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Table of contents :
PART-1-VOL-20-JOCAAA-2016
BLOCK-1-V20-JOCAAA-2016
FACE-1-JOCAAA-2016-VOL-20
JCAAA-2016-V20-front-1
MIDDLE-JOCAAA-2016
SCOPE--JOCAAA--2016
EB--JOCAAA--2016
Instructions--JOCAAA--2016
Binder1-JOCAAA-2016-V20-h
1-2016-FNL-Jin-Woo Park-JOCAAA--2-21-15
3-2016-FNL- Choonkil Park-JOCAAA-1-20-2015
5-2016-FNL-Cihangir Alaca-JOCAAA--2-2-2015
1. Introduction
2. Preliminaries
3. Modular S-metric spaces
4. Fixed Point Theorems
References
6-2016-FEILONG CAO--JOCAAA--7-30-2014
7-2016-Ick-Soon Chang-JOCAAA--7-30-14
8-2016-FNL-Wang-Liang-JOCAAA-22-1-2015
9-2016-FNL-Liu Yang-JOCAAA--1-27-2015
10-2016-fnl-kim-shin-jocaaa--8-14-2015
11-2016-Xiaoqiang Zhou-JOCAAA-7-30-14
12-2016-Nak Eun Cho-JOCAAA--7-30-14
13-2016-Seog-Hoon Rim-JOCAAA--7-30-14
14-2016-Farhadabadi -Shin-Park-JOCAAA--7-31-14
15-2016-LUPAS-OROS-JOCAAA--8-24-2014
16-2016-FNL-Quan Zheng -JOCAAA--1-24-2015
17-2016-Taekyun Kim-JOCAAA--8-24-2014
18-2016-FNL-Changyou Wang-JOCAAA--2-3-2015
19-2016-FNL-Bulut-JOCAAA--7-10-2015
BLANK-JoCAAA-2016
BLANK-JoCAAA-2016
PART-2-VOL-20-JOCAAA-2016
BLOCK-2-V20-JOCAAA-2016
FACE-2-JOCAAA-2016-VOL-20
JCAAA-2016-V20-front-2
MIDDLE-JOCAAA-2016
SCOPE--JOCAAA--2016
EB--JOCAAA--2016
Instructions--JOCAAA--2016
Binder2-JOCAAA-2016-V20-h
20-2016-ANASTASSIOU-IATAN-JOCAAA--8-24-2014
21-2016-ELMETWALLY-JOCAAA--8-28-2014
22-2016-FNL- Zhifeng Dai-JOCAAA-1-27-2015
Introduction
Conditional value-at-risk (CVaR)
Worst-Case Conditional value-at-risk (CVaR)
Computing WSCVaR and its application in portfolio management
Computational Experiments
Experiments with Simulated Data
Experiments with Hedge Funds
Conclusion
23-2016-REV-LEE-CHAE-JANG-JOCAAA--12-12-2014
24-2016-Gang Lu-JOCAAA--8-31-2014
1. Introduction and preliminaries
2. Main results
Acknowledgments
References
25-2016-Elaiw-et-al-JOCAAA--9-2-14
26-2016-d-kang-jocaaa--9-6-2014
27-2016-Lingqiang Li-JOCAAA--9-9-2014
28-2016-Seok-Zun Song--JOCAAA--9-10-14
Introduction
Preliminaries
Uni-soft filters
Uni-soft G-filters
29-2016-FNL-Noura-AlShamrani-JOCAAA--2-6-2015
30-2016-FNL-Jongsung Choi -JOCAAA--2-4-2015
31-2016-FNL-Mohiuddine -JOCAAA-2-1-2015
32-2016-FNL-Xianjiu Huang-JOCAAA-1-22-2015
Introduction and preliminaries
Main results
33-2016-FNL-Mahmoud Belaghi-JOCAAA--10-1-2015
34-2016-Jae-Hyeong Bae --JOCAAA--9-15-14
BLANK-JoCAAA-2016
BLANK-JoCAAA-2016
BLANK-JoCAAA-2016
PART-3-VOL-20-JOCAAA-2016
BLOCK-3-V20-JOCAAA-2016
FACE-3-JOCAAA-2016-VOL-20
JCAAA-2016-V20-front-3
MIDDLE-JOCAAA-2016
SCOPE--JOCAAA--2016
EB--JOCAAA--2016
Instructions--JOCAAA--2016
Binder3-JOCAAA-2016-V20-h
35-2016-arshad-ZIA-JOCAAA--9-15-14
36-2016-FNL-Jin-Woo Park-JOCAAA--2-21-2015
37-2016-Seog-Hoon Rim--JOCAAA--9-18-2014
38-2016-REV-Seog-Hoon Rim--JOCAAA--9-20-14
39-2016-Jiang zhijie --JOCAAA--9-20-2014
40-2016-FNL-Wu Li-Xiaoqiang Zhou-Guanqi Guo-JOCAAA-1-28-2015
41-2016-FNL-Jin-Woo Park-JOCAAA--2-21-15
42-2016-ucar-aydogan-jocaaa--9-24-14
44-2016-Lingling Lv-jocaaa--9-25-14
45-2016-FNL-WentaoCheng-JoCAAA-2-11-2015
Introduction
Notions and notations
Main results
Auxiliary lemmas
Proofs of the theorems
Proof of Theorem 3.1
Proof of Theorem 3.2
Proof of Theorem 3.3
Acknowledgement
46-2016-FNL-AHMAD-ET-AL-JOCAAA-01-19-2015
47-2016-FNL-Zhihua Zhang-JOCAAA--1-21-2015
48-2016--Jianling Li--JOCAAA--9-29-2014
Introduction
Preliminaries
Convergence results
Existence of multipliers
Concluding remarks
49-2016-FNL-Huaping Huang-JOCAAA--1-30-2015
50-2016-Taekyun Kim-jocaaa--10-4-14
52-2016-FNL-Yan-Lan Zhang-JOCAAA-1-24-2015
BLANK-JoCAAA-2016
BLANK-JoCAAA-2016
PART-4-VOL-20-JOCAAA-2016
BLOCK-4-V20-JOCAAA-2016
FACE-4-JOCAAA-2016-VOL-20
JCAAA-2016-V20-front-4
MIDDLE-JOCAAA-2016
SCOPE--JOCAAA--2016
EB--JOCAAA--2016
Instructions--JOCAAA--2016
Binder4-JOCAAA-2016-V20-h
51-2016-Biancamaria Della Vecchia-JOCAAA--10-4-2014
53-2016-FNL-Jun-Ahn-JOCAAA-1-26-2015
54-2016-FNL-NTOUYAS-TARIBOON-JoCAAA-1-20-2015
55-2016-FNL-KARACA-JOCAAA--2-4-2015
56-2016-FNL-Zhiyong Liu--JOCAAA--3-10-2015
57-2016-Jong Soo Jung-JOCAAA--10-14-2014
58-2016-FNL-EL-SAYED-AHMED-JOCAAA--2-3-2015
59-2016-Xiang Wang-JOCAAA-10-15-2014
60-2016-fnl-Xianghu Liu-jocaaa--1-22-2015
61-2016-Haidong Zhang-JOCAAA--10-16-2014
62-2016-Kelin Li-JOCAAA--10-17-2014
BLANK-JoCAAA-2016
BLANK-JoCAAA-2016
BLANK-JoCAAA-2016
PART-5-VOL-20-JOCAAA-2016
BLOCK-5-V20-JOCAAA-2016
FACE-5-JOCAAA-2016-VOL-20
JCAAA-2016-V20-front-5
MIDDLE-JOCAAA-2016
SCOPE--JOCAAA--2016
EB--JOCAAA--2016
Instructions--JOCAAA--2016
Binder5-JOCAAA-2016-V20-h
63-2016-Taekyun Kim-JOCAAA--10-20-2014
64-2016-FNL-Choonkil Park- JOCAAA-1-20-2015
65-2016-BALEANU-JOCAAA--10-31-2014
66-2016-ryoo-kang-JOCAAA--11-1-2014
67-2016-Komashynska-jocaaa--11-2-2014
68-2016-FNL-Huaping Huang-JOCAAA--1-30-2015
69-2016-FNL-Jun-Song-Roh-Ahn-JOCAAA-1-26-2015
70-2016-FNL-Heng-you Lan-JOCAAA--1-24-2015
71-2016-MALIK-JOCAAA--11-8-2014
73-2016-FNL-KEFENG-DUAN-JOCAAA-23-01-2015
74-2016-FNL-Liu-Yang-JoCAAA-1-20-2015
1 Introduction and preliminaries
2 Weighted q-Cebyšev type inequalities for double integrals
3 Weighted q-Ostrowski type inequalities for double integrals
75-2016-CHOI-KIM-ANASTASSIOU-PARK-JOCAAA--11-12-2014
76-2016-ALOTAIBI-J-RASSIAS-MOHIUDDINE-JOCAAA-11-13-2014
77-2016-THANIN-Sitthiwirattham-JOCAAA--11-14-2014
78-2016-FNL-Jin Han Park-JOCAAA--2-22-2015
BLANK-JoCAAA-2016
BLANK-JoCAAA-2016
PART-6-VOL-20-JOCAAA-2016
BLOCK-6-V20-JOCAAA-2016
FACE-6-JOCAAA-2016-VOL-20
JCAAA-2016-V20-front-6
MIDDLE-JOCAAA-2016
SCOPE--JOCAAA--2016
EB--JOCAAA--2016
Instructions--JOCAAA--2016
Binder6-JOCAAA-2016-V20-h
79-2016-FNL-Seung Ki Yoo-JOCAAA--9-20-2015
80-2016-ANASTASSIOU-SAADATI-CHO-YANG-JOCAAA--11-17-2014
81-2016-FNL-Jin Tu-JOCAAA--2-16-2015
82-2016-FNL-DONG-QIU-JOCAAA--1-20-2015
83-2016-FNL-Xu-Zhou-JOCAAA-22-1-2015
84-2016-FNL-Hanying Feng-JOCAAA--1-21-2015
85-2016-Kamaleldin Abodayeh-JOCAAA-11-26-2014-accepted-paid
86-2016-FNL-QingboCai-JoCAAA-2-9-2015
87-2016-Xiaobin Zhang-JOCAAA-12-1-2014
88-2016-Zewen Wang-JOCAAA--12-2-2014
89-2016-Ningxin Xie-JOCAAA-12-2-2014
90-2016-Sheng Luo-JOCAAA-12-2-2014
91-2016-FNL-Wan Se Kim-JOCAAA--2-9-2015
92-2016-FNL--Hong Yan Xu-Cai Feng Yi-JoCAAA-8-28-2015
BLANK-JoCAAA-2016
BLANK-JoCAAA-2016
PART-7-VOL-20-JOCAAA-2016
BLOCK-7-V20-JOCAAA-2016
FACE-7-JOCAAA-2016-VOL-20
JCAAA-2016-V20-front-7
MIDDLE-JOCAAA-2016
SCOPE--JOCAAA--2016
EB--JOCAAA--2016
Instructions--JOCAAA--2016
Binder7-JOCAAA-2016-V20-h
93-2016-FNL--MERAJ-KHAN-JOCAAA--3-6-2015
94-2016-FNL-Stevic-JOCAAA--1-28-2015
95-2016-Arif Rafiq-JOCAAA--12-9-2014
96-2016-jang-kim-jocaaa-12-9-2014
97-2016-fnl- Guangji Yu-jocaaa--3-3-15
98-2016-ALACA-EGE-PARK-JOCAAA-12-10-2014
99-2016-Zhidong Teng -JOCAAA--12-12-2014
100 -2016-FNL-Jun He- JOCAAA--9-8-2015
101-2016-BALEANU-JOCAAA--12-14-2014
Introduction
Preliminaries
Existence results
Existence results: The convex case
Existence results: Nonconvex case
Application
102-2016-fnl-malik-jocaaa--10-21-2015
103-2016-FNL-HongYan Xu-Hua Wang-JoCAAA-8-28-2015
104-2016-FNL-Jun-Ahn-JOCAAA-8-24-2015
105-2016-FNL--Jongsung Choi-- JOCAAA--9-1-2015
106-2016-Xiaodong Cao-JOCAAA--12-25-2014
107-2016-M-A-Obaid-JOCAAA--12-27-2014
108-2016-ALZER-JOCAAA-1-1-2015
BLANK-JoCAAA-2016
BLANK-JoCAAA-2016
Volume 20, Number 1 ISSN:1521-1398 PRINT,1572-9206 ONLINE
January 2016
Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC
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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 (fourteen 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,Lenoir-Rhyne University,Hickory,NC
28601, 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 $700, Electronic OPEN ACCESS. Individual:Print $350. For any other part of the world add $130 more(handling and postages) to the above prices for Print. No credit card payments. Copyright©2016 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
Partial Differential Equations, Semigroups of Operators H. H. Gonska Department of Mathematics University of Duisburg Duisburg, D-47048 Germany 011-49-203-379-3542 e-mail: [email protected] Approximation Theory, Computer Aided Geometric Design
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.
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
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
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
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
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
Christodoulos A. Floudas Department of Chemical Engineering Princeton University Princeton,NJ 08544-5263 609-258-4595(x4619 assistant) e-mail: [email protected] Optimization Theory&Applications, Global Optimization
Margareta Heilmann Faculty of Mathematics and Natural Sciences, University of Wuppertal Gaußstraße 20 D-42119 Wuppertal, Germany, [email protected]
J .A. Goldstein Department of Mathematical Sciences The University of Memphis Memphis, TN 38152 901-678-3130 [email protected]
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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
Approximation Theory (Positive Linear Operators) Xing-Biao Hu Institute of Computational Mathematics AMSS, Chinese Academy of Sciences Beijing, 100190, CHINA [email protected] 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
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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
Approximation Theory, Banach spaces, Classical Analysis 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. Anfithea - Paleon Faliron GR-175 64 Athens, Greece [email protected] Numerical Analysis
Choonkil Park Department of Mathematics Hanyang University Seoul 133-791 S. Korea, [email protected] Functional Equations
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 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
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 Boris Shekhtman Department of Mathematics University of South Florida Tampa, FL 33620, USA Tel 813-974-9710 [email protected]
Roberto Triggiani Department of Mathematical Sciences University of Memphis Memphis, TN 38152 PDE, Control Theory, Functional
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Analysis, [email protected]
USA. Tel 334-844-6557 office 678-642-8703 home Fax 334-844-6555 [email protected] Approximation Theory, Chebychev Systems, Wavelet Theory
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] 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
Lotfi A. Zadeh Professor in the Graduate School and Director, Computer Initiative, Soft Computing (BISC) Computer Science Division University of California at Berkeley Berkeley, CA 94720 Office: 510-642-4959 Sec: 510-642-8271 Home: 510-526-2569 FAX: 510-642-1712 [email protected] Fuzzyness, Artificial Intelligence, Natural language processing, Fuzzy logic Richard A. Zalik Department of Mathematics Auburn University Auburn University, AL 36849-5310
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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.
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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 20, NO.1, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
ON THE λ-DAEHEE POLYNOMIALS WITH q-PARAMETER JIN-WOO PARK
Abstract. In this paper, we consider the generalization of Daehee polynomials with q-parameter and investigate some properties of those polynomials.
1. Introduction Let p be a fixed prime number. Throughout this paper, Zp , Qp , and Cp will respectively denote the ring of p-adic rational integers, the field of p-adic rational numbers and the completions of algebraic closure of Qp . The p-adic norm is defined |p|p = p1 . When one talks of q-extension, q is variously considered as an indeterminate, a complex q ∈ C, or p-adic number q ∈ Cp . If q ∈ C, one normally assumes that 1 |q| < 1. If q ∈ Cp , then we assume that |q − 1|p < p− p−1 so that q x = exp(x log q) for each x ∈ Zp . Throughout this paper, we use the notation : [x]q =
1 − qx . 1−q
Note that limq→1 [x]q = x for each x ∈ Zp . Let U D(Zp ) be the space of uniformly differentiable functions on Zp . For f ∈ U D(Zp ), the p-adic invariant integral on Zp is defined by Kim as follows : Z pn −1 1 X I (f ) = f (x) dµ0 (x) = lim n f (x) , (see [4, 5, 6]). (1.1) n→∞ p Zp x=0 Let f1 be the translation of f with f1 (x) = f (x + 1) . Then, by (1.1), we get df (x) I (f1 ) = I (f ) + f 0 (0) , where f 0 (0) = . (1.2) dx x=0
As it is well-known fact, the Stirling number of the first kind is defined by (x)n = x (x − 1) · · · (x − n + 1) =
n X
S1 (n, l) xl ,
(1.3)
l=0
and the Stirling number of the second kind is given by the generating function to be ∞ X m tl (1.4) et − 1 = m! S2 (l, m) , l! l=m
(see [1, 10]). 1991 Mathematics Subject Classification. 05A19, 11B65, 11B83. Key words and phrases. Bernoulli polynomials, Daehee polynomials with q-parameter, p-adic invariant integral. 1
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Unsigned Stirling numbers of the first kind is given by xn = x(x + 1) · · · (x + n − 1) =
n X
|S1 (n, l)|xl .
(1.5)
l=0
Note that if we replace x to −x in (1.3), then (−x)n =(−1)n xn =
n X
S1 (n, l)(−1)l xl
l=0 n
=(−1)
n X
(1.6) l
|S1 (n, l)|x .
l=0
Hence S1 (n, l) = |S1 (n, l)|(−1)n−l . For r ∈ N, the Bernoulli polynomials of order r are defined by the generating function to be r ∞ X t tn xt (r) e = , (see [7, 8, 11]). (1.7) B (x) n et − 1 n! n=0 (r)
(r)
When x = 0, Bn = Bn (0) are called the Bernoulli numbers of order r, and in the (1) special case, r = 1, Bn (x) = Bn (x) are called the ordinary Bernoulli polynomials. For n ∈ N, let Tp be the p-adic locally constant space defined by Tp = ∪ Cpn = lim Cpn , n→∞
n≥1
pn
where Cpn = ω|ω = 1 is the cyclic group of order pn . 1 We assume that q is an indeterminate in Cp with |1 − q|p < p− p−1 . Then we define the q-analogue of falling factorial sequence as follows : (x)n,q = x(x − q)(x − 2q) · · · (x − (n − 1)q), (n ≥ 1), (x)0,q = 1. Note that lim (x)n,q = (x)n =
q→1
n X
S1 (n, l)xl .
l=0
Recently, D. S. Kim and T. Kim introduced the Daehee polynomials as follows : Z Dn (x) = (x + y)n dµ0 (y), (n ≥ 0), (see [2, 5, 9]). (1.8) Zp
When x = 0, Dn = Dn (0) are called the n’s Daehee numbers. From (1.8), we can derive the generating function to be ∞ X log(1 + t) tn (1 + t)x = Dn (x) , (see [2]). (1.9) t n! n=0 In addition, D. S. Kim et. al. consider the Daehee polynomials with q-parameter which is defined by the generating function to be ∞ X n=0
Dn,q
x tn log(1 + qt) , (see [3]). = (1 + qt) q 1 n! q (1 + qt) q − 1
(1.10)
When x = 0, Dn,q = Dn,q (0) are called the Daehee numbers with q-parameter.
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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 20, NO.1, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
ON THE λ-DAEHEE POLYNOMIALS WITH q-PARAMETER
3
In the viewpoint of generalization of the Daehee polynomials with q-parameter, we consider the λ-Daehee polynomials with q-parameter are defined to be ∞ X n=0
Dn,q (λ|x)
x tn λ log(1 + qt) . = (1 + qt) q λ n! q (1 + qt) q − 1
(1.11)
When x = 0, Dn,q (λ) = Dn,q (λ|0) are called the λ-Daehee numbers with qparameter. In particular, the case λ = 1 is the Daehee polynomials with qparameter. In this paper, we give a p-adic integral representation of the λ- Daehee polynomials with q-parameter, which are called the Witt-type formula for the λ-Daehee polynomials with q-parameter. We can derive some interesting properties related to the λ-Daehee polynomials with q-parameter.
2. Witt-type formula for the n-th λ-Daehee polynomials with q-parameter 1
In this section, we assume that t, q ∈ Cp with |t|p < |q|p p− p−1 and λ ∈ Zp . First, we consider the following integral representation associated with falling factorial sequences : Z (x + λy)n,q dµ0 (y), where n ∈ Z+ = N ∪ {0} .
(2.1)
Zp
By (2.1), ∞ Z X n=0
∞
(x + λy)n,q dµ0 (y)
Zp
tn X n = q n! n=0
tn x + λy dµ0 (y) q n! Zp n Z ∞ x+λy X q = qn dµ0 (y)tn n Zp n=0 Z x+λy = (1 + qt) q dµ0 (y) Z
(2.2)
Zp 1
1
where t ∈ Cp with |t|p < |q|p p− p−1 . For t ∈ Cp with |t|p < |q|p p− p−1 , we get Z x+λy x λ log(1 + qt) (1 + qt) q dµ0 (y) =(1 + qt) q λ Zp q (1 + qt) q − 1 ∞ X
tn = Dn,q (λ|x) . n! n=0
(2.3)
By (2.2) and (2.3), we obtain the following theorem. Theorem 2.1. For n ≥ 0, we have Z Dn,q (λ|x) =
(x + λy)n,q dµ0 (y). Zp
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In (2.3), by replacing t by
(et − 1), we have
λ ∞ n X x Dn,q (λ|x) (et − 1) qt qt = e λ n q n! eqt − 1 n=0
=
∞ X
Bn
x λn tn
n=0
λ
q n n!
(2.4) ,
and ∞ ∞ ∞ X n X Dn,q (λ|x) 1 t tm Dn,q (λ|x) X e S (m, n) − 1 = 2 qn n! qn m! m=n n=0 n=0 ! ∞ m X X Dm,q (λ|x) tn S (n, m) = . 2 qm n! n=0 m=0
(2.5)
By (2.4) and (2.5), we obtain the following corollary. Corollary 2.2. For n ≥ 0, we have n x X Bn = Dm,q (λ|x)q n−m λ−n S2 (n, m). λ m=0 By the Theorem 2.1, Z Dn,q (λ|x) =
(x + λy)n,q dµ0 (y) Zp
Z
=q n
Zp n X
=q n
x + λy q
1 S1 (n, l) ql
l=0
dµ0 (y)
(2.6)
n
Z
(x + λy)l dµ0 (y).
Zp
By (1.2), we can derive easily that Z e(x+λy)t dµ0 (y) =
∞ x (λt)n X λt xt e = Bn λt e −1 λ n! n=0 Z ∞ X tl = (x + λy)l dµ0 (y) , l! Zp
Zp
(2.7)
l=0
and so Bn
x λ
Z =
x λ
Zp
+y
n
dµ0 (y), (n ≥ 0).
(2.8)
By (1.6), (2.7) and (2.8), we obtain the following corollary. Corollary 2.3. For n ≥ 0, we have Dn,q (λ|x) = =
n X l=0 n X
q n−l S1 (n, l)λl Bl
x λ
|S1 (l, n)|(−q)n−l λl Bl
l=0
14
x λ
.
JIN-WOO PARK 11-20
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 20, NO.1, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
ON THE λ-DAEHEE POLYNOMIALS WITH q-PARAMETER
5
From now on, we consider λ-Daehee polynomials of order k(∈ N) with q-parameter. λ-Daehee polynomials of order k with q-parameter are defined by the multivariant p-adic invariant integral on Zp : Z Z (k) Dn,q (λ|x) = ··· (λ(x1 + · · · + xk ) + x)n,q dµ0 (x1 ) · · · dµ0 (xk ) (2.9) Zp
Zp (k)
where n is an nonnegative integer and k ∈ N. In the special case, x = 0, Dnq (λ) = (k) Dn,q (λ|0) are called the λ-Daehee numbers of order k with q-parameter. (k) From (2.9), we can derive the generating function of Dn,q (x) as follows: ∞ X
=
n=0 ∞ X
tn n! Z λ(x1 +···+xk )+x
(k) Dn,q (λ|x)
q
n
Z
n=0
Z Zp
n
Zp
Z ···
=
q
··· Zp
(1 + qt)
λ(x1 +···+xk )+x q
dµ0 (x1 ) · · · dµ0 (xk )tn
dµ0 (x1 ) · · · dµ0 (xk )
(2.10)
Zp
Z
x
Z ···
= (1 + qt) q
(1 + qt)
Zp
λ(x1 +···+xk ) q
dµ0 (x1 ) · · · dµ0 (xk )
Zp
k λ log(1 + qt) . = (1 + qt) λ q (1 + qt) q − 1
x q
Note that, by (2.9), (k) Dn,q (λ|x)
Z Z n X (2.11) S1 (n, m) ··· (λ(x1 + · · · + xk ) + x)m dµ0 (x1 ) · · · dµ0 (xk ). =q m q Zp Zp m=0 n
Since Z
Z
e(x1 +···+xk +x)t dµ0 (x1 ) · · · dµ0 (xk )
··· Zp
= we can derive easily Z Bn(k) (x) = Zp
Zp
t et − 1
k
ext =
∞ X n=0
Z
Bn(k) (x)
tn , n!
(x1 + · · · + xk + x)n dµ0 (x1 ) · · · dµ0 (xk ).
···
(2.12)
Zp
Thus, by (2.11) and (2.12), we have (k) Dn,q (λ|x) =q n
= =
n X S1 (n, m) m (k) x λ Bm qm λ m=0
n X m=0 n X
(k) q n−m S1 (n, m)Bm
x
(k) |S1 (n, m)|(−q)n−m Bm
m=0
15
(2.13)
λ x λ
.
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In (2.10), by replacing t by 1q (et − 1), we get ∞ X
x (et − 1)n (k) =e q t Dn,q (λ|x) n n! q n=0
=
∞ X n=0
e
λt q λ qt
!k
−1
(2.14)
(k) x n n Bn λ t , λ n q n!
and ∞ ∞ ∞ (k) (k) X X Dn,q (λ|x) 1 t tl Dn,q (λ|x) X n (e S2 (l, n) − 1) = n n q n! q l! n=0 n=0 l=n
∞ X
=
! m (k) X Dn,q (λ|x) tm S (m, n) . 2 qn m! n=0
m=0
(2.15)
By (2.13), (2.14) and (2.15), we obtain the following theorem. Theorem 2.4. For n ≥ 0 and k ∈ N, we have n X (k) (k) x Dn,q (λ|x) = q n−m S1 (n, m)Bm λ m=0 =
n X
(k) |S1 (n, m)|(−q)n−m Bm
m=0
and Bn(k)
x λ
n X
= λ−n
x λ
,
(k) Dm,q (λ|x)q n−m S2 (n, m).
m=0
Now, we consider the λ-Daehee polynomials of the second kind with q-parameter as follows : Z b n,ξ,q (λ|x) = D (−λy + x)n,q dµ0 (y), (n ≥ 0). (2.16) Zp
b n,q (λ) = D b n,q (λ|0) are called the λ-Daehee numbers of In the special case, x = 0, D the second kind with q-parameter. By (2.16), we have Z −λy + x n b Dn,q (λ|x) = q dµ0 (y), (2.17) q Zp n b n,q (x) by (1.1) as follows : and so we can derive the generating function of D Z ∞ ∞ X tn X n −λy + x tn b Dn,q (λ|x) = q dµ0 (y) n! n=0 q n! Zp n n=0 Z ∞ −λy+x X q = qn dµ0 (y)tn n Z p n=0 (2.18) Z −λy+x = (1 + qt) q dµ0 (y) Zp
= (1 + qt)
16
x+λ q
λ log(1 + qt) . λ q (1 + qt) q − 1
JIN-WOO PARK 11-20
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 20, NO.1, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
ON THE λ-DAEHEE POLYNOMIALS WITH q-PARAMETER
From (1.3), (1.6) and (2.17), we get Z −λy + x b n,q (λ|x) =q n D dµ0 (y) q Zp n Z X n S1 (n, l) (−λy + x)l dµ0 (y) =q n l q Zp l=0 Z n X x l l dµ0 (y)q n−l = S1 (n, l)(−λ) y− λ Zp =
l=0 n X l=0
7
(2.19)
x q n−l S1 (n, l)(−λ)l Bl − λ
=(−1)n
n X l=0
x |S1 (n, l)|λl Bl − q n−l . λ
By replacing qt to et − 1 in the equation (2.18), we have ∞ X
λ t (x+λ)t b n,q (λ|x) 1 et − 1 n = q e q D λ t n! q eq − 1 n=0
=
∞ X n=0
Bn
(2.20)
x n −n tn 1+ , λ q λ n!
and, by (1.4), ∞ X
∞ X b n,q (λ|x) 1 et − 1 n = D n! n=0 n=0
n X
! b m,q (λ|x)S2 (n, m) D
m=0
tm . m!
(2.21)
Note that , by (1.10), it is easy to show that Bn (−x) = (−1)n Bn (x + 1). Thus, from (2.19), (2.20) and (2.21), we have the following theorem. Theorem 2.5. For n ≥ 0, we have b n,q (λ|x) = D
n X l=0
x S1 (n, l)(−λ)l Bl − q n−l λ
=(−1)n
n X l=0
x |S1 (n, l)|λl q n−l Bl − . λ
and n X x b m,q (λ|x)S2 (n, m). = qn D λn Bn 1 + λ m=0
By Theorem 2.5, we obtain the following corollary. Corollary 2.6. For n ≥ 0, b n,q (λ|x) = q n D
n X l X
b m,q (λ|x)S1 (n, l)S2 (l, m). D
(2.22)
l=0 m=0
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Now, we observe that q −n (−1)n
Z x+λy Dn,q (λ|x) q =(−1)n dµ0 (y) n! n Zp Z x+λy − q +n−1 dµ0 (y) = n Zp Z −x−λy n X n−1 q = dµ0 (y) n − m m Z p m=1 n X b m,q (λ| − x) n − 1 q −n D = , m! n−m m=1
and, by the similar method to (2.23), we have n X b n,q (λ|x) D n − 1 Dn,q (λ| − x) −n q −n (−1)n = q . n! m! n−m m=1
(2.23)
(2.24)
Hence, by (2.23) and (2.24), we obtain the following theorem. Theorem 2.7. For n ≥ 1, we have q −n (−1)n
b n X n−1 D Dn,q (λ|x) m,q (λ| − x) −n = q n! m! n−m m=1
q −n (−1)n
n X b n,q (λ|x) D n − 1 Dn,q (λ| − x) −n = q . n−m n! m! m=1
and
Now, we consider higher-order λ-Daehee polynomials of second kind with qparameter. Higher-order λ-Daehee polynomials of second kind with q-parameter are defined by the multivariant p-adic invariant integral on Zp : Z Z b (k) (λ|x) = D · · · (−λ(x1 + · · · + xk ) + x)n,q dµ0 (x1 ) · · · dµ0 (xk ) (2.25) n,ξ,q Zp
Zp (k)
b n,q (λ) = where n is an nonnegative integer and k ∈ N. In the special case, x = 0, D (k) b Dn,q (λ|0) are called the higher-order λ-Daehee numbers of second kind with qparameter. (k) b n,q (λ|x) as follows: From (2.25), we can derive the generating function of D ∞ X
n
t (k) b n,q D (λ|x) n! n=0 −λ(x1 +···+xk )+x Z Z ∞ X n q = q ··· dµ0 (x1 ) · · · dµ0 (xk )tn n Z Z p p n=0 Z Z −λ(x1 +···+xk )+x q = ··· (1 + qt) dµ0 (x1 ) · · · dµ0 (xk ) Zp
(2.26)
Zp
k λ log(1 + qt) . λ q (1 + qt) q − 1
= (1 + qt)
x+λk q
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ON THE λ-DAEHEE POLYNOMIALS WITH q-PARAMETER
9
By (2.25), (k) b n,q D (λ|x)
Z Z n X S1 (n, m) · · · (−λ(x1 + · · · + xk ) + x)m dµ0 (x1 ) · · · dµ0 (xk ) m q Zp Zp m=0 Z Z n X x m S1 (n, m) m n (−λ) · · · x + · · · + x − dµ0 (x1 ) · · · dµ0 (xk ) =q 1 k qm λ Zp Zp m=0 =q n
=q n
n x X S1 (n, m) m (k) (−λ) B − m qm λ m=0
=(−1)n
n X
x (k) q n−m λm |S1 (n, m)|Bm − . λ m=0 (2.27) (k)
(k)
From (1.10), we know that Bn (−x) = (−1)n Bn (k + x). Hence, by (2.27), we obtain the following theorem. Theorem 2.8. For n ≥ 0, we have (k) b n,q D (λ|x) =
n X
x (k) S1 (n, m)q n−m (−λ)m Bm − λ m=0
=(−1)n
n X
x (k) (−λ)m q n−m |S1 (n, m)|Bm k+ . λ m=0
In (2.26), by replacing t by 1q (et − 1), we get ∞ X
t n b (k) (λ|x) (e − 1) =e qt (x+λk) D n,q n q n! n=0
λt q
!k
λt
e q −1
(k) ∞ X λn Bn λx + k tn = , qn n! n=0
(2.28)
and ∞ b (k) ∞ b (k) ∞ X n X Dn,q (λ|x) 1 t Dn,q (λ|x) X tl e − 1 = S2 (l, n) n n q n! q l! n=0 n=0 l=n
=
∞ X m=0
! m b (k) X Dn,q (λ|x) tm S (m, n) . 2 qn m! n=0
(2.29)
By (2.28) and (2.29), we obtain the following theorem. Theorem 2.9. For n ≥ 0 and k ∈ N, we have Bn(k)
x λ
n X b (k) (λ|x)q n−m S2 (n, m). + k = λ−n D m,q m=0
By Theorem 2.8 and Theorem 2.9, we obtain the following corollary.
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Corollary 2.10. For n ≥ 0, we have n X m X b (k) (λ|x) = b (k) (λ|x)q n−l S1 (n, m)S2 (m, l). D D n,q l,q m=0 l=0
References [1] L. Comtet, Advanced Combinatorics, Reidel, Dordrecht, 1974. [2] D. S. Kim and T. Kim, Daehee numbers and polynomials, Appl. Math. Sci., 7 (2013), no. 120, 5969-5976. [3] D. S. Kim, T. Kim, H. I. Kwon and J. J. Seo, Daehee polynomials with q-parameter, Adv. Studies Theor. Phys., 8 (2014), no. 13, 561-569. [4] T. Kim, On q-analogye of the p-adic log gamma functions and related integral, J. Number Theory, 76 (1999), no. 2, 320-329. [5] T. Kim, An invariant p-adic integral associated with Daehee numbers, Integral Transforms Spec. Funct., 13 (2002), no. 1, 65-69. [6] T. Kim, q-Volkenborn integration, Russ. J. Math. Phys., 9 (2002), no. 3, 288-299. [7] Q. L. Luo, Some recursion formulae and relations for Bernoulli numbers and Euler numbers of higher order, Adv. Stud. Contemp. Math. 10 (2005), no. 1, 63-70. [8] H. Ozden, I. N. Cangul and Y. Simsek, Remarks on q -Bernoulli numbers associated with Daehee numbers, Adv. Stud. Contemp. Math., 18 (2009), no. 1, 41-48. [9] J. W. Park, S. H. Rim and J. Kim, The twisted Daehee numbers and polynomials, Adv. Difference Equ., 2014, 2014:1. [10] S. Roman, The umbral calculus, Dover Publ. Inc. New York, 2005. [11] Y. Simsek,Generating functions of the twisted Bernoulli numbers and polynomials associated with their interpolation functions, Adv. Stud. Contemp. Math. 16 (2008), no. 2, 251-278. Department of Mathematics Education, Daegu University, Jillyang, Gyeongsan, Gyeongbuk 712-714, Republic of Korea. E-mail address: [email protected]
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Stability of ternary quadratic derivation on ternary Banach algebras: revisited Choonkil Park Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Korea Abstract. In [6], Shagholi et al. defined ternary quadratic derivations on ternary Banach algebras and proved the HyersUlam stability of ternary quadratic derivations on ternary Banach algebras. But the definition is not well-defined and so the proofs of the main results are wrong. In this paper, we correct the definition of ternary quadratic derivation and the proofs of the main results.
1. Introduction The study of stability problems for functional equations is related to a question of Ulam [7] concerning the stability of group homomorphisms and affirmatively answered for Banach spaces by Hyers [3]. Subsequently, the result of Hyers was generalized by Aoki [1] for additive mappings and by Th. M. Rassias [4] for linear mappings by considering an unbounded Cauchy difference. The functional equation f (x + y) + f (x − y) = 2f (x) + 2f (y)
(1.1)
is called a quadratic functional equation. In particular, every solution of the quadratic equation (1.1) is said to be a quadratic mapping. The Hyers-Ulam stability problem for the quadratic functional equation (1.1) was proved by Skof for mappings f : A → B, where A is a normed space and B is a Banach space (see [5]). In [6], Shagholi et al. defined a ternary quadratic derivation D from a ternary Banach algebra A into a ternary Banach algebra B such that D[x, y, z] = [D(x), y 2 , z 2 ] + [x2 , D(y), z 2 ] + [x2 , y 2 , D(z)] for all x, y, z ∈ A. But x2 , y 2 , z 2 are not defined and the brackets of the right side are not defined, since A is not an algebra and D(x) ∈ B and y 2 , z 2 ∈ A. So we correct them as follows. Definition 1.1. Let A be an algebra and ternary Banach algebra with norm k · k. A mapping D : A → A is called a ternary quadratic derivation if (1) D is a quadratic mapping, (2) D[x, y, z] = [D(x), y 2 , z 2 ] + [x2 , D(y), z 2 ] + [x2 , y 2 , D(z)] for all x, y, z ∈ A. In this paper, the proofs of the main results given in [6] are corrected. 2. Stability of ternary quadratic derivations Let A be an algebra and ternary Banach algebra with norm k · k. Theorem 2.1. Let f : A → A be a mapping for which there exists a function φ : A × A × A → [0, ∞) such that ˜ y, z) := φ(x,
∞ X 1 φ(2j x, 2j y, 2j z) < ∞ j 4 j=0
kf (x + y) + f (x − y) − 2f (x) − 2f (y)k ≤ φ(x, y, 0), 2
2
2
2
2
2
(2.1) (2.2)
kf ([x, y, z]) − [f (x), y , z ] − [x , f (y), z ] − [x , y , f (z)])k ≤ φ(x, y, z)
(2.3)
for all x, y, z ∈ A. Then there exists a unique ternary quadratic derivation D : A → A such that 1˜ kf (x) − D(x)k ≤ φ(x, x, 0), 4 for all x ∈ A.
(2.4)
0
2010 Mathematics Subject Classification: 39B52, 13N15, 47B47. Keywords: Hyers-Ulam stability; quadratic functional equation; ternary Banach algebra; ternary quadratic derivation. 0 E-mail: [email protected] 0
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C. Park Proof. Putting x = y = 0 in (2.2), we get f (0) = 0. If we replace y in (2.2) by x and multiply both sides of (2.2) by 14 , we get φ(x, x, 0) f (2x) k − f (x)k ≤ (2.5) 4 4 for all x ∈ A. Now we use the Rassias’ method on inequality (2.5) (see [2]). One can use induction on n to show that k
n−1 f (2n x) 1 X φ(2j x, 2j x, 0) − f (x)k ≤ 2n 2 4 j=0 4j
(2.6)
for all x ∈ A and all nonnegative integers n. Hence k
f (2n+m x) f (2m x) 1 − k≤ 22m 4 22(n+m)
n+m−1 X j=m
φ(2j x, 2j x, 0) 4j n
x) } is Cauchy. for all nonnegative integers n and m with n ≥ m and all x ∈ A. It follows from (2.1) that the sequence { f (2 22n Due to the completeness of A, this sequence is convergent. So one can define the mapping D : A → A by
D(x) := lim
n→∞
f (2n x) 22n
for all x ∈ A. Replacing x, y by 2n x, 2n y, respectively, in (2.2) and multiplying both sides of (2.2) by
(2.7) 1 , 22n
we get
kD(x + y) + D(x − y) − 2D(x) − 2D(y)k 1 kf (2n (x + y)) + f (2n (x − y)) − 2f (2n x) − 2f (2n y)k 22n φ(2n x, 2n y, 0) ≤ lim =0 n→∞ 22n for all x, y ∈ A and all nonnegative integers n. So = lim
n→∞
D(x + y) + D(x − y) = 2D(x) + 2D(y) for all x, y ∈ A. Moreover, it follows from (2.6) and (2.7) that kf (x) − D(x)k ≤
1˜ φ(x, x, 0) 4
for all x ∈ A. It follows from (2.3) we get kD([x, y, z]) − [D(x), y 2 , z 2 ] − [x2 , D(y), z 2 ] − [x2 , y 2 , D(z)]k 1 kf ([2n x, 2n y, 2n z]) − [f (2n x), (2n y)2 , (2n z)2 ] − [(2n x)2 , f (2n y), (2n z)2 ] − [(2n x)2 , (2n y)2 , f (2n z)]k 43n φ(2n x, 2n y, 2n z) φ(2n x, 2n y, 2n z) ≤ lim ≤ lim =0 3n n→∞ n→∞ 4 4n for all x, y, z ∈ A. So D[x, y, z] = [D(x), y 2 , z 2 ] + [x2 , D(y), z 2 ] + [x2 , y 2 , D(z)] ≤ lim
n→∞
for all x, y, z ∈ A. Now, let D0 : A → A be another ternary quadratic derivation satisfying (2.4). Then we have 1 kD(2n x) − D0 (2n x)k 22n 1 ≤ 2n (kD(2n x) − f (2n x)kB + kf (2n x) − D0 (2n x)k) 2 2 ≤ 2n φ(2n x, 2n x, 0) 2 which tends to zero as n → ∞ for all x ∈ A. So we can conclude that D(x) = D0 (x) for all x ∈ A. This proves the uniqueness of D. Thus, the mapping D : A → A is a unique ternary quadratic derivation satisfying (2.4). kD(x) − D0 (x)k =
Theorem 2.2. Let f : A → A be a mapping for which there exists a function φ : A × A × A → [0, ∞) satisfying (2.2), (2.3) and ∞ X x y z 43j φ( j , j , j ) < ∞ (2.8) 2 2 2 j=0 22
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Ternary quadratic derivation on ternary Banach algebras for all x, y, z ∈ A. Then there exists a unique ternary quadratic derivation D : A → A such that ˜ x , x , 0), kf (x) − D(x)k ≤ φ( 2 2 for all x ∈ A. Here, ∞ X x y z ˜ y, z) := φ(x, 4j φ( j , j , j )9 2 2 2 j=0
(2.9)
for all x, y, z ∈ A. Proof. It follows from (2.5) that
x x x kf (x) − 4f ( )k ≤ φ( , , 0) 2 2 2
for all x ∈ A. By the same reasoning as in the proof od Theorem 2.1, one can define a quadartic unique mapping D : A → A by x (2.10) D(x) := lim 22n f ( n ) n→∞ 2 for all x ∈ A. It follows from (2.8) and (2.10) that kD([x, y, z]) − [D(x), y 2 , z 2 ] − [x2 , D(y), z 2 ] − [x2 , y 2 , D(z)]k x y z x y z x y z x y z ≤ lim 43n kf ([ n , n , n ]) − [f ( n ), ( n )2 , ( n )2 ] − [( n )2 , f ( n ), ( n )2 ] − [( n )2 , ( n )2 , f ( n )]k n→∞ 2 2 2 2 2 2 2 2 2 2 2 2 x y z ≤ lim 43n φ( n , n , n ) = 0 n→∞ 2 2 2 for all x, y, z ∈ A. So D[x, y, z] = [D(x), y 2 , z 2 ] + [x2 , D(y), z 2 ] + [x2 , y 2 , D(z)] for all x, y, z ∈ A. Thus the mapping D : A → A is a unique ternary quadratic derivation satisfying (2.9).
From Theorems 2.1 and 2.2, we obtain the following corollary concerning the Hyers-Ulam stability of the functional equation (1.1). Corollary 2.3. Let p and θ be nonnegative real numbers with p 6= 2, and let f : A → A be a mapping such that kf (x + y) + f (x − y) − 2f (x) − 2f (y)k ≤ θ(kxkp + kykp ), kf ([x, y, z]) − [f (x), y 2 , z 2 ] − [x2 , f (y), z 2 ] − [x2 , y 2 , f (z)])k ≤ θ(kxkp + kykp + kzkp ), for all x, y, z ∈ A. Then there exists a unique ternary quadratic derivation D : A → A such that 2θ kf (x) − D(x)k ≤ kxkp 4 − 2p holds for all x ∈ X, where p < 2, and the inequality 2θ kf (x) − D(x)k ≤ p kxkp 2 −4 holds for all x ∈ X, where p > 6.
References [1] [2] [3] [4] [5] [6]
T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), 431–434. D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci, 27 (1941), 222–224. Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. F. Skof, Propriet locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53 (1983), 113–129. S. Shagholi, M. Eshaghi Gordji and M. B. Savadkouhi, Stability of ternary quadratic derivation on ternary Banach algebras. J. Comput. Anal. Appl. 13 (2011), 1097–1105. [7] S. M. Ulam, Problems in Modern Mathematics, Chapter V I, Science ed., Wiley, New York, 1940.
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SOME PROPERTIES OF MODULAR S -METRIC SPACES AND ITS FIXED POINT RESULTS MELTEM ERDEN EGE AND CIHANGIR ALACA
Abstract. In this paper, we introduce modular
†
S -metric spaces and deal with their some properties. S -metric spaces.
We also prove some xed point theorems on complete modular
1.
Introduction
Fixed point theory in metric spaces begins with the Banach Contraction Principle which is published in 1922 [6]. Since it is simple and useful, it has become a very popular tool to solve existence problems in mathematical analysis. There are some authors introduced the generalization of metric spaces such as Gahler [16], which is called 2-metric space, and Dhage [14], which is called D-metric space. In 2013, Mustafa and Sims [24] found that the fundamental topology properties of the metric spaces are incorrect. They [25] introduced a generalization of metric spaces which is called G-metric spaces. The concept of S -metric spaces was rstly introduced by Sedghi et al. [28] in 2012. Sedghi and Dung [29] proved a general xed point theorem in S -metric spaces which is a generalization [[28], Theorem 3.1]. Gupta [17] introduced the concepts of cyclic contraction on S -metric space and proved some xed point theorems on S -metric spaces. Chouhan [12] proved a common unique xed point theorem for expansive mappings in S -metric space. Hieu et al. [18] gave a xed point theorem for a class of maps depending on another map on S -metric spaces. The notion of modular space was rstly introduced by Nakano [26] and developed by Koshi, Shimogaki, Yamamuro (see [22, 30]) and others. Recently, many researchers have been interested in xed point of modular space. In 2008, Chistyakov [7] introduced the notion of modular metric space generated by F -modular and developed the theory of this space. He also dened the notion of a modular on an arbitrary set and the modular metric spaces in 2010 [8]. Abdou [1] studied and proved some new xed points theorems for pointwise and asymptotic pointwise contraction mappings in modular metric spaces. Azadifer et. al. [3] introduced the notion of modular G-metric spaces and proved some xed point theorems of contractive in this space. Many authors studied on modular metric spaces [4],[5],[10],[11],[19],[20],[21]. In this paper we introduce the concept of modular S -metric spaces and their properties. Then we give xed point theorems for self mappings on complete modular S -metric spaces. 2.
Preliminaries
Denition 2.1. [27]. A modular on a real linear space X is a functional ρ : X −→ [0, ∞] satisfying the followings: (A1) ρ(0) = 0; (A2) If x ∈ X and ρ(αx) = 0 for all numbers α > 0, then x = 0; (A3) ρ(−x) = ρ(x) for all x ∈ X ; (A4) ρ(αx + βy) ≤ ρ(x) + ρ(y) for all α, β ≥ 0 with α + β = 1 and x, y ∈ X . Let X be a non-empty set and λ ∈ (0, ∞). We remark that the function ω : (0, ∞)×X ×X −→ [0, ∞] is denoted by ωλ (x, y) = ω(λ, x, y) for all λ > 0 and x, y ∈ X . Mathematics Subject Classication. 46A80, 47H10, 54E35. Key words and phrases. modular S -metric space, s-contraction, xed point. 2010
† :Corresponding
Author.
1
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MELTEM ERDEN EGE AND CIHANGIR ALACA
Denition 2.2. [8]. Let X be a non-empty set, a function ω : (0, ∞) × X × X
−→ [0, ∞] is said to
be a metric modular on X if satisfying, for all x, y, z ∈ X the following conditions hold: (i) ωλ (x, y) = 0 for all λ > 0 ⇔ x = y ; (ii) ωλ (x, y) = ωλ (y, x) for all λ > 0; (iii) ωλ+µ (x, y) ≤ ωλ (x, z) + ωµ (z, y) for all λ, µ > 0.
Denition 2.3. [28] Let
X be a non-empty set. An S -metric on X is a function S : X 3 → [0, ∞) that satises the following conditions, for each x, y, z, a ∈ X , (i) S(x, y, z) ≥ 0; (ii) S(x, y, z) = 0 if and only if x = y = z ; (iii) S(x, y, z) ≤ S(x, x, a) + S(y, y, a) + S(z, z, a). The pair (X, S) is called an S -metric space.
3.
Modular
S -metric
spaces
We dene a new concept combining with S -metric and modular metric space.
Denition 3.1. Let X be a non-empty set. An modular S -metric on X is a function sλ : (0, ∞) × X × X × X → [0, ∞]
that satises the following conditions for all x, y, z ∈ X and λ > 0 : (S1) sλ (x, y, z) ≥ 0; (S2) sλ (x, y, z) = 0 if and only if x = y = z ; (S3) sλ+µ+ν (x, y, z) ≤ sλ (x, x, a) + sµ (y, y, a) + sν (z, z, a) for all λ, µ, ν > 0 and a ∈ X .
Example 3.2. (1) sλ (x, y, z) = 0 if x = y = z and sλ (x, y, z) = ∞ if x 6= y 6= z . (2) If S is an modular S -metric on X , we can get:
(a) sλ (x, y, z) = 0 if λ > S(x, y, z) and sλ (x, y, z) = ∞ if λ ≤ S(x, y, z). (b) sλ (x, y, z) = 0 if λ ≥ S(x, y, z) and sλ (x, y, z) = ∞ if λ < S(x, y, z). S(x, y, z) ; where ϕ : (0, ∞) → (0, ∞) is non-decreasing function. (c) sλ (x, y, z) = ϕ(λ)
Lemma 3.3. If the function have sλ (x, x, y) = sλ (y, y, x).
0 < λ → sλ (x, y, z) is continuous on (0, ∞) where x, y, z ∈ X , then we
Proof. There exists ε > 0 such that sλ (x, x, y) ≤ sε (x, x, x) + sε (x, x, x) + sλ−2ε (y, y, x).
If we take limit as ε → 0, we get sλ (x, x, y) ≤ sλ (y, y, x). Similarly sλ (y, y, x) ≤ sλ (x, x, y). So we get sλ (x, x, y) ≤ sλ (y, y, x) ≤ sλ (x, x, y)
and sλ (x, x, y) = sλ (y, y, x).
Remark 3.4. The function sλ (x, y, z) for λ > 0 is non-increasing on (0, ∞) where x, y, z ∈ X , if it is continuous on (0, ∞). In fact if 0 < ν < µ < λ, (S3) implies sλ (x, x, y) ≤ sλ−µ (x, x, x) + sµ−ν (x, x, x) + sν (y, y, x)
and we have sλ (x, x, y) ≤ sν (y, y, x)
from (S2). From Lemma 3.3, we conclude that sλ (x, x, y) ≤ sν (x, x, y). From that inequality the function sλ (x, y, z) is non-increasing on (0, ∞). It follows that at each point λ > 0 the right limit sλ+0 (x, y, z) = lim sµ (x, y, z) µ→λ+0
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SOME PROPERTIES OF MODULAR
S -METRIC
SPACES AND ITS FIXED POINT RESULTS
3
and the left limit sλ−0 (x, y, z) = lim sλ−ε (x, y, z) ε→0
exist in [0, ∞] and the following two inequalities hold: sλ+0 (x, y, z) ≤ sλ (x, y, z) ≤ sλ−0 (x, y, z).
Denition 3.5. Let sλ be a modular S -metric on X . The binary relation ∼s on X dened for x, y ∈ X
by
(3.1)
s
x ∼ y ⇔ lim sλ (x, x, y) = 0 λ→∞
is an equivalence relation. Indeed x ∼ x is clear by virtue of (S2). From Lemma 3.3, we have s
s
s
x ∼ y ⇔ lim sλ (x, x, y) = 0 = lim sλ (y, y, x) ⇔ y ∼ x. λ→∞
λ→∞
If x ∼ y and y ∼ z , we get lim sλ (x, x, y) = 0 and lim sλ (y, y, z) = 0. By (S3) and Lemma 3.3, λ→∞ λ→∞ we conclude that s
s
lim s3λ (x, x, z) ≤ lim sλ (x, x, y) + lim sλ (x, x, y) + lim sλ (y, y, z)
λ→∞
λ→∞
λ→∞
λ→∞
=0 + 0 + 0.
It is clear that
s
lim s3λ (x, x, z) = 0 ⇔ x ∼ z
λ→∞
by (S1). The equivalence class of the element x ∈ X in the quotient set X ∼ is dened by s
s
Xs ≡ Xs (x) = {y ∈ X : y ∼ x}.
For x0 ∈ X , the set Xs∗ is dened as follows: Xs∗ ≡ Xs∗ (x0 ) = {x ∈ X : ∃λ = λ(x) > 0 such that sλ (x, x, x0 ) < ∞}.
From Remark 3.4, the function ∼
s
s
s
S : (X ∼) × (X ∼) × (X ∼) → [0, ∞]
given by
∼
S(Xs (x), Xs (x), Xs (y)) = sλ (x, x, y) is well-dened and satises the axioms of S -metric.
Theorem 3.6. If
sλ is a modular S -metric on X , then the modular set Xs is an modular S -metric space with S -metric given by S ◦ (x, x, y) = inf{λ > 0 : sλ (x, x, y) ≤ λ},
for all x, y ∈ Xs . Proof. Since x ∼ y , there exists λ0 > 0 such that s
sλ (x, x, y) ≤ 1
for all λ ≥ λ0 by (3.1). Taking λ1 = max{1, λ0 }, we get sλ1 (x, x, y) ≤ 1 ≤ λ1
which together with the denition of S (x, x, y) gives ◦
S ◦ (x, x, y) ≤ λ1 < ∞.
Given x ∈ Xs , (S2) implies that sλ (x, x, x) = 0 for all λ > 0, and so, S ◦ (x, x, x) = 0. Let sλ satisfy (S2), x, y ∈ Xs and S ◦ (x, x, y) = 0. Then sµ (x, x, y) does not exceed µ for all µ > 0. Hence for any λ > 0 and 0 < µ < λ, from Remark 3.4 we have sλ (x, x, y) ≤ sµ (x, x, y) ≤ µ → 0 as µ → +0. It follows that sλ (x, x, y) = 0 for all λ > 0. Thus axiom (S2) implies x = y .
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MELTEM ERDEN EGE AND CIHANGIR ALACA
It is clear from (S1) that S ◦ (x, x, y) ≥ 0. Now we show the triangle inequality: S ◦ (x, x, y) ≤ 2S ◦ (x, x, z) + S ◦ (y, y, z)
for some z ∈ Xs . In fact by the denition of S ◦ for any λ > S ◦ (x, x, z) and µ > S ◦ (y, y, z), we nd sλ (x, x, z) ≤ λ and sµ (y, y, z) ≤ µ. As a result, we get s2λ+µ (x, x, y) ≤ 2sλ (x, x, z) + sµ (y, y, z) ≤ 2λ + µ
by the axiom (S3). It follows from the denition of S ◦ that S ◦ (x, x, y) ≤ 2λ + µ and it remains to pass limit as λ → S ◦ (x, x, z) and µ → S ◦ (y, y, z).
Theorem 3.7. Let sλ be a modular S -metric on a set X and S 1 (x, x, y) = inf{λ + sλ (x, x, y) : λ > 0}
be dened for all x, y ∈ Xs . Then S 1 is an S -metric on Xs such that S ◦ ≤ S 1 ≤ 2S ◦ . Proof. Since, for x, y ∈ Xs , the value sλ (x, x, y) is nite due to (3.1) for λ > 0 large enough, then the set {λ + sλ (x, x, y) : λ > 0} ⊂ R+ is non-empty and bounded from below, therefore S 1 (x, x, y) ∈ R+ . Since sλ (x, x, x) = 0, then from the denition of S 1 , S 1 (x, x, x) = inf{λ + sλ (x, x, x) : λ > 0} = 0. |
{z 0
}
Let sλ satisfy (S2), x, y ∈ Xs and S 1 (x, x, y) = 0. The equality x = y will follow from (S2) if we show that sλ (x, x, y) = 0 for all λ > 0. On the contrary, suppose that sλ0 (x, x, y) > 0 for some λ0 > 0. Then for λ ≥ λ0 we nd λ + sλ (x, x, y) ≥ λ0 , and if 0 < λ < λ0 , then 0 < sλ0 (x, x, y) ≤ sλ (x, x, y) ≤ λ + sλ (x, x, y)
from Remark 3.4. Thus, λ + sλ (x, x, y) ≥ λ1 = min{λ0 , sλ0 (x, x, y)} for all λ > 0. By the denition of S 1 , S 1 (x, x, y) ≥ λ1 > 0, which contradicts the assumption. Now let us show that triangle inequality: S 1 (x, x, y) ≤ 2S 1 (x, x, z) + S 1 (y, y, z). For any ε > 0 we nd λ = λ(ε) > 0 and µ = µ(ε) > 0 such that λ + sλ (x, x, z) ≤ S 1 (x, x, z) + ε and µ + sµ (y, y, z) ≤ S 1 (y, y, z) + ε
from the denition of S 1 . Applying axiom (S3), S 1 (x, x, y) ≤(2λ + µ) + s2λ+µ (x, x, y) ≤ 2λ + µ + 2sλ (x, x, z) + sµ (y, y, z) ≤2S 1 (x, x, z) + 2ε + S 1 (y, y, z)
and it remains to take into account the arbitrariness of ε > 0. Let us prove that metrics S ◦ and S 1 are equivalent on Xs . In order to obtain the left-hand side inequality, suppose that λ > 0 is arbitrary. If sλ (x, x, y) ≤ λ, then the denition of S ◦ implies S ◦ (x, x, y) ≤ λ. If sλ (x, x, y) > λ, then S ◦ (x, x, y) ≤ sλ (x, x, y). Setting µ = sλ (x, x, y) we nd µ > λ. Thus it follows from Remark 3.4 that sµ (x, x, y) ≤ sλ (x, x, y) = µ.
Hence
S ◦ (x, x, y) ≤ µ = sλ (x, x, y).
Therefore for any λ > 0 we have S ◦ (x, x, y) ≤ max{λ, sλ (x, x, y)} ≤ λ + sλ (x, x, y).
Taking the inmum over all λ > 0, we get the inequality S ◦ (x, x, y) ≤ S 1 (x, x, y)
To obtain the right-hand side inequality, we note that given λ > 0 such that S ◦ (x, x, y) < λ by the denition of S ◦ . We get sλ (x, x, y) ≤ λ. So S 1 (x, x, y) ≤ λ + sλ (x, x, y) ≤ 2λ. Passing to the limit as λ → S ◦ (x, x, y), we get S 1 (x, x, y) ≤ 2S ◦ (x, x, y).
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Theorem 3.8. Let sλ be a modular S -metric on a set X , x, y ∈ Xs and λ > 0. We have (a) (b) (c) (d)
If S ◦ (x, x, y) < λ, then sλ (x, x, y) ≤ S ◦ (x, x, y) < λ. If sλ (x, x, y) = λ, then S ◦ (x, x, y) = λ. If λ = S ◦ (x, x, y) > 0, then sλ+0 (x, x, y) ≤ λ ≤ sλ−0 (x, x, y). If the function µ → sµ (x, x, y) is continuous from the right on (0, ∞), then along with (a) − (c) we have: S ◦ (x, x, y) ≤ λ ⇔ sλ (x, x, y) ≤ λ.
(e) If the function µ → sµ (x, x, y) is continuous from the left on (0, ∞), then along with (a) − (c) we have: S ◦ (x, x, y) < λ ⇔ sλ (x, x, y) < λ.
(f) If the function µ → sµ (x, x, y) is continuous on (0, ∞), then along with (a) − (e) we have: S ◦ (x, x, y) = λ ⇔ sλ (x, x, y) = λ.
Proof. (a) For any µ > 0 such that S ◦ (x, x, y) < µ < λ by the denition of S ◦ and Remark 3.4, we have sµ (x, x, y) ≤ µ and sλ (x, x, y) ≤ sµ (x, x, y). Hence sλ (x, x, y) ≤ µ and it remains to pass to the limit as µ → S ◦ (x, x, y). (b) By the denition, S ◦ (x, x, y) ≤ λ and item (a) implies S ◦ (x, x, y) = λ. (c) For any µ > λ = S ◦ (x, x, y), the denition of S ◦ implies sµ (x, x, y) ≤ µ and so sλ+0 (x, x, y) = lim sµ (x, x, y) ≤ lim µ = λ. µ→λ+0
µ→λ+0
For any 0 < µ < λ we nd sµ (x, x, y) > µ and so sλ−0 (x, x, y) = lim sµ (x, x, y) ≥ lim µ = λ. µ→λ−0
µ→λ−0
(d) The sucient condition follows from the denition of S ◦ . Let us prove the reverse implication. If S ◦ (x, x, y) < λ, then by virtue of item (a), sλ (x, x, y) < λ and if S ◦ (x, x, y) = λ, then sλ (x, x, y) = sλ+0 (x, x, y) ≤ λ
which is a consequence of the continuity from the right of the function µ → sµ (x, x, y) and item (c). (e) By item (a), it suces to prove the sucient condition. The denition of S ◦ gives S ◦ (x, x, y) ≤ λ but if S ◦ (x, x, y) = λ, then by item (c) we would have sλ (x, x, y) = sλ−0 (x, x, y) ≥ λ
which contradicts the assumption. (f ) Sucient condition follows from (b). For the reverse asertion the two inequalities sλ (x, x, y) ≤ λ ≤ sλ (x, x, y)
follows from (c).
Denition 3.9. Let sλ be a modular S -metric on a set X . (1) A sequence (xn ) ⊂ Xs∗ converges to x ∈ Xs∗ if sλ (xn , xn , x) → 0 as n → ∞. That is, for each ε > 0, there exists n0 ∈ N such that for all n ≥ n0 we have sλ (xn , xn , x) < ε. We write xn → x. (2) A sequence (xn ) ⊂ Xs∗ is a s-Cauchy if sλ (xn , xn , xm ) → 0 as m, n → ∞. That is, for each ε > 0, there exists n0 ∈ N such that for all n ≥ n0 we have sλ (xn , xn , xm ) < ε. (3) The modular S -metric space Xs∗ is s-complete if every s-Cauchy is a s-convergent in Xs∗ .
Lemma 3.10. Let sλ be a modular S -metric on a set X . If xn → x and yn → y, then sλ (xn , xn , yn ) → sλ (x, x, y).
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Proof. Since xn → x and yn → y , then for each ε > 0 there exist n1 , n2 ∈ N such that ∀n ≥n1 , sλ (xn , xn , x) < ε ∀n ≥n2 , sλ (yn , yn , y) < ε.
Without loss of generality we can assume ε 4 ε ∀n ≥n2 , sδ (yn , yn , y) < ε(δ) = . 4 If we set n0 = max{n1 , n2 }, therefore for every n ≥ n0 we get ∀n ≥n1 , sδ (xn , xn , x) < ε(δ) =
sλ (xn , xn , yn ) ≤2sδ (xn , xn , x) + sλ−2δ (yn , yn , x) ≤2sδ (xn , xn , x) + 2sδ (yn , yn , y) + sλ−4δ (x, x, y)
for λ > δ > 0 by triangle inequality. If we take δ → 0, we have ε ε sλ (xn , xn , yn ) ≤ + + sλ (x, x, y) 2 2 sλ (xn , xn , yn ) ≤ε + sλ (x, x, y) sλ (xn , xn , yn ) − sλ (x, x, y) ≤ ε.
On the other hand we get sλ (x, x, y) ≤2sδ (x, x, xn ) + sλ−2δ (y, y, xn ) ≤2sδ (x, x, xn ) + 2sδ (y, y, yn ) + sλ−4δ (xn , xn , yn ).
From Lemma 3.3 and taking the limit as δ → 0 we have: ε ε sλ (x, x, y) ≤ + + sλ (xn , xn , yn ) 2 2 ≤ε + sλ (xn , xn , yn ) sλ (x, x, y) − sλ (xn , xn , yn ) ≤ ε.
So we get from that inequalities |sλ (xn , xn , yn ) − sλ (x, x, y)| < ε, that is, sλ (xn , xn , yn ) → sλ (x, x, y).
4.
Fixed Point Theorems
In this section we introduce some xed point theorems on modular S -metric space.
Denition 4.1. Let
sλ be a modular S -metric on a set X . A map T : Xs∗ → Xs∗ is said to be a s-contraction if there exists a constant 0 ≤ k < 1 such that sλ (T x, T x, T y) ≤ ksλ (x, x, y)
for all x, y ∈ X .
Corollary 4.2. Let Xs∗ , Ys∗ modular S -metric spaces and f at x ∈ Xs∗ if and only if f (xn ) → f (x) where xn → x.
Theorem 4.3. Let Xs∗ be a s-complete and T point u ∈
Xs∗
.
: Xs∗ → Ys∗ be a map. Then f is continuous
: Xs∗ → Xs∗ be s-contraction. Then T has a unique xed
Proof. First, we show uniqueness. Suppose that there exist x, y ∈ Xs∗ with x = T x and y = T y . Then sλ (x, x, y) = sλ (T x, T x, T y) ≤ ksλ (x, x, y) .
Therefore sλ (x, x, y) = 0.
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To show the existence, we select x ∈ Xs∗ and show that (T n x) is a Cauchy sequence. For n = 0, 1, 2, · · · , we get by induction sλ (T n x, T n x, T n+1 x) ≤ksλ (T n−1 x, T n−1 x, T n x)
.. . ≤k n sλ (x, x, T x).
Taking the limit as n → ∞, we get lim sλ (T n x, T n x, T n+1 x) = 0.
n→∞
Thus there exists ε > 0 such that sλ (T n x, T n x, T n+1 x) ≤ ε.
Without loss of generality, we can assume that there exists m−2 X
sλ (T n x, T n x, T m x) ≤2
s
λ m−n
ε m−n
(T i x, T i x, T i+1 x) + s
for λ m−n
λ m−n
such that
(T m−1 x, T m−1 x, T m x)
i=n m−2 X
ki s
≤2
λ m−n
(x, x, T x) + k m−1 s
λ m−n
(x, x, T x)
i=n
ε ε +···+ ) m−n m−n ≤2ε.
≤2(
That is for m > n, sλ (T n x, T n x, T m x) ≤ 2ε.
This shows that (T n x) is a Cauchy sequence and since Xs∗ is s-complete, there exists u ∈ Xs∗ with lim T n x = u. n→∞ From the continuity of T , we get u = lim T n+1 x = lim T (T n x) = T u. n→∞
n→∞
Therefore u is a xed point of T .
Let M be the family of all continuous functions of ve variables M : R5+ → R+ . For some k ∈ [0, 1), we consider the following conditions: (C1) For all x, y, z ∈ R+ , if y ≤ M (x, x, 0, z, y) with z ≤ 2x + y , then y ≤ kx. (C2) If y ≤ M (y, 0, y, y, 0) for all y ∈ R+ , then y = 0.
Theorem 4.4. Let T be a self-map on s-complete Xs∗ and (4.1)
sλ (T x, T x, T y) ≤ M (sλ (x, x, y), sλ (T x, T x, x), sλ (T x, T x, y), s3λ (T y, T y, x), sλ (T y, T y, y))
for all x, y, z ∈ Xs∗ and some M ∈ M. Then we have (1) If M satises the condition (C1), then T has a xed point. (2) If M satises the condition (C2) and T has a xed point x, then the xed point is unique. Proof. (1) For each x0 ∈ Xs∗ and n ∈ N, we take xn+1 = T xn . It follows from (4.1) and Lemma 3.3 that sλ (xn+1 , xn+1 , xn+2 ) =sλ (T xn , T xn , T xn+1 ) ≤M (sλ (xn , xn , xn+1 ), sλ (xn+1 , xn+1 , xn ), sλ (xn+1 , xn+1 , xn+1 ), s3λ (xn+2 , xn+2 , xn ), sλ (xn+2 , xn+2 , xn+1 )) =M (sλ (xn , xn , xn+1 ), sλ (xn , xn , xn+1 ), 0, s3λ (xn , xn , xn+2 ), sλ (xn+1 , xn+1 , xn+2 )).
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By triangle inequality and Lemma 3.3, we have (4.2)
s3λ (xn , xn , xn+2 ) ≤ 2sλ (xn , xn , xn+1 ) + sλ (xn+1 , xn+1 , xn+2 )
From (4.2), we see that z ≤ 2x + y . Since M satises the condition (C1), there exists k ∈ [0, 1) such that (4.3)
sλ (xn+1 , xn+1 , xn+2 ) ≤ ksλ (xn , xn , xn+1 ) ≤ · · · ≤ k n+1 sλ (x0 , x0 , x1 ).
Taking the limit as n → ∞, we get lim sλ (xn , xn , xn+1 ) = 0.
n→∞
Hence there exists ε > 0 for λ > 0 such that sλ (xn , xn , xn+1 ) ≤ ε.
Without loss of generality, we can assume that there exists s
λ m−n
(xn , xn , xn+1 ) ≤
ε m−n
for
λ m−n
> 0 such that
ε . m−n
Thus for all n < m by using (S3), Remark 3.4 and (4.3) we have sλ (xn , xn , xm ) ≤2s λ (xn , xn , xn+1 ) + s λ (xm , xm , xn+1 ) 3
3
≤2s λ (xn , xn , xn+1 ) + s λ (xn+1, xn+1 , xm )
.. .
3
3
ε ε ε + +···+ ) m−n m−n m−n ≤2ε.
≤2(
This proves that (xn ) is s-Cauchy in the s-complete Xs∗ . Then (xn ) converges an x ∈ Xs∗ . Now we prove that x is a xed point of T . By using (4.1), we get sλ (xn+1 , xn+1 , T x) =sλ (T xn , T xn , T x) ≤M (sλ (xn , xn , x), sλ (T xn , T xn , xn ), sλ (T xn , T xn , x), s3λ (T x, T x, xn ), sλ (T x, T x, x)).
Since M ∈ M, then using Lemma 3.10 and taking the limit as n → ∞, we obtain sλ (x, x, T x) ≤ M (0, 0, 0, s3λ (T x, T x, x), sλ (T x, T x, x)).
From Remark 3.4, we can rewrite s3λ (T x, T x, x) ≤ sλ (T x, T x, x).
Then the inequality can be written as follows: sλ (x, x, T x) ≤ M (0, 0, 0, sλ (T x, T x, x), sλ (T x, T x, x)).
Since M satises the condition (C1), then sλ (x, x, T x) ≤ k.0 = 0. This proves that x = T x.
(2) Let x, y be xed points of T . We prove that x = y. It follows from (4.1) that sλ (x, x, y) =sλ (T x, T x, T y)
≤M (sλ (x, x, y), sλ (T x, T x, x), sλ (T x, T x, y), s3λ (T y, T y, x), sλ (T y, T y, y)) ≤M (sλ (x, x, y), 0, sλ (x, x, y), s3λ (y, y, x), 0).
From Remark 3.4 and Lemma 3.3, we get sλ (x, x, y) ≤ M (sλ (x, x, y), 0, sλ (x, x, y), sλ (x, x, y), 0).
Since M satises the condition (C2), sλ (x, x, y) = 0 ⇐⇒ x = y.
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Remark 4.5. Theorem 4.3 is a corollary of Theorem 4.4 when we take k ∈ [0, 1) and x, y, z, s, t ∈ R+ .
9
M (x, y, z, s, t) = k.x for
Now we will give a new corollary of Theorem 4.4.
Corollary 4.6. Let T be a self map on s-complete Xs∗ and sλ (T x, T x, T y) ≤ a(sλ (T x, T x, x) + sλ (T y, T y, y))
for some a ∈ [0,
1 2)
and all x, y ∈ Xs∗ . Then T has a unique xed point in Xs∗ .
Proof. We must show that M (x, y, z, s, t) = a(y + t) satises conditions (C1) and (C2). First, we have M (x, x, 0, z, y) = a(x + y).
So, if y ≤ M (x, x, 0, z, y) with z ≤ 2x + y , then y ≤M (x, x, 0, z, y) = a(x + y) y ≤ax + ay a y≤ x 1−a a
∈ [0, 1). Therefore, M satises condition (C1). with 1−a If y ≤ M (y, 0, y, y, 0) = 0, then y = 0. Therefore, M satises the condition (C2). Since sλ (T x, T x, T y) ≤M (sλ (x, x, y), sλ (T x, T x, x), sλ (T x, T x, y), sλ (T y, T y, x), sλ (T y, T y, y)) =a(sλ (T x, T x, x) + sλ (T y, T y, y)), T has a unique xed point in Xs∗ by Theorem 4.4.
Open problems How can obtain some similar results for the papers (see [2, 15]) in fuzzy metric spaces with the help of this technique? References [1] A.A.N. Abdou, On asymptotic pointwise contractions in modular metric spaces, Abstract and Applied Analysis, Article ID 501631, 2013, 1-7. [2] C. Alaca, Fixed point results for mappings satisfying a general contractive condition of operator type in dislocated fuzzy quasi-metric spaces, J. Computational Analysis and Applications, 12 (1-b), 361-368 (2010). [3] B. Azadifar, M. Maramaei, Gh. Sadeghi, On the modular G-metric spaces and xed point theorems, J. Nonlinear Sci. Appl. 6, 293-304 (2013). [4] B. Azadifar, M. Maramaei and Gh. Sadeghi, Common xed point theorems in modular G-metric spaces, Nonlinear Anal. Appl (2013) 1-9. [5] B. Azadifar, Gh. Sadeghi, R. Saadati and C. Park, Integral type contractions in modular metric spaces. Journal of Inequalities and Applications, 2013(1), 483. [6] S. Banach, Sur les operations dans les ensembles abstraits et leurs applications aux equations integrales, Fund. Math., 3, 133-181 (1922). [7] V.V. Chistyakov, Modular metric spaces generated by
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[8] V.V. Chistyakov, Modular metric spaces I. basic conceps, Nonlinear Anal., 72, 1-14 (2010). [9] V.V. Chistyakov, Fixed points of modular contractive maps, Dokl. Math., 86, 515-518 (2012). [10] V.V. Chistyakov, Modular contractions and their application, In Models, Algorithms, and Technologies for Network Analysis (pp. 65-92), Springer New York, (2013). [11] Y.J. Cho, R. Saadati and G. Sadeghi, Quasi-contractive mappings in modular metric spaces, J. Appl. Math., 907951 (2012). [12] P. Chouhan, A common unique xed point theorem for expansive type mappings in
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Mathematical Forum, 8, 1287-1293 (2013). [13] H. Dehghan, M.E. Gordji and A. Ebadian, Comment on xed point theorems for contraction mappings in modular metric spaces, Fixed Point Theory and Applications, 144 (2012). [14] B.C. Dhage, Generalized metric spaces and mappings with xed point, Bulletin Calcutta Mathematical Society, 84(4), 329-336 (1992). [15] H. Efe, C. Alaca, C. Yldz, Fuzzy multi-metric spaces, J. Computational Analysis and Applications, 10 (3), 367-375 (2008).
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Thai Journal of Mathematics (2014). [19] N. Hussain and P. Salimi, Implicit contractive mappings in modular metric and fuzzy metric spaces, The Scientic World Journal 2014 (2014). [20] E. Kilinc and C. Alaca, A xed point theorem in modular metric spaces, Adv. Fixed Point Theory, 4(2), 199-206 (2014). [21] E. Kilinc, C. Alaca, Fixed point results for commuting mappings in modular metric spaces, J. Applied Functional Analysis, 10(3-4), 204-210 (2015). [22] S. Koshi, T. Shimogaki, On
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Department of Mathematics, Institute of Natural and Applied Sciences, Celal Bayar University, Muradiye Campus 45140 Manisa, Turkey
E-mail address : [email protected]
Department of Mathematics, Faculty of Science and Arts, Celal Bayar University, Muradiye Campus 45140 Manisa, Turkey
E-mail address, Corresponding author: [email protected]
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The strong converse inequality for de la Vall´ee Poussin means on the sphere ∗ Chunmei Ding
Ruyue Yang
Feilong Cao†
Department of Mathematics, China Jiliang University, Hangzhou 310018, Zhejiang Province, P R China
Abstract This paper discusses the approximation by de la Vall´ee Poussin means Vn f on the unit sphere. Especially, the lower bound of approximation is studied. As a main result, the strong converse inequality for the means is ( ) established. Namely, it is( proved ) that there are constants 1 1 √ √ C1 and C2 such that C1 ω f, n ≤ ∥Vn f − f ∥p ≤ C2 ω f, n for any p-th Lebesgue p
p
integrable or continuous function f defined on the sphere, where ω(f, t)p is the modulus of smoothness of f . MSC(2000): 41A25, 42C10 Keywords: sphere; de la Vall´ee Poussin means; approximation; modulus of smoothness; lower bound
1
Introduction
Motivated by geoscience, meteorology, and oceanography, sphere-oriented mathematics has gained increasing attention in recent decades. As main tools, spherical positive polynomial operators play prominent roles in the approximation and the interpolation on the sphere by means of orthonormal spherical harmonics. Several authors such as Ditzian [5], Dai and Ditzian [4], Bernes and Li [3], Wang and Li [16], Nikol’skiˇı and Lizorkin [10, 8] introduced and studied some spherical versions of some known one-dimensional polynomial operators, for example, spherical Jackson operators [8], spherical de la Vall´ee Poussin operators [3, 16], spherical delay mean operators [13] and best approximation operators [5, 4, 16] etc.. The main aim of the present paper is to study the approximation by the de la Vall´ee Poussin means on the unit sphere. For to formulate our results, we first give some notations. Let Rd , d ≥ 3, be ∑dthe Euclidean space of the points x := (x1 , x2 , . . . , xd ) endowed with the scalar product x · x′ = j=1 xj x′j (x, x′ ∈ Rd ) and let σ := σ d−1 be the unit sphere in Rd consisting of the points x satisfying x2 = x · x = 1. We shall denote the points of σ by µ, and the elementary surface piece on σ by dσ. If it is necessary, we shall write dσ :≡ dσ(µ) referring to the variable of integration. The surface area of d ∫ 2π 2 σ d−1 is denoted by |σ d−1 |, and it is easy to deduce that |σ d−1 | = σ dσ = Γ( d . ) 2
By C(σ) and Lp (σ), 1 ≤ p < +∞, we denote the space of continuous, real valued functions and the space of (the equivalence classes of ) p-integrable functions defined on σ endowed with (∫ )1/p (1 ≤ p < ∞). In the the respective norms ∥f ∥∞ := maxµ∈σ |f (µ)|, ∥f ∥p := σ |f (µ)|p dσ(µ) following, Lp (σ) will always be one of the spaces Lp (σ) for 1 ≤ p < ∞, or C(σ) for p = ∞. Now we state some properties of spherical harmonics (see [16], [7], [9]). For integer k ≥ 0, the restriction of a homogeneous harmonic polynomial of degree k on the unit sphere is called a spherical harmonic of degree k. The class of all spherical harmonics of degree k will be denoted by Hk , and the class of all spherical harmonics of degree k ≤ n will be denoted by Πdn . Of course, ∗ The
research was supported by the National Natural Science Foundation of China (No. 61272023) author: Feilong Cao, E-mail: [email protected]
† Corresponding
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C. M. Ding et al.: The strong converse inequality for de la Vall´ee Poussin means on the sphere ⊕n Πdn = k=0 Hk , and it comprises the restriction to σ of all algebraic polynomials in d variables of total degree not exceeding n. The dimension of Hk is given by { 2k+d−2 (k+d−2) , k ≥ 1; k+d−2 k ddk := dim Hkd := 1, k = 0, ∑ n and that of Πdn is k=0 ddk . The spherical harmonics have an intrinsic characterization. To describe this, we first introduce the Laplace-Beltrami operators (see [9]) to sufficiently smooth functions f defined on σ, which ∑d ∂ 2 is the restriction of Laplace operator ∆ := i=1 ∂x 2 on the sphere σ, and can be expressed as i ( ) µ Df (µ) := ∆f |µ| . Clearly, the operator D is an elliptic, (unbounded) selfadjoint operator µ∈σ
on L2 (σ), is invariant under arbitrary coordinate changes, and its spectrum comprises distinct eigenvalues λk := −k(k + d − 2), k = 0, 1, . . . , each having finite multiplicity. The space Hk can be characterized intrinsically as the eigenspace corresponding to λk , i.e. Hk = {Ψ ∈ C ∞ (σ) : DΨ = −k(k + d − 2)Ψ}. Since the λk ’s are distinct, ⊕ and the operator is selfadjoint, the spaces Hk are mutually orthogonal; also, L2 (σ) = closure { k Hk }. Hence, if we choose an orthogonal basis {Yk,l : l = 1, . . . , ddk } for each Hk , then the set {Yk,l : k = 0, 1, . . . , l = 1, . . . , ddk } is an orthogonal basis for L2 (σ). The orthogonal projection Yk : L1 (σ) → Hk is given by ∫ Γ(λ)(k + λ) Yk (f ; µ) := Pkλ (µ · ν)f (ν)dσ(ν), 2π λ+1 σ (or Gegenbauer) polynomials defined by the where 2λ = d − 2, and Pkλ are the ultraspherical ∑∞ generating equation (1 − 2r cos θ + r2 )−λ = k=0 rk Pkλ (cos θ)(0 ≤ θ ≤ π). The further details for the ultraspherical polynomials can be found in [15]. For an arbitrary number θ, 0 < θ < π, we define the spherical translation operator of the function f ∈ Lp (σ) with a step θ by the aid of the following equation (see [12], [2]): ∫ 1 Sθ (f ) := Sθ (f ; µ) := f (ν)dσ(ν), (1.1) |σ d−2 | sind−2 θ µ·ν=cos θ where |σ d−2 | means the (d − 2)-dimensional surface area of the unit sphere of Rd−1 . Here we integrate over the family of points ν ∈ σ whose spherical distance from the given point µ ∈ σ (i.e. the length of minor arc between µ and ν on the great circle passing through them) is equal to θ. Thus Sθ (f ; µ) can be interpreted as the mean value of the function f on the surface of (d − 2)-dimensional sphere with radius sin θ. The properties of spherical translation operator (1.1) are well known; see e.g., [2]. In particular, it can be expressed as the following series Sθ (f ; µ) =
∞ ∑ P λ (cos θ) k
k=0
Pkλ (1)
Yk (f ; µ) :=
∞ ∑
Qλk (cos θ)Yk (f ; µ)
k=0
Pkλ (cos θ) Pkλ (1)
where Qλk (cos θ) := , and for any f ∈ Lp (σ), ∥Sθ (f )∥p ≤ ∥f ∥p , limθ→0 ∥Sθ f − f ∥p = 0. We usually apply the translation operator to define spherical modulus of smoothness of a function f ∈ Lp (σ), i.e. (see [16]) ω(f, t)p := sup0 n.
)
∞ ∑ P λ (cos θ) k
k=0
Pkλ (1)
) Yk (f ; µ) sin2λ θdθ
sin2λ θdθ Yk (f ; µ),
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C. M. Ding et al.: The strong converse inequality for de la Vall´ee Poussin means on the sphere
it is sufficient to prove
∫π 0
vn (θ)
Pkλ (cos θ) Pkλ (1)
(λ)
sin2λ θ dθ = ωn,k (k ≥ 0). Indeed, when k = 1, one has
) ( ∫ π P λ (cos θ) θ vn (θ) 1 λ sin2λ θdθ sin2λ θdθ = vn (θ) 1 − 2 sin2 2 P1 (1) 0 0 (∫ ( ) )2(n+λ) )2(n+λ) ∫ π( π 22λ θ θ 2λ θ 2(λ+1) θ cos sin dθ − 2 cos sin dθ In,d 2 2 2 2 0 0 ( ( ) ( )) 22λ+1 1 1 1 1 1 B λ + ,n + λ + − B λ + 1 + ,n + λ + In,d 2 2 2 2 2 n n!(n + 2λ)! (λ) = = ωn,1 , n + 2λ + 1 (n − 1)!(n + 1 + 2λ)!
∫
= = =
π
where B(a, b) is Beta function. Now, we suppose for k ≤ n that the relation (see page 81 of [15])
∫π 0
vn (θ)
Pkλ (cos θ) Pkλ (1)
(λ)
sin2λ θdθ = ωn,k . Then for k + 1 we first recall
λ λ (k + 1)Pk+1 (x) − 2(λ + k)xPkλ (x) + (2λ + k − 1)Pk−1 (x) = 0
i.e., λ (cos θ) = Pk+1
Then, ∫
(k ≥ 1),
) 1 ( λ (cos θ) . 2(λ + k) cos θPkλ (cos θ) − (2λ + k − 1)Pk−1 k+1
( ∫ π λ Pk+1 (cos θ) 1 2λ sin θdθ = 2(λ + k) vn (θ) cos θPkλ (cos θ) sin2λ θdθ λ (1) λ (1)(k + 1) Pk+1 Pk+1 0 0 ) ∫ π 1 2λ λ −(2λ + k − 1) (2(λ + k)J2 − J1 ) . vn (θ)Pk−1 (cos θ) sin θdθ := λ P (1)(k + 1) 0 k+1 π
vn (θ)
By the assumption, we obtain ∫ π P λ (cos θ) (2λ + k − 1)!n!(n + 2λ)! λ J1 = (2λ + k − 1)Pk−1 (1) vn (θ) k−1 sin2λ θdθ = . λ (1) Γ(2λ)(k − 1)!(n − k + 1)!(n + k − 1 + 2λ)! P 0 k−1 For J2 we have J2
( )2n ( ) θ 2 θ cos 2 cos − 1 Pkλ (cos θ) sin2λ θdθ In,d 0 2 2 ∫ π ∫ π 2In+1,d λ P λ (cos θ) P λ (cos θ) Pk (1) vn+1 (θ) k λ sin2λ θdθ − Pkλ (1) vn (θ) k λ sin2λ θdθ In,d Pk (1) Pk (1) 0 0 J21 − J22 , 1
= = :=
∫
π
which implies from the assumption that J22 = J21 =
Γ(k+2λ) n!(n+2λ)! k!Γ(2λ) (n−k)!(n+k+2λ)! ,
and
2Γ(n + 1 + λ + 1/2)Γ(n + 2λ + 1) Γ(k + 2λ) (n + 1)!(n + 1 + 2λ)! . Γ(n + λ + 1/2)Γ(n + 1 + 2λ + 1) k!Γ(2λ) (n + 1 − k)!(n + 1 + k + 2λ)!
Therefore, J2 =
Γ(k + 2λ) n!(n + 2λ)! (n(n + 1) + k(2λ + k)) . k!Γ(2λ) (n + 1 − k)!(n + 1 + k + 2λ)!
vn (θ)
λ (cos θ) Pk+1 n!(n + 2λ)! (λ) sin2λ θdθ = = ωn,k+1 . λ (n − k − 1)!(n + k + 1 + 2λ)! Pk+1 (1)
So, ∫ 0
π
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C. M. Ding et al.: The strong converse inequality for de la Vall´ee Poussin means on the sphere
(λ)
On the other hand, it is clear that for k > n, ωn,k = 0. Hence, de la Vall´ee Poussin means Vn (f ; µ) have the form of multiplier expression given in (2.1). Now we give some properties for the de la Vall´ee Poussin kernel vn . Lemma 2.1. Let vn (t) be the kernel of de la Vall´ee Poussin defined by (1.4), 2λ = d − 2 and d ≥ 3. Then there hold ∫ π λ θ−λ vn (θ) sin2λ θdθ ≤ C(d)n 2 , (2.2) 0
∫
and
π
θ− m vn (θ) sin2λ θdθ ≤ C(d)n m , 2
1
m = 1, 2, . . . .
(2.3)
0
Proof. We only prove (2.2). The proof of (2.3) is similar. First, a direct computation implies In,d = C(d) Then,
∫π 0
θ−λ vn (θ) sin2λ θdθ =
( )2n sin2λ dθ = θ−λ cos θ2
1
∫π
In,d
0
(−λ)
=
∫ Jn,d
d−1 (2n + d − 3)!! ≈ n− 2 . (2n + 2d − 4)!!
π 2
θ−
d−2 2
0
(−λ)
Jn,d In,d
, where
θ sind−2 θ cos2n dθ. 2
So, we have d
(−λ)
∫
π 2
≤ 22
Jn,d
sin
d−2 2
t cos2n+d−2 tdt = 2 2 −1 B d
0
( d−2 + 1 2n + d − 2 + 1 ) 2 , 2 2
Γ(n + d−1 d d d Γ(n + d−1 2 ) 2 ) = C(d) ≈ n− 4 . = 2 2 −1 Γ( ) 3d−2 4 Γ(n + 4 ) Γ(n + 3d−2 ) 4 Therefore
∫
(−2)
π
θ
−λ
2λ
vn (θ) sin
θdθ =
Jn,d
0
In,d
n− 4
d
≤ C(d)
d−1 n− 2
= C(d)n
d−2 4
.
The proof of Lemma 2.1 is completed. Lemma 2.2. For the kernel of de la Vall´ee Poussin vn (t) defined by (1.4), we have ∫ π θ4 vn (θ) sin2λ θdθ ≤ C(d)n−2 . 0
Proof. Since ∫ (4) Jn,d
π 4
θ cos
= 0
{ =
∫
θ sin2λ θdθ = 2d−1 π 4 2
∫
π 2
sind+2 θ cos2n+d−2 θdθ
0
2d−2 π 5 (2n+d−3)!!(d+1)!! , (2n+2d)!!
if d is even;
2d−1 π 4 (2n+d−3)!!(d+1)!! , (2n+2d)!!
if d is odd
= C(d) we have
2n
(2n + d − 3)!! , (2n + 2d)!! (4)
π
θ4 vn (θ) sin2λ θdθ = 0
This finishes the proof of Lemma 2.2.
Jn,d In,d
= C(d)
(2n + 2d − 4)!! ≤ C(d)n−2 . (2n + 2d)!!
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C. M. Ding et al.: The strong converse inequality for de la Vall´ee Poussin means on the sphere
3
Lower bound of approximation for de la Vall´ ee Poussin means
In this section we prove the main result of this paper, which can be stated as follows. Theorem 3.1. Let Vn (f ; µ) be de la Vall´ee Poussin means on the sphere given by (1.5). Then for f ∈ Lp (σ), 1 ≤ p ≤ +∞, there exists a constant C which is independent of f and n, such that ( ) 1 ω f, √ ≤ C∥Vn f − f ∥p . n p In order to prove the result, we first prove the following lemma. Lemma 3.1. For any g, Dg, D2 g ∈ Lp (σ), 1 ≤ p ≤ ∞, there exist the constants A, B and C2 which are are independent of n and g, such that ∥Vn g − g − α(n)Dg∥p ≤ C2 n−2 ∥D2 g∥p , where B 0< A n ≤ α(n) ≤ n . ∫θ ∫t Proof. Since (see (3.6) of [11]) Sθ (g; µ) − g(µ) = 0 sin−2λ t dt 0 sin2λ u Su (Dg; µ)du, we have ∫ u ∫ γ −2λ Su (Dg; µ) − Dg(µ) = sin γdγ sin2λ ν Sν (D2 g; µ)dν. 0
0
Observing that ∫ Vn (g; µ) − g(µ)
∫
π
θ
vn (θ) sin2λ θdθ
= 0
=
0
∫
2λ
vn (θ) sin 0
vn (θ) sin
∫
θ
t
sin2λ u Su (Dg; µ)du 0
θ
−2λ
sin 0
θdθ
0
:=
θdθ
∫
π 2λ
+
∫
π
Dg(µ) ∫
sin−2λ tdt
−2λ
sin
∫
∫
sin2λ udu 0
t
tdt
0
t
tdt
( ) sin2λ u Su (Dg; µ) − Dg(µ) du
0
Dg(µ)α(n) + Ψ(g; µ),
where α(n) = C(d)n−1 satisfies 0 < An−1 ≤ C(d)n−1 ≤ Bn−1 , we obtain that from the H¨olderMinkowski’s inequality and the contractility of translation operator ∫ π ∫ θ ∫ t ∫ u ∫ γ ∥Ψg∥p ≤ ∥D2 g∥p vn (θ) sin2λ θdθ sin−2λ tdt sin2λ udu sin−2λ γdγ sin2λ νdν 0 0 0 0 0 ∫ π 2λ 2 4 ≤ C3 ∥D g∥p vn (θ)θ sin θdθ. 0
Thus, from Lemma 2.2 it follows that ∥Ψg∥p ≤ C4 n−2 ∥D2 g∥p . The Lemma 3.1 has been proved. Now we turn to the proof of Theorem 3.1. We first introduce an operator Vnm given by Vnm (f ; µ)
=
n (∫ ∑ k=0
π
vn (θ)Qλk (cos θ) sin2λ θdθ
)m Yk (f ; µ).
0
Then, form the orthogonality of projection operator Yk , it follows that Vnm+l f
= =
n (∫ ∑
π
vn (θ)Qλk (cos θ) sin2λ θdθ
)m
0 k=0 m l Vn (Vn f ).
Yk
n (∫ (∑ s=0
π
)l ) vn (θ)Qλs (cos θ) sin2λ θdθ Ys f
0
Thus, we take g = Vnm f and obtain that ∥f − g∥p = ∥f − Vnm f ∥p ≤
m ∑
∥Vnk−1 f − Vnk f ∥p ≤ m∥f − Vn f ∥p ,
k=1
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C. M. Ding et al.: The strong converse inequality for de la Vall´ee Poussin means on the sphere
where Vn0 f = f. A Next, we prove the estimate: ∥DVnm f ∥p ≤ 2C C1 n∥f ∥p , where A and C2 are the same as that 2 in Lemma 3.1. In fact, we have
n
∑
(∫ π )m
∥DVnm f ∥p ≤ k(k + d − 2) vn (θ) Qλk (cos θ) sin2λ θdθ Yk (f ) .
0 k=0
Since (see [1])
p
P λ (cos θ) ( ) |Qλk (cos θ)| ≡ k λ ≤ C5 min (kθ)−λ , 1 , Pk (1)
we use (2.2) and obtain for kθ ≥ 1 and θ ≤ π2 , that
n
∑
(∫ π )m
2λ −λ m − d−2 m k(k + d − 2)k 2 vn (θ)θ sin θdθ Yk (f ) ∥DVn f ∥p ≤ C6
0 k=0
≤ C7 n
d−2 4 m
∥f ∥p
∞ ∑
k
2− d−2 2 m
p
.
k=0
For 2 −
d−2 2 m
< −1, i.e. m >
6 d−2 ,
it is clear that the series ∥DVnm f ∥p ≤ C8 n
d−2 4 m
∑∞ k=0
k 2−
d−2 2 m
is convergence. Thus
∥f ∥p .
For kθ ≤ 1, then (2.3) implies that
n ∫
∑ ( π
)m 1 2
2λ m 2 −m λ ∥DVn f ∥p ≤ (θ k(k + d − 2)) m Qk (cos θ) sin θdθ Yk (f ) vn (θ)θ
0 k=0 p
n ∫
∞
∑ ( π
) m 2 A
∑
C1 n∥f ∥p , ≤ C9 vn (θ)θ− m sin2λ θdθ Yk (f ) ≤ C10 n Yk (f ) =
2C2 0 k=0
k=0
p
p
where A and C2 are the same as that in Lemma 3.1. Therefore, when m > ∥DVnm f ∥p ≤
6 d−2 ,
we have
A C1 n∥f ∥p . 2C2
In the next, without loss generality, we assume m1 > Lemma 3.1 we see that
6 d−2 ,
and m >
6 d−2
α(n)∥DVnm f ∥p ≤ ∥Vnm f − f ∥p + C2 n−2 ∥D2 Vnm f ∥p ≤ m∥Vn f − f ∥p +
+ m1 . According to
AC1 −1 n ∥DVnm−m1 f ∥p 2
AC1 −1 AC1 −1 n ∥DVnm f ∥p + n ∥DVnm−m1 (Vnm1 f − f )∥p 2 2 AC1 C11 m1 AC1 ≤ m∥Vn f − f ∥p + ∥DVnm f ∥p + ∥Vn f − f ∥p 2n 2 AC1 = C12 ∥Vn f − f ∥p + ∥DVnm f ∥p . 2n ≤ m∥Vn f − f ∥p +
Taking α(n) =
AC1 n ,
one has 1 2C12 ∥DVnm f ∥p ≤ ∥Vn f − f ∥p . n AC1
So from the definition of K-functional it follows ( ) ( 1 )2 1 K f, √ ≤ ∥f − Vnm f ∥p + √ ∥DVnm f ∥p n n 2C12 ≤ m∥f − Vn f ∥p + ∥f − Vn f ∥p ≤ C14 ∥f − Vn f ∥p , AC1
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C. M. Ding et al.: The strong converse inequality for de la Vall´ee Poussin means on the sphere
which together with (1.3) implies ( ) 1 ω f, √ ≤ C∥f − Vn f ∥p . n p This finishes the proof of Theorem 3.1. From (1.6) and Theorem 3.1, the following Corollary 3.1 follows directly. Corollary 3.1. For any f ∈ Lp (σ), 1 ≤ p ≤ ∞, there holds ( ) 1 ∥Vn f − f ∥p ≈ ω f, √ . n p
References [1] E. Belinsky, F. Dai, Z. Ditzian, Multivariate approximation averages, J. Approx. Theory, 125 (2003), 85-105. [2] H. Berens, P. L. Butzer, S. Pawelke, Limitierungsverfahren von Reihen mehrdimensionaler Kugelfunktionen und deren Saturationsverhalten, Publ. RIMS, Kyoto Univ., Ser. A, 4 (1968), 201-268. [3] H. Berens, L. Q. Li, On the de la Vall´ee Poussin means on the sphere, Results in Math., 24 (1993), 12-26. [4] F. Dai, Z. Ditzian, Jackson theorem in Lp , 0 < p < 1, for functions on the sphere, J. Approx. Theory, (2009) doi: 10. 1016/jat. 2009. 06. 003. [5] Z. Ditzian, Jackson-type inequality on the sphere, Acta Math. Hungar, 102 (1-2) (2004), 1-35. [6] Z. Ditzian, K. G. Ivanov, Strong converse inequalities, Jour. D’Analyse Math., 61 (1993), 61-111. [7] W. Freeden, T. Gervens, M. Schreiner, Constructive approximation on the sphere, Oxford University Press Inc., New York, 1998. [8] P. I. Lizorkin, S. M. Nikol’skiˇl, A theorem concerning approximation on the sphere, Anal. Math., 9 (1983), 207-221. [9] C. M¨ uller, Spherical harmonics, Lecture Notes in Mathematics, Vol. 17, Springer, Berlin, 1966. [10] S. M. Nikol’skiˇl, P. I. Lizorkin, Approximation theory on the sphere, Proc. Steklov Inst. Math. 172 (1985), 295-302. ¨ [11] S. Pawelke, Uber die approximationsordnung bei kugelfunktionen und algebraischen polynomen, Tˆohoku Math. J., 24 (3) (1972), 473-486. [12] W. Rudin, Uniqueness theory for Laplace series, Trans. Amer. Math. Soc., 68 (1950), 287-303. [13] E. M. Stein, Interpolation in polynomial classes and Markoff’s inequality, Duke Math. J., 24 (1957), 467-476. [14] E. M. Stein G. Weiss, Introduction of Functions of Real Variable, Princeton University Press, Princeton N. J., 1971. [15] G. Szeg¨o, Orthogonal polynomials, Amer. Math. Soc. Coll. Publ., Vol. 23, 2003. [16] K. Wang, L. Li, Harmonic analysis and approximation on the unit sphere, Science Press, Beijing, 2000.
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On the fixed point method for stability of a mixed type AQ-functional equation Ick-Soon Changa and Yang-Hi Leeb a
Department of Mathematics, Chungnam National University, Daejeon 305-764, Korea.
b
Department of Mathematics Education, Gongju National University of Education, Gongju 314-711, Korea.
Abstract In this article, we take into account the stability for the following functional equation of additive-quadratic type f (x − y) − f (−x + y) − 4f (x) + f (2x) − f (−y) + f (y) = 0 with the fixed point method. Keywords : Stability ; Fixed point method ; Additive-quadratic mapping. AMS Mathematics Subject Classification (2000) : 39B52, 39B82, 47H10.
1
Introduction
Ulam [9] proposed the following question concerning the stability of homomorphisms : Let G1 be a group and let G2 be a metric group with the metric d(·, ·). Given ε > 0, does there exist a δ > 0 such that if a function h : G1 → G2 satisfies the inequality d(h(xy), h(x)h(y)) < δ for all x, y ∈ G1 , then there exists a homomorphism H : G1 → G2 with d(h(x), H(x)) < ε for all x ∈ G1 ? Hyers [5] answers the problem of Ulam under the assumption that the groups are Banach spaces. A generalized version of the theorem of Hyers for approximately additive mappings was given by Aoki [1], and for approximately linear mappings was presented by Rassias [7] by considering an unbounded Cauchy difference. Thereafter, many interesting results of the stability of several functional equation have been extensively investigated. On the contrary, C˘adariu and Radu [2] observed that the existence of the solution for a functional equation and the estimation of the difference with the given mapping can be obtained from the fixed point alternative. This method is called a fixed point method. In particular, they [3, 4] applied this method to prove the stability theorems of the additive functional equation and the quadratic functional equation by using the fixed point method. Now we consider the stability of the following mixed type additive-quadratic functional equation (briefly, AQ–functional equation) f (x − y) − f (−x + y) − 4f (x) + f (2x) − f (−y) + f (y) = 0.
(1.1)
by using the fixed point method. In this case, every solution of the functional equation (1.1) is said to be an additive-quadratic mapping. a Corresponding
author. E-mail address : [email protected] (I.-S. Chang), [email protected] (Y.-H. Lee) The first author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (no. 2013R1A1A2A10004419).
1
42
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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 20, NO.1, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC I. Chang and Y. Lee / Stability of a mixed type AQ-functional equation functional equation
2
2
Stability of Eq. (1.1) and its applications
Throughout this article, let V be a real or complex linear space and Y a Banach space. For a given mapping f : V → Y, we use the following abbreviation Df (x, y) := f (x − y) − f (−x + y) − 4f (x) + f (2x) − f (−y) + f (y) for all x, y ∈ V. We first prove the following lemma. Lemma 2.1 Let f : V → Y be a mapping with f (0) = 0 such that Df (x, y) = 0 for all x, y ∈ V \{0}. Then f is an additive-quadratic mapping. Proof. Since f (0) = 0, we get Df (x, 0) = Df (x, x) = 0 for all x ∈ V \{0}, and Df (0, y) = 0 for all y ∈ V . This completes the proof. For explicitly later use, we state the following theorem : Theorem 2.2 (The alternative of fixed point) ([6] or [8]) Suppose that a complete generalized metric space (X, d), which means that the metric d may assume infinite values, and a strictly contractive mapping J : X → X with the Lipschitz constant 0 < L < 1 are given. Then, for each given element x ∈ X, either d(J n x, J n+1 x) = +∞, ∀n ∈ N ∪ {0}, or there exists a nonnegative integer k such that : (1) d(J n x, J n+1 x) < +∞ for all n ≥ k ; (2) the sequence {J n x} is convergent to a fixed point y ∗ of J ; (3) y ∗ is the unique fixed point of J in Y := {y ∈ X, d(J k x, y) < +∞} ; (4) d(y, y ∗ ) ≤ (1/(1 − L))d(y, Jy) for all y ∈ Y. Now, by the use of fixed point method, we obtain the main results as follow. Theorem 2.3 Let φ : (V \{0})2 → [0, ∞) be a function with φ(x, y) = φ(−x, −y) for all x, y ∈ V \{0}. Suppose that a mapping f : V → Y satisfies ∥Df (x, y)∥ ≤ φ(x, y)
(2.1)
for all x, y ∈ V \{0} with f (0) = 0. If there exists a constant 0 < L < 1 such that a function φ has the property φ(2x, 2y) ≤ 2Lφ(x, y)
(2.2)
for all x, y ∈ V \{0}, then there exists a unique additive-quadratic mapping F : V → Y such that ∥f (x) − F (x)∥ ≤
φ(x, x) 2(1 − L)
(2.3)
for all x ∈ V \{0}. In particular, F is represented by ( ) f (2n x) + f (−2n x) f (2n x) − f (−2n x) F (x) = lim + n→∞ 2 · 4n 2n+1
(2.4)
for all x ∈ V. Proof. Consider the set S := {g : g : V → Y, g(0) = 0} and introduce a generalized metric on S by d(g, h) = inf{K ∈ R+ : ∥g(x) − h(x)∥ ≤ Kφ(x, x) for all x ∈ V \{0}}. It is easy to see that (S, d) is a generalized complete metric space.
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Now we define a mapping J : S → S by g(2x) − g(−2x) g(2x) + g(−2x) + 4 8
Jg(x) := for all x ∈ V. Note that
g(2n x) − g(−2n x) g(2n x) + g(−2n x) + 2n+1 2 · 4n for all n ∈ N and all x ∈ V. Let g, h ∈ S and let K ∈ [0, ∞] be an arbitrary constant with d(g, h) ≤ K. From the definition of d, we have J n g(x) =
1 3 ∥Jg(x) − Jh(x)∥ = ∥g(2x) − h(2x)∥ + ∥g(−2x) − h(−2x)∥ 8 8 1 ≤ Kφ(2x, 2x) 2 ≤ KLφ(x, x) for all x ∈ V \{0}, which implies that
d(Jg, Jh) ≤ Ld(g, h)
for any g, h ∈ S, that is, J is a strictly contractive self-mapping of S with the Lipschitz constant L. Moreover, by (2.1), we see that ∥f (x) − Jf (x)∥ =
1 φ(x, x) ∥ − 3Df (x, x) + Df (−x, −x)∥ ≤ 8 2
for all x ∈ V \{0}. It means that d(f, Jf ) ≤ 12 < ∞ by the definition of d. Therefore, according to Theorem 2.2, the sequence {J n f } converges to the unique fixed point F : V → Y of J in the set T = {g ∈ S : d(f, g) < ∞}, which is represented by (2.4). Note that 1 1 d(f, F ) ≤ d(f, Jf ) ≤ , 1−L 2(1 − L) which implies (2.3). By the definition of F, together with (2.1) and (2.4), we find that
Df (2n x, 2n y) − Df (−2n x, −, 2n y)
∥DF (x, y)∥ = lim n→∞ 2n+1 n n Df (2 x, 2 y) + Df (−2n x, −2n y)
+
2 · 4n n 2 +1 ≤ lim (φ(2n x, 2n y) + φ(−2n x, −2n y)) n→∞ 2 · 4n =0 for all x, y ∈ V \{0}. By Lemma 2.1, we have proved that DF (x, y) = 0 for all x, y ∈ V. This completes the proof. We continue our investigation with the following theorem. Theorem 2.4 Let φ : (V \{0})2 → [0, ∞) with φ(x, y) = φ(−x, −y) for all x, y ∈ V \{0}. Suppose that f : V → Y satisfies the inequality ∥Df (x, y)∥ ≤ φ(x, y) for all x, y ∈ V \{0} with f (0) = 0. If there exists 0 < L < 1 such that the mapping φ has the property Lφ(2x, 2y) ≥ 4φ(x, y)
(2.5)
for all x, y ∈ V \{0}, then there exists a unique additive-quadratic mapping F : V → Y such that ∥f (x) − F (x)∥ ≤
Lφ(x, x) 4(1 − L)
(2.6)
for all x ∈ V \{0}. In particular, F is given by ( ( (x) ( x )) 4n ( ( x ) ( x ))) F (x) = lim 2n−1 f n − f − n + f n +f − n n→∞ 2 2 2 2 2
(2.7)
for all x ∈ V.
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Proof. Let (S, d) be the set as in the proof of Theorem 2.3, and we consider the mapping J : S → S defined by (x) ( x) ( (x) ( x )) Jg(x) := g −g − +2 g +g − 2 2 2 2 for all g ∈ S and all x ∈ V. Observe that ( (x) ( x )) 4n ( ( x ) ( x )) J n g(x) = 2n−1 g n − g − n + g n +g − n 2 2 2 2 2 and J 0 g(x) = g(x) for all x ∈ V. Let g, h ∈ S and let K ∈ [0, ∞] be an arbitrary constant with d(g, h) ≤ K. The definition of d yields
(x) (x) ( x) ( x)
−h −h − ∥Jg(x) − Jh(x)∥ = 3 g
+ g −
2 2 2 2 (x x) ≤ 4Kφ , 2 2 ≤ LKφ(x, x) for all x ∈ V \{0}. So we get
d(Jg, Jh) ≤ Ld(g, h)
for any g, h ∈ S, that is, J is a strictly contractive self-mapping of S with the Lipschitz constant L. Also we see that
( x x ) (x x) L
∥f (x) − Jf (x)∥ = Df , , ≤ φ(x, x)
≤φ 2 2 2 2 4 for all x ∈ V \{0}, which implies that d(f, Jf ) ≤ L4 < ∞. Therefore, according to Theorem 2.2, the sequence {J n f } converges to the unique fixed point F of J in the set T := {g ∈ S : d(f, g) < ∞}, which is given by (2.7). Since 1 L d(f, F ) ≤ d(f, Jf ) ≤ 1−L 4(1 − L) the inequality (2.6) holds. From the definition of F with (2.1) and (2.5), we have
( (x y ) ( x y ))
∥DF (x, y)∥ = lim 2n−1 Df n , n − Df − n , − n n→∞ 2 2 2 2 (x y ) ( x y )) 4n (
Df n , n + Df − n , − n +
2 2 2 2 2 ) ( n n ( ( 2 +4 x y x y )) ≤ lim φ n, n + φ − n,− n n→∞ 2 2 2 2 2 =0 for all x, y ∈ V \{0}. So, by Lemma 2.1, F is an additive-quadratic mapping, which completes the proof. From now on, given a mapping f : V → Y, we set Af (x, y) :=f (x + y) − f (x) − f (y), Qf (x, y) :=f (x + y) + f (x − y) − 2f (x) − 2f (y) for all x, y ∈ V. Using Theorem 2.3 and Theorem 2.4, we will prove the stability of the additive functional equation Af ≡ 0, and the quadratic functional equation Qf ≡ 0 in the following results. Corollary 2.5 Let fi : V → Y, i = 1, 2, be mappings for which there exist functions ϕi : (V \{0})2 → [0, ∞), i = 1, 2, such that ∥Afi (x, y)∥ ≤ ϕi (x, y)
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(2.8)
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for all x, y ∈ V \{0}. If fi (0) = 0, ϕi (x, y) = ϕi (−x, −y), i = 1, 2, for all x, y ∈ V \{0}, and there exists 0 < L < 1 such that 1 ϕ1 (x, y) ≤ ϕ1 (2x, 2y) ≤ 2Lϕ1 (x, y), L ϕ2 (2x, 2y) ≤ Lϕ2 (2x, 2y)
(2.9) (2.10)
for all x, y ∈ V \{0}, then there exist unique additive mappings Fi : V → Y, i = 1, 2, such that ϕ1 (x, x) + 3ϕ1 (x, −x) , 2(1 − L)
(2.11)
L(ϕ2 (x, x) + 3ϕ2 (x, −x)) 4(L − 1)
(2.12)
∥f1 (x) − F1 (x)∥ ≤
∥f2 (x) − F2 (x)∥ ≤
for all x ∈ V \{0}. In particular, the mappings Fi , i = 1, 2, are represented by f1 (2n x) , n→∞ 2n( ) x F2 (x) = lim 2n f2 n n→∞ 2 F1 (x) = lim
(2.13) (2.14)
for all x ∈ V. Proof. We first note that Dfi (x, y) = Afi (x, −y) − Afi (−x, y) + Afi (x, x) + Afi (x, −x) for all x, y ∈ V and i = 1, 2. Put φi (x, y) := ϕi (x, −y) + ϕi (−x, y) + ϕi (x, x) + ϕi (x, −x) for all x, y ∈ V \{0} and i = 1, 2, then φ1 satisfies (2.2) and φ2 fulfills (2.5). Therefore ∥Dfi (x, y)∥ ≤ φi (x, y) for all x, y ∈ V \{0} and i = 1, 2. According to Theorem 2.3, there exists a unique mapping F1 : V → Y satisfying (2.11), which is represented by (2.4). Observe that, by virtue of (2.8) and (2.9),
f1 (2n x) + f1 (−2n x)
f1 (2n x) + f1 (−2n x) − f1 (0)
lim lim
= n→∞
n→∞ 2n+1 2n+1 1 = lim n+1 ∥Af1 (2n x, −2n x)∥ n→∞ 2 1 ≤ lim n+1 ϕ1 (2n x, −2n x) n→∞ 2 Ln ≤ lim ϕ1 (x, −x) = 0 n→∞ 2 and
n n
f1 (2n x) + f1 (−2n x)
≤ lim 2 L ϕ1 (x, −x) = 0 lim
n n→∞ n→∞ 2 · 4n 2·4
for all x ∈ V \{0}. This inequality and (2.4) guarantees (2.13). Moreover, we have
Af1 (2n x, 2n y) ϕ1 (2n x, 2n y)
≤ ≤ Ln ϕ1 (x, y)
2n 2n for all x, y ∈ V \{0}. Sending the limit as n → ∞ in the above inequality, and using F1 (0) = 0, we get AF1 (x, y) = 0 for all x, y ∈ V. On the other hand, according to Theorem 2.4, we see that there exists a unique mapping F2 : V → Y satisfying (2.12), which is given by (2.7).
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Notice that, by (2.8) and (2.11), ( )
(x)
(x −x x )
2n−1 2n−1 , − lim 2 = lim 2
f2 n + f2
Af
2 n→∞ n→∞ 2 2n 2n 2n ) (x x ≤ lim 22n−1 ϕ2 n , − n n→∞ 2 2 Ln ϕ2 (x, −x) = 0. ≤ lim n→∞ 2 as well as ( )
(x) −x Ln
lim 2n−1 f2 n + f2 ϕ2 (x, −x) = 0 ≤ lim
n→∞ n→∞ 2n+1 2 2n for all x ∈ V \{0}. From these and (2.7), we obtain (2.14). Moreover, we have
( x y ) ( x y ) Ln
n
2 Af2 n , n ≤ 2n ϕ2 n , n ≤ n ϕ2 (x, y) 2 2 2 2 2 for all x, y ∈ V \{0}. Taking the limit as n → ∞ in the above inequality, and using F2 (0) = 0, we fee that AF2 (x, y) = 0 for all x, y ∈ V. The proof is ended. Corollary 2.6 Let fi : V → Y, i = 1, 2, be mappings for which there exist functions ϕi : (V \{0})2 → [0, ∞), i = 1, 2, such that ∥Qfi (x, y)∥ ≤ ϕi (x, y) for all x, y ∈ V \{0}. If fi (0) = 0, ϕi (x, y) = ϕi (−x, −y), i = 1, 2, for all x, y ∈ V \{0}, and there exists 0 < L < 1 such that the mapping ϕ1 satisfies (2.9) and ϕ2 satisfies (2.10) for all x, y ∈ V \{0}, then there exist unique quadratic mappings Fi : V → Y, i = 1, 2, such that 3ϕ1 (x, x) + 5ϕ1 (x, −x) , 4(1 − L)
(2.15)
L(3ϕ2 (x, x) + 5ϕ2 (x, −x)) 8(1 − L)
(2.16)
∥f1 (x) − F1 (x)∥ ≤
∥f2 (x) − F2 (x)∥ ≤
for all x ∈ V \{0}. In particular, the mappings Fi , i = 1, 2, are given by f1 (2n x) , n→∞ 4n( ) x F2 (x) = lim 4n f2 n n→∞ 2 F1 (x) = lim
(2.17) (2.18)
for all x ∈ V. Proof. Note that 1 1 Dfi (x, y) = Qfi (x, y) − Qfi (y, −x) + fi (x, −x) + Qfi (y, −y) − Qfi (y, y) 2 2 for all x, y ∈ V and i = 1, 2. Put φi (x, y) := ϕi (x, y) + ϕi (y, −x) + ϕi (x, −x) + 12 ϕi (y, y) + 12 ϕi (y, −y) for all x, y ∈ V \{0} and i = 1, 2, then φ1 (resp. φ2 ) satisfies (2.2) (resp. (2.5)). Moreover, ∥Dfi (x, y)∥ ≤ φi (x, y) for all x, y ∈ V \{0} and i = 1, 2. It follows from Theorem 2.3 that there exists a unique mapping F1 : V → Y satisfying (2.15), which is represented by (2.4). Observe that
f (2n x) − f (−2n x)
1
1 1 lim
= lim n+1 Qf1 (2n−1 x, −2n−1 x) − Qf1 (−2n−1 x, 2n−1 x) n+1 n→∞ 2 n→∞ 2 ) 1 ( ≤ lim n+1 ϕ1 (2n−1 x, −2n−1 x) + ϕ1 (−2n−1 x, 2n−1 x) n→∞ 2 ( x x )) Ln ( ( x x ) ≤ lim ϕ1 ,− + ϕ1 − , n→∞ 2 2 2 2 2 =0
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and
7
(x x) ( x x )) n (
f1 (2n x) − f1 (−2n x)
≤ lim L lim ϕ , − + ϕ − , =0 1 1
n→∞ 2n+1 n→∞ 2 · 4n 2 2 2 2
for all x ∈ V \{0}. Due to this fact and (2.4), we get (2.17). Moreover, we have
Qf1 (2n x, 2n y) ϕ1 (2n x, 2n y) Ln
≤ ≤ n ϕ1 (x, y)
n n 4 4 2 for all x, y ∈ V \{0}. As n → ∞ in the above inequality, we see that QF1 (x, y) = 0 for all x, y ∈ V \{0}. By using F1 (0) = 0, then we have (y y) ( y y) QF1 (x, 0) = 0, QF1 (0, y) = − QF1 ,− + QF1 − , =0 2 2 2 2 for all x, y ∈ V \{0}. Therefore, QF1 (x, y) = 0 for all x, y ∈ V. On the other hand, Theorem 2.4 guarantees that there exists a unique mapping F2 : V → Y satisfying (2.16), which is represented by (2.7). Observe that
(x)
( x ) ( ( x x ) x x )
4n f2 n − f2 − n =4n Qf2 n+1 , − n+1 − Qf2 − n+1 , n+1 2 2 2 2 2 2 ( ( x ( x ) x x )) ≤4n ϕ2 n+1 , − n+1 + ϕ2 − n+1 , n+1 2 2 ( (2x x )2 ( x x )) n ≤L ϕ2 ,− + ϕ2 − , 2 2 2 2 for all x ∈ V \{0}. It leads us to get ( (x) ( x )) ( (x) ( x )) lim 4n f2 n − f2 − n = 0, lim 2n f2 n − f2 − n =0 n→∞ n→∞ 2 2 2 2 for all x ∈ V \{0}. Based on these facts and (2.7), we obtain (2.18). Moreover, we have
( x y ) (x y )
n
4 Qf2 n , n ≤ 4n ϕ2 n , n ≤ Ln ϕ2 (x, y) 2 2 2 2 for all x, y ∈ V \{0}. Going the limit as n → ∞ in the previous inequality, and using F2 (0) = 0, we get QF2 (x, y) = 0 for all x, y ∈ V, which complete the proof. Now, we obtain the stability in the framework of normed spaces using Theorem 2.3 and Theorem 2.4. Corollary 2.7 Let X be a normed space and Y a Banach space. Suppose that the mapping f : X → Y satisfies the inequality ∥Df (x, y)∥ ≤ θ(∥x∥p + ∥y∥p ) for all x, y ∈ X\{0} with f (0) = 0, where θ ≥ 0 and p ∈ (−∞, 1) ∪ (2, ∞). Then there exists a unique quadratic-additive mapping F : X → Y such that { 2θ p if p > 2, 2p −4 ∥x∥ ∥f (x) − F (x)∥ ≤= 2θ p ∥x∥ if p < 1, 2−2p for all x ∈ X\{0}. Proof. This follows from Theorem 2.3 and Theorem 2.4 by putting φ(x, y) := θ(∥x∥p + ∥y∥p ) for all x, y ∈ X\{0} with L = 2p−1 < 1 if p < 1 and L = 22−p < 1 if p > 2. Corollary 2.8 Let X be a normd space and Y a Banach space. Suppose that the mapping f : X → Y satisfies the inequality ∥Df (x, y)∥ ≤ θ∥x∥p ∥y∥q
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for all x, y ∈ X\{0} with f (0) = 0, where θ ≥ 0 and p + q ∈ (−∞, 1) ∪ (2, ∞). Then there exists a unique quadratic-additive mapping F : X → Y such that { θ∥x∥p+q if p + q > 2, 2p+q −4 ∥f (x) − F (x)∥ ≤= θ∥x∥p+q if p + q < 1 2(2−2p+q ) for all x ∈ X\{0}. Proof. By considering φ(x, y) := θ∥x∥p ∥y∥q for all x, y ∈ X\{0} with L = 2p+q−1 < 1 if p + q < 1 and L = 22−p−q < 1 if p + q > 2, then by Theorem 2.3 and Theorem 2.4, we arrive at the conclusion of the corollary.
References [1] T. Aoki, On the stability of the linear mapping in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. [2] L. C˘adariu and V. Radu, Fixed points and the stability of Jensen’s functional equation, J. Inequal. Pure Appl. Math. 4 (2003), Art. 4. [3] L. C˘adariu and V. Radu, Fixed points and the stability of quadratic functional equations, An. Univ. Timisoara Ser. Mat.-Inform. 41 (2003), 25–48. [4] L. C˘adariu and V. Radu, On the stability of the Cauchy functional equation : a fixed point approach in iteration theory, Grazer Mathematische Berichte, Karl-Franzens-Universit¨aet, Graz, Graz, Austria 346 (2004), 43–52. [5] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27 (1941), 222–224. [6] B. Margolis and J.B. Diaz, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 74 (1968), 305–309. [7] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [8] I.A. Rus, Principles and applications of fixed point theory, Ed. Dacia, Cluj-Napoca, (1979) (in Romanian). [9] S.M. Ulam, A collection of mathematical problems, Interscience, New York, (1968).
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DIFFERENCES OF COMPOSITION OPERATORS FROM LIPSCHITZ SPACE TO WEIGHTED BANACH SPACES IN POLYDISK CHANG-JIN WANG SCHOOL OF SCIENCE, JIMEI UNIVERSITY, XIAMEN FUJIAN 361021, P.R. CHINA. [email protected] YU-XIA LIANG∗ SCHOOL OF MATHEMATICAL SCIENCES, TIANJIN NORMAL UNIVERSITY, TIANJIN 300387, P.R. CHINA. [email protected]
Abstract. Let ϕ and ψ be holomorphic self-maps of the unit polydisk Dn in the n-dimensional complex space C n , denote by Cϕ and Cψ the induced composition operators. In this paper, we estimate the essential norm of the differences of composition operators Cϕ − Cψ from Lipschitz space to weighted Banach space in the unit polydisk.
1. Introduction The algebra of all holomorphic functions on domain Ω will be denoted by H(Ω), where Ω is a bounded domain in C n , where n ≥ 1 is a fixed integer. Let Dn = {z = (z1 , ..., zn ) ∈ C n , |zi | < 1, 1 ≤ i ≤ n} be the open unit polydisk of the complex ndimensional Euclidean space C n and H(Dn ) be the space of all holomorphic functions on Dn . For z = (z1 , ..., zn ) and w = (w1 , ..., wn ) in C n , the inner product of z and w isg hz, wi = z1 w1 + ... + zn wn , where h., .i denotes the inner product. Moreover, g z = maxj {|zj |} stands for the supremum norm on Dn . For z, w ∈ D, the pseudo-hyperbolic distance between z and w is defined by ρ(z, w) = |(z − w)/(1 − wz)|. It is well known that if f ∈ H(D), then ρ(f (z), f (w)) ≤ ρ(z, w). The Bergman metric on the unit polydisk is given by n X uj vj Hz (u, v) = . (1 − |zj |2 )2 j=1 The Kobayashi distance kDn on Dn is defined by g g 1 + φz (w) 1 g g, kDn (z, w) = log 2 1 − φz (w) where φz : Dn → Dn is the automorphism of Dn given by w −z wn − zn 1 1 φz (w) = , ..., . 1 − z1 w1 1 − zn wn
(1.1)
The work was supported in part by the National Natural Science Foundation of China (Grant Nos. 11371276; 11301373; 11201331). ∗ Corresponding author. 2010 Mathematics Subject Classification.Primary: 47B33; Secondary: 47B38, 32A37, 32H02. Key words and phrases. composition operator, Lipschitz space, weighted Banach space, polydisk. 1
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Wang and Liang:: Differences of composition operators Let v be a strictly positive bounded continuous function (weight) on the open unit polydisk Dn in C n , n ≥ 1. We first introduce the weighted Banach spaces of analytic functions of the following form: o n Hv∞ := f ∈ H(Dn ); kf kv = sup v(z)|f (z)| < ∞ z∈Dn
endowed with the sup-norm k.kv . Spaces of this type appear in the study of growth conditions of analytic functions and have been studied in various articles, see, e.g. [2, 8, 10]. For 0 ≤ α < 1, an f ∈ H(Dn ) belongs to the Lipschitz space Lipα (Dn ), if n X ∂f (z) (1 − |zl |2 )1−α < ∞. ∂zl z∈Dn
kf kα = |f (0)| + sup
(1.2)
l=1
It is easy to show that Lipα (Dn ) is a Banach space endowed with the norm k.kα (see, e.g.[13, 14]). Let ϕ = (ϕ1 (z), ..., ϕn (z)) and ψ(z) = (ψ1 (z), ..., ψn (z)) be holomorphic self-maps of Dn . The composition operator Cϕ induced by ϕ is defined by (Cϕ )f (z) = f (ϕ(z)) for z ∈ Dn and f ∈ H(Dn ) (see, e.g.[3]). The essential norm of a continuous linear operator T is the distance from T to the set of all compact operators, that is, kT ke = inf{kT − Kk : K is compact }. Notice that kT ke = 0 if and only if T is compact, so estimates on kT ke lead to conditions for T to be compact (see, e.g.[6, 14, ?]). In the past few years, many authors have been interested in studying the mapping properties of the differences of two composition operators, that is, an operator of the form T = Cϕ − Cψ . The primary motivation for this has been the desire to understand the topological structure of the whole set of composition operators. Most papers in this area have focused on the classical reflexive spaces, but some classical nonreflexive spaces in the unit disc in the complex plane have also recently been discussed. We refer the readers to the recent papers [1, 4, 5, 6, 7, 9, 12] to learn more about the propertied about the differences. Building on the above foundations we estimate the essential norm for the differences of composition operators induced by ϕ and ψ acting from Lipschitz space to weighted Banach space in the unit polydisk Dn , where ϕ and ψ are two holomorphic self-maps of the unit polydisk in n-dimensional complex space C n . The paper is organized as following: Some lemmas are given in section 2. Section 3 is devoted to the main results. Throughout the remainder of this paper, C will denote a positive constant, the exact value of which will vary from one appearance to the next. 2. Some Lemmas Lemma 1. Assume that f ∈ Lipα (Dn ), then |f (z) − f (w)| ≤ nkf kα kDn (z, w) for any z, w ∈ Dn .
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Wang and Liang:: Differences of composition operators Proof. Empolying the definitions in (1.1) and (1.2) we have that n Z 1 Z 1 0, and we denote the second term in (3.5) by I. Empolying lemma 1, it follows that I
≤
sup v(z)(|f (ϕ(z)) − f (ψ(z))| + |f (rϕ(z)) − f (rψ(z))|)
sup
kf kα ≤1 z∈Eδ
≤
sup v(z)nkf kα (kDn (ϕ(z), ψ(z)) + kDn (rϕ(z), rψ(z)))
sup
kf kα ≤1 z∈Eδ
≤ 2n sup v(z)kDn (ϕ(z), ψ(z)),
(3.6)
z∈Eδ
the last inequality is obtained from kDn (rϕ(z), rψ(z)) ≤ kDn (ϕ(z), ψ(z)). Firstly letting r → 1 and then δ → 0, the upper estimate yeilds. The lower estimate. For l = 1, 2, .., n, set Eδl = {z ∈ Dn : max(|ϕl (z)|, |ψ1 (z)|) > 1 − δ}. Sn It is easy to see that Eδ = l=1 Eδl . For a fixed l (1 ≤ l ≤ n), define al = lim sup v(z)(1 − |ϕl (z)|2 )α |ϕψl (z) (ϕl (z))|. δ→0 z∈E l
δ
If we put δm = 1/m, then δm → 0 as m → ∞. For the case kϕl k∞ = 1 or kψl k∞ = 1, then for large enough m with Eδl m 6= ∅, there exists z m ∈ Eδl m such that lim v(z m )(1 − |ϕl (z m )|2 )α |ϕψl (zm ) (ϕl (z m ))| = al .
m→∞
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(3.7)
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Wang and Liang:: Differences of composition operators Since z m ∈ Eδl m implies that |ϕl (z m )| > 1 − δm or |ψl (z m )| > 1 − δm , without loss of generality we assume that |ϕl (z m )| → 1. Set fm (z) =
1 − |ϕl (z m )|2 (1 − ϕl (z m )zl )1−α
·
hϕψl (zm ) (z), ϕψl (zm ) (ϕl (z m ))i . |ϕψl (zm ) (ϕl (z m ))|
We can easily obtain that (fm )m∈N converges to zero uniformly on compact subsets of Dn as m → ∞ and sup kfm kα ≤ C. Thus for any compact operator K : Lipα → Hv∞ , k∈N
we get kKfm kHv∞ → 0, m → ∞. Moreover, it is obvious that fm (ϕ(z m )) = (1 − |ϕl (z m )|2 )α |ϕψl (zm ) (ϕl (z m ))|, fm (ψ(z m )) = 0.
(3.8)
Thus using the above results, (3.7) and (3.8), it is clear that kCϕ − Cψ − KkLipα →Hv∞ ≥ C lim sup k(Cϕ − Cψ − K)fm kHv∞ m→∞
≥ C lim sup(k(Cϕ − Cψ )fm kHv∞ − kKfm kHv∞ ) m→∞
= C lim sup k(Cϕ − Cψ )fm kHv∞ m→∞
= C lim sup sup v(z)|fm (ϕ(z)) − fm (ψ(z))| m→∞ z∈Dn m
≥ C lim sup v(z )|fm (ϕ(z m )) − fm (ψ(z m ))| m→∞
= C lim sup v(z m )|(1 − |ϕl (z m )|2 )α |ϕψl (zm ) (ϕl (z m ))| m→∞
= Cal = C lim sup v(z)(1 − |ϕl (z)|2 )α |ϕψl (z) (ϕl (z))| δ→0 z∈E l
δ
From the above inequality we obtain that kCϕ − Cψ ke,Lipα →Hv∞ ≥ C lim sup v(z)(1 − |ϕl (z)|2 )α |ϕψl (z) (ϕl (z))|. (3.9) δ→0 z∈E l
δ
If both kϕl k∞ < 1 and kψl k∞ < 1, in this condition, when δ is small enough, Eδl is empty, and without loss of generality we may assume that lim sup v(z)(1 − |ϕl (z)|2 )α |ϕψl (z) (ϕl (z))| = 0.
(3.10)
δ→0 z∈E l
δ
Since the above inequality (3.9) and (3.10) holds for every 1 ≤ l ≤ n, thus we obtain that kCϕ − Cψ ke,Lipα →Hv∞ ≥ C max lim sup v(z)|(1 − |ϕl (z)|2 )α |ϕψl (z) (ϕl (z))|. (3.11) 1≤l≤n δ→0 z∈E l
δ
Now for each l = 1, 2, ..., n, we define bl = lim sup v(z)(1 − |ϕl (z)|2 )α |ϕψl (z) (ϕl (z))|. δ→0 z∈Eδ
Then for any ε > 0, there exists a δ0 with 0 < δ0 < 1 such that v(z)(1 − |ϕl (z)|2 )α |ϕψl (z) (ϕl (z))| > bl − ε
(3.12)
whenever z ∈ Eδ0 and l = 1, 2, ..., n. From the above definition we know that z ∈ Eδl 0 implies that z ∈ Eδ0 , then by (3.11) and (3.12) we obtain that kCϕ − Cψ ke,Lipα →Hv∞ ≥ C max (bl − ε) 1≤l≤n
= C max lim sup v(z)|(1 − |ϕl (z)|2 )α |ϕψl (z) (ϕl (z))| − Cε. 1≤l≤n δ→0 z∈Eδ
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Wang and Liang:: Differences of composition operators Now letting ε → 0 in the above inequality we obtain that kCϕ − Cψ ke,Lipα →Hv∞ ≥ C max lim sup v(z)(1 − |ϕl (z)|2 )α |ϕψl (z) (ϕl (z))|. (3.13) 1≤l≤n δ→0 z∈Eδ
Using the similar proof of (3.13) we can get kCϕ − Cψ ke,Lipα →Hv∞ ≥ C max lim sup v(z)|(1 − |ψl (z)|2 )α |ψϕl (z) (ψl (z))|. (3.14) 1≤l≤n δ→0 z∈Eδ
Combining (3.13) and (3.14), we get the lower estimate for the essential norm of the differences. References [1] J. Bonet, M.Lindstr¨ o, E. Wolf, Differences of composition operators between weighted Banach spaces of holomorphic functions, J. Austral. Math. Soc. 84 (2008) 9-20. [2] K.D. Bierstedt, W.H. Summers, Biduals of weighted Banach spaces of analytic functions, J. Austral. Math. Soc. (Series A) 54 (1993) 70-79. [3] C. C. Cowen, B. D. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, FL, 1995. [4] Z. S. Fang, H. Z. Zhou, Differences of composition operators on the Bloch space in the polydisc, Bull. Aust. Math. Soc. 79 (2009) 465-471. [5] Z. S. Fang, H .Z. Zhou, Differences of composition operators on the space of bounded analytic functions in the polydisc, Abstr. Appl. Anal. Volume 2008, Article ID 983132, 10 pages. [6] P. Gorkin and B. D. MacCluer, Essential norms of composition operators, Integral Equations Operator Theory, 48 (2004) 27-40. [7] T. Hosokawa and S. Ohno, Topologicial structures of the set of composition operators on the Bloch space, J. Math. Anal. Appl. 34 (2006) 736-748. [8] W. Lusky, On weighted spaces of harmonic and holomorphic functions, J. London Math. Soc. 51(1995) 309-320. [9] J. Moorhouse, Compact differences of composition operators, J. Funct. Anal. 219 (2005) 70-92. [10] A. L. Shields, D. L. Williams, Bounded projections and the growth of harmonic conjugates in the disc, Michigan Math. J. 29 (1982) 3-25. [11] S. Li, S. Stevi´ c, Riemann-Stieltjes operators on Hardy spaces in the unit ball of Cn , Bull. Belg. Math. Soc. Simon Stevin 14 (2007) 621-628. [12] S. Stevi´ c , E. Wolf, Differences of composition operators between weighted-type spaces of holomorphic functions on the unit ball of C N , Appl. Math. Comput. 215 (2009) 1752-1760. [13] Z. Zhou, Composition operators on the Lipschitz space in polydiscs, Sci. China Ser. A, vol. 46 (1) 33-38. [14] Z. Zhou and Y. Liu, The essential norms of composition operators between generalized Bloch spaces in the polydisc and their applications, J. Inequal. Appl. vol. 2006, Article ID 90742, pages 1-22.
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THE PATH COMPONENT OF THE SET OF GENERALIZED COMPOSITION OPERATORS ON THE BLOCH TYPE SPACES LIU YANG Abstract. In this note, we give a characterization of the path component of the set of generalized composition operator on Bloch type spaces. Keywords: Path component, composition operator, Bloch type spaces
1. INTRODUCTION Let D be the unit disk of the complex plane C, and H(D) be the space of all analytic functions in D. f ∈ H(D) belongs to the Bloch type space Bα , if ∥f ∥Bα = |f (0)| + sup (1 − |z|2 ) |f ′ (z)| < ∞, α
z∈D
where 0 < α < ∞. It is known that B is a Banach space under the ∥ · ∥Bα norm. If α = 1, B α is just the well-known Bloch space. More details about properties on Bloch type space are given in [4], [32] and [16]. We denote S(D) be the set of analytic self-maps of D. Every analytic self-map φ ∈ S(D) induces a linear composition operator Cφ from H(D) to itself. A general and concerning problem in the investigation of composition operator is to characterize operator theoretic properties of Cφ in terms of function theoretic properties of φ. To learn more conclusions about the composition operator, see [6]. For φ ∈ S(D) and g ∈ H(D), Li and Stevic [10] defined the generalized composition operator Cφg as follows: ∫ z g Cφ (f )(z) = f ′ (φ(w))g(w)dw, f ∈ H(D). α
0
The boundedness and compactness of the generalized composition operator from Zygmund spaces to Bloch-type spaces were considered in [10]. Lindstrom and Sanatpour [15] gave the characterization of the generalized composition operator between Zygmund spaces. We can also refer to [11–14], [21–30] for the study of the operator Cφg and its generalizations. The composition operators between Bloch type spaces have been studied by several authors, for example [1–3, 5, 17, 19]. Recently, lots of researchers are interested in the difference of two composition operators, that is, an operator of the form T = Cφ − Cψ , where φ, ψ ∈ S(D). For example, Shapiro The work is supported by NSF of China (No. 11471202). Department of Mathematics, Shantou University, Guangdong Shantou 515063, P. R. China. e-mail:[email protected]. 1
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and Sundberg [20] studied the difference of composition operators on Hardy spaces. In [18], MacCluer, Ohno and Zhao considered it on H ∞ . In [7] and [8], Hosokawa and Ohno investigated it on Bloch spaces. The purpose of studying the difference of composition operators is to investigate the topological structure of the set of composition operators acting on a given function space. Li [9] gave the sufficient and necessary conditions for the boundedness and compactness of the differences of generalized composition operator on the Bloch space. Yang, Luo, and Zhu [31] generalized Li’s results between Bloch type spaces, which help us to study the topological structure of the set of generalized composition operators on the Bloch type spaces. In fact, we give a sufficient condition for the path component of the set of generalized composition operator on Bloch type spaces.
2. NOTATIONS AND AUXILIARY RESULTS For w, z ∈ D, the pseudo-hyperbolic distance between z and w is defined by w−z ρ(w, z) = | |. 1 − wz ¯ Let us (z, w) = (1 − s)z + sw, ϕs (φ(z), ψ(z)) = (1 − s)φ(z) + sψ(z), where s ∈ [0, 1], w ∈ D, φ, ψ ∈ S(D) and simply denote ϕs (φ(z), ψ(z)) by ϕs (z). Let Γ(φ) = {{zn } ∈ D : |φ(zn )| → 1}, Γ(ψ) = {{zn } ∈ D : |ψ(zn )| → 1}. Obviously, Γ(ϕs ) ⊂ Γ(φ) ∩ Γ(ψ). Define β g(z) (1 − |z|2 ) φ,g Dαφ,g (z) = , D (z) = α g(z), α,β (1 − |φ(z)|2 )α (1 − |φ(z)|2 ) β
Dαψ,h (z)
h(z) (1 − |z|2 ) ψ,h = , D (z) = α h(z), α,β (1 − |ψ(z)|2 )α (1 − |ψ(z)|2 )
and ∫ Cϕs f (z) =
Dαϕs (z) z
1 − |z|α = [(1 − s)g(z) + sh(z)], (1 − |ϕ(z)|2 )α
f ′ ((1 − s)φ(w) + sψ(w))[(1 − s)g(w) + sh(w)]dw, f ∈ B α .
0
Let φ,g I1 (z) = Dα,β ρ(φ(z), ψ(z)), ψ,h I2 (z) = Dα,β ρ(φ(z), ψ(z)),
and φ,g ψ,h I3 (z) = Dα,β (z) − Dα,β (z).
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Lemma 2.1. ( [7, Lemma 4.1] ) Let z, w ∈ D and ρ(z, w) = λ < 1. Then the map s 7→ ρ(us , w) is continuous and decreasing on [0, 1]. Lemma 2.2. ( [31, Theorem 1.] ) The following statements are equivalent: (1) Cφg − Cψh : B α → B β is bounded. (2) supz∈D |I1 (z)| < ∞ and supz∈D |I3 (z)| < ∞. (3) supz∈D |I2 (z)| < ∞ and supz∈D |I3 (z)| < ∞. Lemma 2.3. ([31, Theorem 4.] ) Let 0 < α, β < ∞ and φ, ψ ∈ S(D), g, h ∈ H(D), if Cφg − Cψh : B α → B β is bounded, and Cφg , Cψh : B α → B β are not compact, then Cφg − Cψh : B α → B β is compact if and only if the following two conditions hold. (1) D(g, φ) = D(h, ψ) ̸= ∅, D(g, φ) ⊂ Γ(ψ). (2) For arbitrary {zn } ⊂ Γ(φ) ∩ Γ(ψ), lim |I1 (zn )| = lim |I2 (zn )| = lim |I3 (zn )| = 0.
n→∞
n→∞
n→∞
Lemma 2.4. If t < 0 or t > 1, then 1 − x ≤ t(1 − x). t
Proof. Let f (x) = 1 − xt − t(1 − x), then f ′ (x) = −tx( t − 1) + t, f ′′ (x) = −t(t − 1)x(t − 2). Obviously, f ′ (1) = 0, f ′′ (1) ̸= 0, f ′′ (x) > 0 for t < 0, f ′′ (x) < 0 for t > 1.
Lemma 2.5. Let φ, ψ be analytic self maps of the unit disk D, then (1) For any z ∈ D, when α < 1, we have φ,g Dα (z) − Dαϕs (z) ≤ Dαφ,g (z) − Dαψ,h (z) + (2 − α) Dαφ,g (z) ρ2 (φ(z), ψ(z)). (2) For any z ∈ D, when α ≥ 1, we have φ,g ) ( Dα (z) − Dαϕs (z) ≤ Dαφ,g (z) − Dαψ,h (z) + α Dαφ,g (z) + Dαψ,h (z) ρ2 (φ(z), ψ(z)). Proof. (1) The lemma is trivially for s = 0 or 1. In the following, we assume 0 < s < 1. For 1−|φ(z)|2 1−|ψ(z)|2 φ,g ψ,h arbitrary z ∈ D, denote ζ = 1−|ϕ 2 and ξ = 1−|ϕ (z)|2 . By the definition of Dα (z), Dα (z) (z)| s s and Dαϕs (z), it is easy to see 1 − |z|α [(1 − s)g(z) + sh(z)] (1 − |ϕ(z)|2 )α (1 − |φ(z)|2 )α φ,g (1 − |ψ(z)|2 )α ψ,h = (1 − s) Dα (z) + s D (z) 2 α (1 − |ϕs (z)| ) (1 − |ϕs (z)|2 )α α
Dαϕs (z) =
= Dαφ,g − (1 − s)ζ α Dαφ,g (z) − sξ α Dαψ,h (z)
and 2 α 2 α φ,g Dα (z) − Dαϕs (z) = Dαφ,g (z) − (1 − s) (1 − |φ(z)| ) Dαφ,g (z) − s (1 − |ψ(z)| ) Dαψ,h (z) (1 − |ϕs (z)|2 )α (1 − |ϕs (z)|2 )α = Dαφ,g (z) − (1 − s)ζ α Dαφ,g (z) − sξ α Dαψ,h (z) = Dαφ,g (z)(1 − (1 − s)ζ α ) − Dαψ,h (1 − (1 − s)ζ α ) + Dαψ,h (1 − (1 − s)ζ α ) − sξ α Dαψ,h (z) ≤ Dαφ,g (z) − Dαψ,h (1 − (1 − s)ζ α ) + Dαψ,h (1 − (1 − s)ζ α ) − sξ α ≤ Dαφ,g (z) − Dαψ,h sζ α + Dαφ,g (1 − (1 − s)ζ α ) − sξ α .
(2.1)
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φ,g Dα − Dαϕs (z) ≤ Dαφ,g − Dαψ,h sζ α + Dαφ,g (1 − (1 − s)ζ α ) − sξ α .
(2.2)
By simply calculating and the proving process of Proposition 4.2 in [7], we get 0≤
s(1 − s)|φ(z) − ψ(z)|2 = 1 − (1 − s)ζ − sξ ≤ ρ2 (φ(z), ψ(z)). 1 − |ϕs (t)|2
(2.3)
Firstly, we consider the case 0 < α < 1. 1−|φ(z)|2 Since sζ = s 1−|ϕ 2 ≤ 1, then s (z)| sζ α ≤ s1−α ≤ 1.
(2.4)
Now, we estimate (1 − (1 − s)ζ α ) − sξ α . Choosing f (ζ) = 1 − (1 − s)ζ α − sξ α − (1 − (1 − s)ζ − sξ),
(2.5)
then f (ζ) = (1 − s)ζ(1 − ζ α−1 ) + sξ(1 − ξ α−1 ) ≤ (α − 1)((1 − s)ζ(1 − ζ) + sξ(1 − ξ))
(2.6)
= (α − 1)((1 − s)ζ 2 − sξ 2 ) − (α − 1)(1 − (1 − s)ζ − sξ). The last inequality above is obtained by Lemma 2.4. Uniting (2.5) and (2.6), we obtain 1 − (1 − s)ζ α − sξ α − (1 − (1 − s)ζ − sξ) ≤(1 − (1 − sζ)α − sξ) − (α − 1)(1 − (1 − s)ζ 2 − sξ 2 ) − (α − 1)(1 − (1 − sζ) − sξ)
(2.7)
=(2 − α)(1 − (1 − s)ζ − sξ) + (α − 1)(1 − (1 − s)ζ − sξ ). 2
2
and 1 − (1 − s)ζ 2 − sξ 2 =
s|ψ(z)|2 (1 − |ψ(z)|2 ) + (1 − s)|φ(z)|2 (1 − |φ(z)|2 )) + s(1 − s)|φ(z) − ψ(z)|2 >0 (1 − |ϕs (z)|2 )2
(2.8) Hence, 1 − (1 − s)ζ α − sξ α ≤ (2 − α)(1 − (1 − s)ζ − sξ) ≤ (2 − α)ρ2 (φ(z), ψ(z)).
(2.9)
Combining (2.1), (2.4) and (2.9), we get φ,g Dα (z) − Dαϕs (z) ≤ Dαφ,g (z) − Dαψ,h (z) + (2 − α) Dαφ,g (z) ρ2 (φ(z), ψ(z)). This complete the proof of (1). Next, we are going to prove (2). If α = 1, then by (2.3), we have 1 − (1 − s)ζ α − sξ α = 1 − (1 − s)ζ − sξ ≤ ρ2 (φ(z), ψ(z)) = αρ2 (φ(z), ψ(z).
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(2.10)
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If α > 1, then 1 − (1 − s)ζ α − sξ α = 1 − s − (1 − s)ζ α + s − sξ α = (1 − s)(1 − ζ α ) + s(1 − ξ α ) ≤ α(1 − s)(1 − ζ) + s(1 − ξ))
(2.11)
= α(1 − (1 − s)ζ − sξ) ≤ αρ2 (φ(z), ψ(z)). The first inequality in (2.11) above is obtained by Lemma 2.4. If ξ ≤ 1, using (2.1) and (2.11), we obtain φ,g Dα (z) − Dαϕs (z) ≤ Dαφ,g (z) − Dαψ,h (z) + α Dαφ,g (z) ρ2 (φ(z), ψ(z)).
(2.12)
If ξ ≥ 1, for any s ∈ (0, 1), we have |ψ(z)| ≤ |ϕs (z)| ≤ (1 − s)|φ(z)| + s|ψ(z)| and 1−|φ(z)|2 |ψ(z)| ≤ |φ(z)|. Then |ϕs (z)| ≤ |φ(z)| and 1−|ϕ 2 = ζ ≤ 1. Combing (2.1), (2.10) with s (z)| (2.11), it is obvious that φ,g ) Dα (z) − Dαϕs (z) ≤ Dαφ,g (z) − Dαψ,h (z) + α Dαψ,h (z) ρ2 (φ(z), ψ(z)). (2.13) Due to (2.11), (2.12) and (2.13) above, we infer that φ,g ) ( Dα (z) − Dαϕs (z) ≤ Dαφ,g (z) − Dαψ,h (z) + α Dαφ,g (z) + Dαψ,h (z) ρ2 (φ(z), ψ(z)). 3. MAIN RESULTS Proposition 3.1. Let φ, ψ be analytic self maps of the unit disk D, g, h ∈ H(D). Suppose that Cφg and Cψh are bounded but not compact on B α . For any s ∈ [0, 1], when Cφg − Cψh is compact on B α , then we have (1) Dϕαs ⊂ Γ(φ) ∩ Γ(ψ), where Dϕαs = {{zn } ⊂ D : |φ(zn )| → 1, |Dϕαs (zn )| ̸→ 1}. (2) For any {z}n ⊂ Γ(φ) ∩ Γ(ψ), we have lim (Dαφ,g (zn ) − Dαϕs (zn )) = lim (Dαφ,g (zn )ρ(φ(zn ), ϕs (zn )) = 0.
n→∞
n→∞
Proof. (1) It is trivial. (2) For any {zn } ⊂ Γ(φ) ∩ Γ(ψ), it follows from Lemma 2.3 that lim Dαφ,g (zn ) − Dαϕs (zn ) = lim (Dαφ,g (zn ) ρ(φ(zn ), ϕs (zn )) n→∞ n→∞ = lim Dαψ,h (zn ) ρ(φ(zn ), ϕs (zn )) n→∞
= 0. Applying Lemma 2.5, lim |Dαφ,g (zn ) − Dαϕs (zn )| = 0,
n→∞
then by Lemma 2.1, |Dαφ,g (zn )|ρ(φ(zn ), ϕs (zn )) ≤ |Dαφ,g |ρ(φ(zn ), ψ(zn )) → 0.
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Equivalently, lim (Dαφ,g (zn ) − Dαϕs (zn )) = lim (Dαφ,g (zn )ρ(φ(zn ), ϕs (zn )) = 0.
n→∞
n→∞
Theorem 3.2. Let φ, ψ be analytic self maps of the unit disk D, g, h ∈ H(D). Suppose that Cφg and Cψh are bounded but not compact on B α . If Cφg − Cψh is compact on Bα , then the following two conclusions are equivalent: (1) For any {zn } ⊂ Γ(ψ)\Γ(φ), Dαφ,g (zn ) → 0 as n → ∞ and for any {zn } ⊂ Γ(φ)\Γ(ψ), Dαϕ,h (zn ) → 0 as n → ∞ . (2) The map s 7→ Cϕs : [0, 1] → Cϕs (Bα ) is continous. Proof. (1) =⇒ (2) We only need to prove the continuity at s = 0. Let t(s) = sup |Dαφ,g (z) − Dαϕs (z)| + sup |Dαφ,g (z)|ρ(φ(z)), ψ(z)). z∈D
Then, it is easy to see that we have
z∈D
∥Cφg
− Cϕs ∥
Bα
≤ t(s). By lemma 2.3 and the conditions of (1),
lim |Dαφ,g (zn ) − Dαψ,h (zn )| = lim |Dαφ,g (zn )|ρ(φ(zn ), ψ(zn )) n→∞
n→∞
= lim |Dαψ,h (zn )|ρ(φ(zn ), ψ(zn )) n→∞
=0
Hence, for any ε > 0, there exists r1 ∈ (0, 1) such that for every z ∈ Γr1 (φ) = {z ∈ D : |φ(z)| > r1 }, ε |Dαφ,g (z) − Dαψ,h (z)| < , 2 and ε |Dαφ,g (z)|ρ(φ(z), ψ(z) < . 2 Applying Lemma 2.5, we obtain that ε 1 |Dαφ,g (z) − Dαψ,h (z)| < + αε = ( + α)ε. (3.1) 2 2 If z ∈ D\Γr1 (φ), Dαφ,g − Dαϕs is uniformly convergence to 0 when s approaches to 0, then there exists an s1 very close to 0 such that for any s < s1 , sup |Dαφ,g (z) − Dαϕs (z)| < ε. z∈D\Γr1
(3.2)
For any s < s1 , uniting (3.1) and (3.2), we get sup |Dαφ,g (z) − Dαϕs (z)| < ε.
(3.3)
sup |Dαφ,g (z) − Dαϕs (z)| → 0 as s → 0.
(3.4)
z∈D
Hence, z∈D
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Next, we are going to prove that sup |Dαφ,g (z)|ρ(φ(z)), ψ(z)) → 0 as s → 0. z∈D
For any {zn } ⊂ Γ(φ), applying Proposition 3.1 and Lemma 2.3, we have lim (Dαφ,g ρ(φ(zn ), ϕs (zn )) = 0.
n→∞
This implies that there exists an r2 ∈ (0, 1), such that for any z ∈ Γr2 (φ) = {z ∈ D : |φ(z)| > r2 }, |Dαφ,g (z)|ρ(φ(z), ψ(z)) ≤ |Dαφ,g (z)|ρ(φ(z), ψ(z)) < ε.
And because ρ(φ(z), ψ(z)) uniformly converges to 0 on D\Γr2 (φ), we can find a sufficiently small positive number s2 , such that for any s < s2 , |Dαφ,g (z)|ρ(φ(z), ϕs (z)) < ε.
sup z∈D\Γr2 (φ)
Then,
sup |Dαφ,g (z)|ρ(φ(z), ϕs (z)) → 0 as s → 0.
(3.5)
z∈D
Combing (3.4) with (3.5), we obtain that t(s) converges to 0 as s approaches to 0, which finishes the proof of continuity. (2) =⇒ (1) Assume there is a sequence {zn } ⊂ Γ(ψ)\Γ(φ), such that Dαφ,g (zn ) → δ ̸= 0 as n → ∞. Let λ ∈ D and λ ̸= 0, define the test function fλ and gλ respectively as follows: fλ (z) = gλ (z) =
1 2α+1
1 − |λ|2 ¯ ¯ α, αλ(1 − λz)
1 − |λ|2 λ−z 1 (¯ + ¯2 α+1 α+1 ¯ ¯ α+1 ). (α + 1)2 λ(1 − λz) αλ (1 − λz)
Then ∥fλ ∥Bα ≤ 1, ∥gλ ∥Bα ≤ 1, ∥Cφg − Cϕs ∥ ≥ ∥(Cφg − Cϕs )gφ(zn ) ∥Bα 1 (1 − |φ(zn )|2 )(1 − |φs (zn )|2 )α ≥ α+1 Dαϕs (zn ) ρ(φ(zn ), ϕs (zn )) . α+1 2 (1 − φ(zn )ϕs (zn ))
(3.6)
Because zn ∈ Γ(ψ)\Γ(φ), then ϕs (zn ) ̸→ 1 and limn→∞ ρ(φ(zn ), ϕs (zn )) ̸= 0. And s 7→ Cϕs is continous at 0, then by (3.6), we have (1 − |φ(zn )|2 )(1 − |φs (zn )|2 )α ϕs ρ(φ(zn ), ϕs (zn ) → 0, n → ∞, s → 0. Dα (zn ) (1 − φ(zn )ϕs (zn ))α+1
By the compactness of Cφg − Cψh , it is bounded. It follows from Lemma 2.1, Lemma 2.2 and lemma 2.5 that Cφg − Cϕs is bounded. So ∥Cφg − Cϕs ∥ ≥ ∥(Cφg − Cϕs )gφ(zn ) ∥Bα (1 − |φ(z )|2 )(1 − |φ (z )|2 )α ) 1 ( n s n ≥ α+1 Dαφ,g (zn ) − Dαϕs (zn ) . α+1 2 (1 − φ(zn )ϕs (zn ))
(3.7)
Letting n → ∞ and s → 0, we have ∥Cφg − Cϕs ∥ ≥
62
δ 2α+1
> 0.
(3.8)
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For φ(zn ) ≡ 0, suppose λ ∈ D, λ ̸= 0 and hλ (z) =
1 2α+1
1 ¯ ¯ α. αλ(1 − λz)
(3.9)
Then hλ ∈ B α and ∥hλ ∥Bα ≤ 1. If s ̸= 0, then ϕs (zn ) → s ̸= 0. Choosing λ = ϕs (zn ), we have ∥(Cφg − Cϕs )hϕs (zn ) ∥Bα ≥ ∥(Cφg − Cϕs )hϕs (zn ) ∥Bα 1 ( Dαϕs (zn ) ) ≥ α+1 |(1 − |zn |2 )α φ′ (zn )|Dαϕs (zn ) − . 2 1 − |ϕs (zn )|α
For Γ(ψ)\Γ(φ), Proposition 3.1 implies that Dαϕs (zn ) → 0. Letting n → ∞ and s → 0, we get ∥Cφg − Cϕs ∥ ≥ δ > 0. (3.10) It follows from (3.8) and (3.10) that the map s 7→ Cϕs is not continuous at 0, which is a contradiction. So we complete the proof. Corollary 3.3. Let φ, ψ be two analytic self maps of the unit disk D, g, h ∈ H(D). Suppose Cφg and Cψh are bounded but not compact on Bα . If Cφg − Cψh is compact on B α , then Cφg and Cψh are in the same path component of B α . References [1] R. Allen and F. Colonna, Compact composition operators on the Bloch space, Trans. Amer. Math. Soc. 347, 2679-2687 (1995). [2] R. Allen and F. Colonna, On the isometric composition operators on the Bloch space in Cn , J. Math. Anal. Appl. 355, 675-688 (2009). [3] R. Allen and F. Colonna, Weighted composition operators from H ∞ to the Bloch space of a bounded homogeneous domain, Integr. Equ. Oper. Theory 66, 21-40 (2010). [4] Anderson, J. M., Clunie, J., and Pommerenke, Ch., On Bloch functions and normal functions, J. Reine Angew. Math. 270, 12-37 (1974). [5] F. Colonna, Characterisation of the isometric composition operators on the Bloch space, Bull. Austral. Math. Soc. 72, 283-290 (2005). [6] C. Cowen and B. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, 1995. [7] T. Hosokawa and S. Ohno, Topological structures of the sets of composition operators on the Bloch spaces, J. Math. Anal. Appl. 314, 736-748 (2006). [8] T. Hosokawa and S. Ohno, Differences of composition operators on the Bloch spaces, J. Oper. Theory. 57, 229-242 (2007). [9] S Li, Differences of generalized composition operators on the Bloch space, J. Math. Anal. Appl. 394,706-711 (2012). [10] S. Li and S. Stevi´c, Generalized composition operators on Zygmund spaces and Bloch type spaces, J. Math. Anal. Appl. 338, 1282-1295 (2008). [11] S. Li and S. Stevi´c, Products of Volterra type operator and composition operator from H ∞ and Bloch spaces to the Zygmund space, J. Math. Anal. Appl. 345, 40-52 (2008).
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[12] S. Li and S. Stevi´c, Products of composition and integral type operators from H ∞ to the Bloch space, Complex Var. Elliptic Equ. 53, 463-474 (2008). [13] S. Li and S. Stevi´c, Products of integral-type operators and composition operators between Bloch-type spaces, J. Math. Anal. Appl. 349, 596-610 (2009). [14] S. Li and S. Stevi´c, On an integral-type operator from iterated logarithmic Bloch spaces into Bloch-type spaces, Appl. Math. Comput. 215, 3106-3115 (2009). [15] M. Lindstrom and A. Sanatpour, Derivative-free characterization of compact generalized composition operators between Zygmund type spaces, Bull. Austral. Math. Soc. 81, 398-408 (2010). [16] Z. Lou, Bloch Type Spaces of Analytic Functions, PhD Thesis, Institute of Mathematics, Academia Sinica, 1998. [17] Z. Lou, Composition operators on Bloch type spaces, Analysis (Munich). 23, 81-95 (2003). [18] B. MacCluer, S. Ohno and R. Zhao, Topological structure of the space of composition operators on H ∞ , Integr. Equ. Oper. Theory 40, 481-494 (2001). [19] S. Ohno, K. Stroethoff and R. Zhao, Weighted composition operators between Bloch type spaces, Rocky Mountain J. Math. 33, 191-215 (2003). [20] J. Shapiro and C. Sundberg, Isolation amongst the composition operators, Pacific J. Math. 145, 117-152 (1990). [21] S. Stevi´c, Generalized composition operators from logarithmic Bloch spaces to mixednorm spaces, Util. Math. 77, 167-172 (2008). [22] S. Stevi´c, On an integral operator from the Zygmund space to the Bloch-type space on the unit ball, Glasg. J. Math. 51, 275-287 (2009). [23] S. Stevi´c, Products of integral-type operators and composition operators from the mixed norm space to Bloch-type spaces, Siberian Math. J. 50, 726-736 (2009). [24] S. Stevi´c, Integral-type operators from a mixed norm space to a Bloch-type space on the unit ball, Siberian Math. J. 50, 1098-1105 (2009). [25] S. Stevi´c, On an integral operator between Bloch-type spaces on the unit ball, Bull. Sci. Math. 134, 329-339 (2010). [26] S. Stevi´c, On an integral-type operator from logarithmic Bloch-type spaces to mixed norm spaces on the unit ball, Appl. Math. Comput. 215, 3817-3823 (2010). [27] S. Stevi´c, On some integral-type operators between a general space and Bloch-type spaces, Appl. Math. Comput. 218, 2600-2618 (2011). [28] S. Stevi´c and A. Sharma, Composition operators from weighted Bergman-Privalov spaces to Zygmund type spaces on the unit disk, Ann. Polon. Math. 105, 77-86 (2012). [29] S. Stevi´c and A. Sharma, Generalized composition operators on weighted Hardy spaces, Appl. Math. Comput. 218, 8347-8352 (2012). [30] S. Stevi´c and S. Ueki, Integral-type operators acting between weighted-type spaces on the unit ball, Appl. Math. Comput. 215, 2464-2471 (2009). [31] W. Yang, Y. Luo and X. Zhu, Differences of generalized composition operators between Bloch type spaces, Math. Inequal. Appl. 17, 977-987 (2014). [32] K. Zhu, Bloch type spaces of analytic functions, Rocky Mountain J. Math. 23, 11431177 (1993).
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THE GENERALIZED HYERS-ULAM STABILITY OF QUADRATIC FUNCTIONAL EQUATIONS ON RESTRICTED DOMAINS CHANG IL KIM AND CHANG HYEOB SHIN*
Abstract. In this paper, we prove the generalized Hyers-Ulam stability for the functional equation f (ax + by) + abf (x − y) = a(a + b)f (x) + b(a + b)f (y) for some real numbers a, b with 2a + b = 1 on a restricted domain using the fixed point theorem. Key words. Generalized Hyers-Ulam stability, Quadratic functional equation, Banach space, Restricted domains, Fixed point theorem
1. Introduction In 1940, S. M. Ulam [15] proposed the following stability problem : “Let G1 be a group and G2 a metric group with the metric d. Given a constant δ > 0, does there exist a constant c > 0 such that if a mapping f : G1 −→ G2 satisfies d(f (xy), f (x)f (y)) < c for all x, y ∈ G1 , then there exists a unique homomorphism h : G1 −→ G2 with d(f (x), h(x)) < δ for all x ∈ G1 ?” In 1941, Hyers [7] answered this problem under the assumption that the groups are Banach spaces. Aoki [1] and Rassias [11] generalized the result of Hyers. Rassias [11] solved the generalized Hyers-Ulam stability of the functional inequality kf (x + y) − f (x) − f (y)k ≤ (kxkp + kykp ) for some ≥ 0 and p with 0 < p < 1 and all x, y ∈ X, where f : X −→ Y is a function between Banach spaces. A generalization of the Rassias theorem was obtained by Gˇ avruta [6] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassis approach. The functional equation (1.1)
f (x + y) + f (x − y) = 2f (x) + 2f (y)
is called a quadratic functional equation and a solution of a quadratic functional equation is called quadratic. The generalized Hyers-Ulam stability problem for a quadratic functional equation was proved by Skof [13] for mappings f : X −→ Y , where X is a normed space and Y is a Banach space. Cholewa [2] noticed that the theorem of Skof is still true if the relevant domain X is replaced by an Abelian group. Czerwik [3] proved the generalized Hyers-Ulam stability for a quadratic functional equation. 2010 Mathematics Subject Classification. 39B52, 39B82. *Corresponding Author. 1
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Skof [14] was the first author to slove the Hyers-Ulam problem for additive mappings on a restricted domain and in 1998, Jung [8] investigated the HyersUlam stability for additive and quadratic mappimgs on resticted domains. In 2002, Rassias [12] proved that if f : X −→ Y satisfies the following inequality (1.2)
kf (x + y) + f (x − y) − 2f (x) − 2f (y)k ≤ δ,
then there exists a unique quadratic mapping which is approximately. Recently, Najati and Jung [9] showed that the functional equation (1.3)
f (ax + by) + abf (x − y) = af (x) + bf (y)
is equivalent to (1.1) if a, b are non-zero real numbers with a+b = 1 and proved that the Hyers-Ulam stability for the functional equation (1.3) on a resticted domain if f is even. Elhoucien and Youssef [5] showed the results in [9] by removing the Najati-Jung’s assumption that f is even. In this paper, we consider the functional equation (1.4)
f (ax + by) + abf (x − y) = a(a + b)f (x) + b(a + b)f (y)
for fixed non-zero real numbers a, b with 2a + b = 1, a 6= 1 and we prove the generalized Hyers-Ulam stability of it on a restricted domain. Throughout this paper, we assume that X is a normed space and Y is a Banach space. 2. Solutions of (1.4) Najati and Jung [9] showed that if an even mapping f : X −→ Y satisfies (1.3), then f is quadratic and that if a, b are rational numbers, then f satisfies (1.3) if and only if f is quadratic. Elhoucien and Youssef [5] showed that if a mapping f : X −→ Y satisfies (1.3), then f is additive-quadratic. In this section, we will show that if a mapping f : X −→ Y satisfies (1.4), then f is quadratic. Theorem 2.1. Let f : X −→ Y be a mapping satisfying (1.4). Then f is a quadratic mapping. Proof. Letting x = y = 0 in (1.4), since 2a + b = 1, we have (a2 + ab + b2 − 1)f (0) = 3a(a − 1)f (0) = 0. Since a 6= 0, 1, f (0) = 0. Letting y = 0 in (1.4), we have f (ax) = a2 f (x)
(2.1)
for all x ∈ X. Letting x = 0 in (1.4), we have (2.2)
f (by) = b(a + b)f (y) − abf (−y)
for all y ∈ X. Let fo (x) = hence by (2.2), we have
f (x)−f (−x) . 2
(2.3)
Then fo satisfies (1.4), (2.1) and (2.2) and
fo (bx) = bfo (x)
for all x ∈ X. By (1.4), we have (2.4)
fo (ax + by) + fo (ax − by) = 2a(a + b)fo (x) − ab[fo (x + y) + fo (x − y)]
for all x, y ∈ X. Letting y = ay in (2.4), by (2.1), we have (2.5)
a[fo (x + by) + fo (x − by)] = 2(a + b)fo (x) − b[fo (x + ay) + fo (x − ay)]
for all x, y ∈ X and letting x = bx in (2.5), by (2.3), we have (2.6)
fo (bx + ay) + fo (bx − ay) = 2(a + b)fo (x) − a[fo (x + y) + fo (x − y)]
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for all x, y ∈ X. Interchanging x and y in (1.4), we have (2.7)
fo (bx + ay) + fo (bx − ay) = 2b(a + b)fo (x) + ab[fo (x + y) + fo (x − y)]
for all x, y ∈ X. By (2.6) and (2.7), since a(a + b) 6= 0, we have fo (x + y) + fo (x − y) − 2fo (x) = 0 for all x, y ∈ X and hence fo is additive. By (2.1), we have a2 fo (x) = afo (x) and since a 6= 0, 1, fo (x) = 0 for all x ∈ X. (−x) Let fe (x) = f (x)+f . Then fe : X −→ Y is an even mapping satisfying (1.4) 2 and so fe satisfies (2.1) and (2.2). Replacing x and y by 2x and x + y in (1.4), we have (2.8)
fe (x + by) + abfe (x − y) − a(a + b)fe (2x) − b(a + b)fe (x + y) = 0
for all x, y ∈ X. Since a(a + b) 6= 0 and fe is even, by (2.8), we have (2.9)
fe (2x) = 4fe (x), fe (bx) = b2 f (x)
for all x ∈ X. Letting x = bx in (2.8), by (2.9), we have (2.10)
bfe (x + y) + afe (bx − y) − 4ab(a + b)fe (x) − (a + b)fe (bx + y) = 0
for all x, y ∈ X. Interchanging x and y in (2.10), we have (2.11)
bfe (x + y) + afe (x − by) − 4ab(a + b)fe (y) − (a + b)fe (x + by) = 0
for all x, y ∈ X. Letting y = −y in (2.8), we have (2.12)
fe (x − by) + abfe (x + y) − 4a(a + b)fe (x) − b(a + b)fe (x − y) = 0
for all x, y ∈ X. Since b(1 − 2a2 − 2ab − b2 ) = 2ab(a + b), by (2.8), (2.11), and (2.12), we have fe (x + y) + fe (x − y) = 2fe (x) + 2fe (y) for all x, y ∈ X and so fe is quadraric. Since f = fo + fe = fe , f is quadratic.
Corollary 2.2. Let f : X −→ Y be a mapping. If a, b are rational numbers, then f is quadratic if and only if f satisfies (1.4). 3. Stability of (1.4) In this section, we investigate the generalized Hyers-Ulam stability of (1.4) on a restricted domain. Jung [8] proved the Hyers-Ulam stability for additive and quadratic mappings on a resticted domain and Najati and Jung [9] proved the Hyers-Ulam syability of (1.3) on a resticted domain if f is an even mapping. Rahimi, Najati and Bae [10] investigated the generalized Hyers-Ulam syability of (1.1) with the bounded function δ + (kxk2p + kyk2p ) + θkxkp kykp on a resticted domain. Theorem 3.1. Let φ : X 2 −→ [0, ∞) be a mapping and M a non-negative real number. Let f : X −→ Y be a mapping with f (0) = 0. Suppose that f satisfies the following inequality (3.1)
kf (ax + by) + abf (x − y) − a(a + b)f (x) − b(a + b)f (y)k ≤ δ + φ(x, y)
for all x, y ∈ X with kxk + kyk ≥ M and for some non-negative real number δ. Then we have (3.2)
kf (2x) − 4f (x)k ≤ Φ(x, y)
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for all x, y ∈ X with kyk ≥ M , where Φ(x, y) = {φ(2x − 2by, x + (1 − b)y) + φ(2x − 2by, x − (1 + b)y) + φ(2x + 2by, x + (1 + b)y) + φ(2x + 2by, x − (1 − b)y) + |b|[φ(2x + 2y, x + 2y) + φ(2x + 2y, x) + φ(2x − 2y, x) + φ(2x − 2y, x − 2y)] + φ(2x, x + 2y) + φ(2x, x − 2y) + 4(|b| + 2)δ} × |2a(a + b)|−1 . Proof. Let x, y ∈ X with kxk + kyk ≥ M . Then k2xk + kx + yk ≥ kxk + kyk ≥ M . Hence by (3.1), we have (3.3)
kf (x + by) + abf (x − y) − a(a + b)f (2x) − b(a + b)f (x + y)k ≤ δ + φ(2x, x + y)
and letting y = −y in (3.3), we have (3.4)
kf (x − by) + abf (x + y) − a(a + b)f (2x) − b(a + b)f (x − y)k ≤ δ + φ(2x, x − y).
By (3.3) and (3.4), we have (3.5)
kf (x + by) − f (x − by) + bf (x − y) − bf (x + y)k ≤ 2δ + φ(2x, x + y) + φ(2x, x − y).
Let x, y ∈ X with kyk ≥ M . Since kx − byk + kyk ≥ M and kx + byk + kyk ≥ M , by (3.5), we have (3.6)
kf (x) − f (x − 2by) + bf (x − (1 + b)y) − bf (x + (1 − b)y)k ≤ 2δ + φ(2x − 2by, x + (1 − b)y) + φ(2x − 2by, x − (1 + b)y)
and (3.7)
kf (x + 2by) − f (x) + bf (x − (1 − b)y) − bf (x + (1 + b)y)k ≤ 2δ + φ(2x + 2by, x + (1 + b)y) + φ(2x + 2by, x − (1 − b)y).
Since kx + yk + kyk ≥ M and kx − yk + k − yk ≥ M , by (3.5), we have (3.8)
kf (x + (1 + b)y) − f (x + (1 − b)y) + bf (x) − bf (x + 2y)k ≤ 2δ + φ(2x + 2y, x + 2y) + φ(2x + 2y, x)
and (3.9)
kf (x − (1 + b)y) − f (x − (1 − b)y) + bf (x) − bf (x − 2y)k ≤ 2δ + φ(2x − 2y, x) + φ(2x − 2y, x − 2y).
Since kxk + k2yk ≥ M , by (3.3) and (3.4), we have (3.10)
kf (x + 2by) + abf (x − 2y) − a(a + b)f (2x) − b(a + b)f (x + 2y)k ≤ δ + φ(2x, x + 2y).
and (3.11)
kf (x − 2by) + abf (x + 2y) − a(a + b)f (2x) − b(a + b)f (x − 2y)k ≤ δ + φ(2x, x − 2y).
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Note that 2a(a + b)[f (2x) − 4f (x)] = −[f (x) − f (x − 2by) + bf (x − (1 + b)y) − bf (x + (1 − b)y)] + [f (x + 2by) − f (x) + bf (x − (1 − b)y) − bf (x + (1 + b)y)] (3.12)
+ b[f (x + (1 + b)y) − f (x + (1 − b)y) + bf (x) − bf (x + 2y)] + b[f (x − (1 + b)y) − f (x − (1 − b)y) + bf (x) − bf (x − 2y)] − [f (x + 2by) + abf (x − 2y) − a(a + b)f (2x) − b(a + b)f (x + 2y)] − [f (x − 2by) + abf (x + 2y) − a(a + b)f (2x) − b(a + b)f (x − 2y)]
for all x, y ∈ X with kyk ≥ M . By (3.6), (3.7), (3.8), (3.9), (3.10), (3.11), we have (3.2). We apply the fixed point method to investigate the generalized Hyers-Ulam stability for the functional equation (1.4). Definition 3.2. Let X be a set. A function d : X × X −→ [0, ∞] is called a generalized metric on X if d satisfies (i) d(x, y) = 0 if and only if x = y, (ii) d(x, y) = d(y, x) for all x, y ∈ X, and (iii) d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ X. Now, we consider the following fixed point theorem : Theorem 3.3. [4] Let (X, d) be a complete generalized metric space and J : X −→ X a strictly contractive mapping with a Lipschitz constant L with 0 < L < 1. Then for each element x ∈ X, either (3.13)
d(J n x, J n+1 x) = ∞
for all nonnegative integers n or there is a nonnegative integer k such that (1) d(J n x, J n+1 x) < ∞ for all n ≥ k, (2) a 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 k x, y) < ∞}, and 1 d(y, Jy) for all y ∈ Y . (4) d(y, y ∗ ) ≤ 1−L Now, we will prove the stability of (1.4) on a restrcted domain. Theorem 3.4. Let φ : X 2 −→ [0, ∞) be a function such that (3.14)
φ(2x, 2y) ≤ Lφ(x, y)
for all x, y ∈ X for some positive real number L with L < 1. Let f : X −→ Y be a mapping with (3.1). Then there exists a unique quadratic mapping Q : X −→ Y such that f satisfies (1.4) and 1 Φ(x, y) (3.15) kQ(x) − f (x)k ≤ 4(1 − L) for all x ∈ X and y ∈ X with kyk ≥ M . Proof. By Theorem 3.1, the following inequality (3.16)
kf (x) − 2−2 f (2x)k ≤ 2−2 Φ(x, y)
holds for all x, y ∈ X with kyk ≥ M .
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CHANG IL KIM AND CHANG HYEOB SHIN
Let Ω = {g : X −→ Y | g(0) = 0}. Define a generalized metric d on Ω by d(g, h) = inf {C ∈ [0, ∞) | kg(x) − h(x)k ≤ CΦ(x, y), ∀x, y ∈ X with kyk ≥ M }. We claim that (Ω, d) is a complete metric space. Let {gn } be a Cauchy sequence in (Ω, d) and > 0. Then there is a positive integer k such that d(gn , gm ) ≤ for all n, m ≥ k. Pick y0 ∈ X with ky0 k ≥ M and let x ∈ X. Since kgn (x) − gm (x)k ≤ Φ(x, y0 ) for all n, m ≥ k, {gn (x)} is a Cauchy sequence in Y and hence we can define a mapping g : X −→ Y by g(x) = limn−→∞ gn (x). Clearly, g ∈ Ω and limn−→∞ gn = g. Thus (Ω, d) is a complete metric space. Define a map J : Ω −→ Ω by Jh(x) = 41 h(2x) for all x ∈ X. Let g, h ∈ Ω. Suppose that C is a positive real number such that kg(x) − h(x)k ≤ CΦ(x, y) for all x, y ∈ X with kyk ≥ M . By (3.14), we have 1 1 1 kJg(x) − Jh(x)k = kg(2x) − h(2x)k ≤ CΦ(2x, 2y) ≤ CLΦ(x, y) 4 4 4 for all x, y ∈ X with kyk ≥ M and hence we have L d(Jg, Jh) ≤ d(g, h) 4 for all g, h ∈ Ω. Since 0 < L < 4, J is a strictly contractive mapping and by (3.16), we have 1 d(Jf, f ) ≤ . 4 By Theorem 3.3, {J n f } converges to the unique fixed element Q of J in Y = {h ∈ Ω | d(f, h) < ∞} and (3.15) holds. Further, we have Q(x) = lim J n f (x) = lim 2−2n f (2n x) n−→∞
n−→∞
for all x ∈ X and we have (3.15). Moreover, Q(0) = 0, because f (0) = 0. Now, we claim that Q satisafies (1.4). First, suppose that x 6= 0 or y 6= 0. Replacing x and y by 2n x and 2n y in (3.1), respectively and deviding both sides of (3.1) by 22n , we have k2−2n f (2n (ax + by)) + 2−2n abf (2n (x − y)) (3.17)
1 n [L φ(x, y) + δ] 4n for all x, y ∈ X and sufficiently large positive integer n. Letting n −→ ∞ in (3.17), Q satisfies (1.4). Clealy, if x = 0 and y = 0, then Q satisfies (1.4). By Theorem 2.1, Q is quadratic. Assume that Q1 : X −→ Y is another quadratic mapping satisfying (1.4) and (3.15). Then we have 1 kQ1 (x) − f (x)k ≤ Φ(x, y) 4(1 − L) − a(a + b)2−2n f (2n x) − b(a + b)2−2n f (2n y)k ≤
for all x ∈ X and y ∈ X with kyk ≥ M and so 1 d(Q1 , f ) ≤ < ∞. 4(1 − L) By (3) of Theorem 3.3, Q = Q1 .
Skof [13](Jung [8], resp.) proved an asymptotic property of aditive (quadratic, resp.) mappings. We consider such property for (1.4).
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Corollary 3.5. A mapping f : X −→ Y satisfies (1.4) if and only if the asymptotic condition kf (ax + by) + abf (x − y) − a(a + b)f (x) − b(a + b)f (y)k −→ 0 as kxk + kyk −→ ∞ holds. Acknowledgements The first author was supported by the research fund of Dankook University in 2014. References [1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2, 64-66(1950). [2] P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27, 76-86(1984). [3] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg. 62, 59-64(1992). [4] J. Diaz and B. Margolis, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 74, 305-309(1968). [5] E. Elhoucien and M. Youssef, On the Paper by A. Najati and S.-M. Jung: The Hyers-Ulam Stability of Approximately Quadratic, Journal of Nonlinear Analysis and application 2012, 1-10(2012). [6] P. Gˇ avruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 18,4 431-436(1994). [7] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27, 222-224(1941). [8] S.-M. Jung, On the Hyers-Ulam stability of the functional equations that have the quadratic property, J. Math. Anal. Appl. 222, 126-137(1998). [9] A. Najati and S. M. Jung, Approximately quadratic mappings on restricted domains, J. Ineq. Appl. 2010, 1-10(2010). http://dx.doi.org/10.1155/2010/503458. [10] A. Rahimi, A. Najati, and J. H. Bae, On the Asymptoticity Aspect of Hyers-Ulam stability of quadratic mappings, J. Ineq. Appl. 2010, 1-14(2010). [11] Th. M. Rassias, On the stability of linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72, 297-300(1978). [12] J. M. Rassias, On the Ulam stability of mixed type mappings on restricted domains, J. Math. Anal. Appl. 276, 747-762(2002). [13] F. Skof, Propriet´ a locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53, 113-129(1983). [14] F. Skof, Sull’ approssinazione delle applicazioni localmente δ-additive, Atti Accad. Sc. Torino 117, 377-389(1983). [15] S. M. Ulam, A Collection of Mathematical Problems, Interscience Publ. New York, 1961. Problems in Modern Mathematics, Wiley, New York, 1964. Department of Mathematics Education, Dankook University, Yongin 448-701, Korea E-mail address: [email protected] Department of Mathematics, Soongsil University, Seoul 156-743, Korea E-mail address: [email protected]
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Hesitant fuzzy soft set and its lattice structures a
Xiaoqiang Zhoua , Qingguo Lib∗ College of Mathematics, Hunan Institute of Science and Technology Yueyang, 414006, P.R.China b College of Mathematics and Econometrics, Hunan University Changsha, 410082, P.R.China
Abstract: Hesitant fuzzy set and soft set were introduced by Torra and Molodtsov, respectively. The two sets have been used successfully as effective mathematical tools for dealing with vagueness and uncertainties. By combining hesitant fuzzy set and soft set, in this paper, we propose a new model named hesitant fuzzy soft set, which can be regarded as an extension of many models, such as hesitant fuzzy set, soft set, fuzzy soft set, interval-valued fuzzy soft set and multi-fuzzy soft set. Some basic operations of hesitant fuzzy soft set are defined and some desirable properties of those operations are investigated. Furthermore, the lattice structures of hesitant fuzzy soft set are discussed. Keywords: Hesitant fuzzy set; soft set; fuzzy soft set; hesitant fuzzy soft set; lattice
1
Introduction
Soft set was firstly proposed by Molodtsov [1], it is a new mathematical tool for modeling vagueness and uncertainty. Since its appearance, soft set theory has attracted more and more attention from many researchers and many important results on soft set have been achieved in theory and application. Maji and Biswas et al. [2] defined some basic operations. Ali et al. [3, 4] gave some new operations on soft sets and studied some algebraic structures of soft sets. Yang and Guo [5] introduced some kernels and closures of soft set relations. Many authors applied soft sets to some algebraic structures such as groups, rings, fields and modules [6–8]. The applications of soft set in decision making and other areas could be found in [9–12]. At the same time, in order to extend the application ranges of soft set, fuzzy extension of soft set theory has become a hot research topic. Maji et al. [13] introduced the notions of fuzzy soft set. Jiang et al. [14] and Majumdar and Samanta [15] further generalized fuzzy soft set to intuitionistic fuzzy soft set and generalised fuzzy soft set, respectively. Yang et al. [16] proposed the concept of interval-valued fuzzy soft set by combining the interval-valued fuzzy set and soft set. Some other generalized models of soft set could be seen in [17–19] Recently, Torra [20] introduced hesitant fuzzy set which is a new extension of fuzzy set. It permits the membership degree of an element to a set to be represented as some possible values between 0 and 1. Presently, work on hesitant fuzzy set is making progress rapidly and lots of results on hesitant fuzzy set have been obtained [21–25]. The main goal of this paper is to combine the hesitant fuzzy set and soft set and obtain a new hybrid model named hesitant fuzzy soft set. It can be viewed as a hesitant fuzzy extension of the soft set or a generalization of the hesitant fuzzy set. The rest of this paper is structured as follows. The following section briefly reviews some basic notions of soft set, fuzzy soft set and hesitant fuzzy set. Two new operations on hesitant fuzzy element are defined, and some of their properties are investigated. In Section 3, the concept of ∗
Corresponding author. Tel./fax: +86 13789003995/+86 731 88822755. E-mail address: [email protected], [email protected]. Mailing address: College of Mathematics, Hunan Institute of Science and Technology, Yueyang, Hunan, 414006, P.R.China
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hesitant fuzzy soft set is first proposed by combining hesitant fuzzy set and soft set. Some operations on hesitant fuzzy soft set are given and some of their properties are studied. In Section 4, we discuss the lattice structures of hesitant fuzzy soft set. The conclusion is finally reached in Section 5.
2
Preliminary
Let U be an initial universe of objects and E the set of parameters in relation to objects in U . Parameters are often attributes, characteristics, or properties of objects. Let P (U ) denote the power set of U and A ⊆ E. Molodtsov [1] first gave the definition of soft set as follows. Definition 2.1. [1] A pair (F, A) is called a soft set over U , where A ⊆ E and F is a set valued mapping given by F : A → P (U ). Maji [13] introduced fuzzy soft set which is an fuzzy extension of soft set. Definition 2.2. [13] Let P(U ) be the set of all fuzzy subsets of U . A pair (F, A) is called a fuzzy soft set over U , where F is a set valued mapping given by F : A → P(U ). As a generalization form of fuzzy set, hesitant fuzzy set (HF S) was first introduced by Torra [20] as follows. Definition 2.3. [20] Let X be a reference set, an HF S on X is in terms of{ a function that } when hH (x) applied to X returns a subset of [0, 1], which can be represented as H = x |x ∈ X , where hH (x) is a set of some values in [0, 1], denoting the possible membership degrees of the element x ∈ X to the set H. For convenience, Xu and Xia [21,22] called hH (x) an hesitant fuzzy element (HF E) with respect to x of H. It is worth noting that the number of values of different HF Es may be different, in this paper, let l(hH (x)) denote the number of values of hH (x). We arrange the values of hH (x) in σ(j) increasing order, and let hH (x) be the jth largest value of hH (x). { } Definition 2.4. [20] Let H = hHx(x) |x ∈ X be an HF S. Then (1) H is said to be an empty hesitant set, denoted by Φ, if hH (x) = 0 for all x ∈ X; (2) H is said to be a full hesitant set, denoted by I, if hH (x) = 1 for all x ∈ X; (3) H is said to be a complete ignorance set, denoted by W, if hH (x) = [0, 1] for all x ∈ X. Definition 2.5. [20] Let λ > 0, h, h1 and h2 be three HF Es, some operations on them are given as follows: (1) h1 ∪ h2 = ∪γ1 ∈h1 ,γ2 ∈h2 {max(γ1 , γ2 )}; (2) h1 ∩ h2 = ∪γ1 ∈h1 ,γ2 ∈h2 {min(γ1 , γ2 )}; (3) hc = ∪γ∈h {1 − γ}. We further define the strict union and the strict intersection for HF Es h1 and h2 , which will be useful in the sequel. + Definition 2.6. Let h1 and h2 be two HF Es, h− i = min{γi |γi ∈ hi } and hi = max{γi |γi ∈ hi }(i = 1, 2). The strict union and the strict intersection of h1 and h2 are defined as follows: + (1) h1 ⊔ h2 = ∪γi ∈hi ,i=1,2 {γi |γi > min(h+ 1 , h2 ) or γ1 = γ2 }; − (2) h1 ⊓ h2 = ∪γi ∈hi ,i=1,2 {γi |γi < max(h− 1 , h2 ) or γ1 = γ2 };
For example, let h1 = {0.2, 0.3, 0.6, 0.8} and h2 = {0.4, 0.5, 0.8, 0.9}, then h1 ⊔ h2 = {0.8, 0.9} ̸= {0.4, 0.5, 0.6, 0.8, 0.9} = h1 ∪ h2 , h1 ⊓ h2 = {0.2, 0.3} ̸= {0.2, 0.3, 0.4, 0.5, 0.6, 0.8} = h1 ∩ h2 . In fact, all the above operations on HF Es can be suitable for HF Ss. Some relationships can be further established for these operations on HF Es. 73
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Theorem 2.7. For three HF Es h, h1 and h2 , the following is valid: (1) hc1 ⊔ hc2 = (h1 ⊓ h2 )c ; (2) hc1 ⊓ hc2 = (h1 ⊔ h2 )c . − Proof. (1) Since h1 ⊓ h2 = ∪γi ∈hi ,i=1,2 {γi |γi < max(h− 1 , h2 ) or γ1 = γ2 }, then − (h1 ⊓ h2 )c = ∪γi ∈hi ,i=1,2 {1 − γi |γi < max(h− 1 , h2 ) or γ1 = γ2 }
Since hc1 = ∪γ1 ∈h1 {1 − γ1 } and hc2 = ∪γ2 ∈h2 {1 − γ2 }, then hc1 ⊔ hc2 = {∪γ1 ∈h1 {1 − γ1 }} ⊔ {∪γ2 ∈h2 {1 − γ2 }}
− = ∪γi ∈hi ,i=1,2 {1 − γi |1 − γi < min(1 − h− 1 , 1 − h2 ) or 1 − γ1 = 1 − γ2 }
− = ∪γi ∈hi ,i=1,2 {1 − γi |γi < 1 − min(1 − h− 1 , 1 − h2 ) or γ1 = γ2 } − = ∪γi ∈hi ,i=1,2 {1 − γi |γi < max(h− 1 , h2 ) or γ1 = γ2 }
Theorem 2.8. For three HF Es h1 , h2 and h3 , the following is valid: (1) (h1 ∪ h2 ) ∪ h3 = h1 ∪ (h2 ∪ h3 ); (2) (h1 ∩ h2 ) ∩ h3 = h1 ∩ (h2 ∩ h3 ); (3) h1 ∪ (h2 ∩ h3 ) = (h1 ∪ h2 ) ∩ (h1 ∪ h3 ); (4) h1 ∩ (h2 ∪ h3 ) = (h1 ∩ h2 ) ∪ (h1 ∩ h3 ). Proof. (2) and (4) are similar to (1) and (3), respectively, so we only prove (1) and (3). (1) Since (h1 ∪ h2 ) = ∪γi ∈hi ,i=1,2 {max(γ1 , γ2 )}, then (h1 ∪ h2 ) ∪ h3 = {∪γi ∈hi ,i=1,2 {max(γ1 , γ2 )}} ∪ h3 = ∪γi ∈hi ,i=1,2,3 {max(max(γ1 , γ2 ), γ3 )} = ∪γi ∈hi ,i=1,2,3 {max(γ1 , γ2 , γ3 )} = ∪γi ∈hi ,i=1,2,3 {max(γ1 , max(γ2 , γ3 ))} = h1 ∪ (h2 ∪ h3 ). (3) Since (h2 ∩ h3 ) = ∪γi ∈hi ,i=2,3 {min(γ2 , γ3 )}, then h1 ∪ (h2 ∩ h3 ) = h1 ∪ {∪γi ∈hi ,i=2,3 {min(γ2 , γ3 )}} = ∪γi ∈hi ,i=1,2,3 {max(γ1 , min(γ2 , γ3 ))} = ∪γi ∈hi ,i=1,2,3 {min(max(γ1 , γ2 ), max(γ2 , γ3 ))} = {∪γi ∈hi ,i=1,2 {max(γ1 , γ2 )}} ∩ {∪γi ∈hi ,i=1,3 {max(γ1 , γ3 )}} = (h1 ∪ h2 ) ∩ (h1 ∪ h3 )
Theorem 2.9. For three HF Es h1 , h2 and h3 , the following is valid: (1) (h1 ⊓ h2 ) ⊓ h3 = h1 ⊓ (h2 ⊓ h3 ); (2) (h1 ⊔ h2 ) ⊔ h3 = h1 ⊔ (h2 ⊔ h3 ); (3) (h1 ⊔ h2 ) ⊓ h1 = h1 ; (4) (h1 ⊓ h2 ) ⊔ h1 = h1 . Proof. (2) and (4) are similar to (1) and (3), respectively, so we only prove (1) and (3). − (1) Since h1 ⊓ h2 = ∪γi ∈hi ,i=1,2 {γi |γi < max(h− 1 , h2 ) or γ1 = γ2 }, then − (h1 ⊓ h2 ) ⊓ h3 = {∪γi ∈hi ,i=1,2 {γi |γi < max(h− 1 , h2 ) or γ1 = γ2 }} ⊓ h3
− − = ∪γi ∈hi ,i=1,2,3 {γi |γi < max(max(h− 1 , h2 ), h3 ) or γ1 = γ2 = γ3 }
− − = ∪γi ∈hi ,i=1,2,3 {max(h− 1 , h2 , h3 ) or γ1 = γ2 = γ3 }
− − = ∪γi ∈hi ,i=1,2,3 {γi |γi < max(h− 1 , max(h2 , h3 )) or γ1 = γ2 = γ3 }
− = h1 ⊓ {∪γi ∈hi ,i=2,3 {γi |γi < max(h− 2 , h3 ) or γ2 = γ3 }}
= h1 ⊓ (h2 ⊓ h3 ). 74
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+ (3) Since h1 ⊔ h2 = ∪γi ∈hi ,i=1,2 {γi |γi > min(h+ 1 , h2 ) or γ1 = γ2 }, + + + + + i) If h1 ≤ h2 , then min(h1 , h2 ) = h1 . It follows that h1 ⊔h2 = ∪γi ∈hi ,i=1,2 {γ2 |γ2 > h+ 1 or γ2 = γ1 }. By Definition 2.6, we have (h1 ⊔ h2 ) ⊓ h1 = h1 . + + + + + ii) If h+ 1 > h2 , then min(h1 , h2 ) = h2 . It follows that h1 ⊔h2 = ∪γi ∈hi ,i=1,2 {γ1 |γ1 > h2 or γ1 = γ2 }. By Definition 2.6, we have (h1 ⊔ h2 ) ⊓ h1 = h1 .
3
Hesitant fuzzy soft set
In this section, we present an extended soft set model which is called hesitant fuzzy soft set by combining the hesitant fuzzy set and soft set. Some operations and their properties on hesitant fuzzy soft set will also be discussed. Definition 3.1. Let HF (U ) be the class of all HF Ss of the universe U , A ⊆ E. A pair (Fe, A) is called a hesitant fuzzy soft set (HF SS), where Fe : A → HF (U ) is a mapping. In other words, a hesitant fuzzy soft set over U is a parameterized family of hesitant fuzzy set of the universe U . To illustrate this idea, let us consider the following example. Example 3.2. Let U = {u1 , u2 , u3 } be a set of mobile telephones and A = {e1 , e2 , e3 } ⊆ E be a set of parameters. The ei (i = 1, 2, 3) stands for the parameters “expensive”, “beautiful” and “multifunctional”, respectively. Let Fe : A → HF (U ) be a function given as follows: } { {0.2, 0.7, 0.8} {0.5, 0.8} {0.4, 0.6, 0.8} e , , , F (e1 ) = u1 u2 u3 { } {0.3, 0.5, 0.7} {0.4, 0.6, 0.9} {0.5, 0.7} e F (e2 ) = , , , u1 u2 u3 } { {0.5, 0.8} {0.3, 0.5, 0.8} {0.5, 0.6, 0.9} , , . Fe(e3 ) = u1 u2 u3 Then (Fe, A) is a hesitant fuzzy soft set. Remark 3.3. (1) If A has only an element, i.e. A = {e}, then hesitant fuzzy soft set becomes hesitant fuzzy set [20]; (2) If hFe(e) (u) has only one value for all e ∈ A and u ∈ U , then hesitant fuzzy soft set degenerates to traditional fuzzy soft set [13]; (3) If hFe(e) (u) is a subinterval of [0, 1] for all e ∈ A and u ∈ U , then hesitant fuzzy soft set reduces to interval-valued fuzzy soft set [17]; (4) For all e ∈ A, if hFe(e) (u) has the same number of values with respect to u ∈ U , then hesitant fuzzy soft set transforms to multi-fuzzy soft set [19]. Definition 3.4. The complement of an HF SS (Fe, A) is denoted by (Fe, A)c {and is defined } by hFe c (e) (u) (Fe, A)c = (Fec , A), where Fec : A → HF (U ) is a mapping given by Fec (e) = |u ∈ U , u ∪ where hFec (e) (u) = γ∈h e (u) {1 − γ}. F (e)
Example 3.5. (continued) The complement of (Fe, A) is following as: { } {0.2, 0.3, 0.8} {0.2, 0.5} {0.2, 0.4, 0.6} Fec (e1 ) = , , , u1 u2 u3 { } {0.3, 0.5, 0.7} {0.1, 0.4, 0.6} {0.3, 0.5} c e , , , F (e2 ) = u1 u2 u3 { } {0.2, 0.5} {0.2, 0.5, 0.7} {0.1, 0.4, 0.5} c e F (e3 ) = , , . u1 u2 u3 75
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Definition 3.6. Let (Fe, A) be an HF SS over U . Then e A , if hF (e) (u) = 0 for all u ∈ U and (1) (Fe, A) is said to be an empty hesitant soft set, denoted by Φ e ∈ A; (2) (Fe, A) is said to be a full hesitant soft set, denoted by IeA , if hF (e) (u) = 1 for all u ∈ U and e ∈ A; fA , if hF (e) (u) = [0, 1] for all u ∈ U (3) (Fe, A) is said to be a complete hesitant soft set, denoted by W and e ∈ A. Proposition 3.7. Let A ⊆ E. Then e c = IeA ; (1) Φ A c =Φ e A; (2) IeA fc = W fA . (3) W A e B) be two HF SSs over U and A, B ⊆ E. We define a mapping Definition 3.8. Let (Fe, A) and (G, e H : A ∪ B → HF (U ) such that for all e ∈ A ∪ B ̸= ∅, Fe(e), if e ∈ A − B, e e H(e) = G(e), if e ∈ B − A, H(e), e if e ∈ A ∩ B. e e e A ∪ B) is called the extended union of (Fe, A) and (G, e B), (1) If H(e) = Fe(e) ∪ G(e), then (H, e e e (G, B). denoted by (F , A)∪ e e e e A ∪ B) is called the extended intersection of (Fe, A) and (G, e B), (2) If H(e) = F (e) ∩ G(e), then (H, e e e (G, B). denoted by (F , A)∩ e e e A ∪ B) is called the extended-strict union of (Fe, A) and (G, e B), (3) If H(e) = Fe(e) ⊔ G(e), then (H, e e e denoted by (F , A)⊔(G, B). e e e A ∪ B) is called the extended-strict intersection of (Fe, A) and (4) If H(e) = Fe(e) ⊓ G(e), then (H, e B), denoted by (Fe, A)⊓ e B). e (G, (G, e e B) = Φ e ∅ , (Fe, A)∩ e B) = Φ e ∅ , (Fe, A)⊔ e B) = Φ e ∅ and e (G, e (G, e (G, If A ∪ B = ∅, then (F , A)∪ e B) = Φ e ∅. e (G, (Fe, A)⊓ e B) be two HF SSs over U and A, B ⊆ E. We define a mapping Definition 3.9. Let (Fe, A) and (G, e : A ∩ B → HF (U ) such that for all e ∈ A ∩ B ̸= ∅, H e e e A ∩ B) is called the strict union of (Fe, A) and (G, e B), denoted (1) If H(e) = Fe(e) ∪ G(e), then (H, e e e (G, B). by (F , A)d e e e A ∩ B) is called the strict intersection of (Fe, A) and (G, e B), (2) If H(e) = Fe(e) ∩ G(e), then (H, e e e denoted by (F , A) (G, B). e e e A ∩ B) is called the strict-strict union of (Fe, A) and (G, e B), (3) If H(e) = Fe(e) ⊔ G(e), then (H, e B). e (G, denoted by (Fe, A)⊎ e e e e A ∩ B) is called the strict-strict intersection of (Fe, A) and (4) If H(e) = F (e) ⊓ G(e), then (H, e B), denoted by (Fe, A)C e B). e (G, (G, e e B) = Φ e ∅ , (Fe, A) e (G, e B) = Φ e ∅ , (Fe, A)⊎ e B) = Φ e ∅ and e (G, If A ∩ B = ∅, then (F , A) d (G, e e e e (G, B) = Φ∅ . (F , A)C e }, θ2 ∈ {∩ e, ⊓ e }, Proposition 3.10. Let A ⊆ E, (Fe, A) be an HF SS over (U, E), θ1 ∈ { e , C e } and θ4 ∈ {∪ e, ⊔ e }. Then e, ⊎ θ3 ∈ {d e e e (1) (F , A) θ1 IE = (F , A) θ2 IeA = (Fe, A); (2) (Fe, A) θ3 IeE = (Fe, A) θ4 IeA = IeA ; e E = (Fe, A) θ2 Φ eE = Φ e A; (3) (Fe, A) θ1 Φ e e e e (4) (F , A) θ3 ΦE = (F , A) θ4 ΦA = (Fe, A); e ϕ = (Fe, A) θ3 Φ eϕ = Φ e ϕ; (5) (Fe, A) θ1 Φ e ϕ = (Fe, A) θ4 Φ e ϕ = (Fe, A). (6) (Fe, A) θ2 Φ
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e B) and (H, e C) be HF SSs e, ⊔ e, C e, ⊓ e, ∩ e, e, ∪ e, d e },A, B, C ⊆ E, (Fe, A), (G, Theorem 3.11. Let α ∈ {⊎ over (U, E). Then the following holds: (1) (Fe, A) α (Fe, A) = (Fe, A); e B) = (G, e B) α (Fe, A); (2) (Fe, A) α (G, e B) α (H, e C)) = ((Fe, A) α (G, e B)) α (H, e C). (3) (Fe, A) α ((G, e , the others can be proved Proof. (1) and (2) are trivial. We only prove (3). For example, let α = ⊎ analogously. e B)⊎ e C)) = (J, e M ) and ((Fe, A)⊎ e B))⊎ e C) = (K, e N ), thus e ((G, e (H, e (G, e (H, Suppose that (Fe, A)⊎ e e e e e e C). e ((G, B) ⊎ e (H, C)) = Φϕ = ((F , A) ⊎ e (G, B)) ⊎ e (H, M = N = A ∩ B ∩ C. If M = ϕ, then (F , A) ⊎ If M ̸= ϕ, then by (2) in Theorem 2.9, we have hF (e) (u) ⊔ (hG(e) (u) ⊔ hH(e) (u)) = (hF (e) (u) ⊔ e ⊔ H(e)) e e hG(e) (u)) ⊔ hH(e) (u) for all e ∈ M and u ∈ U . It follows that Fe(e) ⊔ (G(e) = (Fe(e) ⊔ G(e)) ⊔ e e B)⊎ e C)) = e , we have (Fe, A)⊎ e ((G, e (H, H(e) for all e ∈ M . By the definition of the operation ⊎ e e e e e ((F , A)⊎(G, B))⊎(H, C). e , e, ∪ e, d e, ⊔ e, C e and ⊓ e are idempotent, e, ⊎ Remark 3.12. Theorem 3.11 shows that the operations ∩ commutative and associative, respectively. e B) be HF SSs over (U, E). Then the following Theorem 3.13. Let A, B ⊆ E, (Fe, A) and (G, holds: e B))c = (Fe, A)c ⊔ e B)c ; e (G, e (G, (1) ((Fe, A)⊓ e B))c = (Fe, A)c ⊓ e B)c ; e (G, e (G, (2) ((Fe, A)⊔ c c e e e e B)c ; e (G, B)) = (F , A) ⊎ e (G, (3) ((F , A)C c c e B)) = (Fe, A) C e B)c ; e (G, e (G, (4) ((Fe, A)⊎ e B))c = (Fe, A)c d e B)c ; e (G, (5) ((Fe, A) e (G, c c e e e e B)c ; e (G, B)) = (F , A) e(G, (6) ((F , A)d e B))c = (Fe, A)c ∪ e B)c ; e (G, e (G, (7) ((Fe, A)∩ c c e e e e B)c , e (G, B)) = (F , A) ∩ e (G, (8) ((F , A)∪ Proof. We only prove (1). By using a similar technique, (2)-(8) can be proved, too. e B) = (H, e C). Then C = A ∪ B, e (G, Suppose that (Fe, A)⊓ e B))c = Φ e ϕ =(Fe, A)c ⊓ e B)c . e (G, e (G, (i) if C = ϕ, then A = ϕ and B = ϕ. Hence ((Fe, A)⊓ (ii) if C ̸= ϕ, then for each e ∈ C and u ∈ U , we have if e ∈ A − B, hFe(e) (u), hG(e) if e ∈ B − A, hH(e) (u) = e (u), e h (u) ⊓ hG(e) e (u), if e ∈ A ∩ B. Fe(e) Then
if e ∈ A − B, hFec (e) (u), h (u), if e ∈ B − A, hHe c (e) (u) = e c (e) G (h c (u) ⊓ hG(e) e (u)) , if e ∈ A ∩ B. Fe(e)
e B)c = (J, e D). Then D = A ∪ B and for each e ∈ D and u ∈ U , we e (G, Again suppose that (Fe, A)c ⊔ have if e ∈ A − B, hFec (e) (u), hGec (e) (u), if e ∈ B − A, hJ(e) e (u) = h (u) ⊔ hGec (e) (u), if e ∈ A ∩ B. Fec (e) c By Theorem 2.7, we have hFec (e) (u) ⊔ hGec (e) (u) = (hFe(e) (u) ⊓ hG(e) e (u)) , i.e., hJ(e) e (u) = hH e c (e) (u) for all e ∈ A and u ∈ U . e C) and (J, e D) are the same HF SSs. It follows that ((Fe, A)⊓ e B))c = e (G, Therefore, (H, c c e e e (G, B) . (F , A) ⊔
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7
Lattice structures of hesitant fuzzy soft set
In this section, we first recall briefly the necessary definitions and notations. For convenience, we give the following axioms on an algebra Q = (X, ∨, ∧): (1) x ∨ x = x, x ∧ x = x; (2) x ∨ y = y ∨ x, x ∧ y = y ∧ x; (3) (x ∨ y) ∨ z = x ∨ (y ∨ z), (x ∧ y) ∧ z = x ∧ (y ∧ z); (4) (x ∨ y) ∧ x = x, (x ∧ y) ∨ x = x; (5) x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z), x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z), where x, y, z ∈ X. The algebra Q is called a quasilattice, if it satisfies the axioms (1),(2) and (3). If a quasilattice further satisfies the axiom (4), then it is called a lattice. If a quasilattice (or lattice ) further satisfies the axiom (5), then it is called a distributive quasilattice (or lattice ). e e For convenience, let S(U, E) denote the set of all HF SSs over U , i.e., S(U, E) = {(Fe, A)|A ⊆ E, Fe : A → HF (U )}. Then based on Theorem 3.11, we have the following property. e e , e, C e, ⊓ e } and β ∈ {∪ e, d e, ⊔ e }, then (S(U, e, ⊎ Proposition 4.1. Let α ∈ {∩ E), α, β) is a quasilattice. e , the distributive laws hold. For the operations e and ∪ e e B), (H, e C) ∈ S(U, Theorem 4.2. Let (Fe, A), (G, E). Then e e e e e e e C)); e e e e (1) ((F , A) (G, B))∪(H, C) = ((F , A) (G, B))∪((F , A) e (H, e e e e e e e e (G, B))e(H, C) = ((F , A)∪ e (G, B))e((F , A)∪ e (H, C)). (2) ((F , A)∪ Proof. we only prove (1). (2) can be proved by using a similar technique. Suppose that e B)∪ e C)) = (J, e M ) and ((Fe, A)e(G, e B))∪ e C)) = (K, e N ). Then M = e (H, e ((Fe, A)e(H, (Fe, A)e((G, A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) = N . For each e ∈ M , it follows that e ∈ A and e ∈ B ∪ C. e = Fe(e) ∩ H(e) e e (i) if e ∈ A, e ∈ / B, e ∈ C, then J(e) = K(e). e = Fe(e) ∩ G(e) e e (ii) if e ∈ A, e ∈ B, e ∈ / C, then J(e) = K(e). (iii) if e ∈ A, e ∈ B, e ∈ C, then by (4) in Theorem 2.8, we have hFe(e) (u) ∩ (hG(e) (u)) = e (u) ∪ hH(e) e e e e e e (h e (u) ∩ h e (u) ∪ (F (e) ∩ h e (u)) for all u ∈ U . It follows that J(e) = F (e) ∩ (G(e) ∪ H(e)) = F (e)
G(e)
H(e)
e ∪ (Fe(e) ∩ (H, e C)) = K(e). e (Fe(e) ∩ G(e) e e e B)∪ e C)) e (H, Thus, (J, M ) and (K, N ) are the same HF SS, i.e., (Fe, A)e((G, e B))∪ e C)). e ((Fe, A)e(H, ((Fe, A)e(G,
=
e e ) is a distributive quasilattice. Corollary 4.3. (S(U, E), e, ∪ e and d e. e have the similar properties with the operations e and ∪ The operations ∩ e e B), (H, e C) ∈ S(U, Theorem 4.4. Let (Fe, A), (G, E). Then e e e e e e e C)); e e e (H, e e (1) ((F , A)∩(G, B))d(H, C) = ((F , A)∩(G, B))d((F , A)∩ e e e e e e e e (H, C) = ((F , A)d e ((F , A)d e (G, B))∩ e (G, B))∩ e (H, C)). (2) ((F , A)d e e ) is a distributive quasilattice. Corollary 4.5. (S(U, E), e , ∪ e and ⊎ e hold. The following theorem shows that the absorption laws with respect to operations ⊓ e e B) ∈ S(U, Theorem 4.6. Let (Fe, A), (G, E). Then e B))⊎ e (G, e (Fe, A) = (Fe, A); (1) ((Fe, A)⊓ e B))⊓ e (G, e (Fe, A) = (Fe, A). (2) ((Fe, A)⊎ e B) = (J, e M) e (G, Proof. We only prove (1) since (2) can be proved similarly. Suppose that (Fe, A)⊓ e e e e e e and ((F , A) ⊓ (G, B)) C (F , A) = (K, N ). Then M = A ∪ B, N = (A ∪ B) ∩ A = A, and for all e ∈ A and u ∈ U , (i) if e ∈ / B, then hJ(e) (u) = hJ(e) e (u) = hFe(e) (u) and hK(e) e e (u) ⊓ hFe(e) (u) = hFe(e) (u). 78
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(ii) if e ∈ B, then hJ(e) (u) = hJ(e) e (u) = hFe(e) (u) ⊔ hG(e) e (u) and hK(e) e e (u) ⊓ hFe(e) (u) = (hFe(e) (u) ⊔ hG(e) e (u)) ⊓ hFe(e) (u). By (3) in Theorem 2.9, we have (hFe(e) (u) ⊔ hG(e) e (u)) ⊓ hFe(e) (u) = hFe(e) (u), i.e. hK(e) (u) = h (u). e Fe(e) e e B))⊎ e (G, e (Fe, A) = (Fe, A). Thus (K, N ) = (Fe, A), i.e. ((Fe, A)⊓ e e, ⊎ e ) is a bounded lattice. Theorem 4.7. (S(U, E), ⊓ e e, ⊎ e ) is a lattice. It is clear that Proof. By Theorem 3.11 and Theorem 4.6, we get that (S(U, E), ⊓ e e ϕ are the maximum element and the minimum element in (S(U, IeE and Φ E), respectively. e and ⊎ e , the operations ⊔ e and C e have also the following properties. Similar to ⊓ e e B) ∈ S(U, Theorem 4.8. Let (Fe, A), (G, E). Then e e e e e (G, B))C e (F , A) = (F , A); (1) ((F , A)⊔ e B))⊔ e (G, e (Fe, A) = (Fe, A). (2) ((Fe, A)C e e, C e ) is a bounded lattice. Theorem 4.9. (S(U, E), ⊔ e e e e, ⊓ e ), (S(U, e, C e ) and (S(U, Remark 4.10. It is worth noting that (S(U, E), ⊔ E), ⊎ E), α, β) are not e , e} and β ∈ {∪ e, d e }. To lattices, as the absorption laws do not hold necessarily, where α ∈ {∩ illustrate this idea, we give an example below. Example 4.11. Let U = {u1 , u2 , u3 } be the universe, E = {e1 , e2 , e3 } the set of parameters, e B) over U are given as: A = {e1 , e2 } and B = {e2 , e3 }. The HF SSs (Fe, A) and (G, { } {0.2, 0.3, 0.7, 0.8} {0.5, 0.8} {0.4, 0.5, 0.6} e F (e1 ) = , , , u1 u2 u3 } { {0.3, 0.4, 0.7} {0.5, 0.7} {0.1, 0.2, 0.4, 0.7} e , , , F (e2 ) = u1 u2 u3 { } e 2 ) = {0.5, 0.6} , {0.4, 0.8, 0.9} , {0.3, 0.5, 0.7, 0.8} , G(e u1 u2 u3 { } {0.1, 0.3, 0.5} {0.5, 0.6, 0.8} {0.6, 0.9} e G(e3 ) = , , . u1 u2 u3 e B))⊓ e M ), then M = A ∪ B = {e1 , e2 , e3 } ̸= A. So (J, e M ) ̸= e (G, e (Fe, A) = (J, (1) Let ((Fe, A)⊔ e B))⊓ e (G, e (Fe, A) ̸= (Fe, A). (Fe, A), i.e. ((Fe, A)⊔ e e e N ), then N = A ∩ B = {e2 } ̸= A, Therefore, (K, e N ) ̸= e e (2) Let ((F , A)⊎ (G, B))C(Fe, A) = (K, e e e e e e (G, B))C e (F , A) ̸= (F , A). (F , A), i.e. ((F , A)⊎ (3) If e2 ∈ A ∩ B, then (hFe(e2 ) (u1 ) ∩ hG(e e 2 ) (u1 )) ∪ hFe(e2 ) (u1 ) = ({0.3, 0.4, 0.7} ∩ {0.5, 0.6}) ∪ {0.3, 0.4, 0.7} = {0.3, 0.4, 0.5, 0.6} ∪ {0.3, 0.4, 0.7} = {0.3, 0.4, 0.5, 0.6, 0.7} ̸= e 2 )) ∪ Fe(e2 ) ̸= Fe(e2 ). Consequently, {0.3, 0.4, 0.7} = hFe(e2 ) (u1 ). It follows that (Fe(e2 ) ∩ G(e e B)) β (Fe, A) ̸= (Fe, A), where α ∈ {∩ e , e} and β ∈ {∪ e, d e }. ((Fe, A) α (G,
5
Conclusion
Considering that soft set and its existing extension models cannot deal with the situations in which the evaluations of parameters have many possible values, in this paper, we have introduced the notion of HF SS as an new extension to the HF S or the soft set model. We have also defined some basic operations on the HF SS and discussed their properties. Finally, The lattice structures of HF SS have been studied in detail based on the proposed operations.
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References [1] Molodtsov D. Soft set theory–First results. Comput Math Appl 1999; 37: 19-31. [2] Maji PK, Biswas R, Roy AR. Soft set theory, Comput Math Appl 2003; 45: 555-562. [3] Ali MI, Feng F, Liu X et al. On some new operations in soft set theory. Comput Math Appl 2009; 57: 1547-1553. [4] Ali MI, Shabir M, Naz M. Algebraic structures of soft sets associated with new operations. Comput Math Appl 2011; 61; 2647-2654. [5] Yang HL, Guo ZL Kernels and closures of soft set relations, and soft set relation mappings. Comput Math Appl 2011; 61: 651-662. [6] Aktas H, Cagman N. Soft sets and soft groups, inform Sciences 2007; 177: 2726-2735. [7] Acar U, Koyuncu F, Tanay B. Soft sets and soft rings. Comput Math Appl 2010; 59: 3458-3463. [8] Atagn AO, Sezgin A. Soft substructures of rings, fields and modules. Comput Math Appli 2011; 61: 592-601. [9] Maji PK, Roy AR, Biswas R. An application of soft sets in a decision making problem. Comput Math Appl 2002; 44: 1077-1083. [10] Cagman N, Enginoglu S. Soft set theory and uni-int decision making. Eur J Oper Res 2010; 207: 848-855. [11] Zou Y, Xiao Z. analysis approaches of soft sets under incomplete information. Knowl-Based Syst 2008; 21: 941-945. [12] Herawan T, Deris MM. A soft set approach for association rules mining. Knowl-Based Syst 2011; 24: 186-195. [13] Maji PK et al. Fuzzy soft sets. J Fuzzy Math 2001; 9: 589-602. [14] Jiang Y, Tang Y, Chen Q. An adjustable approach to intuitionistic fuzzy soft sets based decision making. Appl Math Model 2011; 35: 824-836. [15] Majumder P, Samanta SK. Generalised fuzzy soft sets. Comput Math Appl 2010; 59: 1425-1432. [16] Yang XB, Lin TY, Yang JY et al. Combination of interval-valued fuzzy set and soft set. Comput Math Appl 2009; 58: 521-527. [17] Xu W, Ma J, Wang S et al. Vague soft sets and their properties. Comput Math Appl 2010; 59: 787-794. [18] Y. Jiang, Y. Tang, Q. Chen, H. Liu, and J. Tang, Interval-valued intuitionistic fuzzy soft sets and their properties, Comput. Math. Appl. 60 (2010) 906-918. [19] Yang Y, Tan X, Meng C. The multi-fuzzy soft set and its application in decision making. Appl Math Model 2013; 37: 4915-4923. [20] Torra V. Hesitant fuzzy sets. Int J Intell Syst 2010; 25: 529-539. [21] Xu ZS, Xia MM. Distance and similarity measures for hesitant fuzzy sets. Inform Sciences 2011; 181: 2128-2138, . [22] Xia MM, Xu ZS. Hesitant fuzzy information aggregation in decision making. Int J Approx Reason 2011; 52: 395-407. [23] Farhadinia B. A Novel Method of Ranking Hesitant Fuzzy Values for Multiple Attribute DecisionMaking Problems. Int J Intell Syst 2013; 28: 752-767. [24] Wei G. Hesitant fuzzy prioritized operators and their application to multiple attribute decision making. Knowl-Based Syst 2012; 31: 176-182. [25] Zhu BZ, Xu ZS, Xia MM. Hesitant fuzzy geometric Bonferroni means. Inform Sciences 2012; 205: 72-85.
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INCLUSION PROPERTIES FOR CERTAIN SUBCLASSES OF ANALYTIC FUNCTIONS ASSOCIATED WITH BESSEL FUNCTIONS N. E. CHO1,∗ , G. MURUGUSUNDARAMOORTHY2 AND T. JANANI3
1
Department of Applied Mathematics Pukyong National University Busan 608-737, KOREA. E-mail: [email protected].
2,3
School of Advanced Sciences, VIT University Vellore - 632014, INDIA. E-mail: [email protected]; [email protected] ∗
Corresponding Author
Abstract: The purpose of the present paper is to investigate some characterization for generalized Bessel functions of first kind to be in the new subclasses G(λ, α) and K(λ, α) of analytic functions. 2010 Mathematics Subject Classification: 30C45. Keywords and Phrases: Starlike functions, Convex functions, Starlike functions of order α, Convex functions of order α, Hadamard product, Bessel function. 1. Introduction Let A be the class of functions f normalized by f(z) = z +
∞ X
an z n
(1.1)
n=2
which are analytic in the open disk U = {z : z ∈ C and |z| < 1}. As usual, we denote by S the subclass of A consisting of functions which are normalized by f(0) = 0 = f 0 (0) − 1 and also univalent in U. Denote by T [16] the subclass of A consisting of functions of the form ∞ X f(z) = z − an z n , an ≥ 0, n = 2, 3, . . . . (1.2) n=2
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Also, for functions f ∈ A given by (1.1) and g ∈ A given by g(z) = z + define the Hadamard product (or convolution) of f and g by (f ∗ g)(z) = z +
∞ X n=2
2
P∞
n=2 bn z
an bn z n , (z ∈ U).
n
, we
(1.3)
The class S ∗(α) of starlike functions of order α (0 ≤ α < 1) may be defined as 0 zf (z) ∗ S (α) = f ∈ A : < > α, z ∈ U . f(z) The class S ∗(α) and the class K(α) of convex functions of order α (0 ≤ α < 1) zf 00 (z) K(α) = f ∈ A : < 1 + 0 > α, z ∈ U f (z) = {f ∈ A : zf 0 ∈ S ∗ (α)}
were introduced by Robertson in [14]. We also write S ∗(0) = S ∗, where S ∗ denotes the class of functions f ∈ A that f(U) is starlike with respect to the origin. Further, K(0) = K is the well-known standard class of convex functions. It is an established fact that f ∈ K(α) ⇐⇒ zf 0 ∈ S ∗(α). A function f ∈ A is said to be in the class α, (z ∈ U). < (1.9) f(z) and also let K(λ, α) the subclass of functions f ∈ A which satisfy the condition z[zf 0 (z) + λz 2 f 00(z)]0 < (1.10) > α, (z ∈ U). zf 0 (z)
Also denote G ∗(λ, α) = G(λ, α) ∩ T and K∗(λ, α) = K(λ, α) ∩ T The study of the generalized Bessel function is a recent interesting topic in geometric function theory (e.g. see the work of [1, 2, 3, 4] and [9]). In this paper, due to Ramesha et al. [13], Padmanabhan [12], and motivated by the works of Srivastava et al. [17], Murugusundaramoorthy and Magesh [11],(see [6, 8, 10, 15]) and by work of Baricz [1, 2, 3, 4], we obtain sufficient conditions for function z(2 − up (z)) in G(λ, α) and K(λ, α) and connections between Rτ (A, B). Remark 1. It is of interest to note that for λ = 0, we have G(λ, α) ≡ S ∗(α) and K(λ, α) ≡ K(α) To prove the main results, we need the following Lemmas. Lemma 1. [18] A function f ∈ A belongs to the class G(λ, α) if ∞ X n=2
(n + λn(n − 1) − α)|an | ≤ 1 − α.
Lemma 2. [18] A function f ∈ A belongs to the class K(λ, α) if ∞ X n=2
n(n + λn(n − 1) − α)|an | ≤ 1 − α.
Further we can easily prove that the conditions are also necessary if f ∈ T . Lemma 3. [18] A function f ∈ T belongs to the class G ∗(λ, α) if and only if ∞ X n=2
(n + λn(n − 1) − α)|an | ≤ 1 − α.
Lemma 4. [18] A function f ∈ T belongs to the class K∗(λ, α) if and only if ∞ X n=2
n(n + λn(n − 1) − α)|an | ≤ 1 − α.
Lemma 5. [4] If b, p, c ∈ C and m 6= 0, −1, −2, . . . then the function up satisfies the recursive relation 4mu0p(z) = −cup+1(z) for all z ∈ C.
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2. Main Results Theorem 1. If c < 0 and m > 0, then z(2 − up(z)) is in G(λ, α) if λu00p (1) + [1 + 2λ]u0p (1) + (1 − α)up(1) ≤ 2(1 − α).
(2.1)
Proof. Since z(2 − up (z)) = z −
∞ X n=2
(−c/4)n−1 zn (m)n−1 (n − 1)!
and by virtue of Lemma 1, it suffices to show that ∞ X L(c, m, λ, α) = (n + λn(n − 1) − α) n=2
(−c/4)n−1 ≤ 1 − α. (m)n−1 (n − 1)!
Writing n2 = (n− 1)(n− 2) + 3(n− 1) + 1 and n = (n− 1) + 1, and by simple computation, we get ∞ X (−c/4)n−1 L(c, m, λ, α) = (n2 λ + n(1 − λ) − α) (m)n−1 (n − 1)! n=2 ∞ X
∞ X (−c/4)n−1 (−c/4)n−1 ≤ λ(n − 1)(n − 2) + (1 + 2λ) (n − 1) (m)n−1 (n − 1)! (m)n−1 (n − 1)! n=2 n=2
+ (1 − α) =λ =λ
∞ X
n=3 ∞ X n=1
∞ X n=2
(−c/4)n−1 (m)n−1 (n − 1)!
∞ ∞ X X (−c/4)n−1 (−c/4)n−1 (−c/4)n−1 + (1 + 2λ) + (1 − α) (m)n−1 (n − 3)! (m)n−1 (n − 2)! (m)n−1 (n − 1)! n=2 n=2 ∞ ∞ X X (−c/4)n+1 (−c/4)n+1 (−c/4)n+1 + (1 + 2λ) + (1 − α) (m)n+1 (n − 1)! (m)n+1 (n)! (m)n+1 (n + 1)! n=0 n=0
∞ ∞ (−c/4)2 X (−c/4)n (−c/4) X (−c/4)n =λ + (1 + 2λ) m(m + 1) n=0 (m + 2)n n! m (m + 1)n n! n=0
+ (1 − α) 2
=λ
∞ X n=0
(−c/4)n+1 (m)n+1 (n + 1)!
(−c/4) (−c/4) up+2 (1) + (1 + 2λ) up+1(1) + (1 − α)[up(1) − 1] m(m + 1) m
= λu00p (1) + (1 + 2λ)u0p (1) + (1 − α)[up (1) − 1]. By a simplification, we see that the last expression is bounded above by 1 − α if (2.1) is satisfied. By taking λ = 0, we state the following corollary.
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Corollary 1. If c < 0 and m > 0, then z(2 − up(z)) is in S ∗(α) if u0p (1) + (1 − α)up (1) ≤ 2(1 − α).
(2.2)
Remark 2. In particular, when c = −1 and b = 1, the condition (2.1) becomes 2p−2 Γ(p + 1) [λIp+2 (1) + [1 + 2λ]Ip+1 (1) + 2(1 − α)Ip (1)] ≤ 1 − α,
(2.3)
which is necessary and sufficient condition for z(2 − ζp (z 1/2) to be in G ∗ (λ, α), where up(z 1/2) = 2p Γ(p + 1)z −p/2 Ip (z 1/2). Theorem 2. If c < 0 and m > 0, then z(2 − up(z)) is in K(λ, α) if 00 0 λu000 p (1) + (1 + 5λ)up (1) + (3 + 4λ − α)up (1) + (1 − α)up (1) ≤ 2(1 − α).
(2.4)
Proof. Since z(2 − up (z)) = z −
∞ X n=2
(−c/4)n−1 zn (m)n−1 (n − 1)!
and by virtue of Lemma 2, it suffices to show that L(c, m, λ, α) =
∞ X n=2
(n3 λ + n2 (1 − λ) − nα)
(−c/4)n−1 ≤ 1 − α. (m)n−1 (n − 1)!
Writing n3 = (n − 1)(n − 2)(n − 3) + 6(n − 1)(n − 2) + 7(n − 1) + 1, n2 = (n − 1)(n − 2) + 3(n − 1) + 1 and n = (n − 1) + 1, we can rewrite the above terms as L(c, m, λ, α) ≤ λ
∞ X
n=2 ∞ X
(n − 1)(n − 2)(n − 3)
(−c/4)n−1 (m)n−1 (n − 1)!
∞ X (−c/4)n−1 (−c/4)n−1 + (1 + 5λ) (n − 1)(n − 2) + (3 + 4λ − α) (n − 1) (m)n−1 (n − 1)! (m)n−1 (n − 1)! n=2 n=2
+ (1 − α) =λ
∞ X n=4
=λ
n=2
(−c/4)n−1 (m)n−1 (n − 1)!
∞ X
(−c/4)n−1 (m)n−1 (n − 1)!
n=2
∞ ∞ X X (−c/4)n−1 (−c/4)n−1 (−c/4)n−1 + (1 + 5λ) + (3 + 4λ − α) (m)n−1 (n − 4)! (m)n−1 (n − 3)! (m)n−1 (n − 2)! n=3 n=2
+ (1 − α) ∞ X
∞ X
n=2
∞ X (−c/4)n+1 (−c/4)n+1 + (1 + 5λ) (m)n+1 (n − 2)! (m)n+1 (n − 1)! n=1
∞ ∞ X X (−c/4)n+1 (−c/4)n+1 + (3 + 4λ − α) + (1 − α) (m)n+1 (n)! (m)n+1 (n + 1)! n=0 n=0
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∞ ∞ X (−c/4)3 (−c/4)n−2 (−c/4)2 X (−c/4)n−1 =λ + (1 + 5λ) m(m + 1)(m + 2) n=2 (m + 3)n−2 (n − 2)! m(m + 1) n=1 (m + 2)n−1 (n − 1)! ∞ ∞ X (−c/4)n (−c/4) X (−c/4)n+1 + (3 + 4λ − α) + (1 − α) m (m + 1)n (n)! (m)n+1 (n + 1)! n=0 n=0 (−c/4)2 (−c/4)3 up+2 (1) up+3(1) + (1 + 5λ) m(m + 1)(m + 2) m(m + 1) (−c/4) + (3 + 4λ − α) up+1 (1) + (1 − α)[up(1) − 1] m 00 0 = λu000 p (1) + (1 + 5λ)up (1) + (3 + 4λ − α)up (1) + (1 − α)[up (1) − 1]. =λ
By a simplification, we see that the last expression is bounded above by 1 − α if (2.4) is satisfied. By taking λ = 0, we state the following corollary. Corollary 2. If c < 0 and m > 0, then z(2 − up(z)) is in ∈ K(α) if u00p (1) + (3 − α)u0p (1) + (1 − α)up (1) ≤ 2(1 − α).
(2.5)
Remark 3. We also note that the function z(2 − up(z)) is in K∗(λ, α) if and only if the condition (2.4) is satisfied. 3. Inclusion Properties Making use of the following lemma, we will study the action of the Bessel function on the classes K(λ, α). Lemma 6. [7] A function f ∈ 0. If f ∈ r + s + t or q1 , · · · , qp < r + s + t. Let f : A → B be a mapping satisfying (2.3) and q kf (xr x∗ s xt ) − f (x)r f (x)∗ s f (x)t kB ≤ θ kxkqA1 + · · · + kxkAp (2.5) for all x ∈ A. Then the mapping f : A → B is an (r, s, t)-J ∗ -homomorphism. q Proof. The proof follows from Theorem 2.4 by taking ϕ(x1 , · · · , xp ) := θ kx1 kqA1 + · · · + kxp kAp with b > 1 for the case q1 , · · · , qp > r +s+t and with b < 1 for the case q1 , · · · , qp < r +s+t.
Corollary 2.6. Let θ be a nonnegative real number and q1 , · · · , qp be positive real numbers such that q1 + · · · + qp 6= r + s + t. Let f : A → B be a mapping satisfying (2.3) and q +···+qp
kf (xr x∗ s xt ) − f (x)r f (x)∗ s f (x)t kB ≤ θkxkA1
for all x ∈ A. Then the mapping f : A → B is an (r, s, t)-J ∗ -homomorphism. q Proof. The proof follows from Theorem 2.4 by taking ϕ(x1 , · · · , xp ) := θ kx1 kqA1 · · · kxp kAp with b > 1 for the case q1 + · · · + qp > r + s + t and with b < 1 for the case q1 + · · · + qp < r + s + t.
3. Hyers-Ulam stability of (r, s, t)-J ∗ -homomorphisms: fixed point method In this section, by using the fixed point method, we prove the Hyers-Ulam stability of (r, s, t)J ∗ -homomorphisms associated with the functional equation (0.2). For a given mapping f : A → B, we define Pp Pp X Pp p j=1 xj − pxi j6=i i=1 xi i=2 xi − x1 %µ f (x1 , · · · , xp ) := f µ + f µ +f µ − µf (x1 ) p−1 p−1 p−1 i=2
for all µ ∈ T1 and all x1 , · · · , xp ∈ A. Lemma 3.1. The mapping f : A → B is a C-linear mapping if and only if %µ f (x1 , · · · , xp ) = 0 for all µ ∈ T1 and all x1 , · · · , xp ∈ A. Proof. The proof is easy and thus omitted.
In the following theorems, we will except the case p = 3. This case will be considered individually. Theorem 3.2. Let ϕ : Ap → [0, ∞) be a function with ϕ(0, · · · , 0) = 0 and p 6= 3 such that there exists an L < 1 with L (3.1) ϕ(x1 , · · · , xp ) < ϕ(kx1 , · · · , kxp ) k 2 for all x1 , · · · , xp ∈ A, where k = p−1 . Suppose that f : A → B is an odd mapping satisfying (2.4) and k%µ f (x1 , · · · , xp )kB ≤ ϕ(x1 , · · · , xp ) (3.2)
for all µ ∈ T1 and all x1 , · · · , xp ∈ A. Then there exists a unique (r, s, t)-J ∗ -homomorphism H : A → B such that L kf (x) − H(x)kB ≤ ϕ (0, x, · · · , x) (3.3) 2(1 − L) for all x ∈ A.
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Proof. We first consider the set S := {g : A → B} and introduce the generalized metric d as follows: d(g, h) = inf C ∈ R+ : kg(x) − h(x)kB ≤ Cϕ (0, x, · · · , x) . x∈A
It is easy to show that (S, d) is complete (see the proof of [35, Lemma 2.1]). Now we define the linear mapping J : S → S such that x J (g(x)) := kg k for all x ∈ A. From (3.2), we can get f (0) = 0. By letting µ = 1, x1 = 0 and x2 = · · · = xp = x in (3.2) and the fact that f (−x) = −f (x), (f is an odd mapping) and then by (3.1), we have
−2
2f (x) + (p − 1)f
≤ ϕ(0, x, · · · , x), x
p − 1 B
x k x x L
− f (x) ≤ ϕ 0, , · · · , ≤ ϕ (0, x, · · · , x)
kf k 2 k k 2 B for all x ∈ A. This means that L (3.4) d(J (f ), f ) ≤ 2 Assume that g, h ∈ S are given with d(g, h) = ε. Then we have
x x x x
kJ (g(x)) − J (h(x))kB = k g −h ≤ kεϕ 0, , · · · ,
k k B k k < Lεϕ (0, x, · · · , x) for all x ∈ A. This implies that d (J (g), J (h)) < Lε = Ld(g, h), which means J is a strictly contractive mapping. By Theorem 1.4, we have the following: (1) J has a fixed point, i.e., there exists a mapping H : A → B, such that J (H) = H. So x (3.5) H(x) = kH k for all x ∈ A. The mapping H is also the unique fixed point of J in the set M = {g ∈ S : d(f, g) < ∞}. This signifies that H is a unique mapping satisfying (3.5), moreover there exists a C ∈ (0, ∞) such that kf (x) − H(x)kB ≤ Cϕ (0, x, · · · , x) for all x ∈ A; (2) The sequence {J n (g)} converges to H, for each given g ∈ S. Thus d (J n (f ), H) → 0, as n → ∞. This implies the equality x H(x) = lim k n f n→∞ kn for all x ∈ A; 1 (3) d(g, H) ≤ 1−L d (g, J (g)), for all g ∈ M. Therefore (3.4) shows us that d(f, H) ≤
1 L d (f, J (f )) ≤ . 1−L 2(1 − L)
By this, we get the inequality (3.3).
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It follows from (3.1) that
x x xp xp
1 1 n k%µ h(x1 , · · · , xp )kB = k n %µ f , · · · , ≤ k ϕ , · · · ,
kn kn B kn kn n < L ϕ (x1 , · · · , xp ) for all x1 , · · · , xp ∈ A, in which the right-hand side tends to zero as n → ∞. Hence by Lemma 3.1, we deduce that H is C-linear. By (3.1) and (2.4), we obtain
h(xr x∗ s xt ) − h(x)r h(x)∗ s h(x)t
B x r x ∗ s x t
x r x ∗s x t (r+s+t)n
=k −f f f
f
n n n n n n k k k k k k B x x (r+s+t)n (r+s+t)n ϕ (x, · · · , x) ≤k ϕ n,··· , n < L k k for all x ∈ A. The right-hand side tends to zero as n → ∞, and so the mapping H : A → B is an (r, s, t)-J ∗ -homomorphism, as desired. Theorem 3.3. Let ϕ : Ap → [0, ∞) be a function with ϕ(0, · · · , 0) = 0 and p 6= 3 such that there exists an L < 1 with x xp 1 ϕ(x1 , · · · , xp ) < kLϕ ,··· , (3.6) k k 2 . Suppose that f : A → B is an odd mapping satisfying for all x1 , · · · , xp ∈ A, where k = p−1 (3.2) and (2.4). Then there exists a unique (r, s, t)-J ∗ -homomorphism H : A → B such that x L x kf (x) − H(x)kB ≤ ϕ 0, , · · · , (3.7) (1 − L)(p − 1) k k
for all x ∈ A. Proof. Let S be the defined set in the proof of Theorem 3.2. Consider the following generalized metric d: n x x o d(g, h) = inf C ∈ R+ : kg(x) − h(x)kB ≤ Cϕ 0, , · · · , . x∈A k k It is easy to show that (S, d) is complete (see the proof of [35, Lemma 2.1]). we define the linear mapping J : S → S such that 1 J (g(x)) := g(kx) k for all x ∈ A. By the same argument as in the proof of Theorem 3.2, we can obtain the mapping H : A → B, as the unique fixed point of J such that H(x) := lim
n→∞
for all x ∈ A. By (3.2) and (3.6), we
f (x) − 1 f (kx) ≤
k B
have x 1 L x ϕ (0, x, · · · , x) ≤ ϕ 0, , · · · , 2 (p − 1) k k
for all x ∈ A. This means that d(f, J (f )) ≤ d(f, H) ≤
1 f (k n x) kn
L (p−1) .
Hence
1 L d (f, J (f )) ≤ 1−L (1 − L)(p − 1)
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which implies that the inequality (3.7) holds. The rest of the proof is similar to the proof of Theorem 3.2.
Theorem 3.4. Let ϕ : A3 → [0, ∞) be a function such that there exists an L < 1 with ϕ(x, y, z)
l. By (4.4), for m − l > 0 and k l x, we have
1
1 1 1 m l m−l l l
k x) − f (k x)
k m f (k x) − k l f (k x) = k l k m−l f (k B
B
≤
≤
1 p−1 1 p−1
m−1 X s=l ∞ X
k −(s+1) ϕ(0, k s x, · · · , k s x)
k −(s+1) ϕ(0, k s x, · · · , k s x)
s=l
for all x ∈ A. By (4.1), the right-hand side tends to zero as l → ∞. Therefore the sequence { k1n f (k n x)} is Cauchy. Since A is a complete space, the sequence { k1n f (k n x)} is convergent and we can define for all x ∈ A, the mapping h : A → B by 1 h(x) := lim n f (k n x). n→∞ k Passing the limit n → ∞ in (4.4) and then by (4.1), we obtain (4.3). It follows from (4.1) and (3.2) that 1 k%µ f (k n x1 , · · · , k n xp )kB n→∞ k n 1 ≤ lim n ϕ (k n x1 · · · , k n xp ) = 0 n→∞ k
k%µ h(x1 , · · · , xp )kB =
lim
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for all µ ∈ T1 and all x1 , · · · , xp ∈ A. So by Lemma 3.1 we deduce that h is C-linear. By (4.2) and substituting x by k n x in (2.4), we obtain
h(xr x∗ s xt ) − h(x)r h(x)∗ s h(x)t B
1 s = lim (r+s+t)n f (k n x)r (k n x)∗ s (k n x)t − f (k n x)r f (k n x)∗ f (k n x)t B n→∞ k 1 ≤ lim (r+s+t)n ϕ (k n x, · · · , k n x) = 0 n→∞ k for all x ∈ A. Hence h(xr x∗ s xt ) = h(x)r h(x)∗ s h(x)t for all x ∈ A. Let g : A → B be another (r, s, t)-J ∗ -homomorphism satisfying (4.3). Then we have kh(x) − g(x)kB ≤ ≤ =
1 1 kf (k n x) − h(k n x)kB + n kf (k n x) − g(k n x)kB n k k 1 2 n n φ (0, k x, · · · , k x) kn p − 1 ∞ 2 X −(s+1) k ϕ (0, k s x, · · · , k s x) p − 1 s=n
for all x ∈ A. By (4.1), the right-hand side tends to zero as n → ∞, which means h is unique. Theorem 4.2. Let ϕ : Ap → [0, ∞) be a function with ϕ(0, · · · , 0) = 0 and p ≥ 4. Denote by φ a function such that ∞ X φ(x1 , · · · , xp ) := k n ϕ k −(n+1) x1 , · · · , k −(n+1) xp < ∞ (4.5) n=0 2 for all x1 , · · · , xp ∈ A, where k = p−1 . Suppose that f : A → B be an odd mapping satisfying (3.2) and (2.4). Then there exists a unique (r, s, t)-J ∗ -homomorphism h : A → B satisfying (4.3).
Proof. It follows from (3.2) that
x
x 1 x
− f (x) ≤ ϕ 0, · · · ,
kf k p−1 k k B for all x ∈ A. By the same method which was done in the proof of Theorem 4.1, we can get the unique and C-linear mapping h(x) := limn→∞ k n f k1n x satisfying (4.3). By (2.4), (4.5) and the fact that k < 1, we have
h(xr x∗ s xt ) − h(x)r h(x)∗ s h(x)t B
x r x ∗ s x t r
x x ∗ s x t (r+s+t)n
f f = lim k − f f
n→∞ kn kn kn kn kn k n B x x x x ≤ lim k (r+s+t)n ϕ n , · · · , n ≤ lim k n ϕ n , · · · , n = 0 n→∞ n→∞ k k k k for all x ∈ A. Hence h(xr x∗ s xt ) = h(x)r h(x)∗ s h(x)t for all x ∈ A.
Corollary 4.3. Let θ be a nonnegative real number and q1 , · · · , qp be positive real numbers such that q1 , · · · , qp > r + s + t or q1 , · · · , qp < 1. Let f : A → B be an odd mapping satisfying (2.5) and q k%µ f (x1 , · · · , xp )kB ≤ θ(kx1 kqA1 + · · · + kxp kAp )
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for all µ ∈ T1 and all x1 , · · · , xp ∈ A. Then there exists a unique (r, s, t)-J ∗ -homomorphism h : A → B such that p q X θkxkAj kf (x) − h(x)kB ≤ 2 |1 − k qj −1 | j=2
for all x ∈ A. q Proof. Defining ϕ(x1 , · · · , xp ) = θ kx1 kqA1 + · · · + kxp kAp and applying Theorem 4.1 for the case q1 , · · · , qp > r + s + t, and Theorem 4.2 for the case q1 , · · · , qp < 1, we get the result.
Theorem 4.4. Let ϕ : A2 → [0, ∞) be a function with ϕ(0, 0) = 0. Denote by φ a function such that ∞ X φ(x, y) := 2−(n+1) ϕ (2n x, 2n y) < ∞ n=0
for all x, y ∈ A. Suppose that f : A → B is an odd mapping satisfying kf (µx + µy) + f (µx − 2µy) + f (µy − µx) − µf (x)kB ≤ ϕ(x, y), r ∗s t
(4.6)
∗s
kf (x x x ) − f (x)r f (x) f (x)t kB ≤ ϕ(x, x) for all µ ∈ such that
T1
and all x, y ∈ A. Then there exists a unique
(4.7)
(r, s, t)-J ∗ -homomorphism
h:A→B
kf (x) − h(x)kB ≤ φ(0, x)
(4.8)
for all x ∈ A. Proof. From (4.6), it follows that
1
f (2x) − f (x) ≤ 1 ϕ(0, x)
2
2 B for all x ∈ A. Using the same method as in the proof of Theorem 4.1, we conclude that the mapping h(x) := limn→∞ 21n f (2n x) is a unique (r, s, t)-J ∗ -homomorphism satisfying (4.8). Theorem 4.5. Let ϕ : A2 → [0, ∞) be a function with ϕ(0, 0) = 0. Denote by φ a function such that ∞ X φ(x, y) := 2n ϕ 2−(n+1) x, 2−(n+1) y < ∞, n=0
lim 2(r+s+t)n ϕ 2−n x, 2−n x = 0
n→∞
for all x, y ∈ A. Suppose that f : A → B is an odd mapping satisfying (4.6) and (4.7). Then there exists a unique (r, s, t)-J ∗ -homomorphism h : A → B satisfying (4.8). Proof. The proof is similar to the proof of Theorem 4.4.
Corollary 4.6. Let θ be a nonnegative real number and q1 , q2 be positive real numbers such that q1 , q2 < 1 or q1 , q2 > r + s + t. Let f : A → B be an odd mapping satisfying kf (µx + µy) + f (µx − 2µy) + f (µy − µx) − µf (x)kB ≤ θ(kxkqA1 + kykqA2 ), kf (xr x∗ s xt ) − f (x)r f (x)∗ s f (x)t kB ≤ θ(kxkqA1 + kxkqA2 ) for all µ ∈ T1 and all x, y ∈ A. Then there exists a unique (r, s, t)-J ∗ -homomorphism h : A → B such that θkxkqA2 kf (x) − h(x)kB ≤ |2 − 2q2 | for all x ∈ A.
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Proof. Defining ϕ(x, y) = θ kxkqA1 + kykqA2 and applying Theorem 4.4 for the case q1 , q2 < 1, and Theorem 4.5 for the case q1 , q2 > r + s + t, we get the result. Theorem 4.7. Let ϕ : A3 → [0, ∞) be a function. Denote by φ a function such that φ(x, y, z) :=
∞ X
2−n ϕ (2n x, 2n y, 2n z) < ∞
n=1
for all x, y, z ∈ A. Suppose that f : A → B is an odd mapping satisfying (3.9) and (3.10). Then there exists a unique (r, s, t)-J ∗ -homomorphism h : A → B such that kf (x) − h(x)kB ≤ φ(x, 0, 0)
(4.9)
for all x ∈ A.
Proof. By (3.9), we get 21 f (2x) − f (x) B ≤ 12 ϕ(2x, 0, 0) for all x ∈ A. The same method as in the proof of Theorem 4.1, leads us to the unique (r, s, t)-J ∗ -homomorphism h(x) := limn→∞ 21n f (2n x)satisfying (4.9). Theorem 4.8. Let ϕ : A3 → [0, ∞) be a function. Denote by φ a function such that φ(x, y, z) :=
∞ X
2n ϕ 2−n x, 2−n y, 2−n z < ∞,
n=0
lim 2(r+s+t)n ϕ 2−n x, 2−n x, 2−n x = 0
n→∞
for all x, y, z ∈ A. Suppose that f : A → B is an odd mapping satisfying (3.9) and (3.10). Then there exists a unique (r, s, t)-J ∗ -homomorphism h : A → B satisfying (4.9). Proof. The proof is similar to the proof of Theorem 4.7.
Corollary 4.9. Let θ be a nonnegative real number and q1 , q2 , q3 be positive real numbers such that q1 , q2 , q3 < 1 or q1 , q2 , q3 > r + s + t. Let f : A → B be an odd mapping satisfying
f µ x + y + z + f µ x + z − 3y + f µ x + y − 3z
2 2 2
y+z−x q1 q2 q3 +f µ − µf (x) ≤ θ kxk + kyk + kzk A A A ,
2 B kf (xr x∗ s xt ) − f (x)r f (x)∗ s f (x)t kB ≤ θ kxkqA1 + kxkqA2 + kxkqA3 for all µ ∈ T1 and all x, y, z ∈ A. Then there exists a unique (r, s, t)-J ∗ -homomorphism h : A → B such that 2 q1 kf (x) − h(x)kB ≤ θkxkqA1 |2 − 2q1 | for all x ∈ A. Proof. Defining ϕ(x, y, z) = θ kxkqA1 + kykqA2 + kzkqA3 and applying Theorem 4.7 for the case q1 , q2 , q3 < 1, and Theorem 4.8 for the case q1 , q2 , q3 > r + s + t, we get the result. Remark 4.10. The obtained results in this paper, could be more remarkable and interesting. In other words, as a consequence including simpler and better results, one can set q1 = · · · = qp = q, as well as r = s = t = 1 (or a fixed n ∈ N) in all the statements. Furthermore, all the obtained results do also hold for (r, s, t)-J ∗ -derivations similarly. The reader can directly verify this point just with a little difference in details.
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Acknowledgments C. Park and D. Y. Shin were supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2012R1A1A2004299) and (NRF-2010-0021792), respectively.
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Differential subordinations obtained by using a generalization of Marx-Strohhäcker theorem Georgia Irina Oros1 , Gheorghe Oros2 , Alina Alb Lupa¸s3 , Vlad Ionescu4 1,2,3 University of Oradea, Department of Mathematics Str. Universita˘¸tii, No.1, 410087 Oradea, Romania 1 [email protected], 2 [email protected], 3 [email protected], 4 [email protected] Abstract In [1] and [6] Marx and Strohhäcker have proved that if f ∈ A is a convex function, then it has the property of starlikeness of order 12 . In [5, Theorem 9.5.6], P. T. Mocanu extended this result to the class A2 for a convex function of order − 12 . In this paper we extend the results proven by Marx and Strohhäcker and by P. T. Mocanu and we’ll prove that, if the function f ∈ An , n ≥ 3, is a close-to-convex function, then it is starlike of order 12 .
Keywords: Analytic function, univalent function, integral operator, close-to-convex function. 2000 Mathematical Subject Classification: 30C45, 30A20, 34A40.
1
Introduction and preliminaries
Let U be the unit disc of the complex plane U = {z ∈ C : |z| < 1}. Let H(U ) be the class of holomorphic functions in U . Also, let An = {f ∈ H(U ) : f (z) = z + an+1 z n+1 + . . . , z ∈ U }, with A1 = A. Let S = {f ∈ A : f univalent in U } be the class of holomorphic and univalent functions in the open unit disc U , with conditions f (0) = 0, f 0 (0) = 1, that is the holomorphic and univalent functions with the following power series development f (z) = z + a2 z 2 + . . . , z ∈ U.o n 00 (z) Denote by K = f ∈ A : Re zff 0 (z) + 1 > 0, z ∈ U the class of normalized convex functions in U and by n o 0 f (z) C = f ∈ A : ∃ ϕ ∈ K, Re ϕ the class of normalized close-to-convex functions in U . 0 (z) > 0, z ∈ U An equivalent formulation would involve the h (not necessarily normalized) ³ existence o ´ n of a starlike function 0 00 (z) (z) 1 1 such that Re zfh(z) > 0, z ∈ U. We consider K − 2γ + 1 > − 2γ , z ∈ U, γ ≥ 1 . = f ∈ An : Re zff 0 (z) n o 0 (z) Let S ∗ = f ∈ A : Re zff (z) > 0, z ∈ U denote the class of starlike functions in U , and n o 0 (z) S ∗ (α) = f ∈ A : Re zff (z) > α, z ∈ U , denote the class of starlike functions of order α, with 0 ≤ α < 1. In order to prove our original results, we use the following lemmas: Lemma 1.1 [2], [3], [4, Theorem 2.3.i, p. 35] Let ψ : C3 × U → C, satisfy the condition Re ψ(is, t) ≤ 0, z ∈ U, for s, t ∈ R, t ≤ − n2 (1 + s2 ). If p(z) = 1 + pn z n + pn+1 z n+1 + . . . satisfies Re [p(z), zp0 (z); z] > 0, then Re p(z) > 0, z ∈ U. More general forms of this lemma can be found in [6]. 0 Lemma 1.2 [5, h Theorem i 4.6.3, p. 84] The function f ∈ A, with f (z) 6= 0, z ∈ U , is close-to-convex if and R θ2 zf 00 (z) only if θ1 Re 1 + f 0 (z) dθ > −π, z = reiθ , for all θ1 , θ2 , with 0 ≤ θ1 < θ2 ≤ 2π and r ∈ (0, 1).
Definition 1.1 [4, Definition ½ 2.2.b, p. 21] We denote ¾ by Q the set of functions q that are analytic and injective on U \ E(q), where E(q) = ζ ∈ ∂U : lim q(z) = ∞ and are such that q 0 (ζ) 6= 0, for ζ ∈ ∂U \ E(q). The set z→ζ
E(q) is called exception set. The subclass of Q for which f (0) = a is denoted by Q(a). 135
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Definition 1.2 [4, Definition 2.3.a, p. 27] Let Ω be a set in C, q ∈ Q and n be a positive integer. The class of admissible functions Ψn [Ω, q] consists of those functions ψ : C × U → C that satisfy the admissibility condition (A)
ψ(r, s, t) 6∈ Ω h i ¡ ¢ 00 (ζ) + 1 , z ∈ U , ζ ∈ ∂U \ E(q), m ≥ n. We write Ψ1 [Ω, q] where r = q(ζ), s = mζq 0 (ζ), R st + 1 ≥ mRe ζqq0 (ζ) as Ψ[Ω, q]. In the special case when Ω is a simply connected domain, Ω 6= C, and h is a conformal mapping of U onto Ω, we denote this class by Ψn [h, q]. If ψ : C2 × U → C, then the admissibility condition (A) reduces to ψ(q(ζ), mζq 0 (ζ); z) 6∈ Ω,
(A0 )
where z ∈ U , ζ ∈ ∂U \ E(q) and m ≥ n. If ψ : C × U → C, then the admissibility condition (A) reduces to (A00 )
ψ(q(ζ); z) 6∈ Ω
where z ∈ U and ζ ∈ ∂U \ E(q). Definition 1.3 [4, p. 36] Let f and F be members of H(U ). The function f is said to be subordinate to F , written f ≺ F or f (z) ≺ F (z), if there exists a function w analytic in U , with w(0) = 0 and |w(z)| < 1, and f (0) = F (0) and f (U ) ⊂ F (U ). Definition 1.4 [4, p. 16] Let ψ : C3 × U → C and let h be univalent in U . If p is analytic in U and satisfies the (second-order) differential subordination (i) ψ(p(z), zp0 (z), z 2 p00 (z); z) ≺ h(z), then p is called a solution of the differential subordination. The univalent function q is called a dominant of the solutions of the differential subordination, or more simply a dominant, if p ≺ q for all p satisfying (i). A dominant qe that satisfies qe ≺ q for all dominant q of (i) is said to be the best dominant of (i). (Note that the best dominant is unique up to a rotation of U ). If we require the more restrictive condition p ∈ [a, n], then p will be called an (q, n)-solution, q an (a, n)dominant, and qe the best (a, n)-dominant,
Lemma 1.3 [4, Theorem 2.3.c, p. 30] Let ψ ∈ Ψn [h, q] with q(0) = a. If p ∈ H[a, n], ψ(p(z), zp0 (z), z 2 p00 (z); z) is analytic in U , and (ii) ψ(p(z), zp0 (z), z 2 p00 (z); z) ≺ h(z), then p(z) ≺ q(z), z ∈ U. ´ ³ 00 (z) + 1 > 0, then Theorem 1.1 [1, 6, Marx-Strohhacker] If f ∈ A and satisfy the condition Re zff 0 (z) £ ¡ ¢¤ 0 (z) (a) Re zff (z) > 12 i.e., f ∈ S ∗ 12 and 1 (b) Re f (z) z > 2 , for z ∈ U .
In [5] has shown that the odd and convex functions of order − 12 are starlike functions of order 12 . ³ 00 ´ (z) +1 Theorem 1.2 [5, Marx-Strohhscker, Theorem 9.5.6, p. 218] If f ∈ A2 and satisfy the condition Re zff 0 (z) £ ¡ ¢¤ 0 (z) > − 12 , then Re zff (z) > 12 i.e., f ∈ S ∗ 12 , for z ∈ U .
2
Main results
We’ll extend the theorem Marx-Strohhäcker for the functions f ∈ An , n ≥ 3, which are close-to-convex functions.
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Theorem 2.1 Let n ≥ 3, γ ≥ 1, f ∈ An , satisfy the condition Re
1 zf 00 (z) +1>− , 0 f (z) 2γ
(2.1)
0
(z) then Re zff (z) > 12 .
Proof. According to Lemma 1.2 we obtain ¸ ∙ Z θ2 Z θ2 Z θ2 1 π zf 00 (z) 1 1 2π dθ ≥ = − > −π, λ ≥ 1. Re 1 + 0 − dθ = − dθ = − (θ2 − θ1 ) > − f (z) 2γ 2γ 2γ 2γ γ θ1 θ1 θ1
(2.2)
From (2.2) we have f ∈ C, hence it is univalent. 0 (z) Let p(z) = 2 · zff (z) − 1. Since f ∈An and f is close-to-convex function (univalent), the function p is analytic in U and p(0) = 1. A simple computation leads to zf 0 (z) p(z) + 1 = . (2.3) 2 f (z) By differentiating (2.3), we obtain zf 00 (z) zf 0 (z) zp0 (z) =1+ 0 − . p(z) + 1 f (z) f (z)
(2.4)
Using (2.3) in (2.4), we have zf 00 (z) p(z) + 1 zp0 (z) + = 1+ 0 . 2 p(z) f (z) h i zp0 (z) 1 + p(z)+1 , which is equivalent to Using (2.1) in (2.5), we obtain Re p(z)+1 > − 2γ 2
(2.5)
¸ zp0 (z) 1 p(z) + 1 + + > 0. Re 2 p(z) + 1 2γ ∙
(2.6)
0
zp (z) 1 s 1 Let ψ : C2 × U → C, ψ(p(z), zp0 (z); z) = p(z)+1 + p(z)+1 + 2γ , where ψ(r, s) = r+1 2 2 + r+1 + 2γ . Then (2.6) is equivalent to Re ψ(p(z), zp0 (z); z) > 0, z ∈ U. ³ ´ t 1 In order to prove Theorem 2.1, we use Lemma 1.1. For that we calculate Re ψ(is, t) = Re is+1 2 + 1+is + 2γ ³ ´ t(1−is) n(1+s2 ) (1−n)γ+1 1 t 1 1 1 1−n 1 = Re is+1 + + ≤ 0. Since n ≥ 3, = 12 + 1+s 2 + 2γ ≤ 2 − 2(1+s2 ) + 2γ = 2 1+s2 2γ 2 + 2γ = 2γ 0
(z) γ ≥ 1. Now, using Lemma 1.1, we get that Re p(z) > 0, z ∈ U , i.e., Re zff (z) > 12 , z ∈ U.
Remark 2.1 Each of the four conditions in the Marx-Strohhäcker theorem can be rewritten in terms of subordination. This leads to the following equivalent form of the theorem. Theorem 2.2 Let n ≥ 3, γ ≥ 1, f ∈ An , satisfies the condition
zf 00 (z) f 0 (z)
Theorem 2.3 Let n ≥ 3, γ ≥ 1, f ∈ An satisfies the conditions ¶ µ 00 1 zf (z) +1 >− Re f 0 (z) 2γ and Re
1 zf 0 (z) > f (z) 2
+1≺
1−( γ1 +1)z , 1+z
then
zf 0 (z) f (z)
≺
1 1+z .
(2.7)
(2.8)
1 then Re f (z) z > 2 , for z ∈ U.
Proof. In order to prove Theorem 2.1, we saw that, if f ∈An , n ≥ 3 and satisfies the condition (2.1) or (2.7), then the function f is close-to-convex (univalent).
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Let p(z) = 2fz(z) − 1. Since f ∈ An , n ≥ 3 and f is close-to-convex function (univalent) then the function p is analytic in U and p(0) = 1. A simple computation leads to f (z) p(z) + 1 = . 2 z
(2.9)
zf 0 (z) zp0 (z) = − 1. p(z) + 1 f (z)
(2.10)
By differentiating (2.9), we obtain
Using (2.8) in (2.10), we have Re
µ
zp0 (z) 1 + p(z) + 1 2 0
¶
> 0, z ∈ U.
(2.11)
zp (z) s + 12 , where ψ(r, s) = 12 + 1+r . Then (2.11) is equivalent to Let ψ : C2 × U → C, ψ(p(z), zp0 (z); z) = 1+p(z) Re ψ(p(z), zp0 (z); z) > 0, z ∈ U. h i t In order to prove Theorem 2.1, we use Lemma 1.1. For that we calculate Re ψ(is, t) = Re 12 + 1+is = h i 2 n(1+s ) t 1 1−n Re 12 + t(1−is) < 0, since n ≥ 3. Therefore, by applying Lemma 1.1 we = 12 + 1+s 2 ≤ 2 − 2(1+s2 ) = 1+s2 2
1 conclude that p satisfies Re p(z) > 0. This is equivalent to Re f (z) z > 2 , z ∈ U. For 0 < γ < 1, n ≥ 3, Theorem 2.2 can be written as the following corollary. h 00 i 0 (z) (z) Corollary 2.4 Let n ≥ 3, 0 < γ < 1, f ∈ An satisfy the conditions Re zff 0 (z) + 1 > − γ2 and Re zff (z) > 12 , 1 then Re f (z) z > 2 , z ∈ U.
Theorem 2.5 Let n ≥ 3, γ ≥ 1, f ∈ An satisfy differential subordination ³ ´ 1 − γ1 + 1 z zf 00 (z) +1≺ , f 0 (z) 1+z and
then
1 zf 0 (z) ≺ f (z) 1+z f (z) z
≺
1 1+z ,
(2.12)
(2.13)
z ∈ U.
Proof. Consider
2f (z) − 1. (2.14) z Since f ∈ An , and f is close-to-convex function (univalent) then the function p is analytic in U , and p(0) = 1. By differentiating (2.14), we obtain zf 0 (z) zp0 (z) +1= . (2.15) p(z) + 1 z p(z) =
Using (2.13) in (2.15), we have 1 zp0 (z) +1≺ . p(z) + 1 1+z
(2.16)
1 Since Re 1+z ≥ 12 , differential subordination (2.16) is equivalent to
Re
µ
1 zp0 (z) + p(z) + 1 2
¶
> 0,
z ∈ U.
(2.17)
zp0 (z) p(z)+1
+ 12 , then (2.17) becomes Re ψ(p(z), zp0 (z); z) > 0, z ∈ U. ³ ´ t In order to prove Theorem 2.5, we use Lemma 1.3. For that we calculate Re ψ(is, t) = Re 1+is + 12 = h i −n(1+s2 ) 1 t 1 1 1−n Re t(1−is) + 2 1+s 2 = 1+s2 + 2 ≤ 2(1+s2 ) + 2 = 2 < 0. Using Definition 1.2, we have ψ ∈ Ψn [h, q]. Therefore Let ψ : C2 × U → C, ψ(p(z), zp0 (z); z) =
by Lemma 1.3, we conclude that p(z) ≺ q(z), i.e.,
f (z) z
≺
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4
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for z ∈ U.
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³ 00 ´ p (z) 1 Theorem 2.6 If f ∈ An , n ≥ 3, γ ≥ 1 and satisfy the condition Re zff 0 (z) + 1 > − 2γ , then Re f 0 (z) > 12 , for z ∈ U. p Proof. Consider p(z) = 2 f 0 (z) − 1, z ∈ U. Since f ∈ An , n ≥ 3, and f is close-to-convex function (univalent) then the function p is analytic in U and p(0) = 1. A simple computation leads to
By differentiating (2.18), we have
2zp0 (z) 1+p(z)
p(z) + 1 p 0 = f (z). 2 +1= Re
If we let ψ : C2 × U → C, ψ(p(z), zp0 (z)) =
∙
zf 00 (z) f 0 (z)
(2.18)
+ 1. Using (2.1), we have
¸ 1 2zp0 (z) +1+ > 0. 1 + p(z) 2γ
2zp0 (z) 1+p(z)
(2.19)
1+2γ 2γ ,
then (2.19) becomes Re ψ(p(z), zp0 (z)) > 0. ³ ´ 2t In order to prove Theorem 2.6, we use Lemma 1.1. For that, we calculate Re ψ(is, t) = Re 1+is + 1+2γ 2γ ³ ´ −n(1+s2 ) 1+2γ 1+2γ 2t = Re 2t(1−is) + + 1+2γ = −2γn+1+2γ = 2γ(1−n)+1 ≤ 0, since n ≥ 3, γ ≥ 1. = 1+s 2 2 + 1+s 2γ 2γ ≤ 1+s2 2γ 2γ p 2γ 1 Using Lemma 1.1, we have Re p(z) > 0, i.e., Re f 0 (z) > 2 . For 0 < γ < 1, n ≥ 3, Theorem 2.6 can be written as the following corollary. ³ 00 ´ p (z) Corollary 2.7 If f ∈ An , n ≥ 3, 0 < γ < 1, satisfy the condition Re zff 0 (z) + 1 > − γ2 , then Re f 0 (z) > 12 , for z ∈ U . +
In differential subordination language Theorem 2.6 can be written as Theorem 2.8 If f ∈ An , n ≥ 3, γ ≥ 1, and satisfy the differential subordination ³ ´ 1 1 − 00 + 1 z γ zf (z) + 1 ≺ , f 0 (z) 1+z p 1 then f 0 (z) ≺ 1+z , for z ∈ U .
(2.20)
References [1] A. Marx, Untersuchungen über schlichte Abbildungen, Math. Ann., 107(1932-1933), 40-67. [2] S. S. Miller, P. T. Mocanu, Differential subordinations and univalent functions, Michig. Math. J., 28(1981), 157-171. [3] S. S. Miller, P. T. Mocanu, Differential subordinations and inequalities in the complex plane, J. of Diff. Eqs., 2(1987), 192-211. [4] S. S. Miller, P. T. Mocanu, Differential subordinations. Theory and applications, Pure and Applied Mathematics, Marcel Dekker, Inc., New York, Basel, 2000. [5] P. T. Mocanu, T. Bulboac˘ a, G. S ¸ t. S˘ al˘ agean, Teoria Geometrica˘ a Func¸tiilor Univalente, Casa C˘ ar¸tii de S ¸ tiin¸ta˘, Cluj-Napoca, 1999. [6] E. Strohhäcker, Beiträge zür Theorie der schhlichten Functionen, Math. Z., 37(1933), 356-380.
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A finite difference method for Burgers’ equation in the unbounded domain using artificial boundary conditions
∗
Quan Zheng†, Yufeng Liu, Lei Fan College of Sciences, North China University of Technology, Beijing 100144, China
Abstract: This paper discusses the numerical solution of one-dimensional Burgers’ equation in the infinite domain. The original problem is converted by Hopf-Cole transformation to the heat equation in the infinite domain, the latter is reduced to an equivalent problem in a finite computational domain with two artificial integral boundary conditions, a finite difference method is constructed for last problem by the method of reduction of order, and therefore the numerical solution of Burgers’ equation is obtained. The method is proved and verified to be uniquely solvable, unconditionally stable and convergent with the order 2 in space and the order 3/2 in time for solving the heat equation as well as Burgers’ equation in the computational domain. Keywords: Burgers’ equation; infinite domain; Hopf-Cole transformation; Artificial boundary condition; Finite difference method
1
Introduction When an analytic solution is not available, or the analytic one is not suitable to be used, a
numerical method is necessary for solving partial differential equations. Therefore, several kinds of exterior problems in the areas of heat transfer, fluid dynamics and other applications were solved numerically by using artificial boundary conditions [1-5]. The artificial boundary methods were established on bounded computational domains for various problems of heat equation on unbounded domains and the feasibility and effectiveness of the methods were shown by the numerical examples [6, 7]. Moreover, for the heat equation in ∗ †
The research is supported by National Natural Science Foundation of China (11471019). E-mail: [email protected] (Q. Zheng).
1
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a semi-unbounded domain [−1, ∞) × [0, ∞), by using an artificial integral boundary condition 1 ux (0, t) = − √ π
∫ 0
t
uλ (0, λ) √ dλ, t−λ
Sun and Wu [8] firstly proved that the finite difference scheme is uniquely solvable, unconditionally stable and convergent with the order 2 in space and the order 3/2 in time under an energy norm. Wu and Zhang [9] also obtained the high-order artificial boundary conditions for the heat equation in unbounded domains, but only proved that the reduced initial-boundary-value problems were stable. Furthermore, Han, Wu and Xu [10] started to consider the nonlinear Burgers’ equation in the unbounded domain as follows: wt + wwx − νwxx = F (x, t), w(x, 0) = f (x), w(x, t) → 0, where ν =
1 Re ,
∀(x, t) ∈ R × (0, T ],
(1.1)
∀x ∈ R,
(1.2)
when |x| → +∞,
∀t ∈ [0, T ],
(1.3)
Re is the Reynolds number, and the given functions F and f are sufficiently
smooth with compact supports supp{F (x, t)} ⊂ [xl , xr ] × [0, T ] and supp{f (x)} ⊂ [xl , xr ]. They obtained nonlinear artificial boundary conditions, constructed a nonlinear difference method with no theoretical convergence analysis, and supported it by numerical examples. Recently, Sun and Wu [11] introduced a function transformation to reduce nonlinear Burgers’ equation to a linear initial boundary value problem, deduced a linear finite difference scheme, and also proved that the finite difference scheme is uniquely solvable, unconditionally stable and convergent with the order 2 in space and 3/2 in time. In this paper, we consider the problem (1.1)-(1.3) with F ≡ 0 and convert it into an initial value problem of heat equation by using Hopf-Cole transformation in the following. Let ∫ ω(x, t) = −
∞
w(y, t)dy,
∀(x, t) ∈ R × (0, T ],
x
we obtain 1 ωt + ωx2 − νωxx = 0, 2 ∫ ∞ ω(x, 0) = − f (y)dy, ∀x ∈ R, x
ω(x, t) → 0,
when |x| → +∞,
∀t ∈ [0, T ].
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Let u = exp(−ω/2ν) − 1, then we have the initial value problem of heat equation: ut − νuxx = 0,
∀(x, t) ∈ R × (0, T ], ∫ ∞ 1 u(x, 0) = ϕ(x) := exp( f (y)dy) − 1, 2ν x u(x, t) → 0,
when |x| → +∞,
(1.4) (1.5)
∀t ∈ [0, T ],
(1.6)
where the sufficiently smooth given function ϕ(x) has compact support supp{ϕ(x)} ⊂ [xl , xr ]. By using artificial linear integral boundary conditions similar to that in [8], we reduce the problem (1.4)-(1.6) to a problem in the bounded computational domain: ut − νuxx = 0,
∀(x, t) ∈ [xl , xr ] × [0, T ],
(1.7)
u(x, 0) = ϕ(x), ∀x ∈ [xl , xr ], ∫ t uλ (xl , λ) 1 √ ux (xl , t) = √ dλ, ∀t ∈ [0, T ], πν 0 t−λ ∫ t 1 uλ (xr , λ) √ ux (xr , t) = − √ dλ, ∀t ∈ [0, T ]. πν 0 t−λ
(1.8) (1.9) (1.10)
In section 2, we construct a finite difference scheme for solving the problem (1.7)-(1.10). Then a new solution of Burgers’ equation is obtained and the difficulty for solving the nonlinear problem is avoided. In section 3, we prove that the finite difference scheme is uniquely solvable, unconditionally stable and convergent with the order 2 in space and 3/2 in time. In section 4, a numerical example confirms the stability and convergence of the finite difference method.
2
The construction of the difference scheme In order to construct the finite difference method, the bounded computational domain is
divided into an M × N uniform mesh. Let h = (xr − xl )/M , xi = xl + ih for 0 ≤ i ≤ M , τ = T /N , tn = nτ for 0 ≤ n ≤ N , r =
ντ , h2
and uni be the numerical solution of u(x, t) at (xi , tn ).
Introduce the notations: 1 uni− 1 = (uni + uni−1 ), 2 2
δx uni− 1 = 2
1 n (u − uni−1 ), h i
n− 21
ui
1 = (uni + un−1 ), i 2
1 n 1 (ui − un−1 ), δx2 uni = 2 (uni+1 − 2uni + uni−1 ), i τ h v v u M u M u ∑ u ∑ n n t n 2 ∥u ∥A = h (ui− 1 ) , ∥δx u ∥ = th (δx uni− 1 )2 . n− 12
δt ui
=
i=1
2
i=1
2
3
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Lemma 2.1 Suppose f (t) ∈ C 2 [0, tn ], then ∫ tn ∫ n ∑ √ 3 f (tk ) − f (tk−1 ) tk dt 1 dt ′ √ − | ≤ (20 2 − 23) max |f ′′ (t)|τ 2 . | f (t) √ 0≤t≤tn τ 12 tn − t tn − t tk−1 0 k=1
Proof
Lemma 2.1 is proved by using
√ √ tn − t − ( tkτ−t tn − tk−1 +
t−tk−1 √ tn τ
− tk ) = 81 (tn −
ξk )− 2 (t − tk−1 )(tk − t) to correct (2.2) and thereupon (2.1) in [8], as corrected in [12]. 3
By introducing a new variable v =
∂u ∂x
to reduce the order of heat equation, the problem
(1.7)-(1.10) is equivalent to the problem of first-order differential equations: ∂u ∂v =ν , ∂x ∂x
∀(x, t) ∈ [xl , xr ] × [0, T ],
(2.1)
∂u = 0, ∂x
∀(x, t) ∈ [xl , xr ] × [0, T ],
(2.2)
v−
u(x, 0) = ϕ(x), xl ≤ x ≤ xr , ∫ t ∂u(xl , λ) 1 1 √ √ dλ, v(xl , t) = ∂λ πν 0 t−λ ∫ t 1 ∂u(xr , λ) 1 √ v(xr , t) = − √ dλ. ∂λ πν 0 t−λ
(2.3) (2.4) (2.5)
Define the grid functions: Uin = u(xi , tn ),
Vin = v(xi , tn ),
0 ≤ i ≤ M,
n ≥ 0.
Using Lemma 2.1, it follows from (2.5) that n ∫ 1 ∑ tk ∂u(xr , λ) dλ n √ VM = − √ ∂λ πν tn − λ k=1 tk−1 ∫ n tk k − U k−1 3 1 ∑ UM dλ M √ = −√ + O(τ 2 ) τ πν tn − λ tk−1 k=1 n 3 2 ∑ k k−1 (UM − UM )an−k + O(τ 2 ) = −√ πν k=1
∑ 3 2 k 0 n (an−k−1 − an−k )UM − an−1 UM ] + O(τ 2 ), = − √ [a0 UM − πν n−1
n = 1, 2, · · · .
k=1
Therefore, we have n− 12
VM
∑ 3 1 n−1 2 n− 1 k− 1 0 n = (VM ] + O(τ 2 ), + VM ) = − √ [a0 UM 2 − (an−k−1 − an−k )UM 2 − an−1 UM 2 πν n−1 k=1
and similarly, n− 1 V0 2
∑ 3 1 2 n− 1 k− 1 = (V0n−1 + V0n ) = √ [a0 U0 2 − (an−k−1 − an−k )U0 2 − an−1 U00 ] + O(τ 2 ). 2 πν n−1 k=1
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Using Taylor expansion, we have n− 1
n− 1
n− 1
δt Ui− 12 − νδx Vi− 1 2 = pi− 12 , 2
2
n− 1
n− 1
n− 1
Vi− 1 2 − δx Ui− 12 = qi− 12 , 2
1 ≤ i ≤ M,
n ≥ 1,
(2.6)
2
2
1 ≤ i ≤ M,
n ≥ 1,
(2.7)
2
0 ≤ i ≤ M,
Ui0 = ϕ(xi ),
(2.8)
∑ 1 2 n− 1 k− 1 = √ [a0 U0 2 − (an−k−1 − an−k )U0 2 − an−1 U00 ] + sn− 2 , πν
n ≥ 1,
(2.9)
∑ 1 2 n− 1 k− 1 0 = − √ [a0 UM 2 − (an−k−1 − an−k )UM 2 − an−1 UM ] + tn− 2 , πν
n ≥ 1,
(2.10)
n− 1 V0 2
n−1 k=1
n− 21
VM
n−1 k=1
where n− 1
|pi− 12 | ≤ c(τ 2 + h2 ), 2
1
n− 1
|qi− 12 | ≤ c(τ 2 + h2 ),
1 ≤ i ≤ M,
n ≥ 1,
(2.11)
2
3
|tn− 2 | ≤ cτ 2 ,
1
3
|sn− 2 | ≤ cτ 2 ,
n ≥ 1,
(2.12)
and c is a constant. Thus, we construct a difference scheme for (2.1)-(2.5) in the following: n− 1
n− 1
δt ui− 12 − νδx vi− 12 = 0, 2
n− 1
n− 1
vi− 12 − δx ui− 12 = 0,
1 ≤ i ≤ M,
u0i = ϕ(xi ),
0 ≤ i ≤ M,
2
n− 12
v0
1 ≤ i ≤ M,
n ≥ 1,
(2.13)
2
n ≥ 1,
(2.14)
2
(2.15)
∑ 2 k− 1 n− 1 (an−k−1 − an−k )u0 2 − an−1 u00 ], = √ [a0 u0 2 − πν n−1
n ≥ 1,
(2.16)
k=1
n− 1 vM 2
∑ 2 k− 1 n− 1 = − √ [a0 uM 2 − (an−k−1 − an−k )uM 2 − an−1 u0M ]. n ≥ 1, πν n−1
(2.17)
k=1
Theorem 2.2 The difference scheme (2.13)-(2.17) is equivalent to the following (2.18)-(2.22): 0 ≤ i ≤ M,
u0i = ϕ(xi ),
1 n− 1 n− 1 n− 1 (δt ui− 12 + δt ui+ 12 ) − νδx2 ui 2 = 0, 2 2 2 n− 12
δt u 1
1 ≤ i ≤ M − 1,
(2.18) n ≥ 1,
∑ 2ν 2 n− 1 k− 1 n− 1 [ √ (a0 u0 2 − (an−k−1 − an−k )u0 2 − an−1 u00 ) − δx u 1 2 ], h πν 2
(2.19)
n−1
+
2
n ≥ 1, (2.20)
k=1
n− 1
δt uM −21 + 2
2ν 2 n− 1 [ √ (a0 uM 2 − h πν
n−1 ∑
k− 12
(an−k−1 − an−k )u0
k=1
n− 1
− an−1 u0M ) + δx uM −21 ],
n ≥ 1, (2.21)
2
5
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where am = √ Proof
1 1 √ =√ √ √ , tm+1 + tm τ ( m + 1 + m)
m = 0, 1, 2, · · · .
(2.22)
Multiplying (2.13) by 21 h and using (2.14) we obtain n− 21
vi
n− 21
vi
n− 1
= δx ui− 12 + 2
n− 1
= δx ui+ 12 − 2
h n− 1 δt ui− 12 , 2ν 2
h n− 1 δt ui+ 12 , 2ν 2
1 ≤ i ≤ M, 0 ≤ i ≤ M − 1,
n ≥ 1,
(2.23)
n ≥ 1,
(2.24)
From (2.23) and (2.24) for i from 1 to M − 1 we obtain n− 1
δx ui− 12 + 2
h h n− 1 n− 1 n− 1 δt ui− 12 = δx ui+ 12 − δt ui+ 12 , 2ν 2ν 2 2 2
1 ≤ i ≤ M − 1,
n ≥ 1,
or 1 n− 1 n− 1 n− 1 (δt ui− 12 + δt ui+ 12 ) − νδx2 ui 2 = 0, 2 2 2
1 ≤ i ≤ M − 1,
n ≥ 1,
which is (2.19). When i = 0, from (2.16) and (2.24), we know that √ n−1 ∑ h n− 1 2 ν k− 1 n− 1 n− 12 √ [a0 u0 (an−k−1 − an−k )u0 2 − an−1 u00 ] = νδx u 1 2 − δt u 1 2 . − 2 π 2 2 k=1
Dividing by h/2 on the both sides we obtain (2.20). Similarly, when i = M , from (2.17) and (2.23), we know that √ n−1 ∑ 2 ν h n− 1 k− 1 n− 1 n− 1 (an−k−1 − an−k )uM 2 − an−1 u0M ] = νδx uM −21 + δt uM −21 , − √ [a0 uM 2 − 2 π 2 2 k=1
Dividing by h/2 on the both sides we obtain (2.21).
The difference scheme (2.18)-(2.21) can be sorted as the following: 1 1 1 1 n−1 ( −r)uni+1 +(1+2r)uni +( −r)uni−1 = ( +r)un−1 +( +r)un−1 i+1 +(1−2r)ui i−1 , 1 ≤ i ≤ M − 1, 2 2 2 2 (2.25) √ √ 4 r n 4 r (1 + 2r + √ )u0 + (1 − 2r)un1 = (1 − 2r − √ )un−1 + (1 + 2r)un−1 1 π π 0 √ n−1 √ 4 rτ ∑ 8 rτ k−1 k + √ (an−k−1 − an−k )(u0 + u0 ) + √ an−1 u00 , (2.26) π π k=1 √ √ 4 r 4 r (1 + 2r + √ )unM + (1 − 2r)unM −1 = (1 − 2r − √ )un−1 + (1 + 2r)un−1 M −1 π π M √ n−1 √ 4 rτ ∑ 8 rτ √ + √ (an−k−1 − an−k )(ukM + uk−1 ) + an−1 u0M . (2.27) M π π k=1
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3
The error estimate of the difference scheme
Lemma 3.1 For any F = {F1 , F2 , F3 , · · · }, we have n l−1 n ∑ ∑ 1 ∑ 2 [a0 Fl − (al−k−1 − al−k )Fk ]Fl ≥ √ Fl , 2 tn l=1
k=1
n = 1, 2, · · · ,
l=1
where am is defined in (2.22). Proof
Let bm = am−1 − am =
1 √ √1 ( √ τ m+ m−1
−
√
1 √ ), m+1+ m
m ≥ 1, then bm > 0, and
n l−1 ∑ ∑ [a0 Fl − (al−k−1 − al−k )Fk ]Fl
= ≥
l=1 n ∑ l=1 n ∑
a0 Fl2 − a0 Fl2 −
l=1
=
=
n ∑ l=1 n ∑
a0 Fl2 − a0 Fl2 −
l=1
≥
n ∑
k=1 n ∑ l−1 ∑
(am−1 − am )Fl−m Fl
l=1 m=1 n ∑ l−1 ∑
1 2
l=1 m=1
n l−1 1∑∑
2 1 2
a0 Fl2 − (
1∑∑ bm Fl2 2 n
2 bl−m Fm −
l=1 m=1 n n ∑ ∑
2 bl−m Fm −
bm )
n ∑
l−1
l=1 m=1 n ∑ l−1 ∑
m=1 l=m+1
n−1 ∑
m=1
l=1
2 bm (Fl−m + Fl2 )
1 2
bm Fl2
l=1 m=1
Fl2
l=1
n ∑ 1 1 1 √ = [ √ − √ (1 − √ )] Fl2 τ τ n + n − 1 l=1
≥
n 1 ∑ 2 √ Fl . 2 tn
l=1
Lemma 3.2 Suppose {uni } be the solution of n− 1
n− 1
n− 1
δt ui− 12 − νδx vi− 12 = Pi− 12 , 2
2
n− 1
n− 1
n− 1
vi− 12 − δx ui− 12 = Qi− 12 , 2
2
n ≥ 1,
(3.1)
1 ≤ i ≤ M,
n ≥ 1,
(3.2)
2
0 ≤ i ≤ M,
u0i = ϕ(xi ), n− 1 v0 2
1 ≤ i ≤ M,
2
(3.3)
∑ 1 2 n− 1 k− 1 = √ [a0 u0 2 − (an−k−1 − an−k )u0 2 − an−1 u00 ] + S n− 2 , πν n−1
n ≥ 1,
(3.4)
k=1
n− 1 vM 2
∑ 1 2 n− 1 k− 1 = − √ [a0 uM 2 − (an−k−1 − an−k )uM 2 − an−1 u0M ] + T n− 2 . πν n−1
n ≥ 1,
(3.5)
k=1
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where Supp{ϕ(x)} ⊂ [x0 , xM ], then
2T ∥un ∥2A ≤ exp( 4−τ )·
+2τ Proof
1 1− τ4
∑n
l=1 (∥P
{∥u0 ∥2A +
l− 21
√ πνtn 2 τ l− 12
∥2A + ∥Q
∑n
l=1 [(T
∥2A )},
l− 12 2 )
1
+ (S l− 2 )2 ] (3.6)
n = 1, 2, · · · .
n− 1
n− 1
Multiplying (3.1) by 2ui− 12 and multiplying (3.2) by 2vi− 12 , then adding the results, 2
2
we have 1 n 2 τ [(ui− 1 ) 2
n− 21
≤ h2 (ui
n− 21 n− 12 vi
n− 1
n− 1 n− 1
− (un−1 )2 ] + 2(vi− 12 )2 = h2 (ui i− 1 2
n− 12
vi
2
n− 12
n− 12
− ui−1 vi−1 ) + 12 (u
n− 12 i− 21
)2 + 2(P
n− 1
n− 1
n− 1
n− 1
− ui−12 vi−12 ) + 2ui− 12 Pi− 12 + 2vi− 12 Qi− 12 2
n− 12 i− 12
n− 21 i− 12
)2 + 12 (v
)2 + 2(Q
2
n− 21 i− 12
2
2
)2 ,
1 ≤ i ≤ M, n ≥ 1.
(3.7)
Multiplying the above inequality by τ h and summing up for i from 1 to M , we obtain n− 1 n− 12
1
(∥un ∥2A − ∥un−1 ∥2A ) + 2τ ∥v n− 2 ∥2A ≤ 2τ (uM 2 vM
n− 21 n− 12 v0 )
− u0
1
1 1 τ τ + ∥un− 2 ∥2A + ∥v n− 2 ∥2A 2 2
1
+2τ ∥P n− 2 ∥2A + 2τ ∥Qn− 2 ∥2A ,
n ≥ 1.
(3.8)
1
Noticing τ2 ∥un− 2 ∥2A ≤ τ4 (∥un ∥2A + ∥un−1 ∥2A ), thus τ + (∥ul ∥2A + ∥ul−1 ∥2A ) 4 l− 12 2 l− 12 2 +2τ ∥P ∥A + 2τ ∥Q ∥A , l = 1, 2, . . . , n. l− 1 l− 1
l− 21 l− 12 v0 )
∥ul ∥2A − ∥ul−1 ∥2A ≤ 2τ (uM 2 vM 2 − u0
Summing up for l from 1 to n, we have ∥un ∥2A
≤
∥u0 ∥2A
+ 2τ
n ∑
l− 1 l− 1
l− 12 l− 12 v0 )
(uM 2 vM 2 − u0
l=1 n−1 n ∑ 1 1 τ∑ l 2 τ + ∥un ∥2A + ∥u ∥A + 2τ (∥P l− 2 ∥2A + ∥Ql− 2 ∥2A ). 4 2 l=0
l=1
Substituting (3.4) and (3.5) into the above inequality, and using Lemma 3.1, we have ∥un ∥2A ≤
∑ l− 1 l− 1 ∑ 1 1 1 l− 12 l− 12 0 2 2 2 [∥u ∥ + 2τ (u v − u v ) + 2τ (∥P l− 2 ∥2A + ∥Ql− 2 ∥2A ) A 0 0 τ M M 1− 4 +
τ 2
n−1 ∑
n
n
l=1
l=1
∥ul ∥2A ]
l=0
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∑ 1 2τ 2 ∑ l− 1 k− 1 l− 1 0 2 ) [a0 uM 2 − (al−k−1 − al−k )uM 2 ]uM 2 τ ∥u ∥A + τ · (− √ 1− 4 1− 4 πν n
=
l−1
l=1
+
+ ≤
2τ 1 − τ4 2τ 1 − τ4
n ∑
l− 1
1
uM 2 T l− 2 −
l=1 n ∑
l− 12
u0
1
S l− 2 +
l=1
k=1
2 2τ · (√ ) 1 − τ4 πν 2τ 1 − τ4
n ∑
l− 12
[a0 u0
−
l−1 ∑
k− 21
(al−k−1 − al−k )u0
l− 12
]u0
k=1
l=1
n n−1 ∑ 1 1 2τ ∑ l 2 (∥P l− 2 ∥2A + ∥Ql− 2 ∥2A ) + ∥u ∥A 4−τ l=1
l=0
n ∑
1 2τ 2 1 2 τ l− 1 0 2 · √ (uM 2 )2 + τ ∥u ∥A − τ · √ τ (√ 1− 4 1− 4 1− 4 πν 2 tn πνtn l=1 √ n n 2 πνtn ∑ l− 1 2 2τ 1 ∑ l− 21 2 √ √ (T 2 ) ) − · (u0 ) · + 2 1 − τ4 πν 2 tn l=1 l=1 √ n n ∑ ∑ 1 1 τ 2 πνtn l− + (u0 2 )2 + (S l− 2 )2 ) τ (√ 1− 4 2 πνtn l=1
n ∑
l− 1
(uM 2 )2
l=1
l=1
n n−1 2τ ∑ 2τ ∑ l 2 l− 12 2 l− 12 2 + (∥P ∥ + ∥Q ∥ ) + ∥u ∥A A A 1 − τ4 4−τ l=1 l=0 √ n 1 τ πνtn ∑ l− 1 2 1 0 2 ∥u ∥ + [(T 2 ) + (S l− 2 )2 ] ≤ A τ τ 1− 4 1− 4 2 l=1
2τ + 1 − τ4
n ∑
1 (∥P l− 2 ∥2A
+
1 ∥Ql− 2 ∥2A )
l=1
n−1 2τ ∑ l 2 ∥u ∥A , + 4−τ
n = 1, 2, · · · .
l=0
Using Gronwall’s lemma, we can obtain (3.6).
Theorem 3.3 The difference scheme (2.18)-(2.22) is uniquely solvable. Proof
By Theorem 2.2, it suffices to prove that the difference scheme (2.13)-(2.17) is solvable u-
niquely. When initial value is homogeneous, by Lemma 3.2, we have ∥un ∥2A = 0, n = 1, 2, · · · .
Theorem 3.4 Let {uni |0 ≤ i ≤ M, n ≥ 1} be the solution of (2.18)-(2.22), then ∥un ∥2A ≤ Proof
2T exp( 4−τ ) 0 2 τ ∥u ∥A , 1− 4
n = 1, 2, · · · .
(3.9)
From Theorem 2.2, it suffices to prove that (3.9) hold for the difference scheme (2.13)-
(2.17). Therefore, (3.9) follows directly from Lemma 3.2.
4,3 Theorem 3.5 Suppose (1.4)-(1.6) have solution u(x, t) ∈ Cx,t (R × [0, T ]). Let {uni } be the
solution of (2.18)-(2.22), and let u ˜ni = Uin − uni , then ∥˜ un ∥2A ≤
3 2T CT √ ( πνT + 4) exp( )(τ 2 + h2 )2 , 4−τ 4−τ
n = 1, 2, · · · , [T /τ ],
(3.10)
where C is a constant independent of τ and h.
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Proof
Subtracting (2.13)-(2.17) from (2.6)-(2.10), respectively, we obtain the error equations: n− 1
n− 1
n− 1
δt u ˜i− 12 − νδx v˜i− 12 = pi− 12 , 2
2
n− 1
n− 1
n− 1
v˜i− 12 − δx u ˜i− 12 = qi− 12 , 2
2
n− 12
n ≥ 1,
(3.11)
1 ≤ i ≤ M,
n ≥ 1,
(3.12)
2
0 ≤ i ≤ M,
u ˜0i = 0, v˜0
1 ≤ i ≤ M,
2
(3.13)
∑ 1 2 n− 1 k− 1 = √ [a0 u ˜0 2 − (an−k−1 − an−k )˜ u0 2 − an−1 u ˜00 ] + sn− 2 , πν n−1
n ≥ 1,
(3.14)
k=1
n− 21
v˜M
∑ 1 2 n− 1 k− 1 = − √ [a0 u ˜M 2 − (an−k−1 − an−k )˜ uM 2 − an−1 u ˜0M ] + tn− 2 , πν n−1
n ≥ 1.
(3.15)
k=1
Using Lemma 3.2 and applying (2.11), (2.12) and (3.13), we obtain √ n 1 2T 1 πνtn ∑ l− 1 2 0 2 n 2 · {∥˜ u ∥ + ∥˜ u ∥A = exp( )· τ [(t 2 ) + (sl− 2 )2 ] A τ 4−τ 1− 4 2 l=1
+2τ
n ∑
1
1
(∥pl− 2 ∥2 + ∥q l− 2 ∥2 )}
l=1
≤
3 CT √ 2T ( πνT + 4) exp( )(τ 2 + h2 )2 , 4−τ 4−τ
n = 1, 2, · · · , [T /τ ].
Theorem 3.5 shows that the convergence order of (2.18)-(2.21) is 2 in space and 3/2 in time for the problem (1.7)-(1.10). Finally, the numerical solution of Burgers’ equation is obtained by win = −
ν uni+1 − uni−1 , h 1 + uni
(3.16)
which keeps the corresponding unique solvability, unconditional stability and convergence.
4
The numerical example 8νx(x −9) For the problem of Burgers’ equation with an initial condition f (x) = − (x 2 −9)2 +1 in the 2
support [xl , xr ] = [−3, 3], the exact solution is w(x, t) = 2ν
∫3
(x−ξ)2 x−ξ 2 2 −3 2νt (ξ −9) exp(− 4νt )dξ ∫3 (x−ξ)2 1+ √1 (ξ 2 −9)2 exp(− 4νt )dξ 2 πνt −3
√1 2 πνt
. The
numerical solutions are obtained by the proposed scheme, then the convergence order w.r.t h is shown in Table 1, and the convergence order w.r.t τ is shown in Table 2. Table 1. Convergence w.r.t. h of the problem for T = 1, ν = 0.1, τ = 0.01 and τ = h4/3 respectively. M 50 100 200 400 800
N 100 100 100 100 100
L∞ -error 2.2705e-3 6.0651e-4 1.6444e-4 5.0024e-5 3.0569e-5
order — 1.9044 1.8830 1.7169 0.7106
L2 -error 2.0737e-3 5.5643e-4 1.4962e-4 4.5653e-5 1.9714e-5
order — 1.8979 1.8949 1.7125 1.2115
N 9 22 54 137 345
L∞ -error 3.1455e-3 7.6893e-4 1.8419e-4 4.5577e-5 1.1295e-5
order — 2.0324 2.0617 2.0148 2.0126
L2 -error 2.5729e-3 6.5174e-4 1.6620e-4 4.1607e-5 1.0393e-5
order — 1.9810 1.9714 1.9980 2.0012
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Table 2. Convergence w.r.t. τ of the problem for T = 1, ν = 0.1, h = 0.002 and h = τ 3/4 respectively. N 20 40 80 160 320
5
M 3000 3000 3000 3000 3000
L∞ -error 1.0398e-3 3.6910e-4 1.0386e-4 2.6518e-5 1.5322e-5
order — 1.4942 1.8294 1.9696 0.7914
L2 -error 2.1342e-4 6.2138e-5 1.8884e-5 6.3713e-6 2.6822e-6
order — 1.7801 1.7183 1.5675 1.2482
M 95 159 267 450 757
L∞ -error 8.7265e-4 2.9735e-4 1.0258e-4 3.5936e-5 1.2623e-5
order — 1.5532 1.5354 1.5132 1.5094
L2 -error 7.2610e-4 2.6197e-4 9.3238e-5 3.2868e-5 1.1614e-5
order — 1.4708 1.4904 1.5042 1.5008
Conclusions In this works, a new finite difference method for Burgers’ equation in the unbounded domain
is presented by (2.18), (2.25)-(2.27) and (3.16) succinctly. The inequality in Lemma 2.1 is slightly stronger than Lemma 1 in [8]. Lemma 3.2 is proved by using Gronwall’s lemma, but for heat equation in the semi-infinite domain, similar Lemma 4 in [8], i.e. Lemma 3.2.4 in [12], was incorrectly proved by not using Gronwall’s lemma, and the lemma can be modified and proved as Lemma 3.2. Finally, the proposed method is clearly proved and verified to be uniquely solvable, unconditionally stable and convergent with the order 2 in space and the order 3/2 in time to solve Burgers’ equation in the unbounded domain.
References [1] B. Enquist, A. Majda, Absorbing boundary conditions for numerical simulation of waves, Math. Comput. 31 (1977) 629-651. [2] K. Feng, Asymptotic radiation conditions for reduced wave equations, J. Comp. Math. 2 (1984) 130-138. [3] D.-H. Yu, Natural Boundary Integral Method and Its Applications, Beijing/Dordrecht/New York/London: Kluwer Academic Publisher/Science Press, 2002. [4] J.M. Strain, Fast adaptive methods for the free-space heat equation, SIAM J. Sci. Comput. 15 (1992) 185-206. [5] D. Givoli, Numerical Methods for Problem in Infinite Domains, Elsevier, Amsterdam, 1992. [6] H.-D. Han, Z.-Y. Huang, A class of artificial boundary conditions for heat equation in unbounded domains, Comput. Math. Appl. 43 (2002) 889-900. [7] H.-D. Han, Z.-Y. Huang, Exact and approximating boundary conditions for the parabolic problems on unbounded domains, Comput. Math. Appl. 44 (2002) 655-666. [8] X.-N. Wu, Z.-Z. Sun, Convergence of difference scheme for heat equation in unbounded domains using artificial boundary conditions, Appl. Numer. Math. 50 (2004) 261-277. [9] X.-N. Wu, J.-W. Zhang, High-order local absorbing boundary conditions for heat equation in unbounded domains, J. Comput. Math. 29 (2011) 74-90. [10] H.-D. Han, X.-N. Wu, Z.-L. Xu, Artificial boundary method for Burgers’ equation using nonlinear boundary conditions, J. Comput. Math. 24 (2006) 295-304. [11] Z.-Z. Sun, X.-N. Wu, A difference scheme for Burgers equation in an unbounded domain, Appl. Math. Comput. 209 (2009) 285-304. [12] H.-D. Han, X.-N. Wu, Artificial Boundary Method, Beijing: Tsinghua University Press/Springer Press, 2012.
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Barnes-type Peters polynomials associated with poly-Cauchy polynomials of the second kind Dae San Kim Department of Mathematics, Sogang University Seoul 121-742, Republic of Korea [email protected]
Taekyun Kim Department of Mathematics, Kwangwoon University Seoul 139-701, Republic of Korea [email protected]
Takao Komatsu Graduate School of Science and Technology, Hirosaki University Hirosaki 036-8561, Japan [email protected]
Hyuck In Kwon Department of Mathematics, Kwangwoon University Seoul 139-701, Republic of Korea [email protected]
Sang-Hun Lee Division of General Education, Kwangwoon University Seoul 139-701, Republic of Korea [email protected] MR Subject Classifications: 05A15, 05A40, 11B68, 11B75, 65Q05
Abstract In this paper, by considering Barnes-type Peters polynomials of the second kind as well as poly-Cauchy polynomials of the second kind, we define and investigate the mixed-type polynomials of these polynomials. From the properties of Sheffer sequences of these polynomials arising from umbral calculus, we derive new and interesting identities.
1
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1
Introduction
In this paper, we consider the polynomials sb(k) b(k) b(k) n (x) = s n (x|λ; µ) = s n (x|λ1 , . . . , λr ; µ1 , . . . , µr ) called the Barnes-type Peters of the second kind and poly-Cauchy of the second kind mixed-type polynomials, whose generating function is given by µj r ∞ Y X (1 + t)λj tn x (k) , (x|λ , . . . , λ ; µ , . . . , µ ) Lif − ln(1 + t) (1 + t) = s b 1 r 1 r k n λj 1 + (1 + t) n! n=0 j=1 (1) where λ1 , . . . , λr , µ1 , . . . , µr ∈ C with λ1 , . . . , λr 6= 0. Here, Lif k (x) (k ∈ Z) is the polyfactorial function ([8]) defined by ∞ X xm . Lif k (x) = m!(m + 1)k m=0 (k)
(k)
(k)
(k)
When x = 0, sbn = sbn (0) = sbn (0|λ; µ) = sbn (0; λ1 , . . . , λr ; µ1 , . . . , µr ) are called the the Barnes-type Peters of the second kind and poly-Cauchy of the second kind mixed-type numbers. Recall that the Barnes-type Peters polynomials of the second kind, denoted by sbn (x|λ1 , . . . , λr ; µ1 , . . . , µr ), are given by the generating function as µj ∞ r X Y tn (1 + t)λj x (1 + t) = s b (x|λ , . . . , λ ; µ , . . . , µ ) . n 1 r 1 r 1 + (1 + t)λj n! n=0 j=1 If r = 1, then sbn (x|λ; µ) are the Peters polynomials of the second kind. Peters polynomials were mentioned in [12, p.128] and have been investigated in e.g. [7]. (k) The poly-Cauchy polynomials of the second kind, denoted by b cn (x) ([6, 9]), are given by the generating function as ∞ X tn x b c(k) (x) . Lif k − ln(1 + t) (1 + t) = n n! n=0 The generalized Barnes-type Euler polynomials En (x|λ1 , . . . , λr ; µ1 , . . . , µr ) are defined by the generating function µj r ∞ Y X tn 2 xt e = E (x|λ , . . . , λ ; µ , . . . , µ ) . n 1 r 1 r λj t 1 + e n! n=0 j=1 If µ1 = · · · = µr = 1, then En (x|λ1 , . . . , λr ) = En (x|λ1 , . . . , λr ; 1, . . . , 1) are called (r) the Barnes-type Euler polynomials. If further λ1 = · · · = λr = 1, then En (x) = En (x|1, . . . , 1; 1, . . . , 1) are called the Euler polynomials of order r. In this paper, by considering Barnes-type Peters polynomials of the second kind as well as poly-Cauchy polynomials of the second kind, we define and investigate the mixedtype polynomials of these polynomials. From the properties of Sheffer sequences of these polynomials arising from umbral calculus, we derive new and interesting identities. 2
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2
Umbral calculus
Let C be the complex number field and let F be the set of all formal power series in the variable t: ) ( ∞ X ak k t ak ∈ C . (2) F = f (t) = k! k=0
∗
Let P = C[x] and let P be the vector space of all linear functionals on P. hL|p(x)i is the action of the linear functional L on the polynomial p(x), and we recall that the vector space operations on P∗ are defined by hL + M |p(x)i = hL|p(x)i + hM |p(x)i, hcL|p(x)i = c hL|p(x)i, where c is a complex constant in C. For f (t) ∈ F, let us define the linear functional on P by setting hf (t)|xn i = an , (n ≥ 0). (3) In particular,
k n t |x = n!δn,k
(n, k ≥ 0),
(4)
where δn,k is the Kronecker’s symbol. P hL|xk i k For fL (t) = ∞ t , we have hfL (t)|xn i = hL|xn i. That is, L = fL (t). The map k=0 k! L 7→ fL (t) is a vector space isomorphism from P∗ onto F. Henceforth, F denotes both the algebra of formal power series in t and the vector space of all linear functionals on P, and so an element f (t) of F will be thought of as both a formal power series and a linear functional. We call F the umbral algebra and the umbral calculus is the study of umbral algebra. The order O f (t) of a power series f (t)(6 = 0) is the smallest integer k k for which the coefficient of t does not vanish. If O f (t) = 1, then f (t) is called a delta series; if O f (t) = 0, then f (t) is called an invertible series. For f (t), g(t) ∈ F with O f (t) = 0, there exists a unique sequence sn (x) (deg sn (x) = n) such
= 1 and O g(t) that g(t)f (t)k |sn (x) = n!δn,k , for n, k ≥ 0. Such a sequence sn (x) is called the Sheffer sequence for g(t), f (t) which is denoted by sn (x) ∼ g(t), f (t) , (see [1, 4-12]). For f (t), g(t) ∈ F and p(x) ∈ P, we have hf (t)g(t)|p(x)i = hf (t)|g(t)p(x)i = hg(t)|f (t)p(x)i and
∞ X
tk f (t)|xk , f (t) = k! k=0
p(x) =
∞ X
tk |p(x)
k=0
(5)
xk k!
(6)
([12, Theorem 2.2.5]). Thus, by (6), we get tk p(x) = p(k) (x) =
dk p(x) dxk
and eyt p(x) = p(x + y).
(7)
Sheffer sequences are characterized in the generating function ([12, Theorem 2.3.4]). Lemma 1 The sequence sn (x) is Sheffer for g(t), f (t) if and only if ∞
X sk (y) 1 ¯ eyf (t) = tk k! g f¯(t) k=0
(y ∈ C) ,
3
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where f¯(t) is the compositional inverse of f (t). For sn (x) ∼ g(t), f (t) , we have the following equations ([12, Theorem 2.3.7, Theorem 2.3.5, Theorem 2.3.9]): f (t)sn (x) = nsn−1 (x) (n ≥ 1), n X 1 D ¯ −1 ¯ j n E j g f (t) f (t) |x x , sn (x) = j! j=0 n X n sn (x + y) = sj (x)pn−j (y) , j j=0
(8) (9) (10)
where pn (x) = g(t)sn (x). Assume that pn (x) ∼ 1, f (t) and qn (x) ∼ 1, g(t) . Then the transfer formula ([12, Corollary 3.8.2]) is given by n f (t) x−1 pn (x) (n ≥ 1). qn (x) = x g(t) For sn (x) ∼ g(t), f (t) and rn (x) ∼ h(t), l(t) , assume that sn (x) =
n X
Cn,m rm (x) (n ≥ 0) .
m=0
Then we have ([12, p.132]) Cn,m
3
1 = m!
*
h g
+ m n f¯(t) l f¯(t) x . f¯(t)
(11)
Main results (k)
From the definition (1), sbn (x|λ1 , . . . , λr ; µ1 , . . . , µr ) is the Sheffer sequence for the pair g(t) =
µ r Y 1 + e λj t j j=1
eλj t
1 Lif k (−t)
and f (t) = et − 1.
So, sb(k) n (x|λ1 , . . . , λr ; µ1 , . . . , µr ) ∼
µ r Y 1 + eλj t j j=1
e λj t
! 1 , et − 1 . Lif k (−t)
(12)
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3.1
Explicit expressions
Let (n)j = n(n − 1) · · · (n − j + 1) (j ≥ 1) with (n)0 = 1. The (signed) Stirling numbers of the first kind S1 (n, m) are defined by (x)n =
n X
S1 (n, m)xm .
m=0
Theorem 1 Let λµ =
Pr
j=1
λj µj . Then, we have
sb(k) n (x|λ1 , . . . , λr ; µ1 , . . . , µr ) n X m X P (−1)l ml − rj=1 µj S1 (n, m)Em−l (x + λµ|λ1 , . . . , λr ; µ1 , . . . , µr ) =2 k (l + 1) m=0 l=0 n X n X n (k) = S1 (l, j)b sn−l xj l j=0 l=j n−j l n X X X n l (k) = S1 (n − l, j)b ci sbl−i (λ1 , . . . , λr ; µ1 , . . . , µr )xj l i j=0 l=0 i=0 n X n (k) = sbn−l (λ1 , . . . , λr ; µ1 , . . . , µr )b cl (x) , l l=0 n X n (k) = sbl (x|λ1 , . . . , λr ; µ1 , . . . , µr )b cn−l . l l=0
(13) (14)
(15) (16) (17)
Proof. Since µ r Y 1 + eλj t j j=1
e λj t
1 sb(k) (x|λ1 , . . . , λr ; µ1 , . . . , µr ) ∼ (1, et − 1) Lif k (−t) n
(18)
and (x)n ∼ (1, et − 1) ,
(19)
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we have sb(k) n (x|λ1 , . . . , λr ; µ1 , . . . , µr ) µj r Y eλj t = Lif k (−t)(x)n λj t 1 + e j=1 µj n r X Y e λj t Lif k (−t)xm = S1 (n, m) λj t 1 + e m=0 j=1 µj X n r m X Y e λj t (−1)l tl m = S1 (n, m) x λj t k 1 + e l!(l + 1) m=0 j=1 l=0 r µj n m l m Y X X (−1) l eλj t = S1 (n, m) xm−l λj t k (l + 1) 1 + e m=0 j=1 l=0 µj n m r X X Pr (−1)l ml λµt Y 2 − j=1 µj S1 (n, m) =2 e xm−l λj t k (l + 1) 1 + e m=0 j=1 l=0 n m m l X X Pr (−1) l Em−l (x + λµ|λ1 , . . . , λr ; µ1 , . . . , µr ) . = 2− j=1 µj S1 (n, m) k (l + 1) m=0 l=0 So, we get (13). By (9) with (12), we get E D −1 j n ¯ ¯ g f (t) f (t) |x * r + µj Y j n (1 + t)λj = Lif k − ln(1 + t) ln(1 + t) x 1 + (1 + t)λj j=1 + * r µj ∞ Y X tl n (1 + t)λj S1 (l, j) x Lif k − ln(1 + t) j! = λj 1 + (1 + t) l! j=1 l=j * + µj n r λj X Y n (1 + t) = j! S1 (l, j) Lif k − ln(1 + t) xn−l λ j l 1 + (1 + t) j=1 l=j + * n ∞ i X X n (k) t n−l = j! S1 (l, j) sbi x i! l i=0 l=j n X n (k) = j! S1 (l, j)b sn−l . l l=j
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On the other hand, D E −1 j n ¯ ¯ g f (t) f (t) |x * r + µj n X Y n−l n (1 + t)λj = j! S1 (l, j) Lif k − ln(1 + t) x 1 + (1 + t)λj l j=1 l=j + * r µj n n−l λj X X Y n n − l (k) (1 + t) n−l−i = j! S1 (l, j) b ci x λ j 1 + (1 + t) l i i=0 j=1 l=j + * ∞ n n−l X X n n − l (k) X tm n−l−i = j! S1 (l, j) b ci sbm (λ1 , . . . , λr ; µ1 , . . . , µr ) x m! l i m=0 i=0 l=j n X n−l X n n−l (k) = j! S1 (l, j)b ci sbn−l−i (λ1 , . . . , λr ; µ1 , . . . , µr ) . l i l=j i=0 Thus, we obtain sb(k) n (x)
=
n X n X n j=0 l=j
l
(k)
S1 (l, j)b sn−l xj
n−j l n X X X n l (k) = S1 (n − l, j)b ci sbl−i (λ1 , . . . , λr ; µ1 , . . . , µr )xj , l i j=0 l=0 i=0
which are the identities (14) and (15).
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Next, sb(k) n (y|λ1 , . . . , λr ; µ1 , . . . , µr ) =
*∞ X
ti (k) sbi (y|λ1 , . . . , λr ; µ1 , . . . , µr ) xn
+
i! * i=0 + µj r Y (1 + t)λj = Lif k − ln(1 + t) (1 + t)y xn λj 1 + (1 + t) j=1 * r + µj Y (1 + t)λj = Lif k − ln(1 + t) (1 + t)y xn λj 1 + (1 + t) j=1 + * r µj X ∞ Y (1 + t)λj tl n (k) = b cl (y) x 1 + (1 + t)λj l! j=1 l=0 + * µj n r X Y n (k) (1 + t)λj n−l = b cl (y) x λj 1 + (1 + t) l j=1 l=0 *∞ + n X n (k) X ti n−l = b cl (y) sbi (λ1 , . . . , λr ; µ1 , . . . , µr ) x i! l i=0 l=0 n X n (k) = b cl (y)b sn−l (λ1 , . . . , λr ; µ1 , . . . , µr ) . l l=0
Thus, we obtain (16). Finally, we obtain that + i t (k) sb(k) sbi (y|λ1 , . . . , λr ; µ1 , . . . , µr ) xn n (y|λ1 , . . . , λr ; µ1 , . . . , µr ) = i! * i=0 + µj r Y (1 + t)λj Lif k − ln(1 + t) (1 + t)y xn = λj 1 + (1 + t) j=1 + * µj r Y (1 + t)λj (1 + t)y xn = Lif k − ln(1 + t) λ j 1 + (1 + t) j=1 * + ∞ X tl n = Lif k − ln(1 + t) sbl (y|λ1 , . . . , λr ; µ1 , . . . , µr ) x l! l=0 n X n−l E n D = sbl (y|λ1 , . . . , λr ; µ1 , . . . , µr ) Lif k − ln(1 + t) x l l=0 + *X n ∞ i X n (k) t n−l = sbl (y|λ1 , . . . , λr ; µ1 , . . . , µr ) b ci x l i! i=0 l=0 n X n (k) = sbl (y|λ1 , . . . , λr ; µ1 , . . . , µr )b cn−l . l l=0 *
∞ X
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Thus, we get the identity (17).
3.2
Sheffer identity
Theorem 2 sb(k) n (x
+ y|λ1 , . . . , λr ; µ1 , . . . , µr ) =
n X n j=0
j
(k)
sbj (x|λ1 , . . . , λr ; µ1 , . . . , µr )(y)n−j .
(20)
Proof. By (12) with pn (x) =
µ r Y 1 + eλj t j j=1
eλj t
1 sbn (x|λ1 , . . . , λr ; µ1 , . . . , µr ) Lif k (−t)
= (x)n ∼ (1, et − 1) , using (10), we have (20).
3.3
Difference relations
Theorem 3 sb(k) b(k) n (x + 1|λ1 , . . . , λr ; µ1 , . . . , µr ) − s n (x|λ1 , . . . , λr ; µ1 , . . . , µr ) (k)
= nb sn−1 (x|λ1 , . . . , λr ; µ1 , . . . , µr ) . (21) Proof. By (8) with (12), we get (k)
(et − 1)b s(k) sn−1 (x|λ1 , . . . , λr ; µ1 , . . . , µr ) . n (x|λ1 , . . . , λr ; µ1 , . . . , µr ) = nb By (7), we have (21).
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3.4
Recurrence
Theorem 4 (k)
sbn+1 (x|λ1 , . . . , λr ; µ1 , . . . , µr ) = (x + λµ)b s(k) n (x − 1|λ1 , . . . , λr ; µ1 , . . . , µr ) n X m X r X P (−1)m−l ml −1− rj=1 µj −2 S1 (n, m) µ λ E (x + λ(µ + ei ) − 1|λ; µ + ei ) k i i l (m − l + 1) m=0 l=0 i=1 n X m m−l m X Pr (−1) l S (n, m)El (x + λµ − 1|λ; µ) (22) − 2− j=1 µj k 1 (m − l + 2) m=0 l=0 = (x + µλ)b s(k) n (x − 1|λ1 , . . . , λr ; µ1 , . . . , µr ) n−j r n 1 XXX n x + λi − 1 (k) j+1 µi λi S1 (n − l, j)b s l Ej − 2 j=0 l=0 i=1 l λi n X m X P (−1)m−l ml − rj=1 µj −2 S (n, m)El (x + λµ − 1|λ; µ) , k 1 (m − l + 2) m=0 l=0 λµ =
Pr
j=1
(23)
λi µi .
Remark. Comparing (22) and (23), r n X m X X (−1)m−l ml 2 µ λ S (n, m)E x + λ(µ + e ) − 1|λ; µ + e i i 1 l i i (m − l + 1)k m=0 l=0 i=1 n−j r n X X X n x + λi − 1 (k) j+1 . µi λi S1 (n − l, j)b s l Ej = λi l j=0 l=0 i=1 −
Pr
j=1 µj
Proof. By applying
g 0 (t) 1 sn+1 (x) = x − sn (x) 0 g(t) f (t)
(24)
([12, Corollary 3.7.2]) with (12), we get (k)
sbn+1 (x|λ1 , . . . , λr ; µ1 , . . . , µr ) −t = xb s(k) n (x − 1|λ1 , . . . , λr ; µ1 , . . . , µr ) − e
g 0 (t) (k) sb (x|λ1 , . . . , λr ; µ1 , . . . , µr ) . g(t) n
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Since 0 g 0 (t) = ln g(t) g(t) !0 X r r X = µi ln(1 + eλi t ) − µi λi t − ln Lif k (−t) i=1
=
i=1
r X µi λi eλi t i=1
1 + e λi t
−
r X
µi λi +
i=1
Lif 0k (−t) , Lif k (−t)
by (13), we have g 0 (t) (k) sb (x) g(t) n ! r r X µi λi eλi t X Lif 0k (−t) sb(k) = − µi λi + n (x) λ t i 1 + e Lif (−t) k i=1 i=1 r µj n X m r X Y Pr (−1)l ml X 2 2 −1− j=1 µj (λµ+λi )t S1 (n, m) =2 µi λi e xm−l λj t k λi t (l + 1) 1 + e 1 + e m=0 l=0 i=1 j=1 µj r n λj t Y X e S1 (n, m) Lif 0k (−t)xm . (25) − λµb s(k) n (x) + λj t 1 + e m=0 j=1 The first term in (25) is P −1− rj=1 µj
2
n X m X r X m=0 l=0 i=1
(−1)l ml S1 (n, m) µ λ E x + λ(µ + e )|λ; µ + e , i i m−l i i (l + 1)k
where λ = (λ1 , . . . , λr ), µ = (µ1 , . . . , µr ) and ei = (0, . . . , 0, 1, 0, . . . , 0) (i = 1, 2, . . . , r). | {z } | {z } i−1
Since
Lif k−1 (−t) − Lif k (−t) =
1 1 − k−1 k 2 2
r−i
t + ··· ,
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the third term in (25) is −
2
Pr
j=1 µj
n X
S1 (n, m)e
λµt
Lif 0k (−t)
m=0
= 2−
Pr
j=1
= −2− −
Pr
×
j=1
n X
S1 (n, m)eλµt
m=0 n X j=1 µj
Pr
= −2
µj
l=0
2 1 + eλj t
µj
xm
Lif k−1 (−t) − Lif k (−t) Em (x|λ; µ) −t
Em+1 (x|λ; µ) S1 (n, m)eλµt Lif k−1 (−t) − Lif k (−t) m+1 m=0
j=1
m+1 X
r Y
µj
n X S1 (n, m) λµt e m+1 m=0
! m+1 X (−1)l tl (−1)l tl Em+1 (x|λ; µ) − Em+1 (x|λ; µ) l!(l + 1)k−1 l!(l + 1)k l=0
n X S1 (n, m) λµt e m + 1 m=0 ! m+1 m+1 X (−1)l m+1 X (−1)l m+1 l l × Em+1−l (x|λ; µ) − Em+1−l (x|λ; µ) k−1 k (l + 1) (l + 1) l=0 l=0 n m+1 l X S1 (n, m) X (−1) m+1 l P − rj=1 µj λµt l = −2 e Em+1−l (x|λ; µ) m+1 (l + 1)k m=0 l=1 m n X X P (−1)m−l ml − rj=1 µj S1 (n, m) =2 E (x + λµ|λ; µ) . k l (m + 2 − l) m=0 l=0
= −2−
Pr
j=1
µj
Thus, we obtain (k)
sbn+1 (x) = (x + λµ)b s(k) n (x − 1) r n m X m−l m XX Pr (−1) l − 2−1− j=1 µj S1 (n, m) µi λi El x + λ(µ + ei ) − 1|λ; µ + ei k (m + 1 − l) m=0 l=0 i=1 n X m X P (−1)m−l ml − rj=1 µj −2 S1 (n, m)El (x + λµ − 1|λ; µ) , (m + 2 − l)k m=0 l=0 which is (22).
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On the other hand, by (14) with (22), we have g 0 (t) (k) sb (x) g(t) n ! r r X µi λi eλi t X Lif 0k (−t) = sb(k) − µi λi + n (x) λi t 1 + e Lif (−t) k i=1 i=1 n−j r n X X 2 n 1X (k) λi t S1 (n − l, j)b s l xj = µi λi e λ t i 2 i=1 1 + e j=0 l=0 l n X m m−l m X Pr (−1) − j=1 µj l − µλb s(k) S (n, m)El (x + λµ|λ; µ) . n (x) + 2 k 1 (m + 2 − l) m=0 l=0
(26)
The first term in (26) is r n n−j X 2 1 XX n (k) S1 (n − l, j)b sl µi λi eλi t xj λi t 2 j=0 l=0 l 1 + e i=1 n n−j r X 1 XX n x (k) λi t j = S1 (n − l, j)b sl µi λi e λi Ej 2 j=0 l=0 l λi i=1 n n−j r X 1 XX n x + λi (k) j+1 = S1 (n − l, j)b sl µi λi Ej . 2 j=0 l=0 l λi i=1
Thus, we obtain (k)
s(k) sbn+1 (x) = (x + µλ)b n (x − 1) n n−j r 1 XXX n x + λi − 1 (k) j+1 − sl E j µi λi S1 (n − l, j)b 2 j=0 l=0 i=1 l λi n X m X P (−1)m−l ml − rj=1 µj −2 S1 (n, m)El (x + λµ − 1|λ; µ) . (m + 2 − l)k m=0 l=0 which is (23).
3.5
Differentiation
Theorem 5 n−1
X (−1)n−l−1 (k) d (k) sbn (x|λ1 , . . . , λr ; µ1 , . . . , µr ) = n! sb (x|λ1 , . . . , λr ; µ1 , . . . , µr ) . dx l!(n − l) l l=0
(27)
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Proof. We shall use n−1 X d n ¯ sbn (x) = f (t)|xn−l sbl (x) dx l l=0
(Cf. [12, Theorem 2.3.12]). Since
f¯(t)|xn−l = ln(1 + t)|xn−l * ∞ + X (−1)m−1 tm = xn−l m m=1 n−l X (−1)m−1 m n−l t |x = m m=1 n−l X (−1)m−1 = (n − l)!δm,n−l m m=1
= (−1)n−l−1 (n − l − 1)! , with (12), we have d (k) sb (x|λ1 , . . . , λr ; µ1 , . . . , µr ) dx n n−1 X n (k) = (−1)n−l−1 (n − l − 1)!b sl (x|λ1 , . . . , λr ; µ1 , . . . , µr ) l l=0 = n!
n−1 X (−1)n−l−1 l=0
l!(n − l)
(k)
sbl (x|λ1 , . . . , λr ; µ1 , . . . , µr ) ,
which is the identity (27).
3.6
A more relation
The classical Cauchy numbers cn of the first kind are defined by ∞ X tn t cn = ln(1 + t) n=0 n! (see e.g. [3, 8]). Theorem 6 sb(k) n (x|λ1 , . . . , λr ; µ1 , . . . , µr ) =
(k) xb sn−1 (x
+
r X
n 1X n (k−1) (k) − 1|λ1 , . . . , λr ; µ1 , . . . , µr ) + cn−l sbl (x − 1) − sbl (x − 1) n l=1 l (k)
µi λi sbn−1 (x − λi − 1|λ; µ + ei ) .
(28)
i=1
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Proof. For n ≥ 1, we have sb(k) n (y|λ1 , . . . , λr ; µ1 , . . . , µr ) + *∞ X (k) tl n = sbl (y|λ1 , . . . , λr ; µ1 , . . . , µr ) x l! * l=0 + µj r Y (1 + t)λj = Lif k − ln(1 + t) (1 + t)y xn λ j 1 + (1 + t) j=1 * ! + µj r Y (1 + t)λj = ∂t Lif k − ln(1 + t) (1 + t)y xn−1 λj 1 + (1 + t) j=1 * + µj ! r Y (1 + t)λj = ∂t Lif k − ln(1 + t) (1 + t)y xn−1 λj 1 + (1 + t) j=1 * r + µj λj Y (1 + t) + ∂t Lif k − ln(1 + t) (1 + t)y xn−1 λ j 1 + (1 + t) j=1 + * r µj Y (1 + t)λj Lif k − ln(1 + t) (∂t (1 + t)y ) xn−1 . + λj 1 + (1 + t) j=1 The third term is * y
µj r Y (1 + t)λj y−1 n−1 Lif k − ln(1 + t) (1 + t) x 1 + (1 + t)λj j=1
+
(k)
= yb sn−1 (y − 1|λ1 , . . . , λr ; µ1 , . . . , µr ) . Since Lif k−1
− ln(1 + t) − Lif k − ln(1 + t) =
1 1 − k−1 k 2 2
t + ··· ,
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the second term is * r + µj Y Lif k−1 − ln(1 + t) − Lif k − ln(1 + t) (1 + t)λj (1 + t)y xn−1 λj 1 + (1 + t) (1 + t) ln(1 + t) j=1 * r µj Y (1 + t)λj = 1 + (1 + t)λj j=1 + Lif k−1 − ln(1 + t) − Lif k − ln(1 + t) t y−1 n−1 (1 + t) x t ln(1 + t) * r µj Y (1 + t)λj = 1 + (1 + t)λj j=1 + ∞ X l Lif k−1 − ln(1 + t) − Lif k − ln(1 + t) t (1 + t)y−1 cl xn−1 t l! l=0 * µj n−1 r X Y n−1 (1 + t)λj = cl l 1 + (1 + t)λj j=1 l=0 + Lif − ln(1 + t) − Lif − ln(1 + t) k k−1 (1 + t)y−1 xn−1−l t
*Y µj r 1 n−1 (1 + t)λj = cl n − l l 1 + (1 + t)λj j=1 l=0 E (1 + t)y−1 Lif k−1 − ln(1 + t) − Lif k − ln(1 + t) xn−l * r + µj n−1 Y X n (1 + t)λj 1 Lif k−1 − ln(1 + t) (1 + t)y−1 xn−l cl = λj n − l l 1 + (1 + t) j=1 l=0 * r +! µj Y (1 + t)λj y−1 n−l − Lif k − ln(1 + t) (1 + t) x 1 + (1 + t)λj j=1 n−1 1X n (k−1) (k) cl sbn−l (y − 1|λ1 , . . . , λr ; µ1 , . . . , µr ) − sbn−l (y − 1|λ1 , . . . , λr ; µ1 , . . . , µr ) = n l=0 l n 1X n (k−1) (k) = cn−l sbl (y − 1|λ1 , . . . , λr ; µ1 , . . . , µr ) − sbl (y − 1|λ1 , . . . , λr ; µ1 , . . . , µr ) . n l=1 l n−1 X
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Since µj r Y (1 + t)λj ∂t 1 + (1 + t)λj j=1 =
r X
µi λi (1 + t)
i=1
−λi −1
µj r (1 + t)λj (1 + t)λi Y , (1 + (1 + t)λi j=1 1 + (1 + t)λj
the first term is * + µj r r λi λj X Y (1 + t) (1 + t) µi λi Lif k − ln(1 + t) (1 + t)y−λi −1 |xn−1 λ j (1 + (1 + t)λi j=1 1 + (1 + t) i=1 =
r X
(k)
µi λi sbn−1 (y − λi − 1|λ; µ + ei ) .
i=1
Therefore, we obtain sb(k) n (x|λ1 , . . . , λr ; µ1 , . . . , µr ) =
(k) xb sn−1 (x
+
r X
n 1X n (k−1) (k) cn−l sbl (x − 1) − sbl (x − 1) − 1|λ1 , . . . , λr ; µ1 , . . . , µr ) + n l=1 l (k)
µi λi sbn−1 (x − λi − 1|λ; µ + ei ) ,
i=1
which is the identity (28).
3.7
A relation including the Stirling numbers of the first kind
Theorem 7 For n − 1 ≥ m ≥ 1, we have n−m X
n (k) m S1 (n − l, m)b sl (λ1 , . . . , λr ; µ1 , . . . , µr ) l l=0 n−m X n − 1 (k) = (m − 1) S1 (n − l − 1, m − 1)b sl (−1|λ1 , . . . , λr ; µ1 , . . . , µr ) l l=0 n−m X n−1 (k−1) (−1|λ1 , . . . , λr ; µ1 , . . . , µr ) + S1 (n − l − 1, m − 1)b sl l l=0 n−m−1 r X X n−1 (k) S1 (n − l − 1, m)µi λi sbl (−λi − 1|λ; µ + ei ) . +m l i=1 l=0
(29)
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Proof. We shall compute * r + µj λj Y (1 + t) m Lif k − ln(1 + t) ln(1 + t) xn λ j 1 + (1 + t) j=1 in two different ways. On the one hand, it is equal to + * r µj Y m n (1 + t)λj Lif k − ln(1 + t) ln(1 + t) x λj 1 + (1 + t) j=1 * r + µj ∞ Y X (1 + t)λj tl n = Lif k − ln(1 + t) m! S1 (l, m) x λj 1 + (1 + t) l! j=1 l=0 * r + µj n X Y n−l n (1 + t)λj = m! S1 (l, m) Lif k − ln(1 + t) x 1 + (1 + t)λj l j=1 l=m *∞ + n X X (k) n ti n−l = m! S1 (l, m) sbi (λ1 , . . . , λr ; µ1 , . . . , µr ) x i! l i=0 l=m n X n (k) = m! S1 (l, m)b sn−l (λ1 , . . . , λr ; µ1 , . . . , µr ) l l=m n−m X n (k) = m! S1 (n − l, m)b sl (λ1 , . . . , λr ; µ1 , . . . , µr ) . l l=0 On the other hand, it is equal to ! + * µj r Y m n−1 (1 + t)λj Lif k − ln(1 + t) ln(1 + t) ∂t x λj 1 + (1 + t) j=1 * + µj ! r Y m n−1 (1 + t)λj = ∂t Lif k − ln(1 + t) ln(1 + t) x λj 1 + (1 + t) j=1 + * r µj Y m n−1 (1 + t)λj ∂t Lif k − ln(1 + t) ln(1 + t) x + 1 + (1 + t)λj j=1 * r + µj Y m n−1 (1 + t)λj + Lif k − ln(1 + t) ∂t ln(1 + t) . x λj 1 + (1 + t) j=1
(30)
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The third term of (30) is equal to * r + µj Y m−1 n−1 (1 + t)λj −1 m Lif k − ln(1 + t) (1 + t) ln(1 + t) x 1 + (1 + t)λj j=1 * r µj Y (1 + t)λj −1 Lif − ln(1 + t) (1 + t) =m k λj 1 + (1 + t) j=1 + ∞ X tl n−1 (m − 1)! S1 (l, m − 1) x l! l=m−1 n−1 X n−1 = m! S1 (l, m − 1) l l=m−1 * r + µj Y (1 + t)λj × Lif k − ln(1 + t) (1 + t)−1 xn−1−l λj 1 + (1 + t) j=1 n−1 X n−1 (k) = m! S1 (l, m − 1)b sn−1−l (−1|λ1 , . . . , λr ; µ1 , . . . , µr ) l l=m−1 n−m X n − 1 (k) = m! S1 (n − l − 1, m − 1)b sl (−1|λ1 , . . . , λr ; µ1 , . . . , µr ) . l l=0 The second term of (30) is equal to + * r ! µj Y m n−1 Lif k−1 − ln(1 + t) − Lif k − ln(1 + t) (1 + t)λj ln(1 + t) x λj 1 + (1 + t) (1 + t) ln(1 + t) j=1 * r + µj Y m−1 n−1 (1 + t)λj −1 = Lif k−1 − ln(1 + t) (1 + t) ln(1 + t) x 1 + (1 + t)λj j=1 * r + µj Y (1 + t)λj m−1 Lif k − ln(1 + t) (1 + t)−1 ln(1 + t) − xn−1 λj 1 + (1 + t) j=1 n−m X n − 1 (k−1) = (m − 1)! S1 (n − l − 1, m − 1)b sl (−1|λ1 , . . . , λr ; µ1 , . . . , µr ) l l=0 n−m X n − 1 (k) − (m − 1)! S1 (n − l − 1, m − 1)b sl (−1|λ1 , . . . , λr ; µ1 , . . . , µr ) . l l=0
19
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The first term of (30) is equal to * + µj ! r Y m n−1 (1 + t)λj ∂t Lif k − ln(1 + t) ln(1 + t) x 1 + (1 + t)λj j=1 * µj r r X (1 + t)λj (1 + t)λi Y = µi λi 1 + (1 + t)λi j=1 1 + (1 + t)λj i=1 m n−1 E −λi −1 Lif k − ln(1 + t) (1 + t) ln(1 + t) x * µj r r X (1 + t)λi Y (1 + t)λj = µi λi 1 + (1 + t)λi j=1 1 + (1 + t)λj i=1 + ∞ X l t Lif k − ln(1 + t) (1 + t)−λi −1 m! S1 (l, m) xn−1 l! l=m r n−1 X X n−1 = m! µi λi S1 (l, m) l l=m + *i=1 µj r (1 + t)λj (1 + t)λi Y Lif k − ln(1 + t) (1 + t)−λi −1 xn−1−l × 1 + (1 + t)λi j=1 1 + (1 + t)λj r n−1 X X n−1 (k) = m! µi λi S1 (l, m)b sn−1−l (−λi − 1|λ; µ + ei ) l i=1 l=m r n−m−1 X X n−1 (k) S1 (n − 1 − l, m)µi λi sbl (−λi − 1|λ; µ + ei ) . = m! l i=1 l=0 Therefore, we get, for n − 1 ≥ m ≥ 1, n−m X
n (k) m! S1 (n − l, m)b sl (λ1 , . . . , λr ; µ1 , . . . , µr ) l l=0 n−m X n − 1 (k) = m! S1 (n − l − 1, m − 1)b sl (−1) l l=0 n−m X n − 1 (k−1) + (m − 1)! S1 (n − l − 1, m − 1)b sl (−1) l l=0 n−m X n − 1 (k) − (m − 1)! S1 (n − l − 1, m − 1)b sl (−1) l l=0 n−m−1 r X X n−1 (k) + m! S1 (n − l − 1, m)µi λi sbl (−λi − 1|λ; µ + ei ) . l i=1 l=0
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Dividing both sides by (m − 1)!, we obtain, for n − 1 ≥ m ≥ 1, n−m X n (k) m S1 (n − l, m)b sl (λ1 , . . . , λr ; µ1 , . . . , µr ) l l=0 n−m X n − 1 (k) = (m − 1) S1 (n − l − 1, m − 1)b sl (−1|λ1 , . . . , λr ; µ1 , . . . , µr ) l l=0 n−m X n−1 (k−1) + S1 (n − l − 1, m − 1)b sl (−1|λ1 , . . . , λr ; µ1 , . . . , µr ) l l=0 n−m−1 r X X n−1 (k) +m S1 (n − l − 1, m)µi λi sbl (−λi − 1|λ; µ + ei ) . l i=1 l=0 Thus, we get (29).
3.8
A relation with the falling factorials
Theorem 8 sb(k) n (x|λ1 , . . . , λr ; µ1 , . . . , µr )
n X n (k) (x)m . sb = m n−m m=0
(31)
P (k) Proof. For (12) and (19), assume that sbn (x|λ1 , . . . , λr ; µ1 , . . . , µr ) = nm=0 Cn,m (x)m . By (11), we have + * m n 1 1 λ ln(1+t) µj Lif k − ln(1 + t) t x Cn,m = 1+e j m! Qr j=1 eλj ln(1+t) * r + µj m n 1 Y (1 + t)λj = Lif k − ln(1 + t) t x m! j=1 1 + (1 + t)λj + *Y µj r n−m n (1 + t)λj = Lif k − ln(1 + t) x m 1 + (1 + t)λj j=1 n (k) = sb . m n−m Thus, we get the identity (31).
3.9
A relation with higher-order Frobenius-Euler polynomials (r)
For α ∈ C with α 6= 1, the Frobenius-Euler polynomials of order r, Hn (x|α) are defined by the generating function r ∞ X 1−α tn xt (r) e = H (x|α) n et − α n! n=0 21
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(see e.g. [10]). Theorem 9 sb(k) n (x|λ1 , . . . , λr ; µ1 , . . . , µr )
=
n X
n−m X n−m−j X
m=0
j=0
−j
×(1 − α) S1 (n − j −
l=0
(k) l, m)b sl
s n−j (n)j j l (s) Hm (x|α) .
(32)
Proof. For (12) and
s et − α ,t , (33) ∼ 1−α P (k) (s) assume that sbn (x|λ1 , . . . , λr ; µ1 , . . . , µr ) = nm=0 Cn,m Hm (x|α). By (11), similarly to the proof of (29), we have ln(1+t) s + * e −α m n 1−α 1 λ ln(1+t) µj Lif k − ln(1 + t) ln(1 + t) x Cn,m = 1+e j m! Qr j=1 eλj ln(1+t) * r + µj Y m 1 (1 + t)λj = Lif k − ln(1 + t) ln(1 + t) (1 − α + t)s xn m!(1 − α)s j=1 1 + (1 + t)λj Hn(s) (x|α)
1 m!(1 − α)s * r + µj min{s,n} Y m X (1 + t)λj s Lif k − ln(1 + t) ln(1 + t) × (1 − α)s−i ti xn λj 1 + (1 + t) i j=1 i=0 n−m X s 1 = (1 − α)s−i (n)i m!(1 − α)s i=0 i + * r µj Y m n−i (1 + t)λj Lif k − ln(1 + t) ln(1 + t) x × 1 + (1 + t)λj j=1 n−m n−m−i X s X 1 n−i (k) s−i = (1 − α) (n)i m! S1 (n − i − l, m)b sl m!(1 − α)s i=0 i l l=0 n−m n−m−i X X s n−i (k) = (n)i (1 − α)−i S1 (n − i − l, m)b sl . i l i=0 l=0
=
Thus, we get the identity (32).
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3.10
A relation with higher-order Bernoulli polynomials (r)
Bernoulli polynomials Bn (x) of order r are defined by r ∞ (r) X t Bn (x) n xt t e = et − 1 n! n=0 (r)
(see e.g. [12, Section 2.2]). In addition, Cauchy numbers of the first kind Cn of order r are defined by r X ∞ (r) t Cn n = t ln(1 + t) n! n=0 (see e.g. [2, (2.1)], [11, (6)]). Theorem 10 sb(k) n (x|λ1 , . . . , λr ; µ1 , . . . , µr ) ! n n−m X X n−m−i X nn − i (s) (k) = Ci S1 (n − i − l, m)b sl B(s) m (x) . i l m=0 i=0 l=0 Proof. For (12) and B(s) n (x) (k)
et − 1 t Pn
∼
s
,t ,
(34)
(35)
(s)
assume that sbn (x|λ1 , . . . , λr ; µ1 , . . . , µr ) = m=0 Cn,m Bm (x). By (11), similarly to the proof of (29), we have ln(1+t) s * + e −1 m n ln(1+t) 1 λ ln(1+t) µj Lif k − ln(1 + t) ln(1 + t) x Cn,m = 1+e j m! Qr j=1 eλj ln(1+t) * r µj s + m 1 Y (1 + t)λj t Lif k − ln(1 + t) ln(1 + t) = xn m! j=1 1 + (1 + t)λj ln(1 + t) * r + µj ∞ i m X 1 Y (1 + t)λj t (s) = Lif k − ln(1 + t) ln(1 + t) Ci xn m! j=1 1 + (1 + t)λj i! i=0 * r + n−m µ j λ X Y j 1 (1 + t) m (s) n = Ci Lif k − ln(1 + t) ln(1 + t) xn−i λ j m! i=0 i 1 + (1 + t) j=1 n−m−i n−m 1 X (s) n X n−i (k) = C m! S1 (n − i − l, m)b sl m! i=0 i i l l=0 n−m X n−m−i X nn − i (s) (k) = Ci S1 (n − i − l, m)b sl . i l i=0 l=0 Thus, we get the identity (34). 23
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Acknowledgements The work reported in this paper was conducted during the sabbatical year of Kwangwoon University in 2014.
References [1] S. Araci,M. Acikgoz, A. Kilicman, Extended p-adic q-invariant integrals on Zp associated with applications of umbral calculus, Adv. Difference Equ. 2013 (2013), 96, 14 pp. [2] L. Carlitz, A note on Bernoulli and Euler polynomials of the second kind, Scripta Math. 25 (1961), 323–330. [3] L. Comtet, Advanced Combinatorics, Reidel, Dordrecht, 1974. [4] R. Dere, Y. Simsek, Applications of umbral algebra to some special polynomials, Adv. Stud. Contemp. Math., 22 (2012), 433-438. [5] Q. Fang, T. Wang, Umbral calculus and invariant sequences, Ars Combinatoria, 101 (2011),257–264. [6] D. S. Kim, T. Kim, Higher-order Cauchy of the second kind and poly-Cauchy of the second kind mixed type polynomials, Ars Combinatoria, 115 (2014), 435-451. [7] D. S. Kim and T. Kim, Poly-Cauchy and Peters mixed-type polynomials, Adv. Difference Equ. 2014, (2014), #4. [8] D. S. Kim, T. Kim, Higher-order Cauchy of the first kind and poly-Cauchy of the first kind mixed type polynomials, Adv. Stud. Contemp. Math. 23 (2013), 621–636. [9] D. S. Kim, T. Kim, S. H. Lee, Poly-Cauchy numbers and polynomials with umbral calculus viewpoint, Int. J. Math. Anal. (Ruse) 7 (2013), 2235–2253. [10] T. Kim, Identities involving Laguerre polynomials derived from umbral calculus , Russ. J. Math. Phys. 21 (2014), 36–45. [11] H. Liang and Wuyungaowa, Identities involving generalized harmonic numbers and other special combinatorial sequences, J. Integer Seq. 15 (2012), Article 12.9.6, 15 pp. [12] S. Roman, The umbral Calculus, Dover, New York, 2005.
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On the solution for a system of two rational difference equations Chang-you Wang, Xiao-jing Fang, Rui Li 1. Key Laboratory of Industrial Internet of Things & Networked Control of Ministry of Education, Chongqing University of Posts and Telecommunications, Chongqing 400065 P.R. China 2. Institute of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing 400065 P. R. China
Abstract: This paper is concerned with the dynamical behavior and the expression of the solution for a system of two rational difference equations xn -3 yn -3 , yn +1 = , n = 0,1, xn +1 = A + xn -3 yn -1 B + yn -3 xn -1
,
where the parameters A, B and the initial conditions x−3 , x−2 , x−1 , x0 , y−3 , y−2 , y−1 , y0 are positive real numbers.
Keywords: difference equations; expression of solutions; recursive sequences, equilibrium
point; asymptotical stability.
1. Introduction Rational difference equations that are one of the most important and practical classes of nonlinear difference equations have applications in various scientific branches such as biology, ecology, physiology, physics, engineering and economics, etc [1-4]. Although difference equations are very simple in form, it is extremely difficult to understand thoroughly the behaviors of their solutions. So recently there has been an increasing interest in the study of qualitative analysis of rational difference equation and systems of difference equations [5-7]. In particular, Papaschinopoulos and Schinas [8] studied the oscillatory behavior, the boundedness of the solutions, and the global asymptotic stability of the positive equilibrium of the system of two nonlinear difference equations
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xn +1 = A +
yn , xn − p
yn +1 = A +
xn , n = 0,1, yn − q
(1.1)
,
where p, q are positive integers. Clark and Kulenovic [9, 10] investigated the global stability properties and asymptotic behavior of solutions of the recursive sequences xn yn xn +1 = yn +1 = n = 0,1, . , , a + cyn b + dxn
(1.2)
where a, b, c, d ∈ (0, ∞) and the initial conditions x0 and y0 are arbitrary nonnegative numbers. The periodicity of the positive solutions of the system of rational difference equations yn 1 , xn +1 = , yn +1 = n = 0,1, , (1.3) yn xn −1 yn −1 was studied by Cinar in [11]. Yalcinkaya [12] has obtained the sufficient conditions for the global asymptotic stability of the system of two nonlinear difference equations x + yn −1 y +x xn +1 = n yn +1 = n n −1 , n = 0,1, . , (1.4) xn yn −1 − 1 yn xn −1 − 1 More recently, Din et al. [13] studied the equilibrium points, local asymptotic stability of an equilibrium point, instability of equilibrium points, periodicity behavior of positive solutions, and global character of an equilibrium point of the following fourth-order system of rational difference equations xn +1 =
α xn −3
β + γ yn yn −1 yn − 2 yn −3
, yn +1 =
α1 yn − 3
β1 + γ 1 xn xn −1 xn − 2 xn −3
, n = 0,1,
.
(1.5)
In [14], Elsayed deals with the form of the solutions of the following rational difference system xn +1 =
xn -1 yn -1 , yn +1 = , n = 0, 1, ±1 + xn -1 yn ∓1 + yn -1 xn
,
(1.6)
with nonzero real number initial conditions. Other related results on the difference equation can be found in references [15-28] and references therein. Based on the above results, we are mainly interested in study the asymptotic behavior and the expression of the solution for the following nonlinear rational difference equations xn -3 yn -3 xn +1 = , yn +1 = , n = 0,1, , (1.7) A + xn -3 yn -1 B + yn -3 xn -1 where the parameters A, B and the initial conditions x−3 , x−2 , x−1 , x0 , y−3 , y−2 , y−1 , y0 are positive real numbers.
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This paper proceeds as follows. In Section 2, we introduce some definitions and preliminary results. The main results and their proofs are given in Section 3.
2. Preliminaries and notations In this section we prepare some materials used throughout this paper, namely notations, the basic definitions and preliminary results. We refer to the monographs of Kocic et al. [5, 29, 30]. Lemma 2.1 Let
Ix , I y
be some intervals of real numbers and f : I x4 × I y4 → I x ,
g : I x4 × I y4 → I y be continuously differentiable functions. Then for every initial conditions
( xi , yi ) ∈ I x × I y , (i = −3, −2, −1, 0) , the system of difference equations ⎧ xn +1 = f (xn ,xn -1 ,xn -2 ,xn -3 ,yn ,yn -1 ,yn -2 ,yn -3 ), n = 0, 1, 2, ⎨ ⎩ yn +1 = g (xn ,xn -1 ,xn -2 ,xn -3 ,yn ,yn -1 ,yn -2 ,yn -3 ),
(2.1)
,
has a unique solution {(xn ,yn )}∞n =−3 . Definition 2.1 A point (x , y ) ∈ I x × I y is called an equilibrium point of system (2.1) if x = f (x, x, x, x, y, y, y, y), y = g ( x.x, x, x, y, y, y, y ) .
That is, ( xn , yn ) = ( x , y ) for all n ≥ 1 when the initial conditions ( x0 , x−1 , x−2 , x−3 , y0 , y−1 , y−2 , y−3 ) = ( x , x , x , x , y , y , y , y ) . Definition 2.2 Let ( x , y ) be an equilibrium point of system (2.1). Then
(1) The equilibrium ( x , y ) of system (2.1) is said to be stable relative to I x × I y if for every ε > 0 ,
there
exits
( xi , yi ) ∈ I x × I y (i =−3, − 2, −1, 0) ,
δ >0 with
such
∑
0
i =−3
that
for
any
initial
xi − x < δ ,
∑
yi − y < δ
0
i =−3
conditions implies
xn − x < ε , yn − y < ε .
(2) The equilibrium ( x , y ) of system (2.1) is called an attractor relative to I x × I y if for all ( xi , yi ) ∈ I x × I y (i = −3, −2, −1,0) , lim n→∞ xn = x and lim n→∞ yn = y hold. (3) The equilibrium ( x , y ) of system (2.1) is called asymptotically stable relative to I x × I y if it is stable and an attractor. (4) The equilibrium ( x , y ) of system (2.1) is called unstable if it is not stable. Definition 2.3 Let ( x , y ) be an equilibrium point of the system (2.1), and f and g are
continuously differentiable functions at ( x , y ) . The linearized system of system (2.1) about
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the equilibrium point ( x , y ) is X n +1 = F ( X n ) = FJ X n where X n = ( xn , xn −1 , xn − 2 , xn −3 , yn , yn −1 , yn − 2 , yn −3 )T and FJ is a Jacobian matrix of the system (2.1) about the equilibrium point ( x , y ) . Lemma 2.2 Assume that X n +1 = F ( X n ) , n = 0, 1,
, is a system of difference equations and
X is the equilibrium point of this system i.e., F ( X ) = X . If all eigenvalues of the Jacobian matrix FJ
about
X
lie inside the open unit disk
λ < 1 , then X
is locally
asymptotically stable. If one of them has a modulus greater than one, then X is unstable.
3. Main results and their proofs It is obviously, if A > 1, B ≠ 1 or B > 1, A ≠ 1 , then (0, 0) is the unique equilibrium point of the system (1.7). Theorem 3.1 Let {xn , yn }∞n =−3 be positive solutions of system (1.7), then for all k ≥ 0 ,
(1)
⎧ x−3 ⎪ Ak +1 , ⎪ ⎪ x−2 , ⎪ k +1 0 ≤ xn ≤ ⎨ A ⎪ x−1 , ⎪ Ak +1 ⎪ x ⎪ k0+1 , ⎩A
n = 4k + 1, n = 4k + 2, (2) n = 4k + 3, n = 4k + 4.
⎧ y−3 ⎪ B k +1 , ⎪ ⎪ y−2 , ⎪ k +1 0 ≤ yn ≤ ⎨ B ⎪ y−1 , ⎪ B k +1 ⎪ y ⎪ k0+1 , ⎩B
n = 4k + 1, n = 4k + 2, (3.1) n = 4k + 3, n = 4k + 4.
Proof. This assertion is true for k = 0 , Assume that it is true for k = m , then for k = m + 1 , we have
x4(m+1)-3 x4m+1 1 x-3 ⎧ x−3 x , n = 4(m +1) +1; ≤ = ≤ = ⎪ 4(m+1)+1 m+1 ( m+1)+1 A A A A A ⎪ x4(m+1)+1-3 x4m+2 1 x-2 x−2 ⎪ x , n = 4(m +1) + 2, ≤ = ≤ = 4( m + 1) + 2 m+1 ( m+1)+1 ⎪⎪ A A A A A xn = ⎨ x4(m+1)+2-3 x4m+3 1 x-1 x ⎪x ≤ = ≤ = (m−+11)+1 , n = 4(m +1) + 3, 4( m+1)+3 + 1 m ⎪ A A AA A ⎪ x4(m+1)+3-3 x4m+4 1 x0 x0 ⎪x = ≤ = , n = 4(m +1) + 4. 4( m+1)+4 ≤ ⎩⎪ A A A Am+1 A(m+1)+1
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y4( m +1)-3 y4 m +1 1 y-3 ⎧ y = ≤ = ( m−+31) +1 , n = 4(m + 1) + 1; ⎪ y4( m +1) +1 ≤ m +1 B B BB B ⎪ y4( m +1) +1-3 y4 m + 2 1 y-2 y ⎪ = ≤ = ( m−+21) +1 , n = 4(m + 1) + 2, m +1 ⎪⎪ y4( m +1) + 2 ≤ B B BB B yn = ⎨ y y 1 y-1 y ⎪y ≤ 4( m +1) + 2-3 = 4 m +3 ≤ = ( m−+11)+1 , n = 4(m + 1) + 3, m +1 ⎪ 4( m +1) +3 B B BB B ⎪ y4( m +1) +3-3 y4 m + 4 1 y0 y ⎪y = ≤ = ( m +01) +1 , n = 4(m + 1) + 4. 4( m +1) + 4 ≤ m +1 ⎪⎩ B B BB B This completes our inductive proof. Corollary 3.1 If A > 1, B > 1 , then by Theorem 3.1 {( xn , yn )}∞n =−3 the solutions of the system
(1.7) exponentially converges to the equilibrium poin( t 0, 0). Theorem 3.2 For the equilibrium point (0, 0) of the system (1.7), the following results
hold: (1) If A > 1, B > 1 , then the equilibrium point (0, 0) of the system (1.7) is locally asymptotically stable. (2) If A < 1 or B < 1 , then the equilibrium point (0, 0) of the system (1.7) is unstable. Proof. We can easily obtain that the linearized system of (1.7) about the equilibrium point (0, 0) is
ϕn +1 = Dϕ n
(3.2)
where
⎡ xn ⎤ ⎢x ⎥ ⎢ n −1 ⎥ ⎢ xn − 2 ⎥ ⎢ ⎥ xn −3 ⎥ ⎢ ϕ n= ⎢ , yn ⎥ ⎢ ⎥ ⎢ yn −1 ⎥ ⎢y ⎥ ⎢ n−2 ⎥ ⎢⎣ yn −3 ⎥⎦
⎡ ⎢0 ⎢ ⎢1 ⎢0 ⎢ ⎢0 D=⎢ ⎢0 ⎢ ⎢0 ⎢ ⎢0 ⎢0 ⎣
1 A 0 0 0 1 0 0 0 1 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
⎤ 0 0 0 0⎥ ⎥ 0 0 0 0⎥ 0 0 0 0⎥ ⎥ 0 0 0 0⎥ 1⎥ ⎥ 0 0 0 B⎥ 1 0 0 0⎥ ⎥ 0 1 0 0⎥ 0 0 1 0 ⎥⎦
the characteristic equation of (3.2) is
1 1 (3.3) f (λ ) = (λ 4 − )(λ 4 − ) = 0 . A B 1 1 (1) If A > 1, B > 1 , then we have | |< 1, | |< 1 , this shows that all the roots of A B characteristic equation (3.3) lie inside unit disk. So the unique equilibrium (0, 0) is
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locally asymptotically stable. (2) It is easy to see that if A < 1 or B < 1 , then there exists at least one root λ of the characteristic equation (3.3) such that λ > 1 . Thus, the equilibrium (0, 0) of the system (1.7) is unstable when A < 1 or B < 1 . By Corollary 3.1 and Theorem 3.2, we have the following result. Corollary 3.2 If A > 1, B > 1 , then the equilibrium point (0, 0) is globally asymptotically stable. Theorem 3.3 If A = B = 1 , then every solution of the system (1.7) is bounded when the initial conditions x−3 , x−2 , x−1 , x0 , y−3 , y−2 , y−1 and y0 are positive real numbers. Proof. It follows from Eq. (1.7) that xn -3 yn -3 xn +1 = ≤ xn -3 , yn +1 = ≤ yn -3 . 1 + xn -3 yn -1 1 + yn -3 xn -1
Then the subsequences { x4 n −3 } ∞n =0 , { x4 n − 2 } ∞n =0 , { x4 n −1 } ∞n =0 , { x4 n } ∞n =0 are decreasing and so are bounded from above by M =max{ x−3 , x−2 , x−1 , x0 }, also, the subsequences { y4 n −3 } ∞n =0 , { y4 n − 2 } ∞n =0 , { y4 n −1 } ∞n =0 , { y4 n } ∞n =0 are decreasing and so are bounded from above by m =max{ y−3 , y−2 , y−1 , y0 }. Hence, every solution of the system (1.7) is bounded for any positive initial conditions. In next section, we study the expressions of the solutions for the systems (1.7) with the parameters A = B . Theorem 3.4 If A = B , suppose that {( xn , yn )}∞n =−3 are solutions of the system (1.7). Also,
assume that x−3 , x−2 , x−1 , x0 , y−3 , y−2 , y−1 and y0 are arbitrary positive numbers and let x−3 = a, x−2 = b, x−1 = c, x0 = d , y−3 = e, y−2 = f , y−1 = g , y0 = h . Then x4 n −3
A2i + A2i −1ag + + Aag + ag , = a∏ 2i +1 + A2i ag + A2i −1ag + + Aag + ag i =0 A
x4 n − 2
A2i + A2i −1bh + + Abh + bh , = b∏ 2i +1 + A2i bh + A2i −1bh + + Abh + bh i =0 A
n −1
n −1
A2i +1 + A2i ce + + Ace + ce , x4 n −1 = c∏ 2i + 2 + A2i +1ce + A2i ce + + Ace + ce i =0 A n −1
(3.4)
A2i +1 + A2i df + + Adf + df , x4 n = d ∏ 2i + 2 + A2i +1df + A2i df + + Adf + df i =0 A n −1
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n −1
y4 n −3 = e∏ i =0
A2i + A2i −1ce + + Ace + ce , A2i +1 + A2i ce + A2i −1ce + + Ace + ce
A2i + A2i −1df + + Adf + df , = f ∏ 2i +1 + A2i df + A2i −1df + + Adf + df i =0 A n −1
y4 n − 2
A2i +1 + A2i ag + + Aag + ag , y4 n −1 = g ∏ 2i + 2 + A2i +1ag + A2i ag + + Aag + ag i =0 A n −1
A2i +1 + A2i bh + + Abh + bh , A2i + 2 + A2i +1bh + A2i bh + + Abh + bh
n −1
y4 n = h∏ i =0
where n = 1, 2,
(3.5)
.
Proof. If A = B , then the system (1.7) is reduced to xn -3 yn -3 , yn +1 = . xn +1 = A + xn -3 yn -1 A + yn -3 xn -1
(3.6)
It is easy to prove that Eqs. (3.4) and (3.5) hold for n = 1 . Now suppose that k ∈ N , k > 1 and that Eqs. (3.4) and (3.5) hold for n = k − 1 . That is, x4 k −7
A2i + A2i −1ag + + Aag + ag , = a∏ 2i +1 + A2i ag + A2i −1ag + + Aag + ag i =0 A
x4 k −6
A2i + A2i −1bh + + Abh + bh , = b∏ 2i +1 + A2i bh + A2i −1bh + + Abh + bh i =0 A
x4 k −5
A2i +1 + A2i ce + + Ace + ce , = c∏ 2 i + 2 + A2i +1ce + A2i ce + + Ace + ce i =0 A
x4 k − 4
A2i +1 + A2i df + + Adf + df , = d ∏ 2i + 2 + A2i +1df + A2i df + + Adf + df i =0 A
k −2
k −2
k −2
k −2
A2i + A2i −1ce + + Ace + ce , A2i +1 + A2i ce + A2i −1ce + + Ace + ce
k −2
y4 k −7 = e∏ i =0
k −2
y4 k − 6 = f ∏ i =0
k −2
y4 k − 5 = g ∏ i =0
k −2
y4 k − 4 = h∏ i =0
A2i + A2i −1df + + Adf + df , A2i +1 + A2i df + A2i −1df + + Adf + df A2i +1 + A2i ag + + Aag + ag , A2i + 2 + A2i +1ag + A2i ag + + Aag + ag
A2i +1 + A2i bh + + Abh + bh . A2i + 2 + A2i +1bh + A2i bh + + Abh + bh
Then, it follows from Eq. (3.6) and our assumptions that
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x4 k − 7 A + x4 k −7 y4 k −5
x4 k −3 =
A2i + A2i −1ag + + Aag + ag 2 i +1 + A2i ag + A2i −1ag + + Aag + ag i =0 A = k −2 k −2 A2i + A2i −1ag + + Aag + ag A2i +1 + A2i ag + + Aag + ag A + a∏ 2i +1 g ∏ + A2i ag + A2i −1ag + + Aag + ag i = 0 A2i + 2 + A2i +1ag + A2i ag + + Aag + ag i =0 A k −2
a∏
A2i + A2i −1ag + + Aag + ag 2 i +1 + A2i ag + A2i −1ag + + Aag + ag i =0 A 1 A + ag 2 k − 2 2 k −1 A + A ag + + Aag + ag
k −2
=
a∏
k −2
= a∏ i =0
k −1
= a∏ i =0
⎛ A2 k − 2 + A2 k −1ag + A2i + A2i −1ag + + Aag + ag ⎜ A2i +1 + A2i ag + A2i −1ag + + Aag + ag ⎝ A2 k −1 + A2 k − 2 ag +
+ Aag + ag ⎞ ⎟ + Aag + ag ⎠
A2i + A2i −1ag + + Aag + ag . A2i +1 + A2i ag + A2i −1ag + + Aag + ag
That is k −1
x4 k −3 = a∏ i =0
A2i + A2i −1ag + + Aag + ag . A2i +1 + A2i ag + A2i −1ag + + Aag + ag
In addition to, by Eq. (3.6) and our assumptions one has
y4k −3 =
y4k −7 A + y4k −7 x4k −5
A2i + A2i−1ce + + Ace + ce e∏ 2i+1 2i + A ce + A2i−1ce + + Ace + ce i =0 A = k −2 k −2 A2i + A2i−1ce + + Ace + ce A2i+1 + A2i ce + + Ace + ce A + e∏ 2i+1 2i c ∏ + A ce + A2i−1ce + + Ace + ce i=0 A2i+2 + A2i+1ce + A2i ce + + Ace + ce i =0 A k −2
A2i + A2i−1ce + + Ace + ce e∏ 2i+1 2i A + A ce + A2i−1ce + + Ace + ce = i=0 1 A + ce 2n−2 2n−1 A + A ce + + Ace + ce k −2 ⎛ A2k −2 + A2k −1ce + A2i + A2i−1ce + + Ace + ce = e∏ 2i+1 2i ⎜ + A ce + A2i−1ce + + Ace + ce ⎝ A2k −1 + A2k −2ce + i =0 A k −2
+ Ace + ce ⎞ ⎟ + Ace + ce ⎠
A2i + A2i−1ce + + Ace + ce . = e∏ 2i+1 2i + A ce + A2i−1ce + + Ace + ce i =0 A k −1
That is,
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A2i + A2i −1ce + + Ace + ce . = e∏ 2i +1 + A2i ce + A2i −1ce + + Ace + ce i =0 A k −1
y4 k − 3
Similarly, one can prove k −1
x4 k − 2 = b∏ i =0
A2i + A2i −1bh + + Abh + bh , A2i +1 + A2i bh + A2i −1bh + + Abh + bh
A2i +1 + A2i ce + + Ace + ce x4 k −1 = c∏ 2i + 2 , + A2i +1ce + A2i ce + + Ace + ce i =0 A k −1
A2i +1 + A2i df + + Adf + df x4 k = d ∏ 2i + 2 , + A2i +1df + A2i df + + Adf + df i =0 A k −1
k −1
y4 k − 2 = f ∏ i =0
k −1
y4 k −1 = g ∏ i =0
k −1
y 4 k = h∏ i =0
A2i + A2i −1df + + Adf + df , A2i +1 + A2i df + A2i −1df + + Adf + df A2i +1 + A2i ag + + Aag + ag , A2i + 2 + A2i +1ag + A2i ag + + Aag + ag
A2i +1 + A2i bh + + Abh + bh . A2i + 2 + A2i +1bh + A2i bh + + Abh + bh
Hence, Eqs. (3.4) and (3.5) hold for n = k . The proof is complete according to the mathematical induction. Corollary 3.3 If A = B = 1 , suppose that {( xn , yn )}∞n =−3 are solutions of the system (1.7).
Also, assume that x−3 , x−2 , x−1 , x0 , y−3 , y−2 , y−1 and y0 are arbitrary positive numbers and let x−3 = a, x−2 = b, x−1 = c, x0 = d , y−3 = e, y−2 = f , y−1 = g , y0 = h , then one has n −1 1 + 2iag 1 + 2ibh , x4 n − 2 = b∏ , i = 0 1 + (2i + 1) ag i = 0 1 + (2i + 1)bh n −1
x4 n −3 = a∏
n −1 1 + (2i + 1)ce 1 + (2i + 1)df , x4 n = d ∏ , i = 0 1 + (2i + 2)ce i = 0 1 + (2i + 2) df n −1
x4 n −1 = c∏
1 + 2ice , i = 0 1 + (2i + 1)ce
1 + 2idf , i = 0 1 + (2i + 1) df
n −1
n −1
y4 n −3 = e∏
y4 n − 2 = f ∏
1 + (2i + 1)ag , i = 0 1 + (2i + 2) ag
n −1
y4 n −1 = g ∏ where n = 1, 2,
1 + (2i + 1)bh , i = 0 1 + (2i + 2)bh
n −1
y4 n = h∏
.
4. Conclusions It is obvious that the system of two rational difference equations (1.7) is the extension of the models in [9, 10, 13, 14]. In this paper, we investigated the globally asymptotically stable of the equilibrium point (0, 0)for the difference equation (1.7) with the parameters A > 1, B > 1 , and the unstable of the equilibrium point (0, 0) with the parameter A < 1 or B < 1 using
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linearization method. Moreover, the expressions of solutions of the system (1.7) with the parameters A = B are obtained according to the mathematical induction. This paper presents the use of a variational iteration method and mathematical induction for systems of nonlinear difference equations. This technique is a powerful tool for solving various difference equations and can also be applied to other nonlinear differential equations in mathematical physics. In addition, the sufficient conditions that we obtained are very simple, which provide flexibility for the application and analysis of nonlinear difference equation. In addition, the system can be used to analyze and describe the pier buffering isolation system.
Acknowledgements This work is supported by the Chongqing Natural Science Fund (Nos. cstc2012jjA20016 and cstc2012jjA40035), the National Nature Science Fund of People’s Republic of China (Nos. 11372366 and 11101298), and Chongqing Outstanding Youth Fund (No. cstc2014jcyjjq 40004).
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[11] C. Cinar, On the positive solutions of the difference equation system xn +1 = 1/ yn , yn +1 = yn / xn −1 yn −1 . Appl. Math. Comput., 158, 303-305 (2004). [12] I. Yalcinkaya, On the global asymptotic behavior of a system of two nonlinear difference equations. ARS Combinatoria, 95, 151-159 (2010). [13] Q. Din, M. N. Qureshi, A. Q. Khan, Dynamics of a fourth-order system of rational difference equations. Adv. Differ. Equ., 2012, 2012: 215. [14] E. M. Elsayed, Solutions of rational difference systems of order two. Math. Comput. Model., 55, 378-384 (2012). [15] C. Y. Wang, S. Wang, W. Wang, Global asymptotic stability of equilibrium point for a family of rational difference equations. Appl. Math. Lett., 24, 714-718 (2011). [16] C. Y. Wang, S. Wang, Z. W. Wang, F. Gong, R. F. Wang, Asymptotic stability for a class of nonlinear difference equation. Dis. Dyn. Nat.Soc., Volume 2010, Article ID 791610, 10pages. [17] C. Y. Wang, F. Gong, S. Wang, L. R. LI, Q. H. Shi, Asymptotic behavior of equilibrium point for a class of nonlinear difference equation. Adv. Differ. Equ., Volume 2009, Article ID 214309. 8pages. [18] E. M. Elabbasy, H. El-Metwally, E. M. Elsayed, Global behavior of the solutions of difference equation, Adv. Differ. Equ., 2011, 2011:28. [19] E. M. Elsayed, Solution and attractivity for a rational recursive sequence, Dis. Dyn. Nat. Soc., Volume 2011, Article ID 982309, 17 pages. [20] Q. Zhang, L. Yang, J. Liu, Dynamics of a system of rational third order difference equation. Adv. Differ. Equ., 2012, 2012: 136. [21] M. Mansour, M. M. El-Dessoky, E. M. Elsayed, The form of the solutions and periodicity of some systems of difference equations, Dis. Dyn. Nat. Soc., Volume 2012, Article ID 406821, 17 pages. [22] Q. H. Shi, Q. Xiao, G. Q. Yuan, X. J. Liu, Dynamic behavior of a nonlinear rational difference equation and generalization. Adv. Diff. Equ., 2011, 2011:36. [23] A. S. Kurbanli, On the behavior of solutions of the system of rational difference equations, Adv. Differ. Equ., 2011, 2011:40. [24] E. M. Elsayed, Behavior and expression of the solutions of some rational difference equations. J. Comput. Anal. Appl., 15, 73-81 (2013). [25] O. Zkan, A. S. Kurbanli, On a system of difference equation. Dis. Dyn. Nat. Soc., Volume 2013, Article ID 970316, 7 pages. [26] L. Alsedà, M. Misiurewicz, A note on a rational difference equation. Journal of
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Difference Equations and Applications, 17, 1711-1713 (2011). [27] T. F. Ibrahim, Periodicity and Global Attractivity of Difference Equation of Higher Order. J. Comput. Anal. Appl., 16, 552-564 (2014). [28] E. M. Elsayed, H. El-Metwally, Stability and Solutions for Rational Recursive Sequence of Order Three, J. Comput. Anal. Appl., 17, 305-315 (2014). [29] H. Sedaghat, Nonlinear Difference Equations: Theory with Applications to Social Science Models, Kluwer Academic, Dordrecht, 2003. [30] M. R. Kulenovic, G. Ladas, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman Hall/CRC, Boca Raton, 2001.
Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this article.
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On Distributions of Discrete Order Statistics Y. Bulut1, M. Güngör2, B. Yüzbaşı3 , F. Özbey4 and E. Canpolat5 Department of Econometrics, Inonu University, 44280 Malatya, Turkey 4 Department of Statistics, Bitlis Eren University, 13000 Bitlis, Turkey 1 [email protected], [email protected], [email protected], [email protected] and 5 [email protected] 1,2,3,5
Abstract. In this study, the joint distributions of order statistics of innid discrete random variables are expressed in the form of an integral. Then, the results related to pf and df are given. 2010 Mathematics Subject Classification: 62G30, 62E15. Key words and phrases: Order statistics, discrete random variable, probability function, distribution function.
1. Introduction The joint probability density function(pdf) and marginal pdf of order statistics of independent but not necessarily identically distributed(innid) random variables was derived by Vaughan and Venables[22]
by means of permanents. In addition,
Balakrishnan[3], and Bapat and Beg[8] obtained the joint pdf and distribution function(df) of order statistics of innid random variables by means of permanents. In the first of two papers, Balasubramanian et al.[5] obtained the distribution of single order statistic in terms of distribution functions of the minimum and maximum order statistics of some subsets of { X 1 , X 2 ,..., X n } where X i ’s are innid random variables. Later, Balasubramanian et al.[6] generalized their previous results[5] to the case of the joint distribution function of several order statistics. Recurrence relationships among the distribution functions of order statistics arising from innid random variables were obtained by Cao and West[10]. Using multinomial arguments, the pdf of X r:n1 (1 r n+1) was obtained by Childs and Balakrishnan[11] by adding another independent random variable to the original n variables X 1, X 2 ,..., X n . Also, Balasubramanian et
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al.[7] established the identities satisfied by distributions of order statistics from nonindependent non-identical variables through operator methods based on the difference and differential operators. In a paper published in 1991, Beg[9] obtained several recurrence relations and identities for product moments of order statistics of innid random variables using permanents. Recently, Cramer et al.[13] derived the expressions for the distribution and density functions by Ryser’s method and the distributions of maxima and minima based on permanents. A multivariate generalization of classical order statistics for random samples from a continuous multivariate distribution was defined by Corley[12]. Guilbaud[17] expressed the probability of the functions of innid random vectors as a linear combination of probabilities of the functions of independent and identically distributed(iid) random vectors and thus also for order statistics of random variables. Expressions for generalized joint densities of order statistics of iid random variables in terms of RadonNikodym derivatives with respect to product measures based on df were derived by Goldie and Maller[16]. Several identities and recurrence relations for pdf and df of order statistics of iid random variables were established by numerous authors including Arnold et al.[1], Balasubramanian and Beg[4], David[14], and Reiss[21]. Furthermore, Arnold et al.[1], David[14], Gan and Bain[15], and Khatri[18] obtained the probability function(pf) and df of order statistics of iid random variables from a discrete parent. Balakrishnan[2] showed that several relations and identities that have been derived for order statistics from continuous distributions also hold for the discrete case. In a paper published in 1986, Nagaraja[19] explored the behavior of higher order conditional probabilities of order statistics in a attempt to understand the structure of discrete order statistics. Later, Nagaraja[20] considered some results on order statistics of a random sample taken from a discrete population. In general, the distribution theory for order statistics is complex when the parent distribution is discrete. In this study, the joint distributions of p order statistics of innid discrete random variables are obtained as an p fold integral.
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As far as we know, these approaches have not been considered in the framework of order statistics from innid discrete random variables. From now on, the subscripts and superscripts are defined in the first place in which they are used and these definitions will be valid unless they are redefined. Let X 1 , X 2 ,..., X n be innid discrete random variables and X 1:n X 2:n ... X n:n be the order statistics obtained by arranging the n X i ’s in increasing order of magnitude. Let Fi and f i be df and pf of X i (i =1, 2,…, n), respectively. For notational convenience we write
x1
,
z1 , z 2 ,..., z p
n rp
m p 0
rp 1 r p1
r3 1 r2
...
k p 0
m2 0
r2 1 r1
k2 0
,
m p , k p ..., m1 , k1
r2 1 r1
r1 1
m1 0
k1 0
and
instead of
Fir ( x2 )
1
, 1
Fir ( x p )
2
... Fi ( x1 ) Fi ( x2 ) r1
p
2
Fir ( x1 ) Fir ( x2 )
xp
x3
z1 0 z2 z1 z3 z2
V
Fir ( x1 )
x2
...
and
Fir ( x p ) p
0
r2
vi(1) r1
,
z p z p1
Fir ( x p ) p
...
in
vi( p1) r p1
the expressions below, respectively ( xi 0 , 1, 2 , . . .) ( z0 0) .
2. Theorems for distribution and probability functions In this section, the theorems related to pf and df of X r1 :n , X r2 :n ,..., X r p :n (1 r1 r2 ... rp n , p=1, 2,…, n) will be given. We will now express the following theorem for the joint pf of order statistics of innid discrete random variables.
Theorem 2.1.
p 1 rw 1 p f r1 , r2 ,...,rp :n ( x1 , x2 ,..., x p ) D [vi(l w ) vi(l w1) ] dvi(rw ) , w P w1 l rw1 1 w1
(2.1)
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where x1 x2 ... x p ,
denotes the sum over all n! permutations (i1 , i2 ,...,in ) of
P
p 1
(1,2,…,n),
D [ (rw rw 1 1)!]1 , r0 0 ,
rp 1 n 1 , vi(l0) 0 , vi(l p 1) 1 and
w1
vi(lw ) [vi(rww) Firw (xw )]
f il ( xw ) f irw ( xw )
Fil (xw ) .
Proof. Consider the event { X r1 :n x1 , X r2 :n x2 ,..., X rp :n x p } .
The above event can be realized mutually exclusive as follows: r1 1 k1 observations are less than x1 , kw 1 mw (w=1, 2,…, p) observations are equal to xw , r 1 k m 1 r 1 ( 2, 3,..., p ) observations are in interval ( x 1, x ) and n m p rp observation exceed x p . The probability function of the above event can be
written as f r1 , r2 ,..., r p :n ( x1 , x2 ,..., x p ) P{ X r1 :n x1, X r2 :n x2 ,..., X r p :n x p } .
(2.2)
(2.2) can be expressed as f r1 , r2 ,...,rp :n ( x1 , x2 ,..., x p )
mp , k p ...,m1 ,k1
C
P
p1 rw 1kw p rw mw [ Fil ( xw ) Fil ( xw1 ) ] fi j ( xw ) , w1 l rw1 mw1 1 w1 j rw kw
(2.3)
p1 p where C [(rw 1 kw mw1 rw1 )!]1 [(kw1 mw )!]1, m0 0, k p 1 0, Fil ( x0 ) 0 , w1 w1 Fil ( x p1 ) 1 , Fil ( xw ) P ( X il xw ) and mw 1 k w rw rw 1 1 (w =1, 2,…, p+1).
(2.3) can be written as
f r1 , r2 ,...,rp :n ( x1 , x2 ,..., x p )
m p ,k p ..., m1 , k1
C
P
p 1 rw 1 kw [ Fil ( xw ) Fil ( xw1 ) ] w 1 l rw1 mw1 1
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p
w1
rw mw 1 kw (k w 1 mw )! rw 1 m fi j ( xw ) firw ( xw ) f i j ( xw ) yw (1 yw ) w dyw . k w ! mw ! j rw kw j rw 1 0
(2.4)
Also, (2.4) can be clearly written as 1 1
f r1 ,r2 ,...,rp:n ( x1 , x2 ,..., x p )
m p ,k p ...,m1 , k1
P
1
... 0 0
0
1 Fi ( x1 ) Fi2 ( x1 )...Fir1 1k1 ( x1 ) y1 f ir1 k1 ( x1 ) y1 f ir1 k11 ( x1 )... y1 f ir1 1 ( x1 )dy1 f ir1 ( x1 ) ( r1 1 k1 )!k1! 1
1 (1 y1 ) f ir11 ( x1 )(1 y1 ) f ir1 2 ( x1 )...(1 y1 ) f ir1 m1 ( x1 )[ Fir1 m11 ( x2 ) Fir1 m11 ( x1 ) ]... ( r2 r1m1k 2 1)! m1! k 2 !
[ Fir2 k2 1 ( x2 ) Fir2 k 2 1 ( x1 ) ] y2 f ir2 k 2 ( x2 ) y 2 f ir2 k 2 1 ( x2 )...y 2 f ir2 1 ( x2 ) dy 2 f ir2 ( x2 )...
1 (1 y p1 ) f irp1 1 ( x p1 )(1y p1 ) f irp 1 2 ( x p1 )...(1y p1 ) f irp 1 m p1 ( x p1 ) (rp rp1 mp1 k p 1)!mp1!k p !
[ Firp1m p 1 1 ( xrp ) Firp1m p11 ( xrp 1 ) ]...[Firp k p 1 ( xrp ) Firp k p 1 ( xrp 1 ) ] y p f irp k p ( x p ) y p f irp k p 1 ( x p )...y p f irp 1 ( x p )dy p firp ( x p )
1 (1 y p ) f irp 1 ( x p )(1 y p ) f irp 2 ( x p )...(1 y p ) f irp m p ( x p )[1 Firp m p 1 ( x p ) ]...[1 Fin ( x p ) ]. (n rp m p )!m p !
The following expression can be written from the last identity.
1 1
f r1 , r2 ,..., rp :n ( x1 , x2 ,..., x p )
m p , k p ..., m1 , k1 P
1 p 1 1 ... 0 0 0 w 1 ( rw 1 k w mw 1 rw 1 )! mw 1! kw!
rw1 m w1 rw 1 k w (1 yw 1 ) f i1 ( xw 1 ) [ Fi 2 ( xw ) Fi 2 ( xw 1 ) ] 1 rw1 1 2 rw1 mw1 1 rw 1 p yw fi ( xw ) fir ( xw )dyw . 3 w 3 rw kw w 1
(2.5)
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In (2.5), if vi(jw) y w f i j ( x w ) Fi j ( xw ) , the following identity is obtained. Fir ( x1 ) Fir ( x2 )
fr1 ,r2 ,...,rp :n ( x1 , x2 ,..., x p )
m p ,k p ...,m1 , k1
Fir ( x p )
p 1 1 P ... ( rw 1 kw mw1 rw1 )! mw1 ! kw ! Fir ( x1 ) Fir ( x2 ) Fir ( x p ) w 1 1 2 p 1
2
p
rw1 mw1 rw1k w ( w 1) [Fi1 ( xw 1 ) v i ] [ Fi 2 ( xw) Fi2 ( xw 1 )] 1 1 rw11 2 rw1 mw11 rw 1 p ( w) [vi Fi ( xw )] dvi(rw ) . 3 w 3 3 rw kw w1
(2.6)
By considering n
n
0
0
(1) n (2) n 1 Gi ( x ) Gi ( x) Gi(3) ( x) P !(n )! ! 3 1 2 1 1 2 1 3 n 1
n 1 [Gi(l1) ( x ) Gi(l 2) ( x) Gi(l3) ( x)] , n ! P l 1
(2.7)
where n and using (2.7) for each mw 1 and kw in (2.6), we get f r1 ,r2 ,..., rp :n ( x1 , x2 ,..., x p ) p 1 1 w1 ( rw rw1 1)! P
rw 1
p 1 w1
p (w) ( w 1) [ F ( x ) F ( x ) v F ( x ) F ( x ) v ] dvi(rw) . il w il w 1 il w il w 1 il il w l rw11 w1
Thus, the proof is completed.
Specially,
in
Theorem
vi(32) [vi(22) Fi2 (x2 )] Fi1 ( x1 )
f1,2:3 ( x1 , x2 )
fi3 ( x2 ) f i2 ( x2 )
2.1,
by
taking
p 2,
n 3,
r1 1 ,
r2 2 ,
Fi3 (x2 ) and for x1 x2 ,
Fi2 ( x2 )
[1 vi(2) ] dvi(2) dvi(1) 3 2 1
P Fi ( x1 ) Fi ( x2 ) 1 2
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fi ( x2 ) 1 fi1 ( x1 ) fi2 ( x2 ) [ fi2 ( x2 ) Fi2 (x2 ) Fi2 (x2 ) fi2 ( x2 ) Fi2 (x2 )] 3 fi2 ( x2 ) Fi3 (x2 ) 2 f i2 ( x2 ) P 1 1 = f1 ( x1 ) f 2 ( x2 ) f3 ( x2 ) F2 ( x2 ) f3 ( x2 ) F2 ( x2 ) f 2 ( x2 ) F3 ( x2 ) 2 2 1 1 f1 ( x1 ) f 3 ( x2 ) f 2 ( x2 ) F3 ( x2 ) f 2 ( x2 ) F3 ( x2 ) f3 ( x2 ) F2 ( x2 ) 2 2 1 1 f 2 ( x1 ) f3 ( x2 ) f1 ( x2 ) F3 ( x2 ) f1 ( x2 ) F3 ( x2 ) f3 ( x2 ) F1 ( x2 ) 2 2 1 1 f 2 ( x1 ) f1 ( x2 ) f3 ( x2 ) F1 ( x2 ) f 3 ( x2 ) F1 ( x2 ) f1 ( x2 ) F3 ( x2 ) 2 2 1 1 f3 ( x1 ) f1 ( x2 ) f 2 ( x2 ) F1 ( x2 ) f 2 ( x2 ) F1 ( x2 ) f1 ( x2 ) F2 ( x2 ) 2 2 1 1 f 3 ( x1 ) f 2 ( x2 ) f1 ( x2 ) F2 ( x2 ) f1 ( x2 ) F2 ( x2 ) f 2 ( x2 ) F1 ( x2 ) . 2 2
Morever, the above identity in the iid case can be expressed as f1,2:3 ( x1 , x2 ) 6 f ( x1 ) f ( x2 ) 6 f ( x1 ) f ( x2 ) F ( x2 ) 3 f ( x1 ) f 2 ( x2 ) .
This result is obtained, if i 1 , j 2 and n 3 in equation (6) in [18]. In case x1 x2 ... x p , vi(1)r vi(r2) ... vi(rp ) is automatically satisfied because of 1
p
2
Fir1 x1 vi(1)r Fir1 x1 , Fir2 x2 vi(r2) Fir2 x2 , …, Firp x p vi(r p ) Firp x p . 1
2
p
Also, in case x1 x2 ... x p x , the integration region is over Fir1 x vi(1)r vi(r2) ... vi(rp ) Firp x , Fir1 x vi(1)r Fir1 x , 1
2
1
p
Fir2 x vi(r2) Fir2 x , …, Firp x vi(rp ) Firp x . 2
p
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So, if x1 x2 ... x p , it should be written
...
Fir ( x1 ) 1
instead of
Fir ( x p )
Fir ( x2 )
p
2
...
Fir ( x1 ) Fir ( x2 ) 1
in (2.1), where
...
Fir ( x p ) p
2
is to be carried out over the region: vi(1)r vi(r2) ... vi(r p ) , 1
p
2
Fir1 x1 vi(1)r Fir1 x1 , Fir2 x2 vi(r2) Fir2 x2 , …, Firp x p vi(r p ) Firp x p . 1
2
p
The proof was given only in case x1 x2 ... x p , the proof for case x1 x2 ... x p is omitted. Specially,
in
Theorem
vi(32) [vi(22) Fi2 (x2 )] Fi1 ( x )
f1,2:3 ( x, x)
P Fi ( x ) 1
fi3 ( x2 ) f i2 ( x2 )
2.1,
by
taking
p 2,
n 3,
r1 1 ,
r2 2 ,
Fi3 (x2 ) and for x1 x2 x ,
Fi2 ( x )
[1 vi(2) ] dvi(2) dvi(1) 3 2 1
vi(1) 1
1 1 Fi2 ( x) f i1 ( x) Fi1 ( x) Fi1 ( x ) fi1 ( x ) Fi22 ( x) f i1 ( x) 2 2 P fi ( x ) fi3 ( x ) 1 Fi2 (x) Fi2 (x ) 1 Fi1 ( x ) Fi1 ( x ) fi1 ( x ) Fi2 (x ) fi2 ( x ) 2
fi3 ( x)
f i ( x) 1 Fi13 ( x) Fi13 ( x ) 3 fi2 ( x) 6 f i2 ( x)
fi3 ( x ) f i2 ( x)
Fi2 ( x) Fi3 (x ) fi1 ( x )
1 Fi1 ( x ) Fi1 ( x ) fi1 ( x ) Fi3 (x ) 2 f ( x) 1 3 f ( x) 1 1 F2 ( x) f1 ( x) F1 ( x) F1 ( x ) f1 ( x ) F22 ( x ) f1 ( x) 3 F1 ( x) F13 ( x ) 3 2 2 f 2 ( x) 6 f 2 ( x) f ( x ) f 3 ( x) 1 f ( x) F2 (x) F2 (x ) 1 F1 ( x ) F1 ( x ) f1 ( x) F2 (x ) 3 F2 ( x ) F3 (x ) f1 ( x ) f 2 ( x) 2 f 2 ( x) 1 F1 ( x) F1 ( x ) f1 ( x ) F3 (x ) 2
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1 1 f ( x) 1 3 f ( x) F3 ( x) f1 ( x ) F1 ( x) F1 ( x ) f1 ( x) F32 ( x) f1 ( x) 2 F1 ( x ) F13 ( x ) 2 2 2 f 3 ( x) 6 f 3 ( x) f ( x) f 2 ( x) 1 f ( x) F3 (x ) F3 (x ) 1 F1 ( x ) F1 ( x ) f1 ( x ) F3 (x ) 2 F3 ( x) F2 (x ) f1 ( x) f3 ( x) 2 f 3 ( x) 1 F1 ( x) F1 ( x ) f1 ( x ) F2 (x ) 2 f ( x) 1 3 f ( x) 1 1 F1 ( x) f 2 ( x ) F2 ( x) F2 ( x ) f 2 ( x ) F12 ( x) f 2 ( x) 3 F2 ( x) F23 ( x ) 3 2 2 f1 ( x) 6 f1 ( x) f ( x) f 3 ( x ) 1 f ( x) F1 (x) F1 (x ) 2 F2 ( x ) F2 ( x ) f 2 ( x ) F1 (x ) 3 F1 ( x ) F3 (x ) f 2 ( x) f1 ( x) 2 f1 ( x ) 1 F2 ( x ) F2 ( x ) f 2 ( x ) F3 (x ) 2 1 1 f ( x) 1 3 f ( x) F3 ( x ) f 2 ( x) F2 ( x) F2 ( x ) f 2 ( x) F32 ( x) f 2 ( x) 1 F2 ( x) F23 ( x ) 1 2 2 f 3 ( x) 6 f 3 ( x) f ( x) f1 ( x) 1 f ( x) F3 (x) F3 (x ) 2 F2 ( x ) F2 ( x ) f 2 ( x ) F3 (x ) 1 F3 ( x) F1 (x ) f 2 ( x) f3 ( x) 2 f3 ( x) 1 F2 ( x ) F2 ( x ) f 2 ( x ) F1 (x ) 2 1 1 f ( x) 1 3 f ( x) F2 ( x) f3 ( x) F3 ( x ) F3 ( x ) f3 ( x ) F22 ( x ) f3 ( x) 1 F3 ( x) F33 ( x ) 1 2 2 f 2 ( x) 6 f 2 ( x) f ( x ) f1 ( x ) 1 f ( x) F2 (x) F2 (x ) 3 F3 ( x ) F3 ( x ) f3 ( x) F2 (x ) 1 F2 ( x) F1 (x ) f3 ( x) f2 ( x) 2 f2 ( x) 1 F3 ( x) F3 ( x ) f3 ( x) F1 (x ) 2 1 1 f ( x) 1 3 f ( x) F1 ( x) f3 ( x) F3 ( x) F3 ( x ) f3 ( x) F12 ( x) f 3 ( x) 2 F3 ( x) F33 ( x ) 2 2 2 f1 ( x) 6 f1 ( x ) f ( x) f 2 ( x ) 1 f ( x) F1 (x) F1 (x ) 3 F3 ( x ) F3 ( x ) f 3 ( x ) F1 (x ) 2 F1 ( x) F2 (x ) f 3 ( x ) f1 ( x) 2 f1 ( x )
1 F3 ( x) F3 ( x) f3 ( x) F2 (x) . 2
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Morever, the above identity in the iid case can be expressed as 6 F ( x ) f ( x ) 3 F ( x ) F ( x) f ( x) 3F 2 ( x) f ( x ) F 3 ( x ) F 3 ( x) 6 F ( x) F ( x) f ( x )
3 F ( x ) F ( x) F ( x) f ( x ) 6 F ( x ) F ( x) f ( x) 3 F ( x ) F ( x) f ( x ) F ( x) 6 F ( x ) f ( x ) 3 F ( x ) f ( x ) 3 F ( x) f ( x) 3 F 2 ( x) f ( x ) F 3 ( x ) F 3 ( x) 3 f 2 ( x ) 3F 2 ( x ) f ( x ) f ( x) 3F 2 ( x) 3F ( x ) f ( x ) f 2 ( x) f 3 ( x) 3 f 2 ( x) 1 F ( x ) .
This result is obtained, if r 1 , s 2 and n 3 in equation (2.4.3) in [14].
We will now express the following theorem to obtain the joint df of order statistics of innid discrete random variables. Theorem 2.2.
p 1 rw 1 Fr1 ,r2 ,..., rp :n ( x1 , x2 ,..., x p ) D [vi(l w ) vi(l w1) ] P V w 1 l rw1 1
p ( w) dvirw . w1
(2.8)
Proof. We have
Fr1 , r2 ,...,rp :n ( x1 , x2 ,..., x p )
f r1 , r2 ,...,rp :n ( z1 , z 2 ,..., z p ) .
(2.9)
z1 , z 2 ,..., z p
Using (2.1) in (2.9), (2.8) is obtained. 3. Results for distribution and probability functions
In this section, the results related to pf and df of
X r1 :n , X r2 :n ,..., X r p :n will be
given. We will express the following result for pf of the rth order statistic of innid discrete random variables. Result 3.1. F (x )
ir1 1 r1 1 1 n 1 vil [1 vil1 ] f r1:n ( x1 ) (r1 1)!(n r1 )! P Fi ( x1 ) l 1 l r1 1 r1
1 dvi . r1
(3.1)
Proof. In (2.1), if p 1 , (3.1) is obtained.
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In Result 3.2 and Result 3.3, the pf ’s of minimum and maximum order statistics of innid discrete random variables are given, respectively. Result 3.2. Fi ( x1 )
1 n 1 f1:n ( x1 ) [1 vil1 ] (n 1)! P Fi (x1 ) l 2 1
1 dvi1 .
(3.2)
Proof. Putting r1 1 in (3.1), one will get (3.2).
Result 3.3. Fi ( x1 )
n n 1 1 1 f n:n ( x1 ) vi (n 1)! P Fi (x1 ) l 1 l n
1 dvin .
(3.3)
Proof. On taking r1 n in (3.1), one will get (3.3).
In the following result, we will give the joint pf of X 1:n , X 2:n ,..., X p:n . Result 3.4. If x1 x2 ... x p , n p ( p) ( w) ... [1 v ] dviw , P l il p 1 w1 (1) is to be carried out over the region: vi vi( 2) ... vi( p ) ,
f1,2,..., p :n ( x1 , x2 ,..., x p )
where
...
1 (n p)!
1
2
(3.4)
p
Fi1 x1 vi(1) Fi1 x1 , Fi2 x2 vi( 2) Fi2 x2 , …, Fi p x p vi(pp ) Fi p x p . 1
2
Proof. On taking rw w for w 1, 2, ..., p and
...
instead of
in (2.1), one
will get (3.4). We will now give three results for the df of single order statistic of innid discrete random variables.
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Result 3.5. 1 Fr1 :n ( x1 ) (r1 1)!(n r1 )! P
Fir ( x1 ) 1
0
r1 1 1 n vil [1 vil1 ] l 1 l r1 1
1 dvi . r1
(3.5)
Proof. On taking p 1 in (2.8), one will get (3.5).
Result 3.6.
1 F1:n ( x1 ) (n 1)! P
Fi1 ( x1 )
0
n [1 vil1 ] l 2
1 dvi1 .
(3.6)
Proof. Putting r1 1 in (3.5), one will get (3.6). Result 3.7.
1 Fn:n ( x1 ) (n 1)! P
Fin ( x1 )
0
n 1 1 vil l 1
1 dvin .
(3.7)
Proof. On taking r1 n in (3.5), one will get (3.7). Specially, in (3.7), by taking n=2 and vi11 [vi21 Fi2 (x1 )]
f i1 ( x1 ) f i2 ( x1 )
Fi1 (x1 ) , the
following identity is obtained. Fi2 ( x1 )
F2:2 ( x1 )
P
vi11 dvi21
0
v1 i2 P 2
2
Fi ( x1 )
2 f ( x ) i 1 vi21 Fi2 ( x1 ) 1 vi21 Fi1 ( x1 ) fi2 ( x1 ) 0
Fi 2 ( x1 ) f i ( x1 ) 2 Fi2 ( x1 ) Fi2 ( x1 ) 1 Fi2 ( x1 ) Fi1 ( x1 ) 2 f i ( x1 ) P 2 F22 ( x1 ) f (x ) F2 ( x1 ) F2 ( x1 ) 1 1 F2 ( x1 ) F1 ( x1 ) 2 f 2 ( x1 ) F 2 (x ) f (x ) 1 1 F1 ( x1 ) F1 ( x1 ) 2 1 F1 ( x1 ) F2 ( x1 ). 2 f1 ( x1 ) 12 198
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Morever, the above identity for iid case can be expressed as F2:2 ( x1 ) F 2 ( x1 ).
Also, the above identity for x1 1 can be written as F2:2 (1) F 2 (1) 2
f (0) f (1) . In the following result, we will give the joint df of X 1:n , X 2:n ,..., X p:n . Result 3.8.
1 F1,2,..., p:n ( x1 , x2 ,..., x p ) (n p )!
Fi p ( x p )
Fi1 ( x1 ) Fi2 ( x2 )
P
0
...
vi(1) 1
vi( p1) p1
n p ( p) ( w) [1 ] v dviw . il l p 1 w1
(3.8)
Proof. On considering rw w for w 1, 2, ..., p from (2.8), one will get (3.8). References B. C. Arnold, N. Balakrishnan and H. N. Nagaraja, A first course in order statistics, John Wiley and Sons Inc., New York, 1992. [2] N. Balakrishnan, Order statistics from discrete distributions, Commun. Statist. Theory Meth. 15 (1986), no.3, 657-675. [3] N. Balakrishnan, Permanents, order statistics, outliers and robustness, Rev. Mat. Complut. 20 (2007), no.1, 7-107. [4] K. Balasubramanian and M. I. Beg, On special linear identities for order statistics, Statistics 37 (2003), no.4, 335-339. [5] K. Balasubramanian, M. I. Beg and R. B. Bapat, On families of distributions closed under extrema, Sankhyā Ser. A 53 (1991), no.3, 375-388. [6] K. Balasubramanian, M. I. Beg and R. B. Bapat, An identity for the joint distribution of order statistics and its applications, J. Statist. Plann. Inference 55 (1996), no.1, 13-21. [7] K. Balasubramanian, N. Balakrishnan and H. J. Malik, Identities for order statistics from nonindependent non- identical variables, Sankhyā Ser. B 56 (1994), no.1, 67-75. [8] R. B. Bapat and M. I. Beg, Order statistics for nonidentically distributed variables and permanents, Sankhyā Ser. A 51 (1989), no.1, 79-93. [9] M. I. Beg, Recurrence relations and identities for product moments of order statistics corresponding to nonidentically distributed variables, Sankhyā Ser. A 53 (1991), no.3, 365-374. [10] G. Cao and M. West, Computing distributions of order statistics, Commun. Statist. Theory Meth. 26 (1997), no.3, 755-764. [11] A. Childs and N. Balakrishnan, Relations for order statistics from non-identical logistic random variables and assessment of the effect of multiple outliers on bias of linear estimators, J. Statist. Plan. Inference 136 (2006), no.7, 2227-2253. [12] H. W. Corley, Multivariate order statistics, Commun. Statist. Theory Meth. 13 (1984), no.10, 1299-1304. [1]
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[13] E. Cramer, K. Herle and N. Balakrishnan, Permanent Expansions and Distributions of Order Statistics in the INID Case, Commun. Statist. Theory Meth. 38 (2009), no.12, 2078-2088. [14] H. A. David, Order statistics, John Wiley and Sons Inc., New York, 1970. [15] G. Gan and L. J. Bain, Distribution of order statistics for discrete parents with applications to censored sampling, J. Statist. Plann. Inference 44 (1995), no.1, 37-46. [16] C. M. Goldie and R. A. Maller, Generalized densities of order statistics, Statist. Neerlandica 53 (1999), no.2, 222-246. [17] O. Guilbaud, Functions of non-i.i.d. random vectors expressed as functions of i.i.d. random vectors, Scand. J. Statist. 9 (1982), no.4, 229-233. [18] C. G. Khatri, Distributions of order statistics for discrete case, Ann. Inst. Statist. Math. 14 (1962), no.1, 167-171. [19] H. N. Nagaraja, Structure of discrete order statistics, J. Statist. Plann. Inference 13 (1986), no.1, 165-177. [20] H. N. Nagaraja, Order statistics from discrete distributions, Statistics 23 (1992), no.3, 189-216. [21] R. -D. Reiss, Approximate distributions of order statistics, Springer-Verlag, New York, 1989. [22] R. J. Vaughan and W. N. Venables, Permanent expressions for order statistics densities, J. Roy. Statist. Soc. Ser. B 34 (1972), no.2, 308-310.
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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 20, NO. 1, 2016
On the λ-Daehee Polynomials With q-Parameter, Jin-Woo Park, ……………………………11 Stability of Ternary Quadratic Derivation on Ternary Banach Algebras: revisited, Choonkil Park,……………………………………………………………………………………………21 Some Properties of Modular S-Metric Spaces and its Fixed Point Results, Meltem Erden Ege and Cihangir Alaca,…………………………………………………………………………………24 The Strong Converse Inequality for de la Vallee Poussin Means on the Sphere, Chunmei Ding, Ruyue Yang, and Feilong Cao,…………………………………………………………………34 On the Fixed Point Method for Stability of a Mixed Type, AQ-Functional Equation, Ick-Soon Chang, and Yang-Hi Lee,……………………………………………………………………..…42 Differences of Composition Operators from Lipschitz Space to Weighted Banach Spaces in Polydisk, Chang-Jin Wang, and Yu-Xia Liang,…………………………………………………50 The Path Component of the Set of Generalized Composition Operators on the Bloch Type Spaces, Liu Yang,……………………………………………………………………………….56 The Generalized Hyers-Ulam Stability of Quadratic Functional Equations on Restricted Domains, Chang Il Kim, and Chang Hyeob Shin,………………………………………………65 Hesitant Fuzzy Soft Set and its Lattice Structures, Xiaoqiang Zhou, and Qingguo Li,…………72 Inclusion Properties for Certain Subclasses of Analytic Functions Associated With Bessel Functions, N. E. Cho, G. Murugusundaramoorthy, and T. Janani,…………………………….81 Barnes-type Narumi of the Second Kind and Poly-Cauchy of the Second Kind Mixed-Type Polynomials, Dae San Kim, Taekyun Kim, Takao Komatsu, Jong-Jin Seo, and Seog-Hoon Rim,……………………………………………………………………………………………...91 Superstability and Stability of (r,s,t)-J*-Homomorphisms: Fixed Point and Direct Methods, Shahrokh Farhadabadi, Choonkil Park, and Dong Yun Shin,…………………………………121 Differential Subordinations Obtained by Using a Generalization of Marx-Strohhäcker Theorem, Georgia Irina Oros, Gheorghe Oros, Alina Alb Lupas, and Vlad Ionescu,……………………135
TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 20, NO. 1, 2016 (continued) A Finite Difference Method for Burgers’ Equation in the Unbounded Domain Using Artificial Boundary Conditions, Quan Zheng, Yufeng Liu, and Lei Fan,………………………………140 Barnes-Type Peters Polynomials Associated with Poly-Cauchy Polynomials of the Second Kind, Dae San Kim, Taekyun Kim, Takao Komatsu, Hyuck In Kwon, and Sang-Hun Lee,…………151 On the Solution for a System of two Rational Difference Equations, Chang-you Wang, Xiao-jing Fang, and Rui Li,………………………………………………………………………………175 On Distributions of Discrete Order Statistics, Y. Bulut, M. Güngör, B. Yüzbaşı, F. Özbey, and E. Canpolat,…………………………………………………………………………………..…..187
Volume 20, Number 2 ISSN:1521-1398 PRINT,1572-9206 ONLINE
February 2016
Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC
203
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 (fourteen 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,Lenoir-Rhyne University,Hickory,NC
28601, USA.
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Editorial Board Associate Editors of Journal of Computational Analysis and Applications Francesco Altomare Dipartimento di Matematica Universita' di Bari Via E.Orabona, 4 70125 Bari, ITALY Tel+39-080-5442690 office +39-080-3944046 home +39-080-5963612 Fax [email protected] Approximation Theory, Functional Analysis, Semigroups and Partial Differential Equations, Positive Operators.
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
Partial Differential Equations, Semigroups of Operators H. H. Gonska Department of Mathematics University of Duisburg Duisburg, D-47048 Germany 011-49-203-379-3542 e-mail: [email protected] Approximation Theory, Computer Aided Geometric Design
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.
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
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
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
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
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
Christodoulos A. Floudas Department of Chemical Engineering Princeton University Princeton,NJ 08544-5263 609-258-4595(x4619 assistant) e-mail: [email protected] Optimization Theory&Applications, Global Optimization
Margareta Heilmann Faculty of Mathematics and Natural Sciences, University of Wuppertal Gaußstraße 20 D-42119 Wuppertal, Germany, [email protected]
J .A. Goldstein Department of Mathematical Sciences The University of Memphis Memphis, TN 38152 901-678-3130 [email protected]
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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
Approximation Theory (Positive Linear Operators) Xing-Biao Hu Institute of Computational Mathematics AMSS, Chinese Academy of Sciences Beijing, 100190, CHINA [email protected] 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
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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
Approximation Theory, Banach spaces, Classical Analysis 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. Anfithea - Paleon Faliron GR-175 64 Athens, Greece [email protected] Numerical Analysis
Choonkil Park Department of Mathematics Hanyang University Seoul 133-791 S. Korea, [email protected] Functional Equations
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 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
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 Boris Shekhtman Department of Mathematics University of South Florida Tampa, FL 33620, USA Tel 813-974-9710 [email protected]
Roberto Triggiani Department of Mathematical Sciences University of Memphis Memphis, TN 38152 PDE, Control Theory, Functional
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Analysis, [email protected]
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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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
Lotfi A. Zadeh Professor in the Graduate School and Director, Computer Initiative, Soft Computing (BISC) Computer Science Division University of California at Berkeley Berkeley, CA 94720 Office: 510-642-4959 Sec: 510-642-8271 Home: 510-526-2569 FAX: 510-642-1712 [email protected] Fuzzyness, Artificial Intelligence, Natural language processing, Fuzzy logic Richard A. Zalik Department of Mathematics Auburn University Auburn University, AL 36849-5310
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A Recurrent Neural Fuzzy Network George A. Anastassiou Department of Mathematical Sciences, University of Memphis Memphis, TN 38152, U.S.A. [email protected] Iuliana F. Iatan Department of Mathematics and Computer Science Technical University of Civil Engineering Bucharest [email protected] Abstract Besides the feedforward neural networks, there are the recurrent networks, where the impulses can be transmitted in both directions due to some reaction connections in these networks. Recurrent neural networks are linear or nonlinear dynamic systems. The dynamic behavior presented by the recurrent neural networks can be described both in continuous time, by differential equations and at discrete times by the recurrence relations (difference equations). The distinction between recurrent (or dynamic) neural networks and static neural networks is due to recurrent connections both between the layers of neurons of these networks and within the same layer, too. The aim of this paper is to describe a Recurrent Fuzzy Neural Network (RFNN) model, whose learning algorithm is based on the Improved Particle Swarm Optimization (IPSO) method. Each particle (candidate solution), which is moving permanently includes the parameters of the membership function and the weights of the recurrent neuralfuzzy network; initially, their values are randomly generated. The RFNN presented in this paper is unlike the others variants of RFNN models, by the number of the evolution directions that they use: in this paper, we update the velocity and the position of all particles along three dimensions, while in [8] are used two dimensions. Keywords: recurrent networks; Improved Particle Swarm Optimization method; fuzzy rules; Wavelet Neural Network; feedback weight; delayed operator.
1
Introduction
Neural network (NN) is one of the important components in Artificial Intelligence (AI). NN architectures used in modelling of the nervous systems can be classified into three categories, each with a different philosophy: feedforward, recurrent (feedback), self-organizing map. Neural networks (NNs) are used in many different application domains in order to solve various information processing problems. For several years now, neural network models have enjoyed wide popularity [4], being applied to problems of regression, classification, computational science, computer vision, data processing and time series analysis. 1
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Figure 1: Schematic diagram of the WNN. The main drawback of the feedforward neural networks is that the updating of the weights can fall [17] in a local minimum. An other major drawback of the feedforward neural networks consists in the fact that their application domain is limited to static problems due their inherent feedforward structure. Since recurrent networks incorporate feedback, they have powerful representation capability and can [17] successfully overcome disadvantages of feedforward networks. This feedback implies that the network has [12] local memory characteristics that is able to store activity patterns and present those patterns to the network more than once, allowing the layer with feedback connections to use its own past activation in its preceding behavior. The Recurrent Neural Network (RNN) has the feedforward and feedback connections contrasted which provides it with nonlinear mapping capacity and dynamical characteristics, so it can be used [22] to simulate dynamical system and solve dynamic problems. Different architectures can be created [12] by adding recurrent connections at different points in the basic feedforward architecture. Recently some researchers have proposed several recurrent neuro- fuzzy networks. Kumar et al. 2004 compares the traditional feedforward approach of RNNs to forecast monthly river flows. Lin & Hsu, 2007 has proposed [10] a recurrent wavelet-based neuro- fuzzy system with the reinforcement hybrid evolutionary learning algorithm for solving various control problems. Carcano et al., 2008 has simulated [3] daily river flows for water resource purposes using the Jordan Recurrent Neural Network. Maraqua et al., 2012 has proposed [12] the use of a recurrent network architecture as a classification engine for automatic ˇ Arabic Sign Language recognition system. Ster, 2013 has introduced [18] an extended architecture of recurrent neural networks (called Selective Recurrent Neural Network ) for dealing with long term dependencies.
1.1
Wavelet Neural Networks
Neural networks employing wavelet neurons are referred to as Wavelet Neural Networks(WNNs) [10]; they are characterized by weights and wavelet bases. Lin & Chin, 2004 was proposed a Recurrent Neural Fuzzy Network (RNFN) where each fuzzy rule corresponding to a WNN (see Figure 1) consists (see [11], [8]) of single-scaling wavelets. The shape and position of the wavelet bases are shown [11] in Figure 2. An ordinary wavelet neural network model is often used to normalize input vectors in the interval [0, 1]. The functions φa.b (xi ) are used to input vectors 2
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Figure 2: Wavelet bases are over-complete and compactly supported. to fire up the wavelet interval; a such value is given in the following equation, which gives the shape of the M wavelet bases φ0.0 , φ1.0 , . . . , φm.m : φ(xi ) = cos(xi ), −0.5 ≤ xi ≤ 0.5 (1) 0 otherwise, φa.b (xi ) = cos(axi − b), b = 1, a, a = 1, m, b being a shifting parameter and a meaning a scaling parameter corresponding to the maximum value of b. A crisp value ϕa.b can be obtained as follows: Pn j=1 φa.b (xi ) ϕa.b = , (2) |X| where |X| represents the number of input dimensions and n is the dimension of the input vector to the model.
1.2
Z- transform
The Z- transform is [20] the discrete- time counterpart of the Laplace transform. The Z- transform can be considered to be an extension of the discrete- time Fourier transform as the Laplace transform can be considered an extension of the Fourier transform. The bilateral Z- transform of a discrete- time sequence x(n) is: ∞ X
Z{x(n)} = X(z) =
x(n)z −n .
(3)
n=−∞
For causal sequences (n ≥ 0) the Z- transform becomes: Z{x(n)} = X(z) =
∞ X
x(n)z −n .
(4)
n=0
The equation (4) is called the unilateral Z- transform; it exists only if the power series from its expression converges. 3
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There are several methods for computing the inverse Z- transform, namely the sequence x(n), given X(z): 1. using the inversion integral : 1 x(n) = 2πj
I
X(z)z n−1 dz,
(5)
Γ
H where Γ means the integration along the closed contour Γ in the counterclockwise closed contour in the region of convergence of X(z); 2. by a power series expansion: expressing X(z) in a power series in z −1 , x(n) can be achieved by identifying it with the coefficient of z −n in the power series expansion; 3. by partial fraction expansion: for a rational functions, can be obtained a partial fraction expansion of X(z) over its poles and the table of Ztransform helps to identify the sequences corresponding to the terms in that partial fraction expansion.
1.3
Application of Genetic Algorithms
The specialists think that the Genetic Algorithms are a computational intelligence application as well as the expert systems, fuzzy systems, neural networks, the intelligent agents, hybrid intelligent systems, electronic voice. The genetic algorithms are some adaptive techniques of heuristic search, based on the genetic and selection natural principles, enunciated by Darwin (the best adapted will survive). The mechanism is similar to the evolutionary biological process. This process has a feature through that only the species which one adapt better to the environment are capable to survive and to develop into generations, while that those less adapted fail to survive and they disappear in time, as a result of the natural selection. The main notions that allow the analogy between the solution of the search problems and the natural evolution are: 1. Population. A population consists in some individuals (chromosomes) that have to live in an environment to which they must adapt. 2. Fitness. Each of the population individuals is adapted more or less to the environment. The fitness is a measure of the degree of adaptation to the environment. 3. Chromosome. It is a ordered set of elements, named genes, whose values establish the individual features. 4. Generation. A stage in a population evolution. If we see evolution as an iterative process in which a population turns to another population, then the generation is an iteration in this process. 5. Selection. The process of natural selection has the survival of individuals with a high environmental fitness (high fitness) as effect.
4
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Figure 3: The RNFN architecture. 6. Reproduction. It is the process through which one passes from one generation to another. The individuals of the new generation inherit some features from their precursors (parents) but they can also get some new features as a result of some processes of mutation that have a random character. In the case when in the reproduction process at least two parents occur, the inherited features of the survivor (son) are obtained by combining (crossover) of the parent features. The remainder of the paper is organized as follows. In Section 2 is discussed and analyzed the RNFN. We follow with the learning algorithm of the recurrent model in Section 3. We conclude in Section 4.
2
RNFN Architecture
The network construction is based on fuzzy rules, each corresponding to a Wavelet Neural Network (WNN). The figure Figure 3 illustrates the RNFN model, whose training algorithm is based on Improved Particle Swarm Optimization (IPSO) method. The nodes from the first layer constitute some input nodes; hence they only pass the input signal to the next layer, namely: (1)
Oi
(1)
= xi .
(6)
The neurons in the second layer act as a membership function, meaning that they determine how an input value belongs to a fuzzy set. The following Gaussian function is chosen as the membership function:
5
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Figure 4: Delayed cell.
2
(2) Oij
−
(2) −mij ) (Iij σ2 ij
=e
,
(7)
where: • mij and σij are the mean and standard deviation, respectively; (2)
• Iij denotes the input of this layer for the discrete time scan: (2)
(2)
Iij = Oi
(f )
+ Oij ,
(8)
where (f )
(2)
Oij = Oij (t − 1)θij .
(9) (2)
The inputs of this layer contain the terms of memory Oij (t − 1), that store network information at a previous time; this information, which is an additional input of the network will be reintroduced at the entrance of the second layer. The weight θij constitutes the feedback weight of the network and z −1 signifies the delayed operator. Figure 4 represents [14] a delayed cell, X(z) being the Z- transform of the signal x[n]. The neurons of the third layer achieve the product operation of their input signals: 2
Oj3
=
n Y
(2) Oij
i=1
=
n Y
−
(2) −mij ) (Iij
e
σ2 ij
,
(10)
i=1
where n is the number of external dimensions. The neurons of the fourth layer receive both the output of a WNN, denoted yˆj and of a neuron from the third layer, namely Oj3 . The mathematical function of each node j is: Oj4 = yˆjp · Oj3 , yˆjp
(11)
being the local output of the WNN for the output yp and the j-th rule: yˆjp =
M X
p wjk ϕa.b ,
(12)
k=1
with ϕa.b from (2), where: • M = m+1 denotes the number of wavelet bases, which equals the number of existing fuzzy rules in the considered model, 6
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p • the link wjk is the output action strength associated with in the p output, j-th rule and k-th ϕa.b .
The fifth layer acts as a defuzzifier namely it provides the nonfuzzy outputs yp of the fuzzy recurrent neural network: 1
yp = 1+e
PM O4 j −λ· Pj=1 M O3 j=1 j
1
= −λ·
1+e
PM p y ˆ ·O 3 j=1 j j PM O3 j=1 j
,
(13)
namely: 1
yp = 1+e
3
PM p p p (w ϕ1.1 +w ϕ2.1 +...+w ϕ )·O 3 j2 j jM m.m −λ· j=1 j1 PM O3 j=1 j
, λ ∈ Fv then Cv = Co C R = Cv Cm = CR Cr = Cm Cu = Cr . (2) Else if Fo > FR and Fo < Fv then CR = Co Cm = CR Cr = Cm Cu = Cr . (3) Else if Fo > Fm and Fo < FR then Cm = Co Cr = Cm Cu = Cr . (4) Else if Fo > Fr and Fo < Fm then Cr = Co Cu = Cr . (5) Else if Fo > Fu and Fo < Fr then Cu = Co . (6) Else if Fo = Fu = Fr = Fm = FR = Fv then Co = Co + Nr (Nr ∈ [0, 1]). (7) Else if Fo 0 and that our assumption holds for n ¡ 1. That is; x6n¡8 = x6n¡6 = x6n¡4 = x6n¡3 =
x¡2 xn¡1 xn¡1 0 ¡3 n¡1 xn¡1 ¡1 x¡4
n¡2 Y
(1+(3i+1)x¡1 x¡4 ) ; (1+(3i+2)x0 x¡3 )
x6n¡7 =
n¡1 xn ¡1 x¡4
xn¡1 xn¡1 0 ¡3
n¡2 Y
(1+(3i+1)x0 x¡3 ) ; (1+(3i+3)x¡1 x¡4 )
i=0 i=0 n¡2 n¡2 Y 1+(3i+2)x x n¡1 Y n xn xn 1+(3i+2)x¡1 x¡4 0 x¡3 ¡1 x¡4 0 ¡3 ; x = ; n¡1 n¡1 n¡1 n¡1 6n¡5 1+(3i+3)x0 x¡3 1+(3i+4)x¡1 x¡4 x¡1 x¡4 x¡2 x0 x¡3 (1+x¡1 x¡4 ) i=0 i=0 n¡2 Y ³ (1+(3i+3)x x ) ´ n xn ¡1 ¡4 0 x¡3 ; n¡1 (1+(3i+4)x0 x¡3 ) xn ¡1 x¡4 (1+x0 x¡3 ) i=0 n¡2 Y ³ (1+(3i+3)x x ) ´ n xn 0 ¡3 ¡1 x¡4 : n¡1 n (1+(3i+5)x¡1 x¡4 ) x0 x¡3 (1+2x¡1 x¡4 ) i=0
Now, it follows from Eq.(2) that x6n¡2 =
x6n¡4 x6n¡7 x6n¡5 (1+x6n¡4 x6n¡7 )
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Q
µ
= µ =
1+
0 @
³
=
³
=
x¡2 xn¡1 xn¡1 0 ¡3 (1+x¡1 x¡4 ) n¡2 Q³
n xn 0 x¡3
n¡1 xn ¡1 x¡4 (1+x0 x¡3 )
i=0
xn xn ¡1 ¡4 n¡1 (1+x x¡2 xn¡1 x ¡1 x¡4 ) 0 ¡3
x¡2 xn¡1 xn¡1 0 ¡3 (1+x¡1 x¡4 ) n xn ¡1 x¡4
(1+(3i+3)x¡1 x¡4 ) (1+(3i+4)x0 x¡3 )
´ n¡2 Q³ i=0
´ n¡2 Q³
n x¡2 xn 0 x¡3 (1+x¡1 x¡4 ) n xn ¡1 x¡4
´
n¡2 Q³
n¡1 xn ¡1 x¡4
xn¡1 xn¡1 0 ¡3
i=0
(1+(3i+4)x¡1 x¡4 ) (1+(3i+2)x0 x¡3 )
(1+(3i+4)x¡1 x¡4 ) (1+(3i+2)x0 x¡3 )
n x¡2 xn 0 x¡3 n xn x ¡1 ¡4
Similarly x6n¡1 =
i=0
n¡1 Y³ i=0
´
´
´¶
(1+(3i+1)x0 x¡3 ) (1+(3i+3)x¡1 x¡4 )
Q
n¡2 µ i=0
´¶
1 ¶ (1+(3i+1)x0 x¡3 ) A (1+(3i+4)x0 x¡3 )
x0 x¡3 µ (1+(3n¡2)x0 x¡3 ) ¶ x0 x¡3 1+ (1+(3n¡2)x0 x¡3 )
1 : (1+(3n¡1)x0 x¡3 )
(1+(3i+1)x¡1 x¡4 ) (1+(3i+2)x0 x¡3 )
´
:
x6n¡3 x6n¡6 x6n¡4 (1 + x6n¡3 x6n¡6 )
Q
Q
¶ n n¡1 n¡2µ ¶ n¡2µ n x0 x xn (1+(3i+3)x0 x¡3 ) (1+(3i+2)x¡1 x¡4 ) ¡1 x¡4 ¡3 n¡1 n¡1 (1+(3i+5)x¡1 x¡4 ) xn¡1 (1+(3i+3)x0 x¡3 ) xn 0 x¡3 (1+2x¡1 x¡4 ) i=0 ¡1 x¡4 i=0
= µ
=
i=0
(1+(3i+2)x0 x¡3 ) (1+(3i+4)x¡1 x¡4 )
n¡2 Q µ (1+(3i+1)x0 x¡3 ) ¶ x0 x¡3 (1+x0 x¡3 ) (1+(3i+4)x0 x¡3 ) i=0 10 n¡2 Q µ (1+(3i+2)x0 x¡3 ) ¶ x0 x¡3 A@1+ 1+(3i+4)x¡1 x¡4 ) (1+x0 x¡3 ) ( i=0
x6n¡2 =
=
n¡2 Q³
n xn ¡1 x¡4
Hence, we have
=
Q
¶ n n¡1 n¡2µ ¶ n¡2 µ n xn (1+(3i+3)x¡1 x¡4 ) x¡1 x¡4 (1+(3i+1)x0 x¡3 ) 0 x¡3 n¡1 n¡1 n¡1 n (1+(3i+4)x0 x¡3 ) x0 x¡3 i=0 (1+(3i+3)x¡1 x¡4 ) x¡1 x¡4 (1+x0 x¡3 ) i=0
1+
n xn ¡1 x¡4
n xn 0 x¡3
n¡1 xn ¡1 x¡4 (1+x0 x¡3 )
n¡1 xn 0 x¡3 (1+2x¡1 x¡4 )
n¡2 Q³ i=0
n xn 0 x¡3 @ n¡1 (1+x x xn x 0 ¡3 ) ¡1 ¡4
³ xn
n¡1 ¡1 x¡4 (1+x0 x¡3 n xn 0 x¡3
i=0
(1+(3i+3)x¡1 x¡4 ) (1+(3i+4)x0 x¡3 )
Q
n¡2 µ
´
n¡1 xn 0 x¡3
n¡1 xn¡1 ¡1 x¡4
¶
(1+(3i+2)x¡1 x¡4 ) (1+(3i+5)x¡1 x¡4 ) 10 µ ¶ n¡2 Q (1+(3i+3)x¡1 x¡4 ) x¡1 x¡4 A@1+ (1+2x¡1 x¡4 ) (1+(3i+4)x0 x¡3 ) i=0
´ n¡2 Q³ )
n¡1 xn ¡1 x¡4 (1+x0 x¡3 ) n xn 0 x¡3
Hence, we have
n¡2 Q³
(1+(3i+3)x0 x¡3 ) (1+(3i+5)x¡1 x¡4 )
x¡1 x¡4 (1+2x¡1 x¡4 )
0
³
µ
i=0
´ n¡2 Q³ i=0
i=0
(1+(3i+4)x0 x¡3 ) (1+(3i+3)x¡1 x¡4 ) (1+(3i+4)x0 x¡3 ) (1+(3i+3)x¡1 x¡4 )
x6n¡1 =
n xn+1 ¡1 x¡4 n n x0 x¡3
n¡1 Q³ i=0
´ ´
´¶
n¡2 Q³ i=0
Q
n¡2µ i=0
(1+(3i+2)x¡1 x¡4 ) (1+(3i+3)x0 x¡3 )
´¶
1 ¶ (1+(3i+2)x¡1 x¡4 ) A (1+(3i+5)x¡1 x¡4 )
x¡1 x¡4 µ (1+(3n¡1)x¡1 x¡4 ) ¶ x¡1 x¡4 1+ (1+(3n¡1)x¡1 x¡4 ) x¡1 x¡4 : (1+(3n)x¡1 x¡4 )
(1+(3i+1)x0 x¡3 ) (1+(3i+3)x¡1 x¡4 )
´
:
Similarly, we can easily obtain the other relations. Thus, the proof is completed. 4
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Theorem 2 Eq.(2) has x = 0 as a unique equilibrium point and it is unstable. Proof: For the equilibrium points of Eq.(2), set x= Then
x2 : x (1 + x2 )
¢ ¡ ¢ ¡ x2 1 + x2 = x2 ; ) x2 1 + x2 ¡ 1 = 0; ) x4 = 0:
Thus the equilibrium point of Eq.(2) is x = 0: Let f : (0; 1)3 ¡! (0; 1) be a function de…ned by f (t; u; v; w) =
vw : u(1+vw)
Thus the linearized equation of Eq.(2) about the equilibrium point x is given by yn+1 =
4 P
i=0
@f (x;x;x;x) : @xn¡i
The proof follows by Theorem A. Numerical examples For con…rming the results of this section, we consider some numerical examples which represent di¤erent types of solutions to Eq.(2). Example 1. Consider Eq.(2) with x¡4 = 0:21; x¡3 = 2; x¡2 = 0:5; x¡1 = 7; x0 = 0:3: See Fig. 1. Example 2. Consider Eq.(2) with x¡4 = 9; x¡3 = 2; x¡2 = 6; x¡1 = 7; x0 = 3. See Fig. 2. plot of x(n+1)= x(n-1)x(n-4)/(x(n-2)(1+x(n-1)x(n-4))
plot of x(n+1)= x(n-1)x(n-4)/(x(n-2)(1+x(n-1)x(n-4))
7
9 8
6 7 5 6 4 x(n)
x(n)
5 4
3
3 2 2 1 1 0
0
10
20
30
40
50
60
70
80
0
90
n
0
10
20
30
40
50
60
70
80
90
n
Figure 1.
Figure 2.
5
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3
On the Equation xn+1 =
xn¡1xn¡4 xn¡2 (¡1+xn¡1 xn¡4 )
In this section we obtain the solution of the di¤erence equation xn¡1 xn¡4 xn+1 = ; n = 0; 1; :::; xn¡2(¡1 + xn¡1 xn¡4 )
(3)
where the initial values are arbitrary non zero real numbers with x¡1 x¡4 6= 1; x¡3 x0 6= 1: Theorem 3 Every solution fxn g1 n=¡4 of Eq.(3) has the form 2n 2n
2n
x0 x¡3 (¡1+x¡1 x¡4 )n n ; 2n¡1 (¡1+x x2n ¡3 x0 ) ¡1 x¡4 2n 2n x x0 x¡3 (¡1+x¡1 x¡4 )n x12n¡2 = ¡2 ; 2n (¡1+x¡3 x0 )n x2n ¡1 x¡4 2n+1 2n n x x ¡1 x¡4 ) x12n = x02n x2n¡3 (¡1+x ; (¡1+x¡3 x0 )n ¡1 ¡4 2n+1 2n+1 x x (¡1+x¡1 x¡4 )n ; x12n+2 = x0 2n+1 x¡32n (¡1+x x )n+1 ¡1
x12n+4 =
x12n¡3 =
¡3 0
¡4
2n+1 x¡2 x02n+1 x¡3 (¡1+x¡1 x¡4 )n+1 ; 2n+1 2n+1 (¡1+x¡3 x0 )n x¡1 x¡4 2n+2 2n+1 n x0 x¡3 (¡1+x¡1 x¡4 )
x12n+6 =
2n+1 2n+1 x¡1 x¡4 (¡1+x¡3 x0 )n+1
2n
x¡1 x¡4 (¡1+x¡3 x0 )n n; 2n¡1 (¡1+x x2n ¡1 x¡4 ) 0 x¡3 2n+1 2n x x (¡1+x¡3 x0 )n x12n¡1 = x¡12n x2n¡4 (¡1+x n; ¡1 x¡4 ) 0 ¡3 2n+1 2n+1 x x¡4 (¡1+x¡3 x0 )n x12n+1 = x¡1 2n x2n n+1 ; x ¡2 0 ¡3 (¡1+x¡1 x¡4 ) 2n+1 2n+1 x x (¡1+x¡3 x0 )n x12n+3 = x¡12n+1 x¡42n (¡1+x n; ¡1 x¡4 ) ¡3 0 2n+2 2n+1 n+1 x x¡4 (¡1+x¡3 x0 ) x12n+5 = x¡1 ; 2n+1 2n+1 x (¡1+x x )n+1
x12n¡4 =
0
;
x12n+7 =
¡3
2n+2 2n+2 x¡1 x¡4
2n+1 x¡2 x2n+1 x¡3 0
¡1 ¡4
(¡1+x¡3 x0 )n : (¡1+x¡1 x¡4 )n+1
Proof: For n = 0 the result holds. Now suppose that n > 0 and that our assumption holds for n ¡ 1. That is; x12n¡16 = x12n¡14 = x12n¡12 = x12n¡10 = x12n¡8 = x12n¡6 =
2n¡2 x02n¡2 x¡3 (¡1+x¡1 x¡4 )n¡1 ; 2n¡2 2n¡3 x¡1 x¡4 (¡1+x¡3 x0 )n¡1
x12n¡15 =
2n¡2 x¡2 x2n¡2 x¡3 (¡1+x¡1 x¡4 )n¡1 0 ; 2n¡2 2n¡2 (¡1+x¡3 x0 )n¡1 x¡1 x¡4 2n¡2 x02n¡1 x¡3 (¡1+x¡1 x¡4 )n¡1 ; 2n¡2 2n¡2 x¡1 x¡4 (¡1+x¡3 x0 )n¡1
x12n¡13 =
x12n¡9 =
2n¡1 x¡2 x2n¡1 x¡3 (¡1+x¡1 x¡4 )n 0 ; 2n¡1 2n¡1 x¡1 x¡4 (¡1+x¡3 x0 )n¡1 2n¡1 x2n (¡1+x¡1 x¡4 )n¡1 0 x¡3 ; 2n¡1 2n¡1 (¡1+x¡3 x0 )n x¡1 x¡4
x12n¡7 =
2n¡1 2n¡2 x¡1 x¡4 (¡1+x¡3 x0 )n¡1 ; 2n¡2 (¡1+x¡1 x¡4 )n¡1 x02n¡2 x¡3 2n¡1 2n¡1 x¡1 x¡4
x12n¡11 =
2n¡1 x02n¡1 x¡3 (¡1+x¡1 x¡4 )n¡1 ; 2n¡1 2n¡2 (¡1+x¡3 x0 )n x¡1 x¡4
2n¡2 2n¡2 x¡1 x¡4 (¡1+x¡3 x0 )n¡1 ; 2n¡3 x02n¡2 x¡3 (¡1+x¡1 x¡4 )n¡1
(¡1+x¡3 x0 )n¡1 n 2n¡2 2n¡2 (¡1+x x¡2 x0 x¡3 ¡1 x¡4 ) 2n¡1 2n¡1 x¡1 x¡4 (¡1+x¡3 x0 )n¡1 2n¡1 2n¡2 x0 x¡3 (¡1+x¡1 x¡4 )n¡1
;
;
2n¡1 x2n ¡1 x¡4
(¡1+x¡3 x0 )n n; 2n¡1 (¡1+x x02n¡1 x¡3 ¡1 x¡4 )
x12n¡5 =
2n x2n ¡1 x¡4
(¡1+x¡3 x0 )n¡1 n : 2n¡1 (¡1+x x¡2 x02n¡1 x¡3 ¡1 x¡4 )
Now, it follows from Eq.(3) that x12n¡4 = =
=
x12n¡6 x12n¡9 x12n¡7 (¡1+x12n¡6 x12n¡9 ) Ã
2n¡1 x2n (¡1+x¡1 x¡4 )n¡1 0 x¡3 2n¡1 2n¡1 (¡1+x¡3 x0 )n x¡1 x¡4 Ã 2n¡1 2n¡1 x2n x2n (¡1+x¡3 x0 )n ¡1 x¡4 0 x¡3 ¡1+ 2n¡1 n 2n¡1 2n¡1 (¡1+x¡1 x¡4 ) x x x x2n¡1 0 ¡3 ¡1 ¡4
Ã
2n¡1 x2n ¡1 x¡4 2n¡1 2n¡1 x0 x¡3
µ
!Ã
!
n¡1 x2n¡1 x2n¡1 (¡1+x ¡3 x0 ) ¡1 ¡4 2n¡1 2n¡2 x0 x¡3 (¡1+x¡1 x¡4 )n¡1 ! 2n¡1 2n¡1 (¡1+x¡1 x¡4 )n¡1 x¡1 x¡4 (¡1+x¡3 x0 )n¡1 2n¡2 (¡1+x¡3 x0 )n x02n¡1 x¡3 (¡1+x¡1 x¡4 )n¡1
¶ x0 x¡3 (¡1+x¡3 x0 ) !µ µ ¶¶ (¡1+x¡3 x0 )n x0 x¡3 ¡1+ n (¡1+x¡1 x¡4 ) (¡1+x¡3 x0 )
=
2n n x2n 0 x¡3 (¡1+x¡1 x¡4 ) n ; 2n¡1 (¡1+x 2n x¡1 x¡4 ¡3 x0 )
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x12n¡3 = = =
x12n¡5 x12n¡8 x12n¡6 (¡1+x12n¡5 x12n¡8 ) 2n¡1 x2n¡1 2n x2n (¡1+x¡3 x0 )n¡1 x¡2 x0 (¡1+x¡1 x¡4 )n ¡1 x¡4 ¡3 n 2n¡1 2n¡1 2n¡1 2n¡1 (¡1+x¡1 x¡4 ) x¡2 x x x x (¡1+x¡3 x0 )n¡1 0 ¡1 ¡4 á3 ! n¡1 2n 2n n¡1 n 2n¡1 2n¡1 x2n¡1 ¡1+x x¡1 x¡4 (¡1+x¡3 x0 ) x2n x¡2 x0 ¡1 x¡4 ) 0 x¡3 (¡1+x¡1 x¡4 ) ¡3 ( ¡1+ 2n¡1 2n¡1 x2n¡1 ¡1+x n n¡1 2n¡1 x2n¡1 ¡1+x x¡1 x¡2 x2n¡1 x¡3 (¡1+x¡1 x¡4 )n x¡1 ¡3 x0 ) ¡3 x0 ) 0 ¡4 ( ¡4 (
2n¡1 2n¡1 x¡1 x¡4 2n¡1 x2n 0 x¡3
(¡1+x¡3 x0 )n x¡1 x¡4 (¡1+x¡1 x¡4 )n¡1 (¡1+x¡1 x¡4 )
=
2n x2n ¡1 x¡4
(¡1+x¡3 x0 )n n; 2n¡1 (¡1+x x2n x ¡1 x¡4 ) 0 ¡3
Similarly, we can easily obtain the other relations. Thus, the proof is completed. p Theorem 4 Eq.(3) has three equilibrium points which are x = 0 and x = § 2and all of them are unstable. Proof: The proof is similar to Theorem 2 and will be omitted. Lemma 1. It is easy to see that every solution of Eq.(3) is unbounded except in the case x¡3 x0 = x¡1 x¡4 . Theorem 5 Eq.(3) has a periodic solution of period twelve i¤ x¡3 x0 = x¡1 x¡4 : Moreover the periodic solution has the following form ( ) x¡1 x¡4 x0 x¡3 x¡4 ; x¡3 ; x¡2 ; x¡1 ; x0 ; x¡2 (¡1+x ; ; x ; x (¡1 + x x ) ; ¡3 ¡2 ¡1 ¡4 ¡1 x¡4 ) x¡1 (¡1+x¡3 x0 ) : x¡1 x¡4 x¡1 ; (¡1+xx0¡3 x0 ) ; x¡2 (¡1+x ; x¡4 ; x¡3 ; ::: ¡1 x¡4 ) Proof: First suppose that there exists a prime period twelve solution of Eq.(3) of the following form ) ( x¡1 x¡4 x0 x¡3 ; ; x ; x (¡1 + x x ) ; x¡4 ; x¡3 ; x¡2 ; x¡1 ; x0 ; x¡2 (¡1+x ¡3 ¡2 ¡1 ¡4 ¡1 x¡4 ) x¡1 (¡1+x¡3 x0 ) : x¡1 x¡4 x¡1 ; (¡1+xx0¡3 x0 ) ; x¡2 (¡1+x ; x ; x ; ::: ¡4 ¡3 ¡1 x¡4 ) Then we see from Theorem 3 that x12n¡4 = x12n¡2 = x12n = x12n+2 = x12n+3 = x12n+4 = x12n+5 =
2n n x2n 0 x¡3 (¡1+x¡1 x¡4 ) n 2n¡1 (¡1+x 2n x ) x¡1 x¡4 ¡3 0
2n n x¡2 x2n 0 x¡3 (¡1+x¡1 x¡4 ) n 2n (¡1+x x ) x2n x ¡3 0 ¡1 ¡4 n x02n+1 x2n ¡3 (¡1+x¡1 x¡4 ) n 2n 2n x¡1 x¡4 (¡1+x¡3 x0 )
x12n¡3 =
=x ;
x
= x0 ; x
2n+1 x02n+1 x¡3 (¡1+x¡1 x¡4 )n 2n+1 2n x¡1 x¡4 (¡1+x¡3 x0 )n+1 2n+1 2n+1 x¡1 x¡4 (¡1+x¡3 x0 )n n (¡1+x x02n+1 x2n ¡1 x¡4 ) ¡3
2n x2n ¡1 x¡4
(¡1+x¡3 x0 )n n 2n¡1 (¡1+x 2n x0 x¡3 ¡1 x¡4 ) 2n+1 2n x¡1 x¡4 (¡1+x¡3 x0 )n n 2n ¡2 12n¡1 x2n 0 x¡3 (¡1+x¡1 x¡4 ) 2n+1 2n+1 x¡1 x¡4 (¡1+x¡3 x0 )n n+1 12n+1 2n x¡2 x0 x2n ¡3 (¡1+x¡1 x¡4 )
= x¡4 ;
=
= x¡3 ;
=
=
=
= x¡1 ; x¡1 x¡4 ; x¡2 (¡1+x¡1 x¡4 )
x0 x¡3 ; x¡1 (¡1+x¡3 x0 )
= x¡3 ;
2n+1 x¡2 x2n+1 x¡3 (¡1+x¡1 x¡4 )n+1 0 2n+1 2n+1 (¡1+x¡3 x0 )n x¡1 x¡4 2n+2 2n+1 x¡1 x¡4 (¡1+x¡3 x0 )n+1 2n+1 2n+1 x0 x¡3 (¡1+x¡1 x¡4 )n+1
= x¡2 (¡1 + x¡1 x¡4 ) ;
= x¡1 ;
x12n+6 =
2n+1 x02n+2 x¡3 (¡1+x¡1 x¡4 )n 2n+1 2n+1 x¡1 x¡4 (¡1+x¡3 x0 )n+1
x12n+7 =
x¡1 x¡4 (¡1+x¡3 x0 )n 2n+1 n+1 x¡2 x2n+1 x (¡1+x ¡1 x¡4 ) 0 ¡3
=
x0 ; (¡1+x¡3 x0 )
2n+2 2n+2
=
x¡1 x¡4 : x¡2 (¡1+x¡1 x¡4 )
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Then we get (¡1 + x¡3 x0 ) = (¡1 + x¡1 x¡4 ) : Second assume that (¡1 + x¡3 x0 ) = (¡1 + x¡1 x¡4 ) : Then we see from the form of the solution of Eq.(3) that x12n¡4 = x¡4 ; x12n¡3 = x¡3 ; x12n¡2 = x¡2 ; x12n¡1 = x¡1 ; x12n = x0 ; x¡1 x¡4 x0 x¡3 x12n+1 = x¡2 (¡1+x ; x12n+2 = x¡1 (¡1+x ; x12n+3 = x¡1xx0 ¡4 = x¡3 ; ¡1 x¡4 ) ¡3 x0 ) x12n+4 = x¡2 (¡1 + x¡1 x¡4 ) ; x12n+5 = x¡1 ; x¡1 x¡4 x0 x12n+6 = ¡1+x ; x12n+7 = x¡2 (¡1+x : ¡3 x0 ¡1 x¡4 )
Thus we have a periodic solution of period twelve and the proof is complete. Theorem 6 n Eq.(3) has a periodic solution of period six oi¤ x¡1 x¡4 = x¡3 x0 = 2 and 2 has the form x¡4 ; x¡3 ; x¡2 ; x¡1 ; x0 ; x¡2 ; x¡4 ; x¡3 ; x¡2 ; ::: :
Proof: The proof is consequently from the previous Theorems and will be omitted. In the following we present some …gures illustrate the behavior of the solutions of Eq.(3) under some di¤erent initial values. plot of x(n+1)= x(n-1)x(n-4)/(x(n-2)(-1+x(n-1)x(n-4))
plot of x(n+1)= x(n-1)x(n-4)/(x(n-2)(-1+x(n-1)x(n-4))
60
3
50 2.5
40
x(n)
x(n)
2 30
1.5 20
1
10
0
0
10
20
30 n
40
50
0.5
60
Figure 3. x¡4 = 3; x¡3 = 2; x¡2 = 5; x¡1 = 4; x0 = 6:
0
10
20
30 n
40
50
60
Figure 4. x¡4 = 3; x¡3 = 2; x¡2 = 3; x¡1 = 2=3; x0 = 1:
The following cases can be treated similarly.
4
On the Equation xn+1 =
xn¡1xn¡4 xn¡2 (1¡xn¡1 xn¡4 )
In this section we get the solution of the third following equation xn+1 =
xn¡1 xn¡4 ; xn¡2 (1¡xn¡1 xn¡4 )
n = 0; 1; :::;
(4)
where the initial values are arbitrary positive real numbers.
8
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Theorem 7 Assume that fxn g1 n=¡4 be a solution of Eq.(4). Then for n = 0; 1; ::: x6n¡2 = x6n = x6n+2 = x6n+3 =
n x¡2 xn 0 x¡3 n n x¡1 x¡4
xn+1 xn ¡3 0 n x¡1 xn ¡4
n¡1 Y³
i=0 n¡1 Y³ i=0
xn+1 xn+1 0 ¡3
(1¡(3i+1)x¡1 x¡4 ) (1¡(3i+2)x0 x¡3 )
1¡(3i+2)x¡1 x¡4 1¡(3i+3)x0 x¡3
n xn+1 ¡1 x¡4 (1¡x0 x¡3 )
n¡1 Y³
; x6n¡1 =
i=0
´
n xn+1 ¡1 x¡4 n n x0 x¡3
n¡1 Y³ i=0
(1¡(3i+1)x0 x¡3 ) (1¡(3i+3)x¡1 x¡4 )
n+1 xn+1 ¡1 x¡4 n n x¡2 x0 x¡3 (1¡x¡1 x¡4 )
; x6n+1 =
(1¡(3i+3)x¡1 x¡4 ) (1¡(3i+4)x0 x¡3 )
i=0 n¡1 Y³
n+1 xn+1 ¡1 x¡4 n+1 n x0 x¡3 (1¡2x¡1 x¡4 )
´
´
n¡1 Y³ i=0
´
;
1¡(3i+2)x0 x¡3 1¡(3i+4)x¡1 x¡4
´
;
;
(1¡(3i+3)x0 x¡3 ) (1¡(3i+5)x¡1 x¡4 )
´
:
Theorem 8 Eq.(4) has the unique equilibrium point x = 0 and it is unstable. Example 3. Consider Eq.(4) with x¡4 = 3; x¡3 = 5; x¡2 = 2; x¡1 = 2=3; x0 = 0:4: See Fig. 5. plot of x(n+1)= x(n-1)x(n-4)/(x(n-2)(1-x(n-1)x(n-4)) 5
4
3
x(n)
2
1
0
-1
-2
-3
0
10
20
30
40
50 n
60
70
80
90
100
Figure 5.
5
On the Equation xn+1 =
xn¡1 xn¡4 xn¡2 (¡1¡xn¡1 xn¡4)
Here we obtain the analytical form of the solutions of the equation xn+1 =
xn¡1 xn¡4 ; xn¡2 (¡1¡xn¡1 xn¡4 )
(5)
n = 0; 1; :::;
where the initial values are arbitrary non zero real numbers with x¡1 x¡4 6= ¡1; x¡3 x0 6= ¡1: Theorem 9 Let fxn g1 n=¡4 be a solution of Eq.(5). Then for n = 0; 1; 2; ::: the solution of Eq.(5) is given by x12n¡4 = x12n¡2 =
2n n x2n 0 x¡3 (¡1¡x¡1 x¡4 ) n ; 2n¡1 (¡1¡x 2n x¡1 x¡4 ¡3 x0 )
x12n¡3 =
2n n x¡2 x2n 0 x¡3 (¡1¡x¡1 x¡4 ) n ; 2n (¡1¡x x ) x2n x ¡3 0 ¡1 ¡4
x12n¡1 =
2n x2n ¡1 x¡4
(¡1¡x¡3 x0 )n n 2n¡1 (¡1¡x 2n x0 x¡3 ¡1 x¡4 ) 2n+1 2n x¡1 x¡4 (¡1¡x¡3 x0 )n n 2n (¡1¡x x2n x ¡1 x¡4 ) 0 ¡3
;
;
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x12n = x12n+2 = x12n+4 = x12n+6 =
n x2n+1 x2n 0 ¡3 (¡1¡x¡1 x¡4 ) n ; 2n (¡1¡x x ) x2n x ¡3 0 ¡1 ¡4
x12n+1 =
2n+1 x2n+1 x¡3 (¡1¡x¡1 x¡4 )n 0 ; 2n+1 2n x¡1 x¡4 (¡1¡x¡3 x0 )n+1
2n+1 2n+1 x¡1 x¡4 (¡1¡x¡3 x0 )n 2n n+1 ; x¡2 x2n x 0 ¡3 (¡1¡x¡1 x¡4 )
x12n+3 =
x¡2 x02n+1 x2n+1 (¡1¡x¡1 x¡4 )n+1 ¡3 ; 2n+1 2n+1 (¡1¡x¡3 x0 )n x¡1 x¡4 2n+1 x2n+2 x¡3 (¡1¡x¡1 x¡4 )n 0 ; 2n+1 2n+1 x¡1 x¡4 (¡1¡x¡3 x0 )n+1
2n+1 x2n+1 (¡1¡x¡3 x0 )n ¡1 x¡4 n; 2n+1 2n (¡1¡x x0 x¡3 ¡1 x¡4 )
x12n+5 =
x12n+7 =
2n+2 2n+1 x¡1 x¡4 (¡1¡x¡3 x0 )n+1 ; 2n+1 2n+1 x0 x¡3 (¡1¡x¡1 x¡4 )n+1
2n+2 x2n+2 ¡1 x¡4
2n+1 x¡2 x02n+1 x¡3
(¡1¡x¡3 x0 )n : (¡1¡x¡1 x¡4 )n+1
Theorem 10 Eq.(5) has x = 0 as a unique equilibrium point which is unstable. Lemma 2. It is easy to see that every solution of Eq.(5) is unbounded except in the case x¡3 x0 = x¡1 x¡4 . Theorem 11 Eq.(5) has a periodic solution of period twelve i¤ x¡3 x0 = x¡1 x¡4 . x¡1 x¡4 Moreover the periodic solution has the form fx¡4 ; x¡3 ; x¡2 ; x¡1 ; x0 ; x¡2 (¡1¡x ; ¡1 x¡4 ) x0 x¡3 ; x¡3 ; x¡2 x¡1 (¡1¡x¡3 x0 )
x¡1 x¡4 x0 (¡1 ¡ x¡1 x¡4 ) ; x¡1 ; ¡1¡x ; ; x¡4 ; :::g ¡3 x0 x¡2 (¡1¡x¡1 x¡4 )
Theorem 12 Eq.(5) has a periodic solution of period six i¤ x¡3 x0 = x¡1 x¡4 = ¡2 and will be taken the form fx¡4 ; x¡3 ; x¡2 ; x¡1 ; x0 ; x¡2 ; x¡4 ; :::g: ¡2 Example 4. Fig. 6 below shows the behavior of the solution of Eq.(5) whenever x¡4 = 3; x¡3 = 5; x¡2 = ¡7; x¡1 = 4; x0 = 2. plot of x(n+1)= x(n-1)x(n-4)/(x(n-2)(-1-x(n-1)x(n-4)) 80 70 60 50
x(n)
40 30 20 10 0 -10
0
50
100
150
n
Figure 6.
Acknowledgements This Project was funded by the Deanship of Scienti…c Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia under grant no. (145/662/1434). The authors, therefore, acknowledge with thanks DSR technical and …nancial support. Last, but not least, sincere appreciations are dedicated to all our colleagues in the Faculty of Science, Rabigh branch for their nice wishes. 10
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References [1] R. P. Agarwal and E. M. Elsayed, On the solution of fourth-order rational recursive sequence, Advanced Studies in Contemporary Mathematics, 20 (4) (2010), 525–545. axn¡1 [2] C. Cinar, On the positive solutions of the di¤erence equation xn+1 = 1+bx ; n xn¡1 Appl. Math. Comp., 156 (2004) 587-590. [3] E. M. Elabbasy , H. El-Metwally and E. M. Elsayed, Global behavior of the solutions of di¤erence equation, Advances in Di¤erence Equations 2011, 2011:28 doi:10.1186/1687-1847-2011-28. [4] E. M. Elabbasy and E. M. Elsayed, Global attractivity and periodic nature of a di¤erence equation, World Applied Sciences Journal, 12 (1) (2011), 39–47. [5] H. El-Metwally, On the dynamics of a higher order di¤erence equation, Discrete Dynamics in Nature and Society, Volume 2012 (2012), Article ID 263053, 8 pages. [6] H. El-Metwally and E. M. Elsayed, Form of solutions and periodicity for systems of di¤erence equations, Journal of Computational Analysis and Applications, 15(5) (2013), 852-857. [7] H. El-Metwally and E. M. Elsayed, Solution and Behavior of a Third Rational Di¤erence Equation, Utilitas Mathematica, 88 (2012), 27–42. [8] E. M. Elsayed, Dynamics of recursive sequence of order two, Kyungpook Mathematical Journal, 50 (2010), 483-497. [9] E. M. Elsayed, Solutions of rational di¤erence system of order two, Mathematical and Computer Modelling, 55 (2012), 378–384. [10] R. Karatas and A. Geli¸sken, Qualitative behavior of a rational di¤erence equation, Ars Combinatoria, 100 (2011), 321–326. [11] R. Karatas, C. Cinar and D. Simsek, On positive solutions of the di¤erence n¡5 equation xn+1 = 1+xxn¡2 ; Int. J. Contemp. Math. Sci., 1(10) (2006), 495-500. xn¡5 [12] V. L. Kocic and G. Ladas, Global Behavior of Nonlinear Di¤erence Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, 1993. [13] M. R. S. Kulenovic and G. Ladas, Dynamics of Second Order Rational Di¤erence Equations with Open Problems and Conjectures, Chapman & Hall / CRC Press, 2001. [14] C. Wang, F. Gong, S. Wang, L. Li and Q. Shi, Asymptotic behavior of equilibrium point for a class of nonlinear di¤erence equation, Advances in Di¤erence Equations, Volume 2009, 2009, Article number 214309 [15] I. Yalç¬nkaya, On the di¤erence equation xn+1 = ® + xn¡m , Discrete Dynamics xkn in Nature and Society, Vol. 2008, Article ID 805460, 8 pages, doi: 10.1155/2008/ 805460. [16] I. Yalç¬nkaya, C. Cinar and M. Atalay, On the solutions of systems of di¤erence equations, Advances in Di¤erence Equations, Vol. 2008, Article ID 143943, 9 pages, doi: 10.1155/2008/ 143943. [17] E. M. E. Zayed and M. A. El-Moneam, On the rational recursive sequence xn+1 = ®+¯xn +°xn¡1 ; Communications on Applied Nonlinear Analysis, 12 (4) (2005), 15–28. A+Bxn +Cxn¡1
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Worse-Case Conditional Value-at-Risk for Asymmetrically Distributed Asset Scenarios Returns Zhifeng Dai1,2 1
Donghui Li3 ,
Fenghua Wen2
∗
College of Mathematics and Computational Science, Changsha University of Science and Technology, Changsha 410114, China. 2
3
College of business, Central South University, Hunan, 410083, China.
School of Mathematical Sciences, South China Normal University Guangzhou, 510631, China.
Abstract: Many studies have reported empirical evidence of asymmetries in asset return distributions. Meanwhile, optimal solutions to the Conditional Value-at-Risk (CVaR) minimization are highly susceptible to estimation error of the risk measure because the estimate depends on only a small portion of sampled scenarios. In this paper, based on the robust optimization techniques Chen et al.(2007)[19], we propose a computationally tractable worst-case Conditional Value-at-Risk (CVaR). In the situation, the sampled scenario returns are generated by a factor model with some asymmetric affine uncertainty set. The remarkable characteristic of the new method is that the robust optimization model retains the complexity of original portfolio optimization problem, i.e., the robust counterpart problem is still a linear programming problem. Moreover, it takes into consideration asymmetries in the distributions of scenarios returns used for defining CVaR. We present some numerical experiments with simulated and real market data to illustrate the behavior of the robust optimization model. Keywords: Portfolio optimization, Conditional value at risk(CVaR), Robust optimization, Linear programming(LP).
1.
Introduction
Portfolio optimization problem is an attractive and important research topic since the pioneering Markowitz work on optimal portfolio selection [1]. It is now well known that while mean-variance optimization is appropriate for symmetrically distributed portfolio returns, it results in unsatisfactory asset allocations when returns are asymmetrically distributed, or when downside risk is more weighted than upside risk. ∗
E-mail: [email protected].
1
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Since the middle of 1990s, Value-at-Risk (VaR, [4]), a new measure of downside risk, has become popular in financial risk management. It has even been recommended as a standard on banking supervision by the Basel Committee. However, Critics have pointed out numerous shortcomings of VaR [5]. On the other hand, Conditional Value-at-Risk (CVaR), defined as the mean of the tail distribution exceeding VaR, has attracted much attention in recent years. As a measure of risk, CVaR exhibits some better properties than VaR. First, it can deal with the asymmetric distribution of asset return better than mean-variance analysis, especially for assets with returns that are heavy-tailed. Secondly, minimizing CVaR usually results in solving a convex programming problem, such as a linear programming problem, which allows the decision maker to deal with a large-scale portfolio problem efficiently [6, 7]. Finally, Artzner et al.[5] demonstrate that CVaR is a coherent measure of risk, which has been widely accepted as a benchmark to evaluate risk measures. All these stimulate the application of CVaR in practice, and CVaR is getting more and more popular in financial management. In fact, it is noted that in the process of portfolio selection, the original data brought to the model are not always accurate, i.e., it may be subject to some errors. Thus the result may be influenced by perturbations in the parameters. As pointed out by Black and Litterman [8], in the classical mean-variance model, the portfolio decision is very sensitive to the mean and the covariance matrix, especially to the mean. Chopra and Ziemba [9] showed that small changes in the input parameters can result in large changes in the optimal portfolio allocation. Thus, the modeling risk arises due to the uncertainty of the underlying probability distribution. Being aware of the importance of robustness in recent years, researchers from both finance and operations research have paid increasing attentions to the robust version of portfolio selection problems. Lobo and Boyd (2000)[10], Goldfarb and Iyengar (2003)[11] studied the robust portfolio problem under the mean-variance framework. Instead of assuming precise information on the mean and the covariance matrix of asset returns, they introduced some types of uncertainties, such as polyhedral uncertainty, box uncertainty and ellipsoidal uncertainty, in the parameters in determining the mean and the covariance matrix, and they then transformed the problem into semidefinite programs(SDP) or second-order cone programs(SOCP), which can be efficiently solved by interior-point algorithms developed in recent years. Halld´orsson and T¨ ut¨ uncu (2004) [12] applied their interior-point method for saddle-point problems to the robust mean-variance portfolio selection under the box uncertainty of the elements in the mean vector and the covariance matrix. El Ghaoui, Oks and Oustry (2003)[13] investigated the robust portfolio optimization problem using worst-case VaR, where only the first- and second-moment information on the distribution is available. Several formulations corresponding to various structures of partial information have been extensively exploited to derive the resulting portfolio selection problems in a form of a semidefinite program(SDP). Natarajan, Pachamanova, and Sim, (2008) [14] proposed a computationally tractable approximation method for minimizing the VaR of a portfolio based on robust optimization techniques in Chen et al.(2007)[19]. The method results in the optimization of a modified VaR
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measure, Asymmetry-Robust VaR, that takes into consideration asymmetries in the distributions of returns and is coherent. Zhu and Fukushima (2009)[15] further investigated the worst-case CVaR risk measure with several structures of uncertainty in the underlying distribution. They focus on the uncertainty in the probability distribution used for defining CVaR. Such a modeling is called distributionally robust modeling. It is true that the probability estimation itself is under uncertainty and we cannot know the true one. However, it is not easy to imagine what form of uncertainty set is proper for the probability measure. In this sense, employing the uncertainty of probability distribution may not provide investors with a satisfactory solution. On the other hand, since the estimate of CVaR is computed by using only an upper tail part of the loss distribution, a large number of samples are required for assuring the statistical reliability of the estimate. Especially when CVaR is employed as the objective of a portfolio optimization, a much larger number of samples are required for ensuring the accuracy of the optimal portfolio. In practice, however, the number of samples which is available for the estimation is limited, and the estimated CVaR and the resulting optimal portfolio may contain considerable estimation error. Meanwhile, many studies have reported empirical evidence of asymmetries and large kurtosis in asset return distributions. Empirically, however, there is evidence that both short- and longhorizon stock returns can be skewed and highly leptokurtic (Fama 1976 [22], Duffee 2002 [23]). Furthermore, the returns of portfolios involving derivatives or credit risky assets can have extremely left-skewed distributions (Sch¨onbucher 2000 [24]). More recently, Ang and Chen (2002)[25] find that the asymmetries in the data reject the null hypothesis of multivariate normal distributions. Conine and Tamarkin (1981) [26] also claim that though diversification can change skewness exposure, the remaining idiosyncratic skewness is relevant in asset pricing and thus portfolio optimization under asymmetric distribution is a significant topic for research. In this paper, we further study the Worse-Case Conditional Value-at-Risk by supposing the sampled scenario returns are generated by a factor model with some asymmetric affine uncertainty set in order to Mitigate the fragility of CVaR-based portfolio optimization problem. Motivated by the works in Chen et al.(2007)[19], we provide a computationally tractable robust optimization method for minimizing the Worse-Case CVaR of a portfolio. Moreover, it takes into consideration asymmetries in the distributions of returns used for defining CVaR. Notations: Throughout this paper, we use boldface letter such as x for vector to distinguish it from scalar x.
2.
Conditional value-at-risk (CVaR)
The conditional value-at-risk (CVaR) has gained growing popularity in financial risk management due to the coherence property and tractability in its optimization. Let f (x , y ) be the loss associated with the decision vector x , to be chosen from a certain subset X of Rn , and the random vector y in Rm . For convenience, the underling probability of y will be
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assumed to have a density function p(.). The probability of f (x , y ) not exceeding a threshold α is then given by Z
Ψ(x , α) =
f (x ,y )≤α
p(y )dy .
(2.1)
As a function of α for fixed x , Ψ(x , α) is the cumulative distribution function for the loss associated with x . For a confidence level β and a fixed x ∈ X the value-at-risk, denoted by VaRβ (x ) is defined as VaRβ (x ) = min{α ∈ R : Ψ(x , α) ≥ β}.
(2.2)
The conditional value-at-risk, denoted by CVaRβ (x ), is defined as the expected value of the loss that exceeds VaRβ (x ), that is, CVaRβ (x ) = (1 − β)
−1
Z f (x ,y )≥VaRβ (x )
f (x , y )p(y )dy .
(2.3)
The CVaR is a coherent risk measure [5]. We note that the problem involved CVaRβ (x ) is difficult to proceed due to its convoluted and implicit version. Rockafellar and Uryasev made a remarkable contribution in [6] by introducing a simpler auxiliary function Fβ on X × R, defined by Fβ (x , α) = α + (1 − β)−1
Z
y
[f (x , y ) − α]+ p(y )dy ,
(2.4)
∈Rm
In practice, the probability density function p(y ) is often not available, or is very difficult to estimate. Instead, we might have T different scenarios Y = (y [1] , y [2] , . . . , y [T ] ) that are sampled from the probability distribution or that have been obtained from computer simulations. Evaluating the auxiliary function F˜β (x , α) using the scenarios Y, we have F˜β (x , α) = α + (1 − β)−1
T X
πt [f (x , y [t] ) − α]+ ,
(2.5)
t=1
where y [t] denotes the tth sample (the subscript [t] is used to distinguish a vector from a scalar) generated by simple random sampling with respect to x according to its density function p(.), and T denotes the number of samples, where πt are probabilities of scenarios y [t] . If πt is equal to T−1 for all t, then (2.5) reduces to F˜α (x , α) = α +
T X 1 [f (x , y [t] ) − α]+ . T (1 − β) t=1
(2.6)
Obviously, F˜α (x , α) is convex and piecewise linear with respect to α. Further, F˜α (x , α) is convex for (x , α) if f (x , y ) is convex (see Theorem 2 in [6]). Replacing [f (x , y [t] ) − α]+ by the auxiliary variables dt along with appropriate constraints, we obtain the equivalent optimization problem min n T (x ,d ,α)∈R ×R ×R s.t.
α+
T X 1 dt , T (1 − β) t=1
x ∈X dt ≥ f (x , y [t] ) − α,
t = 1, . . . , T,
(2.7)
d ≥ 0. 4
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Generally, the loss and return functions of portfolio allocation are chosen by: f (x , y ) = −x T y ,
Rp (x) = Ep [x T y ] = x T Ep [y ] = x T r ,
(2.8)
in which y is the vector of the assets’ return, r is the vector of the expected assets’ return, and x T r is the mean return of the portfolio. Hence, adding an auxiliary variable θ ∈ R, the minimization model of CVaR (2.9) becomes the following linear programming (LP) problem with variables (x , d , α, θ) ∈ Rn × RT × R × R. min
θ
s.t.
x ∈X α+
T X 1 dt ≤ θ, T (1 − β) t=1
dt ≥ −x T y [t] − α,
(2.9)
t = 1, . . . , T,
d ≥ 0. Portfolio optimization tries to find an optimal trade-off between the risk and the return according to the investor’s preference. Thus, the portfolio selection problem using CVaR as a risk measure can be represented as min CVaRβ (x ) x ∈X where X denotes the constraint on the portfolio position, which usually includes the budget constraint and no short sales constraint x T 1 = 1,
x ≥ 0.
(2.10)
Let µ be the smallest expected return of the portfolio required by the investor. From (2.8), this return requirement can be represented as x T r ≥ µ.
(2.11)
Therefore, the feasible decision set of portfolios can be denoted as X = {x |
x T 1 = 1,
x T r ≥ µ}.
x ≥ 0,
(2.12)
From (2.9) and 2.12, the mean-CVaR Portfolio optimization can be be written as the following linear program min
θ
s.t.
α+
T X 1 dt ≤ θ, T (1 − β) t=1
dt ≥ −x T y [t] − α,
t = 1, . . . , T,
(2.13)
d ≥ 0. x T 1 = 1,
x ≥ 0, 5
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3.
Worst-Case Conditional value-at-risk (CVaR)
However, optimal solutions to the CVaR minimization are highly susceptible to estimation error of the risk measure because the estimate depends on only a small portion of sampled scenarios, for example Y = (y [1] , y [2] , . . . , y [T ] ). A practical way to alleviate the effect of such a perturbation is to employ a statistical model. For example, Konno, Waki and Yuuki (2002) replace the observed returns Y = (y [1] , y [2] , . . . , y [T ] ) in 2.6 with values estimated by a regression approach. Based on the robust optimization techniques in Chen et al.(2007)[19], we suppose that future asset returns r˜ are generated by the following factor model r = r 0 + ∆r z , z ∈ C
(3.1)
in which r 0 is a vector of expected returns, and ∆r is a matrix of factor loadings. The factors z are stochastically independent with following support set n
o
−1 −1 C = z : ∃v, w ∈ RN + , z = v − w , kP v + Q wk ≤ Ω ,
(3.2)
and P = diag(p1 , . . . , pN ), Q = diag(q1 , . . . , qN ). The parameters pj > 0 and qj > 0 are the ”forward”and the ”backward” deviations of random variable zj , j = 1, . . . , N , respectively. The uncertainty set C is convex, and its size is controlled by Ω. Intuitively speaking, the uncertain factors z are decomposed into two random variables: v = max{z , 0} and w = max{−z , 0}, so that z = v − w . The multipliers
1 pj
and
1 qj
normalize the effective perturbation contributed by
both v and w such that the norm of the aggregated values falls within the budget of uncertainty. Therefore, considered sampling error of the samples, we present the Sample-based Worst-Case CVaR, its mathematical definition is as follows: WSCVaRβ (x ) =
sup
CVaRβ (x ), )
(3.3)
(r 1 ,...,r T )∈SΩ
where n
o
SΩ = r t : r t = r 0t + ∆r t z t , z t ∈ Ct ) , n
(3.4) o
−1 −1 Ct = z t : ∃v, w ∈ RN + , z t = v t − w t , kPt vt + Qt wt k ≤ Ω .
(3.5)
Next, we prove the WSCVaR 3.3 is a coherent risk measure. Theorem 3.1 If (r1 , . . . , rT ) ∈ SΩ §then WSCVaR is a coherent risk measure. Proof. Letting ρ(x ) = CVaRβ (x ), ρw (x ) = WSCVaRβ (x ), we have ρw (x ) =
sup
ρ(x ).
(r 1 ,...,r T )∈SΩ
As CVaRβ (x ) is a coherent risk measure, so ρ(x ) satisfies four axioms of Coherent risk measure. In what following, we prove ρw (x ) also satisfies four axioms of Coherent risk measure. 6
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• Monotonicity: if x < y , then ρ(x ) < ρ(y ). Therefore ρw (x ) =
sup
ρ(x )
0, we have ρw (λx ) =
sup
ρ(λx ) = λ
(r 1 ,...,r T )∈SΩ
sup
ρ(x ) = λρw (x );
(r 1 ,...,r T )∈SΩ
• translation invariance: for any constant a ∈ R, we have ρw (x + a) =
sup
ρ(x + a) =
(r 1 ,...,r T )∈SΩ
sup
ρ(x ) + a = ρw (x ) + a.
(r 1 ,...,r T )∈SΩ
Therefore, the theorem is true. Chen, Sim and Sun [19] stated the uncertainty set SΩ is convex, and its size is determined by Ω. Therefore, SΩ is a compact convex set. Let f (x , y ) = −x T r be the loss associated with the decision vector x , to be chosen from a certain subset X of Rn , and the random vector r in Rm . So, from 2.6, WSCVaR can be converted to the following form: WSCVaRβ (x ) =
T n o X 1 max min α + max{−r Tt x − α, 0} . T (1 − β) t=1 (r 1 ,...,r T )∈SΩ
(3.6)
Next, we will show the WSCVaR enjoys an important nature, in the process the dual-norm kuk∗ , (see Bertsimas and Sim [18]) is required. It is defined as: kuk∗ = max u T x . {kx k≤1} Theorem 3.2 If (r1 , . . . , rT ) ∈ SΩ , we have T X Ω WSCVaRβ (x) = CVaRβ (x) + kut k∗ . T (1 − β) t=1
(3.7)
Proof. From 3.6, we can obtain T o X 1 WSCVaRβ (x ) = max min α + max{−r Tt x − α, 0} T (1 − β) t=1 (r 1 ,...,r T )∈SΩ
n
T o X 1 = max min α + max{−(r 0t )T x − (∆r t z t )T x − α, 0} z t ∈Ct T (1 − β) t=1
n
T n o X 1 (∆r t z t )T x = CVaRβ (x ) + max max z t ∈Ct T (1 − β) t=1
= CVaRβ (x ) +
T n o X 1 max z Tt y t , y t = ∆r Tt x . T (1 − β) t=1 z t ∈Ct
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Observe that max z Tt y t
{z t ∈Ct }
= {vt ,wt ∈R
(v t − w t )T y t max −1 −1 Pt vt +Qt wt k≤Ω}
N + :k
max (Pt y t )T v t − (Qt y t )T w t {vt ,wt ∈R v t +w t k≤Ω} ∗ = Ωkut k =
N + :k
where ut = max{Pt y t , −Qt y t , 0} = max{Pt y t , −Qt y t } Note: Theorem 3.7 indicates that the WSCVaR can be seen as the original CVaR plus a regular item. It is easy to know that CVaRβ (x ) ≤ WSCVaRβ (x ). Obvious, WSCVaR is more cautious than the original CVaR.
4.
Computing WSCVaR and its application in portfolio management
By the Chen, Sim and Sun [19] Theorem 2 and Theorem 3.2, adding an auxiliary variable ht ∈ R, t = 1, 2, . . . , T , the WSCVaR (3.7) can be transformed into the following form T T X X 1 Ω dt + + ht , T (1 − β) t=1 T (1 − β) t=1
min
α+
s.t.
ku t k∗ ≤ ht , t = 1, 2, . . . T, ut ≥ −Pt ∆r Tt x , t = 1, 2, . . . T,
(4.1)
ut ≥ Qt ∆r Tt x , t = 1, 2, . . . T, dt ≥ (r 0t )T x − α,
t = 1, . . . , T,
d ≥ 0. The complete formulation and complexity class of the robust counterpart depends on the representation of the dual norm constraint, ku t k∗ ≤ ht , t = 1, 2, . . . T . Table 1 lists the common choices of norms, the representation of their dual norms which is come from reference [18](See page 14, Table 2). Table 1: Representation of the dual norm for u ≥ 0. Norms
ktk
kuk∗ ≤ h
l2
ktk2
kuk2 ≤ h
l1
ktk1
uj ≤ h, ∀j = {1, . . . , N }
l∞
ktk∞
N P
uj ≤ h
j=1
l1
T
l∞
max{ Ω1 ktk1 , ktk∞ }
Ωδ +
N P j=1
N vj ≤ h; vj + δ ≥ uj , ∀j ∈ N ; δ ∈ R+ , v ∈ R+
8
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In [18], Bertsimas and Sim discussed the nature and size of the proposed robust conic problem. In terms of keeping the model linear and simplicity in size, the l1 norm also is an attractive choice. In this paper, we adopt l1 norm. So under l1 norm, the constraints ku t k∗ ≤ ht , t = 1, 2, . . . T in (4.1) is equivalent to ujt ≤ ht , ∀j = {1, . . . , N }, t = 1, 2, . . . T.
(4.2)
Hence, the resulting problem (4.2) is still a linear constraint. For the constraint term ut ≥ −Pt ∆r Tt x , t = 1, 2, . . . T in (4.1), as discussed in [18], when all the data entries of the problem have independent random perturbation, we can further reduce the size of the robust model. In this article, we assume that the dimension of x and u is identical (n=N), that is, zjt in (3.4)is the independent random variable associated with the j-th data element, and ∆r j contains mostly zeros except at the entries corresponding to the data element, such as ∆r jt = (0, . . . , 0, ∆rtj , 0, . . . , 0)T . Then ujt ≥ −pjt (∆r jt )T x will reduce to ujt ≥ −pjt ∆rtj · xj . Then, the constraint term ut ≥ −Pt ∆r Tt x , t = 1, 2, . . . T in (4.1) can be transformed into the following form ujt ≥ −pjt ∆rtj · xj , j = 1, . . . , n, t = 1, 2, . . . T.
(4.3)
Based on investor preferences, portfolio optimization try to find the balance between risk and return. Therefore, the WSCVaR-based portfolio problem can be expressed as min x ∈X
WSCVaRβ (x ),
where X denotes the constraint on the portfolio position, which usually includes the budget constraint, no short sales constraint, and the return requirement. Therefore, the feasible decision set of portfolios can be denoted as X = {x |
x T 1 = 1,
x ≥ 0,
x T r ≥ µ}.
(4.4)
From (4.1) (4.2) and (4.3), adding an auxiliary variable θ ∈ R, the AWCVaR-based robust portfolio selection problem can be written as the following linear programming problem with variables (x , d , u t , ht , θ, α) min
θ
s.t.
α+
T T X X Ω 1 dt + + ht ≤ θ, T (1 − β) t=1 T (1 − β) t=1
ujt ≤ ht , ∀j = {1, . . . , N }, t = 1, 2, . . . T, ujt ≥ −pjt ∆rtj · xj , j = 1, . . . , n, t = 1, 2, . . . T,
(4.5)
ujt ≥ qtj ∆rtj · xj , j = 1, . . . , n, t = 1, 2, . . . T, dt ≥ (r 0t )T x − α,
t = 1, . . . , T,
d ≥ 0, u t ≥ 0, x T 1 = 1,
x ≥ 0, 9
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5.
Computational Experiments
We compare the performance of minimizing-portfolio WSCVaR under our approach with the initial CVaR method [6]. Firstly, we use simulated asset returns and show that our WSCVaR approach performs well for negatively-skewed returns. Secondly, we compare initial CVaR method and the robust portfolio optimization methods by employing a widely available data set of Hedge Funds returns, from http://www.hedgeindex.com. In our numerical experiments, the methods have the following meanings: • ”CVaR” stands for the initial mean-CVaR Portfolio optimization model (2.13)[6]; • ”WSCVaR” stands for the robust mean-WSCVaR Portfolio optimization model (4.5). We utilize Matlab2012 to solve models CVaR and WSCVaR, which are linear programming problems.
5.1.
Experiments with Simulated Data
Consider a portfolio of n = 20 assets with uncertain returns r˜it , i = 1, . . . , n, t = 1, . . . , T . Each return r˜it is determined by a simple single factor model r˜it = rˆit + z˜(ωit ), where rˆit = 1. The factors z˜t (ωi ) are independent and distributed as follows: √ t ωi (1−ωit ) , with probability t √ωit z˜(ωit ) = t) (1−ω ω i i − , with probability 1−ω t i
ωit ,
(5.1)
1 − ωit .
Note that the mean and the standard deviation of z˜(ωit ) are the same for all ωit ∈ (0, 1) - they are 0 and 1, respectively. However, the degree of symmetry of z˜t (ωit ) can be different. Higher values for ωit (e.g., ωit = 0.9) result in larger negative skew. We generate values for ωit as follows: ωit =
i 1 1+ , i = 1, . . . , n, t = 1, . . . , T. 2 N +t
(5.2)
Therefore, the return distributions for stocks with high index numbers in the portfolio are more negatively skewed than those for stocks with low index numbers. We use exact values for the parameters in the CVaR and WSCVaR optimization problems. These parameters include the standard deviation and average returns for the CVaR approaches, and the backward and forward deviations for the WSCVaR approach are set to ptj = 1.5, qjt = 2. ∆rjt is set to the vector of standard deviation of asset returns estimated by the T samples. We use a training set of 1,000 simulated returns from the above distributions that is T = 1000. The optimal portfolio allocations resulting from the five approximate CVaR optimization approaches for β = 1% are shown in Figure 1.
10
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0.1 WSCVaR CVaR
0.09 0.08
Asset weight
0.07 0.06 0.05 0.04 0.03 0.02 0.01
0
5
10 Asset number
15
20
Figure 1-Optimal portfolio weights (as proportions) for assets numbered 1 through 20 resulting from different optimization formulations. The behavior of the CVaR approach is erratic. In fact, the optimal weights for the portfolios found by the CVaR approach vary widely from sample to sample. WSCVaR is able to detect the asymmetry in the distributions, and allocates less in assets with more negatively skewed return distributions (those with high index numbers).
5.2.
Experiments with Hedge Funds
We select 12 Credit Suisse/Tremont Hedge Fund Indices (listed in Table 2) as the candidates for constructing hedge fund portfolios. Monthly returns of these indices, from January 1994 to December 2012 (240 samples in total) are used as the data set, which can be freely downloaded from http://www.hedgeindex.com. Table 2: Credit Suisse/Tremont Hedge Fund Indices 1
Convertible Arbitrage
2
Dedicated Short Bias
3
Emerging Markets
4
Equity Market Neutral
5
Event Driven
6
Distressed
7
Multi-Strategy
8
Risk Arbitrage
9
Fixed Income Arbitrage
10
Global Macro
11
Long/Short Equity
12
Managed Futures
11
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To construct an optimal portfolio with an accuracy to certain degree, we need to generate adequate scenarios with the given 240 samples. A question we face first in scenario generation is which distribution the asset returns follow. Statistic test shows that most of the distributions of returns of these hedge fund indices are skewed and exhibit a high kurtosis. Thus, the returns should not be modeled by a normal distribution. Table 3 shows the means and standard deviations of these 12 asset returns within three different but overlapped time periods. Each of these three time periods covers 100 months. The beginning and the end dates for each time period are specified in Table 3. We find that, for most assets, there exist remarkable differences among three periods for both the mean and the standard deviation, especially for the mean. For example, the mean of asset 4 during the time period of 1/31/1994-4/30/2002 is 15 times of that during the time period of 6/30/2002-9/30/2010. Table 3: Mean and standard deviation of asset returns within different time periods Time
1/31/1994-4/30/2002
Asset
Mean
Std
Mean
Std
Mean
Std
1
0.0084
0.0143
0.0066
0.0147
0.0046
0.0254
2
0.0005
0.0534
-0.0003
3
0.0061
0.0554
0.0048
0.0448
0.0091
4
0.0090
0.0076
0.0070
0.0006
5
0.0094
0.0178
0.0080
0.0178
0.0072
0.0175
6
0.0110
0.0202
0.0092
0.0193
0.0070
0.0182
7
0.0086
0.0193
0.0073
0.0192
0.0074
0.0184
8
0.0078
0.0130
0.0056
0.0134
0.0041
0.0109
9
0.0057
0.0117
0.0039
0.0117
0.0029
0.0214
10
0.0117
0.0381
0.0088
0.0270
0.0087
0.0160
11
0.0107
0.0342
0.0096
0.0320
0.0062
0.0226
12
0.0038
0.0332
0.0062
0.0358
0.0075
0.0347
0.0094
8/31/1997-11/30/2005
0.0534
6/30/2002-9/30/2010
-0.0036
0.0454 0.0296 0.0424
Since the distribution of asset returns is unknown, we adopt a distribution free method to generate scenarios given in Topaloglou et al. (2002)[20] and Zhu et al. (2013)[16]. We use back test method to check the performances of the robust approaches and the traditional approach in portfolio management, and the initial wealth is set at 1. Firstly, asset returns of the first N=162 (from 1/31/1994 to 7/31/2007) months are used to generate T=500 scenarios. Portfolio optimization models of the CVaR, and WSCVaR are then, respectively, solved to generate the traditional and the robust portfolio strategies. In month N+1, the two portfolios are constructed according to the derived strategies. At the beginning of month N+2, the scenarios are reproduced using the data from month 2 up to month N+1. The portfolio models are then re-solved, respectively, using the updated scenarios to generate new portfolio strategies for month N+1. The above procedure repeats until the end of the data set. In this experiments, we also use exact values for the parameters in the CVaR, WSCVaR optimization problems. These parameters include the standard deviation and average returns for the CVaR, and the backward and forward deviations for the WSCVaR approach are set to 12
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pjt = 1.5, qtj = 2. ∆rtj is set to the vector of standard deviation of asset returns estimated by the i − th T samples. 1.5
1.4
Portfolio Values
1.3 WSCVaR
1.2
1.1
CVaR
1
0.9
0.8
0
10
20
30 40 Time Period
50
60
70
Figure 2-Portfolio Values for Out-of-Sample Observations When a Simple Buy-and-Hold Strategy is Employed From Figure 2, we can see the optimal portfolio allocation based on the WSCVaR approach tends to result in stable returns, whereas, for example, the behavior of the optimal portfolio obtained with the CVaR approach is some erratic. In addition, the portfolio Values for generated by the WSCVaR model is better than the initial CVaR model at the end of investment period. But, during the gradually declining period from June to October, 2008, robust portfolio strategies perform better than the traditional ones in most cases.
6.
Conclusion
With an asymmetric affine uncertainty set based on the factor model, which is often employed in practice for estimating the asset return distribution, we propose a computationally tractable robust optimization method for minimizing the Worse-Case CVaR of a portfolio. The remarkable characteristic of the new method is that the robust optimization model retains the complexity of original portfolio optimization problem, i.e., the robust counterpart problem is still a linear programming problem. Specially in the new method, we incorporate information about asymmetries in the distributions of uncertainties. We present some numerical experiments with simulated and real market data to illustrate the behavior of robust optimization model. Acknowledgments This work is supported by the NSF of China Grants 11301041, 11371154, 71371065, and 71371195, Project funded by China Postdoctoral Science Foundation(2014M560654), Natural Science Foundation of Hunan Province(2015JJ3015), A Project Supported by Scientific Research Fund of Hunan Provincial Education Department. 13
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References [1] H.M. Markowitz, Portfolio selection, Journal of Finance. 7 (1952) 77-91. [2] H.M. Markowitz, Portfolio Selection: Efficient diversification of investment, New York: John Wiley & Sons, 1959. [3] Levy, H. 1992. Stochastic dominance and expected utility: Survey and analysis. Management Sci. 38(4) 555õ593. [4] T.J. Linsmeier and N. D. Pearson, Risk Measurement: An introduction to value-at-risk. Technical report 96-04, OFOR, University of Illinois, Urbana-Champaign, IL, 1996. [5] P. Artzner, F. Delbaen, J. M. Eber and D. Heath, Coherence measures of risk, Math. Finance, 9 (1999) 203-228. [6] R.T. Rockafellar and S. Uryasev, Optimization of conditional Valueat-Risk, J. Risk, 2 (2000) 21-41. [7] R.T. Rockafellar and S. Uryasev, Conditional Value-at-Risk for general loss distributions, J. Banking and Finance, 26 (2002) 1443-1471. [8] F. Black and R. Litterman, Global portfolio optimization, J. Financial Analysts, 48 (1992) 28-43. [9] V.K. Chopra and W.T. Ziemba, The effect of errors in means, variance and covariances on optimal portfolio choice, Journal Portfolio Management, 19 (1993) 6-11. [10] M.S. Lobo and S. Boyd. The worst-case risk of a portfolio,
Technical Report,
http://faculty.fuqua.duke.edu/∼ mlobo/bio/researchfiles/rsk-bnd.pdf, 2000. [11] D. Goldfarb and G. Iyengar, Robust portfolio selection problems, Mathematics of Operations Research, 28 (2003) 1-38. [12] R.H. T¨ ut¨ unc¨ u and M. Koenig, Robust asset allocation, Annals of Operations Research, 132 (2004) 157-187. [13] L. El Ghaoui, M. Oks, and F. Oustry, Worst-Case Value-at-Risk and Robust Portfolio Optimization: A Conic Programming Approach. Operations Research, 51 (2003) 543-556. [14] K. Natarajan, D. Pachamanova and M. Sim, Incorporating Asymmetric Distributional Information in Robust Value-at-Risk Optimization, Management Science, 54 (2008) 573-585. [15] S.S. Zhu and M. Fukushima, Worst-Case Conditional Value-at-Risk with application to robust portfolio management, Operations Research, 57 (2009) 1155-1168.
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[16] S. S. Zhu, X. D. Ji and D. Li, Robust set-valued scenario approach for handling modeling risk in portfolio optimization, Technical Report, Sun Yat-Sen Business School, Sun Yat-Sen University , 2013. [17] H. Konno, A. Yuuki, H. Waki. Portfolio Optimization under Lower Partial Risk Measures. Asia-Pacific Financial Markets 9 (2002) 127õ140. [18] D. Bertsimas and M. Sim, Tractable approximations to robust conic optimization problems, Math. Program, 107 (2006) 5-36. [19] X. Chen, M. Sim and P. Sun, A robust optimization perspective of stochastic programming, Operations Research, 55 (2007) 1058-1077. [20] N. Topaloglou, H. Vladimirou and S.A. Zenios, CVaR models with selective hedging for international asset allocation, Journal of Banking and Finance, 26 (2002) 1535-1561. [21] L.Y. Han and C. L. Zheng, Fuzzy options with application to default risk analysis for municipal bonds in China, Nonlinear Analysis, Theory, Methods and Applications, 2005, 63, 2353-2365. [22] E. Fama, Foundations of Finance. Basic Books, New York, 1976. [23] G. Duffee, The long-run behavior of firms. stock returns: Evidence and interpretations. Working paper, Haas School of Business, University of California at Berkeley, Berkeley, CA, 2002. [24] P. Sch¨onbucher, Factor models for portfolio credit risk. Working paper, University of Bonn, Germany. http://www.gloriamundi.org, 2000. [25] A. Andrew and J. Chen, Asymmetric Correlations of Equity Portfolios, Journal of Financial Economics, 63(3) (2002)443-494. [26] T.E. Conine, and M. J. Tamarkin, On diversification given asymmetry in returns, Journal of Finance 36 (1981)1143-1155. [27] Z.F. Dai, D.H. Li, F.H. Wen, Robust Conditional value-at-risk optimization for Asymmetrically Distributed Asset Returns, Pacific Journal of Optimization, 8 (2012) 429-445. [28] Z.F. Dai, F.H. Wen, Robust CVaR-based portfolio optimization under a genal affine data perturbation uncertainty set, Journal of Computational Analysis and Application, 16(2014) 93-102.
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A NOTE ON THE INTERVAL-VALUED SIMILARITY MEASURE AND THE INTERVAL-VALUED DISTANCE MEASURE INDUCED BY THE CHOQUET INTEGRAL WITH RESPECT TO AN INTERVAL-VALUED CAPACITY JEONG GON LEE AND LEE-CHAE JANG
Division of Mathematics and Informational Statistics, and Nanoscale Science and Technology Institute, Wonkwang University, Iksan 570-749, Republic of Korea E-mail : [email protected], Phone:082-63-850-6189 General Education Institute, Konkuk University, Chungju 138-701, Republic of Korea E-mail : [email protected], Phone:082-43-840-3591 Abstract. In this paper, we introduce an interval-valued capacity which is motivated by the goal to represent reasonable capacity and to define the Choquet integral with respect to an interval-valued capacity. We also investigate some properties of the Choquet integral with respect to an interval-valued capacity on the space of fuzzy sets and discuss their applications, for examples, interval-valued similarity measure and interval-valued distance measure induced by the Choquet integral with respect to an interval-valued capacity.
1. Introduction The theory of fuzzy sets defined by Zadeh (1965) has been researching many new approaches and theories, for examples, entropy, similarity measures, distance measures, Choquet integrals, fuzzy sets, and intuitionistic fuzzy sets which are applied to theories treating reasonability and uncertainty. Note that measuring the similarity between fuzzy sets is important in pattern recognition research and decision making. Balopoulos-Hatzimichailidis-Papadopoulos [2], Fan-Ma-Xie [5], Hong-Lee [6], Li-Sheng [13], Liu [11], Turksen [22], Wang-Li [23], Wei-Chen [25], Xu-Xia [26], Zeng-Li [27], Zeng-Guo [28], and Zhang-Zhang-Mei [29] have studied some properties and applications of similarity measures, entropy, and distance measures on interval-valued fuzzy sets (or fuzzy set), and Choquet [3], Murofushi-Sugeno [15,16], and Narukawa-Murofushi-Sugeno [18,19] have studied the theory of fuzzy measures(or capacity) and Choquet integrals. Couso-Montes-Gil [4], Jang [12], Murofushi-Sugen0-Suzaki [17], Pedrycz-Yang-Ha [20], and Wang [24] have studied various convergence properties of the Choquet integral with respect to a capacity. By using interval-valued functions, we have studied the Choquet integral with respect to a fuzzy measure of interval-valued functions which are able to better handle the representation of decision making and information theory (see [7-11]). Recently, we studied some convergence properties of the Choquet integral with respect to an interval-valued capacity functional (see 1991 Mathematics Subject Classification. 28E10, 28E20, 03E72, 26E50 11B68. Key words and phrases. Choquet integral, fuzzy set, interval-valued capacity, interval-valued similarity measure. 1
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2
LEE-CHAE JANG
[12]). Main purpose of this paper is to provide some applications of the Choquet integral with respect to an interval-valued capacity on the space of all fuzzy sets. In section 2, we define an interval-valued similarity measure and an interval-valued distance measure, and discuss some basic properties of them. In section 3, we define an interval-valued capacity and the Choquet integral with respect to an interval-valued capacity of a fuzzy set, and discuss some properties of them. In section 4, we prove that an interval-valued mapping induced by the Choquet integral with respect to a continuous from below intervalvalued capacity is an interval-valued similarity measure on the space of fuzzy sets, and discuss their applications, for examples, the interval-valued similarity measure and the interval-valued distance measure. In section 5, we discuss various convergence properties of the interval-valued distance measure induced by the Choquet integral with respect to an interval-valued capacity. In section 6, we give a brief summary results and some conclusions. 2. Choquet integrals and interval-valued similarity measures In this section, we consider the Choquet integral with respect to a capacity and discuss their properties. Let [0, 1] be the unit interval in the set of real numbers and Ω be a σ-algebra on a set X. Definition 2.1. ([14-17]) (1) A real-valued set function µ : Ω −→ [0, 1] is called a capacity if it satisfies the following properties: (i) µ(∅) = 0 and µ(X) = 1, and (ii) µ(E1 ) ≤ µ(E2 ) whenever E1 , E2 ∈ Ω and E1 ⊂ E2 . (2) A capacity µ is said to be continuous from below if for each increasing sequence {En } ⊂ Ω, µ(∪∞ n=1 En ) = limn→∞ µ(En ). (3) A capacity µ is said to be continuous from above if for each decreasing sequence {En } ⊂ Ω, µ(∩∞ n=1 En ) = limn→∞ µ(En ). (4) A capacity µ is said to be continuous if it is continuous from above and continuous from below. (5) A capacity µ is said to be subadditive if µ(E1 ∪ E2 ) ≤ µ(E1 ) + µ(E2 ) whenever E1 , E2 ∈ Ω and E1 ∩ E2 = ∅. We consider the Choquet integral with respect to a capacity which was introduced by Murofushi at el ([15-17]). Throughout this paper, we assume that the membership function of a fuzzy set A is a measurable function ηA from X to [0, 1]. Definition 2.2. ([14-17]) (1) The Choquet integral with respect to a capacity µ of a fuzzy set A is defined by ∫ ∫ 1 (C) Adµ = µηA (r)dr (1) 0
where µηA (r) = µ({x ∈ X|ηA (x) > r}) for all r ∈ [0, 1] and the integral on the right-hand side is the Lebesgue integral of µηA . (2) A fuzzy set A is said to be µ-integrable if the Choquet integral of A on X exists. We note that if A, B are fuzzy sets on X, then A ≤ B means ηA (x) ≤ ηB (x) for all x ∈ X and that ηA∨B (x) = ηA (x) ∨ ηB (x) and ηA∧B (x) = ηA (x) ∧ ηB (x) for all x ∈ X. Theorem 2.1. ([14-17]) Let ∫ A and B be ∫ µ-integrable fuzzy sets. (1) If A ≤ B, then (C) Adµ ≤ (C) Bdµ. ∫ ∫ (2) If E1 , E2 ∈ Ω and E1 ⊂ E2 , then (C) E1 Adµ ≤ (C) E2 Adµ.
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3
(3) If we define ηA∨B = ηA (x) ∨ ηB (x) and ηA∧B (x) = ηA (x) ∧ ηB (x) for all x ∈ X, then ∫ ∫ ∫ (C) A ∨ Bdµ ≥ (C) Adµ ∨ (C) Bdµ, and
∫ (C)
∫ A ∧ Bdµ ≤ (C)
∫ Adµ ∧ (C)
Bdµ.
Let [[0, 1]] is the set of all closed intervals in [0, 1] as follows: [[0, 1]] = {¯ a = [a− , a+ ]|a− , a+ ∈ [0, 1] and a− ≤ a+ }. For any a ∈ [0, 1], we define a = [a, a]. Obviously, a ∈ [[0, 1]](see [7-13, 21-223, 25, 27-29]). + Definition 2.3. Let I be an index set. If a ¯ = [a− , a+ ], ¯b = [b− , b+ ], a ¯n = [a− n , an ] ∈ [[0, 1]] for all n ∈ N and k ∈ [0, 1], then we define arithmetic, minimum, maximum, order, and inclusion operations as follows: (1) k¯ a = [ka− , ka+ ], ¯ (2) a ¯b = [a− b− , a+ b+ ], (3) a ¯ ∧ ¯b = [a− ∧ b− , a+ ∧ b+ ], (4) a ¯ ∨ ¯b = [a− ∨ b− , a+ ∨ b+ ], (5) a ¯ ≤ ¯b if and only if a− ≤ b− and a+ ≤ b+ , (6) a ¯ < ¯b if and only if a ¯ ≤ ¯b and a ¯ ̸= ¯b, − − ¯ (7) a ¯ ⊂ b if and only if b ≤ a and a+ ≤ b+ , (8) 1 − a ¯ = [1 − a+ , 1 − a− ], + (9) supn∈I a ¯n = [supn∈I a− n , supn∈I an ], and + − (10) inf n∈I a ¯n = [inf n∈I an , inf n∈I an ].
Theorem 2.2. For a ¯, ¯b, c¯ ∈ [[0, 1]], we have (1) idempotent law: a ¯∧a ¯=a ¯ and a ¯∨a ¯=a ¯, (2) commutative law: a ¯ ∧ ¯b = ¯b ∧ a ¯ and a ¯ ∨ ¯b = ¯b ∨ a ¯, (3) associative law: (¯ a ∧ ¯b) ∧ c¯ = a ¯ ∧ (¯b ∧ c¯) and (¯ a ∨ ¯b) ∨ c¯ = a ¯ ∨ (¯b ∨ c¯), ¯ ¯ (4) absorptive law: a ¯ ∧ (¯ a ∨ b) = a ¯ ∨ (¯ a ∧ b) = a ¯, and (5) distributive law: a ¯ ∧ (¯b ∨ c¯) = (¯ a ∧ ¯b) ∨ (¯ a ∧ c¯) and a ¯ ∨ (¯b ∧ c¯) = (¯ a ∨ ¯b) ∧ (¯ a ∨ c¯). Let F(X) be the family of all fuzzy sets A of X with the membership measurable function ηA : X → [0, 1]. Recall that for A, B ∈ F(X), A ≡ B means µ({x ∈ X|ηA (x) ̸= ηB (x)}) = 0, where µ is a capacity on X. We introduce the definitions of similarity measures and distance measures on F(X), and some characterizations of them(see [2,5,6,14,26-29]). Definition 2.4. (1) A real-valued function s : F(X) × F(X) −→ [0, 1] is called a similarity measure if it satisfies the following properties: (i) s(A, Ac ) = 0 if A is a crisp set, (ii) for A, B ∈ F(X), s(A, B) = 1 if and only if A ≡ B, (iii) for A, B ∈ F(X), s(A, B) = s(B, A), and (iv) if A, B, C ∈ F(X) and A ≤ B ≤ C, then s(A, C) ≤ s(A, B) and s(A, C) ≤ s(B, C). (2) A real-valued function d : F(X) × F(X) −→ [0, 1] is called a distance measure if it satisfies the following properties: (i) d(A, Ac ) = 1 if A is a crisp set, (ii) for A, B ∈ F(X), d(A, B) = 0 if and only if A ≡ B, (iii) for A, B ∈ F(X), d(A, B) = d(B, A), and (iv) if A, B, C ∈ F(X) and A ≤ B ≤ C, then d(A, C) ≥ d(A, B) and d(A, C) ≥ d(B, C).
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It is easy to see that if s is a similarity measure and we define l1 = 1 − s, then l1 is a distance measure and that if d is a distance measure and we define l2 = 1 − d, then l2 is a similarity measure. Definition 2.5. (1) An interval-valued function S = [s− , s+ ] : F(X) × F(X) −→ [[0, 1]] is called an interval-valued similarity measure if it satisfies the following properties: (i) S(A, Ac ) = 0 if A is a crisp set, (ii) for A, B ∈ F(X), S(A, B) = 1 if and only if A ≡ B, (iii) for A, B ∈ F(X), S(A, B) = S(B, A), and (iv) if A, B, C ∈ F(X) and A ≤ B ≤ C, then S(A, C) ≤ S(A, B) and S(A, C) ≤ S(B, C). (2) An interval-valued function D = [d− , d+ ] : F(X) × F(X) −→ [[0, 1]] is called a distance measure if it satisfies the following properties: (i) D(A, Ac ) = 1 if A is a crisp set, (ii) for A, B ∈ F(X), D(A, B) = 0 if and only if A ≡ B, (iii) for A, B ∈ F(X), D(A, B) = D(B, A), and (iv) if A, B, C ∈ F(X) and A ≤ B ≤ C, then D(A, C) ≥ D(A, B) and D(A, C) ≥ D(B, C). By the definitions of an interval-valued similarity measure and an interval-valued distance measure, we can obtain the following theorem. Theorem 2.3. (1) An interval-valued function S = [s− , s+ ] is an interval-valued similarity measure if and only if real-valued functions s− and s+ are real-valued similarity measures, and 0 ≤ s− ≤ s+ ≤ 1. (2) An interval-valued function D = [d− , d+ ] is an interval-valued distance measure if and only if real-valued functions d− and d+ are real-valued distance measures, and 0 ≤ d− ≤ d+ ≤ 1. (3) If S is an interval-valued similarity measure and we define H = 1−S = [1−s+ , 1−s− ], then H is an interval-valued distance measure. (4) If D is an interval-valued distance measure and we define L = 1 − D = [1 − d+ , 1 − d− ], then L is an interval-valued similarity measure. Proof. (1) (=⇒) Suppose that S is an interval-valued similarity measure. If A is a crisp set, then 0 = S(A, Ac ) = [s− (A, Ac ), s+ (A, Ac )]. Thus s− (A, Ac ) = 0 and s+ (A, Ac ) = 0. Since S(A, B) = S(B, A) for all A, B ∈ F(X), s− (A, B) = s− (B, A) and s+ (A, B) = S + (B, A). Let A, B, C ∈ F(X) and A ≤ B ≤ C. Then we have S(A, C) ≤ S(A, B) and S(A, C) ≤ S(B, C). Thus, we have
s− (A, C) ≤ s− (A, B) and s− (A, C) ≤ s− (B, C),
and s+ (A, C) ≤ s+ (A, B) and s+ (A, C) ≤ s+ (B, C), Therefore, we obtain that s− and s+ are real-valued similarity measures and 0 ≤ s− ≤ s+ ≤ 1. (⇐=) The proof is similar to the proof of (=⇒). (2) The proof is similar to the proof of (1). (3) Let S be an interval-valued similarity measure and we define H = 1−S = [1−s+ , 1−s− ]. If A is a crisp set, then S(A, Ac ) = 0. Thus, H(A, Ac ) = 1 − S(A, Ac ) = 1 − 1 = 0. Let A, B ∈ F(X). Then, A ≡ B if and only if S(A, B) = 1, that is, H(A, B) = 1 − S(A, B) = 0. If A, B ∈ F(X), then S(A, B) = S(B, A). Then, H(A, B) = 1 − S(A, B) = 1 − S(B, A) = H(B, A).
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If A, B, C ∈ F(X) and A ≤ B ≤ C, then S(A, C) ≤ S(A, B) and S(A, C) ≤ S(B, C). Thus, we have H(A, C) = 1 − S(A, C) ≥ 1 − S(A, B) = H(A, B) and H(A, C) = 1 − S(A, C) ≥ 1 − S(B, C) = H(B, C). Therefore, H is an interval-valued distance measure. (4) The proof is similar to the proof of (3). 3. The Choquet integral with respect to an interval-valued capacity In this section, we define an interval-valued capacity and the Choquet integral with respect to an interval-valued capacity of a fuzzy set. Note that a mapping dH : [[0, 1]] × [[0, 1]] −→ [0, ∞) is the Hausdorff metric defined by { } dH (A, B) = max sup inf |x − y|, sup inf |x − y| (2) x∈A y∈B
y∈B x∈A
for all A, B ∈ [[0, 1]], and ([[0, 1]], dH ) is a metric space. By the definition of the Hausdorff metric, it is easy to see that for any a ¯ = [a− , a+ ], ¯b = [b− , b+ ] ∈ [[0, 1]], we have { } dH (¯ a, ¯b) = max |a− − b− |, |a+ − b+ | . (3) We recall that for any {¯ an } ⊂ [[0, 1]] and a ¯ ∈ [[0, 1]], dH − lim a ¯n = a ¯ means n→∞
lim dH (¯ an , a ¯) = 0.
(4)
n→∞
We define an interval-valued capacity µ ¯ = [µ− , µ+ ] : Ω −→ [[0, 1]] on a measurable space (X, Ω) as follows: Definition 3.1. (1) An interval-valued set function µ ¯ : Ω −→ [[0, 1]] is called an intervalvalued capacity if it satisfies the following properties: (i) µ ¯(∅) = 0 and µ ¯(X) = 1, and (ii) µ ¯(E1 ) ≤ µ ¯(E2 ) whenever E1 , E2 ∈ Ω and E1 ⊂ E2 . (2) An interval-valued capacity µ ¯ is said to be continuous from above if for each increasing ¯(En ). sequence {En } ⊂ Ω, µ ¯(∪∞ n=1 En ) = dH − limn→∞ µ (3) An interval-valued capacity µ ¯ is said to be continuous from below if for each decreasing sequence {En } ⊂ Ω, µ ¯(∩∞ ¯(En ). n=1 En ) = dH − limn→∞ µ (4) An interval-valued capacity µ ¯ is said to be continuous if it is continuous from above and continuous from below. (5) An interval-valued capacity µ ¯ is said to be subadditive if µ ¯(E1 ∪ E2 ) ≤ µ ¯(E1 ) + µ ¯(E2 ), whenever E1 , E2 ∈ Ω and E1 ∩ E2 = ∅. It is easy to see that for each increasing sequence {En } ⊂ Ω with E = ∪∞ n=1 En , lim dH (¯ µ(En ), µ ¯(E)) = 0 if and only if lim µ− (En ) = µ− (E) and lim µ+ (En ) = µ+ (E), (5)
n→∞
n→∞
n→∞
and for each decreasing sequence {En } ⊂ Ω with F =
∩∞ n=1 En ,
lim dH (¯ µ(En ), µ ¯(F )) = 0 if and only if lim µ− (En ) = µ− (F ) and lim µ+ (En ) = µ+ (F ). (6)
n→∞
n→∞
n→∞
By (5) and (6), we can directly derive the following theorem.
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Theorem 3.1. (1) An interval-valued set function µ ¯ = [µ− , µ+ ] : Ω −→ [[0, 1]] is an interval− + valued capacity if and only if µ and µ are capacities and µ− ≤ µ+ . (2) An interval-valued capacity µ ¯ = [µ− , µ+ ] is continuous from below if and only if µ− + and µ are continuous from below and µ− ≤ µ+ . (3) An interval-valued capacity µ ¯ = [µ− , µ+ ] is continuous from above if and only if µ− + and µ are continuous from above and µ− ≤ µ+ . (4) An interval-valued capacity µ ¯ = [µ− , µ+ ] is continuous if and only if µ− and µ+ are − + continuous and µ ≤ µ . (5) An interval-valued capacity µ ¯ = [µ− , µ+ ] is subadditive if and only if µ− and µ+ are − + subadditive and µ ≤ µ . Recall that if ([0, 1], M, m) is the Lebesgue measure space and C([0, 1]) is the family of all closed subsets of I, then the Aumann integral of a closed set-valued function G : [0, 1] −→ C([0, 1]) is defined by {∫ } ∫ (A) Gdm = gdm| g ∈ S(G) , (7) where S(G) is the set of all integrable selections of G, that is, ∫ S(G) = {g : [0, 1] −→ [0, 1]| gdm < ∞ and g(r) ∈ G(r) m − a.e.}.
(8)
We note that m − a.e. means almost everywhere in the Lebesgue measure m (see[1,16]). Then, we introduce the following theorems which are used to define the Choquet integral with respect to an interval-valued capacity of a fuzzy set. Theorem 3.2. ([13, Lemma 2.1]) If a closed set-valued function G : [0, 1] −→ C([0, 1]) is ∫ M-measurable, then (A) Gdm is convex in [0, 1]. Theorem 3.3. ([13, Lemma 2.2]) If a closed set-valued function G : [0, 1] −→ C([0, 1]) is Mmeasurable and integrably bounded, that is, there exists a integrable function φ : [0, 1] −→ [0, 1] such that sup x ≤ φ(r)
for r ∈ [0, 1],
(9)
x∈G(r)
∫ then (A) Gdm is nonempty compact convex in [0, 1]. ∫ From Theorem 3.3, we can see that (A) Gdm is a nonempty bounded and closed subset in [0, 1] under the same assumption of G. Thus, we obtain the following corollary (see [12,13,21]). Corollary 3.4. If an interval-valued function G = [g − , g + ] : I −→ [[0, 1]] is M-measurable and integrably bounded, then g − , g + ∈ S(F ) and [∫ ] ∫ ∫ − + (A) Gdm = g dm, g dm , (10) where the integrals on the right-hand side are the Lebesgue integral with respect to m. ∫ ∫1 We write gdm = 0 g(r)dm(r) for all measurable functions g. By using an interval-valued capacity, we define the Choquet integral with respect to an interval-valued capacity of a fuzzy set A. Definition 3.2. (1) The Choquet integral with respect to an interval-valued capacity µ ¯ of a fuzzy set A ∈ F is defined by ∫ ∫ 1 (C) Ad¯ µ = (A) µ ¯A (r)dr, (11) 0
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where ηA is the membership measurable function of A, µ ¯A (r) = µ ¯({x ∈ X|ηA (x) > r}) for all r ∈ [0, 1], and the integral on the right-hand side is the Aumann integral in (7). ∫ (2) A fuzzy set A ∈ F is said to be µ ¯-integrable if (C) Ad¯ µ ∈ [[0, 1]]. Note that if an interval-valued capacity µ ¯ is continuous from below and A ∈ F(X), then µ ¯A : I −→ [[0, 1]] is continuous from below on [0, 1]. Thus, we obtain that µ ¯A is M-measurable and integrably bounded on [0, 1]. Thus, by Definition 3.2 and Corollary 3.4, we can easily obtain the following theorem. Theorem 3.5. If an interval-valued capacity µ ¯ is continuous from below and A ∈ F, then we have [ ] ∫ ∫ ∫ (C) Ad¯ µ = (C) Adµ− , (C) Adµ+ , (12) where the integrals on the right-hand side are Choquet integrals. Proof. By Definition 3.2 and Corollary 3.4, we can derive ∫ ∫ 1 (C) Ad¯ µ = (A) µ ¯A (r)dr ∫0 1 + = (A) [µ− A (r), µA (r)]dr [∫ 1 0 ] ∫ 1 − + = µA (r)dr, µA (r)dr 0 [ 0 ∫ ] ∫ = (C) Adµ− , (C) Adµ+ . By Theorem 3.5, we can easily obtain the following basic properties of the Choquet integrals with respect to a continuous from below interval-valued capacity of a fuzzy set. Theorem 3.6. Let (X, Ω) be a measurable space. Assume that an interval-valued µ ¯ is continuous from below. (1) If A, B ∈ F(X) and A ≤ B, then ∫ ∫ (C) Ad¯ µ ≤ (C) Bd¯ µ. (2) If A, B ∈ F(X) and we define η(A∨B) (x) = ηA (x) ∨ ηB (x) for all x ∈ X, then ∫ ∫ ∫ (C) A ∨ Bd¯ µ ≥ (C) Ad¯ µ ∨ (C) Ad¯ µ. (3) If A, B ∈ F(X) and we define η(A∧B) (x) = ηA (x) ∧ ηB (x) for all x ∈ X, then ∫ ∫ ∫ (C) A ∧ Bd¯ µ ≤ (C) Ad¯ µ ∧ (C) Ad¯ µ. 4. Interval-valued similarity measures induced by the Choquet integral In this section, we discuss some applications of the Choquet integral with respect to a continuous from below interval-valued capacity of a fuzzy set. Theorem 4.1. Assume that an interval-valued µ ¯ is continuous from below and µ ¯(X) = ¯{µ}(X) = 1. If we define an interval-valued function Sµ¯ : F × F −→ [[0, 1]] as following ∫ Sµ¯ (A, B) = 1 − (C) |ηA − ηB |d¯ µ (13) for all A, B ∈ F(X), then Sµ¯ is an interval-valued similarity measure.
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Proof. (i) If A is a crisp measurable set, then the membership measurable function ηA of a fuzzy set A is defined by { 1 if x ∈ A ηA (x) = 0 if x ∈ Ac = I \ A. We note that if the membership measurable function ηAc of the complement of a fuzzy set A, then { 0 if x ∈ A ηAc (x) = 1 if x ∈ Ac = I \ A. Thus, we have |ηA (x) − ηAc (x)| = 1 for all x ∈ X. Therefore, we have ∫ Sµ¯ (A, Ac ) = 1 − (C) |ηA − ηAc |d¯ µ ∫ 1 = 1− µ ¯({x ∈ X| |ηA (x) − ηAc (x)| > r})dr 0 ∫ 1 = 1− µ ¯(X)dr = 0. 0
(ii) If A ≡ B, then ηA = ηB µ ¯ − a.e. on X. Thus, we have ∫ Sµ¯ (A, B) = 1 − (C) |ηA − ηB |d¯ µ ∫ 1 = 1− µ ¯({x ∈ X| |ηA (x) − ηB (x)| > r})dr ∫0 1 = 1− µ ¯(∅)dr = 1. 0
If Sµ¯ (A, B) = 1, then
∫
1
µ ¯({x ∈ X| |ηA (x) − ηB (x)| > r})dr = 0. 0
Then, it is easy to see that µ ¯({x ∈ X| |ηA (x) − ηB (x)| > r}) = 0 m − a.e. on I.
(14)
From (14), we have µ ¯({x ∈ X| |ηA (x) − ηB (x)| ̸= 0}) = 0, that is, ηA = ηB µ ¯ − a.e. on X and hence A ≡ B. (iii) If A, B ∈ F(X), then we have ∫ Sµ¯ (A, B) = 1 − (C) |ηA − ηB |d¯ µ ∫ = 1 − (C) |ηB − ηA |d¯ µ = Sµ¯ (B, A). (iv) If A, B, C ∈ F(X) and A ≤ B ≤ C, then ηA ≤ ηB ≤ ηC . Thus, we have |ηA (x) − ηB (x)| ≤ |ηA (x) − ηC (x)| and |ηB (x) − ηC (x)| ≤ |ηA (x) − ηC (x)|,
(15)
for all x ∈ X. By (15) and Theorem 2.2 (1), we have ∫ Sµ¯ (A, C) = 1 − (C) |ηA − ηC |d¯ µ ∫ ≤ 1 − (C) |ηA − ηB |d¯ µ = Sµ¯ (A, B),
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and
∫ Sµ¯ (A, C)
=
1 − (C)
≤ 1 − (C)
∫
|ηA − ηC |d¯ µ |ηB − ηC |d¯ µ = Sµ¯ (B, C),
By (i),(ii),(iii), and (iv), we see that Sµ¯ is an interval-valued similarity measure. By Theorem 4.1 and Theorem 2.3(3), we can easily obtain the following corollary. Corollary 4.2. Assume that an interval-valued µ ¯ is continuous from below and µ ¯(X) = ¯{µ}(X) = 1. If we define an interval-valued function Dµ¯ = 1 − Sµ¯ = (C) ∫ |ηA − ηB |d¯ µ for all A, B ∈ F(X), then Dµ¯ is an interval-valued distance measure. In order to illustrate the proposed similarity measure are reasonable, we give the following example. Example 4.1. Let X = {x1 , x2 , x3 } and Ω = ℘(X) be the power set of X. Suppose that µ ¯ : Ω −→ [[0, 1]] is defined by µ ¯(E) = [µ− (E), µ+ (E)], (16) )2 ( m(E) m(E) where m(E) is the cardinality of E ∈ Ω, µ− (E) = m(X) , and µ+ (E) = m(X) . Since X is a finite set, clearly, we see that µ ¯ is a continuous from below interval-valued capacity on a measurable space (X, Ω) and µ ¯(X) = ¯{µ}(X) = 1. . The three patterns are denoted as follows: A1 = {(x1 , 0.3), (x2 , 0.2), (x3 , 0.1)}, A2 = {(x1 , 0.2), (x2 , 0.2), (x3 , 0.2)}, and A3 = {(x1 , 0.4), (x2 , 0.4), (x3 , 0.4)}. Assume that a sample B = {(x1 , 0.3), (x2 , 0.2), (x3 , 0.1)} is given. In order to interpret the measure of similarity of B with these patterns, we calculate the proposed interval-valued similarity measure Sµ¯ as follows: Sµ¯ (A1 , B) = 1 −
3 ∑ (|ηA1 (x(i)) − ηB (x(i)) |)(¯ µ(A(i) ) = 1,
(17)
i=1
[ ] 3 ∑ 14 43 (|ηA2 (x(i)) − ηB (x(i)) |)(¯ µ(A(i) ) = , , and 15 45 i=1 ] [ 3 ∑ 4 38 , . Sµ¯ (A3 , B) = 1 − (|ηA3 (x(i)) − ηB (x(i)) |)(¯ µ(A(i) ) = 5 45 i=1
Sµ¯ (A2 , B) = 1 −
(18)
(19)
By (17), (18), and (19), we interpret that B is equal(or, absolutely similar) to A1 and B is more similar to A2 than similar to A3 . Example 4.2. Let X = {x1 , x2 , x3 } and Ω = ℘(X) be the power set of X. Suppose that ν¯ : Ω −→ [I] is defined by ν¯(E) = [ν − (E), ν + (E)], (20) ( )3 ( )2 m(E) m(E) where m(E) is the cardinality of E ∈ Ω, ν − (E) = m(X) , and ν + (E) = m(X) . The three patterns are denoted as follows: A1 = {(x1 , 0.3), (x2 , 0.2), (x3 , 0.1)}, A2 = {(x1 , 0.2), (x2 , 0.2), (x3 , 0.2)}, and A3 = {(x1 , 0.4), (x2 , 0.4), (x3 , 0.4)}.
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Assume that a sample B = {(x1 , 0.3), (x2 , 0.2), (x3 , 0.1)} is given. In order to interpret the measure of similarity of B with these patterns, we calculate the proposed interval-valued similarity measure Sν¯ as follows: [ ] [ ] 38 13 43 131 , , and Sν¯ (A3 , B) = , . (21) Sν¯ (A1 , B) = 1, Sν¯ (A2 , B) = 45 135 45 15 Thus, we can see that there is an interpretation of the notions of these patterns under two different interval-valued capacity µ ¯ and ν¯ as follows: Sµ¯ (A1 , B) = 1 = Sν¯ (A1 , B), [ ] [ ] 14 43 43 131 Sµ¯ (A2 , B) = , < , = Sν¯ (A2 , B), and 15 45 45 135 ] [ ] [ 38 13 4 38 , < , = Sν¯ (A3 , B). Sµ¯ (A3 , B) = 5 45 45 15 Therefore, this means that ν¯ has more positive sense than µ ¯. 5. Convergence in the interval-valued distance measure Throughout this section, we assume that µ ¯ = [µ− , µ+ ] is continuous from below. At first, we introduce uniformly µ-integrability and convergence in the interval-valued distance measure on F(X). Definition 5.1. ([26]) Let µ be a capacity on a measurable space (X, Ω), {An } be a sequence of fuzzy sets and A be a fuzzy set. (1) A sequence {An } converges to A almost everywhere on X if there exist a null set N ∈ Ω with µ(N ) = 0 such that ηA (x) = lim ηAn (x), n→∞
for all x ∈ N c .
(22)
(2) A sequence {An } converges in the distance measure dµ to A if lim dµ (ηAn , ηA ) = 0,
where dµ (ηAn , ηA ) = (C)
∫
(23)
n→∞
|ηAn (x) − ηA (x)|dµ for all n ∈ N.
Remark that convergence in the distance measure dµ is equal to convergence in µ-mean(see [4 ]) Definition 5.2. ([4]) Let µ be a capacity on a measurable space (X, Ω) and I ⊂ N be an index set. A class {An }n∈I of fuzzy sets is said to be uniform µ-integrable if (i) sup dµ (An , 0) < ∞,
(24)
n∈I
(ii) ∀ε > 0, ∃δ(ε) > 0 such that dE,µ (An , 0) < ε if E ∈ Ω and µ(E) < δ(ε), ∫ where dE,µ (An , 0) = (C) E |ηAn |dµ for all n ∈ N.
(25)
We also introduce various convergence properties of the Choquet integral on F(X) as follows: Theorem 5.1. ([4]) Let a capacity µ be subadditive and {An } a sequence of fuzzy sets in F(X). Then {An } is an uniformly µ-integrable if and only if lim sup d[|ηAn |>a],µ (An , 0) = 0.
a→∞ n∈N
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Theorem 5.2. ([4]) Let a capacity µ be subadditive and a sequence {An } of fuzzy sets in F(X) converges to a fuzzy set A in F(X) µ-almost everywhere on X and An ≤ B for some µ-integrable fuzzy set B, then we have (1) An and A are µ-integrable for all n ∈ N, and (2) {An } converges to A in the distance measure dµ , that is, lim dµ (An , 0) = 0.
(27)
n→∞
We assume that an interval-valued capacity µ ¯(X) = [µ− , µ+ ] is continuous from below. Then we define convergence in the interval-valued distance measure Dµ¯ and uniform µ ¯integrability on f(X). It is easy to see that Dµ¯ (A, B) = [dµ− (A, B), dµ+ (A, B)], for all A, B ∈ F(X).
(28)
Definition 5.3. Let I ⊂ N be an index set. (1) A sequence {An } converges in the interval-valued distance measure Dµ¯ to A if dH − lim Dµ¯ (An , A) = 0,
(29)
n→∞
where dH − lim Dµ¯ (An , A) = lim dH {Dµ¯ (An , A), 0} n→∞
n→∞
and Dµ¯ (An , A) = [dµ− (An , A), dµ− (An , A)] for all n ∈ N. (2) A class {An }n∈I of fuzzy sets in F(X) is said to be µ ¯-integrable if (i) sup Dµ¯ (An , 0) < ∞,
(30)
n∈I
(ii) ∀ε > 0, ∃δ(ε) > 0 such that DE,¯µ (An , 0) < ε if E ∈ Ω and µ ¯(E) < δ(ε), ∫ where DE,¯µ (An , 0) = (C) E |ηAn |d¯ µ for all n ∈ N.
(31)
By (3), it is easy to see that (29) holds if and only if lim max{dµ− (An , A), dµ+ (An , A)}) = 0,
(32)
n→∞
By Definition 5.1 and Definition 5.3, we obtain various convergence properties of the intervalvalued distance measure Dµ¯ as follows: Theorem 5.3. Let I ⊂ N be an index set. (1) A class {An }n∈I is uniformly µ ¯-integrable if and only if it is uniformly µ− -integrable + − and uniformly µ -integrable, and µ ≤ µ+ . (2) A sequence {An } of fuzzy sets in F(X) converges to a fuzzy set A ∈ F(X) in the intervalvalued distance measure Dµ¯ if and only if {An } converges to A in the distance measures dµ− and dµ+ , and dµ− ≤ dµ+ . Proof. (1) Let {An } be a sequence of fuzzy sets in F(X). If {An } converges to A in the interval-valued distance measure Dµ¯ , then, by (12) and (29), lim dµ− (An , A)
n→∞
≤ =
lim (max{dµ− (An , A), dµ+ (An , A)})
n→∞
lim dH (Dµ¯ (An , A), 0) = 0.
(33)
n→∞
As in the same method with (33), we obtain lim dµ+ (An , A) = 0.
(34)
n→∞
Thus, by (33) and (34), {An } converges to A in the distance measure dµ− and dµ+ .
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Conversely, if we take an interval-valued distance measure µ ¯ = [µ− , µ+ ], then, similarly, we can obtain the converse result. (2) Suppose that {An }n∈I is uniformly µ ¯-integrable and µ ¯ is continuous from below. By (12) and Definition2.3 (9), we have sup Dµ¯ (An , 0)
=
sup[dµ− (An , 0), dµ+ (An , 0)]
=
[sup dµ− (An , 0), sup dµ+ (An , 0)] < ∞,
n∈I
n∈I
n∈I
(35)
n∈I
and for arbitrary ε > 0 and E ∈ Ω, there exists δ(ε) > 0 such that sup DE,¯µ (An , 0)
=
sup[dE,µ− (An , 0), dE,µ+ (An , 0)]
=
[sup dE,µ− (An , 0), sup dE,µ+ (An , 0)] < ε,
n∈I
n∈I n∈I
(36)
n∈I
if µ ¯ < δ(ε). By (35) and (36), {An } converges to A in the distance measures dµ− and dµ+ , nd dµ− ≤ dµ+ . Conversely, if we take an interval-valued distance measure µ ¯ = [µ− , µ+ ], then, similarly, we can obtain the converse result. Theorem 5.4. Let an interval-valued capacity µ ¯ be subadditive and {An } a sequence of fuzzy sets in F(X). Then, {An } is an uniformly µ ¯-integrable if and only if lim sup D[|ηAn |>a],¯µ (An , 0) = 0.
(37)
a→∞ n∈N
Proof. Since an interval-valued capacity µ ¯ = [µ− , µ+ ] is subadditive, by Theorem 3.1(5), + µ and µ are subadditive. From Theorem 5.3 (1), {An } is an uniformly µ ¯-integrable if and only if {An } is an uniformly µ− -integrable and an uniformly µ+ -integrable. Thus, by Theorem 5.1, {An } is an uniformly µ− -integrable if and only if −
lim sup d[|ηAn |>a],µ− (An , 0) = 0
(38)
a→∞ n∈N
and {An } is an uniformly µ+ -integrable if and only if lim sup d[|ηAn |>a],µ+ (An , 0) = 0
(39)
a→∞ n∈N
By (38) and (39), and (12), we have lim sup dH (D|ηAn |>a],¯µ (An , 0), 0)
a→∞ n∈N
=
lim sup max{d[|ηAn |>a],µ− (An , 0), d[|ηAn |>a],µ+ (An , 0)} = 0.
a→∞ n∈N
(40)
Conversely, by the similar method of the above proof, we can obtain the converse result. Lemma 5.5. Assume that an interval-valued capacity µ ¯ = [µ− , µ+ ] is continuous from below. Then {An } is µ ¯-integrable if and only if {An } is µ− -integrable and µ+ -integrable Proof. The proof is trivial. Theorem 5.6. Let an interval-valued capacity µ be subadditive. If a sequence {An } of fuzzy sets in F(X) converges to a fuzzy set A in F(X) µ-almost everywhere on X and An ≤ B for some µ ¯-integrable fuzzy set B, then we have (1) An and A are µ ¯-integrable for all n ∈ N, and (2) {An } converges to A in the interval-valued distance measure Dµ¯ , that is, dH − lim Dµ¯ (An , 0) = 0. n→∞
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Proof. Since B is µ ¯-integrable fuzzy set and An ≤ B, by Theorem 5.3 (1), we have (i)An and A are µ− -integrable and µ− -integrable for all n ∈ N, and (ii) {An } converges to A in the distance measure dµ− and in the distance measure dµ+ . Thus, by Lemma 5.5 and Theorem 5.3 (1) and (12), we obtain (1) An and A are µ ¯-integrable for all n ∈ N, and (2) {An } converges to A in the interval-valued distance measure Dµ¯ , that is, dH − lim Dµ¯ (An , 0) = 0.
(42)
n→∞
6. Conclusions In this paper, we define the concept of interval-valued capacity which means reasonable capacity. By using Aumann integral of integrably bounded interval-valued functions in Corollary 3.4, we consider the Choquet integral with respect to a continuous interval-valued capacity of a fuzzy set. From Definitions 2.3, 3.1, 3.2 and Theorems 3.5, 3.6, we discuss interval-valued similarity measures induced by the Choquet integral with respect to a continuous interval-valued capacity on F(X). By Examples 4.1 and 4.2, it is possible that we interpret the interval-valued measure of similarity of a sample with the three patterns. From Definitions 5.1, 5.2, 5.3, and Theorems 5.3, 5.4, and 5.6, we can provide the concept of convergence in the interval-valued distance measure and discuss various convergence properties of the interval-valued distance3 measure on the space of fuzzy sets for the Choquet integral. In the future, by using these results of this paper, we can develop various problems and models for representing uncertain similarity measures and uncertain distance measures in pattern recognition research, information theory, decision making, and fuzzy risk analysis, etc. Acknowledgement This paper was supported by Wonkwang University in 2013.
References [1] R.J. Aumann, Integrals of set-valued functions, J. Math. Anal. Appl. 12 (1965) 1-12. [2] V. Balopoulos, A.G. Hatzimichailidis, B.K. Papadopoulos, Distance and similarity measures for fuzzy operators, J. Math. Anal. Appl. 12 (1965) 1-12. [3] G. Choquet, Theory of capacities, Ann. Inst. Fourier 5 (1953) 131-295. [4] I. Couso, S. Montes, P. Gil, Stochastic convergence, uniform integrability and convergence in mean on fuzzy measure spaces, Fuzzy Sets and Systems 129 (2002) 95-104. [5] Jin-Lum Fan, Yuan-Liang Ma, and Wei-Xin Xie, On some properties of distance measures, Fuzzy Sets and Systems , 117 (2001), 355-361. [6] D.H. Hong, S.H. Lee, Some algebraic properties and distance measures for interval-valued fuzzy numbers, Information Sciences, 148 (2002), 1-10. [7] L.C. Jang, B.M. Kil, Y.K. Kim, J.S. Kwon, Some properties of Choquet integrals of set-valued functions, Fuzzy Sets and Systems, 91 (1997), 61-67. [8] L.C. Jang, J.S. Kwon, On the representation of Choquet integrals of set-valued functions and null sets, Fuzzy Sets and Systems, 112 (2000), 233-239. [9] L.C. Jang, Interval-valued Choquet integrals and their apllications, J. Appl. Math. and Computing, 16(12) (2004), 429-445. [10] L.C. Jang, A note on the monotone interval-valued set function defined by the interval-valued Choquet integral, Commun. Korean Math. Soc., 22 (2007), 227-234. [11] L.C. Jang, On properties of the Choquet integral of interval-valued functions, Journal of Applied Mathematics, 2011 (2011), Article ID 492149, 10pages. [12] L.C. Jang, A note on convergence properties of interval-valued capacity functionals and Choquet integrals, Information Sciences, 183 (2012), 151-158.
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[13] L.S. Li, Z. Sheng, The fuzzy set-valued measures generated by fuzzy random variables, Fuzzy Sets and Systems, 97 (1998), 203-209. [14] X. Liu, Entropy, distance measure and similarity measure of fuzzy sets and their relations, Fuzzy Sets and Systems, 52 (1992), 305-318. [15] T. Murofushi, M. Sugeno, An interpretation of fuzzy measures and the Choquet integral as an integral with respect to a fuzzy measure, Fuzzy Sets and Systems, 29 (1989), 201-227. [16] T. Murofushi, M. Sugeno, A theory of fuzzy measures: representations, the Choquet integral, and null sets, J. Math. Anal. Appl., 159 (1991), 532-549. [17] T. Murofushi, M. Sugeno,M. Suzaki, Autocontinuity, convergence in measure, and convergence in distribution, Fuzzy Sets and Systems 92(2)(1997) 197-203. [18] Y. Narukawa, T. Murofushi, M. Sugeno, Regular fuzzy measure and representation of comonotonically additive functional, Fuzzy sets and Systems, 112 (2000), 177-186. [19] Y. Narukawa, T. Murofushi, M. Sugeno, Extension and representation of comonotonically additive functional, Fuzzy sets and Systems, 121 (2001), 217-226. [20] W. Pedrycz, L. Yang, M. Ha, On the fundamental convergence in the (C) mean in problems of information fusion, J. Math. Anal. Appl. 358 (2009) 203-222. [21] P. Pucci, G. Vitillaro, A representation theroem for Aumann integrals, J. Math. Anal. Appl. , 102 (1984), 86-101. [22] I.B. Turksen, Non-specificity and interval-valued fuzzy sets, Fuzzy sets and Systems, 80 (1996), 87-100. [23] G. Wang and X. Li, The applications of interval-valued fuzzy numbers and interval-distribution numbers, Fuzzy Sets and Systems, 98 (1998), 331-335. [24] Z. Wang, Convergence theorems for sequences of Choquet integral, Int. Gen. Syst. 26 (1997) 133-143. [25] S.H. Wei, S.M. Chen, Fuzzy risk analysis based on interval-valued fuzzy sets, Expert Systems with Applications, 36(2009), 2285-2299. [26] Z. Xu, M. Xia, Distance and similarity measures for hesitant fuzzy sets, Information Sciences 181(2011), 2128-2138. [27] W. Zeng and H. Li, Relationship between similarity measure and entropy of interval-valued fuzzy sets, Fuzzy Sets and Systems, 157(2004), 1447-1484. [28] W. Zeng and P. Guo, Normalized distance, similarity measure, inclusion measure and entropy of intervalvalued fuzzy sets and their relationship, Information Sciences 179(2008), 1334-1342. [29] H. Zhang, W. Zhang, C. Mei, Entropy of interval-valued fuzzy sets based on distance and its relationship with similarity mesaure, Knowledge-Based Systems, 22(2009), 449-454.
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n-JORDAN ∗-DERIVATIONS ON INDUCED FUZZY C ∗ -ALGEBRAS GANG LU, YANDUO WANG, AND PENGYU YE Abstract. Using the fixed point alternative theorem, we investigate the Hyers-Ulam stability of of n-Jordan ∗-derivations on induced fuzzy C ∗ -algebras associated with the following functional equation f (y − x) + f (x − z) + f (3x − y + z) = f (3x).
1. Introduction and preliminaries The stability problem of functional equations originated from a question of Ulam [43] concerning the stability of group homomorphisms. Hyers [22] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [1] for additive mappings and by Rassias [38] for linear mappings by considering an unbounded Cauchy difference. Those results have been recently complemented in [9]. A generalization of the Aoki and Rassias theorem was obtained by G˘avruta [21], who used a more general function controlling the possibly unbounded Cauchy difference in the spirit of Rassias’ approach. The stability problems for several functional equations or inequalities have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [8, 15], [23]–[31], [39]–[41]). We recall a fundamental result in fixed point theory. 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 (see [14, 18]). Let (X, d) be a complete generalized metric space and let J : X → X be a strictly contractive mapping with Lipschitz constant L < 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) < ∞, for all n ≥ n0 ; (2) the sequence {J n x} converges to a fixed point y ∗ of J; (3) y ∗ is the unique fixed point of J in the set Y = {y ∈ X|d(J n0 x, y) < ∞}; 1 (4) d(y, y ∗ ) ≤ 1−L d(y, Jy) for all y ∈ Y . By using the fixed point method, the stability problems of several functional equations have been extensively investigated by a number of authors (see [10, 13, 14, 17, 19, 28, 33, 34, 37, 46]). 2010 Mathematics Subject Classification. Primary 39B62, 39B52, 46B25. Key words and phrases. Fuzzy normed space; additive functional equation; Hyers-Ulam stability; induced fuzzy C ∗ -algebra. 1
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In 1984, Katsaras [27] defined a fuzzy norm on a linear space and at the same year Wu and Fang [44] also introduced a notion of fuzzy normed space and gave the generalization of the Kolmogoroff normalized theorem for fuzzy topological linear space. In [7], Biswas defined and studied fuzzy inner product spaces in linear space. Since then some mathematicians have defined fuzzy metrics and norms on a linear space from various points of view [6, 20, 30, 42, 45]. In 1994, Cheng and Mordeson introduced a definition of fuzzy norm on a linear space in such a manner that the corresponding induced fuzzy metric is of Kramosil and Michalek type [29]. In 2003, Bag and Samanta [6] modified the definition of Cheng and Mordeson [16] by removing a regular condition. They also established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy norms (see [3]). Following [2], we give the employing notion of a fuzzy norm. Let X be a real linear space. A function N : X × R → [0, 1](the so-called fuzzy subset) is said to be a fuzzy norm on X if for all x, y ∈ X and all a, b ∈ R: (N1 ) N (x, a) = 0 for a ≤ 0; (N2 ) x = 0 if and only if N (x, a) = 1 for all a > 0; b ) if a 6= 0; (N3 ) N (ax, b) = N (x, |a| (N4 ) N (x + y, a + b) ≥ min{N (x, a), N (y, b)}; (N5 ) N (x, .) is a non-decreasing function on R and lima→∞ N (x, a) = 1; (N6 ) For x 6= 0, N (x, .) is (upper semi) continuous on R. The pair (X, N ) is called a fuzzy normed linear space. One may regard N (x, a) as the 0 truth value of the statement the norm of x is less than or equal to the real number a . Definition 1.2. Let (X, N ) be a fuzzy normed linear space. Let xn be a sequence in X. Then xn is said to be convergent if there exists x ∈ X such that limn→∞ N (xn − x, a) = 1 for all a > 0. In that case, x is called the limit of the sequence xn and we denote it by N -limn→∞ xn = x. Definition 1.3. A sequence xn in X is called Cauchy if for each > 0 and each a > 0 there exists n0 such that for all n ≥ n0 and all p > 0, we have N (xn+p − xn , a) > 1 − . It is known that every convergent sequence in fuzzy normed space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed space is called a fuzzy Banach space. We say that a mapping f : X → Y between fuzzy normed vector space X, Y is continuous at point x0 ∈ X if for each sequence {xn } converging to x0 in X, then the sequence {f (xn )} converges to f (x0 ). If f : X → Y is continuous at each x ∈ X, then f : X → Y is said to be continuous on X(see [2]) Definition 1.4. [36] Let X be a ∗-algebra and (X, N ) a fuzzy normed space. (1) The fuzzy normed space (X, N ) is called a fuzzy normed ∗-algebra if N (xy, st) ≥ N (x, s) · N (y, t) and N (x∗ , t) = N (x, t). (2) A complete fuzzy normed ∗-algebra is called a fuzzy Banach ∗-algebra. Example 1.5. Let (X, k.k) be a normed ∗-algebras. Let a , a > 0 , x ∈ X, a+kxk N (x, a) = 0, a ≤ 0, x ∈ X Then N (x, t) is a fuzzy norm on X and (X, N (x, t)) is a fuzzy normed ∗-algebra.
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Definition 1.6. Let (X, k · k) be a C ∗ -algebra and N a fuzzy norm on X. (1) The fuzzy normed ∗-algebra (X, N ) is called an induced fuzzy normed ∗-algebra. (2) The fuzzy Banach ∗-algebra (X, N ) is called an induced fuzzy C ∗ -algebra. Definition 1.7. Let (X, k · k) be an induced fuzzy normed ∗-algebra. Then a C-linear mapping D : (X, N ) → (X, N ) is called a fuzzy n-Jordan ∗-derivation if D(xn ) = D(x)xn−1 + xD(x)xn−2 + · · · + xn−2 D(x)x + xn−1 D(x), D(x∗ ) = D(x)∗ for all x ∈ X. Throughout this paper, assume that (X, N ) is an induced fuzzy C ∗ -algebra. 2. Main results Lemma 2.1. Let (Z, N ) be a fuzzy normed vector space and f : X → Z be a mapping such that t (2.1) N (f (y − x) + f (x − z) + f (3x − y + z) , t) ≥ N f (3x) , 2 for all x, y, z ∈ X and all t > 0. Then f is additive. Proof. Letting x = y = z = 0 in (2.1), we get t t N (3f (0), t) = N f (0), ≥ N f (0), 3 2 for all t > 0. By (N5 ) and (N6 ), N (f (0), t) = 1 for all t > 0. It follows from (N2 ) that f (0) = 0. Letting x = z = 0 in (2.1), we get t N (f (y) + f (0) + f (−y), t) ≥ N f (0), =1 2 for all t > 0. It follows from (N2 ) that f (−y) + f (y) = 0 for all y ∈ X. Thus f (−y) = −f (y) for all y ∈ X. Letting x = 0 and replacing z by −z in (2.1), we get t N (f (y) + f (z) + f (−y − z), t) ≥ N f (0), =1 2 for all t > 0. It follows from (N2 ) that f (y) + f (z) + f (−y − z) = 0 for all y, z ∈ X. Thus f (y + z) = f (y) + f (z) for all y, z ∈ X, as desired.
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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) 3 3 3 3 for all x, y, z ∈ X. Let f : X → X be a mapping such that N (f (µ(y − x)) + f (µ(x − z)) + f (µ(3x − y + z)) − µf (3x) , t) t , ≥ t + φ(x, y, z)
(2.3)
N f (wn ) − f (w)wn−1 − wf (w)wn−2 − · · · − wn−2 f (w)w − wn−1 f (w) t +f (v ∗ ) − f (v)∗ , t) ≥ t + φ(w, v, 0)
(2.4)
for all x, y, z, w, v ∈ X, all t > 0 and all µ ∈ T1 := {c ∈ C : |c| = 1}. Then the limit A(x) = N − limn→∞ 3n f 3xn exists for each x ∈ X and the mapping A : X → X is a fuzzy n-Jordan ∗-derivation satisfying N (f (x) − A(x), t) ≥
3(1 − L)t 3(1 − L)t + Lφ (x, 2x, 0)
(2.5)
for all x ∈ X and all t > 0. Proof. Letting µ = 1, y = 2x , z = 0 in (2.3), we have t N (3f (x) − f (3x), t) ≥ t + φ (x, 2x, 0)
(2.6)
and so x t t = N 3f − f (x), t ≥ L x 2x 3 t + 3 φ (x, 2x, 0) t + φ 3, 3 , 0 for all x ∈ X. Thus x L N 3f − f (x), t ≥ 3 3
L t 3 L t 3
+
L φ (x, 2x, 0) 3
=
t t + φ (x, 2x, 0)
(2.7)
for all x ∈ X. Consider the set G := {g : X → X} and introduce the generalized metric on G: d(g, h) := inf{a ∈ R+ : N (g(x) − h(x), at) ≥
t } t + φ (x, 2x, 0)
for all x ∈ X and all t > 0, where inf φ = +∞. It is easy to show that (S, d) is complete (see the proof of [32, Lemma 2.1] Now, we consider the linear mapping Q : G → G such that x Qg(x) := 3g 3 for all x ∈ X.
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Let g, h ∈ G be given such that d(g, h) = ε. Then N (g(x) − h(x), εt) ≥
t t + φ (x, 2x, 0)
for all x ∈ X and all t > 0. Hence x x x L x N (Qg(x) − Qh(x), Lεt) = N 3g − 3h , Lεt = N g −h , εt 3 3 3 3 3 ≥ =
Lt 3
+φ
Lt 3 x 2x , ,0 3 3
≥
Lt 3 Lt 3
+ L3 φ (x, 2x, 0)
t t + φ (x, 2x, 0)
for all x ∈ X and all t > 0. Thus d(g, h) = ε implies that d(Qg, Qh) ≤ Lε. This means that d(Qg, Qh) ≤ Ld(g, h) for all g, h ∈ G. It follows from (2.7) that d(f, Qf ) ≤ L3 . By Theorem 1.1, there exists a mapping A : X → X satisfying the following: (1) A is a fixed point of Q, i.e., x 1 A = A(x) 3 3
(2.8)
for all x ∈ X. The mapping A is a unique fixed point of Q in the set M = {g ∈ G : d(f, g) < ∞}. This implies that A is a unique mapping satisfying (2.8) such that there exists an a ∈ (0, ∞) satisfying N (f (x) − A(x), at) ≥
t t + φ (x, 2x, 0)
for all x ∈ X. (2) d(Qk f, A) → 0 as k → ∞. This implies the equality x N − lim 3k f k = A(x) k→∞ 3 for all x ∈ X; (3) d(f, A) ≤
1 d(f, Qf ), 1−L
which implies the inequality d(f, A) ≤
L . 3(1 − L)
This implies that the inequality (2.5) holds.
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Next we show that A is additive. It follows from (2.2) that ∞ X
3k φ
k=0
x
x y z x y z y z 2 = φ(x, y, z) + 3φ , , + 3 φ + ··· , , , , 3k 3k 3k 3 3 3 32 32 32 ≤ φ(x, y, z) + Lφ(x, y, z) + L2 φ(x, y, z) + · · · 1 = φ(x, y, z) < ∞ 1−L
for all x, y, z ∈ X. By (2.3), x−z 3x − y + z 3 y−x k k k k +3 f µ k +f µ − 3 µf N 3 f µ k x ,3 t 3 3 3k 3k t ≥ x y t + φ 3k , 3k , 3zk and so y−x x−z 3x − y + z 3 k k k k x ,t N 3 f µ k +3 f µ k +3 f µ − 3 µf 3 3 3k 3k ≥
t 3k
+φ
t 3k x y z , , k k k 3 3 3
=
t t + 3k φ
x y . , z 3k 3k 3k
for all x, y, z ∈ X, all t > 0 and all µ ∈ T1 . Since limk→∞ x, y, z ∈ X and all t > 0,
t t+3k φ(
x y , , z 3k 3k 3k
)
= 1 for all
N (A (µ(y − x)) + A (µ(x − z)) + A (µ(3x − y + z)) − µA (3x) , t) = 1 for all x, y, z ∈ X, all t > 0 and all µ ∈ T1 . So A (µ(y − x)) + A (µ(x − z)) + A (µ(3x − y + z)) = µA (3x)
(2.9)
for all x, y, z ∈ X, all t > 0 and all µ ∈ T1 . Letting x = y = z = 0 in (2.9), we have A(0) = 0. Let µ = 1, x = 0 and replace z by −z in (2.9). By the same reasoning as in the proof of Lemma 2.1, one can easily show that A is additive. Letting y = 2x, z = 0 in (2.9), we get x µA(x) = 3A µ = A(µx) 3 for all x ∈ X and µ ∈ T1 . The mapping A : X → X is C-linear by [35, Theorem 2.1]. By (2.4) and letting v = 0 in (2.4), we get n w w n−1 w w n−2 w nk nk nk w N 3 f − 3 f − 3 f − ··· 3nk 3k 3k 3k 3k 3k w n−2 w w n−1 w t nk nk nk f k w−3 f k ,3 t ≥ −3 k k 3 3 3 3 t + φ( 3wk , 0, 0)
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for all w ∈ X and all t > 0. Thus n w w n−1 w w n−2 w nk nk nk w N 3 f − 3 f − 3 f − ··· 3nk 3k 3k 3k 3k 3k t w n−1 w w n−2 w nk nk 3nk w − 3 , t ≥ −3 f f t 3k 3k 3k 3k + φ( 3wk , 0, 0) 3nk t ≥ n−1 t + (3 L)k φ(w, 0, 0) for all w ∈ X and all t > 0. Since limk→∞ t > 0, we get
t t+(3n−1 L)k φ(w,0,0)
= 1 for all w ∈ X and all
N (D(wn ) − D(w)wn−1 − wD(w)wn−2 − · · · − wn−2 D(w)w − wn−1 D(w), t) = 1 for all x ∈ X and all t > 0. So D(wn ) − D(w)wn−1 − wD(w)wn−2 − · · · − wn−2 D(w)w − wn−1 D(w) = 0 for all w ∈ X. Letting w = 0 in (2.4), similarly, we get D(v ∗ ) − D(v)∗ = 0 for all v ∈ X. Therefore, the mapping D : X → X is a fuzzy n-Jordan ∗-derivation.
Corollary 2.3. Let p be a real number with p > 1 , θ ≥ 0, and X be a normed vector space with norm k · k. Let f : X → X be a mapping satisfying N (f (µ(y − x)) + f (µ(x − z)) + f (µ(3x − y + z)) − µf (3x) , t) t , ≥ p t + θ(kxk + kykp + kzkp )
(2.10)
N f (wn ) − f (w)wn−1 − wf (w)wn−2 − · · · − wn−2 f (w)w − wn−1 f (w) t +f (v ∗ ) − f (v)∗ , t) ≥ t + θ(kwkp + kvkp )
(2.11)
for all x, y, w, v ∈ X, all t > 0 and all µ ∈ T1 . Then the limit A(x) = N −limn→∞ 3n f 3xn exists for each x ∈ X and the mapping A : X → X is a fuzzy n-Jordan ∗-derivation satisfying (3p − 3)t N (f (x) − A(x), t) ≥ p (3 − 3)t + θ(1 + 2p )kxkp for all x ∈ X and all t > 0. Proof. The proof follows from Theorem 2.2 by taking φ(x, y, z) = θ(kxkp + kykp + kzkp ) and L = 31−p .
Theorem 2.4. Let φ : X 3 → [0, ∞) be a function such that there exists an L < 1 with x y z 3Lφ , , ≤ φ(x, y, z) (2.12) 3 3 3
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for all x, y, z ∈ X. Let f : X → X be a mapping satisfying (2.3) and (2.4). Then the limit A(x) = N − limn→∞ 31n f (3n x) exists for each x ∈ X and the mapping A : X → X is a fuzzy n-Jordan ∗-derivation satisfying N (f (x) − A(x), t) ≥
3(1 − L)t 3(1 − L)t + φ (x, 2x, 0)
(2.13)
for all x ∈ X and all t > 0. Proof. Let (G, d) be generalized metric space defined in the proof of Theorem 2.2. Consider the linear mapping Q : G → G such that 1 Qg(x) := g(3x) 3 for all x ∈ X. It follow from (2.6) that 1 1 t N f (x) − f (3x), t ≥ 3 3 t + φ (x, 2x, 0) for all x ∈ X and all t > 0. Thus d(f, Qf ) ≤ 13 . Hence d(f, A) ≤
1 , 3(1 − L)
which implies that the inequality (2.13) holds. The rest of the proof is similar to the proof of Theorem 2.2.
Corollary 2.5. Let θ ≥ 0 and let p be a positive real number with p < 1. Let X be a normed vector space with normed k · k. Let f : X → X be a mapping satisfying (2.10) and (2.11). Then A(x) = N − limn→∞ 31n f (3n x) exists for each x ∈ X and defines a fuzzy n-Jordan ∗-derivation A : X → X such that N (f (x) − A(x), t) ≥
(3 − 3p )t (3 − 3p )t + θ(1 + 2p )kxkp
for every x ∈ X and all t > 0. Proof. The proof follows from Theorem 2.4 by taking φ(x, y, z) = θ(kxkp + kykp + kzkp ) and L = 3p−1 .
Acknowledgments G. Lu was supported by Doctoral Science Foundation of Liaoning Province, China, by Hall of Liaoning Province Science and Technology (No. 2012-1055), Shengyang University of Technology(No.521101302) and the Project Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry. C. Park was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2012R1A1A2004299).
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[27] A. K. Katsaras, Fuzzy topological vector spaces II, Fuzzy Sets Syst. 12 (1984), 143–154. [28] H. Khodaei, R. Khodabakhsh and M. Eshaghi Gordji, Fixed points, Lie ∗-homomorphisms and Lie ∗-derivations on Lie C ∗ -algebras, Fixed Point Theory 14 (2013), 387–400. [29] I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica 11 (1975), 326–334. [30] S. V. Krishna and K. K. M. Sarma, Separation of fuzzy normed linear spaces, Fuzzy Sets Syst. 63 (1994), 207–217. [31] G. Lu and C. Park, Hyers-Ulam stability of additive set-valued functional equations, Appl. Math. Lett. 24 (2011), 1312–1316. [32] D. Mihet and V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl. 343 (2008), 567–572. [33] F. Moradlou and M. Eshaghi Gordji, Approximate Jordan derivations on Hilbert C ∗ -modules, Fixed Point Theory 14 (2013), 413–425. [34] C. Park, Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras, Fixed Point Theory Appl. 2007, Article ID 50175 (2007). [35] C.Park, Homomorphisms between Poisson JC ∗ -algebras, Bull. Braz. Math. Soc. 36 (2005), 79–97. [36] C. Park, K. Ghasemi and S. Ghaleh, Fuzzy n-Jordan ∗-derivations on induced fuzzy C ∗ -algebras, J. Comput. Anal. Appl. 16 (2014), 494–502. [37] C. Park and J. M. Rassias, Stability of the Jensen-type functional equation in C ∗ -algebras: A fixed point approach, Abs. Appl. Anal. 2009, Article ID 360432 (2009). [38] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [39] Th. M. Rassias (Ed.), Functional Equations and Inequalities, Kluwer Academic, Dordrecht, 2000. [40] Th. M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 251 (2000), 264–284. [41] Th .M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Math. Appl. 62 (2000), 23–130. [42] B. Shieh, Infinite fuzzy relation equations with continuous t-norms, Inform. Sci. 178 (2008), 1961– 1967. [43] S.M. Ulam, Problems in Modern Mathematics, Chapter VI, Science ed., Wiley, New York, 1940. [44] C. Wu and J. Fang, Fuzzy generalization of Klomogoroff ’s theorem, J. Harbin Inst. Technol. 1 (1984), 1–7. [45] J. Z. Xiao and X.-H. Zhu, Fuzzy normed spaces of operators and its completeness, Fuzzy Sets Syst. 133 (2003), 389–399. [46] T. Z. Xu, J. M. Rassias and W. X. Xu, A fixed point approach to the stability of a general mixed additive-cubic functional equation in quasi fuzzy normed spaces, Internat. J. Phys. Sci. 6 (2011), 313–324.
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Gang Lu 1. Department of Mathematics, Zhejiang University, Hangzhou 310027, People’s Republic of China 2.Department of Mathematics, School of Science, ShenYang University of Technology, Shenyang 110178, P. R. China E-mail address: [email protected] Yanduo Wang Department of Mathematics, School of Science, ShenYang University of Technology, Shenyang 110178, P. R. China E-mail address: [email protected] Pengyu Ye Department of Mathematics, School of Science, ShenYang University of Technology, Shenyang 110178, P. R. China E-mail address: yuxiang1 [email protected]
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Global stability analysis of a delayed viral infection model with antibodies and general nonlinear incidence rate A. M. Elaiw, N. H. AlShamrani and M. A. Alghamdi Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia. Email: [email protected] (A. Elaiw).
Abstract In this paper, we study the global properties of a viral infection model with antibody immune response. The incidence rate is given by a general function of the populations of the uninfected target cells, infected cells and free viruses. The model contains two types of intracellular discrete time delays to describe the time required for viral contacting an uninfected target cell and viral emission. We have established a set of conditions on the general incidence rate function and determined two threshold parameters R0 (the basic infection reproduction number) and R1 (the antibody immune response activation number) which are su¢ cient to determine the global behavior of the model. The global asymptotic stability of the equilibria of the model has been proven by using direct Lyapunov method and applying LaSalle’s invariance principle.
Keywords: Virus dynamics; Intracellular delay; global stability; antibody immune response; Lyapunov functional.
1
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1
Introduction
In recent years, several works have been devoted to study and develop mathematical models of the virus dynamics such as human immunode…ciency virus (HIV) (see e.g. [1]-[14]), hepatitis B virus (HBV) [15]-[18], hepatitis C virus (HCV) [19]-[21] and human T cell leukemia HTLV [22], etc. Mathematical models of viral infection can help for understanding the viral dynamics and developing antiviral drug therapies. In reality, the immune response needs an indispensable components to do its job such as antibodies, cytokines, natural killer cells, and T cells. The antibody immune response is a part of the adaptive system in which the body responds to pathogens by primarily using antibodies that produced from the B cells. While the other part is the Cytotoxic T Lymphocytes (CTL) immune response where the CTL attacks and kills the infected cells [4]. In some infections such as in malaria, the CTL immune response is less e¤ective than the antibody immune response [23]. Mathematical models of viral infection with antibody immune response have been proposed and analyzed in ([24]-[29]). The basic model of viral infection with antibody immune response has introduced by Murase et. al. [24] and Shi… Wang [29] as: x(t) _ =s
dx(t)
y(t) _ = v(t)x(t) v(t) _ = ky(t)
v(t)x(t); ay(t);
bz(t)v(t)
z(t) _ = rz(t)v(t)
(1) (2)
cv(t);
(3)
z(t);
(4)
where x(t), y(t), v(t) and z(t) denote the populations of uninfected target cells, infected cells, free virus particles and antibody immune cells at time t, respectively. Parameters s, k and r represent, respectively, the rate at which new healthy cells are generated from the source within the body, the generation rate constant of free viruses produced from the infected cells and the proliferation rate constant of antibody immune cells. Parameters d, a, c and
are the natural death rate constants of the uninfected cells, infected
cells, free virus particles and antibody immune cells, respectively. Parameter
is the infection rate constant
and b is the removal rate constant of the virus due to the antibodies. All the parameters given in model (1)-(4) are positive. 2
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The intracellular time delay between the time of the virus contacting the target cells and the time of generating new infectious viruses has been neglected in system (1)-(4). In fact, the intracellular delay in the infection process is actually exists (see e.g. [8]-[12]). Note that, the infection rate in model (1)-(4) is presented to be bilinear in x and v, which can not be completely describe the interaction between the uninfected target cells and viruses. Nevertheless, there are many types of improved incidence rates which are more commonly used due to their bene…t for helping us gain the uni…cation theory through passing over the unessential details (see e.g. [30] and [31]). Variety of viral infection models with antibody immune response have been considered with di¤erent forms of the incidence rate such as saturated incidence rate, 0, [27], Beddington-DeAngelis functional response,
xv 1+ x+ v ,
;
xv 1+ v
0 [26], and general form,
where (x; v)v
[28]. In [28], a discrete time delay has been incorporated within the model. However, the infection rate does not depend on the infected cells y. In some viral infections such as HBV, the infection rate depends on x, y and v [17], [16]. In [32], the infection rate is given by (x; y; v)v, however the antibody immune response has been neglected. Our aim in this paper is to investigate the global stability analysis of a viral infection model with general incidence rate function and antibody immune response taking into consideration two types of discrete time delays. The rest of the paper is designed as follows. In the next section, we introduce the model and discuss the non-negativity and boundedness of the solutions. In Section 3, we de…ne two threshold parameters and discuss the existence of the model’s equilibria. In Section 4, we study the global asymptotic stability of the equilibria using suitable Lyapunov functional and applying LaSalle’s invariance principle. Finally, conclusion is given in Section 5.
3
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2
The mathematical model
In this section, we consider the following viral infection model with general incidence rate taking into consideration the antibody immune response. x(t) _ =s
dx(t)
y(t) _ =e
1 1
v(t) _ = ke
2 2
(x(t); y(t); v(t))v(t);
(x(t y(t
2)
z(t) _ = rz(t)v(t) where time t
1
and 1,
2
1 ); y(t
1 ); v(t
bz(t)v(t)
1)
ay(t);
(6)
cv(t);
(7) (8)
are the delay parameters. We assume that, the virus contacts an uninfected target cell at
the cell becomes infected at time t. The term e
addition, we assume that a cell infected at time t 2 2
1 ))v(t
z(t);
contacted cell during the time delay interval, where
term e
(5)
2
1
1 1
represents the probability of surviving the
is the death rate constant of the contacted cells. In
starts to generate new infectious viruses at time t. The
denotes the probability of surviving the infected cell during the time delay interval, where
2
is a constant. The de…nitions of all variables and parameters are identical to those given in Section 1. The incidence rate of infection is presented by a general function in the form (x; y; v)v, where
is continuously
di¤erentiable and satis…es the following assumptions [28] and [32]: Assumption A1.
(0; y; v) = 0 for all y; v
0 and
(x; y; v) > 0 for all x > 0, y
0, v
0.
@ (x; y; v) > 0 for all x > 0, y 0 and v 0: @x @ (x; y; v) @ (x; y; v) Assumption A3. < 0, < 0 for all x; y; v > 0: @y @v @ ( (x; y; v)v) Assumption A4. > 0 for all x; y; v > 0: @v Assumption A2.
Let the initial states of system (5)-(8) be given as: x( ) = j(
)
j (0)
where
= maxf 1 ;
2 g;
( 1 ( );
1(
0;
); y( ) = 2[
2(
); v( ) =
3(
); z( ) =
4(
);
; 0); j = 1; :::; 4;
> 0; j = 1; :::; 4; 2(
);
3(
);
4(
(9)
)) 2 C([
; 0]; R4 0 ). We denote by C = C([
; 0]; R4 0 ) the
4
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Banach space of continuous functions mapping the interval [ for
; 0] into R4 0 ; with norm k k =
sup j ( )j 0
2 C. We note that the system (5)-(8) with initial states (9) has a unique solution [33].
2.1
Non-negativity and boundedness of solutions
In this section, we show that the solutions of model (5)-(8) with initial states (9) are non-negative and ultimately bounded. Proposition 1. Assume that Assumption A1 is satis…ed. Then the solutions of (5)-(8) with the initial states (9) are non-negative and ultimately bounded. Proof. At the beginning, we show that x(t) is positive for all t x(t)
0 on the time interval [0; ] where
0. Let us assume in contrary that
is a constant, and let where t 2 [0; ] be such that x(t) = 0,
= s > 0. Thus, for su¢ ciently small " > 0, we have x(t) > 0 for some . Then from Eq. (5) we get x(t) _ t 2 t; t + " . This contradicts our assumption and then x(t) > 0, 8 t y(t) = y(0)e
at
+e
1 1
Zt
e
a(t
)
(x(
1 ); y(
0: Now from Eqs. (6)-(8) we get
1 ); v(
1 ))v(
1 )d
;
0
v(t) = v(0)e
Rt
z(t) = z(0)e
Rt
(c+bz( ))d 0
0
Zt
Rt
e
0
(
which yield y(t), v(t), z(t) all t
+ ke
2 2
rv( ))d
(c+bz( ))d
y(
2 )d
;
;
0 for all t 2 [0; ]. By a recursive argument, we get that y(t); v(t); z(t)
0 for
0:
Next we prove the ultimate bound of the solutions of system (5)-(8). From Eq. (5) we get x(t) _ s d.
and thus lim supt!1 x(t) T_1 (t) = e
1 1
+e
1 1
(s
= se
1 1
= se
1 1
Let T1 (t) = e dx(t
(x(t de
1)
1 ); y(t 1 1
1 T1 (t)
x(t s
1 1
x(t
1)
(x(t
1)
+ y(t), then
1 ); y(t
1 ); v(t
1 ))v(t
ay(t)
s dx(t)
se
1 ); v(t 1) 1 1
1 ))v(t
1 ))
ay(t); 1
e
1 1
x(t
1)
+ y(t)
1 T1 (t);
5
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where
1
= minfd; ag. Hence lim supt!1 T1 (t)
then lim supt!1 y(t)
where
2
L1 , where L1 =
s
. Since x(t) and y(t) are non-negative,
1
L1 . Moreover, let T2 (t) = v(t) + rb z(t), then
T_2 (t) = ke
2 2
y(t
2)
= ke
2 2
L1
2 T2 (t)
b z(t) r
cv(t)
kL1
ke
L1
2 (v(t)
b + z(t)) r
2 T2 (t);
= minfc; g. It follows that, lim supt!1 T2 (t)
non-negative, then lim supt!1 v(t)
2 2
L2 , where L2 =
kL1
. Since v(t) and z(t) are
2
L2 and lim supt!1 z(t)
L3 , where L3 =
r b L2 .
Therefore, all the
state variables of the model are ultimately bounded.
2.2
The equilibria and threshold parameters
At any equilibrium we have s e
dx
(x; y; v)v
1 1
ke
(x; y; v)v = 0;
2 2
y
bvz
(10)
ay = 0;
(11)
cv = 0;
(12)
)z = 0:
(13)
(rv
From Eq. (13), either z = 0 or z 6= 0. If z = 0, then from Eqs. (10)-(12) we get s dx c = v; 1 1 ae ke 2 2
y=
v=
k(s ace 1
dx)
:
(14)
v = 0:
(15)
1+ 2 2
Substituting from Eq. (14) into Eq. (11) we get: x;
s dx k(s ; ae 1 1 ace 1
ac e k
dx) 1+ 2 2
1 1+ 2 2
Eq. (15) has two possible solutions v = 0 or v 6= 0. If v = 0; then from Eqs. (10) and (11), we get x = s=d and y = 0 which leads to the infection-free equilibrium E0 (x0 ; 0; 0; 0) where x0 = s=d. If v 6= 0; then we have x;
s dx k(s ; ae 1 1 ace 1
dx)
ac e k
1+ 2 2
1 1+ 2 2
= 0:
Let 1
(x) =
x;
s dx k(s ; ae 1 1 ace 1
dx) 1+ 2 2
ac e k
1 1+ 2 2
= 0;
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then, we have 0 1
@ @x
(x) =
@ @y
d ae
1 1
0 1
Because of Assumptions A2 and A3, we have
kd ace
1 1+ 2 2
@ : @v
(x) > 0 which implies that function
1 (x)
is strictly
increasing w.r.t. x. Moreover, 1 (0)
=
1 (x0 )
=
Therefore, if
0;
s ae
1 1
;
ac e k
(x0 ; 0; 0)
k (x0 ; 0; 0) e ac
ks
1 1
2 2
ac e k
1 1+ 2 2
ace
1 1+ 2 2
ac e k
=
1 1+ 2 2
1 1+ 2 2
ac e k
=
1 1+ 2 2
k (x0 ; 0; 0) e ac
< 0;
1 1
2 2
1 :
> 1; then there exist a unique x1 2 (0; x0 ) such that
It follows from (12) and (14) that y1 =
d(x0 ae
x1 )
> 0 and v1 =
1 1
1 (x1 )
kd(x0 x1 ) > 0. ace 1 1 + 2 2
= 0.
It means
that, a chronic-infection equilibrium without antibody immune response E1 (x1 ; y1 ; v1 ; 0) exists when k (x0 ; 0; 0) e ac
1 1
2 2
> 1. Let us de…ne the basic infection reproduction number as: R0 =
k (x0 ; 0; 0) e ac
1 1
2 2
:
The parameter R0 determines whether a chronic-infection can be established. The other possibility of Eq. (13) is z 6= 0 which leads to v2 =
r
. From Eq. (10) we let 2 (x)
=s
dx
(x;
According to Assumptions A2 and A3, we know that and
2 (x0 )
=
2
s dx ; v2 )v2 = 0: ae 1 1 is a decreasing function of x. Clearly,
(x0 ; 0; v2 )v2 < 0. Thus, there exists a unique x2 2 (0; x0 ) such that
from Eq. (14) that, y2 =
d(x0 ae
x2 ) 1 1
> 0 and z2 =
k (x2 ; y2 ; v2 ) abe 1 1 + 2 2
c c = b b
2 (x2 )
k (x2 ; y2 ; v2 ) ace 1 1 + 2 2
2 (0)
=s>0
= 0. It follows 1 . Then, if
k (x2 ; y2 ; v2 ) > 1 then z2 > 0. Now we de…ne the antibody immune response activation number as ace 1 1 + 2 2 R1 =
k (x2 ; y2 ; v2 ) ; ace 1 1 + 2 2
which determines whether a persistent antibody immune response can be established. Hence, z2 can be c rewritten as z2 = (R1 b
1). It follows that, there is a chronic-infection equilibrium with antibody immune
response E2 (x2 ; y2 ; v2 ; z2 ) when R1 > 1. Clearly from Assumptions A2 and A3, we have R1 =
k (x2 ; y2 ; v2 ) k (x0 ; y2 ; v2 ) k (x0 ; 0; 0) < < = R0 : + + 1 1 2 2 1 1 2 2 ace ace ace 1 1 + 2 2 7
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2.3
Global stability analysis
In this section, the global asymptotic stability of the three equilibria of model (5)-(8) will be established by using direct Lyapunov method and applying LaSalle’s invariance principle. In the remaining parts of the paper we shall use the following function: H : (0; 1) ! [0; 1), H(u) = u
1
ln u:
Theorem 1. Let Assumptions A1-A3 be hold true and R0
1; then the infection-free equilibrium E0 is
globally asymptotically stable (GAS). Proof. We construct a Lyapunov functional as: U0 = x
Z
x0
x
(x0 ; 0; 0) d +e ( ; 0; 0)
1 1
y+
(x( ); y( ); v( ))v( )d + ae
1 1
x0
+ t dU0 dt
We calculate dU0 = dt
1 1
t
1
1 1+ 2 2
ab e rk
v+
1 1+ 2 2
z
y( )d :
(16)
2
along the solutions of model (5)-(8) as: (x0 ; 0; 0) (x; 0; 0)
1
+ ae
Zt
a e k Zt
y(t
2)
+ (x; y; v)v
(s ac e k
(x(t
dx
(x; y; v) v) + (x(t
1 1+ 2 2
ab e k
v
1 ); y(t
1 1+ 2 2
1 ); v(t
=s 1
(x0 ; 0; 0) (x; 0; 0)
1
x x0
+
=s 1
(x0 ; 0; 0) (x; 0; 0)
1
x x0
+
From Assumptions A2-A3 we know that
ac e k
zv +
1 ))v(t
(x; y; v)
1 ); y(t
ab e k
1)
1 1+ 2 2
+ ae
(x0 ; 0; 0) (x; 0; 0)
1 ); v(t
1 1
ac e k
(x; y; v) R0 (x; 0; 0)
1 1+ 2 2
1 ))v(t
ab e rk
zv
(y
y(t
1 1+ 2 2
1 v
1)
1 1+ 2 2
ae
1 1
y
z
2 ))
ab e rk
v ab e rk
1 1+ 2 2
1 1+ 2 2
z:
z (17)
(x; y; v) is an increasing function of x and decreasing function of
y and v. Then, the …rst term of Eq. (17) is less than or equal zero and (x; y; v) < (x; 0; 0),
x; y; v > 0:
It follows that dU0 dt
s 1
(x0 ; 0; 0) (x; 0; 0)
1
x x0
+
ac e k
1 1+ 2 2
(R0
1) v
ab e rk
1 1+ 2 2
z:
(18)
8
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Therefore, if R0 to
1, then
dU0 dt
0 for all x; y; v; z > 0. We note that the solutions of system (5)-(8) converge
, the largest invariant subset of
The set
dU0 dt
= 0 [33]. From (18), we have
is invariant and for any element belongs to
dU0 dt
= 0 i¤ x = x0 , v = 0 and z = 0.
satis…es v = 0 and z = 0. We can see from Eq. (7)
that v_ = 0 = ke It follows that, y = 0. Hence
dU0 dt
2 2
y(t
2 ):
= 0 i¤ x = x0 and y = v = z = 0. Using LaSalle’s invariance principle, we
derive that E0 is GAS. Assumption A5. 1
(x; y; v) (x; yi ; vi )
(x; yi ; vi ) (x; y; v)
v vi
0; i = 1; 2 for all x; y; v > 0:
Theorem 2. Let Assumptions A1-A5 be hold true and R1
1 < R0 , then the chronic-infection equilibrium
without antibody immune response E1 is GAS. Proof. De…ne: U1 = x
x1
Z
x
(x1 ; y1 ; v1 ) d +e ( ; y1 ; v1 )
x1
+
a e k
1 1+ 2 2
v v1
v1 H
+ (x1 ; y1 ; v1 )v1 t
Zt
H
+
ab e rk
1 1
y1 H
1 1+ 2 2
y y1
z
(x( ); y( ); v( ))v( ) (x1 ; y1 ; v1 )v1
1
d + ae
1 1
y1 t
Zt
H
y( ) y1
d :
(19)
2
9
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Calculating the time derivative of U1 along the trajectories of system (5)-(8), we obtain dU1 = dt +e +
(x1 ; y1 ; v1 ) (x; y1 ; v1 )
1
a e k
y1 y
1
1 1
1 1+ 2 2
e
=
y
1 1
y1 (x(t y ac e k ab + e k
y(t
v
1 1+ 2 2
v1 z
y1 ln
(s
1 ); v(t
2)
cv
1 ); v(t
1 ); v(t (x; y; v)v
y(t
2)
bvz +
1)
ab e rk
ay
1 1+ 2 2
(rvz
z)
1)
1 ))v(t
1)
2)
y
1 ))v(t
v1 e v
1 ))v(t
1 ))v(t
dx) + (x1 ; y1 ; v1 )
ab e rk (x(t 2)
y(t
1 ); y(t
1 ); v(t
y
1 ); y(t
2 2
+ y1 ln
ay(t
y(t
(x; y; v) v)
1 ); y(t
1 ); y(t
1 1+ 2 2
1 1
ke
2)
+ (x1 ; y1 ; v1 )v1 ln + ae
(x(t
(x(t
(x1 ; y1 ; v1 ) (x; y1 ; v1 )
1
1 1
(x(t
+ (x1 ; y1 ; v1 )v1 ln + ae
dx
v1 v
1
+ (x; y; v)v
(s
1 1
1 1+ 2 2
+
1)
ac e k
(x; y; v)v (x; y1 ; v1 )
+ ay1 e
1 1+ 2 2
1 1
v1
z
1 ); y(t
1 ); v(t (x; y; v)v
1 ))v(t
1)
:
(20)
Using the equilibrium conditions for E1 : s = dx1 + ae
1 1
y1 ;
(x1 ; y1 ; v1 )v1 = ae
1 1
y1 =
ac e k
1 1+ 2 2
v1 ;
10
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we obtain dU1 = dx1 1 dt
(x1 ; y1 ; v1 ) (x; y1 ; v1 ) (x1 ; y1 ; v1 ) + ae (x; y1 ; v1 )
ae
1 1
y1
ae
1 1
y1
ae
1 1
y1
+ ae
1 1
y1 ln
+ ae
1 1
y1 ln
y1 (x(t v v1
x x1
1 1 1
+ 3ae
1 ); v(t y (x1 ; y1 ; v1 )v1
1 1
y1
(x(t
v1 y(t vy1 1 ); y(t
y(t
2)
+
y
y1
(x; y; v)v (x; y1 ; v1 )v1
y1
1 ); y(t
ae
1 1
1 ))v(t
1)
2)
1 ); v(t (x; y; v)v
ab e k
1 ))v(t
1 1+ 2 2
v1
r
1)
z:
(21)
Using the following equalities: ln
(x(t
1 ); y(t
1 ); v(t
1 ))v(t
1)
y1 (x(t
= ln
(x; y; v)v
y(t
2)
y
1 ); v(t
1 ))v(t
1)
y (x1 ; y1 ; v1 )v1 (x1 ; y1 ; v1 ) (x; y1 ; v1 )
+ ln ln
1 ); y(t
= ln
vy1 v1 y
y1
(x; y; v)v (x; y1 ; v1 )v1
+ ln
(x; y1 ; v1 ) (x; y; v)
+ ln v1 y(t vy1
2)
+ ln
v1 y vy1
;
;
we get dU1 = dx1 1 dt ae + + +
1 1
(x1 ; y1 ; v1 ) (x; y1 ; v1 ) (x1 ; y1 ; v1 ) (x; y1 ; v1 )
y1
y1 (x(t v1 y(t vy1
1
x x1
1
1 ); v(t y (x1 ; y1 ; v1 )v1
(x; y1 ; v1 ) (x; y; v)
1
ln
1
ln
v1 y(t vy1
1 1
v v1
1+
(x; y1 ; v1 ) (x; y; v)
(x1 ; y1 ; v1 ) (x; y1 ; v1 )
ln
1 ); y(t
2)
+ ae
1 ))v(t
1)
1
ln
y1 (x(t
1 ); y(t
1 ); v(t y (x1 ; y1 ; v1 )v1
1 ))v(t
1)
2)
(x; y1 ; v1 ) (x; y; v)
+
ab e k
1 1+ 2 2
v1
r
z:
(22)
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Eq. (22) can be simpli…ed as: dU1 = dx1 1 dt
(x1 ; y1 ; v1 ) (x; y1 ; v1 )
+ ae
1 1
y1 1
(x; y; v) (x; y1 ; v1 )
ae
1 1
y1 H
(x1 ; y1 ; v1 ) (x; y1 ; v1 )
y1 (x(t
+H +
ab e k
1 1+ 2 2
x x1
1
(x; y1 ; v1 ) (x; y; v) v1 y(t vy1
+H
1 ); y(t
1 ); v(t y (x1 ; y1 ; v1 )v1
v1
r
v v1
1 ))v(t
2)
1)
+H
(x; y1 ; v1 ) (x; y; v)
z:
(23)
From Assumptions A1 and A5, we get that the …rst and second terms of Eq. (23) are less than or equal zero. Now we show that if R1
1 then v1
sgn(x2
r
x1 ) = sgn(v1
= v2 . Let R0 > 1, then we want to show that v2 ) = sgn(y1
y2 ) = sgn(R1
1):
From Assumptions A2-A4, for x1 ; x2 ; y1 ; y2 ; v1 ; v2 > 0, we have ( (x2 ; y2 ; v2 )
(x1 ; y2 ; v2 ))(x2
x1 ) > 0;
(24)
( (x1 ; y1 ; v1 )
(x1 ; y2 ; v1 ))(y2
y1 ) > 0;
(25)
( (x1 ; y1 ; v1 )
(x1 ; y1 ; v2 ))(v2
v1 ) > 0;
(26)
( (x2 ; y2 ; v2 )v2 First, we claim sgn(x2
x1 ) = sgn(v1
(x2 ; y2 ; v1 )v1 )(v2
v1 ) > 0:
v2 ). Suppose this is not true, i.e., sgn(x2
(27) x1 ) = sgn(v2
v1 ).
Using the conditions of the equilibria E1 and E2 we have (s
dx2 )
(s
dx1 ) = (x2 ; y2 ; v2 )v2 = ae
1 1
(y2
(x1 ; y1 ; v1 )v1
y1 ):
(28)
Then, sgn(x2
x1 ) = sgn(y1
y2 )
(29)
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Moreover, (s
dx2 )
(s
dx1 ) = (x2 ; y2 ; v2 )v2
(x1 ; y1 ; v1 )v1
= ( (x2 ; y2 ; v2 )v2
(x2 ; y2 ; v1 )v1 ) + ( (x2 ; y2 ; v1 )v1
+ ( (x1 ; y2 ; v1 )v1
(x1 ; y1 ; v1 )v1 ):
(x1 ; y2 ; v1 )v1 )
Therefore, from inequalities (24) and (29) we get: sgn (x1 which leads to contradiction. Thus, sgn (x2 we have
k (x1 ;y1 ;v1 ) ace 1 1 + 2 2
x2 ) = sgn (x2 x1 ) = sgn (v1
k (x2 ; y2 ; v2 ) k (x1 ; y1 ; v1 ) ace 1 1 + 2 2 ace 1 1 + 2 2 k = e 1 1 2 2 [ (x2 ; y2 ; v2 ) (x2 ; y2 ; v1 ) + (x2 ; y2 ; v1 ) ac
1=
(x1 ; y2 ; v1 ) + (x1 ; y2 ; v1 ) 1) = sgn(v1
(x1 ; y1 ; v1 )] :
v2 ): Hence, if R0 > 1; then x1 ; y1 ; v1 > 0, and if R1
follows from the above discussion that to
v2 ) : Using the equilibrium conditions for E1
= 1, then R1
We get sgn(R1
x1 ) ;
dU1 dt
1, then v1
v2 =
r.
It
0 for all x; y; v; z > 0. The solutions of system (5)-(8) converge
, the largest invariant subset of (x; y; v; z) :
dU1 dt
= 0 [33]. We have
dU1 dt
= 0 i¤ x = x1 ; v = v1 ; z = 0
and H = 0 i.e. y1 (x(t
1 ); y(t
1 ); v(t y (x1 ; y1 ; v1 )v1
1 ))v(t
1)
From Eq. (30), if v = v1 then y = y1 and hence
=
v1 y(t vy1
dU1 dt
2)
= 1 for almost all
i
2 [0; ]; i = 1; 2:
= 0 i¤ x = x1 ; y = y1 ; v = v1 and z = 0. So
(30)
contains a
unique point, that is E1 . Thus, the global asymptotic stability of the chronic-infection equilibrium without antibody immune response E1 follows from LaSalle’s invariance principle. Theorem 3. Let Assumptions A1-A5 be hold true and R1 > 1, then the chronic-infection equilibrium with antibody immune response E2 is GAS.
13
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Proof. We construct a Lyapunov functional as follows: U2 = x
Z
x2
x
(x2 ; y2 ; v2 ) d +e ( ; y2 ; v2 )
x2
+
a e k
1 1+ 2 2
v v2
v2 H Zt
+ (x2 ; y2 ; v2 )v2 t
ab e rk
+
1 1
y2 H
1 1+ 2 2
y y2 z z2
z2 H
(x( ); y( ); v( ))v( ) (x2 ; y2 ; v2 )v2
H
d + ae
1 1
y2 t
1
Zt
H
y( ) y2
1
z2 (rvz z
d :
(31)
2
Function U2 satis…es: dU2 = dt +e +
(x2 ; y2 ; v2 ) (x; y2 ; v2 )
1 1 1
a e k
y2 y
1
1 1+ 2 2
e 1
+ (x; y; v)v
1 1
dU2 =d 1 dt
+ ae +
1 1
ab e k
y2
(x(t ke
y(t
1 1
1 ); y(t
y(t
1 ); y(t
2)
1 ); v(t
2)
cv
1 ); v(t
1 ); y(t
1 ))v(t
bvz +
1 ))v(t
1 ); v(t (x; y; v)v
y(t
+ y2 ln
2)
ab e rk
1)
ay
1 1+ 2 2
1)
1 ))v(t
1)
:
y
z)
(32)
y2 , we get (x2
(x2 ; y2 ; v2 ) (x; y2 ; v2 ) ac e k
1 1+ 2 2
(x; y; v)v)
2 2
(x(t
(x2 ; y2 ; v2 ) (x; y2 ; v2 )
+ (x; y; v)v
1 1
(x(t
y
Applying s = dx2 + ae
dx
v2 v
+ (x2 ; y2 ; v2 )v2 ln + ae
(s
1 1+ 2 2
v2 z
+ (x2 ; y2 ; v2 )v2 ln
x) + ae
1 1
y2
(x2 ; y2 ; v2 )v2 v
ab e rk (x(t
ae
1 1
y(t
ae
y2 (x(t
2)
(x2 ; y2 ; v2 ) (x; y2 ; v2 )
y2
1 ); y(t
1 ); v(t y (x2 ; y2 ; v2 )v2
ac v2 + e v k
ab 1 1 + e k 1 ); y(t 1 ); v(t (x; y; v)v
1 1+ 2 2
1 1
z
2 2
1 1+ 2 2
z2 v +
1 ))v(t
1 ))v(t
1)
v2
ab e rk 1)
1 1+ 2 2
+ ae
1 1
z2
y2 ln
y(t
2)
y
(33)
By using the equilibrium conditions of E2 (x2 ; y2 ; v2 )v2 = ae
1 1
y2 ;
cv2 = ke
2 2
y2
bv2 z2 ;
= rv2 ;
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and the following equalities cv = cv2 ln
(x(t
1 ); y(t
1 ); v(t (x; y; v)v
1 ))v(t
1)
v = ke v2 y2 (x(t
= ln
y(t
2)
vy2 v2 y
= ln
y
+ ln
y2
bv2 z2
v ; v2
1 ); y(t
1 ); v(t y (x2 ; y2 ; v2 )v2
(x2 ; y2 ; v2 ) (x; y2 ; v2 )
+ ln ln
2 2
1 ))v(t
(x; y2 ; v2 ) (x; y; v)
+ ln v2 y(t vy2
2)
1)
v2 y vy2
+ ln
;
;
we obtain dU2 =d 1 dt ae + +
(x2 ; y2 ; v2 ) (x; y2 ; v2 )
1 1
(x2
(x2 ; y2 ; v2 ) (x; y2 ; v2 )
y2
y2 (x(t v2 y(t vy2
x) + ae 1
1 ); v(t y (x2 ; y2 ; v2 )v2
2)
1
ln
v2 y(t vy2
(x; y; v)v (x; y2 ; v2 )v2
y2
v v2
1+
(x; y2 ; v2 ) (x; y; v)
(x2 ; y2 ; v2 ) (x; y2 ; v2 )
ln
1 ); y(t
1 1
1 ))v(t
2)
1)
1
y2 (x(t
ln
(x; y2 ; v2 ) (x; y; v)
+
1
ln
1 ); y(t
1 ); v(t y (x2 ; y2 ; v2 )v2
(x; y2 ; v2 ) (x; y; v)
1 ))v(t
1)
:
(34)
We can rewrite (34) as dU2 = dx2 1 dt ae +H
1 1
(x2 ; y2 ; v2 ) (x; y2 ; v2 )
(x2 ; y2 ; v2 ) (x; y2 ; v2 )
y2 H
v2 y(t vy2
1
2)
+H
x x2
+ ae
1 1
y2 (x(t
+H
(x; y2 ; v2 ) (x; y; v)
(x; y2 ; v2 ) (x; y; v)
(x; y; v) (x; y2 ; v2 )
y2 1 1 ); y(t
1 ); v(t y (x2 ; y2 ; v2 )v2
1 ))v(t
v v2
1)
:
(35)
We note that from Assumptions A2 and A5, the …rst and second terms of Eq. (35) are less than or equal zero. Noting that x; y; v; z > 0, we have that largest invariant subset of (x; y; v; z) : y2 (x(t
1 ); y(t
1 ); v(t y (x2 ; y2 ; v2 )v2
dU2 dt
1 ))v(t
dU2 dt
0. The solutions of model (5)-(8) converge to
= 0 [33]. We have 1)
=
v2 y(t vy2
If v = v2 , then from Eq. (36) we get y = y2 . The set v = v2 =
r.
2)
dU2 dt
= 0 i¤ x = x2 ; v = v2 and H = 0 i.e.,
= 1 for almost all
i
2 [0; ]; i = 1; 2:
is invariant and for any element belongs to
From Eq. (7) we get z = z2 . Therefore,
dU2 dt
, the
(36) satis…es
= 0 i¤ x = x2 ; y = y2 ; v = v2 and z = z2 . The
global asymptotic stability of the chronic-infection equilibrium with antibody immune response E2 follows from LaSalle’s invariance principle. 15
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3
Conclusion
In this paper, we have proposed a delayed viral infection model with general incidence rate function and antibody immune response. The model has been incorporated with two kinds of discrete time delays representing the time needed for infecting an uninfected target cell and viral production. We have derived a set of conditions on the general functional response and have determined two threshold parameters R0 and R1 to prove the existence and the global stability of the model’s equilibria. The global asymptotic stability of the three equilibria, infection-free, chronic-infection without antibody immune response and chronic-infection with antibody immune response has been proven by using direct Lyapunov method and LaSalle’s invariance principle.
4
Con‡ict of Interests
The authors declare that there is no con‡ict of interests regarding the publication of this article.
5
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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[4] M.A. Nowak, and R.M. May, “Virus dynamics: Mathematical Principles of Immunology and Virology,” Oxford Uni., Oxford, 2000. [5] D.S. Callaway, and A.S. Perelson, HIV-1 infection and low steady state viral loads, Bull. Math. Biol., 64 (2002), 29-64. [6] P.W. Nelson, J. Murray, and A. Perelson, A model of HIV-1 pathogenesis that includes an intracellular delay, Math. Biosci., 163 (2000), 201-215. [7] P.W. Nelson, and A.S. Perelson, Mathematical analysis of delay di¤ erential equation models of HIV-1 infection, Math. Biosci., 179 (2002), 73-94. [8] R.V. Culshaw, and S. Ruan, A delay-di¤ erential equation model of HIV infection of CD4 + T-cells, Math. Biosci., 165 (2000), 27-39. [9] A.M. Elaiw, I. A. Hassanien, and S. A. Azoz, Global stability of HIV infection models with intracellular delays, J. Korean Math. Soc., 49 (2012), 779-794. [10] A.M. Elaiw, Global dynamics of an HIV infection model with two classes of target cells and distributed delays, Discrete Dyn. Nat. Soc., 2012, Article ID 253703. [11] A.M. Elaiw and A. S. Alsheri, Global Dynamics of HIV Infection of CD4+ T Cells and Macrophages, Discrete Dyn. Nat. Soc., 2013, Article ID 264759. [12] N.M. Dixit, and A.S. Perelson, Complex patterns of viral load decay under antiretroviral therapy: In‡uence of pharmacokinetics and intracellular delay, J. Theoret. Biol., 226 (2004), 95-109. [13] A. M. Elaiw, Global properties of a class of virus infection models with multitarget cells, Nonlinear Dynam., 69 (2012) 423-435. [14] A.M. Elaiw, Global properties of a class of HIV models, Nonlinear Anal. Real World Appl., 11 (2010), 2253–2263.
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[15] M.A. Nowak, C.R.M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79. [16] S. Eikenberry, S. Hews, J. D. Nagy and Y. Kuang, The dynamics of a delay model of HBV infection with logistic hepatocyte growth, Math. Biosc. Eng., 6, (2009), 283-299. [17] S. A. Gourley,Y. Kuang and J. D. Nagy, Dynamics of a delay di¤ erential equation model of hepatitis B virus infection, J. Biological Dynamics, 2, (2008), 140-153 [18] J. Li, K. Wang, Y. Yang, Dynamical behaviors of an HBV infection model with logistic hepatocyte growth, Mathematical and Computer Modelling, 54 (2011), 704-711. [19] R. Qesmi, J. Wu, J. Wu and J.M. He¤ernan, In‡uence of backward bifurcation in a model of hepatitis B and C viruses, Math. Biosci. 224 (2010) 118–125. [20] R. Qesmi, S. ElSaadany, J.M. He¤ernan and J. Wu, A hepatitis B and C virus model with age since infection that exhibit backward bifurcation, SIAM J. Appl. Math., 71 (4) (2011) 1509–1530. [21] A. U. Neumann, N. P. Lam, H. Dahari, D. R. Gretch, T. E. Wiley, T. J, Layden, A. S. Perelson, Hepatitis C viral dynamics in vivo and the antiviral e¢ cacy of interferon-alpha therapy, Science, 282 (1998), 103-107. [22] M. Y. Li, H. Shu, Global dynamics of a mathematical model for HTLV-I infection of CD4+ T cells with delayed CTL response, Nonlinear Anal. Real World Appl., 13 (2012), 1080-1092. [23] J.A. Deans, S. Cohen, Immunology of malaria, Ann. Rev. Microbiol. 37 (1983), 25-49. [24] A. Murase, T. Sasaki, and T. Kajiwara, Stability analysis of pathogen-immune interaction dynamics, J. Math. Biol., 51 (2005), 247-267. [25] W. Dominik, R. M. May, M. A. Nowak, T he role of antigen-independent persistence of memory cytotoxic T lymphocytes, Int. Immunol. 12 (4) (2000), 467–477.
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[26] A. M. Elaiw, A. Alhejelan, and M. A. Alghamdi, A delayed viral infection model with antibody immune response, Life Science Journal 10(4) (2013) 695-700. [27] A. M. Elaiw, A. Alhejelan, and M. A. Alghamdi, Global dynamics of virus infection model with antibody immune response and distributed delays, Discrete Dynamics in Nature and Society, 2013, Article ID 781407, 2013. [28] T. Wang, Z. Hu, F. Liao, Wanbiao, Global stability analysis for delayed virus infection model with general incidence rate and humoral immunity, Math. Comput. Simulation, 89 (2013), 13-22. [29] S. Wang, D. Zou, Global stability of in host viral models with humoral immunity and intracellular delays, J. Appl. Math. Mod., 36 (2012), 1313-1322. [30] A. Korobeinikov, Global properties of infectious disease models with nonlinear incdence, Bull. Math. Biol., 69 (2007), 1871-1886. [31] G. Huang, Y. Takeuchi, and W. Ma, Lyapunov functionals for delay di¤ erential equations model of viral infection, SIAM J. Appl. Math., 70 (2010), 2693-2708. [32] K. Hattaf, N. Yous…, A. Tridane, Stability analysis of a virus dynamics model with general incidence rate and two delays, Applied Mathematics and Computation, 221 (2013) 514-521. [33] J.K. Hale, and S. Verduyn Lunel, Introduction to functional di¤ erential equations, Springer-Verlag, New York, 1993.
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STABILITY OF GENERALIZED CUBIC SET-VALUED FUNCTIONAL EQUATIONS DONGSEUNG KANG Abstract. We will show the general solution of the functional equation f (ax + by) + f (bx − ay) + (a + b)2 (a − b)f (y) 2
= a bf (x + y) + ab2 f (x − y) + (a + b)(a − b)2 f (x) and investigate the Hyers-Ulam stability of cubic set-valued functional equation when b = 1 .
1. Introduction The theory of set-valued functions in Banach spaces is connected to the control theory and the mathematical economics. Aumann [4] and Debreu [8] wrote papers that were motivated from the topic. We refer the reader to the papers by [1], [18], [10], [3], [17], [7] and [9]. The stability problem of functional equations originated from a question of Ulam [25] concerning the stability of group homomorphisms. Hyers [11] gave a first affirmative partial answer to the question of Ulam. Afterwards, the result of Hyers was generalized by Aoki [2] for additive mapping and by Rassias [23] for linear mappings by considering a unbounded Cauchy difference. Later, the result of Rassias has provided a lot of influence in the development of what we call Hyers-Ulam stability or Hyers-Ulam-Rassias stability of functional equations. For further information about the topic, we also refer the reader to [13], [12], [5] and [6]. Jun and Kim [15] introduced the following cubic functional equation: f (2x + y) + f (2x − y) = 2f (x + y) + 2f (x − y) + 12f (x) and established a general solution. Najati [20] investigated the following generalized cubic functional equation: (1.1)
f (ax + y) + f (ax − y) = af (x + y) + af (x − y) + 2(a3 − a)f (x) .
In this paper, we deal with the following functional equation: (1.2)
f (ax + by) + f (bx − ay) + (a + b)2 (a − b)f (y) = a2 bf (x + y) + ab2 f (x − y) + (a + b)(a − b)2 f (x)
2000 Mathematics Subject Classification. 39B55; 47B47; 39B72. Keywords : Hyers-Ulam-Rassias Stability, Cubic Mapping, Set-Valued Functional Equation, Closed and Convex Subset, Cone, Fixed Point. 1
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for all x , y ∈ X and integers a, b (a > b ≥ 1) . We will show the general solution of the functional equation (1.2) and investigate the Hyers-Ulam stability of cubic set-valued functional equation when b = 1 . 2. A generalized cubic functional equation In this section let X and Y be vector spaces and we investigate the general solution of the functional equation (1.2). Theorem 2.1. A function f : X → Y satisfies the functional equation (1.1) if and only if it satisfies the functional equation (2.1)
f (ax + y) + f (x − ay) − a2 f (x + y) − af (x − y) = (a − 1)(a2 − 1)f (x) − (a + 1)(a2 − 1)f (y)
Proof. See [16, Theorem 2.1].
Theorem 2.2. A function f : X → Y satisfies the functional equation (1.1) if and only if it satisfies the functional equation (1.2). Proof. Suppose that f satisfies the equation (1.1). Since f satisfies the equation (1.1), it is easy to show f (0) = 0 , f (x) = −f (−x) and f (ax) = a3 f (x) for all x ∈ X and integer a (a 6= 0 , ±1). Replacing x and y in the equation (1.1), we obtain (2.2)
f (x + ay) − f (x − ay) = a[f (x + y) − f (x − y)] + 2a(a2 − 1)f (y)
for all x , y ∈ X and an integer a (a 6= 0 , ±1). By letting x = ax in the equation (2.2), we have (2.3)
f (ax + y) − f (ax − y) = a2 [f (x + y) − f (x − y)] + 2(1 − a2 )f (y)
for all x , y ∈ X and an integer a (a 6= 0 , ±1). By replacing x and y in the equation (2.3), we get (2.4)
f (x + ay) + f (x − ay) = a2 [f (x + y) + f (x − y)] + 2(1 − a2 )f (x)
for all x , y ∈ X and an integer a (a 6= 0 , ±1). Replacing a by b in the equation (1.1), we have (2.5)
f (bx + y) + f (bx − y) = bf (x + y) + bf (x − y) + 2(b3 − b)f (x)
Letting y = by in the equation (1.1), (2.6) f (ax + by) + f (ax − by) = af (x + by) + af (x − by) + 2(a3 − a)f (x) Letting y = ay in equation (2.5), (2.7) f (bx + ay) + f (bx − ay) = bf (x + ay) + bf (x − ay) + 2(b3 − b)f (x) Replacing x and y in the equation (2.7), (2.8) f (ax + by) − f (ax − by) = bf (ax + y) − bf (ax − by) + 2(b3 − b)f (y) Replacing x and y in equation (2.6), (2.9) f (bx + ay) − f (bx − ay) = af (bx + y) − af (bx − y) + 2(a3 − a)f (y)
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Adding two equations (2.6) and (2.8), we obtain (2.10)
2f (ax + by) = af (x + by) + af (x − by) + 2(a3 − a)f (x) +bf (ax + y) − bf (ax − y) + 2(b3 − b)f (y)
Subtracting (2.9) from (2.7), we have (2.11)
2f (bx − ay) = bf (x + ay) + bf (x − ay) + 2(b3 − b)f (x) −af (bx + y) + af (bx − y) − 2(a3 − a)f (y)
Now, adding two equations (2.10) and (2.11), we get (2.12)
2[f (ax + by) + f (bx − ay)] = a[f (x + by) + f (x − by)] +b[f (ax + y) − f (ax − y)] + 2(a3 − a)f (x) + 2(b3 − b)f (y) +b[f (x + ay) + f (x − ay)] − a[f (bx + y) − f (bx − y)] +2(b3 − b)f (x) − 2(a3 − a)f (y)
The desired result is obtained from the equation (2.12) by using the equations (2.3) and (2.4). Conversely, suppose that f satisfies the equation (1.2). Letting b = 1 in the equation (1.2), we have the equation (2.1). The remains follow from Theorem 2.1. If f satisfies the equation (1.2), we call f a generalized cubic mapping. 3. Stability of the generalized cubic set-valued functional equation In this section, we first introduce some definitions and notations which are needed to prove the main theorems. Let Y be a Banach space. The family of all closed subsets, containing 0 , of Y will be denoted by Cz (Y ) . Let A , B be nonempty subsets of a real vector space X and λ a real number. We define A + B = {a + b ∈ X | a ∈ A , b ∈ B} λA = {λa ∈ X | a ∈ A} . Lemma 3.1 ( [21]). Let λ and µ be real numbers. If A and B are nonempty subset of a real vector space, then λ(A + B) = λA + λB (λ + µ)A ⊆ λA + µA . Moreover, if A is a convex set and λµ ≥ 0 , then we have (λ + µ)A = λA + µA .
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A subset A ⊆ X is said to be a cone if A + A ⊆ A and λA ⊆ A for all λ > 0 . If the zero vector in X belongs to A , then we say that A is a cone with zero. Let Cb (Y ) be the set of all closed bounded subsets of Y , Cc (Y ) the set of all closed convex subsets of Y and Ccb (Y ) the set of all closed bounded convex subsets of Y . For elements A , B of Cc (Y ) and positive real values λ , µ , we denote A⊕B =A+B. For a subset A of Y , the distance function d(· , A) and the support function s(· , A) are defined by d(x , A) := inf {||x − y|| | y ∈ A} for all x ∈ Y s(x∗ , A) := sup {hx∗ , xi|| | x ∈ A} for all x∗ ∈ Y ∗ . For A , A0 ∈ Cb (Y ) , the Hausdorff distance h(A , A0 ) between A and A0 is defined by h(A , A0 ) := inf {α ≥ 0 | A ⊆ A0 + αBY , A0 ⊆ A + αBY } , where BY is the closed unit ball in Y . Castaing and Valadier [7] proved that (Ccb (Y ) , ⊕ , h) is a complete metric semigroup. Radstr¨ ˙ om [22] showed that (Ccb (Y ) , ⊕ , h) is isometrically embedded in a Banach space. The following remark is directly obtained from the notion of the Hausdorff distance. Remark 3.2. Let A , A0 , B , B 0 , C ∈ Ccb (Y ) and α > 0 . Then the following properties hold: (1) h(A ⊕ A0 , B ⊕ B 0 ) ≤ h(A , B) + h(A0 , B 0 ) (2) h(αA , αB) = αh(A , B) (3) h(A , B) = h(A ⊕ C , B ⊕ C) . First, let X be a real vector space , A ⊂ X a cone with zero and Y a Banach space. Theorem 3.3. If f : A + (−1)A → Cz (Y ) is a set-valued mapping with f (0) = {0} satisfying (3.1)
f (ax + y) + f (x − ay) + (a2 − 1)(a + 1)f (y) ⊆ a2 f (x + y) + af (x − y) + (a2 − 1)(a − 1)f (x)
and sup{diam(f (x)) | x ∈ A} < ∞ for all x , y ∈ A and an integer a (a ≥ 2) , then there exists a unique generalized cubic mapping C : A + (−1)A → Y such that C(x) ∈ f (x) for all x ∈ A. Proof. Letting y = 0 in (3.1), we have (3.2)
f (ax) ⊆ a3 f (x)
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for all x ∈ A and an integer a (a ≥ 2) . Replacing x by an x , n ∈ N in (3.2) , we get f (an+1 x) ⊆ a3 f (an x) and 1 1 f (an+1 x) ⊆ 3n f (an x) 3(n+1) a a 1 for all x ∈ A and an integer a (a ≥ 2) . Let fn (x) = a3n f (an x) for each x ∈ A , n ∈ N . Then {fn (x)}n≥0 is a decreasing sequence of closed subsets of the Banach space Y . Also, we obtain 1 diam(fn (x)) = 3n diam(f (an x)) . a Since sup{diam(f (x)) | x ∈ A} < ∞ , we have lim diam(fn (x)) = 0 .
n→∞
Using the Cantor theorem for the sequence {fn (x)}n≥0 , we get that ∩n≥0 fn (x) is a singleton set and we denote this intersection by C(x) for all x ∈ A . Hence we obtain a map C : A + (−1)A → Y and C(x) ∈ f0 (x) = f (x) for all x ∈ A . We claim that C is generalized cubic. We note that fn (ax + y) + fn (x − ay) + (a2 − 1)(a + 1)fn (y) f (an (ax + y)) f (an (x − ay)) (a2 − 1)(a + 1)f (an y) = + + a3n a3n a3n n 2 2 n a f (a (x + y)) af (a (x − y)) (a − 1)(a − 1)f (an x) + + ⊆ a3n a3n a3n 2 2 = a fn (x + y) + afn (x − y) + (a − 1)(a − 1)fn (x) for all x ∈ A and an integer a (a ≥ 2) . By the definition of C , we obtain C(ax + y) + C(x − ay) + (a2 − 1)(a + 1)C(y) 2 = ∩∞ f (ax + y) + f (x − ay) + (a − 1)(a + 1)f (y) n n n n=0 2 2 ⊆ ∩∞ a f (x + y) + af (x − y) + (a − 1)(a − 1)f (x) n n n n=0 for all x ∈ A and an integer a (a ≥ 2) . Hence we have ||C(ax + y) + C(x − ay) + (a2 − 1)(a + 1)C(y) −a2 C(x + y) − aC(x − y) − (a2 − 1)(a − 1)C(x)|| ≤ a2 diam(fn (x + y)) + a diam(fn (x − y)) + (a2 − 1)(a − 1)diam(fn (x)) , which tends to zero as n → ∞ . Thus C satisfies the equality (1.2). Hence C is a generalized cubic, as claimed. Next, let us prove the uniqueness of C . Assume f has two generalized cubic functional equations C1 and C2 from A + (−1)A into Y . Then we have (an)3 Ci (x) = Ci (anx) ∈ f (anx)
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for all x ∈ X , n ∈ N and i ∈ {1 , 2} . Then we have (an)3 ||C1 (x) − C2 (x)|| = ||(an)3 C1 (x) − (an)3 C2 (x)|| = ||(C1 (anx) − C2 (anx)|| ≤ diam(f (anx)) for all x ∈ X , n ∈ N . Since sup{diam(f (x)) | x ∈ A} < ∞ , C1 (x) = C2 (x) , for all x ∈ X . Definition 3.4. Let f : X → Ccb (Y ) . The generalized cubic set-valued functional equation is defined by (3.3)
f (ax + y) ⊕ f (x − ay) ⊕ (a2 − 1)(a + 1)f (y)
= a2 f (x + y) ⊕ af (x − y) ⊕ (a2 − 1)(a − 1)f (x) for all x ∈ A and an integer a (a ≥ 2) . Every solution of the generalized cubic set-valued functional equation is called a generalized cubic set-valued mapping. Theorem 3.5. Let φ : X × X → [0, ∞) be a function such that ∞ X 1 e y) := (3.4) φ(x, φ(aj x, aj y) < ∞ a3j j=0
for all x, y ∈ X and an integer a (a ≥ 2) . Suppose that f : X → (Ccb (Y ), h) is a mapping with f (0) = {0} satisfying (3.5) h f (ax + y) ⊕ f (x − ay) ⊕ (a2 − 1)(a + 1)f (y), a2 f (x + y) ⊕ af (x − y) ⊕ (a2 − 1)(a − 1)f (x) ≤ φ(x, y) for all x, y ∈ X and an integer a (a ≥ 2) . Then there exists a unique generalized cubic set-valued mapping C : X → (Ccb (Y ), h) such that 1 e 0) (3.6) h(f (x), C(x)) ≤ 3 φ(x, a for all x, y ∈ X and an integer a (a ≥ 2) . Proof. Let y = 0 in the inequality (3.5). Since f (x) is convex, we have h f (ax) ⊕ f (x), a2 f (x) ⊕ af (x) ⊕ (a2 − 1)(a − 1)f (x) ≤ φ(x, 0) , that is, 1 1 h f (x), 3 f (ax) ≤ 3 φ(x, 0) a a for all x ∈ X . Replacing x by ak x , k ∈ N , we get 1 1 h f (ak x), 3 f (ak+1 x) ≤ 3 φ(ak x, 0) a a and 1 1 1 h 3k f (ak x), 3(k+1) f (ak+1 x) ≤ 3(k+1) φ(ak x, 0) a a a
(3.7)
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for all x ∈ X . Using the induction on k , we obtain (3.8)
n−1 1 1 X 1 n h f (x), 3n f (a x) ≤ 3 φ(ak x, 0) a a a3k
k=0
for all x ∈ X and n ∈ N . Dividing the inequality (3.8) by a3m and replacing x by am x , we have (3.9)
n−1 1 1 1 X 1 1 h 3m f (am x), 3(n+m) f (an+m x) ≤ 3 m φ(am+k x, 0) a a a a3k a k=0
for all x ∈ X and n , m ∈ N . Since the right-hand side of the inequality (3.9) 1 tends to zero as m → ∞ , the sequence { a3n f (an x)} is a Cauchy sequence in (Ccb (Y ), h) . By the completeness of Ccb (Y ) , we can define C(x) := lim
n→∞
1 f (an x) a3n
for all x ∈ X and an integer a (a ≥ 2) . We note that f (an (ax + y)) f (an (x − ay)) (a2 − 1)(a + 1)f (an y) h ⊕ ⊕ , a3n a3n a3n a2 f (an (x + y)) af (an (x − y)) (a2 − 1)(a − 1)f (an x) ⊕ ⊕ a3n a3n a3n 1 ≤ 3n φ(an x, an y) a for all x, y ∈ X and an integer a (a ≥ 2) . By the definition of C , we have h C(ax + y) ⊕ C(x − ay) ⊕ (a2 − 1)(a + 1)C(y), a2 C(x + y) ⊕ aC(x − y) ⊕ (a2 − 1)(a − 1)C(x) 1 φ(an x, an y) = 0 . a3n Hence C is a generalized cubic set-valued mapping. Now, by taking n → ∞ in the inequality (3.8), we have the inequality (3.6). It remains to show the uniqueness of C . Assume C 0 : X → (Ccb (Y ), h) is another generalized cubic set-valued mapping satisfying the inequality (3.6). Then 1 n 0 n h C(a x), C (a x) h C(x), C 0 (x) = a3n 1 1 n n n 0 n ≤ h C(a x), f (a x) + h f (a x), C (a x) a3n a3n 2 e n ≤ φ(a x, 0) a3(n+1) 2 e n x, 0) → 0 as n → ∞ , we may conclude that for all x ∈ X . Since a3(n+1) φ(a the generalized cubic set-valued mapping C is unique. ≤ lim
n→∞
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Corollary 3.6. Let 0 < p < 3 , θ ≥ 0 be real numbers and let X be a real normed space. Suppose that f : X → (Ccb (Y ), h) is a mapping with f (0) = {0} satisfying h f (ax + y) ⊕ f (x − ay) ⊕ (a2 − 1)(a + 1)f (y), a2 f (x + y) ⊕ af (x − y) ⊕ (a2 − 1)(a − 1)f (x) ≤ θ(||x||p + ||y||p ) for all x, y ∈ X and an integer a (a ≥ 2) . Then there exists a unique generalized cubic set-valued mapping C : X → (Ccb (Y ), h) satisfying h(f (x), C(c)) ≤
a3
θ ||x||p − ap
for all x, y ∈ X and an integer a (a ≥ 2) . Proof. It follows from Theorem 3.5 by letting φ(x, y) = θ(||x||p + ||y||p ) for all x, y ∈ X . 4. Stability of set-valued functional equation by the fixed point method Now, we will investigate the stability of the given functional equation (3.3) using the alternative fixed point method. Before proceeding the proof, we will state the theorem, the alternative of fixed point; see [19] and [24]. Definition 4.1. 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 4.2. [ The alternative of fixed point [19], [24] ] Suppose that we are given a complete generalized metric space (Ω, d) and a strictly contractive mapping T : Ω → Ω with Lipschitz constant L . Then for each given x ∈ Ω , either d(T n x, T n+1 x) = ∞ for all n ≥ 0 , or there exists a natural number n0 such that (1) d(T n x, T n+1 x) < ∞ for all n ≥ n0 ; (2) The sequence (T n x) is convergent to a fixed point y ∗ of T ; (3) y ∗ is the unique fixed point of T in the set 4 = {y ∈ Ω|d(T n0 x, y) < ∞} ; (4) d(y, y ∗ ) ≤
1 1−L
d(y, T y) for all y ∈ 4 .
Theorem 4.3. Suppose that f : X → (Ccb (Y ), h) is a mapping with f (0) = {0} satisfying (4.1) h f (ax + y) ⊕ f (x − ay) ⊕ (a2 − 1)(a + 1)f (y),
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9
a2 f (x + y) ⊕ af (x − y) ⊕ (a2 − 1)(a − 1)f (x) ≤ φ(x, y) for all x, y ∈ X and an integer a (a ≥ 2) and there exists a constant L with 0 < L < 1 for which the function φ : X 2 → R+ satisfies φ(ax, 0) ≤ a3 Lφ(x, 0)
(4.2)
for all x ∈ X . Then there exists a unique generalized cubic set-valued mapn x) ping C : X → (Ccb (Y ), h) given by C(x) = limn→∞ f (a such that a3n (4.3)
h(f (x), C(x)) ≤
1 e 0) φ(x, − L)
a3 (1
for all x, y ∈ X and an integer a (a ≥ 2) . Proof. Consider the set Ω = {g | g : X → Ccb (Y ) , g(0) = {0}} and introduce the generalized metric on Ω defined by d(g1 , g2 ) = inf {µ ∈ (0, ∞) | h(g1 (x) , g2 (x)) ≤ µφ(x, 0) , for all x ∈ X} . We note that inf ∅ := ∞ . It is easy to show that (Ω, d) is complete; see [14]. Now we define a function T : Ω → Ω by 1 (4.4) T (g)(x) = 3 g(ax) a for all x ∈ X . Note that for all g1 , g2 ∈ Ω , let µ ∈ (0, ∞) be an arbitrary constant with d(g1 , g2 ) = µ . Then 1 µ 1 (4.5) h( 3 g1 (ax) , 3 g2 (ax)) ≤ 3 φ(ax, 0) a a a for all x ∈ X . By using (4.2), we have 1 1 (4.6) h( 3 g1 (ax) , 3 g2 (ax)) ≤ µ Lφ(x, 0) a a for all x ∈ X . Hence we obtain d(T g1 , T g2 ) ≤ Ld(g1 , g2 ) for all g1 , g2 ∈ Ω , that is, T is a strictly self-mapping of Ω with the Lipschitz constant L . Letting y = 0 in the inequality (4.1), we get 1 1 h( 3 f (ax) , f (x)) ≤ 3 φ(x, 0) a a for all x ∈ X . This means that 1 d(T f, f ) ≤ 3 . a By Theroem 4.2, there exists a fixed point C : X → (Ccb (Y ), h) of T in {g ∈ Ω | d(g1 , g2 ) < ∞} such that {T k f } → 0 ad k → ∞ . Hence we have (4.7)
f (an x) , n→∞ a3n
C(x) = lim
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DONGSEUNG KANG
for all x ∈ X . Also, we have 1 1 1 d(T f, f ) ≤ 3 . 1−L a 1−L This implies that the inequality (4.3) holds for all x ∈ X . By the inequalities (4.1) and (4.2), we have h C(ax + y) ⊕ C(x − ay) ⊕ (a2 − 1)(a + 1)C(y), a2 C(x + y) ⊕ aC(x − y) ⊕ (a2 − 1)(a − 1)C(x) d(f, C) ≤
≤ lim Ln φ(an x, an y) = 0 n→∞
for all x, y ∈ X and an integer a (a ≥ 2) . Thus C is a unique generalized cubic set-valued mapping. Corollary 4.4. Let 0 < p < 3 and θ ≥ 0 be real numbers and let X be a real normed space. Suppose that f : X → (Ccb (Y ), h) is a mapping with f (0) = {0} satisfying (4.8) h f (ax + y) ⊕ f (x − ay) ⊕ (a2 − 1)(a + 1)f (y), a2 f (x + y) ⊕ af (x − y) ⊕ (a2 − 1)(a − 1)f (x) ≤ θ(||x||p + ||y||p ) for all x, y ∈ X and an integer a (a ≥ 2) . Then there exists a unique generalized cubic set-valued mapping C : X → (Ccb (Y ), h) such that (4.9)
h(f (x), C(x)) ≤
a3
θ ||x||p − ap
for all x ∈ X and an integer a (a ≥ 2) . Proof. It follows from Theorem 4.3 by letting φ(x, y) = θ(||x||p + ||y||p ) for all x, y ∈ X . Then we can choose L = ap−3 and hence we have the desired result. References [1] K.J. Arrow and G. Debreu, Existence of anequilibrium for a competitive economy, Econometrica 22 (1954), 265–290. [2] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. [3] J.P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhauser, Boston (1990). [4] R.J. Aumann, Integrals of set-valued functions, J. Math. Anal.Appl. 12 (1965) 1–12. [5] N. Brillou¨et-Belluot, J. Brzd¸ek and K. Ciepli´ nski, Fixed Point Theory and the Ulam Stability, Abstract and Applied Analysis 2014, Article ID 829419, 16 pages (2014). [6] J. Brzd¸ek, L. Cˇ adariu and K. Ciepli´ nski, On Some Recent Developments in Ulam’s Type Stability, Abstract and Applied Analysis 2012, Article ID 716936, 41 pages (2012). [7] C. Castaing, M. Valadier, Convex Analysis and Measurable Multifunctions, in: Lect. Notes in Math., 580, Springer, Berlin (1977).
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[8] G. Debreu, Integration of correspondences, in: Proceedings of Fifth Berkeley Symposium on Mathematical Statistics and Probability, vol. II, 1966, pp. 351–372. Part I. [9] C. Hess, Set-valued Integration and Set-valued Probability Theory: an Overview, in: Handbook of Measure Theory, vols. I, II, North-Holland, Amsterdam (2002). [10] W. Hindenbrand, Core and Equilibria of a Large Economy, Princeton Univ. Press, Princeton (1974). [11] D. H. Hyers, On the stability of the linear equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222–224. [12] D. H. Hyers, G. Isac, and T. M. Rassias, Stability of Functional Equations in Several Variables, Birkhauser, Boston, Mass, USA (1998). [13] S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, vol. 48 of Springer Optimization and Its Applications, Springer, New York, NY, USA, (2011). [14] S.-M. Jung and Z.-H. Lee, A fixed point approach to the stability of quadratic functional equation with involution, Fixed Point Theory Appl. 2008 Article ID 732086, 11 pages (2008). [15] K.-W. Jun and H.-M. Kim, The generalized Hyers-Ulam-Rassias stability of a cubic functional equation, J. Math. Anal. Appl. 274 (2002), 867–878. [16] D. Kang, On the Stability of Lie ∗-Derivations of Cubic Functional Equations, Abstract and Applied Analysis 2014, Article ID 808042, 6 pages (2014). [17] E. Klein and A. Thompson, Theory of Correspondence, Wiley, New York (1984). [18] L.W. McKenzie, On the existence of general equilibrium for a competitive market, Econometrica 27 (1959) 54–71. [19] B. Margolis and J.B. Diaz, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 126, 74(1968), 305–309. [20] A. Najati, The Generalized Hyers-Ulam-Rassias Stability of a Cubic Functional Equation, Turk. J. Math. 31 (2007) 395–408. [21] K. Nikodem, K-Convex and K-Concave Set-Valued Functions, Zeszyty Naukowe Nr., 559, Lodz (1989). [22] H. Radstr¨ ˙ om, An embedding theorem for spaces of convex sets, Proc. Amer. Math. Soc. 3 (1952) 165–169. [23] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978) 297–300. [24] I.A. Rus, Principles and Appications of Fixed Point Theory, Ed. Dacia, ClujNapoca, 1979 (in Romanian). [25] S. M. Ulam, Problems in Morden Mathematics, Wiley, New York (1960). Department of Mathematical Education, Dankook University, 152, Jukjeon, Suji, Yongin, Gyeonggi, South Korea 448-701 E-mail address: [email protected] (D. Kang)
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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 20, NO.2, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
A new regularity (p-regularity) of stratified L-generalized convergence spaces Lingqiang Lia,b , Qingguo Lia∗ a College
of Mathematics and Econometrics, Hunan University Changsha, 410082, P.R.China
b Department
of Mathematics, Liaocheng University,
Liaocheng, 252059, P.R.China
Abstract: In the classical theory of convergence spaces, both regularity (p-regularity) and topologicallness (p-topologicallness) are important notions. It is well known that topologicallness (p-topologicallness) can be described by a sophisticated Fischer-type diagonal condition, and regularity (p-regularity) can be described by dualizing that diagonal condition. Additionally, regularity (p-regularity) can also be characterized by the notion of closures of filters. In this paper, for stratified L-generalized convergence spaces, a new regularity (p-regularity) is defined by duzlizing a Fischer-type diagonal condition, which is used to describe the L-topologicallness of stratified L-convergence spaces (a subcategory of stratified L-generalized convergence spaces). Additionally, a characterization on this new regularity (p-regularity) by a notion of closures of stratified L-filters, is also presented. Keywords: Topology; Lattice-valued topology; Lattice-valued convergence space; regularity; Diagonal condition
1
Introduction
p-topologicallness [17] and p-regularity [11] are dual notions in the classical theory of convergence spaces [16]. For a set X, let F(X) denote the set of all filters on X. Let q and p be convergence structures on a set X. Then the space (X, q) is called p-topological if it satisfies either of the two conditions below. q q (1) Up (F) → x whenever F → x, where Up (F) is the neighborhood of F w.r.t p. ∗
Corresponding author. Tel./fax: +86 15206506635/+86 635 8258028. E-mail address: [email protected], [email protected]. Mailing address: Department of Mathematics, Liaocheng University, Liaocheng, 252059, P.R.China
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(2) (Fischer-type diagonal condition) Let J be any set, ψ : J −→ X, and let p σ : J −→ F(X) have the condition that σ(j) → ψ(j), for all j ∈ J. If F ∈ F(J) is S T q q such that ψ(F) → x, then kσF → x. Here, kσF = σ(j) ∈ F(X) is called the F ∈F j∈F
compression of F relative to σ. The space (X, q) is called p-regular if it satisfies either of the two conditions below. q q (1) Fp → x whenever F → x, where Fp is the closure of F w.r.t p. (2) (Dual Fischer-type diagonal condition) Let J be any set, ψ : J −→ X, and let p σ : J −→ F(X) have the condition that σ(j) → ψ(j), for all j ∈ J. If F ∈ F(J) is such q q that kσF → x, then ψ(F) → x. When p = q, p-topologicallness and p-regularity are refereed to topologicallness and regularity [1, 3, 12], respectively. Stratified L-generalized convergence spaces defined by J¨ager [7] are lattice- valued extensions of convergence spaces. In [9], J¨ager studied a regularity of stratified Lgeneralized convergence spaces both by a dual Fischer-type diagonal condition and a notion of α-lever closures of stratified L-filters. Later, Li and Jin [14] generalized J¨ager’s regularity to p-regularity. Quite recently, by modifying J¨ager’s Fischer-type diagonal condition, the first author and his co-author [15] introduced a new Fischertype diagonal condition, and proved that this condition happens to characterize the topologicallness of stratified L-convergence spaces [4, 13] (a subcategory of stratified L-generalized convergence spaces). In this paper, by dualizing that diagonal condition, a new regularity (p-regularity) of stratified L-generalized convergence spaces is defined, and a characterization on this new regularity (p-regularity) by the notion of closures of stratified L-filters, is also presented. The contents are arranged as follows. Section 2 fixes some notions and notations used in this note. Section 3 recalls the Fischer-type diagonal notion such that stratified L-convergence spaces are L-topological. Section 4 presents the main results. That is, by dualizing a Fischer-type diagonal condition in Section 3, we define a new regularity (p-regularity) of stratified L-generalized convergence spaces and then present a characterization on that regularity (p-regularity) by a notion of closures of stratified L-filters. In this paper, if not otherwise specified, L = (L, ≤) is always a complete lattice with a top element 1 and a bottom element 0, which satisfies the distributive law W W α ∧ ( i∈I βi ) = i∈I (α ∧ βi ). A lattice with these conditions is called a complete W Heyting algebra or a frame. The operation →: L × L −→ L given by α → β = {γ ∈ L : α ∧ γ ≤ β}, is called the residuation with respect to ∧. For the properties of ∧ and →, please refer to the literatures [6, 7, 14]. For a set X, the set LX of functions from X to L with the pointwise order becomes a complete lattice. Each element of LX is called an L-set (or a fuzzy subset) of X. And
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we make no difference between a constant function and its value since no confusion will arise. Let f : X −→ Y be a function. We definef ← : LY −→ LX [6] by f ← (µ) = µ ◦ f for µ ∈ LY . Let X be a set. A fuzzy partial order (or, an L-partial order) on X [2] is a reflexive, transitive and antisymmetric fuzzy relation on X. The pair (X, R) is called an L-partially ordered set. Let [LX ] : LX × LX −→ L be a function defined by V [LX ](λ, µ) = x∈X (λ(x) → µ(x)). Then [LX ] is an L-partial order on LX [2, 19]. The value [LX ](λ, µ) ∈ L is interpreted as the degree that λ is contained in µ. In the sequel, we use the symbol [λ, µ] to denote [LX ](λ, µ) for simplicity. The following lemma is useful to the subsequent section. Lemma 1.1. [14] Let f : X −→ Y be an function. For any λ, µ, ν ∈ LX and any {λi }i∈I , {µi }i∈I ⊆ LX , we have (1) λ ≤ µ implies [λ, ν] ≥ [µ, ν]; (2) [λ, ∧i∈I µi ] = ∧i∈I [λ, µi ]; (3) λ ∧ [λ, µ] ≤ µ; (4) [∨i∈I λi , µ] = ∧i∈I [λi , µ]; (5) [λ, µ] ≤ [f → (λ), f → (µ)]. A stratified L-filter [6] on a set X is a function F : LX −→ L such that: (F0) F(0) = 0, (F1) F(1) = 1, (F2) ∀λ, µ ∈ LX , F(λ) ∧ F(µ) = F(λ ∧ µ), (Fs) ∀α ∈ L, F(α) ≥ α. The set FLs (X) of all stratified L-filters on X is ordered by F ≤ G ⇔ ∀λ ∈ LX , F(λ) ≤ G(λ). There is a natural fuzzy partial order on FLs (X) inherited X X from L(L ) . Precisely, for all F, G ∈ FLs (X), if we let [FLs (X)](F, G) = [LL ](F, G) = V s λ∈LX (F(λ) → G(λ)), then [FL (X)] is an L-partially order. Example 1.2. (1) For each point x in a set X, the function [x] : LX −→ L, [x](λ) = V λ(x) is a stratified L-filter on X. (2) If {Fj |j ∈ J} ⊆ FLs (X), then j∈J Fj ∈ FLs (X). (3) Let f : X −→ Y be a function. If F ∈ FLs (X), then the function f ⇒ (F) ∈ FLs (Y ), where f ⇒ (F) : LY −→ L is defined by λ 7→ F(λ ◦ f ) = F(f ← (λ)).
2
Fischer-type diagonal condition of stratified Lconvergence spaces
In this section, we shall recall the Fischer-type diagonal condition such that a stratified L-convergence space is L-topological. Definition 2.1. A stratified L-generalized convergence structure [7, 18] on a set X is a function limq : FLs (X) −→ LX satisfying (LC1) ∀x ∈ X, limq [x](x) = 1; and (LC2) ∀F, G ∈ FLs (X), F ≤ G =⇒ limq F ≤ limq G. The pair (X, limq ) is called a stratified Lgeneralized convergence space. The pair (X, limq ) is called a stratified L-convergence space [13] (or, a stratified L-ordered convergence space in [4]) if lim : FLs (X) −→ LX is a function satisfying (LC1) and (LC20 ) ∀F, G ∈ FLs (X), [FLs (X)](F, G) ≤ [LX ](limq F, limq G). Because (LC2)0 ⇒(LC2), a stratified L-convergence space is a
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stratified L-generalized convergence space. A function f : X −→ X 0 between two 0 stratified L-generalized convergence spaces (X, limq ), (X 0 , limq ) is called continuous if 0 for all F ∈ FLs (X) and all x ∈ X we have limq F(x) ≤ limq f ⇒ (F)(f (x)). fi
For a given source (X −→ (Xi , limqi ))i∈I , the initial structure, limq on X is defined V by ∀F ∈ FLs (X), ∀x ∈ X, limq F(x) = limqi fi⇒ (F)(fi (x)). i∈I
qi
fi
For a given sink ((Xi , lim ) −→ X)i∈I , the final structure, limq on X is defined by ( 1, F ≥ [x]; W limq F(x) = qi i∈I,xi ∈Xi ,Gi ∈F s (Xi ),fi (xi )=x,f ⇒ (Gi )≤F lim Gi (xi ), F 6≥ [x]. i
L
When X = ∪i∈I fi (Xi ), the final structure limq can be simplified as [14] _ limq F(x) = limqi Gi (xi ). s (X ),f (x )=x,f ⇒ (G )≤F i∈I,xi ∈Xi ,Gi ∈FL i i i i i
In the theory of convergence spaces, Fischer-type diagonal condition is formulated by the aid of the notion of compression. The situation with lattice-valued convergence is similar. In [8], J¨ager introduced an lattice-valued version of compression, which first appeared in [5] with a slightly different formalization. Let σ : J −→ FLs (X) be a function and F ∈ FLs (X). Then the function kL σF : LX −→ L defined by ∀λ ∈ LX , kL σF(λ) := F(b σ (λ)), where σ b(λ) = σ(−)(λ) ∈ LJ forms a stratified L-filter on X; and it is called the compression of F w.r.t σ. In [15], the first author and his co-author modified J¨ager’s compression and introduced a Fischer-type diagonal condition. It was proved that a stratified L-convergence space with this diagonal condition is L-topological. Note that when a function σ : J −→ FLs (X) being given, that means an L-filter σ(j) is selected for each j ∈ J. In this sense, we call σ : J −→ FLs (X) an L-filter select function. The definition below generalizes that notion. Definition 2.2. [15] A function σ = (σ1 , σ2 ) : J −→ FLs (X)×L0 , where L0 = L−{0}, is said to be an L-filter select degree function. For any j ∈ J, the value σ2 (j) ∈ L is interpreted as the degree to which the stratified L-filter σ1 (j) is selected. Obviously, an L-filter select function can be regarded as an L-filter select degree function with σ2 ≡ 1. Definition 2.3. [15] Let σ : J −→ FLs (X) × L0 be an L-filter select degree function and F ∈ FLs (J). If the function kL σF : LX −→ L defined by ∀λ ∈ LX , kL σF(λ) := F(b σ (λ)), where σ b(λ) = σ2 (−) → σ1 (−)(λ) ∈ LJ
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forms a stratified L-filter on X, then we call such F compressible w.r.t σ and call kL σF as the compression of F w.r.t σ. It is easily seen that kL σF satisfies (F1), (F2) and (Fs) for any F ∈ FLs (J). If σ : J −→ FLs (X) × L0 is an L-filter select function, then kL σF ∈ FLs (X) for any F ∈ FLs (J). In this case, kL σF coincides with J¨ager’s compression. Thus, Definition 2.3 generalizes J¨ager’s compression. Theorem 2.4. [15] Let (X, limq ) be a stratified L-convergence spaces. Then (X, limq ) is L-topological if and only if it satisfies the following condition (Lf ). (Lf ) Let J be any set, ψ : J −→ X, and let σ : J −→ FLs (X) × L0 . If F ∈ FLs (J) is compressible w.r.t σ, then for each x ∈ X, limq ψ ⇒ (F)(x) ∗
^
limq σ(j)(ψ(j)) ≤ limq kL σF(x),
j∈J
where limq σ(j)(ψ(j)) := σ2 (j) → limq σ1 (j)(ψ(j)). Obviously, the condition (Lf ) implies the following condition (Lfw). (Lfw): Let J be any set, ψ : J −→ X, and let σ : J −→ FLs (X) × L0 have the condition ∀j ∈ J, σ2 (j) = limq σ1 (j)(ψ(j)) (which means that limq σ(j)(ψ(j)) ≡ 1). If F ∈ FLs (J) is compressible w.r.t σ, then limq ψ ⇒ (F)(x) ≤ limq kL σF(x) for each x ∈ X. Note that in the proof of the sufficiency of Theorem 2.4, the selected σ, ψ satisfies the condition σ2 (j) = limq σ1 (j)(ψ(j)) (see Theorem 4.9 in [15]). It follows immediately that (Lf )⇔(Lfw). In addition, the characterization on L-topologicallness of stratified L-convergence spaces by the notion of neighborhoods of stratified L-filters, was presented in [10].
3
regularity
and
p-regularity
of
stratified
L-
generalized convergence spaces In this section, by dualizing the condition (Lfw) we define a new regularity (pregularity) of stratified L-generalized convergence spaces. Then we also present a characterization on that regularity (p-regularity) by a notion of closures of stratified Lfilters. Let (X, limp , limq ) be a pair of stratified L-generalized convergence spaces. p-(DLfw): Let J be any set, ψ : J −→ X, and let σ : J −→ FLs (X) × L0 have the condition ∀j ∈ J, σ2 (j) = limp σ1 (j)(ψ(j)). If F ∈ FLs (J) is compressible w.r.t σ, then limq kL σF(x) ≤ limq ψ ⇒ (F)(x) for each x ∈ X.
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When limp = limq , the condition p-(DLfw) is denoted as (DLfw). Obviously, the condition (DLfw) is obtained by dualizing the condition (Lfw). It is easily seen that when L = {0, 1}, the condition p-(DLfw) is equivalent to the crisp dual Fischer-type diagonal condition. Definition 3.1. Let (X, limp , limq ) be a pair of stratified L-generalized convergence spaces. Then (X, limq ) is called p-regular if it satisfies the dual Fischer-type diagonal condition p-(DLfw). When limp = limq , then (X, limq ) is called regular if it is pregular. In the following, we shall give a characterization on regularity (p-regularity) by the notion of closures of stratified L-filters. Definition 3.2. Let (X, limp ) be a stratified L-generalized convergence space, and let λ ∈ LX . Then the L-set λp ∈ LX defined by _ ∀x ∈ X, λp (x) = (limp F(x) → F(λ)) s (X): limp F (x)6=0 F∈FL
is called the closure of λ w.r.t (X, limp ). Remark 3.3. When L = {0, 1}, a stratified L-generalized convergence space reduces p to a convergence space. Then it is easily seen that x ∈ λp ⇔∃F → x s.t. λ ∈ F. This shows that closure is precisely the crisp closure in [16] when L = {0, 1}. Lemma 3.4. Let (X, limp ) be a stratified L-generalized convergence space. Then for all λ, µ ∈ LX and all α ∈ L we get (1) λ ≤ λp ; (2) λ ≤ µ implies λp ≤ µp ; (3) αp ≥ α. Proof. (1) For each x ∈ X, by limp [x](x) = 1 we get λp (x) ≥ [x](λ) = λ(x). So, λ ≤ λp . Take λ = 1 in (1), we obtain 1p = 1. (2) It follows from the property (F2) of stratified L-filters. (3) For each x ∈ X we have αp (x) =
_
_
(Fs)
(limp F(x) → F(α)) ≥
lim F (x)6=0
(limp F(x) → α) ≥ α.
lim F (x)6=0
Theorem 3.5. Let (X, limp ) be a stratified L-generalized convergence space. For each W F ∈ FLs (X), the function F p : LX −→ L, defined by ∀λ ∈ LX , F p (λ) = µ∈LX (F(µ) ∧ [µp , λ]), is a stratified L-filter, called the closure of F w.r.t (X, limp ). Proof. (F1) That F p (1) = 1 is obvious. That F p (0) = 0 follows by F p (λ) =
_ µ∈LX
_
(F(µ) ∧ [µp , λ]) ≤
(F(µ) ∧ [µ, λ]) ≤ F(λ).
µ∈LX
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(F2) Obviously, F p (λ ∧ µ) ≤ F p (λ) ∧ F p (µ). Conversely, _ _ F p (λ) ∧ F p (µ) = (F(a) ∧ [ap , λ]) ∧ (F(b) ∧ [bp , µ]) a∈LX
b∈LX
_
=
(F(a) ∧ F(b) ∧ [ap , λ] ∧ [bp , µ])
a,b∈LX
_
≤
(F(a ∧ b) ∧ [(a ∧ b)p , λ ∧ µ])
a,b∈LX
≤
_
(F(c) ∧ [cp , λ ∧ µ]) = F p (λ ∧ µ).
c∈LX
(Fs) By 1p = 1, it follows that F p (α) =
W
µ∈LX (F(µ)
∧ [µp , α]) ≥ F(1) ∧ α = α.
Remark 3.6. When L = {0, 1}, a stratified L-generalized convergence space reduces to a convergence space. It is easily seen that F p is precisely the filter generated by {A : A ∈ F} as a filterbasis [16]. Lemma 3.7. Let J, X, σ, ψ satisfy the condition in p-(DLfw). Then for any λ, µ ∈ LX ˆ we have [µp , λ] ≤ [φ(µ), ψ ← (λ)]. Proof. Note that ∀j ∈ J, σ2 (j) = limp σ1 (j)(ψ(j)) 6= 0. Then ^ _ [µp , λ] = ( (limp G(x) → G(µ)) → λ(x)) s (X):limp G(x)6=0 x∈X G∈FL
=
^
^
((limp G(x) → G(µ)) → λ(x))
s (X):limp G(x)6=0 x∈X G∈FL
≤
^
(limp σ1 (j)(ψ(j)) → σ1 (j)(µ)) → λ(ψ(j)))
j∈J
≤
^
(σ2 (j) → σ1 (j)(µ)) → ψ ← (λ)(j))
j∈J
=
^
(ˆ σ (µ)(j) → ψ ← (λ)(j)) = [ˆ σ (µ), ψ ← (λ)].
j∈J
Lemma 3.8. Let J, X, σ, ψ satisfy the condition in p-(DLfw), and let F ∈ FLs (X). W Then the function F σ : LJ −→ L, defined by F σ (λ) = µ∈LX (F(µ)∧[ˆ σ (µ), λ]), satisfies σ (F1), (F2), (Fs) and kL σF ≥ F. Proof. (F1): It is obvious. (F2): Obviously, F σ (λ ∧ µ) ≤ F σ (λ) ∧ F σ (µ). Conversely, _ _ F σ (λ) ∧ F σ (µ) = (F(a) ∧ [b σ (a), λ]) ∧ (F(b) ∧ [b σ (b), µ]) a∈LX
=
b∈LX
_
(F(a) ∧ F(b) ∧ [b σ (a), λ] ∧ [b σ (b), µ])
a,b∈LX
≤
_
(F(a ∧ b) ∧ [b σ (a ∧ b), λ ∧ µ])
a,b∈LX
≤
_
(F(c) ∧ [b σ (c), λ ∧ µ]) = F σ (λ ∧ µ).
c∈LX
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(Fs): For any β ∈ L, we have _ F σ (β) = (F(µ) ∧ [b σ (µ), β]) ≥ F(1) ∧ [b σ (1), β] = 1 ∧ β = β. µ∈LX
It follows by the following inequality that kL σF σ ≥ F. For any λ ∈ LX , _ kL σF σ (λ) = F σ (ˆ σ (λ)) = (F(µ) ∧ [ˆ σ (µ), σ ˆ (λ)]) ≥ F(λ). µ∈LX
Theorem 3.9. Let (X, limp , limq ) be a pair of stratified L-generalized convergence spaces. Then (X, limq ) is p-regular if and only if limq F ≤ limq F p for any F ∈ FLs (X). Proof. Necessity: Let J = {(G, y) ∈ FLs (X) × X|limp G(y) 6= 0}, ψ : J −→ X, (G, y) 7→ y, σ : J −→ FLs (X) × L0 , (G, y) 7→ (G, limp G(y)). Then (1) σ2 (j) = limp σ1 (j)(ψ(j)) 6= 0. (2) For any F ∈ FLs (X) we have F σ ∈ FLs (J). Indeed, by Lemma 3.8, we need only to check that F σ (0) = 0. ^ _ _ σ (µ)(j) → 0)) (F(µ) ∧ (ˆ (F(µ) ∧ [ˆ σ (µ), 0]) = F σ (0) = µ∈LX
≤
_
(F(µ) ∧ (
_
(F(µ) ∧ (
=
(F(µ) ∧ (
_
^
((limp [y](y) → [y](µ)) → 0))
^
(µ(y) → 0)) =
y∈X
µ∈LX
≤
(ˆ σ (µ)([y], y) → 0))
y∈X
µ∈LX
_
j∈J
µ∈LX
y∈X
µ∈LX
=
^
_
(F(µ) ∧ [µ, 0])
µ∈LX
F(µ ∧ [µ, 0]) ≤ F(0) = 0.
µ∈LX
(3) ψ ⇒ (F σ ) = F p . For any λ, µ ∈ LX , ^ _ ( [µp , λ] =
(limp G(x) → G(µ)) → λ(x))
s (X):limp G(x)6=0 x∈X G∈FL
=
^
^
((limp G(x) → G(µ)) → λ(x))
s (X):limp G(x)6=0 x∈X G∈FL
=
^
(limp σ1 (j)(ψ(j)) → σ1 (j)(µ)) → λ(ψ(j)))
j∈J
=
^
(σ2 (j) → σ1 (j)(µ)) → ψ ← (λ)(j))
j∈J
=
^
(ˆ σ (µ)(j) → ψ ← (λ)(j)) = [ˆ σ (µ), ψ ← (λ)].
j∈J
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It follows that _
ψ ⇒ (F σ )(λ) = F σ (ψ ← (λ)) =
(F(µ)∧[ˆ σ (µ), ψ ← (λ)]) =
µ∈LX
_
(F(µ)∧[µp , λ]) = F p (λ).
µ∈LX
(4) F σ is compressible w.r.t. σ. For any λ, µ ∈ LX , ^ [b σ (λ), σ b(µ)] = (b σ (λ)(j) → σ b(µ)(j)) j∈J
^
=
((σ2 (j) → σ1 (j)(λ)) → (σ2 (j) → σ1 (j)(µ))
p
(G,y):lim G(y)6=0
^
≤
((limp [y](y) → [y](λ)) → (limp [y](y) → [y](µ)))
([y],y):y∈X
=
^
(λ(y) → µ(y)) = [λ, µ].
y∈X
Therefore, for any λ ∈ LX , kL σF σ (λ) = F σ (ˆ σ (λ)) =
_
(F(µ) ∧ [ˆ σ (µ), σ ˆ (λ)]) ≤
_
(F(µ) ∧ [µ, λ]) ≤ F(λ).
µ∈LX
µ∈LX
By Lemma 3.8, we have kL σF σ = F ∈ FLs (X). Thus kL σF σ is compressible w.r.t. σ. Applying (1)-(4) in p-(DLfw) we have limq F ≤ limq F p . Sufficiency: Let J, X, σ, ψ satisfy the condition in (DLfw). Then for any F ∈ FLs (J), by (X, limq ) is p-regular we have that limq kL σF ≤ limq kL σF p (x). For any λ ∈ LX , by Lemma 3.7 we have _ _ kL σF p (λ) = (kL σF(µ) ∧ [µp , λ]) = (F(ˆ σ (µ)) ∧ [µp , λ]) µ∈LX
≤
_
µ∈LX
(F(ˆ σ (µ)) ∧ [ˆ σ (µ), ψ ← (λ)]) ≤ F(ψ ← (λ)) = ψ ⇒ (F)(λ).
µ∈LX
So, kL σF p ≤ ψ ⇒ (F), and hence limq ψ ⇒ (F) ≥ limq kL σF p ≥ limq kL σF, i.e., the condition p-(DLfw) holds. The next two theorems show that p-regularity behave reasonably well relative to initial and final structures. Definition 3.10. Let f : (X, limq ) −→ (Y, limp ) be a function between stratified Lgeneralized convergence spaces. Then f is said to be a closure function if f → (λq ) ≥ f → (λ)p for all λ ∈ LX . Lemma 3.11. Let f : (X, limq ) −→ (Y, limp ) be a function between stratified Lgeneralized convergence spaces, and let F ∈ FLs (X). (1) If f is continuous, then f ⇒ (F q ) ≥ f ⇒ (F)p . (2) If f is a closure function, then f ⇒ (F q ) ≤ f ⇒ (F)p .
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A new regularity (p-regularity) of stratified L-generalized convergence spaces 10
Proof. (1) Let f be a continuous function. Then for each λ ∈ LY we check below that (f ← (λ))q ≤ f ← (λp ). Indeed, for each x ∈ X, _ ¡ q ¢ (f ← (λ))q (x) = lim G(x) → G(f ← (λ)) s (X):limq G(x)6=0 G∈FL
_
≤
¡
s (X):limq G(x)6=0 G∈FL
_
≤
¢ limp f ⇒ (G)(f (x)) → f ⇒ (G)(λ)
¡
¢ limp H(f (x)) → H(λ) = f ← (λp )(x),
s (Y ):limp H(x)6=0 H∈FL
where the first inequality holds for the continuity of f . Then for each F ∈ FLs (X) and each λ ∈ LY _ ([µq , f ← (λ)] ∧ F(µ)) f ⇒ (F q )(λ) = F q (f ← (λ)) = ≥
_
µ∈LX
ν∈LY
≥
_
_
([(f ← (ν))q , f ← (λ)] ∧ F(f ← (ν))) ≥
([f ← (ν p ), f ← (λ)] ∧ F(f ← (ν)))
ν∈LY
([ν p , λ] ∧ f ⇒ (F)(ν)) = f ⇒ (F)p (λ).
ν∈LY
Thus f ⇒ (F q ) ≥ f ⇒ (F)p . (2) Let f be a closure function. Then for each λ ∈ LY , _ _ f ⇒ (F)p (λ) = (f ⇒ (F)(µ) ∧ [µp , λ]) = (F(f ← (µ)) ∧ [µp , λ]) µ∈LY
≥
_
µ∈LY
(F(f ← f → (ν)) ∧ [f → (ν)p , λ]) ≥
ν∈LX
≥
_
(F(ν) ∧ [f → (ν q ), λ]) =
ν∈LX
_
_
(F(ν) ∧ [f → (ν)p , λ])
ν∈LX
(F(ν) ∧ [ν q , f ← (λ)])
ν∈LX
= F q (f ← (λ)) = f ⇒ (F q )(λ), where the third inequality holds for f being a closure function, and the third equality follows from Lemma 1.1 (7). By the arbitrariness of λ, we get f ⇒ (F q ) ≤ f ⇒ (F)p . Theorem 3.12. Let {(Xi , limqi , limpi )}i∈I be pairs of stratified L-generalized convergence spaces and let limq (resp., limp ) be the initial structure on X relative to the source fi fi (X −→ (Xi , limqi ))i∈I (resp., (X −→ (Xi , limpi ))i∈I ). If each limqi is pi -regular, then (X, limq ) is p-regular. Proof. Let F ∈ FLs (X) and x ∈ X. Then by Lemma 3.11 (1) we have fi⇒ (F p ) ≥ fi⇒ (F)pi for all i ∈ I. It follows by each (Xi , limqi ) being pi -regular that ^ ^ limq F p (x) = limqi fi⇒ (F)pi (fi (x)) limqi fi⇒ (F p )(fi (x)) ≥ i∈I
≥
^
i∈I qi
lim
fi⇒ (F)(fi (x))
= limq F(x).
i∈I q
Thus (X, lim ) is p-regular.
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Theorem 3.13. Let {(Xi , limqi , limpi )}i∈I be pairs of stratified L-generalized converfi gence spaces, and let limq be the final structure on X w.r.t. the sink ((Xi , limqi ) −→ X)i∈I with X = ∪i∈I fi (Xi ). If each limqi is pi -regular and limp is a stratified Lgeneralized convergence structure on X such that each fi : (Xi , limpi ) −→ (X, limp ) is a closure function, then (X, limq ) is p-regular. Proof. Let F ∈ FLs (X) and x ∈ X. Then _
limq F(x) =
limqi Gi (xi )
s (X ),f (x )=x,f ⇒ (G )≤F i∈I,xi ∈Xi ,Gi ∈FL i i i i i
_
≤
limqi Gipi (xi )
s (X ),f (x )=x,f ⇒ (G )≤F i∈I,xi ∈Xi ,Gi ∈FL i i i i i
_
≤
limqi Gipi (xi )
s (X ),f (x )=x,f ⇒ (G ) ≤F i∈I,xi ∈Xi ,Gi ∈FL p i p i i i i
_
=
limqi Gipi (xi )
⇒ s (X ),f (x )=x,f ⇒ (G i∈I,xi ∈Xi ,Gi ∈FL ipi )≤fi (Gi )p ≤F p i i i i
_
≤
limqi Hi (xi ) = limq F p (x),
s (X ),f (x )=x,f ⇒ (H )≤F i∈I,xi ∈Xi ,Hi ∈FL p i i i i i
where the first inequality holds for pi -regularity of limqi , the second equality follows from Lemma 3.11 (2). Then limq is p-regular by limq F ≤ limq F p . The regularity has similar characterization and properties, we omit them here.
4
Conclusion
In this paper, by dualizing the Fischer-type diagonal condition (Lfw), which is used to describe the L-topologicallness of stratified L-convergence spaces, we define a new regularity (p-regularity) of stratified L-generalized convergence spaces. Then we also present a characterization on that regularity (p-regularity) by the notion of closures of stratified L-filters. The regularity (p-regularity) is proved to behave reasonably well relative to initial and final structures.
References [1] H.J. Biesterfeld, Regular convergence spaces, Indag. Math. 28 (1966) 605–607. [2] R. Bˇelohl´avek, Fuzzy Relational Systems: Foundations and Principles, Kluwer Academic Publishers, New York, 2002. [3] C.H. Cook, H.R. Fischer, Regular Convergence Spaces, Math. Annalen 174 (1967) 1–7.
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[4] J.M. Fang, Stratified L-ordered convergence structures, Fuzzy Sets and Systems 161 (2010) 2130–2149. [5] W. G¨ahler, Monadic convergence structures, In: Topological and Algebraic Structures in Fuzzy Sets, (S.E. Rodabaugh, E.P. Klement, eds.), Kluwer Academic Publishers, Dordrecht, 2003. [6] U. H¨ohle, S.E. Rodabaugh (Eds.), Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory, The Handbooks of Fuzzy Sets Series, Vol.3, Kluwer Academic Publishers, Boston, Dordrecht, London, 1999. [7] G. J¨ager, A category of L-fuzzy convergence spaces, Quaestiones Mathematicae 24 (2001) 501–517. [8] G. J¨ager, Fischer’s diagonal condition for lattice-valued convergence spaces, Quaestiones Mathematicae 31 (2008) 11–25. [9] G. J¨ager, Lattice-valued convergence spaces and regularity, Fuzzy Sets and Systems 159 (2008) 2488–2502. [10] G. J¨ager, G¨ahler’s neighbourhood condition for lattice-valued convergence spaces, Fuzzy Sets and Systems 204 (2012) 27–39. [11] D.C. Kent, G.D. Richardson, p-regular convergence spaces, Math. Nachr. 149 (1990) 215–222. [12] D.C. Kent, G.D. Richardson, Convergence spaces and diagonal conditions, Topology and its Applications 70 (1996) 167–174. [13] L. Li, Many-Valued Convergence, Many-Valued Topology, and Many-Valued Order Structure, PhD Thesis, Sichuan University, 2008 (In Chinese). [14] L. Li, Q. Jin, p-Topologicalness and p-Regularity for Lattice-Valued Convergence Spaces, Fuzzy Sets and Systems 238 (2014) 26–45. [15] L. Li, Q. Jin, K. Hu, On Stratified L-Convergence Spaces: Fischer’s Diagonal Axiom, DOI: 10.1016/j.fss.2014.09.001. [16] G. Preuss, Fundations of Topology, Kluwer Academic Publishers, London, 2002. [17] S.A. Wilde, D.C. Kent, p-topological and p-regular: dual notions in convergence theory, Internat. J. Math. & Math. Sci. 22 (1999) 1–12. [18] W. Yao, On many-valued stratified L-fuzzy convergence spaces, Fuzzy Sets and Systems 159 (2008) 2503-2519. [19] D. Zhang, An enriched category approach to many valued topology, Fuzzy Sets and Systems 158 (2007) 349–366.
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Uni-soft filters and uni-soft G-filters in residuated lattices Young Bae Jun Department of Mathematics Education, Gyeongsang National University, Jinju 660-701, Korea e-mail: [email protected] Seok Zun Song∗ Department of Mathematics, Jeju National University, Jeju 690-756, Korea e-mail: [email protected] Abstract The notions of uni-soft filters and uni-soft G-filters in residuated lattices are introduced, and their relations, properties and characterizations are investigated. Conditions for a uni-soft filter to be a uni-soft G-filter are provided. Keywords: Residuated lattice, Uni-soft filter, Uni-soft G-filter. 2010 Mathematics Subject Classification. 06F35, 03G25, 06D72.
1
Introduction
Non-classical logic has become a formal and useful tool in dealing with fuzzy and uncertain informations. Various logical algebras have been proposed as the semantical systems of non-classical logic systems. Residuated lattices are important algebraic structures which are basic of BL-algebras, M V -algebras, M T L-algebras, G¨odel algebras, R0 -algebras, lattice implication algebras, and so forth. The (fuzzy) filter theory in the logical algebras has an important role in studying these algebras and completeness of the corresponding non-classical logics, and it is studied in the papers [1], [2], [3], [9], [12], [13] and [14]. Filter theory, which is an important notion, in residuated lattices is studied by Shen and Zhang [11] and Zhu and Xu [16]. Uncertainty is an attribute of information. As a new mathematical tool for dealing with uncertainties, Molodtsov [10] introduced the concept of soft sets. Since then several authors studied (fuzzy) algebraic structures based on soft *Corresponding author.
1
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set theory in several algebraic structures. In order to lay a foundation for providing a soft algebraic tool in considering many problems that contain uncertainties, Jun [7] discussed the union soft sets with applications in BCK/BCI-algebras. Also, Jun et al. [8] discussed uni-soft sets applied to commutative BCI-ideals. In this paper, we introduce uni-soft filters and uni-soft G-filters in residuated lattices, and investigate their properties. We consider characterizations of uni-soft filters and unisoft G-filters. We provide conditions for a uni-soft filter to be a uni-soft G-filter.
2
Preliminaries
Definition 2.1 ([1, 5, 6]). A residuated lattice is an algebra L := (L, ∨, ∧, ⊙, →, 0, 1) of type (2, 2, 2, 2, 0, 0) such that (1) (L, ∨, ∧, 0, 1) is a bounded lattice. (2) (L, ⊙, 1) is a commutative monoid. (3) ⊙ and → form an adjoint pair, that is, (∀x, y, z ∈ L) (x ≤ y → z ⇔ x ⊙ y ≤ z) . In a residuated lattice L, the ordering ≤ is defined as follows: (∀x, y ∈ L) (x ≤ y ⇔ x ∧ y = x ⇔ x ∨ y = y ⇔ x → y = 1) and x′ will be reserved for x → 0, and x′′ = (x′ )′ , etc. for all x ∈ L. Proposition 2.2 ([1, 5, 6, 12, 13]). In a residuated lattice L, the following properties are valid. 1 → x = x, x → 1 = 1, x → x = 1, 0 → x = 1, x → (y → x) = 1.
(2.1)
x → (y → z) = (x ⊙ y) → z = y → (x → z).
(2.2)
x ≤ y ⇒ z → x ≤ z → y, y → z ≤ x → z.
(2.3)
z → y ≤ (x → z) → (x → y), z → y ≤ (y → x) → (z → x).
(2.4)
Definition 2.3 ([11]). A nonempty subset F of a residuated lattice L is called a filter of L if it satisfies the conditions: (∀x, y ∈ L) (x, y ∈ F ⇒ x ⊙ y ∈ F ) .
(2.5)
(∀x, y ∈ L) (x ∈ F, x ≤ y ⇒ y ∈ F ) .
(2.6)
2
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Proposition 2.4 ([11]). A nonempty subset F of a residuated lattice L is a filter of L if and only if it satisfies: 1 ∈ F.
(2.7)
(∀x ∈ F ) (∀y ∈ L) (x → y ∈ F ⇒ y ∈ F ) .
(2.8)
Definition 2.5 ([15]). A nonempty subset F of L is called a G-filter of L if it is a filter of L that satisfies the following condition: (∀x, y ∈ L) ((x ⊙ x) → y ∈ F ⇒ x → y ∈ F ) .
(2.9)
A soft set theory is introduced by Molodtsov [10], and C ¸ aˇgman et al. [4] provided new definitions and various results on soft set theory. In what follows, let U be an initial universe set and E be a set of parameters. Let P(U ) denotes the power set of U and A, B, C, · · · ⊆ E. Definition 2.6 ([4, 10]). A soft set (f˜, A) over U is defined to be the set of ordered pairs { } ˜ ˜ ˜ (f , A) := (x, fA (x)) : x ∈ E, fA (x) ∈ P(U ) , where f˜A : E → P(U ) such that f˜(x) = ∅ if x ∈ / A. The soft set (f˜, A) is simply denoted by f˜A . For a soft set f˜A over U and a subset τ of U, the τ -exclusive set of f˜A , denoted by e(f˜A ; τ ), is defined to be the set { } e(f˜A ; τ ) := x ∈ A | f˜A (x) ⊆ τ .
3
Uni-soft filters
In what follows, we take a residuated lattice L as a set of parameters. Definition 3.1. A soft set f˜L over U is called a uni-soft filter of L if it satisfies: ( ) (∀x, y ∈ L) x ≤ y ⇒ f˜L (x) ⊇ f˜L (y) , ( ) (∀x, y ∈ L) f˜L (x) ∪ f˜L (y) ⊇ f˜L (x ⊙ y) .
(3.1) (3.2)
3
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Proposition 3.2. Every uni-soft filter f˜L of L satisfies: ( ) (∀x ∈ L) f˜L (x) ⊇ f˜L (1) . ( ) (∀x, y ∈ L) f˜L (x) ∪ f˜L (x → y) ⊇ f˜L (y) .
(3.3) (3.4)
Proof. Let x, y ∈ L. Since x ≤ 1, we have f˜L (x) ⊇ f˜L (1) by (3.1). Since x ⊙ (x → y) ≤ y, it follows from (3.2) and (3.1) that f˜L (x) ∪ f˜L (x → y) ⊇ f˜L (x ⊙ (x → y)) ⊇ f˜L (y). This completes the proof. Lemma 3.3. If a soft set f˜L over U satisfies two conditions (3.3) and (3.4), then ( ) (∀x, y, z ∈ L) x ≤ y → z ⇒ f˜L (x) ∪ f˜L (y) ⊇ f˜L (z) , (3.5) ( ) (∀x, y, z ∈ L) x ⊙ y ≤ z ⇒ f˜L (x) ∪ f˜L (y) ⊇ f˜L (z) . (3.6) Proof. Assume that x ≤ y → z for all x, y, z ∈ L. Then x → (y → z) = 1, and so ( ) ˜ ˜ ˜ ˜ fL (x) ∪ fL (y) = fL (x) ∪ fL (1) ∪ f˜L (y) ( ) = f˜L (x) ∪ f˜L (x → (y → z)) ∪ f˜L (y) ⊇ f˜L (y) ∪ f˜L (y → z) ⊇ f˜L (z). Since x ≤ y → z ⇔ x ⊙ y ≤ z, we know that (3.5) induces (3.6). We consider characterizations of uni-soft filters. Theorem 3.4. A soft set f˜L over U is a uni-soft filter of L if and only if it satisfies two conditions (3.3) and (3.4). Proof. The necessity is from Proposition 3.2. Conversely, let f˜L be a soft set over U that satisfies (3.3) and (3.4). Let x, y ∈ L be such that x ≤ y. Then x → y = 1 and so f˜L (x) = f˜L (x) ∪ f˜L (1) = f˜L (x) ∪ f˜L (x → y) ⊇ f˜L (y). Since x ⊙ y ≤ x ⊙ y for all x, y ∈ L, it follows from (3.6) that f˜L (x) ∪ f˜L (y) ⊇ f˜L (x ⊙ y) for all x, y ∈ L. Therefore f˜L is a uni-soft filter of L. 4
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Theorem 3.5. A soft set f˜L over U is a uni-soft filter of L if and only if it satisfies the condition (3.5). Proof. The necessity is from Lemma 3.3 and Theorem 3.4. Conversely let f˜L be a soft set over U satisfying (3.5). Since x ≤ x → 1 and x → y ≤ x → y for all x, y ∈ L, it follows from (3.5) that f˜L (x) = f˜L (x) ∪ f˜L (x) ⊇ f˜L (1) and f˜L (x) ∪ f˜L (x → y) ⊇ f˜L (y) for all x, y ∈ L. Hence f˜L is a uni-soft filter of L by Theorem 3.4. Proposition 3.6. Every uni-soft filter f˜L of L satisfies the following condition: ( ) (∀x, y, z ∈ L) f˜L (x → (y → z)) ∪ f˜L (x → y) ⊇ f˜L (x → (x → z)) .
(3.7)
Proof. Let x, y, z ∈ L. Using (2.2) and (2.4), we have x → (y → z) = y → (x → z) ≤ (x → y) → (x → (x → z)). It follows from Theorem 3.5 that f˜L (x → (y → z)) ∪ f˜L (x → y) ⊇ f˜L (x → (x → z)). This completes the proof. Theorem 3.7. A soft set f˜L over U is a uni-soft filter of L if and only if f˜L satisfies the condition (3.3) and ( ) ˜ ˜ ˜ (∀x, y, z ∈ L) fL (x → (y → z)) ∪ fL (y) ⊇ fL (x → z) . (3.8) Proof. Assume that f˜L is a uni-soft filter of L. Then the condition (3.3) is valid. Using (3.4) and (2.2), we have f˜L (x → z) ⊆ f˜L (y) ∪ f˜L (y → (x → z)) = f˜L (y) ∪ f˜L (x → (y → z)) for all x, y, z ∈ L. Conversely, let f˜L be a soft set over U satisfying (3.3) and (3.8). Taking x := 1 in (3.8) and using (2.1), we have f˜L (z) = f˜L (1 → z) ⊆ f˜L (1 → (y → z)) ∪ f˜L (y) = f˜L (y → z) ∪ f˜L (y) for all y, z ∈ L. Thus f˜L is a uni-soft filter of L by Theorem 3.4. 5
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Proposition 3.8. Every uni-soft filter f˜L of L satisfies the following condition: ( ) (∀a, x ∈ L) f˜L (a) ⊇ f˜L ((a → x) → x) .
(3.9)
Proof. If we take y := (a → x) → x and x := a in (3.4), then f˜L ((a → x) → x) ⊆ f˜L (a) ∪ f˜L (a → ((a → x) → x)) = f˜L (a) ∪ f˜L ((a → x) → (a → x)) = f˜L (a) ∪ f˜L (1) = f˜L (a). This completes the proof. Theorem 3.9. A soft set f˜L over U is a uni-soft filter of L if and only if it satisfies the following conditions: ( ) (∀x, y ∈ L) f˜L (x) ⊇ f˜L (y → x) , (3.10) ( ) (∀x, a, b ∈ L) f˜L (a) ∪ f˜L (b) ⊇ f˜L ((a → (b → x)) → x) . (3.11) Proof. Assume that f˜L is a uni-soft filter of L. Using (3.4), (2.1) and (3.3), we have f˜L (y → x) ⊆ f˜L (x) ∪ f˜L (x → (y → x)) = f˜L (x) ∪ f˜L (1) = f˜L (x) for all x, y ∈ L. Using (3.8) and (3.9), we get f˜L ((a → (b → x)) → x) ⊆ f˜L ((a → (b → x)) → (b → x)) ∪ f˜L (b) ⊆ f˜L (a) ∪ f˜L (b) for all a, b, x ∈ L. Conversely, let f˜L be a soft set over U satisfying two conditions (3.10) and (3.11). If we take y := x in (3.10), then f˜L (x) ⊇ f˜L (x → x) = f˜L (1) for all x ∈ L. Using (3.11) induces f˜L (y) = f˜L (1 → y) = f˜L (((x → y) → (x → y)) → y) ⊆ f˜L (x → y) ∪ f˜L (x) for all x, y ∈ L. Therefore f˜L is a uni-soft filter of L by Theorem 3.4. Theorem 3.10. A soft set f˜L over U is a uni-soft filter of L if and only if the nonempty τ -exclusive set of f˜L is a filter of L for all τ ∈ P(U ).
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Proof. Assume that f˜L is a uni-soft filter of L and let τ ∈ P(U ) be such that e(f˜L ; τ ) ̸= ∅. Let x, y ∈ L be such that x ∈ e(f˜L ; τ ) and x → y ∈ e(f˜L ; τ ). Then τ ⊇ f˜L (x) and τ ⊇ f˜L (x → y). It follows from (3.3) and (3.4) that f˜L (1) ⊆ f˜L (x) ⊆ τ and f˜L (y) ⊆ f˜L (x) ∪ f˜L (x → y) ⊆ τ . Hence 1 ∈ e(f˜L ; τ ) and y ∈ e(f˜L ; τ ), and therefore e(f˜L ; τ ) is a filter of L by Proposition 2.4. Conversely, suppose that e(f˜L ; τ ) is a filter of L for all τ ∈ P(U ) with e(f˜L ; τ ) ̸= ∅. For any x ∈ L, let f˜L (x) = δ. Then x ∈ e(f˜L ; δ) and e(f˜L ; δ) is a filter of L. Hence 1 ∈ e(f˜L ; δ) and so f˜L (x) = δ ⊇ f˜L (1). For any x, y ∈ L, let f˜L (x) = δx and f˜L (x → y) = δx→y . If we take δ = δx ∪ δx→y , then x ∈ e(f˜L ; δ) and x → y ∈ e(f˜L ; δ) which imply that y ∈ e(f˜L ; δ). Thus f˜L (x) ∪ f˜L (x → y) = δx ∪ δx→y = δ ⊇ f˜L (y). Therefore f˜L is a uni-soft filter of L by Theorem 3.4. Theorem 3.11. For a soft set f˜L over U , let f˜L∗ be a soft set over U which is given as follows: { f˜L (x) if x ∈ e(f˜L ; τ ), f˜L∗ : L → P(U ), x 7→ U otherwise, where τ ∈ P(U ) with τ ̸= U . If f˜L is a uni-soft filter of L, then so is f˜L∗ . Proof. Suppose that f˜L is a uni-soft filter of L. Then e(f˜L ; τ ) is a filter of L for all τ ∈ P(U ) with e(f˜L ; τ ) ̸= ∅ by Theorem 3.10. Thus 1 ∈ e(f˜L ; τ ), and so f˜L∗ (1) = f˜L (1) ⊆ f˜L (x) ⊆ f˜L∗ (x) for all x ∈ L. Let x, y ∈ L. If x ∈ e(f˜L ; τ ) and x → y ∈ e(f˜L ; τ ), then y ∈ e(f˜L ; τ ). Hence f˜L∗ (x) ∪ f˜L∗ (x → y) = f˜L (x) ∪ f˜L (x → y) ⊇ f˜L (y) = f˜L∗ (y). If x ∈ / e(f˜L ; τ ) or x → y ∈ / e(f˜L ; τ ), then f˜L∗ (x) = U or f˜L∗ (x → y) = U . Thus f˜L∗ (x) ∪ f˜L∗ (x → y) = U ⊇ f˜L∗ (y). Therefore f˜L∗ is a uni-soft filter of L. Theorem 3.12. If f˜L is a uni-soft filter of L, then the set La := {x ∈ L | f˜L (a) ⊇ f˜L (x)} is a filter of L for every a ∈ L. 7
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Proof. Since f˜L (1) ⊆ f˜L (a) for all a ∈ L, we have 1 ∈ La . Let x, y ∈ L be such that x ∈ La and x → y ∈ La . Then f˜L (x) ⊆ f˜L (a) and f˜L (x → y) ⊆ f˜L (a). Since f˜L is a uni-soft filter of L, it follows from (3.4) that f˜L (a) ⊇ f˜L (x) ∪ f˜L (x → y) ⊇ f˜L (y) so that y ∈ La . Hence La is a filter of L by Proposition 2.4. Theorem 3.13. Let a ∈ L and let f˜L be a soft set over U. Then (1) If La is a filter of L, then f˜L satisfies the following condition: (∀x, y ∈ L) (f˜L (a) ⊇ f˜L (x) ∪ f˜L (x → y) ⇒ f˜L (a) ⊇ f˜L (y)).
(3.12)
(2) If f˜L satisfies (3.3) and (3.12), then La is a filter of L. Proof. (1) Assume that La is a filter of L. Let x, y ∈ L be such that f˜L (a) ⊇ f˜L (x) ∪ f˜L (x → y). Then x → y ∈ La and x ∈ La . Using (2.8), we have y ∈ La and so f˜L (a) ⊇ f˜L (y). (2) Suppose that f˜L satisfies (3.3) and (3.12). Then 1 ∈ La by (3.3). Let x, y ∈ L be such that x ∈ La and x → y ∈ La . Then f˜L (a) ⊇ f˜L (x) and f˜L (a) ⊇ f˜L (x → y), which imply that f˜L (a) ⊇ f˜L (x) ∪ f˜L (x → y). Thus f˜L (a) ⊇ f˜L (y) by (3.12), and so y ∈ La . Therefore La is a filter of L by Proposition 2.4.
4
Uni-soft G-filters
Definition 4.1. A soft set f˜L over U is called a uni-soft G-filter of L if it is a uni-soft filter of L that satisfies: ( ) ˜ ˜ (∀x, y ∈ L) fL ((x ⊙ x) → y) ⊇ fL (x → y) . (4.1) Note that the condition (4.1) is equivalent to the following condition: ( ) (∀x, y ∈ L) f˜L (x → (x → y)) ⊇ f˜L (x → y) .
(4.2)
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Example 4.2. Let L := [0, 1] (unit interval). For any a, b ∈ L, define a ∨ b = max{a, b}, a ∧ b = min{a, b}, { 1 if a ≤ b, a→b= and a ⊙ b = min{a, b}. b otherwise, Then L := (L, ∨, ∧, ⊙, →, 0, 1) is a residuated lattice. Let f˜L be a soft set over U defined by { τ if x ∈ [ 21 , 1], f˜L : L → P(U ), x 7→ U otherwise, where τ ∈ P(U ) with τ ̸= U . Then f˜L is a uni-soft G-filter of L. Theorem 4.3. Let f˜L be a soft set over U . Then f˜L is a uni-soft G-filter of L if and only if it is a uni-soft filter of L that satisfies the following condition: ( ) ˜ ˜ ˜ (∀x, y, z ∈ L) fL (x → (y → z)) ∪ fL (x → y) ⊇ fL (x → z) . (4.3) Proof. Assume that f˜L is a uni-soft G-filter of L. Then f˜L is a uni-soft filter of L. Note that x ≤ 1 = (x → y) → (x → y), and thus x → y ≤ x → (x → y) for all x, y ∈ L. It follows from (3.1) that f˜L (x → y) ⊇ f˜L (x → (x → y)). Combining this and (4.2), we have f˜L (x → y) = f˜L (x → (x → y))
(4.4)
for all x, y ∈ L. Using (3.7) and (4.4), we have f˜L (x → (y → z)) ∪ f˜L (x → y) ⊇ f˜L (x → z) for all x, y, z ∈ L. Conversely, let f˜L be a uni-soft filter of L that satisfies the condition (4.3). If we put y = x and z = y in (4.3) and use (2.1) and (3.3), then f˜L (x → y) ⊆ f˜L (x → (x → y)) ∪ f˜L (x → x) = f˜L (x → (x → y)) ∪ f˜L (1) = f˜L (x → (x → y)) for all x, y ∈ L. Therefore f˜L is a uni-soft G-filter of L.
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Theorem 4.4. Let f˜L be a soft set over U that satisfies the condition (3.3) and ( ) (∀x, y, z ∈ L) f˜L (x) ∪ f˜L ((y → z) → (x → y)) ⊇ f˜L (y) .
(4.5)
Then f˜L is a uni-soft G-filter of L. Proof. If we take z := 1 in (4.5) and use (2.1), then f˜L (x) ∪ f˜L (x → y) = f˜L (x) ∪ f˜L (1 → (x → y)) = f˜L (x) ∪ f˜L ((y → 1) → (x → y)) ⊇ f˜L (y). Hence f˜L is a uni-soft filter of L by Theorem 3.4. Let x, y, z ∈ L. Since x → (y → z) ≤ (x → y) → (x → (x → z)) by (2.2) and (2.4), we have f˜L (x → (y → z)) ⊇ f˜L ((x → y) → (x → (x → z))) by (3.1). It follows from (3.1), (3.3), (3.4), (2.4) and (4.5) that f˜L (x → y) ∪ f˜L (x → (y → z)) ⊇ f˜L (x → y) ∪ f˜L ((x → y) → (x → (x → z))) ⊇ f˜L (x → (x → z)) ⊇ f˜L (((x → z) → z) → (x → z)) = f˜L (((x → z) → z) → (1 → (x → z))) ⊇ f˜L (x → z). Therefore f˜L is a uni-soft G-filter of L by Theorem 4.3. The following example shows that any uni-soft G-filter may not satisfy the condition (4.5). Example 4.5. The uni-soft G-filter f˜L of L in Example 4.2 does not satisfy the condition (4.5) since f˜L ( 32 ) ∪ f˜L (( 31 → 14 ) → ( 32 → 31 )) = f˜L ( 23 ) ∪ f˜L (1) = τ ⊉ U = f˜L ( 13 ). Proposition 4.6. For a uni-soft filter f˜L of L, the condition (4.5) is equivalent to the following condition. ( ) (∀x, y ∈ L) f˜L ((x → y) → x) ⊇ f˜L (x) . (4.6) 10
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Proof. Assume that the condition (4.5) is valid. It follows from (3.3) and (2.1) that f˜L ((x → y) → x) = f˜L (1) ∪ f˜L ((x → y) → x) = f˜L (1) ∪ f˜L ((x → y) → (1 → x)) ⊇ f˜L (x) for all x, y ∈ L. Conversely, suppose that the condition (4.6) is valid. It follows from (2.2) and (3.4) that f˜L (x) ∪ f˜L ((y → z) → (x → y)) = f˜L (x) ∪ f˜L (x → ((y → z) → y)) ⊇ f˜L ((y → z) → y) ⊇ f˜L (y) for all x, y ∈ L. Combining Theorem 4.4 and Proposition 4.6, we have the following result. Theorem 4.7. Every uni-soft filter satisfying the condition (4.6) is a uni-soft G-filter. Proposition 4.8. Every uni-soft filter f˜L of L with the condition (4.5) satisfies the following condition. ( ) (∀x, y ∈ L) f˜L ((x → y) → y) ⊇ f˜L ((y → x) → x) . (4.7) Proof. Let f˜L be a uni-soft filter of L that satisfies the condition (4.5) and let x, y ∈ L. Since x → ((y → x) → x) = (y → x) → (x → x) = (y → x) → 1 = 1, that is, x ≤ (y → x) → x, we have ((y → x) → x) → y ≤ x → y by (2.3). It follows from (2.4), (2.2) and (2.3) that (x → y) → y ≤ (y → x) → ((x → y) → x) = (x → y) → ((y → x) → x) ≤ (((y → x) → x) → y) → ((y → x) → x). Using (3.1), (3.3), (2.1), (2.2) and (4.5), we have f˜L ((x → y) → y) ⊇ f˜L ((((y → x) → x) → y) → ((y → x) → x)) = f˜L (1) ∪ f˜L (1 → ((((y → x) → x) → y) → ((y → x) → x))) = f˜L (1) ∪ f˜L ((((y → x) → x) → y) → (1 → ((y → x) → x))) ⊇ f˜L ((y → x) → x). Hence the condition (4.7) is valid. 11
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Corollary 4.9. Every uni-soft filter f˜L of L with the condition (4.6) satisfies the condition (4.7). Proposition 4.10. Every uni-soft G-filter f˜L of L with the condition (4.7) satisfies the condition (4.5). Proof. Let f˜L be a uni-soft G-filter of L that satisfies the condition (4.7). For any x, y, z ∈ L, we have f˜L (z) ∪ f˜L ((x → y) → (z → x)) = f˜L (z) ∪ f˜L (z → ((x → y) → x)) ⊇ f˜L ((x → y) → x) ⊇ f˜L ((x → y) → ((x → y) → y)) ⊇ f˜L ((x → y) → y) ⊇ f˜L ((y → x) → x) by (2.2), (3.4), (3.1), (2.4), (4.2) and (4.7). Since (x → y) → x ≤ y → x ≤ z → (y → x), it follows from (3.1) that f˜L ((x → y) → x) ⊇ f˜L (z → (y → x)) and so from (3.4) that f˜L (z) ∪ f˜L ((x → y) → (z → x)) ⊇ f˜L (z) ∪ f˜L ((x → y) → x) ⊇ f˜L (z) ∪ f˜L (z → (y → x)) ⊇ f˜L (y → x). Therefore f˜L (z) ∪ f˜L ((x → y) → (z → x)) ⊇ f˜L (y → x) ∪ f˜L ((y → x) → x) ⊇ f˜L (x). Hence the condition (4.5) is valid. Theorem 4.11. Let f˜L be a uni-soft filter of L. Then f˜L is a uni-soft G-filter of L if and only if the following condition holds: ( ) (∀x ∈ L) f˜L (x → (x ⊙ x)) = f˜L (1) . (4.8) Proof. Suppose that f˜L is a uni-soft G-filter of L. Since x → (x → (x ⊙ x)) = 1 for all x ∈ L, we have f˜L (x → (x → (x ⊙ x))) = f˜L (1). It follows from (4.3) and (2.1) that f˜L (x → (x ⊙ x)) ⊆ f˜L (x → (x → (x ⊙ x))) ∪ f˜L (x → x) = f˜L (1) and so from (3.3) that f˜L (x → (x ⊙ x)) = f˜L (1) for all x ∈ L. 12
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Conversely, let f˜L be a uni-soft filter of L which satisfies the condition (4.8) and let x, y ∈ L. Since x → (x → y) = (x ⊙ x) → y ≤ (x → (x ⊙ x)) → (x → y) by (2.2) and (2.4), it follows from (3.1) that f˜L (x → (x → y)) ⊇ f˜L ((x → (x ⊙ x)) → (x → y)). Hence, we have f˜L (x → y) ⊆ f˜L ((x → (x ⊙ x)) → (x → y)) ∪ f˜L (x → (x ⊙ x)) ⊆ f˜L (x → (x → y)) ∪ f˜L (x → (x ⊙ x)) = f˜L (x → (x → y)) ∪ f˜L (1) = f˜L (x → (x → y)) by using (3.4), (4.8) and (3.3). Hence f˜L is a uni-soft G-filter of L. Theorem 4.12. A soft set f˜L over U is a uni-soft G-filter of L if and only if it is a uni-soft filter of L with an additional condition: ( ) ˜ ˜ (∀x, y ∈ L) fL (x → y) = fL (x → (x → y)) . (4.9) Proof. Suppose that f˜L is a uni-soft G-filter of L. Then f˜L is a uni-soft filter of L. Let x, y ∈ L. Since x → y ≤ x → (x → y), we have f˜L (x → y) ⊇ f˜L (x → (x → y)) by (3.1). Hence f˜L (x → y) = f˜L (x → (x → y)) by using (4.2). Conversely, let f˜L be a uni-soft filter of L with the condition (4.9). It follows from Proposition 3.6 that f˜L (x → (y → z)) ∪ f˜L (x → y) ⊇ f˜L (x → (x → z)) = f˜L (x → z) for all x, y, z ∈ L. Therefore f˜L is a uni-soft G-filter of L by Theorem 4.3. Proposition 4.13. Every uni-soft G-filter f˜L of L satisfies the following conditions: ( ) ˜ ˜ (∀x, y, z ∈ L) fL (x → (y → z)) ⊇ fL ((x → y) → (x → z)) . (4.10) ( ) (∀x, y, z ∈ L) f˜L (x → (y → z)) = f˜L ((x → y) → (x → z)) . (4.11)
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Proof. Let f˜L be a uni-soft G-filter of L. Using (2.2), (4.3), (2.4) and (3.3), we have f˜L ((x → y) → (x → z)) = f˜L (x → ((x → y) → z)) ⊆ f˜L (x → (y → z)) ∪ f˜L (x → ((y → z) → ((x → y) → z))) = f˜L (x → (y → z)) ∪ f˜L ((y → z) → ((x → y) → (x → z))) = f˜L (x → (y → z)) ∪ f˜L (1) = f˜L (x → (y → z)) for all x, y, z ∈ L. Thus (4.10) holds. Since (x → y) → (x → z) ≤ x → (y → z) for all x, y, z ∈ L, it follows from (3.1) that f˜L ((x → y) → (x → z)) ⊇ f˜L (x → (y → z)) and so that f˜L (x → (y → z)) = f˜L ((x → y) → (x → z)) for all x, y, z ∈ L by using (4.10). Proposition 4.14. Assume that L satisfies the divisibility, that is, x ∧ y = x ⊙ (x → y) for all x, y ∈ L. If f˜L is a uni-soft G-filter of L satisfying (4.11), then the following equality is true. ( ) (∀x, y, z ∈ L) f˜L ((x ⊙ y) → z) = f˜L ((x ∧ y) → z) . (4.12) Proof. Using the divisibility and (2.2), we have (x ∧ y) → z = (x ⊙ (x → y)) → z = (x → y) → (x → z) for all x, y, z ∈ L. It follows from (2.2) and (4.11) that f˜L ((x ⊙ y) → z) = f˜L (x → (y → z)) = f˜L ((x → y) → (x → z)) = f˜L ((x ∧ y) → z) for all x, y, z ∈ L. Theorem 4.15. Let L satisfy the divisibility, that is, x ∧ y = x ⊙ (x → y) for all x, y ∈ L. Then every uni-soft filter f˜L of L satisfying the condition (4.12) is a uni-soft G-filter of L. Proof. Using Proposition 3.6, (2.2) and (4.12), we have f˜L (x → (y → z)) ∪ f˜L (x → y) ⊇ f˜L (x → (x → z)) = f˜L ((x ⊙ x) → z) = f˜L ((x ∧ x) → z) = f˜L (x → z) for all x, y, z ∈ L. Therefore f˜L is a uni-soft G-filter of L by Theorem 4.3. 14
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Theorem 4.16. Let f˜L and g˜L be uni-soft filters of L such that f˜L (1) = g˜(1) and f˜L ⊇ g˜L , i.e., f˜L (x) ⊇ g˜L (x) for all x ∈ L. If f˜L is a uni-soft G-filter of L, then so is g˜L . Proof. Assume that f˜L is a uni-soft G-filter of L. Using (2.2) and (2.1), we have x → (x → ((x → (x → y)) → y)) = (x → (x → y)) → (x → (x → y)) = 1 for all x, y ∈ L. Thus g˜(x → ((x → (x → y)) → y)) ⊆ f˜L (x → ((x → (x → y)) → y)) = f˜L (x → (x → ((x → (x → y)) → y))) = f˜L (1) = g˜(1) by hypotheses and (4.4), and so g˜(x → ((x → (x → y)) → y)) = g˜(1) for all x, y ∈ L by (3.3). Since g˜L is a uni-soft filter of L, it follows from (3.4), (2.2) and (3.3) that g˜(x → y) ⊆ g˜(x → (x → y)) ∪ g˜((x → (x → y)) → (x → y)) = g˜(x → (x → y)) ∪ g˜(x → ((x → (x → y)) → y)) = g˜(x → (x → y)) ∪ g˜(1) = g˜(x → (x → y)) for all x, y ∈ L. Therefore g˜L is a uni-soft G-filter of L.
References [1] R. Belohlavek, Some properties of residuated lattices, Czechoslovak Math. J. 53(123) (2003) 161–171. [2] K. Blount and C. Tsinakis, The structure of residuated lattices, Internat. J. Algebra Comput. 13(4) (2003) 437–461. [3] R. A. Borzooei, S. Khosravi Shoar and R. Americ, Some types of filters in MTLalgebras, Fuzzy Sets and Systems 187 (2012) 92–102. [4] N. C ¸ aˇgman and S. Engino˘glu, Soft set theory and uni-int decision making, Eur. J. Oper. Res. 207 (2010) 848–855. 15
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[5] F. Esteva and L. Godo, Monoidal t-norm based logic: towards a logic for leftcontinuous t-norms, Fuzzy Sets and Systems 124 (2001) 271–288. [6] P. H´ajek, Metamathematics of Fuzzy Logic, Kluwer Academic Press, Dordrecht, 1998. [7] Y. B. Jun, Union soft sets with applications in BCK/BCI-algebras, Bull. Korean Math. Soc. 50 (2013) 1937–1956. [8] Y. B. Jun, S. Z. Song and S. S. Ahn, Union soft sets applied to commutative BCIideals, J. Comput. Anal. Appl. 16 (2014) 468–477. [9] K. H. Kim, Q. Zhang and Y. B. Jun, On fuzzy filters of MTL-algebras, J. Fuzzy Math. 10 (2002), no. 4, 981–989. [10] D. Molodtsov, Soft set theory - First results, Comput. Math. Appl. 37 (1999) 19–31. [11] J. G. Shen and X. H. Zhang, Filters of residuated lattices, Chin. Quart. J. Math. 21 (2006) 443–447. [12] E. Turunen, BL-algebras of basic fuzzy logic, Mathware & Soft Computing 6 (1999), 49–61. [13] E. Turunen, Boolean deductive systems of BL-algebras, Arch. Math. Logic 40 (2001) 467–473. [14] X. H. Zhang, On filters in MTL-algebras, Adv. Syst. Sci. Appl. 7 (2007) 32–38. [15] X. H. Zhang and W. H. Li, On fuzzy logic algebraic system MTL, Adv. Syst. Sci. Appl. 5 (2005) 475–483. [16] Y. Q. Zhu and Y. Xu, On filter theory of residuated lattices, Inform. Sci. 180 (2010) 3614–3632.
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Mathematical analysis of a general viral infection model with immune response N. H. AlShamrani, A. M. Elaiw and M. A. Alghamdi Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia. Emails: [email protected]. (N. AlShamrani), [email protected] (A. Elaiw).
Abstract In this paper, we study the global dynamics of a viral infection model with antibody immune response. The incidence rate is given by a general function of the population of the uninfected target cells, infected cells and free viruses. We have established a set of conditions on the general incidence rate function and determined two threshold parameters R0 (the basic infection reproduction number) and R1 (the antibody immune response activation number) which are su¢ cient to determine the global behavior of the model. The global asymptotic stability of the equilibria of the model has been proven by using direct Lyapunov method and applying LaSalle’s invariance principle.
Keywords: Virus dynamics; global stability; antibody immune response; Lyapunov functional. Mathematics Subject Classi…cation: 34D20; 34D23; 37N25; 92D30
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1
Introduction
Several works have been devoted to propose mathematical models of viral infectious dynamics such as human immunode…ciency virus (HIV) (see, for example, [1]-[22]), hepatitis B virus (HBV) [23][26], hepatitis C virus (HCV) [27]-[29] and human T cell leukemia HTLV [30], etc. Mathematical models of viral infection can help for understanding the viral dynamics and developing antiviral drug therapies. In reality, the immune response needs an indispensable components to do its job such as antibodies, cytokines, natural killer cells, and T cells. The antibody immune response is a part of the adaptive system in which the body responds to pathogens by primarily using antibodies that produced from the B cells. While the other part is the Cytotoxic T Lymphocytes (CTL) immune response where the CTL attacks and kills the infected cells [7]. In some infections such as malaria, the CTL immune response is less e¤ective than the antibody immune response [31]. Mathematical models of viral infection with antibody immune response have been proposed and analyzed in ([32]-[39]). The basic model of viral infection with antibody immune response has introduced by Murase et. al. [32] and Shi… Wang [39] as:
x_ = s
dx
y_ = vx v_ = ky z_ = rzv
vx;
(1)
ay; bzv
(2) cv;
(3)
z;
(4)
where x, y, v and z denote the populations of uninfected target cells, infected cells, free virus particles and antibody immune cells at time t, respectively. Parameters s, k and r represent, respectively, the rate at which new healthy cells are generated from the source within the body, the generation rate constant of free viruses produced from the infected cells and the proliferation
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rate constant of antibody immune cells. Parameters d, a, c and
are the natural death rate
constants of the uninfected target cells, infected cells, free virus particles and antibody immune cells, respectively. Parameter
is the infection rate constant and b is the removal rate constant of
the viruses due to the antibodies. All the parameters given in model (1)-(4) are positive. Note that, the infection rate in model (1)-(4) is presented to be bilinear in x and v, which can not be completely describe the interaction between the uninfected target cells and viruses. Nevertheless, there are many types of an improved incidence rate which are more commonly used due to their bene…t for helping us gain the uni…cation theory through passing over the unessential details (see e.g. [40] and [41]). Variety of viral infection models with antibody immune response have been considered di¤erent forms of the incidence rate such as saturated incidence rate, where
0 [42], [37], [35], Beddington-DeAngelis functional response,
general form,
xv 1+ x+ v ,
;
xv 1+ v
0 [36], and
(x; v)v [38].
However the infection rate does not depend on the infected cells y. In some viral infections such as HBV, the infection rate depends on x, y and v [25], [24]. In [43], the infection rate is given by (x; y; v)v, however the antibody immune response has been neglected. Our aim in this paper is to investigate the global stability analysis of the viral infection model with general incidence rate function and antibody immune response. The rest of the paper is designed as follows. In the next section, we introduce the model and discuss the non-negativity and boundedness of the solutions. In Section 3, we de…ne two threshold parameters and discuss the existence of the model’s equilibria. In Section 4, we study the global asymptotic stability of the equilibria using suitable Lyapunov functional and applying LaSalle’s invariance principle. Finally, conclusion is given in Section 5.
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2
The mathematical model
In this section, we consider the following viral infection model with general incidence rate taking into consideration the antibody immune response. x_ = s
dx
(x; y; v)v;
y_ = (x; y; v)v v_ = ky
bzv
z_ = rzv
(5)
ay;
(6)
cv;
(7)
z:
(8)
The de…nitions of all variables and parameters are identical to those given in Section 1. The incidence rate of infection is presented by a general function in the form
(x; y; v)v, where
is
continuously di¤erentiable and satis…es the following assumptions (see [38] and [43]): Assumption A1. v
(x; y; v) > 0 for all x > 0, y
0, v
0, and
(0; y; v) = 0 for all y
0,
0. @ (x; y; v) > 0 for all x > 0, y 0 and v 0: @x @ (x; y; v) @ (x; y; v) < 0, < 0 for all x > 0, y > 0 and v > 0: Assumption A3. @y @v @ ( (x; y; v)v) Assumption A4. > 0 for all x > 0, y > 0 and v > 0: @v Assumption A2.
2.1
Positive invariance
In the following proposition, we show that the non-negative orthant R4 0 is the positively invariant and there exists a compact set which is positively invariant for model (5)-(8). Proposition 1. Assume that Assumption A1 is satis…ed. Then there exist positive numbers Li , i = 1; 2; 3, such that the compact set = (x; y; v; z) 2 R4 0 : 0
x; y
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L1 ; 0
v
L2 ; 0
z
L3
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is positively invariant. Proof. First, we prove that the orthant R4 0 is positively invariance for system (5)-(8). We have
x_ jx=0 = s > 0; y_ jy=0 = (x; 0; v)v v_ jv=0 = ky
0 for all x > 0; v
0 for all y
0;
0;
z_ jz=0 = 0: Hence, all the solutions are nonnegative. Next we show that the solutions of system are bounded. Let T1 (t) = x(t) + y(t), then T_1 (t) = (s =s
where
1
x(t); y(t)
= minfd; ag. Hence 0 L1 for all t
dx dx
(x; y; v)v) + (x; y; v)v ay
T1 (t)
0 if x(0) + y(0)
s s 1
1 (x
+ y) = s
for all t
L1 , where L1 =
ay; 1 T1 (t);
0 if T1 (0) s 1
s 1
. It follows that, 0
. Moreover, let T2 (t) = v(t) + rb z(t),
then T_2 (t) = ky where 0
v(t)
2
cv
= minfc; g. Hence 0 L2 and 0
z(t)
b z r T2 (t)
L3 for all t
kL1
2 (v
b + z) = kL1 r
L2 for all t 0 if v(0) + rb z(0)
2 T2 (t);
0 when T2 (0)
L2 . It follows that
L2 , where L2 =
kL1 2
and L3 = rb L2 .
Therefore, x(t); y(t); v(t) and z(t) are all bounded.
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2.2
The equilibria and threshold parameters
At any equilibrium we have
s
dx
(x; y; v)v = 0;
(9)
(x; y; v)v
ay = 0;
(10)
ky
bzv
cv = 0;
(11)
rzv
z = 0:
(12)
From Eq. (12), either z = 0 or z 6= 0. If z = 0, then from Eqs. (9)-(11) we get s
y=
dx
=
a
c v; k
v=
k(s
dx) ac
:
(13)
Substituting from Eq. (13) into Eq. (10) we get:
x;
s a
dx k(s ;
dx) ac
ac v = 0: k
(14)
Eq. (14) has two possible solutions v = 0 or v 6= 0. If v = 0; then from Eqs. (9) and (10), we get x = s=d and y = 0 which leads to the infection-free equilibrium E0 (x0 ; 0; 0; 0) where x0 = s=d. If v 6= 0; then we have x;
s a
dx k(s ;
dx)
ac = 0: k
ac
Let 1 (x)
=
x;
s a
dx k(s ;
dx) ac
ac = 0: k
Then, we have 0 1 (x)
=
@ @x
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d@ a @y
kd @ : ac @v
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Because of Assumptions A2 and A3, we have
0 (x) 1
> 0 which implies that function
1 (x)
is
strictly increasing w.r.t. x. Moreover, 1 (0)
=
1 (x0 )
=
ac = k
s ks 0; ; a ac
ac < 0; k k (x0 ; 0; 0) ac
ac ac = k k
(x0 ; 0; 0)
1 :
k (x0 ; 0; 0) > 1; then there exists a unique x1 2 (0; x0 ) such that 1 (x1 ) = 0. ac d(x0 x1 ) kd(x0 x1 ) Therefore from Eq. (13) we obtain y1 = > 0 and v1 = > 0. It follows a ac k (x0 ; 0; 0) > 1, then there exists a chronic-infection equilibrium without antibody immune that, if ac Therefore, if
response E1 (x1 ; y1 ; v1 ; 0). Let us de…ne the basic reproduction number as: R0 =
k (x0 ; 0; 0) : ac
The parameter R0 determines whether a chronic-infection can be established. The other possibility of Eq. (12) is z 6= 0 which leads to v2 = 2 (x)
=s
2 (x0 )
=
. From Eq. (9) we let
dx
Assumptions A2 and A3 provide that and
r
2
x;
k (x2 ; y2 ; v2 ) ac
dx a
; v2 v2 = 0:
is a decreasing function of x. Clearly,
2 (0)
(x0 ; 0; v2 )v2 < 0. Thus, there exists a unique x2 2 (0; x0 ) such that
It follows from Eqs. (11) and (13) that, y2 = c b
s
d(x0
x2 )
> 0 and z2 =
= s > 0
2 (x2 )
k (x2 ; y2 ; v2 ) ab
= 0. c = b
a k (x2 ; y2 ; v2 ) 1 . Then if > 1 then z2 > 0. Now we De…ne the antibody ac
immune response activation number as: R1 =
k (x2 ; y2 ; v2 ) ; ac
which determines whether a persistent antibody immune response can be established. Hence, z2 c can be rewritten as z2 = (R1 b
1). It follows that, there is a chronic-infection equilibrium with
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antibody immune response E2 (x2 ; y2 ; v2 ; z2 ) i¤ R1 > 1. Clearly from Assumptions A2 and A3, we have
R1 =
2.3
k (x2 ; y2 ; v2 ) k (x0 ; y2 ; v2 ) k (x0 ; 0; 0) < < = R0 : ac ac ac
Global stability analysis
In this section, the global asymptotic stability of the three equilibria of model (5)-(8) will be established by using direct Lyapunov method and applying LaSalle’s invariance principle. Let us de…ne the function H : (0; 1) ! [0; 1) as H(w) = w
1
ln w:
Theorem 1. Let Assumptions A1-A3 be hold true and R0
1; then the infection-free equilibrium
E0 is globally asymptotically stable (GAS). Proof. We construct a Lyapunov functional as:
U0 = x
x0
Z
x
x0
We calculate
dU0 dt
(x0 ; 0; 0) a ab d + y + v + z: ( ; 0; 0) k rk
(15)
along the solutions of model (5)-(8) as:
dU0 =d 1 dt
(x0 ; 0; 0) (x; 0; 0)
(x0
=s 1
(x0 ; 0; 0) (x; 0; 0)
1
x) + x x0
(x; y; v) +
ac k
(x0 ; 0; 0) (x; 0; 0)
(x; y; v) R0 (x; 0; 0)
ac k 1 v
v
ab z rk
ab z: rk
(16)
From Assumptions A2 and A3 we know that (x; y; v) is an increasing function of x and decreasing function of y and v. Then the …rst term of Eq. (16) is less than or equal zero and
(x; y; v) < (x; 0; 0),
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x; y; v > 0:
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It follows that dU0 dt Therefore, if R0
1, then
(5)-(8) converge to
(x0 ; 0; 0) (x; 0; 0)
s 1
x x0
1
+
ac (R0 k
ab z: rk
1) v
(17)
dU0 dt
0 for all x; y; v; z > 0. We note that the solutions of system n o 0 0 , the largest invariant subset of dU = 0 [44]. From (17), we have dU dt dt = 0
i¤ x = x0 , v = 0 and z = 0. The set
is invariant and for any element belong to
satis…es v = 0
and z = 0. We can see from Eq. (7) that v_ = 0 = ky: dU0 dt
It follows that, y = 0. Hence
= 0 i¤ x = x0 and y = v = z = 0. Using LaSalle’s invariance
principle, we derive that E0 is GAS. Assumption A5 (x; yi ; vi ) (x; y; v)
(x; y; v) (x; yi ; vi )
1
v vi
0; i = 1; 2 for all x; y; v > 0:
Theorem 2. Assume that Assumptions A1-A5 are satis…ed and R1
1 < R0 , then the chronic-
infection equilibrium without antibody immune response E1 is GAS. Proof. De…ne a Lyapunov functional as: U1 = x
x1
Z
x
x1
(x1 ; y1 ; v1 ) d + y1 H ( ; y1 ; v1 )
y y1
a + v1 H k
v v1
+
ab z: rk
Calculating the time derivative of U1 along the trajectories of system (5)-(8), we obtain dU1 = dt
1
+
a 1 k
=
1
(x1 ; y1 ; v1 ) (x; y1 ; v1 )
(s
dx
(x; y; v) v) + 1
y1 y
( (x; y; v)v
ay)
v1 ab z) (ky bzv cv) + (rzv v rk (x1 ; y1 ; v1 ) (x; y; v)v (s dx) + (x1 ; y1 ; v1 ) (x; y1 ; v1 ) (x; y1 ; v1 )
y1 (x; y; v)v + ay1 y
ac v k
ay
v1 ac ab + v1 + v1 z v k k
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ab z: rk
(18)
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Using the equilibrium conditions for E1 : s = dx1 + ay1 ;
(x1 ; y1 ; v1 )v1 = ay1 =
ac v1 ; k
we obtain dU1 =d 1 dt ay1
(x1 ; y1 ; v1 ) (x; y1 ; v1 )
(x1
y1 (x; y; v)v y (x1 ; y1 ; v1 )v1
ay1
x) + 3ay1 v v1
ay1
ay1
(x1 ; y1 ; v1 ) (x; y; v)v + ay1 (x; y1 ; v1 ) (x; y1 ; v1 )v1
v1 y ab + v1 vy1 k
r
z:
(19)
Collecting terms of Eq. (19) we get (x1 ; y1 ; v1 ) (x; y1 ; v1 )
dU1 = dx1 1 dt + ay1
v v1
(x; y; v)v (x; y1 ; v1 )v1 (x1 ; y1 ; v1 ) (x; y1 ; v1 )
+ ay1 4 +
1
ab v1 k
r
x x1 1+
(x; y1 ; v1 ) (x; y; v)
y1 (x; y; v)v y (x1 ; y1 ; v1 )v1
v1 y vy1
(x; y1 ; v1 ) (x; y; v)
z:
(20)
Eq. (20) can be simpli…ed as: dU1 = dx1 1 dt
(x1 ; y1 ; v1 ) (x; y1 ; v1 )
1
+ ay1 1
(x; y; v) (x; y1 ; v1 )
(x; y1 ; v1 ) (x; y; v)
+ ay1 4
(x1 ; y1 ; v1 ) (x; y1 ; v1 )
+
ab v1 k
r
x x1 v v1
y1 (x; y; v)v y (x1 ; y1 ; v1 )v1
v1 y vy1
(x; y1 ; v1 ) (x; y; v)
z:
(21)
From Assumptions A1 and A5, we get that the …rst and second terms of Eq. (21) is less than or equal zero. Since the geometrical mean is less than or equal to the arithmetical mean, then the third term of Eq. (21) is also less than or equal zero. Now we show that if R1 sgn(x2
1 then v1 x1 ) = sgn(v1
r
= v2 . Let R0 > 1, then we want to show that v2 ) = sgn(y1
344
y2 ) = sgn(R1
1):
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From Assumptions A2-A4, for x1 ; x2 ; y1 ; y2 ; v1 ; v2 > 0, we have ( (x2 ; y2 ; v2 )
(x1 ; y2 ; v2 ))(x2
x1 ) > 0;
(22)
( (x1 ; y1 ; v1 )
(x1 ; y2 ; v1 ))(y2
y1 ) > 0
(23)
( (x1 ; y1 ; v1 )
(x1 ; y1 ; v2 ))(v2
v1 ) > 0;
(24)
( (x2 ; y2 ; v2 )v2
(x2 ; y2 ; v1 )v1 )(v2
v1 ) > 0:
(25)
First, we claim sgn(x2 x1 ) = sgn(v1 v2 ). Suppose this is not true, i.e., sgn(x2 x1 ) = sgn(v2 v1 ). Using the conditions of the equilibria E1 and E2 we have (s
dx2 )
(s
dx1 ) = (x2 ; y2 ; v2 )v2 = a(y2
then sgn(x1 (s
x2 ) = sgn(y2
dx2 )
(s
(x1 ; y1 ; v1 )v1
y1 );
(26)
y1 ). Moreover
dx1 ) = (x2 ; y2 ; v2 )v2
(x1 ; y1 ; v1 )v1
= ( (x2 ; y2 ; v2 )v2
(x2 ; y2 ; v1 )v1 ) + ( (x2 ; y2 ; v1 )v1
+ ( (x1 ; y2 ; v1 )v1
(x1 ; y1 ; v1 )v1 ):
(x1 ; y2 ; v1 )v1 )
Therefore, from inequalities (22)-(26) we get: sgn (x1 which leads to contradiction. Thus, sgn (x2 for E1 we have
k (x1 ;y1 ;v1 ) ac
R1
x2 ) = sgn (x2 x1 ) = sgn (v1
x1 ) ; v2 ) : Using the equilibrium conditions
= 1, then k (x2 ; y2 ; v2 ) k (x1 ; y1 ; v1 ) ac ac k = ( (x2 ; y2 ; v2 ) (x2 ; y2 ; v1 ) + (x2 ; y2 ; v1 ) ac
1=
(x1 ; y2 ; v1 ) + (x1 ; y2 ; v1 )
345
(x1 ; y1 ; v1 )):
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We get sgn(R1 v1
v2 =
r.
1) = sgn(v1
v2 ): Hence, if R0 > 1; then x1 ; y1 ; v1 > 0, and if R1
It follows from the above discussion that
i¤ x = x1 ; y = y1 ; v = v1 and z = 0. So
dU1 dt
1, then
0 for all x; y; v; z > 0 and
dU1 dt
=0
contains a unique point, the equilibrium E1 . Thus, we
prove the global asymptotic stability of the chronic-infection equilibrium without antibody immune response E1 by using LaSalle’s invariance principle. Theorem 3. Let Assumptions A1-A5 be hold true and R1 > 1, then the chronic-infection equilibrium with antibody immune response E2 is GAS. Proof. We construct a Lyapunov functional as follows: U2 = x
x2
Z
x
x2
(x2 ; y2 ; v2 ) d + y2 H ( ; y2 ; v2 )
y y2
a + v2 H k
v v2
+
ab z2 H rk
y2 y
( (x; y; v)v
z z2
:
(27)
Function U2 satis…es: dU2 = dt +
(x2 ; y2 ; v2 ) (x; y2 ; v2 )
1 a 1 k
v2 (ky v
(s
bzv
dx
(x; y; v)v) + 1
cv) +
ab 1 rk
z2 (rzv z
z):
ay) (28)
Applying s = dx2 + ay2 , we get dU2 =d 1 dt ay2
(x2 ; y2 ; v2 ) (x; y2 ; v2 )
x) + ay2
(x2 ; y2 ; v2 ) (x2 ; y2 ; v2 ) + (x; y; v)v (x; y2 ; v2 ) (x; y2 ; v2 )
(x2 ; y2 ; v2 )v2 +
(x2
y2 (x; y; v)v + ay2 y (x2 ; y2 ; v2 )v2
ac ab v2 + v2 z k k
ab z rk
ac v k
ay
v2 v
ab ab z2 v + z2 : k rk
(29)
By using the equilibrium conditions of E2 (x2 ; y2 ; v2 )v2 = ay2 ;
cv2 = ky2
bv2 z2 ;
= rv2 ;
and the following equality cv = cv2
v v = (ky2 v2 v2
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we obtain dU2 =d 1 dt
(x2 ; y2 ; v2 ) (x; y2 ; v2 )
(x2
(x; y; v)v (x; y2 ; v2 )v2
x) + ay2
(x2 ; y2 ; v2 ) (x; y2 ; v2 )
y2 (x; y; v)v y (x2 ; y2 ; v2 )v2
(x2 ; y2 ; v2 ) (x; y2 ; v2 )
1
+ ay2 4
v2 y vy2
v v2
1+
(x; y2 ; v2 ) (x; y; v)
(x; y2 ; v2 ) : (x; y; v)
(30)
We can simplify (30) as: dU2 = dx2 1 dt
(x2 ; y2 ; v2 ) (x; y2 ; v2 )
+ ay2 4
x x2
+ ay2 1 v2 y vy2
y2 (x; y; v)v y (x2 ; y2 ; v2 )v2
(x; y; v) (x; y2 ; v2 ) (x; y2 ; v2 ) : (x; y; v)
(x; y2 ; v2 ) (x; y; v)
v v2 (31)
We note that from assumptions A2, A5 and the relationship between the arithmetical and geometrical means, we have
dU2 dt
0. One can easily see that
dU2 dt
= 0 at E2 . The global asymptotic stability
of the chronic-infection equilibrium with antibody immune response E2 follows from LaSalle’s invariance principle.
3
Conclusion
In this paper, we have proposed a viral infection model with general incidence rate function and antibody immune response. We have derived a set of conditions on the general functional response and have determined two thresholds parameters R0 and R1 to prove the existence and global stability of the model’s equilibria. The global asymptotic stability of the three equilibria, infection-free, chronic-infection without antibody immune response and chronic-infection with antibody immune response has been proven by using direct Lyapunov method and LaSalle’s invariance principle.
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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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NEWTON’S METHOD FOR COMPUTING THE FIFTH ROOTS OF p-ADIC NUMBERS Y.H. KIM, H.M. KIM, AND J. CHOI
Abstract We consider Newton’s method to compute the fifth root of a p-adic number in Qp . We have the sufficient conditions for the convergence of Newton’s method and the speed of its convergence. We also calculate the number of iterations to obtain a number of corrected digits in the approximation.
1. Introduction Let p be a prime and Qp be the field of p-adic numbers. The theory of the field of p-adic numbers introduced by Hensel has been related to several areas of mathematics including number theory, analysis and other modern mathematics, and recently to physics. The study of this field has been an important area of research in mathematics([9]). The application of classical methods in numerical analysis to padic numbers and polynomials and the analysis of their convergence in Qp have been a recent development([2-3], [5], [7], [10-11]). Newton’s method is the most often used method to find zeros of polynomials. In [7], the authors applied Newton’s method to compute the cubic root of a p-adic number. In [2-3], the authors also used Newton-Raphson method to compute square and cube roots of p-adic numbers in Qp . Computing the q-th root of a p-adic number is useful in the field of computer science and cryptography, specially when q is a prime. In [6], Kim-Choi give the conditions for the existence of the q-th roots of p-adic numbers in Qp when (p, q) = 1, and also have the condition for the existence the fifth roots including p = q. In this paper, we use Newton’s method to compute the fifth root of a p-adic number in Qp . We have the sufficient conditions for the convergence of Newton’s method and the speed of its convergence. We also calculate the number of iterations to obtain a number of corrected digits in the approximation. 2010 Mathematics Subject Classification: 11E95, 26E30, 65H04 Key words and phrases: Newton’s method, p-adic roots Correspondence should be addressed to Jongsung Choi, [email protected]. The present research has been conducted by the Research Grant of Kwangwoon University in 2014. 1
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2. Preliminaries The following definitions and results are needed for our discussion. See [4] and [8] for details. Definition 1. Let p ∈ N be a prime number and x ∈ Q (x 6= 0). The p-adic order of x, ordp x, is defined by the highest power of p which divides x, if x ∈ Z, ordp x = ordp a − ordp b, if x = ab , a, b ∈ Z, b 6= 0. Consider a map | · |p : Q → R+ as follows. Definition 2. Let p ∈ N be a prime number and x ∈ Q. The p-adic norm | · |p of x is defined by −ord x p p , if x 6= 0, |x|p = 0, if x = 0. The field of p-adic numbers Qp is the completion of Q with respect to the p-adic norm |·|p of Definition 2. The elements of Qp are equivalence classes of Cauchy sequences in Q with respect to the extension of the p-adic norm defined by |a|p = lim |an |p , n→∞
where {an } is a Cauchy sequence in Q representing a ∈ Qp . Theorem 1. Every equivalence class a in Qp satisfying |a|p ≤ 1 has exactly one representative Cauchy sequence {ai } such that (1) ai ∈ Z, 0 ≤ ai < pi for i = 1, 2, . . ., (2) ai ≡ ai+1 (mod pi ) for i = 1, 2, . . . . From this, every p-adic number a ∈ Qp has a unique representation a=
∞ X
an p n ,
n=−m
where a−m 6= 0 and an ∈ {0, 1, 2, . . . , p − 1} for n ≥ −m. We represent the given p-adic number a as a fraction in the base p as follows: a = . . . an . . . a2 a1 a0 .a−1 . . . a−m . This representation is called the canonical p-adic expansion of a. P∞ i Definition 3. Let Zp = {a ∈ QP p| a = i=0 ai p } be the set of p-adic ∞ × i integers and Zp = {a ∈ Qp | a = i=0 ai p , a0 6= 0} be the set of p-adic units.
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From Definition 3, it is easy to see that Zp = {a ∈ Qp | |a|p ≤ 1} and Z× p = {a ∈ Qp | |a|p = 1}. Hence the following theorem follows. Theorem 2. Let a be a p-adic number of norm p−n . Then a = pn u for some u ∈ Z× p. From now, we discuss the conditions for the existence of p-adic roots. Definition 4. A p-adic number x ∈ Qp is said to be a q-th root of a ∈ Qp of order k ∈ N if and only if xq ≡ a (mod pk ). When q = 5, the q-th root of a ∈ Qp is called the fifth root of a. The following lemmata are essential for our discussions([4]). Lemma 3. Let a, b ∈ Qp . Then a and b are congruent modulo pk and write a ≡ b (mod pk ) if and only if |a − b|p ≤ 1/pk . Lemma 4. Let a, b ∈ Qp . If |a − b|p < |b|p , then |a|p = |b|p . The next theorem is the basis for the existence of p-adic roots([8]). Theorem 5. (Hensel’s lemma) Let F (x) = c0 + c1 x + · · · + cn xn be a polynomial whose coefficients are p-adic integers. Let F 0 (x) = c1 + c2 x + 3c3 x2 + · · · + ncn xn be the derivative of F (x). Let a0 be a p-adic integer such that F (a0 ) ≡ 0 (mod p) and F 0 (a0 ) 6≡ 0 (mod p). Then there exists a unique p-adic integer a such that F (a) = 0 and
a ≡ a0 (mod p).
The following theorem follows from Theorem 5, and provides the condition between p-adic numbers and congruence([4]). Theorem 6. A polynomial with integer coefficients has a root in Zp if and only if it has an integer root modulo pk for any k ≥ 1. Some results of the existence of square roots of p-adic numbers are obtained from Theorem 6([4]). In [6], we have the conditions for the existence of the fifth roots of p-adic numbers in Qp as followings. Theorem 7. A rational integer a not divisible by p has a fifth root in Zp (p 6= 5) if and only if a is a fifth residue modulo p. From Theorem 7, we have the following theorem([6]). Theorem 8. Let p be a prime number. Then we have: (1) If p 6= 5, then a = pordp a u ∈ Qp for some u ∈ Z× p has a fifth root in Qp if and only if ordp a = 5m for m ∈ Z and u = v 5 for some unit v ∈ Z× p. (2) If p = 5, then a = 5ord5 a u ∈ Q5 for some u ∈ Z× 5 has a fifth root in Q5 if and only if ord5 a = 5m for m ∈ Z and u ≡ 1 (mod 25) or u ≡ k (mod 5) for some k (2 ≤ k ≤ 4).
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3. Newton’s Method Newton’s method is a well known numerical method to find zeros of a polynomial f (x) in R([1]). The iterative formula for this method is given by f (xn ) xn+1 = xn − 0 , n = 0, 1, 2, . . . . (3.1) f (xn ) To seek the fifth root of a is to find the zero of f (x) = x5 − a. The iteration (3.1) for Newton’s method becomes the recurrence relation xn+1 =
4x5n + a , 5x4n
n = 0, 1, 2, . . . .
(3.2)
Like for real numbers, we can show that Newton’s method also converges quadratically for convergence. Let a(6= 0) ∈ Qp be a p-adic number such that |a|p = p−ordp a = p−5m ,
m ∈ Z.
The following theorem is the result when p 6= 5. Theorem 9. Let p 6= 5 and {xn } be the sequence of p-adic numbers obtained from the Newton’s iteration (3.2). If x0 is a fifth root of a of order r with |x0 |p = p−m and r > 5m, then (1) |xn |p = p−m , n = 1, 2, . . . , n n (2) x5n ≡ a (mod p2 r−5m(2 −1) ), (3) {xn } converges to the fifth root of a. Proof. We will prove (1) and (2) by induction. (i) First, we prove it when p > 5. Let n = 1. By assumption, we have x50 = a + bpr
(0 < b < p).
(3.3)
From (3.2), (3.3) and Lemma 4, we have |x1 |p =
|4x50 + a|p |5a + 4bpr |p max{|5a|p , |4bpr |p } = = = p−m . (3.4) 4 4 4 |5x0 |p |5x0 |p |5x0 |p
Also by (3.2), we have x51 − a =
(x50 − a)2 3 10 2 5 (1024x15 0 + 203ax0 + 22a x0 + a ). 3125x20 0
(3.5)
To calculate the p-adic norm of x51 − a, we let h(x) = 1024x15 + 203ax10 + 22a2 x5 + a3 .
(3.6)
From (3.3), we have h(x0 ) = 1250a3 + 3500a2 bpr + 3275ab2 p2r + 1024b3 p3r .
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(3.7)
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Using the strong triangle inequality, we have from (3.7) that |h(x0 )|p ≤ max |2 · 54 a3 |p , |22 53 7a2 bpr |p , |52 131ab2 p2r |p , |210 b3 p3r |p = max p−15m , p−10m−r , p−5m−2r , p−3r
(3.8)
= p−15m . Also the p-adic norm of the denominator of the right hand of (3.5) is −20m 5 20 . |3125x20 0 |p = |5 x0 |p = p
(3.9)
Since x0 is a fifth root of a of order r, we have |(x50 − a)2 |p = p−2r .
(3.10)
By (3.5), (3.8), (3.9) and (3.10), we have |x51 − a|p ≤ p5m−2r . By Lemma 3, x51 − a ≡ 0 (mod p2r−5m ). Hence (1) and (2) is true when n = 1. Now assume that |xn−1 |p = p−m , (3.11) n−1 r−5m(2n−1 −1)
x5n−1 = a (mod p2
),
(3.12)
and so n−1
n−1 −1)
x5n−1 = a + bp2 r−5m(2 From (3.2), (3.11) and (3.13), we have
(0 < b < p).
n−1
|4x5n−1 + a|p |5a + 4bp2 r−5m(2 |xn |p = = |5x4n−1 |p |5x4n−1 |p n−1
n−1 −1)
max{|5a|p , |4bp2 r−5m(2 = |5x4n−1 |p
|p }
n−1 −1)
=p
(3.13)
|p (3.14)
−m
.
Thus (1) is proved by (3.4), (3.11) and (3.14). Also from (3.2), it follows that (x5 − a)2 x5n − a = n−1 h(xn−1 ). (3.15) 55 x20 n−1 Let Q = p2
n−1 r−5m(2n−1 −1)
for simplicity. From (3.13),
h(xn−1 ) = 2 · 54 a3 + 22 · 53 · 7a2 bQ + 52 · 131ab2 Q2 + 210 b3 Q3 . (3.16) Since r > 5m, the p-adic norm of h(xn−1 ) in (3.16) is n−1 (r−5m)
|h(xn−1 )|p ≤ max{p−15m , p−15m−2 p
−15m−2n (r−5m)
,p
,
−15m−3·2n−1 (r−5m)
}
(3.17)
= p−15m .
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Since xn−1 is a fifth root of a of order 2n−1 r − 5m(2n−1 − 1), we have from (3.12), (3.15) and (3.17) that n r+5m(2n −1)
|x5n − a|p ≤ p−2
.
n r−5m(2n −1)
By Lemma 3, we have x5n − a ≡ 0 (mod p2 for all n ∈ N.
). Thus (2) is true
(ii) When p < 5, there are two cases, p = 3 and p = 2. The proof is the same with (i) when the first case p = 3, because 3 is no factor of any coefficients of terms of h(x0 ) in (3.7). It means that |h(x0 )|p ≤ p−15m , and so x51 ≡ a (mod p2r−5m ). By assuming x5n−1 ≡ a n−1 n−1 n n (mod p2 r−5m(2 −1) ), we have x5n ≡ a (mod p2 r−5m(2 −1) ) using the same process of (i). Moreover we can check easily |xn |3 = 3−m by induction. The other case is p = 2. Let n = 1, |x1 |p = p−m is obtained easily from (3.4). And we have x51 − a =
(x50 − a)2 h(x0 ), 3125x20 0
(3.18)
where h(x) is the polynomial in (3.6). Since r > 5m, we have |h(x0 )|p ≤ max{p−15m−1 , p−10m−r−2 , p−5m−2r , p−3r−10 } ≤ p−15m . (3.19) In (3.18), we have −20m |3125x20 , (3.20) 0 |p = p and, by assumption, |(x50 − a)2 |p = p−2r .
(3.21) −2r+5m
− a|p ≤ p , and so x51 ≡ Also (3.19), (3.20) and (3.21) imply a (mod p2r−5m ). Thus (1) and (2) are true when n = 1 if p = 2. n−1 n−1 Assume that |xn−1 |p = p−m and x5n−1 ≡ a (mod p2 r−5m(2 −1) ). That is, n−1 n−1 x5n−1 = a + bp2 r−5m(2 −1) (0 < b < p). (3.22) |x51
It follows (3.15) and (3.16), and so we have n−1 (r−5m)
|h(xn−1 )|p ≤ max{p−15m−1 , p−15m−2−2 p
−15m−2n (r−5m)
, n−1 (r−5m)
, p−15m−10−3·2
}
(3.23)
≤ p−15m . By (3.15), (3.17), (3.20) and (3.23), we have n r+5m(2n −1)
|x5n − a|p ≤ p−2
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n
Hence we have that for all n ∈ N, x5n ≡ a (mod p2 r−5m(2 −1) ). We note that |xn |2 = 2−m is obtained easily from (3.14). So we complete the proof of (1) and (2). From (2), we have |x5n − a|p ≤ p−2
n r+5m(2n −1)
(3.24)
for each prime p(6= 5). (3) follows immediately from the inequality (3.24) as n → ∞. When p = 5, we have the following theorem. Theorem 10. Let p = 5 and {xn } be the sequence of p-adic numbers obtained from the Newton’s iteration (3.2). If x0 is a fifth root of a of order r with |x0 |p = p−m and r > 5m + 1, then (1) |xn |p = p−m , n = 1, 2, . . . , n n (2) x5n ≡ a (mod p2 r−(5m+1)(2 −1) ), (3) {xn } converges to the fifth root of a. Proof. (1) and (2) will be proved by induction. Let n = 1. By assumption x50 ≡ a (mod pr ), and from (3.2) and Lemma 4, we have |x1 |p =
|5a + 4bpr |p max{|5a|p , |4bpr |p } p−5m−1 = p−m . = = |5x40 |p |5x40 |p p−4m−1
By calculating the p-adic norms of h(x0 ) in (3.7), we have |h(x0 )|p ≤ max{p−15m−4 , p−10m−r−3 , p−5m−2r−2 , p−3r } = p−15m−4 , −20m−5 . Thus since r > 5m + 1. Also we have |3125x20 0 |p = p
|x51 − a|p ≤ p−2r+5m+1 , and so x51 ≡ a (mod p2r−(5m+1) ) by Lemma 3. Hence it is true when n = 1. Now we assume that |xn−1 |p = p−m and n−1 r−(5m+1)(2n−1 −1)
x5n−1 ≡ a (mod p2
).
In the similar manner as (3.14), (3.16) and (3.17), we have n−1
n−1 −1)
|4x5n−1 + a|p |5a + 4bp2 r−(5m+1)(2 |xn |p = = |5x4n−1 |p |5x4n−1 |p n−1
n−1 −1)
max{|5a|p , |4bp2 r−(5m+1)(2 = |5x4n−1 |p
359
|p }
=
|p
p−5m−1 = p−m p−4m−1
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and n−1 [r−(5m+1)]
|h(xn−1 )|p ≤ max{p−15m−4 , p−15m−4−2 n [r−(5m+1)]
p−15m−4−2 =p
−15m−4
, p−15m−3−3·2
n−1 [r−(5m+1)]
}
.
And so we have |x5n − a|p ≤ p−2
n r+(5m+1)(2n −1)
.
(3.25)
It follows that (1) and (2) are true for all n ∈ N. (3) follows from the inequality (3.25) as n → ∞.
To determine the rate of convergence of the sequence {xn } given by (3.2), we consider the sequence {en } defined by en = xn+1 − xn ,
∀n ∈ N.
(3.26)
From Theorem 9 and Theorem 10, we obtain the following theorem. Theorem 11. If x0 is the fifth root of a of order r, then the sequence {en } in (3.26) is en ≡ 0 (mod pαn ), where 2n r − 5m · 2n + m, if p 6= 5, αn = n n 2 r − (5m + 1) · 2 + m, if p = 5. Proof. (i) First, let p 6= 5. Then, from the Newton’s iteration formula (3.2), we have 1 en = xn+1 − xn = 4 (a − x5n ), ∀n ∈ N. (3.27) 5xn By computing the p-adic norms of each side of the equation (3.27), we have from Theorem 8 that 1 n n |en |p = |xn+1 − xn |p = 4 · |a − xn |p ≤ p−2 r+5m·2 −m . 5x n p
Hence en ≡ 0 (mod pαn ) by Lemma 3. (ii) Let p = 5. By a similar way as (i), we have from Theorem 9 that 1 n n |en |p = 4 · |a − xn |p ≤ p−2 r+(5m+1)·2 −m . 5x n p
αn
Hence en ≡ 0 (mod p ) by Lemma 3. This completes the proof.
From Theorem 11, we have that the rate of convergence of the sequence {xn } is of order αn . Thus the number of correct digits in the approximation increases by αn for every iteration. We can compute the number of iterations to obtain certain finite digits. From Theorem 9 and Theorem 10, we have the following corollary.
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Corollary 12. (1) For p 6= 5, let {xn } be the sequence of approximation in Theorem 9. Then the number of iterations to obtain at least M correct digits is " # −4m ln Mr−5m n= . (3.28) ln 2 (2) Let p = 5 and {xn } be the sequence of approximation in Theorem 10. Then the number of iterations to obtain at least M correct digits is M −(4m+1) ln r−(5m+1) . n= (3.29) ln 2 Proof. (1) Since we need M correct digits in the approximation, we must set the order to M + m to find the number of iterations with M correct digits. That is, 2n r − 5m(2n − 1) = M + m.
(3.30)
From (3.30), we have M − 4m . r − 5m Since {xn } converges to the fifth root of a by Theorem 8 (3) and r > 5m, we have the equation (3.28). (2) As in the proof of (1), we set 2n =
2n r − (5m + 1)(2n − 1) = M + m.
(3.31)
From (3.31), we have 2n =
M − 4m − 1 . r − (5m + 1)
Since r > 5m + 1, the result follows from (3.32).
(3.32)
The numbers in (3.28) and (3.29) are sufficient numbers of iterations to provide at least M correct digits in the approximation. References [1] R. L. Burden, J. D. Faires, Numerical analysis (5th ed.), PWS Publishing, 1993. [2] P. S. Ignacio, On the square and cube roots of p-adic numbers, J. Math. Comput. Sci., 3 (2013), No. 4, 993–1003. [3] P. S. Ignacio, J. M. Addawe, W. V. Alangui, J. A. Nable, Computation of square and cube roots of p-adic numbers via Newton-Raphson method, J. Math. Research, 5 (2013), No. 2, 31–38. [4] S. Katok, p-Adic analysis compared with real, American Math. Soc., 2007
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[5] M. Keicies, T.Zerzaihi, General approach of the root of a p-adic number, Filomat, 27 (2013), No. 3, 431–436. [6] Y.-H. Kim, J. Choi, On the existence of p-adic roots, accepted in J. of Chungcheong Math. Soc. 28 (2015), No. 2. [7] M. Knapp, C. Xenophontos, Numerical analysis meets number theory using rootfinding method to calculate inverses mod pn , Appl. Anal. Discrete Math., 4 (2010), 23–31. [8] N. Koblitz, p-Adic numbers, p-adic analysis and zeta functions (2nd ed.), Springer-Verlag, 1984. [9] V. S. Vladimirov, I. V. Volvich, E. I. Zelenov, p-Adic analysis and mathematical physics, Norld Scientific, 1994. [10] T. Zerzaihi, M. Kecies, Computation of the cubic root of a p-adic number, J. Math. Research, 3 (2011), No. 3, 40–47. [11] T. Zerzaihi, M. Kecies, M. Knapp, Hensel codes of square roots of p-adic numbers, Appl. Anal. Discrete Math., 4 (2010), 32–44. Young-Hee Kim. Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea, Hyun-Mee Kim. Mathematics Education Major, Graduate School of Education, Kookmin University, Seoul 136-702, Republic of Korea, Jongsung Choi. Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea,
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Solution of the Ulam stability problem for Euler-Lagrange (α, β; k)-quadratic mappings S.A. Mohiuddine1, John Michael Rassias2 and Abdullah Alotaibi1 1
Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia 2 National and Capodistrian University of Athens, Pedagogical Department, Mathematics and Informatics, 4, Agamemnonos Str., Aghia Paraskevi, Attikis 15342, Greece Email: 1 [email protected]; 2 [email protected]; 1 [email protected]
Abstract. In 1940 S. M. Ulam proposed at the University of Wisconsin the problem: “Give conditions in order for a linear mapping near an approximately linear mapping to exist”. In 1982-2013, the second author solved the above Ulam problem for a variety of quadratic mappings. Interesting stability results have been achieved by S. A. Mohiuddine et al., since 2009. In this paper, we solve the Ulam stability problem for Euler-Lagrange (α, β; k) quadratic mapping. The other authors of this research area have established important results also on functional inequalities. Keywords and phrases: Quartic functional equations and inequalities; Various normed spaces; Ulam stability. AMS subject classification (2000): 39B.
1. Introduction In 1940 S. M. Ulam [36] proposed the famous “Ulam stability problem”, which was solved by D. H. Hyers [4], in 1941, for additive mappings. In 1950 T. Aoki [3] solved this Ulam problem for weaker additive mappings. In 1978 Th. M. Rassias [33] generalized the theorem of Hyers for linear mappings. In 1982-1999, J. M. Rassias ( [23–30]) generalized this problem. For more detail of Ulam stability problem, we refer to [5, 6, 8–11, 19, 20, 32, 34] and references therein. In 1992, the second author [23, 24] introduced the term “Euler-Lagrange functional equation” and “Euler-Lagrange quadratic mappings”, of satisfying Q(x + y) + Q(x − y) = 2 Q(x) + Q(y)
(1.1)
and then solved the Ulam stability problem of the Euler-Lagrange quadratic functional equation (1.1). In 1996, J. M. Rassias [30] established the Ulam stablity of the general Euler-Lagrange quadratic functional equation Q(αx + βy) + Q(βx − αy) = (α2 + β 2 ) Q(x) + Q(y) . (1.2) In 2009-2014, S. A. Mohiuddine et al. ( [1,2,12–18]) solved this problem in several normed spaces. In
2008-2012 J. M. Rassias et al. ( [21, 22, 31,37]) solved the generalized Ulam problem via various methods. In 2010, M. E. Gordji et al [7] established Ulam stabilities on Banach algebras. Also J. R¨atz [35] results are interesting on orthogonal mappings. In this paper, we solve the Ulam stability problem for the Euler-Lagrange (α, β; k) quadratic mapping satisfying kQ(αx + βy) + Q(kβx − αy) = (α2 + kβ 2 ) kQ(x) + Q(y) . 363
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(1.3)
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Let us note that Q(x) = |x|2 satisfies equation (1.3) because the following Euler-Lagrange quadratic identity k|αx + βy|2 + |kβx − αy|2 = (α2 + kβ 2 ) k|x|2 + |y|2 (1.4) holds with any fixed reals α, β and k.
Definition 1.1. Let X be a normed linear space and let Y be a real complete normed linear space. Then a non-linear mapping Q : X → Y is called Euler-Lagrange quadratic if equation (1.3) holds for all 2-dimensional vectors (x, y) ∈ X 2 , and any fixed reals α, β and k. We note that Q may be called quadratic because the above Euler-Lagrange identity (1.4) holds and because the functional equation Q(mn x) = (mn )2 Q(x)
(1.5)
m = α2 + kβ 2 .
(1.6)
holds for all x ∈ X, all n ∈ N :
Assume m ∈ R − {0, 1} and k ∈ R − {−1, 0}. In fact, substitution of x = y = 0 in equation (1.3) yields (k + 1)(1 − m)Q(0) = 0, or Q(0) = 0,
m 6= 1 (and k 6= −1).
(1.7)
Substituting x = x, y = 0 in (1.3), one gets that kQ(αx) + Q(kβx) = kmQ(x) + mQ(0),
(1.8)
or
1 m Q(kβx) = mQ(x) + Q(0), k k holds for all x ∈ X, and any fixed real k 6= 0. Employing (1.7), we obtain from (1.8) that Q(αx) +
Q(αx) + Q(kβx) = kmQ(x).
(1.9)
(1.10)
Moreover, substitution x → αx, y = kβx in (1.3), we find that
or
kQ(mx) + Q(0) = m kQ(αx) + Q(kβx) , kQ(αx) + Q(kβx) = km−1 Q(mx) +
1 Q(0), m
(1.11)
or
1 1 (1.12) Q(kβx) = m−1 Q(mx) + Q(0) k km holds for all x ∈ X, and any fixed reals k 6= 0, m 6= 0. Functional Equations (1.8) and (1.11), or (1.9) and (1.12) yield 1 km−1 Q(mx) + Q(0) = kmQ(x) + mQ(0), m or 1 −2 km Q(x) − m Q(mx) = − m Q(0), m Q(αx) +
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or
1 − m2 Q(0), m 1 1 − m2 Q(0). Q(x) − m−2 Q(mx) = k m2
km Q(x) − m2 Q(mx) =
or
(1.13)
Employing (1.7), one gets Q(x) = m−2 Q(mx),
(1.14)
Q(mx) = m2 Q(x)
(1.15)
or
Replaying x → mx in (1.15), we find Q(m2 x) = m2 Q(mx), or Q(m2 x) = m4 Q(x)
(1.16)
Then by induction on n ∈ N with x → mn−1 x yields equation (1.5). Definition 1.2. Let X be a normed linear space and let Y be a real complete normed linear space. Then ¯ : X → Y, a 2-dimensional quadratic weighted mean if we call the non-linear mapping Q kQ(αx) + Q(kβx) ¯ Q(x) = km
(1.17)
holds for all x ∈ X and any fixed reals k, m 6= 0. Let us note that from (1.8) and (1.17), one get kmQ(x) + mQ(o) ¯ Q(x) = , km or
1 ¯ = Q(x) + Q(o), Q(x) k for all x ∈ X, and any fixed real k = 6 0. From (1.7) and (1.18), we obtain
(1.18)
¯ Q(x) = Q(x),
(1.19)
for all x ∈ X.
2. Stability for Euler-Lagrange quadratic mappings Let us introduce the Euler-Lagrange (α, β; k) quadratic functional inequality
kf(αx + βy) + f(kβx − αy) − (α2 + kβ 2 ) kf(x) + f(y) ≤ c,
(2.1)
for all 2-dimensional vectors (x, y) ∈ X 2 and any fixed reals α, β and k as well as m = α2 + kβ 2 , with m ∈ R − {0, 1} (k ∈ R − {−1, 0}), and c(:= constant inde of x, y) ≥ 0. Then we prove the following theorem. Theorem 2.1. Let X be a normed linear space and let Y be a real complete normed linear space. Let us denote, kf(αx) + f(kβx) f¯(x) = (2.2) km 365
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holds for all x ∈ X and any fixed reals k, m 6= 0. Also let us assume m : |m| > 1. Then the limit Q(x) = lim m−2n f(mn x),
(2.3)
n→∞
exists for all x ∈ X, all n ∈ N, and any fixed real m : |m| > 1 and Q : X → Y is the unique quadratic mapping satisfying functional equation (1.3) such that
f(x) − Q(x) ≤ c3 =
where
c2 = m2 c1 =
c2 , |m| > 1, m2 − 1
(2.4)
|k + 1| 1 + |m| + |1 + m| c, |k| |k + 1| c2 c3 = 2 . m −1
Moreover , identity Q(x) = m−2n Q(mn x)
(2.5)
holds for all x ∈ X all n ∈ N, and any fixed reals: α, β; k, m : |m| > 1 with m ∈ R − {0, 1}, (k ∈ R − {−1, 0}). Proof of Existence in Theorem 2.1. In fact, substitution of x = y = 0 in equality (2.1) yields
|k + 1| |1 − m| f(0) ≤ c, or
f(0) ≤
c , k 6= −1, m 6= 1. |k + 1| |1 − m|
(2.6)
Substituting x = x, y = 0 in (2.1), one gets that
kmf(x) − [kf(αx) + f(kβx)] + mf(0) ≤ c, or
f(x) − f¯(x) + 1 f(0) ≤ c , k 6= 0, m 6= 0, |m| > 1 k |k||m|
(2.7)
from (2.2). Moreover substitution x → αx, y = kβx in (2.1), we find that
kf(mx) + f(0) − m[kf(αx) + f(kβx)] ≤ c, or
or
kf(αx) + f(kβx) − km−1 f(mx) − 1 f(0) ≤ c , m |m|
f(x) ¯ − m−2 f(mx) −
1 c f(0) ≤ , 2 km |k| m2
(2.8)
Functional inequalities (2.6),(2.7),(2.8) and triangle inequality yields
f(x) − m−2 f(mx)
1 1 ≤ f(x) − f¯(x) + f(0) + f¯(x) − m−2 f(mx) − f(0) 2 k km
1
1
+ f(0) − f(0) km2 k
c c |1 − m2 |
f(0) ≤ + + 2 2 |k| |m| |k| m |k|m 2
1 + |m| |1 − m |
= c + f(0) |k| m2 |k|m2 366
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≤ = = or where
1 + |m| |1 − m2 | 1 + c |k| m2 |k|m2 |k + 1||1 − m| |1 + m| 1 + |m| + c |k| m2 |k||k + 1|m2 |k + 1| 1 + |m| + |1 + m| c1 = c, |k||k + 1|m2
f(x) − m−2 f(mx) ≤ c1 = c2 , m2 |k + 1| 1 + |m| + |1 + m| 2 c, c2 = m c1 = |k||k + 1|
(2.9)
(2.10)
holds for fixed k, m 6= 0, m 6= 1, m > 1. Replacing x → mx in (2.9) and then multiplying by m−2 , we find
−2
m f(mx) − m−4 f(m2 x) ≤ m−2 c1 , m 6= 0 (2.11) From (2.9) and (2.11), one gets
f(x) − m−4 f(m2 x) ≤ f(x) − m−2 f(mx) + m−2 f(mx) − m−4 f(m2 x) ≤ 1 + m−2 c1 , or
f(x) − m−4 f(m2 x) ≤ (1 + m−2 )c1 , m 6= 0.
(2.12)
Employing (2.9) and (2.12) without induction, we obtain
f(x) − m−2n f(mn x) ≤ f(x) − m−2 f(mx) + m−2 f(mx) − m−4 f(m2 x) + · · ·
+km−2(n−1)f(mn−1 x) − m−2n f(mn x) ≤ 1 + m−2 + · · · + m−2(n−1) c1 , or
−2n
m2
f(x) − m−2n f(mn x) ≤ 1 − m c = 1 − m−2n c1 , 1 −2 2 1−m m −1 or the general inequality:
f(x) − m−2n f(mn x) ≤
where |m| > 1, c2 = m2 c1 . Claim now that the sequence
m2
1 1 − m−2n c2 , −1
fn (x) , fn (x) = m−2n f(mn x)
(2.13)
(2.14)
(2.15)
converges. Note that from the general inequality (2.14) and the completeness of Y , one proves that the above sequence (2.15) is a Cauchy sequence. In fact, if i > j > 0, then
fi (x) − fj (x) = m−2i f(mi x) − m−2j f(mj x)
= m−2j m−2(i−j)f(mi x) − f(mj x)
= m−2j f(mj x) − m−2(i−j)f(mi−j · mj x) 1 ≤ m−2j · 2 (1 − m−2(i−j))c2 , m −1 or
fi (x) − fj (x) ≤
m2
1 (m−2j − m−2i )c2 , |m| > 1, −1
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or 0≤ or
lim fi (x) − fj (x) ≤ 0,
i>j→∞
lim fi (x) − fj (x) = 0,
(2.17)
i>j→∞
completing the proof that the sequence fn (x) converges. Hence Q = Q(x) is well-defined via the formula (2.3). This means that the limit (2.3) exists for all x ∈ X. In addition claim that mapping Q satisfies the functional equation (1.3) for all vectors (x, y) ∈ X 2 . In fact, it is clear from functional inequality (2.1) and the limit (2.3) that inequality
k lim m−2n f mn (αx + βy) + lim m−2n f mn (kβx − αy) n→∞ n→∞
−(α2 + kβ 2 ) k lim m−2n f(mn x) + lim m−2n f(mn y) n→∞
≤
c( lim m−2n ) = 0, n→0
n→∞
|m| > 1,
(2.18)
or
kQ(αx + βy) + Q(kβx − αy) − (α2 + kβ 2 )[kQ(x) + Q(y) = 0,
or mapping Q satisfies the functional equation (1.3) for all x, y ∈ X, and |m| > 1. Thus Q is a 2dimensional quadratic mapping. It is now clear from general inequality (2.14), n → ∞, and the formula (2.3) that inequality (2.4) holds in X, completing the existence proof of this Theorem 2.1. Proof of Uniqueness in Theorem 2.1. Let Q0 : X → Y be another 2-dimensional quadratic mapping satisfying equation (1.3), such that
f(x) − Q0 (x) ≤ c3 = c2 , (2.4)0 m2 − 1
for all x ∈ X, and any fixed real m : |m| > 1. To prove the above-mentioned uniqueness employ (2.5) for Q and Q0 , as well, so that Q0 (x) = m−2n Q0 (mn x)
(2.5)0
holds for all x ∈ X, all n ∈ N, and any fixed real m : |m| > 1. Moreover, the triangle inequality and functional inequalities (2.4)-(2.4)0 yield
or
Q(mn x) − Q0 (mn x) ≤ Q(mn x) − f(mn x) + f(mn x) − Q0 (mn x) ,
Q(mn x) − Q0 (mn x) ≤ 2c3 ,
(2.19)
for all x ∈ X, all n ∈ N, and any fixed real m : |m| > 1. Then from (2.5)-(2.5)0, and (2.19), one proves that
or
Q(x) − Q0 (x) = m−2n Q(mn x) − m−2n Q0 (mn x) ,
Q(x) − Q0 (x) ≤ 2m−2n c3 ,
(2.20)
holds for all x ∈ X, all n ∈ N, and any fixed real m : |m| > 1. Therefore from (2.20), and n → ∞, one establishes
0 ≤ lim Q(x) − Q0 (x) ≤ 2 lim m−2n c3 = 0, |m| > 1, n→∞
n→∞
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or
Q(x) − Q0 (x) = 0,
or
Q(x) = Q0 (x),
|m| > 1,
(2.21)
for all x ∈ X, completing the proof of uniqueness and thus the stability of Theorem 2.1. Theorem 2.2. Let X be a normed linear space and let Y be a real complete normed linear space. Let us denote 1 k m ¯ f(x) = m2 f¯(m−1 x) = (2.2)0 kf αx +f βx k m m holds for all x ∈ X and any fixed reals k, m 6= 0. Also let us assume |m| < 1. Then the limit Q(x) = lim m2n f(m−n x),
(2.3)0
n→∞
exists for all x ∈ X, all n ∈ N, and any fixed real m : |m| < 1, and Q : X → Y is the unique quadratic mapping satisfying functional equation (2.3)0 , such that
f(x) − Q(x) ≤ c4 = c1 . 1 − m2 Moreover, identity Q(x) = m2n Q(m−n x) (2.5)0 holds for all x ∈ X, n ∈ N and |m| < 1, m 6= 0. From (2.7) with x → m−1 x(m 6= 0, |m| < 1) and multiplying by m2 , one find
2
2 ¯ + m f(0)
m f(m−1 x) − f(x)
≤ |m| c, (2.22)
|k| k where
¯ = m2 f¯(m−1 x) = m kf m−1 αx + f k βx , m 6= 0, |m| < 1. f(x) k m
(2.23)
From (2.8) with x → m−1 x (m 6= 0, |m| < 1), one obtains
c
f¯(m−1 x) − m−2 f(x) − 1 f(0) ≤ .
2 km |k| m2
Multiplying by m2 , we get
¯
f(x) − f(x) − 1 f(0) ≤ c .
|k| k
(2.24)
Functional inequalities (2.6),(2.23),(2.24) and triangle inequality yield
¯
m2 ¯ + 1 f(0) 2 −1
f(x) − m2 f(m−1 x) ≤ f(x) − f(x)
+ f(x) − m f(m x) − f(0)
k k
2
m
1
+
k f(0) − k f(0)
c |m| |m2 − 1|
f(0) ≤ + c+ |k| |k| |k| = ≤ =
1 + |m| |1 − m2 | c+ kf(0)k |k| |k| 1 + |m| |1 + m| c + |k| |k||k + 1| |k + 1| 1 + |m| + |1 + m| c = c2 , |k||k + 1| 369
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or
f(x) − m2 f(m−1 x) ≤ c2 ,
where c2 =
|k + 1| 1 + |m| + |1 + m| c, |k||k + 1|
(2.25)
|m| < 1, k 6= 0, k 6= −1, m 6= 0.
Replacing x → m−1 x in (2.25) and multiplying by m2 , we get m2 f(m−1 x) − m4 f(m−2 x)k ≤ m2 c2 ,
(2.26)
From (2.25)-(2.26), one finds
f(x) − m4 f(m−2 x) ≤ f(x) − m2 f(m−1 x) + m2 f(m−1 x) − m4 f(m−2 x) ≤ (1 + m2 )c2 , or
f(x) − m4 f(m−2 x) ≤ 1 + m2 c2 , m 6= 0.
(2.27)
Employing (2.25) and (2.27), without induction, we get
f(x) − m2n f(m−n x) ≤ f(x) − m2 f(m−1 x) + m2 f(m−1 x) − m4 f(m−2 x) + · · ·
+ m2(n−1)f(m−(n−1) x) − m2n f(m−n x) ≤ 1 + m2 + · · · + m2(n−1) c2 or
2(n−1)
c2
f(x) − m2n f(m−n x) ≤ 1 − m c2 = (1 − m2(n−1)), 1 − m2 1 − m2 or the general inequality:
f(x) − m2n f(m−n x) ≤ c2 , 1 − m2 where |m| < 1, m 6= 0. Rest of the proof is similar to the proof of Theorem 2.1.
(2.28) (2.29)
Assume the following condition on f: f(0) = 0.
(2.30)
From (2.30) and (2.7)-(2.8), we get
f(x) − f¯(x) ≤
and
¯ − m−2 f(mx) ≤
f(x)
c , |k| |m|
(2.31)
c , k 6= 0, m 6= 0, |m| > 1. |k| m2
(2.32)
From (2.31)-(2.32), one obtains
f(x) − m−2 f(mx) ≤ f(x) − f¯(x) + f¯(x) − m−2 f(mx) , or
Thus
f(x) − m−2 f(mx) ≤ c0 = |m| + 1 c, k 6= 0, m 6= 0, |m| > 1. 1 |k| m2
f(x) − m−2n f(mn x) ≤ ≤
(2.33)
f(x) − m−2 f(mx) + m−2 f(mx) − m−4 f(m2 x)
+ · · · + m−2(n−1)f(mn−1 x) − m−2n f(mn x) 1 + m−2 + · · · + m−2(n−1) c01 , 370
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or
−2n
m2
f(x) − m−2n f(mn x) ≤ 1 − m c01 = 2 1 − m−2n c01 , −2 1−m m −1
or
f(x) − m−2n f(mn x) ≤
where
m2
1 1 − m−2n c02 , −1
|m| > 1, with c02 = m2 c01 =
(2.34)
|m| + 1 c. |k|
Therefore the following Theorem 2.1a holds. Theorem 2.1a. Let X be a normed linear space and let Y be a real complete normed linear space. Then the limit (2.3) exists for all x ∈ X, all n ∈ N, |m| > 1 and Q : X → Y is the unique quadratic mapping satisfying equation (1.3), such that
f(x) − Q(x) ≤
c02 |m| + 1 1 = 2 c, k 6= 0, |m| > 1. −1 m − 1 |k|
(2.35)
m2
The proof of this Theorem 2.1a is similar to the proof of the previous Theorem 2.1. Alternatively: |m| < 1, f(0) = 0: From (2.30) and (2.22) , (2.24), we get
¯
f(x) − f(x)
≤ c , |k|
and
(2.36)
¯
f − m2 f(m−1 x) ≤ |m| c, |k|
(2.37)
k 6= 0, m 6= 0, |m| < 1. From (2.36)-(2.37), one obtains
or
¯
¯
f(x) − m2 f(m−1 x) ≤ f(x) − f(x)
+ f(x) − m2 f(m−1 x)
k 6= 0, m 6= 0, |m| < 1. Thus
f(x) − m2 f(m−1 x) ≤ c0 = |m| + 1 c 2 |k|
f(x) − m2n f(m−n x) ≤ ≤ or
(2.38)
f(x) − m−2 f(mx)
+ · · · + m2(n−1)f(m−(n−1) x) − m2n f(m−n x) 1 + m2 + · · · + m2(n−1) c02 ,
f(x) − m2n f(m−n x) ≤
1 1 − m2n c02 , 2 1−m
(2.39)
where |m| < 1, m 6= 0. Therefore the following Theorem 2.2a (analogous to Theorem 2.1a) holds for |m| < 1, m 6= 0. Theorem 2.2a. Let X be a normed linear space, and Y a real complete normed linear space. Then the limit (2.3)0 exists for all x ∈ X, n ∈ N, |m| < 1; m 6= 0, and Q : X → Y is the unique quadratic mapping satisfying equation (1.3), such that
f(x) − Q(x) ≤
c02 1 + |m| 1 = c, 1 − m2 1 − m2 |k|
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(2.40)
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k 6= 0, |m| < 1; m 6= 0. Special case: Replacing α = β = 1 in equation (1.3) and (2.1), one gets kf(x + y) + f(kx − y) = (k + 1) kf(x) + f(y) , k ∈ R − {−1, 0}.
Thus
(2.41)
m = k + 1 ∈ R − {0, 1}. Also
kf(x + y) + f(kx − y) − (k + 1) kf(x) + f(y) ≤ c, k ∈ R − {−1, 0}.
(2.42)
Acknowledgement. The authors gratefully acknowledge the financial support from King Abdulaziz University, Jeddah, Saudi Arabia.
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Math. Anal. Appl. 184 (1994) 431-436. [7] M.E. Gordji, N. Ghobadipour, Generalized Ulam-Hyers stabilities of quartic derivations on Banach algebras, Proyecciones J. Math., 29 (2010) 209-226. [8] K.W. Jun, H.M. Kim, The generalized Hyers-Ulam-Rassias stability of a cubic functional equation, J. Math. Anal. Appl. 274 (2002) 867-878. [9] S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor (2001) [10] S.-M. Jung, On the Hyers-Ulam stability of the functional equations that have the quadratic property, J. Math. Anal. Appl. 222 (1998) 126-137. [11] M. Mirmostafaee, M.S. Moslehian, Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Sets Syst. 159 (2008) 720-729. [12] S.A. Mohiuddine, H. Sevli, Stability of Pexiderized quadratic functional equation in intuitionistic fuzzy normed space, J. Comp. Appl. Math. 235 (2011) 2137-2146. [13] S.A. Mohiuddine, M. Cancan, H. S ¸ evli, Intuitionistic fuzzy stability of a Jensen functional equation via fixed point technique, Math. Comput. Modelling 54 (2011) 2403-2409. [14] S.A. Mohiuddine, Stability of Jensen functional equation in intuitionistic fuzzy normed space, Chaos, Solitons Fract. 42 (2009) 2989-2996. [15] S.A. Mohiuddine, A. Alotaibi, Fuzzy stability of of a cubic functional equation via fixed point technique, Adv. Difference Equ. 2012, 2012:48. [16] S.A. Mohiuddine, A. Alotaibi, M. Obaid, Stability of of various functional equations in non-Archimedean intuitionistic fuzzy normed spaces, Discrete Dynamics Nature Soc. Volume 2012, Article ID 234727, 16 pages. [17] S.A. Mohiuddine, M.A. Alghamdi, Stability of a functional equation obtained through a fixed-point alternative, Adv. Difference Equ. 2012, 2012:141.
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[18] M. Mursaleen, S.A. Mohiuddine, On stability of a cubic functional equation in intuitionistic fuzzy normed spaces, Chaos, Solitons Fract. 42 (2009) 2997-3005. [19] M. Mursaleen, K.J. Ansari, Stability results in intuitionistic fuzzy normed spaces for a cubic functional equation, Appl. Math. Inf. Sci. 7(5) (2013) 1685-1692. [20] A. Najati, C. Park, On the stability of an n-dimensional functional equation originating from quadratic forms, Taiwan. J. Math. 12 (2008) 1609-1624. [21] C. Park, J.M. Rassias, Cubic derivations and quartic derivations on Banach modules, in: “Functional Equations, Difference Inequalities and Ulam Stability Notions” (F. U. N.), Editor: J.M. Rassias, 2010, 119-129, ISBN 978-1-60876-461-7, Nova Science Publishers, Inc. [22] M.M. Pourpasha, J.M. Rassias, R. Saadati, S.M. Vaezpour, A fixed point approach to the stability of Pexider quadratic functional equation with involution, J. Inequal. Appl. 2010, Art. ID 839639, 18 pp. [23] J.M. Rassias, On the stability of the Euler-Lagrange functional equation, C. R. Acad. Bulgare Sci. 45 (1992) 17-20. [24] J.M. Rassias, On the stability of the Euler-Lagrange functional equation, Chinese J. Math. 20 (1992) 185-190. [25] J.M. Rassias, Solution of a problem of Ulam, J. Approx. Theory 57 (1989) 268-273. [26] J.M. Rassias, On a new approximation of approximately linear mappings by linear mappings, Discuss, Math. 7 (1985) 193-196. [27] J.M. Rassias, On approximation of approximately linear mappings by linear mappings, Bull. Sci. Math. (2) 108 (1984) 445-446. [28] J.M. Rassias, On approximation of approximately linear mappings by linear mappings, J. Funct. Anal. 46 (1982) 126-130. [29] J.M. Rassias, Solution of the Ulam stability problem for quartic mappings, Glas. Mat. Ser. III 34(54) (1999) 243-252. [30] J.M. Rassias, On the stability of the general Euler-Lagrange functional equation, Demonstratio Math. 29 (1996) 755-766. [31] J.M. Rassias, H.-M. Kim, Approximate homomorphisms and derivations between C ∗ -ternary algebras. J. Math. Phys. 49 (2008), no. 6, 063507, 10 pp. [32] T.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Am. Math. Soc. 72 (1978) 297-300. [33] T.M. Rassias, On a modified Hyers-Ulam sequence, J. Math. Anal. Appl. 158 (1) (1991) 106-113. [34] T.M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math. 62 (2000) 123-130. [35] J. R¨ atz, On the orthogonal additive mappings, Aequationes Math. 28 (1985) 35-49. [36] S.M. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, No.8, Interscience. Publ., New York , 1960; “Problems in Modern Mathematics”, Ch. VI, Science Ed., Wiley, 1940. [37] T.Z. Xu, J.M. Rassias, W.X. Xu, A generalized mixed Quadratic-Quartic functional equation, Bull. Malays. Math. Sci. Soc. 35(3) (2012) 633-649.
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Some integral inequalities via (h − (α, m))−logarithmically convexity Jianhua Chen, Xianjiu Huang∗ Department of Mathematics, Nanchang University, Nanchang, 330031, P. R. China Abstract. In this paper, we introduce the concept of (h − (α, m))−logarithmically convex functions and establish some new integral inequalities of these classes of functions.
Keywords:
Hermite’s inequalities; m−logarithmically convex; (α, m)−logarithmically
convex; (h − (α, m))−logarithmically convex; MR(2010) Subject Classification: Primary 26D15, Secondary 26A51
1
Introduction and preliminaries The mathematical inequalities play an important role in the mathematical branches and their
enormous application can not be underestimated. Afterwards, many researchers[1-13] studied the properties of convexity and achieve some different integral inequalities. The purpose of this paper is to introduce the definition of (h − (α, m))−logarithmically convex functions and establish some new integral inequalities of these classes of functions. Before stating our results, we need recall some notions. Throughout this paper, by ℜ, we denote the set of all real numbers. Definition 1.1 Let f : I ⊂ ℜ → ℜ be a function define on interval I of real numbers. Then f is called convex (see[4]) if f (tx + (1 − t)y) ≤ tf (x) + (1 − t)f (y) †
To whom correspondence should be addressed. E-mail:[email protected](J. Chen), [email protected] (X.
Huang). † This work has been supported by the National Natural Science Foundation of China (11461043, 11361042 and 11326099) and supported partly by the Provincial Natural Science Foundation of Jiangxi, China (20114BAB201003 and 20142BAB201005) and the Science and Technology Project of Educational Commission of Jiangxi Province, China (GJJ11346).
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for all x, y ∈ I and t ∈ [0, 1]. In [2] , Toader gave the definition of m−convexity as follows. Definition 1.2 The function f : [a, b] → ℜ, 0 ≤ a < b is said to be m−convex , where m ∈ [0, 1], if f (tx + m(1 − t)y) ≤ tf (x) + m(1 − t)f (y) holds for all x, y ∈ [0, 1] and t ∈ [0, 1]. We say that f is m−concave if −f is m−convex. In [3] , Mihesan gave the definition of (α, m)−convexity as follows. Definition 1.3 The function f : [a, b] → ℜ, 0 ≤ a < b is said to be (α, m)−convex , where (α, m) ∈ [0, 1]2 , if f (tx + m(1 − t)y) ≤ tα f (x) + m(1 − tα )f (y) holds for all x, y ∈ [0, 1] and t ∈ [0, 1]. ¨ In [1], Ozedemir et al. gave the definition of (h − (α, m))−convexity as follows. Definition 1.4 Let h : K ⊂ ℜ → ℜ be a nonnegative function, h ̸= 0. The function f : L ⊂ ℜ → ℜ is said to be (h − (α, m))−convex function if f is non-negative and for all x, y ∈ [0, 1] and t ∈ (0, 1) for (α, m) ∈ [0, 1]2 , we have f (tx + m(1 − t)y) ≤ hα (t)f (x) + m(1 − hα (t))f (y). In [5], Bai gave the definition of m− and (α, m)−logarithmically convex functions as follows. Definition 1.5 The function f : [a, b] → (0, ∞),0 ≤ a < b is said to be m−logarithmically convex, where m ∈ (0, 1], if f (tx + m(1 − t)y) ≤ [f (x)]t [f (y)]m(1−t) holds for all x, y ∈ [0, 1] and t ∈ [0, 1]. Definition 1.6 The function f : [a, b] → (0, ∞),0 ≤ a < b is said to be (α, m)−logarithmically convex, where (α, m) ∈ (0, 1]2 , if α
f (tx + m(1 − t)x) ≤ [f (x)]t [f (x)]m(1−t
α)
holds for all x, y ∈ [0, 1] and t ∈ [0, 1].
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2
Main results In this section, we will introduce the concept of (h − (α, m))−logarithmically convex functions. We
give some new integral inequalities of these classes of functions. First, we present the definition of (h − (α, m))−logarithmically convex functions as follow. Definition 2.1 Let h : K ⊂ ℜ → ℜ be a nonnegative function, h ̸= 0. The function f : L ⊂ ℜ → ℜ is said to be (h − (α, m))−logarithmically convex function if f is nonnegative and for all x, y ∈ L and t ∈ (0, 1) for (α, m) ∈ (0, 1]2 , we have hα (t)
f (tx + m(1 − t)y) ≤ [f (x)]
(
)
m 1−hα (t)
[f (y)]
.
Obviously, if h(t) = t, then (h−(α, m))−logarithmically convex function is a (α, m)−logarithmically convex function; if h(t) = t,α = 1, then (h−(α, m))−logarithmically convex function is a m−logarithmically convex function. ¨ Before giving our results, we need the following lemma which is proved by Ozdemir et al. [13]. Lemma 2.1 Let f : [a, b] → ℜ, 0 ≤ a < b be continuous on [a, b] such that f ∈ L([a, b]). Then the equality
∫
b
∫
1
(x − a)p (x − b)q f (x)dx = (b − a)p+q+1
a
(1 − t)p tq f (tx + (1 − t)y)dt
0
holds for some fixed p, q > 0. Theorem 2.1 Let f : [a, b] → ℜ, 0 ≤ a < b be continuous on [a, b] such that f ∈ L([a, b]). If the mapping f is (h − (α, m))−logarithmically convex on [a, b] for all t ∈ (0, 1) and (α, m) ∈ (0, 1]2 , then ∫
b
a
q p (x − a)p (x − b)q f (x)dx ≤ (b − a)p+q+1 [β( + 1, + 1)]1−m 1 − m 1 − m {∫ 1 }m hα (t) b α [f (a)] m f ( )1−h (t) dt × m 0
∫1
where β(x, y) =
0
(2.1)
(t)x−1 (1 − t)y−1 dt.
Proof. Using Lemma 2.1 , we have ∫
b
∫ (x − a) (x − b) f (x)dx = (b − a) p
q
p+q+1
a
1
(1 − t)p tq f (ta + (1 − t)b)dt.
(2.2)
0
Since f is (h − (α, m))−logarithmically convex on [a, b], we know that for every t ∈ (0, 1) ( ) α b b α f (ta + (1 − t)b) = f (ta + m(1 − t)( )) ≤ [f (a)]h (t) [f ( )]m 1−h (t) . m m
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(2.3)
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From (2.1), (2.2), (2.3) and H¨older inequality, we can conclude that ∫
b
(x − a)p (x − b)q f (x)dx a ∫ 1 p+q+1 = (b − a) (1 − t)p tq f (ta + (1 − t)b)dt ∫0 1 b p+q+1 = (b − a) (1 − t)p tq f (ta + m(1 − t) )dt m( ∫0 1 ) b m 1−hα (t) p+q+1 p q hα (t) ≤ (b − a) (1 − t) t [f (a)] [f ( )] dt m{ ∫ }1−m }m {0∫ 1 1{ 1 b m(1−hα (t)) } m1 hα (t) p+q+1 p q 1−m dt [f (a)] [f ( ) ] dt ≤ (b − a) [(1 − t) t ] m 0 0 }1−m { ∫ 1 }m {∫ 1 { } q p hα (t) b α [f (a)] m [f ( )1−h (t) ] dt ≤ (b − a)p+q+1 [(1 − t) 1−m t 1−m dt m 0 0 {∫ }m 1{ hα (t) b 1−hα (t) } q p p+q+1 1−m ≤ (b − a) [β( + 1, + 1)] [f (a)] m [f ( ) ] dt . 1−m 1−m m 0 Hence, the proof of theorem 2.1 is completed. Remark 2.1 If α = 1, then we can conclude the following inequality: ∫
b
a
p q + 1, + 1)]1−m (x − a)p (x − b)q f (x)dx ≤ (b − a)p+q+1 [β( 1 − m 1 − m }m {∫ 1 { h(t) b 1−h(t) } m [f (a)] [f ( ) ] dt . × m 0
Theorem 2.2 Let f : [a, b] → ℜ, 0 ≤ a < b be continuous on [a, b] such that f ∈ L([a, b]). If k
the mapping |f | k−1 (k > 1) is (h − (α, m))−logarithmically convex on [a, b] for all t ∈ (0, 1) and (α, m) ∈ (0, 1]2 , then ∫
b
(x − a) (x − b) f (x)dx ≤ (b − a) p
q
p+q+1
a
[∫ × 0
where β(x, y) =
∫1 0
[β(kq + 1, kp + 1)]
1 k
[∫
1
|f (a)|
k2 hα (t) k−1
] k−1 2 dt
k
0
1
k2 m (1−hα (t)) (k−1)2
b |f ( )| m
(2.4)
2 ] (k−1) 2 k
(t)x−1 (1 − t)y−1 dt.
Proof. Using Lemma 2.1 , we have ∫
b
∫ (x − a) (x − b) f (x)dx = (b − a) p
q
p+q+1
a
1
(1 − t)p tq f (ta + (1 − t)b)dt.
(2.5)
0 k
Taking into account that |f | k−1 is (h − (α, m))−logarithmically convex on [a, b], we deduce that k
|f (ta + (1 − t)b)| k−1 = |f (ta + m(1 − t)(
k k k α α b k−1 b ))| ≤ |f (a)| k−1 h (t) |f ( )| k−1 m(1−h (t)) . m m
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(2.6)
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Hence, from (2.4), (2.5), (2.6) and H¨older inequality, we can achieve the following inequality: ∫ b (x − a)p (x − b)q f (x)dx a ∫ 1 p+q+1 = (b − a) (1 − t)p tq f (ta + (1 − t)b)dt 0 k } k−1 [∫ 1 ]1 {∫ 1 k k k−1 b p+q+1 kp kq f (ta + m(1 − t) ) dt ≤ (b − a) (1 − t) t dt m 0 k } k−1 { ∫ 10 k 1 b k−1 p+q+1 k f (ta + m(1 − t) = (b − a) [β(kq + 1, kp + 1)] ) dt m [ ∫ 01 ] k−1 k k k α (t) α (t)) 1 b h m(1−h p+q+1 . ≤ (b − a) [β(kq + 1, kp + 1)] k |f (a)| k−1 |f ( )| k−1 dt m 0
(2.7)
Using H¨older inequality again, we have [∫
] k−1 k k b k−1 m(1−hα (t)) |f (a)| |f ( )| dt m 0 {[ ]1 [∫ 1 ] k−1 } k−1 ∫ 1 k k k [ k2 k α α (t)) ] k b h (t) m(1−h k−1 |f (a)| k−1 ≤ |f ( )| k−1 dt dt m {[ ∫0 1 ] 1 [ ∫0 1 ] k−1 } k−1 2 k k k [ k k α (t) 2 m(1−hα (t)) ] b h ( ) ≤ |f (a)| k−1 dt |f ( )| k−1 dt m 0 0 1
k hα (t) k−1
(2.8)
Combining with (2.7) and (2.8), we can conclude that (2.4) holds. Hence, the proof of theorem 2.2 is completed. Remark 2.2 If α = 1, then we can conclude the following inequality: ∫
b
(x − a) (x − b) f (x)dx ≤ (b − a) p
q
p+q+1
a
[∫
1
× 0
[β(kq + 1, kp + 1)]
k2 m b (1−h(t)) |f ( )| (k−1)2 m
]
1 k
[∫
(k−1)2 k2
1
|f (a)|
k2 h(t) k−1
] k−1 k2 dt
0
.
Theorem 2.3 Let f : [a, b] → ℜ, 0 ≤ a < b be continuous on [a, b] such that f ∈ L([a, b]). If the mapping |f |l (l ≥ 1) is (h − (α, m))−logarithmically convex on [a, b] for all t ∈ (0, 1) and (α, m) ∈ (0, 1]2 , then ∫
b
(x − a) (x − b) f (x)dx ≤ (b − a) p
q
a
p+q+1
[∫ ×
1
0
|f (
[β(q + 1, p + 1)]
b )| m
l2 m(1−hα (t)) (l−1)2
l−1 l
[ ] 12 [ ∫ l β(ql + 1, pl + 1)
2 ] (l−1) 3
dt
1
|f (a)|
l3 hα (t) l−1
] l−1 3 dt
0
l
(2.9) where β(x, y) =
∫1 0
(t)x−1 (1 − t)y−1 dt.
Proof. Using Lemma 2.1 , we have ∫ b ∫ 1 p q p+q+1 (x − a) (x − b) f (x)dx = (b − a) (1 − t)p tq f (ta + (1 − t)b)dt. a
(2.10)
0
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Since, |f |l is (h − (α, m))−logarithmically convex on [a, b], we have |f (ta + (1 − t)b)|l = |f (ta + m(1 − t)(
b l b α α ))| ≤ |f (a)|lh (t) |f ( )|lm(1−h (t)) . m m
From (2.9), (2.10), (2.11) and H¨older inequality, we can achieve the following inequality: ∫ b (x − a)p (x − b)q f (x)dx a ∫ 1 b p+q+1 = (b − a) (1 − t)p tq f (ta + m(1 − t)( ))dt m ∫0 1 l−1 1 b ≤ (b − a)p+q+1 [(1 − t)p (t)q ] l [(1 − t)p (t)q ] l f (ta + m(1 − t)( ))dt m 0 [∫ ] l−1 { ∫ 1 }1 1 l l b l p+q+1 p q p q ≤ (b − a) (1 − t) (t) dt [(1 − t) (t) ]|f (ta + m(1 − t)( ))| dt m 0 {∫ 1 0 }1 l l−1 b l p+q+1 p q = (b − a) [β(q + 1, p + 1)] l [(1 − t) (t) ]|f (ta + m(1 − t)( ))| dt . m 0 Using H¨older inequality again, we have {∫ 1 }1 l b l p q [(1 − t) (t) ]|f (ta + m(1 − t)( ))| dt m {0 ∫ 1 }1 l b lm(1−hα (t)) p q lhα (t) [(1 − t) (t) ]|f (a)| ≤ |f ( )| dt m { 0∫ 1 ] l }1 ∫ 1[ l 1 { } { b lm(1−hα (t)) l−1 } l−1 p q l lhα (t) l l ≤ [(1 − t) (t) ] dt dt |f (a)| |f ( )| m 0 [ ] 12 [ ∫ 1 0 ] l−1 2 m(1−hα (t)) l2 hα (t) l l l2 b l−1 ≤ β(ql + 1, pl + 1) |f (a)| l−1 |f ( )| dt m 0 2 l−1 [ ∫ ] ] (l−1) [ ] 12 [ ∫ 1 1 l3 m(1−hα (t)) l3 hα (t) l l3 l3 b |f (a)| l−1 dt |f ( )| (l−1)2 dt . ≤ β(ql + 1, pl + 1) m 0 0
(2.11)
(2.12)
(2.13)
By (2.12) and (2.13), we can achieve that (2.9) holds. Hence, the proof of theorem 2.3 is completed. Remark 2.3 If α = 1, then we can conclude the following inequality: ∫
b
(x − a) (x − b) f (x)dx ≤ (b − a) p
a
q
p+q+1
[∫ × 0
1
|f (
[β(q + 1, p + 1)]
b )| m
l2 m(1−h(t)) (l−1)2
l−1 l
[ ] 12 [ ∫ l β(ql + 1, pl + 1)
l
|f (a)|
l3 h(t) l−1
] l−1 3 dt
l
0
2 ] (l−1) 3
dt
1
.
References ¨ [1] M. E. Ozdemir, H. Kavurmaci and M. Avci, Hermite-Hadamard Type Inequalities for (h − (α, m))−convex Functions, RGMIA Research Report Collection, 14(2011)Article 31. [ONLINE: http://http://rgmia.org/papers/v14/v14a31.pdf] [2] G. H. Toader, Some generalizations of the convexity, Proceedings of the Colloquium on Approximation and Optimization, 1984: 329-338. 6 379
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[3] V. G.Mihesan, A generalization of the convexity, Seminar on Functional Equations, Approx. and Convex. 1993. [4] D. S. Mitrinovi´c, J. E. Pe˘car´c and A. M. Fink, Classical and new inequalities in analysis, Mathematics and its Applications (East European Series), 61, Kluwer Acad. Publ., Dordrecht, 1993. [5] R. F. Bai, F. Qi, and B. Y. Xi, Hermite-Hadamard type inequalities for the m- and (α, m)− logarithmically convex functions, Filomat 27 (2013), no.1, 1-7. [6] W. J. Liu, New integral inequalities via (α, m)−convexity and quasi-convexity, arXiv:1201.6226v1 [math.FA] [7] Z. P. Ji, T. Y. Zhang, Integral inqualities of Hermite-Hadamard type for (α, m)−GA-convex functions,http://arxiv.org/abs/1306.0852v1[math.FA] [8] S. S. Dragomir, On some new inequalities of Hermite-Hadamard type for m-convex functions, Tamkang J. Math. 33 (2002) 45-55. [9] T. Y. Zhang, A. P. Ji, F. Qi, On integral inequalities of Hermite-Hadamard type for sgeometrically convex functions, Abstr. Appl. Anal. 2012 (2012), Article ID 560586, 14 pages; Available online at http://dx.doi.org/10.1155/2012/560586 [10] S. H. Wang, B. Y. Xi, F. Qi, Some new inequalities of Hermite-Hadamard type for n-time differentiable functions which are m-convex, Analysis (Munich) 32 (2012) 247-262; Available online at http://dx.doi.org/10.1524/anly.2012.1167 [11] M. Iqbal, M. I. Bahtti and M. Muddassar, Hadamard-type inequalities for h-Convex functions, Pakistan Journal of Science (ISSN 1016-2526), Vol. 63 No. 3 September 2011 pp. 170-175. [12] M. Muddassar, M. I. Bhatti and M. Iqbal, Some New s-Hermite Hadamard Type Ineqalities for Differentiable Functions and Their Applications, Proceedings of the Pakistan Academy of Sciences 49(1) (2012), 9-17. ¨ [13] M. E. Ozdemir, E. Set and M. Alomari, Integral inequalities via several kinds of convexity, Creat. Math. Inform. 20 (2011), no. 1, 62-73.
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On Gosper’s q-Trigonometric Function Mahmoud Jafari Shah Belaghi Bah¸ce¸sehir University, Istanbul, Turkey [email protected] Nuri Kuruo˘ glu ˙ Istanbul Geli¸sim University, Istanbul, Turkey [email protected] Abstract. In this paper, we study about periodicity of q-trigonometric function which was introduced by Gosper and also we rewrite the q-analogue of Legendres duplication formula with the same bases. Furthermore, we modify some identities involving q-shifted factorial. Keywords. Gosper’s q-trigonometric function, q-Gamma function, Legendres duplication formula. Mathematics Subject Classification. 11B65, 33D05.
1
Introduction
The q-shifted factorial [1, 3] is defined by ( 1 (a; q)n = Qn−1 m m=0 (1 − aq )
n = 0, n = 1, 2, ....
(1)
and it is assumed that a 6= q −m , m = 0, 1, .... The q-shifted factorial [1, 3] is also defined for any complex number α, (a; q)α =
(a; q)∞ , (aq α ; q)∞
(2)
Qn where (a; q)∞ := limn→∞ m=0 (1 − aq m ) and the principal value of q α is taken and it is assumed that 0 < q < 1. The q-Gamma function was introduced by Thomae [6] and Jackson [5], (see [3], page 20) (q; q)∞ Γq (x) = x (1 − q)1−x , 0 < q < 1. (3) (q ; q)∞ A q-analogue of Legendre’s duplication formula [5, 7] has the form 1 1 Γq (2x)Γq2 ( ) = (1 + q)2x−1 Γq2 (x)Γq2 (x + ). 2 2 Gosper [4] defined q-trigonometric functions as follows: (q 2z ; q 2 )∞ (q 2−2z ; q 2 )∞ , 0 < q < 1, (q; q 2 )2∞ 1+2z 2 2 (q ; q )∞ (q 1−2z ; q 2 )∞ cosq (πz) := q z , 0 < q < 1. (q; q 2 )2∞ 2
sinq (πz) := q (z−1/2)
(4)
(5) (6)
1
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It can be seen [4] that cosq (z) = sinq (
π − z). 2
(7)
By using (5), (6) and (7), one can see that, for the cases x = 0 and x = sinq (x) and cosq (x) are; sinq ( π2 ) = 1, cosq ( π2 ) = 0.
sinq (0) = 0, cosq (0) = 1,
π 2,
(8)
There are many identities involving q-shifted factorial [1, 3], but in this paper we are using the following identities; For all a ∈ C and n ∈ N, following equations hold (q 2a ; q 2 )n = (q a ; q)n (−q a ; q)n , 2
2
(a; q)2n = (a; q )n (aq; q )n ,
(10) n 2
(q 1−a−n ; q)n = (q a ; q)n (−1)n q −( )−an .
2
(9)
(11)
Main result
In the next lemma we show that the equations (9) and (10) are also valid for any complex number α, Lemma 1. For all a, α ∈ C, the following equations hold (q 2a ; q 2 )α = (q a ; q)α (−q a ; q)α , 2
2
(a; q)2α = (a; q )α (aq; q )α .
(12) (13)
Proof. To prove (12) we use (2), then we have (q 2a ; q 2 )α =
(q 2a ; q 2 )∞ . (q 2a+2α ; q 2 )∞
By using the definition of q-shifted factorial (1), we obtain (q 2a ; q 2 )∞ (q 2a+2α ; q 2 )∞ Q∞ (1 − q 2a+2i ) = Q∞i=0 2a+2α+2i ) i=0 (1 − q Q∞ (1 − q a+i )(1 + q a+i ) = Q∞ i=0 a+α+i )(1 + q a+α+i ) i=0 (1 − q Q Q∞ ∞ a+i (1 − q a+i ) ) i=0 (1 + q Q = Q∞i=0 ∞ a+α+i a+α+i (1 − q ) (1 + q ) i=0 i=0 a a (q ; q)∞ (−q ; q)∞ = a+α (q ; q)∞ (−q a+α ; q)∞ = (q a ; q)α (−q a ; q)α .
(q 2a ; q 2 )α =
2
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The proof of (12) is complete. To Prove the next equation, we use (1) and (2), then we have (a; q 2 )α (aq; q 2 )α (a; q 2 )∞ (aq; q 2 )∞ (aq 2α ; q)∞ = (a; q)2α (aq 2α ; q 2 )∞ (aq 2α+1 ; q 2 )∞ (a; q)∞ (aq 2α ; q)∞ (a; q 2 )∞ (aq; q 2 )∞ , = 2α (a; q)∞ (aq ; q 2 )∞ (aq 2α+1 ; q 2 )∞ 1
1
each fraction in the last line is equal 1, since (c; q)∞ (cq 2 ; q)∞ = (c; q 2 )∞ (see [8] , page 13). The proof of (13) is complete. In the next lemma, we want to modify the equation (11). Lemma 2. For all α and β ∈ C, the following equation holds (q 1−α−β ; q)α = (q β ; q)α
sin√q π(α + β) −(α)−αβ q 2 , sin√q π(β)
(14)
where sinq is defined as in (5). Proof. After applying the equation (2) for both numerator and denominator of the left hand side of the following equation, we obtain that (q 1−α−β ; q)α (q 1−α−β ; q)∞ (q α+β ; q)∞ = (q β ; q)α (q 1−β ; q)∞ (q β ; q)∞ and by using the definition of sinq which is written in (5), we have sin√q π(α + β) −(α)−αβ (q 1−α−β ; q)∞ (q α+β ; q)∞ . q 2 = 1−β β (q ; q)∞ (q ; q)∞ sin√q π(β) Therefore proof is complete. Theorem 1. For all n ∈ N and x ∈ C, the following equations hold sinq (x + nπ) = (−1)n sinq (x), cosq (x + nπ) = (−1)n cosq (x), tanq (x + nπ) = tanq (x), cotq (x + nπ) = cotq (x).
(15) (16) (17) (18)
Proof. We use lemma 2 for prove the equation (15). Taking any arbitrary n ∈ N and a ∈ C, then we have (q 1−n−a ; q)n = (q a ; q)n
sin√q π(a + n) −(n)−na q 2 . sin√q π(a)
(19)
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By comparing the equations (11) and (19), we obtain sin√q π(a + n) = (−1)n . sin√q π(a) √ Substituting q with q and x with aπ completes the proof of equation (15). By using (7) and (15), one can shows that (16) is valid for all n ∈ N and sin (x) cos (x) x ∈ C, and the last two equations (17) and (18) comes from cosqq (x) and sinqq (x) , respectively. Remark 1. The cosq (x) is an even function, its come from the definition (6) directly. And the sinq (x) is an odd function, since by using (7), we can write sinq (x) = cosq ( π2 −x) and also we know that cosq (x) is an even function then we have sinq (x) = cosq (x− π2 ), again apply (7), we obtain cosq (x− π2 ) = sinq (π−x). Now by using the Theorem 1, we obtain sinq (π − x) = −sinq (−x). Therefore sinq (x) = −sinq (−x). Lemma 3. For all k ∈ Z, zeroes of q-sine and q-cosine functions are kπ and (2k+1)π , respectively. 2 Proof. Since sinq is an odd function, therefore its enough to prove the lemma for positive value of k. We prove the lemma for positive value of k by induction. For k = 1 and using (15), we have sinq (π) = sinq (0 + π) = sinq (0) = 0, since sinq (0) = 0 comes from definition of sinq . Then lemma is valid for k = 1. Assume that sinq (nπ) = 0 is true. We need to show that sinq ((n + 1)π) = 0 is also true. By using (15), we have sinq ((n + 1)π) = sinq (nπ + π) = (−1)sinq (nπ) = 0. Therefore zeroes of sinq (x) are kπ, for all k ∈ Z. About the zeroes of cosq (x), take any arbitrary k ∈ Z, and by using the (7), we have cosq (
(2k + 1)π π ) = cosq (kπ + ) = sinq (−kπ) = 0. 2 2
Therefore zeroes of cosq (x) are
(2k+1)π , 2
for all k ∈ Z.
Lemma 4. For all z ∈ C, the following equation holds 1
1
1
(q z+1 ; q)z = (−q 2 ; q 2 )2z (q 2 ; q)z .
4
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1
Proof. Taking a = 1, α = 2z and substituting q with q 2 in (12), and applying 1 1 to (−q 2 ; q 2 )2z , we obtain 1
1
(q 2 ; q)z (q; q)2z (q 2 ; q)z = 1 1 . (−q ; q )2z z+1 (q ; q)z (q 2 ; q 2 )2z (q z+1 ; q)z 1 2
1 2
(20)
By using (2), the right hand side of (20) can be written as 1
(q; q)2z (q 2 ; q)z = (q z+1 ; q)z (q 21 ; q 12 )2z
(q;q)∞ (q 2z+1 ;q)∞ (q z+1 ;q)∞ (q 2z+1 ;q)∞
1
(q 2 ; q)z 1 2
1
(q ; q 2 )2z
,
1
=
(q 2 ; q)z (q; q)∞ , z+1 (q ; q)∞ (q 12 ; q 12 )2z 1
= (q; q)z
(q 2 ; q)z 1
1
(q 2 ; q 2 )2z
1
.
(21)
1
By substituting q with q 2 and then taking a = q 2 in equation (13), one can see that the right hand side of (21) is equal 1 and this completes the proof. Lemma 5. For all z ∈ C, the following equation holds (q
1 2 −z
; q)z = (q
z+1
; q)z
q−
z2 2 1
1
(−q 2 ; q 2 )2z
cos√q (πz).
(22)
Proof. By using the Lemma 4 the equation (22) can be written as 1
1
(q 2 −z ; q)z = (q 2 ; q)z q −
z2 2
cos√q (πz).
(23)
The equation (23) is a special case of lemma 2 when β = 21 , since cosq (z) = sinq ( π2 − z) and also cosq is an even function. Corollary 1. For the positive integers value of n, Lemma 5 deduce to q−
1
(q 2 −n ; q)n = (−1)n (q n+1 ; q)n
1
n2 2 1
(−q 2 ; q 2 )2n
.
Proof. The result is obtained by using Theorem 1. Euler (see [2], page 271 or [3], page 222) found the following formula in connection with partitions, (−q; q)∞ (q; q 2 )∞ = 1. In the next lemma, we want to generalize this Euler’s formula. 5
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Theorem 2. For all z ∈ C, the following equation holds (q z+1 ; q)z = (−q; q)z (q; q 2 )z . 1
1
Proof. By substituting q with q 2 and then taking a = −q 2 in equation (13), we have 1 1 1 (−q 2 ; q 2 )2z = (−q 2 ; q)z (−q; q)z , now, we apply the result to Lemma 5 and obtain 1
1
(q z+1 ; q)z = (−q 2 ; q)z (−q; q)z (q 2 ; q)z .
(24)
Taking a = 12 in the equation (12) and then applying to the right hand side of the equation (24) completes the proof. Theorem 3. For all x ∈ C, the following equation holds, 1 1 1 1 Γq (2x)Γq ( ) = Γq (x)Γq (x + )(−q 2 ; q 2 )2x−1 . 2 2
Proof. By using the definition of q-Gamma function (3) and then applying the equation (2), we can write 1
Γq (2x)Γq ( 21 ) (q x ; q)∞ (q x+ 2 ; q)∞ (q x ; q)x = = , 1 1 Γq (x)Γq (x + 21 ) (q 2x ; q)∞ (q 2 ; q)∞ (q 2 ; q)x the last equation holds since (q x ; q)x =
(q x ;q)∞ (q 2x ;q)∞
1
and (q 2 ; q)x =
(25) 1
(q 2 ;q)∞ 1 (q 2 +x ;q)∞
. Tak-
1 2
ing β = in Lemma 2 and applying for the denominator of the last fraction in (25), we get (q x ; q)x 1
(q 2 ; q)x
= =
(q x ; q)x sin√q π( 12 + x) − x2 q 2, 1 (q 2 −x ; q)x sin√q π( 12 ) (q x ; q)x (q
1 2 −x
; q)x
cos√q (πx) q −
x2 2
.
Now, by using Lemma 4, we have (q x ; q)x (q
1 2 −x
; q)x
cos√q (πx) q −
x2 2
=
1 1 (q x ; q)x (−q 2 ; q 2 )2x . (q x+1 ; q)x
Making use of (2), we have 1 1 1 1 (q x ; q)x (q x ; q)∞ (q 2x+1 ; q)∞ 2;q2) (−q = (−q 2 ; q 2 )2x . 2x (q x+1 ; q)x (q 2x ; q)∞ (q x+1 ; q)∞
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After expanding the first and second fractions and then a simplification, yields 1 1 1 1 (q x ; q)∞ (q 2x+1 ; q)∞ 1 − qx (−q 2 ; q 2 )2x , (−q 2 ; q 2 )2x = 2x x+1 2x (q ; q)∞ (q ; q)∞ 1−q 1 1 1 (−q 2 ; q 2 )2x , = 1 + qx 1
1
= (−q 2 ; q 2 )2x−1 .
References [1] Ernst, T., A Comprehensive Treatment of Q-calculus, Springer, 2012. [2] Euler, L., Introductio in Analysin Infinitorum, Vol. 1, Lausanne, 1748. [3] Gasper, G., Rahman, M., Basic hypergeometric series. Vol. 96, Cambridge university press, 2004. [4] Gosper, R. W., Experiments and discoveries in q-trigonometry, in Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics, Editors: F. G. Garvan and M. E. H. Ismail. Kluwer, Dordrecht, Netherlands, pp. 79–105, 2001. [5] Jackson, F. H., A Generalisation of the Functions Γ(n) and xn , Proceedings of the Royal Society of London, pp. 64–72, 1904. [6] Thomae, J., Beitr¨ age zur Theorie der durch die Heinesche Reihe:... darstellbaren Functionen, Journal f¨ ur die reine und angewandte Mathematik,Vol. 70, pp. 258–281, 1869. [7] Jackson, F. H., The basic gamma-function and the elliptic functions, Proceedings of the Royal Society of London. Series A, Vol. 76, no. 508, pp. 127– 144, 1905. [8] Berndt, B. C., Ramanujan’s Notebooks, Part III, Springer-Verlag, New York, 1991.
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APPROXIMATE QUADRATIC FORMS ON RESTRICTED DOMAINS WON-GIL PARK AND JAE-HYEONG BAE* Abstract. Let r, s be nonzero real numbers with r + s = 1. In [9], Najati and Jung investigated a quadratic functional equation g(rx + sy) + rs g(x − y) = rg(x) + sg(y). We introduce a functional equation f (rx + sy, rz + sw) + rsf (x − y, z − w) = rf (x, z) + sf (y, w) and investigate the relation between the above two functional equations. And we find out the general solution and the Hyers-Ulam stability of the latter on restricted domains.
1. Introduction In 1940 and in 1968, Ulam [12] proposed the general Ulam stability problem: “When is it true that by slightly changing the hypotheses of a theorem one can still assert that the thesis of the theorem remains true or approximately true?” In 1941, Hyers [7] solved this problem for linear mappings. In 1950, Aoki [2] provided a generalization of the Hyers’ theorem for additive mappings. This stability concept is also applied to the case of other functional equations. For more results on the stability of functional equations (see [5, 6, 11]). In 1998, S.-M. Jung [8] investigated the Hyers-Ulam stability for additive and quadratic mappings on restricted domains. Let X and Y be real vector spaces. For a mapping g : X → Y , consider the quadratic functional equation: (1.1)
g(x + y) + g(x − y) = 2g(x) + 2g(y).
In 1989, J. Aczel [1] solved the solution of the equation (1.1). Later, many different quadratic functional equations were solved by numerous authors [3, 8, 10]. In recent, A. Najati and S.-M. Jung [9] introduced a generalized quadratic functional equation (1.2)
g(rx + sy) + rs g(x − y) = rg(x) + sg(y),
where r, s are nonzero real numbers with r + s = 1. In 2007, the authors [4] solved the solution of the 2-variable quadratic functional equation (1.3)
f (x + y, z + w) + f (x − y, z − w) = 2f (x, z) + 2f (y, w).
Consider a generalized 2-variable quadratic functional equation (1.4)
f (rx + sy, rz + sw) + rsf (x − y, z − w) = rf (x, z) + sf (y, w),
where r, s are nonzero real numbers with r + s = 1. 2000 Mathematics Subject Classification. Primary 39B52, 39B72. Key words and phrases. Solution, Stability, Approximate quadratic form. * Corresponding author. 1
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WON-GIL PARK AND JAE-HYEONG BAE
2
In this paper, we investigate the relation between (1.2) and (1.4) by the same method as the proofs of Theorem 1 and Theorem 2 in [4]. And we find out the general solution and the Hyers-Ulam stability of (1.4) in the spirit of Najati and Jung [9]. 2. Relation between (1.2) and (1.4) The functional equation (1.4) induces the quadratic functional equation (1.2) as follows. Theorem 2.1. Let f : X × X → Y be a mapping satisfying (1.4) and let g : X → Y be the mapping given by (2.1)
g(x) := f (x, x)
for all x ∈ X, then g satisfies (1.2). Proof. By (1.4) and (2.1), we obtain g(rx + sy) + rsg(x − y) = f (rx + sy, rx + sy) + rsf (x − y, x − y) = rf (x, x) + sf (y, y) = rg(x) + sg(y) for all x, y ∈ X. Example 1. Let X be a real algebra and D : X → X a derivation on X. Define a mapping f : X × X → X by f (x, y) := D(xy) = xD(y) + D(x)y for all x, y ∈ X. Then we see that f (rx + sy, rz + sw) + rsf (x − y, z − w) = D[(rx + sy)(rz + sw)] + rsD[(x − y)(z − w)] = (rx + sy)D(rz + sw) + D(rx + sy)(rz + sw) + rs[(x − y)D(z − w) + D(x − y)(z − w)] = (rx + sy)[rD(z) + sD(w)] + [rD(x) + sD(y)](rz + sw) ( ) + rs (x − y)[D(z) − D(w)] + [D(x) − D(y)](z − w) = r2 xD(z) + s2 yD(w) + r2 D(x)z + s2 D(y)w + rsxD(z) + rsyD(w) + rsD(x)z + rsD(y)w = r[xD(z) + D(x)z] + s[yD(w) + D(y)w] = rD(xz) + sD(yw) = rf (x, z) + sf (y, w) for all x, y, z, w ∈ X. Thus f satisfies (1.4). Define a mapping g : X → X by g(x) := D(x2 ) = xD(x) + D(x)x for all x ∈ X. Then g satisfies (2.1). By Theorem 2.1, g satisfies (1.2). The quadratic functional equation (1.2) induces the functional equation (1.4) with an additional condition. Theorem 2.2. Let a, b, c ∈ R and g : X → Y be a mapping satisfying (1.2). If f : X × X → Y is the mapping given by b (2.2) f (x, y) := ag(x) + [g(x + y) − g(x − y)] + cg(y) 4
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for all x, y ∈ X, then f satisfies (1.4). Furthermore, (2.1) holds if r is a rational number and a + b + c = 1. Proof. By (1.2) and (2.2), we see that f (rx + sy, rz + sw) + rsf (x − y, z − w) ) ( )] b[ ( = ag(rx + sy) + g r(x + z) + s(y + w) − g r(x − z) + s(y − w) + cg(rz + sw) 4 ) ( ] b[ +rs ag(x − y) + g(x − y + z − w) − g(x − y − z + w) + cg(z − w) 4 ) ( )] b[ ( = ag(rx + sy) + rsag(x − y) + g r(x + z) + s(y + w) + rsg (x + z) − (y + w) 4 ) ( )] b[ ( − g r(x − z) + s(y − w) + rsg (x − z) − (y − w) + cg(rz + sw) + rscg(z − w) 4 [ ] b[ ] = a g(rx + sy) + rsg(x − y) + rg(x + z) + sg(y + w) 4 ] [ ] b[ − rg(x − z) + sg(y − w) + c g(rz + sw) + rsg(z − w) 4 ] [ ]) b( [ = a[rg(x) + sg(y)] + r g(x + z) − g(x − z) + s g(y + w) − g(y − w) + c[rg(z) + sg(w)] 4 = rf (x, z) + sf (y, w) for all x, y, z, w ∈ X. Let r be a rational number. Since g satisfies (1.2), it also satisfies (1.1) (see Theorem 2.3. in [9]). Letting x = y = 0 and y = x in (1.1), respectively, g(0) = 0 and g(2x) = 4g(x) for all x ∈ X. By (2.2) and the above two equalities, b f (x, x) = ag(x) + [g(2x) − g(0)] + cg(x) 4 = (a + b + c)g(x) = g(x) for all x ∈ X. Example 2. Consider the function g : R2 → R given by g(x) := xT Ax for all x ∈ R2 , where A is a 2 × 2 real matrix. Then we see that g(rx + sy) + rs g(x − y) = (rx + sy)T A(rx + sy) + rs(x − y)T A(x − y) = (rxT + syT )A(rx + sy) + rs(xT − yT )A(x − y) = r2 xT Ax + rs(xT Ay + yT Ax) + s2 yT Ay + rs(xT Ax − xT Ay − yT Ax + yT Ay) = r(r + s)xT Ax + s(r + s)yT Ay = rg(x) + sg(y)
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for all x, y ∈ R2 , where r, s are nonzero real numbers with r + s = 1. Thus g satisfies (1.2). Let a, b, c ∈ R and define f (x, y) := ag(x) + 4b [g(x + y) − g(x − y)] + cg(y) for all x, y ∈ R2 . By Theorem 2.2, the function f satisfies (1.4). In fact, ( )T ( )( ) x a 2b Ax f (x, y) = b y Ay c 2 for all x, y ∈ R2 . Example 3. Let Mn be the algebra of n × n real matrices. Consider the mapping g : Mn → Mn given by g(A) := A2 for all A ∈ Mn . Then we see that g(rA + sB) + rs g(A − B) = (rA + sB)2 + rs(A − B)2 = r2 A2 + rs(AB + BA) + s2 B 2 + rs(A2 − AB − BA + B 2 ) = r2 A2 + rsAB + rsBA + s2 B 2 + rsA2 − rsAB − rsBA + rsB 2 = r(r + s)A2 + s(r + s)B 2 = rg(A) + sg(B) for all A, B ∈ R2 , where r, s are nonzero real numbers with r + s = 1. Thus g satisfies (1.2). Let a, b, c ∈ R and define f (A, B) := aA2 + bA ◦ B + cB 2 , where A ◦ B the Jordan product 21 (AB + BA) of A and B for all A, B ∈ Mn . Then the mapping f : Mn × Mn → Mn satisfies (2.2). By Theorem 2.2, the mapping f satisfies (1.4). 3. Solution of the equation (1.4) We recall that r, s are nonzero real numbers with r + s = 1. In the following theorem, we find out the general solution of the functional equation (1.4). Theorem 3.1. Let f : X × X → Y be a mapping such that f (x, y) = f (−x, −y) for all x, y ∈ X. Then f satisfies (1.3) if it satisfies (1.4). If r and s are rational numbers and f satisfies (1.3), then it also satisfies (1.4). Proof. Letting x = y = z = w = 0 in (1.4), we gain f (0, 0) = 0. Putting y = w = 0 in (1.4), we get f (rx, rz) = r2 f (x, z) for all x, z ∈ X. Replacing x by x + y and z by z + w in (1.4), we have (3.1)
f (rx + y, rz + w) = rf (x + y, z + w) + sf (y, w) − rsf (x, z)
for all x, y, z, w ∈ X. Replacing y by −y and w by −w in (3.1), we obtain f (rx − y, rz − w) = rf (x − y, z − w) + sf (y, w) − rsf (x, z) for all x, y, z, w ∈ X. Adding (3.1) to the above equation, we see that (3.2) f (rx + y, rz + w) + f (rx − y, rz − w) = r[f (x + y, z + w)+ f (x − y, z − w)] + 2sf (y, w) − 2rsf (x, z) for all x, y, z, w ∈ X. Replacing y by x + ry and w by z + rw in (3.1), we obtain (3.3)
f (r(x + y) + x, r(z + w) + z) = rf (2x + ry, 2z + rw) + sf (x + ry, z + rw) − rsf (x, z)
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for all x, y, z, w ∈ X. Replacing x, y, z, w by 2x, ry, 2z, rw in (3.1), respectively, we obtain (3.4)
rf (2x + ry, 2z + rw) = r2 f (2x + y, 2z + w) − r2 sf (y, w) + rsf (2x, 2z)
for all x, y, z, w ∈ X. Replacing y by ry and w by rw in (3.1), we obtain (3.5)
sf (x + ry, z + rw) = rsf (x + y, z + w) − rs2 f (y, w) + s2 f (x, z)
for all x, y, z, w ∈ X. Replacing x, y, z, w by x + y, x, z + w, z in (3.1), respectively, we obtain (3.6)
f (r(x + y) + x, r(z + w) + z) = rf (2x + y, 2z + w) + sf (x, z) − rsf (x + y, z + w)
for all x, y, z, w ∈ X. By (3.3), (3.4), (3.5) and (3.6), we see that (3.7)
f (2x + y, 2z + w) + 2f (x, z) + f (y, w) = 2f (x + y, z + w) + f (2x, 2z)
for all x, y, z, w ∈ X. Putting y = −x and w = −z in (3.7), we get f (2x, 2z) = 4f (x, z) for all x, z ∈ X. Therefore, it follows from (3.7) that f (2x + y, 2z + w) + f (y, w) = 2f (x + y, z + w) + 2f (x, z) for all x, y, z, w ∈ X. Replacing y by y − x and w by w − z in the above equation, we have f (x + y, z + w) + f (y − x, w − z) = 2f (x, z) + 2f (y, w) for all x, y, z, w ∈ X. Hence f satisfies (1.3). Conversely, let r and s be rational numbers and let f satisfy (1.3). Then there exist two symmetric bi-additive mappings S1 , S2 : X × X → Y and a bi-additive mapping B : X × X → Y such that f (x, y) = S1 (x, x) + B(x, y) + S2 (y, y) for all x, y ∈ X (see [4]). Since r and s are rational numbers, rf (x, z) + sf (y, w) − rsf (x − y, z − w) = r2 S1 (x, x) + 2rsS1 (x, y) + s2 S1 (y, y) + r2 B(x, z) + rsB(x, w) + rsB(y, z) + s2 B(y, w) + r2 S2 (z, z) + 2rsS2 (z, w) + s2 S2 (w, w) = S1 (rx, rx) + 2S1 (rx, sy) + S1 (sy, sy) + B(rx, rz) + B(rx, sw) + B(sy, rz) + B(sy, sw) + S2 (rz, rz) + 2S2 (rz, sw) + S2 (sw, sw) = S1 (rx + sy, rx + sy) + B(rx + sy, rz + sw) + S2 (rz + sw, rz + sw) = f (rx + sy, rz + sw) for all x, y, z, w ∈ X. Therefore f satisfies (1.4).
4. Stability of the equation (1.4) From now on, let X be a real normed space and Y a Banach space. The authors proved a generalized Hyers-Ulam stability theorem on a functional equation (1.3). The following theorem is a particular case of Theorem 4 in [4]. Theorem 4.1 Let δ ≥ 0 be fixed. If a mapping f : X × X → Y satisfies the inequality (4.1)
∥f (x + y, z + w) + f (x − y, z − w) − 2f (x, z) − 2f (y, w)∥ ≤ δ
for all x, y, z, w ∈ X, then there exists a unique 2-variable quadratic mapping F : X × X → Y such that ∥f (x, y) − F (x, y)∥ ≤ 31 δ for all x, y ∈ X.
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Using a similar method used in the paper [8], we obtain the following theorem. Theorem 4.2 Let d > 0 and δ ≥ 0 be fixed and let X ̸= {0}. If a mapping f : X × X → Y satisfies the inequality (4.1) for all x, y, z, w ∈ X with ∥x + z∥ + ∥y + w∥ ≥ d, then there exists a unique 2-variable quadratic mapping F : X × X → Y such that 5 ∥f (x, y) − F (x, y)∥ ≤ δ 3
(4.2) for all x, y ∈ X.
Proof. Assume that ∥x + z∥ + ∥y + w∥ < d. Let ) d 1( t= 1+ (x + z) 2 ∥x + z∥ ) 1( d 1+ (y + w) t= 2 ∥y + w∥
if if
∥x + z∥ ≥ ∥y + w∥; ∥x + z∥ < ∥y + w∥.
If x + z = y + w = 0, then one can choose a t ∈ X with ∥t∥ = d2 . Note that 2∥t∥ = ∥x + z∥ + d ≥ d
if
∥x + z∥ ≥ ∥y + w∥;
2∥t∥ = ∥y + w∥ + d > d
if
∥x + z∥ < ∥y + w∥.
Clearly, we see that ∥x + z − 2t∥ + ∥y + w + 2t∥ ≥ 4∥t∥ − (∥x + z∥ + ∥y + w∥) ≥ 2d − (∥x + z∥ + ∥y + w∥) ≥ 2d > d, ∥x + z − y − w∥ + 4∥t∥ ≥ ∥x + z − y − w∥ + 2d ≥ 2d > d, ∥x + z + 2t∥ + ∥ − y − w + 2t∥ ≥ max{∥x + z + 2t∥, ∥ − y − w + 2t∥} ≥ d, (4.3)
∥x + z∥ + 2∥t∥ ≥ 2∥t∥ ≥ d,
2∥t∥ + ∥y + w∥ ≥ 2∥t∥ ≥ d,
4∥t∥ ≥ 2d > d.
These inequalities (4.3) come from the corresponding substitutions attached between the right-hand sided parentheses of the following functional identity. Besides from (4.1) with x = y = z = w = 0 we get ∥f (0, 0)∥ ≤ 2δ . Therefore from (4.1), (4.3) and the new functional identity 2[f (x + y, z + w) + f (x − y, z − w) − 2f (x, z) − 2f (y, w) − f (0, 0)] = [f (x + y, z + w) + f (x − y − 2t, z − w − 2t) − 2f (x − t, z − t) − 2f (y + t, w + t)] − [f (x − y − 2t, z − w − 2t) + f (x − y + 2t, z − w + 2t) − 2f (x − y, z − w) − 2f (2t, 2t)] + [f (x − y + 2t, z − w + 2t) + f (x + y, z + w) − 2f (x + t, z + t) − 2f (−y + t, −w + t)] + 2[f (x + t, z + t) + f (x − t, z − t) − 2f (x, z) − 2f (t, t)] + 2[f (t + y, t + w) + f (t − y, t − w) − 2f (t, t) − 2f (y, w)] − 2[f (2t, 2t) + f (0, 0) − 4f (t, t)],
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we get 2∥f (x + y, z + w) + f (x − y, z − w) − 2f (x, z) − 2f (y, w) − f (0, 0)∥ ≤ δ + δ + δ + 2δ + 2δ + 2δ = 9δ, or 9 ∥f (x + y, z + w) + f (x − y, z − w) − 2f (x, z) − 2f (y, w)∥ ≤ δ + ∥f (0, 0)∥ ≤ 5δ. 2 Applying now Theorem 4.1 and the above inequality, there exists a unique 2-variable quadratic mapping F : X × X → Y satisfying (4.2) such that F (x, y) = limn→∞ 2−2n f (2n x, 2n y), completing the proof. (4.4)
We recall that r, s are nonzero real numbers with r + s = 1. Theorem 4.3. Let d > 0 and δ ≥ 0 be given. Assume that a mapping f : X × X → Y such that f (x, y) = f (−x, −y) and (4.5)
∥f (rx + sy, rz + sw) + rsf (x − y, z − w) − rf (x, z) − sf (y, w)∥ ≤ δ
for all x, y, z, w ∈ X with ∥x + z∥ + ∥y + w∥ ≥ d. Then there exists K > 0 such that f satisfies (4.6)
∥f (x + y, z + w) + f (x − y, z − w) − 2f (x, z) − 2f (y, w)∥ ≤
4(2 + |r| + |s|) δ |rs|
for all x, y, z, w ∈ X with ∥x + z∥ + ∥y + w∥ ≥ K. Proof. Let x, y, z, w ∈ X with ∥x+z∥+∥y+w∥ ≥ 2d. Since 2∥y+w∥ = ∥x+y+z+w+y+w−x−z∥ ≤ ∥x + y + z + w∥ + ∥y + w∥ + ∥x + z∥, we get 2∥y + w∥ − ∥x + z∥ ≤ ∥x + y + z + w∥ + ∥y + w∥. Since ∥x + z∥ = ∥x + y + z + w − y − w∥ ≤ ∥x + y + z + w∥ + ∥y + w∥, we have (4.7)
max{∥x + z∥, 2∥y + w∥ − ∥x + z∥} ≤ ∥x + y + z + w∥ + ∥y + w∥.
If ∥x + z∥ < d, then, since ∥x + z∥ + ∥y + w∥ ≥ 2d, we get 2∥y + w∥ > 2d = d + d > d + ∥x + z∥ and 2∥y + w∥ − ∥x + z∥ > d. So we have (4.8)
max{∥x + z∥, 2∥y + w∥ − ∥x + z∥} ≥ d.
By (4.7) and (4.8), we have ∥x + y + z + w∥ + ∥y + w∥ ≥ d. So it follows from (4.5) that (4.9)
∥f (rx + y, rz + w) + rsf (x, z) − rf (x + y, z + w) − sf (y, w)∥ ≤ δ
for all x, y, z, w ∈ X with ∥x + z∥ + ∥y + w∥ ≥ 2d. So (4.10)
∥f (ry + x, rw + z) + rsf (y, w) − rf (x + y, z + w) − sf (x, z)∥ ≤ δ
for all x, y, z, w ∈ X with ∥x + z∥ + ∥y + w∥ ≥ 2d.( ) Let x, y, z, w ∈ X with ∥x + z∥ + ∥y + w∥ ≥ 4d 1/|r| + 1 − 1/|r| . If ∥y + w∥ > 2d/|r|, then (4.11)
∥x + z∥ + ∥x + ry + z + rw∥ ≥ |r|(∥y + w∥) ≥ 2d.
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If ∥y + w∥ ≤ 2d/|r|, then ∥x + z∥ ≥ 2d(1/|r| + 2|1 − 1/|r||) and (4.12)
∥x + z∥ + ∥x + ry + z + rw∥ ≥ 2∥x + z∥ − |r| · ∥y + w∥ ≥
(
) 1 2 + 4 1 − − 1 ≥ 2d. |r| |r|
Therefore we get that ∥x + z∥ + ∥x + ry + z + rw∥ ≥ 2d from (4.11) and (4.12). Hence by (4.9) we have (4.13) ∥f (r(x + y) + x, r(z + w) + z) + rsf (x, z) − rf (2x + ry, 2z + rw) − sf (x + ry, z + rw)∥ ≤ δ ) ) ( ( for all x, y, z, w ∈ X with ∥x + z∥ + ∥y + w∥ ≥ 4d 1/|r| + 1 − 1/|r| . Set M := 4d 1/|r| + 1 − 1/|r| . Then M ≥ 2d, ∥2x + 2z∥ + ∥y + w∥ ≥ M ≥ 4d (4.14) ∥x + y + z + w∥ + ∥x + z∥ ≥ 2 for all x, y, z, w ∈ X with ∥x + z∥ + ∥y + w∥ ≥ M . From (4.9) and (4.10), we get the following inequalities: ∥f (r(x + y) + x, r(z + w) + z) + rsf (x + y, z + w) − rf (2x + y, 2z + w) − sf (x, z)∥ ≤ δ, ∥rf (ry + 2x, rw + 2z) + r2 sf (y, w) − r2 f (2x + y, 2z + w) − rsf (2x, 2z)∥ ≤ δ|r|, ∥sf (ry + x, rw + z) + rs2 f (y, w) − rsf (x + y, z + w) − s2 f (x, z)∥ ≤ δ|s|. Using (4.13) and the above three inequalities, we get (4.15)
∥f (2x + y, 2z + w) + 2f (x, z) + f (y, w) − 2f (x + y, z + w) − f (2x, 2z)∥ ≤
2 + |r| + |s| δ |rs|
for all x, y, z, w ∈ X with ∥x + z∥ + ∥y + w∥ ≥ M . If x, y, z, w ∈ X with ∥x + z∥ + ∥y + w∥ ≥ 2M , then ∥x + z∥ + ∥y + w − x − z∥ ≥ M . So it follows from (4.15) that (4.16)
∥f (x + y, z + w) + 2f (x, z) + f (y − x, w − z) − 2f (y, w) − f (2x, 2z)∥ ≤
2 + |r| + |s| δ |rs|
for all x, y, z, w ∈ X with ∥x + z∥ + ∥y + w∥ ≥ 2M . Letting y = 0 and w = 0 in (4.16), we get ∥4f (x, z) − f (2x, 2z) − 2f (0, 0)∥ ≤
(4.17)
2 + |r| + |s| δ |rs|
for all x, z ∈ X with ∥x+z∥ ≥ 2M . Letting x = 0 and z = 0 (and y, w ∈ X with ∥y∥ ≥ 2M, ∥w∥ ≥ 2M ) in (4.16), we get ( ) (4.18) ∥f (0, 0)∥ ≤ (2 + |r| + |s|)/|rs| δ. Therefore it follows from (4.16), (4.17) and (4.18) that ∥f (x + y,z + w) + f (y − x, w − z) − 2f (x, z) − 2f (y, w)∥ ≤ ∥f (x + y, z + w) + 2f (x, z) + f (y − x, w − z) − 2f (y, w) − f (2x, 2z)∥ + ∥4f (x, z) − f (2x, 2z) − 2f (0, 0)∥ + 2∥f (0, 0)∥ ≤
4(2 + |r| + |s|) δ |rs|
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for all x, y, z, w ∈ X with ∥x + z∥ ≥ 2M . Since f (x, y) = f (−x, −y) for all x, y ∈ X, the above inequality holds for all x, y, z, w ∈ X with ∥y + w∥ ≥ 2M . Therefore ∥f (x + y, z + w) + f (y − x, w − z) − 2f (x, z) − 2f (y, w)∥ ≤
4(2 + |r| + |s|) δ |rs|
for all x, y, z, w ∈ X with ∥x + z∥ + ∥y + w∥ ≥ 4M . This completes the proof by letting K := 4M .
Theorem 4.4 Let d > 0 and δ ≥ 0 be given. Assume that a mapping f : X × X → Y such that f (x, y) = f (−x, −y) and (4.5) for all x, y, z, w ∈ X with ∥x + z∥ + ∥y + w∥ ≥ d. Then there exists K > 0 such that f satisfies ∥f (x + y, z + w) + f (x − y, z − w) − 2f (x, z) − 2f (y, w)∥ ≤
19(2 + |r| + |s|) δ |rs|
for all x, y, z, w ∈ X. Proof. By Theorem 4.3, there exists K > 0 such that f satisfies (4.6) for all x, y, z, w ∈ X with ∥x + z∥ + ∥y + w∥ > K. By (4.4) and (4.18), we get that 18(2 + |r| + |s|) δ + ∥f (0, 0)∥ |rs| 19(2 + |r| + |s|) δ ≤ |rs|
∥f (x + y, z + w) + f (x − y, z − w) − 2f (x, z) − 2f (y, w)∥ ≤
for all x, y, z, w ∈ X.
Theorem 4.5 Let d > 0 and δ ≥ 0 be given. Assume that a mapping f : X × X → Y such that (4.5) and f (x, y) = f (−x, −y) for all x, y, z, w ∈ X with ∥x + z∥ + ∥y + w∥ ≥ d. Then there exists a unique quadratic mapping F : X × X → Y such that F (x, y) = limn→∞ 4−n f (2n x, 2n y) and ∥f (x, y) − Q(x, y)∥ ≤
19(2 + |r| + |s|) δ 3|rs|
for all x, y ∈ X.
Proof. The result follows from Theorem 4.1 and Theorem 4.4.
Corollary 4.6. Let r and s be rational numbers and a mapping f : X × X → Y satisfy f (x, y) = f (−x, −y) for all x, y ∈ X. Then f is quadratic if and only if the asymptotic condition (4.19) ∥f (rx + sy, rz + sw) + rsf (x − y, z − w) − rf (x, z) − sf (y, w)∥ → 0 as ∥x + z∥ + ∥y + w∥ → ∞ holds true. Proof. The asymptotic condition (4.19) is equivalent to the condition that there exists a sequence {δn } monotonically decreasing to 0 such that (4.20)
∥f (rx + sy, rz + sw) + rsf (x − y, z − w) − rf (x, z) − sf (y, w)∥ ≤ δn
for all x, y, z, w ∈ X with ∥x + z∥ + ∥y + w∥ ≥ n.
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WON-GIL PARK et al 388-397
J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 20, NO.2, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
WON-GIL PARK AND JAE-HYEONG BAE
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It follows from (4.20) and Theorem 4.4 that there exists a unique quadratic mapping Qn : X × X → Y such that 19(2 + |r| + |s|) (4.21) ∥f (x, y) − Qn (x, y)∥ ≤ δn |rs| for all x, y ∈ X. Since {δn } is monotonically decreasing, the quadratic mapping Qm satisfies (4.21) for all m ≥ n. The uniqueness of Qn implies Qm = Qn for all m ≥ n. By letting n → ∞ in (4.21), we conclude that f is quadratic. The converse is trivial. Acknowledgment This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology(grant number 2014014135) References [1] J. Acz´el, and J. Dhombres, Functional Equations in Several Variables, Cambridge Univ. Press, Cambridge, 1989. [2] T. Aoki, On the stability of the linear transfomation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. [3] J.-H. Bae and K.-W. Jun, On the generalized Hyers-Ulam-Rassias stability of an n-dimensional quadratic functional equation, J. Math. Anal. Appl. 258 (2001), 183–193. [4] J.-H. Bae and W.-G. Park, A functional equation originating from quadratic forms, J. Math. Anal. Appl. 326 (2007), 1142–1148. [5] D. G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math. Soc. 57 (1951), 223– 237. [6] P. M. Gruber, Stability of isometries, Trans. Amer. Math. Soc. 245 (1978), 263–277. [7] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. 27 (1941), 222–224. [8] S.-M. Jung , On the Hyers-Ulam stability of the functional equations that have the quadratic property, J. Math. Anal. Appl. 222 (1998), 126–137. [9] A. Najati and S.-M. Jung , Approximate quadratic mappings on restricted domains, J. Inequal. Appl. 2010 (2010), Art. No. 503458. [10] W.-G. Park and J.-H. Bae, On a bi-quadratic functional equation and its stability, Nonlinear Anal. 62 (2005), 643–654. [11] Th. M. Rassias, On the stability of linear mappings in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [12] S. M. Ulam, A Collection of Mathematical Problems, Interscience Publishers, New York, 1968, p.63. Won-Gil Park, Department of Mathematics Education, College of Education, Mokwon University, Daejeon 302-729, Republic of Korea E-mail address: [email protected] Jae-Hyeong Bae, Humanitas College, Kyung Hee University, Yongin 446-701, Republic of Korea E-mail address: [email protected]
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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 20, NO. 2, 2016
A Recurrent Neural Fuzzy Network, George A. Anastassiou, and Iuliana F. Iatan,…………213 Qualitative Behavior of some Rational Difference Equations, H. El-Metwally, and E. M. Elsayed,……………………………………………………………………………………….226 Worse-Case Conditional Value-at-Risk for Asymmetrically Distributed Asset Scenarios Returns, Zhifeng Dai, Donghui Li, and Fenghua Wen,………………………………………..……….237 A Note on the Interval-Valued Similarity Measure and the Interval-Valued Distance Measure Induced by the Choquet Integral with Respect to an Interval-Valued Capacity, Jeong Gon Lee and Lee-Chae Jang,……………………………………………………………………………252 n-Jordan *-Derivations on Induced Fuzzy C*-Algebras, Gang Lu, Yanduo Wang, and Pengyu Ye,………………………………………………………………………………………………266 Global Stability Analysis of a Delayed Viral Infection Model With Antibodies and General Nonlinear Incidence Rate, A. M. Elaiw, N. H. AlShamrani, and M. A. Alghamdi,……………277 Stability of Generalized Cubic Set-Valued Functional Equations, Dongseung Kang,…………296 A New Regularity (p-Regularity) of Stratified L-Generalized Convergence Spaces, Lingqiang Li, and Qingguo Li,………………………………………………………………………………..307 Uni-Soft Filters and Uni-Soft G-Filters in Residuated Lattices, Young Bae Jun, and Seok Zun Song,……………………………………………………………………………………………319 Mathematical Analysis of a General Viral Infection Model With Immune Response, N. H. AlShamrani, A. M. Elaiw and M. A. Alghamdi,……………………………………………….335 Newton's Method for Computing the Fifth Roots of p-Adic Numbers, Y.H. Kim, H.M. Kim, and J. Choi,………………………………………………………………………………………….353 Solution of the Ulam Stability Problem for Euler-Lagrange (α, β; k)-Quadratic Mappings, S.A. Mohiuddine, John Michael Rassias, and Abdullah Alotaibi,………………………………….363 Some Integral Inequalities via (h−(α,m))−Logarithmically Convexity, Jianhua Chen, and Xianjiu Huang,…………………………………………………………………………………………374 On Gosper's q-Trigonometric Function, Mahmoud Jafari Shah Belaghi, and Nuri Kuruoglu,.381
TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 20, NO. 2, 2016 (continued) Approximate Quadratic Forms on Restricted Domains, Won-Gil Park and Jae-Hyeong Bae,.388
Volume 20, Number 3 ISSN:1521-1398 PRINT,1572-9206 ONLINE
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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 20, NO.3, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC
FIXED POINTS IN TOPOLOGICAL VECTOR SPACE(tvs)VALUED CONE METRIC SPACES Muhammd Arshad([email protected]) Department of mathematics, International Islamic University, H-10, Islamabad-44000, Pakistan. Abstract: We use the notion of topological vector space valued cone metric space and generalized a common …xed point theorem of a pair of mappings satisfying a generalized contractive type condition. Our results extend some well-known recent results in the literature. _____________________________ 2010 Mathematics Subject Classi…cation: 47H10; 54H25. Keywords and Phrases: Topological vector space valued;cone metric space; nonnormal cones; …xed point; common …xed point. ______________________________
1
Introduction and Preliminaries
Many authors [1, 3, 4, 6, 17, 10, 11, 12, 13, 14, 15, 16, 17, 18, 21] studied …xed points results of mappings satisfying contractive type condition in Banach space valued cone metric spaces. The class of tvs-cone metric spaces is bigger than the class of cone metric spaces studied in [2, 7, 8, 19, 20]. Recently Azam et al. [5] obtain common …xed points of mappings satisfying a generalized contractive type condition in tvs-cone metric spaces. In this paper we continue these investigations to generalize the results in [1, 10]. Let (E; ) be always a topological vector space (tvs) and P a subset of E. Then, P is called a cone whenever (i) P is closed, non-empty and P 6= f0g, (ii) ax + by 2 P for all x; y 2 P and non-negative real numbers a; b, (iii) P \ ( P ) = f0g. For a given cone P E, we can de…ne a partial ordering with respect to P by x y if and only if y x 2 P . x < y will stand for x y and x 6= y, while x y will stand for y x 2 intP , where intP denotes the interior of P . De…nition 1 Let X be a non-empty set. Suppose the mapping d : X X ! E satis…es (d1 ) 0 d(x; y) for all x; y 2 X and d(x; y) = 0 if and only if x = y, (d2 ) d(x; y) = d(y; x) for all x; y 2 X, (d3 ) d(x; y) d(x; z) + d(z; y) for all x; y; z 2 X. Then d is called a topological vector space-valued cone metric on X and (X; d) is called a topological vector space-valued cone metric space. 1
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If E is a real Banach space then (X; d) is called (Banach space valued) cone metric space [1, 6, 17, 10, 21] De…nition 2 [7] Let (X; d) be a tvs-cone metric space, x 2 X and fxn gn 1 a sequence in X. Then (i) fxn gn 1 converges to x whenever for every c 2 E with 0 c there is a natural number N such that d(xn ; x) c for all n N . We denote this by limn!1 xn = x or xn ! x. (ii) fxn gn 1 is a Cauchy sequence whenever for every c 2 E with 0 c there is a natural number N such that d(xn ; xm ) c for all n; m N . (iii) (X; d) is a complete cone metric space if every Cauchy sequence is convergent. Lemma 3 [7] Let (X; d) be a tvs-cone metric space, P be a cone. Let fxn g be a sequence in X and fan g be a sequence in P converging to 0. If d(xn ; xm ) an for every n 2 N with m > n, then fxn g is a Cauchy sequence. The …xed point theorems and other results, in the case of cone metric spaces with non-normal solid cones, cannot be proved by reducing to metric spaces. Further, the vector valued function cone metric is not continuous in the general case. Remark 4 [7] Let A; B; C; D; E be non negative real numbers with A + B + C + D + E < 1; B = C or D = E: If = (A + B + D)(1 C D) 1 and = (A + C + E)(1 B E) 1 , then < 1.
2
Common Fixed Points
The following theorem improves/generalizes the results in [1, 7]. Theorem 5 Let (X; d) be a complete topological vector space-valued cone metric space, P be a cone and m; n be positive integers. If mappings F; G : X ! X satis…es: d(F x; Gy)
A d(x; y)+B d(x; F x)+Cd(y; Gy)+D d(x; Gy)+E d(y; F x) (2.1)
for all x; y 2 X, where A; B; C; D; E are non negative real numbers with A + B + C + D + E < 1; B = C or D = E: Then F and G have a unique common …xed point. Proof. For x0 2 X and k
0, de…ne x2k+1 x2k+2
= F x2k = Gx2k+1 :
2
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Then, d(x2k+1 ; x2k+2 )
= d(F x2k ; Gx2k+1 ) Ad(x2k ; x2k+1 ) + Bd(x2k ; F x2k ) + Cd(x2k+1 ; Gx2k+1 ) +Dd(x2k ; Gx2k+1 ) + Ed(x2k+1 ; F x2k ) [A + B] d(x2k ; x2k+1 ) + Cd(x2k+1 ; x2k+2 ) + D d(x2k ; x2k+2 ) [A + B + D] d(x2k ; x2k+1 ) + [C + D] d(x2k+1 ; x2k+2 ):
It implies that [1
C
D]d(x2k+1 ; x2k+2 )
[A + B + D] d(x2k ; x2k+1 ):
That is, d(x2k+1 ; x2k+2 ) where
=
d(x2k ; x2k+1 );
A+B+D . Similarly, 1 C D
d(x2k+2 ; x2k+3 )
= d(F x2k+2 ; Gx2k+1 ) Ad(x2k+2 ; x2k+1 ) + B d(x2k+2 ; F x2k+2 ) + Cd(x2k+1 ; Gx2k+1 ) +Dd(x2k+2 ; Gx2k+1 ) + E d(x2k+1 ; F x2k+2 ) A d(x2k+2 ; x2k+1 ) + B d(x2k+2 ; x2k+3 ) + Cd(x2k+1 ; x2k+2 ) +D d(x2k+2 ; x2k+2 ) + E d(x2k+1 ; x2k+3 ) [A + C + E] d(x2k+1 ; x2k+2 ) + [B + E] d(x2k+2 ; x2k+3 );
which implies d(x2k+2 ; x2k+3 ) with
=
d(x2k+1 ; x2k+2 )
A+C +E . Now by induction, we obtain for each k = 0; 1; 2; : : : 1 B E d(x2k+1 ; x2k+2 )
d(x2k ; x2k+1 ) ( ) d(x2k 1 ; x2k ) ( ) d(x2k 2 ; x2k 1 ) ( )k d(x0 ; x1 )
and d(x2k+2 ; x2k+3 )
d(x2k+1 ; x2k+2 ) (
k+1
)
d(x0 ; x1 ):
3
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For p < q and by Remark 1.4, we have d(x2p+1 ; x2q+1 )
d(x2p+1 ; x2p+2 ) + d(x2p+2 ; x2p+3 ) + d(x2p+3 ; x2p+4 ) + + d(x2q ; x2q+1 ) 2 3 q 1 q X X i i 4 ( ) + ( ) 5 d(x0 ; x1 ) i=p
i=p+1
p
( )p+1 d(x0 ; x1 ) 1 1 ( )p (1 + ) d(x0 ; x1 ): 1 (
)
+
In analogous way, we deduce d(x2p ; x2q+1 ) d(x2p ; x2q )
(1 + )
(1 + )
and d(x2p+1 ; x2q )
(1 + )
)p
( 1
)p
( 1
d(x0 ; x1 ); d(x0 ; x1 )
)p
( 1
d(x0 ; x1 ):
Hence, for 0 < n < m d(xn ; xm )
an
p
( ) d(x0 ; x1 ) with p the integer part of n=2: Fix 0 c 1 and choose a symmetric neighborhood V of 0 such that c + V intP . Since an ! 0 as n ! 1, by Lemma 1.3, we deduce that fxn g is a Cauchy sequence. Since X is a complete, there exist u 2 X such that xn ! u: Fix 0 c and choose n0 2 N be such that where an = (1 + )
d(u; x2n ) for all n
c ; 3K
d(x2n
1 ; x2n )
c ; 3K
d(u; x2n
c 3K
1)
n0 , where K = max
1+D A+E ; ; 1 B E 1 B E 1
C B
E
:
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Now, d(u; F u)
d(u; x2n ) + d(x2n ; F u) d(u; x2n ) + d(Gx2n 1 ; F u) d(u; x2n ) + A d(u; x2n 1 ) + B d(u; F u) + Cd(x2n 1 ; Gx2n 1 ) +D d(u; Gx2n 1 ) + E d(x2n 1 ; F u) d(u; x2n ) + A d(u; x2n 1 ) + B d(u; F u) + Cd(x2n 1 ; x2n ) +D d(u; x2n ) + E d(x2n 1 ; u) + E d(u; F u)] (1 + D) d(u; x2n ) + (A + E) d(u; x2n 1 ) + Cd(x2n 1 ; x2n ) +(B + E) d(u; F u):
So, d(u; F u)
K d(u; x2n ) + K d(u; x2n c c c + + =c 3 3 3
Hence d(u; F u)
1)
+ K d(x2n
1 ; x2n )
c p
for every p 2 N. From
c d(u; F u) 2 intP; p being P closed, as p ! 1, we deduce d(u; F u) 2 P and so d(u; F u) = 0. This implies that u = F u: Similarly, by using the inequality, d(u; Gu)
d(u; x2n+1 ) + d(x2n+1 ; Gu);
we can show that u = Gu; which in turn implies that u is a common …xed point of F; G and, that is u = F u = Gu: For uniqueness, assume that there exists another point u in X such that u = T u = Gu for some u in X: From d(u; u )
= d(F u; Gu ) Ad(u; u ) + Bd(u; F u) + Cd(u ; Gu ) +Dd(u; Gu ) + Ed(u ; F u) Ad(u; u ) + Bd(u; u) + Cd(u ; u ) +D d(u; u ) + Ed(u; u ) (A + D + E)d(u; u );
we obtain that u = u: By substituting D = E = 0 in the Theorem 2.1, we obtain the following result. 5
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Corollary 6 Let (X; d) be a complete topological vector space-valued cone metric space, P be a cone and m; n be positive integers. If mappings F; G : X ! X satis…es: d(F x; Gy)
A d(x; y) + B d(x; F x) + Cd(y; Gy)
(2.2)
for all x; y 2 X, where A; B; C are non negative real numbers with A+B+C < 1: Then F and G have a unique common …xed point. By substituting B = C = 0 in the Theorem 2.1, we obtain the following result. Corollary 7 Let (X; d) be a complete topological vector space-valued cone metric space, P be a cone and m; n be positive integers. If mappings F; G : X ! X satis…es: d(F x; Gy)
A d(x; y) + D d(x; Gy) + E d(y; F x)
(2.3)
for all x; y 2 X, where A; D; E are non negative real numbers with A+D+E < 1: Then F and G have a unique common …xed point. By substituting F = T m ; G = T n in the Theorem 2.1, we obtain the following result. Corollary 8 [7] Let (X; d) be a complete topological vector space-valued cone metric space, P be a cone and m; n be positive integers. If a mapping T : X ! X satis…es: d(T m x; T n y)
A d(x; y)+B d(x; T m x)+Cd(y; T n y)+D d(x; T n y)+E d(y; T m x) (2.4) for all x; y 2 X, where A; B; C; D; E are non negative real numbers with A + B + C + D + E < 1; B = C or D = E: Then T has a unique …xed point.
Corollary 9 [1] Let (X; d) be a complete Banach space-valued cone metric space, P be a cone. If a mapping F; G : X ! X satis…es: d(F x; Gy)
pd(x; y) + q [d(x; F x) + d(y; Gy)] + r [d(x; Gy) + E d(y; F x)] (2.5) for all x; y 2 X, where p; q; r are non negative real numbers with p + 2q + 2r < 1: Then F and G have a unique common …xed point.
3
Multivalued Fixed point results in tvs-valued cone metric spaces
In the sequel, let E be a locally convex Hausdor¤ tvs with its zero vector , P be a proper, closed and convex pointed cone in E with int P 6= ; and 4 denotes the induced partial ordering with respect to P . 6
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According to [5] let (X; d) be a tvs-valued cone metric space with a solid cone P and CB(X) be a collection of nonempty closed and bounded subsets of X. Let T : X ! CB(X) be a multi-valued mapping. For any x 2 X; A 2 CB(X), de…ne a set Wx (A) as follows: Wx (A) = fd(x; a) : a 2 Ag: Thus, for any x; y 2 X, we have Wx (T y) = fd(x; u) : u 2 T yg: De…nition 10 [9] Let (X; d) be a cone metric space with the solid cone P . A multi-valued mapping S : X ! 2E is said to be bounded from below if, for any x 2 X, there exists z(x) 2 E such that Sx
z(x)
P:
De…nition 11 [9] Let (X; d) be a cone metric space with the solid cone P . A cone P is said to be complete if, for any bounded from above and nonempty subset A of E, sup A exists in E. Equivalently, a cone P is complete if, for any bounded from below and nonempty subset A of E, inf A exists in E. De…nition 12 [5] Let (X; d) be a tvs-valued cone metric space with the solid cone P: A multi-valued mapping T : X ! CB(X) is said to have the lower bound property ( l.b. property) on X if, for any x 2 X, the multi-valued mapping Sx : X ! 2E de…ned by Sx (y) = Wx (T y) is bounded from below, that is, for any x; y 2 X, there exists an element `x (T y) 2 E such that Wx (T y) `x (T y) P: `x (T y) is called the lower bound of T associated with (x; y) : De…nition 13 [5] Let (X; d) be a tvs-valued cone metric space with the solid cone P: A multi-valued mapping T : X ! CB(X) is said to have the greatest lower bound property (for short, g.l.b. property) on X if the greatest lower bound of Wx (T y) exists in E for all x; y 2 X: We denote d(x; T y) by the greatest lower bound of Wx (T y); that is, d(x; T y) = inffd(x; u) : u 2 T yg: According to [20], we denote s (p) = fq 2 E : p 4 qg for all q 2 E and s (a; B) = [ s (d (a; b)) = [ fx 2 E : d (a; b) 4 xg b2B
b2B
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for all a 2 X and B 2 CB(X). For any A; B 2 CB(X), we denote s (A; B) =
\ s (a; B) \
a2A
\ s (b; A) :
b2B
Remark 14 [20] Let (X; d) be a tvs-valued cone metric space. If E = R and P = [0; +1); then (X; d) is a metric space. Moreover, for any A; B 2 CB(X), H(A; B) = inf s(A; B) is the Hausdor¤ distance induced by d: Now we present the following theorem regarding the common …xed point of multivalued mapping with g.l.b property. Theorem 15 Let (X; d) be a complete tvs-valued cone metric space with the solid (normal or non-normal) cone P and let S; T : X ! CB(X) be multivalued mappings with g.l.b property such that A d(x; y)+B d(x; Sx)+Cd(y; T y)+Dd(x; T y)+Ed(y; Sx)) 2 s (Sx; T y) (2.6) ¤or all x; y 2 X, where A; B; C; D; E are non negative real numbers with A+ B + C + D + E < 1: Then S and T have common …xed point. Proof. Let x0 be an arbitrary point in X and x1 2 Sx0 : From (2.6), we have Ad (x0 ; x1 )+B(x0 ; Sx0 )+Cd(x1 ; T x1 )+Dd(x0 ; T x1 )+Ed(x1 ; Sx0 ) 2 s (Sx0 ; T x1 ) : This implies that Ad (x0 ; x1 )+B(x0 ; Sx0 )+Cd(x1 ; T x1 )+Dd(x0 ; T x1 )+Ed(x1 ; Sx0 ) 2
\
x2Sx0
s (x; T x1 )
and Ad (x0 ; x1 )+B(x0 ; Sx0 )+Cd(x1 ; T x1 )+Dd(x0 ; T x1 )+Ed(x1 ; Sx0 ) 2 s (x; T x1 ) f or all x 2 Sx0 : Since x1 2 Sx0 ; so we have Ad (x0 ; x1 )+B(x0 ; Sx0 )+Cd(x1 ; T x1 )+Dd(x0 ; T x1 )+Ed(x1 ; Sx0 ) 2 s (x1 ; T x1 ) and Ad (x0 ; x1 )+B(x0 ; Sx0 )+Cd(x1 ; T x1 )+Dd(x0 ; T x1 )+Ed(x1 ; Sx0 ) 2 s (x1 ; T x1 ) =
[ s (d (x1 ; x)) :
x2T x1
So there exists some x2 2 T x1 ; such that Ad (x0 ; x1 )+B(x0 ; Sx0 )+Cd(x1 ; T x1 )+Dd(x0 ; T x1 )+Ed(x1 ; Sx0 ) 2 s (d(x1 ; x2 )): That is d(x1 ; x2 )
Ad (x0 ; x1 )+B(x0 ; Sx0 )+Cd(x1 ; T x1 )+Dd(x0 ; T x1 )+Ed(x1 ; Sx0 ):
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By using the greatest lower bound property (g.l.b property) of S and T; we get d (x1 ; x2 )
Ad (x0 ; x1 ) + B(x0 ; x1 ) + Cd(x1 ; x2 ) + Dd(x0 ; x2 ) + Ed(x1 ; x1 );
which implies that d (x1 ; x2 )
(A + B + D)d (x0 ; x1 ) + (C + D)d(x1 ; x2 )
which further implies that d (x1 ; x2 )
A+B+D d (x0 ; x1 ) : 1 C D
Similarly from (2.6), we get Ad (x1 ; x2 )+B(x2 ; Sx2 )+Cd(x1 ; T x1 )+Dd(x2 ; T x1 )+Ed(x1 ; Sx2 ) 2 s (T x1 ; Sx2 ) : This implies that Ad (x1 ; x2 )+B(x2 ; Sx2 )+Cd(x1 ; T x1 )+Dd(x2 ; T x1 )+Ed(x1 ; Sx2 ) 2
\
x2T x1
s (x; Sx2 )
and Ad (x1 ; x2 )+B(x2 ; Sx2 )+Cd(x1 ; T x1 )+Dd(x2 ; T x1 )+Ed(x1 ; Sx2 ) 2 s (x; Sx2 ) for all x 2 T x1 : Since x2 2 T x1 ; so we have Ad (x1 ; x2 )+B(x2 ; Sx2 )+Cd(x1 ; T x1 )+Dd(x2 ; T x1 )+Ed(x1 ; Sx2 ) 2 s (x2 ; Sx2 ) and Ad (x1 ; x2 )+B(x2 ; Sx2 )+Cd(x1 ; T x1 )+Dd(x2 ; T x1 )+Ed(x1 ; Sx2 ) 2 s (x2 ; Sx2 ) =
[ s (d (x2 ; x)) :
x2Sx2
So there exists some x3 2 Sx2 ; such that Ad (x1 ; x2 )+B(x2 ; Sx2 )+Cd(x1 ; T x1 )+Dd(x2 ; T x1 )+Ed(x1 ; Sx2 ) 2 s (d(x2 ; x3 )): That is d(x2 ; x3 )
Ad (x1 ; x2 )+B(x2 ; Sx2 )+Cd(x1 ; T x1 )+Dd(x2 ; T x1 )+Ed(x1 ; Sx2 ):
By using the greatest lower bound property (g.l.b property) of S and T; we get d(x2 ; x3 )
Ad (x1 ; x2 ) + B(x2 ; x3 ) + Cd(x1 ; x2 ) + Dd(x2 ; x2 ) + Ed(x1 ; x3 ):
which implies that d(x2 ; x3 )
(A + C + E)d (x1 ; x2 ) + (B + E)(x2 ; x3 ):
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This further implies A+C +E d (x1 ; x2 ) : 1 B E
d(x2 ; x3 )
A+C+E Let = maxf A+B+D < 1: Thus inductively, one can easily 1 C D ; 1 B E g. Then construct a sequence fxn g in X such that
x2n+1 2 Sx2n ; and
x2n+2 2 T x2n+1
d(x2n ; x2n+1 ) 4 d(x2n
1 ; x2n ):
for each n 0. We assume that xn 6= xn+1 for each n 0. Otherwise, there exists n such that x2n = x2n+1 : Then x2n 2 Sx2n and x2n is a …xed point of S and hence a …xed point of T: Similarly, if x2n+1 = x2n+2 for some n, then x2n+1 is a common …xed point of T and S: Similarly, one can show that d(x2n+1 ; x2n+2 ) 4 d(x2n ; x2n+1 ): Thus we have d(xn ; xn+1 ) 4 d(xn for each n
1 ; xn )
4
2
d(xn
2 ; xn 1 )
4
4
n
d(x0 ; x1 )
0. Now, for any m > n; consider
d(xm ; xn ) 4 d(xn ; xn+1 ) + d(xn+1 ; xn+2 ) + + d(xm n n+1 m 1 4 + + + d(x0 ; x1 )
for all n
xm )
n
4 Let c+V
1;
d(x0 ; x1 ):
1
c be given and choose a symmetric neighborhood h Vn iof such that intP . Also, choose a natural number k1 such that 1 d(x0 ; x1 ) 2 V k1 . Then
n
1
d(x1 ; x0 )
d(xm ; xn ) 4
c for all n
k1 . Thus we have
n
d(x0 ; x1 )
1
c
for all m > n. Therefore, fxn g is a Cauchy sequence. Since X is complete, there exists 2 X such that xn ! : Choose a natural number k2 such that 1+E d( ; x2n+1 ) 1 C for all n
c ; 3
A 1
k2 . Then, for all n
C
d(x2n ; v)
c B and d(x2n ; x2n ) 3 1 C
c 3 (2.7)
k2 , we have
Ad(x2n ; v)+Bd(x2n ; Sx2n )+Cd(v; T v)+Dd(x2n ; T )+Ed( ; Sx2n ) 2 s (Sx2n ; T ) :
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This implies that Ad(x2n ; v)+Bd(x2n ; Sx2n )+Cd(v; T v)+Dd(x2n ; T )+Ed( ; Sx2n ) 2
\
x2Sx2n
s (x; T v)
and we have Ad(x2n ; v)+Bd(x2n ; Sx2n )+Cd(v; T v)+Dd(x2n ; T )+Ed( ; Sx2n ) 2 s (x; T v) f or all x 2 Sx2n : Since x2n+1 2 Sx2n ; so we have Ad(x2n ; v)+Bd(x2n ; Sx2n )+Cd(v; T v)+Dd(x2n ; T )+Ed( ; Sx2n ) 2 s x2n+1 ; T v : By de…nition, we obtain Ad(x2n ; v)+Bd(x2n ; Sx2n )+Cd(v; T v)+Dd(x2n ; T )+Ed( ; Sx2n ) 2 s x2n+1 ; T v = There exists some
n
[ s d x2n+1 ; u0
u0 2T u
2 T v such that
Ad(x2n ; v)+Bd(x2n ; Sx2n )+Cd(v; T v)+Dd(x2n ; T )+Ed( ; Sx2n ) 2 s x2n+1 ; T v 2 s d(x2n+1 ; that is d(x2n+1 ;
n)
Ad(x2n ; v)+Bd(x2n ; Sx2n )+Cd(v; T v)+Dd(x2n ; T )+Ed( ; Sx2n ):
By using the greatest lower bound property (g.l.b property) of S and T; we have d(x2n+1 ; vn )
Ad(x2n ; v)+Bd(x2n ; x2n )+Cd(v;
n )+Dd(x2n ;
n )+Ed(
; x2n+1 ):
Now by using the triangular inequality, we get d (x2n+1 ;
n)
Ad(x2n ; v)+Bd(x2n ; x2n+1 )+Cd(v; x2n+1 )+Dd(x2n ;
n )+Ed(
; x2n+1 )
and it follows that d (x2n+1 ;
n)
A 1
C
d(x2n ; v) +
B 1
C
d(x2n ; x2n )) +
C +E d( ; x2n+1 ): 1 C
By using again triangular inequality, we get d( ;
n)
d( ; x2n+1 ) + d(x2n+1 ; n ) A B C +E d( ; x2n+1 ) + d(x2n ; v) + d(x2n ; x2n )) + d( ; x2n+1 ) 1 C 1 C 1 C 1+E A B d( ; x2n+1 ) + d(x2n ; v) + d(x2n ; x2n ) 1 C 1 C 1 C c c c + + =c 3 3 3
Thus, we get
c m c c for all m 1 and so m d(v; vn ) 2 P for all m 1. Since m ! as m ! 1 and P is closed, it follows that d(v; vn ) 2 P: But d(v; vn ) 2 P . Therefore, d(v; vn ) = and vn ! v 2 T v; since T v is closed. This implies that v is a common point of S and T . This completes the proof. d(v; vn )
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:
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Corollary 16 [5] Let (X; d) be a complete tvs-valued cone metric space with the solid (normal or non-normal) cone P and let S; T : X ! CB(X) be multivalued mappings with g.l.b property such that B d(x; Sx) + Cd(y; T y) 2 s (Sx; T y) ¤or all x; y 2 X, where B; C are non negative real numbers with B + C < 1: Then S and T have common …xed point. Theorem 17 [5] Let (X; d) be a complete tvs-valued cone metric space with the solid (normal or non-normal) cone P and let S; T : X ! CB(X) be multivalued mappings with g.l.b property such that Dd(x; T y) + Ed(y; Sx)) 2 s (Sx; T y) ¤or all x; y 2 X, where D; E are non negative real numbers with D + E < 1: Then S and T have common …xed point.
References [1] M. Abbas and B.E. Rhoades, Fixed and periodic point results in cone metric spaces, Appl. Math. Lett., 22 (2009) 511–515. [2] M. Abbas, Y.J. Cho and T. Nazir, Common …xed point theorems for four mappings in tvs-valued cone metric spaces, J. Math. Inequal., 5(2011), 287–299. [3] M. Abbas and G. Jungck, Common …xed point results for non-commuting mappings without continuity in cone metric spaces, J. Math. Anal. Appl. 341(2008), 416–420. [4] M. Arshad, A. Azam and P. Vetro, Some common …xed point results in cone metric spaces, Fixed Point Theory Appl., 2009, Article ID 493965 (2009), 11 pp. [5] A. Azam, N. Mehmood, Multivalued Fixed Point Theorems in tvs-Cone Metric Spaces, Fixed Point Theory and Appl., 2013, 2013:184. DOI: 10.1186/1687-1812-2013-184. [6] A. Azam, M. Arshad and I. Beg, Common …xed points of two maps in cone metric spaces, Rend. Circ. Mat. Palermo, 57(2008), 433–441. [7] A. Azam, I. Beg and M. Arshad, Fixed point in topological vector spacevalued cone metric spaces, Fixed Point Theory and Appl., 2010, Article ID 604084 (2010), 9 pp. [8] I. Beg, A. Azam and M. Arshad, Common …xed points for maps on topological vector space valued cone metric spaces, Interant. J. Math. Math. Sci., 2009, Article ID 604084 (2009), 8 pp. 12
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[9] S.H. Cho and J.S. Bae, Fixed point theorems for multivalued maps in cone metric spaces, Fixed Point Theory Appl., 87 (2011). [10] L. Huang and X. Zhang, Cone metric spaces and …xed point theorems of contractive mappings, J. Math. Anal. Appl., 332 (2007) 1468–1476. [11] D. Ili´c and V. Pavlovi´c, Common …xed points for maps on cone metric space, J. Math. Anal. Appl., 341(2008), 876–882. [12] Z. Kadelburg and S. Jankovi´c and S. Radenovi´c, A note on the equivalence of some metric and cone metric …xed point results, Appl. Math. Lett. 24 (2011), 370–374. [13] S. Jankovi´c, Z. Kadelburg and S. Radenovi´c, On cone metric spaces, A survey, Nonlinear Anal., 74(2011), 2591–260. [14] M. Khani and M. Pourmahdian, On the metrizability of cone metric spaces, Topology Appl., 158(2011), 190–193. [15] A. Latif and F.Y. Shaddad, Fixed point results for multivalued maps in cone metric spaces, Fixed Point Theory Appl., 2010 (2010), Article ID 941371. [16] S. Radenovic and B.E. Rhoades, Fixed point theorem for two non-self mappings in cone metric spaces, Comp. Math. Appl., 57 (2009), 1701–1707. [17] S. Rezapour and R. Hamlbarani, Some notes on paper “Cone metric spaces and …xed point theorems of contractive mappings", J. Math. Anal. Appl., 345(2008), 719–724. [18] S. Rezapour and R.H. Haghi, Fixed points of multifunctions on cone metric spaces, Numer. Funct. Anal. Optim., 30(2009), 1–8. [19] S. Rezapour, H. Khandani and S.M. Vaezpour, E¢ cacy of cones on topological vector spaces and application to common …xed points of multifunctions, Rend. Circ. Mat. Palermo, 59(2010), 185–197. ´ [20] W. Shatanawi, V. Cojbaš i´c, S. Radenovi´c and A. Al-Rawashdeh, Mizoguchi-Takahashi-type theorems in tvs-cone metric spaces, Fixed Point Theory Appl., 2012, 2012:106. [21] P. Vetro, Common …xed points in cone metric spaces, Rend. Circ. Mat. Palermo, 56(2007), 464–468.
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ON THE TWISTED q-CHANGHEE POLYNOMIALS OF HIGHER ORDER JIN-WOO PARK
Abstract. The q-Changhee polynomials and numbers are introduced by T. Kim et al in [3]. Some interesting properties of those polynomials are derived from umbral calculus (see [4]). In this paper, we consider Witt-type formula for the n-th twisted q-Changhee numbers and polynomials of higher order and derive some new interesting identities and properties of those polynomials and numbers from the Witt-type formula which are related to special polynomials and numbers.
1. Introduction Let p be an odd prime number. Zp , Qp and Cp will denote the ring of p-adic integers, the field of p-adic numbers and the completion of algebraic closure of Qp . The p-adic norm | · |p is normalized by |p|p = p1 . Let C(Zp ) be the space of continuous functions on Zp . For f ∈ C(Zp ), the fermionic p-adic integral on Zp is defined by T.Kim to be N
pX −1 1 I−q (f ) = f (x)dµ−q (x) = lim f (x)(−q)x , (see [6, 7, 9]). N →∞ [pN ]−q Zp x=0
Z
(1.1)
Let f1 (x) = f (x + 1). Then, by (1.1), we get qI−q (f1 ) + I−q (f ) = [2]q f (0), (see [6, 7]).
(1.2)
By (1.2), we easily see that n
q I−q + (−1)
n−1
I−q = [2]q
n−1 X
(−1)n−1−l f (l),
(1.3)
l=0
where fn (x) = f (x + n) and n ≥ 0. It is well known that the twisted q-Euler polynomials are defined by the generating function to be ∞ X [2]q tn xt e = En,ε,q (x) , (see [13]). t 1 + qεe n! n=0
(1.4)
When x = 0, En,ε,q = En,ε,q (0) are called the n−th twisted q-Euler numbers. For ε = 1, En,1,q (x) = En,q (x) are the n-th q-Euler polynomials, and x = 0 , En,1,q (0) = En,q (0) are the n-th q-Euler numbers. 2000 Mathematics Subject Classification. 11S80, 11B68, 05A30. Key words and phrases. Euler numbers, q-Changhee numbers, twisted q-Changhee numbers of higher order. 1
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Indeed, we note that En,1,q (x) = Hn (x| − q), where Hn (x|λ) are the FrobeniusEuler polynomials which are defined by the generating function to be ∞ 1 − λ tx X tn e = Hn (x|λ) , (see [1]). t e −λ n! n=0
Recently, the q-Changhee polynomials are defined by the generating function to be
∞ X [2]q tn x (1 + t) = Ch (x) , (see [10]). n,q 1 + q εt n! n=0
(1.5)
When x = 0, Chn,ε,q = Chn,ε,q (0) are called the q-Changhee numbers, (see [3]). The Stirling number of the first kind is defined by (x)n = x(x − 1) · · · (x − n + 1) =
n X
S1 (n, l)xl , (see [3]).
(1.6)
l=0
The q-Changhee numbers and polynomials are introduced by T. Kim et. al. in [3], and found interesting identities in [5, 8, 11, 12]. In this paper, we consider the twisted q-Changhee numbers and polynomials of order k which are derived from the multivariate fermionic p-adic q-integral of higher order on Zp , and give some relationship between twisted q-Changhee polynomials and numbers of higher-order and special polynomials and numbers. 2. Twisted q-Changhee numbers and polynomials of higher-order For n ∈ N, let Tp be the p-adic locally constant space defined by Tp = ∪ Cpn = lim Cpn , n→∞
n≥1
n
where Cpn = ω|ω p = 1 is the cyclic group of order pn . 1 For ε ∈ Tp , let us take f (x) = (1 + εt)x for |t|p < p− p−1 . Then by (1.2), we get Z ∞ X [2]q tn (1 + εt)x dµ−q (x) = Chn,ε,q (2.1) = qεt + [2]q n! Zp n=0 where Chn,ε,q are called the n-th twisted q-Changhee numbers. From (2.1), we can derive the following equation: Z ∞ X [2]q tn (1 + εt)x+y dµ−q (y) = Chn,ε,q (x) , (1 + εt)x = qεt + [2]q n! Zp n=0
(2.2)
where Chn,ε,q (x) are called the n-th twisted q-Changhee polynomials. Note that Chn,ε,q (0) = Chn,ε,q are n-th twisted q-Changhee numbers. Since Z Z ∞ X x+y x+y n (1 + εt) dµ−q (y) = ε dµ−q (y)tn n Zp Z p n=0 (2.3) Z ∞ X tn n = ε (x + y)n dµ−q (y) , n! Zp n=0 by (2.2) and (2.3), we obtained the following theorem.
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Theorem 2.1. For n ≥ 0, we have Chn,ε,q (x) = εn
Z (x + y)n dµ−q (y). Zp
From (2.1), we note that ∞ X
εn
n ∞ X x [2]q qε dµ−q (x)tn = = tn . − n qεt + [2] [2] q q Zp n=0
Z
n=0
(2.4)
Thus, by comparing the coefficients on the both sides, we obtain the following theorem. Theorem 2.2. For n ≥ 0, we have n Z x q . dµ−q (x) = − [2]q Zp n Replacing t by ∞ X
et −1 ε
in (2.2), we get
En,q (x)
n=0
n ∞ tn [2]q xt X 1 et − 1 = t e = Chn,ε,q (x) , n! qe − 1 n! ε n=0
where En,q is the n-th q-Euler polynomials and n X ∞ ∞ X 1 et − 1 1 Chn,ε,q (x) = Chn,ε,q (x) ε−n n! n! ε n! n=0 n=0 ∞ X m X
=
∞ X
tm S2 (m, n) m! m=n −n t
Chn,ε,q (x)S2 (m, n)ε
m=0 n=0
(2.5)
!
m
m!
(2.6)
,
where S2 (m, n) is the Striling number of the second kind. By comparing the coefficients on the both sides of (2.5) and (2.6), we obtain the following theorem. Theorem 2.3. For n ≥ 0, we have n X En,q (x) = Chm,ε,q (x)S2 (n, m)ε−m . m=0
By Theorem 2.1, we easily get Chn,ε,q (x) =εn =εn
Z (x + y)n dµ−q (y) Zp n X l=0
Z S1 (n, l)
(x + y)l dµ−q (y) = εn
Zp
n X
(2.7) S1 (n, l)El,q (x).
l=0
Therefore, by (2.7), we obtain the following theorem. Theorem 2.4. For n ≥ 0, we have Chn,ε,q (x) = εn
n X
S1 (n, l)El,q (x).
l=0
where S1 (n, l) is the Stirling number of the first kind.
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In viewpoint of (2.3), the n−th twisted q-Changhee numbers of the first kind with order k are defined by the generating function to be Z Z n Ch(k) = ε · · · (x1 + · · · + xk )n dµ−q (x1 ) . . . dµ−q (xk ), (2.8) n,ε Zp
Zp
where n is a positive integer. By (2.8), we easily get Z Z X ∞ ∞ n X x1 + · · · + xk (k) t ··· Chn,ε,q = (εt)n dµ−q (x1 ) · · · dµ−q (xk ) n! n Z Z p p n=0 (2.9) Z Z n=0 = ··· (1 + εt)n dµ−q (x1 ) · · · dµ−q (xk ). Zp
Zp
From (2.1) and (2.9), we have ∞ X
tn = n!
Ch(k) n,ε,q
n=0
and
[2]q qεt + [2]q
k =
∞ X
X
n=0
l1 +···+lk =n
[2]q qεt + [2]q
k ,
(2.10)
! n tn . Chl1 ,ε,q · · · Chlk ,ε,q l1 , . . . , l k n!
By simple calculation. we easily see that k X n n ∞ [2]q q n k+n−1 t . = n!ε − qεt + [2]q [2]q n! n n=0
(2.11)
(2.12)
Thus, by (2.10) and (2.12), we get n+k−1 n =(−q)n εn (k + n − 1)n
n n [2]nq Ch(k) n,ε,q =(−q) n!ε
=(−q)n εn
n X
(2.13)
S1 (n, l)(k + n − 1)l .
l=0
Therefore, by (2.10), (2.11) and (2.13), we obtain the following theorem. Theorem 2.5. For n ≥ 0, we have X (k) [2]nq Ckn,ε,q =[2]nq
n Chi1 ,ε,q · · · Chlk ,ε,q l1 , . . . , l k
l1 +···+lk =n n X =(−q)n εn S1 (n, l)(k
+ n − 1)l .
l=0
From (2.8), we have Z Z (k) n Chn,ε,q =ε ··· (x1 + · · · + xk )n dµ−q (x1 ) · · · dµ−q (xk ) Zp n
=ε
n X l=0
Zp
Z ···
S1 (n, l) Zp
(2.14)
Z
l
(x1 + · · · + xk ) dµ−q (x1 ) · · · dµ−q (xk ). Zp
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ON THE TWISTED q-CHANGHEE POLYNOMIALS OF HIGHER ORDER
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Now, we observe that k X Z ∞ n [2]q (x1 +···+xk )t (k) t ··· = e dµ−q (x1 ) · · · dµ−q (xk ) = E , (2.15) n,q qet + 1 n! Zp Zp n=0
Z
(k)
where En,q are the q-Euler numbers of order k. From (2.14) and (2.15), we obtain the following theorem. Theorem 2.6. For n ≥ 0, we have n Ch(k) n,ε,q = ε
n X
(k)
S1 (n, l)El,q .
l=0 t
e −1 ε ,
we get t n k X ∞ n e −1 [2]q (k) 1 (k) t = = , Chn,ε,q E n,q n! ε qet + 1 n! n=0 n=0
Replacing t by ∞ X
(2.16)
and ∞ X n=0
Ch(k) n,ε,q
1 n!
et − 1 ε
n =
∞ X
m X
m=0
n=0
! ε
−n
Ch(k) n,ε,q S2 (m, n)
tm . m!
(2.17)
Thus, by (2.16) and (2.17), we obtain the following theorem. Theorem 2.7. For n ≥ 0, we have n X (k) ε−m Ch(k) En,q = m,ε,q S2 (n, m). m=0
Now we define the twisted q-Changhee polynomials of the first kind with order k as follows: Z Z (k) n Chn,ε,q (x) = ε ··· (x1 + · · · + xk + x)n dµ−q (x1 ) · · · dµ−q (xk ), (2.18) Zp
Zp
where n ≥ 0 and k ∈ N. From (2.18), we can derive the generating function of the twisted q-Changhee polynomials as follows: Z Z ∞ X tn Ch(k) (x) (1 + εt)x1 +···+xk +x dµ−q (x1 ) · · · dµ−q (xk ) = · · · n,ε,q n! Zp Zp n=0 (2.19) k [2]q = (1 + εt)x . qεt + [2]q It is easy to show that k ∞ X [2]q (1 + εt)x = qεt + [2]q n=0
! n tn (k) ε (x)m Chn−m,ε,q . n! m m=0 n X
m
(2.20)
By (2.20), we get Ch(k) n,ε,q (x)
n X
x n! (k) = ε Chn−m,ε,q m (n − m)! m=0 n X x n! n−m = ε Ch(k) m,ε,q . n − m m! m=0 m
428
(2.21)
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From (2.18), we have Z Z (k) n Chn,ε,q (x) =ε ··· (x1 + · · · + xk + x)n dµ−q (x1 ) · · · dµ−q (xk ) Zp n
=ε
=εn
n X l=0 n X
Zp
Z
Z
(x1 + · · · + xk + x)l dµ−q (x1 ) · · · dµ−q (xk )
···
S1 (n, l) Zp
Zp
(k)
S1 (n, l)El,q (x).
l=0
(2.22) Hence, by (2.22), we obtain the following theorem. Theorem 2.8. For n ≥ 0, we have n n X X x n! (k) n (k) m Ch(k) S1 (n, l)El,q (x). Chn,ε,q (x) = ε m,ε,q = ε n − m m! m=0 l=0
where
(k) El,q
are the q-Euler polynomials of order k.
Now, we consider the twisted q-Changhee polynomials of second kind with order k as follows: Z Z (k) n c Ch (x) = ε · · · (−x1 − · · · − xk + x)n dµ−q (x1 ) · · · dµ−q (x)k . (2.23) n,ε,q Zp
Zp
By (2.23), we have ∞ X
n
t c = Ch n,ε,q (x) n! n=0 (k)
Z
Z
(1 + εt)−x1 −···−xk +x dµ−q (x1 ) · · · dµ−q (xk )
··· Zp
=
Zp
[2]q εt + [2]q
(2.24)
k
k+x
(1 + εt)
,
where k is positive integer. Hence, (k)
c Ch n,ε,q (x) Z Z n =ε ··· (−x1 − · · · − xk + x)n dµ−q (x1 ) · · · dµ−q (xk ) Zp
=ε
n
=εn
n X l=0 n X
Zp l
Z
Z ···
S1 (n, l)(−1)
Zp
(x1 + · · · + xk − x)l dµ−q (x1 ) · · · dµ−q (xk )
(2.25)
Zp
(k)
S1 (n, l)(−1)l El,q (−x).
l=0
Therefor, by (2.25), we obtain the following theorem. Theorem 2.9. For n ≥ 0, we have (k)
n c Ch n,ε,q (x) = ε
n X
(k)
S1 (n, l)(−1)l El,q (−x).
l=0
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Now, we consider the n-th twisted q-Changhee polynomials of the first kind relate to n-th twisted q-Changhee polynomials of second kind. (k)
c (−1)n Ch n,ε,q (x) n!Z Z
−x1 − · · · − xk + x =(−1) ε ··· dµ−q (x1 ) · · · dµ−q (xk ) n Zp Zp Z Z x1 + · · · + xk − x + n − 1 =εn ··· dµ−q (x1 ) · · · dµ−q (xk ) n Zp Zp Z Z ∞ X n−1 x1 + · · · + xk − x n =ε ··· dµ−q (x1 ) · · · dµ−q (xk ) n − m Zp m Zp m=0 Z Z n X n − 1 ε−m x1 + · · · + xk − x =εn m!εm ··· dµ−q (x1 ) · · · dµ−q (xk ) m − 1 m! m Zp Zp m=1 n (k) X n − 1 n−m Chm,ε,q (−x) . = ε m! m−1 m=1 (2.26) n n
By (2.26) and proceeding similar to (2.26), we have the following theorem. Theorem 2.10. For n ≥ 0, we have (k) n (k) c X (−1)n Ch n − 1 n−m Chm,ε,q (−x) n,ε,q (x) ε = , m−1 n! m! m=1
and (k) n (k) c X (−1)n Chn,ε,q (x) n − 1 n−m Ch m,ε,q (−x) = , ε n! m! m−1 m=1
By (2.25), c n,ε,q (x) Ch =εn
n X
S1 (n, l)(−1)l
=ε
n
l=0
Z
S1 (n, l)
l X
(x1 + · · · + xk − x)l dµ−q (x1 ) · · · dµ−q (xk )
··· Zp
l=0 n X
Z
Zp l+m
(−1)
m=0
l (k) E xm , m l−m
and thus we obtain the following theorem. Theorem 2.11. For n ≥ 0, we have c n,ε,q (x) = εn Ch
n X l X
l+m
(−1)
l=0 m=0
l (k) S1 (n, l)El−m xm . m
References [1] S. Araci and M. Acikgoz, A note on the Frobenius-Euler numbers and polynomials associated with Bernstein polynomials, Adv. Stud. Contemp. Math., 22 (2012), no.3, 399-406.
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[2] J. Choi, D. S. Kim, T. Kim and Y. H. Kim, Some arithmetic identities on Bernoulli and Euler numbers arising from the p-adic integrals on Zp , Adv. Stud. Contemp. Math. 22 (2012) 239-247. [3] D. Kim, T. Mansour, S. H. Rim and J. J. Seo, A Note on q-Changhee Polynomials and Numbers, Adv.Studies Theor. Phys., Vol. 8, 2014, no. 1, 35-41. [4] T. Kim, D. S. Kim, T. Mansour, S,-H. Rim and M. Schork Umbral calculus and Sheffer sequences of polynomials, J. Math. Phys. 54, 083504 (2013); doi:10.1063/1.4817853. [5] T. Kim, S.-H. Rim, New Changhee q-Euler numbers and polynomials associated with p-adic q-integrals, Comput. Math.Appl. 54 (2007), no. 4, 484-489. [6] T. Kim, On q-analogye of the p-adic log gamma functions and related integral, J. Number Theory, 76 (1999), no. 2, 320-329. [7] T. Kim, q-Volkenborn integration, Russ. J. Math. Phys., 9 (2002), no. 3, 288-299. [8] T. Kim, Non-Archimedean q-integrals assoiciated with multiple Changhee q-Bernoulli polynomials, Russ. J. Math. Phys. 10 (2003), 91-98. [9] T. Kim, p-adic q-integrals associated with the Changhee-Barnes’ q-Bernoulli polynomials, Integral Transforms Spec. Funct. 15 (2004), no. 5, 415-420. [10] T. Kim, An invariant p-adic q-integral on Zp , Applied Mathematics Letters, 21 (2008), no. 2, 105-108. [11] S. H. Lee, W. J. Kim and Y. S. Jang, Higher-order q-Changhee polynomials, to appear. [12] S. H. Rim, J. W. Park, S. S¿ Pyo and J. Kwon, On the twisted Changhee polynomials and numbers, to appear. [13] C. S. Ryoo, A note on the twisted q-Euler numbers and polynomials with weak weight α, Adv. Studies Theor. Phys., 6 (2012), no. 22, 1109-1116. [14] Y. Simsek, T. Kim, I. S. Pyung, Barnes’ type multiple Changhee q-zeta functions, Adv. Stud. Contemp. Math. 10 (2005), no. 2, 121-129. Department of Mathematics Education, Daegu University, Jillyang, Gyeongsan, Gyeongbuk 712-714, Republic of Korea. E-mail address: [email protected]
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SOME SYMMETRY IDENTITIES FOR THE (h, q)-BERNOULLI POLYNOMIALS UNDER THE THIRD DIHEDRAL GROUP D3 ARISING FROM q-VOLKENBORN INTEGRAL ON Zp S.-H. RIM, T. G. KIM, S. H. LEE
Abstract. In this paper, we give some new identities of symmetry for the (h, q)-Bernoulli polynomials arising from q-Volkenborn integral on Zp .
1. Introduction let p be a fixed prime number. Throughout this paper, Zp , Qp and Cp will, respectively, denote the ring of p-adic integers, the field of p-adic rational numbers and the completion of algebraic closure of Qp . Let vp be the normalized exponential valuation of Cp with |p|p = p−vp (p) = 1/p and let q be an indeterminate in Cp with x 1 |1 − q|p < p− p−1 . The q-extension of x is defined by [x]q = 1−q 1−q . Note that limq→1 [x]q = x. Suppose that f is a uniformly differentiable function on Zp . Then the p-adic q-Vollenborn integral is defined by Kim to be Z Iq (f ) = (1)
f (x)dµq (x) = lim
N →∞
Zp
= lim
N →∞
N pX −1
f (x)µq (x + pN Zp )
x=0
1 [pN ]q
N pX −1
f (x)q x .
x=0
As is well known, Carlitz’s q-Bernoulli numbers are defined by ( 1 if n = 1 n β0,q = 1, q(qβ + 1) − βn,q = 0 if n > 1, with the usual convention about replacing βqn by βn,q (see [1,8,10]). The q-Bernoulli polynomials are given by n X n βn,q (x) = [x]n−l q lx βl,q q l l=0 n X 1 n l+1 = (−1)l q lx , (see [10]). (1 − q)n l [l + 1]q l=0
In 1999, Kim gave the formula which is given by Z βn,q (x) = [x + y]nq dµq (x), (n ∈ N ∪ {0}, )
(see [1-15]).
Zp 1
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For h ∈ Z, we consider (h, q)-Bernoulli polynomials as follows: Z (h) βn,q (x) = q (h−1)x [x + y]nq dµq (x), (n ∈ Z≥0 ) Zp
(2)
n X n lx h+1 1 q (−1)l , = n (1 − q) l [h + l]q
(see [8,10]).
l=0
(h)
(h)
When x = 0, βn,q = βn,q (0) are called the (h, q)-Bernoulli numbers. In this paper, we consider the symmetric identities for the (h, q)-Bernoulli polynomials under the third Dihedral group D3 which are derive from p-adic q-Volkenborn integral on Zp . 2. Symmetric identities for the (h, q)-Bernoulli polynomials Let w1 , w2 , w3 be positive integers. Then we observe that (3) Z
q (h−1)w2 w3 y e[w2 w3 y+w1 w2 w3 x+w1 w3 i+w1 w2 j]q t dµqw2 w3 (y)
Zp N
pX −1 1 q (h−1)w2 w3 y e[w2 w3 y+w1 w2 w3 x+w1 w3 i+w1 w2 j]q t q w2 w3 y = lim N →∞ [pN ]q w2 w3 y=0 N
−1 wX 1 −1 pX 1 = lim q hw2 w3 (k+w1 y) e[w2 w3 (k+w1 y)+w1 w2 w3 x+w1 w3 i+w1 w2 j]q t . N →∞ [w1 pN ]q w2 w3 y=0 k=0
By (3), we get wX 2 −1 w 3 −1 X 1 q (w1 w3 i+w1 w2 j)h [w2 w3 ]q i=0 j=0 Z × q (h−1)w2 w3 y e[w2 w3 y+w1 w2 w3 x+w1 w3 i+w1 w2 j]q t dµqw2 w3 (y)
(4)
Zp wX 2 −1 w 3 −1 w 1 −1 X X 1 q h(w1 w3 i+w1 w2 j+w2 w3 k)+hw1 w2 w3 y N →∞ [w1 w2 w3 pN ]q i=0 j=0
= lim
k=0
×e
[w2 w3 (k+w1 y)+w1 w2 w3 x+w1 w3 i+w1 w2 j]q t
.
From (4), we note that the expression is invariant under any permutation of w1 , w2 , w3 in third Dihedral group D3 . Therefore, by (4), we obtain the following theorem. Theorem 2.1. Let w1 , w2 , w3 be positive integers. Then, the following expressions wσ(2) −1 wσ(3) −1 X X 1 q h(wσ(1) wσ(3) i+wσ(1) wσ(2) j) [wσ(2) wσ(3) ]q i=0 j=0 Z × q (h−1)wσ(2) wσ(3) y e[wσ(2) wσ(3) y+wσ(1) wσ(2) wσ(3) x+wσ(1) wσ(3) i+wσ(1) wσ(2) j]q t dµqwσ(2) wσ(3) (y) Zp
are the same for any σ ∈ D3 .
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Now, we note that (5) w1 w1 [w2 w3 y + w1 w2 w3 x + w1 w3 i + w1 w2 j]q = [w2 w3 ]q y + w1 x + i+ j w2 w3 qw2 w3 Therefore, by (2), Theorem 1 and (5), we obtain the following theorem. Theorem 2.2. For w1 , w2 , w3 ∈ N, the following expressions wσ(2) −1 wσ(3) −1
[wσ(2) wσ(3) ]n−1 q
X
X
i=0
j=0
q h(wσ(1) wσ(3) i+wσ(1) wσ(2) j) (h)
× βn,qwσ(2) wσ(3) wσ(1) x +
wσ(1) wσ(1) i+ j wσ(2) wσ(3)
are the same for any σ ∈ D3 . It is not difficult to show that (6) w1 w1 1 − q w1 w3 i+w1 w2 j + q w1 w3 i+w1 w2 j [y + w1 x]qw2 w3 i+ j y + w1 x + = w2 w 3 q w2 w3 1 − q w2 w3 =
[w1 ]q [w3 i + w2 j]qw1 + q w1 w3 i+w1 w2 j [y + w1 x]qw2 w3 [w2 w3 ]q
From (6), we have n Z w1 w1 i+ j q (h−1)w2 w3 y dµqw2 w3 (y) y + w1 x + w2 w3 qw2 w3 Zp (7) n−k n X n [w1 ]q k(w1 w3 i+w1 w2 j) (h) = [w3 i + w2 j]n−k βk,qw2 w3 (w1 x). q w1 q [w2 w3 ]q k k=0
Thus, by Theorem 2 and (7), we get (8) [w2 w3 ]n−1 q
wX 2 −1 w 3 −1 X i=0
q h(w1 w3 i+w1 w2 j)
j=0
n w1 w1 q (h−1)w2 w3 y y + w1 x + i+ j dµqw2 w3 (y) w2 w3 qw2 w3 Zp
Z
wX 2 −1 w 3 −1 X n k−1 n−k (h) = [w2 w3 ]q [w1 ]q βk,qw2 w3 (w1 x) q (k+h)(w1 w3 i+w1 w2 j) [w3 i + w2 j]n−k q w1 k i=0 j=0 k=0 n X n (h) (h) = [w2 w3 ]k−1 [w1 ]n−k βk,qw2 w3 (w1 x) Tn,qw1 (w2 , w3 |k), q q k n X
k=0
where (9)
(h) Tn,q (w1 , w2 |k) =
wX 1 −1 w 2 −1 X i=0
q (k+h)(w2 i+w1 j) [w2 i + w1 j]n−k . q
j=0
As this expression is invariant under the third Dihedral group D3 , we have the following theorem.
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Theorem 2.3. For n ≥ 0, w1 , w2 , w3 ∈ N, the following expressions n X n (h) (h) [wσ(2) wσ(3) ]k−1 [wσ(1) ]n−k βk,qwσ(2) wσ(3) (wσ(1) x) Tn,qwσ(1) (wσ(2) , wσ(3) |k) q q k k=0
are all the same for any σ ∈ D3 . ACKNOWLEDGEMENTS. The present Research has been supported by Jangjeon Research Institute for Mathematics and Physics and has been conducted by the Research Grant of Kwangwoon University in 2014.
References [1] J. Choi, T. Kim, Arithmetic properties for the q-Bernoulli numbers and polynomials, Proc. Jangjeon Math. Soc. 15 (2012), no. 2, 137–143. [2] S. Gaboury, R. Tremblay, B. J. Fugere, Some explicit formulas for certain new classes of Bernoulli, Euler and Genocchi polynomials, Proc. Jangjeon Math. Soc. 17 (2014), no. 1, 115–123. [3] D.V. Dolgy, D. S. Kim, T. Kim, J.-J. Seo, Identities of Symmetry for Carlitz q-Bernoulli Polynomials Arising from q-Volkenborn Integrals on Zp under Symmetry Group S3, Advanced Studies in Theoretical Physics 8(2014), no. 17 , 737 - 744 [4] D. S. Kim, T. Kim, q-Bernoulli polynomials and q-umbral calculus, Sci. China Math. 57 (2014), no. 9, 1867–1874. [5] D. S. Kim, N. Lee, J. Na, K. H. Park, Abundant symmetry for higher-order Bernoulli polynomials (II), Proc. Jangjeon Math. Soc. 16 (2013), no. 3 , 359–378 [6] D. S. Kim, D. V. Dolgy, T. Kim, S.-H. Identities involving Bernoulli and Euler polynomials arising from Chebyshev polynomials, Proc. Jangjeon Math. Soc. 15 (2012), no. 4, 361–370. [7] H. M. Kim, D. S. Kim, T. Kim,S. H. Lee, D. V. Dolgy, B. Lee, Identities for the Bernoulli and Euler numbers arising from the p-adic integral on Zp , Proc. Jangjeon Math. Soc. 15 (2012), no. 2, 155–161 [8] T. Kim, S.H. Rim, Generalized Carlitz’s q-Bernoulli numbers in the p-adic number field, Adv. Stud. Contemp. Math. (Pusan) 2 (2000), 9–19. [9] T. Kim, On the weighted q-Bernoulli numbers and polynomials, Adv. Stud. Contemp. Math. 21 (2011), no. 2, 207–215. [10] T. Kim, J. Choi, Y.-H. Kim, On extended Carlitz’s type q-Euler numbers and polynomials, Adv. Stud. Contemp. Math. 20 (2010), no. 4, 499–505. [11] T. Kim, Y.-H. Kim, B. Lee, Note on Carlitz’s type q-Euler numbers and polynomials, Proc. Jangjeon Math. Soc. 13 (2010), no. 2, 149–155. [12] T. Kim, Y.-H. Kim, K.-W. Hwang, On the q-extensions of the Bernoulli and Euler numbers, related identities and Lerch zeta function, Proc. Jangjeon Math. Soc. 12 (2009), no. 1, 77–92. [13] J.-W. Park, S.-H. Rim, J. Seo, J. Kwon, A note on the modified q-Bernoulli polynomials, Proc. Jangjeon Math. Soc. 16 (2013), no. 4, 451–456. [14] S. H. Rim, J. Joung, J.-H. Jin, S.-J. Lee, A note on the weighted Carlitz’s type q-Euler numbers and q-Bernstein polynomials, Proc. Jangjeon Math. Soc. 15 (2012), no. 2, 195–201.
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[15] J.-J. Seo, S.-H. Rim, S.-H. Lee, D. V. Dolgy, T. Kim, q-Bernoulli numbers and polynomials related to p-adic invariant integral on Zp , Proc. Jangjeon Math. Soc. 16 (2013), no. 3, 321–326
Seog-Hoon Rim Department of Mathematics Education, Kyungpook National University, Tagegu 702-701, S. Korea E-mail: [email protected] Tae Gyun Kim Jangjeon Research Institute for Mathematics and Physics, Hapcheon 678-800, S. Korea Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea E-mail: [email protected] Sang Hun Lee Division of General Education, Kwangwoon University, Seoul 139-701, Republic of Korea E-mail: [email protected]
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SOME IDENTITIES OF BELL POLYNOMIALS ASSOCIATED WITH p-ADIC INTEGRAL ON Zp SEOG-HOON RIM, HONG KYUNG PAK, J.K. KWON, AND TAE GYUN KIM Abstract. In this paper, we investigate some identities of Bell polynomials associated with special polynomials which are derived from p-adic integral on Zp .
1. Introduction Let p be a fixed odd prime number. Throughout this paper, Zp , Qp and Cp will denote the ring of p-adic integers, the field of p-adic rational numbers and the completion of algebraic closure of Qp . Let νp be the normalized exponential valuation 1 − p−1
of Cp with |p|p = p−νp (p) = p1 . Let q be an indeterminate in Cp with |1 − q|p < p x
and let the q-extension of number x is defined as [x]q = 1−q 1−q . The Euler polynomials of order r are defined by the generating function to be ∞ 2 r X tn xt (r) , (see [1 − 18]) e = E (x) n et + 1 n! n=0
and the higher-order Bernoulli polynomials of order r are given by ∞ t r X tn xt (r) e = B (x) , (see [9 − 10]). n et − 1 n! n=0
(r) Bn
(r) Bn (0),
(r)
(r)
When x = 0, = En = En (0) are called higher-order Bernoulli numbers and Euler numbers. Let f (x) be a uniformly continuous function on Zp . Then the bosonic p-adic integral on Zp is defined by Z pN −1 1 X (1) f (x)dµ0 (x) = lim N f (x), (see [12]), N →∞ p Zp x=0
and the fermieuic p-adic integral on Zp is given by Z pN −1 1 X (2) f (x)dµ−1 (x) = lim N f (x)(−1)x , N →∞ p Zp
(see [12]).
x=−1
Thus, we have Z
Z f (x + 1)dµ0 (x) −
(3) Zp
f (x)dµ0 (x) = f 0 (0),
Zp
and Z
Z f (x + 1)dµ−1 (x) +
(4) Zp
f (x)dµ−1 (x) = 2f (0). Zp 1
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As is well know, the higher-order Changhee polynomials are given by (5)
∞ 2 r X tn , (1 + t)x = Ch(r) (x) n t+2 n!
(see [11 − 15]),
n=0
and the higher-order Daehee polynomials are defined by the generating function to be (6)
log(1 + t) r t
x
(1 + t) =
∞ X
Dn(r) (x)
n=0
(r)
(r)
(r)
tn , n!
(see [11 − 15]).
(r)
When x = 0, Chn = Chn (0) and Dn = Dn (0) are called the Changhee numbers and the Daehee numbers with order r. Finally, we introduce the Bell polynomials which are given by the generating function to be (et −1)x
(7)
e
=
∞ X
Beln (x)
n=0
tn , n!
(see [4, 14, 16]).
The purpose of this paper is to given some identities of Bell polynomials associated with special polynomials arising from p-adic integral on Zp . 2. Some identities of Bell polynomials From (2), we note that Z
e(e
t −1)(x+y)
dµ0 (y)
Zp
=
(8)
=
∞ Z X
(x + y)n dµ0 (y)
n=0 Zp ∞ X n X
Bk (x)S2 (n, k)
n=0 k=0
(et − 1)n n! tn n!
,
where S2 (n, k) is the Stirling number of the second kind. On the other hand, Z (9)
(et −1)(x+y)
e
dµ0 (y) =
Zp
∞ Z X
Beln (x + y)dµ0 (y)
n=0 Zp
tn . n!
Thus, by (8) and (9), we get Z (10)
Beln (x + y)dµ0 (y) = Zp
n X
Bk (x)S2 (n, k).
k=0
By the same method as (10), we get Z (11)
Beln (x + y)dµ−1 (y) = Zp
n X
Ek (x)S2 (n, k).
k=0
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Note that Z
Z
(1 + t)(x1 +···+xr +x) dµ0 (x1 ) · · · dµ0 (xr ) =
··· (12)
log(1 + t) r t
Zp
Zp
=
∞ X
Dn(r) (x)
n=0 t −1
By replacing t by ee Z
Z
Zp
tn . n!
− 1, we get t −1)(x +···+x +x) r 1
e(e
···
(1 + t)x
dµ0 (x1 ) · · · dµ0 (xr )
Zp
∞ ∞ et − 1 r t X t l X tm (r) (e − 1) (e −1)x Bel (x) e = B m l l! m! eet −1 − 1 m=0
=
∞ X
=
l=0 ∞ X
(13) =
∞ X (r)
Bl
S2 (k, l)
k=l k X
l=0 ∞ k X t
k!
m=0
tm Belm (x) m!
∞ tk X
(r)
Bl S2 (k, l)
k=0 l=0 ∞ X n X
k!
m=0
Belm (x)
tm m!
n−m X
tn Belm (x)n! (r) Bl S2 (n − m, l) m!(n − m)! n! n=0 m=0 l=0 ∞ X n n−m tn X X (r) n = Belm (x) Bl S2 (n − m, l) . m n!
=
n=0 m=0
l=0
On the other hand, Z
Z
e(e
··· (14)
Zp
=
t −1)(x
1 +···+xr +x)
dµ0 (x1 ) · · · dµ0 (xr )
Zp
∞ Z X
Z Beln (x1 + · · · + xr + x)dµ0 (x1 ) · · · dµ0 (xr )
··· Zp
n=0 Zp
tn . n!
Therefore, we obtain the following theorem. Theorem 1. For n ≥ 0, we have Z
Z ···
Zp
Beln (x1 + · · · + xr + x)dµ0 (x1 ) · · · dµ0 (xr ) Zp
n n−m X (r) X n = Belm (x) Bl S2 (n − m, l). m m=0
l=0
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From (12), we note that Z Z t ··· e(e −1)(x1 +···+xr +x) dµ0 (x1 ) · · · dµ0 (xr ) Zp
=
(15)
= = =
∞ X
Zp
Dn(r) (x)
n=0 ∞ X m X m=0 k=0 ∞ X m X
∞ ∞ n X X 1 (et −1) (et − 1)m (r) e −1 = Dk (x) S2 (m, k) n! m! k=0
(r)
Dk (x)S2 (m, k)
1 t (e − 1)m m! ∞ X
(r)
Dk (x)S2 (m, k)
S2 (n, m)
n=m
m=0 k=0 ∞ nX n X m X
m=k
tn n!
o tn (r) . Dk (x)S2 (m, k)S2 (n, m) n!
n=0 m=0 k=0
Therefore, by Theorem 1 and (15), we obtain the following theorem. Theorem 2. For n ≥ 0, we have n n−m X X (r) n Bl S2 (n − m, l) Belm (x) m m=0
l=0
n X m X
=
(r)
Dk (x)S2 (m, k)S2 (n, m).
m=0 k=0
From (7), we note that xt
e
= =
(16)
∞ X
=
m=0 ∞ X
Belm (x) Belm (x)
m=0 ∞ X n X n=0 m=0
m 1 log(1 + t) m! ∞ X
S1 (n, m)
n=m
tm m!
tn Belm (x)S1 (n, m) , n!
where S1 (n, m) is the Stirling number of the first kind. Therefore, by (16), we obtain the following theorem. Theorem 3. For n ≥ 0, we have n
x =
n X
Belm (x)S1 (n, m).
m=0
It is easy to show that Z (17)
∞
X tn t = Bn . e dµ0 (x) = t e −1 n! Zp xt
n=0
Thus, by (17), we have Z
xn dµ0 (x) = Bn ,
(n ≥ 0).
Zp
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From Theorem 3, we can derive the following equation: Z Z n X n x dµ0 (x) = Belm (x)dµ0 (x), (18) Bn = S1 (n, m) Zp
5
(n ≥ 0).
Zp
m=0
Therefore, by (10) and (18), we obtain the following theorem. Theorem 4. For n ≥ 0, we have Bn =
n X m X
S1 (n, m)S2 (m, k)Bk .
m=0 k=0
It is not difficult to show that Z ext dµ−1 (x) = (19) Zp
∞
X 2 tn = . E n et + 1 n! n=0
Thus, by (19), we get Z
xn dµ−1 (x) = En ,
(20)
(n ≥ 0).
Zp
From Theorem 3 and (20), we have Z Z n X n x dµ−1 (x) = S1 (n, m) (21) En = Zp
Belm (x)dµ−1 (x).
Zp
m=0
Therefore, by (11) and (21), we obtain the following theorem. Theorem 5. For n ≥ 0, we have En =
n X m X
S1 (n, m)S2 (m, k)Ek .
m=0 k=0
Now, we consider the following equation. (x+x1 +···+xr )t
e
(22)
= = =
∞ X m=0 ∞ X
Belm (x1 + · · · + xr + x) Belm (x1 + · · · + xr + x)
m=0 ∞ X n X n=0 m=0
(log(1 + t))m m! ∞ X n=m
S1 (n, m)
tn n!
tn Belm (x1 + · · · + xr + x)S1 (n, m) . n!
Thus, by (22), we have the following theorem. Theorem 6. For n ≥ 0, we have (x + x1 + · · · + xr )n =
n X
Belm (x1 + · · · + xr + x)S1 (n, m).
m=0
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From (4), we can easily derive the following equation: Z Z e(x1 +···+xr +x)t dµ−1 (x1 ) · · · dµ−1 (xr ) = ··· Zp
Zp
(23)
=
2 r xt e et + 1
∞ X
En(r) (x)
n=0
tn . n!
Thus, by (23), we get Z Z (x1 + · · · + xr + x)n dµ−1 (x1 ) · · · dµ−1 (xr ) = En(r) (x). ··· (24) Zp
Zp
By (3), we easily get Z Z e(x1 +···+xr +x)t dµ0 (x1 ) · · · dµ0 (xr ) = ··· Zp
Zp
(25)
=
t r xt e et − 1
∞ X
Bn(r) (x)
n=0
tn . n!
From (25), we have Z Z (26) ··· (x1 + · · · + xr + x)n dµ0 (x1 ) · · · dµ0 (xr ) = Bn(r) (x). Zp
Zp
From Theorem 6, (24) and (26), we have Z Z n X (r) (27) Bn (x) = S1 (n, m) ··· Belm (x + x1 + · · · + xr )dµ0 (x1 ) · · · dµ0 (xr ) Zp
m=0
Zp
and (28) En(r) (x) =
n X
Z Zp
m=0
Now, we observe that Z ∞ Z X ··· n=0 Zp
Z Zp ∞ X
= =
Belm (x + x1 + · · · + xr )dµ−1 (x1 ) · · · dµ−1 (xr ). Zp
Beln (x + x1 + · · · + xr )dµ0 (x1 ) · · · dµ0 (xr )
Zp
Z
e(e
···
= (29)
Z ···
S1 (n, m)
t −1)(x +···+x +x)t r 1
tn n!
dµ0 (x1 ) · · · dµ0 (xr )
Zp (r) Bm (x)
m=0 ∞ X n X n=0 m=0
m 1 t e −1 m!
tn (r) Bm (x)S2 (n, m) . n!
Thus, by (29), we get Z Z n X (r) (30) ··· Beln (x1 + · · · + xr + x)dµ0 (x1 ) · · · dµ0 (xr ) = Bm (x)S2 (n, m). Zp
Zp
m=0
Therefore, by (27) and (30), we obtain the following theorem.
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7
Theorem 7. For n ≥ 0, we have Bn(r) (x)
=
n X m X
(r)
S1 (n, m)S2 (m, k)Bk (x).
m=0 k=0
By the same method of (29), we get Z ∞ Z X tn Beln (x + x1 + · · · + xr )dµ−1 (x1 ) · · · dµ−1 (xr ) ··· n! Zp n=0 Zp Z Z t = ··· e(e −1)(x1 +···+xr +x)t dµ−1 (x1 ) · · · dµ−1 (xr ) (31) =
Zp Zp ∞ n X X n=0 m=0
tn (r) Em (x)S2 (n, m) . n!
From (31), we have Z Z n X (r) (32) ··· Beln (x1 +· · ·+xr +x)dµ−1 (x1 ) · · · dµ−1 (xr ) = Em (x)S2 (n, m). Zp
Zp
m=0
Therefore, by Theorem 6 and (32), we obtain the following theorem. Theorem 8. For n ≥ 0, we have En(r) (x)
=
n X m X
(r)
S1 (n, m)S2 (m, k)Ek (x).
m=0 k=0
From (4), we have Z
Z
(1 + t)(x1 +···+xr +x) dµ−1 (x1 ) · · · dµ−1 (xr )
··· Zp
Zp
(33)
∞ 2 r X tn = (1 + t)x = Ch(r) (x) . n 1+t n! n=0
t
By replacing t by e(e −1) − 1, we get Z Z t ··· e(e −1)(x1 +···+xr +x) dµ−1 (x1 ) · · · dµ−1 (xr ) Zp
=
(34)
= = =
Zp
∞ Z X
Z ···
(x1 + · · · + xr + x)dµ−1 (x1 ) · · · dµ−1 (xr )
Zp m=0 Zp ∞ X 1 (r) Em (x) (et − 1)m m! m=0 ∞ ∞ X X tn (r) Em (x) S2 (n, m) n! n=m m=0 ∞ X n tn X (r) , Em (x)S2 (n, m) n! n=0 m=0
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1 t (e − 1)m m!
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and
t −1)r
2r e−(e
e(e
t −1)x
= 2r
∞ X
∞
Bell (−r)
tl X tm Belm (x) l! m! m=0
l=0 ∞ n X X
Belm (x)Beln−m (−r)n! tn m!(n − m)! n! n=0 m=0 ∞ n tn X X n = . 2r Belm (x)Beln−m (−r) n! m
= 2r
(35)
n=0
m=0
Therefore, by (33),(34) and (35), we obtain the following theorem. Theorem 9. For n ≥ 0, we have
n X
(r) Em (x)S2 (n, m)
m=0
n X n Belm (x)Beln−m (−r). =2 m r
m=0
Now, we observe that
∞ X
Ch(r) m (x)
m=0
= (36) =
∞ X
∞ ∞ m X X 1 (et −1) (et − 1)k (r) e −1 = Chm (x) S2 (k, m) m! k! m=0
k X
k=0 m=0 ∞ X k X
Ch(r) m (x)S2 (k, m) Ch(r) m (x)S2 (k, m)
k=0 m=0
=
∞ X n X k X
k=m
1 t (e − 1)k k! ∞ X
S2 (n, k)
n=k
tn n!
Ch(r) m (x)S2 (k, m)S2 (n, k)
n=0 k=0 m=0
tn n!
.
Therefore, by (33), (34) and (36), we obtain the following theorem. Theorem 10. For n ≥ 0, we have
n X m=0
(r) Em (x)S2 (n, m) =
n X k X
Ch(r) m (x)S2 (k, m), S2 (n, k).
k=0 m=0
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From (4), we have Z Z ···
t −1)(x +···+x +x) r 1
dµ−1 (x1 ) · · · dµ−1 (xr )
Zp
Zp
=
e(e
9
2
r
et −1
e(e
t −1)x
e +1 ∞ ∞ X t m X tl (r) (e − 1) = Em Bell (x) m! l! m=0
= (37)
∞ X
l=0
(r) Em
m=0
=
=
=
∞ X k X
∞
∞ X k=m
tl tk X Bell (x) S2 (k, m) k! l! l=0
(r) Em S2 (k, m)
k=0 m=0 ∞ n X k XnX
∞ tk X
k!
Bell (x)
l=0
(r) Em (x)S2 (k, m)Beln−k (x)
n=0 k=0 m=0 ∞ nX n X
k X
n=0 k=0
m=0
n k
tl l!
o tn n! k!(n − k)! n!
(r) Em (x)S2 (k, m)Beln−k (x)
o tn n!
Therefore, by (34) and (37), we obtain the following theorem. Theorem 11. For n ≥ 0, we have n X k=0
(r) Ek (x)S2 (n, k)
=
k n X X n k=0
k
(r) Em (x)S2 (k, m)Beln−k (x).
m=0
ACKNOWLEDGEMENTS. The present Research has been conducted by the Research Grant of Kwangwoon University in 2015 References [1] S. Araci, X. Kong, M. Acikgoz, E. Sen, A new approach to multivariate q-Euler polynomials using the umbral calculus, J. Integer Seq. 17 (2014), no. 1, Article 14.1.2, 10 pp. [2] G. E. Andrews, The theory of partitions, Reprint of the 1976 original. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1998. xvi+255 pp. ISBN: 0-521-63766-X. [3] A. Bayad, Modular properties of elliptic Bernoulli and Euler functions, Adv. Stud. Contemp. Math. 20 (2010), no. 3, 389–401. [4] Bell, E. T. ”Exponential Polynomials.” Ann. Math. 35(1934), 258-277. [5] Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, Dordrecht, Netherlands: Reidel, 1974. [6] S. Gaboury, R. Tremblay, B.-J. Fug´ere, Some explicit formulas for certain new classes of Bernoulli, Euler and Genocchi polynomials, Proc. Jangjeon Math. Soc. 17 (2014), no. 1, 115–123 [7] H. W. Gould, T. He, Characterization of (c)-Riordan arrays, Gegenbauer-Humbert-type polynomial sequences, and (c)-Bell polynomials, J. Math. Res. Appl. 33 (2013), no. 5, 505–527. [8] D. S. Kim, T. Kim, Higher-order cauchy of the second kind and poly-cauchy of the second kind mixed type polynomials, Ars Combinatoria 115(2014), pp.435-451. [9] D. S. Kim, D.V. Dolgy, T. Kim, S.-H. Rim, Identities involving Bernoulli and Euler polynomials arising from Chebyshev polynomials, Proc. Jangjeon Math. Soc. 15 (2012), no. 4, 361–370 [10] D. S. Kim, T. Kim, C.S. Ryoo, Sheffer sequences for the powers of sheffer pairs under umbral composition, Adv. Stud. Contemp. Math. 23 (2013), no. 2, 275–285. [11] T. Kim, Identities involving Laguerre polynomials derived from umbral calculus. Russ. J. Math. Phys. 21 (2014), no. 1, 36–45.
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[12] T. Kim, D. V. Dolgy, D.S. Kim, S.-H. Rim, A note on the identities of special polynomials, Ars Combin. 113A (2014), 97–106. [13] Q.-M. Luo, F. Qi, Relationships between generalized Bernoulli numbers and polynomials and generalized Euler numbers and polynomials, Adv. Stud. Contemp. Math. 7 (2003), no. 1, 11–18 [14] T. Mansour, M. Shattuck, A recurrence related to the Bell numbers, Integers 12 (2012), no. 3, 373–384. [15] J. Riordan, An Introduction to Combinatorial Analysis, New York: Wiley, 1980. [16] S. Roman, The umbral calculus. Pure and Applied Mathematics, 111. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1984. x+193 pp. ISBN: 0- 12-594380-6 [17] Z. Zhang, H. Yang, Some closed formulas for generalized Bernoulli-Euler numbers and polynomials, Proc. Jangjeon Math. Soc. 11 (2008), no.2, 191–198 [18] Z. Zhang, J. Yang, Notes on some identities related to the partial Bell polynomials, Tamsui Oxf. J. Inf. Math. Sci. 28 (2012), no. 1, 39–48. Department of Mathematics, Kyungpook National University, Taegu 702-701, S. Korea E-mail address: [email protected] Department of Computer Science, Daegu Haany University, Kyungsan 712-715, S. Korea E-mail address: [email protected] Department of Mathematics, Kyungpook National University, Taegu 702-701, S. Korea E-mail address: [email protected] Jangjeon Research Institute for Mathematics and Physics, Hapcheon 678-800, S. Korea, Department of mathematics, Kwangwoon University, Seoul 139-701, S. Korea E-mail address: [email protected]
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ON A PRODUCT-TYPE OPERATOR FROM WEIGHTED BERGMAN-NEVANLINNA SPACES TO WEIGHTED ZYGMUND SPACES ON THE UNIT DISK ZHI JIE JIANG, HONG BIN BAI, AND ZUO AN LI
Abstract. Let D = {z ∈ C : |z| < 1} be the open unit disk, ϕ an analytic self-mapping of D and ψ an analytic function in D. Let D be the differentiation operator and Wϕ,ψ the weighted composition operator. The boundedness and compactness of the product-type operator Wϕ,ψ D from weighted BergmanNevanlinna spaces to weighted Zygmund spaces on D are characterized.
1. Introduction Let C be the complex plane, D = {z ∈ C : |z| < 1} the open unit disk in C, H(D) the class of all holomorphic functions on D, ϕ a holomorphic self-mapping of D and ψ ∈ H(D). Weighted composition operator Wϕ,ψ on H(D) is defined by Wϕ,ψ f (z) = ψ(z) · f (ϕ(z)), z ∈ D. If ψ ≡ 1 the operator is reduced to, so called, composition operator and usually denote by Cϕ . If ϕ(z) = z, it is reduced to, so called, multiplication operator and usually denote by Mψ . Standard problem is to provide function theoretic characterizations when ϕ and ψ induce a bounded or compact weighted composition operator. Weighted composition operators between various spaces of holomorphic functions on different domains have been studied by numerous authors, see, e.g., [1, 2, 8, 9, 11, 13–17, 19, 21, 23, 28, 34, 35, 45, 49, 50, 53] and the references therein. Let D be the differentiation operator on H(D), that is, Df (z) = f 0 (z), z ∈ D. The product-type operator Cϕ D has been studied, for example, in [4, 18, 20, 25, 26, 29, 41, 44, 46]. In [31] Sharma has studied the following operators from BergmanNevanlinna spaces to Bloch-type spaces: Mψ Cϕ Df (z) = ψ(z)f 0 (ϕ(z)), Mψ DCϕ f (z) = ψ(z)ϕ0 (z)f 0 (ϕ(z)), Cϕ Mψ Df (z) = ψ(ϕ(z))f 0 (ϕ(z)), and Cϕ DMψ f (z) = ψ 0 (ϕ(z))f (ϕ(z)) + ψ(ϕ(z))f 0 (ϕ(z)), 2000 Mathematics Subject Classification. Primary 47B38; Secondary 47B33, 47B37. Key words and phrases. Weighted Bergman-Nevanlinna spaces, product-type operators, weighted Zygmund spaces, little weighted Zygmund spaces. 1
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for z ∈ D and f ∈ H(D). These operators on weighted Bergman spaces, were also studied in [51] and [52] by Stevi´c, Sharma and Bhat. If we consider the product-type operator Wϕ,ψ D, then it is clear that Mψ Cϕ D = Wϕ,ψ D, Mψ DCϕ = Wϕ,ψ·ϕ0 D, Cϕ Mψ D = Wϕ,ψ◦ϕ D and Cϕ DMψ = Wϕ,ψ0 ◦ϕ + Wϕ,ψ◦ϕ D. Quite recently, the present author has considered operator Wϕ,ψ D from weighted Bergman spaces to weighted Zygmund spaces in [10]. This paper is devoted to characterizing the boundedness and compactness of operator Wϕ,ψ D from weighted Bergman-Nevanlinna spaces to weighted Zygmund spaces. It can be regarded as a continuation of the investigation of operators from weighted Bergman-Nevanlinna spaces to other spaces (see. e.g., [12] and [30]). Next we introduce the needed spaces and some facts. Let dA(z) = π1 dxdy be the normalized Lebesgue measure on D. For α > −1, let dAα (z) = (α + 1)(1−|z|2 )α dA(z) be the weighted Lebesgue measure on D. The weighted BergmanNevanlinna space Aα log on D consists of all f ∈ H(D) such that Z log(1 + |f (z)|)dAα (z) < ∞. kf kAαlog = D
It is a Fr´echet space with the translation invariant metric d(f, g) = kf − gkAαlog . For some details of this space, see, e.g., [6], [7], [47] and [54]. For β > 0, the weighted-type Aβ consists of all f ∈ H(D) such that sup(1 − |z|2 )β |f (z)| < ∞. z∈D
This space is a non-separable Banach space with the norm defined by kf kAβ = sup(1 − |z|2 )β |f (z)|. z∈D
The closure of the set of polynomials in Aβ is denoted by Aβ,0 , which is a separable Banach space and consists exactly of those functions f in Aβ satisfying the next condition lim − (1 − |z|2 )β |f (z)| = 0. |z|→1
For β > 0, the weighted Bloch space is defined by Bβ = f ∈ H(D) : sup(1 − |z|2 )β |f 0 (z)| < ∞ . z∈D
Under the norm kf kBβ = |f (0)| + sup(1 − |z|2 )β |f 0 (z)|, z∈D
it is a Banach space. For more detail on the space, see, e.g. [55]. The closure of the set of polynomials in Bβ is called the little weighted Bloch space and is denoted by Bβ,0 . For a good source for such spaces, we refer to [55]. For β > 0, the weighted Zygmund space Zβ consists of all f ∈ H(D) such that sup(1 − |z|2 )β |f 00 (z)| < ∞. z∈D
It is a Banach space with the norm kf kZβ = |f (0)| + |f 0 (0)| + sup(1 − |z|2 )β |f 00 (z)|. z∈D
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The little weighted Zygmund space Zβ,0 consists those functions f in Zβ satisfying lim (1 − |z|2 )β |f 00 (z)| = 0,
|z|→1−
and it is a closed subspace of the weighted Zygmund space. For weighted-type spaces, weighted Bloch spaces and weighted Zygmund spaces on the unit disk, the upper half plane, the unit ball, the unit polydisk and some operators, see, e.g. [5, 11, 16, 22–24, 27, 28, 32, 33, 36–40, 42, 43, 48] and the references therein. Since the weighted Bergman-Nevanlinna space is a Fr´echet space and not a Banach space, it is necessary to introduce several definitions needed in this paper. Let X and Y be topological vector spaces whose topologies are given by translation invariant metrics dX and dY , respectively, and let L : X → Y be a linear operator. It is said that L is metrically bounded if there exists a positive constant K such that dY (Lf, 0) ≤ KdX (f, 0) for all f ∈ X. When X and Y are Banach spaces, the metrical boundedness coincides with the usual definition of bounded operators between Banach spaces. Recall that L : X → Y is metrically compact if it maps bounded sets into relatively compact sets. When X and Y are Banach spaces, the metrical compactness coincides with the usual definition of compact operators between Banach spaces. When X = Aα log and Y is a Banach space, we define kLkAαlog →Y =
kLf kY ,
sup kf kAα ≤1 log
and we often write kLkAαlog →Y by kLk. Throughout this paper, an operator is bounded (respectively, compact), if it is metrically bounded (respectively, metrically compact). Constants are denoted by C, they are positive and may differ from one occurrence to the next. The notation a b means that there exists a positive constant C such that a/C ≤ b ≤ Ca. 2. The operator Wϕ,ψ D : Aα log → Zβ (Zβ,0 ) Our first lemma characterizes the compactness in terms of sequential convergence. Since the proof is standard, it is omitted (see, e.g., Proposition 3.11 in [3]). Lemma 2.1. Let α > −1, β > 0 and Y ∈ {Zβ , Zβ,0 }. Then the bounded operator Wϕ,ψ D : Aα log → Y is compact if and only if for every bounded sequence (fn )n∈N in Aα such that fn → 0 uniformly on every compact subset of D as n → ∞, it log follows that lim kWϕ,ψ Dfn kY = 0. n→∞
The next result can be found, for example, in [54]. Lemma 2.2. Let α > −1 and n ∈ N0 = N ∪ {0}. Then for all f ∈ Aα log and z ∈ D, there exists a positive constant C independent of f such that (1 − |z|2 )n |f (n) (z)| ≤ exp
Ckf kAαlog (1 − |z|2 )α+2
.
Now we consider the boundedness of operator Wϕ,ψ D : Aα log → Zβ .
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Theorem 2.3. Let α > −1, β > 0, ϕ be an analytic self-map of D and ψ ∈ H(D). Then for all c > 0, the following statements are equivalent: (i) The operator Wϕ,ψ D : Aα log → Zβ is bounded. (ii) The operator Wϕ,ψ D : Aα log → Zβ is compact. (iii) ψ ∈ Zβ , M0 = sup(1 − |z|2 )β |ψ(z)||ϕ0 (z)|2 < ∞, z∈D
M1 = sup(1 − |z|2 )β |ψ(z)ϕ00 (z) + 2ψ 0 (z)ϕ0 (z)| < ∞, z∈D
(1 − |z|2 )β 00 c |ψ (z)| exp = 0, (1 − |ϕ(z)|2 )α+2 ϕ(z)→∂D 1 − |ϕ(z)|2 lim
(1 − |z|2 )β c ψ(z)ϕ00 (z) + 2ψ 0 (z)ϕ0 (z) exp = 0, 2 2 (1 − |ϕ(z)|2 )α+2 ϕ(z)→∂D (1 − |ϕ(z)| ) lim
and (1 − |z|2 )β c |ψ(z)||ϕ0 (z)|2 exp = 0. 2 3 (1 − |ϕ(z)|2 )α+2 ϕ(z)→∂D (1 − |ϕ(z)| ) lim
Proof. Suppose that (i) holds. Take the functions f (z) = z and f (z) = z 2 , respectively. Since the operator Wϕ,ψ D : Aα log → Zβ is bounded, we have sup(1 − |z|2 )β |ψ 00 (z)| ≤ kWϕ,ψ DzkZβ ≤ CkWϕ,ψ Dk
(1)
z∈D
and sup(1 − |z|2 )β ψ 00 (z)ϕ(z) + 2ψ 0 (z)ϕ0 (z) + ψ(z)ϕ00 (z) ≤ CkWϕ,ψ Dk.
(2)
z∈D
Inequality (1) shows that ψ ∈ Zβ . Also by (1) and the boundedness of ϕ, sup(1 − |z|2 )β |ψ 00 (z)||ϕ(z)| < ∞.
(3)
z∈D
Then by (2), (3) and the boundedness of ϕ, M1 = sup(1 − |z|2 )β |ψ(z)ϕ00 (z) + 2ψ 0 (z)ϕ0 (z)| < ∞.
(4)
z∈D
Let the function f (z) = z 3 . Then sup(1 − |z|2 )β ψ 00 (z)ϕ(z)2 +2ψ(z)ϕ0 (z)2 + 4ψ 0 (z)ϕ0 (z)ϕ(z) + 2ψ(z)ϕ00 (z)ϕ(z) z∈D
≤ CkWϕ,ψ Dk.
(5)
By (1), (4) and (5), M0 = sup(1 − |z|2 )β |ψ(z)||ϕ0 (z)|2 ≤ CkWϕ,ψ Dk < ∞.
(6)
z∈D
For w ∈ D, we choose the function f1 (z) = c1
(1 − |ϕ(w)|2 )α+2
+ c2
(1 − |ϕ(w)|2 )α+4
(1 − ϕ(w)z)2(α+2) (1 − ϕ(w)z)2(α+2)+2 (1 − |ϕ(w)|2 )α+5 (1 − |ϕ(w)|2 )α+6 + c3 − (1 − ϕ(w)z)2(α+2)+3 (1 − ϕ(w)z)2(α+2)+4
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where
5
48α3 + 460α2 + 1398α + 1340 , 24α3 + 214α2 + 655α + 682 16α2 + 104α + 164 c3 = , 6α2 + 37α + 62
c2 = −
and c1 = 1 − c2 − c3 . We also choose the function 2α + 7 (1 − |ϕ(w)|2 )α+2 6α + 21 (1 − |ϕ(w)|2 )α+4 g1 (z) = − 2(α+2) 4α + 8 (1 − ϕ(w)z) 4α + 12 (1 − ϕ(w)z)2(α+2)+2 +
(1 − |ϕ(w)|2 )α+5 (1 − ϕ(w)z)2(α+2)+3
.
For the functions f1 and g1 , we have f1 (ϕ(w)) = f100 (ϕ(w)) = f1000 (ϕ(w)) = 0
(7)
g10 (ϕ(w)) = g100 (ϕ(w)) = 0.
(8)
and
Consequently, (7) and (8) make the function f (z) = f1 (z) exp cg1 (z) to satisfy f 00 (ϕ(w)) = f 000 (ϕ(w)) = 0 and f 0 (ϕ(w)) = C
ϕ(w) c exp , 2 α+3 (1 − |ϕ(w)| ) (1 − |ϕ(w)|2 )α+2
where C = 2c2 + 3c3 − 4. By the boundedness of the operator Wϕ,ψ D : Aα log → Zβ , we find |ϕ(w)|(1 − |w|2 )β 00 c exp ≤ C. ψ (w) (1 − |ϕ(w)|2 )α+3 (1 − |ϕ(w)|2 )α+2 Thus (1 − |w|2 )β 00 c exp = 0. ψ (w) (1 − |ϕ(w)|2 )α+2 ϕ(w)→∂D 1 − |ϕ(w)|2 lim
For w ∈ D, we choose the functions f2 (z) =
3α + 8 (1 − |ϕ(w)|2 )α+2 6α + 22 (1 − |ϕ(w)|2 )α+4 − 3α + 10 (1 − ϕ(w)z)2(α+2) 3α + 10 (1 − ϕ(w)z)2(α+2)+2 +
(1 − |ϕ(w)|2 )α+6 6α + 24 (1 − |ϕ(w)|2 )α+5 − , 3α + 10 (1 − ϕ(w)z)2(α+2)+3 (1 − ϕ(w)z)2(α+2)+4
and g2 (z) =
α + 3 (1 − |ϕ(w)|2 )α+2 (1 − |ϕ(w)|2 )α+4 − . α + 2 (1 − ϕ(w)z)2(α+2) (1 − ϕ(w)z)2(α+2)+2
Then f2 (ϕ(w)) = f20 (ϕ(w)) = f2000 (ϕ(w)) = 0
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(9)
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and g20 (ϕ(w)) = 0. From this and (9), for the function g(z) = f2 (z) exp cg2 (z) we have g 0 (ϕ(w)) = g 000 (ϕ(w)) = 0 and 2 c ϕ(w) 00 g (ϕ(w)) = C exp , 2 α+4 (1 − |ϕ(w)| ) (1 − |ϕ(w)|2 )α+2 where 24α + 120α + 141 C=− . 3α + 10 By the boundedness of Wϕ,ψ D : Aα log → Zβ , kWϕ,ψ DgkZβ ≤ CkWϕ,ψ Dk, and from which we obtain c |ϕ(w)|2 (1 − |w|2 )β ψ(w)ϕ00 (w) + 2ψ 0 (w)ϕ0 (w) exp ≤ C. 2 α+4 (1 − |ϕ(w)| ) (1 − |ϕ(w)|2 )α+2 This shows that (1 − |w|2 )β c lim = 0. ψ(w)ϕ00 (w) + 2ψ 0 (w)ϕ0 (w) exp (1 − |ϕ(w)|2 )α+2 ϕ(w)→∂D (1 − |ϕ(w)|2 )2 For w ∈ D, we choose the functions f3 (z) =
1 (1 − |ϕ(w)|2 )α+2 (1 − |ϕ(w)|2 )α+4 −2 3 (1 − ϕ(w)z)2(α+2) (1 − ϕ(w)z)2(α+2)+2 (1 − |ϕ(w)|2 )α+6 8 (1 − |ϕ(w)|2 )α+5 − + 3 (1 − ϕ(w)z)2(α+2)+3 (1 − ϕ(w)z)2(α+2)+4
and g3 (z) =
(1 − |ϕ(w)|2 )α+2 (1 − ϕ(w)z)2(α+2)
.
From a calculation, we obtain f3 (ϕ(w)) = f30 (ϕ(w)) = f300 (ϕ(w)) = 0.
(10)
Define the function h(z) = f3 (z) exp cg3 (z). Then by (10), h0 (ϕ(w)) = h00 (ϕ(w)) = 0, and by a direct calculation, 3
ϕ(w) c h (ϕ(w)) = C exp , 2 α+5 (1 − |ϕ(w)| ) (1 − |ϕ(w)|2 )α+2 000
where C = −30(α + 2)2 − 8. Since Wϕ,ψ D : Aα log → Zβ is bounded, we have kWϕ,ψ DhkZβ ≤ CkWϕ,ψ Dk, and so (1 − |z|2 )β |(Wϕ,ψ Dh)00 (z)| ≤ CkWϕ,ψ Dk,
(11)
for all z ∈ D. Letting z = w in (11) yields to (1 − |w|2 )β c |ψ(w)||ϕ0 (w)|2 |ϕ(w)|3 exp ≤ CkWϕ,ψ Dk. (1 − |ϕ(w)|2 )α+5 (1 − |ϕ(w)|2 )α+2
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Thus (1 − |w|2 )β c C(1 − |ϕ(w)|2 )α+2 |ψ(w)||ϕ0 (w)|2 exp ≤ . 2 3 2 α+2 (1 − |ϕ(w)| ) (1 − |ϕ(w)| ) |ϕ(w)|3
(12)
Taking limit as ϕ(w) → ∂D in (12) gives (1 − |w|2 )β c |ψ(w)||ϕ0 (w)|2 exp = 0. (1 − |ϕ(w)|2 )α+2 ϕ(w)→∂D (1 − |ϕ(w)|2 )3 lim
The proof of the implication is finished. α (iii) ⇒ (ii). Let (fn )n∈N be a sequence in Aα log with supn∈N kfn kAlog ≤ M and fn → 0 uniformly on every compact subset of D as n → ∞. We have that for the constant C in Lemma 2.2, for any ε > 0 there exits a constant δ ∈ (0, 1) such that whenever δ < |ϕ(z)| < 1, it follows that (1 − |z|2 )β 00 C |ψ (z)| exp < ε, 1 − |ϕ(z)|2 (1 − |ϕ(z)|2 )α+2 (1 − |z|2 )β C 00 0 0 exp ψ(z)ϕ (z) + 2ψ (z)ϕ (z) < ε, (1 − |ϕ(z)|2 )2 (1 − |ϕ(z)|2 )α+2 and (1 − |z|2 )β C |ψ(z)||ϕ0 (z)|2 exp < ε. (1 − |ϕ(z)|2 )3 (1 − |ϕ(z)|2 )α+2 Then by Lemma 2.2, for a fixed δ ∈ (0, 1) we have kWϕ,ψ Dfn kZβ = (ψ · fn0 ◦ ϕ)(0) + (ψ · fn0 ◦ ϕ)0 (0) + sup(1 − |z|2 )β (ψ(z)fn0 (ϕ(z))00 z∈D 0 0 0 00 = ψ(0) fn (ϕ(0)) + ψ (0)fn (ϕ(0)) + ψ(0)fn (ϕ(0))ϕ(0) + sup(1 − |z|2 )β ψ 00 (z)fn0 (ϕ(z)) + ψ(z)ϕ00 (z) + 2ψ 0 (z)ϕ0 (z) fn00 (ϕ(z)) + ψ(z)ϕ0 (z)2 fn000 (ϕ(z)) z∈D ≤ ψ(0) + ψ 0 (0) fn0 (ϕ(0)) + ϕ(0) ψ(0) fn00 (ϕ(0)) + sup(1 − |z|2 )β ψ 00 (z) fn0 (ϕ(z)) z∈D
2 + sup(1 − |z|2 )β ψ(z)ϕ00 (z) + 2ψ 0 (z)ϕ0 (z) fn00 (ϕ(z)) + sup(1 − |z|2 )β ψ(z) ϕ0 (z) fn000 (ϕ(z)) z∈D z∈D 00 0 0 ≤ ψ(0) + ψ (0) fn (ϕ(0)) + ϕ(0) ψ(0) fn (ϕ(0)) + sup (1 − |z|2 )β ψ 00 (z) fn0 (ϕ(z)) + sup (1 − |z|2 )β ψ 00 (z) fn0 (ϕ(z)) |ϕ(z)|≤δ
δ s(h2 ), then h1 > h2 ; if s(h1 ) = s(h2 ), then h1 = h2 . Xia and Xu [11, 12] further gave some hesitant fuzzy aggregation operators as follows: Let hj (j = 1, 2, · · · , n) be a collection of HFEs, ω = (ω1 , ω2 , · · · , ωn )T be the weight vector of hj (j = n ∑ 1, 2, · · · , n) with ωj ∈ [0, 1] and ωj = 1, then j=1
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Hesitant fuzzy Maclaurin symmetric mean operators
3
(1) The hesitant fuzzy weighted averaging (HF W A) operator n n ⊕ ∪ ∏ HF W A(h1 , h2 , · · · , hn ) = (ωj hj ) = 1− (1 − γj )ωj γj ∈hj , i=1,··· ,n
j=1
j=1
Especially, if ω = (1/n, 1/n, · · · , 1/n)T , then the HF W A operator reduces to the hesitant fuzzy averaging (HF A) operator n ∪ ∏ HF A(h1 , h2 , · · · , hn ) = 1− (1 − γj )1/n (1) γj ∈hj , i=1,··· ,n
j=1
(2) The hesitant fuzzy weighted geometric (HF W G) operator HF W G(h1 , h2 , · · · , hn ) =
n ⊗
∪
hj ωj =
γj ∈hj , i=1,··· ,n
j=1
n ∏
γj ωj
j=1
Especially, if ω = (1/n, 1/n, · · · , 1/n)T , then the HF W G operator becomes to the hesitant fuzzy geometric (HF G) operator n ∪ ∏ HF G(h1 , h2 , · · · , hn ) = γj 1/n (2) γj ∈hj , i=1,··· ,n
2.2
j=1
Maclaurin symmetric mean
The M SM introduced by Maclaurin [18] is a useful technique characterized by the ability to capture the interrelationship among the multi-input arguments. The definition of M SM is given as follows. Definition 2.3. [18] Let ai (i = 1, 2, · · · , n) be a collection of nonnegative real numbers and r = 1, 2, · · · , n. If
∑
r ∏
1≤i1 0: Keywords: Oscillation, Stochastic di¤erential equations, Zeros of solutions, Wiener process, Itô integral AMS Subject Classi…cation: 60H10; 60H25; 34K11 ————————————————————————————————————————————————–
1
Introduction
During the past few decades, stochastic di¤erential equations (SDEs) are becoming increasingly important as models of stochastic phenomena that play a prominent role in a diverse range of application areas, including mathematical modeling in engineering and physics, geophysical sciences, stochastic control, mechanics, environmental processes, mathematical biology, molecular dynamics for chemistry, epidemiology, economic modeling, industrial mathematics and mathematical …nance [1-10]. Indeed, these models can be stochastic for di¤erent reasons. Therefore, numerous studies have been performed to understanding their dynamical behaviors, particularly in relation to problems of the speci…cation of the stochastic processes governing the behaviors of an underlying quantity, as well as fundamental microscopic laws generate stochastic behaviors in the case of coarse-graining and modeling error and so on [11-16]. However, a complete understanding of SDEs theory requires familiarity with advanced probability and stochastic processes, whereas solutions of such models are themself stochastic processes. Further, in particular, second-order di¤erential equations with random coe¢ cients have found wide variety applications in branches of science. Typically, they are mathematical models of objects under the in‡uence of random forces such that the presence of in…nite set of zeros of solutions for these equations indicates that the evolution of investigated objects is oscillatory. Recently, research work about oscillation phenomenon occupies an important place in di¤erential stochastic theory due to the sensitivity of stochastic forces and behaviors. Moreover, the stochastic theory for these equations, as well as the theory of oscillatory solutions of deterministic equations have been studied extensively and are well-developed. Oscillation and nonoscillatition conditions for both linear and nonlinear di¤erential equations, di¤erence equations and delay equations have been investigated in [17-22]. The oscillating properties for solutions of di¤erence equations can be found in the excellent monograph of Agarwal et al. [23]. Besides, the authors in the monograph [24] were devoted to the problem of relationship between oscillation behavior of solutions for di¤erential equations and the corresponding di¤erence equations. On the contrary, the theory of the oscillation of stochastic system is not well-developed. Incidentally, Mao in [25] considered the stochastic equation of the following form _ (t); x • + kx = hW
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_ (t) is a Wiener process which is nowhere di¤erentiable. It was proved that the solution with initial values where W x(0) = 1; x(0) _ = 0 has in…nitely many zeros, all simple, on each half-line [t0 ; 1) for every t0 0. The …rst two moments of the …rst zero were estimated. In contrast, the more general equation of the form _ (t); x • + k(t; x; x) _ = hW was studied in [26]. The author there demonstrated that this equation has in…nitely many zeros with probability 1. Consequently, the explicit upper and lower estimates for the expected values of these zeros were obtained. However, the Itô stochastic equations of the form _ (t))x = 0 x • + (p(t) + q(t)W was considered by method of asymptotic equivalence in [27,28], whereas the oscillation of solutions was analyzed. In the monographs [29,30], the oscillatory properties of solutions for both linear and nonlinear stochastic delay di¤erential equations with multiplicative noise are given. It was shown that such noise induces an oscillation in solutions. Besides, the oscillation of solutions of …rst order nonlinear stochastic di¤erence equations is investigated in [31]. The purpose of this paper is to study an asymptotic behavior, as t ! 1, of solutions of a second order stochastic Itô equation. Meanwhile, we investigate existence of zeros of its solutions with probability 1. In the sequel, unless otherwise speci…ed, we say that a solution is oscillatory if it has in…nitely many zeros with probability 1 on the half-line [0; 1). A solution which is not oscillatory is called nonoscillatory. This paper is organized in …ve sections including the introduction. In the next section, we present some necessary de…nitions and preliminary results that will be used in this work. In the same time, statement of a second order SDEs is introduced. In Section 3, the discussion of a solution for linear case of second-order SDEs is presented, as well as the conditions of nonoscillatory behavior of its solutions for nonlinear case of SDEs are constructed. Finally, the conclusions are drawn in Section 4.
2
Statement of the problem and auxiliary results
The material in this section is basic in some sense. For the reader’s convenience, we present some necessary de…nitions and auxiliary results related to the SDEs theory that will be used in the remainder of this paper. Let us consider a nonlinear second-order stochastic equation of the following form _ (t) = 0; t x • + p(t; x; x) _ + q(t; x; x) _ W
0:
(1)
While the corresponding system of stochastic Itô equations will be written as dx1 = x2 dt; dx2 =
p(t; x1 ; x2 )dt
(2) q(t; x1 ; x2 )dW (t);
where x 2 R1 ; t 0; W (t) is a standard Wiener process de…ned on the probability space ( ; F; P ), fFt ; t 0g is the family of -algebras adapted to W (t); and the functions p(t; x1 ; x2 ) and q(t; x1 ; x2 ) are continuous with respect to x1 ; x2 2 R1 for t 0, as well as satisfy the Lipschitz condition with respect to x1 , x2 together with linear growth condition. Without loss of generality, we assume that p(t; 0; 0) = q(t; 0; 0) = 0. It should be noted that the presence of stochastic in equation (1) causes new di¢ culties in studying the oscillation of solutions. In this regard, we mention here the following remark: Firstly, solutions of equation (1) are random processes, so their zeros are random variables with certain properties. As a consequence, we need to introduce a new de…nition of zero which is di¤erent of the deterministic case (q = 0). Secondly, from the Strook-Varadham support theorem, it follows that solutions of equation (1) can be nonoscillatory on …nite intervals. Therefore, the oscillatory solutions should be considered only on in…nite intervals. Thirdly, since solutions of equation (1) have only …rst derivative, so we can not use a second derivative to apply the concavity property of the solution between two successive zeros. It is well known that this method is used in the deterministic case. 861
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Nevertheless, system (2) is a particular case of general second-order system, it would seem that this simpli…es its investigation, as well it is the system with a degenerate di¤usion that completes its investigation by probability methods. Subsequently, under the above assumptions of equation (1) and the corresponding system (2), we assume that the solution x(t) = (x1 (t); x2 (t)) of system (2) subject to the initial condition x(t0 ) = x0 satisfy all necessary requirements of the existence of a unique solution for t t0 ; whereas x0 is an Ft0 measurable random variable. In addition, the process x(t) will never reach the origin (0,0), for more details see Lemma 2.3 in [32]. In our notation, let x1 (t) = x(t). Throughout this paper, a solution x(t) of equation (1) is called a nontrivial solution if it satisfy the following condition P fx(t) = 0; t > t0 g = 0 On the other hand, for any nontrivial solution x(t) of equation (1), where t can be de…ned as follows 8 < inf ft > t0 j x1 (t) = 0g ; if ft > t0 j x1 (t) = 0g 6= ?; 1 = : 1; otherwise.
t0
0; the random variable
1
(3)
Now, we will introduce the de…nition of zeros of a solution x(t) on the half-line t > 0.
De…nition 2.1 The random variable with probability 1.
1
is called the …rst zero of a solution x(t) on the interval t
t0 ; if
1
1 j x1 (t) = 0g ; if ft > 1 j x1 (t) = 0g 6= ?; = 2 : 1; otherwise.
(4)
Here, the random variable 2 is called the second zero of a solution x(t) on the interval t t0 ; if 2 < 1 with probability 1. Correspondingly, one can de…ne by induction a sequence of zeros f n g of a solution x(t) on the interval t t0 . Particularly, if t0 = 0: Then, we deal with zeros on the half-line t > 0. De…nition 2.2 A nontrivial solution x(t) of equation (1) is called oscillatory on the half-line t > 0; if it has in…nitely many zeros there. Otherwise, it is called nonoscillatory.
3
Main results and behavior solutions of the SDEs
In this section, some de…nitions and results are brie‡y reviewed to establish and generalize the results to the main equation in this work. Meanwhile, we study the behavior of the zeros of solutions for a class of second-order SDEs subject to some initial conditions, as well as we detect the conditions of nonoscillatory behavior of its solutions.
3.1
Linear stochastic Itô equation
Consider the following equation _ (t); x • + x = f (t)W
(5)
subject to the initial conditions x(0) = 1; x(0) _ = 0;
(6)
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where f (t) is a nonrandom function de…ned on t Zt
4
0 such that a stochastic Itô integral
f (s)dW (s)
0
is de…ned for any t > 0. Note that equation (5) is special case of equation (1), so it satis…es all arguments mentioned in the previous part of our work. Further, we give the following theorem regarding to study the behavior of the zeros of solutions for equation (5) with initial conditions (6). Theorem 3.1 Assume that f (t) satis…es the following conditions: 1. f (t) is di¤erentiable function for t s)f (s))0
2. (sin(t
0 for 0
s
0 such that f (0)
t
0;
2:
Then, the solution of equation (5) subject to initial conditions (6) oscillates on the half-line t mathematical expectation 1 of the …rst zero satis…es the estimation E
1
0: Besides, a
1 2t? ( p ? ); t
(7)
where t is the solution of the equation f (0) = cot(t) on [0; 2 ] and
(z) =
(8) 1 2
Zz
u2 2
e
du:
0
Proof. From Itô formula, the representation of the solution of (5) with initial conditions (6) is given by
x(t) = cos(t) +
Zt
f (s) sin(t
s)dW (s):
(9)
0
Which implies that Zt x(t) = cos(t) + sin(t) f (s) cos(s)dW (s) 0
Zt cos(t) f (s) sin(s)dW (s): 0
Accordingly, the process x(t) can be written as x(t) = cos(t) + W1 (p(t)) sin(t) + W2 (q(t)) cos(t); where p(t) =
Zt
2
2
f (s) cos (s)ds; q(t) =
0
Zt
f 2 (s) sin2 (s)ds;
0
and W1 ; W2 are Wiener processes. In contrast, if we consider x(t) at the times tm = (2m + 12 ) fYm g by Ym = x((2m + 21 ) )
x((2(m
for m = 1; 2; 3; :::; and de…ne a sequence 0 1 (2m+ 12 ) Z B C cos2 (s)f 2 (s)dsA : 1) + 12 ) ); whereas x((2m + 12 ) ) = W1 @ 0
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Then, Y0 = W ( 4 ); Y1 = W1 ( 54 )
W1 ( 4 ); :::; is a sequence of random variables with mean zero and variance
(2m+ 12 )
Z
cos2 (s)f 2 (s)ds:
(2(m 1)+ 12 )
Here, it is worth to mention that 1 x((2m + ) ) = Y0 + Y1 + ::: + Ym : 2
(10)
By the familiar theorems on the limits of sums of independent random variables (e.g. the law of the iterated logarithm), it follows that the sequence fx((2m+ 12 ) )g has in…nitely many switches of sign. Since x(t) is continuous on [0; 1), so it has in…nitely many zeros on [0; 1): Therefore, it oscillates on [0; 1): Now, let us prove the estimation (7) for the …rst zero of the oscillation. By applying the integration-by-parts formula to (9), we obtain Zt
x(t) = cos(t)
0
(sin(t
s)f (s)) dW (s)
cos(t) +
Zt
(sin(t
s)f (s))0 ds;
(11)
0
0
for ! 2 ; where W (t)
1 and 0
s
t
2.
From properties of a Wiener process, it follows that P
where
! j max W (t) > t2[0;T ]
(z) =
Zz
p1 2 2
e
u2 2
1
1 = 2 ( p ); 2 T
(12)
du:
0
As a result, from equation (11), we obtain the estimate x(t)
cos(t)
f (0) sin(t) > 0;
(13)
for t 2 [0; t ); where t is solution of equation (8). Hence, from equations (12) and (13), we have ! 1 1 p P cot (f (0)) 2 ; cot 1 (f (0))
for the …rst zero
3.2
1.
(14)
By using Chebyshev’s inequality, it yields that E
1
2t? ( p1t? ). The proof is complete.
Nonlinear stochastic Itô equation
Let P (a; n) = fx 2 Rm j (x is any set of the form \ P (a ; n );
a; n)
0g, where a; n 2 Rm , and ( ; ) is the usual scalar product. Thus, a polyhedron (15)
2I
where I = f1; :::; N g is a …nite subset of N: Now, consider a system of SDEs dx = f (t; x)dt + g(t; x)dW (t)
(16)
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where x 2 Rm ; f : [0; 1) Rm ! Rm ; g = [gij ] : [0; 1] r dimensional Wiener process.
Rm ! Rm
Rr are mappings, and W (t) is an
De…nition 3.1 A set K 2 Rm is said to be stochastically invariant for system (16), if for any x(0) 2 K and every solution x(t) of equation (1), then P fx(t) 2 K; t > 0g = 1. The next theorem states conditions of an invariance of the set (15) for system (16). Theorem 3.2 [33] Let K =
\
P (a ; n ) be a polyhedron in Rm . Suppose that the coe¢ cients f (t; x) and g(t; x)
2I
0; x 2 Rm ; and satisfy the following conditions:
of system (16) are de…ned for t
1. for each T > 0; there exists a constant KtT > 0 such that for all x 2 K and t 2 [0; T ); 2
2
2
kf (t; x)k + kg(t; x)k
KT (1 + jxj );
2. for all T > 0; x 2 K; y 2 K and t 2 [0; T ); kf (t; x)
f (t; y)k + kg(t; x)
g(t; y)k
KT jx
yj ;
3. for each x 2 K; the functions f ( ; x) and g( ; x); de…ned for t
0; are continuous.
Then, the set K is invariant for the system (16) if and only if the following condition holds: (a) for all
2 I and x 2 K such that (x
(f (t; x); n ) where t
a ; n ) = 0, we have
0 and (gj (t; x); n ) = 0;
0; j = 1; r; and gj is the j
th column of the matrix g = [gij ]:
Now, we use the above theorem to …nd the conditions of nonoscillatory behavior of the solutions of equation (1). As well, we state the following theorem: Theorem 3.3 Suppose that the functions p and q in equation (1) satisfy the conditions (1)-(3) of Theorem 3.2. Moreover, if (1) p(t; x1 ; 0)
0; x1 < 0; t
0;
(2) p(t; x1 ; 0)
0; x1 > 0; t
0;
(3) q(t; x1 ; 0) = 0; t
(17)
0; x1 2 R1 :
Then, all solutions of equation (1) with nonrandom initial values such that x(0) > 0; x(0) _ are not oscillate on the half-line [0; 1).
0 or x(0) < 0; x(0) _
0
Proof. We consider any solution of equation (1) with initial values x(0) > 0; x(0) _ 0. It corresponds to the solution (x1 ; x2 ) of system (2) with initial values x1 (0) > 0; x2 (0) 0. Obviously, there exists > 0 such that 0< x1 (0): Let M be a set such that M = f(x1 ; x2 ) j x1 ; x2 0g. It is a polyhedron, if we set a1 = ; n1 = l1 = (1; 0)T ; \ a2 = 0; n2 = l2 = (0; 1)T . Then, M = P (a ; n ); where I = f1; 2g: Consequently, the boundaries of this 2I
polyhedron are lines 1
= f(x1 ; x2 ) j x1 = ; x2
2
= f(x1 ; x2 ) j x1
0g ;
; x2 = 0g : 865
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Therefore, by using Theorem 3.2, the functions f and q have the form f (t; x1 ; x2 ) = (x2 ; p(t; x1 ; x2 ))T ; g(t; x1 ; x2 ) = (0; q(t; x1 ; x2 ))T ; Next, we verify the conditions of Theorem 3.2. On the boundary (f; n1 ) = (f; l1 ) = x2
1,
we have
0; and (g; n1 ) = (g; l1 ) = 0:
From condition (3) of equation (17), we have (f; n2 ) = (f; l2 ) =
p(t; x1 ; 0)
0;
(g; n2 ) = (g; l2 ) =
q(t; x1 ; 0) = 0:
and
on the boundary
2:
Again from Theorem 3.2, it follows that the set M is the invariant set for the solutions of system (2). Thus, the curve (x1 (t); x2 (t)) does not intersect with probability 1 the line x1 = 0: This means that the solutions of equation (1) with initial values x(0) > 0; x(0) _ 0 do not oscillate. It remains to consider the case with initial values x(0) < 0; x(0) _ 0: We only introduce the polyhedron M1 = f(x1 ; x2 ) j x1 ; x2 0g instead the set M: Hence, the proof is complete.
4
Concluding remarks
The use of SDEs is a natural way to model real-world phenomena under stochastic processes. In this paper, we study the qualitative behavior of nonlinear second order stochastic di¤erential equations. Interest focuses on solutions of such equations which are oscillatory. A nontrivial solution is called oscillatory if it has in…nitely many zeros with probability 1 on half-line. Otherwise, it is called nonoscillatory. The su¢ cient conditions for the oscillation and nonoscillation of solutions are obtained.
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