.

It follows that f is convex on R, i.e., for each α, β ∈ R and ω ∈ (0, 1) we have x(1−ω)α+ωβ + x−((1−ω)α+ωβ) 6 (1 − ω)(xα + x−α ) + ω(xβ + x−β ).

(3.2)

Applying the functional calculus on the spectrum of A yields the desired inequality (3.1). Note also that f ′ (α) = α(xα−1 − x−α−1 ) for each α ∈ R. Hence, f is increasing on [0, ∞), decreasing on (−∞, 0] and attaining its minimum at α = 0. Similarly, applying the functional calculus yields the desired results. A proof of a part of this fact in matrix context was given in [13], using diagonalization. Theorem 3.2. Let (At )t∈Ω be a continuous field of positive operators in B(H) such that sp(At ) ⊆ [m, M ] ⊆ (0, ∞) for all t ∈ Ω. Let (Xt )t∈Ω be a bounded continuous field of operators in B(H). Then the map ∫ α 7→

Xt∗ Aα t Xt dµ(t) ◦

Ω

∫

Xt∗ A−α t Xt dµ(t)

(3.3)

Ω

is convex on R, increasing on [0, ∞), decreasing on (−∞, 0] and attaining its minimum at α = 0. Proof. Denote this map by F . Proposition 2.3 asserts the Bochner integrability of the map t 7→ Xt∗ Aα t Xt for each α ∈ R. For each α ∈ R, we have by Proposition 2.6 and Fubini’s theorem for Bochner integrals [2] that ∫ ( F (α) =

Xt∗ Aα t Xt

∫Ω∫ ∫∫

Ω2

1 2

Xs∗ A−α s Xs

◦

) dµ(s)

dµ(t)

Ω

= =

∫

∗ −α (Xt∗ Aα t Xt ◦ Xs As Xs ) dµ(s) dµ(t)

Ω2

(3.4)

∗ −α ∗ −α ∗ α (Xt∗ Aα t Xt ◦ Xs As Xs ) + (Xt At Xt ◦ Xs As Xs ) dµ(s) dµ(t).

5

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Then, appealing the isometry Z defined by (1.2), we have ∫∫ [ ] 1 ∗ −α ∗ −α ∗ α F (α) = Z ∗ (Xt∗ Aα t Xt ⊗ Xs As Xs ) + (Xt At Xt ⊗ Xs As Xs ) Z 2 Ω2 dµ(s) dµ(t) ∫∫ [ ] 1 α −1 −α = Z ∗ (Xt ⊗ Xs )∗ (At ⊗ A−1 (Xt ⊗ Xs )Z s ) + (At ⊗ As ) 2 Ω2 dµ(s) dµ(t). (3.5) Now, for each α, β ∈ R and ω ∈ (0, 1), we have from Lemma 3.1 and (3.5) that F ((1 − ω)α + ωβ) ∫∫ [ 1 α −1 −α 6 Z ∗ (Xt ⊗ Xs )∗ (1 − ω){(At ⊗ A−1 } s ) + (At ⊗ As ) 2 2 Ω ] β −1 −β + ω{(At ⊗ A−1 } (Xt ⊗ Xs )Zdµ(s) dµ(t) s ) + (At ⊗ As ) ∫∫ [ 1 ∗ −α ∗ −α ∗ α Z ∗ (1 − ω)(Xt∗ Aα = t Xt ⊗ Xs As Xs + Xt At Xt ⊗ Xs As Xs ) 2 2 Ω ] ∗ −β ∗ β + ω(Xt∗ Aβt Xt ⊗ Xs∗ A−β s Xs + Xt At Xt ⊗ Xs As Xs ) Zdµ(s) dµ(t) ∫∫ 1−ω ∗ −α ∗ −α ∗ α = (Xt∗ Aα t Xt ◦ Xs As Xs + Xt At Xt ◦ Xs As Xs )dµ(s) dµ(t) 2 2 ∫ ∫Ω ω ∗ β + (Xt∗ Aβt Xt ◦ Xs∗ As−β Xs + Xt∗ A−β t Xt ◦ Xs As Xs )dµ(s) dµ(t) 2 Ω2 = (1 − ω)F (α) + ωF (β). Therefore, F is convex. In the rest, it suffices to show that F is increasing on [0, ∞) since the Hadamard product is commutative. It follows from (3.5) and Lemma 3.1 that for 0 6 α 6 β, ∫∫ ] [ 1 β −1 −β F (α) 6 (Xt ⊗ Xs )Z dµ(s) dµ(t) Z ∗ (Xt ⊗ Xs )∗ (At ⊗ A−1 s ) + (At ⊗ As ) 2 Ω2 ∫∫ [ ] 1 ∗ −β ∗ β = Z ∗ (Xt∗ Aβt Xt ⊗ Xs∗ A−β X ) + (X A X ⊗ X A X ) Z dµ(s) dµ(t) s t s s t t s s 2 2 ∫∫ Ω = (Xt∗ Aβt Xt ◦ Xs∗ A−β s Xs ) dµ(s) dµ(t). Ω2

From (3.4), the right-hand side is equal to F (β). Thus, F is increasing on [0, ∞). In the rest of section, we discuss certain special cases of Theorem 3.2. Corollary 3.3. Let (At )t∈Ω and (Bt )t∈Ω be two bounded continuous field of positive operators in B(H) such that sp(At ) ⊆ [m, M ] ⊆ (0, ∞) and At Bt = Bt At for all t ∈ Ω. Then the map ∫ ∫ α 7→ Aα B dµ(t) ◦ A−α (3.6) t Bt dµ(t) t t Ω

Ω

6

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is convex on R, increasing on [0, ∞), decreasing on (−∞, 0] and attaining its minimum at α = 0. 1/2

Proof. Set Xt = Bt for each t ∈ Ω. Then (Xt )t∈Ω is a continuous field by Lemma 2.1. The family (Xt )t∈Ω is bounded due to the boundedness of (Bt )t∈Ω . The result now follows from Theorem 3.2. An interesting special case of Corollary 3.3 is when Bt = f (At ) where f is a complex-valued continuous function on [m, M ]. In this case, the field (Bt )t∈Ω is bounded since ∥f (At )∥ 6 ∥f ∥∞,[m,M ] for all t ∈ Ω. Hence we obtain monotonicity information of the map ∫ ∫ α 7→ Aα f (A ) dµ(t) ◦ A−α t t f (At ) dµ(t). t Ω

Ω

In particular, when f (x) = xλ , we get the following result. Corollary 3.4. For any λ ∈ R, the map ∫ ∫ α 7→ Aλ+α dµ(t) ◦ Aλ−α dµ(t) t t Ω

Ω

is convex on R, increasing on [0, ∞), decreasing on (−∞, 0] and attaining its minimum at α = 0. The next result is also a special case of Theorem 3.2 in which the weights are scalars. Corollary 3.5. Let (At )t∈Ω be a continuous field of positive operators in B(H) such that sp(At ) ⊆ [m, M ] ⊆ (0, ∞) for all t ∈ Ω. For any bounded continuous function w : Ω → [0, ∞), the map ∫ ∫ α α 7→ w(t)At dµ(t) ◦ w(t)A−α dµ(t) (3.7) t Ω

Ω

is convex on R, increasing on [0, ∞), decreasing on (−∞, 0] and attaining its minimum at α = 0. √ Proof. Set Xt = w(t)I for all t ∈ Ω in Theorem 3.2. We see that (Xt )t∈Ω is a bounded continuous field of operators. Corollary 3.6. Let f : Ω → C and g : Ω → [0, ∞) be bounded continuous functions such that Range(f ) ⊆ [m, M ] ⊆ (0, ∞). Then the map ∫ ∫ α 7→ gf α dµ gf −α dµ Ω

Ω

is convex on R, increasing on [0, ∞), decreasing on (−∞, 0] and attaining its minimum at α = 0. 7

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Proof. Set H = C in Corollary 3.3. A discrete version of Theorem 3.2 is obtained in the next corollary, which is an operator extension of Theorem 1.2. Corollary 3.7. For each i = 1, 2, . . . , n, let Ai , Xi ∈ B(H) be such that Ai is positive and invertible. Then the map α 7→

n ∑

Xi∗ Aα i Xi ◦

i=1

n ∑

Xi∗ A−α i Xi

i=1

is convex on R, increasing on [0, ∞), decreasing on (−∞, 0] and attaining its minimum at α = 0. Proof. In Theorem 3.2, set Ω to be the finite space {1, 2, . . . , n} equipped with the counting measure.

4

AM-GM, Jensen, and Fiedler type inequalities

From the main result (Theorem 3.2), we get three interesting inequalities. The first consequence is an integral version of the weighted arithmetic-harmonic mean inequality for bounded continuous function defined on a locally compact Hausdorff space: Corollary 4.1. Let f be a bounded continuous function defined on Ω such that ∫Range(f ) ⊆ [m, M ] ⊆ (0, ∞). Let w : Ω → [0, ∞) be a weight function, i.e., w dµ = 1. We obtain the following bound for the weight integral of f : Ω ∥wf ∥1 >

1 . ∥w/f ∥1

(4.1)

Here, ∥·∥1 denotes the L1 -norm on Ω. Proof. Setting H = C in Corollary 3.5 yields that the function ∫ ∫ w α 7→ wf α dµ dµ α f Ω Ω is increasing on [0, ∞). In particular, this implies that ∫

∫ wf dµ Ω

Now, since

∫

w f

dµ >

∫

Ω

w dµ > f

(∫

)2 w dµ

= 1.

Ω

wM −1 dµ = M −1 > 0, the inequality (4.1) follows.

The second consequence is a Jensen type inequality for a continuous field of strictly positive operators. 8

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Corollary 4.2. Let (At )t∈Ω be a continuous field of positive operators in B(H) such that sp(At ) ⊆ [m, M ] ⊆ (0, ∞) for all t ∈ Ω. Suppose that µ(Ω) = 1. Then ∫ ∫ ∫ A2t dµ(t) ◦ I > At dµ(t) ◦ At dµ(t). Ω

Ω

Ω

Proof. Form Corollary 3.4, the map ∫ ∫ 1+α At dµ(t) ◦ At 1−α dµ(t) G(α) ≡ Ω

Ω

is increasing on [0, ∞). In particular, we have G(1) > G(0), which is the desired inequality. Corollary 4.2 may be regarded as a Jensen type inequality for continuous field of operators (cf. [9]). Indeed, the case H = C of this corollary can be described as follows. Suppose that (Ω, µ) is a probability space. For any continuous function f : Ω → (0, ∞), we have (∫

∫

)2

f 2 dµ >

f dµ

Ω

,

Ω

which is Jensen’s inequality for the convex function ϕ(x) = x2 . The final result is an operator extension of Fiedler’s inequality (Theorem 1.1). Corollary 4.3. For each invertible positive operator A, we have A ◦ A−1 > I.

(4.2)

Proof. The case n = 1 in Corollary 3.7 says that the map α 7→ Aα ◦ A−α has a minimum at α = 0. It follows that Aα ◦ A−α > I for any α ∈ R. Replacing A with A1/α yields the inequality (4.2).

Acknowledgements This research is supported by King Mongkut’s Institute of Technology Ladkrabang Research Fund.

References [1] C. D. Aliprantis, K. C. Border, Infinite Dimensional Analysis, SpringerVerlag, New York, 2006. [2] W. M. Bogdanowicz, Fubini’s theorem for generalized Lebesgue-BochnerStieltjes integral. Proc. Jpn. Acad., 42, 979-983 (1966). 9

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[3] P. Chansangiam, Certain integral inequalities involving tensor products, positive linear maps, and operator means. J. Inequal. Appl., Article 121, (2016), DOI 10.1186/ s13660-016-1063-7. [4] P. Chansangiam, Kantorovich type integral inequalities for tensor products of continuous fields of positive operators. J. Comput. Anal. Appl., 25, 1385-1397 (2018). [5] R. Drnovˇsek, Inequalities on the spectral radius and the operator norm of Hadamard products of positive operators on sequence spaces. Banach J. Math. Anal., 10(4), 800-814 (2016). [6] M. Fiedler, Uber eine ungleichung fur positiv definite matrizen. Math. Nachr., 23, 197-199 (1961) [7] J. I. Fujii, The Marcus-Khan theorem for Hilbert space operators. Mathematica Japonica, 41, 531-535 (1995). [8] J. I. Fujii, M. Nakamura, Y. Seo, Ando’s theorem for Hadamard products and operator means. Sci. Math. Jpn., e-2006, 603-608 (2006). [9] F. Hansen, J. Peˇcari´c, I. Peri´c, Jensen’s operator inequality and its converses. Math. Scand., 100, 61-73 (2007). [10] K. Kitamura, R. Nakamoto, Schwarz inequalities on Hadamard products. Sci. Math. Jpn., 1(2), 243-246 (1998). [11] J. S. Matharu, J. S. Aujla, Hadamard product versions of the Chebyshev and Kantorovich inequalities, J. Inequal. Pure Appl. Math., 10, Article 51 (2009). [12] M. S. Moslehian, Chebyshev type inequalities for Hilbert space operators. J. Math. Anal. Appl., 420(1), 737-749 (2014). [13] M. S. Moslehian, J. S. Matharu, J. S. Aujla, Non-commutative Callebaut inequality. Linear Algebra Appl., 436(9), 3347-3353 (2012). [14] G. K. Pedersen, Analysis Now, Springer-Verlag, New York, 1989.

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Fibonacci periodicity and Fibonacci frequency Hee Sik Kim1 , J. Neggers2 and Keum Sook So3,∗ 1

Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul, 04763, Korea 2 Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487-0350, U. S. A 3,∗ Department of Statistics and Financial Informatics, Hallym University, Chuncheon, 24252, Korea Abstract. In this paper we introduce the notion of Fibonacci periodicity modulo n, denoting this period by the function Fb(n). We note that Fb(n) is an integral multiple of a fundamental frequency fb(n), where the ratio Fb(n)/fb(n) is a power of 2 for a collection of observed values of n. It is demonstrated that if a, b are natural numbers with gcd(a, b) = 1, then Fb(n) = lcm{Fb(a), Fb(b)} and thus that Fb is a non-trivial example of a function which we refer to as radical. From observations it also seems clear that

b(ps+1 ) F b(ps ) F

= p for primes p.

1. Introduction and Preliminaries Fibonacci-numbers have been studied in many different forms for centuries and the literature on the subject is consequently incredibly vast. One of the amazing qualities of these numbers is the variety of mathematical models where they play some sort of role and where their properties are of importance in elucidating the ability of the model under discussion to explain whatever implications are inherent in it. The fact that the ratio of successive Fibonacci numbers approaches the Golden ratio (section) rather quickly as they go to infinity probably has a good deal to do with the observation made in the previous sentence. Surveys and connections of the type just mentioned are provided in [1] and [2] for a very minimal set of examples of such texts, while in [3] an application (observation) concerns itself with a theory of a particular class of means which has apparently not been studied in the fashion done there by two of the authors the present paper. Surprisingly novel perspectives are still available. Kim and Neggers [6] showed that there is a mapping D : M → DM on means such that if M is a Fibonacci mean so is DM , that if M is the harmonic mean, then DM is the arithmetic mean, and if M is a Fibonacci mean, then limn→∞ Dn M is the golden section mean. In [5] Han et al. discussed Fibonacci functions on the real numbers R, i.e., functions f : R → R such that for all x ∈ R, f (x+2) = f (x+1)+f (x), and developed the notion of Fibonacci functions using the concept of f -even and f -odd functions. Moreover, they showed that if f is a Fibonacci function then limx→∞

f (x+1) f (x)

=

√ 1+ 5 2 .

The present

authors [8] discussed Fibonacci functions using the (ultimately) periodicity and we also discuss the exponential Fibonacci functions. Especially, given a non-negative real-valued function, we obtain several exponential Fibonacci functions. The present authors [9] introduced the notions of Fibonacci (co-)derivative of real-valued functions, and found general solutions of the equations 4(f (x)) = g(x) and (4 + I)(f (x)) = g(x). Moreover, they [10] defined and studied a function F : [0, ∞) → R and extensions F : R → C, Fe : C → C which are continuous and such that if n ∈ Z, the set of all integers, then F (n) = Fn , the nth Fibonacci number based on F0 = F1 = 1. If x is not an integer and x < 0, then F (x) may be a complex number, e.g., F (−1.5) = 0∗

1 2

+ i. If z = a + bi,

Correspondence: Tel.: +82 33 248 2011, Fax: +82 33 256 2011 (K. S. So). 874

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Hee Sik Kim, J. Neggers and Keum Sook So∗ then Fe(z) = F (a) + iF (b − 1) defines complex Fibonacci numbers. In connection with this function (and in general) they defined a Fibonacci derivative of f : R → R as (4f )(x) = f (x + 2) − f (x + 1) − f (x) so that if e is given as (4f )(x) ≡ 0 for all x ∈ R, then f is a (real) Fibonacci function. A complex Fibonacci derivative 4 e (a + bi) = 4f (a) + i 4 f (b − 1) and its properties are discussed in same detail. 4f The notion of Fibonacci means was introduced by M (x, y) =

a(x + y) + 2bxy 2a + b(x + y)

2ax + 2bx2 = x provided 2a + 2bx 6= 0 ([6]). 2a + 2bx Particular cases are a > 0, b = 0, M (x, y) = x+y 2 , the average (arithmetic mean), a = 0, b > 0, M (x, y) =

where M (x, x) =

the harmonic mean, and if q =

√ 1+ 5 2 ,

Mq (x, y) =

q(x+y)+2xy 2q+(x+y) ,

2xy x+y ,

the golden section mean. Hence both the harmonic

mean, the arithmetic mean and golden section mean are special cases of the Fibonacci mean. The golden section mean Mq (x, y) is defined by Mq (x, y) = by

q ∗ (x+y)+2xy 2q ∗ +(x+y)

∗

where q =

√ 1− 5 2 ,

q(x+y)+2xy 2q+(x+y)

where q =

√ 1+ 5 2 ,

and we define Mq∗ (x, y)

which is called a conjugate golden section mean. a(x+y)+2bxy 2a+b(x+y)

It was shown that: if M (x, y) =

is a Fibonacci mean and if M (x, y) = DM (x, y), then either

M (x, y) = Mq (x, y) or M (x, y) = Mq∗ (x, y). 2. Fibonacci frequency Given a positive integer n ≥ 2, let Fb(n) = m provided Fk ≡ Fk+m (mod n) for all positive integers k, where m is the smallest positive integer with this property and Fk is the k th Fibonacci number relative to arbitrary inputs F1 = a, F2 = b, non-negative integers. For example, for n = 2 we have with inputs a = 1, b = 1: 1, 1, 0, 1, 1, 0, · · · whence Fb(2) ≥ 3. Also, a, b, a + b, a, b, a + b, · · · shows that Fb(2) ≤ 3. Thus, combining these observations we establish that: Proposition 2.1. Fb(2) = 3. For n = 3, a = 1, b = 1 yields a lower bound computation is: 1, 1, 2, 0, 2, 2, 1, 0, 1, 1, 2, 0, 2, 2, 1, 0, 1, · · · and Fb(3) ≥ 8. Also, a, b, a + b, a + 2b, 2a, 2b, 2a + 2b, 2a + b, a, b, · · · and Fb(3) ≤ 8. Hence, it follows that: Proposition 2.2. Fb(3) = 8. When we consider the method of proof of the above two propositions, we note that the pattern 0, 1, 1, 2, · · · corresponds to a pattern αa + βb, a, b, a + b, · · · in the second sequence where α + β ≡ 0 (mod n), and that if · · · , s, · · · is any term in the a = 1, b = 1 sequence, then · · · , s, · · · corresponds to · · · , λa + µb, · · · where λ + µ ≡ s (mod n). Hence if s = 1, then we have λ ≡ 1 (mod n), µ ≡ 0 (mod n) or λ ≡ 0 (mod n), µ ≡ 1 (mod n), i.e., in the first case we find the input a, whereas in the second case we find the input b. Notice that αa+βb, a, b, a+b, · · · with α+β ≡ 0 (mod n) means (α+1)a+βb ≡ b (mod n), β ≡ −α (mod n), (α+1)(a−b) ≡ 0 875

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Fibonacci periodicity and Fibonacci frequency (mod n), and a − b arbitrary means α + 1 ≡ 0 (mod n), i.e., α ≡ n − 1 (mod n), β ≡ 1 (mod n). Hence, the sequence looks like (n − 1)a + b, a, b, · · · . If we continue the construction by including one more term, then λa + µb, (n − 1)a + b, a, b, · · · yields (λ + (n − 1))a + (µ + 1)b ≡ a (mod n) and (λ + n − 2)a + (µ + 1)b ≡ 0 (mod n). Hence a = 0 yields µ ≡ n − 1 (mod n) and λ + n − 2 ≡ λ − 2 ≡ 0 (mod n), i.e., λ ≡ 2 (mod n), i.e., the sequence is · · · , 2a + (n − 1)b, (n − 1)a + b, a, b, · · · . Letting a = b = 1, the corresponding pattern is · · · , 2 + (n − 1) ≡ 1, (n − 1) + 1 ≡ 0, 1, 1, · · · . Thus, if this occurs at the mth location, then Fb(n) ≥ m from a = 1, b = 1 and Fb(n) ≤ m from a, b unspecified, whence Fb(n) = m. We thus obtain: Theorem 2.3. To determine Fb(n) it suffices to take a = 1, b = 1, and construct the Fibonacci sequence modulo n until the pattern · · · , 1, 0 is obtained. If the sequence has m terms, then Fb(n) = m. Suppose for example that we wish to determine Fb(4). Using Theorem 2.3 we let a = 1, b = 1, whence the sequence is 1, 1, 2, 3, 1, 0, and Fb(4) = 6. Note that Fb(4)/Fb(2) = 6/3 = 2. As another example consider the computation of Fb(9). Again, using Theorem 3.3 and a = 1, b = 1, we obtain the following sequence: 1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 0, 8, 8, 7, 6, 4, 1, 5, 6, 2, 8, 1, 0 and Fb(9) = 24. Note that Fb(9)/Fb(3) = 24/8 = 3. In this case we also note that the first 0 shows up after 12 steps. Accordingly we take fb(9) = 12 and Fb(9) = fb(9)m(9), b Fb(9) = 2. We consider fb(9) to be the fundamental frequency and m(9) b to be the multiplicity, with Fb(9) the Fibonacci frequency of the integer 9 (≥ 2). Fibonacci numbers have been studied in great detail over many years and the literature on the subject is quite substantial with entire books on the subject dedicated to their study and the study of these numbers also meriting chapters in books on number theory ([1, 2]). Recently, two of the authors of this paper were able to make a different but not entirely surprising connection between Fibonacci numbers and the Golden Section than the usual one ([3]). If a = b = 1, then it is well-known that Fm | Fn if and only if m | n for example. Another known result is the following: For any prime p, there are infinitely many Fibonacci numbers which are divisible by p and these are equally spaced in the Fibonacci sequence. The case fb(3) = 4 is an instance of this observation. Our point of view allows us to consider a more general situation and obtain some results on relationships connecting various values of Fb(n) and to make some conjectures on these relationships which appear to be interesting.

3. Radical functions In the number-theoretical setting, a function f on the natural numbers is multiplicative if gcd(a, b) = 1 implies f (ab) = f (a)f (b). Certainly any function for which f (xy) = f (x)f (y) is multiplicative. The Euler-phi-function is multiplicative in the number-theoretical sense without being multiplicative in the strict sense. 876

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Hee Sik Kim, J. Neggers and Keum Sook So∗ Given a natural number m = pr11 · · · prnn , where the pi ’s are distinct primes in the factorization of m, we let rad(m) = p1 · · · pn , according to conventional ring-theoretical practice. Thus, for natural numbers m1 , m2 , we find that rad(m1 m2 ) = lcm{rad(m1 ), rad(m2 )}. This function is an example of functions on the natural numbers satisfying the following “multiplicative” condition: a function f on the natural numbers is a radical function if gcd(a, b) = 1 implies f (ab) = lcm{f (a), f (b)}. When we check the table in the previous section, we observe that in the available examples it is true that gcd(a, b) = 1 implies Fb(ab) = lcm{Fb(a), Fb(b)}. For example, Fb(4) = 6, Fb(5) = 20, Fb(20) = 60, gcd(4, 5) = 1 and Fb(20) = lcm{6, 20} 6= 120, i.e., Fb is not a multiplicative function in the number-theoretical sense. Thus, it is our goal in this section to prove that Fb is a radical function in the sense described above. Lemma 3.1. If d|n, then Fb(d) ≤ Fb(n). Proof. Since d|n, there exists an integer q such that n = dq. If we let Fb(n) = m, then Fk ≡ Fk+m (mod n) for any integer k, so that Fk+m − Fk = nu = dqu for some u ∈ Z, i.e., d|Fk+m − Fk . This means that Fb(d) ≤ m = Fb(n).  Lemma 3.2. If d|n, then Fb(d)|Fb(n). Proof. Using Division Algorithm, we have Fb(n) = Fb(d)q + r for some q, r ∈ Z, where 0 ≤ r < Fb(d). Let Fb(n) := m and Fb(d) := t. Then m = qt + r and hence Fk ≡ Fk+m (mod n), so that n|Fk+m − Fk . Since d|n, we have d| Fk+m − Fk . We claim that Fk+r ≡ Fk+qt+r (mod d). Since Fb(d) := t, Fk ≡ Fk+t

(mod d)

(1)

for any natural number k ∈ N . If we take k := k + t in (1), then Fk+t ≡ F(k+t)+t ≡ Fk+2t (mod d). Similarly, we obtain Fk ≡ Fk+(q−1)t

(mod d)

(2)

for any natural number k ∈ N and natural number q > 1. If we replace k by k+r in (2), then Fk+r ≡ Fk+r+(q−1)t ≡ Fk+r+qt (mod d). Hence Fk ≡ Fk+m ≡ Fk+qt+r ≡ Fk+r (mod d). Since Fb(d) = t is the smallest positive integer with this property and 0 ≤ t < t, we have r = 0, i.e., m = qt, proving the assertion.



Theorem 3.3. If n = ab, where a, b are natural numbers with gcd(a, b) = 1, then Fb(n) = lcm{Fb(a), Fb(b)}. Proof. Suppose that n = ab, where a, b are natural numbers with gcd(a, b) = 1. Then Fb(a)|Fb(n), Fb(b)|Fb(n) by Lemma 4. Hence lcm{Fb(a), Fb(b)} ≤ Fb(n). If we let m := lcm{Fb(a), Fb(b)}, then there exists natural numbers m r, s such that m = rFb(a) = sFb(b) where gcd(r, s) = 1. Let α := m r , β := s . Then Fk ≡ Fk+α (mod a), Fk ≡ Fk+β (mod b) for any positive integer k. Hence Fk ≡ Fk+rα ≡ Fk+r mr ≡ Fk+m (mod a). Similarly, Fk ≡ Fk+sβ ≡ Fk+s ms ≡ Fk+m (mod b) for any positive integer k. This means that a|Fk − Fk+m , b|Fk − Fk+m . Since gcd(a, b) = 1, it follows that Fk ≡ Fk+m (mod ab), i.e., Fk+m ≡ Fk (mod n), so that Fb(n) ≤ m by the minimality property of Fb, proving the theorem.



Corollary 3.4. Let a, b, c are natural numbers which are relatively prime. Then Fb(abc) = lcm{Fb(a), Fb(b), Fb(c)}. 877

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Fibonacci periodicity and Fibonacci frequency Corollary 3.5. Let a, b are natural numbers which are relatively prime. Then Fb(a)Fb(b) Fb(ab)

gcd{Fb(a), Fb(b)} =

Example 3.6. Fb(1147) = Fb(1517) = 760 using the table above along with Theorem 3.3. Indeed, 1147 = 31 · 37 and 1517 = 41 · 37, Fb(31) = Fb(41) = 40, Fb(37) = 76 and lcm{40, 76} = 760. It is of course true that F760 is not a small integer.

4. Powers of primes From the table given above, it is not immediately clear that there is any pattern to the values of Fb(p), where p is a prime. However, in all cases we have seen, the following properties holds: Conjecture 4.1. For any prime p, Fb(ps+1 ) =p Fb(ps ) Thus, for example Fb(27)/Fb(9) = 72/24 = 3 and Fb(25)/Fb(5) = 5. Accepting Conjecture 4.1 as true, we note that if η(n) = Fb(n)/n, then η(ps+1 ) = Fb(ps+1 )/ps+1 = pFb(ps )/ps+1 = Fb(ps )/ps = η(ps ) = · · · = η(p). For example, η(27) = η(3) = Fb(3)/3 = 8/3 = 72/27. If n = pr+1 q s+1 , then

Fb(n) = lcm{Fb(pr+1 , Fb(q s+1 } = lcm{pr Fb(p), q s Fb(q)} =

pr q s Fb(p)Fb(q) gcd{pr Fb(p), q s Fb(q)}

=

nFb(p)Fb(q) gcd{Fb(pr+1 ), Fb(q s+1 )}pq

η(n) =

Fb(p)Fb(q) pq gcd{Fb(pr+1 ), Fb(q s+1 )}

and thus

Now, pq = rad(n). Continuing in the same fashion, if m = pr11 · · · prnn , then we find that η(m) =

Fb(p1 ) · · · Fb(pn ) rad(m) gcd{Fb(pr1 ), · · · , Fb(prnn )} 1

Global properties of the function Fb(n) may then be gathered in the function: Zη (s) = so that Zη (1) =

P∞

n=2

b(n) F n

=

P∞

n=2

∞ b X F (n) , ns n=2

η(n), which does not appear to converge for s = 1, but may well converge

for complex variables with Re(s) sufficiently large. Other generating functions may also be constructed such as P∞ b P∞ P∞ b zn n n n=2 F (n)z , n=2 η(n)z , n=2 F (n) n! , etc.. 878

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Hee Sik Kim, J. Neggers and Keum Sook So∗ Given Fb(n) for n ≥ 2, define Fb(1) = 1 and for positive integers a, b with gcd(a, b) = 1, let Fb( ab ) satisfying the following equation: a lcm{Fb(a), Fb(b)} Fb( ) = b lcm{Fb(b), Fb(b2 )} Thus, if b = 1, then Fb( a1 ) = Fb(a)/1 = Fb(a). Also, Fb( 1b ) = Fb(b)/lcm{Fb(b), Fb(b2 )}. Thus, if b = p is a prime and if Conjecture 4.1 is accepted, then Fb( 1 ) = Fb(p)/pFb(p) = 1/p. The meaning or interpretation of the values of Fb p

on fractions is not quite clear. It does however demonstrate that the function Fb defined on integers n ≥ 2 has extensions to the positive rationals, the one described here being one of them. Since mn = (−m)(−n), it makes sense to define Fb(q) = Fb(−q) for rationals q > 0. Also, since we expect Fb( a ) to be “near zero” if a is “near zero”, b

b

Fb(0) = 0 appears to be a sensible decision also. For irrational values α, the definition of Fb(α) could be as follows: if we define S(n, α) := sup{F (q) | q ∈ Q ∩ [α − 1 1 n , α + n ]},

then 0 ≤ S(n + 1, α) ≤ S(n, α) and hence limn→∞ S(n, α) = inf n∈ω S(n, α). Since ∩n∈ω S(n, α) = {α}, it follows that this permits us to define Fb(α) for α an irrational number. If Fb(α) = ∞, then S(n, α) = ∞ for all integers n. Thus, if this is the case, there is a sequence of rational numbers {qi }∞ i=1 such that limi→∞ qi = α and b b at the same time limi→∞ F (qi ) = α and at the same time limi→∞ F (qi ) = ∞. We conjecture the following: b Conjecture 4.2. There is no sequence {qi }∞ i=1 of rational numbers such that limi→∞ qi = α and limi→∞ F (qi ) = ∞. Given that the conjecture holds, Fb(α) is defined for irrational values of α as well, i.e., the domain of Fb is the real numbers. 5. Comments In this paper we have considered several aspects of the sequence of Fibonacci numbers with inputs a, b arbitrary related to the periodicity of such a sequence modulo n. Because of the plenitude of relations known to exist among various Fibonacci numbers it was not surprising that patterns would be observed. We were pleased to discover that there were numerous relationships to be found, even if not all of them are explainable. The most mysterious values are those for Fb(p) where p is an arbitrary prime. Thus, Fb(29) = 14, Fb(31) = 40, which insists on announcing that from the “Fibonacci point of view” there is a “big difference’ between these two primes in the twin-prime couple. Also, given an integer n, then fact that Fb(n2 )/Fb(n) 6= n, suffices to identify it as a composite number without knowing anything about any factorization of n. Thus, e.g., Fb(36)/Fb(6) = 24/24 = 1. Since Fibonacci numbers grow rather quickly, this observation may prove useful in the exercise of primality testing. Also, if Fb(n2 )/Fb(n) = n, then, although this does not guarantee (maybe) that n is a prime, it seems that it ought to greatly improve the probability that it is.

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Fibonacci periodicity and Fibonacci frequency 6. Appendix

n Fb(n) fb(n) 2 3 3 6 24 12 10 60 15 14 48 24 18 24 12 22 30 30 26 84 21 30 120 60 34 36 9 38 18 18 42 48 24 46 48 24 50 300 75 54 72 36 58 42 42 62 30 30 66 120 60 70 240 120 74 228 57 78 168 84 82 120 60 86 264 132 90 120 60 94 96 48 98 336 168

n Fb(n) fb(n) 3 8 4 7 16 8 11 10 10 15 40 20 19 18 18 23 48 24 27 72 36 31 30 30 35 80 40 39 56 28 43 88 44 47 32 16 51 72 36 55 20 10 59 58 58 63 48 24 67 136 68 71 70 70 75 200 100 79 78 78 83 168 84 87 56 28 91 112 56 95 180 90 99 120 60

n Fb(n) fb(n) 4 6 6 8 12 6 12 24 12 16 24 12 20 60 30 24 24 12 28 48 24 32 48 24 36 24 12 40 60 30 44 30 30 48 24 12 52 84 42 56 48 24 60 120 60 64 96 48 68 36 18 72 24 12 76 18 18 80 120 60 84 48 24 88 60 30 92 48 24 96 48 24 100 300 150

n Fb(n) fb(n) 5 20 5 9 24 12 13 28 7 17 36 9 21 16 8 25 100 25 29 14 14 33 40 20 37 76 19 41 40 20 45 120 60 49 112 56 53 108 27 57 72 36 61 60 15 65 140 35 69 48 24 73 148 37 77 80 40 81 216 108 85 180 45 89 44 11 93 120 60 97 196 49

References [1] K. Atanassove et. al, New Visual Perspectives on Fibonacci numbers, World Sci. Pub. Co., New Jersey, 2002. [2] R. A. Dunlap, The Golden Ratio and Fibonacci Numbers, World Scientific, New Jersey, 1997. [3] J. S. Han, H. S. Kim, J. Neggers, The Fibonacci norm of a positive integer n- observations and conjectures -, Int. J. Number Th. 6 (2010), 371-385. [4] J. S. Han, H. S. Kim and J. Neggers, Fibonacci sequences in groupoids, Advances in Difference Equations 2012 2012:19 (doi:10.1186/1687-1847-2012-19). [5] J. S. Han, H. S. Kim and J. Neggers, On Fibonacci functions with Fibonacci numbers, Advances in Difference Equations 2012 2012:126 (doi:10.1186/1687-1847-2012-126). [6] H. S. Kim and J. Neggers, On Fibonacci Means and Golden Section Mean, Computers and Mathematics with Applications 56 (2008), 228-232. [7] H. S. Kim, J. Neggers and K. S. So, Generalized Fibonacci sequences in groupoids, Advances in Difference Equations 2013 2013:26 (doi:10.1186/1687-1847-2013-26). 880

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Hee Sik Kim, J. Neggers and Keum Sook So∗ [8] H. S. Kim, J. Neggers and K. S. So, On Fibonacci functions with periodicity, Advances in Difference Equations 2014 2014:293. (doi:10.1186/1687-1847-2014-293). [9] H. S. Kim, J. Neggers and K. S. So, On continuous Fibonacci functions, J. Comput. & Appl. Math. 24 (2018), 1482-1490. [10] H. S. Kim, J. Neggers and K. S. So, On Fibonacci derivative equations, J. Comput. & Appl. Math. 24 (2018), 628-635.

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The weighted moving averages for a series of fuzzy numbers based on non-additive measures with σ − λ rules† Zeng-Tai Gonga,∗ , Wen-Jing Leia,b a College of Mathematics and Statistics, Northwest Normal University, Lanzhou, 730070, China b School of Economics and Management, Tongji University, Shanghai, 200092, China

Abstract Non-additive measure theory is an important mathematical tool to deal with inter-dependent or interactive information. The concept of fuzzy number provides an effective means of describing vague and uncertain system. The aim of this study is to integrate moving average with non-additive measures with σ − λ rules under fuzzy environment. That is, the moving average for a series of fuzzy numbers based on non-additive measures with σ − λ rules is proposed. Further, its specific calculation is invested and some properties are discussed. In particular, triangular fuzzy numbers about this method are also discussed. Finally, an example is given to illustrate our results. Keywords: Fuzzy number; Fuzzy measure; Moving average. 1. Introduction Non-additive measure theory, as an extension of classical measure theory for the study of interdependent or interactive information, was proposed by Sugeon [18] by replacing additivity with monotonicity. Many studies have focused on theoretical aspects and applications of non-additive measures. Asahina [1] studied implication relationship among six continuity conditions and two null-additivity conditions with respect to non-additive measures. Li [8] discussed four versions of Egoroff’s theorem in non-additive measure theory by using special condition. In particular, the Choquet integral with respect to non-additive measures has lbeen successfully applied in decision-making [23, 19], information fusion [6], economic theory [17] and so on. Considering the inherent uncertain and imprecise of information in practical life, another key mathematical structure is introduced to model uncertain and incomplete systems, which is called fuzzy number, proposed by Zadeh [25], on the basis of fuzzy sets [24]. Fuzzy number has been investigated intensively by researches from various aspects. Gong [5] generalized convexity from vector-valued maps to n-dimensional fuzzy number-valued functions. Saeidifar [16] introduced the concepts of nearest weighted interval and point approximations of a fuzzy number. And Wang [22] applied triangular fuzzy number to study management knowledge performance evaluation. Moving average is that, given a series of numbers and fixed subset size, the first element of the moving average is obtained by taking the average of the initial fixed subset of the number series [2]. The moving average has been widely applied in time series analysis [20], cloud computing [14] signal processing and financial mathematics, etc. However, when we use moving average to make forecasting, it is not reasonable to assume that all data are real data before we elicit them from practical data, fuzzy data may exit, such as in financial and sociological application. So we need to take the vagueness of the universe of importance. Furthermore, there is interaction among data in real application. The aim of this paper is to propose the moving average for a series of fuzzy numbers based on non-additive measures with σ − λ rules. In particular, triangular fuzzy numbers about this method are also discussed. The structure of this paper is as follows. In Section 2, we review some basic cponcepts and properties about non-additive measure with σ − λ rules and fuzzy numbers. And the definition of conduct between a non-negative matrix and fuzzy number vector is given to make our analysis possible. In Section 3, we propose the moving average for a series of fuzzy numbers based on non-additive measures with σ −λ rules. † ∗

Supported by National Natural Science Fund of China (61763044, 11461062). Corresponding author. Tel.: +8613993196400. Email addresses: [email protected](Zeng-Tai Gong), [email protected](Wen-Jing Lei). 882

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Zeng-Tai Gong, Wen-Jing Lei: The weighted moving averages for a series of fuzzy-numbers based on...

In Section 4, the calculation of the weighted moving averages for fuzzy-number based on a non-additive measure with σ − λ rules is invested and some properties are discussed. The paper ends with conclusion in In Section 5. 2. Preliminaries Throughout this study, Rm denotes the m-dimension real Euclidean space and R+ = (0, ∞). Definition 2.1 [18, 10, 3]. Let X denote a nonempty set and A a σ− algebra on the X. A set function µ is referred to as a regular fuzzy measure if (1) µ(∅) = 0; (2) µ(X) = 1; (3) for every A and B ∈ A such that A ⊆ B, µ(A) ≤ µ(B). Definition 2.2 [18, 10, 3]. gλ is called a fuzzy measure based on σ − λ rules if it satisfies )  (∞  1 Y  [1 + λgλ (Ai )] − 1 , λ 6= 0,  !   ∞  λ i=1 [ gλ Ai = ∞  X  i=1   gλ (Ai ), λ = 0,   i=1

S 1 , ∞) {0} , {Ai } ⊂ A , Ai ∩ Aj = ∅ for all i, j = 1, 2, · · · and i 6= j. where λ ∈ (− supµ Particularly, if λ = 0, then gλ is a classic probability measure. A regular fuzzy measure µ is called Sugeno measure based on σ − λ rules if µ satisfies σ − λ rules, briefly denoted as gλ . The fuzzy measure denoted in this paper is Sugeno measure. Remark 2.1. In the Definition, if n = 2, then ( µ(A) + µ(B) + λµ(A)µ(B), λ 6= 0, µ (A ∪ B) = µ(A) + µ(B), λ = 0. Remark 2.2. If X be a finite set, for any subset A of X, then )  ( Y  1   [1 + λgλ ({x})] − 1 ,    λ x∈A gλ (A) = ∞    X   gλ ({x}), 

λ 6= 0,

λ = 0.

i=1

Remark 2.3 [3]. If X be a finite set, then the parameter λ of a regular Sugeno measure based on σ − λ rules is determined by the equation n Y (1 + λgλi ) = 1 + λ. i=1

Let gλ ({xi }) = gi , i = 1, 2, ..., m, then gλ (Ai ) is obtained from the following recurrence relation gλ (Am ) = gλ ({xm }) = gm , gλ (Ai ) = gλ (Ai+1 ) + λgi gλ (Ai+1 ), 1 ≤ i < m. ˜ ˜ r ∈ (0, 1] and [A] ˜ r = {x ∈ R : u ˜ (X) ≥ r}. If A˜ satisfies Let A(x) ∈ E, A (1) A˜ is a normal fuzzy set, i.e., an x0 ∈ R exists such that uA˜ (x0 ) = 1; (2) A˜ is a convex fuzzy set, i.e., uA˜ (λx + (1 − λ)y) ≥ min {uA˜ (x), uA˜ (y)} for any x, y ∈ R, and λ ∈ (0, 1]; (3) A˜ is a upper semicontinuous fuzzy set; S (4) [A]0 = X ∈ R : uA˜ (x) > 0 = r∈(0,1] [A]r is compact, where A¯ denotes the closure of A. ˜ to denote the fuzzy number space [9]. Then, A˜ is called a fuzzy number. We use E 883

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Zeng-Tai Gong, Wen-Jing Lei: The weighted moving averages for a series of fuzzy-numbers based on...

It is clear that each x ∈ R can be consider as a fuzzy number A˜ defined by ( 1, x = A, uA˜ (x) = 0, otherwise. Given any two fuzzy numbers A˜1 , A˜2 , k, k1 k2 ≥ 0, the operational rules are as follows: (1) k(A˜1 + A˜2 ) = k A˜1 + k A˜2 , (2) k1 (k2 A˜1 ) = (k1 k2 )A˜1 , (3) (k1 + k2 )A˜1 ) = k1 A˜1 + k2 A˜1 . ˜ it satisfy the following equation Lemma 2.1 [11, 12, 9]. For a fuzzy set A, \ [ ˜ r ), (r∗ [A] A˜ = r∈[0,1]

where r∗ denotes the fuzzy set whose membership function is constant function r. ˜ B ˜ ∈ E, ˜ k ∈ R, the addition and scalar conduct are defined by Let A, ˜ r = [A] ˜ r + [B] ˜ r, [A˜ + B]

˜ r = k[A] ˜ r, [k A]

˜ r = {x : u ˜ (x) > r} = [A− (r), A+ (r)], for any r ∈ (0, 1]. respectively, where [A] A ˜ then Lemma 2.2 [11, 12, 9]. If A˜ ∈ E, r ˜ is a nonempty bounded closed interval for any r ∈ (0, 1]; (1) [A] ˜ r1 ⊃ [A] ˜ r2 where 0 6 r1 6 r2 6 1; (2) [A] (3) if rn > 0 and {rn } converging increasingly to r ∈ (0, 1], then ∞ \

˜ rn = [A] ˜ r. [A]

n=1

Conversely, if for any r ∈ [0, 1], there exists Br ⊂ R satisfying (1) − (3), then there exists a unique ˜ such that [A] ˜ r = Ar , r ∈ (0, 1], and [A] ˜0=S ˜r A˜ ∈ E r∈(0,1] [A] ⊂ B0 . Definition 2.3 [21]. A triangle fuzzy number A˜ is a fuzzy number with piecewise linear membership function A˜ defined by  x − al  , al ≤ x ≤ am ,     am − al uA˜ (x) = an − x  , am < x ≤ an ,     an − am 0, otherwise, which can be indicated as a triplet (al , am , an ). Given any two triangle fuzzy numbers x ˜i = (xi − δi,1 , xi , xi + δi,1 ), x ˜j = (xj − δj,1 , xj , xj + δj,1 )), and k ≥ 0, the operational rules are as follows: (1) x ˜i + x ˜j = (xi − δi,1 + xj − δj,1 , xi + xj , xi + δi,1 + xj + δj,1 ), (2) k · x ˜i = (kxi − kδi,1 , kxi , kxi + kδi,2 ). ˜ if P ∈ Rm×m and Definition 2.4. Given a nonnegative matrix P = [pij ] and a fuzzy-number vector X, + ˜ = [˜ ˜ m (The | denotes the conjugate transpose of a vector or a matrix.), then the X x1 , x ˜2 , · · · , x ˜m ]| ∈ E product of P and X is defined as follows:  P  m p x ˜  j=1 ij j     .. ˜ n−1 =  PX  . .  m    P pmj x ˜j j=1

884

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3. The weighted moving averages for fuzzy-number based on a non-additive measure with σ − λ rules ˜ m , (t1 , t2 , · · · , tm ) ∈ Rm , and gλ be fuzzy measure satisfying Definition 3.1. Let (˜ x1 , x ˜2 , · · · , x ˜m ) ∈ E δ − λ rules. Denote Ai = {ti , ti+1 , · · · , tm }, i = 1, 2, ..., m, Am+1 = ∅. Then the weighted moving averages for fuzzy-number based on a non-additive measure with σ − λ rules is defined as follows: x ˜n = (gλ (A1 ) − gλ (A2 ))˜ xn−m + (gλ (A2 ) − gλ (A3 ))˜ xn−m+1 + · · · + (gλ (Am ) − gλ (Am+1 ))˜ xn−1 , where n > m. When we use moving average to make forecasting, it is not reasonable to assume that all data are real data before we elicit them from practical data, fuzzy data may exit, such as in financial and sociological application. So we need to take the vagueness of the universe of importance. Furthermore, there is interaction among data in real application. Remark 3.1. If λ = 0, and x˜i is a special fuzzy number, namely, real number, i = 1, 2, · · · , the weighted moving averages for a series of fuzzy numbers based on non-additive measures with σ−λ rules degenerates to the classic weighted moving average in Ref. [2]. ˜ m , (t1 , t2 , · · · , tm ) ∈ Rm , and gλ be fuzzy measure satisfying Theorem 3.1. Let (˜ x1 , x ˜2 , · · · , x ˜m ) ∈ E ˜ n = [˜ δ − λ rules. Let Ai = {ti , ti+1 , · · · , tm }, i = 1, 2, ..., m, Am+1 = ∅, X xn , x ˜n+1 , · · · , x ˜n+m−1 ]| , then ˜ n = PX ˜ n−1 = P2 X ˜ n−2 = · · · = Pn−1 X ˜1, X n = 1, 2, 3, ..., where     P=  

0 0 .. .

1 0

0 1

··· ···

··· ··· ··· 0 0 0 ··· gλ (A1 ) − gλ (A2 ) gλ (A2 ) − gλ (A3 ) gλ (A3 ) − gλ (A4 ) · · ·

0 0 .. . 1 gλ (Am ) − gλ (Am+1 )

Proof. Based on Definition 3.1 and the operational rules of fuzzy numbers, we have   0 1 ··· 0 x ˜n−1   0 0 · · · 0 x ˜n      . . .. ˜ .. .. PXn−1 =   ··· ··· .     x 0 0 ··· 1 ˜n+m−3 gλ (A1 ) − gλ (A2 ) gλ (A2 ) − gλ (A3 ) · · · gλ (Am ) − gλ (Am+1 ) x ˜n+m−2

    .  

    =  

[˜ xn , x ˜n+1 , · · · , x ˜n+m−3 , (gλ (A1 )−gλ (A2 ))˜ xn−1 +(gλ (A2 )−gλ (A3 ))˜ xn +· · ·+(gλ (Am )−gλ (Am+1 ))˜ xn+m−2 ]| . And we know that (gλ (A1 ) − gλ (A2 ))˜ xn−1 + (gλ (A2 ) − gλ (A3 ))˜ xn + · · · + (gλ (Am ) − gλ (Am+1 ))˜ xn+m−2 = x ˜n+m−1 , This follows that ˜ n−1 = X ˜n. PX The proof is complete.  ˜ m , (t1 , t2 , · · · , tm ) ∈ Rm , and gλ be fuzzy measure satisfying Theorem 3.1. Let (˜ x1 , x ˜2 , · · · , x ˜m ) ∈ E ˜ n = [˜ δ − λ rules. Let Ai = {ti , ti+1 , · · · , tm }, i = 1, 2, ..., m, Am+1 = ∅, X xn , x ˜n+1 , · · · , x ˜n+m−1 ]| , and − + ˜ ˜ Xn (r) and Xn (r) as follows | ˜ − (r) = [˜ X x− ˜− ˜− n n (r), x n+1 (r), · · · , x n+m−1 (r)] , | ˜ + (r) = [˜ X x+ ˜+ ˜+ n n (r), x n+1 (r), · · · , x n+m−1 (r)] , + where [x˜i ]r = [x− i (r), xi (r)]. Then

˜ − (r) = PX ˜ − (r) = P2 X ˜ − (r) = · · · = Pn−1 X ˜ − (r), X n n−1 n−2 1 885

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˜ + (r), ˜ + (r) = · · · = Pn−1 X ˜ n+ (r) = PX ˜ + (r) = P2 X X 1 n−2 n−1 where n = 1, 2, 3, ..., and P is the same matrix in Theorem 3.1. Proof. Based on Theorem 3.1, we have  0 1 ··· 0  0 0 · · · 0  .. .. ˜ − (r) =  PX  . ··· ··· . n−1   0 0 ··· 1 gλ (A1 ) − gλ (A2 ) gλ (A2 ) − gλ (A3 ) · · · gλ (Am ) − gλ (Am+1 )



x ˜− n−1 (r) x ˜− n (r) .. .

     −  x ˜n+m−3 (r) x ˜− n+m−2 (r)

    =  

| ˜− [˜ x− x− x− n (r), · · · , x n+m−3 (r), (gλ (A1 ) − gλ (A2 ))˜ n−1 (r) + · · · + (gλ (Am ) − gλ (Am+1 ))˜ n+m−2 (r)] .

By Definition 3.1, we get (gλ (A1 ) − gλ (A2 ))˜ xn−1 + (gλ (A2 ) − gλ (A3 ))˜ x− x− ˜− n (r) + · · · + (gλ (Am ) − gλ (Am+1 ))˜ n+m−2 (r) = x n+m−1 (r). This follows that ˜ − (r). ˜ − (r) = X PX n n−1 Similarly, we can prove that ˜ n+ (r). ˜ + (r) = X PX n−1 The proof is complete.  ˜ m , (t1 , t2 , · · · , tm ) ∈ Rm , and gλ be fuzzy measure satisfying Theorem 3.2. Let (˜ x1 , x ˜2 , · · · , x ˜m ) ∈ E ˜ n = [˜ δ − λ rules. Let Ai = {ti , ti+1 , · · · , tm }, i = 1, 2, ..., m, Am+1 = ∅, X xn , x ˜n+1 , · · · , x ˜n+m−1 ]| , If x ˜i is a triangle fuzzy number, and x˜i = (xi − δi,1 , xi , xi + δi,2 ), i = 1, 2, · · · , then | ˜ n− (r) = [˜ ˜− X x− ˜− n (r), x n+m−1 (r)] n+1 (r), · · · , x

= [δn,1 r + xn − δn,1 , δn+1,1 r + xn − δn+1,1 , · · · , δn+m−1,,1 r + xn+m−1 − δn+m−1,1 ]| , | ˜ + (r) = [˜ ˜+ X x+ ˜+ n n (r), x n+m−1 (r)] n+1 (r), · · · , x

= [−δn,2 r + xn + δn,2 , −δn+1,2 r + xn+1 + δn+1,2 , · · · , −δn+m−1,2 r + xn+m−1 + δn+m−1,2 ]| . Proof. Based on the operational rules we have | ˜ n− (r) = [˜ X x− ˜− ˜− n (r), x n+m−1 (r)] n+1 (r), · · · , x

= [δn,1 r + xn − δn,1 , δn,1 r + xn − δn,1 , · · · , δn+m−1,1 r + xn+m−1 − δn+m−1,1 ]| , | ˜ + (r) = [˜ ˜+ X x+ ˜+ n n (r), x n+m−1 (r)] n+1 (r), · · · , x

= [−δn,2 r + xn + δn,2 , −δn,2 r + xn + δn,2 , · · · , −δn+m−1,2 r + xn+m−1 + δn+m−1,2 ]| . The proof is complete.



4. The calculation of the weighted moving averages for fuzzy-number based on a nonadditive measure with σ − λ rules m and set Lemma 4.1 [15]. Let(d1 , d2 , · · · , dm ) ∈ R+

q(x) = xm − d1 xm−1 − · · · − dm . Suppose that gcd{k ∈ {1, 2, · · · , m} : dk > 0} = 1, where the greatest common division of a set S is denoted by gcd(S). Then q has a unique positive rootr. Moreover, the algebraic multiplicity of r is 1, coinciding with the geometric multiplicity of r, and the modulus of every other root of q is strictly less than r. Lemma 4.2 [13]. LetB ∈ C m×m , where C denotes plural numbers. Then the following holds 886

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(1) {B n } converges to nonzero matrix if and only if 1 is a eigenvalue of B, whose algebraic multiplicities and geometric multiplicities coincide, and every other eigenvalues of B has modulus strictly less than 1; (2) If ρ(B) = max |λ| = 1 is a eigenvalue of B whose algebraic multiplicity and geometric multiplicity λ∈σ(B)

of 1 coincide, equal to 1, with right-hand and left-hand eigenvalue x and y | respectively, then lim B n =

n→∞

xy | , y|x

where σ(B) is the set of eigenvalues of B. ˜ m , (t1 , t2 , · · · , tm ) ∈ Rm , and gλ be fuzzy measure satisfying Theorem 4.1. Let (˜ x1 , x ˜2 , · · · , x ˜m ) ∈ E ˜ n = [˜ δ − λ rules. Let Ai = {ti , ti+1 , · · · , tm }, i = 1, 2, ..., m, Am+1 = ∅, X xn , x ˜n+1 , · · · , x ˜n+m−1 ]| , For the matrix P satisfying the following recurrence relation in Theorem 3.1 ˜ n = PX ˜ n−1 = P2 X ˜ n−2 = · · · = Pn−1 X ˜1, X if gcd {i ∈ {1, 2, · · · , m}|gλ (Ai ) − gλ (Ai+1 ) > 0} = 1, then lim Pn exists, and n→∞

lim Pn =

n→∞

where e =

m P

ea| = eb| , a| e

ek = [1, 1, · · · , 1]| ∈ Rm , ek is the ith standard unit column vector,

i=1

a = [a1 , a2 , · · · , am ]| , b = [b1 , b2 , · · · , bm ]| , ak =

k P

(gλ (Ai ) − gλ (Ai+1 )),

i=1

bk =

a| ek a| e

=

ak m P ai i=1

=

gλ (A1 )−gλ (Ak+1 ) ,k m P mgλ (A1 )− gλ (Ai )

= 1, 2, 3, ..., m.

i=2

Proof. For matrix P, its characteristic polynomial is p(t) = det(tId − P), where Id is the unit matrix of order m . It is easy to obtain p(t) = tm − (gλ (Am ) − gλ (Am+1 ))tm−1 − · · · − (gλ (A2 ) − gλ (A3 ))t − (gλ (A1 ) − gλ (A2 )). Since

m P

(gλ (Ai ) − gλ (Ai+1 )) = gλ (A1 ) = 1, t = 1 is a positive root of p(t). Note that

i=1

gcd{k ∈ {1, 2, · · · , m} : gλ (Ai ) − gλ (Ai+1 ) > 0} = 1. According to Lemma 4.1, we can obtain t = 1 is the unique root of p(t), whose algebraic multiplicity and geometric multiplicity of 1 are both equal to 1, and the modulus of every other root of q is strictly less than r. Let x be the right-hand eigenvector of matrix P with respect to eigenvalue 1, then Px = x. By using the elementary line transformation and the first elementary column transformation to matrix P, we can obtain Id − P    =  

1 −1 0 0 1 −1 ··· ··· ··· 0 0 0 gλ (A2 ) − gλ (A1 ) gλ (A3 ) − gλ (A2 ) gλ (A4 ) − gλ (A3 )  1 0 0 ···  0 1 0 ···  · · · · · · · ·· ··· → ··· →    0 0 0 ··· 0 0 0 ··· 887

··· ··· ··· ··· ··· −1 −1 ··· −1 0

0 0 ··· −1 1 − (gλ (Am+1 ) − gλ (Am ))

     

   .   Zeng-Tai Gong ET AL 882-891

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Hence, a basic systeme of solutions for homogeneous linear equation set (Id − P)x = [0, 0, · · · , 0]| is determined. It follows that the right-hand eigenvalue of P with respect to 1 is x = [1, 1, · · · , 1]| = e. Let y | be the left-hand eigenvector of matrix P with respect to eigenvalue 1, then y | P = y | . By using the elementary line transformation and the first elementary column transformation to matrix Id − P| , we can obtain   1 0 0 ··· gλ (A2 ) − gλ (A1 )   0 1 0 ··· gλ (A3 ) − gλ (A2 )     .. Id − P| =  ... · · · · · · · · ·  .     0 0 0 ··· gλ (Am ) − gλ (Am−1 ) 0 0 0 · · · 1 − (gλ (Am ) − gλ (Am+1 ))   1 0 0 · · · gλ (A2 ) − gλ (A1 )  0 1 0 · · · gλ (A3 ) − gλ (A1 )      .. .. → ··· →  . ··· ··· ··· . .    0 0 0 · · · gλ (Am ) − gλ (A1 )  0 0 0 ··· 0 Thus, a basic system of solutions for homogeneous linear equation set (Id − P| )y = [0, 0, · · · , 0] is determined as follows: [gλ (A1 ) − gλ (A2 ), gλ (A1 ) − gλ (A3 ), · · · , gλ (A1 ) − gλ (Am )]| , It follows that the left-hand eigenvalue of P with respect to 1 is a| = [a1 , a2 , · · · , am ], ak =

k P

(gλ (Ai ) −

i=1

gλ (Ai+1 )), k = 1, 2, 3, ..., m. According to Lemma 4.2(1), we know that {Pn } converges to a nonzero matrix. Combing Lemma 4.2(2), we can get lim Pn =

n→∞

ea| = eb| . a| e

The proof is complete.  ˜ m , (t1 , t2 , · · · , tm ) ∈ Rm , and gλ be fuzzy measure satisfying Theorem 4.2. Let (˜ x1 , x ˜2 , · · · , x ˜m ) ∈ E ˜ n = [˜ δ − λ rules. Let Ai = {ti , ti+1 , · · · , tm }, i = 1, 2, ..., m, Am+1 = ∅, X xn , x ˜n+1 , · · · , x ˜n+m−1 ]| . For the matrix P satisfying the recurrence relation in Theorem 3.1 ˜ n−2 = · · · = Pn−1 X ˜1, ˜ n = PX ˜ n−1 = P2 X X if gcd {i ∈ {1, 2, · · · , m} : gλ (Ai ) − gλ (Ai+1 ) > 0} = 1, then lim x ˜n exists, and n→∞

lim x ˜n =

n→∞

where e =

m P

m X

bi x ˜i ,

i=1

ek = [1, 1, · · · , 1]| ∈ Rm×1 , ek is the ith standard unit column vector,

i=1

a = [a1 , a2 , · · · , am ]| , b = [b1 , b2 , · · · , bm ]| , ak =

k P

(gλ (Ai ) − gλ (Ai+1 )),

i=1

bk =

a| ek a| e

=

ak m P ai

=

gλ (A1 )−gλ (Ak+1 ) ,k m P mgλ (A1 )− gλ (Ai )

i=1

= 1, 2, 3, ..., m.

i=2

Proof. Since ˜ n = PX ˜ n−1 = P2 X ˜ n−2 = · · · = Pn−1 X ˜1, X we have lim [˜ xn , x ˜n+1 , · · · , x ˜n+m−1 ]| = lim Pn−1 [˜ xn , x ˜n+1 , · · · , x ˜n+m−1 ]| .

n→∞

n→∞

then, by Theorem 4.2, we can get lim [˜ xn , x ˜n+1 , · · · , x ˜n+m−1 ]| = er| [˜ xn , x ˜n+1 , · · · , x ˜n+m−1 ]| ,

n→∞

888

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˜ 1 = [˜ i.e. lim x ˜n is determined by the operation of the first row of lim Pn−1 and X x1 , x ˜2 , · · · , x ˜m ]| . It n→∞ n→∞ follows that m P ai x ˜i X m | a x i=1 lim x ˜n = | = m = bi x ˜i . P n→∞ a e i=1 ai i=1

The proof is complete.  In moving weighted average, the weight of the information contained in the data is not the same, and is independent of each other, so to identify the data of each phase is not reasonable. And introducing the non-additive measure into the moving weighted average is of practical significance. Example 4.1. Given a closing stock prices system over 5 days. The closing prices of each day is ˜ 5 , and every x denoted as x ˜i , (˜ x1 , x ˜2 , · · · , x ˜5 ) ∈ E ˜i is a triangle fuzzy number, x ˜i = (xi − δi,1 , xi , xi + δi,2 ), i = 1, 2, · · · , 5. Suppose (t1 , t2 , · · · , t5 ) ∈ R5 , Ai = {ti , ti+1 , · · · , t5 }, i = 1, 2, ..., 5, A6 = ∅. The value and the weight of each x ˜i is shown in Table 1, i = 1, 2, · · · , 5, then we can get the closing stock price over 10 days and some relevant results. Day 1 2 3 4 5

Closing stock price (19,20,21) (21,22,23) (23,24,25) (24,25,26) (22,23,24)

gλ 0.1 0.2 0.3 0.15 0.175

Table 1: The closing stock prices over 5 days. According to Remark 2.3 again, we know that

5 Q

(1 + λgλi ) = 1 + λ, hence we can gain λ = 0.218.

i=1

Then, by Remark 2.3, we(have ) 5 1 Y gλ (A1 ) = 1, gλ (A2 ) = [1 + λgλ ({x})] − 1 = 0.88, λ i=2 ( 5 ) ( 5 ) 1 Y 1 Y gλ (A3 ) = [1 + λgλ ({x})] − 1 = 0.65, gλ (A4 ) = [1 + λgλ ({x})] − 1 = 0.33, λ λ i=3 i=4 gλ (A5 ) = gλ ({x5 }) = 0.175, gλ (A6 ) = 0. By Definition 3.1, we have 5 X x ˜6 = ( ((xi − δi,1 )(gλ (Ai ) − gλ (Ai+1 )), i=1 5 X

5 X

i=1

i=1

xi (gλ (Ai ) − gλ (Ai+1 )),

(xi + δi,2 )(gλ (Ai ) − gλ (Ai+1 ))),

= (22.04, 23.04, 24.04). Similarly, we can also calculate x ˜n , n = 7, 8, 9, 10, with respect to fuzzy measure gλ on A, shown in Table 2. And by Theorem 4.1 and Theorem 4.3, we have ea| lim Pn = | n→∞ a e  0.12  0.12  1  0.12 = 0.12 + 0.35 + 0.67 + 0.825 + 1   0.12 0.12 889

0.35 0.35 0.35 0.35 0.35

0.67 0.67 0.67 0.67 0.67

0.825 0.825 0.825 0.825 0.825

1 1 1 1 1

     

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Day 1 2 3 4 5 6 7 8 9 10

Closing stock price (19,20,21) (21,22,23) (23,24,25) (24,25,26) (22,23,24) (22.04,23.04,24.04) (22.76,23.76,24.76) (22.72,23,72,24.72) (22.5,23.5,24.5) (22.45,23.45,24.45)

gλ 0.1 0.2 0.3 0.15 0.175

Table 2: The closing stock prices over 10 days.    =  

lim x ˜n =

n→∞

where e =

0.04 0.04 0.04 0.04 0.04

0.11 0.11 0.11 0.11 0.11

0.23 0.23 0.23 0.23 0.23

0.28 0.28 0.28 0.28 0.28

0.34 0.34 0.34 0.34 0.34

   ,  

˜1 a| X 1 = (66.84, 69.865, 72.77) = (22.54, 23.56, 24.54), | a e 0.12 + 0.35 + 0.67 + 0.825 + 1

5 P

ek = [1, 1, · · · , 1]| ∈ R5×1 , ek is the ith standard unit column vector,

i=1

a1 = gλ (A1 ) − gλ (A2 ) = 0.12, a2 =

2 P

(gλ (Ai ) − gλ (Ai+1 )) = 0.35, a3 =

i=1

0.67, a4 =

4 P

(gλ (Ai ) − gλ (Ai+1 )) = 0.825, a5 =

i=1

3 P

(gλ (Ai ) − gλ (Ai+1 )) =

i=1 5 P

(gλ (Ai ) − gλ (Ai+1 )) = 0.1.

i=1

Here when n is infinite, the forecasting value of xn will become a stable value (22.54,23.56,24.54) by the weighted moving averages for a series of fuzzy numbers based on non-additive measures with σ − λ rules. 5. Conclusion In this paper, the moving average for a series of fuzzy numbers was proposed by means of non-additive measures with σ − λ rules and fuzzy number. Meanwhile, the special case, i,e. the moving average for a series of triangular fuzzy numbers based on non-additive measures with σ−λ were also discussed. Further, the calculation of the weighted moving averages for fuzzy-number based on a non-additive measure with σ − λ rules was invested and some properties were discussed. Finally, an example was given to illustrate the practical importance of the main results. References [1] S. Asahina, K. Uchino, T. Murofushi, Relationship among continuity conditions and null-additivity conditions in non-additive measure theory. 157(5) (2004) 691-698. [2] H.H. Bauschke, J. Sarada, X. Wang, On moving averages, Journal of Convex Analysis 21 (2014) 219-235. [3] L. Chen, Z.T. Gong, Genetic algorithm optimization for determing fuzzy measures from fuzzy data, Journal of Applied Mathematics 2013(3) (2013) 1-11. [4] G. Choquet. Theory of Capacities[J]. Annual instuitute Fourier, 1954, 5: 131-295. [5] Z.T. Gong, S.X. Hai, Convexity of n-dimensional fuzzy number-valued functions and its applications, Fuzzy Sets and Systems 295 (2016) 2016 19-36. [6] Z.T. Gong, L. Chen, G. Duan, Choquet Integral of Fuzzy-Number-Valued Functions: The Differentiability of the Primitive with respect to Fuzzy Measures and Choquet Integral Equations, Abstract and Applied Analysis 2014(3) (2014) 1-11. 890

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Zeng-Tai Gong, Wen-Jing Lei: The weighted moving averages for a series of fuzzy-numbers based on...

[7] M. Grabisch, New algorithm for identifying fuzzy measures and its application to pattern recognition, In Proceedings of the IEEE International Conference on Fuzzy Systems (IFES .95) Yokohama, Japan (1995) 145-150. [8] J. Li, M. Yasuda, On Egoroff’s theorems on finite monotone non-additive measure spaceFuzzy Sets and Systems 153(1) (2005) 71-78 . [9] M. Ma, On embedding problems of fuzzy number spaces: part 5, Fuzzy Sets Syst 55 (1993) 313õ318. [10] T. Murofushi, M. Sugeno, An interpretation of fuzzy measures and the Choquet integral as an integral with respect to a fuzzy measure, Fuzzy Sets and Systems 29 (1989) 201-227. [11] C.V. Negoita, D.A. Ralescu, Application of Fuzzy Sets to Systems Analysis Wiley, New York, 1975. [12] O. Kaleva, Fuzzy differential equations, Fuzzy Sets Syst 24 (1987) 301õ317. [13] C.D. Meyer, Matrix Analysis and Applied linear Algebra, SIAM, Philadelphia, 2000. [14] P. V, C. Nelson Kennedy Babu, Moving average fuzzy resource scheduling for virtualized cloud data services, Computer Standards and Interfaces 50 (2017) 251õ257. [15] A.M. Ostrowski, Solution of Equation and Systems of Equations, Academic Press, New York and London 1966. [16] A. Saeidifar, E. Pasha, The possibilistic moments of fuzzy numbers and their applications, Journal of Computational and Applied Mathematics 223(2) (2009) 1028-1042. [17] D. Schmeidler, Subjective probability and expected utility without additivity, Econometrica 57(3) (1989) 571õ587. [18] M. Sugeno, Theory of fuzy integral and its application, Doctorial Dissertation, Tokyo Institute of Technology, 1974. [19] C.Q. Tan, X.H. Chen. Intuitionistic fuzzy Choquet integral operator for multi-criteria decision making, Expert Systems with Applications 37 (2010) 149-157. [20] C.P. Tsokos, K-th moving, weighted and exponential moving average for time series forecasting models, European Journal of Pure and Applied Mathematics 3 (2010) 406-416. [21] P.Z. Wang, Fuzzy Set Theory and Application, Shanghai Science and Technology Press, Shanghai, 1983(in Chinese). [22] Y.L. Wang, J.G. Zheng, Knowledge management performance evaluation based on triangular fuzzy number, Procedia Engineering 7 (2010) 38-45. [23] J.Q. Wang, Overview on fuzzy multi-criteria decision-making approach, Control and Decision 23 (2008) 601606. [24] L.A. Zadeh, Fuzzy Sets, Information and Control 8(3) (1965) 338-353. [25] L.A. Zadeh, The concept of a linguistic variable and its application approx- imate reasoning, Information Sciences 8(3) (1975) 199-249.

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A Periodic Observer Based Stabilization Synthesis Approach for LDP Systems based on iteration ∗ Lingling Lv †, Wei He ‡, Zhe Zhang §, Lei Zhang ¶, Xianxing Liu

∥

Abstract The stabilization problem of state observer based for linear discrete-time periodic (LDP) system and its robust consideration are discussed in this paper. It is proved that the periodic controller and the full-dimensional periodic state observer can be designed separately. Based on the well-known CGalgorithm for matrix equation Ax = b as well as applying the lifting technique and algebraic operations, an iterative algorithm for both periodic observer gains and periodic state feedback gains can be generated simultaneously. By optimizing the free parameter matrix in the proposed algorithm, a robust stabilization algorithm based on periodic observer for LDP systems is presented. One numerical example is worked out to illustrate the effect of the proposed approaches. Keywords: Linear discrete-time periodic (LDP) systems; periodic state observers; stabilization; iterative method.

1

Introduction

The controller design requires us to master the state characteristics of the system. However, it is impractical to direct measure all state variables precisely in practical applications. So it requires us to make reliable estimates of the states that cannot be measured directly. The state observer is also called state reconstruction. The basic design idea is to design a state equivalent to the original system and use the designed state equivalent to the original state (see [1]-[2] and references therein). Especially, full-dimensional state observer in the construction idea is based on the original observed coefficient matrix in accordance with the same structure to establish a copy system. The difference between the observed system y and the copy system output yb is taken as a fixed variable and fed back to the input of the integrator group in the copy system to form a closed-loop system (see [3]-[5] and references therein). The design of observer has always been a research hot topic in control theory and control engineering, one can see [6, 7, 8] and references therein for instance. Because of its extensive applications in cyclostationary process, multirate digital control, economics and management, biology, etc., and advantages of improving control performance by using periodic controllers, linear discrete periodic systems have been paid renewed attentions in the control theory community(see [9][11] and the references therein). The stabilization problem of dynamic systems has a fundamental importance in engineering, and hence it is among the most studied problems in modern control theory. Particularly, the ∗ This work is supported by the Programs of National Natural Science Foundation of China (Nos. U1604148, 11501200, 61402149), Innovative Talents of Higher Learning Institutions of Henan (No. 17HASTIT023), China Postdoctoral Science Foundation (No. 2016M592285). † 1. College of Environment and Planning, Henan University, Kaifeng, 475004, P. R. China. 2. Institute of Electric power, North China University of Water Resources and Electric Power, Zhengzhou 450011, P. R. China. Email: lingling [email protected] (Lingling Lv). ‡ Institute of electric power, North China University of Water Resources and Electric Power, Zhengzhou 450011, P. R. China. Email: [email protected] (Wei He). § Institute of electric power, North China University of Water Resources and Electric Power, Zhengzhou 450011, P. R. China. Email: zhe [email protected] (Zhe Zhang) ¶ Institute of Data and Knowledge Engineering, School of Computer and Information Engineering, Henan University, Kaifeng 475004, P.R. China. Email: [email protected] (Lei Zhang). ∥ Computer and Information Engineering College, Henan University, Kaifeng 475004, P. R. China. Email: [email protected] (Xianxing Liu). Corresponding author.

1

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stabilization of periodic motions of dynamic systems has drawn much attention over the past years (see [12][16] and references therein). In [14], LMI based conditions for stabilization via static periodic state feedback as well as via static periodic output feedback are presented, and the problem of quadratic stabilization in the presence of either norm-bounded or polytopic parameter uncertainty is also treated. The output stabilization problem for discrete-time linear periodic systems is solved in [15], where both the state-feedback control law and the state-predictor are based on a suitable time-invariant state-sampled reformulation associated with a periodic system. In addition, utilizing parametric poles assignment algorithm and robust performance index, an algorithm of robust stabilization based on periodic observers is proposed in [16]. In this paper, the problem of stabilization of discrete-time periodic systems based on state observer is transformed into the solution of the corresponding matrix equations, and a neat iterative algorithm is given based on the well-known conjugate gradient algorithm. Initially, we consider the stabilization problem for linear discrete-lime periodic systems without disturbances and give the expected algorithm. On this basis, in case that uncertain disturbances exist in the system parameters, a robust control algorithm for purpose of stabilization is also derived. Notation 1 The superscripts ”T” and ”−1” stand for matrix transposition and matrix inverse, respectively; Rn denotes the n-dimensional Euclidean space; i, j represents the integer set {i, i + 1, . . . , j − 1, j}, tr(A) means the trace of matrix A. Norm ∥A∥ is a Frobenius norm of matrix A. Λ(A) means the eigenvalue set of matrix A and ΨA denotes the monodromy matrix AT −1 AT −2 · · · A0 with period T .

2

Preliminaries

Consider the completely observable and completely reachable LDP systems with the following state space representation { xt+1 = At xt + Bt ut (1) yt = Ct xt where t ∈ Z, the set of integers, xt ∈ Rn , ut ∈ Rr and yt ∈ Rm are respectively the state vector, the input vector and the output vector, At , Bt , Ct are matrices of compatible dimensions satisfying At+T = At , Bt+T = Bt , Ct+T = Ct . In case that the state of system (1) can be measured, by periodic feedback control law ut = −Kt xt + v(t),

Kt+T = Kt ,

Kt ∈ Rr×n

(2)

where vt is the reference input, we can obtain the following combined system with period T { xt+1 = (At − Bt Kt )xt + Bt vt yt = Ct xt

(3)

When there exists some restrictions in practice, the state of system (1) can not be gotten by hardware, but the input ut and the output yt can be measured. In this case, we need build another periodic system which can give an asymptotic estimation of system states. The system with the following form can be adopted: x ˆt+1 = At x ˆt + Bt ut − Lt (Ct x ˆ − yt ) where x ˆ ∈ R and L(t) ∈ R equivalent presentation: n

n×m

(4)

, t ∈ Z are real matrices of period T. Obviously, equation 4 has the following x ˆt+1 = (At − Lt Ct )ˆ xt + Bt ut + Lt yt

(5)

Integrating (4) and (3) gives the following augmented system:  [ ][ ] [ ] [ ] At Bt K t xt+1 xt Bt   = + vt  et − Bt Kt x ˆt+1 x ˆt Bt L[t Ct ]A [ ] xt    yt = Ct 0 x ˆt

(6)

et = At − Lt Ct . where A Then the problem of stabilization based on periodic observer for LDP system (1) can be represented as 2

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.5, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

Problem 1 Given a completely reachable and completely observable LDP system (1), find periodic matrix K(t) ∈ Rr×n , t ∈ 0, T − 1 and L(t) ∈ Rn×m , t ∈ 0, T − 1, such that the augmented system (6) is asymptotically stable. When the system is disturbed by external environment, the closed loop system matrix will deviate from the et , which can be generally expressed as nominal matrix A At − Bt Kt 7→ At + ∆a,t − (Bt + ∆b,t ) Kt , t ∈ 0, T − 1, At + Lt Ct 7→ At + ∆a,t + Lt (Ct + ∆c,t ) , t ∈ 0, T − 1, in which ∆a,t ∈ Rn×n , ∆b,t ∈ Rn×r , ∆c,t ∈ Rm×n , t ∈ 0, T − 1 are random small perturbations. Thus, the problem of robust observer design for linear discrete-time periodic system (1) can be portrayed as Problem 2 Consider the completely observable and completely reachable linear discrete-time periodic system (1), seek the periodic matrix K(t) ∈ Rr×n , t ∈ 0, T − 1 and Lt ∈ Rn×m , t ∈ 0, T − 1, such that the following conditions are met: 1. The augmented system (6) is asymptotically stable; 2. Eigenvalues of the augmented system (6) are as insensitive as possible to small perturbations on systems matrices.

3

Main result

The first thing to consider is the existence condition for a periodic state observer and a periodic state feedback controller. To do this, we would like to give the following theorem firstly. Theorem 1 For a given completely observable and completely reachable LDP system (1), the transfer function of the closed-loop system (6) is equal to the transfer function of the closed-loop system (3). Proof. It is easy to calculate that the transfer function of the closed-loop system (3) is: G(s) = Ct (sI − At − Bt Kt )−1 Bt [

Let Pt = It is easily computed that

Pt−1 =

I −I [

I I

0 I 0 I

(7)

] . ] .

Noticing the coefficient matrices of system (6), we can obtain that ] [ [ A t + Bt K t At Bt Kt −1 P = Pt t et + Bt Kt 0 −Lt Ct A

Bt Kt et A

] ,

[ P [

Ct

] [ ] Bt Bt = , Bt 0 ] [ ] 0 Pt−1 = Ct 0 .

Obviously, system (6) is algebra equivalent to the following system: ] [ ([ ] [ At + Bt Kt Bt Kt Bt , , Ct e 0 0 At

0

) ]

(8)

3

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Since the systems which are algebra equivalent to each other have the same transfer function, we only need to prove that the transfer function of system (8) is as shown in (7). By noticing [ ]−1 [ ] sI − At − Bt Kt Bt Mt (sI − At − Bt Kt )−1 ∗ = (9) et et )−1 0 sI − A 0 (sI − A the transfer function corresponding to (8) can be calculated as [ ]−1 [ ] [ ] sI − At − Bt Kt Bt Kt Bt ¯ Ct 0 G(s) = et 0 0 SI − A = Ct (sI − At − Bt Kt )−1 Bt which is exactly equal to the transfer function of system (6). Thus the proof is accomplished. According to theorem 1, the introduction of periodic state observer has no influence on the desired poles of the closed-loop systems via periodic state feedback. Similarly, the introduction of periodic state feedback has no influence on the designed poles of observer. In this point, the LDP systems keep pace with the linear time invariant systems. Therefore, for the stabilization problem of LDP systems based on periodic observer, the periodic state feedback controller and periodic observer can be designed separately. In the following, poles assignment techniques are adopt to realize the desired purpose. Let Γ1 and Γ2 be the predetermined set of poles of the close-loop system (3) and (5) respectively, which are both symmetric with respect to the real axis. Let F¯jK , F¯jL ∈ Rn×n be the T -periodic matrix satisfying Λ(ΨF¯ K ) = Γ1 and Λ(ΨF¯ L ) = Γ2 , respectively. Clearly, to make system (3) and (5) possess the pole set Γ1 and Γ2 if and only if there exists a T -periodic invertible matrix Xj and Yj such that

and

−1 Xj+1 (Aj − Bj Kj )Xj = −FjK .

(10)

−1 L T T Yj+1 (AT j − Cj Lj )Yj = −Fj .

(11)

where FjK = −F¯jK , FjL = −F¯jL , j ∈ 0, T − 1. Obviously, equations (10) and (11) can be rewritten as the following periodic Sylvester matrices: Aj Xj − Bj Kj Xj = −Xj+1 FjK ,

(12)

T T L AT j Yj − Cj Lj Yj = −Yj+1 Fj ,

(13)

and

Next, an iterative algorithm of stabilization problem based on periodic observer via periodic state feedback is presented firstly, and its correctness will be strictly verified in the subsequence. Algorithm 1 (Periodic CG-based Algorithm of problem 1) 1. Let FjK ∈ Rn×n , FjL ∈ Rn×n , j ∈ 0, T − 1 be a real periodic matrix, which satisfies Λ(ΨFjK ) = Γ1 ∩ ∩ and Λ(ΨFjK ) Λ(ΨAj ) = 0; Λ(ΨFjL ) = Γ2 and Λ(ΨFjL ) Λ(ΨATj ) = 0. Further, let Gj = Kj Xj ∈

m×n Rr×n , Dj = LT are real parametric matrix such that periodic matrix pair (FjK , Gj ) and j Yj ∈ R L (Fj , Dj ) is completely observable.

2. Set tolerance ε; Choose arbitrary initial periodic matrix Xj (0) ∈ Rn×n , Yj (0) ∈ Rn×n , j ∈ 0, T − 1; Calculated as follows: Qj (0) = Bj Gj − Aj Xj (0) − Xj+1 (0)FjK , L Wj (0) = CjT Dj − AT j Yj (0) − Yj+1 (0)Fj ; K T Rj (0) = AT j Qj (0) + Qj−1 (0)(Fj−1 ) ; L Nj (0) = Aj Wj (0) + Wj−1 (0)(Fj−1 )T ;

Pj (0) = −Rj (0); Hj (0) = −Nj (0); t := 0. 4

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3. If

∑T −1 j=0

4. While

∥Rj (t)∥ ≤ ε and

∑T −1 j=0

∑T −1 i=0

∥Nj (t)∥ ≤ ε, stop; else, go to next step.

∑T −1

∥Rj (t)∥ ≥ ε and

i=0

∥Nj (t)∥ ≥ ε, calculate [ ] tr PjT (t)Rj (t)

∑T −1 j=0

αj (t) = ∑T −1

; 2 ∥Aj Pj (t) + Pj+1 (t)Bj ∥ ] ∑T −1 [ T j=0 tr Hj (t)Nj (t) βj (t) = ∑T −1

; T 2

T j=0 Aj Hj (t) + Hj+1 (t)Cj j=0

Xj (t + 1) = Xj (t) + αj (t)Pj (t); Yj (t + 1) = Yj (t) + βj (t)Hj (t); Qj (t + 1) = Bj Gj − Aj Xj (t + 1) − Xj+1 (t + 1)FjK ; Wj (t + 1) = CjT Dj − ATj Yj (t + 1) − Yj+1 (t + 1)FjL ; K T Rj (t + 1) = AT j Qj (t + 1) + Qj−1 (t + 1)(Fj−1 ) ,

Nj (t + 1) = Aj Wj (t + 1) + Wj−1 (t + 1)(FjL )T ; ∑T −1 2 j=0 ∥Rj (t + 1)∥ Pj (t + 1) = −Rj (t + 1) + ∑T −1 Pj (t); 2 j=0 ∥Rj (t)∥ ∑T −1 2 j=0 ∥Nj (t + 1)∥ Hj (t + 1) = −Nj (t + 1) + ∑T −1 Hj (t); 2 j=0 ∥Nj (t)∥ t = t + 1; 5. Let Xj = Xj (t),Yj = Yj (t). The real periodic matrix Kj and Lj can be obtained as Kj

= Gj Xj−1 , j ∈ 0, T − 1,

Lj

=

(Dj Yj−1 )T , j ∈ 0, T − 1.

Remark 1 The main part of the algorithm does not contain nested loops, so the computational complexity of the algorithm is O(n). Next, the convergence and correctness of the algorithm are proved. Lemma 1 For sequences {Rj (k)}, {Pj }(k),{Nj (k)}, {Hj (k)}, j ∈ 0, T − 1, the following relations hold for k ≥ 0: T −1 ∑

[ ] tr RjT (k + 1)Pj (k) = 0,

j=0 T −1 ∑ j=0

∑ k>0

T −1 ∑

[ ] tr NjT (k + 1)Hj (k) = 0,

(14)

j=0 −1 [ ] T∑ 2 tr RjT (k)Pj (k) + ∥Rj (k)∥ = 0,

(∑

j=0

T −1 j=0

∥Rj (k)∥

j=0

∥Pj (k)∥

∑T −1

2

)2

2

< ∞,

∑

(∑

T −1 ∑ j=0

T −1 j=0

∑T −1 j=0

k>0

−1 [ ] T∑ 2 tr NjT (k)Hj (k) + ∥Nj (k)∥ = 0

∥Nj (k)∥

2

)2

2

∥Hj (k)∥

(15)

j=0

0 such that T −1 ∑

2

∥Rj (k)∥ ≥ δ

j=0

for all k ≥ 0. It follows from (23) and (24) that t(k + 1) ≤ t(k) +

k+1 1 ≤ · · · ≤ t(0) + , δ δ 8

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which means

1 δ ≥ . t(k + 1) δt(0) + k + 1

So we have

∞ ∞ ∑ ∑ 1 δ ≥ = ∞. t(k) δt(0) + k + 1

k=1

k=1

However, according to Equation (16) that

∞ ∑ 1 < ∞. t(j) j=1

This gives a contradiction. Thus, there holds T −1 ∑

lim

k→∞

2

∥Rj (k)∥ = 0,

j=0

Similarity, we have lim

T −1 ∑

k→∞

2

∥Rj (k)∥ = 0,

j=0

which indicates that the matrix sequence {Xj (k)}, {Yj (k)}, j ∈ 0, T − 1, generated by Algorithm 1 are convergent to matrices {Xj }, {Yj }, j ∈ 0, T − 1, which are respectively the solutions to the two periodic Sylvester equations (12) and (13). According to the poles assignment theory as previously mentioned, matrix Lj , Kj derived from Algorithm 1 are solutions to Problem 1.

3.1

Minimum norm and robust consideration

In this section, we will consider robust poles assignment problem raised in problem 2. In previous work, we have discussed the sensitivity of the closed-loop LDP systems with respect to parameter uncertainties. Here, we revisit it in the following lemma. Lemma 2 [17] Let Ψ = A(T − 1)A(T − 2) · · · A(0) ∈ Rn×n be diagonalizable and Q ∈ Cn×n be a nonsingular matrix such that Ψ = Q−1 ΛQ ∈ Rn×n , where Λ = diag{λ1 , λ2 , · · · , λn } is the Jordan canonical form of matrix Ψ. For a real scalar ε > 0, ∆i (ε) ∈ Rn×n , i ∈ 0, T − 1, are matrix functions of ε satisfying lim

ε→0+

∆i (ε) = ∆i , ε

where ∆i ∈ Rn×n , i ∈ 0, T − 1 are constant matrices. Then for any eigenvalue λ of matrix Ψ(ε) = (A(T − 1) + ∆T −1 (ε)) (A(T − 2) + ∆T −2 (ε)) · · · (A(0) + ∆0 (ε)) , the following relation holds: min{|λi − λ|} ≤ εnκF (Q) i

(T −1 ∑

) T −1 ∥A(i)∥F

i=0

max{∥∆i ∥F } + O(ε2 ). i

(25)

According to Lemma 2, combining the Algorithm 1, one could take the robust performance index of problem 2 as T −1 T −1 ∑ ∑

T

T −1 T −1

Aj + CjT LT

J(Gj , Dj ) = κF (X0 ) ∥Aj + Bj Kj ∥F + κF (Y0 ) (26) j F j=0

j=0

Based on the above discussion, the algorithm for robust stabilization based on observer design for LDP systems can be presented as follows.

9

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Algorithm 2 (Robust stabilization based on periodic observer) 1. Perform the operations of step 1-4 of Algorithm 1. 2. Based on gradient-based search methods and the index (26), solve the optimization problem Minimize J(Gj , Dj ), opt and denote the optimal decision matrix by Gopt j , Dj , j ∈ 0, T − 1. opt 3. Substituting Gopt into steps 2-4 of algorithm 1 gives optimization matrices Xjopt , Yjopt . j , Dj

4. The robust controller and observer gains can be obtained as ( )T opt −1 Kjopt = Gopt , Lopt = Djopt (Yjopt )−1 , j ∈ 0, T − 1. j (Xj ) j

4

A Numerical Example

Consider LDP system (1) with parameters as follows: [ ] [ ] [ ] 1 2 −1 2 −2 1 A(0) = , A(1) = , A(2) = −2 3 3 −1 1 3 [ ] [ ] [ ] −1 −1 −1 B(0) = , B(1) = , B(2) = 1 1 1 [ ] [ ] [ ] 2 −1 1 C(0) = , C(1) = , C(2) = −1 −1 2 This is a diverging system and it is easy to prove that the system is completely reachable and completely observable. Hence, we can claim that it can be stabilized by a periodic state feedback law based on a full-dimensional state observer. Without loss of generality, let the pole set the system (3) and (5) be Γ1 = {−0.3, 0.3} and Γ2 = {−0.4, 0.4}, respectively. According to algorithm 1, by choosing parameter matrices G and D randomly, we obtain a group of solutions as follows:  [ ]T  rand [ ] rand = −2.5762 −1.4737 = [ 1.7699 −1.8268 ]   L0  K0 [ ]T K rand = −1.9615 −2.3782 ] , = 0.1217 2.0509 Lrand 1   1rand [ [ ]T  rand = −1.1669 −0.8084 K2 = −1.1765 −1.6305 L2 Furthermore, employing the robust stabilization algorithm 2, we obtain a group of solution as follows:  [ ]T  robu [ ] robu  = −0.6456 0.9869 = [ 1.8432 −3.5251 ]  L0  K0 [ ] T K robu = −3.1085 1.4631 ] , Lrobu = 0.3933 1.3929 1   1robu [ [ ]T  robu K2 = −1.1128 −2.4826 = −1.0894 −1.7176 L2 Let discrete reference input v(t) = 0.1 sin( π2 + t) and the initial values of state and the observer state be [ ]T [ ]T x0 = −1 1 ,x ˆ0 = 0 0 . We depict the trajectory of state variable x for the original system (1), state variable x and its estimated state of system (6) under (Krand , Lrand ), state variable x and its estimated state of system (6) under (Krobu , Lrobu ) in Fig.1 respectively, where the red line denote the histories of xL and the green line denote the histories the observed state x ˆ. From the simulation results, we can see the good performance of the controller and observer generated by the proposed algorithm.

5

Conclusion

A stabilizing controller design method for LDP systems based on periodic full-dimensional state observer is introduced in this paper. As similar with linear time variant systems, the periodic state feedback controller 10

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11

x2 of the original system

5 0 −5

0

5

10

15 20 steps

25

30

35

x2 and its estimator by Krobu and Lrobux2 and its estimator by Krand and Lrand

x1 and its estimator by Krobu and Lrobu x1 and its estimator by Krand and Lrand

x1 of the original system

11

x 10

10

400 200 0

0

5

10

15 20 steps

25

30

10 5 0 −5

0

5

10

15

20

25

30

35

20

25

30

35

20

25

30

35

steps 100 0

−200

35

100 50 0 −50

x 10

−100

−200 −400

15

0

5

10

15

20

25

30

35

steps

0

5

10

15 steps

10 0 −10 −20

0

5

10

15 steps

Figure 1: Comparison of state and the observed state under different cases and periodic observer are designed separately, based on the periodic poles assignment technique. An iterative algorithm is presented to generate periodic observer gains and periodic controller gains simultaneously. In addition, robust stabilization problem is also discussed in this paper, and the corresponding algorithm is derived. The effectiveness of the proposed algorithms are shown by simulation results on an illustrate example.

References [1] Wang L Y, Xu G, Yin G. State reconstruction for linear time-invariant systems with binary-valued output observations[J]. Systems & Control Letters, 2008, 57(11): 958-963. [2] Wang L Y, Li C, Yin G G, et al. State observability and observers of linear-time-invariant systems under irregular sampling and sensor limitations[J]. IEEE Transactions on Automatic Control, 2011, 56(11): 2639-2654. [3] S. P. Xu, S. R. Wang, X. H. Su, The Research of Direct Torque Control Based on Full-Dimensional State Observer[J]. Advanced Materials Research, 2012, 433-440:1576-1581. [4] Wen G L, Xu D. Observer-based control for full-state projective synchronization of a general class of chaotic maps in any dimension[J]. Physics Letters A, 2004, 333(5-6): 420-425. [5] Wenli L, Leiting Z, Kan D. Performance analysis of re-adhesion optimization control based on fulldimension state observer[J]. Procedia Engineering, 2011, 23: 531-536.

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[6] Angulo M T, Moreno J A, Fridman L. On functional observers for linear systems with unknown inputs and hosm differentiators[J]. Journal of the Franklin Institute, 2014, 351(4): 1982-1994. [7] Zhang P, Ding S X, Wang G Z, et al. Fault detection of linear discrete-time periodic systems[J]. IEEE Transactions on Automatic Control, 2005, 50(2): 239-244. [8] Tadeo F, Rami M A. Selection of time-after-injection in bone scanning using compartmental observers[C]//World Congress on Engineering. 2010. [9] Arzelier D, Peaucelle D, Farges C, et al. Robust analysis and synthesis of linear polytopic discrete-time periodic systems via LMIs[C]//Decision and Control, 2005 and 2005 European Control Conference. CDC-ECC’05. 44th IEEE Conference on. IEEE, 2005: 5734-5739. [10] Lavaei J, Sojoudi S, Aghdam A G. Pole assignment with improved control performance by means of periodic feedback[J]. IEEE Transactions on Automatic Control, 2010, 55(1): 248-252. [11] Varga A. Robust and minimum norm pole assignment with periodic state feedback[J]. IEEE Transactions on Automatic Control, 2000, 45(5): 1017-1022. [12] Zhou B, Zheng W X, Duan G R. Stability and stabilization of discrete-time periodic linear systems with actuator saturation[J]. Automatica, 2011, 47(8): 1813-1820. [13] Zhou B, Duan G R. Periodic Lyapunov equation based approaches to the stabilization of continuous-time periodic linear systems[J]. IEEE Transactions on Automatic Control, 2012, 57(8): 2139-2146. [14] De Souza C E, Trofino A. An LMI approach to stabilization of linear discrete-time periodic systems[J]. International Journal of Control, 2000, 73(8): 696-703. [15] Colaneri P. Output stabilization via pole placement of discrete-time linear periodic systems[J]. IEEE Transactions on Automatic Control, 1991, 36(6): 739-742. [16] Lv L L, Zhang L. Robust Stabilization Based on Periodic Observers for LDP Systems[J]. Journal of Computational Analysis & Applications, 2016, 20(3): 487-498. [17] Lv L, Duan G, Zhou B. Parametric pole assignment and robust pole assignment for discrete-time linear periodic systems[J]. SIAM Journal on Control and Optimization, 2010, 48(6): 3975-3996.

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SUBORDINATION AND SUPERORDINATION PROPERTIES FOR CERTAIN FAMILY OF INTEGRAL OPERATORS ASSOCIATED WITH MULTIVALENT FUNCTIONS M. K. AOUF, H. M. ZAYED, AND N. E. CHO

Abstract. The object of the present paper is to obtain subordination, superordination and sandwich-type results related to a certain family of integral operators defined on the space of multivalent functions in the open unit disk. Also we point out relevant connections of the results presented here with those obtained in earlier.

Keywords and phrases: p−valent function, differential subordination, superordination, subordination chain, integral operator. 2010 Mathematics Subject Classification: 30C45, 30C50. 1. Introduction Let H = H(U) be the class of functions analytic in U = {z ∈ C : |z| < 1} and H[a, n] be the subclass of H(U) consisting of functions of the form f (z) = a + an z n + an+1 z n+1 + . . . and denote H0 := H[0, 1] and H := H[1, 1]. Let P denote the class of functions P = {h ∈ H[0, 1] : h(z)h0 (z) 6= 0, z ∈ U∗ := U \ {0}} , and A(p) be the class of all functions of the form ∞ X p f (z) = z + ak+p z k+p (p ∈ N = {1, 2, ...}),

(1)

(2)

k=1

which are analytic in U. We note that A(1) = A. For f, g ∈ H(U), the function f (z) is said to be subordinate to g(z) or g(z) is superordinate to f (z), if there exists a function ω(z) analytic in U with ω(0) = 0 and |ω(z)| < 1 (z ∈ U), such that f (z) = g(ω(z)). In such a case we write f (z) ≺ g(z). If g is univalent, then f (z) ≺ g(z) if and only if f (0) = g(0) and f (U) ⊂ g(U) (see [14, 15]). Let φ : C2 × U → C and h (z) be univalent in U. If p (z) is analytic in U and satisfies the first order differential subordination: φ (p (z) , zp0 (z) ; z) ≺ h (z) ,

(3)

then p (z) is a solution of the differential subordination (3). The univalent function q (z) is called a dominant of the solutions of the differential subordination (3) if p (z) ≺ q (z) 1

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for all p (z) satisfying (3). A univalent dominant ˜q that satisfies ˜q ≺ q for all dominants of (3) is called the best dominant. If p (z) and φ (p (z) , zp0 (z) ; z) are univalent in U and if p(z) satisfies the first order differential superordination: h (z) ≺ φ (p (z) , zp0 (z) ; z) ,

(4)

then p (z) is a solution of the differential superordination (4). An analytic function q (z) is called a subordinant of the solutions of the differential superordination (4) if q (z) ≺ p (z) for all p (z) satisfying (4). A univalent subordinant ˜q that satisfies q ≺ ˜q for all subordinants of (4) is called the best subordinant (see [14, 15]). For the functions fi (z) ∈ A(p) (p ∈ N, i = 2, 3, ..., m), h(z) ∈ P and the parameters p,m β, α1 , α2 , ..., αm ∈ C with β 6= 0, we introduce the integral operator Ih;α : A(p) → 1 ,αi ,β A(p) as follows:   β1 m P ! z Z Y m  α1 + p i=2 αi  p,m αi α1 −1 0   . Ih;α1 ,αi ,β [fi ](z) =  (t) h (t)h (t)dt f (5) m i P  z

α1 −pβ+p

αi

i=2

i=2

0

(All powers are principal ones). We note the next special cases of the above defined integral operator: (i) For p = 1, m = 2, α1 = γ, α2 = β and f2 (t) = f (t), we obtain   β1 Zz β+γ Ih;β,γ (f )(z) =  γ f β (t)hγ−1 (t)h0 (t)dt , z 0

where the operator Ih;β,γ was introduced and studied by Cho and Bulboac˘a [6]. (ii) For p = 1, m = 2, α1 = γ, α2 = β, f2 (t) = f (t) and h(t) = t, we obtain   β1 Zz β+γ f β (t)tγ−1 (t)dt , Iβ,γ (f )(z) =  γ z 0

where the operator Iβ,γ was introduced by Miller et al. [16] and studied by Bulboac˘a [3–5]. To prove our results, we need the following definitions and lemmas. Definition 1. [14] Denote by Q the set of all functions q(z) that are analytic and injective on U\E(q) where   E(q) =

ζ ∈ ∂U : lim q(z) = ∞ z→ζ

0

and are such that q (ζ) 6= 0 for ζ ∈ ∂U\E(q). Further, denote by Q(a) the subclass of Q for which q(0) = a.

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3

Definition 2. [14] A function L (z, t) (z ∈ U, t ≥ 0) is said to be a subordination chain (or L¨ owner chain) if L (., t) is analytic and univalent in U for all t ≥ 0, L (z, .) is continuously differentiable on [0, ∞) for all z ∈ U and L (z, s) ≺ L (z, t) for all 0 ≤ s ≤ t. Lemma 1. [17] The function L (z, t) : U × [0, ∞) −→ C of the form L (z, t) = a1 (t) z + a2 (t) z 2 + ... (a1 (t) 6= 0; t ≥ 0) and lim |a1 (t)| = ∞ is a subordination chain if and only if t→∞   ∂L (z, t)   z  ∂z Re > 0 (z ∈ U; t ≥ 0) ,   ∂L (z, t)   ∂t and |L (z, t)| ≤ K0 |a1 (t)| (|z| < r0 < 1; t ≥ 0), for some positive constants K0 and r0 . Lemma 2. [10] Suppose that the function H : C2 → C satisfies the condition Re {H (is; t)} ≤ 0 for all real s and for all t ≤ −n (1 + s2 ) /2, n ∈ N. If the function p(z) = 1 + pn z n + pn+1 z n+1 + ... is analytic in U and Re {H (p(z); zp0 (z))} > 0 (z ∈ U) , then Re {p(z)} > 0 for z ∈ U. Lemma 3. [11] Let κ, γ ∈ C with κ 6= 0 and let h ∈ H(U) with h(0) = c. If Re {κh(z) + γ} > 0 (z ∈ U) , then the solution of the following differential equation: q (z) +

zq0 (z) = h (z) (z ∈ U; q(0) = c) κq(z) + γ

is analytic in U and satisfies Re {κq(z) + γ} > 0 for z ∈ U. Lemma 4. [14] Let p ∈ Q(a) and let q(z) = a + an z n + an+1 z n+1 + ... be analytic in U with q (z) 6= a and n ≥ 1. If q is not subordinate to p, then there exists two points z0 = r0 eiθ ∈ U and ζ0 ∈ ∂U\E(q) such that 0

q(Ur0 ) ⊂ p(U), q(z0 ) = p(ζ0 ) and z0 p (z0 ) = mζ0 q 0 (ζ0 ) (m ≥ n) . Lemma 5. [15] Let q ∈ H[a; 1] and ϕ : C2 → C. Also set ϕ (q (z) , zq 0 (z)) = h (z) . If L (z, t) = ϕ (q (z) , tzq 0 (z)) is a subordination chain and q ∈ H[a, 1] ∩ Q(a), then h (z) ≺ ϕ (q (z) , zq 0 (z)) , implies that q (z) ≺ p (z). Furthermore, if ϕ (q (z) , zq 0 (z)) = h (z) has a univalent solution q ∈ Q(a), then q is the best subordinant.

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Let c ∈ C with Re (c) > 0 and N = N (c) =

|c|

p

1 + 2Re (c) + Im (c) . Re (c)

2N z −1 If R = R(z) = 1−z (c), then the open door function 2 is a univalent function and b = R Rc (z) is defined by   z+b (z ∈ U). Rc (z) = R 1 + bz The function Rc is univalent in U, Rc (0) = c and Rc (U) = R (U) is the complex plane slit along the half lines Re (w) = 0, Im (w) ≥ N and Re (w) = 0, Im (w) ≤ −N.

Lemma 6. (Integral Existence Theorem [12–14]) Let φ, Φ ∈ H with φ(z) 6= 0, Φ(z) 6= 0 for z ∈ U. Let α, β, γ, δ ∈ C with β 6= 0, α + δ = β + γ and Re (α + δ) > 0. If the function g(z) ∈ A and zg 0 (z) zφ0 (z) α + + δ ≺ Rα+δ (z), g(z) φ(z) then   β1 Zz β+γ G(z) =  γ g α (t)φ(t)tδ−1 (t)dt ∈ A, z Φ(z) 0 G(z) z

6= 0 (z ∈ U) and zG0 (z) zΦ0 (z) Re β + +γ G(z) Φ(z) 

 > 0 (z ∈ U) .

(All powers are principal ones). Indeed, Lemma 6 is extended for p-valent functions as follows: Lemma 7. [18] (see also [1]) Let p ∈ N, φ, Φ ∈ H with φ(z) 6= 0, Φ(z) 6= 0 for z ∈ U. Let α, β, γ, δ ∈ C with β 6= 0, pα + δ = pβ + γ and Re (pα + δ) > 0. If the function f (z) ∈ A(p) and   zf 0 (z) zφ0 (z) Ap,α,δ = f (z) ∈ A(p) : α + + δ ≺ Rpα+δ (z) , f (z) φ(z) then  F (z) = 

pβ + γ z γ Φ(z)

Zz

 β1 f α (t)φ(t)tδ−1 dt = z p + ... ∈ A(p),

0

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SUBORDINATION AND SUPERORDINATION PROPERTIES F (z) zp

5

6= 0 (z ∈ U) and zF 0 (z) zΦ0 (z) + +γ Re β F (z) Φ(z) 

 > 0 (z ∈ U) .

(All powers are principal ones). 2. Main results Unless otherwise mentioned,  we assume throughout this paper that h ∈ P, β, α1 , α2 , ..., αm ∈ m P C with β 6= 0 such that Re α1 + p αi > 0 and all powers are principal ones. i=2

Using similar arguments to Lemma 7, we obtain the following lemma. Lemma 8. If fi ∈ Ap,h;α1 ,αi (i = 2, 3, · · · , m), where ( m X zf 0 (z) zh00 (z) fi (z) ∈ A(p) : αi i Ap,h;α1 ,αi = +1+ 0 fi (z) h (z) i=2 ) zh0 (z) m ≺R + (α1 − 1) (z) , P α1 +p αi h(z)

(6)

i=2

I p,m

[fi ](z)

p,m 6= 0 and then Ih;α [fi ](z) ∈ A(p), h;α1 ,αzi ,β p 1 ,αi ,β " #
0 p,m m X z Ih;α [f ](z) i ,α ,β 1 i Re β + α1 + p αi − pβ > 0 (z ∈ U) , p,m [fi ](z) Ih;α 1 ,αi ,β i=2 p,m where Ih;α is the integral operator defined by (5). 1 ,αi ,β

Theorem 1. Let fi , gi ∈ Ap,h;α1 ,αi (i = 2, 3, · · · , m) and   zφ00 (z) > −δ Re 1 + 0 φ (z) ! α  α −1 m  Y gi (z) i h(z) 1 h0 (z); z ∈ U , φ (z) = z p z z i=2

(7)

where δ is given by 2

δ=

2

1 + |a| − |1 − a | 4Re{a}

a = α1 + p

m X

! αi − 1, Re{a} > 0 .

(8)

i=2

Then the subordination condition: α −1 α  m  Y fi (z) i h(z) 1 z h0 (z) ≺ φ (z) p z z i=2

908

(9)

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implies that   p,m  p,m Ih;α [fi ](z) β Ih;α1 ,αi ,β [gi ](z) β 1 ,αi ,β z ≺z zp zp   I p,m [gi ](z) β is the best dominant. and the function z h;α1 ,αzi ,β p 

(10)

Proof. Define the functions Ψ(z) and Φ(z) in U by  p,m   p,m  Ih;α1 ,αi ,β [fi ](z) β Ih;α1 ,αi ,β [gi ](z) β Ψ(z) = z and Φ(z) = z (z ∈ U) . zp zp

(11)

From Lemma 8, it follows that these two functions are well defined. We first show that, if q (z) = 1 +

zΦ00 (z) (z ∈ U) , Φ0 (z)

(12)

then Re {q (z)} > 0 (z ∈ U) . From (5) and the definitions of the functions φ(z) and Φ(z), we obtain ! ! m m X X α1 + p αi φ (z) = zΦ0 (z) + α1 + p αi − 1 Φ (z) . i=2

(13)

i=2

Hence, it follows that zφ00 (z) 1+ 0 = q (z) + φ (z)

0

zq (z) = h(z) (z ∈ U) . m P q (z) + α1 + p αi − 1

(14)

i=2

It follows from (7) and (14) that ( Re h (z) + α1 + p

m X

) αi − 1

> 0 (z ∈ U) .

(15)

i=2

Moreover, by using Lemma 3, we conclude that the differential equation (14) has a solution q (z) ∈ H (U) with h (0) = q (0) = 1. Let v + δ, H (u, v) = u + m P u + α1 + p αi − 1 i=2

where δ is given by (8). From (14) and (15), we obtain Re {H (q(z); zq 0 (z))} > 0 (z ∈ U) . To verify the condition   1 + s2 Re {H (is; t)} ≤ 0 s ∈ R; t ≤ − , (16) 2

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SUBORDINATION AND SUPERORDINATION PROPERTIES

7

we proceed as follows:  Re {H (is; t)} = Re is +

t +δ is + a

 =δ+

tRe{a} Eδ (s) , 2 ≤ − |is + a| 2 |a + is|2

where
Eδ (s) = (Re{a} − 2δ) s2 − 4δ (Ima) s + Re{a} − 2δ |a|2 . (17) 2 For δ given by (8), the coefficient of s in the quadratic expression Eδ (s) given by (17) is positive or equal to zero and Eδ (s) ≥ 0. Thus, we see that Re {H (is; t)} ≤ 0 for all 2 s ∈ R and t ≤ − 1+s . Thus, by using Lemma 2, we conclude that 2 Re {q (z)} > 0 (z ∈ U) , that is, that Φ(z) defined by (11) is convex (univalent) in U. Next, we prove that the subordination condition (9) implies that Ψ (z) ≺ Φ (z) , for Ψ(z) and Φ(z) defined by (11). Without loss of generality, we assume that Φ(z) is analytic, univalent on U and Φ0 (ζ) 6= 0 (|ζ| = 1) . If not, then we replace Ψ(z) and Φ(z) by Ψ(ρz) and Φ(ρz), respectively, with 0 < ρ < 1. These new functions have the desired properties on U, so we can use them in the proof of our result and the result would follow by letting ρ → 1. Consider the function L (z, t) given by    L (z, t) =  1 −

1 α1 + p

m P

αi

  Φ (z) + 

(1 + t) zΦ0 (z) (0 ≤ t < ∞; z ∈ U) . m P α1 + p αi

i=2

(18)

i=2

We note that 



 ∂L (z, t) = 1 + ∂z z=0

t α1 + p

m P

αi

 0  Φ (0) 6= 0 (0 ≤ t < ∞; z ∈ U) . 

i=2

This show that the function L (z, t) = a1 (t) z + a2 (t)z 2 + ..., satisfy the conditions lim |a1 (t)| = ∞ and a1 (t) 6= 0 (0 ≤ t < ∞) . Further, we have t→∞   ∂L (z, t)  (   ) m z  00 X zΦ (z) ∂z Re = Re α1 + p αi − 1 + (1 + t) 1 + 0 >0 ∂L (z, t)   Φ (z)   i=2 ∂t (0 ≤ t < ∞; z ∈ U) ,

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 since Φ (z) is convex and Re α1 + p

m P

 αi − 1 > 0, by using the well-known growth

i=2

and distortion sharp inequalities for convex functions (see [8]), the second inequality of Lemma 1 is satisfied and so L (z, t) is a subordination chain. It follows from the definition of subordination chain that    φ (z) =  1 −

1 α1 + p

m P

αi

  Φ (z) + 

1 α1 + p

i=2

m P

zΦ0 (z) = L (z, 0) αi

i=2

and L (z, 0) ≺ L (z, t) (0 ≤ t < ∞) , which implies that L (ζ, t) ∈ / L (U, 0) = φ (U) (0 ≤ t < ∞; ζ ∈ ∂U) .

(19)

If Ψ(z) is not subordinate to Φ(z), by using Lemma 4, we know that there exist two points z0 ∈ U and ζ0 ∈ ∂U such that Ψ (z0 ) = Φ (ζ0 ) and z0 Ψ0 (z0 ) = (1 + t) ζ0 Φ0 (ζ0 ) (0 ≤ t < ∞) .

(20)

Hence, by using (10), (18), (20) and (8), we have    L (ζ0 , t) =  1 −

1 α1 + p

m P

αi

  Φ (ζ0 ) + 

(1 + t) ζ0 Φ0 (ζ0 ) m P α1 + p αi

i=2

i=2



  =  1 −

1 α1 + p

m P

αi

i=2 αi

= z0

m  Y fi (z0 ) i=2

z0p

  Ψ (z0 ) + 

1 α1 + p

m P

z0 Ψ0 (z0 ) αi

i=2



h(z0 ) z0

α1 −1

h0 (z0 ) ∈ φ (U) .

This contradicts (19). Thus, we deduce that Ψ ≺ Φ. Considering Ψ = Φ, we see that the function Φ is the best dominant. This completes the proof of Theorem 1. We now derive the following superordination result. Theorem 2. Let fi , gi ∈ Ap,h;α1 ,αi (i = 2, 3, · · · , m) and   zφ00 (z) Re 1 + 0 > −δ φ (z)

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SUBORDINATION AND SUPERORDINATION PROPERTIES

φ (z) = z

α −1 α  m  Y gi (z) i h(z) 1 zp

i=2

z

9

! h0 (z); z ∈ U ,

where δ is given by (8). If the function α  α −1 m  Y fi (z) i h(z) 1 z h0 (z) p z z i=2 is univalent in U and z

 I p,m

h;α1 ,αi ,β [fi ](z) zp

φ (z) ≺ z

β

∈ H[0, 1]∩Q. Then the superordination condition

α  α −1 m  Y fi (z) i h(z) 1 zp

i=2

z

h0 (z)

(21)

implies that  p,m   p,m Ih;α [gi ](z) β Ih;α1 ,αi ,β [fi ](z) β 1 ,αi ,β z ≺z zp zp  I p,m β h;α1 ,αi ,β [gi ](z) is the best subordinant. and the function z zp 

(22)

Proof. Suppose that the functions Ψ(z), Φ(z) and q(z) are defined by (11) and (12), respectively. We will use similar method as in the proof of Theorem 1. As in Theorem 1, we have    φ (z) =  1 −

1 α1 + p

m P

  Φ (z) + 

αi

1 α1 + p

i=2

m P

zΦ0 (z) = ϕ (G (z) , zG0 (z)) αi

i=2

and we obtain Re {q (z)} > 0 (z ∈ U) . Next, to obtain the desired result, we show that Φ(z) ≺ Ψ(z). For this, we suppose that the function    L (z, t) =  1 −

1 α1 + p

m P

αi

  Φ (z) + 

t α1 + p

i=2

m P

zΦ0 (z) (0 ≤ t < ∞; z ∈ U) . αi

i=2

We note that  ∂L (z, t) ∂z

z=0

 = 1 −

 1 α1 + p

m P

αi

 0  Φ (0) 6= 0 (0 ≤ t < ∞; z ∈ U) . 

i=2

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This show that the function L (z, t) = a1 (t) z + a2 (t)z 2 + ... satisfy the conditions lim |a1 (t)| = ∞ and a1 (t) 6= 0 (0 ≤ t < ∞) . Further, we have t→∞   ∂L (z, t)  (  )  m z  00 X zΦ (z) ∂z Re >0 = Re α1 + p αi − 1 + t 1 + 0 ∂L (z, t)   Φ (z)   i=2 ∂t (0 ≤ t < ∞; z ∈ U) ,   m P since Φ (z) is convex and Re α1 + p αi − 1 > 0. By using the well-known growth i=2

and distortion sharp inequalities for convex functions (see [8]), the second inequality of Lemma 1 is satisfied and so L (z, t) is a subordination chain. Therefore, by using Lemma 5, we conclude that the superordination condition (21) must imply the superordination given by (22). Moreover, since the differential equation has a univalent solution Φ, it is the best subordinant. This completes the proof of Theorem 2. Combining Theorems 1 and 2, the following sandwich-type results are derived. Theorem 3. Let f, gj ∈ Ap,h;α1 ,αi , (i = 2, 3, · · · , m; j = 1, 2) and   zφ00j (z) Re 1 + 0 > −δ φj (z) φj (z) = z

α −1 α  m  Y gi,j (z) i h(z) 1 i=2

zp

z

! h0 (z); z ∈ U ,

where δ is given by (8). If the function α  α −1 m  Y fi (z) i h(z) 1 z h0 (z) p z z i=2 is univalent in U and z

 I p,m

h;α1 ,αi ,β [fi ](z) zp

φ1 (z) ≺ z

β

∈ H[0, 1] ∩ Q. Then

α  α −1 m  Y fi (z) i h(z) 1 i=2

zp

z

h0 (z) ≺ φ2 (z)

implies that  p,m   p,m   p,m  Ih;α1 ,αi ,β [fi ](z) β Ih;α1 ,αi ,β [gi,2 ](z) β Ih;α1 ,αi ,β [gi,1 ](z) β z ≺z ≺z . zp zp zp

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11

 I p,m   I p,m  [gi,1 ](z) β [gi,2 ](z) β Moreover, the functions z h;α1 ,αiz,βp and z h;α1 ,αiz,βp are, respectively, the best subordinant and the best dominant. We note that the assumption of Theorem 3 that the functions α  α −1  p,m  m  Y Ih;α1 ,αi ,β [fi ](z) β fi (z) i h(z) 1 0 z h (z) and z zp z zp i=2 need to be univalent in U, may be replaced as in the following corollary. Corollary 1. Let f, gj ∈ Ap,h;α1 ,αi , (i = 2, 3, ·, m; j = 1, 2) and   zφ00j (z) > −δ Re 1 + 0 φj (z) φj (z) = z

α  α −1 m  Y gi,j (z) i h(z) 1 i=2

and

zp

z

! h0 (z); z ∈ U

 zΘ00 (z) > −δ Re 1 + 0 Θ (z) ! α  α −1 m  Y fi (z) i h(z) 1 0 Θ (z) = z h (z); z ∈ U , zp z i=2 

(23)

where δ is given by (8). Then φ1 (z) ≺ z

α −1 α  m  Y fi (z) i h(z) 1 i=2

zp

z

h0 (z) ≺ φ2 (z)

implies that  p,m   p,m   p,m  Ih;α1 ,αi ,β [fi ](z) β Ih;α1 ,αi ,β [gi,2 ](z) β Ih;α1 ,αi ,β [gi,1 ](z) β z ≺z ≺z . zp zp zp Proof. To prove Corollary 1, we have to show that condition (23) implies the univalence  I p,m β h;α1 ,αi ,β [fi ](z) of Θ (z) and Ψ(z) = z . Since 0 ≤ δ < 21 , it follows that Θ (z) is close to zp convex function in U (see [9]) and hence Θ (z) is univalent in U. Also, by using the same techniques as in the proof of Theorem 1, we can prove that Ψ is convex (univalent) in U, and so the details may be omitted. Therefore, by applying Theorem 3, we obtain the desired result.

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Remark 1. (i) Putting p = 1, m = 2, α1 = γ, α2 = β and f2 (t) = f (t) in Theorem 1, 2, 3 and Corollary 1, we obtain the results by Cho and Bulboac˘ a [6] and the results by Al-Kharsani et al. [2]; (ii) If we take α1 = 0 in the results mentioned above, then we have those by Aouf et. al [1]. Moreover, putting p = 1, m = 2, α1 = 0, α2 = β and f2 (t) = f (t) in Theorem 1, 2, 3 and Corollary 1, we obtain the results by Cho and Kim [7]. Acknowledgements The third author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2016R1D1A1A09916450). References [1] M. K. Aouf, A. O. Mostafa and H. M. Zayed, Certain family of integral operators associated with multivalent functions preserving subordination and superordination, to appear in Filomat. [2] H. A. Al-Kharsani, N. M. Al-Areefi and J. Sokol, A class of Integral operators preserving subordination and superordination for analytic functions, ISRN Math. Anal., 2012 (2012), 1-17. [3] T. Bulboac˘ a, Integral operators that preserve the subordination, Bull. Korean Math. Soc., 32 (1997), 627–636. [4] T. Bulboac˘ a, On a class of integral operators that preserve the subordination, Pure Math. Appl., 13 (2002), 87–96. [5] T. Bulboac˘ a, A class of superordination-preserving integral operators. Indag. Math. (N. S.), 13 (2002), 301–311. [6] N. E. Cho and T. Bulboac˘ a, Subordination and superordination properties for a class of integral operators, Acta Math. Sin. (Engl. Ser.), 26 (2010), no. 3, 515–522. [7] N. E. Cho and I. H. Kim, A class of Integral operators preserving subordination and superordination, J. Inequal. Appl., 2008 (2008), 1-14. [8] D. J. Hallenbeck and T. H. MacGregor, Linear Problems and Convexity Techniques in Geometric Function Theory, Pitman, London, 1984. [9] W. Kaplan, Close-to-convex schlicht functions, Michigan Math. J., 2 (1952), 169–185. [10] S. S. Miller and P. T. Mocanu, Differential subordinations and univalent functions, Michigan Math. J., 28 (1981), no. 2, 157–172. [11] S. S. Miller and P. T. Mocanu, Univalent solutions of Briot-Bouquet differential equations, J. Differential Equations, 56 (1985), no. 3, 297-309. [12] S. S. Miller and P. T. Mocanu, Integral Operators on Certain Classes of Analytic Functions, Univalent Functions, Fractional Calculus and their Applications, New York: Halstead Press, J Wiley & Sons, 1989: 153–166. [13] S. S. Miller and P. T. Mocanu, Classes of univalent integral operators, J. Math. Anal. Appl., 157 (1991), no. 1, 147–165. [14] S. S. Miller and P. T. Mocanu, Differential Subordinations: Theory and Applications, Series on Monographs and Textbooks in Pure and Applied Mathematics, Vol. 225, Marcel Dekker, New York and Basel, 2000. [15] S. S. Miller and P. T. Mocanu, Subordinants of differential superordinations, Complex Variables Theory Appl., 48 (2003), no. 10, 815–826.

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13

[16] S. S. Miller and P. T. Mocanu and M. O. Reade, Starlike integral operators, Pacific J. Math., 79 (1978), 157-168. [17] Ch. Pommerenke, Univalent Functions, Vandenhoeck and Ruprecht, G¨ottingen, 1975. [18] H. M. Srivastava, M. K. Aouf, A. O. Mostafa and H. M. Zayed, Certain subordination-preserving family of integral operators associated with p−valent functions, Appl. Math. Inf. Sci. 11 (4) (2017) 951-960. (M. K. Aouf) Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt E-mail address: [email protected] (H. M. Zayed) Department of Mathematics, Faculty of Science, Menofia University, Shebin Elkom 32511, Egypt E-mail address: hanaa [email protected] (N. E. Cho) Corresponding Author, Department of Applied Mathematics, Pukyong National University, Busan 48513, Korea E-mail address: [email protected]

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.5, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

ADDITIVE s-FUNCTIONAL INEQUALITIES AND DERIVATIONS ON BANACH ALGEBRAS TAEKSEUNG KIM, YOUNGHUN JO∗ , JUNHA PARK, JAEMIN KIM, CHOONKIL PARK∗ , AND JUNG RYE LEE Abstract. In this paper, we introduce the following new additive s-functional inequalities kf (x − y) + f (y) + f (−x)k ≤ ks (f (x + y) − f (x) − f (y)) k,

(0.1)

kf (x + y) − f (x) − f (y)k ≤ ks (f (x − y) + f (y) + f (−x)) k,

(0.2)

where s is a fixed complex number with |s| < 1, and prove the Hyers-Ulam stability of linear derivations on Banach algebras associated to the additive s-functional inequalities (0.1) and (0.2).

1. Introduction and preliminaries The stability problem of functional equations originated from a question of Ulam [20] concerning the stability of group homomorphisms. Hyers [6] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [1] for additive mappings and by Rassias [15] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Rassias theorem was obtained by G˘avruta [3] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias’ approach. The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [8, 9, 13, 14, 17, 18, 19]). Gil´anyi [4] showed that if f satisfies the functional inequality k2f (x) + 2f (y) − f (x − y)k ≤ kf (x + y)k

(1.1)

then f satisfies the Jordan-von Neumann functional equation 2f (x) + 2f (y) = f (x + y) + f (x − y). See also [16]. Fechner [2] and Gil´anyi [5] proved the Hyers-Ulam stability of the functional inequality (1.1). Park, Cho and Han [12] investigated the Cauchy additive functional inequality kf (x) + f (y) + f (z)k ≤ kf (x + y + z)k and the Cauchy-Jensen additive functional inequality

 

x+y

kf (x) + f (y) + 2f (z)k ≤ 2f +z

2

(1.2)

(1.3)

2010 Mathematics Subject Classification. Primary 39B52, 39B62. Key words and phrases. derivation on Banach algebra; additive s-functional inequality; direct method; HyersUlam stability. ∗ Corresponding authors.

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and proved the Hyers-Ulam stability of the functional inequalities (1.2) and (1.3) in Banach spaces. Park [10, 11] defined additive ρ-functional inequalities and proved the Hyers-Ulam stability of the additive ρ-functional inequalities in Banach spaces and non-Archimedean Banach spaces. This paper is organized as follows: In Section 2, we prove the Hyers-Ulam stability of linear derivations on Banach algebras associated to the additive s-functional inequality (0.1). In Section 3, we prove the Hyers-Ulam stability of linear derivations on Banach algebras associated to the additive s-functional inequality (0.2). Throughout this paper, assume that s is a fixed complex number with |s| < 1. 2. Stability of linear derivations on Banach algebras associated to the functional inequality (0.1) In this section, we prove the Hyers-Ulam stability of linear derivations on Banach algebras associated to the additive s-functional inequality (0.1). Theorem 2.1. Let θ ≥ 0 and p be real numbers with p > 2. Let f : B → B be a mapping satisfying kf (λ(x − y)) + λf (y) + λf (−x)k ≤ ks (f (x + y) − f (x) − f (y)) k + θ (kxkp + kykp ) ,

(2.1)

kf (xy) − xf (y) − yf (x)k ≤ θ (kxkp + kykp )

(2.2)

for all λ ∈ S1 := {µ ∈ C||µ| = 1} and all x, y ∈ B. Then there exists a unique C-linear derivation D : B → B such that 2θ kxkp (2.3) kf (x) − D(x)k ≤ p (2 − 2)(1 − |s|) for all x ∈ B. Proof. Letting x = y = 0 and λ = −1 ∈ S1 in (2.1), we get kf (0)k ≤ ksf (0)k and so we get f (0) = 0. Replacing y by x and letting λ = 1 in (2.1), we get kf (x) + f (−x)k ≤ ks (f (2x) − 2f (x)) k + 2θkxkp

(2.4)

for all x ∈ B. Replacing x by −x and y by x and letting λ = −1 in (2.1), we get kf (2x) − 2f (x)k ≤ ks (f (x) + f (−x)) k + 2θkxkp

(2.5)

for all x ∈ B. From (2.4) and (2.5), we get kf (2x) − 2f (x)k ≤ |s|2 kf (2x) − 2f (x)k + 2(1 + |s|)θkxkp and so kf (2x) − 2f (x)k ≤

2θ kxkp 1 − |s|

(2.6)

for all x ∈ B. So one can obtain that

 x  2θ

kxkp

f (x) − 2f

≤ p 2 2 (1 − |s|)

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and hence

x  x  2 · 2n(1−p) θ

n kxkp

2 f n − 2n+1 f n+1 ≤ p 2 2 2 (1 − |s|) for all x ∈ B. So we get

 x  n−1

X 2 · 2l(1−p) θ kxkp

f (x) − 2n f n ≤ 2 2p (1 − |s|)

(2.7)

l=0

for all x ∈ B. For positive integers n and m with n > m,

 x  n−1 x

X 2 · 2l(1−p) θ

n kxkp ,

2 f n − 2m f m ≤ 2 2 2p (1 − |s|) l=m

which tends to zero as m → ∞. So {2n f ( 2xn )} is a Cauchy sequence for all x ∈ B. Since B is complete, the sequence {2n f ( 2xn )} converges for all x ∈ B. We can define a mapping D : B → B by x D(x) = lim 2n f n (2.8) n→∞ 2 for all x ∈ B. Letting x = 0 in (2.1), we get kf (λx) + λf (−x)k ≤ θkxkp for all λ ∈ S1 and all x ∈ B. By (2.8), we get

   x  n

n λx n

≤ lim 2 θkxkp = 0

+ 2 λf − kD(λx) + λD(−x)k = lim 2 f n→∞ 2n 2n n→∞ 2pn for all x ∈ B and all λ ∈ S1 . Hence D(λx) + λD(−x) = 0

(2.9)

D(x) + D(−x) = 0

(2.10)

D(λx) = λD(x)

(2.11)

S1 .

for all x ∈ B and all λ ∈ Letting λ = 1 in (2.9), we get for all x ∈ B. Hence S1 .

for all x ∈ B and all λ ∈ Let λ = 1 in (2.1). By (2.1), (2.8) and (2.10), we get

  y  x 

n x−y n n

kD(x − y) − D(x) + D(y)k = lim 2 f +2 f n +2 f − n n n→∞ 2 2 2          

x+y x y n n n

+ 2n(1−p) θ(kxkp + kykp ) ≤ lim s 2 f − 2 f − 2 f n→∞ 2n 2n 2n = ks (D(x + y) − D(x) − D(y)) k for all x, y ∈ B. Hence kD(x − y) − D(x) + D(y)k ≤ ks(D(x + y) − D(x) − D(y))k

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for all x, y ∈ B. Replacing y by −y in (2.12), we get kD(x + y) − D(x) − D(y)k ≤ ks(D(x − y) − D(x) + D(y))k

(2.13)

for all x, y ∈ B. It follows from (2.12) and (2.13) that kD(x + y) − D(x) − D(y)k ≤ ks2 (D(x + y) − D(x) − D(y))k for all x, y ∈ B. Since |s| < 1, we get kD(x + y) − D(x) − D(y)k = 0 for all x, y ∈ B. So one can obtain that D is additive. Moreover, by passing to the limit in (2.7) as n → ∞, we get the inequality (2.3). Now let S : B → B be another additive mapping satisfying kf (x) − S(x)k ≤

2p θkxkp 2p − 2

for all x ∈ B.

x  x 

kD(x) − S(x)k = 2l D l − S l 2 2

x

x  x   x 



≤ 2l D l − f l + 2l f l − S l 2 2 2 2 2p 2l+1 × θkxkp , ≤ 2p − 2 2lp which tends to zero as l → ∞. Thus D(x) = S(x) for all x ∈ B. This proves the uniqueness of D. λ Now let µ ∈ C (µ 6= 0) and M an integer greater than 4|λ|. Then | M | < 14 < 1 − 32 = 31 . λ By [7, Theorem 1], there exist three elements µ1 , µ2 , µ3 ∈ S1 such that 3 M = µ1 + µ2 + µ3 . By (2.11),       λ 1 λ M λ M D(λx) = D ·3 x =M ·D ·3 x = D 3 x 3 M 3 M 3 M M M = D(µ1 x + µ2 x + µ3 x) = (D(µ1 x) + D(µ2 x) + D(µ3 x)) 3 3 M M λ = (µ1 + µ2 + µ3 )D(x) = · 3 D(x) 3 3 M = λD(x) for all x ∈ B. Hence D(αx + βy) = D(αx) + D(βy) = αD(x) + βD(y) for all α, β ∈ C(α, β 6= 0) and all x, y ∈ B. And D(0x) = 0 = 0D(x) for all x ∈ B. So the unique additive mapping D : B → B is a C-linear mapping. It follows from (2.2) and (2.8) that

x y y  x 

kD(xy) − xD(y) − yD(x)k = lim 22n f n n − 2n xf n − 2n yf n n→∞ 2 2 2 2 ≤ lim 2n(2−p) θ(kxkp + kykp ) = 0 n→∞

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for all x, y ∈ B. Hence D(xy) = xD(y) + yD(x) for all x, y ∈ B. Hence the mapping D : B → B is a C-linear derivation satisfying (2.3).



Theorem 2.2. Let θ ≥ 0 and p be real numbers with 0 < p < 1. Let f : B → B be a mapping satisfying (2.1) and (2.2). Then there exists a unique C-linear derivation D : B → B such that 2θ kf (x) − D(x)k ≤ kxkp (2.14) p (2 − 2 )(1 − |s|) for all x ∈ B.

θ Proof. It follows from (2.6) that f (x) − 12 f (2x) ≤ 1−|s| kxkp and hence

1

2pn θ p

f (2n x) − 1 f (2n+1 x) ≤

2n

2n (1 − |s|) kxk 2n+1 for all x ∈ B. For positive integers n and m with n > m,

n−1

X

1 1 2pl θ n m

≤

f (2 x) − f (2 x) kxkp , (2.15)

2n 2m 2l (1 − |s|) l=m

{ 21n f (2n x)}

is a Cauchy sequence for all x ∈ B. Since B is which tends to zero as m → ∞. So complete, the sequence { 21n f (2n x)} converges for all x ∈ B. We can define a mapping D : B → B by D(x) = limn→∞ 21n f (2n x) for all x ∈ B. Moreover, by letting m = 0 and passing to the limit in (2.15) as n → ∞, we get (2.14). The rest of the proof is similar to the proof of Theorem 2.1.  3. Stability of linear derivations on Banach algebras associated to the functional inequality (0.2) In this section, we prove the Hyers-Ulam stability of linear derivations on Banach algebras associated to the additive s-functional inequality (0.2). Theorem 3.1. Let θ ≥ 0 and p be real numbers with p > 2. Let f : B → B be a mapping satisfying (2.2) and kf (λ(x + y)) − λf (x) − λf (y)k ≤ ks (f (x − y) + f (y) + f (−x)) k + θ (kxkp + kykp ) for all λ ∈ that

S1

(3.1)

and all x, y ∈ B. Then there exists a unique C-linear derivation D : B → B such kf (x) − D(x)k ≤

(2p

(2 − |s|)θ kxkp − 2)(1 − |s|)

(3.2)

for all x ∈ B. Proof. Letting x = y = 0 and λ = −1 ∈ S1 in (3.1), we get k3f (0)k ≤ k3sf (0)k and so we get f (0) = 0. Letting y = 0 and λ = −1 in (3.1), we get kf (−x) + f (x)k ≤ ks (f (x) + f (−x)) k + θkxkp

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and so kf (−x) + f (x)k ≤

1 θkxkp 1 − |s|

for all x ∈ B. Letting y = x and λ = 1 in (3.1), we get kf (2x) − 2f (x)k ≤ ks (f (x) + f (−x)) k + 2θkxkp |s| 2 − |s| ≤ θkxkp + 2θkxkp = θkxkp 1 − |s| 1 − |s|

(3.3)

for all x ∈ B. So one can obtain that

 x  (2 − |s|)θ

kxkp

f (x) − 2f

≤ p 2 2 (1 − |s|) and hence

x  x  (2 − |s|)2n(1−p) θ

n

kxkp

2 f n − 2n+1 f n+1 ≤ 2 2 2p (1 − |s|) for all x ∈ B. So we get

 x  n−1

X (2 − |s|)2l(1−p) θ

n kxkp

f (x) − 2 f n ≤ 2 2p (1 − |s|)

(3.4)

l=0

for all x ∈ B. For positive integers n and m with n > m,

x  x  n−1

n

X (2 − |s|)2l(1−p) θ kxkp ,

2 f n − 2m f m ≤ 2 2 2p (1 − |s|) l=m

{2n f ( 2xn )}

which tends to zero as m → ∞. So is a Cauchy sequence for all x ∈ B. Since B is x n complete, the sequence {2 f ( 2n )} converges for all x ∈ B. We can define a mapping D : B → B by x D(x) = lim 2n f n (3.5) n→∞ 2 for all x ∈ B. It follows from (3.1) and (3.5) that

  x  y 

n

x + y n n kD(λ(x + y)) − λD(x) − λD(y)k = lim − 2 λf n − 2 λf n 2 f λ n

n→∞ 2 2 2        y

x−y −x n n n(1−p) p p n

+2 f n +2 f θ(kxk + kyk ) ≤ lim s 2 f

+2 n→∞ 2n 2 2n = ks (D(x − y) + D(y) + D(−x)) k for all λ ∈ S1 and all x, y ∈ B. Hence kD(λ(x + y)) − λD(x) − λD(y)k = ks (D(x − y) + D(y) + D(−x)) k

(3.6)

for all λ ∈ S1 and all x, y ∈ B. Letting λ = −1 and x = y = 0 in (3.6), we get k3D(0)k ≤ k3sD(0)k and so D(0) = 0. Replacing x by −x and letting y = −x and λ = −1 in (3.6), we get kD(2x) + 2D(−x)k ≤ ks(D(−x) + D(x))k

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ADDITIVE s-FUNCTIONAL INEQUALITIES AND DERIVATIONS ON BANACH ALGEBRAS

for all x ∈ B. Letting y = −x and λ = 1 in (3.6), we get kD(x) + D(−x)k ≤ ks(D(2x) + 2D(−x))k ≤ |s|2 kD(x) + D(−x)k and so D(−x) = −D(x) for all x ∈ B. Replacing y by −y and letting λ = 1 in (3.6), we get kD(x − y) − D(x) + D(y)k ≤ ks (D(x + y) − D(y) − D(x)) k for all x, y ∈ B. Letting λ = 1 in (3.6), we get kD(x + y) − D(x) − D(y)k ≤ ks (D(x − y) + D(y) + D(−x)) k ≤ |s|2 kD(x + y) − D(x) − D(y)k for all x, y ∈ B. Thus D(x + y) = D(x) + D(y) for all x, y ∈ B. Letting y = 0 in (3.6), we get kD(λx) − λD(x)k ≤ 0 and so D(λx) = λD(x) for all λ ∈ S1 and x ∈ B. The rest of the proof is similar to the proof of Theorem 2.1.



Theorem 3.2. Let θ ≥ 0 and p be real numbers with 0 < p < 1. Let f : B → B be a mapping satisfying (3.1) and (2.2). Then there exists a unique C-linear derivation D : B → B such that kf (x) − D(x)k ≤

(2 − |s|)θ kxkp (2 − 2p )(1 − |s|)

(3.7)

for all x ∈ B. Proof. It follows from (3.3) that



f (x) − 1 f (2x) ≤ (2 − |s|)θ kxkp

2(1 − |s|) 2 and hence

pn

1

f (2n x) − 1 f (2n+1 x) ≤ (2 − |s|)2 θ kxkp

2n

2n+1 (1 − |s|) 2n+1 for all x ∈ B. For positive integers n and m with n > m,

n−1

1

X (2 − |s|)2pl θ

f (2n x) − 1 f (2m x) ≤ kxkp ,

2n

2m 2l+1 (1 − |s|)

(3.8)

l=m

which tends to zero as m → ∞. So { 21n f (2n x)} is a Cauchy sequence for all x ∈ B. Since B is complete, the sequence { 21n f (2n x)} converges for all x ∈ B. We can define a mapping D : B → B by D(x) = limn→∞ 21n f (2n x) for all x ∈ B. Moreover, by letting m = 0 and passing to the limit in (3.8) as n → ∞, we get (3.7). The rest of the proof is similar to the proofs of Theorems 2.1 and 3.1. 

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T. KIM, Y. JO, J. PARK, J. KIM, C. PARK, AND J. R. LEE

Acknowledgments This work was supported by the Seoul Science High School R&E Program in 2017. C. Park was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2017R1D1A1B04032937). References [1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. [2] W. Fechner, Stability of a functional inequalities associated with the Jordan-von Neumann functional equation, Aequationes Math. 71 (2006), 149–161. [3] P. G˘ avruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436. [4] A. Gil´ anyi, Eine zur Parallelogrammgleichung ¨ aquivalente Ungleichung, Aequationes Math. 62 (2001), 303– 309. [5] A. Gil´ anyi, On a problem by K. Nikodem, Math. Inequal. Appl. 5 (2002), 707–710. [6] D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222–224. [7] R.V. Kadison and G. Pedersen, Means and convex combinations of unitary operators, Math. Scand. 57 (1985), 249–266. [8] H. Kim, M. Eshaghi Gordji, A. Javadian and I. Chang, Homomorphisms and derivations on unital C ∗ -algebras related to Cauchy-Jensen functional inequality, J. Math. Inequal. 6 (2012), 557–565. [9] J. Lee, C. Park and D. Shin, An AQCQ-functional equation in matrix normed spaces, Results Math. 27 (2013), 305–318. [10] C. Park, Additive ρ-functional inequalities and equations, J. Math. Inequal. 9 (2015), 17–26. [11] C. Park, Additive ρ-functional inequalities in non-Archimedean normed spaces, J. Math. Inequal. 9 (2015), 397–407. [12] C. Park, Y. Cho and M. Han, Stability of functional inequalities associated with Jordan-von Neumann type additive functional equations, J. Inequal. Appl. 2007, Art. ID 41820 (2007). [13] C. Park, K. Ghasemi, S. G. Ghaleh and S. Jang, Approximate n-Jordan ∗-homomorphisms in C ∗ -algebras, J. Comput. Anal. Appl. 15 (2013), 365-368. [14] C. Park, A. Najati and S. Jang, Fixed points and fuzzy stability of an additive-quadratic functional equation, J. Comput. Anal. Appl. 15 (2013), 452–462. [15] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [16] J. R¨ atz, On inequalities associated with the Jordan-von Neumann functional equation, Aequationes Math. 66 (2003), 191–200. [17] S. Shagholi, M. Eshaghi Gordji and M. Bavand Savadkouhi, Stability of ternary quadratic derivation on ternary Banach algebras, J. Comput. Anal. Appl. 13 (2011), 1097–1105. [18] D. Shin, C. Park and Sh. Farhadabadi, On the superstability of ternary Jordan C ∗ -homomorphisms, J. Comput. Anal. Appl. 16 (2014), 964–973. [19] D. Shin, C. Park and Sh. Farhadabadi, Stability and superstability of J ∗ -homomorphisms and J ∗ -derivations for a generalized Cauchy-Jensen equation, J. Comput. Anal. Appl. 17 (2014), 125–134. [20] S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960. Taekseung Kim, Younghun Jo, Junha Park, Jaemin Kim Mathematics Branch, Seoul Science High School, Seoul 03066, Korea E-mail address: [email protected]; [email protected]; [email protected]; [email protected] Choonkil Park Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea E-mail address: [email protected] Jung Rye Lee Depatment of Mathematics, Daejin University, Kyunggi 11159, Korea E-mail address: [email protected]

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On modulus of convexity of quasi-Banach spaces Shin Min Kang1,2 , Hussain Minhaj Uddin Ahmad Qadri3 , Waqas Nazeer4,∗ and Absar Ul Haq5

1

Department of Mathematics and RINS, Gyeongsang National University, Jinju 52828, Korea e-mail: [email protected] 2

Center for General Education, China Medical University, Taichung 40402, Taiwan 3

4

5

Aitchison College, Lahore 54000, Pakistan e-mail: minhaj [email protected]

Division of Science and Technology, University of Education, Lahore 54000, Pakistan e-mail: [email protected]

Department of Mathematics, University of Management and Technology, Sialkot Campus, Lahore 51410, Pakistan e-mail: [email protected] Abstract The aim of this report is to study modulus of convexity δB of a quasi-Banach space B. We prove that δB is convex, continuous, nondecreasing and for arbitrary uniformly q 2

2

convex quasi-Banach space B, δB () = 1 − C1 1 −  4C . We also prove that a quasiBanach space B is uniformly convex if and only if δB () ≥ 0. Moreover we prove that a non-trivial quasi-Banach space B is uniformly non-square if and only if δB () > 0.

2010 Mathematics Subject Classification: 47H05, 46B20, 46E30 Key words and phrases: modulus of convexity, uniformly convex, uniformly nonsquare, quasi Banach space.

1

Introduction

Many of the geometric constants for Banach spaces have been investigated so far. These constants play an important role in the description of various geometric structures of Banach spaces. In 1899 Jung [10] was the first who introduced a geometric constant for Banach spaces. In 1936 and 1937, Clarkson [4,5] introduced classical modulus of convexity to define a uniformly convex space. A great number of such moduli have been defined and introduced since then. The theory of the geometry of a Banach space has evolved very rapidly over the past fifty years. By contrast the study of a quasi-Banach space has lagged far behind, even though the first research papers in the subject appeared in the early 1940’s [2, 4–6]. There are very sound reasons to develop the understanding of these space, but the absence of one of the fundamental tools of functional analysis, the Hahn-Banach ∗

Corresponding author

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theorem, has proved a very significant stumbling block. However, there has been some progress in the non-convex theory and arguably it has contributed to our appreciation of Banach space theory. A systematic study of a quasi-Banach space only really started in the late 1950’s and early 1960’s with the work of several authors. The efforts of these researchers tended to go in rather separate directions. The subject was given great impetus by the paper of Duren et al. [7] in 1969 which demonstrated both the possibilities for using quasi-Banach spaces in classical function theory and also high-lighted some key problems related to the Hahn-Banach theorem. This opened up many new directions of research. The 1970’s and 1980’s saw a significant increase in activity with a number of authors contributing to the development of a coherent theory. An important breakthrough was the work of Roberts [13, 14] who showed that the Krein-Milman Theorem fails in general quasi-Banach spaces by developing powerful new techniques. Quasi-Banach spaces (H pspace when p < 1) were also used significantly in Alexandrov’s solution of the inner function problem in 1982 [1]. During this period three books on the subject appeared by Turpin [16], Rolewicz [15] (actually an expanded version of a book first published in 1972 and the author, Roberts [14]. In the 1990’s it seems to the author that while more and more analysts find that quasi-Banach spaces have uses in their research, paradoxically the interest in developing a general theory has subsided somewhat. The strictly convex Banach spaces were introduced in 1936 by Clarkson, [4], who also studied the concept of uniform convexity. The uniform convexity of Lp spaces, 1 < p < ∞, was established by Clarkson [4]. The concept of duality map was introduced in 1962 by Beurling and Livingston [3] and was further developed by many others and, De Figueiredo [8]. General properties of the duality map can be found in De Figueiredo [8]. In this paper we aim study modulus of convexity in the setting of quasi Banach spaces.

2

Preliminaries

Throughout this paper SB is a closed unit ball in a quasi Banach space. Definition 2.1. A uniformly convex space is a normed vector space so that, for every 0 <  ≤ 2 there is some δ > 0 so that

for

any two vectors with kxk = 1 and kyk = 1, the

≤ 1 − δ. Intuitively, the center of a line segment condition kx − yk ≥ ε implies that x+y 2 inside the unit ball must lie deep inside the unit ball unless the segment is short. Definition 2.2. A quasi-Banach space B is said to be uniformly non-square if there exists a positive number δ < 2 such that for any x1 , x2 ∈ SB , we have



 

x1 + x2 x1 + x2

,

≤ δ. min C C

Definition 2.3. Let  ∈ [0, 2] and C ≥ 1. For a quasi-Banach space B, the modulus of convexity is a function δB : (0, 2] −→ [0, 1] defined as   kx1 + x2 k kx1 − x2 k δB () = inf 1 − : x1 , x2 ∈ SB ; ≥ . (2.1) 2C C A characteristic or related coefficient of this modulus is δ0 (B) = sup { ∈ [0, 2] : δB () = 0} .

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3

Main results

Lemma 3.1. ([9]) Every convex function f with convex domain in R is continuous. Lemma 3.2. Let B be a quasi-Banach space, and x1 , x2 ∈ SB . Then   kx1 + x2 k kx1 − x2 k ≤ 1 − δB . 2C C

(3.1)

Proof. Let dim(B) < ∞. Let  ∈ [0, 2] and choose u, v ∈ SB such that ku−vk is maximal C ku−vk subject to C = . So, here this is enough to prove that kuk = kvk = 1. The case  = 0 is trivial. ku+vk Assume that  6= 0. Let x∗ ∈ X ∗ satisfying kx∗ k = 1 and x∗ (u + v) = 2C . It ∗ would be suffices to prove that if, say, kvk < 1, then x (v − u) =  and kuk < 1. Indeed an analogous reasoning would then yields, x∗ (u − v) =  and hence  = −, which is a contradiction. ku+vk kw−uk To this end, let A = {w ∈ B : C = }. If w ∈ A ∩ SB , then by maximality of 2C we get ku + wk ku + vk x∗ (u + w) ≤ ≤ ≤ x∗ (u + v). 2C 2C Hence, if we had kvk < 1, then x∗ would attain at v local maximum on A. Consequently, x∗ would norm the vector v − u, that is, x∗ (v − u) = kv−uk = . And also C  1  ku + vk + ku − vk 2C i 1 h ∗ x (u + v) + x∗ (u − v) < 2C = x∗ (v) < 1

kuk
0 we can choose xk , yk ∈ N (u, v) such that (for k = 1, 2) xk + yk ≥ λk

and

δ(u, v, λk) +

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The choice of (xk , yk ) is possible because of the definition of δ(u, v, r) in (3.2) as infimum. Now, for λ ∈ (0, 1) we assume x3 = λx1 + (1 − λ)x2 ,

y3 = λy1 + (1 − λ)y2 .

(3.3)

kx3 k = λkx1 k + (1 − λ)kx2 k because x1 , x2 ∈ SB (0). Similarly, (xk , yk ) ∈ N (u, v) implies that there exist constants such that (for k = 1, 2) xk − yk = αk u, xk − yk = βk v.

(3.4)

From equation (3.3) we have x3 − y3 = λx1 + (1 − λ)x2 − λy1 − (1 − λ)y2 = λ[x1 − y1 ] + (1 − λ)[x2 − y2 ] = λ[α1 u] + (1 − λ)[α2 u] = [λα1 + (1 − λ)α2 ]u. Similarly, x3 − y3 = λx1 + (1 − λ)x2 − λy1 + (1 − λ)y2 = λ[x1 − y1 ] + (1 − λ)[x2 − y2 ] = λ[β1v] + (1 − λ)[β2 v] = [λβ1 + (1 − λ)β2 ]v. Now we have kx3 − y3 k = [λα1 + (1 − λ)α2 ]kuk.

(3.5)

kx3 − y3 k = [λβ1 + (1 − λ)β2 ]kvk.

(3.6)

Similarly, Therefore, using (3.5) and (3.6), generally, we get, kx3 − y3 k = λ1 + (1 − λ)2 .

(3.7)

Now we have kx3 + y3 k 2C λkx1 + y1 k + (1 − λ)kx2 + y2 k =1−   2C  kx2 + y2 k kx1 + y1 k ≤λ 1− + (1 − λ) 1 − 2C 2C = λ[δ(u, v, 1] + (1 − λ)[δ(u, v, 2].

δ(u, v, [λ(1) + (1 − λ)2 ]) ≤ 1 −

Belonging to some N (u, v) since δB (u, v, ) is convex, which shows that δB () is convex. Since δB () is convex, so is continuous by Lemma 3.1. −x2 k +x2 k (b) Let 0 ≤ 1 ≤ 2 ≤ 2 and x1 , x2 ∈ SB satisfying kx1C ≤ 2 and kx12C ≤ 2 −1 1 − δB (2 ). Then letting E = 22 and x = x1 + E(x2 − x1 ) and y = x2 − E(x2 − x1 ) we have x, y ∈ SB and

kx−yk C

≤ 1 .

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Now applying Lemma 3.2 we get δB (1 ) ≤ 1 −

x+y x1 + x2 ≤ 1− ≤ δB (2 ), 2C 2C

which shows that δB () is a non-decreasing function. (c) Fix η ∈ (0, 2] with η < . Let x1 , x2 ∈ B such that kx1 k = kx2 k = 1 and Here, it will be suffices to show that

kx1−x2 k C

= .

δB (η) δB () ≤ . η  Consider

 η u1 = x1 + 1 −   η u2 = x2 + 1 − 

then u1 − u2 = And

thus

η (x1 − x2 ) 

  η  x1 + x2 ,  kx1 + x2 k   η  x1 + x2 ,  kx1 + x2 k =⇒

ku1 − u2 k = C. η

   u1 + u2 x1 + x2 η ηkx1 + x2 k = 1− + , 2 kx1 + x2 k  2    u1 + u2 x1 + x2 1 η ηkx1 + x2 k = − + . 2C kx1 + x2 k C C 2C

This implies that

 

x1 + x2

u + u η ηkx + x k 1 1 2 1 2

kx + x k − 2C = 1 − C − C + 2C 1 2 ku1 + u2 k =1− . 2C Here note that

 

x1 + x2 x1 + x2 1 1

kx + x k − 2C = kx1 + x2 k kx + x k − 2C 1 2 1 2 kx1 + x2 k = 1− . 2C Now we have

x1 +x2 kx1 +x2 k

−

u1 +u2 2C

ku1 − u2 k

  C η ηkx1 + x2 k = − η C 2C   1 kx1 + x2 k = 1−  2C

x +x

1 2 − x1 +x2 2C kx1+x2 k = , kx1 − x2 k

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then

x +x

+u2 k

1 2 − u1 +u2 1 − ku12C δ(η) 2c kx1 +x2 k = = η ku1 − u2 k ku − u2 k

x +x 1

1 2 − x1 +x2 2C kx1 +x2 k = kx1 − x2 k   +x2 k 1 − kx12C 1 kx1 + x2 k = ≤ 1− kx1 − x2 k  2C
1 = δB () .  This completes the proof. Proposition 3.4. Let B be a quasi-Banach space. Then ∀0 ≤ 1 ≤ 2 ≤ 2 we have δB (2 ) − δB (1 ) 1 − δB (1 ) ≤ . 2 − 1 2 − 1

(3.8)

Proof. Let 2 = 2



2 − 1 2 − 1



+ 1



2 − 1 1− 2 − 1



.

Then we have      2 − 1 2 − 1 δB (2 ) = δB 2 + 1 1 − 2 − 1 2 − 1     2 − 1 2 − 1 ≤ δB (2) + δB (1 ) 1 − 2 − 1 2 − 1     2 − 1 2 − 1 ≤ δB (2) + δB (1 ) − δB (1 ) 2 − 1 2 − 1   2 − 1 = [δB (2) − δB (1 )] + δB (1 ). 2 − 1 Now δB (2 ) − δB (1 ) ≤ Hence

This completes the proof.



2 − 1 2 − 1



[1 − δB (1 )] .

δB (2 ) − δB (1 ) 1 − δB (1 ) ≤ . 2 − 1 2 − 1

Theorem 3.5. Let B be a uniformly convex space. Then for every d > 0, ε > 0, and for −x2 k arbitrary vectors, x1 , x2 ∈ B with kx1 k ≤ d, kx2 k ≤ d and kx1C ≥ ε, there exists δ > 0 such that  ε i kx1 + x2 k h ≤ 1−δ d. 2C d

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Proof. For any arbitrary x1 , x2 ∈ B we assume that x2 x1 ε z1 = , z2 = , and set  = . d d d Obviously  > 0. Moreover, with kx1 k ≤ 1 and kx2 k ≤ 1 we have kz1 − z2 k =

ε 1 kx1 − x2 k ≥ = . d d

Now, for uniform convexity, we have ε > 0 and δ=δ d

kz1 + z2 k ≤ 1 − δ(), 2C

which implies that ε kx1 + x2 k ≤ 1−δ , 2dC d

thus This completes the proof.

 ε i kx1 + x2 k h ≤ 1−δ d. 2C d

Theorem 3.6. A quasi-Banach space B is uniformly convex iff δB () ≥ 0. Proof. If X is uniformly convex, then, for given  > 0 there exists δ > 0 such that −x2 k ∀x1 , x2 ∈ B with kx1 k = 1, kx2 k = 1 and kx1 C ≥ 1−

kx1 + x2 k ≥δ 2C

=⇒

δB () > 0.

Conversely, assume that δB () > 0 for every  ∈ (0, 2]. Let fix  ∈ (0, 2] and then take −x2 k x1 , x2 ∈ B with kx1 k = 1, kx2 k = 1 and kx1 C ≥ . Then 0 < δB () ≤ 1 −

kx1 + x2 k . 2C

kx +x k

This implies that 1 − 12C 2 ≤ 1 − δ with δ = δB (), which does not depends upon either x1 or x2 . This completes the proof. Theorem 3.7. For arbitrary uniformly convex quasi-Banach space B, r 1 2 C 2 δB () = 1 − 1− . C 4 −x2 k Proof. Let x1 , x2 ∈ B with kx1 k = 1, kx2 k = 1 and kx1C = . Then using the parallelogram identity,
kx1 + x2 k2 + kx1 − x2 k2 = 2 kx1 k2 + kx2 k2 ,

thus


kx1 + x2 k2 = 2 kx1 k2 + kx2 k2 − kx1 − x2 k2 = 2(12 + 12 ) − kx1 − x2 k2

= 2(2) − (C)2 = 4 − 2 C 2 ,

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hence kx1 + x2 k = thus we have

p 4 − 2 C 2 ,

kx1 + x2 k 1− =1− 2C

√

4 − 2 C 2 , 2C

which implies that r   1 2 C 2 kx1 + x2 k 1− inf 1 − = 1− . 2C C 4 Hence, we get 1 δB () = 1 − C

r

1−

2 C 2 . 4

This completes the proof. Theorem 3.8. A non-trivial quasi-Banach space B is uniformly non-square if and only if δB () > 0. Proof. Let B be uniformly non-square. Set  = 2 − 2δ,  ∈ (0, 2). Then δB () ≥ 1 −

 > 0. 2

Conversely, let there is 0 ∈ (0, 2) such that δB () > 0, that is, δB () ≥ η0 > 0 for some η0 ∈ (0, 1). Let 2 − 2δ =  ∈ [0 , 2). Then δ ∈ (0, 1 − 0 /2] and

δB (2 − 2δ) = δB () ≥ η0 > 0.

This indicates that for any x, y ∈ SB , if kx − yk ≥ 2 − 2δ, C then

kx + yk ≥ η0 . 2C Let δ 0 = min{δ, η0 }. Then of course δ 0 ∈ (0, 1). kx−yk Now we just need to show that either 2C ≤ 1 − δ 0 or 1−

kx+yk 2C

≤ 1 − δ 0 . If

kx + yk ≤ 1 − δ0 , 2C then we are done: Let we consider

kx + yk > 1 − δ0 . 2C

Then kx − yk > 2C(1 − δ 0 ) ≥ 2C(1 − δ).

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By this assumption we get 1−

kx + yk ≥ η0 , 2C

which implies that kx + yk ≤ 1 − η0 ≤ (1 − δ 0 ), 2C which shows that B is uniformly non-square. This completes the proof. Proposition 3.9. Let B be a quasi-Banach space and H be a Hilbert space. Then δB () ≤ δH (),

∀ ∈ [0, 2].

(3.9)

Proof. From Theorem 3.7 we can easily prove the result.

References [1] A. B. Alexandrov, The existence of inner functions in a ball, Mat. Sb. (N.S.), 118(160) (1982), 147–163. [2] T. Aoki, Locally bounded linear topological spaces, Proc. Imp. Acad. Tokyo, 18 (1942), 588–594. [3] A. Beurling and A. E. Livingston, A theorem on duality mappings in Banach spaces, Ark. Mat., 4 (1962), 405–411. [4] J. A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc., 40 (1936), 396– 414. [5] J. A. Clarkson, The von Neumann-Jordan constant for the Lebesgue spaces, Ann. of Math., 38 (1937), 114–115. [6] M. M. Day, The spaces Lp with 0 < p < 1, Bull. Amer. Math. Soc., 46 (1940), 816–823. [7] P. L. Duren, B. W. Romberg, and A. L. Shields, Linear functionals on H p spaces with 0 < p < 1, J. Reine Angew. Math., 238 (1969), 32–60. [8] D. G. de Figueiredo, Topics in Nonlinear Functional Analysis (Vol. 48), University of Maryland, Institute for Fluid Dynamics and Applied Mathematics, 1967. [9] H. M. U. A. Qadri and Q. Mehmood, On moduli and constants of quasi-Banach space, Open J. Math. Sci., (in press) ¨ [10] H. W. E. Jung, Uber die kleinste kugel die eine r¨aumliche figur einschliesst, University of California Libraries, 1899. [11] Y. C. Kwun, Q. Mehmood, W. Nazeer, A. U. Haq and S. M. Kang, Relations between generalized von Neumann-Jordan and James constants for quasi-Banach spaces, J. Inequal. Appl., 2016 (2016), Article ID 171, 10 pages.

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[12] W. Nazeer, Q. Mehmood, S. M. Kang, and A. U. Haq, Generalized von NeumannJordan and James constants for quasi-Banach spaces, J. Comput. Anal. Appl., 25 (2018), 1043–1052. [13] J. W. Roberts, Pathological compact convex sets in lp[0, 1], 0 ≤ p < 1. In The Altgeld Book, University of Illinois Functional Analysis Seminar, 1975-76. [14] J. W. Roberts, A nonlocally convex F -space with the Hahn-Banach approximation property, Banach spaces of analytic functions (Proc. Pelczynski Conf., Kent State Univ., Kent, Ohio, 1976). Lecture Notes in Math. 604, Springer Berlin, 1977, pp. 76– 81. [15] S. Rolewicz, On a certain class of linear metric spaces, Bull. Acad. Polon. Sci., 5 (1957), 471–473. [16] P. Turpin, Convexit´es dans les espaces vectoriels topologiques g´en´eraux, Dissertationes Math. (Rozprawy Mat.), 131 (1976), 221 pages.

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO. 5, 2019

Fixed point theorems for F-contractions on closed ball in partial metric spaces, Muhammad Nazam, Choonkil Park, Aftab Hussain, Muhammad Arshad, and Jung-Rye Lee,………759 Lacunary sequence spaces defined by Euler transform and Orlicz functions, Abdullah Alotaibi, Kuldip Raj, Ali H. Alkhaldi, and S. A. Mohiuddine,……………………………………770 Oscillation analysis for higher order difference equation with non-monotone arguments, Özkan Öcalan and Umut Mutlu Özkan,…………………………………………………………781 On Orthonormal Wavelet Bases, Richard A. Zalik,……………………………………790 Neutrosophic sets applied to mighty filters in BE-algebras, Jung Mi Ko and Sun Shin Ahn,798 Coupled fixed point of firmly nonexpansive mappings by Mann’s iterative processes in Hilbert spaces, Tamer Nabil,………………………………………………………………………807 Dynamics of the zeros of analytic continued the second kind q-Euler polynomial, Cheon Seoung Ryoo,………………………………………………………………………………………822 Remarks on the blow-up for damped Klein-Gordon equations with a gradient nonlinearity, Hongwei Zhang, Jian Dang, and Qingying Hu,……………………………………………831 The γ-fuzzy topological semigroups and γ-fuzzy topological ideals, Cheng-Fu Yang,……838 The Behavior and Closed Form of the Solutions of Some Difference Equations, E. M. Elsayed and Hanan S. Gafel,…………………………………………………………………………849 Convexity and Monotonicity of Certain Maps Involving Hadamard Products and Bochner Integrals for Continuous Fields of Operators, Pattrawut Chansangiam,……………………864 Fibonacci periodicity and Fibonacci frequency, Hee Sik Kim, J. Neggers, and Keum Sook So,……………………………………………………………………………………………874 The weighted moving averages for a series of fuzzy numbers based on non-additive measures with 𝜎𝜎 − 𝜆𝜆 rules, Zeng-Tai Gong and Wen-Jing Lei,……………………………………….882

A Periodic Observer Based Stabilization Synthesis Approach for LDP Systems based on iteration, Lingling Lv, Wei He, Zhe Zhang, Lei Zhang, and Xianxing Liu,………………..892

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO. 5, 2019 (continued) Subordination and superordination properties for certain family of integral operators associated with multivalent functions, M. K. Aouf, H. M. Zayed, and N. E. Cho,……………………904 Additive s-functional inequalities and derivations on Banach algebras, Taekseung Kim, Younghun Jo, Junha Park, Jaemin Kim, Choonkil Park, and Jung Rye Lee,………………917 On modulus of convexity of quasi-Banach spaces, Shin Min Kang, Hussain Minhaj Uddin Ahmad Qadri, Waqas Nazeer, and Absar Ul Haq,………………………………………….925

Volume 27, Number 6 ISSN:1521-1398 PRINT,1572-9206 ONLINE

November 15, 2019

Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.6, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

Asymptotic behavior of equilibrium point for a system of fourth-order rational difference equations Ping Liu 1, Changyou Wang 1, 2*, Yonghong Li 1*, Rui Li 3 1. College of Science, Chongqing University of Posts and Telecommunications, Chongqing, 400065, People’s Republic of China 2. College of Applied Mathematics, Chengdu University of Information Technology, Chengdu, Sichuan 610225, People’s Republic of China 3. College of Automation, Chongqing University of Posts and Telecommunications, Chongqing 400065, People’s Republic of China Abstract Our aim in this paper is to investigate the dynamics of a system of fourth-order rational difference equations xn 1 

xn -3  yn -1 y x ， yn 1  n -3 n -1 ， n  0, 1,  , A  xn -3 yn -1 A  yn -3 xn -1

where the parameter A is arbitrary positive real number and the initial conditions x3 , x2 , x1 , x0 , y3 , y2 , y1 , y0 are arbitrary nonnegative real numbers. By using new iteration

method for the more general nonlinear difference equations and inequality skills, we establish some sufficient conditions which guarantee the existence, unstability and global asymptotic stability of the equilibriums for this nonlinear system. Numerical examples to the difference system are given to verify our theoretical results. Keywords: difference system; equilibrium point; asymptotical stability; unstability

1. Introduction Because of the necessity for some techniques that can be used in mathematical models describing real situations, nonlinear difference equations have been studied in the fields of population biology, economics, probability theory, genetics, psychology etc (see, e.g., [1-4] and the references therein). In recent years, with the dramatically development of *Corresponding authors at: College of Science, Chongqing University of Posts and Telecommunications, Chongqing, 400065, People’s Republic of China Email addresses: [email protected] (C.Y. Wang), [email protected] (Y.H. Li ). 1 947

Ping Liu ET AL 947-961

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.6, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

computer-based computational techniques, difference equations are found to be much appropriate mathematical representations for computer simulation and experiment (see, e.g., [5-8]). However, it is more interesting to investigate the behavior of solutions of a system of higher-order rational difference equations and to discuss the asymptotic stability of their equilibrium points (for example, see [9-19]). Recently, Bajo and Liz [20] described the asymptotic behavior and the stability properties of the solution to the following nonlinear second-order difference equation xn 1 

xn 1 , n  0,1, . a  bxn xn 1

(1.1)

for all values of the real parameters a, b, and any initial condition ( x1 , x0 )  R 2 . In [21], Kurbanli, Cinar, and Yalcinkaya investigated the positive solutions of the following difference equations xn 1 

xn 1 yn 1 , yn 1  , yn xn 1  1 xn yn 1  1

n  0,1,  ,

(1.2)

where ( xk , yk )  [0, ) for k  1, 0 . Moreover, Touafek and Elsayed [22] deal with the periodic nature and the form of the solutions of the following systems of rational difference equations. xn 1 

xn 3 yn  3 , yn 1  , n  0,1,  , 1  xn 3 yn 1 1  yn 3 xn 1

(1.3)

with a nonzero real number’s initial conditions. As an extension of (1.3), Elsayed [23] continuously dealt with the existence of solutions and the periodicity character of the following systems of rational difference equations xn 1 

xn yn 1 yn xn 1 , yn 1  , n  0,1, , y n ( xn yn 1  1) xn ( yn xn 1  1)

(1.4)

where the initial conditions x1 , x0 , y1 and y0 are nonzero real numbers. More recently, Khan and Qureshi [24] study the equilibrium points, local asymptotic stability of equilibrium point, unstability of equilibrium points, global character of equilibrium point, periodicity behavior of positive solutions and rate of convergence of positive solutions of the following systems xn 1 

 xn 1 1 yn 1 , yn 1  , n  0,1, ,    yn yn 1 1   1 xn xn 1

(1.5)

and 2 948

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.6, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

xn 1 

ayn 1 a1 xn 1 , yn 1  , n  0,1, . b  cxn xn 1 b1  c1 yn yn 1

(1.6)

Especially, Yalçınkaya [25] investigated the sufficient condition for the global asymptotic stability of the following systems of difference equations xn 1 

xn  yn 1 y x , yn 1  n n 1 , xn yn 1  1 yn xn 1  1

(1.7)

n  0,1, .

where the initial conditions x1 , x0 , y1 and y0 are nonzero real numbers. Motivated by works [20-25], our aim in this paper is to investigate the dynamics of a system of fourth-order rational difference equations xn 1 

xn -3  yn -1 y x ， yn 1  n -3 n -1 , A  xn -3 yn -1 A  yn -3 xn -1

n  0,1,  ,

(1.8)

where A (0, ) and ( xn , yn )  [0, )  [0, ) for n  3,  2,  1, 0,  . For more related work, one can refer to [26-35] and references therein.

2. Some preliminary results To prove the main results in this paper we first give some definitions and preliminary results [36-38] which are basically used throughout this paper. Lemma 2.1 Let

g : I xk 1  I yl 1  I y

Ix , I y

be some intervals of real numbers and let f : I xk 1  I yl 1  I x ,

be continuously differentiable functions. Then for every set of initial

conditions ( xi , y j )  I x  I y , (i  k ,  k  1, , 0, j  l ,  l  1, , 0) , the following system of difference equations  xn 1  f (xn , xn -1 , , xn -k , yn , yn -1 , , yn -l ),   yn 1  g (xn , xn -1 , , xn -k , yn , yn -1 , , yn -l ),

(2.1)

n  0, 1, 2, ,

 , 

has a unique solution {(xi ,y j )}i  k , j l . Definition 2.1 A point (x , y )  I x  I y is called an equilibrium point of system (2.1) if x  f ( x , x , , x , y , y , , y ) , y  g  x .x , , x , y , y , , y  .

That is, ( xn , yn )  ( x , y ) for n  0 is the solution of difference system (2.1), or equivalently, ( x , y ) is a fixed point of the vector map ( f , g ) . 3 949

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Definition 2.2 Assume that ( x , y ) be an equilibrium point of the system (2.1). Then, we have (i) An equilibrium point ( x , y )

is called locally stable if for every   0 , there exits

  0 such that for any initial conditions ( x i , y i )  I x  I y (i   k , , 0, j  l , , 0) , with



0 ik

xi  x   ,



0 j l

y j  y   , we have xn  x   , yn  y   for any n  0 .

(ii) An equilibrium point ( x , y ) is called attractor if limn xn  x , limn yn  y for any initial conditions ( xi , y i )  I x  I y (i   k , , 0, j  l , , 0) . (iii) An equilibrium point ( x , y ) is called asymptotically stable if it is stable, and ( x , y ) is also attractor. (iv) An equilibrium point ( x , y ) is called unstable if it is not locally stable. Definition 2.3 Let (x, y) be an equilibrium point of the vector map F ( f , xn,, xnk , g, yn,, ynl ) , where f and g are continuously differentiable functions at ( x , y ) . The linearized system of (1.8) about the equilibrium point ( x , y ) is X n 1  F ( X n )  F j  X n , where FJ

is the

Jacobian matrix of the system (1.8) about ( x , y ) and X n  ( xn ,, xnk , yn ,, ynl )T . Definition 2.4 let p, q, s, t be four nonnegative integers such that p  q  n, s  t  m . Splitting x  (x1, x2 ,, xn ) into x  ( xp ,  xq ) and y  ( y1, y2 ,, ym ) into y  ( ys ,  yt ) , where  x  denotes

a vector with  -components of x . We say that the function f ( x1 , x2 ,, xn , y1 , y2 ,, ym ) possesses a mixed monotone property in subsets I xn  I ym of R n  R m if f ([x]p ,[x]q ,[ y]s ,[ y]t ) is monotone non-decreasing in each component of ([ x] p ,[ y ]s ) , and is monotone non-increasing in each component of ( x q ,[ y ]t ) for ( x, y)  I xn  I ym . In particular, if q  0, t  0 , then it is said to be monotone non-decreasing in Ixn  I ym . Lemma 2.2 Assume that X (n  1)  F ( X (n)) , n  0,1,  , is a system of difference equations and X is the equilibrium point of this system i.e., F ( X )  X . Then we have (i) If all eigenvalues of the Jacobian matrix J F about X

lie inside the open unit disk

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  1 , then X is locally asymptotically stable. (ii) If one of eigenvalues of the Jacobian matrix J F about X has norm greater than one, then X is unstable. Lemma 2.3 Assume that X (n  1)  F ( X (n)) , n  0,1,  , is a system of difference equations and X is the equilibrium point of this system, the characteristic polynomial of this system about the equilibrium point X

is P ( )  a0  n  a1 n -1    an -1  an  0 , with the real

coefficients and a0  0 . Then all roots of the polynomial P ( ) lie inside the open unit disk

  1 if and only if  k  0 for k  1, 2, , n ,

(2.2)

where  k is the principal minor of order k of the n  n matrix  a1 a3 a5 a a a 2 4  0  n   0 a1 a3     0 0 0 

 0   0   0 .      an 

3. Main results and their proofs In this section, we shall investigate the qualitative behavior of the system (1.8). Let (x, y)

be an equilibrium point of system (1.8), then the system (1.8) has a unique

equilibrium （0, 0） when 0  A  2 , and the system (1.8) has following three equilibrium points P0  (0, 0) , P1  ( A  2,  A  2) , and P2  ( A  2, A  2) if A  2 . To construct corresponding linearlized form of the nonlinear system (1.8), we consider the transformation ( xn , xn 1 , xn  2 , xn 3 , yn , yn 1 , yn  2 , yn 3 )  ( f , f1 , f 2 , f 3 , g , g1 , g 2 , g 3 ) ,

(3.1)

where f 

xn -3  yn -1 y x , fi  xni 1, g  n -3 n -1 , gi  yni 1, i  1, 2, 3 . A  xn -3 yn -1 A  yn -3 xn -1 5 951

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The Jacobian matrix about the equilibrium point ( x , y ) under the transformation (3.1) is given by 0 1  0  0 FJ ( x , y )   0  0 0   0

where 1 

0 0 1 0  2 0 0  0 0 0 0 0 0 0  1 0 0 0 0 0 0  0 1 0 0 0 0 0 , 3 0 0 0 0 0  4   0 0 0 1 0 0 0 0 0 0 0 1 0 0  0 0 0 0 0 1 0 

A y2 A x2 A x2 A y2 , , , .       2 2 4 ( A  x y )2 ( A  x y )2 ( A  x y )2 ( A  x y )2

Theorem 3.1 If A  1 , then the equilibrium point （0, 0） of the system (1.8) is locally asymptotically stable. Proof: We can easily obtain that the linearlized system of (1.8) about the equilibrium point （0, 0） is

 n 1  D n

(3.2)

where  xn  x   n 1   xn  2    xn 3   ,  n  yn     yn 1  y   n2   yn 3 

1 1   0 0 0 A 0  A 0 0    1 0 0 0 0 0 0 0  0 1 0 0 0 0 0 0    0 0 1 0 0 0 0 0  D  1 1 0  0 0 0 0 0  A A  0 0 0 0 1 0 0 0    0 0 0 0 0 1 0 0  0 0 0 0 0 0 1 0   

The characteristic equation of (3.2) is 1 f ( )  ( 4  ) 2  0 . (3.3) A In view of A  1 , it is clear that all roots of characteristic equation (3.3) lie inside unit disk.

Hence the equilibrium （0, 0） is locally asymptotically stable by Lemma 2.1.

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Theorem 3.2 Let I x , I y be some intervals of real numbers and assume that f : I xk 1  I yl 1  I x and g : I xk 1  I yl 1  I y be continuously differentiable functions satisfying mixed monotone property. If there exits  m0  min{x k , , x0 , y  l , , y0 }  max{x k , , x0 , y  l , , y0 }  M 0 ,   n0  min{x k , , x0 , y  l , , y0 }  max{x k , , x0 , y  l , , y0 }  N 0 ,

(3.4)

such that  m0  f ([m0 ] p ,[M 0 ]q ,[n0 ]s ,[ N 0 ]t )  f ([M 0 ] p ,[m0 ]q ,[ N 0 ]s ,[n0 ]t )  M 0 ,   n0  g ([m0 ] p1 ,[M 0 ]q1 ,[n0 ]s1 ,[ N 0 ]t1 )  g ([M 0 ] p1 ,[m0 ]q1 ,[ N 0 ]s1 ,[n0 ]t1 )  N 0 ,

(3.5)

then there exit (m, M ) [m0 , M 0 ] 2 and (n, N )  [n0 , N 0 ] 2 satisfying  M  f ([M ] p ,[m ]q ,[ N ]s ,[n ]t ), m  f ([m ] p ,[M ]q ,[n ]s ,[ N ]t ),   N  g ([M ] p1 ,[m ]q1 ,[ N ]s1 ,[n ]t1 ), n  g ([m ] p1 ,[M ]q1 ,[n ]s1 ,[ N ]t1 ).

(3.6)

Moreover, if m  M , n  N , then equation (2.1) has a unique equilibrium point ( x , y )  [m0 , M0 ][n0 , N0 ] and every solution of (2.1) converges to ( x , y ) .

Proof. Using m0 , M 0 , n0 and N 0

as two couples of initial iterations, we construct four

sequences {mi }, {M i }, {ni } , and {N i } (i  1, 2, ) from the following equations  mi  f ([mi 1 ] p ,[M i 1 ]q ,[ni 1 ]s ,[ N i 1 ]t ), M i  f ([M i 1 ] p ,[mi 1 ]q ,[ N i 1 ]s ,[ni 1 ]t ),   ni  g ([mi 1 ] p1 ,[M i 1 ]q1 ,[ni 1 ]s1 ,[ N i 1 ]t1 ), N i  g ([M i 1 ] p1 ,[mi 1 ]q1 ,[ N i 1 ]s1 ,[ni 1 ]t1 ).

(3.7)

It is obvious from the mixed monotone property of f and g that the sequences {mi }, {M i } , {ni } and {N i } possess the following monotone property  m0  m1    mi    M i    M 1  M 0 ,   n0  n1    ni    N i    N1  N 0 ,

(3.8)

where i =0, 1, 2, , and mi  xu  M i , ni  yv  N i , for u  (k  1)i  1, v  (l  1)i  1, i  0, 1, 2, .

(3.9)

Set m  lim mi , M  lim M i , n  lim ni , N  lim N i . i 

i 

i 

i 

(3.10)

Then

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m  lim inf xi  lim sup xi  M , n  lim inf yi  lim sup yi  N . i 

i 

i 

i 

(3.11)

By the continuity of f and g , one has  M  f ([M ] p ,[m ]q ,[ N ]s ,[n ]t ), m  f ([m ] p ,[M ]q , [n ]s ,[ N ]t ),   N  g ([M ] p1 ,[m ]q1 , [ N ]s1 ,[n ]t1 ), n  g ([m ] p1 ,[M ]q1 , [n ]s1 ,[ N ]t1 ).

(3.12)

Moreover, if m  M , n  N , then m  M  lim xi  x , n  N  lim yi  y , and then the proof i 

i 

is complete. Theorem 3.3 If 1  A , then the equilibrium point (0, 0)

of the system (1.8) is global

attractor for any condition ( x3 , x2 x1 , x0 , y3 , y2 , y1 , y0 ) (0, ) 8 . Proof: Let ( f , g ) : (0, ) 4  (0, ) 4  (0, )  (0, ) be a function defined by f (xn ,xn -1 ,xn -2 ,xn-3 ,yn ,yn -1 ,yn -2 ,yn -3 )=

xn-3  yn-1 , A  xn-3 yn-1

(3.13)

g (xn ,xn -1 ,xn -2 ,xn-3 ,yn ,yn -1 ,yn -2 ,yn -3 )=

yn-3  xn-1 . A  xn 1 yn-3

(3.14)

and

Set f=

x y yx , , g= A  xy A  xy

(3.15)

we can obtain that fx 

A  y2 A  y2    0, g 0, x ( A  xy ) 2 ( A  xy ) 2

(3.16)

 A  x2 A  x2    0, fy  g 0, x ( A  xy ) 2 ( A  xy ) 2

which implies that f and g possess a mixed monotone property. Let M 0  N 0  max{x3 , x2 x1 , x0 , y3 , y2 , y1 , y0 } and A/ M0  m0  n0  M0 /(1 A) . Thus, we have m0 

m0  N 0 M 0  n0 n  M0 N  m0   M 0 , n0  0  0  N0 . A  m0 N 0 A  M 0 n0 A  n0 M 0 A  N 0 m0

(3.17)

Moreover, from (1.8) and Theorem 3.2, one can derive that there exists m, M  [m0 , M 0 ] ,

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n, N  [n0 , N 0 ] satisfying m

mN nM M n N m . ,n  ,M  ,N  A  mN A  nM A  nM A  Nm

(3.18)

Hence, we have M  N mn0.

According to Lemma 2.2 and Theorem 3.2, If 1  A , the unique equilibrium （0, 0） is not only locally asymptotically stable, but also a global attractor. The proof is complete. Theorem 3.4 If A  1 , then the equilibrium point (0, 0) is unstable. Proof: It is easy to see that there exist roots of characteristic equation (3.3) lie outside unit disk when A  1 . According to Lemma 2.2, the equilibrium point (0, 0) is unstable. Theorem 3.5 The equilibrium points p1  ( A  2,  A  2) , and p2  ( A  2, A  2) of the system (1.8) are locally asymptotically stable when 2  A  3 . And the equilibrium points p1 and p2 of the system (1.8) are unstable when A  3 . Proof: We can easily obtain that the linear equations of the system (1.8) about the equilibrium point p1  ( A  2,  A  2) is

 n 1  D* n , where  xn  x   n 1   xn  2    xn 3   ,  n  yn     yn 1  y   n2   yn 3 

 0 0  1 0 0 1  0 0 D  1 A 0 2  0 0  0 0 0 0 

A 1 1 A 0 0 2 2 0 0 0 0 0 0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

1

0

0

0

0

0

1

   0  0   0  . A  1  2  0   0  0  0

(3.19)

The characteristic equation of (3.18) is f ( )  ( 4 

A 1 2 ) 0. 2

(3.20)

Hence, we have that the equilibrium point p1 of the system (1.8) is locally asymptotically stable when 2  A  3 , and the equilibrium point p1 of the system (1.8) is unstable when 9 955

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A  3 . The stability and unstability of the equilibrium point p2 can be proved similarly.

4. Numerical simulations In this section some numerical examples are given in order to confirm the results of the previous sections and to support our theoretical discussions. These examples represent different types of qualitative behavior of solutions of the system (1.8). As examples, we consider the following difference equations xn 1 

xn -3  yn -1 y x ， yn 1  n -3 n -1 ， n  0, 1,  , 3  xn -3 yn -1 3  yn -3 xn -1

(4.1)

xn 1 

xn -3  yn -1 y x ， yn 1  n -3 n -1 ， n  0, 1,  , 5  xn -3 yn -1 5  yn -3 xn -1

(4.2)

xn 1 

xn -3  yn -1 y x ， yn 1  n -3 n -1 ， n  0, 1,  . 0.5  xn -3 yn -1 0.5  yn -3 xn -1

(4.3)

and

By employing the software package MATLAB7.0, we can solve the numerical solutions of the system (4.1), (4.2) and (4.3) which are shown respectively in Figures 4.1-Figure 4.4. More precisely, it is obvious that the equations (4.1) satisfy the conditions of Theorems 3.1 and Figure 4.1 shows that the solution of the difference system (1.8) is local stability if A  3 and the initial conditions x3  6, x2  4, x1  2, x0  8, y3  1, y2  4 , y1  2 and y0  3 . We can also see that the equations (4.1) satisfy the conditions of Theorems 3.2 and Theorem 3.3, and Figure 4.2 shows that the solution of the difference system (1.8) is globally asymptotically stable where A  3 , n0  m0  0.9, N 0  M 0  0.9 and the initial conditions x3  0.01, x2  0.02, x1  0.01, x0  0.03, y3  0.2, y2  0.4 , y1  0.8 and y0  0.7 . It can

be noticed that the equations (4.2) satisfy the conditions of Theorems 3.1, Theorems 3.2 and Theorem 3.3, and Figure 4.3 shows that the solution of the difference system (1.8) is globally asymptotically stable where A  5 , n0  m0  0.3, N 0  M 0  0.5 and the initial conditions x3  0.2, x2  0.06, x1  0.4, x0  0.08, y3  0.02, y2  0.04 , y1  0.01 and y0  0.1 . It is

clear that the equations (4.3) satisfy the conditions of Theorem 3.4, and Figure 4.4 shows that the solution of the difference system (1.8) is unstable where

A  0.5 , and the initial

conditions x3  1.6, x2  1, x1  1.5, x0  1.8, y3  1, y2  4 , y1  2 and y0  3 . 10 956

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8 7

x(n),A=3 y(n),A=3

6 5

x(n)/y(n)

4 3 2 1 0 -1 -2 -20

0

20

40 n

60

80

100

Figure 4.1. Solutions of (4.1) with A=3 and the initial conditions x3  6, x2  4, x1  2, x0  8, y3  1, y2  4 , y1  2 and y0  3

1 x(n),A=3 M(n),A=3 m(n),A=3 y(n),A=3

0.8 0.6 0.4

x(n),y(n)

0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -20

0

20

40 n

60

80

100

Figure 4.2. Solutions of (4.1) with A  3 , n0  m0 0.9, N0  M0 0.9 and the initial conditions x3  0.01, x2  0.02, x1  0.01, x0  0.03, y3  0.2, y2  0.4 , y1  0.8 and y0  0.7

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0.5 x(n) A=5 M(n) A=5 m(n) A=5 y(n) A=5

0.4 0.3

x(n),y(n)

0.2 0.1 0 -0.1 -0.2 -0.3 -20

0

20

40 n

60

80

100

Figure 4.3. Solutions of (4.2) with A  5 , n0  m0 0.3, N0  M0  0.5 and the initial conditions x3  0.2, x2  0.06, x1  0.4, x0  0.08, y3  0.02, y2  0.04 , y1  0.01 and y0  0.1

x(n)A=0.5 y(n)A=0.5

200

x(n)/y(n)

150

100

50

0

-50 0

50

100 n

150

200

Figure 4.4. Solutions of (4.3) with A  0.5 , and the initial conditions

x3  1.6, x2  1, x1  1.5, x0  1.8, y3  1, y2  4 , y1  2 and y0  3 12 958

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5. Conclusions This paper presents the use of a variational iteration method for systems of nonlinear difference equations. This technique is a powerful tool for solving various difference equations and can also be applied to other nonlinear differential equations in mathematical physics. The numerical simulations show that this method is an effective and convenient one. The variational iteration method provides an efficient method to handle the nonlinear structure. Computations are performed using the software package MATLAB7.0. We have dealt with the problem of global asymptotic stability analysis for a class of nonlinear high order difference equations. The general sufficient conditions have been obtained to ensure the existence, unstability and global asymptotic stability of the equilibrium point for the nonlinear difference equations. These criteria generalize and improve some known results. In particular, some illustrate examples are given to show the effectiveness of the obtained results. In addition, the sufficient conditions that we obtained are very simple, which provide flexibility for the application and analysis of nonlinear difference equation.

Acknowledgements This work is supported Science Fund for Distinguished Young Scholars (cstc2014j cyjjq40004) of China, the National Nature Science Fund (Project nos.11372366 and 61503053) of China, the Natural Science Foundation Project of CQ CSTC (Grant nos. cstc2015jcyjA00034, cstc2015jcyjBX0135 and cstc2015jjA20016) of China.

References [1] E.C. Pielou, Population and Community Ecology, Gordon and Breach, London, 1975. [2] E.P. Popov, Automatic Regulation and Control, Nauka, Moscow, 1966 (in Russian). [3] S. Stević, Behaviour of the positive solutions of the generalized Beddington-Holt equation, PanAm. Math. J. 10 (4) (2000) 77-85. [4] E.M. Elsayed, On the solutions and periodic nature of some systems of difference equations, Int. J. Biomath. 7 (6) (2014), Article ID:1450067. [5] S. Stević, Global stability and asymptotics of some classes of rational difference equations, J. Math. Anal. Appl. 316 (1) (2006) 60-68. [6] S. Stević, Asymptotics of some classes of higher-order difference equations, Discrete Dyn. Nat. Soc. 2007 (2007), Article ID: 56813. [7] D.B. Iricanin, S. Stević, Some systems of nonlinear difference equations of higher order with periodic solutions, Dyn. Contin. Discret. Impuls. Syst. Ser. A-Math Anal. 13 (3) (2006) 499-507. 13 959

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[8] E.M. Elsayed, Behavior and expression of the solutions of some rational difference equations, J. Comput. Anal. Appl. 15 (1) (2013) 73-81. [9] L.X. Hu, W.S. He, H.M. Xia, Global asymptotic behavior of a rational difference equation, Appl. Math. Comput. 218 (15) (2012) 7818-7828. [10] G. Papaschinopoulos, M. Radin, C. J. Schinas, Study of the asymptotic behavior of the solutions of three systems of difference equations of exponential form, Appl. Math. Comput. 218 (9) (2012) 5310-5318. [11] Y. Muroya, E. Ishiwata, N. Guglielmi, Global stability for nonlinear difference equations with variable coefficients, J. Math. Anal. Appl. 334 (1) (2007) 232-247. [12] M. Galewski, A note on the existence of a bounded solution for a nonlinear system of difference equations, J. Differ. Equ. Appl. 16 (1) (2010) 121-124. [13] E.A. Grove, G. Ladas, Periodicities in Nonlinear Difference Equations, Chapman and Hall/CRC Press, Boca Raton, 2004. [14] C.Y. Wang, S. Wang, Z.W. Wang, F. Gong, R. Wang, Asymptotic stability for a class of nonlinear difference equation, Discrete Dyn. Nat. Soc. 2010 (2010), Article ID 791610. [15] C.Y. Wang, Q.H. Shi, S. Wang, Asymptotic behavior of equilibrium point for a family of rational difference equation, Adv. Differ. Equ. 2010 (2010), Article ID 505906. [16] C.Y. Wang, S. Wang, L.R. LI, Q.H. Shi, Asymptotic behavior of equilibrium point for a class of nonlinear difference equation, Adv. Differ. Equ. 2009 (2009), Article ID 214309. [17] C.Y. Wang, S. Wang, W. Wang, Global asymptotic stability of equilibrium point for a family of rational difference equations, Appl. Math. Lett. 24 (5) (2011) 714-718. [18] M.M. El-Dessoky, E.M. Elsayed, E.O. Alzahrani, The form of solutions and periodic nature for some rational difference equations systems, J. Nonlinear Sci. Appl. 9 (10) (2016) 5629-5647. [19] E.M. Elsayed, Solutions of rational difference system of order two, Math. Comput. Model. 55 (2012) 378-384. [20] I. Bajo, E. Liz, Global behaviour of a second-order nonlinear difference equation, J. Differ. Equ. Appl. 17 (10) (2011) 1471-1486. [21] A.S. Kurbanli, C. Cinar, I. Yalcinkaya, On the behavior of positive solutions of the system of rational difference equations xn 1 

xn 1 yn 1 , yn 1  , Math. Comput. yn xn 1  1 xn yn 1  1

Model. 53 (2011) 1261-1267. [22] N. Touafek, E.M. Elsayed, On the solutions of systems of rational difference equations, Math. Comput. Model. 55 (2012) 1987-1997 [23] E.M. Elsayed, Solution for systems of difference equations of rational form of order two, Comput. Appl. Math. 33 (3) (2014) 751-765. [24] A.Q. Khan, M.N. Qureshi, Global dynamics of some systems of rational difference 14 960

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equations, J. Egypt. Math. Soc. 24 (1) (2016) 30-36. [25] I. Yalçınkaya, On the global asymptotic behavior of a system of two nonlinear difference equations, ARS Comb. 95 (2010) 151-159. [26] H. Sedaghat, Reduction of order, periodicity and boundedness in a class of nonlinear, higher order difference equations, Comput. Math. Appl. 66 (11) (2013) 2231-2238. [27] T.H. Thai, V.V. Khuong, Global asymptotic stability of a second-order system of difference equations, Indian J. Pure Appl. Math. 45 (2) (2014) 185-198. [28] A. Khaliq, F. Alzahranib, E.M. Elsayed, Global attractivity of a rational difference equation of order ten, J. Nonlinear Sci. Appl. 9 (6) (2016) 4465-4477. [29] M.M. El-Dessoky, E.M. Elsayed, E.O. Alzahrani, The form of solutions and periodic nature for some rational difference equations systems, J. Nonlinear Sci. Appl. 9 (10) (2016) 5629-5647. [30] M.M. El-Dessoky, On the dynamics of a higher order rational difference equations, J. Egypt. Math. Soc. 25 (1) (2017) 28-36. [31] R. Abo-Zeid, On the oscillation of a third order rational difference equation, J. Egypt. Math. Soc. 23 (1) (2015) 62-66. [32] M. Saleh, N. Alkoumi, Aseel Farhat, On the dynamics of a rational difference equation xn 1 

   xn   xn k , Chaos Solitons Fractals 96 (1) (2017) 76-84. Bxn  Cxn k

[33] A. Khaliq, F. Alzahranib, E. M. Elsayed, Global attractivity of a rational difference equation of order ten, J. Nonlinear Sci. Appl. 9 (6) (2016) 4465-4477. [34] C.Y. Wang, X.J. Fang, R. Li, On the solution for a system of two rational difference equations, J. Comput. Anal. Appl. 20 (1) (2016) 175-186 [35] C.Y. Wang, X.J. Fang, R. Li, On the dynamics of a certain four-order fractional difference equations, J. Comput. Anal. Appl. 22 (5) (2017) 968-976. [36] V.L. Kocic, G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic, Dordrecht, 1993. [37] H. Sedaghat, Nonlinear Difference Equations: Theory with Applications to Social Science Models, Kluwer Academic Publishers, Dordrecht, 2003. [38] E. Camouzis, G. Ladas, Dynamics of Third-order Rational Difference Equations: With Open Problems and Conjectures, Chapman and Hall/HRC, Boca Raton, 2007.

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A version of the Hadamard inequality for Caputo fractional derivatives and related results Shin Min Kang1,2 , Ghulam Farid3, Waqas Nazeer4,∗ and Saira Naqvi5

1

Department of Mathematics and RINS, Gyeongsang National University, Jinju 52828, Korea e-mail: [email protected] 2 3

Department of Mathematics, Comsats Institute of Information Technology, Attock 43600, Pakistan e-mail: [email protected] 4

5

Center for General Education, China Medical University, Taichung 40402, Taiwan

Division of Science and Technology, University of Education, Lahore 54000, Pakistan e-mail: [email protected]

Department of Mathematics, Comsats Institute of Information Technology, Attock 43600, Pakistan e-mail: [email protected] Abstract In this paper we are interested to give the Hadamard inequality for n-times differentiable convex functions via Caputo fractional derivatives. We also find bounds of a difference of this inequality. 2010 Mathematics Subject Classification: 26A51, 26D10, 26D15 Key words and phrases: convex functions, Hadamard inequality, Caputo Fractional derivatives

1

introduction

Fractional calculus was mainly a study kept for the finest minds in mathematics. The history of fractional calculus is as old as the history of differential calculus. It does indeed provide several potentially useful tools for solving differential and integral equations, and various other problems involving special functions of mathematical physics as well as their extensions and generalizations in one and more variables. Fourier, Eulern and Laplace are among those mathematicians who showed a casual interest by fractional calculus and ∗

Corresponding author

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mathematical consequences. A lot of them established definitions by means of their own notion and style. Most renowned of these definitions are the Grunwald-Letnikov and Riemann-Liouville definitions [6–8]. In the following we give the definition of Caputo fractional derivatives [6]. Definition 1.1. Let α > 0 and α ∈ / {1, 2, 3, ...}, n = [α] + 1, f ∈ AC n [a, b], the space of functions having nth derivatives absolutely continuous. The right-sided and left-sided Caputo fractional derivatives of order α are defined as follows: (

C

α Da+ f )(x)

1 = Γ(n − α)

Z

x

α Db− f )(x)

(−1)n = Γ(n − α)

Z

b

a

f (n) (t) dt, x > a (x − t)α−n+1

(1.1)

f (n) (t) dt, x < b. (t − x)α−n+1

(1.2)

and (

C

x

If α = n ∈ {1, 2, 3, ...} and usual derivative f (n) (x) of order n exists, then Caputo fractional α f )(x) coincides with f (n) (x), whereas (C D α f )(x) coincides with f (n) (x) derivative (C Da+ b− with exactness to a constant multiplier (−1)n . In particular we have 0 0 (C Da+ f )(x) = (C Db− f )(x) = f (x)

where n = 1 and α = 0.

Definition 1.2. ([7]) Let f ∈ L[a, b]. Then Riemann-Liouville fractional integrals of f of order α are defined as follows Z x 1 α Ia+ f (x) = (x − t)α−1 f (t)dt, x > a Γ(α) a and Ibα− f (x)

1 = Γ(α)

Z

b

(t − x)α−1 f (t)dt, x < b.

x

In [10], Sarikaya et al. proved following Hadamard-type inequalities for RiemannLiouville fractional integrals: Theorem 1.3. Let f : [a, b] → R be a positive function with 0 ≤ a < b and f ∈ L1 [a, b]. If f is a convex function on [a,b], then the following inequalities for fractional integrals hold f



a+b 2



i 2α−1 Γ(α + 1) h α α + I f (b) f (a) I a+b a+b ( 2 )− ( 2 )+ (b − a)α f (a) + f (b) ≤ 2 ≤

(1.3)

with α > 0.

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Theorem 1.4. Let f : [a, b] → R be a differentiable mapping on (a, b) with a < b. If |f 0 |q is convex on [a, b] for q ≥ 1, then the following inequality for fractional integrals holds

with α > 0.

α−1   2 Γ(α + 1) α a + b α [I f (a)] + I f (b) −f (b − a)α ( a+b )+ )− ( a+b 2 2 2  1 h q
1 1 b−a ≤ (α + 1)|f 0 (a)|q + (α + 3)|f 0 (b)|q q 4(α + 1) 2(α + 2)
1 i + (α + 3)|f 0 (a)|q + (α + 1)|f 0 (b)|q q .

(1.4)

Theorem 1.5. Let f : [a, b] → R be a differentiable mapping on (a, b) with a < b. If |f 0 |q is convex on [a, b] for q > 1, then the following inequality for fractional integral holds α−1   2 Γ(α + 1) α a + b α f (b) + I( a+b )− f (a)] − f (b − a)α [I( a+b )+ 2 2 2 1  1  0  1 !  p 1 b−a |f (a)|q + 3|f 0 (b)|q q 3|f 0 (a)|q + |f 0 (b)|q q ≤ + 4 αp + 1 4 4  1 p 4 b−a ≤ [|f 0 (a)| + |f 0 (b)|], 4 αp + 1 where

1 p

+

1 q

(1.5)

= 1.

In recent days many researchers have focused their attention in establishing inequalities of Hadamard type via utilization of fractional integral operators, (see, [1–5, 9]) and references therein. In this paper we are interested to give versions of inequalities (1.3), (1.4) and (1.5) for n-times differentiable convex functions via Caputo fractional derivatives. In the whole paper C n [a, b] denotes the space of n-times differentiable functions such that f (n) are continuous on [a, b].

2

Hadamard-type inequalities for Caputo fractional derivatives

In this section we give a version of the Hadamard inequality via Caputo fractional derivatives. First we prove the following lemma. Lemma 2.1. Let g : [a, b] → R be a function such that g ∈ C n [a, b], also let g (n) is integrable and symmetric to a+b 2 . Then we have α α (C Da+ g)(b) = (−1)n (C Db− g)(a) 1 α α = [(C Da+ g)(a)]. g)(b) + (−1)n (C Db− 2

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Proof. By symmetricity of g (n) we have g (n) (a+b−x) = g (n)(x), where x ∈ [a, b]. Replacing x with a + b − x in the following integral we have Z b g (n)(x) 1 C α ( Da+ g)(b) = dx Γ(n − α) a (b − x)α−n+1 Z b (n) 1 g (a + b − x) dx = Γ(n − α) a (x − a)α−n+1 Z b g (n)(x) 1 = dx Γ(n − α) a (x − a)α−n+1 α = (−1)n(C Db− g)(a).

Theorem 2.2. Let f : [a, b] → R be a positive function with 0 ≤ a < b and f ∈ C n [a, b]. If f (n) is a convex function on [a, b], then the following inequalities for Caputo fractional derivatives hold   a+b f (n) 2 i n−α−1 2 Γ(n − α + 1) h C α n C α + (−1) ( D ≤ ( D f )(b) f )(a) (2.1) ( a+b ( a+b )+ )− (b − a)n−α 2 2 ≤

f (n) (a) + f (n) (b) . 2

Proof. From convexity of f (n) we have   f (n) (x) + f (n) (y) (n) x + y f ≤ . 2 2 Setting x = 2t a + inequality gives 2f

(n)

(2−t) 2 b, y



a+b 2

=



(2−t) 2 a

≤f

(n)



+ 2t b for t ∈ [0, 1]. Then x, y ∈ [a, b] and above    t 2−t t (n) 2 − t a+ b +f a+ b , 2 2 2 2

multiplying both sides of above inequality with tn−α−1 and integrating over [0, 1] we have  Z 1 (n) a + b 2f tn−α−1 dt 2 0     Z 1 Z 1 t 2−t t n−α−1 (n) n−α−1 (n) 2 − t ≤ t f a+ b dt + t f a + b dt 2 2 2 2 0 0 h i n−α 2 Γ(α) C α = ( D( a+b )+ f )(b) + (−1)n (C D(αa+b )− f )(a) , 2 2 (b − a)n−α from which one can have   (n) a + b f 2 i n−α−1 2 Γ(n − α + 1) h C α n C α + (−1) ( D ≤ ( D f )(b) f )(a) . a+b a+b ( 2 )− ( 2 )+ (b − a)n−α

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(2.2)

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On the other hand convexity of f (n) gives 

   t 2−t t (n) 2 − t f a+ b +f a+ b 2 2 2 2 t (n) 2 − t (n) 2 − t (n) t f (b) + f (a) + f (n) (b), ≤ f (a) + 2 2 2 2 (n)

multiplying both sides of above inequality with tn−α−1 and integrating over [0, 1] we have 1

   Z 1 t 2−t t n−α−1 (n) 2 − t t f a+ b dt + t f a + b dt 2 2 2 2 0 0 iZ 1 h tn−α−1 dt, ≤ f (n) (a) + f (n) (b)

Z

n−α−1 (n)



0

from which one can have i 2n−α−1 Γ(n − α + 1) h C α n C α + (−1) ( D f )(b) f )(a) ( D a+b a+b ( 2 )− ( 2 )+ (b − a)n−α f (n) (a) + f (n) (b) ≤ . 2

(2.3)

Combining inequality (2.2) and inequality (2.3) we get inequality (2.1).

3

Caputo fractional inequalities related to the Hadamard inequality

In this section we give the bounds of a difference of the Hadamard inequality proved in previous section. For our results we use the following lemma. Lemma 3.1. Let f : [a, b] → R be a differentiable mapping on (a, b) with a < b. If f ∈ C n+1 [a, b], then the following equality for Caputo fractional derivatives holds i 2n−α−1 Γ(n − α + 1) h C α n C α + (−1) ( D ( D f )(b) f )(a) a+b a+b ( 2 )− ( 2 )+ (b − a)n−α   a+b − f (n) 2  Z 1  t b−a 2−t tn−α f (n+1) = a+ b dt 4 2 2 0    Z 1 2−t t − tn−α f (n+1) a + b dt . 2 2 0

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(3.1)

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Proof. One can note that  Z 1   b−a t 2−t n−α (n+1) t f a+ b dt 4 2 2 0    b − a n−α 2 2−t t (n) = f a+ b |10 t 4 a−b 2 2   Z 1 2−t t n−α−1 2 (n) − αt f a+ b a−b 2 2 0    b−a 2 (n) a + b = f − 4 b−a 2 # n−α−1 Z a+b  2 2 2α 2 (b − x) f (n) (x)dx − (a − b) b b−a a−b     b−a 2 2n−α+1 Γ(n − α + 1) n C α (n) a + b = − f + (−1) ( D( a+b )− f )(b) . 4 b−a 2 (b − a)n−α+1 2

(3.2)

Similarly  Z 1   b−a t n−α (n+1) 2 − t − a + b dt t f 4 2 2 0     n−α+1 b−a 2 2 Γ(n − α + 1) C α (n) a + b =− f − ( D( a+b )+ f )(a) . 4 b−a 2 (b − a)n−α+1 2

(3.3)

Combining (3.2) and (3.3) one can have (3.1).

Using the above lemma we give the following Caputo fractional Hadamard-type inequality. Theorem 3.2. Let f : [a, b] → R be a differentiable mapping on (a, b) with a < b and f ∈ C n+1 [a, b]. If |f (n+1) |q is convex on [a, b] for q ≥ 1, then the following inequality for Caputo fractional derivatives holds n−α−1 i 2 Γ(n − α + 1) h C α n C α ( D( a+b )+ f )(b) + (−1) ( D( a+b )− f )(a) (b − a)n−α 2 2   a + b −f (n) 2   1 hh q b−a 1 ≤ (n − α + 1) |f (n+1)(a)|q 4(n − α + 1) 2(n − α + 2) i1 h q + (n − α + 3) |f (n+1) (b)|q + (n − α + 3) |f (n+1) (a)|q i1 (n+1) q q + (n − α + 1) |f (b)| .

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.6, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

Proof. From Lemma 3.1 and convexity of |f (n+1) | and for q = 1 we have

n−α−1 i 2 Γ(n − α + 1) h C α C α f )(b) + ( D f )(a) ( D a+b a+b ( 2 )− ( 2 )+ (b − a)n−α   a + b −f (n) 2       Z 1 (n+1) 2 − t 2 − t t t b−a n−α (n+1) t a+ b dt + f a + b dt. ≤ f 4 2 2 2 2 0 i h b−a = |f (n+1)(a)| + |f (n+1) (b)| . 4 (n − α + 1)

For q > 1 using Lemma 3.1 we have

n−α−1 2 Γ(n − α + 1) C α [( D( a+b )+ f )(b) + (−1)n (C D(αa+b )− f )(a)] (b − a)n−α 2 2   a + b −f 2  Z 1  t 2 − t b−a n−α (n+1) a+ b dt ≤ t f 4 2 2 0    Z 1 2−t t + a + b dt . tn−α f (n+1) 2 2 0 Using power mean inequality we get

n−α−1 i 2 Γ(n − α + 1) h c α n C α + (−1) ( D ( D f )(b) f )(a) ( a+b ( a+b )+ )− (b − a)n−α 2 2   a + b −f (n) 2    1 " Z 1   1q p t b−a 1 2 − t q n−α (n+1) ≤ t a+ b dt f 4 n−α+1 2 2 0  Z 1   q1 # t q n−α (n+1) 2 − t + t a + b dt . f 2 2 0 968

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Convexity of |f (n+1) |q gives

n−α−1 i 2 Γ(n − α + 1) h C α n C α f )(a) + (−1) ( D ( D f )(b) )+ )− ( a+b ( a+b (b − a)n−α 2 2   a + b −f (n) 2  1 "Z 1   q1   p 2 − t 1 b−a t q q |f (n+1)(a)| + |f (n+1)(b)| dt tn−α ≤ 4 n−α+1 2 2 0   q1 # Z 1  t (n+1) q q n−α 2 − t (n+1) (a)| + |f (b)| dt |f + t 2 2 0  #1   1 " (n+1) q q q q (n+1) (n+1) p (b)| (a)| (b)| b−a 1 |f |f |f  = + − 4 n−α+1 2(n − α + 2) n − αk + 1 2(n − α + 2) " #1  q q q q (n+1) (n+1) (n+1) (a)| (a)| (b)| |f |f |f , + − + n−α+1 2(n − α + 2) 2(n − α + 2) which after a little computation gives the required result.

Theorem 3.3. Let f : [a, b] → R be a function such that f ∈ C n+1 [a, b], a < b. If |f (n+1) |q is convex on [a, b] for q > 1, then the following inequality for Caputo fractional derivatives holds n−α−1 i 2 Γ(n − α + 1) h C α n C α + (−1) ( D f )(b) f )(a) ( D a+b a+b ( 2 )− ( 2 )+ (b − a)n−α   a + b −f (n) 2  !1 1  q (n+1) q q (n+1) p (a)| + 3|f (b)| b−a 1 |f  ≤ 4 np − αp + 1 4  !1 q 3|f (n+1)(a)|q + |f (n+1)(b)|q  + 4 b−a ≤ 4

with

1 p

+

1 q



4 3(np − αp + 1)

1

p

(3.4)

[|f (n+1)(a)| + |f (n+1)(b)|],

= 1.

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Proof. From Lemma 3.1 we have n−α−1 i 2 Γ(n − α + 1) h C α n C α f )(b) + (−1) ( D ( D f )(a) ( a+b ( a+b )+ )− (b − a)n−α 2 2   a + b −f (n) 2   Z 1 t 2 − t b−a n−α (n+1) a+ b dt t ≤ f 4 2 2 0    Z 1 t 2−t + tn−α f (n+1) a + b dt . 2 2 0

From H¨older’s inequality we get

n−α−1 i 2 Γ(n − α + 1) h C α n C α + (−1) ( D ( D f )(b) f )(a) a+b a+b ( 2 )− ( 2 )+ (b − a)n−α   a + b −f (n) 2 "Z  p1 Z 1    q1 1 (n+1) t b−a 2 − t q np−αp dt ≤ a+ b dt t f 4 2 2 0 0   p1 Z 1 Z 1  q  q1 # (n+1) 2 − t t np−αp dt f dt + a + b . t 2 2 0 0 Convexity of |f (n+1) |q gives n−α−1 i 2 Γ(n − α + 1) h C α n C α + (−1) ( D ( D f )(b) f )(a) ( a+b ( a+b )+ )− (b − a)n−α 2 2   a + b −f (n) 2  1 " Z 1    q1  p 2 − t (n+1) b−a 1 t (n+1) q q (a)| + (b)| dt ≤ |f |f 4 np − αp + 1 2 2 0   q1 # Z 1  t (n+1) 2 − t (n+1) q q (a)| + |f (b)| dt + |f 2 2 0  #1  1 " (n+1)  q + 3|f (n+1) (b)|q q p (a)| b−a 1 |f  = 4 np − αp + 1 4 " #1 q (n+1) q (n+1) (a)| + |f (b)| q  3|f + . 4

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Shin Min Kang ET AL 962-972

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For second inequality of (3.4) we use Minkowski’s inequality as follows n−α−1 i 2 Γ(n − α + 1) h C α n C α f )(b) + (−1) ( D ( D f )(a) a+b a+b ( 2 )− ( 2 )+ (b − a)n−α   a + b −f (n) 2   1 h i1 p 4 b−a q |f (n+1) (a)|q + 3|f (n+1) (b)|q ≤ 16 np − αp + 1 i1  h q + 3|f (n+1) (a)|q + |f (n+1) (b)|q 1

b−a ≤ 16



4 np − αp + 1

b−a 4



4 3(np − αp + 1)

≤

p

1

(3 q + 1)(|f (n+1)(a)| + |f (n+1) (b)|) 1

p

(|f (n+1)(a)| + |f (n+1) (b)|).

Acknowledgement The research work of author Ghulam Farid is supported by Higher Education Commission of Pakistan under NRPU 2016, Project No. 5421.

References [1] G. Abbas and G. Farid, Some integral inequalities for m-convex functions via generalized fractional integral operator containing generalized Mittag-Leffler function, Cogent Math., 3 (2016), Article ID 1269589, 12 pages. [2] G. Abbas, K. A. Khan, G. Farid and A. Ur Rehman, Generalizations of some fractional integral inequalities via generalized Mittag-Leffler function, J. Inequal. Appl., 2017 2017, Paper No. 121, 10 pages. [3] G. Farid, A. Ur Rehman and B. Tariq, On Hadamard-type inequalities for m-convex functions via Riemann-Liouville fractional integrals, Stud. Univ. Babe¸s-Bolyai Math., 62 (2017), 141–150. [4] G. Farid, A. Ur Rehman and M. Zahara, On Hadamard inequalities for k-fractional integrals, Nonlinear Funct. Anal. Appl., 21 (2016), 463–478. [5] G. Farid, U. N. Katugampola and M. Usman, Ostrowski type fractional integral inequalities for S-Godunova-Levin functions via Katugampola fractional integrals, Open J. Math. Sci., 1 (2017), 97–110. [6] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, Amsterdam, 2006.

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[7] A. Loverro, Fractional Calculus: History, Definitions and Applications for the Engineer, Department of Aerospace and Mechanical Engineering, University of Notre Dame, 2004. [8] S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, John Wiley and Sons, Inc., New York, 1993. [9] M. Z. Sarikaya, E. Set, H. Yaldiz and N. Basak, Hermite-Hadamard’s inequalities for fractional integrals and related fractional inequalities, Math. Comput. Model., 57 (2013), 2403–2407. [10] M. Z. Sarikaya and H. Yildirim, On Hermite-Hadamard type inequalities for RiemannLiouville fractional integrals, Miskolc Math. Notes, 17 (2016), 1049–1059.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.6, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

A hesitant fuzzy ordered information system Haidong Zhang1∗, Yanping He2 1. School of Mathematics and Computer Science, Northwest MinZu University Lanzhou, Gansu, 730030, P. R. China 2. School of Electrical Engineering, Northwest MinZu University Lanzhou, Gansu, 730030, P. R. China

Abstract Hesitant fuzzy information systems are generalized types of traditional information systems. First, a dominance relation is defined by the score function of hesitant fuzzy value in hesitant fuzzy information systems. By introducing the dominance relation to hesitant fuzzy information systems, we then establish a dominance-based rough set model by replacing the indiscernibility relation in classic rough set theory with the dominance relation, and develop a ranking approach for all objects based on dominance classes. Furthermore, to simplify the knowledge representation, we provide an attribute reduction approach to eliminate the redundant information. And an example is provided to illustrate the validity of this approach. Key words: Dominance relation; Dominance-based rough set; Hesitant fuzzy information systems; Reduction

1

Introduction

As a mathematical approach to handle imprecision, vagueness and uncertainty in data analysis, rough set theory introduced by Pawlak [22, 23] is a valid means of granular computing [24]. In Pawlak’s rough set model, the equivalence relation is a key tool and can represent information systems or decision tables. However, the equivalence relation is a very stringent condition that may limit the application of rough sets in practical problems. Therefore many researchers have generalized the notion of Pawlak’s rough set by replacing the equivalence relation with other binary relations. It may be a fuzzy, intuitionistic fuzzy, interval-valued fuzzy, hesitant fuzzy or other indiscernibility one within the generalized rough sets [1, 3, 4, 15, 21, 27, 31, 34, 39, 40, 42–51, 54, 55, 59]. The aforementioned rough sets, such as fuzzy rough set [1,3,21,27,34,39], intuitionistic fuzzy rough set [15, 54, 55], hesitant fuzzy rough set [4, 40, 46], and so on, do not consider ∗

Corresponding author. Address: School of Mathematics and Computer Science Northwest MinZu University, LanZhou, Gansu, 730030, P.R.China. E-mail:[email protected]

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attributes with preference-ordered domains. However, in many real-life situations, we are always faced with some problems in which the ordering of properties of the considered attributes plays a key role. In such case, to take into consideration the ordering properties of criteria, Greco et al. [8–11] generalized the notion of Pawlak’s rough set and initiated the dominance-based rough sets approach (DRSA) by replacing the indiscernibility relation with a dominance relation. In DRSA, the knowledge approximated is a collection of upward and downward unions of classes and the dominance classes are sets of objects defined by a dominance relation in which condition attributes are the criteria and classes are preference ordered. Up to now, many fruitful results in DRSA have been achieved [5, 12, 13, 25, 29, 30, 52]. Hesitant fuzzy (HF) set theory, initiated by Torra and Narukawa [32] and Torra [33] as one of the extensions of Zadeh’s fuzzy set [56], permits the membership degree of an element to a set having several possible different values. Because HF set can express the hesitant information more comprehensively than other extensions of fuzzy set, it has been applied in dealing with lots of decision making problems successfully [2,6,17,18,28,35–38, 57]. Although rough sets and HF sets both capture particular facets of the same notionimprecision, studies on the combination of rough set theory and HF set theory are rare. In [40], Yang et al. proposed the concept of HF rough sets by integrating HF sets with rough sets. However, Zhang et al. [46] pointed out that hesitant fuzzy subset based on the hesitant fuzzy rough sets is not necessarily antisymmetric. To remedy this defect, they introduced an HF rough set over two universes and give a new decision making approach in uncertainty environment using the model. Subsequently, Zhang et al. [47] extended the rough set into interval-valued hesitant fuzzy environment and introduced the concept of interval-valued hesitant fuzzy rough sets. In typical hesitant fuzzy background, Zhang and Yang [53] studied the constructive approach to rough set approximation operators and proposed a typical hesitant fuzzy rough set. By combining the hesitant fuzzy linguistic term set and rough set, Zhang et al. [41] developed a general framework for the study of hesitant fuzzy linguistic rough sets over two universes. On the one hand, hybrid models integrating an HF set with a rough set are rarely developed despite the above mentioned research efforts. Knowledge reduction is also an important task in classic and generalized rough set theory. However, the issue has rarely been discussed under the hesitant fuzzy environment. On the other hand, it is well known that the rough set data analysis starts from information systems which contain data about objects of interest, characterized by a finite set of attributes. As an important type of data tables, information systems on decision problems have been widely studied [7, 14, 16, 19, 20, 25, 26]. However, in general, we may not have enough expertise or possess a sufficient level of knowledge to precisely express our preferences over the objects by using a value or a single term, and then, we may usually have a certain hesitancy between a few different values. In such a case, the traditional information system can not express our preferences or assessments by only a single term or value. Considering the facts, it is natural for

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us to investigate information systems in the context of hesitant fuzzy settings which is called hesitant fuzzy information systems. So how to make a decision by a dominance relation is an urgent need in hesitant fuzzy information systems. The aim of this paper is to introduce a dominance relation to hesitant fuzzy information systems and establish a rough set approach by replacing the indiscernibility relation with the dominance relation. Then we develop a reduction approach in hesitant fuzzy ordered information systems for eliminating redundant information from the perspective of the ordering of objects. The rest of the paper is organized as follows. In Section 2, by reviewing some basic concepts, a dominance relation is introduced to hesitant fuzzy information systems and some properties are discussed. Section 3 establishes a dominance-based rough set approach in hesitant fuzzy ordered information systems by replacing the indiscernibility relation with a dominance relation. In Section 4, a ranking approach is established through the notions of dominance degree and whole dominance degree. Section 5 proposes a reduction approach in hesitant fuzzy ordered information system for eliminating redundant information from the perspective of the ordering of objects. Finally, we conclude the paper in Section 6.

2

Dominance relation in hesitant fuzzy information systems In [32, 33], Torra and Narukawa introduced the notions related to HF sets.

Definition 2.1 ( [32, 33]) Let U be a fixed set, a hesitant fuzzy set A on U is in terms of a function hA (x) that when applied to U returns a subset of [0,1]. To be easily understood, Xia and Xu [35] denoted the HF set by a mathematical symbol: A = {< x, hA (x) > |x ∈ U }, where hA (x) is a set of some different values in [0,1], standing for the possible membership degrees of the element x ∈ U to A. For convenience, Xia and Xu [35] called hA (x) an HF element, and denoted the set of all HF sets on U by HF (U ). To compare the HF elements, Xia and Xu [35] defined the following comparison laws: 1 P Definition 2.2 ( [35]) For an HF element h, s(h) = #h γ∈h γ is called the score function of h, where #h is the number of the elements in h. For two HF elements h1 and h2 , if s(h1 ) > s(h2 ), then h1  h2 ; if s(h1 ) = s(h2 ), then h1 = h2 .

An HF information system is a quadruple I = (U, AT, V, f ), where • U is a non-empty finite set of objects called the universe; • AT is a non-empty finite set of attributes; S • V is the domain of all attributes, i.e., V = VAT = a∈AT Va ; • f : U ×AT −→ V is a total function such that f (x, a) ∈ Va for every a ∈ AT, x ∈ U , called 975

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U x1 x2 x3 x4 x5 x6 x7 x8

a1 {0.4,0.6,0.7} {0.4,0.5,0.6} {0.5,0.6,0.7} {0.4,0.7,0.8} {0.4,0.6,0.8} {0.2,0.3,0.4} {0.1,0.4,0.5} {0.2,0.5,0.7}

Table 1: An a2 {0.4,0.5,0.6} {0.0,0.4,0.5} {0.5,0.7,0.8} {0.4,0.6,0.7} {0.4,0.5,0.8} {0.2,0.3,0.6} {0.5,0.6,0.7} {0.2,0.6,0.8}

HF information system a3 a4 {0.3,0.4,0.6} {0.1,0.3,0.4} {0.5,0.6,0.7} {0.4,0.6,0.7} {0.6,0.8,0.9} {0.5,0.7,0.8} {0.5,0.7,0.8} {0.8,0.9,1.0} {0.4,0.6,0.7} {0.5,0.7,0.8} {0.3,0.4,0.5} {0.4,0.6,0.9} {0.4,0.6,0.7} {0.3,0.7,0.8} {0.3,0.4,0.5} {0.4,0.6,0.8}

a5 {0.4,0.5,0.8} {0.2,0.3,0.4} {0.6,0.8,0.9} {0.7,0.8,0.9} {0.4,0.6,0.8} {0.3,0.6,0.7} {0.6,0.8,0.9} {0.4,0.5,0.8}

an information function, where Va is a set of HF elements. Denote as f (x, a) = ha (x), then we call it the HF value of x under the attribute a. In particular, if the information function f (x, a) contains only one real number, the HF information system degenerates into a traditional information system [29]. Example 2.3 An HF information system is given in Table 1, where U = {x1 , x2 , x3 , x4 , x5 , x6 , x7 , x8 }, AT = {a1 , a2 , a3 , a4 , a5 }. In practical decision analysis, we always consider a dominance relation between objects that are possibly dominant in terms of values of an attributes set in an HF information system. Generally, an increasing preference and a decreasing preference can be considered by a decision maker. If the domain of an attribute is ordered by a decreasing or increasing preference, then the attribute is a criterion. Definition 2.4 An HF information system is called an HF ordered information system (HFOIS) if all attributes are criterions. On the basis of Definition 2.2, we develop an approach to rank two objects whose attribute characters are described by HF values. Definition 2.5 Let I = (U, AT, V, f ) be an HFOIS. For x, y ∈ U , denote as x A y ⇐⇒ ∀a ∈ A, f (x, a)  f (y, a) ⇐⇒ ∀a ∈ A, f (x, a)  f (y, a) ∨ f (x, a) = f (y, a), then we say that x dominates y with respect to A ⊆ AT if x A y, denoted by xR A y. Where  RA = {(y, x) ∈ U × U |y A x} is called a dominance relation in HFOIS. Analogously, we call the relation R A a dominated relation in HFOIS, which can be defined as follows: R A = {(y, x) ∈ U × U |x A y}. From Definitions 2.5 and 2.2, we can easily obtain the following theorem.

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Theorem 2.6 Let I = (U, AT, V, f ) be an HFOIS and A ⊆ AT , then (1) R R A and A are reflexive,Ttransitive and unsymmetric; T    (2) RA = a∈A R a∈A R{a} . {a} , RA = The dominance class induced by the dominance relation R A is the set of objects dominating x, i.e., [x] A = {y ∈ U |f (y, a)  f (x, a) ∨ f (y, a) = f (x, a)(∀a ∈ A)} = {y ∈ U |(y, x) ∈ R A }, where [x] A describes the set of objects that may dominate x and is called the A-dominating set with respect to x ∈ U . Similarly, the dominance class induced by the dominated relation R A is the set of objects dominated by x, i.e., [x] A = {y ∈ U |f (x, a)  f (y, a) ∨ f (x, a) = f (y, a)(∀a ∈ A)} = {y ∈ U |(y, x) ∈ R A }, where [x] A describes the set of objects that may be dominated by x and is called the A-dominated set with respect to x ∈ U .  Let U/R A denote classification on the universe, which is the family set {[x]A |x ∈ U }. Any element from U/R A is called a dominance class with respect to A. Dominance classes  in U/RA do not constitute a partition of U , but constitute a covering of U . In the text that follows, without loss of generality, we adopt the dominance relation R A for investigating HFOIS and consider attributes with increasing preference. Theorem 2.7 Let I = (U, AT, V, f ) be an HFOIS and A, B ⊆ AT .   (1) If B ⊆ A ⊆ AT, then R B ⊇ RA ⊇ RAT .   (2) If B ⊆ A ⊆ AT, then [x] B ⊇ [x]A ⊇ [x]AT . S     (3) If xj ∈ [xi ] {[xj ] A , then [xj ]A ⊆ [xi ]A and [xi ]A = A : xj ∈ [xi ]A }.   (4) [xi ]A = [xj ]A iff f (xi , a) = f (xj , a)(∀a ∈ A). Proof. (1) and (2) are straightforward. (3) If xj ∈ [xi ] A , by Definition 2.5, then f (xj , a)  f (xi , a) for all a ∈ A. Similarly, for  all x ∈ [xj ]A , we have f (x, a)  f (xj , a). According to the transitivity of the dominance    relation R A , then f (x, a)  f (xi , a), i.e. x ∈ [xi ]A . Thus [xj ]A ⊆ [xi ]A . Consequently, S  [xi ] {[xj ] A = A : xj ∈ [xi ]A }.    (4) “⇒” Assume that [xi ] A = [xj ]A , then [xi ]A ⊆ [xj ]A . Based on the result (3), for all a ∈ A, we have f (xi , a)  f (xj , a). Similarly, we can conclude that f (xj , a)  f (xi , a). Consequently, f (xi , a) = f (xj , a)(∀a ∈ A). “⇐” It can be directly derived from the definition of the set of objects dominating x. 2 977

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Example 2.8 (Continued from Example 2.3). Compute the classification induced by the dominance relation R AT in Table 1. From Table 1, we have    U/R AT = {[x1 ]AT , [x2 ]AT , . . . , [x8 ]AT }, where    [x1 ] AT = {x1 , x3 , x4 , x5 }, [x2 ]AT = {x2 , x3 , x4 }, [x3 ]AT = {x3 }, [x4 ]AT = {x4 },    [x5 ]AT = {x3 , x4 , x5 }, [x6 ]AT = {x3 , x4 , x5 , x6 }, [x7 ]AT = {x3 , x7 }, [x8 ] AT = {x3 , x4 , x5 , x8 }. From Example 2.8, it is evident that dominance classes in U/R AT do not constitute a partition of U , but constitute a covering of U .

3

Rough set approach to HFOIS

In this section, we shall investigate the problems of set approximation and roughness measure with respect to the dominance relation R A in HFOIS. Definition 3.1 Let I = (U, AT, V, f ) be an HFOIS. For any X ⊆ U and A ⊆ AT , the lower and upper approximations of the set X with respect to the dominance relation R A are defined as follows:  R A (X) = {x ∈ U |[x]A ⊆ X},  R A (X) = {x ∈ U |[x]A ∩ X 6= ∅}.

From Definition 3.1, we can easily obtain the following theorem. Theorem 3.2 Let I = (U, AT, V, f ) be an HFOIS and X, Y ⊆ U , then    (1) R A (X) =∼ RA (∼ X), RA (X) =∼ RA (∼ X);  (2) R A (X) ⊆ X ⊆ RA (X);    (3) A ⊆ AT =⇒ R A (X) ⊆ RAT (X), RA (X) ⊇ RAT (X);    (4) X ⊆ Y =⇒ R A (X) ⊆ RA (Y ), RA (X) ⊆ RA (Y );      (5) R A (X ∩ Y ) = RA (X) ∩ RA (Y ), RA (X ∪ Y ) = RA (X) ∪ RA (Y );      (6) R A (X ∪ Y ) ⊇ RA (X) ∪ RA (Y ), RA (X ∩ Y ) ⊆ RA (X) ∩ RA (Y );    (7) R A (∅) = RA (∅) = ∅, RA (U ) = RA (U ) = U ;      (8) R A (RA (X)) = RA (X), RA (RA (X)) = RA (X).  Theorem 3.3 Let I = (U, AT, V, f ) be an HFOIS and A ⊆ AT . If R A = RAT , then    R A (X) = RAT (X) and RA (X) = RAT (X).

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Proof. It is directly derived from Definitions 2.5 and 3.1. 2 Generally speaking, the uncertainty of a set is due to the existence of the borderline region. The wider the borderline region of a set is, the lower the accuracy of the set is. To express the idea precisely, some basic measures (accuracy and roughness) are defined to depict the quality of the rough approximation of a set. In the following, we introduce the concepts of roughness measure and accuracy measure to measure the imprecision of rough sets induced by dominance relation R A in HFOIS. Definition 3.4 Let I = (U, AT, V, f ) be an HFOIS, X ⊆ U and A ⊆ AT . Then the R

roughness measure ρXA of the set X with respect to the dominance relation R A is defined as follows: |R R A (X)| A ρX = 1 − , |R (X)| A R

R

A A where |·| denotes the cardinality of a set. If R A (X) = ∅, we define ρX = 0. ηX =

is referred to as the accuracy measure of X with respect to the dominance

|R A (X)|

|R A (X)| relation R A. R

According to Definition 3.4 and Theorem 3.2(2), we observe that 0 ≤ ρXA ≤ 1 and R

0 ≤ ηXA ≤ 1. Obviously, by Theorem 3.3 and Definition 3.4, we can draw the following conclusion.  Theorem 3.5 Let I = (U, AT, V, f ) be an HFOIS and A ⊆ AT . If R A = RAT , then R

R

R

R

ρXA = ρXAT and ηXA = ηXAT . Theorem 3.6 Let I = (U, AT, V, f ) be an HFOIS, X ⊆ U and A ⊆ AT , then the following holds: R

R

R

R

(1) ρXAT ≤ ρXA , (2) ηXAT ≥ ηXA .   Proof. (1) Since A ⊆ AT , by Theorem 3.2(3) we have R A (X) ⊆ RAT (X) and RA (X) ⊇

R AT (X). It implies that 1−

|R A (X)| |R A (X)|

≥1−

|R AT (X)| |R AT (X)|

|R A (X)| |R A (X)| R AT

≤

|R AT (X)| |R AT (X)| R AT

= ρX , i.e., ρX

R

. According to Definition 3.4, then ρXA = R

≤ ρXA .

(2) It is directly derived from the result (1) and Definition 3.4.

2

Example 3.7 Consider HFOIS in Table 1. Let A = {a1 , a4 , a5 } ⊆ AT and X = {x2 , x3 , x5 , x7 }.  Now we compute the rough sets of X induced by U/R AT and U/RA , respectively. 979

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 By Definition 3.1 and Example 2.8, the rough set (R AT (X), RAT (X)) can be obtained as follows:  R AT (X) = {x3 , x7 }, RAT (X)) = {x1 , x2 , x3 , x5 , x6 , x7 , x8 }.

Then we compute the classification set induced by the dominance relation U/R A . By Table 1, we have    U/R A = {[x1 ]A , [x2 ]A , . . . , [x8 ]A }, where    [x1 ] A = {x1 , x3 , x4 , x5 }, [x2 ]A = {x2 , x3 , x4 , x5 }, [x3 ]A = {x3 , x4 }, [x4 ]A = {x4 },     [x5 ]A = {x3 , x4 , x5 }, [x6 ]A = {x3 , x4 , x5 , x6 }, [x7 ]A = {x3 , x4 , x7 }, [x8 ]A = {x3 , x4 , x5 , x8 }.  Similarly, by Definition 3.1, we calculate the rough set (R A (X), RA (X)) as follows:  R A (X) = ∅, RA (X)) = {x1 , x2 , x3 , x5 , x6 , x7 , x8 }. Therefore, we have R

ρXA = 1 −

|R A (X)| |R A (X)|

R

ηXA = R

R

= 1,

|R A (X)| |R A (X)|

R

R

ρXAT = 1 −

|R AT (X)| |R AT (X)|

R

ηXAT =

= 0,

|R AT (X)| |R AT (X)|

=1−

2 5 = , 7 7

2 = . 7

R

Thus, ρXAT ≤ ρXA and ηXAT ≥ ηXA .

4

Ranking for all objects in HFOIS

In [58], Zhang et al. defined the concept of dominance degrees for ranking all objects in classical ordered information systems. Inspired by the idea, we introduce a dominance degree between two objects in HFOIS as follows: Definition 4.1 Let I = (U, AT, V, f ) be an HFOIS and A ⊆ AT . Dominance degree between two objects with respect to the dominance relation R A is defined as DA (xi , xj ) =

 | ∼ [xi ] A ∪ [xj ]A | , |U |

where | · | denotes the cardinality of a set, xi , xj ∈ U. Theorem 4.2 Dominance degree DA (xi , xj ) satisfies the following properties: (1) |U1 | ≤ DA (xi , xj ) ≤ 1; (2) if (xj , xk ) ∈ R A , then DA (xi , xj ) ≤ DA (xi , xk ) and DA (xj , xi ) ≥ DA (xk , xi ). Proof. (1) It is straightforward.

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  (2) Assume that (xj , xk ) ∈ R A . By Theorem 2.7, then [xj ]A ⊆ [xk ]A . Therefore, we have

1    (| ∼ [xi ] A ∪ [xj ]A | − | ∼ [xi ]A ∪ [xk ]A |) |U | 1    ≤ (| ∼ [xi ] A ∪ [xk ]A | − | ∼ [xi ]A ∪ [xk ]A |) |U |

DA (xi , xj ) − DA (xi , xk ) =

= 0, 1    (| ∼ [xj ] A ∪ [xi ]A | − | ∼ [xk ]A ∪ [xi ]A |) |U | 1    (| ∼ [xk ] ≥ A ∪ [xi ]A | − | ∼ [xk ]A ∪ [xi ]A |) |U |

DA (xj , xi ) − DA (xk , xi ) =

= 0. That is, DA (xi , xj ) ≤ DA (xi , xk ) and DA (xj , xi ) ≥ DA (xk , xi ). 2 According to Definition 4.1, we may construct a dominance relation matrix with respect to A induced by the dominance relation R A . Based on the dominance relation matrix, the whole dominance degree of each object can be calculated by the following formula DA (xi ) =

X 1 DA (xi , xj ), xi , xj ∈ U. |U | − 1

(1)

j6=i

Obviously, by the concepts of dominance degree and whole dominance degree, the following theorem holds.  Theorem 4.3 Let I = (U, AT, V, f ) be an HFOIS and A ⊆ AT . If R A = RAT , then DA (xi , xj ) = DAT (xi , xj ) and DA (xi ) = DAT (xi ).

By employing the whole dominance degree of each object on the universe, we may rank all objects by the values of DA (xi ). The following example is given to demonstrate the application of this method. Example 4.4 (Continued from Example 2.8). Rank all objects in U based on the dominance relation R AT . By the concept of dominance degree, we obtain the dominance relation matrix   as follows 1 0.75 0.625 0.625 0.875 0.875 0.625 0.875  0.875 1 0.75 0.75 0.875 0.875 0.75 0.875        1 1 1 0.875 1 1 1 1     1 1 0.875 1 1 1 0.875 1  .   1 0.875 0.75 0.75 1 1 0.75 1     0.625 0.875   0.875 0.75 0.625 0.625 0.875 1    0.875 0.875 0.875 0.75 0.875 0.875 1 0.875  0.875 0.875 0.625 0.625 0.875 0.875 0.625 1 981

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Therefore, by Equation 1, the whole dominance degree of each object xi can be calculated as follows: DAT (x1 ) = 0.75, DAT (x2 ) = 0.82, DAT (x3 ) = 0.98, DAT (x4 ) = 0.96, DAT (x5 ) = 0.875, DAT (x6 ) = 0.75, DAT (x7 ) = 0.857, DAT (x8 ) = 0.768. An object with larger value implies a better object. Therefore, based on the values of DAT (xi ), we can rank all objects as follows: ! x1 x3  x4  x5  x7  x8  . x6

5

Attribute reduction in HFOIS

In order to simplify knowledge representation in HFOIS, it is necessary for us to reduce some dispensable attributes in the context of dominance relations. In this section, we will develop an approach to attribute reduction in a given HFOIS. Definition 5.1 Let I = (U, AT, V, f ) be an HFOIS and A ⊆ AT . For any B ⊂ A, if    R A = RAT and RB 6= RAT , then we call A an attribute reduction of I. By Definition 5.1, we can easily verify the following conclusion holds. Theorem 5.2 Let I = (U, AT, V, f ) be an HFOIS and A ⊆ AT . If A is an attribute reduction of I, then DA (xi , xj ) = DAT (xi , xj ), xi , xj ∈ U. In what follows we define several special attributes in HFOIS as follows:  Definition 5.3 Let I = (U, AT, V, f ) be an HFOIS. If R AT = R(AT −{a}) , an attribute

a ∈ AT is called dispensable with respect to the dominance relation R AT ; otherwise, a is called indispensable. The set of all indispensable attributes is called a core with respect to the dominance relation R AT . Definition 5.4 Let I = (U, AT, V, f ) be an HFOIS and A ⊆ AT . Denote by Dis(x, y) = {a ∈ A|(x, y) ∈ / R {a} }, then we call Dis(x, y) a discernibility attribute set between x and y, and DIS = (Dis(x, y) : x, y ∈ U ) a discernibility matrix of the HFOIS. Theorem 5.5 Let I = (U, AT, V, f ) be an HFOIS and A ⊆ AT . Suppose that Dis(x, y) is  the discernibility attribute set of I; then R AT = RA iff A ∩ Dis(x, y) 6= ∅ (Dis(x, y) 6= ∅).    Proof. “=⇒” Assume that R AT = RA , for any y ∈ U then [y]AT = [y]A . If some   x∈ / [y]AT , then x ∈ / [y]A . Therefore, there exists a ∈ A such that (x, y) ∈ / R {a} . Thus, a ∈ Dis(x, y). Consequently, if Dis(x, y) 6= ∅, we have A ∩ Dis(x, y) 6= ∅. “⇐=” Based on Definition 5.4, we can observe that if (x, y) ∈ / R AT for any (x, y) ∈ U × U, then Dis(x, y) 6= ∅. Since A ∩ Dis(x, y) 6= ∅, there exists a ∈ A such that a ∈ Dis(x, y),   i.e., (x, y) ∈ / R / R A . Consequently, RAT ⊇ RA . On the other hand, note {a} . Thus (x, y) ∈    that A ⊆ AT , then we have R AT ⊆ RA . Hence, RAT = RA .

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U x1 x2 x3 x4 x5 x6 x7 x8

x1 ∅ a1 a2 a5 ∅ ∅ ∅ a1 a2 a3 a5 a1 a1 a3

Table 2: The discernibility matrix of Table 1 x2 x3 x4 x5 x6 a3 a4 a1 a2 a3 a4 a5 a1 a2 a3 a4 a5 a1 a2 a3 a4 a5 a4 ∅ a1 a2 a3 a4 a5 a1 a2 a3 a4 a5 a1 a2 a4 a5 a2 a4 a5 ∅ ∅ a1 a4 a5 ∅ ∅ ∅ a2 a3 ∅ ∅ ∅ a3 a2 a3 a5 a1 a3 a4 a5 ∅ ∅ a1 a3 a1 a2 a3 a4 a5 a1 a2 a3 a4 a5 a1 a2 a3 a4 a5 ∅ a1 a3 a1 a2 a3 a4 a1 a3 a4 a5 a1 a4 a4 a1 a3 a1 a2 a3 a4 a5 a1 a2 a3 a4 a5 a1 a2 a3 a4 a5 a4

x7 a2 a3 a4 a5 a2 a4 a5 ∅ a2 a2 a5 a1 a2 a3 a5 ∅ a2 a3 a5

x8 a2 a4 a2 a4 a5 ∅ ∅ ∅ a1 a2 a5 a1 ∅

Definition 5.6 Let I = (U, AT, V, f ) be an HFOIS, A ⊆ AT and Dis(x, y) the discernibility attributes set of I with respect to R AT . Denote as M=

^ n_

o {a : a ∈ Dis(x, y)|x, y ∈ U } ,

then we call M a discernibility function. From the definition of minimal disjunctive normal form of the discernibility function and Theorem 5.5, we can easily verify the following conclusion. Theorem 5.7 Let I = (U, AT, V, f ) be an HFOIS. The minimal disjunctive normal form of M is ! qk t _ ^ M= a is . k=1

s=1

Denoted by Bk = {ais : s = 1, 2, . . . , qk }, then {Bk : k = 1, 2, . . . , t} are the family of all attribute reductions of I. By Theorem 5.7, a practical approach to attribute reductions of HFOIS is provided. In the following, we shall illustrate how to obtain attribute reductions of an HFOIS by an example. Example 5.8 (Continued from Example 2.3). According to Definition 5.4, we obtain the discernibility matrix of Table 1 (see Table 2). Thus, we have M = (a1 ∨ a2 ∨ a5 ) ∧ (a1 ∨ a2 ∨ a3 ∨ a5 ) ∧ a1 ∧ (a1 ∨ a3 ) ∧ (a3 ∨ a4 ) ∧ a3 ∧ (a1 ∨ a2 ∨ a3 ∨ a4 ∨ a5 ) ∧ (a2 ∨ a3 ) ∧ (a2 ∨ a3 ∨ a5 ) ∧ (a1 ∨ a2 ∨ a3 ∨ a4 ) ∧ (a1 ∨ a4 ∨ a5 ) ∧ (a1 ∨ a3 ∨ a4 ∨ a5 ) ∧ (a1 ∨ a2 ∨ a4 ∨ a5 ) ∧ (a1 ∨ a4 ) ∧ a4 ∧ (a2 ∨ a4 ∨ a5 ) ∧ (a2 ∨ a3 ∨ a4 ∨ a5 ) ∧ a2 ∧ (a2 ∨ a5 ) ∧ (a2 ∨ a4 ) = a1 ∧ a2 ∧ a3 ∧ a4 983

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Therefore, there is only one attribute reduction for the HFOIS, which is {a1 , a2 , a3 , a4 }. From the perspective of the ordering of objects, the attributes a1 , a2 , a3 and a4 are indispensable in Table 1.

6

Conclusions

Although the conventional rough set theory is a powerful and useful mathematical tool to deal with uncertainty information, it can not deal with ordering objects instead of classifying objects. In this situation, we have investigated information systems in the context of hesitant fuzzy settings, which is called hesitant fuzzy information systems. The hesitant fuzzy information system is an important type of data tables, which is generalized from the traditional information systems. First, based on the score function of hesitant fuzzy value, a dominance relation has been introduced to hesitant fuzzy information systems. Then we have established a rough set approach in HFOIS by replacing the indiscernibility relation with the dominance relation, and given a ranking approach to all objects by employing the whole dominance degree of each object. Finally, from the perspective of the ordering of objects, we have also developed a reduction approach in HFOIS for eliminating redundant information.

Acknowledgements The authors would like to thank the anonymous referees for their valuable comments and suggestions. This work is supported by the Natural Science Foundation of Gansu Province (No. 1606RJZA003), the Research Project Funds for Higher Education Institutions of Gansu Province (No. 2016B-005), the Fundamental Research Funds for the Central Universities of Northwest MinZu University (No. 31920170010) and the first-class discipline program of Northwest Minzu University.

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THE STABILITY OF CUBIC FUNCTIONAL EQUATIONS WITH INVOLUTION IN MODULAR SPACES CHANGIL KIM AND GILJUN HAN∗

Abstract. In this paper, we prove the generalized Hyers-Ulam stability for the following cubic functional equation with involution f (2x + y) + f (2x + σ(y)) − 2f (x + y) − 2f (x + σ(y)) − 12f (x) = 0 in modular spaces by using a fixed point theorem.

1. Introduction and preliminaries In 1940, Ulam proposed the following stability problem (cf. [21]): “Let G1 be a group and G2 a metric group with the metric d. Given a constant δ > 0, does there exist a constant c > 0 such that if a mapping f : G1 −→ G2 satisfies d(f (xy), f (x)f (y)) < c for all x, y ∈ G1 , then there exists a unique homomorphism h : G1 −→ G2 with d(f (x), h(x)) < δ for all x ∈ G1 ?” In the next year, Hyers [6] gave the first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ theorem was generalized by Aoki [2] for additive mappings and by Rassias [17] for linear mappings by considering an unbounded Cauchy difference, the latter of which has influenced many developments in the stability theory. This area is then referred to as the generalized Hyers-Ulam stability. A generalization of the Rassias’ theorem was obtained by Gˇavruta [5] by replasing the unbounded Cauchy difference by a general control function in the spirit of Rassias’ approach. A problem that mathematicians has dealt with is ”how to generalize the classical function space Lp ”. A first attempt was made by Birnhaum and Orlicz in 1931. This generalization found many applications in differential and intergral equations with kernls of nonpower types. The more abstract generalization was given by Nakano [14] in 1950 based on replacing the particular integral form of the functional by an abstract one that satisfies some good properties. This functional was called modular. Since then, these have been thoroughly developed by several mathematicians, for example, Amemiya [1], Koshi and Shimogaki [9], Yamamuro [23], Orlicz [15], Mazur [11], Musielak [12], Luxemburg [10], Turpin [20]. This idea was refined and generalized by Musielak and Orlicz [13] in 1959. Recently, Sadeghi [18] presented a fixed point method to prove the generalized Hyers-Ulam stability of functional equations in modular spaces with the 42 condition, Wongkum, Chaipunya, and Kumam [22] proved the fixed point theorem and the generalized Hyers-Ulam stability for quadratic mappings in a modular 2010 Mathematics Subject Classification. 39B52, 39B72, 47H09. Key words and phrases. Fixed point theorem, Hyers-Ulam stability, cubic functional equations, modular spaces. * Corresponding author. 1

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space whose modular is convex, lower semi-continuous but do not satisfy the 42 condition, and Park, Bodaghi, and Kim [16] proved the generalized Hyers-Ulam stability for additive mappings in a modular space with 42 -conditions. Let X and Y be real vector spaces. For an additive mapping σ : X −→ X with σ(σ(x)) = x for all x ∈ X, σ is called an involution of X. For a given involution σ : X −→ X, the functional equation (1.1)

f (x + y) + f (x + σ(y)) = 2f (x)

is called an additive functional equation with involution and a solution of (1.1) is called an additive mapping with involution. For a given involution σ : X −→ X, the functional equation (1.2)

f (x + y) + f (x + σ(y)) = 2f (x) + 2f (y)

is called the quadratic functional equation with involution and a solution of (1.2) is called a quadratic mapping with involution. The functional equation (1.2) has been studied by Stetkær [19] and the generalized Hyers-Ulam stability for (1.2) has been obtained by Bouikhalene et al. [3, 4, 7]. In this paper, we prove the generalized Hyers-Ulam stability for the following cubic functional equation with involution (1.3)

f (2x + y) + f (2x + σ(y)) − 2f (x + y) − 2f (x + σ(y)) − 12f (x) = 0

in modular spaces without the 42 -condition and the convexity by using a fixed point theorem. Unlike Banach spaces and F -spaces, due to the triangle inequlity in modular spaces, we need subtle calculation in the proofs of Theorem 2.1 and Theorem 2.2 Definition 1.1. Let X be a vector space over a field K(R, C, or N). (1) A generalized functional ρ : X −→ [0, ∞] is called a modular if (M1) ρ(x) = 0 if and only if x = 0 , (M2) ρ(αx) = ρ(x) for every scalar α with |α| = 1, and (M3) ρ(z) ≤ ρ(x) + ρ(y) whenever z is a convex combination of x and y. (2) If (M3) is replaced by (M4) ρ(αx + βy) ≤ αρ(x) + βρ(y) for all x, y ∈ V and for all nonnegative real numbers α, β with α + β = 1, then we say that ρ is convex. For any modular ρ on X, the modular space Xρ is defined by Xρ = {x ∈ X | ρ(λx) → 0 as λ → 0} and the modular space Xρ can be equipped with a norm called the Luxemburg norm, defined by n x o ≤1 . kxkρ = inf λ > 0 | ρ λ Let Xρ be a modular space and {xn } a sequence in Xρ . Then (i) {xn } is called ρ-Cauchy if for any  > 0, one has ρ(xn − xm ) <  for sufficiently large m, n ∈ N, (ii) {xn } is called ρ-convergent to a point x ∈ Xρ if ρ(xn − x) → 0 as n → ∞, and (iii) a subset K of Xρ is called ρ-complete if each ρ-Cauchy sequence is ρ-convergent to a point in K. Another unnatural behavior one usually encounter is that the convergence of a sequence {xn } to x does not imply that {cxn } converges to cx for some c ∈ K.

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THE STABILITY OF CUBIC FUNCTIONAL EQUATIONS WITH INVOLUTION...

3

Thus, many mathematicians imposed some additional conditions for a modular to meet in order to make the multiples of {xn } converge naturally. Such preferences are referred to mostly under the term related to 42 -condition. A modular space Xρ is said to satisfy the 42 -condition if there exists k ≥ 2 such that Xρ (2x) ≤ kXρ (x) for all x ∈ X. Some authors varied the notion so that only k > 0 is required and called it the 42 -type condition. In fact, one may see that these two notions coincide. There are still a number of equivalent notions related to the 42 -condition. In [8], Khamsi proved a series of fixed point theorems in modular spaces where the modulars do not satisfy 42 -conditions. His results exploit one unifying hypothesis in which the boundedness of an orbit is assumed. Example 1.2. A convex function ζ defined on the interval [0, ∞), nondecreasing and continuous, such that ζ(0) = 0, ζ(α) > 0 for α > 0, ζ(α) → ∞ as α → ∞, is called an Orlicz function. Let (Ω, Σ, µ) be a measure space and L0 (µ) the set of all measurable real valued (or complex valued) functions on Ω. Deine the Orlicz modular ρζ on L0 (µ) by the formula Z ρζ (f ) = ζ(|f |)dµ. Ω

The associated modular space with respect to this modular is called an Orlicz space, and will be denoted by (Lζ , Ω, µ) or briefly Lζ . In other words, Lζ = {f ∈ L0 (µ) | ρζ (λf ) < ∞ for some λ > 0}. It is known that the Orlicz space Lζ is ρζ -complete. Moreover, (Lζ , k · kρζ ) is a Banach space, where the Luxemburg norm k · kρζ is defined as follows Z  |f |  o n dµ ≤ 1 . kf kρζ = inf λ > 0 ζ λ Ω Further, if µ is the Lebesgue measure on R and ζ(t) = et − 1, then ρζ does not satisfy the 42 -condition. For a modular space Xρ , a nonempty subset C of Xρ , and a mapping T : C −→ C, the orbit of T at x ∈ C is the set O(x) = {x, T x, T 2 x, · · ·}. The quantity δρ (x) = sup{ρ(u − v) | u, v ∈ O(x)} is called the orbital diameter of T at x and if δρ (x) < ∞, then one says that T has a bounded orbit at x. Khamsi [8] proved a series of fixed point theorems in modular spaces where the modulars do not satisfy 42 -conditions. His results exploit one unifying hypothesis in which the boundedness of an orbit is assumed. Lemma 1.3. [8] Let Xρ be a modular space whose induced modular is lower semicontinuous and let C ⊆ Xρ be a ρ-complete subset. If T : C −→ C is a ρcontraction, that is, there is a constant L ∈ [0, 1) such that ρ(T x − T y) ≤ Lρ(x − y), ∀x, y ∈ C and T has a bounded orbit at a point x0 ∈ C, then the sequence {T n x0 } is ρconvergent to a point w ∈ C.

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For any modular ρ on X and any linear space V , we define a set M M := {g : V −→ Xρ | g(0) = 0} and the generalized function ρe on M by for each g ∈ M, ρe(g) := inf{c > 0 | ρ(g(x)) ≤ cψ(x, 0), ∀x ∈ V }, where ψ : V 2 −→ [0, ∞) is a mapping. The proof of the following lemma is similar to the proof of Lemma 10 in [22]. Lemma 1.4. Let V be a linear space, Xρ a ρ-complete modular space where ρ is lower semi-continuous and f : V −→ Xρ a mapping with f (0) = 0. Let ψ : V 2 −→ [0, ∞) be a mapping such that (1.4)

ψ(2n x, 2n y) = 0, ψ(2x, 2x) ≤ 8Lψ(x, x) n→∞ 8n lim

for all x, y ∈ V and some L with 0 ≤ L < 1. Then we have the following : (1) M is a linear space, (2) ρe is a modular on M, (3) if ρ is convex, then ρe is convex, (4) Mρe = M and Mρe is ρe-complete, and (5) ρe is lower semi-continuous. Proof. (1), (2), and (3) are trivial. (4) By the definition of Mρe, Mρe = M. Let  > 0. Take any ρe-Cauchy sequence {gn } in Mρe. Then there is an l ∈ N such that for n, m ∈ N with n, m ≥ l, (1.5)

ρ(gn (x) − gm (x)) ≤ ψ(x, 0)

for all x ∈ V . Hence {gn (x)} is a ρ-Cauchy sequence in Xρ for all x ∈ V . Since Xρ is a ρ-complete modular space, there is a mapping g : V −→ Xρ such that ρ(gn (x) − g(x)) −→ 0 as n → ∞ for all x ∈ V . Since each gn ∈ M, there is an m ∈ N such that ρ(gm (0) − g(0)) = ρ(g(0)) ≤  and hence g ∈ Mρe. Since ρ is lower semi-continuous, by (1.5), we have ρ(gn (x) − g(x)) ≤ lim inf ρ(gn (x) − gm (x)) ≤ ψ(x, 0) m→∞

for all x ∈ V . Hence Mρe is ρe-complete. (5) Suppose that {gn } is a sequence in Mρe which is ρe-convergent to g ∈ Mρe. Let  > 0. Then for any n ∈ N, there is a positive real number cn such that ρe(gn ) ≤ cn ≤ ρe(gn ) +  and so ρ(g(x)) ≤ lim inf ρ(gn (x)) (1.6)

n→∞

≤ lim inf cn ψ(x, 0) ≤ n→∞



 lim inf ρ(gn (x)) +  ψ(x, 0) n→∞

for all x ∈ V . Hence ρe is lower semi-continuous.

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2. The generalized Hyers-Ulam stability for (1.3) in modular spaces Throughout this section, we assume that every modular is lower semi-continuous. In this section, we prove the generalized Hyers-Ulam stability for (1.3). For any f : V −→ Xρ and any involution σ : V −→ V , let Df (x, y) = f (2x + y) + f (2x + σ(y)) − 2f (x + y) − 2f (x + σ(y)) − 12f (x). Theorem 2.1. Let V be a linear space, Xρ a ρ-complete modular space and f : V −→ Xρ a mapping with f (0) = 0. Let φ : V 2 −→ [0, ∞) be a mapping such that (2.1)

φ(2x, 2y) ≤ 8Lφ(x, y), φ(x + σ(x), y + σ(y)) ≤ 8Lφ(x, y)

for all x, y ∈ V and some L with 0 < L < (2.2)

1 16

and

ρ(Df (x, y)) ≤ φ(x, y)

for all x, y ∈ V . Then there exists a unique cubic mapping F : V −→ Xρ with involution such that   1 2 (2.3) ρ F (x) − f (x) ≤ φ(x, 0) 4 1 − 8L for all x ∈ V . Proof. Let ψ(x, y) = φ(x, y) + φ(y, x) for all x, y ∈ V . Then ψ satisfies (1.4) and hence, by Lemma 1.4, ρe is a lower semi-continuous convex modular on Mρe, Mρe = M, and Mρe is ρe-complete. Define T : Mρe −→ Mρe by  1 g(2x) + g(x + σ(x)) T g(x) = 8 for all g ∈ Mρe and all x ∈ V . Let g, h ∈ Mρe. Suppose that ρe(g − h) ≤ c for some nonnegative real number c. Then by (2.1), we have  1  1 ρ(T g(x) − T h(x)) ≤ ρ [g(2x) − h(2x)] + ρ [g(x + σ(x)) − h(x + σ(x))] 4 4 ≤ 16Lcψ(x, 0) for all x ∈ V and so ρe(T g − T h) ≤ 16Le ρ(g − h). Hence T is a ρe-contraction. By (2.2), we get (2.4)

ρ(f (x) + f (σ(x))) ≤ φ(0, x)

and (2.5)

  ρ f (2x) − 8f (x) ≤ ρ(2f (2x) − 16f (x)) ≤ φ(x, 0)

for all x ∈ X. Letting x = x + σ(x) in (2.4), by (M3), we have (2.6)

ρ(f (x + σ(x))) ≤ ρ(2f (x + σ(x))) ≤ φ(0, x + σ(x)) ≤ 8Lφ(0, x),

for all x ∈ X and by (2.5) and (M3), we get 1  (2.7) ρ 3 f (2x) − f (x) ≤ ρ(f (2x) − 8f (x)) ≤ φ(x, 0) 2 for all x ∈ X. Now, we claim that T has a bounded orbit at 41 f . By the definition of T , we have 1 (2.8) T n f (x + σ(x)) = 2n f (2n (x + σ(x))) 2

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for all x ∈ V and for all n ∈ N. Hence by (2.1), (2.6), and (2.8), we have ρ(T n f (x + σ(x))) ≤ ρ(f (2n (x + σ(x)))) ≤ (8L)n+1 φ(0, x) for all x ∈ V and for all n ∈ N. By (2.7), for any nonnegative integer n, we obtain 1 1 ρ( T n f (x) − f (x)) 2 2   1 1 n ≤ ρ T f (x) − 3 f (2x) + ρ( 3 f (2x) − f (x)) 2 2 1    1 n−1 ≤ρ T f (2x) − f (2x) + ρ T n−1 f (x + σ(x)) + φ(x, 0) 2 2 1  1 n−1 ≤ρ T f (2x) − f (2x) + (8L)n φ(0, x) + φ(x, 0) 2 2 for all x ∈ V and by induction, we have ρ (2.9)

n−1  n−1 X X 1 T n f (x) − f (x) ≤ (8L)n−i φ(0, 2i x) + φ(2i x, 0) 2 2 i=0 i=0

1

≤ n(8L)n φ(0, x) +

1 φ(x, 0) 1 − 8L

for all x ∈ V and all n ∈ N. Hence by (2.9), we get 1  1 2 (2.10) ρ T n f (x) − T m f (x) ≤ φ(x, 0) + [n(8L)n + m(8L)m ]φ(0, x) 4 4 1 − 8L 1 , by (2.10), for all x ∈ V and all all nonnegative integers n, m and since 0 < L < 16 we have 1  1 ρ T n f (x) − T m f (x) ≤ 4φ(x, 0) + φ(0, x) ≤ 4ψ(x, 0) 4 4 for all x ∈ V and all nonnegative integers n, m. Hence we have  1 1  ρe T n f − T m f ≤ 4 4 4

all nonnegative integers n, m and thus T has a bounded orbit at 14 f . By Lemma 1.3, there is an F ∈ Mρe such that {T n 41 f } is ρe-convergent to F . Since ρe is lower semi-continuous, we get   1  1  0 ≤ ρe(T F − F ) ≤ lim inf ρe T F − T n+1 f ≤ lim inf 16Le ρ F − Tn f = 0 n→∞ n→∞ 4 4 and hence F is a fixed point of T in Mρe. By induction, we can easily show that T n f (x) = =

n−1 1 1 X i n f (2 x) + 2 f (2n−1 (x + σ(x))) 23n 23n i=0

1 2n − 1 n f (2 x) + f (2n−1 (x + σ(x))) 23n 23n

for all x ∈ X and n ∈ N. Moreover, we have 1  1h i 1  1 1 (2.11) ρ 8 DF (x, y) ≤ ρ 7 DF (x, y) − T n Df (x, y) + ρ 7 T n Df (x, y) 2 2 4 2 4

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for all x, y ∈ V and all n ∈ N. Note that 1h i 1 ρ 7 DF (x, y) − T n Df (x, y) 2 4 1h i 1h i 1 n1 ≤ ρ 6 F (2x + y) − T f (2x + y) + ρ 4 F (2x + σ(y)) − T n f (2x + σ(y)) 2 4 2 4 1h i 1h i n1 n1 + ρ 4 2F (x + y) − T 2f (x + y) + ρ 4 2F (x + σ(y)) − T 2f (x + σ(y)) 2 4 2 4 1h i n1 + ρ 4 12F (x) − T 12f (x) 2 4 for all x, y ∈ V and all n ∈ N. Since {T n 14 f } is ρe-convergent to F , we get 1h i 1 (2.12) lim ρ 7 DF (x, y) − T n Df (x, y) = 0 n→∞ 2 4 for all x, y ∈ V . Further, we have 1  1  1 ρ 7 T n Df (x, y) = ρ 9 T n Df (x, y) 2 4 2   2n − 1   1 ≤ ρ 3n+8 Df (2n x, 2n y) + ρ 3n+8 Df (2n−1 (x + σ(x)), 2n−1 (y + σ(y))) 2 2 ≤ φ(2n x, 2n y) + φ(2n−1 (x + σ(x)), 2n−1 (y + σ(y))) ≤ 2(8L)n φ(x, y) for all x, y ∈ V and all n ∈ N. Letting n → ∞ in the last inequality, we get 1  1 (2.13) lim ρ 7 T n Df (x, y) = 0 n→∞ 2 4 for all x, y ∈ V . By (2.11), (2.12), (2.13), and (M1), we obtain DF (x, y) = 0 for all x, y ∈ V and hence F is a cubic mapping with involution. Moreover, since ρ is lower semi-continuous, by (2.10), we get   1 2 ρ F (x) − f (x) ≤ φ(x, 0) 4 1 − 8L for all x ∈ X.  If ρ is covex, then Theorem 2.1 can replaced by the following theorem. Theorem 2.2. All conditions of Theorem 2.1 are assumed. Further, suppose that ρ is a convex modular and 0 < L < 21 . Then there exists a unique cubic mapping F : V −→ Xρ with involution such that   1 1 (2.14) ρ F (x) − f (x) ≤ 4 φ(x, 0) 4 2 (1 − L) for all x ∈ V . Proof. Let ψ(x, y) = φ(x, y) + φ(y, x) for all x, y ∈ V . Then ψ satisfies (1.4) and hence, by Lemma 1.4, ρe is a lower semi-continuous convex modular on Mρe, Mρe = M, and Mρe is ρe-complete. Define T : Mρe −→ Mρe by  1 T g(x) = g(2x) + g(x + σ(x)) 8

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for all g ∈ Mρe and all x ∈ V . Let g, h ∈ Mρe. Suppose that ρe(g − h) ≤ c for some nonnegative real number c. Then by (2.1), we have  1 1  1 1 ρ(T g(x) − T h(x)) ≤ ρ [g(2x) − h(2x)] + ρ [g(x + σ(x)) − h(x + σ(x))] 2 4 2 4 ≤ 2Lcψ(x, 0) for all x ∈ V and so ρe(T g − T h) ≤ 2Le ρ(g − h). Hence T is a ρe-contraction. By (2.2), we get ρ(f (x) + f (σ(x))) ≤ φ(0, x)

(2.15) and

  1 1 ρ f (2x) − 8f (x) ≤ ρ(2f (2x) − 16f (x)) ≤ φ(x, 0) 2 2 for all x ∈ X. Letting x = x + σ(x) in (2.15), by (M3), we have

(2.16)

1 1 ρ(2f (x + σ(x))) ≤ φ(0, x + σ(x)) ≤ 4Lφ(0, x), 2 2 for all x ∈ X and by (2.16) and (M3), we get 1  1 1 (2.18) ρ 3 f (2x) − f (x) ≤ 3 ρ(f (2x) − 8f (x)) ≤ 4 φ(x, 0) 2 2 2 for all x ∈ X. Now, we claim that T has a bounded orbit at 41 f . By the definition of T , we have 1 (2.19) T n f (x + σ(x)) = 2n f (2n (x + σ(x))) 2 for all x ∈ V and for all n ∈ N. Hence by (2.1), (2.15), and (2.19), we have (2.17)

ρ(f (x + σ(x))) ≤

1 ρ(f (2n (x + σ(x)))) ≤ 2(2L)n+1 φ(0, x) 22n for all x ∈ V and for all n ∈ N. By (2.18), for any nonnegative integer n, we obtain 1  1 ρ T n f (x) − f (x) 2 2  1 1  1  n 1 ≤ ρ T f (x) − 3 f (2x) + ρ 3 f (2x) − f (x) 2 2 2 2   1 1  1  1 n−1 1 f (2x) − f (2x) + 4 ρ T n−1 f (x + σ(x)) + 5 φ(x, 0) ≤ 3ρ T 2 2 2 2 2  (2L)n 1 1 1 1 ≤ 3 ρ T n−1 f (2x) − f (2x) + φ(0, x) + 5 φ(x, 0) 3 2 2 2 2 2 for all x ∈ V and by induction, we have ρ(T n f (x + σ(x))) ≤

n−1  n−1 X (2L)n−i 1 1 X 1 i ρ T f (x) − f (x) ≤ φ(0, 2 x) + 5 φ(2i x, 0) 3i 3(i+1) 2 2 2 2 2 i=0 i=0

1

(2.20)

n

≤

(2L)n 1 φ(0, x) + 5 φ(x, 0) 4 2 (1 − L)

for all x ∈ V and all n ∈ N. Hence by (2.20), we get 1  1 1 1 (2.21) ρ T n f (x) − T m f (x) ≤ 4 φ(x, 0) + [(2L)n + (2L)m ]φ(0, x) 4 4 2 (1 − L) 4

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for all x ∈ V and all all nonnegative integers n, m and since 0 < L < 12 , by (2.21), we have ρ

 1 1 1 T n f (x) − T m f (x) ≤ 4 φ(x, 0) + φ(0, x) 4 4 2 (1 − L) 2 1 1 ≤ φ(x, 0) + φ(0, x) 8 2 1 ≤ ψ(x, 0) 2

1

for all x ∈ V and all nonnegative integers n, m. Hence we have  1 1  1 ρe T n f − T m f ≤ 4 4 2 all nonnegative integers n, m and thus T has a bounded orbit at 14 f . By Lemma 1.3, there is an F ∈ Mρe such that {T n 41 f } is ρe-convergent to F . Since ρe is lower semi-continuous, we get   1  1  ρ F − Tn f = 0 0 ≤ ρe(T F − F ) ≤ lim inf ρe T F − T n+1 f ≤ lim inf 2Le n→∞ n→∞ 4 4 and hence F is a fixed point of T in Mρe. By induction, we can easily show that T n f (x) = =

n−1 1 X i 1 n f (2 x) + 2 f (2n−1 (x + σ(x))) 23n 23n i=0

1 2n − 1 n f (2 x) + f (2n−1 (x + σ(x))) 23n 23n

for all x ∈ X and n ∈ N. Moreover, we have (2.22) 1  1 1h i 1  1  1 1 ρ 8 DF (x, y) ≤ ρ 7 DF (x, y) − T n Df (x, y) + ρ 7 T n Df (x, y) 2 2 2 4 2 2 4 for all x, y ∈ V and all n ∈ N. Note that 1h i 1 ρ 7 DF (x, y) − T n Df (x, y) 2 4 i i 1 1h 1 1 1h 1 ≤ ρ 6 F (2x + y) − T n f (2x + y) + 3 ρ 4 F (2x + σ(y)) − T n f (2x + σ(y)) 2 2 4 2 2 4 i i 1 1h 1 1h n1 n1 + 3 ρ 4 2F (x + y) − T 2f (x + y) + 3 ρ 4 2F (x + σ(y)) − T 2f (x + σ(y)) 2 2 4 2 2 4 i 1 1h n1 + 3 ρ 4 12F (x) − T 12f (x) 2 2 4 for all x, y ∈ V and all n ∈ N. Since {T n 14 f } is ρe-convergent to F , we get (2.23)

lim ρ

n→∞

1h i n1 DF (x, y) − T Df (x, y) =0 27 4

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for all x, y ∈ V . Further, we have 1  1  1 ρ 7 T n Df (x, y) = ρ 9 T n Df (x, y) 2 4 2  1  2n − 1  1  1 n n ≤ ρ 3n+8 Df (2 x, 2 y) + ρ 3n+8 Df (2n−1 (x + σ(x)), 2n−1 (y + σ(y))) 2 2 2 2 2n − 1 1 n n n−1 (x + σ(x)), 2n−1 (y + σ(y))) ≤ 3n+9 φ(2 x, 2 y) + 3n+9 φ(2 2 2 (2L)n ≤ φ(x, y) 29 for all x, y ∈ V and all n ∈ N. Letting n → ∞ in the last inequality, we get 1  1 (2.24) lim ρ 7 T n Df (x, y) = 0 n→∞ 2 4 for all x, y ∈ V . By (2.22), (2.23), (2.24), and (M1), we obtain DF (x, y) = 0 for all x, y ∈ V and hence F is a cubic mapping with involution. Moreover, since ρ is lower semi-continuous, by (2.21), we get   1 1 ρ F (x) − f (x) ≤ 4 φ(x, 0) 4 2 (1 − L) for all x ∈ X.



It is well-known that every normed space is a modular space with ρ(x) = kxk. Using Theorem 2.2, we have the following corollary. Corollary 2.3. Let X and Y be normed spaces and , θ, and p be real numbers with  ≥ 0, θ ≥ 0, and 0 < p < 32 . Let f : X −→ Y be a mapping with involution σ such that f (0) = 0 and kDf (x, y)k ≤  + θ(kxk2p + kyk2p + kxkp kykp ) and kx + σ(x)k ≤ 2kxk for all x, y ∈ X. Then there is a cubic mapping F : X −→ Y with involution such that 1 kF (x) − f (x)k ≤ ( + θkxk2p ) 2(8 − 22p ) for all x ∈ X. Proof. Let ρ(z) = kzk for all y ∈ Y and φ(x, y) =  + θ(kxk2p + kyk2p + kxkp kykp ) for all x, y ∈ V . Then ρ is a convex modular on a normed space Y , Y = Yρ , and φ(2x, 2y) ≤ 22p φ(x, y), φ(x + σ(x), y + σ(y)) ≤ 22p φ(x, y) for all x, y ∈ V . By Theorem 2.2, we have the results.



Using Example 1.3, we get the following example.

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Example 2.4. Let , θ, and p be real numbers with  ≥ 0, θ ≥ 0, and 0 < p < 23 . Let ζ be an Orlicz function and Lζ the Orlicz space. Let f : V −→ Lζ be a mapping with involution σ such that f (0) = 0 and Z ζ(|f (2x + y) + f (2x + σ(y)) − 2f (x + y) − 2f (x + σ(y)) − 12f (x)|)dµ Ω

≤  + θ(kxk2p + kyk2p + kxkp kykp ) and kx + σ(x)k ≤ 2kxk for all x, y ∈ X. Then there is a cubic mapping F : X −→ Y with involution such that Z 1 1 ( + θkxk2p ) ζ(|F (x) − f (x)|)dµ ≤ 4 2(8 − 22p ) Ω for all x ∈ X. 1

Define a mapping ρ2 : R −→ R by ρ2 (x) = |x| 2 . Then clearly, ρ2 is a modular on R and Rρ2 = R. Note that √ 12 1 √ 1 1 1 1 1 2 + 1 = × |2| 2 + × |4| 2 . × 2 + × 4 = 3 > 2 2 2 2 2 Hence ρ2 is not convex. Moreover, since (R, | · |) is a complete normed space, we can easily show that (R, ρ2 ) is a complete modular space. Using these and Theorem 2.1, we have the following example. Example 2.5. Let , θ, and p be real numbers with  ≥ 0, θ ≥ 0, and 0 < p < 23 . Let f : V −→ R be a mapping with involution σ such that f (0) = 0 and 1

|f (2x + y) + f (2x + σ(y)) − 2f (x + y) − 2f (x + σ(y)) − 12f (x)| 2 ≤  + θ(kxk2p + kyk2p + kxkp kykp ) and

kx + σ(x)k ≤ 2kxk for all x, y ∈ X. Then there is a cubic mapping F : X −→ Y with involution such that 1 1 2 |F (x) − f (x)| 2 ≤ ( + θkxk2p ) 4 1 − 22p for all x ∈ X. References [1] I. Amemiya, On the representation of complemented modular lattices, J. Math. Soc. Japan. 9(1957), 263-279. [2] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2(1950), 64-66. [3] B. Boukhalene, E. Elqorachi, and Th. M. Rassias, On the generalized Hyers-Ulam stability of the quadratic functional equation with a general involution, Nonlinear Funct. Anal. Appl. 12. no 2 (2007), 247-262. [4] , On the Hyers-Ulam stability of approximately pexider mappings, Math. Ineqal. Appl. 11 (2008), 805-818. [5] P. Gˇ avruta, A generalization of the Hyer-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184(1994), 431-436. [6] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. 27(1941), 222-224.

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[7] S. M. Jung, Z. H. Lee, A fixed point approach to the stability of quadratic functional equation with involution, Fixed Point Theory Appl. 2008. [8] M. A. Khamsi, Quasicontraction mappings in modular spaces without 2-condition, Fixed Point Theory and Applications, 2008(2008), 1-6. [9] S. Koshi and T. Shimogaki, On F-norms of quasi-modular spaces, J. Fac. Sci., Hokkaido Univ., Ser. 1 15(1961), 202-218. [10] W. A. Luxemburg, Banach function spaces. PhD thesis, Delft University of Technology, Delft, The Netherlands 1959. [11] B. Mazur, Modular curves and the Eisenstein ideal. Publ. Math. IHS 47(1978), 33-186. [12] J. Musielak, Orlicz Spaces and Modular Spaces. Lecture Notes in Mathematics, vol. 1034. Springer, Berlin 1983. [13] J. Musielak and W. Orlicz, On modular spaces, Studia Mathematica, 18(1959), 591-597. [14] H. Nakano, Modular semi-ordered spaces, Tokyo, Japan, 1959. [15] W. Orlicz, Collected Papers, vols. I, II. PWN, Warszawa 1988. [16] C. Park, A. Bodaghi, and S. O. Kim, A fixed point approach to stability for additive mappings in a modular space with 42 -conditions, J. Comput. Anal. Appl. 24 (2018), 1036-1048. [17] Th. M. Rassias, On the stability of the linear mapping in Banach sapces, Proc. Amer. Math. Sco. 72(1978), 297-300. [18] G. Sadeghi, A fixed point approach to stability of functional equations in modular spaces, Bulletin of the Malaysian Mathematical Sciences Society. Second Series, 37(2014), 333-344. [19] H. Stetkær, Functional equations on abelian groups with involution, Aequationes Math. 54 (1997), 144-172. [20] P. Turpin, Fubini inequalities and bounded multiplier property in generalized modular spaces, Comment. Math. 1(1978), 331-353. [21] S. M. Ulam, Problems in Modern Mathematics, Wiley, New York; 1964. [22] K. Wongkum, P. Chaipunya, and P. Kumam, On the generalized Ulam-Hyers-Rassias stability of quadratic mappings in modular spaces without 42 -conditions, 2015(2015), 1-6. [23] S. Yamamuro, On conjugate spaces of Nakano spaces, Trans. Am. Math. Soc. 90(1959), 291-311. Department of Mathematics Education, Dankook University, 152, Jukjeon-ro, Sujigu, Yongin-si, Gyeonggi-do, 448-701, KOREA E-mail address: [email protected] Department of Mathematics Education, Dankook University, 152, Jukjeon-ro, Sujigu, Yongin-si, Gyeonggi-do, 448-701, Korea E-mail address: [email protected]

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A nonstandard finite difference method applied to a mathematical cholera model with spatial diffusion Shu Liao a

Weiming Yang a

a

School of Mathematics and Statistics Chongqing Technology and Business University, Chongqing, 400067,China Abstract In this paper, we propose a nonstandard finite difference (NSFD) scheme to solve numerically a cholera epidemic model with spatial diffusion. Through constructing discrete Lyapunov functions, we prove the globally asymptotical stabilities of the diseasefree equilibrium and the chronic infection equilibrium, which coincide with the corresponding continuous model . Finally, numerical simulations are provided to illustrate the theoretical results.

Cholera, partial differential equation, nonstandard finite difference scheme, Lyapunov function, global stability.

1

Introduction

Cholera is an infection of the intestines caused by the bacterium called Vibrio cholerae and can spread rapidly in areas with inadequate treatment of sewage and drinking water. The World Health Organization (WHO) has warned that there are an estimated 3-5 million infected cases and 28,800-130,000 deaths worldwide due to cholera every year. Since 1817, seven cholera pandemics have spread in many places, with periodic outbreaks such as the latest one in Yemen in October 2016, which is the worst cholera outbreak in the world. The total cases in Yemen have exceeded half a million, with nearly 2,000 deaths reported, since the outbreak began to spread rapidly at the end of April 2017 due to the deteriorating hygiene and sanitation conditions and the disrupted water supply across the country. There have been massive outbreaks of cholera in many developing countries of Africa and South-east Asia, including Congo (2008), Iraq (2008), Zimbabwe (2008-2009), Vietnam (2009), Nigeria (2010), Haiti (2010), Mexico (2013), South Sudan (2014), and Somalia (2017). In recent years, many epidemic models have been proposed to a better understanding of the transmission of cholera. In 2001, Code¸co [1] proposed a SIRB epidemic model to study the transmission of cholera in which B represents the V. cholerae concentration in water. Hartley Morris and Smith [2] in 2006 discovered a representative hyperinfectious state of the pathogen, which is the ’explosive’ infectivity of freshly shed V. cholerae based on the laboratory results. Tien and Earn [3] proposed a water-borne disease model with multiple transmission pathways: both direct human-to-human and indirect water-to-human transmissions, and identified how these transmission routes influence disease dynamics. Mukandavire et al. [4] simplified Hartley’s model to understand transmission dynamics of cholera outbreak in Zimbabwe. Liao and Wang [5] in 2011 conducted a dynamical analysis of the Hartley’s model to study the stability of both the disease-free and endemic equilibria so as to explore the complex epidemic and endemic dynamics of the disease. Bertuzzo et al. [6] based on 1 1000

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the Codeco’s work and developed a partial differential equation model for cholera epidemics. Their results suggested that cholera outbreaks may be triggered by time scales of disease dynamics. In a recently study, Safi et al. [7] designed a new two-strain model to assess the impact of basic control measures and dose-structured mass vaccination on cholera transmission dynamics in a population. More papers in the field of cholera epidemic models are presented in ( [8–11]). Nowadays, more and more researchers consider to discretize the continuous models for practical purposes. One of the reasons is that most numerical methods like traditional Euler, Runge-Kutta and some standard procedures of MATLAB software will fail to solve nonlinear systems generating oscillations, chaos, and unsteady states if the time step size increases to a critical size. The other reason is that the results of the discrete time models are more accurate and convenient to describe infectious diseases and can preserve as much as possible the qualitative properties of the corresponding continuous models. The nonstandard finite difference (NSFD) scheme developed by Mickens ( [12–14]) performs well and has been applied to many articles. An NSFD discretization must satisfy one of the following two conditions ( [15, 16]): nonlocal approximation is used and discretization of derivative must be a denominator function. Cui et al. [17] employed an NSFD scheme to discuss a class of SIR epidemic model with vaccination and treatment. The dynamical properties of their discretized model were analysed to demonstrate that the discretized epidemic model maintains essential properties of the corresponding continuous model, such as positivity property, boundness of solutions, equilibrium points and their local stability properties. Suryanto et al. [18] constructed an NSFD scheme to solve a SIR epidemic model with modified saturated incidence rate. From their numerical simulations, the NSFD scheme allowed large time step size to save the computational cost. Qin et al. [19] proposed an NSFD method for an epidemic model which described the hepatitis B virus infection with spatial dependence. They have shown that the NSFD method is unconditionally positive by using the M-matrix theory. Moveover, asymptotical stabilities of the steady-state solutions were fully determined by constructing discrete Lyapunov functions independent of the time and space step sizes. Manna and Chakrabarty [20] analysed a spatiotemporal model for HBV infection by using an NSFD scheme, and studied the global stability properties of the discretized model. The simulation results demonstrated the advantages of the usage of NSFD method over the other schemes. For more investigations on NSFD scheme can be found in ( [21–24]). In 2015, Wang and Wang [25] proposed a PDE model to simulate cholera infection with spatial diffusion, taking multiple transmission ways into account among the human host, the pathogen, and the environment. The model in their paper assumes that both the human population and the bacteria undergo a diffusion process and is given by the following system of PDEs: ∂S W (x, t)S(x, t) = Λ − βW − βh S(x, t)I(x, t) − µS(x, t) + D1 △S, dt κ + W (x, t) ∂I W (x, t)S(x, t) = βW + βh S(x, t)I(x, t) − (γ + µ + u1 )I(x, t) + D2 △I, dt κ + W (x, t) ∂W = ξI(x, t) − δW (x, t) + D3 △W, dt

(1) (2) (3)

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∂R = γI(x, t) − µR(x, t) + D4 △R, dt

(4)

where S(x, t), I(x, t), R(x, t) and W (x, t) denote the susceptible, the infected, the recovered populations and the density of V. cholerae at location x and time t, respectively. The parameters βh and βW denote the concentrations of the hyperinfectious (HI) and less-infectious (LI) vibrios, respectively. µ represents the natural death rate that is not related to the disease, u1 defines the rate of disease-related death, κ is the concentration of vibrios in contaminated water, ξ the natural decay rate of V. cholerae, δ the bacterial death rate, γ the recovery rate, and Di (i = 1, 2, 3, 4) are the diffusion coefficients. Ω is a bounded in Rn with smooth boundary ∂Ω, △ is the Laplacian operator, ∑n domain ∂2 that is △ = i=1 ∂x2 with n is the number of spatial dimensions of the domain Ω. The i Neumann boundary conditions of the model system are: ∂S ∂I ∂W ∂R = = = = 0, x ∈ ∂Ω. dt dt dt dt

(5)

In the case that the diffusion coefficients Di are all equal to zero, according to Wang and Wang’s [25], we know that the basic reproduction number is given by: R0 =

Λ (ξβW + δκβh ). µδκ(γ + µ + u1 )

(6)

And the disease-free equilibrium E0 (S0 , I0 , W0 , R0 ) is ( Λµ , 0, 0, 0), the endemic equilibrium E ∗ (S ∗ , I ∗ , W ∗ , R∗ ) is determined by: S∗ =

Λ (γ + µ + u1 )I ∗ ∗ βe S ∗ δκ ξI ∗ ∗ γI ∗ ∗ − ,I = − , W = ,R = . µ µ γ + µ + u 1 − βh S ∗ ξ δ µ

Wang and Wang’ paper [25] also established the following results: Theorem 1 Assume Di = 0, then for model system (1-4), (1) the disease-free equilibrium E0 is locally and globally asymptotically stable if R0 < 1; and (2) if R0 > 1, the unique chronic infection equilibrium E ∗ is globally asymptotically stable. In this paper, we consider the cholera spatially dependent model proposed in Wang and wang [25] and construct an NSFD scheme for this model. As far as we know, there are few studies on the continuous cholera models designed as discrete equations. The rest of the paper is organized as follows. In the next section, we construct a discretized cholera model with diffusion from the continuous model by using the nonstandard finite difference method. In Section 3 and Section 4, the global asymptotic stability analysis of the equilibria is performed by using discrete Lyapunov functions. In Section 5, we carry out the numerical study of the discrete model, which confirms our theoretical results. Finally, the conclusions are summarized in Section 6.

2

A discretized model

be the space Assume Ω = [a, b] with a, b ∈ R, let △t be the time step size and △x = (b−a) N step size, tk = k△t for k ∈ N be the time mesh point, where N is the set of all non-negative 3 1002

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integers. The space mesh point is Xn = n∆x for n ∈ {0, 1, · · · , N }. At each point, we denote approximations of S(xn , tk ), I(xn , tk ), W (xn , tk ) and R(xn , tk ) by Snk , Ink , Wnk and Rnk , respectively. For the sake of convenience, a (N + 1)− dimensional vector U k = (U0k , U1k , · · · , UNk )T

(7)

is used to represent all the approximation solutions at the time tk . The notation (·)T denotes the transposition of a vector, and all components of a vector U are non-negative. We construct the following NSFD method for model system (1-4): Snk+1 − Snk △t k+1 In − Ink △t Wnk+1 − Wnk △t k+1 Rn − Rnk △t

k+1 k+1 Sn+1 − 2Snk+1 + Sn−1 Snk+1 Wnk k+1 k+1 k + D − µS I − β S , (8) 1 h n n n κ + Wnk (△x)2 k+1 k+1 − 2Ink+1 + In−1 In+1 S k+1 Wnk k+1 k+1 k + D − (γ + µ + u )I I = βW n + β S (9) , 2 1 n h n n κ + Wnk (△x)2 k+1 W k+1 − 2Wnk+1 + Wn−1 = ξInk+1 − δWnk+1 + D3 n+1 , (10) (△x)2 k+1 Rk+1 − 2Rnk+1 + Rn−1 = γInk+1 − µRnk+1 + D4 n+1 , (11) (△x)2

= Λ − βW

with discrete initial value conditions Sn0 = ψ1 (xn ), In0 = ψ2 (xn ), Wn0 = ψ3 (xn ), Rn0 = ψ4 (xn ), for n ∈ {0, 1, · · · , N }, and discrete boundary condition is given as: k k k k k k k k k k k k k k k S−1 = S0k , SN = SN +1 , I−1 = I0 , IN = IN +1 , W−1 = W , WN = WN +1 , R−1 = R0 , RN = RN +1 .

It is easy to check that the solutions of the discrete system (8-11) are positive, and have the disease-free equilibrium E0 and the chronic infection equilibrium E ∗ , which are the same as that of the model (1-4).

3

Global stability of the disease-free equilibrium

Since R does not appear in the first three equations of the system (8-11), we only need to study the system (8-10). In this section, we establish the global stability of the disease-free equilibrium of system (8-10) by constructing a discrete Lyapunov function. Theorem 2 If R0 < 1, the disease-free equilibrium E0 of the model system (8-10) is globally asymptotically stable. Proof Define a discrete Lyapunov function L

k

N ∑ 1 Sk (γ + µ + u1 )(1 + δ∆t) k = [S0 g( n ) + Ink + Wn ], ∆t S0 ξ n=0

(12)

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where the function g(x) = x − 1 − lnx, x ∈ R+ , clearly, g(x) ≥ 0 with equality only if x = 1. Thus we have Lk ≥ 0 with equality if and only if Snk = S0 , Ink = 0 and Wnk = 0 for all n ∈ {0, 1, . . . , N }. Then, along the trajectory of (8-10), we have L

k+1

N ∑ 1 k+1 Snk γ + µ + u1 −L = [Sn − Snk + S0 ln k+1 + Ink+1 − Ink + (Wnk+1 − Wnk )] ∆t S ξ n n=0 k

δ(γ + µ + u1 ) (Wnk+1 − Wnk ) ξ N k+1 k+1 ∑ Sn+1 − 2Snk+1 + Sn−1 βW Snk+1 Wnk k+1 k k+1 = [2Λ − − β S I − µS + D h n 1 n n κ + Wnk (△x)2 n=0

+

k+1 k+1 − 2Snk+1 + Sn−1 Λβh Ink ΛβW Wnk ΛD1 Sn+1 Λ2 + + − − µSnk+1 µ(κ + Wnk ) µ µSnk+1 (△x)2 k+1 k+1 In+1 − 2Ink+1 + In−1 βW Snk+1 Wnk k+1 k k+1 + + β S I − (γ + µ + u )I + D h n 1 n 2 n κ + Wnk (△x)2 k+1 k+1 − 2Wnk+1 + Wn−1 (γ + µ + u1 ) Wn+1 δ(γ + µ + u1 ) k+1 k+1 Wn + D3 ] + (γ + µ + u1 )In − ξ ξ (△x)2 δ(γ + µ + u1 ) + (Wnk+1 − Wnk ) ξ N ∑ Λ µSnk+1 ≤ [Λ(2 − ) + (γ + µ + u1 )Ink (R0 − 1)] − k+1 µSn Λ n=0 k+1 k+1 k+1 k+1 k+1 SN INk+1 S0k+1 − S−1 I0k+1 − I−1 +1 − SN +1 − IN + D + D + D 1 2 2 (△x)2 (△x)2 (△x)2 (△x)2 k+1 k+1 (γ + µ + u1 ) WNk+1 (γ + µ + u1 ) W0k+1 − W−1 +1 − WN + D3 + D3 ξ (△x)2 ξ (△x)2 N ∑ Λ µSnk+1 = [Λ(2 − − ) + (γ + µ + u1 )Ink (R0 − 1)]. k+1 µS Λ n n=0

+ D1

Since 2 −

Λ k+1 µSn k+1

−

k+1 µSn Λ k

≤ 0 by the arithmetic-geometric inequality, it then follows that if

R0 < 1, L − L < 0, for all k ∈ N and the equality holds if and only if Snk+1 = Λµ . This yields that {Lk } is a monotone decreasing sequence. Thus, there exists a constant L0 such that limk→+∞ (Lk+1 − Lk ) = 0. Therefore, we have limk→+∞ Snk = Λµ , limk→+∞ Ink = 0, limk→+∞ Wnk = 0, for all n ∈ {0, 1, . . . N }. Hence, E0 is globally asymptotically stable when R0 < 1. This completes the proof.

4

Global Stability of the chronic infection equilibrium

In this section we concern with the global stability of the chronic infection steady state of system (8-10) when R0 > 1.

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Theorem 3 If R0 > 1, the chronic infection equilibrium E ∗ of the model system (8-10) is globally asymptotically stable. Using the expression for S ∗ along with system (8-10) and discrete boundary conditions, we first have N ∑ 1 S k+1 Sk [g( n ∗ ) − g( n∗ )] ∆t S S n=0 N ∑ S k+1 − S ∗ 1 ≤ [(Snk+1 − Snk )( n ∗ k+1 )] ∆t S Sn n=0 N k+1 k+1 ∑ Sn+1 − 2Snk+1 + Sn−1 1 βW Snk+1 Wnk S∗ k+1 k k+1 = [(Λ − + β S I − µS + D )(1 − )] h n 1 n n ∗ k 2 k+1 S κ + W (△x) S n n n=0 N ∑ 1 βW S ∗ W ∗ βW Snk+1 Wnk S∗ ∗ ∗ ∗ k+1 k k+1 = [( + β S I + µS − − β S I − µS )(1 − )] h h n n n ∗ ∗ k k+1 S κ + W κ + W S n n n=0

+

=

N k+1 k+1 ∑ Sn+1 − 2Snk+1 + Sn−1 S∗ 1 [(D )(1 − )] 1 ∗ 2 k+1 S (△x) S n n=0 N ∑

[−

n=0

µ(Snk+1 − S ∗ )2 S∗ βW W ∗ (κ + W ∗ )Snk+1 Wnk (1 − + )(1 − ) S ∗ Snk+1 κ + W∗ Snk+1 (κ + Wnk )S ∗ W ∗

k+1 ∑ (Sn+1 − Snk+1 )2 Snk+1 Ink S∗ . + βh I (1 − k+1 )(1 − ∗ ∗ )] − D1 k+1 k+1 Sn S I (△x)2 Sn+1 Sn n=0 ∗

N −1

In the same way, we have N ∑ 1 I k+1 Ik [g( n ∗ ) − g( n∗ )] ∆t I I n=0 N ∑ 1 I k+1 − I ∗ ≤ [(Ink+1 − Ink )( n ∗ k+1 )] ∆t I In n=0

=

N ∑ 1 βW Snk+1 Wnk [ + βh Snk+1 Ink − (γ + µ + u1 )Ink+1 ∗ k I κ + W n n=0

k+1 k+1 In+1 − 2Ink+1 + In−1 I∗ + D2 ( )(1 − )] (△x)2 Ink+1 N ∑ βW I ∗ S k+1 Wnk Ink+1 S ∗ W ∗ I ∗ Snk+1 Ink Ink+1 ∗ = [ ∗ (1 − k+1 )( n − ) + β S (1 − )( − ∗ )] h I In κ + Wnk (κ + W ∗ )I ∗ Ink+1 S ∗ I ∗ I n=0 N k+1 I k+1 − 2Ink+1 + In−1 1 ∑ I∗ + ∗ [(D2 n+1 )(1 − )] I n=0 (△x)2 Ink+1

=

N ∑ βW I ∗ S k+1 Wnk Ink+1 S ∗ W ∗ I ∗ Snk+1 Ink Ink+1 ∗ [ ∗ (1 − k+1 )( n − ) + β S (1 − )( ∗ ∗ − ∗ )] h k ∗ )I ∗ k+1 I I κ + W (κ + W I S I I n n n n=0

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− D2

N −1 ∑ n=0

k+1 (In+1 − Ink+1 )2 . k+1 k+1 (△x)2 In+1 In

Similarly, by letting ξI ∗ = δW ∗ , we obtain: N ∑ W k+1 Wk 1 [g( n ∗ ) − g( n∗ )] ∆t W W n=0 N k+1 ∑ − W∗ 1 k Wn k+1 ≤ [(Wn − Wn )( )] ∆t W ∗ Wnk+1 n=0 N k+1 k+1 ∑ Wn+1 − 2Wnk+1 + Wn−1 1 W∗ k+1 k+1 + D = − δW [(ξI )(1 − )] 3 n n W∗ (△x)2 Wnk+1 n=0 N N k+1 k+1 ∑ ∑ − 2Wnk+1 + Wn−1 δ W ∗ W ∗ Ink+1 1 Wn+1 W∗ k+1 = [ ∗ (1 − k+1 )( − W )] + [( )(1 − )] n W Wn I∗ W∗ (△x)2 Wnk+1 n=0 n=0 N N −1 k+1 ∑ ∑ (Wn+1 − Wnk+1 )2 δ W ∗ W ∗ Ink+1 k+1 = [ ∗ (1 − k+1 )( − W )] − D . 3 n k+1 W Wn I∗ (△x)2 Wn+1 Wnk+1 n=0 n=0

We then define the following Lyapunov function: Hk =

N ∑ 1 1 Snk 1 Ink βW Wnk [ g( ) + g( ) + g( )]. ∗ ∗ ∗ ∗ ∗ ∗ ∆t β I S β S I β δI W h h h n=0

(13)

Thus, H k ≥ 0 for all k ∈ N, with equality if and only if Snk = S ∗ , Ink = I ∗ and Wnk = W ∗ for all n ∈ {0, 1, . . . N }. The difference of H k is: H

k+1

−H = k

N ∑ n=0

+

βW ( δβh I ∗

− D2

N −1 ∑

n=0 N ∑

[

Snk 1 Ink+1 − Ink Ink 1 Snk+1 − Snk ( + ln ) + ( + ln ) βh I ∗ S∗ Snk+1 βh S ∗ I∗ Ink+1

k+1 ∑ (Sn+1 − Snk+1 )2 − Wnk Wnk + ln )] − D 1 k+1 k+1 W∗ Wnk+1 (∆x)2 Sn+1 Sn n=0 N −1

Wnk+1

k+1 k+1 ∑ (Wn+1 (In+1 − Ink+1 )2 − Wnk+1 )2 − D 3 k+1 k+1 k+1 (∆x)2 In+1 In (∆x)2 Wn+1 Wnk+1 n=0 N −1

S∗ Ink+1 Snk+1 Ink Ink+1 µ(Snk+1 − S ∗ )2 ≤ {− + (2 − k+1 − ∗ − k+1 ∗ + ∗ ) βh Snk+1 S ∗ I ∗ Sn I In S I n=0 βW W ∗ S∗ Ink+1 Snk+1 Wnk I ∗ (κ + W ∗ ) Wnk (κ + W ∗ ) [ + + k+1 − − 2] βh I ∗ (κ + W ∗ ) Snk+1 I∗ In (κ + Wnk )S ∗ W ∗ (κ + Wnk )W ∗ βW W ∗ Wnk+1 Ink+1 W ∗ Ink+1 − ( + k+1 ∗ − ∗ − 1)} βh I ∗ (κ + W ∗ ) W ∗ Wn I I N −1 N −1 N −1 k+1 k+1 k+1 ∑ ∑ ∑ (Sn+1 − Snk+1 )2 (In+1 − Ink+1 )2 (Wn+1 − Wnk+1 )2 − D1 − D − D 2 3 k+1 k+1 k+1 k+1 k+1 (∆x)2 Sn+1 Sn (∆x)2 In+1 In (∆x)2 Wn+1 Wnk+1 n=0 n=0 n=0

−

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N ∑ S∗ Ink+1 Snk+1 Ink Ink+1 µ(Snk+1 − S ∗ )2 − [g( ) + ) + g( − ln ] ≤ {− βh Snk+1 S ∗ I ∗ Snk+1 Ink+1 S ∗ I∗ Ink n=0

βW W ∗ S∗ Ink+1 Snk+1 I ∗ (κ + W ∗ ) κ + W ∗ − [ + + − 2] βh I ∗ (κ + W ∗ ) Snk+1 I∗ Ink+1 S ∗ W ∗ W∗ Wnk+1 Ink+1 W ∗ Ink+1 βW W ∗ ( + k+1 ∗ − ∗ − 1)} − βh I ∗ (κ + W ∗ ) W ∗ Wn I I N −1 N −1 N −1 k+1 k+1 k+1 ∑ ∑ ∑ (Sn+1 − Snk+1 )2 (In+1 − Ink+1 )2 (Wn+1 − Wnk+1 )2 − D1 − D2 − D3 k+1 k+1 k+1 k+1 k+1 (∆x)2 Sn+1 Sn (∆x)2 In+1 In (∆x)2 Wn+1 Wnk+1 n=0 n=0 n=0 −

N ∑ S∗ µ(Snk+1 − S ∗ )2 Snk+1 Ink − g( ) ≤ {− ) − g( βh Snk+1 S ∗ I ∗ Snk+1 Ink+1 S ∗ n=0

βW W ∗ S∗ Snk+1 Ink+1 (κ + W ∗ ) Wnk+1 Ink+1 W ∗ [g( ) + g( ) + g( ) + g( )]} βh I ∗ (κ + W ∗ ) Snk+1 Ink+1 S ∗ W ∗ W∗ Wnk+1 I ∗ N −1 N −1 N −1 k+1 k+1 k+1 ∑ ∑ ∑ (In+1 − Ink+1 )2 (Wn+1 − Wnk+1 )2 (Sn+1 − Snk+1 )2 − D − D . − D1 2 3 2 S k+1 S k+1 2 I k+1 I k+1 2 W k+1 W k+1 (∆x) (∆x) (∆x) n+1 n+1 n+1 n n n n=0 n=0 n=0

−

It is easy to see H k+1 − H k ≤ 0 for all k ∈ N. Then there exists a constant H ∗ such that limk→+∞ (H k+1 − H k ) = 0, which implies limk→+∞ Snk = S ∗ . Combined with system (8-10), we have limk→+∞ Ink = I ∗ and limk→+∞ Wnk = W ∗ as well, for all n ∈ {0, 1, . . . N }. Hence, E ∗ is globally asymptotically stable when R0 > 1. This completes the proof.

5

Numerical results

In this section, we propose numerical simulations to verify the stability properties of the NSFD scheme. We use the data regarding the course of the cholera in Zimbabwe during 2008-2009, which is the worst outbreak in Africa in the past 30 years with over 100,000 humans have been infected and more than 4,300 killed. The total population in Zimbabwe is 12,347,240, for mathematical simplicity, we scale down all data numbers by a factor of 1,200. All epidemiological parameter values for cholera in literature are given as: Λ = 4.5, µ = 0.000442, ξ = 70, δ = 0.2333, u1 = 0.04, γ = 1.4, κ = 1000000 ( [2, 4, 5, 9]). In addition, the initial values are taken as I(x, 0) = 10 × exp(−x), S(x, 0) = 1000 × exp(−x), W (x, 0) = 10 × exp(−x), and R(x, 0) = 10 × exp(−x), where x ∈ [0, 50]. Let the grid sizes used in the simulation are ∆x = 0.5 and ∆t = 0.1, respectively, and the diffusion coefficients Di are all fixed as 0.01. The discussions in ( [2,4,5,9]) indicate that parameters βW and βh are sensitive and vary from place to place, so we first set βW = 0.0001 and βh = 0.0001, which renders R0 = 0.7070 < 1. Hence, model system has a diseasefree equilibrium in this case, the number of infectious decreases quickly and the disease dies out. It can be observed from Figure 1, where the steady state approaches to E0 = (0.6, 0, 0). For the other case, we choose βW = 0.0001, βh = 0.000236 and do not change the other parameter values, which gives R0 = 1.6683 > 1, the chronic infection steady state is E ∗ = (0.5135, 1899.14, 8200.46) by calculation, the infected steady state is stable as can

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be observed numerically in Figure 2. We then examine the case with different sets of initial conditions when R0 > 1, also obtain almost the same patterns. Figure 3 compares the profile when we choose two different combinations of Di , as, (0.01, 0.05, 0.01, 0.05) and (0.05, 0.1, 0.05, 0.1) for R0 > 1. Only the distribution of the density of I(x, t) is depicted, similar results for the other two variables S(x, t) and R(x, t) are not presented here. Comparing Fig. 3 and Fig 1.(a), we can find that diffusion coefficients have no effect on the convergence of solutions, but the larger diffusion coefficients will deduce the number of infected population and speed up the arrived time at the chronic infection equilibrium. In a addition, we perform numerical simulations of a standard finite difference (SFD) scheme to compare the results with NSFD scheme using the same discrete boundary conditions and parameter values in Figure 4. The stronger competitiveness of NSFD scheme has been proved by its succsess in preserving the global stability of equilibrium and the failure of the SFD method.

Numbers Infectious

10 8 6 4 2 0 60 50

40

40 30

20

t

20 0

10 0

x

(a)

(b)

(c)

Figure 1: Graphs of the numerical solutions of the NSFD method when R0 < 1.

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6

Conclusions and discussions

In this article, we derive a discrete cholera infection model with spatial diffusion by using an NSFD method. We show that the disease-free steady state of the discrete model is globally

10000

100

Susceptible Numbers

Numbers Infectious

120

80 60 40 20

8000 6000 4000 2000

0 60

0 60

50

40

t

50

40

40

40

30

20

20 0

10 0

30

20

20

t

x

10

0

(a)

0

x

(b)

Recovered Numbers

10000 8000 6000 4000 2000 0 60 50

40

40 30

20

20

t

10

0

0

x

(c)

60

30

50

25

Numbers Infectious

Numbers Infectious

Figure 2: Graphs of the numerical solutions of the NSFD method when R0 > 1.

40 30 20 10 0 60

20 15 10 5 0 60

50

40

40 30

20

t

10 0

40 30

20

20 0

50

40

t

x

(a)

20 0

10 0

x

(b)

Figure 3: Dynamics of infected population when R0 > 1 for two different sets of Di . 10 1009

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asymptotically stable if the basic reproduction number R0 < 1, and the chronic infection equilibrium is globally asymptotically stable when R0 > 1. In a word, our results (Theorem 2 and Theorem 3) imply that the discretization scheme (8-11) is dynamically consistent with the continuous system with respect to the globally asymptotical stability of the steady-state solutions. Our simulation results also conclude that the diffusion coefficients have no relation to the global stability of such cholera epidemic. Finally, numerical results show the advantage of our method in comparison to an SFD method. Application of this method to the general delayed discrete epidemic models is our future work.

Acknowledgments This work was partially supported by the Natural Science Foundation of China (11401059), the Natural Science Foundation of CQ (cstc2015jcyjA00024, cstc2017jcyjAX0067), Scientific and Technological Research Program of Chongqing Municipal Education Commission (KJ1600610, KJ1706163).

Numbers Infectious

1000 800 600 400 200 0 60 50

40

40 30

20

t

20 0

10 0

x

(a)

(b)

(c)

Figure 4: Graphs of the numerical solutions of the SFD method.

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References [1] Code¸co CT, Endemic and epidemic dynamics of cholera: the role of the aquatic reservoir, BMC Infectious Diseases, 1. 1, 2001. [2] Hartley DM, Morris JG and Smith DL, Hyperinfectivity: A critical element in the ability of V. cholerae to cause epidemics?, PLoS Medicine, 3(1): 63-69, 2006. [3] Tien JH and Earn DJD, Multiple transmission pathways and disease dynamics in a waterborne pathogen model, Bulletin of Mathematical Biology, 72(6): 1502-1533, 2010. [4] Mukandavire Z, Liao S, Wang J, Gaff H, Smith DL and Morris JG, Estimating the reproductive numbers for the 2008–2009 cholera outbreaks in Zimbabwe, Proceedings of the National Academy of Sciences of the United States of America, 108(21): 8767-8772, 2011. [5] Liao S and Wang J, Stability analysis and application of a mathematical cholera model, Mathematical Biosciences and Engineering, 8(3): 733-752, 2011. [6] Bertuzzo E, Casagrandi R, Gatto M, Rodriguez-Iturbe I and Rinaldo A, on spatially explicit models of cholera epidemics, Journal of the Royal Society Interface, 7(43): 321333, 2010. [7] Safi MA, Melesse DY and Gumel AB, Analysis of a Multi-strain Cholera Model with an Imperfect Vaccine, Bulletin of Mathematical Biology, 75(7): 1104-1137, 2013. [8] Misra AK and Singh V, A delay mathematical model for the spread and control of water borne diseases, Journal of Theoretical Biology, 301(5): 49-56, 2012. [9] Liao S and Yang W, On the dynamics of a vaccination model with multiple transmission ways, International Journal of Applied Mathematics and Computer Science, 23(4): 761772, 2013. [10] Tuite AR, Tien JH, Eisenberg MC, Earn DJD, Ma J and Fisman DN, Cholera Epidemic in Haiti: Using a Transmission Model to Explain Spatial Spread of Disease and Identify Optimal Control Interventions, Annals of Internal Medicine, 154(2011): 293-302, 2010. [11] Capone F, De CV and De LR, Influence of diffusion on the stability of equilibria in a reactionCdiffusion system modeling cholera dynamic, Mathematical Biology, 71(5): 1107-1131, 2015. [12] Mickens RE, Exact solutions to a finite difference model of a nonlinear reactionadvection equation: implications for numerical analysis, Numerical Methods for Partial Differential Equations, 5: 313-325, 1989. [13] Mickens RE, Nonstandard Finite difference models of differential equations, World Scientific, Singapore, 1994.

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[14] Mickens RE, Dynamic consistency: a fundamental principle for constructing nonstandard finite difference schemes for differential equations, Journal of Difference Equations & Applications, 11(7): 645-653, 2005. [15] Mickens RE, Calculation of denominator functions for nonstandard finite difference schemes for differential equations satisfying a positivity condition, Numerical Methods for Partial Differential Equations, 23(3): 672-691, 2006. [16] Mickens RE, Discretizations of nonlinear differential equations using explicit nonstandard methods, Journal of Computational and Applied Mathematics, 110(1): 181-185, 1999. [17] Cui Q, Yang X and Zhang Q, An NSFD scheme for a class of SIR epidemic models with vaccination and treatment, Journal of Difference Equations and Applications, 20(3): 416-422, 2014. [18] Suryanto A, Kusumawinahyu WM, Darti I and Yanti I, Dynamically consistent discrete epidemic model with modified saturated incidence rate, Applied Mathematics and Computation, 32(2): 373-383, 2013. [19] Qin W, Wang L and Ding X, A non-standard finite difference method for a hepatitis B virus infection model with spatial diffusion, Journal of Difference Equations and Applications, 20(12): 1641-1651, 2014. [20] Manna K and Chakrabarty SP, Global stability and a non-standard finite difference scheme for a diffusion driven HBV model with capsids, Journal of Difference Equations and Applications, 21(10): 918-933, 2015. [21] Villanueva R, Arenas A and Gonzalez-Parra G, A nonstandard dynamically consistent numerical scheme applied to obesity dynamics, Journal of Applied Mathematics, Article ID 640154, 2008. [22] Jodar L, Villanueva RJ, Arenas AJ and Gonzalez GC, Nonstandard numerical methods for a mathematical model for influenza disease, Mathematics and Computers in Simulation, 79(3): 622-633, 2008. [23] Arenas AJ, Gonzalez-Parra G and Chen-Charpentier BM, A nonstandard numerical scheme of predictor-corrector type for epidemic models, Computers and Mathematics with Applications, 59(12): 3740-3749, 2010. [24] Garba SM, Gumel AB and Lubuma JMS, Dynamically-consistent non-standard finite difference method for an epidemic model, Mathematical and Computer Modelling, 53(12): 131-150, 2011. [25] Wang X and Wang J, Analysis of cholera epidemics with bacterial growth and spatial movement, Journal of Biological Dynamics, 9(sup1): 233-261, 2014.

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On the Higher Order Di®erence Equation xn+1 = ®xn + ¯xn¡l + °xn¡k +

1

axn xn¡k bxn + cxn¡l + dxn¡k

M. M. El-Dessoky1;2 and K. S. Al-Basyouni1 King Abdulaziz University, Faculty of Science, Mathematics Department, P. O. Box 80203, Jeddah 21589, Saudi Arabia. 2 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt. E-mail: [email protected]; [email protected] ABSTRACT

The main objective of this paper is to investigate the global stability of the solutions, the boundedness and the periodic character of the nonlinear di¤erence equation xn+1 = ®xn + ¯xn¡l + °xn¡k +

axn xn¡k ; bxn + cxn¡l + dxn¡k

n = 0; 1; :::;

where the parameters ®; ¯; °; a; b; c and d are positive real numbers and the initial conditions x¡s ; x¡s+1 ; :::; x¡1 , x0 are positive real numbers where s = maxfl; kg. Some numerical examples will be given to explicate our results. Keywords: Di¤erence equations, Stability, Global stability, Boundedness, Periodic solutions. Mathematics Subject Classi…cation: 39A10 —————————————————

1. INTRODUCTION Our goal is to study some qualitative behavior of the solutions of the di¤erence equation xn+1 = ®xn + ¯xn¡l + °xn¡k +

axn xn¡k ; bxn + cxn¡l + dxn¡k

n = 0; 1; :::;

(1)

where the parameters ®; ¯; °; a; b; c and d are positive real numbers and the initial conditions x¡s ; x¡s+1 ; :::; x¡1 , x0 are positive real numbers where s = maxfl; kg. Recently there has been a great interest in studying the qualitative properties of rational di¤erence equations. For the systematical studies of rational and nonrational di¤erence equations, one can refer to the papers [1-270] and references therein. Ibrahim [4] investigated the global attractivity of the positive solutions of the di¤erence equation xn+1 =

xn¡(2k+1) 1+xn¡k xn¡(2k+1) ;

n = 0; 1; ::::

Zayed et al. et al. [5] studied the periodicity, the boundedness and the global stability of the positive solution of the di¤erence equation, ®xn +¯xn¡1 +°xn¡2 +±xn¡3 xn+1 = Ax ; n = 0; 1; :::: n +Bxn¡1 +Cxn¡2 +Dxn¡3 In [6] El-Dessoky investigated the global stability character and the periodicity of solutions of the recursive sequence n¡l +bxn¡k xn+1 = ax c+dxn¡l xn¡k ; n = 0; 1; :::: Guo-Mei Tang et al. [7] obtained the global behavior of solutions of the following nonlinear di¤erence equation xn+1 =

®+xn A+Bxn +xn¡k ;

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Papaschinopoulos et al. [8] studied the asymptotic behavior and the periodicity of the positive solutions of the nonautonomous di¤erence equation p

xn+1 = An +

xn¡1 ; xqn

n = 0; 1; ::::

El-Dessoky [9] obtained the global stability, the boundedness and the periodicity of the nonlinear di¤erence equation dxn¡s xn+1 = axn + bxn¡k + cxn¡l ¡ exn¡s n = 0; 1; :::: ¡®xn¡t ; Nirmaladevi et al. [10] studied the periodicity solution and the global stability of nonlinear di¤erence equation yn+1 = P yn + Qyn¡k + Ryn¡l +

byn¡k dyn¡k ¡eyn¡l ;

n = 0; 1; ::::

"Let I be some interval of real numbers and let F : I s+1 ! I; be a continuously di¤erentiable function. Then for every set of initial conditions x¡s ; x¡s+1 ; :::; x0 2 I; the di¤erence equation xn+1 = F (xn ; xn¡1 ; :::; xn¡s ); n = 0; 1; :::; (2) has a unique solution fxn g1 n=¡s .

Definition 1.1. (Equilibrium Point) A point x 2 I is called an equilibrium point of the di¤erence equation (2) if x = F (x; x; :::; x).

That is, xn = x for n ¸ 0; is a solution of the di¤erence equation (2), or equivalently, x is a …xed point of F:

Definition 1.2. (Stability) Let x 2 (0; 1) be an equilibrium point of the di¤erence equation (2). Then, we have (i) The equilibrium point x of the di¤erence equation (2) is called locally stable if for every ²> 0; there exists ± > 0 such that for all x¡s ; :::; x¡1 ; x0 2 I with jx¡t ¡ xj + ::: + jx¡1 ¡ xj + jx0 ¡ xj < ±; we have jxn ¡ xj < ²

for all n ¸ ¡t:

(ii) The equilibrium point x of the di¤erence equation (2) is called locally asymptotically stable if x is locally stable solution of equation (2) and there exists °> 0; such that for all x¡t ; :::; x¡1 ; x0 2 I with jx¡s ¡ xj + ::: + jx¡1 ¡ xj + jx0 ¡ xj < °; we have lim xn = x:

n!1

(iii) The equilibrium point x of the di¤erence equation (2) is called global attractor if for all x¡s ; :::; x¡1 ; x0 2 I; we have lim xn = x: n!1

(iv) The equilibrium point x of the di¤erence equation (2) is called globally asymptotically stable if x is locally stable, and x is also a global attractor of the di¤erence equation (2). (v) The equilibrium point x of the di¤erence equation (2) is called unstable if x is not locally stable.

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Definition 1.3. (Periodicity) 1 A sequence fxn g1 n=¡s is said to be periodic with period p if xn+p = xn for all n ¸ ¡t: A sequence fxn gn=¡s is said to be periodic with prime period p if p is the smallest positive integer having this property. Definition 1.4. Equation (2) is called permanent and bounded if there exists numbers M and m with 0 < m < M < 1 such that for any initial conditions x¡s ; :::; x¡1 ; x0 2 (0; 1) there exists a positive integer N which depends on these initial conditions such that m · xn · M

for all n > N:

Definition 1.5. The linearized equation of the di¤erence equation (2) about the equilibrium x is the linear di¤erence equation s X @F (x; x; :::; x) yn+1 = yn¡i : (3) @xn¡i i=0

Now, assume that the characteristic equation associated with (3) is

p(¸) = p0 ¸s + p1 ¸s¡1 + ::: + ps¡1 ¸+ ps = 0;

(4)

where pi =

@F (x;x;:::;x) : @xn¡i

Theorem 1.6. [1]: Assume that pi 2 R; i = 1; 2; :::; s and s is non-negative integer. Then s X i=1

jpi j < 1;

is a su¢cient condition for the asymptotic stability of the di¤erence equation yn+s + p1 yn+s¡1 + ::: + ps yn = 0; n = 0; 1; ::: :

Theorem 1.7. [1]: Consider the the di¤erence equation (2) where F 2 C(I t+1 ; R) and I is an open interval of real numbers. Let x be an equilibrium point of the di¤erence equation (2). Finally, suppose that F satis…es the following two conditions: (i) F is nondecreasing in each of its argements. (ii) F satis…es the negative feedback property [F (x; x; :::; x) ¡ x] (x ¡ x) < 0; for all x 2 I ¡ f0g : Then the equilibrium point x isa global attractor of all solutions of the di¤erence equation (2)."

2. LOCAL STABILITY In this section, we study the local stability character of the equilibrium point of equation (1). Equation (1) has equilibrium point and is given by x = ®x + ¯x + °x +

ax2 bx+cx+dx ;

[(1 ¡ ® ¡ ¯¡ °) (b + c + d) ¡ a] x2 = 0: If (1 ¡ ® ¡ ¯¡ °) (b + c + d) 6= a, then the equilibrium point of the di¤erence equation (1) is x = 0.

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Let f : (0; 1)3 ¡! (0; 1) be a continuous function de…ned by f (u; v; w) = ®u + ¯v + °w +

auw bu+cv+dw :

Therefore, it follows that @f(u; v; w) @u

aw(cv+dw) ; @f (u;@vv; w) (bu+cv+dw)2

= ®+

= ¯¡

acuw (bu+cv+dw)2

and

@f(u; v; w) @w

= °+

au(bu+cv) : (bu+cv+dw)2

Theorem 2.1. The zero equilibrium x of the di¤erence equation (1) is locally asymptotically stable if (® + ¯+ °) (b + c + d) + a < 1:

(5)

Proof: So, we can write Eq. (6) at zero equilibrium point x = 0 @f(x; x; x) @u

and

@f(x; x; x) @w

= ®+

a(c+d) (b+c+d)2

= p1 ;

= °+

a(b+c) (b+c+d)2

= p3 :

@f (x; x; x) @v

= ¯¡

ac (b+c+d)2

= p2

Then the linearized equation of equation (1) about x is yn+1 ¡ p1 yn¡k ¡ p2 yn¡l ¡ p3 yn¡s = 0; It follows by Theorem 1 that, equation (1) is asymptotically stable if and only if jp1 j + jp2 j + jp3 j < 1: Thus,

¯ ¯ ¯® +

and so

¯

a(c+d) ¯ ¯ (b+c+d)2

®+

a(c+d) (b+c+d)2

¯ ¯ + ¯¯¡ + ¯¡

¯

¯ ac ¯ (b+c+d)2 ac (b+c+d)2

® + ¯+ °+

¯ ¯ + ¯°+

+ °+

a(b+c+d) (b+c+d)2

¯

a(b+c) ¯ ¯ (b+c+d)2

a(b+c) (b+c+d)2

< 1;

< 1;

< 1;

(®+ ¯+ °) (b + c + d) + a < 1: The proof is complete. Example 1. Consider l = 2; k = 3; ® = 0:3; ¯ = 0:02; ° = 0:01; a = 0:1; b = 0:2; c = 0:3 and d = 0:7 and the initial conditions x¡3 = 0:2; x¡2 = 0:4; x¡1 = 0:6 and x0 = 0:1; the zero solution of the di¤erence equation (1) is local stability (see Fig. 1). plot of x(n+1)= alfa X(n)+ beta X(n-l)+ gamma X(n-k)+(a X(n)X(n-k)/(b X(n)+ c X(n-l)+ d X(n-k))) 0.7

0.6

0.5

x(n)

0.4

0.3

0.2

0.1

0

0

5

10

15

20

25 n

30

35

40

45

50

Figure 1. Sketch the behavior of zero solution of equation (1) is local stable.

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Example 2. The solution of the di¤erence equation (1) is unstable if l = 2; k = 3; ® = 0:3; ¯ = 0:2; °= 0:1; a = 0:5; b = 0:2; c = 0:3 and d = 0:7 and the initial conditions x¡3 = 0:2; x¡2 = 0:4; x¡1 = 0:6 and x0 = 0:1. (See Fig. 2). plot of x(n+1)= alfa X(n)+ beta X(n-l)+ gamma X(n-k)+(a X(n)X(n-k)/(b X(n)+ c X(n-l)+ d X(n-k))) 0.6 0.55 0.5 0.45

x(n)

0.4 0.35 0.3 0.25 0.2 0.15 0.1

0

10

20

30

40

50 n

60

70

80

90

100

Figure 2. Draw the behavior of the solution of equation (1) is unstable.

3. GLOBAL STABILITY In this section, the global asymptotic stability of equation (1) is studied. Theorem 3.1. The equilibrium point x is a global attractor of Eq. (1) if ® + ¯+ °6= 1: Proof: Suppose that ³and ´ are real numbers and assume that F : [³; ´]3 ¡! [³; ´] is a function de…ned by F (x; y; z) = ®x + ¯y + °z +

axz bx+cy+dz :

Then @F (x; y; z) @x

= ®+

az(cy+dz) ; @F (x;@yy; z) (bx+cy+dz)2

= ¯¡

acxz (bx+cy+dz)2

and

@F (x; y; z) @z

= °+

ax(bx+cy) : (bx+cy+dz)2

Now, we can see that the function F (x; y; z) nondecreasing in x; y and z: Then h i ax2 [F (x; x; x) ¡ x] (x ¡ x) = ®x + ¯x + °x + bx+cx+dx ¡ x (x ¡ x) ·µ ¶ ¸ a = ¡ 1 ¡ ® ¡ ¯¡ °¡ x (x ¡ 0) b+c+d ¶ µ a x2 < 0 = ¡ 1 ¡ ® ¡ ¯¡ °¡ b+c+d If ® + ¯+ °+

a b+c+d

< 1; then F (x; y; z) satis…es the negative feedback property [F (x; x; x) ¡ x] (x ¡ x0 ) < 0; for x0 = 0:

According to Theorem 2, then x is a global attractor of Eq. (1). This completes the proof. Example 3. The solution of the di¤erence equation (1) is global stability when l = 2; k = 3; ® = 0:03; ¯ = 0:02; ° = 0:01; a = 0:1; b = 0:2; c = 0:3 and d = 0:7 and the initial conditions x¡3 = 0:2; x¡2 = 0:4; x¡1 = 0:6 and x0 = 0:1. (See Fig. 3).

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plot of x(n+1)= alfa X(n)+ beta X(n-l)+ gamma X(n-k)+(a X(n)X(n-k)/(b X(n)+ c X(n-l)+ d X(n-k))) 0.7

0.6

0.5

x(n)

0.4

0.3

0.2

0.1

0

0

5

10

15

20

25 n

30

35

40

45

50

Figure 3. Plot the behavior of the solution of equation (1) is global stability.

4. BOUNDEDNESS OF THE SOLUTIONS In this section, we investigate the boundedness nature of the positive solutions of equation (1). Theorem 4.1. Every solution of Equation (1) is bounded if one of the following conditions holds: a < 1; ¯ < 1 and °< 1: d a (ii) ® < 1; ¯ < 1 and °+ < 1: b

(i) ® +

Proof: First we prove every solution of Equation (1) is bounded if ® + be a solution of Equation (1). It follows from Equation (1) that xn+1

= ®xn + ¯xn¡l + °xn¡k +

a d

(6) (7)

< 1; ¯ < 1 and °< 1: Let fxn g1 n=¡s

axn xn¡k bxn +cxn¡l +dxn¡k ; axn xn¡k dxn¡k

6 ®xn + ¯xn¡l + °xn¡k + ¡ ¢ = ® + ad xn + ¯xn¡l + °xn¡k < xn + xn¡l + xn¡k : Then

xn+1 < xn + xn¡l + xn¡k for all n ¸ 0: So every solution of Eq. (1) is bounded from above by M = x0 + x¡l + x¡k . Second we prove every solution of Equation (1) is bounded if ® < 1; ¯ < 1 and °+ a solution of Equation (1). It follows from Equation (1) that xn+1

= ®xn + ¯xn¡l + °xn¡k +

a b

< 1: Let fxn g1 n=¡s be

axn xn¡k bxn +cxn¡l +dxn¡k ; axn xn¡k bxn

6 ®xn + ¯xn¡l + °xn¡k + ¡ ¢ = ®xn + ¯xn¡l + °+ ab xn¡k < xn + xn¡l + xn¡k : Then

xn+1 < xn + xn¡l + xn¡k for all n ¸ 0: So every solution of Eq. (1) is bounded from above by M = x0 + x¡l + x¡k . Theorem 4.2. Every solution of Equation (1) is unbounded if ® > 1or ¯ > 1 or °> 1:

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Proof: Let fxn g1 n=¡s be a solution of Equation (1).Then from Equation (1) we see that xn+1 = ®xn + ¯xn¡l + °xn¡k +

axn xn¡k bxn +cxn¡l +dxn¡k

> ®xn for all n ¸ 0:

We see that the right hand side can be written as follows zn+1 = ®zn¡l : then zln+i = ®n zl+i + const:;

i = 0; 1; :::; l;

and this equation is unstable because ® > 1, and lim zn = 1:Then by using ratio test fxn g1 n=¡s is unbounded n!1 from above. Similarly we can prof that every solution of Eq. (1) is unbounded if ¯ > 1 or ° > 1. Thus, the proof is now completed. Example 4. We assume l = 2; k = 3; ® = 1:3; ¯ = 0:2; ° = 0:1; a = 0:1; b = 0:2; c = 0:3 and d = 0:7 and the initial conditions x¡3 = 0:2; x¡2 = 0:4; x¡1 = 0:6 and x0 = 0:1; the solution of the di¤erence equation (1) is unbounded (see Fig. 4). 15

x 10

6 plot

of x(n+1)= alfa X(n)+ beta X(n-l)+ gamma X(n-k)+(a X(n)X(n-k)/(b X(n)+ c X(n-l)+ d X(n-k)))

x(n)

10

5

0

0

5

10

15

20

25 n

30

35

40

45

50

Figure 4. Plot the behavior of the solution of equation (1) is unbounded.

5. EXISTENCE OF PERIODIC SOLUTIONS Theorem 5.1. Suppose that l and k are even positive integers, then equation (1) has no prime period two solutions. Proof: First suppose that there exists a prime period two solution :::P; Q; P; Q; :::; of equation (1). We see from equation (1) when l and k are an even, then xn = xn¡l = xn¡k : It follows equation (1) that aQ 2 P = ®Q + ¯Q + °Q + bQ+cQ+dQ ; and Q = ®P + ¯P + °P +

aP 2 bP +cP +dP

:

Therefore, (b + c + d) P = (b + c + d) (® + ¯+ °)Q + aQ;

(8)

(b + c + d) Q = (b + c + d) (® + ¯+ °)P + aP;

(9)

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Subtracting (9) from (8) gives (b + c + d) (P ¡ Q) = ((b + c + d) (® + ¯+ °) + a) (Q ¡ P ) (P ¡ Q) [(b + c + d) (1 + ® + ¯+ °) + a] = 0 Since (b + c + d) (1 + ® + ¯+ °) + a 6= 0, then p = q. This is a contradiction. Thus, the proof is completed. Theorem 5.2. Let l is even and k is odd positive integers, then equation (1) has no positive prime period two solutions. Proof: First suppose that there exists a prime period two solution :::P; Q; P; Q; :::; of equation (1). We see from equation (1) when l is an even and k is an odd, then xn = xn¡l and xn+1 = xn¡k : It follows equation (1) that aQP P = ®Q + ¯Q + °P + bQ+cQ+dP ; and Q = ®P + ¯P + °Q +

aP Q bP +cP +dQ :

Therefore, (b + c) (1 ¡ °) P Q + dP 2 = (b + c) (® + ¯)Q2 + d(® + ¯)P Q + aQP;

(10)

(b + c) (1 ¡ °) P Q + dQ2 = (b + c) (® + ¯)P 2 + d(® + ¯)P Q + aP Q;

(11)

Subtracting (11) from (10) gives d(P 2 ¡ Q2 ) = (b + c) (® + ¯)(Q2 ¡ P 2 ) (P 2 ¡ Q2 ) (d + (b + c) (® + ¯)) = 0 Then P = §Q. This is a contradiction. Thus, the proof is completed. Theorem 5.3. Suppose that l is odd and k is even positive integers, then equation (1) has no positive prime period two solutions. Proof: First suppose that there exists a prime period two solution :::P; Q; P; Q; :::; of equation (1). We see from equation (1) when k is an even and l is an odd, then xn = xn¡k and xn+1 = xn¡l : It follows equation (1) that aQ2 P = ®Q + ¯P + °Q + bQ+cP +dQ ; and Q = ®P + ¯Q + °P +

aP 2 bP +cQ+dP

:

Therefore, (b + d) (1 ¡ ¯) P Q + cP 2 = (b + d) (® + °)Q2 + c(® + °)P Q + aQ2 ;

(12)

(b + d) (1 ¡ ¯) P Q + cQ2 = (b + d) (® + °)P 2 + c(® + °)P Q + aP 2 ;

(13)

Subtracting (13) from (12) gives c(P 2 ¡ Q2 ) = ((b + d) (® + °) + a) (Q2 ¡ P 2 )

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(P 2 ¡ Q2 ) [c ((b + d) (® + °) + a)] = 0 Then P = §Q. This is a contradiction. Thus, the proof is completed. Theorem 5.4. Let l, k are odd positive integers. If (1 + ® ¡ ¯¡ °) (b + c + d) ¡ a 6= 0; then Eq. (1) has no prime period two solution. Proof: First suppose that there exists a prime period two solution :::P; Q; P; Q; :::; of equation (1). We see from equation (1) when l and k are an odd, then xn+1 = xn¡l = xn¡k : It follows equation (1) that aP 2 P = ®Q + ¯P + °P + bP +cP +dP ; and Q = ®P + ¯Q + °Q +

aQ 2 bQ+cQ+dQ :

Therefore, (1 ¡ ¯¡ °) (b + c + d) P = ®(b + c + d) Q + aP;

(14)

(1 ¡ ¯¡ °) (b + c + d) Q = ®(b + c + d) P + aQ;

(15)

Subtracting (15) from (14) gives (1 ¡ ¯¡ °) (b + c + d) (P ¡ Q) = ®(b + c + d) (Q ¡ P ) + a(P ¡ Q) (P ¡ Q) [(1 + ® ¡ ¯¡ °) (b + c + d) ¡ a] = 0 Since (1 + ® ¡ ¯¡ °) (b + c + d) ¡ a 6= 0, then p = q. This is a contradiction. Thus, the proof is completed.

Acknowledgements

This article was funded by the Deanship of Scienti…c Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR technical and …nancial support.

REFERENCES 1. V. L. Kocic, and G. Ladas, Global Behavior of Nonlinear Di¤erence Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, 1993. 2. M. R. S. Kulenovic and G. Ladas, Dynamics of Second Order Rational Di¤erence Equations with Open Problems and Conjectures, Chapman & Hall / CRC Press, 2001. 3. E. A. Grove and G. Ladas, Periodicities in nonlinear di¤erence equations, Vol. 4, Chapman & Hall / CRC Press, 2005. 4. I. Yalcinkaya, On The Global Attractivity of Positive Solutions of A Rational Di¤erence Equation, Selçuk J. Appl. Math., 9(2), (2008), 3-8. ®xn +¯xn¡1 +°xn¡2 +±xn¡3 5. E. M. E. Zayed, M. A. El-Moneam, On the Rational Recursive Sequence xn+1 = Ax , n +Bxn¡1 +Cxn¡2 +Dxn¡3 J. Appl. Math. Comput., 22, (2006), 247-262. 6. M. m. El-Dessoky, Qualitative behavior of rational di¤erence equation of big Order, Discrete Dyn. Nat. Soc., 2013, (2013), Article ID 495838, 6 pages. 7. Guo-Mei Tang, Lin-Xia Hu, and Xiu-Mei Jia, Dynamics of a Higher-Order Nonlinear Di¤erence Equation, Discrete Dyn. Nat. Soc., 2010, (2010), Article ID 534947, 15 pages.

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8. G. Papaschinopoulos, C. J. Schinas, Stefanidou, G., On the nonautonomous di¤erence equation xn+1 = xp An + n¡1 , Appl. Math. Comput., 217(12), (2011), 5573-5580. xqn 9. M. M. El-Dessoky, Dynamics and Behavior of the Higher Order Rational Di¤erence equation, J. Comput. Anal. Appl., 21(4), (2016), 743-760. 10. S. Nirmaladevi, and N. Karthikeyan, Dynamics and Behavior of Higher Order Nonlinear Rational Di¤erence Equation, International Journal Of Advance Research And Innovative Ideas In Education, 3 (4) (2017), 2395-4396. 11. Y. Yazlik, D. T. Tollu, N. Taskara, On the Behaviour of Solutions for Some Systems of Di¤erence Equations, J. Comp. Anal. Appl., 18(1), (2015), 166-178. 12. M. A. El-Moneam, S. O. Alamoudy, On Study of the Asymptotic Behavior of Some Rational Di¤erence Equations, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 21, (2014), 89-109. n xn¡k 13. R. Abu-Saris, C. Cinar, I. Yalç¬nkaya, On the asymptotic stability of xn+1 = a+x xn +xn¡k , Comput. Math. Appl., 56, (2008), 1172–1175. 14. C. J. Schinas, G. Papaschinopoulos, G. Stefanidou, On the Recursive Sequence xn+1 = A + Di¤er. Equ., 2009, (2009), Article ID 327649, 11 page.

xp n¡1 , xqn

Adv. xp

, 15. Mehmet G½um½u¸s, The Periodicity of Positive Solutions of the Nonlinear Di¤erence Equation xn+1 = ®+ n¡k xp n Disc. Dyn. Nat. Soc., 2013, (2013), Article ID 742912, 3 pages. ®¡¯xn , J. 16. M. T. Aboutaleb, M. A. El-Sayed, A. E. Hamza, Stability of the Recursive Sequence xn+1 = °+x n¡1 Math. Anal. Appl., 261(1), (2001), 126–133. ½ ½ , Disc. Dyn. 17. Mehmet G½um½u¸s and Ozkan Ocalan, Some Notes on the Di¤erence Equation xn+1 = ® + xn¡1 xk n Nat. Soc., 2012, (2012), Article ID 258502, 12 pages. 18. A. Brett, E. J. Janowski, M. R. S. Kulenov´c, Global Asymptotic Stability for Linear Fractional Di¤erence Equation, Journal of Di¤erence Equations, 2014, (2014), Article ID 275312, 11 pages. p °y p 19. I· lhan Öztürk, Saime Zengin, On the di¤erence equation yn+1 = ®yp n ¡ n¡1 p , Mathematica Slovaca, 61(6), ¯yn¡1

20. 21. 22. 23. 24. 25. 26. 27. 28.

¯yn

(2011), 921-932. E. M. Elsayed, M. M. El-Dessoky, Dynamics and global behavior for a fourth-order rational di¤erence equation, Hacettepe J. Math. and Stat., 42(5), (2013), 479–494. R. Abo-Zeid, Global behavior of a higher order di¤erence equation, Mathematica Slovaca, 64(4), (2014), 931-940. E. M. Elsayed, M. M. El-Dessoky, Asim Asiri, Dynamics and Behavior of a Second Order Rational Di¤erence equation, J. Comput. Anal. Appl., 16(4), (2014), 794-807. E. M. Elsayed, M. M. El-Dessoky, E. O. Alzahrani, The Form of The Solution and Dynamic of a Rational Recursive Sequence, J. Comput. Anal. Appl., 17(1), (2014), 172-186. I. Yalcinkaya, A. E. Hamza, C. Cinar, Global Behavior of a Recursive Sequence, Selçuk J. Appl. Math., 14(1), (2013), 3-10. M. A. El-Moneam, On the Dynamics of the Higher Order Nonlinear Rational Di¤erence Equation, Math. Sci. Lett. 3(2), (2014), 121-129. M. M. El-Dessoky and M. A. El-Moneam, On the Higher Order Di¤erence equation xn+1 = Axn + Bxn¡l + °xn¡k ; J. Comput. Anal. Appl., 25(2), (2018), 342-354. Cxn¡k + Dxn¡s +Exn¡t M. M. El-Dessoky, On the dynamics of a higher Order rational di¤erence equations, J. Egypt. Math. Soc. 25(1), (2017), 28-36. M. M. El-Dessoky and Aatef Hobiny, Dynamics of a Higher Order Di¤erence Equations xn+1 = ®xn + axn¡l +bxn¡k ¯xn¡l + °xn¡k + cxn¡l +dxn¡k , J. Comput. Anal. Appl., 24(7), (2018), 1353-1365.

®+cx , J. Comput. 29. M. M. El-Dessoky and Aatef Hobiny, On the Di¤erence equation xn+1 = axn + bxn¡1 + ¯+dxn¡2 n¡2 Anal. Appl., 24(4), (2018), 644-655. cxn¡s 30. M. M. El-Dessoky, On the Di¤erence equation xn+1 = axn¡l + bxn¡k + dxn¡s ¡e , Math. Meth. Appl. Sci., 40(3), (2017), 535–545.

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BEST PROXIMITY POINT OF CONTRACTION TYPE MAPPING IN METRIC SPACE Kyung Soo Kim Graduate School of Education, Mathematics Education Kyungnam University, Changwon, Gyeongnam, 51767, Republic of Korea e-mail: [email protected] Abstract. The purpose of this article, we consider the existence of a unique best proximity point x∗ ∈ A such that d(x∗ , T x∗ ) = dist(A, B) for generalized ϕ-weak contraction mapping T : A → B, where A, B(6= ∅) are subsets of a metric space (X, d).

1. Introduction Let (X, d) be a metric space. A mapping T : X → X is a contraction if there exists a constant α ∈ (0, 1) such that d(T x, T y) ≤ α · d(x, y),

∀ x, y ∈ X.

A mapping T : X → X is a ϕ-weak contraction if there exists a continuous and nondecreasing function ϕ : [0, ∞) → [0, ∞) with ϕ−1 (0) = {0} and limt→∞ ϕ(t) = ∞ such that d(T x, T y) ≤ d(x, y) − ϕ(d(x, y)),

∀ x, y ∈ X.

(1.1)

If X is bounded, then the infinity condition can be omitted. The concept of the ϕ-weak contraction was introduced by Alber and GuerreDelabriere [1] in 1997, who proved the existence of fixed points in Hilbert spaces. Later Rhoades [14] in 2001, who extended the results of [1] to metric spaces. Theorem 1.1. ([14]) Let (X, d) be a complete metric space, T : X → X be a ϕ-weak contractive self-map on X. The T has a unique fixed point p in X. Remark 1.2. Theorem 1.1 is one of generalizations of the Banach contraction principle because it takes ϕ(t) = (1 − α)t for α ∈ (0, 1), then ϕ-weak contraction contains contraction as special cases. Next, we present a brief discussion about best proximity point which is a interesting topic in best proximity theory. 0

2010 Mathematics Subject Classification: 54H25, 47H09, 47H10, 41A65. Keywords: Optimal solution, best proximity point, P -property, generalized ϕ-weak contraction mapping, fixed point, metric space. 0

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Kyung Soo Kim

Let (X, d) be a metric space and A(6= ∅) be a subset of (X, d). Consider a mapping T : A → X. The solutions to the fixed point equation T x = x are called fixed points of the mapping T. It is clear that T (A) ∩ A 6= ∅ is a necessary (but not sufficient) condition for the existence of a fixed point for the mapping T : A → X. If the necessary condition fails, then d(x, T x) > 0, for all x ∈ A. This means that the mapping T : A → X does not have any fixed point, i.e., T x = x has no solution. This point of view, it give us to think of a point x ∈ A which is closest to T x in some sense. Best approximation theory and best proximity point theory are relevant in this perspective. One of the most interesting best approximation theorem is due to Fan [3]. Theorem 1.3. ([3]) Let C(6= ∅) be a compact convex subset of a normed linear space V and F : C → V be a continuous function. Then there exists a point p ∈ C such that kp − F pk = d(F p, C) = inf{kF p − ck : c ∈ C}. Such an element p ∈ C in Theorem 1.3 is called a best approximant point of T in C. Although a best approximation point acts as an approximate solution of the equation F p = p, the value kp − F pk need not be the optimum, i.e., a best approximant point is not an optimal solution in the sense that min kp − F pk. p∈A

Naturally, let us consider nonempty subsets A, B of a metric space (X, d) and a mapping T : A → B. Then one can think of finding an element x∗ ∈ A such that d(x∗ , T x∗ ) = min{d(x, T x) : x ∈ A}. Since d(x, T x) ≥ dist(A, B) = inf{d(a, b) : a ∈ A, b ∈ B} for all x ∈ A, the optimal solution of minx∈A d(x, T x) is one for which the value dist(A, B) is attained. A point x∗ ∈ A is said to be a best proximity point of T : A → B if d(x∗ , T x∗ ) = dist(A, B). So a best proximity point of the mapping T is an approximate solution of the equation T x = x which is optimal solution in the sense that min d(x, T x). x∈A

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Remark 1.4. It is trivial that all best proximity point theorems work as a natural generalization of fixed point theorems if the mapping T is a selfmapping. Recently Sultana and Vetrivel [15] obtained the following best proximity point theorem for mapping satisfies (1.1). Theorem 1.5. ([15], Theorem 3.4) Let A, B(6= ∅) be two closed subsets of a complete metric space (X, d) such that the pair (A, B) has the P -property and A0 6= ∅ and T : A → B be a mapping such that T (A0 ) ⊆ B0 and it satisfies (1.1). Then there exists a unique p ∈ A such that d(p, T p) = dist(A, B). In 2016, Xue [16] introduced a new contraction type mapping as follows. Definition 1.6. ([16]) A mapping T : X → X is a generalized ϕ-weak contraction if there exists a continuous and nondecreasing function ϕ : [0, ∞) → [0, ∞) with ϕ(0) = 0 such that d(T x, T y) ≤ d(x, y) − ϕ(d(T x, T y)),

∀ x, y ∈ X

(1.2)

holds. We notice immediately that if T : X → X is ϕ-weak contraction, then T satisfies the following inequality d(T x, T y) ≤ d(x, y) − ϕ(d(T x, T y)),

∀ x, y ∈ X.

However, the converse is not true in general. Example 1.7. Let X = (−∞, +∞) be endowed with the Euclidean metric d(x, y) = |x−y| and let T x = 25 x for each x ∈ X. Define ϕ : [0, +∞) → [0, +∞) by ϕ(t) = 43 t. Then T satisfies (1.2), but T does not satisfy inequality (1.1). Indeed, 2 2 d(T x, T y) = x − y 5 5 4 2 · |x − y| 3 5 = d(x, y) − ϕ(d(T x, T y)) ≤ |x − y| −

and 2 2 d(T x, T y) = x − y 5 5 4 ≥ |x − y| − |x − y| 3 = d(x, y) − ϕ(d(x, y)) for all x, y ∈ X.

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Example 1.8. ([16]) Let X = (−1, +∞) be endowed by d(x, y) = |x − y| and x t2 let T x = 1+x for each x ∈ X. Define ϕ : [0, +∞) → [0, +∞) by ϕ(t) = 1+t . Then x y |x − y| d(T x, T y) = − = 1+x 1+y (1 + x)(1 + y) |x − y|2 |x − y| = |x − y| − 1 + |x − y| 1 + |x − y| = d(x, y) − ϕ(d(x, y))

≤

holds for all x, y ∈ X. So T is a ϕ-weak contraction. However T is not a contraction. Remark 1.9. The above examples show that the class of generalized ϕ-weak contractions properly includes the class of ϕ-weak contractions and the class of ϕ-weak contractions properly includes the class of contractions. In fact, let T : X → X be a contraction, there exists α ∈ (0, 1) such that d(T x, T y) ≤ α · d(x, y), ∀ x, y ∈ X. Then d(T x, T y) ≤ α · d(x, y) = d(x, y) − (1 − α)d(x, y) = d(x, y) − ϕ(d(x, y)), where, ϕ(d(x, y)) = (1 − α)d(x, y). So, T is a ϕ-weak contraction. Moreover, let T be a ϕ-weak contraction, from property of ϕ, we have d(T x, T y) ≤ d(x, y) and ϕ(d(T x, T y)) ≤ ϕ(d(x, y)). From (1.1), d(T x, T y) ≤ d(x, y) − ϕ(d(x, y)) ≤ d(x, y) − ϕ(d(T x, T y)),

∀ x, y ∈ X.

Therefore, T is a generalized ϕ-weak contraction. In the meantime, if T is a ϕ-weak contractive self mapping for one mapping ϕ so we do not expect that the ϕ-weak contractivity should be satisfied with the same function ϕ. Let us suppose that T is a ϕ-weak contractive self mapping and consider ϕ(x) ˜ = min {ϕ(x/2); x/2} . Then, if d(T x, T y) > 12 d(x, y), we have 

1 d(T x, T y) ≤ d(x, y) − ϕ(d(T x, T y)) ≤ d(x, y) − ϕ d(x, y) 2

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on account of monotonocity of ϕ and finally d(T x, T y) ≤ d(x, y) − ϕ(d(x, ˜ y)). On the other hand, if d(T x, T y) < 12 d(x, y), we get 1 d(T x, T y) < d(x, y) − d(x, y) ≤ d(x, y) − ϕ(d(x, ˜ y)). 2 So T is just the ϕ-weak ˜ contractive mapping. The continuity and monotonocity of ϕ˜ follows directly from properties of min function, ϕ and the metric. For related results, please see [9], [10], [11] and the references therein ([5], [6], [7], [8]). The purpose of this article, we consider the existence of a unique best proximity point x∗ ∈ A such that d(x∗ , T x∗ ) = dist(A, B) for generalized ϕ-weak contraction mapping T : A → B, where A, B(6= ∅) are subsets of a metric space (X, d). 2. Preliminaries Let A, B be two nonempty subsets of a metric space (X, d). Let us define the following notation which will be need throughout this article: A0 = {x ∈ A : d(x, y) = dist(A, B)

for some y ∈ B},

B0 = {y ∈ B : d(x, y) = dist(A, B)

for some x ∈ A}.

In [12], the authors discussed sufficient conditions which guarantee the nonemptiness of A0 and B0 . Also, in [2], the authors proved that A0 is contained in the boundary of A. Let us define the notion of nonself generalized ϕ-weak contraction mapping as follows. Definition 2.1. Let A, B be two nonempty subsets of a metric space (X, d). A mapping T : A → B is said to be a generalized ϕ-weak contraction if d(T x, T y) ≤ d(x, y) − ϕ(d(T x, T y)),

∀ x, y ∈ A,

(2.1)

where ϕ : [0, ∞) → [0, ∞) is a continuous and nondecreasing function such that ϕ is positive on (0, ∞), ϕ−1 (0) = {0} and limt→∞ ϕ(t) = ∞. If A is bounded, then the infinity condition can be omitted. The notion called the P -property was introduced in [13]. Definition 2.2. ([13]) Let (A, B) be a pair of nonempty subsets of a metric space (X, d) with A0 6= ∅. Then the pair (A, B) is said to be has the P -property

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if and only if for any x1 , x2 ∈ A0 and y1 , y2 ∈ B0 , d(x1 , y1 ) = dist(A, B) = d(x2 , y2 )

⇒

d(x1 , x2 ) = d(y1 , y2 ).

Now we recall the following results from [4] and [15]. Lemma 2.3. ([15]) Let ϕ : [0, ∞) → [0, ∞) be a function such that ϕ−1 (0) = {0} and ϕ is either nondecreasing or continuous. Then ϕ(µn ) → 0

⇒

µn → 0

for any bounded sequence {µn } of positive reals. Lemma 2.4. ([4]) For a given subset D of {(x, y) ∈ R2 : x, y ≥ 0}, the following statements are equivalent: (i) for any ε > 0, there exist δ > 0 and γ ∈ (0, ε) such that ⇒

u 0

v ≤ φ(u), ∀ (u, v) ∈ D.

and

3. Main results Lemma 3.1. Let A and B be two nonempty subsets of a metric space (X, d) and ϕ : [0, ∞) → [0, ∞) be a function such that ϕ−1 (0) = {0} and ϕ(tn ) → 0

⇒

tn → 0

(3.1)

for any bounded sequence {tn } of positive reals. Let T : A → B be a genearlized ϕ-weak contraction mapping satisfying (2.1). Then, for any ε > 0, there exist δ > 0 and γ ∈ (0, ε) such that d(x, y) < ε + δ

⇒

d(T x, T y) ≤ γ

for all x, y ∈ A. Proof. Suppose that there exists an ε0 > 0 such that for any δ > 0, γ ∈ (0, ε0 ) and there exist x, y ∈ A such that d(x, y) < ε0 + δ

⇒

Let δn =

1 n2

and γn =

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Then there exist {xn } and {yn } in A such that d(xn , yn ) < ε0 +

1 n2

⇒

d(T xn , T yn ) >

n2 ε0 . 1 + n2

(3.2)

From (2.1), we have n2 ε0 < d(T xn , T yn ) 1 + n2 ≤ d(xn , yn ) − ϕ(d(T xn , T yn )) 1 < ε0 + 2 − ϕ(d(T xn , T yn )). n That is ϕ(d(T xn , T yn )) < ε0 +

1 n2 1 ε0 − ε0 = 2 + . 2 2 n 1+n n 1 + n2

Hence ϕ(d(T xn , T yn )) → 0 as n → ∞. Since d(T xn , T yn ) ≤ d(xn , yn ) and {d(xn , yn )} is bounded, we get {d(T xn , T yn )} is bounded. By the given hypothesis (3.1), d(T xn , T yn ) → 0

as

n → ∞.

On the other hand, from (3.2), lim d(T xn , T yn ) ≥ ε0 > 0.

n→∞

This is a contradiction. Thus Lemma 3.1 holds.



The following theorem is main result which gives sufficient conditions for the existence of a unique best proximity point for generalized ϕ-weak contraction mapping. Theorem 3.2. Let (A, B) be a pair of two nonempty closed subsets of a complete metric space (X, d) such that A0 6= ∅. Let T : A → B be a generalized ϕ-weak contraction mapping such that T (A0 ) ⊆ B0 . Assume that the pair (A, B) has the P -property. Then there exists a unique x∗ in A such that d(x∗ , T x∗ ) = dist(A, B). Proof. Let x0 ∈ A0 . Since T x0 ∈ T (A0 ) ⊆ B0 , there exists x1 ∈ A0 such that d(x1 , T x0 ) = dist(A, B). Again, since T x1 ∈ T (A0 ) ⊆ B0 , there exists x2 ∈ A0 such that d(x2 , T x1 ) = dist(A, B). Continuing this process, we can find a sequence {xn } in A0 such that d(xn+1 , T xn ) = dist(A, B),

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Since (A, B) has the P -property, from (3.3), we obtain ∀ n ∈ N.

d(xn , xn+1 ) = d(T xn−1 , T xn ),

(3.4)

By the definition of T and (3.4), we have d(xn , xn+1 ) = d(T xn−1 , T xn ) ≤ d(xn−1 , xn ) − ϕ(d(T xn−1 , T xn )) ≤ d(xn−1 , xn ),

∀ n ∈ N.

Therefore, the sequence {d(xn , xn+1 )} is monotone nonincresing and bounded. Hence it converges. If we set λn = d(xn , xn+1 ) and L be the limit of λn , i.e., lim λn = lim d(xn , xn+1 ) = L ≥ 0.

n→∞

n→∞

Now, we claim that L = 0. Suppose to the contrary that L > 0. Since {λn } is nonincreasing sequence, i.e., λn ≥ λn+1 ≥ · · · ≥ L > 0,

∀n ∈ N

and ϕ is nondecreasing, we obtain ϕ(λn ) ≥ ϕ(L) > 0,

∀ n ∈ N.

(3.5)

From the inequality λn = d(xn , xn+1 ) ≤ d(xn−1 , x) − ϕ(d(T xn−1 , T xn )) = λn−1 − ϕ(d(T xn−1 , T xn )), (3.4) and (3.5), we have λn ≤ λn−1 − ϕ(d(xn , xn+1 )) = λn−1 − ϕ(λn ) ≤ λn−1 − ϕ(L),

∀ n ∈ N.

Since ϕ is continuous, we get L ≤ L − ϕ(L). That is ϕ(L) ≤ 0 which contradicts condition of ϕ. Hence lim d(xn , xn+1 ) = L = 0.

n→∞

Now we apply Lemma 2.3 and Lemma 2.4 to the set D = {(d(x, y), d(T x, T y)) : x, y ∈ A} on Lemma 3.1, one knows that there exists a function φ : [0, ∞) → [0, ∞) such that φ is continuous and nondecreasing with φ(t) < t, ∀ t > 0

and d(T x, T y) ≤ φ(d(x, y)), ∀ x, y ∈ A.

(3.6)

Thus for a given ε > 0, there exists N ∈ N such that d(xn , xn+1 ) ≤ ε − φ(ε),

∀ n ≥ N.

(3.7)

Next we show that {xn } is a Cauchy sequence.

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Best proximity point of contraction type mapping in metric space

9

Denotes the closed ball with center x and radius ε by B[x, ε], we will claim the following relations. Claim I. T (B[xN , ε] ∩ A) ⊆ B[T xN −1 , ε]. Let x ∈ B[xN , ε] ∩ A, i.e., d(xN , x) ≤ ε, from (3.4), (3.6) and (3.7), then d(T x, T xN −1 ) ≤ d(T x, T xN ) + d(T xN , T xN −1 ) ≤ φ(d(x, xN )) + d(xN +1 , xN ) ≤ φ(ε) + {ε − φ(ε)} = ε, which implies that T x ∈ B[T xN −1 , ε]. Claim II. y ∈ B[T xN −1 , ε] with d(x, y) = dist(A, B) for some x ∈ A0 implies x ∈ B[xN , ε] ∩ A. Let y ∈ B[T xN −1 , ε] with d(x, y) = dist(A, B) for some x ∈ A0 . From (3.3), we have d(xN , T xN −1 ) = dist(A, B). Therefore, by using the P -property of (A, B), we obtain that d(xN , x) = d(T xN −1 , y) ≤ ε. Hence Claim II holds. From (3.7), it is clear that xN +1 ∈ B[xN , ε] ∩ A. And by Claim I, we have T xN +1 ∈ B[T xN −1 , ε]. From (3.3), d(xN +2 , T xN +1 ) = dist(A, B) with xN +2 ∈ A0 . From Claim II, xN +2 ∈ B[xN , ε] ∩ A. Again by Claim I, T xN +2 ∈ B[T xN −1 , ε] and by (3.3), d(xN +3 , T xN +2 ) = dist(A, B) with xN +3 ∈ A0 . Again by Claim II, xN +3 ∈ B[xN , ε] ∩ A. Continuing this process, we can conclude that xN +m ∈ B[xN , ε] ∩ A,

∀ m ∈ N,

i.e., d(xN , xN +m ) ≤ ε. Hence the sequence {xn } is a Cauchy sequence. Since A is closed subset of the complete metric space (X, d), there exists an element x∗ ∈ A such that limn→∞ xn = x∗ . By the definition of T, we have d(T x, T y) ≤ d(x, y) for all x, y ∈ A which implies that T is continuous in A. Therefore we obtain lim T xn = T x∗ .

n→∞

Also, from the continuity of the distance function d, we have lim d(xn , T xn ) = d(x∗ , T x∗ ).

n→∞

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10

Kyung Soo Kim

Equation (3.3), it means that the sequence {d(xn+1 , T xn )} is a constant sequence with the value dist(A, B). Hence d(x∗ , T x∗ ) = dist(A, B), i.e., x∗ is a best proximity point of T. Finally, we show that x∗ is unique best proximity point of T. Suppose that x1 and x2 are two best proximity points of T in A with x1 6= x2 . Since x1 and x2 are two best proximity points of T, we have d(x1 , T x1 ) = dist(A, B) = d(x2 , T x2 ). By the P -property of (A, B), we obtain d(x1 , x2 ) = d(T x1 , T x2 ). Since x1 and x2 are distinct elements in A, one can have ϕ(d(x1 , x2 )) > 0.

(3.8)

From the definition of T and (3.8), d(x1 , x2 ) = d(T x1 , T x2 ) ≤ d(x1 , x2 ) − ϕ(d(T x1 , T x2 )) = d(x1 , x2 ) − ϕ(d(x1 , x2 )) < d(x1 , x2 ). This is a contradiction. Therefore the uniqueness of the best proximity point follows.  The following example illustrates that Theorem 3.2 holds. Example 3.3. Let X = (−∞, +∞) be endowed with the Euclidean metric d(x, y) = |x − y|. Then (X, d) is a complete metric space. Let A = [−1, 1] and B = [0, 2] be two subsets of (X, d). Define T : A → B by 2 Tx = x 5 for each x ∈ A. Define ϕ(t) : [0, +∞) → [0, +∞) by 4 ϕ(t) = t. 3 Then, by Example 1.7, T satisfies (1.2). It is easy to check that A and B are closed subsets of complete metric space (X, d), ∅ 6= A0 = [0, 1] = B0 and T (A0 ) = [0, 25 ] ⊆ [0, 1] = B0 . Moreover (A, B) has the P -property. Indeed, let d(x1 , T x1 ) = dist(A, B) = d(x2 , T x2 ). By 0 = dist(A, B) = inf{d(a, b) : a ∈ A, b ∈ B},

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Best proximity point of contraction type mapping in metric space

11

we have x1 = T x1 and x2 = T x2 . Thus d(x1 , x2 ) = d(T x1 , T x2 ). Hence (A, B) has the P -property. Therefore all the assumption of Theorem 3.2 hold and note that x∗ = 0 is the unique best proximity point. Acknowledgments This work was supported by Kyungnam University Research Fund, 2017. References [1] Y.I. Alber and S. Guerre-Delabriere, Principle of weakly contractive maps in Hilbert spaces, in: I. Gohberg, Yu. Lyubich(Eds.), New Results in Operator Theory, in: Advances and Appl., vol. 98, Birkh¨ auser, Basel, 1997, 7–22. [2] S.S. Basha and P. Veeramani, Best proximity pair theorems for multifunctions with open fibres, J. Approx. Theory, 103(1) (2000), 119–129. [3] K. Fan, Extension of two fixed point theorems of F.E. Browder, Math. Z., 122 (1969), 234–240. [4] M. Heged¨ us and T. Szil´ agyi, Equivalent conditions and a new fixed point theorem in the theory of contractive type mappings, Math. Japon., 25 (1980), 147–157. [5] J.K. Kim, K.H. Kim and K.S. Kim, Three-step iterative sequences with errors for asymptotically quasi-nonexpansive mappings in convex metric spaces, Proc. of RIMS Kokyuroku, Kyoto Univ., 1365 (2004), 156–165. [6] J.K. Kim, K.S. Kim and S.M. Kim, Convergence theorems of implicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings in convex metric spaces, Proc. of RIMS Kokyuroku, Kyoto Univ., 1484 (2006), 40–51. [7] J.K. Kim, K.S. Kim and Y.M. Nam, Convergence and stability of iterative processes for a pair of simultaneously asymptotically quasi-nonexpansive type mappings in convex metric spaces, J. of Compu. Anal. Appl., 9(2) (2007), 159–172. [8] K.S. Kim, Some convergence theorems for contractive type mappings in CAT (0) spaces, Abstract and Applied Analysis, 2013, Article ID 381715, 9 pages, http://dx.doi.org/10.1155/2013/381715 [9] K.S. Kim, Convergence and stability of generlaized ϕ-weak contraction mapping in CAT (0) spaces, Open Mathematics, 15 (2017), 1063–1074. [10] K.S. Kim, Equivalence between some iterative schemes for generalized ϕ-weak contraction mappings in CAT (0) spaces, East Asian Math. J., 33(1) (2017), 11–22. [11] K.S. Kim, H. Lee, S.J. Park, S.Y. Yu, J.H. Ahn and D.Y. Kwon, Convergence of generlaized ϕ-weak contraction mapping in convex metric spaces, Far East J. Math. Sci., 101(7) (2017), 1437–1447. [12] W.A. Kirk, S. Reich and P. Veeramani, Proximal retracts and best proximity pair theorems, Numer. Funct. Anal. Optim., 24(7-8) (2003), 851–862. [13] V.S. Raj, A best proximity point theorem for weakly contractive non-self-mappings, Nonlinear Anal., 74 (2011), 4804–4808. [14] B.E. Rhoades, Some theorems on weakly contractive maps, Nonlinear Anal., 47 (2001), 2683–2693. [15] A. Sultana and V. Vetrivel, On the existence of best proximity points for generalized contractions, Appl. Gen. Topol., 15(1) (2014), 55–63. [16] Z. Xue, The convergence of fixed point for a kind of weak contraction, Nonlinear Func. Anal. Appl., 21(3) (2016), 497–500.

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Explicit viscosity rule of nonexpansive mappings in CAT(0) spaces Shin Min Kang1,2 , Absar Ul Haq3, Waqas Nazeer4,∗, Iftikhar Ahmad5 and Maqbool Ahmad6 1

Department of Mathematics and RINS, Gyeongsang National University, Jinju 52828, Korea e-mail: [email protected] 2 3

Center for General Education, China Medical University, Taichung 40402, Taiwan

Department of Mathematics, University of Management and Technology, Sialkot Campus, Lahore 51410, Pakistan e-mail: [email protected] 4

Division of Science and Technology, University of Education, Lahore 54000, Pakistan e-mail: [email protected]

5

Department of Mathematics and Statistics, University of Lahore, Lahore 54000, Pakistan e-mail: [email protected]

6

Department of Mathematics and Statistics, University of Lahore, Lahore 54000, Pakistan e-mail: [email protected] Abstract In this paper, we present a explicit viscosity technique of nonexpansive mappings in the framework of CAT(0) spaces. The strong convergence theorem of the proposed technique is proved under certain assumptions imposed on the sequence of parameters. The results presented in this paper extend and improve some recent announced in the current literature. 2010 Mathematics Subject Classification: 47J25, 47N20, 34G20, 65J15 Key words and phrases: viscosity rule, CAT(0) space, nonexpansive mapping, variational inequality.

1

Introduction

The study of spaces of nonpositive curvature originated with the discovery of hyperbolic spaces, and flourished by pioneering works of Hadamard and Cartan, etc. in the first decades of the twentieth century. The idea of nonpositive curvature geodesic metric spaces could be traced back to the work of Busemann and Alexandrov, etc. in the 50’s. Later on Gromov [9] restated some features of global Riemannian geometry solely based on the so-called CAT(0) inequality. For through discussion of CAT(0) spaces and of fundamental role they play in geometry, we refer the reader to Bridson and Haefliger [5]. ∗

Corresponding author

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As we know, iterative methods for finding fixed points of nonexpansive mappings have received vast investigations due to its extensive applications in a variety of applied areas of inverse problem, partial differential equations, image recovery, and signal processing; see [1–3,7,14–17] and the references therein. One of the difficulties in carrying out results from Banach space to complete CAT(0) space setting lies in the heavy use of the linear structure of the Banach spaces. Berg and Nikolaev [4] introduce the notion of an inner product-like notion (quasi-linearization) in complete CAT(0) spaces to resolve these difficulties. Fixed-point theory in CAT(0) spaces was first studied by Kirk [10,11]. He showed that every nonexpansive (single-valued) mapping defined on a bounded closed convex subset of a complete CAT(0) space always has a fixed point. Since then, the fixed-point theory for single-valued and multivalued mappings in CAT(0) spaces has been rapidly developed. In 2000, Moudaf’s [13] introduce viscosity approximation methods as following Theorem 1.1. Let C be a nonempty closed convex subset of the real Hilbert space X. Let T be a nonexpansive mapping of C into itself such that F ix(T ) is nonempty. Let f be a contraction of C into itself with coefficient θ ∈ [0, 1). Pick any x0 ∈ [0, 1), let {xn } be a sequence generated by xn+1 =

γn 1 f (xn ) + T (xn ), 1 + γn 1 + γn

n ≥ 0,

where {γn } is a sequence in (0, 1) satisfying the following conditions: (1) P limn→∞ γn = 0, (2) ∞ γ = ∞, Pn=0 n 1 1 (3) ∞ | n=0 γn+1 − γn | = 0. Then {xn } converges strongly to a fixed point x∗ of the mapping T , which is also the unique solution of the variational inequality hx − f (x), x − yi ≥ 0,

∀y ∈ F ix(T ).

In other words, x∗ is the unique fixed point of the contraction PF ix(T )f , that is, PF ix(T )f (x∗ ) = x∗ . Shi and Chen [15] studied the convergence theorems of the following Moudaf’s viscosity iterations for a nonexpansive mapping in CAT(0) spaces. xn+1 = tf (xn ) ⊕ (1 − t)T (xn ),

(1.1)

xn+1 = αn f (xn ) ⊕ (1 − αn )T (xn ).

(1.2)

They proved that {xn } defned by (1.1) and {xn } defined by (1.2) converged strongly to a fxed point of T in the framework of CAT(0) space. Zhao et al. [18] applied viscosity approximation methods for the implicit midpoint rule for nonexpansive mappings   xn ⊕ xn+1 xn+1 = αn f (xn ) ⊕ (1 − αn )T , ∀n ≥ 0. 2 Motivated and inspired by the idea of Kwun et al. [12], in this paper, we extend and study the explicit viscosity rules of nonexpansive mappings in CAT(0) spaces ( xn+1 = (1 − αn )f (xn ) ⊕ αn T (yn ), (1.3) yn = (1 − βn )xn ⊕ βn T (xn ).

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2

Preliminaries

Let(X, d) be a metric space. A geodesic path joining x ∈ X to y ∈ X (or, more briefly, a geodesic from x to y) is a map c from a closed interval [0, l] ⊂ R to X such that c(0)= x, c(l) = y, and d(c(t), c(t0)) = |t − t0 | for all t, t0 ∈ [0, l]. In particular, c is an isometry and d(x, y) = l. The image α of c is called a geodesic (or metric) segment joining x and y. When it is unique, this geodesic segment is denoted by [x, y]. The space (X, d) is said to be a geodesic space if every two points of X are joined by a geodesic, and X is said to be uniquely geodesic if there is exactly one geodesic joining x and y for each x, y ∈ X. A subset Y ⊂ X is said to be convex if Y includes every geodesic segment joining any two of its points. A geodesic triangle 4(x1 , x2 , x3 ) in a geodesic metric space (X, d) consists of three points x1 , x2,and x3 in X (the vertices of 4) and a geodesic segment between each pair of vertices (the edges of 4). A comparison triangle for the geodesic triangle 4(x1 , x2 , x3 in (X, d) is a triangle 4(x1 , x2 , x3 ) := 4(x1 , x2 , x3 ) in the Euclidean plane E2 such that dE2 d(xi, xj ) = d(xi, xj )for i, j = 1, 2, 3. A geodesic space is said to be a CAT(0) space if all geodesic triangles satisfy the following comparison axiom. Let 4 be a geodesic triangle in X, and let 4 be a comparison triangle for 4 . Then, 4 is said to satisfy the CAT(0) inequality if for all x, y ∈ 4 and all comparison points x, y ∈ 4, d(x, y) = dE2 (x, y) (2.1) Let x, y ∈ X and by the Lemma 2.1(iv) of [8] for each t ∈ [0, 1], there exists a unique point z ∈ [x, y] such that d(x, z) = td(x, y),

d(y, z) = (1 − t)d(x, y).

(2.2)

From now on, we will use the notation (1 − t)x ⊕ ty for the unique fixed point z satisfying the above equation. We now collect some elementary facts about CAT(0) spaces which will be used in the proofs of our main results. Lemma 2.1. ([8]) Let X be a CAT(0) spaces. (a) For any x, y, z ∈ X and t ∈ [0, 1], d((1 − t)x ⊕ ty, z) ≤ (1 − t)d(x, z) + td(y, z).

(2.3)

(b) For any x, y, z ∈ X and t ∈ [0, 1], d2 ((1 − t)x ⊕ ty, z) ≤ (1 − t)2 d(x, z) + td2 (y, z) − t(1 − t)d2 (x, y).

(2.4)

Complete CAT(0) spaces are often called Hadamard spaces (see [5]). If x, y1 , y2 are points of a CAT(0) spaces and y0 is the midpoint of the segment [y1 , y2 ], which we will 2) denoted by (y1 ⊕y , then the CAT(0) inequality implies 2   y1 ⊕ y2 1 1 1 2 d x, ≤ d2 (x, y1 ) + d2 (x, y2 ) − d2 (y1 , y2 ). (2.5) 2 2 2 4 This inequality is the (CN) inequality of Bruhat and Tits [6]. In fact, a geodesic metric space is a CAT(0) space if and only if it satisfies the (CN) inequality (cf. [5], page 163).

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Definition 2.2. Let X be a CAT(0) space and T : X → X be a mapping. Then T is called nonexpensive if d(T (x), T (y)) ≤ d(x, y), x, y ∈ C Definition 2.3. Let X be a CAT(0) space and T : X → X be a mapping. Then T is called contraction if d(T (x), T (y)) ≤ θd(x, y),

x, y ∈ C, θ ∈ [0, 1)

Berg and Nikolaev [4] introduce the concept of quasi-linearization as follow. Let us − → denote the pair (a, b) ∈ X × X by the ab and call it a vector. Then, quasi-linearization is defined as a mapping h·, ·i : (X × X) × (X × X) −→ R defined as

− → → − 1 hab, cdi = (d2 (a, d) + d2 (b, c) − d2 (a, c) − d2 (b, d)) (2.6) 2 − → → − − → → − − → − → − → → − → − → → − → − it is easy to see that hab, cdi = hcd, abi, hab, cdi = −hba, cdi and h− ax, cdi + hxb, cdi = − → → − hab, cdi for all a, b, c, d ∈ X. We say that X satisfies the Cauchy-Schwarz inequality if − → → − hab, cdi ≤ d(a, b)d(a, c) for all a, b, c, d ∈ X. It is well-known [4] that a geodesically connected metric space is a CAT(0) space of and only if it satisfy the Cauchy-Schwarz inequality. Let C be a non-empty closed convex subset of a complete CAT(0) space X. The metric projection Pc : X → C is defined by u = Pc (x) ⇐⇒ inf{d(y, x) : y ∈ C},

∀x ∈ X

Definition 2.4. Let Pc : X → C is called the metric projection if for every x ∈ X there exist a unique nearest point in C, denoted by Pc x, such that d(x, Pcx) ≤ d(x, y), y ∈ C The following theorem gives you the conditions for a projection mapping to be nonexpensive. Theorem 2.5. Let C be a non-empty closed convex subset of a real CAT(0) space X and Pc : X → X a metric projection. Then −−−→ → − (1) d(Pc x, Pc y) ≤ h− xy, Pc xPc yi for all x, y ∈ X, (2) Pc is nonexpensive mapping, that is, d(x, pcx) ≤ d(x, y) for all y ∈ C, −−−→ −−→ (3) hxPc x, yPcyi ≤ 0 for all x ∈ X and y ∈ C. Further if, in addition, C is bounded, then F ix(T ) is nonempty. The following lemmas are very useful for proving our main results: Lemma 2.6. (The demiclosedness principle) Let C be a nonempty closed convex subset of the real CAT(0) space X and T : C → C such that xn * x ∗ ∈ C

and

(I − T )xn → 0.

Then x∗ = T x∗ . Here → and * denote strong and weak convergence, respectively.

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Moreover, the following result gives the conditions for the convergence of a nonnegative real sequences. Lemma 2.7. Assume that {an } is a sequence of nonnegative real numbers such that an+1 ≤ (1 − γn )an + δn , ∀n ≥ 0, where {βn } is a sequence in (0, 1) and {δn } is a sequence with P (1) ∞ n=0 γn = ∞, P (2) limn→∞ sup γδnn ≤ 0 or ∞ n=0 |δn | < ∞. Then limn→∞ an → 0.

3

The main result

Theorem 3.1. Let C be a nonempty closed convex subset of a complete CAT(0) space X. Let T : C → C be a nonexpansive mapping with F (T ) 6= φ and f : C → C be a contraction with coefficient θ ∈ [0, 1). Pick any x0 ∈ C, let {xn } be a sequence generated by (1.3), where{αn } and {βn } are the sequence in (0, 1) satisfying the following conditions: (1) limn→∞ αn = 1 and limn→∞ βn = 1, P P (2) ∞ αn = ∞ and ∞ = ∞, n=0 n=0 βn P P∞ (3) n=0 |αn+1 − αn | < ∞ and ∞ n=0 |βn+1 − βn | < ∞∀n ≥ 0, (4) limn→∞ d(xn , T (xn)) = 0. Then {xn } converges strongly to a fixed point x∗ of the nonexpansive mapping T which is also the unique solution of the variational inequality −−−→ → hxf (x), − yxi ≥ 0,

∀y ∈ F (T ).

In other words, x∗ is the unique fixed point of the contraction PF ix(T )f, that is, PF ix(T )f (x∗ ) = x∗ . Proof. We divide the proof into the following five steps. Step 1. First, we show that xn is bounded. Indeed, take p ∈ F (T ) arbitrarily, we have d(xn+1 , p) = d((1 − αn )f (xn ) ⊕ αn T (yn ), p) ≤ (1 − αn )d(f (xn), p) + αn d(T (yn), p) ≤ (1 − αn )d(f (xn), f (p)) + (1 − αn )d(f (p), p) + αn d(yn , p)

(3.1)

≤ (1 − αn )θd(xn , p) + (1 − αn )d(f (p), p) + αn d(yn , p). Now consider d(yn , p) = ((1 − βn )xn ⊕ βn T (xn ), p) ≤ (1 − βn )d(xn, p) + βn d(T (xn), p) ≤ (1 − βn )d(xn, p) + βn d(xn, p) ≤ d(xn , p).

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Using this in (3.1) we have d(xn+1 , p) ≤ (1 − αn )θd(xn , p) + (1 − αn )d(f (p), p) + αn d(xn , p) = [(1 − αn )θ + αn ]d(xn, p) + (1 − αn )d(f (p), p) = [1 − 1 + α + (1 − αn )θ]d(xn, p) + (1 − αn )d(f (p), p)

thus we have

= [1 − (1 − α) + (1 − αn )θ]d(xn , p) + (1 − αn )d(f (p), p)   1 = [1 − (1 − αn )(1 − θ)]d(xn , p) + (1 − αn )(1 − θ) d(f (p), p) , 1−θ 

1 d(xn+1 , p) ≤ max d(xn, p), d(f (p), p) 1−θ

similarly

From this





,

   1 d(xn , p) ≤ max d(xn−1 , p), . d(f (p), p) 1−θ d(xn+1 , p)    1 ≤ max d(xn , p), d(f (p), p) 1−θ    1 ≤ max d(xn−1 , p), d(f (p), p) 1−θ .. .    1 , ≤ max d(x0 , p), d(f (p), p) 1−θ

which shows that {xn } is bounded. From this we deduce immediately that {f (xn )}, {T (xn)} are bounded. Step 2. Next, we want to prove that limn→∞ d(xn+1 , xn) = 0. For this consider d(xn+1 , xn) = d((1 − αn )f (xn ) ⊕ αn T (yn ), (1 − αn−1 )f (xn−1 ) ⊕ αn−1 T (yn−1 ))

(3.2)

≤ (1 − αn )θd(xn , xn−1 ) + |αn − αn−1 |d(T (yn−1 ), f (xn−1 )) + αn d(yn , yn−1 ). Now consider d(yn , yn−1 ) = d((1 − βn )xn ⊕ βn T (xn ), (1 − βn−1 )xn−1 ⊕ βn−1 T (xn−1 )) ≤ (1 − βn )d(xn , xn−1 ) + |βn − βn−1 |d(T (xn−1 ), xn−1 ) + βn d(xn , xn−1 ) ≤ d(xn , xn−1 ) + |βn − βn−1 |d(T (xn−1 ), xn−1 ). Using this in (3.2) we get d(xn+1 , xn) ≤ (1 − αn )θd(xn , xn−1 ) + |αn − αn−1 |d(T (yn−1 ), f (xn−1 )) + αn d(xn , xn−1 ) + αn |βn − βn−1 |d(T (xn−1 ), xn−1 ) = [(1 − αn )θ + αn ]d(xn, xn−1 ) + |αn − αn−1 |d(T (yn−1 ), f (xn−1 )) + αn |βn − βn−1 |d(T (xn−1 ), xn−1 ).

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P P Let λn = (1 − αn ) so λn ∈ (0, 1), since αn ∈ (0, 1) ∞ λ = ∞, ∞ n=0 n=0 |αn − αn−1 | < ∞ P∞ and n=0 |βn − βn−1 |. By using Lemma 2.7, we get limn→∞ d(xn+1 , xn ) = 0. Step 3. Now, we want to prove thatlimn→∞ d(xn , T (yn)) → 0 d(xn , T (yn )) ≤ d(xn, T (xn )) + d(T (xn), T (yn)) ≤ d(xn, T (xn )) + d(xn, yn ) = d(xn, T (xn )) + d(xn, (1 − βn )xn ⊕ βn T (xn )) ≤ d(xn, T (xn )) + βn d(xn , T (xn)) ≤ (1 + βn )d(xn, T (xn )) → 0 (n → ∞). −−−−−→ −−→ Step 4. In this step, we claim that lim supn→∞ hx∗ f (x∗ ), x∗ xn i ≤ 0, where x∗ = PF (T )f (x∗ ). Indeed, we take a subsequence {xni } of {xn } which converges weakly to a fixed point p of T . Without loss of generality, we may assume that {xni } * p. From limn→∞ d(xn , T xn ) = 0 and Lemma 2.6 we have p = T (p). This together with the property of the metric projection implies that −−−−−→ −−→ −−−−−→ −−−→ −−−−−→ −→ lim suphx∗ f (x∗ ), x∗ xn i = lim suphx∗ f (x∗ ), x∗ xni i = hx∗ f (x∗ ), x∗ pi ≤ 0.

n→∞

n→∞

Step 5. Finally, we show that xn → x∗ as n → ∞. Here again x∗ ∈ F ix(T ) is the unique fixed point of the contraction PF ix(T )f . Consider d2 (xn+1 , x∗ ) = d2 ((1 − αn )f (xn ) ⊕ αn T (yn ), x∗ ) = (1 − αn )2n d2 (f (xn ), x∗) + (1 − αn )d2 (T (xn ), x∗) −−−−−→ −−−−−→ + 2αn (1 − αn )hf (xn )x∗ , T (yn )x∗ i −−−−−−−→ −−−−−→ ≤ α2n d2 (yn , x∗ ) + (1 − αn )2 d2 (f (xn ), x∗) + 2αn (1 − αn )hf (xn )f (x∗ ), T (yn)x∗ i −−−−−→ −−−−−→ + 2αn (1 − αn )hf (x∗ )x∗ , T (yn )x∗ i

(3.3)

≤ α2n d2 (yn , x∗ ) + (1 − αn )2 d2 (f (xn ), x∗) + 2αn (1 − αn )d(f (xn ), f (x∗))d(T (yn), x∗) −−−−−→ −−−−−→ + 2αn (1 − αn )hf (x∗ )x∗ , T (yn )x∗ i ≤ α2n d2 (yn , x∗ ) + 2αn (1 − αn )θd(xn , x∗ )d(yn , x∗ ) + (1 − αn )2 d2 (f (xn ), x∗ ) −−−−−→ −−−−−→ + 2αn (1 − αn )hf (x∗ )x∗ , T (yn )x∗ i, now consider d(yn , x∗ ) = d((1 − βn )xn ⊕ βn T (xn ), x∗ ) ≤ (1 − βn )d(xn , x∗ ) + βn d(T (xn), x∗ ) ≤ (1 − βn )d(xn , x∗ ) + βn d(xn , x∗ )

(3.4)

≤ d(xn , x∗ ),

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using (3.2) in (3.3) we get d2 (xn+1 , x∗ ) ≤ α2n d2 (xn , x∗) + 2αn (1 − αn )θd(xn , x∗)d(xn , x∗ ) + (1 − αn )2 d2 (f (xn ), x∗ ) −−−−−→ −−−−−→ + 2αn (1 − αn )hf (x∗ )x∗ , T (yn)x∗ i ≤ α2n d2 (xn , x∗) + 2αn (1 − αn )θd(xn , x∗)d(xn , x∗ ) + (1 − αn )2 d2 (f (xn ), x∗ ) −−−−−→ −−−−−→ + 2αn (1 − αn )hf (x∗ )x∗ , T (yn)x∗ i

(3.5)

≤ [α2n + 2αn (1 − αn )θ]d2 (xn , x∗ ) + (1 − αn )2 d2 (f (xn ), x∗) + 2αn (1 − αn )hf (x∗ ) − x∗ , T (yn ) − x∗ i. Note that αn θ < αn since αn ∈ (0, 1) and θ ∈ [0, 1) 2αn θ < 2αn , which implies that α2n + 2αn θ(1 − αn ) < α2n + 2αn (1 − αn ), therefore, we have d2 (xn+1 , x∗ ) ≤ [α2n + 2αn (1 − αn )]d2 (xn , x∗ ) + (1 − αn )2 d2 (f (xn ), x∗ ) −−−−−→ −−−−−→ + 2αn (1 − αn )hf (x∗ )x∗ , T (yn)x∗ i ≤ [2αn − α2n )]d2(xn , x∗ ) + (1 − αn )2 d2 (f (xn ), x∗) −−−−−→ −−−−−→ + 2αn (1 − αn )hf (x∗ )x∗ , T (yn)x∗ i 2

2 2

∗

(3.6)

∗

≤ 2αn d (xn , x ) + (1 − αn ) d (f (xn ), x ) −−−−−→ −−−−−→ + 2αn (1 − αn )hf (x∗ )x∗ , T (yn)x∗ i ≤ 2[1 − (1 − αn )]d2(xn , x∗ ) + (1 − αn )2 d2 (f (xn ), x∗) −−−−−→ −−−−−→ + 2αn (1 − αn )hf (x∗ )x∗ , T (yn)x∗ i, as by limn→∞ αn = 1 we have −−−−−→ −−−−−→ (1 − αn )2 d2 (f (xn ), x∗ ) + 2αn (1 − αn )hf (x∗ )x∗ , T (yn )x∗ i lim sup n→∞ (1 − αn ) −−−−−→ −−−−−→ = lim sup[(1 − αn )d2 (f (xn ), x∗) + 2αn hf (x∗ )x∗ , T (yn )x∗ i]

(3.7)

n→∞

≤ 0. From (3.6), (3.7), and Lemma 2.7 we have lim d2 (xn+1 , xn ) = 0,

n→∞

which implies that xn → x∗ as n → ∞. This completes the proof.

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References [1] I. Ahmad and M. Ahmad, An implicit viscosity technique of nonexpansive mapping in Cat(0) spaces, Open J. Math. Anal., 1 (2017), 1–12. [2] I. Ahmad and M. Ahmad, On the viscosity rule for common fixed points of two nonexpansive mappings in CAT(0) spaces, Open J. Math. Anal., 2 (2018) (in press). [3] M. A. Alghamdi, M. A. Alghamdi, N. Shahzad and H. K. Xu, The implicit midpoint rule for nonexpansive mappings, Fixed Point Theory Appl. 2014 (2014), Paper No. 96, 9 pages [4] I. D. Berg and I. G. Nikolaev, Quasilinearization and curvature of Aleksandrov spaces, Geom. Dedicata, 133 (2008), 195–218. [5] M. R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Grundlehren der mathematischen Wissenschaften, vol. 319, Springer-Verlag, Berlin, 1999. ´ [6] F. Bruhat and J. Tits, Groupes r´eductifs sur un corps local, Inst. Hautes Etudes Sci. Publ. Math., 41 (1972), 5–251. [7] H. Dehghan, J. Rooin, A characterization of metric projection in CAT(0) spaces, In: International Conference on Functional Equation, Geometric Functions and Applications (ICFGA 2012), Payame Noor University, Tabriz, 2012, pp. 41-43. [8] S. Dhompongsa and B. Panyanak, On δ-convergence theorems in CAT(0) spaces, Comput. Math. Appl., 56 (2008), 2572–2579. [9] M. Gromov, CAT(κ)-spaces: construction and concentration, J. Math. Sci., 119 (2004), 178–200. [10] W. A. Kirk, Geodesic geometry and fixed point theory, In Seminar of Mathematical Analysis (Malaga/Seville, 2002/2003), 64 (2003), 195–225. [11] W. A. Kirk, Geodesic geometry and fixed point theory, II, International Conference on Fixed Point Theory and Applications, Yokohama Publ., Yokohama, 2004, pp. 113– 142. [12] Y. C. Kwun, W. Nazeer, M. Munir and S. M. Kang, Explicit viscosity rules and applications of nonexpansive mappings, J. Comput. Anal. Appl., 24 (2018), 1541–1552. [13] A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl., 241 (2000), 46–55. [14] S. F. A. Naqvi and M. S. Khan, On the viscosity rule for common fixed points of two nonexpansive mappings in Hilbert spaces, Open J. Math. Sci., 1 (2017), 111–125. [15] L. Y. Shi and R. D. Chen, Strong convergence of viscosity approximation methods for nonexpansive mappings in CAT(0) spaces, J. Appl. Math., 2012 (2012), Article ID 421050, 11 pages.

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[16] K. Shimoji and W. Takahashi, Strong convergence to common fixed points of infinite nonexpansive mappings and applications, Taiwanese J. Math., 5 (2001) 387–404. [17] H. K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl., 298 (2004) 279–291. [18] L. Zhao, S. S. Chang, L. Wang and G. Wang, Viscosity approximation methods for the implicit midpoint rule of nonexpansive mappings in CAT(0) Spaces, J. Nonlinear Sci. Appl., 10 (2017), 386–394.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.6, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

The generalized viscosity implicit rules of asymptotically nonexpansive mappings in CAT(0) spaces Shin Min Kang1,2 , Absar Ul Haq3 , Waqas Nazeer4,∗ and Iftikhar Ahmad5

1

Department of Mathematics and RINS, Gyeongsang National University, Jinju 52828, Korea e-mail: [email protected] 2 3

Center for General Education, China Medical University, Taichung 40402, Taiwan

Department of Mathematics, University of Management and Technology, Sialkot Campus, Lahore 51410, Pakistan e-mail: [email protected] 4

5

Division of Science and Technology, University of Education, Lahore 54000, Pakistan e-mail: [email protected]

Department of Mathematics and Statistics, University of Lahore, Lahore 54000, Pakistan e-mail: [email protected] Abstract In this paper, we establish the generalized viscosity implicit rules of asymptotically nonexpansive mappings in CAT(0) spaces. The strong convergence theorems of the implicit rules proposed are proved under certain assumptions imposed on the control parameters. The results presented in this paper improve and extend some recent corresponding results announced. 2010 Mathematics Subject Classification: 47J25, 47N20, 34G20, 65J15 Key words and phrases: viscosity rule, CAT(0) space, nonexpansive mapping, asymptotically nonexpansive mapping, variational inequality

1

Introduction

The study of spaces of nonpositive curvature originated with the discovery of hyperbolic spaces, and flourished by pioneering works of Hadamard and Cartan, etc. in the first decades of the twentieth century. The idea of nonpositive curvature geodesic metric spaces could be traced back to the work of Busemann and Alexandrov, etc. in the 50’s. Later on Gromov [11] restated some features of global Riemannian geometry solely based on the so-called CAT(0) inequality. For through discussion of CAT(0) spaces and of fundamental role they play in geometry, we refer the reader to Bridson and Haefliger [6]. ∗

Corresponding author

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As we know, iterative methods for finding fixed points of nonexpansive mappings have received vast investigations due to its extensive applications in a variety of applied areas of inverse problem, partial differential equations, image recovery, and signal processing; see [1–4, 8, 9, 16, 18–21] and the references therein. One of the difficulties in carrying out results from Banach space to complete CAT(0) space setting lies in the heavy use of the linear structure of the Banach spaces. Berg and Nikolaev [5] introduce the notion of an inner product-like notion(quasilinearization) in complete CAT(0) spaces to resolve these difficulties. Fixed-point theory in CAT(0) spaces was frsc studied by Kirk [13,14]. He showed that every nonexpansive (single-valued) mapping defined on a bounded closed convex subset of a complete CAT(0) space always has a fxed point. Since then, the fxed-point theory for single-valued and multivalued mappings in CAT(0) spaces has been rapidly developed. In 2000, Moudaf’s [15] introduce viscosity approximation methods as following Theorem 1.1. Let C be a nonempty closed convex subset of the real Hilbert space X. Let T be a nonexpansive mapping of C into itself such that F ix(T ) = {x : T (x) = x} is nonempty. Let f be a contraction of C into itself with coefficient θ ∈ [0, 1). Pick any x0 ∈ [0, 1), let {xn } be a sequence generated by xn+1 =

γn 1 f (xn ) + T (xn ), 1 + γn 1 + γn

n ≥ 0,

where {γn } is a sequence in (0, 1) satisfying the following conditions: (1) limn→∞ γn = 0, P (2) P∞ n=0 γn = ∞, ∞ 1 (3) n=0 γn+1 − γ1n = 0. Then {xn } converges strongly to a fixed point x∗ of the mapping T , which is also the unique solution of the variational inequality hx − f (x), x − yi ≥ 0,

∀y ∈ F ix(T ).

In other words, x∗ is the unique fixed point of the contraction PF ix(T )f , that is, PF ix(T )f (x∗ ) = x∗ . Shi and Chen [17] studied the convergence theorems of the following Moudaf’s viscosity iterations for a nonexpansive mapping in CAT(0) spaces. xn+1 = tf (xn ) ⊕ (1 − t)T (xn ),

(1.1)

xn+1 = αn f (xn ) ⊕ (1 − αn )T (xn ).

(1.2)

They proved that {xn } defned by (1.1) and {xn } defned by (1.2) converged strongly to a fxed point of T in the framework of CAT(0) space. Zhao et al. [22] applied viscosity approximation methods for the implicit midpoint rule for nonexpansive mappings   xn ⊕ xn+1 xn+1 = αn f (xn ) ⊕ (1 − αn )T , ∀n ≥ 0. 2 Motivated by He et al. [12], in this paper, we study the generalized viscosity implicit rules of asymptotically nonexpansive mappings in the framework of CAT(0) spaces.

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More precisely, we consider the following implicit iterative algorithm xn+1 = αn f (xn ) ⊕ (1 − αn )T n (βn xn ⊕ (1 − βn )xn+1 )

(1.3)

under suitable conditions, we proved that the sequence {xn } converge strongly to a fixed point of the asymptotically nonexpansive mapping T .

2

Preliminaries

Let(X, d) be a metric space. A geodesic path joining x ∈ X to y ∈ X (or, more briefly, a geodesic from x to y) is a map c from a closed interval [0, l] ⊂ R to X such that c(0)= x, c(l) = y, and d(c(t), c(t0)) = |t − t0 | for all t, t0 ∈ [0, l]. In particular, c is an isometry and d(x, y) = l. The image α of c is called a geodesic (or metric) segment joining x and y. When it is unique, this geodesic segment is denoted by [x, y]. The space (X, d) is said to be a geodesic space if every two points of X are joined by a geodesic, and X is said to be uniquely geodesic if there is exactly one geodesic joining x and y for each x, y ∈ X. A subset Y ⊂ X is said to be convex if Y includes every geodesic segment joining any two of its points. A geodesic triangle 4(x1 , x2 , x3 ) in a geodesic metric space (X, d) consists of three points x1 , x2,and x3 in X (the vertices of 4) and a geodesic segment between each pair of vertices (the edges of 4). A comparison triangle for the geodesic triangle 4(x1 , x2 , x3 in (X, d) is a triangle 4(x1 , x2 , x3 ) := 4(x1 , x2 , x3 ) in the Euclidean plane E2 such that dE2 d(xi, xj ) = d(xi, xj )for i, j = 1, 2, 3. A geodesic space is said to be a CAT(0) space if all geodesic triangles satisfy the following comparison axiom. Let 4 be a geodesic triangle in X, and let 4 be a comparison triangle for 4 . Then, 4 is said to satisfy the CAT(0) inequality if for all x, y ∈ 4 and all comparison points x, y ∈ 4, d(x, y) = dE2 (x, y). (2.1) Let x, y ∈ X and by the Lemma 2.1(iv) of [10] for each t ∈ [0, 1], there exist a unique point z ∈ [x, y] such that d(x, z) = td(x, y),

d(y, z) = (1 − t)d(x, y).

(2.2)

From now on, we will use the notation (1 − t)x ⊕ ty for the unique fixed point z satisfying the above equation. We now collect some elementary facts about CAT(0) spaces which will be used in the proofs of our main results. Lemma 2.1. ([10]) Let X be a CAT(0) spaces. (a) For any x, y, z ∈ X and t ∈ [0, 1], d((1 − t)x ⊕ ty, z) ≤ (1 − t)d(x, z) + td(y, z).

(2.3)

(b) For any x, y, z ∈ X and t ∈ [0, 1], d2 ((1 − t)x ⊕ ty, z) ≤ (1 − t)2 d(x, z) + td2 (y, z) − t(1 − t)d2 (x, y).

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(2.4)

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Complete CAT(0) spaces are often called Hadamard spaces (see [6]). If x, y1 , y2 are points of a CAT(0) spaces and y0 is the midpoint of the segment [y1 , y2 ], which we will 2 denoted by y1 ⊕y 2 , then the CAT(0) inequality implies   1 y1 ⊕ y2 1 1 2 (2.5) d x, ≤ d2 (x, y1 ) + d2 (x, y2 ) − d2 (y1 , y2 ). 2 2 2 4 This inequality is the (CN) inequality of Bruhat and Tits [7]. In fact, a geodesic metric space is a CAT(0) space if and only if it satisfies the (CN) inequality (cf. [6], page 163). Definition 2.2. Let X be a CAT(0) space and T : X → X be a mapping. Then T is called nonexpensive if d(T (x), T (y)) ≤ d(x, y), x, y ∈ C. Definition 2.3. Let X be a CAT(0) space and T : X → X be a mapping. Then T is called contraction if d(T (x), T (y)) ≤ θd(x, y),

x, y ∈ C θ ∈ [0, 1).

Berg and Nikolaev [5] introduce the concept of quasi-linearization as follow. Let us − → denote the pair (a, b) ∈ X × X by the ab and call it a vector. Then, quasi-linearization is defined as a mapping h·, ·i : (x × X) × (X × X) −→ R defined as

− → → − 1 hab, cdi = (d2 (a, d) + d2 (b, c) − d2 (a, c) − d2 (b, d)), (2.6) 2 − → → − − → → − − → − → − → → − → − → → − → − it is easy to see that hab, cdi = hcd, abi, hab, cdi = −hba, cdi and h− ax, cdi + hxb, cdi = − → → − hab, cdi for all a, b, c, d ∈ X. We say that X satisfies the Cauchy-Schwarz inequality if − → → − hab, cdi ≤ d(a, b)d(a, c) for all a, b, c, d ∈ X. It is well-known [5] that a geodesically connected metric space is a CAT(0) space of and only if it satisfy the Cauchy-Schwarz inequality. Let C be a non-empty closed convex subset of a complete CAT(0) space X. The metric projection Pc : X → C is defined by u = Pc (x) ⇐⇒ inf{d(y, x) : y ∈ C},

∀x ∈ X

Definition 2.4. Let Pc : X → C is called the metric projection if for every x ∈ X there exist a unique nearest point in C, denoted by Pc x, such that d(x, Pcx) ≤ d(x, y),

y ∈ C.

The following theorem gives you the conditions for a projection mapping to be nonexpensive. Theorem 2.5. Let C be a non-empty closed convex subset of a real CAT(0) space X and Pc : X → X a metric projection. Then −−−→ → − (1) d(Pc x, Pc y) ≤ h− xy, Pc xPc yi for all x, y ∈ X, (2) Pc is nonexpensive mapping, that is, d(x, pcx) ≤ d(x, y) for all y ∈ C, −−−→ −−→ (3) hxPc x, yPcyi ≤ 0 for all x ∈ X and y ∈ C.

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Definition 2.6. A mapping T : C → C is called asymptotically nonexpensive if there exist a sequence a sequence {kn } ⊂ [0, ∞) with limn→∞ kn = 1 such that d(T n x − T n y) ≤ kn d(x, y),

∀x, y ∈ C, n ≥ 1.

(2.7)

It is well known that if T is an asymptotically nonexpansive, then F ix(T ) is always closed and convex. Further if, in addition, C is bounded, then F ix(T ) is nonempty. The following lemmas are very useful for proving our main results: Lemma 2.7. (The demiclosedness principle) Let C be a nonempty closed convex subset of the real CAT(0) space X and T : C → C such that xn * x ∗ ∈ C

and

(I − T )xn → 0.

Then x∗ = T x∗ . Here → and *) denote strong and weak) convergence, respectively. Moreover, the following result gives the conditions for the convergence of a nonnegative real sequence. Lemma 2.8. Assume that {an } is a sequence of nonnegative real numbers such that an+1 ≤ (1 − βn )an + δn , ∀n ≥ 0, where {βn } is a sequence in (0, 1) and {δn } is a sequence with P (1) ∞ n=0 βn = ∞, P (2) lim supn→∞ sup βδnn ≤ 0 or ∞ n=0 |βn| < ∞. Then limn→∞ an → 0.

3

The main results

Theorem 3.1. Let C be a non-empty closed convex subset of a complete CAT(0) space X and T : C → C be an asymptotically nonexpensive mapping with sequence {kn } ⊂ [0, +∞) with limn→∞ kn = 1 and F ix(T ) 6= ∅. Let f : C −→ C be a contraction with coefficient θ ∈ [0, 1). For arbitrary initial point x0 ∈ C, let {xn } be a sequence generated by (1.3), where {αn } and {βn} are the Psequence in (0, 1) satisfying the following conditions: (1) limn→∞ αn = 0 and ∞ n=1 αn = ∞, 2 −1 kn (2) limn→∞ αn = 0, (3) 0 < τ < βn < βn+1 < 1, for all n ≥ 0, (4) limn→∞ d(T n (xn ), (xn)) = 0. Then {xn } converges strongly to the point x∗ = PF ix(T )f (x∗ ) of the mapping T , which is also the unique solution of the variational inequality −−−→ → hxf (x), − xyi ≥ 0,

∀y ∈ Fix(T ).

In other words, x∗ is the unique fixed point of the contraction PF ix(T )f , that is, PF ix(T )f (x∗ ) = x∗ . Proof. We have divided the proof into four steps. Step 1: First, we show that the generalized viscosity implicit rule (??) is well-defined Sn (x) = αn f (xn ) ⊕ (1 − αn )T n (βnxn ⊕ (1 − βn )xn+1 ).

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Consider d(Sn (x), Sn(y)) = d(αn f (xn ) ⊕ (1 − αn )T n (βn xn ⊕ (1 − βn )x), αn f (yn ) ⊕ (1 − αn )T n (βn yn ⊕ (1 − βn )y)) = (1 − αn )d(T n(βn xn ⊕ (1 − βn )x), T n(βn yn ⊕ (1 − βn )y)) ≤ (1 − αn )kn (1 − βn )d(x, y). Since lim αn = 0 and n→∞

P∞

n=1

αn = ∞, limn→∞

2 −1 kn αn

= 0, or n ≥ 0. We may assume that

(1 − αn )kn (1 − βn ) ≤ 1 − τ for all n ≥ 0. This implies that Sn is a contraction for each n. Therefore there exists a unique fixed point for Sn by contraction principle, which also implies that (1.3) is well-defined. Step 2: Now, we show that the sequence {xn } is bounded. Indeed take p ∈ F ix(T ) arbitrary, we have d(xn+1 , p) = d(αn f (xn ) ⊕ (1 − αn )T n (βn xn ⊕ (1 − βn )xn+1 ), p) ≤ αn d((f (xn), p) + (1 − αn )d((βnxn ⊕ (1 − βn )xn+1 ), p) ≤ αn d((f (xn), f (p)) + αn d((f (p), p) + (1 − αn )kn d((βnxn ⊕ (1 − βn )xn+1 ), p) ≤ αn θd(xn , p) + αn d((f (p), p) + (1 − αn )kn βn d(xn , p) + (1 − βn )kn(1 − βn )d(xn+1 , p) ≤ (αn θ + (1 − αn )kn βn )d(xn, p) + αn d((f (p), p) + (1 − βn )kn(1 − βn )d(xn+1 , p), it follows that [1 − (1 − αn )kn (1 − βn )]d(xn+1 , p) = (αn θ + (1 − αn )kn βn )d(xn , p) + αn d((f (p), p).

(3.1)

Since {αn } and {βn } are the sequence in (0, 1) lim inf (1 − αn )kn (1 − βn ) ≤ 1 n→∞

for any given positive number , 0 <  < 1 − θ, there exists a sufficient large positive integer n0 , such that for any n > n0 , we have kn2 − 1 ≤ βn αn and kn − 1 ≤

kn + 1 k2 − 1 (kn − 1) ≤ n ≤ αn . βn βn

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Moreover, by (3.1) αn θ + (1 − αn )kn βn αn d(xn , p) + d(f (p), p) 1 − (1 − αn )kn (1 − βn ) 1 − (1 − αn )kn (1 − βn )   αn (kn − θ) − (kn − 1) = 1− d(xn , p) 1 − (1 − αn )kn(1 − βn ) αn d(f (p), p) + 1 − (1 − αn )kn (1 − βn )   αn (kn − θ − ) ≤ 1− d(xn , p) 1 − (1 − αn )kn(1 − βn )   αn (kn − θ − ) 1 + d(f (p), p) 1 − (1 − αn )kn (1 − βn ) (kn − θ − )   1 ≤ max d(xn , p), d(f (p), p) kn − θ −    1 d(x , p), ≤ max d(f (p), p) . n 1−θ−

d(xn+1 , p) =

By applying induction, we obtain  d(xn+1 , p) ≤ max d(x0 , p),

 1 d(f (p), p) . 1−θ−

Hence, we conclude that {xn } is bounded. Consequently, we deduce immediately from it that {f (xn )} and {T n (βn xn ⊕ (1 − βn )xn+1 } are bounded. Step 3: Now, we prove that lim d(xn+1 , xn ) = 0 n→∞

d(xn+1 , xn ) ≤ d(xn+1 , T n xn ) + d(T n xn , xn) = d(αn f (xn ) ⊕ (1 − αn )T n (βn xn ⊕ (1 − βn )xn+1 ), T nxn ) + d(T n xn , xn) ≤ αn d((f (xn), T nxn ) + (1 − αn )d(T n(xn (βn xn ⊕ (1 − βn )xn+1 )), T nxn ) + d(T nxn , xn) ≤ αn d((f (xn), T nxn ) + (1 − αn )knd((βn xn ⊕ (1 − βn )xn+1 ), xn) + d(T nxn , xn) ≤ αn d((f (xn), T nxn ) + (1 − αn )kn(1 − βn )d(xn+1 , xn ) + d(T n xn , xn ) ≤ αn M1 + (1 − αn )kn (1 − βn )d(xn+1 , xn ) + d(T n xn , xn ), where M1 = sup{d((f (xn), T nxn ), n ≥ 1} is constant such that 1 − (1 − αn )kn (1 − βn )d(xn+1 , xn ) ≤ αn M1 + d(T n xn , xn ) It gives d(xn+1 , xn ) ≤

αn M1 1 − (1 − αn )kn (1 − βn ) 1 + d(T nxn , xn ) 1 − (1 − αn )kn (1 − βn )

Since 1 − (1 − αn )kn (1 − βn ) ≥ τ by virtue of the conditions (1) and (4), we have lim d(xn+1 , xn ) = 0.

n→∞

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(3.2)

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Step 4: Now we show that lim d(xn, T xn ) = 0. n→∞

d(xn , T n−1 xn ) = d(αn−1 f (xn−1 ) ⊕ (1 − αn−1 )T n−1 (βn−1 xn−1 ⊕ (1 − βn−1 )xn ), T n−1 xn ) ≤ αn−1 d((f (xn−1 ), T n−1 xn ) + (1 − αn−1 )knd((βn−1 xn−1 ⊕ (1 − βn−1 )xn ), xn) ≤ αn−1 d((f (xn−1 ), T n−1 xn ) + (1 − αn−1 )knβn−1 d(xn, xn−1 ) ≤ αn−1 M1 + (1 − αn−1 )kn βn−1 d(xn, xn−1 ) by condition (1) and (3.2) we have lim d(xn , T n−1 xn ) = 0.

n→∞

Hence we get d(xn , T xn ) ≤ d(xn , T nxn ) + d(T n xn , T xn ) ≤ d(xn , T nxn ) + k1 d(T n−1 xn , xn)

(3.3)

→ 0 (n → ∞) Then, it follows from (3.2) and (3.3) that d(T n (βn xn ⊕ (1 − βn )xn+1 , xn ) ≤ d(T n (βn xn ⊕ (1 − βn )xn+1 , T xn ) + d(T xn , xn) ≤ kn d((βn xn ⊕ (1 − βn )xn+1 , xn ) + d(T xn , xn) ≤ kn (1 − βn )d(xn+1 , xn ) + d(T xn , xn) ≤ kn d(xn+1 , xn ) + d(T xn , xn ) →0

(n → ∞).

Step 5: In this step, we claim that −−−−−→ −−→ lim suphx∗ f (x∗ ), x∗ xn i ≤ 0, x→∞

where x∗ = PF ix(T )f (x∗ ). Indeed, we take a subsequence {xni } of {xn } which converges weakly to a fixed point p of T . Without loss of generality, we may assume that {xni } * p. From limn→∞ d(xn , T (xn) = 0 and the Lemma 2.7 we have p = T (p). This together, with the property of metric projection implies that −−−−−→ −−→ −−−−−→ −−−→ lim suphx∗ f (x∗ ), x∗ xn i = lim suphx∗ f (x∗ ), x∗ xni i x→∞

x→∞

−−−−−→ −→ = lim suphx∗ f (x∗ ), x∗ pi x→∞

≤ 0. Step 6: Finally, we show that xn → x∗ as n → ∞.

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Now, we prove that limn→∞ d(xn+1 , xn ) = 0. Now, we again take x∗ ∈ F ix(T ) is the unique fixed point of the contraction PF ix(T )f . Consider d2 (xn , xn) = d2 (αn f (xn ) ⊕ (1 − αn )T n (βn xn ⊕ (1 − βn )xn+1 ), x∗) = α2n d2 (f (xn ), x∗ ) + (1 − α2n )d2 (T n (βn xn ⊕ (1 − βn )xn+1 ), x∗ ) −−−−−→ −−−−−−−−−−−−−−−−−−−−→ + 2αn (1 − αn )hf (xn)x∗ , T n(βn xn ⊕ (1 − βn )xn+1 )x∗ i ≤ α2n d2 (f (xn ), x∗ ) + (1 − α2n )kn2 d2 (βn xn ⊕ (1 − βn )xn+1 ), x∗ ) −−−−−−−→ −−−−−−−−−−−−−−−−−−−−→ + 2αn (1 − αn )hf (xn)f (x∗ ), T n (βn xn ⊕ (1 − βn )xn+1 )x∗ i −−−−−→ −−−−−−−−−−−−−−−−−−−−→ + 2αn (1 − αn )hf (x∗ )x∗ , T n (βnxn ⊕ (1 − βn )xn+1 )x∗ i ≤ (1 − α2n )kn2 d2 (βn xn ⊕ (1 − βn )xn+1 ), x∗) + 2αn (1 − αn )d(f (xn)f (x∗ ))d(T n(βn xn ⊕ (1 − βn )xn+1 )x∗ ) + Kn ≤ (1 − α2n )kn2 d2 (βn xn ⊕ (1 − βn )xn+1 ), x∗) + 2θαn (1 − αn )kn d(xn , x∗ )d(T n(βn xn ⊕ (1 − βn )xn+1 )x∗ ) + Kn , where −−−−−→ −−−−−−−−−−−−−−−−−−−−→ Kn = α2n d2 (f (xn ), x∗) + 2αn (1 − αn )hf (x∗ )x∗ , T n (βn xn ⊕ (1 − βn )xn+1 )x∗ i, it becomes (1 − α2n )kn2 d2 (βn xn ⊕ (1 − βn )xn+1 ), x∗ ) + 2θαn (1 − αn )kn d(xn, x∗ )d(T n (βn xn ⊕ (1 − βn )xn+1 )x∗ ) + Kn + d2 (xn , xn) ≥ 0. Solving this quadratic inequality for d((βn xn ⊕ (1 − βn )xn+1 )x∗ ) yields d((βn xn ⊕ (1 − βn )xn+1 )x∗ ) n 1 ≥ − 2θαn (1 − αn )kn d(xn , x∗ ) 2(1 − αn )2 kn2 o p + 4θ2 α2n (1 − αn )2 kn2 d2 (xn , x∗) − 4(1 − αn )2 kn2 (Kn − d2 (xn , x∗ )) p −θαn d(xn, x∗ ) + θ2 α2n d2 (xn , x∗ ) − Kn + d2 (xn+1 , x∗ ) = . (1 − αn )kn This implies that βn d(xn , x∗ ) + (1 − βn )d(xn+1 , x∗ ) p −θαn d(xn , x∗ ) + θ2 α2n d2 (xn , x∗ ) − Kn + d2 (xn+1 , x∗ ) ≥ , (1 − αn )kn namely, [(1 − αn )kn βn ]d(xn, x∗ ) + (1 − αn )kn (1 − βn )d(xn+1 , x∗) p ≥ θ2 α2n d2 (xn , x∗ ) − Kn + d2 (xn+1 , x∗). 1052

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Then θ2 α2n d2 (xn , x∗ ) − Kn + d2 (xn+1 , x∗ ) ≤ [(1 − αn )kn βn + θαn ]2 d2 (xn , x∗ ) + (1 − αn )2 kn2 (1 − βn )2 d2 (xn+1 , x∗ ) + 2[(1 − αn )kn βn + θαn ](1 − αn )kn (1 − βn )d(xn , x∗ )d(xn+1 , x∗ ) ≤ [(1 − αn )kn βn + θαn ]2 d2 (xn , x∗ ) + (1 − αn )2 kn2 (1 − βn )2 d2 (xn+1 , x∗ ) + ((1 − αn )knβn + θαn )(1 − αn )kn(1 − βn )(d2(xn , x∗ ) + d2 (xn+1 , x∗ )), which is reduced to the inequality [1 − (1 − αn )2 kn2 (1 − βn )2 − ((1 − αn )kn βn + θαn )(1 − αn )kn (1 − βn )]d2 (xn+1 , x∗ ) ≤ [((1 − αn )kn(1 − βn ))2 + (1 − αn )kn(1 − βn )(1 − αn )kn (1 − βn ) − θ2 α2n ]d2 (xn , x∗ ) + Kn , that is, [1 − (kn + αn (θ − kn ))(1 − αn )(1 − βn )kn ]d2 (xn+1 , x∗ ) ≤ [((1 − αn )kn βn + θαn )(kn + αn (θ − kn )) − θ2 α2n ]d2 (xn , x∗ ) + Kn

(3.4)

it follows from (3.4) that d2 (xn+1 , x∗ ) ≤

[((1 − αn )kn βn + θαn )(kn + αn (θ − kn )) − θ2 α2n ]d2 (xn , x∗ ) [1 − (kn + αn (θ − kn ))(1 − αn )(1 − βn )kn ] Kn + . [1 − (kn + αn (θ − kn ))(1 − αn )(1 − βn )kn ]

(3.5)

Let   1 ((1 − αn )kn βn + θαn )(kn + αn (θ − kn )) − θ2 α2n wn = 1− αn 1 − (kn + αn (θ − kn ))(1 − αn )(1 − βn )kn 2 1 1 − kn − 2αn kn (θ − kn ) − α2n (θ − kn ) − θ2 α2n = αn 1 − (kn + αn (θ − kn ))(1 − αn )(1 − βn )kn −βn  − 2kn (θ − kn ) − αn (θ − kn )2 + θ2 αn ≤ 1 − (kn + αn (θ − kn ))(1 − αn )(1 − βn )kn since 0 <  < 1 − θ and the sequence {βn } satisfies 0 < τ ≤ βn ≤ βn+1 < 1 for all n ≥ 0 and lim βn exists, assume that n→∞

lim βn = β ∗ > 0.

n→∞

Then lim wn ≤

n→∞

(2 − β ∗ )(1 − θ) > 0. β∗

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(2−β ∗ )(1−θ)

. Then there exists an sufficiently large integer N1 such that Let 0 < λ1 < β∗ wn > λ1 for all n > N1 . Hence, we have ((1 − αn )kn βn + θαn )(kn + αn (θ − kn )) − θ2 α2n ≤ 1 − λ1 αn , 1 − (kn + αn (θ − kn ))(1 − αn )(1 − βn )kn

∀n ≥ N1 .

(3.6)

It turns out from (3.5) that d2 (xn+1 , x∗) ≤ (1 − λ1 αn )d2 (xn , x∗ ) +

Kn . (3.7) [1 − (kn + αn (θ − kn ))(1 − αn )(1 − βn )kn ]

From (3.5), limn→∞ αn = 0 and Step 4 we have Kn n→∞ αn λ1 [1 − (kn + αn (θ − kn ))(1 − αn )(1 − βn )kn ] −−−−−→ −−−−−−−−−−−−−−−−−−−−→ α2 d2 (f (xn ), x∗) + 2αn (1 − αn )hf (x∗ )x∗ , T n(βn xn ⊕ (1 − βn )xn+1 )x∗ i = lim sup n αn λ1 [1 − (kn + αn (θ − kn ))(1 − αn )(1 − βn )kn] n→∞ −−−−−→ −−−−−−−−−−−−−−−−−−−−→ αn d2 (f (xn ), x∗) + 2(1 − αn )hf (x∗ )x∗ , T n (βnxn ⊕ (1 − βn )xn+1 )x∗ i = lim sup λ1 [1 − (kn + αn (θ − kn ))(1 − αn )(1 − βn )kn ] n→∞ ≤ 0.

lim sup

(3.8)

From (3.7) and (3.8) and the Lemma 2.8 we have lim d(xn+1 , x∗ ) = 0.

n→∞

This implies that xn → x∗ as n → ∞. This complete the proof. The following result is an immediate consequence of the Theorem 3.1. Theorem 3.2. Let C be a non-empty closed convex subset of a complete CAT(0) space X and T : C → C be an nonexpensive mapping with F ix(T ) 6= ∅. Let f : C → C be a contraction with coefficient θ ∈ [0, 1) and for arbitrary initial point x0 ∈ C. Let {xn } be a sequence generated by xn+1 = αn f (xn ) ⊕ (1 − αn )T (βn xn ⊕ (1 − βn )xn+1 ),

(3.9)

where {αn } and {βn} are the sequence in (0, 1) satisfying the condition of Theorem 3.1. Then {xn } converges strongly to the point x∗ = PF ix(T )f (x∗ ) of the mapping T , which is also the unique solution of the variational inequality −−−→ → hxf (x), − xyi ≥ 0,

∀y ∈ Fix(T ).

In other words, x∗ is the unique fixed point of the contraction PF ix(T )f , that is, PF ix(T )f (x∗ ) = x∗ .

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References [1] I. Ahmad and M. Ahmad, An implicit viscosity technique of nonexpansive mapping in Cat(0) spaces, Open J. Math. Anal., 1 (2017), 1–12. [2] I. Ahmad and M. Ahmad, On the viscosity rule for common fixed points of two nonexpansive mappings in CAT(0) spaces, Open J. Math. Anal., 2 (2018) (in press). [3] M. A. Alghamdi, M. A. Alghamdi, N. Shahzad and H. K. Xu, The implicit midpoint rule for nonexpansive mappings, Fixed Point Theory Appl. 2014 (2014), Paper No. 96, 9 pages [4] H. Attouch, Viscosity solutions of minimization problems, SIAM J. Optim., 6 (1996), 769–806. [5] I. D. Berg and I. G. Nikolaev, Quasilinearization and curvature of Aleksandrov spaces, Geom. Dedicata, 133 (2008), 195–218. [6] M. R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Grundlehren der mathematischen Wissenschaften, vol. 319, Springer-Verlag, Berlin, 1999. ´ [7] F. Bruhat and J. Tits, Groupes r´eductifs sur un corps local, Inst. Hautes Etudes Sci. Publ. Math., 41 (1972), 5–251. [8] H. Dehghan and J. Rooin, A characterization of metric projection in CAT(0) spaces, In: International Conference on Functional Equation, Geometric Functions and Applications (ICFGA 2012), Payame Noor University, Tabriz, 2012, pp. 41-43. [9] S. Dhompongsa and W. A. Kirk, B. Panyanak,Nonexpansive set-valued mappings in metric and Banach spaces, J. Nonlinear Convex Anal., 8 (2007), 35–45. [10] S. Dhompongsa and B. Panyanak, On δ-convergence theorems in CAT(0) spaces, Comput. Math. Appl., 56 (2008), 2572–2579. [11] M. Gromov, CAT(κ)-spaces: construction and concentration, J. Math. Sci., 119 (2004), 178–200. [12] S. He, Y. Mao, Z. Zhou, and J. Q. Zhang, The generalized viscosity implicit rules of asymptotically nonexpansive mappings in Hilbert spaces, Appl. Math. Sci., 11 (2017), 549–560. [13] W. A. Kirk, Geodesic geometry and fixed point theory II, International Conference on Fixed Point Theory and Applications, Yokohama Publ., Yokohama, 2004, pp. 113– 142. [14] W. A. Kirk and B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal., 68 (2008), 3689–3696. [15] A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl., 241 (2000), 46–55.

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[16] S. F. A. Naqvi and M. S. Khan, On the viscosity rule for common fixed points of two nonexpansive mappings in Hilbert spaces, Open J. Math. Sci., 1 (2017), 111–125. [17] L. Y. Shi and R. D. Chen, Strong convergence of viscosity approximation methods for nonexpansive mappings in CAT(0) spaces, J. Appl. Math., 2012 (2012), Article ID 421050, 11 pages. [18] K. Shimoji and W. Takahashi, Strong convergence to common fixed points of infinite nonexpansive mappings and applications, Taiwanese J. Math., 5 (2001) 387–404. [19] D. Wu, S. S. Chang and G. X. Yuan, Approximation of common fixed points for a family of finite nonexpansive mappings in Banach space, Nonlinear Anal., 63 (2005), 987–999. [20] H. K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl., 298 (2004) 279–291. [21] Y. Yao and N. Shahzad, New methods with perturbations for non-expansive mappings in Hilbert spaces, Fixed Point Theory Appl., 2011 (2011), Paper No. 79, 9 pages. [22] L. Zhao, S. S. Chang, L. Wang and G. Wang, Viscosity approximation methods for the implicit midpoint rule of nonexpansive mappings in CAT(0) Spaces, J. Nonlinear Sci. Appl., 10 (2017), 386–394.

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On some sixth-order rational recursive sequences M. Folly-Gbetoula

∗

and D. Nyirenda

†

Abstract We study the sixth-order recursive sequences of the form xn+1 =

xn−5 xn , xn−4 (an + bn xn−5 xn )

where an and bn are sequences of real numbers, via the technique of Lie group analysis. Symmetry generators associated with the group of transformations that map solutions onto themselves are obtained and exact solutions derived. The ‘final constraint’ when finding the symmetries, is used to split the solution into different categories. The result of this work generalizes a recent work by Elsayed et al.

Keywords Difference equation; Symmetry; Group invariant solutions PACS 39A10; 39A13; 39A90

1

Introduction

Among the numerous well-known techniques for solving differential equations, is the powerful Lie symmetry approach. In the nineteenth century, the Norwegian mathematician Sophus Lie [12] developed a systematic algorithm based on the invariance of the ordinary differential equations under a group of transformations (symmetry). In the twentieth century, Maeda [13, 14] demonstrated that this approach can be extended to ordinary difference equations and recently, Hydon [6] used a similar approach to come up with some interesting results. It is now known that Lie’s method can be implemented to find symmetries, first integrals (conservation laws) and closed form solutions of difference equations, even in the context of variational equations. ∗

School of Mathematics, University of the Witwatersrand, Johannesburg, X3, Wits 2050, South Africa Tel.: +27(0)117176289 Email: [email protected] † School of Mathematics, University of the Witwatersrand, Johannesburg, X3, Wits 2050, South Africa Tel.: +27(0)117176224 Email: [email protected]

1

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In this paper, we obtain symmetry generators admitted by the difference equations of the form xn−5 xn xn+1 = , (1) xn−4 (an + bn xn−5 xn ) where an and bn are random sequences, and then proceed to find the solutions in closed form via the invariance of the group of transformations admitted by (2). We first present the solutions in a unified manner and then split them into different categories based on some properties of the ‘final constraint’. This work generalizes the work by Elsayed et. al. [3], where the authors obtained the formulas of the solutions of the difference equations xn−5 xn , n = 0, 1, . . . , (2) xn+1 = xn−4 (± + ±xn−5 xn ) in which the initial conditions x−5 , x−4 , x−3 , x−2 , x−1 , x0 are arbitrary nonzero real numbers. For similar work on the symmetry approach, see [4, 5, 7, 15, 16] and on different methods, see [1, 2, 8, 10, 19].

1.1

Preliminaries

In this section, we shortly present key elements of Lie group analysis of difference equations. For more understanding of the concepts and notation, we refer the reader to [6, 17] where our definitions and most of our notation are taken from. Let x? = X(x; ε) (3) be a one parameter Lie group of transformations. Definition 1.1 An infinitely differentiable function F is an invariant function of the Lie group of point transformation (3) if and only if, for any group transformations, F (x) = F (x? ).

(4)

Definition 1.2 The infinitesimal generator of the one-parameter Lie group of point transformation (3) is the operator n X ∂ , (5) ξi (x) X = X(x) = ξ(x) × ∆ = ∂x i i=1 where ∆ is the gradient operator. 2

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Theorem 1.1 F (x) is invariant under the Lie group of transformations (3) if and only if XF (x) = 0.

(6)

Now, consider a general kth-order difference equation un+k = ω(n, un , un+1 , . . . , un+k−1 )

(7)

for some smooth function ω. We are seeking a one-parameter Lie group of point transformations n∗ =n, u∗n =un + εξ(n, un )+)(ε2 ), .. .

(8a) (8b)

u∗n+k =un+k + εS k ξ(n, un )+)(ε2 ),

(8c)

where ξ denotes the characteristic, ε (ε is small enough) is the group parameter and S : n 7→ n + 1 stands for the shift forward operator . The symmetry criterion is given by u∗n+k = ω(n, u∗n , u∗n+1 , . . . , u∗n+k−1 ),

(9)

whenever (7) holds, and further the substitution of (8) in (9) yields the linearized symmetry condition: S k ξ(n, un ) − Xω = 0

(10)

where X, the corresponding prolonged symmetry operator of the group of transformations (8), is given by X = ξ(n, un )

∂ ∂ ∂ + Sξ(n, un ) + · · · + S k−1 ξ(n, un ) . ∂un ∂un+1 ∂un+k−1

(11)

The characteristics are obtained by solving the functional equation (10). As simple as (10) may look, its solution is found after a series of steps that require a set of cumbersome calculations. In this work, we employ the well-known choice of canonical coordinate [9] Z dun Sn = (12) ξ(n, un ) to reduce the order of the difference equation under investigation. 3

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2

Main results

Consider the sixth-order difference equations of the form (2). Let un+6 = ω =

un un+5 , un+1 (An + Bn un un+5 )

(13)

where An and Bn are random sequences, be the forward difference equation equivalent to (2). The linearized symmetry condition (10) imposed on (13) leads to un un+5 ξ(n + 1, un+1 ) An un ξ(n + 5, un+5 ) + un+1 (An + Bn un un+5 )2 u2n+1 (An + Bn un un+5 ) An un+5 ξ(n, un ) = 0. − un+1 (An + Bn un un+5 )2

ξ(n + 6, ω) −

(14)

By the means of the first-order partial differential operator L=

∂ ωun ∂ − , ∂un ωun+5 ∂un+5

we can get rid of the first term in (14). This yields the following: An un+5 ξ 0 (n + 5, un+5 ) An un+5 ξ 0 (n, un ) An ξ (n + 5, un+5 ) − − 2 2 un+1 (An + Bn un un+5 ) un+1 (An + Bn un un+5 ) un+1 (An + Bn un un+5 )2 An un+5 ξ (n, un ) + = 0. (15) un un+1 (An + Bn un un+5 )2 Here, it is important to simplify the equation in order to minimize the number of derivations. Thus, we clear fractions in (15) and divide the resulting equation by un un+5 to get ξ 0 (n + 5, un+5 ) −

1 un+5

ξ (n + 5, un+5 ) − ξ 0 (n, un ) +

1 ξ (n, un ) = 0. un

Differentiating (16) with respect to un , keeping un+5 fixed, leads to   d 1 0 −ξ (n, un ) + ξ (n, un ) = 0. dun un

(16)

(17)

Clearly, the solution of (17) is ξ (n, un ) = f (n)un + g(n)un ln un

(18)

4

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for some arbitrary functions f and g of n. To ease the computation we shall assume that g is zero. Using the expression of the characteristic given in (18), equation (14) becomes Bn f (n + 1)un un+5 + Bn f (n + 6)un un+5 − An f (n) + An f (n + 1) − An f (n + 5) + An f (n + 6) = 0. (19) which splits into 1: un un+5 :

f (n) + f (n + 5) = 0 f (n + 1) + f (n + 6) = 0.

(20) (21)

The system above reduces to the final constraint: f (n) + f (n + 5) = 0.

(22)

Solving (22) for f , we obtain five independent solutions given by (−1)n , exp(±nπ/5) and exp(±3nπ/5). Therefore, the characteristics are ξ1 =(−1)n un , ξ2 = β n un , ξ3 = β¯n un , ξ4 = θn un , ξ5 = θ¯n un ,

(23)

and so the prolonged infinitesimal generators admitted by (13) are X1 =(−1)n un ∂un + (−1)n+1 un+1 ∂un+1 + (−1)n+2 un+2 ∂un+2 + (−1)n+3 un+3 ∂un+3 + (−1)n+4 un+4 ∂un+4 + (−1)n+5 un+5 ∂un+5 ,

(24a)

X2 =β n un ∂un + β n+1 un+1 ∂un+1 + β n+2 un+2 ∂un+2 + β n+3 un+3 ∂un+3 + β n+4 un+4 ∂un+4 + β n+5 un+5 ∂un+5 ,

(24b)

X3 =β¯n un ∂un + β¯n+1 un+1 ∂un+1 + β¯n+2 un+2 ∂un+2 + β¯n+3 un+3 ∂un+3 + β¯n+4 un+4 ∂un+4 + β¯n+5 un+5 ∂un+5 ,

(24c)

X4 =θn un ∂un + θn+1 un+1 ∂un+1 + θn+2 un+2 ∂un+2 + θn+3 un+3 ∂un+3 + θn+4 un+4 ∂un+4 + θn+5 un+5 ∂un+5 ,

(24d)

X5 =θ¯n un ∂un + θ¯n+1 un+1 ∂un+1 + θ¯n+2 un+2 ∂un+2 + θ¯n+3 un+3 ∂u + θ¯n+4 un+4 ∂u + θ¯n+5 un+5 ∂u

(24e)

n+3

n+4

n+5

.

5

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Note that β = exp(π/5) and θ = exp(3π/5) Using the generator X2 , we have the canonical coordinate Z dun 1 = n ln |un |. Sn = (25) n β un β Taking advantage of the form of the relation (22), we construct the invariant function V˜n V˜n = Sn β n + Sn+5 β n+5

(26)

X1 V˜n =(−1)n + (−1)n+5 = 0, X2 V˜n =β n + β n+5 = 0, X3 V˜n =β¯n + β¯n+5 = 0,

(27a)

in view of the fact that

X4 V˜n =θn + θn+5 = 0,

(27b) (27c) (27d) (27e)

and X5 V˜n = θ¯n + θ¯n+5 = 0.

(27f)

For rational difference equations, it is convenience to use |Vn | = exp{−V˜n },

(28)

i.e., Vn = ±1/(un un+5 ) but we will be using the plus sign. Substituting (28) into equation (13), we reduce it to Vn+1 = An Vn + Bn .

(29)

We iterate (29) to get its solution in closed form as ! j−1 ! j−1 j−1 Y X Y Vj =V0 Ak 1 + Bl Ak2 . k1 =0

l=0

(30)

k2 =l+1

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From (25), (26) and (28), we have |un | = exp (βn Sn ) "

n−1 1X n n n ¯ ¯ = exp (−1) c1 + β c2 + β c3 + θ c4 + θ c5 − (−1)n−k1 |V˜k1 | 5 k =0 n

n

1

−

−

n−1 X

1 5k

β n β¯k2 |V˜k2 | −

2 =0

n−1 X

1 5k

3

n−1 X

1 β¯n β k3 |V˜k3 | − 5k =0

θn θ¯k4 |V˜k4 |

4 =0

#

n−1 X

1 θ¯n θk5 |V˜k5 | 5 k =0 5 "

n−1 1X n n n ¯ ¯ = exp (−1) c1 + β c2 + β c3 + θ c4 + θ c5 + (−1)n−k1 ln |Vk1 | 5 k =0 n

n

1

n−1 X

n−1 1 1 X n ¯k4 n k3 ¯ β β ln |Vk3 | + β β ln |Vk2 | + θ θ ln |Vk4 | 5 k =0 5 k =0 2 =0 3 4 # n−1 1 X ¯n k5 θ θ ln |Vk5 | + 5 k =0

1 + 5k

n ¯k2

n−1 X

5

# n−1  1 X n−k (−1) + 2Re(γ1 (n, k) + γ2 (n, k)) ln |Vk | , = exp Hn + 5 k=0 "

(31)

where Hn = (−1)n c1 + β n c2 + β¯n c3 + θn c4 + θ¯n c5 , γ1 (n, k) = β n β¯k and γ2 (n, k) = θn θ¯k . The following properties hold: ¯ γ1 (0, 3) = θ, ¯ γ1 (0, 5) = −1, γ1 (0, 7) = θ, γ1 (1, 0) = β, γ1 (0, 1) = β, ¯ γ1 (n + 9, k) = γ1 (n, k + 1), γ1 (3, 0) = θ, γ1 (5, 0) = −1, γ1 (7, 0) = θ, γ1 (n, k + 9) = γ1 (n + 1, k), γ1 (10n, k) = γ1 (0, k), γ1 (n, 10k) = γ1 (n, 0); ¯ γ2 (0, 3) = β, γ2 (0, 5) = −1, γ2 (0, 7) = β, ¯ γ2 (1, 0) = θ, γ2 (0, 1) = θ, ¯ γ2 (5, 0) = −1, γ2 (7, 0) = β, γ2 (n + 9, k) = γ2 (n, k + 1), γ2 (3, 0) = β, γ2 (n, k + 9) = γ2 (n + 1, k), γ2 (10n, k) = γ2 (0, k), γ2 (n, 10k) = γ2 (n, 0). (32) From the expression of un given in (31) and from the above properties (32), 7

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it is clear that 1 |u10n+j | = exp Hj + 5

10n+j−1

! X   (−1)k + 2Re(γ1 (0, k) + γ2 (0, k)) ln |Vk1 | .

k1 =0

(33) For j = 0, we have that |u10n | = exp(H0 + ln |V0 | − ln |V5 | + . . . + ln |V10n−10 | − ln |V10n−5 |) n−1 Y V10s (34) = exp(H0 ) V10s+5 . s=0 By setting n = 0 in (31), we get exp(H0 ) = u0 and so n−1 Y V10s |u10n | =|u0 | V10s+5 . s=0

(35)

It can be shown, using (28), that we need not the absolute value function in (36). Similarly, for any j = 0, 1, . . . , 9, we obtain the following: u10n+j =uj

n−1 Y s=0

V10s+j . V10s+j+5

(36)

Thus, using (30), 10s+j−1 Q

u10n+j

10s+j−1 P

 10s+j−1  Q V Ak1 + Bl Ak2 n−1 Y 0 k1 =0 l=0 k2 =l+1 = uj 10s+j+4  10s+j+4  10s+j+4  Q P Q s=0 V Ak1 + Bl Ak2 0 

k1 =0

= uj

n−1 Y s=0

10s+j−1 Q

l=0

 Ak 1

+ u0 u5

k1 =0

10s+j+4 Q

k2 =l+1

10s+j−1 P

 Bl

l=0

 Ak 1

+ u0 u5

k1 =0

10s+j+4 P l=0

10s+j−1 Q

 Ak2

k2 =l+1

 Bl

10s+j+4 Q

. Ak2

k2 =l+1

Hence, the solution to our equation (2) is 10s+j−1   10s+j−1  10s+j−1 Q P Q ak1 + x−5 x0 bl ak 2 n−1 Y k1 =0 l=0 k2 =l+1 x10n+j−5 = xj−5 10s+j+4   10s+j+4  10s+j+4 P Q Q s=0 ak1 + x−5 x0 bl ak 2 k1 =0

l=0

(37)

k2 =l+1

8

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where j = 0, 1, 2, . . . , 9, whenever the denominators do not vanish. In the following section, we turn to the special case where an and bn are constant sequences.

3

The case when an and bn are constant sequences

In this case, let an = a and bn = b where a, b ∈ R.

3.1

The case a 6= 1

Using (37), the solution is given by x10n+j−5 = x¯j

10s+j

n−1 Y

a10s+j + bx−5 x0 1−a1−a

s=0

a10s+j+5 + bx−5 x0 1−a1−a

10s+j+5

,

(38)

where j = 0, 1, 2, 3, . . . , 9, x¯j is defined as  xj−5 , 0 ≤ j ≤ 5; x¯j = x x −5 0  x aj−5 +x−5 x0 b 1−aj−5  , 6 ≤ j ≤ 9, j−10

1−a

and for all (j, s) ∈ {0, 1, 2, . . . , 9} × {0, 1, 2, . . . , n − 1}, (1 − a)a10s+j + bx−5 x0 (1 − a10s+j ) 6= 0. 3.1.1

The case a = −1

In this case, we have x10n+j−5 = x¯j where x¯j =

j

n−1 Y

(−1)j + bx−5 x0 1−(−1) 2

s=0

(−1)j+1 + bx−5 x0 1−(−1) 2

j+1

 xj−5 , x−5 x0

x

j−10

 (−1)j+1 +x−5 x0 b

,

0 ≤ j ≤ 5;  , 6 ≤ j ≤ 9. 1−(−1)j+1 2

9

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Evaluating the above, we obtain the following solution which, for b = ±1, appears in [3] (see Theorems 3.1 and 5.1). x10n−5 = x−5 (−1 + bx−5 x0 )−n ,

x10n−4 = x−4 (−1 + bx−5 x0 )n ,

x10n−3 = x−3 (−1 + bx−5 x0 )−n ,

x10n−2 = x−2 (−1 + bx−5 x0 )n ,

x10n−1 = x−1 (−1 + bx−5 x0 )−n ,

x10n = x0 (−1 + bx−5 x0 )n ,

x10n+1 =

x−5 x0 , x−4 (−1 + bx−5 x0 )n+1

x10n+2 =

x−5 x0 (−1 + bx−5 x0 )n , x−3

x10n+3 =

x−5 x0 , x−2 (−1 + x−5 x0 b)n+1

x10n+4 =

x−5 x0 (−1 + bx−5 x0 )n , x−1

where bx−5 x0 6= 1. However, the solution can be written in a more compact form, i.e., ( j+1 xj−5 (−1 + bx−5 x0 )(−1) n , 0 ≤ j ≤ 5; x10n−j+5 = x−5 x0 1−(−1)j j+1 (−1 + bx−5 x0 ) 2 + (−1) n , 6 ≤ j ≤ 9; xj−10 as long as bx−5 x0 6= 1.

3.2

The case a = 1

Using (37), the solution, which for b = ±1 appears in [3] (see Theorems 2.1 and 4.1), is given by

x10n−5 = x−5

n−1 Y s=0

x10n−3 = x−3

n−1 Y s=0

n−1 Y 1 + (10s + 1)bx−5 x0 1 + 10sbx−5 x0 , x10n−4 = x−4 , 1 + (10s + 5)bx−5 x0 1 + (10s + 6)bx x −5 0 s=0 n−1 Y 1 + (10s + 3)bx−5 x0 1 + (10s + 2)bx−5 x0 , x10n−2 = x−2 , 1 + (10s + 7)bx−5 x0 1 + (10s + 8)bx−5 x0 s=0

10

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x10n−1 = x−1

n−1 Y s=0

n−1 Y 1 + (10s + 5)bx−5 x0 1 + (10s + 4)bx−5 x0 , x10n = x0 , 1 + (10s + 9)bx−5 x0 1 + (10s + 10)bx−5 x0 s=0 n−1

x10n+1

Y 1 + (10s + 6)bx−5 x0 x−5 x0 = , x−4 (1 + bx−5 x0 ) s=0 1 + (10s + 11)bx−5 x0

x10n+2

Y 1 + (10s + 7)bx−5 x0 x−5 x0 = , x−3 (1 + 2bx−5 x0 ) s=0 1 + (10s + 12)bx−5 x0

x10n+3

Y 1 + (10s + 8)bx−5 x0 x−5 x0 = , x−2 (1 + 3bx−5 x0 ) s=0 1 + (10s + 13)bx−5 x0

x10n+4

Y 1 + (10s + 9)bx−5 x0 x−5 x0 = , x−1 (1 + 4bx−5 x0 ) s=0 1 + (10s + 14)bx−5 x0

n−1

n−1

n−1

where jbx−5 x0 6= −1 for all j = 5, 6, 7, . . . , 10n + 4. More compactly, the solution can be written as  n−1 Q 1+(10s+j)bx−5 x0   , 0 ≤ j ≤ 5; xj−5 1+(10s+j+5)bx−5 x0 s=0 x10n+j−5 = n−1 Q 1+(10s+j)bx−5 x0  x−5 x0  , 6 ≤ j ≤ 9.  xj−10 (1+b(j−5)x 1+(10s+j+5)bx−5 x0 −5 x0 ) s=0

4

Conclusion

In this paper, we derived symmetry generators for the difference equations (2) and explicit formulas for the solutions of the equations were obtained. As a recent result, Theorems 2.1, 3.1, 4.1 and 5.1 of Elsayed et al. [3] were generalized.

References [1] C. Cinar, On the positive solutions of the difference equation xn+1 = axn−1 /(1 + bxn xn−1 ), Applied Mathematics and Computational 156, 587590 (2004). 11

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[2] E. M. Elsayed and T.F. Ibrahim, Periodicity and solutions for some systems of nonlinear rational difference equations, Hacet. J. Math. Stat. 44:6, 1361–1390 (2015). [3] E.M. Elsayed, F. Alzahrani and H.S. Alayachi, Formulas and properties of some class of nonlinear difference equations, J. Computational Analysis and Applications 24:8, (2018). [4] M. Folly-Gbetoula and A.H. Kara, Symmetries, conservation laws, and ’integrability’ of difference equations, Advances in Difference Equations 2014, (2014). [5] M. Folly-Gbetoula , Symmetry, reductions and exact solutions of the difference equation un+2 = (aun )/(1 + bun un+1 ), Journal of Difference Equations and Applications 23:6 (2017). [6] P. E. Hydon, Difference Equations by Differential Equation Methods, Cambridge University Press, Cambridge, 2014. [7] P. E. Hydon, Symmetries and first integrals of ordinary difference equations, Proc. Roy. Soc. Lond. A 456, 2835-2855 (2000). [8] T. F. Ibrahim and M. A. El-Moneam, Global stability of a higher-order difference equation, Iran J. Sci. Technol. Trans. Sci. 41:1, 51–58 (2017). [9] N. Joshi and P. Vassiliou, The existence of Lie Symmetries for First-Order Analytic Discrete Dynamical Systems, Journal of Mathematical Analysis and Applications 195, 872-887 (1995). [10] A. Khaliq and E.M. Elsayed, The dynamics and solution of some difference equations, J. Nonlinear Sci. Appl. 9, 1052–1063 (2016). [11] D. Levi, L. Vinet and P. Winternitz, Lie group formalism for difference equations, J. Phys. A: Math. Gen. 30, 633-649 (1997). [12] S. Lie, Classification und Integration von gewohnlichen Differentialgleichungen zwischen xy, die eine Gruppe von Transformationen gestatten I , Math. Ann. 22 , 213–253 (1888). [13] S. Maeda, Canonical structure and symmetries for discrete systems, Math. Japonica 25, 405–420 (1980). 12

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[14] S. Maeda, The similarity method for difference equations, IMA J. Appl. Math.38, 129-134 (1987). [15] N. Mnguni, D. Nyirenda and M. Folly-Gbetoula, On solutions of some fifth-order difference equations, Far East Journal of Mathematical Sciences 102:12, 3053-3065 (2017). [16] D. Nyirenda and M. Folly-Gbetoula, Invariance analysis and exact solutions of some sixth-order difference equations, J. Nonlinear Sci. Appl. 10, 6262-6273 (2017). [17] P. J. Olver, Applications of Lie Groups to Differential Equations, Second Edition, Springer, New York, 1993. [18] G. R. W. Quispel and R. Sahadevan, Lie symmetries and the integration of difference equations, Physics Letters A, 184, 64-70 (1993). [19] I. Yalcinkaya, On the global attractivity of positive solutions of a rational difference equation, Selcuk J. Appl. Math., 9:2 (2008) 3-8.

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO. 6, 2019

Asymptotic behavior of equilibrium point for a system of fourth-order rational difference equations, Ping Liu, Changyou Wang, Yonghong Li, and Rui Li,…………………………947 A version of the Hadamard inequality for Caputo fractional derivatives and related results, Shin Min Kang, Ghulam Farid, Waqas Nazeer, and Saira Naqvi,………………………………962 A hesitant fuzzy ordered information system, Haidong Zhang and Yanping He,…………973 The stability of cubic functional equations with involution in modular spaces, Changil Kim and Giljun Han,…………………………………………………………………………………988 A nonstandard finite difference method applied to a mathematical cholera model with spatial diffusion, Shu Liao and Weiming Yang,………………………………………………….1000 On the Higher Order Difference Equation 𝑥𝑥𝑛𝑛+1 = 𝛼𝛼𝑥𝑥𝑛𝑛 + 𝛽𝛽𝑥𝑥𝑛𝑛−𝑙𝑙 + 𝛾𝛾𝑥𝑥𝑛𝑛−𝑘𝑘 + 𝑏𝑏𝑥𝑥

𝑎𝑎𝑥𝑥𝑛𝑛 𝑥𝑥𝑛𝑛−𝑘𝑘

𝑛𝑛 +𝑐𝑐𝑥𝑥𝑛𝑛−𝑙𝑙 +𝑑𝑑𝑥𝑥𝑛𝑛−𝑘𝑘

,

M. M. El-Dessoky and K. S. Al-Basyouni,……………………………………………….1013

Best proximity point of contraction type mapping in metric space, Kyung Soo Kim,……1023 Explicit viscosity rule of nonexpansive mappings in CAT(0) spaces, Shin Min Kang, Absar Ul Haq, Waqas Nazeer, Iftikhar Ahmad, and Maqbool Ahmad,……………………………1034 The generalized viscosity implicit rules of asymptotically nonexpansive mappings in CAT(0) spaces, Shin Min Kang, Absar Ul Haq, Waqas Nazeer, and Iftikhar Ahmad,……………1044 On some sixth-order rational recursive sequences, M. Folly-Gbetoula and D. Nyirenda,.1057

Volume 27, Number 7 ISSN:1521-1398 PRINT,1572-9206 ONLINE

December 2019

Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC

Journal of Computational Analysis and Applications ISSNno.’s:1521-1398 PRINT,1572-9206 ONLINE SCOPE OF THE JOURNAL An international publication of Eudoxus Press, LLC (fifteen times annually) Editor in Chief: George Anastassiou Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152-3240, U.S.A [email protected] http://www.msci.memphis.edu/~ganastss/jocaaa The main purpose of "J.Computational Analysis and Applications" is to publish high quality research articles from all subareas of Computational Mathematical Analysis and its many potential applications and connections to other areas of Mathematical Sciences. Any paper whose approach and proofs are computational,using methods from Mathematical Analysis in the broadest sense is suitable and welcome for consideration in our journal, except from Applied Numerical Analysis articles. Also plain word articles without formulas and proofs are excluded. The list of possibly connected mathematical areas with this publication includes, but is not restricted to: Applied Analysis, Applied Functional Analysis, Approximation Theory, Asymptotic Analysis, Difference Equations, Differential Equations, Partial Differential Equations, Fourier Analysis, Fractals, Fuzzy Sets, Harmonic Analysis, Inequalities, Integral Equations, Measure Theory, Moment Theory, Neural Networks, Numerical Functional Analysis, Potential Theory, Probability Theory, Real and Complex Analysis, Signal Analysis, Special Functions, Splines, Stochastic Analysis, Stochastic Processes, Summability, Tomography, Wavelets, any combination of the above, e.t.c. "J.Computational Analysis and Applications" is a peer-reviewed Journal. See the instructions for preparation and submission of articles to JoCAAA. Assistant to the Editor: Dr.Razvan Mezei,[email protected], Madison,WI,USA.

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Editorial Board Associate Editors of Journal of Computational Analysis and Applications Francesco Altomare Dipartimento di Matematica Universita' di Bari Via E.Orabona, 4 70125 Bari, ITALY Tel+39-080-5442690 office +39-080-3944046 home +39-080-5963612 Fax [email protected] Approximation Theory, Functional Analysis, Semigroups and Partial Differential Equations, Positive Operators.

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Ravi P. Agarwal Department of Mathematics Texas A&M University - Kingsville 700 University Blvd. Kingsville, TX 78363-8202 tel: 361-593-2600 [email protected] Differential Equations, Difference Equations, Inequalities

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George A. Anastassiou Department of Mathematical Sciences The University of Memphis Memphis, TN 38152,U.S.A Tel.901-678-3144 e-mail: [email protected] Approximation Theory, Real Analysis, Wavelets, Neural Networks, Probability, Inequalities.

Jerry L. Bona Department of Mathematics The University of Illinois at Chicago 851 S. Morgan St. CS 249 Chicago, IL 60601 e-mail:[email protected] Partial Differential Equations, Fluid Dynamics

J. Marshall Ash Department of Mathematics De Paul University 2219 North Kenmore Ave. Chicago, IL 60614-3504 773-325-4216 e-mail: [email protected] Real and Harmonic Analysis

Luis A. Caffarelli Department of Mathematics The University of Texas at Austin Austin, Texas 78712-1082 512-471-3160 e-mail: [email protected] Partial Differential Equations George Cybenko Thayer School of Engineering

Dumitru Baleanu Department of Mathematics and Computer Sciences, Cankaya University, Faculty of Art and Sciences, 06530 Balgat, Ankara, Turkey, [email protected]

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Dartmouth College 8000 Cummings Hall, Hanover, NH 03755-8000 603-646-3843 (X 3546 Secr.) e-mail:[email protected] Approximation Theory and Neural Networks

011-49-203-379-3542 e-mail: [email protected] Approximation Theory, Computer Aided Geometric Design John R. Graef Department of Mathematics University of Tennessee at Chattanooga Chattanooga, TN 37304 USA [email protected] Ordinary and functional differential equations, difference equations, impulsive systems, differential inclusions, dynamic equations on time scales, control theory and their applications

Sever S. Dragomir School of Computer Science and Mathematics, Victoria University, PO Box 14428, Melbourne City, MC 8001, AUSTRALIA Tel. +61 3 9688 4437 Fax +61 3 9688 4050 [email protected] Inequalities, Functional Analysis, Numerical Analysis, Approximations, Information Theory, Stochastics.

Weimin Han Department of Mathematics University of Iowa Iowa City, IA 52242-1419 319-335-0770 e-mail: [email protected] Numerical analysis, Finite element method, Numerical PDE, Variational inequalities, Computational mechanics

Oktay Duman TOBB University of Economics and Technology, Department of Mathematics, TR06530, Ankara, Turkey, [email protected] Classical Approximation Theory, Summability Theory, Statistical Convergence and its Applications

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Common fixed point theorems in Gb-metric space ∗

Youqing Shen, Chuanxi Zhu , Zhaoqi Wu Department of Mathematics, Nanchang University, Nanchang, 330031, P. R. China [email protected] (Y. Q. Shen), [email protected] (C. X. Zhu)

Abstract

In this paper, we introduce a new type of common fixed point for three mappings in

Gb -complete Gb -metric space. On the other hand, we prove that the theory is also established in G-metric space and several corollaries and examples are listed. Keywords: Gb -metric space; common fixed point; G-metric space

1

Preliminaries Mustafa and Sims [1] generalized the concept of metric space and Mustafa [2,3,7] obtained some fixed point

theorems in his papers. After that, many authors established fixed point and common fixed point theorems for different contractive-type condition in G-metric space. In 1998, Czerwik [10] introduced the notion of b-metric space, and then Aghajani [12] based on the notion gave the concept of Gb -metric space and some authors obtained the existence and uniqueness fixed point in Gb -metric space [7,11]. Fixed point theory has a large number of applications in many branches of nonlinear analysis and has been extended in many different directions. Let A, B and C are self mappings of a nonempty set X, if there exists a p ∈ X, such that Ap = Bp = Cp = p, then we call p is a common fixed point of A, B and C. For a mapping T on nonempty set X to itself, we have T x = x, and x is unique then we call x is a Picard operator. In this paper, we mainly obtain a unique common fixed point for three mappings in Gb -metric space. First, we recall some basic properties of Gb -metric space. Let R = (−∞, ∞), R+ = [0, ∞) and N be the set of all natural numbers. Denote N+ the set of all positive integers. Definition 1.1 ([12]) Let X be a nonempty set and s ≥ 1 be a given real number, and let the function G : X × X × X → [0, ∞) satisfy the following properties: (Gb 1) G(x, y, z) = 0 if x = y = z whenever x, y, z ∈ X ; (Gb 2) 0 ≤ G(x, x, y) for all x, y ∈ X with x ̸= y; (Gb 3) G(x, x, y) ≤ G(x, y, z) for all x, y, z ∈ X with y ̸= z; (Gb 4) G(x, y, z) = G(p{x, y, z}), where p is a permutation of x, y, z; (Gb 5) G(x, y, z) ≤ s(G(x, a, a) + G(a, y, z)) for all x, y, z ∈ X. †∗ Correspondence

author. Chuanxi Zhu. Email address: [email protected]. Tel:+8613970815298.

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Then G is called a Gb -metric on X, and (X, G) is called a Gb -metric space. Definition 1.2 ([12]) A Gb -metric space G is said to be symmetric if G(x, x, y) = G(y, x, x) for all x, y ∈ X. Proposition 1.3 ([12]) Let X be a Gb -metric space, then for each x, y, z, a ∈ X it follows that: (1) if G(x, y, z) = 0 then x = y = z; (2) G(x, y, z) ≤ sG(G(x, y, y) + G(x, x, z)); (3) G(x, y, y) ≤ 2s(G(y, x, x)); (4) G(x, y, z) ≤ s(G(x, a, a) + G(a, y, z)). Definition 1.4 ([12]) Let X be a Gb -metric space. A sequence {xn } in X is said to be: (1) Gb -Cauchy if for each ε > 0, there exists a positive integer n0 such that for all m, n, l ≥ n0 , G(xn , xm , xl ) < ε; (2) Gb -convergent to a point x ∈ X if for each ε > 0, there exists a positive integer n0 such that for all m, n, ≥ n0 , G(xn , xm , x) < ε; Definition 1.5 ([12]) A Gb -metric space X is called complete if every Gb -Cauchy sequence is Gb -convergent in X. lemma 1.6 ([11]) Let (X, , G) be a Gb -metric space with s > 1. (1) Suppose that {xn }, {yn } and {zn } are Gb -convergent to x, y and z, respectively. Then we have 1 G(x, y, z) ≤ lim inf G(xn , yn , zn ) ≤ lim sup G(xn , yn , zn ) ≤ s3 G(x, y, z). n→∞ s3 n→∞ (2) If {zn } = c is constant, then 1 G(x, y, c) ≤ lim inf G(xn , yn , c) ≤ lim sup G(xn , yn , c) ≤ s2 G(x, y, c). n→∞ s2 n→∞ (3) If {yn } = b and {zn } = c are constant, then 1 G(x, b, c) ≤ lim inf G(xn , b, c) ≤ lim sup G(xn , b, c) ≤ sG(x, b, c). n→∞ s n→∞

2

Common fixed point theorems in Gb -metric space Theorem 2.1 Let (X, G) be a Gb -complete Gb -metric space and A, B and C are mappings from X to itself.

Suppose that A, B and C satisfy the following condition: G(Ax, By, Cz) ≤

G(x, Ax, Ax) + G(x, By, By) + G(z, Cz, Cz) )G(x, y, z) G(x, Ax, By) + G(y, By, Cz) + G(z, Cz, Ax) + 1

(2.1)

for all x, y, z ∈ X. Then either one of A, B and C has a fixed point, or, A, B and C have a unique common fixed point. Proof. Define the sequence {xn } as x3n+1 = Ax3n , x3n+2 = Bx3n+1 , x3n+3 = Bx3n+2 for all n = 0, 1, 2, · · · . 2 1084

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If x3n = x3n+1 , then x3n is a fixed point of A. If x3n+1 = x3n+2 , then x3n+1 is a fixed point of B. If x3n+2 = x3n+3 , then x3n+2 is a fixed point of C. If the above conclusions are not true, then we assume that xn ̸= xn+1 for all n. Let dn = G(xn , xn+1 , xn+2 ), then for (2.1) we have G(Ax3n , Bx3n+1 , Cx3n+2 ) G(x3n , Ax3n , Ax3n ) + G(x3n+1 , Bx3n+1 , Bx3n+1 ) + G(x3n+2 , Cx3n+2 , Cx3n+2 ) )G(x3n , x3n+1 , x3n+2 ) G(x3n , Ax3n , Bx3n+1 ) + G(x3n+1 , Bx3n+1 , Cx3n+2 ) + G(x3n+2 , Cx3n+2 , Ax3n ) + 1 G(x3n , x3n+1 , x3n+1 ) + G(x3n+1 , x3n+2 , x3n+2 ) + G(x3n+2 , x3n+3 , , x3n+3 ) = )G(x3n , x3n+1 , x3n+2 ) G(x3n , x3n+1 , x3n+2 ) + G(x3n+1 , x3n+2 , x3n+3 ) + G(x3n+2 , x3n+3 , x3n+1 ) + 1 G(x3n , x3n+1 , x3n+2 ) + G(x3n+1 , x3n+2 , x3n+3 ) + G(x3n+1 , x3n+2 , , x3n+3 ) ≤ )G(x3n , x3n+1 , x3n+2 ) G(x3n , x3n+1 , x3n+2 ) + G(x3n+1 , x3n+2 , x3n+3 ) + G(x3n+2 , x3n+3 , x3n+1 ) + 1

≤

so we have d3n+1 ≤

d3n + 2d3n+1 d3n d3n + 2d3n+1 + 1

Let α3n =

d3n + 2d3n+1 d3n + 2d3n+1 + 1

so we have d3n+1 ≤ α3n d3n by introduction, we have d3n+1 ≤ α3n α3n−1 · · · α1 d1 It is obvious that for any natural number n ∈ N, we have 0 < αn < 1,and so dn ≤ dn−1 then we have dn ≤ dn−1 ⇒ dn + dn+1 ≤ dn−1 + dn 1 1 ≤1+ dn−1 + 2dn dn + 2dn+1 1 1 ⇒ ≤ αn−1 αn ⇒1+

Hence, we can get αn−1 ≥ αn so we can obtain α3n α3n−1 · · · α1 ≤ α1 3n 3 1085

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taking the limit as n → ∞, so we have lim d3n+1 ≤ lim α3n α3n−1 · · · α1 d1 ≤ lim α1 3n d1 = 0

n→∞

n→∞

n→∞

so lim G(xn , xn+1 , xn+2 ) = 0.

n→∞

Next,we will show that {xn } is a Gb -Cauchy sequence. on the other hand, according to (Gb 3) we have lim G(xn , xn+1 , xn+1 ) ≤ lim G(xn , xn+1 , xn+2 ) = 0

n→∞

(2.2)

n→∞

for any n, m ∈ N, m > n, using (Gb 5), so we have G(xn , xm , xm ) ≤ sG(xn , xn+1 , xn+1 ) + sG(xn+1 , xm , xm ) ≤ sG(xn , xn+1 , xn+1 ) + s2 G(xn+1 , xn+2 , xn+2 ) + s2 G(xn+2 , xm , xm ) ≤ sG(xn , xn+1 , xn+1 ) + s2 G(xn+1 , xn+2 , xn+2 ) + · · · + sm−n G(xm−1 , xm−1 , xm ) ≤ sG(xn , xn+1 , xn+2 ) + s2 G(xn+1 , xn+2 , xn+3 ) + · · · + sm−n G(xm−1 , xm , xm+1 ) = d1 (sα1 n + s2 α1 n+1 + · · · + sm−n α1 m−1 ) = d1

sα1 n (1 − (sα)m−n−1 ) 1 − sα1 n

taking the limit as n → ∞, then we have lim G(xn , xm , xm ) ≤ lim d1

n→∞

n→∞

sα1 n (1 − (sα)m−n−1 ) =0 1 − sα1 n

so {xn } is a Gb -Cauchy sequence. Since X is complete, so there exists a p ∈ X, such that {xn } is a Gb -Cauchy sequence and Gb -converges to p such that lim x3n+1 = lim Ax3n = lim x3n+2 = lim Bx3n+1

n→∞

n→∞

n→∞

n→∞

= lim x3n+3 = lim Cx3n+2 = p. n→∞

n→∞

Now we prove that p is a common fixed point of A, B and C. Using Lemma 1.6 and (2.1), taking the upper limit as n → ∞, we get G(Ap, p, p) ≤ s2 lim sup G(Ap, Bx3n+1 , Cx3n+2 ) n→∞

G(p, Ap, Ap) + G(x3n+1 , Bx3n+1 , Bx3n+1 ) + G(x3n+2 , Cx3n+2 , Cx3n+2 ) G(p, x3n+1 , x3n+1 ) G(p, Ap, Bx3n+1 ) + G(x3n+1 , Bx3n+1 , Cx3n+2 ) + G(x3n+2 , Cx3n+2 , Ap) + 1 G(p, Ap, Ap) + G(x3n+1 , Bx3n+1 , Bx3n+1 ) + G(x3n+2 , Cx3n+2 , Cx3n+2 ) ≤ s4 lim sup G(p, p, p) n→∞ G(p, Ap, Bx3n+1 ) + G(x3n+1 , Bx3n+1 , Cx3n+2 ) + G(x3n+2 , Cx3n+2 , Ap) + 1

≤ s2 lim sup n→∞

=0 then we get G(Ap, p, p) = 0. Hence by (1) of Proposition 1.1, we can get Ap = p. Similarly, letting x = x3n , y = p, z = x3n+2 and x = x3n , y = x3n+1 , z = p we can get Bp = p and Cp = p respectively, so we have Ap = Bp = Cp = p. 4 1086

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Now, we show that the common fixed point of A, B and C is unique. Assume on contrary that q is another fixed point, i.e. Aq = Bq = Cq = q such that p ̸= q. Then, by our assumption, we apply (2.1) to obtain G(p, p, q) = G(Ap, Bp, Cq) G(p, Ap, Ap) + G(p, Bp, Bp) + G(q, Cq, Cq) G(p, p, q) G(p, Ap, Bp) + G(p, Bp, Cq) + G(q, Cq, Ap) + 1 G(p, p, p) + G(p, p, p) + G(q, q, q) = G(p, p, q) G(p, p, p) + G(p, p, q) + G(q, q, p) + 1 ≤

=0 so by the Proposition 1.1, we have G(p, p, q) = 0, then p = q. Corollary 2.2 Let (X, G) be a Gb -complete Gb -metric space and T be a mapping from X to itself. Suppose that T satisfy the following condition: G(T x, T y, T z) ≤

G(x, T x, T x) + G(x, T x, T x) + G(x, T x, T x) )G(x, y, z) G(x, T x, T y) + G(y, T y, T z) + G(z, T z, T x) + 1

for all x, y, z ∈ X. Then T has a unique fixed point. Proof. Taking A = B = C = T , the result follow from Theorem 2.1. Theorem 2.3 Let (X, G) be a Gb -complete Gb -metric space and A, B and C are mappings from X to itself. Suppose that A, B and C satisfy the following condition: G(Ax, By, Cz) ≤ α

min{G(y, By, By), G(z, Cz, Cz)} G(x, Ax, Ax) + βG(x, y, z) G(z, Cz, Ax) + 1

(2.3)

for all x, y, z ∈ X, where α + β ≤ 1. Then either one of A, B and C has a fixed point, or, A, B and C have a unique common fixed point. Proof. Let dn = G(xn , xn+1 , xn+2 ), then for (2.3) we have G(Ax3n , Bx3n+1 , Cx3n+2 ) ≤ α

min{G(x3n+1 , Bx3n+1 , Bx3n+1 ), G(x3n+2 , Cx3n+2 , Cx3n+2 )} ) G(x3n+2 , Cx3n+2 , Ax3n ) + 1

G(x3n , Ax3n , Ax3n ) + βG(x3n , x3n+1 , x3n+2 ) =α

min{G(x3n+1 , x3n+2 , x3n+2 ), G(x3n+2 , x3n+3 , x3n+3 )} ) G(x3n+2 , x3n+3 , x3n+1 ) + 1

G(x3n , x3n+1 , x3n+1 ) + βG(x3n , x3n+1 , x3n+2 ) d3n+1 d3n + βd3n d3n+1 + 1 d3n+1 + β)d3n = (α d3n+1 + 1 ≤α

d3n+1 Since α d3n+1 +1 + β ≤ 1, the following proof is similar to Theorem 2.1.

Corollary 2.4 Let (X, G) be a Gb -complete Gb -metric space and T be mapping from X to itself. Suppose that T satisfy the following condition: G(T x, T y, T z) ≤ α(

min{G(y, T y, T y), G(z, T z, T z)} )G(x, T x, T x) + βG(x, y, z) G(z, T z, T x) + 1

for all x, y, z ∈ X, where α + β ≤ 1. 5 1087

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Then T has a unique fixed point. Proof. Taking A = B = C = T , the result follow from Theorem 2.4. Theorem 2.5 Let (X, G) be a G-complete G-metric space and A, B and C are mappings from X to itself. Suppose that A, B and C satisfy the following condition: G(Ax, By, Cz) ≤

G(x, Ax, Ax) + G(y, By, By) + G(z, Cz, Cz) )G(x, y, z) G(x, Ax, By) + G(y, By, Cz) + G(z, Cz, Ax) + 1

for all x, y, z ∈ X. Then either one of A, B and C has a fixed point, or, A, B and C have a unique common fixed point. Proof. The proof is similar to Theorem 2.1. There is a little difference between them. First, when we prove that {xn } is a Gb -Cauchy sequence, we have G(xn , xm , xm ) ≤ G(xn , xn+1 , xn+1 ) + G(xn+1 , xm , xm ) ≤ G(xn , xn+1 , xn+1 ) + G(xn+1 , xn+2 , xn+2 ) + G(xn+2 , xm , xm ) ≤ G(xn , xn+1 , xn+1 ) + G(xn+1 , xn+2 , xn+2 ) + · · · + G(xm−1 , xm−1 , xm ) ≤

m−n ∑

G(xn+i , xn+i+1 , xn+i+1 )

i=0

so we can get G(xn , xm , xm ) ≤ ≤ =

m−n ∑ i=0 m−n ∑ i=0 m−n ∑

G(xn+i , xn+i+1 , xn+i+1 ) G(xn+i , xn+i+1 , xn+i+2 ) dn+i

i=0

taking the limit as n → ∞, then we have lim G(xn , xm , xm ) ≤ lim

n→∞

n→∞

m−n ∑

α1 n+i d1 = 0

i=0

so {xn } is a G-Cauchy sequence. Secondly, since that G-metric space is continuous so when we prove that p is a common fixed point in G-metric space we have G(Ap, p, p) ≤

G(p, Ap, Ap) + G(p, Bp, Bp) + G(p, Cp, Cp) G(p, p, p) = 0 G(p, Ap, Bp) + G(p, Bp, Cp) + G(p, Cp, Ap) + 1

Corollary 2.6 Let (X, G) be a G-complete G-metric space and T be a mapping from X to itself. Suppose that T satisfy the following condition: G(T x, T y, T z) ≤

G(x, T x, T x) + G(x, T x, T x) + G(x, T x, T x) )G(x, y, z) G(x, T x, By) + G(y, T y, T z) + G(z, T z, T x) + 1

for all x, y, z ∈ X. Then T has a unique fixed point. Proof. Taking A = B = C = T , the result follow from Theorem 2.6. 6 1088

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3

An example Example 3.1 Let G(x, y, z) = (max{|x − y|, |y − x|, |z − x|})2 for all x, y, z ∈ X then G is a Gb -metric on

X where s = 2. Define self-mappings A, B and C on x by A(x) = 1, B(x) = 1, C(x) =

7+x 8

Then we have G(Ax, By, Cz) = (max{|1 − 1|, |1 −

7+z 7+z 2 |, |1 − |}) 8 8

1 z = ( − )2 8 8 and we also have G(x, Ax, Ax) = (1 − x)2 , G(y, By, By) = (1 − y)2 G(z, Cz, Cz) = (

7 − 7z 2 ) , G(x, Ax, By) = (1 − x)2 8

G(y, By, Cz) = max{(1 − y)2 , (

1−z 2 7+z ) ,( − y)2 } 8 8

G(z, Cz, Ax) = (1 − z)2 G(x, y, z) = max{|x − y|2 , |y − z|2 , |z − x|2 } Case1: when z ≤ x, z ≤ y α ≥

1 64 , β

= 0 and y ≤

1+7z 8 ,

we have the (2.3) established. Then x = y = z = 1

is a common fixed point. Case2: when y ≥

1+7z 8

α = 1, β = 0 and z ≥ −6 + 7x, we have the (2.3) established. Then x = y = z = 1 is

a common fixed point.

Acknowledgements This work is supported by the Natural Science Foundation of China (11771198, 11361042, 11071108, 11461045, 11701259), the Natural Science Foundation of Jiangxi Province of China (20132BAB201001,20142BAB211016) and the Scientific Program of the Provincial Education Department of Jiangxi (GJJ150008) and the Innovation Program of the Graduate student of Nanchang University(colonel-level project).

References [1] Z. Mustafa, B. Sims, A new approach to generalized metric space. J. Nonlinear Convex Anal. 7, 289-297, (2006). [2] Z. Mustafa, B. Sims, Fixed point theorems for contractive mappings in complete G-metric space, Fixed Point Theory Appl. 2009, 977175, (2013). [3] H. Obiedat, Z. Mustafa, Fixed point result on a nonsymmetric G-metric space, Jordan J. Math. Stat. 3, 65-79, (2010).

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[4] A. Azam, N.Mehmood, Fixed point theorems for multivalued mappings in G-cone metric space, J. Inequal. Appl. 2013, 354, (2013). [5] Y. U. Caba, Fixed point theorems in G-metric space, J. Math. Anal. Appl. 455, 528-537, (2017). [6] P. N. Dutta, B. S. Choudhury, K. Das, Some fixed point results in Menger spaces, Surv. Math. Appl. 4, 41-52, (2009). [7] Z. Mustafa, J. R. Roshan, Coupled coincidence point result for (ψ, φ)-weakly contractive mappings in partially ordered Gb -metric spaces. Fixed Point Theory Appl. 2013 , 206, (2013). [8] J. R. Roshan, V. Sedghi, Common fixed point of almost generalized (ψ, φ)s -contractive mappings in ordered b-metric space. Fixed Point Theory Appl. 2013, 159, (2013). [9] V.Parvaneh, J. R. Radenovi´c, Existence of tripled coincidence points in ordered b-metric spaces and an application to a system of integral equations, Fixed Point Theory Appl. 2013, 130, (2013). [10] S. Czerwik, Nonlinear set-valued contraction mappings in b-metric space, Atti Sem Mat Fis Univ Modena. 46, 236-276, (1998). [11] R. R. Jamal, S. Nabioliah, Common fixed point theorems for three maps in discontinuous Gb -metric spaces, Act Math Sci. 34, 1643-1654, (2014). [12] A. Aghajani, M. Abbas, J. R. Roshan, C common fixed point of generalized weak contractive mappings in partially ordered Gb -metric space. Filomat. 28, 1087-1101, (2014). [13] I. A. Baskhtin, The contraction mapping principle in quasimetric spaces, Func Anal Unianowsk Gos Ped Inst. 30, 26-37, (1989). [14] C. X. Zhu, Several nonlinear operator problems in Menger PN space, Nonlinear Anal. 65 1281-1284 (2006). [15] C. X. Zhu, Research on some problems for nonlinear operator, Nonlinear Anal. 71, 4568-4571, (2009).

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A modified collocation method for weakly singular Fredholm integral equations of second kind∗ Guang Zenga, b†, Chaomin Chena,‡, Li Leia, b§, Xi Xub,¶ a

Fundamental Science on Radioactive Geology and

Exploration Technology Laboratory, East China University of Technology, Nanchang, Jiangxi, 330013, P.R. China b

School of Scinence, East China University of Technology, Nanchang, Jiangxi, 330013, P.R. China

Abstract In this paper, a collocation method with high precision by using the polynomial basis functions is proposed to solve the Fredholm integral equation of second kind with weakly singular kernel. We introduce the polynomial basis functions and use it to reduce the given equation to a system of linear algebraic equation. Thus, we can simplify the solving of the equation. The error analysis are given. Numerical examples are given to illustrate the efficiency of our method. Keyword : Weakly Singular · Fredholm Integral Equation · Polynomial basis function Method AMS subject classification: 65D10 · 65D32

1

Introduction

This paper is concerned with collocation method for weakly singular Fredholm integral equations of the second kind as follows ∗

The work is supported by National Natural Science Foundation of China(11661005,11301070). Corresponding author: [email protected] (G. Zeng) ‡ [email protected] (C. Chen) § [email protected] (L. Lei) ¶ [email protected] (X. Xu) †

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Z

b

κ(x, t)φ(t)dt = f (x), 0 ≤ x ≤ 1,

φ(x) + λ

(1.1)

a H(x,t) where κ(x, t) = |x−t| α , 0 < α < 1, H(x, t), f (x) are continue and bounded functions and φ(x) is the function to be determined. Numerical methods for weakly singular Fredholm integral equations of the second kind have been developed by many scholars in recent years because of their important applications in science and engineering. These methods can be classified into two types. One type is through making approximations to the analytical solutions directly. For instance, Tricomi used successive approximations method to solve the integral equations in his book [1]. Variational iteration method and Adomian decomposition method were introduced in [2] and [3] respectively. Also, The homotopy analysis method was proposed by Liao [4] and has applied it in [5] et. Another type is through shifting the equations into a form which easier to solve than the original equations. For example, Taylor expansion collocation methods are presented to solve integral equations in [6-8]. In [9], the orthogonal triangular basis functions were used by Babolian et al. to solve some integral equations systems. And Legendre wavelets method was proposed by Jafari et al. in [10] to find the numerical solutions of linear integral equations systems. Moreover, in [12] architecture artificial neural networks was suggested to approximate the solutions of linear integral equations systems. Furthermore, Jafarian et al. [13] using the Bernstein polynomials to obtain the numerical solutions of linear Fredholm and Volterra integral equations systems of the second kind. And application of Bernstein polynomial have been made by scholar for solving both differential equations and integral equations, see [11]. And piecewise polynomial collocation method were applied to solve the Volterra integro-differential equations with weakly singular kernel in [14] respectively. And the stability of piecewise polynomial collocation methods for solving weakly singular integral equations of the second kind has been discussed by Kangro et al. in [15]. Besides, Baratella et al. [16] had proposed an approach with product integration to solve the weakly singular Volterra integral equations. Kolk et al. And Pallaw et al. [17] used the quadratic spline collocation to solve the smoothed weakly singular Fredholm integral equations. However, these methods introduced above do not provide a good accuracy in the solution near the singular points. In this paper, we are going to use polynomial basis functions collocation method to approximate the solution of singular Fredholm integral equations of the second kind. The proposed approach converted the given equation with unique solution into a system of linear algebraic equations in general case. To do this, first the polynomial basis functions of certain degree n of unknown functions are substituted in the given integral equations. So that the solution of the unknown function of given equations have converted into the solutions of the coefficients of the unknown polynomial basis functions, such that we can solve the integral equations in a convenient way.

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The layout of this paper is as follows: In section 2 we presented the procedure of the polynomial basis functions collocation method to obtain the approximate solution of the weakly singular Fredholm integral equation. In section 3, we had demonstrated that the proposed method is convergent to all the weakly singular Fredholm integral equations of second kind. In section 4, we give numerical example to test the effectiveness and efficiency of the method. Finally, Numerical examples are given to illustrate the efficiency of our method.

2

The Polynomial Basis Function Method

We are going to use the polynomial basis functions to solve the eq.(1.1). The form of the functions are as follows: m−1 X xk , U= k=0

where the polynomial basis functions 1, x, x2 , · · · , xm−1 are linear independent. Since eq.(1.1) is a weakly singular integral equation, the singularity of the equation must be removed such that the procedure of solving the problem can be move on. But since the proposed method of this paper is belong to the collocation method, which can smooth the singular points of the discretion, so that we can use the method directly. Then we provided the procedure of using polynomial basis functions to solve the kind of the integral equations proposed in this paper concretely as follows: Step 1. Choosing the basis functions u = [1, x, x2 , ..., xk ], (k = 0, 1, 2, ..., m − 1) the unknown function φ(x) is substituted by the following polynomials φ(x) ≈ φm (x) =

m−1 X

ak xk ,

(2.1)

k=0

Step 2. Substituting (2.1) into (1.1) we have m−1 X

k

ak x + λ

k=0

m−1 X

ak x

k

b

Z

κ(x, t)tk dt = f (x),

(2.2)

a

k=0

Step 3. Discrete the interval [a,b] into n sections uniformly, we obtained the systems of the coefficient ak as follows m−1 X k=0

ak xkj

+λ

m−1 X k=0

ak xkj

Z

b

κ(x, t)tk dt = f (xj ),

(2.3)

a

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where j = 1, 2, ..., n, xj = a + j(b − a)/n. We transformed the equations into the form of linear matrix as follows (U + KU )A = f , where



    a0 1 x1 · · · xm−1 1  ..  , A =  ..  , f =  U =  ... ... . . .  .   .  m−1 1 xn · · · xn am−1

(2.4)  f1 .. , .  fn

(2.5)

Rb and K = a κ(x, t)dt which is the integral operator. Step 4. Solve the system we obtained the solutions of the coefficients of ak as follows a0 , a1 , ..., am−1 . Substituting them into eq.(2.1) we obtained the approximate solution φm (x).

3

Convergence and Error Analysis

In this section, we are going to prove that the approximate method we proposed in this paper is convergent to the analytic solution of eq.(1.1). Firstly, we rewrite the form of the weakly singular kernel as follows K(x, t) =

H(x, t) . |x − t|α

Let 0 < α ≤ 12 , and H(x, t) is continuously bounded. Then the eigenvalue integral equation with weakly singular kernel is as follows Z b λφ(x) = K(x, t)φ(t)dt, 0 ≤ x ≤ 1, (3.1) a

where K(x, t) is the weakly singular kernel, λ is the eigenvalue of the K(x, t), φ(x) is the eigenfunction of λ. Lemma 3.1 [14]. If x1 , x2 ∈ C m,v (0, T ], m ∈ N, v < 1, then x1 x2 ∈ C m,v (0, T ], and kx1 x2 km,v ≤ ckx1 km,v kx2 km,v with a constant c which is independent of x1 and x2 . Proof. See [14]. Lemma 3.2 [18] Suppose that the function φm (x) obtained by the polynomial basis function is the approximation of eq.(1) and eq.(1) is with bounded first

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derivative, then eq.(1) can be expanded as an infinite sum of the polynomial basis m−1 X functions, that is, φ(x) = ck xk , and the coefficients ck are bounded as k=0

K

ck
0} and K := {f ∈ S : Re(1 + zf 00 (z)/f 0 (z)) > 0}. Thomas [15], in 1967, introduced a general form of the class of starlike functions. Thomas [15], for a starlike functions g, defined the class Bα := {f ∈ S : Re(zf 0 (z)f (z)α−1 /g(z)α ) > 0}. This class is popularly known as the class of Bazileviˇc functions of type α. In 1973, Singh [12] investigated a special case of Bα . For α ≥ 0 and setting g(z) = z, he considered a subclass of Bα defined by ) ( α−1 !  f (z) >0 . B1 (α) := f ∈ A : Re f 0 (z) z In his paper, he obtained the sharp radius estimates for certain integral operator to be a member of the class B1 (α) and he also obtained the sharp upper bound on the first four initial coefficients. He also investigated the sharp bound on the Fekete-Szeg¨o functional for functions in this class. It should be noted that the class B1 (1) is a subclass of close-to-convex 2010 Mathematics Subject Classification. 30C45, 30C50. Key words and phrases. Univalent function, Coefficient bound, Hankel determinant. 1

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functions and hence univalent in D. Moreover, B1 (0) = S ∗ . In 2015, Thomas [13] proved the sharp bound |a2 a4 − a23 | ≤ 4/(2 + α)2 for functions in the class B1 (α) for α ∈ [0, 1]. In 2017, Marjono et al. [6] investigated the sharp upper bound on fifth and sixth coefficients. They also conjectured that if f ∈ B1 (α), then 2 (n = 2, 3, 4, · · · ) n−1+α holds for all α ≥ 1. This conjecture for the fifth coefficient, for certain range of α, was recently settled by Cho and Kumar [1]. For many results related to the Bazileviˇc functions we refer the reader to the papers [11, 14, 15, 17] and the references cited therein. A class B(α, β) with stronger conditions was considered by Ponnusamy [8]. For α > 0 and 0 < β < 1, he defined ( )  α−1 f (z) B(α, β) := f ∈ A : f 0 (z) − 1 < β . z |an | ≤

For the negative value of α ∈ (−1, 0), the class B(α, β) can be rewritten as ( )  α+1 z ¯ β) := f ∈ A : f 0 (z) B(α, − 1 < β . f (z) This class was introduced and investigated by Obradovi´c, in 1998. He obtained the conditions on the parameter β that embeds this class into the class of starlike functions. Later in 2002, Tuneski and Darus [16], for 0 < α < 1, considered the class ) ( α+1 !  z ¯ >0 . B(α) := f ∈ A : Re f 0 (z) f (z) This class, as mentioned by Obradovi´c in the conference “Computational Methods and Function Theory 2001” is called to be class of functions of non-Bazileviˇc type, see [16]. Tuneski and Darus investigated the sharp bounds on |a2 | and the Fekete-Szeg¨o functional |a3 − µa22 |. Some typographical errors in the result [16, Theorem 1, p. 64] were reported by Kumar and Kumar [5]. For a more general result and the correct version of their result one can refer to [5]. Starlikeness of multivalent non-Bazileviˇc functions were investigated by Guo et al. [2]. Estimate on the second Hankel determinant for the class of functions f ∈ A satisfying Re (f 0 (z) (z/f (z))α ) > 0 for α ∈ (0, 1/3] was obtained by Krishna and Reddy [4]. Motivated by the above works, in this paper, sharp bound on the third to eighth coefficients ¯ of functions in the class B(α) are investigated. Moreover, sharp bound on the functional ¯ |a2 a3 − a4 | for functions in the class B(α) is also obtained. Let P be the class of analytic functions having the Taylor series of the form p(z) = 1 + p1 z + p2 z 2 + p3 z 3 + · · · and mapping the unit disk D onto the right-half of the complex plane i.e. satisfying the condition Re p(z) > 0 (z ∈ D). Let B be the class of Schwarz functions consisting of analytic functions of the form w(z) = c1 z + c2 z 2 + c3 z 3 + · · · (z ∈ D)

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ˇ FUNCTIONS SHARP COEFFICIENT ESTIMATES FOR NON-BAZILEVIC

3

and satisfying the condition |w(z)| < 1 for z ∈ D. The following correspondence between the classes B and P holds: p(z) − 1 ∈ B. (1.2) p ∈ P if and only if w(z) = p(z) + 1 Comparing coefficients in (1.2), we have p1 2p2 − p21 4p3 − 4p1 p2 + p31 8p4 − 8p1 p3 − 4p22 + 6p21 p2 − p41 , c2 = , c3 = , c4 = . (1.3) 2 4 8 16 Lemma 1.1. [3](see also [10]) If p ∈ P, then, for any complex number ν, c1 =

|p2 − νp21 | ≤ 2 max{1; |2ν − 1|} and the equality holds for the functions given by 1 + z2 1+z p(z) = and p(z) = . 2 1−z 1−z Consider the functional Ψ(µ, ν) = |c3 + µc1 c2 + νc31 | for w ∈ B and µ, ν ∈ R. Let us assume that the symbols Ωk ’s are defined as follows:  Ω1 := (µ, ν) ∈ R2 : |µ| ≤ 1/2, |ν| ≤ 1 ,   4 3 2 1 (|µ| + 1) − (|µ| + 1) ≤ ν ≤ 1 , Ω2 := (µ, ν) ∈ R : ≤ |µ| ≤ 2, 2 27     1 2 2 2 Ω3 := (µ, ν) ∈ R : |µ| ≤ , ν ≤ −1 , Ω4 := (µ, ν) ∈ R : |µ| ≥ 1/2, ν ≤ − (|µ| + 1) , 2 3    1 2 2 2 Ω5 := (µ, ν) ∈ R : |µ| ≤ 2, ν ≥ 1 , Ω6 := (µ, ν) ∈ R : 2 ≤ |µ| ≤ 4, ν ≥ (µ + 8) , 12   2 Ω7 := (µ, ν) ∈ R2 : |µ| ≥ 4, ν ≥ (|µ| − 1) , 3   2 4 2 1 3 Ω8 := (µ, ν) ∈ R : ≤ |µ| ≤ 2, − (|µ| + 1) ≤ ν ≤ (|µ| + 1) − (|µ| + 1) , 2 3 27   2 2|µ|(|µ| + 1) 2 Ω9 := (µ, ν) ∈ R : |µ| ≥ 2, − (|µ| + 1) ≤ ν ≤ 2 , 3 µ + 2|µ| + 4   2|µ|(|µ| + 1) 1 2 2 Ω10 := (µ, ν) ∈ R : 2 ≤ |µ| ≤ 4, 2 ≤ ν ≤ (µ + 8) , µ + 2|µ| + 4 12   2|µ|(|µ| + 1) 2|µ|(|µ| − 1) 2 ≤ν≤ 2 , Ω11 := (µ, ν) ∈ R : |µ| ≥ 4, 2 µ + 2|µ| + 4 µ − 2|µ| + 4   2|µ|(|µ| − 1) 2 Ω12 := (µ, ν) ∈ R2 : |µ| ≥ 4, 2 ≤ ν ≤ (|µ| − 1) . µ − 2|µ| + 4 3

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The following result is due to Prokhorov and Szynal [9] which we need in our investigation. Lemma 1.2. [9, Lemma 2, p. 128] If w ∈ B, then  1,       |ν|,      1/2  |µ|+1 2 (|µ| + 1) 3(|µ|+ν+1) , |Ψ(µ, ν)| ≤ 3       1/2  2 −4 µ2 −4 1   ν µµ2 −4ν ,  3 3(ν−1)      1/2  |µ|−1  2 (|µ| − 1) , 3

3(|µ|−ν−1)

for any real numbers µ and ν, we have (µ, ν) ∈ Ω1 ∪ Ω2 ∪ {(2, 1)}; 7 S (µ, ν) ∈ Ωk ; k=3

(µ, ν) ∈ Ω8 ∪ Ω9 ; (µ, ν) ∈ Ω10 ∪ Ω11 \ {(2, 1)}; (µ, ν) ∈ Ω12 .

The extremal functions, up to rotations, are of the form z(t1 − z) z(t2 + z) w1 (z) = z 3 , w2 (z) = z, w3 (z) = , w4 (z) = 1 − t1 z 1 + t2 z 2 3 and w5 (z) = c1 z + c2 z + c3 z + · · · , where the parameters t1 , t2 and the coefficients ci are given by  1/2  1/2  1/2 |µ| + 1 |µ| − 1 2ν(µ2 + 2) − 3µ2 t1 = , t2 = , c1 = , 3(|µ| + ν + 1) 3(|µ| − ν − 1) 3(ν − 1)(µ2 − 4ν) "  1/2 # 2 2 µ ν(µ + 8) − 2(µ + 2) . c2 = (1 − c21 )eiθ0 , c3 = −c1 c2 eiθ0 , θ0 = ± arccos 2 2ν(µ2 + 2) − 3µ2 2. Coefficient Estimates The following theorem gives the sharp estimates on |a3 |, |a4 | and on the functional |a2 a3 − ¯ a4 | for functions in the class B(α). Theorem 2.1. Let α0 ≈ 2.36, α1 ≈ 2.68 and α2 ≈ 2.71 are the smallest positive roots of the equations 3α4 − 11α3 + α2 + 11α + 20 = 0, α6 − 11α5 + 56α4 − 138α3 + 151α2 − 7α − 148 = 0 ¯ has the from (1.1). Then, the and α3 − 5α2 + 11α − 13 = 0, respectively. Let f ∈ B(α) following sharp inequalities hold: ( 2 , if α ∈ (0, 3] \ {1, 2}; α−2 |a3 | ≤ (2.1) 2(α−3) , if α > 3, (α−2)(α−1)2  2 α4 −5α3 +11α2 −19α+36 ( )  ,  3(α−1)(α−2)(α−3)    4(|a|−1)3/2 , 3(α−3)(|a|−b−1)1/2 |a4 | ≤ 2 2 3/2  2(α−1) (a −4)   2 −4b)(3(b−1))1/2 ,   (α−3)(a 2 , α−3

if α ∈ (0, α0 ] \ {1, 2} or α2 ≤ α < 3; if α0 ≤ α ≤ α1 ;

(2.2)

if α1 ≤ α ≤ α2 ; if 3 < α,

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ˇ FUNCTIONS SHARP COEFFICIENT ESTIMATES FOR NON-BAZILEVIC

5

where a and b are given by a := −

2(α − 5) α4 − 5α3 + 11α2 − 19α + 36 and b := . (α − 1)(α − 2) 3(α − 1)3 (α − 2)

¯ Proof. Since f ∈ B(α), it follows that there exists p(z) = 1 + p1 z + p2 z 2 + p3 z 3 + · · · ∈ P such that  α+1 z 0 f (z) = p(z). (2.3) f (z) Comparing coefficients of like-power terms in (2.3), we get a2 = −

p1 (α − 2)(α + 1)p21 − 2(α − 1)2 p2 and a3 = . α−1 2(α − 2)(α − 1)2

(2.4)

Now consider (α − 2)(α + 1)p21 − 2(α − 1)2 p2 2(α − 2)(α − 1)2   1 (α − 2)(α + 1) 2 = − p1 . p2 − α−2 2(α − 1)2

a3 =

(2.5)

An application of Lemma 1.1 on (2.5), gives   |α − 3| 2 max 1; |a3 | ≤ α−2 (α − 1)2 which equivalently can be written as ( |a3 | ≤

2 , α−2 2(α−3) , (α−2)(α−1)2

α ∈ (0, 3] \ {1, 2}; α > 3.

This is the required bound on third coefficient as stated in the theorem. In the first case of ¯ (2.1), equality occurs for the function f0 ∈ B(α) defined by  α+1 z 1 + z2 f00 (z) = , (2.6) f0 (z) 1 − z2 ¯ whereas in the second case of (2.2), equality holds for the function f˜0 ∈ B(α) defined by  α+1 z 1+z 0 f˜0 (z) = . (2.7) 1−z f˜0 (z) Next we shall find the estimate on |a4 |. From (2.3), we have a4 =

−(α − 3)(α − 2)(α + 1)(2α + 1)p31 + 6(α − 1)2 (α − 3)(α + 1)p1 p2 − 6(α − 2)(α − 1)3 p3 . 6(α − 1)3 (α − 2)(α − 3) (2.8)

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In view of the interconnections in (1.2) and (1.3), Eqn. (2.8) can be rewritten as: 2 [(α4 − 5α3 + 11α2 − 19α + 36) c31 − 6(α − 5)(α − 1)2 c1 c2 + 3(α − 2)(α − 1)3 c3 ] a4 = − 3(α − 1)3 (α − 2)(α − 3) (2.9) or equivalently  2  a4 = − c3 + ac1 c2 + bc31 , α−3 where the parameters a and b are given by 2(α − 5) α4 − 5α3 + 11α2 − 19α + 36 a := − and b := . (α − 1)(α − 2) 3(α − 1)3 (α − 2)

(2.10)

Assume that Ωi ’s are defined as in Lemma 1.2 with the settings µ = a and ν = b. We now proceed further in the proof with the following steps: √ (1) Assume that α ≥ ( 73 − 1)/2 ≈ 3.772. In this case, we see that −1/2 ≤ a ≤ 1/2 holds. Moreover, b ≤ 1 holds if √ and only if α4 − 5α3 + 8α2 − α − 15 ≥ 0, which holds for all α ≥ 3. Thus for all α ≥ ( 73 √− 1)/2, we conclude that (a, b) ∈ Ω1 . (2) Next assume that 3 < α ≤ ( 73 − 1)/2. Then, we see that the condition −1/2 ≤ a ≤ 2 holds for √ all such α and (4/27)(a + 1)3 − (a + 1) ≤ b ≤ 1 all α > 3. Therefore, for 3 < α ≤ ( 73 − 1)/2, we must have (a, b) ∈ Ω2 . (3) Let ! q √ 1 3 8 α2 := 53 + 9 41 − p √ + 5 ≈ 2.71 3 3 53 + 9 41 and 1 11 + α0 := 12 12

with

r 4

(22/3 )

q 3

√

q

8989 + 9 14717 + 4

3

√

1 35956 − 36 14717 + 113 − 2

q

ˆ ≈ 2.36 Cˆ + D

q q √ √ 113 1 2/3 3 13 ˆ C := − (2 ) 8989 + 9 14717 − 35956 − 36 14717, 18 9 9

and ˆ := D

407 q p p √ √ 3 3 18 4 (22/3 ) 8989 + 9 14717 + 4 35956 − 36 14717 + 113

are the smallest positive roots of the equations α3 − 5α2 + 11α − 13 = 0 and 3α4 − 11α3 + α2 + 11α + 20 = 0, respectively. Now assume that 0 < α < 1 or 2 < α ≤ α0 . Then a ≥ 4 and b ≥ 2(a − 1)/3 hold and hence (a, b) ∈ Ω7 . Moreover, a ≤ −1/2 and b ≤ −2(−a + 1)/3 holds whenever 1 < α < 2. Therefore, (a, b) ∈ Ω4 whence 1 < α < 2. Also it can be easily seen that 2 ≤ a ≤ 4 and b ≥ (a2 + 8)/12 hold for α2 ≤ α < 3.

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(4) Let 2.69 ≈ (5 +

√

7

33)/4 ≤ α ≤ α0 . Then a and b satisfy 2 ≤ a ≤ 4 and

2a(a + 1) a2 + 8 ≤ b ≤ . a2 + 2a + 4 12 Therefore, for this range of α, we see that (a, b) ∈ Ω10 . Let α1 ≈ 2.68 is the smallest 6 5 4 3 2 positive root √ of α − 11α + 56α − 138α + 151α − 7α − 148 = 0. Further, when α1 ≤ α ≤ ( 33 + 5)/4, the parameters a and b satisfy a ≥ 4 and 2a(a + 1) 2a(a − 1) ≤b≤ 2 . 2 a + 2a + 4 a − 2a + 4 Hence, in view of Lemma 1.2, we have (a, b) ∈ Ω11 . (5) Assume that α0 ≤ α ≤ α1 . In this case, it is a simple matter to check that a ≥ 4 and 2(a − 1) 2a(a − 1) ≤ b ≤ . a2 − 2a + 4 3 Therefore, Lemma 1.2 gives (a, b) ∈ Ω12 . In the light of the above discussions, an application of Lemma 1.2 gives the desired estimates on |a4 |. In the first case of (2.2), the equality holds for the function f0 defined in (2.6), whereas in the forth case of (2.2), the equality holds for the function function f˜0 defined in (2.7). In the case third of (2.2), the extremal function f1 is given by  α+1 z 1 + w(z) 0 f1 (z) = (2.11) f1 (z) 1 − w(z) with choice of the Schwarz function (up to rotation) w(z) = c1 z + c2 z 2 + c3 z 3 + · · · ∈ B, where the coefficients ci are given by  1/2 2b(a2 + 2) − 3a2 c1 = , c2 = (1 − c21 )eiθ0 , c3 = −c1 c2 eiθ0 , 3(b − 1)(a2 − 4b) with "  1/2 # a b(a2 + 8) − 2(a2 + 2) , θ0 = ± arccos 2 2b(a2 + 2) − 3a2 where a and b are given by (2.10). Finally, in the second case of (2.2), the equality holds for the function f˜1 defined by  α+1 z 1 + w(z) 0 ˜ f1 (z) = (2.12) ˜ 1 − w(z) f1 (z) with the Schwarz function given by w(z) = z(κ + z)/(1 + κz), where  1/2 |a| − 1 κ := . 3 (|a| − b − 1) This completes the proof.

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The following theorem provides sharp bound on the fifth, sixth, seventh and eighth coeffi¯ cients for functions in the class B(α). Theorem 2.2. Let us denote Ψ := 2α6 − 28α5 + 137α4 − 331α3 + 437α2 − 433α + 360, ˆ := −6α9 +96α8 −674α7 +2836α6 −8942α5 +22504α4 −40886α3 +45124α2 −30132α+21600, Ψ χ := 23α12 − 756α11 + 10218α10 − 77686α9 + 376014α8 − 1243398α7 + 2969824α6 − 5401638α5 + 7729083α4 − 8432486α3 + 6389238α2 − 3333636α + 1360800, and χˆ := −(45α15 − 1530α14 + 23641α13 − 221500α12 + 1438032α11 − 7061480α10 + 27696314α9 − 88000680α8 + 222370901α7 − 435300650α6 + 653299149α5 − 763502860α4 + 703545502α3 − 473136900α2 + 206026416α − 76204800). ¯ If f ∈ B(α) has the from (1.1), then for 0 < α < 1, the following sharp inequalities hold: 2Ψ |a5 | ≤ , 3(α − 4)(α − 3)(α − 2)2 (α − 1)4 ˆ Ψ |a6 | ≤ , 15(α − 5)(α − 4)(α − 3)(α − 2)2 (α − 1)5 2χ |a7 | ≤ 45(α − 6)(α − 5)(α − 4)(α − 3)2 (α − 2)3 (α − 1)6 and 2χˆ . |a8 | ≤ 315(α − 7)(α − 6)(α − 5)(α − 4)(α − 3)2 (α − 2)3 (α − 1)7 Proof. From (2.3), on comparing the coefficients, we have a5 =

τ1 p4 + τ2 p21 p2 + τ3 p22 + τ4 p1 p3 + τ5 p41 , 24(α − 4)(α − 3)(α − 2)2 (α − 1)4

(2.13)

where τi ’s are given by τ1 := −24(α − 3)(α − 2)2 (α − 1)4 , τ2 := −12(α − 4)(α − 3)(α − 2)(α − 1)2 (α + 1)(2α + 1), τ3 := 12(α − 3)(α − 4)(α − 1)4 (α + 1), τ4 := 24(α − 4)(α − 2)2 (α − 1)3 (α + 1), τ5 := (α − 4)(α − 3)(α − 2)2 (α + 1)(2α + 1)(3α + 1). Similarly, the sixth coefficient is given by a6 = −

τˆ1 p5 + τˆ2 p22 p1 + τˆ3 p2 p3 + τˆ4 p31 p2 + τˆ5 p21 p3 + τˆ6 p1 p4 + τˆ7 p51 , 120(α − 5)(α − 4)(α − 3)(α − 2)2 (α − 1)5

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where τˆi ’s are defined by τˆ1 := 120(α − 4)(α − 3)(α − 2)2 (α − 1)5 , τˆ2 := 60(α − 5)(α − 4)(α − 3)(α − 1)4 (α + 1)(2α + 1), τˆ3 := −120(α − 5)(α − 4)(α − 2)(α − 1)5 (α + 1), τˆ4 := −20(α − 5)(α − 4)(α − 3)(α − 2)(α − 1)2 (α + 1)(2α + 1)(3α + 1), τˆ5 := 60(α − 5)(α − 4)(α − 2)2 (α − 1)3 (α + 1)(2α + 1), τˆ6 := −120(α − 5)(α − 3)(α − 2)2 (α − 1)4 (α + 1), τˆ7 := (α − 5)(α − 4)(α − 3)(α − 2)2 (α + 1)(2α + 1)(3α + 1)(4α + 1). To find the estimate on |a5 |, we observe from (2.13) that the coefficients τi (i = 1, 2, 3, 4, 5, 6, 7) of p4 , p21 p2 , p22 , p1 p3 and p41 are positive. Hence applying triangle inequality in (2.13) and using the fact that |pj | ≤ 2, we get the required estimate on |a5 |. A similar argument can be used to obtained the estimates on |a6 |, |a7 | and |a8 |. In all the cases, equality hold for the function f˜0 given by (2.7). This completes the proof. The following theorem gives the sharp bound on the functional |a2 a3 − a4 | for the functions ¯ in the class B(α). ¯ Theorem 2.3. Let f ∈ B(α) has the form (1.1). Then, the following sharp result holds:  2(α3 −4α2 +α+18)   3(α−1)2 (α−2)(α−3) , if α ∈ (0, 2) \ {1}; 2(α3 −4α2 +α+18) (2.15) |a2 a3 − a4 | ≤ if 2 < α < 3; 2 (α−2)(3−α) ,   3(α−1) 2 , if α > 3. α−3 Proof. Proceeding as in the proof of previous theorem and using (2.4) and (2.9), we can write a2 a3 − a4 =

2 [(α3 − 4α2 + α + 18) c31 + 12(α − 1)c1 c2 + 3(α − 2)(α − 1)2 c3 ] . 3(α − 3)(α − 2)(α − 1)2

(2.16)

By setting s :=

4 α3 − 4α2 + α + 18 and t := (α − 1)(α − 2) 3(α − 2)(α − 1)2

the expression in (2.16) can be written as a2 a3 − a4 =

 2  c3 + sc1 c2 + tc31 . α−3

Assume that the symbols Ωi ’s are as defined in Lemma 1.2 with the settings µ = s and ν = t. Now the proof is accomplished in the following steps:

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√ 33)/2 ≤ α. Then it can be easily verified that 1 1 − ≤s≤ and − 1 ≤ t ≤ 1. 2 2 √ 33)/2 ≤ α, we have (s, t) ∈ Ω1 . Further, when 3 < α ≤ Therefore, for the range (3 + √ (3 + 33)/2, wee see that (s, t) ∈ Ω2 . (2) Let 0√< α < 1 or 1 < α < 2. Then in a similar way we have √ (s, t) ∈ Ω4 . Further if (3 + 5)/2 ≤ α < 3, then (s, t) ∈ Ω6 and when 2 < α ≤ (3 + 5)/2, then (s, t) ∈ Ω7 .

(1) Let (3 +

In the light of the above discussions, an application of Lemma 1.2, establish the required estimate on |a2 a3 − a4 |. In the first two cases of (2.15), the equality hold for the function ¯ f˜0 ∈ B(α) defined by (2.7). In the third case of (2.15), the equality holds for the function f2 defined by  α+1 z 1 + z3 0 f2 (z) = . (2.17) f2 (z) 1 − z3 This completes the proof. Remark 2.4. It would be interesting to find out the sharp bound on |ai | (i = 5, 6, 7, 8) for the ¯ functions f˜ ∈ B(α) in the case when α > 1. Acknowledgement The second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2016R1D1A1A09916450). References [1] N. E. Cho and V. Kumar, On a coefficient conjecture for Bazileviˇc functions, preprint. [2] L. Guo, Y. Ling and G. Bao, On the starlikeness for the class of multivalent non-Bazilevic functions, South Asian Journal of Mathematics 3 (2013), no. 1, 67–70. [3] F. R. Keogh and E. P. Merkes, A coefficient inequality for certain classes of analytic functions, Proc. Amer. Math. Soc. 20 (1969), 8–12. [4] D. V. Krishna and T. R. Reddy, An upper bound to the second Hankel functional for non-Bazilevic functions, Far East J. Math. Sci. 67 (2012), no. 2, 187–199. [5] S. S. Kumar and V. Kumar, Fekete-Szeg¨o problem for a class of analytic functions defined by convolution, Tamkang J. Math. 44 (2013), no. 2, 187–195. [6] Marjono, J. Sok´ ol and D. K. Thomas, The fifth and sixth coefficients for Bazileviˇc functions B1 (α), Mediterr. J. Math. 14 (2017), no. 4, Art. ID. 158, 11 pp. [7] M. Obradovi´c, A class of univalent functions, Hokkaido Math. J. 27 (1998), no. 2, 329–335. [8] S. Ponnusamy, Convolution properties of some classes of meromorphic univalent functions, Proc. Indian Acad. Sci. Math. Sci. 103 (1993), no. 1, 73–89. [9] D. V. Prokhorov and J. Szynal, Inverse coefficients for (α, β)-convex functions, Ann. Univ. Mariae Curie-Sklodowska Sect. A 35 (1981), 125–143.

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[10] V. Ravichandran, Y. Polatoglu, M. Bolcal and A. Sen, Certain subclasses of starlike and convex functions of complex order, Hacet. J. Math. Stat. 34 (2005), 9–15. [11] T. Sheil-Small, On Bazileviˇc functions, Quart. J. Math. Oxford Ser. 23 (1972), no. 2, 135–142. [12] R. Singh, On Bazileviˇc functions, Proc. Amer. Math. Soc. 38 (1973), 261–271. [13] D. K. Thomas, On the coefficients of Bazileviˇc functions with logarithmic growth, Indian J. Math. 57 (2015), no. 3, 403–418. [14] D. K. Thomas, On a subclass of Bazileviˇc functions, Internat. J. Math. Math. Sci. 8 (1985), no. 4, 779–783. [15] D. K. Thomas, On starlike and close-to-convex univalent functions, J. London Math. Soc. 42 (1967), 427–435. [16] N. Tuneski and M. Darus, Fekete-Szeg¨o functional for non-Bazileviˇc functions, Acta Math. Acad. Paedagog. Nyh´ azi. (N.S.) 18 (2002), no. 2, 63–65. [17] J. Zamorski, On Bazileviˇc schlicht functions, Ann. Polon. Math. 12 (1962), 83–90. (J. H. Park) Department of Applied Mathematics, Pukyong National University, Busan 48513, South Korea E-mail address: [email protected] (V. Kumar) Department of Applied Mathematics, Pukyong National University, Busan 48513, South Korea E-mail address: [email protected] (N. E. Cho) Corresponding Author, Department of Applied Mathematics, Pukyong National University, Busan 48513, South Korea E-mail address: [email protected]

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A new extragradient method for the split feasibility and fixed point problems ∗ Ming Zhao1†and Yunfei Du2 1 School

of Science, China University of Geosciences(Beijing), Beijing 100083, China

2 LMIB-School

of Mathematics and Systems Science, Beihang University, Beijing 100191, China

Abstract: In this paper, we propose a new extragradient method with regularization for finding a common element of the solution set Γ of the split feasibility problem and the set Fix(S) of fixed points of a nonexpansive mapping S in infinitedimensional Hilbert spaces, combining the regularization method and the technique of averaged operator, we prove the sequences generated by the proposed algorithm ∩ converge weakly to an element of Fix(S) Γ under mild conditions. Keywords: split feasibility problem , extragradient, regularization.

1. Introduction Throughout this paper, let H be a Hilbert space, ⟨·, ·⟩ denotes the inner product, and ∥ · ∥ denotes for the corresponding norm. The split feasibility problem (SFP) which was first introduced by Censor and Elfving [1] in 1994 for modeling inverse problems arising from phase retrievals and in medical image reconstruction. Let C and Q be closed convex sets in the infinite-dimensional real Hilbert spaces H1 and H2 , respectively. The SFP is to find a vector x∗ satisfying x∗ ∈ C such that Ax∗ ∈ Q, (1.1) where A ∈ B(H1 , H2 ) which denotes the family of all bounded linear operators from H1 to H2 . Some related work in the infinite-dimensional setting can be found in [2, 3, 4, 5, 9, 10, 12] and the references therein. Many methods have been developed to solve the SFP, The basic algorithm have CQ algorithm proposed by Byrne [2], the relaxed CQ algorithm proposed by Yang [9], the half-space relaxation projection method proposed by Qu and Xiu [11], the variable Krasnosel skii-Mann algorithm proposed by Xu [12]. The projections of a point onto C and Q are difficult to compute when C and Q fail to have closed-form expressions, though theoretically we can prove the (weak) convergence of the algorithm. Very recently, Xu [6] gave a continuation of the study on the CQ algorithm and its convergence. He applied Mann’s algorithm to the SFP and proposed an averaged CQ algorithm which was proved to be weakly convergent to a solution of the SFP. On the other hand, Korpelevich ∗ †

This work was supported by the Fundamental Research Funds for the Central Universities. Corresponding Author. Email address: [email protected](M.Zhao)

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[7] introduced the so-called extragradient method for finding a solution of a saddle point problem. He proved that the sequences generated by the proposed iterative algorithm converge to a solution of a saddle. Motivated by the idea of an extragradient method, Nadezhina and Takahashi [8] introduced an iterative algorithm for finding a common element of the set of fixed points of a nonexpansive mapping and the solution set of a variational inequality problem [13] for a monotone, Lipschitz continuous mapping in a real Hilbert space. They obtained a weak convergence theorem for two sequence generated by the proposed algorithm. In our paper, we introduce and analyze a new extragradient iterative algorithm to find a common element of the solution set Γ of the split feasibility problem and the set Fix(S) of fixed points of a nonexpansive mapping S in infinite-dimensional Hilbert spaces, furthermore, we prove its convergence. The results of this paper represent the improvement of the corresponding results in [6] and [14].

2. Preliminaries Throughout this paper, we use xn → x and xn ⇀ x to denote strong and weak convergence to x of the sequence xn , respectively. Let K be a nonempty closed convex subset of H. Recall that the projection (nearest point or metric) from H onto K, denoted by PK , is defined in such a way that, for each x ∈ H, PK x is the unique point in K with the property ∥ x − PK x ∥= inf ∥ x − y ∥=: d(x, K), y∈K

i.e. PK (x) = argmin{∥ x − y ∥| y ∈ K}. Some important properties of projections are gathered in the following Lemma. Lemma 2.1 For given x ∈ H and z ∈ K, the following properties hold: (1) x ∈ K ⇔ PK (x) = x; (2) ⟨x − PK (x), z − PK (x)⟩ ≤ 0, ∀x ∈ H and ∀z ∈ K; (3) ⟨x − y, PK (x) − PK (y)⟩ ≥ ∥PK (x) − PK (y)∥2 , ∀x, y ∈ H; (4) ∥PK (x) − z∥2 ≤ ∥x − z∥2 − ∥PK (x) − x∥2 , ∀x ∈ H and ∀z ∈ K; (5) ∥PK (x) − PK (y)∥ ≤ ∥x − y∥, ∀x, y ∈ H. Proof. See Facchinei and Pang [15]. Definition 2.1 Let T be a mapping from K ⊆ H into H, then (a) T is called monotone on K if ⟨T (x) − T (y), x − y⟩ ≥ 0, ∀ x, y ∈ K. (b) T is called strongly monotone on K if there is a µ > 0, such that ⟨T (x) − T (y), x − y⟩ ≥ µ∥x − y∥2 , ∀ x, y ∈ K. (c) F is called co-coercive (or ν-inverse strongly monotone) on K if there is a ν > 0, such that ⟨T (x) − T (y), x − y⟩ ≥ ν∥T (x) − T (y)∥2 , ∀ x, y ∈ K. (d) F is called pseudo-monotone on K if ⟨T (y), x − y⟩ ≥ 0 ⇒ ⟨T (x), x − y⟩ ≥ 0, ∀ x, y ∈ K. (e) T is called Lipschitz continuous on K if there exists a constant L > 0 such that ∥T (x) − T (y)∥ ≤ L∥x − y∥, ∀ x, y ∈ K.

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Definition 2.2 A mapping T : H → H is said to be: (a) nonexpansive if ∥ T x − T y ∥≤∥ x − y ∥, ∀x, y ∈ H; (b) firmly nonexpansive if 2T − I is nonexpansive, or equivalently, ⟨x − y, T x − T y⟩ ≥∥ T x − T y ∥, ∀x, y ∈ H, or alternatively, T is firmly nonexpansive if and only if T can be expressed as 1 T = (I + S) 2 where S : H → H is nonexpansive. Remark 2.1 From Lemma 2.1 and Definition 2.1-2.2, we can infer that if S is nonexpansive, then I-S is monotone; A monotone mapping is pseudo-monotone mapping; An inverse strongly monotone mapping is monotone and Lipschitz continuous; A Lipschitz continuous and strongly monotone mapping is an inverse strongly monotone mapping; The projection operator is 1-ism and nonexpansive. Lemma 2.2 A mapping T is 1-ism if and only if the mapping I-T is 1-ism, where I is the identity operator. Proof. See [16, Lemma 2.3]. Remark 2.2 If T is an inverse strongly monotone mapping, then T is a nonexpansive mapping. Definition 2.3 A mapping T : H → H is said to be an averaged mapping if it can be written as the average of the identity I and a nonexpansive mapping S, that is, T = (1 − α)I + αS

(2.1)

where α ∈ (0, 1) and S : H → H is nonexpansive. More precisely, when (2.1) holds, we say that T is α-averaged. Thus firmly nonexpansive mappings (for example, projections) are 12 -averaged mappings. Proposition 2.1 ([16]). Let T : H → H be a given mapping: (1) T is nonexpansive if and only if the complement I-T is 21 -ism. (2) If T is µ-ism, then for γ > 0, γT is γν -ism. (3) T is averaged if and only if the complement I-T is ν-ism for some ν > 12 . Indeed, for 1 α ∈ (0, 1), T is α-averaged if and only if I-T is 2α -ism. Proposition 2.2 ([16, 17]). Let S, T, V : H → H be given operators. (1) If T = (1 − α)S + αV for some α ∈ (0, 1) and if S is averaged and V is nonexpansive, then T is averaged. (2) T is firmly nonexpansive if and only if the complement I-T is firmly nonexpansive. (3) If T = (1 − α)S + αV for some α ∈ (0, 1) and if S is firmly nonexpansive and V is nonexpansive, then T is averaged. (4) The composite of finitely many averaged mappings is averaged. That is, if each of the mappings {Ti }N i=1 is averaged, then so is the composite T1 ◦ · · · ◦ TN . In particular, if T1 is α1 averaged and T2 is α2 -averaged, where α1 , α2 ∈ (0, 1), then the composite T1 ◦ T2 is α-averaged, where α = α1 + α2 − α1 α2 . (5) If the mapping {Ti }N i are averaged and have a common fixed point, then N ∩

Fix(Ti ) = Fix(T1 ◦ · · · ◦ TN ).

i=1

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The notation Fix(T ) denotes the set of all fixed points of the mapping T , that is Fix(T ) = {x ∈ H : T x = x}. The so-called demiclosedness principle plays an important role in our argument. Definition 2.4 Let T : H → H be an operator. We say that I-T is demiclosed (at zero), if for any sequence xn in H, there holds the following implication: xn ⇀ x and (I-T )xn → 0 ⇒ (I-T )x = 0. Lemma 2.3 ([18]). Let H be a Hilbert space. Then for all x, y ∈ H and λ ∈ [0, 1], ∥ λx + (1 − λ)y ∥2 = λ ∥ x ∥2 +(1 − λ) ∥ y ∥2 −λ(1 − λ) ∥ x − y ∥2 . ∞ ∞ Lemma 2.4 ([19]). Let {an }∞ n=1 , {bn }n=1 and {δn }n=1 be sequences of nonnegative real numbers satisfying the inequality an+1 ≤ (1 + δn )an + bn , ∀n ≥ 1.

∑

∑

∞ If ∞ n=1 δn < ∞ and n=1 bn < ∞, then limn→∞ an exists. ∞ Corollary 2.1 ([20]). Let {an }∞ n=1 and {bn }n=1 be two sequences of nonnegative real numbers satisfying the inequality an+1 ≤ an + bn , ∀n ≥ 1.

∑∞

n=1 bn < ∞, then limn→∞ an exists. Recall that a Banach space X is said to satisfy the Opial condition [22] if for any sequence {xn } in X the condition that {xn } converges weakly to x ∈ X implies that the inequality

If

lim inf ∥ xn − x ∥< lim inf ∥ xn − y ∥ n→∞

n→∞

holds for every y ∈ X with y ̸= x. It is well-known that every Hilbert space satisfies the Opial condition.

3. Main results Throughout this paper, we assume that the SFP is consistent, that is, the solution set Γ of the SFP is nonempty. It is easy to see that SFP is equivalent to the following minimization problem 1 min f (x) := ∥Ax − PQ Ax∥2 , x∈C 2

(3.1)

where f : H1 → R is a continuous differentiable function, however it is ill-posed. Therefore, Xu [6] considered the following Tikhonov regularized problem: 1 1 min fα (x) := ∥Ax − PQ Ax∥2 + α∥x∥2 , x∈C 2 2

(3.2)

where α > 0 is the regularization parameter. We observe that the gradient ∇fα (x) = ∇f (x) + αI = A∗ (I − PQ )A + αI

(3.3)

is (α + ∥A∥2 )-Lipschitz continuous and α-strongly monotone. Proposition 3.1 ([21]) Given x∗ ∈ H1 , the following statements are equivalent: (1) x∗ solves the SFP; (2) x∗ solves the fixed point equation PC (I − λ∇f ) = PC [I − λA∗ (I − PQ )A]x∗ = x∗

1117

(3.4).

Ming Zhao ET AL 1114-1123

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.7, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

(3) x∗ solves the variational inequality problem (VIP) of finding x∗ ∈ C such that ⟨∇f (x∗ ), x − x∗ ⟩ ≥ 0, ∀x ∈ C,

(3.5)

where ∇f = A∗ (I − PQ )A and A∗ is the adjoint of A. Remark 3.1. It is clear from Proposition 3.1 that Γ = Fix(PC (I − λ∇f )) = V I(C, ∇f ) for any λ > 0, where Fix(PC (I − λ∇f )) and V I(C, ∇f ) denote the set of fixed points of PC (I − λ∇f ) and the solution set of VIP(3.5). Next, we will present our method for solving the SFP and prove its convergence. Theorem 3.1 Let S : C → C be a nonexpansive mapping such that Fix(S) ∩ Γ ̸= ∅ in Hilbert space. Let {xn }, {yn } and {zn } be the sequences in C generated by the following extragradient algorithm:   x0 ∈ C chosen arbitrarily,    z = (1 − γ )x + γ P (I − λ ∇f )x , n n n n C n αn n (3.6)  yn = (1 − βn )zn + βn SPC (I − λn fαn )zn ,    xn+1 = (1 − µn )yn + µn SPC (I − λn ∇fαn )yn , ∀n > 0, where the sequences of parameters {αn }, {βn }, {γn } and {µn } satisfy the following conditions: ∑ (a) ∞ n=1 αn ( < ∞; ) 1 1 and 0 < lim inf n→∞ λn ≤ lim supn→∞ λn < ∥A∥ (b) {λn } ⊂ 0, ∥A∥ 2 2; (c) {γn } ⊂ (0, 1), and 0 < lim inf n→∞ γn ≤ lim supn→∞ γn < 1; (d) {βn } ⊂ (0, 1), and 0 < lim inf n→∞ βn ≤ lim supn→∞ βn < 1; (e) {µn } ⊂ (0, 1), and 0 < lim inf n→∞ µn ≤ lim supn→∞ µn < 1. ∩ Then, the sequences {xn }, {yn } and {zn } are all converge weakly to an element ¯ ∈ Fix(S) Γ. ( x ) 2 Proof. It [21] has been proved PC (I − λ∇fα ) is ζ-averaged for each λ ∈ 0, α+∥A∥2 , where ( ) 2+λ(α+∥A∥2 ) 1 , so P (I − λ∇f ) is nonexpansive. Furthermore, for {λ } ⊂ 0, , we have ζ= α n C 4 ∥A∥2 0 < lim inf λn ≤ lim sup λn < n→∞

n→∞

1 1 = lim . ∥A∥2 0→∞ αn + ∥A∥2

Without loss of generality, we may assume that 0 < lim inf λn ≤ lim sup λn < n→∞

n→∞

1 , ∀n ≥ 0. αn + ∥A∥2

Consequently, PC (I − λn ∇fαn ) is ζn -averaged for each integer n ≥ 0, where ζn =

2 + λn (αn + ∥A∥2 ) ∈ (0, 1). 4

This implies that PC (I − λn ∇fαn ) is nonexpansive for all n ≥ 0. Next, we show the sequences {xn }, {yn }, {zn } generated in Theorem 3.1 are bounded. ∩ Indeed, take a fixed p ∈ Fix(S) Γ arbitrarily. Then, we get Sp = p and PC (I − λ∇f )p = p for

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.7, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

(

)

1 λ ∈ 0, ∥A∥ 2 . From (3.6), it follows that

∥zn − p∥ = ∥(1 − γn )(xn − p) + γn [PC (I − λn ∇fαn )xn − p]∥ ≤ (1 − γn )∥xn − p∥ + γn ∥PC (I − λn ∇fαn )xn − p∥ = (1 − γn )∥xn − p∥ + γn ∥PC (I − λn ∇fαn )xn − PC (I − λn ∇f )p∥ = (1 − γn )∥xn − p∥ + γn ∥PC (I − λn ∇fαn )xn − PC (I − λn ∇fαn )p +PC (I − λn ∇fαn )p − PC (I − λn ∇f )p∥ ≤ (1 − γn )∥xn − p∥ + γn (∥PC (I − λn ∇fαn )xn − PC (I − λn ∇fαn )p∥ +∥PC (I − λn ∇fαn )p − PC (I − λn ∇f )p∥) ≤ (1 − γn )∥xn − p∥ + γn (∥xn − p∥ + ∥(I − λn ∇fαn )p − (I − λn ∇f )p∥) = ∥xn − p∥ + λn αn γn ∥p∥,

(3.7)

∥yn − p∥ = ∥(1 − βn )(zn − p) + βn [SPC (I − λn ∇fαn )zn − p]∥ ≤ (1 − βn )∥zn − p∥ + βn ∥PC (I − λn ∇fαn )zn − p∥ = (1 − βn )∥zn − p∥ + βn ∥PC (I − λn ∇fαn )zn − PC (I − λn ∇f )p∥ = (1 − βn )∥zn − p∥ + βn ∥PC (I − λn ∇fαn )zn − PC (I − λn ∇fαn )p +PC (I − λn ∇fαn )p − PC (I − λn ∇f )p∥ ≤ (1 − βn )∥zn − p∥ + βn (∥PC (I − λn ∇fαn )zn − PC (I − λn ∇fαn )p∥ +∥PC (I − λn ∇fαn )p − PC (I − λn ∇f )p∥) ≤ (1 − βn )∥zn − p∥ + βn (∥zn − p∥ + ∥(I − λn ∇fαn )p − (I − λn ∇f )p∥) = ∥zn − p∥ + λn αn βn ∥p∥,

(3.8)

∥xn+1 − p∥ = ∥(1 − µn )(yn − p) + µn [SPC (I − λn ∇fαn )yn − p]∥ ≤ (1 − µn )∥yn − p∥ + µn ∥PC (I − λn ∇fαn )yn − p∥ = (1 − µn )∥yn − p∥ + µn ∥PC (I − λn ∇fαn )yn − PC (I − λn ∇f )p∥ = (1 − µn )∥yn − p∥ + µn ∥PC (I − λn ∇fαn )xn − PC (I − λn ∇fαn )p +PC (I − λn ∇fαn )p − PC (I − λn ∇f )p∥ ≤ (1 − µn )∥yn − p∥ + µn (∥PC (I − λn ∇fαn )zn − PC (I − λn ∇fαn )p∥ +∥PC (I − λn ∇fαn )p − PC (I − λn ∇f )p∥) ≤ (1 − µn )∥yn − p∥ + µn (∥yn − p∥ + ∥(I − λn ∇fαn )p − (I − λn ∇f )p∥) = ∥yn − p∥ + λn αn µn ∥p∥ ≤ ∥xn − p∥ + λn αn (γn + βn + µn )∥p∥,

(3.9)

and

where the last inequality follows from (3.7) and (3.8). Since Σ∞ n=1 αn < ∞, and {λn }, {γn }, {βn }, {µn } are bounded, then from Corollary 2.1, we conclude that ∩ lim ∥xn − p∥ exists for each p ∈ Fix(S) Γ. (3.10) n→∞

Hence {xn } is bounded and so are {yn } and {zn }. In the following, we will show lim ∥xn − yn ∥ = lim ∥yn − zn ∥ = lim ∥xn − un ∥ = lim ∥yn − Swn ∥ = lim ∥zn − Svn ∥ = 0,

n→∞

n→∞

n→∞

n→∞

n→∞

where un = PC (I − λn ∇fαn )xn , vn = PC (I − λn ∇fαn )zn , wn = PC (I − λn ∇fαn )yn . Note that ∥un − p∥ = ∥PC (I − λn ∇fαn )xn − p∥ = ∥PC (I − λn ∇fαn )xn − PC (I − λn ∇fαn )p +PC (I − λn ∇fαn )p + PC (I − λn ∇f )p∥ ≤ ∥PC (I − λn ∇fαn )xn − PC (I − λn ∇fαn )p∥ +∥PC (I − λn ∇fαn )p + PC (I − λn ∇f )p∥ ≤ ∥xn − p∥ + λn αn ∥p∥.

1119

(3.11)

Ming Zhao ET AL 1114-1123

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.7, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

Similarly, we can obtain that ∥vn − p∥ ≤ ∥zn − p∥ + λn αn ∥p∥

(3.12)

∥wn − p∥ ≤ ∥yn − p∥ + λn αn ∥p∥.

(3.13)

and Indeed, observe that ∥zn − p∥2 = ∥(1 − γn )(xn − p) + γn (un − p)∥2 = (1 − γn )∥xn − p∥2 + γn ∥un − p)∥2 − γn (1 − γn )∥xn − un ∥2 ≤ (1 − γn )∥xn − p∥2 + γn (∥xn − p∥ + λn αn ∥p∥)2 − γn (1 − γn )∥xn − un ∥2 = (1 − γn )∥xn − p∥2 + γn (∥xn − p∥2 + 2λn αn ∥p∥∥xn − p∥ + λ2n αn2 ∥p∥2 ) −γn (1 − γn )∥xn − un ∥2 = ∥xn − p∥2 + αn γn (2λn ∥p∥∥xn − p∥ + αn λ2n ∥p∥2 ) − γn (1 − γn )∥xn − un ∥2 ≤ ∥xn − p∥2 + αn M1 − γn (1 − γn )∥xn − un ∥2 , (3.14) where M1 = supn≥0 {γn (2λn ∥p∥∥xn − p∥ + αn λ2n ∥p∥2 )} < ∞ and the first inequality follows from (3.11). Also, observe that ∥yn − p∥2 = ∥(1 − βn )(zn − p) + βn (Svn − p)∥2 = (1 − βn )∥zn − p∥2 + βn ∥Svn − p)∥2 − βn (1 − βn )∥zn − Svn ∥2 ≤ (1 − βn )∥zn − p∥2 + βn ∥vn − p)∥2 − βn (1 − βn )∥zn − Svn ∥2 ≤ (1 − βn )∥zn − p∥2 + βn (∥zn − p∥ + λn αn ∥p∥)2 − βn (1 − βn )∥zn − Svn ∥2 = (1 − βn )∥zn − p∥2 + βn (∥zn − p∥2 + 2λn αn ∥p∥∥zn − p∥ + λ2n αn2 ∥p∥2 ) −βn (1 − βn )∥zn − Svn ∥2 = ∥zn − p∥2 + αn βn (2λn ∥p∥∥zn − p∥ + αn λ2n ∥p∥2 ) − βn (1 − βn )∥zn − Svn ∥2 ≤ ∥zn − p∥2 + αn M2 − βn (1 − βn )∥zn − Svn ∥2 , (3.15) { } where M2 = supn≥0 βn (2λn ∥p∥∥zn − p∥ + αn λ2n ∥p∥2 ) < ∞ and the second inequality follows from (3.12). And ∥xn+1 − p∥2 = ∥(1 − µn )(yn − p) + µn (Swn − p)∥2 = (1 − µn )∥yn − p∥2 + µn ∥Swn − p)∥2 − µn (1 − µn )∥yn − Swn ∥2 ≤ (1 − µn )∥yn − p∥2 + µn ∥wn − p)∥2 − µn (1 − µn )∥yn − Swn ∥2 ≤ (1 − µn )∥yn − p∥2 + µn (∥yn − p∥ + λn αn ∥p∥)2 − µn (1 − µn )∥yn − Swn ∥2 = (1 − µn )∥yn − p∥2 + µn (∥yn − p∥2 + 2λn αn ∥p∥∥yn − p∥ + λ2n αn2 ∥p∥2 ) −µn (1 − µn )∥yn − Swn ∥2 = ∥yn − p∥2 + αn µn (2λn ∥p∥∥yn − p∥ + αn λ2n ∥p∥2 ) − µn (1 − µn )∥yn − Swn ∥2 ≤ ∥yn − p∥2 + αn M3 − µn (1 − µn )∥yn − Swn ∥2 , (3.16) where M3 = supn≥0 {µn (2λn ∥p∥∥yn − p∥ + αn λ2n ∥p∥2 )} < ∞ and the second inequality follows from (3.13). Substitute (3.14) and (3.15) into (3.16), we have ∥xn+1 − p∥2 ≤ ∥xn − p∥2 + αn (M1 + M2 + M3 ) − γn (1 − γn )∥xn − un ∥2 −βn (1 − βn )∥zn − Svn ∥2 − µn (1 − µn )∥yn − Swn ∥2 .

1120

(3.17)

Ming Zhao ET AL 1114-1123

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.7, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

Hence, it follows that γn (1 − γn )∥xn − un ∥2 + βn (1 − βn )∥zn − Svn ∥2 + µn (1 − µn )∥yn − Swn ∥2 ≤ ∥xn − p∥2 − ∥xn+1 − p∥2 + αn (M1 + M2 + M3 ).

(3.18)

Since Σ∞ n=1 αn < ∞, 0 < lim inf n→∞ γn ≤ lim supn→∞ γn < 1, 0 < lim inf n→∞ βn ≤ lim supn→∞ βn < 1, and 0 < lim inf n→∞ µn ≤ lim supn→∞ µn < 1, we deuce from the existence of limn→∞ ∥xn − p∥ that lim ∥xn − un ∥ = lim ∥yn − Swn ∥ = lim ∥zn − Svn ∥ = 0.

n→∞

n→∞

n→∞

(3.19)

Then, utilizing (3.6) we get lim ∥zn − xn ∥ = lim γn ∥un − xn ∥ = 0,

(3.20)

lim ∥yn − zn ∥ = lim βn ∥Svn − zn ∥ = 0,

(3.21)

lim ∥xn − yn ∥ = lim µn ∥Swn − yn ∥ = 0.

(3.22)

n→∞

n→∞

n→∞

n→∞

and n→∞

n→∞

This implies that lim ∥xn − yn ∥ = lim ∥yn − zn ∥ = lim ∥xn − un ∥ = lim ∥yn − Swn ∥ = lim ∥zn − Svn ∥ = 0.

n→∞

n→∞

n→∞

n→∞

n→∞

Furthermore, note that ∥Svn − vn ∥ ≤ ∥Svn − zn ∥ + ∥zn − xn ∥ + ∥xn − un ∥ + ∥un − vn ∥ = ∥Svn − zn ∥ + ∥zn − xn ∥ + ∥xn − un ∥ +∥PC (I − λn ∇fαn )xn − PC (I − λn ∇fαn )zn ∥ ≤ ∥Svn − zn ∥ + ∥zn − xn ∥ + ∥xn − un ∥ + ∥xn − zn ∥. From (3.20-3.22), we can get that lim ∥un − vn ∥ = lim ∥Svn − vn ∥ = 0.

(3.23)

lim ∥un − wn ∥ = lim ∥Swn − wn ∥ = 0.

(3.24)

n→∞

n→∞

Similarly, we can prove n→∞

n→∞

As {xn } is bounded, there is a subsequence {xni } of {xn } that converges weakly to some x ¯. ∩ Next, we will show x ¯ ∈ Fix(S) Γ. We first show x ¯ ∈ Γ, let T = PC (I − λn ∇f ), then ∥xn − T xn ∥ ≤ ∥xn − un ∥ + ∥un − T xn ∥ = ∥xn − un ∥ + ∥PC (I − λn ∇fαn )xn − PC (I − λn ∇f )xn ∥ ≤ ∥xn − un ∥ + ∥(I − λn ∇fαn )xn − (I − λn ∇f )xn ∥ = ∥xn − un ∥ + λn αn ∥xn ∥.

(3.25)

From limn→∞ ∥xn − un ∥ = 0, limn→∞ ∥αn ∥ = 0 and {λn }, {xn } are bounded, we can get that limn→∞ ∥xn − T xn ∥ = 0. Taking into account xni ⇀ x ¯ and Definition 2.4, we obtain x ¯ ∈ Fix(T ). Thus, utilizing Remark 3.1, we have x ¯ ∈ Γ. On the other hand, since lim ∥xn − un ∥ = lim ∥un − vn ∥ = lim ∥Svn − vn ∥ = 0,

n→∞

n→∞

n→∞

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there is subsequence vnj of vn that converges weakly to x ¯ and limn→∞ ∥Svnj − vnj ∥ = 0. Then ∩ from Definition 2.4, we have x ¯ ∈ Fix(S). Therefore, we get x ¯ ∈ Fix(S) Γ. ∩ Let {xnj } be another subsequence of {xn } such that xnj ⇀ x ˜. Then, x ˜ ∈ Fix(S) Γ. Next, we prove x ˜=x ¯. Assume that x ˜ ̸= x ¯. From the Opial condition [22], we have limn→∞ ∥xn − x ˜∥ = lim inf i→∞ ∥xni − x ˜∥ < lim inf i→∞ ∥xni − x ¯∥ = limn→∞ ∥xn − x ¯∥ = lim inf j→∞ ∥xnj − x ¯∥ < lim inf j→∞ ∥xnj − x ˜∥ = limn→∞ ∥xn − x ˜∥, ∩

which is a contradiction. Thus, we have x ˜=x ¯. This implies xn ⇀ x ¯ ∈Fix(S) Γ. Furthermore, from limn→∞ ∥xn − yn ∥ = limn→∞ ∥zn − xn ∥ = 0, we can get yn ⇀ x ¯ and zn ⇀ x ¯. This shows ∩ that the sequences {xn }, {yn } and {zn } are all converge weakly to an element x ¯ ∈ Fix(S) Γ. Theorem 3.2 Let S : C → C be a nonexpansive mapping such that Fix(S) ∩ Γ ̸= ∅ in Hilbert space. Let {xn }, {yn } and {zn } be the sequences in C generated by the following extragradient algorithm:   x0 ∈ C chosen arbitrarily,   

zn = (1 − γn )xn + γn PC (I − λn ∇f )xn ,  yn = (1 − βn )zn + βn SPC (I − λn ∇f )zn ,    xn+1 = (1 − µn )yn + µn SPC (I − λn ∇f )yn , ∀n > 0,

(3.26)

where the sequences )of parameters {βn }, {γn } and {µn } satisfy the following condition: ( 1 1 (a) {λn } ⊂ 0, ∥A∥2 and 0 < lim inf n→∞ λn ≤ lim supn→∞ λn < ∥A∥ 2; (b) {γn } ⊂ (0, 1), and 0 < lim inf n→∞ γn ≤ lim supn→∞ γn < 1; (c) {βn } ⊂ (0, 1), and 0 < lim inf n→∞ βn ≤ lim supn→∞ βn < 1; (d) {µn } ⊂ (0, 1), and 0 < lim inf n→∞ µn ≤ lim supn→∞ µn < 1. ∩ Then, the sequences {xn }, {yn } and {zn } are all converge weakly to an element x ¯ ∈ Fix(S) Γ. Proof. Let αn =0 in Theorem 3.1, then we can obtain the desired result. Remark 3.2. Our iteration method improves the corresponding results of [6], [8] and [14].

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Ming Zhao ET AL 1114-1123

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.7, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

Behavior of Meromorphic Solutions of Composite Functional-Difference Equations ∗† Man-Li Liua and Ling-Yun Gaob† a

b

School of Mathematics, Shandong University Jinan, Shandong, 250100,P.R.China e-mail: [email protected] Department of Mathematics,Jinan University Guangzhou,Guangdong, 510632,P.R.China e-mail:[email protected]

Abstract In view of Nevanlinna value distribution theory, we will investigate the behavior of meromorphic solutions of four types of composite functional-difference equations, and a type of system of composite functional-difference equations, some results are obtained. Moreover, we also give some examples to show that the conditions of our theorems are accurate. Key words: meromorphic solutions; composite functional-difference equations; behavior; growth order MR(2010) Subject Classification: 30D35,39B32

1.Introduction Recently, with the establishment of the difference analogues of Nevanlinna value distribution theory, researchers obtained many interesting theorems about the existence and growth of solutions of difference equations, functional equations and so on([3-6]). To state the results, a number of basic definition and standard notations should be introduced. We shall assume that the reader is familiar with the standard notations and results of Nevanlinna value distribution theory such as m(r, f (z)), n(r, f (z)), N (r, f (z)) and T (r, f (z))([15,18,22]) denote the proximity function, the non-integrated counting function, the counting function and the characteristic function of f (z), respectively. For the integrated counting function for distinct poles of f (z) we use the notations N (r, f (z)), and N1 (r, f ) = N (r, f ) − N (r, f ). In this article, a meromorphic function means meromorphic in the whole complex plane. Given a meromorphic function f (z), recall that a meromorphic function h(z) is said to be a small function of f (z), if T (r, h(z)) = S(r, f ),where S(r, f ) is used to denote ∗ This work was partially supported by NSFC of China(No.11271227,11271161), PCSIRT(No.IRT1264), and the Fundamental Research Funds of Shandong University (No.2017JC019). † Corresponding author:Gao Lingyun

1

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Man-Li Liu ET AL 1124-1141

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.7, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

any quantity that satisfies S(r, f ) = o(T (r, f )) as r → ∞, possibly outside of a set of r of finite logarithmic measure. Let c be a fixed, non-zero complex number, ∆c f (z) = f (z + c) − f (z), and ∆nc f (z) = ∆c (∆n−1 f (z)) = ∆n−1 f (z + c) − ∆n−1 f (z) for each integer n ≥ 2.Equations written with c c c the above difference operators ∆nc f (z) are difference equations. Let E be a subset on the positive real axis. We define the logarithmic measure of E to be Z

log(E) = E∩(1,+∞)

dr . r

A set E ∈ (1, +∞) is said to have finite logarithmic measure if log(E) < ∞. Difference equations have been studied in many aspects see e.g.,[1],[5-6],[17]. Some expositions consider (system of) difference equations in real domains, or discrete domain. So far, the previous researches are only on complex differential equations (systems) or difference equations (systems)[5,6], but not on composite functional-difference equations (systems). Therefore, it is very important and meaningful to study the cases of composite functional-difference equations (systems). That will be an innovative contribution of this paper. The remainder of the paper is organised as follows. In section 2, we will study the existence of meromorphic solutions or the form on some type of composite functionaldifference equations, and obtain three theorems, some examples are give to show that our results hold. In section 3, we will discuss the growth order of meromorphic solutions on some types of composite functional-difference equations or system of composite functionaldifference equations, which extend the result of Theorem B.

2. Existence of meromorphic solutions of difference equations and form of difference equations In 2003, H.Silvennoinen[21] was devoted to considering many types of composite functional equations, he got some good results, for example, the following theorem A is one of his results. Theorem A([21]) The composite functional equation f (p(z)) =

a0 (z) + a1 (z)f (z) b0 (z) + b1 (z)f (z)

where the coefficients ai , bj are of growth S(r, f ) such that a0 (z)b1 (z) − a1 (z)b0 (z) 6= 0 and p(z) is a polynomial of deg p(z) = k ≥ 2, does not have meromorphic solutions. A question is,whether or not the assertion of Theorem A remains valid, if we replace the equation a0 (z) + a1 (z)f (z) f (p(z)) = b0 (z) + b1 (z)f (z) with the following form X

a(i) (z)(f (z))i0 (∆c f (z))i1 · · · (∆nc f (z))in =

(i)

a0 (z) + a1 (z)f (p(z)) . b0 (z) + b1 (z)f (p(z))

2

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Man-Li Liu ET AL 1124-1141

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.7, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

In this section, the authors will pay attention to considering the properties of meromorphic solutions on three types of composite functional difference equations in complex domain, and extend the results obtained by H.Silvennoinen [21] to types of composite functional-difference equations (1)-(3) of the following forms, which are different from the complex differential equations or systems of complex difference equations. At this point we pause briefly to introduce the notation used in this paper. Let I be a finite set of multi-indexes i = (i0 , ..., in ),J be a finite set of multi-indexes j = (j0 , ..., jn ). Difference polynomials Ω1 (z, f ), Ω2 (z, f ) of a meromorphic function f (z) are defined as Ω1 (z, f ) =

X

a(i) (f (z))i0 (∆c f (z))i1 · · · (∆nc f (z))in ,

(i)∈I

Ω2 (z, f ) =

X

b(j) (f (z))j0 (∆c f (z))j1 · · · (∆nc f (z))jn ,

(j)∈J

where each {a(i) (z)}, {b(j) (z)} is a small meromorphic function with respect to f . We denote that n X

n X

l=0

l=0

(l + 1)il }, u2 = max{

u1 = max{

(l + 1)jl }.

First, we will investigate the existence of meromorphic solutions of a type of composite functional-difference equations of the form X

a(i) (z)(f (z))i0 (∆c f (z))i1 · · · (∆nc f (z))in =

(i)

a0 (z) + a1 (z)f (p(z)) , b0 (z) + b1 (z)f (p(z))

(1)

where the coefficients {ai (z)},{bj (z)}(i, j = 0, 1) and {a(i) (z)} are of growth S(r, f ) such that a0 (z)b1 (z) − a1 (z)b0 (z) 6≡ 0, p(z) = ck z k + · · · + c0 ,deg p(z) ≥ 2. For the composite functional-difference equations (1), the main theorem can be stated as follows. Theorem 2.1 Let u1 < k. The composite function-difference equation (1) does not have meromorphic solutions. Remark 1 The example 1 shows that Theorem 2.1 does not hold if at least ai (z), bj (z) and a(i) (z) are not of growth S(r, f ),there may exist a rational solution. 1 Example 1 Let p(z) = z 2 , c = 1.Then function f (z) = z−1 is a solution of the following equation 1 − z 2 f (p(z)) z(z − 1) = f ∆c f. (1 + (z − 1)f (p(z)) 2+z Second, we will study the properties of p(z) of composite functional-difference equations of the following l X Ω1 (z, f ) ai (z)f (p(z))i = , (2) Ω2 (z, f ) i=0 where p(z) is an entire function,{ai (z)}, {a(i) (z)}, {b(j) (z)} are small functions. We obtain the following result Theorem 2.2 Let f be a non-constant meromorphic solution of the composite functional-difference equations (2).Then p(z) is a polynomial. 3

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Man-Li Liu ET AL 1124-1141

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.7, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

Third, we shall consider the growth and characteristic estimate of meromorphic solutions of the following composite functional-difference equation X

a(i) (z)f i0 (∆c f )i1 · · · (∆nc f )in =

m X

ai (z)(f (p(z)))i ,

(3)

i=0

(i)∈I

where {ai (z)} are meromorphic functions, a(i) ≡ 6 0, am (z) 6= 0, p(z) is a polynomial of degree k ≥ 2. We get the main result below. Theorem 2.3 Let f (z) be a finite order transcendental meromorphic solution of (3),{a(i) (z)} be polynomials, T (r, ai ) < KT (rs , f ), i = 0, 1, 2, · · · , m, where K and s are positive constants, r is large enough. If s < k, then for given ε > 0, T (r, f ) = O((log r)α+ε ), where α=

log((m + 1)K + log ks

and α= where u1 = max{

n P

u1 ms )

, if 1 ≤ s < k,

log u1 +m(m+1)Ks m , if s < 1 < k, log k

(l + 1)il }.

l=0

Remark 2 The example 2 shows that the condition s < k in Theorem 2.3 is best possible. Example 2 Let p(z) = ck z k + · · · + c0 ,deg p(z) ≥ 2, i ai (z) = Cm

Then

m X

e2z , i = 0, 1, 2, ..., m. (1 + ep(z) )m

ai (z)f (p(z))i =

i=0

m X e2z C i f (p(z))i , (1 + ep(z) )m i=0 m

ez

f = is a transcendental meromorphic solution of the composite functional-difference equation of the form 1+z(ec −1)2 (∆c f )(∆2c f )2 − f (∆c f )2 − z(∆c f )2 (∆2c f ) (ec −1)3 m P −f (∆2c f )2 + f 2 = ai (z)(f (p(z)))i . i=0

In this case, f (z) satisfirs T (r, f (z)) =

+ (ec − 1)3 f 2 (∆c f )

r + O(1). π

However, by k ≥ 2,we have T (r, ai (z)) = (1 + o(1))

m|ck |rk , π

4

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Man-Li Liu ET AL 1124-1141

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.7, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

it shows that Theorem 2.3 does not hold if s = k. To prove Theorem 2.1-2.3, we need some lemmas as follows. Lemma 2.1([13]) Let f be a transcendental meromorphic function and p(z) = ak z k + ak−1 z k−1 + . . . + a1 (z) + a0 , ak 6= 0, k ≥ 1,be a polynomial of degree k.Given 0 < δ < |ak |,let λ = |ak | + δ, µ = |ak | − δ.Then, given ε > 0,for any a ∈ C ∪ {∞} and for r large enough,we have kn(µrk , N (µrk ,

1 1 1 ) ≤ n(r, ) ≤ kn(λrk , ), f −a f (p) − a f −a

1 1 1 ) + O(log r) ≤ N (r, ) ≤ N (λrk , ) + O(log r), f −a f (p) − a f −a (1 − ε)T (µrk , f ) ≤ T (r, f (p)) ≤ (1 + ε)T (λrk , f ).

Lemma 2.2([12]) Let ψ:[r0 , +∞) → (0, +∞) be positive and bounded in every finite interval. Suppose that ψ(µrm ) ≤ Aψ(r) + B, (r ≥ r0 ), where µ > 0, m > 1, A > 1 and B are real constants.Then ψ(r) = O((log r)α ), where α= p P

log A . log m ai (z)f i

i=0 q

Lemma 2.3([18]) Let R(z, f ) = P

be an irreducible rational function in bj (z)f j

j=0

f (z) with the meromorphic coefficients {ai (z)} and {bj (z)}.If f (z) is a meromorphic function,then T (r, R(z, f )) = max{p, q}T (r, f ) + O{

X

T (r, ai ) +

X

T (r, bj )}.

Lemma 2.4([3]) Let f be a non-constant meromorphic function and let g be a transcendental entire function.Then there exists an increasing sequence,rn → ∞,such that 1 r T (r, f (g(z))) ≥ T ((M ( , g)) 30 , f ) 4

holds for r = rn . Lemma 2.5([18]) Let g:(0, +∞) → R, h:(0, +∞) → R be monotone increasing functions such that g(r) ≤ h(r) outside of an exceptional set E of finite linear measure.Then,for any α > 1,there exists r0 such that g(r) ≤ h(αr) for all r > r0 . Lemma 2.6([17]) Let T : [0, +∞) → [0, +∞) be a non-decreasing continuous function,let δ ∈ (0, 1),and let s ∈ (0, ∞).If T is of finite order,i.e., log T (r) < ∞, r→∞ log r lim

5

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Man-Li Liu ET AL 1124-1141

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.7, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

then

T (r) ), rδ where r runs to infinity outside of a set of finite logarithmic measure. Lemma 2.7 Let f be a meromorphic function of finite order, T (r + s) = T (r) + o(

Ω1 (z, f ) =

X

a(i) (z)f i0 (∆c f )i1 · · · (∆nc f )in ,

(i)∈I

Ω2 (z, f ) =

X

b(j) (z)f j0 (∆c f )j1 · · · (∆nc f )jn .

(j)∈I

Then T (r, Ω1 (z, f )) ≤ u1 T (r, f ) + S1 (r, f ) +

X

T (r, a(i) ),

(i)∈I

and T (r,

X X Ω1 (z, f ) ) ≤ (u1 + u2 )T (r, f ) + S1 (r, f ) + T (r, a(i) ) + T (r, b(j) ), Ω2 (z, f ) (i)∈I (j)∈J

where u1 = max{

n P

(l + 1)il }, u2 = max{

l=0

S(r, f ) is of finite logarithmic measure

n P

(l + 1)jl }, the exceptional set E associated to

R l=0 dr E r

< +∞.

Proof It follows from ∆nc f (z) = ∆c (∆n−1 f (z)) = ∆n−1 f (z + c) − ∆n−1 f (z) c c c that ∆m c f (z) =

m X

i Cm (−1)m−i f (z + ci).

i=0

Similar to the proof of Lemma 4.2 in [16](pp. 181-182), we have m(r, Ω(z, f )) = λm(r, f ) + S(r, f ), where λ =

n P

il .

l=0

In order to estimate the poles of Ω(z, f ), we consider the term of Ω(i) (z, f ) = a(i) (z)f i0 (∆c f )i1 · · · (∆nc f )in . Noting that n(r, f (z + c)) ≤ n(r + C, f ) + S(r, f ) = n(r, f ) + S(r, f ), C = |lc|, it is easy to get that n(r, Ω(i) (z, f )) ≤

n X

il (l + 1)n(r, f (z + lc)) + n(r, a(i) (z)).

l=0

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Man-Li Liu ET AL 1124-1141

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.7, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

Hence, we get n(r, Ω(z, f )) ≤ max(

n X

il (l + 1))n(r, f (z)) + S(r, f ) +

l=0

X

n(r, a(i) (z)).

(i)

By the above equality, we get T (r, Ω1 (z, f )) ≤ uT (r, f ) + S(r, f ) +

X

T (r, a(i) (z)),

(i) n P

where u1 = max{

(l+1)il },r runs to infinity outside of a set of finite logarithmic measure.

l=0

Further,we have T (r, Ω2 (z, f )) ≤ u2 T (r, f ) + S(r, f ) +

X

T (r, b(j) ),

(j)∈J

where u2 = max{

n P

(l + 1)jl }.

l=0

Hence,we obtain Ω1 (z,f ) 1 ) ≤ T (r, Ω1 (z, f )) + T (r, Ω2 (z,f T (r, Ω )) P 2 (z,f ) P ≤ (u1 + u2 )T (r, f ) + S(r, f ) + T (r, a(i) ) + T (r, b(j) ). (i)∈I

Lemma 2.8([21]) Let P (z, f ) =

p P

(j)∈I

ai (z)f i be polynomial in f (z) with the mero-

i=0

morphic coefficients {ai (z)}.If f (z) is a meromorphic function,then T (r, P (z, f )) ≤ pT (r, f ) +

p X

T (r, ai ) + O(1),

i=0

T (r, P (z, f )) ≥ p(T (r, f ) −

p X

T (r, ai )) + O(1).

i=0

Lemma 2.9([21]) Let f be a meromorphic function.Then T (r, f ) is an increasing (r,f ) function of log r and convex function of log r, Tlog r is an incresing function of r. Proof of Theorem 2.1 First, we suppose that there is a transcendental meromorphic solution f (z) of composite functional-difference equation (1) . For a sufficiently small ε > 0, by Lemma 2.1, Lemma 2.3 and Lemma 2.7, we get (1 − ε)T (µrk , f ) ≤ T (r, f (p(z))) ≤ (u1 + ε)T (r, f ), where u1 = max{

n P

(l + 1)il }, µ = |ck |(1 − ε), outside a possible exceptional set of finite

l=0

logarithmic measure. Hence, for α > 1 and for r large enough (1 − ε)T (µrk , f ) ≤ (u1 + ε)T (αr, f ). 7

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Man-Li Liu ET AL 1124-1141

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Set t = αr. Then T(

u1 + ε µ k t , f) ≤ T (t, f ). k α (1 − ε)

By Lemma 2.2 we obtain T (t, f ) = O((log t)α1 ), where α1 =

u1 +ε log (1−ε)

log k

< 1,

there is a contradiction. Second, we suppose that f (z) is a rational solution of (1). Then the coefficients a(i) (z), a0 (z), a1 (z), b0 (z), b1 (z) must be constants. Set P (z) αp z p + αp−1 z p−1 + · · · + α0 , f (z) = = Q(z) βq z q + βq−1 z q−1 + · · · + β0 where αp 6= 0, βq 6= 0, deg w(z) = max{p, q} = l. 1 (z)f (p(z)) If p 6= q, we immediately have deg( ab00 (z)+a (z)+b1 (z)f (p(z)) ) = kl. If p = q, we have a0 (z)+a1 (z)f (p(z)) b0 (z)+b1 (z)f (p(z))

=

a0 +a1 f (p(z)) b0 +b1 f (p(z))

αp (p(z))p +αp−1 (p(z))p−1 +···+α0 βq (p(z))q +βq−1 (p(z))q−1 +···+β0 αp (p(z))p +αp−1 (p(z))p−1 +···+α0 b0 +b1 βq (p(z))q +βq−1 (p(z))q−1 +···+β0 (a0 βq +a1 αp )(p(z))p +(a0 βq−1 +a1 αp−1 )(p(z))p−1 +···+(a0 β0 +a1 α0 ) (b0 βq +b1 αp )(p(z))p +(b0 βq−1 +b1 αp−1 )(p(z))p−1 +···+(b0 β0 +b1 α0 )

a0 +a1

= =

.

It follows from the equation above that a0 βq + a1 αp = 0 and b0 βq + b1 αp = 0 can not (z)+a1 (z)w(p(z)) hold at the same time. Otherwise ab00 (z)+b = c, c is a constant. 1 (z)w(p(z)) Hence, we get 1 (z)f (p(z)) kl = deg( ab00 (z)+a (z)+b (z)f (p(z)) ) 1 P = deg( (i) a(i) (z)(f (z))i0 (∆c f (z))i1 · · · (∆nc f (z))in ) ≤ max{i0 + 2i1 + · · · + (n + 1)in }l = u1 l.

So, u1 ≥ k, there is also a contradiction. Thus, f (z) is not a rational solution of (1). Combined with the first and second steps above, the assertion follows. Proof of Theorem 2.2

Suppose that p(z) is transcendental entire function, we have lim inf r→∞

log M (r, p(z)) = ∞. log r

Hence, for any given K > 30 and for r large enough M (r, p) > rK . There exists an increasing sequence rn → ∞, as in Lemma 2.4, for any n such that M(

rn rn , p) > ( )K . 4 4 8

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Applying Lemma 2.3 and Lemma 2.7 to equation (2), we have lT (r, f (p(z))) ≤ (u1 + u2 )T (r, f ) + S(r, f ), outside a possible exceptional set of finite linear measure. According to Lemma 2.5, for ∀α > 1, r ≥ rα , we obtain (u1 + u2 )(1 + o(1)) T (αr, f ). l

T (r, f (p(z))) ≤

(4)

It follows from Lemma 2.4 that T (rn , f (p(z))) ≥ T (( Note that

T (r,f ) log r

rn K ) 30 , f ). 4

(5)

is an increasing function of r. As (

rn K ) 30 > αrn , 4

for sufficiently large n, we have T ((

K/30(log rn − log 4) K rn K ) 30 , f ) > T (αrn , f ) > T (αrn , f ), 4 log rn + log α 40

(6)

as n → ∞. By (4),(5) and (6), we get (u1 + u2 )(1 + o(1)) K rn K T (αrn , f ) ≥ T (( ) 30 , f ) > T (αrn , f ), l 4 40

(7)

as n → ∞. Because K can be arbitrarily large, this is a contradiction in (7). This shows that p(z) is a polynomial. Proof of Theorem 2.3

By the equation (3), Lemma 2.7 and Lemma 2.8, we have

mT (r, f (p(z))) − m

m X

T (r, ai (z)) ≤ (u1 + ε)T (r, f ),

i=0

i.e., mT (r, f (p(z))) ≤ (u1 + ε)T (r, f ) + m

m X

T (r, ai (z)).

(8)

i=0

Combining (8) and T (r, ai (z)) < KT (rs , f ), i = 0, 1, 2, · · · , m, we obtain T (r, f (p(z))) ≤

u1 + ε T (r, f ) + (m + 1)KT (rs , f ), m

(9)

where K is a positive constant.

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(r,f ) Case (1): If s ≥ 1, by Lemma 2.9, we have Tlog r is increasing functions of r, we can obtain for any positive constant C and any t ≥ 1

log C + t log r T (Crt , f ) ≥ > (1 − ε)t. T (r, f ) log r Hence, for r sufficiently large, 1 T (r, f ) < T (Crt , f ). (1 − ε)t Let s = t, C = 1. Then 1 T (rs , f ). T (r, f ) < (1 − ε)s It follows from (9)and (10) that

(10)

u1 +ε T (r, f (p)) ≤ (m + 1)KT (rs , f ) + (1−ε)ms T (rs , f ) u1 + ε1 )T (rs , f ). ≤ ((m + 1)K + ms

By Lemma 2.1 (1 − ε)T (µrk , f ) ≤ ((m + 1)K +

u1 + ε1 )T (rs , f ). ms

From the above inequality we further get k

(1 − ε)T (µr s , f ) ≤ ((m + 1)K +

u1 + ε2 )T (r, f ). ms

(11)

Since k > s, then by (11) and Lemma 2.2, we obtain T (r, f (z)) = O((log r)α1 +ε ), where α1 =

log((m + 1)K +

u1 ms )

log ks

Case (2):If s < 1, by Lemma 2.9, since

T (r,f ) log r

.

is increasing function of r, we obtain

T (r, f ) T (rs , f ) ≥ , log r log rs i.e.

T (r, f ) 1 ≥ . s T (r , f ) s

(12)

From (9) and (12) we get T (r, f (p(z))) ≤ (

u1 + m(m + 1)Ks + ε3 )T (r, f ). m

According to Lemma 2.1, we obtain u1 + m(m + 1)Ks + ε4 )T (r, f ). T (µrk , f ) ≤ ( m We obtain from Lemma 2.2 T (r, f (z)) = O((log r)α2 +ε ), where

log u1 +m(m+1)Ks m . log k Combining case (1) and case (2), we get the proof of Theorem 2.3. α2 =

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3. Growth of meromorphic solutions Since the 1970’s, R.Goldstein[10-13], W.Bergweiler[2-4], J.Heittokangas[16] et al had investigated the existence and growth of meromorphic solutions on composite functional equations in the whole complex plane and a number of important results were obtained. Particularly, J.Rieppo [20] discussed the growth on meromorphic solutions of many types of functional equations, he also obtained some interesting results, for example, the following theorem B is one of his some results. For the following functional equations Q(z, f (az + b)) = R(z, f (z)),

(∗)

where Q(z, f ), R(z, f ) are rational functions in f with small meromorphic coefficients R relative to f such that 0 < q = degQ 6 1. f ≤ d = degf and a, b ∈ C, a 6= 0 and |a| = He obtained Theorem B([20]) Suppose that f is a transcendental meromorphic solution of the equation (∗). Then log d − log q µ(f ) = ρ(f ) = . log |a| It is known that when treating the meromorphic solutions of difference equations, the basic task is to estimate their growth order, while in the case of complex composite functional difference equations, considering the growth order of them is also an interesting task. Hence, this section is devoted to investigating the growth order of meromorphic solutions on two types of composite functional-difference equations (3), (13) and systems of difference equations (14) in complex domain. As regards the growth order of meromorphic solutions of complex composite functionaldifference equations (3), we obtain Theorem 3.1. Theorem 3.1 Let {ai (z)}, {a(i) (z)} be of growth order of S(r, f ), u1 ≥ km. Then the lower order and the order of meromorphic solution f of the equation (3) satisfy ρ(f ) = µ(f ) = 0. In the following, we will also investigate the growth of meromorphic solutions about a type of composite functional-difference equations of the form l P

di f (a1i z + b1i )i

i=0 t P

= ej f (a2j z + b2j )j

Ω1 (z, f ) , Ω2 (z, f )

(13)

j=0

where {a1i }, {a2i }, {b1j }, {b2j }, {di }, {ej } are constants ,{a(i) (z)}, {b(j) (z)} are small functions and a(i) (z) 6≡ 0, b(j) (z) 6≡ 0. For complex composite functional-difference equations (13), we obtain the following main result. Theorem 3.2 Suppose that f is a transcendental meromorphic solution of composite functional-difference equations (13), a1i , a2j , b1i , b2j ∈ C, |a1i | > 1, |a2j | > 1, and the coefficients a(i) (z) are of growth S(r, f ).

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(i). If l > t, then ρ(f ) ≤

2 log u1 +u l ; log |a1l |

ρ(f ) ≤

2 log u1 +u t ; log |a2t |

ρ(f ) ≤

2 log u1 +u l , log |a|

(ii). If l < t, then

(iii). If l = t, then

where |a| = max{|a1l |, |a2t |}. Remark 3 The example 3 shows that the upper bound in Theorem 3.2 can be reached. Example 3 f (z) = ez is a meromorphic solution of the following equation (ec − 1)2 f (6z + c) f ∆2c f = . ec f (5z + c) ∆c f + f u +u

1 2 log max{l,t} log |a12 |

log

6

log 6 = log 61 = log We see that u1 = 4, u2 = 2, ρ(f ) = 1 = 6. By using the Nevanlinna value distribution theory of meromorphic functions, difference equation theory, a large number of papers also have considered the properties of meromorphic solutions of some types of system of functional equations, and obtained some results([7-9]). Now, we consider the problem of the growth order on a class of system of composite functional equations as follows

 m P1   a1µ (z)f2 (z)µ  l  P  µ=0 i   di f1 (c1i z + d1i ) = P , n1    a2ν (z)f2 (z)ν  i=0 ν=0

m P2   b1s (z)f1 (z)s  t  P  s=0 j   ej f2 (c2j z + d2j ) = P , n2    b2k (z)f1 (z)k  j=0

(14)

k=0

where {c1i }, {c2j }, {d1i }, {d2j }, di , ej are constants,{a1µ (z)}, {a2ν (z)}, {b1s (z)}, {b2k (z)} are small functions,|c1l | > 1, |c2t | > 1. The growth order of meromorphic solutions (f1 , f2 ) of (14) is defined by ρ(f1 , f2 ) = max{ρ(f1 ), ρ(f2 )}, ρ(fk ) = lim sup r→∞

log+ T (r, fk ) , k = 1, 2. log r

The lower order of meromorphic function fi , i = 1, 2 are defined by µ(fk ) = lim inf r→∞

log+ T (r, fk ) , k = 1, 2. log r

As regards the complex composite functional-difference equation (14), we obtain Theorem 3.3 and Theorem 3.4 as follows. 12

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Theorem 3.3 Suppose that f is a transcendental meromorphic solution of the system (14), cij , dij ∈ C, |c1l | > 1, |c2t | > 1, and the coefficients aij (z) and bij (z) are of growth S(r, fi ).Then log max{m1 ,n1 }ltmax{m2 ,n2 } ρ(f1 , f2 ) ≤ . log |c1l ||c2t | Example 4 Let b ∈ C be a constant such that b 6= mπ 2 ,where m ∈ Z. We see that (f1 (z), f2 (z)) = (tan z, − tan z) is a meromorphic solution of the following system of composite functional equations of the form   f1 (2z + b) =  f (2z + b) = 2

−2f2 (z)−C(1−f22 ) , 1−f22 −2Cf2 2f1 (z)−C(1−f12 ) , 1−f12 +2Cf1

where C = − tan b 6= 0, ∞. In this case, |a1l ||a2t | = 4, max{m1 , n1 } max{m2 , n2 }} = 4, lt = 1, thus, ρ(f1 , f2 ) = 1 =

log max{m1 ,n1 }ltmax{m2 ,n2 } log 4 = . log |a1l ||a2t | log 4

It shows that the upper bound in Theorem 3.3 can be reached. Theorem 3.4 Let (f1 , f2 ) be a transcendental meromorphic solution of the system (14), and µ(f1 ), µ(f2 ) be the lower order of f1 , f2 , respectively. Then µ(f1 ) + µ(f2 ) ≥

log max{m1 ,n1 }ltmax{m2 ,n2 } , log |c1l ||c2t |

where {a1µ (z)}, {a2ν (z)}, {b1s (z)}, {b2k (z)} are small functions are small functions. In order to prove Theorems 3.1-3.4, we need the following Lemmas. Lemma 3.1([14]) Let Φ : (1, ∞) → (0, ∞) be a monotone increasing function,and let f be a nonconstant meromorphic function.If for some real constant α ∈ (0, 1),there exist real constants K1 > 0 and K2 ≥ 1 such that T (r, f ) ≤ K1 Φ(αr) + K2 T (αr, f ) + S(αr, f ), then ρ(f ) ≤

log K2 log Φ(r) + lim sup . − log α log r r→∞

Lemma 3.2([3]) Suppose that a meromorphic function f has finite lower order λ.Then for every constant c > 1 and a given ε there exists a sequence rn = rn (c, ε) → ∞ such that T (crn , f ) ≤ cλ+ε T (rn , f ).

Proof of Theorem 3.1 For a sufficiently small ε > 0, by Lemma 2.1 and Lemma 2.3, we get m(1 − ε)T (µrk , f ) ≤ mT (r, f (p(z))) ≤ (u1 + ε)T (r, f ),

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where µ = |ck |(1 − ε), u1 = max{

n P

(l + 1)il }, outside a possible exceptional set of finite

l=0

logarithmic measure of r. Hence, for α > 1 and for r large enough m(1 − ε)T (µrk , f ) ≤ (u1 + ε)T (αr, f ). Set t = αr. Then T(

u1 + ε µ k t , f) ≤ T (t, f ). k α m(1 − ε)

By Lemma 2.2 we obtain T (t, w) = O((log t)α1 ), where α1 =

log um1 + ε1 . log k

From the above equation, we can obtain that ρ(f ) = lim sup r→∞

µ(f ) = lim inf r→∞

log+ T (r, f ) = 0, log r log+ T (r, f ) = 0. log r

Thus, we have completed the proof of Theorem 3.1. Proof of Theorem 3.2 Applying Lemma 2.3 and Lemma 2.7 to equation (13),we get l P

max{l, t}T (r, f (ask z + bsk )) = T (r,

di f (a1i z + b1i )i

i=0 t P

) ≤ (u1 + u2 )T (r, f ) + S(r, f ), ej f (a2j z + b2j

)j

j=0

where s = 1 or 2, k = max{l, t}. Applying Lemma 2.1 to equation (13), we get (1 − ε) max{l, t}T (µr, f ) ≤ (u1 + u2 )T (r, f ) + S(r, f ), that is T (µr, f ) ≤

u1 + u2 T (r, f ) + S(r, f ), (1 − ε) max{l, t}

where µ = |a| − δ > 1, |a| = max{|a1k |, |a2k |}, δ > 0. Denoting α = µ1 , we have 0 < α < 1, and we deduce that T (r, f ) ≤

u1 + u2 T (αr, f ) + S(αr, f ). (1 − ε) max{l, t}

By Lemma 3.1, we obtain ρ(f ) ≤

u1 +u2 log (1−ε) max{l,t}

− log α

.

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Let ε → 0 and δ → 0. Then ρ(f ) ≤ Proof of Theorem 3.3

u1 +u2 log max{l,t}

log |a|

.

Applying Lemma 2.3 to system (14), we get

lT (r, f1 (c1l z + d1l )) = max{m1 , n1 }T (r, f2 ) + S(r, f2 ).

(15)

tT (r, f2 (c2t z + d2t )) = max{m2 , n2 }T (r, f1 ) + S(r, f1 ).

(16)

Applying Lemma 2.1 to equations (15) and (16), we get (1 − ε)lT (µ1 r, f1 ) ≤ max{m1 , n1 }T (r, f2 ) + S(r, f2 ), (1 − ε)tT (µ2 r, f2 ) ≤ max{m2 , n2 }T (r, f1 ) + S(r, f1 ), that is T (µ1 r, f1 ) ≤

max{m1 , n1 } T (r, f2 ) + S(r, f2 ), (1 − ε)l

T (µ2 r, f2 ) ≤

max{m2 , n2 } T (r, f1 ) + S(r, f1 ), (1 − ε)t

where µ1 = |c1l | − δ1 > 1, δ1 > 0,µ2 = |c2t | − δ2 > 1, δ2 > 0. Denoting α1 = µ11 , α2 = µ12 , we have 0 < α1 < 1, 0 < α2 < 1, and we deduce that T (r, f1 ) ≤

max{m1 , n1 } T (α1 r, f2 ) + S(α1 r, f2 ), (1 − ε)l

(17)

T (r, f2 ) ≤

max{m2 , n2 } T (α2 r, f1 ) + S(α2 r, f1 ), (1 − ε)t

(18)

outside a possible exceptional set of finite logarithmic measure of r. Combining (17) and (18), it yields T (r, f1 ) ≤

(1 + o(1)) max{m1 , n1 } max{m2 , n2 } T (α1 α2 r, f1 ) + S(α1 α2 r, f1 ), (1 − ε)2 lt

outside a possible exceptional set of finite logarithmic measure of r. By Lemma 3.1, we obtain ρ(f1 ) ≤

1 } max{m2 ,n2 } log max{m1 ,n (1−ε)2 lt

− log α1 α2

.

By a similar reasoning as to above, we also can get ρ(f2 ) ≤

1 } max{m2 ,n2 } log max{m1 ,n (1−ε)2 lt

− log α1 α2

.

Let ε → 0 and δi → 0, i = 1, 2. Then Theorem 3.3 is proved.

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Proof of Theorem 3.4 We assume conversely that f1 , f2 are transcendental meromorphic functions. By Lemma 2.3 and T (r, f (z + c)) ≤ (1 + o(1))T (r + |c|, f ) + M ([17]), where M is a constant, we have  1l  ))) + S(r, f2 ) max{m1 , n1 }T (r, f2 ) ≤ lT (r, f1 (c1l (z + dc1l     1l ≤ (1 + o(1))lT (|c1l |r + | dc1l |, f1 ) + S(r, f2 ), d2t  max{m2 , n2 }T (r, f1 ) ≤ tT (r, f2 (c2t (z + c2t ))) + S(r, f1 )     ≤ (1 + o(1))tT (|c |r + | d2t |, f ) + S(r, f ). 2t

c2t

2

(19)

1

There are two constants c1 = |c1l | + ε1 , c2 = |c2t | + ε2 , εi > 0, i = 1, 2, such that T (|c1l |r + |

d2t d1l |, f1 ) ≤ T (c1 r, f1 ), T (|c2t |r + | |, f2 ) ≤ T (c2 r, f2 ). c1l c2t

(20)

When r is large enough, we can obtain from (19) and (20) (

max{m1 , n1 }T (r, f2 ) ≤ (1 + o(1))lT (c1 r, f1 ) + S(r, f2 ), max{m2 , n2 }T (r, f1 ) ≤ (1 + o(1))tT (c2 r, f2 ) + S(r, f1 ),

outside a possible exceptional set of finite linear measure of r. According to Lemma 2.5, for given σ1 > 1, σ2 > 1, (

max{m1 , n1 }T (r, f2 ) ≤ (1 + o(1))lT (σ1 c1 r, f1 ) + S(r, f2 ), max{m2 , n2 }T (r, f1 ) ≤ (1 + o(1))tT (σ2 c2 r, f2 ) + S(r, f1 ).

(21)

Let µ(f1 ), µ(f2 ) be the finite lower order in f1 , f2 ,respectively. By Lemma 3.2, for any given εi > 0, i = 1, 2, there exists a sequence rn → ∞ such that for rn > r0 µ(f1 )+ε1

T (c1 rn , f1 ) ≤ c1

µ(f2 )+ε2

T (rn , f1 ), T (c2 rn , f2 ) ≤ c2

T (rn , f2 ).

By (21) (

max{m1 , n1 }T (rn , f2 ) ≤ (1 + o(1))l(σ1 c1 )µ(f1 )+ε1 T (rn , f1 ) + S(rn , f2 ), max{m2 , n2 }T (rn , f1 ) ≤ (1 + o(1))t(σ2 c2 )µ(f2 )+ε2 T (rn , f2 ) + S(rn , f1 ).

(22)

From (22), we get   max{m1 , n1 } ≤ (1 + o(1))l(σ1 c1 )µ(f1 )+ε1 T (rn ,f1 ) + T (rn ,f2 )  max{m2 , n2 } ≤ (1 + o(1))t(σ2 c2 )µ(f2 )+ε2 T (rn ,f2 ) + T (rn ,f1 )

Taking lower limit as n → ∞, and lim inf n→∞

S(rn ,fi ) T (rn ,fi )

S(rn ,f2 ) T (rn ,f2 ) , S(rn ,f1 ) T (rn ,f1 ) .

(23)

= 0, i = 1, 2. Then (23) becomes

max{m1 , n1 } max{m2 , n2 } ≤ lt(σ1 c1 )µ(f1 )+ε3 (σ2 c2 )µ(f2 )+ε3 , where ε3 = max{ε, ε1 , ε2 }, ε3 → 0, σ1 → 1, σ2 → 1. Hence µ(f1 ) + µ(f2 ) ≥

log max{m1 ,n1 }ltmax{m2 ,n2 } . log |c1l ||c2t |

Thus, we have completed the proof of Theorem 3.4. 16

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Reference [1] Ablowitz,M.J. Halburd R,Herbst B,On the extension of the Painleve property to difference equations.Nonlinearity, 2000, 13:889-905 [2] Bergweiler,W. Untersuchungen des Wachstums Zusammengesetzter meromorpher Funktionen, Dissertation, Aachen, 1986. [3] Bergweiler,W. Ishizaki,K., Yanagihara,N. Growth of meromorphic solutions of some functional equations I, Aequations Math., 2002, 63(1-2):140-151 [4] Bergweiler,W. Ishizaki,K., Yanagihara,N. Meromorphic solutions of some functional equations, Methods Appl.Anal., 1998, 5(3):248-258 [5] Chen Zongxuan,Growth and zeros of meromorphic solution of some linear difference equations.Journal of Mathematical Analysis and Applications,2011,373:235–241. [6] Chen Z.X.,K.H.Shon,On zeros and fixed points of differences of meromorphic functions, J. Math. Anal. Appl. 2008,344:373-383. [7] Gao Lingyun. On meromorphic solutions of a type of system of composite functional equations, Acta Mathematica Scientia, 2012, 32B(2):800-806 [8] Gao Lingyun. On solutions of a type of system of complex differential-difference equations, Chinese Journal of Contemporary Mathematics, 2017, 381: 23-30 [9] Gao Lingyun. On admissible solutions of two types of systems of differential equations in the complex plane. Acta Mathematica Sinica,2000,43(1):149-156 [10] Goldstein,R. On certain compositions of functions of a complex variable, Aequations Math, 1970, 4:103-126 [11] Goldstein,R. On meromorphic solutions of a functional equations, Aequations Math, 1972, 8:82-94 [12] Goldstein,R. On meromorphic solutions of certain functional equations, Aequations Math, 1978, 18:112-157 [13] Goldstein,R. Some results on factorisation of meromorphic functions, J.London Math.Soc., 1971, 4(2):357-364 [14] Gundersen,R.Heittokangas,J.,Laine,I.,Rieppo,J.,D.Yang,Meromorphic solutions of generalized Schroder equations,Aequationes Math.,2002,63(1-2):110-135 [15] He Yuzan, Xiao Xiuzhi. Algebroid function and ordinary differential equations. Beijing:Science Press, 1988 [16] Heittokangas,J. Laine,I., Rieppo,J., D.Yang. Meromorphic solutions of some linear functional equations, Aequations Math., 2000, 60:148-166 [17] Korhonen,R.A new Clunie type theorem for difference polynomials,J. Difference Equ.Appl.,2011,17(3):387-400 17

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[18] Laine,I. Nevanlinna theory and complex differential equations. Berlin:Walter de Gruyter, 1993 [19] Mokhonko A. Z and Mokhonko V. D. Estimates for the Nevanlinna characteristics of some classes of meromorphic functions and their aplications to differential equations, Siberian Math. J., 1974,15, 921-934. [20] Rieppo J. On a class of complex functional equations, Ann.Acad.Sci.Fenn., 2007,32:151-170 [21] Silvennoinen H. Meromorphic solutions of some composite functional equations. Ann Acad Sci Fenn, Helsinki: Mathematica Dissertations, 2003, 133 [22] Yi Hongxun, Yang C C. Theory of the uniqueness of meromorphic functions(in Chinese). Beijing:Science Press, 1995

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Locally and globally small Riemann sums and Henstock-Stieltjes integral for n-dimensional fuzzy-number-valued functions a

Muawya Elsheikh Hamida,b ∗ School of Mathematical Science, Yangzhou University, Yangzhou 225002, China b Faculty of Engineering, University of Khartoum, Khartoum, Sudan

Abstract: In this paper, we study locally and globally small Riemann sums with respect to α for n-dimensional fuzzynumber-valued functions. And we prove that a fuzzy-number-valued functions in n-dimensional is Henstock-Stieltjes (HS) integrable on [a, b] if and only if it has (LSRS) with respect to α on [a, b]. Also we shall prove that a fuzzynumber-valued functions in n-dimensional is Henstock-Stieltjes (HS) integrable on [a, b] if and only if it has (GSRS) with respect to α on [a, b]. Keywords: Fuzzy-number-valued functions in E n ; Henstock-Stieltjes integral (HS); locally small Riemann sums (LSRS); globally small Riemann sums (GSRS).

1

Introduction

Since the concept of fuzzy sets was firstly introduced by Zadeh in 1965 [13], it has been studied extensively from many different aspects of the theory and applications, such as fuzzy topology, fuzzy analysis, fuzzy decision making and fuzzy logic, information science and so on. The locally and globally small Riemann sums have been introduced by many authors from different points of views including [3, 4, 5, 7, 8, 10, 11]. In 1986, Schurle characterized the Lebesgue integral in (LSRS) (locally small Riemann sums) property [10]. The (LSRS) property has been used to characterized the Perron (P ) integral on [a, b] [11]. By considering the equivalency between the (P ) integral and the Henstock-Kurzweil (HK) integral, the (LSRS) property has been used to characterized the (HK) integral on [a, b] [8]. In 2015, Indrati [7] introduced a countably Lipschitz condition of a function which is simpler than the ACG∗ , and proved that the (HK) integrable function or it, s primitive could be characterized in countably Lipschitz condition. Also, by considering the characterization of the (HK) integral in the (GSRS) property, it showed that the relationship between (GSRS) property and countably Lipschitz condition of an (HK) integrable function on [a, b]. In 2018, Hamid et al. [5] introduced locally and globally small Riemann sums for fuzzy-number-valued functions and established two main theorems: (i) A fuzzy-number-valued functions f˜(x) is (HS) integrable on [a, b] iff f˜(x) has (LSRS). (ii) A fuzzy-number-valued functions f˜(x) is (HS) integrable on [a, b] iff f˜(x) has (GSRS). In this paper, the concept of locally small Riemann sums for n-dimensional fuzzy-number-valued functions with respect to α is introduced and discussed. Furthermore, we provide a characterizations of globally small Riemann sums in n-dimensional fuzzy-number-valued functions with respect to α. The rest of this paper is organized as follows. To make our analysis possible, in Section 2 we shall review the relevant concepts and properties of fuzzy-number-valued functions in E n and the definition of Henstock-Stieltjes (HS) integral for fuzzy-number-valued functions in E n . In Section 3, we introduce the support function characterizations of locally small Riemann sums and (HS) integral for fuzzy-number-valued functions in E n . In section 4, we shall discuss the support function characterizations of globally small Riemann sums and (HS) integral for fuzzy-number-valued functions in E n . The last section provides the Conclusions.

2

Preliminaries

 In this paper the close interval [a, b] denotes a compact interval on R. The set of intervals-point ([a1 , b1 ], ξ1 ), ([a2 , b2 ], ξ2 ), · · · , ([ak , bk ], ξk ) is called a division of [a, b] that is ξ1 , ξ2 , · · · , ξk ∈ [a, b], intervals [a1 , b1 ], [a2 , b2 ], · · · , [ak , bk ] k  S are non-intersect and [ai , bi ] = [a, b]. Marking the division of [a, b] as P = ([a1 , b1 ], ξ1 ), ([a2 , b2 ], ξ2 ), · · · , ([ak , bk ], ξk ) , i=1  shortening as P = [u, v]; ξ [9]. ∗ Corresponding author. Tel.: +8613218977118. E-mail address: [email protected], [email protected] (M.E. Hamid). 1142 HAMID 1142-1149

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M.E. Hamid: Locally and globally small Riemann sums and Henstock-Stieltjes integral for n-dimensional...

Definition 2.1 [6, 8] Let δ : [a, b] → R+ be a positive real-valued function. P = {[xi−1 , xi ]; ξi } is said to be a δ-fine division, if the following conditions are satisfied: (1) a = x0 < x1 < x2 < ... < xn = b; (2) ξi ∈ [xi−1 , xi ] ⊂ (ξi − δ(ξi ), ξi + δ(ξi ))(i = 1, 2, · · · , n). For brevity, we write P = {[u, v]; ξ}, where [u, v] denotes a typical interval in P and ξ is the associated point of [u, v]. Definition 2.2 [12] E n is said to be a fuzzy number space if E n = {u : Rn → [0, 1] : u satisfies (1)-(4) below}: (1) u is normal, i.e., there exists a x0 ∈ Rn such that u(x0 ) = 1; (2) u is a convex fuzzy set, i.e., u(rx + (1 − r)y) > min(u(x), u(y)), x, y ∈ Rn , r ∈ [0, 1]; (3) u is upper semi-continuous; S (4) [u]0 = {x ∈ Rn : u(x) > 0} is compact, for 0 < r ≤ 1, denote [u]r = {x : x ∈ Rn and u(x) > r}, [u]0 = r∈(0,1] [u]r . From (1)-(4), it follows that for any u ∈ E n and r ∈ [0, 1] the r−level set [u]r is a compact convex set. For any u, v ∈ E n D(u, v) = sup d([u]r , [v]r ), (1) r∈[0,1]

where d is Hausdorff metric. It is well known that (E n , d) is an metric space [12]. The norm of fuzzy number u ∈ E n is defined by kuk = D(u, ˜ 0) = sup |α|, (2) α∈[u]0

where the k · k is norm on E , ˜ 0 is fuzzy number on E and ˜ 0 = χ{0} . n

n

Definition 2.3 [12] For A ∈ Pk (Rn ), x ∈ S n−1 , define the support function of A as σ(x, A) = sup hy, xi, where S n−1 is y∈A

the unit sphere of Rn , i.e., S n−1 = {x ∈ Rn : kxk = 1}, h·, ·i is the inner product in Rn . Definition 2.4 [2] Let α : [a, b] → R be an increasing function. A fuzzy-number-valued function f˜ : [a, b] → E n is said to be fuzzy Henstock-Stieltjes (F HS) integrable with respect to α on [a, b], if there exists A˜ ∈ E n , for every ε > 0, there is a function δ(ξ) > 0, such that for any δ-fine division P = {[u, v], ξ} of [a, b], we have X
D f˜(ξ)[α(v) − α(u)], A˜ < ε. (3) (P )

We write (F HS)

Rb

e f˜(x)dα = A.

a

Lemma 2.1 [12] If u, v ∈ E n , k ∈ R, for any r ∈ [0, 1], we have [u + v]r = [u]r + [v]r , [ku]r = k[u]r . Lemma 2.2 [12] Suppose u ∈ E n , then (1) u∗ (r, x + y) ≤ u∗ (r, x) + u∗ (r, y), (2) if u, v ∈ E n , r ∈ [0, 1], then d([u]r , [v]r ) =

(4)

sup |u∗ (r, x) − v ∗ (r, x)|,

(5)

x∈S n−1

(3) (u + v)∗ (r, x) = u∗ (r, x) + v ∗ (r, x), (4) (ku)∗ (r, x) = ku∗ (r, x), k ≥ 0. Lemma 2.3 [1, 12] Given u, v ∈ E n the distance D : E n × E n → [0, +∞) between u and v is defined by the equation D(u, v) = sup d([u]r , [v]r ), then r∈[0,1]

(1) (2) (3) (4) (5) (6) Where

(E n , D) is a complete metric space, D(u + w, v + w) = D(u, v), D(u + v, w + e) 6 D(u, w) + D(v, e), D(ku, kv) = |k|D(u, v), k ∈ R, D(u + v, ˜ 0) 6 D(u, ˜ 0) + D(v, ˜ 0), D(u + v, w) 6 D(u, w) + D(v, ˜ 0). u, v, w, e, e 0 ∈ En, e 0 = X({0}) .

Lemma 2.4 [2] Let α : [a, b] → R be an increasing function. A fuzzy-number-valued function Fe : [a, b] → E n is (F HS) integrable with respect to α on [a, b] if and only if F ∗ (t)(r, x) is (RHS) integrable with respect to α on [a, b] uniformly for any r ∈ [0, 1] and x ∈ S n−1 , we have Zb

 (F HS)

Zb

∗ Fe(t)dα

(r, x) = (RHS)

(6)

a

a

Uniformly for any r ∈ [0, 1].

F ∗ (t)(r, x)dα.

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M.E. Hamid: Locally and globally small Riemann sums and Henstock-Stieltjes integral for n-dimensional...

3

Support function characterizations of locally small Riemann sums and (HS) integral for fuzzy-number-valued functions in E n

In this section, we shall define locally small Riemann sums or in short (LSRS) with respect to α on [a, b] by using support function f ∗ (ξ)(r, x) and show that it is the necessary and sufficient condition for f˜ to be (HS) integrable with respect to α on [a, b]. Definition 3.1 Let α : [a, b] → R be an increasing function. A fuzzy-number-valued function f˜ : [a, b] → E n is said to be have locally small Riemann sums or (LSRS) with respect to α on [a, b] if for every ε > 0 there is a δ(ξ) > 0 such that for every t ∈ [a, b], we have

X

˜(ξ)[α(v) − α(u)]

f (7)

n < ε, E

whenever P = {[u, v]; ξ} is a δ-fine division of an interval C ⊂ (t − δ(t), t + δ(t)), t ∈ C and Σ sums over P . (Where C = [y, z]). The following Theorem 3.1 shows that f˜ has (LSRS) with respect to α on [a, b] is equal to the type of it, s support functions. Theorem 3.1 Let α : [a, b] → R be an increasing function and let f˜ : [a, b] → E n be a fuzzy-number-valued function, the support-function-wise f ∗ (ξ)(r, x) of f˜ has locally small Riemann sums or (LSRS) with respect to α on [a, b] if and only if for every ε > 0, there is a δ(ξ) > 0 such that for every t ∈ [a, b], we have X ∗ < ε, f (ξ)(r, x)[α(v) − α(u)] (8) uniformly for any r ∈ [0, 1] and x ∈ S n−1 , whenever P = {[u, v]; ξ} is a δ-fine division of an interval C ⊂ (t−δ(t), t+δ(t)), t ∈ C and Σ sums over P. Proof Let ˜ 0 ∈ E n denote the (F HS) integral of f˜ with respect to α on [a, b]. Given ε > 0 there is a δ(ξ) > 0 such that for any δ-fine division P = {[u, v]; ξ} of [a, b], we have X  D f˜(ξ)[α(v) − α(u)], ˜ 0 < ε. (9) That is sup d

 X 

r ˜ r f˜(ξ)[α(v) − α(u)] , [0]

 < ε.

(10)

r∈[0,1]

By Lemma 2.2 we have sup

X
∗ ˜ sup f (ξ)[α(v) − α(u)] (r, x) − σ(x, 0) < ε. n−1

(11)

r∈[0,1] x∈S

Furthermore, by σ(x, A) = sup hy, xi, we have y∈A

sup

X ∗ sup f (ξ)(r, x)[α(v) − α(u)] − σ(x, 0) < ε. n−1

(12)

r∈[0,1] x∈S

Hence, for any r ∈ [0, 1], x ∈ S n−1 and for any δ-fine division P we have X ∗ f (ξ)(r, x)[α(v) − α(u)] < ε.

(13)

Where σ(x, 0) = 0. This completes the proof.



Lemma 3.1 (Henstock Lemma). Let α : [a, b] → R be an increasing function and let f˜ : [a, b] → E be a fuzzy-numbervalued function and (HS) integrable to A˜ with respect to α on [a, b]. Then, the support-function-wise f ∗ (ξ)(r, x) of f˜ on [a, b] is (HS) integrable to A∗ (r, x) with respect to α on [a, b] uniformly for any r ∈ [0, 1], x ∈ S n−1 and A˜ ∈ E n , i.e., for every ε > 0 there is a positive function δ(ξ) > 0, for δ-fine division P = {[u, v]; ξ} of [a, b] and for any x ∈ S n−1 , we have X ∗ ∗ f (ξ)(r, x)[α(v) − α(u)] − A (r, x) < ε. (14) P P Furthermore, for any sum of parts from we have n

1

X ∗ ∗ < ε. f (ξ)(r, x)[α(v) − α(u)] − A (r, x) 1

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M.E. Hamid: Locally and globally small Riemann sums and Henstock-Stieltjes integral for n-dimensional...

Proof Let A˜ ∈ E n denote the (F HS) integral of f˜ with respect to α on [a, b]. Given ε > 0 there is a δ(ξ) > 0 such that for any δ-fine division P = {[u, v]; ξ} of [a, b], we have X  D f˜(ξ)[α(v) − α(u)], A˜ < ε. (16) That is sup d

 X 

r ˜ r f˜(ξ)[α(v) − α(u)] , [A]

 < ε.

(17)

r∈[0,1]

By Lemma 2.2 we have X
∗ sup f˜(ξ)[α(v) − α(u)] (r, x) − A∗ (r, x) < ε. n−1

sup

(18)

r∈[0,1] x∈S

Furthermore, by A∗ (r, x) = sup hy, xi, we have y∈[A]r

sup

X sup f ∗ (ξ)(r, x)[α(v) − α(u)] − A∗ (r, x) < ε. n−1

(19)

r∈[0,1] x∈S

Hence, for any r ∈ [0, 1], x ∈ S n−1 and for any δ-fine division P we have X ∗ ∗ f (ξ)(r, x)[α(v) − α(u)] − A (r, x) < ε.

(20)

For proof X ∗ ∗ < ε, f (ξ)(r, x)[α(v) − α(u)] − A (r, x)

(21)

1

the proof is similar to the Theorem 3.7 in [8]. This completes the proof.



Theorem 3.2 Let α : [a, b] → R be an increasing function and let f˜ : [a, b] → E n be a fuzzy-number-valued function. If f˜ is (HS) integrable to F˜ ([a, b]) with respect to α on [a, b], then f˜ has LSRS with respect to α on [a, b]. Proof Since f˜ is (HS) integrable to F˜ ([a, b]) with respect to α on [a, b], by Theorem 3.1 the support-function-wise f ∗ (ξ)(r, x) of f˜ on [a, b] is (HS) integrable to F ∗ ([a, b])(r, x) with respect to α on [a, b] uniformly for any r ∈ [0, 1], x ∈ S n−1 , i.e., for every ε > 0 there is a positive function δ(ξ) > 0, for δ-fine division P = {[u, v]; ξ} of [a, b] and for any x ∈ S n−1 , we have X ∗ ∗ < ε. (22) f (ξ)(r, x)[α(v) − α(u)] − F ([a, b])(r, x) 2 For each t ∈ [a, b], there is a closed interval C = [y, z] ⊂ (t − δ(t), t + δ(t)) such that ∗ F ([y, z])(r, x) < ε . 2 According to Henstock Lemma, for each X ∗ ≤ f (ξ)(r, x)[α(v) − α(u)]
0 there is a positive function δ(ξ) > 0, such that for any [u, v] ⊂ [a, b] with α(v) − α(u) < δ(ξ), we have



Z



˜dα

(F HS)

F˜ ([u, v]) < ε. (24) f =



n

En

[u,v]

En

Proof The continuity follows from Lemma 3.1 and the following inequality:

 

˜ (u), F˜ (v)

F˜ ([u, v]) = D F

n E

 

˜(ξ)[α(v) − α(u)] ≤ D F˜ ([u, v]), f˜(ξ)[α(v) − α(u)] + f

< We only need set δ(ξ) < 2(kf˜(ξ)kε n +1) . E This completes the proof.

En

ε. 1145

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M.E. Hamid: Locally and globally small Riemann sums and Henstock-Stieltjes integral for n-dimensional...

Theorem 3.3 Let α : [a, b] → R be an increasing function and let a fuzzy-number-valued function f˜ : [a, b] → E n has LSRS with respect to α on [a, b], then f˜ is (F HS) integrable with respect to α on [a, b]. Proof Given any ε > 0 and P = {([a, b], ξ)} = {([a1 , b1 ], ξ1 ), ([a2 , b2 ], ξ2 ), · · · , ([an , bn ], ξn )} is a δ-fine partition of [a, b]. For each i(i = 1, 2, · · · , n) there is a positive function δi with Pi = {([ui , vi ], ξi )} is a δi -fine partition of [ai , bi ]. Since f˜ has LSRS with respect to α on [ai , bi ], then we have

X

ε ˜(ξ)[α(v) − α(u)]

(25) f

n < 2n . E P i

Taken η = max{δ(ξ), ξ ∈ [a, b]}, according to the Lemma 3.2 we have



Z



˜dα

F˜ ([ai , bi ])

(F HS) = f

n

[ai ,bi ]

E

Therefore, for any δi -fine partition Pi = {([ui , vi ], ξi )} X  f˜(ξ)[α(v) − α(u)], F˜ ([ai , bi ]) ≤
0 and for every δn -fine division P = {[u, v]; ξ} of [a, b], we have

X

˜(ξ)[α(v) − α(u)]

f < ε, (27)

En

kf˜(ξ)kE n >n

where the

P

is taken over P and for which f˜(ξ) E n >n .

The following Theorem 4.1 shows that f˜ has (GSRS) with respect to α on [a, b] is equal to the type of it, s support functions. Theorem 4.1 Let α : [a, b] → R be an increasing function and let f˜ : [a, b] → E n be a fuzzy-number-valued function, the support-function-wise f ∗ (ξ)(r, x) of f˜ has globally small Riemann sums or (GSRS) with respect to α on [a, b] if and only if for every ε > 0, there exists a positive integer N such that for every n > N there is a δn (ξ) > 0 and for every δn -fine division P = {[u, v]; ξ} of [a, b], we have X ∗ < ε, f (ξ)(r, x)[α(v) − α(u)] (28) |f ∗ (ξ)(r,x)|>n

uniformly for any r ∈ [0, 1] and x ∈ S n−1 , where the

P

is taken over P and for which f ∗ (ξ)(r, x) > n. 1142-1149 1146 HAMID



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M.E. Hamid: Locally and globally small Riemann sums and Henstock-Stieltjes integral for n-dimensional...

Proof First, we can prove the following statements are equivalent: (1) f˜(ξ) E n > n. (2) f ∗ (ξ)(r, x) > n. In fact




f˜(ξ) n > n = sup d [f˜(ξ)]r , [˜ 0]r E r∈[0,1]

=

sup

sup f ∗ (ξ)(r, x) .

r∈[0,1] x∈S n−1

Second, let ˜ 0 ∈ E n denote the (F HS) integral of f˜ with respect to α on [a, b]. Given ε > 0 there exists a positive integer N such that for every n > N there is a δn (ξ) > 0 and for every δn -fine division P = {[u, v]; ξ} of [a, b], we have   X D f˜(ξ)[α(v) − α(u)], ˜ 0 < ε. (29) kf˜(ξ)kE n >n

That is

  sup d r∈[0,1]

r  r f˜(ξ)[α(v) − α(u)] , ˜ 0

X

 < ε.

(30)

kf˜r (ξ)kE n >n

By Lemma 2.2 we have sup n−1

sup

r∈[0,1] x∈S

X |f ∗ (ξ)(r,x)|>n


∗ f (ξ)[α(v) − α(u)] (r, x) − σ(x, 0) < ε.

(31)

f ∗ (ξ)(r, x)[α(v) − α(u)] − σ(x, 0) < ε.

(32)

Furthermore, by σ(x, A) = sup hy, xi, we have y∈A

sup

sup n−1

r∈[0,1] x∈S

X |f ∗ (ξ)(r,x)|>n

Hence, for any r ∈ [0, 1], x ∈ S n−1 and for any δ-fine division P we have X ∗ < ε. f (ξ)(r, x)[α(v) − α(u)]

(33)

|f ∗ (ξ)(r,x)|>n

Where σ(x, 0) = 0. This completes the proof.



Theorem 4.2 Let α : [a, b] → R be an increasing function and let f˜ : [a, b] → E n be a fuzzy-number-valued function. If f˜ has GSRS with respect to α on [a, b] then f˜ is (HS) integrable with respect to α on [a, b]. Proof Because f˜ has GSRS with respect to α on [a, b], then by Theorem 4.1 for every ε > 0, there exists a positive integer N such that for every n > N there is a δn (ξ) > 0 and for every δn -fine division P = {[u, v]; ξ} of [a, b], we have X ∗ f (ξ)(r, x)[α(v) − α(u)] < ε. (34) |f ∗ (ξ)(r,x)|>n

P uniformly for any r ∈ [0, 1] and x ∈ S n−1 , where the is taken over P and for which f ∗ (ξ)(r, x) > n. For each two δ-fine divisions P1 = {[u1 , v1 ]; ξ1 }, P2 = {[u2 , v2 ]; ξ2 } of [a, b], we have X ∗ X ∗ f (ξ )(r, x)[α(v ) − α(u )] − f (ξ )(r, x)[α(v ) − α(u )] 1 1 1 2 2 2 X ∗ X ∗ f (ξ2 )(r, x)[α(v2 ) − α(u2 )] ≤ f (ξ1 )(r, x)[α(v1 ) − α(u1 )] + X X ≤ f ∗ (ξ1 )(r, x)[α(v1 ) − α(u1 )] + f ∗ (ξ1 )(r, x)[α(v1 ) − α(u1 )] |f ∗ (ξ1 )(r,x)|>n

+




n

|f ∗ (ξ1 )(r,x)|≤n

f ∗ (ξ2 )(r, x)[α(v2 ) − α(u2 )] +

X |f ∗ (ξ2 )(r,x)|≤n

According to the properties of Cauchy, f˜ is (HS) integrable on [a, b]. This completes the proof. 1147

f ∗ (ξ2 )(r, x)[α(v2 ) − α(u2 )]

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M.E. Hamid: Locally and globally small Riemann sums and Henstock-Stieltjes integral for n-dimensional...

Theorem 4.3 Let α : [a, b] → R be an increasing. Given a fuzzy-number-valued function f˜ : [a, b] → E n , for each r ∈ [0, 1] and x ∈ S n−1 defined the support function fn∗ (ξ)(r, x) of f˜n by the formula: ( f ∗ (ξ)(r, x), ξ ∈ [a, b] if |f ∗ (ξ)(r, x)| ≤ n, ∗ fn (ξ)(r, x) = 0, others. A fuzzy-number-valued function f˜ is (HS) integrable with respect to α on [a, b] if and only if f˜ has GSRS with respect to α on [a, b] and F˜n ([a, b]) → F˜ ([a, b]) as n → ∞. (Where F˜ ([a, b]) and F˜n ([a, b]) the integral of f˜ and f˜n with respect to α on [a, b] respectively). Proof First we shall prove the necessity. Because a fuzzy-number-valued function f˜ is (HS) integrable with respect to α on [a, b] uniformly for any r ∈ [0, 1] and x ∈ S n−1 , i.e., for every ε > 0 there is a positive function δ ∗ , for δ ∗ -fine division P = {[u, v]; ξ} of [a, b], we have X ∗ ∗ < ε. f (ξ)(r, x)[α(v) − α(u)] − F ([a, b])(r, x) (35) 3 For each n ∈ N, there is a positive function δn , for δn -fine division P = {[u, v]; ξ} of [a, b], we have X ∗ ∗ < ε, f (ξ)(r, x)[α(v) − α(u)] − F ([a, b])(r, x) n n 3

(36)

for each r ∈ [0, 1] and x ∈ S n−1 . Because {Fn∗ ([a, b])(r, x)} converge to F ∗ ([a, b])(r, x) of [a, b] then there is a positive number N so if n ≥ N we have ∗ Fn ([a, b])(r, x) − F ∗ ([a, b])(r, x) < ε . (37) 3 For n ≥ N, defined a positive function δ on [a, b] by the formula: δ(ξ) = min{δ ∗ (ξ), δn (ξ)}.

(38)

Therefor, for each δ-fine division P = {[u, v]; ξ} of [a, b], we have X ∗ f (ξ)(r, x)[α(v) − α(u)] |f ∗ (ξ)(r,x)|>n

= ≤ +
0, there exists a positive integer N such that for every n > N there is a δn (ξ) > 0 and for every δn -fine division P = {[u, v]; ξ} of [a, b], we have X ∗ < ε, f (ξ)(r, x)[α(v) − α(u)] (39) |f ∗ (ξ)(r,x)|>n

P uniformly for any r ∈ [0, 1] and x ∈ S n−1 , where the is taken over P and for which f ∗ (ξ)(r, x) > n. Note that f˜n , is Henstock-Stieltjes integrable with respect to α on [a, b] for all n. Choose N so that whenever n, m > N we have ∗ ∗ < ε. Fn ([a, b])(r, x) − Fm ([a, b])(r, x) (40) Then for n, m > N and a suitably chosen δ-fine division P = {[u, v]; ξ}, we have ∗ ∗ Fn ([a, b])(r, x) − Fm ([a, b])(r, x) X X ∗ ≤ Fn ([a, b])(r, x) − f ∗ (ξ)(r, x)[α(v) − α(u)] + f ∗ (ξ)(r, x)[α(v) − α(u)] |f ∗ (ξ)(r,x)|≤n

+
n

∗ f ∗ (ξ)(r, x)[α(v) − α(u)] − Fm ([a, b])(r, x) + 1148

X |f ∗ (ξ)(r,x)|>m

f ∗ (ξ)(r, x)[α(v) − α(u)] HAMID 1142-1149

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M.E. Hamid: Locally and globally small Riemann sums and Henstock-Stieltjes integral for n-dimensional...

That is, {Fn∗ ([a, b])(r, x)} converge to F ∗ ([a, b])(r, x), as n → ∞. Again, for suitably chosen N and δ(ξ) and for every δ-fine division P = {[u, v]; ξ}, we have X ∗ ∗ f (ξ)(r, x)[α(v) − α(u)] − F ([a, b])(r, x) X ∗ ∗ ∗ ≤ f (ξ)(r, x)[α(v) − α(u)] − FN ([a, b])(r, x) + FN ([a, b])(r, x) − F ∗ ([a, b])(r, x) X X ∗ ≤ f ∗ (ξ)(r, x)[α(v) − α(u)] − FN ([a, b])(r, x) + f ∗ (ξ)(r, x)[α(v) − α(u)] |f ∗ (ξ)(r,x)|≤N

+

∗ FN ([a, b])(r, x) − F ∗ ([a, b])(r, x)


N

That is, f˜ is (HS) integrable on [a, b]. This completes the proof.

5



conclusions

In this paper, the notions of locally and globally small Riemann sums modifications with respect to fuzzy-numbervalued functions in E n are introduced and studied. The basic properties and characterizations are presented. In particular, it is proved that a fuzzy-number-valued functions in E n is (HS) integrable on [a, b] iff it has (LSRS), and also it is proved that a fuzzy-number-valued functions in E n is (HS) integrable on [a, b] iff it has (GSRS).

References [1] S.X. Hai and Z.T. Gong, On Henstock integral of fuzzy-number-valued functions in Rn , International Journal of Pure and Applied Mathematics, 7(1)(2003), 111-121. [2] M.E. Hamid and Z.T Gong, The Henstock-Stieltjes Integral for n-dimensional Fuzzy-Number-Valued Functions, International Journal of Mathematics And its Applications, 5(1-B)(2017), 171-185. [3] M.E. Hamid, L.S. Xu and Z.T. Gong, Locally and globally small Riemann sums and Henstock integral of fuzzynumber-valued functions, Journal of Computational Analysis and Applications, 25(1)(2018), 11-18. [4] M.E. Hamid, L.S. Xu, Locally and globally small Riemann sums and Henstock integral of fuzzy- number-valued functions in En , Journal of Computational Analysis and Applications, in Press. [5] M.E. Hamid, L.S. Xu and Z.T. Gong, Locally and globally small Riemann sums and Henstock-Stieltjes integral of fuzzy- number-valued functions, Journal of Computational Analysis and Applications, 25(6)(2018), 1107-1115. [6] R. Henstock, Theory of Integration, Butterworth, London, 1963. [7] C.R. Indrati, Some Characteristics of the Henstock-Kurzweil in Countably Lipschitz Condition, The 7th SEAMSUGM Conference 2015. [8] P.Y. Lee, Lanzhou Lectures on Henstock Integration, World Scientific, Singapore, 1989. [9] P.Y. Lee and R. Vyborny, The Integral: An Easy Approach after Kurzweil and Henstock, Cambridge University Press, 2000. [10] A.W. Schurle, A new property equivalent to Lebesgue integrability, Proceedings of the American Mathematical Society, 96(1)(1986), 103-106. [11] A.W. Schurle, A function is Perron integrable if it has locally small Riemann sums, Journal of the Australian Mathematical Society (Series A), 41(2)(1986), 224-232. [12] C.X. Wu, M. Ma and J.X. Fang, Structure Theory of Fuzzy Analysis, Guizhou Scientific Publication (1994), In Chinese. [13] L.A. Zadeh, Fuzzy sets, Information Control, 8(1965), 338-353.

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Solving Systems of Nonhomogeneous Coupled Linear Matrix Differential Equations in Terms of Mittag-Leffler Matrix Functions Rungpailin Kongyaksee, Pattrawut Chansangiam∗ Department of Mathematics, Faculty of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand.

Abstract In this paper, we investigate systems of nonhomogeneous coupled linear matrix differential equations. Applying Kronecker products, the vector operator, and matrix convolution product, we obtain explicit formula of the general solution to this system in terms of matrix series concerning exponentials and Mittag-Leffler functions.

Keywords: linear matrix differential equation, Kronecker product, vector operator, matrix convolution product, Mittag-Leffler function. Mathematics Subject Classifications 2010: 15A16, 15A69, 33E12, 34A30, 44A35.

1

Introduction

Theory of linear matrix differential equations can be applied in a broad range of scientific fields, e.g. statistics [2, 6, 8], game theory [4], ecometrics and Leondief model [6, 8, 11], control and system theory [3, 7]. The simplest first-order homogeneous linear matrix differential equation with time-invariant coefficient is given by X ′ (t) = AX(t).

(1.1)

Here, A is a given square matrix and X(t) is an unknown matrix-valued function to be solved. The system (1.1) has been widely studied, and the solution relies on the computation of etA ; see more information in [12, 13]. The nonhomogeneous case appears in the form X ′ (t) = AX(t) + U (t), ∗ Corresponding

(1.2)

author. Email: [email protected]

1

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here U (t) is a given matrix-valued function. In fact, the equation (1.2) has a general solution given by a one-parameter matrix-valued function X(t) = e(t−t0 )A X(t0 ) + etA ∗ U (t),

(1.3)

where ∗ denotes the matrix convolution product. See related works on nonhomogeneous case in [10, 15] and references therein. Coupled matrix differential equations have numerous applications in pure and applied mathematics. For example, to obtain the solution of an optimal control problem with performance index we need to solve the system [7] X ′ (t) = AX(t) + BY (t), Y ′ (t) = CX(t) − AT Y (t). A general system of nonhomogeneous coupled linear matrix differential equations with time-invariant coefficient takes the form X ′ (t) = AX(t)B + CY (t)D + U (t), Y ′ (t) = EX(t)F + GY (t)H + V (t).

(1.4)

In [5], a homogeneous case of (1.4) when E = C, F = D, G = A, H = B was investigated under the assumption that AC = CA and BD = DB. In this case, the solution is given in terms of Kronecker products, the vector operator, and matrix series concerning exponentials and hyperbolic functions. A nonhomogeneous case of (1.4) was discussed in [1]. In this work, we investigate the system (1.4) under the assumption that AC = CG, GE = EA, DB = HD, F H = BF . We apply Kronecker products and the vector operator to reduce our complex system to the simplest form. Thus, an explicit formula of the general solution to this system is obtained in terms of Mittag-Leffler matrix functions. In particular, we obtain general solution of several special cases of the main system. When initial conditions are imposed to these problems, its solution is uniquely determined. Our results also include the previous works [1, 5]. This paper is structured as follows. In Section 2, we supply useful facts for solving linear matrix differential equations, including matrix functions defined by power series, Kronecker product, vector operator, and matrix convolution product. The main part of the paper, Section 3, deals with solving the system (1.4) and its interesting special cases. In Sections 4, we treat an initial value problem related to (1.4) and illustrate it with a numerical example.

2

Preliminaries

In this section, we provide adequate tools for solving system of linear matrix differential equations. We shall denote the set of all m-by-n complex matrices by Mm,n , and we set Mn = Mn,n . 2

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2.1

Functions of a matrix defined by power series

Consider A ∈ Mn and a holomorphic function f defined on a region in the complex plane containing the origin and the spectrum of A. Let R > 0 be such that f admits the Taylor series expansion f (z) =

∞ ∑

ak z k

for |z| < R,

k=0

where a0 = f (0) and ak = f (k) (0)/k! for any∑ k ∈ N. If the spectral radius of A ∞ is less than R, then the matrix power series k=0 ak Ak converges, denoted by f (A). Hence if f is an entire function then f (A) is a well-defined matrix for any A ∈ Mn . In particular, the following matrix series converge for any A ∈ Mn : sinh(A) =

∞ ∑ k=0

1 A2k+1 , (2k + 1)!

cosh(A) =

∞ ∑ k=0

1 A2k . (2k)!

Recall that the two-parameter Mittag-Leffler functions (e.g. [14]) is defined by Eα,β (z) =

∞ ∑ k=0

zk Γ(αk + β)

(2.1)

where Γ is the Gamma function. The power series (2.1) converges for all complex numbers z. The Mittag-Leffler function of a matrix A ∈ Mn with parameters α > 0 and β > 0 is defined by Eα,β (A) =

∞ ∑ k=0

1 1 1 Ak = In + A+ A2 + · · · . Γ(αk + β) Γ(α + β) Γ(2α + β)

The class of matrix Mittag-Leffler functions include the following functions: ∞ ∑ 1 k A = eA , k!

E1,1 (A) =

∞ ∑

E2,1 (A2 ) =

k=0

k=0

( ) ∑∞ An expansion shows that E2,2 (A2 ) A = k=0

1 A2k = cosh(A). (2k)!

1 A2k+1 = sinh(A). (2k + 1)!

Lemma 2.1 (see e.g. [9]). If (A, B) is a pair of commuting complex matrices, then eA+B = eA eB . The next lemma is useful for deriving explicit formulas of solutions for system of linear matrix differential equations in Section 3. Lemma 2.2. For any A ∈ Mn (C) and B ∈ Mn (C), we have 

0 B



e



A 0

[ =

E ( 2,1 (AB) ) E2,2 (BA) B

(

) ] E2,2 (AB) A . E2,1 (BA)

3

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Proof. A computation using matrix analysis reveals that  

e

0 B

=

=

=

=

2.2



A 0

[ ]k ∞ ∑ 1 0 A = k! B 0 k=0 ] ∑ ] [ [ ∞ ∞ ∑ 1 1 (AB)k 0 0 (AB)k A + 0 0 (BA)k (2k)! (2k + 1)! (BA)k B k=0 k=0  ∞ ∑ 1 (AB)k 0    k=0 (2k)!   ∞   ∑ 1 k  0 (BA) (2k)! k=0   ∞ ∑ 1 k 0 (AB) A  (2k + 1)!   k=0  +  ∞ ∑  1 k   0 (BA) B (2k + 1)! k=0  ∞  ∞ ∑ ∑ 1 1 k k (AB) (AB) A   Γ(2k + 1) Γ(2k + 2)   k=0 k=0  ∞  ∞  ∑  ∑ 1 1 k k   (BA) B (BA) Γ(2k + 2) Γ(2k + 1) k=0 k=0 ( ) ] [ E (AB) E (AB) A 2,1 2,2 ( ) . E2,2 (BA) B E2,1 (BA)

Kronecker product and vector operator

Given two matrices A = [aij ] ∈ Mm,n and B = [bij ] ∈ Mp,q the Kronecker product of A and B is defined by A ⊗ B = [aij B]ij ∈ Mmp,nq . The the vector operator Vec : Mm,n → Cmn is defined for each A = [aij ] by Vec A = [a11 . . . am1 . . . a12 . . . am2 . . . a1m . . . amn ]T . It is clear that Vec is a linear isomorphism. Algebraic properties of the Kronecker product and the vector operator used in this paper are as follows: Lemma 2.3 (see e.g. [9]). The map (A, B) 7→ A ⊗ B is bilinear. The following properties hold for matrices of appropriate sizes: 1. Im ⊗ In = Imn , 2. (A ⊗ B)(C ⊗ D) = AC ⊗ BD, 3. Vec(AXB) = (B T ⊗ A) Vec X. 4

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The Kronecker product is compatible with holomorphic functions in the following sense. Lemma 2.4 (see e.g.[9]). Let f be a holomorphic function defined on a region including the origin and the spectrum of A ∈ Mn . Then f (I ⊗ A) = I ⊗ f (A) and f (A ⊗ I) = f (A) ⊗ I. In particular, the following relations hold for any A ∈ Mn : Eα,β (A ⊗ I) = Eα,β (A) ⊗ I sinh(A ⊗ I) = sinh(A) ⊗ I cosh(A ⊗ I) = cosh(A) ⊗ I

2.3

and and and

Eα,β (I ⊗ A) = I ⊗ Eα,β (A), sinh(I ⊗ A) = I ⊗ sinh(A), cosh(I ⊗ A) = I ⊗ cosh(A).

Matrix convolution product

Let Ω = [0, ∞) or Ω = [0, b] for some b > 0. The convolution is a binary operation assigned to each pair of integrable function f and g defined by ∫

t

(f ∗ g)(t) =

f (τ )g(t − τ )dτ,

t ∈ Ω.

0

The convolution is bilinear and commutative. Given two integrable matrixvalued functions A : Ω → Mm,n (R), A(t) = [aij (t)] and B : Ω → Mn,p (R), B(t) = [bij (t)], we define the matrix convolution product of A and B by [ n ] ∑ aik (t) ∗ bkj (t) ∈ Mm,p (R), t ∈ Ω. (A ∗ B)(t) = k=1

We may write A(t) ∗ B(t) for (A ∗ B)(t). The matrix convolution product is bilinear, but not commutative in general.

3

General solutions of systems of nonhomogeneous coupled linear matrix differential equations

From now on, let A, B, C, D, E, F, G, H, J, K ∈ Mn (C) be given constant matrices and let U, V : Ω → Mn (C) be given matrix-valued functions. We wish to solve certain systems of linear matrix differential equations in unknown matrixvalued functions X, Y : Ω → Mn (C). Theorem 3.1. Assume that DB = HD, AC = CG, F H = BF , GE = EA. Then the general solution of the system of nonhomogeneous coupled linear matrix differential equations: X ′ (t) = AX(t)B + CY (t)D + U (t), Y ′ (t) = EX(t)F + GY (t)H + V (t)

(3.1)

5

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is given by {( ) T Vec X(t) = e(t−t0 )(B ⊗A) E2,1 ((t − t0 )2 M ) Vec X(t0 ) ( )( )( ) + t − t0 E2,2 ((t − t0 )2 M ) DT ⊗ C Vec Y (t0 ) ( ) + E2,1 ((t − t0 )2 M ) ∗ Vec U (t) ( )( )( ) } + t − t0 E2,2 ((t − t0 )2 M ) DT ⊗ C ∗ Vec V (t) , { ( )( ) T Vec Y (t) = e(t−t0 )(H ⊗G) (t − t0 ) E2,2 ((t − t0 )2 N F T ⊗ E Vec X(t0 ) ( ) + E2,1 ((t − t0 )2 N ) Vec Y (t0 ) ( )( )( ) + t − t0 E2,2 ((t − t0 )2 N ) F T ⊗ E ∗ Vec U (t) ( ) } + E2,1 ((t − t0 )2 N ) ∗ Vec V (t) , (3.2) where M = (F D)T ⊗ CE and N = (DF )T ⊗ EC. Proof. Using Lemma 2.3, we can transform the system (3.1) into the vector form: [ ] [ T ][ ] [ ] Vec X ′ (t) B ⊗A DT ⊗ C Vec X(t) Vec U (t) = + . Vec Y ′ (t) Vec Y (t) Vec V (t) FT ⊗ E HT ⊗ G [ T ] [ ] B ⊗A 0 0 DT ⊗ C Let us denote P = and Q= . 0 HT ⊗ G FT ⊗ E 0 From (1.3), this system has the following solution: [ ] [ ] [ ] Vec X(t) Vec U (t) (t−t0 )S Vec X(t0 ) (t−t0 )S =e +e ∗ , Vec Y (t) Vec Y (t0 ) Vec V (t) where S=P + Q. Now, we will compute eS . Since DB = HD, AC = CG, F H = BF and GE = EA, by Lemma 2.3 we have P Q = QP . From which it follows from Lemma 2.1 that eS = eP +Q = eP eQ . By expanding the power series of matrix exponential, we have [ T ] eB ⊗A 0 P e = . T 0 eH ⊗G By Lemma 2.2, we have [ E (M ) )( Q ) e = ( 2,1 E2,2 (N ) F T ⊗ E Thus S

[

( )( ) ] E2,2 (M ) DT ⊗ C . E2,1 (N )

][

( )( T ) ] E ( 2,1 (M ) )( T ) E2,2 (M ) D ⊗ C E2,2 (N ) F ⊗ E E2,1 (N ) 0 eH ⊗G [ ( )( ) ] T T eB ⊗A E2,1 (M ) eB ⊗A E2,2 (M ) DT ⊗ C ( )( ) H T ⊗G . = T eH ⊗G E2,2 (N ) F T ⊗ E e E2,1 (N )

e =

eB

T

⊗A

0

T

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Denoting T

⊗A)

T

⊗G)

E2,1 ((t − t0 )2 M ), ( )( )( ) T R2 = e(t−t0 )(B ⊗A) t − t0 E2,2 ((t − t0 )2 M ) DT ⊗ C , ( )( )( ) T R3 = e(t−t0 )(H ⊗G) t − t0 E2,2 ((t − t0 )2 N ) F T ⊗ E , R1 = e(t−t0 )(B

R4 = e(t−t0 )(H

E2,1 ((t − t0 )2 N ),

we obtain [ ] [ Vec X(t0 ) R1 = e(t−t0 )S Vec Y (t0 ) R3

R2 R4

We also have [ ] [ Vec U (t) R1 (t−t0 )S e ∗ = Vec V (t) R3

] [ ] [ ] R2 Vec U (t) R1 ∗ Vec U (t) + R2 ∗ Vec V (t) ∗ = . R4 Vec V (t) R3 ∗ Vec U (t) + R4 ∗ Vec V (t)

][ ] [ ] Vec X(t0 ) R1 Vec X(t0 ) + R2 Vec Y (t0 ) = . Vec Y (t0 ) R3 Vec X(t0 ) + R4 Vec Y (t0 )

Therefore, the general solution of (3.1) is given by (3.2). Corollary 3.2. Assume that DB = HD, AC = CG, F H = BF , GE = EA. Then the general solution of the system X ′ (t) = AX(t)B + CY (t)D, Y ′ (t) = EX(t)F + GY (t)H is given by {( ) T Vec X(t) = e(t−t0 )(B ⊗A) E2,1 ((t − t0 )2 M ) Vec X(t0 ) ( )( )( ) + t − t0 E2,2 ((t − t0 )2 M ) DT ⊗ C Vec Y (t0 ), {( )( )( ) T Vec Y (t) = e(t−t0 )(H ⊗G) t − t0 E2,2 ((t − t0 )2 N ) F T ⊗ E Vec X(t0 ) ( ) } + E2,1 ((t − t0 )2 N ) Vec Y (t0 ) (3.3) where M = (F D)T ⊗ CE and N = (DF )T ⊗ EC. Proof. Put U (t) = V (t) = 0 in (3.2) and then use Lemma 2.3. The next result was firstly established in [1]. Corollary 3.3. The general solution of the system X ′ (t) = AX(t)B + CY (t)D + U (t), Y ′ (t) = CX(t)D + AY (t)B + V (t) under the assumption that AC = CA and BD = DB, is given by { T Vec X(t) =e(t−t0 )(B ⊗A) cosh L Vec X(t0 ) + sinh L Vec Y (t0 ) } + cosh L ∗ Vec U (t) + sinh L ∗ Vec V (t) , { T Vec Y (t) =e(t−t0 )(B ⊗A) sinh L Vec X(t0 ) + cosh L Vec Y (t0 ) } + sinh L ∗ Vec U (t) + cosh L ∗ Vec V (t) ,

(3.4)

(3.5)

7

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where L = (t − t0 )(DT ⊗ C). Proof. Put E = C, F = D, G = A and H = B in (3.2), and use Lemma 2.3. The corresponding homogeneous system of (3.4) is given by X ′ (t) = AX(t)B + CY (t)D, Y ′ (t) = CX(t)D + AY (t)B.

(3.6)

If AC = CA and BD = DB, then the general solution of (3.6) is reduced to { } T Vec X(t) = e(t−t0 )(B ⊗A) cosh L Vec X(t0 ) + sinh L Vec Y (t0 ) , { } T Vec Y (t) = e(t−t0 )(B ⊗A) sinh L Vec X(t0 ) + cosh L Vec Y (t0 ) . This result was firstly obtained in [5]. Corollary 3.4. The general solution of the system X ′ (t) = AX(t)B + CY (t) + U (t), Y ′ (t) = EX(t) + GY (t)B + V (t) under the condition AC = CG, GE = EA, is given by {( } ) ( )( ) T Vec X(t) = e(t−t0 )(B ⊗A) Vec E2,1 (K1 ) X(t0 ) + t − t0 E2,2 (K1 ) CY (t0 ) {( ) T + e(t−t0 )(B ⊗A) In ⊗ E2,1 (K1 ) ∗ Vec U (t) } ( ( )( ) ) + In ⊗ t − t0 E2,2 (K1 ) C ∗ Vec V (t) , {( } )( ) ( ) T Vec Y (t) = e(t−t0 )(B ⊗G) Vec t − t0 E2,2 (K2 ) EX(t0 ) + E2,1 (K2 ) Y (t0 ) {( ( )( ) ) T + e(t−t0 )(B ⊗G) In ⊗ t − t0 E2,2 (K2 ) E ∗ Vec U (t) } ( ) + In ⊗ E2,1 (K2 ) ∗ Vec V (t) , where K1 = (t − t0 )2 CE and K2 = (t − t0 )2 EC. Proof. Put H = B, D = F = In in (3.2) and then use Lemmas 2.3 and 2.4. Corollary 3.5. The general solution of the system X ′ (t) = AX(t)B + Y (t) + U (t), Y ′ (t) = X(t) + AY (t)B + V (t) is given by T

⊗A)

{

cosh(t − t0 ) Vec X(t0 ) + sinh(t − t0 ) Vec Y (t0 ) ) ( )} + cosh(t − t0 ) In2 ∗ Vec U (t) + sinh(t − t0 ) In2 ∗ Vec V (t) , { T Vec Y (t) = e(t−t0 )(B ⊗A) sinh(t − t0 ) Vec X(t0 ) + cosh(t − t0 ) Vec Y (t0 ) ( ) ( )} + sinh(t − t0 ) In2 ∗ Vec U (t) + cosh(t − t0 ) In2 ∗ Vec V (t) .

Vec X(t) = e(t−t0 )(B

(

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Proof. Put C = D = In in (3.5) and then use Lemma 2.3. Corollary 3.6. The general solution of the system X ′ (t) = AX(t)B + U (t), Y ′ (t) = EX(t)F + GY (t)H + V (t) under the condition F H = BF and GE = EA, is given by { } T Vec X(t) = e(t−t0 )(B ⊗A) Vec X(t0 ) + I ∗ Vec U (t) , { } T Vec Y (t) = e(t−t0 )(H ⊗G) Vec (t − t0 )EX(t0 )F + Y (t0 ) { } T + e(t−t0 )(H ⊗G) (t − t0 )(F T ⊗ E) ∗ Vec U (t) + I ∗ Vec V (t) . Proof. Put C = D = 0 in (3.2) and then use Lemma 2.3. Corollary 3.7. The general solution of equation X ′ (t) = AX(t)B + U (t) is { } T given by Vec X(t) = e(t−t0 )(B ⊗A) Vec X(t0 ) + I ∗ Vec U (t) . Proof. Put E = F = 0 in Corollary 3.6.

4

Unique solution of initial value problem and a numerical example

Consider the following initial value problem associated with the system (3.1): X ′ (t) = AX(t)B + CY (t)D + U (t), Y ′ (t) = EX(t)F + GY (t)H + V (t) subject to initial conditions X(0) = J and Y (0) = K. Suppose DB = HD, AC = CG, F H = BF , GE = EA. In this case, the solution of this problem is unique and given by {( ) ( )( ) T Vec X(t) = et(B ⊗A) E2,1 (t2 M ) Vec J + t E2,2 (t2 M ) DT ⊗ C Vec K ( ) ( )( ) } + E2,1 (t2 M ) ∗ Vec U (t) + t E2,2 (t2 M ) DT ⊗ C ∗ Vec V (t) , {( )( ) ( ) T Vec Y (t) = et(H ⊗G) t E2,2 (t2 N F T ⊗ E Vec J + E2,1 (t2 N ) Vec K ( )( ) ( ) } + t E2,2 (t2 N ) F T ⊗ E ∗ Vec U (t) + E2,1 (t2 N ) ∗ Vec V (t) , where M = (F D)T ⊗ CE and N = (DF )T ⊗ EC. Let us see a numerical example. Example 4.1. The initial value problem X ′ (t) = AX(t)B + Y (t)

+ U (t),

Y ′ (t) = X(t) + AY (t)B + V (t) X(0) = J and Y (0) = K 9

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[ [ ] [ ] [ ] ] 0 −1 2 −1 3 1 1 2 with A = ,J = ,K = , ,B = 1 1[ 1 0 1 −1 [ 32t 4 ] ] −e 1 1 e2t U (t) = , V (t) = has a unique solution given by 1 sin t cos t sin 2t Vec X(t) = etW Vec

[ ] w1 (t) cosh t + w2 (t) sinh t w3 (t) cosh t + w4 (t) sinh t , w5 (t) cosh t + w6 (t) sinh t w7 (t) cosh t + w8 (t) sinh t [

] w2 (t) cosh t + w1 (t) sinh t w4 (t) cosh t + w3 (t) sinh t . w6 (t) cosh t + w5 (t) sinh t w8 (t) cosh t + w7 (t) sinh t   0 0 1 2  0 0 3 4  Here, W =  −1 −2 1 2, −3 −4 3 4 1 w1 (t) = 2 (5 − e2t ), w2 (t) = 3 + t, w3 (t) = −1 + t, w4 (t) = 12 (1 + e2t ), w5 (t) = 1 + t, w6 (t) = 1 + sin t, w7 (t) = 1 − cos t, w8 (t) = − 12 (1 + cos 2t). Vec Y (t) = etW Vec

Acknowledgements The authors would like to thank King Mongkut’s Institute of Technology Ladkrabang Research Fund for financial supports.

References [1] Z. Al-Zhour, Efficient solutions of coupled matrix and matrix differential equations, Intell. Cont. Autom., 3(2), 176-184 (2012). [2] G. N. Boshnakov, The asymptotic covariance matrix of the multivariate serial correlations, Stoch. Proc. Appl., 65, 251-258 (1996). [3] T. Chen, B. A. Francis, Optimal Sampled-Data Control Systems, Springer, London, 1995. [4] J. B. Cruz, C. I. Chen Jr., Series Nash solution of two-person nonzero sum linear differential games, J. Optimal. Theory, 7(4), 240-257 (1971). [5] A. Kilicman, Z. Al-Zhour, The general common exact solutions of coupled linear matrix and matrix differential equations, J. Anal. Comput., 1(1), 15-30 (2005). [6] J. R. Magnus, H. Neudecker, Matrix Differential Calculus with Applications in Statistics and Econometrics, John Wiley & Sons, 1975. [7] S. G. Mouroutsos, P. D. Sparis, Taylor series approach to system identification, analysis and optimal control, J. Frankin Inst., 319(3), 359-371 (1985). 10

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[8] C. R. Rao, M. B. Rao, Matrix Algebra and Its Applications to Statistics and Econometrics, World Scientific, Singapore, 1998. [9] W. H. Steeb, Y. Hardy, Matrix Calculus and Kronecker Product: A Practical Approach to Linear and Multilinear Algebra, World Scientific, Singapore, 2011. [10] Z. Al-Zhour, The general (vector) solutions of such linear (coupled) matrix fractional differential equations by using Kronecker structures, Appl. Math. Comput., 232, 498-510 (2014). [11] S. L. Campbell, Singular systems of differential equations II., Pitman, San Francisco, 1982. [12] R. Ben Taher, M. Rachidi, Linear recurrence relations in the algebra of matrices and applications, Linear Algebra Appl., 330, 15-24 (2001). [13] H-W. Cheng, SS-T. Yau, More explicit formulas for the matrix exponential, Linear Algebra Appl., 262, 131-163 (1997). [14] B. Ross, Fractional Calculus and Its Applications, Springer-Verlag, Berlin, 1975. [15] Z. Al-Zhour, New techniques for solving some matrix and matrix differential equations, Ain Shams Engineering Journal, 6, 347-354 (2015).

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Expressions of the solutions of some systems of di®erence equations M. M. El-Dessoky12  E. M. Elsayed12 , E. M. Elabbasy2 and Asim Asiri1 King Abdulaziz University, Faculty of Science,Mathematics Department, P. O. Box 80203, Jeddah 21589, Saudi Arabia. 2 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt. E-mail: [email protected]; [email protected]; [email protected]; [email protected]

1

ABSTRACT In this paper, we deal with the form of the solutions and the periodicity character of the following systems of nonlinear di¤erence equations of order two +1 =

 ¡1  ¡1  +1 =  § § ¡1 § § ¡1

where the initial conditions ¡1  0  ¡1 and 0 are nonzero real numbers. Keywords: recursive sequences, di¤erence equations, periodic solution, solution of di¤erence equation, system of di¤erence equations. Mathematics Subject Classi…cation: 39A10. ———————————————————

1. INTRODUCTION Through this paper, we will obtain the form of the solutions of some nonlinear di¤erence equations systems of order two of the following form    +1 = §¡1 +1 = §¡1 §¡1   §¡1 where the initial conditions ¡1  0  ¡1 and 0 are nonzero real numbers. We will then investigate the periodicity character of the solutions of the systems under study. Finally we will present some numerical examples and some …gures will be given to explain the behavior of the obtained solutions. The study of di¤erence equations is a very rich research …eld, and di¤erence equations have been applied in several mathematical models in biology, population dynamics, genetics, economics, medicine, and so forth. Solving di¤erence equations and studying the asymptotic behavior of their solutions has attracted the attention of many authors, see for example [1-39]. El-Dessoky et al. [6] studied the periodic nature and the form of the solutions of nonlinear di¤erence equations systems of order four  ¡3  ¡3  +1 = ¡2 (§1§  +1 = ¡2 (§1§  ¡3 )  ¡3 ) Grove et al. [7] obtained the existence and behavior of solutions of the rational system +1 =

 

+

  

+1 =

 

+

  

Mansour et al. [8] investigated the periodic nature and get the form of the solutions of the following systems of rational di¤erence equations ¡1 ¡1 +1 = §¡1  ¡  +1 = §¡1  ¡ 

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El-Dessoky [9] studied the solutions of the rational equation systems +1 =

¡1 ¡2  (§1§¡1 ¡2 ) 

+1 =

¡1 ¡2  (§1§¡1 ¡2 ) 

Touafek et al. [10] investigated the periodic nature and gave the form of the solutions of the following systems of rational second order di¤erence equations +1 =

 ¡1 (§1§ ) 

+1 =

 ¡1 (§1§ ) 

Yang et al. [11] studied global behavior of the system of the two nonlinear di¤erence equations +1 =

   1+

+1 =

  1+ 

Din et al. [6] studied the behavior of the solutions of the following system of di¤erence equations +1 =

¡3 + ¡1 ¡2 ¡3 

+1 =

1 ¡3  1 + 1  ¡1 ¡2 ¡3 

De…nition 1. (Periodicity) A sequence f g1 =¡ is said to be periodic with period  if + =  for all  ¸ ¡

De…nition 2. (Fibonacci Sequence)

The sequence f g1 =1 = f1 2 3 5 8 13 21 g i.e. +1 =  + ¡1   ¸ 0 ¡1 = 0 0 = 1 is called Fibonacci Sequence.

2. THE FIRST SYSTEM: +1 =

  ¡1  ¡ ¡1 

+1 =

 ¡1  ¡¡1

In this section, we investigate the solutions of the two di¤erence equations system +1 =

 ¡1  ¡¡1 

+1 =

 ¡1  ¡¡1 

(1)

where  2 N0 and the initial conditions ¡1  0  ¡1 and 0 are arbitrary nonzero real numbers Theorem 2.1. Assume that f   g are solutions of system (1). Then for  = 0 1 2  we see that all solutions of system (1) are given by the following formulae 2¡1 = ¡1

¡1 Y

(2¡2 0 ¡2¡1 ¡1 )(2¡1 0 ¡2 ¡1 ) (2¡1 0 ¡2 ¡1 )(2 0 ¡2+1 ¡1 ) 

2 = 0

2¡1 = ¡1

¡1 Y

(2¡1 0 ¡2 ¡1 )(2¡2 0 ¡2¡1 ¡1 ) (2 0 ¡2+1 ¡1 )(2¡1 0 ¡2 ¡1 ) 

2 = 0

and

where

f g1 =¡2

=0

=0

= f1 0 1 1 2 3 5 8 13 g

¡1 Y

(2 0 ¡2+1 ¡1 )(2¡1 0 ¡2 ¡1 ) (2+1 0 ¡2+2 ¡1 )(2 0 ¡2+1 ¡1 ) 

¡1 Y

(2¡1 0 ¡2 ¡1 )(2 0 ¡2+1 ¡1 ) (2 0 ¡2+1 ¡1 )(2+1 0 ¡2+2 ¡1 ) 

=0

=0

Proof: For  = 0 the result holds. Now suppose that   0 and that our assumption holds for  ¡ 1. that is, 2¡3 2¡3

= ¡1 = ¡1

¡2 Y

=0 ¡2 Y =0

(2¡2 0 ¡2¡1 ¡1 )(2¡1 0 ¡2 ¡1 ) (2¡1 0 ¡2 ¡1 )(2 0 ¡2+1 ¡1 ) 

2¡2 = 0

(2¡1 0 ¡2 ¡1 )(2¡2 0 ¡2¡1 ¡1 ) (2 0 ¡2+1 ¡1 )(2¡1 0 ¡2 ¡1 ) 

2¡2 = 0

1162

¡2 Y

=0 ¡2 Y =0

(2 0 ¡2+1 ¡1 )(2¡1 0 ¡2 ¡1 ) (2+1 0 ¡2+2 ¡1 )(2 0 ¡2+1 ¡1 ) 

(2¡1 0 ¡2 ¡1 )(2 0 ¡2+1 ¡1 ) (2 0 ¡2+1 ¡1 )(2+1 0 ¡2+2 ¡1 ) 

El-Dessoky ET AL 1161-1172

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Now we …nd from system (1) that 2¡1

= =

2¡2 2¡3 2¡2 ¡2¡3    ¡2 ¡2  (2 0 ¡2+1 ¡1 )(2¡1 0 ¡2 ¡1 )  (2¡1 0 ¡2 ¡1 )(2¡2 0 ¡2¡1 ¡1 ) 0 ¡1 (2+1 0 ¡2+2 ¡1 )(2 0 ¡2+1 ¡1 ) (2 0 ¡2+1 ¡1 )(2¡1 0 ¡2 ¡1 ) =0 =0     ¡2 ¡2  (2¡1 0 ¡2 ¡1 )(2 0 ¡2+1 ¡1 )  (2¡1 0 ¡2 ¡1 )(2¡2 0 ¡2¡1 ¡1 ) 0 ¡  ¡1 (  ¡  )(  ¡  ) (  ¡  )(  ¡  ) 2 0

=0

=



0

= 0

=

=



¡¡1

³

2¡1 0

2+1 ¡1

2 ¡1

¡2  (2 0 ¡2+1 ¡1 )(2¡1 0 ¡2 ¡1 ) 0 ¡1 (2+1 0 ¡2+2 ¡1 )(2 0 ¡2+1 ¡1 ) =0   (¡1 0 ¡0 ¡1 )(2¡4 0 ¡2¡3 ¡1 ) 0 ( ¡¡1  ¡  )(  ¡  ) 2¡3 0

(2 0 ¡2+1 ¡1 )(2¡1 0 ¡2 ¡1 ) (2+1 0 ¡2+2 ¡1 )(2 0 ¡2+1 ¡1 ) (2¡3 0 ¡2¡2 ¡1 )

(  ¡  ) ¡ (2¡4 0 ¡2¡3 ¡1 ) ¡1 2¡3 0 2¡2 ¡1

2¡3 0 ¡2¡2 ¡1 2¡3 0 ¡2¡2 ¡1

2¡2 ¡1

¡2 0

¡1 ¡1

´

¡2¡4 0 +2¡3 ¡1 ¡2¡3 0 +2¡2 ¡1

¡2 Y

(2 0 ¡2+1 ¡1 )(2¡1 0 ¡2 ¡1 ) (2¡3 0 ¡2¡2 ¡1 ) (2+1 0 ¡2+2 ¡1 )(2 0 ¡2+1 ¡1 ) (¡2¡2 0 +2¡1 ¡1 )

=0

(2¡2 0 ¡2¡1 ¡1 )(2¡1 0 ¡2 ¡1 ) (2¡1 0 ¡2 ¡1 )(2 0 ¡2+1 ¡1 ) 

2¡2 2¡3 2¡2 ¡2¡3    ¡2 ¡2  (2¡1 0 ¡2 ¡1 )(2 0 ¡2+1 ¡1 )  (2¡2 0 ¡2¡1 ¡1 )(2¡1 0 ¡2 ¡1 ) 0 ¡1 (2 0 ¡2+1 ¡1 )(2+1 0 ¡2+2 ¡1 ) (2¡1 0 ¡2 ¡1 )(2 0 ¡2+1 ¡1 ) =0 =0     ¡2 ¡2  (2 0 ¡2+1 ¡1 )(2¡1 0 ¡2 ¡1 )  (2¡2 0 ¡2¡1 ¡1 )(2¡1 0 ¡2 ¡1 ) 0 ¡  ¡1 (  ¡  )(  ¡  ) (  ¡  )(  ¡  ) 2+1 0

=0

¡1 0

= 0

¡2 Q =0

2+2 ¡1

2 0

2+1 ¡1

=0

2¡1 0

2 ¡1

2 0

2+1 ¡1

(2¡1 0 ¡2 ¡1 )(2 0 ¡2+1 ¡1 ) (2 0 ¡2+1 ¡1 )(2+1 0 ¡2+2 ¡1 )

¡2 Q

(2¡1 0 ¡2 ¡1 )(2 0 ¡2+1 ¡1 ) (2+1 0 ¡2+2 ¡1 )(2¡2 0 ¡2¡1 ¡1 )

¡2 Q

(2¡1 0 ¡2 ¡1 )(2 0 ¡2+1 ¡1 ) (2 0 ¡2+1 ¡1 )(2+1 0 ¡2+2 ¡1 )

=0

=

(2 0 ¡2+1 ¡1 )(2¡1 0 ¡2 ¡1 ) (2+1 0 ¡2+2 ¡1 )(2 0 ¡2+1 ¡1 )

¡2 

= ¡1

=

2 0

=0

(2 0 ¡2+1 ¡1 )(2¡1 0 ¡2 ¡1 ) (2+1 0 ¡2+2 ¡1 )(2 0 ¡2+1 ¡1 )

=0 ¡1 Y

2¡1

2+2 ¡1

¡2 

=0

=

2+1 0

(2¡1 0 ¡2 ¡1 )(2 0 ¡2+1 ¡1 ) (2+1 0 ¡2+2 ¡1 )(2¡2 0 ¡2¡1 ¡1 ) =0

=0

=

=0

¡2 

0

0

¡2 

0 ¡1

2+1 ¡1

¡ ¡1

¡2 Q

(2¡1 0 ¡2 ¡1 )(2 0 ¡2+1 ¡1 ) (2 0 ¡2+1 ¡1 )(2+1 0 ¡2+2 ¡1 ) =0 ¡1 0 ¡0 ¡1 )(2¡4 0 ¡2¡3 ¡1 ) 0 ((2¡3 0 ¡2¡2 ¡1 )(¡2 0 ¡¡1 ¡1 ) ¡ ¡1

¡1 0

0

=0

= 0

= = 0

¡2  =0

¡2 Y

2¡4 0 ¡2¡3 ¡1 ) ¡ ( (2¡3 0 ¡2¡2 ¡1 ) ¡ 1

2¡3 0 ¡ 2¡2 ¡1 2¡3 0 ¡ 2¡2 ¡1

¶

(2¡1 0 ¡2 ¡1 )(2 0 ¡2+1 ¡1 ) (2 0 ¡2+1 ¡1 )(2+1 0 ¡2+2 ¡1 ) (2¡3 0 ¡2¡2 ¡1 )

¡2¡4 0 +2¡3 ¡1 ¡2¡3 0 +2¡2 ¡1

(2¡1 0 ¡2 ¡1 )(2 0 ¡2+1 ¡1 ) (2¡3 0 ¡2¡2 ¡1 ) (2 0 ¡2+1 ¡1 )(2+1 0 ¡2+2 ¡1 ) (¡2¡2 0 +2¡1 ¡1 )

=0 ¡1 Y

= ¡1

µ

=0

(2¡1 0 ¡2 ¡1 )(2¡2 0 ¡2¡1 ¡1 ) (2 0 ¡2+1 ¡1 )(2¡1 0 ¡2 ¡1 ) 

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Also, we infer from system (1) that 2

= =

2¡1 2¡2 2¡1 ¡2¡2    ¡1 ¡2  (2¡2 0 ¡2¡1 ¡1 )(2¡1 0 ¡2 ¡1 )  (2¡1 0 ¡2 ¡1 )(2 0 ¡2+1 ¡1 ) ¡1 0 (2¡1 0 ¡2 ¡1 )(2 0 ¡2+1 ¡1 ) (2 0 ¡2+1 ¡1 )(2+1 0 ¡2+2 ¡1 ) =0 =0     ¡1 ¡2  (2¡1 0 ¡2 ¡1 )(2¡2 0 ¡2¡1 ¡1 )  (2¡1 0 ¡2 ¡1 )(2 0 ¡2+1 ¡1 ) ¡1 ¡  0 (  ¡  )(  ¡  ) (  ¡  )(  ¡  ) 2 0

=0

=

=

2+1 ¡1

2¡1 0

2 ¡1

=0

2 0

¡1  (2¡2 0 ¡2¡1 ¡1 )(2¡1 0 ¡2 ¡1 ) (2¡1 0 ¡2 ¡1 )(2 0 ¡2+1 ¡1 ) =0  ¡1  (2¡1 0 ¡2 ¡1 ) ¡2  (2 0 ¡2+1 ¡1 ) ¡ (2 0 ¡2+1 ¡1 ) (2¡1 0 ¡2 ¡1 ) ¡1 =0 =0

2+2 ¡1

¡1



¡1  (2¡2 0 ¡2¡1 ¡1 )(2¡1 0 ¡2 ¡1 ) (2¡1 0 ¡2 ¡1 )(2 0 ¡2+1 ¡1 ) =0 (  ¡  ) ¡ (2¡3 0 ¡2¡2 ¡1 ) ¡1 2¡2 0 2¡1 ¡1

¡1

³

(2¡2 0 ¡2¡1 ¡1 ) (2¡2 0 ¡2¡1 ¡1 )

´

¡1

¡1 Q

(2¡2 0 ¡2¡1 ¡1 )(2¡1 0 ¡2 ¡1 ) (2¡1 0 ¡2 ¡1 )(2 0 ¡2+1 ¡1 )

= ¡1

¡1 Y

(2¡2 0 ¡2¡1 ¡1 )(2¡1 0 ¡2 ¡1 ) (2¡2 0 ¡2¡1 ¡1 ) (2¡1 0 ¡2 ¡1 )(2 0 ¡2+1 ¡1 ) (¡2¡1 0 +2 ¡1 )

=

2+1 0

2+1 ¡1

= 0

=0

=0 ¡1 Y

(2¡2 0 ¡ 2¡1 ¡1 )

¡2¡3 0 + 2¡2 ¡1 ¡ 2¡2 0 + 2¡1 ¡1

(2 0 ¡2+1 ¡1 )(2¡1 0 ¡2 ¡1 ) (2+1 0 ¡2+2 ¡1 )(2 0 ¡2+1 ¡1 ) 

=0

and so, 2

= =

2¡1 2¡2 2¡1 ¡2¡2    ¡1 ¡2  (2¡1 0 ¡2 ¡1 )(2¡2 0 ¡2¡1 ¡1 )  (2 0 ¡2+1 ¡1 )(2¡1 0 ¡2 ¡1 ) ¡1  0 (2 0 ¡2+1 ¡1 )(2¡1 0 ¡2 ¡1 ) (2+1 0 ¡2+2 ¡1 )(2 0 ¡2+1 ¡1 ) =0 =0     ¡1 ¡2  (2¡2 0 ¡2¡1 ¡1 )(2¡1 0 ¡2 ¡1 )  (2 0 ¡2+1 ¡1 )(2¡1 0 ¡2 ¡1 ) ¡1 ¡  0 (  ¡  )(  ¡  ) (  ¡  )(  ¡  ) =0

=





¡1

¡

=

=

2¡1 0

2 ¡1

2 0

2+1 ¡1

=0

¡1  (2¡1 0 ¡2 ¡1 )(2¡2 0 ¡2¡1 ¡1 ) (2 0 ¡2+1 ¡1 )(2¡1 0 ¡2 ¡1 ) =0

¡1  (2¡1 0 ¡2 ¡1 ) ¡2  (2 0 ¡2+1 ¡1 ) (2 0 ¡2+1 ¡1 ) (2¡1 0 ¡2 ¡1 ) =0 =0

¡1

= ¡1 = ¡1 = 0

¡1  (2¡1 0 ¡2 ¡1 )(2¡2 0 ¡2¡1 ¡1 ) (2 0 ¡2+1 ¡1 )(2¡1 0 ¡2 ¡1 ) =0

¡1 Y =0 ¡1 Y

=0 ¡1 Y =0

2 0

2+1 ¡1

¡1

=0

2¡3 0 ¡2¡2 ¡1 ) ¡ ( (2¡2 0 ¡2¡1 ¡1 ) ¡ 1

2+2 ¡1



¶ µ ¡1 Q (2¡1 0 ¡2 ¡1 )(2¡2 0 ¡2¡1 ¡1 ) ¡1 ³ (2 0 ¡2+1 ¡1 )(2¡1 0 ¡2 ¡1 )



2+1 0



(2¡2 0 ¡2¡1 ¡1 ) (2¡2 0 ¡2¡1 ¡1 )



´

(2¡2 0 ¡2¡1 ¡1 )

¡2¡3 0 +2¡2 ¡1 ¡2¡2 0 +2¡1 ¡1

(2¡1 0 ¡2 ¡1 )(2¡2 0 ¡2¡1 ¡1 ) (2¡2 0 ¡2¡1 ¡1 ) (2 0 ¡2+1 ¡1 )(2¡1 0 ¡2 ¡1 ) (¡2¡1 0 +2 ¡1 ) (2¡2 0 ¡2¡1 ¡1 ) (2¡2 0 ¡2¡1 ¡1 ) 2¡1 0 ¡2 ¡1 ) (¡2¡1 0 +2 ¡1 )

(2¡1 0 ¡2 ¡1 )(2 0 ¡2+1 ¡1 ) (2 0 ¡2+1 ¡1 )(2+1 0 ¡2+2 ¡1 ) 

The proof is complete.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.7, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

Example 1. For con…rming the results of this section, we consider numerical example for the di¤erence system (1) with the initial conditions ¡1 = 03 0 = 04 ¡1 = 015 and 0 = ¡01. (See Fig. 1). plot of z(n+1)=t(n-1)z(n)/(t(n)-t(n-1)),t(n+1)=z(n-1)t(n)/(z(n)-z(n-1)) 0.4 z(n) t(n)

0.3 0.2

z(n),t(n)

0.1 0 -0.1 -0.2 -0.3 -0.4

0

2

4

6

8

10 n

12

14

16

18

20

Figure 1. Plot the behavior of the solution of the system (1).

3. THE SECOND SYSTEM: +1 =

  ¡1  ¡ ¡1 

+1 =

 ¡1 ¡ ¡ ¡1

We obtain, in this section, the form of the solutions of the di¤erence equations system +1 =

 ¡1  ¡1  +1 =   ¡ ¡1 ¡ ¡ ¡1

(2)

where  2 N0 and the initial conditions ¡1  0  ¡1  0 are arbitrary non zero real numbers with ¡1 6= ¡0 

+1 +1 Theorem 3.1. Let f   g+1 =¡1 be solutions of system (2).Then f g=¡1 and f g=¡1 are given by the formulae for  = 0 1 2 

4

=

4+1

=

4+2

=

4+3

=

¡¡1 0 ¡1 0 (0 +¡1 ) (2¡2 0 +2 ¡1 )(2¡1 0 +2+1 ¡1 )(2¡1 0 ¡2¡2 ¡1 )(2 0 ¡2¡1 ¡1 )  ¡1 0 ¡1 0 (0 +¡1 ) (2¡2 0 +2 ¡1 )(2¡1 0 +2+1 ¡1 )(2 0 ¡2¡1 ¡1 )(2+1 0 ¡2 ¡1 )  ¡¡1 0 ¡1 0 (0 +¡1 ) (2¡1 0 +2+1 ¡1 )(2 0 +2+2 ¡1 )(2 0 ¡2¡1 ¡1 )(2+1 0 ¡2 ¡1 ) 

¡1 0 ¡1 0 (0 +¡1 ) (2¡1 0 +2+1 ¡1 )(2 0 +2+2 ¡1 )(2+1 0 ¡2 ¡1 )(2+2 0 ¡2+1 ¡1 ) 

and 4

=

(2¡2 0 +2 ¡1 )(2 0 ¡2¡1 ¡1 )  (0 +¡1 )

4+2

=

(2¡1 0 +2+1 ¡1 )(2+1 0 ¡2 ¡1 )  (0 +¡1 )

4+1 =

¡(2¡1 0 +2+1 ¡1 )(2 0 ¡2¡1 ¡1 )  (0 +¡1 )

4+3 =

¡(2 0 +2+2 ¡1 )(2+1 0 ¡2 ¡1 )  (0 +¡1 )

Proof: For  = 0 the result holds. Now suppose that   0 and that our assumption holds for  ¡ 1. that is, 4¡4

=

4¡3

=

4¡2

=

4¡1

=

¡¡1 0 ¡1 0 (0 +¡1 ) (2¡4 0 +2¡2 ¡1 )(2¡3 0 +2¡1 ¡1 )(2¡3 0 ¡2¡4 ¡1 )(2¡2 0 ¡2¡3 ¡1 )  ¡1 0 ¡1 0 (0 +¡1 ) (2¡4 0 +2¡2 ¡1 )(2¡3 0 +2¡1 ¡1 )(2¡2 0 ¡2¡3 ¡1 )(2¡1 0 ¡2¡2 ¡1 )  ¡¡1 0 ¡1 0 (0 +¡1 ) (2¡3 0 +2¡1 ¡1 )(2¡2 0 +2 ¡1 )(2¡2 0 ¡2¡3 ¡1 )(2¡1 0 ¡2¡2 ¡1 )  ¡1 0 ¡1 0 (0 +¡1 ) (2¡3 0 +2¡1 ¡1 )(2¡2 0 +2 ¡1 )(2¡1 0 ¡2¡2 ¡1 )(2 0 ¡2¡1 ¡1 ) 

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.7, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

4¡4

=

(2¡4 0 +2¡2 ¡1 )(2¡2 0 ¡2¡3 ¡1 )  (0 +¡1 )

4¡3 =

¡(2¡3 0 +2¡1 ¡1 )(2¡2 0 ¡2¡3 ¡1 )  (0 +¡1 )

4¡2

=

(2¡3 0 +2¡1 ¡1 )(2¡1 0 ¡2¡2 ¡1 )  (0 +¡1 )

4¡1 =

¡(2¡2 0 +2 ¡1 )(2¡1 0 ¡2¡2 ¡1 )  (0 +¡1 )

Now, we obtain from system (2) that ³ 4+1

4 4¡1 4 ¡4¡1

=



=

=

=

(2¡1 0 ¡2¡2 ¡1 )+(2 0 ¡2¡1 ¡1 )

4 4¡1 ¡4 ¡4¡1

 (

=

   

¡

 (

´

³

(2¡2 0 +2 ¡1 )(2 0 ¡2¡1 ¡1 ) (0 +¡1 ) ´ ³ ¡1 0 ¡1 0 (0 +¡1 ) (2¡3 0 +2¡1 ¡1 )(2¡2 0 +2 ¡1 )(2¡1 0 ¡2¡2 ¡1 )(2 0 ¡2¡1 ¡1 )

´  ¡¡1 0 ¡1 0 (0 +¡1 ) ¡  ³ (2¡2 0 +2 ¡1 )(2¡1 0 +2+1 ¡1 )(2¡1 0 ¡2¡2 ¡1 )(2 0 ¡2¡1 ¡1 ) ´   ¡1 0 ¡1 0 (0 +¡1 ) (2¡3 0 +2¡1 ¡1 )(2¡2 0 +2 ¡1 )(2¡1 0 ¡2¡2 ¡1 )(2 0 ¡2¡1 ¡1 )   (   ¡  )  +  )(  ¡  ) ³

2¡2 0 +2 ¡1 )(2 0 2¡1 ¡1 (0 +¡1 ) (  +  ) ¡1+ (2¡3 0 +2¡1 ¡1 ) 2¡1 0 2+1 ¡1

= ¡ =

¡1 0 ¡1 0 0 ¡1 (2¡2 0 +2 ¡1 )(2¡1 0 +2+1 ¡1 )(2 0 ¡2¡1 ¡1 )

¡1 0 ¡1 0 (0 +¡1 ) (2¡2 0 +2 ¡1 )(2¡1 0 +2+1 ¡1 )(2 0 ¡2¡1 ¡1 )(2+11 0 ¡2 ¡1 ) 

=

4+1

=

´

¡¡1 0 ¡1 0 (0 +¡1 ) (2¡2 0 +2 ¡1³)(2¡1 0 +2+1 ¡1 )(2¡1 0 ¡2¡2 ¡1 )( ´ 2 0 ¡2¡1 ¡1 ) ¡(2¡2 0 +2 ¡1 )(2¡1 0 ¡2¡2 ¡1 ) (0 +¡1 )  (   ¡(  2¡2 0 +2 ¡1 )(2 0 ¡2¡1 ¡1 ) 2¡2 0 +2 ¡1 )(2¡1 0 ¡2¡2 ¡1 ) ¡ (0 +¡1 ) (0 +¡1 )      ( + )

2¡2 0 +2 ¡1 )(2 0 ¡2¡1 ¡1 ) (0 +¡1 ) (  +  ) 1¡ (2¡3 0 +2¡1 ¡1 ) 2¡1 0 2+1 ¡1

2¡2 0

=¡



=¡

 (

2 ¡1 2 0 2¡1 ¡1 (0 +¡1 ) (  +  ) 1¡ (2¡3 0 +2¡1 ¡1 ) 2¡1 0 2+1 ¡1

2¡2 0 +2 ¡1 )(2 0 ¡2¡1 ¡1 ) (0 +¡1 ) (2¡2 0 +2 ¡1 ) (2¡1 0 +2+1 ¡1 )



¡(2¡1 0 +2+1 ¡1 )(2 0 ¡2¡1 ¡1 )  (0 +¡1 )

Also, we can prove the other relations. This completes the proof. Example 2. We assume that the initial conditions for the di¤erence system (2) are ¡1 = 038 0 = ¡17 ¡1 = 085 and 0 = 126. (See Fig. 2). plot of z(n+1)=t(n-1)z(n)/(t(n)-t(n-1)),t(n+1)=z(n-1)t(n)/(-z(n)-z(n-1)) 120 z(n) t(n)

100 80 60

z(n),t(n)

40 20 0 -20 -40 -60 -80

0

2

4

6

8 n

10

12

14

16

Figure 2. Sketch the behavior of the solution of the system (2).

1166

El-Dessoky ET AL 1161-1172

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.7, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

4. PERIODICITY OF THE SYSTEMS In this section, we study the periodicity nature of the solutions of the following systems of the di¤erence equations +1 =

 ¡1  ¡¡1 

+1 =

 ¡1  +¡1 

(3)

+1 =

 ¡1  +¡1 

+1 =

 ¡1  ¡¡1 

(4)

 ¡1 ¡ ¡¡1 

+1 =

 ¡1 ¡ ¡¡1 

(5)

+1 =

Where  = 0 1 2  and the initial conditions ¡1  0  ¡1 and 0 are arbitrary nonzero real numbers. Theorem 4.1. Suppose that f   g are solutions of di¤erence equation system (3) with 0 6= ¡¡1  0 6= ¡1  Then all solutions of system (3) are periodic with period six and for  = 0 1 2  6¡1 = ¡1  6 = 0  6+1 =

0 ¡1 0 ¡¡1 

6+2 =

¡1 (0 +¡1 ) (¡1 ¡0 ) 

6+3 =

6¡1 = ¡1  6 = 0  6+1 =

¡1 0 0 +¡1 

6+2 =

¡1 (0 ¡¡1 ) (0 +¡1 ) 

6+3 =

0 (0 +¡1 ) (0 ¡¡1 ) 

6+4 =

¡1 0 (¡1 ¡0 ) 

and 0 (¡1 ¡0 ) (0 +¡1 ) 

6+4 =

0 ¡1 (0 +¡1 ) 

Proof: For  = 0 the result holds. Now suppose that   0 and that our assumption holds for  ¡ 1. that is, 6¡7 = ¡1  6¡6 = 0  6¡5 =

0 ¡1 0 ¡¡1 

6¡4 =

¡1 (0 +¡1 ) (¡1 ¡0 ) 

6¡3 =

6¡7 = ¡1  6¡6 = 0  6¡5 =

¡1 0 0 +¡1 

6¡4 =

¡1 (0 ¡¡1 ) (0 +¡1 ) 

6¡3 =

0 (0 +¡1 ) (0 ¡¡1 ) 

6¡2 =

¡1 0 (¡1 ¡0 ) 

and 0 (¡1 ¡0 ) (0 +¡1 ) 

6¡2 =

0 ¡1 (0 +¡1 ) 

Now, we obtain from system (3) that 6¡1

=

6¡2 6¡3 6¡2 ¡6¡3

=

6¡1

=

6¡2 6¡3 6¡2 +6¡3

=

6

=

6¡1 6¡2 6¡1 ¡6¡2

=

6

=

6¡1 6¡2 6¡1 +6¡2

=

  0 (¡1 ¡0 ) ¡1 0 (¡1 ¡0 ) (0 +¡1 )     0 ¡1 0 (¡1 ¡0 ) ¡ (0 +¡1 ) (0 +¡1 )    0 ¡1 0 (0 +¡1 ) (0 +¡1 ) (0 ¡¡1 )     ¡1 0 0 (0 +¡1 ) (¡1 ¡0 ) + (0 ¡¡1 ) 

0 ¡1 ¡1 ( + ) 0 ¡1 0 ¡1 ¡1 ¡ ( + ) 0 ¡1 ¡1 0 ¡1 ( ¡ ) ¡1 0 ¡1 0 ¡1 + ( ¡ ) ¡1 0

=

¡1 0 0 0 ¡1 ¡0 (¡1 ¡0 )

=

0 ¡1 0 ¡¡1 0 +0 (0 +¡1 )

=

¡1 0 ¡1 ¡1 (0 +¡1 )¡0 ¡1

= 0 

=

¡1 ¡1 0 ¡1 (¡1 ¡0 )+¡1 0

= 0 

=

¡1 0 0 0 0

=

= ¡1 

0 ¡1 0 0 0

= ¡1 

We can prove the other relations similarly. The proof is completed. Theorem 4.2. If f   g are solutions of system (4) with 0 6= ¡1  0 6= ¡¡1  Then all solutions of system (4) are periodic with period six and given by the formulae 6¡1

= ¡1  6 = 0  6+1 =

6¡1

= ¡1  6 = 0  6+1 =

0 ¡1 0 +¡1 

¡1 0 0 ¡¡1 

6+2 = 6+2 =

1167

¡1 (0 ¡¡1 ) (0 +¡1 )  ¡1 (0 +¡1 ) (¡1 ¡0 ) 

6+3 = 6+3 =

0 (¡1 ¡0 ) (0 +¡1 ) 

0 (0 +¡1 ) (0 ¡¡1 ) 

6+4 =

6+4 =

¡1 0 0 +¡1 

0 ¡1 ¡1 ¡0 

El-Dessoky ET AL 1161-1172

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.7, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

Theorem 4.3. Assume that f   g are solutions of di¤erence equation system (5) with 0 6= ¡¡1  0 6= ¡¡1  Then all solutions of system (5) are periodic with period six and for  = 0 1 2  6¡1 6¡1

¡1 = ¡1  6 = 0  6+1 = ¡ 00+  6+2 = ¡1 0 = ¡1  6 = 0  6+1 = ¡ 0¡1 +¡1  6+2 =

¡1 (0 +¡1 ) (0 +¡1 )  ¡1 (0 +¡1 ) (0 +¡1 ) 

6+3 = 6+3 =

0 (0 +¡1 ) (0 +¡1 ) 

0 (¡1 +0 ) (0 +¡1 ) 

0 6+4 = ¡ 0¡1 +¡1 

¡1 6+4 = ¡ 00+  ¡1

Example 3. See Figure (3) where we take system (3) with the initial conditions ¡1 = 018 0 = 017 ¡1 = 05 and 0 = 086. plot of z(n+1)=t(n-1)z(n)/(t(n)-t(n-1)),t(n+1)=z(n-1)t(n)/(z(n)+z(n-1)) 1 z(n) t(n)

z(n),t(n)

0.5

0

-0.5

0

5

10

15

20

25

n

Figure 3. Draw the behavior of the solution of the system (3).

5. OTHER SYSTEMS In this section, we obtain the form of the solutions of the follwing systems of the di¤erence equations. Theorem 5.1. If f   g are solutions of system +1 =

 ¡1 ¡ +¡1 

+1 =

 ¡1 ¡ ¡¡1 

(6)

where  2 N0 and the initial conditions ¡1  0  ¡1 and 0 are arbitrary non zero real numbers, then for  = 0 1 2  2¡1 2¡1

such that

¡1 Q

= ¡1 = ¡1

¡1 Y

=0 ¡1 Y =0

(2¡2 0 +2¡1 ¡1 )(2¡1 0 ¡2 ¡1 ) (2¡1 0 +2 ¡1 )(2 0 ¡2+1 ¡1 ) 

2 = 0

(2¡1 0 +2 ¡1 )(2¡1 ¡1 ¡2¡2 0 ) (2 0 +2+1 ¡1 )(2 ¡1 ¡2¡1 0 ) 

2 = 0

¡1 Y

=0 ¡1 Y =0

(2 0 +2+1 ¡1 )(2¡1 0 ¡2 ¡1 ) (2+1 0 +2+2 ¡1 )(2 0 ¡2+1 ¡1 ) 

(2¡1 0 +2 ¡1 )(2 0 ¡2+1 ¡1 ) (2 0 +2+1 ¡1 )(2+1 0 ¡2+2 ¡1 ) 

 = 1.

=0

1168

El-Dessoky ET AL 1161-1172

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.7, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

Theorem 5.2. The solutions of system +1 =

 ¡1 ¡ ¡¡1 

+1 =

 ¡1 ¡ +¡1 

(7)

are given by the relations 2¡1 2¡1

= ¡1 = ¡1

¡1 Y

=0 ¡1 Y =0

(2¡1 ¡1 ¡2¡2 0 )(2¡1 0 +2 ¡1 ) (2 ¡1 ¡2¡1 0 )(2 0 +2+1 ¡1 ) 

2 = 0

(2¡1 0 ¡2 ¡1 )(2¡2 0 +2¡1 ¡1 ) (2 0 ¡2+1 ¡1 )(2¡1 0 +2 ¡1 ) 

2 = 0

¡1 Y

=0 ¡1 Y =0

(2 0 ¡2+1 ¡1 )(2¡1 0 +2 ¡1 ) (2+1 0 ¡2+2 ¡1 )(2 0 +2+1 ¡1 ) 

(2¡1 0 ¡2 ¡1 )(2 0 +2+1 ¡1 ) (2 0 ¡2+1 ¡1 )(2+1 0 +2+2 ¡1 ) 

where  2 N0 and the initial conditions ¡1  0  ¡1 and 0 are arbitrary non zero real numbers and

¡1 Q

 = 1.

=0

Theorem 5.3. Suppose that f   g+1 =¡1 are solutions of system +1 =

 ¡1 ¡ ¡¡1 

+1 =

 ¡1  +¡1 

(8)

where  2 N0 and the initial conditions ¡1  0  ¡1 and 0 are arbitrary non zero real numbers with ¡1 6= ¡0 . +1 Then f g+1 =¡1 and f g=¡1 are given by the formula for  = 0 1 2  4

=

¡1 0 ¡1 0 (0 +¡1 ) (2¡2 0 +2 ¡1 )(2¡1 0 +2+1 ¡1 )(2¡1 ¡1 +2¡2 0 )(2 ¡1 +2¡1 0 ) 

4+1

=

¡¡1 0 ¡1 0 (0 +¡1 ) (2¡2 0 +2 ¡1 )(2¡1 0 +2+1 ¡1 )(2 ¡1 +2¡1 0 )(2+1 ¡1 +2 0 ) 

4+2

=

¡1 0 ¡1 0 (0 +¡1 ) (2¡1 0 +2+1 ¡1 )(2 0 +2+2 ¡1 )(2 ¡1 +2¡1 0 )(2+1 ¡1 +2 0 ) 

4+3

=

¡¡1 0 ¡1 0 (0 +¡1 ) (2¡1 0 +2+1 ¡1 )(2 0 +2+2 ¡1 )(2+1 ¡1 +2 0 )(2+2 ¡1 +2+1 0 ) 

and 4

=

(2¡2 0 +2 ¡1 )(2 0 +2¡1 ¡1 )  (0 +¡1 )

4+2

=

(2¡1 0 +2+1 ¡1 )(2+1 0 +2 ¡1 )  (0 +¡1 )

4+1 =

(2¡1 0 +2+1 ¡1 )(2 0 +2¡1 ¡1 )  (0 +¡1 )

4+3 =

(2 0 +2+2 )(2+1 0 +2 ¡1 )  (0 +¡1 )

+1

Theorem 5.4. Let f   g=¡1 be solutions of system +1 =

 ¡1 ¡ +¡1 

+1 =

 ¡1  ¡¡1 

(9)

+1 Then f g+1 =¡1 and f g=¡1 are given by the following expressions for  = 0 1 2 

4

=

4+1

=

4+2

=

4+3

=

¡1 0 ¡1 0 (0 ¡¡1 ) (2¡2 0 ¡2 ¡1 )(2¡1 0 ¡2+1 ¡1 )(2¡1 0 ¡2¡2 ¡1 )(2 0 ¡2¡1 ¡1 )  ¡1 0 ¡1 0 (0 ¡¡1 ) (2¡2 0 ¡2 ¡1 )(2¡1 0 ¡2+1 ¡1 )(2 0 ¡2¡1 ¡1 )(2+1 0 ¡2 ¡1 )  ¡¡1 0 ¡1 0 (0 ¡¡1 ) (2¡1 0 ¡2+1 ¡1 )(2 0 ¡2+2 ¡1 )(2 0 ¡2¡1 ¡1 )(2+1 0 ¡2 ¡1 ) 

¡¡1 0 ¡1 0 (0 ¡¡1 ) (2¡1 0 ¡2+1 ¡1 )(2 0 ¡2+2 ¡1 )(2+1 0 ¡2 ¡1 )(2+2 0 ¡2+1 ¡1 ) 

and 4

=

4+2

=

(2¡2 0 ¡2 ¡1 )(2 0 ¡2¡1 ¡1 )  (0 ¡¡1 )

4+1 =

(2¡1 0 ¡2+1 ¡1 )(2+1 0 ¡2 ¡1 )  (0 ¡¡1 )

1169

¡(2¡1 0 ¡2+1 ¡1 )(2 0 ¡2¡1 ¡1 )  (0 ¡¡1 )

4+3 =

¡(2 0 ¡2+2 )(2+1 0 ¡2 ¡1 )  (0 ¡¡1 )

El-Dessoky ET AL 1161-1172

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.7, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

where  2 N0 and the initial conditions ¡1  0  ¡1 and 0 are arbitrary non zero real numbers with ¡1 6= 0 . Theorem 5.5. Let f   g+1 =¡1 be solutions of system +1 =

 ¡1 ¡ ¡¡1 

+1 =

 ¡1  ¡¡1 

(10)

where  2 N0 and the initial conditions ¡1  0  ¡1 and 0 are arbitrary non zero real numbers with ¡1 6= ¡0 . +1 Then f g+1 =¡1 and f g=¡1 are given by the following relations for  = 0 1 2  4

=

(2 0 ¡2¡1 ¡1 )(2¡2 0 +2 ¡1 )  0 +¡1

4+2

=

(2+1 0 ¡2 ¡1 )(2¡1 0 +2+1 ¡1 )  0 +¡1

(2 0 ¡2¡1 ¡1 )(2¡1 0 +2+1 ¡1 )  0 +¡1

4+1 =

4+3 =

(2+1 0 ¡2 ¡1 )(2 0 +2+2 ¡1 )  0 +¡1

and 4

=

4+1

=

4+2

=

4+3

=

¡0 ¡1 0 ¡1 (0 +¡1 ) (2¡1 0 ¡2¡2 ¡1 )(2 0 ¡2¡1 ¡1 )(2¡2 0 +2 ¡1 )(2¡1 0 +2+1 ¡1 )  0 ¡1 0 ¡1 (0 +¡1 ) (2 0 ¡2¡1 ¡1 )(2+1 0 ¡2 ¡1 )(2¡2 0 +2 ¡1 )(2¡1 0 +2+1 ¡1 )  ¡0 ¡1 0 ¡1 (0 +¡1 ) (2 0 ¡2¡1 ¡1 )(2+1 0 ¡2 ¡1 )(2¡1 0 +2+1 ¡1 )(2 0 +2+2 ¡1 ) 

0 ¡1 0 ¡1 (0 +¡1 ) (2+1 0 ¡2 ¡1 )(2+2 0 ¡2+1 ¡1 )(2¡1 0 +2+1 ¡1 )(2 0 +2+2 ¡1 ) 

Theorem 5.6. Suppose that f   g+1 =¡1 be solutions of system +1 =

 ¡1  ¡¡1 

+1 =

 ¡1 ¡ +¡1 

(11)

+1 Then f g+1 =¡1 and f g=¡1 are given by the following relations for  = 0 1 2 

4

=

4+2

=

(2 0 ¡2¡1 ¡1 )(2¡2 0 ¡2 ¡1 )  0 ¡¡1

4+1 =

(2+1 0 ¡2 ¡1 )(2¡1 0 ¡2+1 ¡1 )  0 ¡¡1

¡(2 0 ¡2¡1 ¡1 )(2¡1 0 ¡2+1 ¡1 )  0 ¡¡1

4+3 =

¡(2+1 0 ¡2 ¡1 )(2 0 ¡2+2 ¡1 )  0 ¡¡1

and 4

=

4+1

=

4+2

=

4+3

=

¡0 ¡1 0 ¡1 (0 ¡¡1 ) (2¡1 0 ¡2¡2 ¡1 )(2 0 ¡2¡1 ¡1 )(2¡2 0 ¡2 ¡1 )(2¡1 0 ¡2+1 ¡1 )  0 ¡1 0 ¡1 (0 ¡¡1 ) (2 0 ¡2¡1 ¡1 )(2+1 0 ¡2 ¡1 )(2¡2 0 ¡2 ¡1 )(2¡1 0 ¡2+1 ¡1 )  ¡0 ¡1 0 ¡1 (0 ¡¡1 ) (2 0 ¡2¡1 ¡1 )(2+1 0 ¡2 ¡1 )(2¡1 0 ¡2+1 ¡1 )(2 0 ¡2+2 ¡1 ) 

0 ¡1 0 ¡1 (0 ¡¡1 ) (2+1 0 ¡2 ¡1 )(2+2 0 ¡2+1 ¡1 )(2¡1 0 ¡2+1 ¡1 )(2 0 ¡2+2 ¡1 ) 

where  2 N0 and the initial conditions ¡1  0  ¡1 and 0 are arbitrary non zero real numbers with ¡1 6= 0 . Theorem 5.7. Let f   g+1 =¡1 be solutions of system +1 =

 ¡1  +¡1 

+1 =

 ¡1 ¡ ¡¡1 

(12)

+1

Then f g+1 =¡1 and f g=¡1 are given by the following relations for  = 0 1 2  4

=

(2 0 +2¡1 ¡1 )(2¡2 0 +2 ¡1 )  0 +¡1

4+2

=

(2+1 0 +2 ¡1 )(2¡1 0 +2+1 ¡1 )  0 +¡1

1170

4+1 =

(2 0 +2¡1 ¡1 )(2¡1 0 +2+1 ¡1 )  0 +¡1

4+3 =

(2+1 0 +2 ¡1 )(2 0 +2+2 ¡1 )  0 +¡1

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.7, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

and 4

=

¡0 ¡1 0 ¡1 (0 +¡1 ) (2¡1 0 +2¡2 ¡1 )(2 0 +2¡1 ¡1 )(2¡2 0 +2 ¡1 )(2¡1 0 +2+1 ¡1 ) 

4+1

=

0 ¡1 0 ¡1 (0 +¡1 ) (2 0 +2¡1 ¡1 )(2+1 0 +2 ¡1 )(2¡2 0 +2 ¡1 )(2¡1 0 +2+1 ¡1 ) 

4+2

=

¡0 ¡1 0 ¡1 (0 +¡1 ) (2 0 +2¡1 ¡1 )(2+1 0 +2 ¡1 )(2¡1 0 +2+1 ¡1 )(2 0 +2+2 ¡1 ) 

4+3

=

0 ¡1 0 ¡1 (0 +¡1 ) (2+1 0 +2 ¡1 )(2+2 0 +2+1 ¡1 )(2¡1 0 +2+1 ¡1 )(2 0 +2+2 ¡1 ) 

where  2 N0 and the initial conditions ¡1  0  ¡1 and 0 are arbitrary non zero real numbers with ¡1 6= ¡0 .

Acknowledgements This article was funded by the Deanship of Scienti…c Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR technical and …nancial support.

REFERENCES 1. P. Cull, M. Flahive, and R. Robson, Di¤erence Equations: From Rabbits to Chaos, Undergraduate Texts in Mathematics, Springer, New York, NY, USA, 2005. 2. R. J. H. Beverton and S. J. Holt, On the Dynamics of Exploited Fish Populations, Fishery Investigations Series II, Volume 19, Blackburn Press, Caldwell, NJ, USA, 2004. 3. M. R. S. Kulenovic and G. Ladas, Dynamics of Second Order Rational Di¤erence Equations with Open Problems and Conjectures, Chapman & Hall / CRC Press, 2001. 4. V. L. Kocic and G. Ladas, Global Behavior of Nonlinear Di¤erence Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, 1993. 5. S. Elaydi, An Introduction to Di¤erence Equations, Undergraduate Texts in Mathematics, Springer, New York, NY, USA, 3 edition, (2005). 6. M. M. El-Dessoky, E. M. Elsayed and M. Alghamdi, Solutions and periodicity for some systems of fourth order rational di¤erence equations, J. Comput. Anal. Appl., Vol. 18(1), (2015), 179-194. 7. E. A. Grove, G. Ladas, L. C. McGrath, and C.T. Teixeira, Existence and behavior of solutions of a rational system, Commun. Appl. Nonlinear Anal., 8 (2001), 1-25. 8. M. Mansour, M. M. El-Dessoky and E. M. Elsayed, On the solution of rational systems of di¤erence equations, J. Comput. Anal. Appl., 15 (5) (2013), 967-976. 9. M. M. El-Dessoky, The form of solutions and periodicity for some systems of third order rational di¤erence equations, Math. Methods Appl. Sci., 39, (2016), 1076-1092. 10. N. Touafek and E. M. Elsayed, On the periodicity of some systems of nonlinear di¤erence equations, Bull. Math. Soc. Sci. Math. Roumanie, Tome 55 (103), No. 2, (2012), 217–224. 11. L. Yang and J. Yang, Dynamics of a system of two nonlinear di¤erence equations, Int. J. Contemp. Math. Sciences, 6 (5) (2011), 209 - 214 12. Q. Din, M. N. Qureshi and A. Qadeer Khan, Dynamics of a fourth-order system of rational di¤erence equations, Adv. Di¤erence Equ., 2012, (2012): 215 doi: 10.1186/1687-1847-2012-215. 13. Q. Din, Asymptotic behavior of an anti-competitive system of second-order di¤erence equations, J. Egyptian Math. Soc., 24, (2016), 37-43. 14. M. M. El-Dessoky, E. M. Elsayed, On a solution of system of three fractional di¤erence equations, J. Comput. Anal. Appl., 19, (2015), 760-769. 15. N. Battaloglu, C. Cinar and I. Yalç¬nkaya, The dynamics of the di¤erence equation, ARS Combinatoria, 97 (2010), 281-288. 16. M. Aloqeili, Dynamics of a rational di¤erence equation, Appl. Math. Comp., 176(2), (2006), 768-774. 17. C. Cinar, I. Yalçinkaya and R. Karatas, On the positive solutions of the di¤erence equation system +1 =   +1 =  ¡1 ¡1  J. Inst. Math. Comp. Sci., 18 (2005), 135-136.

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18. S. E. Das and M. Bayram, On a system of rational di¤erence equations, World Applied Sciences Journal, 10(11) (2010), 1306–1312. 19. Q. Din, Dynamics of a discrete Lotka-Volterra model, Adv. Di¤erence Equ., 2013, (2013): 95. 20. E. O. Alzahrani, M. M. El-Dessoky, E. M. Elsayed and Y. Kuang, Solutions and Properties of Some Degenerate Systems of Di¤erence Equations, J. Comput. Anal. Appl., Vol. 18(2), (2015), 321-333. 21. A. Q. Khan, M. N. Qureshi, Global dynamics of some systems of rational di¤erence equations, J. Egyptian Math. Soc., 24, (2016), 30-36. 22. E. M. Elabbasy, H. El-Metwally and E. M. Elsayed, Global behavior of the solutions of di¤erence equation, Adv. Di¤er. Equ., 2011, 2011:28. 23. M. M. El-Dessoky, M. Mansour, E. M. Elsayed, Solutions of some rational systems of di¤erence equations, Utilitas Mathematica, 92, (2013), 329-336. 24. M. M. El-Dessoky, On the solutions and periodicity of some nonlinear systems of di¤erence equations, J. Nonlinear Sci. Appl., 9(5), (2016), 2190-2207. 25. E. M. Elsayed, A. M. Ahmed, Dynamics of a three-dimensional systems of rational di¤erence equations, Math. Methods Appl. Sci., 39, (2016), 1026-1038. 26. N. Touafek and E. M. Elsayed, On a second order rational systems of di¤erence equations, Hokkaido Math. J., 44, (2015), 29–45. 27. Y. Yazlik, D. T. Tollu, N. Taskara, On the Behaviour of Solutions for Some Systems of Di¤erence Equations, J. Comput. Anal. Appl., 18 (1), (2015), 166-178. 28. Qianhong Zhang, Jingzhong Liu, and Zhenguo Luo, Dynamical Behavior of a System of Third-Order Rational Di¤erence Equation, Discrete Dyn. Nat. Soc., 2015, (2015), Article ID 530453, 6 pages. 29. M. M. El-Dessoky, On a solvable for some systems of rational di¤erence equations, J. Nonlinear Sci. Appl., Vol. 9(6), (2016), 3744-3759. 30. Wenqiang Ji, Decun Zhang and Liying Wang, Dynamics and behaviors of a third-order system of di¤erence equation, Mathematical Sciences, 2013, (2013),:34. 31. Q. Zhang and W. Zhang, On a system of two high-order nonlinear di¤erence equations, Adv. Math. Phys., 2014, (2014), Article ID 729273, 8 pages. 32. Mehmet Gümü¸s and Yüksel Soykan, Global Character of a Six-Dimensional Nonlinear System of Di¤erence Equations, Discrete Dyn. Nat. Soc., 2016, (2016), Article ID 6842521, 7 pages. 33. M. M. El-Dessoky, Solution of a rational systems of di¤erence equations of order three, Mathematics,4(3), (2016), 1-12. 34. A. Gelisken, On A System of Rational Di¤erence Equations, J. Comput. Anal. Appl., Vol. 23(4), (2017), 593-606. 35. N. Haddad, N. Touafek, Julius Fergy T. Rabago, Solution form of a higher-order system of di¤erence equations and dynamical behavior of its special case, Math. Methods Appl. Sci.,40(10), (2017), 3599-3607. 36. Chang-you Wang, Xiao-jing Fang, Rui Li, On the dynamics of a certain four-order fractional di¤erence equations , J. Comput. Anal. Appl., Vol. 22(5), (2017), 968-976. 37. M. M. El-Dessoky, E. M. Elsayed and E. O. Alzahrani, The form of solutions and periodic nature for some rational di¤erence equations systems, J. Nonlinear Sci. Appl., Vol., 9(10), (2016), 5629–5647. 38. M. M. El-Dessoky, Abdul Khaliq and Asim Asiri, On some rational systems of di¤erence equations, J. Nonlinear Sci. Appl., Vol. 11(1), (2018), 49-72. 39. Asim Asiri, M. M. El-Dessoky and E. M. Elsayed, Solution of a third order fractional system of di¤erence equations , J. Comput. Anal. Appl., Vol., 24(3), (2018), 444-453.

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Hardy type inequalities for Choquet integrals George A. Anastassiou Department of Mathematical Sciences University of Memphis Memphis, TN 38152, U.S.A. [email protected] Abstract Here we present Hardy type integral inequalities for Choquet integrals. These are very general inequalities involving convex and increasing functions. Initially we collect a rich machinery of results about Choquet integrals needed next, and we prove also results of their own merit such as, Choquet-Hölder’s inequalities for more than two functions and a multivariate Choquet-Fubini’s theorem. The main proving tool here is the property of comonotonicity of functions. We …nish with independent estimates on left and right Riemann-Liouville-Choquet fractional integrals.

2010 AMS Mathematics Subject Classi…cation: 26A33, 26D10 26D15, 26E50, 28E10. Keywords and Phrases: Choquet integral, Hardy inequality, comonotonicity, fractional integral, convexity.

1

Introduction

To motivate the work in this article we mention the Riemann-Liouville fractional integrals, see [9]. Let [a; b], ( 1 < a < b < 1) be a …nite interval on the real axis R. The left and right Riemann-Liouville fractional integrals Ia+ f and Ib f (respectively) of order > 0 are de…ned by Z x 1 1 Ia+ f (x) = f (t) (x t) dt, (x > a) ; ( ) a Z b 1 1 Ib f (x) = f (t) (t x) dt; (x < b) ; ( ) x where is the Gamma function. We mention a basic property of the operators Ia+ f and Ib f of order > 0, see also [11]: It holds that the fractional integral operators Ia+ f and Ib f are 1

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bounded in Lp (a; b), 1 Ia+ f

p p

1, that is K kf kp ,

where K=

Ib f

(b

p

K kf kp ;

a) : ( )

The …rst inequality that is the result involving the left-sided fractional integral, was proved by H.G. Hardy in one of his …rst papers, see [7]. He did not write down the constant, but the calculation of the constant was hidden inside his proof. General Hardy inequalities of the above type were derived also in [8] and [1]. We continue this kind of research for Choquet integrals based on the comonotonicity property of functions and convexity. We derive a wide range of Choquet integral inequalities of Hardy type.

2

Background

In this section we give some de…nitions and basic properties of Choquet integral essential for this work. De…nition 1 ([15]) Let X be a non-empty set, F X and : F ! [0; 1] be a nonnegative real-valued a fuzzy measure i¤ : (1) (?) = 0; (2) for any A; B 2 F, A B implies (A) (3) for fAn g F, A1 A2 ::: An

be a -algebra of subsets of set function, is said to be

(B) (monotonicity), :::, implies lim (An ) =

([1 n=1 An ) (continuity from below) (4) for fAn g F, A1 A2 ::: An :::, lim (An ) = (\1 A ) (continuity from above). n=1 n

n!1

(A1 ) < 1;

implies

n!1

If is a fuzzy measure from F to [0; 1] with (X) = 1, is called a regular fuzzy measure. If is a fuzzy measure, (X; F; ) is called a fuzzy measure space and (X; F) is a fuzzy measurable space. Clearly is not necessarily an additive measure. Let F be the set of all real-valued nonnegative measurable functions de…ned on X. De…nition 2 ([10]) Let (X; F; ) be a fuzzy measure space, submodular (supermodular) if (A \ B) + (A [ B)

( ) (A) + (B) ; 8 A; B

is said to be

F:

(1)

De…nition 3 ([4]) Let f; g 2 F , f and g are said to be comonotonic i¤ f (x) < f (x0 ) implies g (x) g (x0 ), 8 x; x0 2 X: 2

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De…nition 4 ([5], [16]) Let (X; F; ) be a fuzzy measure space, f 2 F and A 2 F. The Choquet integral of f with respect to on A is de…ned by Z Z 1 (C) fd = (A \ fxjf (x) g) d : (2) A

0

R

If (C) X f d < 1, we call f (C)-integrable, L1 ( ) is the set of all (C)integrable function. R R Clearly (C) X f d < 1, implies (C) A f d < 1:

Theorem 5 ([14]) Let (X; F; ) be a fuzzy measurable space, ff1 ; f2 ; f g A; B 2 F and c 0 constant. Then, R (1) if (A) = 0, then (C) A f d = 0; R (2) (C) A cd = c (A) ; (3) if f1 f2 , then Z Z (C) f1 d (C) f2 d ; A

F,

(3)

A

R R (4) if A B, then (C) A f d (C) B f d ; R R (5) (C) A (f + c) d = (C) A f d + c (A) ; R R (6) (C) A cf d = c (C) A f d :

Theorem 6 ([5]) Let (X; F; ) be a fuzzy measure space and f; g 2 F . Then (1) if f; g are comonotonic, then for any A 2 F, Z Z Z (C) (f + g) d = (C) f d + (C) gd ; (4) A

(2) if

A

A

is submodular, then for any A 2 F, Z Z Z (C) (f + g) d (C) f d + (C) gd : A

A

(5)

A

The Jensen’s inequality for Choquet integrals follows: Theorem 7 ([13]) Let (X; F; ) be a fuzzy measure space and f 2 L1 ( ). If is a regular fuzzy measure and : [0; 1) ! [0; 1) is a convex function, then Z Z (C) fd (C) (f ) d : (6) X

X

Corollary 8 ([13]) Let (X; F; ) be a fuzzy measure space and f 2 L1 ( ). If is a regular fuzzy measure, then Z Z p (C) fd (C) f pd ; (7) X

X

for any 1 < p < 1: 3

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Theorem 9 ([13]) (Hölder’s inequality) Let (X; F; ) be a fuzzy measure space and f; g 2 F . If is a submodular fuzzy measure and 1 < p; q < 1 with 1 1 p + q = 1, then (C)

Z

f gd

(C)

X

Z

f pd

1 p

(C)

X

Z

1 q

gq d

:

(8)

X

Theorem 10 ([13]) (Minkowski inequality) Let (X; F; ) be a fuzzy measure space and f; g 2 F . If is a submodular fuzzy measure and 1 p < 1, then 1 1 1 Z Z Z p p p p p p (C) f d + (C) g d : (9) (C) (f + g) d X

X

X

We give Theorem 11 (Hölder’s inequality for three functions) Let (X; F; ) be a fuzzy measure space and f1 ; f2 ; f3 2 F . If is a submodular fuzzy measure and 1 1 1 1 < p1 p2 p3 < 1 with p1 + p2 + p3 = 1, then (C)

Z

f1 f2 f3 d

(C)

X

Z

1 p1

f1p1 d

(C)

X

Z

1 p2

f2p2 d

(C)

X

Z

1 p3

:

X

Proof. Let p = p3p3 1 > 1 and q = p3 . Notice that p1 + 1q = 1: We apply (8) as follows 1 Z Z Z p p (C) f1 f2 f3 d (C) (f1 f2 ) d (C) f3p3 d X

f3p3 d

X

(10)

1 p3

:

(11)

X

We see that p p + =p p1 p2

1 1 + p1 p2

=p 1

Clearly it holds pp1 ; pp2 > 1. Therefore we get Z Z p (8) p 1 p p (C) f1 f2 d (C) f1 p d X

Z

f1p1 d

p p1

(C)

X

That is (C)

X

p

(f1 f2 ) d

p3 1 p3

=p

p p1

(C)

X

(C) Z

1 p3

Z

p f2

p2 p

d

= 1:

p p2

(12)

=

(13)

:

(14)

X

Z

f2p2 d

p p2

:

X

1 p

(C)

Z

f1p1 d

X

1 p1

(C)

Z

f2p2 d

1 p2

X

Combining (11) and (14), we produce (10). In general we have 4

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Theorem 12 (Hölder’s inequality for n functions) Let (X; F; ) be a fuzzy measure space and fi 2 F , i = 1; :::; n 2 N. If is a submodular fuzzy measure and n P 1 1 < p1 p2 ::: pn < 1 with pi = 1, then i=1

(C)

Z Y n

n Y

fi d

X i=1

(C)

Z

X

i=1

fipi d

1 pi

:

(15)

Proof. By induction. Remark 13 Let A be a -algebra, and let fAk gk2N A be a family of pairwise disjoint sets. Here P is a probability measure on (X; A) with only the …nite additivity property valid: i.e., P

([nk=1 Ak )

=

n X

k=1

P (Ak ) , 8 n 2 N.

We observe that n P ([1 k=1 Ak ) = lim P ([k=1 Ak ) = lim n!1

n!1

n X

P (Ak ) =

k=1

1 X

P (Ak ) :

(16)

k=1

That is, the countable additivity property holds, hence P is a usual probability measure. Notice that a -algebra on X is also an algebra of subsets of X. De…nition 14 ([3], [6]) For every space and algebra A of subsets of a setfunction : A ! R is called a (normalized) capacity if it satis…es the following: (i) (?) = 0; ( ) = 1; (17) (ii) 8 A; B 2 A : A B ) (A) (B) : From (i) and (ii) we get that the range of

is contained in [0; 1] :

In general the Choquet integral is de…ned as follows: De…nition 15 ([3], [12]) Let ( ; A) be an algebra and f : ! R is a bounded A-measurable function and is any (normalized) capacity on we de…ne the Choquet integral of f with respect to to be the number Z Z 1 (C) f (!) d (!) = (f! 2 : f (!) g) d + (18) 0

Z

0 1

[ (! 2

: f (!)

g)

1] d ;

5

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where the integrals are taken in the sense of Riemann. A (normalized) capacity is called probability ([6]) i¤ 8 A; B 2 A :

(A [ B) + (A \ B) =

(A) + (B) :

(19)

Notice that since the integrands are monotone, the Choquet integral always exists, and if is a probability it collapses to a usual Lebesgue integral. De…nition 16 ([6]) Let f; g : ! R be two bounded A-measurable functions. We say that f and g are comonotonic, if for every !; ! 0 2 , (f (!)

f (! 0 )) (g (!)

g (! 0 ))

0:

(20)

A class of functions F is said to be comonotonic if for every f; g 2 F , f and g are comonotonic. Proposition 17 ([6]) If and are (normalized) capacities on the algebra ( ; A) ; and f; g : ! R are bounded A-measurable functions then: (i) Z (C)

1A d =

(A) , 8 A 2 A;

where 1A is the characteristic function on A, (ii) (positive homogeneity) Z Z (C) pf d = p (C) fd , for every p (iii) (monotonicity) f

(iv) (C)

Z

g implies Z (C) fd

(C)

(f + p) d = (C)

Z

Z

gd ;

(21)

0;

(22)

(23)

f d + p, 8 p 2 R,

(24)

(v) (comonotonic additivity) If f; g are comonotonic then Z Z Z (C) (f + g) d = (C) f d + (C) gd .

(25)

We need the very important Lemma 18 ([6]) Let ( ; A) be an algebra. Suppose that F is a comonotonic class of bounded and A-measurable functions from into R and is a (normalized) capacity on ( ; A). Then there exists a probability measure P on ( ; A) such that for every f 2 F Z Z fd = f dP: (26) Here

R

f dP is a standard integral of Lebesgue type. 6

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Based on Remark 13, Lemma 18 is still valid in case that ( ; A) is a algebra.

-

De…nition 19 ([6]) Let X; Y be two sets and Z = X Y . Let f : Z ! R. We say that f has comonotonic x-sections if for every x; x0 2 X, f (x; ) : Y ! R, and f (x0 ; ) : Y ! R are comonotonic functions. Comonotonicity of y-sections is similarly de…ned. We call f slice-comonotonic if it has both comonotonic x-sections and y-sections. Remark 20 Notice that De…nitions 14-16 and Proposition 17, are still valid when ( ; A) is a -algebra. Next we mention Fubini’s theorem for Choquet integrals. Theorem 21 ([2]) Let ( 1 ; 1 ), ( 2 ; 2 ) be -algebras. Let ui , i = 1; 2 be submodular (or supermodular) regular fuzzy measures on i , respectively. Let = 1 -algebra = 1 2 be endowed with the product 2 . Let f : -measurable mapping, then: 1 2 ! R be a slice-comonotonic bounded R 1) f ( ; ! 2 ) is 1 -measurable and ! 2 2 2 ! (C) 1 f (s; ! 2 ) du1 (s) is bounded and 2 -measurable, R f (! 1 ; ) is 2 -measurable and ! 1 2 1 ! (C) 2 f (! 1 ; t) du2 (t) is bounded and 1 -measurable, R R R R 2) the iterated integrals (C) 2 1 f du1 du2 , (C) 1 2 f du2 du1 exist and are equal: Z Z Z Z (C) (C) f (! 1 ; ! 2 ) du1 du2 = (C) (C) f (! 1 ; ! 2 ) du2 du1 : 2

1

1

2

(27)

We give De…nition 22 Let f :

n Q

i=1

i

! R, n 2 N. If the i-sections

f (x1 ; :::; xi 1 ; ; xi+1 ; :::; xn ) and f x01 ; :::; x0i 1 ; ; x0i+1 ; :::; x0n are comonotonic functions, for all i = 1; :::; n; where the vectors (x1 ; :::; xi 1 ; xi+1 ; :::; xn ) ; nQ1 x01 ; :::; x0i 1 ; x0i+1 ; :::; x0n 2 j are di¤ erent, for all i = 1; 2; :::; n; we call f j=1 j6=i

slice-n-comonotonic function. We denote by a permutation of the set f1; 2; :::; ng into itself, n 2 N. There are n! permutations.

In [2] is mentioned that Theorem 21 can be generalized for n spaces. Next we state in brief Fubini’s theorem for n Choquet iterated integrals.

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Theorem 23 Let ( i ; i ) be -algebras, i = 1; 2; :::; n 2 N. Let ui , i = 1; 2; :::; n be submodular (or supermodular) regular fuzzy measures on i , respectively. Let n n Q Q = = ni=1 i . Let f : i be endowed with the product -algebra i ! i=1

i=1

R be a slice-comonotonic bounded -measurable mapping, then Z Z Z (C) ::: f du1 du2 :::dun = n

(C)

Z

(n)

Z

n

1

::: (n

1)

1

Z

f du

(1) du (2) :::du (n) ;

(28)

(1)

for any permutation on the set f1; :::; ng. All the iterated Choquet integrals in (28) exist and are equal. Proof. By induction, (23) and using Theorem 21. Remark 24 If is a countably additive bounded measure, then the Choquet R integral (C) A f d reduces to the usual Lebesgue type integral (see, e.g. [5], p. 62, or [17], p. 226), above it is A :

3

Main Results

This section is motivated by [8]. Let the fuzzy measure spaces ( 1 ; 1 ; 1 ) and ( 2 ; 2 ; 2 ), where 1 ; 2 are regular fuzzy measures, furthermore 1 ; 2 are assumed to be submodular. Let k : 1 2 ! R+ which is a bounded measurable function and k (x; y) is slice comonotonic and belongs to a comonotonic class F1 as a f unction of y: Consider the function Z K (x) = (C) k (x; y) d 2 (y) , x 2 1 ; (29) 2

and assume that K (x) > 0: Notice that K is bounded. Denote by W (k) the class of functions g : 1 ! R+ , such that Z g (x) = (C) k (x; y) f (y) d 2 (y) ;

(30)

2

where f : 2 ! R+ is a bounded measurable function, such that k (x; y) f (y) is slice comonotonic and belongs to a comonotonic class F2 as a f unction of y: Notice that g is also bounded. We give

8

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Theorem 25 Let u be a nonnegative measurable function on u(x) on 2 by 1 . De…ne K(x) is bounded on (y) = (C)

Z

1

u (x) k (x; y) d K (x)

1

1.

Assume that

(x) ;

(31)

which is bounded. Let : R+ ! R+ be a convex and increasing function, such that k (x; y) (f (y)) is x-section comonotonic with comonotonic class F3 . Assume here that (F1 [ F2 [ F3 ) F , where F is one comonotonic class of 1 functions on 2 . Assume further that u (x) (K (x)) k (x; y) (f (y)) is slicecomonotonic. Then Z Z g (x) u (x) d 1 (x) (C) (32) (C) (y) (f (y)) d 2 (y) ; K (x) 1 2 holds for all g 2 W (k), with f as in (30). Proof. We observe that Z (C) (C)

Z

1

1 (C) K (x)

u (x) 1

g (x) K (x)

u (x) Z

d

1

(x) =

k (x; y) f (y) d

2

(y) d

1

(x) =

(33)

2

(next we use Lemma 18, where P is a probability measure on 2 ) Z Z 1 (C) u (x) (C) k (x; y) f (y) dP (y) d 1 (x) K (x) 2 1 R (we can also write K (x) = 2 k (x; y) dP (y) ; hence by classic Jensen’s inequality) Z Z 1 (C) u (x) (K (x)) (C) k (x; y) (f (y)) dP (y) d 1 (x) = (34) 1

2

(again by Lemma 18) Z (C) u (x) (K (x)) 1

(C)

Z

(C) 1

Z

1

(C)

Z

u (x) (K (x))

k (x; y)

(f (y)) d

2

(y) d

1

(x) =

k (x; y)

(f (y)) d

2

(y) d

1

(x) =

2

1

2

1

(since the functions (f (y)) and u (x) (K (x)) k (x; y) (f (y)) are bounded and the second one is slice-comonotonic, we can apply Fubini’s Theorem 21) Z Z 1 (C) (C) u (x) (K (x)) k (x; y) (f (y)) d 1 (x) d 2 (y) = 2

1

9

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(C)

Z

(f (y)) (C) 2

Z

u (x) (K (x))

1

k (x; y) d

1

(x) d

(31)

2

(y) =

(35)

1

(C)

Z

(f (y)) (y) d

2

(y) ;

2

proving the claim. We also give Corollary 26 All as in Theorem 25, with = identity mapping. Then Z Z u (x) (C) g (x) d 1 (x) (C) (y) f (y) d 2 (y) ; K (x) 1 2

(36)

holds for all g 2 W (k), with f as in (30). Corollary 27 All as in Theorem 25, with (x) = xp , 8 x 2 R+ , p > 1. Then Z Z u (x) p (C) g (x) d 1 (x) (C) (y) f p (y) d 2 (y) ; (37) K p (x) 1 2 holds for all g 2 W (k), with f as in (30). Corollary 28 All as in Theorem 25, with (x) = ex , 8 x 2 R+ . Then Z Z g(x) (y) ef (y) d 2 (y) ; (C) u (x) e K(x) d 1 (x) (C)

(38)

2

1

holds for all g 2 W (k), with f as in (30). Corollary 29 All as in Theorem 25, with K (x). Then Z Z (C) g (x) d 1 (x) (C)

= identity mapping and u (x) =

(y) f (y) d

2

holds for all g 2 W (k), with f as in (30). Here (y) = (C)

R

1

Corollary 30 All as in Theorem 25, with u (x) = K p (x). Then Z Z (C) g p (x) d 1 (x) (C) 1

(y) ;

(39)

2

1

k (x; y) d

1

(x) is bounded:

(x) = xp , 8 x 2 R+ , p > 1, and (y) f p (y) d

2

(y) ;

(40)

2

holds for all g 2 W (k), with f as in (30). Here Z (y) = (C) K p 1 (x) k (x; y) d 1 (x) is bounded:

(41)

1

10

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Remark 31 (on Corollary 30) Let us assume that k (x; y) M , M > 0, 8 p (x; y) 2 1 , then K (x) M . And from (41), (y) M . 2 Consequently, from (40), it holds Z Z (C) g p (x) d 1 (x) M p (C) f p (y) d 2 (y) ; (42) 1

2

and even better written Z (C) g p (x) d

1 p

1

(x)

M

(C)

1

Z

1 p

p

f (y) d

2

(y)

:

(43)

2

Next we rewrite the result of (43) in detail. Theorem 32 Assume that k (x; y) M , M > 0, 8 (x; y) 2 p > 1. De…ne Z (y) = (C) K p 1 (x) k (x; y) d 1 (x) ,

1

2,

and let (44)

1

p

which is bounded. Here k (x; y) (f (y)) is x-section comonotonic with comonotonic class F3 . Assume that (F1 [ F2 [ F3 ) F , where F one comonotonic class p 1 p on 2 . Assume further that (K (x)) k (x; y) (f (y)) is slice-comonotonic. Then 1 1 Z Z p p p p (C) g (x) d 1 (x) M (C) f (y) d 2 (y) ; (45) 1

2

holds for all g 2 W (k), with f as in (30). Remark 33 Assume that k (x; y) M , M > 0, 8 (x; y) 2 directly by (30) we get Z g (x) M (C) f (y) d 2 (y) , 8 x 2 1 :

1

2.

Hence

2

Therefore

Z

g (x) d

1 (x)

M

(C)

1

Z

f (y) d

2

(y) ;

(46)

2

holds for all g 2 W (k), with f as in (30).

R Theorem 34 De…ne on 2 by (y) = (C) 1 k (x; y) d 1 (x) ; which is bounded. p Let p > 1. Here k (x; y) (f (y)) is slice comonotonic and belongs to a comonotonic class F3 as a function of y. Assume that (F1 [ F2 [ F3 ) F , where F one comonotonic class on 2 . Then Z Z 1 p p (K (x)) (y) f p (y) d 2 (y) ; (47) (C) g (x) d 1 (x) (C) 1

2

holds for all g 2 W (k), with f as in (30). 11

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Proof. By Theorem 25, take f (x) = xp , x

0, p > 1, and u (x) = K (x) :

Corollary 35 All as in Theorem 34. Then (C)

Z

g p (x) d

1 p

M

1 (x)

(C)

1

Z

1 p

f p (y) d

2 (y)

:

(48)

2

holds for all g 2 W (k), with f as in (30). Here k (x; y) (x; y) 2 1 2:

M , M > 0, 8

Proof. Since p > 1, 1 p < 0. Hence the left hand side of (47) is greater R 1 p equal to M 1 p (C) 1 g p (x) d 1 (x) , by K (x) M and (K (x)) M 1 p. R And the right hand side of (47) is less equal to M (C) 2 f p (y) d 2 (y) , by (y) M . Therefore Z Z 1 p p M (C) g (x) d 1 (x) M (C) f p (y) d 2 (y) ; (49) 1

2

proving the claim.

4

Appendix

Here B stands for the Borel -algebra on [a; b] : Let the fuzzy measure spaces ([a; b] ; B; 1 ) and ([a; b] ; B; 2 ), where [a; b] R and 1 ; 2 are bounded fuzzy measures with 2 submodular. Let p; q > 1 such that p1 + 1q = 1. Let f : [a; b] ! R+ which is bounded and B-measurable. We de…ne the left and right Riemann-Liouville-Choquet fractional integrals of order > 1 (respectively): Z x 1 1 (C) (x t) f (t) d 2 (t) ; (50) Ia+ f (x) = ( ) a and 1 (C) ( )

Ib f (x) =

Z

b

(t

x)

1

f (t) d

2

(t) ;

(51)

x

8 x 2 [a; b], where is the gamma function. We assume that Ia+ f and Ib f are B-measurable functions. Clearly Ia+ f; Ib f are nonnegative and bounded over [a; b] : Remark 36 By Theorem 9 we obtain Ia+ f (x)

1 ( )

(C)

Z

x

(x

p(

t)

1)

a

1 p

d

2 (t)

(C)

Z

1 q

x

f q (t) d

2 (t)

a

(52)

12

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1 (b ( )

p(

1)

a)

([a; b])

2

1 p

(C)

Z

! q1

b q

f (t) d

2

(t)

a

:

Hence it holds Ia+ f (x)

1 p (b ( ( ))

p

8 x 2 [a; b] : Therefore

Z

(C)

p(

1)

a)

2

([a; b]) (C)

Z

! pq

b q

f (t) d

2

(t)

a

b

p

Ia+ f (x)

d

1

;

(53)

(x)

a

1 ([a; b]) p (b ( ( ))

p(

1)

a)

2

([a; b]) (C)

Z

! pq

b

f q (t) d

(t)

2

a

:

(54)

We have proved that (C)

Z

b

p

Ia+ f (x)

d

1

! p1

(x)

a

(

1

([a; b])

1

2

([a; b])) p (b ( )

(

a)

Z

1)

(C)

! q1

b q

f (t) d

2

(t)

a

:

(55)

Similarly, we have (8)

Ib f (x)

1 ( )

(C)

Z

b

p(

(t

x)

1)

d

2

! p1

(t)

x

1 (b ( )

p(

1)

a)

2

1 p

([a; b])

(C)

Z

(C)

Z

b q

f (t) d

x

f q (t) d

2

:

! q1

:

(t)

a

(t)

(56)

! q1

b

2

! q1

As before we obtain (C)

Z

b

Ib f (x)

p

d

1

! p1

(x)

a

(

1

([a; b])

1

2

([a; b])) p (b ( )

(

a)

1)

(C)

Z

a

b q

f (t) d

2

(t)

(57)

We have proved

13

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Theorem 37 Here > 1 and the rest are as in this section. It holds 8 ! p1 9 ! p1 Z b Z b < = p p max (C) Ia+ f (x) d 1 (x) ; (C) Ib f (x) d 1 (x) : ; a a (

1

1

([a; b])

2

(

([a; b])) p (b ( )

Z

1)

a)

(C)

! q1

b

q

f (t) d

2

Z

q

(t)

a

:

(58)

Remark 38 From (52) we get 1 (x ( )

Ia+ f (x)

p(

1)

a)

1 p

([a; x])

2

(C)

! q1

b

f (t) d

2

(t)

a

; (59)

and from (56) we derive (by exchanging the roles of p and q) 1 (b ( )

Ib f (x)

q(

1)

x)

1 q

([x; b])

2

(C)

Z

! p1

b p

f (t) d

2

(t)

a

: (60)

Therefore by multiplying (59), (60) we get 1

Ia+ f (x) Ib f (x)

(b

q(

x)

1) 2

1 q

([x; b])

2

( ( ))

(C)

Z

p(

(x

! q1

b q

f (t) d

2

(t)

a 1

p(

(x

a)

2

1)

a p

+ qb )

2 ([a; x])

(b

x)

(C)

Z

b

f q (t) d

2

! q1

(C)

(t)

a

+

Z

1 p

b p

f (t) d

q(

1)

2 ([x; b])

q

Z

(61)

2

a

1

p

( ( ))

2 ([a; x])

(C)

0, a p b q

(using Young’s inequality for a; b 1

1)

a)

b

f p (t) d

2

! p1

(t)

a

! p1

(t)

!

:

(62)

We have that h 1 2

( ( ))

Ia+ f (x) Ib f (x) (x a)p(

1)

2 ([a;x])

p

(C)

Z

a

b

f q (t) d

2

(b x)q(

+

1)

2 ([x;b])

q

! q1

(t)

(C)

Z

a

b

i

f p (t) d

2

! p1

(t)

:

(63)

Notice that the denominator of left hand side of (63) is never zero. 14

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Integrating (63) with respect to x we obtain: Theorem 39 Here Z (C)

a

1

([a; b]) 2

( ( ))

> 1 and the rest are as in this section. It holds b

h

Ia+ f (x) Ib f (x) d (x a)p(

(C)

1)

2 ([a;x])

p

Z

a

b p

f (t) d

2

+

! p1

(t)

1

(b x)q(

(x)

1)

2 ([x;b])

q

(C)

Z

a

i

b q

f (t) d

2

! q1

(t)

:

(64)

Inequality (64) is a Hilbert-Pachpatte type inequality for Choquet fractional integrals.

References [1] G. Anastassiou, Intelligent Comparisons: Analytic Inequalities, Springer, Heidelberg, New York, 2016. [2] A. Chateauneuf, J.P. Lefort, Some Fubini theorems on product sigmaalgebras for non-additive measures, Internat. J. Approx. Reason, 48 (2008), no. 3, 686-696. [3] G. Choquet, Theory of capacities, Ann. Inst. Fourier, 5 (1953), 131-295. [4] L.M. de Campos, M.J. Bolanos, Characterization and comparison of Sugeno and Choquet integrals, Fuzzy Sets Syst., 52 (1992), 61-67. [5] D. Denneberg, Nonadditive Measure and Integral, Kluwer Academic, Dordrecht, 1994. [6] P. Ghirardato, On independence for non-additive measures, with a Fubini theorem, J. Economic Theory, 73 (1997), 261-291. [7] H.G. Hardy, Notes on some points in the integral calculus, Messenger of Mathematics, vol. 47, no. 10, 1918, 145-150. [8] S. Iqbal, K. Krulic, J. Pecaric, On an inequality of G. Hardy, J. of Inequalities and Applications, Vol. 2010, Article ID 264347, 23 pages. [9] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Di¤ erential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier, New York, NY, USA, 2006. [10] E. Pap, Null-Additive Set Functions, Kluwer Academic, Dordrecht, 1995.

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[11] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integral and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, Switzerland, 1993. [12] D. Schmeidler, Subjective probability and expected utility without additivity, Econometrica, 57 (1989), 571-587. [13] Rui-Sheng Wang, Some inequalities and convergence theorems for Choquet integrals, J. Appl. Math. Comput. 35 (2011), 305-321. [14] Z. Wang, Convergence theorems for sequences of Choquet integrals, Int. J. Gen. Syst. 26 (1997), 133-143. [15] Z. Wang, G. Klir, Fuzzy Measure Theory, Plenum, New York, 1992. [16] Z. Wang, G.J. Klir, W. Wang, Monotone set functions de…ned by Choquet integrals, Fuzzy Sets Syst. 81 (1996), 241-250. [17] Z. Wang, G.J. Klir, Generalized Measure Theory, Springer, New York, 2009.

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO. 7, 2019

Common fixed point theorems in 𝐺𝐺𝑏𝑏 -metric space, Youqing Shen, Chuanxi Zhu, and Zhaoqi Wu,……………………………………………………………………………………1083 A modified collocation method for weakly singular Fredholm integral equations of second kind, Guang Zeng, Chaomin Chen, Li Lei, and Xi Xu,…………………………………….1091 Sharp coefficient estimates for non-Bazilevič functions, Ji Hyang Park, Virendra Kumar, and Nak Eun Cho,……………………………………………………………………………1103 A new extragradient method for the split feasibility and fixed point problems, Ming Zhao and Yunfei Du,…………………………………………………………………………………1114 Behavior of Meromorphic Solutions of Composite Functional-Difference Equations, Man-Li Liu and Ling-Yun Gao,…………………………………………………………………………1124 Locally and globally small Riemann sums and Henstock-Stieltjes integral for n-dimensional fuzzy-number-valued functions, Muawya Elsheikh Hamid,……………………………….1142 Solving Systems of Nonhomogeneous Coupled Linear Matrix Differential Equations in Terms of Mittag-Leffler Matrix Functions, Rungpailin Kongyaksee and Pattrawut Chansangiam,…1150 Expressions of the solutions of some systems of difference equations, M. M. El-Dessoky, E. M. Elsayed, E. M. Elabbasy, and Asim Asiri,………………………………………………….1161 Hardy type inequalities for Choquet integrals, George A. Anastassiou,……………………1173