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: feilongcao@gmail.com

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



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



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



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)



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 .



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

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 : malek_-ghazel@yahoo.fr

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

E-mail address : tshassan@mans.edu.eg

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

E-mail address : ahmedmetwally77@hotmail.com

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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: feilongcao@gmail.com

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



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

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 mansour@mans.edu.eg,

hegazi@mans.edu.eg,

− t− amer@yahoo.com,

4 heshammoustafa14@gmail.com

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

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

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

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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: dessokym@mans.edu.eg; eoalzahrani@gmail.com; nalmohammadi0010@stu.kau.edu.sa 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: tkkim@kw.ac.kr 2

Department of Mathematics, Sogang University, Seoul, 121-742, Republic of Korea E-mail address: dskim@sogang.ac.kr 3 Department of Mathematics, Kwangwoon University, Seoul, 139-701, Republic of Korea E-mail address: sura@kw.ac.kr 4,∗ Department of Mathematics Education and RINS, Gyeongsang National University, Jinju, Gyeongsangnamdo, 52828, Republic of Korea E-mail address: mathkjk26@gnu.ac.kr

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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 lv@163.com (Lingling Lv). ‡ Institute of electric power, North China University of Water Resources and Electric Power, Zhengzhou 450011, P. R. China. Email: zhe Zhang5218@163.com (Zhe Zhang) § Computer and Information Engineering College, Henan University, Kaifeng 475004, P. R. China. Email: zhanglei@henu.edu.cn (Lei Zhang). ¶ Computer and Information Engineering College, Henan University, Kaifeng 475004, P. R. China. Email: liuxianxing@henu.edu.cn (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: baak@hanyang.ac.kr (Choonkil Park), dyshin@uos.ac.kr (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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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: moeini145523@gmail.com 2

Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran E-mail address: analsisamirmath2@gmail.com 3

Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea E-mail address: baak@hanyang.ac.kr 4

Department of Mathematics, University of Seoul, Seoul 02504, Korea E-mail address: dyshin@uos.ac.kr

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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: nazeer.waqas@ue.edu.pk (W.N); mmunir@ue.edu.pk (M.M) 2

3

Department of Mathematics, Lahore Leads University, Lahore 54810, Pakistan e-mail: fabbas27@gmail.com

Department of Business Administration, Gyeongsang National University, Jinju 52828, Korea e-mail: bb5734@gnu.ac.kr 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: smkang@gnu.ac.kr 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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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: dskang@dankook.ac.kr 2 Department

of Mathematics, Oregon State University, Corvallis, Oregon 97331

E-mail address: kimho@math.oregonstate.edu

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, kimho@math.oregonstate.edu. 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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3

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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GENERALIZED STABILITY OF CUBIC FUNCTIONAL EQUATIONS

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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5

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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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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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.

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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 math@yahoo.com; baak@hanyang.ac.kr; ssyun@hs.ac.kr

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:

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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

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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:

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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].

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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).

15

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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: haishexiang@lut.cn.

† 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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Shexiang Hai and Fangdi Kong: The Differentiability and Gradient for Fuzzy Mappings Based on ...

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: zhangzh@bnu.edu.cn

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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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.2, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

(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

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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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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.2, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

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



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



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: pattrawut.ch@kmitl.ac.th

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)

317

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.2, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

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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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.2, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

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

Gang Wang ET AL 326-344

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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

Gang Wang ET AL 326-344

(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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(3.5)

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.2, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

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→∞

9 334

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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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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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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.2, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

T. Kim, D. S. Kim, G.-W. Jang, J. Kwon

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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T. Kim, D. S. Kim, G.-W. Jang, J. Kwon

9

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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T. Kim, D. S. Kim, G.-W. Jang, J. Kwon

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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16

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: tkkim@kw.ac.kr 2

Department of Mathematics, Sogang University, Seoul, 121-742, Republic of Korea E-mail address: dskim@sogang.ac.kr 3 Department of Mathematics, Kwangwoon University, Seoul, 139-701, Republic of Korea E-mail address: jgw5687@naver.com 4,∗ Department of Mathematics Education and ERI, Gyeongsang National University, Jinju, Gyeongsangnamdo, 52828, Republic of Korea E-mail address: mathkjk26@gnu.ac.kr

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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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C. PARK, D.Y. SHIN, AND G.A. ANASTASSIOU

(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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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.2, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

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

369

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

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.

370

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.

372

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)

373

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

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

374

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

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.

377

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

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[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: baak@hanyang.ac.kr Dong Yun Shin Department of Mathematics, University of Seoul, Seoul 02504, Republic of Korea E-mail address: dyshin@uos.ac.kr George A. Anastassiou Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA E-mail address: ganastss@memphis.edu

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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

September 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 ganastss@memphis.edu 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,mezei_razvan@yahoo.com, 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 altomare@dm.uniba.it 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 carlo.bardaro@unipg.it 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 Agarwal@tamuk.edu 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: ganastss@memphis.edu Approximation Theory, Real Analysis, Wavelets, Neural Networks, Probability, Inequalities.

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Luis A. Caffarelli Department of Mathematics The University of Texas at Austin Austin, Texas 78712-1082 512-471-3160 e-mail: caffarel@math.utexas.edu 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, dumitru@cankaya.edu.tr

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Dartmouth College 8000 Cummings Hall, Hanover, NH 03755-8000 603-646-3843 (X 3546 Secr.) e-mail:george.cybenko@dartmouth.edu Approximation Theory and Neural Networks

011-49-203-379-3542 e-mail: heiner.gonska@uni-due.de Approximation Theory, Computer Aided Geometric Design John R. Graef Department of Mathematics University of Tennessee at Chattanooga Chattanooga, TN 37304 USA John-Graef@utc.edu 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 sever.dragomir@vu.edu.au 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: whan@math.uiowa.edu 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, oduman@etu.edu.tr 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 the@iwu.edu 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: selaydi@trinity.edu Ordinary Differential Equations, Difference Equations

Margareta Heilmann Faculty of Mathematics and Natural Sciences, University of Wuppertal Gaußstraße 20 D-42119 Wuppertal, Germany, heilmann@math.uni-wuppertal.de Approximation Theory (Positive Linear Operators)

J .A. Goldstein Department of Mathematical Sciences The University of Memphis Memphis, TN 38152 901-678-3130 jgoldste@memphis.edu Partial Differential Equations, Semigroups of Operators

Xing-Biao Hu Institute of Computational Mathematics AMSS, Chinese Academy of Sciences Beijing, 100190, CHINA hxb@lsec.cc.ac.cn

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 ram.mohapatra@ucf.edu Real and Complex Analysis, Approximation Th., Fourier Analysis, Fuzzy Sets and Systems

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Robert Kozma Department of Mathematical Sciences The University of Memphis Memphis, TN 38152, USA rkozma@memphis.edu Neural Networks, Reproducing Kernel Hilbert Spaces, Neural Percolation Theory Mustafa Kulenovic Department of Mathematics University of Rhode Island Kingston, RI 02881,USA kulenm@math.uri.edu Differential and Difference Equations

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Hrushikesh N. Mhaskar Department Of Mathematics California State University Los Angeles, CA 90032 626-914-7002 e-mail: hmhaska@gmail.com Orthogonal Polynomials, Approximation Theory, Splines, Wavelets, Neural Networks

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Choonkil Park Department of Mathematics Hanyang University Seoul 133-791 S. Korea, baak@hanyang.ac.kr Functional Equations

Anfithea - Paleon Faliron GR-175 64 Athens, Greece tsimos@mail.ariadne-t.gr 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 harimsri@math.uvic.ca 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.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.3, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

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: Kkanyarat.cheaw@gmail.com; beawrock@hotmail.com

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

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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.

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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 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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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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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: nazim254.butt@gmail.com; marshadzia@iiu.edu.pk 2 Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea

e-mail: baak@hanyang.ac.kr 3 Department of Mathematics, Faculty of Science and Arts, Mersin University, 33343, Yeni¸sehir, Mersin, Turkey

e-mail: ozlemacar@mersin.edu.tr 4 Department of Financial Mathematics, Hanshin University, Gyeonggi-do 18101, Korea

e-mail: ssyun@hs.ac.kr 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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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.3, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

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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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.3, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

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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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.3, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

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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Nazam ET AL 417-426

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 27, NO.3, 2019, COPYRIGHT 2019 EUDOXUS PRESS, LLC

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 s