JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS VOLUME 13, 2011 0651098283

Table of contents :
PART-1-JOCAAA-11.pdf
I-JOCAAA-11.pdf
JCAAA-2011-front-1.pdf
SCOPE--JOCAAA--2011.pdf
JOCAAA-2011--Editorial Board.pdf
Instructions--JOCAAA--2011.pdf
Binder-2011-JOCAAA-VOL-13-NO-1.pdf
1-Anastas-Gal-jocaaa--11.pdf
2-ZALIK--JOCAAA--11.pdf
3-PANDEY-JOCAAA--11.pdf
4-anastas-mursaleen-jocaaa--11.pdf
5-PANDEY--JOCAAA--11.pdf
6-Anast_Gal--jocaaa--11.pdf
7-ALB-LUPAS--JOCAAA--11.pdf
8-alina-lupas-jocaaa--11.pdf
9-ALINA-LUPAS---JOCAAA-11.pdf
10-alina-lupas-jocaaa--11.pdf
11-Anastas-Gal---jocaaa-11.pdf
12--ZHAOWEN-LI---JOCAAA--11.pdf
13-Anastas-DUM-jocaaa--11.pdf
14-huang---jocaaa--11.pdf
BLANK-JOCAAA-2011.pdf
TOC-2011-JOCAAA-VOL-13-N0-1.pdf
PART-2-JOCAAA-11.pdf
II-JOCAAA-11.pdf
JCAAA-2011-front-2.pdf
SCOPE--JOCAAA--2011.pdf
JOCAAA-2011--Editorial Board.pdf
Instructions--JOCAAA--2011.pdf
Binder-2011-JOCAAA-VOL-13-NO-2.pdf
15-Coroianu-Gal-jocaaa--11.pdf
16-Cakalli--jocaaa--11.pdf
17-nguerekata---jocaaa--11.pdf
18-RADENOVIC--jocaaa--11.pdf
19-CALIN-DUBAU--JOCAAA-11.pdf
20-BUNPOG--JOCAAA--11.pdf
21-chaofeng-Shi--jocaaa--11.pdf
22-CHOI--JOCAAA-11.pdf
23--ChoParkRassiasSaadati-JoCAAA-11.pdf
24-ChoParkSaadati-JoCAAA-11.pdf
25-EL-SAYED---AL-AMRI---JOCAAA--11.pdf
26-HENG-YOU-LAN-JOCAAA--11.pdf
27-JITPEERA-KUMAM--JOCAAA--11.pdf
28--SINTUNAVARAT-KUMAM--JOCAAA---11.pdf
29--poom-kumam--jocaaa--11.pdf
30--LEE-CHAE-JANG---JOCAAA--11.pdf
31--MARIN---JOCAAA--11.pdf
TOC-2011-JOCAAA-VOL-13-N0-2.pdf
PART-3-JOCAAA-11.pdf
III-JOCAAA-11.pdf
JCAAA-2011-front-3.pdf
SCOPE--JOCAAA--2011.pdf
JOCAAA-2011--Editorial Board.pdf
Instructions--JOCAAA--2011.pdf
Binder-2011-JOCAAA-VOL-13-NO-3.pdf
32-MOHSEN--TORABI--JOCAAA--11.pdf
33-POOM-KUMAM--JOCAAA-11.pdf
34--NEDRET-OGDUK--JOCAAA--11.pdf
35--Rahbarnia---SAADATI---JOCAAA--11.pdf
36-RATCHAGIT---JOCAAA---11.pdf
37--Tunc -Korkmaz ---JOCAAA--11.pdf
38--Tunc-Ayhan--JOCAAA---11.pdf
39--YANG-XIAOYUAN---JOCAAA--11.pdf
40--YILMAZ-ARAL-TUNCA--JOCAAA--11.pdf
41--ZHENHAI-LIU-JOCAAA--11.pdf
42-RADENOVIC--JOCAAA--11.pdf
43-CHO-SAADATI--JOCAAA--11.pdf
44-KREANGRI-RACHAGIT--JOCAAA--11.pdf
45-SIMSEK--JOCAAA--11.pdf
46--THABET-ABDELJAWAD--JOCAAA--11.pdf
47--XIANJIU-HUANG---JOCAAA--11.pdf
48--YOUNG-HEE-KIM-JOCAAA--11.pdf
BLANK-JOCAAA-2011.pdf
TOC-2011-JOCAAA-VOL-13-N0-3.pdf
PART-4-JOCAAA-11.pdf
IV-JOCAAA-11.pdf
JCAAA-2011-front-4.pdf
SCOPE--JOCAAA--2011.pdf
JOCAAA-2011--Editorial Board.pdf
Instructions--JOCAAA--2011.pdf
Binder-2011-JOCAAA-VOL-13-NO-4.pdf
49-ABDELJAWAD--JOCAAA-11.pdf
50-ABDELJAWAD---JOCAAA---11.pdf
51-AGARWAL-SOMAYEH--JOCAAA--11.pdf
52-Kim-Jang-lee-Jocaa--11.pdf
53-AMINATAEI---JOCAAA---11.pdf
54-catas---jocaaa--11.pdf
55-BUYUKYAZICI-JOCAAA--11.pdf
56-CEM-KAANOGLU-JOCAAA--11.pdf
57-CENAP-DUYAR---JOCAAA--11.pdf
58-TIAN-XIAO-HE--JOCAAA--11.pdf
59-ESHAGHI---JOCAAA--11.pdf
60-ESHAGHI----JOCAAA--11.pdf
61--GORDJI--JOCAAA--11.pdf
62-CHENG-FU-YANG--JOCAAA--11.pdf
63-coskun---JOCAAA--11.pdf
64-EBADIAN---JOCAAA-11.pdf
65-EBADIAN---JOCAAA--11.pdf
66-elezovic-buric--jocaaa--11.pdf
BLANK-JOCAAA-2011.pdf
TOC-2011-JOCAAA-VOL-13-N0-4.pdf
PART-5-JOCAAA-11.pdf
V-JOCAAA-11.pdf
JCAAA-2011-front-5.pdf
SCOPE--JOCAAA--2011.pdf
JOCAAA-2011--Editorial Board.pdf
Instructions--JOCAAA--2011.pdf
Binder-2011-JOCAAA-VOL-13-NO-5.pdf
67-ERYILMAZ--JOCAAA--11.pdf
68-F-CAO--JOCAAA-11.pdf
69--FEILONG-CAO--JOCAAA-11.pdf
70--II--W-SULAIMAN--JOCAAA--11.pdf
71--I---W-SULAIMAN--JOCAAA--11.pdf
72--LAMPRET---JOCAAA-11.pdf
73--NADARAJAH--SARALLEES---JOCAAA--11.pdf
74--GE-YING--JOCAAA--11.pdf
75--HIKMET-OZARSLAN--JOCAAA--11.pdf
76--i-canak-jocaaa--11.pdf
77-KHONGTHAM---JOCAAA--11.pdf
78--Milovanovic---JOCAAA--11.pdf
79--OZDEMIR---JOCAAA--11.pdf
80--LouisPaquette---jocaaa---11.pdf
81--S-JAFAROV-JOCAAA-11.pdf
82--turkoglu--jocaaa--11.pdf
83-RYOOCS--jocaaa--11.pdf
84--RYOOCS--JOCAAA--11.pdf
85--ryoocs--JOCAAA--11.pdf
TOC-2011-JOCAAA-VOL-13-N0-5.pdf
PART-6-JOCAAA-11.pdf
VI-JOCAAA-11.pdf
JCAAA-2011-front-6.pdf
SCOPE--JOCAAA--2011.pdf
JOCAAA-2011--Editorial Board.pdf
Instructions--JOCAAA--2011.pdf
Binder-2011-JOCAAA-VOL-13-NO-6.pdf
86--ryoocs---jocaaa--11.pdf
87-RYOO--JOCAAA--11.pdf
88--RYOO---jocaaa--11.pdf
89--VAEZPOUR--jocaaa--11.pdf
90--VAEZPOUR---JOCAAA--11.pdf
91--Rashid--Himat---JOCAAA--11.pdf
92--Tunc-JOCAAA--11.pdf
93--ZHI-JE-JIANG--JOCAAA--11.pdf
94--Zhu-Zeng-JOCAAA--11.pdf
95--SIMIN-KABOLI--JOCAAA--11.pdf
96--SAEED-SHAGHOOLI--JOCAAA--11.pdf
1. Introduction
2. Stability of ternary quadratic derivations
References
97-saeed-shagholi--jocaaa-11.pdf
1. Introduction
2. Stability of ternary cubic homomorphisms
References
98--UGUR-YUSKEL-JOCAAA--11.pdf
1. Introduction and Preliminaries
2. Main Results
References
99--JIANDONG-Yin--JOCAAA--11.pdf
100--JIANLEI-LI-JOCAAA--11.pdf
101--jinghong-liu-jocaaa--11.pdf
102--JIN-HAN-PARK-JOCAAA--11.pdf
103--jin-han-park--jocaaa--11.pdf
104--KESONG-YAN--JOCAAA--11.pdf
BLANK-JOCAAA-2011.pdf
TOC-2011-JOCAAA-VOL-13-N0-6.pdf
PART-7-JOCAAA-11.pdf
VII-JOCAAA-11.pdf
JCAAA-2011-front-7.pdf
SCOPE--JOCAAA--2011.pdf
JOCAAA-2011--Editorial Board.pdf
Instructions--JOCAAA--2011.pdf
Binder-2011-JOCAAA-VOL-13-NO-7.pdf
105--Kim_Jang_Kim_jocaaa--11.pdf
106--Kim-Kim-Lee--JOCAA--11.pdf
107--LEE-CHAE-JANG---JOCAAA---11.pdf
108--NAN---JING-HUANG--JOCAAA-11.pdf
109--Sababheh--JOCAAA--11.pdf
1. Introduction
2. Main Results
3. uniquely remotal sets in Hilbert spaces
References
110--SEOG-HOON-RIM--JOCAAA--11.pdf
111--SHENG-KUN-LI--JOCAAA--11.pdf
112--shrim-KIM-LEE-JOCAAA--11.pdf
113--SICHUN-WANG--JOCAAA--11.pdf
114--TAIXIANG-SUN--JOCAAA--11.pdf
115--TIAN-ZHOU-XU--JoCAAA--11.pdf
116--WANG-XIANG--JOCAAA--11.pdf
117--T-KIM-JOCAAA-11.pdf
118--WANMEI-TANG--JOCAAA--11.pdf
119--XIAOYUAN-YANG--JOCAAA--11.pdf
120--XIAO-YUNHAI---JOCAAA---11.pdf
121--YOUNG-HEE-KIM-KIM--JOCAAA--11.pdf
122--yu-chen-jocaaa--11.pdf
123--Zhensheng-Yu-JOCAAA--11.pdf
BLANK-JOCAAA-2011.pdf
TOC-2011-JOCAAA-VOL-13-N0-7.pdf

Citation preview

Volume 13, Number 1 ISSN:1521-1398 PRINT,1572-9206 ONLINE

January 2011

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(seven 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. Webmaster:Ray Clapsadle 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 $470,Electronic $300,Print and Electronic $500.Individual:Print $150,Electronic $100,Print &Electronic $200.For any other part of the world add $50 more to the above prices for Print.No credit card payments. Copyright©2011 by Eudoxus Press,LLCAll 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 20) Hrushikesh N.Mhaskar 1) George A. Anastassiou Department of Mathematical Sciences Department Of Mathematics California State University The University of Memphis Los Angeles,CA 90032 Memphis,TN 38152,U.S.A 626-914-7002 Tel.901-678-3144 e-mail: [email protected] e-mail: [email protected] Approximation Theory,Real Analysis, Orthogonal Polynomials, Approximation Theory,Splines, Wavelets, Neural Wavelets, Neural Networks Networks,Probability, Inequalities. 21) M.Zuhair Nashed Department Of Mathematics 2) J. Marshall Ash University of Central Florida Department of Mathematics PO Box 161364 De Paul University Orlando, FL 32816-1364 2219 North Kenmore Ave. e-mail: [email protected] Chicago,IL 60614-3504 Inverse and Ill-Posed problems, 773-325-4216 Numerical Functional Analysis, e-mail: [email protected] Integral Equations,Optimization, Real and Harmonic Analysis Signal Analysis 3) Mark J.Balas 22) Mubenga N.Nkashama Department Head and Professor Department OF Mathematics Electrical and Computer Engineering University of Alabama at Dept. Birmingham College of Engineering Birmingham,AL 35294-1170 University of Wyoming 205-934-2154 1000 E. University Ave. e-mail: [email protected] Laramie, WY 82071 Ordinary Differential Equations, 307-766-5599 Partial Differential Equations e-mail: [email protected] Control Theory,Nonlinear Systems, 23) Charles E.M.Pearce Neural Networks,Ordinary and Applied Mathematics Department Partial Differential Equations, University of Adelaide Functional Analysis and Operator Adelaide 5005, Australia Theory e-mail: [email protected] 4) Drumi D.Bainov Stochastic Processes, Probability Department of Mathematics Theory, Harmonic Analysis, Medical University of Sofia Measure Theory, P.O.Box 45,1504 Sofia,Bulgaria Special Functions, Inequalities e-mail: [email protected] 24) Josip E. Pecaric e-mail:[email protected] Differential Equations/Inequalities Faculty of Textile Technology University of Zagreb Pierottijeva 6,10000 5) Carlo Bardaro Zagreb,Croatia Dipartimento di Matematica e e-mail: [email protected] Informatica Inequalities,Convexity Universita di Perugia Via Vanvitelli 1

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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. 6) 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 7) 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 8) George Cybenko Thayer School of Engineering Dartmouth College 8000 Cummings Hall, Hanover,NH 03755-8000 603-646-3843 (x 3546 Secr.) e-mail: [email protected] Approximation Theory and Neural Networks 9) 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 10) Sever S.Dragomir School of Computer Science and Mathematics, Victoria University,

25) Svetlozar T.Rachev Department of Statistics and Applied Probability University of California at Santa Barbara, Santa Barbara,CA 93106-3110 805-893-4869 e-mail: [email protected] and Chair of Econometrics, Statistics and Mathematical Finance School of Economics and Business Engineering University of Karlsruhe Kollegium am Schloss, Bau II,20.12, R210 Postfach 6980, D-76128, Karlsruhe,GERMANY. Tel +49-721-608-7535, +49-721-608-2042(s) Fax +49-721-608-3811 [email protected] Probability,Stochastic Processes and Statistics,Financial Mathematics, Mathematical Economics. 26) 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 27) Ervin Y.Rodin Department of Systems Science and Applied Mathematics Washington University,Campus Box 1040 One Brookings Dr.,St.Louis,MO 63130-4899 314-935-6007 e-mail: [email protected] Systems Theory, Semantic Control, Partial Differential Equations, Calculus of Variations,Optimization and

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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. 11) 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 12) Augustine O.Esogbue School of Industrial and Systems Engineering Georgia Institute of Technology Atlanta,GA 30332 404-894-2323 e-mail: [email protected] Control Theory,Fuzzy sets, Mathematical Programming, Dynamic Programming,Optimization

Artificial Intelligence, Operations Research, Math.Programming 28) 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 29) I. P. Stavroulakis Department of Mathematics University of Ioannina 451-10 Ioannina, Greece [email protected] Differential Equations Phone +3 0651098283 30) 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

13) Christodoulos A.Floudas Department of Chemical Engineering Princeton University Princeton,NJ 08544-5263 609-258-4595 (x4619 assistant) e-mail: [email protected] 31) Gilbert G.Walter OptimizationTheory&Applications, Department Of Mathematical Global Optimization Sciences University of Wisconsin14) J.A.Goldstein Department of Mathematical Sciences Milwaukee,Box 413, Milwaukee,WI 53201-0413 The University of Memphis 414-229-5077 Memphis,TN 38152 e-mail: [email protected] 901-678-3130 Distribution e-mail:[email protected] Functions,Generalised Functions, Partial Differential Equations, Wavelets Semigroups of Operators 15) H.H.Gonska Department of Mathematics University of Duisburg Duisburg, D-47048

32) Halbert White Department of Economics University of California at San Diego

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Germany 011-49-203-379-3542 e-mail: [email protected] Approximation Theory, Computer Aided Geometric Design 16) 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 17) Christian Houdre School of Mathematics Georgia Institute of Technology Atlanta,Georgia 30332 404-894-4398 e-mail: [email protected] Probability,Mathematical Statistics,Wavelets 18) V. Lakshmikantham Department of Mathematical Sciences Florida Institute of Technology Melbourne, FL 32901 e-mail: [email protected] Ordinary and Partial Differential Equations, Hybrid Systems, Nonlinear Analysis 19) 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 36) Ahmed I. Zayed Department Of Mathematical Sciences DePaul University 2320 N. Kenmore Ave. Chicago, IL 60614-3250 773-325-7808 e-mail: [email protected]

La Jolla,CA 92093-0508 619-534-3502 e-mail: [email protected] Econometric Theory,Approximation Theory, Neural Networks 33) 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 34) Xiang Ming Yu Department of Mathematical Sciences Southwest Missouri State University Springfield,MO 65804-0094 417-836-5931 e-mail: [email protected] Classical Approximation Theory,Wavelets 35) 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 e-mail: [email protected] Fuzzyness, Artificial Intelligence, Natural language processing, Fuzzy logic

Shannon sampling theory, Harmonic analysis and wavelets, Special functions and orthogonal polynomials, Integral transforms.

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JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.1, 11-19, 2011, COPYRIGHT 2011 EUDOXUS PRESS, 11 LLC

Approximation Properties of Some Multivariate Generalized Singular Integrals in the Unit Polydisk∗ George A. Anastassiou and Sorin G. Gal Department of Mathematical Sciences The University of Memphis Memphis, TN 38152, U.S.A. [email protected] and Department of Mathematics and Computer Science University of Oradea Str. Universitatii No. 1 410087 Oradea, ROMANIA [email protected] Abstract The aim of this paper is to obtain several results in approximation by Jackson-type generalizations of multi-complex Picard, Poisson–Cauchy and Gauss–Weierstrass singular integrals in terms of higher order moduli of smoothness in polydisks.

AMS 2000 Mathematics Subject Classification : 30E10, 32E30, 41A25, 41A35. Key words and phrases: Generalized multi-complex singular integrals, Jacksontype estimates, global smoothness preservation.

1

Introduction

Let us consider the open polydisk Dm , where D = {z ∈ C : |z| < 1}, m ∈ N, and n m m A(D ) = f : D → C; f is analytic with respect to any variable z1 , . . . , zm ∈ o ³ m´ m D, continuous on D . Therefore, if f ∈ A D , then according to e. g. [3], ∗ This paper was written during the 2009 Spring Semester when the second author was a Visiting Professor at the Department of Mathematical Sciences, The University of Memphis, TN, U.S.A.

1

12

ANASTASSIOU, GAL: SINGULAR INTEGRALS

Theorem 2, p. 65, we can write ∞ X

f (z1 , . . . zm ) =

im αi1 ,...,im z1i1 . . . zm ,

i1 ,...,im =0

for all z1 , . . . , zm ³ ∈mD, ´ m ∈ N. For f ∈ A D and ξj ∈ R, ξj > 0, j = 1, . . . , m, let us consider the multivariate variants of the Jackson-type generalizations of Picard, PoissonCauchy and Gauss-Weierstrass singular integrals given by Pn,ξ1 ,...,ξm (f ) (z1 , . . . , zm ) µ ¶Z ∞ Z ∞ n+1 X ¡ ¢ 1 n+1 (−1)k ... f z1 eiku1 , . . . , zm eikum k −∞ −∞ j=1 (2ξj )

= − Qm

k=1

·

m Y

e−|uj |/ξj du1 . . . dum ,

j=1

Qn,ξ1 ,...,ξm (f ) (z1 , . . . zm ) µ ¶Z π Z π n+1 X n+1 1 h ³ ´i (−1)k ... = −Q m 2 π k −1 −π −π k=1 j=1 ξj tan ξj ¡ iku ¢ f z1 e 1 , . . . , zm eikum ¢ Qm ¡ 2 du1 . . . dum , 2 j=1 uj + ξj Wn,ξ1 ,...,ξm (f )(z1 , . . . , zm ) µ ¶Z ∞ Z ∞ n+1 X 1 k n+1 (−1) = − m/2 Qm ... k π −∞ −∞ j=1 ξj k=1 m ¡ ¢Y 2 2 f z1 eiku1 , . . . , zm eikum e−uj /ξj du1 . . . dum , j=1

S

zi ∈ D, i = 1, . . . , m, with n ∈ N {0}, m ∈ N. In [1] we have obtained approximation results for the above singular integrals in the univariate case m = 0. On the other hand, in [2] we proved approximation properties of the above complex multivariate integrals in the case when n = 0. The aim of the present paper is to obtain similar results for arbitrary n, m ∈ N.

2

ANASTASSIOU, GAL: SINGULAR INTEGRALS

2

13

Multi-Complex Generalized Picard Integrals

In this section we study the approximation properties of the multicomplex singular integral Pn,ξ1 ,...,ξm (f )(z ³ 1 , . .´. , zm ). Theorem 1. Let f ∈ A D

m

and ξj > 0, j = 1, . . . , m. We have : m

(i) Pn,ξ1 ,...,ξm (f )(z1 , . . . , zm ) is continuous on D ¢ and analytic on Dm . ¡ n+1 (ii) ω1 (Pn,ξ1 ,...,ξm (f ); δ1 , . . . , δm )Dm ≤ 2 − 1 ω1 (f ; δ1 , . . . , δm )Dm , for all ξj > 0; j = 1, . . . , m, where ω1 (f ; δ1 , . . . , δm )Dm = sup {|f (u1 , . . . , um ) − f (v1 , . . . , vm )|; |uj − vj | ≤ δj , j = 1, . . . , m} . (iii) It holds |Pn,ξ1 ,...,ξm (f )(z1 , . . . , zm ) − f (z1 , . . . , zm )| "n+1 µ #m X n + 1¶ ≤ k! ωn+1 (f ; ξ1 , . . . , ξm )(∂D)m , k k=0

for all ξj > 0, j = 1, . . . , m, and zj ∈ D, where ∂D denotes the boundary of the disk D and we define ωn+1 (f ; ξ1 , . . . , ξm )(∂D)m ¯ ©¯ n+1 ª ix = sup ¯∆u1 ,...,um f (e 1 , . . . , eixm )¯ : |xj | ≤ π, |uj | ≤ ξj ; j = 1, . . . , m , with i2 = −1 and

ix1 ∆n+1 , . . . , eixm ) u1 ,...,um f (e µ ¶ ³ n+1 ´ X n+1−j n + 1 = (−1) f ei(x1 +ju1 ) , . . . , ei(xm +jum ) , n ∈ N. j j=0

Proof. (i) Because Pn,ξ1 ,...,ξm (f ) can be written as a linear combination of usual multivariate-complex Picard singular integrals proved in [2] to be continm m uous on D and analytic on Dm , the continuity on D and analyticity on Dm follows immediately. (ii) Let z1,j , z2,j ∈ D, |z1,j − z2,j | ≤ δj , for j = 1, . . . , m. We obtain |Pn,ξ1 ,...,ξm (f ) (z1,1 , . . . , z1,m ) − Pn,ξ1 ,...,ξm (f ) (z2,1 , . . . , z2,m )| Z ∞ n+1 X µn + 1¶ Z ∞ ¯ ¡ ¢ 1 ¯f z1,1 eiu1 , . . . , z1,m eium ... k −∞ −∞ j=1 (2ξj )

≤ Qm

k=1

m ¡ ¢¯ Y −f z2,1 eiu1 , . . . , z2,m eium ¯ e−|uj |/ξj du1 . . . dum j=1

≤ ω1 (f ; |z1,1 − z2,1 |, . . . , |z1,m − z2,m |)Dm

Ãn+1 µ ! X n + 1¶ k=1

3

k

14

ANASTASSIOU, GAL: SINGULAR INTEGRALS

¡ ¢ ≤ 2n+1 − 1 ω1 (f ; δ1 , . . . , δm )Dm , proving the claim. (iii) By the multivariate maximum modulus principle, see e.g. [4], p. 23, Corollary 1.2.5, we can take |zj | = 1 for j = 1, . . . , m, i.e. zj = eixj , |xj | ≤ π, and it is enough to prove the result on the boundary (∂D)m . We have that f (z1 , . . . , zm ) − Pn,ξ1 ,...,ξm (f ) (z1 , . . . , zm ) Z ∞ Z ∞ 1 = Qm ... (−1)n+1 ∆n+1 u1 ,...,um f (z1 , . . . , zm ) −∞ j=1 (2ξj ) −∞ ·

m Y

e−|uj |/ξj du1 . . . dum .

j=1

Hence 1 j=1 (2ξj )

≤ Qm

Z

|Pn,ξ1 ,...,ξm (f )(z1 , . . . , zm ) − f (z1 , . . . , zm )| Z ∞ m ∞ ¯ n+1 ¯Y ¯∆u ,...,u f (z1 , . . . , zm )¯ ... e−|uj |/ξj du1 . . . dum 1 m

−∞

−∞

j=1

µ ¶ 1 ξ1 |u1 | |um | ≤ Qm ··· ωn+1 f ; , . . . , ξm ξ1 ξm (∂D)m −∞ j=1 (2ξj ) −∞ Z

Z

∞

·

m Y

∞

e−|uj |/ξj du1 . . . dum

j=1

n+1  Z ∞ m X ωn+1 (f ; ξ1 , . . . , ξm )(∂D)m Z ∞ |uj |  1 + Qm = ... ξj (2ξ ) j −∞ −∞ j=1 j=1 Z ∞ ωn+1 (f ; ξ1 , . . . , ξm )(∂D)m Z ∞ Q · e du1 . . . dum ≤ ... m −∞ −∞ j=1 (2ξj ) j=1  n+1 ¶ m µ m Y Y |uj |   · 1+ e−|uj |/ξj du1 . . . dum ξ j j=1 j=1 " # ¶n+1 m Y 1 Z ∞µ |uj | −|uj |/ξj = ωn+1 (f ; ξ1 , . . . , ξm )(∂D)m 1+ e duj 2ξj −∞ ξj j=1 # " Z µ ¶n+1 m Y uj 1 ∞ −uj /ξj 1+ e duj = ωn+1 (f ; ξ1 , . . . , ξm )(∂D)m ξj 0 ξj j=1 m Y

−|uj |/ξj

(according to e. g. [1, p. 427]) "n+1 µ #m X n + 1¶ = ωn+1 (f ; ξ1 , . . . , ξm )(∂D)m k! , k k=0

proving the claim.

¤ 4

ANASTASSIOU, GAL: SINGULAR INTEGRALS

3

15

Multi-Complex Generalized Poisson-Cauchy Integrals

In this section we study the approximation properties of the multicomplex singular integral Qn,ξ1 ,...,ξm (f )(z ³ 1 , . .´. , zm ). Theorem 2. Let f ∈ A D

m

and ξj > 0, j = 1, . . . , m. We have : m

(i) Qn,ξ1 ,...,ξm (f )(z1 , . . . , zm ) is continuous on D ¢ and analytic on Dm . ¡ n+1 (ii) ω1 (Qn,ξ1 ,...,ξm (f ); δ1 , . . . , δm )Dm ≤ 2 − 1 ω1 (f ; δ1 , . . . , δm )Dm , for all ξj > 0, j = 1, . . . , m. (iii) It holds |Qn,ξ1 ,...,ξm (f )(z1 , . . . , zm ) − f (z1 , . . . , zm )|   m Y ≤ K(n, ξj ) ωn+1 (f ; ξ1 , . . . , ξm )(D)m , j=1

for all ξj > 0 and zj ∈ D, j = 1, . . . , m. Here R π/ξj (u+1)n+1 K(n, ξj ) =

0

u2 +1

³ ´

du

π ξj

tan−1

, j = 1, . . . , m.

Proof. (i) Because Qn,ξ1 ,...,ξm (f ) can be written as a linear combination of usual multivariate-complex Poisson-Cauchy singular integrals proved in [2] to m m be continuous on D and analytic on Dm , the continuity on D and analyticity on Dm follows immediately. (ii) Let z1,j , z2,j ∈ D, |z1,j − z2,j | ≤ δj , for j = 1, . . . , m. We obtain |Qn,ξ1 ,...,ξm (f ) (z1,1 , . . . , z1,m ) − Qn,ξ1 ,...,ξm (f ) (z2,1 , . . . , z2,m )| h ≤Q m 2 j=1

1

i −1 ( π ) tan ξj ξj

n+1 Xµ k=1

n+1 k

¶Z

Z

π

π

··· −π

−π

|f (z1,1 eiu1 , . . . , z1,m eium ) − f (z2,1 eiu1 , . . . , z2,m eium )| ¢ Qm ¡ 2 du1 . . . dum 2 j=1 uj + ξj ω1 (f ; |z1,1 − z2,1 | , . . . |z1,m − z2,m |)Dm ³ ´i Qm h 2 π −1 tan j=1 ξj ξj Ãn+1 µ ! Z π X n + 1¶ Z π du . . . du Qm 1 2 m2 · ··· k −π −π j=1 (uj + ξj ) k=1 ¡ n+1 ¢ = 2 − 1 ω1 (f ; |z1,1 − z2,1 |, . . . , |z1,m − z2,m |)Dm ≤

5

16

ANASTASSIOU, GAL: SINGULAR INTEGRALS

·

m Y

Ã

! ÃZ

1

π

duj 2 uj + ξj2

!

( ξ2j )tan−1 (π/ξj ) −π ¡ n+1 ¢ = 2 − 1 ω1 (f ; |z1,1 − z2,1 |, . . . , |z1,m − z2,m |)Dm ¡ ¢ ≤ 2n+1 − 1 ω1 (f ; δ1 , . . . , δm )Dm , j=1

proving the claim. (iii) By the multivariate maximum modulus principle, see e.g. [4], p. 23, Corollary 1.2.5, we can take |zj | = 1 for j = 1, . . . , m, i.e. zj = eixj , |xj | ≤ π, and it is enough to prove the result on the boundary (∂D)m . We observe that f (z1 , . . . , zm ) − Qn,ξ1 ,...,ξm (f ) (z1 , . . . , zm ) Z π Z π 1 ³ ´i h =Q ··· m 2 π −1 −π −π j=1 ξj tan ξj n+1

(−1)

∆n+1 u1 ,...,um f (z1 , . . . , zm ) Qm du1 . . . dum . 2 2 j=1 (uj + ξj )

Hence |Qn,ξ1 ,...,ξm (f )(z1 , . . . , zm ) − f (z1 , . . . , zm )| Z π Z π 1 ³ ³ ´´ ≤Q ... m 2 π −1 −π −π j=1 ξj tan ξj ¯ n+1 ¯ Z π Z π ¯∆u ,...,u f (z1 , . . . , zm )¯ 1 1 ³ ³ ´´ Qmm 2 du . . . du ≤ . . . 1 m Qm 2 2 −π −π j=1 (uj + ξj ) tan−1 π j=1

³

m| ωn+1 f ; ξ1 |uξ11 | , . . . , ξm |uξm Qm 2 2 j=1 (uj + ξj )

≤ ³

´

ξj

(∂D)m

ξj

du1 . . . dum

Z π ωn+1 (f ; ξ1 , . . . , ξm )(∂D)m Z π ³ ³ ´´ . . . Qm 2 π −1 −π −π j=1 ξj tan ξj

Pm u ´n+1 Z π 1 + j=1 ξjj ωn+1 (f ; ξ1 , . . . , ξm )(∂D)m Z π ¢ ³ ´´ ³ Qm ¡ 2 . . . du . . . du ≤ 1 m Qm 2 2 −π −π j=1 uj + ξj tan−1 π j=1

³

Qm

|u |

´n+1

ξj

ξj

1 + j=1 ξjj ¢ du1 . . . dum Qm ¡ 2 2 j=1 uj + ξj   ³ ´n+1 Z π 1 + |uj| m Y ξj 1  ³ ³ ´´ duj  = ωn+1 (f ; ξ1 , . . . , ξm )(∂D)m  2 + ξ2 2 π −1 u −π j j j=1 ξj tan ξj 6

ANASTASSIOU, GAL: SINGULAR INTEGRALS

17

(according to e. g. [1], p. 428)  = ωn+1 (f ; ξ1 , . . . , ξm )(∂D)m 

m Y

 K(n, ξj ) ,

j=1

proving the claim.

4

¤

Multi-Complex Generalized Gauss-Weierstrass Integrals

In this section we study the approximation properties of the multicomplex singular integral Wn,ξ1 ,...,ξm (f )(z ³ 1 , .´. . , zm ). Theorem 3. Let f ∈ A D

m

and ξj > 0, j = 1, . . . , m. We have : m

(i) Wn,ξ1 ,...,ξm (f )(z1 , . . . , zm is continuous on D ¢ and analytic on Dm . ¡ n+1 − 1 ω1 (f ; δ1 , . . . , δm )Dm , (ii) ω1 (Wn,ξ1 ,...,ξm (f ); δ1 , . . . , δm )Dm ≤ 2 for all ξj > 0; j = 1, . . . , m. (iii) It holds ¶m µ Z ∞ 2 2 (1 + u)n+1 e−u du |Wn,ξ1 ,...,ξm (f )(z1 , . . . , zm ) − f (z1 , . . . , zm )| ≤ √ π 0 ·ωn+1 (f ; ξ1 , . . . , ξm )(∂D)m , for all ξj > 0 and zj ∈ D, j = 1, . . . , m,. Proof. (i) Because Wn,ξ1 ,...,ξm (f ) can be written as a linear combination of usual multivariate-complex Gauss-Weierstrass singular integrals proved in [2] m m to be continuous on D and analytic on Dm , the continuity on D and the m analyticity on D follows at once. (ii) Let z1,j , z2,j ∈ D, |z1,j − z2,j | ≤ δj , for j = 1, . . . , m. We obtain |Wn,ξ1 ,...,ξm (f ) (z1,1 , . . . , z1,m ) − Wn,ξ1 ,...,ξm (f ) (z2,1 , . . . , z2,m )| ≤

π

n+1 Xµ

1 Qm m/2

j=1 ξj k=1

n+1 k

¶Z

Z

∞

∞

|f (z1,1 eiu1 , . . . , z1,m eium )

... −∞

−f (z2,1 eiu1 , . . . , z2,m eium )|

−∞ m Y

2

2

e−uj /ξj du1 . . . dum

j=1

" ≤ ω1 (f ; |z1,1 − z2,1 |, . . . , |z1,m − z2,m |)Dm Z

Z

∞

··· −∞

∞

m Y

π

Ãn+1 µ ! X n + 1¶

1 Qm m/2

j=1 ξj

 −u2j /ξj2

e

−∞ j=1

7

du1 . . . dum 

k=1

k

18

ANASTASSIOU, GAL: SINGULAR INTEGRALS

¡ ¢ = 2n+1 − 1 ω1 (f ; |z1,1 − z2,1 |, . . . , |z1,m − z2,m |)Dm ¡ ¢ ≤ 2n+1 − 1 ω1 (f ; δ1 , . . . , δm )Dm , proving the claim. (iii) By the multivariate maximum modulus principle, see e.g. [4], p. 23, Corollary 1.2.5, we can take |zj | = 1 for j = 1, . . . , m, i.e. zj = eixj , |xj | ≤ π, and it is enough to prove result on the boundary (∂D)m . We have that f (z1 , . . . , zm ) − Wn,ξ1 ,...,ξm (f ) (z1 , . . . , zm ) µ ¶ n+1 X 1 k n+1 = f (z1 , . . . , zm ) + m/2 Qm (−1) k π j=1 ξj k=1 Z

Z

∞

∞

... −∞

=

m ¡ ¢Y 2 2 f z1 eiku1 , . . . , zm eikum e−uj /ξj du1 . . . dum

−∞

π m/2

Z

1 Qm

j=1 ξj

j=1

Z

∞

∞

... −∞

−∞

m Y

·

2

(−1)n+1 ∆n+1 u1 ,...,um f (z1 , . . . , zm )

2

e−uj /ξj du1 . . . dum .

j=1

Hence

≤

π m/2 ≤

1 Qm

j=1 ξj

π m/2

|Wn,ξ1 ,...,ξm (f ) (z1 , . . . , zm ) − f (z1 , . . . , zm )| Z ∞ Z ∞ m ¯ n+1 ¯Y 2 2 ¯∆u ,...,u f (z1 , . . . , zm )¯ ... e−uj /ξj du1 . . . dum 1 m −∞

1 Qm

j=1 ξj

−∞

Z

j=1

¶ µ |um | |u1 | , . . . , ξm ... ωn+1 f ; ξ1 ξ1 ξm (∂D)m −∞ −∞ Z

∞

·

m Y

∞

2

2

e−uj /ξj du1 . . . dum

j=1

 n+1 Z ∞ m X ωn+1 (f ; ξ1 , . . . , ξm )(∂D)m Z ∞ |u | j 1 +  Qm ≤ ... ξj π m/2 j=1 ξj −∞ −∞ j=1 ·

m Y

2

2

e−uj /ξj du1 . . . dum

j=1

  ¶ n+1 Z ∞ Y m µ ωn+1 (f ; ξ1 , . . . , ξm )(∂D)m Z ∞ |u | j   Qm ≤ ... 1+ ξj π m/2 j=1 ξj −∞ −∞ j=1

8

ANASTASSIOU, GAL: SINGULAR INTEGRALS

·

m Y

2

19

2

e−uj /ξj du1 . . . dum

j=1

= ωn+1 (f ; ξ1 , . . . , ξm )(∂D)m

m Y

"

j=1

= ωn+1 (f ; ξ1 , . . . , ξm )(∂D)m

m Y j=1

1 √ πξj

"

2 √ πξj

# µ ¶n+1 |uj | −u2j /ξj2 1+ e duj ξj −∞

Z

∞

Z

∞ 0

# µ ¶n+1 uj −u2j /ξj2 e duj 1+ ξj

( according to [1], p. 428) µ ¶m Z ∞ 2 n+1 −u2 = ωn+1 (f ; ξ1 , . . . , ξm )(∂D)m √ (1 + u) e du , π 0 proving the claim.

¤

References [1] G.A. Anastassiou and S. G. Gal, Geometric and approximation properties of generalized singular integrals in the unit disk, J. Korean Math. Soc., 43(2006), No. 2, 425-443. [2] G.A. Anastassiou and S. G. Gal, Approximation properties of some multicomplex singular integrals in the unit polydisk, submitted for publication. [3] C. Andreian Cazacu, Theory of Functions of Several Complex Variables (in Romanian), Edit. Didact. Pedag., Bucharest, 1971. [4] G. Kohr, Basic Topics in Holomorphic Functions of Several Complex Variables, University Press, Cluj-Napoca, 2003.

9

JOURNAL 20 OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.1, 20-26, 2011, COPYRIGHT 2011 EUDOXUS PRESS, LLC

SOME PROPERTIES OF CHEBYSHEV SYSTEMS RICHARD A. ZALIK Abstract. We study Chebyshev systems defined on an interval, whose constituent functions are either complex or real–valued, and focus on problems that may have have an application in the theory of differential equations and cannot be solved by a mere rewording of existing proofs, specifically those dealing with the existence of an adjoined function, the extension of the interval of definition, and the problem of embedding a set of functions into an Extended Complete Chebyshev System.

1. Introduction A system of functions F = (f0 , f1 , . . . , fn ) of complex–valued functions defined on a proper interval I is called a Chebyshev system, or Tchebycheff system, or T–system, if the determinant (1)

D(f0 , . . . fn ; t0 , . . . tn ) := det(fj (tk ); 0 ≤ j, k ≤ n)

does not vanish for any choice of points {tk ; 0 ≤ k ≤ n} in I. It is called a Complete Chebyshev system or CT–system or Markov system, if (f0 , f1 , . . . , fk ) is a T –system for all k = 0, . . . , n. If the functions fj are sufficiently smooth, we can extend the definition of D(f0 , . . . fn ; t0 , . . . tn ), so as to allow for equalities amongst the tk : if t0 ≤ · · · ≤ tn is any set of points of I, then D∗ (f0 , . . . fn ; t0 , . . . tn ) is defined to be the determinant on the right hand of (1), where for each set of consecutive tk , the corresponding columns are replaced by the successive derivatives evaluated at the point. For example, f0 (t0 ) f0 (t1 ) f0′ (t1 ) D∗ (f0 , f1 , f2 ; t0 , t1 , t1 ) = f1 (t0 ) f1 (t1 ) f1′ (t1 ) , f2 (t0 ) f2 (t1 ) f2′ (t1 )

and D∗ (f0 , f1 , f2 ; t, t, t) = W (f0 , f1 , f2 )(t). With this definition, the system F is called an Extended Chebyshev system or ET–system on I, provided that for any set t0 ≤ · · · ≤ tn of points of I, D∗ (f0 , . . . fn ; t0 , . . . tn ) does not vanish, and it is called an Extended Complete Chebyshev system or ECT–system on I, if (f0 , f1 , . . . , fk ) is an ET–system on I for all k = 0, . . . , n. Chebyshev systems are of considerable importance in approximation theory, in particular in the study of spline functions, as well as in the theory of finite moments. 2010 Mathematics Subject Classification. 30C15; 26A51; 26C10; 26E05; 34C07; 34C08. Key words and phrases. Chebyshev systems; Extended Chebyshev systems; Extended complete Chebyshev systems. 1

21

2

RICHARD A. ZALIK

Examples of T–systems include eigenfunctions of Sturm–Liouville operators. These topics are discussed, for example, in Karlin and Studden’s classical monograph [2]. Results on spline functions have appeared in a plethora of later publications. For more recent results in the theory of real–valued T–systems, the reader is referred to the article by Carnicer, Pe˜ na and the author [1], and references thereof. Lately, there has been renewed interest in Chebyshev systems because of their applications in the theory of differential equations. For example P. Mardˇesi´c in his memoir [3], which develops the theory of versal unfolding of cusps of order n, emphasizes the development of results on T–systems for the study of unfolding singularities of vector fields, whereas in [4] Ma˜ nosas and Villadelprat use ECT– systems in their study of the period functions of centers of potential systems. It is therefore useful to study properties of T-systems that may be applied in the study of differential equations, and that may have been previously overlooked. The following theorem is well known, although it is usually stated for real–valued functions. Theorem 1. Let F = (f0 , f1 , . . . , fn ) be a system of complex–valued functions defined on a proper interval I. Then (1) (f0 , f1 , . . . , fn ) is a T–system on I if and only if any nontrivial linear combination of the functions of F has at most n zeros. (2) (f0 , f1 , . . . , fn ) is an ET–system on I if and only if any nontrivial linear combination of the functions of F has at most n zeros counting multiplicities. (3) (f0 , f1 , . . . , fn ) is an ECT–system on I if and only if for any k, 0 ≤ k ≤ n, (f0 , f1 , . . . , fk ) is an ET– system. Note that if F = (f0 , . . . , fn ) is a real–valued T –system on a proper interval I, a continuity argument shows that, multiplying if needed fn by −1, there is no essential loss of generality if we assume that for any set t0 < · · · < tn of points of I the determinants D(f0 , . . . fn ; t0 , . . . tn ) are strictly positive. Moreover, if F is an ET – system for which D(f0 , . . . fn ; t0 , . . . tn ) > 0 for any set t0 < · · · < tn of points of I then, proceeding as in [2, pp. 6–8], we deduce that for any set t0 ≤ · · · ≤ tn of points of [a, b], the determinants D∗ (f0 , . . . fn ; t0 , . . . tn ) are strictly positive. This in turn implies that if F is an ECT–system for which D(f0 , . . . fk ; t0 , . . . tk ) > 0 for any 0 ≤ k ≤ n and any set t0 < · · · < tn of points of I then, for any 0 ≤ k ≤ n and any set t0 ≤ · · · ≤ tn of points of I, the determinants D∗ (f0 , . . . fk ; t0 , . . . tk ) are strictly positive for 0 ≤ k ≤ n. We shall call such systems positive. Thus we may speak of positive T–systems, positive ET–systems, and positive ECT–systems. Positive ECT–systems, as we define them here, are the ECT–systems of Karlin and Studden [2]. They are called full differentiable ECT–systems by Mardˇesi´c [3]. We emphasize that all functions in a positive systems, and in particular positive ECT– systems, are assumed to be real–valued. In the theory of real–valued ECT–systems defined on a closed interval [a, b], the following theorem is of fundamental importance. A proof can be found in [2, pp. 376–379]. We have adapted the statement to our definition of T –systems. Theorem 2. Let f0 , f1 , . . . , un be real–valued functions of class C n [a, b]. The following two conditions are equivalent. (1) (f0 , . . . fn ) is a positive ECT –system on [a, b].

22

SOME PROPERTIES OF CHEBYSHEV SYSTEMS

3

(2) W (f0 , . . . , fk )(t) > 0 on [a, b] for 0 ≤ k ≤ n. If, in addition, the functions fk satisfy the initial conditions (p)

(2)

fk (a) = 0,

0 ≤ p ≤ k − 1;

1 ≤ k ≤ n,

then (a) and (b) are equivalent to (3) There are functions wk , strictly positive on [a, b] and of continuity class C n−k [a, b], such that

(3)

f0 (t)

=

w0 (t)

f1 (t)

=

w0 (t)

f2 (t) .. .

=

w0 (t)

Rt

w1 (s1 )

R s1

w2 (s2 ) ds2 ds1

fn (t)

=

w0 (t)

Rt

w1 (s1 )

R s1

w2 (s2 ) · · ·

Rt a

a

a

w1 (s1 ) ds1 a

a

R sn−1 a

wn (sn ) dsn · · · ds1 .

From [2, p. 380, (1.12) and (1.13)] we also know that if (f0 , . . . fn ) has the representation (3), then W (f0 , f1 , · · · fk ) = w0k+1 w1k · · · wk ,

(4) which implies that

w0 = f0 , (5) wk =

w1 =

W (f0 , f1 ) , f02

W (f0 , · · · , fk )W (f0 , · · · fk−2 ) , [W (f0 , · · · fk−1 )]2

2 ≤ k ≤ n.

To prove that (c) implies (a) in Theorem 2, Rolle’s theorem is used. Thus, the proof is not valid for complex–valued functions. The other parts of the statement still hold. 2. Existence of Adjoined Functions In this section we discuss the existence of adjoined functions i.e., given a T– system (f0 , . . . , fn ), whether there exists a function fn+1 such that (f0 , . . . , fn , fn+1 ) is a T–system. For dense subsets of open intervals this was answered in the affirmative by Zielke [7], and for any interval by the author [5]. The question has been raised of whether the same is true for complex–valued T–systems and whether to a T–system of analytic functions can be adjoined an analytic function. Although the methods usually used for real–valued functions cannot be applied in this setting, but we can still give an answer for real analytic functions. Theorem 3. Let (f0 , . . . , fn ) be an ECT–system on a proper interval I. Assume, moreover, that the functions fk are analytic on an open region D that contains I, and that they are real–valued on I. Then there is a function fn+1 , analytic on an open region D1 that contains I and real–valued on I, such that (f0 , . . . , fn , fn+1 ) is an ECT–system on I.

23

4

RICHARD A. ZALIK

Proof. The hypotheses imply that the Wronskians W (f0 , . . . , fk ), 1 ≤ k ≤ n do not vanish on I. Multiplying the functions fk by −1 if necessary, we may assume without essential loss of generality that these Wronskians are strictly positive on I. Let a < b be points in I. Subtracting if necessary from each function fk a suitable linear combination of its predecessors we obtain a system (u0 , . . . , un ) that satisfies the initial conditions (2). Thus, from Theorem 2 we know that (u0 , . . . , un ) has a representation of the form (3) on [a, b]. It follows from (5) that the functions wk are strictly positive on I and analytic on some open region D1 that contains I. Let 2 wn+1 be an entire function strictly positive on I (eg. e−t ), and define Z t Z sn−1 Z sn un+1 (t) := w0 (t) w1 (s1 ) · · · wn (sn ) wn+1 (sn+1 ) dsn+1 dsn · · · ds1 . a

a

a

Clearly un+1 is analytic on D1 . From (4) we deduce that W (u0 , u1 , · · · un+1 ) = w0n+2 w1n+1 · · · wn+1 > 0 on I, and by another application of Theorem 2 we deduce that (u0 , . . . , un , un+1 ) is an ECT system on [a, b]. Since a and b are arbitrary, the assertion follows.  3. Extending the Domain of Definition The problem of extending the domain of definition of a T– system has been studied extensively (see [1]). Here we consider the problem of extending the domain of definition of an ECT–system of complex–valued functions. Theorem 2 cannot be applied in this case, which makes the arguments more involved. Theorem 4. Let F = (f0 , . . . , fn ) be an ECT–system of complex–valued functions defined on a proper interval I with endpoints a and b. Assume, moreover, that the functions fk are of class C n (α, β), where α < a < b < β. If a ∈ I there is a c < a such that F is an ECT–system on (c, a) ∪ I, whereas if b ∈ I there is a d > b such that F is ECT–system on I ∪ (b, d). Proof. It suffices to assume that a ∈ I: the other case readily follows by the change of variables t → −t. Let Ik denote the set of integers from 0 to k. A partition of Ik is a family {Sr ; 0 ≤ m} of sets of integers such that Sm (1) r=0 Sr = Ik . (2) If α is the largest number in Sr and β is the smallest number in Sr+1 ,then β = α + 1. The preceding definition implies that the Sr are sets of consecutive integers. A simple inductive argument shows that there are 2k+1 different partitions of Ik . If P is a partition of Ik and S is a set in P , then S is called a component of P . A set of integers t0 ≤ t1 · · · ≤ tk is called a configuration associated with P if, whenever α and β belong to the same component of P , tα = tβ , and whenever α and β belong to different components, then tα 6= tβ . Thus, any set t0 ≤ t1 · · · ≤ tk of points of Ik belongs to one of 2k+1 configurations. For each configuration, D∗ (f0 , . . . fk ; t0 , . . . tk ) is a continuous function of the free variables involved. For example, if t0 < t1 < t2 , then D∗ (f0 , f1 , f2 ; t0 , t1 , t2 ) is a continuous function of t1 , t2 and t3 , whereas if t0 < t1 = t2 , then D∗ (f0 , f1 , f2 ; t0 , t1 , t1 ) is a continuous function of t0 and t1 . It follows that for an

24

SOME PROPERTIES OF CHEBYSHEV SYSTEMS

5

arbitrary k, 0 ≤ k ≤ n, if P is a partition of Ik having m sets and S is a configuration associated with P , then D∗ (f0 , . . . fk ; t0 , . . . tk ) is a continuous nonvanishing function in the m–fold cartesian product of I with itself. Therefore there is a number ck (P ) < a such that D∗ (f0 , . . . fk ; t0 , . . . tk ) 6= 0 whenever t0 ≤ t1 ≤ · · · ≤ tk is a configuration associated with P and the points tk are in (ck (P ), a) ∪ I. Setting ck to be the largest of the ck (P ) and c to be the largest of the ck , the assertion follows.  4. Embedding Given a finite set of functions, the embedding problem consists in finding necessary and sufficient conditions for the existence of a T–system whose linear span contains them. For a single real–valued function, this problem was solved by the author in [6], whereas in [4, Proposition 2.2 and Proposition 2.3] Ma˜ nosas and Villadelprat show how to embed an analytic function into an ECT–system of analytic functions defined on an interval. The problem in its full generality remains unsolved. We can give an answer in a particular case, but first we need to prove some auxiliary propositions. Lemma 5. Let (f0 , · · · , fn ), n ≥ 1, be a positive ECT–system on a closed interval [a, b] such that the functions fk satisfy (3), and let (c(k); 0 ≤ k ≤ m) be a strictly increasing sequence with 0 ≤ c(0) < c(n) ≤ n. Then (fc(k) ; 0 ≤ k ≤ m) is a positive ECT–system on (a, b]. Proof. Let D − 0 = f /w0 , Dr f (t) :=

d dt



f (t) wr (t)



,

1 ≤ r ≤ k,

and Lr f (t) := Dr Dr−1 · · · D0 f (t). We proceed by induction. The assertion is obvious for n = 1. To prove the inductive step we proceed as follows: Let α = c0 . Since (Lα fk ; α + 1 ≤ k ≤ n) has a representation of the form (3), the inductive hypothesis implies that (Lα fc(k) ; c1 ≤ k ≤ m) is a positive ECT–system on (a, b]. By repeated application of the inverse −1 operators Dα−1 , Dα−1 . . . D0−1 to (Lα fc(k) ; c1 ≤ k ≤ m) and using Rolle’s theorem at each step, we deduce that (fc(k) ; 0 ≤ k ≤ m) is a positive ECT–system on (a, b].  Lemma 6. Let (f0 , · · · , fn ), n ≥ 1, be a positive ECT– system on an interval I(a, b) having endpoints a < b, let c ∈ I(a, b), assume that the functions fk satisfy initial conditions of the form (2) at the point c, that f0′ (t) > 0 on I(a, b), and let gk (t) := (t − c)fk (t). Then (g0′ , · · · , gn′ ) is a positive ECT– system on I(a, b]. Proof. We may assume, without loss of generality, that I(a, b) is a closed interval. Assume first that c = a. Let k be arbitrary but fixed and f := (f0 , · · · fk )T . Then W (g0′ )(t) > 0, and for k>0 W (g0′ , · · · , gk′ )(t) = det((t − a)f ′ (t) + f (t), · · · , (t − a)f (k) (t) + f (k−1) (t)). Let c = (c(r); 0 ≤ r ≤ k) be a sequence of zeros and ones, let Λ denote the set of all such sequences, and for 0 ≤ r ≤ k let qr (1, t) := (t − a)f (r+1) (t), qr (2, t) := (r + 1)f (r) (t). Then Wk (t) := det (q0 (1, t) + q0 (2, t), · · · , qk (1, t) + qk (2, t)) (t) =

25

6

RICHARD A. ZALIK

X

det (qℓ (c(r), t); 0 ≤ ℓ ≤ k) =

X

Dc (t).

c∈Λ

c∈Λ

For a given 0 ≤ ℓ ≤ k − 1 there are four possibilities: If c(ℓ) = 1 and c(ℓ + 1) = 2 then qℓ (c(ℓ), t) = (t − a)f (ℓ+1) (t) and qℓ+1 (c(ℓ + 1), t) = (ℓ + 2)f (ℓ+1) (t). (This implies that Dc (t) = 0). If c(ℓ) = 2 and c(ℓ + 1) = 1 then qℓ (c(ℓ), t) = (ℓ + 1)f (ℓ) (t) and qℓ+1 (c(ℓ + 1), t) = (t − a)f (ℓ+2) (t). If c(ℓ) = 1 and c(ℓ + 1) = 1 then qℓ (c(ℓ), t) = (t − a)f (ℓ+1) (t) and qℓ+1 (c(ℓ + 1), t) = (t − a)f (ℓ+2) (t). If c(ℓ) = 2 and c(ℓ + 1) = 2 then qℓ (c(ℓ), t) = (ℓ)f (ℓ) (t) and qℓ+1 (c(ℓ + 1) = (ℓ + 1)f (ℓ+1) (t). In summation, there are constants α > 0 and β ≥ 0, and a sequence a0 ≤ a1 ≤ · · · ≤ ak such that Dc (t) = α(t − a)β det(f (a0 ) (t), f (a1 ) (t), · · · f (ak ) (t)), whence Lemma 5 implies that Dc (t) ≥ 0. In particular, if c(ℓ) = 2 for all ℓ, then   Dc (t) = det (ℓ + 1)f (ℓ) (t); 0 ≤ ℓ ≤ k = (k + 1)!W (f0 , · · · , fk )(t) > 0.

Thus W (g0′ , · · · , gk′ )(t) > 0 on I for 0 ≤ k ≤ n. If c = b, the assertion follows by the change of variable t → −t. In the general case, whether t < c or t ≥ c, the preceding discussion insures that W (g0′ , · · · , gk′ )(t) > 0 on I for 0 ≤ k ≤ n.  Theorem 7. Let 0 ≤ k ≤ n and assume that the functions fk+1 , . . . , fn are in C n (a, b), that for k+1 ≤ r ≤ n, any linear combination of the functions fk+1 , . . . , fr has at most r zeros counting multiplicities and there is at least one linear combination of these functions having exactly r zeros counting multiplicities. Assume, moreover, that fr(p) (a) = 0,

k + 1 ≤ r ≤ n,

0 ≤ p ≤ r − 1.

Then there are functions f0 , . . . , fk such that (f0 , . . . , fn ) is an ECT system on [a, b]. Proof. Without essential loss of generality we may assume that each function fr has exactly r zeros counting multiplicities. We proceed by induction. Assume first that n = 1. The hypotheses imply that f1 (t) = (t − a)g(t), where g(t) is nonvanishing and continuously differentiable on [a, b]. Setting f0 (t) := g(t), the assertion follows. To prove the inductive step, let gr−1 (t) := (t − a)−1 fr (t). Then for k ≤ r ≤ n − 1 every nontrivial linear combination of the functions gk , . . . , gr has at most r zeros counting multiplicities. By inductive hypothesis there are functions g0 , . . . gk−1 such that (g0 , · · · , gn−1 ) is an ECT–system on I. Since W (exp(α ·)f0 , . . . , exp(α ·)fr )(t) = exp(r α t)W (f0 , . . . , fr )(t), multiplying if necessary the functions fr by exp(α t) with α sufficiently large we may assume, without loss of generality that g0′ is strictly positive on I. If fℓ (t) := (t − a)gℓ+1 , 1 ≤ ℓ ≤ k, we see that (f1′ , · · · , fn′ ) is a positive ECT–system on I. Defining f0 (t) := 1 we see that for 1 ≤ k ≤ n, W (f0 , · · · , fk )(t) = W (f1′ , · · · , fk′ )(t) > 0,

26

SOME PROPERTIES OF CHEBYSHEV SYSTEMS

whence the assertion follows from Theorem 2.

7



References [1] J. M. Carnicer, J. M. Pe˜ na and R.A. Zalik, Strictly Totally Positive Systems, J. Approx. Theory 92 (1998) 411–441. [2] S. Karlin and W. Studden, Tchebycheff Systems: With Applications in Analysis and Statistics, Interscience, New York, 1966. [3] P. Mardˇ esi´ c, Chebyshev systems and the versal unfolding of the cusps of order n, Travaux en Cours 57. Hermann, Paris, 1998. [4] F. Ma˜ nosas and J. Villadelprat, Criteria to bound the number of Critical Periods, J. Differential Equations 246 (2009), 2415–2433. [5] R. A. Zalik, Existence of Tchebycheff Extensions, J. Math. Anal. Appl. 51 (1975). [6] R. A. Zalik, Embedding a Function into a Haar Space, J. Approx. Theory 55 (1988), 61–64. [7] R. Zielke, Alternation Properties of Tchebyshev–Systems and the Existence of Adjoined Functions, J. Approx. Theory 10 (1974), 172–184. Department of Mathematics and Statistics, Auburn University, AL 36849–5310. E-mail address: [email protected]

JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.1, 27-36, 2011, COPYRIGHT 2011 EUDOXUS PRESS, 27 LLC

S o lu t io n t o a S in g u la r In t e g r o -D iffe r e n t ia l E qu a t io n III J. N. Pandey School of Mathematics and Statistics, Carleton University Ottawa, Ontario K1S 5B6, Canada email: [email protected] Abstract In this paper we solve the singular integro-differential equation  dy
λti + e + eµti Hy = eνti dt

(i)

in the space of semi-almost periodic distributions where λ, µ, ν are positive and rationally independent numbers and H is the Hilbert transform operator defined on the real line. Solution to the general problem dy + a(t)Hy = f (t) dt

(ii)

when a(t), f (t) are any almost periodic functions still remains an open problem. It also remains an open problem to associate a physical problem with (i) and (ii). Nevertheless solutions to problems (i) and (ii) are important from the point of view of operator theory. AMS Subject Classification [2010]: Primary: 26A33, 42A75; Secondary: 46F12 Keywords: Almost periodic functions and distributions, Hilbert transform of almost periodic functions and distributions, singular integral equations.

Definitions and P r eliminar ies In [4, 5, 6] we have considered the case when the exponents of the exponential terms appearing in a(t) and f (t) of (ii) are rationally dependent. We are now dealing with the case when those terms of a(t) and f (t) are rationally independent. We have also noticed that when a(t) is a constant the system (ii) can be reduced to a simple differential equation with constant coefficients by eliminating the operator H of Hilbert transformation. Here (Hf )(x) is defined as  ∞ 1 f (t) (Hf )(x) = (P ) dt. π x −t −∞ To solve (i) or (ii) when a(t) is not a constant cannot be reduced to linear d.e. with constant coefficients. In order to eliminate H we used the properties that d (Hf )(x) = H f ′ (x) dt in the space of differentiable functions as well as in the spaces of generalized functions which are Hilbert transferable. However this technique does not work when a(t) is not a constant. In the

1

28

PANDEY: INTEGRO-DIFFERENTIAL EQUATIONS

process of eliminating H we come across an identity. For the detailed description of the Hilbert transform of periodic function one may look into [2] and [3] and for the Hilbert transform of almost periodic distribution one may look into [4] and the subsequent references [5] and [8]. The space of almost periodic functions and distributions. We denote by L the set of  trigonometric polynomials k Ak eiλk t with real λk and finitely many coefficients Ak . Let M be the set of all continuous functions defined on R which are the uniform limit of the sequence of trigonometric polynomials then the set of functions LU M is the space of almost periodic functions defined on the real line IR, where an almost periodic function is defined as follows (Bohr). We say that a continuous function f defined on IR is an almost periodic function if for ε > 0 there exists a number ℓ = ℓ(ε) > 0 such that in every interval of length ℓ there exists at least one τ > 0 such that |f (t + τ ) − f (t)| < ε ∀ t ∈ (−∞, ∞). The set LU M is denoted by B and H. Bohr proved that a continuous function f defined on R is an almost periodic function if and only if it belongs to B. So an almost periodic function is either a trigonometric polynomial or a uniform limit of the sequence of trigonometric polynomials. The linear space (system) L is metrized as follows: f (t) =

m 

Ar e

iλr t

,

g(t) =

r= 1

h 

Bs eiµs t

s= 1

then the scalar product of the trigonometric polynomials f and g are defined by  T 1 f, g = lim f (t)g(t)dt T→ ∞ 2T −T  T n m   1 = lim Ar B s eit(λr −µs ) dt T→ ∞ 2T −T r= 1 s= 1 =

m  n 

δ(λr , µs )Ar B s

(1)

r= 1 s= 1

f 2 = f, f  =

m 

|Ar |2 .

(2)

r= 1

It is a simple exercise to verify that the norm defined in (2) is actually a norm. The norm (2) can be easily extended over the space B by the continuity property. Let B 2 be the completion of the space L with respect to the norm defined by (2). The space B 2 is not separable as there exist uncountably many orthonormal systems {eλt }λ∈ R in B 2 . [1] Thus the space B 2 is an example of non-separable Hilbert space. Let f and g  be two almost ∞ periodic (a.p.) functions belonging to the closed linear span of the orthonormal set eiλn t n= 1 . It is a simple exercise to show that if f (t) =

∞ 

am e

iλm t

,

and g(t) =

m= 1

∞ 

bm eiλm t

m= 1

then f (t) = g(t), if and only if am = bm ∀ m = 1, 2, 3, . . . .

(3)

Property (3) will be used in solving the system of equations (1) and (2) in the space of almost periodic functions and distributions and also in the space of semi-almost periodic distributions.

2

PANDEY: INTEGRO-DIFFERENTIAL EQUATIONS

29

 The Weierstrass function bn cos(an πx), 0 < b < 1 and ab > 1+ 3π , a is an odd positive integer 2  1 is an a.p. function differentiable nowhere on the real line but the function φ(t) = ∞n= 1 n12 e n ti is an a.p. function differentiable to any order on the real line. Each of the derivatives φ(k) (t) is also an a.p. function. The function f (t) =

∞ 

an eλn ti

(4)

n= 1

∞ k if n= 1 |an ||λn | is convergent for each k = 0, 1, 2, . . . is an infinitely differentiable function on the real line converging uniformly  there (on the real line), along with all its derivatives. Under these conditions itfollows that ∞n= 1 |an |2 |λn |2k is also convergent for each k = 0, 2, 3, 4, and that ∞ Hf 2 = f 2 = n= 1 |an |2 |λ2k n . Thus f (t) along with all its derivatives is an a.p. function. Semi-almost Periodic Distributions

The testing function space Φ: Let B be the space of a.p. functions defined on IR equipped with the norm topology defined by (2). This space is not a complete inner product space. Therefore, we generate completion space of B with the norm defined by (2) and denote this completion space (Hilbert space) by B 2 . The space B is dense in B 2 . Next let Φ be a testing function space consisting of all the infinitely differentiable almost periodic functions defined on IR. The topology over Φ is generated by the separating collection of seminorms {γk }∞k= 0 defined by   T 1 (k) γk (φ) = γ(φ (t)) = lim |φ(k) |2 dt T→ ∞ 2T −T (k)

so a sequence {φν }∞ν= 1 → φ in Φ if and only if for each k = 0, 1, 2, . . . , γk (φν −φ) = γ(φν −φ(k) ) → 0 as ν → ∞. The space Φ is a subspace of the space B and we now define the spaces Φ ⊇ Φ1 ⊇ Φ2 ⊃ Φ3 · · · by Φk = {Ψ · Ψ = φ(k) (t), φ(t) ∈ Φ} i.e. the space Φk is obtained by differentiating each element of Φ k times. The topology over each of the subspaces Φk is generated by the seminorm on γ defined by (2), i.e.   T 1 lim |φ|2 dt, φ ∈ Φk . γ(φ) = T→ ∞ 2T −T Define the sequence of seminorms {ρk }∞k=

0

by

ρ0 (φ) = γ(φ) ≡ γ0 (φ) ρ1 (φ) = γ0 (φ) + γ1 (φ) ρ2 (φ) = γ0 (φ) + γ1 (φ) + γ2 (φ) and ρn (φ) = γ0 (φ) + γ1 (φ) + · · · γn (φ). The two sequences of seminorms which are separating generate the same topology [9] and if f is a continuous linear functional over Φ then there exists k ≥ 0 such that |f, φ| ≤ γ0 (φ) + γ1 (φ) + · · · φk (φ) ≤ γ(φ) + γ(φ(1) ) + · · · γ(φ(k) ).

3

30

PANDEY: INTEGRO-DIFFERENTIAL EQUATIONS

Since the completion space of each of the subspaces Φ0 , Φ1 , Φ2 , · · · Φk with respect to the seminorm γ (in fact γ is a norm on B) is contained in the space B 2 ; therefore in view of Riesz representation theorem and Hahn Banach theorem there exist g0 , g1 , · · · gk belonging to B 2 such that f, φ = g0 , φ + g1 , φ(1)  + g2 , φ(2)  + · · · gk , φ(k) 

k  (m) m gk (−1) , φ , ∀ φ ∈ Φ. = m= 0

Since Φ is dense in B 2 we have f, ψ = so f =



k 

(m) gm (−1)m , ψ

m= 10



∀ ψ ∈ B2

(m) m m= 0 gm (−1) .

k

Corollary 1. In a special case when gm are almost periodic functions f is an a.p. distribution. [7] Corollary 2. When the series obtained by differentiating gm when gm are a.p. functions are convergent, then f is an a.p. function. [7] Thus we see that Φ′ contains the space of a.p. functions as well as the space of a.p. distributions. We call the space Φ′ as the space of semi-almost periodic distributions. Corollary 3. Using the Riesz theorem in Hilbert space one can show that f (t) = 1 also belongs to the Hilbert space B (2) . So this unit function is a semi-almost periodic distribution. To solve (1) we first solve the associated homogeneous differential equation  dy
λti + e + eµti Hy = 0. dx

(5)

Since λ, µ are rationally independent λ is not a rational multiple of µ and vice-versa. The solution space of (5) must satisfy the following properties (i) The space containing the solution to (5) must be closed with respect to the operator of the Hilbert transformation. (ii) It must also be closed with respect to the operation of multiplication by eλti as well as eµti . (iii) It must also be closed with respect to the operator of differentiation. (iv) It must also be closed with respect to the operation of addition (or subtraction). Therefore a suggested solution to (5) should be in the closed linear span of  ∞ e(mλ+ nµ)ti . m,n= 0

Hence, a suggested solution to (5) is of the form y=

∞ ∞  

am,n e(mλ+

nµ)ti

n= 0 m= 0

=

∞ 

m,n= 0

4

am,n e(mλ+

nµ)ti

(6)

PANDEY: INTEGRO-DIFFERENTIAL EQUATIONS

31

Since we are looking for solutions in the form (6) in the space of a.p. functions or a.p. distributions the terms in (6) can be arranged in any order. So we have written the solution (6) in the form y=

∞ 

am,n e(mλ+

nµ)ti

.

(7)

m,n= 0

Substituting for y from (7) into (5) we get ∞ 

iam,n (mλ+nµ)e(mλ+

nµ)ti

= i(eλti +eµti )

m, n = 0 (m, n)
= (0, 0) ∞ 

∞ 

am,n e(mλ+

nµ)ti

m, n = 0 (m, n)
= (0, 0)

am,n (mλ+nµ)e(mλ+

nµ)ti

= (eλti + eµti )

m, n = 0 (m, n)
= (0, 0)

∞ 

am,n e(mλ+

nµ)ti

(8)

m, n = 0 (m, n)
= (0, 0)

We will now equate the coefficients of the like exponential terms to determine aij . Coefficients of e(λ+

µ)ti

: (λ + µ)a1,1 = a1,0 + a0,1 a1,0 + a0,1 = a1,1 .

Equating the coefficients of e

(mλ+ nµ)ti

we get

am,n (mλ + nµ) = am−1,n + am,n−1 am−1,n |m=

0

m, n = 0, 1, 2, 3, . . .

(9)

= a−1,n .

There is no such coefficients in (9). Therefore, a−1,n is taken to be zero. For the same reason am,n−1 |n=

0

= am,−1 = 0.

Also, in the recurrence relation (9) it is improper to take m and n both zero simultaneously, as the coefficient a0,0 vanishes both after differentiation as well as the operation of Hilbert transformation. So in (9) we will avoid the situation m, n being taken to be zero simultaneously. But it is legitimate to take only one of m or n to be zero at a time with the interpretation that if one of the lower suffixes in am,n is negative then am,n is interpreted to be zero. am,n (mλ + nµ) = am−1,n + am,n−1 , with m = 2, n = 0; a2,0 2λ = a1,0 + a2,−1 = a1,0 a1,0 a2,0 = , 2λ with m = 1, n = 0; a1,0 (λ) = a0,0 + a1,−1 = 0

so a1,0 = 0,

a0,1 (µ) = a−1,1 + a0,0 = 0,

so a0,1 = 0.

with m = 1, n = 1; we get,

5

32

PANDEY: INTEGRO-DIFFERENTIAL EQUATIONS

Note that in (9) a0,0 should not appear so it is taken to be zero in the recurrence relation only. a2,0 (2λ) = a1,0 = 0,

so a2,0 = 0.

Thus by induction am,0 = 0 ∀ m = 1, 2, 3, . . . . By induction we can also show that a0,n = 0,

∀ n = 1, 2, 3, . . . .

With m = 2, n = 1 we get a1,1 (λ + µ) = a0,1 + a1,0 = 0

so a1,1 = 0.

a2,1 (2λ + µ) = a1,1 + a2,0 = 0

so a2,1 = 0

a3,1 (3λ + µ) = a2,1 + a3,0 = 0

so by induction am,1 = 0 ∀ m = 0, 1, 2, 3 . . . .

Thus, continuing by induction we can also show that am,n = 0 ∀ m, n = 0, 1, 2, 3, . . . , (m, n) ± (0, 0). Therefore the solution in this case is y = a0,0 , where a0,0 is an arbitrary constant. We now proceed to find a solution to (1) i.e. solution to  dy
λti + e + eµti Hy = eνti dt

We assume that there is a solution to (10) of the form    y= b0,0 + 

∞ 

bm,n e

m, n = 0 (m, n)
= (0, 0)

Using the formula Heλti = −i sgn(λ)eλti we get    νti Hy = −iy = −i  b0,0 e + 

(10)



  e 

(mλ+ nµ)ti  νti

∞ 

bm,n e(mλ+

m, n = 0 (m, n)
= (0, 0)

.

(11)



  . 

nµ+ ν)ti 

Therefore, from (10) and (11) we get b0,0 ν +

∞ 

bm,n (mλ + nµ + ν)e(mλ+

nµ)ti

m, n = 0 (m, n)
= (0, 0)

= (e

λti

+e

µti



  ) b0,0 + 

∞ 

m, n = 0 (m, n)
= (0, 0)

6

bm,n e



   − i. 

(mλ+ nµ)ti 

(12)

PANDEY: INTEGRO-DIFFERENTIAL EQUATIONS

33

Equating the coefficients of the like exponential powers of (14) we get b0,0 ν = −i i b1,0 (λ + ν) = b0,0 = − , ν

i.e.

b0,0 = −

b1,0 = −

i b0,1 (µ + ν) = b0,0 = − , ν

i ν

i −i = ν(λ + ν) ν(ν + λ)

b0,1 =

−i ν(µ + ν)

bm,n (mλ + nµ + ν) = bm−1,n + bm,n−1 ,

(13)

with m = 2, n = 0 we get b1,0 −i = ν + 2λ ν(ν + λ)(ν + 2λ) b2,0 −i = = . (ν + 3λ) ν(ν + λ)(ν + 2λ)(ν + 3λ)

b2,0 = b3,0 By induction we get

bn,0 =

−i ν(ν + λ)(ν + 2λ) · · · (ν + nλ)

b0,2 (2µ + ν) = b0,1 b0,1 −i b0,2 = = . 2µ + ν ν(ν + µ)(ν + 2µ) By induction we get b0,n =

−i ν(ν + µ)(ν + 2µ) · · · (ν + nµ)

From (13) with m = n = 1 we get −i i b0,1 + b1,0 ν(ν+ µ) − ν(ν+ λ) = λ+µ λ+µ+ν −i i = − . ν(ν + µ)(λ + µ + ν) ν(ν + λ)(λ + µ + ν)

b1,1 = b1,1

From (13) with m = 1, n = 2 we get   b0,2 + b1,1 1 −i i i b1,2 = = − − . λ + 2µ + ν (λ + 2µ + ν) ν(ν + µ)(ν + 2µ) ν(ν + λ)(λ + µ) ν(ν + µ)(ν + λ + µ) Proceeding this way we can calculate b1n . 1 [b1,1 + b2,0 ] (2λ + µ + ν)   1 −i i i = − − (2λ + µ + ν) ν(ν + µ)(ν + µ + λ) ν(ν + λ)(ν + λ + µ) ν(ν + λ)(ν + 2λ)

b2,1 =

7

34

PANDEY: INTEGRO-DIFFERENTIAL EQUATIONS

Thus each term like bi,j can be worked out. Substituting for bi,j in the series     y =k+  

∞ 

bm,n e

(λm+ µn)it

m, n = 0 (m, n)
= (0, 0)

  νti + b0,0  e 

(14)

we get the required solution, k is an arbitrary constant, a replacement for a0,0 . Note that this a0,0 is different from a0,0 that has appeared in the recurrence relation. Thus our solution has only one arbitrary constant. The general term of the coefficients bmn is hard to work out this way. We have given a method below giving the solutions to (i) in closed form and the coefficients bmn can be worked out from this solution. It is a simple exercise to show that the solution (14) to the system (i) is absolutely convergent trigonometric series and so it is a uniformly convergent trigonometric series and therefore represents an a.p. function. But there are cases when a series thus obtained is divergent. In that case our solution is an a.p. distribution or some other kind of function, called semi-almost periodic distributions. The number of linearly independent solutions of the associated homogeneous system also depends upon the basis vectors we have chosen.  If we find the ∞ solution to the associated homogeneous linear system of (i) in the closed linear span of e(mλ+ nµ)ti m,n= −∞ there may be three arbitrary constants appearing in our solution, including one of the constants as a0,0 . It is particularly true when λ and µ are of different signs. The foregoing method is the general one and applies in all such cases whether or not λ, µ, µ are all of the same sign. We will now take advantage of the special situation mentioned by us and solve the problem using the technique of solving a linear differential equation of the order one. We first solve the associated homogeneous differential equation of (i)

We are seeking solutions of the form y=

 dy
λti + e + eµti Hy = 0. dt

∞ 

amn e(mλ+

nµ)ti

m,n= 0

= a00 +

∞ 

am,n e(mλ+

nµ)ti

m, n = 0 (m, n)
= (0, 0)

= a00 + v(t). Hy = −i(y − a00 ) = −iy + ia00 . So now (5) takes the form dy − i(eλti + eµti )y = −i(eλti + eµti )a00 . dt This is linear in y and its general solution is y = a00 + Ce( a00 and C are arbitrary constants.

8

eλti λ

+

eµti µ

)

,

(15)

PANDEY: INTEGRO-DIFFERENTIAL EQUATIONS

If we plug in y from (15) in (5) we get
 i C eλti + eµti = 0

35

so C = 0.

Therefore the general solution in this case of (5) is

y = a00 , a00 can be chosen arbitrarily and so y=k

where a00 = k = arbitrary constant.

So the general solution to (5) is y = k. We now solve  dy
λti + e + eµti Hy = eνti . dt Let us use the substitution y = (a00 + v)eνti

(16)

where ∞ 

v=

am,n e(λm+

µn)ti

(17)

m, n = 0 (m, n)
= (0, 0)

in the d.e. (16). We now have

or


 dv νti e + (a00 + v)eνti νi + eλti + eµti (−i)(a00 + v)eνti = eνti dt   dv + v νi − (eλti + eµti )i = 1 + ia00 (eλti + eµti − ν) dt   ³ λti ³ λti ´ ´ µti µti   d νit− e λ + e µ νit− e λ + e µ ve = e 1 + ia00 (eλti + eµti − ν) dt  ³ λti ´ µti eλti eµti eλti eµti νit− e λ + e µ v · eνit− λ − µ = −a00 · e + eνit−( λ + µ ) dt + c v = −a00 + e−νit+

eλti λ

+



eµti µ

eνit−(

eλti λ

+

eµti µ

)

 dt + c .

In order that v should be of the form (17) we must have c = a00 = 0. So, 
λti µti   λti µti e −νit+ + eµ νit−( e λ + e µ ) λ v=e e dt . Therefore, the general solution to (16) or (i) in this case is    λti µti eλti −νit+ ( e λ + e µ ) νti y = (v + a00 )e + k = e eνit−( λ +

eµti µ

or

y=



e

eλti λ

+

eµti µ

 

eνit−(

eλti λ

+

eµti µ



dt + k.

(18)

)

dt



eνti + k,

(19)

y = k is the general solution to the associated homogeneous differential equation already found and (19) satisfies the d.e. (i) or solution of (i) or (16) that we have found belongs to  (16). This general ∞ the closed linear span of e(ℓλ+ mµ+ nν)ti ℓ,m,n= 0 .

9

36

PANDEY: INTEGRO-DIFFERENTIAL EQUATIONS

Refer ences [1] Akhiezer, N.I. and Glazman, I.M., Theory of linear operators in Hilbert space, Vol. I, Frederick Ungar Publishing Company New York, 1961. [2] Pandey, J.N., The Hilbert transform of periodic distributions, Journal of Integral Transforms and Special Functions, Vol. 5, (1997), No. 2, pp. 117-142. [3] Pandey, J.N., The Hilbert transform of Schwartz Distributions and Applications, John Wiley and Sons, Inc., January (1996). [4] J.N. Pandey, The Hilbert transform of almost periodic functions and distributions, Journal of Computational Analysis and Applications, Vol. 6, (2004), No. 3, pp. 199-210. [5] Pandey, J.N. and Fayez Seifeddine, Solution to a Singular Integro-Differential Equation. Journal of Computational Analysis and Applications, 14 pages to appear in December 2009 issue. [6] Pandey, J.N. and Fayez Seifeddine, Solution to some Singular Integro-Differential Equations II, Advanced Studies in Contemporary Mathematics 13 (2006), No. 1, pp. 101-113. [7] Schwartz, Laurent, Th´eorie des distributions, Herman, Paris, 1966. [8] Titchmarsh, E.C., The Theory of Functions, Oxford University Press, 1939. [9] Zemanian, A.H., Generalized Integral Transformation, Interscience Publishers, a division of John Wiley and Sons Inc., 1968 [10] Zygmund, A., Trigonometric Series, Vol. 1, Cambridge University Press, Cambridge, 1979.

10

JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.1, 37-46, 2011, COPYRIGHT 2011 EUDOXUS PRESS, 37 LLC

Some approximation theorems for functions of two variables through almost convergence of double sequences G. A. Anastassioua , M. Mursaleenb and S. A. Mohiuddineb a

) Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152, USA b ) Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India [email protected]; [email protected]; [email protected]

Abstract. The idea of almost convergence for double sequences was introduced by Moricz and Rhoades [Math. Proc. Camb. Phil. Soc., 104 (1988) 283-294]. In this paper, we use this concept to prove a Korovkin type approximation theorem for functions of two variables along with a supporting example. Further, we present some consequences of our main theorem. Keywords and phrases: Double sequence; almost convergence; Korovkin type approximation theorem. AMS subject classification (2000): 41A10, 41A25, 41A36, 40A05, 40A30.

1. Introduction and preliminaries Let c and l∞ denote the spaces of all convergent and bounded sequences, respectively, and note that c ⊂ l∞ . In the theory of sequence spaces, a beautiful application of the well known Hahn-Banach Extension Theorem gave rise to the concept of the Banach limit. That is, the lim functional defined on c can be extended to the whole of l∞ and this extended functional is known as the Banach limit [1] which was used by Lorentz [4] to define a new type of convergence, known as the almost convergence A double sequence x = (xjk ) of real or complex numbers is said to be bounded if ∥x∥∞ = supj,k |xjk | < ∞. The space of all bounded double sequences is denoted by Mu . A double sequence x = (xjk ) is said to converge to the limit L in Pringsheim’s sense (shortly, p-convergent to L) [9] if for every ε > 0 there exists an integer N such that |xjk − L| < ε whenever j, k > N . In this case L is called the p-limit of x. If in addition x ∈ Mu , then x is said to be boundedly convergent to L in P ringsheim′ s sense (shortly, bp-convergent to L). Let Ω denote the vector space of all double sequences with the vector space operations defined coordinatewise. Vector subspaces of Ω are called double sequence spaces. In addition to above-mentioned double sequence spaces we consider the double sequence space { } ∑ Lu := x ∈ Ω | ∥x∥1 := |xjk | < ∞ j,k

1

38

ANASTASSIOU ET AL: APPROXIMATION THEOREMS

of all absolutely summable double sequences. All considered double sequence spaces are supposed to contain Φ := span{ejk | j, k ∈}, where

{

1 ; if (j, k) = (i, ℓ), 0 ; otherwise. ∑ ∑ ∑ We denote the pointwise sums j,k ejk , j ejk (k ∈), and k ejk (j ∈) by e, ek and ej respectively. The idea of almost convergence for double sequences was introduced and studied by Moricz and Rhoades [5]. A double sequence x = (xjk ) of real numbers is said to be almost convergent to a limit L if m+p−1 n+q−1 1 ∑ ∑ lim sup xjk − L = 0 (∗) p,q→∞ m,n>0 pq j=m k=n ejk il

=

In this case L is called the F2 -limit of x and we shall denote by F2 the space of all almost convergent double sequences. Note that a convergent double sequence need not be almost convergent. However every bounded convergent double sequence is almost convergent and every almost convergent double sequence is bounded. Example 1.1 The double sequence z = (zmn ) defined by  1 if m = n odd,      −1 if m = n even, zmn =      0 (m ̸= n);

(1.1.1)

is almost convergent to zero but not convergent.

For recent developments on almost convergent double sequences and matrix transformations, we refer to [6, 7, 8, 10]. If m = n = 1 in (*) then we get (C, 1, 1)-convergence, and in this case we write xjk → ℓ(C, 1, 1); where ℓ = (C, 1, 1)-lim x. Let C[a, b] be the space of all functions f continuous on [a, b]. We know that C[a, b] is a Banach space with norm ∥f ∥∞ := sup |f (x)|, f ∈ C[a, b]. x∈[a,b]

The classical Korovkin approximation theorem states as follows [3]:

2

ANASTASSIOU ET AL: APPROXIMATION THEOREMS

39

Let (Tn ) be a sequence of positive linear operators from C[a, b] into C[a, b]. Then limn ∥Tn (f, x)−f (x)∥∞ = 0, for all f ∈ C[a, b] if and only if limn ∥Tn (fi , x)−fi (x)∥∞ = 0, for i = 0, 1, 2, where f0 (x) = 1, f1 (x) = x and f2 (x) = x2 . Quite recently, such type of approximation theorems are proved in [2] for functions of two variables by using statistical convergence. In this paper, we use the notion of almost convergence to prove approximation theorems for functions of two variables. 2. Korovkin type approximation theorem The following is the F2 -version of the classical Korovkin approximation theorem followed by an example to show its importance. Let C(I 2 ) be the space of all two dimensional continuous functions on I × I, where I = [a, b]. Suppose that Tm,n : C(I 2 ) → C(I 2 ). We write Tm,n (f ; x, y) for Tm,n (f (s, t); x, y); and we say that T is a positive operator if T (f ; x, y) ≥ 0 for all f (x, y) ≥ 0. Theorem 2.1. Let (Tj,k ) be a double sequence of positive linear operators from C(I 2 ) m+p−1 ∑ n+q−1 ∑ 1 into C(I 2 ) and Dm,n,p,q (f ; x, y) = pq Tj,k (f ; x, y). Then for all f ∈ C(I 2 ) j=m

k=n



= 0, i.e. F2 - lim T (f ; x, y) − f (x, y) j,k

j,k→∞

∞



lim

Dm,n,p,q (f ; x, y) − f (x, y) = 0, uniformly in m, n.

p,q→∞

if and only if

(2.1.0)

∞



lim Dm,n,p,q (1; x, y) − 1

= 0 uniformly in m, n, p,q→∞

(2.1.1)

∞



lim Dm,n,p,q (s; x, y) − x

= 0 uniformly in m, n, p,q→∞ ∞



lim

Dm,n,p,q (t; x, y) − y = 0 uniformly in m, n. p,q→∞

(2.1.2) (2.1.3)

∞



2 2 2 2

lim Dm,n,p,q (s + t ; x, y) − (x + y ) = 0 uniformly in m, n. p,q→∞

(2.1.4)

∞

Proof. Since each 1, x, y, x2 + y 2 belongs to C(I 2 ), conditions (2.1.1)-(2.1.4) follow immediately from (2.1.0). By the continuity of f on I 2 , we can write |f (x, y)| ≤ M, − ∞ < x, y < ∞, where M = ∥f ∥∞ . Therefore, |f (s, t) − f (x, y)| ≤ 2M, − ∞ < s, t, x, y < ∞.

3

(2.1.5)

40

ANASTASSIOU ET AL: APPROXIMATION THEOREMS

Also, since f ∈ C(I 2 ), for every ϵ > 0, there is δ > 0 such that |f (s, t) − f (x, y)| < ϵ, ∀ |s − x| < δ and |t − y| < δ.

(2.1.6)

Using (2.1.5), (2.1.6), putting ψ1 = ψ1 (s, x) = (s − x)2 and ψ2 = ψ2 (t, y) = (t − y)2 , we get |f (s, t) − f (x, y)| < ϵ +

2M (ψ1 + ψ2 ), ∀ |s − x| < δ and |t − y| < δ. δ2

This is, 2M 2M (ψ1 + ψ2 ) < f (s, t) − f (x, y) < ϵ + 2 (ψ1 + ψ2 ). 2 δ δ Now, operating Tj,k (1; x, y) to this inequality since Tj,k (f ; x, y) is monotone and linear. We obtain ( ) ( ) 2M 2M Tj,k (1; x, y) −ϵ− 2 (ψ1 +ψ2 ) < Tj,k (1; x, y)(f (s, t)−f (x, y)) < Tj,k (1; x, y) ϵ+ 2 (ψ1 +ψ2 ) . δ δ −ϵ −

Note that x and y are fixed and so f (x, y) is constant number. Therefore −ϵTj,k (1; x, y)−

2M Tj,k (ψ1 +ψ2 ; x, y) < Tj,k (f ; x, y)−f (x, y)Tj,k (1; x, y) δ2 < ϵTj,k (1; x, y) +

2M Tj,k (ψ1 + ψ2 ; x, y). (2.1.7) δ2

But Tj,k (f ; x, y)−f (x, y) = Tj,k (f ; x, y)−f (x, y)Tj,k (1; x, y)+f (x, y)Tj,k (1; x, y)−f (x, y) = [Tj,k (f ; x, y)−f (x, y)Tj,k (1; x, y)]+f (x, y)[Tj,k (1; x, y)−1]. (2.1.8) Using (2.1.7) and (2.1.8), we have Tj,k (f ; x, y) − f (x, y) < ϵTj,k (1; x, y) +

2M Tj,k (ψ1 + ψ2 ; x, y) + f (x, y)(Tj,k (1; x, y) − 1). δ2 (2.1.9)

Now Tj,k (ψ1 + ψ2 ; x, y) = Tj,k ((s − x)2 + (t − y)2 ; x, y) = Tj,k (s2 − 2sx + x2 + t2 − 2ty + y 2 ; x, y) = Tj,k (s2 + t2 ; x, y) − 2xTj,k (s; x, y) − 2yTj,k (t; x, y) +(x2 + y 2 )Tj,k (1; x, y) = [Tj,k (s2 + t2 ; x, y) − (x2 + y 2 )] − 2x[Tj,k (s; x, y) − x] − 2y[Tj,k (t; x, y) − y] + (x2 + y 2 )[Tj,k (1; x, y) − 1]. 4

ANASTASSIOU ET AL: APPROXIMATION THEOREMS

41

Using (2.1.9), we obtain Tj,k (f ; x, y) − f (x, y) < ϵTj,k (1; x, y) +

2M {[Tj,k (s2 + t2 ; x, y) − (x2 + y 2 )] δ2

− 2x[Tj,k (s; x, y) − x] − 2y[Tj,k (t; x, y) − y] + (x2 + y 2 )[Tj,k (1; x, y) − 1]} + f (x, y)(Tj,k (1; x, y) − 1) 2M {[Tj,k (s2 + t2 ; x, y) − (x2 + y 2 )] δ2 − 2x[Tj,k (s; x, y) − x] − 2y[Tj,k (t; x, y) − y]

= ϵ[Tj,k (1; x, y) − 1] + ϵ +

+ (x2 + y 2 )[Tj,k (1; x, y) − 1]} + f (x, y)(Tj,k (1; x, y) − 1). Since ϵ is arbitrary, we can write Tj,k (f ; x, y) − f (x, y) ≤ ϵ[Tj,k (1; x, y) − 1] +

2M {[Tj,k (s2 + t2 ; x, y) − (x2 + y 2 )] δ2

− 2x[Tj,k (s; x, y) − x] − 2y[Tj,k (t; x, y) − y] + (x2 + y 2 )[Tj,k (1; x, y) − 1]} + f (x, y)(Tj,k (1; x, y) − 1). Similarly, Dm,n,p,q (f ; x, y)−f (x, y) ≤ ϵ[Dm,n,p,q (1; x, y)−1]+

2M {[Dm,n,p,q (s2 +t2 ; x, y)−(x2 +y 2 )] δ2

− 2x[Dm,n,p,q (s; x, y) − x] − 2y[Dm,n,p,q (t; x, y) − y] + (x2 + y 2 )[Dm,n,p,q (1; x, y) − 1]} + f (x, y)(Dm,n,p,q (1; x, y) − 1) and therefore

) ( 2 2



2M (a + b )

Dm,n,p,q (f ; x, y) − f (x, y) ≤ ϵ +

Dm,n,p,q (1; x, y) − 1 + M



δ2 ∞ ∞

4M a Dm,n,p,q (s; x, t) − x − 2

δ ∞

4M b Dm,n,p,q (t; x, y) − y − 2

δ ∞

2M 2 2 2 2

. + 2 D (s + t ; x, y) − (x + y ) m,n,p,q

δ ∞

Letting p, q → ∞ and using (2.1.1), (2.1.2), (2.1.3), (2.1.4), we get



lim Dm,n,p,q (f ; x, y) − f (x, y)

= 0, uniformly in m, n. p,q→∞

∞

5

42

ANASTASSIOU ET AL: APPROXIMATION THEOREMS

This completes the proof of the theorem. In the following we give an example of a double sequence of positive linear operators satisfying the conditions of Theorem 2.1 but does not satisfy the conditions of the Korovkin theorem. Example 2.1. Consider the sequence of classical Bernstein polynomials of two variables )( )( ) ( m ∑ n ∑ m n j j k Bm,n (f ; x, y) := f , x (1 − x)m−j y k (1 − y)n−k ; 0 ≤ x, y ≤ 1. m n j k j=0 k=0 Let Pm,n : C(I 2 ) → C(I 2 ) be defined by Pm,n (f ; x, y) = (1 + zmn )Bm,n (f ; x, y), where (zmn ) is a double sequence defined as above. Then Bm,n (1; x, y) = 1, Bm,n (s; x, y) = x, Bm,n (t; x, y) = y, x − x2 y − y 2 + , m n and a double sequence (Pm,n ) satisfies the conditions (2.1.1), (2.2.2), (2.1.3) and (2.1.4). Hence we have F2 - lim ∥Pm,n (f ; x, y) − f (x, y)∥∞ = 0. Bm,n (s2 + t2 ; x, y) = x2 + y 2 +

m,n→∞

On the other hand, we get Pm,n (f ; 0, 0) = (1+zmn )f (0, 0), since Bm,n (f ; 0, 0) = f (0, 0), and hence ∥Pm,n (f ; x, y) − f (x, y)∥∞ ≥ |Pm,n (f ; 0, 0) − f (0, 0)| = zmn |f (0, 0)|. We see that (Pm,n ) does not satisfy the classical Korovkin theorem, since lim zmn m,n→∞

does not exist. 3. Some consequences Now we present here some consequences of Theorem 2.1. Theorem 3.1. Let (Tm,n ) be a double sequence of positive linear operators on C(I 2 ) such that lim ∥Tm+1,n+1 − Tm,n+1 − Tm+1,n + Tm,n ∥ = 0. (3.1.1) m,n

6

ANASTASSIOU ET AL: APPROXIMATION THEOREMS

43

If F2 - lim ∥Tm,n (tν ; x, y) − tν ∥∞ = 0 (ν = 0, 1, 2, 3), m,n

(3.1.2)

where t0 (x, y) = 1, t1 (x, y) = x, t2 (x, y) = y and t3 (x, y) = x2 + y 2 . Then for any function f ∈ C(I 2 ), we have lim ∥Tm,n (f ; x, y) − f (x, y)∥∞ = 0. m,n

(3.1.3)

Proof. From Theorem 2.1, we have that if (3.1.2) holds then lim ∥Dm,n,p,q (f ; x, y) − f (x, y)∥∞ = 0, uniformly in m, n. p,q

(3.1.4)

We have the following inequality ∥Tm,n (f ; x, y) − f (x, y)∥∞ ≤ ∥Dm,n,p,q (f ; x, y) − f (x, y)∥∞ ) m+p−1 n+q−1 ( j k ∑ ∑ 1 ∑ ∑ ∥Tα,β − Tα−1,β − Tα,β−1 + Tα−1,β−1 ∥ + pq j=m+1 k=n+1 α=m+1 β=n+1 ≤ ∥Dm,n,p,q (f ; x, y) − f (x, y)∥∞ { } p−1q−1 + sup ∥Tj,k − Tj−1,k − Tj,k−1 + Tj−1,k−1 ∥ . 2 2 j≥m,k≥n

(3.1.5)

Hence using (3.1.1) and (3.1.4), we get (3.1.3). This completes the proof of the theorem. We know that double almost convergence implies (C, 1, 1) convergence. This motivates us to further generalize our main result by weakening the hypothesis or to add some condition to get more general result. Theorem 3.2. Let (Tm,n ) be a double sequence of positive linear operators on C(I 2 ) such that (C, 1, 1) − lim ∥Tm,n (tν , x) − tν ∥∞ = 0 (ν = 0, 1, 2, 3), (3.2.1) m,n

and

}

mn

σm+p−1,n+q−1 (f ; x, y) − σm−1,n−1 (f ; x, y) = 0, lim sup

p,q m≥p,n≥q pq ∞ {

where

m

n

∑∑ 1 σm,n (f ; x, y) = Tj,k (f ; x, y). (m + 1)(n + 1) j=0 k=0 7

(3.2.2)

44

ANASTASSIOU ET AL: APPROXIMATION THEOREMS

Then for any function f ∈ C(I 2 ), we have



F2 - lim

Tm,n (f ; x, y) − f (x, y) = 0. m,n→∞

∞

Proof. For m ≥ p ≥ 1; n ≥ q ≥ 1 it is easy to show that Dm,n,p,q (f ; x, y) = σm+p−1,n+q−1 (f ; x, y)+

mn (σm+p−1,n+q−1 (f ; x, y)−σm−1,n−1 (f ; x, y)), pq

which implies



sup D (f ; x, y) − σ (f ; x, y) m,n,p,q m+p−1,n+q−1

m≥p,n≥q

∞



mn

σm+p−1,n+q−1 (f ; x, y) − σm−1,n−1 (f ; x, y)) . (3.2.3) = sup

m≥p,n≥q pq ∞

Also by Theorem 2.1, Condition (3.2.1) implies that



= 0. (C, 1, 1)- lim T (f ; x, y) − f (x, y) m,n

m,n→∞

(3.2.4)

∞

Using (3.2.1)-(3.2.4) and the fact that almost convergence implies (C, 1, 1) convergence, we get the desired result. This completes the proof of the theorem. Theorem 3.3. Let (Tm,n ) be a double sequence of positive linear operators on C(I 2 ) such that s+m−1 t+n−1 1 ∑ ∑ ∥Tm,n − Tj,k ∥ = 0. lim sup m,n s,t mn j=s k=t If F2 - lim ∥Tm,n (tν , x) − tν ∥∞ = 0 (ν = 0, 1, 2, 3). m,n

(3.3.1)

Then for any function f ∈ C(I 2 ), we have lim ∥Tm,n (f ; x, y) − f (x, y)∥∞ = 0. m,n

(3.3.2)

Proof. From Theorem 2.1, we have that if (3.3.1) holds then F2 - lim ∥Tm,n (f ; x, y) − f (x, y)∥∞ = 0, m,n

which is equivalent to lim sup ∥Ds,t,m,n (f ; x, y) − f (x, y)∥∞ = 0. m,n s,t

8

(3.3.3)

ANASTASSIOU ET AL: APPROXIMATION THEOREMS

Now Tm,n − Ds,t,m,n

45

s+m−1 t+n−1 1 ∑ ∑ = Tm,n − Tj,k mn j=s k=t s+m−1 t+n−1 1 ∑ ∑ = (Tm,n − Tj,k ). mn j=s k=t

Therefore sup ∥Tm,n − Ds,t,m,n ∥∞ s,t

s+m−1 t+n−1 1 ∑ ∑ ≤ sup ∥Tm,n − Tj,k ∥. s,t mn j=s k=t

Now, by using the hepothesis we get lim sup ∥Tm,n (f ; x, y) − Ds,t,m,n (f ; x, y)∥∞ = 0. m,n s,t

(3.3.4)

By the triangle inequality, we have ∥Tm,n (f ; x, y)−f (x, y)∥∞ ≤ ∥Tm,n (f ; x, y)−Ds,t,m,n (f ; x, y)∥∞ +∥Ds,t,m,n (f ; x, y)−f (x, y)∥∞ , and hence from (3.3.3) and (3.3.4), we get lim ∥Tm,n (f ; x, y) − f (x, y)∥∞ = 0, m,n

that is (3.3.2) holds. This completes the proof of the theorem.

References [1] S. Banach, Th´eorie des Operations Lineaires, Warszawa, 1932. [2] F. Dirik and K. Demirci, Korovkin type approximation theorem for functions of two variables in statistical sense, Turk. J. Math. 33(2009)1-11. [3] A. D. Gad˘ziev, The convergence problems for a sequence of positive linear operators on unbounded sets, and theorems analogous to that of P.P.Korovkin, Soviet Math. Dokl. 15(1974)1433-1436. [4] G.G. Lorentz, A contribution to theory of divergent sequences, Acta Math. 80(1948)167-190. [5] F. Moricz and B.E. Rhoades, Almost convergence of double sequences and strong regularity of summability matrices, Math. Proc. Camb. Phil. Soc. 104(1988)283294. 9

46

ANASTASSIOU ET AL: APPROXIMATION THEOREMS

[6] M. Mursaleen, Almost strongly regular matrices and a core theorem for double sequences, J. Math. Anal. Appl. 293(2004)523-531. [7] M. Mursaleen and O.H.H. Edely, Almost convergence and a core theorem for double sequences, J. Math. Anal. Appl. 293(2004)532-540. [8] M. Mursaleen and S.A. Mohiuddine, Almost bounded variation of double sequences and some four dimensional summability matrices, Publ. Math. Debrecen 75(2009)495-508. [9] A. Pringsheim, Zur theorie der zweifach unendlichen Zahlenfolgen, Math. Z. 53(1900)289-321. [10] M. Zeltser, M. Mursaleen and S.A. Mohiuddine, On almost conservative matrix methods for double sequence spaces, Publ. Math. Debrecen 75(2009)387-399.

10

JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.1, 47-83, 2011, COPYRIGHT 2011 EUDOXUS PRESS, 47 LLC

Wavelet Transforms of Schwartz Distributions J.N. Pandey School of Mathematics and Statistics Carleton University, Ottawa e-mail: [email protected]

Abstract: Let D be the Schwartz testing function space consisting of infinitely differentiable functions defined on IR with compact supports. Let ψ(t) be the Shannon wavelet defined as ψ(t) = 2φ(2t) − φ(t) where the scaling function φ(t) is defined as φ(t) =

sin πt πt

t 6= 0

=1 n

t = 0. o∞

m

It is well known that the sequence of wavelets 2 2 ψ(2m t − n) set in

L2 (R).

The discrete wavelet transforms of a function f D

f (t), 2

m 2

ψ(2m t

E

− n)

Z

∞

=

m,n=−∞ ∈ L2 (R)

is a complete orthonormal is defined as

m

f (t)2 2 ψ(2m t − n)dt

Z−∞ ∞

= −∞

f (t)ψm,n (t)dt

m

where ψm,n (t) = 2 2 ψ(2m t − n) which exist for m, n ∈ Z. But if f ∈ D0 then hf (t), ψm,n (t)i, m, n ∈ Z is not defined. However for a fixed ω(t) ∈ D the weighted discrete wavelet transform of the distribution f is defined by αm,n = hf (t), ω(t)ψm,n (t)i, m, n ∈ Z, the function ω(t) is termed as a weight function. We prove that 1a.

lim

m→∞

∞ D X

E

E

f (t), ω(t)φm,n (t) φm,n (t), ξ(t)

n=−∞

=

hf (t), ω(t)ξ(t)i

=

hω(t)f (t), ξ(t)i

exists in the weak distributional sense,

1

48

PANDEY: WAVELET AND DISTRIBUTIONS

1b.

∞ X

D m−1 X

lim

m→∞

E

hf (t), ω(t)ψk,n (t)iψk,n (t), ξ(t)

k=−∞ n=−∞

= hf (t), ω(t)ξ(t)i ∀ ξ(t) ∈ D = hω(t)f (t), ξ(t)i ∀ ξ(t) ∈ D

i.e.

lim

∞ X

m−1 X

m→∞

hf (t), ω(t)ψk,n (t)iψk,n (t) = w(t)f (t),

k=−∞ n=−∞

in the weak distributional sense. 2a.

Let f be a tempered distribution defined on R i.e. f ∈ S 0 and let ω(t) be an element of

Schwartz testing function space S then 2a.

lim

∞ D X

m→∞

E

hf (t), ω(t)φm,n (t)iφm,n (t), η(t)

n=−∞

= hf (t), ω(t)η(t)i ∀ η(t) ∈ S = hω(t)f (t), η(t)i ∀ η(t) ∈ S

2b.

m−1 X

lim

m→∞

∞ X

hhf (t), ω(t)ψk,n (t)iψk,n (t), η(t)i

k=−∞ n=−∞

= hf (t), ω(t)η(t)i ∀ η(t) ∈ S = hω(t)f (t), η(t)i ∀ η(t) ∈ S.

3a.

We define a testing function space DL2 (R) (F ) as a subspace of the testing function space

0 DL2 (R) and equip the space with an appropriate topology and show that ∀ f ∈ DL 2 (R) (F )

lim

m→∞

3b.

lim

m→∞

∞ D X

E

hf (t), φm,n (t)iφm,n (t), η(t) = hf (t), η(t)i ∀ η(t) ∈ DL2 (R) (F ).

n=−∞ m−1 X

∞ D X

E

hf (t), ψk,n (t)iψk,n (t), η(t) = hf, ηi ∀ η(t) ∈ DL2 (R) (F ).

k=−∞ n=−∞

4. Let us now assume that ψ(x) is a wavelet satisfying the following two requirements: 2

(A) ψ(x) is continuous and has an exponential decay i.e. ψ(x) ≤ me−cx for some constants c and m. 2

PANDEY: WAVELET AND DISTRIBUTIONS

R∞

(B) The integral of ψ is zero i.e.

−∞ ψ(x)dx

49

= 0.

Then the continuous wavelet transform of f ∈ L2 (IR) is defined to be the function Wf : R2 → R given by

=

x−b )dx a f (ya + b)ψ(y)dy; a, b ∈ IR.

f (x)ψ(

|a|

p

∞

Z

Wf (a, b) = √1

|a|

Z−∞ ∞ −∞

Therefore, we assume that Wf (a, b) = 0 for a = 0. The following inversion formula is well known: 1 f (x) = cψ where cφ = 2π

R∞

−∞

2 ˆ |ψ(λ)| |λ| dλ,

Z

∞

Z

∞

|a|−1/2 ψ

−∞ −∞

x − b

a

Wf (a, b)

dbda a2

ˆ ψ(λ) = (F ψ)(λ).

Now imposing further condition on ψ(x) that it is infinitely differentiable on R the inversion 0 formula is extended to the Schwartz distribution space DL 2 (R) i.e. we can now define the wavelet

transform of f ∈ *

1 cψ

Z

0 DL 2 (R)

∞

Z



as Wf (a, b) = f (x),

√1 ψ( x−b a ) |a|



and prove the inversion formula +

∞

−1/2

|a|

−∞ −∞

x−b dbda ψ( )Wf (a, b) 2 , φ(x) a a

= hf (x), φ(x)i∀φ ∈ DL2 (R) .

Some applications of our results are discussed. 2010, AMS Subject classification: Primary: 42C40, 46F12 Secondary: 42C15, 44A05, 44A55 Key Words and phrases:

Wavelet transforms, generalized integral transform, Schwartz

distributions, orthogonal series expansion of distributions, generalized functions, distribution spaces.

Some Known Definitions and Preliminaries A seminorm γ on a linear space V is a functional satisfying (1)

γ(αφ) = |α|γ(φ), α ∈ C and φ ∈ V

(2)

γ(φ + ψ) ≤ γ(φ) + γ(ψ), φ, ψ ∈ V.

From (1) and (2) one can easily derive that γ(0) = 0, γ(φ) ≥ 0 ∀ φ ∈ V and γ(φ − ψ) ≥ |γ(φ) − γ(ψ)| ∀ φ, ψ ∈ V . 3

50

PANDEY: WAVELET AND DISTRIBUTIONS

If the above mentioned seminorm γ satisfies (3)

γ(φ) = 0 =⇒ φ = 0 then it becomes a norm. One can easily see that if γ1 , γ2 , . . . , γn are seminorms defined on V then (γ1 + γ2 + · · · γn )

which is defined by (γ1 + γ2 + · · · γn )(φ) = γ1 (φ) + γ2 (φ) + · · · + γn (φ) is also a seminorm on V . 4

[Max(γ1 + γ2 + · · · γn )](φ) = Max[γ1 (φ), γ2 (φ), . . . , γn (φ)] is also a seminorm. If a > 0 then the functional aγ defined on V by [aγ](φ) = aγ(φ) ∀ φ ∈ V is a seminorm too on V . Two seminorms γ and ρ defined on V are said to be equivalent if there exist positive numbers a and b satisfying aγ(φ) ≤ ρ(φ) ≤ bγ(φ) ∀ φ ∈ V. One can easily see that the seminorms [γ1 + γ2 + · · · + γn ] and Max[γ1 , γ2 , · · · γn ] are equivalent seminorms. Now let S = {γµ }µ∈A be a collection of seminorms on V where index µ traverses a finite or infinite set A. A collection of seminorms S is said to be separating (or S separates V ) if for every φ 6= ∅ in V there is at least one γµ (φ) 6= 0. In other words S is separating if only the zero element in V has a number zero assigned to it by every seminorm in S. In this case S is called a multinorm. A sufficient condition that a collection of seminorms {γµ }µ∈A be a multinorm is that at least one of γµ may be a norm. Let {γνk }nk=1 be a non-void finite subset of the multinorm S and ε1 , ε2 , . . . εn the arbitrary positive numbers, then we define a balloon centered at ψ where ψ is a fixed point in V as [φ ∈ V : γνk (φ − ψ) ≤ εk ,

k = 1, 2, . . . , n].

Clearly intersection of two balloons centered at ψ is a balloon centered at ψ. A neighbourhood in V is any set in V that contains a balloon and a heighbourhood of ψ ∈ V is any set that contains a balloon centered at ψ. A neighbourhood of origin ∅ is called a neighbourhood of zero. We refer to the collection of all neighbourhoods in V as a topology in V . The collection of neighbourhoods determines and is determined by a collection of open sets [13]. The neighbourhoods of any ψ ∈ V are simply translations through the element ψ of the neighbourhoods of zero. Thus the topology of V is simply the collection of all possible translations of all neighbourhoods of zero. 4

PANDEY: WAVELET AND DISTRIBUTIONS

51

A multinorm space V is a linear space having a topology generated by a multinorm S (i.e. by a separating collection of seminorms). If S is countable V is called a countably multinormed space. A Cauchy sequence in a multinormed space V is a sequence {φν }∞ ν=1 of elements in V such that for every neighbourhood Ω of zero there exists a positive integer N for which φν − φµ is in Ω whenever ν > N , µ > N . This fact is also expressed by saying that φν − φµ is eventually in Ω. We can easily show that a sequence {φν }∞ ν=1 in V is a Cauchy sequence in V if and only if for each γ ∈ S, γ(φν − φµ ) → 0 as ν and µ tend to infinity independently. One can also derive quite readily that a sequence {φν }∞ ν=1 in V tends to φ ∈ V with respect to the topology of V if and only if for each γ ∈ S, γ(φν − φ) → 0 as ν → ∞. When every Cauchy sequence in V is convergent we say that V is a complete countably multinormed space. The word countable here means that S is countable and not the space V . A complete countably multinormed space is called a Fr´echet space. As an example the space DK (Rn ) consisting of C ∞ functions defined in Rn with supports contained in a compact subset K of Rn and equipped with the topology generated by a separating collection of seminorms {γm }∞ m=0 where γm is defined by γm (φ) =

|Ds φ(t)|,

sup

φ ∈ DK

(1)

t∈K

s∈{s; |s|=m}

s = (s1 , s2 , s3 , . . . , sn )

and |s| = s1 + s2 + · · · sn dsi i = (

Ds = D1s1 , D2s2 , . . . Dnsn ;

∂ si ) ∂xi

in a complete countably multinormed space, and therefore DK (Rn ) is a Fr´echet space. The fact that DK (Rn ) is non-empty can be seen from the example given below; we construct C ∞ function ψ on Rn with a compact support {t : −1 ≤ t1 ≤ 1, − 1 ≤ t2 ≤ 1, as follows:

 1  e− 1−|t|2 ψ(t) =  0

for |t| =

− 1 ≤ tn ≤ 1}

q

t21 + t22 + · · · t2n < 1

(2)

for |t| ≥ 1.

For the proof of the sequential completeness of the space DK (Rn ) one can look into references [22, 23, 24]. The fact that the collection of seminorms {γk }∞ |k|=0 is separating follows from the fact that γ0 is a norm.

Countable Union Spaces Let {Vm }∞ m=1 be a sequence of countably multinormed spaces such that V1 ⊂ V2 ⊂ V3 ⊂ · · · . 5

52

PANDEY: WAVELET AND DISTRIBUTIONS

Assume also that the topology of Vm is stronger than the topology induced on it by Vm+1 . Let V =

∞ [

Vm .

m=1

V is clearly a linear space. A sequence {φν }∞ ν=1 in V is said to converge to φ ∈ V if all the φν and φ belong to the same Vm and the sequence φν converges to φ in Vm and therefore in Vm+1 , Vm+2 , Vm+3 . . . as well. The space V under these circumstances is called a countable union space. This space was introduced by Gelfand and Shilove [11, 1923]. A sequence {φν }∞ ν=1 is said to be a Cauchy sequence in V if it is a Cauchy sequence in one of the spaces Vm . When all the Cauchy sequences in V are convergent, V is called a complete countable union space. A countable-union space V =

S∞

m=1 Vm

will be called a strict countable union space if for each m the

topology of Vm is identical with the topology induced on Vm by Vm+1 . This situation arises when a separating collection of seminorms {γk }∞ k=0 under consideration is defined over all the elements of V and the topology over each Vm is generated by these seminorms. We will see that the space D(Rn ) of L. Schwartz contained in C ∞ (Rn ) is a complete strict countable union space. This can be seen as follows. Define γm (φ) = sup |Ds φ(t)| t∈Rn

s ∈ {s; |s| = m}, s = (s1 , s2 , . . . sn ) and |s| = s1 + s2 . . . , +sn The components s1 , s2 , . . . , sn are non-negative integers Ds = D1s1 , D2s2 , . . . Dnsn Disi



=

∂ ∂xi

si

.

n such that K Let {Ki }∞ 1 ⊂ K2 ⊂ K3 . . . and i=1 be a sequence of compact subsets of R

S∞

i=1 Ki

= Rn and that DK1 ⊂ DK2 ⊂ DK3 . . . ∞ [

and

DKi = D(Rn ).

i=1

The topology on every DKi is generated by the sequence of seminorms {γi }∞ i=1 clearly the topology of DKi is the same as the topology induced on DKi by that of DKi+1 . The countable union space D(Rn ) =

S∞

i=1 DKi

is complete as each of the spaces DKi is a complete countably multinormed

space. Thus the space D(Rn ) is a complete strict countably union space and is said to be strict inductive limit of Dki [18, 22].

6

PANDEY: WAVELET AND DISTRIBUTIONS

53

Duals of Countably Multinormed Spaces Space of all continuous linear functional over countably multinormed space DK (Rn ) is denoted 0 (Rn ) and in usual terminology it is a vector space. by DK

If the topology over the space DK (Rn ) is generated by the sequence of seminorms {γk }∞ k=1 this topology over DK (Rn ) can also be described by the sequence of seminorms {ρk }∞ k=1 where ρk (φ) = γ0 (φ) + γ1 (φ) + γ2 (φ) + · · · + γk (φ) ∀ φ ∈ DK (Rn ). Clearly ρ0 (φ) ≤ ρ1 (φ) ≤ ρ2 (φ), ρ0 = γ0 (φ). If f is a continuous linear functional over DK (Rn ) then there exists a constant c > 0 and an integer r ≥ 0 such that |hf, φi| ≤ cρr (φ) ∀ φ ∈ DK (Rn ).

(3)

|hf, φi| ≤ c[γ0 (φ) + γ1 (φ) + · · · γv (φ)].

(4)

or

n If (3) is not true then there exists a sequence of functions {φν }∞ ν=1 in DK (R ) such that

|hf, φν i| > νρν (φν ) But for k < ν, ρk



φν νρν (φν )




1. νρν (φν )

(5)

→ 0 in DK (Rn ) ν → ∞. Therefore letting ν → ∞ in (5)

we get a contradiction 0 ≥ 1, which proves (3). Definition: Convergence in the sense of D(IRn ). n n Let {φν (t)}∞ ν=1 be a sequence in D(IR ). We say that φν (t) → φ in the sense of D(R ) iff

(i) Support of all {φν (t)} is contained in a compact subset of IRn . (k)

n (k) (ii) The sequence {φν (t)}∞ ν=1 converges to φ (t) uniformly on IR for each |k| = 0, 1, 2, . . ..

This definition is a consequence of the convergence in the countable union space D(IRn ). Lemma: ρ : DK (Rn ) × DK (Rn ) → R defined by ρ(φ, ψ) =

∞ X 1 k=0

2k

γk (φ − ψ) 1 + γk (φ − ψ)

(6)

is really a metric. The fact that ρ(φ, ψ) follows triangle inequality can be readily seen. γk (φ − ψ) γk (φ − θ + θ − ψ) γk (φ − θ) + γk (θ − ψ) = ≤ 1 + γk (φ − ψ) 1 + γk (φ − θ + θ − ψ) 1 + γk (φ − θ) + γk (θ − ψ) This follows by virtue of the fact that γk (φ − ψ) 1 + γk (φ − ψ)

x 1+x

is an increasing function of x for x ≥ 0. Hence,

≤

γk (φ − θ) γk (θ − ψ) + . 1 + γk (φ − θ) 1 + γk (θ − ψ) 7

(7)

54

PANDEY: WAVELET AND DISTRIBUTIONS

Therefore, ρ(φ, ψ) ≤ ρ(φ, θ) + ρ(θ, ψ). The other properties of a metric are trivially satisfied. Therefore it can be verified that as ν→∞ φν (t) → φ in DK (Rn ) ⇐⇒ ρ(φν , φ) → 0. But this convergence property is not true for D(Rn ). Verification of the fact is done by the following counter-example given by Gelfand in [11, p.8]. Define φ(x, m) =

 

e



0

−

m2 m2 −x2

for |x| < m, m > 0 for |x| ≥ m.

Define a sequence {ψm,ν }∞ ν=1 by ψm,ν (x) =

1 φ(x, m), ν

m = 1, 2, 3, . . .

Now consider the following subsequences for m = 1, 2, 3, . . . ψ11 ψ12 ψ13 · · · → 0 in D ψ21 ψ22 ψ23 · · · → 0 in D .. . ψm1 ψm2 ψm3 · · · → 0 in D Let D be metrizable by a metric ρ. From the first row above we can choose ψ1,ν1 ν1 large and greater than 1 such that ρ(ψ1,ν1 , 0) < ε. From the second row we can choose ν2 > 2 and large such that ρ(ψ2,ν2 , 0)
0, j = 1, . . . , m. Proof. (i) We observe that f (z1 , . . . , zm ) − Pn,ξ1 ,...,ξm (f )(z1 , . . . , zm ) = f (z1 , . . . , zm ) +

n+1 X

2

1 Qm m

j=1 ξj k=0

(−1)k

µ ¶Z ∞ Z ∞ n+1 ··· f (z1 + kt1 , . . . , zm + ktm ) k −∞ −∞ m Y

·

e−|tj |/ξj dt1 . . . dtm

j=1

=

2

1 Qm m

j=1 ξj

Z

Z

∞

∞

··· −∞

·

m Y

−∞

(−1)n+1 ∆n+1 t1 ,t2 ,...,tm f (z1 , . . . , zm )

e−|tj |/ξj dt1 . . . dtm .

j=1

4

,

88

ANASTASSIOU, GAL: GENERALIZED SINGULAR INTEGRALS

Hence |Pn,ξ1 ,...,ξm (f )(z1 , . . . , zm ) − f (z1 , . . . , zm )| Z ∞ Z ∞ 1 ≤ m Qm ··· ωn+1 (f ; |t1 |, . . . , |tm |)×m Sd j=1 j 2 −∞ j=1 ξj −∞ ·

m Y

e−|tj |/ξj dt1 . . . dtm

j=1

Z

1

= Qm m Y

∞

...

j=1 ξj

·

Z

∞ 0

0

e−tj /ξj dt1 . . . dtm ≤

j=1

µ ¶ t1 tm ωn+1 f ; ξ1 , . . . , ξm ξ1 ξm × m

j=1 Sdj

Z ∞ Z ∞ ωn+1 (f ; ξ1 , . . . , ξm )×m j=1 Sdj Qm ... 0 0 j=1 ξj



n+1 m m Y X t j 1 + e−tj /ξj dt1 . . . dtm ξ j j=1 j=1 ≤

Z ∞ ωn+1 (f ; ξ1 , . . . , ξm )×m Sd Z ∞ j=1 j Qm ··· 0 0 j=1 ξj



 ¶ n+1 Y m µ m Y t j   1+ e−tj /ξj dt1 . . . dtm = ωn+1 (f ; ξ1 , . . . , ξm )×m Sd j=1 j ξ j j=1 j=1 ·

m Y

"

j=1

1 ξj

Z

µ

#

¶n+1

ωn+1 (f ; ξ1 , . . . , ξm )×m Sd j=1 ¢ j dtj = ¡R ∞ n+1 e−u du m (1 + u) 0 0 Ãn+1 µ !m X n + 1¶ = ωn+1 (f ; ξ1 , . . . , ξm )×m Sd k! , j=1 j k ∞

tj 1+ ξj

e

−tj /ξj

k=0

proving the claim. (ii) We notice that f (z1 , . . . , zm ) − Qn,ξ1 ,...,ξm (f )(z1 , . . . , zm ) h =Q m 2 j=1

ξj

Z

1 tan−1

π ξj

Z

π

³ ´i

π

... −π

−π

(−1)n+1 ∆n+1 f (z1,...,zm ) m Qmt1 ,...,t dt1 . . . dtm . 2 2 j=1 (tj + ξj )

Therefore ³ ≤Q m 2 j=1

ξj

|Qn,ξ1 ,...,ξm (f )(z1 , . . . , zm ) − f (z1 , . . . , zm )| Z π Z π 1 |∆t1 ,...,tm f (z1 , . . . , zm )| ³ ´´ Qm 2 ... dt1 . . . dtm 2 π −1 −π −π j=1 (tj + ξj ) tan ξj

5

ANASTASSIOU, GAL: GENERALIZED SINGULAR INTEGRALS

³ ≤Q m 2 j=1

ξj

Z

1 tan−1

Z

π

³ ´´

π

...

π ξj

−π

−π

ωn+1 (f ; |t1 |, . . . , |tm |)×m Sd j=1 j Qm 2 dt1 . . . dtm 2) (t + ξ j j=1 j

³ =Q m 1 j=1

89

ξj

1 tan−1

³ ´´ π ξj

³ ´ t1 tm ω f ; ξ , . . . , ξ n+1 1 ξ1 m ξm π π ×m j=1 Sdj ¢ Qm ¡ 2 dt1 . . . dtm · ... 2 0 −0 j=1 tj + ξj ³ ´n+1 Z π 1 + Pm tj ωn+1 (f ; ξ1 , . . . , ξm )×m Sd Z π j=1 ξj ³ ³ j=1 ´´ j ¢ dt1 . . . dtm Qm ¡ ≤ Q ... m 1 π − tj 2 + ξj2 0 0 tan 1 j=1 j=1 ξj ξj ³ ´n+1  tj Z πY m ωn+1 (f ; ξ1 , . . . , ξm )×m Sd Z π 1 + ξj   ³ ³ j=1 ´´ j ≤ Q ···   dt1 . . . dtm 2 + ξ2 m 1 π −1 t 0 0 j=1 j j tan j=1 ξj ξj Z

Z

 Ã m Y  j=1

= ωn+1 (f ; ξ1 , . . . , ξm )×m Sd j=1 j   ´ ³ n+1 ! Z t π/ξj 1 + j ξj ξj   ¡2 ¢ dtj    −1 π tan ξj tj + ξj2 0 m Y

= ωn+1 (f ; ξ1 , . . . , ξm )×m

j=1 Sdj

"

j=1

ÃZ

1 tan−1

π ξj

π/ξj 0

!# (1 + u)n+1 , du (1 + u2 )

proving the first claim. For the second claim of (ii) we get

³ ≤Q m 2 j=1

ξj

³ =Q m 2 j=1

ξj

|Qn,ξ1 ,...,ξm (f )(z1 , . . . , zm ) − f (z1 , . . . , zm )| ´ ³ Z π Z π Pm |tj |αj (n+1) j=1 K ³ ´´ Qm 2 ... dt1 . . . dtm 2 −π −π j=1 (tj + ξj ) tan−1 π ξj

Z π m Z π X |tj |αj (n+1) ´ ¢ dt1 . . . dtm Qm ¡ 2 ··· 2 0 tan−1 (π/ξj ) j ∗ =1 0 j=1 tj + ξj K

³ =Q m 1 j=1

ξj

m X

K

"ÃZ

π

α

´ 0 tan−1 (π/ξj ) j ∗ =1   m Z π Y dtj  ¡ ¢  · 2  2 t + ξ 0 j j j=1 j6=j ∗

6

∗ (n+1)

t ∗j ¡ j2 ¢ dtj ∗ tj ∗ + ξj2∗

!

90

ANASTASSIOU, GAL: GENERALIZED SINGULAR INTEGRALS

"Ã

m X

=K

j ∗ =1



m

Y · j=1 j6=j ∗

"Ã

m X

=K

ξj tan−1 (π/ξj ) α

ξj ∗j

∗ (n+1)

tan−1

j ∗ =1 m

≤K

"Ã

m X

∗ (n+1)

m X

"Ã

α (n+1)

dtj   (t2j + ξj2 )

π/ξj ∗

∞

! µZ

π ξj

π/ξj

0

!

! uαj∗ (n+1) du (u2 + 1)  du   + 1)

(u2

0

0

ξj j

tan−1

j=1

! µZ

π ξj ∗

tan−1

j ∗ =1

=K

α

ξj ∗j

π

0

Z

∗ (n+1)

tj ∗j dtj ∗ (t2j ∗ + ξj2∗ ) 

0

π ξj ∗

1 −1 tan (π/ξj )

j=1 j6=j ∗

Z

α

π

0

! ÃZ



Y ·

Z

ξj ∗ tan−1 (π/ξj )

# ¶ uαj∗ (n+1) du Πj=1,j6=j ∗ 1 (u2 + 1) ∞

¶# uαj (n+1) du < ∞, (u2 + 1)

proving the second claim of (ii). (iii) Reasons as in the proof of (ii) we get |Q∗n,ξ1 ,...,ξm (f )(z1 , . . . , zm ) − f (z1 , . . . , zm )| ¯ n+1 ¯ ¯∆ +t1 ,...,tm f (z1 , . . . , zm )¯ Qm 2 dt1 . . . dtm ... 2 −∞ −∞ j=1 (tj + ξj ) ´ ³ Qm Z Z ∞ Pm |tj |αj (n+1) K j=1 ξj ∞ j=1 ¢ dt1 . . . dtm Qm Qm ¡ 2 ≤ ··· 2 −∞ −∞ j=1 tj + ξj µ ¶m X µZ ∞ αj (n+1) ¶¸ m · 2 u α (n+1) ≤K ξj j du < ∞, π (u2 + 1) 0 j=1

Qm

j=1 ξj πm

≤

Z

Z

∞

∞

proving the claim. (iv) We have that f (z) − Wn,ξ1 ,...,ξm (f )(z1 , . . . , zm ) = f (z)+

2

1 Qm m

j=1

n+1 X

C(ξj )

(−1)k

k=1

·

m Y

µ ¶Z π Z π n+1 ··· f (z1 +kt1 , . . . , zm +ktm ) k −π −π 2

2

e−tj /ξj dt1 . . . dtm

j=1

7

ANASTASSIOU, GAL: GENERALIZED SINGULAR INTEGRALS

=

Z

1 Qm m

2

Z

π

π

···

j=1 C(ξj )

−π

·

m Y

−π 2

(−1)n+1 ∆n+1 t1 ,...,tm f (z1 , . . . , zm )

2

e−tj /ξj dt1 . . . dtm .

j=1

Therefore reasoning as in the proof of (i) we get |Wn,ξ1 ,...,ξm (f ) (z1 , . . . , zm ) − f (z1 , . . . , zm )| Z π ωn+1 (f ; ξ1 , . . . , ξm )×m Sd Z π j=1 j Qm ··· 0 0 j=1 C (ξj )  n+1 ¶ m µ m Y Y 2 2 tj   1+ e−tj /ξj dt1 . . . dtm ξj j=1 j=1 ≤

³Q =

m j=1 ξj

´

  ¶ n+1 Z π Y m µ ωn+1 (f ; ξ1 , . . . , ξm )×m Sd Z π t j  j=1 j  Qm 1+ ··· ξ C (ξ ) j j 0 0 j=1 j=1 ·

³Q

j=1

´

j=1

³Q ≤

2

2

e−tj /ξj

dt1 dtm ... ξ1 ξm

Z π/ξm ωn+1 (f ; ξ1 , . . . , ξm )×m Sd Z π/ξ1 j=1 j Qm ··· 0 0 j=1 C (ξj )  n+1 m m Y Y 2  (1 + uj ) e−uj du1 . . . dum

m j=1 ξj

=

m Y

j=1

´

Z ∞ ωn+1 (f ; ξ1 , . . . , ξm )×m Sd Z ∞ j=1 j Qm ··· 0 0 j=1 C (ξj )  n+1 m m Y Y 2  (1 + u) e−uj du1 . . . dum

m j=1 ξj

j=1

j=1

¶m ωn+1 (f ; ξ1 , . . . , ξm )×m Sd µZ ∞ j=1 j n+1 −u2 ¡R π ¢m ≤ (1 + u) e du , e−u2 du 0 0 proving the claim. (v) We observe that ∗ f (z1 , . . . , zm ) − Wn,ξ (f )(z1 , . . . , zm ) = f (z1 , . . . , zm ) 1 ,...,ξm

+ Qm

j=1

µ ¶Z ∞ Z ∞ n+1 (−1) ··· f (z1 + kt1 , . . . , zm + ktm ) k −∞ −∞

n+1 X

1 p

ξj π

k=1

k

8

91

92

ANASTASSIOU, GAL: GENERALIZED SINGULAR INTEGRALS

· = Qm

1 p

j=1

Z ξj π

m Y j=1

∞

2

e−tj /ξj dt1 . . . dtm Z

∞

··· −∞

·

−∞ m Y

(−1)n+1 ∆n+1 t1 ,t2 ,...,tm f (z1 , . . . , zm )

2

e−tj /ξj dt1 . . . dtm .

j=1

Hence 1 ≤ Qm p j=1

ξj π

∗ |Wn,ξ (f )(z1 , . . . , zm ) − f (z1 , . . . , zm )| 1 ,...,ξm Z ∞ Z ∞ m Y 2 ... ωn+1 (f ; |t1 |, . . . , |tm |)×m Sd e−tj /ξj dt1 . . . dtm −∞

j=1

−∞

j

j=1

µ ¶ p t1 p 2 tm = Qm p ··· ωn+1 f ; ξ1 √ , . . . , ξm √ ξ1 ξ m × m Sd ξj π −∞ −∞ j=1 j=1 j ¡ ¢ √ √ m Z ∞ m 2 ωn+1 f ; ξ1 , . . . , ξm ×m S Z ∞ Y 2 j=1 dj · e−tj /ξj dt1 . . . dtm ≤ ··· Qm p ξj π 0 0 j=1 j=1 m

Z

∞

Z

∞



n+1 m m Y X 2 t j 1 + p  etj /ξj dt1 . . . dtm ξj j=1 j=1 ¡ ¢ √ √ Z ∞ 2m ωn+1 f ; ξ1 , . . . , ξm ×m S Z ∞ j=1 dj ≤ ··· Qm p ξj π 0 0 j=1 n+1  ! Ã m m Y Y 2 tj   e−tj /ξj dt1 . . . dtm 1+ p ξj j=1 j=1 ³ p p ´ = 2m ωn+1 f ; ξ1 , . . . , ξm m m Y



 p1 πξj j=1 µ =

2 √ π

¶m

×j=1 Sdj

Z

∞ 0

Ã

tj 1+ p ξj

³ p p ´ ωn+1 f ; ξ1 , . . . , ξm

!n+1 2

e−tj /ξj dtj  µZ

×m j=1 Sdj



∞

n+1 −u2

(1 + u)

e

¶m du ,

proving the claim. ¤ In the next, for Pn,ξ1 ,...,ξm (f ; (z1 , . . . , zm ) and Wn,ξ1 ,...,ξm (f ; z1 , . . . , zm ) we will consider the weighted approximation on ×m j=1 Sdj , which seems to be more m natural because ×m j=1 Sdj is unbounded in C . For this purpose, first we need some general notations. Let w : ×m j=1 Sdj → R+ be a continuous weighted functions in ×m j=1 Sdj , with the properties that 9

ANASTASSIOU, GAL: GENERALIZED SINGULAR INTEGRALS

w(z1 , . . . , zm ) > 0 for all (z1 , . . . , zm ) ∈ ×m j=1 Sdj and Define the space

lim

|zj |→0, j=1,...,m

w(z1 , . . . , zm ) = 0.

¡ ¢ Cw × m j=1 Sdj

© ª m = f : ×m j=1 Sdj → C; f is continuous in ×j=1 Sdj and kf kw < ∞ , where © ª kf kw := sup w(z1 , . . . , zm )|f (z1 , . . . , zm )|; (z1 , . . . , zm ) ∈ ×m j=1 Sdj . ¡ ¢ Also, for f ∈ Cw ×m j=1 Sdj define the weighted modulus of smoothness ωn+1,w (f ; t1 , . . . , tm )×m

j=1 Sdj

=

sup |hj |≤tj , j=1,...,m

¯ ¯ n n ¯ ¯ sup w(z1 , . . . , zm ) ¯∆n+1 f (z , . . . , z ) 1 m ¯ : h1 ,...,h,m

ªª zj , zj + hj , . . . , zj + (n + 1)hj ∈ Sdj , j = 1, . . . , m , Remark. This modulus of smoothness has the properties: a) it is increasing as function of tj , j = 1, . . . , m; b) ωn+1,w (f ; 0, . . . , 0)×m Sd = 0; j=1

c)

j

ωn+1,w (f ; λ1 t1 , . . . , λm tm )×m

j=1 Sdj

 ≤ 1 +

m X

n+1 λj 

ωn+1,w (f ; t1 , . . . , tm )×m

j=1 Sdj

j=1

,

for all λj , tj ≥ 0, j = 1, . . . , m. We present Theorem 2. Let dj > 0, j = 1, . . . , m, and suppose that f : ×m j=1 Sdj → C is continuous in the polystrip. Let the Freud-type multiweight w(z 1¢, . . . , zm ) = ¡ m Qm −qj |zj | e with q > 0, j = 1, . . . , m, fixed and f ∈ C × S j w j=1 dj . j=1 Then (i) Ãn+1 µ !m X n + 1¶ kPn,ξ1 ,...,ξm (f ) − f kw ≤ k! ωn+1,w (f ; ξ1 , . . . , ξm )×m Sd , j=1 j k k=0

for 0 < ξj < (ii)

1 (n+1)qj ,

j = 1, . . . , m, and

∗ kWn,ξ (f ) − f kw ¶m µ Z ∞ ³ p p ´ 2 2 (1 + u)n+1 e−u du , ωn+1,w f ; ξ1 , . . . , ξm m ≤ √ π 0 ×j=1 Sdj

for all 0 < ξj ≤ 1, j = 1, . . . , m. 10

93

94

ANASTASSIOU, GAL: GENERALIZED SINGULAR INTEGRALS

Proof. The continuity of f in ×m j=1 Sdj implies the continuity of the gener∗ alized singular integrals Pn,ξ1 ,...,ξm (f ) and Wn,ξ (f ) in ×m j=1 Sdj . 1 ,...,ξm (i) We have |w(z1 , . . . , zm ) · Pn,ξ1 ,...,ξm (f )(z1 , . . . , zm )| ¯ µ ¶Z ∞ Z ∞ n+1 ¯ X 1 ¯ k n+1 Q =¯ m m (−1) ··· ¯2 k −∞ −∞ j=1 ξj k=1

w(z1 + kt1 , . . . , zm + ktm )f (z1 + kt1 , . . . , ktm )

¯ ¯ m Y ¯ w(z1 , . . . , zm ) −|tj |/ξj · e dt1 . . . dtm ¯¯ w(z1 + kt1 , . . . , zm + ktm ) j=1 ¯   ≤ kf kw

n+1 Xµ

1 Qm m

2

j=1 ξj k=1

n+1 k

¶Z

Z

∞

··· −∞

∞

m Y

e

´ ³ −|tj | kqj − ξ1 j

  dt1 . . . dtm

−∞ j=1



≤ Cξ1 ,...,ξm ; n;q1 ,...,gm kf kw < ∞. Passing to the supremum over (z1 , . . . , zm ) ∈ ×m j=1 Sdj , it follows that ¡ ¢ Pn,ξ1 ,...,ξm (f ) ∈ Cw ×m S , for all 0 < ξ < (n + 1)q , j = 1, . . . , m. d j j j j=1 Also, for all zj ∈ Sdj , j = 1, . . . , m, we obtain w(z1 , . . . , zm )[f (z1 , . . . , zm ) − Pn,ξ1 ,...,ξm (f )(z1 , . . . , zm )] Z ∞ Z ∞ 1 = m Qm ··· (−1)n+1 w(z1 , . . . , zm )∆n+1 t1 ,...,tm f (z1 , . . . , zm ) 2 ξ −∞ j=1 j −∞ m Y

·

e−|tj |/ξj dt1 . . . dtm ,

j=1

which gives us w(z1 , . . . , zm ) |Pn,ξ1 ,...,ξm (f )(z1 , . . . , zm ) − f (z1 , . . . , zm )| Z ∞ Z ∞ ¯ ¯ 1 ¯ Q ≤ m m ··· w(z1 , . . . , zm ) ¯∆n+1 t1 ,...,tm f (z1 , . . . , zm ) 2 −∞ j=1 ξj −∞ m Y

· ≤

2

1 Qm m

j=1 ξj

Z

j=1 ∞

e−|tj |/ξj dt1 . . . dtm Z

∞

··· −∞

·

−∞ m Y

ωn+1,w (f ; |t1 |, . . . , |tm |)×m

j=1 Sdj

e−|tj |/ξj dt1 . . . dtm

j=1

11

ANASTASSIOU, GAL: GENERALIZED SINGULAR INTEGRALS

Z

1 = Qm

j=1 ξj

Z

∞

∞

··· 0

0

·

95

µ ¶ t1 tm ωn+1,w f ; ξ1 , . . . , ξm ξ1 ξm × m

j=1 Sdj

m Y

e−tj /ξj dt1 . . . dtm ,

j=1

 n+1 Z ∞ m ωn+1,w (f ; ξ1 , . . . , ξm )×m Sd Z ∞ X tj  j=1 j 1 + Qm ≤ ··· ξ ξ 0 0 j=1 j j=1 j ·

m Y

e−tj /ξj dt1 . . . dtm

j=1

  ¶ n+1 Z ∞ Y m µ ωn+1,w (f ; ξ1 , . . . , ξm )×m Sd Z ∞ t j=1 j  j  Qm ··· ≤ 1+ ξ ξ j j 0 0 j=1 j=1 ·

m Y

e−tj /ξj dt1 . . . dtm

j=1

= ωn+1,w (f ; ξ1 , . . . , ξm )×m

j=1 Sdj

m Y

"

j=1

1 ξj

Z

∞ 0

µZ = ωn+1,w (f ; ξ1 , . . . , ξm )×m

j=1 Sdj

∞

# µ ¶n+1 tj −tj /ξj 1+ e dtj ξj ¶m

(1 + u)n+1 e−u du

0

Ãn+1 µ !m X n + 1¶ = k! ωn+1,w (f ; ξ1 , . . . , ξm )×m Sd , j=1 j k k=0

proving the claim. (ii) We have ¯ ∗ ¯w(z1 , . . . , zm )Wn,ξ

1 ,...,ξm

¯ (f )(z1 , . . . , zm )¯

¯ µ ¶Z ∞ Z ∞ n+1 ¯ X 1 ¯ k n+1 p (−1) ··· = ¯ Qm ¯ j=1 ξj π k −∞ −∞ k=1

ω(z1 + kt1 , . . . , zm + ktm )f (z1 + kt1 + kt1 , . . . , zm + ktm ) ¯ ¯ m Y ¯ 2 w(z1 , . . . , zm ) −|tj |/ξj · e dt1 . . . dtm ¯¯ w(z1 + kt1 , . . . , zm + ktm ) j=1 ¯ ( Z ∞ n+1 X µn + 1¶ Z ∞ 1 ··· ≤ kf kw Qm p k ξj π −∞ −∞ j=1 k=1

12

96

ANASTASSIOU, GAL: GENERALIZED SINGULAR INTEGRALS

m Y

e

 

³ ´ |t | −|tj | kqj − ξj

dt1 . . . dtm

j

j=1

≤ Cξ∗1 ,...,ξm ;n; q1 ,...,gm kf kw < ∞.



Passing to the supremum over (z1 , . . . , zm ) ∈ ×m j=1 Sdj , it follows that ¢ ¡ m ∗ Wn,ξ1 ,...,ξm (f ) ∈ Cw ×j=1 Sdj , for all 0 < ξj ≤ 1 j = 1, . . . , m. Notice here that ´ ´ Z ∞ ³ Z kqj +1 ³ t t tj kqj − ξj tj kqj − ξj j dt = j dt e e j j 0

Z

0 ´ ³ t tj kqj − ξj

∞

+

e

j

dtj ,

kqj +1

³ and for 0 < ξj ≤ 1 and tj ≥ kqj + 1 we get tj kqj − e

³ ´ t tj kqj − ξj j

≤ e−tj , which implies ³ ´ Z ∞ Z t tj kqj − ξj j dt ≤ e j

∞

tj ξj

´ ≤ −tj

and

e−tj dtj = e−(kqj +1) .

kqj +1

kqj +1

So that

Z

∞

e

³ ´ t tj kqj − ξj j

dtj < ∞.

0

Also, for all zj ∈ Sdj , j = 1, . . . , m, we obtain ∗ w(z1 , . . . , zm )[f (z1 , . . . , zm ) − Wn,ξ (f )(z1 , . . . , zm )] 1 ,...,ξm Z ∞ Z ∞ 1 = m Qm p ··· (−1)n+1 w(z1 , . . . , zm )∆n+1 t1 ,...,tm f (z1 , . . . , zm ) 2 ξj π −∞ −∞ j=1 m Y

·

2

e−tj /ξj dt1 . . . dtm ,

j=1

which gives us ¯ ∗ ¯ w(z1 , . . . , zm ) ¯Wn,ξ (f )(z1 , . . . , zm ) − f (z1 , . . . , zm )¯ 1 ,...,ξm Z ∞ Z ∞ ¯ n+1 ¯ 1 ¯∆t ,...,t f (z1 , . . . , zm )¯ ··· ≤ Qm p 1 m ξj π −∞ −∞ j=1 · ≤ Qm

1 p

Z

j=1

ξj π

m Y j=1

∞

2

e−tj /ξj dt1 . . . dtm Z

∞

··· −∞

·

−∞ m Y

ωn+1,w (f ; |t1 |, . . . , |tm |)×m

j=1 Sdj

2

e−tj /ξj dt1 . . . dtm

j=1

13

ANASTASSIOU, GAL: GENERALIZED SINGULAR INTEGRALS

Z

2m = Qm p j=1

·

m Y

ξj π

µ ¶ p t1 p tm ωn+1,w f ; ξ1 √ , . . . , ξm √ ξ1 ξm ×m Sd 0 j=1 j ¡ ¢ √ √ m Z Z 2 ωn+1,w f ; ξ1 , . . . , ξm ×m S ∞ ∞ j=1 dj ≤ ··· Qm p ξj π 0 0 j=1 Z

∞

97

∞

··· 0

2

e−tj /ξj dt1 . . . dtm

j=1



n+1 m m X Y 2 t 1 + pj  e−tj /ξj dt1 . . . dtm ξ j j=1 j=1 ¡ √ √ ¢ Z ∞ 2m ωn+1,w f ; ξ1 , . . . , ξm ×m S Z ∞ j=1 dj ≤ ··· Qm p ξj π 0 0 j=1   Ã ! n+1 m m Y Y 2 t j   1+ p e−tj /ξj dt1 . . . dtm ξ j j=1 j=1 µ

¶m ³ p p ´ 2 = √ ωn+1,w f ; ξ1 , . . . , ξm m π ×j=1 Sdj   !n+1 Z ∞à m Y 2 tj  p1 1+ p e−tj /ξj dtj  · ξ ξ 0 j j j=1 µ =

2 √ π

¶m

³ ωn+1,w f ;

p

ξ1 , . . . ,

p

µZ

´ ξm

proving the claim.

×m j=1 Sdj

∞

n+1 −u2

(1 + u)

e

¶m du

,

0

¤

References [1] G. A. Anastassiou and S. G. Gal, Convergence of generalized singular integrals to the unit, multivariate case, in : Applied Mathematics Reviews (G. A. Anastassiou ed.) , vol. 1, World Scientific, Singapore, New Jersey, London, Hong Kong, 2000, pp. 1-8. [2] G. A. Anastassiou and S. G. Gal, Quantitative estimates in the overconvergence of some generalized singular integrals, submitted for publication. [3] G. A. Anastassiou and S. G. Gal, Quantitative estimates in the overconvergence of some singular integrals, Communications in Applied Analysis, 14(2010), No. 1, 13-20. [4] S.G. Gal, Degree of approximation of continuous functions by some singular integrals, Revue d’Analyse Num´er. Th´eor. l’Approx. (Cluj), XXVII(1998), No. 2, 251-261.

14

JOURNAL 98 OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.1, 98-107, 2011, COPYRIGHT 2011 EUDOXUS PRESS, LLC

On special differential subordinations using a generalized Sa˘la˘gean operator and Ruscheweyh derivative Alina Alb Lupa¸s Department of Mathematics and Computer Science University of Oradea str. Universitatii nr. 1, 410087 Oradea, Romania [email protected]

Abstract In the present paper we define a new operator using the generalized Sa˘la˘gean and Ruscheweyh n n n the operator given by RDλ,α : A → A, RDλ,α f (z) = (1 − operators. Denote by RDλ,α n n n α)R f (z) + αDλ f (z), for z ∈ U, where R f (z) denote the Ruscheweyh derivative, Dλn f (z) is the generalized S˘ al˘ agean operator and An = {f ∈ H(U ) : f (z) = z + an+1 z n+1 + . . . , z ∈ U } is the class of normalized analytic functions with A1 = A. A certain subclass, denoted by RDn (δ, λ, α) , of analytic functions in the open unit disc is introduced by means of the new operator. By making use of the concept of differential subordination we will derive various properties and characteristics of the class RDn (δ, λ, α) . Also, several differential subordinations are n . established regardind the operator RDλ,α

Keywords: differential subordination, convex function, best dominant, differential operator, generalized Sa˘la˘gean operator, Ruscheweyh derivative. 2000 Mathematical Subject Classification: 30C45, 30A20, 34A40.

1

Introduction

Denote by U the unit disc of the complex plane, U = {z ∈ C : |z| < 1} and H(U ) the space of holomorphic functions in U . Let An = {f ∈ H(U ) : f (z) = z + an+1 z n+1 + . . . , z ∈ U } with A1 = A and H[a, n] = {f ∈ H(U ) : f (z) = a + annz n + an+1 z n+1 + . . . , z ∈ U } for oa ∈ C and n ∈ N. 00 (z) + 1 > 0, z ∈ U , the class of normalized convex functions Denote by K = f ∈ A : Re zff 0 (z) in U . If f and g are analytic functions in U , we say that f is subordinate to g, written f ≺ g, if there is a function w analytic in U , with w(0) = 0, |w(z)| < 1, for all z ∈ U such that f(z) = g(w(z)) for all z ∈ U . If g is univalent, then f ≺ g if and only if f (0) = g(0) and f (U ) ⊆ g(U ). Let ψ : C3 × U → C and h an univalent function in U . If p is analytic in U and satisfies the (second-order) differential subordination ψ(p(z), zp0 (z), z 2 p00 (z); z) ≺ h(z),

for z ∈ U,

(1.1)

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 (1.1). A dominant qe that satisfies qe ≺ q for all dominants q of (1.1) is said to be the best dominant of (1.1). The best dominant is unique up to a rotation of U . 1

LUPAS: DIFFERENTIAL SUBORDINATIONS

99

Definition 1.1 (Al Oboudi [2]) For f ∈ A, λ ≥ 0 and n ∈ N, the operator Dλn is defined by Dλn : A → A, Dλ0 f (z) = f (z) Dλ1 f (z) = (1 − λ) f (z) + λzf 0 (z) = Dλ f (z) ...

= (1 − λ) Dλn f (z) + λz (Dλn f (z))0 = Dλ (Dλn f (z)) , for z ∈ U. P P∞ n j n j Remark 1.1 If f ∈ A and f(z) = z + ∞ j=2 aj z , then Dλ f (z) = z + j=2 [1 + (j − 1) λ] aj z , for z ∈ U . Dλn+1 f(z)

Remark 1.2 For λ = 1 in the above definition we obtain the Sa ˘la˘gean differential operator [5]. Definition 1.2 (Ruscheweyh [4]) For f ∈ A, n ∈ N, the operator Rn is defined by Rn : A → A, R0 f (z) = f (z) R1 f (z) = zf 0 (z) ... f (z) = z (Rn f (z))0 + nRn f (z) , for z ∈ U. P P∞ j n n j Remark 1.3 If f ∈ A, f (z) = z + ∞ j=2 aj z , then R f (z) = z + j=2 Cn+j−1 aj z , for z ∈ U . (n + 1) R

n+1

Lemma 1.1 (Hallenbeck and Ruscheweyh [3, Th. 3.1.6, p. 71]) Let h be a convex function with h(0) = a, and let γ ∈ C\{0} be a complex number with Re γ ≥ 0. If p ∈ H[a, n] and p(z) +

1 0 zp (z) ≺ h(z), γ

for z ∈ U,

then where g(z) =

γ nz γ/n

Rz 0

p(z) ≺ g(z) ≺ h(z),

h(t)tγ/n−1 dt, for z ∈ U.

for z ∈ U,

Lemma 1.2 (Miller and Mocanu [3]) Let g be a convex function in U and let h(z) = g(z)+nαzg 0 (z), for z ∈ U, where α > 0 and n is a positive integer. If p(z) = g(0) + pn z n + pn+1 z n+1 + . . . , for z ∈ U, is holomorphic in U and p(z) + αzp0 (z) ≺ h(z),

for z ∈ U,

then p(z) ≺ g(z),

for z ∈ U,

and this result is sharp.

2

Main results

n the operator given by RDn : A → A, Definition 2.1 Let α, λ ≥ 0, n ∈ N. Denote by RDλ,α λ,α n f (z) = (1 − α)Rn f (z) + αDλn f (z), RDλ,α

for z ∈ U.

P j Remark 2.1 If f ∈ A, f (z) = z + ∞ j=2 aj z , then n o P∞ n n f (z) = z + n j + (1 − α) C α [1 + (j − 1) λ] RDλ,α n+j−1 aj z , for z ∈ U. j=2 2

100

LUPAS: DIFFERENTIAL SUBORDINATIONS

n f (z) = Rn f (z), where z ∈ U and for α = 1, RD n f (z) = D n f (z), Remark 2.2 For α = 0, RDλ,0 λ,1 λ where z ∈ U. n f (z) = Ln f (z) which was studied in [1]. For λ = 1, we obtain RD1,α α 0 f (z) = (1 − α) R0 f (z) + αD0 f (z) = f (z) = R0 f (z) = D 0 f (z), where z ∈ U. For n = 0, RDλ,α λ λ

Definition 2.2 Let δ ∈ [0, 1), α, λ ≥ 0 and n ∈ N. A function f ∈ A is said to be in the class RDn (δ, λ, α) if it satisfies the inequality ¢0 ¡ n f (z) > δ, for z ∈ U. (2.1) Re RDλ,α

Theorem 2.1 The set RDn (δ, λ, α) is convex. Proof. Let the functions fj (z) = z +

∞ P

ajk z j ,

for k = 1, 2,

j=2

z ∈ U,

be in the class RDn (δ, λ, α). It is sufficient to show that the function h (z) = η 1 f1 (z) + η 2 f2 (z) is in the class RDn (δ,Pλ, α) , with η 1 and η2 nonnegative such that η 1 + η 2 = 1. j Since h (z) = z + ∞ j=2 (η 1 aj1 + η 2 aj2 ) z , for z ∈ U, then n RDλ,α h (z) = z +

∞ © ª P n α [1 + (j − 1) λ]n + (1 − α) Cn+j−1 (η 1 aj1 + η 2 aj2 ) z j , for z ∈ U. (2.2)

j=2

Differentiating obtain ³ ´0 (2.2) we o P∞ n n n RDλ,α h (z) = 1 + j=2 α [1 + (j − 1) λ]n + (1 − α) Cn+j−1 (η 1 aj1 + η 2 aj2 ) jz j−1 , for z ∈ U. Hence à ! ∞ © ¡ ª ¢0 P n n n j−1 Re RDλ,α h (z) = 1 + Re η 1 j α [1 + (j − 1) λ] + (1 − α) Cn+j−1 aj1 z (2.3) Ã

j=2

! ∞ © ª P n aj2 z j−1 . j α [1 + (j − 1) λ]n + (1 − α) Cn+j−1 + Re η2 j=2

Taking into account that f1 , f2 ∈ RDn (δ, λ, α) we deduce à ! ∞ © ª P n n j α [1 + (j − 1) λ] + (1 − α) Cn+j−1 Re η k ajk z j−1 > η k (δ − 1) ,

for k = 1, 2.

(2.4)

j=2

Using (2.4) we get from (2.3) ¡ ¢0 n Re RDλ,α h (z) > 1 + η 1 (δ − 1) + η2 (δ − 1) = δ,

for z ∈ U,

which is equivalent that RDn (δ, λ, α) is convex.

1 Theorem 2.2 Let g be a convex function in U and let h (z) = g (z) + c+2 zg 0 (z) , where z ∈ U, c > 0. Rz c If f ∈ RDn (δ, λ, α) and F (z) = Ic (f ) (z) = zc+2 c+1 0 t f (t) dt, for z ∈ U, then ¢0 ¡ n f (z) ≺ h (z) , for z ∈ U, (2.5) RDλ,α

implies

and this result is sharp.

¡ ¢0 n RDλ,α F (z) ≺ g (z) , 3

for z ∈ U,

LUPAS: DIFFERENTIAL SUBORDINATIONS

Proof. We obtain that z

c+1

F (z) = (c + 2)

Z

z

101

tc f (t) dt.

(2.6)

0

Differentiating (2.6), with respect to z, we have (c + 1) F (z) + zF 0 (z) = (c + 2) f (z) and ¡ ¢0 n n n F (z) + z RDλ,α F (z) = (c + 2) RDλ,α f (z) , for z ∈ U. (c + 1) RDλ,α

Differentiating (2.7) we have ¡ ¢0 ¢00 ¡ ¢0 ¡ 1 n n n RDλ,α F (z) + F (z) = RDλ,α f (z) , for z ∈ U. z RDλ,α c+2 Using (2.8), the differential subordination (2.5) becomes ¡ ¢0 ¢00 ¡ 1 1 n n z RDλ,α zg 0 (z) . F (z) + F (z) ≺ g (z) + RDλ,α c+2 c+2 If we denote ¡ ¢0 n p (z) = RDλ,α F (z) , for z ∈ U,

(2.7)

(2.8)

(2.9) (2.10)

then p ∈ H [1, 1] . Replacing (2.10) in (2.9) we obtain

1 1 zp0 (z) ≺ g (z) + zg 0 (z) , for z ∈ U. c+2 c+2 Using Lemma 1.2 we have ¢0 ¡ n F (z) ≺ g (z) , for z ∈ U, p (z) ≺ g (z) , for z ∈ U, i.e. RDλ,α p (z) +

and g is the best dominant. ¡ ¢ Example 2.1 If f ∈ RD1 1, 1, 12 , then f 0 (z) + zf 00 (z) ≺ Rz where F (z) = z32 0 tf (t) dt.

3−2z 3(1−z)2

implies F 0 (z) + zF 00 (z) ≺

, where δ ∈ [0, 1) and c > 0. Theorem 2.3 Let h (z) = 1+(2δ−1)z 1+z Rz c If α, λ ≥ 0, n ∈ N and Ic (f ) (z) = zc+2 c+1 0 t f (t) dt, for z ∈ U, then

Ic [RDn (δ, λ, α)] ⊂ RDn (δ ∗ , λ, α) , R 1 c+1 where δ ∗ = 2δ − 1 + (c + 2) (2 − 2δ) 0 tt+1 dt.

1 1−z ,

(2.11)

Proof. The function h is convex and using the same steps as in the proof of Theorem 2.2 we get from the hypothesis of Theorem 2.3 that p (z) +

1 zp0 (z) ≺ h (z) , c+2

where p (z) is defined in (2.10). Using Lemma 1.1 we deduce that p (z) ≺ g (z) ≺ h (z) , where

i.e.

¡

¢0 n F (z) ≺ g (z) ≺ h (z) , RDλ,α

Z Z (c + 2) (2 − 2δ) z tc+1 c + 2 z c+1 1 + (2δ − 1) t t dt = 2δ − 1 + dt. z c+2 0 1+t z c+2 0 t+1 Since g is convex and g (U ) is symmetric with respect to the real axis, we deduce Z 1 c+1 ¢0 ¡ t n ∗ dt. Re RDλ,α F (z) ≥ min Re g (z) = Re g (1) = δ = 2δ − 1 + (c + 2) (2 − 2δ) |z|=1 0 t+1 g (z) =

From (2.12) we deduce inclusion (2.11).

4

(2.12)

102

LUPAS: DIFFERENTIAL SUBORDINATIONS

Theorem 2.4 Let g be a convex function, g(0) = 1 and let h be the function h(z) = g(z) + zg 0 (z), for z ∈ U. If α, λ ≥ 0, n ∈ N, f ∈ A and satisfies the differential subordination ¡ ¢0 n RDλ,α f (z) ≺ h(z), for z ∈ U, (2.13) then

n f (z) RDλ,α

z

≺ g(z),

for z ∈ U,

and this result is sharp. n , we have Proof. By using the properties of operator RDλ,α n f (z) = z + RDλ,α

∞ X © ª n α [1 + (j − 1) λ]n + (1 − α) Cn+j−1 aj z j , j=2

Consider p(z) =

n f (z) RDλ,α z

=

U.

P n n j z+ ∞ j=2 {α[1+(j−1)λ] +(1−α)Cn+j−1 }aj z z

for z ∈ U.

= 1+p1 z+p2 z 2 +..., for z ∈

We deduce that p ∈ H[1, 1].

³ ´0 n f (z) = zp(z), for z ∈ U. Differentiating we obtain RDn f (z) Let RDλ,α = p(z) + zp0 (z), for λ,α z ∈ U. Then (2.13) becomes p(z) + zp0 (z) ≺ h(z) = g(z) + zg 0 (z),

for z ∈ U.

By using Lemma 1.2, we have p(z) ≺ g(z),

for z ∈ U,

n f (z) RDλ,α

i.e.

z

≺ g(z),

for z ∈ U.

Theorem 2.5 Let h be an holomorphic function which satisfies the inequality Re − 12 ,

for z ∈ U, and h(0) = 1. If α, λ ≥ 0, n ∈ N, f ∈ A and satisfies the differential subordination ¡ ¢0 n RDλ,α f (z) ≺ h(z), for z ∈ U,

then

n f (z) RDλ,α

where q(z) =

1 z

Proof. Let

Rz 0

z

≺ q(z),

³ 1+

z+

for z ∈ U,

o P∞ n n n j + (1 − α) C α [1 + (j − 1) λ] n+j−1 aj z j=2

= z z ∞ ∞ X X © ª n pj z j−1 , = 1+ α [1 + (j − 1) λ]n + (1 − α) Cn+j−1 aj z j−1 = 1 +

p(z) =

j=2

j=2

5

´

>

(2.14)

h(t)dt. The function q is convex and it is the best dominant.

n f (z) RDλ,α

zh00 (z) h0 (z)

LUPAS: DIFFERENTIAL SUBORDINATIONS

103

for z ∈ U, p ∈ H[1, 1].

´0 ³ n f(z) = p(z) + zp0 (z), for z ∈ U, and (2.14) becomes Differentiating, we obtain RDλ,α p(z) + zp0 (z) ≺ h(z),

for z ∈ U.

Using Lemma 1.1, we have p(z) ≺ q(z), for z ∈ U,

i.e.

n f (z) RDλ,α

z

≺ q(z) =

1 z

and q is the best dominant.

Z

z

0

h(t)dt, for z ∈ U,

Theorem 2.6 Let g be a convex function such that g (0) = 1 and let h be the function h (z) = g (z) + zg 0 (z), for z ∈ U . If α, λ ≥ 0, n ∈ N, f ∈ A and the differential subordination !0 Ã n+1 zRDλ,α f (z) ≺ h (z) , for z ∈ U (2.15) n f (z) RDλ,α holds, then n+1 RDλ,α f (z) n f (z) RDλ,α

≺ g (z) ,

for z ∈ U,

and this result is sharp. P j Proof. For f ∈ A, f (z) = z + ∞ j=2 aj z we have n o P∞ n n f (z) = z + n j RDλ,α j=2 α [1 + (j − 1) λ] + (1 − α) Cn+j−1 aj z , for z ∈ U. Consider o P∞ n n+1 n+1 j n+1 z + + (1 − α) C α [1 + (j − 1) λ] RDλ,α f (z) j=2 n+j aj z n o p(z) = . P n f (z) = n RDλ,α n j z+ ∞ + (1 − α) C z α [1 + (j − 1) λ] a j n+j−1 j=2 µ ¶0 0 n+1 n+1 n f (z) 0 f (z)) zRDλ,α f (z) (RDλ,α (RDλ,α ) 0 0 We have p (z) = RDn f (z) −p (z)· RDn f (z) and we obtain p (z)+z ·p (z) = RDn f (z) . λ,α

λ,α

λ,α

Relation (2.15) becomes

p(z) + zp0 (z) ≺ h(z) = g(z) + zg 0 (z),

for z ∈ U.

By using Lemma 1.2, we have p(z) ≺ g(z),

for z ∈ U,

i.e.

n+1 f (z) RDλ,α n f (z) RDλ,α

≺ g(z),

for z ∈ U.

Theorem 2.7 Let g be a convex function such that g(0) = 0 and let h be the function h(z) = g(z) + zg 0 (z), for z ∈ U. If α, λ ≥ 0, n ∈ N, f ∈ A and the differential subordination

holds, then

n+1 n (n + 1) RDλ,α f (z) − (n − 1) RDλ,α f (z) − µ ¶ ¤ 1 £ n+1 α n+1− Dλ f (z) − Dλn f (z) ≺ h(z), for z ∈ U λ

This result is sharp.

n RDλ,α f (z) ≺ g(z),

6

for z ∈ U.

(2.16)

104

LUPAS: DIFFERENTIAL SUBORDINATIONS

Proof. Let

=z+

∞ X © j=2

n f (z) = (1 − α)Rn f (z) + αDλn f (z) p(z) = RDλ,α

(2.17)

ª n α [1 + (j − 1) λ]n + (1 − α) Cn+j−1 aj z j = p1 z + p2 z 2 + ....

We deduce that p ∈ H[0, 1]. n , Rn and D n , after a short calculation, we obtain By using the properties of operators RDλ,α λ ¡ ¢ £ n+1 ¤ n+1 n f (z) − α n + 1 − 1 n f (z) . D f (z) − D p (z) + zp0 (z) = (n + 1) RDλ,α f (z) − (n − 1) RDλ,α λ λ λ Using the notation in (2.17), the differential subordination becomes p(z) + zp0 (z) ≺ h(z) = g(z) + zg 0 (z). By using Lemma 1.2, we have p(z) ≺ g(z),

for z ∈ U,

n i.e. RDλ,α f (z) ≺ g(z),

for z ∈ U,

and this result is sharp. Theorem 2.8 Let h(z) = 1+(2β−1)z be a convex function in U , where 0 ≤ β < 1. 1+z If α, λ ≥ 0, n ∈ N, f ∈ A and satisfies the differential subordination

then

n+1 n (n + 1) RDλ,α f (z) − (n − 1) RDλ,α f (z) − µ ¶ ¤ 1 £ n+1 α n+1− Dλ f (z) − Dλn f (z) ≺ h(z), for z ∈ U, λ n f (z) ≺ q(z), RDλ,α

(2.18)

for z ∈ U,

, for z ∈ U. The function q is convex and it is where q is given by q(z) = 2β − 1 + 2(1 − β) ln(1+z) z the best dominant. Proof. Following the same steps as in the proof of Theorem 2.7 and considering p(z) = n f (z), the differential subordination (2.18) becomes RDλ,α p(z) + zp0 (z) ≺ h(z) =

1 + (2β − 1)z , 1+z

for z ∈ U.

By using Lemma 1.1 for γ = 1 and n = 1, we have p(z) ≺ q(z), i.e., Z Z 1 z 1 z 1 + (2β − 1)t 1 n h(t)dt = dt = 2β−1+2(1−β) ln(z+1), RDλ,α f (z) ≺ q(z) = z 0 z 0 1+t z

for z ∈ U.

h Theorem 2.9 Let h be an holomorphic function which satisfies the inequality Re 1 +

zh00 (z) h0 (z)

− 12 ,

for z ∈ U, and h (0) = 0. If α, λ ≥ 0, n ∈ N, f ∈ A and satisfies the differential subordination

i

>

n+1 n (n + 1) RDλ,α f (z) − (n − 1) RDλ,α f (z) −

then

¶ µ ¤ 1 £ n+1 Dλ f (z) − Dλn f (z) ≺ h(z), α n+1− λ

where q is given by q(z) =

1 z

Rz 0

n f (z) ≺ q(z), RDλ,α

for z ∈ U,

for z ∈ U,

h(t)dt. The function q is convex and it is the best dominant. 7

(2.19)

LUPAS: DIFFERENTIAL SUBORDINATIONS

105

n and considering p (z) = RD n f (z), we obtain Proof. Using the properties of operator RDλ,α λ,α n+1 n p(z) + zp0 (z) = (n + 1) RDλ,α f (z) − (n − 1) RDλ,α f (z) − ¶ µ ¤ 1 £ n+1 Dλ f (z) − Dλn f (z) , for z ∈ U. α n+1− λ

Then (2.19) becomes

p(z) + zp0 (z) ≺ h(z),

for z ∈ U.

Since p ∈ H[0, 1], using Lemma 1.1, we deduce p(z) ≺ q(z), for z ∈ U,

i.e.

n f (z) RDλ,α

1 ≺ q(z) = z

Z

z

h(t)dt,

0

for z ∈ U,

and q is the best dominant. Theorem 2.10 Let g be a convex function such that g(0) = 1 and let h be the function h(z) = g(z) + zg 0 (z), for z ∈ U. If α, λ ≥ 0, n ∈ N, f ∈ A and the differential subordination (n + 1) (n + 2) (n + 1) (2n + 1) n2 n+2 n+1 n f (z) − f (z) + RDλ,α f (z) − RDλ,α RDλ,α z z z i i h h α (n + 1) (2n + 1) − 2(1−λ) α (n + 1) (n + 2) − λ12 2 λ Dλn+2 f (z) + Dλn+1 f (z) − z z h i 2 α n2 − (1−λ) 2 λ Dλn f (z) ≺ h(z), for z ∈ U z holds, then n [RDλ,α f(z)]0 ≺ g(z), for z ∈ U.

(2.20)

This result is sharp. Proof. Let

=1+

¡ ¢0 n p(z) = RDλ,α f (z) = (1 − α) (Rn f (z))0 + α (Dλn f (z))0

∞ X © ª n α [1 + (j − 1) λ]n + (1 − α) Cn+j−1 jaj z j−1 = 1 + p1 z + p2 z 2 + ....

(2.21)

j=2

We deduce that p ∈ H[1, 1]. n , Rn and D n , after a short calculation, we obtain By using the properties of operators RDλ,α λ 2 (n+1)(n+2) n+2 n+1 n f (z) − RDλ,α f (z) − (n+1)(2n+1) RDλ,α f (z) + nz RDλ,α z z ¸ ∙ 2 i i h h 2(1−λ) α n2 − (1−λ) α (n+1)(n+2)− 12 α (n+1)(2n+1)− 2 λ λ λ2 Dλn+2 f (z) + Dλn+1 f (z) − Dλn f (z) . z z z

p (z) + zp0 (z) =

Using the notation in (2.21), the differential subordination becomes p(z) + zp0 (z) ≺ h(z) = g(z) + zg 0 (z). By using Lemma 1.2, we have p(z) ≺ g(z),

and this result is sharp.

for z ∈ U,

i.e.

¢0 ¡ n f (z) ≺ g(z), RDλ,α 8

for z ∈ U,

106

LUPAS: DIFFERENTIAL SUBORDINATIONS

Example 2.2 If n = 1, α = 1, λ = 1, f ∈ A, we deduce that f 0 (z) + 3zf 00 (z) + z 2 f 000 (z) ≺ g(z) + zg 0 (z), which yields that f 0 (z) + zf 00 (z) ≺ g(z), for z ∈ U. Theorem 2.11 Let h(z) = 1+(2β−1)z be a convex function in U, where 0 ≤ β < 1. 1+z If α, λ ≥ 0, n ∈ N, f ∈ A and satisfies the differential subordination (n + 1) (n + 2) (n + 1) (2n + 1) n2 n+2 n+1 n f (z) − f (z) + RDλ,α f (z) − RDλ,α RDλ,α z z z i i h h α (n + 1) (2n + 1) − 2(1−λ) α (n + 1) (n + 2) − λ12 2 λ Dλn+2 f (z) + Dλn+1 f (z) − z z h i 2 (1−λ) α n2 − λ2 Dλn f (z) ≺ h(z), for z ∈ U, z then

¡

¢0 n RDλ,α f (z) ≺ q(z),

(2.22)

for z ∈ U,

, for z ∈ U. The function q is convex and it is where q is given by q(z) = 2β − 1 + 2(1 − β) ln(1+z) z the best dominant. Following the same steps as in the proof of Theorem 2.10 and considering p(z) = ³ Proof. ´ 0 n RDλ,α f (z) , the differential subordination (2.22) becomes p(z) + zp0 (z) ≺ h(z) =

1 + (2β − 1)z , 1+z

for z ∈ U.

By using Lemma 1.1 for γ = 1 and n = 1, we have p(z) ≺ q(z), i.e., Z Z ¡ ¢0 1 z 1 z 1 + (2β − 1)t 1 n RDλ,α f(z) ≺ q(z) = h(t)dt = dt = 2β−1+2(1−β) ln(z+1), for z ∈ U. z 0 z 0 1+t z h Theorem 2.12 Let h be an holomorphic function which satisfies the inequality Re 1 +

zh00 (z) h0 (z)

− 12 ,

for z ∈ U, and h (0) = 1. If α, λ ≥ 0, n ∈ N, f ∈ A and satisfies the differential subordination

(n + 1) (n + 2) (n + 1) (2n + 1) n2 n+2 n+1 n f (z) − f (z) + RDλ,α f (z) − RDλ,α RDλ,α z z z h i i h α (n + 1) (2n + 1) − 2(1−λ) α (n + 1) (n + 2) − λ12 λ2 Dλn+2 f (z) + Dλn+1 f (z) − z z h i 2 α n2 − (1−λ) 2 λ Dλn f (z) ≺ h(z), for z ∈ U, z then where q is given by q(z) =

1 z

Rz 0

¡

¢0 n RDλ,α f (z) ≺ q(z),

for z ∈ U,

h(t)dt. The function q is convex and it is the best dominant.

9

i

>

(2.23)

LUPAS: DIFFERENTIAL SUBORDINATIONS

107

³ ´0 n n f (z) , we Proof. Using the properties of operator RDλ,α and considering p (z) = RDλ,α obtain (n + 1) (n + 2) (n + 1) (2n + 1) n+2 n+1 p(z) + zp0 (z) = RDλ,α RDλ,α f (z) − f (z) + z z h i 1 α (n + 1) (n + 2) − 2 n λ2 n f (z) − RDλ,α Dλn+2 f (z) + z z h i i h 2 2 − (1−λ) α n α (n + 1) (2n + 1) − 2(1−λ) 2 2 λ λ Dλn+1 f (z) − Dλn f (z) , for z ∈ U. z z Then (2.23) becomes p(z) + zp0 (z) ≺ h(z), for z ∈ U. Since p ∈ H[1, 1], using Lemma 1.1, we deduce p(z) ≺ q(z), for z ∈ U, and q is the best dominant.

i.e.

¡

¢0 n f (z) RDλ,α

1 ≺ q(z) = z

Z

0

z

h(t)dt,

for z ∈ U,

References [1] A. Alb Lupa¸s, On special differential subordinations using Sa˘la ˘gean and Ruscheweyh operators, Mathematical Inequalities and Applications, Volume 12, Issue 4, October 2009 (to appear). [2] F.M. Al-Oboudi, On univalent functions defined by a generalized Sa˘la˘gean operator, Ind. J. Math. Math. Sci., 27 (2004), 1429-1436. [3] S.S. Miller, P.T. Mocanu, Differential Subordinations. Theory and Applications, Marcel Dekker Inc., New York, Basel, 2000. [4] St. Ruscheweyh, New criteria for univalent functions, Proc. Amet. Math. Soc., 49(1975), 109-115. [5] G. St. Sa˘la˘gean, Subclasses of univalent functions, Lecture Notes in Math., Springer Verlag, Berlin, 1013(1983), 362-372.

10

JOURNAL 108 OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.1, 108-115, 2011, COPYRIGHT 2011 EUDOXUS PRESS, LLC

Certain special differential superordinations using multiplier transformation and Ruscheweyh derivative Alina Alb Lupa¸s Department of Mathematics and Computer Science University of Oradea str. Universitatii nr. 1, 410087 Oradea, Romania [email protected] Abstract In the present paper we establish several differential superordinations regardind the new operator α defined by using the multiplier transformation I (m, λ, l) f (z) and Ruscheweyh derivative Rm f (z), RIm,λ,l α α m f(z) = (1 − α)R (m, λ, l) f (z), z ∈ U, where m, n ∈ N, λ, α, l ≥ 0 RIm,λ,l : An → An , RIm,λ,l P f (z) + αI j and f ∈ An = {f ∈ H(U ) : f (z) = z + ∞ j=n+1 aj z , z ∈ U }. A number of interesting consequences of some of these superordination results are discussed. Relevant connections of some of the new results obtained in this paper with those in earlier works are also provided.

Keywords: differential superordination, convex function, best subordinant, differential operator. 2000 Mathematical Subject Classification: 30C45, 30A20, 34A40.

1

Introduction

Denote by U the unit disc of the complex plane, U = {z ∈ C : |z| < 1} and H(U ) the space of holomorphic functions in U . P j Let A (p, n) = {f ∈ H(U ) : f (z) = z p + ∞ j=p+n aj z , z ∈ U }, with A (1, n) = An and H[a, n] = n {f an+1 z n+1 + . . . , z ∈ U }, where p, n ∈ N, a ∈ C. Denote by K = n ∈ H(U ) : f 00(z) = a + an z + o zf (z) f ∈ An : Re f 0 (z) + 1 > 0, z ∈ U the class of normalized convex functions in U . If f and g are analytic functions in U , we say that f is superordinate to g, written g ≺ f , if there is a function w analytic in U , with w(0) = 0, |w(z)| < 1, for all z ∈ U such that g(z) = f (w(z)) for all z ∈ U . If f is univalent, then g ≺ f if and only if f (0) = g(0) and g(U ) ⊆ f (U ). Let ψ : C2 × U → C and h analytic in U . If p and ψ (p (z) , zp0 (z) ; z) are univalent in U and satisfies the (first-order) differential superordination h(z) ≺ ψ(p(z), zp0 (z); z),

z ∈ U,

(1.1)

then p is called a solution of the differential superordination. The analytic function q is called a subordinant of the solutions of the differential superordination, or more simply a subordinant, if q ≺ p for all p satisfying (1.1). An univalent subordinant qe that satisfies q ≺ qe for all subordinants q of (1.1) is said to be the best subordinant of (1.1). The best subordinant is unique up to a rotation of U .

Definition 1.1 (Ruscheweyh [13]) For f ∈ An , m ∈ N, the operator Rm is defined by Rm : An → An , R0 f (z) = f (z) R1 f (z) = zf 0 (z) ... 0 m+1 (m + 1) R f (z) = z (Rm f (z)) + mRm f (z) , z ∈ U. P∞ P∞ m aj z j , z ∈ U . Remark 1.1 If f ∈ An , f (z) = z + j=n+1 aj z j , then Rm f (z) = z + j=n+1 Cm+j−1 1

LUPAS: SPECIAL DIFFERENTIAL SUBORDINATIONS

109

Definition 1.2 For f ∈ A(p, n), p, n ∈ N, m ∈ N∪ {0}, λ, l³≥ 0, the operator Ip (m, λ, l) f (z) is defined by ´m P∞ p+λ(j−p)+l p j aj z . the following infinite series Ip (m, λ, l) f (z) := z + j=p+n p+l

Remark 1.2 It follows from the above definition that Ip (0, λ, l) f (z) = f (z), (p + l) Ip (m + 1, λ, l) f (z) = [p(1 − λ) + l] Ip (m, λ, l) f (z) + λz (Ip (m, λ, l) f (z))0 , z ∈ U.

Remark 1.3 If p = 1, we have I1 (m, λ, l) f (z) = I (m, λ, l) and (l + 1) I (m + 1, λ, l) f (z) = [l + 1 − λ] I (m, λ, l) f (z) + λz (I (m, λ, l) f (z))0 , z ∈ U. ³ ´m P P∞ 1+λ(j−1)+l j aj z j , Remark 1.4 If f ∈ An , f (z) = z + ∞ j=n+1 aj z , then I (m, λ, l) f (z) = z + j=n+1 l+1 z ∈ U. Remark 1.5 For l = 0, λ ≥ 0, the operator Dλm = I (m, λ, 0) was introduced and studied by Al-Oboudi [9], which reduced to the Sa˘la˘gean differential operator S m = I (m, 1, 0) [14] for λ = 1. The operator Ilm = I (m, 1, l) was studied recently by Cho and Srivastava [10] and Cho and Kim [11]. The operator Im = I (m, 1, 1) was studied by Uralegaddi and Somanatha [15], the operator Dλδ = I (δ, λ, 0), with δ ∈ R, δ ≥ 0, was introduced by Acu and Owa [1]. Definition 1.3 We denote by Q the set of functions that are analytic and injective on U \E (f ), where E (f ) = {ζ ∈ ∂U : lim f (z) = ∞}, and are such that f 0 (ζ) 6= 0 for ζ ∈ ∂U \E (f ). The subclass of Q for z→ζ

which f (0) = a is denoted by Q (a). We will use the following lemmas. Lemma 1.1 (Miller and Mocanu [12, Th. 3.1.6, p. 71]) Let h be a convex function with h(0) = a and let γ ∈ C\{0} be a complex number with Re γ ≥ 0. If p ∈ H[a, n] ∩ Q, p(z) + γ1 zp0 (z) is univalent in U and Rz h(z) ≺ p(z) + γ1 zp0 (z), z ∈ U, then q(z) ≺ p(z), z ∈ U, where q(z) = nzγγ/n 0 h(t)tγ/n−1 dt, z ∈ U. The function q is convex and is the best subordinant.

Lemma 1.2 (Miller and Mocanu [12]) Let q be a convex function in U and let h(z) = q(z) + γ1 zq 0 (z), z ∈ U, where Re γ ≥ 0. If p ∈ H [a, n] ∩ Q, p(z) + γ1 zp0 (z) is univalent in U and q(z) + γ1 zq 0 (z) ≺ p(z) + γ1 zp0 (z) , Rz z ∈ U, then q(z) ≺ p(z), z ∈ U, where q(z) = nzγγ/n 0 h(t)tγ/n−1 dt, z ∈ U. The function q is the best subordinant.

2

Main results

α α Definition 2.1 [6] Let α, λ, l ≥ 0, n, m ∈ N. Denote by RIm,λ,l the operator given by RIm,λ,l : An → An , α m RIm,λ,l f (z) = (1 − α)R f (z) + αI (m, λ, l) f (z), z ∈ U. P Remark 2.1 If f ∈ An , f (z) = z + ∞ aj z j , then j=n+1 n ³ ´ o m P 1+λ(j−1)+l α m f (z) = z + ∞ + (1 − α) Cm+j−1 aj z j , for z ∈ U. RIm,λ,l j=n+1 α l+1 0 1 f (z) = Rm f (z), where z ∈ U and for α = 1, RIm,λ,l f (z) = I (m, λ, l) f (z), Remark 2.2 For α = 0, RIm,λ,l α m f (z) which was studied where z ∈ U , which was studied in [3], [7]. For l = 0, we obtain RIm,λ,0 f (z) = RD1,α α f (z) = Lm f (z) which was studied in [2], [4]. in [5], [8] and for l = 0 and λ = 1, we obtain RIm,1,0 α α f (z) = (1 − α) R0 f (z) + αI (0, λ, l) f (z) = f (z) = R0 f (z) = I (0, λ, l) f (z), where For m = 0, RI0,λ,l z ∈ U.

Theorem 2.1 Let h be a convex function in U with h (0) = 1. Let m ∈ N, λ, α, l ≥ 0, f ∈ An , ³ ´0 Rz c α t f (t) dt, z ∈ U , Re c > −2, and suppose that RI f (z) is univalent F (z) = Ic (f ) (z) = zc+2 c+1 m,λ,l 0 ´0 ³ α F (z) ∈ H [1, n] ∩ Q and in U , RIm,λ,l ¡ α ¢0 h (z) ≺ RIm,λ,l f (z) ,

then where q(z) =

c+2 nz

c+2 n

Rz 0

h(t)t

c+2 n −1

¢0 ¡ α F (z) , q (z) ≺ RIm,λ,l

z ∈ U,

z ∈ U,

dt. The function q is convex and it is the best subordinant. 2

(2.1)

110

LUPAS: SPECIAL DIFFERENTIAL SUBORDINATIONS

Proof. We have z c+1 F (z) = (c + 2)

Rz

tc f (t) dt and differentiating it, with respect to z, we obtain ³ ´0 α α α F (z)+z RIm,λ,l F (z) = (c + 2) RIm,λ,l f (z) , z ∈ (c + 1) F (z)+zF 0 (z) = (c + 2) f (z) and (c + 1) RIm,λ,l U. Differentiating the last relation we have ¡ α ¢0 RIm,λ,l F (z) +

0

¡ α ¢00 ¡ α ¢0 1 z RIm,λ,l F (z) = RIm,λ,l f (z) , z ∈ U. c+2

(2.2)

Using (2.2), the differential superordination (2.1) becomes ¢0 ¡ α F (z) + h (z) ≺ RIm,λ,l

Denote

¡ α ¢00 1 z RIm,λ,l F (z) . c+2

(2.3)

¢0 ¡ α F (z) , z ∈ U. p (z) = RIm,λ,l

(2.4)

1 Replacing (2.4) in (2.3) we obtain h (z) ≺ p (z) + c+2 zp0 (z), z ∈ U. Using Lemma 1.1 for γ = c + 2, we ³ ´0 Rz c+2 α have q (z) ≺ p (z) , z ∈ U, i.e. q (z) ≺ RIm,λ,l F (z) , z ∈ U, where q(z) = c+2 h(t)t n −1 dt. The c+2 0 nz

function q is convex and it is the best subordinant.

n

1 Theorem 2.2 Let q be a convex function in U and let h (z) = q (z) + c+2 zq 0 (z) , where z ∈ U, Re c > −2. ³ ´0 R z c α Let m ∈ N, λ, α, l ≥ 0, f ∈ An , F (z) = Ic (f ) (z) = zc+2 t f (t) dt, z ∈ U and suppose that RI f (z) c+1 m,λ,l 0 ³ ´0 α is univalent in U , RIm,λ,l F (z) ∈ H [1, n] ∩ Q and

¢0 ¡ α f (z) , h (z) ≺ RIm,λ,l

then where q(z) =

c+2 nz

c+2 n

Rz 0

h(t)t

c+2 n −1

¢0 ¡ α F (z) , q (z) ≺ RIm,λ,l

z ∈ U,

(2.5)

z ∈ U,

dt. The function q is the best subordinant.

´0 ³ α F (z) , Proof. Following the same steps as in the proof of Theorem 2.1 and considering p (z) = RIm,λ,l

1 1 zq 0 (z) ≺ p (z) + c+2 zp0 (z), z ∈ U, the differential superordination (2.5) becomes h (z) = q (z) + c+2 ³ ´0 α Using Lemma 1.2 for γ = c + 2, we have q (z) ≺ p (z) , z ∈ U, i.e. q (z) ≺ RIm,λ,l F (z) , Rz c+2 h(t)t n −1 dt. The function q is the best subordinant. where q(z) = c+2 c+2 0 nz

z ∈ U.

z ∈ U,

n

Theorem 2.3 Let h (z) = 1+(2β−1)z , where β ∈ [0, 1). Let m ∈ N, λ, α, l ≥ 0, f ∈ An , F (z) = Ic (f ) (z) = 1+z ³ ´0 ³ ´0 Rz c c+2 α α t f (t) dt, z ∈ U, Re c > −2, and suppose that RI f (z) is univalent in U , RI F (z) ∈ c+1 m,λ,l m,λ,l z 0 H [1, n] ∩ Q and ¡ α ¢0 h (z) ≺ RIm,λ,l f (z) , z ∈ U, (2.6)

then

¡ α ¢0 q (z) ≺ RIm,λ,l F (z) ,

where q is given by q(z) = 2β − 1 + (c+2)(2−2β) c+2 nz

subordinant.

n

Rz 0

c+2 t n −1

t+1

z ∈ U,

dt, z ∈ U. The function q is convex and it is the best

³ ´0 α Proof. Following the same steps as in the proof of Theorem 2.1 and considering p(z) = RIm,λ,l F (z) , 1+(2β−1)z 1+z Rz = c+2 c+2 0 nz n

the differential superordination (2.6) becomes h(z) = Lemma 1.1 for γ = c+2, we have q(z) ≺ p(z), i.e., q(z) ³ ´0 R z t c+2 −1 n α dt ≺ RI F (z) , = 2β − 1 + (c+2)(2−2β) c+2 m,λ,l t+1 0

subordinant.

nz

n

3

≺ p (z) +

h(t)t

c+2 n −1

1 0 z ∈ U. By using c+2 zp (z) , R z 1+(2β−1)t c+2 −1 c+2 dt = c+2 0 t n dt 1+t nz n

z ∈ U. The function q is convex and it is the best

LUPAS: SPECIAL DIFFERENTIAL SUBORDINATIONS

111

Theorem 2.4 Let h be a convex function, h(0) = 1. Let m ∈ N, λ, α, l ≥ 0, f ∈ An and suppose that ´0 ³ RI α f (z) α f (z) is univalent and m,λ,l ∈ H [1, n] ∩ Q. If RIm,λ,l z ¢0 ¡ α f (z) , h(z) ≺ RIm,λ,l

then

where q(z) =

1 1

nz n

Rz 0

q(z) ≺

α f (z) RIm,λ,l , z

z ∈ U,

(2.7)

z ∈ U,

1

h(t)t n −1 dt. The function q is convex and it is the best subordinant.

α Proof. By using the properties of operator RIm,λ,l , we have n ³ ´m o P ∞ 1+λ(j−1)+l α m j + (1 − α) C RIm,λ,l f (z) = z + j=n+1 α m+j−1 aj z , l+1 α RIm,λ,l f (z)

= Consider p(z) = z U. We deduce that p ∈ H[1, n].

z ∈ U.

P 1+λ(j−1)+l m m z+ ∞ ) +(1−α)Cm+j−1 }aj z j j=n+1 {α( l+1 z

= 1+pn z n +pn+1 z n+1 +...,

z∈

³ ´0 α α f (z) = zp(z), z ∈ U. Differentiating we obtain RIm,λ,l f (z) = p(z) + zp0 (z), z ∈ U. Let RIm,λ,l Then (2.7) becomes h(z) ≺ p(z)+zp0 (z), z ∈ U. By using Lemma 1.1 for γ = 1, we have q(z) ≺ p(z), Rz 1 RI α f (z) z ∈ U, i.e. q(z) ≺ m,λ,l , z ∈ U, where q(z) = 1 1 0 h(t)t n −1 dt. The function q is convex and it z nz n is the best subordinant. Theorem 2.5 Let q be convex in U and let h be defined by h (z) = q (z) + zq 0 (z) . If m ∈ N, λ, α, l ≥ 0, ³ ´0 RI α f (z) α f (z) is univalent and m,λ,l ∈ H [1, n] ∩ Q and satisfies the differential f ∈ An , suppose that RIm,λ,l z superordination ¡ α ¢0 f (z) , z ∈ U, (2.8) h(z) = q (z) + zq 0 (z) ≺ RIm,λ,l

then

where q(z) =

1 1 nz n

Rz 0

q(z) ≺

α f (z) RIm,λ,l , z

z ∈ U,

1

h(t)t n −1 dt. The function q is the best subordinant. RI α

f (z)

Proof. Following the same steps as in the proof of Theorem 2.4 and considering p(z) = m,λ,l , the z differential superordination (2.8) becomes q(z) + zq 0 (z) ≺ p(z) + zp0 (z) , z ∈ U. Using Lemma 1.2 for Rz 1 RI α f (z) γ = 1, we have q(z) ≺ p(z), z ∈ U, i.e. q(z) = 1 1 0 h(t)t n −1 dt ≺ m,λ,l , z ∈ U, and q is the z nz n best subordinant. Theorem 2.6 Let h(z) = 1+(2β−1)z be a convex function in U , where 0 ≤ β < 1. Let m ∈ N, λ, α, l ≥ 0, 1+z ³ ´0 RI α f (z) α f ∈ An and suppose that RIm,λ,l f (z) is univalent and m,λ,l ∈ H [1, n] ∩ Q. If z then

¢0 ¡ α f (z) , h(z) ≺ RIm,λ,l q(z) ≺

where q is given by q(z) = 2β − 1 + subordinant.

2−2β 1 nz n

Rz 0

α f (z) RIm,λ,l , z

z ∈ U,

(2.9)

z ∈ U,

1

t n −1 t+1 dt,

z ∈ U. The function q is convex and it is the best RI α

f (z)

, the Proof. Following the same steps as in the proof of Theorem 2.4 and considering p(z) = m,λ,l z 0 differential superordination (2.9) becomes h(z) = 1+(2β−1)z ≺ p(z) + zp (z), z ∈ U. By using Lemma R1+z Rz 1 1 z t n −1 dt = 2β − 1 + 1.1 for γ = 1, we have q(z) ≺ p(z), i.e., q(z) = 1 1 0 h(t)t n −1 dt = 1 1 0 1+(2β−1)t 1+t nz n nz n α R 1 RIm,λ,l f (z) 2−2β z t n −1 dt ≺ , z ∈ U. The function q is convex and it is the best subordinant. 1 z 0 t+1 nz n

4

112

LUPAS: SPECIAL DIFFERENTIAL SUBORDINATIONS

Theorem 2.7 Let h be a convex function, h(0) = 1. Let m ∈ N, λ, α, l ≥ 0, f ∈ An and suppose that ´ ³ α zRIm+1,λ,l f (z) 0 RI α f (z) is univalent and RIm+1,λ,l α α RI f (z) f (z) ∈ H [1, n] ∩ Q. If m,λ,l

m,λ,l

h(z) ≺

Ã

α zRIm+1,λ,l f (z) α RIm,λ,l f (z)

then

1 1 nz n

Rz 0

,

z ∈ U,

α f (z) RIm+1,λ,l , α RIm,λ,l f (z)

q(z) ≺ where q(z) =

!0

(2.10)

z ∈ U,

1

h(t)t n −1 dt. The function q is convex and it is the best subordinant.

P∞ Proof. For f ∈ An , f (z) = z + j=n+1 aj z j we have n ³ ´m o P∞ α m f (z) = z + j=n+1 α 1+λ(j−1)+l + (1 − α) C aj z j , z ∈ U. RIm,λ,l m+j−1 l+1 Consider p(z) = We have p0 (z) =

α RIm+1,λ,l f (z) α RIm,λ,l f (z)

(

=

0 α RIm+1,λ,l f (z) α RIm,λ,l f (z)

)

n o P 1+λ(j−1)+l m+1 m+1 aj z j z+ ∞ +(1−α)Cm+j ) j=n+1 α( l+1 . P∞ 1+λ(j−1)+l m m z+ j=n+1 {α( ) +(1−α)Cm+j−1 }aj zj l+1 0 α f (z)) (RIm,λ,l

− p (z) ·

α RIm,λ,l f (z) 0

and we obtain p (z) + z · p0 (z) =

³ zRI α

m+1,λ,l f (z) α RIm,λ,l f (z)

´0

.

Relation (2.10) becomes h(z) ≺ p(z) + zp (z), z ∈ U. By using Lemma 1.1 for γ = 1, we have Rz 1 RI α f (z) q(z) ≺ p(z), z ∈ U, i.e. q(z) ≺ RIm+1,λ,l , z ∈ U, where q(z) = 1 1 0 h(t)t n −1 dt. The function q α m,λ,l f (z) n nz is convex and it is the best subordinant. Theorem 2.8 Let q be a convex function and h be defined by h (z) = q (z) + zq 0 (z) . Let λ, α, l ≥ 0, m ∈ N, ³ zRI α ´0 RI α f (z) m+1,λ,l f (z) is univalent and RIm+1,λ,l f ∈ An and suppose that α α RI f (z) f (z) ∈ H [1, n] ∩ Q. If m,λ,l

m,λ,l

0

h(z) = q (z) + zq (z) ≺ then q(z) ≺ where q(z) =

1 1 nz n

Rz 0

Ã

α zRIm+1,λ,l f (z) α RIm,λ,l f (z)

α f (z) RIm+1,λ,l , α RIm,λ,l f (z)

!0

,

z ∈ U,

(2.11)

z ∈ U,

1

h(t)t n −1 dt. The function q is the best subordinant.

Proof. Following the same steps as in the proof of Theorem 2.7 and considering p(z) = 0

0

α RIm+1,λ,l f (z) α RIm,λ,l f (z) ,

the

differential superordination (2.11) becomes h (z) = q(z) + zq (z) ≺ p(z) + zp (z), z ∈ U. By using Lemma Rz 1 RI α f (z) 1.2 for γ = 1, we have q(z) ≺ p(z), z ∈ U, i.e. q(z) = 1 1 0 h(t)t n −1 dt ≺ RIm+1,λ,l , z ∈ U, and α m,λ,l f (z) nz n q is the best subordinant. Theorem 2.9 Let h(z) = 1+(2β−1)z be a convex function in U , where 0 ≤ β < 1. Let λ, α, l ≥ 0, m ∈ N, 1+z ³ ´ α zRIm+1,λ,l f (z) 0 RI α f (z) f ∈ An and suppose that is univalent and RIm+1,λ,l α RI α f (z) f (z) ∈ H [1, n] ∩ Q. If m,λ,l

m,λ,l

Ã

α zRIm+1,λ,l f (z) α RIm,λ,l f (z)

q(z) ≺

α f (z) RIm+1,λ,l , α RIm,λ,l f (z)

h(z) ≺ then

where q is given by q(z) = 2β − 1 + subordinant.

2−2β 1 nz n

Rz 0

!0

,

z ∈ U,

(2.12)

z ∈ U,

1

t n −1 t+1 dt,

5

z ∈ U. The function q is convex and it is the best

LUPAS: SPECIAL DIFFERENTIAL SUBORDINATIONS

Proof. Following the same steps as in the proof of Theorem 2.7 and considering p(z) = the differential superordination (2.12) becomes h(z) =

113 α RIm+1,λ,l f (z) α RIm,λ,l f (z) ,

1+(2β−1)z ≺ p(z) + zp0 (z), z ∈ U. By using 1+z R Rz 1 1 z t n −1 dt = = 1 1 0 h(t)t n −1 = 1 1 0 1+(2β−1)t 1+t nz n nz n

Lemma 1.1 for γ = 1, we have q(z) ≺ p(z), i.e., q(z) R z t n1 −1 RI α f (z) dt ≺ RIm+1,λ,l z ∈ U. The function q is convex and it is the best subordinant. 2β − 1 + 2−2β α 1 f (z) , 0 t+1 nz n

m,λ,l

Theorem 2.10 Let h ∈ H (U ) be a convex function in U with h(0) = 1 and let λ, α, l ≥ 0, m ∈ N, f ∈ An , 2 (m+1)(m+2) α α α RIm+2,λ,l f (z) − (m+1)(2m+1) RIm+1,λ,l f (z) + mz RIm,λ,l f (z) + z z ∙ ¸ i h ¯ 2 α (l+1) α 2(l+1−λ)(l+1) − (m + 1) (m + 2) I (m + 2, λ, l) f (z) − − (m + 1) (2m + 1) I (m + 1, λ, l) f (z) + 2 2 z z λ λ h i 2 α (l+1−λ) m − m2 I (m, λ, l) f (z) is univalent and [RDλ,α f (z)]0 ∈ H [1, n] ∩ Q. If z λ2 (m + 1) (2m + 1) α (m + 1) (m + 2) α RIm+2,λ,l f (z) − RIm+1,λ,l f (z) + (2.13) z z # " 2 α (l + 1) m2 α RIm,λ,l f (z) + − (m + 1) (m + 2) I (m + 2, λ, l) f (z) − z z λ2 # # " " ¡ ¢ α (l + 1 − λ)2 α 2 (l + 1 − λ) ¯l + 1 2 − (m + 1) (2m + 1) I (m + 1, λ, l) f (z) + − m I (m, λ, l) f (z) , z z λ2 λ2 h(z) ≺

z ∈ U , holds, then where q(z) =

1 1

nz n

Rz 0

h(t)t

1 n −1

α f (z)]0 , q(z) ≺ [RIm,λ,l

z ∈ U,

. The function q is convex and it is the best subordinant.

P Proof. For f ∈ An , f (z) = z + ∞ aj z j we have j=n+1 n ³ ´ o m P 1+λ(j−1)+l α m f (z) = z + ∞ + (1 − α) Cm+j−1 aj z j , z ∈ U. RIm,λ,l j=n+1 α l+1 Let ¢0 ¡ α f (z) (2.14) p(z) = RIm,λ,l ¶ ½ µ ¾ ∞ m X 1 + λ (j − 1) + l m α jaj z j−1 = 1 + pn z n + pn+1 z n+1 + .... + (1 − α) Cm+j−1 = 1+ l + 1 j=n+1

α By using the properties of operators RIm,λ,l , Rm and I (m, λ, l), after a short calculation, we obtain 2

α α α RIm+2,λ,l f (z) − (m+1)(2m+1) RIm+1,λ,l f (z) + mz RIm,λ,l f (z) + p (z) + zp0 (z) = (m+1)(m+2) z z∙ ¸ i h ¯ 2 α (l+1) α 2(l+1−λ)(l+1) − (m + 1) (m + 2) I (m + 2, λ, l) f (z) − − (m + 1) (2m + 1) I (m + 1, λ, l) f (z) + z z λ2 λ2 h i 2 α (l+1−λ) − m2 I (m, λ, l) f (z) . z λ2 Using the notation in (2.14), the differential superordination becomes h(z) ≺ p(z) + zp0 (z). By using ´0 ³ α f (z) , z ∈ U, where Lemma 1.1 for γ = 1, we have q(z) ≺ p(z), z ∈ U, i.e. q(z) ≺ RIm,λ,l Rz 1 q(z) = 1 1 0 h(t)t n −1 . The function q is convex and it is the best subordinant. nz n

Theorem 2.11 Let q be a convex function in U and h (z) = q (z) + zq 0 (z). Let λ, α, l ≥ 0, m ∈ N, f ∈ An , 2 α α suppose that (m+1)(m+2) RIm+2,λ,l f (z) − (m+1)(2m+1) RI α f (z) + mz RIm,λ,l f (z) + z z ∙ m+1,λ,l ¸ i h ¯ 2 α (l+1) α 2(l+1−λ)(l+1) − (m + 1) (m + 2) I (m + 2, λ, l) f (z) − − (m + 1) (2m + 1) I (m + 1, λ, l) f (z) + z z λ2 λ2 h i 2 α (l+1−λ) α − m2 I (m, λ, l) f (z) is univalent in U and [RIm,λ,l f (z)]0 ∈ H [1, n] ∩ Q. If z λ2 (m + 1) (m + 2) α (m + 1) (2m + 1) α RIm+2,λ,l f (z) − RIm+1,λ,l f (z) + z z # " 2 α (l + 1) m2 α RIm,λ,l f (z) + − (m + 1) (m + 2) I (m + 2, λ, l) f (z) − z z λ2

h(z) = q (z) + zq 0 (z) ≺

6

(2.15)

114

LUPAS: SPECIAL DIFFERENTIAL SUBORDINATIONS

# # " " ¡ ¢ α (l + 1 − λ)2 α 2 (l + 1 − λ) ¯l + 1 2 − (m + 1) (2m + 1) I (m + 1, λ, l) f (z) + − m I (m, λ, l) f (z) , z z λ2 λ2 z ∈ U, then where q(z) =

1 1

nz n

Rz 0

h(t)t

1 n −1

¢0 ¡ α f (z) , q(z) ≺ RIm,λ,l

z ∈ U,

. The function q is the best subordinant.

´0 ³ α f (z) , Proof. Following the same steps as in the proof of Theorem 2.10 and considering p(z) = RIm,λ,l the differential superordination (2.15) becomes h(z) = q (z) + zq 0 (z) ≺ p(z) + zp0 (z), z ∈ U. By using ³ ´0 Rz 1 α f (z) , z ∈ U. The Lemma 1.2 for γ = 1, we have q(z) ≺ p(z), i.e., q(z) = 1 1 0 h(t)t n −1 ≺ RIm,λ,l nz n function q is the best subordinant. 1+(2β−1)z be a convex function in U , where 0 ≤ β 1+z (m+1)(m+2) α α RIm+2,λ,l f (z) − (m+1)(2m+1) RIm+1,λ,l f m ∈ N, f ∈ An , suppose that z z ∙ i h ¯ 2 2(l+1−λ) l+1 ( ) α (l+1) − (m + 1) (m + 2) I (m + 2, λ, l) f (z) − αz − (m + 1) (2m + z λ2 λ2

Theorem 2.12 Let h(z) =

α z

h

(l+1−λ)2 λ2

< 1. Let λ, α, l ≥ 0, 2

α (z) + mz RIm,λ,l f (z) + ¸ 1) I (m + 1, λ, l) f (z) +

i α − m2 I (m, λ, l) f (z) is univalent in U and [RIm,λ,l f (z)]0 ∈ H [1, n] ∩ Q. If

(m + 1) (2m + 1) α (m + 1) (m + 2) α RIm+2,λ,l f (z) − RIm+1,λ,l f (z) + (2.16) z z # " m2 α α (l + 1)2 RIm,λ,l f (z) + − (m + 1) (m + 2) I (m + 2, λ, l) f (z) − z z λ2 # # " " ¡ ¢ α (l + 1 − λ)2 α 2 (l + 1 − λ) ¯l + 1 2 − (m + 1) (2m + 1) I (m + 1, λ, l) f (z) + − m I (m, λ, l) f (z) , z z λ2 λ2 h(z) ≺

z ∈ U,

then

¢0 ¡ α f (z) , q(z) ≺ RIm,λ,l

where q is given by q(z) = 2β − 1 + subordinant.

2−2β 1

nz n

Rz 0

1 t n −1

t+1

z ∈ U,

dt, z ∈ U. The function q is convex and it is the best

³ ´0 α Proof. Following the same steps as in the proof of Theorem 2.10 and considering p(z) = RIm,λ,l f (z) ,

the differential superordination (2.16) becomes h(z) =

1+(2β−1)z ≺ p(z) + zp0 (z), z ∈ U. By using 1+z R Rz 1 1 z t n −1 dt = = 1 1 0 h(t)t n −1 = 1 1 0 1+(2β−1)t 1+t nz n nz n

Lemma 1.1 for γ = 1, we have q(z) ≺ p(z), i.e., q(z) ³ ´0 R z t n1 −1 α dt ≺ RIm,λ,l f (z) , z ∈ U. The function q is convex and it is the best subordinant. 2β − 1 + 2−2β 1 0 t+1 nz n

References [1] M. Acu, S. Owa, Note on a class of starlike functions, RIMS, Kyoto, 2006. [2] Alina Alb Lupa¸s, On a certain subclass of analytic functions defined by Salagean and Ruscheweyh operators, Journal of Mathematics and Applications, No. 31, (2009), p. 67-76. [3] Alina Alb Lupa¸s, A new comprehensive class of analytic functions defined by multiplier transformation, Bull. Sci. Math. Roum., submitted 2009. [4] Alina Alb Lupa¸s, Adriana C˘ ata¸s, Certain special differential superordinations using Sa˘la˘gean and Ruscheweyh operators, Automation, Computers, Applied Mathematics, submitted 2010. [5] Alina Alb Lupa¸s, On a certain subclass of analytic functions defined by a generalized Sa˘la˘gean operator and Ruscheweyh derivative, Carpathian Journal of Mathematics, submitted 2009. 7

LUPAS: SPECIAL DIFFERENTIAL SUBORDINATIONS

115

[6] Alina Alb Lupa¸s, On a certain subclass of analytic functions defined by multiplier transformation and Ruscheweyh derivative, American Journal of Mathematics, submitted 2010. [7] Alina Alb Lupa¸s, Certain special differential superordinations using multiplier transformation, IJOPCA, submitted 2010. [8] Alina Alb Lupa¸s, Certain special differential superordinations using a generalized Sa˘la˘gean operator and Ruscheweyh derivative, Analele Universitatii din Oradea, Fasc. Math . (to appear) 2011. [9] F.M. Al-Oboudi, On univalent functions defined by a generalized Sa˘la˘gean operator, Ind. J. Math. Math. Sci., 27 (2004), 1429-1436. [10] N.E. Cho, H.M. Srivastava, Argument estimates of certain analytic functions defined by a class of multiplier transformations Math. Comput. Modelling, 37 (1-2) (2003), 39-49. [11] N.E. Cho, T.H. Kim, Multiplier transformations and strongly close-to-close functions, Bull. Korean Math. Soc., 40 (3) (2003) 399-410. [12] S.S. Miller, P.T. Mocanu, Subordinants of Differential Superordinations, Complex Variables, vol. 48, no. 10, 815-826, October, 2003. [13] St. Ruscheweyh, New criteria for univalent functions, Proc. Amet. Math. Soc., 49(1975), 109-115. [14] G. St. Sa˘la˘gean, Subclasses of univalent functions, Lecture Notes in Math., Springer Verlag, Berlin, 1013(1983), 362-372. [15] B.A. Uralegaddi, C. Somanatha, Certain classes of univalent functions, Current topics in analytic function theory, World. Sci. Publishing, River Edge, N.Y., (1992), 371-374.

8

JOURNAL 116 OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.1, 116-120, 2011, COPYRIGHT 2011 EUDOXUS PRESS, LLC

On a subclass of analytic functions defined by Ruscheweyh derivative and multiplier transformations Alina Alb Lupa¸s Department of Mathematics and Computer Science University of Oradea 1 Universitatii Street, 410087 Oradea, Romania [email protected]

Abstract P j Let A (p, n) = {f ∈ H(U ) : f (z) = z p + ∞ j=p+n aj z , z ∈ U }, with A (1, 1) = A. We consider in this paper the operator RI γ (m, λ, l) : A → A, defined by RI γ (m, λ, l)f (z) := (1 − γ) Rm f (z) + P∞ h 1+λ(j−1)+l im aj z j and (m + 1)Rm+1 f (z) = γI(m, λ, l)f (z) where I(m, λ, l)f (z) = z + j=2 l+1 z(Rm f (z))0 + mRm f (z), m ∈ N0 , N0 = N ∪ {0}, λ ∈ R, λ ≥ 0, l ≥ 0 is the Ruscheweyh operator. By making use of the above mentioned differential operator, a new subclass of univalent functions in the open unit disc is introduced. The new subclass is denoted by RI γ (m, µ, α, λ, l). Parallel results, for some related classes including the class of starlike and convex functions respectively, are also obtained.

Keywords: Analytic function, starlike function, convex function, Ruscheweyh derivative, multiplier transformations. 2000 Mathematical Subject Classification: 30C45

1

Introduction and definitions

Denote by U the unit disc of the complex plane, U = {z ∈ C : |z| < 1} and H(U ) the space of holomorphic functions in U . P j Let A (p, n) = {f ∈ H(U ) : f (z) = z p + ∞ j=p+n aj z , z ∈ U }, with A (1, n) = An , A (1, 1) = n A1 = A and H[a, n] = {f ∈ H(U ) : f(z) = a + an z + an+1 z n+1 + . . . , z ∈ U }, here p, n ∈ N, a ∈ C. Let S denote the subclass of functions that are univalent in U . By S ∗ (α) we denote ³ 0 a´subclass of A consisting of starlike univalent functions of order α, 0 ≤ α < 1 (z) > α, z ∈ U. which satisfies Re zff (z) ³ 00 ´ (z) Further, a function f belonging to S is said to be convex of order α in U , if and only if Re zff 0 (z) +1 > α, z ∈ U for some α, (0 ≤ α < 1) . We denote by K(α) the class of functions in S which are convex of order α in U and denote by R(α) the class of functions in A which satisfy Re f 0 (z) > α, z ∈ U. It is well known that K(α) ⊂ S ∗ (α) ⊂ S. If f and g are analytic functions in U , we say that f is subordinate to g, written f ≺ g, if there is a function w analytic in U , with w(0) = 0, |w(z)| < 1, for all z ∈ U such that f (z) = g(w(z)) for all z ∈ U. If g is univalent, then f ≺ g if and only if f (0) = g(0) and f (U ) ⊆ g(U ). 1

LUPAS: ANALYTIC FUNCTIONS

117

Definition 1.1 [5] For f ∈ A(p, n), p, n ∈ N, m ∈ N∪ {0}, λ, l ≥ 0, the operator Ip (m, λ, l) f (z) is defined by the following infinite series ¶ µ ∞ P p + λ (j − 1) + l m p aj z j . Ip (m, λ, l) f (z) := z + p+l j=p+n Remark 1.2 It follows from the above definition that Ip (0, λ, l) f (z) = f (z), (p + l) Ip (m + 1, λ, l) f (z) = [p(1 − λ) + l] Ip (m, λ, l) f (z) + λz (Ip (m, λ, l) f (z))0 , for z ∈ U. Remark 1.3 If p = 1, n = 1, we have A(1, 1) = A1 = A, I1 (m, λ, l) f (z) = I (m, λ, l) and (l + 1) I (m + 1, λ, l) f (z) = [l + 1 − λ] I (m, λ, l) f (z) + λz (I (m, λ, l) f (z))0 , for z ∈ U. Remark 1.4 If f ∈ A, f (z) = z + for z ∈ U .

P∞

j j=2 aj z , then I (m, λ, l) f (z) = z +

P∞ ³ 1+λ(j−1)+l ´m j=2

l+1

aj z j ,

Remark 1.5 For l = 0, λ ≥ 0, the operator Dλm = I (m, λ, 0) was introduced and studied by Al-Oboudi [4], which reduced to the Sa˘la˘gean differential operator S m = I (m, 1, 0) [12] for λ = 1. The operator Ilm = I (m, 1, l) was studied recently by Cho and Srivastava [7] and Cho and Kim [8]. The operator Im = I (m, 1, 1) was studied by Uralegaddi and Somanatha [13], the operator Dλδ = I (δ, λ, 0), with δ ∈ R, δ ≥ 0, was introduced by Acu and Owa [1]. Definition 1.6 [11] Ruscheweyh has defined the operator Rm : A → A, R0 f (z) = f (z) R1 f (z) = zf 0 (z) (m + 1)Rm+1 f (z) = z [Rm f (z)]0 + mRm f (z), To prove our main theorem we shall need the following lemma. Lemma 1.7 [10] Let u be analytic in U with u(0) = 1 and suppose that ¶ µ 3α − 1 zu0 (z) , z ∈ U. > (1.1) Re 1 + u(z) 2α Then Re u(z) > α for

z ∈ U and 1/2 ≤ α < 1.

2

z ∈ U.

118

LUPAS: ANALYTIC FUNCTIONS

2

Main results

Definition 2.1 For a function f ∈ A we define the differential operator (2.1)

RI γ (m, λ, l)f (z) = (1 − γ) Rm f (z) + γI(m, λ, l)f (z)

where m ∈ N0 , N0 = N ∪ {0}, λ ∈ R, λ ≥ 0, γ ≥ 0, l ≥ 0. Remark 2.2 For l = 0 and λ = 1 the above defined operator was introduced in [2]. Definition 2.3 We say that a function f ∈ A is in the class RI γ (m, µ, α, λ, l), m ∈ N, µ ≥ 0, α ∈ [0, p), γ ≥ 0 if ¯ ¯ ¶µ ¯ RI γ (m + 1, λ, l)f (z) µ ¯ z ¯ ¯ (2.2) − 1 z ∈ U. ¯ ¯ < 1 − α, ¯ ¯ z RI γ (m, λ, l)f (z)

Remark 2.4 The family RI γ (m, µ, α, λ, l) is a new comprehensive class of analytic functions which includes various new classes of analytic univalent functions as well as some very well-known ones. For example, RI 1 (m, µ, α, λ, l) was studied in [6], RI 1 (0, 1, α, 1, 0) = S ∗ (α) , BI 1 (1, 1, α, 1, 0) = K (α) and BI 1 (0, 0, α, 1, 0) = R (α). Another interesting subclass is the special case RI 1 (0, 2, α, 1, l)=B (α) which has been introduced by Frasin and Darus [9] and also the class RI 1 (0, µ, α, 1, 0) = B(µ, α) which has been introduced by Frasin and Jahangiri [10]. In this note we provide a sufficient condition for functions to be in the class RI(m, µ, α, λ, l). Consequently, as a special case, we show that convex functions of order 1/2 are also members of the above defined family. Theorem 2.5 For the function f ∈ A, m ∈ N, µ ≥ 0, 1/2 ≤ α < 1 if

(2.3)

RI γ (m + 1, λ, l)f (z) (m + 2)RI γ (m + 2, λ, l)f (z) − µ(m + 1) + RI γ (m + 1, λ, l)f (z) RI γ (m, λ, l)f (z) µ ¶ µ ¶ l+1 I(m + 2, λ, l)f (z) l+1 I(m + 1, λ, l)f (z) +γ −m−2 + +γµ −m−1 − λ RI γ (m + 1, λ, l)f (z) λ RI γ (m, λ, l)f (z) ¸ ∙ I(m + 1, λ, l)f (z) l+1 −m−2 + −γ λ RI γ (m + 1, λ, l)f(z) ¸ ∙ I(m, λ, l)f (z) l+1 −m−1 + (m + 1)(µ − 1) ≺ 1 + βz, z ∈ U, +γµ λ RI γ (m, λ, l)f (z)

where β=

3α − 1 2α

then f ∈ RI γ (m, µ, α, λ, l). Proof. If we consider (2.4)

u(z) =

RI γ (m + 1, λ, l) f (z) z 3

µ

z RI γ (m, λ, l) f(z)

¶µ

LUPAS: ANALYTIC FUNCTIONS

119

then u(z) is analytic in U with u(0) = 1. Taking into account the relation (l + 1)I(m + 1, λ, l)f (z) = (1 − λ + l) I(m, λ, l)f (z) + λz (I(m, λ, l)f (z))0 a simple differentiation yields (2.5)

(m + 2)RI γ (m + 2, λ, l)f (z) RI γ (m + 1, λ, l)f(z) zu0 (z) = − µ(m + 1) + u(z) RI γ (m + 1, λ, l)f (z) RI γ (m, λ, l)f (z) µ ¶ µ ¶ l+1 I(m + 2, λ, l)f (z) l+1 I(m + 1, λ, l)f (z) +γ −m−2 + +γµ −m−1 − γ λ RI (m + 1, λ, l)f (z) λ RI γ (m, λ, l)f (z) ¸ ∙ I(m + 1, λ, l)f (z) l+1 −m−2 + −γ λ RI γ (m + 1, λ, l)f(z) ¸ ∙ I(m, λ, l)f (z) l+1 −m−1 + (m + 1)(µ − 1) − 1. +γµ λ RI γ (m, λ, l)f (z)

Using (2.3) we get

¶ µ 3α − 1 zu0 (z) > . Re 1 + u(z) 2α

Thus, from Lemma 1.7 we deduce that ( µ ¶µ ) RI γ (m + 1, λ, l)f (z) z Re > α. z RI γ (m, λ, l)f (z) Therefore, f ∈ RI γ (m, µ, α, λ, l), by Definition 2.3. As a consequence of the above theorem we have the following interesting corollaries [3]. ¾ ½ 00 n o 0 2zf (z)+z 2 f 00 (z) zf 00 (z) zf 00 (z) 1 − , z ∈ U, then Re 1 + > − Corollary 2.6 If f ∈ A and Re > 2 f 0 (z)+zf 00 (z) f 0 (z) f 0 (z) ¢ ¡ 1 z ∈ U. That is, f is convex of order 12 , or f ∈ RI 1 1, 1, 12 , 1, 0 . 2, Corollary 2.7 If f ∈ A and Re that is Re {f 0 (z) + zf 00 (z)} > 12 ,

½

0

0

2zf 0 (z)+z 2 f 00 (z) f 0 (z)+zf 00 (z)

z ∈ U.

o

n Corollary 2.8 If f ∈ A and Re 1 +

words, if the function f is convex of Corollary 2.9 If f ∈ A and Re

hence f ∈ RI 1 (0, 1, 12 , 1, 0).

¾

zf 00 (z) > 12 , f 0 (z) order 12 then f ∈

n

zf 00 (z) f 0 (z)

−

zf 0 (z) f (z)

o

> − 12 ,

z ∈ U,

¢ ¡ then f ∈ RI 1 1, 0, 12 , 1, 0 ,

z ∈ U, then Re f 0 (z) > 12 , ¡ ¢ RI 1 (0, 0, 12 , 1, 0) ≡ R 12 .

> − 32 ,

z ∈ U. In another

z ∈ U, then f is starlike of order

1 2,

References [1] M. Acu, S. Owa, Note on a class of starlike functions, RIMS, Kyoto, 2006. [2] A. Alb Lupa¸s, On special differential subordinations using Sa ˘la˘gean and Ruscheweyh operators, Mathematical Inequalities and Applications, Volume 12, Issue 4, October 2009 (to appear). 4

120

LUPAS: ANALYTIC FUNCTIONS

[3] A. Alb Lupa¸s, A subclass of analytic functions defined by Ruscheweyh derivative, Acta Universitatis Apulensis, nr. 19/2009, 31-34. [4] F.M. Al-Oboudi, On univalent functions defined by a generalized Sa˘la ˘gean operator, Ind. J. Math. Math. Sci., 27 (2004), 1429-1436. [5] A. Ca˘ta¸s, On certain class of p-valent functions defined by new multiplier transformations, Proceedings Book of the International Symposium on Geometric Function Theory and Applications, August 20-24, 2007, TC Istanbul Kultur University, Turkey, 241-250. [6] A. Ca˘ta¸s and A. Alb Lupa¸s, A New Comprehensive Class of Analytic Functions Using Multiplier Transformations, 2009, (submitted). [7] N.E. Cho, H.M. Srivastava, Argument estimates of certain analytic functions defined by a class of multiplier transformations Math. Comput. Modelling, 37 (1-2) (2003), 39-49. [8] N.E. Cho, T.H. Kim, Multiplier transformations and strongly close-to-close functions, Bull. Korean Math. Soc., 40 (3) (2003) 399-410. [9] B.A. Frasin and M. Darus, On certain analytic univalent functions, Internat. J. Math. and Math. Sci., 25(5), 2001, 305-310. [10] B.A. Frasin and Jay M. Jahangiri, A new and comprehensive class of analytic functions, Analele Universita˘¸tii din Oradea, Tom XV, 2008, 61-64. [11] St. Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math. Soc., 49(1975), 109-115. [12] G. St. Sa˘la˘gean, Subclasses of univalent functions, Lecture Notes in Math., Springer Verlag, Berlin, 1013(1983), 362-372. [13] B.A. Uralegaddi, C. Somanatha, Certain classes of univalent functions, Current topics in analytic function theory, World. Sci. Publishing, River Edge, N.Y., (1992), 371-374.

5

JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.1, 121-126, 2011, COPYRIGHT 2011 EUDOXUS PRESS, 121 LLC

On special differential superordinations using multiplier transformation Alina Alb Lupa¸s Department of Mathematics and Computer Science University of Oradea str. Universitatii nr. 1, 410087 Oradea, Romania [email protected] Abstract In the present paper we establish several differential superordinations regardind the multiplier transformations µ ¶m ∞ P 1 + λ (j − 1) + l I (m, λ, l) f (z) = z + aj z j , l+1 j=2 where m ∈ N∪ {0}, λ, l ≥ 0 and f ∈ A,

A = {f ∈ H(U ) : f(z) = z +

∞ P

j=2

aj z j , z ∈ U }.

A number of interesting consequences of some of these superordination results are discussed. Relevant connections of some of the new results obtained in this paper with those in earlier works are also provided.

Keywords: differential superordination, convex function, best subordinant, differential operator. 2000 Mathematical Subject Classification: 30C45, 30A20, 34A40.

1

Introduction

Denote by U the unit disc of the complex plane, U = {z ∈ C : |z| < 1} and H(U ) the space of holomorphic functions in U . P∞ Let A (p, n) = {f ∈ H(U ) : f (z) = z p + j=p+n aj z j , z ∈ U }, with A (1, n) = An , A (1, 1) = A1 = A and H[a, ) : f (z) = a + aon z n + an+1 z n+1 + . . . , z ∈ U }, where p, n ∈ N, a ∈ C. Denote by n n] = {f ∈ H(U zf 00 (z) K = f ∈ A : Re f 0 (z) + 1 > 0, z ∈ U the class of normalized convex functions in U . If f and g are analytic functions in U , we say that f is superordinate to g, written g ≺ f , if there is a function w analytic in U , with w(0) = 0, |w(z)| < 1, for all z ∈ U such that g(z) = f (w(z)) for all z ∈ U . If f is univalent, then g ≺ f if and only if f (0) = g(0) and g(U ) ⊆ f (U ). Let ψ : C2 × U → C and h analytic in U . If p and ψ (p (z) , zp0 (z) ; z) are univalent in U and satisfies the (first-order) differential superordination h(z) ≺ ψ(p(z), zp0 (z); z),

z ∈ U,

(1.1)

then p is called a solution of the differential superordination. The analytic function q is called a subordinant of the solutions of the differential superordination, or more simply a subordinant, if q ≺ p for all p satisfying (1.1). An univalent subordinant qe that satisfies q ≺ qe for all subordinants q of (1.1) is said to be the best subordinant of (1.1). The best subordinant is unique up to a rotation of U .

Definition 1.1 For f ∈ A(p, n), p, n ∈ N, m ∈ N∪ {0}, λ, l³≥ 0, the operator Ip (m, λ, l) f (z) is defined by ´m P∞ p+λ(j−p)+l p j aj z . the following infinite series Ip (m, λ, l) f (z) := z + j=p+n p+l

Remark 1.1 It follows from the above definition that Ip (0, λ, l) f (z) = f (z), (p + l) Ip (m + 1, λ, l) f (z) = 0 [p(1 − λ) + l] Ip (m, λ, l) f (z) + λz (Ip (m, λ, l) f (z)) , for z ∈ U. 1

122

LUPAS: USING MULTIPLIER TRANSFORMATION

Remark 1.2 If p = 1, n = 1, we have A(1, 1) = A1 = A, I1 (m, λ, l) f (z) = I (m, λ, l) and (l + 1) I (m + 1, λ, l) f (z) = [l + 1 − λ] I (m, λ, l) f (z) + λz (I (m, λ, l) f (z))0 , for z ∈ U. P P∞ ³ 1+λ(j−1)+l ´m j aj z j , for Remark 1.3 If f ∈ A, f (z) = z + ∞ j=2 aj z , then I (m, λ, l) f (z) = z + j=2 l+1 z ∈ U. Remark 1.4 For l = 0, λ ≥ 0, the operator Dλm = I (m, λ, 0) was introduced and studied by Al-Oboudi [2], which reduced to the Sa˘la˘gean differential operator S m = I (m, 1, 0) [4] for λ = 1. Definition 1.2 We denote by Q the set of functions that are analytic and injective on U \E (f ), where E (f ) = {ζ ∈ ∂U : lim f (z) = ∞}, and are such that f 0 (ζ) 6= 0 for ζ ∈ ∂U \E (f ). The subclass of Q for z→ζ

which f (0) = a is denoted by Q (a). We will use the following lemmas. Lemma 1.1 (Miller and Mocanu [3, Th. 3.1.6, p. 71]) Let h be a convex function with h(0) = a, and let γ ∈ C\{0} be a complex number with Re γ ≥ 0. If p ∈ H[a, n] ∩ Q, p(z) + γ1 zp0 (z) is univalent in U and Rz h(z) ≺ p(z) + γ1 zp0 (z), z ∈ U, then q(z) ≺ p(z), z ∈ U, where q(z) = nzγγ/n 0 h(t)tγ/n−1 dt, z ∈ U. The function q is convex and is the best subordinant. Lemma 1.2 (Miller and Mocanu [3]) Let q be a convex function in U and let h(z) = q(z) + γ1 zq 0 (z), z ∈ U, where Re γ ≥ 0. If p ∈ H [a, n] ∩ Q, p(z) + γ1 zp0 (z) is univalent in U and q(z) + γ1 zq 0 (z) ≺ p(z) + γ1 zp0 (z) , Rz z ∈ U, then q(z) ≺ p(z), z ∈ U, where q(z) = nzγγ/n 0 h(t)tγ/n−1 dt, z ∈ U. The function q is the best subordinant.

2

Main results

Theorem 2.1 LetR h be a convex function in U with h (0) = 1. Let m ∈ N, λ, l ≥ 0, f ∈ A, F (z) = z c t f (t) dt, z ∈ U , Re c > −2, and suppose that (I (m, λ, l) f (z))0 is univalent in U , Ic (f ) (z) = zc+2 c+1 0 0 (I (m, λ, l) F (z)) ∈ H [1, 1] ∩ Q and h (z) ≺ (I (m, λ, l) f (z))0 , then where q(z) =

0

c+2 z c+2

Rz

q (z) ≺ (I (m, λ, l) F (z)) ,

c+1

z ∈ U,

(2.1)

z ∈ U,

dt. The function q is convex and it is the best subordinant. Rz Proof. We have z c+1 F (z) = (c + 2) 0 tc f (t) dt and differentiating it, with respect to z, we obtain 0 (c + 1) F (z)+zF 0 (z) = (c + 2) f (z) and (c + 1) I (m, λ, l) F (z)+z (I (m, λ, l) F (z)) = (c + 2) I (m, λ, l) f (z) , z ∈ U. Differentiating the last relation we have 0

h(t)t

0

(I (m, λ, l) F (z)) +

1 00 0 z (I (m, λ, l) F (z)) = (I (m, λ, l) f (z)) , z ∈ U. c+2

(2.2)

Using (2.2), the differential superordination (2.1) becomes 0

h (z) ≺ (I (m, λ, l) F (z)) + Denote

1 00 z (I (m, λ, l) F (z)) . c+2 0

p (z) = (I (m, λ, l) F (z)) , z ∈ U.

(2.3)

(2.4)

1 Replacing (2.4) in (2.3) we obtain h (z) ≺ p (z)+ c+2 zp0 (z), z ∈ U. Using Lemma 1.1 for n = 1 and γ = c+2, Rz we have q (z) ≺ p (z) , z ∈ U, i.e. q (z) ≺ (I (m, λ, l) F (z))0 , z ∈ U, where q(z) = zc+2 h(t)tc+1 dt. c+2 0 The function q is convex and it is the best subordinant.

2

LUPAS: USING MULTIPLIER TRANSFORMATION

123

1 Theorem 2.2 Let q be a convex function in U and let h (z) = q (z) + c+2 zq 0 (z) , where z ∈ U, Re c > −2. R z 0 Let m ∈ N, λ, l ≥ 0, f ∈ A, F (z) = Ic (f ) (z) = zc+2 tc f (t) dt, z ∈ U and suppose that (I (m, λ, l) f (z)) c+1 0 is univalent in U , (I (m, λ, l) F (z))0 ∈ H [1, 1] ∩ Q and

h (z) ≺ (I (m, λ, l) f (z))0 , then where q(z) =

0

c+2 z c+2

Rz 0

q (z) ≺ (I (m, λ, l) F (z)) ,

z ∈ U,

(2.5)

z ∈ U,

h(t)tc+1 dt. The function q is the best subordinant.

0

Proof. Following the same steps as in the proof of Theorem 2.1 and considering p (z) = (I (m, λ, l) F (z)) , 1 1 zq 0 (z) ≺ p (z) + c+2 zp0 (z), z ∈ U. z ∈ U, the differential superordination (2.5) becomes h (z) = q (z) + c+2 0 Using Lemma 1.2 for n = R1 and γ = c + 2, we have q (z) ≺ p (z) , z ∈ U, i.e. q (z) ≺ (I (m, λ, l) F (z)) , z c+2 c+1 z ∈ U, where q(z) = zc+2 0 h(t)t dt. The function q is the best subordinant.

Theorem 2.3 Let h (z) = 1+(2β−1)z , where β ∈ [0, 1). Let m ∈ N, λ, l ≥ 0, f ∈ A, F (z) = Ic (f ) (z) = 1+z Rz c 0 0 c+2 t f (t) dt, z ∈ U, Re c > −2, and suppose that (I (m, λ, l) f (z)) is univalent in U , (I (m, λ, l) F (z)) ∈ z c+1 0 H [1, 1] ∩ Q and (2.6) h (z) ≺ (I (m, λ, l) f (z))0 , z ∈ U,

then

0

q (z) ≺ (I (m, λ, l) F (z)) , z ∈ U, R z tc+1 where q is given by q(z) = 2β − 1 + (c+2)(2−2β) dt, z ∈ U. The function q is convex and it is the best z c+2 0 t+1 subordinant. Proof. Following the same steps as in the proof of Theorem 2.1 and considering p(z) = (I (m, λ, l) F (z))0 , 1 the differential superordination (2.6) becomes h(z) = 1+(2β−1)z ≺ p (z) + c+2 zp0 (z) , z ∈ U. By 1+z Rz c+1 using Lemma 1.1 for γ = c + 2 and n = 1, we have q(z) ≺ p(z), i.e., q(z) = zc+2 h(t)t dt = c+2 0 R R c+1 z z 0 1+(2β−1)t c+1 (c+2)(2−2β) c+2 t t dt = 2β − 1 + dt ≺ (I (m, λ, l) F (z)) , z ∈ U. The function q z c+2 0 1+t z c+2 0 t+1 is convex and it is the best subordinant. Theorem 2.4 Let h be a convex function, h(0) = 1. Let m ∈ N, λ, l ≥ 0, f ∈ A and suppose that (z) (I (m, λ, l) f (z))0 is univalent and I(m,λ,l)f ∈ H [1, 1] ∩ Q. If z 0

h(z) ≺ (I (m, λ, l) f (z)) , then

where q(z) =

R 1 z

z

0

q(z) ≺

I (m, λ, l) f (z) , z

z ∈ U,

(2.7)

z ∈ U,

h(t)dt. The function q is convex and it is the best subordinant.

Proof. By using the properties of operator I (m, λ, l), we have P∞ ³ 1+λ(j−1)+l ´m aj z j , z ∈ U. I (m, λ, l) f (z) = z + j=2 l+1 P∞ 1+λ(j−1)+l m j z+ j=2 ( ) aj z (z) l+1 = = 1 + p1 z + p2 z 2 + ..., z ∈ U. We deduce that Consider p(z) = I(m,λ,l)f z z p ∈ H[1, 1]. 0 Let I (m, λ, l) f (z) = zp(z), z ∈ U. Differentiating we obtain (I (m, λ, l) f (z)) = p(z) + zp0 (z), z ∈ U. 0 Then (2.7) becomes h(z) ≺ p(z) + zp (z), z ∈ U. By using Lemma 1.1 for n = 1 and γ = 1, we have Rz (z) q(z) ≺ p(z), z ∈ U, i.e. q(z) ≺ I(m,λ,l)f , z ∈ U, where q(z) = z1 0 h(t)dt. The function q is convex z and it is the best subordinant. Theorem 2.5 Let q be convex in U and let h be defined by h (z) = q (z) + zq 0 (z) . (z) If m ∈ N, λ, l ≥ 0, f ∈ A, suppose that (I (m, λ, l) f (z))0 is univalent and I(m,λ,l)f ∈ H [1, 1] ∩ Q and z satisfies the differential superordination h(z) = q (z) + zq 0 (z) ≺ (I (m, λ, l) f (z))0 , 3

z ∈ U,

(2.8)

124

LUPAS: USING MULTIPLIER TRANSFORMATION

then

I (m, λ, l) f (z) , z ∈ U, z R z where q(z) = z1 0 h(t)dt. The function q is the best subordinant. q(z) ≺

(z) , the Proof. Following the same steps as in the proof of Theorem 2.4 and considering p(z) = I(m,λ,l)f z 0 0 differential superordination (2.8) becomes q(z) + zq (z) ≺ p(z) + zp (z) , z ∈ U. Using Lemma 1.2 for Rz (z) n = 1 and γ = 1, we have q(z) ≺ p(z), z ∈ U, i.e. q(z) = z1 0 h(t)dt ≺ I(m,λ,l)f , z ∈ U, and q is the z best subordinant.

Theorem 2.6 Let h(z) =

1+(2β−1)z 1+z 0

be a convex function in U , where 0 ≤ β < 1. Let m ∈ N, λ, l ≥ 0, f ∈ A

and suppose that (I (m, λ, l) f (z)) is univalent and

I(m,λ,l)f (z) z

∈ H [1, 1] ∩ Q. If

h(z) ≺ (I (m, λ, l) f (z))0 , then q(z) ≺

z ∈ U,

I (m, λ, l) f (z) , z

(2.9)

z ∈ U,

, z ∈ U. The function q is convex and it is the best where q is given by q(z) = 2β − 1 + 2(1 − β) ln(1+z) z subordinant. (z) , Proof. Following the same steps as in the proof of Theorem 2.4 and considering p(z) = I(m,λ,l)f z 1+(2β−1)z 0 the differential superordination (2.9) becomes h(z) = ≺ p(z) + zp (z), z ∈ U. By using 1+z Rz Rz dt = Lemma 1.1 for γ = 1 and n = 1, we have q(z) ≺ p(z), i.e., q(z) = z1 0 h(t)dt = z1 0 1+(2β−1)t 1+t

2β − 1 + 2(1 − β) z1 ln(z + 1) ≺

I(m,λ,l)f (z) , z

z ∈ U. The function q is convex and it is the best subordinant.

Theorem 2.7 Let h be a convex function, h(0) = 1. Let m ∈ N, λ, l ≥ 0, f ∈ A and suppose that ´0 ³ zI(m+1,λ,l)f (z) (z) is univalent and I(m+1,λ,l)f I(m,λ,l)f (z) I(m,λ,l)f (z) ∈ H [1, 1] ∩ Q. If h(z) ≺

µ

zI (m + 1, λ, l) f (z) I (m, λ, l) f (z)

then

where q(z) =

1 z

Rz

q(z) ≺

¶0

,

I (m + 1, λ, l) f (z) , I (m, λ, l) f (z)

z ∈ U,

(2.10)

z ∈ U,

h(t)dt. The function q is convex and it is the best subordinant. P P∞ ³ 1+λ(j−1)+l ´m j Proof. For f ∈ A, f (z) = z + ∞ a z we have I (m, λ, l) f (z) = z + aj z j , z ∈ U. j=2 j j=2 l+1 P∞ 1+λ(j−1)+l m+1 j z+ j=2 ( aj z ) (z) l+1 P Consider p(z) = I(m+1,λ,l)f I(m,λ,l)f (z) = z+ ∞ ( 1+λ(j−1)+l )m aj z j . l+1 j=2 ³ ´0 (I(m+1,λ,l)f (z))0 (I(m,λ,l)f (z))0 (z) 0 We have p (z) = I(m,λ,l)f (z) − p (z) · I(m,λ,l)f (z) and we obtain p (z) + z · p0 (z) = zI(m+1,λ,l)f . I(m,λ,l)f (z) Relation (2.10) becomes h(z) ≺ p(z) + zp0 (z), z ∈ U. By using Lemma 1.1 for n = 1 and γ = 1, we Rz (z) have q(z) ≺ p(z), z ∈ U, i.e. q(z) ≺ I(m+1,λ,l)f z ∈ U, where q(z) = z1 0 h(t)dt. The function q I(m,λ,l)f (z) , is convex and it is the best subordinant. 0

Theorem 2.8 Let q be a convex function and h be defined by h (z) = q (z) + zq 0 (z) . Let λ, l ≥ 0, m ∈ N, ´0 ³ (z) (z) is univalent and I(m+1,λ,l)f f ∈ A and suppose that zI(m+1,λ,l)f I(m,λ,l)f (z) I(m,λ,l)f (z) ∈ H [1, 1] ∩ Q. If 0

h(z) = q (z) + zq (z) ≺ then

where q(z) =

1 z

Rz 0

q(z) ≺

µ

zI (m + 1, λ, l) f (z) I (m, λ, l) f (z)

I (m + 1, λ, l) f (z) , I (m, λ, l) f (z)

h(t)dt. The function q is the best subordinant. 4

¶0

z ∈ U,

,

z ∈ U,

(2.11)

LUPAS: USING MULTIPLIER TRANSFORMATION

125

(z) Proof. Following the same steps as in the proof of Theorem 2.7 and considering p(z) = I(m+1,λ,l)f I(m,λ,l)f (z) , the differential superordination (2.11) becomes h (z) = q(z) + zq 0 (z) ≺ p(z) + zp0 (z), z ∈ U. By using Lemma Rz (z) 1.2 for n = 1 and γ = 1, we have q(z) ≺ p(z), z ∈ U, i.e. q(z) = z1 0 h(t)dt ≺ I(m+1,λ,l)f z ∈ U, I(m,λ,l)f (z) , and q is the best subordinant.

Theorem 2.9 Let h(z) = 1+(2β−1)z be a convex function in U , where 0 ≤ β < 1. Let λ, l ≥ 0, m ∈ N, f ∈ A 1+z ³ ´0 (z) (z) and suppose that zI(m+1,λ,l)f is univalent and I(m+1,λ,l)f I(m,λ,l)f (z) I(m,λ,l)f (z) ∈ H [1, 1] ∩ Q. If ¶0

µ

zI (m + 1, λ, l) f (z) I (m, λ, l) f (z)

q(z) ≺

I (m + 1, λ, l) f (z) , I (m, λ, l) f (z)

h(z) ≺ then

,

z ∈ U,

(2.12)

z ∈ U,

where q is given by q(z) = 2β − 1 + 2(1 − β) ln(1+z) , z ∈ U. The function q is convex and it is the best z subordinant. Proof. Following the same steps as in the proof of Theorem 2.7 and considering p(z) =

I(m+1,λ,l)f (z) I(m,λ,l)f (z) ,

1+(2β−1)z 1+z

≺ p(z) + zp0 (z), z ∈ U. By using Rz Rz dt = Lemma 1.1 for γ = 1 and n = 1, we have q(z) ≺ p(z), i.e., q(z) = z1 0 h(t)dt = z1 0 1+(2β−1)t 1+t

the differential superordination (2.12) becomes h(z) = 2β − 1 + 2(1 − β) z1 ln(z + 1) ≺

I(m+1,λ,l)f (z) I(m,λ,l)f (z) ,

z ∈ U. The function q is convex and it is the best subordinant.

Theorem 2.10 Let h ∈¡H (U ) be ¢ a convex function in U with h(0) = 1 and let 0 λ, l ≥ 0, m ∈ N, f ∈ A, l+1 l+1 I (m, λ, l) f (z) is univalent and [I (m, λ, l) f (z)] ∈ H [1, 1] ∩ Q. If I (m + 1, λ, l) f (z) + 2 − λ λ µ ¶ l+1 l+1 I (m + 1, λ, l) f (z) + 2 − I (m, λ, l) f (z) , z∈U (2.13) h(z) ≺ λ λ holds, then R 1 z

q(z) ≺ [I (m, λ, l) f (z)]0 ,

z ∈ U,

h(t)dt. The function q is convex and it is the best subordinant. P∞ ³ 1+λ(j−1)+l ´m P j Proof. For f ∈ A, f (z) = z + ∞ aj z j , z ∈ U. j=2 aj z we have I (m, λ, l) f (z) = z + j=2 l+1 Let ¶m ∞ µ X 1 + λ (j − 1) + l p(z) = (I (m, λ, l) f (z))0 = 1 + jaj z j−1 = 1 + p1 z + p2 z 2 + .... (2.14) l + 1 j=2

where q(z) =

z

0

We obtain p (z) + z · p0 (z) = I (m, λ, l) f (z) + z (I (m, λ, l) f (z))0 = I (m, λ, l) f (z) + ¢ ¡ (l+1)I(m+1,λ,l)f (z)−(l+1−λ)I(m,λ,l)f (z) l+1 I (m, λ, l) f (z) . = l+1 λ λ I (m + 1, λ, l) f (z) + 2 − λ Using the notation in (2.14), the differential superordination becomes h(z) ≺ p(z) + zp0 (z). By using 0 Lemma 1.1 for Rn = 1 and γ = 1, we have q(z) ≺ p(z), z ∈ U, i.e. q(z) ≺ (I (m, λ, l) f (z)) , z ∈ U, z where q(z) = z1 0 h(t)dt. The function q is convex and it is the best subordinant. Example 2.1 [1] If m = 1, α = 1, f ∈ A, we deduce that q(z) + zq 0 (z) ≺ f 0 (z) + 3zf 00 (z) + z 2 f 000 (z), which yields that q(z) ≺ f 0 (z) + zf 00 (z), z ∈ U.

Theorem 2.11 Let q be a convex function U¢ and h (z) = q (z) + zq 0 (z). Let λ, l ≥ 0, m ∈ N, f ∈ ¡ inl+1 l+1 A, suppose that λ I (m + 1, λ, l) f (z) + 2 − λ I (m, λ, l) f (z) is univalent in U and [I (m, λ, l) f (z)]0 ∈ H [1, 1] ∩ Q. If µ ¶ l+1 l+1 0 I (m + 1, λ, l) f (z) + 2 − I (m, λ, l) f (z) , z ∈ U, (2.15) h(z) = q (z) + zq (z) ≺ λ λ then where q(z) =

1 z

Rz 0

q(z) ≺ (I (m, λ, l) f (z))0 ,

h(t)dt. The function q is the best subordinant. 5

z ∈ U,

126

LUPAS: USING MULTIPLIER TRANSFORMATION 0

Proof. Following the same steps as in the proof of Theorem 2.10 and considering p(z) = (I (m, λ, l) f (z)) , 0 the differential superordination (2.15) becomes h(z) = q (z) + zq 0 (z) R ≺ p(z) + zp (z), z ∈ U.0 By using 1 z Lemma 1.2 for γ = 1 and n = 1, we have q(z) ≺ p(z), i.e., q(z) = z 0 h(t)dt ≺ (I (m, λ, l) f (z)) , z ∈ U. The function q is the best subordinant. Theorem 2.12 Let h(z) = 1+(2β−1)z be a convex function in U , where 0 ≤ β < 1. Let λ, l ≥ 0, m ∈ N, 1+z ¢ ¡ I (m, λ, l) f (z) is univalent in U and [I (m, λ, l) f (z)]0 ∈ f ∈ A, suppose that l+1 I (m + 1, λ, l) f (z)+ 2 − l+1 λ λ H [1, 1] ∩ Q. If µ ¶ l+1 l+1 I (m + 1, λ, l) f (z) + 2 − I (m, λ, l) f (z) , z ∈ U, (2.16) h(z) ≺ λ λ then

q(z) ≺ (I (m, λ, l) f (z))0 ,

z ∈ U,

where q is given by q(z) = 2β − 1 + 2(1 − β) ln(1+z) , z ∈ U. The function q is convex and it is the best z subordinant. Proof. Following the same steps as in the proof of Theorem 2.10 and considering p(z) = (I (m, λ, l) f (z))0 , the differential superordination (2.16) becomes h(z) = 1+(2β−1)z ≺ p(z) + zp0 (z), z ∈ U. By using 1+z Rz Rz dt = Lemma 1.1 for γ = 1 and n = 1, we have q(z) ≺ p(z), i.e., q(z) = z1 0 h(t)dt = z1 0 1+(2β−1)t 1+t 2β − 1 + 2(1 − β) z1 ln(z + 1) ≺ (I (m, λ, l) f (z))0 , z ∈ U. The function q is convex and it is the best subordinant.

References [1] Alina Alb Lupa¸s, A special comprehensive class of analytic functions defined by multiplier transformation, Journal of Computational Analysis and Applications, Vol. 12, No. 2, 2010, 387-395 . [2] F.M. Al-Oboudi, On univalent functions defined by a generalized Sa˘la˘gean operator, Ind. J. Math. Math. Sci., 27 (2004), 1429-1436. [3] S.S. Miller, P.T. Mocanu, Subordinants of Differential Superordinations, Complex Variables, vol. 48, no. 10, 815-826, October, 2003. [4] G. St. Sa˘la˘gean, Subclasses of univalent functions, Lecture Notes in Math., Springer Verlag, Berlin, 1013(1983), 362-372.

6

JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.1, 127-156, 2011, COPYRIGHT 2011 EUDOXUS PRESS, 127 LLC

Approximation Properties of Some Multicomplex Singular Integrals in the Unit Polydisk∗ George A. Anastassiou and Sorin G. Gal Department of Mathematical Sciences The University of Memphis Memphis, TN 38152, U.S.A. [email protected] and Department of Mathematics and Computer Science University of Oradea Str. Universitatii No. 1 410087 Oradea, ROMANIA [email protected] Abstract The aim of this paper is to obtain several results in approximation by the multi-complex Picard, Poisson–Cauchy and Gauss–Weierstrass singular integrals in terms of the modulus of continuity in polydisks. Also, exact estimates are proved.

AMS 2000 Mathematics Subject Classification : 30E10, 32E30, 41A25, 41A35. Key words and phrases : Multi-complex singular integrals, Jackson-type estimates, exact estimates, global smoothness preservation.

1

Introduction

Let us consider the open polydisk Dm , where D = {z ∈ C : |z| < 1}, m ∈ m m N, and A(D ) = {f : D → C; f is analytic with respect to ³ any variable ´ m m z1 , z2 , . . . , zm ∈ D and continuous on D }. Therefore, if f ∈ A D , then according to e. g. [2], Theorem 2, p. 65, we can write f (z1 , . . . , zm ) =

∞ X

im αi1 ,i2 ...,im z1i1 z2i2 . . . zm ,

i1 ,i2 ...,im =0 ∗ This

paper was written during the 2009 Spring Semester when the second author was a Visiting Professor at the Department of Mathematical Sciences, The University of Memphis, TN, U.S.A.

1

128

ANASTASSIOU, GAL: MULTICOMPLEX SINGULAR INTEGRALS

³ m´ for all z1 , . . . , zm ∈ D, m ∈ N and αi1 ,i2 ...,im ∈ C. For f ∈ A D , i ∈ C with i2 = −1 and ξj ∈ R, ξj > 0, let us consider the complex multi-singular integrals Pξ1 ,...,ξm (f ) (z1 , . . . , zm ) 1 = m Qm 2 j=1 ξj

Z

Z

∞

∞

... −∞

¡

iu1

f z1 e

ium

, . . . , zm e

m ¢Y

−∞

e−|uj |/ξj du1 . . . dum ,

j=1

Qξ1 ,...,ξm (f ) (z1 , . . . zm ) ¢ Z π ¡ iu1 f z1 e , . . . , zm eium j=1 ξj ¢ Qm ¡ 2 du1 . . . dum , = ... 2 πm −π −π j=1 uj + ξj Qm Z Z ∞ 2m j=1 ξj3 ∞ Rξ1 ,...,ξm (f )(z1 , . . . , zm ) = . . . πm −∞ −∞ ¡ iu ¢ iu f z1 e 1 , . . . , zm e m Qm du1 . . . dum , 2 2 2 j=1 (uj + ξj ) Z ∞ Z ∞ ¡ ¢ 1 Wξ1 ,...,ξm (f )(z1 , . . . , zm ) = Qm p ... f z1 eiu1 , . . . , zm eium πξj −∞ −∞ j=1 Qm

Z

π

·

m Y

2

e−uj /ξj du1 . . . dum ,

j=1

zj ∈ D, j = 1, . . . , m. Here Pξ1 ,...,ξm (f ) is called of Picard type, Qξ1,... ξm (f ) and Rξ1 ,...,ξm (f ) are called of Poisson-Cauchy type and Wξ1 ,...,ξm (f ), is called of Gauss-Weierstrass type. In [1] we have obtained upper estimates in approximation and global smoothness preservation properties for the univariate analogs of the above integrals, while in [3] exact estimates are given. In this paper we extend these mentioned results to the multivariate case. For our purpose, the following result called the Cauchy’s formula on polydisk is needed. Theorem A. (see e.g. [4]) Let i2 = −1, a = (a1 , ..., ap ) ∈ Cp , R = (R1 , ..., Rp ), Rj > 0, j = 1, ..., p and f : P (a; R) → C be a holomorphic function in P (a; R) = {(z1 , ..., zp ) ∈ Cp ; |zj − aj | < Rj , j = 1, ..., p} and continuous in P (a; R). Then for all z = (z1 , ..., zp ) ∈ P (a; R) and all S kj ∈ N {0}, j = 1, ..., p, we have : (i) Z Z 1 f (u1 , ..., up )du1 ...dup ··· . f (z1 , ..., zp ) = (2πi)p |up −zp |=Rp |u1 −z1 |=R1 (u1 − z1 )...(up − zp )

2

ANASTASSIOU, GAL: MULTICOMPLEX SINGULAR INTEGRALS

(ii) ∂ k1 +...+kp f k

=

2

(k1 !)...(kp !) (2πi)p

∂z1k1 ...∂zp p Z ···

Z

|up −zp |=Rp

(z1 , ..., zp )

|u1 −z1 |=R1

f (u1 , ..., up )du1 ...dup . (u1 − z1 )k1 +1 ...(up − zp )kp +1

Multi-Complex Picard Integrals

In this section we study the approximation properties of Pξ1 ,...,ξm (f )(z1 , . . . , zm ). ³ ´ Theorem 1. Let f ∈ A D

m

and ξi > 0, i = 1, . . . , m. We have m

(i) Pξ1 ,...,ξm (f )(z1 , . . . , zm ) is continuous on D and analytic on Dm , moreover we can write Pξ1 ,...,ξm (f )(z1 , . . . , zm )  ! Ã ∞ m X Y 1 im  z1i1 z2i2 . . . zm , = αi1 ,i2 ,...,im  2ξ2 1 + i j j i ,i ,...,i =0 j=1 1

2

m

where f (z1 , . . . , zm ) =

∞ X

im , αi1 ,i2 ,...,im z1i1 . . . zm

i1 ,i2 ...,im =0

for all z1 , . . . , zm ∈ D. (ii) ω1 (Pξ1 ,...,ξm (f ); δ1 , δ2 , . . . , δm )Dm ≤ ω1 (f ; δ1 , δ2 , . . . , δm )Dm , for all δ1 , δ2 , . . . δm ≥ 0, where ω1 (f ; δ1 , . . . , δm )Dm = sup{|f (u1 , . . . , um ) − f (v1 , . . . , vm )|; |uj − vj | ≤ δj , j = 1, . . . , m}; (iii) |Pξ1 ,...,ξm (f )(z1 , . . . , zm ) − f (z1 , . . . , zm )| ≤ (m + 1)ω1 (f ; ξ1 , . . . , ξm )Dm , for all zj ∈ D, ξj > 0; j = 1, . . . , m. (iv) Denote Dr ³= {z´∈ C; |z| < r} (clearly D1 = D) and for R > 1 let us m suppose that f ∈ A DR , that is we can write f (z1 , . . . , zm ) =

∞ X

im αi1 ,i2 ...,im z1i1 z2i2 . . . zm ,

i1 ,i2 ...,im =0

for all z1 , . . . , zm ∈ DR , m ∈ N and αi1 ,i2 ...,im ∈ C. 3

129

130

ANASTASSIOU, GAL: MULTICOMPLEX SINGULAR INTEGRALS

If the Taylor’s development of f (z1 , z2 , ..., zm ) contains at least one term of pm the form αp1 ,p2 ,...,pm z1p1 z2p2 ...zm with p1 , p2 , ..., pm ≥ 1 and αp1 ,p2 ,...,pm 6= 0 for q = 0, and at least one term of the form km +pm αk1 +p1 ,k2 +p2 ,...,km +pm z1k1 +p1 z2k2 +p2 ...zm with p1 , p2 , ..., pm ≥ 1,

such that αk1 +p1 ,k2 +p2 ,...,km +pm 6= 0 and k1 + k2 + ... + km = q forPq ≥ 1, then m for all 1 ≤ r < r1 < R, ξj ∈ (0, 1], j = 1, ..., m, kj ∈ N ∪ {0} with j=1 kj = q, q ∈ N ∪ {0}, we have ° ° m ° ∂ k1 +...+km P X ∂ k1 +...+km f ° ° ° ξ1 ,...,ξm (f ) − ∼ ξj2 , ° ° km km ° ° ∂z1k1 ...∂zm ∂z1k1 ...∂zm j=1 r where the constants in the equivalence depend only on f , q, r and r1 . Here kf kr = sup{|f (z1 , ..., zm )|; |zj | ≤ r, j = 1, ..., m}. m Proof. (i) Let z0 , zn ∈ D be with lim zn,j = z0,j , j = 1, . . . , m. We get n→∞

|Pξ1 ,...,ξm (f ) (zn,1 , . . . , zn,m ) − Pξ1 ,...,ξm (f )(z0,1 , . . . , z0,m )| Z ∞ Z ∞ ¡ ¢ 1 ≤ m Qm ... |f zn,1 eiu1 , . . . , zn,m eium 2 −∞ j=1 ξj −∞ m ¡ ¢ Y −f z0,1 eiu1 , . . . , z0,m eium | · e−|uj |/ξj du1 . . . dum

Z

1 Qm m

≤

2

j=1 ξj

¡

j=1 ∞

Z

∞

... −∞

−∞

¢ ω1 f ; |zn,1 eiu1 − z0,1 eiu1 |, . . . , |zn,m eium − z0,m eium | Dm · = ·

2

Z

1 Qm m

m Y

j=1 ξj

m Y

e−|uj |/ξj du1 . . . dum

j=1

Z

∞

∞

... −∞

−∞

ω1 (f ; |zn,1 − z0,1 |, . . . , |zn,m − z0,m |)Dm

e−|uj |/ξj du1 . . . dum = ω1 (f ; |zn,1 − z0,1 |, . . . , |zn,m − z0,m |)Dm

j=1

·

2

1 Qm m

j=1 ξj

Z

Z

∞

∞

... −∞

m Y

e−|uj |/ξj du1 . . . dum

−∞ j=1

= ω1 (f ; |zn,1 − z0,1 |, . . . , |zn,m − z0,m |)Dm . Passing to limit with n → ∞,it follows that Pξ1 ,...,ξm (f )(z1 , . . . , zm ) is conm m tinuous at z0 ∈ D , since f is continuous on D . It remains to prove that m m Pξ1 , . . . , ξn is analytic on D . For f ∈ A(D ), we can write f (z1 , . . . , zm ) = 4

ANASTASSIOU, GAL: MULTICOMPLEX SINGULAR INTEGRALS

P∞

m

im αi1 ,i2 ,...,im z1 . . . zm , for all z1 , . . . , zm ∈ D . For fixed z ∈ D the form z = (z1 , . . . , zm ), we can write ¡ ¢ f z1 eiu1 , . . . , zm eium

131

m

i1 ,i2 ,...,im =0

=

of

¡ ¢ ³ i(u1 i1 +...+um im ) ´ im αi1 ,i2 ,...,im z1i1 z2i2 . . . zm e

∞ X i1 ,i2 ,...,im =0

i(u1 i1 +...+um im ) and since |αi1 ,iP | = |αi1 ,i2 ,...,im |, for uj ∈ R, j = 1, . . . , m, 2 ,...,im e ∞ im and the series i1 ,i2 ...,im =0 αi1 ,i2 ,...,im z1i1 z2i2 . . . zm is absolutely convergent, it follows that the series above is uniformly convergent with respect to u1 , . . . , um ∈ R. This immediately implies that the series can be integrated term by term, i.e. Ã ! ∞ X 1 Q Pξ1 ,...,ξm (z1 , . . . , zm ) = αi ,i ,...,im m 2m j=1 ξj i ,i ,...,i =0 1 2 1 2 m   Z ∞ Z ∞ m Y im  ... ei(u1 i1 +...+um im ) e−|uj |/ξj du1 . . . dum  . z1i1 z2i2 . . . zm −∞

−∞

j=1

Moreover we can write Z ∞ Z ∞ m Y 1 i(u1 i1 +um im ) Q . . . e e−|uj |/ξj du1 . . . dum m 2m j=1 ξj −∞ −∞ j=1 =

¶ Z ∞ m µ Y 1 eiij uj e−|uj |/ξj duj (by [1], see the proof al Theorem 2.3) 2ξj −∞ j=1 ! Ã m Y 1 . = 1 + i2j ξj2 j=1

That is completing the proof of (i). (ii) Let z1,j , z2,j ∈ D, |z1,j − z2,j | ≤ δj , for j = 1, . . . , m. We obtain |Pξ1 ,...,ξm (f ) (z1,1 , . . . , z1,m ) − Pξ1 ,...,ξm (f ) (z2,1 , . . . , z2,m )| Z ∞ Z ∞ ¯ ¡ ¢ 1 ¯f z1,1 eiu1 , . . . , z1,m eium ≤ m Qm ... 2 −∞ j=1 ξj −∞ m ¡ ¢¯ Y −f z2,1 eiu1 , . . . , z2,m eium ¯ e−|uj |/ξj du1 , . . . , dum j=1

≤ ω1 (f ; |z1,1 − z2,1 |, |z1,2 − z2,2 |, . . . , |z1,m − z2,m |)Dm ≤ ω1 (f ; δ1 , . . . , δm )Dm , passing to sup with |z1,j − z2,j | ≤ δj , j = 1, . . . , m, it follows the desired inequality. 5

132

ANASTASSIOU, GAL: MULTICOMPLEX SINGULAR INTEGRALS

(iii) We have |Pξ1 ,...,ξm (f ) (z1 , . . . , zm ) − f (z1 , . . . , zm ) | Z ∞ Z ∞ ¯ ¡ iu ¢ 1 ¯f z1 e 1 , . . . , zm eium Q ≤ m m ... 2 ξ −∞ j=1 j −∞ m Y

−f (z1 , . . . , zm )| ≤

Z

1 2m Πm j=1 ξj

j=1

Z

∞

∞

... −∞

−∞

·

e−|uj |/ξj du1 . . . dum

m Y

¡ ¢ ω1 f ; |z1 eiu1 − z1 |, . . . , |zm eium − zm | Dm e−|uj |/ξj du1 . . . dum .

j=1

By the Maximum Modulus Principle (see e.g. [4], p. 23, Corollary 1.2.5) we can take |zj | = |, for j = 1, . . . , m. Thus we have |Pξ1 ,...,ξm (f )(z1 , . . . , zm ) − f (z1 , . . . , zm )| Z ∞ Z ∞ ¡ ¢ 1 ... ω1 f ; |eiu1 − 1|, . . . , |eium − 1| Dm ≤ m Qm 2 −∞ j=1 ξj −∞ Z ∞ Z ∞ m ¯ ³ Y 1 u1 ¯¯ ¯ e−|uj |/ξj du1 . . . dum ≤ m Qm ... ω1 f ; 2 ¯sin ¯ , 2 2 −∞ j=1 ξj −∞ j=1 m ¯ ¯ u ¯´ Y u2 ¯¯ ¯ ¯ m¯ e−|uj |/ξj du1 . . . dum 2 ¯sin ¯ , . . . , 2sin ¯ ¯ m 2 2 D j=1

≤

2

1 Qm m

Z

j=1 ξj

Z

∞

∞

... −∞

−∞

ω1 (f ; |u1 |, |u2 |, . . . , |um |)Dm

m Y

e−|uj |/ξj du1 . . . dum

j=1

≤ ω1 (f ; ξ1 , ξ2 , . . . , ξm )Dm   Z ∞ Z ∞ m m X Y 2m u j 1 +  · m Qm ... e−uj /ξj du1 . . . dum ξ 2 j 0 j=1 ξj 0 j=1 j=1 =

ω1 (f ; ξ1 , ξ2 , . . . , ξm )D Qm j=1 ξj ·

m Y

m

Z



Z

∞

∞

... 0

0

1 +

m X uj j=1

ξj

e−uj /ξj du1 . . . dum = ω1 (f ; ξ1 , . . . , ξm )Dm

j=1

1 · Qm

j=1 ξj

Z

Z

∞

... 0

0

m ∞Y j=1

6

e−uj /ξj du1 . . . dum

 

ANASTASSIOU, GAL: MULTICOMPLEX SINGULAR INTEGRALS

Z

1

+ Qm

Z

∞

...

j=1 ξj

0

0

∞

   m m X Y u j  e−uj /ξj  du1 . . . dum ξ j j=1 j=1

= ω1 (f ; ξ1 , . . . , ξm )Dm = (m + 1)ω1 (f ; ξ1 , . . . , ξm )Dm , proving the claim, and finishing the proof of (iii). (iv) For the simplicity of calculation, we will prove the case m = 2. In our reasonings will be useful the following simple relationship Z ∞ Z ∞ F (u)du = [F (u) + F (−u)]du. −∞

0

Also, we will need the following two partial moduli of smoothness it 1 ω2,z1 (f ; ξ)(∂Dr )2 = sup{|∆2,z u f (re , z2 )|; |t| ≤ π, |u| ≤ ξ, |z2 | ≤ r},

and it 2 ω2,z2 (f ; η)(∂Dr )2 = sup{|∆2,z u f (z1 , re )|; |t| ≤ π, |u| ≤ η, |z1 | ≤ r},

where it i(t+u) 1 ∆2,z , z2 ) − 2f (reit , z2 ) + f (rei(t−u) , z2 ) u f (re , z2 ) = f (re

and it i(t+u) 2 ) − 2f (z1 , reit ) + f (z1 , rei(t−u) ). ∆2,z u f (z1 , re ) = f (z1 , re

Applying twice this relationship we easily get Pξ1 ,ξ2 (f )(z1 , z2 ) − f (z1 , z2 ) =

1 22 ξ

1 = · 2ξ2

=

Z

Z

1 ξ2

∞ −∞

½

∞

−∞

1 1 · 2ξ1 2ξ2

[f (z1 eiu1 , z2 eiu2 ) − f (z1 , z2 )]e−|u1 |/ξ1 e−|u2 |/ξ2 du1 du2

1 2ξ1

Z

∞

0

[f (z1 eiu1 , z2 eiu2 ) − 2f (z1 , z2 ) + f (z1 e−iu1 , z2 eiu2 )] o ·e−u1 /ξ1 du1 · e−|u2 |/ξ2 du2

( changing the order of the two integrals) ½Z ∞ [f (z1 eiu1 , z2 eiu2 ) − 2f (z1 , z2 ) + f (z1 e−iu1 , z2 eiu2 )]

Z

∞ 0

+[f (z1 e

0

iu1

ª , z2 e−iu2 ) − 2f (z1 , z2 ) + f (z1 e−iu1 , z2 e−iu2 )] ·e−u1 /ξ1 e−u2 /ξ2 du1 du2 .

Simple calculation shows [f (z1 eiu1 , z2 eiu2 ) − 2f (z1 , z2 ) + f (z1 e−iu1 , z2 eiu2 )] 7

133

134

ANASTASSIOU, GAL: MULTICOMPLEX SINGULAR INTEGRALS

+[f (z1 eiu1 , z2 e−iu2 ) − 2f (z1 , z2 ) + f (z1 e−iu1 , z2 e−iu2 )] = [f (z1 eiu1 , z2 eiu2 ) − 2f (z1 , z2 eiu2 ) + f (z1 e−iu1 , z2 eiu2 )] +[f (z1 eiu1 , z2 e−iu2 ) − 2f (z1 , z2 e−iu2 ) + f (z1 e−iu1 , z2 e−iu2 )] +2[f (z1 , z2 eiu2 ) − 2f (z1 , z2 ) + f (z1 , z2 e−iu2 )]. Passing to absolute value, we obtain

≤

Z

∞

0

|Pξ1 ,ξ2 (f )(z1 , z2 ) − f (z1 , z2 )| Z ∞ ∞ [2ω2,z1 (f ; u1 )(∂Dr )2 + 2ω2,z2 (f ; u2 )(∂Dr )2 ] 0

·e−u1 /ξ1 e−u2 /ξ2 du1 du2 1 1 ≤ · 2ξ1 2ξ2

Z

∞

· 0

1 1 · 2ξ1 2ξ2

Z

0

[2(1 + u1 /ξ1 )2 ω2,z1 (f ; ξ1 )(∂Dr )2 + 2(1 + u2 /ξ2 )2 ω2,z2 (f ; ξ2 )(∂Dr )2 ] ·e−u1 /ξ1 e−u2 /ξ2 du1 du2 ≤ C[ω2,z1 (f ; ξ1 )(∂Dr )2 + ω2,z2 (f ; ξ2 )(∂Dr )2 ].

Since by the mean value theorem for divided difference in Complex Analysis (see e.g. [5], p. 258, Exercise 4.20), for all |z1 | ≤ r, |z2 | ≤ r we have ° 2 ° ° ° it 2 °∂ f ° 2 |∆2,z f (z , re )| ≤ u 1 u ° ∂z 2 ° 2 r and

° 2 ° ° ° it 2 °∂ f ° 1 f (re , z )| ≤ u |∆2,z 2 u ° ∂z 1 ° , 2 r

for all ξ1 , ξ2 ∈ (0, 1] we immediately obtain kPξ1 ,ξ2 (f ) − f kr ≤ Cr (f )[ξ12 + ξ22 ]. Now, denoting by γ the circle of radius r1 > 1 and center 0, since for any |z| ≤ r and u, v ∈ γ, we have |v − z| ≥ r1 − r and |u − z| ≥ r1 − r, by the Cauchy’s formula in Theorem A, it follows that for all |z1 | ≤ r, |z2 | ≤ r, ξ1 , ξ2 ∈ (0, 1], we have ¯ ¯ ¯ ∂ k1 +k2 P ¯ ∂ k1 +k2 f ¯ ¯ ξ1 ,ξ2 (f ) (z , z ) − (z , z ) ¯ ¯ 1 2 1 2 k1 k1 k2 k2 ¯ ¯ ∂z1 ∂z2 ∂z1 ∂z2 ¯ ¯Z Z ¯ Pξ1 ,ξ2 (f )(u, v) − f (u, v) (k1 )!(k2 )! ¯¯ ¯ dudv = ¯ ¯ 2 k +1 k +1 1 4π (v − z2 ) 2 γ γ (u − z1 ) ≤ Cr (f )[ξ12 + ξ22 ]

(k1 !)(k2 !) 2πr1 2πr1 · · , 2 k +1 1 4π (r1 − r) (r1 − r)k2 +1 8

ANASTASSIOU, GAL: MULTICOMPLEX SINGULAR INTEGRALS

which proves the upper estimate. It remains to prove the lower estimate for ° ° ° ∂ k1 +k2 P ∂ k1 +k2 f ° ° ° ξ1 ,ξ2 (f ) − k1 k2 ° . ° k1 k2 ° ∂z1 ∂z2 ∂z1 ∂z2 °r For this purpose, take z1 = reiϕ1 , z2 = reiϕ2 and p1 , p2 ∈ N ∪ {0}. Taking into account Theorem 1, (i) too, for k1 + k2 = q, (q ∈ N ∪ {0}), we have " # ∂ k1 +k2 Pξ1 ,ξ2 (f ) 1 ∂ k1 +k2 f (z1 , z2 ) − k1 k2 (z1 , z2 ) e−ip1 ϕ1 e−ip2 ϕ2 4π 2 ∂z1k1 ∂z2k2 ∂z1 ∂z2 ∞ ∞ 1 X X αi1 ,i2 i1 (i1 − 1)...(i1 − k1 + 1)ri1 −k1 eiϕ1 (i1 −k1 −p1 ) 4π 2 i1 =k1 i2 =k2 · ¸ 1 1 i2 −k2 iϕ1 (i2 −k2 −p2 ) · . ·i2 (i2 − 1)...(i2 − k2 + 1)r e 1− 1 + ξ12 i21 1 + ξ22 i22

=

Integrating twice from −π to π, we obtain # Z π " k1 +k2 1 ∂ k1 +k2 f ∂ Pξ1 ,ξ2 (f ) (z1 , z2 ) − k1 k2 (z1 , z2 ) e−ip1 ϕ1 e−ip2 ϕ2 dϕ1 dϕ2 4π 2 −π ∂z1k1 ∂z2k2 ∂z1 ∂z2 = αk1 +p1 ,k2 +p2 (k1 + p1 )...(p1 + 1)(k2 + p2 )...(p2 + 1)rp1 +p2 · ¸ 1 1 · 1− · . 1 + ξ12 (k1 + p1 )2 1 + ξ22 (k2 + p2 )2 Passing now to absolute value, we easily obtain |αk1 +p1 ,k2 +p2 |(k1 + p1 )...(p1 + 1)(k2 + p2 )...(p2 + 1)rp1 +p2 · ¸ 1 1 · 1− · 1 + ξ12 (k1 + p1 )2 1 + ξ22 (k2 + p2 )2 ° ° ° ∂ k1 +k2 P ∂ k1 +k2 f ° ° ° ξ1 ,ξ2 (f ) ≤° − k1 k2 ° . ° ∂z1k1 ∂z2k2 ∂z1 ∂z2 ° r

First consider q = 0, that is k1 = k2 = 0. Let us denote Eξ1 ,ξ2 ,p1 ,p2 = 1 − 1 · 1 and 1+ξ12 p21 1+ξ22 p22 Vξ1 ,ξ2 = inf Eξ1 ,ξ2 ,p1 ,p2 . p1 ≥1,p2 ≥1

Clearly we get Vξ1 ,ξ2

µ = 1−

1 1 · 2 1 + ξ1 1 + ξ22

¶ =

9

ξ12 + ξ22 + ξ12 ξ22 ξ12 + ξ22 . ≥ (1 + ξ12 )(1 + ξ22 ) 4

135

136

ANASTASSIOU, GAL: MULTICOMPLEX SINGULAR INTEGRALS

By the above lower estimate for kPξ1 ,ξ2 (f ) − f kr , for all p1 , p2 ≥ 1 and ξ1 , ξ2 ∈ (0, 1], it follows 4 kPξ1 ,ξ2 (f ) − f kr kPξ1 ,ξ2 (f ) − f kr kPξ1 ,ξ2 (f ) − f kr ≥ ≥ ≥ |αp1 ,p2 |rp1 +p2 . 2 2 ξ1 + ξ2 Vξ1 ,ξ2 Eξ1 ,ξ2 ,p1 ,p2 (1)

(2)

This implies that if would exist subsequences in (0, 1], (ξk )k , (ξk )k with (1) (2) ξk , ξk → 0 (as k → ∞) and such that ° ° ° ° °Pξ(1) ,ξ(2) (f ) − f ° k k r lim = 0, (1) (2) k→∞ [ξk ]2 + [ξk ]2 then we would have αp1 ,p2 = 0 for all p1 , p2 ≥ 1. Therefore, if the Taylor’s development of f (z1 , z2 ) contains at least one term of the form αp1 ,p2 z1p1 z2p2 with p1 , p2 ≥ 1 and αp1 ,p2 6= 0, then kPξ1 ,ξ2 (f ) − f kr > 0, ξ12 + ξ22 ξ1 ,ξ2 ∈(0,1] inf

which implies that there exists a constant Cr (f ) > 0 such that Cr (f ), for all ξ1 , ξ2 ∈ (0, 1], that is

kPξ1 ,ξ2 (f )−f kr ξ12 +ξ22

kPξ1 ,ξ2 (f ) − f kr ≥ Cr (f )[ξ12 + ξ22 ], for all ξ1 , ξ2 ∈ (0, 1]. Now, consider k1 + k2 = q ≥ 1 and denote Eξ1 ,ξ2 ,p1 ,p2 ,k1 ,k2 = 1 −

1 1+

Vk1 ,k2 ,ξ1 ,ξ2 =

ξ12 (k1

inf

p1 ,p2 ≥1

+ p1

)2

·

1 1+

ξ22 (k2

+ p2 )2

,

Eξ1 ,ξ2 ,p1 ,p2 ,k1 ,k2 .

Evidently that we have Vk1 ,k2 ,ξ1 ,ξ2 = 1 − ≥1−

1 1 · 1 + ξ12 (k1 + 1)2 1 + ξ22 (k2 + 1)2

1 1 ξ12 + ξ22 · ≥ 1 + ξ12 1 + ξ22 4

and reasoning as in the case q = 0 we obtain ° ° ° ° ° ∂ k1 +k2 Pξ ,ξ (f ) ° ° ° ∂ k1 +k2 Pξ ,ξ (f ) ∂ k1 +k2 f ° ∂ k1 +k2 f ° 1 2 1 2 ° ° 4° − k1 k2 ° − k1 k2 ° k k ° ∂z1k1 ∂z2k2 ∂z1 1 ∂z2 2 ∂z1 ∂z2 ∂z1 ∂z2 r r ≥ ξ12 + ξ22 Vk1 ,k2 ,ξ1 ,ξ2 ° ° ° ∂ k1 +k2 Pξ ,ξ (f ) ° ∂ k1 +k2 f ° 1 2 ° − k1 k2 ° ° ∂z1k1 ∂z2k2 ∂z1 ∂z2 r ≥ Eξ1 ,ξ2 ,p1 ,p2 ,k1 ,k2 10

≥

ANASTASSIOU, GAL: MULTICOMPLEX SINGULAR INTEGRALS

≥ |αk1 +p1 ,k2 +p2 |(k1 + p1 )...(p1 + 1)(k2 + p2 )...(p2 + 1)rp1 +p2 , for all p1 , p2 ≥ 1 and ξ1 , ξ2 ∈ (0, 1]. (1) (2) This implies that if would exist subsequences in (0, 1], (ξk )k , (ξk )k with (1) (2) ξk , ξk → 0 (as k → ∞) and such that ° k +k ° ° ∂ 1 2 P (1) (2) (f ) ° k1 +k2 ξ ,ξ ° ° k k − ∂ k1 kf2 ° ° k1 k2 ∂z1 ∂z2 ∂z1 ∂z2 ° ° r lim = 0, (1) (2) k→∞ [ξk ]2 + [ξk ]2 then we would have αk1 +p1 ,k2 +p2 = 0 for all p1 , p2 ≥ 1 and k1 + k2 = q ≥ 1. Therefore, if the Taylor’s development of f (z1 , z2 ) contains at least one term of the form αk1 +p1 ,k2 +p2 z1k1 +p1 z2k2 +p2 with p1 , p2 ≥ 1, k1 + k2 = q ≥ 1 and αk1 +p1 ,k2 +p2 6= 0, then ° ° ° ° ∂ k1 +k2 Pξ ,ξ (f ) ∂ k1 +k2 f ° 1 2 ° − k1 k2 ° ° ∂z1k1 ∂z2k2 ∂z1 ∂z2 r > 0, inf ξ12 + ξ22 ξ1 ,ξ2 ∈(0,1] which implies that there exists a constant Cr,k1 ,k2 (f ) > 0 such that ° ° ° ∂ k1 +k2 Pξ ,ξ (f ) ° ∂ k1 +k2 f ° 1 2 ° − k1 k2 ° ° ∂z1k1 ∂z2k2 ∂z1 ∂z2 r ≥ Cr,k1 ,k2 (f ), for all ξ1 , ξ2 ∈ (0, 1], ξ12 + ξ22 that is ° ° ° ∂ k1 +k2 P ∂ k1 +k2 f ° ° ° ξ1 ,ξ2 (f ) − k1 k2 ° ≥ Cr,k1 ,k2 (f )[ξ12 + ξ22 ], for all ξ1 , ξ2 ∈ (0, 1]. ° k1 k2 ° ∂z1 ∂z2 ∂z1 ∂z2 °r ¤ Remark. For m = 1, Theorem 1, (iv) was obtained by [3], p. 207, Theorem 3.2.1, (iv).

3

Multi-Complex Poisson-Cauchy Integrals

In this section we study the approximation properties of Qξ1 ,...,ξm (f )(z1 , . . . , zm ) and Rξ1 ,...,ξm (f )(z1 , . . . , zm ).³ ´ m

and ξi > 0, i = 1, . . . , m. We have Theorem 2. Let f ∈ A D (i) Qξ1 ,...,ξm (f )(z1 , . . . , zm ) and Rξ1 ,...,ξm (f )(z1 , . . . , zm ) are continuous on m D and analytic on Dm . (ii) ω1 (Qξ1 ,...,ξm (f ); δ1 , . . . , δm )Dm ≤ ω1 (f ; δ1 , . . . , δm )Dm , ω1 (Rξ1 ,...,ξm (f ); δ1 , . . . , δm )Dm ≤ ω1 (f ; δ1 , . . . , δm )Dm , 11

137

138

ANASTASSIOU, GAL: MULTICOMPLEX SINGULAR INTEGRALS

for all δ1 , . . . , δm ≥ 0. (iii) · ¸ 2m |Rξ1 ,...,ξm (f )(z1 , . . . , zm ) − f (z1 , . . . , zm )| ≤ 2m + ω1 (f ; ξ1 , . . . , ξm )Dm , π for any ξj > 0 , j = 1, . . . , m. (iv)³ Denote Dr = {z ∈ C; |z| < r} and for R > 1 let us suppose that ´ m f ∈ A DR , that is we can write f (z1 , . . . , zm ) =

∞ X

im αi1 ,i2 ...,im z1i1 z2i2 . . . zm ,

i1 ,i2 ...,im =0

for all z1 , . . . , zm ∈ DR , m ∈ N and αi1 ,i2 ...,im ∈ C. If the Taylor’s development of f (z1 , z2 , ..., zm ) contains at least one term of pm the form αp1 ,p2 ,...,pm z1p1 z2p2 ...zm with p1 , p2 , ..., pm ≥ 1 and αp1 ,p2 ,...,pm 6= 0 for q = 0, and at least one term of the form km +pm with p1 , p2 , ..., pm ≥ 1, αk1 +p1 ,k2 +p2 ,...,km +pm z1k1 +p1 z2k2 +p2 ...zm

such that αk1 +p1 ,k2 +p2 ,...,km +pm 6= 0 and k1 + k2 + ... + km = q forPq ≥ 1, then m for all 1 ≤ r < r1 < R, ξj ∈ (0, 1], j = 1, ..., m, kj ∈ N ∪ {0} with j=1 kj = q, q ∈ N ∪ {0}, we have ° ° m ° ∂ k1 +...+km R k1 +...+km ° X ∂ f (f ) ° ° ξ1 ,...,ξm − ∼ ξj2 , ° ° km km ° ° ∂z1k1 ...∂zm ∂z1k1 ...∂zm j=1 r where the constants in the equivalence depend only on f , q, r and r1 . Here kf kr = sup{|f (z1 , ..., zm )|; |zj | ≤ r, j = 1, ..., m}. m Proof. (i) Let z0 , zn ∈ D be with lim zn,j = z0,j , j = 1, . . . , m. n→∞ We get |Qξ1 ,...,ξm (f ) (zn,1 , . . . , zn,m ) − Qξ1 ,...,ξm (f )(z0,1 , . . . , z0,m )| Qm Z π Z π j=1 ξj ≤ ... πm −π −π ¡ ¢ ¡ ¢ iu1 ium iu1 |f zn,1 e , . . . , zn,m e − f z0,1 e , . . . , z0,m eium | Qm du1 . . . dum 2 2 j=1 (uj + ξj ) Qm Z π Z π j=1 ξj ... ≤ πm −π −π ¡ ¢ iu1 iu1 ium ω1 f ; |zn,1 e , −z0,1 e |, . . . , |zn,m e − z0,m eium | Dm ¢ Qm ¡ 2 du1 . . . dum 2 j=1 uj + ξj

12

ANASTASSIOU, GAL: MULTICOMPLEX SINGULAR INTEGRALS

Qm

j=1 ξj πm

=

Z

Z

π

π

... −π

−π

139

ω1 (f ; |zn,1 − z0,1 |, . . . , |zn,m − z0,m |)Dm ¢ Qm ¡ 2 du1 . . . dum 2 j=1 uj + ξj

= ω1 (f ; |zn,1 − z0,1 |, . . . , |zn,m − z0,m |)Dm Qm Z π Z π du . . . du j=1 ξj Qm 1 2 m2 · . . . m π −π −π j=1 (uj + ξj ) = ω1 (f ; |zn,1 − z0,1 , . . . , |zn,m − z0,m |)Dm "µ !# ¶ ÃZ π m Y 2ξj duj · 2 2 π 0 uj + ξj j=1 = ω1 (f ; |zn,1 − z0,1 |, . . . , |zn,m − z0,m |)Dm

m · Y 2 j=1

π

tan

−1

π ξj

¸

≤ ω1 (f ; |zn,1 − z0,1 |, . . . , |zn,m − z0,m |)Dm . So we proved that |Qξ1 ,...,ξm (f ) (zn,1 , . . . , zn,m ) − Qξ1 ,...,ξm (f )(z0,1 , . . . , z0,m )| ≤ ω1 (f ; |zn,1 − z0,1 |, . . . , |zn,m − z0,m |)Dm . Passing to limit with n → ∞, it follows that Qξ1 ,...,ξm (f )(z1 , . . . , zm ) is conm m tinuous at z0 ∈ D , since f is continuous on D . It remains to prove that m m Qξ1 ,...,ξm (f ) is analytic on D . For f ∈ A(D ), we can write f (z1 , . . . , zm ) = P∞ m i1 im i1 ,...,im =0 αi1 ,...,im z1 . . . zm , for all z1 , . . . , zm ∈ D . As in the proof of Theorem 1, (i) we can write Qm j=1 ξj Qξ1 ,...,ξm (f ) (z1 , . . . , zm ) = πm Z π Z π i(u1 i1 +...+um im ) ∞ X e im ¢ du1 . . . dum . Qm ¡ 2 αi1 ,...,im z1i1 . . . zm ... 2 −π −π j=1 uj + ξj i ,...,i =0 1

m

Hence

Qm |Qξ1 ,...,ξm (f ) (z1 , . . . , zm ) | ≤ Z

∞ X

|αi1 ,...,im | |z1 |i1 . . . |zm |im

i1 ,...,im =0

=

Qm

j=1 ξj πm

Z

π

π

... −π

−π

j=1 ξj πm

|ei(u1 i1 +...+um im ) | ¢ du1 . . . dum Qm ¡ 2 2 j=1 uj + ξj Z

∞ X

i1

im

|αi1 ,...,im | |z1 | . . . |zm |

i1 ,...,im =0

Qm ¡ j=1

1 ¢ du1 . . . dum u2j + ξj2

13

2

m

Z

π

π

... 0

0

140

ANASTASSIOU, GAL: MULTICOMPLEX SINGULAR INTEGRALS

≤

∞ X

i

i

|αi1 ,...,im | |z1 | 1 . . . |zm | m < ∞.

i1 ,...,im =0 m

The last implies that Qξ1 ,...,ξm (f ) is analytic on D . The proof of the continuity property of Rξ1 ,...,ξm (f )(z1 , . . . , zm ) follows in a similar manner as the proof de continuity of Qξ1 ,...,ξm (f )(z1 , . . . , zn ). Fort the analyticity property of Rξ1 ,...,ξm (f )(z1 , . . . , zm ) we can write Qm 2m j=1 ξj3 Rξ1 ,...,ξm (f )(z1 , . . . , zm ) = πm Z ∞ Z ∞ i(u1 i1 +...+um im ) ∞ X e i1 im · αi1 ,...,im z1 . . . zm ... ¢ du1 . . . dum Qm ¡ 2 2 2 −∞ −∞ i1 ,...,im =0 j=1 uj + ξj Qm Z ∞ Z ∞ ∞ X 2m j=1 ξj3 i1 im = α z . . . z . . . i ,...,i m 1 m 1 πm −∞ −∞ i1 ,...,im =0 ! Ã m Y eiuj ij · ¡ 2 ¢2 du1 . . . dum uj + ξj2 j=1 " # Z m ∞ Y X 2ξj3 ∞ eiij uj i1 im = αi1 ,...,im z1 . . . zm ¢ duj ¡ π −∞ u2 + ξ 2 2 j=1 i1 ,...,im =0 j j # " Z ∞ m X Y 4ξj3 ∞ cos(ij uj ) i1 im = αi1 ,...,im z1 . . . zm ¡ 2 ¢2 duj . π 0 u + ξ2 i ,...,i =0 j=1 1

j

m

j

Therefore |Rξ1 ,...,ξm (f )(z1 , . . . , zm )| ≤ 2m

∞ X

i

i

|αi1 , . . . , im | |z1 | 1 . . . |zm | m < ∞.

i1 ,...,im =0

The last implies that Rξ1 ,...,ξm (f ) is analytic on Dm . (ii) Let z1,j , z2,j ∈ D, |z1,j − z2,j | ≤ δj , for j = 1, . . . , m. We obtain Qm |Qξ1 ,...,ξm (f ) (z1,1 , . . . , z1,m ) − Qξ1 ,...,ξm (f ) (z2,1 , . . . , z2,m )| ≤ Z

π

Z

π

|f (z1,1 eiu1 , . . . , z1,m eium ) − f (z2,1 eiu1 , . . . , z2,m eium )| ¢ Qm ¡ 2 du1 . . . dum 2 −π j=1 uj + ξj Qm j=1 ξj ω1 (f ; |z1,1 − z2,1 | , . . . |z1,m − z2,m |)Dm ≤ πm Z π Z π 1 m Qm · ... 2 2 du1 . . . dum ≤ ω1 (f ; δ1 , . . . , δm )D , (u −π −π j=1 j + ξj )

...

· −π

j=1 ξj πm

14

ANASTASSIOU, GAL: MULTICOMPLEX SINGULAR INTEGRALS

141

passing to sup with |z1,j − z2,j | ≤ δj , j = 1, . . . , m, it follows the desired inequality, proving global smoothness for Qxi1 ,...,ξm (f ). The global smoothness property for Rξ1 ,...,ξm (f ) follows similarly as of Qξ1 ,...,ξm (f ). (iii) We have |Rξ1 ,...,ξm (f ) (z1 , . . . , zm ) − f (z1 , . . . , zm )| ¯ Z ∞ ¯¯ 3 Z ∞ 2 f (z1 eiu1 , . . . , zm eium ) − f (z1 , . . . , zm )¯ j=1 ξj Qm ≤ ... du1 . . . dum 2 2 2 πm −∞ −∞ j=1 (uj + ξj ) Qm 2m j=1 ξj3 ≤ πm ¡ ¢ Z ∞ Z ∞ iu1 ω1 f ; |z1 e − z1 |, . . . , |zm eium − zm | Dm · ... du1 . . . dum . ¢ Qm ¡ 2 2 2 −∞ −∞ j=1 uj + ξj Qm m

By the Maximum Modulus Principle (see,e.g [4], p. 23, Corollary 1.2.5) we can take |zj | = |, for j = 1, . . . , m. Thus we have |Rξ1 ,...,ξm (f ) (z1 , . . . , zm ) − f (z1 , . . . , zm )| ¡ ¢ Z ∞ 3 Z ∞ 2 ω1 f ; |eiu1 − 1|, . . . , |eium − 1| Dm j=1 ξj ≤ ... du1 . . . dum ¢ Qm ¡ 2 2 2 πm −∞ −∞ j=1 uj + ξj ´ ³ Qm Z Z ∞ ω1 f ; ξ1 |u1 | , . . . , ξm |um | 2m j=1 ξj3 ∞ m ξ1 ξm D du1 . . . dum ≤ ... ¡ 2 ¢ Q 2 m m 2 π −∞ −∞ j=1 uj + ξj à !Z Qm Z ∞ (1 + Pm uj ) ∞ 22m j=1 ξj3 j=1 ξj ≤ ω1 (f ; ξ1 , . . . , ξm )Dm ... ¢ du1 . . . dum Qm ¡ 2 2 2 πm 0 0 j=1 uj + ξj ! "Z à Qm Z ∞ ∞ 22m j=1 ξj3 du1 . . . dum = ω1 (f ; ξ1 , . . . , ξm )Dm . . . ¢ Qm ¡ 2 2 2 πm 0 0 j=1 uj + ξj  Z ∞ m Z ∞ X uj ∗ 1 + · Qm ¡ ... ¢2 du1 . . . dum  ∗ ξ j 0 u2 + ξ 2 j ∗ =1 0 Qm m

j

j=1

"

à m

= ω1 (f ; ξ1 , . . . , ξm )Dm 2 +

·

Ã m  Z X j ∗ =1



∞ 0

1 uj ∗ ·¡ ¢2 duj ∗ 2 ξj ∗ uj ∗ + ξj2∗ "

= ω1 (f ; ξ1 , . . . , ξm )Dm 2m +

22m

Qm

!

15

2

m Y j=1,j6=j ∗

3 j=1 ξj

πm

j

2m

Qm

3 j=1 ξj πm

ÃZ 0

∞

!

!  duj  ¡ 2 ¢ 2  u + ξ2 j

Ã ! m  X 1 ·  2ξj3∗ j ∗ =1

j

m Y j=1,j6=j ∗

 π  4ξj3 

142

ANASTASSIOU, GAL: MULTICOMPLEX SINGULAR INTEGRALS

· ¸ 2m = ω1 (f ; ξ1 ; . . . , ξm )Dm 2m + , π proving the claim, and finishing the proof of (iii). (iv) For the simplicity of calculation, we will prove the case m = 2. In our reasonings will be useful the following simple relationship Z ∞ Z ∞ F (u)du = [F (u) + F (−u)]du. −∞

0

Also, we will need the following two partial moduli of smoothness it 1 ω2,z1 (f ; ξ)(∂Dr )2 = sup{|∆2,z u f (re , z2 )|; |t| ≤ π, |u| ≤ ξ, |z2 | ≤ r},

and it 2 ω2,z2 (f ; η)(∂Dr )2 = sup{|∆2,z u f (z1 , re )|; |t| ≤ π, |u| ≤ η, |z1 | ≤ r},

where it i(t+u) 1 , z2 ) − 2f (reit , z2 ) + f (rei(t−u) , z2 ) ∆2,z u f (re , z2 ) = f (re

and it i(t+u) 2 ∆2,z ) − 2f (z1 , reit ) + f (z1 , rei(t−u) ). u f (z1 , re ) = f (z1 , re

Applying twice this relationship we easily get Rξ1 ,ξ2 (f )(z1 , z2 ) − f (z1 , z2 ) Z 22 ξ13 ξ23 ∞ 1 [f (z1 eiu1 , z2 eiu2 ) − f (z1 , z2 )] 2 2 )2 (u2 + ξ 2 )2 du1 du2 π2 (u + ξ −∞ 1 1 2 2 ½ 3Z ∞ 3 Z ∞ 2ξ 2ξ1 = 2 · [f (z1 eiu1 , z2 eiu2 ) − 2f (z1 , z2 ) + f (z1 e−iu1 , z2 eiu2 )] π π −∞ 0 ¾ 1 1 du du2 · 2 1 · 2 2 2 (u1 + ξ1 ) (u2 + ξ22 )2 =

=

22 ξ13 ξ23 π2

Z

∞

( changing the order of the two integrals) ½Z ∞ [f (z1 eiu1 , z2 eiu2 ) − 2f (z1 , z2 ) + f (z1 e−iu1 , z2 eiu2 )]

0

+[f (z1 e

0 iu1

ª , z2 e−iu2 ) − 2f (z1 , z2 ) + f (z1 e−iu1 , z2 e−iu2 )] ·

1 du1 du2 . (u21 + ξ12 )2 (u22 + ξ22 )2

Simple calculation shows [f (z1 eiu1 , z2 eiu2 ) − 2f (z1 , z2 ) + f (z1 e−iu1 , z2 eiu2 )] 16

ANASTASSIOU, GAL: MULTICOMPLEX SINGULAR INTEGRALS

+[f (z1 eiu1 , z2 e−iu2 ) − 2f (z1 , z2 ) + f (z1 e−iu1 , z2 e−iu2 )] = [f (z1 eiu1 , z2 eiu2 ) − 2f (z1 , z2 eiu2 ) + f (z1 e−iu1 , z2 eiu2 )] +[f (z1 eiu1 , z2 e−iu2 ) − 2f (z1 , z2 e−iu2 ) + f (z1 e−iu1 , z2 e−iu2 )] +2[f (z1 , z2 eiu2 ) − 2f (z1 , z2 ) + f (z1 , z2 e−iu2 )]. Passing to absolute value, we obtain |Rξ1 ,ξ2 (f )(z1 , z2 ) − f (z1 , z2 )| ≤

22 ξ13 ξ23 π2

Z

∞Z 0

∞

0

·

[2ω2,z1 (f ; u1 )(∂Dr )2 + 2ω2,z2 (f ; u2 )(∂Dr )2 ] 1

(u21

+

ξ12 )2 (u22 ≤

Z

∞

Z

∞

· 0

0

+ ξ22 )2

du1 du2

22 ξ13 ξ23 π2

[2(1 + u1 /ξ1 )2 ω2,z1 (f ; ξ1 )(∂Dr )2 + 2(1 + u2 /ξ2 )2 ω2,z2 (f ; ξ2 )(∂Dr )2 ] ·

1 (u21

+

ξ12 )2 (u22

+ ξ22 )2

du1 du2

≤ C[ω2,z1 (f ; ξ1 )(∂Dr )2 + ω2,z2 (f ; ξ2 )(∂Dr )2 ]. (For the last inequality see the univariate case in [3], p. 217) Since by the mean value theorem for divided difference in Complex Analysis (see e.g. [5], p. 258, Exercise 4.20), for all |z1 | ≤ r, |z2 | ≤ r we have ° 2 ° ° ° it 2 °∂ f ° 2 |∆2,z f (z , re )| ≤ u 1 u ° ∂z 2 ° 2 r and

° 2 ° ° ° it 2 °∂ f ° 1 f (re , z )| ≤ u |∆2,z 2 u ° ∂z 1 ° , 2 r

for all ξ1 , ξ2 ∈ (0, 1] we immediately obtain kRξ1 ,ξ2 (f ) − f kr ≤ Cr (f )[ξ12 + ξ22 ]. Now, denoting by γ the circle of radius r1 > 1 and center 0, since for any |z| ≤ r and u, v ∈ γ, we have |v − z| ≥ r1 − r and |u − z| ≥ r1 − r, by the Cauchy’s formula in Theorem A, it follows that for all |z1 | ≤ r, |z2 | ≤ r, ξ1 , ξ2 ∈ (0, 1], we have ¯ ¯ ¯ ∂ k1 +k2 R ¯ ∂ k1 +k2 f ¯ ¯ ξ1 ,ξ2 (f ) (z , z ) − (z , z ) ¯ ¯ 1 2 1 2 ¯ ¯ ∂z1k1 ∂z2k2 ∂z1k1 ∂z2k2 ¯ ¯Z Z ¯ Rξ1 ,ξ2 (f )(u, v) − f (u, v) (k1 )!(k2 )! ¯¯ ¯ dudv = ¯ ¯ 2 k +1 k +1 1 4π (v − z2 ) 2 γ γ (u − z1 ) 17

143

144

ANASTASSIOU, GAL: MULTICOMPLEX SINGULAR INTEGRALS

≤ Cr (f )[ξ12 + ξ22 ]

(k1 !)(k2 !) 2πr1 2πr1 · · , 4π 2 (r1 − r)k1 +1 (r1 − r)k2 +1

which proves the upper estimate. It remains to prove the lower estimate for ° ° ° ∂ k1 +k2 R ∂ k1 +k2 f ° ° ° ξ1 ,ξ2 (f ) − k1 k2 ° . ° k1 k2 ° ∂z1 ∂z2 ∂z1 ∂z2 °r For this purpose, take z1 = reiϕ1 , z2 = reiϕ2 and p1 , p2 ∈ N ∪ {0}. Taking into account the representation of Rξ1 ,ξ2 (f )(z1 , z2 ) in the proof of Theorem 2, (i) too, for k1 + k2 = q, (q ∈ N ∪ {0}), we have " # 1 ∂ k1 +k2 Rξ1 ,ξ2 (f ) ∂ k1 +k2 f (z1 , z2 ) − k1 k2 (z1 , z2 ) e−ip1 ϕ1 e−ip2 ϕ2 4π 2 ∂z1k1 ∂z2k2 ∂z1 ∂z2 =

∞ ∞ 1 X X αi1 ,i2 i1 (i1 − 1)...(i1 − k1 + 1)ri1 −k1 eiϕ1 (i1 −k1 −p1 ) 4π 2 i1 =k1 i2 =k2

£ ¤ ·i2 (i2 −1)...(i2 −k2 +1)ri2 −k2 eiϕ1 (i2 −k2 −p2 ) 1 − (1 + i1 ξ1 )e−i1 ξ1 (1 + i2 ξ2 )e−i2 ξ2 . Integrating twice from −π to π, we obtain # Z π " k1 +k2 1 ∂ Rξ1 ,ξ2 (f ) ∂ k1 +k2 f (z1 , z2 ) − k1 k2 (z1 , z2 ) e−ip1 ϕ1 e−ip2 ϕ2 dϕ1 dϕ2 4π 2 −π ∂z1k1 ∂z2k2 ∂z1 ∂z2 = αk1 +p1 ,k2 +p2 (k1 + p1 )...(p1 + 1)(k2 + p2 )...(p2 + 1)rp1 +p2

h i · 1 − (1 + (k1 + p1 )ξ1 )e−(k1 +p1 )ξ1 (1 + (k2 + p2 )ξ2 )e−(k2 +p2 )ξ2 . Passing now to absolute value, we easily obtain |αk1 +p1 ,k2 +p2 |(k1 + p1 )...(p1 + 1)(k2 + p2 )...(p2 + 1)rp1 +p2 h i · 1 − (1 + (k1 + p1 )ξ1 )e−(k1 +p1 )ξ1 (1 + (k2 + p2 )ξ2 )e−(k2 +p2 )ξ2 ° ° ° ∂ k1 +k2 P ∂ k1 +k2 f ° ° ° ξ1 ,ξ2 (f ) − k1 k2 ° . ≤° k1 k2 ° ∂z1 ∂z2 ∂z1 ∂z2 °r First consider q = 0, that is k1 = k2 = 0. Let us denote Eξ1 ,ξ2 ,p1 ,p2 = 1 − (1 + ξ1 p1 )e−p1 ξ1 (1 + ξ2 p2 )e−p2 ξ2 and Vξ1 ,ξ2 =

inf

p1 ≥1,p2 ≥1

Eξ1 ,ξ2 ,p1 ,p2 .

Clearly we get Vξ1 ,ξ2 = 1 − (1 + ξ1 )e−ξ1 (1 + ξ2 )e−ξ2 ≥ 18

ξ12 + ξ22 . 8

ANASTASSIOU, GAL: MULTICOMPLEX SINGULAR INTEGRALS

Indeed, by [3], p. 219, we have 1 − (1 + ξ)e−ξ ≥

ξ2 4 ,

145

which implies

1 − (1 + ξ1 )e−ξ1 (1 + ξ2 )e−ξ2 ≥ 1 − (1 + ξ1 )e−ξ1 ≥

ξ12 , 4

1 − (1 + ξ1 )e−ξ1 (1 + ξ2 )e−ξ2 ≥ 1 − (1 + ξ2 )e−ξ2 ≥

ξ22 , 4

which gives the required inequality. By the above lower estimate for kRξ1 ,ξ2 (f ) − f kr , for all p1 , p2 ≥ 1 and ξ1 , ξ2 ∈ (0, 1], it follows 8 kRξ1 ,ξ2 (f ) − f kr kRξ1 ,ξ2 (f ) − f kr kRξ1 ,ξ2 (f ) − f kr ≥ ≥ ≥ |αp1 ,p2 |rp1 +p2 . 2 2 ξ1 + ξ2 Vξ1 ,ξ2 Eξ1 ,ξ2 ,p1 ,p2 (1)

(2)

This implies that if would exist subsequences in (0, 1], (ξk )k , (ξk )k with (1) (2) ξk , ξk → 0 (as k → ∞) and such that ° ° ° ° °Rξ(1) ,ξ(2) (f ) − f ° k k r = 0, lim (1) 2 (2) 2 k→∞ [ξk ] + [ξk ] then we would have αp1 ,p2 = 0 for all p1 , p2 ≥ 1. Therefore, if the Taylor’s development of f (z1 , z2 ) contains at least one term of the form αp1 ,p2 z1p1 z2p2 with p1 , p2 ≥ 1 and αp1 ,p2 6= 0, then kRξ1 ,ξ2 (f ) − f kr > 0, ξ12 + ξ22 ξ1 ,ξ2 ∈(0,1] inf

which implies that there exists a constant Cr (f ) > 0 such that Cr (f ), for all ξ1 , ξ2 ∈ (0, 1], that is

kRξ1 ,ξ2 (f )−f kr ξ12 +ξ22

≥

kRξ1 ,ξ2 (f ) − f kr ≥ Cr (f )[ξ12 + ξ22 ], for all ξ1 , ξ2 ∈ (0, 1]. Now, consider k1 + k2 = q ≥ 1 and denote Eξ1 ,ξ2 ,p1 ,p2 ,k1 ,k2 = 1 − (1 + (k1 + p1 )ξ1 )e−(k1 +p1 )ξ1 (1 + (k2 + p2 )ξ2 )e−(k2 +p2 )ξ2 , Vk1 ,k2 ,ξ1 ,ξ2 =

inf

p1 ,p2 ≥1

Eξ1 ,ξ2 ,p1 ,p2 ,k1 ,k2 .

Evidently that we have Vk1 ,k2 ,ξ1 ,ξ2 = 1 − (1 + (k1 + 1)ξ1 )e−(k1 +1)ξ1 (1 + (k2 + 1)ξ2 )e−(k2 +ξ2 ≥ 1 − (1 + ξ1 )e−ξ1 (1 + ξ2 )e−ξ2 ≥

ξ12 + ξ22 8

and reasoning as in the case q = 0 we obtain ° ° ° ° ° ∂ k1 +k2 Rξ ,ξ (f ) ° ° ° ∂ k1 +k2 Rξ ,ξ (f ) ∂ k1 +k2 f ° ∂ k1 +k2 f ° 1 2 1 2 ° ° 8° − k1 k2 ° − k1 k2 ° k k ° ∂z1k1 ∂z2k2 ∂z1 1 ∂z2 2 ∂z1 ∂z2 ∂z1 ∂z2 r r ≥ ξ12 + ξ22 Vk1 ,k2 ,ξ1 ,ξ2 19

146

ANASTASSIOU, GAL: MULTICOMPLEX SINGULAR INTEGRALS

≥

° ° ∂ k1 +k2 Rξ ,ξ (f ) 1 2 ° − ° ∂z1k1 ∂z2k2

∂ k1 +k2 f k k ∂z1 1 ∂z2 2

° ° ° °

r

Eξ1 ,ξ2 ,p1 ,p2 ,k1 ,k2

≥ |αk1 +p1 ,k2 +p2 |(k1 + p1 )...(p1 + 1)(k2 + p2 )...(p2 + 1)rp1 +p2 , for all p1 , p2 ≥ 1 and ξ1 , ξ2 ∈ (0, 1]. (1) (2) This implies that if would exist subsequences in (0, 1], (ξk )k , (ξk )k with (1) (2) ξk , ξk → 0 (as k → ∞) and such that ° ° k +k ° ∂ 1 2 R (1) (2) (f ) ° ξ ,ξ ° ∂ k1 +k2 f ° k k − k1 k2 ° ° k1 k2 ∂z1 ∂z2 ∂z1 ∂z2 ° ° r lim = 0, (1) 2 (2) 2 k→∞ [ξk ] + [ξk ] then we would have αk1 +p1 ,k2 +p2 = 0 for all p1 , p2 ≥ 1 and k1 + k2 = q ≥ 1. Therefore, if the Taylor’s development of f (z1 , z2 ) contains at least one term of the form αk1 +p1 ,k2 +p2 z1k1 +p1 z2k2 +p2 with p1 , p2 ≥ 1, αk1 +p1 ,k2 +p2 6= 0, k1 +k2 = q ≥ 1, then ° ° ° ∂ k1 +k2 Rξ ,ξ (f ) ° ∂ k1 +k2 f ° 1 2 ° − k1 k2 ° ° ∂z1k1 ∂z2k2 ∂z1 ∂z2 r > 0, inf ξ12 + ξ22 ξ1 ,ξ2 ∈(0,1] which implies that there exists a constant Cr,k1 ,k2 (f ) > 0 such that ° ° ° ∂ k1 +k2 Rξ ,ξ (f ) ° ∂ k1 +k2 f ° 1 2 ° − k1 k2 ° ° ∂z1k1 ∂z2k2 ∂z1 ∂z2 r ≥ Cr,k1 ,k2 (f ) for all ξ1 , ξ2 ∈ (0, 1], ξ12 + ξ22 that is ° ° ° ∂ k1 +k2 R ∂ k1 +k2 f ° ° ° ξ1 ,ξ2 (f ) − k1 k2 ° ≥ Cr,k1 ,k2 (f )[ξ12 + ξ22 ], for all ξ1 , ξ2 ∈ (0, 1]. ° ° ∂z1k1 ∂z2k2 ∂z1 ∂z2 °r ¤ Remark. For m = 1, Theorem 2, (iv) was obtained by [3], p. 214, Theorem 3.2.5, (iv).

4

Multi-Complex Gauss-Weierstrass Integrals

In this section we study the approximation properties for the multicomplex singular integral Wξ1 ,...,ξm (f³)(z1 ,´. . . , zm ). Theorem 3. Let f ∈ A D

m

and ξi > 0, i = 1, . . . , m. We have :

(i) Wξ1 ,...,ξm (f )(z1 , . . . , zm ) is continuous on D In addition if f (z1 , . . . , zm ) =

∞ X i1 ,i2 ,...,im =0

20

m

and analytic on Dm .

im αi1 ,i2 ,...,im z1i1 . . . zm ,

ANASTASSIOU, GAL: MULTICOMPLEX SINGULAR INTEGRALS

then 

∞ X

Wξ1 ,...,ξm (f )(z1 , . . . , zm ) =

αi1 ,...,im 

i1 ,...,im =0

m Y

 e

−i2j ξj /4

im  z1i1 . . . zm ,

j=1

for all z1 , . . . , zm ∈ D. (ii) ω1 (Wξ1 ,...,ξm (f ); δ1 , . . . , δm )Dm ≤ ω1 (f ; δ1 , . . . , δm )Dm , for all δ1 , . . . , δm ≥ 0. (iii) |Wξ1 ,...,ξm (f )(z1 , . . . , zm ) − f (z1 , . . . , zm )| µ ¶ ³ p p ´ m m ≤ 2 +√ ω1 f ; ξ1 , . . . , ξm m , π D for all zj ∈ D, ξj > 0; j = 1, . . . , m. (iv)³ Denote Dr = {z ∈ C; |z| < r} and for R > 1 let us suppose that ´ m

f ∈ A DR , that is we can write f (z1 , . . . , zm ) =

∞ X

im , αi1 ,i2 ...,im z1i1 z2i2 . . . zm

i1 ,i2 ...,im =0

for all z1 , . . . , zm ∈ DR , m ∈ N and αi1 ,i2 ...,im ∈ C. If the Taylor’s development of f (z1 , z2 , ..., zm ) contains at least one term of pm with p1 , p2 , ..., pm ≥ 1 and αp1 ,p2 ,...,pm 6= 0 for the form αp1 ,p2 ,...,pm z1p1 z2p2 ...zm q = 0, and at least one term of the form km +pm with p1 , p2 , ..., pm ≥ 1, αk1 +p1 ,k2 +p2 ,...,km +pm z1k1 +p1 z2k2 +p2 ...zm

such that αk1 +p1 ,k2 +p2 ,...,km +pm 6= 0 and k1 + k2 + ... + km = q forPq ≥ 1, then m for all 1 ≤ r < r1 < R, ξj ∈ (0, 1], j = 1, ..., m, kj ∈ N ∪ {0} with j=1 kj = q, q ∈ N ∪ {0}, we have ° ° m ° ∂ k1 +...+km W X ∂ k1 +...+km f ° ° ° ξ1 ,...,ξm (f ) − ∼ ξj , ° ° km km ° ° ∂z1k1 ...∂zm ∂z1k1 ...∂zm r

j=1

where the constants in the equivalence depend only on f , q, r and r1 . Here kf kr = sup{|f (z1 , ..., zm )|; |zj | ≤ r, j = 1, ..., m}. m Proof. (i) Let zo , zn ∈ D be with lim zn,j = z0,j , j = 1, . . . , m. We get n→∞

|Wξ1 ,...,ξm (f )(zn,1 , . . . , zn,m ) = −Wξ1 ,...,ξm (f )(z0,1 , . . . , z0,m )| Z ∞ Z ∞ 1 ≤ Qm p ... |f (zn,1 eiu1 , . . . , zn,m eium ) πξj −∞ −∞ j=1

21

147

148

ANASTASSIOU, GAL: MULTICOMPLEX SINGULAR INTEGRALS

−f (z0,1 eiu1 , . . . , z0,m eium )| Z

1 ≤ Qm p

ium

. . . , |zn,m e Z

1 = Qm p

− z0,m e Z

∞ −∞

ium

|)D

m Y m

2

e−uj /ξj du1 . . . dum

j=1 ∞

−∞

·

ω1 (f ; |zn,1 eiu1 − z0,1 eiu1 |,

−∞

...

πξj

j=1

−∞

2

e−uj /ξj du1 . . . dum

j=1 ∞

...

πξj

j=1

Z

∞

m Y

m Y

ω1 (f ; |zn,1 − z0,1 |, . . . , |zn,m − z0,m |)Dm 2

e−uj /ξj du1 . . . dum

j=1

Z

1 = ω1 (f ; |zn,1 − z0,1 |, . . . , |zn,m − z0,m |)Dm Qm p j=1

·

m Y

Z

∞

∞

...

πξj

−∞

−∞

2

e−uj /ξj du1 . . . dum

j=1

= ω1 (f ; |zn,1 − z0,1 |, . . . , |zn,m − z0,m |)Dm . Passing to limit with n → ∞, it follows that Wξ1 ,...,ξm (f )(z1 , . . . , zm ) is conm tinuous at z0 ∈ Dm , since f is continuous on D . It remains to prove that m m Wξ1 ,...,ξm (f ) is analytic on D . For f ∈ A(D ), we can write f (z1 , . . . , zm ) =

∞ X

im , αi1 ,i2 ,...,im z1i1 . . . zm

i1 ,...,im =o m

for all z1 , . . . , zm ∈ D . As in the proof of Theorem 1, (i) we can write for all (zi ∈ D, i = 1, . . . , m), 1 Wξ1 ,...,ξm (f )(z1 , . . . , zm ) = Qm p j=1

Z

Z

∞

∞

... −∞

=

−∞

i1 ,...,im =0 ∞ X i1 ,...,im =0

im αi1 ,...,im z1i1 . . . zm

i1 ,...,im =0

m Y

2

e−uj /ξj du1 . . . dum

j=1 m Y

1 p πξj j=1

im  αi1 ,...im z1i1 . . . zm

=

πξj

ei(u1 i1 +...+um im ) 

∞ X

∞ X

 αi1 ,...,im 

m Y

j=1

22

Z

 ∞

ei ij uj e

−u2j /ξj

−∞

 e

−i2j ξj /4

im  z1i1 . . . zm ,

duj 

ANASTASSIOU, GAL: MULTICOMPLEX SINGULAR INTEGRALS

149

proving the analyticity of Wξ1 ,...,ξm (f ). (ii) Let z1,j , z2,j ∈ D, |z1,j − z2,j | ≤ δj , for j = 1, . . . , m. We obtain |Wξ1 ,...,ξm (f ) (z1,1 , . . . , z1,m ) − Wξ1 ,...,ξm (f ) (z2,1 , . . . , z2,m )| Z ∞ Z ∞ 1 ... |f (z1,1 eiu1 , . . . , z1,m eium ) ≤ Qm p πξ −∞ j −∞ j=1 −f (z2,1 eiu1 , . . . , z2,m eium )|

m Y

2

e−uj /ξj du1 . . . dum

j=1

≤ ω1 (f ; |z1,1 − z2,1 | , |z1,2 − z2,2 | , . . . , |z1,m − z2,m |)Dm ≤ ω1 (f ; δ1 , . . . , δm )Dm , passing to sup with |z1,j − z2,j | ≤ δj , j = 1, . . . , m, it follows the desired inequality. (iii) We have |Wξ1 ,...,ξm (f ) (z1 , . . . , zm ) − f (z1 , . . . , zm )| Z ∞ Z ∞ ¯ 1 ¯f (z1 eiu1 , . . . , zm eium ) ≤ Qm p ... πξ −∞ −∞ j j=1 −f (z1 , . . . , zm )| Z

1 ≤ Qm p

−∞

2

e−uj /ξj du1 . . . dum

j=1 ∞

¡

ω1 f ; |z1 eiu1 − z1 |, . . . , |zm eium − zm |

...

πξj

j=1

Z

∞

m Y

¢

−∞ m Y

D

m

2

e−uj /ξj du1 . . . dum .

j=1

By the Maximum Modulus Principle (see e.g [4], p. 23, Corollary 1.2.5) we can take |zj | = |, for j = 1, . . . , m. Thus we have |Wξ1 ,...,ξm (f ) (z1 , . . . , zm ) − f (z1 , . . . , zm )| Z ∞ Z ∞ ¡ ¢ 1 ≤ Qm p ... ω1 f ; |eiu1 − 1|, . . . , |eium − 1| Dm πξj −∞ −∞ j=1 · ≤ Qm

1 p

Z

j=1

πξj

m Y

2

e−uj /ξj du1 . . . dum

j=1

Z

∞

... −∞

·

m Y

³ u1 um ´ ω1 f ; 2|sin |, . . . , 2|sin | 2 2 Dm −∞ ∞

2

e−uj /ξj du1 . . . dum

j=1

23

150

ANASTASSIOU, GAL: MULTICOMPLEX SINGULAR INTEGRALS

Z

1 ≤ Qm p

...

πξj

j=1

µ ¶ p |u1 | p |um | ω1 f ; ξ1 √ , . . . , ξm √ ξ1 ξm D m −∞

Z

∞

−∞

·

∞

m Y

2

e−uj /ξj du1 . . . dum

j=1

³ p p ´ ≤ ω1 f ; ξ1 , . . . , ξm

·

m Y

D

Z

2m Qm p

m

 ∞

...

πξj

j=1

Z

∞ 0

0

 m X u j 1 + p  ξj j=1

³ p p ´ 2 e−uj /ξj du1 . . . dum = 2m ω1 f ; ξ1 , . . . , ξm

j=1



Z

1 ·  Qm p 1 + Qm p j=1

Z

0

0

0

2

e−uj /ξj dui . . . dum

j=1

  m m X Y 2 u  p j∗  e−uj /ξj du1 . . . dum  ξ j∗ j ∗ =1 j=1

∞

...

πξj

0

³ p p ´ = 2m ω1 f ; ξ1 , . . . , ξm  1 · 1 + Qm p j=1

m Z X

πξj

Z

∞

0

j ∗ =1

0

= 2m ω1 f ; · 1 +

m Z X

1 π m/2

Z

∞

· 1 +

m X

1 π m/2

j=1,j6=j∗

 · 1 +

1 π m/2

p

ξ1 , . . . ,

∞

0

j ∗ =1

m Y

·

"ÃZ

ÃZ

D

m Y

m

 2

e−uj /ξj du1 . . . dum 

j=1

p ´ ξm

D

m

D

m

 m uj∗ Y −(uj /√ξj )2 du1 dum  p √ ... √ e ξ1 ξm ξj∗ j=1

0

= 2m ω1 f ; 

ξ1 , . . . ,

∞

³

u p j∗ ξj∗

p

... 0

j ∗ =1

∞

... ³



∞

p ´ ξm

√ 2 du u pj∗ e−(uj∗ / ξj∗ ) p j∗ ξj ∗ ξj ∗

e−(uj /

0

√

2

ξj )

du pj ξj

0

j=1,j6=j∗

24

!

! 

³ p p ´ = 2m ω1 f ; ξ1 , . . . , ξm m D  µZ ¶ µZ m m ∞ X Y −u2j ∗  uj∗ e duj∗ j ∗ =1

m



Z

∞

m ∞Y

...

πξj

j=1

Z

∞

D

∞

e 0

−u2j

 ¶ duj 

ANASTASSIOU, GAL: MULTICOMPLEX SINGULAR INTEGRALS

³ p p ´ = 2m ω1 f ; ξ1 , . . . , ξm

 D

m

1 +

³ p p ´ = 2m ω1 f ; ξ1 , . . . , ξm

1 π m/2

·

" µ √ ¶m−1 # 1 π  2 2 ∗ j =1 m X

m−1 m ·π 2 m m m/2 2 D π · ¸ ³ p ´ p m m = ω1 f ; ξ1 , . . . , ξm m 2 + √ , π D

1+

151

1

¸

·

proving the claim and finishing the proof of (iii). (iv) For the simplicity of calculation, we will prove the case m = 2. In our reasonings will be useful the following simple relationship Z ∞ Z ∞ F (u)du = [F (u) + F (−u)]du. −∞

0

Also, we will need the following two partial moduli of smoothness it 1 ω2,z1 (f ; ξ)(∂Dr )2 = sup{|∆2,z u f (re , z2 )|; |t| ≤ π, |u| ≤ ξ, |z2 | ≤ r},

and it 2 ω2,z2 (f ; η)(∂Dr )2 = sup{|∆2,z u f (z1 , re )|; |t| ≤ π, |u| ≤ η, |z1 | ≤ r},

where it i(t+u) 1 , z2 ) − 2f (reit , z2 ) + f (rei(t−u) , z2 ) ∆2,z u f (re , z2 ) = f (re

and it i(t+u) 2 ∆2,z ) − 2f (z1 , reit ) + f (z1 , rei(t−u) ). u f (z1 , re ) = f (z1 , re

Applying twice this relationship we easily get Wξ1 ,ξ2 (f )(z1 , z2 ) − f (z1 , z2 ) Z

∞ 2 2 1 √ [f (z1 eiu1 , z2 eiu2 ) − f (z1 , z2 )]e−u1 /ξ1 e−u2 /ξ2 du1 du2 π ξ1 ξ2 −∞ Z ∞½ Z ∞ 1 1 √ √ = · [f (z1 eiu1 , z2 eiu2 ) − 2f (z1 , z2 ) + f (z1 e−iu1 , z2 eiu2 )] πξ2 −∞ πξ1 0 o 2 2 ·e−u1 /ξ1 du1 · e−u2 /ξ2 du2

=

=

1 √ π ξ1 ξ2

Z

∞

( changing the order of the two integrals) ½Z ∞ [f (z1 eiu1 , z2 eiu2 ) − 2f (z1 , z2 ) + f (z1 e−iu1 , z2 eiu2 )]

0

+[f (z1 e

0 iu1

ª , z2 e−iu2 ) − 2f (z1 , z2 ) + f (z1 e−iu1 , z2 e−iu2 )] 25

152

ANASTASSIOU, GAL: MULTICOMPLEX SINGULAR INTEGRALS

2

2

·e−u1 /ξ1 e−u2 /ξ2 du1 du2 . Simple calculation shows [f (z1 eiu1 , z2 eiu2 ) − 2f (z1 , z2 ) + f (z1 e−iu1 , z2 eiu2 )] +[f (z1 eiu1 , z2 e−iu2 ) − 2f (z1 , z2 ) + f (z1 e−iu1 , z2 e−iu2 )] = [f (z1 eiu1 , z2 eiu2 ) − 2f (z1 , z2 eiu2 ) + f (z1 e−iu1 , z2 eiu2 )] +[f (z1 eiu1 , z2 e−iu2 ) − 2f (z1 , z2 e−iu2 ) + f (z1 e−iu1 , z2 e−iu2 )] +2[f (z1 , z2 eiu2 ) − 2f (z1 , z2 ) + f (z1 , z2 e−iu2 )]. Passing to absolute value, we obtain

≤

1 √ π ξ1 ξ2

|Wξ1 ,ξ2 (f )(z1 , z2 ) − f (z1 , z2 )| Z ∞Z ∞ [2ω2,z1 (f ; u1 )(∂Dr )2 + 2ω2,z2 (f ; u2 )(∂Dr )2 ] 0

0

2

2

·e−u1 /ξ1 e−u2 /ξ2 du1 du2 Z ∞Z ∞ p p · [2(1 + u1 / ξ1 )2 ω2,z1 (f ; ξ1 )(∂Dr )2

1 √ π ξ1 ξ2 0 0 p 2 p 2 2 +2(1 + u2 / ξ2 ) ω2,z2 (f ; ξ2 )(∂Dr )2 ] · e−u1 /ξ1 e−u2 /ξ2 du1 du2 p p ≤ C[ω2,z1 (f ; ξ1 )(∂Dr )2 + ω2,z2 (f ; ξ2 )(∂Dr )2 ]. ≤

(For the last inequality see the univariate case in [3], p. 226). Since by the mean value theorem for divided difference in Complex Analysis (see e.g. [5], p. 258, Exercise 4.20), for all |z1 | ≤ r, |z2 | ≤ r we have ° 2 ° ° ° it 2 °∂ f ° 2,z2 |∆u f (z1 , re )| ≤ u ° 2 ° ∂z2 r and

° 2 ° ° ° it 2 °∂ f ° 1 |∆2,z f (re , z )| ≤ u 2 u ° ∂z 1 ° , 2 r

for all ξ1 , ξ2 ∈ (0, 1] we immediately obtain kWξ1 ,ξ2 (f ) − f kr ≤ Cr (f )[ξ1 + ξ2 ]. Now, denoting by γ the circle of radius r1 > 1 and center 0, since for any |z| ≤ r and u, v ∈ γ, we have |v − z| ≥ r1 − r and |u − z| ≥ r1 − r, by the Cauchy’s formula in Theorem A, it follows that for all |z1 | ≤ r, |z2 | ≤ r, ξ1 , ξ2 ∈ (0, 1], we have ¯ ¯ ¯ ∂ k1 +k2 W ¯ ∂ k1 +k2 f ¯ ¯ ξ1 ,ξ2 (f ) (z , z ) − (z , z ) ¯ ¯ 1 2 1 2 ¯ ¯ ∂z1k1 ∂z2k2 ∂z1k1 ∂z2k2 ¯ ¯Z Z ¯ Wξ1 ,ξ2 (f )(u, v) − f (u, v) (k1 )!(k2 )! ¯¯ ¯ dudv = ¯ ¯ 2 k +1 k +1 1 4π (v − z2 ) 2 γ γ (u − z1 ) 26

ANASTASSIOU, GAL: MULTICOMPLEX SINGULAR INTEGRALS

≤ Cr (f )[ξ12 + ξ22 ]

(k1 !)(k2 !) 2πr1 2πr1 · · , 4π 2 (r1 − r)k1 +1 (r1 − r)k2 +1

which proves the upper estimate. It remains to prove the lower estimate for ° ° ° ∂ k1 +k2 W ∂ k1 +k2 f ° ° ° ξ1 ,ξ2 (f ) − k1 k2 ° . ° k1 k2 ° ∂z1 ∂z2 ∂z1 ∂z2 °r For this purpose, take z1 = reiϕ1 , z2 = reiϕ2 and p1 , p2 ∈ N ∪ {0}. Taking into account Theorem 3, (i) too, for k1 + k2 = q, (q ∈ N ∪ {0}), we have " # 1 ∂ k1 +k2 Wξ1 ,ξ2 (f ) ∂ k1 +k2 f (z1 , z2 ) − k1 k2 (z1 , z2 ) e−ip1 ϕ1 e−ip2 ϕ2 4π 2 ∂z1k1 ∂z2k2 ∂z1 ∂z2 ∞ ∞ 1 X X αi1 ,i2 i1 (i1 − 1)...(i1 − k1 + 1)ri1 −k1 eiϕ1 (i1 −k1 −p1 ) 4π 2 i1 =k1 i2 =k2 h i 2 2 ·i2 (i2 − 1)...(i2 − k2 + 1)ri2 −k2 eiϕ1 (i2 −k2 −p2 ) 1 − e−i1 ξ1 /4 e−i2 ξ2 /4 .

=

Integrating twice from −π to π, we obtain # Z π " k1 +k2 ∂ k1 +k2 f 1 ∂ Wξ1 ,ξ2 (f ) (z1 , z2 ) − k1 k2 (z1 , z2 ) e−ip1 ϕ1 e−ip2 ϕ2 dϕ1 dϕ2 4π 2 −π ∂z1k1 ∂z2k2 ∂z1 ∂z2 = αk1 +p1 ,k2 +p2 (k1 + p1 )...(p1 + 1)(k2 + p2 )...(p2 + 1)rp1 +p2 h i 2 2 · 1 − e−(k1 +p1 ) ξ1 /4 e−(k2 +p2 ) ξ2 /4 . Passing now to absolute value, we easily obtain |αk1 +p1 ,k2 +p2 |(k1 + p1 )...(p1 + 1)(k2 + p2 )...(p2 + 1)rp1 +p2 h i 2 2 · 1 − e−(k1 +p1 ) ξ1 /4 e−(k2 +p2 ) ξ2 /4 ° ° ° ∂ k1 +k2 W k1 +k2 ° ∂ f (f ) ° ° ξ1 ,ξ2 ≤° − k1 k2 ° . k1 k2 ° ∂z1 ∂z2 ∂z1 ∂z2 °r First consider q = 0, that is k1 = k2 = 0. Let us denote Eξ1 ,ξ2 ,p1 ,p2 = 1 − 2 2 e−p1 ξ1 /4 e−p2 ξ2 /4 and Vξ1 ,ξ2 =

inf

p1 ≥1,p2 ≥1

Eξ1 ,ξ2 ,p1 ,p2 .

Clearly we get ´ ³ ξ1 + ξ2 Vξ1 ,ξ2 = 1 − e−ξ1 /4 · e−ξ2 /4 = 1 − e−(ξ1 +ξ2 )/4 ≥ . 8 27

153

154

ANASTASSIOU, GAL: MULTICOMPLEX SINGULAR INTEGRALS

(For the last inequality see [3], the last line on the page 227). By the above lower estimate for kWξ1 ,ξ2 (f ) − f kr , for all p1 , p2 ≥ 1 and ξ1 , ξ2 ∈ (0, 1], it follows kWξ1 ,ξ2 (f ) − f kr 8 kWξ1 ,ξ2 (f ) − f kr kWξ1 ,ξ2 (f ) − f kr ≥ ≥ ≥ |αp1 ,p2 |rp1 +p2 . ξ1 + ξ2 Vξ1 ,ξ2 Eξ1 ,ξ2 ,p1 ,p2 (1)

(2)

This implies that if would exist subsequences in (0, 1], (ξk )k , (ξk )k with (1) (2) ξk , ξk → 0 (as k → ∞) and such that ° ° ° ° °Wξ(1) ,ξ(2) (f ) − f ° k k r lim = 0, (1) (2) k→∞ ξk + ξk then we would have αp1 ,p2 = 0 for all p1 , p2 ≥ 1. Therefore, if the Taylor’s development of f (z1 , z2 ) contains at least one term of the form αp1 ,p2 z1p1 z2p2 with p1 , p2 ≥ 1 and αp1 ,p2 6= 0, then kWξ1 ,ξ2 (f ) − f kr > 0, ξ1 + ξ2 ξ1 ,ξ2 ∈(0,1] inf

which implies that there exists a constant Cr (f ) > 0 such that Cr (f ), for all ξ1 , ξ2 ∈ (0, 1], that is

kWξ1 ,ξ2 (f )−f kr ξ1 +ξ2

kWξ1 ,ξ2 (f ) − f kr ≥ Cr (f )[ξ1 + ξ2 ], for all ξ1 , ξ2 ∈ (0, 1]. Now, consider k1 + k2 = q ≥ 1 and denote 2

Eξ1 ,ξ2 ,p1 ,p2 ,k1 ,k2 = 1 − e−(k1 +p1 ) Vk1 ,k2 ,ξ1 ,ξ2 =

inf

p1 ,p2 ≥1

ξ1 /4 −(k2 +p2 )2 ξ2 /4

e

,

Eξ1 ,ξ2 ,p1 ,p2 ,k1 ,k2 .

Evidently that we have 2

Vk1 ,k2 ,ξ1 ,ξ2 = 1 − e−(k1 +1)

ξ1 /4 −(k2 +1)2 ξ2 /4

≥ 1 − e−ξ1 /4 e−ξ2 /4 ≥

e

ξ1 + ξ2 8

and reasoning as in the case q = 0 we obtain ° ° ° ° ° ∂ k1 +k2 Wξ ,ξ (f ) ° ° ° ∂ k1 +k2 Wξ ,ξ (f ) ∂ k1 +k2 f ° ∂ k1 +k2 f ° 1 2 1 2 ° ° − k1 k2 ° − k1 k2 ° 8° k k ° ∂z1k1 ∂z2k2 ∂z1 1 ∂z2 2 ∂z1 ∂z2 ∂z1 ∂z2 r r ≥ ξ12 + ξ22 Vk1 ,k2 ,ξ1 ,ξ2 ° ° ° ∂ k1 +k2 Wξ ,ξ (f ) ° ∂ k1 +k2 f ° 1 2 ° − k1 k2 ° ° ∂z1k1 ∂z2k2 ∂z1 ∂z2 r ≥ Eξ1 ,ξ2 ,p1 ,p2 ,k1 ,k2 28

≥

ANASTASSIOU, GAL: MULTICOMPLEX SINGULAR INTEGRALS

≥ |αk1 +p1 ,k2 +p2 |(k1 + p1 )...(p1 + 1)(k2 + p2 )...(p2 + 1)rp1 +p2 , for all p1 , p2 ≥ 1 and ξ1 , ξ2 ∈ (0, 1]. (1) (2) This implies that if would exist subsequences in (0, 1], (ξk )k , (ξk )k with (1) (2) ξk , ξk → 0 (as k → ∞) and such that ° k +k ° ° ∂ 1 2 W (1) (2) (f ) ° k1 +k2 ξ ,ξ ° ° k k − ∂ k1 kf2 ° ° k1 k2 ∂z1 ∂z2 ∂z1 ∂z2 ° ° r lim = 0, (1) (2) k→∞ ξk + ξk then we would have αk1 +p1 ,k2 +p2 = 0 for all p1 , p2 ≥ 1 and k1 + k2 = q ≥ 1. Therefore, if the Taylor’s development of f (z1 , z2 ) contains at least one term of the form αk1 +p1 ,k2 +p2 z1k1 +p1 z2k2 +p2 with p1 , p2 ≥ 1, k1 + k2 = q ≥ 1 and αk1 +p1 ,k2 +p2 6= 0, then ° ° ° ° ∂ k1 +k2 Wξ ,ξ (f ) ∂ k1 +k2 f ° 1 2 ° − k1 k2 ° ° ∂z1k1 ∂z2k2 ∂z1 ∂z2 r inf > 0, ξ1 + ξ2 ξ1 ,ξ2 ∈(0,1] which implies that there exists a constant Cr,k1 ,k2 (f ) > 0 such that ° ° ° ∂ k1 +k2 Wξ ,ξ (f ) ° ∂ k1 +k2 f ° 1 2 ° − k1 k2 ° ° ∂z1k1 ∂z2k2 ∂z1 ∂z2 r ≥ Cr,k1 ,k2 (f ) for all ξ1 , ξ2 ∈ (0, 1], ξ1 + ξ2 that is ° ° ° ∂ k1 +k2 W ∂ k1 +k2 f ° ° ° ξ1 ,ξ2 (f ) − k1 k2 ° ≥ Cr,k1 ,k2 (f )[ξ1 + ξ2 ], for all ξ1 , ξ2 ∈ (0, 1]. ° k1 k2 ° ∂z1 ∂z2 ∂z1 ∂z2 °r ¤ Remark. For m = 1, Theorem 3, (iv) was obtained by [3], p. 224, Theorem 3.2.8, (iv).

References [1] G.A. Anastassiou and S. G. Gal, Geometric and approximation properties of some singular integrals in the unit disk, J. Ineq. Appl., vol. 2006, Article ID 17231, 19 pages, 2006. [2] C. Andreian Cazacu, Theory of Functions of Several Complex Variables (in Romanian), Edit. Didact. Pedag., Bucharest, 1971. [3] S.G. Gal, Approximation by Complex Bernstein and Convolution Type Operators, World Scientific Publishing Company, New Jersey, London, Singapore, Beijing, Shanghai, Hong Kong, Taipei, Chennai, 2009. 29

155

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[4] G. Kohr, Basic Topics in Holomorphic Functions of Several Complex Variables, University Press, Cluj-Napoca, 2003. [5] D.D. Stancu, Course of Numerical Analysis (in Romanian), Faculty of Mathematics and Mechanics, ”Babes-Bolyai” University, Cluj, 1977.

30

JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.1, 157-166, 2011, COPYRIGHT 2011 EUDOXUS PRESS, LLC

Spaces with σ-compact finite weak-bases

∗

Zhaowen Li1 and Qingguo Li2 1

College of Mathematics and Computer Science, Guangxi University for Nationalities, Nanning, Guangxi 530006, P.R.China E-mail: [email protected]

2

College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, P.R.China

Abstract: In this paper, we give some characterizations of spaces with σ-compact finite weak-bases and prove that spaces have σ-compact finite weak-bases if and only if they are 1-sequence-covering and quotient msk-images of metric spaces. Keywords and Phrases: Weak-bases; cs-networks; k-networks; Fr´ echet spaces; g-first countable spaces; msk-mappings; 1-sequence-covering mappings. 2000 Mathematics Subject Classification:

54E35; 54E40

1. Introduction and definitions Weak-bases were introduced by A.V.Arhangel’skii[1] , and they are an important concept in generalized metric spaces. In this paper, we characterize spaces with σ-compact finite weakbases. As its application, we give a metrization theorem and partially answer one question posed by C. Liu in [2]. At fast, we prove that spaces have σ-compact finite weak-bases if and only if they are 1-sequence-covering and quotient msk-images of metric spaces. In this paper, all spaces are assumed to be T2 , and all mappings are assumed to be continuous and onto. N denotes the set of all natural numbers. Denote {f (P ) : P ∈ P} by f (P). Definition 1.1.

Let f : X → Y be a mapping.

(1) f is called a msk-mapping[18] (i.e., metrizably stratified k-mapping) if there exists a ∗

The work is supported by the NSF of China (No. 10771056, 10671211), the NSF of Hunan Province in

China (No. 09JJ6005).

1

158

LI, LI: SPACES WITH FINITE WEAK BASES

subspace X of the usual product space

Q i∈N

Xi of the collection {Xi : i ∈ N } of metric spaces

satisfying the following condition: for each compact subset K of Y and i ∈ N , cl(πi f −1 (K)) is compact in Xi , and X is a metric space. (2) f is called a 1-sequence-covering mapping[12] if for each y ∈ Y , there exists x ∈ f −1 (y) satisfying the following condition: whenever {yn } is a sequence of Y converging to a point y in Y , there exists a sequence {xn } of X converging to a point x in X such that each xn ∈ f −1 (yn ). Definition 1.2.

Let P be a collection of subsets of a space X.

(1) P is called a k-network

[8] for

X if for each compact subset K of X, its open neigh-

borhood V in X, there exists a finite subcollection P ∗ ⊂ P such that K ⊂

S

P∗ ⊂ V .

(2) P is called a cs-network[6] for X if for each x ∈ X, for its open neighborhood V in X, and for a sequence {xn } in X which converges to x, there are P ∈ P and m ∈ N such S

that {xn : n ≥ m} {x} ⊂ P ⊂ V . (3) X is called an ℵ-space (resp. ℵ0 -space) if X has a σ-locally finite (resp. a countable) k-network. Definition 1.3[14] .

Let X be a space, and P ⊂ X. Then,

(1) A sequence {xn } in X is called eventually in P if {xn } converges to x, and there S

exists m ∈ N such that {x} {xn : n ≥ m} ⊂ P . (2) P is called a sequential neighborhood at x in X if whenever a sequence {xn } in X converges to x, then {xn } is eventually in P . (3) P is called sequential open in X if P is a sequential neighborhood at each of its points. (4) X is called a sequential space if any sequential open subset of X is open in X. (5) X is called a Fr´ echet space if whenever x ∈ cl(A), then x ∈ cl(A ∩ C) for some compact subset C of X. Definition 1.4.

Let P =

S

{Px : x ∈ X} be a collection of subsets of a space X

satisfying that for each x ∈ X, (a) Px is a network at x in X (i.e., x ∈ X, P ⊂ U for some P ∈ Px ), (b)

If U, V ∈ Px , then W ⊂ U

T

T

Px and for each neighborhood U of x in

V for some W ∈ Px .

(1) P is called a weak-base[1] for X if for G ⊂ X, G is open in X if and only if for each 2

LI, LI: SPACES WITH FINITE WEAK BASES

159

x ∈ G, there exists P ∈ Px such that P ⊂ G, here Px is called a weak-base at x in X. (2) P is called an sn-network[12] (i.e., an sequential neighborhood network) for X if each element of Px is a sequential neighborhood at x in X, here Px is called an sn-network at x in X. (3) X is called g-metrizable (resp. g-second countable)[4] if X has a σ-locally finite (resp. a countable) weak-base. (4) X is called g-first countable[1] if X has a weak-base P = ∪{Px : x ∈ X} such that Px is countable for every x ∈ X. Definition 1.5[15] .

X is called a countably bi-k-space if whenever (Fn ) is a decreasing

sequence of subsets of X with a common cluster point x, then there exists a decreasing sequence (An ) of subsets of X such that x ∈ Fn

T

An for any n ∈ N , K =

T

{An : n ∈ N } is

compact in X and each neighborhood of K contains some An . Remark 1.6.

(1) For a space, base =⇒ weak-base =⇒ sn-network =⇒ cs-network,

and base =⇒ k-network. An sn-network for a sequential space is a weak-base (see [12] ). (2) ℵ-spaces ⇐⇒ spaces with σ-locally finite cs-networks (see [7, Theorem 4]). (3) g-metrizable spaces =⇒ spaces with σ-compact finite weak-bases =⇒ spaces with point-countable weak-bases =⇒ g-first countable spaces =⇒ sequential spaces =⇒ k-spaces (see [10]). (4) First countable spaces ⇐⇒ Fr´ echet, g-first countable spaces (see [4] or [9]). (5) Every countably bi-k-space is characterized as a countably bi-quotient image of a paracompact M -space. Every locally compact space and every first countable space are a countably bi-k-space, every countably bi-k-space is a k-space (see [15]). 2. The characterizations of spaces with σ-compact finite weak-bases. Lemma 2.1. Proof.

Every compact-countable cs-network for a space X is a k-network for X.

Let P be a compact-countable cs-network for X. We will show that P is a

k-network for X. Suppose K ⊂ V with K non-empty compact and V open in X. Put A = {P ∈ P : P ∩ K 6= ∅ and P ⊂ V }, then A =

S

{An : n ∈ N } is countable. Denote A = {Pi : i ∈ N }, then K ⊂

some n ∈ N . Otherwise, K 6⊂

S i≤n

Pi for each n ∈ N , so choose xn ∈ K \ 3

S i≤n

S i≤n

Pi for

Pi . Because

160

LI, LI: SPACES WITH FINITE WEAK BASES

{P ∩ K : P ∈ P} is a countable cs-network for a subspace K and a compact space with a countable network is metrizable, then K is a compact metrizable space. Thus {xn } has a convergent subsequence {xnk }, where xnk → x. Obviously, x ∈ K, so V is an open neighborhood of x in X. Since P is a cs-network for X, then there exist m ∈ N and P ∈ P S

such that {xnk : k ≥ m} {x} ⊂ P ⊂ V . Now, P = Pj for some j ∈ N . Take l ≥ m such that nl ≥ j, then xnl ∈ Pj . This is a contradiction. Remark 2.2. By Lemma 2.1, a space X has a compact-countable cs-network ⇒ X has a compact-countable k-network. But X has a point-countable cs-network 6⇒ X has a compactcountable k-network because X has a point-countable sn-network 6⇒ X has a point-countable ˘ k-network, for example, the Stone-Cech compactification βN . Theorem 2.3.

For a space X, (1) ⇐⇒ (2) ⇐⇒ (3) =⇒ (4) below hold.

(1) X has a σ-compact finite weak-base. (2) X is a k-space with a σ-compact finite sn-network. (3) X is a g-first countable space with a σ-compact finite cs-network. (4) X is a g-first countable space with a σ-compact finite k-network. Proof.

(1) =⇒ (2) and (1) =⇒ (3) are obvious.

(3) =⇒ (4) holds by Lemma 2.1. (2) =⇒ (1). Suppose X is a k-space with a σ-compact finite sn-network P, then P is a σ-compact finite cs-network for X. By Lemma 2.1, P is a σ-compact finite k-network for X. Since a k-space with a point countable k-network is sequential (see [6, Corollary 3.4]), then X is a sequential space. Thus P is a weak-base for X. Hence X has a σ-compact finite weak-base. (3) =⇒ (1). Let P =

S

Suppose X is a g-first countable space with σ-compact finite cs-network.

{Pm : m ∈ N } be a σ-compact finite cs-network for X, where each Pm is compact

finite in X which is closed under finite intersections and X ∈ Pm ⊂ Pm+1 , and for each x ∈ X, let {B(n, x) : n ∈ N } be a decreasing weak neighborhood sequence of x in X. Put Fm,x = {P ∈ Pm : B(n, x) ⊂ P for some n ∈ N }, Fx = Fm = F=

S

{Fm,x : m ∈ N }

S

S

{Fm,x : x ∈ X}

{Fx : x ∈ X}

we will show that F is a σ-compact finite weak-base for X. 4

LI, LI: SPACES WITH FINITE WEAK BASES

It is easy to check that F satisfies Definition 1.4 (a),(b). Suppose G is an open subset of X, then for each x ∈ G, there exists P ∈ Fx with P ⊂ G. Otherwise, denote {P ∈ P : x ∈ P ⊂ G} = {P (m, x) : m ∈ N }. Then B(n, x) 6⊂ P (m, x) for each n, m ∈ N , so choose xnm ∈ B(n, x)\P (m, x). For n ≥ m, let xnm = yk , where k = m+

n(n−1) . 2

Then sequence {yk : k ∈ N } converges to the point x. Since P is a csS

network for X, thus there exist m, i ∈ N such that {yk : k ≥ i} {x} ⊂ P (m, x) ⊂ G. Take j ≥ i with yj = xnm for some n ≥ m. Then xnm ∈ P (m, x). This is a contradiction. On the other hand. If G ⊂ X satisfies that for each x ∈ G there exists P ∈ Fx with P ⊂ G, then B(n, x) ⊂ G for some n ∈ N . Thus G is open in X. Hence F is a weak-base for X. For each m ∈ N , Fm ⊂ Pm , then Fm is compact finite in X. Thus F = ∪{Fm : m ∈ N } is σ-compact finite in X. Therefore, (3) =⇒ (1) holds. Corollary 2.4.

The following are equivalent for a regular space X.

(1) X is a metrizable space. (2) X has a σ-compact finite base. (3) X is a Fr´ echet space with a σ-compact finite weak-base. (4) X is a Fr´ echet space with a σ-compact finite sn-network. (5) X is a first countable space with a σ-compact finite cs-network. (6) X is a first countable space with a σ-compact finite k-network. (7) X is a countably bi-k-space with a σ-compact finite k-network. (8) X is a Fr´ echet, g-metrizable space. Proof.

(1) ⇐⇒ (2) holds by Theorem 2.5.17 in [11].

(1) =⇒ (3) and (6) =⇒ (7) are obvious. (7) =⇒ (6).

Suppose X is a countably bi-k-space with a σ-compact finite k-network.

Note every countably bi-k-space with a point-countable k-network has a point-countable base (see [5]). Thus X is first countable. (3) ⇐⇒ (4) ⇐⇒ (5) =⇒ (6) hold by Theorem 2.3 and Corollary 2.1 in [9]. (6) =⇒ (8).

Suppose X is a first countable space with a σ-compact finite k-network.

Obviously, X is Fr´ echet. By Theorem 3 in [2], X is a Lasn˘ ev space (i.e., a close image of a metric space). Since a close image of a g-metrizable space is g-metrizable if and only if it is g-first countable (see [9, Theorem 3.1]), then X is g-metrizable.

5

161

162

LI, LI: SPACES WITH FINITE WEAK BASES

(1) ⇐⇒ (8) holds by Theorem 1.13 in [4]. Remark 2.5.

Corollary 2.4 partially answers the following problem posed by C. Liu in

[2] (i.e. Problem 3.9.8 in [19]): Can we characterize a g-metrizable space with a σ-compact finite weak-base? Example 2.6.

A g-first countable space with a σ-compact finite k-network 6=⇒ g-

metrizable. Let

½

I = [0, 1],

S1 =

1 :n∈N n

¾

∪ {0},

and let S(x) be a homeomorphism of S1 for each x ∈ I. Put M

M =I ⊕(

S(x)),

x∈I

then M is a locally compact metric space. Let X be the quotient space obtained from M by identifying the limit point of S(x) with x for each x ∈ I, and we denote this quotient mapping by f . Then, from Example 2.9.27 in [11] or Example 9.8 in [5], f is a compactcovering and compact mappings, and X has no point-countable cs-networks. It is easy to see that X is g-first countable and has a σ-compact-finite k-network (see [13]). Since X has no point-countable cs-networks, then X does not have a σ-compact finite weak-base. Thus X is not g-metrizable. This example illustrates: (4) 6=⇒ (1) in Theorem 2.3. Example 2.7.

Perfect mappings do not preserve spaces with σ-compact finite weak-

bases. S2 and Sω denote the Arens space and the sequential fan respectively, by Example 1.5.1 in [10], we have the following facts: (a) S2 is a g-second countable space; (b) Sω is the perfect image of S2 , but Sω is not a g-first countable space; then S2 has a σ-compact finite weak-base, but Sω do not have. Example 2.8. A space X has a σ-compact-finite k-network 6⇒ X has a σ-compact-finite cs-network. snev space and does not have a pointLet X be Sω1 , by Proposition 2.7.21 in [11], X is a La˘ countable cs-network. Then X has a σ-hereditarily closure-preserving k-network, so X has a

6

LI, LI: SPACES WITH FINITE WEAK BASES

σ-compact-finite k-network (see [2, Proposition 2]). But X does not have a σ-compact-finite cs-network. On the other hand. By Lemma 2.1, a space X has a σ-compact finite cs-network ⇒ X has a σ-compact finite k-network. Example 2.9.

A space X has a σ-locally countable base 6⇒ X has a σ-compact-finite

k-network (or cs-network). ˘ There exists a Cech-complete space X with a σ-locally countable base (see [16]), but X is not a σ-space (i.e., a space with σ-locally finite network), and X does not have a σ-compactfinite k-network. By Lemma 2.1, X does not have a σ-compact-finite cs-network. Example 2.10.

Fr´ echet, ℵ0 6=⇒ first countable.

Let X be a sequential fan Sω (see [11, Example 1.8.7]), then X is a regular, non-first countable, Fr´ echet, ℵ0 -space. Since X is an ℵ-space, X has a σ-compact finite k-network (or cs-network ). Thus, we have that the condition “first countable” or “countably bi-k” in Corollary 2.4 can not be replaced by “Fr´ echet”. Example 2.11.

g-second countable 6=⇒ Fr´ echet.

Let X be a Arens space S2 (see [11, Example 1.8.6]), then X is a regular, non-Fr´ echet, g-second countable space. It is obvious that a space is g-second countable if and only if it is Lindel¨of, g-metrizable. So X has a σ-compact finite weak-base (or sn-network). Thus, we have that the condition “Fr´ echet” in Corollary 2.4 can not be omitted. 3. The relationship between spaces with σ-compact finite weak-bases and metric spaces In [18], we introduce the concept of msk-mappings and establish the relationships between spaces with σ-compact finite strong k-networks ( cs-networks, cs∗ -networks ) and metric spaces by means of some covering mappings and msk-mappings. Below, we consider the relationship between spaces with σ-compact finite weak-bases and metric spaces. Lemma 3.1[18] .

Suppose f : X → Y is a msk-mapping, then there exists a base B for

X such that f (B) is a σ-compact finite network for Y . Lemma 3.2.

The following are equivalent for a space X.

(1) X has a σ-compact finite sn-network.

7

163

164

LI, LI: SPACES WITH FINITE WEAK BASES

(2) X is a 1-sequence-covering and msk-image of a metric space. Proof.

(1) =⇒ (2). Suppose X has a σ-compact finite sn-network. Let P =

S

{Pi :

i ∈ N } be a σ-compact finite sn-network for X, where each Pi = {Pα : α ∈ Ai } is a compact finite collection of subsets of X which is closed under finite intersections and X ∈ Pi ⊂ Pi+1 . By the proof of Theorem 2.2 in [18], there exist a metric space M and a msk-mapping f : M → X. We will prove that f is 1-sequence-covering. For each x ∈ X, by the idea of P, there exists (αi ) ∈ β = (αi ), then β ∈

Q

Ai such that {Pαi : i ∈ N } ⊂ P is a sn-network of x in X. Let

i∈N f −1 (x).

For each n ∈ N , let Rn = {(γi ) ∈ M : if i ≤ n, then γi = αi }.

Then {Rn : n ∈ N } is a monotonic decreasing neighborhood base of β in M . For each n ∈ N , it is easy to check that f (Rn ) =

T

i≤n

Pαi . Now suppose xj → x in X. For each n ∈ N , since

f (Bn ) is a sequential neighborhood of x in X, there exists i(n) ∈ N such that if i ≥ i(n), then xi ∈ f (Rn ). Thus f −1 (xi ) ∩Rn 6= φ. We may assume 1 < i(n) < i(n + 1). For each j ∈ N , let

  f −1 (xj ), if j < i(1), βj ∈  f −1 (x ) ∩ R , if i(n) ≤ j < i(n + 1), n ∈ N. j n

It is easy to show that the sequence {βj } converges to β in M . Hence f is 1-sequence-covering. (2) =⇒ (1). Suppose f : M → X is a 1-sequence-covering msk-map, where M is a metric space. From Lemma 3.1, there exists a base B for M such that f (B) is σ-compact finite in X. For each x ∈ X, there exists βx ∈ f −1 (x) satisfying Definition 1.1(2). Put Px = {f (B) : βx ∈ B ∈ B}, P=

[

{Px : x ∈ X},

it is easy to prove that P is a sn-network for X. Since P ⊂ f (B), then P is σ-compact finite in X. Thus X has a σ-compact finite sn-network P. Theorem 3.3.

The following are equivalent for a space X.

(1) X has a σ-compact finite weak-base. (2) X is a 1-sequence-covering and quotient msk-image of a metric space. Proof. (1) =⇒ (2). Suppose X has a σ-compact finite weak-base, then X is a sequential space with a σ-compact finite sn-network. By Lemma 3.2, X is a 1-sequence-covering mskimage of a metric space. Let f : M → X be a 1-sequence-covering msk-mapping, where M is a metric space. Since a 1-sequence-covering mapping is sequence-covering and X is a

8

LI, LI: SPACES WITH FINITE WEAK BASES

sequential space, then f is quotient (see [11, Proposition 2.1.16(2)]). Thus X is a 1-sequencecovering and quotient msk-image of a metric space. (2) =⇒ (1).

Suppose X is a 1-sequence-covering and quotient msk-image of a metric

space, then X is a sequential space with a σ-compact finite sn-network P by Lemma 3.2. It is easy to prove that P is a σ-compact finite weak-base for X.

References

[1] A. V. Arhangel’skii, Mappings and spaces, Russian Math. Surveys, 21 (1966), 115-162. [2] C. Liu, Spaces with a σ-compact finite k-spaces, Q & A in Gen. Top., 10(1992), 81-87. [3] C. Liu, Y. Tanaka, Spaces having σ-compact finite k-networks, and related matters, Top. Proc., 21 (1996), 173-200. [4] F. Siwiec, On defining a space by a weak-base, Pacific J. Math., 52 (1974), 233-245. [5] G. Gruenhage, E. Michael, Y. Tanaka, Spaces determined by point-countable covers, Pacific J. Math., 113(1984), 303-332. [6] J. A. Guthrie, A characterization of ℵ0 -spaces, Gen. Top. Appl, 1(1971), 105-110. [7] L. Foged, Characterizations of ℵ-spaces, Pacific J. Math., 110(1984), 59-63. [8] P. O’Meara, On paracompactness in function space with open topology, Proc. Amer. Math. Soc., 29(1971), 183-189. [9] S. Lin, On g-metrizable spaces, Chinese Ann. Math., 13(1992), 403-409. [10] S. Lin, Point-countable covers and sequence-covering mappings, Chinese Science Press, Beijing, 2002. [11] S. Lin, Generalized metric spaces and mappings, Chinese Science Press, Beijing, 1995. [12] S. Lin, On sequence-covering s-maps, Adv. in Math., 25(1996), 548-551. [13] S. Lin Y. Tanaka, Point-countable k-network, closed maps and related results, Top. Appl., 59(1994), 79-86. 9

165

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LI, LI: SPACES WITH FINITE WEAK BASES

[14] S. P. Franklin, Spaces in which sequences suffice, Fund. Math., 57(1965), 107-115. [15] E. Michael, A quintuple quotient quest, Gen. Top. Appl., 2(1972), 91-138. ˘ [16] S. W. Davis, A nondevelopable Cech-complete space with a point-countable base, Proc. Amer. Math. Soc., 78(1980), 139-142. [17] Y. Tanaka, Z. Li, Certain covering-maps and k-networks, and related matters, Topology Proc., 27(2003), 317-334. [18] Z. Li, S. Jiang, On msk-images of metric spaces, Georgian Math. J., 12(2005), 515-524. [19] S. Lin, Generalized metric spaces and mappings (second ed.), Chinese Science Press, Beijing, 2007.

10

JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.1, 167-187, 2011, COPYRIGHT 2011 EUDOXUS PRESS, 167 LLC

STATISTICAL Lp -CONVERGENCE OF DOUBLE SMOOTH PICARD SINGULAR INTEGRAL OPERATORS GEORGE A. ANASTASSIOU AND OKTAY DUMAN

Abstract. In this paper, we give some statistical approximation results for the double smooth Picard singular integral operators de…ned on Lp -spaces, which are not positive in general. Also, displaying a nontrivial example we demonstrate that our statistical Lp -approximation is stronger than the ordinary one.

1. Introduction Some of approximating operators, such as, Picard, Poisson-Cauchy and GaussWeierstrass singular integral operators do not need to be positive. In this case, the classical Korovkin theorem fails for such operators. However, it is possible to approximate a function by means of these operators (see, e.g., [1, 2, 3, 7, 8, 9, 10, 16]). Of course, by the term “approximation”we mean the ordinary convergence of a sequence of operators to a function. If we use the method of A-statistical convergence in the approximation, then we can get more powerful results than the classical aspects since the A-statistical convergence is weaker than the ordinary convergence. Actually, so far, this concept has already been used in the approximation by positive linear operators (see, e.g., [4, 11, 12, 13]). But, in this paper, we consider this convergence method in the approximation by double Picard singular integral operators de…ned on Lp -spaces, which are not positive in general. We should recall that, in our recent work [5], we have already studied uniform statistical approximation properties of these operators. First of all, we give some basic de…nitions and notations used in the present paper. Let A := [ajn ]; j; n = 1; 2; :::; be an in…nite summability matrix and assume P1 that, for a given sequence x = (xn )n2N ; the series n=1 ajn xn converges for every j 2 N. Then, by the A-transform of x; we mean the sequence Ax = ((Ax)j )j2N such that, for every j 2 N, 1 X (Ax)j := ajn xn : n=1

A summability matrix A is said to be regular (see [17]) if for every x = (xn )n2N for which limn!1 x = L we get limj!1 (Ax)j = L: Now, …x a non-negative regular summability matrix A: In [15] Freedman and Sember introduced a convergence method, the so-called A-statistical convergence, as in the following way. A given Key words and phrases. A-statistical convergence, statistical approximation, Picard singular integral operators, Lp -spaces. 2000 Mathematics Subject Classi…cation. Primary: 41A36; Secondary: 62L20. 1

168

2

GEORGE A. ANASTASSIOU AND OKTAY DUM AN

sequence x = (xn )n2N is said to be A-statistically convergent to L if, for every " > 0; X lim anj = 0: j!1

n : jxn Lj "

This limit is denoted by stA limn xn = L: Observe that if A = C1 = [cjn ]; the Cesáro matrix of order one de…ned to be cjn = 1=j if 1 n j; and cjn = 0 otherwise, then C1 -statistical convergence coincides with the concept of statistical convergence, which was …rst introduced by Fast [14]. In this case, we use the notation st lim instead of stC1 lim (see Section 5 for this situation). Notice that every convergent sequence is A-statistically convergent to the same value for any non-negative regular matrix A; however, the converse is not always true. Not all properties of convergent sequences hold true for A-statistical convergence (or statistical convergence). For instance, although it is well-known that a subsequence of a convergent sequence is convergent, this is not always true for A-statistical convergence. Another example is that every convergent sequence must be bounded, however it does not need to be bounded of an A-statistically convergent sequence. Of course, with these properties, the usage of A-statistical convergence in the approximation theory provides us many advantages. 2. Construction of the operators As usual, by Lp R2 we denote the space of all functions f de…ned on R2 for which Z1 Z1 p jf (x; y)j dxdy < 1; 1 p < 1: 1

1

In this case, the Lp -norm of a function f in Lp R2 ; denoted by kf kp , is given by 0

kf kp = @

Z1 Z1 1

1

11=p

p

jf (x; y)j dxdy A

:

Throughout the paper, for r 2 N and m 2 N0 ; we use 8 r > > j m if j = 1; 2; :::; r; < ( 1)r j j [m] (2.1) := r j;r P r > > ( 1)r j j m if j = 0: : 1 j j=1

and

[m] k;r

(2.2)

:=

r X

[m] k j;r j ;

j=1

k = 1; 2; :::; m 2 N:

We observe that (2.3)

r X j=0

[m] j;r

= 1 and

r X j=1

( 1)r

j

r j

= ( 1)r

r : 0

169

STATISTICAL Lp -CONVERGENCE OF DOUBLE PICARD OPERATORS

3

Then, we consider the following double smooth Picard singular integral operators: 0 1 1 1 Z Z r p X 1 2 2 [m] @ [m] (2.4) Pr;n (f ; x; y) = f (x + sj; y + tj) e s +t = n dsdtA ; j;r 2 2n j=0 1

1

where (x; y) 2 R2 , n; r 2 N, m 2 N0 , f 2 Lp R2 ; 1 a bounded sequence of positive real numbers.

p < 1; and also (

n )n2N

is

Remarks. [m]

The operators Pr;n are not in general positive. For example, consider the non-negative function f (u; v) = u2 + v 2 and also take r = 2; m = 3; x = 0 and y = 0 in (2.4). [m] It is not hard to see that the operators Pr;n preserve the constant functions in two variables. We observe, for any > 0; that Z1 Z1

(2.5)

1

(2.6) Z1 Z1 1

e

p

s2 +t2 =

dsdt = 2

2

:

1

Let k 2 N0 . Then, it holds, for each ` = 0; 1; :::; k and for every n 2 N, that sk

` `

te

p ( s2 +t2 )=

n

0

dsdt =

2B

1

k `+1 `+1 ; 2 2

k+2 n (k

if k is odd + 1)! if k is even

3. Estimates for the operators (2.4) Let C (m) R2 denote the space of all functions having m times continuous partial derivatives with respect to the variables x and y: If f 2 Lp R2 , then the rth (double) Lp -modulus of smoothness of f is given by (see, e.g., [6]) (3.1)

! r (f ; h)p := p sup

u2 +v 2

r u;v (f ) p

h

< 1; h > 0; 1

p < 1;

where (3.2)

r u;v

(f (x; y)) =

r X

( 1)r

j

j=0

r f (x + ju; y + jv): j

Throughout this paper we use the notation @ r;s f (x; y) :=

@ m f (x; y) for r; s = 0; 1; :::; m with r + s = m: @ r x@ s y

We assume the following conditions (3.3)

f 2 C (m) (R2 ) and @ m

`;`

f (x; y) 2 Lp R2 ; for each ` = 0; 1; :::; m:

170

4

GEORGE A. ANASTASSIOU AND OKTAY DUM AN

3.1. Estimates in the case of m 2 N. In this subsection, we only consider the case of m 2 N. For r 2 N and f satisfying (3.3); let [m] Hr;n (x; y)

:

[m] = Pr;n (f ; x; y) f (x; y) Z1 Z1 X [m] k m X 1 k k;r sk 2 k! k ` 2 n k=1 `=0 1

` ` k `;`

t@

2

2 n

Z1 Z1 1

[m] k;r

m X

1

[m=2]

1 X

=

(2i + 1)

k X

k!

k=1

f (x; y) dsdt:

1

By (2.6), since, for every r; n; m 2 N, 1

!

[m] 2i 2i;r n

i=1

`=0

k k

( 2i X `=0

k ` ` k `;`

`

s

t@

2i @ 2i 2i `

`;`

!

f (x; y) dsdt 2i

f (x; y)B

)

`+1 `+1 ; 2 2

;

we have [m]

[m]

Hr;n (x; y) = Pr;n (f ; x; y) (3.4)

2i P

`=0

We need

P 1 [m=2]

f (x; y)

2i @ 2i 2i `

`;`

1 1 0 p s2 +t2 = n ` `

te

w)m

1

[m] 2i 2i;r n

`+1 `+1 ; 2 2

2i

f (x; y)B

Lemma 3.1. For every r; n; m 2 N, we have 0 1 Z m Z1 Z1 X 1 [m] @ (1 Hr;n (x; y) = 2 2n (m 1)! `=0 m sm m `

(2i + 1)

i=1

r sw;tw

@m

`;`

:

1

f (x; y) dwA

dsdt:

Proof. Let (x; y) 2 R2 be …xed. By Taylor’s formula, one can obtain that r X

[m] j;r

(f (x + js; y + jt)

f (x; y))

=

j=0

m X

k=1

+

[m] k;r

k X

k!

`=0

1 (m

1)!

Z1

k

sk

k

`

(1

w)m

` ` k `;`

t@

1

f (x; y)

'[m] x;y (w; s; t)dw;

0

where '[m] x;y (w; s; t)

:

=

r X

[m] m j;r j

j=0

[m] m;r

(m X `=0

m X `=0

m sm m `

m sm m `

` ` m `;`

` ` m `;`

t@

t@

f (x; y):

)

f (x + jsw; y + jtw)

171

STATISTICAL Lp -CONVERGENCE OF DOUBLE PICARD OPERATORS

We can also get that '[m] x;y (w; s; t)

r X

=

( 1)

j=0

m X

=

`=0

m X

=

`=0

(m X

r j

r j

m sm m `

`=0

m sm m `

` `

m sm m `

` `

t

t

8 r 1 such that p1 + 1q = 1 and f as in (3:3). Then ! p1 m X p m [m] m `;` Hr;n !r @ f; n p ; C n p

`=0

for some positive constant C depending on m; p; q; r: Proof. By Lemma 3.1, we …rst obtain that p

[m] Hr;n (x; y) 8 0 1 Z m Z1 Z1 0; there exists a > 0 such that ! r (f ; h)p < " whenever 0 < h < : Hence, ! r (f ; h)p " implies that h : Now replacing h by n ; for every " > 0; we see that fn : ! r (f ;

n )p

"g

which guarantees that, for each j 2 N, X n:! r (f ;

n )p

Also, by (4:1), we get lim j

fn : X

ajn

n:

"

X

n:

n

g;

ajn :

n

ajn = 0:

n

The last equality implies that X

lim j

n:! r (f ;

ajn = 0;

n )p "

which gives (4.2). So, the proof is completed. 4.1. Statistical Lp -approximation in the case of m 2 N. Combining Theorems 3.3 and 3.5 we immediately get the next result. Corollary 4.2. Let 1 p < 1 and m 2 N: Then, for every f as in (3:3) with @ 2i `;` f 2 Lp (R2 ); 1 i [m=2]; ` = 0; 1; :::; 2i; we have (m ) p1 [m=2] X X p m 2i [m] m `;` Pr;n (f ) f M1 ! @ f; + M 2 n p n r n p

i=1

`=0

for some positive constants M1 ; M2 depending on m; p; q; r; where M1 :=

D (as in Theorem 3:5) if p = 1 C (as in Theorem 3:3) if 1 < p < 1 with (1=p) + (1=q) = 1

M2 :=

E (as in Theorem 3:5) if p = 1 B (as in Theorem 3:3) if 1 < p < 1 with (1=p) + (1=q) = 1:

and

Now we are ready to give our …rst statistical Lp -approximation result. Theorem 4.3. Let m; r 2 N and A = [ajn ] be a non-negative regular summability matrix, and let ( n )n2N be a bounded sequence of positive real numbers for which (4:1) holds. Then, for all f as in (3:3) with @ 2i `;` f 2 Lp (R2 ); 1 i [m=2]; ` = 0; 1; :::; 2i; 1 p < 1; we have (4.3)

stA

lim kPr;n (f ) n

f kp = 0:

185

STATISTICAL Lp -CONVERGENCE OF DOUBLE PICARD OPERATORS

19

Proof. From (4:1) and Lemma 4.1 we may write that (4.4)

stA

lim n

m n !r

@m

`;`

p

f;

= 0 for each ` = 0; 1; :::; m

n p

and (4.5)

stA

lim n

2i n

= 0 for each i = 1; 2; :::;

Now, for a given " > 0; de…ne the following sets: S

:

=

[m] (f ) n 2 N : Pr;n

S`

:

=

n2N:

m n !r

f

@m

p

`;`

2

:

" ; p

f;

hmi

" (m + [m=2] + 1) M1

n p

;

(` = 0; 1; :::; m); " (m + [m=2] + 1) M2 Then, it follows from Corollary 4.2 that Si+m :=

n2N:

2i n

(i = 1; 2; :::;

hmi 2

):

m+[m=2]

[

S

Sk ;

k=0

which implies, for every j 2 N, that X

m+[m=2]

X

ajn

n2S

X

ajn :

n2Sk

k=0

Now, taking limit as j ! 1 in the both sides of the last inequality and also using (4.4), (4.5), we conclude that X ajn = 0; lim j

n2S

which gives (4.3). Hence, the proof is completed.

4.2. Statistical Lp -approximation in the case of m = 0. In this subsection, we …rst combining Theorems 3.6 and 3.7 as follows: Corollary 4.4. Let 1

p < 1 and r 2 N: Then, for every f 2 Lp R2 ; we have [0] Pr;n (f )

f

p

N ! r (f;

n )p

for some positive constant N depending on p; r; where N :=

L (as in Theorem 3:7) if p = 1 K (as in Theorem 3:6) if 1 < p < 1 with (1=p) + (1=q) = 1:

Now we can state our second statistical Lp -approximation result. Theorem 4.5. Let r 2 N and A = [ajn ] be a non-negative regular summability matrix, and let ( n )n2N be a bounded sequence of positive real numbers for which (4:1) holds. Then, for all f 2 Lp R2 with 1 p < 1; we have (4.6)

stA

[0] lim Pr;n (f ) n

f

p

= 0:

186

20

GEORGE A. ANASTASSIOU AND OKTAY DUM AN

Proof. It follows from Corollary 4.4 that, for every " > 0; n [0] " n 2 N : ! r (f; n 2 N : Pr;n (f ) f p

Hence, for each j 2 N, we have X

X

ajn

[0]

n: Pr;n (f ) f

n:! r (f;

" p

n )p

n )p

"o : N

ajn : " N

Now, letting j ! 1 in the last inequality and also considering using Lemma 4.1, we obtain that X ajn = 0; lim j

[0]

n: Pr;n (f ) f

"

p

which proves (4.6). 5. Concluding remarks In this section, we give some special cases of our approximation results obtained in the previous section. In particular, we …rst consider the case of A = C1 ; the Cesáro matrix of order one. In this case, from Theorems 4.3 and 4.5 we have the following result immediately. Corollary 5.1. Let m 2 N0 ; r 2 N, and let ( n )n2N be a bounded sequence of positive real numbers for which st limn n = 0 holds. Then, for all f as in (3:3) with @ 2i `;` f 2 Lp (R2 ); 1 i [m=2]; ` = 0; 1; :::; 2i; 1 p < 1; we have st

[m] lim Pr;n (f ) n

f

p

= 0:

The second result is the case of A = I; the identity matrix. Then, the next approximation theorem is a direct consequence of Theorems 4.3 and 4.5. Corollary 5.2. Let m 2 N0 ; r 2 N, and let ( n )n2N be a bounded sequence of positive real numbers for which limn n = 0 holds. Then, for all f as in (3:3) `;` with @ 2i o f 2 Lp (R2 ); 1 i [m=2]; ` = 0; 1; :::; 2i; 1 p < 1; the sequence n [m]

Pr;n (f )

n2N

is uniformly convergent to f with respect to the Lp -norm.

Finally, de…ne a sequence (

(5.1)

n

:=

n )n2N n 1+n ; n 1+n2 ;

as follows:

if n = k 2 ; k = 1; 2; ::: otherwise.

Then, observe that st limn n = 0: So, if use this sequence ( n )n2N in the de…nition [m] of the operator Pr;n , then, we obtain from Corollary 5.1 (or, Theorems 4.3 and 4.5) [m]

that st limn Pr;n (f )

f

p

= 0 holds for all f as in (3:3) with @ 2i

`;`

f 2 Lp (R2 );

1 i [m=2]; ` = 0; 1; :::; 2i; 1 p < 1. However, since the sequence ( n )n2N given by (5.1) is non-convergent, the classical Lp -approximation to a function f [m] by the operators Pr;n (f ) is impossible, i.e., Corollary 5.2 fails for these operators. We should remark that Theorems 4.3 and 4.5, and Corollary 5.1 are also valid when lim n = 0 because every convergent sequence is A-statistically convergent, and so statistically convergent. But, as in the above example, our theorems still work although ( n )n2N is non-convergent. Therefore, this nontrivial example clearly

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STATISTICAL Lp -CONVERGENCE OF DOUBLE PICARD OPERATORS

21

shows that our statistical Lp -approximation results in Theorems 4.3 and 4.5, and also in Corollary 5.1 are stronger than Corollary 5.2. References [1] G.A. Anastassiou, Lp convergence with rates of smooth Picard singular operators, Di¤ erential & Di¤ erence Equations and Applications, 31-45, Hindawi Publ. Corp., New York, 2006. [2] G.A. Anastassiou, Basic convergence with rates of smooth Picard singular integral operators, J. Comput. Anal. Appl. 8 (2006) 313-334. [3] G.A. Anastassiou, Global smoothness and uniform convergence of smooth Picard singular operators, Comput. Math. Appl. 50 (2005) 1755-1766. [4] G.A. Anastassiou and O. Duman, A Baskakov type generalization of statistical Korovkin theory, J. Math. Anal. Appl. 340 (2008) 476-486. [5] G.A. Anastassiou and O. Duman, Statistical approximation by double Picard singular integral operators, Studia Univ. Babe¸ s -Bolyai Math.; (accepted for publication). [6] G.A. Anastassiou and S.G. Gal, Approximation Theory: Moduli of Continuity and Global Smoothness Preservation, Birkhäuser Boston, Inc., Boston, MA, 2000. [7] G.A. Anastassiou and S. Gal, Convergence of generalized singular integrals to the unit, multivariate case, Applied Mathematics Reviews, Vol. 1, 1-8, World Sci. Publ., River Edge, NJ, 2000. [8] G.A. Anastassiou and S. Gal, Convergence of generalized singular integrals to the unit, univariate case, Math. Inequal. Appl. 4 (2000) 511-518. [9] G.A. Anastassiou and S. Gal, General theory of global smoothness preservation by singular integrals, univariate case, J. Comput. Anal. Appl. 1 (1999) 289-317. [10] A. Aral, Pointwise approximation by the generalization of Picard and Gauss-Weierstrass singular integrals, J. Concr. Appl. Math. 6 (2008) 327-339. [11] O. Duman, E. Erku¸s and V. Gupta, Statistical rates on the multivariate approximation theory, Math. Comput. Modelling 44 (2006) 763-770. [12] O. Duman, M.A. Özarslan and O. Do¼ g ru, On integral type generalizations of positive linear operators, Studia Math. 174 (2006) 1-12. [13] E. Erku¸s and O. Duman, A Korovkin type approximation theorem in statistical sense, Studia Sci. Math. Hungar. 43 (2006) 285-294. [14] H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951), 241-244. [15] A.R. Freedman and J.J. Sember, Densities and summability, Paci…c J. Math. 95 (1981), 293-305. [16] S.G. Gal, Degree of approximation of continuous functions by some singular integrals, Rev. Anal. Numér. Théor. Approx. (Cluj ), 27 (1998) 251-261. [17] G.H. Hardy, Divergent Series, Oxford Univ. Press, London, 1949.

George A. Anastassiou Department of Mathematical Sciences The University of Memphis Memphis, TN 38152, USA E-mail: [email protected] Oktay Duman TOBB Economics and Technology University, Faculty of Arts and Sciences, Department of Mathematics, Sö¼ gütözü TR-06530, Ankara, TURKEY E-mail: [email protected]

JOURNAL 188 OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.1, 188-198, 2011, COPYRIGHT 2011 EUDOXUS PRESS, LLC

On the computation of the rank of triangular banded block Toeplitz matrices∗ Jie Huang†, Ting-Zhu Huang School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan, 611731, P. R. China

Abstract A fast and effective numerical algorithm for computing the rank of block bidiagonal Toeplitz (BBT) matrices has been recently proposed by Triantafyllou and Mitrouli [On the computation of the rank of block bidiagonal Toeplitz matrices, J. Comput. Appl. Math., 227(2009): 126-135]. In this paper, we extend the modified resultant block bidiagonal Toeplitz (MRBBT) factorization algorithm to deal with a triangular banded block Toeplitz (TBBT) sequence of matrices. Compared with the complexity of O(k3 n3 ) required by the classic methods for computing the rank of TBBT matrices with the bandwidth set to be 3, the extended approach need O(kn3 ) only, which is therefore fast and efficient for large k. A numerical example is presented here to demonstrate the performance of the modified approach. Key words: Triangular banded; Block Toeplitz matrix; Rank; LU factorization AMSC: 65F30; 15B05

1

Introduction

The computation of the rank of a matrix [1-3, 6] arises naturally from the application of numerical methods to many computational fields of science. It is well known that the most efficient way of computing the rank of a dense matrix A ∈ Rm×n is the singular value decomposition (SVD). In the present paper, we are concerned with the problem of computing the rank of a triangular banded block Toeplitz (TBBT) sequence of matrices, based on the modified resultant block bidiagonal Toeplitz factorization (MRBBT) algorithm proposed by ∗ †

This research was supported by NSFC (10926190, 60973015). E-mail address: [email protected]

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HUANG, HUANG: TOEPLITZ MATRICES

189

Triantafyllou and Mitrouli for computing the rank of block bidiagonal Toeplitz (BBT) matrices [4]. For simplicity of exposition, we consider only the bandwidth of the TBBT matrices is 3 for the moment, which can be then naturally extended to any arbitrary bandwidth. Definition: (A, B, C) sequence Let A, B, C be three matrices of m × n, and let Γi , i = 1, 2, · · · , k, · · · be the following sequence of matrices:      A 0 0 0  !  A 0 0   B A 0 0  A 0    , · · · , Γ0 = A, Γ1 = , Γ2 =  B A 0  , Γ3 =   B A C B A 0     C B A 0 C B A     A   B A    . Γk =  C B A   .. .. ..   . . .   C B A The above sequence is defined as a (A, B, C) sequence of matrices. Obviously when C = 0, it becomes the (A, B) sequence, which has been already considered in [4]. In general, for computing the rank of a matrix, the Gauss-Jordan (GJ) [2, 5] factorization, the QR factorization with column pivoting (QRCP) [11], the Rank Revealing QR (RRQR) [1], the Singular Value Decomposition (SVD) [2] and the Partial SVD (PSVD) [12, 13] are the most known techniques. When the number k of block [C B A] increases, however, all the methods above become inefficient for implementation, which has been mentioned in [4]. In this way, for C = 0, taking advantage of the special form of the (A, B) sequence, [4] introduced reliable and efficient algorithms for the computation of rank of BBT matrices. In this paper, thanks to the special form of Γk , we extend the algorithm in [4] to deal with the rank of Γk efficiently without the requirement on the matrix C. We will show that if the bandwidth of a TBBT matrix is 3, then the complexity of O(kn3 ) is obtained, comparing with O(k3 n3 ) by the classic methods for the rank of the TBBT matrix. The outline of the paper is as follows. Section 2 briefly describes the application of the displacement rank of a matrix [7, 8, 10, 14-16] to the computation of the rank of TBBT matrices. The heart of the article is section 3, where an extended MRBBT algorithm for TBBT matrices and the resulting complexity are derived. Section 4 shows a numerical example to demonstrate the efficiency of the extended approach. Finally, the concluding section reviews the main idea.

2

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2

The displacement rank

Initially, we briefly describe the displacement rank method for BBT matrices. More references and information concerning the method, mentioned in [4], include [7-8, 14-18]. The (lower) shift matrix Z has the following form:    0     I 0  Z =   , .. .. . .     I 0 where I is the n × n identity matrix. The name ’(lower) shift matrix’ is derived from the fact that if we multiply an array by Z, the entries are shifted down, and if we multiply by Z T , the entries are shifted up. Let T be a kn × kn block Toeplitz matrix. It is showed in [8] that the displacement rank of T : ∇T = T − ZT Z T is at most 2n. In particular, exploiting the special form of a BBT matrix: many zero blocks, the authors in [4] have obtained its displacement rank is at most equal to n. That is T ∇T = T  − ZT Z  A  B A =  .. .. . .   B   A 0 · · ·  B 0 · · ·   =  0 0 · · ·  .. ..  . .  0 0 ···

A ··· ··· ··· ···

    0     I 0  −  .. .. . .     I 0  0  0   0  , ..  .   0

   A   B A   .. ..   . .   B A

    0 I     .. ..  . .      0 I     0

and the block [B A] is an n × 2n submatrix, the rank of ∇T is therefore less or equal to n. Similarly, given    A   B A     , T =  C B A   ... ... ...     C B A

3

HUANG, HUANG: TOEPLITZ MATRICES

then we have

 0 · · · · · · 0  0 · · · · · · 0   0 · · · · · · 0  , .. ..  . .   0 0 ··· ··· 0 and consequently, the rank of ∇T is less or equal to n. Kailath and Chun in [7] have proposed a Generalized Schur algorithm for fast triangular and orthogonal factorizations of Toeplitz matrices and Toeplitz-derived matrices (or Toeplitz-like, close-to-Toeplitz). Their procedure for an m × n Toeplitz matrix T with a r × r leading principal submatrix, requires O(αpr) flops, where p = max(m, n), α is the length of the generator (any matrix pair X, Y such that ∇T = XY T , where X = [x1 , · · · , xα ], Y = [y1 , · · · , yα ] is called generator of T [7]). Instead of matrix operations, required by two look-ahead Schur algorithms [10], the extended method treats block-Toeplitz matrices in the same way as scalar Toeplitz matrices, using only elementary scalar operations just as the algorithm in [4].

3 3.1

     ∇T =    

191

A B C .. .

The modified resultant TBBT factorization The MRTBBT algorithm

In this subsection, we use a similar approach in [4] to make use of the special structure of TBBT: reorder its blocks by moving the first 2n rows to the end of the matrix, taking the modified TBBT (MTBBT). In the (k + 2)-th term of the sequence we will have:      A   C B A   B A    . . . .. .. ..      C B A    ˜ Γk+1 =  −→ Γk+1 =    . C B A     . . .  . . . . . .    A  C B A B A It is a known result that the rank of a matrix is invariant under row interchanges. Therefore, we consider the rank of the matrix Γ˜ k+1 instead of Γk+1 in order to exploit the special form of the modified matrix Γ˜ k+1 : there are k same [C B A] blocks. We first triangularize the block [C B A] using LU with partial pivoting factorization: LU([C B A]) −→ [U (0) B(0) A(0) ], where U (0) is an upper triangular matrix. If G1 and G2 are matrices such that: G1 · [C B A] = [U (0) B(0) A(0) ], ! ! A 0 U1(0) X (0) , G2 · = B A 0 U2(0) 4

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HUANG, HUANG: TOEPLITZ MATRICES

where U (0) , U1(0) , U2(0) are upper triangular matrices, G2 is a matrix of 2n × 2n, then by multiplying the Γ˜ k term with the matrix:    G1    . ..   G =   ,   G1   G2 we have G · Γ˜ k

    C B A     . . . .. .. ..        ·  C B A       A   G2 B A  A(0)   .. ..  . . (0) (0) (0)  U B A  .  

  G1  ...  =   G1   (0)  U    =   (0)  U1  0

B(0) .. . X (0) U2(0)

After the block [C B A] is triangularized using the LU factorization, we shall update the other entries (the k − 2 blocks of [C B A]) without additional calculations. Here we have a huge saving in floating point operations, since we need only one LU factorization to be applied to the n × 3n block [C B A] instead of the whole matrix Γ˜ k . Now the resultant matrix is in almost upper triangular form (only the last 2n rows: the ! A 0 block , is non-triangular). In the following we only need to apply k − 1 more B A times the LU factorization to the matrix G · Γ˜ k , in fact, apply LU factorization to the following matrices:  (0) B(0) A(0)  U  (0) (0) 0  U1 X (0) 0 U2 0

    ,

 (0) B(0) A(0)  U  (1) (1) 0  U1 X (1) 0 U2 0

5

    , . . . ,

 (0) B(0) A(0)  U  (k−2) X (k−2) 0  U1 0 U2(k−2) 0

    .

HUANG, HUANG: TOEPLITZ MATRICES

Therefore, the final step could be performed as follows.  (0)   (1) B(1) B(0) A(0)  U  U       U (0) B(0) A(0) U (0)      .. .. ..  LU   . . .   −→  Γ˜ (1) = k (0) (0) (0)     U B A  (0)    (0)  0  U U1(1) X 0    1  0 0 0 U2(0) 0  (1)  B(1) A(1)  U    (2) (2) (2) U B A   (3) (3) (3)   U B A   LU LU  ... ... ...  −→ · · · −→     (0) (0) (0)  U B A    (k−2) (k−2)  U X 0  1  U2(k−2) 0 0  (1)  B(1) A(1)   U   (2) (2) (2) U B A   (3) (3) (3)   U B A   LU .. .. ..  . −→  . . .    (k−1) (k−1) (k−1)  U B A   (k−1) (k−1)    0 U X 1   0 0 U2(k−1)

193

A(1) B(0) .. .

A(0) .. .

..

. U (0) B(0) A(0)

X (1) U2(1)

We note that if some zero rows arise in the first factorization of the block [C B A] or the above step, we delete them, and therefore, it reduces the required flops more. Based on the above analysis, the algorithm, called the MRTBBT algorithm, for computing the rank of Γk can be summarized as follows. The MRTBBT Algorithm

! A 0 S tep 1: Triangularize the first block [C B A] and of Γ˜ k using LU factorizaB A tion with partial pivoting, and then delete any zero rows; S tep 2: Update the entries of the last k − 2 blocks [C B A] without additional calculations forming matrix Γ˜ k ; S tep 3: Compute the upper triangular matrix U, U = LU(Γ˜ k ). Obvious that, the number of non-zero rows of the resultant U is the rank of Γk . In practice, if the given k is not big enough, in order to avoid any floating point operation errors and without significantly increasing the required executable time, the previous algorithm can be implemented symbolically. Notice that this technique holds for rectangular m × n 6

         

194

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blocks A, B and C as well.

3.2 Computational cost In this subsection, we analyze the computational costs of our method and compare it with other methods. First, the LU factorization of [C B A] requires O( 38 n3 ), where [C B A] is an n × 3n ! ! A 0 U1(0) X (0) matrix. Then the triangularization of to demands O( 16 n3 ). Fi(0) 3 B A 0 U2 nally, since the upper left block is already in an upper triangular form, the factorization   (0) B(0) A(0)   U   of every 3n × 3n block  U1(i) X (i) 0  demands fewer flops, and the complexity of   0 U2(i) 0 every one of the last k − 1 calling of the LU factorization is O(6n3 ). Hence the LU factorizations in Step 3 demand O(6(k − 1)n3 ) flops. To sum up, the total complexity is n3 + 6(k − 1)n3 ) = O(2(3k + 1)n3 ) flops. O( 83 n3 + 16 3 In comparison, the complexity of the classic methods for computing the rank of a 3 3 TBBT matrix is O(k3 n3 ): the Gauss-Jordan: O( k 2n ) flops, the Classical SVD: O(2k3 n3 ) flops. Thus our method is fast and effective for large k. Moreover, taking into account the fact that we delete any zero rows that appears during the process, the required flops can be less than the above estimation in practise, reducing the floating point operations further. It is should be noted that our algorithm bases on the LU factorization, yet makes the numerical implementation of the method theoretically not stable, but it is known that Gaussian elimination with partial pivoting is utterly stable in practice [2]. We also note that by implementing the the Singular Value Decomposition (SVD) [5] or the Partial Singular Value Decomposition (PSVD) [12, 13] to the first phase of the previous algorithm we can obtain a numerical result more efficiently. In addition, if we use symbolical arithmetic, there will be no round off errors during the computations.

4 Example In this section, to illustrate the MRTBBT algorithm in section 3, we now present an example. Let        1 −3 2   1 −5 2   1 −5 3        A =  1 0 −1  , B =  1 −1 0  , C =  1 −1 0  .       2 −6 4 −2 10 −4 −2 10 −6

7

HUANG, HUANG: TOEPLITZ MATRICES

195

Then for the (A, B, C) sequence, the forth term Γ3 :           Γ3 =         

1 1 2 1 1 −2 1 1 −2 0 0 0

−3 0 −6 −5 −1 10 −5 −1 10 0 0 0

2 −1 4 2 0 −4 3 0 −6 0 0 0

and the modified fourth term Γ˜ 3 :   1 −5 3  1 −1 0   −2 10 −6   0 0 0   0 0 0  ˜Γ3 =  0 0 0  1 −3 2   1 0 −1  2 −6 4   1 −5 2   1 −1 0  −2 10 −4

0 0 0 1 1 2 1 1 −2 1 1 −2

0 0 0 −3 0 −6 −5 −1 10 −5 −1 10

0 0 0 2 −1 4 2 0 −4 3 0 −6

0 0 0 0 0 0 1 1 2 1 1 −2

0 0 0 0 0 0 −3 0 −6 −5 −1 10

0 0 0 0 0 0 2 −1 4 2 0 −4

0 0 0 0 0 0 0 0 0 1 1 2

0 0 0 0 0 0 0 0 0 −3 0 −6

0 0 0 0 0 0 0 0 0 2 −1 4

1 1 −2 1 1 −2 0 0 0 1 1 2

−5 −1 10 −5 −1 10 0 0 0 −3 0 −6

2 0 −4 3 0 −6 0 0 0 2 −1 4

1 1 2 1 1 −2 0 0 0 0 0 0

−3 0 −6 −5 −1 10 0 0 0 0 0 0

2 −1 4 2 0 −4 0 0 0 0 0 0

0 0 0 1 1 2 0 0 0 0 0 0

0 0 0 −3 0 −6 0 0 0 0 0 0

0 0 0 2 −1 4 0 0 0 0 0 0

Applying the LU factorization to blocks [C B A] and    LU([C B A]) = LU     −2  −→  0  0

           ,         

           .         

! A 0 , we have: B A

1 −5 3 1 −5 2 1 −3 2 1 −1 0 1 −1 0 1 0 −1 −2 10 −6 −2 10 −4 2 −6 4  10 −6 −2 10 −4 2 −6 4   4 −3 0 4 −2 2 −3 1  ,  0 0 0 0 0 2 −6 4

8

   

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HUANG, HUANG: TOEPLITZ MATRICES

and

LU

     LU    

 0   0  !! 0  A 0 =  B A 2   −1   4   0 0 0   2 −6 4   0 4 0 2 −6 4    0 0 −3 −1.5 4.5 −3   . −→  2 −6 4   0 0 0   0 0 0 0 3 −3    0 0 0 0 0 0 1 1 2 1 1 −2

−3 0 −6 −5 −1 10

2 −1 4 2 0 −4

0 0 0 1 1 2

0 0 0 −3 0 −6

Therefore the updated Γ˜ (1) 3 , after deleting the last line which is zeroed, becomes:

Γ˜ (1) 3

Γ˜ (2) 3

          =         

          =         

−2 10 −6 −2 0 4 −3 0 0 0 0 0 0 0 0 −2 0 0 0 0 0 0 0 0 2 −6 4 0 0 4 0 2 0 0 −3 −1.5 0 0 0 2 0 0 0 0

10 4 0 10 4 0 0 −6 4.5 −6 3

−4 2 −2 2 0 2 −6 −2 −3 0 0 0 0 0 4 0 −3 0 4 0 −3 0

 −6 4 0 0 0   −3 1 0 0 0   −6 4 0 0 0   10 −4 2 −6 4   4 −2 2 −3 1   0 0 2 −6 4  ,  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

 −2 10 −6 −2 10 −4 2 −6 4 0 0 0   0 4 −3 0 4 −2 2 −3 1 0 0 0   0 0 3 2 −10 6 −2 3 −1 0 0 0   0 0 0 −2 10 −6 −2 10 −4 2 −6 4   0 0 0 0 4 −3 0 4 −2 2 −3 1   0 0 0 0 0 0 0 0 0 2 −6 4   0 0 0 −2.667 9.333 −4 0.667 −4 3.333 0 0 0   0 0 0 0 −3.75 2.25 −1.875 2.25 −0.375 0 0 0  0 0 0 0 0 1.6 0 −2.4 2.4 0 0 0   0 0 0 0 0 0 2 −6 4 0 0 0   0 0 0 0 0 0 0 −4.5 4.5 0 0 0

9

HUANG, HUANG: TOEPLITZ MATRICES

197

and

Γ˜ (3) 3

          =         

−2 0 0 0 0 0 0 0 0 0 0

10 4 0 0 0 0 0 0 0 0 0

−6 −2 10 −4 2 −6 4 0 0 0 −3 0 4 −2 2 −3 1 0 0 0 3 2 −10 6 −2 3 −1 0 0 0 0 −2.667 9.333 −4 0.667 −4 3.333 0 0 0 0 0 4 −3 0 4 −2 2 −3 1 0 0 0 1.6 0 −2.4 2.4 0 0 0 0 0 0 0 −2.5 8.875 −3.875 0.5 −3.75 3.25 0 0 0 0 0 −4.5 4.5 0 0 0 0 0 0 0 0 0 2 0.4 −3 2.6 0 0 0 0 0 0 0 2 −6 4 0 0 0 0 0 0 0 0 4.5 −4.5

           .       

The number of non zero rows is 11 and thus the rank of Γ3 is equal to 11.

5

Conclusions

In this paper we have presented a modified resultant TBBT factorization technique for the rank of triangular banded block Toeplitz matrices. Our method is favorable for a large k, since it requires O(kn3 ) in complexity, comparing with the complexity of O(k3 n3 ) by other classic methods. The performance of our approach, which significantly reduces the required flops, is illustrated with a numerical example.

References [1] T. Chan, Rank revealing QR factorizations, Linear Algebra Appl., 88/89 (1987) 67-82. [2] B.N. Datta, Numerical Linear Algebra and Applications, second edition, Brooks/Cole Publishing Company, United States of America, 1995. [3] L. Foster, Rank and null space calculations using matrix decomposition without column interchanges, Linear Algebra Appl.,74(1984) 47-71. [4] D. Triantafyllou, M. Mitrouli, On the computation of the rank of block bidiagonal Toeplitz matrices, J. Comput. Appl. Math., 227(2009) 126-135 [5] M.K. Anstreicher, G.U. Rothblum, Using Gauss-Jordan elimination to compute the index, generalized nullspaces, and Drazin inverse, Linear Algebra Appl., 85(1987) 221-239. [6] P.Y. Yalamov, M. Mitrouli, A fast algorithm for index annihilation computations, J. Comput. Appl. Math., 108(1999) 99-111.

10

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[7] T. Kailath, J. Chun, Generalized displacement structure for block-Toeplitz, Toeplitz-block, and Toeplitz-derived matrices, SIAM J. Matrix Anal. Appl., 15(1994) 114-128. [8] T. Kailath, S.Y. Kung, M. Morf, Displacement ranks of matrices and linear equations, J. Math. Anal. Appl., 68(1979) 395-407 [9] M. Stewart, P.V. Dooren, Stability issues in the factorization of structured matrices, SIAM J. Matrix Anal. Appl., 18(1997) 104-118. [10] K.A. Gallivan, S. Thirumalai, P.V. Dooren, V. Vermaut, High performance algorithms for Toeplitz and block Toeplitz matrices, Linear Algebra Appl., 242/243 (1996) 343-388. [11] G.H. Golub, C.F. Van Loan, Matrix Computations, third edition, The John Hopkins Univercity Press, Baltimore, London, 1989. [12] V. Huffel, J. Vandewalle, An efficient and reliable algorithm for computing the singular subspace of a matrix, associated with its smallest singular values, J. Comput. Appl. Math., 19(1987) 313-330. [13] V. Huffel, Partial singular value decomposition algorithm, J. Comput. Appl. Math., 33(1990) 105-112. [14] H.A. Sayed, T. Kailath, A look-ahead block Schur algorithm for Toeplitz-like matrices, SIAM J. Matrix Anal. Appl., 16(1995) 388-414. [15] M. Stewart, P.V. Dooren, Stability issues in the factorization of structured matrices, SIAM J. Matrix Anal. Appl., 18(1997) 104-118. [16] T. Kailath, H.A. Sayed, Displacement structure: Theory and applications, SIAM Rev., 37(1995) 297-386. [17] S. Chandrasekaran, H.A. Sayed, A fast stable solver for nonsymmetric Toeplitz and QuasiToeplitz systems of linear equations, SIAM J. Matrix Anal. Appl., 19 (1998) 107-139. [18] N. Mastronardi, D. Kressner, V. Sima, P.V. Dooren, V. Huffel, A fast algorithm for statespace identification via exploitation of the displacement structure, J. Comput. Appl. Math., 132(2001) 71-81.

11

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL.13, NO.1, 2011

Approximation Properties of Some Multivariate Generalized Singular Integrals in the Unit Polydisk, George A. Anastassiou and Sorin G. Gal,…….……………………………………..11 Some Properties of Chebyshev Systems, Richard A. Zalik,……….…………………………..20 Solution to a Singular Integro-Differential Equation III, J.N. Pandey,………………………...27 Some approximation theorems for functions of two variables through almost convergence of double sequences, G.A. Anastassiou, M. Mursaleen and S.A. Mohiuddine,…..……………....37 Wavelet Transforms of Schwartz Distributions, J.N. Pandey,………………………………....47 Quantitative Estimates in the Overconvergence of Some Complex Multivariate Generalized Singular Integrals in Polystrips, George A. Anastassiou and Sorin G. Gal,.…………………...84 On special differential subordinations using a generalized Salagean operator and Ruscheweyh derivative, Alina Alb Lupas,……………………………………………………………………98 Certain special differential superordinations using multiplier transformation and Ruscheweyh derivative, Alina Alb Lupas,……………………………………………………………………108 On a subclass of analytic functions defined by Ruscheweyh derivative and multiplier transformations, Alina Alb Lupas,……………………………………………………………...116 On special differential superordinations using multiplier transformation, A. Alb Lupas,….......121 Approximation Properties of Some Multicomplex Singular Integrals in the Unit Polydisk, George A. Anastassiou and Sorin G. Gal,……………………………………………………....127 Spaces with σ-compact finite weak-bases, Zhaowen Li and Qingguo Li,………………….....157 Statistical Lp-Convergence of Double Smooth Picard Singular Integral Operators, George A. Anastassiou and Oktay Duman,………………………………………………………………..167 On the computation of the rank of triangular banded block Toeplitz matrices, Jie Huang and Ting-Zhu Huang,……………………………………………………………………………….188  

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Volume 13, Number 2 ISSN:1521-1398 PRINT,1572-9206 ONLINE

February 2011

Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC

202

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(seven 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. Webmaster:Ray Clapsadle 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 $470,Electronic $300,Print and Electronic $500.Individual:Print $150,Electronic $100,Print &Electronic $200.For any other part of the world add $50 more to the above prices for Print.No credit card payments. Copyright©2011 by Eudoxus Press,LLCAll 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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13) Christodoulos A.Floudas Department of Chemical Engineering Princeton University Princeton,NJ 08544-5263 609-258-4595 (x4619 assistant) e-mail: [email protected] 31) Gilbert G.Walter OptimizationTheory&Applications, Department Of Mathematical Global Optimization Sciences University of Wisconsin14) J.A.Goldstein Department of Mathematical Sciences Milwaukee,Box 413, Milwaukee,WI 53201-0413 The University of Memphis 414-229-5077 Memphis,TN 38152 e-mail: [email protected] 901-678-3130 Distribution e-mail:[email protected] Functions,Generalised Functions, Partial Differential Equations, Wavelets Semigroups of Operators 15) H.H.Gonska Department of Mathematics University of Duisburg Duisburg, D-47048

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Germany 011-49-203-379-3542 e-mail: [email protected] Approximation Theory, Computer Aided Geometric Design 16) 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 17) Christian Houdre School of Mathematics Georgia Institute of Technology Atlanta,Georgia 30332 404-894-4398 e-mail: [email protected] Probability,Mathematical Statistics,Wavelets 18) V. Lakshmikantham Department of Mathematical Sciences Florida Institute of Technology Melbourne, FL 32901 e-mail: [email protected] Ordinary and Partial Differential Equations, Hybrid Systems, Nonlinear Analysis 19) 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 36) Ahmed I. Zayed Department Of Mathematical Sciences DePaul University 2320 N. Kenmore Ave. Chicago, IL 60614-3250 773-325-7808 e-mail: [email protected]

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JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.2, 211-224, 2011, COPYRIGHT 2011 EUDOXUS PRESS, LLC

Approximation by nonlinear Lagrange interpolation operators of max-product kind on Chebyshev knots of second kind Lucian Coroianu and Sorin G. Gal Department of Mathematics and Computer Science, The University of Oradea, Universitatii 1, 410087, Oradea, Romania E-mail: [email protected] and [email protected] Abstract The aim of this note is to associate to the Lagrange interpolatory polynomials on Chebyshev nodes of second kind plus −1 and 1, a continuous piecewise rational interpolatory operator of the so-called max-product kind, uniformly convergent to the function f , with an essentially better, Jackson-type rate of approximation.

AMS 2000 Mathematics Subject Classification: 41A05, 41A25, 41A20. Keywords and phrases: Nonlinear Lagrange interpolation operators of max-product kind, Chebyshev nodes of second kind, degree of approximation.

1

Introduction

Based on the Open Problem 5.5.4, pp. 324-326 in [7], in a series of recent papers [1, 2, 3, 4, 5], we have introduced and studied the so-called max-product operators attached to the Bernstein polynomials and to other linear Bernsteintype operators, like those of Favard-Sz´asz-Mirakjan operators (truncated and nontruncated case), Baskakov operators (truncated and nontruncated case) and Bleimann-Butzer-Hahn operators. applied, for example, to the linear Bernstein Bn (f )(x) = ¡n¢idea ¡n¢ k operators PnThis k n−k n−k x (1 − x) f (k/n), (where p (x) = x (1 − x) ) works as n,k k=0 k k P n n k n−k f (k/n) k=0 ( k )x (1−x) Pn follows. Writing in the equivalent form Bn (f )(x) = n k (1−x)n−k x k=0 ( k ) and the sum operator Σ by the maximum¡ operator ¢ W then replacing Pn ¡ ¢everywhere n k n−k (that is k=0 nk xk (1−x) f (k/n) by the maximum ¡max k=0,..,n { k x (1− ¡ ¢ ¢ Pn x)n−k f (k/n)} and k=0 nk xk (1−x)n−k by maxk=0,...,n { nk xk (1−x)n−k }), one

1

212

COROIANU, GAL: LAGRANGE INTERPOLATION OPERATOR

obtains the nonlinear Bernstein operator of max-product kind n _

Bn(M ) (f )(x)

=

pn,k (x)f

¡k¢ n

k=0 n _

, pn,k (x)

k=0

for which, surprisingly nice approximation and shape preserving properties were found. For example, it is proved that for some classes of functions (like those of monotonous concave functions), the order of approximation given by the maxproduct Bernstein operators, are essentially better than the approximation order of their linear counterparts. In the recent paper [6], suggested by the fact that the fundamental HermiteFe´jer polynomials are positive exactly as the fundamental Bernstein polynomials are, this idea is applied to the Hermite-Fej´er interpolation based on the Chebyshev nodes of first kind, obtaining a max-product interpolation operator which, in general, (for example, in the class of positive Lipschitz functions) approximates essentially better than the corresponding Hermite-Fej´er polynomials. The aim of the present paper is to use the same idea in the case of the Lagrange interpolation polynomials based on the Chebyshev knots of second kind in [−1, 1] plus the endpoints. Surprisingly, although in this case the fundamental Lagrange polynomials are not anymore positive, however, the corresponding max-product Lagrange operators obtained have essentially better approximation properties than the corresponding Lagrange interpolation polynomials. First we present the problem in general frame. Thus, let I ⊂ R be a bounded or unbounded interval, f : I → R, xn,k ∈ I, k ∈ {1, ..., n}, xn,1 < ... < xn,n , and consider the Lagrange interpolation polynomial of degree ≤ n − 1 attached to f and to the nodes (xn,k )k , Ln (f )(x) =

n X

ln,k (x)f (xn,k ),

k=1

with (x − xn,1 )...(x − xn,k−1 )(x − xn,k+1 )...(x − xn,n ) . (xn,k − xn,1 )...(xn,k − xn,k−1 )(xn,k − xn,k+1 )...(xn,k − xn,n ) Pn It is well known that k=1 ln,k (x) = 1, for all x ∈ R, which allows us to write Pn l (x)f (xn,k ) Pnn,k Ln (f )(x) = k=1 , for all x ∈ I. k=1 ln,k (x) ln,k (x) =

Therefore, its corresponding max-product interpolation operator will be given

2

COROIANU, GAL: LAGRANGE INTERPOLATION OPERATOR

by

n _ ) L(M n (f )(x)

=

ln,k (x)f (xn,k )

k=1 n _

, x ∈ I. ln,k (x)

k=1

The plan of the paper goes as follows : in Section 2 we present some auxiliary results, while in Section 3 we prove the main approximation result for the Lagrange interpolation based on Chebyshev knots of second kind in [−1, 1] plus the endpoints, essentially improving the order of approximation given by the corresponding Lagrange interpolation polynomials.

2

Auxiliary Results

Let us define the space CB+ (I) = {f : I → R+ ; f is continuous and bounded on I}. (M )

Remark. Firstly, it is clear that Ln (f )(x) Pn is a well-defined function for all x ∈ R and it is continuous on R. Indeed, by k=1 ln,k (x) = 1, for all x ∈ R, for any xWthere exists an index k ∈ {1, ..., n} such that ln,k (x) > 0 (which implies n that k=1 ln,k (x) > 0), because contrariwise would follow Pn that ln,k (x) ≤ 0 for all k and therefore we would obtain the contradiction k=1 ln,k (x) ≤ 0. Also, by the property ln,k (xn,j ) = 1 if k = j and ln,k (xn,j ) = 0 if k 6= j, (M ) we immediately obtain that Ln (f )(xn,j ) = f (xn,j ), for all j ∈ {1, ..., n}. In (M ) addition, Ln (e0 )(x) = 1, where e0 (x) = 1, for all x ∈ R. (M ) In what follows we will see that for f ∈ CB+ [a, b], the Ln (f ) operator (M ) fulfils similar properties with those of the Bn (f ) operator in [1], [2]. Lemma 2.1. Let I ⊂ R be a bounded or unbounded interval. (M ) (i) Then Ln : CB+ (I) → CB+ (I), for all n ∈ N : (M ) (M ) (ii) If f, g ∈ CB+ (I) satisfy f ≤ g then Ln (f ) ≤ Ln (g) for all n ∈ N ; (M ) (M ) (M ) (iii) Ln (f + g) ≤ Ln (f ) + Ln (g) for all f, g ∈ CB+ (I) ; (iv) For all f, g ∈ CB+ (I), n ∈ N and x ∈ I we have ) (M ) (M ) |L(M n (f )(x) − Ln (g)(x)| ≤ Ln (|f − g|)(x);

(M )

(M )

(M )

(v) Ln is positive homogenous, that is Ln (λf ) = λLn (f ) for all λ ≥ 0 and f ∈ CB+ (I). (M ) Proof. (i) The continuity of Ln (f )(x) on I follows from the previous (M ) Remark. Also, by the formula of definition for Ln (f )(x), if f is bounded by the (M ) constant C > 0 on I, then it easily follows that Ln is bounded on I by the same (M ) constant. It remains to prove the positivity of Ln (f ). So let f : I → R+ and fix x ∈ I. Reasoning exactly as in the above Remark, there exists k ∈ {1, ..., n} 3

213

214

COROIANU, GAL: LAGRANGE INTERPOLATION OPERATOR

such that ln,k (x) > 0. Therefore, denoting In+ (x) = {k ∈ {1, ..., n}; ln,k (x) > 0}, clearly In+ (x) is nonempty and for f ∈ CB+ (I) we get that W + ln,k (x)f (xn,k ) k∈In (M ) W (x) Ln (f )(x) = ≥ 0. (1) + k∈In (x) ln,k (x) (ii) Let f, g ∈ CB+ (I) be with f ≤ g and fix x ∈ I. Since In+ (x) is indepen(M ) (M ) dent of f and g, by (1) we immediately obtain Ln (f )(x) ≤ Ln (g)(x). W (iii) By (1) and by the sublinearity of , it is immediate. (iv) Let f, g ∈ CB+ (I). We have f = f − g + g ≤ |f − g| + g, which by (M ) (M ) (M ) (i) − (iii) successively implies Ln (f )(x) ≤ Ln (|f − g|)(x) + Ln (g)(x), that (M ) (M ) (M ) is Ln (f )(x) − Ln (g)(x) ≤ Ln (|f − g|)(x). Writing now g = g − f + f ≤ |f − g| + f and applying the above reasonings, (M ) (M ) (M ) it follows Ln (g)(x) − Ln (f )(x) ≤ Ln (|f − g|)(x), which combined with (M ) (M ) (M ) the above inequality gives |Ln (f )(x) − Ln (g)(x)| ≤ Ln (|f − g|)(x). (v) By (1) it is immediate. ¤ (M ) Remark. By (1) it is easy to see that instead of (ii), Ln satisfies the stronger condition Ln (f ∨ g)(x) = Ln (f )(x) ∨ Ln (g)(x), f, g ∈ CB+ (I). Corollary 2.2. For all f ∈ CB+ (I), n ∈ N and x ∈ I we have ¸ · 1 (M ) (M ) L (ϕx )(x) + 1 ω1 (f ; δ)I , |f (x) − Ln (f )(x)| ≤ δ n where δ > 0, ϕx (t) = |t − x| for all t ∈ I, x ∈ I and ω1 (f ; δ)I = max{|f (x) − f (y)|; x, y ∈ I, |x − y| ≤ δ}. Proof. Indeed, denoting e0 (x) = 1, from the identity ) (M ) (M ) (M ) L(M n (f )(x) − f (x) = [Ln (f )(x) − f (x) · Ln (e0 )(x)] + f (x)[Ln (e0 )(x) − 1],

by Lemma 2.1 it easily follows ) |f (x) − L(M n (f )(x)| ≤ (M ) (M ) ) |L(M n (f (x))(x) − Ln (f (t))(x)| + |f (x)| · |Ln (e0 )(x) − 1| ≤ ) (M ) L(M n (|f (t) − f (x)|)(x) + |f (x)| · |Ln (e0 )(x) − 1|.

Now, since for all t, x ∈ I we have ·

¸ 1 |f (t) − f (x)| ≤ ω1 (f ; |t − x|)I ≤ |t − x| + 1 ω1 (f ; δ)I , δ (M )

replacing above and taking into account that Ln (e0 ) = 1, for all x ∈ I, we immediately obtain the estimate in the statement. ¤ 4

COROIANU, GAL: LAGRANGE INTERPOLATION OPERATOR

215

(M )

Remarks. 1) Therefore, to get an approximation property for Ln enough to obtain a good estimate for n _

En (x) :=

) L(M n (ϕx )(x)

=

ln,k (x) |xn,k − x|

k=1 n _

W

+ k∈In (x) ln,k (x)|xn,k

W

= ln,k (x)

, it is

− x|

+ k∈In (x) ln,k (x)

.

k=1

2) The results in Lemma 2.1 and Corollary 2.2 remain valid if we replace the space CB+ (I) by the space C+ (I) = {f : I → R+ ; f is continuous on I}.

3

Main Results (M )

In this section we will study the properties of Ln (f )(x), ³ n ≥´2, in the case of Lagrange interpolation based on the knots xn,k = cos n−k n−1 π , k = 1, ..., n. It is known that xn,k are the roots of ωn (x) = sin(n − 1)tsint, x = cos t (which represents in fact the Chebyshev polynomial of second kind of degree n − 2 multiplied by 1 − x2 ) and that in this case for the fundamental Lagrange polynomials we can write (see [9], p. 377) ln,k (x) =

(−1)k−1 ωn (x) , n ≥ 2, (1 + δk,1 + δk,n )(n − 1)(x − xn,k )

k = 1, ..., n,

(2)

where ωn (x) = Πnk=1 (x − xn,k ) and δi,j denotes the Kronecker’s symbol, that is δi,j = 1 if i = j and δi,j = 0 if i 6= j. Wn For our purpose it will be useful to exactly calculate k=1 ln,k (x) for x ∈ [−1, 1]. In this sense, we need the following. Lemma 3.1. Let n ∈ N, n ≥ 2. (i) Let j ∈ {1, ..., n − 1} be fixed. For all k ∈ {1, ..., n} and x ∈ (xn,j , xn,j+1 ) we have : sign[ln,k (x)] = (−1)n+k−j−1 if k ≤ j, and sign[ln,k (x)] = (−1)n+k−j if k > j. (ii) For all j ∈ {1, ..., n − 1} and x ∈ (xn,j , xn,j+1 ) we have sign[ln,j (x)] = sign[ln,j+1 (x)] = (−1)n−1 . (iii) For all x ∈ R it follows : 3x−(2x +xn,2 ) n (x) If k = 1 then ln,k (x) − ln,k+1 (x) = ωn−1 · 2(x−xn,1n,1 )(x−xn,2 ) ; If 1 < k ≤ n − 2 then ln,k (x) − ln,k+1 (x) = ;

ωn (x)(−1)k−1 n−1 n−2

·

2x−(xn,k +xn,k+1 ) (x−xn,k )(x−xn,k+1 )

3x−(x

+2x

)

n,n−1 n,n If k = n − 1 then ln,k (x) − ln,k+1 (x) = ωn (x)(−1) · (x−xn,n−1 n−1 )(x−xn,n ) . (iv) Suppose that n ∈ iN, n ≥ 3 is an odd number. We have h: i h 2xn,1 +xn,2 2xn,1 +xn,2 then l (x) ≥ l (x) ≥ 0 and if x ∈ , x if x ∈ xn,1 , n,1 n,2 n,2 3 3 then 0 ≤ ln,1 (x) ≤ ln,2 (x) ; h i

x

+x

for any 1 < k ≤ n − 2, if x ∈ xn,k , n,k 2 n,k+1 then ln,k (x) ≥ ln,k+1 (x) ≥ 0 h i x +x and if x ∈ n,k 2 n,k+1 , xn,k+1 then 0 ≤ ln,k (x) ≤ ln,k+1 (x) ; 5

216

COROIANU, GAL: LAGRANGE INTERPOLATION OPERATOR

h i x +2x for k = n − 1, if x ∈ xn,n−1 , n,n−13 n,n then ln,n−1 (x) ≥ ln,n (x) ≥ 0 and h i x +2x if x ∈ n,n−13 n,n , xn,n then 0 ≤ ln,n−1 (x) ≤ ln,n (x). Proof. (i) By ωn (x) = [(x−xn,1 )...(x−xn,j )]·[(x−xn,j+1 )...(x−xn,n )], we get that sign[ω(x)] = (−1)n−j , for all x ∈ (xn,j , xn,j+1 ). Suppose first that k ≤ j. Since sign[(x − xk )] = +1, by (2) we easily obtain that sign[ln,k (x)] = (−1)k−1 · (−1)n−j = (−1)n+k−j−1 . Suppose now that k > j. Since sign[(x − xk )] = −1, k−1 ·(−1)n−j by (2) we easily obtain that sign[ln,k (x)] = (−1) −1 = (−1)n+k−j . (ii) Choosing k = j and k = j + 1 in (i) we immediately get the desired conclusion. (iii) By using (2) we immediately obtain ln,k (x) − ln,k+1 (x) =

ωn (x)(−1)k−1 · n−1

½

x[1 + δk+1,1 + δk+1,n + 1 + δk,1 + δk,n ] (1 + δk,1 + δk,n )(1 + δk+1,1 + δk+1,n )(x − xn,k )(x − xn,k+1 ) ¾ xn,k (1 + δk,1 + δk,n ) + xn,k+1 (1 + δk+1,1 + δk+1,n ) − . (1 + δk,1 + δk,n )(1 + δk+1,1 + δk+1,n )(x − xn,k )(x − xn,k+1 ) Now, since 1 + δk+1,1 + δk+1,n + 1 + δk,1 + δk,n = 3 if k = 1, k = n − 1 and 1 + δk+1,1 + δk+1,n + 1 + δk,1 + δk,n = 2 if 1 < k ≤ n − 2, by simple calculation we get the desired conclusions. (iv) Since n is odd number, by (ii) we get ln,k (x) ≥ 0 and ln,k+1 (x) ≥ 0 on [xn,k , xn,k+1 ], for all k ∈ {1, ..., n − 1}. Let k = 1. Because sign[ωn (x)] = (−1)n−1 = +1 and sign[x − xn,2 ] = −1 for all x ∈ (xn,1 , xn,2 ), by (iii) it follows that the sign of ln,1 (x) − ln,2 (x) on [xn,1 , xn,2 ] is opposite to the sign of the expression 3x − (2xn,1 + xn,2 )/3, which implies the desired conclusion. Now let 1 < k ≤ n − 2. Since sign[ωn (x)] = (−1)n−k for x ∈ (xn,k , xn,k+1 ), by (iii) we easily get that the sign of ln,k (x) − ln,k+1 (x) on [xn,k , xn,k+1 ] is opposite to the sign of the expression x − (xn,k + xn,k+1 )/2, which implies the desired conclusion. Finally, let k = n − 1. Since sign[ωn (x)] = (−1)n−(n−1) = −1 for x ∈ (xn,n−1 , xn,n ), by (iii) we easily get that the sign of ln,n−1 (x) − ln,n (x) on [xn,n−1 , xn,n ] is opposite to the sign of the expression x − (xn,n−1 + 2xn,n )/3, which implies the desired conclusion. ¤ Lemma 3.2. Let n ∈ N, n ≥ 3 be odd number. Wn (i) If x ∈ [xn,1 , (2xn,1 +Wxn,2 )/3] then k=1 ln,k (x) = ln,1 (x) and if x ∈ n [(2xn,1 + xn,2 )/3, xn,2 ] then k=1 ln,k (x) = ln,2 (x) ; Wn (ii) For 1 < j ≤ n − 2, if x ∈ [xn,j , (xn,j + xn,j+1 Wn )/2] then k=1 ln,k (x) = ln,j (x) and if x ∈ [(xn,j + xn,j+1 )/2, xn,j+1 ] thenW k=1 ln,k (x) = ln,j+1 (x) ; n (iii) If x ∈ [xn,n−1 , (xn,n−1 + 2xW n,n )/3] then k=1 ln,k (x) = ln,n−1 (x) and if n x ∈ [(xn,n−1 + 2xn,n )/3, xn,n ] then k=1 ln,k (x) = ln,n (x). Proof. Let j ∈ {1, ..., n − 1} be fixed and suppose that x ∈ [xn,j , xn,j+1 ]. Taking into account Lemma 3.1, (iv) and denoting An,j := {k ∈ {1, ..., n}} \

6

COROIANU, GAL: LAGRANGE INTERPOLATION OPERATOR

217

{j, j + 1}, it suffices to prove that ln,k (x) ≤ max{ln,j (x), ln,j+1 (x)}, for all k ∈ An,j .

(3)

By Lemma 3.1, (ii) we get ln,j (x), ln,j+1 (x) ≥ 0 for all x ∈ [xn,j , xn,j+1 ]. It follows that (3) is immediate for those k ∈ An,j which satisfies ln,k (x) ≤ 0 for all x ∈ [xn,j , xn,j+1 ]. But according to Lemma 3.1, (i), we distinguish two cases : 1) k ≤ j ; 2) k > j. Case 1). Since sign[ln,k (x)] = (−1)k−j on (xn,j , xn,j+1 ), it follows that we can eliminate those k ≤ j in An,j for which k − j is odd number. Case 2). Since sign[ln,k (x)] = (−1)k−j+1 on (xn,j , xn,j+1 ), it follows that we can eliminate those k > j in An,j for which k − j − 1 is odd number. Therefore, in what follows let us suppose that k ∈ An,j excepting those k in the above Cases 1) and 2). Firstly, taking into account the above Case 1), let us suppose that k ∈ An,j , k ≤ j and j − k is an even number. By using (2) we can write ln,j (x) − ln,k (x) =

ωn (x) n−1

½

¾ (−1)j−1 (−1)k−1 − (1 + δj,1 + δj,n )(x − xn,j ) (1 + δk,1 + δk,n )(x − xn,k ) ½ ¾ 1 1 ωn (x)(−1)j−1 − . = n−1 (1 + δj,1 + δj,n )(x − xn,j ) (1 + δk,1 + δk,n )(x − xn,k ) ·

Note that here we have that j 6= 1 and j 6= n. Indeed, first j ≤ n − 1 < n, then if j = 1 then k = 1 ∈ An,1 , which is impossible. It immediately follows δj,1 = δj,n = δk,1 = 0. Suppose first that k 6= 1. It follows δk,1 = 0 and therefore we obtain ½ ¾ ωn (x)(−1)j−1 1 1 ln,j (x) − ln,k (x) = − n−1 x − xn,j x − xn,k =

ωn (x)(−1)j−1 (xn,j − xn,k ). (n − 1)(x − xn,k )(x − xn,j )

Since for x ∈ (xn,j , xn,j+1 ) we have sign[ωn (x)] = (−1)n−j , we immediately obtain that on (xn,j , xn,j+1 ), we get sign[ln,j (x) − ln,k (x)] = (−1)n−1 = +1, which is the desired conclusion. Now if k = 1 then δk,1 = 1 and we obtain ½ ¾ ωn (x)(−1)j−1 1 1 ln,j (x) − ln,1 (x) = − . n−1 x − xn,j 2(x − xn,1 ) 1 Because x − xn,1 > x − xn,j > 0 for x ∈ (xn,j , xn,j+1 ), it follows 2(x−x < n,1 ) 1 1 n−1 = +1 (x−xn,1 ) < (x−xn,j ) and we get again that sign[ln,j (x)−ln,k (x)] = (−1) on (xn,j , xn,j+1 ).

7

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COROIANU, GAL: LAGRANGE INTERPOLATION OPERATOR

Taking into account the above Case 2), let us suppose that k ∈ An,j , k > j and k − j − 1 is an even number. By using (2) we can write ln,j+1 (x) − ln,k (x) = ½ ·

1 (1 + δj+1,1 + δj+1,n )(x − xn,j+1 )

−

ωn (x)(−1)j n−1 1

¾

(1 + δk,1 + δk,n )(x − xn,k )

.

Note here that we cannot have j = n − 1, because contrariwise, by k > j we get k = n = j + 1, contradicting that k ∈ An,j . Therefore, we can suppose that 1 ≤ j ≤ n − 2. We get δj+1,1 = δj+1,n = δk,1 = 0. We have two subcases : a) j < k < n ; b) j < k = n. Subcase a). It follows δk,n = 0 and we can write ½ ¾ ωn (x)(−1)j 1 1 ln,j+1 (x) − ln,k (x) = − n−1 x − xn,j+1 x − xn,k =

ωn (x)(−1)j (xn,j+1 − xn,k ). (n − 1)(x − xn,j+1 )(x − xn,k )

Since k ∈ An,j we also have k 6= j + 1 and by the above equality we easily get that on (xn,j , xn,j+1 ) we have sign[ln,j+1 (x) − ln,k (x)] = (−1)n−j (−1)j (−1)3 = (−1)n+3 = +1. Subcase b). It follows δk,n = 1 and since k 6= j + 1 we obtain j + 1 < n and therefore δj+1,n = δj+1,1 = 0 and by using (2) it follows · ¸ ωn (x)(−1)j 1 1 ln,j+1 (x) − ln,n (x) = − n−1 x − xn,j+1 2(x − xn,n ) =

ωn (x)(−1)j [(x − xn,n ) + (xn,j+1 − xn,n )]. 2(n − 1)(x − xn,j+1 )(x − xn,n )

Since x ∈ (xn,j , xn,j+1 ) and k = n > j + 1, we immediately get that sign[ln,j+1 (x) − ln,k (x)] = (−1)n−j (−1)j (−1)3 = (−1)n+3 = +1. This proves the lemma. ¤ Remarks. 1) Let n ∈ N, n ≥ 4 be an even number and let j ∈ {2, ..., n − 2}. For any x ∈ [xn,j , xn,j+1 ] we have ln,j (x) ≤ 0, ln,j+1 (x) ≤ 0 and ln,j−1 (x) ≥ 0, ln,j+2 (x) ≥ 0. Also, for all x ∈ [xn,1 , xn,2 ] we have ln,1 (x) ≤ 0, ln,2 (x) ≤ 0, ln,3 (x) ≥ 0, and for all x ∈ [xn,n−1 , xn,n ] we have ln,n−1 (x) ≤ 0, ln,n (x) ≤ 0, ln,n−2 (x) ≥ 0. Indeed, firstly let j ∈ {1, ..., n − 1}. Taking k = j and k = j + 1 in Lemma 3.1, (i), we get that ln,j (x) ≤ 0, ln,j+1 (x) ≤ 0, for all x ∈ [xn,j , xn,j+1 ]. Also, for j = 1 and k = 3, from the same Lemma 3.1, (i) we obtain that ln,3 (x) ≥ 0 for all x ∈ [xn,1 , xn,2 ]. Finally, for j = n − 1 and k = n − 2, from the same Lemma 3.1, (i), we get that ln,n−2 (x) ≥ 0 for all x ∈ [xn,n−1 , xn,n ]. 8

COROIANU, GAL: LAGRANGE INTERPOLATION OPERATOR

Therefore, it follows that in the case when n is even number, Lemma 3.2 dose not apply. Because of this reason, the cases when n is an odd number and n is an even number will be treated separately. (M ) 2) Since Ln (f )(xn,j ) − f (xn,j ) = 0 for all n ∈ N and j = 1, ..., n, we note that in the next notations, proofs and statements of the all approximation results, in fact we always may suppose that x ∈ [−1, 1] and x 6= xn,j , for all j = 1, ..., n. Theorem 3.3. If f : [−1, 1] → R+ is continuous and n ∈N, n ≥ 3 is an odd number then we have the estimate µ ¶ 1 |Ln(M ) (f )(x) − f (x)| ≤ 4ω1 f ; , for all x ∈ [−1, 1], n−1 where ω1 (f ; δ) = sup{|f (x) − f (y)|; x, y ∈ [−1, 1], |x − y| ≤ δ}. Proof. First, let us observe that for all x ∈ (−1, 1) we have _

ln,k (x)|xn,k − x| =

+ (x) k∈In

|ωn (x)| n−1

_ + (x) k∈In

1 . (1 + δk,1 + δk,n )

Since n ≥ 3, it is easy to check that for each x ∈ (−1, 1) there exists k ∈ {2, 3, ..., n − 1} such that k ∈ In+ (x). This relation implies _

ln,k (x)|xn,k − x| =

+ (x) k∈In

|ωn (x)| n−1

for all x ∈ (−1, 1). In order to prove the conclusion of the theorem we distinguish the following cases: 1) x ∈ (xn,1 , xn,2 ); 2) x ∈ (xn,j , xn,j+1 ) with j ∈ {2, 3, ..., n− 2} and 3) x ∈ (xn,n−1 , xn,n ). Case 1) If x ∈ (xn,1 , (2xn,1 + xn,2 )/3], then from Lemma 3.2, (i), we get _ ln,k (x) = ln,1 (x) + (x) k∈In

and it follows that

µ ¶ |ωn (x)| 2xn,1 + xn,2 = 2(x − xn,1 ) ≤ 2 − xn,1 (n − 1)ln,1 (x) 3 2 2 n−2 = · (xn,2 − xn,1 ) = · (cos( π) + 1) 3 3 n−1 π n−2 4 sin2 ( 2(n−1) 4 cos2 ( 2(n−1) π) ) π2 = ≤ . = 3 3 3(n − 1)2

En (x) =

If x ∈ ((2xn,1 + xn,2 )/3, xn,2 ), then from Lemma 3.2, (i), we get _ ln,k (x) = ln,2 (x) + k∈In (x)

9

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and it follows that |ωn (x)| 2xn,1 + xn,2 = xn,2 − x ≤ xn,2 − (n − 1)ln,2 (x) 3 2 2 π = · (xn,2 − xn,1 ) ≤ . 3 3(n − 1)2

En (x) =

Case 2) If x ∈ (xn,j , (xn,j + xn,j+1 )/2], then from Lemma 3.2, (ii), we get _ ln,k (x) = ln,j (x) + k∈In (x)

and it follows that xn,j + xn,j+1 |ωn (x)| = x − xn,j ≤ − xn,j (n − 1)ln,j (x) 2 1 1 n−j−1 n−j = · (xn,j+1 − xn,j ) = · [cos( π) − cos( π)] 2 2 n−1 n−1 π (2n − 2j − 1) π = sin( ) sin( π) ≤ . 2(n − 1) 2(n − 1) 2(n − 1)

En (x) =

If x ∈ ((xn,j + xn,j+1 )/2, xn,j+1 ), then from Lemma 3.2, (ii), we get _ ln,k (x) = ln,j+1 (x) + (x) k∈In

and it follows that |ωn (x)| xn,j + xn,j+1 = xn,j+1 − x ≤ xn,j+1 − (n − 1)ln,j+1 (x) 2 1 π = · (xn,j+1 − xn,j ) ≤ . 2 2(n − 1)

En (x) =

Case 3) If x ∈ (xn,n−1 , (xn,n−1 + 2xn,n )/3], then from Lemma 3.2, (iii), we get

_

ln,k (x) = ln,n−1 (x)

+ (x) k∈In

and it follows that |ωn (x)| xn,n−1 + 2xn,n = x − xn,n−1 ≤ − xn,n−1 (n − 1)ln,n−1 (x) 3 2 2 π 4 π = · (xn,n − xn,n−1 ) = · (1 − cos( )) = · sin2 ( ) 3 3 n−1 3 2(n − 1) π2 ≤ . 3(n − 1)2

En (x) =

10

COROIANU, GAL: LAGRANGE INTERPOLATION OPERATOR

If x ∈ ((xn,n−1 + 2xn,n )/3, xn,n ), then from Lemma 3.2, (iii), we get _ ln,k (x) = ln,n (x) + k∈In (x)

and it follows that µ ¶ |ωn (x)| xn,n−1 + 2xn,n = 2(xn,n − x) ≤ 2 xn,n − (n − 1)ln,n (x) 3 2 2 π = · (xn,n − xn,n−1 ) ≤ . 3 3(n − 1)2

En (x) =

immediately

π2 3(n−1)2

π π π π π 3 · 2(n−1) and 2(n−1) ≤ 3 · 2(n−1) , it 2 π follows that En (x) ≤ 6(n−1) for all x ∈ (−1, 1) and n ≥ 3. 2 2 π δn = 6(n−1) in Corollary 2.2, since [ π6 ] = 1, from the property

Since for n ≥ 3 we have

≤

Now, taking ω1 (f, λδ) ≤ ([λ] + 1) ω1 (f, δ), we immediately obtain the desired conclusion. ¤ Remark In what follows we prove that the order of uniform approximation in Theorem 3.3 cannot be improved. Indeed, for each odd number n ∈ N, n ≥ 3 x +x let us denote n0 = n−1 and let us denote yn = n,n0 2 n,n0 +1 . By Lemma 3.2, 2 (ii), it follows that _ ln,k (yn ) = ln,n0 (yn ) = ln,n0 +1 (yn ) + (yn ) k∈In

and we get |ωn (yn )| xn,n0 + xn,n0 +1 = yn − xn,n0 = − xn,n0 (n − 1)ln,n0 (yn ) 2 xn,n0 +1 − xn,n0 1 n − n0 − 1 n − n0 = = · [cos( π) − cos( π)] 2 2 n−1 n−1 2n − 2n0 − 1 π nπ π ) sin( π) = sin( ) sin( ). = sin( 2(n − 1) 2(n − 1) 2(n − 1) 2(n − 1)

En (yn ) =

By the well known inequality sin x ≥ π2 x for all x ∈ [0, π/2] we get En (yn ) ≥ 1 nπ nπ n−1 sin( 2(n−1) ). Since lim sin( 2(n−1) ) = 1, it follows that for n sufficiently n→∞

1 large we have En (yn ) ≥ n−1 and we obtain the desired conclusion. In what follows, for the case when n is an even number, we present a similar result. Theorem 3.4. If f : [−1, 1] → R+ is continuous and n ∈N, n ≥ 4 is an even number, then we have the estimate ¶ µ 1 (M ) , for all x ∈ [−1, 1]. |Ln (f )(x) − f (x)| ≤ 28ω1 f ; n−1

11

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COROIANU, GAL: LAGRANGE INTERPOLATION OPERATOR

Proof. First, let us observe that in the same manner as in the proof of Theorem 3.3 we obtain _

ln,k (x)|xn,k − x| =

+ k∈In (x)

|ωn (x)| n−1

for all x ∈ (−1, 1). Then, let us observe that from the definition of En it follows that if x ∈ (−1, 1) and k ∈ In+ (x), then W + |ωn (x)| k∈In (x) ln,k (x)|xn,k − x| En (x) ≤ = . ln,k (x) (n − 1)ln,k (x) We distinguish the following four cases: 1) x ∈ (xn,1 , xn,2 ); 2) x ∈ (xn,2 , xn,3 ); 3) x ∈ (xn,j , xn,j+1 ) with j ∈ {3, 4, ..., n − 2} and 4) x ∈ (xn,n−1 , xn,n ). Case 1). From the Remark after the proof of Lemma 3.2 it follows that for k = 3 we have k ∈ In+ (x) which implies |ωn (x)| = xn,3 − x ≤ xn,3 − xn,1 (n − 1)ln,3 (x) n−3 π 2π 2 = cos( π) − cos π = 2 sin2 ( )≤ . n−1 n−1 (n − 1)2

En (x) ≤

Case 2). From the Remark after the proof of Lemma 3.2 it follows that for k = 1 we have k ∈ In+ (x) which implies En (x) ≤

|ωn (x)| 4π 2 = 2(x − xn,1 ) ≤ 2(xn,3 − xn,1 ) ≤ . (n − 1)ln,1 (x) (n − 1)2

Case 3). From the Remark after the proof of Lemma 3.2 it follows that for k = j − 1 we have k ∈ In+ (x) which implies |ωn (x)| = x − xn,j−1 ≤ xn,j+1 − xn,j−1 (n − 1)ln,j−1 (x) n−j+1 π n−j 2π n−j−1 = cos( π) − cos( π) = 2 sin( ) sin( π) ≤ . n−1 n−1 n−1 n−1 n−1

En (x) ≤

Case 4). From the Remark after the proof of Lemma 3.2 it follows that for k = n − 2 we have k ∈ In+ (x) which implies |ωn (x)| = x − xn,n−2 ≤ xn,n − xn,n−2 (n − 1)ln,n−2 (x) n − (n − 2) π 2π 2 = cos(0) − cos( π) = 2 sin2 ( )≤ . n−1 n−1 (n − 1)2

En (x) ≤

2

4π 2π 2π 2 Since for n ≥ 4 we have (n−1) 2 ≤ 3 · n−1 and since [4π /3] = 13 we easily get the estimate in the statement of the theorem. ¤

12

COROIANU, GAL: LAGRANGE INTERPOLATION OPERATOR

Remark. Note that in the same manner as in the remark after the proof of Theorem 3.3 we get that the order of uniform approximation in Theorem 3.4 cannot be improved. Corollary 3.5. Let f : [−1, 1] → R+ be continuous. Then we have the estimate µ ¶ 1 ) |L(M (f )(x) − f (x)| ≤ 28ω f, , n ∈ N, n ≥ 3, x ∈ [−1, 1]. 1 n n−1 Proof. The proof is immediate by Theorems 3.3 and 3.4. ¤ Remark. Starting from a standard technique in interpolation (see [10]), in the proof of Theorem 4, p. 410 in [8], for the Lagrange polynomials Ln (f )(x), based on the Chebyshev nodes of second kind plus the endpoints ±1, it is proved the uniform estimate µ ¶ 1 kLn (f ) − f k ≤ Cω1 f ; ln(n), n ∈ N, n where k · k denotes the uniform norm on C[−1, 1] and ln(n) denotes the natural logarithm of n. If f is a Lipschitz function and positive ´on [−1, 1], it follows that the approx³ ln(n) , while from Corollary 3.5 it follows imation order given by Ln (f ) is O n ¡ ¢ (M ) that the approximation order given by Ln (f ) is O n1 , which is an essential improvement.

References [1] Bede B. and Gal, S.G., Approximation by nonlinear Bernstein and FavardSz´asz-Mirakjan operators of max-product kind, Journal of Concrete and Applicable Mathematics, 8(2010), No. 2, 193–207. [2] Bede, B., Coroianu L. and Gal, S.G., Approximation and shape preserving properties of the Bernstein operator of max-product kind, Intern. J. Math. and Math. Sci., volume 2009, Article ID 590589, 26 pages, doi:10.1155/2009/590589. [3] Bede, B., Coroianu, L. and Gal, S.G., Approximation by truncated nonlinear Favard-Sz´asz-Mirakjan operators of max-product kind, Demonstratio Mathematica, (accepted for publication). [4] Bede, B., Coroianu L. and Gal, S.G., Approximation and shape preserving properties of the nonlinear Baskakov operator of max-product kind, Studia Univ. ”Babes-Bolyai”, ser. math., (accepted for publication). [5] Bede, B., Coroianu, L. and Gal, S.G., Approximation and shape preserving properties of the nonlinear Bleimann-Butzer-Hahn operators of maxproduct kind, Comment. Math. Univ. Carol., (accepted for publication). 13

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[6] Coroianu L. and Gal, S.G., Approximation by nonlinear Hermite-Fej´er interpolation operators of max-product kind on Chebyshev nodes, submitted. [7] Gal, S.G., Shape-Preserving Approximation by Real and Complex Polynomials, Birkh¨auser, Boston-Basel-Berlin, 2008. [8] Gal, S.G. and Szabados, J., On the preservation of global smoothness by some interpolation operators, Studia Sci. Math. Hung., 35(1999), 393–414. [9] Mastroianni G. and Szabados, J., Jackson order of approximation by Lagrange interpolation, Suppl. Rend. Circ. Mat. Palermo, ser. II, No. 33(1993), 375–386. [10] Szabados J. and V´ertesi, P., Interpolation of Functions, World Scientific, Singapore, New Jersey, London, Hong Kong, 1990.

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JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.2, 225-230, 2011, COPYRIGHT 2011 EUDOXUS PRESS, 225 LLC

FORWARD CONTINUITY ¨ ˙ C HUSEY IN ¸ AKALLI

Abstract. A real function f is continuous if and only if (f (xn )) is a convergent sequence whenever (xn ) is convergent and a subset E of R is compact if any sequence x = (xn ) of points in E has a convergent subsequence whose limit is in E where R is the set of real numbers. These well known results suggest us to introduce a concept of forward continuity in the sense that a function f is forward continuous if limn→∞ ∆f (xn ) = 0 whenever limn→∞ ∆xn = 0 and a concept of forward compactness in the sense that a subset E of R is forward compact if any sequence x = (xn ) of points in E has a subsequence z = (zk ) = (xnk ) of the sequence x such that limk→∞ ∆zk = 0 where ∆zk = zk+1 −zk . We investigate forward continuity and forward compactness, and prove related theorems.

1. Introduction A subset E of R is compact if any open covering of E has a finite subcovering where R is the set of real numbers. This is equivalent to the statement that any sequence x = (xn ) of points in E has a convergent subsequence whose limit is in E. A real function f is continuous if and only if (f (xn )) is a convergent sequence whenever (xn ) is convergent. Regardless of limit, this is equivalent to the statement that (f (xn )) is a Cauchy sequence whenever (xn ) is. Using the idea of continuity of a real function and the idea of compactness in terms of sequences, we introduce a concept of forward continuity in the sense that a function f is forward continuous if it transforms forward convergent to 0 sequences to forward convergent to 0 sequences, i.e. (f (xn )) is forward convergent to 0 whenever (xn ) is forward convergent to 0, and a concept of forward compactness in the sense that a subset E of R is forward compact if any sequence x = (xn ) of points in E has a subsequence z = (zk ) = (xnk ) of the sequence x such that limk→∞ ∆zk = 0 where ∆zk = zk+1 − zk . Before we begin, some definitions and notation will be given in the following. Throughout this paper, N will denote the set of all positive integers. We will use boldface letters x, y, z, ... for sequences x = (xn ), y = (yn ), z = (zn ), ... of terms in R. c and ∆ will denote the set of all convergent sequences and the set of all forward convergent to 0 sequences of points in R where a sequence x = (xn ) is called forward convergent to 0 if limn→∞ ∆xn = 0. Following the idea given in a 1946 American Mathematical Monthly problem [4], a number of authors Posner [12], Iwinski [10], Srinivasan [16], Antoni [1], Antoni and Salat [2], Spigel and Krupnik [15] have studied A-continuity defined by a regular ¨ urk [11], Sava¸s and Das [13], Borsik summability matrix A. Some authors, Ozt¨ Date: September 23, 2010. 2000 Mathematics Subject Classification. Primary: 40A05; Secondary: 26A05. Key words and phrases. Continuity, sequences, series, summability, compactness, 1

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and Salat [3]) have studied A-continuity for methods of almost convergence or for related methods. Fast [8] introduced the definition of statistical convergence. Recall that for a subset M of N the asymptotic density of M , denoted by δ(M ), is given by 1 δ(M ) = lim |{k ≤ n : k ∈ M }|, n→∞ n if this limit exists, where |{k ≤ n : k ∈ M }| denotes the cardinality of the set {k ≤ n : k ∈ M }. A sequence (xn ) is statistically convergent to ` if δ({n : |xn − `| > }) = 0, for every  > 0. In this case ` is called the statistical limit of x. Schoenberg [14] studied some basic properties of statistical convergence and also studied the statistical convergence as a summability method. Fridy [9] gave characterizations of statistical convergence. Connor and Grosse-Erdman [5] gave sequential definitions of continuity for real functions calling G-continuity instead of A-continuity and their results covers the earlier works related to A-continuity where a method of sequential convergence, or briefly a method, is a linear function G defined on a linear subspace of s, denoted by cG , into R. A sequence x = (xn ) is said to be G-convergent to ` if x ∈ cG and G(x) = `. In particular, lim denotes the limit function lim x = limn xn on the linear space c and st − lim denotes the statistical limit function st − lim x = st − limn xn on the linear space st(R). A function f is called G-continuous at a point u provided that whenever a sequence x = (xn ) of terms in the domain of f is G-convergent to u, then the sequence f (x) = (f (xn )) is G-convergent to f (u). A method G is called regular if every convergent sequence x = (xn ) is G-convergent with G(x) = lim x. A method is called subsequential if whenever x is G-convergent with G(x) = `, then there is a subsequence (xnk ) of x with limk xnk = `. Recently, Cakalli gave new sequential definitions of compactness and slowly oscillating compactness in [6] and [7]. The purpose of this note is to introduce a concept of forward continuity of a function and a concept of forward compactness of a subset of R and prove that any forward continuous function on a forward compact subset E of R is uniformly continuous. 2. forward continuity We say that a sequence x = (xn ) is forward convergent to a number ` if limk→∞ ∆xk = ` where ∆xk = xk+1 − xk . Now we give the definition of forward compactness of a subset of R. Definition 1. A subset E of R is called forward compact if whenever x = (xn ) is a sequence of points in E there is a forward convergent to 0 subsequence z = (zk ) = (xnk ) of x. Firstly, we note that any finite subset of R is forward compact, union of two forward compact subsets of R is forward compact and intersection of any forward compact subsets of R is forward compact. Furthermore any subset of a forward compact set is forward compact and any bounded subset of R is forward compact. Any compact subset of R is also forward compact, and the set N is not forward compact. We note that any slowly oscillating compact subset of R is forward compact (see [7] for the definition of slowly oscillating compactness).

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FORWARD CONTINUITY

3

We note that this definition of forward compactness can not be obtained by any G-sequential compactness, i.e. by any summability matrix A, even by the summability matrix A = (ank ) defined by ank = −1 if k = n and akn = 1 if k = n + 1 and ∞ X G(x) = lim Ax = lim akn xn = lim ∆xk (1) k→∞

n=1

k→∞

(see [6] for the definition of G-sequential compactness). Despite that G-sequential compact subsets of R should include the singleton set {0}, forward compact subsets of R do not have to include the singleton {0}. A real function f is continuous if and only if, for each point x0 in the domain, limn→∞ f (xn ) = f (x0 ) whenever limn→∞ xn = x0 . This is equivalent to the statement that (f (xn )) is a convergent sequence whenever (xn ) is. This is also equivalent to the statement that (f (xn )) is a Cauchy sequence whenever (xn ) is Cauchy. These well known results for continuity for real functions in terms of sequences might suggest us to give a new type continuity, namely, forward continuity: Definition 2. A function f is called forward continuous on E if the sequence (f (xn )) is forward convergent to 0 whenever x = (xn ) is a forward convergent to 0 sequence of terms in E. We note that this definition of continuity can not be obtained by any A-continuity, i.e. by any summability matrix A, even by the summability matrix A = (ank ) defined by (1) however for this special summability matrix A if A-continuity of f at the point 0 implies forward continuity of f , then f (0) = 0; and if forward continuity of f implies A-continuity of f at the point 0, then f (0) = 0. We also note that sum of two forward continuous functions is forward continuous and composite of two forward continuous functions is forward continuous but the product of two forward continuous functions need not be forward continuous as it can be seen by considering product of the forward continuous function f (x) = x with itself. We note that if f and g are forward continuous functions, then so are max{f, g} and min{f, g}. More generally, if (fn ) is a sequence of forward continuous functions, then so are supfn and inf fn . In connection with forward convergent to 0 sequences and convergent sequences the problem arises to investigate the following types of continuity of functions on R. (δ): (xn ) ∈ ∆ ⇒ (f (xn )) ∈ ∆ (δc): (xn ) ∈ ∆ ⇒ (f (xn )) ∈ c (c): (xn ) ∈ c ⇒ (f (xn )) ∈ c (d): (xn ) ∈ c ⇒ (f (xn )) ∈ ∆ We see that (δ) is forward continuity of f and (c) states the ordinary continuity of f . It is easy to see that (δc) implies (δ), and (δ) does not imply (δc); and (δ) implies (d), and (d) does not imply (δ); (δc) implies (c) and (c) does not imply (δc); and (c) is eqivalent to (d). We see that (c) can be replaced by statistical continuity, i.e. st−limn→∞ f (xn ) = f (`) whenever x = (xn ) is a statisctically convergent sequence with st−limn→∞ xn ) = `. More generally (c) can be replaced by G-sequential continuity of f for any regular subsequential method G (see Corollary to Theorem 5 on page 106 of [5]).

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¨ ˙ C HUSEY IN ¸ AKALLI

Now we give the implication (δ) implies (c), i.e. any forward continuous function is continuous in the ordinary sense. Theorem 1 If f is forward continuous on a subset E of R, then it is continuous on E in the ordinary sense. Proof. Let (xn ) be any convergent sequence with limk→∞ xk = x0 . Then (f (x1 ), f (x0 ), f (x2 ), f (x0 ), ..., f (xn ), f (x0 ), ...) is forward convergent to 0. It follows from this that the sequence (f (xn )) converges to f (x0 ). This completes the proof of the theorem. 2 The converse √ is not always true for the function f (x)√= x is an example since the sequence ( n) is forward convergent to 0 while (f ( n)) = (n) is not forward convergent to 0.  Corollary 2 If f is forward continuous, then it is G-continuous for any regular subsequential method G. Now we state the following straightforward result related to statistical continuity. Corollary 3 If f is forward continuous, then it is statistically continuous. Theorem 4 Forward continuous image of any forward compact subset of R is forward compact. Proof. Write yn = f (xn ) where xn ∈ E for each n ∈ N. Forward compactness of E implies that there is a subsequence z = (zk ) = (xnk ) of x with limk→∞ ∆zk = 0. Write (tk ) = f (z) = (f (zk )). (tk ) is a subsequence of the sequence f (x) with limk→∞ ∆tk = 0. This completes the proof of the theorem.  Corollary 5 Forward continuous image of any compact subset of R is compact. The proof of this theorem follows from the preceding theorem. Theorem 6 If a function f on a subset E of R is uniformly continuous, then it is forward continuous on E. Proof. To prove that (f (xn )) is forward convergent to 0 whenever (xn ) is, take any ε > 0 . Uniform continuity of f implies that there exists a δ > 0, depending on ε, such that |f (x) − f (y)| < ε whenever |x − y| < δ. For this δ > 0, there exists an N = N (δ) = N1 (ε) such that |∆xn | < δ whenever n > N . Hence |∆f (xn )| < ε if n > N . It follows from this that (f (xn )) is forward convergent to 0. This completes the proof of the theorem.  It is well known that any continuous function on a compact subset E of R is also uniformly continuous on E. It is also true for a regular subsequential method G that any forward continuous function on a G-sequentially compact subset E of R is also uniformly continuous on E (see [6]). Furthermore, for forward continuous functions, we have the following: Theorem 7 If a function is forward continuous on a forward compact subset E of R, then it is uniformly continuous on E. Proof. Suppose that f is not uniformly continuous on E so that there exist an 0 > 0 and sequences (xn ) and (yn ) of points in E such that |xn − yn | < 1/n and |f (xn ) − f (yn )| ≥ 0

229

FORWARD CONTINUITY

5

for all n ∈ N. Since E is forward compact, There is a subsequence of (xnk ) of (xn ) that is forward convergent to 0. Since E is forward compact, On the other hand there is a subsequence of (ynkj ) of (ynk ) that is forward convergent to 0. It is clear that the corresponding sequence (xnkj ) is also forward convergent to 0, since (ynkj ) is forward convergent to 0 and |xnkj − xnkj+1 | ≤ |xnkj − ynkj | + |ynkj − ynkj+1 | + |ynkj+1 − xnkj+1 |. Now define a sequence z = (zj ) by setting z1 = xnk1 , z2 = ynk1 , z3 = xnk2 , z4 = ynk2 , z5 = xnk3 , z6 = ynk3 , and so on. Thus the sequence z = (zj ) defined in this way is forward convergent to 0 while f (z) = (f (zj )) is not forward convergent to 0. Hence this establishes a contradiction so this completes the proof of the theorem.  It is a well known result that uniform limit of a sequence of continuous functions is continuous. This is also true in case forward continuity, i.e. uniform limit of a sequence of forward continuous functions is forward continuous. Theorem 8 If (fn ) is a sequence of forward continuous functions defined on a subset E of R and (fn ) is uniformly convergent to a function f , then f is forward continuous on E. Proof. Let ε > 0. Then there exists a positive integer N such that |fn (x)−f (x)| < 3ε for all x ∈ E whenever n ≥ N . There exists a positive integer N1 , depending on ε, and greater than N such that |fN (xn+1 ) − fN (xn )| < 3ε for n ≥ N1 . Now for n ≥ N1 we have |f (xn+1 )−f (xn )| ≤ |f (xn+1 )−fN (xn+1 )|+|fN (xn+1 )−fN (xn )|+|fN (xn )−f (xn )| ε ε ε ≤ + + = ε. 3 3 3 This completes the proof of the theorem.  A definition of double continuity of a real function can be given in a natural way and equivalence of double continuity and ordinary continuity can be seen easily. A question arises if double forward continuity is equivalent to forward continuity or not when double forward continuity is defined in a natural way. We note that the study in this paper can be extended to first countable Hausdorff topological groups. We also note that fuzzy version of this work can be done as a further study. References [1] J.Antoni, On the A-continuity of real functions II, Math. Slovaca, 36, No.3, (1986), 283-287. MR 88a:26001 [2] J.Antoni and T.Salat, On the A-continuity of real functions, Acta Math. Univ. Comenian. 39, (1980), 159-164. MR 82h:26004 [3] J.Borsik and T.Salat, On F -continuity of real functions, Tatra Mt. Math. Publ. 2, 1993, 37-42. MR 94m:26006 [4] R.C.Buck, Solution of problem 4216 Amer. Math. Monthly 55, (1948), 36. MR 15: 26874 [5] J.Connor, Grosse-Erdmann K.-G. Sequential definitions of continuity for real functions, Rocky Mountain J. Math., 33, (1), (2003), 93-121. MR 2004e:26004 [6] H.C ¸ akallı, Sequential definitions of compactness, Applied Mathematics Letters, 21 , No:6, 594-598, (2008). [7] .............., Slowly oscillating continuity, Abstract and Applied Analysis, Volume 2008, Article ID 485706, (2008).MR 2393124 [8] H.Fast, Sur la convergence statistique, Colloq. Math. 2, (1951), 241-244. MR 14:29c [9] J.A.Fridy, On statistical convergence, Analysis, 5, (1985), 301-313 . MR 87b:40001

230

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[10] T.B.Iwinski, Some remarks on Toeplitz methods and continuity, Comment.Math. Prace Mat. 17, 1972, 37-43. MR 48759 ¨ urk, On almost-continuity and almost A-continuity of real functions, Comm.Fac.Sci. [11] E.Ozt¨ Univ.Ankara Ser. A1 Math. 32, 1983, 25-30. MR 86h:26003 [12] E.C.Posner, Summability preserving functions, Proc.Amer.Math.Soc. 12, 1961, 73-76. MR 2212327 ˙ [13] E. Sava¸s and G. Das, On the A-continuity of real functions, Istanbul Univ. Fen Fak. Mat Derg. 53, (1994), 61-66. MR 97m:26004 [14] I.J.Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly 66, 1959, 361-375. MR 21:3696 [15] E. Spigel and N. Krupnik, On the A-continuity of real functions, J.Anal. 2, (1994), 145-155. MR 95h:26004 [16] V.K.Srinivasan, An equivalent condition for the continuity of a function, Texas J. Sci. 32, 1980, 176-177. MR 81f :26001 ¨ ˙ C ˙ Im ˙ ˘ It HUsey In ¸ akallı, Department of Mathematics, Maltepe University, Marmara Eg ˙ ¨ yu ¨ , TR 34857, Maltepe, Istanbul-Turkey Ko Phone:(+90216)6261050 ext:1206, fax:(+90216)6261113 E-mail address: [email protected]

JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.2, 231-242, 2011, COPYRIGHT 2011 EUDOXUS PRESS, 231 LLC

Almost periodic solutions to abstract semilinear evolution equations with Stepanov almost periodic coefficients∗ Hui-Sheng Ding a , Wei Long a

a

and Gaston M. N’Gu´er´ekata

b,†

College of Mathematics and Information Science, Jiangxi Normal University Nanchang, Jiangxi 330022, People’s Republic of China b

Department of Mathematics, Morgan State University

1700 E. Cold Spring Lane, Baltimore, M.D. 21251, USA

Abstract In this paper, almost periodicity of the abstract semilinear evolution equation u0 (t) = A(t)u(t) + f (t, u(t)) with Stepanov almost periodic coefficients is discussed. We establish a new composition theorem of Stepanov almost periodic functions; and, with its help, we study the existence and uniqueness of almost periodic solutions to the above semilinear evolution equation. Our results are even new for the case of A(t) ≡ A. Keywords: almost periodic, Stepanov almost periodic, semilinear evolution equations, Banach space. 2000 Mathematics Subject Classification: 43A60, 34G20.

∗

Hui-Sheng Ding and Wei Long acknowledge support from the NSF of China (10826066), the NSF of

Jiangxi Province of China (2008GQS0057), and the Youth Foundation of Jiangxi Provincial Education Department(GJJ09456). † Corresponding author. E-mail addresses: [email protected] (H.-S. Ding), [email protected] (W. Long), Gaston.N’[email protected] (G. M. N’Gu´er´ekata).

1

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1

Introduction

Recall that almost periodic functions is a generalization of continuous periodic functions. However, sometimes, the requirement of continuity is a drawback for applications of almost periodic functions. Thus, in 1920s, Stepanov [12] introduced a generalization of almost periodic functions—Stepanov almost periodic functions, which is not necessarily continuous. Since Stepanov almost periodic functions is not necessarily continuous, the study of Stepanov almost periodic problems may be more interesting in terms of applications, and be more difficult considering complexity of Stepanov almost periodic functions. Therefore, the study of Stepanov almost periodic problems is a meaningful and interesting work. On the other hand, recently, since the work of [10] by N’Gu´er´ekata and Pankov, Stepanov almost automorphic problems have widely been investigated and many interesting results are established (see, e.g., [3, 5–7] and references therein). Especially, in the above works, existence and uniqueness of almost automorphic solutions to the abstract semilinear evolution equations u0 (t) = A(t)u(t) + f (t, u(t))

(1.1)

with Stepanov almost automorphic coefficients are studied. However, there seems to be no corresponding results for Eq. (1.1) with Stepanov almost periodic coefficients. In this paper, we consider this probelm. We first establish a composition theorem of Stepanov almost periodic functions, and then, we investigate the existence and uniqueness of almost periodic solutions to the abstract linear evolution equations u0 (t) = A(t)u(t) + f (t)

(1.2)

with Stepanov almost periodic coefficients. At last, combining the composition theorem and the existence and uniqueness theorem for Eq. (1.2), we discuss the existence of almost periodic solution to Eq. (1.1) with Stepanov almost periodic coefficients. As one will see, the composition theorem plays a key role in this paper, and is more complex than composition theorem of almost periodic functions. 2

DING ET AL: SEMILINEAR EVOLUTION EQUATIONS

233

Throughout this paper, we denote by R the set of real numbers, by mesE the Lebesgue measure for any subset E ⊂ R, and by X a Banach space. Moreover, we assume 1 ≤ p < +∞ if there is no special statement. First, let us recall some definitions and basic results of almost periodic functions and Stepanov almost periodic function (for more details, see [1, 8, 11]). Definition 1.1. A set E ⊂ R is called relatively dense if there exists a number l > 0 such that (a, a + l) ∩ E 6= ∅,

∀a ∈ R.

Definition 1.2. A continuous function f : R → X is called almost periodic if for each ε > 0 there exists a relatively dense set P (ε, f ) ⊂ R such that sup kf (t + τ ) − f (t)k < ε,

∀τ ∈ P (ε, f ).

t∈R

We denote the set of all such functions by AP (X). Definition 1.3. The Bochner transform f b (t, s), t ∈ R, s ∈ [0, 1], of a function f (t) on R, with values in X, is defined by f b (t, s) := f (t + s). Definition 1.4. The space BS p (X) of all Stepanov bounded functions, with the exponent p, consists of all measurable functions f on R with values in X such that ³ Z t+1 ´1/p kf kS p := sup kf (τ )kp dτ < +∞. t∈R

t

It is obvious that Lp (R; X) ⊂ BS p (X) ⊂ Lploc (R; X) and BS p (X) ⊂ BS q (X) whenever p ≥ q ≥ 1. Definition 1.5. A function f ∈ BS p (X) is called Stepanov almost periodic if f b ∈ ¡ ¢ AP Lp (0, 1; X) , that is, ∀ε > 0, there exists a relatively dense set P (ε, f ) ⊂ R such that

³Z sup t∈R

1

kf (t + s + τ ) − f (t + s)kp ds

´1

p

0

We denote the set of all such functions by AP S p (X). 3

< ε,

∀τ ∈ P (ε, f ).

234

DING ET AL: SEMILINEAR EVOLUTION EQUATIONS

Remark 1.6. It is clear that AP (X) ⊂ AP S p (X) ⊂ AP S q (X) for p ≥ q ≥ 1. Next, we introduce a notion about Stepanov almost periodic functions from R × X to X. Definition 1.7. A function f : R × X → X, (t, u) 7→ f (t, u) with f (·, u) ∈ BS p (R, X) for each u ∈ X, is said to be Stepanov almost periodic in t ∈ R uniformly for u ∈ X, if for each ε > 0 and each compact set K ⊂ X, there exists a relatively dense set P (ε, f, K) ⊂ R such that

³Z

1

sup t∈R

kf (t + s + τ, u) − f (t + s, u)kp ds

´1

p

< ε,

0

for each τ ∈ P (ε, f, K) and each u ∈ K. We denote by AP S p (R × X, X) the set of all such functions. Similar to Remark 1.6, it is easy to show that AP S p (R × X, X) ⊂ AP S q (R × X, X) for p ≥ q ≥ 1. Next, let us recall some notations about evolution family and exponential dichotomy. For more details, we refer the reader to [4]. Definition 1.8. A set {U (t, s) : t ≥ s, t, s ∈ R} of bounded linear operator on X is called an evolution family if (a) U (s, s) = I, U (t, s) = U (t, r)U (r, s) for t ≥ r ≥ s and t, r, s ∈ R, (b) {(τ, σ) ∈ R2 : τ ≥ σ} 3 (t, s) 7−→ U (t, s) is strongly continuous. Throughout the rest of this paper, we suppose that A(t) generates an evolution family U (t, s). Definition 1.9. An evolution family U (t, s) is called hyperbolic (or has exponential dichotomy) if there are projections P (t), t ∈ R, being uniformly bounded and strongly continuous in t, and constants M , ω > 0 such that (a) U (t, s)P (s) = P (t)U (t, s) for all t ≥ s, (b) the restriction UQ (t, s) : Q(s)X → Q(t)X is invertible for all t ≥ s (and we set UQ (s, t) = UQ (t, s)−1 ), 4

DING ET AL: SEMILINEAR EVOLUTION EQUATIONS

235

(c) kU (t, s)P (s)k ≤ M e−ω(t−s) and kUQ (s, t)Q(t)k ≤ M e−ω(t−s) for all t ≥ s. Here and below Q := I − P . Exponential dichotomy is a classical concept in the study of long-term behaviour of evolution equations, see e.g. [2, 4]. If U (t, s) is hyperbolic, then   U (t, s)P (s), t ≥ s, t, s ∈ R, Γ(t, s) :=  −UQ (t, s)Q(s), t < s, t, s ∈ R, is called Green’s function corresponding to U (t, s) and P (·), and   M e−ω(t−s) , t ≥ s, t, s ∈ R, kΓ(t, s)k ≤  M e−ω(s−t) , t < s, t, s ∈ R.

(1.3)

Next, we recall some definitions. Definition 1.10. A continuous function u : R → X is called a mild solution of Eq. (1.2) if

Z

t

u(t) = U (t, s)u(s) +

U (t, τ )f (τ )dτ,

t ≥ s.

s

Definition 1.11. A continuous function u : R → X is called a mild solution of Eq. (1.1) if

Z u(t) = U (t, s)u(s) +

t

U (t, τ )f (τ, u(τ ))dτ,

t ≥ s.

s

2

Main results

First, we establish a composition theorem of Stepanov almost periodic functions. Throughout the rest of this paper, for convenience, we denote the norm of Lp (0, 1; X) by k · kp . Theorem 2.1. Assume that p > 2 and the following conditions hold: (a) f ∈ AP S p (R × X, X), and there exists a function L ∈ BS p (X) such that kf (t, u) − f (t, v)k ≤ L(t)ku − vk, 5

∀t ∈ R, u, v ∈ X;

(2.1)

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DING ET AL: SEMILINEAR EVOLUTION EQUATIONS

(b) x ∈ AP S p (X), and there exists a set E ⊂ R with mesE = 0 such that K := {x(t) : t ∈ R \ E} is compact in X. p

Then f (·, x(·)) ∈ AP S 2 (X). p

Proof. First, we claim that f (·, x(·)) ∈ BS 2 (X). In fact, by (2.1), we have µZ

¶2/p

t+1

kf (τ, x(τ ))k

p/2

dτ

t

µZ

t+1

≤

¶2/p p/2

L

(τ )kx(τ )k

t

µZ ≤

t+1

p/2

dτ

µZ +

kf (τ, 0)k t

¶1/p µZ

t+1

p

L (τ )dτ

¶2/p

t+1

¶1/p

p

kx(τ )k dτ

t

µZ

p/2

dτ ¶1/p

t+1

+

p

kf (τ, 0)k dτ

t

t

≤ kLkS p kxkS p + kf (·, 0)kS p < +∞,

∀t ∈ R.

Noticing that K is a compact set, ∀ε > 0, there exist x1 , . . . , xk ∈ K such that K⊂

k [ i=1

µ

ε B xi , 6kLkS p/2

¶ .

(2.2)

Since f ∈ AP S p (R × X, X) and x ∈ AP S p (X), by [8, page 6, Property 7], for the above ε > 0, there exists a common relatively dense set P (ε) ⊂ R such that ε , 6k

(2.3)

ε , 2kLkS p

(2.4)

kf (t + τ + ·, u) − f (t + ·, u)kp < and kx(t + τ + ·) − x(t + ·)kp < for all τ ∈ P (ε), t ∈ R and u ∈ K. By (2.1), we obtain that kf (t + s + τ, x(t + s + τ )) − f (t + s, x(t + s))k ≤

L(t + s + τ )kx(t + s + τ ) − x(t + s)k + kf (t + s + τ, x(t + s)) − f (t + s, x(t + s))k

:= I + J. 6

DING ET AL: SEMILINEAR EVOLUTION EQUATIONS

237

For I, by (2.4), we have µZ kIkp/2 =

1

¶2/p p/2

L

(t + s + τ )kx(t + s + τ ) − x(t + s)k

0

µZ

1

≤

¶1/p µZ p

p/2

ds ¶1/p

1

L (t + s + τ )ds

p

kx(t + s + τ ) − x(t + s)k ds

0

0

≤ kLkS p · kx(t + τ + ·) − x(t + ·)kp < ε/2, for all τ ∈ P (ε) and t ∈ R. Next, let us consider J. Fix t ∈ R. Define Et = {s ∈ [0, 1] : t + s ∈ / E}. It follows that ³ ´ mes [0, 1] \ Et = 0 and x(t + s) ∈ K for all s ∈ Et . Then, by (2.2), for each s ∈ Et , there exists i(s) ∈ {1, 2, . . . , k} such that kxi(s) − x(t + s)k
0, i.e., ∀ε > 0 and ∀h > 0, there exists a common relatively dense set P (ε) ⊂ R such that sup kΓ(t + r + τ, s + r + τ ) − Γ(t + r, s + r)k < ε, r∈R

for all τ ∈ P (ε) and t, s ∈ R with |t − s| ≥ h. Remark 2.2. For some general conditions which can ensure that (H2) holds, we refer the reader to [9, Theorem 4.5]. In addition, in the case of A(t) ≡ A, P (t) ≡ I, and A generating a semigroup T (t), Γ(t, s) = T (t − s) for t ≥ s. Then, (H2) obviously holds. Theorem 2.3. Assume that f ∈ AP S p (X) with p > 1 and (H1), (H2) hold. Then the equation (1.2) has a unique almost periodic mild solution given by Z Γ(t, s)f (s)ds, t ∈ R. u(t) = R

8

DING ET AL: SEMILINEAR EVOLUTION EQUATIONS

Proof. For each k ∈ N and t ∈ R, let Z t−k+1 Z Φk (t) := Γ(t, s)f (s)ds = t−k

239

k

Γ(t, t − s)f (t − s)ds.

k−1

We claim that Φk ∈ AP (X). In fact, it follows from f ∈ AP S 1 (X) and (H2) that ∀ε > 0 and taking

½ µ h0 = min 1,

ε 6M kf kS p

¶q ¾ ,

1 1 + = 1, p q

there exists a common relatively dense set P (ε) ⊂ R such that Z 1 ε sup kf (t + s + τ ) − f (t + s)kds < , 3M t∈R 0

(2.7)

for all τ ∈ P (ε), and sup kΓ(t + r + τ, s + r + τ ) − Γ(t + r, s + r)k < r∈R

ε , 3kf kS 1

(2.8)

for all τ ∈ P (ε) and t, s ∈ R with |t − s| ≥ h0 . Then, we deduce by (1.3), (2.7) and (2.8), for all τ ∈ P (ε) and t ∈ R, kΦ1 (t + τ ) − Φ1 (t)k Z 1 ≤ kΓ(t + τ, t + τ − s)f (t + τ − s) − Γ(t, t − s)f (t − s)kds 0 Z 1 ≤ kΓ(t + τ, t + τ − s) − Γ(t, t − s)k · kf (t − s)kds 0 Z 1 + kΓ(t + τ, t + τ − s)k · kf (t + τ − s) − f (t − s)kds 0 Z h0 ≤ kΓ(t + τ, t + τ − s) − Γ(t, t − s)k · kf (t − s)kds 0 Z 1 Z kΓ(t + τ, t + τ − s) − Γ(t, t − s)k · kf (t − s)kds + M + 0

h0

Z

1

kf (t + τ − s) − f (t − s)kds

Z 1 Z 1 ε ≤ 2M kf (t − s)kds + kf (t − s)kds + M kf (t − 1 + s + τ ) − f (t − 1 + s)kds 3kf kS 1 h0 0 0 µZ h 0 ¶1/p Z 1 ε ε 1/q p ≤ 2M h0 kf (t − s)k ds + kf (t − s)kds + 3kf kS 1 0 3 0 ε ε 1/q ≤ 2M h0 kf kS p + + ≤ ε. 3 3 h0

9

240

DING ET AL: SEMILINEAR EVOLUTION EQUATIONS

For Φk with k ≥ 2, the proof is similar to that of Φ1 (but simpler). On the other hand, since Z t−k+1 kΦk (t)k ≤ M e−ω(t−s) · kf (s)kds ≤ M e−ω(k−1) · kf kS 1 , t−k

we deduce that

∞ P k=1

Φk (t) is uniformly convergent on R. So Z

t

Γ(t, s)f (s)ds = −∞

∞ X

Φk (t) ∈ AP (X).

k=1

Analogously, one can show that Z +∞ Γ(t, s)f (s)ds ∈ AP (X). t

Then, it follows that

Z u(t) :=

Γ(t, s)f (s)ds ∈ AP (X). R

At last, one can prove that u(t) is just the unique almost periodic mild solution to equation (1.2) (see, e.g., [2, Theorem 4.28]). Now, we are ready to present our results for Eq. (1.1). Theorem 2.4. Assume that f ∈ AP S p (R × X, X) with p > 2, and there exists a function L ∈ BS p (X) such that (2.1) holds. Moreover, (H1) and (H2) hold. Then the equation (1.1) has a unique almost periodic mild solution provided that µ ¶1 q 1 − e−ω ωq kLkS p < · , 2M 1 − e−ωq where

1 p

+

1 q

(2.9)

= 1.

Proof. Let u ∈ AP (X). Then u ∈ AP S p (X) and it is well known (cf. [8]) that K := {u(t) : t ∈ R} is compact in X. Let Z Γ(t, s)f (s, u(s))ds, F (u)(t) :=

t ∈ R.

R

By Theorem 2.1 and Theorem 2.3, we have F (u) ∈ AP (X). So F maps AP (X) into AP (X). For u, v ∈ AP (X), by using the H¨older’s inequality, we have Z kΓ(t, s)k · kf (s, u(s)) − f (s, v(s))kds kF (u)(t) − F (v)(t)k ≤ R

10

DING ET AL: SEMILINEAR EVOLUTION EQUATIONS

Z

Z

t

−ω(t−s)

≤

Me

≤ ≤

Z

t−k+1

−ω(t−s)

Me

M e−ω(s−t) L(s)ds · ku − vk ¶ −ω(s−t)

Me

L(s)ds · ku − vk

t+k−1

µ

2M 1 − e−ω

t t+k

L(s)ds +

t−k

k=1

+∞

L(s)ds · ku − vk +

−∞

∞ µZ X

241

1 − e−ωq ωq

¶1 q

kLkS p · ku − vk,

for all t ∈ R, i.e., 2M kF (u) − F (v)k ≤ 1 − e−ω

µ

1 − e−ωq ωq

¶1 q

kLkS p · ku − vk.

Then, in view of (2.9), we know that F has a unique fixed point u ∈ AP (X), which satisfies

Z Γ(t, s)f (s, u(s))ds,

u(t) =

t ∈ R.

R

Next, similar to the corresponding proof of Theorem 2.3, one can prove that u(t) is just the unique almost periodic mild solution to (1.1). Remark 2.5. To the best of our knowledge, Theorem 2.4 are even new for the case of A(t) ≡ A. In addition, for f ∈ AP S p (R × X, X), the condition (2.1) is more natural than the usual Lipschitz assumption, i.e., there exists a constant L > 0 such that kf (t, u) − f (t, v)k ≤ Lku − vk for all t ∈ R and u, v ∈ X.

References [1] L. Amerio, G. Prouse, Almost-periodic functions and functional equations, Van Nostrand Reinhold Co., 1971. [2] C. Chicone and Y. Latushkin, Evolution semigroups in dynamical systems and differential equations, Amer. Math. Soc., 1999. [3] H. S. Ding, J. Liang, T. J. Xiao, Some properties of Stepanov-like almost automorphic functions and applications to abstract evolution equations, Appl. Anal. 88 (2009), 1079–1091. 11

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[4] K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, 2000. [5] S. Fatajou, N. Van Minh, G. M. N’Gu´er´ekata, and A. Pankov, Stepanov-like almost automorphic solutions for nonautonomous evolution equations, Electron. J. Differ. Equ. 121 (2007), 1–11. [6] H. Lee, H. Alkahby, Stepanov-like almost automorphic solutions of nonautonomous semilinear evolution equations with delay, Nonlinear Anal. 69 (2008), 2158–2166. [7] H. Lee, H. Alkahby, G. M. N’Gu´er´ekata, Stepanov-like almost automorphic solutions of semilinear evolution equations with deviated argument, Int. J. Evol. Equ. 3 (2008), 217–224. [8] B. M. Levitan, V. V. Zhikov, Almost Periodic Functions and Differential Equations, Cambrige University Press, 1982. [9] L. Maniar and R. Schnaubelt, Almost periodicity of inhomogeneous parabolic evolution equations, Lecture Notes in Pure and Appl. Math. 234, Dekker, New York, 2003, 299–318. [10] G. M. N’Gu´er´ekata, A. Pankov, Stepanov-like almost automorphic functions and monotone evolution equations, Nonlinear Anal. 68 (2008), 2658–2667. [11] A. Pankov, Bounded and almost periodic solutions of nonlinear operator differential equations, Kluwer, Dordrecht, 1990. ¨ [12] W. Stepanov, Uber einige Verallgemeinerungen der fastperiodischen Funktionen, Math. Ann. 95 (1926), 473–498.

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JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.2, 243-253, 2011, COPYRIGHT 2011 EUDOXUS PRESS, 243 LLC

1

Abstract metric spaces and approximating fixed points of a pair of contractive type mappings

M. Abbas,1 Mirko Jovanovi´c,2 S. Radenovi´c,3 Aleksandra Sretenovi´c,3 Suzana Simi´c,4 Department of Mathematics, Lahore University of Management Sciences, 54792-Lahore, Pakistan 2 University of Belgrade, Faculty of Electrical Engineering, Bul. Kralja Aleksandra 73, 11000 Beograd, Serbia 3 University of Belgrade, Faculty of Mechanical Engineering, Kraljice Marije 16, 11 120 Beograd, Serbia 4 Faculty of Sciences, Department of Mathematics, Radoja Domanovi´ca 12, 34 000 Kragujevac, Serbia 1

1

––––––––––––––––––––––––––––––—

Abstract

Recently, Chao Wang, Jinghao Zhu, Boško Damjanovi´c, Liang-gen Hu [Approximating fixed points of a pair of contractive type mappings in generalized convex metric spaces, Applied Mathematics and Computation 215 (2009) 1522-1525] proved some results for an Ishikawa type iteration process with errors, which converges to the unique common fixed point of a pair of contractive type mappings in complete generalized convex metric spaces. The aim of this paper is to consider this in the frame of abstract (cone) metric spaces. We extend some fixed point results of these mappings from complete generalized convex metric spaces to convex abstract metric spaces with the solid cone. Also, we have improved some results. Four examples are included. 2000 Mathematical Subject Classification 47H10, 54H25 Keywords: Convex abstract metric spaces; Ishikawa type iteration process with errors; Normal and non-normal cone; Uniformly quasi-Lipschitzian mappings.

––––––––––––––––––––––––––––––– 1. I     Ordered normed spaces and cones have applications in applied mathematics, for instance, in using Newton’s approximation method [13]-[15], [25], [30], and in optimization theory [5]. K−metric and K−normed spaces were introduced in the mid-20th century ([14], see also [25], [30]) by replacing an ordered Banach space instead of the set of real numbers, as the codomain for a metric. L.G. Huang and X. Zhang [7] re-introduced such spaces under the name of cone metric spaces, but went further, defining convergent and Cauchy sequences in the terms of interior points of the underlying cone. In such a way, non-normal cones can be used as well (although they used only normal cones), paying attention to the fact that Sandwich theorem and continuity of the metric may not hold. These and other authors ([1], [2], [9], [19], [20], [30]) proved some fixed point theorems for contractive-type mappings in cone metric spaces. Consistent with [5] (see also [7], [14], [16], [25]) the following definitions and results will be needed in the sequel. Let E be a real Banach space. A subset P of E is called a cone whenever the following conditions hold: 1 E-mail address: [email protected] (M. Abbas); [email protected] (M.Jovanovi´c); [email protected] (S. Radenovi´c); [email protected] (A.Sretenovi´c); [email protected] (S.Simi´c ). The authors are thankful to the Ministry of Science and Environmental Protection of Serbia.

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2 (a) P is closed, nonempty and P = {θ} ; (b) a, b ∈ R, a, b ≥ 0, and x, y ∈ P imply ax + by ∈ P ; (c) P ∩ (−P ) = {θ} . Given a cone P ⊂ E, we define a partial ordering with respect to P by x y if and only if y − x ∈ P. We shall write x ≺ y to indicate that x y but x = y, while x ≪ y will stand for y − x ∈ intP (interior of P ). If intP = ∅ then P is called a solid cone (see [25]). There exist two kinds of cones- normal (with the normal constant k) and non-normal ones [5]. Let E be a real Banach space, P ⊂ E a cone and partial ordering defined by P. Then P is called normal if there is a number k > 0 such that for all x, y ∈ P, θ x y imply x ≤ k y ,

(1.1)

or equivalently, if xn yn zn and lim xn = lim zn = x imply lim yn = x.

n→∞

n→∞

n→∞

(1.2)

The least positive number k satisfying (1.1) is called the normal constant of P. It is clear that k ≥ 1. Most of ordered Banach spaces used in applications posses a normal cone with the normal constant k = 1. For details see [5]. Example 1.1.[25] Let E = CR1 [0, 1] with x = x ∞ + x′ ∞ on P = {x ∈ E : x (t) ≥ 0} . tn 1 This cone is not normal. Consider, for example, and
tnx
n (t) =
n n−1
yn1 (t) = n . Then
= + 1 > 1; hence θ xn yn , and lim yn = θ, but xn = max
n
+ max
t n n→∞

t∈[0,1]

t∈[0,1]

xn does not converge to zero. It follows by (1.2) that P is a non-normal cone.. Definition 1.2.([7], [30]) Let X be a nonempty set. Suppose that the mapping d : X × X → E satisfies: (d1) θ d (x, y) for all x, y ∈ X and d (x, y) = θ if and only if x = y; (d2) d (x, y) = d (y, x) for all x, y ∈ X; (d3) d (x, y) d (x, z) + d (z, y) for all x, y, z ∈ X. Then d is called a cone metric [7] or K−metric [30] on X and (X, d) is called a cone metric [7] or K−metric space [30] (we shall use the first terms). The concept of a cone metric space is more general than of a metric space, because each metric space is a cone metric space where E = R and P = [0, +∞). Examples 1.3.   10 [7] Let E = R2 , P = (x, y) ∈ R2 : x ≥ 0, y ≥ 0 , X = R and d : X × X → E defined by d (x, y) = (|x − y| , α |x − y|) , where α ≥ 0 is a constant. Then (X, d) is a cone metric space [7] with normal cone P where k = 1. 20 For other examples of a cone metric spaces, i.e., P −metric spaces one can see [30], pp. 853-854. Now we give definition of a cone normed spaces which is generalization of a norm spaces. Definition 1.4. [17], [24], [22] Let X be a real vector spaces and E a real Banach space ordered with cone P. Suppose that mapping . P : X → E satisfies: (cn1) x P = θE if and only if x = θX ; (cn2) αx P = |α| x P for any scalar α and any x ∈ X; (cn3) x + y P x P + y P for all x, y ∈ X. Then . P is called a cone norm on X, and we call (X, . P ) a cone normed space. One can see that from (cn3) it follows x P ∈ P for all x ∈ X. It is clear that d (x, y) =

x − y P is a cone metric.

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3 Definition 1.5.[7] Let (X, d) be a cone metric space. We say that {xn } is: (i) a Cauchy sequence if for every c in E with θ ≪ c, there is an N such that for all n, m > N, d (xn , xm ) ≪ c; (ii) a convergent sequence if for every c in E with θ ≪ c, there is an N such that for all n > N, d (xn , x) ≪ c for some fixed x in X (for the relation ”≪” see [18]). A cone metric space X is said to be complete if every Cauchy sequence in X is convergent in X. Let (X, d) be a cone metric space. Then the following properties are often used, particularly when dealing with cone metric spaces in which the cone need not be normal (for details see [12]). p1 ) If u v and v ≪ w, then u ≪ w. p2 ) If θ u ≪ c for each c ∈ intP, then u = θ. p3 ) If E is a real Banach space with a cone P and if a λa where a ∈ P and 0 ≤ λ < 1, then a = θ. p4 ) If c ∈ intP, θ an and an → θ, then there exists n0 such that for all n > n0 we have an ≪ c. It follows from Example 1.1. that the converse is not true in general. Indeed, in the mentioned Example, xn  θ but xn ≪ c for n sufficiently large. In the sequel we assume that E is a real Banach space and that P is a solid cone in E, that is cone with intP = ∅. The last assumption is necessary in order to obtain reasonable results connected with convergence and continuity. The partial ordering induced by the cone P will be denoted by . 2. C      Let (X, d) be a cone metric space with solid cone P . A mapping f : X → X is called asymptotically nonexpansive if there exists kn ∈ [1, ∞), limn→∞ kn = 1, such that d (f n x, f n y) kn d (x, y) for all x, y ∈ X. Let F (f) = {x ∈ X : f x = x} ; if F (f ) = ∅, then f is called asymptotically quasi-nonexpansive if there exists kn ∈ [1, ∞), limn→∞ kn = 1, such that d (f n x, p) kn d (x, p) for all x ∈ X, p ∈ F (f) . Moreover, it is uniformly quasi-Lipschitzian if there exists L > 0 such that d (f n x, p) Ld (x, p) for all x ∈ X, p ∈ F (f) . From the above definition, if F (f) = ∅, it follows that an asymptotically nonexpansive mapping must be asymptotically quasi-nonexpansive, and an asymptotically quasinonexpansive must be uniformly quasi-Lipschitzian (L = supn≥0 {kn } < ∞). However, the inverse relation does not hold (for details see [21]). In recent years, asymptotically nonexpansive mappings and asymptotically quasi-nonexpansive mappings have been studied extensively in the setting of convex metric spaces ([6], [8], [29]). In 1970, Takahashi [23] first introduced a notion of convex metric space which is more general space. It should be pointed out that each linear normed space is a special example of convex metric space, but there exist some convex metric spaces which cannot be embedded into normed space [23]. In the sequel, we shall need the following definitions and results: Definition 2.1. [21] Let (X, d) be a cone metric space, and I = [0, 1] . A mapping w : X 2 × I → X is said to be convex structure on X, if for any (x, y, λ) ∈ X 2 × I and u ∈ X, the following inequality holds: d (w (x, y, λ) , u) λd (x, u) + (1 − λ) d (y, u) . If (X, d) is a cone metric space with a convex structure w, then (X, d) is called a convex abstract metric space or convex cone metric space (see also [9], [20]). Moreover, a non-

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4 empty subset K of X is said to be convex if w (x, y, λ) ∈ K, for all (x, y, λ) ∈ K 2 × I. For convex cone-normed spaces see [17] Definition 2.2. [21] Let (X, d) be a cone metric space, I = [0, 1] , and {an } , {bn } , {cn } real sequences in [0, 1] with an + bn + cn = 1. A mapping w : X 3 × I 3 → X is said to be convex structure on X, if for any (x, y, z, an , bn , cn ) ∈ X 3 × I 3 and u ∈ X, the following inequality holds: d (w (x, y, z, an , bn , cn ) , u) an d (x, u) + bn d (y, u) + cn d (z, u) . If (X, d) is a cone metric space with a convex structure w, then (X, d) is called a generalized convex cone metric space. Moreover, a nonempty subset K of X is said to be convex if w (x, y, z, an , bn , cn ) ∈ K, for all (x, y, z, an , bn , cn ) ∈ K 3 × I 3 . Example 2.3. [17], [21] Let (X, d) be a cone metric space as in example 1.3., 10 . If w (x, y) =: λx+(1 − λ) y, then (X, d) is a convex cone metric space. Hence, this concept is more general than that of convex metric space. Also, every cone normed space (X, . P ) is convex cone normed space. Indeed, if for x, y ∈ X, λ ∈ [0, 1], w (x, y) =: λx + (1 − λ) y, then for all u ∈ X we have

u − w (x, y) P

= u − (λx + (1 − λ) y) P = λ (u − x) + (1 − λ) (u − y) P λ u − x P + (1 − λ) u − y P .

Definition 2.4. [21]. Let (X, d) be a cone metric space with a convex structure w : X 3 × I 3 → X, f, g : X → X be uniformly quasi-Lipschitzian mappings with L > 0 and L′ > 0, {an } , {bn } , {cn } , {a′n } , {b′n } , {c′n } be six sequences in [0, 1] with an + bn + cn = a′n + b′n + c′n = 1, n = 0, 1, 2, ..., . For any given x0 ∈ X, define a sequence {xn } as follows:  xn+1 = w (xn , f n yn , un , an , bn , cn ) , (2.1) yn = w (xn , g n xn , vn , a′n , b′n , c′n ) , where {un } , {vn } are two sequences in X satisfying the following condition: for any nonnegative integers n, m, 0 ≤ n < m, if δ (Anm ) > 0, then max { d (x, y) : x ∈ {ui , vi } , y ∈ {xj , yj , fyj , gxj , uj , vj }} < δ (Anm ) ,

n≤i,j≤m

where Anm = {xi , yi , f yi , gxi , ui , vi : n ≤ i ≤ m} , δ (Anm ) = sup

(2.2)

d (x, y) .

x,y∈Anm

Then {xn } is called the Ishikawa type iteration process with errors of two uniformly quasi-Lipschitzian mappings f and g in convex cone metric space (X, d) . Definition 2.5. [26] Let (X, d) be a metric space and K a nonempty closed convex subset of X. Two mappings S, T : K → K are said to be a pair of contractive type mappings if there exists h ∈ [0, 1) such that d (S n x, T n y) ≤ h · max {d (x, y) , d (x, S n x) , d (y, T n y) , d (x, T n y) , d (y, S n x)}

(2.3)

for all x, y ∈ K, n ≥ 1. It is easy to see that a pair of contractive type mappings are very general, and include ´ c’s quasi-contractive mapping [3] (that is., S = T and n = 1). In this frame of cone Ciri´ metric spaces the set of these mappings include f −quasi-contractive and quasi-contractive mappings of Ili´c’s-Rakoˇcevi´c’s type [10],[11], ( i.e., S = T and n = 1) Since, in the case of

ABBAS ET AL: ABSTRACT METRIC SPACES...

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5 cone metric spaces the set of vectors {d (x, y) , d (x, S n x) , d (y, T n y) , d (x, T n y) , d (y, S n x)} need not even have the supremum in ordered Banach space E then, as in [10],[11], we use ”∈”. It is clear that ”∈” can be used in metric spaces, while ” ” can not be used in general in cone metric spaces. In this case we introduce: Definition 2.6. Let (X, d) be a cone metric space. Two mappings S, T : X → X are said to be a pair of contractive type mappings if there exists h ∈ [0, 1) such that for every x, y ∈ X there exists u (x, y) ∈ {d (x, y) , d (x, S n x) , d (y, T n y) , d (x, T n y) , d (y, S n x)} such that d (S n x, T n y) h · u (x, y) .

(2.4)

Theorem 2.7. [26] Let K be a nonempty closed convex subset of a complete generalized convex metric space X and S, T : K → K a pair of contractive type mappings with F = F (S) ∩ F (T ) = ∅. For any x1 ∈ K, {xn } is the Ishikawa type iteration process with errors defined by (2.1), where {un } , {vn } ⊂ K satisfy (2.2) and {an } , {bn } , {cn } , {a′n } , {b′n } , {c′n } are six sequences in [0, 1] satisfying an + bn + cn = a′n + b′n + c′n = 1 and

∞ 

(an + bn ) < ∞.

n=0

Then, {xn } converges to the unique common fixed point of S and T if and only if lim inf n→∞ d (xn , F ) = 0. Corollary 2.8. [26] Let K be a nonempty closed convex subset of a Banach space X, and S, T : K → K a pair of contractive type mappings, that is,

S n x − T n y ≤ h max { x − y , x − S n x , y − T n y , x − T n y , y − S n x } for h ∈ [0, 1), all x, y ∈ K and n ≥ 1. Let F = F (S) ∩ F (T ) = ∅. For any x1 ∈ K, {xn } is the Ishikawa type iteration process with errors defined by xn+1 yn

= an xn + bn S n yn + cn yn = a′n xn + b′n T n xn + c′n vn ,

(2.5)

where {un } , {vn } ⊂ K are two bounded sequences and {an } , {bn } , {cn } , {a′n } , {b′n } , {c′n } are six sequences in [0, 1] satisfying an + bn + cn = a′n + b′n + c′n = 1 and

∞ 

(an + bn ) < ∞.

n=0

Then {xn } converges to the unique common fixed point of S and T if and only if lim inf n→∞ d (xn , F ) = 0. Theorem 2.9. [26]Let K be a nonempty closed convex subset of a complete generalized convex metric space X and S, T : K → K a pair of contractive type mappings with h < 12 . For any x1 ∈ K, {xn } is the Ishikawa type iteration process with errors defined by (2.1), where {un } , {vn } ⊂ K satisfy (2.2) and {an } , {bn } , {cn } , {a′n } , {b′n } , {c′n } are six sequences in [0, 1] satisfying an + bn + cn = a′n + b′n + c′n = 1,

∞ 

n=1

bn < ∞,

∞ 

n=1

cn < ∞ and lim c′n = 0. n→∞

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6 If F = F (S) ∩ F (T ) = ∅, then {xn } converges to the unique common fixed point of S and T. Corollary 2.10. [26] Let K be a nonempty closed convex subset of Banach space X, and S, T : K → K a pair of contractive type mappings with h < 12 . For any x1 ∈ K, {xn } is the Ishikawa type iteration process with errors defined by (2.1), where {un } , {vn } ⊂ K are two bounded sequences and {an } , {bn } , {cn } , {a′n } , {b′n } , {c′n } are six sequences in [0, 1] satisfying an + bn + cn = a′n + b′n + c′n = 1,

∞ 

n=1

bn < ∞,

∞ 

cn < ∞ and lim c′n = 0.

n=1

n→∞

If F = F (S) ∩ F (T ) = ∅, then {xn } converges to the unique common fixed point of S and T. 3. M    Now we give our main results of this paper by using only the fact that (X, d) is a cone metric space with a solid cone P. Hence, we do not impose the normality condition for the cones. The following theorem extends and improves Theorem 2.2. of [26]. Theorem 3.1. Let (X, d) be a cone metric space with a solid cone P ; S, T : X → X be a pair of contractive type mappings satisfying (2.4). If h ∈ [0, 12 ) then S and T have a unique common fixed point. Moreover, in this case, S and T are two uniformly h quasi-Lipschitzian mappings (with L = L′ = 1−h ). Remark 3.2. It is worth noticing that if S and T satisfy (2.4) with 0 ≤ h < 12 and u (x, y) ∈ {d (x, y) , d (x, S n x) , d (y, T n y) , d (x, T n y) , d (y, S n x)} , then S and T satisfy (2.4) with 0 ≤ h < 1 and u (x, y) satisfy more general condition:   d (x, T n y) + d (y, S n x) n n . u (x, y) ∈ d (x, y) , d (x, S x) , d (y, T y) , 2 Indeed, we have d (x, T n y) + d (y, S n x) d (x, T n y) + d (y, S n x) d (x, T n y) 2h · =λ· , 2 2 2 d (x, S n y) d (x, T n y) + d (y, S n x) d (x, T n y) + d (y, S n x) h · d (x, S n y) = 2h · 2h · =λ· , 2 2 2

h · d (x, T n y) = 2h ·

where 2h = λ ∈ [0, 1) if and only if h ∈ [0, 12 ). We now give the proof of Theorem 3.1. Proof. Taking S n = f, T n = g in the condition (2.4) we obtain the following inequality: d (fx, gy) h · u, (3.1) where u ∈ {d (x, y) , d (x, f x) , d (y, gy) , d (x, gy) , d (y, fx)} . Suppose x0 is an arbitrary point of X, and define {xn } by x2n+1 = fx2n , x2n+2 = gx2n+1 , n = 0, 1, 2, .... We first show that d (xn , xn+1 )

h d (xn−1 , xn ) , for n ≥ 1. 1−h

(3.2)

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7 Now, we have that d (x2n+1 , x2n+2 ) = d (f x2n , gx2n+1 ) h · u1 ,

(3.3)

where u1

∈ {d (x2n , x2n+1 ) , d (x2n , f x2n ) , d (x2n+1 , gx2n+1 ) , d (x2n , gx2n+1 ) , d (x2n+1 , f x2n )} = {θ, d (x2n , x2n+1 ) , d (x2n+1 , x2n+2 ) , d (x2n , x2n+2 )} .

Now we have to consider the following four cases. If u1 = θ or u1 = d (x2n+1 , x2n+2 ) then d (x2n+1 , x2n+2 ) = θ, and (3.2) is immediate. If u1 = d (x2n , x2n+1 ) then clearly (3.2) holds. Finally, suppose that u1 = d (x2n , x2n+2 ) . Now, d (x2n , x2n+2 ) h · (d (x2n , x2n+1 ) + d (x2n+1 , x2n+2 )) . h Hence, d (x2n+1 , x2n+2 ) 1−h d (x2n , x2n+1 ) , and we proved (3.2). Similarly it can be obtain that

d (x2n+3 , x2n+2 ) = d (f x2n+2 , gx2n+1 ) h · u2 ,

(3.4)

where u2

∈

{d (x2n+1 , x2n+2 ) , d (x2n+2 , f x2n+2 ) , d (x2n+1 , gx2n+1 ) , d (x2n+2 , gx2n+1 ) , d (x2n+1 , f x2n+2 )} = {θ, d (x2n+1 , x2n+2 ) , d (x2n+2 , x2n+3 ) , d (x2n+1 , x2n+3 )} .

Now repetition of the argument for case u1 leads to d (x2n+3 , x2n+2 ) Hence, the inequality (3.2) holds for every n = 1, 2, 3, .... Now we have that for n = 1, 2, 3, ... d (xn , xn+1 ) λn d (x0 , x1 ) , λ =

h 1−h d (x2n+2 , x2n+1 ) .

h ∈ [0, 1), 1−h

(3.5)

because h ∈ [0, 12 ). We will show that {xn } is a Cauchy sequence. For n > m we have d (xn , xm ) d (xn , xn−1 ) + d (xn−1 , xn−2 ) + · · · + d (xm+1 , xm )   λn−1 + λn−2 + · · · + λm d (x1 , x0 ) λm d (x0 , x1 ) → θ, as m → ∞. 1−λ

(3.6)

From p4 ) it follows that for θ ≪ c and large m : λm (1 − λ)−1 d (x0 , x1 ) ≪ c; thus according to p1 ) d (xn , xm ) ≪ c. Hence, by Definition 1.4 (i), {xn } is a Cauchy sequence. Since X is complete, there exists u in X such that xn → u as n → ∞. Let us show that f u = gu = u. For this we have d (f x2n , gu) h · un , where un ∈ {d (x2n , u) , d (x2n , f x2n ) , d (u, gu) , d (x2n , gu) , d (u, f x2n )} .

(3.7)

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8 Let θ ≪ c. Clearly at least one of the following five cases holds for infinitely many n. (Case 10 ) c d (f x2n , gu) h · d (x2n , u) ≪ h · = c. (3.8) h (Case 20 ) d (f x2n , gu) h · d (x2n , fx2n ) h · d (x2n , u) + h · d (u, fx2n ) ≪ h ·

c c +h· = c. (3.9) 2h 2h

(Case 30 ) d (f x2n , gu) h · d (u, gu) h · d (u, fx2n ) + h · d (f x2n , gu) ≪ h ·

c c +h· = c. (3.10) 2h 2h

(Case 40 ) h · d (x2n , gu) h · d (x2n , u) + h · d (u, f x2n ) + h · d (f x2n , gu) , i.e., h h d (f x2n , gu) d (x2n , u) + d (u, f x2n ) 1−h 1−h

h 1−h 1−h ≪ c+ c = c. (3.11) 1−h 2h 2h d (f x2n , gu)



(Case 50 )

c = c. (3.12) h In all cases (according to Definition 1.4 (ii)) we obtain that fx2n → gu, that is xn → gu. The uniqueness of a limit in a cone metric space implies that u = gu. Now, we shall prove that f u = gu. We have d (f x2n , gu) h · d (u, f x2n ) ≪ h ·

d (f u, u) = d (fu, gu) h · u, where u ∈ {d (u, u) , d (u, f u) , d (u, gu) , d (u, gu) , d (u, fu)} = {0, d (u, fu)} . Hence, we get the following cases: d (fu, u) h · θ = θ and d (f u, u) h · d (f u, u) . According to p3 ) it follows that fu = u, that is., u is a common fixed point of f and g. Uniqueness of the common fixed point follows easily from (3.1). Now we get that S n u = T n u = u, i.e., u is a unique common fixed point of S and T. This completes the proof of Theorem 3.1.
We now list some corollaries of Theorem 3.1. Corollary 3.3. Let us remark that in Theorem 3.1., setting E = R, P = [0, ∞), d (x, y) = |x − y| , x, y ∈ R (that is . = |.|), we get that for h ∈ [0, 12 ) a pair of contractive type mappings defined in [26] in the frame of metric spaces, have a unique fixed point, i.e., the set F = F (S) ∩ F (T ) is a singleton. Corollary 3.4. The hypothesis that F = F (S) ∩ F (T ) = ∅ is superfluous in Theorem 2.9 and Corollary 2.10 (see Theorem 2.2 and Corollary 2.2 in [26]). Corollary 3.5. If h ∈ [0, 12 ) then the mappings S and T from Definition 2.6. are h quasi-Lipschitziang with K = K ′ = 1−h ∈ [0, 1).

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9 Proof. According to Theorem 3.1. we have that F = F (S) ∩ F (T ) = {u} . We shall h h prove that d (S n x, u) 1−h d (x, u) and d (T n x, u) 1−h d (x, u) for all x ∈ X. For this we have d (S n x, u) = d (S n x, T n u) h · u, where u

∈ {d (x, u) , d (x, S n x) , d (u, T n u) , d (x, T n u) , d (u, S n x)} = {d (x, u) , d (x, S n x) , θ, d (x, u) , d (u, S n x)} = {θ, d (x, u) , d (u, S n x) , d (x, S n x)} .

Hence, we get the following four cases: h d (x, u) . 10 : If d (S n x, u) h · θ = θ, then d (S n x, u) = θ 1−h h 0 n 2 : If d (S x, u) h · d (x, u) 1−h · d (x, u) . h 30 : If d (S n x, u) h · d (u, S n x) , then d (S n x, u) = θ 1−h d (x, u) . 40 : If d (S n x, u) h · d (x, S n x) , then d (S n x, u) h · d (x, u) + h · d (u, S n x) from h which it follows that d (S n x, u) 1−h d (x, u) . In all cases we obtain that d (S n x, u) h h n 1−h d (x, u) . Similarly, we also obtain d (T x, u) 1−h d (x, u) , that is., we have that S h and T are two uniformljy quasi-Lipschitzian mappings (with L = L′ = 1−h > 0).
Theorem 3.6. Let K be a nonempty closed convex subset of a complete convex cone metric space X with a normal solid cone P, f, g : K → K a pair of contractive type mappings with λ ∈ [0, 12 ). For any x1 ∈ K, {xn } is the Ishikawa type iteration process with errors defined by (2.1), where {un } , {vn } ⊂ K satisfy (2.2) and {an } , {bn } , {cn } , {a′n } , {b′n } , {c′n } are six sequences in [0, 1] satisfying an + bn + cn = a′n + b′n + c′n = 1 and

∞ 

(an + bn ) < ∞.

n=0

Then, {xn } converges to a fixed point u of f and g if and only if lim inf d (xn , u) = 0. n→∞

Proof. The proof follows from ([21], Theorem 3.1.) and the previous theorem (see also [27]).
The next example (where the idea is taken from [4]) shows that the condition (3.1) alone is not sufficient to obtain the conclusion of Theorem 3.1. We shall stay in the setting of metric spaces-it would be easy to adapt it to the setting of cone metric spaces. Example 3.6. Let X = {a, b, u, v} , where a = (0, 0, 0) , b = (4, 0, 0) , u = (2, 2, 0) , v = (2, −2, 1) , and let d be the Euclidean metric in R3 . Consider the mappings f and g, defined by fa = u, f b = v, f u = v, f v = u and ga = b, gb = a, gu = b, gv = a. By a careful computation it is easy to obtain that d (fx, gy) ≤

3 max {d (x, y) , d (x, f x) , d (y, gy) , d (x, gy) , d (y, f x)} , 4

for all x, y ∈ X. However, f and g have not a common fixed point. Remark 3.7. Also, hypothesis that F = F (S) ∩ F (T ) = ∅ is superfluous in Theorem 2.1, Corollary 2.3. and Corollary 2.4. in [28].

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10

R  [1] M. Abbas, G. Jungck, Common fixed point results for noncommuting mappings without continuity in cone metric spaces, J. Math. Anal. Appl. 341 (2008) 416-420. [2] V. Berinde, Common fixed points of noncommuting almost contractions in cone metric spaces, Math. Commun., 15 (1) (2010) 229-241. ´ c, A generalization on Banach principle, Proc. Amer. Math. Soc. 45 (1974) [3] Lj. B. Ciri´ 727-730. ´ c, Fixed point theory contraction mapping principle, Faculty of Mechanical [4] Lj. B. Ciri´ Engineering, Beograd, 2003. [5] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, 1985. [6] H. Fukhar-ud-Din, S. H. Khan, Convergence of iterates with errors of asymptotically quasi-nonexpansive mappings and applications, J. Math. Appl. 328 (2007) 821-829. [7] L. G. Huang, X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings. J. Math. Anal. Appl. 332 (2) (2007) 1468-1476. [8] Jae Ug Jeong, Soo Hwan Kim, Weak and strong convergence of the Ishikawa iteration process with errors for two asymptotically nonexpansive mappings, Applied Mathematics and Computation 181 (2006) 1394-1401. [9] S. Jankovi´c, Z. Kadelburg, S. Radenovi´c and B. E. Rhoades, Assad-Fixed-Type Point Theorems for a Pair of Nonself Mappings on Cone Metric Spaces, Fixed Point Theory and Applications, Volume 2009, Article ID 761086, 16 pages, doi:10.1155/2009/761086 [10] D. Ili´c, V. Rakoˇcevi´c, Common fixed points for maps on cone metric space, J. Math. Anal. Appl. 341 (2008) 876-882. [11] D. Ili´c, V. Rakoˇcevi´c, Quasi-contraction on cone metric space, Appl. Math. Lett. 22 (2009) 728-731. [12] Slobodanka Jankovi´c, Zorana Golubovi´c, Stojan Radenovi´c, Compatible and weakly compatible mappings in cone metric spaces, Mathematical and Computer Modeling (2010), doi: 10.1016/j.mcm.2010.06.043. [13] L. V. Kantorovich, The method of successive approximations for functional equations, Acta Math., 71 (1939) 63-77. [14] L. V. Kantorovich, The majorant principle and Newton’s method, Dokl. Akad. Nauk SSSR (N.S.), 76 (1951) 17-20. [15] L. V. Kantorovich, On some further applications of the Newton approximation method, Vestnik Leningrad Univ. Ser. mat. Meh. Astr., 12 (7) (1957), 68-103. [16] M. A. Krasnoseljski and P. P. Zabreiko, Geometrical Methods in Nonlinear Analysis, Springer, 1984. [17] Erdal Karapinar, Duran Turkoglu, Best approximations theorem for a couple in cone Banach space, Fixed Point Theory and Applications, 2010, in press.

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11 [18] M. G. Krein and M. A. Rutman, Linear operators leaving invariant a cone in a Banach spaces, Uspehi Math. Nauk (N.S), 3 (1) (1948) 3-95. [19] S. Radenovi´c, Common fixed points under contractive conditions in cone metric spaces, Computers and Mathematics with Applications 58 (2009) 1273-1278. [20] S. Radenovi´c, B. E. Rhoades, Fixed point theorem for two non-self mappings in cone metric spaces, Computers and Mathematics with Applications, 57 (2009) 1701-1707. [21] S. Radenovi´c, Convergence of iterates with errors of uniformly quasi-Lipschitzian mappings in cone metric spaces, submitted. [22] Ayse Sonmez, Huseyin Cakalli, Cone normed spaces and weighted means, Mathematical and Computer Modeling (2010), doi:10.1016/j.mcm.2010.06.032 [23] W. Takahashi, A convexity in metric space and nonexpansive mappings, Kodai. Math. Sem. Rep. 22 (1970) 142-149. [24] Thabet Abdeljawad, Duran Turkoglu, Muhib Abuloha, Some Theorems and Examples of Cone Banach Spaces, Journal of Computational Analysis and Applications, 12 (4) (2010) 739-753. [25] J. Vandergraft, Newton method for convex operators in partially ordered spaces, SIAM J. Numer. Anal. 4 (1967) 406-432. [26] C. Wang, Jinghao Zhu, Boško Damjanovi´c, Liang-gen Hu, Approximating fixed points of a pair of contractive type mappings in generalized convex metric spaces, Appl. Math. Comput., 215 (2009) 1522-1525 [27] C. Wang, Li-wei Liu, Convergence theorems for fixed points of uniformly quasiLipschitzian mappings in convex metric spaces, Nonlinear Analysis 70 (2009) 20672071. [28] C. Wang, Jin Li, and Daoli Zhu, Convergence Theorems for the Unique Common Fixed Point of a Pair of Asymptotically Nonexpansive Mappings in Generalized Convex Metric Space, Fixed Point Theory and Applications, Volume 2010, Article ID 281890, 6 pages, doi: 10.1155/2010/281890. [29] You-Xian Tian, Convergence of an Ishikawa Type Iterative Scheme for Asymptotically Quasi-Nonexpansive Mappings, Computers and Mathematics with Applications 49 (2005) 1905-1912. [30] P. P. Zabreiko, K-metric and K-normed spaces: survey, Collect. Math 48 (1997) 852-859.

JOURNAL 254 OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.2, 254-263, 2011, COPYRIGHT 2011 EUDOXUS PRESS, LLC

Optimal property of the shape of aeolian blade profile using cubic splines C˘alin Dub˘au Email: calin [email protected] University of Oradea Faculty of Environmental Protection 26, Gen. Magheru Street Oradea, Romania

Abstract A new method to obtain natural cubic splines generated by initial conditions is constructed. An application to aeolian blade profile is proposed.

Key words and phrases: cubic splines generated by initial conditions, optimal property, aeolian blade profile. AMS 2000 Subject Classification: 65D05.

1

Introduction

Within aero-electrical aggregates the wine turbine is the component that ensures the conversion of kinetic energy of wind into mechanical energy useable to turbine shaft, through the interaction between air current and moving blade. Wind turbine is composed mainly of an rotator fixed on a support shaft, comprising a hub and a moving blade consisting of one or more blades. Active body of aeolian turbines which made the quantity of converted energy is the blade. The achieving of aerodynamic performances, kinematics and energy curves of the aeolian turbines depend on the choice of a certain geometry. Wind energy conversion is achieved by the interaction between air currents and solid surface of the blade. For design of the blade profile are used optimized shape (aerodynamic profile) selected and positioned such that the obtained performances for certain conditions, proper to the location, to be optimal. Interaction moment between moving blade and fluid 1

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255

current is deduced from lifted aerodynamic forces and resistance. Calculation of the blade profile was realized by determining the geometric contour of the aerodynamic profile deduced by an analytical method, combining two mathematical functions, a framework function and a gauge function, respectively. Presented model was synthesized by a section calculation which set the final shape of the blade, using a computer algorithm, taking into account the behavior of aerodynamic profile which was placed in a stream of air. The aerodynamic profile behavior depends mainly on profile position compared to the air current speed, through the angle of incidence value. It is known that cubic splines has applications in various technical domains, due to its smooth interpolation and uniform approximation properties. Usually, for a partition ∆ of an interval [a, b], ∆ : a = x0 < x1 < . . . < xn−1 < xn = b a cubic spline is a function s : [a, b] → R with the following properties: i) s is two times differentiable with continuous second derivative on [a, b]; ii) the restriction si of S to an arbitrary interval [xi−1 , xi ], i = 1, n, is third order polynomial function. For giving data yi , i = 0, n, a cubic spline of interpolation of the points (xi , yi ), i = 0, n has the following supplementary property: s(xi ) = yi , i = 0, n. If there exist a function f : [a, b] → R such that f (xi ) = yi , i = 0, n we say that s interpolates f on the knots xi , i = 0, n. Usually, a cubic spline of interpolation is obtained integrating the differential equation, (1.1)

′′

si (x) = Mi−1 +

Mi − Mi−1 · (x − xi−1 ), x − xi−1

x ∈ [xi−1 , xi ]

on each interval [xi−1 , xi ], i = 1, n. Here we have used the classical notations: s(xi ) = yi ,

s′′ (xi ) = mi ,

s(xi ) = Mi ,

i = 0, n.

To obtain unique solution, two conditions must be imposed. These can be two point boundary conditions, or initial value conditions. Frequently, in the literature, two point boundary conditions are considered si (xi−1 ) = yi−1 , si (xi ) = yi 2

256

DUBAU: USING CUBIC SPLINES

and leads to the well-known cubic spline. With the supplementary conditions M0 = Mn = 0 natural cubic splines it obtains (see [5], [9]). In that follows, we impose the initial value conditions si (xi−1 ) = yi−1 ′

si (xi−1 ) = mi−1 and integrating in (1.1) we get the cubic spline of interpolation (introduced by C. Iancu in [7]) and used to data fitting in [8]): (1.2) Mi − Mi−1 Mi−1 si (x) = ·(x−xi−1 )3 + ·(x−xi−1 )2 +mi−1 ·(x−xi−1 )+yi−1 , 6hi 2 x ∈ [xi−1 , xi ], i = 1, n. Since S has continuous second derivative on [a, b], we impose the conditions: si (xi ) = yi , s′ (xi ) = mi , i = 1, n which lead to the relations: (1.3)

Mi + 2Mi−1 =

6 (yi − yi−1 − mi−1 · hi ) h2i

2 (mi − mi−1 ), i = 1, n hi where hi = xi − xi−1 , i = 1, n. From (1.3) we obtain (see [7]) the recurrent relations: Mi + Mi−1 =

(1.4)

Mi =

6 6mi−1 − 2Mi−1 (yi − yi−1 ) − hi h2i

3 Mi−1 hi (yi − yi−1 ) − 2mi−1 − , i = 1, n. hi 2 The cubic spline having the restrictions in (1.2) can be named cubic splines generated by initial conditions. The recurrent relations (1.4) permit to obtain the values mi , Mi , i = 1, n, starting from yi , i = 0, n, m0 and M0 . In this sense, in [7] was obtained: mi =

Proposition 1.1 (lemma 2.1 in [7])): For given y0 , y1 , . . . , yn , m0 , M0 , there exists an unique cubic spline of interpolation generated by initial conditions which satisfies: s(xi ) = yi , i = 0, n s′ (x0 ) = m0 s′′ (x0 ) = M0 . 3

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257

Remark 1.1 From the above Proposition follows that for given values y0 , y1 , . . . , yn obtained by measurements, any choice of the values m0 and M0 determines the properties of the corresponding cubic spline (so, these two values remains free). For instance, in [1] these values are established such that the cubic spline to realize a suboptimal fitting of data affected by small and unknown errors. In [2], the values m0 and M0 are obtained such that for the corresponding cubic spline we have minimal quadratic oscillation in average (this notion was introduced in [4]). In [3], m0 and M0 are established according to the values of the solution and of its first derivative on the initial point, in the aim to derive a combined numerical method for second order ODE’S with retarded argument. In that follows, we obtain m0 and M0 such that the corresponding cubic spline to be natural.

2

Natural cubic splines

The notion of natural cubic spline was introduced by Holladay in 1957 starting to the requirement to construct a smooth curve interpolating given points in plane and having minimal curvature. Consider the notation C 2 [a, b] = {f : [a, b] → R/f has continuous second derivative on [a,b]}. Proposition 2.1 If f ∈ C 2 [a, b] and s ∈ C 2 [a, b] is cubic spline of interpolation such that s(xi ) = f (xi ), ∀i = 0, n and s′′ (a) = s′′ (b) = 0, then ∫ b ∫ b ∫ b ′′ 2 ′′ 2 (2.1) [f (x)] dx = [s (x)] dx + [f ′′ (x) − s′′ (x)]2 dx. a

a

a

Proof. We see [f ′′ (x)]2 = (s′′ (x) + [f ′′ (x) − s′′ (x)])2 ,

∀x ∈ [a, b]

and ∫

b

s′′ (x) · [f ′′ (x) − s′′ (x)]dx =

a

=

n ∫ ∑

′′

si (x) · [f ′ (x) − s′ (x)]′ dx =

xi−1

i=1

n [ ∑

xi

′′

si (xi ) · (f ′ (xi ) − s′ (xi )) − si (xi−1 ) · (f ′ (xi−1 ) − s′ (xi−1 ))−

i=1

−

n ∫ ∑ i=1

xi

′ ′′ ′ ′ s′′′ i (x) · [f (x) − s(x)] dx = s (xn ) · [f (xn ) − s (xn )] ] −

xi−1

4

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DUBAU: USING CUBIC SPLINES

−s′′ (x0 ) · [f ′ (x0 ) − s′ (x0 )] after integrating by parts. Since s′′ (x0 ) = s′′ (xn ) = 0, we get (2.1). This above orthogonality property leads to the following result.  Corollary 2.1 If s ∈ C 2 [a, b] is cubic spline interpolating the values yi = f (xi ), i = 0, n and s′′ (a) = s′′ (b) = 0, then considering the functional J : g ∈ C 2 [a, b]/g(xi ) = yi , ∀ i = 0, n → R, J(g) =

∫

b

[g ′′ (x)]2 dx,

a

we have J(s) = min{J(g) : g ∈ C 2 [a, b], g(xi ) = yi , ∀ i = 0, n}. Proof. J(s) =

∫

b

[s′′ (x)]2 dx =

a

≤

∫

b

∫

b

[g ′′ (x)]2 dx −

a

∫

b

[g ′′ (x) − s′′ (x)]2 dx ≤

a

[g ′′ (x)]2 dx = J(g), ∀g ∈ C 2 [a, b]

a

with g(xi ) = yi , ∀ i = 0, n. This minimal property represent after Halladay, the minimal curvature of s. Therefore, if the cubic spline s has this property then it is named natural cubic spline. 

3

The natural cubic spline generated by initial conditions

In the following, we present a new method to obtain the above minimal curvature property for the cubic spline generated by initial conditions (1.2). After Corollary 2.1, this property follows imposing the conditions s′′ (a) = s′′ (b) = 0, that is M0 = Mn = 0. For M0 = n, from relations (1.4) we determine the value of m0 such that Mn = 0.

5

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259

Theorem 3.1 Consider M0 = n and m0 = − dcnn , where recurrently, an = −2an−1 −

hn · cn−1 , 2

cn = −2cn−1 −

6 · an−1 , hn

3 hn · (yn − yn−1 ) − 2 · bn−1 − · dn−1 , hn 2 6 6 dn = 2 · (yn − yn−1 ) − 2 · dn−1 − · bn−1 , n ≥ 2, hn hn bn =

starting from a1 = −2, c1 = − h61 , b1 = h31 · (y1 − y0 ), d1 = h62 · (y1 − y0 ). 1 Then the corresponding spline (as in (1.2)) generated by initial conditions is natural cubic spline. Proof. Using the recurrent relations (1.4), with M0 = 0 and with notation z = m0 , we obtain m1 = M1 =

3 · (y1 − y0 ) − 2z = a1 z + b1 h1

6 6 · (y1 − y0 ) − · z = c1 z + d1 2 h1 h1

where a1 = −2, c1 = − h61 , b1 = h31 · (y1 − y0 ), d1 = Applying again the relations (1.4) we get,

6 h21

· (y1 − y0 ).

m2 = a2 z + b2 M2 = c2 z + d2 with

3h2 , c2 = 12 · a2 = 4 + h1

(

)

,

b2 =

3 6 3h2 · (y2 − y1 ) − · (y1 − y0 ) − 2 · (y1 − y0 ), h2 h1 h1

d2 =

6 18 12 · (y2 − y1 ) − · (y1 − y0 ) − 2 · (y1 − y0 ). h2 h1 h1 h1

By induction it proves that (3.1)

1 1 + h1 h2

mn = an z + bn 6

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Mn = cn z + dn , ∀n ∈ N∗ (3.2)

an = −2an−1 −

hn · cn−1 , 2

6 · an−1 , hn 3 hn bn = · (yn − yn−1 ) − 2 · bn−1 − · dn−1 , hn 2 6 6 dn = 2 · (yn − yn−1 ) − 2 · dn−1 − · bn−1 , n ≥ 2. hn hn cn = −2cn−1 −

For z = − dcnn , in (3.1) we obtain Mn = 0 and together M0 = 0, the cubic spline as in (3.1) is natural cubic spline. So, the interpolation conditions s(xi ) = yi , i = 0, n s′ (x0 ) = −

dn cn

s′′ (x0 ) = 0 uniquely determine the natural cubic spline, s, generated by initial conditions which has the optimal property ∫ b ∫ b ′′ 2 [s (x)] dx = min{ [g ′′ (x)]2 dx : g ∈ C 2 [a, b], g(xi ) = yi , ∀ i = 0, n}, a

a

that is minimal curvature. Now, we prove that ck ̸= 0, ∀ k = 1, n. F From (3.2) it obtains ) ( ( ) ( ) 2 h2n an an−1 (3.3) = (−1) · · 6 cn cn−1 5 hn

∀n ∈ N, n ≥ 2.

Since a1 = −2 < 0, c1 = − h61 < 0, follows 12 12 h2 > 0, c1 = + > 0, h1 h1 h2 h2 h3 h3 a3 = −8 − 6 − 6 − 6 < 0, h1 h1 h2 24 24 24 h2 c3 = − − − − 18 0, cn > 0 for even n. So, an ̸= 0, cn ̸= 0, ∀ n ∈ N∗ .  Remark 3.1 For the natural cubic spline of interpolation of a function s ∈ C 2 [a, b] on the knots xi , i = 0, n, the following error estimation holds: √ |f (x) − s(x)| ≤ ∥f ′′ (x)∥2 · h h √ |f ′ (x) − s′ (x)| ≤ ∥f ′′ (x)∥2 · h, ∀x ∈ [a, b] where h = max{hi : i = 1, n}. Indeed, for φ = f − s, according to f (xi ) = s(xi ), ∀i = 0, n, we have φ(xi ) = 0, ∀i = 0, n. Using the Rolle’s theorem we infer that for any i = 1, n there exist ηi ∈ (xi−1 , xi ) such that φ′ (ηi ) = 0, that is f ′ (ηi ) = s′ (ηi ). For arbitrary x ∈ [a, b] we find j ∈ 1, 2, . . . , n such that x ∈ [xj−1 , xj ]. If ηj < x we have, ∫ x ∫ x ′′ ′′ ′ ′ |[f ′′ (t) − s′′ (t)]|dt ≤ [f (t) − s (t)]dt| ≤ |f (x) − s (x)| = | ηj

ηj

≤

≤

(∫

a

b

(∫

x ηj

)1/2 (∫ [f (t) − s (t)] dt · ′′

′′

2

dt

ηj

)1/2 (∫ [f (t) − s (t)] dt · ′′

x

′′

xj

2

dt xj−1

)1/2

)1/2

≤

≤ ∥f ′′ − s′′ ∥2 ·

√

h.

According to (2.1), ∥f ′′ − s′′ ∥2 ≤ ∥f ′′ ∥2 and thus, √ |f ′ (x) − s′ (x)| ≤ ∥f ′′ ∥2 · h, ∀x ∈ [a, b]. Finally, |f (x) − s(x)| ≤

∫

x

′

′

|[f (t) − s (t)]|dt ≤

xj−1

∫

x

∥f ′′ ∥2 ·

√

h≤

xj−1

√ ≤ ∥f ′′ ∥2 · h h,

∀x ∈ [a, b].

This error estimation holds even for the natural cubic spline generated by initial conditions.

8

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DUBAU: USING CUBIC SPLINES

4

Application

The procedure presented in the proof of Theorem 3.1 was implemented in the following on an exempla from aerodynamics. For the aerodynamic profile of the aeolian blade we consider at an cross section the points xi , i = 0, 10 situated on the axis of the section and the corresponding values on the surface of the blade yi , i = 0, 10. There are: x0 = 0, x1 = 35.5, x2 = 71.00, x3 = 106.50, x4 = 142.00, x5 = 177.50, x6 = 213.00, x7 = 248.50, x8 = 284.00, x9 = 319.50, x10 = 355.00 (for top shape) with the algorithm presented above we obtain the derivatives on these knots: m[0] = 1.502952, M [0] = 0 (for top shape), m[0] = 0.1854874, M [0] = 0 (for middle contour), m[0] = −1.131526, M [0] = 0 (for bottom contour) and the profile in Figure 1.

Figure 1: Aeolian blade profile

5

Conclusions

The main result of this paper in Theorem 3.1 presenting a new method to obtain natural cubic spline generated by initial conditions. This result is applied to the aerodynamic profile of the aeolian blade.

References [1] A. Bica, C. Iancu, On a delay integral equation in biomathematics, J. of Concrete and Applicabile Math. 4, No. 2, 2006, 153-170.

9

DUBAU: USING CUBIC SPLINES

[2] A.M. Bica, V.A. C˘au¸s, I. Fechete, S. Mure¸san, Application of the Cauchy-Buniakovski-Schwarz’s inequality to an optimal property for cubic splines, J. of Comput. Analysis and Appl. 9, No. 1, 2007, 43-53. [3] A.M. Bica, M. Curil˘a, S. Curil˘a, Approximating the solution of second order differential equation with retarded argument, J. of Comput. Analysis and Appl. 12, No. 1-A, 2010, 37-47. [4] A.M. Bica, Iterative numerical methods for operatorial equations, University of Oradea Press, 2006, (in Romanian). [5] C. de Boor, A practical guide splines, Applied Math. Sciences, 27, Springer Verlag, Berlin, 1978. [6] C. Dub˘au, Utilizarea microagregatelor eoliene in componenta unor sisteme complexe, Ph. D. Thesis, Timisoara Politechnic Press, 2007, (in Romanian). [7] C. Iancu, On the cubic spline of interpolation, Seminar on Functional Analysis and Num. Methods, 4, 1981, 52-71. [8] C. Iancu, Data analysis and processing with spline functions, Ph. D. Thesis, Babes-Bolyai University, Cluj-Napoca, 1983, (in Romanian). [9] G. Micula, S. Micula, Handbook of splines, Mathematics and its Applications 462, Kluwer Academic Publishers, Dordrecht, 1999.

10

263

JOURNAL 264 OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.2, 264-271, 2011, COPYRIGHT 2011 EUDOXUS PRESS, LLC

Cauchy problem of the

4(2k ) operator related to the

Diamond operator and the Laplace operator iterated

k

times

Chalermpon Bunpog

∗

Department of Mathematics, Chiang Mai University, Chiang Mai, 50200 Thailand. Centre of Excellence in Mathematics, CHE, Sri Ayutthaya Road, Bangkok 10400, Thailand.

Abstract Given the Laplace operator 4 is defined by 4=

∂2 ∂2 ∂2 + + . . . + , ∂x2n ∂x21 ∂x22

(0.1)

the Ultra-hyperbolic operator iterated k times 2k is defined by à 2k =

∂2 ∂2 ∂2 ∂2 ∂2 ∂2 + + . . . + − − . . . − − ∂x2p ∂x2p+1 ∂x2p+2 ∂x21 ∂x22 ∂x2p+q

!k ,

(0.2)

where p + q = n is the dimension of the Euclidean space Rn . In this paper, we study Cauchy problem and fundamental solution of the 4(2k ) operator by using Green’s identity, In particular, 4(2k ) reduces to the Diamond operator if k = 1. Moreover, for q = 0 the ultra-hyperbolic operator 2 reduces to 4, and 4(2k−1 ) reduces to the Laplace operator 4k iterated k times.

Keywords: Cauchy problem, Fundamental solution, Distribution.

1

Introduction

It is well known that the Green’s identity of the Lapace operator 4 is given by the formula, ¶ Z Z Z µ ∂v ∂u −u v4udx = u4vdx + v dS (1.1) ∂n ∂n Ω Ω ∂Ω where u, v ∈ C 2 (Ω) and normal to ∂Ω.

∂ ∂n

indicates differentiation in the direction of the exterior

∗

Corresponding address: Department of Mathematics, Chiang Mai University, Chiang Mai, 50200 Thailand. E-mail address: [email protected]

1

BUNPOG: CAUCHY PROBLEM

265

Let x, ξ ∈ Rn . Denote that

K(x, ξ) = s where kx − ξk = It is clear that

 kx − ξk2−n     (2 − n)ωn , for n > 2;     lnkx − ξk , 2π

(1.2)

for n = 2,

n X (xi − ξi )2 , ωn denotes the surface area of the unit sphere in Rn . i=1

4K(x, ξ) = 0, r > 0

(1.3)

4K(x, ξ) = δξ .

(1.4)

Hence K(x, ξ) is the fundamental solution of the Laplace operator. In ([1], p.79) F. John studied the Cauchy problem of the Laplace operator by using Green’s identity. He obtain the solution of such problem is, µ ¶ ∂u(x) ∂K(x, ξ) u(ξ) = K(x, ξ)4udx − K(x, ξ) − u(x) dSx ∂nx ∂nx Ω ∂Ω Z

Z

(1.5)

for ξ ∈ Ω, the subscript x in Sx and ∂n∂ x indicates the variable of integration and differentiation respectively. In [2] C. Bunpog and A. Kananthai study the Cauchy problem of the Diamond operator, denote by ♦ and defined by µ ♦=

∂2 ∂2 ∂2 + + . . . + ∂x21 ∂x22 ∂x2p

µ

¶2 −

∂2 ∂2 ∂2 + + . . . + ∂x2p+1 ∂x2p+2 ∂x2p+q

¶2 ,

they obtain the solution of such problem is, ¶ Z Z µ ∂u(x) ∂(2M (x, ξ)) u(ξ) = M (x, ξ)♦udx − 2M (x, ξ) − u(x) ∂n ∂n Ω ∂Ω µ ¶ ∂(4u) ∂M (x, ξ) − M (x, ξ) − 4u dSx ∂n∗ ∂n∗

(1.6)

(1.7)

where M (x, ξ) = M1 (x, ξ) is defined by (2.8) with k = 1. In this paper we study the solution of the Cauchy problem of the operator 4(2k ) by using Green’s identity, we obtain µ ¶ Z ∂2M1 ∂u k −u u(ξ) = Mk 4(2 )udx − 2M1 (x, ξ) dSx ∂n ∂n Ω ¶ µ k Z X ∂Mi ∂(4(2i−1 u) i−1 − 4(2 u) dSx , k ∈ N (1.8) − Mi ∂n ∂n ∗ ∗ ∂Ω i=1 is the solution of such problem, where Mk is defined by (2.8). Moreover, we obtain that Mk (x, ξ) is the fundamental solution of the operator 4(2k ). 2

266

BUNPOG: CAUCHY PROBLEM

2

Preliminaries

Definition 2.1 Let C0∞ (Ω) be the space of all infinitely differentiable functions on Ω with compact support. A distribution is a linear functional < f, φ > defined for all D = C0∞ (Ω) which is continuous on D in the following sense: Let the φr be a sequence in D. Then limk→∞ < f, φk (x) >= 0, provided (1) all φk vanish outside the same compact subset of Ω, and m (2) limk→∞ Dα φk (x) = 0 uniformly in x for each α, where Dα = ∂xα1∂...∂xαn , |α| = n 1 α1 + α2 + αn = m. Note that each continuous(or even locally integrable) function f (x) generates a distribution Z < f, φ >= f (x)φ(x)dx. More generally, we write any distribution f symbolically as Z < f, φ >= f (x)φ(x)dx. Ω

Definition 2.2 The Dirac delta distribution with singularity ξ, denoted by δξ ,which is defined by < δξ , φ >= φξ . It is given symbolically by

Z δξ (x)φ(x)dx = φ(ξ). Ω

Definition 2.3 Let L be an operator and u be a distribution defined for all φ ∈ C0∞ (Ω). We call u a fundamental solution of the operator L if it satisfy the equation Lu = δξ . Definition 2.4 Let x = (x1 , x2 , . . . , xn ), ξ = (ξ1 , ξ2 , . . . , ξn ) be be a point of Rn , the following function is defined by E(x, ξ) = where ωn =

kx − ξk2−n , (2 − n)ωn

n>2

(2.1)

(2π n/2 ) is a surface area of the unit sphere in Rn . Γ(n/2)

Definition 2.5 Let x = (x1 , x2 , . . . , xn ), ξ = (ξ1 , ξ2 , . . . , ξn ) be a point of Rn , put y = x − ξ = (x1 − ξ1 , x2 − ξ2 , . . . , xn − ξn ) and write v u p p+q X uX 2 t V = yi − yj2 , p + q = n. i=1

J=p+1

Γ+ = {y ∈ Rn : y1 > 0, V > 0} designates the interior of the forward cone and denotes Γ+ by its closure and the following function is defined by  V α−n   , for y ∈ Γ+ ;  Kn (α) Rα (y) = (2.2)    0, for y 6∈ Γ+ , 3

BUNPOG: CAUCHY PROBLEM

267

where the constant Kn (α)is defined by Kn (α) =

π

n−1 2

Γ( α+2−n )Γ( 1−α )Γ(α) 2 2 . α+2−p p−α Γ( 2 )Γ( 2 )

Definition 2.6 Let Ω be a bounded open subset of Rn , and ∂Ω is the boundary of Ω, n denote the exterior unit normal vector ζ = (ζ1 , ζ2 , . . . , ζn ) of ∂Ω. We define the vector n∗ in Rn by n∗ · ei = n · ei , i = 1, 2, . . . , p and n∗ · ej = −n · ej , j = p + 1, p + 2, . . . , p + q th

where ei = (0, 0, . . . , 1i , . . . , 0). We note that n∗ known as the transversal to ∂Ω. Lemma 2.1 (Green’s identity of the Ultra-hyperbolic operator). Let Ω is an bounded open subset of Rn and u, v ∈ C 2 (Ω). Then Green’s identity of the Ultra-hyperbolic operator is ¶ Z Z µ Z ∂u ∂v v u2vdx + v2udx = −u dSx (2.3) ∂n∗ ∂n∗ ∂Ω Ω Ω where 2 is the ultra-hyperbolic operator defined by (0.2) for k = 1, n∗ is the transversal to ∂Ω and ∂n∂ ∗ denotes derivative in the transversal direction (see [3], p.41). Proof. See ([2], p.32). Lemma 2.2 Let Ω is an open subset of Rn and u ∈ C 4 (Ω), v ∈ C 2 (Ω). Then ¶ Z Z Z µ ∂(4u) ∂v v4(2u)dx = 2v4udx + v − 4u dSx ∂n∗ ∂n∗ Ω Ω ∂Ω where

∂ ∂n∗

(2.4)

denotes derivative in the transversal direction.

Proof. We replace u in (2.3) by 4u, and use the property that 24 = 42. Hence (2.4) is obtained. Lemma 2.3 Given the equation 2k Lk (x, ξ) = δξ ,

(2.5)

for x, ξ ∈ Rn , where 2k , k ∈ N is defined by (0.2). Then Lk (x, ξ) = R2k (y), where R2k (y) is defined by (2.2) with α = 2k, k ∈ N. Proof.See ([4], p.11). Lemma 2.4 Let Rα (y) and Rβ (y) be defined by (2.2), then Rα (y) ∗ Rβ (y) = Rα+β (y) where α and β are positive even integers with α + β = 2k. 4

(2.6)

268

BUNPOG: CAUCHY PROBLEM

Proof. See([5], p.103 ). Lemma 2.5 Given the equation 2k Mk (x, ξ) = E(x, ξ),

k∈N

(2.7)

where x, ξ ∈ Rn , E(x, ξ) is defined by (2.1) and 2k is defined by (0.2). Then Mk (x, ξ) = R2k (y) ∗ E(x, ξ),

k∈N

(2.8)

and 2Mk (x, ξ) = Mk−1 (x, ξ),

k = 2, 3, . . .

(2.9)

where R2k (y) is defined by (2.2) with α = 2k. Proof. First, to prove (2.8). Convolving both sides of (2.7) by the function R2k (y). We have R2k (y) ∗ 2k Mk (x, ξ) = R2k (y) ∗ E(x, ξ), by the property of convolution, this equation becomes 2k R2k (y) ∗ Mk (x, ξ) = R2k (y) ∗ E(x, ξ), by Lemma (2.3) we obtain δξ ∗ Mk (x, ξ) = R2k (y) ∗ E(x, ξ), then Mk (x, ξ) = R2k (y) ∗ E(x, ξ). Next, to prove (2.9). From (2.8), 2Mk (x, ξ) = = = =

2R2k (y) ∗ E(x, ξ), 2R2 (y) ∗ R2(k−1) (y) ∗ E(x, ξ), δξ ∗ Mk−1 (x, ξ), Mk−1 (x, ξ).

(by Lemma2.4)

That completes the proof.

3

Main Results

Theorem 3.1 Let Ω be an bounded open subset of Rn , x ∈ Ω u ∈ C 2k+2 (Ω) and Mk (x, ξ) be a function which satisfy (2.8). Then for ξ ∈ Ω, ¶ Z Z µ ∂2M1 ∂u k −u u(ξ) = Mk 4(2 )udx − 2M1 (x, ξ) dSx ∂n ∂n Ω ∂Ω ¶ µ k Z X ∂Mi ∂(4(2i−1 u) i−1 − 4(2 u) dSx , k ∈ N (3.1) − Mi ∂n ∂n ∗ ∗ ∂Ω i=1 is the solution of the Cauchy problem of the operator 4(2k ). 5

BUNPOG: CAUCHY PROBLEM

269

Proof. Since M1 is defined by (2.8) satisfy (2.7) with k = 1, then we can replace K(x, ξ) in (1.5) by 2M1 , we obtain ¶ Z Z µ ∂2M1 ∂u u(ξ) = 2M1 4udx − 2M1 −u dSx , ∂n ∂n Ω ∂Ω or ¶ Z Z µ ∂2M1 ∂u 2M1 4udx = u(ξ) + 2M1 −u dSx . (3.2) ∂n ∂n Ω ∂Ω From (2.4), replace v by M1 , thus ¶ Z Z Z µ ∂M1 ∂(4u) M1 4(2u)dx = 2M1 4udx + M1 − 4u dSx , ∂n∗ ∂n∗ Ω Ω ∂Ω from this equation and (3.2), we have ¶ Z Z µ ∂u ∂2M1 M1 4(2u)dx = u(ξ) + 2M1 dSx −u ∂n ∂n Ω ∂Ω ¶ Z µ ∂(4u) ∂M1 + M1 − 4u dSx . ∂n∗ ∂n∗ ∂Ω

(3.3)

From (2.4) again, replace u and v by 2u and M2 respectively, that is ¶ Z Z Z µ ∂M2 ∂4(2u) 2 M2 4(2 u)dx = 2M2 4(2u)dx + − 4(2u) M2 dSx , (3.4) ∂n∗ ∂n∗ Ω Ω ∂Ω where M2 defined by (2.8) with k = 2. From (2.9) with k = 2 we have 2M2 = M1 . Thus (3.4) becomes ¶ Z Z Z µ ∂M2 ∂4(2u) 2 M2 4(2 u)dx = M1 4(2u)dx + − 4(2u) M2 dSx , ∂n∗ ∂n∗ Ω Ω ∂Ω from this equation and (3.3), we have Z Z 2 M2 4(2 u)dx = u(ξ) + Ω

∂Ω

µ

2 Z X

+

i=1

µ

∂Ω

∂2M1 ∂u −u 2M1 ∂n ∂n

¶ dSx

∂4(2i−1 u) ∂Mi Mi − 4(2i−1 u) ∂n∗ ∂n∗

¶ dSx .

By method of induction, we obtain ¶ Z Z µ ∂u ∂2M1 k Mk 4(2 u)dx = u(ξ) + 2M1 −u dSx ∂n ∂n Ω ∂Ω ¶ µ k Z X ∂Mi ∂4(2i−1 u) i−1 − 4(2 u) dSx , + Mi ∂n ∂n ∗ ∗ ∂Ω i=1 Thus

µ ¶ ∂2M1 ∂u −u u(ξ) = Mk 4(2 )udx − 2M1 dSx ∂n ∂n Ω ∂Ω µ ¶ k Z X ∂(4(2i−1 u) ∂Mi i−1 Mi − 4(2 u) dSx , k ∈ N, − ∂n∗ ∂n∗ i=1 ∂Ω Z

Z

k

6

k ∈ N.

270

BUNPOG: CAUCHY PROBLEM

as required. In particular, 4(2k ) reduces to the Diamond operator if k = 1. Thus (3.1) is reduces to (1.7). Moreover, for q = 0 the ultra-hyperbolic operator 2 reduces to 4, and 4(2k−1 ) reduces to the Laplace operator iterated k times 4k . Therefore (3.1) reduces to ¶ Z Z µ ∂u ∂4M1 k u(ξ) = −u dSx Mk 4 udx − 4M1 ∂n ∂n Ω ∂Ω µ ¶ k Z X ∂(4i u) ∂Mi i − Mi − (4 u) dSx , k ∈ N, ∂n ∂n ∗ ∗ ∂Ω i=1 That completes the proof. Theorem 3.2 Let Ω is a bounded open subset of Rn , u ∈ C 2k+2 (Ω). Then there is a fundamental solution Gk (x, ξ) of the operator 4(2k ) such that ! Z Ã Z k X ∂(2G (x, ξ)) ∂G (x, ξ) k k u(ξ) = Gk (x, ξ)4(2k )udx+ u(x) + 4(2i−1 u) dSx ∂n ∂n ∗ ∗ Ω ∂Ω i=1 (3.5) Proof. Define Gk (x, ξ) = Mk (x, ξ) − Vk (x, ξ) for x ∈ Ω, ξ ∈ Ω, x 6= ξ, where Mk (x, ξ) satisfy 4(2k )Mk (x, ξ)) = δξ and Vk (x, ξ) ∈ C 2 (Ω) is a solution of 4(2k )Vk (x, ξ) = 0 for ξ ∈ Ω, such that Mk (x, ξ) = Vk (x, ξ) for x ∈ ∂Ω, then we have Gk (x, ξ) satisfy 4(2k )Gk (x, ξ)) = δξ , Gk (x, ξ) = 0 and 2G1 (x, ξ) = 0 for x ∈ ∂Ω. Thus we can replace Mk (x, ξ) by Gk (x, ξ) in (3.4), we obtain ! Z Z Ã k X ∂(2G (x, ξ)) ∂G (x, ξ) k k u(ξ) = Gk (x, ξ)4(2k )udx+ u(x) + 4(2i−1 u) dSx , ∂n ∂n ∗ ∗ Ω ∂Ω i=1 as required. In the special case, let Ω = B(0, a) = {x, |x| < a} we have that the sphere Ω is the locus of point x for which the ratio of distances r = |x − ξ| and r∗ = |x − ξ ∗ | from certain points and is constant. Here we can choose any point ξ ∈ Ω, then ξ ∗ is the point obtained from ξ by reflection with respect to the sphere ∂Ω. a2 1 1 2−n That is ξ ∗ = |ξ| and K(x, ξ ∗ ) = (2−n)ω r∗2−n are 2 ξ, such that K(x, ξ) = (2−n)ω r n n ∗ the fundamental solutions of Laplace operator with poles ξ and ξ respectively, thus ³ ´ for x ∈ ∂Ω, K(x, ξ ∗ ) =

a |ξ|

2−n

K(x, ξ). Then the function µ

G(x, ξ) = K(x, ξ) − vanishes for x ∈ ∂Ω(See [1], p. 106-107). We define à G∗k (x, ξ) = R2k (y) ∗

|ξ| a

µ

K(x, ξ) −

¶2−n

|ξ| a

K(x, ξ ∗ )

K(x, ξ ∗ ) µ

7

!

¶2−n

= R2k (y) ∗ K(x, ξ) − R2k (y) ∗

(3.6)

|ξ| a

¶2−n K(x, ξ ∗ ),

(3.7)

BUNPOG: CAUCHY PROBLEM

271

then we have G∗k (x, ξ) = 0 for x ∈ ∂Ω, moreover G∗k (x, ξ) satisfy 4(2k )G∗k (x, ξ)) = δξ , G∗k (x, ξ) = 0 and 2G∗1 (x, ξ) = 0 for x ∈ ∂Ω. Hence G∗k (x, ξ) is the fundamental solution of 4(2k ) for Ω = B(0, a) = {x, |x| < a}. Theorem 3.3 Let Ω is an open subset of Rn , φ ∈ C0∞ (Ω) and Mk (x, ξ) is defined by (2.8), then Mk (x, ξ) is the fundamental solution of the 4(2k ) operator. Proof. We replace u(ξ) in (3.1) by φ(ξ) and by properties of φ we obtain Z φ(ξ) = Mk (x, ξ)4(2k )φdx Ω

=< Mk (x, ξ), 4(2k )φ > =< Mk (x, ξ), 2k (4φ) > =< 2k Mk (x, ξ), (4φ) > =< 4(2k )Mk (x, ξ), φ > . Hence 4(2k )Mk (x, ξ) = δξ . That is Mk (x, ξ) is the fundamental solution of the 4(2k ) operator. Acknowledgement. This research is supported by the Centre of Excellence in Mathematics, Thailand.

References [1] F. John, Partial Differential Equations, 4th Edition, Springer-Verlag, New York, 1982. [2] C. Bunpog and A. Kananthai, Green’s Identity, Fundamental Solution and Dirichlet Problem of the Diamand Operator, Far East Journal. . [3] I. Stakgold, Boundary Value Problems of Mathematical Physics, Volume II, The Macmillan Company, New York, 1968. [4] S. E. Trione, On Marcel Riesz’s Ultra-hyperbolic kernel, Studies Appl. Math. 79(1988) 185–191. [5] A. Kananthai, On The distribution related to the ultra-hyperbolic equation, Comput. Appl. Math. 84(1997)101-106.

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JOURNAL 272 OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.2, 272-281, 2011, COPYRIGHT 2011 EUDOXUS PRESS, LLC

Some algorithms for a class of set-valued variational inclusions in Frechet Spaces Chaofeng Shi, Nan-jing Huang Abstract. A new inequality in Frechet spaces is given in this paper, which can be regarded as the Frechet space versions of the well-known polarization identity occurring in Hilbert spaces and some inequalities for norms in Banach spaces. By using this inequality, we give a convergence analysis result of Mann iteration scheme for approximating the fixed point of contractive mapping in a Frechet space. We also suggest and analyze some algorithms for solving a class of set-valued variational inclusions in Frechet spaces. Key Words and Phrases: Frechet spaces, Inequality, Mann iteration schemes, Fixed point, Variational inclusions. 2000 Mathematics Subject Classification: 46B05, 46C05.

1

Introduction

Let H be a Hilbert space with an inner h·, ·i and norm k · k. It is well known that, for any x, y ∈ H, kx + yk2 + kx − yk2 = 2(kxk2 + kyk2 ),

(1.1)

or equivalently the polarization identity kx + yk2 = kxk2 + 2Rehx, yi + kyk2 .

(1.2)

In 1970, Petryshyn [16] generalized (1.1) and (1.2) to Banach spaces and obtained the following inequality: kx + ykp ≤ kxkp + phy, jp i,

∀jp ∈ Jp (x + y),

(1.3)

where Jp : X → X ∗ is the duality mapping and p ≥ 2. Recently, many iterative methods for approximating the fixed points of nonlinear operators and the solutions of the nonlinear operator equations or the variational inequalities (inclusions) have been studied in Hilbert spaces and Banach spaces by using equality (1.2) and inequality (1.3) under certain conditions (see, for example, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 17, 20] and the references therein). Among these methods, Mann and Ishikawa iterative schemes are two important ones, which have been employed by many authors for approximating the solutions of nonlinear problems. Some authors also studied the iterative approximation problems for solutions of some kinds of nonlinear problems by using the Mann and Ishikawa iterative sequences with errors or mixed errors in Hilbert spaces or Banach spaces (see [2]). As an generalization of the Banach space, the Frechet space plays an important role in functional analysis and some related fields. It is an interesting problem to generalize inequality (1.3) from the Banach space to the Frechet space, which can be applied to solve some nonlinear problems in Frechet spaces. In this paper, we establish a new inequality in Frechet spaces, which is a generalization of inequality (1.3). By using this inequality, we give a convergence analysis result of Mann iteration scheme for approximating the fixed point of contractive mapping in the Frechet space, which generalized the corresponding result of Isac and Li [10]. We also suggest and analyze some algorithms for solving a class of set-valued variational inclusions in Frechet spaces, which improved and unified some corresponding results of Chang [1], Ding [5], Huang [8, 9], Shi and Liu [17], and Zeng [20]. 1

SHI, HUANG: SET-VALUED VARIATIONAL INCLUSIONS

2

273

A New Inequality in Frechet Spaces

Let F be a Frechet space with paranorm | · | and topological dual F ∗ . Let h·, ·i be the duality pair between F and F ∗ . Proposition 2.1 F ∗ is a paranorm space. Proof. Let |f | = sup x6=0

|hx, f i| , |x|

∀x ∈ F, f ∈ F ∗ .

(2.1)

Then, for any f, g ∈ F ∗ , |f + g| =

sup x6=0

≤ sup x6=0

|hx, f i + hx, gi| |x| |hx, gi| |hx, f i| + sup |x| |x| x6=0

= |f | + |g|.

(2.2)

If {tn } is a sequence of scalars with tn → t and {fn } ⊂ F ∗ with |fn − f | → 0, then |tn fn − tf | = ≤

sup x6=0

|hx, tn fn i − hx, tf i| |x|

|tn − t| sup x6=0

|hx, fn i| |hx, fn − f i| + |t| sup |x| |x| x6=0

= |tn − t||fn | + |t||fn − f |

(2.3)

and so limn→∞ |tn fn − tf | = 0. From (2.2) and (2.3), it is easy to see that | · | defined by (2.1) is a paranorm. This completes the proof. ∗ Let p ≥ 2 be a positive integer and Jp : F → 2F be a paranormalized duality mapping of F defined by Jp (x) = {x∗ ∈ F ∗ : hx, x∗ i = |x| · |x∗ |, |x∗ | = |x|p−1 , x ∈ F }. Remark 2.1 Similar to the classical Hahn-Banach theorem in Banach spaces, one can easily show that Jp (x) is nonempty. Lemma 2.1 Jp (x) ⊂ ∂ψ(x), where ψ(x) = p−1 |x|p , for all x ∈ F. Proof. If f ∈ Jp (x), then for any y ∈ F , hf, y − xi = hf, yi − hf, xi ≤ |f ||y| − |x||f |.

(2.4)

We first show p|x|p−1 |y| ≤ |y|p + (p − 1)|x|p ,

p = 2, 3, · · · .

(2.5)

For p = 2, we have 2|x||y| ≤ |x|2 + |y|2 and so inequality (2.5) holds. Assume that k|x|k−1 |y| ≤ |y|k + (k − 1)|x|k

(2.6)

holds for p = k. Then for p = k + 1, (|x| − |y|)(|x|k − |y|k ) ≥ 0, which implies that

(2.7)

|y|k+1 + |x|k+1 ≥ |x|k |y| + |x||y|k .

It follows from (2.6) and (2.7) that (k + 1)|x|k |y| = |x|k |y| + k|x|k |y| ≤ |x|k |y| + |x||y|k + (k − 1)|x|k+1 ≤ |y|k+1 + k|x|k+1 . 2

(2.8)

274

SHI, HUANG: SET-VALUED VARIATIONAL INCLUSIONS

Thus, the inequality (2.5) holds for all p = 2, 3, · · · . Next we prove that f ∈ ∂ψ(x). Since |f | = |x|p−1 , it follows from (2.5) that p|f ||y| − p|x|p ≤ |y|p − |x|p , i.e., |f ||y| − |x||f | ≤ p−1 |y|p − p−1 |x|p ,

(2.9)

which together with (2.4) imply hf, y − xi ≤ p−1 |y|p − p−1 |x|p = ψ(y) − ψ(x) and so f ∈ ∂ψ(x). This completes the proof. ∗ Theorem 2.1 Let F be a Frechet space and Jp : F → 2F be a paranormalized duality mapping of F . Then |x + y|p ≤ |x|p + phy, jp (x + y)i, for all x, y ∈ F and jp (x + y) ∈ Jp (x + y). Proof. By Lemma 2.1, we know that Jp (x) ⊂ ∂ψ(x), where ψ(x) = p−1 |x|p , for all x ∈ F. It follows from the definition of subdifferential of ψ that ψ(x) − ψ(x + y) ≥ hx − (x + y), jp i,

∀jp ∈ Jp (x + y).

(2.10)

Substituting ψ(x) by p−1 |x|p in (2.10) and simplifying it, we have |x + y|p ≤ |x|p + phy, jp i,

∀jp ∈ Jp (x + y).

This completes the proof. Remark 2.2 Theorem 2.1 extended the inequality (1.3) to the Frechet space. ∗ Corollary 2.1 Let F be a Frechet space and J : F → 2F be a paranormalized duality mapping of F . Then |x + y|2 ≤ |x|2 + 2hy, j(x + y)i for all x, y ∈ F and j(x + y) ∈ J(x + y).

3

A Fixed Point Theorem for Contractive Mappings in Frechet Spaces

In this section, we shall study the convergence analysis of Mann iteration scheme for approximating the fixed point of contractive mapping in a Frechet space. Let K ⊂ F be a close convex subset and T : K → K be a mapping. For any x0 ∈ K, the Mann iteration scheme is defined by xn+1 = (1 − αn )xn + αn T xn ,

n = 0, 1, 2, · · · ,

(3.1)

where 0 ≤ αn ≤ 1 satisfying some conditions. Theorem 3.1 Let K be a close convex subset of a Frechet space F and T : K → K be a contractive mapping with a constant 0 < k < 1, that is, |T x − T y| ≤ k|x − y|, 2

∀x, y ∈ K.

If there exists a positive integer N such that k 2+1 < αn for all n ≥ N , then for any x0 ∈ K, the Mann iteration scheme of T defined by (3.1) converges to a fixed point of T . 3

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275

Proof. By using the similar proof method of Banach fixed point theorem, we can show that T has a unique fixed point in F . Let u ∈ F be a fixed point of T . From Corollary 2.1 and (3.1), we have |xn+1 − u|2

= |(1 − αn )(xn − u) + αn (T xn − T u)|2 ≤ |αn (T xn − T u)|2 + 2h(1 − αn )(xn − u), j(xn+1 − u)i ≤ k 2 |xn − u|2 + 2(1 − αn )|xn − u||xn+1 − u|, (3.2)

which implies |xn+1 − u|

p ≤ ( k 2 + (1 − αn )2 + (1 − αn ))|xn − u| = c|xn − u|,

where c=

p

(3.3)

k 2 + (1 − αn )2 + (1 − αn ).

From the assumption, we know that 0 < c < 1. Thus, it follows from (3.3) that |xn − u| → 0. This completes the proof.

4

Set-valued Variational Inclusions in Frechet Spaces

In this section, we consider the problem of approximating of solutions for a class of set-valued variational inclusions in Frechet spaces. Based on the result due to Nadler [13], we constructed some algorithms for solving the set-valued variational inclusions in Frechet spaces. We also prove the existence of solutions of the set-valued variational inclusions and the convergence of iterative sequences generated by Algorithms. We first give some basic definitions and notions. Definition 4.1 Let F be a Frechet space with a paranorm | · |. A Frechet space F is said to be convex Pn if, for any u, x1 , x2 , · · · , xn ∈ F and a1 , a2 , · · · , an with i=1 ai = 1 and 0 ≤ ai ≤ 1, |u −

n X

ai xi | ≤

i=1

n X

ai |u − xi |.

i=1

It is worth mentioning that, any convex Banach space is a convex Frechet space. However, there exist some convex Frechet spaces which can not be injected to any linear norm space (see [18]). Let F be a convex Frechet space, T, V : F → CB(F ) be two set-valued mappings, N (·, ·) : F × F → F and g : F → F be two single-valued mappings, and A : D(A) ⊂ F → 2F be a set-valued mapping. We consider the following problem of finding p ∈ F, w0 ∈ T p, y ∈ V p such that 0 ∈ N (w0 , y) + A(g(p)).

(4.1)

This problem is called the set-valued variational inclusions in convex Frechet spaces. If F is a Banach space, then problem (4.1) was studied by many authors (see, for example, Chang, Cho and Zhou [2] and the references therein). Remark 4.1 For a suitable choice of the mappings T, V, N, g, A and the space F , a number of known and new variational inequalities, variational inclusions, and related optimization problems introduced and studied by Chang [1], Ding [5], Huang [8, 9], and Zeng [20] can be obtained from (2.1). Let F be a Frechet space with a paranorm | · | and CB(F ) denote the set of all nonempty closed and bounded subsets of F . For any A, B ∈ CB(F ), we define the Hausdorff metric H(·, ·) as follows: H(A, B) = max{sup inf |a − b|, sup inf |a − b|}. a∈A b∈B

b∈B a∈A

Definition 4.3 A set-valued mapping T : F → CB(F ) is said to be µ-Lipschitz continuous if, for any x, y ∈ F , H(T x, T y) ≤ µ|x − y|, 4

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where µ > 0 is a constant. Lemma 4.1 [13] Let E be a complete metric space and T : E → CB(E) be a set-valued mapping. Then for any given ε > 0, x, y ∈ E and u ∈ T x, there exists v ∈ T y such that d(u, v) ≤ (1 + ε)H(T x, T y). Next we will suggest and analyze some algorithms for solving variational inclusion (4.1). Algorithm 4.1 For any given x0 ∈ F , u0 ∈ T x0 , z0 ∈ V x0 , compute the sequences {xn }, {yn }, {un }, {zn }, {wn } and {vn } by the iterative schemes such that  rn ∈ N (un , zn ) + A(g(xn )) + xn ,     yn = a0n xn + b0n rn + c0n fn ,    1   wn ∈ T yn , |wn − wn+1 | ≤ (1 + n+1 )H(T yn , T yn+1 ),   1   vn ∈ V yn , |vn − vn+1 | ≤ (1 + n+1 )H(V yn , V yn+1 ), pn ∈ N (wn , vn ) + A(g(yn )) + yn , (4.2)   x = a x + b p + c e ,  n+1 n n n n n n   1  un ∈ T xn , |un − un+1 | ≤ (1 + n+1 )H(T xn , T xn+1 ),    1   z ∈ V xn , |zn − zn+1 | ≤ (1 + n+1 )H(V xn , V xn+1 ),   n n = 0, 1, 2, · · · , where an , bn , cn , a0n , b0n , c0n ∈ [0, 1] with an + bn + cn = 1,

a0n + b0n + c0n = 1,

n = 0, 1, 2, · · ·

and {en } ⊂ F and {fn } ⊂ F satisfy the following conditions: max{|x − y| : x ∈ {ei , fi : n ≤ i ≤ m}, y ∈ {xj , yj , pj , rj , ej , fj : n ≤ j ≤ m}} ≤ max{|xn − pi |, |xn − ri | : n ≤ i ≤ m}. The sequence {xn } defined by (4.2) is called the Ishikawa iterative sequence with errors. If b0n = c0n = 0 for all n ≥ 0, then Algorithm 4.1 reduces to the following algorithm. Algorithm 4.2 For any given x0 ∈ F, w0 ∈ T x0 , v0 ∈ V x0 , compute the sequences {xn }, {wn } and {vn } by the iterative schemes such that  pn ∈ N (wn , vn ) + A(g(xn )) + xn ,      xn+1 = an xn + bn pn + cn en 1 wn ∈ T xn , |wn − wn+1 | ≤ (1 + n+1 )H(T xn , T xn+1 ) (4.3)  1  v ∈ V x , |v − v | ≤ (1 + )H(V x , V x ),  n n n n+1 n n+1 n+1   n = 0, 1, 2, · · · . The sequence {xn } is called the Mann iterative sequence with errors. Remark 4.2 If F is a convex Banach space, then F is a convex Frechet space. Thus, the Ishikawa iterative sequence considered by Chang [1] is a special case of (4.2). Theorem 4.1 Let F be a convex Frechet space, T, V, A three set-valued mappings, g : F → F and N (·, ·) : F × F → F two single-valued mappings satisfying the following conditions: (1) N (·, ·) is Lipschitz continuous with respect to the first argument and a constant β > 0; (2) N (·, ·) is Lipschitz continuous with respect to the second argument and a constant γ > 0; (3) T : F → CB(F ) is µ− Lipschitz continuous; (4) V : F → CB(F ) is ξ− Lipschitz continuous; (5) I + A ◦ g is contractive and continuous with a constant 0 < k < 1; p (6) 0 < φ = (µβ + γξ)2 + k 2 + k < 1.

5

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277

Then the sequences {xn }, {un } and {zn } generated by Algorithm 4.1 strongly converge to a solution p ∈ F, w0 ∈ T p, y ∈ V p of the varational inclusion (4.1) in F. Proof. Let N be the set of all nonnegative integers. For any n, m ∈ N with 0 ≤ n < m, let δ(An,m ) =

sup

|x − y|,

(4.4)

x,y∈An,m

where An,m = {xi , yi , pi , ri , ui , vi : n ≤ i ≤ m}. Then δ(An,m ) = max{D1 , D2 , D3 , D4 , D5 , D6 }, where D1 D2 D3 D4 D5 D6

= = = = = =

max{|xn − pi |, |xn − ri | : n ≤ i ≤ m}, max{|pi − pj |, |pi − rj |, |ri − rj | : n ≤ i, j ≤ m}, max{|xi − pj |, |xi − rj | : n < i ≤ m, n ≤ j ≤ m}, max{|yi − rj |, |yi − pj | : n ≤ i, j ≤ m}, max{|xi − xj |, |xi − yj |, |yi − yj | : n ≤ i, j ≤ m}, max{|x − y| : x ∈ {ei , fi : n ≤ i ≤ m}, y ∈ {xj , yj , pj , rj , ej , fj : n ≤ j ≤ m}}.

Next we prove that δ(An,m ) = D1 . (I) From (4.2), Corollary 2.1 and the assumptions, there exist hi ∈ A(g(xi )) and hj ∈ A(g(xj ) such that pi = N (wi , vi ) + hi + xi , pj = N (wj , vj ) + hj + xj and |pi − pj |2

= ≤ ≤ ≤

|N (wi , vi ) + hi + xi − N (wj , vj ) − hj − xj |2 |N (wi , vi ) − N (wj , vj )|2 + 2hhi − hj + xi − xj , j(pi − pj )i (µβ + γξ)2 |xi − xj |2 + 2|hi − hj + xi − xj ||pi − pj | (µβ + γξ)2 |xi − xj |2 + 2k|xi − xj ||pi − pj |

which implies that p |pi − pj | ≤ ( (µβ + γξ)2 + k 2 + k)|xi − xj | = φ|xi − xj | ≤ φδ(An,m ). Also, we have |pi − rj | ≤ φδ(An,m ) and |ri − rj | ≤ φδ(An,m ). It follows that D2 ≤ φδ(An,m ). (II) By Algorithm 4.1 and (4.5), for any n < i ≤ m, n ≤ j ≤ m, we have |xi − pj | = |ai−1 xi−1 + bi−1 pi−1 + ci−1 ei−1 − pj | ≤ ai−1 |xi−1 − pj | + bi−1 |pi−1 − pj | + ci−1 |ei−1 − pj | ≤ max{|xi−1 − pj |, φδ(An,m ), D6 }. When i − 1 > n, it follows that |xi−1 − pj | ≤ max{|xi−2 − pj |, φδ(An,m ), D6 }. 6

(4.5)

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By induction, for any n < i ≤ m, n ≤ j ≤ m, we have |xi − pj | ≤ ≤ ≤ ≤

max{|xi−1 − pj |, φδ(An,m ), D6 } max{|xi−2 , pj |, φδ(An,m ), D6 } ··· max{|xn − pj |, φδ(An,m ), D6 }.

Similarly, for any n < i ≤ m, n ≤ j ≤ m, we can get |xi − rj | ≤ max{|xn − rj |, φδ(An,m ), D6 }. Thus, we have D3

= max{|xi − rj |, |xi − pj | : n < i ≤ m, n ≤ j ≤ m} ≤ max{|xn − rj |, |xn − pj |, φδ(An,m ), D6 : n ≤ j ≤ m} = max{D1 , φδ(An,m ), D6 }.

(4.6)

(III) For any n ≤ i, j ≤ m, it follows from (4.5) and (4.6) that |yi − pj | = |a0n xi + b0n ri + c0n fi − pj | ≤ a0n |xi − pj | + b0n |ri − pj | + c0n |fi − pj | ≤ max{D1 , φδ(An,m ), D6 } and so D4

= max{|yi − pj |, |yi − rj | : n ≤ i, j ≤ m} ≤ max{D1 , φδ(An,m ), D6 }.

(4.7)

(IV) Now we consider the case for D5 = max{|xi − xj |, |xi − yj |, |yi − yj | : n ≤ i, j ≤ m}. a) firstly, we estimate max{|xi − xj | : n ≤ i, j ≤ m}. Denote A1 = max{|xi − xj | : n ≤ i, j ≤ m}. Then there exist k and l with n ≤ k ≤ l ≤ m such that A1 = |xk − xl | and |xk − xl−1 | < |xk − xl | = A1 .

(4.8)

It follows that A1

= |xk − xl | = |xk − (al−1 xl−1 + bl−1 pl−1 + cl−1 el−1 )| ≤ al−1 |xk − xl−1 | + bl−1 |xk − pl−1 | + cl−1 |xk − el−1 | ≤ al−1 |xk − xl−1 | + bl−1 D3 + cl−1 D6 .

(4.9)

If al−1 = 0, then (4.9) implies that A1 ≤ max{D3 , D6 }. If al−1 6= 0, then it follows from (4.8) and (4.9) that A1

≤ al−1 |xk − xl−1 | + bl−1 D3 + cl−1 D6 ≤ max{D3 , D6 }.

(4.10)

b) Now we make estimation for max{|xi − yj | : n ≤ i, j ≤ m}. Let A2 = max{|xi − yj | : n ≤ i, j ≤ m}. Since yj = a0j xj + b0j rj + c0j fj , from (4.6) and (4.10), we have A2

= max{|xi − (a0j xj + b0j rj + c0j fj )|} ≤ max{a0j |xi − xj | + b0j |xi − rj | + c0j |xi − fj | : n ≤ i, j ≤ m} ≤ max{D1 , D6 , φδ(An,m )}. 7

(4.11)

SHI, HUANG: SET-VALUED VARIATIONAL INCLUSIONS

279

c) Finally, we made estimation for max{|yi − yj | : n ≤ i, j ≤ m}. Denote A3

= max{|yi − (a0j xj + b0j rj + c0j fj )|} ≤ max{a0j |yi − xj | + b0j |yi − rj | + c0j |yi − fj | : n ≤ i, j ≤ m} ≤ max{D1 , D6 , φδ(An,m )}.

(4.12)

From (4.10), (4.11) and (4.12), we obtain D5

= max{|xi − xj |, |xi − yj |, |yi − yj | : n ≤ i, j ≤ m} ≤ max{D1 , D6 , φδ(An,m )}.

(4.13)

Now combining the cases (I)-(IV), we have δ(An,m ) = max{D1 , D2 , D3 , D4 , D5 , D6 } ≤ max{D1 , D6 , φδ(An,m )}.

(4.14)

If D1 < φδ(An,m ), then D6 < D1 < φδ(An,m ) < δ(An,m ). From (4.14), we have δ(An,m ) < δ(An,m ) which is a contradiction. Therefore, D1 ≥ φδ(An,m ) and so (4.14) implies that δ(An,m ) ≤ D1 . Obviously, D1 ≤ δ(An,m ), so D1 = δ(An,m ), i.e., δ(An,m ) = D1 . Letting n = 0 in δ(An,m ) = D1 , we have δ(A0,m ) = max{|x0 − pj |, |x0 − rj | : 0 ≤ j ≤ m} ≤ |x0 − p0 | + max{|p0 − pj |, |p0 − rj | : 0 ≤ j ≤ m} ≤ |x0 − p0 | + φδ(A0,m ). Thus, δ(A0,m ) ≤ (1 − φ)−1 (|x0 − p0 |), (m ≥ 0) which implies that the sequence {δ(A0,m )} is bounded. On the other hand, for any positive integers n and m with 1 ≤ n ≤ m, we have δ(An,m )

= max{d(xn , pj ), d(xn , rj ) : n ≤ j ≤ m} ≤ max{|an−1 xn−1 + bn−1 rn−1 + cn−1 en−1 − pj |, |an−1 xn−1 , bn−1 rn−1 , cn−1 en−1 − rj |} ≤ an−1 δ(An−1,m ) + bn−1 φδ(An−1,m ) + cn−1 δ(An−1,m ) = (1 − bn−1 )δ(An−1,m ) + bn−1 φδ(An−1,m ) = (1 − bn−1 (1 − φ))(δ(An−1,m )),

which implies that δ(An,m ) ≤ Since

P∞

j=0 bj

n−1 Y

(1 − bj (1 − φ))(δ(A0,m )).

j=0

= ∞, it follows that lim δ(An,m ) = 0,

n,m→∞

which implies the sequence {xn } is a Cauchy sequence in F and lim |xn − rn | = 0.

n→∞

Since F is complete, we can suppose that xn → p ∈ F . Thus, lim xn = lim rn = p.

n→∞

n→∞

8

(4.15)

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SHI, HUANG: SET-VALUED VARIATIONAL INCLUSIONS

From (4.2) and condition (3), we have 1 )H(T xn , T xn+1 ) n+1 1 ≤ µ(1 + )|xn − xn+1 |, n+1

|un − un+1 | ≤

(1 +

which implies that the sequence {un } is a Cauchy sequence in F . Let un → w0 ∈ F . Since d(w0 , T p) ≤

|w0 − un | + H(T xn , T p) 1 )|xn − p| → 0, ≤ |w0 − un | + µ(1 + n+1

we know that w0 ∈ T p. Similarly, we have zn → y ∈ V p. Since rn ∈ N (un , zn ) + A(g(xn )) + xn , letting n → ∞, we get p ∈ N (w0 , y) + A(g(p)) + p, i.e.,

0 ∈ N (w0 , y) + A(g(p)).

This complete the proof. Remark 4.3 Theorem 4.1 improved and unified some corresponding results of Chang [1], Chang, Kim and Kim [3], Ding [5], Huang [8, 9], Shi and Liu [17], and Zeng [20]. From Theorem 4.1, we can obtain the convergence analysis for the Mann iterative sequence as follows. Theorem 4.2 Let F be a convex Frechet space, T, V, A three set-valued mappings, g : F → F and N (·, ·) : F × F → F two single-valued mappings satisfying the conditions (1)-(6) of Theorem 4.1. Then the sequences {xn }, {wn } and {vn } generated by Algorithm 4.2 strongly converge to a solution p ∈ F, w0 ∈ T p, y ∈ V p of the variational inclusion (4.1) in F . Acknowledgement This research is supported by the Science Foundation Project of Education Department of Shaanxi Province (No. 2010JK896) and the Natural Science Founation of Shaanxi Province (No.2010JQ1013).

References [1] S.S. Chang, Set-valued variational inclusions in Banach spaces, J. Math. Anal. Appl. 248 (2000), 438-454. [2] S.S. Chang, Y.J. Cho and H.Y. Zhou, Iterative Methods for Nonlinear Operator Equations in Banach Spaces, Nova Science, New York, 2002. [3] S.S. Chang, J.K. Kim and K.H. Kim, On the existence and iterative approximation problems of solutions for set-valued variational inclusions in Banach spaces, J. Math. Anal. Appl. 268 (2002), 89-108. [4] C.E. Chidume, H. Zegeye and K.R. Kazmi, Existence and convergence theorems for a class of multivalued variational inclusions in Banach spaces, Nonlinear Anal. TMA 59 (2004), 649-656. [5] X.P. Ding, Iterative process with errors of nonlinear equation involving m-accretive operator, J. Math. Anal. Appl. 209 (1997), 191-201. [6] Y.P. Fang and N.J. Huang, A new system of variational inclusions with (H, η)-monotone operators in Hilbert spaces, Computers Math. Appl. 49 (2005), 365-374. [7] Y.P. Fang and N.J. Huang, H-monotone operator and resolvent operator technique for variational inclusions, Appl. Math. Comput. 145(2003), 795-803. [8] N.J. Huang, Generalized nonlinear variational inclusions with noncompact valued mappings, Appl. Math. Lett. 9 (1996), 25-29. 9

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[9] N.J. Huang, On the generalized implicit quasi-variational inequalities, J. Math. Anal. Appl. 216 (1997), 197-210. [10] G. Isac and J.L. Li, The convergence property of Ishikawa iterative schemes in noncompact subsets of Hilbert spaces and its applications to complementarity theory, Comput. Math. Appl. 47(2004), 1745-1751. [11] L.S. Liu, Ishikawa and Mann iterative processes with errors for nonlinear strongly accretive mappings in Banach spaces, J. Math. Anal. Appl. 194(1995), 114-125. [12] Z.Q. Liu and S.M. Kang, Convergence theorems for φ-strongly accretive and φ-hemicontractive operators, J. Math. Anal. Appl. 253(2001), 35-49. [13] S.B. Naddler, Multivalued contraction mappings, Pacific J. Math. 30 (1969), 175-488. [14] M.A. Noor, K.I. Noor and T.M. Rassias, Set-valued resolvent equations and mixed variational inequalities, J. Math. Anal. Appl. 220(1998), 741-759. [15] M.A. Noor, T.M. Rassias and Z.Y. Huang, Three-step iterations for nonlinear accretive operator equations, J. Math. Anal. Appl. 274 (2002), 59-68. [16] W.V. Petryshyn, A characterization of strictly comvexity of Banach spaces and other uses of duality mappings, J. Func. Anal. 6(1970), 282-291. [17] C.F. Shi and S.Y. Liu, Generalized set-valued variational inclusion in q-uniformly smooth Banach spaces, J. Math. Anal. Appl. 296(2004), 553-562. [18] W. Takahashi, A convexity in metric space and nonexpansive mappings, Kodai Math. Sem. Rep. 22(1970), 142-149. [19] A. Wilansky, Modern Methods in Topological Vector Spaces, McGraw-Hill, New York, 1978. [20] L.C. Zeng, Iterative algorithm for finding approximate solutions to completely generalized strongly nonlinear quasi-variational inequality, J. Math. Anal. Appl. 201(1996), 180-191.

Chaofeng Shi Chongqing Jiaotong University Chongqing, 400074, P. R. China and School of Mathematics and Information Science Xianyang Normal University Xianyang, Shaanxi 712000, P. R. China E-mail: [email protected] Nan-jing Huang Department of Mathematics Sichuan University Chengdu, Sichuan 610064, P. R. China E-mail: [email protected]

10

JOURNAL 282 OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.2, 282-295, 2011, COPYRIGHT 2011 EUDOXUS PRESS, LLC

ON THE EXTENDED KIM’S q-BERNSTEIN POLYNOMIALS S.-H. RIM, L.C. JANG, J. CHOI, Y. H. KIM, B. LEE, AND T. KIM

Abstract The purpose of this paper is to present a systemic study of some families of the Kim’s q-Bernstein polynomials. By using double fermionic p-adic integral representation on Zp , we give some interesting and new formulae related to q-Euler numbers. 1. Introduction Let p be a fixed odd prime number. Thoughout this paper Zp , Qp , C, and Cp will, respectively, denote the ring of p-adic rational integers, the field of p-adic rational numbers, the complex number field and the completion of algebraic closure of Qp . Let N be the set of the natural numbers and Z+ = N ∪ {0}. Let νp be the normalized exponential valuation of Cp with |p|p = p−νp (p) = p1 (see [3-10]). Let q be regarded as either a complex number q ∈ C or a p-adic number q ∈ Cp . If q ∈ C, then we always assume that |q| < 1. If q ∈ Cp , we usually assume that |1 − q|p < 1 x (see [3-5]). In this paper, the q-numbers are defined by [x]q = 1−q 1−q (see [1-12]). Assume that U D(Zp ) denotes the space of uniformly differentiable functions on Zp . For f ∈ U D(Zp ), the fermionic p-adic q-inetgral on Zp is defined by N

pX −1 1 f (x)dµ−q (x) = lim I−q (f ) = f (x)(−q)x , N →∞ [pN ]−q Zp x=0

Z

(see [3,8]).

The R ordinary fermionic p-adic integral on Zp is defined by I−1 (f ) = limq→1 I−q (f ) = f (x)dµ−1 (x) (see [3]). As a well known definition, the Euler polynomials are Zp defined as ∞ X tn 2 xt En (x) , (see [3,7,8]). e = t e +1 n! n=0 In the special case, x = 0, En (0) = En are called the n-th Euler numbers. From the definition of Euler numbers, the recurrence formula for En is given by E0 = 1, and (E + 1)n + En = 1 if n > 0,

(see [3,9]),

with the usual convention of replacing E n by En . As the q-extension of the Euler numbers, En,q , Kim defined the q-Euler numbers as follows: E0,q = 1, and (qEq + 1)n + En,q = 1 if n > 0, with the usual convention of replacing

Eqn

(see [3]),

by En,q .

2000 Mathematics Subject Classification : 11B68, 11S80, 41A30. Key words and phrases : Bernstein type q-polynomial, Laurent series, q-Euler numbers and polynomials, fermionic p-adic integral. 1

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283

2

For n ∈ N, write fn (x) = f (x + n). We have Z n−1 X I−1 (fn ) = f (x + n)dµ−1 (x) = (−1)n I−1 (f ) + 2 (−1)n−1−l f (l). Zp

l=0

Thus, we see that Z e(x+y)t dµ−1 (y) = Zp

∞ X 2 tn xt e = , E (x) n et + 1 n! n=0

This is equivalent to Z (x + y)n dµ−1 (y) = En (x),

(see [3, 13-20]).

(see [3,8,9]).

Zp

In [3], the q-Euler polynomials are also given by Z n   X n (−q x )l 2 = (q x Eq + [x]q )n , En,q (x) = [x + y]nq dµ−1 (y) = n l (1 − q) l 1 + q Zp l=0

with the usual convention of replacing Eqn by En,q . In the special case, x = 0, R n En,q (0) = En,q = Zp [x]q dµ−1 (x). Let C[0, 1] denote the set of continuous functions on [0, 1]. For f ∈ C[0, 1], Bernstein introduced the following well known linear operators (see [2, 9]):   n n X X k n k k Bn (f |x) = f( ) x (1 − x)n−k = f ( )Bk,n (x). n k n k=0

k=0

Here Bn (f |x) is called the Bernstein operator of order n for f . For n, k ∈ Z+ , the Bernstein polynomials of degree n are defined by Bk,n (x) = xk (1 − x)n−k ,

(see [1,2,6,11,12]).

In [9], Kim defined the new q-extension of Bernstein operator of order n for f as follows:   n n X X k n k Bn,q (f |x) = f( ) [x]kq [1 − x]n−k = f ( )Bk,n (x, q). q −1 n k n k=0

k=0

The Kim’s q-Bernstein polynomials of degree n are also given by   n Bk,n (x, q) = [x]kq [1 − x]n−k q −1 , for n, k ∈ Z+ and x ∈ [0, 1]. k Thus, we note that Bn,q (1|x) = 1, Bn,q (t|x) = [x]q , and Bn,q (t2 |x) =

n − 1 2 [x]q [x]q + , n n

(see [9]).

That is, Kim’s q-Bernstein operator of order n for f is uniformly converge to f ([x]q ). In this paper, we present a systemic study of some families of the Kim’s qBernstein polynomials. By using double fermionic p-adic integral representation on Zp , we derive some interesting identities closely related to q-Euler numbers from some families of the Bernstein type q-polynomials.

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2. On the extended Bernstein type q-polynomials For α, β ∈ R and f ∈ C[0, 1], we consider the linear operator of order n for f as follows:   n X k n Bn,q (f : x1 , x2 |α, β) = f( ) [x1 ]kqα [1 − x2 ]n−k q −β n k =

k=0 n X

k=0

k f ( )Bk,n (x1 , x2 : q|α, β), n

(1)

where x1 , x2 ∈ [0, 1] and n, k ∈ Z+ . Here Bk,n (x1 , x2 : q|α, β) is called the extended Kim’s q-Bernstein polynomials of degree n with weight α, β. By (1), we get the following generating function for Bk,n (x1 , x2 : q|α, β) : (k)

Fq;α,β (x1 , x2 |t)

(t[x1 ]qα )k exp(t[1 − x2 ]q−β ) k! ∞ X [x1 ]kqα [1 − x2 ]nq−β n+k = t k!n! n=0 =

= = =

∞ X [x1 ]kqα [1 − x2 ]n−k n! tn q −β n=k ∞  X n=k ∞ X

k!(n − k)!

n!

n tn [x1 ]kqα [1 − x2 ]n−k −β q k n! 

Bk,n (x1 , x2 : q|α, β)

n=k

where x1 , x2 ∈ [0, 1], n, k ∈ Z+ and α, β ∈ R. From (2), we note that ( n
n−k k k [x1 ]q α [1 − x2 ]q −β Bk,n (x1 , x2 : q|α, β) = 0

tn , n!

(2)

if n ≥ k, (3) if k > n.

By (3), we easily get Bn−k,n (1 − x2 , 1 − x1 : q| − β, −α) = Bk,n (x1 , x2 : q|α, β).

(4)

For 0 ≤ k ≤ n, we have [1 − x2 ]q−β Bk,n−1 (x1 , x2 : q|α, β) + [x1 ]qα Bk−1,n−1 (x1 , x2 : q|α, β)     n−1 n−1 n−1−k n−k k α = [1 − x2 ]q−β [x1 ]qα [1 − x2 ]q−β + [x1 ]q [x1 ]k−1 q α [1 − x2 ]q −β k k−1   n−k n k = [x1 ]qα [1 − x2 ]q−β = Bk,n (x1 , x2 : q|α, β). (5) k The partial derivatives of extended Kim’s q-Bernstein polynomials of degree n are also q-polynomials of degree n − 1: ∂ log q α αx1 Bk,n (x1 , x2 : q|α, β) = α q nBk−1,n−1 (x1 , x2 : q|α, β), ∂x1 q −1

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4

and ∂ log q β βx2 q nBk,n−1 (x1 , x2 : q|α, β). Bk,n (x1 , x2 : q|α, β) = ∂x2 1 − qβ Thus, we obtain the following theorem. Theorem 1. For k, n ∈ Z+ and x1 , x2 ∈ [0, 1], we have [1 − x2 ]q−β Bk,n−1 (x1 , x2 : q|α, β) + [x1 ]qα Bk−1,n−1 (x1 , x2 : q|α, β) = Bk,n (x1 , x2 : q|α, β), ∂ log q α αx1 q nBk−1,n−1 (x1 , x2 : q|α, β), Bk,n (x1 , x2 : q|α, β) = α ∂x1 q −1 and ∂ log q β βx2 q nBk,n−1 (x1 , x2 : q|α, β). Bk,n (x1 , x2 : q|α, β) = ∂x2 1 − qβ Let f (t) = 1. From (1), we note that Bn,q (1 : x1 , x2 |α, β)

= =

n X

k=0

=

Bk,n (x1 , x2 : q|α, β)

k=0 n  X

 n [x1 ]kqα [1 − x2 ]n−k q −β k

(1 + [x1 ]qα − [x2 ]qβ )n .

(6)

Thus, we have 1 Bn,q (1 : x1 , x2 |α, β) = 1. (1 + [x1 ]qα − [x2 ]qβ )n Let f (t) = t. Then we have Bn,q (t : x1 , x2 |α, β)

n   X k

  n n k k=0   n X n−1 = [x1 ]kqα [1 − x2 ]n−k q −β k − 1 k=1   n−1 X n−1 = [x1 ]qα [x1 ]kqα [1 − x2 ]n−k−1 , q −β k =

[x1 ]kqα [1 − x2 ]n−k q −β

k=0

where n ∈ N and x1 , x2 ∈ [0, 1], α, β ∈ R. By (7), we get 1 Bn,q (t : x1 , x2 |α, β) = [x1 ]qα . (1 + [x1 ]qα − [x2 ]qβ )n−1 By the same method, we see that Bn,q (t2 : x1 , x2 |α, β) n−1 [x1 ]qα = [x1 ]2qα (1 + [x1 ]qα − [x2 ]qβ )n−2 + (1 + [x1 ]qα − [x2 ]qβ )n−1 . n n For α = β and x1 = x2 = x, we have n−1 2 [x]qα Bn,q (t2 : x, x|α, α) = [x]qα + −→ [x]2qα as n → ∞. n n

(7)

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Thus, we obtain the following theorem. Theorem 2. For n ∈ Z+ , α ∈ R and x ∈ [0, 1], we have Bn,q (1 : x, x|α, α) = 1, Bn,q (t : x, x|α, α) = [x]qα , and Bn,q (t2 : x, x|α, α) =

n−1 2 [x]qα [x]qα + −→ [x]2qα as n → ∞. n n

It is easy to show that [1 −

x2 ]n−k q −β

=

n−k X l=0

 n−k (−1)l [x2 ]lqβ . l

(8)

By (1) and (8), we get   n X k Bk,n (x1 , x2 : q|α, β) Bn,q (f : x1 , x2 |α, β) = f n k=0    n n−k X X n − k  k n k = f [x1 ]qα (−1)j [x2 ]jqβ . (9) n k j j=0 k=0

By simple calculation, we easily see that for m = k + j,       n n−k n m = . k j m k

(10)

From (9) and (10), we note that Bn,q (f : x1 , x2 |α, β)   k n   m   X X n m k [x1 ]qα m−k [x2 ]m (−1) f = . β q k n [x2 ]qβ m m=0

(11)

k=0

Thus, we obtain the following theorem. Theorem 3. For f ∈ C[0, 1], n ∈ Z+ , x1 , x2 ∈ [0, 1], and α, β ∈ R, we have Bn,q (f : x1 , x2 |α, β) k   n   m   X X [x1 ]qα n m k m m−k = . [x2 ]qβ (−1) f m k n [x2 ]qβ m=0 k=0

The second kind stirling numbers are defined by k   ∞ X (et − 1)k 1 X k tn = (−1)k−l elt = S(n, k) , k! k! l n! m=0

(12)

l=0

where k ∈ N (see [3,5]). Let ∆ be the shift difference operator with ∆f (x) = f (x + 1) − f (x). From the definition of ∆, we derive n   X n n ∆ f (0) = (−1)n−k f (k). (13) k k=0

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By (12) and (13), we get 1 k n ∆ 0 = S(n, k). k! Hence, we obtain the following theorem. Theorem 4. For f ∈ C[0, 1], n ∈ Z+ , x1 ∈ [0, 1], and α ∈ R, we have   n   X 0 n k k Bn,q (f : x1 , x1 |α, α) = . [x1 ]qα ∆ f n k k=0

In the special case, f (t) = tm (m ∈ Z+ ), we have n   X n m m n Bn,q (t : x1 , x1 |α, α) = [x1 ]kqα ∆k 0m , k k=0

where x1 ∈ [0, 1] and m, n ∈ Z+ , α ∈ R. For x, t ∈ C and n ∈ Z+ with n ≥ k, consider Z 1 dt n! (t[x1 ]qα )k exp(t[1 − x2 ]q−β ) n+1 , 2πi C k! t

(14)

where C is a circle around the origin and integration is in the positive direction. By (2) and (14), we get Z ([x1 ]qα t)k exp(t[1 − x2 ]q−β ) dt k! tn+1 C Z ∞ X Bk,m (x1 , x2 : q|α, β) dt = m! tn+1 m=0 C = 2πi

Bk,n (x1 , x2 : q|α, β) . n!

Thus, we have Z ([x1 ]qα t)k exp(t[1 − x2 ]q−β ) dt n! = Bk,n (x1 , x2 : q|α, β). 2πi C k! tn+1

(15)

(16)

From Laurent series and (2), we have Z (t[x1 ]qα )k exp(t[1 − x2 ]q−β ) dt k! tn+1 C ! ∞ m Z k X [1 − x2 ]q−β [x1 ]qα m−n−1+k = t dt k! m=0 m C = 2πi

[x1 ]kqα [1 − x2 ]n−k q −β k!(n − k)! k

n−k

2πi n![x1 ]qα [1 − x2 ]q−β = . n! k!(n − k)! By (16) and (17), we get   n Bk,n (x1 , x2 : q|α, β) = [x1 ]kqα [1 − x2 ]n−k , q −β k

(17)

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where x1 , x2 ∈ [0, 1] and n, k ∈ Z+ , α, β ∈ R+ . From the definition of Bk,n (x1 , x2 : q|α, β), we have n−k k+1 Bk,n (x1 , x2 : q|α, β) + Bk+1,n (x1 , x2 : q|α, β) n n (n − 1)! (n − 1)! n−k−1 = [x1 ]kqα [1 − x2 ]n−k [x1 ]k+1 + q α [1 − x2 ]q −β q −β k!(n − k − 1)! k!(n − k − 1)! = ([x1 ]qα + [1 − x2 ]q−β )Bk,n−1 (x1 , x2 : q|α, β) = (1 − [x2 ]qβ + [x1 ]qα )Bk,n−1 (x1 , x2 : q|α, β), (18) where k ∈ Z+ , n ∈ N and x1 , x2 ∈ [0, 1], α, β ∈ R. Thus, we obtain the following theorem. Theorem 5. For k ∈ Z+ , n ∈ N and x1 , x2 ∈ [0, 1], α, β ∈ R, we have k+1 n−k Bk,n (x1 , x2 : q|α, β) + Bk+1,n (x1 , x2 : q|α, β) n n = (1 − [x2 ]qβ + [x1 ]qα )Bk,n−1 (x1 , x2 : q|α, β). In the special case, α = β and x1 = x2 = x, we have n−k k+1 Bk,n (x, x : q|α, α) + Bk+1,n (x, x : q|α, α) = Bk,n−1 (x, x : q|α, α). (19) n n From the definition of Bk,n (x1 , x2 : q|α, β) and binomial coefficient, we can derive the following equation (20).   n−k X n − k  n k Bk,n (x1 , x2 : q|α, β) = [x1 ]qα (−1)l [x2 ]lqβ k l l=0  k X n    [x1 ]qα l n (−1)l−k [x2 ]lqβ . (20) = [x2 ]qβ k l l=k

By (20), we get the following theorem. Theorem 6. For α, β ∈ R and n, k ∈ Z+ , x1 , x2 ∈ [0, 1], we have k X  n    l n [x1 ]qα (−1)l−k [x2 ]lqβ . Bk,n (x1 , x2 : q|α, β) = k l [x2 ]qβ

(21)

l=k

In the special case, x1 = x2 = x and α = β, n    X l n (−1)l−k [x]lqα . Bk,n (x, x : q|α, α) = k l l=k

[x1 ]kqα

It is possible to write as a linear combination of Bk,n (x1 , x2 : q|α, β) by using the degree evaluation formulae and mathematical induction:
 n n  k X X n−1 1
[x1 ]kqα [1 − x2 ]n−k n Bk,n (x1 , x2 : q|α, β) = q −β k − 1 k=1 1 k=1 n−1 X n − 1 n−k−1 = [x1 ]k+1 q α [1 − x2 ]q −β k k=0

=

[x1 ]qα (1 + [x1 ]qα − [x2 ]qβ )n−1 .

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Thus, we have (1 + [x1 ]qα

n X 1 − [x2 ]qβ )n−1

k=1

By the same method, we get
n k X 2
n Bk,n (x1 , x2 : q|α, β) k=2

k 1
n Bk,n (x1 , x2 1




n X

=

2

k=2

k   2
n [x1 ]kqα [1 n k 2

n−2 X

=

k=0

: q|α, β) = [x1 ]qα .




− x2 ]n−k q −β

 n−2 n−2−k [x1 ]k+2 q α [1 − x2 ]q −β k

[x1 ]2qα (1 + [x1 ]qα − [x2 ]qβ )n−2 .

= Thus, we obtain

(k2) B (x , x : q|α, β) (n2 ) k,n 1 2 = [x1 ]2qα . (1 + [x1 ]qα − [x2 ]qβ )n−2

Pn

k=2

Continuing this process, we obtain the following theorem. Theorem 7. For j ∈ Z+ , α, β ∈ R and x1 , x2 ∈ [0, 1], we have Pn (kj) k=j (n) Bk,n (x1 , x2 : q|α, β) j = [x1 ]jqα . (1 + [x1 ]qα − [x2 ]qβ )n−j In [3,5,9], the q-stirling numbers of the second kind are defined by k   k q −(2) X j (2j ) k [k − j]nq , (−1) q Sq (n, k) = [k]q ! j=0 j q where

k j q




=

[k]q ! [j]q ![k−j]q !

[x1 ]nqα

(22)

(23)

and [k]q ! = [k]q [k − 1]q · · · [2]q [1]q . For n ∈ Z+ , we have

=

n X k=0

q

α(k 2)



x1 k

 [k]qα !Sqα (k, n − k).

(24)

qα

By (22) and (24), we get the following corollary. Corollary 8. For n, j ∈ Z+ , x1 , x2 ∈ [0, 1], and α, β ∈ R, we have Pn (kj)   j X k=j (n) Bk,n (x1 , x2 : q|α, β) x1 j α(k ) 2 = q [k]qα !Sqα (k, j − k), n−j α (1 + [x1 ]q − [x2 ]qβ ) k qα k=0

3. On p-adic integral representation for Bk,n (x1 , x2 : q|α, β) In this section, we assume that q ∈ Cp with |1 − q|p < 1. From the definition of fermionic p-adic integral on Zp , we note that Z Z En,q−1 (1 − x) = [1 − x + y]nq−1 dµ−1 (y) = (−1)n q n [x + y]nq dµ−1 (y), (25) Zp

Zp

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and Z

[1 − x]nq−1 dµ−1 (x)

Zp n

n

Z

[x − 1]nq dµ−1 (x) = (−1)n q n En,q (−1) = En,q−1 (2).

= q (−1)

(26)

Zp

By the definition of q-Euler numbers and polynomials, we get En,q (2) = 2 + En,q , if n > 0.

(27)

From (26) and (27), we have the following theorem. Theorem 9. For n ∈ N, we have Z [1 − x]nq−1 dµ−1 (x) Zp

Z =2+

[x]nq−1 dµ−1 (x) = 2 + En,q−1 .

Zp

Taking double fermionic p-adic integral on Zp , we get Z Z Bk,n (x1 , x2 : q|α, β)dµ−1 (x1 )dµ−1 (x2 ) Zp

Zp

 Z Z n dµ−1 (x2 ) [1 − x2 ]n−k [x1 ]kqα dµ−1 (x1 ) = q −β k Zp Zp   n = Ek,qα (2 + En−k,q−β ), k where n > k, k ∈ Z+ and α, β ∈ Q . Thus, we obtain the following theorem.

(28)

Theorem 10. For n > k, k ∈ Z+ and α, β ∈ Q, we have Z Z Bk,n (x1 , x2 : q|α, β)dµ−1 (x1 )dµ−1 (x2 ) Zp

Zp

  n = Ek,qα (2 + En−k,q−β ). k By the q-symmetric properties for Bk,n (x1 , x2 : q|α, β), we get Z Z Bk,n (x1 , x2 : q|α, β)dµ−1 (x1 )dµ−1 (x2 ) Zp

=

Zp

k  X

Zl=0 =

 Z Z k (−1)k+l [1 − x1 ]k−l dµ (x ) [1 − x2 ]n−k dµ−1 (x2 ) −α −1 1 q q −β l Zp Zp

[1 − x2 ]n−k dµ−1 (x2 ) q −β

Zp k−1 X

 Z Z k + (−1)k+l [1 − x1 ]k−l dµ (x ) [1 − x2 ]n−k dµ−1 (x2 ) −α −1 1 q q −β l Z Z p p l=0 k−1 X k  = En−k,q−β (2) + (−1)k+l (2 + Ek−l,q−α )(2 + En−k,q−β ), (29) l l=0

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where n, k ∈ Z+ with n > k. Thus, we obtain the following theorem. Theorem 11. For n, k ∈ Z+ with n > k, we have Z

Z Bk,n (x1 , x2 : q|α, β)dµ−1 (x1 )dµ−1 (x2 )

Zp

Zp

= En−k,q−β (2) +

k−1 X l=0

 k (−1)k+l (2 + Ek−l,q−α )(2 + En−k,q−β ). l

From (28) and (29), we have the following corollary. Corollary 12. For n, k ∈ Z+ with n > k, we have   n Ek,qα (2 + En−k,q−β ) k k−1 X k  = En−k,q−β (2) + (−1)k+l (2 + Ek−l,q−α )(2 + En−k,q−β ). l

(30)

l=0

For m, n, k ∈ Z+ with m + n > 2k, we have Z

Z Bk,n (x1 , x2 : q|α, β)Bk,m (x1 , x2 : q|α, β)dµ−1 (x1 )dµ−1 (x2 )

Zp

Zp

   Z Z n m [x1 ]2k dµ (x ) [1 − x2 ]n+m−2k dµ−1 (x2 ) −1 1 qα q −β k k Zp Zp    n m E2k,qα (2 + En+m−2k,q−β ). = k k

=

Thus, we obtain the following theorem. Theorem 13. For m, n, k ∈ Z+ with m + n > 2k, we have Z

Z Bk,n (x1 , x2 : q|α, β)Bk,m (x1 , x2 : q|α, β)dµ−1 (x1 )dµ−1 (x2 )

Zp

Zp

   n m = E2k,qα (2 + En+m−2k,q−β ). k k

(31)

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From the q-symmetric properties of Bk,n (x1 , x2 : q|α, β), we note that Z Z Bk,n (x1 , x2 : q|α, β)Bk,m (x1 , x2 : q|α, β)dµ−1 (x1 )dµ−1 (x2 ) Zp

Zp

2k  X

 Z Z 2k (−1)2k+l [1 − x1 ]2k−l dµ (x ) [1 − x2 ]qn+m−2k dµ−1 (x2 ) −α −1 1 −β q l Z Z p p l=0 Z 2k−1 X 2k  n+m−2k = [1 − x2 ]q−β dµ−1 (x2 ) + (−1)2k+l l Zp Z Z l=0 2k−l × [1 − x1 ]q−α dµ−1 (x1 ) [1 − x2 ]qn+m−2k dµ−1 (x2 ) −β

=

Zp

Zp

= En+m−2k,q−β (2) 2k−1 X 2k  + (−1)l+2k (2 + E2k−l,q−α )(2 + En+m−2k,q−β ). l

(32)

l=0

Thus, we obtain the following theorem. Theorem 14. For n, m, k ∈ Z+ with n + m > 2k and α, β ∈ R, we have Z Z Bk,n (x1 , x2 : q|α, β)Bk,m (x1 , x2 : q|α, β)dµ−1 (x1 )dµ−1 (x2 ) Zp

Zp

= En+m−2k,q−β (2) 2k−1 X 2k  + (−1)l+2k (2 + E2k−l,q−α )(2 + En+m−2k,q−β ). l l=0

From (31) and (32), we have the following corollary. Corollary 15. For n, m, k ∈ Z+ with n + m > 2k and α, β ∈ R, we have    n m E2k,qα (2 + En+m−2k,q−β ) k k = En+m−2k,q−β (2) 2k−1 X 2k  + (−1)l+2k (2 + E2k−l,q−α )(2 + En+m−2k,q−β ). l

(33)

l=0

Continuing this process, we obtain the following equation. For n1 , n2 , · · · , ns , k ∈ Z+ (s ∈ N) with n1 + n2 + · · · + ns > sk, we get Z Z Y s Bk,ni (x1 , x2 : q|α, β)dµ−1 (x1 )dµ−1 (x2 ) Zp

= =

Zp i=1

s  Y i=1 s  Y i=1

ni k

Z

[x1 ]sk q α dµ−1 (x1 )

Zp

Z

1 +···+ns −sk [1 − x1 ]qn−α dµ−1 (x2 )

Zp

 ni Esk,q−β (2 + En1 +···+ns −sk,q−β ). k

Thus, we obtain the following theorem.

(34)

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Theorem 16. For n1 , · · · , ns , k ∈ Z+ (s ∈ N) with n1 +· · ·+ns > sk, and α, β ∈ R, x1 , x2 ∈ [0, 1], we have Z Z Y s Bk,ni (x1 , x2 : q|α, β)dµ−1 (x1 )dµ−1 (x2 ) Zp

Zp i=1

s  Y

=

i=1

 ni Esk,q−β (2 + En1 +···+ns −sk,q−β ). k

By using q-symmetric properties of Bk,n (x1 , x2 : q|α, β), we see that Z Z Y s Bk,ni (x1 , x2 : q|α, β)dµ−1 (x1 )dµ−1 (x2 ) Zp

=

Zp i=1

sk   X sk

l

l=0

Z

sk+l

Z [1 −

(−1)

x1 ]qsk−l −α dµ−1 (x1 )

Zp

Z

1 +···+ns −sk [1 − x2 ]qn−β dµ−1 (x2 )

Zp

1 +···+ns −sk [1 − x2 ]nq−β dµ−1 (x2 )

= Zp

+

sk−1 X l=0

 Z Z sk sk−l sk+l 1 +···+ns −sk (−1) [1 − x1 ]q−α dµ−1 (x1 ) [1 − x2 ]qn−β dµ−1 (x2 ) l Zp Zp

= En1 +···+ns −sk,q−β (2) sk−1 X sk  (−1)l+sk (2 + Esk−l,q−α )(2 + En1 +···+ns −sk,q−β ). + l

(35)

l=0

Thus, we have the following theorem. Theorem 17. For n1 , · · · , ns , k ∈ Z+ (s ∈ N) with n1 +· · ·+ns > sk, and α, β ∈ R, x1 , x2 ∈ [0, 1], we have Z Z Y s Bk,ni (x1 , x2 : q|α, β)dµ−1 (x1 )dµ−1 (x2 ) Zp

Zp i=1

= En1 +···+ns −sk,q−β (2) sk−1 X sk  + (−1)l+sk (2 + Esk−l,q−α )(2 + En1 +···+ns −sk,q−β ). l l=0

From (34) and (35), we obtain the following corollary. Corollary 18. For n1 , · · · , ns , k ∈ Z+ (s ∈ N) with n1 +· · ·+ns > sk, and α, β ∈ R, x1 , x2 ∈ [0, 1], we have s   Y ni Esk,qα (2 + En1 +···+ns −sk,q−β ) k i=1 = En1 +···+ns −sk,q−β (2) sk−1 X sk  + (−1)l+sk (2 + Esk−l,q−α )(2 + En1 +···+ns −sk,q−β ). l l=0

Acknowledgement. This Research was supported by Kyungpook National University Research Fund, 2010

294

RIM ET AL: KIM q-BERNSTEIN POLYNOMIALS

13

References [1] M. Acikgoz, S. Araci, A study on the integral of the product of several type Bernstein polynomials, IST Transaction of Applied Mathematics-Modelling and Simulation, 2010. [2] S. Bernstein, Demonstration du theoreme de weierstrass, fondee surle calcul des probabilities, Commun. Soc. Math. Kharkow (2) 13 (1912-13), 1-2. [3] T. Kim, Some identities on the q-Euler polynomials of higher order and qstirling numbers by the fermionic p-adic integral on Zp , Russ. J. Math. Phys. 16 (2009), 484-491. [4] T. Kim, q-Volkenborn integration, Russ. J. Math. Phys. 9 (2002), 288-299. [5] T. Kim, q-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients, Russ. J. Math. Phys. 15 (2008), 51-57. [6] T. Kim, L.-C. Jang, H. Yi, A note on the modified q-Bernstein polynomials, Discrete Dynamics in Nature and Society 2010 (2010), Article ID 706483, 12 pages. [7] T. Kim, Note on the Euler q-zeta functions, J. Number Theory 129 (2009), 1798-1804. [8] T. Kim, Barnes-type multiple q-zeta functions and q-Euler polynomials, J. Physics A:Math. Theor. 43 (2010), 255201, 11pp. [9] T. Kim, A note on q-Bernstein polynomials, Russ. J. Math. Phys. (accepted) [10] H. Ozden, Y. Simsek, S.-H. Rim, I. N. Cancul, A note on p-adic q-Euler measure, Adv. Stud. Contemp. Math. 14 (2007), 233-239. [11] G. M. Phillips, Bernstein polynomials based on the q-integrals, Annals of Numerical Analysis 4 (1997), 511-514. [12] Y. Simsek, M. Acikgoz, A new generating function of q-Bernstein-type polynomials and their interpolation function, Abstr. Appl. Anal. 2010 (2010), Article ID 769095, 12 pages. [13] T. Kim, q-Euler numbers and polynomials associated with p-adic q-integrals, J. Nonlinear Mathematical Physics 14 (2007), 15-27. [14] T. Kim, On the q-extension of Euler and Genocchi numbers, J. Math. Anal. Appl. 326 (2007), 1458–1465. pages. [15] T. Kim, On a q-analogue of the p-adic log gamma functions and related integrals, J. Number Theory 76 (1999), 320–329. [16] T. Kim, Symmetry properties of the generalized higher-order Euler polynomials, Proc. Jangjeon Math. Soc. 13 (2010), 13-16. [17] T. Kim, On the multiple q-Genocchi and Euler numbers, Russ. J. Math. Phys. 15 (2008), 481–486. [18] V. Gupta, T. Kim, J. Choi, Y.-H. Kim, Generating function for q-Bernstein, qMeyer-Konig-Zeller and q-Beta basis, Automation Computers Applied Mathematics 19 (2010), 7-11 pages. [19] T. Kim, Some identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials, Adv. Stud. Contemp. Math. 20 (2010), 23-28. [20] A. Bayad, T. Kim, Identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials, Adv. Stud. Contemp. Math. 20 (2010), 247-253.

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14

Seog-Hoon Rim. Department of Mathematics Education, Kyungpook National University, Taegu 702-701, Republic of Korea, E-mail address: [email protected] Lee-Chae Jang. Department of Mathematics and Computer Science, KonKuk University, Chungju 380-701, Republic of Korea, E-mail address: [email protected] Jongsung Choi. Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea, E-mail address: [email protected] Young-Hee Kim. Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea, E-mail address: [email protected] B. Lee. Department of Wireless Communications Engineering, Kwangwoon University, Seoul 139-701, Republic of Korea, E-mail address: bj− lee @kw.ac.kr Taekyun Kim. Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea, E-mail address: [email protected]

JOURNAL 296 OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.2, 296-304, 2011, COPYRIGHT 2011 EUDOXUS PRESS, LLC

JoCAAA 1 March 10

INNER PRODUCT SPACES AND FUNCTIONAL EQUATIONS YEOL JE CHO1 , CHOONKIL PARK2 , THEMISTOCLES M. RASSIAS3 AND REZA SAADATI4

1 Department

of Mathematics Education and the RINS, Gyeongsang National University, Chinju 660-701, Korea 2 Department

3 Department

of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Korea of Mathematics, National Technical University of Athens, Zografou Campus, 15780 Athens, Greece

4 Department of Mathematics and Computer Science, Amirkabir University of Technology, 424 Hafez Avenue, Tehran 15914, Iran

Abstract. In [7], Th.M. Rassias introduced the following equality n X

kxi − xj k2 = 2n

i,j=1

n X

n X

kxi k2 ,

xi = 0

i=1

i=1

for a fixed integer n ≥ 3. Let V, W be real vector spaces. In this paper, we show that, if a mapping f : V → W satisfies n X

n X

f (xi − xj ) = 2n

Pn

i,j=1

f (xi )

(0.1)

i=1

x = 0, then the mapping f : V → W is realized as the sum of an additive mapping for all x1 , · · · , xn ∈ V with i=1 i and a quadratic mapping. Furthermore, we prove the generalized Hyers-Ulam stability of the functional equation (0.1) in real Banach spaces. Keywords: additive mapping, quadratic mapping, functional equation associated with inner product space, generalized Hyers-Ulam stability. 2010 AMS Subject Classification: Primary 39B72, 46C05.

1. Introduction The stability problem of functional equations is originated from a question of Ulam [15] concerning the stability of group homomorphisms. Hyers [5] 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 Th.M. Rassias [6] for linear mappings by considering an unbounded Cauchy difference. The paper of Th.M. Rassias [6] has provided a lot of influence in the development of what we call generalized Hyers-Ulam stability of functional equations. A generalization of the Th.M. Rassias theorem was obtained by G˘avruta [4] by replacing the unbounded Cauchy difference by a general control function in the spirit of Th.M. Rassias’ approach. 0

The corresponding author: [email protected] (Reza Saadati) 1

297 2

Y. J. CHO, C. K. PARK, TH.M. RASSIAS AND R. SAADATI

A square norm on an inner product space satisfies the following parallelogram equality: kx + yk2 + kx − yk2 = 2kxk2 + 2kyk2 . The functional equation f (x + y) + f (x − y) = 2f (x) + 2f (y) is called a quadratic functional equation. A generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [14] 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. In [3], Czerwik proved the generalized Hyers-Ulam stability of the quadratic functional equation. A square norm on an inner product space satisfies the following equality: 3 X

2

kxi − xj k = 6

i,j=1

3 X

kxi k2

i=1

for all x1 , x2 , x3 ∈ R with x1 + x2 + x3 = 0 (see [7]). From the above equality, we can define the functional equation h(x − y) + h(2x + y) + h(x + 2y) = 3h(x) + 3h(y) + 3h(x + y), which is called a quadratic functional equation. In fact, h(x) = ax2 in R satisfies the quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. Several functional equations have been investigated in [8]–[13]. Throughout this paper, assume that n is a fixed integer greater than 2. Let X be a real normed vector space with norm k · k and Y a real Banach space with norm k · k. In this paper, we investigate the functional equation (0.1) and prove the generalized Hyers-Ulam stability of the functional equation (0.1) in real Banach spaces. 2. Functional equations associated with inner product spaces We investigate the functional equation (0.1). Lemma 2.1. Let V and W be real vector spaces. If a mapping f : V → W satisfies n X

f (xi − xj ) = 2n

i,j=1

for all x1 , · · · , xn ∈ V with

Pn i=1

n X

f (xi )

(2.1)

i=1

xi = 0, then the mapping f : V → W is realized as the sum of an

additive mapping and a quadratic mapping. Proof. Let g(x) :=

f (x)−f (−x) 2

and h(x) :=

f (x)+f (−x) 2

for all x ∈ V . Then g(x) is an odd mapping

and h(x) is an even mapping satisfying f (x) = g(x) + h(x) and (2.1).

298 INNER PRODUCT SPACES AND FUNCTIONAL EQUATIONS

3

Letting x1 = x, x2 = y, x3 = −x − y and x4 = · · · = xn = 0 in (2.1) for the mapping g, we get g(x) + g(y) − g(x + y) = g(x) + g(y) + g(−x − y) = 0 for all x, y ∈ V . So g(x) is an additive mapping. Letting x1 = x, x2 = y, x3 = −x − y and x4 = · · · = xn = 0 in (2.1) for the mapping h, we get h(x − y) + h(2x + y) + h(x + 2y) = 3h(x) + 3h(y) + 3h(x + y) for all x, y ∈ V . So h(x) is a quadratic mapping. This completes the proof.

¤

For a given mapping f : X → Y , we define Df (x1 , · · · , xn ) :=

n X

f (xi − xj ) − 2n

i,j=1

n X

f (xi )

i=1

for all x1 , · · · , xn ∈ X. Let g(x) :=

f (x)−f (−x) 2

and h(x) :=

f (x)+f (−x) 2

for all x ∈ X. Then g(x) is an odd mapping and h(x)

is an even mapping satisfying f (x) = g(x) + h(x). If Df (x1 , · · · , xn ) = 0, then Dg(x1 , · · · , xn ) = 0 and Dh(x1 , · · · , xn ) = 0. Now, we prove the generalized Hyers-Ulam stability of the functional equation Df (x1 , · · · , xn ) = 0 in real Banach spaces. Theorem 2.2. Let f : X → Y be a mapping satisfying f (0) = 0 for which there exists a function ϕ : X n → [0, ∞) such that ϕ(x e 1 , · · · , xn ) :=

∞ X j=1

4j ϕ

³x

1 ,··· 2j

,

xn ´ < ∞, 2j

(2.2)

kDh(x1 , · · · , xn )k ≤ ϕ(x1 , · · · , xn ) (2.3) Pn for all x1 , · · · , xn ∈ X with i=1 xi = 0. Then there exists a unique quadratic mapping Q : X → Y such that kh(x) − Q(x)k ≤

1 ϕ(x, e −x, 0, · · · , 0 ) | {z } 8

(2.4)

n−2 times

for all x ∈ X. Proof. Letting x1 = x, x2 = −x and x3 = · · · = xn = 0 in (2.3), we get k2h (2x) − 8h(x)k ≤ ϕ(x, −x, 0, · · · , 0 ) | {z } n−2 times

for all x ∈ X. It follows from (2.5) that

  ° ³ x ´° 1 x x ° ° °h(x) − 4h ° ≤ ϕ  , − , 0, · · · , 0  2 2 2 2 | {z } n−2 times

(2.5)

299 4

Y. J. CHO, C. K. PARK, TH.M. RASSIAS AND R. SAADATI

for all x ∈ X and so

  m X x x 1 x x k4l h( l ) − 4m h( m )k ≤ 4j ϕ  j , − j , 0, · · · , 0  2 2 8 2 2 | {z } j=l+1

(2.6)

n−2 times

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (2.2) and (2.6) that the sequence {4k h( 2xk )} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence {4k h( 2xk )} converges and so one can define the mapping Q : X → Y by ³x´ Q(x) := lim 4k h k k→∞ 2 for all x ∈ X. By (2.2) and (2.3), we have kDQ(x1 , · · · , xn )k

= ≤

for all x1 , · · · , xn ∈ X with

Pn i=1

° ³x xn ´° ° ° 1 lim 4k °Dh k , · · · , k ° k→∞ 2 2 ³x xn ´ 1 lim 4k ϕ k , · · · , k = 0 k→∞ 2 2

xi = 0. So DQ(x1 , · · · , xn ) = 0. Since h : X → Y is even,

Q : X → Y is even and so the mapping Q : X → Y is quadratic. Moreover, letting l = 0 and passing the limit m → ∞ in (2.6), we get (2.4). Hence there exists a quadratic mapping Q : X → Y satisfying (2.4). Now, let Q0 : X → Y be another quadratic mapping satisfying (2.4). Then we have ° ³x´ ³ x ´° ° ° kQ(x) − Q0 (x)k = 4q °Q q − Q0 q ° 2 2 ° ³x´ ° ³x´ ³ x ´° ³ x ´° ° ° ° ° ≤ 4q °Q q − h q ° + 4q °Q0 q − h q ° 2 2 2 2   x x ≤ 2 · 4q ϕ e  q , − q , 0, · · · , 0  , 2 2 | {z } n−2 times

which tends to zero as q → ∞ for all x ∈ X. Therefore, we can conclude that Q(x) = Q0 (x) for all x ∈ X. This proves the uniqueness of Q. This completes the proof.

¤

Corollary 2.3. Let p > 2 and θ be positive real numbers and f : X → Y be a mapping such that kDh(x1 , · · · , xn )k ≤ θ for all x1 , · · · , xn ∈ X with

n X

||xi ||p

(2.7)

i=1

Pn i=1

xi = 0. Then there exists a unique quadratic mapping Q : X → Y

such that kh(x) − Q(x)k ≤

θ ||x||p 2p − 4

for all x ∈ X.

Proof. Define ϕ(x1 , · · · , xn ) = θ

Pn i=1

||xi ||p and apply Theorem 2.2, then we get the conclusion.

¤

300 INNER PRODUCT SPACES AND FUNCTIONAL EQUATIONS

5

Theorem 2.4. Let f : X → Y be a mapping satisfying f (0) = 0 for which there exists a function ϕ : X n → [0, ∞) satisfying (2.3) such that ϕ(x e 1 , · · · , xn ) :=

∞ X

4−j ϕ(2j x1 , · · · , 2j xn ) < ∞

(2.8)

j=0

for all x1 , · · · , xn ∈ X. Then there exists a unique quadratic mapping Q : X → Y such that 1 kh(x) − Q(x)k ≤ Φ(x, −x, 0, · · · , 0 ) | {z } 8 n−2 times

for all x ∈ X. Proof. It follows from (2.5) that ° ° ° ° °h (x) − 1 h(2x)° ≤ 1 ϕ(x, −x, 0, · · · , 0 ) ° 8 ° | {z } 4 n−2 times

for all x ∈ X. The rest of the proof is similar to the proof of Theorem 2.2.

¤

Corollary 2.5. Let p < 2 and θ be positive real numbers and f : X → Y be a mapping satisfying (2.7). Then there exists a unique quadratic mapping Q : X → Y such that kh(x) − Q(x)k ≤

θ kxkp 4 − 2p

for all x ∈ X. Proof. Define ϕ(x1 , · · · , xn ) = θ

Pn i=1

||xi ||p and apply Theorem 2.4, then we get the conclusion.

¤

Theorem 2.6. Let f : X → Y be a mapping for which there exists a function ϕ : X n → [0, ∞) such that Φ(x1 , · · · , xn ) :=

∞ X j=1

2j ϕ

³x

1 ,··· 2j

,

xn ´ < ∞, 2j

(2.9)

kDg(x1 , · · · , xn )k ≤ ϕ(x1 , · · · , xn ) (2.10) Pn for all x1 , · · · , xn ∈ X with i=1 xi = 0. Then there exists a unique additive mapping A : X → Y such that kg(x) − A(x)k ≤

1 Φ(x, x, −2x, 0, · · · , 0 ) | {z } 4n n−3 times

for all x ∈ X. Proof. Letting x1 = x, x2 = x, x3 = −2x and x4 = · · · = xn = 0 in (2.10), we get k4ng (x) − 2ng (2x)k ≤ ϕ(x, x, −2x, 0, · · · , 0 ) | {z }

(2.11)

n−3 times

for all x ∈ X. The rest of the proof is similar to the proof of Theorem 2.2.

¤

301 6

Y. J. CHO, C. K. PARK, TH.M. RASSIAS AND R. SAADATI

Corollary 2.7. Let p > 1 and θ be positive real numbers and f : X → Y be a mapping such that n X kDg(x1 , · · · , xn )k ≤ θ ||xi ||p (2.12) for all x1 , · · · , xn ∈ X with

i=1

Pn

i=1 xi = 0. Then there exists a unique additive mapping A : X → Y

such that kg(x) − A(x)k ≤

(2p + 2)θ kxkp 2n(2p − 2)

for all x ∈ X. Proof. Define ϕ(x1 , · · · , xn ) = θ

Pn i=1

||xi ||p and apply Theorem 2.6, then we get the conclusion.

¤

Theorem 2.8. Let f : X → Y be a mapping for which there exists a function ϕ : X n → [0, ∞) satisfying (2.10) such that Φ(x1 , · · · , xn ) : for all x1 , · · · , xn ∈ X with

=

∞ X

¡ ¢ 2−j ϕ 2j x1 , · · · , 2j xn < ∞

(2.13)

j=0

Pn i=1

xi = 0. Then there exists a unique additive mapping A : X → Y

such that kg(x) − A(x)k ≤

1 Φ(x, x, −2x, 0, · · · , 0 ) | {z } 4n n−3 times

for all x ∈ X. Proof. It follows from (2.11) that ° ° ° ° °g (x) − 1 g (2x)° ≤ 1 ϕ(x, x, −2x, 0, · · · , 0 ) ° 4n ° | {z } 2 n−3 times

for all x ∈ X. The rest of the proof is similar to the proof of Theorem 2.2.

¤

Corollary 2.9. Let p < 1 and θ be positive real numbers and f : X → Y be a mapping satisfying (2.12). Then there exists a unique additive mapping A : X → Y such that kg(x) − A(x)k ≤

(2 + 2p )θ ||x||p 2n(2 − 2p )

for all x ∈ X. Proof. Define ϕ(x1 , · · · , xn ) = θ

Pn i=1

kxi kp and apply Theorem 2.8, then we get the conclusion. ¤

From kDf (x1 , · · · , xn )k ≤ ϕ(x1 , · · · , xn ), we have 1 1 kDh(x1 , · · · , xn )k ≤ ϕ(x1 , · · · , xn ) + ϕ(−x1 , · · · , −xn ) 2 2 and 1 1 ϕ(x1 , · · · , xn ) + ϕ(−x1 , · · · , −xn ). kDg(x1 , · · · , xn )k ≤ 2 2

302 INNER PRODUCT SPACES AND FUNCTIONAL EQUATIONS

Note that ∞ X

2j ϕ

³x

j=1

1 ,··· 2j

7

xn ´ X j ³ x1 xn ´ ≤ 4 ϕ , · · · , . 2j 2j 2j j=1 ∞

,

Thus, combining Theorem 2.2 and Theorem 2.6, we obtain the following result: Theorem 2.10. Let f : X → Y be a mapping satisfying f (0) = 0 for which there exists a function ϕ : X n → [0, ∞) satisfying (2.2) and kDf (x1 , · · · , xn )k for all x1 , · · · , xn ∈ X with

Pn i=1

≤

ϕ(x1 , · · · , xn )

(2.14)

xi = 0. Then there exist a unique additive mapping A : X → Y

and a unique quadratic mapping Q : X → Y such that

≤

kf (x) − A(x) − Q(x)k 1 1 ϕ(x, e −x, 0, · · · , 0 ) + ϕ(−x, e x, 0, · · · , 0 ) | {z } | {z } 16 16 n−2 times

n−2 times

1 1 + Φ(x, x, −2x, 0, · · · , 0 ) + Φ(−x, −x, 2x, 0, · · · , 0 ) | {z } | {z } 8n 8n n−3 times

n−3 times

for all x ∈ X, where ϕ e and Φ are defined in (2.2) and (2.9), respectively.

Corollary 2.11. Let p > 2 and θ be positive real numbers and f : X → Y be a mapping such that kDf (x1 , · · · , xn )k ≤ θ for all x1 , · · · , xn ∈ X with

Pn i=1

n X

||xi ||p

(2.15)

i=1

xi = 0. Then there exist a unique additive mapping A : X → Y

and a unique quadratic mapping Q : X → Y such that µ ¶ 1 2p + 2 k2f (x) − A(x) − Q(x)k ≤ + θ||x||p 2p − 4 2n(2p − 2) for all x ∈ X.

Proof. Define ϕ(x1 , · · · , xn ) = θ

Pn i=1

||xi ||p and apply Theorem 2.10, then we get the conclusion. ¤

Note that ∞ X j=0

4−j ϕ(2j x1 , · · · , 2j xn ) ≤

∞ X

2−j ϕ(2j x1 , · · · , 2j xn ).

j=1

Thus, combining Theorem 2.4 and Theorem 2.8, we obtain the following result: Theorem 2.12. Let f : X → Y be a mapping satisfying f (0) = 0 for which there exists a function ϕ : X n → [0, ∞) satisfying (2.13) and (2.14). Then there exist a unique additive mapping A : X → Y

303 8

Y. J. CHO, C. K. PARK, TH.M. RASSIAS AND R. SAADATI

and a unique quadratic mapping Q : X → Y such that

≤

kf (x) − A(x) − Q(x)k 1 1 ϕ(x, e −x, 0, · · · , 0 ) + ϕ(−x, e x, 0, · · · , 0 ) | {z } | {z } 16 16 n−2 times

n−2 times

1 1 + Φ(x, x, −2x, 0, · · · , 0 ) + Φ(−x, −x, 2x, 0, · · · , 0 ) | {z } | {z } 8n 8n n−3 times

n−3 times

for all x ∈ X, where ϕ e and Φ are defined in (2.8) and (2.13), respectively. Corollary 2.13. Let p < 1 and θ be positive real numbers and f : X → Y be a mapping satisfying (2.15). Then there exist a unique additive mapping A : X → Y and a unique quadratic mapping Q : X → Y such that

µ kf (x) − A(x) − Q(x)k ≤

1 2 + 2p + p 4−2 2n(2 − 2p )

¶ θ||x||p

for all x ∈ X. Proof. Define ϕ(x1 , · · · , xn ) = θ

Pn i=1

||xi ||p and apply Theorem 2.12, then we get the conclusion. ¤ Acknowledgements

The first author was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00050) and the second author was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2009-0070788). References [1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. [2] P.W. Cholewa, Remarks on the stability of functional equations, Aequat. Math. 27 (1984), 76–86. [3] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59–64. [4] P. G˘ avruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436. [5] D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA 27 (1941), 222–224. [6] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [7] Th.M. Rassias, On characterizations of inner product spaces and generalizations of the H. Bohr inequality, in Topics in Mathematical Analysis (ed. Th.M. Rassias), World Scientific Publ. Co., Singapore, 1989, pp. 803–819. [8] Th.M. Rassias, On the stability of the quadratic functional equation and its applications, Studia Univ. BabesBolyai XLIII (1998), 89–124. [9] Th.M. Rassias, The problem of S.M. Ulam for approximately multiplicative mappings, J. Math. Anal. Appl. 246 (2000), 352–378. [10] Th.M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 251 (2000), 264–284. [11] Th.M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math. 62 (2000), 23–130. ˇ [12] Th.M. Rassias and P. Semrl, On the Hyers-Ulam stability of linear mappings, J. Math. Anal. Appl. 173 (1993), 325–338.

304 INNER PRODUCT SPACES AND FUNCTIONAL EQUATIONS

9

[13] Th.M. Rassias and K. Shibata, Variational problem of some quadratic functionals in complex analysis, J. Math. Anal. Appl. 228 (1998), 234–253. [14] F. Skof, Propriet` a locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53 (1983), 113–129. [15] S.M. Ulam, Problems in Modern Mathematics, Wiley, New York, 1960.

JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.2, 305-320, 2011, COPYRIGHT 2011 EUDOXUS PRESS, 305 LLC

FUZZY FUNCTIONAL INEQUALITIES YEOL JE CHO, CHOONKIL PARK, AND REZA SAADATI Abstract. In this paper, we investigate the following functional inequalities kf (x) + f (y) + f (z)k

≤

and kf (x) + f (y) + 2f (z)k

≤

kf (x + y + z)k

(0.1)

° ³ ´° x+y ° ° +z ° °2f

(0.2) 2 in fuzzy normed vector spaces and prove the generalized Hyers-Ulam stability of the functional inequalities (0.1) and (0.2) in fuzzy normed vector spaces in the spirit of the Th.M. Rassias stability approach.

1. Introduction and preliminaries The stability problem of functional equations originated from a question of Ulam [32] concerning the stability of group homomorphisms. Hyers [9] 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 Th.M. Rassias [22] for linear mappings by considering an unbounded Cauchy difference. The paper of Th.M. Rassias [22] has provided a lot of influence in the development of what we call the generalized Hyers-Ulam stability of functional equations. A generalization of the Th.M. Rassias theorem was obtained by G˘avruta [6] by replacing the unbounded Cauchy difference by a general control function in the spirit of Th.M. Rassias’ approach. During the last two decades a number of papers and research monographs have been published on various generalizations and applications of the generalized Hyers-Ulam stability to a number of functional equations and mappings (see [11], [16]–[20]). Gil´ anyi [7] showed that if f satisfies the functional inequality k2f (x) + 2f (y) − f (x − y)k ≤ kf (x + y)k

(1.1)

then f satisfies the Jordan-von Neumann functional equation 2f (x) + 2f (y) = f (x + y) + f (x − y). See also [25]. Fechner [4] and Gil´anyi [8] proved the generalized Hyers-Ulam stability of the functional inequality (1.1). Park, Cho and Han [21] investigated the Cauchy additive 0

The corresponding author: [email protected] (Reza Saadati) 2000 Mathematics Subject Classification. Primary 46S40; Secondary 39B72, 39B52, 46S50, 26E50. Key words and phrases. fuzzy Banach space, additive functional inequality, additive functional equation, generalized Hyers-Ulam stability. 1

306 2

YEOL JE CHO, CHOONKIL PARK, AND REZA SAADATI

functional inequality kf (x) + f (y) + f (z)k ≤ kf (x + y + z)k

(1.2)

and the Cauchy-Jensen additive functional inequality

° µ ¶° ° ° x+y ° kf (x) + f (y) + 2f (z)k ≤ °2f +z ° ° 2

(1.3)

and proved the generalized Hyers-Ulam stability of the functional inequalities (1.2) and (1.3) in Banach spaces. Katsaras [13] defined a fuzzy norm on a vector space to construct a fuzzy vector topological structure on the space. Some mathematicians have defined fuzzy norms on a vector space from various points of view [15, 33]. In particular, Saadati and Vaezpour [27], following Cheng and Mordeson [2], gave an idea of fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michalek type [14]. They established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces [28, 29, 31]. We use the definition of fuzzy (random) normed spaces given in [26, 27, 30] to investigate a fuzzy version of the generalized Hyers-Ulam stability for the Cauchy functional inequality (1.2) and the Cauchy-Jensen functional inequality (1.3) in the fuzzy normed vector space setting. Definition 1.1. Let X be a real vector space. A function N : X × R → [0, 1] is called a fuzzy norm on X if for all x, y ∈ X and all s, t ∈ R, (N1 ) N (x, t) = 0 for t ≤ 0; (N2 ) x = 0 if and only if N (x, t) = 1 for all t > 0; t (N3 ) N (cx, t) = N (x, |c| ) if c 6= 0; (N4 ) N (x + y, s + t) ≥ min{N (x, s), N (y, t)}; (N5 ) N (x, ·) is a non-decreasing function of R and limt→∞ N (x, t) = 1; (N6 ) for x 6= 0, N (x, ·) is continuous on R. The pair (X, N ) is called a fuzzy normed vector space. The properties of fuzzy normed vector spaces and examples of fuzzy norms are given in [27]. Definition 1.2. Let (X, N ) be a fuzzy normed vector space. A sequence {xn } in X is said to be convergent or converges if there exists x ∈ X such that limn→∞ N (xn −x, t) = 1 for all t > 0. In this case, x is called the limit of the sequence {xn } and we denote it by N -limn→∞ xn = x. Definition 1.3. Let (X, N ) be a fuzzy normed vector space. A sequence {xn } in X is called a Cauchy sequence if, for any ε > 0 and t > 0, there exists n0 ∈ N such that, for all n ≥ n0 and p > 0, N (xn+p − xn , t) > 1 − ε.

307 FUZZY FUNCTIONAL INEQUALITIES

3

It is well-known that every convergent sequence in a fuzzy normed vector space is a Cauchy sequence. If every Cauchy sequence is convergent, then the fuzzy normed space (X, N ) is said to be complete and the complete fuzzy normed vector space is called a fuzzy Banach space. We say that a mapping f : X → Y between fuzzy normed vector spaces X and Y is continuous at a point x0 ∈ X if, for each sequence {xn } converging to x0 in X, the sequence {f (xn )} converges to f (x0 ). If f : X → Y is continuous at every x ∈ X, then f : X → Y is said to be continuous on X (see [27]). This paper is organized as follows: In Section 2, we investigate the functional inequalities (1.2) and (1.3) in fuzzy normed vector spaces, and prove the generalized Hyers-Ulam stability of the functional inequalities (1.2) and (1.3) in fuzzy Banach spaces. Throughout this paper, assume that X is a vector space and that (Y, N ) is a fuzzy Banach space. 2. Generalized Hyers-Ulam stability of functional inequalities in fuzzy normed vector spaces In this section, we investigate the functional inequalities (1.2) and (1.3) in fuzzy normed vector spaces and prove the generalized Hyers-Ulam stability of the functional inequalities (1.2) and (1.3) in fuzzy Banach spaces. Lemma 2.1. Let (Z, N ) be a fuzzy normed vector space. Let f : X → Z be a mapping such that µ ¶ t N (f (x) + f (y) + f (z), t) ≥ N f (x + y + z), (2.1) 2 for all x, y, z ∈ X and t > 0. Then f is Cauchy additive, that is, f (x + y) = f (x) + f (y) for all x, y ∈ X. Proof. Letting x = y = z = 0 in (2.1), we get µ

¶

µ

¶

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 y = −x and z = 0 in (2.1), we get µ ¶ µ ¶ t t N (f (x) + f (−x), t) ≥ N f (0), = N 0, =1 2 2 for all t > 0. It follows from (N2 ) that f (x) + f (−x) = 0 for all x ∈ X and so f (−x) = −f (x) for all x ∈ X. Letting z = −x − y in (2.1), we get N (f (x) + f (y) − f (x + y), t) = N (f (x) + f (y) + f (−x − y), t) µ ¶ µ ¶ t t ≥ N f (0), = N 0, =1 2 2

308 4

YEOL JE CHO, CHOONKIL PARK, AND REZA SAADATI

for all x, y ∈ X and t > 0. By (N2 ), N (f (x) + f (y) − f (x + y), t) = 1 for all x, y ∈ X and t > 0 and so f (x + y) = f (x) + f (y) for all x, y ∈ X. This completes the proof.

¤

Lemma 2.2. Let (Z, N ) be a fuzzy normed vector space. Let f : X → Z be a mapping such that µ µ ¶ ¶ x+y 2 N (f (x) + f (y) + 2f (z), t) ≥ N 2f +z , t (2.2) 2 3 for all x, y, z ∈ X and all t > 0. Then f is Cauchy additive, that is, f (x + y) = f (x) + f (y) for all x, y ∈ X. Proof. Letting x = y = z = 0 in (2.2), we get µ

N (4f (0), t) = N f (0),

t 4

¶

µ

¶

µ

2 t ≥ N 2f (0), t = N f (0), 3 3

¶

for all t > 0. By (N5 ) and (N6 ), N (f (0), t) = 1 for all t > 0 and so f (0) = 0. Letting y = −x and z = 0 in (2.2), we get µ

¶

µ

¶

2 2 N (f (x) + f (−x), t) ≥ N 2f (0), t = N 0, t = 1 3 3 for all t > 0. It follows from (N2 ) that f (x) + f (−x) = 0 for all x ∈ X and so f (−x) = −f (x) for all x ∈ X. Letting z = − x+y 2 in (2.2), we get µ

N (f (x) + f (y) − 2f

x+y ,t 2

¶

µ

µ

¶

x+y = N f (x) + f (y) + 2f − ,t 2 ¶ µ ¶ µ 2 2 ≥ N 2f (0), t = N 0, t = 1 3 3

¶

for all x, y ∈ X and t > 0. By (N2 ), N (f (x) + f (y) − 2f ( x+y 2 ), t) = 1 for all x, y ∈ X and t > 0 and so µ ¶ x+y 2f = f (x) + f (y) 2 for all x, y ∈ X. Since f (0) = 0, µ

f (x + y) = 2f for all x, y ∈ X. this completes the proof.

x+y 2

¶

= f (x) + f (y) ¤

Now, we prove the generalized Hyers-Ulam stability of the Cauchy functional inequality (1.2) in fuzzy Banach spaces.

309 FUZZY FUNCTIONAL INEQUALITIES

5

Theorem 2.3. Let ϕ : X 3 → [0, ∞) be a function such that e ϕ(x, y, z) :=

∞ X

2−n ϕ(2n x, 2n y, 2n z) < ∞

(2.3)

n=0

for all x, y, z ∈ X. Let f : X → Y be an odd mapping such that lim N (f (x) + f (y) + f (z), tϕ(x, y, z)) = 1

(2.4)

t→∞

n

uniformly on X 3 . Then L(x) := N -limn→∞ f (22n x) exists for any x ∈ X and defines a Cauchy additive mapping L : X → Y such that, if there exist δ > 0 and α > 0 satisfying N (f (x) + f (y) + f (z), δϕ(x, y, z)) ≥ α for all x, y, z ∈ X, then

µ

(2.5)

¶

δ e N f (x) − L(x), ϕ(x, x, −2x) ≥ α 2 for all x ∈ X. Furthermore, the additive mapping L : X → Y is a unique mapping such that e lim N (f (x) − L(x), tϕ(x, x, −2x)) = 1

(2.6)

t→∞

uniformly on X. Proof. Since f is an odd mapping, f (−x) = −f (x) for all x ∈ X and f (0) = 0. For any ε > 0, by (2.4), we can find some t0 > 0 such that N (f (x) + f (y) + f (z), tϕ(x, y, z)) ≥ 1 − ε for all t ≥ t0 . By induction on n, we show that Ã

n

n

N 2 f (x) − f (2 x), t

n−1 X

n−k−1

2

(2.7) !

k

k

k+1

ϕ(2 x, 2 x, −2

x)

≥1−ε

(2.8)

k=0

for all t ≥ t0 , x ∈ X and n ∈ N. In fact, letting y = x and z = −2x in (2.7), we get N (2f (x) − f (2x), tϕ(x, x, −2x)) ≥ 1 − ε for all x ∈ X and t ≥ t0 and so we get (2.8) for n = 1. Assume that (2.8) holds for n ∈ N. Then we have à n+1

N 2

(

f (x) − f (2

n+1

x), t

!

2

n−k

k

k

k+1

ϕ(2 x, 2 x, −2

x)

k=0

Ã

≥ min N 2

n X

n+1

n

f (x) − 2f (2 x), t0

n−1 X k=0

! n−k

2

n

n

ϕ(2 x, 2 x, −2 o

N (2f (2n x) − f (2n+1 x), t0 ϕ(2n x, 2n x, −2n+1 x)) ≥ min{1 − ε, 1 − ε} = 1 − ε. This completes the induction argument.

n+1

x) ,

310 6

YEOL JE CHO, CHOONKIL PARK, AND REZA SAADATI

Letting t = t0 and replacing n and x by p and 2n x in (2.8), respectively, we get 



p−1 f (2n x) f (2n+p x) t0 X p−k−1 N − , 2 ϕ(2n+k x, 2n+k x, −2n+k+1 x) 2n 2n+p 2n+p k=0

≥1−ε

(2.9)

for all n ≥ 0 and p > 0. It follows from (2.3) and the equality p−1 X

2

−n−k−1

ϕ(2

n+k

n+k

x, 2

x, −2

n+k+1

k=0

n+p−1 1 X −k x) = 2 ϕ(2k x, 2k x, −2k+1 x) 2 k=n

that, for some δ > 0, there exists n0 ∈ N such that n+p−1 t0 X −k 2 ϕ(2k x, 2k x, −2k+1 x) < δ 2 k=n

for all n ≥ n0 and p > 0. Now, we deduce from (2.9) that Ã

N

f (2n x) f (2n+p x) − ,δ 2n 2n+p

!



≥N

f (2n x) 2n

−

f (2n+p x) 2n+p

,

t0 2n+p

p−1 X



2p−k−1 ϕ(2n+k x, 2n+k x, −2n+k+1 x)

k=0

≥1−ε n

for all n ≥ n0 and p > 0. Thus the sequence { f (22n x) } is a Cauchy sequence in Y . Since Y n is a fuzzy Banach space, the sequence { f (22n x) } converges to a point L(x) ∈ Y and so we can n define a mapping L : X → Y by L(x) := N -limn→∞ f (22n x) , that is, for any t > 0 and x ∈ X, n limn→∞ N ( f (22n x) − L(x), t) = 1. Let x, y, z ∈ X and fix t > 0, 0 < ε < 1. Since limn→∞ 2−n ϕ(2n x, 2n y, 2n z) = 0, there n exists n1 > n0 such that t0 ϕ(2n x, 2n y, 2n z) < 24 t for all n ≥ n1 . Hence, for all n ≥ n1 , we have N (L(x) + L(y) + L(z), t) ½

µ

¶

≥ min N L(x) − 2−n f (2n x), µ

¶

µ

¶

t t , N L(y) − 2−n f (2n y), , 16 16 µ

¶

t t , N L(x + y + z) − 2−n f (2n (x + y + z)), , 16 16 µ ¶ µ ¶¾ 2n t t N f (2n (x + y + z)) − f (2n x) − f (2n y) − f (2n z), , N L(x + y + z), . 4 2 The first four terms on the right-hand side of the above inequality tend to 1 as n → ∞ and the fifth term is greater than N L(z) − 2−n f (2n z),

N (f (2n (x + y + z)) − f (2n x) − f (2n y) − f (2n z), t0 ϕ(2n x, 2n y, 2n z)), which is greater than or equal to 1 − ε. Thus it follows that ½

µ

¶

t ,1 − ε N (L(x) + L(y) + L(z), t) ≥ min N L(x + y + z), 2

¾

311 FUZZY FUNCTIONAL INEQUALITIES

for all t > 0 and 0 < ε < 1 and so

µ

N (L(x) + L(y) + L(z), t) ≥ N L(x + y + z),

7

t 2

¶

for all t > 0 or N (L(x) + L(y) + L(z), t) ≥ 1 − ε for all t > 0. For the former case, by Lemma 2.1, the mapping L : X → Y is Cauchy additive. For the latter case, N (L(x) + L(y) + L(z), t) = 1 for all t > 0 and hence N (3L(x), t) = 1 for all t > 0 and x ∈ X. By (N2 ), L(x) = 0 for all x ∈ X. Thus the mapping L : X → Y is Cauchy additive, that is, L(x + y) = L(x) + L(y) for all x, y ∈ X. Now, for some positive δ and α, assume that (2.5) holds. Let ϕn (x, y, z) :=

n−1 X

2−k−1 ϕ(2k x, 2k y, 2k z)

k=0

for all x, y, z ∈ X. For any x ∈ X, by the same reasoning as in the beginning of the proof, one can deduce from (2.5) that Ã

N 2n f (x) − f (2n x), δ

n−1 X

!

2n−k−1 ϕ(2k x, 2k x, −2k+1 x)

≥α

(2.10)

k=0

for all n ∈ N. For any t > 0, we have N (f (x) − L(x), δϕn (x, x, −2x) + t) ½ µ ¶ µ ¶¾ f (2n x) f (2n x) ≥ min N f (x) − , δϕn (x, x, −2x) , N − L(x), t . 2n 2n

(2.11)

n

Thus, combining (2.10) and (2.11), it follows the fact that limn→∞ N ( f (22n x) − L(x), t) = 1 that N (f (x) − L(x), δϕn (x, x, −2x) + t) ≥ α for large enough n ∈ N. From the continuity of the function N (f (x) − L(x), ·), we see that e N (f (x) − L(x), 2δ ϕ(x, x, −2x) + t) ≥ α. Letting t → 0, we conclude that µ

¶

δ e N f (x) − L(x), ϕ(x, x, −2x) ≥ α. 2 Finally, it remains to prove the uniqueness assertion. Let T be another additive mapping satisfying (2.6). Fix c > 0. For any ε > 0, by (2.6) for L and T , we can find some t0 > 0 such that µ ¶ t e N f (x) − L(x), ϕ(x, x, −2x) ≥ 1 − ε, 2 µ ¶ t e N f (x) − T (x), ϕ(x, x, −2x) ≥ 1−ε 2 for all x ∈ X and t ≥ t0 . Fix some x ∈ X and find some integer n0 such that t0

∞ X k=n

2−k ϕ(2k x, 2k x, −2k+1 x)
0. Thus L(x) = T (x) for all x ∈ X. This completes the proof. ¤ Corollary 2.4. Let θ ≥ 0 and p be a real number with 0 < p < 1. Let f : X → Y be an odd mapping such that lim N (f (x) + f (y) + f (z), tθ(kxkp + kykp + kzkp )) = 1

t→∞

(2.12)

n

uniformly on X 3 . Then L(x) := N -limn→∞ f (22n x) exists for any x ∈ X and defines a Cauchy additive mapping L : X → Y such that, if there exist δ > 0 and α > 0 satisfying N (f (x) + f (y) + f (z), δθ(kxkp + kykp + kzkp )) ≥ α for all x, y, z ∈ X, then

µ

¶

2 + 2p δθkxkp ≥ α 2 − 2p for all x ∈ X. Furthermore, the additive mapping L : X → Y is a unique mapping such that N f (x) − L(x),

lim N (f (x) − L(x),

t→∞

uniformly on X.

2 + 2p 2tθkxkp ) = 1 2 − 2p

313 FUZZY FUNCTIONAL INEQUALITIES

9

Proof. Define ϕ(x, y, z) := θ(kxkp + kykp + kzkp ) and apply Theorem 2.3, then we get the conclusion. ¤ Similarly, we can obtain the following and so omit the proofs. Theorem 2.5. Let ϕ : X 3 → [0, ∞) be a function such that ∞ X

µ

x y z e ϕ(x, y, z) := 2 ϕ n, n, n 2 2 2 n=1

¶

n

0 and α > 0 satisfying N (f (x) + f (y) + f (z), δϕ(x, y, z)) ≥ α for all x, y, z ∈ X, then

µ

¶

δ e N f (x) − L(x), ϕ(x, x, −2x) ≥ α 2 for all x ∈ X. Furthermore, the additive mapping L : X → Y is a unique mapping such that e lim N (f (x) − L(x), tϕ(x, x, −2x)) = 1

t→∞

uniformly on X. Corollary 2.6. Let θ ≥ 0 and p be a real number with p > 1. Let f : X → Y be an odd mapping satisfying (2.12). Then L(x) := N -limn→∞ 2n f ( 2xn ) exists for any x ∈ X and defines a Cauchy additive mapping L : X → Y such that, if there exist δ > 0 and α > 0 satisfying N (f (x) + f (y) + f (z), δθ(kxkp + kykp + kzkp )) ≥ α for all x, y, z ∈ X, then

µ

¶

2p + 2 δθkxkp ≥ α N f (x) − L(x), p 2 −2 for all x ∈ X. Furthermore, the additive mapping L : X → Y is a unique mapping such that lim N (f (x) − L(x),

t→∞

2p + 2 2tθkxkp ) = 1 2p − 2

uniformly on X. Proof. Define ϕ(x, y, z) := θ(kxkp + kykp + kzkp ) and apply Theorem 2.5, then we get the conclusion. ¤ Finally, we prove the generalized Hyers-Ulam stability of the Cauchy-Jensen functional inequality (1.3) in fuzzy Banach spaces.

314 10

YEOL JE CHO, CHOONKIL PARK, AND REZA SAADATI

Theorem 2.7. Let ϕ : X 3 → [0, ∞) be a function satisfying (2.3). Let f : X → Y be an odd mapping such that lim N (f (x) + f (y) + 2f (z), tϕ(x, y, z)) = 1

(2.14)

t→∞

n

uniformly on X 3 . Then L(x) := N -limn→∞ f (22n x) exists for any x ∈ X and defines a Cauchy additive mapping L : X → Y such that, if there exist δ > 0 and α > 0 satisfying N (f (x) + f (y) + 2f (z), δϕ(x, y, z)) ≥ α for all x, y, z ∈ X, then

µ

(2.15)

¶

δ e −2x, x) ≥ α N f (x) − L(x), ϕ(0, 2 for all x ∈ X. Furthermore, the additive mapping L : X → Y is a unique mapping such that e −2x, x)) = 1 lim N (f (x) − L(x), tϕ(0,

(2.16)

t→∞

uniformly on X. Proof. Since f is an odd mapping, f (−x) = −f (x) for all x ∈ X and f (0) = 0. For any ε > 0, by (2.14), we can find some t0 > 0 such that N (f (x) + f (y) + 2f (z), tϕ(x, y, z)) ≥ 1 − ε for all t ≥ t0 . By induction on n, we show that Ã

n

n

N 2 f (x) − f (2 x), t

n−1 X

n−k−1

2

(2.17)

!

ϕ(0, −2

k+1

k

x, 2 x) ≥ 1 − ε

(2.18)

k=0

for all t ≥ t0 , x ∈ X and n ∈ N. In fact, letting x = 0, y = −2x and z = x in (2.17), we get N (2f (x) − f (2x), tϕ(0, −2x, x)) ≥ 1 − ε for all x ∈ X and t ≥ t0 and so we get (2.18) for n = 1. Assume that (2.18) holds for n ∈ N. Then we have Ã

n+1

N 2

(

f (x) − f (2

n+1

x), t

n X

!

2

n−k

ϕ(0, −2

k+1

k

x, 2 x)

k=0

à n+1

≥ min N 2

n

f (x) − 2f (2 x), t0

n−1 X k=0

! n−k

2

k+1

ϕ(0, −2

k

x, 2 x) ,

o

N (2f (2n x) − f (2n+1 x), t0 ϕ(0, −2n+1 x, 2n x)) ≥ min{1 − ε, 1 − ε} = 1 − ε.

This completes the induction argument. Letting t = t0 and replacing n and x by p and 2n x in (2.18), respectively, we get 



p−1 f (2n x) f (2n+p x) t0 X p−k−1 2 ϕ(0, −2n+k+1 x, 2n+k x) N − , 2n 2n+p 2n+p k=0

≥1−ε

(2.19)

315 FUZZY FUNCTIONAL INEQUALITIES

11

for all n ≥ 0 and p > 0. It follows from (2.3) and the equality p−1 X

2−n−k−1 ϕ(0, −2n+k+1 x, 2n+k x) =

k=0

n+p−1 1 X −k 2 ϕ(0, −2k+1 x, 2k x) 2 k=n

that, for a given δ > 0, there exists n0 ∈ N such that n+p−1 t0 X −k 2 ϕ(0, −2k+1 x, 2k x) < δ 2 k=n

for all n ≥ n0 and p > 0. Now, we deduce from (2.19) that Ã

N

f (2n x) f (2n+p x) − ,δ 2n 2n+p

!





p−1 f (2n x) f (2n+p x) t0 X p−k−1 ≥N − , 2 ϕ(0, −2n+k+1 x, 2n+k x) 2n 2n+p 2n+p k=0

≥1−ε n

for all n ≥ n0 and p > 0. Thus the sequence { f (22n x) } is a Cauchy sequence in Y . Since Y n is a fuzzy Banach space, the sequence { f (22n x) } converges to a point L(x) ∈ Y and so we can n define a mapping L : X → Y by L(x) := N -limn→∞ f (22n x) , that is, for any t > 0 and x ∈ X, n limn→∞ N ( f (22n x) − L(x), t) = 1. Let x, y, z ∈ X and fix t > 0, 0 < ε < 1. Since limn→∞ 2−n ϕ(2n x, 2n y, 2n z) = 0, there n exists n1 > n0 such that t0 ϕ(2n x, 2n y, 2n z) < 212t for all n ≥ n1 . Hence, for all n ≥ n1 , we have N (L(x) + L(y) + 2L(z), t) ½

µ

¶

t , 16 µ ¶ µ ¶ t t −n n −n n N L(y) − 2 f (2 y), , N 2L(z) − 2 f (2 z), , 16 16 µ µ ¶ µ µ ¶¶ ¶ t x+y x+y N 2L + z − 2−n+1 f 2n +z , , 2 2 16 µ µ µ ¶¶ ¶ 2n t n x+y n n n N 2f 2 +z − f (2 x) − f (2 y) − f (2 z), , 2 12 µ µ ¶ ¶¾ x+y 2 N 2L +z , t . 2 3

≥ min N L(x) − 2−n f (2n x),

The first four terms on the right-hand side of the above inequality tend to 1 as n → ∞ and the fifth term is greater than µ

µ

N 2f 2n

µ

x+y +z 2

¶¶

¶

− f (2n x) − f (2n y) − 2f (2n z), t0 ϕ(2n x, 2n y, 2n z) ,

which is greater than or equal to 1 − ε. Thus we have ½

µ

µ

¶

¶

¾

x+y 2 N (L(x) + L(y) + 2L(z), t) ≥ min N 2L + z , t ,1 − ε 2 3

316 12

YEOL JE CHO, CHOONKIL PARK, AND REZA SAADATI

for all t > 0 and 0 < ε < 1 and so

µ

N (L(x) + L(y) + 2L(z), t) ≥ N 2L

µ

¶

x+y 2 +z , t 2 3

¶

for all t > 0 or N (L(x) + L(y) + 2L(z), t) ≥ 1 − ε for all t > 0. For the former case, by Lemma 2.2, the mapping L : X → Y is Cauchy additive. For the latter case, N (L(x) + L(y) + 2L(z), t) = 1 for all t > 0 and so N (4L(x), t) = 1 for all t > 0 and x ∈ X. By (N2 ), L(x) = 0 for all x ∈ X. Thus the mapping L : X → Y is Cauchy additive, that is, L(x + y) = L(x) + L(y) for all x, y ∈ X. Now, for some positive δ and α, assume that (2.15) holds. Let ϕn (x, y, z) :=

n−1 X

2−k−1 ϕ(2k x, 2k y, 2k z)

k=0

for all x, y, z ∈ X. For any x ∈ X, by the same reasoning as in the beginning of the proof, one can deduce from (2.15) that Ã

N 2n f (x) − f (2n x), δ

n−1 X

!

2n−k−1 ϕ(0, −2k+1 x, 2k x)

≥α

(2.20)

k=0

for all n ∈ N. For any t > 0, we have N (f (x) − L(x), δϕn (0, −2x, x) + t) ½ µ ¶ µ ¶¾ f (2n x) f (2n x) ≤ min N f (x) − , δϕn (0, −2x, x) , N − L(x), t . 2n 2n

(2.21)

n

Thus, combining (2.20) and (2.21), it follows from limn→∞ N ( f (22n x) − L(x), t) = 1 that N (f (x) − L(x), δϕn (0, −2x, x) + t) ≥ α for large enough n ∈ N. From the continuity of the function N (f (x) − L(x), ·), we see that e −2x, x) + t) ≥ α. Letting t → 0, we conclude that N (f (x) − L(x), 2δ ϕ(0, µ

¶

δ e −2x, x) ≥ α. N f (x) − L(x), ϕ(0, 2 Finally, it remains to prove the uniqueness assertion. Let T be another additive mapping satisfying (2.16). Fix c > 0. For any ε > 0, by (2.16) for L and T , we can find some t0 > 0 such that ¶ µ t e −2x, x) ≥ 1 − ε, N f (x) − L(x), ϕ(0, 2 µ ¶ t e −2x, x) N f (x) − T (x), ϕ(0, ≥ 1−ε 2 for all x ∈ X and t ≥ t0 . Fix some x ∈ X and find some integer n0 such that t0

∞ X k=n

2−k ϕ(0, −2k+1 x, 2k x)
0. Thus L(x) = T (x) for all x ∈ X. This completes the proof. ¤ Corollary 2.8. Let θ ≥ 0 and l p be a real number with 0 < p < 1. Let f : X → Y be an odd mapping such that lim N (f (x) + f (y) + 2f (z), tθ(kxkp + kykp + kzkp )) = 1

t→∞

(2.22)

n

uniformly on X 3 . Then L(x) := N -limn→∞ f (22n x) exists for any x ∈ X and defines a Cauchy additive mapping L : X → Y such that, if there exist δ > 0 and α > 0 satisfying N (f (x) + f (y) + 2f (z), δθ(kxkp + kykp + kzkp )) ≥ α for all x, y, z ∈ X, then

¶

µ

1 + 2p δθkxkp ≥ α 2 − 2p for all x ∈ X. Furthermore, the additive mapping L : X → Y is a unique mapping such that N f (x) − L(x), µ

lim N f (x) − L(x),

t→∞

uniformly on X.

¶

1 + 2p 2tθkxkp = 1 2 − 2p

318 14

YEOL JE CHO, CHOONKIL PARK, AND REZA SAADATI

Proof. Define ϕ(x, y, z) := θ(kxkp +kykp +kzkp ) and apply Theorem 2.7, we get the conclusion. ¤ Similarly, we can obtain the following results and so we omit the proofs. Theorem 2.9. Let ϕ : X 3 → [0, ∞) be a function satisfying (2.13). Let f : X → Y be an odd mapping satisfying (2.14). Then L(x) := N -limn→∞ 2n f ( 2xn ) exists forany x ∈ X and defines a Cauchy additive mapping L : X → Y such that, if there exist δ > 0 and α > 0 satisfying N (f (x) + f (y) + 2f (z), δϕ(x, y, z)) ≥ α for all x, y, z ∈ X, then

µ

¶

δ e −2x, x) ≥ α N f (x) − L(x), ϕ(0, 2 for all x ∈ X. Furthermore, the additive mapping L : X → Y is a unique mapping such that e −2x, x)) = 1 lim N (f (x) − L(x), tϕ(0,

t→∞

uniformly on X. Corollary 2.10. Let θ ≥ 0 and p be a real number with p > 1. Let f : X → Y be an odd mapping satisfying (2.22). Then L(x) := N -limn→∞ 2n f ( 2xn ) exists for any x ∈ X and defines a Cauchy additive mapping L : X → Y such that, if there exist δ > 0 and α > 0 satisfying N (f (x) + f (y) + 2f (z), δθ(kxkp + kykp + kzkp )) ≥ α for all x, y, z ∈ X, then

µ

¶

2p + 1 N f (x) − L(x), p δθkxkp ≥ α 2 −2 for all x ∈ X. Furthermore, the additive mapping L : X → Y is a unique mapping such that ¶

µ

2p + 1 lim N f (x) − L(x), p 2tθkxkp = 1 t→∞ 2 −2 uniformly on X. Proof. Define ϕ(x, y, z) := θ(kxkp +kykp +kzkp ) and apply Theorem 2.9, we get the conclusion. ¤ Acknowledgements The first author was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00050) and the second author was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2009-0070788).

319 FUZZY FUNCTIONAL INEQUALITIES

15

References [1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. [2] S.C. Cheng and J.M. Mordeson, Fuzzy linear operators and fuzzy normed linear spaces, Bull. Calcutta Math. Soc. 86 (1994), 429–436. [3] P. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific Publishing Company, New Jersey, Hong Kong, Singapore and London, 2002. [4] W. Fechner, Stability of a functional inequalities associated with the Jordan-von Neumann functional equation, Aequationes Math. 71 (2006), 149–161. [5] Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), 431–434. [6] P. Gˇ avruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436. [7] A. Gil´ anyi, Eine zur Parallelogrammgleichung ¨ aquivalente Ungleichung, Aequat. Math. 62 (2001), 303–309. [8] A. Gil´ anyi, On a problem by K. Nikodem, Math. Inequal. Appl. 5 (2002), 707–710. [9] D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA 27 (1941), 222–224. auser, [10] D.H. Hyers, G. Isac and Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkh¨ Basel, 1998. [11] S. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press lnc., Palm Harbor, Florida, 2001. [12] R.V. Kadison and G. Pedersen, Means and convex combinations of unitary operators, Math. Scand. 57 (1985), 249–266. [13] A.K. Katsaras, Fuzzy topological vector spaces II, Fuzzy Sets and Systems 12 (1984), 143–154. [14] I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica 11 (1975), 326–334. [15] S.V. Krishna and K.K.M. Sarma, Separation of fuzzy normed linear spaces, Fuzzy Sets and Systems 63 (1994), 207–217. [16] C. Park, On the stability of the linear mapping in Banach modules, J. Math. Anal. Appl. 275 (2002), 711–720. [17] C. Park, Modified Trif ’s functional equations in Banach modules over a C ∗ -algebra and approximate algebra homomorphisms, J. Math. Anal. Appl. 278 (2003), 93–108. [18] C. Park, On an approximate automorphism on a C ∗ -algebra, Proc. Amer. Math. Soc. 132 (2004), 1739– 1745. [19] C. Park, Lie ∗-homomorphisms between Lie C ∗ -algebras and Lie ∗-derivations on Lie C ∗ -algebras, J. Math. Anal. Appl. 293 (2004), 419–434. [20] C. Park, Homomorphisms between Poisson JC ∗ -algebras, Bull. Braz. Math. Soc. 36 (2005), 79–97. [21] C. Park, Y. Cho and M. Han, Stability of functional inequalities associated with Jordan-von Neumann type additive functional equations, J. Inequal. Appl. 2007, Article ID 41820 (2007). [22] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [23] Th.M. Rassias, Problem 16; 2, Report of the 27th International Symp. on Functional Equations, Aequationes Math. 39 (1990), 292–293; 309. ˇ [24] Th.M. Rassias and P. Semrl, On the behaviour of mappings which do not satisfy Hyers-Ulam stability, Proc. Amer. Math. Soc. 114 (1992), 989–993. [25] J. R¨ atz, On inequalities associated with the Jordan-von Neumann functional equation, Aequat. Math. 66 (2003), 191–200. [26] R. Saadati, S.M. Vaezpour and C. Park, The stability of the cubic functional equation in various spaces, Math. Commun. in press. [27] R. Saadati and S.M. Vaezpour, Some results on fuzzy Banach spaces, J. Appl. Math. Comput. 17 (2005), 475–484. [28] R. Saadati and J. Park, On the intuitionistic fuzzy topological spaces, Chaos, Solitons and Fractals 27 (2006), 331–344. [29] R. Saadati, On the L-Fuzzy Topological Spaces, Chaos, Solitons and Fractals, 37 (2008), 1419–1426. [30] R. Saadati, S.M. Vaezpour and Y.J. Cho, A note on the ”On the stability of cubic mappings and quadratic mappings in random normed spaces”, J. Inequal. Appl., Volume 2009, Article ID 214530, 6 pages.

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[31] R. Saadati, A note on “Some results on the IF-normed spaces”, Chaos, Solitons and Fractals 41 (2009), 206–213. [32] S.M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960. [33] J.Z. Xiao and X.H. Zhu, Fuzzy normed spaces of operators and its completeness, Fuzzy Sets and Systems 133 (2003), 389–399. Yeol Je Cho,, Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju 660-701, Korea E-mail address: [email protected] Choonkil Park, Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Republic of Korea E-mail address: [email protected] Reza Saadati, Department of Mathematics and Computer Science, Amirkabir University of Technology, 424 Hafez Avenue, Tehran 15914, Iran E-mail address: [email protected]

JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.2, 321-334, 2011, COPYRIGHT 2011 EUDOXUS PRESS, 321 LLC

A class of weighted holomorphic Bergman spaces A. El-Sayed Ahmed and H. Al-Amri Taif University, Faculty of Science, Mathematics Department Kingdom of Saudi Arabia e-mail: [email protected]

Abstract In this paper, we introduce the class NK,ϕa ,p of analytic functions in the unit disc D of the complex plane C. By using functions with lacunary Taylor series, it is shown that different values for p give raise to different spaces. Finally, we study composition operators acting between NK,ϕa ,p -type classes.

1

Introduction

Let D = {z : |z| < 1} be the unit disk in the complex plane C, ∂D its boundary and H(D) be the class of all holomorphic functions in D. For 0 < p < ∞, the Bergman space Ap is the set of analytic functions f in the unit disk D with Z 1 |f (z)|p dA(z) < ∞, kf kpAp = π D

where dA(z) denotes Lebesgue area measure. If p > 1, Ap is a Banach space with the norm k.kAp . If 0 < p < 1, it is a complete metric space, where the metric is given by d(f, g) = kf − gkpAp . A2 is a Hilbert space with inner product Z 1 hf, gi = f (z)g(z) dA(z) π D

and reproducing kernel ka (z) = (1−¯1az)2 at a ∈ D. It actually turns out that this holds for f ∈ Ap , 1 ≤ p < ∞. For more information about Bergman spaces we refer to [7, 8, 9, 14, 15, 23, 26]. A space closely related to Ap is A−n (n > 0), which consists of functions f analytic in D with kf kA−n = sup |f (z)|(1 − |z|2 )n < ∞. z∈D

−

1

A−n is also a Banach space. It is easy to see that for any δ > 0, A p−δ ⊆ Ap . One can also −2 check that Ap ⊆ A p . The space A−n has been studied extensively (see [9, 19, 21, 22] and others). If n = 1 we call that f ∈ A−1 . AMS: 32A36, 30B10 , 46 E 15. Key words and phrases: Bergman spaces, Hadamard gaps, composition operators.

1

322

EL-SAYED AHMED, AL-AMRI: HOLOMORPHIC BERGMAN SPACES

2

It should be remarked here that a function f is in the Bloch space if the derivative of f is in A−1 (see [9]). Palmberg in [19] introduced the Np −spaces, (with p ∈ (0, ∞)) consist of f ∈ H(D) such that Z 2 kf kNp = sup |f (z)|2 (1 − |ϕa (z)|2 )p dA(z) < ∞, a∈D

D

when p = 1, Np coincides N1 . The N1 −space was introduced in [16]. Very recently El-Sayed and Bakhit [3, 4] introduced the space NK of holomorphic functions. This space consists of f ∈ H(D) such that Z 2 kf kNK = sup |f (z)|2 K(g(z, a))dA(z) < ∞. a∈D

If

D

Z lim

|a|→1 D

|f (z)|2 K(g(z, a))dA(z) = 0,

then f is said to belong to NK,0 . Clearly, if K(t) = tp , then NK = Np ; since g(z, a) ≈ (1 − |ϕa (z)|2 ). For K(t) = 1 it gives the Bergman space A2 . It is easy to check that k · kNK is a complete semi-norm on NK . If NK consists of just the constant functions, we say that it is trivial. In this paper, we introduce the new space NK,ϕa ,p by the right continuous and nondecreasing function K : [0, ∞) → [0, ∞). The NK,ϕa ,p space consists of f ∈ H(D) such that Z 1 p 2 2p kf kNK,ϕ ,p = sup(1 − |a| ) |f ◦ ϕa |p (1 − |z|2 )p−2 K(log )dA(z) < ∞. a |z| a∈D D If

Z 2 2p

lim (1 − |a| )

|a|→1

D

|f ◦ ϕa |p (1 − |z|2 )p−2 K(log

1 )dA(z) = 0, |z|

then f is said to belong to NK,ϕa ,p,0 . We assume from now on that all K : [0, ∞) → [0, ∞) to appear in this paper are right-continuous and nondecreasing functions such that the integral Z 1/e Z ∞ K(log(1/ρ))ρ dρ = K(t)e−2t dt is convergent. 0

1

From a change of variable we see that the coordinate function z belongs to NK,ϕa ,p space if and only if µ ¶ Z (1 − |a|2 )3p (1 − |z|2 )p−2 2 sup dA(z) < ∞, where |a| > r, K log |1 − a ¯z|2p |a| − |z| a∈D D simplifying the above integral in polar coordinates, we conclude that NK,ϕa ,p space is nontrivial if and only if µ ¶ Z 1 (1 − t)3p (1 − |z|2 )p−2 2 sup K log rdr < ∞. (1) (1 − tr2 )p |a| − r t∈(0,1) 0 Throughout this paper we always assume that condition (1) is satisfied, so that the NK,ϕa ,p space, which we study is not trivial. An important tool in the study of NK,ϕa ,p space is the auxiliary function φK defined by K(st) , 0 < s < ∞. 0 0 such that K(2t) ≤ CK(t) for all 0 ≤ 2t ≤ 1. Lemma 1.1 [25] If K satisfies condition (2), then the function Z ∞ ds ∗ K (t) = t φK (s) 2 (where, 0 < t < ∞), s 1 has the following properties: (A) K ∗ is nondecreasing on (0, ∞). (B) K ∗ (t)/t is nondecreasing on (0, ∞). (C) K ∗ (t) ≥ K(t) for all t ∈ (0, ∞). (D)K ∗ . K on (0, 1]. If K(t) = K(1) for t ≥ 1, then we also have (E) K ∗ (t) = K ∗ (1) = K(1) for t ≥ 1, so K ∗ ≈ K on (0, ∞). Theorem 1.1 [25] If K satisfies condition (2) then for any α ≥ 1 and 0 ≤ β < 1 we have Z

1

r 0

α−1

µ ¶ µ ¶ 1 −β 1 1 − β 1−β 1−β (log ) K(log )dr ≈ C(β) K , r r α α

(3)

where C(β) is a constant depending only on β. Now, we give the following proposition. Proposition 1.1 If K satisfies condition (2), then we can find another nonnegative weight function given by Z ∞ ds ∗ K (t) = t φK (s) 2 where, 0 < t < ∞, s 1 such that NK,ϕa ,p = NK ∗ ,ϕa ,p and that the new function K ∗ has the following properties: (a) K ∗ is nondecreasing on (0, ∞). (b) K ∗ satisfies condition 1. (c) K ∗ satisfies condition 2. (d) K ∗ (t)/t is nondecreasing on (0, ∞).

323

324

EL-SAYED AHMED, AL-AMRI: HOLOMORPHIC BERGMAN SPACES

4

(e) K ∗ (2t) ≈ K ∗ (t) on (0, ∞). (f ) K ∗ is concave on (0, ∞). (g) K ∗ is differentiable (up to any given order) on (0, ∞). (h) K ∗ (t) = K ∗ (1) for all t ≥ 1. (i) K ∗ . K on (0, 1]. Proof: The proof of this proposition can be obtained from Lemma 1.1, and also using some similar steps to that used in the proof of proposition 4 in [25], so it will be omitted. Throughout this paper, C stands for positive constants, which may indicate different constants from one occurrence to the next.

2

Holomorphic NK,ϕa ,p space

In this section we prove some basic properties of NK,ϕa ,p −space. Let D(a, r) = {z : |ϕa (z)| < r} be the pseudo-hyperbolic disk with radius r, where 0 < r < 1. For a point a ∈ D and 0 < r < 1, we can get that D(a, r) with pseudo-hyperbolic center a and (1−r2 )a (1−|a|2 )r pseudo-hyperbolic radius r, its Euclidean center and Euclidean radius are 1−r 2 |a|2 and 1−r 2 |a|2 , respectively (see [20]). Now, we give the following results: Theorem 2.1 For each nondecreasing function K : [0, ∞) → [0, ∞) and 0 < p < ∞. Then (i) NK,ϕa ,p ⊂ A−4 (ii) NK,ϕa ,p = A−4 , if ¶ µ Z 1 2 p 2 −3p−2 rdr < ∞, (4) sup (1 − |a|) (1 − r ) K log |a| − r a∈D 0 where |a| > r. Proof: Suppose that f ∈ NK,ϕa ,p , and let C be a constant such that Z 1 2 2p sup(1 − |a| ) |f ◦ ϕa |p (1 − |z|2 )p−2 K(log )dA(z) = C |z| a∈D D By the fact K is nondecreasing, for all r, 0 < r < 1, we have Z 1 C ≥ (1 − |a|2 )2p |f ◦ ϕa |p (1 − |z|2 )p−2 K(log )dA(z) |z| D Z 2 p 1 (1 − |ϕa (z)| ) K(log )dA(z) ≥ (1 − |a|2 )2p |f (z)|p 2 2 (1 − |z| ) |ϕa (z)| D(a,r) Z 1 (1 − |a|2 )p (1 − |z|2 )p 2 2p ≥ (1 − |a| ) K(log ) |f (z)|p dA(z) r D(a,r) |1 − a ¯z|2p (1 − |z|2 )2 Z 1 1 ≥ K(log ) |f (z)|p (1 − |a|2 )3p (1 − |z|2 )p−2 dA(z) (4)p r D(a,r) Z p−2 1 1 2 3p 2 (1 − |a| ) |f (z)|p dA(z) ≥ K(log )|D(a, r)| (4)p r D(a,r) ≥

p−2 πr2 1 2 4p 2 (1 − |a| ) K(log )|D(a, r)| |f (a)|p , (4)p r

EL-SAYED AHMED, AL-AMRI: HOLOMORPHIC BERGMAN SPACES

5

where (1 − |z|2 )2 ≈ |D(a, r)| and (1 − |a|) < |1 − a ¯z| < (1 + |a|) for any fixed r0 ∈ (0, 1), (see [20, 24]), then C (1 − |a|2 )4p |f (a)|p ≤ p−2 . 1 2 r0 K(log r0 )|D(a, r0 )| 2 Thus f ∈ A−4 in D. Hence, NK,ϕa ,p ⊂ A−4 . The proof of (i) is therefore completed. Now, we show that A−4 ⊂ NK,ϕa ,p , provided that K satisfies condition (4). For f ∈ A−4 , we have that, ¶ µ Z 1 2 2p p 2 p−2 (1 − |a| ) |f ◦ ϕa (z)| (1 − |z| ) K log dA(z) |z| D µ ¶ Z 2 p 1 2 2p p (1 − |ϕa (z)| ) = (1 − |a| ) |f (z)| K log dA(z) (1 − |z|2 )2 |ϕa (z)| D µ ¶ Z 1 (1 − |a|2 )3p (1 − |z|2 )p−2 K log dA(z) ≤ |f (z)|p 2p (1 − |a|) |ϕa (z)| D ¶ µ Z 2 4p−2 1 3p p p (1 − |z| ) = (2) (1 − |a|) |f (z)| dA(z) K log (1 − |z|2 )3p |ϕa (z)| D ¶ µ Z 1 p 3p p 2 −3p−2 = (2) (1 − |a|) kf (z)kA−4 (1 − |z| ) dA(z) K log |ϕa (z)| ¶ µ ZD |1 − a ¯z| dA(z) ≤ (2)3p (1 − |a|)p kf (z)kpA−4 (1 − |z|2 )−3p−2 K log |a| − |z| D ¶ µ Z 1 2 p 3p p 2 −3p−2 dA(z) ≤ (2) (1 − |a|) kf (z)kA−4 (1 − r ) K log |a| − r 0 µ ¶ Z 1 2 p 3p p 2 −3p−2 ≤ (2) kf (z)kA−4 sup (1 − |a|) (1 − r ) K log dA(z) |a| − r a∈D 0 ≤ Ckf (z)kpA−4 , Hence f ∈ NK,ϕa ,p , then we have proved that (4) is a sufficient condition, so that NK,ϕa ,p = A−4 . Theorem 2.2 Let K1 (1) > 0, K2 (t) = inf{K1 (t), K1 (1)}, and 0 < p < ∞. Also, we suppose that kf kNK ,ϕa ,p = kf kA−4 , then NK1 ,ϕa ,p = NK2 ,ϕa ,p Proof: Since K2 < K1 and K2 is nondecreasing, it is clear that NK1 ,ϕa ,p ⊂ NK2 ,ϕa ,p . It remains to prove that NK2 ,ϕa ,p ⊂ NK1 ,ϕa ,p . We note that g(z, a) > 1, z ∈ D(a, 1/e), g(z, a) ≤ 1, z ∈ D\D(a, 1/e). Thus K1 (g(z, a)) = K2 (g(z, a)) in D\D(a, 1/e). It suffices to deal with integrals over D(a, 1/e). If f ∈ NK2 ,ϕa ,p , then f ∈ A−4 , hence µ ¶ Z 1 2 2p p 2 p−2 (1 − |a| ) |f ◦ ϕa | (1 − |z| ) K1 log dA(z) |z| D(a,1/e)

325

326

EL-SAYED AHMED, AL-AMRI: HOLOMORPHIC BERGMAN SPACES

µ ¶ − |ϕa (z)|2 )p 1 (1 − |a| ) |f (z)| K1 log dA(z) (1 − |z|2 )2 |ϕa (z)| D(a,1/e) µ ¶ Z 2 4p−2 (1 − |a|2 )p 1 2 2p p (1 − |z| ) (1 − |a| ) |f (z)| K1 log dA(z) |1 − a ¯z|2p (1 − |z|2 )3p |ϕa (z)| D(a,1/e) µ ¶ Z (1 − |z|2 )−3p−2 (1 − |a|2)3p 1 p kf (z)kA−4 K1 log dA(z) (1 − |a|)2p |ϕa (z)| D(a,1/e) µ ¶ Z 1 p 3p p 2 −3p−2 (2) kf (z)kA−4 (1 − |a|) (1 − |z| ) K1 log dA(z) |ϕa (z)| D(a,1/e) ¯¶ µ ¯ Z ¯1 − a ¯z ¯¯ p 2 −3p−2 3p ¯ (2) kf (z)kA−4 (1 − |z| ) K1 log¯ dA(z) a−z ¯ D(a,1/e) µ ¶ Z 2 p 2 −3p−2 3p (1 − |z| ) K1 log (2) kf (z)kA−4 dA(z) |a| − z D(a,1/e) µ ¶ Z 1/e 2 p 2 −3p−2 3p (1 − r ) K1 log rdr (2) kf (z)kA−4 |a| − r a Ckf (z)kpNK ,ϕ ,p Z

= = ≤ = = ≤ ≤ ≤

6

p (1

2 2p

2

a

The right-hand member gives abound for the supremum over a ∈ D of the first term in this chain of inequalities. Hence f ∈ NK1 ,ϕa ,p and the theorem is proved. Theorem 2.3 let K : [0, ∞) → [0, ∞) be a nondecreasing function and let 0 < p < ∞.Then (i) NK,ϕa ,p,0 ⊂ A−4 0 (ii) NK,ϕa ,p,0 = A−4 0 if (4) holds. Proof: (i) Without loss of generality, we note that if f ∈ NK,ϕa ,p,0 and K(1) > 0, then Z 2 2p

|f (z)|p

K(1)(1 − |a| )

D(a,1/e)

(1 − |ϕa (z)|2 )p dA(z) (1 − |z|2 )2

Z ≤ (1 − |a|2 )2p D

|f ◦ ϕa (z)|p (1 − |z|2 )p−2 K(log

1 )dA(z). |z|

For a fixed r ∈ (0, 1), let E(a, r) = {z ∈ D, |z − a| < r(1 − |a|)}. We know that E(a, r) ⊂ D(a, r) and for any z ∈ E(a, r), we have (1 − r)(1 − |a|) ≤ 1 − |z| ≤ (1 + r)(1 − |a|), which means that 1 − |z|2 ≈ 1 − |a|2 for any z ∈ E(a, r). Since |f (z)|p is subharmonic, for a ∈ D, Z 1 2 2p (1 − |a| ) |f ◦ ϕa (z)|p (1 − |z|2 )p−2 K(log )dA(z) |z| D 1 ≤ π(1/e)2 2p K(1)(1 − |a|2 )4p |f (a)|p (2) Hence, NK,ϕa ,p,0 ⊂ A−4 0 .

EL-SAYED AHMED, AL-AMRI: HOLOMORPHIC BERGMAN SPACES

7

(ii) We only need to prove that A−4 0 ⊂ NK,ϕa ,p,0 . Assume that

Z p

L = sup(1 − |a|)

µ

1

2 −3p−2

(1 − r )

a∈D

0

¶ 2 K log rdr < ∞. |a| − r

For a given ² > 0 there exists an r1 , 0 < r1 < 1, such that µ ¶ Z 1 2 p 2 −3p−2 (1 − |a|) (1 − r ) K log rdr < ². |a| − r r1 Thus,

Z 2 2p

(1 − |a| )

D\D(a,r)

|f ◦ ϕa |p (1 − |z|2 )p−2 K(log

Z = (1 − |a|2 )2p

1 (1 − |ϕa (z)|2 )p K(log )dA(z) 2 2 (1 − |z| ) |ϕa (z)|

|f (z)|p

(1 − |a|2 )p (1 − |z|2 )p−2 1 K(log )dA(z) 2p |1 − a ¯z| |ϕa (z)|

D\D(a,r)

= (1 − |a|2 )2p D\D(a,r)

Z

1 )dA(z) |z|

|f (z)|p Z

(5)

(1 − |a|2 )3p (1 − |z|2 )4p−2 1 )dA(z) K(log 2p 2 3p (1 − |a|) (1 − |z| ) |ϕa (z)| D\D(a,r) ¯¶ µ ¯ Z ¯1 − a ¯z ¯¯ p 3p p 2 −3p−2 ¯ ≤ (2) kf (z)kA−4 (1 − |a|) dA(z) (1 − |z| ) K log¯ a−z ¯ D\D(a,r) ¶ µ Z 2 p 3p p 2 −3p−2 dA(w) ≤ (2) kf (z)kA−4 (1 − |a|) (1 − |w| ) K log |a| − |w| r≤|w| 1 such that nkn+1 ≥ c for all k ∈ N. It is well known that a lacunary series belongs k to BMOA space if and only if it is in the Hardy space H 2 ; see [6]. It is also well known that a lacunary series in the Bloch space if and only if its Taylor coefficients are bounded; see [26] for instance. In [18], Miao gave characterizations of Besov space by the help of a lacunary series. These characterizations extended to higher dimensions using several complex variables and quaternion sense (see [1, 5, 12]). In this section, we obtain some results for functions to be in NK,ϕa ,p classes in terms of Taylor coefficients. Theorem 3.1 If K satisfies condition (2), 0 < p < ∞ and f (z) =

∞ X

an z n−1 ,

n=1

then

Z a 2p

(1 − |a| )

D

< C 0 (1 − |a|)p

∞ X

|f ◦ ϕa |p (1 − |z|2 )p−2 K(log µ

|an |p

n=1

Proof: Write

p−1 p(n − 1) + 2

¶p−1 µ K

1 )dA(z) |z| p−1 p(n − 1) + 2

¶

Z (1 − |ϕa (z)|2 )p 1 I(f ) = (1 − |a|2 )2p )dA(z) |f (z)|p K(log 2 )2 (1 − |z| |ϕ a (z)| D Z 1 (1 − |a|2 )p (1 − |z|2 )p−2 2 2p )dA(z) = (1 − |a| ) K(log |f (z)|p 2p |1 − a ¯z| |ϕa (z)| D Z 1 ≤ (2)3p (1 − |a|)p )dA(z). |f (z)|p (1 − |z|2 )p−2 K(log |ϕ a (z)| D

Integrating in polar coordinates leads to 3p

p

I(f ) ≤ (2) (1 − |a|)

∞ X

Z p

|an |

n=1

0

1

1 rp(n−1)+1 (1 − r2 )p−2 K(log )dr, r

where r ∈ (0, 1). By the inequality 1 − r2 ≤ 2 log( 1r ), we have Z 1 ∞ X 1 1 4p−2 p p I(f ) ≤ (2) (1 − |a|) |an | rp(n−1)+1 (log )p−2 K(log )dr. r r 0 n=1

We apply Theorem 1.1, with β = 2 − p and α = p(n − 1) + 2, we obtain µ ¶p−1 µ ¶ ∞ X p−1 p−1 4p−2 p p I(f ) ≈ (2) C(2 − p)(1 − |a|) |an | K p(n − 1) + 2 p(n − 1) + 2 n=1

I(f ) ≈ (1 − |a|)

p

∞ X n=1

µ p

|an |

p−1 p(n − 1) + 2

¶p−1 µ K

p−1 p(n − 1) + 2

¶

EL-SAYED AHMED, AL-AMRI: HOLOMORPHIC BERGMAN SPACES

9

Theorem 3.2 If K satisfies condition (2) and f (z) =

∞ X

bj z

nj −1

,

j=1

is in the Hadamard gap class, then f ∈ NK,ϕa ,p if µ ¶ ∞ X 1 |bj |p K < ∞. nj

(6)

j=1

Proof: First assume that condition (6) holds. We write z = reiθ in polar form and observe that |f (z)| ≤

∞ X

|bj |rnj −1 .

j=1

Then, Z kf kpNK,ϕ

a ,p

= sup(1 − |a|2 )2p a∈D

ZD

|f ◦ ϕa |p (1 − |z|2 )p−2 K(log

1 )dA(z) |z|

1 )dA(z) |f (z)|p (1 − |ϕa (z)|2 )p (1 − |z|2 )−2 K(log |ϕa (z)| a∈D D Z |1 − a ¯z|p ≤ (2)3p sup(1 − |a|)p K(g(z, a))dA(z) |f (z)|p (1 − |z|2 )p−2 |1 − a ¯z|p a∈D D Z 4p ≤ (2) sup |f (z)|p (1 − |z|2 )p−2 K(g(z, a))dA(z)

= sup(1 − |a|2 )2p

a∈D 4p

≤ (2) sup

D

Z 1µ X ∞

a∈D

0

|bj |r

nj −1

¶p

µ 2 p−2

(1 − r )

j=1

1 2π

Z

2π

¶ K(g(re , a))dθ rdr. iθ

0

By Proposition 1.1, we may well assume that K is concave. Then µ Z 2π ¶ Z 2π 1 1 iθ iθ K(g(re , a))dθ ≤ K g(re , a)dθ . 2π 0 2π 0 by Jensen’s formulae (see [11]). We proceed to estimate the integral ¯ ¯ Z 2π Z 2π ¯1 − a 1 1 ¯reiθ ¯¯ I(a) = g(reiθ , a)dθ = log¯¯ iθ dθ. 2π 0 2π 0 re − a ¯ Hence,

 1  log( |a| ) ,

I(a) ≤



In particular, 1 2π

I(a) =

log( 1r ) , Z

2π

0

0 < r ≤ |a|; |a| < r < 1.

1 g(reiθ , a)dθ ≤ log( ). r

From this we have, Z kf kpNK,ϕ ,p a

4p

≤ (2) sup a∈D

1

r 0

−p+1

·X ∞ j=1

|bj |r

nj

¸p

¶ 1 dr. K log r µ

2 p−2

(1 − r )

(7)

329

330

EL-SAYED AHMED, AL-AMRI: HOLOMORPHIC BERGMAN SPACES

10

Using the Cauchy-Schwarz inequality (see [11]) to produce ·X ¸p ·X ¸p · X ¸p ∞ ∞ X ∞ X nj nj 2n |bj |r = |bj |r ≤ |bj |r n=0 nj ∈In

j=1

·X ∞

≤

n=0 nj ∈In

n/2 2n 1−1/p

(2

r )

n=0

·X ∞

≤

n

r2 2((1−p)/2)n

(r 2

)

nj ∈In

n=0

nj ∈In

1 ≤ C log r

¶−(p−1)/2 X ∞

¸p |bj |

¸p−1 ¶p ¸· X ∞ n |bj | 2n/2 r2

µ X

n=0

µ

X

2n (1−p)n/2 1/p

2n ((1−p)/2)n

r 2

µ X

n=0

¶p |bj |

nj ∈In

where In = {j : 2n ≤ j < 2n+1 , j ∈ N}. To this end, we combine the elementary estimates: ∞ Z 2n+1 ∞ X √ X 1 t n 2n 2 = 2 t− 2 r 2 dt 2 r n=0

≤

√ 2

n n=0 2 Z ∞ − 12

t

t

r 2 dt

0

µ ¶ 1 1 1 −2 ≤ 2Γ( ) log 2 r This very useful tool can now be applied to the calculation above to obtain kf kpNK,ϕ ,p a

≤C

∞ X

n (1−p)/2

(2 )

n=0

· X

¸p Z |bj |

nj ∈In

0

µ

1

r

2n −p+1

1 log r

This together with (7) and Theorem 1.1 for α = 2n − p + 2, β = kf kpNK,ϕ ,p a

≤ C

∞ · X X n=0 nj ∈In

≤ C

∞ · X X n=0 nj ∈In

≤C

¶ p−3

3−p 2 ,

2

µ ¶ 1 K log dr r

we obtain

¶ p−1 µ ¶ ¸p µ ¶ p−1 µ 2 2 p−1 p−1 1 K n+1 |bj | 2n 2n+1 − 2(p + 2) 2 − 2(p + 2) ¸p µ ¶ p−1 µ ¶ p−1 µ ¶ 2 2 1 1 1 |bj | K n n n 2 2 2

∞ · X X n=0 nj ∈In

¸p µ ¶p−1 µ ¶ 1 1 K n |bj | 2n 2

(8)

If nj ∈ In , then nj < 2n < 2n+1 . It follows from the monotonicity of k and K(2t) ≤ CK(t) for all 0 ≤ 2t ≤ 1, such that µ ¶p−1 µ ¶ µ ¶ 1 1 1 (p−1) K n < nj K . n 2 2 nj Combining this with (8), we obtain kf kpNK,ϕ ,p a

.

∞ · X X n=0 nj ∈In

¸p µ ¶ 1 p−1 |bj | nj K nj

(9)

EL-SAYED AHMED, AL-AMRI: HOLOMORPHIC BERGMAN SPACES

11

Since f is in the Hadamard gap class, there exists a constant c such that nj+1 ≥ cnj for all j ∈ N. Hence, the Taylor series of f (z) has at most ([logc 2] + 1) terms aj z nj such that nj ∈ In . By (9) and H o¨lder0 s inequality, kf kpNK,ϕ

a ,p

. (logc 2 + 1)p−1

∞ X X n=0 nj ∈In

p−1

= (logc 2 + 1)

∞ X n=1

µ |bj |p K

1 nj

¶

µ

¶ 1 |bj | K . nj p

Then, f ∈ NK,ϕa ,p .

4

Composition operators on NK,ϕa ,p classes

There have been several attempts to study compactness and boundedness of composition operators in many function spaces (see, e.g., [2, 10, 13, 16, 17, 19] and others). In this section we study composition operators acting on NK,ϕa ,p space. Now, we let φ ∈ H(D) to denote a non-constant function satisfying φ(D) ⊆ D. The composition operator Cφ : H(D) → H(D) is defined by Cφ = f ◦ φ. First, in the following result, we describe boundedness for our NK,ϕa ,p classes. Lemma 4.1 (Test function in NK,ϕa ,p For w ∈ D, we define hw (z) =

1 − |w|2 . (1 − wz) ¯ 2

Then hw (z) ∈ NK,ϕa ,p and sup khw (z)kNK,ϕa ,p ≤ 1. Proof: Trivially hω (z) ∈ H(D), then Z khw kpNK,ϕ

a ,p

= sup (1 − |w|2 )2p w∈D

D

|ϕ0ω ◦ ϕw (z)|p (1 − |z|2 )p−2 K(log

1 )dA(z) ≤ 1. |z|

Theorem 4.1 Let K : [0, ∞) → [0, ∞) be a nondecreasing function and α ∈ (0, ∞). Then Cφ : NK,ϕa ,p → Aα is bounded if and only if sup z∈D

(1 − |z|2 )α < ∞. (1 − |φ(z)|2 )4

Proof: First assume that condition (10) holds. Then kCφ f kAα

= sup |f (φ(z))|(1 − |z|2 )α z∈D

≤ sup z∈D

(1 − |z|2 )α sup |f (φ(z))|(1 − |φ(z)|2 )4 (1 − |φ(z)|2 )4 z∈D

(1 − |z|2 )α 4 z∈D (1 − |φ(z)|) C kf kNK,ϕa ,p ,

≤ kf kA−4 sup ≤

(10)

331

332

EL-SAYED AHMED, AL-AMRI: HOLOMORPHIC BERGMAN SPACES

12

Conversely, assume that Cφ : NK,ϕa ,p → Aα is bounded, then kCφ f kAα ≤ kf kNK,ϕa ,p . Fix a point z0 ∈ D, and let hw be the test function in Lemma 4.1 with w = φ(z0 ). Then, 1 ≥ khw kNK,ϕa ,p ≥ C khw ◦ (φ)kAα ≥ C

1 − |w|2 (1 − |z0 |2 )α (1 − |z0 |2 )α ≥ C . 2 |1 − ω ¯ φ(z0 )| (1 − |φ(z0 )|2 )4

Theorem 4.2 Cφ : NK,ϕa ,p → Aα is compact if and only if (1 − |z|2 )α = 0. r→1 |φ(z)|>r (1 − |φ(z)|2 )4 lim sup

Proof: First assume that Cφ : NK,ϕa ,p → Aα is compact and suppose that there exists ²0 > 0 a sequence {zn } ⊂ D such that (1 − |zn |2 )α 1 ≥ ²0 whenever |φ(zn )| > 1 − . 2 4 (1 − |φ(z)| ) n 2

1−|wn | Clearly, we can assume that wn = φ(zn ) tends to w0 ∈ ∂D as n → ∞. Let hwn = (1− be the w¯n z)2 function in Lemma 4.1. Then hwn → hw0 with respect to the compact-open topology. Define fn = hwn − hw0 . Then kfn kNK,ϕa ,p ≤ 1 (see Lemma 4.1) and fn → 0 uniformly on compact subsets of D. Thus, fn ◦ φ → 0 in Aα by assumption. But, for n big enough,

kCφ fn kAα

≥ |hwn (φ(zn )) − hw0 (φ(zn ))|(1 − |zn |2 )α ¯ ¯ (1 − |wn |2 )(1 − |w0 |2 ) ¯¯ (1 − |zn |2 )α ¯¯ 1− ≥ ¯, 1 − |φ(zn )|2 ¯ |1 − w¯0 wn | | {z }| {z } ≥²0

=1

which is a contradiction. Conversely, assume that for all ² > 0 there exists δ ∈ (0, 1) such 2 )α that (1−|z| < ² whenever |φ(z)| > δ. Let {fn } be a bounded sequence in NK,ϕa ,p norm which 1−|φ(z)|2 converges to zero on compact subsets of D. Clearly, we may assume that |φ(z)| > δ. Then kCφ fn kAα

= sup |fn (φ(z))|(1 − |z|2 )α z∈D

= sup z∈D

(1 − |z|2 )α |f (φ(z))|(1 − |φ(z)|2 )4 . (1 − |φ(z)|2 )4

It is not hard to show that kf kA−4 . kf kNK,ϕa ,p . Thus, we obtain that kCφ fn kAα ≤ ²kfn kA−4 ≤ ²kfn kNK,ϕa ,p ≤ ². It follows that Cφ is a compact operator. This completes the proof of the theorem.

EL-SAYED AHMED, AL-AMRI: HOLOMORPHIC BERGMAN SPACES

13

References [1] A. El-Sayed Ahmed, Lacunary series in quaternion Bp,q spaces, Complex Variables and Elliptic Equations, Vol.54(7)(2009), 705-723. [2] A. El-Sayed Ahmed and M.A. Bakhit, Composition operators on some holomorphic Banach function spaces, Math. Scand. 104(2)(2009), 275-295. [3] A. El-Sayed Ahmed and M. A. Bakhit, Holomorphic NK and Bergman-type spaces, in JJ Grobler, LE Labuschagne and M. M¨oller (Eds), Proceedings of IWOTA 2007 conference, Birkh¨auser Series on Operator Theory, Advances and Applications, (Birkh¨auser Verlarg Publisher Basel/Switzerland), Vol 195 (2009), 121-138. [4] A. El-Sayed Ahmed and M. A. Bahkit, Hadamard products and NK space, Mathematical and computer Modelling, 51(1-2)(2010), 33-43. [5] A. Avetsiyan, Hardy-Bloch type spaces and lacunary series in the polydisk, Glasg. Math. J. 49 No.2(2007) 345-356. [6] A. Baernstein , Analytic functions of Bounded mean oscillation, Aspects of contemporary complex Analysis II, in: D. Brannan, J. Clunie (Eds.) Academic Press London, (1980) 3-36. [7] A. Borichev, H. Hedenmalm and K. Zhu, Bergman spaces and related topics in complex Analysis, American Mathematical Society, (2006). [8] B. R. Choe, H. Koo and M. Stessin, Carleson measures for Bergman spaces and their dual Berezin transforms, Proc. Am. Math. Soc. 137(12)(2009), 4143-4155. [9] P. Duren and A. Schuster, Bergman spaces, Mathematical surveys and Monographs 100, American Mathematical Society, Providence RI, (2004). [10] P. Galindo, T. W. Gamelin and M. Lindstr¨om, Fredholm composition operators on algebras of analytic functions on Banach spaces, J. Funct. Anal. 258(5)(2010), 1504-1512. [11] J. Garnett, Bounded analytic functions, Academic Press, New York, (1981). [12] K. G¨ urlebeck and A. El-Sayed Ahmed, On series expansions of hyperholomorphic Bq functions, In Tao Qian et al. (Eds), Trends in Mathematics Advances in Analysis and Geometry, (Basel/Switzerland : Birkh¨auser Verlarg Publisher) (2004),113-129. [13] M. Kotilainen, Studies on composition operators and function spaces, Report Series. Department of Mathematics, University of Joensuu 11. Joensuu: (Dissertation). 27 p. (2007). [14] E. G. Kwon, Quantities equivalent to the norm of a weighted Bergman space, J. Math. Anal. Appl. 338(2)(2008), 758-770. [15] H. Hedenmalm, B. Korenblum and K. Zhu, Theory of Bergman spaces, Springer, New York, (2000). [16] M. Lindstr¨om and N. Palmberg, Spectra of composition operators on BMOA, Integr. Equ. Oper. Theory, 53 (2005), 7586. [17] M. Lindstr¨om and A. H. Sanatpour, Derivative-free characterizations of compact generalized composition operators between Zygmund type spaces, Bull. Aust. Math. Soc. 81(3)(2010), 398-408.

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[18] J. Miao, A property of Analytic functions with Hadamard gaps, Bull. Austral. Math. Soc. 45(1992), 105-112. [19] N. Palmberg, Composition operators acting on Np -spaces, Bull. Belg. Math. Soc.- Simon Stevin, 14 (2007) 545-554. [20] K. Stroethoff, Besov-type characterisations for the Bloch space, Bull. Austral. Math. Soc. 39 (1989), 405420. [21] A. P. Schuster, The homogeneous approximation property in the Bergman space, Houston J. Math. 24(4)(1998), 707-722. [22] K. Seip, Beurling type density theorems in the unit disk, Invent. Math. 113(1)(1993), 21-39. [23] N. L. Vasilevski, Commutative algebras of Toeplitz operators on the Bergman space. Operator Theory: Advances and Applications 185. Basel: Birkh¨auser. xxix (2008). [24] H. Wulan and J. Zhou, QK type spaces of analytic functions. J. Funct. Spaces Appl. 4(1)(2006), 73-84 . [25] H. Wulan and K. Zhu, Lacunary spaces in QK spaces, Studia Math., 178 (2007) 217-230. [26] K. Zhu, Operator theory in function spaces 2nd ed., Mathematical Surveys and Monographs 138. Providence, RI: American Mathematical Society (AMS). xvi (2007).

JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.2, 335-344, 2011, COPYRIGHT 2011 EUDOXUS PRESS, 335 LLC

Existence and Iterative Approximations of Solutions for Nonlinear Implicit Fuzzy Resolvent Operator Systems of (A, η)-monotone Type

1

Heng-you Lan2 , Yongming Li and Jianfang Tang Department of Mathematics, Sichuan University of Science & Engineering, Zigong, Sichuan 643000, P. R. China

Abstract. The purpose of this paper is to introduce a new system of nonlinear implicit fuzzy resolvent operators of (A, η)-monotone type and to develop a new iterative algorithm to approximate the solutions of the nonlinear implicit fuzzy resolvent operator systems. Further, by using Rus’s results and and some new analytic techniques, we also prove the existence of solutions and the convergence of the sequences generated by the algorithms. The results presented in this paper improve and generalize the corresponding results of recent works. Key Words and Phrases. Nonlinear implicit fuzzy resolvent operator system, (A, η)-monotone and relaxed cocoercive operator, new iterative algorithm, existence and convergence. AMS Subject Classification. 47J20, 49J40, 47J06.

1

Introduction

In this paper, for given operators a : H1 → [0, 1], b : H2 → [0, 1] and c : H3 → [0, 1], we shall consider the following system of nonlinear implicit fuzzy resolvent operators: Find u, x ∈ H1 , v, y ∈ H2 and w, z ∈ H3 such that x ∈ (Gu )a(u) , y ∈ (Sv )b(v) , z ∈ (Tw )c(w) and  A1 ,ρ   g(u) = Rη1 ,M1 [A1 (f (u)) − ρN1 (p(u), y, w)], ,λ (1.1) v = RηA22,M [A2 (v) − λN2 (u, v, z)], 2   A3 ,σ w = Rη3 ,M3 [A3 (w) − σN3 (x, v, w)],

where for i = 1, 2, 3, Hi is a real Hilbert space, 2Hi denotes the family of all the nonempty ˆ i ) is the collection of all fuzzy sets on Hi , RAi ,t = (Ai + tMi )−1 : Hi → subsets of Hi , F(H ηi ,Mi Hi is the corresponding resolvent operator associated with nonlinear operator Mi , t > 0 is an any constant, Ni : H1 × H2 × H3 → Hi , f, g, p : H1 → H1 and ηi : Hi × Hi → Hi are ˆ 1 ), S : H2 → F(H ˆ 2 ) and T : H3 → F(H ˆ 3 ) are five single-valued operators, G : H1 → F(H H i three fuzzy operators, Ai : Hi → Hi and Mi : Hi → 2 are two any nonlinear (in general) operators such that f (u) ∈ dom(M1 ) for all u ∈ H1 , and ρ, λ, σ > 0 are three constants. Example 1.1 If f ≡ g, then the problem (1.1) reduces to the following implicit resolvent operator system with fuzzy operators: ,ρ g(u) = RηA11,M [A1 (g(u)) − ρN1 (p(u), y, w)], 1 A2 ,λ ,σ v = Rη2 ,M2 [A2 (v) − λN2 (u, v, z)], w = RηA33,M [A3 (w) − σN3 (x, v, w)], 3 1

This work was supported by the Sichuan Youth Science and Technology Foundation (08ZQ026-008) and the Open Foundation of Artificial Intelligence of Key Laboratory of Sichuan Province (2008RQ002, 2009RZ001). 2 The corresponding author: [email protected] (H. Y. Lan).

1

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which is equivalent to the following inclusion systems formula 0 ∈ N1 (p(u), y, w) + M1 (g(u)), 0 ∈ N2 (u, v, z) + M2 (v), 0 ∈ N3 (x, v, w) + M3 (w).

(1.2)

The problem (1.2) is called a system of nonlinear implicit fuzzy variational inclusions. Example 1.2 Furthermore, if G : H1 → 2H1 , S : H2 → 2H2 and T : H3 → 2H3 are three classical set-valued operators, by using G, S and T , we can define three fuzzy operators as follows: ˆ 1 ), u 7→ χG(u) , S : H2 → F(H ˆ 2 ), v 7→ χS(v) , T : H3 → F(H ˆ 3 ), w 7→ χT (w) , G : H1 → F(H where χG(u) , χS(v) and χT (w) are the characteristic functions of the set G(u), S(v) and T (w), respectively. It is easy to see that G, S and T both are closed fuzzy mappings satisfying condition (C) with constant functions a(u) = 1, b(v) = 1 and c(w) = 1 for all u ∈ H1 , v ∈ H2 and w ∈ H3 , respectively. Furthermore, (Gu )a(u) = χG(u) 1 = {ν ∈ H1 | χG(u) (ν) = 1} = G(u),

(Sv )b(v) = χS(v) 1 = {ν ∈ H2 | χS(v) (ν) = 1} = S(v),

(Tw )c(w) = χT (w) 1 = {ν ∈ H3 | χT (w) (ν) = 1} = T (w).

Clearly, the fuzzy operators include set-valued operators as special cases. Thus, the problem (1.2) is equivalent to finding u, x ∈ H1 , v, y ∈ H2 and w, z ∈ H3 such that x ∈ G(u), y ∈ S(v), z ∈ T (w) and 0 ∈ N1 (p(u), y, w) + M1 (g(u)), 0 ∈ N2 (u, v, z) + M2 (v), 0 ∈ N3 (x, v, w) + M3 (w).

(1.3)

The problem (1.3) is called a nonlinear implicit set-valued variational inclusion system, which includes a number of quasi-variational inclusions, generalized quasi-variational inclusions, quasi-variational inequalities, implicit quasi-variational inequalities, variational inclusion systems studied by many authors as special cases, see, for example, [1-12] and the references therein. The study of such types of problems is motivated by an increasing interest in the variational inequality theory introduced by Stampacchia [4] in early sixties. Variational inequality theory has witnessed an explosive growth in theoretical advances, algorithmic developments and applications across all disciplines of the pure and applied sciences. In recent years, much attention has been given to develop efficient and implementable numerical method for solving variational inequalities, operator equations and related optimization problems. It has been shown in [1, 3] that the three-step schemes give better numerical results than the two-step and one-step approximation iterations. The concept of variational inequalities can be defined differently depending upon the area in which one seeks to study these concepts. In [5], the author introduced first a new concept of (A, η)-monotone operators, which generalizes the (H, η)-monotonicity (see [3, 11, 13]) and A-monotonicity (see [7, 10]) in Hilbert spaces and other existing monotone operators as special cases, and studied some properties of (A, η)-monotone operators and defined resolvent operators associated with (A, η)-monotone operators. Then, by using the new resolvent operator technique, which is a very important method to find solutions of variational inequality and variational inclusion 2

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problems, the author constructed some new iterative algorithms to approximate the solutions of a new class of nonlinear (A, η)-monotone operator inclusion problems with relaxed cocoercive operators and also proved the existence of solutions and the convergence of the sequences generated by the algorithms in Hilbert spaces. For more detail, we can refer to [3, 5, 7-15] and the references therein. On the other hand, in 1989, Chang and Zhu [16] were the first to introduce and study a class of variational inequalities for fuzzy mappings. Inspired and motivated by the research work going on this different field, several kinds of variational inequalities, variational inclusions and complementarity problems for fuzzy operators in different contexts were considered and studied by many authors, see, for example, [13-17] and the references therein. Motivated and inspired by the works of [1, 3, 6, 12, 13, 15, 17], in this paper, we shall introduce and study a new system of nonlinear implicit fuzzy resolvent operators of (A, η)monotone type. Further, by using Rus’s results and and some new analytic techniques, a new class of iterative algorithm is presented for approximate solvability of above system of nonlinear implicit fuzzy resolvent operators of (A, η)-monotone type.

2

Preliminaries

Throughout this paper, let H be a real Hilbert space, whose norm and inner product are denoted by ∥ · ∥ and ⟨·, ·⟩, respectively. In the sequel, we denote 2H , CB(H) and ˆ F(H) by 2H = {B| B ∈ H}, CB(H) = {B ⊂ H| B is nonempty, bounded and closed} and ˆ F(H) = {p : H → [0, 1]} (which is called the collection of all fuzzy sets over H), respectively. ˆ ˆ An operator G from H into F(H) is called a fuzzy operator. If G : H → F(H) is a fuzzy operator, then for u ∈ H, the set G(u) (we denote it by Gu in the sequel) is a fuzzy set in ˆ F(H) and Gu (v), for all v ∈ H is the membership function of v in Gu . Before proceeding further, we need following definitions and lemmas. ˆ Definition 2.1 (1) A fuzzy operator G : H → F(H) is said to be closed, if for each u ∈ H, the function v 7→ Gu (v) is upper semicontinuous, that is, for any given net {vα }α∈Γ ⊂ H satisfying vα → v0 ∈ H, we have lim supα∈Γ Gu (vα )) ≤ Gu (v0 )). ˆ Let p ∈ F(H), λ ∈ [0, 1]. Then the set (p)λ = {u ∈ H| p(u) ≥ λ} is called a λ-cut set of p. For details and fundamental concepts see [18]. ˆ (2) A closed fuzzy operator G : H → F(H) is said to satisfy the condition (C), if there exists an operator a : H → [0, 1] such that for all u ∈ H the set (Gu )a(u) := {v ∈ H| Gu (v) ≥ a(u)} is nonempty bounded subset of H. Remark 2.2 It is worth mentioning that if G is a closed fuzzy operator satisfying condition (C), then for each u ∈ H, the set (Gu )a(u) ∈ CB(H). In fact, let {vα }α∈Γ ⊂ (Gu )a(u) be a net and vα → v0 ∈ H, then (Gu )vα ≥ a(u) for all α ∈ Γ. Since the fuzzy operator G is closed, we have a(u) ≤ lim sup Gu (vα )) ≤ Gu (v0 )). α∈Γ

This implies that v0 ∈ (Gu )a(u) and so (Gu )a(u) ∈ CB(H). 3

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ˆ Definition 2.3 Let G : Ω × H → F(H) be closed fuzzy operator satisfying the condition (C) with function a : H → [0, 1]. Then G is called γ-H-Lipschitz continuous, if there exists a positive constant γ such that H((Gu )a(u) , (Gv )a(v) ) ≤ γ∥u − v∥, ∀u, v ∈ H, where H(·, ·) is the Hausdorff pseudo-metric on 2H defined as follows: { } H(B, D) = max sup inf d(u, v), sup inf d(u, v) , ∀B, D ∈ 2H . u∈B v∈D

v∈D u∈B

Note that if the domain of H is restricted to closed subset CB(H), then H is the Hausdorff metric. Definition 2.4 An operator f : H → H is called (ζ, ϖ)-relaxed cocoercive if, there exist constants ζ, ϖ > 0 such that for all u, v ∈ H ⟨f (u) − f (v), u − v⟩ ≥ −ζ∥f (u) − f (v)∥2 + ϖ∥u − v∥2 . Definition 2.5 Let H be a real Hilbert space, η : H × H → H and A : H → H be singlevalued operators. Then set-valued operator M : H → 2H is said to be (i) ξ-H-Lipschitz continuous, if there exists a constant ξ > 0 such that H(M (u), M (v)) ≤ ξ∥u − v∥,

∀u, v ∈ H.

(ii) monotone if ⟨x − y, u − v⟩ ≥ 0, ∀u, v ∈ H, x ∈ M (u), y ∈ M (v); (iii) m-relaxed η-monotone if there exists a constant m > 0 such that ⟨x − y, η(u, v)⟩ ≥ −m∥u − v∥2 ,

∀u, v ∈ H, x ∈ M (u), y ∈ M (v);

(iv) maximal monotone if M is monotone and (I + ρM )(H) = H for all ρ > 0, where I denotes the identity operator on H; (v) (A, η)-monotone if M is m-relaxed η-monotone and (A + ρM )(H) = H for every ρ > 0. Remark 2.6 (1) If m = 0 or A = I or η(u, v) = u − v for all u, v ∈ H, (v) of Definition 2.2 reduces to the definition of (H, η)-monotone operators, maximal η-monotone operators, H-monotone operators, classical maximal monotone operators, A-monotone operators (see [3, 5, 8-11, 13]). (2) Further, operator M is said to be generalized maximal monotone (in short GMM − monotone) if M is monotone and A + ρM is maximal monotone or pseudomonotone for ρ > 0. Example 2.7 (1) An r-strongly monotone operator M : H → 2H is m-relaxed monotone for m ∈ (1, r + r2 ) and r > 0.618. (2) Let f : H → H be an nonlinear (in general) operator such that f (u) = 21 u for all u ∈ H. Then f is (ζ, ϖ)-relaxed cocoercive for ζ = 12 and ϖ = 34 . 4

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Example 2.8 Let H = R = (−∞, +∞), M (x) = 2x, A(x) = x3 and η(x, y) = y − x for all x, y ∈ H. Then M is an (A, η)-monotone operator. Definition 2.9 Let A : H → H be a strictly η-monotone operator and M : H → 2H be A,ρ an (A, η)-monotone operator. Then the corresponding resolvent operator Rη,M : H → H is defined by A,ρ Rη,M (u) = (A + ρM )−1 (u), ∀u ∈ H. Lemma 2.10 ([5]) Let η : H × H → H be τ -Lipschitz continuous, A : H → H be an r-strongly η-monotone operator and M : H → 2H be an (A, η)-monotone operator. Then A,ρ τ the resolvent operator Rη,M : H → H is r−ρm -Lipschitz continuous, i.e., A,ρ A,ρ ∥Rη,M (u) − Rη,M (v)∥ ≤

τ ∥u − v∥, r − ρm

∀u, v ∈ H,

where ρ ∈ (0, r/m) is a constant. Lemma 2.11 ([19]) Let (X , d) be a complete metric space and let B1 , B2 ∈ CB(X ) and t > 1 be any real number. Then, for every b1 ∈ B1 there exists b2 ∈ B2 such that d(b1 , b2 ) ≤ t H(B1 , B2 ).

3

Existence and Iterative Approximations

In this section, we present our main result for approximate solvability of the problem (1.1). By Lemma 2.11, we now introduce a new three step iterative algorithm for approximation solvability of the problem (1.1). Algorithm 3.1 Step 1. Choose an arbitrary initial point u0 ∈ H1 , v 0 ∈ H2 , w0 ∈ H3 , a : H1 → [0, 1], b : H2 → [0, 1] and c : H3 → [0, 1]. Step 2. Take {(xk , y k , z k )} ⊂ H1 × H2 × H3 such that xk ∈ (Guk )a(uk ) , ∥xk − xk+1 ∥ ≤ (1 + (1 + k)−1 )H1 ((Guk )a(uk ) , (Guk+1 )a(uk+1 ) ), y k ∈ (Svk )b(vk ) , ∥y k − y k+1 ∥ ≤ (1 + (1 + k)−1 )H2 ((Svk )b(vk ) , (Svk+1 )b(vk+1 ) ), z k ∈ (Twk )c(wk ) , ∥z k − z k+1 ∥ ≤ (1 + (1 + k)−1 )H3 ((Twk )c(wk ) , (Twk+1 )c(wk+1 ) ),

(3.1)

where Hi : 2Hi × 2Hi → (−∞, +∞) ∪ {+∞} is the Hausdorff pseudo-metric for i = 1, 2, 3. Step 3. Choose sequences {ek } ⊂ H1 is error to take into account a possible inexact computation of the resolvent operator point, which satisfies the following conditions: lim ∥e ∥ = 0, k

k→∞

∞ ∑

∥ek − ek−1 ∥ < ∞,

(3.2)

k=1

Step 4. Let {(uk , v k , wk )} ⊂ H1 × H2 × H3 satisfy ,ρ uk+1 = (1 − ι)uk + ι{uk − g(uk ) + RηA11,M [A1 (f (uk )) − ρN1 (p(uk ), y k , wk )]} + ek , 1 (3.3) A ,λ ,σ v k+1 = Rη22,M2 [A2 (v k ) − λN2 (uk , v k , z k )], wk+1 = RηA33,M [A3 (wk ) − σN3 (xk , v k , wk )]. 3

where ρ, λ, σ > 0 are constants and ι ∈ (0, 1] is a size constant. Step 5. If uk , v k , wk , xk , y k , z k and ek satisfy (3.1) and (3.3) to sufficient accuracy, stop; otherwise, set k := k + 1 and return to Step 2. 5

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Remark 3.2 By choosing suitable and appropriate choice of ρ, λ, σ, ι and ek in Algorithm 3.1, one can obtain a number of known iterative algorithms for approximate solvability of nonlinear variational inclusions and variational inequalities. ˆ 1 ), S : H2 → Theorem 3.3 Let Hi be a real Hilbert space for i = 1, 2, 3, G : H1 → F(H ˆ ˆ F(H2 ) and T : H3 → F(H3 ) be three closed fuzzy operators satisfying the condition (C) with functions a : H1 → [0, 1], b : H2 → [0, 1] and c : H3 → [0, 1], respectively, and let ξ1 -H1 -Lipschitz continuous, ξ2 -H2 -Lipschitz continuous and ξ3 -H3 -Lipschitz continuous, respectively. Let g : H1 → H1 be (ζ, ϖ)-relaxed cocoercive and ε-Lipschitz continuous, f, p : H1 → H1 be ϵ-Lipschitz continuous and κ-Lipschitz continuous, respectively. Suppose that for i = 1, 2, 3, ηi : Hi × Hi → Hi is τi -Lipschitz continuous, Ai : Hi → Hi is ri strongly ηi -monotone and αi -Lipschitz continuous, Mi : Hi → 2Hi is (Ai , ηi )-monotone with constant mi , Ni : H1 × H2 × H3 → Hi is βij -Lipschitz continuous in the jth variable for j = 1, 2, 3, N1 is (δ1 , s1 )-relaxed cocoercive with respect to f1 in the first variable, where f1 : H1 → H1 is defined by f1 (u) = A1 ◦ f (u) = A1 (f (u)) for all u ∈ H1 , and Nl is (δl , sl )relaxed cocoercive with respect to Al in the lth variable for l = 2, 3. If, in addition, there exist constants ρ ∈ (0, r1 /m1 ), λ ∈ (0, r2 /m2 ), σ ∈ (0, r3 /m3 ) and ι ∈ (0, 1] such that σβ32 τ3 ρτ1 β13 λτ2 β23 ξ3 31 ξ1 τ3 1 β12 ξ2 + σβ < ι(1 − θ1 ), ρτ r1 −ρm1 +√ r3 −σm3 < 1 − θ2 , r1 −ρm1 + r2 −λm2 < 1 − θ3 , √r3 −σm3 τ1 2 + ρ2 κ2 β 2 < 1, ϵ2 α12 − 2ρs1 + 2ρδ1 κ2 β11 θ1 = 1 − 2ϖ + 2ζε2 + ε2 + r1 −ρm (3.4) 11 1 √ 2 √ 2 +λ2 β 2 2 +σ 2 β 2 τ2 α2 −2λs2 +2λδ2 β22 τ3 α23 −2σs3 +2σδ3 β33 22 33 θ2 = < 1, θ3 = < 1, r2 −λm2 r3 −σm3

λβ21 τ2 r2 −λm2

then the iterative sequence (uk , v k , wk , xk , y k , z k ) defined by Algorithm 3.1 converges strongly to (u∗ , v ∗ , w∗ , x∗ , y ∗ , z ∗ ) ∈ H1 × H2 × H3 × H1 × H2 × H3 , and (u∗ , v ∗ , w∗ , x∗ , y ∗ , z ∗ ) is a solution of the problem (1.1). Proof. Now define ∥ · ∥∗ on H1 × H2 × H3 by ∥(u, v, w)∥∗ = ∥u∥ + ∥v∥ + ∥w∥,

∀(u, v, w) ∈ H1 × H2 × H3 .

It is easy to see that (H1 × H2 × H3 , ∥ · ∥∗ ) is a Banach space. Since p is κ-Lipschitz continuous, g is (ζ, ϖ)-relaxed cocoercive and ε-Lipschitz continuous, G is ξ1 -H1 -Lipschitz continuous, S is ξ2 -H2 -Lipschitz continuous, T is ξ3 -H3 -Lipschitz continuous, A1 is α1 -Lipschitz continuous, f is ϵ-Lipschitz continuous, N1 is (δ1 , s1 )-relaxed cocoercive with respect to f1 and β11 -Lipschitz continuous in the first variable, N1 is β12 Lipschitz continuous and β13 -Lipschitz continuous in the second and third variable, respectively, we know that ∥uk − uk−1 − [g(uk ) − g(uk−1 )]∥2 = ∥uk − uk−1 ∥2 + ∥g(uk ) − g(uk−1 )∥2 − 2⟨g(uk ) − g(uk−1 ), uk − uk−1 ⟩ ≤ ∥uk − uk−1 ∥2 + ε2 ∥uk − uk−1 ∥2 − 2(−ζ∥g(uk ) − g(uk−1 )∥2 + ϖ∥uk − uk−1 ∥2 ) ≤ [1 − 2ϖ + 2ζε2 + ε2 ]∥uk − uk−1 ∥2 , ∥A1 (f (uk )) − A1 (f (uk−1 )) − ρ[N1 (p(uk ), y k , wk ) − N1 (p(uk−1 ), y k , wk )]∥ √ 2 + ρ2 κ2 β 2 ∥uk − uk−1 ∥ ≤ ϵ2 α12 − 2ρs1 + 2ρδ1 κ2 β11 11 6

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and ∥N1 (p(uk−1 ), y k , wk ) − N1 (p(uk−1 ), y k−1 , wk )∥ ≤ β12 ∥y k − y k−1 ∥ ≤ (1 + k −1 )β12 H1 (S(v k ), S(v k−1 )) ≤ (1 + k −1 )β12 ξ2 ∥v k − v k−1 ∥, ∥N1 (p(uk−1 ), y k−1 , wk ) − N1 (p(uk−1 ), y k−1 , wk−1 )∥ ≤ β13 ∥wk − wk−1 ∥. Hence, it follows from Algorithm 3.1 and Lemma 2.5 that ∥uk+1 − uk ∥ ≤ (1 − ι)∥uk − uk−1 ∥ + ι∥uk − uk−1 − [g(uk ) − g(uk−1 )]∥ + ∥ek − ek−1 ∥ ,ρ +ι∥RηA11,M [A1 (f (uk )) − ρN1 (p(uk ), y k , wk )] 1 ,ρ −RηA11,M [A1 (f (uk−1 )) − ρN1 (p(uk−1 ), y k−1 , wk−1 )]∥ + ∥ek − ek−1 ∥ 1

≤ (1 − ι)∥uk − uk−1 ∥ + ι∥uk − uk−1 − [g(uk ) − g(uk−1 )]∥ ιτ1 + {∥A1 (f (uk )) − A1 (f (uk−1 )) r1 − ρm1 −ρ[N1 (p(uk ), y k , wk ) − N1 (p(uk−1 ), y k , wk )]∥ +ρ∥N1 (p(uk−1 ), y k , wk ) − N1 (p(uk−1 ), y k−1 , wk )∥ +ρ∥N1 (p(uk−1 ), y k−1 , wk ) − N1 (p(uk−1 ), y k−1 , wk−1 )∥} ≤ (1 − ι + ιθ1 )∥uk − uk−1 ∥ + ∥ek − ek−1 ∥ ρτ1 [(1 + k −1 )β12 ξ2 ∥v k − v k−1 ∥ + β13 ∥wk − wk−1 ∥], + r1 − ρm1 √ √ τ1 2 + ρ2 κ 2 β 2 . where θ1 = 1 − 2ϖ + 2ζε2 + ε2 + r1 −ρm ϵ2 α12 − 2ρs1 + 2ρδ1 κ2 β11 11 1 Similarly, we have ∥v k+1 − v k ∥

≤ θ2 ∥v k − v k−1 ∥ λτ2 [β21 ∥uk − uk−1 ∥ + (1 + k −1 )β23 ξ3 ∥wk − wk−1 ∥] + r2 − λm2

(3.5)

(3.6)

and ∥wk+1 − wk ∥

≤ θ3 ∥wk − wk−1 ∥ στ3 + [(1 + k −1 )β31 ξ1 ∥uk − uk−1 ∥ + β32 ∥v k − v k−1 ∥], (3.7) r3 − σm3 √ 2 √ 2 2 2 τ α −2λs +2λδ β 2 +λ2 β22 τ α −2σs +2σδ β 2 +σ 2 β33 where θ2 = 2 2 r22−λm22 22 and θ3 = 3 3 r33−σm33 33 . Thus, it follows from (3.5)-(3.7) that ∥(uk+1 , v k+1 , wk+1 ) − (uk , v k , wk )∥∗ = ∥uk+1 − uk ∥ + ∥v k+1 − v k ∥ + ∥wk+1 − wk ∥ ≤ ϑk ∥(uk , v k , wk ) − (uk−1 , v k−1 , wk−1 )∥∗ + ∥ek − ek−1 ∥,

(3.8)

where ϑk

λβ21 τ2 (1 + k −1 )σβ31 ξ1 τ3 + , r2 − λm2 r3 − σm3 σβ32 τ3 ρτ1 β13 (1 + k −1 )λτ2 β23 ξ3 (1 + k −1 )ρτ1 β12 ξ2 + θ2 + , + + θ3 }. r1 − ρm1 r3 − σm3 r1 − ρm1 r2 − λm2

= max{(1 − ι + ιθ1 ) +

7

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Letting ϑ

λβ21 τ2 σβ31 ξ1 τ3 + , r2 − λm2 r3 − σm3 ρτ1 β12 ξ2 σβ32 τ3 ρτ1 β13 λτ2 β23 ξ3 + θ2 + , + + θ3 }, r1 − ρm1 r3 − σm3 r1 − ρm1 r2 − λm2

= max{(1 − ι + ιθ1 ) +

then, we know that ϑk ↓ ϑ as k → ∞. By condition (3.4), now we know that 0 < ϑ < 1. Using the same arguments as those used in Page 9 of Lan [5] (we can also refer to [8, 10]), it follow that {(uk , v k , wk )} and {xk }, {y k }, {z k } are Cauchy sequences. Thus, there exists (u∗ , v ∗ , w∗ ), (x∗ , y ∗ , z ∗ ) ∈ H1 × H2 × H3 such that (uk , v k , wk ) → (u∗ , v ∗ , w∗ ) and (xk , y k , z k ) → (x∗ , y ∗ , z ∗ ), respectively. Now, we will show that x∗ ∈ (Gu∗ )a(u∗ ) , y ∗ ∈ (Sv∗ )b(v∗ ) and z ∗ ∈ (Tw∗ )c(w∗ ) , it follows from the definition of Hausdorff pseudo-metric that d(x∗ , (Gu∗ )a(u∗ ) ) = inf{∥xk − ν∥ | ν ∈ (Gu∗ )a(u∗ ) } ≤ ∥x∗ − xk ∥ + ξ1 ∥uk − u∗ ∥ → 0. Hence, d(x∗ , (Gu∗ )a(u∗ ) ) = 0. Therefore x∗ ∈ (Gu∗ )a(u∗ ) . Similarly, we can prove that y ∗ ∈ (Sv∗ )b(v∗ ) and z ∗ ∈ (Tw∗ )c(w∗ ) . By continuity and (3.2), we know that u∗ , v ∗ , w∗ , x∗ , y ∗ , z ∗ satisfy ,ρ g(u∗ ) = RηA11,M [A1 (f (u∗ )) − ρN1 (p(u∗ ), y ∗ , w∗ )], 1 A2 ,λ ,σ ∗ ∗ ∗ v = Rη2 ,M2 [A2 (v ) − λN2 (u , v ∗ , z ∗ )], w∗ = RηA33,M [A3 (w∗ ) − σN3 (x∗ , v ∗ , w∗ )]. 3

It follows that (u∗ , v ∗ , w∗ , x∗ , y ∗ , z ∗ ) ∈ H1 × H2 × H3 × H1 × H2 × H3 is a solution of the problem (1.1). This completes the proof. 2 Remark 3.4 Condition (3.4) of Theorem 3.3 holds for some suitable value of constants, for example, ι = 0.9, λ = ρ = σ = 0.06, ϵ = ε = 0.3, ζ = 0.4, ϖ = 0.5, κ = 0.04, βij = 0.05, τi = 0.1, ri = 0.2, mi = 0.01, αi = 0.95, si = 0.02, δi = 0.3, ξi = 0.3 for i, j = 1, 2, 3. From Theorem 3.3, we have the following result. Corollary 3.5 Assume that ηi , Ai , p, g, Mi , N2 , N3 and Hi are the same as in Theorem 3.1 for i = 1, 2, 3,. Let G : H1 → CB(H1 ) be ξ1 -H1 -Lipschitz continuous, S : H2 → CB(H2 ) be ξ2 -H2 -Lipschitz continuous, T : H3 → CB(H3 ) be ξ3 -H3 -Lipschitz continuous, N1 : H1 × H2 × H3 → H1 be β11 -Lipschitz continuous and (δ1 , s1 )-relaxed cocoercive with respect to g1 in the first variable, where g1 : H1 → H1 is defined by g1 (u) = A1 ◦ g(u) = A1 (g(u)) for all u ∈ H1 . For any given (u0 , v 0 , w0 ) ∈ H1 × H2 × H3 , define an iterative sequence {(uk , v k , wk , xk , y k , z k )} ⊂ H1 × H2 × H3 × H1 × H2 × H3 as follows  ,ρ  uk+1 = (1 − ι)uk + ι{uk − g(uk ) + RηA11,M [A1 (g(uk )) − ρN1 (p(uk ), y k , wk )]} + ek ,  1   ,λ ,σ   v k+1 = RηA22,M [A2 (v k ) − λN2 (uk , v k , z k )], wk+1 = RηA33,M [A3 (wk ) − σN3 (xk , v k , wk )], 2 3 (3.9) xk ∈ G(uk ), ∥xk − xk+1 ∥ ≤ (1 + (1 + k)−1 )H1 (G(uk ), G(uk+1 )),   k k k k+1 −1 k k+1  y ∈ S(v ), ∥y − y ∥ ≤ (1 + (1 + k) )H2 (S(v ), S(v )),    k z ∈ T (wk ), ∥z k − z k+1 ∥ ≤ (1 + (1 + k)−1 )H3 (T (wk ), T (wk+1 )), where ρ, λ, σ, ι and ek are the same as in Algorithm 3.1. 8

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If, in addition, there exist constants ρ ∈ (0, r1 /m1 ), λ ∈ (0, r2 /m2 ), σ ∈ (0, r3 /m3 ) and ι ∈ (0, 1] such that λβ21 τ2 r2 −λm2

σβ32 τ3 ρτ1 β13 λτ2 β23 ξ3 31 ξ1 τ3 1 β12 ξ2 + σβ < ι(1 − θ1 ), ρτ r1 −ρm1 +√ r3 −σm3 < 1 − θ2 , r1 −ρm1 + r2 −λm2 < 1 − θ3 , √r3 −σm3 τ1 2 + ρ2 κ2 β 2 < 1, θ1 = 1 − 2ϖ + 2ζε2 + ε2 + r1 −ρm ε2 α12 − 2ρs1 + 2ρδ1 κ2 β11 11 1 √ √ 2 2 +λ2 β 2 2 +σ 2 β 2 τ3 α23 −2σs3 +2σδ3 β33 τ2 α2 −2λs2 +2λδ2 β22 22 33 < 1, θ3 = < 1, θ2 = r2 −λm2 r3 −σm3

then the iterative sequence {(uk , v k , wk , xk , y k , z k )} defined by (3.10) converges strongly to a solution (u∗ , v ∗ , w∗ , x∗ , y ∗ , z ∗ ) of the problem (1.3). Remark 3.6 (1) If we define the operators presented in the problems (1.1) and (1.3) by Ni : Ωi × H1 × H2 × H3 → Hi , f, g, p : Ω1 × H1 → H1 , ηi : Ωi × Hi × Hi → Hi , ˆ 1 ), S : Ω2 × H2 → F(H ˆ 2 ), T : Ω3 × H3 → F(H ˆ 3 ), Ai : Ωi × Hi → Hi G : Ω1 × H1 → F(H H i and Mi : Ωi × Hi → 2 for all i = 1, 2, 3, where Ω1 , Ω2 and Ω3 are three different sets, then we can obtain the corresponding results of Theorem 3.3 and Corollary 3.5. (2) In Theorem 3.3 and Corollary 3.5, if g is strongly monotone or Ni is strongly monotone in the ith variable for i = 1, 2, 3, then we can obtain the corresponding results. Our results improve and generale the known results in [3, 5-11, 13, 14, 16]. Remark 3.7 Using the same arguments as those used in the proof of Theorem 3.3, we can consider the following general system of nonlinear implicit fuzzy resolvent operators ∏ of (A, η)-monotone type in real Hilbert space ni=1 Hi : Find ui , xi ∈ Hi such that xi ∈ (Gi (ui ) )ai (ui ) and  ,ρ1  g(u1 ) = RηA11,M [A1 (f (u1 )) − ρ1 N1 (p(u1 ), x2 , u3 , u4 , · · · , un−1 , un )],  1   A2 ,ρ2    u2 = Rη2 ,M2 [A2 (u2 ) − ρ2 N2 (u1 , u2 , x3 , u4 , · · · , un−1 , un )], ··· ,  ,ρn−1   un−1 = RηAn−1,M [An−1 (un−1 ) − ρn−1 Nn−1 (u1 , u2 , u3 , u4 , · · · , un−1 , xn )],  n−1 n−1    u = RAn ,ρn [A (u ) − ρ N (x , u , u , u , · · · , u n n n n n 1 2 3 4 n−1 , un )], ηn ,Mn

ˆ i ) is a closed fuzzy operators satisfying the condition (C) with function where Gi : Hi → F(H ,ρi ai : Hi → [0, 1], ρi > 0 is a constant and RηAii,M = (Ai + ρi Mi )−1 is the corresponding i resolvent operator associated with (Ai , ηi )-monotone operator Mi for i = 1, 2, · · · , n.

References [1] S.S. Chang, J.K. Kim, Y.M. Nam, K.H. Kim, Remark on the three-step iteration for nonlinear operator equations and nonlinear variational inequalities, J. Comput. Anal. Appl. 8(2) (2006) 139–149. [2] S.S. Chang, H.W.J. Lee, C.K. Chan, J.K. Kim, On a new system of nonlinear variational inequalities and algorithms, J. Comput. Anal. Appl. 11(1) (2009) 119–130. [3] J.W. Peng, D.L. Zhu, Three-step iterative algorithm for a system of set-valued variational inclusions with (H, η)-monotone operators, Nonlinear Anal.: TMA 68(1) (2008) 139-153. [4] G. Stampacchia, Formes bilin´eaires coercitives sur les ensembles convexes, C. R. Acad. Sci. Paris 258 (1964) 4413-4416. (in French)

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[5] H.Y. Lan, A class of nonlinear (A, η)-monotone operator inclusion problems with relaxed cocoercive mappings, Adv. Nonlinear Var. Inequal. 9(2) (2006) 1-11. [6] H.Y. Lan, Iterative schemes with mixed errors for general nonlinear resolvent operator equations, J. Appl. Funct. Anal. 1(2) (2006) 153-164. [7] H.Y. Lan, Y.J. Cho and R. U. Verma, On solution sensitivity of generalized relaxed cocoercive implicit quasivariational inclusions with A-monotone mappings, J. Comput. Anal. Appl. 8(1) (2006) 75-87. [8] H.Y. Lan, J. I. Kang and Y. J. Cho, Nonlinear (A, η)-monotone operator inclusion systems involving non-monotone set-valued mappings, Taiwanese Math. J. 11(3) (2007) 683-701. [9] H.Y. Lan, A stable iteration procedure for relaxed cocoercive variational inclusion systems based on (A, η)-monotone operators, J. Comput. Anal. Appl. 9(2) (2007) 147-157. [10] H.Y. Lan, J.H. Kim and Y.J. Cho, On a new system of nonlinear A-monotone multivalued variational inclusions, J. Math. Anal. Appl. 327(1) (2007) 481-493. [11] Ram U. Verma, Approximation solvability of a class of nonlinear set-valued variational inclusions involving (A, η)-monotone mappings, J. Math. Anal. Appl. 337(2) (2008) 969-975. [12] Q.B. Zhang, X.P. Ding, C.Z. Cheng, Resolvent operator technique for generalized implicit variational-like inclusion in Banach space, J. Math. Anal. Appl. 361(2) (2010) 283-292. [13] G.Q. Wu, A new system of generalized nonlinear fuzzy variational inclusions involving (H, η)monotone mappings, Adv. Nonlinear Var. Inequal. 11(1) (2008) 31–39. [14] J.Y. Park, J.U. Jeong, A perturbed algorithm of variational inclusions for fuzzy mappings, Fuzzy Sets and Systems 115(3) (2000) 419-424. [15] H.Y. Lan, R.U. Verma, Iterative algorithms for nonlinear fuzzy variational inclusion systems with (A, η)-accretive mappings in Banach spaces, Adv. Nonlinear Var. Inequal. 11(1) (2008) 15-30. [16] S.S. Chang, Y. Zhu, On variational inequalities for fuzzy mappings, Fuzzy Sets and System 32 (1989) 259-267. [17] P. Kuman, N. Petrot, Mixed variational-like inequality for fuzzy mappings in reflexive Banach spaces, J. Inequal. Appl. 2009 (2009) Art. ID 209485, 15 pp. [18] H.J. Zimmermann, Fuzzy Set Theory and its Aplications (4th ed.), Kluwer Academic Publishers Group, Boston, 2001. [19] I.A. Rus, Generalized Contractions and Applications, Cluj University Press, Cluj-Napoca, Romania, 2001.

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JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.2, 345-361, 2011, COPYRIGHT 2011 EUDOXUS PRESS, 345 LLC

A COMPOSITE ITERATIVE METHOD FOR GENERALIZED MIXED EQUILIBRIUM PROBLEMS AND VARIATIONAL INEQUALITY PROBLEMS† THANYARAT JITPEERA AND POOM KUMAM‡ Abstract. In this paper, we introduce a new composite iterative scheme for finding the common element of the set of fixed points of a nonexpansive mapping, the set of solution of generalized mixed equilibrium problems and the set of variational inequality problems for an α-inverse-strongly monotone mapping with the viscosity approximation method in a real Hilbert space. We show that the sequence converges strongly to a common element of the above three sets under some parameter controlling conditions. Application to optimization problems which is one of the main result in this work is also given. The results in this paper generalize and improve some recent results of Jung [J. S. Jung, Strong convergence of composite iterative methods for equilibrium problems and fixed point problems. Appl. Math. Comput. 213 (2009), 498–505], Jaiboon et al. [C. Jaiboon, P. Kumam and U.W. Humphries, Convergence theorems by the viscosity approximation methods for equilibrium problems and variational inequality problems. J. Comput. Math. Optim. 5 (2009), 29–56], Su et al. [Y. Su, M. Shang and X. Qin, An iterative method of solution for equilibrium and optimization problems. Nonlinear Anal. 69 (2008), 2709–2719], Yao et al. [Y. Yao, Y.C Liou and R. Chen, Convergence theorems for fixed point problems and variational inequality problems. J. Nonlinear Convex Anal. 9 (2008), 239–248] and connected with Jung [J. S. Jung, Strong convergence of composite iterative methods for equilibrium problems and fixed point problems. J. Comput. Anal. Appl. Vol.12, NO.1-A (2010), 124-140].

1. Introduction Let C be a closed convex subset of a real Hilbert space H with the inner product h·, ·i and the norm k · k. Let F be a bifunction of C × C into R, where R is the set of real numbers, B : C → H be a nonlinear mapping and ϕ : C → R be a real-valued function. The generalized mixed equilibrium problem is to find x ∈ C such that F (x, y) + hBx, y − xi + ϕ(y) − ϕ(x) ≥ 0,

∀y ∈ C.

(1.1)

The set of solutions of (1.1) is denoted by GM EP (F, ϕ, B), that is GM EP (F, ϕ, B) = {x ∈ C : F (x, y) + hBx, y − xi + ϕ(y) − ϕ(x) ≥ 0,

∀y ∈ C}.

In particular, if B ≡ 0, the problem (1.1) is reduced into the mixed equilibrium problem [7] for finding x ∈ C such that F (x, y) + ϕ(y) − ϕ(x) ≥ 0, ∀y ∈ C. (1.2) The set of solutions of (1.2) is denoted by M EP (F, ϕ). If ϕ ≡ 0, the problem (1.1) is reduced into the generalized equilibrium problem [25] for finding x ∈ C such that F (x, y) + hBx, y − xi ≥ 0, ∀y ∈ C. (1.3) The set of solutions of (1.3) is denoted by GEP (F, B), which this problem was studied by Takahashi and Takahashi [26]. If F ≡ 0, the problem (1.1) is reduced into the mixed variational inequality of Browder type [2] for finding x ∈ C such that hBx, y − xi + ϕ(y) − ϕ(x) ≥ 0, ∀y ∈ C. (1.4) 2000 Mathematics Subject Classification. : 46C05, 47D03, 47H09, 47H10, 47H20. Key words and phrases. Viscosity method; Nonexpansive mapping; α-inverse-strongly monotone mappings; Variational inequality problem; Generalized Mixed Equilibrium problem; Fixed points. † This project was supported by the National Research Council of Thailand. ‡ Corresponding author email: [email protected].(P. Kumam). 1

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The set of solutions of (1.4) is denoted by M V I(C, ϕ, B). If B ≡ 0 and ϕ ≡ 0, the problem (1.1) is reduced into the equilibrium problem [3] for finding x ∈ C such that F (x, y) ≥ 0, ∀y ∈ C. (1.5) The set of solutions of (1.5) is denoted by EP (F ). This problem contains fixed point problems, includes as special cases numerous problems in physics, optimization, and economics. Some methods have been proposed to solve the equilibrium problem, please consult [6, 9, 13, 14, 26]. If F ≡ 0 and ϕ ≡ 0, the problem (1.1) is reduced into the Hartmann-Stampacchia variational inequality [11] for finding x ∈ C such that hBx, y − xi ≥ 0, ∀y ∈ C. (1.6) The set of solutions of (1.6) is denoted by V I(C, B). The variational inequality has been extensively studied in the literature. See, e.g. [29, 31] and the references therein. If F ≡ 0 and B ≡ 0, the problem (1.1) is reduced into the minimize problem for finding x ∈ C such that ϕ(y) − ϕ(x) ≥ 0, ∀y ∈ C. (1.7) The set of solutions of (1.7) is denoted by Argmin(ϕ). Recall that the following definitions. (1) A mapping A of C into H is called monotone if hAu − Av, u − vi ≥ 0,

for all

u, v ∈ C.

(1.8)

(2) A is called α-inverse-strongly monotone [4, 15] if there exists a positive real number α such that hAu − Av, u − vi ≥ αkAu − Avk2 ,

for all

u, v ∈ C.

(1.9)

Clearly, every α-inverse-strongly monotone is monotone. (3) A is said to be β-strongly monotone if there exists a positive real number β such that hAu − Av, u − vi ≥ βku − vk2 ,

for all

u, v ∈ C.

(1.10)

(4) A is called L-Lipschitz continuous if there exists a positive real number L such that kAu − Avk ≤ Lku − vk,

for all

u, v ∈ C.

(1.11) 1 α –Lipschitz

It is easy to see that if A is an α-inverse-strongly monotone mapping of C into H, then A is continuous. (5) A mapping f : C −→ C is said to be a contraction if there exists a coefficient k (0 < k < 1) such that kf (x) − f (y)k ≤ kkx − yk, for all x, y ∈ C. (1.12) To find an element of F (S) ∩ V I(C, A), Takahashi and Toyoda [25] introduced the following iterative scheme: xn+1 = αn xn + (1 − αn )SPC (xn − λn Axn ), (1.13) for every n ≥ 1, where x1 = x ∈ C, {αn } is a sequence in (0, 1) and {λn } is a sequence in (0, 2α). Recently, Nadezhkina and Takahashi [16] and Zeng and Yao [32] proposed some new iterative schemes for finding elements in F (S) ∩ V I(C, A). In 2007, Chen et al. [8] introduced the following iterative scheme: xn+1 = αn f (xn ) + (1 − αn )SPC (xn − λn Axn ), (1.14) for every n ≥ 1, where x1 = x ∈ C, {αn } is a sequence in (0, 1), {λn } is a sequence in (0, 2α), f is a contraction of C into itself, S is a nonexpansive self-mapping of a closed convex subset C of a Hilbert space H. They proved that such a sequence {xn } converges strongly to a common element of the set of fixed points of nonexpansive mapping and the set of solutions of the variational inequality for an inverse-strongly-monotone mapping which solves some variational inequality problems. Recently, many authors studied the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of an equilibrium problem in the framework of Hilbert spaces and Banach spaces, respectively; see, for instance, [1, 12, 19, 20, 25, 26, 27] and the references therein. On the other hand, for finding an element of F (S) ∩ V I(C, A) ∩ EP (F ), Su et al. [24] introduced the following iterative scheme by the viscosity approximation method in a Hilbert space: x1 ∈ H ( F (un , y) + r1n hy − un , un − xn i ≥ 0, ∀y ∈ C, (1.15) xn+1 = αn f (xn ) + (1 − αn )SPC (un − λn Aun ),

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for all n ∈ N, where {αn } ⊂ [0, 1) and {rn } ⊂ (0, ∞) satisfy some appropriate conditions. Furthermore, they proved {xn } and {un } converge strongly to the same point z ∈ F (S) ∩ V I(C, A) ∩ EP (F ) where z = PF (S)∩V I(C,A)∩EP (F ) f (z). Very recently, Jung [13] also introduced the following iterative scheme:  1   F (un , y) + rn hy − un , un − xn i ≥ 0, ∀y ∈ C, (1.16) yn = αn f (xn ) + (1 − αn )Sun ,   xn+1 = (1 − βn )yn + βn Syn , for every n ≥ 1, where {αn }, {βn } ⊂ [0, 1] and {rn } ⊂ (0, ∞). He proved that, if F (S) ∩ EP (F ) 6= ∅, then the sequence {xn } and {un } generate by (1.16) converges strongly to F (S) ∩ EP (F ). In this paper motivated by the iterative schemes considered in (1.14), (1.15) and (1.16), we will introduce a new composite iterative schemes for finding a common element of the set of fixed points of a nonexpansive mapping, the set of solution of generalized mixed equilibrium problems and the set of variational inequality problems for an α-inverse-strongly monotone mapping in a real Hilbert space. Consequently, we obtain strong convergence theorems under the some mild conditions on parameters. As applications, we also apply our results to study the optimization problem and we next utilize our main results for a class of strictly pseudo-contraction mappings. The results present in this paper improved and connected with some recent results of Chen et al. [8], Su et al. [24], Yao et al. [30], Jung [13], Jaiboon et al. [12] and many authors. 2. Preliminaries Let H be a real Hilbert space with inner product h·, ·i and the norm k · k. Let C be a closed convex subset of H. It is well known that for all x, y ∈ H and λ ∈ [0, 1] there holds kλx + (1 − λ)yk2 = λkxk2 + (1 − λ)kyk2 − λ(1 − λ)kx − yk2 .

(2.1)

For every point x ∈ H, there exists a unique nearest point in C, denoted by PC x, such that kx − PC xk ≤ kx − yk,

for all y ∈ C.

PC is called the metric projection of H onto C. It is well known that PC is a nonexpansive mapping of H onto C and satisfies hx − y, PC x − PC yi ≥ kPC x − PC yk2 , (2.2) for every x, y ∈ H. Moreover, PC x is characterized by the following properties: PC x ∈ C and hx − PC x, y − PC xi ≤ 0,

(2.3)

kx − yk2 ≥ kx − PC xk2 + ky − PC xk2

(2.4)

for all x ∈ H, y ∈ C. It is easy to see that the following is true: u ∈ V I(A, C) ⇔ u = PC (u − λAu), λ > 0.

(2.5)

We note that, for all u, v ∈ C and λ > 0, k(I − λA)u − (I − λA)vk2

=

k(u − v) − λ(Au − Av)k2

=

ku − vk2 − 2λhu − v, Au − Avi + λ2 kAu − Avk2

≤

ku − vk2 + λ(λ − 2α)kAu − Avk2 .

(2.6)

So, if λ ≤ 2α, then I − λA is a nonexpansive mapping from C to H. A set-valued mapping T : H −→ 2H is called monotone if for all x, y ∈ H, f ∈ T x and g ∈ T y imply hx − y, f − gi ≥ 0. A monotone mapping T : H −→ 2H is maximal if the graph of G(T ) of T is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping T is maximal if and only if for (x, f ) ∈ H × H, hx − y, f − gi ≥ 0 for every (y, g) ∈ G(T ) implies f ∈ T x. Let A be an inverse-strongly monotone mapping of C into H and let NC v be the normal cone to C at v ∈ C, i.e., NC v = {w ∈ H : hv − u, wi ≥ 0, ∀u ∈ C}. Define ½ Av + NC v, v ∈ C; Tv = ∅, v∈ / C.

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Then T is the maximal monotone and 0 ∈ T v if and only if v ∈ V I(C, A); see [21, 22]. It is also known that H satisfies the Opial condition; for any sequence {xn } with xn * x, the inequality lim inf kxn − xk < lim inf kxn − yk, n−→∞

(2.7)

n−→∞

holds for every y ∈ H with y 6= x. The following lemmas will be useful with proving the convergence result of this paper. Lemma 2.1. [23] Let {xn } and {yn } be bounded sequences in a Banach space X and let {αn } be a sequence in [0, 1] with 0 < lim inf n−→∞ αn ≤ lim supn−→∞ αn < 1. Suppose xn+1 = (1 − αn )yn + αn xn for all integers n ≥ 0 and lim supn−→∞ (kyn+1 −yn k−kxn+1 −xn k) ≤ 0. Then, limn−→∞ kyn −xn k = 0. Lemma 2.2. [28] Assume {an } is a sequence of nonnegative real numbers such that an+1 ≤ (1 − γn )an + δn , n ≥ 1, where {γn } and {δn }satisfy the following conditions: P∞ Q∞ (a) {γn } ⊂ [0, 1] and n=1 γn = ∞ or, equivalently, n=0 (1 − γn ) = 0, P∞ (b) lim supn−→∞ γδnn ≤ 0 or n=1 |δn | < ∞. Then limn−→∞ an = 0. For solving the generalized mixed equilibrium problem and the mixed equilibrium problem, let us give the following assumptions for the bifunction F , the function ϕ and the set C: (A1) (A2) (A3) (A4) (A5) (B1)

F (x, x) = 0 for all x ∈ C; F is monotone, i.e., F (x, y) + F (y, x) ≤ 0 for all x, y ∈ C; for each y ∈ C, x 7→ F (x, y) is weakly upper semicontinuous; for each x ∈ C, y 7→ F (x, y) is convex; for each x ∈ C, y 7→ F (x, y) is lower semicontinuous; For each x ∈ H and r > 0, there exist a bounded subset Dx ⊆ C and yx ∈ C ∩ dom(ϕ) such that for any z ∈ C \ Dx , 1 F (z, yx ) + ϕ(yx ) + hyx − z, z − xi < ϕ(z); r

(B2) C is a bounded set. Lemma 2.3. [3] Let C be a nonempty closed convex subset of H and let F be a bifunction of C × C into R satisfying (A1)-(A5). Let r > 0 and x ∈ H. Then, there exists z ∈ C such that 1 F (z, y) + hy − z, z − xi ≥ 0, for all y ∈ C. r The following lemma was also given in [5]. Lemma 2.4. [5] Assume that F : C × C −→ R satisfies (A1)-(A5). For r > 0 and x ∈ H, define a mapping Tr : H −→ C as follows: 1 Tr (x) = {z ∈ C : F (z, y) + hy − z, z − xi ≥ 0, r Then, the following hold:

∀y ∈ C}.

(1) Tr is single-valued; (2) Tr is firmly nonexpansive, i.e., for any x, y ∈ H, kTr x − Tr yk2 ≤ hTr x − Tr y, x − yi; (3) F (Tr ) = EP (F ); and (4) EP (F ) is closed and convex. By similar argument as in the proof of Lemma 2.3 in [17] (see also in [6]), we have the following result. Lemma 2.5. [7, 17, 18] Let C be a nonempty closed convex subset of real Hilbert space H. Let F : C × C −→ R be a bifunction and ϕ : C −→ R ∪ {+∞} is a proper lower semicontinuous and convex

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A COMPOSITE ITERATIVE METHOD FOR GENERALIZED MIXED EQUILIBRIUM PROBLEMS

5

function satisfies (A1)-(A5). Assume that either (B1) or (B2) holds. For r > 0 and x ∈ H, define a mapping Kr : H −→ C as follows: 1 Kr (x) = {z ∈ C : F (z, y) + ϕ(y) − ϕ(z) + hy − z, z − xi ≥ 0, ∀y ∈ C}, r for all x ∈ H. Then, the following hold: (1) For each x ∈ H, Kr (x) is nonempty; (2) Kr is single-valued; (3) Kr is firmly nonexpansive, i.e., for any x, y ∈ H, kKr (x) − Kr (y)k2 ≤ hKr (x) − Kr (y), x − yi; (4) F (Kr ) = M EP (F, ϕ); and (5) M EP (F, ϕ) is closed and convex. 3. Main Results In this section, we prove a strong convergence theorem for finding a common element of the set of solutions for a generalized mixed equilibrium problem, the set variational inequality and the set of fixed points of a nonexpansive mapping by the composite viscosity approximation method in Hilbert spaces. Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from C × C −→ R satisfying (A1)-(A5), ϕ : C → R be a lower semicontinuous and convex function and let A, B be two α, β-inverse-strongly monotone mapping of C into H, respectively. Let f : C → C be a contraction with coefficient k, (0 < k < 1) and S be a nonexpansive mappings from C into itself such that Θ := F (S) ∩ V I(C, A) ∩ GM EP (F, ϕ, B) 6= ∅. Assume that either (B1) or (B2) holds. Suppose x1 ∈ C and {xn }, {yn } and {un } are given by  F (un , y) + hBxn , y − un i + ϕ(y) − ϕ(un ) + r1n hy − un , un − xn i ≥ 0, ∀y ∈ C,     z = P (u − λ Au ), n C n n n (3.1)  y = β f (x ) + (1 − βn )Szn , n n n    xn+1 = (1 − αn )yn + αn Syn , ∀n ≥ 1, where {αn }, {βn } are two sequence in (0, 1), λn ∈ [a, b] ⊂ (0, 2α) and rn ∈ [c, d] ⊂ (0, 2β) satisfy the following conditions: P∞ (C1) limn−→∞ βn = 0 and n=1 βn = ∞, (C2) limn−→∞ αn = 0, (C3) limn−→∞ |λn+1 − λn | = 0, (C4) lim inf n−→∞ rn > 0 and limn−→∞ |rn+1 − rn | = 0. Then {xn } and {un } converge strongly to q ∈ Θ, where q = PΘ f (q). Proof. Let Q = PΘ . Then Qf is a contraction of H into itself. In fact, there exists k ∈ (0, 1) such that kf (x) − f (y)k ≤ k||x − yk for all x, y ∈ H. So, we have that kQf (x) − Qf (y)k ≤ kf (x) − f (y)k ≤ k||x − yk, for all x, y ∈ H. This implies that Qf is a contraction on H into itself. Since H is Hilbert spaces, there exists a unique q ∈ H, such that q = Qf (q). Such a q is an element of C. The unique fixed point of the mapping Qf is denoted by q in the statement of Theorem 3.1. We proceed with following steps: Step 1. We show that {xn } is bound. Further, we take x∗ ∈ Θ and let {Krn } be a sequence of mappings defined as in Lemma 2.5. Then x∗ = Sx∗ = PC (x∗ − λn Ax∗ ) = Krn (x∗ − rn Bx∗ ) and un = Krn (xn − rn Bxn ), it follows that kun − x∗ k

= kKrn (xn − rn Bxn ) − Krn (x∗ − rn Bx∗ )k ≤ k(xn − rn Bxn ) − (x∗ − rn Bx∗ )k = k(I − rn B)xn − (I − rn B)x∗ k ≤ kxn − x∗ k.

By the fact that PC and I − λn B are nonexpansive and x∗ = PC (x∗ − λn Bx∗ ), we get kzn − x∗ k

= kPC (un − λn Aun ) − PC (x∗ − λn Ax∗ )k

350

6

T. JITPEERA AND P. KUMAM

≤ = ≤ ≤

k(un − λn Aun ) − (x∗ − λn Ax∗ )k k(I − λn A)un − (I − λn A)x∗ k kun − x∗ k kxn − x∗ k,

and kyn − x∗ k ≤ ≤ ≤

βn kf (xn ) − x∗ k + (1 − βn )kSzn − x∗ k βn kf (xn ) − x∗ k + (1 − βn )kzn − x∗ k βn kf (xn ) − x∗ k + (1 − βn )kxn − x∗ k,

and hence kxn+1 − x∗ k ≤ ≤ = ≤ ≤ ≤

(1 − αn )kyn − x∗ k + αn kSyn − x∗ k (1 − αn )kyn − x∗ k + αn kyn − x∗ k kyn − x∗ k βn kf (xn ) − x∗ k + (1 − βn )kxn − x∗ k βn kf (xn ) − f (x∗ )k + βn kf (x∗ ) − x∗ k + (1 − βn )kxn − x∗ k βn kkxn − x∗ k + (1 − βn )kxn − x∗ k + βn kf (x∗ ) − x∗ k kf (x∗ ) − x∗ k ≤ {1 − (1 − k)βn }kxn − x∗ k + βn (1 − k) 1−k ∗ ∗ kf (x ) − x k ≤ max{kxn − x∗ k, }. 1−k ∗

∗

)−x k By induction on n, we obtain kxn+1 − x∗ k ≤ max{kx1 − x∗ k, kf (x1−k } for all n ∈ N, x1 ∈ C and hence {xn } is bounded. Consequently, the sequences {zn },{yn },{un }, {Szn },{Syn } and {f (xn )} are also bounded. Step 2. We show that limn→∞ kxn+1 −xn k = 0, limn→∞ kyn+1 −yn k = 0 and limn→∞ kun+1 −un k = 0. Since I − λn A and PC are nonexpansive, we have

kzn − zn−1 k

= kPC (un − λn Aun ) − PC (un−1 − λn−1 Aun−1 )k ≤ k(un − λn Aun ) − (un−1 − λn−1 Aun−1 )k = k(un − λn Aun ) − (un−1 − λn Aun−1 ) − (λn − λn−1 )Aun−1 k = k(I − λn A)un − (I − λn A)un−1 − (λn − λn−1 )Aun−1 k ≤

kun − un−1 k + |λn − λn−1 |kAun−1 k.

(3.2)

On the other hand, from un−1 = Krn−1 (xn−1 − rn−1 Bxn−1 ) and un = Krn (xn − rn Bxn ), it follows that 1 F (un−1 , y) + hBxn−1 , y − un−1 i + ϕ(y) − ϕ(un−1 ) + hy − un−1 , un−1 − xn−1 i ≥ 0, ∀y ∈ C (3.3) rn−1 and

1 hy − un , un − xn i ≥ 0, ∀y ∈ C. (3.4) rn Substituting y = un in to (3.3) and y = un−1 in to (3.4), we have 1 F (un−1 , un ) + hBxn−1 , un − un−1 i + ϕ(un ) − ϕ(un−1 ) + hun − un−1 , un−1 − xn−1 i ≥ 0 rn−1 F (un , y) + hBxn , y − un i + ϕ(y) − ϕ(un ) +

and F (un , un−1 ) + hBxn , un−1 − un i + ϕ(un−1 ) − ϕ(un ) +

1 hun−1 − un , un − xn i ≥ 0. rn

From (A2), we have hun − un−1 , Bxn−1 − Bxn +

un−1 − xn−1 un − xn − i ≥ 0, rn−1 rn

and then hun − un−1 , rn−1 (Bxn−1 − Bxn ) + un−1 − xn−1 −

rn−1 (un − xn )i ≥ 0, rn

351

A COMPOSITE ITERATIVE METHOD FOR GENERALIZED MIXED EQUILIBRIUM PROBLEMS

so hun − un−1 , rn−1 Bxn−1 − rn−1 Bxn + un−1 − un + un − xn−1 −

7

rn−1 (un − xn )i ≥ 0. rn

It follows that hun − un−1 , (I − rn−1 B)xn − (I − rn−1 B)xn−1 + un−1 − un + un − xn −

rn−1 (un − xn )i ≥ 0, rn

rn−1 )(un − xn )i ≥ 0. rn Without loss of generality, let us assume that there exists a real number c such that rn−1 > c > 0, for all n ∈ N. Then, we have rn−1 kun − un−1 k2 ≤ hun − un−1 , xn − xn−1 + (1 − )(un − xn )i r ¯ nr ¯ n o ¯ n−1 ¯ ≤ kun − un−1 k kxn − xn−1 k + ¯1 − ¯kun − xn k rn and hence 1 kun − un−1 k ≤ kxn − xn−1 k + |rn − rn−1 |kun − xn k rn M1 ≤ kxn − xn−1 k + |rn − rn−1 |, (3.5) c where M1 = sup{kun − xn k : n ∈ N}. Substituting (3.5) into (3.2), we have hun − un−1 , un−1 − un i + hun − un−1 , xn − xn−1 + (1 −

kzn − zn−1 k

≤

kxn − xn−1 k +

M1 |rn − rn−1 | + |λn − λn−1 |kAun−1 k. c

(3.6)

Note that kyn−1 − yn k = kβn−1 f (xn−1 ) + (1 − βn−1 )Szn−1 − βn f (xn ) − (1 − βn )Szn k = kβn−1 (f (xn−1 ) − f (xn )) − (1 − βn−1 )(Szn − Szn−1 ) + (βn−1 − βn )f (xn ) − (βn−1 − βn )Szn k ≤ βn−1 kf (xn−1 ) − f (xn )k + (1 − βn−1 )kSzn − Szn−1 k + |βn−1 − βn |kf (xn )k + |βn−1 − βn |kSzn k = βn−1 kf (xn−1 ) − f (xn )k + (1 − βn−1 )kSzn − Szn−1 k + |βn−1 − βn |(kf (xn )k + kSzn k) ≤ βn−1 kkxn−1 − xn k + (1 − βn−1 )kzn − zn−1 k + |βn−1 − βn |(kf (xn )k + kSzn k) M1 ≤ βn−1 kkxn−1 − xn k + (1 − βn−1 ){ kxn − xn−1 k + |rn − rn−1 | + |λn−1 − λn |kAun−1 k} c + (kf (xn )k + kSzn k)|βn−1 − βn | M1 ≤ (1 − (1 − k)βn−1 )kxn−1 − xn k + |rn − rn−1 | + |λn−1 − λn |kAun−1 k + M2 |βn−1 − βn |, (3.7) c where M2 = sup{kf (xn )k + kSzn k : n ∈ N}. Also, simple calculations that xn+1 − xn = (1 − αn )(yn − yn−1 ) + αn (Syn − Syn−1 ) + (αn − αn−1 )(Syn−1 − yn−1 ). It follows that kxn+1 − xn k ≤ ≤

(1 − αn )kyn − yn−1 k + αn kSyn − Syn−1 k + |αn − αn−1 |kSyn−1 − yn−1 k (1 − αn )kyn − yn−1 k + αn kyn − yn−1 k + kSyn−1 − yn−1 k|αn − αn−1 |

≤ kyn − yn−1 k + M3 |αn − αn−1 | ≤

M1 |rn − rn−1 | + |λn−1 − λn |kAun−1 k c +M2 |βn−1 − βn | + M3 |αn − αn−1 |, (3.8) ((1 − (1 − k)βn−1 )kxn − xn−1 k +

where M3 = sup{kSyn−1 − yn−1 k : n ∈ N}. From the condition (C1), it easy to see that lim (1 − k)βn = 0,

n−→∞

∞ X

(1 − k)βn = ∞

n=1

and by (C1)-(C4), that ∞ ³ ´ X M1 |rn − rn−1 | + |λn−1 − λn |kAun−1 k + M2 |βn−1 − βn | + M3 |αn − αn−1 | < ∞ c n=1

352

8

T. JITPEERA AND P. KUMAM

Applying Lemma (2.2) to (3.8), we have lim kxn+1 − xn k

=

0.

(3.9)

lim kun+1 − un k

=

0.

(3.10)

lim kyn − yn−1 k

= 0.

(3.11)

n→∞

Moreover, from (3.5) it follows that n→∞

By( 3.7), we also have n→∞

Step 3. We show that limn−→∞ kxn − Szn k = 0. Indeed, from (C2), we get kxn+1 − yn k = αn kSyn − yn k → 0 as n → ∞. It follows that kxn − yn k ≤ kxn − xn+1 k + kxn+1 − yn k → 0 as n → ∞. Notice that yn − Szn = βn (f (xn ) − Szn ), and (C1), that lim kyn − Szn k = 0.

n−→∞

(3.12)

On the other hand, we observe that kxn − Szn k

≤

kxn − yn k + kyn − Szn k,

hence, we get lim kxn − Szn k = 0.

n−→∞

(3.13)

Step 4. We show that kxn − un k −→ 0, as n → ∞. For each x∗ ∈ Θ, note that Krn is firmly nonexpansive, then we have kun − x∗ k2

kKrn (xn − rn Bxn ) − Krn (x∗ − rn Bx∗ )k2 hKrn (xn − rn Bxn ) − Krn (x∗ − rn Bx∗ ), un − x∗ i h(xn − rn Bxn ) − (x∗ − rn Bx∗ ), un − x∗ i 1 = {k(xn − rn Bxn ) − (x∗ − rn Bx∗ )k2 + kun − x∗ k2 2 −k(xn − rn Bxn ) − (x∗ − rn Bx∗ ) − (un − x∗ )k2 } 1 ≤ {kxn − x∗ k2 + kun − x∗ k2 − kxn − un − rn (Bxn − Bx∗ )k2 } 2 1 ≤ {kxn − x∗ k2 + kun − x∗ k2 − kxn − un k2 2 +2rn hxn − un , Bxn − Bx∗ i − rn2 kBxn − Bx∗ k2 }, = ≤ =

which imply that kun − x∗ k2

≤ kxn − x∗ k2 − kxn − un k2 + 2rn kxn − un kkBxn − Bx∗ k.

(3.14)

2

Therefore, from the convexity of k · k and (3.1), we have kzn − x∗ k2

= kPC (un − λn Aun ) − PC (x∗ − λn Ax∗ )k2 ≤ k(un − λn Aun ) − (x∗ − λn Ax∗ )k2 = k(I − λn A)un − (I − λn A)x∗ k2 ≤ kun − x∗ k2 ,

(3.15)

from (3.14), we get kyn − x∗ k2

≤ βn kf (xn ) − x∗ k2 + (1 − βn )kSzn − x∗ k2 ≤ βn kf (xn ) − x∗ k2 + (1 − βn )kzn − x∗ k2 ≤ βn kf (xn ) − x∗ k2 + (1 − βn )kun − x∗ k2

(3.16)

≤ βn kf (xn ) − x∗ k2 + (1 − βn ){kxn − x∗ k2 − kxn − un k2 + 2rn kxn − un kkBxn − Bx∗ k} ≤ βn kf (xn ) − x∗ k2 + kxn − x∗ k2 − (1 − βn )kxn − un k2

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A COMPOSITE ITERATIVE METHOD FOR GENERALIZED MIXED EQUILIBRIUM PROBLEMS

9

+2(1 − βn )rn kxn − un kkBxn − Bx∗ k.

(3.17)

On the other hand, we note that kun − x∗ k2

=

kKrn (xn − rn Bxn ) − Krn (x∗ − rn Bx∗ )k2

≤

k(xn − rn Bxn ) − (x∗ − rn Bx∗ )k2

=

k(xn − x∗ ) − rn (Bxn − Bx∗ )k2

≤

kxn − x∗ k2 − 2rn hxn − x∗ , Bxn − Bx∗ i + rn2 kBxn − Bx∗ k2

≤

kxn − x∗ k2 − 2rn βkBxn − Bx∗ k2 + rn2 kBxn − Bx∗ k2 .

(3.18)

Using (3.16) and (3.18), we have kxn+1 − x∗ k2

≤

(1 − αn )kyn − x∗ k2 + αn kSyn − x∗ k2

≤

(1 − αn )kyn − x∗ k2 + αn kyn − x∗ k2

=

kyn − x∗ k2

≤

βn kf (xn ) − x∗ k2 + (1 − βn )kun − x∗ k2

≤ ∗ 2

kxn+1 − x k

∗ 2

∗ 2

∗ 2

∗ 2

(3.19) ∗ 2

βn kf (xn ) − x k + (1 − βn ){kxn − x k − 2rn βkBxn − Bx k +

rn2 kBxn ∗ 2

≤

βn kf (xn ) − x k + (1 − βn ){kxn − x k + rn (rn − 2β)kBxn − Bx k }

≤

βn kf (xn ) − x∗ k2 + kxn − x∗ k2 + (1 − βn )rn (rn − 2β)kBxn − Bx∗ k2 .

− Bx∗ k2 }

Then, we have (1 − βn )c(2β − d)kBxn − Bx∗ k2

≤

(1 − βn )rn (2β − rn )kBxn − Bx∗ k2

≤

βn kf (xn ) − x∗ k2 + kxn − x∗ k2 − kxn+1 − x∗ k2

≤

βn kf (xn ) − x∗ k2 + kxn − xn+1 k(kxn − x∗ k + kxn+1 − x∗ k).

From conditions (C1), {rn } ⊂ [c, d] ⊂ (0, 2β) and limn−→∞ kxn+1 − xn k = 0, we obtain lim kBxn − Bx∗ k = 0.

n−→∞

(3.20)

From (3.14) and (3.19), we have kxn+1 − x∗ k2

≤ βn kf (xn ) − x∗ k2 + (1 − βn )kun − x∗ k2 ≤ βn kf (xn ) − x∗ k2 + (1 − βn ){kxn − x∗ k2 − kxn − un k2 +2rn kxn − un kkBxn − Bx∗ k} ≤ βn kf (xn ) − x∗ k2 + kxn − x∗ k2 − (1 − βn )kxn − un k2 +2rn (1 − βn )kxn − un kkBxn − Bx∗ k

and (1 − βn )kxn − un k2

≤

βn kf (xn ) − x∗ k2 + kxn − x∗ k2 − kxn+1 − x∗ k2 +2rn (1 − βn )kxn − un kkBxn − Bx∗ k

≤

βn kf (xn ) − x∗ k2 + kxn − xn+1 k(kxn − x∗ k + kxn+1 − x∗ k) +2rn (1 − βn )kxn − un kkBxn − Bx∗ k.

Since kxn+1 − xn k −→ 0 , βn −→ 0 and kBxn − Bx∗ k −→ 0, n −→ ∞, then we have lim kxn − un k = 0.

n−→∞

(3.21)

Since lim inf n−→∞ rn > 0, we obtain xn − un 1 k = lim kxn − un k = 0. n−→∞ rn rn Step 5. We show that limn−→∞ kSzn − zn k = 0. For x∗ ∈ Θ, we compute lim k

n−→∞

kyn − x∗ k2

≤ βn kf (xn ) − x∗ k2 + (1 − βn )kSzn − x∗ k2 ∗ 2

∗ 2

≤ βn kf (xn ) − x k + (1 − βn )kzn − x k

= βn kf (xn ) − x∗ k2 + (1 − βn )kPC (un − λn Aun ) − PC (x∗ − λn Ax∗ )k2 ≤ βn kf (xn ) − x∗ k2 + (1 − βn )k(un − λn Aun ) − (x∗ − λn Ax∗ )k2 ≤ βn kf (xn ) − x∗ k2 + (1 − βn )k(un − x∗ ) − λn (Aun − Ax∗ )k2

(3.22)

(3.23)

354

10

T. JITPEERA AND P. KUMAM

= βn kf (xn ) − x∗ k2 + (1 − βn ){kun − x∗ k2 − 2λn hun − x∗ , Aun − Ax∗ i + λ2n kAun − Ax∗ k2 } ≤ βn kf (xn ) − x∗ k2 + (1 − βn ){kun − x∗ k2 − 2λn αkAun − Ax∗ k2 + λ2n kAun − Ax∗ k2 } ≤ βn kf (xn ) − x∗ k2 + (1 − βn ){kun − x∗ k2 + λn (λn − 2α)kAun − Ax∗ k2 } ≤ βn kf (xn ) − x∗ k2 + (1 − βn )kun − x∗ k2 + (1 − βn )λn (λn − 2α)kAun − Ax∗ k2 ≤ βn kf (xn ) − x∗ k2 + kxn − x∗ k2 + (1 − βn )λn (λn − 2α)kAun − Ax∗ k2 .

(3.24)

Using (3.1) and (3.23), we get kxn+1 − x∗ k2

≤

(1 − αn )kyn − x∗ k2 + αn kSyn − x∗ k2

≤

(1 − αn )kyn − x∗ k2 + αn kyn − x∗ k2

=

kyn − x∗ k2

≤

βn kf (xn ) − x∗ k2 + kxn − x∗ k2 + (1 − βn )λn (λn − 2α)kAun − Ax∗ k2 .

So, we get (1 − βn )a(2α − b)kAun − Ax∗ k2

≤

(1 − βn )λn (2α − λn )kAun − Ax∗ k2

≤

βn kf (xn ) − x∗ k2 + kxn − x∗ k2 − kxn+1 − x∗ k2

≤

βn kf (xn ) − x∗ k2 + kxn+1 − xn k(kxn − x∗ k + kxn+1 − x∗ k).

From conditions (C1), {λn } ⊂ [a, b] ⊂ (0, 2α) and limn−→∞ kxn+1 − xn k = 0, we obtain lim kAun − Ax∗ k = 0.

n−→∞

(3.25)

From (3.1) and (2.2), we observe that kzn − x∗ k2

=

kPC (un − λn Aun ) − PC (x∗ − λn Ax∗ )k2

≤

h(un − λn Aun ) − (x∗ − λn Ax∗ ), zn − x∗ i 1 {k(un − λn Aun ) − (x∗ − λn Ax∗ )k2 + kzn − x∗ k2 2 −k(un − λn Aun ) − (x∗ − λn Ax∗ ) − (zn − x∗ )k2 } 1 {kun − x∗ k2 + kzn − x∗ k2 − k(un − zn ) − λn (Aun − Ax∗ )k2 } 2 1 {kun − x∗ k2 + kzn − x∗ k2 − kun − zn k2 2 +2λn hun − zn , Aun − Ax∗ i − λ2n kAun − Ax∗ k2 } 1 {kxn − x∗ k2 + kzn − x∗ k2 − kun − zn k2 2 +2λn hun − zn , Aun − Ax∗ i − λ2n kAun − Ax∗ k2 },

=

= ≤

≤

and hence kzn − x∗ k2

≤ kxn − x∗ k2 − kun − zn k2 + 2λn hun − zn , Aun − Ax∗ i.

(3.26)

Therefore, form (3.1) and (3.26), we have kyn − x∗ k2

≤ βn kf (xn ) − x∗ k2 + (1 − βn )kSzn − x∗ k2 ≤ βn kf (xn ) − x∗ k2 + (1 − βn )kzn − x∗ k2 ≤ βn kf (xn ) − x∗ k2 + (1 − βn ){kxn − x∗ k2 − kun − zn k2 + 2λn hun − zn , Aun − Ax∗ i} ≤ βn kf (xn ) − x∗ k2 + kxn − x∗ k2 − (1 − βn )kun − zn k2 + 2λn (1 − βn )hun − zn , Aun − Ax∗ i,

and hence kxn+1 − x∗ k2

≤ (1 − αn )kyn − x∗ k2 + αn kSyn − x∗ k2 ≤ (1 − αn )kyn − x∗ k2 + αn kyn − x∗ k2 ≤ kyn − x∗ k2 ≤ βn kf (xn ) − x∗ k2 + kxn − x∗ k2 − (1 − βn )kun − zn k2 +2λn (1 − βn )hun − zn , Aun − Ax∗ i,

(3.27)

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A COMPOSITE ITERATIVE METHOD FOR GENERALIZED MIXED EQUILIBRIUM PROBLEMS

11

which imply that (1 − βn )kun − zn k2

≤

βn kf (xn ) − x∗ k2 + kxn − x∗ k2 − kxn+1 − x∗ k2 +2λn (1 − βn )kun − zn kkAun − Ax∗ k

≤

βn kf (xn ) − x∗ k2 + kxn+1 − xn k(kxn − x∗ k + kxn+1 − x∗ k) +2λn (1 − βn )kun − zn kkAun − Ax∗ k.

Since kxn+1 − xn k → 0, βn → 0 and kAun − Ax∗ k → 0, n −→ ∞, then lim kun − zn k = 0.

(3.28)

n−→∞

Since kSzn − zn k ≤ kSzn − xn k + kxn − un k + kun − zn k. By (3.13), (3.21) and (3.28), we conclude that lim kSzn − zn k = 0.

(3.29)

n−→∞

Step 6. We show that lim suphf (q) − q, Szn − qi ≤ 0. n−→∞

Indeed, we choose a subsequence {zni } of {zn } such that lim suphf (q) − q, Szn − qi = lim hf (q) − q, Szni − qi, i−→∞

n−→∞

where q = PΘ f (q). Without loss of generality, we may assume that {zni } converges weakly to z ∈ C. From kSzn − zn k −→ 0, we obtain Szni * z. Now, we will show that z ∈ Θ := F (S) ∩ V I(C, A) ∩ GM EP (F, ϕ, B). Firstly, we will show z ∈ F (S). Assume that z 6∈ F (S). Since zni * z and Sz 6= z. By the Opial’s condition, we obtain lim inf kzni − zk < n−→∞

lim inf kzni − Szk i−→∞

=

lim inf kzni − Szni + Szni − Szk

≤

lim inf (kzni − Szni k + kSzni − Szk)

=

lim inf kSzni − Szk

≤

lim inf kzni − zk.

i−→∞ i−→∞

i−→∞ i−→∞

This is a contradiction. Thus, we have z ∈ F (S). Next, let us show that z ∈ V I(C, A). Let ½ Aw1 + NC w1 , T w1 = ∅,

w1 ∈ C; w1 ∈ / C.

Then T is maximal monotone (see [22]). Let (w1 , w2 ) ∈ G(T ). Since w2 − Aw1 ∈ NC (w1 ) and zn ∈ C, we have hw1 − zn , w2 − Aw1 i ≥ 0. On the other hand, from zn = PC (un − λn Aun ), we have hw1 − zn , zn − (un − λn Aun )i ≥ 0 that is, hw1 − zn ,

(3.30)

zn − un + Aun i ≥ 0. λn

(3.31)

Therefore, we obtain hw1 − zni , w2 i

≥

hw1 − zni , Aw1 i ≥ hw1 − zni , Aw1 i − hw1 − zni ,

zni − uni + Auni i λn i

zni − uni i λni = hw1 − zni , Aw1 − Azni i + hw1 − zni , Azni − Auni i zn − uni −hw1 − zni , i i λn i zn − uni ≥ hw1 − zni , Azni − Auni i − hw1 − zni , i i, λn i = hw1 − zni , Aw1 − Auni −

(3.32)

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which together limn−→∞ kzn − un k = 0 and A is α-inverse-strongly monotone imply that hw1 − z, w2 i ≥ 0. Since T is maximal monotone, we have z ∈ T −1 0, and hence z ∈ V I(C, A). Finally, we show that z ∈ GM EP (F, ϕ, B). Since un = Krn (xn − rn Bxn ), we have 1 F (un , y) + hBxn , y − un i + ϕ(y) − ϕ(un ) + hy − un , un − xn i ≥ 0, ∀y ∈ C. rn From (A2), we also have 1 hBxn , y − un i + ϕ(y) − ϕ(un ) + hy − un , un − xn i ≥ F (y, un ), ∀y ∈ C. rn and hence un − xni hBxni , y − uni i + ϕ(y) − ϕ(uni ) + hy − uni , i i ≥ F (y, uni ), ∀y ∈ C. rni

(3.33)

For t with 0 < t ≤ 1 and y ∈ C, let yt = ty + (1 − t)z. Since y ∈ C and z ∈ C, we have yt ∈ C. So, from (3.33), we have un − xni hyt − uni , Byt i ≥ hyt − uni , Byt i − ϕ(yt ) + ϕ(uni ) − hyt − uni , Bxni i − hyt − uni , i i + F (yt , uni ) rni = hyt − uni , Byt − Buni i + hyt − uni , Buni − Bxni i − ϕ(yt ) + ϕ(uni ) un − xni −hyt − uni , i i + F (yt , uni ). rni Since kuni −xni k → 0, we have kBuni −Bxni k → 0. Further, from the inverse strongly monotonicity of B, we have hyt − uni , Byt − Buni i ≥ 0. So, from (A4),(A5), and the weak lower semicontinuity of u −x ϕ, nirn ni → 0 and un * z, we have at the limit i

hyt − z, Byt i ≥ −ϕ(yt ) + ϕ(z) + F (yt , z)

(3.34)

as i → ∞. From (A1),(A4) and (3.34), we also get 0

= ≤ = ≤ = 0 ≤

F (yt , yt ) + ϕ(yt ) − ϕ(yt ) tF (yt , y) + (1 − t)F (yt , z) + tϕ(y) − (1 − t)ϕ(z) − ϕ(yt ) t[F (yt , y) + ϕ(y) − ϕ(yt )] + (1 − t)[F (yt , z) + ϕ(z) − ϕ(yt )] t[F (yt , y) + ϕ(y) − ϕ(yt )] + (1 − t)hyt − z, Byt i t[F (yt , y) + ϕ(y) − ϕ(yt )] + (1 − t)thy − z, Byt i, F (yt , y) + ϕ(y) − ϕ(yt ) + (1 − t)hy − z, Byt i.

Letting t → 0, we have, for each y ∈ C, F (z, y) + ϕ(y) − ϕ(z) + hy − z, Bzi ≥ 0. This implies that z ∈ GM EP (F, ϕ, B). Therefor z ∈ Θ. Since q = PΘ f (q), which implies that lim suphf (q) − q, Szn − qi n→∞

=

lim hf (q) − q, Szni − qi

n→∞

= hf (q) − q, z − qi ≤ 0.

(3.35)

From (3.12) and (3.35) it follows that lim suphf (q) − q, yn − qi ≤ n→∞

lim hf (q) − q, yn − Szni i + lim hf (q) − q, Szni − qi ≤ 0. (3.36)

n→∞

n→∞

Step 7. We prove that {xn } converge strongly to q. From (3.1), we also have kyn − qk2

≤ kβn (f (xn ) − q) + (1 − βn )kSzn − qk2 ≤

(1 − βn )2 kSzn − qk2 + 2βn hf (xn ) − q, yn − qi

≤

(1 − βn )2 kzn − qk2 + 2βn [hf (xn ) − f (q), yn − qi + hf (q) − q, yn − qi]

≤

(1 − βn )2 kxn − qk2 + 2βn kkxn − qkkyn − qk + 2βn hf (q) − q, yn − qi

≤

(1 − βn )2 kxn − qk2 + βn k(kxn − qk2 + kyn − qk2 ) + 2βn hf (q) − q, yn − qi

= ((1 − βn )2 + βn k)kxn − qk2 + βn kkyn − qk2 + 2βn hf (q) − q, yn − qi,

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13

that is

³ (1 − β )2 + β k ´ 2βn n n kxn − qk2 + hf (q) − q, yn − qi. 1 − βn k 1 − βn k From (3.1) and (3.37), we obtain kyn − qk2

kxn+1 − qk2

(3.37)

≤

(1 − αn )kSyn − qk2 + αn kyn − qk2

=

kyn − qk2 ³ (1 − β )2 + β k ´ 2βn n n kxn − qk2 + hf (q) − q, yn − qi 1 − βn k 1 − βn k ³ 1 − 2β + β 2 + β k ´ 2βn n n n kxn − qk2 + hf (q) − q, yn − qi 1 − βn k 1 − βn k ³ 1 + β k − 2β ´ 2βn βn2 n n kxn − qk2 + kxn − qk2 + hf (q) − q, yn − qi 1 − βn k 1 − βn k 1 − βn k ³ 2(1 − k)βn ´ βn 1− (βn kxn − qk2 + 2hf (q) − q, yn − qi) kxn − qk2 + 1 − βn k 1 − βn k ³ 2(1 − k)βn ´ hf (q) − q, yn − qi ´ 2(1 − k)βn ³ βn kxn − qk2 1− kxn − qk2 + + . 1 − βn k 1 − βn k 2(1 − k) (1 − k)

≤ = = = = Put γn = That is

≤

2(1−k)βn 1−βn k

and δn =

βn 2(1−k) kxn

− qk2 +

1 1−k hf (q)

− q, yn − qi.

kxn+1 − qk2 ≤ (1 − γn )kxn − qk2 + γn δn . (3.38) P∞ It is easy to seen that γn −→ 0, n=1 γn = ∞, and lim supn−→∞ δn ≤ 0 by (3.36). Applying Lemma 2.2 to (3.38), the sequence {xn } converges strongly to q = PΘ f (q). Consequently, also {un } converges strongly to q. This completes the proof. ¤ Corollary 3.2. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from C × C −→ R satisfying (A1)-(A5), ϕ : C → R be a lower semicontinuous and convex function and let f : C −→ C be a contraction with coefficient k (0 < k < 1). Let B be a β-inversestrongly monotone mapping of C into H and let S be a nonexpansive mappings of C into itself such that F (S) ∩ GM EP (F, ϕ, B) 6= ∅. Assume that either (B1) or (B2) holds. Suppose x1 ∈ C and {xn }, {yn } and {un } are given by  1   F (un , y) + hBxn , y − un i + ϕ(y) − ϕ(un ) + rn hy − un , un − xn i ≥ 0, ∀y ∈ C, yn = βn f (xn ) + (1 − βn )Sun , ∀n ≥ 1,   xn+1 = (1 − αn )yn + αn Syn , where {αn }, {βn } are two sequence in (0, 1) and rn ∈ [c, d] ⊂ (0, 2β) satisfying the conditions: P∞ (C1) limn−→∞ βn = 0, n=1 βn = ∞, (C2) limn−→∞ αn = 0, (C3) lim inf n−→∞ rn > 0 and limn−→∞ |rn+1 − rn | = 0. Then {xn } and {un } converge strongly to q ∈ F (S)∩M EP (F, ϕ, B), where q = PF (S)∩M EP (F,ϕ,B) f (q). Proof. Putting A ≡ 0 then PC = I, we get zn = un . By Theorem 3.1, we have the desired result easily. ¤ Corollary 3.3. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from C × C −→ R satisfying (A1)-(A5) and let f : C −→ C be a contraction with coefficient k (0 < k < 1) and let S be a nonexpansive mappings of C into itself such that F (S) ∩ EP (F ) 6= ∅. Suppose x1 ∈ C and {xn }, {yn } and {un } are given by  1   F (un , y) + rn hy − un , un − xn i ≥ 0, ∀y ∈ C, yn = βn f (xn ) + (1 − βn )Sun , ∀n ≥ 1,   xn+1 = (1 − αn )yn + αn Syn , where {αn }, {βn } are two sequence in (0, 1) and {rn } ⊂ (0, ∞) satisfying the conditions: P∞ (C1) limn−→∞ βn = 0, n=1 βn = ∞,

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(C2) limn−→∞ αn = 0, (C3) lim inf n−→∞ rn > 0 and limn−→∞ |rn+1 − rn | = 0. Then {xn } and {un } converge strongly to q ∈ F (S) ∩ EP (F ), where q = PF (S)∩EP (F ) f (q). Proof. Taking ϕ ≡ 0, B ≡ 0 and if putting A ≡ 0 then PC = I, we have zn = un .

¤

Remark 3.4. (i) Corollary 3.2 improves and generalizes the result of Jung [13]. (ii) Corollary 3.3 extends Theorem 3.1 of Jaiboon et al. [12] from equilibrium problems to more generalized mixed equilibrium problems. 4. Applications In this section, we will utilize the results presented in this paper to study the following optimization problem: min ϕ(y), (4.1) y∈C

where C is a nonempty bounded closed convex subset of a Hilbert space and ϕ : C −→ R is a proper convex and lower semicontinuous function. We denote by Argmin(ϕ) the set of solutions in (4.1). Let F (x, y) = 0 for all x, y ∈ C in Theorem 3.1, then GM EP (F, ϕ, B) = Argmin(ϕ). It follows from Theorem 3.1 that the iterative sequence {xn } defined by  x1 = x ∈ C chosen arbitrarily,     u = Argmin 1 2 n y∈C {ϕ(y) + 2rn ky − xn k }, (4.2)  yn = βn f (xn ) + (1 − βn )SPC (un − λn Aun ),    xn+1 = (1 − αn )yn + αn Syn , n ≥ 1, where {αn }, {βn } are two sequence in (0, 1), λn ∈ [a, b] ⊂ (0, 2α) and rn ∈ [c, d] ⊂ (0, 2β) satisfy the conditions (C1)-(C4) in Theorem 3.1. Then the sequence {xn } converges strongly to a solution q = PF (S)∩V I(C,A)∩Argmin(ϕ) f q. Let F (x, y) = 0 for all x, y ∈ C, S = I, A ≡ 0 and f = x in Theorem 3.1, then GM EP (F, ϕ, B) = Argmin(ϕ). It follows from Theorem 3.1 that the iterative sequence {xn } defined by  x1 = x ∈ C chosen arbitrarily,     un = argminy∈C {ϕ(y) + 2r1n ky − xn k2 }, (4.3)  yn = βn x + (1 − βn )Sun ,    xn+1 = (1 − αn )yn + αn Syn , n ≥ 1, where {αn }, {βn } are two sequence in (0, 1) and rn ⊂ (0, ∞) satisfy the conditions (C1),(C2) and (C4), respectively in Theorem 3.1. Then the sequence {xn } converges strongly to a solution q = PF (S)∩Argmin(ϕ) q. We remark that the algorithms (4.2) and (4.3) are variants of the proximal method for optimization problems introduced and studied by Rockafellar [22], Ferris [10] and many others. Next, we prove two theorem in Hilbert spaces by using Theorem 3.1. A mapping T : C −→ C is called strictly pseudo-contraction if there exists a constant 0 ≤ κ < 1 such that kT x − T yk2 ≤ kx − yk2 + κk(I − T )x − (I − T )yk2 ,

∀x, y ∈ C.

If κ = 0, then T is nonexpansive. Put A = I − T . Then, we have k(I − A)x − (I − A)yk2 ≤ kx − yk2 + κkAx − Ayk2 ,

∀x, y ∈ C.

Observe that k(I − A)x − (I − A)yk2 = kx − yk2 + kAx − Ayk2 − 2hx − y, Ax − Ayi,

∀x, y ∈ C.

Hence we obtain hx − y, Ax − Ayi ≥ Then, A is 1−κ 2 -inverse-strongly monotone. Now we get the following result.

1−κ kAx − Ayk2 , 2

∀x, y ∈ C.

(4.4)

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15

Theorem 4.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from C × C −→ R satisfying (A1)-(A5), TA be an κα -strict pseudo-contraction of C into inself and TB be an κβ -strict pseudo-contraction of C into C Let f : C −→ C be a contraction with coefficient k (0 < k < 1). Let B be an β-inverse-strongly monotone mapping of C into H and let S be a nonexpansive mappings of C into itself such that F (S) ∩ F (TA ) ∩ GM EP (F, ϕ, I − TB ) 6= ∅. Assume that either (B1) or (B2) holds. Suppose x1 ∈ C and {xn }, {yn } and {un } are given by  1   F (un , y) + h(I − TB )xn , y − un i + ϕ(y) − ϕ(un ) + rn hy − un , un − xn i ≥ 0, ∀y ∈ C, yn = βn f (xn ) + (1 − βn )S((1 − λn )un + λn TA un ),   xn+1 = (1 − αn )yn + αn Syn , ∀n ≥ 1, where {αn }, {βn } are two sequence in (0, 1), λn ∈ [a, b] for some a, b with 0 < a < b < 1 − κα and rn ∈ [c, d] for some 0 < c < d < 1 − κβ satisfy the following conditions: P∞ (C1) limn−→∞ βn = 0 and n=1 βn = ∞, (C2) lim P∞n−→∞ αn = 0, (C3) n=1 |λn+1 − λn | < ∞ (C4) lim inf n−→∞ rn > 0 and limn−→∞ |rn+1 − rn | = 0, then {xn } and {un } converge strongly to q ∈ F (S) ∩ F (TA ) ∩ GM EP (F, ϕ, I − TB ), where q = PF (S)∩F (TA )∩GM EP (F,ϕ,I−TB ) f (q). α Proof. Put A = (I−TA ) and B = (I−TB ) from (4.4) we know that A is 1−κ 2 -inverse-strongly monotone 1−κβ mapping and B is 2 -inverse-strongly monotone mapping, respectively. We have F (TA ) = V I(C, A) and zn = PC (un − λn Aun ) = PC ((1 − λn )un + λn TA un ) = (1 − λn )un + λn TA un ∈ C. So, from Theorem 3.1, we obtain the desired result. ¤

Let E be a Banach space. An operator A with domain D(A) and R(A) is accretive, if for each xi ∈ D(A) and yi ∈ Axi (i = 1, 2), there exists j(x2 −x1 ) ∈ J(x2 −x1 ) such that hy2 −y1 , j(x2 −x1 )i ≥ 0. Where J is the duality map from E to the dual space E ∗ . This is defined by J(x) = {x∗ ∈ E ∗ : hx, x∗ i = kxk2 = kx∗ k2 }, x ∈ E. An accretive operator A is m-accretive if R(I + rA) = E for each r > 0. Throughout this article we always assume that A is m-accretive and has a zero(i.e., the inclusion 0 ∈ A(z) is solvable). The set zeros of A is denoted by Γ. Hence, Γ = {z ∈ D(A) : 0 ∈ A(z)} = A−1 (0). For each r > 0, we denote Jr the resolvent of A, i.e., Jr = (I + rA)− 1. Note that if A is m-accretive, then Jr : E → E is single-valued and nonexpansive and F (Jr ) = Γ for all r > 0. We also denote by Ar the Yosida approximation of A, i.e., Ar = 1r (I − J − r). It is know that Jr is nonexpansive mapping from E to C := D(A) which will be assumed convex. The following theorem is connected with the problem of obtaining of a common element of the set of zeroes of a maximal monotone operator and an ξ-inverse-strongly monotone operator. Theorem 4.2. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from C × C −→ R satisfying (A1)-(A5), A : C −→ H be an α-inverse-strongly monotone mapping and B : C −→ H be a β-inverse-strongly monotone mapping. Let f : C −→ C be a contraction with coefficient k (0 < k < 1) and Ψ : H → 2H be a maximal monotone operator such that F (S) ∩ A−1 (0) ∩ GM EP (F, ϕ, B) 6= ∅. Let JrΨ be the resolvent of Ψ for each r > 0. Suppose x1 ∈ C and {xn }, {yn } and {un } are given by  1   F (un , y) + hBxn , y − un i + ϕ(y) − ϕ(un ) + rn hy − un , un − xn i ≥ 0, ∀y ∈ C, yn = βn f (xn ) + (1 − βn )S((1 − λn )un + λn Aun ),   xn+1 = (1 − αn )yn + αn JrΨ yn , ∀n ≥ 1, where {αn }, {βn } are two sequence in (0, 1) , λn ∈ [a, b] for some a, b with 0 < a < b < 2α and rn ∈ [c, d] for some 0 < c < d < 2β satisfy the following conditions: P∞ (C1) limn−→∞ βn = 0 and n=1 βn = ∞, (C2) limn−→∞ αn = 0,

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P∞ (C3) n=1 |λn+1 − λn | < ∞ (C4) lim inf n−→∞ rn > 0 and limn−→∞ |rn+1 − rn | = 0, Then {xn } and {un } converge strongly to q ∈ F (S) ∩ A−1 (0) ∩ GM EP (F, ϕ, B), where q = PF (S)∩A−1 (0)∩GM EP (F,ϕ,B) f (q). Proof. Since A−1 (0) is the solution set of V I(H, A) i.e, A−1 (0) = V I(H, A), we can obtain the conclusion by Theorem 3.1 and we get that F (S) ∩ A−1 (0) ⊂ V I(F (S), A). ¤ 5. Acknowledgement The authors would like to express their thank to the National Research Council of Thailand and the Faculty of Science, King Monkut’s University of Technology Thonburi for their financial support. References 1. K. Aoyama and W. Takahashi, Weak convergence theorems by Ces` aro means for a nonexpansive mapping and an equilibrium problem, Pacific J. Optimaization. 3 (2007) 501–509. 2. F.E. Browder, Existence and approximation of solutions of nonlinear variational inequalities, Proc. Natl. Acad. Sci. USA. 56 (1966) 1080–1086. 3. E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student. 63 (1994) 123–145. 4. F.E. Browder and W.V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl. 20 (1967) 197–228. 5. P.L. Combettes and S.A. Hirstoaga, Equilibrium programming using proximal-like algorithms, Math. Program. 78 (1997) 29–41. 6. P. L. Combettes and S.A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal. 6 (2005) 117–136. 7. L.C. Ceng and J.C. Yao, A hybrid iterative scheme for mixed equilibrium problems and fixed point problems, J. Comput. Appl. Math. 214 (2008) 186–201. 8. J. Chen, L. Zhang and T. Fan , Viscosity approximation methods for nonexpansive mappings and monotone mappings. J. Math. Anal. Appl. 334 (2007) 1450–1461. 9. S. D. Flam and A. S. Antipin, Equilibrium programming using proximal-link algolithms, Math. Program. 78 (1997) 29–41. 10. M.C. Ferris, Finite termination of the proximal point algorithm, Math. Program. 50 (1991) 359-366. 11. P. Hartman and G. Stampacchia, On some nonlinear elliptic differential functional equations, Acta Math. 115 (1966) 271–310. 12. C. Jaiboon, P. Kumam and U.W. Humphries, Convergence theorems by the viscosity approximation method for equilibrium problems and variational inequality problems, J. Comput. Math. Optim. 5 (2009) 29–56. 13. J. S. Jung, Strong convergence of composite iterative methods for equilibrium problems and fixed point problems, Appl. Math. Comput. 213 (2009) 498–505. 14. J. S. Jung, Strong convergence of composite iterative methods for equilibrium problems and fixed point problems, J. Comput. Anal. Appl. Vol.12, NO.1-A (2010) 124-140. 15. F. Liu and M.Z. Nashed, Regularization of nonlinear Ill-posed variational inequalities and convergence rates, SetValued Anal. 6 (1998) 313–344. 16. N. Nadezhkina and W. Takahashi, Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl. 128 (2006) 191–201. 17. J. W. Peng and J. C. Yao, A new hybrid-extragradient method for generalized mixed equilibrium problems and fixed point problems and variational inequality problems, Taiwanese J. Math. 12 (6) (2008) 1401-1432. 18. J. W. Peng and J. C. Yao, Two extragradient methods for generalized mixed equilibrium problems, nonexpansive mappings and monotone mappings, Comput. Math. with Appl. 58 (2009) 1287–1301. 19. S. Plubtieng and R. Punpaeng, A new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings, Appl. Math. Comput. 197 (2008) 548–558. 20. S. Plubtieng and R. Punpaeng, A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 336 (2007) 455–469. 21. R.T. Rockafellar, On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc. 149 (1970) 75–88. 22. R.T. Rockafellar, Monotone operators and proximal point algorithm, SIAM J. Control Optim. 14 (1976) 877–898. 23. T. Suzuki, Strong convergence of krasnoselskii and manns type sequences for one-parameter nonexpansive semigroups without bochner integrals. J. Math. Anal. Appl. 305 (2005) 227–239 24. Y. Su, M. Shang and X. Qin, An iterative method of solution for equilibrium and optimization problems, Nonlinear Anal. 69 (2008) 2709–2719. 25. W. Takahashi and M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl. 118 (2003) 417–428. 26. S. Takahashi and W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2007) 506–515.

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27. A. Tada and W. Takahashi, Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem, J. Optim. Theory Appl. 133 (2007) 359–370. 28. H.K. Xu, An iterative approach to quadratic optimization, J. Optim. Theory Appl. 116 (2003) 659–678. 29. J.-C. Yao, O. Chadli, Pseudo monotone complementarity problems and variational inequalities, in: J.P. Crouzeix, N. Haddjissas, S. Schaible (Eds.), Handbook of Generalized Convexity and Monotonicity, (2005) 501–558. 30. Y. Yao, Y.C Liou and R. Chen, Convergence theorems for fixed point problems and variational inequality problems, J. Nonlinear Convex Anal. 9 (2008) 239–248. 31. L.C. Zeng, S. Schaible and J.C. Yao, Iterative algorithm for generalized set-valued strongly nonlinear mixed variational-like inequalities, J. Optim. Theory Appl. 124 (2005) 725–738. 32. L.C. Zeng and J.C. Yao, Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems, Taiwanese J. Math. 10 (2006) 1293–1303. (Thanyarat Jitpeera) Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi, Bangmod, Bangkok 10140, Thailand. E-mail address: [email protected](T.Jitpeera) (Thanyarat Jitpeera) Department of Mathematics, Faculty of Science, Rajamangala University of Technology Lanna, Phan, Chiangrai 57120. Thailand. E-mail address: [email protected] (T.Jitpeera) (Poom Kumam) Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi, Bangmod, Bangkok 10140, Thailand. E-mail address: [email protected](P.Kumam) (Corresponding author.)

JOURNAL 362 OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.2, 362-367, 2011, COPYRIGHT 2011 EUDOXUS PRESS, LLC

COINCIDENCE AND COMMON FIXED POINTS FOR GENERALIZED CONTRACTION MULTI-VALUED MAPPINGS† WUTIPHOL SINTUNAVARAT AND POOM KUMAM‡ Abstract. In this paper, we prove the existence of coincidence and common fixed point theorems for generalized multi-valued f -contraction and generalized f - nonexpansive maps. In our results, we can drop the assumption of T-weakly commuting which improved and extend the recent results of Al-Thagafi and Shahzad [M.A. Al-Thagafi, Naseer Shahzad, Coincidence points, generalized Inonexpansive multimaps, and applications, Nonlinear Analysis 67 (2007) 2180- 2188.], Shahzed and Hussain [Naseer Shahzad, Nawab Hussain, Deterministic and random coincidence point results for f -nonexpansive maps, J. Math. Anal. Appl. 323 (2006) 1038-1046.] and many authors.

1. Introduction and preliminaries Let (X, d) be a metric space. We denote by CL(X) (resp. CB(X)) the class of all nonempty (resp. bounded) closed subsets of X. The Hausdorff metric induced by d on CL(X) is given by H(A, B) = max{sup d(a, B), sup d(b, A)} a∈A

b∈B

for every A, B ∈ CL(X), where d(a, B) = inf{d(a, b) : b ∈ B} is the distance from a to B ⊆ X. Let f : X → X and T : X → CL(X). A point x ∈ X is said to be a fixed point of f (resp. T ) if x = f x (resp. x ∈ T x). The set of all fixed points of f (resp. T ) is denoted by F (f ) (resp. F (T )). A point x ∈ X is said to be a coincidence point of f and T if f x ∈ T x. The set of all coincidence points of f and T is denoted by C(f, T ). A point x ∈ X is said to be a common fixed point of f and T if x = f x ∈ T x. The set of all common fixed points of f and T is denoted by F (f, T ). The first important result on fixed points for contractive-type mappings was the well-known Banach contraction principle, published for the first time in 1922 in [7] (see also [9]). Banach contraction principle plays an important role in several branches of mathematics. For instance, it has been used to study the existence of solutions for nonlinear integral equations, nonlinear integro-differential equations in Banach space and to prove the convergence of algorithms in computational mathematics. Because of its importance for mathematical theory, Banach contraction principle has been extended in many different directions, see [22, 8, 33, 6, 10, 1, 32], etc. In 1969, Nadler extended the Banach contraction principle to multi-valued contractive mappings in complete metric spaces. In 1973, the study of fixed points for multi-valued contractions using the Hausdorff metric was initiated by Markin [21]. Afterward, an interesting and rich fixed point theory for such maps was developed. The theory of multi-valued maps has application in optimization problem, control theory, differential equations and economics. Recent fixed point results for multivalued mappings can be found in [34, 23, 24, 31, 32, 13] and references therein. In 2004, Kamran [18] defined the property “f is T -weakly commuting” as follows: Definition1.1 ([18]). Assume that (X, d) is a metric space and x ∈ X. Let f : X → X and T : X → CB(X). The map f is said to be T -weakly commuting at x ∈ X if f f x ∈ T f x. Afterward, Al-Thagafi and Shahzed [4] established some coincidence and common fixed point theorems for a multimaps satisfying generalized f -contraction type conditions which used the assumption of “f is T -weakly commuting” at coincidence point. 2000 Mathematics Subject Classification. : 47H09, 47H10. Key words and phrases. Generalized multi-value f -contraction mapping, Generalized multi-value f -nonexpansive mapping, T -weakly commuting, fixed point, coincidence point, common fixed point. † This research was partially supported by the Thailand Research Fund and the Commission on Higher Education under Grant No. MRG5380044. ‡ Corresponding author email: [email protected] (P. Kumam). 1

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Recently, Sintunavarat and Kumam [30] showed that can dropped the assumption of “f is T weakly commuting” at coincidence point in Theorem 3.1, 3.6 [30]. The aim of this paper is combine ideas of Sintunavarat and Kumam with a generalized f -contraction mappings in Theorem 2.1, 2.3 of Al-Thagafi and establish common fixed point theorem for a generalized f -nonexpansive which extends main results of Shahzad and Hussain [29]. Several invariant approximation results are obtained as applications. Our results unify, extend and complement several well-known results. 2. Generalized f -contraction mappings Theorem 2.1. Let (X, d) be a metric space, f : X → X, T : X → CB(X), T (X) ⊆ f (X), and T (X) be complete. If ½ ¾ 1 H(T x, T y) ≤ k max d(f x, f y), d(f x, T x), d(f y, T y), [d(f x, T y) + d(f y, T x)] (2.1) 2 for all x, y ∈ X where k ∈ [0, 1). Then C(f, T ) 6= ∅. Moreover, if f f v = f v for some v ∈ C(f, T ) then F (f, T ) 6= ∅. Proof. Let x0 be an arbitrary point of X. Since T (X) ⊆ T (X) ⊆ f (X), we construct a sequence {xn } in X such that f xn ∈ T xn−1 ⊆ T (X) ⊆ f (X) for all n ≥ 1. By Theorem 2.1 [4] claims that {f xn } is a Cauchy sequence in T (X). It follows from the completeness of T (X) that f xn → z ∈ T (X) ⊆ f (X), where z = f u for some u ∈ X. Using the fact (2.1), we know that for every n ≥ 1, we get ½ ¾ 1 H(T xn−1 , T u) ≤ k max d(f xn−1 , f u), d(f xn−1 , T u), d(f u, T u), [d(f xn−1 , T u) + d(f u, T xn−1 )] 2 ½ ¾ 1 ≤ k max d(f xn−1 , z), d(f xn−1 , T u), d(z, T u), [d(f xn−1 , T u) + d(z, T xn−1 )] . 2 Since f xn ∈ T xn−1 , we have d(f xn , T u) ≤ H(T xn−1 , T u) which implies ½ ¾ 1 d(f xn , T u) ≤ k max d(f xn−1 , z), d(f xn−1 , T u), d(z, T u), [d(f xn−1 , T u) + d(z, T xn−1 )] 2 for every n ≥ 1. Letting n → ∞, we have d(z, T u) ≤ kd(z, T u). Then, z = f u ∈ T u and, hence, C(f, T ) is nonempty. Since there exists v ∈ C(I, T ) such that f f v = f v. Let t =: f v. So t = f v = f f v = f t ∈ T v. It follows that d(t, T t) ≤ ≤ ≤ ≤

H(T v, T t) ¾ ½ 1 k max d(f v, f t), d(f v, T v), d(f t, T t), [d(f v, T t) + d(f t, T v)] 2 ½ ¾ 1 k max d(t, t), d(t, T v), d(t, T t), [d(t, T t) + d(t, T v)] 2 kd(t, T t).

Then t ∈ T t and so t = f t ∈ T t. Hence F (f, T ) 6= ∅.

¤

Remark 2.2. Theorem 2.1 generalizes and cover than Theorem 2.1 of Al-Thagafi and Shahzad [4], Theorem 2.1 of Al-Thagafi [2], the main results of Jungck [15], and Theorem 2.1 of Shahzad [27]. Corollary 2.3. Let (X, d) be a metric space, T : X → CB(X), and T (X) be complete. If ½ ¾ 1 H(T x, T y) ≤ k max d(x, y), d(x, T x), d(y, T y), [d(x, T y) + d(y, T x)] 2

(2.2)

for all x, y ∈ X where k ∈ [0, 1). Then F (T ) 6= ∅. Proof. Take f as the identity mapping from X into X in Theorem 2.1 to get F (T ) 6= ∅.

¤

Remark 2.4. The Banach Contraction Principle [7], Nadler’s Contraction Principle [22], Theorem 2.4 of Daffer and Kaneko [11], and many results in literature are special cases of Corollary 2.3.

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3

3. Generalized f -nonexpansive mapping A subset D of a normed space X := (X, k · k) is called (i) convex if kx + (1 − k)y ∈ D for all x, y ∈ D and all k ∈ [0, 1]; and (ii) q-starshaped if kx + (1 − k)q ∈ D for all x ∈ D and all k ∈ [0, 1]. The map f : D → D is called (iii) affine if D is convex and f (kx + (1 − k)y) = kf x + (1 − k)f y for all x, y ∈ D and all k ∈ [0, 1]; and q-affine if D is (iv) q-starshaped and f (kx + (1 − k)q) = kf x + (1 − k)f q for all x, y ∈ D and all k ∈ [0, 1]. We note that (i)⇒(ii) and (ii)⇒(iv). Let D be a q-starshaped subset of a normed space X, f : D → D, and T : D → CL(D). The pair (f, T ) satisfies the coincidence point condition (in short, CPC) [4] on A ∈ CL(D) if whenever {xn } is a sequence in A such that d(f xn , T xn ) → 0, then f z → T z for some z ∈ A. The map T satisfies the fixed point condition (in short, FPC) [4] on A ∈ CL(D) if whenever {xn } is a sequence in A such that d(xn , T xn ) → 0, then z → T z for some z ∈ A. Theorem 3.1. Let D be a q-starshaped subset of a normed space X, f : D → D, T : D → CB(D), T (D) ⊆ f (D), and T (D) be complete, f(D)=D, and the pair (f,T) satisfies the CPC on D. If ½ ¾ 1 1 H(T x, T y) ≤ max kf x − f yk, [ρ(f x, T x) + ρ(f y, T y)], [ρ(f x, T y) + ρ(f y, T x)] (3.1) 2 2 for all x, y ∈ X, where ρ(f x, T y) := inf{kf x − Tk yk : 0 ≤ k ≤ 1}. Then C(I, T ) 6= ∅. Moreover, if ffv=fv for some v ∈ C(f, T ), then F (f, T ) 6= ∅. Proof. Let {kn } be a sequence in (0, 1) such that kn → 1. For n ≥ 1, define Tn : D → D by Tn x = kn T x + (1 − kn )q for all x ∈ D. As D is q−starshaped, T (D) ⊆ f (D) = D, and T (D) is complete that Tn (D) ⊆ f (D) and for every Tn (D) is complete. Using the fact (3.1), we know that for every n ≥ 1 H(Tn x, Tn y)

= kn H(T x, T y) ¾ ½ 1 1 ≤ kn max kf x − f yk, [ρ(f x, T x) + ρ(f y, T y)], [ρ(f x, T y) + ρ(f y, T x)] 2 2 ½ ¾ 1 1 ≤ kn max kf x − f yk, [d(f x, Tn x) + d(f y, Tn y)], [d(f x, Tn y) + d(f y, Tn x)] 2 2 ½ ¾ 1 1 ≤ kn max kf x − f yk, [2 max{d(f x, Tn x), d(f y, Tn y)}], [d(f x, Tn y) + d(f y, Tn x)] 2 2 ½ ¾ 1 ≤ kn max kf x − f yk, max{d(f x, Tn x), d(f y, Tn y)}, [d(f x, Tn y) + d(f y, Tn x)] 2 ½ ¾ 1 ≤ kn max kf x − f yk, d(f x, Tn x), d(f y, Tn y), [d(f x, Tn y) + d(f y, Tn x)] . 2

It follows from Theorem 2.1 that there exist {zn } in D such that zn ∈ C(f, Tn ) for all n ≥ 1. So f zn ∈ Tn zn = kn T zn + (1 − kn )q which implies that f zn = kn an + (1 − kn )q for some an ∈ T zn ⊆ T (D). Since T (D) is bounded, kn → 1, and kf zn − an k

= =

kkn an + (1 − kn )q − an k kkn an + q − kn q − an k

= ≤

k(1 − kn )(q − an )k (1 − kn )(kq − an k),

(f zn − an ) → 0 whenever n → ∞. Since d(f zn , T zn ) ≤ kf zn − an k, d(f zn , T zn ) → 0. As the pair (f, T ) satisfies the CP C on D that there exists v ∈ D such that f v ∈ T v. Therefore C(f, T ) 6= ∅. Since there exists v ∈ C(f, T ) such that f f v = f v. Let t := f v. So t = f v = f f v = f t ∈ T v. Since d(t, T t) ≤ ≤

H(T v, T t) ¾ ½ 1 1 k max d(f v, f t), [d(f v, T v) + d(f t, T t)], [d(f v, T t) + d(f t, T v)] 2 2

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½ ¾ 1 1 k max d(t, t), [d(t, T v) + d(t, T t)], [d(t, T t) + d(t, T v)] 2 2 1 ≤ d(t, T t). 2 So t ∈ T t which implies t = f t ∈ T t. Hence F (f, T ) 6= ∅. ≤

¤

Remark 3.2. Theorem 3.1 generalizes and extends Theorem 2.1, 2.2, 2.4, 2.6, 2.7, 2.8, 2.9, 2.11 of Shahzad and Hussain [29], Corollaries 3.2, 3.4 of Jungck [16]. Corollary 3.3. Let D be a q-starshaped subset of a normed space X, T : D → CB(D), T (D) ⊆ D, and T (D) be complete, and T satisfies the FPC on D. If ½ ¾ 1 1 H(T x, T y) ≤ max kx − yk, [ρ(x, T x) + ρ(y, T y)], [ρ(x, T y) + ρ(y, T x)] , (3.2) 2 2 for all x, y ∈ X where ρ(x, T y) := inf{kx − Tk yk : 0 ≤ k ≤ 1}. Then F (T ) 6= ∅. Proof. Take f as the identity mapping from X into X in Theorem 3.1 to get F (T ) 6= ∅.

¤

Remark 3.4. Corollary 3.3 generalizes and extends Theorem 1, 2 of Doston [12], Theorem 3.2 of Lami Dozo [19]. 4. Invariant approximations results Invariant approximations for non-commuting maps was considered first time by Shahzad [26, 28]. Let M be a subset of a normed space X and p ∈ X. The set BM (p) := {x ∈ M : kx − pk = d(p, M )} is called the set of best M -approximants to p ∈ X out of M . Theorem 4.1. Let M be a subset of a normed space X, f : X → X and T : X → CB(X) such that satisfy the following conditions: (i) BM (p) is q-starshaped. (ii) T (BM (p)) ⊆ f (BM (p)) and T (BM (p)) is complete. (iii) Equation (3.1) holds on BM (p). (iv) ffv=fv for v ∈ C(f, T ) ∩ BM (p). (v) f (BM (p)) = BM (p). (vi) The pair (f,T) satisfies the CPC on BM (p). (vii) sup ky − pk ≤ kf x − pk for all x ∈ BM (p). y∈T x

Then F (f, T ) ∩ BM (p) 6= ∅. Proof. Let x ∈ BM (p) and z ∈ T x. Since f (BM (p)) = BM (p), f x ∈ BM (p) for all x ∈ BM (p). It follows from the definition of BM (p) that kf x − pk = d(p, M ). Since kz − pk ≤ sup ky − pk ≤ kf x − pk = d(p, M ), y∈T x

z ∈ BM (p). Thus T x ⊆ BM (p) for all x ∈ BM (p). Since T x is closed for all x ∈ X, so T x is closed for all x ∈ BM (p). Therefore f |BM (p) : BM (p) → BM (p), T |BM (p) : BM (p) → CB(BM (p)). Clearly, F (f |BM (p) , T |BM (p) ) = F (I, T ) ∩ BM (p). Now the result follows from Theorem 3.1 with D = BM (p). ¤ Remark 4.2. Theorem 4.1 extends Theorem 3.1 of Al-Thagafi [4], Theorem 3.2, 3.3 of Al-Thagafi [2], Theorem 3.1, 3.3 of Al-Thagafi [3], Theorem 3 of Sahab, Khan and Sessa [25], Theorem 2.12, 2.13 of Shahzad and Hussain [29], results of Hicks and Humphries [14], Theorem 7 of Jungck and Sessa [17], Theorem 3 of Latif and Bano [20], and results of many authors. Corollary 4.3. Let M be a subset of a normed space X, T : X → CB(X) such that satisfy the following conditions: (i) BM (p) is q-starshaped. (ii) T (BM (p)) ⊆ BM (p) and T (BM (p)) is complete. (iii) Equation (3.2) holds on BM (p). (iv) The map T satisfies the FPC on BM (p).

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(v) sup ky − pk ≤ kx − pk for all x ∈ BM (p). y∈T x

Then F (T ) ∩ BM (p) 6= ∅. Proof. Take f as the identity mapping from X into X in Theorem 4.1 to get F (T ) ∩ BM (p) 6= ∅.

¤

Remark 4.4. Corollary 4.3 extends Theorem 1.1 of Al-Thagafi [3]. Results of Hicks and Humphries [14] is special cases of Corollary 4.3. 5. Acknowledgments The first author would like to thank the Research Professional Development Project Under the Science Achievement Scholarship of Thailand (SAST) for financial support during the preparation of this manuscript. The second author was supported by the Thailand Research Fund and the Commission on Higher Education under Grant No. MRG5380044. References [1] R.P. Agarawal, D.O. ORegan, N. Shahzad, Fixed point theorem for generalized contractive maps of MeirKeeler type, Math. Nachr. 276 (2004) 3-22. [2] M.A. Al-Thagafi, Common fixed points and best approximation, J. Approx. Theory 85 (1996) 318-323. [3] M.A. Al-Thagafi, N. Shahzad, Noncommuting selfmaps and invariant approximations, Nonlinear Anal. 64 (2006) 2778-2786 [4] M.A. Al-Thagafi, Naseer Shahzad, Coincidence points, generalized I-nonexpansive multimaps, and applications, Nonlinear Analysis 67 (2007) 2180-2188. [5] N.A. Assad, W.A. Kirk, Fixed point theorems for setvalued mappings of contractive type, Pacific J. Math. 43 (1972) 553-562. [6] J.P. Aubin, J. Siegel, Fixed point and stationary points of dissipative multi-valued maps, Proc. Amer. Math. Soc. 78 (1980) 391-398. [7] S. Banach, Sur les op´ erations dans les ensembles abstraits et leurs applications aux ´ equations int´ egrales, Fund. Math. 3 (1922) 133-181. [8] A. Branciari, A fixed point theorem for mappings satisfying a general contractive condition of integral type, Int. J. Math. Math. Sci. 29 (2002) 531-536. [9] R. Caccioppoli, Un teorema generale sull esistenza di elementi uniti in una trasformazione funzionale, Rend. Accad. dei Lincei 11 (1930), 794-799 (Italian). [10] H. Covitz, S.B. Nadler Jr., Multi-valued contraction mappings in generalized metric space, Israel J. Math. 8 (1970) 5-11. [11] P.Z. Daffer, H. Kaneko, Applications of f-contraction mappings to nonlinear integral equations, Bull. Inst. Math. Acad. Sinica 22 (1994) 69-74. [12] W.J. Dotson Jr., Fixed point theorems for nonexpansive mappings on starshaped subsets of Banach spaces, J. London Math. Soc. 4 (1972) 408-410. [13] Y. Feng, S. Liu, Fixed point theorems for multi-valued operators in partial ordered spaces, Soochow J. Math. 30 (2004) 461-469. [14] T.L. Hicks, M.D. Humphries, A note on fixed point theorems, J. Approx. Theory 34 (1982) 221-225. [15] G. Jungck, Commuting mappings and fixed points, Amer. Math. Monthly 83 (1976) 261-263. [16] G. Jungck, Coincidence and fixed points for compatible and relatively nonexpansive maps, Int. J. Math. Math. Sci. 16 (1993) 95-100. [17] G. Jungck, S. Sessa, Fixed point theorems in best approximation theory, Math. Japonica 42 (1995) 249-252. [18] T. Kamran, Coincidence and fixed points for hybrid strict contractions, J. Math. Anal. Appl. 299 (2004) 235-241. [19] E. Lami Dozo, Multi-valued nonexpansive mappings and Opials condition, Proc. Amer. Math. Soc. 38 (1973) 286-292. [20] A. Latif, A. Bano, A result on invariant approximation, Tamkang J. Math. 33 (2002) 89-92. [21] J. T. Markin, Continuous dependence of fixed point sets, Proc. Amer. Math. Soc., 38 (1973), 545-547. [22] S.B. Nadler Jr., Multi-valued contraction mappings, Pacific J. Math. 30 (1969) 475-487. [23] S.V.R. Naidu, Fixed-point theorems for a broad class of multimaps, Nonlinear Anal. 52 (2003) 961-969. [24] A. Petru sel, A. Sntˇ amˇ arian, Single-valued and multi-valued Caristi type operators, Publ. Math. Debrecen 60 (2002) 167-177. [25] S.A. Sahab, M.S. Khan, S. Sessa, A result in best approximation theory, J. Approx. Theory 55 (1988) 349-351. [26] N. Shahzad, A result on best approximation, Tamkang J. Math. 29 (1998) 223-226; corrections: Tamkang J. Math. 30 (1999) 165. [27] N. Shahzad, Invariant approximations, generalized I -contractions and R-subweakly commuting maps, Fixed Point Theory Appl. 1 (2005) 79-86. [28] N. Shahzad, Invariant approximation and R-subweakly commuting maps, J. Math. Anal. Appl. 257 (2001) 39-45. [29] N. Shahzad, N. Hussain, Deterministic and random coincidence point results for f -nonexpansive maps, J. Math. Anal. Appl. 323 (2006) 1038-1046.

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[30] W. Sintunavart, P. Kumam, Coincidence and common fixed points for hybrid strict contractions without the weakly commuting condition, Appl. Math. Lett. 22 (2009) 1877-1881. [31] L. Van Hot, Fixed point theorems for multi-valued mapping, Comment. Math. Univ. Carolin. 23 (1982) 137-145. [32] P. Vijayaraju, B.E. Rhoades, R. Mohanraj, A fixed point theorem for a pair of maps satisfying a general contractive condition of integral type, Int. J. Math. Math. Sci. 15 (2005) 2359-2364. [33] T. Wang, Fixed point theorems and fixed point stability for multivalued mappings on metric spaces, J. Nanjing Univ. Math. Baq. 6 (1989) 16-23. [34] C.K. Zhong, J. Zhu, P.H. Zhao, An extension of multi-valued contraction mappings and fixed points, Proc. Amer. Math. Soc. 128 (2000) 2439-2444. (Wutiphol Sintunavarat) Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangkok 10140, Thailand. E-mail address: poom [email protected] (Poom Kumam) Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangkok 10140, Thailand. E-mail address: [email protected]

JOURNAL 368 OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.2, 368-375, 2011, COPYRIGHT 2011 EUDOXUS PRESS, LLC

Random Common Fixed Points of Single-Valued and Multivalued Random Operators in a Uniformly Convex Banach Space∗ Poom Kumam1,2 1

Department of Mathematics, Faculty of Science,

King Mongkut’s University of Technology Thonburi, KMUTT, Bangmod, Bangkok 10140. Thailand. 2

Centre of Excellence in Mathematics,

CHE, Si Ayutthaya Rd., Bangkok 10400, Thailand E-mail: [email protected]

Abstract Let (Ω, Σ) be a measurable space with Σ a sigma-algebra of subsets of Ω. Let M be a nonempty closed bounded convex and separable subset of a uniformly convex Banach space X, and let f : Ω × M → M, T : Ω × M → KC(M ) be a single valued and a multivalued nonexpansive commuting random operators, where KC(M ) is the family of all nonempty compact convex subset of M with the Hausdorff metric induced by the norm of X. It is shown that every random operator T and f has a common random fixed point. Moreover, we also derive a random coincidence point for a pair of multi-valued and single-valued commuting random operators in a uniformly convex Banach space. 2000 Mathematics Subject Classification: 47H10, 47H09, 47H40. Key words and phrases: random coincidence point,random common fixed point, multi-valued random operators, random fixed point

1

Introduction

Random common fixed point theorems are stochastic generalizations of classical common fixed point theorems. The study of random fixed point theorems was initiated by the Prague school of probability in the 1950s. Random fixed point theorems for contraction mappings in Polish space ˇ cek [19], Hanˇs [5, 6, 7], etc. For a brief survey of them were proved by Spaˇ and relate results, please refer to Bharucha-Reid [2]. Random fixed point ∗ This research is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand.

1

KUMAM: RANDOM COMMON FIXED POINTS...

theory has received much attention for the last two decades because of its importance in probabilistic functional analysis; the reader is referred to Beg and Shahzad [1], Khan et al. [9], Nashine and Shrivastava [14], Li and Duan [15], Tan and Yaun [20] and the reference therein. On the other hand, Nadler [18] extended the Banach Contraction Principle to multivalued contractive mappings in complete metric spaces. From then on, many researchers have studied the possibility of extending classical fixed point theorems for single valued nonexpansive mappings to the setting of multivalued nonexpansive mappings. In 1972, Itoh [8] obtained a random fixed point theorem for a multivalued contraction mapping in a Polish space. In 1974, Lim [16] proved that every multivalued nonexpansive mapping has a fixed point on a uniformly convex Banach space. Afterwards, Xu [22] obtained random versions of there results. In 1996, Beg and Shahzad [1] studied the structure of common random fixed points and randon coincidence point of a pair of compatible random operators. Recently, Kumam and Plubtieng [11, 12, 13] obtained a random fixed point theorem for a multivalued non-self nonexpansive mapping in a separable Banach space. Very recently, Dompongsa et al. [4] proved a common fixed point theorem for two nonexpansive commuting mappings t : E → E and T : E → KC(E) denoted the class of all compact convex subsets of X where X is a uniformly convex Banach space. The purpose of this paper is to prove the random version of the following celebrated deterministic result due to Dhompongsa et al. [4, Theorem 4.2]. Theorem 1.1. (Dhompongsa et al. [4, Theorem 4.2]) Let E be a nonempty bounded closed convex subset of a uniformly convex Banach space X, t : E → E, T : E → KC(E) a single-valued and a multi-valued nonexpansive mapping,respectively. Assume that t and T are commuting. Then t and T have a common fixed point, i.e., there exists a point x in E such that x = tx ∈ T x. We obtained a common random fixed point for two single valued and multivalued nonexpansive commuting random operators in a uniformly convex Banach space. Furthermore, we also derive a random coincidence point for a pair of multi-valued and single-valued commuting random operators in a uniformly convex Banach space. Our results extend and improve of the results of Xu [22], Kumam and Plubtieng [11, 12] and many authors.

2

Preliminaries

We begin with establishing some preliminaries. Let (Ω, Σ) be a measurable space with Σ is a sigma-algebra of subset of Ω. Let (X, d) be a complete metric space. We denote by CL(X)(resp.CB(X), K(X), KC(X)) the family of all nonempty closed (resp. nonempty closed bounded, nonempty compact, nonempty compact convex) subset of X, and by H the

2

369

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KUMAM: RANDOM COMMON FIXED POINTS...

Hausdorff metric on CB(X) induced by d, i.e., H(A, B) = max sup d(a, B), sup d(b, A) a∈A

b∈B

for A, B ∈ CB(X), where d(x, E) = inf{d(x, y) : y ∈ E} is the distance from x to E ⊂ X. A mapping T : Ω → 2X is said to be measurable if T −1 (B) ∈ Σ for each open subset B of X, where T −1 (B) = {ω ∈ Ω : T (ω) ∩ B 6= φ}. Let C be a subset of X. A mapping T : Ω × C → 2X is said to be a random operator if, for every x ∈ C, T (·, x) is measurable. A mapping ξ : Ω → C is said to be a deterministic fixed point of a random operator T : Ω × C → 2X if, for each ω ∈ Ω, ξ(ω) ∈ T (ω, ξ(ω)) and a random fixed point of a random operator T : Ω × C → 2X if ξ is a measurable map such that for each ω ∈ Ω, ξ(ω) ∈ T (ω, ξ(ω)). Obviously if a random operator T : Ω × C → 2X has a random fixed point, then for each ω ∈ Ω, T (ω, ·) has a (deterministic) fixed point in C. We will denote by F (ω) the fixed point set of T (ω, ·), i.e., F (ω) := {x ∈ C : x ∈ T (ω, x)} . Note that if we do not assume the existence of fixed point for the deterministic mapping T (ω, ·) : C → 2X , F (ω) may be empty. A random operator f : Ω × C → C is said to be nonexpansive if, for fixed ω ∈ Ω the map f (ω, ·) : C → C is nonexpansive. A measurable mapping ξ : Ω → X is a random coincidence point of random operators T : Ω × X → CB(X) and f : Ω × X → X if for every ω ∈ Ω, f (ω, ξ(ω)) ∈ T (ω, ξ(ω)) (respectively, common random fixed point if for every ω ∈ Ω, ξ(ω) = f (ω, ξ(ω)) ∈ T (ω, ξ(ω))). Let C be a nonempty bounded closed subset of Banach spaces X and {xn } bounded sequence in X, we use r(C, {xn }) and A(C, {xn }) to denote the asymptotic radius and the asymptotic center of {xn } in C, respectively, i.e. r(C, {xn })

= inf

A(C, {xn })

=

lim sup kxn − xk : x ∈ C

,

n

x ∈ C : lim sup kxn − xk = r(C, {xn }) . n

If D is a bounded subset of X, the Chebyshev radius of D relative to C is defined by rC (D) := inf {sup{kx − yk : y ∈ D} : x ∈ C} . Obviously, the convexity of C implies that A(C, {xn }) is convex. Notice that A(C, {xn }) is a nonempty weakly compact set if C is weakly compact, or C is a closed convex subset of a reflexive Banach space X. Let {xn } and C be a nonempty bounded closed subset of Banach spaces X. Then {xn } is called regular with respect to C if r(C, {xn }) = r(C, {xni }) for all subsequences {xni } of {xn }; while {xn } is called asymptotically uniform with respect to C if A(C, {xn }) = A(C, {xni }) for all subsequences {xni } of {xn }.

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KUMAM: RANDOM COMMON FIXED POINTS...

Definition 2.1. (Dhompongsa et al. [4]) Let E be a nonempty bounded closed convex subset of a Banach spaces X, f : E → X, and T : E → CB(X). Then f and T are said to be commuting if for every x, y ∈ E such that x ∈ T y and ty ∈ E, there holds tx ∈ T ty. Lemma 2.2. (Deimling [3]) Let E be a nonempty bounded closed convex subset of a Banach space X and T : E → F C(X) an upper semicontinuous and χ−condensing mapping. Assume T x ∩ IE (x) 6= ∅ for all x ∈ E. Then T has a fixed point. Theorem 2.3. (Wagner [21]). Let (X, d) be a complete separable metric spaces and F : Ω → CL(X) a measurable map. Then F has a measurable selector. Lemma 2.4. ( Tan and Yuan [20]). Let X be a separable metric space and Y a metric space. If f : Ω × X → Y is a measurable in ω ∈ Ω and continuous in x ∈ X, and if x : Ω → X is measurable, then f (·, x(·)) : Ω → Y is measurable. As an easy application of Proposition 3 of Itoh[8] we have the following Lemma: Lemma 2.5. Let C be a closed separable convex subset of a Banach space X and (Ω, Σ) be a measurable space. Let T : Ω×C → CB(X) be a random continuous operator and F : Ω → 2C a measurable closed-valued operator. Then for any s > 0, the operator G : Ω → 2C given by G(ω) = {x ∈ F (ω) : kx − T (ω, x)k < s} , ω ∈ Ω is measurable.

3

The main results

We obtain the following result which extends and improves random version of [4, Theorem 4.2]. Our proof need not use the ultra power technique. Theorem 3.1. Let (Ω, Σ) be a measurable space with Σ a sigma-algebra of subsets of Ω. Let M be a nonempty closed bounded convex and separable subset of a uniformly convex Banach space X, and let f : Ω × M → M, T : Ω × M → KC(M ) be a single valued and a multivalued nonexpansive random operators respectively. Assume that f and T are commuting. Then T and f have a common random fixed point. Proof. For each ω ∈ Ω, set F (ω) = {x ∈ M : x = f (ω, x)}. From Browder-G¨ ohde Theorem, F (ω) is nonempty for each ω ∈ Ω. Let x ∈ F (ω). Since f and T are commuting, we have f (ω, y) ∈ T (ω, x) for each y ∈ T (ω, x) and for each ω ∈ Ω. Thus we can considered f (ω, ·) as a nonexpansive mapping from T (ω, x) to T (ω, x) for each ω ∈ Ω. Since T (ω, x) is compact convex, f has a fixed point y(ω) ∈ F (ω). This implies

4

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KUMAM: RANDOM COMMON FIXED POINTS...

that T (ω, x) ∩ F (ω) 6= ∅ for all ω ∈ Ω. For a fixed element x0 ∈ F, we let x0 (ω) = x0 for each ω ∈ Ω. For each n, we define a random operator Tn : Ω × F (ω) → KC(M ) by Tn (ω, x) =

1 1 x0 + (1 − )T (ω, x), x ∈ F (ω). n n

It is easy to see that Tn is a multivalued random contraction operator for all n ∈ N. Since for T (ω, x) ∩ F 6= ∅, for each x ∈ F (ω), It follows the convexity of F (ω) that implies Tn (ω, x) ∩ F (ω) 6= ∅. Then as in the proof of the proof of Theorem 4.2 in [4] we note that Tn (ω, ·) : F (ω) → KC(M ) is χ-condensing for each ω ∈ Ω. Hence, by Lemma 2.2, Tn (ω, ·) has a fixed point zn (ω) ∈ F (ω) for each ω ∈ Ω. Also it is easily seen that d(zn (ω), T (ω, zn (ω))) ≤ n1 diamC → 0 as n → ∞. This implies that the set 1 Fn (ω) = {x ∈ F (ω) : dist(x, T (ω, x)) ≤ } n is nonempty closed for each n ≥ 1. Furthermore, by Lemma 2.5, each Fn is measurable. Then, by Lemma 2.3, each Fn admits a measurable selector xn (ω) such that 1 diamC → 0 as n → ∞. n

d(xn (ω), T (ω, xn (ω))) ≤

Define a function f1 : Ω × M → R+ := [0, ∞) by f1 (ω, x) = lim sup kxn (ω) − xk, ω ∈ Ω. n

Applying Lemma 2.4, it not difficult to see that f1 (·, x) is measurable. In addition f1 is continuous in x ∈ C and convex. Therefore it is a weakly lower semicontinuous function. On the other hand, since the space X is uniformly convex and F (ω) is closed and convex and hence weakly compact, there exists a unique point x(ω) ∈ F (ω) such that f1 (ω, x(ω)) =

inf

x∈F (ω)

f1 (ω, x) =: r(ω).

Note that x(ω) is an asymptotic center of the sequence {xn (ω)} with respect to F (ω), i.e., x(ω) = A(F (ω), {xn }). Lim [17], and Kirk and Massa [10] actually proved that for each ω ∈ Ω, x(ω) is a fixed point of the map T (ω, ·). By using the same argument as in the proof of Xu [22, p.1091], we obtain x(ω) is measurable. Therefore x(ω) is a random fixed point of T, since x(ω) ∈ F (ω) and so f and T have a common random fixed point x(ω), i.e., x(ω) = f (ω, x(ω)) ∈ T (ω, x(ω)). The proof of the theorem, therefore, is complete. From Theorem 3.1 we immediately got the following Theorems: Theorem 3.2. Let (Ω, Σ) be a measurable space with Σ a sigma-algebra of subsets of Ω. Let M be a nonempty closed bounded convex and separable subset of a uniformly convex Banach space X, and let f : Ω × M → M, T : Ω × M → KC(M ) be a single valued and a multivalued nonexpansive random operators respectively. Assume that f and T are commuting. Then T and f have a random coincidence point.

5

KUMAM: RANDOM COMMON FIXED POINTS...

Theorem 3.3. Let (Ω, Σ) be a measurable space with Σ a sigma-algebra of subsets of Ω. Let M be a nonempty closed bounded convex and separable subset of a uniformly convex Banach space X, and let T : Ω × M → CB(M ) be multivalued nonexpansive random operator. Then T have a random fixed point. Proof. If f (ω, x) = x for all (ω, x) ∈ Ω × X, in Theorem 3.1, then we get the desired result. Corollary 3.4. (Xu’s Theorem cf. [22]) Let (Ω, Σ) be a measurable space with Σ a sigma-algebra of subsets of Ω. Let M be a nonempty closed bounded convex and separable subset of a uniformly convex Banach space X, and let T : Ω × M → KC(M ) be a multivalued nonexpansive random operator. Then T has a random fixed point. Corollary 3.5. Let M be a separable nonempty closed bounded convex and separable subset of a uniformly convex Banach space X, and let f : Ω × M → M, T : Ω × M → M be two single valued nonexpansive random operators. Then there is a measurable map ξ : Ω → M such that T (ω, ξ(ω)) = f (ω, ξ(ω)) = ξ(ω) for each ω ∈ Ω.

4

Deterministic results

By using Theorem 3.1, we have the followings Theorem. Theorem 4.1. (Dhompongsa et al. [4, Theorem 4.2]) Let M be a nonempty bounded closed convex subset of a uniformly convex Banach space X, t : M → M, T : M → KC(M ) a single-valued and a multi-valued nonexpansive mapping,respectively. Assume that t and T are commuting. Then t and T have a common fixed point. As an application of Theorem 3.3 or Corollary 3.4, we immediately obtain the following result: Corollary 4.2. (Lim’s Theorem [16]) Let X be a uniformly convex Banach space, let M be a nonempty bounded closed convex subset of X Then every multivalued nonexpansive mapping T : C → K(M ) has a fixed point. Remark 4.3. Theorem 3.1, Theorem 3.3 and Corollary 3.5 are stochastic version of [4, Theorem 4.2] and Lim’s Theorem in [16]. Acknowledgement. The author would like to thank Professor Somyot Plubiteng for providing valuable suggestions and comments.

References [1] I. Beg and N. Shahzad, On random approximation and coincidence point theorems for multivalued operators, Nonlinear Anal. 26 6 (1996) 1035–1041. [2] A. T. Bharucha-Reid, Random Integral Equations. Bull. Academic Press, New York and London, (1972).

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[3] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin (1974). [4] S. Dhompongsa, A. Kaewcharoen, and A. Kaewkhao, The Dominguez-Lorenzo condition and multivalued nonexpansive mappings, Nonlinear Anal. 64 (2006) 958–970. [5] O. Han˘s, Random fixed point theorems, in: Transactions of the First Prague Conference on Information Theory, Statistical Decision Functions, Random Processes (Liblice, Prague, 1956), pp. 105–125, Czechoslovak Academy of Sciences, Prague, Czech Republic, (1957). [6] O. Han˘s, Random operator equations, in: Proc. of the 4th Berkeley Symposium on Mathematical Statistics and Probability, vol. II, Part I, pp. 185–201, University of California Prees, Berkeley, (1966). [7] O. Han˘s, Reduzierende Zuf¨ allige Transformationen, Czechoslovak Math. J. vol. 7 (1957) 154–158. [8] S. Itoh, Random fixed point theorem for a multivalued contraction mapping, Pacific J. Math. 68 (1977) 85–90. [9] A. R. Khan F. Akbar, N. Sultana and N. Hussain, Coincidence and invariant approximation theorems for generalized f -nonexpansive multivalued mappings, Int. J. Math and Math. Sci., Volum 2006, Article ID 17637 (2006) 1–18. [10] W.A. Kirk, S. Massa, Remarks on asymptotic and Chebyshev centers”, Houston J. Math. 16 (1990) 357–364. [11] P. Kumam and S. Plubtieng,“The Characteristic of noncompact convexity and random fixed point theorem for set-valued operators”, Czechoslovak Mathematical Journal, 57(132) (2007), 269-279. [12] P. Kumam and S. Plubtieng, “Some random fixed point theorems for non-self nonexpansive random operators” Turkish Journal of Mathematics. 30 (2006), 359-372. [13] P. Kumam and S. Plubtieng, “Random fixed point theorems for multivalued nonexpansive non-self random operators”, Journal of Applied Mathematics and stochastic Analysis, Volume 2006, Article ID43796, 9 Pages. [14] H. K. Nashine and R. Shrivastava, “Random fixed point and random best approximation”, J. Comput. Anal. Appl., 11 2 (2009) 338–345. [15] G. Li and H. Duan, On random fixed point theorems of random monotone operators, Appl. Math. Let. 18 (2005) 1019–1026. [16] T.C. Lim, A fixed point theorem for multivalued nonexpansive mappings in a uniformly convex Banach space, Bull. Amer. Math. Soc. 80 (1974) 1123–1126. [17] T.C. Lim, Remark on some fixed point theorems, Proc. Amer. Math. Soc. 60 (1976) 179-182. [18] S. B. Nadler, Jr., Multivalued contraction mapping, Pacific J. Math. 30 (1969) 475–488. ˇ cek, Zuf¨ [19] A. Spaˇ allige Gleichungen, Czechoslovak Math. J.5 (1955) 462–466.

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[20] K.-K. Tan and X.Z. Yuan, Some random fixed point theorem , in: K.K. Tan (Ed), Fixed Point Theory and Applications, Wold Sciedtific, pp 334–345, (1992). [21] D.-H. Wagner, Survey of measurable selection theorems, SIAM J. Control Optim. 15 (1977) 859–903. [22] H. K. Xu, A random theorem for multivalued nonexpansive operators in Uniformly convex Banach spaces, Proc. Amer. Math. Soc. 117 (1993) 1089–1092.

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JOURNAL 376 OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.2, 376-387, 2011, COPYRIGHT 2011 EUDOXUS PRESS, LLC

A FAMILY OF BARNES-TYPE MULTIPLE TWISTED q-EULER NUMBERS AND POLYNOMIALS RELATED TO FERMIONIC p-ADIC INVARIANT INTEGRALS ON Zp

Lee-Chae Jang Department of Mathematics and Computer Science, KonKuk University, Chungju, S. Korea [email protected]

Abstract. In this paper, by using the fermionic p-adic invariant integrals, we consider the twisted q-extension of the Barnes-type multiple q-Euler polynomials and numbers and discuss some properties of them. In particular, we present a systemic study of the generalized Barnes-type multiple twisted q-Euler polynomials of higher order and give Barnes-type multiple twisted zeta functions which interpolate the generalized Barnestype multiple twisted q-Euler polynomials

§1. Introduction Let p be a fixed odd 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 p-adic completion of the algebraic closure of Qp . Let N be the set of natural numbers and Z+ = N ∪ {0}. The p-adic absolute value in Cp is normalized so that |p|p = p1 . When one talks of q-extension, q is variously considered as an indeterminate, a complex number q ∈ C or a p-adic number q ∈ Cp . If q ∈ C, one normally assumes |q| < 1. If q ∈ Cp , one normally assumes |1 − q|p < 1. We use the notation 1 − qx 1 − (−q)x [x]q = , and [x]−q = 1−q 1+q

see [1-17]

2000 AMS Subject Classification: 11B68, 11S80 keywords and phrases : Euler polynomials and numbers, Fermionic p-adic invariant integrals, q-Euler polynomials and numbers, the Barnes-type multiple twisted q-Euler polynomials and numbers

1

Typeset by AMS-TEX

JANG: MULTIPLE TWISTED q-EULER NUMBERS

377

for all x ∈ Zp . For a fixed d ∈ N with (p, d) = 1,d ≡ 1(mod2), we set X = Xd = lim Z/dpN Z, X ∗ = ← − N

∪

0 0, a1 , · · · , ar ∈ C and ξ ∈ Tp , we define Barnes-type multiple twisted q-zeta function as follows: ζξ,q,r (s, x|w1 , · · · , wr ; a1 , · · · , ar ) ∑r ∑r ∞ ∑ j=1 mj (ξq) i=1 ai mi (−1) = 2r . [x + w1 m1 + · · · + wr mr ]sq m ,··· ,m =0 1

(28)

r

Note that ζξ,q,r (s, x|w1 , · · · , wr ; a1 , · · · , ar ) is meromorphic function in the whole complex s-plane. By using the Mellin transformation and the Cauchy residue theorem, we obtain the following theorem. Theorem 6. For x ∈ C with R(x) > 0, n ∈ Z+ , and ξ ∈ Tp , we have (r)

ζξ,q,r (−n, x|w1 , · · · , wr ; a1 , · · · , ar ) = En,χ,ξ,q (x|w1 , · · · , wr ; a1 , · · · , ar ). Let χ be a Dirichlet’s character with conductor f ∈ N, with f ≡ 1(mod2). From (25), we can define the generalized Barnes-type multiple twisted q-Euler polynomials attached to χ in C as follows: (r)

Fχ,ξ,q (t, x|w1 , · · · , wr ; a1 , · · · , ar )   ∞ r ∑r ∑ ∏ ∑r n = 2r (−1) j=1 mj  χ(xj ) (ξq) i=1 ai mi e[x+w1 m1 +···+wr mr ]q t m1 ,··· ,mr =0

=

∞ ∑

j=1

(r)

En,χξ,q (x|w1 , · · · , wr ; a1 , · · · , ar )

n=0

tn , n! (r)

(29)

From (29) and Mellin transformation of Fχ,ξ,q (t, x|w1 , · · · , wr ; a1 , · · · , ar ), we can drive the following equation (29). I ∞ 1 (r) ts−1 Fχ,ξ,q (−t, x|w1 , · · · , wr ; a1 , · · · , ar )dt Γ(s) 0   ∑r ∑r ∞ r ∑ ∏ j=1 mj (ξq) i=1 ai mi (−1)  . (30) = 2r χ(xj ) s [x + w m + · · · + w m ] 1 1 r r q m ,··· ,m =0 j=1 1

r

10

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JANG: MULTIPLE TWISTED q-EULER NUMBERS

For s, x ∈ C with R(x) > 0, we also define Barnes-type multiple twisted q − l-function as follows: (r)

lχ,ξ,q (s, x|w1 , · · · , wr ; a1 , · · · , ar )   ∑r ∑r ∞ r ∑ ∏ j=1 mj (ξq) i=1 ai mi (−1) r  χ(xj ) =2 . [x + w1 m1 + · · · + wr mr ]sq m ,··· ,m =0 j=1 1

(31)

r

(r)

Note that lχ,ξ,q (s, x|w1 , · · · , wr ; a1 , · · · , ar ) is meromorphic function in the whole complex s-plane. By using (29), (30),(31), and the Chquchy residue theorem, we obtain the following theorem. Theorem 7. For x ∈ C with R(x) > 0, n ∈ Z+ , and ξ ∈ Tp , we have (r)

(r)

lχ,ξ,q (−n, x|w1 , · · · , wr ; a1 , · · · , ar ) = En,χ,ξ,q (x|w1 , · · · , wr ; a1 , · · · , ar ).

Acknowledgement. This paper was supported by Konkuk University in 2011. References [1] [2] [3]

[4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

E.W. Barnes, On the theory of multiple gamma function, Trans. Camb. Philos. Soc. 18 (2009), 127-133. M. Cenkci, The p-adic generalized twisted (h, q)-Euler-l-function and its applications, Adv. Stud. Contemp. Math. 15(1) (2007), 37-47. M. Cenkci, M. Can, V Kurt, p-adic interpolation function and kummer-type congruences for q-twisted and q-generalized twisted Euler numbers, Adv. Stud. Contemp. Math. 9(2) (2004), 203-216. T. Kim, Barnes-type multiple q-zeta functions and q-Euler polynomials, J. Phis. A: Math. Theor. 43 (2010), 255201(11 pp). T. Kim, New approach to q-Euler polynomials of higher order, Russian J. of Math. Phys. 17(2) (2010), 201-207. T. Kim, Note on the Euler q-zeta functions, J. Number Theory 129 (2009), 798-804. T. Kim, A note on some formulae for the q-Euler numbers and polynomials, Proc. Jangjeon Math. Soc. 9(2) (2006), 227-232. T. Kim, A note on p-adic q-integral on Zp associated with q-Euler numbers, Adv. Stud. Contemp. Math. 15(2) (2007), 133-138. T. Kim, On the analogs of Euler numbers and polynomials associated with p-adic q-integral on Zp at q = −1, J. Math. Anal. Appl. 331 (2007), 779-792. T. Kim, On Euler-Barnes multiple zeta functions, Russ. J. Math. Phys. 10 (2003), 261-267. T. Kim and S.H. Rim, On the twisted q-Euler numbers and polynomials associated with basic q − l-functions, J. Math. Anal. Appl. 336 (2007), 738-744. T. Kim, q-Volkenborn integration, Russ. J. Math. Phys. 9(3) (2002), 288-299. T. Kim, On a q-Analogue of the p-Adic Log Gamma Functions and Related Integrals, J. Number Theory 76(2) (1999), 320-329. L.C. Jang, Multiple twisted q-Euler numbers and polynomials associated with p-adic qintegrals, Advances in Difference Equations Article ID 738603 (2008), 5 pages.

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[15] [16]

[17]

387

L.C. Jang, A study on the distribution of twisted q-Genocchi polynomials, Adv. Stud. Contemp. Math. 18 (2009), 181-189. H. Ozden, Y. Simsek and I.N. Cangul, Remarks on sum of products of (h, q) twisted Euler polynomials and numbers, J. of Inequalities and Applications Article ID 816129 (2008), 8pages. Y. Simsek, Generating functions of the twisted Bernoulli numbers and polynomials associated with their interpolation functions, Adv. Stud. Contemp. Math. 16(2) (2008), 251–278.

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JOURNAL 388 OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.2, 388-398, 2011, COPYRIGHT 2011 EUDOXUS PRESS, LLC

Some results on analyticity with applications Marin Marin Dept. of Mathematics, University of Brasov, Romania

Abstract. The aim of our study is to obtain a criterion of analyticity. In fact we have obtained the conditions for what a complex continuous function, defined on the closed unit disc, is an analytic function on the open unit disc. As a consequence, we have deduced, in a new manner, the Schwartz-Villat formula and the Joukowski function. Keywords: complex function; analytic function; criterion of analyticity; Schwartz-Villat’s formula; Joukowski’s function 2010 MSC: 30B50, 30E20, 30E25, 32A55, 32S05

1. Introduction The study of analytic functions is always going to be a topic of interest for mathematicians and not only. The researchers assume that more than 90 % of the practical applications of the mathematical analysis deal with analytical functions. It is each to ascertain that the methods based on the complex function theory are quite efficient in many problems of numerical differentiation, numerical integration and so on. Also, to investigate what kind of analyticity the state function of a material has, or in other words, where and what kind of singularities exist in the complex plane, is an important subject in the field of recent statistical mechanics. So, we can deduce that is more realistic to study in what region the function is regular rather than to establish the differentiability. The analytic functions are used in various branches of Mathematics. For instance, in the paper [5], the system of four linear partial differential equations of Lord-Shulman model of thermoelasticity with one relaxation constant is trasformed, by using some complex variables, in a complex system of equations. This offer the possibility to express the solution using a holomorphic expansion with analytic functions. In the paper [1], some integral representations are constructed for analytic functions in a strip. Using these representations, an effective solution of Calerman type problem is given for a strip. 1

MARIN: ANALYTICITY

389

In our study [4] we use some analytic functions to obtain a minimum principle in Thermoelasticity of dipolar bodies with stretch. There are a number of criteria regarding the analyticity of a function, starting with the well known criterion due to the Cauchy-Riemann’s conditions (see . Regarding our statements above we propose a new criterion in our study which is easy to understand and applicable in solving actual problems like the examples we present below. Denote by A the space of all complex functions which are analytic on the open unit disc and continuous on the closed unit disc. Also, denote by C the space of all complex functions which are continuous on the closed unit disc and are analytic on the open unit disc, except certain points. ˜ be the space of complex functions which are H˝older continuous on the unit Let H ˜ of restrictions of functions in A. circle and let H be the subspace of H In the following, our goals are, first, to find the conditions nedeed for a function from ˜ to belong to the subspace H and, second, when a function from the space the space H ˜ is a restriction of a function from C. H ˜ which can be written in the form: Let us consider a function ϕ ∈ H 



ϕ eis = a(s) + i b(s),

(1)

where i is the complex unit and a and b are H˝older continuous functions on the interval [−π, π], satisfying the boundary conditions: a(−π) = a(π), b(−π) = b(π). We can assume, without restricting the generality, that Z

π

a(s)ds =

−π

Z

π

b(s)ds = 0.

(2)

−π

It is well known the fact that the function f , defined by the Cauchy’s Formula, f (z) =

ϕ(ξ) 1 I dξ 2πi |ξ|=1 ξ − z

(3)

is an analytic function inside and outside the unit circle. Also, we know that the above integral exists for z on the unit circle in the sense of Cauchy principal value. In this case, we have the following Sokhotski-Plemelj formulas for inside and outside limits, denoted by fi (z) and fe (z) (see [3], [6]). For a fixed complex number σ such that |σ| = 1, we have 1 I ϕ(ξ) ϕ(σ) lim f (z) = fi (σ) = dξ + z→σ, |z|1 2πi |ξ|=1 ξ − σ 2 2

(4)

390

MARIN: ANALYTICITY

2. Basic results In our following main result, we will prove the relationship between the inside limit ˜ value of the function f , defined by the Cauchy’s Formula (3), and the function ϕ ∈ H, defined in (1). Theorem 1. The function f , defined by (3), satisfies the condition fi (σ) = ϕ(σ)

(5)

if and only if the functions a and b, defined in (1) satisfy the conditions: s−t 1Zπ a(t) cot dt, π −π 2 1Zπ s−t a(s) = − b(t) cot dt. π −π 2 b(s) =

(6)

In this case if the function f is defined by (3), then f ∈ A and f (z) = 0, f or |z| > 1. In particular, for all σ on the unit circle, that is, |σ| = 1), we have fe (σ) = 0. Proof. Let us first prove the necessity of conditions (6). If the function f , defined by (3), satisfies (4), then by using the relations (4), we have fe (σ) = 0 for all σ on the unit circle. Then, by (4)2 ) we deduce 1 I ϕ(σ) ϕ(ξ) = dξ. 2 2πi |ξ|=1 ξ − σ Therefore, we have a(s) + ib(s) = =

1 Z π a(t) + i b(t) dt = 2π −π 1 − ei(s−t)

a(t) + i b(t) 1 Zπ dt. 2 2π −π 2 sin (s − t)/2 − 2 i sin(s − t)/2 cos2 (s − t)/2

After some simple calculations we obtain   1 Zπ s−t a(s) + ib(s) = a(t) − b(t) cot dt + 2π −π 2   s−t 1 Zπ + b(t) + a(t) cot dt. 2π −π 2

3

MARIN: ANALYTICITY

391

From this equality and the relation (2) we deduce the conditions (6). Now, we will prove that the conditions (6) lead to the condition (5). For this, we use the Fourier expansions of the functions a and b. By using the conditions (5) we can deduce the following relationships between the Fourier coefficients of the functions a and b: 1Zπ 1Zπ a(t) cos nt dt = b(t) sin nt dt, π −π π −π 1Zπ 1Zπ a(t) sin nt dt = − b(t) cos nt dt. π −π π −π

(7)

Taking into account the geometric series expansion ∞ X 1 ξk = 1 − ξ/z k=0 z k

for any z such that |z| > 1, we have

f (z) = −

∞ X ϕ(ξ) 1 I ξk 1 I dξ = − ϕ(ξ) dξ = k 2πiz |ξ|=1 1 − ξ/z 2πiz |ξ|=1 k=0 z

Z π  ∞ 1 X 1 = [a(t) cos(k + 1)t − b(t) sin(k + 1)t] dt + 2π k=0 z k+1 −π Z π  ∞ 1 X 1 + [a(t) sin(k + 1)t + b(t) sin(k + 1)t] dt . 2π k=0 z k+1 −π

From this relations and considering the relationships (7), we can draw to the conclusion f (z) = 0. Thus, for all σ on the unit circle, we have fe (σ) = 0, such that, by using the condition (4), we obtain the desired conditions (5). This concludes the sufficiency of conditions (6) and Theorem 1. In a similar way we can prove the relationship between the outside limit value of the ˜ defined in (1). function f , defined by the Cauchy’s Formula (3), and the function ϕ ∈ H, Theorem 2. The function f , defined by (3), satisfies the condition fe (σ) = ϕ(σ) 4

392

MARIN: ANALYTICITY

if and only if the functions a and b, defined in (1) satisfy the conditions: s−t 1Zπ a(t) cot dt, b(s) = − π −π 2 1Zπ s−t dt. a(s) = b(t) cot π −π 2

(8)

In this case the function f defined by (3) is analytic for all z, |z| = 6 1, continuous in all points z such that |z| ≥ 1 and f (z) = 0, f or |z| < 1. In particular, for all σ on the unit circle, that is, |σ| = 1, we have fi (σ) = 0. A pair of functions (a, b) which satisfy conditions (6) or (8) will be called an inside conjugate pair or an outside conjugate pair, respectively. As a consequence of the above results, we want to solve two problems, having a practical significance. First, we will solve a boundary value problem of Hilbert type and then we will obtain the Schwartz-Villat formula, by using a new procedure. Regarding the boundary value problem, let us consider the given functions a, b and c assumed to be H˝older continuous on the interval [−π, π] and satisfying the boundary conditions a(−π) = a(π), b(−π) = b(π), c(−π) = c(π). We would like to find a complex function F , F (z) = u(z) + iv(z) ∈ A, such that for all s ∈ [−π, π] we have 







a(s)u eis + b(s)v eis = c(s). Let’s assume that the function ϕ ∈ H satisfies the condition (1). Then there exists a function f (z) ∈ A such that f (z) ≡ ϕ(z), on the unit circle, i. e. |z| = 1. We consider the function c0 such that c0 (s) =

c(s) . + b2 (s)

a2 (s)

5

(9)

MARIN: ANALYTICITY

393

Also, we will consider the function c¯0 , which is the inside conjugate of the function c0 , defined by 1Zπ s−t c0 (t) cot dt. c¯0 (s) = π −π 2 Clearly, there exists a function p(z) ∈ A such that 



p eis = c0 (s) + i c¯0 (s). Then we can take the function F (z) in the form F (z) = f (z) p(z). This form of the function F is the desired function which solves our problem, since: 











F eis = u eis + i v eis = [a(s)c0 (s) − b(s)c¯0 (s)] + i [a(s)c¯0 (s)(s) − b(s)c0 (s)] , and 







h

i

a(s)u eis + b(s)v eis = c0 (s) a2 (s) + b2 (s) = c(s). The last equality is a consequence of the relation (9). Now, let us derive the Schwartz-Villat formula, as another application of the previous theoretical result. Theorem 3. Given a function a on the interval [−π, π], satisfying the condition a(−π) = a(π), there exists a function f ∈ A such that f (z) =

1 Z π a(s)(ξ + z) ds 2π −π ξ−z

which is the Schwartz-Villat formula. Proof. Given a function a having the above properties and using the result of Theorem 1, we can deduce that there exists a function f ∈ A such that, for all s ∈ [π, π], we have 



Re f eis = a(s). We will denote by b the inside conjugated function of a. By virtue of Theorem 1, the ˜ satisfies the condition required function f is given by (3), where the function ϕ ∈ H 



ϕ eis = a(s) + i b(s). 6

394

MARIN: ANALYTICITY

If we use the notation ξ = eis , then we obtain a(s) + i b(s) 1 Z π a(s) + i b(s) 1 I dξ = ξ ds = 2πi |ξ|=1 ξ−z 2π −π ξ−z

f (z) =

1 Z π i b(s)ξ − za(s) 1 Z π a(s)(ξ + z) ds + ds. = 2π −π ξ−z 2π −π ξ−z

(10)

We will prove that the last integral in the right-side of relation (10) is equal to zero for z inside of the unit disc, i.e. |z| < 1. For this we will use a geometric series expansion and the Fourier series for the functions a(s) and b(s). Let us denote by an and bn the Fourier coefficients of function a(s). Using the relation (2) we deduce that a0 = 0. Also, by virtue of the relations (7) we obtain the Fourier expansion for the function b(s) as well: ∞ X

a(s) = b(s) =

n=1 ∞ X

(an cos ns + bn sin ns) , (−bn cos ns + an sin ns) .

n=1

For z such that |z| < 1 we have 1 Z π i b(s) − a(s)z/ξ 1 Z π i b(s)ξ − za(s) ds = ds = 2π −π ξ−z 2π −π 1 − ξ/z ! ∞ 1 Zπ z X zk = i b(s) − a(s) ds = 2π −π ξ k=0 ξ k

(11)

! ∞ ∞ X X 1 Zπ zk zk i b(s) − a(s) ds = k k 2π −π k=0 ξ k=1 ξ

Using the relation ξ −n = cos(ns) − i sin(ns) and the orthogonality of the Fourier basis, the integrant of the last integral of the right side of (11) take the following expression: ∞  X



−i bn cos2 (ns) + an sin2 (ns) − an cos2 (ns) + i bn sin2 (ns) z n =

n=1

=

∞ h X







i

−an cos2 (ns) − sin2 (ns) − i bn cos2 (ns) − sin2 (ns)

n=1

=−

∞ X

(an + i bn ) cos(2ns)z n .

n=1

7

zn =

MARIN: ANALYTICITY

395

By integrating the last term, on the interval [−π, π], we obtain zero and then from (11) we deduce 1 Z π i b(s)ξ − za(s) ds = 0, 2π −π ξ−z and, from (10) it results 1 Z π a(s)(ξ + z) f (z) = ds 2π −π ξ−z that is, the Schwartz-Villat formula and the proof of Theorem 3 is concluded. We will conclude our study with a result, similar to the classical Milne-Thomson’s result. Theorem 4. Consider a pair of functions (a, b), which are H˝ older continuous on the interval [−π, π] and satisfy the boundary conditions: a(−π) = a(π), b(−π) = b(π). Then there exists a function f which is analytic inside of the open unit disc, |z| < 1, except certain points, such that 



f eis = a(s) + i b(s). Proof. Let us remark that if we find the functions F a (z) and F b (z) which are analytic on the open unit disc, except certain points, and satisfying the relations 







F a eis = a(s), F b eis = a(s), then the function f (z) = F a (z) + i F b (z) can be the desired function for our problem. Let’s consider the function α(s) defined by α(s) =

a(s) 2

and its inside conjugate α ¯ (s). Define the function f a (z) ∈ A by 1 I α(s) + i α ¯ (s) f (z) = dξ 2πi |ξ|=1 ξ−z a

Then 



fia eis = α(s) + i α ¯ (s). 8

(12)

396

MARIN: ANALYTICITY

Also, for |z| > 1 we have fa (z) = 0. Now, let’s consider the outside conjugate s−t 1 Zπ ¯ α(t) cot dt. β(s) = − 2π −π 2 Then there exists a function g a (z), which is analytic outside of the unit disc, such that 



¯ = α(s) − i α gea eis = α(s) + i β(s) ¯ (s). Let gpa be the analytic continuation of the function g a to the interior of the unit disc. Generally speaking, the function gpa may and must have singularities. Otherwise the function f a + gpa would be analytic inside the open disc, continuous for |z| ≥ 1 and would have zero imaginary part. Such a function must be constant. As a consequence, we can take the function F a in the form F a = f a + gpa .

(13)

Similarly, we obtain the function F b and the desired function from the enunciation is f (z) = F a (z) + i F b (z)

(14)

and this concludes the proof of the Theorem 4. We must outline that the above decomposition (13) is similar to the one of the classical Milne-Thomson’s Theorem. As an application of the result from Theorem 4, we can obtain the well known Joukowski function. Let us consider the particular pair of functions (a, b) from the above Theorem 4 of the form     a(s) = 1 + k 2 cos s, b(s) = 1 − k 2 sin s. It is easy to see that a(s) + i b(s) 6∈ H. Using Theorem 4 we find that 1 + k2 a(s) = cos s, 2 2 1 Z π 1 + k2 s−t 1 + k2 β(s) = cos t cot dt = sin s. 2π −π 2 2 2 α(s) =

9

MARIN: ANALYTICITY

397

After some calculations we deduce that 1 + k2 . 2z Now we will consider the outside conjugate f a (z)

Z π 1 + k2 s−t 1 + k2 ¯ =− 1 β(s) cos t cot dt = − sin s = −β(s). 2π −π 2 2 2 So we obtain 1 + k2 z, g a (z) 2 which is an analytic function outside the unit disc and satisfies the relation

1 + k2 1 1 + k2 (cos s − i sin s). = 2 eis 2 We can continue the function g a (z) to a function which is analytic on the unit disc, except the origin. We have 



g a eis =

1 + k2 1 F (z) = z+ . 2 z 

a



(15)

Similarly, we find that the function b(s) has the form 



b(s) = 1 − k 2 sin s. Also, it is easy to find the other functions regarding the function b(s): 1 − k2 sin s, 2 1 Z π 1 − k2 s−t 1 − k2 β(s) = sin t cot dt = − cos s, 2π −π 2 2 2 1 − k2 z, f b (z) = −i 2 1 − k2 1 g b (z) = −i . 2 z Finaly, we obtain α(s) =

F b (z) = −i

1 − k2 1 z+ . 2 z 



(16)

Taking into account (14)-(16) we obtain 1 1 − k2 1 1 + k2 z+ + i −i z+ f (z) = 2 z 2 z 





which is the Joukowski function. 10

!

=z+

k2 , z

398

MARIN: ANALYTICITY

4. Conclusion With the help of our new criterion of analyticity (in Theorem 1 and Theorem 2), we can solve a Hilbert type problem and, as a consequence, we deduce the Schwartz-Villat formula. Also, a specific boundary value problem can be solved and, as a concequence, we obtain the Joukowski function. Reference 1. R. Bantsuri, Boundary value problems of theory of analytic functions with displacements, Georgian Mathematical Journal, 3(1999), 213-232 2. J. B. Conway, Functions of complex variable Springer, Berlin, New-York, 1996 3. S. G. Krantz, Handbook of Complex variables Birkh˝auser, Boston, 1999 4. M. Marin, On the minimum principle for dipolar bodies with stretch, Nonlinear Analysis: RWA, Vol. 10, 3(2009), 1572-1578 5. A. Rodionov, Explicit solutions for partial differential equations of Lord-Shulman thermoelasticity, Teoret. Appl. Mech. Vol. 36, 2(2009), 137-156 6. E. T. Whittaker,G. N. Watson, A course of Modern Analysis Cambridge University Press, 1990

11

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL.13, NO.2, 2011

Approximation by nonlinear Lagrange interpolation operators of max-product kind on Chebyshev knots of second kind, Lucian Coroianu and Sorin G. Gal,……………………….211 Forward Continuity, Huseyin Cakalli,……….……………………….……………………….225 Almost periodic solutions to abstract semilinear evolution equations with Stepanov almost periodic coefficients, Hui-Sheng Ding, Wei Long and Gaston M. N'Guerekata……………..231 Abstract metric spaces and approximating fixed points of a pair of contractive type mappings, M. Abbas, Mirko Jovanovic, S. Radenovic, Aleksandra Sretenovic, Suzana Simic,……………..243 Optimal property of the shape of aeolian blade profile using cubic splines, Calin Dubau,…...254 Cauchy problem of the Δ ( k) operator related to the Diamond operator and the Laplace operator iterated k times, Chalermpon Bunpog,…………………………………………………………264 Some algorithms for a class of set-valued variational inclusions in Frechet Spaces, Chaofeng Shi, Nan-jing Huang,………………………………………………………………………………..272 On the Extended Kim’s q-Bernstein Polynomials, S.-H. Rim, L.C. Jang, J. Choi, Y. H. Kim, B. Lee, and T. Kim,………………………………………………………………………………..282 Inner Product Spaces and Functional Equations, Yeol Je Cho, Choonkil Park, Themistocles M. Rassias and Reza Saadati,………………………………………………………………………296 Fuzzy Functional Inequalities, Yeol Je Cho, Choonkil Park and Reza Saadati,……………......305 A class of weighted holomorphic Bergman spaces, A. El-Sayed Ahmed and H. Al-Amri,…....321 Existence and Iterative Approximations of Solutions for Nonlinear Implicit Fuzzy Resolvent Operator Systems of (A, η)-monotone Type, Heng-you Lan, Yongming Li and Jianfang Tang,……………………………………………………………………………………………335 A Composite Iterative Method for Generalized Mixed Equilibrium Problems and Variational Inequality Problems, Thanyarat Jitpeera and Poom Kumam,…………………………………..345 Coincidence and Common Fixed Points for Generalized Contraction Multi-Valued Mappings, Wutiphol Sintunavarat and Poom Kumam,…………………………………………………….362 Random Common Fixed Points of Single-Valued and Multivalued Random Operators in a Uniformly Convex Banach Space, Poom Kumam,…………………………………………….368

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL.13, NO.2, 2011 (continues)

A Family of Barnes-Type Multiple Twisted q-Euler Numbers and Polynomials Related to Fermionic p-Adic Invariant Integrals on Ζp, Lee-Chae Jang,………………………………376 Some results on analyticity with applications, Marin Marin,………………………………388

401

Volume 13, Number 3 ISSN:1521-1398 PRINT,1572-9206 ONLINE

April 2011

Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC

402

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(seven 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. Webmaster:Ray Clapsadle 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 $470,Electronic $300,Print and Electronic $500.Individual:Print $150,Electronic $100,Print &Electronic $200.For any other part of the world add $50 more to the above prices for Print.No credit card payments. Copyright©2011 by Eudoxus Press,LLCAll 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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407

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JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.3, 411-424, 2011, COPYRIGHT 2011 EUDOXUS PRESS, 411 LLC

Analytical and Numerical Solution of Non-Fourier Heat Conduction in Cylindrical Coordinates Seyfolah Saedodin 1, Mohsen Torabi 2, Hadi Eskandar 3, Mohammad Akbari 4 1

Assistant Professor, Department of Mechanical Engineering, Faculty of Engineering, Semnan University, Semnan, Iran, E-mail: [email protected]

2

M.S. Student, Department of Mechanical Engineering, Faculty of Engineering, Semnan University, Semnan, Iran, E-mail: [email protected]

3

M.S. Student, Department of Mechanical Engineering, Faculty of Engineering, Semnan University, Semnan, Iran, E-mail: [email protected]

4

PH.D. Student, Department of Mechanical Engineering, Faculty of Engineering, Semnan University, Semnan, Iran, E-mail: [email protected] Abstract The purpose of the present paper is to carry out the Non-Fourier effect subjected to heat flux boundary condition. The governing equation expressed in cylindrical coordinates. Equations are solved by deriving the analytical and the numerical solution. The temperature layers and profiles of sample calculations performed. It is found from these calculations that the numerical solution is in good agreement with the analytical solution. Also, the temperature layers and profiles of sample calculations show that, as much as the Vernotte number is higher, the point can get to higher temperature during the process. Also, it can be perceived from temperature profiles that, it is possible that the temperature of different points of object become even lower than initial temperature. Keywords: Non-Fourier – Heat Conduction – Relaxation Time – Analytical Solution – Numerical Solution

1

412

SAEDODIN ET AL:ANALYTICAL AND NUMERICAL SOLUTION...

Nomenclature

A, B, C, D, an , c1 , c2 , C12 , C fg

Constant coefficients

Specific heat capacity Fourier number

Greek symbols

α

Thermal diffusivity

Heat flux vector

ρ

Mass density

Thermal conductivity

Laplace's differential operator

Radius of cylinder

∆ ∇ τ

Radius of heat flux

θ

Dimensionless temperature

Ve

Vernotte number

ξ ,ω

Dimensionless spatial coordinate

r, z

Spatial coordinate

∆ξ , ∆ω

Increment coordinate

X (ξ ), Z (ω )

Function employed in Eqs.(16) and (28)

ψ (ξ , ω , Fo)

Function employed in Eq. (11)

t

Temporal coordinate

φ (ξ , ω )

Function employed in Eqs. (11)

T

Temperature

ξ1

Dimensionless radius of heat flux

T (Fo)

Function employed in Eq. (28)

β n , γ f ,η g

Eigenvalues

T∞

Ambient temperature

κ

Parameter difinded by Eq. (41a)

Ti

Initial temperature

κi

Parameter difinded by Eq. (41b)

ϑl

Parameter difinded by Eq. (39)

c

Fo q

k L R r1

Height of the cylinder

Square ratio of height to radius of cylinder Increment of the Fourier number

M ∆Fo

Gradient operator Thermal relaxation time

of

the

dimensionless

spatial

1. Introduction During the past few years there has been research concerned with departures from Fourier's heat conduction law when unsteady processes are involved. In order to eliminate these departures, Cattaneo [1] and Vernotte [2], independently proposed a modification of Fourier's law. Which, is now well known as Cattaneo-Vernotte's constitutive equation: ∂q q +τ = −k∇T (1) ∂t Where q is the heat flux vector, τ is the thermal relaxation time, k is the constant thermal conductivity of the material and ∇ T is the temperature gradient. If equation (1), combined with the conservation of energy gives the non-Fourier heat conduction equation: ∂T ∂ 2T + τ 2 = α∆T ∂t ∂t

Where α =

k

ρc

(2)

, ρ , c and ∆ are thermal diffusivity, mass density, specific heat capacity and Laplace’s differential

operator, respectively. Equation (2) is a hyperbolic partial differential equation and causes the propagation speed, reach a limit amount

α τ , in τ > 0 . Many heat transfer researchers have attached much importance to the

potentially feasible values of the non-Fourier heat conduction in many applications, such as rapid metal thawing and solidifying process, temperature control of superconductor, freezing surgery, rapid drying, and also in situations which include extremely large heat fluxes such as surface thermal processing by laser, rapid prototyping, laser

2

SAEDODIN ET AL:ANALYTICAL AND NUMERICAL SOLUTION...

413

surgery, etc. Hence, the non-Fourier heat conduction has become one of noteworthy subjects in the field of heat transfer. First, we review literatures that have been applied non-Fourier heat conduction Eq. analytically. There are many methods to solve Eq. (2) analytically. Most researches have applied non-Fourier heat conduction equation in onedimensional [3-9]. Lewandowska and Malinowski [3] solved the case of a thin film subjected to a symmetrical heating on both side. Moosaie investigated the non-Fourier heat conduction in a finite medium with the arbitrary source term [4] and arbitrary initial condition [5]. Tang and Araki [6] computed the non-Fourier fin problems under the periodic thermal conditions. Zhang et al. [7] presented a non-Fourier model with heat source. Saleh and Al-Nimr [8] employed Laplace transforms, software package MATLAP and Taylor series, to solve the one-dimensional nonFourier equation. To the authors’ knowledge, there is only one paper that solved multi-dimensional non-Fourier equation analytically. Barletta and Zanchini [9] analytically investigated the non-Fourier equation, using threedimensional rectangular coordinates. Also, there are a lot of literatures that applied this equation numerically. Chen and Lin [10] applied a hybrid numerical technique to problem in one spatial dimension. Fan and Lu [11] derived a new numerical method to solve non-Fourier equation. Liu [12] applied the hybrid method of the Laplace transform technique and a modified discretization scheme to analyze the non-linear non-Fourier heat conduction problems in a semi-infinite domain. He assumed either linearly or exponentially temperature-dependent thermal conductivity. Chen [13] combined the Laplace transform, weighting function scheme and the non-Fourier equation, with a conservation term. Zhou et al. [14] presented a thermal wave model of bioheat transfer, together with a seven-flux model, for light propagation and a rate process equation for tissue damage. Yang [15] applied a forward difference method to solved two-dimensional non-Fourier equation. Also, he proved the stable condition for the problem. In this paper, both analytical and numerical expression of temperature field is obtained for a cylinder. Therefore, we solved non-Fourier equation in cylindrical coordinates. Using our solutions, we performed sample calculation of temperature surfaces and profiles. Also, on those examples, we mention the reflection of the thermal wave. 2. Problem statement Consider a cylinder, as shown as Fig. 1. The heat flux is applied normally to the upper surface ( Z = L ) of the cylinder but only for r < r . 1

Fig. 1. The cylinder configuration 2.1. Governing differential equation For this case, the non-Fourier heat equation without any heat generation, the governing equation can then be expressed as:

τ ∂ 2T 1 ∂T ∂ 2T 1 ∂T ∂ 2T + = + + 2 2 α ∂t α ∂t r ∂r ∂r ∂z 2

(3)

2.2. Boundary conditions 3

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SAEDODIN ET AL:ANALYTICAL AND NUMERICAL SOLUTION...

For this case the boundary conditions are: ∂T ( 0, z , t ) = 0 ∂r ∂T ( R, z , t ) + h[T ( R, z , t ) − T∞ ] = 0 k ∂r T (r ,0, t ) = T∞

k

q ∂T ( r , L, t ) =  ∂z 0

(4a) (4b) (4c)

rr

1

2.3. Initial conditions Consider the solid initially has been at the ambient temperature. Then: (5)

Ti = T∞

Hence the initial conditions are: T (r , z ,0) = T

(6a)

∞

∂T ( r , z ,0 ) = 0 ∂t

(6b)

3. Analytical solution For convenience of subsequent analysis, we introduce the following dimensionless quantities: αt hR T − T∞ r r z L Bi = θ =k , M = ( )2 , ξ = , ω = , Fo = L2 , Ve = ατ2 , ξ1 = 1 k Lq R L R R L

(7)

Where θ is dimensionless temperature and ξ , ω are dimensionless coordinates. Fo is the Fourier number, Ve is the Vernotte number, M is Square ratio of height to radius of cylinder, ξ 1 is dimensionless radius of heat flux and

Bi is the Biot number. By introducing the dimensionless quantities, the normalized temperature of the cylinder obeys the Eq. (8): ∂ 2θ M ∂θ ∂ 2θ ∂ 2θ ∂θ = + + + M ∂Fo 2 ∂Fo ∂ξ 2 ξ ∂ξ ∂ω 2 Also, the boundary conditions are: ∂θ (0, ω , Fo ) = 0 ∂ξ ∂θ (1,ω, Fo) + Biθ (1, ω, Fo) = 0 ∂ξ θ (ξ ,0, Fo ) = 0 Ve 2

1 ∂θ (ξ ,1, Fo ) =  ∂ω 0

(8)

(9a) (9b) (9c)

ξ ≤ξ (9d) ξ >ξ and the initial conditions are: ∂θ (ξ , ω,0) = 0 (10a) ∂Fo θ (ξ , ω,0) = 0 (10b) If we want to apply the well-known separation of variables method, first we should split up eq. (8) with the boundary (9) and the initial conditions (10) into a set of simpler problems. Carslaw [16] and Özisik [17] determined the solution of Eq. (8) from: θ (ξ , ω, Fo) = ψ (ξ , ω, Fo) + φ (ξ , ω ) (11) Where the temperature φ (ξ , ω ) is taken as the solution of the following Eqs.: 1

1

4

SAEDODIN ET AL:ANALYTICAL AND NUMERICAL SOLUTION...

M

∂ 2φ M ∂φ ∂ 2φ + + =0 ∂ξ 2 ξ ∂ξ ∂ω 2

(12)

∂ φ (0, ω ) = 0 ∂ξ ∂ φ (1, ω ) + Biφ (1, ω ) = 0 ∂ξ φ (ξ ,0) = 0 1 ∂ φ (ξ ,1) =  ∂ω 0

ξ ≤ξ ξ >ξ

415

(13a) (13b) (13c) 1

(13d)

1

Where the temperature ψ (ξ , ω , Fo) is taken as the solution of the following Eqs.: Ve 2

∂ 2ψ ∂ψ ∂ 2ψ M ∂ψ ∂ 2ψ + =M + + 2 ξ ∂ξ ∂ω 2 ∂Fo ∂Fo ∂ξ 2

(14)

∂ ψ (0, ω , Fo) = 0 ∂ξ ∂ ψ (1, ω, Fo) + Biψ (1, ω, Fo) = 0 ∂ξ ψ (ξ ,0, Fo ) = 0 ∂ ψ (ξ ,1, Fo) = 0 ∂ω ∂ ψ (ξ , ω ,0) = 0 ∂Fo ψ (ξ , ω ,0) = −φ (ξ , ω ) Solving the partial differential equation (12), we should use the following separation ansantz: φ (ξ , ω ) ≡ X (ξ ) Z (ω ) By substituting eq. (16) into eq. (12) and subtracting to (16): M(

1 d2X 1 dX 1 d 2Z + ) = − = ±β 2 X dξ 2 Xξ dξ Z dω 2

(15a) (15b) (15c) (15d) (15e) (15f) (16) (17)

Here − β 2 is suitable to our problem. Finally, the problem separately expressed in ξ - and ω -directions as follows: d 2 X 1 dX + + m2 X = 0 d ξ 2 ξ dξ d X ( 0) = 0 dξ d X (1) + BiX (1) = 0 dξ

(18) (19a) (19b)

d 2Z − β 2Z = 0 dω 2 Z ( 0) = 0 Where m2 =

(20) (21)

β2

M By solving Eqs. (18) and (20) according to conditions (19) and (21): X (ξ ) = C J 0 (ξmn ) Z (ω ) = B sinh( β nω )

5

(22) (23) (24)

416

SAEDODIN ET AL:ANALYTICAL AND NUMERICAL SOLUTION...

Where β n is eigenvalue of Eq. Bi J 0 (

βn

β

β

) − n J 1 ( n ) = 0 . The first eight eigenvalues of this Eq. are given in M M M Table 1 for Bi = 2 and M = 16 . By substituting the Eqs. (23) and (24) into Eq. (16), we obtain following Eq. as the solution of the Eq. (12): ∞

φ (ξ , ω ) = ∑ a n sinh( β nω ) J 0 ( mnξ )

(25)

n =1

Using boundary condition (13d) and orthogonality condition, we find the following Eq.: ξ1

∫ξ J

an β n cosh β n =

0

( m n ξ ) dξ

0 1

(26)

2 ∫ ξ J 0 (mnξ )dξ 0

Finally, the constant a n is given as following Eq.:

2ξ1 J 1 ( an = M cosh β n [J 0 (

βn M

βn

β

)]2 × [ Bi 2 +

M

1 6.3977

n

ξ1 )

2 17.1638

βn2 M

(27)

]

Table 1. First eight values of β n 3 4 5 29.1535 41.4633 53.8875

6 66.3641

7 78.8688

8 91.3902

To solve the partial differential equation (14), we should use the following separation ansantz: ψ (ξ , ω , Fo) ≡ X (ξ ) Z (ω )T ( Fo) By substituting the Eq. (28) into the Eq. (14) and subtracting to (28):

(28)

1 dT 1 d2X 1 dX 1 d 2Z Ve 2 d 2T + = M( + )+ = ±η 2 2 2 T dFo T dFo X dξ Xξ d ξ Z dω 2

(29)

Here − η 2 is suitable to our problem. Finally, the problem separately expressed in ξ - and ω - and Fo -directions as follows: d 2Z +η 2Z = 0 dω 2 Z ( 0) = 0

(30) (31a)

∂ Z (1) = 0 ∂ω d 2 X 1 dX γ 2 + + X =0 dξ 2 ξ dξ M d X ( 0) = 0 dξ d X (1) + BiX (1) = 0 dξ Ve 2

(31b) (32) (33a) (33b)

d 2T dT + + (η 2 + γ 2 )T = 0 dFo 2 dFo

(34)

d T (0) = 0 dFo Solving the Eqs. (30), (32) and (34) according to conditions (31), (33) and (35): 6

(35)

SAEDODIN ET AL:ANALYTICAL AND NUMERICAL SOLUTION...

417

Z (ω) = Asin(η g ω) γf

X (ξ ) = B J 0 (

M

(36)

ξ)

γf

Where η g and

(37)

are eigenvalues of Eqs. cosη g = 0 and Bi J 0 (

γf M

)−

γf M

J1 (

γf

) = 0 , respectively. For the

M

Eq. (34), if 1 − 4Ve 2ϑl 2 > 0 , we obtain:

T ( Fo) = e

−

Fo 2Ve 2

(c1 sinh(

κFo 2Ve

2

) + c 2 cosh(

κFo 2Ve 2

(38)

))

Where

ϑl 2 = λ f 2 + η g 2

(39) 2

2

And if 1 − 4Ve ϑl < 0 T ( Fo) = e

−

Fo 2Ve 2

(c1 sin(

κ i Fo 2Ve

2

) + c2 cos(

κ i Fo 2Ve 2

(40)

))

Where

κ = 1 − 4Ve 2ϑl 2

(41a)

κ = iκ i By substituting the Eqs. (38) and (40) into initial condition (35) to eliminating c1 or c 2 :

(41b)

 − Fo2  1 κFo κFo  ) + cosh( ) κ = real e 2Ve  sinh( 2 2Ve 2Ve 2  κ   T ( Fo) = C12  Fo − e 2Ve 2  1 sin( κ i Fo ) + cos( κ i Fo ) κ = iκ i  2Ve 2 2Ve 2  κ i   Substituting the Eqs. (36), (37) and (42) into (28), the following equation for ψ (ξ , ω , Fo) obtain: F

G

ψ (ξ , ω , Fo) = ∑∑ C fg exp(− f =1 g =0

(42)

γf κFo κFo  Fo 1 ξ) ) ×  sinh( ) + cosh( ) sin(η g ω ) J 0 ( 2 2 2  2Ve 2Ve 2Ve  κ M

(43) γf κ i Fo κ i Fo  Fo  1 C fg exp(− + )  sin( ) + cos( )  sin(η g ω ) J 0 ( ξ) 2Ve 2  κ i 2Ve 2 2Ve 2  M F +1 G +1 Where F , G are maximum value of f , g when the κ is real for each loop, respectively. Using boundary condition ∞

∞

∑∑

(15f) and orthogonality condition, we find the following Eq.: 1

∫

− a f sinh(γ f ω ) sin(η g ω )dω C fg =

0

(44)

1

∫ sin

2

(η g ω )dω

0

Finally

ξ1 J 1 ( C fg = −4 M [J 0 (

γf M

γf M 2

ξ1 )γ f 2

)] × [ Bi +

(−1) g

(45)

γ f 2 (η g 2 + γ f 2 ) M

]

7

418

SAEDODIN ET AL:ANALYTICAL AND NUMERICAL SOLUTION...

4. Numerical solution To solve this problem numerically, Eq. (8) should be discretized. The discretization can be done in many ways, using Finite Element Method (FEM) or Control Volume Method (CVM). In this work we adopted an implicit Finite Difference Method (FDM) with non-uniform grids. In implicit methods, the finite difference approximations of the individual exact partial derivatives in the partial differential equation are evaluated at the solution time level n + 1 . The implicit schemes are unconditionally stable for any of time step, but the accuracy of the solution is only firstorder in time. A forward difference representation is used for time derivative and the central difference representation is used for space derivative. Therefore Eq. (8) can be discretized as the follows: θ in, +j 1 − 2θ in, j + θ in, −j 1 θ in, +j 1 − θ in, j θ in++11,j − 2θ in, +j 1 + θ in−+11,j θ in++11,j − θ in−+11,j θ in, +j +11 − 2θ in, +j 1 + θ in, +j −11 + = ( + Ve 2 M ) + (46) ∆Fo 2 ∆Fo ∆ξ i ∆ξ i +1 ξ i , j (∆ξ i + ∆ξ i +1 ) ∆ω j ∆ω j +1

Arranging the Eq. (46) gives: 1 2M 2 Ve 2 M M M M [ ]θ in, +j 1 + [− ]θ in−+11,j + [− ]θ in++11,j + + + + − 2 ∆Fo ∆Fo ∆ξ i ∆ξ i +1 ∆ωi ∆ωi +1 ∆ξ i ∆ξ i +1 ξ i , j (∆ξ i + ∆ξ i +1 ) ∆ξ i ∆ξ i +1 ξ i , j (∆ξ i + ∆ξ i +1 ) +

2Ve 2 1 −1 Ve 2 (θ in, +j +11 + θ in, +j −11 ) = [ ]θ in, j + [ ]θ in, −j 1 + 2 ∆ωi ∆ω i +1 ∆Fo ∆Fo ∆Fo 2

(47)

Where

∆ξ i = ξ i , j − ξ i −1, j ∆ξ i +1 = ξ i +1, j − ξ i , j

(48)

∆ω i = ω i , j − ω i −1, j ∆ω i +1 = ω i +1, j − ω i , j The above system of linear algebraic equations can be written in matrix equation as fallowing:

[ A]{θ }n+1 = [ B]{θ }n + [C ]{θ }n−1

(49)

Where [ A] is five-diagonal matrix, [ B ] and [C ] are just diagonal matrix. At the center, ξ = 0 , we have lim ( ξ →0

Ve 2

∂θ ∂ 2θ by LʼHospitalʼs Rule. Then, Eq (8) takes the form: )= ξ∂ξ ∂ξ 2

∂ 2θ ∂θ ∂ 2θ ∂ 2θ + = 2M + 2 ∂Fo ∂Fo ∂ξ 2 ∂ω 2

(50)

Hence, Eq (50) should be discretized for ξ = 0 . By using inverse method, the dimensionless temperature distribution at each time step can be determined. The numerical solution corresponds to the mesh size of ∆Fo = 0.001 . Also, Computational grids tested were 55× 55 . Detailed flow chart of the numerical solution for the cylinder temperature profile is shown in Fig. 2.

8

SAEDODIN ET AL:ANALYTICAL AND NUMERICAL SOLUTION...

419

Start

Input parameters: Ve, Fo, ξ1 , M

Set initial conditions

Set boundary conditions

Calculate [ A], [ B ],[C ]

Inverse method

Store every θ at any Fo

Fo = Fo + ∆Fo

If Fo equal

No

to Fo final ? Yes

Stop Fig. 2. Flow chart of numerical solution 5. Results and discussion

Using our solution, we performed sample numerical computations of temperature surfaces and profiles in the cylinder for the considered type of the heat source, based on Eqs. (11), (25), (43) and numerical solution. These calculations are obtained for ξ1 = 0.2 , Bi = 2 and M = 16 . The results of calculations are presented in Figs. 3-6. Fig. 3 shows surface temperature profiles for the three cases. The Vernotte number that we simulated for is 0.7. It can be perceived from Fig. 3 that, the numerical solution and the analytical solution are in good agreement with each 9

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other. Also, it can be observed from Fig. 2 that, in the wake of the non-Fourier effect, the heat wave cannot touch the other side of the cylinder at the moment. Furthermore, due to the non-Fourier effects, heat waves can be seen clearly in Fig. 3.

Fo = 0.05, Analytical

Fo = 0.05, Numerical

θ

θ

ω

ω

ξ

ξ

Fo = 0.25, Numerical

Fo = 0.25, Analytical

θ

θ

ω

ω

ξ

ξ

Fo = 0.5, Numerical

Fo = 0.5, Analytical

θ

θ

ω

ω

ξ

Fig. 3. The surface temperature evolution with Ve = 0.7 for non-Fourier model

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Fig. 4 shows temperature profiles along the ω direction at Fo = 0.5 and ξ = 0 . This Fig. shows that, according to the amount of Vernotte number for a specific Fourier number, it is possible that the temperature of different points of object become even lower than initial temperature but the conservation of energy for whole the cylinder remains indefeasible. In addition, it can be seen that due to the non-Fourier effects, the temperature for great number of points in the object remain steady for some moments.

θ

θ

ω

ω

θ

ω Fig. 4. The distribution for the non-Fourier model with the same Fourier number along the ω direction Fig. 5 shows temperature profiles along the ξ direction at Fo = 0.5 and ω = 1 . It can be perceived again from Fig. 5 that, the numerical solution is in good agreement with the analytical solution. So, it can be deduced from Fig. 5 that, our numerical method is accurate and the error of the numerical method is negligible. It can be expressed that the proposed method can be implemented in the non-Fourier heat equations.

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θ

θ

ξ

ξ

θ

ξ

Fig. 5. The distribution for the non-Fourier model with the same Fourier number along the ξ direction Fig. 6 shows surface temperature profiles for the five cases, using numerical method. It can be seen from Fig. 6 that, the higher Vernotte number causes each point to remain more at initial temperature. As it is observed, as much as the Vernotte number increases, the Fourier number that the whole cylinder needs, in order to reach the equilibrium temperature, increases. Regarding Fig. 5, the thermal wave reflection causes the existence of a fracture in the surface temperature profiles of the cylinder. Also, it can be seen that, because of the reflection of heat waves, the temperature of specific points can become lower than initial temperature, especially with Ve = 0.9 . This interesting behavior does not appear under the Fourier heat conduction model.

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Ve = 0.1, Fo = 0.04

θ

ω

ξ

Ve = 0.5, Fo = 0.7

Ve = 0.5, Fo = 0.04

θ

θ

ω

ω

ξ

Ve = 0.9, Fo = 0.7

ξ

Ve = 0.9, Fo = 2.5

θ

θ

ω

ω

ξ

ξ

Fig. 6. The surface temperature evolution with different Fourier and Vernotte number for non-Fourier model

5. Conclusion In this paper, the two-dimensional non-Fourier haet equation was solved analytically and numerically for the case of a cylinder. The separation of variables method has been employed for analytical solution. The implicit FDM with non-uniform grids has been employed for numerical solution. Four different examples have been analyzed. We observed that, the numerical solution and the analytical solution are in good agreement with each other. Also, we 13

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deduced that, the more the Vernotte number, the more the Foureir number passed for the point to sense the thermal wave. We also perceived that, the more the Vernotte number, the more the Fourier number the cylinder needs to reach an equilibrium temperature. Acknowledgment The authors gratefully acknowledge the support of the department of mechanical engineering and the office of gifted of Semnan University for funding the current research grant. References [1] C. Cattaneo, Sur une forme de l’équation de la chaleur éliminant le paradoxe d’une propagation instantanée, C.R. Acad. Sci. 247 (1958) 431–433. [2] P. Vernotte, Les paradoxes de la théorie continue de l’équation de la chaleur, C.R. Acad. Sci. 246 (1958) 3154– 3155. [3] M. Lewandowska, L. Malinowski, An analytical solution of the hyperbolic heat conduction equation for the case of a finite medium symmetrically heated on both sides, Int. Commun. Heat Mass Transf. 33 (2006) 61– 69. [4] A. Moosaie, Non-Fourier heat conduction in a finite medium with arbitrary source term and initial conditions, Forsch. Ing.wes. 71 (2007) 163–169. [5] A.Moosaie, Non-Fourier heat conduction in a finite medium with insulated boundaries and arbitrary initial conditions, Int. Commun. Heat Mass Transf. 35 (2008) 103–111. [6] D.W. Tang, N. Araki, Non-Fourier heat conduction in a finite medium under periodic surface thermal disturbance, Int. J. Heat Mass Transf. 39 (1996) 1585–1590. [7] Duanming Zhang, Li Li, Zhihua Li, Li Guan, Xinyu Tan, Non-Fourier conduction model with thermal source term of ultra short high power pulsed laser ablation and temperature evolvement before melting, Physica B 364 (2005) 285–293. [8] A. Saleh, M. Al-Nimr, Variational formulation of hyperbolic heat conduction problems applying Laplace transform technique, Int. Commun. Heat Mass Transf. 35 (2008) 204–214. [9] A. Barletta, E. Zanchini, Three-dimensional propagation of hyperbolic thermal waves in a solid bar with rectangular cross-section, Int. J. Heat Mass Transf. 42 (1999) 219-229. [10] H.T. Chen, J.Y. Lin, Numerical analysis for hyperbolic heat conduction, Int. J. Heat Mass Transf. 36 (1992) 2891–2898. [11] Q.M. Fan, W. Q. Lu, A new numerical method to simulate the non-Fourier heat conduction in a single-phase medium, Int. J. Heat Mass Transf. 45 (13) (2002) 2815-2821. [12] K. C. Liu, Numerical simulation for non-linear thermal wave, Appl. Math. Comput. 175 (2) (2006) 1385-1399. [13] Tzer-Ming Chen, Numerical solution of hyperbolic heat conduction in thin surface layers, Int. J. Heat Mass Transf. 50 (2007) 4424–4429. [14] Jianhua Zhou, Yuwen Zhang, J. K. Chen, Non-Fourier Heat Conduction Effect on Laser-Induced Thermal Damage in Biological Tissues, Numer. Heat Transf. Part A 54 (2008) 1–19. [15] Ching-yu Yang, Direct and inverse solutions of the two-dimensional hyperbolic heat conduction problems, App. Math. Modell. 33 (2009) 2907–2918. [16] H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd ed., Oxford University Press, New York, 2000. [17] M. N. Ozisik, Heat conduction, 2nd ed., John Wiley &Sons, New York, 1993.

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JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.3, 425-449, 2011, COPYRIGHT 2011 EUDOXUS PRESS, 425 LLC

Convergence Theorem for Mixed Equilibrium Problems and Variational Inequality Problems for Relaxed Cocoercive Mappings Rabian Wangkeeree1 , Narin Petrot1 , Poom Kumam2,∗and Chaichana Jaiboon2,3 1

Department of Mathematics, Faculty of Science, Naresuan University Phitsanulok 65000. Thailand

e-mail: [email protected] (R. Wangkeeree) and [email protected] (N. Petrot) 2

Department of Mathematics, Faculty of Science,

King Mongkut’s University of Technology Thonburi, KMUTT, Bangkok 10140. Thailand. e-mail: [email protected] (C. Jaiboon)and [email protected] (P. Kumam) 3

Department of Mathematics, Faculty of Applied Liberal Arts,

Rajamangala University of Technology Rattanakosin, RMUTR, Bangkok 10100. Thailand. e-mail: [email protected](C. Jaiboon)

Abstract In this paper, we introduce a new iterative scheme for finding the common element of the set of common fixed points of countable family of nonexpansive mappings, the set of solution of a mixed equilibrium problem and the set of solution of the variational inequality problems for a relaxed (u, v)-cocoercive and µ-Lipschitz continuous mapping in Hilbert spaces. We show that the sequence converges strongly to a common element of the above three sets under some parameter controlling conditions. Our results improve and extend the recent ones announced by [Y.J. Cho, X.Q. Qin, M. Kang, Some results for equilibrium problems and fixed point problems in Hilbert spaces, J. Comput. Anal. Appl. 11(2009) 294–316] and many others. 2000 Mathematics Subject Classification: 47H09, 47H10, 47H17 Key words and phrases: Nonexpansive mapping, Relaxed cocoercive mapping, Variational inequality, Fixed points, Mixed equilibrium problem

1

Introduction

Let H be a real Hilbert space with inner product h·, ·i and norm k · k, let C be a nonempty closed convex subset of H and PC is the metric This work is supported by the Thailand National Research Universities Project, KMUTT, CSEC Project No. E01008. ∗ The corresponding author: [email protected] (P. Kumam)

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projection of H onto C. We denote weak convergence and strong convergence by notations * and →, respectively. A mapping S : C → C is called nonexpansive if kSx − Syk ≤ kx − yk for all x, y ∈ C. We denote F (S) = {x ∈ C : Sx = x} be the set of fixed points of S. Recall that a self-mapping f : C → C is contraction if there exists a constant α ∈ [0, 1) such that kf (x) − f (y)k ≤ αkx − yk, ∀x, y ∈ C. Let ϕ : C → R ∪ {+∞} be a proper extended real-valued function and let φ be a bifunction of C × C into R such where R  that C ∩ domϕ 6= ∅, is the set of real numbers and domϕ = x ∈ C : ϕ(x) < +∞ . Ceng and Yao [6] considered the following the mixed equilibrium problem for finding x ∈ C such that φ(x, y) + ϕ(y) − ϕ(x) ≥ 0,

∀y ∈ C.

(1.1)

The set of solutions of (1.1) is denoted by M EP (φ, ϕ), i.e.,  M EP (φ, ϕ) = x ∈ C : φ(x, y) + ϕ(y) − ϕ(x) ≥ 0, ∀y ∈ C . (1.2) of problem (1.1) implies that x ∈ domϕ =  We see that x is a solution x ∈ C : ϕ(x) < +∞ . If ϕ = 0, then the mixed equilibrium problem (1.1) becomes the following the equilibrium problem is to find x ∈ C such that φ(x, y) ≥ 0,

∀y ∈ C.

(1.3)

The set of solutions of (1.3) is denoted by EP (φ). If ϕ = 0 and let φ(x, y) = hAx, y−xi, for all x, y ∈ C, where A : C → H is a nonlinear mapping, then problem (1.1) becomes the following the variational inequality problem is to find x ∈ C such that hAx, y − xi ≥ 0,

∀y ∈ C.

(1.4)

The set of solutions of (1.4) is denoted by V I(C, A). The variational inequality has been extensively studied in the literature. See, e.g. [2, 32, 35] and the references therein. The mixed equilibrium problems include fixed point problems, variational inequality problems, optimization problems, Nash equilibrium problems and the equilibrium problem as special cases. Numerous problems in physics, optimization and economics reduce to find a solution of (1.1). Some methods have been proposed to solve the M EP (φ, ϕ) and EP (φ); see, for instance [3, 6, 9, 10, 14, 11, 12, 13, 20, 33, 34]. In 1997, Combettes and Hirstoaga [3] introduced an iterative scheme of finding the best approximation to initial data when EP (φ) is nonempty and proved a strong convergence theorem. Definition 1.1. Let A : C → H be a nonlinear mapping. Then A is called (1) monotone if hAx − Ay, x − yi ≥ 0,

∀x, y ∈ C,

(2) v-strongly monotone if there exists a positive real number v such that hAx − Ay, x − yi ≥ vkx − yk2 , ∀x, y ∈ C, for constant v > 0. This implies that kAx − Ayk ≥ vkx − yk, that is, A is v-expansive and when v = 1, it is expansive, (3) µ-Lipschitz continuous if there exists a positive real number µ such that kAx − Ayk ≤ µkx − yk, ∀x, y ∈ C, (4) u-cocoercive [29, 30], if there exists a positive real number u such that hAx − Ay, x − yi ≥ ukAx − Ayk2 , ∀x, y ∈ C. Clearly, every u-cocoercive map A is u1 -Lipschitz continuous,

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(5) relaxed u-cocoercive, if there exists a positive real number u such that hAx − Ay, x − yi ≥ (−u)kAx − Ayk2 , ∀x, y ∈ C, (6) relaxed (u, v)-cocoercive, if there exists a positive real number u, v such that hAx−Ay, x−yi ≥ (−u)kAx−Ayk2 +vkx−yk2 , ∀x, y ∈ C, for u = 0, A is v-strongly monotone. This class of maps is more general that the class of strongly monotone maps. It is easy to see that we have the following implication: v-strongly monotonicity implying relaxed (u, v)-cocoercivity. For finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of variational inequalities for u-cocoercive map, Takahashi and Toyoda [28] introduced the following iterative scheme: ( x0 ∈ C chosen arbitrary, (1.5) xn+1 = αn xn + (1 − αn )SPC (xn − λn Axn ), ∀n ≥ 0, where A is a u-cocoercive mapping, {αn } is a sequence in (0, 1), and {λn } is a sequence in (0, 2u). They showed that, if F (S) ∩ V I(C, A) is nonempty, then the sequence {xn } generated by (1.5) converges weakly to some q ∈ F (S) ∩ V I(C, A). On the other hand, Shang et al. [25] introduced a new iterative process for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequalities for relaxed (u, v)-cocoercive maps in a real Hilbert space. Let S : C → C be a nonexpansive mapping. Starting with arbitrary initial x1 ∈ C, define sequences {xn } recursively by xn+1 = αn f (xn ) + βn xn + γn SPC (I − λn A)xn ,

∀n ≥ 1.

(1.6)

They proved that under certain appropriate conditions imposed on {αn }, {βn }, {γn } and {λn }, the sequence {xn } converges strongly to q ∈ F (S) ∩ V I(C, A), where q = PF (S)∩V I(C,A) f (q). Further, Takahashi and Takahashi [27] introduced an iterative scheme by using the viscosity approximation method for finding a common element of the set of the solution (1.3) and the set of fixed points of a nonexpansive mapping in a real Hilbert space. Let S : C → C be a nonexpansive mapping. Starting with an arbitrary initial x1 ∈ C, define sequences {xn } and {un } recursively by ( φ(un , y) + r1n hy − un , un − xn i ≥ 0, ∀y ∈ C, (1.7) xn+1 = αn f (xn ) + (1 − αn )Sun , ∀n ≥ 1. They proved that under certain appropriate conditions imposed on {αn } and {rn }, the sequences {xn } and {un } converge strongly to q ∈ F (S) ∩ EP (φ), where q = PF (S)∩EP (φ) f (q). In 2009, Cho et al. [5] introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of the solution (1.3), the set of common fixed points of a finite family of nonexpansive mapping and the set of solutions of variational inequalities for relaxed cocoercive mapping and proved a strong convergence theorem in a real Hilbert space. In 2008, Ceng and Yao [6] introduced an iterative scheme for finding a common fixed point of a finite family of nonexpansive mappings and the set of solution of problem (1.1) in Hilbert spaces and obtained the strong convergence theorem which used the following condition:

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(E) K : C → R is η-strongly convex with constant σ > 0 and its derivative K 0 is sequentially continuous from the weak topology to the strong topology. We note that the condition (E) for the function K : C → R is a very strong condition. We also note that the condition (E) does not cover 2 the case K(x) = kxk and η(x, y) = x − y for each (x, y) ∈ C × C. In 2009, 2 Peng and Yao [20] introduced a new iterative scheme based on only the extragradient method for finding a common element the set of solution of a (1.1), the set of fixed point of a finite family of nonexpansive mappings and the set of solution of (1.4). They obtained a strong convergence theorem except the condition (E) for the sequences generated by these processes. Moreover, Aoyama et al.[1] introduced an iterative scheme for finding a common fixed point of a countable family of nonexpansive mappings in Banach spaces and obtained the strong convergence theorem for such scheme. In 2008, Plubtieng and Thammathiwat [18] introduced an iterative scheme for finding a common fixed point of a countable family of nonexpansive mappings in Hilbert spaces and obtained the strong convergence theorem for the sequences generated by these processes applying to variational inequilities and equilibrium problems. In this paper, motivated by Cho et al. [5], Ceng and Yao [6], Plubtieng and Thammathiwat [18], Peng and Yao [20], Shang et al. [25] and Takahashi and Takahashi [27], we will introduce an itertive scheme which is mixed the iterative schemes considered in (1.6) and (1.7) for finding a common element of the set of common fixed points of a countable family of nonexpansive mappings, the set of solutions of an mixed equilibrium problem, and the solution set of the classical variational inequality problem for relaxed (u, v)-cocoercive and µ-Lipschitz continuous mappings in a real Hilbert space. Then, the strong convergence theorem is proved under some parameter controlling conditions. The results obtained in this paper improve and extend the recent ones announced by Cho et al. [5], Peng and Yao [20], Shang et al. [25], Takahashi and Takahashi [27] and many others.

2

Preliminaries

Let H be a real Hilbert space and C be a nonempty closed convex subset of H. In a real Hilbert space H, it is well known that kλx + (1 − λ)yk2 = λkxk2 + (1 − λ)kyk2 − λ(1 − λ)kx − yk2 , for all x, y ∈ H and λ ∈ [0, 1]. For every point x ∈ H, there exists a unique nearest point in C, denoted by PC x, such that kx − PC xk ≤ kx − yk for all y ∈ C. PC is called the metric projection of H onto C. It is well known that PC is a nonexpansive mapping of H onto C and satisfies hx − y, PC x − PC yi ≥ kPC x − PC yk2

(2.8)

for every x, y ∈ H. Moreover, PC x is characterized by the following properties: PC x ∈ C and hx − PC x, y − PC xi ≤ 0,

(2.9)

kx − yk2 ≥ kx − PC xk2 + ky − PC xk2 , for all x ∈ H, y ∈ C. For more details see [26].

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(2.10)

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A set-valued mapping T : H → 2H is called monotone if for all x, y ∈ H, f ∈ T x and g ∈ T y imply hx − y, f − gi ≥ 0. A monotone mapping T : H → 2H is maximal if the graph of G(T ) of T is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping T is maximal if and only if for (x, f ) ∈ H × H, hx − y, f − gi ≥ 0 for every (y, g) ∈ G(T ) implies f ∈ T x. Let A be an ρ-inverse-strongly monotone mapping of C into H and let NC ϑ be the normal cone to C at ϕ ∈ C, i.e., NC ϑ = {w ∈ H : hϑ − ξ, wi ≥ 0, ∀ξ ∈ C} and define  Aϑ + NC ϑ, ϑ ∈ C; Tϑ = ∅, ϑ∈ / C. Then T is the maximal monotone and 0 ∈ T ϑ if and only if ϑ ∈ V I(C, A); see [21]. The following lemmas will be useful for proving the convergence result of this paper. Lemma 2.1. [4] Let H be Hilbert space, let C be a nonempty closed convex subset of H and let A be a mapping of C into H. Let p ∈ C. Then for λ > 0, p ∈ V I(A, C) ⇐⇒ p = PC (p − λAp), where PC is the metric projection of H onto C. It is clear from Lemma 2.1 that the variational inequality and fixed point problem are equivalent. This alternative equivalent formulation has played a significant role in the studies of the variational inequalities and related optimization problems. Lemma 2.2. [16]. Each Hilbert space H satisfies Opial’s condition, i.e., for any sequence {xn } ⊂ H with xn * x, the inequality lim inf kxn − xk < lim inf kxn − yk, n→∞

n→∞

hold for each y ∈ H with y 6= x. Lemma 2.3. [17] Let (C, h., .i) be an inner product space. Then, for all x, y, z ∈ C and α, β, γ ∈ [0, 1] with α + β + γ = 1, we have kαx+βy+γzk2 = αkxk2 +βkyk2 +γkzk2 −αβkx−yk2 −αγkx−zk2 −βγky−zk2 . Lemma 2.4. [23] Let {xn } and {zn } be bounded sequences in a Banach space X and let {βn } be a sequence in [0, 1] with 0 < lim inf n→∞ βn ≤ lim supn→∞ βn < 1. Suppose xn+1 = (1−βn )zn +βn xn for all integers n ≥ 1 and lim supn→∞ (kzn+1 − zn k − kxn+1 − xn k) ≤ 0. Then, limn→∞ kzn − xn k = 0. Lemma 2.5. [31]. Assume {an } is a sequence of nonnegative real numbers such that an+1 ≤ (1 − ln )an + %n , n ≥ 1, where {ln } is a sequence in (0, 1) and {%n } is a sequence in R such that P∞ (1) n=1 ln = ∞ P (2) lim supn→∞ %lnn ≤ 0 or ∞ n=1 |%n | < ∞. Then limn→∞ an = 0. Lemma 2.6. [18] Let C be a nonempty bounded closed convex subset of a Hilbert space H and let {Tn } be a sequence of mappings of C into itself. Suppose that lim ρkl = 0,

(2.11)

k,l→∞

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where ρkl = sup{kTk z − Tl zk : z ∈ C} < ∞, for all k, l ∈ N. Then for each x ∈ C, {Tn x} converges strongly to some point of C. Moreover, let T be a mapping of C into itself defined by T x = lim Tn x

for all x ∈ C.

n→∞

Then limn→∞ sup{kT z − Tn zk : z ∈ C} = 0. From Lemma 2.6, it easy to see that T is nonexpansive. For solving the mixed equilibrium problem, let us give the following assumptions for the bifunction F, ϕ and the set C : (A1) φ(x, x) = 0 for all x ∈ C; (A2) φ is monotone, i.e., φ(x, y) + φ(y, x) ≤ 0 for all x, y ∈ C; (A3) for each x, y, z ∈ C, limt→0 φ(tz + (1 − t)x, y) ≤ φ(x, y); (A4) for each x ∈ C, y 7→ φ(x, y) is convex and lower semicontinuous; (A5) for each y ∈ C, x 7→ φ(x, y) is weakly upper semicontinuous; (B1) for each x ∈ H and r > 0, there exist abounded subset Dx ⊆ C and yx ∈ C such that for any z ∈ C \ Dx , φ(z, yx ) + ϕ(yx ) +

1 hyx − z, z − xi < ϕ(z); r

(B2) C is a bounded set. By similar argument as in the proof of Lemma 2.7 in [19], we have the following lemma appearing. Lemma 2.7. Let C be a nonempty closed convex subset of H. Let φ : C × C → R be a bifunction satisfies (A1)-(A5) and let ϕ : C → R ∪ {+∞} be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For r > 0 and x ∈ H, define a mapping Tr : H → C as follows: n 1 Tr (x) = z ∈ C : φ(z, y) + ϕ(y) + hy − z, z − xi ≥ ϕ(z), r

o ∀y ∈ C

for all z ∈ H. Then, the following hold: 1. For each x ∈ H, Tr (x) 6= ∅; 2. Tr is single-valued; 3. Tr is firmly nonexpansive, i.e., kTr x − Tr yk2 ≤ hTr x − Tr y, x − yi,

∀x, y ∈ H;

4. F (Tr ) = M EP (φ, ϕ); 5. M EP (φ, ϕ) is closed and convex.

3

The results

In this section, we prove a strong convergence theorem which is our main result.

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431

Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H and let φ be a bifunction from C × C to R satisfying (A1)-(A4) and let ϕ : E → R ∪ {+∞} be a proper lower semicontinuous and convex function. Let A : C → H be relaxed (u, v) − cocoercive and µ-Lipschitz continuous mapping, let f : C → C be a contraction with coefficient α (0 < α < 1) and {Sn } be a sequence of nonexpansive mappings of C into itself such that satisfies condition (2.11) and Θ := ∩∞ n=1 F (Sn ) ∩ V I(C, A) ∩ M EP (φ, ϕ) 6= ∅. Assume that either (B1) or (B2). Let {xn }, {yn } and {un } be sequences generated by  x1 = x ∈ C,     φ(u , y) + ϕ(y) − ϕ(u ) + 1 hy − u , u − x i ≥ 0, ∀y ∈ C, n n n n n rn  y = δ u + (1 − δ )P (u − λ Au ), n n n n C n n n    xn+1 = αn f (xn ) + βn xn + γn Sn PC (yn − λn Ayn ), ∀n ≥ 1, (3.12) where {αn }, {βn }, {γn } and {δn } are sequences in (0, 1) and {rn } is a sequence in (0, ∞). If the following conditions are satisfied: (C1) αn + βn + γn = 1 and 0 < δn ≤ e < 1, P (C2) limn→∞ αn = 0 and ∞ n=1 αn = ∞, (C3) 0 < lim inf n→∞ βn ≤ lim supn→∞ βn < 1, (C4) lim inf n→∞ rn > 0 and limn→∞ |rn+1 − rn | = 0, (C5) limn→∞ |λn+1 − λn | = limn→∞ |δn+1 − δn | = 0, (C6) {λn } ∈ [a, b] for some a, b with 0 ≤ a ≤ b ≤

2(v−uµ2 ) , µ2

v > uµ2 .

Let S be a mapping of C into itself defined by Sx = limn→∞ Sn x for all x ∈ C and suppose that F (S) = ∩∞ n=1 F (Sn ). Then, {xn } converges strongly to the same point q ∈ Θ := ∩∞ n=1 F (Sn ) ∩ V I(C, A) ∩ M EP (φ, ϕ), where q = PΘ f (q). Proof. Let Q = P∩∞ . Since f is a contraction n=1 F (Sn )∩V I(C,A)∩M EP (φ,ϕ) with α ∈ (0, 1), we obtain kQf (x) − Qf (y)k ≤ kf (x) − f (y)k ≤ αkx − yk, ∀x, y ∈ C. Therefore, Qf is a contraction of C into itself, which implies that there exists a unique element q ∈ C such that q = Qf (q). For all x, y ∈ C and A : C → H be relaxed (u, v)-cocoercive and µ-Lipschitz continuous mappings, we note that k(I − λn A)x − (I − λn A)yk2 =

k(x − y) − λn (Ax − Ay)k2

= ≤

kx − yk2 − 2λn hx − y, Ax − Ayi + λ2n kAx − Ayk2 n o kx − yk2 − 2λn −ukAx − Ayk2 + vkx − yk2 + λ2n kAx − Ayk2

≤

kx − yk2 + 2λn uµ2 kx − yk2 − 2λn vkx − yk2 + λ2n µ2 kx − yk2

=

(1 + 2λn uµ2 − 2λn v + λ2n µ2 )kx − yk2

≤

kx − yk2 ,

7

432

WANGKEEREE ET AL: MIXED EQUILIBRIUM PROBLEMS

which implies that I − λn A is nonexpansive. Then we divide the proof into five steps. Step 1. We claim that {xn }, {yn } and {un } are bounded. Indeed, let p ∈ ∩∞ n=1 F (Sn ) ∩ V I(C, A) ∩ M EP (φ, ϕ) and let {Trn } be the sequence of mappings defined as in Lemma 2.7. Then p = PC (p − λn Ap) = Trn p. From un = Trn xn ∈ C, we have kun − pk

=

kTrn xn − Trn pk ≤ kxn − pk.

(3.13)

Putting vn = PC (yn − λn Ayn ), we obtain kvn − pk

=

kPC (yn − λn Ayn ) − PC (p − λn Ap)k

≤

k(yn − λn Ayn ) − (p − λn Ap)k

=

k(I − λn A)yn − (I − λn A)p)k ≤ kyn − pk

and kyn − pk

≤

δn kun − pk + (1 − δn )kPE (un − λn Aun ) − pk

≤

δn kun − pk + (1 − δn )kun − p)k = kun − pk ≤ kxn − pk,

and hence kvn − pk ≤ kyn − pk ≤ kun − pk ≤ kxn − pk

(3.14)

which yields that kxn+1 − pk ≤

αn kf (xn ) − pk + βn kxn − pk + γn kSn vn − pk

≤

αn kf (xn ) − pk + βn kxn − pk + γn kvn − pk

≤

αn kf (xn ) − pk + βn kxn − pk + γn kxn − pk

≤

αn kf (xn ) − f (p)k + αn kf (p) − pk + βn kxn − pk + γn kxn − pk

≤

αn αkxn − pk + (1 − αn )kxn − pk + αn kf (p) − pk|

≤

(1 − αn (1 − α))kxn − pk + αn kf (p) − pk kf (p) − pk (1 − αn (1 − α))kxn − pk + αn (1 − α) . (1 − α)

=

It follows from induction that   kf (p) − pk kxn − pk ≤ max kx1 − pk, , 1−α

∀n ≥ 1.

(3.15)

Therefore, {xn } is bounded. Hence, so are {vn }, {yn } and {un }. Step 2. We claim that limn→∞ kxn+1 − xn k = 0. Let vn = PC (yn − λn Ayn ) and ψn = PC (un − λn Aun ). Thus, we compute kvn+1 − vn k =

kPC (yn+1 − λn+1 Ayn+1 ) − PC (yn − λn Ayn )k

≤

k(yn+1 − λn+1 Ayn+1 ) − (yn − λn Ayn )k

=

k(yn+1 − λn+1 Ayn+1 ) − (yn − λn+1 Ayn ) + (λn − λn+1 )Ayn k

≤

k(yn+1 − λn+1 Ayn+1 ) − (yn − λn+1 Ayn )k + |λn − λn+1 |kAyn k

=

k(I − λn+1 A)yn+1 − (I − λn+1 A)yn k + |λn − λn+1 |kAyn k

≤

kyn+1 − yn k + |λn − λn+1 |kAyn k.

8

(3.16)

WANGKEEREE ET AL: MIXED EQUILIBRIUM PROBLEMS

433

Similarly, we have kψn+1 − ψn k

=

kPC (un+1 − λn+1 Aun+1 ) − PC (un − λn Aun )k

≤

kun+1 − un k + |λn − λn+1 |kAun k.

(3.17)

Observing that ( yn = δn un + (1 − δn )ψn yn+1 = δn+1 un+1 + (1 − δn+1 )ψn+1 , we obtain yn − yn+1 = δn (un − un+1 ) + (1 − δn )(ψn − ψn+1 ) + (ψn+1 − un+1 )(δn+1 − δn ), which yields that kyn − yn+1 k ≤

(3.18)

δn kun − un+1 k + (1 − δn )kψn − ψn+1 k + kψn+1 − un+1 k|δn+1 − δn |.

Substitution of (3.17) into (3.18) yields that kyn − yn+1 k ≤

n o δn kun − un+1 k + (1 − δn ) kun+1 − un k + |λn − λn+1 |kAun k + kψn+1 − un+1 k|δn+1 − δn |

=

kun − un+1 k + (1 − δn )|λn − λn+1 |kAun k + kψn+1 − un+1 k|δn+1 − δn |

≤

kun − un+1 k + M1 (|λn − λn+1 | + |δn − δn+1 |),

(3.19)

where M1 is an appropriate constant such that n o M1 = max sup kAun k, sup kψn − un k . n≥1

n≥1

On the other hand, from un = Trn xn ∈ dom ϕ and un+1 = Trn+1 xn+1 ∈ dom ϕ, we note that φ(un , y) + ϕ(y) − ϕ(un ) +

1 hy − un , un − xn i ≥ 0, rn

∀y ∈ C (3.20)

and φ(un+1 , y) + ϕ(y) − ϕ(un+1 ) +

1 hy − un+1 , un+1 − xn+1 i ≥ 0 rn+1

∀y ∈ C. (3.21)

Putting y = un+1 in (3.20) and y = un in (3.21), we obtain φ(un , un+1 ) + ϕ(un+1 ) − ϕ(un ) +

1 hun+1 − un , un − xn i ≥ 0 rn

and φ(un+1 , un ) + ϕ(un ) − ϕ(un+1 ) +

1 hun − un+1 , un+1 − xn+1 i ≥ 0. rn+1

From (A2) we have * un+1 − un ,

+ un − xn un+1 − xn+1 − rn rn+1

9

≥0

434

WANGKEEREE ET AL: MIXED EQUILIBRIUM PROBLEMS

and hence *

+

un+1 − un , un − un+1 + un+1 − xn −

rn rn+1

(un+1 − xn+1 )

≥ 0.

Without loss of generality, let us assume that there exists a real number c such that rn > c > 0 for all n ∈ N. Then, we have * +  rn  2 kun+1 − un k ≤ un+1 − un , xn+1 − xn + 1 − (un+1 − xn+1 ) rn+1 ( ) rn ≤ kun+1 − un k kxn+1 − xn k + 1 − kun+1 − xn+1 k rn+1 and hence kun+1 − un k

≤ ≤

1 |rn+1 − rn |kun+1 − xn+1 k rn+1 M2 kxn+1 − xn k + |rn+1 − rn |, (3.22) c

kxn+1 − xn k +

where M2 = sup{kun − xn k : n ∈ N}. It follows from (3.19) and the last inequality that kyn − yn+1 k

≤

(3.23)
M2 kxn+1 − xn k + |rn+1 − rn | + M1 |λn − λn+1 | + |δn − δn+1 | . c

Substitution of (3.23) into (3.16) yields that kvn+1 − vn k

≤

kyn+1 − yn k + |λn − λn+1 |kAyn k

≤

kxn+1 − xn k + M |rn+1 − rn | + 2|λn − λn+1 |
+|δn − δn+1 | , (3.24)

where M is an appropriate constant such that n M2 o M = max sup kAyn k, M1 , . c n≥1 x

−β x

)+γn Sn vn n n Setting zn = n+1 = αn f (xn1−β , we obtain xn+1 = (1 − 1−βn n βn )zn + βn xn for all n ∈ N. Thus, we have

kzn+1 − zn k

= =

≤

≤

αn f (xn ) + γn Sn vn

αn+1 f (xn+1 ) + γn+1 Sn+1 vn+1 −

1 − βn+1 1 − βn


γn+1

αn+1 f (xn+1 ) − f (xn ) + (Sn+1 vn+1 − Sn vn )

1 − βn+1 1 − βn+1

 α   αn γn+1 γn 

n+1 + − f (xn ) + − Sn v 1 − βn+1 1 − βn 1 − βn+1 1 − βn αn+1 γn+1 kf (xn+1 ) − f (xn )k + kSn+1 vn+1 − Sn vn k 1 − βn+1 1 − βn+1 αn αn+1 + − kf (xn ) − Sn vn k 1 − βn+1 1 − βn γn+1 ααn+1 kxn+1 − xn k + kSn+1 vn+1 − Sn vn k 1 − βn+1 1 − βn+1 αn αn+1 + − (3.25) kf (xn ) − Sn vn k. 1 − βn+1 1 − βn

10

WANGKEEREE ET AL: MIXED EQUILIBRIUM PROBLEMS

435

It follows from (3.24), that kSn+1 vn+1 − Sn vn k

≤

kSn+1 vn+1 − Sn+1 vn k + kSn+1 vn − Sn vn k

≤

kvn+1 − vn k + kSn+1 vn − Sn vn k

≤

kxn+1 − xn k + M |rn+1 − rn | + 2|λn − λn+1 | + |δn − δn+1 | +kSn+1 vn − Sn vn k.




(3.26)

Combining (3.25) and (3.26), we have

≤

≤

kzn+1 − zn k ααn+1 γn+1 n kxn+1 − xn k + kxn+1 − xn k + M |rn+1 − rn | + 2|λn − λn+1 | 1 − βn+1 1 − βn+1 o α
αn n+1 +|δn − δn+1 | + kSn+1 vn − Sn vn k + − kf (xn ) − Sn vn k 1 − βn+1 1 − βn
kxn+1 − xn k + K |rn+1 − rn | + 2|λn − λn+1 | + |δn − δn+1 | γn+1 αn αn+1 + kSn+1 vn − Sn vn k + − kf (xn ) − Sn vn k. 1 − βn+1 1 − βn+1 1 − βn

It follows that kzn+1 − zn k − kxn+1 − xn k

≤

≤


γn+1 K |rn+1 − rn | + 2|λn − λn+1 | + |δn − δn+1 | + kSn+1 vn − Sn vn k 1 − βn+1 αn αn+1 + − kf (xn ) − Sn vn k 1 − βn+1 1 − βn
K |rn+1 − rn | + 2|λn − λn+1 | + |δn − δn+1 | n o γn+1 + sup kSn+1 z − Sn zk : z ∈ {vn } 1 − βn+1 αn αn+1 + − kf (xn ) − Sn vn k, 1 − βn+1 1 − βn γ

n+1 where K = M ( 1−β ) is a constant. This together with (C1)-(C6) and n+1 (2.11), imply that

lim sup(kzn+1 − zn k − kxn+1 − xn k) ≤ 0. n→∞

Hence, by Lemma 2.4, we obtain kzn − xn k → 0 as n → ∞. It follows that lim kxn+1 − xn k = lim (1 − βn )kzn − xn k = 0.

n→∞

n→∞

(3.27)

So, we also get lim kun+1 − un k = lim kyn+1 − yn k = lim kvn+1 − vn k = 0.

n→∞

n→∞

n→∞

Step 3. We claim that limn→∞ kSvn − vn k = 0. Indeed, pick any p ∈ ∩∞ n=1 F (Sn ) ∩ V I(C, A) ∩ M EP (φ, ϕ), to obtain kun − pk2

=

kTrn xn − Trn pk2

≤

hTrn xn − Trn p, xn − pi = hun − p, xn − pi  1 kun − pk2 + kxn − pk2 − kxn − un k2 . 2

=

11

436

WANGKEEREE ET AL: MIXED EQUILIBRIUM PROBLEMS

Therefore, kun − pk2 ≤ kxn − pk2 − kxn − un k2 . From Lemma 2.3, (3.14) and (3.12), we obtain kxn+1 − pk2 ≤

αn kf (xn ) − pk2 + βn kxn − pk2 + γn kvn − pk2

≤ ≤

αn kf (xn ) − pk2 + βn kxn − pk2 + γn kun − pk2 n o αn kf (xn ) − pk2 + βn kxn − pk2 + γn kxn − pk2 − kxn − un k2

=

αn kf (xn ) − pk2 + (1 − αn )kxn − pk2 − γn kxn − un k2

and hence γn kxn − un k2

≤

αn kf (xn ) − pk2 + kxn − pk2 − kxn+1 − pk2

≤

αn kf (xn ) − pk2 + kxn − xn+1 k(kxn − pk + kxn+1 − pk).

It is easy to see that lim inf n→∞ γn > 0, (C2) and (3.27), we arrive that lim kxn − un k = 0.

(3.28)

n→∞

Since lim inf n→∞ rn > 0, we obtain lim k

n→∞

xn − un 1 k = lim kxn − un k = 0. n→∞ rn rn

(3.29)

For p ∈ ∩∞ n=1 F (Sn ) ∩ V I(C, A) ∩ M EP (φ, ϕ), we have kψn − pk2 =

kPC (un − λn Aun ) − PC (p − λn Ap)k2

≤

k(un − λn Aun ) − (p − λn Ap)k2

=

k(un − p) − λn (Aun − Ap)k2

=

kun − pk2 − 2λn hun − p, Aun − Api + λ2n kAun − Apk2 n o kun − pk2 − 2λn −ukAun − Apk2 + vkun − pk2 + λ2n kAun − Apk2

≤ = ≤ = ≤

kun − pk2 + 2λn ukAun − Apk2 − 2λn vkun − pk2 + λ2n kAun − Apk2 2λn v kun − pk2 + 2λn ukAun − Apk2 − kAun − Apk2 + λ2n kAun − Apk2 µ2   2λn v kun − pk2 + 2λn u + λ2n − kAun − Apk2 µ2   2λn v kxn − pk2 + 2λn u + λ2n − kAun − Apk2 . (3.30) µ2

Similarly, we have   2λn v kAyn − Apk2 . (3.31) kvn − pk2 ≤ kxn − pk2 + 2λn u + λ2n − µ2 It follows that kyn − pk2 ≤

δn kxn − pk2 + (1 − δn )kψn − pk2 ( 2

 2

λ2n

≤

δn kxn − pk + (1 − δn ) kxn − pk + 2λn u +

≤

  2λn v kAun − Apk2 . kxn − pk2 + 2λn u + λ2n − µ2

12

)  2λn v 2 − kAun − Apk µ2 (3.32)

WANGKEEREE ET AL: MIXED EQUILIBRIUM PROBLEMS

On the other hand, we have kxn+1 − pk2 ≤

αn kf (xn ) − pk2 + βn kxn − pk2 + γn kSn vn − pk2

≤

αn kf (xn ) − f (p)k2 + αn kf (p) − pk2 + 2αn kf (xn ) − f (p)kkf (p) − pk + βn kxn − pk2 + γn kvn − pk2

≤

αn kf (xn ) − f (p)k2 + αn kf (p) − pk2 + 2αn kf (xn ) − f (p)kkf (p) − pk + βn kxn − pk2 + γn kyn − pk2

≤

αn kf (xn ) − f (p)k2 + αn kf (p) − pk2 + 2αn kf (xn ) − f (p)kkf (p) − pk    2λn v  2 + βn kxn − pk2 + γn kxn − pk2 + 2λn u + λ2n − kAu − Apk n µ2

≤

(1 − αn )kxn − pk2 + ααn kxn − pk2 + αn kf (p) − pk2  2λn v  +2ααn kxn − pkkf (p) − pk + γn 2λn u + λ2n − kAun − Apk2 µ2

≤

kxn − pk2 + ααn kxn − pk2 + αn kf (p) − pk2  2λn v  +2ααn kxn − pkkf (p) − pk + 2λn u + λ2n − γn kAun − Apk2 . µ2

It follows from condition (C6) that  2av  − 2bu − b2 γn kAun − Apk2 2 µ  2λ v  n ≤ − 2λn u − λ2n γn kAun − Apk2 2 µ ≤

kxn − pk2 − kxn+1 − pk2 + ααn kxn − pk2 + αn kf (p) − pk2 +2ααn kxn − pkkf (p) − pk

≤


kxn − xn+1 k kxn − pk + kxn+1 − pk + ααn kxn − pk2 + αn kf (p) − pk2 + 2ααn kxn − pkkf (p) − pk.

Since limn→∞ αn = 0, lim inf n→∞ γn > 0 and (3.27), we otain lim kAun − Apk = 0.

(3.33)

n→∞

In a similar way, we can prove lim kAyn − Apk = 0.

(3.34)

n→∞

Observe that kvn − pk2 = ≤ =

≤ ≤

kPC (yn − λn Ayn ) − PC (p − λn Ap)k2

 (I − λn A)yn − (I − λn A)p, vn − p 1n k(I − λn A)un − (I − λn A)pk2 + kvn − pk2 2 o

−k(I − λn A)yn − (I − λn A)p − (vn − p)k2 o 1n kyn − pk2 + kvn − pk2 − k(yn − vn ) − λn (Ayn − Ap)k2 2 1n kxn − pk2 + kvn − pk2 − kyn − vn k2 − λ2n kAyn − Apk2 2 o + 2λn hyn − vn , Ayn − Api

13

437

438

WANGKEEREE ET AL: MIXED EQUILIBRIUM PROBLEMS

and hence kvn − pk2 ≤ kxn − pk2 − kyn − vn k2 + 2λn kyn − vn kkAyn − Apk. (3.35) Similar, we can prove kψn − pk2 ≤ kxn − pk2 − kun − ψn k2 + 2δn kun − ψn kkAun − Apk. (3.36) Therefore, from (3.33) and (3.35), we also have kxn+1 − pk2 ≤

αn kf (xn ) − f (p)k2 + αn kf (p) − pk2 + 2αn kf (xn ) − f (p)kkf (p) − pk + βn kxn − pk2 + γn kvn − pk2

≤

αn kf (xn ) − f (p)k2 + αn kf (p) − pk2 + 2αn kf (xn ) − f (p)kkf (p) − pk n o + βn kxn − pk2 + γn kxn − pk2 − kyn − vn k2 + 2λn kyn − vn kkAyn − Apk

≤

(1 − αn )kxn − pk2 + ααn kxn − pk2 + αn kf (p) − pk2 +2ααn kxn − pkkf (p) − pk − γn kyn − vn k2 + 2λn γn kyn − vn kkAyn − Apk

≤

kxn − pk2 + ααn kxn − pk2 + αn kf (p) − pk2 + 2ααn kxn − pkkf (p) − pk − γn kyn − vn k2 + 2λn γn kyn − vn kkAyn − Apk.

It follows that γn kyn − vn k2 ≤

kxn − pk2 − kxn+1 − pk2 + ααn kxn − pk2 + αn kf (p) − pk2 + 2ααn kxn − pkkf (p) − pk + 2λn γn kyn − vn kkAyn − Apk

≤

kxn − xn+1 k(kxn − pk − kxn+1 − pk) + ααn kxn − pk2 + αn kf (p) − pk2 + 2ααn kxn − pkkf (p) − pk + 2λn γn kyn − vn kkAyn − Apk.

Since αn → 0, lim inf n→∞ γn > 0, kxn − xn+1 k → 0 and kAyn − Apk → 0 as n → ∞, we have lim kyn − vn k = 0.

(3.37)

n→∞

Using (3.33) again, we have kxn+1 − pk2 ≤

αn kf (xn ) − f (p)k2 + αn kf (p) − pk2 + 2αn kf (xn ) − f (p)kkf (p) − pk + βn kxn − pk2 + γn kvn − pk2

≤

αn kf (xn ) − f (p)k2 + αn kf (p) − pk2 + 2αn kf (xn ) − f (p)kkf (p) − pk + βn kxn − pk2 + γn kyn − pk2

≤

αn kf (xn ) − f (p)k2 + αn kf (p) − pk2 + 2αn kf (xn ) − f (p)kkf (p) − pk h i + βn kxn − pk2 + γn δn kxn − pk2 + (1 − δn )kψn − pk2 . (3.38)

14

WANGKEEREE ET AL: MIXED EQUILIBRIUM PROBLEMS

Substituting (3.36) into (3.38) yields that kxn+1 − pk2 ≤

αn kf (xn ) − f (p)k2 + αn kf (p) − pk2 + 2αn kf (xn ) − f (p)kkf (p) − pk " n 2 + βn kxn − pk + γn δn kxn − pk2 + (1 − δn ) kxn − pk2 − kun − ψn k2 o + 2δn kun − ψn kkAun − Apk

#

=

αn kf (xn ) − f (p)k2 + αn kf (p) − pk2 + 2αn kf (xn ) − f (p)kkf (p) − pk h + βn kxn − pk2 + γn kxn − pk2 − (1 − δn )kun − ψn k2 i +2δn (1 − δn )kun − ψn kkAun − Apk

=

αn kf (xn ) − f (p)k2 + αn kf (p) − pk2 + 2αn kf (xn ) − f (p)kkf (p) − pk + βn kxn − pk2 + γn kxn − pk2 − (1 − δn )γn kun − ψn k2 +2δn γn (1 − δn )kun − ψn kkAun − Apk

≤

(1 − αn )kxn − pk2 + ααn kxn − pk2 + αn kf (p) − pk2 +2ααn kxn − pkkf (p) − pk − (1 − δn )γn kun − ψn k2 +2δn γn (1 − δn )kun − ψn kkAun − Apk

≤

kxn − pk2 + ααn kxn − pk2 + αn kf (p) − pk2 + 2ααn kxn − pkkf (p) − pk − (1 − δn )γn kun − ψn k2 + 2δn γn (1 − δn )kun − ψn kkAun − Apk.

It follows from 0 < δn ≤ e < 1 that (1 − e)γn kun − ψn k2 ≤

(1 − δn )γn kun − ψn k2

≤

kxn − pk2 − kxn+1 − pk2 + ααn kxn − pk2 + αn kf (p) − pk2 + 2ααn kxn − pkkf (p) − pk + 2δn γn (1 − δn )kun − ψn kkAun − Apk

≤

kxn − xn+1 k(kxn − pk − kxn+1 − pk) + ααn kxn − pk2 + αn kf (p) − pk2 + 2ααn kxn − pkkf (p) − pk + 2δn γn (1 − δn )kun − ψn kkAun − Apk

Observing that (3.27), (3.33) and αn → 0 as n → ∞, we have lim kun − ψn k = 0.

(3.39)

n→∞

Noting that yn − ψn

=

δn (un − ψn ).

By (3.39), we obtain limn→∞ kyn − ψn k = 0.

(3.40)

Furthermore, by the triangular inequality we also have kxn − yn k ≤ kxn − un k + kun − ψn k + kψn − yn k, from (3.28), (3.39) and (3.40), we have limn→∞ kxn − yn k = 0.

(3.41)

From (3.37) and (3.41), we have limn→∞ kxn − vn k = 0.

(3.42)

15

439

440

WANGKEEREE ET AL: MIXED EQUILIBRIUM PROBLEMS

Since xn+1 = αn f (xn ) + βn xn + γn Sn vn and αn + βn + γn = 1, we obtain
xn+1 − xn = αn f (xn ) − xn + γn (Sn vn − xn ). By condition (C2), lim inf n→∞ γn > 0 and (3.27), we have limn→∞ γn kSn vn − xn k =

limn→∞ (kxn+1 − xn k − αn kf (xn ) − xn k) = 0.

(3.43)

We note that kSn vn − vn k

≤

kSn vn − xn k + kxn − yn k + kyn − vn k.

(3.44)

From (3.37), (3.41) and (3.43), we obtain limn→∞ kSn vn − vn k = 0.

(3.45)

Applying Lemma 2.6 and (3.45), we have kSvn − vn k

≤ ≤

kSvn − Sn vn k + kSn vn − vn k n o sup kSz − Sn zk : z ∈ {vn } + kSn vn − vn k → 0.

It follows from the last inequality and (3.42) that kxn − Svn k ≤ kxn − vn k + kvn − Svn k → 0 as n → ∞. Step 4. We claim that lim supn→∞ hf (q) − q, xn − qi ≤ 0. Indeed, we choose a subsequence {vni } of {vn } such that lim suphf (q) − q, Svn − qi = lim hf (q) − q, Svni − qi. i→∞

n→∞

Without loss of generality, we may assume that {vni } converges weakly to z ∈ C. From kSvn − vn k → 0, we obtain Svni * z. Now, we will show that z ∈ Θ := ∩∞ n=1 F (Sn )∩V I(C, A)∩M EP (φ, ϕ). Firstly, we will show z ∈ F (S) = ∩∞ n=1 F (Sn ). Assume z ∈ / F (S). Since uni * z and z 6= Sz, it follows by the Opial’s condition (Lemma 2.2) that lim inf kuni − zk i→∞

< ≤

lim inf kuni − Szk i→∞  lim inf kuni − Suni k + kSuni − Szk

≤

lim inf kuni − zk,

i→∞

i→∞

which derives a contradiction. Thus, we have z ∈ F (S) = ∩∞ n=1 F (Sn ). Next we will show that z ∈ V I(C, A). Let  Aw1 + NC w1 , w1 ∈ C; T w1 = ∅, w1 ∈ / C. Since A is relaxed (u, v)-cocoercive and condition (C6), we have hAx−Ay, x−yi ≥ (−u)kAx−Ayk2 +vkx−yk2 ≥ (v −uµ2 )kx−yk2 ≥ 0, which yields that A is monotone. Then T is maximal monotone (see [21]). Let (w1 , w2 ) ∈ G(T ). Since w2 − Aw1 ∈ NC (w1 ) and vn ∈ C, we have hw1 − vn , w2 − Aw1 i ≥ 0. On the other hand, from vn = PC (yn − λn Ayn ), we have

 w1 − vn , vn − (yn − λn Ayn ) ≥ 0

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WANGKEEREE ET AL: MIXED EQUILIBRIUM PROBLEMS

and hence, *

+

vn − yn w1 − v n , + Ayn λn

≥ 0.

Therefore, we obtain hw1 − vni , w2 i ≥

hw1 − vni , Aw1 i

≥

hw1 − vni , Aw1 i −

+

*

* = =

≥ =

vni − yni + Ayni λni + vni − yni − λni

w1 − vni ,

w1 − vni , Aw1 − Ayni

hw1 − vni , Aw1 − Avni i + hw1 − vni , Avni * + vni − yni − Ayni i − w1 − vni , λni * + vni − yni + Ayni hw1 − vni , Avni i − w1 − vni , λni * + vn − yni hw1 − vni , Avni − Ayni i − w1 − vni , i λni

(3.46)

which together with kvn − yn k → 0 and A is relaxed (u, v)-cocoercive implies that hw1 − z, w2 i ≥ 0. Since T is maximal monotone, we have z ∈ T −1 0 and hence z ∈ V I(C, A). Finally, we will show z ∈ M EP (φ, ϕ). Indeed, we observe that un = Trn xn ∈ dom ϕ, we have φ(un , y) + ϕ(y) − ϕ(un ) +

1 hy − un , un − xn i ≥ 0, ∀y ∈ C. rn

From (A2), we also have ϕ(y) − ϕ(un ) +

1 hy − un , un − xn i ≥ −φ(un , y) ≥ φ(y, un ), ∀y ∈ C rn

and hence

+

*

ϕ(y) − ϕ(un ) +

un − xni y − uni , i rni

≥ φ(y, uni ), ∀y ∈ C.

From kun − xn k → 0, kxn − Sn vn k → 0, and kSn vn − vn k → 0, we get u −x uni * z. Since nirn ni → 0, it follows by (A4) and the weakly lower i semicontinuity of ϕ that φ(y, z) + ϕ(z) − ϕ(y) ≤ 0, ∀y ∈ C. For t with 0 < t ≤ 1 and y ∈ C, let yt = ty + (1 − t)z. Since y ∈ C and z ∈ C, we obtain yt ∈ C and hence φ(yt , z) + ϕ(z) − ϕ(yt ) ≤ 0. So, from (A1), (A4) and the convexity of ϕ, we have 0

=

φ(yt , yt ) + ϕ(yt ) − ϕ(yt )

≤

tφ(yt , y) + (1 − t)φ(yt , z) + tϕ(y) + (1 − t)ϕ(z) − ϕ(yt )
t φ(yt , y) + ϕ(y) − ϕ(yt ) .

≤

17

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WANGKEEREE ET AL: MIXED EQUILIBRIUM PROBLEMS

Dividing by t, we get φ(yt , y) + ϕ(y) − ϕ(yt ) ≥ 0. From (A3) and the weakly lower semicontinuity of ϕ, we have φ(z, y) + ϕ(y) − ϕ(z) ≥ 0 for all y ∈ C and hence z ∈ M EP (φ, ϕ). Hence z ∈ Θ. Therefore, we have lim suphf (q) − q, xn − qi

=

lim suphf (q) − q, Svn − qi

=

lim hf (q) − q, Svni − qi

n→∞

n→∞

=

i→∞

hf (q) − q, z − qi ≤ 0.

(3.47)

Step 5. We claim that limn→∞ kxn − qk = 0. From (3.12), we observe that

= = ≤

≤

≤

kxn+1 − qk2

 αn f (xn ) + βn xn + γn Sn vn − q, xn+1 − q

 αn f (xn ) − q, xn+1 − q + βn hxn − q, xn+1 − qi +γn hSn vn − q, xn+1 − qi  1   1  βn kxn − qk2 + kxn+1 − qk2 + γn kvn − qk2 + kxn+1 − qk2 2 2

  + αn f (xn ) − f (q), xn+1 − q + αn f (q) − q, xn+1 − q   1  1 (1 − αn ) kxn − qk2 + kxn+1 − qk2 + αn kf (xn ) − f (q)k2 2 2 

 2 +kxn+1 − qk + αn f (q) − q, xn+1 − q
 1 1 1 − αn 1 − α2 kxn − qk2 + (1 − αn )kxn+1 − qk2 2 2

 1 2 + αn kxn+1 − qk + αn f (q) − q, xn+1 − q , 2

which implies that 
 kxn+1 −qk2 ≤ 1−αn 1−α2 kxn −qk2 +2αn hf (q)−q, xn+1 −qi. (3.48) Taking

 %n = 2αn f (q) − q, xn+1 − q and ln = αn (1 − α2 ), we have lim supn→∞ %lnn ≤ 0. Applying Lemma 2.5 to (3.48), we conclude that {xn } converges strongly to q in norm. Finally, noticing kun − zk = kTrn xn − Trn zk ≤ kxn − zk. We also conclude that un → z in norm. This completes the proof. Corollary 3.2. Let C be a nonempty closed convex subset of a real Hilbert space H and let φ be a bifunction from C × C to R satisfying (A1)-(A4) and let ϕ : E → R ∪ {+∞} be a proper lower semicontinuous and convex function. Let A : C → H be relaxed (u, v) − cocoercive and µ-Lipschitz continuous mapping and let f : C → C be a contraction with coefficient α (0 < α < 1) such that Θ := V I(C, A) ∩ M EP (φ, ϕ) 6= ∅.

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WANGKEEREE ET AL: MIXED EQUILIBRIUM PROBLEMS

Assume that either (B1) or (B2). Let {xn }, {yn } and {un } be sequences generated by  x1 = x ∈ C,     φ(u , y) + ϕ(y) − ϕ(u ) + 1 hy − u , u − x i ≥ 0, ∀y ∈ C, n n n n n rn  y = δ u + (1 − δ )P (u − λ Au ), n n n n C n n n    xn+1 = αn f (xn ) + βn xn + γn PC (yn − λn Ayn ), ∀n ≥ 1, where {αn }, {βn }, {γn } and {δn } are sequences in (0, 1) and {rn } is a sequence in (0, ∞). If the following conditions are satisfied: (C1) αn + βn + γn = 1 and 0 < δn ≤ e < 1, P (C2) limn→∞ αn = 0 and ∞ n=1 αn = ∞, (C3) 0 < lim inf n→∞ βn ≤ lim supn→∞ βn < 1, (C4) lim inf n→∞ rn > 0 and limn→∞ |rn+1 − rn | = 0, (C5) limn→∞ |λn+1 − λn | = limn→∞ |δn+1 − δn | = 0, (C6) {λn } ∈ [a, b] for some a, b with 0 ≤ a ≤ b ≤

2(v−uµ2 ) , µ2

v > uµ2 .

Then, {xn } converges strongly to the same point q ∈ Θ := V I(C, A) ∩ M EP (φ, ϕ), where q = PΘ f (q). Proof. Putting Sn = I for all n ∈ N in Theorem 3.1. Then, the sequence {xn } generated in Corallary 3.2 converges strongly to PV I(C,A)∩M EP (φ,ϕ) f (q).

Corollary 3.3. Let C be a nonempty closed convex subset of a real Hilbert space H and let A : C → H be relaxed (u, v) − cocoercive and µ-Lipschitz continuous mapping. Let f : C → C be a contraction with coefficient α (0 < α < 1) and {Sn } be a sequence of nonexpansive mappings of C into itself such that satisfies condition (2.11) and Θ := ∩∞ n=1 F (Sn ) ∩ V I(C, A) 6= ∅. Let {xn } and {yn } be sequences generated by    x1 = x ∈ C, yn = δn xn + (1 − δn )PC (xn − λn Axn ),   xn+1 = αn f (xn ) + βn xn + γn Sn PC (yn − λn Ayn ),

∀n ≥ 1,

where {αn }, {βn }, {γn } and {δn } are sequences in (0, 1). If the following conditions are satisfied: (C1) αn + βn + γn = 1 and 0 < δn ≤ e < 1, P (C2) limn→∞ αn = 0 and ∞ n=1 αn = ∞, (C3) 0 < lim inf n→∞ βn ≤ lim supn→∞ βn < 1, (C4) limn→∞ |λn+1 − λn | = limn→∞ |δn+1 − δn | = 0, (C5) {λn } ∈ [a, b] for some a, b with 0 ≤ a ≤ b ≤

2(v−uµ2 ) , µ2

v > uµ2 .

Let S be a mapping of C into itself defined by Sx = limn→∞ Sn x for all x ∈ C and suppose that F (S) = ∩∞ n=1 F (Sn ). Then, {xn } converges strongly to the same point q ∈ Θ := ∩∞ n=1 F (Sn ) ∩ V I(C, A), where q = PΘ f (q).

19

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WANGKEEREE ET AL: MIXED EQUILIBRIUM PROBLEMS

Proof. Put ϕ = 0, φ(x, y) = 0 for all x, y ∈ C and rn = 1 for all n ∈ N in Theorem 3.1. Then, we have un = PC xn = xn . So, from Theorem 3.1. the sequence {xn } generated in Corollary 3.3 converges strongly to P∩∞ f (q). n=1 F (Sn )∩V I(C,A) Corollary 3.4. ([25, Theorem 3.1]) Let C be a nonempty closed convex subset of a real Hilbert space H. Let A : C → H be relaxed (u, v) − cocoercive and µ-Lipschitz continuous mapping, let f : C → C be a contraction with coefficient α (0 < α < 1) and S be a nonexpansive mapping of C into itself such that Θ := F (S) ∩ V I(C, A) 6= ∅. Let {xn } be sequence generated by (

x1 = x ∈ C, xn+1 = αn f (xn ) + βn xn + γn SPC (xn − λn Axn ),

∀n ≥ 1,

where {αn }, {βn } and {γn } are sequences in (0, 1). If the following conditions are satisfied: (C1) αn + βn + γn = 1, (C2) limn→∞ αn = 0 and

P∞ n=1

αn = ∞,

(C3) 0 < lim inf n→∞ βn ≤ lim supn→∞ βn < 1, (C4) limn→∞ |λn+1 − λn | = 0, (C5) {λn } ∈ [a, b] for some a, b with 0 ≤ a ≤ b ≤

2(v−uµ2 ) , µ2

v > uµ2 .

Then, {xn } converges strongly to the same point q ∈ Θ := F (S)∩V I(C, A), where q = PΘ f (q). Corollary 3.5. ([27, Theorem 3.1]) Let C be a nonempty closed convex subset of a real Hilbert space H . Let φ be a bifunction from C × C to R satisfying (A1)-(A4), let f : C → C be a contraction with coefficient α (0 < α < 1) and S be a nonexpansive mapping of C into itself such that Θ := F (S) ∩ EP (φ) 6= ∅. Let {xn } and {un } be sequences generated by    x1 = x ∈ C, φ(un , y) + r1n hy − un , un − xn i ≥ 0, ∀y ∈ C,   xn+1 = αn f (xn ) + (1 − αn )Sun , ∀n ≥ 1, where {αn } is a sequence in (0, 1) and {rn } is a sequence in (0, ∞). If the following conditions are satisfied: P (C1) limn→∞ αn = 0 and ∞ n=1 αn = ∞, (C2) lim inf n→∞ rn > 0 and limn→∞ |rn+1 − rn | = 0. Then, {xn } converges strongly to the same point q ∈ Θ := F (S) ∩ EP (φ), where q = PΘ f (q).

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WANGKEEREE ET AL: MIXED EQUILIBRIUM PROBLEMS

4

Applications

In this section, we will utilize the results presented in this paper to study the following Optimization Problem (for short, OP)(see [8, 15, 22] for more detalis): OP : min ϕ(y)

(4.49)

y∈C

where C is a nonempty bounded closed convex subset of a Hilbert space and ϕ : C → R∪{+∞} be a proper convex and lower semicontinuous function. We denote by Argmin(ϕ) the set of solutions in (4.49). Let φ(x, y) = 0 for all x, y ∈ C in Corollary 3.2, then M EP (φ, ϕ) =Argmin(ϕ). It follows from Corollary 3.2 that the iterative sequence {xn } defined by  o n   un = argminy∈C ϕ(y) + 2r1n ky − xn k2 , yn = δn un + (1 − δn )PC (un − λn Aun ),   xn+1 = αn f (xn ) + βn xn + γn PC (yn − λn Ayn ),

∀n ≥ 1,

where {αn }, {βn }, {γn } and {δn } are sequences in (0, 1) and {rn } is a sequence in (0, ∞) satisfying the conditions (C1)-(C6) in Theorem 3.1. Then, {xn } converges strongly to q = PV I(C,A)∩Argmin(ϕ) f (q). If φ(x, y) = g(y) − g(x) for all x, y ∈ C, where g : C → R is function, then the mixed equilibrium problem (1.1) becomes a problem of finding x ∈ C which is a solution of following optimization problem (for short, OP1):  OP 1 : min ϕ(y) + g(y) . (4.50) y∈C

The set of solution of (4.50) is denoted by Argmin(g, ϕ). By taking φ(x, y) = g(y) − g(x) for all x, y ∈ C, in Corollary 3.2, we obtain the following results. Theorem 4.1. Let C be a nonempty closed convex subset of a real Hilbert space H and let φ be a bifunction from C × C to R satisfying (A1)-(A4). Let A : C → H be relaxed (u, v) − cocoercive and µ-Lipschitz continuous mapping, let f : C → C be a contraction with coefficient α (0 < α < 1) such that Θ := V I(C, A) ∩ Argmin(g, ϕ) 6= ∅. Assume that either (B1) or (B2). Let {xn }, {yn } and {un } be sequences generated by    x1 = x ∈ C,   g(y) + ϕ(y) − ϕ(u ) − g(u ) + 1 hy − u , u − x i ≥ 0, ∀y ∈ C, n n n n n rn  y = δ u + (1 − δ )P (u − λ Au ), n n n n C n n n    xn+1 = αn f (xn ) + βn xn + γn PC (yn − λn Ayn ), ∀n ≥ 1, where {αn }, {βn }, {γn } and {δn } are sequences in (0, 1) and {rn } is a sequence in (0, ∞). If the following conditions are satisfied: (C1) αn + βn + γn = 1 and 0 < δn ≤ e < 1, P (C2) limn→∞ αn = 0 and ∞ n=1 αn = ∞, (C3) 0 < lim inf n→∞ βn ≤ lim supn→∞ βn < 1, (C4) lim inf n→∞ rn > 0 and limn→∞ |rn+1 − rn | = 0, (C5) limn→∞ |λn+1 − λn | = limn→∞ |δn+1 − δn | = 0,

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(C6) {λn } ∈ [a, b] for some a, b with 0 ≤ a ≤ b ≤

2(v−uµ2 ) , µ2

v > uµ2 .

Then, {xn } converges strongly to the same point q ∈ Θ := V I(C, A) ∩ Argmin(g, ϕ), where q = PΘ f (q). Let { Tn : C → C } be a family of infinitely nonexpansive mappings and let {µn } be a nonnegative real sequence with 0 ≤ µn ≤ 1, for each n ≥ 1. Shimoji and Takahashi [24], define a mapping Wn of C into itself as follows: Un,n+1

=

I,

Un,n

=

µn Tn Un,n+1 + (1 − µn )I,

Un,n−1

= .. .

µn−1 Tn−1 Un,n + (1 − µn−1 )I,

Un,k

=

µk Tk Un,k+1 + (1 − µk )I,

Un,k−1

= .. .

µk−1 Tk−1 Un,k + (1 − µk−1 )I,

(4.51)

Un,2

=

µ2 T2 Un,3 + (1 − µ2 )I,

Wn = Un,1

=

µ1 T1 Un,2 + (1 − µ1 )I.

Such a mappings Wn is nonexpansive from C to C and it is called the W -mapping generated by T1 , T2 , ..., Tn and µ1 , µ2 , ..., µn . For each n, k ∈ N, let the mapping Un,k be defined by (4.51). Then we can have the following crucial conclusions concerning Wn . You can find them in [24]. Now we only need the following similar version in Hilbert spaces. Lemma 4.2. [24]. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T1 , T2 , ... be a family of infinitely nonexpansive mappings of C into itself such that ∩∞ n=1 F (Tn ) is nonempty, let µ1 , µ2 , ... be real numbers such that 0 ≤ µn ≤ b < 1 for every n ≥ 1. Then: (1) Wn is nonexpansive and F (Wn ) = ∩∞ n=1 F (Tn ) for each n ≥ 1, (2) for each x ∈ C and for each positive integer k, the limit limn→∞ Un,k x exists ; (3) the mapping W : C → C define by W x = lim Wn x = lim Un,1 x, n→∞

n→∞

(4.52)

is a nonexpansive mapping satisfying F (W ) = ∩∞ n=1 F (Tn ) and it is called the W -mapping generated by T1 , T2 , ... and µ1 , µ2 , .... Lemma 4.3. [7]. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T1 , T2 , ... be a family of infinitely nonexpansive mappings of C into itself such that ∩∞ n=1 F (Tn ) is nonempty, let µ1 , µ2 , ... be real numbers such that 0 ≤ µn ≤ b < 1 for every n ≥ 1. If K is any bounded subset of C, then lim

sup kWm x − Wn xk = 0.

m,n→∞ x∈K

Setting Sn = Wn in Theorem 3.1 and using the previous lemma, then we obtain the following theorem.

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WANGKEEREE ET AL: MIXED EQUILIBRIUM PROBLEMS

Theorem 4.4. Let C be a nonempty closed convex subset of a real Hilbert space H and let φ be a bifunction from C × C to R satisfying (A1)-(A4). Let A : C → H be relaxed (u, v) − cocoercive and µ-Lipschitz continuous mapping, let f : C → C be a contraction with coefficient α (0 < α < 1) and let {Tn } be a family of infinitely nonexpansive mappings of C into itself such that the set Θ := ∩∞ n=1 F (Tn ) ∩ V I(C, A) ∩ M EP (φ, ϕ) 6= ∅. Assume that either (B1) or (B2). Let {xn }, {yn } and {un } be sequences generated by    x1 = x ∈ C,   φ(u , y) + ϕ(y) − ϕ(u ) + 1 hy − u , u − x i ≥ 0, ∀y ∈ C, n n n n n rn  y = δ u + (1 − δ )P (u − λ Au ), n n n n C n n n    xn+1 = αn f (xn ) + βn xn + γn Wn PC (yn − λn Ayn ), ∀n ≥ 1, where {Wn } is the sequence generated by (4.51) and {αn }, {βn }, {γn } and {δn } are sequences in (0, 1) and {rn } is a sequence in (0, ∞). If the following conditions are satisfied: (C1) αn + βn + γn = 1 and 0 < δn ≤ e < 1, P (C2) limn→∞ αn = 0 and ∞ n=1 αn = ∞, (C3) 0 < lim inf n→∞ βn ≤ lim supn→∞ βn < 1, (C4) lim inf n→∞ rn > 0 and limn→∞ |rn+1 − rn | = 0, (C5) limn→∞ |λn+1 − λn | = limn→∞ |δn+1 − δn | = 0, (C6) {λn } ∈ [a, b] for some a, b with 0 ≤ a ≤ b ≤

2(v−uµ2 ) , µ2

v > uµ2 .

Then, {xn } converges strongly to the same point q ∈ Θ := ∩∞ n=1 F (Tn ) ∩ V I(C, A) ∩ M EP (φ, ϕ), where q = PΘ f (q). Acknowledgement. This paper was supported by the Thailand National Research Universities Project, KMUTT, CSEC Project No. E01008. Moreover, C. Jaiboon is also supported by King Mongkut’s Diamond scholarship for Ph.D. Program at KMUTT.

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[6] L.C. Ceng, J.C. Yao, A hybrid iterative scheme for mixed equilibrium problems and fixed point problems, J. Comput. Appl. Math. 214 (2008) 186–201. [7] S.S. Chang, H.W. Joseph Lee, C.K. Chan, A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization, Nonlinear Analysis. 70(2009) 3307–3319 [8] M. C. Ferris, Finite termination of the proximal point algorithm, Math. Program. 50 (1991) 359–366. [9] X. Gao, Y. Guo, Strong convergence theorem of a modified iterative algorithm for mixed equilibrium problems in Hilbert spaces, Journal of Inequalities and Applications. Article ID 621621 (2008), 18 pages. [10] C. Jaiboon, and P. Kumam, A hybrid extragradient viscosity approximation method for solving equilibrium problems and fixed point problems of infinitely many nonexpansive mappings, Fixed Point Theory and Applications. Article ID 374815 (2009), 32 pages. [11] C. Jaiboon , P. Kumam and H.W. Humphries, Weak Convergence Theorem by an Extragradient Method for Variational Inequality, Equilibrium and Fixed Point Problems, Bull. Malays. Math. Sci.Soc. (2)32(2)(2009) 173–185. [12] C. Jaiboon, P. Kumam and H.W. Humphries, Convergence Theorems by the Viscosity Approximation Method for Equilibrium Problems and Variational Inequality Problems, Comput. Math. Optim. 5(1)(2009) 29–56. [13] A. Kangtunyakarn and S. Suantai, A new mapping for finding common solutions of equilibrium problems and fixed point problems of finite family of nonexpansive mappings, Nonlinear Analysis. 71(2009), 4448–4460. [14] P. Kumam and C. Jaiboon, A new hybrid iterative method for mixed equilibrium problems and variational inequality problem for relaxed cocoercive mappings with application to optimization problems, Nonlinear Anal: Hybrid Systems. 3(2009)510–530. [15] B. Martinet, Perturbation des methodes d’optimisation, RAIRO Anal. Numer. 12(1978) 153-171. [16] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73(1967) 595–597. [17] M.O. Osilike and D.I. Igbokwe, Weak and strong convergence theorems for fixed points of pseudocontractions and solutions of monotone type operator equations, Comput. Math. Appl, 40(2000) 559–567. [18] S. Plubtieng and T. Thammathiwat, A Viscosity approximation method for nding a common xed point of nonexpansive and firmly nonexpansive mappings in Hilbert spaces, Thai. J. Math. 6(2)(2008) 377–390. [19] J.W. Peng and J.C. Yao, A new hybrid-extragradient method for generalized mixed equilibrium problems and fixed point problems and variational inequality problems, Taiwanese. J. Math.12(6)(2008) 1401– 1432.

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[20] J.W. Peng and J.C. Yao, Strong convergence theorems of iterative scheme based on the extragradient method for mixed equilibrium problems and fixed point problems, Math. and Comput. Model. 49(2009)1816–1828. [21] R.T. Rockafellar, On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc. 149 (1970) 75–88. [22] R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM Journal on Control and Optimization. 14(5) (1976) 877-898. [23] T. Suzuki, Strong convergence of krasnoselskii and manns type sequences for one-parameter nonexpansive semigroups without bochner integrals, J. Math. Anal. Appl. 305(2005) 227–239. [24] K. Shimoji and W. Takahashi, Strong convergence to common fixed points of infinite nonexpansive mappings and applications, Taiwanese Journal of Mathematics. 5(2001)387–404. [25] M. Shang, Y. Su and X. Qin, Strong convergence theorem for nonexpansive mappings and relaxed cocoercive mappings, Int. J. Appl. Math. and Mech. 3(4)(2007) 24–34. [26] W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, 2000 [27] S. Takahashi and W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 311,(1) (2007) 506–515. [28] W. Takahashi and M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl. 118(2003) 417–428. [29] RU. Verma, Generalized system for relaxed cocoercive variational inequality and its projection method, J. Optim. Theory Appl. 121(2004) 203–210. [30] RU. Verma, General convergence analysis for two–step projection methods and application to variational problems, J. Optim. Theory Appl. 18, (11) (2005) 1286–1292. [31] H.K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298(2004) 279–291. [32] J.C. Yao and O. Chadli, Pseudomonotone complementarity problems and variational inequalities, in: J.P. Crouzeix, N. Haddjissas, S. Schaible (Eds.), Handbook of Generalized Convexity and Monotonicity, (2005) 501–558. [33] Y. H. Yao, Y. C. Liou and J. C. Yao, A New Hybrid Iterative Algorithm for Fixed-Point Problems, Variational Inequality Problems, and Mixed Equilibrium Problems, Fixed Point Theory and Applications. Article ID 417089 (2008), 15 pages. [34] L. Yang, J.A. Liu, and Y.X. Tian, Proximal methods for equilibrium problem in Hilbert spaces, Math. Commun. 13(2008) 253–263. [35] L.C. Zeng, S. Schaible, and J.C. Yao, Iterative algorithm for generalized set-valued strongly nonlinear mixed variational-like inequalities, J. Optim. Theory Appl. 124(2005) 725–738.

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JOURNAL 450 OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.3, 450-457, 2011, COPYRIGHT 2011 EUDOXUS PRESS, LLC

A STUDY ON ALMOST INCREASING SEQUENCES ¨ GD ¯ UK ¨ AYBEK H. N. O Abstract. In this paper a general theorem dealing with | A, pn ; δ |k summa¨ gd¨ bility method has been proved. This theorem also includes a result of O¯ uk [5] concerning the | A, pn |k summability method.

1. Introduction

∑

Let an be a given infinite series with the partial sums (sn ), and let A = (anv ) be a normal matrix, i.e., a lower triangular matrix of non-zero diagonal entries. Then A defines the sequence-to-sequence transformation, mapping the sequence s = (sn ) to As = (An (s)), where An (s) = The series

∑

n ∑

anv sv ,

n = 0, 1, ...

(1.1)

v=0

an is said to be summable | A |k , k ≥ 1, if (see [8]) ∞ ∑

nk−1 | ∆An (s) |k < ∞.

(1.2)

n=1

where ∆An (s) = An (s) − An−1 (s). Let (pn ) be a sequence of positive numbers such that Pn =

n ∑

pv → ∞

as

n → ∞,

(P−i = p−i = 0, i ≥ 1).

(1.3)

v=0

The sequence-to-sequence transformation tn =

n 1 ∑ pv sv Pn v=0

(1.4)

¯ , pn ) mean of the sequence (sn ), generated by defines the sequence (tn ) of the (N ∑ the sequence of coefficients (pn ) (see [3]). The series an is said to be summable ¯ , pn |k , k ≥ 1, if (see [2]) |N )k−1 ∞ ( ∑ Pn | tn − tn−1 |k < ∞, (1.5) p n n=1 2010 AMS Subject Classification: 40F05, 40G99. Keywords: Absolute summability, summability factors, almost increasing sequences. 1

451

¨ GD ¯ UK ¨ AYBEK H. N. O

2

and it is said to be summable | A, pn |k , k ≥ 1, if (see [7]) )k−1 ∞ ( ∑ Pn | ∆An (s) |k < ∞. (1.6) p n n=1 ∑ The series an is said to be summable | A, pn ; δ |k , k ≥ 1 and δ ≥ 0, if (see [6]) )δk+k−1 ∞ ( ∑ Pn (1.7) | ∆An (s) |k < ∞. p n n=1 In the special case when δ = 0, | A, pn ; δ |k summability reduces to | A, pn |k summability. Also if we take δ = 0 and anv = Ppvn , then | A, pn ; δ |k summability is ¯ , pn |k summability. the same as | N A positive sequence (bn ) is said to be an almost increasing sequence if there exists an increasing sequence (cn ) and positive constants A and B such that Acn ≤ bn ≤ Bcn (see [1]). Obviously, every increasing sequence is an almost increasing sequence. However, the converse need not be true as can be seen by taking the n example, say bn = e(−1) n. 2. Known results. Given a normal matrix A = (anv ), we may associate two lower semi-matrices b = (b A = (anv ) and A anv ) as follows: anv =

n ∑

ani , n, v = 0, 1, ...,

(2.1)

i=v

and anv = anv − an−1,v , n = 1, 2, ... b a00 = a00 = a00 ; b

(2.2)

b are the well-known matrices of series-to-sequence It may be noted that A and A and series-to-series transformations, respectively. Then, we have An (s) = =

n ∑ v=0 n ∑ i=0

∆An (s)

=

n ∑

anv sv =

n ∑

v ∑

anv

v=0

ai

n ∑

anv =

v=i

ani ai −

i=0

= ann an +

ai

i=0 n ∑

ani ai

i=0 n−1 ∑

an−1,i ai

i=0 n−1 ∑

(ani − an−1,i )ai

i=0

= b ann an +

n−1 ∑ i=0

b ani ai =

n ∑

b ani ai .

i=0

For any double sequence (unv ), we write ∆n unv = unv − un+1,v .

(2.3)

452

A STUDY ON ALMOST INCREASING SEQUENCES

3

Before we state our main result, we show A = (anv ) to be of class U if the following hold; A is lower triangular anv ≥ 0, n, v = 0, 1, ...;

(2.4)

an−1,v ≥ anv f or n ≥ v + 1;

(2.5)

an0 = 1, n = 0, 1, ... . ¨ gd¨ Quite recently, O¯ uk [5] has obtained the sufficient conditions for be summable | A, pn |k , k ≥ 1 as follows. Theorem 2.1. Let A ∈ U satifsying ann = O n−1 ∑ v=1

(

pn Pn

(2.6) ∑

an λn to

) ,

b an,v+1 = O(ann ), n → ∞. v

(2.7)

(2.8)

If (Xn ) is an almost increasing sequence such that | λn | Xn = O(1), n → ∞, m ∑

nXn | ∆2 λn |= O(1), m → ∞,

(2.9) (2.10)

n=1

then the series

∑

m ∑ pn | tn |k = O(Xm ), m → ∞, P n n=1

(2.11)

m ∑ | tn |k = O(Xm ), m → ∞, n n=1

(2.12)

an λn is summable | A, pn |k , k ≥ 1. 3. The main result

The main purpose of this paper is to prove a more general theorem which includes the above-mentioned result as a special case. Now, we shall prove the following theorem. Theorem 3.1. Let A ∈ U satisfying the conditions (2.7) and (2.8) and let (λn ) be a sequence of real numbers satisfying the conditions (2.9) and (2.10). If (Xn ) is an almost increasing sequence such that {( ) } m+1 δk ∑ ( Pn )δk Pv | ∆v b anv |= O avv , m → ∞, pn pv n=v+1

(3.1)

453

¨ GD ¯ UK ¨ AYBEK H. N. O

4

)δk−1 m ( ∑ Pv pv

v=1

| tv |k = O(Xm ), m → ∞,

(3.2)

)δk m ( ∑ Pv | tv |k = O(Xm ), m → ∞, pv v v=1 (

m+1 ∑

then

∑

n=v+1

Pn pn

((

)δk b an,v+1 = O

Pv pv

(3.3)

)δk ) , m → ∞,

(3.4)

an λn is summable | A, pn ; δ |k , k ≥ 1, 0 ≤ δ < 1/k.

It may be noticed that if we take δ = 0, then we get Theorem 2.1. We need the following lemma for the proof of our theorem. Lemma 3.2. ([4]) If (Xn ) is an almost increasing sequence, then under the conditions (2.9) and (2.10), we have that ∞ ∑

Xn | ∆λn |< ∞,

(3.5)

n=1

nXn | ∆λn |= O(1), n → ∞. Proof of Theorem 3.1 Let (Tn ) denotes A−transform of the series by definition, we have ∆Tn =

n ∑

(3.6) ∑

an λn . Then,

b anv av λv .

v=0

Applying Abel’s transformation, we can write ∆Tn =

n ∑ b anv λv v=1

=

n−1 ∑ v=1

+

n−1 ∑ v=1

v

vav

n−1 ∑ v+1 v+1 ∆v b anv λv tv + b an,v+1 ∆λv tv v v v=1

b an,v+1 λv+1 tv n+1 + ann λn tn v n

= Tn (1) + Tn (2) + Tn (3) + Tn (4), say. Since | Tn (1) + Tn (2) + Tn (3) + Tn (4) |k ≤ 4k (| Tn (1) |k + | Tn (2) |k + | Tn (3) |k + | Tn (4) |k ), to complete the proof, it is sufficient to show that )δk+k−1 ∞ ( ∑ Pn | Tn (r) |k < ∞, f or r = 1, 2, 3, 4. p n n=1

(3.7)

454

A STUDY ON ALMOST INCREASING SEQUENCES

5

′

Applying H¨older’s inequality with indices k and k , where k > 1 and we have that

I1 =

m+1 ∑( n=2

= O(1) = O(1)

×

(n−1 ∑

Pn pn

1 k

+

1 k′

= 1,

)δk+k−1 | Tn (1) |k

m+1 ∑( n=2

Pn pn

n=2

Pn pn

m+1 ∑(

)δk+k−1 (n−1 ∑

)k | ∆v b anv || λv || tv |

v=1

)δk+k−1 (n−1 ∑

| ∆v b anv || λv | | tv |

k

v=1

)k−1

| ∆v b anv |

) k

.

v=1

By (2.5), ∆v b anv = b anv − b an,v+1 = anv − an−1,v − an,v+1 + an−1,v+1 = anv − an−1,v ≤ 0. Thus using (2.6), n−1 ∑

| ∆v b anv |=

v=0

n−1 ∑

(an−1,v − anv ) = ann .

(3.8)

v=0

From (2.9), it follows that λn = O(1). Using (2.7), (3.1) and property (3.5) of Lemma 3.2, we have

I1 =

m+1 ∑( n=2

= O(1) = O(1) = O(1) = O(1)

)δk (

Pn pn

m ∑

v=1

Pv pv

v=1

Pv pv

n−1 ∑(

(

Pn pn

| ∆v b anv || λv |k−1 | λv || tv |k )δk | ∆v b anv |

avv | λv || tv |k )δk−1

v=1

= O(1)

v=1

n=v+1

)δk

∆ | λv |

m−1 ∑

)k−1 n−1 ∑

m+1 ∑

| λv || tv |k

v=1 m ( ∑

n−1 ∑

Pn ann pn

| λv || tv |k )δk−1 v ( ∑ Pi i=1

pi

| ti |k +O(1) | λm |

| ∆λv | Xv + O(1) | λv | Xm

v=1

= O(1), m → ∞.

)δk−1 m ( ∑ Pv v=1

pv

| tv |k

455

¨ GD ¯ UK ¨ AYBEK H. N. O

6

Again using H¨older’s inequality, as in I1 m+1 ∑(

=

I2

n=2

= O(1) = O(1)

)δk+k−1

m+1 ∑(

| Tn (2) |k

n=2

Pn pn

n=2

Pn pn

m+1 ∑(

(n−1 ∑

×

Pn pn

)δk+k−1 (n−1 ∑

)k b an,v+1 | ∆λv || tv |

v=1

)δk+k−1 (n−1 ∑ )k−1

b an,v+1 | ∆λv |

) b an,v+1 | ∆λv || tv |k

v=1

.

v=1

It is easy to see that

n−1 ∑

b an,v+1 | ∆λv |≤ M ann .

v=1

Thus, we have that

I2 = O(1)

m+1 ∑( n=2

= O(1) = O(1)

m ∑

v=1

= O(1)

)δk n−1 ∑

Pv pv

= O(1)

m+1 ∑

(

)δk b an,v+1

| ∆λv || tv |k

∆(v | ∆λv |)

)δk )δk v ( m ( ∑ ∑ Pv Pi | ti |k | tv |k + O(1)m | ∆λm | pi i pv v v=1 i=1

v | ∆2 λv | Xv + O(1)

v=1

Pn pn

n=v+1

)δk

v=1 m ∑

b an,v+1 | ∆λv || tv |k

v=1

| ∆λv || tv |k

v=1 m ( ∑

m ∑

Pn pn

m ∑

| ∆λv+1 | Xv+1 + O(1)m | ∆λm | Xm

v=1

= O(1), m → ∞.

by virtue of the hypothesis of Theorem 3.1 and Lemma 3.2.

456

A STUDY ON ALMOST INCREASING SEQUENCES

7

Taking account of (2.8) and (3.4) I3 =

m+1 ∑( n=2

= O(1)

Pn pn

)δk+k−1 | Tn (3) |k

m+1 ∑( n=2

= O(1)

m+1 ∑( n=2

×

)δk+k−1 (n−1 ∑

Pn pn

v=1

)δk+k−1 (n−1 ∑

Pn pn )k−1

v=1

(n−1 ∑b an,v+1 v=1

v +1 ( ∑

m

= O(1)

n=2

= O(1) = O(1)

m ∑

Pn pn

)δk (

| tv | v

| λv+1 |

v=1 m ( ∑ v=1

Pv pv

Pn ann pn

)δk

b an,v+1 | λv+1 || tv | v

b an,v+1 | λv+1 |k−1 | λv+1 || tv |k v

)k−1 (n−1 ∑ (

m+1 ∑

k

n=v+1

| λv+1 |

)k

v=1

Pn pn

b an,v+1 | λv+1 || tv |k v

)

)

)δk b an,v+1

| tv |k . v

Using Abel’s transformation and Lemma 3.2, I3 =

m−1 ∑

∆ | λv+1

v=1

= O(1)

m−1 ∑

)δk )δk v ( m ( ∑ ∑ Pi | ti |k | tv | Pv | + O(1) | λm+1 | pi i pv v v=1 i=1

| ∆λv | Xv + O(1) | λm | Xm

v=1

= O(1), m → ∞. Finally, as in Tn (1), I4 =

)δk+k−1 m ( ∑ Pn n=1

= O(1) = O(1) = O(1)

pn

| Tn (4) |k

)δk+k−1 m ( ∑ Pn n=1 m ( ∑ n=1 m ( ∑ n=1

pn Pn pn Pn pn

aknn | λn |k | tn |k

)δk−1 | λn |k−1 | λn || tn |k )δk−1 | λn || tn |k

= O(1), m → ∞. Thus, we obtain (3.7). This completes the proof of the Theorem.

457

8

¨ GD ¯ UK ¨ AYBEK H. N. O

References [1] S. Aljancic and D. Arandelovic, 0− regularly varying functions, Publ. Inst. Math. 22 (1977) 5–22. [2] H. Bor, On two summability methods, Math. Proc. Cambridge Philos. Soc. 97 (1985) 147–149. [3] G. H. Hardy, Divergent series, Oxford Univ. Press., Oxford (1949). [4] S. M. Mazhar, Absolute summability factors of infinite series, Kyungpook Math. J. 39 (1999) 67–73. ¨ gd¨ [5] H. N. O¯ uk, A summability factor theorem by using an almost increasing sequence, J. Comp. Anal. Appl. 11, no.1 (2009) 45–53. ¨ ¨ gd¨ [6] H. S. Ozarslan, H. N. O¯ uk, Generalizations of two theorems on absolute summability methods, Aust. J. Math. Anal. Appl. 1 (2) (2004), Art.13, 7 pp. [7] W. T. Sulaiman, Inclusion theorems for absolute matrix summability methods of an infinite series (IV), Indian. J. Pure Appl. Math. 34 (11) (2003) 1547–1557. [8] N. Tanovic-Miller, On strong summability, Glasnik Mathematicki 34 (14) (1979) 87–97. ¨ GD ¯ UK ¨ AYBEK H. N. O Faculty of Education, Department of Mathematics Education, University of Mersin, 33169 Mersin, Turkey E-mail address: [email protected]

JOURNAL 458 OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.3, 458-462, 2011, COPYRIGHT 2011 EUDOXUS PRESS, LLC

FORTI’S APPROACH IN FIXED POINT THEORY AND THE STABILITY OF A FUNCTIONAL EQUATION ON METRIC AND ULTRA METRIC SPACES F. RAHBARNIA, TH. M. RASSIAS, R. SAADATI, AND GH. SADEGHI

Abstract. In this paper, using the fixed point method, we prove the stability of a generalized functional equation in the Forti spirit.

1. INTRODUCTION AND PRELIMINARIES A basic question in the theory of functional equations is as follows: ’when is it true that a function, which approximately satisfies a functional equation must be close to an exact solution of the equation’ ? If the problem accepts a solution, we say the equation is stable. The first stability problem concerning group homomorphisms was raised by Ulam [20] in the year 1940 and it was partially solved by Hyers [8] in the year 1941. The result of Hyers was generalized by Aoki [1] for approximate additive function and by Rassias [16] for approximate linear functions by allowing the difference Cauchy equation kf (x + y) − f (x) − f (y)k to be controlled by ε(k x kp + k y kp ). Taking into consideration a lot of influence of Ulam, Hyers and Rassias on the development of stability problems of functional equations, the stability phenomenon that was proved by Rassias is called the Hyers-Ulam-Rassias stability. In 1994, a generalization of Rassias theorem was obtained by Gˇ avruta [6], who replaced ε(k x kp + k y kp ) by a general control function ϕ(x, y). We refer the reader to the survey articles [5, 8, 17] and monographs [3, 9, 12, 14, 18] and references therein. For a nonempty set X , a function d : X × X → [0, ∞] is called a generalized metric on X if d satisfies (M1 ) d(x, y) = 0 if and only if x = y; (M2 ) d(x, y) = d(y, x) for all x, y ∈ X ; (M3 ) d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ X . Obviously the only one difference of the generalized metric from the usual metric is that the range of the former is permitted to include infinity. We now introduce one of fundamental results of fixed point theory. For the proof, we refer to [13]. This theorem will play an important role in proving our main theorem. Theorem 1.1. (The alternative of fixed point) Suppose we are given a complete generalized metric space (X , d) and a strictly contractive mapping J : X → X ,with the Lipschitz constant L. Then for each x ∈ X, either d(J n x, J n+1 x) = +∞,

f or all n ≥ 0,

(1.1)

f or all n ≥ 0;

(1.2)

or, There exists a natural number n0 such that: d(J n x, J n+1 x) < +∞,

The sequence {J n x} is convergent to a fixed point y ∗ of J; y ∗ is the unique fixed point of J in the set Y = {y ∈ X , d(J n0 x, y) < +∞}; d(y, y ∗ ) ≤

1 d(y, Jy), 1−L

f or all y ∈ Y.

2000 Mathematics Subject Classification. Primary 39B22; Secondary 39B82, 46S10. Key words and phrases. Generalized Hyers–Ulam stability; functional equation. 1

(1.3)

459 2

RAHBARNIA, RASSIAS, SAADATI, AND SADEGHI

2. Main result let (E, d) be a complete metric space, S, F are sets, G : S → S and H : E → F , be three given function and f : S → E. In this section by using the idea of Forti [4], we will apply the fixed point method for proving stability the following functional equation H{f [G(x)]} = f (x).

(2.1)

And also assume that δ : S → R+ , φ : R+ → R+ , which φ is non-decreasing function. Theorem 2.1. Assume that f : S → E is a function satisfying the inequality d(H{f [G(x)]}, f (x)) ≤ δ(x).

f or all x ∈ S.

(2.2)

f or all x, y ∈ E,

(2.3)

If the function H : E → E satisfies the inequality d(H(x), H(y)) ≤ φ(d(x, y)) and δ(x) ≤ φ(δ(x))

f or all x ∈ S,

(2.4)

φ[αφ(δ(G(x)))] ≤ αLφ(δ(x)),

(2.5)

for some (Lipschitz constant) L < 1 and α ∈ R+ . Then there exists a unique function F : S → Esolution of the functional equation H{F [G(x)]} = F (x), and satisfying the following inequality d(f (x), F (x)) ≤

1 φ(δ(x)) 1−L

(2.6)

for all x ∈ S. Proof. First, let us define X to be the set of all functions g : S → E and introduce the generalized metric on X : dX (g, h) = inf{c ∈ R+ , d(g(x), h(x)) ≤ cϕ(δ(x)). It is easy to see that (X , dX ) is complete. Now we will consider the mapping, J :X →X Jg(x) = HgG(x). Let g, h ∈ X and dX (g, h) ≤ c, then we have d(g(x), h(x)) ≤ cφ(δ(x)). Since φ is non-decreasing function we get φ{d(g(x), h(x))} ≤ φ{cφ(δ(x))} or φ{d(gG(x), hG(x))} ≤ φ{cφ(δ(G(x))} from (2.3) and (2.5) we have d{HgG(x), HhG(X)} ≤ φ{d(gG(x), hG(x))}

Therefore d(Jg(x), Jh(x)) ≤ cLφ(δ(x))

≤

φ{cφ(δ(G(x))}

≤

cLφ(δ(x)).

460 FORTI’S APPROACH

3

i.e., dX (Jg, Jg) ≤ cL and dX (Jg, Jh) ≤ LdX (g, h), which imply that J is a strictly contractive self-mapping on X with Lipschitz constant L. On the other hand from (2.3) we get d(Jf (x), J 2 f (x)) = d(Hf G(x), HHf GG(X)) ≤ φ(d(f G(x), Hf G(x)). Now, from (2.2) we get d(f G(x), KHf G(x)) ≤ δ(G(x)) and ϕ{d(f G(x), Hf G(x)} ≤ φ(δ(G(x))) which imply that d(Jf (x), J 2 f (x)) ≤ Lφ(δ(x)) and dX (Jf, J 2 f ) ≤ L < ∞. Now, we can apply the fixed point alternative, and we obtain the existence of mapping F : S → X such that F is a fixed point of J, that is JF (x) = F (x) i.e., HF G(x) = F (x). The mapping F is the unique fixed point of J in the set Y = {g ∈ X ;

dX (f, g) < ∞},

then F is the unique mapping which ∃c ∈ (0, ∞) such that d(F (x), f (x)) ≤ cφ(δ(x)). We also dX (J n f, F ) → 0 as n → ∞, which implies the equality limn→∞ (H)n {f [Gn (x)]} = F (x), and dX (f, F ) ≤

1 d (f, Jf ), 1−L X

f or all x ∈ S

which implies the inequality dX (f, F ) ≤

1 . 1−L

Since δ(x) ≤ φ(δ(x)), d(f (x), F (x)) ≤

1 ϕ(δ(x)). 1−L ¤

3. Stability of functional equations on ultra metric spaces An ultra metric (or non-Archimedean metric) on a set X is function d : X ×X → R≥0 with the following properties, (N1) d(x, y) = 0 if and only if x = y, (N2) d(x, y) = d(y, x), (N3) d(x, y) ≤ max{d(x, z) = d(z, y)}, ( strong triangle inequality) for all x, y, z ∈ X . Note that if d(x, z) 6= d(z, y), then in fact d(x, y) = max{d(x, z) = d(z, y)}. By the using strong triangle inequality sequence {xn }∞ n=1 is a cauchy sequence in X if for ε > 0, there exists n0 ∈ N such that for all n ≥ n0 , d(xn+1 , xn ) < ε In this section we let (X , d) be a complete ultra metric space. Theorem 3.1. Assume that f : S → X is a function satisfying the inequality d(H{f [G(x)]}, f (x)) ≤ δ(x).

(3.1)

If the function H : X → X is continuous and satisfies the inequality d(H(x), H(y)) ≤ φ(d(x, y))

x, y ∈ X

(3.2)

for a certain non-decreasing function φ : R+ → R+ and lim φn (δ[Gn (x)]) = 0.

(3.3)

n→∞

Then there exists a function F : S → X solution of the functional equation (2.1), and satisfying following inequality, d(F (x), f (x)) ≤ sup{φn (δ[Gn (x)]) :

n ∈ N}

(3.4)

461 4

RAHBARNIA, RASSIAS, SAADATI, AND SADEGHI

Moreover, if limn→∞ max{φi (δ[Gi (x)]) :

0 ≤ i ≤ n} = δ[G(x)], then F is a unique function satisfying

(2.1). Proof. Replacing in (3.1) x by G(x), we get d(H{f [G2 (x)]}, f (x)) ≤ δ[G(x)] then by (3.2) we have d((H)2 {f [G2 (x)]}, H{f [G(x)]})

≤

φ(d(H{f [G2 (x)]}, f [G(x)]))

≤

φ(δ[G(x)])

since φ is non-decreasing. Then by induction for each integer n, we obtain d((H)n+1 {f [Gn+1 (x)]}, (H)n {f [Gn (x)]}) ≤ φn (δ[Gn (x)])

x ∈ S.

(3.5)

Now we set Qn (x) := H n {f [Gn (x)]})

x ∈ S.

We show that the sequence {Qn (x)} is a Cauchy sequence for every x ∈ S. Since the ultra metric space X is complete then sequence {Qn (x)} is conference. By (3.1) we have d(Qn+1 (x), Qn (x)) ≤ φn (δ[Gn (x)]) the right of previous inequality tend to zero as n → ∞. Thus {Qn (x)} is a Cauchy sequence. Put F (x) = limn→∞ Qn (x). d(Qn (x), f (x)) = d((H)n {f [Gn (x)]}, f (x))

≤

max{d((H)i {f [Gi (x)]}, (H)i−1 {f [Gi−1 (x)]}

≤

max{φi−1 (δ[Gi−1 (x)]) :

1 ≤ i ≤ n}.

Taking the limits as n → ∞ we obtain (3.1). By the continuity of function H we have the following chain equalities. H{F [G(x)]}

= =

H{ lim Qn [G(x)]} = lim H{Qn [G(x)]} n→∞

lim (H)n+1 {f [Gn+1 (x)]} = F (x).

n→∞

Suppose that another function T satisfies in (2.1) and (3.1).Thus d(T (x), Qn (x))

=

d(H n {T [Gn (x)]}, H n {f [Gn (x)]}) ≤ φn (d(T [Gn (x)], f [Gn (x)]))

≤

φn ( lim max{φi (δ[Gn+i (x)]) :

≤

n

n→∞ n

0 ≤ i ≤ n})

φ (δ[G (x)])

Taking the limits as n → ∞ since the last term goes to zero we obtain, limn→∞ d(T (x), Qn (x)) = d(T (x), F (x)) = 0.

¤ References

[1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. [2] L.M. Arriola and W.A. Beyer, Stability of the Cauchy functional equation over p-adic fields, Real Analysis Exchange 31 (2005/2006), 125–132. [3] S. Czerwik, Stability of Functional Equations of Ulam–Hyers–Rassias Type, Hadronic Press, Palm Harbor, Florida, 2003. [4] G.L. Forti, Comments the core of the direct method for proving Hyers-Ulam stability of functional equations , J. Math. Anal. Appl. 295 (2004), 127–133. [5] G.L. Forti, Hyers-Ulam stability of functional equations in several variables Aequationes Math. 50 (1995), no. 1-2, 143–190. [6] P. Gˇ avruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994) 431-436.

462 FORTI’S APPROACH

5

¨ [7] K. Hensel, Uber eine neue Begr¨ undung der Theorie der algebraischen Zahlen, Jahresber. Deutsch. Math. Verein 6 (1897), 83–88. [8] D.H. Hyers and Th.M. Rassias, Approximate homomorphisms, Aequationes Math. 44 (1992), 125–153. [9] D.H. Hyers, G. Isac and Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkh¨ auser, Basel, 1998. [10] 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. [11] K.W. Jun and H.M. Kim, On the stability of Euler–Lagrange type cubic mappings in quasi-Banach spases, J. Math. Anal. Appl. (to appear). [12] S.-M. Jung, Hyers–Ulam–Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press lnc., Palm Harbor, Florida, 2001. [13] B. Margolis and J. Diaz, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc., 74 (1968), 305-309. [14] M.S. Moslehian, Approximately vanishing of topological cohomology groups, J. Math. Anal. Appl. 318 (2006), no. 2, 758- 771. [15] M.S. Moslehian and Th.M. Rassias, Stability of functional equations in non-Archimedian spaces, Appl. Anal. Disc. Math. 1(2007) 325–334. [16] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978) 297–300. [17] Th.M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math. 62 (2000), no. 1, 23–130. [18] Th.M. Rassias, Functional Equations, Inequalities and Applications, Kluwer Academic Publishers, Dordrecht, Boston and London, 2003. [19] A.M. Robert, A Course in p-adic Analysis, Springer–Verlag, New York, 2000. [20] S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960. F. Rahbarnia, Department of Mathematics, Ferdowsi University, P. O. Box 1159, Mashhad 91775, Iran; E-mail address: [email protected] Themistocles M. Rassias, Department of Mathematics, National Technical University of Athens, Zografou Campus, 15780 Athens, Greece E-mail address: trassias@@math.ntua.gr Reaz Saadati, Department of Mathematics,Science & Research Branch, Islamic Azad University, Post Code 14778, Ashrafi Esfahani Ave,Tehran, I.R. Iran. E-mail address: [email protected] Ghadir Sadeghi, Department of Mathematics and Computer Sciences, Sabzevar Tarbiat Moallem University, Sabzevar, IRAN E-mail address: [email protected], [email protected]

JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.3, 463-469, 2011, COPYRIGHT 2011 EUDOXUS PRESS, 463 LLC

New delay-dependent conditions for the robust stability of linear polytopic discrete-time systems K. RATCHAGIT Department of Mathematics Maejo University, Chiangmai, 50290 Thailand email: [email protected] Abstract This paper addresses the robust stability for a class of linear polytopic discretetime systems with interval time-varying delays. Based on the parameter-dependent Lyapunov-Krasovskii functional, new delay-dependent conditions for the robust stability are established in terms of linear matrix inequalities. Keywords: Robust stability, discrete-time systems, polytopic uncertainties, time delay, linear matrix inequalities.

AMS (MOS) Subject Classification. 34D20, 93D20, 37C75.

1

Introduction

Stability problems of continuous-time and discrete-time systems with time delays have been tackled and interesting results have been reported in the literature, see, e.g; [2, 3] and the references therein. Most of the results reported in the literature are for linear continuous time-delay systems and only few are for the class of discrete-time system with constant delays [4,5]. The main reason for this is that such systems can be transformed into equivalent systems without time delay and then one can use the known results on stability. Recently delay-dependent stability for discrete-time systems with time-varying delay was investigated in [3,5], where discrete Lyapunov functtional method were proposed to derived sufficient conditions for the stability. For more information on some of the results developed for discrete-time systems with interval delays, we refer the reader to [1,2], where the stability conditions were presented in terms of linear matrix inequalities (LMIs). Theoretically, stability analysis of the systems with time-varying delays is more complicated, especially for the case where the system matrices belong to some convex 1

464

RATCHAGIT: POLYTOPIC DISCRETE-TIME SYSTEMS

polytope. In this case, the parameter-dependent Lyapunov-Krasovskii functionals are constructed as the convex combination of a set of functions assures the robust stability of the nominal systems and the stability conditions must be solved upon a grid on the parameter space, which results in testing a finite number of linear matrix inequalities (LMI) [4, 5]. To the best of the authors’ knowledge, the stability for the class of linear discrete-time systems with both time-varying delays and polytopic uncertainties has not been investigated and this will be the subect of this paper. The aim of this paper is to develop new delay-dependent conditions for stability for linear polytopic discrete-time systems with interval time-varying delays. In this paper, we propose parameter-dependent LyapunovKrasovskii functional. The stability conditions are derived in terms of LMI, being thus solvable by the numeric technology available in the literature to date. By these delaydependent conditions, the stabilization of this class of discrete-time systems is derived. The paper is organized as follows. Section 2 introduces the main notations, definitions and some lemmas needed for the development of the main results. In Section 3, conditions are derived for stability of linear discrete-time systems with time-varying delays and polytopic uncertainties. They are followed by some remarks.

2

Preliminaries

The following notations will be used throughout this paper. R+ denotes the set of all real non-negative numbers; Rn denotes the n-dimensional space with the scalar product h., .i and the vector norm k.k; Rn×r denotes the space of all matrices of (n × r)− dimension. AT denotes the transpose of A; a matrix A is symmetric if A = AT . Matrix A is semi-positive definite (A ≥ 0) if hAx, xi ≥ 0, for all x ∈ Rn ; A is positive definite (A > 0) if hAx, xi > 0 for all x 6= 0; A ≥ B means A − B ≥ 0. Consider a delayed discrete-time systems with polytopic uncertainties of the form x(k + 1) = A(ξ)x(k) + D(ξ)x(k − h(k)), x(k) = vk , k = −h2 , −h2 + 1, ..., 0,

k = 0, 1, 2, ... (Σξ )

where x(k) ∈ Rn is the state, the state-space data are subjected to uncertainties and belong to the polytope Ω given by Ω = {[A, D](ξ) :=

p X

ξi [Ai , Di ],

i=1

p X

ξi = 1, ξi ≥ 0},

i=1

where Ai , Di , i = 1, 2, ..., p, are constant matrices with appropriate dimensions. The timevarying function h(k) satisfies the condition: 0 < h1 ≤ h(k) ≤ h2 , Let kvk = max{kvi k, i = −k, −k + 1, ..., 0}. 2

∀k = 0, 1, 2, ....

RATCHAGIT: POLYTOPIC DISCRETE-TIME SYSTEMS

465

Definition 2.1. The system Σξ , where u(k) = 0 is robustly stable if the zero solution of the system is asymptotically stable.

3

Main results

In this section, we present delay-dependent conditions for robust stability of system (Σξ ). Let us set kxk k = sup kx(k + s)k, s∈[−h2 ,0]

 RiT Dj (h2 − h1 + 1)Qi − Pi − ATj Ri − RiT Aj RiT − ATj Si Ri − SiT Aj Pi + Si + SiT −SiT Dj  , Mij (P, Q, R, S) =  T Dj Ri DjT Si −Qi   p p S 0 0 X X   S = 0 0 0 , P (ξ) = ξi Pi , Q(ξ) = ξi Qi , i=1 i=1 0 0 0 

R(ξ) =

p X

ξi Ri ,

S(ξ) =

i=1

p X

ξi Si .

i=1

Theorem 3.1. The system Σξ is robustly stable if there exist symmetric matrices Pi > 0, Qi > 0, S > 0, Ri , Si , i = 1, 2..., p satisfying the following LMIs (i) Mii (P, Q, R, S) + S < 0, i = 1, 2, ..., p. (ii) Mij (P, Q, R, S) + Mji (P, Q, R, S)
0 , there exist a T (α o ) > 0 such that

whenever X 0 = X (t , t 0 , X 0 ) < α 0 then

X (t , t 0 , X 0 ) < B for all t0 ≥ 0 and t ≥ t 0 + T (α o ) . We now give a lemma which will play an important role in the proof of our results. Lemma 1. Suppose that there exists a Lyapunov function V (t , X (t )) defined on ℜ + , X ≥ K , where K may be large, which satisfies the following conditions: (i) a ( X (t ) ) ≤ V (t , X (t )) ≤ b( X (t ) ),

where a ( r ), b( r ) are continuous and increasing and a ( r ) → ∞ as r → ∞, d 1 (ii) V( 4) (t , X (t )) ≡ lim sup [V (t + h, X (t ) + F (t , X (t ))) − V (t , X (t ))] dt h h →0 + ≤ −[c − λ1 (t )] V (t , X (t )) + λ2 (t ) V β (t , X (t )) , (0 ≤ β < 1),

where c > 0 is constant and λi ≥ 0 (i = 1, 2) are continuous functions satisfying 1 t +v lim sup ∫ λ1 ( s )ds < c (t , v )→(∞ , ∞ ) v t and

(5)

(6)

t +1

sup ∫ λ 2 ( s)ds < ∞. t ≥0

(7)

t

Then the solutions of (4) are uniformly ultimately bounded (see [17, Lemma 2.1]). 3. MAIN RESULTS Theorem 1. Suppose that a, b, b1 , c, δ 0 and P0 are positive constants, p ≡ p (t ) and that

3

TUNC, AYHAN: NONLINEAR DIFFERENTIAL EQUATIONS

h( x ) , ( x ≠ 0), h ′( x) ≤ c for all x, x

(i) h(0) = 0 , δ 0 ≤

g ( x, y ) ≤ b1 , ( y ≠ 0), g x ( x, y ) ≤ 0 for all x, y, y f ( x, y , z ) (iii) a ≤ , ( z ≠ 0), for all x, y, z, z (ii) b ≤

t

(iv)

∫ p( µ ) dµ ≤ P

0

< ∞.

0

Then for any given finite constants x0 , y 0 , z 0 there exists a positive constant D = D ( x0 , y 0 , z 0 ), such that any solution ( x(t ), y (t ), z (t )) of the system (3) determined by x(0) = x 0 , y (0) = y 0 , z (0) = z 0 for t = 0, satisfies x(t ) ≤ D,

y (t ) ≤ D,

z (t ) ≤ D

(8)

for all t ≥ 0. The proof of Theorem 1 and subsequent results depend on some certain fundamental properties of a continuously differentiable Lyapunov function V (t ) = V ( x(t ), y (t ), z (t )) defined by y

x

2V (t ) = 2a ∫ h(ξ )dξ + 2 ∫ g ( x,τ )dτ + 2 yh( x) + αbx 2 + (α + a 2 ) y 2 0

0 2

+ z + 2αaxy + 2αxz + 2ayz,

(9)

where α a is positive fixed constant satisfying 0 < α < b − ca −1 .

(10)

The Lyapunov function in (9) and its time derivatives satisfy some fundamental inequalities as will be seen later. In what follows, we shall state and prove some results that would be useful in the proof of our main results. Lemma 2. Under the hypotheses of Theorem 1, there exist positive constants Di (i = 0, 1) such that for all ( x, y, z ) ∈ ℜ 3

D0 ( x 2 (t ) + y 2 (t ) + z 2 (t )) ≤ V (t ) ≤ D1 ( x 2 (t ) + y 2 (t ) + z 2 (t )). Proof. We observe that the Lyapunov function in (9) can be rewritten as

2V (t ) = V1 + V2 , where x

y

0

0

V1 = 2a ∫ h(ξ ) dξ + 2 ∫ g ( x,τ )dτ + 2 yh( x)

and V2 = αbx 2 + (α + a 2 ) y 2 + z 2 + 2αaxy + 2αxz + 2ayz.

(11)

479

480

TUNC, AYHAN: NONLINEAR DIFFERENTIAL EQUATIONS

4

In view of hypothesis (ii) of Theorem 1, we have g ( x, y ) ≥ by for all y ≠ 0. Hence y

y

2 ∫ g ( x,τ )dτ + 2 yh( x) ≥ 2 ∫ bτdτ + yh( x) 0

0

= (by + h( x)) 2 b −1 − b −1 h 2 ( x) ≥ −b −1 h 2 ( x).

(12)

Moreover, hypotheses (i) and (ii) of Theorem 1 imply that x

2a ∫ h(ξ )dξ = 2b 0

x

−1

∫ (ab − h′(ξ ))h(ξ )dξ + b

−1

h 2 ( x) ≥ (ab − c)b −1δ 0 x 2 + b −1 h 2 ( x) .

(13)

0

Combining the estimates (12) and (13), we obtain

V1 ≥ (ab − c)b −1δ 0 x 2 = k1 x 2 , (k1 = (ab − c)b −1 > 0)

(14)

for all x. Next, V2 can be rearranged as V2 = XQ0 X T ,

 αb αa  where X = ( x, y, z ), Q0 =  αa α + a 2 α a 

α

 a  and det Q0 = α 2 (b − α ) > 0 since b − α > 0, 1 

(which follows from (10)). Thus, we get V2 ≥ α 2 ( x 2 + y 2 + z 2 )

(15)

for all ( x, y, z ) ∈ ℜ 3 with α > 0. Combining the estimates (14) and (15), the lower inequality in (11) is obtained. Now to obtain the upper inequality in (11), we proceed as follows: Since h(0) = 0, hypothesis (i) of Theorem 1 implies that h( x) ≤ cx for all x ≠ 0. It follows from hypotheses (i) and (ii) of Theorem 1 and the inequality 2 m n ≤ m 2 + n 2 that V1 ≤ acx 2 + b1 y 2 + c 2 x 2 + y 2 = c(a + c) x 2 + (b1 + 1) y 2

(16)

and V2 ≤ α (b + 2α ) x 2 + (α + 3a 2 ) y 2 + 3z 2 .

(17)

In view of the estimates (16) and (17), the upper inequality in (11) can be easily obtained. From (9) it is clear that V (0,0,0) = 0, and the lower inequality in the inequalities (11) implies that V ( x, y, z ) > 0 as x 2 + y 2 + z 2 ≠ 0, hence, it follows that V ( x, y, z ) → ∞ as x 2 + y 2 + z 2 → ∞.

(18)

TUNC, AYHAN: NONLINEAR DIFFERENTIAL EQUATIONS

5

Now, the inequality (11) together with (18) establishes condition (i) of Lemma 1. Lemma 3. Under the hypotheses of Theorem 1, there are positive constants Di , (i = 2, 3, 4, 5) such that if ( x(t ), y (t ), z (t )) is any solution of the system (3), then

d V& = V ( x(t ), y (t ), z (t )) ≤ −( D2 x 2 + D3 y 2 + D4 z 2 ) + D5 ( x + y + z ) p(t ) . dt

(19)

Proof. Let ( x, y, z ) = ( x(t ), y (t ), z (t )) be a solution of (3). Along this solution, it follows from (9) and (3) that V& (t ) = −αxh( x) − {ayg ( x, y) − y 2 h ′( x)} − α {g ( x, y ) − by}x − (αx + ay + z ){ f ( x, y, z ) − az} y

+ y ∫ g x ( x,τ )dτ + (αx + ay + z ) p(t ) + αYQ1Y T ,

(20)

0

a 1

where Y = ( y, z ) Q1 = 

1 , and det Q1 = −1. 0 

In view of hypotheses of Theorem 1, we have 1 7 1 V& (t ) ≤ − αδ 0 x 2 − (αa + ab − c) y 2 − αz 2 − W j + (αx + ay + z ) p (t ), 2 8 2

(21)

( j = 1, 2, 3), where 1 1 W1 = α { δ 0 x 2 + ( g ( x, y ) − by ) x + (αa + ab − c) y 2 }, 4 16α

(22)

1 1 W2 = α { δ 0 x 2 + ( f ( x, y, z ) − az ) x + z 2 }, 4 4

(23)

1 α 2 z }, W3 = a{ (αa + ab − c) y 2 + ( f ( x, y, z ) − az ) y + 16a 4a

(24)

Using the estimates (22)-(24) and taking into consideration the following inequalities {g ( x, y ) − by}2
0 such that hAu, ui ≥ Ckuk2V ,

∀u ∈ V.

Moreover we have hAu, vi = hAv, ui, 3

∀u, v ∈ V,

488

DUAN, YANG: STOCHASTIC NAVIER-STOKES EQUATIONS

where h·, ·i denotes the dual pairing. Meanwhile the operator B : V × V → V ∗ is a bilinear operator which satisfies < B(u, v), v >= 0,

∀u, v ∈ V.

(2.7)

And there exists a constant C > 0 such that | < B(u, v), w > |2 ≤ CkukkukV kvkkvkV kwk2V ,

∀u, v, w ∈ V.

(2.8)

In our paper, we define the stochastic equation in the generalized sense as an evolution equation. Using relevant operators, stochastic equation (2.1) and corresponding conditions can be expressed equivalently as the following evolution equation. By studying the evolution equation, we can consider weak solutions of martingale type or strong solutions. ∂u ∂W + Au + B(u, u) = σ(u) , u ∈ H, t ∈ [0, T ], (2.9) ∂t ∂t Now we illustrate a definition of the correlative space and its norm. In order to investigate the error estimations, several important properties are also obtained. Let E denote the expectation. For any Hilbert space H, we define ½ ¾ Z 2 2 L2 (Ω; H) = v : EkvkH = kv(ω)kH dP(ω) < ∞ , (2.10) Ω

with norm kvkL2 (Ω;H) = (Ekvk2H )1/2 , see Yubin Yan [16]. Rt Then the representation of stochastic integral 0 v(s) dW (s) can be defined with v ∈ L2 (Ω; H). Meanwhile the isometry property holds as follows, more details see Yubin Yan [17]. °Z t °2 Z t ° ° ° E ° v(s) dW (s)° kEv(s)k2L2 (Ω;H) ds (2.11) ° = 0

0

With the above-mentioned items, a strong solution of stochastic Navier-Stokes equation RT 2 (2.1) is an adapted V -valued process (u(t)t∈[0,T ] ) with Eku(t)k < ∞ and E 0 ku(t)k2V ds < ∞. Moreover the strong solution satisfies the following form, see H.Breckner [9] for more details. Z t Z t (u(t), v) + < Au(s), v > ds + < B(u(s), u(s) >, v) ds 0 0 Z t = (u0 , v) + (σ(u(s)), v) dW (s) (2.12) 0

for all v ∈ V , t ∈ [0, T ]. The stochastic integral is understood in the IT oˆ sense. At the same time the evolution equation (2.9) admits a unique mild solution as follow. Z t Z t u(t) = E(t)u0 + E(t − s)σ(u) dW (s) − E(t − s)B(u, u) ds. (2.13) 0 −tA

where E(t) = e

0

is the analytic semigroup generated by −A. 4

DUAN, YANG: STOCHASTIC NAVIER-STOKES EQUATIONS

3

489

Discretization of the Stochastic Equation

Let Ωh be a polygonal approximation to Ω with the S boundary ∂Ωh . Then we consider a h family {T }h>0 of finite element spaces satisfied Ωh = K∈T h K, for all h > 0. Herein hK denotes the maximum diameter of the element K in T h , and h = maxK∈T h hK . It is convenient to assume the family {T h }h>0 to be shape regular. Herein we consider the finite element space Ωh . Moreover we assume that {T h } ⊂ H01 = {u ∈ L2 (Ω), ∇u ∈ L2 (Ω), u|∂Ω = 0}. To illustrate the finite element formulation of stochastic equation (2.9), we introduce the generalized L2 -projection operator Ph defined by (Ph σ, χ) =< σ, χ >,

∀χ ∈ Ωh .

See K. Chrysafinos and L. S. Hou [11], Yubin Yan [16], [17] for more details. Then, we assume for ∀ σ ∈ H l (Ω) k(I − Ph )σk ≤ chl kσkl .

(3.1)

By the generalized divergence free version of L2 projection, the semidiscrete finite element approximation corresponding to (2.9) holds: Find uh (·, t) ∈ Ωh such that ∂W ∂uh + Ah uh + Bh (uh , uh ) = Ph σ(uh ) , ∂t ∂t

t ∈ [0, T ],

(3.2)

where Ah : Ωh → Ωh is the discrete analogue of the operator A, and in the same way we have the definition of Bh . Furthermore it follows the scheme (2.13) that Z t uh (t) = Eh (t)u0 − Eh (t − s)Bh (uh (s), uh (s)) ds 0 Z t + Eh (t − s)Ph σ(uh (s)) dW (s). (3.3) 0

Let τ be a time step and tn = nτ with n ≥ 1. Considering the semidiscrete problem, the full discrete scheme of the main equation can be shown. We apply a linearized version of backward Euler scheme to (2.9): unh − un−1 n h + Ah unh + Bh (un−1 h , uh ) τZ 1 tn = Ph σ(un−1 n ≥ 1, u0 = Ph u0 , h ) dW (s), τ tn−1

(3.4)

which determines a sequence of solutions of finite dimensional evolution equations unh (n = 1, 2, . . . ). In order to give the proof of our main conclusion, we now introduce some priori estimates. First of all we recall a discrete version of the uniform Gronwall lemma which will be useful in our discussion, see R.Temam [13]. 5

490

DUAN, YANG: STOCHASTIC NAVIER-STOKES EQUATIONS

Lemma 3.1 Let τ , B, and aj , bj , cj , γj , for integers j ≥ 0, be nonnegative numbers such that an + τ

n X

bj ≤ τ

j=0

n X

γj aj + τ

j=0

n X

cj + B,

f or n ≥ 0.

cj + B),

f or n ≥ 0.

j=0

Suppose that τ γj < 1 for all j, then an + τ

n X

τ

bj ≤ e

Pn

γj j=0 1−τ γj

(τ

j=0

n X j=0

4 The Error Estimate From the prior approximation point of view, it is convenient to consider the error approximation of stochastic Navier-Stokes equation. It follows from equation (2.13) that Z tn E(tn − s)B(u(s), u(s)) ds u(tn ) = E(tn )u0 − 0 Z tn E(tn − s)σ(u(s)) dW (s). (4.1) + 0

Meanwhile we define Eτ h = I+τ1Ah , where I denoting the identity. Moreover Eτnh represents the n-th power of Eτ h . Then considering the full discrete scheme (3.4), there exists n X j n n uh = Eτ h u0 − Eτn−j+1 τ Bh (uj−1 h h , uh ) +

j=1 Z n X tj j=1

tj−1

Eτn−j+1 Ph σ(uj−1 h h ) dW (s),

(4.2)

where tj denotes the jth time step such that tj = τ j, j = 0, 1, 2, . . . , n. Now we can give the definition of the error estimation en = unh − u(tn ). For the sake of simplicity, it divides into three parts, denoted by Ik , k = 1, 2, 3 respectively. Generally speaking, the I1 and I2 are treated similarly with respect to the corresponding deterministic case. For I3 containing stochastic error, we will prove it in a different way. Herein the main error approximation can be shown as follows. en = [Eτnh − E(tn )]u0 Z tn n X j τ Bh (uj−1 E(tn − s)B(u(s), u(s)) ds − Eτn−j+1 + h , uh ) h 0

+

n Z tj X j=1

tj−1

j=1

Eτn−j+1 Ph σ(uj−1 h h )

Z

tn

E(tn − s)σ(u(s)) dW (s)

dW (s) − 0

= I1 + I2 + I3 .

(4.3) 6

DUAN, YANG: STOCHASTIC NAVIER-STOKES EQUATIONS

491

First of all we deal with the second part I2 of (4.3). The analysis is based on the approximation properties of the operators and related norm. Herein the estimation is illustrated by means of the derived results step by step. Theorem 4.1 Let I2 be the second part of the main error estimation en in (4.3). Then there exists a constant C = C(T ) such that à ! n X kI2 k2L2 (Ω;H) ≤ C h2l + τ E||ej−1 ||21 + τ 2 . j=1

Proof. We investigate the second part I2 as follows. Z

tn

I2 =

E(tn − s)B(u(s), u(s)) ds − 0

" =

+ +

j=1 n X

#

E(tn − s)[B(u(s), u(s)) − B(u(tj ), u(tj ))] ds

tj−1

"Z n X

j Eτn−j+1 τ Bh (uj−1 h h , uh )

j=1

n Z tj X j=1

n X

tj

tj−1

# E(tj − s) ds − Eτn−j+1 τ B(u(tj ), u(tj )) h

Eτn−j+1 τ [B(u(tj ), u(tj )) − B(u(tj−1 ), u(tj ))] h

j=1

" +

n X

Eτn−j+1 τ B(u(tj−1 ), u(tj )) − h

j=1

n X

# j Eτn−j+1 τ Bh (uj−1 h h , uh )

j=1

= I2,1 + I2,2 + I2,3 + I2,4 . First of all we give the estimation of the first section I2,1 . Making use of (2.8), there exists kB(u(s), u(s)) − B(u(tj ), u(tj ))k ≤ C(s − tj ). Therefore with respect to the L2 (Ω; H) norm, we can hold °2 ° n Z ° °X tj ° ° E(tn − s)[B(u(s), u(s)) − B(u(tj ), u(tj ))] ds° kI2,1 k2L2 (Ω;H) = ° ° ° j=1 tj−1 L2 (Ω;H) °2 ° n Z ° °X tj ° ° E(tn − s)[B(u(s), u(s)) − B(u(tj ), u(tj ))] ds° = E° ° ° j=1 tj−1 n Z tj X ≤ C (s − tj )2 ds ≤ Cτ 2 . j=1

tj−1

7

492

DUAN, YANG: STOCHASTIC NAVIER-STOKES EQUATIONS

For I2,2 , with the definition of E(t), there holds kI2,2 k2L2 (Ω;H)

°"Z °2 # n ° ° tj X ° ° = ° E(tj − s) ds − Eτn−j+1 τ B(u(t ), u(t )) j j ° h ° tj−1 ° j=1

n Z tj °X

° ≤ E°

j=1

L2 (Ω;H)

£ E(tj − s) − E(tj − tj−1 )

tj−1

°2 ¤ ° +E(tj − tj−1 ) − Eτn−j+1 dsB(u(t ), u(t )) j j ° h ° °2 n Z tj °X ° ° ° [E(tj − s) − E(tj − tj−1 )] dsB(u(tj ), u(tj ))° ≤ E° ° ° j=1 tj−1 °2 ° n Z ° °X tj ° ° n−j+1 [E(tj − tj−1 ) − Eτ h ] dsB(u(tj ), u(tj ))° +E ° ° ° tj−1 j=1

2

≤ Cτ . For I2,3 , considering kEτ h k ≤ 1, there exists

kI2,3 k2L2 (Ω;H)

° °2 n °X ° ° ° = ° Eτn−j+1 τ [B(u(t ) − u(t ), u(t ))] ° j j−1 j h ° ° j=1 L2 (Ω;H) ° °2 n °X ° ° ° n−j+1 ≤ E° Eτ h τ [B(u(tj ) − u(tj−1 ), u(tj ))]° ° ° j=1 2

≤ Cτ .

Then we deal with the last part I2,4 . Since u0 = u(t0 ) = u0 ,

8

DUAN, YANG: STOCHASTIC NAVIER-STOKES EQUATIONS

kI2,4 k2L2 (Ω;H) = =

≤

≤

493

° °2 n n °X ° X ° n−j+1 n−j+1 j−1 j ° Eτ h τ B(u(tj−1 ), u(tj )) − Eτ h τ Bh (uh , uh )° ° ° ° j=1 j=1 L2 (Ω;H) ° n °X £ ° Eτn−j+1 τ B(u(tj−1 ), u(tj )) − Bh (uj−1 , uj ) ° h ° j=1 °2 ° ¤ j ° j−1 j−1 j−1 +Bh (uj−1 , uj ) − Bh (uh , uj ) + Bh (uh , uj ) − Bh (uh , uh ) ° ° L2 (Ω;H) ° °2 ° °2 n n °X ° °X ° ° ° ° ° τ (I − Ph )B(uj−1 , uj )° +° τ Bh (ej−1 , uj )° ° ° ° ° ° j=1 j=1 L2 (Ω;H) L2 (Ω;H) ° °2 n−1 ° °X ° ° +° τ Bh (uj−1 , e ) + cτ 2 j ° h ° ° j=1 L2 (Ω;H) " # n X C h2l + τ E||ej−1 ||21 + τ 2 . j=1

The proof is now completed. Then we deal with the third part I3 of (4.3). The analysis is based on the properties with respect to the projection operators Ph and the stochastic integral. Theorem 4.2 Let I3 be the third part of the main error estimation en in (4.3). Then there exists à n ! X kI3 k2L2 (Ω;H) ≤ C τ Ekej−1 k2 + τ 2 . j=1

Proof. Considering the definitions and relevant properties of E(t) and Eτnh , we deal with I3 as following.

9

494

DUAN, YANG: STOCHASTIC NAVIER-STOKES EQUATIONS

I3 = =

n Z X

tj

j=1 tj−1 n Z tj X j=1

+ + +

tj−1

Z Eτn−j+1 Ph σ(uj−1 h h )

tn

E(tn − s)σ(u(s)) dW (s) 0

Eτn−j+1 Ph (σ(uj−1 h h ) − σ(u(tj−1 ))) dW (s)

n Z tj X j=1 tj−1 n Z tj X j=1 tj−1 n Z tj X j=1

dW (s) −

Eτn−j+1 Ph (σ(u(tj−1 )) − σ(u(s))) dW (s) h Ph − E(tn − tj−1 ))σ(u(s)) dW (s) (Eτn−j+1 h (E(tn − tj−1 ) − E(tn − s))σ(u(s)) dW (s)

tj−1

= I3,1 + I3,2 + I3,3 + I3,4 .

(4.4)

For I3,1 , it follows the isometry property (2.11) and Lipschitz conditon (2.2) that

kI3,1 k2L2 (Ω;H)

° n Z °2 °X tj ° ° ° j−1 = E° Eτn−j+1 P (σ(u ) − σ(u(t ))) dW (s) ° h j−1 h h ° ° tj−1 j=1

n X ° £ ¤°2 ° ≤ τ °E Eτn−j+1 Ph (σ(uj−1 h h ) − σ(u(tj−1 ))) j=1 n X ° ° °2 ° ≤ τ °Eτn−j+1 Ph ||2 E°σ(uj−1 h h ) − σ(u(tj−1 )) j=1

≤ C

n X

τ Ekuj−1 − u(tj−1 )k2 h

j=1

≤ C

n X

τ Ekej−1 k2 .

(4.5)

j=1

10

DUAN, YANG: STOCHASTIC NAVIER-STOKES EQUATIONS

495

For I3,2 , it is convenient to obtain the following scheme similarly to (4.5). ° °2 n Z tj °X ° ° ° n−j+1 2 kI3,2 kL2 (Ω;H) = E ° Eτ h (σ(u(tj−1 )) − σ(u(s))) dW (s)° ° ° j=1 tj−1 n Z tj X ° £ n−j+1 ¤°2 °E E ° = (σ(u(t )) − σ(u(s))) ds j−1 τh L2 (Ω;H) tj−1

j=1

≤ C

à n Z X

!

tj

(s − tj−1 )2 ds

≤ Cτ 2 .

(4.6)

tj−1

j=1

In order to estimate I3,3 and I3,4 , we only need to focus on the important operators E and Eτ h. Since for every j = 1, 2, · · · , n, there holds 1 )n−j+1 − etn −tj−1 I + τ Ah = (I + τ Ah )−(n−j+1) − e−τ A(n−j+1) ≤ C(T, j)τ

Eτn−j+1 − E(tn − tj−1 ) = ( h

Hence kI3,3 k2L2 (Ω;H)

° °2 n Z tj °X ° ° ° (Eτn−j+1 = E° − E(t − t ))σ(u(s)) dW (s) ° n j−1 h ° ° j=1 tj−1 n Z tj X Ek(Eτn−j+1 = − E(tn − tj−1 ))σ(u(s))k2 ds h tj−1

j=1

≤ C

n Z X

tj

τ 2 ds = Cτ 2 .

(4.7)

tj−1

j=1

Similarly, we can get the following conclusion. ° °2 n Z tj °X ° ° ° kI3,4 k2L2 (Ω;H) = E ° (E(tn − tj−1 ) − E(tn − s))σ(u(s)) dW (s)° ° ° j=1 tj−1 n Z tj X = EkE(tn − s)(I − E(s − tj−1 ))σ(u(s))k2 ds j=1

≤ C

tj−1

n Z X j=1

tj

τ 2 ds = Cτ 2 .

(4.8)

tj−1

Together these estimations above, there holds the inequality (4.4). Now the proof is complete. With the prior estimations, we can arrive at the main error approximation about the stochastic Navier-Stokes equation. The main results of this paper are given in the following theorem. 11

496

DUAN, YANG: STOCHASTIC NAVIER-STOKES EQUATIONS

Theorem 4.3 Let u(tn ) and unh be the solutions of (2.9) and (3.4) respectively. There exists a constant C = C(T ) satisfied ken kL2 (Ω;H) ≤ C(τ + hl ). Proof. Now we consider the first part I1 of (4.3). Herein I1 can be illustrated as follows. kI1 k2L2 (Ω;H) = k[Eτnh − E(tn )]u0 k2L2 (Ω;H) = Ek[Eτnh − E(tn )]u0 k2 = Ek[(I + τ Ah )−n − e−nτ A ]u0 k2 ≤ Cτ 2 .

(4.9)

Hence by the conclusions in Theorems (4.1) and (4.2), we can obtain ken k2L2 (Ω;H) ≤ kI1 k2L2 (Ω;H) + kI2 k2L2 (Ω;H) + kI3 k2L2 (Ω;H) ≤

kI1 k2L2 (Ω;H) Ã

≤ C

+

4 X

kI2,k k2L2 (Ω;H)

k=1 n−1 X

τ 2 + h2l +

kI3,k k2L2 (Ω;H)

k=1

! τ Ekej k21

+

4 X

.

j=1

Then it follows the discrete Gronwall lemma that ken k2L2 (Ω;H) ≤ C(τ 2 + h2l ), which implies that ken kL2 (Ω;H) ≤ C(τ + hl ). The proof is now complete.

5

Conclusion

In this paper we consider the error estimation of a stochastic equation of Navier-Stokes type. At first, we introduce the basic properties of our main equation and some related operators. Meanwhile the wild solution is illustrate by the defined operators. The L2 space and its norm are referred in order to study the stochastic part. Then we consider the discretization of the main equation with respect to the finite element approximation. Herein the semidiscrete scheme and the full discrete scheme are obtained, and the prior estimates are proved. Furthermore we give the proof of some preliminary conclusions. In our main theorem, the optimal convergence error approximation is shown.

12

DUAN, YANG: STOCHASTIC NAVIER-STOKES EQUATIONS

497

References [1] A. Bensoussan, Stochastic Navier-Stokes Equations, Acta Appl. Math. 38:3, (1995), pp.267-304. [2] A. Bensoussan and R. Temam, Equations Stochastiques du Type Navier-Stokes, Journal of Functional Analysis 13, no. 3, (1973), pp.195-222. [3] M. Capinski and N.J. Cutland, Nonstandard Methods for Stochastic Fluid Mechanics, World Scientific Publishing Co. (1995). [4] M. Capinski and D. Gatarek, Stochastic Equations in Hilbert Space with Application to Navier-Stokes Equations in any Dimension, J. Funct. Anal. 126:1, (1994), pp.26-35. [5] C. Foias, Statistical study of Navier-Stokes equations, I. Rend. Sem. Math. Padova 48, (1972), pp.219-348. [6] E. Hopf, Statistical Hydromechanics and Functional Calculus, J. Rational Mech. and Anal. 1, (1952), pp.87-123. [7] W. Grecksch and P.E. Kloeden, Time-discretized Galerkin approximations of stochastic parabolic PDEs, Bull. Austral. Math. Soc. 54, (1996), pp.79-85. [8] I. Gy¨ongy, On the approximation of stochastic partial differential equations I, II., Stochastics 25:2 (1988), 53-85, Stochastics 26:3, (1989), pp.129-164. [9] H. Breckner, Galerkin Approximation and the Strong Solution of the Navier-Stokes Equation, Journal of Applied Mathematics and Stochastic Analysis, 13:3, (2000), pp.239-259. [10] Jonathan C. Mattingly, On Recent Progress for the Stochastic Navier Stokes Equations, ´ Journ´ees Equations aux d´eriv´ees partielles, (Forges-les-Eaux), (2003), Exp. No. XI. [11] K. Chrysafinos and L.S. Hou, Error estimates for semidiscrete finite element approximations of linear and semilinear parabolic equations under minimal regularity assumptions, SIAM J. Numer. Anal., 40, (2002), pp.282-306. [12] A.I. Komech and M.I. Vishik, Statistical solutions of the Navier-Stokes and Nuler Equations, Adv. in Mechanics 5:1-2, (1982), pp.65-120. [13] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Second Edition, New York: Springer-Verlag, (1997). [14] R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, North-Holland, Amsterdam, (1979). [15] M. Viot, Solution faible d’´equation aux deriv´ees partielles stochastiques non lin´eares, Th´ese doct. Sci. Math. Paris, (1976). 13

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DUAN, YANG: STOCHASTIC NAVIER-STOKES EQUATIONS

[16] Yubin Yan, Galerkin Finete Element Methods for Stochastic Parabolic Partial Differential Equations, SIAM J. NUMER. ANAL. Vol. 43, No.4, (2005), pp.1363-1384. [17] Yubin Yan, Semidiscrete Galerkin Approximation for a Linear Stochastic Parabolic Partial Differential Equation Driven by an Additive Noise, BIT Numerical Mathematics, 44,(2004), pp.829-847.

14

JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.3, 499-513, 2011, COPYRIGHT 2011 EUDOXUS PRESS, 499 LLC

WEIGHTED APPROXIMATION PROPERTIES OF GENERALIZED PICARD OPERATORS BA¸ S AR YILMAZ, ALI ARAL, AND GÜLEN BA¸ S CANBAZ-TUNCA Abstract. In this work, we continue the study of generalized Picard operator P ; ([2]) depending on nonisotropic -distance, in the direction of weighted approximation process. For this purpose, we …rst de…ne weighted n-dimensional Lp space by involving weight depending on nonisotropic distance. Then we introduce a new weighted -Lebesgue point depending on nonisotropic distance and study pointwise approximation of P ; to the unit operator at these points. Also, we compare the order of convergence at the weighted -Lebesgue point with the order of convergence of the operators to the unit operator. Finally, we show that this type of convergence also occurs with respect to nonisotropic weighted norm.

1. Introduction In some recent papers, various results for the q modi…cation of approximation operators have been increasingly studied. For the brief knowledge, it may be useful to refer to the work of Anastassiou and Aral [2] and references therein. As it is known, one of the central research directions of approximation theory is singular integral operators. Among others, we are interested in Picard singular operator in multivariate setting de…ned as (1.1)

P (f ) (x) = Q n

i=1

1 (2 i )

Z1 1

:::

Z1 1

f (x1 + t1 ; :::; xn + tn ) dt1 :::dtn n jti j Q e i i=1

for = ( 1 ; :::; n ) > 0; which means that each component i (i = 1; :::; n) is positive and x = (x1 ; :::; xn ) 2 Rn : For a general framework related to this operator, [3], [4], [5], [12], and [13] may be referred. In [9] and [10], multivariate Picard and Gauss-Weierstrass operators with kernels including nonisotropic distance were introduced and pointwise convergence results were given. Yet, q generalization of Picard and Gauss-Weierstrass singular integral operators have been stated and some approximation properties in weighted space have been discussed, also complex variants of them have been studied ([6], [7], [8]). Recently, another interesting improvement related to the multivariate q Picard singular operators depending on nonisotropic norm, P ; , has been subsequently stated in [2]. Here, the authors have investigated pointwise convergence of the family of P ; (f ) to f at the so called Lebesgue points depending on nonisotropic distance. Moreover, they 2000 Mathematics Subject Classi…cation. Primary 42B20; Secondary 45P05. Key words and phrases. Generalized Picard singular integral; Nonisotropic distance; Lebesgue point. 1

500

2

BA S ¸ AR YILM AZ, ALI ARAL, AND GÜLEN BA S ¸ CANBAZ-TUNCA

have introduced a suitable modulus of continuity, depending on nonisotropic distance with supremum norm to measure the rate of convergence. Also they have proved the global smoothness preservation property of these operators. In this work, for a weight depending on nonisotropic distance, we give analogues de…nitions of n dimensional nonisotropic weighted Lp space and nonisotropic weighted Lebesgue point at which we obtain a pointwise convergence result for the family of P ; (f ) to f for f belonging to this weighted space. We also give the measure of the rate of this pointwise convergence. Convergence in the norm of this space is also discussed. Suppose that Rn is the n dimensional Euclidean space of vectors x = (x1 ; :::; xn ) with real components and let n 2 N and = ( 1; 2; ; n ) 2 Rn with each component is positive, i.e. i > 0 (i = 1; :::; n) : Using standard notation, we denote j j = 1 + 2 + + n . Recall that the nonisotropic -distance between x and 0 is de…ned as 1 1

jjxjj = jx1 j

+

+ jxn j

j j n

1 n

; x 2 Rn :

Note that, for t > 0; jjxjj is homogeneous, namely t 1 x1

1 1

+

+ t

n

xn

1 n

j j n

=t

j j n

jjxjj

and has the following properties: (1) jjxjj = 0 , x = 0;

(2)

t x

=t

(3) jjx + yjj

j j n

M

jjxjj ; jjxjj + jjyjj

; 1+

1

j j

n min where min = min f 1 ; 2 ; : : : n g and M = 2 (see [15]). 1 We should note here that when i = 2 (i = 1; 2; : : : ; n) ; the nonisotropic distance jjxjj becomes the ordinary Euclidean distance jxj and also that jj:jj does not satisfy the triangle inequality. For convenience of exposition, we present here some known results concerning q numbers. Suppose we are given a q > 0; then q number is ( 1 q q 6= 1 1 q ; [ ]q = ; q=1

for all nonnegative : If is an integer, i.e. = n for some n; we write [n]q and call it q integer. The q extension of exponential function ex is de…ned as (1.2)

Eq (x) :=

n(n 1 X q 2

n=0

where (a; q)n =

n Q

k=0

1

1)

(q; q)n

xn = ( x; q)1 ;

aq k and ( x; q)1 =

Also, we de…ne a q factorial as [n]q ! =

[n]q [n 1

1]q

1 Q

1 + xq k :

k=0

[1]q ;

n2N : n=0

501

W EIGHTED APPROXIM ATION

For integers 0

k

3

n; the q binomial coe¢ cients are given by n k

[n]q !

=

[k]q ! [n

q

k]q !

:

For more information we refer to [14]. Recall from [11] and [1] that the q extension of Euler integral representation for the Gamma function is de…ned as Z1 tx 1 1 q x(x 1) 2 q dt ; Re x > 0 (1.3) cq (x) q (x) = 1 ln q Eq ((1 q) t) 0

for 0 < q < 1; where

q

(x) is the q extension of Gamma function de…ned by

q

(x) =

(q; q)1 (1 (q x ; q)1

1 x

q)

; 0 0; n 2 N and + n : The generalized q-Picard i 2 (0; 1) (i = 1; 2; : : : ; n) with j j = 1 + 2 + singular integral depending on nonisotropic -distance, attached to f is de…ned as P (1.7)

;

(f ; q; x)

P :

=

(f ; x) Z c(n; ; q)

;

j j

[ ]q

Rn

f (x + t) P ( ; t) dt;

where P ( ; t) and c(n; ; q) are de…ned as in (1.5) and (1.6), respectively.

502

4

BA S ¸ AR YILM AZ, ALI ARAL, AND GÜLEN BA S ¸ CANBAZ-TUNCA

The case i = 12 (i = 1; 2; : : : ; n) clearly gives the operators P ; 12 (f ; q; x) introduced in [7]. Letting q ! 1 imply that P ; 21 (f ; 1; x) will be the classical multivariate Picard singular integrals (1.1). Now, we present the following de…nition. De…nition 2. Let p; 1 p < 1; be …xed. By Lp; (Rn ) we denote the weighted space with nonisotropic distance of real valued functions f de…ned on Rn for which f (x) n 1+jjxjj is p-absolutely Lebesgue integrable on R such that the norm 0

kf kp; = @

Z

f (x) 1 + jjxjj

Rn

is …nite. For the case p = 1; we also have ( f (x) kf k1; = sup 1 + jjxjj

1 p1

p

dxA

: x = (x1 ; :::; xn ) 2 R

n

)

:

Lemma 2. Let > 0; n 2 N and i 2 (0; 1) (i = 1; 2; : : : n ) with j j = 1+ 2+ + n . P ; (f ) is a linear positive operator from the space Lp; (Rn ) into Lp; (Rn ) : That is kP

(f )kp;

;

K (n; ; q) kf kp; ;

where K (n; ; q) = max f1; M g 1 +

j j n c(n; ; q) [ ]qn ! 2j j

;n 1 q

(n + 1)

ln q 1 q 1 q

n(n+1) 2

in which M is the number appeared in the property 3 of nonisotropic distance. Proof. Using generalized Minkowsky inequality we have 0 1 p1 p Z P ; (f ; x) kP ; (f ; x)kp; = @ dxA 1 + jjxjj Rn

=

(1.8)

0 1 p1 p Z Z c(n; ; q) @ 1 f (x + t) P ( ; t) dt dxA j j 1 + jjxjj [ ]q Rn Rn 0 11 p p Z Z c(n; ; q) @ f (x + t) dx A P ( ; t) dt: j j 1 + jjxjj [ ] q

Rn

Rn

From the property 3 we have 1 + jjx + tjj 1 + jjxjj

(1.9)

1+M

jjxjj + jjtjj 1 + jjxjj

max f1; M g

1 + jjxjj

1 + jjtjj

1 + jjxjj

:

;

503

W EIGHTED APPROXIM ATION

5

Taking into account (1.9) and Lemma 1; the inequality (1.8) reduces to kP

;

(f ; x)kp; c(n; ; q)

p

1 p1

f (x + t) dx A P ( ; t) 1 + jjtjj 1 + jjx + tjj [ Rn Rn 0 1 Z c(n; ; q) max f1; M g @1 + P ( ; t) jjtjj dtA : j j [ ]q n

max f1; M g kf kp;

0 Z @

Z

j j ]q

dt

R

The substitution t = [ ]q x it follows that 0 kP

;

j j n

kf kp; max f1; M g @1 + c(n; ; q) [ ]q

(f ; x)kp;

Z

1

jjxjj

E (1 Rn q

dxA :

q) jjxjj

Now, we shall use generalized -spherical coordinates ([15]) by taking the following transformation into account x1

=

x2

=

xn

(u cos

(u sin .. . = (u sin

1

xn

=

(u sin

2 1) 1

cos

2 2)

2

1

sin

2

sin

n 2

1

sin

2

sin

2 n 1)

where 0 ; n 2 ;0 1; 2; transformation is denoted by J (u; J (u; ( ) = 2n

where

1

:::

n

1; : : : ;

nQ1

1

(cos

n 1) 2 j)

j=1

(1.10)

!

;n 1

=

j

Z

Sn

is …nite. Here S kP

;

kf kp; kf kp;

n 1

n

= u2j 1

n

n

1

;

2 ; u 0: The Jacobian of this ) and obtained as 1

n 1 1; : : : ;

2 n 1)

cos

(sin

j 1

j)

j+1 P

k=j

( ); 2

k

1

. Clearly the integral

( )d 1

n

is the unit sphere in R : Thus we have

(f ; x)kp;

0

1 ( ) d du A max f1; M g @1 + c(n; ; q) [ ]q 2j j n E (1 q) u q 0 Sn 1 0 1 Z1 n j j n u du A max f1; M g @1 + c(n; ; q) [ ]qn ! ; n 1 2j j Eq ((1 q) u) j j n

Z1 Z

u2j

j+

2j j n

1

0

kf kp;

j j n max f1; M g 1 + c(n; ; q) [ ]qn ! 2j j

The lemma is proved.

;n 1 q

(n + 1)

ln q 1 q 1 q

n(n+1) 2

:

504

6

BA S ¸ AR YILM AZ, ALI ARAL, AND GÜLEN BA S ¸ CANBAZ-TUNCA

2. Pointwise convergence This section provides a result related to pointwise convergence. Below, we …rst give the de…nition of the points at which pointwise convergence will be observed. De…nition 3. Let f 2 Lp; (Rn ) ; 1 p < 1; and i 2 (0; 1) (i = 1; :::; n) with j j = 1 + 2 + ::: + n : We say that x 2Rn is weighted Lebesgue point of f provided 9 p1 8 > > > > p > > Z = < 1 f (x + t) f (x) dt lim = 0: 2j j > h!0 > 1 + ktk h > > n > > ; : 2j j ktk

P

0; there exists an > 0 such that h < implies that 8 9 p1 > > > > p > > Z < 1 = f (x + t) f (x) dt < ; > > 1 + ktk h2j j > > n > > : ; 2j j ktk

which clearly means that Z ktk

n 2j j

p = < R c(n; ;q) f (x+t) f (x) P ( ; t) dt jP ; (f ; q; x) f (x)j j j 1+ktk > > n ; : [ ]q 2j j ktk < 9 q1 8 > > = < q R c(n; ;q) P ( ; t) dt 1 + ktk j j > > n ; : [ ]q 2j j < ktk R + c(n;j ;q) jf (x + t) f (x)j P ( ; t) dt j [ ]q

ktk

1+

n 2j j

2j j n

+ c(n;j ;q) j [ ]q

ktk

R

8 >
[ ]jq j

:

ktk

R

n 2j j

p f (x+t) f (x) 1+ktk
= > ;

= J1 ( ) + J2 ( ) : Now passing to the generalized spherical coordinates, J1 ( ) gives rise to (2.2) 8 9 p1 p Z Z < = x+ (u ) f (x) f 2j j ( ) u2j j 1 P 0 ( ; u) d du ; J1 ( ) = 1 + n 2j j : ; 1+u n n 1 0 S

where (2.3)

P 0 ( ; u) =

c (n; ; q) j j ]q Eq

[

2j j n j j n ]q

(1 q)u [

!:

Therefore taking into account (2.1) ; (2.2) can be expressed as 8 9 p1

: [ ]q

P ( ; t)

1

) q1

sup ktk

+

1 + ktk

n 2j j

c (n; ; q) j j

[ ]q

jf (x)j

9 q1 > = > ;

P ( ; t)

j j

[ ]q

=

sup ktk

c (n; ; q) j j

[ ]q

1 + ktk

n 2j j

1 + ktk

sup ktk

P ( ; t)

n 2j j

Eq

j j

[ ]q n j j

c (n; ; q) j j

[ ]q

(2.5)

[ ]qn

1 1

!

(1 q)ktk

sup

q

ktk

j j

c (n; ; q) [ ]qn

8 > >

> :

which clearly tends to zero as

n Q

n 2j j

(n+1)

1

q + (1

j j

[ ]qn + (1

k=0 1 1 q

n Q

k=0

! 0:

j j

[ ]qn + (1

j j n

[ ]q

+ (1

q)

q) q k

q) ktk

q) q k ktk 2j j n

2j j n

9 > > = > > ;

;

> ;

P ( ; t)

The …rst factor including supremum norm on the right hand side of (2.4) tends to zero as ! 0; indeed c (n; ; q)

9 p1 > =

1

:

508

10

BA S ¸ AR YILM AZ, ALI ARAL, AND GÜLEN BA S ¸ CANBAZ-TUNCA

For the second factor in (2.4) we have c (n; ; q) j j

[ ]q

1 + ktk

P ( ; t)

c(n; ; q) 1

j j

[ ]q

jjtjj

Z

Z

= c(n; ; q) jjtjj

1 + jjtjj

n 2j j

n 2j j

1 Eq (1 p

+ c(n; ; q) [ ]qn

jjtjj

dt

q) jjtjj

[ ]q

j j

(2.6)

P ( ; t) dt

Z

jjtjj Eq (1

n 2j j

p

q) jjtjj

[ ]q

1 is integrable on [0; 1) the …rst and last terms of ((1 q)jjtjj ) (2.6) tend to zero as ! 0: Finally, from the …nal proof of Theorem 2 in [2], the last term of (2.4) also approaches zero as ! 0: Hence we obtain the assertion of the theorem.

Since the function E

q

3. Order of pointwise convergence Now, we shall discuss the order of pointwise convergence that we have already presented above. For this purpose we give the following generalization of the concept of weighted Lebesgue point. De…nition 4. Let f 2 Lp; (Rn ) ; 1 p < 1; and i ; 2 (0; 1) (i = 1; :::; n) with j j = 1 + 2 + ::: + n : We say that x 2Rn is weighted ; Lebesgue point of f provided

lim

8 > > >
>h

> :

ktk

Z

n 2j j

> > = > > > ;

= 0:

2 (0; 1) (i = 0; 1; :::; n) and ( ! 0)

< min

n

2j j n ;

o p :

Proof. Let x be a weighted ; Lebesgue point of f; then for any exists an > 0 such that h < implies that (3.1) ktk

Z

n 2j j

0; there

dt:

509

W EIGHTED APPROXIM ATION

Transforming into the generalized

ktk

Z

n 2j j

Zh Z

=

0 Sn

spherical coordinates, if h
; 1

;

where c (n; ; q)

(3.4)

j j

[ ]q

sup ktk

P ( ; t) = o [ ]q2

1 + ktk

n 2j j

( ! 0) :

Indeed, from (2.5) we can reach to the following inequality c (n; ; q) j j

[ ]q2 [ ]q

sup ktk

1 + ktk

n 2j j

c (n; ; q) [ ]qn

2

j j n

1

[ ]q

1 q

n Q > > :

+ (1

j j

[ ]qn + (1

k=0

which gives that (3.4) holds for (3.5)

j j

P ( ; t)

8 > >

> = > > ;

( ! 0) :

n

and making the substitution t =[ ]q x; it Actually, by taking into account the fact that ktk 2j j follows that Z c (n; ; q) c (n; ; q) dt ! jf (x)j P ( ; t) 1 = jf (x)j j j j j (1 q)ktk [ ]q2 [ ]q [ ]q2P [ ]q n Eq 2j j j j ktk [ ]q n

c (n; ; q) j j

[ ]q2P [ ]q

jf (x)j ktk

=

c (n; ; q)

jf (x)j jjtjj

n

Z

ktk 2j j dt

n 2j j

Z

n 2j j

(1 q)ktk

Eq

j j

[ ]q n P

!

n

[ ]q 2P ktk 2j j dt Eq (1 p

q) ktk

[ ]q

tends to zero as ! 0; which indicates that (3.5) is satis…ed. Finally for the other factor of J2 ( ) we get lim

!0

c (n; ; q) j j

[ ]q

1 + ktk

P ( ; t)

1

=0

;

511

W EIGHTED APPROXIM ATION

13

as in (2.6) and this completes the proof.

4. Norm convergence This section we give a convergence result in the norm of Lp; (Rn ) : Theorem 3. Let f 2 Lp; (Rn ) ; 1 p < 1; with j j = 1 + 2 + ::: + n : If the following condition (4.1)

lim

h!0

1 h2j

j jjtjj

Z

n 2j j

jjf (x + t)

i

2 (0; 1) (i = 0; 1; :::; n) and

f (x)jjp; dt = 0;

0; there exists a > 0 such that h < implies that Z jjf (x + t) f (x)jjp; dt < h2j j : jjtjj

n 2j j

0 and L > 0, there exists δ > 0 such that for all x, y ∈ X, !p !p ∞ ∞ X X 1 X 1 X |x(i) + y(i)| − |x(i)| < ε, λ λ n n n=1 n=1 i∈In

i∈In

whenever ∞ X n=1

1 X |x(i)| λn i∈In

!p 0 and (xn ) ⊂ B(Vp (λ)) with sep ({xn }) > ε. Let xm n = (0, 0, ..., xn (m), ∞ xn (m+1), ...) for each m ∈ N. Since for each i ∈ N, {xn (i)}i=1 is bounded therefore using the diagonal method one can find a subsequence {xnk } of {xn } such that the sequence {xnk (i)} converges for each i ∈ N. Therefore, there exists an increasing sequence of positive integer (km ) such that ∞ sep({xm nk }k>km ) ≥ ε. Hence there is a sequence of positive integers (nm )m=1 with n1 < n2 < n3 < ... such that

m ε

xn ≥ m 2

(3.1) for all m ∈ N. Write Ip (x) =

∞ X n=1

!p 1 λn

X

|x(i)|

and put ε1 =

kp−1 −1 ε p 2kp (k−1) ( 2 ) .

Then by Lemma

i∈In

3.1, there exists δ > 0 such that |Ip (x + y) − Ip (x)| < ε1

(3.2)

whenever Ip (x) ≤ 1 and Ip (y) ≤ δ (see[6]). 1 There exists m1 ∈ N such that Ip (xm 1 ) ≤ δ. Next there exists m2 > m1 such m2 that Ip (x2 ) ≤ δ. In such a way, there exists m2 < m3 < ... < mk−1 such that m Ip (xj j ) ≤ δ for all j = 1, 2, ..., k − 1. Define mk = mk−1 + 1. By condition (3.1),
ε p k there exists nk ∈ N such that Ip (xm nk ) ≥ 2 . Put ni = i for 1 ≤ i ≤ k − 1. Then in virtue of (3.1), (3.2) and convexity of the function f (u) = |u|p , we get m1 X xn + xn2 + ... + xnk Ip ( 1 )= k n=1

!p 1 X xn1 (i) + xn2 (i) + ... + xnk (i) + λn k i∈In

!p 1 X xn1 (i) + xn2 (i) + ... + xnk (i) + λn k n=m1 +1 i∈In !p m1 k X 1 X 1 X ≤ xnj (i) k j=1 λn n=1 i∈In !p ∞ X 1 X xn2 (i) + xn3 (i) + ... + xnk (i) + + ε1 λn k n=m1 +1 i∈In !p !p m1 m2 k X X xn (i) + xn (i) + ... + xn (i) X 1 X 1 X 1 2 3 k + = xnj (i) + k j=1 λn λn k n=1 n=m1 +1 i∈In i∈In !p m3 X 1 X xn2 (i) + xn3 (i) + ... + xnk−1 (i) + xnk (i) + + ε1 λn k n=m +1 ∞ X

2

i∈In

570

˙ S ˙ ¸ EK NECIP ¸ IMS

6

m1 k X 1X ≤ k j=1 n=1 ∞ X

+

n=m2 +1

!p 1 X + xnj (i) + λ n n=m1 i∈In i∈In !p 1 X xn3 (i) + xn4 (i) + ... + xnk−1 (i) + xnk (i) + 2ε1 λn k

1 X xnj (i) λn

!p

m2 X

k 1X k j=2 +1

i∈In

··· ≤

≤

=

≤ ≤ =

!p
mk−1 Ip (xn1 ) + ... + Ip xnk−1 1 X 1 X |xnk (i)| + + k k n=1 λn i∈In !!p ∞ X X xn (i) 1 k + (k − 1)ε1 + k λn n=mk−1 +1 i∈In !p !p mk−1 ∞ X k−1 1 X 1 1 X 1 X |xnk (i)| + p |xnk (i)| + (k − 1)ε1 + k k n=1 λn k n=m +1 λn i∈In i∈In k−1  !p  !p ∞ ∞ X X X 1 1 X 1 1 1+ 1− |xnk (i)|  + p |xnk (i)| + (k − 1)ε1 k λn k n=m +1 λn n=mk−1 +1 i∈In i∈In k−1 !p  p−1  ∞ X k −1 1 X 1 + (k − 1)ε1 − |xnk (i)| kp λn n=mk−1 +1 i∈In   p−1 k − 1  ε p 1 + (k − 1)ε1 − kp 2  p−1   p 1 k −1 ε 1− 2 kp 2

Under the condition (3.1), Vp (λ) is (k − N U C) for any integer k ≥ 2. Theorem 3.3. For any (1 < p < ∞), the space Vp (λ) has the uniform Opial property. Proof. Let ε > 0 and ε0 ∈ (0, ε). Also let x ∈ X and ||x|| ≥ ε. There exists n1 ∈ N such that ∞ X n=n1 +1

!p  ε p 1 X 0 |x(i)| < . λn 4 i∈In

Hence we have

∞

X

ε

0 x(i)ei < ,

4 i=n +1 1

571

ON SOME GEOMETRIC AND TOPOLOGICAL PROPERTIES OF SEQUENCE SPACES

7

i th

where ei = (0, 0, ..., 1 , 0, 0, ...). Furthermore, we have !p n1 ∞ X X 1 X p ε ≤ |x(i)| + λn n=1 n=n1 +1 i∈In !p n 1  ε p X 1 X 0 < |x(i)| + λ 4 n n=1 i∈In !p n 1  ε p X 1 X 0 p ε − |x(i)| < 4 λn n=1

!p 1 X |x(i)| λn i∈In

i∈In

whence p

ε −

 ε p 0

4

≤

n1 X n=1

1 X |x(i)| λn

!p .

i∈In

Since xm (i) → 0 for i = 1, 2, ... , we choose any weakly null sequences {xm } such that lim inf kxm k ≥ 1. Then there exists m0 ∈ N such that m→∞

n

1

X

ε

0 xm (i)ei <

4 i=1 when m > m0 . Therefore,

n ∞ 1

X X

(xm (i) + x(i)) ei (xm (i) + x(i)) ei + kxm + xk =

i=n1 +1 i=1

∞



n

n1 ∞ 1

X

X

X X



x(i)ei xm (i)ei − x(i)ei + xm (i)ei − ≥



i=n1 +1 i=1 i=1 i=n1 +1

n

∞ 1

X

ε X

0 x(i)ei + xm (i)ei − ≥

2 i=1 i=n +1 1

Moreover;

n

p ∞ 1

X

X

x(i)ei + xm (i)ei

i=1

i=n1 +1

!p !p ∞ X 1 X 1 X = |x(i)ei | + |xm (i)| λn λn n=1 n=n1 +1 i∈In i∈In  ε p 0 . ≥ 1 + εp − 2 4 1
p p
Since 2 ε40 − 1 + (1 + ε0 ) p > ε0 for 1 < p < ∞, we can choose ε ≥ 2 and we have n1 X

n

∞ 1

X

  ε p  p1 X

0 x(i)ei + xm (i)ei ≥ 1 + εp − 2

4 i=1

i=n1 +1

≥ 1 + ε0


ε0 p 4

1

p
p

− 1 + (1 + ε0 )

572

8

˙ S ˙ ¸ EK NECIP ¸ IMS

Therefore, combining this result with the previous inequality, we get

n

∞ 1

X

ε X

0 x(i)ei + kxm + xk ≥ xm (i)ei −

2 i=1

i=n1 +1

ε0 ε0 ≥ 1 + ε0 − =1+ . 2 2 This means that Vp (λ) has the uniform Opial property. From the Theorem 3.2, we get that Vp (λ) is (k − N U C). Clearly (k − N U C) Banach spaces are (N U C), and (N U C) implies property (H) and reflexivity holds, [10]. Also, Huff proved that X is (N U C) if and only if X is reflexive and (U KK) (see in [10]). On the other hand, it is well known that (U C) ⇒ (kR) ⇒ (k + 1)R, and (kR) spaces are reflexive and rotund, and it is easy to see that (k − N U C) ⇒ (kR). By the facts presented in the introduction and the just above; we get the following corollaries: Corollary 3.4. The space Vp (λ) (1 < p < ∞) is (N U C) and then is reflexive. Corollary 3.5. The space Vp (λ) (1 < p < ∞) is (U KK). Corollary 3.6. The space Vp (λ) (1 < p < ∞) is (kR). Corollary 3.7. The space Vp (λ) (1 < p < ∞) is rotund. References [1] F. Basar, B. Altay and M. Mursaleen, Some generalizations of the space bvp of p-bounded variation sequences, Nonlinear Analysis:Theory, Methods&Applications, Volume 68, (2), (2008) 273-287. [2] W. L. Bynum, Normal structure coefficient for Banach spaces, J. Math., 86(1980), 427-436. [3] S. T. Chen, Geometry of Orlicz spaces, Dissertationes M ath., 356 (1996). [4] Y. Cui, H. Hudzik, Some geometric properties related to fixed point theory in Ces´ aro spaces, Collect. Math. 50, 3(1999), 277-288. [5] Y. Cui, H. Hudzik, Packing constant for Ces´ aro sequence spaces, Nonlinear Analysis, 47 (2001) 2695-2702. [6] Y. Cui, R. Pluciennik, Local uniform nonsquareness in Ces´ aro sequence spaces, Comment. Math., 37(1997), 47-58. [7] M. Et, Spaces of Ces´ aro difference sequences of order r defined by a modulus function in a locally convex space, Taiwanese J. Math., 10 (2006) no. 4, 865-879. [8] K. Fan and I. Glicksberg, Fully convex normed linear spaces, Proc. Nat. Acad. Sci. USA. 41(1955), 947-953. [9] C. Franchetti, Duality mapping and homeomorphisms in Banach theory, in ”Proceedings of Research Workshop on Banach Spaces Theory”, University of Iowa, 1981.

573

ON SOME GEOMETRIC AND TOPOLOGICAL PROPERTIES OF SEQUENCE SPACES

9

[10] R. Huff, Banach spaces which are nearly uniformly convex, Rocky Mountain J. Math., 10(1980), 473-479. [11] V. Karakaya, Some geometric properties of sequence spaces involving Lacunary sequence, Journal of Inequalities and Applications, Volume 2007, Article ID 81028, 8 pages doi:10.1155/2007/81028. [12] D. N. Kutzarova, k-β and k-nearly uniformly convex Banach spaces, J. Math. Anal. Appl. 162(1991), 322-338. ¨ [13] L. Leindler, Uber die verallgemeinerte de la Vall´ ee-Poussinsche summierbarkeit allgemeiner Orthogonalreihen, Acta Math. Acad. Sci. Hungar., 16, no.3-4 (1965) 375–387. [14] Y. Q. Liu, B. E. Wu and Y. P. Lee, M ethod of sequence spaces, Guangdong of Science and Technology Press, (1996) (in Chinese). [15] I. J. Maddox, On Kuttners theorem, J. London Math. Soc., 43 (1968), 285-290. [16] E. Malkowsky, E. Sava¸s, Some λ-sequence space defined by a modulus, Arch.Math. (BRNO), 36 (2000), 219-228. [17] M. Mursaleen, F. Ba¸sar, B. Altay, On the Euler sequence spaces which include the spaces lp and l∞ II, Nonlinear Analysis, 65 (2006) 707-717. [18] M. Mursaleen, Rifat C ¸ olak and Mikail Et, Some geometric inequalities in a new Banach sequence space, Journal of Inequalities and Applications, Volume 2007 (2007), Article ID 86757, 6 pages, doi:10.1155/2007/86757. [19] J. Musielak, Orlicz spaces and modular spaces, Lecture Notes in Mathematics, Vol.1034, Springer-Berlin, 1983. [20] A. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73(1967), 591-597. [21] S. Prus, Banach spaces with uniform Opial property, Nonlinear Anal., 8(1992), 697-704. [22] E. Sava¸s, V. Karakaya, N. S ¸ im¸sek, Some `(p)-type new sequence spaces and their geometric properties, Abstract and Applied Analysis, Volume 2009, Article ID 696971, 12 pages doi:10.1155/2009/696971. [23] E. Sava¸s, R. Sava¸s, Some λ-sequence spaces defined by Orlicz functions, Indian J. Pure Appl. Math., 34 (2003), no. 12, 1673–1680. [24] J. S. Shue, Ces´ aro sequence spaces, Tamkang J. Math., 1 (1970) 143-150. [25] N. S ¸ im¸sek and V. Karakaya, On some geometrial properties of generalized modular spaces of Ces´ aro type defined by weighted means, Journal of Inequalities and Applications, Volume 2009, Article ID 932734, 13 pages doi:10.1155/2009/932734. [26] N. S ¸ im¸sek, E. Sava¸s and V. Karakaya, Some geometric and topological properties of a new sequence space defined by de la Vall´ ee-Poussin mean, Journal of Computational Analysis and Applications, Vol.12, No.4 (2010), 768-779. ˙ ¨ u ˙ ¨ dar, Istanbul, ISTANBUL COMMERCE UNIVERSITY, Department of Mathematics, Usk TURKEY. E-mail address: [email protected]

JOURNAL 574 OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.3, 574-582, 2011, COPYRIGHT 2011 EUDOXUS PRESS, LLC

Fractional Differences and Integration by Parts Thabet Abdeljawad and Dumitru Baleanu∗ Department of Mathematics and Computer Science C ¸ ankaya University, 06530 Ankara, Turkey March 9, 2010

Abstract In this paper we define the right fractional sum and difference following the delta time scale calculus and obtain results on them analogous to those obtained for the left ones studied in [6], [7], [8]. In addition of that a formula for the integration by parts was obtained. The obtained formula is used to obtain a discrete Euler-Lagrange equation in fractional calculus. Key Words: left fractional sum, right fractional sum, left and right fractional differences, integration by parts, Euler-Lagrange equation .

1

Introduction

The fractional calculus, which is the calculus of derivative and integrals of arbitrary orders, gained importance during the last decade [1, 2, 3]. The idea of using the left and the right fractional derivatives gained importance among scientists (see for example Refs. [4, 5] and the references therein). In [8] the authors, starting with a linear difference equation, were able to define the left fractional sums and differences and obtained the Leibniz rule and the law of exponents. Also the fractional differences of certain special functions were computed there. Then the authors in [6] continued on studying those discrete fractional operators and developed a transform of solution. Then, very recently, the same authors in [7] carefully developed the commutativity properties of the fractional sum and the fractional difference operators, where they also solved a nonlinear problem with an initial condition. In this article, we define the right fractional sum and difference operators and obtained many of their properties similar to that obtained for the forward ∗ On

leave of absence from Institute of 23, R 76900,Maturely-Bucharest, Romania, [email protected]

1

Space Sciences, P.O.BOX, MAGEmails: [email protected],

ABDELJAWAD, BALEANU: FRACTIONAL DIFFERENCES

575

operators studied in [6], [7], [8]. Then by using those properties we obtain a by-part formula analogous to that in usual fractional calculus. For a natural number n, the factorial polynomial is defined by, t(n) =

n−1 Y

(t − j) =

j=0

Γ(t + 1) Γ(t + 1 − n)

(1)

where Γ denotes the special gamma function and the product is zero when t + 1 − j = 0 for some j. More generally, for arbitrary α, define t(α) =

Γ(t + 1) , Γ(t + 1 − α)

(2)

where the convention t(α) = 0 when we divide over a pole. Given that the forward and backward difference operators are defined by ∆f (t) = f (t + 1) − f (t), ∇f (t) = f (t) − f (t − 1)

(3)

respectively, we define iteratively the operators ∆m = ∆(∆m−1 ) and ∇m = ∇(∇m−1 ), where m is a natural number. Here are some the properties of the above factorial function Lemma 1. ([6]) Assume the following factorial functions are well defined. (i) ∆t(α) = αt(α−1) (ii) (t − µ)t(µ) = t(µ+1) , where µ ∈ R. (iii) µµ = Γ(µ + 1). (iv) If t ≤ r, then t(α) ≤ t(α) for any α > r (v) If 0 < α < 1, then t(αν) ≥ (t(ν) )α . (vi) t(α+β) = (t − β)(α) t(β) Also, for our purposes we list down the following two identities, which can be easily proved ∇s (s − t)(α−1) = (α − 1)(ρ(s) − t)(α−2)

(4)

∇t (ρ(s) − t)(α−1) = −(α − 1)(ρ(s) − t)(α−2)

(5)

For two real numbers a and b, we write Na = {a, a + 1, a + 2, ...} and = {b, b − 1, b − 2, ...}. If α > 0 and σ(s) = s + 1. Then, the α − th left fractional sum of f is defined (as done in [8] and used in [6] and [7]) by bN

t−α

∆−α f (t) ,

1 X (t − σ(s))(α−1) f (s) Γ(α) s=a 2

(6)

576

ABDELJAWAD, BALEANU: FRACTIONAL DIFFERENCES

Note that ∆−α maps functions defined on Na to functions defined on Na+α . Also to be noted that (i) u(t) = ∆−n f (t), n ∈ N, satisfies the initial value problem ∆n u(t) = f (t), t ∈ Na , u(a + j − 1) = 0, j = 1, 2, ..., n

(7)

(n−1)

vanishes at s = t − (n − 1), ..., t − 1. (ii) the Cauchy function (t−σ(s)) (n−1)! If α > 0 and ρ(s) = s − 1. Then, we define the α − th right fractional sum of f by b X 1 −α (ρ(s) − t)(α−1) f (s) (8) ∇ f (t) , Γ(α) s=t+α Note that ∇−α maps functions defined on b N to functions defined on Also to be noted that (i) u(t) = ∇−n f (t), n ∈ N, satisfies the initial value problem ∇n u(t) = (−1)n f (t), t ∈

b N,

u(b − j + 1) = 0, j = 1, 2, ..., n

b−α N.

(9)

(n−1)

vanishes at s = t + 1, t + 2, ..., t + (n − 1). (ii) the Cauchy function (ρ(s)−t) (n−1)! As used to be done in usual fractional calculus, the Riemann left and the right fractional differences are to be, respectively, defined by ∆α f (t) , ∆n ∆−(n−α) f (s) and ∇α f (t) , (−1)n ∇n ∇−(n−α) f (s)

(10)

where n = [α] + 1. It is clear that, the fractional left difference operator ∆α maps functions defined on Na to functions defined on Na+n−α , while the fractional right difference operator ∇α maps functions defined on b N to functions defined on b−(n−α) N . We used the nabla symbols to define the right fractional sums and differences just to simplify the notation and it does not mean a nabla time scale analysis. Throughout this article, for simplicity we write ∆α and ∇α in place of ∆α a and ∇α b , respectively, where α ∈ R. Otherwise, we point to the end points up to which we take the fractional sum or difference. However, one has to note that if α = n ∈ N, then ∆na f (t) = ∆n f (t) and ∇nb f (t) = (−1)n ∇n f (t).

(11)

The ν −th left fractional sum behaves well in composition. In fact, Theorem 2.2 in [6] states Lemma 2. Let f be a real-valued function, and let µ, ν > 0. Then, for all t such t = a + µ + ν, mod(1),we have ∆−ν [∆−µ f (t)] = ∆−(µ+ν) f (t) = ∆−µ [∆−ν f (t)].

3

(12)

ABDELJAWAD, BALEANU: FRACTIONAL DIFFERENCES

577

Theorem 2.1 and Remark 2.2 in [7] are to be summarized in the following lemma: Lemma 3. For any ν ∈ R, we have ∆−ν ∆f (t) = ∆∆−ν f (t) −

(t − a)(ν−1) f (a) Γ(ν)

(13)

where f is defined on Na Following inductively, Theorem 2.2 in [7] states Lemma 4. For any α ∈ R and any positive integer p, the following equality holds: −α

∆

p

p

−α

∆ f (t) = ∆ ∆

p−1 X (t − a)(α−p+k ) k f (t) − ∆ f (a) Γ(α + k − p + 1)

(14)

k=0

where f is defined on Na Lemma 5. For any α > 0, the following equality holds: ∇−α ∇b f (t) = ∇b ∇−α f (t) −

(b − t)(α−1) f (b) Γ(α)

(15)

where f is defined on b N . Proof. First, by the help of (4) and the difference calculus, the following by-parts version is valid: ∇s ((s − t)α−1 f (s)) = (ρ(s) − t)α−1 ∇s f (s) + (α − 1)(ρ(s) − t)(α−2) f (s) (16) Using (16) and that ∇b f (s) = −∇f (s) we obtain ∇−α ∇b f (t) = b X 1 [(α − 1) (ρ(s) − t)(α−2) f (s) + (α − 1)(α−1) f (t + α − 1) − (b − t)(α−1) f (b)] Γ(α) t+α (17) Then, (iii) of Lemma 1 leads to

∇−α ∇b f (t) =

b X 1 (b − t)(α−1) (ρ(s) − t)(α−2) f (s) − f (b) Γ(α − 1) t+α−1 Γ(α)

(18)

By (5) and that (α − 2)(α−1) = 0, we see that ∇b ∇−α f (t) =

b X 1 (ρ(s) − t)(α−2) f (s) Γ(α − 1) t+α−1

which completes the proof. 4

(19)

578

ABDELJAWAD, BALEANU: FRACTIONAL DIFFERENCES

Remark 6. Let α > 0 and n = [α] + 1. Then, by the help of Lemma 5 we can have ∇b ∇α f (t) = ∇b ∇nb (∇−(n−α) f (t)) = ∇nb (∇b ∇−(n−α) f (t)) or ∇b ∇α f (t) = ∇nb [∇−(n−α) ∇b f (t) +

(b − t)(n−α−1) f (b)] Γ(n − α)

(20)

(21)

Then, using the identity ∇nb

(b − t)(−α−1) (b − t)(n−α−1) = Γ(n − α) Γ(−α)

(22)

we infer that (15) is valid for any real α. Using Lemma 5 and Remark 6 we reach the following result Theorem 7. For any real number α and any positive integer p, the following equality holds: ∇−α ∇pb f (t) = ∇pb ∇−α f (t) −

p−1 X (b − t)(α−p+k) ∇k f (b) Γ(α + k − p + 1) b

(23)

k=0

where f is defined on b N and we remind that ∇kb f (t) = (−1)k ∇k f (t). Proof. Proceeding as in the proof of Theorem 2.2 in ([7]), we can inductively obtain the result, by the help of (5) and Lemma 5. In order to prove the commutative property for the right fractional sums, we need the following power rule: Lemma 8. Let α > 0, µ > 0. Then, (µ) ∇−α = b−µ (b − t)

Γ(µ + 1) (b − t)(µ+α) Γ(µ + α + 1)

(24)

Proof. The proof can be achieved by checking that both sides of the identity (24) verify the difference equation ((b − (µ + α)) − t + 1)∇b g(t) = (µ + α)g(t), g(b − (µ + α)) = Γ(µ + 1) (25) Through the verification we use (ii) and (iii) of Lemma 1 and modify the steps followed in the proof of Lemma 2.3 in [6]. Actually, formula (24) is the analogous of (µ) ∆−α = a+µ (t − a)

Γ(µ + 1) (t − a)(µ+α) Γ(µ + α + 1) 5

ABDELJAWAD, BALEANU: FRACTIONAL DIFFERENCES

579

Theorem 9. Let α > 0, µ > 0. Then, for all t such that t ≡ b − (µ + α) (mod 1), we have ∇−α [∇−µ f (t)] = ∇−(µ+α) f (t) = ∇−µ [∇−α f (t)]

(26)

where f is defined on b N . Proof. By direct substitution we have ∇−µ [∇−α f (t) =

b−α b X X 1 (ρ(s) − t)(µ−1) (ρ(r) − s)(α−1) f (r) (27) Γ(α)Γ(µ) s=t+µ r=s+α

Interchange the order of sums to get ∇−µ [∇−α f (t) =

b r−α X X 1 1 [ (ρ(s) − t)(µ−1) (ρ(r) − s)(α−1) ]f (r) Γ(α) r=t+µ+α Γ(µ) s=t+µ

(28) rewrite ρ(r) − s = (b − (s − ρ(r) + b)) and make the change of variable x = s − ρ(r) + b to conclude that ∇−µ [∇−α f (t) =

b X 1 ∇−µ (b − u)(α−1) |u=b+t−ρ(r) f (r) Γ(α) r=t+µ+α b−(α−1)

(29)

Then, the result follows by the help of Lemma 8. As a consequence of Lemma 4, Lemma 2, (7) and that ∆−(n−α) f (a + n − Pa−1 α − 1) = 0 ( the convention that s=a g(s) = 0 is made), we state Proposition 10. For α > 0, and f defined in a suitable domain Na , we have for t ∈ Na+n ⊂ Na −α ∆α f (t) = f (t), a+α ∆

(30)

α ∆−α / N, a+n−α−1 ∆ f (t) = f (t), when α ∈

(31)

and

∆−α ∆α f (t) = f (t) −

n−1 X k=0

(t − a)(k) k ∆ f (a),when α = n ∈ N k!

(32)

Similarly, by the help of Lemma 5, Theorem 9, (9) and that ∇−(n−α) f (b − (n − α) + 1) = 0, we can, for the right sums and differences, obtain

6

580

ABDELJAWAD, BALEANU: FRACTIONAL DIFFERENCES

Proposition 11. For α > 0, and f defined in a suitable domain Na , we have for t ∈ b−n N ⊂ b N −α ∇α f (t) = f (t), b−α ∇

(33)

α ∇−α / N, b−(n−α)+1 ∇ f (t) = f (t), when α ∈

(34)

and

∇

−α

∇α b f (t)

= f (t) −

n−1 X k=0

2

(b − t)(k) k ∇b f (b),when α = n ∈ N. k!

(35)

Integration by Parts for Fractional Differences

After the left and right fractional sums are defined above, it becomes ready to obtain a by-part formula for them. Namely Proposition 12. Let α > 0, a, b ∈ R such that a < b and b ≡ a + α (mod 1). If f is defined on Na and g is defined on b N , then we have b X

(∆−α f )(s)g(s) =

s=a+α

b−α X

f (s)∇−α g(s)

(36)

s=a

Proof. By direct substitution we have b X

(∆

−α

s=a+α

s−α b X X 1 ( (s − σ(t))(α−1) f (t)) g(s) f )(s)g(s) = Γ(α) s=a+α t=a

(37)

Then, the result follows by interchanging the order of summation in (37) and noting that (s − σ(t)) = (ρ(s) − t). Obeying the above by- parts formula for fractional sums, we can obtain a by-parts formula also for fractional differences. Proposition 13. Let α > 0 be non-integer and assume that b ≡ a + (n − α) (mod 1). If f is defined on b N and g is defined on Na , then b−n+1 X

b−(n−α)+1

f (s)∆α g(s) =

X

g(s)∇α f (s)

(38)

s=a+n−1

s=a+(n−α)−1

Proof. By the help of equation (34) of Proposition 11 we can write b−n+1 X s=a+(n−α)−1

f (s)∆α g(s) =

b−n+1 X s=a+(n−α)−1

7

α α ∇−α b−(n−α)+1 (∇b f (s))∆ g(s)

(39)

ABDELJAWAD, BALEANU: FRACTIONAL DIFFERENCES

581

and by Proposition 12 we have b−(n−α)+1

b−n+1 X

X

f (s)∆α g(s) =

α ∇α f (s)∆−α a+(n−α)−1 ∆ g(s)

(40)

s=a+n−1

s=a+(n−α)−1

But then the result follows by equation (31) of Proposition 10. When α is an integer it is possible to obtain, by the help of equation (35) of Proposition 11 and equation (32) of Proposition 10, the usual by-parts formula in difference calculus. For example, if α = 1 we obtain b−1 X

f (s)∆g(s) =

b X

f (s)g(s)|ba −

s=a

g(s)∇f (s) =

f (s)g(s)|ba −

b−1 X

g(s + 1)∆f (s)

s=a

s=a+1

(41)

3

Application

The obtained by-parts formula in the previous section is very useful in discrete fractional calculus. Here, for example, we show how it is used to obtain EulerLagrange equations for a discrete variational problem in fractional calculus. We consider the functional J : S → R, b X

J(y) =

L(s, y(s), ∆α y(s))

(42)

s=a−α

where a, b ∈ R, 0 < α < 1 L : (Na−α ∩

b+α N )

× (Rn )2 → R, b ≡ a + α (mod 1)

and S = {y : Na−α ∩

b+α N

→ Rn : y(a) = y0 and y(b + α) = y1 }

Moreover, we assume that the function y fits the discrete time scales Na and Na+n−α . That is, y(s) = y(s + n − α) for all s ∈ Na . We shall shortly write : L(s) ≡ L(s, y(s), ∆α y(s)) We calculate the first variation of the functional J on the linear manifold S: Let η ∈ H = {h :→ Rn : h(a) = h(b + α) = 0}, then δJ(y(x), η(x)) =

d J(y(x) + η(x))|=0 d 8

582

ABDELJAWAD, BALEANU: FRACTIONAL DIFFERENCES

b X ∂L(s) ∂L(s) α [ η(s) + ∆ η(s)]. αy ∂y ∂∆ s=a−α

(43)

Then we apply Proposition with 0 < α < 1 and n = 1 to get δJ(y(x), η(x)) =

b+α X

[

s=a

∂L(s) ∂L(s) + ∇α ]η(s) = 0, ∂y ∂∆α y

(44)

and by applying a suitable discrete fundamental lemma in calculus of variations we obtain the Euler-Lagrange equation: ∂L(s) ∂L(s) + ∇α = 0. ∂y ∂∆α y

References [1] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional integrals and derivatives. Theory and applications. Gordon and Breach, Switzerland, 1993. [2] I. Podlubny, Fractional differential equations, Academic Press, San Diego, 1999. [3] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, Netherlands, 2006. [4] T. M. Atanackovic, B. Stankovic, On a class of differential equations with left and right fractional derivatives, Z. Angew. Math. Mech. 87(7) 537– 546, 2007. [5] T. Abdeljawad, D. Baleanu, F. Jarad, Existence and uniqueness theorem for a class of delay differential equations with left and right Caputo fractional derivatives, J. Math. Phys., 49(8) 083507 (2008). [6] F. M. Atıcı, P.W. Eloe, A Transform method in discrete fractional calculus , Int. J. Diff. Eq. 2(2) 165-176 (2007). [7] F. M. Atıcı, P. W. Eloe, Initial value problems in discrete fractional calculus, Proc. Amer. Math. Soc., 137, 981-989 (2009). [8] K. S. Miller, B. Ross, Fractional difference calculus, Proceedings of the International Symposium on Univalent Functions, Fractional Calculus and Their Applications, Nihon University, Koriyama, Japan, 139-152 (1989).

9

JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.3, 583-589, 2011, COPYRIGHT 2011 EUDOXUS PRESS, 583 LLC

Common fixed point theorem under contractions in partial metric spaces

∗

Xi Wen1 ; Xianjiu Huang2† 1. Department of Computer Science, Nanchang University, Nanchang, 330031, Jiangxi, P.R.China 2. Department of Mathematics, Nanchang University, Nanchang, 330031, Jiangxi, P.R.China

Abstract In 1994, Matthews introduced the notion of a partial metric space. Later on, O’Neill generalized Matthews’ notion of partial metric, in order to establish connections between these structures and the topological aspects of domain theory. The aim of this paper is to present coincidence point result for two mappings in complete partial metric spaces in the sense of O’Neill which satisfy new contractive conditions. Our result generalizes Banach’s fixed point theorem for dualistic partial metric spaces.

Keywords: Dualistic partial metric; Partial metric; Common fixed point; Domain theory AMS(2000) Subject Classification : 47H10; 54H25; 68Q55

1

Introduction In recent years many works on domain theory have been made in order to equip semantics

domain with a notion of distance. In particular, Matthews ([5]) introduced the notion of a partial metric space as a part of the study of denotational semantics of dataflow networks. They generalize the concept of a metric space in the sense that the self-distance from a point to itself need not be equal to zero. They are useful in modeling partially defined information, which often appears in computer science. Further applications of partial metrics to problems in Theoretical Computer Science were discussed in [1-2, 8-11]. Recently, an extension of the Banach contraction mapping theorem have been proved in the dualistic partial metric context (see [6]) in such a way that the Matthews contraction mapping theorem can be deduced as a special case of such a result. In this paper, we present coincidence point result for two mappings in complete partial metric spaces in the sense of O’Neill which ∗

Project supported by the National Natural Science Foundation of China(10461007 and 10761007) and sup-

ported partly by the Provincial Natural Science Foundation of Jiangxi, China (2008GZS0076 and 2007GZS2051). †

Corresponding author. E-mail addresses: [email protected]

1

584

WEN, HUANG: COMMON FIXED POINT THEOREM

satisfy new contractive conditions. Our result generalizes some known results in partial metric spaces.

2

Preliminaries Throughout this paper the letters R, R+ , N will denote the set of real numbers, of nonnegative

real numbers and natural numbers, respectively. Consistent with Mattews [3], the following definitions and results will be needed in the sequel. Definition 2.1 A function p : X × X → R+ is called a partial metric if and only if for all x, y, z ∈ X such that (p1 ) x = y ⇔ p(x, x) = p(x, y) = p(y, y); (p2 ) p(x, x) ≤ p(x, y); (p3 ) p(x, y) = p(y, x); (p4 ) p(x, z) ≤ p(x, y) + p(y, z) − p(y, y). A partial metric space is a pair (X, p) such that X is a nonempty set and p is a partial metric on X. In [7], O’Neill proposed one significant change to Matthews’ definition of the partial metric, and that was extend their range from R+ to R. According to [6], the partial metrics in the O’Neill sense will be called dualistic partial metric and a pair (X, p) such that X is a nonempty set and p is a dualistic partial metric on X will be called a dualistic partial metric space. In this way, O’Neill developed several connections between partial metrics and the topological aspects of domain theory. Moreover, the pair (R, p), where p(x, y) = x ∨ y for all x, y ∈ R, provides a paradigmatic example of a dualistic partial metric space that is not a partial metric space. Each dualistic partial metric p on X generates a T0 topology T (p) on X which has as a base the family of open p-balls {Bp (x, ε) : x ∈ X; ε > 0}, where {Bp (x, ε) = {y ∈ X : p(x, y) < p(x, x) + ε} for all x ∈ X and ε > 0. From this fact it immediately follows that a sequence {xn }n in a dualistic partial metric space (X, p) converges to a point x ∈ X if and only if p(x, x) = lim p(x, xn ). Following [7] n→∞

(compare [5]), a sequence {xn }n in a dualistic partial metric space (X, p) is called a Cauchy sequence if there exists

lim p(xn , xm ). A dualistic partial metric space (X, p) is said to be

n,m→∞

complete if every Cauchy sequence {xn }n in X converges, with respect to T (p), to a point x ∈ X such that p(x, x) = Definition

lim p(xn , xm ).

n,m→∞ 2.2 [3−4] A

function d : X × X → R+ is called a quasi-metric if and only if for

all x, y, z ∈ X such that (d1 ) d(x, y) = d(y, x) = 0 ⇔ x = y; (d2 ) d(x, y) ≤ d(x, z) + d(z, y).

2

WEN, HUANG: COMMON FIXED POINT THEOREM

585

A quasi-metric space is a pair (X, d) such that X is a nonempty set and d is a quasi-metric on X. Each quasi-metric d on X generates a T0 topology T (d) on X which has as a base the family of open d-balls {Bd (x, ε) : x ∈ X; ε > 0}, where {Bd (x, ε) = {y ∈ X : d(x, y) < ε} for all x ∈ X and ε > 0. If d is a quasi-metric on X, then the function ds defined on X×X by ds (x, y) = max{d(x, y), d(y, x)}, is a metric on X. The proof of the following auxiliary results can be found in [6] (compare [5] and [7]). Lemma 2.3 If (X, p) is a dualistic partial metric space, then the function dp : X × X → R+ defined by dp (x, y) = p(x, y) − p(x, x) is a quasi-metric on X such that T (p) = T (dp ). As a consequence of Lemma 2.1 a mapping between dualistic partial metric spaces (X, p) and (Y, q) is continuous if it is continuous between the associated quasi-metric spaces. Lemma 2.4 Let (X, p) be a dualistic partial metric space. Then the following assertions are equivalent: (1) (X, p) is complete; (2) The induced metric space (X, dsp ) is complete. Furthermore lim dsp (a, xn ) = 0 if and n→∞

only if p(a, a) = lim p(a, xn ) = n→∞

3

lim p(xn , xm ).

n,m→∞

Main result We now state and prove our main result. Theorem 3.1 Let (X, p) be a complete dualistic partial metric space. Suppose that the

commuting mappings f, g : X → X are such that for some constant k ∈ (0, 1) and for all x, y ∈ X, |p(f x, f y)| ≤ k|p(gx, gy)|.

(3.1)

If the range of g contains the range of f and g is continuous, then f and g have a unique common fixed point. Proof Let x0 ∈ X be arbitrary and let x1 ∈ X be chosen such that y0 = f (x0 ) = g(x1 ). This can be done, since f (X) ⊂ g(X). Let x2 ∈ X be chosen such that y1 = f (x1 ) = g(x2 ). Continuing this process, having chosen xn ∈ X such that yn = f (xn ) = g(xn+1 ). Then it is clear that for each n ∈ N we have |p(yn , yn )| = |p(f (xn ), f (xn ))| ≤ k n |p(y0 , y0 )|, and |p(yn , yn+1 )| = |p(f (xn ), f (xn+1 ))| ≤ k n |p(y0 , y1 )|. Since, by Lemma 2.3, dp (yn , yn+1 ) + p(yn , yn ) = p(yn , yn+1 ), 3

586

WEN, HUANG: COMMON FIXED POINT THEOREM

we deduce that dp (yn , yn+1 ) + p(yn , yn ) ≤ k n |p(y0 , y1 )|. Hence dp (yn , yn+1 ) ≤ k n |p(y0 , y1 )| − p(yn , yn ) ≤ k n |p(y0 , y1 )| + |p(yn , yn )| ≤ k n (|p(y0 , y1 )| + |p(y0 , y0 )|). Now let n, l ∈ N . Then dp (yn , yn+l ) ≤ dp (yn , yn+1 ) + · · · + dp (yn+l−1 , yn+l ) ≤ (k n + · · · + k n+l−1 )(|p(y0 , y1 )| + |p(y0 , y0 )|) ≤

kn (|p(y0 , y1 )| + |p(y0 , y0 )|). 1−k

Similarly, we obtain that dp (yn+l , yn ) ≤

kn (|p(y0 , y1 )| + |p(y0 , y0 )|). 1−k

Consequently {yn }n is a Cauchy sequence in the metric space (X, dsp ), which is complete by Lemma 2.4. So there is a ∈ X such that lim ds (a, yn ) n→∞ p

= 0.

(3.2)

Now, we show that a is the unique common fixed point of f and g. First note that, by Lemma 2.4, we have p(a, a) = lim p(a, yn ) = n→∞

lim p(yn , ym ).

n,m→∞

(3.3)

Moreover, since lim dp (yn , ym ) = lim p(yn , yn ) = 0,

n,m→∞

n→∞

we deduce, from Lemma 2.3, that lim p(yn , ym ) = 0.

n,m→∞

Therefore, p(a, a) = lim p(a, f (xn )) = 0. n→∞

Since g is continuous, from (3.3) we get p(ga, ga) = lim p(ga, gf (xn )) = n→∞

lim p(gf (xn ), gf (xm )).

n,m→∞

Since f and g commute (i.e. g ◦ f = f ◦ g), from (3.1) we have |p(gf (xn ), gf (xn ))| = |p(f g(xn ), f g(xn ))| ≤ k|p(gg(xn ), gg(xn ))| 4

WEN, HUANG: COMMON FIXED POINT THEOREM

587

= k|p(gf (xn−1 ), gf (xn−1 ))| = k|p(f g(xn−1 ), f g(xn−1 ))| ≤ k 2 |p(gg(xn−1 ), gg(xn−1 ))| ≤ · · · ≤ k n |p(gg(x1 ), gg(x1 ))|. Thus, lim p(gf (xn ), gf (xn )) = 0.

n→∞

Moreover, Since lim dp (gf (xn ), gf (xm )) = 0,

n,m→∞

we deduce, from Lemma 2.3, that lim p(gf (xn ), gf (xm )) = 0.

n,m→∞

Therefore, p(ga, ga) = lim p(ga, gf (xn )) = 0. Now, since n→∞

|p(f a), f a)| ≤ k|p(ga, ga)|, as 0 < k < 1 it follows that p(f a, f a) = p(ga, ga) = 0. On the other hand, from (3.2), we have lim f (xn ) = lim g(xn+1 ) = a.

n→∞

n→∞

Since g is continuous and f and g commute we get ga = g( lim g(xn )) = lim g 2 (xn ), n→∞

n→∞

ga = g( lim f (xn )) = lim gf (xn ) = lim f g(xn ). n→∞

n→∞

n→∞

From (3.1) we get |p(f g(xn ), f a)| ≤ k|p(g 2 (xn ), ga)|. Taking the limit as n → ∞ we obtain |p(ga, f a)| ≤ k|p(ga, ga)|. Hence, as 0 < k < 1 and p(ga, ga) = 0, we have p(ga, f a) = 0. Thus, p(f a, f a) = p(ga, ga) = p(ga, f a) = 0, that is f a = ga. Again from (3.1) it follows that |p(f xn , f a)| ≤ k|p(g(xn ), ga)|. Taking the limit as n → ∞ we obtain |p(a, f a)| ≤ k|p(a, ga)| = k|p(a, f a)|.

5

588

WEN, HUANG: COMMON FIXED POINT THEOREM

Hence, p(f a, a) = p(a, a) = p(f a, f a) = 0, that is f a = a. Thus, we proved that f a = ga = a. The uniqueness of the common fixed point a follows from (3.1). Indeed, let b be another common fixed point of f and g. Then |p(a, b)| = |p(f a, f b)| ≤ k|p(ga, gb)| = k|p(a, b)|. As 0 < k < 1 it follows that p(a, b) = 0, i.e. a = b. This concludes the proof. Corollary 3.2 Let (X, p) be a complete partial metric space. Suppose that the commuting mappings f, g : X → X are such that for some constant k ∈ (0, 1) and for all x, y ∈ X, p(f x, f y) ≤ kp(gx, gy).

(3.4)

If the range of g contains the range of f and g is continuous, then f and g have a unique common fixed point. Remark 3.3 In the light of the preceding corollary one can ask if the contractive condition (3.1) in the statement of our theorem can be replaced by the corresponding contraction condition (3.4) above. The following easy example shows that it is not the case. Example 3.4 Let X = (−∞, 2], and let p be the dualistic partial metric on X given by p(x, y) = x ∨ y for all x, y ∈ X. Since (X, dsp ) is a complete metric space, (X, p) is a complete dualistic partial metric space. Let f, g be the mappings from X into itself defined by f (x) = x − 1 and g(x) = x −

1 2

for

all x ∈ (−∞, 2], respectively. It can be easily seen that g is continuous, g(X) ⊃ f (X) and p(f x, f y) ≤ 32 p(gx, gy), for all x, y ∈ X. However, f and g have no any common fixed point. Remark 3.5 Let IX be the identity mapping on X. Setting g = IX in Theorem 3.1 and Corollary 3.2, we obtain the following results: Corollary 3.6 (Oltra and Valero) Let f be a mapping of a complete dualistic partial metric space (X, p) into itself such that there is a real number k with 0 < k < 1, such that |p(f (x), f (y))| ≤ k|p(x, y)|, for all x, y ∈ X. Then f has a unique fixed point. Corollary 3.7 (Matthews) Let f be a mapping of a complete partial metric space (X, p) into itself such that there is a real number k with 0 < k < 1, such that p(f (x), f (y)) ≤ kp(x, y), for all x, y ∈ X. Then f has a unique fixed point. Acknowledgments The authors would like to express their sincere appreciation to the referees for their very helpful suggestions and many kind comments.

6

WEN, HUANG: COMMON FIXED POINT THEOREM

589

References [1] M. A. Bukatin, J. S. Scott, Towards computing distances between programs via Scott domains, in: Logical Foundations of Computer Sicence, Lecture Notes in Computer Science (eds. S. Adian and A. Nerode), vol. 1234, Springer (Berlin, 1997), 33-43. [2] M. A. Bukatin, S. Y. Shorina, Partial metrics and co-continuous valuations, in: Foundations of Software Science and Computation Structures, Lecture Notes in Computer Science (ed. M. Nivat), vol. 1378, Springer (Berlin, 1998), 33-43. [3] P. Fletcher, W. F. Lindgren, Quasi-Uniform Spaces, Marcel Dekker, New York, 1982. [4] H.P.A. Kunzi, Nonsymmetric distances and their associated topologies: About the origins of basic ideas in the area of asymmetric topology, in: Handbook of the History of General Topology (eds. C.E. Aull and R. Lowen), vol. 3, Kluwer Acad. Publ. (Dordrecht, 2001), 853-968. [5] S. G. Matthews, Partial metric topology, in: Proc. 8th Summer Conference on General Topology and Applications. Ann. New York Acad. Sci. 728 (1994), 183-197. [6] S. Oltra, O. Valero, Banach’s fixed point theorem for partial metric spaces, Rend. Ist. Mat. Univ. Trieste 36 (2004), 17-26. [7] S. J. O’Neill, Partial metrics, valuations and domain theory, in: Proc. 11th Summer Conference on General Topology and Applications. Ann. New York Acad. Sci. 806 (1996), 304-315. [8] S. Romaguera, M. Schellekens, Duality and quasi-normability for complexity spaces, Appl. General Topology 3 (2002), 91-112. [9] S. Romaguera, M. Schellekens, Weightable quasi-metric semigroups and semilattices, In: Proc. MFCSIT2000, Electronic Notes in Theoretical Computer Science 40 (2003), 12 pages. [10] M.P. Schellekens, A characterization of partial metrizability: domains are quantifiable, Theor. Comput. Sci. 305 (2003), 409-432. [11] M.P. Schellekens, The correspondence between partial metrics and semivaluations, Theor. Comput. Sci. 315 (2004), 135-149.

7

JOURNAL 590 OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.3, 590-595, 2011, COPYRIGHT 2011 EUDOXUS PRESS, LLC

A NOTE ON CARLITZ’S q-BERNOULLI MEASURE TAEKYUN KIM, YOUNG-HEE KIM∗ , AND BYUNGJE LEE

Abstract In this paper, we consider the q-extension of Nasybullin’s lemma for studying the p-adic Carlitz’s q-Bernoulli measure. 1. Introduction Let p be a fixed prime number. Throughout this paper, the symbol Z, Q, Zp , Qp , and Cp will denote the ring of rational integers, the field of rational numbers, the ring of p-adic rational integers, the field of p-adic rational numbers, and the completion of algebraic closure of Qp , respectively. Let N be the set of natural numbers and Z+ = N ∪ {0}. Let νp be the normalized exponential valuation of Cp such that |p|p = p−νp (p) = p1 . When one talks of q-extension, q is variously considered as an indeterminate, a complex number q ∈ C or a p-adic number q ∈ Cp . In this paper, we assume that q ∈ Cp with |1 − q|p < 1, and we use the x notation [x]q = 1−q 1−q (see [5, 6, 22-27]). The usual Bernoulli numbers Bn are defined as ∞ X n=0

Bn

tn t = t , n! e −1

t which can be written symbolically as eBt = et −1 , interpreted to mean that B k must be replaced by Bk when we expand on the left (see [2-4, 7]). This relation can also be written e(B+1)t − eBt = t, or if we equate powers of t, ( 1 if k = 1, B0 = 1, (B + 1)k − B k = 0 if k > 1,

where we first expand and then replace B i by Bi (see [7-9]). The Bernoulli polynomials are defined by n µ ¶ n µ ¶ X X n n n n−k Bn (x) = (B + x) = Bk x = Bn−k xk , (see [12-19]). k k k=0

k=0

In the complex case, the Carlitz’s q-Bernoulli numbers βk,q can be determined inductively by ( 1 if k = 1, k k β0,q = 1, q(qβ + 1) − β = 0 if k > 1, ∗

Correponding author, [email protected]. 1

KIM ET AL: CARLITZ'S q-BERNOULLI MEASURE

591

2

with the usual convention of replacing β k by βk,q (see [1], [7], [21]). Thus we have βk,q

k µ ¶ X 1 k j+1 = (−1)j . k (1 − q) j=0 j [j + 1]q

Note that limq→1 βk,q = Bk . The Carlitz’s q-Bernoulli polynomials are given as (q x β + [x]q )k , that is, βk,q (x) =

k µ ¶ X k βl,q [x]k−l q , l

(see [1], [7], [8], [21]).

l=0

For f ∈ N, let f be the least common multiple of f and p. We set Xf

= lim Z/f pN Z, ← −

Xf∗

=

a + f pn Zp

=

X1 = Zp ,

N

∪ 0 1 it. Let us now take any g 2 W p;q . Then m2Z Qm f pq = s < 1, and for any " > 0 there exists a positive integer N such that s

N X

Qm f

pq

=

N 1 X

Qm f

+ pq

m= 1

m= N

1 X

Qm f pq

< ":

m=N +1

We de…ne a function fN by f (x) ; 0;

fN (x) =

~ := [N Qm . Then we get P where Q N m2Z kf

fN kW p;q

=

max t2Q

2

~ if x 2 Q ~ if x 2 =Q Qm

X

(f

; fN )

pq

Qm Lt

(f

fN )

(f

fN )

pq

< ", and hence pq

m2Z

X

m2Z

6

Qm

< 2":

698

SAGIR, DUYAR: SEGAL ALGEBRAS

On the other hand, since ~f Q

kfN kpq =

N X

pq

X

Qm f pq

Qm f pq

m2Z

N

< 1;

fN is in L (p; q) (Rn ). Also since Cc (Rn ) is dense in L (p; q) (Rn ), there is a g 2 Cc (Rn ) such that " ~ kfN gkpq < and sup pg Q. 2N Hence we have X

Qm (fN

g)

pq

=

m2Z n

N X

Qm

(fN

g)

N X

~ Q

(fN

g)

N

and so kfN

2

gkW p;q

X

Qm

(fN

g)

pq

pq

< ";

< 2":

m2Z n

Consequently we get kf

pq

N

gkW p;q

kf

fN kW p;q + kfN

gkW p;q < 4":

This completes the proof. Proposition 9 The mapping t ! Lt f of Rn ,for all f 2 W p;q , into W p;q is continuous. Proof. Since Cc (Rn ) is dense in W p;q , for any f 2 W p;q and " > 0 there is a g 2 Cc (Rn ) such that kf gkW p;q < 3" . Hence tere exists a neighborhood U of 0 2 U such that kf

Lt f kW p;q

=

kf gkW p;q + kg Lt gkW p;q + kLt (g 2 kf gkW p;q + kg Lt gkW p;q < ":

f )kW p;q

Corollary 10 W p;q is dense in L1 (Rn ). Proposition 11 If f; g 2 W p;q , then kf p;q

gkW p;q

kf kW p;q kgkW p;q . p;q

Proof. For all f 2 W and arbitrary g 2 W the vector-valued integral R g (y) Ly f dy exists as an element of W p;q , and Z g f = g (y) Ly f dy; and also ([12]).

kg f kW p;q

kgk1 kf kW p;q

7

kf kW p;q kgkW p;q ;

SAGIR, DUYAR: SEGAL ALGEBRAS

699

Corollary 12 W p;q is a Segal algebra. De…nition 13 The space of all functions belonging to locally L (r; s) (G) ; 1 < r < 1; 1 s 1 such that for every " > 0 there exists a compact set K in Rn such that = K is denoted by (Lr;s Qt f rs < " for all t 2 loc )0 . This space is also a normed space by the norm kf k = sup

t2Rn

Qt f rs

:

The space of all L (r; s) (Q)-valued functions on Z n vanishing at in…nity is denoted by c0 (L (r; s) (Q)). Brie‡y we write c0 (L (r; s) (Q)) = f : f : Z n ! L (r; s) (Q) ; f 2 C 0 (Z n ) : This space is a Banach space by the norm kf k0 = sup kf (m)kL(r;s)(Q) ; m2Z n

and also dual space of it is isometrically ispmorphic to l1 (L (p; q) (Q)), where 1 1 1 1 p + r = 1 and q + s = 1. If it is de…ned a mapping ' as in Proposition 4 of r;s 0 (Lloc )0 to c (L (r; s) (Q)), then it is easily seen that this spaces are isomorphic. In this case we write the following proposition by help of Proposition 4 and hence the following corollary. Proposition 14 Let 1 < p; q < 1; 1 < r; s < 1, p1 + p;q Then the dual space of (Lr;s . loc )0 is isomorphic to W Corollary 15 For 1 < p; q < 1; 1 < r; s < 1,

1 p

+

1 r

1 r

= 1 and

= 1 and

r;s p;q HomL1 (W p;q ; W p;q ) u HomL1 (Lr;s loc )0 ; (Lloc )0 u W

1 q

+

L1

1 q

1 s

+

1 s

= 1.

= 1,

(Lr;s loc )0

:

We will make use of the work done by Burnham and Goldberg in [ 3] about multipliers from L1 to Segal algebras to characterize HomL1 L1 ; W p;q . Applying Theorem 2.6 in [3] and Theorem 1.3 in [8] to the Segal algebras W p;q we have the following proposition: Proposition 16 Let 1 < p; q < 1; 1 < r; s < 1, p1 + 1r = 1 and 1q + 1s = 1. Then v v HomL1 L1 ; W p;q = (W p;q ) , where (W p;q ) denotes the relative completion v of W p;q . In this case if, for f 2 (W p;q ) , we de…ne Tf 2 HomL1 L1 ; W p;q by Tf (g) = f

g 2 L1 :

g;

v

Then the correspondence f ! Tf is an isometric isomorphism of (W p;q ) HomL1 L1 ; W p;q .

on

Proof. W p;q is a subalgebra L1 and W p;q satis…es properties (M1) and (M2) in [8]. Hence we have HomL1 L1 ; W p;q L1 by Theorem 1.3 in [8]. 8

700

SAGIR, DUYAR: SEGAL ALGEBRAS

References [1] Bennett, C. and Sharpley, R.: Interpolation of Operators, Academic Press 1988. [2] Blozinski, A. P.: Convolution of L (p; q) functions, Proc. of the Amer. Math. Soc. 32-1, 237-240, (1972). [3] Burnham, J. T. and Goldberg, R. R.: Multipliers from L1 (G) to a Segal algebra, Bull. Inst. Math. Acad. Sinica 2, 153-164, (1974). [4] Chen, Y. K. and Lai, H. C.: Multipliers of Lorentz spaces, Hokkaido Math. J., 4, 247-260, (1975). [5] Ery¬lmaz, I. and Duyar, C.: Basic Properties and Multipliers Spaces on L1 (G) \ L (p; q) (G) Spaces, Turk. J. Math., 32, 235-243, (2008) [6] Hunt, R. A.: On L (p; q) spaces, L’enseigment Mathematique, TXII-4, 249276, (1966). [7] Krogstad, H. E.: Multipliers of Segal Algebras, Math. Scand. 38, 285-303, (1976). [8] Quek, T. S. and Yap, L. Y. H.: Multipliers from L1 (G) to Lipschitz space, J. Math. Anal. App. 69, 531-539, (1979). [9] Rie¤el, M. A.: Induced Banach representation of Banach algebras and locally compact goups, J. Funct. Anal. 1, 444-491, (1967). [10] — — — — — –: Multipliers and Tensor Products of Lp -spaces of locally compact groups, Studia Math. 33, 71-82, (1969). [11] Saeki, S. and Thome, E. L.: Lorentz as L1 modules and multipliers, Hokkaido Math. J. 23, 55-92, (1994). [12] Unni, K. R.: Parameasures and multipliers of Segal algebras, Funct. Anal. and its App., Internat. Conf., Madras, 511-528, (1973). [13] Warner, C. R.: Closed ideals in group algebra L1 (G)\L2 (G), Trans. Amer. Math. Soc. 121, 408-423, (1966). [14] Yap, L. Y. H.: Ideals in subalgebras of the group algebras, Studia Math. 35, 165-175, (1972). [15] — — — — — -: On two classes of subalgebras of L1 (G), Proc. Japan Acad. 48, 315-319, (1972). ¼ Birsen SAGIR, Cenap DUYAR Ondokuz MAy¬s University, Faculty of Science and Arts, Department of Mathematics, 55139 Kurupelit, Samsun, TURKEY e-mail: [email protected] . [email protected] 9

JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.4, 701-723, 2011, COPYRIGHT 2011 EUDOXUS PRESS, 701 LLC

Characterizations of Orthogonal Generalized Gegenbauer-Humbert Polynomials and Orthogonal Sheffer-type Polynomials Tian-Xiao He Department of Mathematics and Computer Science Illinois Wesleyan University Bloomington, IL 61702-2900, USA

Dedicated to Professor Leetsch C. Hsu on the Occasion of his 90th Birthday

Abstract We present characterizations of the orthogonal generalized Gegen-bauerHumbert polynomial sequences and the orthogonal Sheffer-type polynomial sequences. Using a new polynomial sequence transformation technique presented in [12], we give a method to evaluate the measures and their supports of some orthogonal generalized GegenbauerHumbert polynomial sequences. AMS Subject Classification: 41A80, 65B10, 33C45, 33D45, 39A70, 42C05. Key Words and Phrases: generalized Gegenbauer-Humbert polynomial sequence, Sheffer-type polynomial sequence, Chebyshev polynomial, Legendre polynomial, Morgan-Voyc polynomial, Fermat polynomial, Dickson polynomial of the second kind, and Laguerre polynomial, measures, supports.

1

Introduction

A system of polynomials {pn (x), n ∈ N}, where pn (x) is a polynomial of exact degree n and N = {0, 1, 2, . . .} or {0, 1, 2, . . . , N } for a finite nonnegative integer N , is an orthogonal system of polynomials with respect to some real positive measure µ on X, if {pn (x)} is a set linearly independent in L2 (X, µ) and satisfies the orthogonality relation 1

702

2

T. X. He

Z hpi , pj iµ :=

pi (x)pj (x)dµ(x) = d2i δij ,

i, j ∈ N,

(1)

S

where S is the support of the measure µ and di are nonzero constants. If these constants di = 1, we say the system is orthonormal. The measure µ usually has a density µ0 (x) = w(x) or is a discrete measure with weights w(i) at the points xi . The relation (1) then becomes Z pi (x)pj (x)w(x)dx = d2i δij , i, j ∈ N, (2) S

in the former case and M X

pi (xn )pj (xn )wn = d2i δij ,

i, j ∈ N,

(3)

n=0

in the latter case where it is possible that M = ∞. In this paper, we shall present a characterization of the orthogonal generalized Gegenbauer-Humbert polynomial sequences and give a method to find the density functions and their supports for a class of orthogonal generalized Gegenbauer-Humbert polynomial sequences. We shall also give a characterization of the orthogonal Sheffer-type polynomial sequences. We now start from a general result on orthogonal polynomial sequences. It is well-known that all orthogonal polynomials {pn (x)} on the real line satisfy a recurrence relation of order 2 (see, for examples, [1], [2], [3], [4]) −xpn (x) = bn pn+1 (x) + γn pn (x) + cn pn−1 (x),

n ≥ 1,

(4)

where bn , cn 6= 0 and cn /bn−1 > 0. Note that if for all n ∈ N, pn (0) = 1, we have γn = −(bn + cn ) and the polynomials pn (x) can be defined by the recurrence relation

−xpn (x) = bn pn+1 (x) − (bn + cn )pn (x) + cn pn−1 (x),

n≥1

(5)

together with p−1 (x) = 0 and p0 (x) = 1. Favard proved a converse result (see, for example, [4]). Theorem 1.1 (Favard’s Theorem) Let An , Bn , and Cn be arbitrary sequences of real numbers and let {pn (x)} be defined by the recurrence relation of order 2 pn+1 (x) = (An x + Bn )pn (x) − Cn pn−1 (x),

n ≥ 0,

(6)

703

characterization of some orthogonal polynomials

3

together with p0 (x) = c 6= 0 and p−1 (x) = 0. Then {pn (x)} is a sequence of orthogonal polynomials if and only if An 6= 0, Cn 6= 0, and Cn An An−1 > 0 for all n ≥ 1. For more references of the orthogonal polynomial sequences, readers may find from a recently published very nice survey, [5], by Chihara. In this paper, we will discuss the characterization of the orthogonal generalized Gegenbauer-Humbert polynomials {Pnλ,y,C (x)}n≥0 , which are defined by the expansion (see, for example, [6], Gould [7], and Shiue, Hsu and the author [8]) Φ(t) ≡ (C − 2xt + yt2 )−λ =

X

Pnλ,y,C (x)tn ,

(7)

n≥0

where λ > 0, y and C 6= 0 are real numbers. As special cases of (7), we consider Pnλ,y,C (x) as follows (see [8]) Pn1,1,1 (x) = Un (x), Chebyshev polynomial of the second kind, Pn1/2,1,1 (x) = ψn (x), Legendre polynomial, Pn1,−1,1 (x) = Pn+1 (x), P ell polynomial, x Pn1,−1,1 = Fn+1 (x), F ibonacci polynomial,  x2  Pn1,1,1 + 1 = Bn (x), M organ − V oyc polynomial ([9] by Koshy), 2 x Pn1,2,1 = Φn+1 (x), F ermat polynomial of the f irst kind, 2 Pn1,2a,2 (x) = Dn (x, a), Dickson polynomial of the second kind, a 6= 0 (see, f or example, [10] by Lidl, M ullen, and T urnwald), where a is a real parameter, and Fn = Fn (1) is the Fibonacci number. In particular, if y = C = 1, the corresponding polynomials are called Gegenbauer polynomials (see [6]). More results on the Gegenbauer-type polynomials can be found in Hsu[11] and Shiue and the author [12], etc. It is interesting that for each generalized Gegenbauer-Humbert polynomial sequence there exists a non-generalized Gegenbauer-Humbert polynomial sequence, for instance, corresponding to the Chebyshev polynomials of the second kind, Pell polynomials, Fibonacci polynomials, Fermat polynomials of the first kind, and the Dickson polynomials of the second kind, we have the Chebyshev polynomials of the first kind, Pell-Lucas polynomials (see [13] by Horadam and Mahon), Lucas polynomials, Fermat polynomials of the second kind (see [14] by Horadam), and the Dickson polynomials of the first kind, respectively. The class of the generalized Gegenbauer-Humbert polynomial sequences satisfy (see [12])

704

4

T. X. He

λ + n − 1 λ,y,C 2λ + n − 2 λ,y,C Pn−1 (x) − y Pn−2 (x) Cn Cn for all n ≥ 2 with initial conditions Pnλ,y,C (x) = 2x

(8)

P0λ,y,C (x) = Φ(0) = C −λ , P1λ,y,C (x) = Φ0 (0) = 2λxC −λ−1 , [12] also obtained the explicit expression of {Pnλ,y,C (x)} as follows. √ Theorem 1.2 ([12]) Let x 6= ± Cy. The generalized Gegenbauer-Humbert polynomials {Pn1,y,C (x)}n≥0 defined by expansion (7) can be expressed as  Pn1,y,C (x) = C −n−2

x+

n+1  n+1 p p x2 − Cy − x − x2 − Cy p . 2 x2 − Cy

(9)

One may write (8) into the form

xPnλ,y,C (x) =

C(n + 1) λ,y,C y(2λ + n − 1) (λ,y,C Pn+1 (x) + Pn−1 (x). 2(λn) 2(λ + n)

(10)

In [2], Dombrowski and Nevai presented properties of the measures associated with orthogonal polynomial sequences {Pn (x) = γn xn + · · · }n≥0 (γ > 0) defined by the following recurrence relation of order 2: xPn (x) = an+1 Pn+1 (x) + bn Pn (x) + an Pn−1 (x),

(11)

n = 0,R 1, . . ., where P−1 (x) = 0, P0 (x) = γ0 , a0 = 0, an = γn−1 /γn and ∞ bn = −∞ xPn2 (x)dµ(x). Comparing (10) and (11), we immediately learn that the polynomial sequences generated by the above recurrence relation and having generating function shown in (7) must be {Pn1,C,C (x)}n≥0 , C 6= 0. In this paper, we shall discuss the characterization of the orthogonal Sheffer-type polynomial sequences, which are polynomial sequences possessing a different type generating functions. Sheffer-type polynomial sequences have applications to variable subjects including L´evy processes, financial mathematics, wavelet analysis, mathematical physics, etc. We now present the definition of Sheffer-type polynomial sequences. Definition 1.3 Let A(t) and g(t) be any given formal power series over the real number field R or complex number field C with A(0) = 1, g(0) = 0 and g 0 (0) 6= 0. Then the polynomials pn (x) (n = 0, 1, 2, · · · ) defined by the generating function (GF )

705

characterization of some orthogonal polynomials

A(t)exg(t) =

X

pn (x)tn

5

(12)

n≥0

are called Sheffer-type polynomials with p0 (x) = 1. Sheffer-type polynomials include a lot of famous polynomials as the special cases such as the Bernoulli polynomials, Euler polynomials, Laguerre polynomials, etc. Here, we present a short list of the Sheffer-type polynomials in terms of different choices of (A(t), g(t)). 1 Bn (x), Bernoulli polynomials, n! 1 F or (2/(et + 1), t), pn (x) = En (x), Euler polynomials, n! F or (et , log(1 + t)), pn (x) = (P C)n (x), P oisson − Charlier polynomials, F or (e−αt (α 6= 0), log(1 + t)), pn (x) = Cˆn(α) (x), Charlier polynomials F or (t/(et − 1), t), pn (x) =

F or (1, log(1 + t)/(1 − t)), pn (x) = (M L)n (x) M ittag − Lef f ler polynomials F or ((1 − t)−1 , log(1 + t)/(1 − t)), pn (x) = (P i)n (x), P idduck polynomials F or ((1 − t)(−p) , t/(t − 1))(p > 0), pn (x) = L(p−1) (x), Laguerre polynomials n F or (eλt (λ 6= 0), 1 − et ), pn (x) = (T os)(λ) n (x), T oscano polynomials F or (1, et − 1), pn (x) = τn (x), T ouchard polynomials F or (1/(1 + t), t/(t − 1)), pn (x) = An (x), Angelescu polynomials F or ((1 − t)/(1 + t)2 , t/(t − 1)), pn (x) = (De)n (x) Denisyuk polynomials F or ((1 − t)−p , et − 1)(p > 0), pn (x) = Tn(p) (x), W eighted − T ouchard polynomials The set of all Sheffer-type polynomial sequences {pn (x) = [tn ]A(t)exg(t) } with an operation, “umbral composition” (cf. [15] and [16]), forms a group called the Sheffer group. Some properties and characterizations of Sheffer group are shown in [17]. In addition, a higher dimensional extension of the Sheffer-type polynomial sequences are discussed in [18]. In Sections 2 and 3, we shall give characterizations of the orthogonal generalized Gegenbauer-Humbert polynomial sequences and the orthogonal Sheffer-type polynomial sequences, respectively. In Section 4, we shall present a method to find the densities of the measures µ(x) and their supports S shown in (1) for generalized Gegenbauer-Humbert polynomial sequences {Pn1,y,C (x)} using a technique of representing a polynomial sequence {pn (x)} generated by a linear recurrence relation of order two in terms of one or two terms of a orthogonal generalized Gegenbauer-Humbert polynomial sequence.

706

6

2

T. X. He

A characterization of the orthogonal generalized Gegenbauer-Humbert polynomials

First, we consider the characterization of the orthogonal generalized GegenbauerHumbert polynomials defined by (8). From Favard’s Theorem, one may obtain the following result. Theorem 2.1 A generalized Gegenbauer-Humbert polynomial sequence defined by (8) is an orthogonal polynomial sequence if and only if yC > 0. Proof. Writing the recurrence relation (8) into the standard form in Theorem 1.1, we have Cn = y

2λ + n − 1 λ+n and An = 2 . C(n + 1) C(n + 1)

Thus from Theorem 1.1, {Pnλ,y,C (x)} is an orthogonal polynomial sequence if and only if Cn An An−1 = 4y

(λ + n)(λ + n − 1)(2λ + n − 1) >0 C 3 n(n + 1)2

for all n ≥ 1. Noting λ > 0 and n ≥ 1, we immediately learn that the above inequality is equivalently yC > 0, which completes the proof.

Example 1 Using Theorem 2.1, we may identify the Chebyshev polynomial sequence of the second kind {Pn1,1,1 (x) = Un (x)} and the Legendre polyno1/2,1,1 mial sequence {Pn (x) = ψn (x)} are orthogonal, while Pell polynomial sequence and Fibonacci polynomial sequence are not orthogonal. MorganVoyc polynomial sequence {Bn (2(x−1)) = Pn1,1,1 (x)} (and {Bn (x)}) and the 1,2,1 sequence of the Fermat polynomials of the first kind, {Φn (2x) = Pn−1 (x)} (and Φn (x)}), are orthogonal polynomial sequences. Dickson polynomials of the second kind are orthogonal when a > 0 and non-orthogonal when a < 0. We will evaluate the measures and their supports for Morgan-Voyc polynomials, Fermat polynomials, and Dickson polynomials of the second kind in Section 4. We need the following lemma to find out the recurrence structure of an orthogonal generalized Gegenbauer-Humbert polynomial sequence. Lemma 2.2 If {pn (x)} is an orthogonal polynomial sequence, then there exist sequences {An }n≥0 , {Bn }n≥0 , and {Cn }n≥1 so that pn+1 (x) = (An x + Bn )pn (x) − Cn pn−1 (x),

(13)

707

characterization of some orthogonal polynomials

7

where kn+1 An hn kn+1 kn−1 hn , and , Cn = = kn An−1 hn−1 kn2 hn−1 Z Z An kn+1 2 = − xpn (x) dµ(x) = − xpn (x)2 dµ(x), hn S kn hn S

An = Bn

kn is the leading coefficient of pn (x), and Z pn (x)2 dµ(x) hn = S

is a structural constant. Proof. The proof can be found in [4] and [3]. However, for the sake of convenience, we present a brief proof as follows. We first determine An so that pn+1 (x) − An xpn (x) ∈ πn , a collection of all polynomials of degree ≤ n. Hence, pn+1 (x) − An xpn (x) =

n X

cj pj (x).

j=0

Using the orthogonality of hpn+1 (x), pj (x)iµ = 0 and hpn (x), xpj (x)iµ = 0 for all j = 0, 1, . . . , n − 2, it is readily seen that cj = 0 for all j = 0, 1, . . . , n − 2. Therefore, (13) follows and the expression of An is a consequence of (13). To obtain the expression of Cn , we take inner product of (13) with pn−1 (x) and consider Z

Z xpn (x)pn−1 (x)dµ(x) − Cn hn−1 ,

pn+1 (x)pn−1 (x)dµ(x) = 0 = An S

S

in which the integral of the right-hand member can be written as Z

pn (x)(kn−1 xn + lower powers)dµ(x) =

S

kn−1 hn hn = . kn An−1

Thus the relation An

hn − Cn hn−1 = 0 An−1

yields the expression of Cn . Taking the inner product with pn (x) on the both sides of (13) yields

708

8

T. X. He

Z xpn (x)pn (x)dµ(x) + Bn hn ,

0 = An S

which implies the expression of Bn . From Lemma 2.2, one may obtain Theorem 2.3 If the generalized Gegenbauer-Humbert polynomial sequence {Pnλ,y,C (x)} defined by (8) is an orthogonal polynomial sequence, then y nhn (λ + n) = (14) C hn−1 (λ + n − 1)(2λ + n − 1) R for all n ≥ 1, where hn = S (Pnλ,y,C (x))2 dµ(x). In addition, every element of the sequence {Pnλ,y,C (x)} satisfies Z xPnλ,y,C (x)2 dµ(x) = 0. (15) S

Proof. From the definition (8) of {Pnλ,y,C (x)} and the expression of Cn in Lemma 2.2, we have  2λ + n − 1 hn (λ + n) hn−1 (λ + n − 1) y =2 2 , C(n + 1) C(n + 1) Cn which implies (14). Comparing (8) and the standard recurrence relation (13), we know Bn = 0 for all n ≥ 0, which is equivalent to (15). Remark 1 From (14) one immediately have hn =

y(λ + n − 1)(2λ + n − 1) hn−1 , nC(λ + n)

which implies hn =

 y n (λ + n − 1)n (2λ + n − 1)n h0 , C n!(λ + n)n

where the falling factorial notation xr (sometimes also denoted (x)r ) is defined by xr = x(x − 1)r−1 (r ≥ 1) with x0 = 1. Using the above equations and equation (15), we may evaluate the measures and their supports. Example 2 For the orthogonal sequence of the Chebyshev polynomials of the second order {Pn1,1,1 (x) = Un (x)}, we have y/C = 1 that implies hn = h1 = π/2 and

709

characterization of some orthogonal polynomials

Z

1

x

9

p 1 − x2 (Un (x))2 dx = 0

−1

for all n ≥ 0. The above equation is obviously true by observing that U2n−1 (x) are odd and U2n (x) are even. 1/2,1,1 For the sequence of the Legendre polynomials {Pn (x) = ψn (x)}, we have hn n − 1/2 = , hn−1 n + 1/2 which implies hn = 2/(2n + 1), and Z

1

x(ψn (x))2 dx = 0

−1

for all n ≥ 0. The last formula holds obviously because ψ2n+1 (x) are odd and ψ2n (x) are even. Example 3 We know both Un (x) and ψn (x) are special cases of Gegenbauer polynomials {Pnλ,1,1 (x)} (λ > 0). From Theorem 2.1, we know {Pnλ,1,1 (x)} (λ > 0) is orthogonal. Using Theorem 2.3, we obtain (λ + n − 1)(2λ + n − 1) hn = , hn−1 n(λ + n) which implies hn =

πΓ(2λ + n) , + n) (Γ(λ))2

22λ−1 n!(λ

where Γ(x) is the gamma function. In addition, we have Z

1

−1

3

 2 x(1 − x2 )λ−1/2 Pnλ,1,1 (x) dx = 0.

A characterization of the orthogonal Sheffer-type polynomial sequences

Meixner determined all sets of monic orthogonal Sheffer-type polynomials in his historic paper [19]. Here, a polynomial is said to be monic if the coefficient of its highest order term is 1. We now use a modified Meixner’s approach to give a characterization of all orthogonal Sheffer-type polynomials. Denote D = d/dx and f = g −1 , the composition inverse of g. Expansion (12) suggests

710

10

T. X. He

f (D)pm (x) = mpm−1 (x)

(16)

because of f (D)A(t)exg(t) = A(t)exg(t) f (g(t)) = tA(t)exg(t) X tn+1 X tn pn (x) npn−1 (x) , = = n! n! n≥0

n≥0

where we have used p−1 (x) = 0. Theorem 3.1 Let A(t) and g(t) be defined as Definition 1.3. Then the polynomial sequence {pn (x)} defined by (12) is orthogonal if and on if it satisfies pn+1 (x) = (A0 x + B0 + nλ)pn (x) − n(C1 + (n − 1)γ)pn−1 (x),

(17)

where A0 6= 0, B0 , C1 , λ, andγ are constant, and C1 , γ > 0. Furthermore, g(t) and A(t) satisfy g 0 (t) =

A0 A0 (t) B 0 − C1 t , and = . 1 − λt + γt2 A(t) 1 − λt + γt2

(18)

Proof. All orthogonal polynomial sequences including orthogonal Sheffertype polynomial sequences, {pn (x)}, satisfy the recurrence relation (13) shown in Lemma 2.2: pn+1 (x) = (An x + Bn )pn (x) − Cn pn−1 (x).

(19)

We now apply f (D) defined by (16) on the both sides of the relation and note that f (0) = 0 and f 0 (0) 6= 0 implies f (D)x = f 0 (D). Thus, (n + 1)pn (x) = f (D)pn+1 (x) = f (D) [(An x + Bn )pn (x) − Cn pn−1 (x)] = An f 0 (D)pn (x) + n(An x + Bn )pn−1 (x) − (n − 1)Cn pn−2 (x), (20) where we need Cn An An−1 > 0, which is a necessary and sufficient condition of the orthogonality of {pn (x)} presented in (19) (See Lemma 2.2). On the other hand, multiplying n to the both sides of relation (13) for pn (x) yields npn (x) = n(An−1 x + Bn−1 )pn−1 (x) − nCn−1 pn−2 (x).

(21)

711

characterization of some orthogonal polynomials

11

Subtracting (21) from (20), we obtain (1 − An f 0 (D))pn (x) = n[(An − An−1 )x + (Bn − Bn−1 )]pn−1 (x)   Cn Cn−1 −n(n − 1) pn−2 (x). (22) − n n−1 Applying f (D) on the leftmost and rightmost sides of (22) yields n(1 − An f 0 (D))pn−1 (x) = n(An − An−1 )f 0 (D)pn−1 (x) +n(n − 1)[(An − An−1 )x + (Bn − Bn−1 )]pn−2 (x)   Cn Cn−1 −n(n − 1)(n − 2) − pn−3 (x). n n−1 By transferring n to n + 1, the above equation implies (1 + (An − 2An+1 )f 0 (D))pn (x) = n[(An+1 − An )x + (Bn+1 − Bn )]pn−1 (x)   Cn Cn+1 − pn−2 (x). −(n)(n − 1) n+1 n (23) From (22) and (23) we have identity −(1 − An f 0 (D))pn (x) + n[(An − An−1 )x + (Bn − Bn−1 )]pn−1 (x)   Cn Cn−1 −n(n − 1) − pn−2 (x) n n−1 = −(1 + (An − 2An+1 )f 0 (D))pn (x) + n[(An+1 − An )x + (Bn+1 − Bn )]pn−1 (x)   Cn+1 Cn −(n)(n − 1) − pn−2 (x). (24) n+1 n Comparing the nth degree terms on the both sides of (24) yields −(1 − An f 0 (D))pn (x) + n(An − An−1 )xpn−1 (x) = −(1 + (An − 2An+1 )f 0 (D))pn (x) + n(An+1 − An )xpn−1 (x). (25) In (25) the constant terms on the both sides are equal, which implies

712

12

T. X. He

−(1 − An f 0 (D))pn (x) = −(1 + (An − 2An+1 )f 0 (D))pn (x), or equivalently, An = An+1 for every n ≥ 0. Hence, (25) holds if and only if An = A0 ,

(26)

a nonzero constant for every n ≥ 0. Comparing the terms of degree n − 1 and n − 2 on the both sides of (24), we have the results Bn+1 − Bn = λ and Cn+1 Cn − =γ n+1 n for every n ≥ 0, where λ and γ are constants. Hence, Bn = B0 + nλ and Cn = n(C1 + (n − 1)γ)

(27)

for all n ≥ 1, where C1 , γ > 0 because of the request Cn An An−1 = Cn A20 > 0 for all n ≥ 1 (see Theorem 2.1). Substituting all of the established relationship of the sequences {An }n≥0 , {Bn }n≥0 , and {Cn }n≥1 into (19) and (22), we obtain, respectively,

pn+1 (x) = (A0 x + B0 + nλ)pn (x) − n(C1 + (n − 1)γ)pn−1 (x),

(28)

where A0 6= 0 and C1 , γ > 0, and (1 − A0 f 0 (D))pn (x) = λf (D)pn (x) − γf 2 (D)pn (x).

(29)

From (29), we further have f 0 (y) =

1 (1 − λf (y) + γf 2 (y)), A0

which implies g 0 (t) =

A0 1 − λt + γt2

by using the inverse function theorem. From (28), we have pn+1 (0) = (B0 + nλ)pn (0) − n(C1 + (n − 1)γ)pn−1 (0).

(30)

713

characterization of some orthogonal polynomials Noting A(t) =

tn n≥0 pn (0) n! ,

P

13

(30) implies

A0 (t) B0 − C1 t = A(t) 1 − λt + γt2 because

A(t)(B0 − C1 t) =

X

(B0 pn (0) − nC1 pn−1 (0))

n≥0

=

X

(pn+1 (0) − nλpn (0) + n(n − 1)γpn−1 (0))

n≥0

= (1 − λt + γt2 )

X

pn+1 (0)

n≥0 2

0

tn n!

tn n!

tn n!

= A (t)(1 − λt + γt ), which completes the proof of the theorem. Let the zeros of the denominator of g 0 (t) shown in (18) be α and β. Then one may solve g(t) and A(t) from (18) as follows. Corollary 3.2 Let A(t) and g(t) be defined as Definition 1.3. Then the polynomial sequence {pn (x)} defined by (12) is orthogonal if and on if ( g(t) =

A0 α−β ln A0 t 1−αt ,



1−βt 1−αt



,

if α 6= β, if α = β.

and  C1 −βB0 C1 −αB0   α(α−β) ln(1 − αt) − β(α−β) ln(1 − βt),   − C1 ln(1 − αt) − C1 −αB0 t , 2 α 1−αt ln f (t) = C α−αB C1 1 0  ln(1 − αt) + t,  2 α α   − C1 t2 + B t, 0 2

if 0 6= α 6= β 6= 0, if α = β = 6 0, if α = 6 β = 0, if α = β = 0,

Example 4 As an example, we set A0 = −1, B0 = C1 = 1, and α = β = 1 in Corollary 3.2 and obtain g(t) =

−t 1−t

and

A(t) =

1 . 1−t

714

14

T. X. He

Thus, from Theorem 3.1, the Laguerre polynomial sequence, {Ln (x)}, generated by (12) in Definition 1.3 with (A(t), g(t)) = (1/(1 − t), −t/(1 − t)) is an orthogonal polynomial sequence. Furthermore, from the expansion of (1 − t)2 , we can read λ = 2 and γ = 1, which implies the following recurrence relation for {Ln (x)}: Ln+1 (x) = (2n + 1 − x)Ln (x) − n2 Ln−1 (x) with the initial conditions L−1 (x) = 0 and L0 (x) = 1. Thus, L1 (x) = 1 − x, L2 (x) = 2 − 4x + x2 , L3 (x) = 6 − 18x + 9x2 − x3 , etc. Using Lemma 2.2, one may check the assumption of B0 = C1 = 1 is satisfied for {Ln (x)}. Since Z Z ∞ 2 h0 = L0 (x)dµ(x) = e−x dx = 1 S

0

and Z h1 =

L21 (x)dµ(x) =

S

Z

∞

(1 − x)2 e−x dx = 1,

0

we have A0 B0 = − h0

Z

xL20 (x)dµ(x)

S

Z

∞

=

e−x dx = 1

0

and C1 =

4

A1 h1 h1 = = 1. A0 h0 h0

Evaluate the measures and their supports of orthogonal sequences {Pn1,y,C (x)}

In this section, we will present a method to find the densities of measures µ(x) and their supports S (see (1)) of orthogonal generalized GegenbauerHumbert polynomial sequences, {Pn1,y,C (x)} (Cy > 0), using a technique of transferring a polynomial sequence defined by a recurrence relation of order two to an orthogonal Gegenbauer-Humbert polynomial sequence. This transfer technique can also give an orthogonal representation of non-orthogonal polynomials satisfying recurrence relation of order 2 in terms of only one or two terms of an orthogonal polynomial sequence. Thus, many useful approximation properties for orthogonal polynomials (for instance, Gaussian quadratures) can be transfered to some non-orthogonal polynomials. Many number and polynomial sequences can be defined, characterized, evaluated, and classified by linear recurrence relations with certain orders.

715

characterization of some orthogonal polynomials

15

A polynomial sequence {an (x)} is called sequence of order 2 if it satisfies the linear recurrence relation of order 2: an (x) = p(x)an−1 + q(x)an−2 (x),

n ≥ 2,

(31)

for some coefficient p(x) 6≡ 0 and q(x) 6≡ 0 and initial conditions a0 (x) and a1 (x). To construct an explicit formula of its general term, one may use a generating function, characteristic equation, or a matrix method (See Comtet [6], Hsu [20], Strang [21], Wilf [22], etc.) [12] presented a new method to construct an explicit formula of {an (x)} generated by (31). For the sake of reader’s convenience, we cite this result as follows (see also Miller and Takloo-Bighash [23] with different approaches). Proposition 4.1 Let {an (x)} be a sequence of order 2 satisfying the linear recurrence relation (31). Then    (  a1 (x)−β(x)a0 (x) n (x) − a1 (x)−α(x)a0 (x) β n (x), if α(x) 6= β(x); α α(x)−β(x) α(x)−β(x) an (x) = na1 (x)αn−1 (x) − (n − 1)a0 (x)αn (x), if α(x) = β(x), (32) where α(x) and β(x) are roots of t2 − p(x)t − q(x) = 0, namely, p p 1 1 α(x) = (p(x) + p2 (x) + 4q(x)), β(x) = (p(x) − p2 (x) + 4q(x)). (33) 2 2 We now give a transfer formula between different generalized GegenbauerHumbert polynomial sequences. This technique can be used to transfer any polynomials defined by recurrence relations of order 2 to a generalized Gegenbauer-Humbert polynomials. 0

0

Theorem 4.2 If {an (x) = Pn1,C ,y (x)}, a generalized Gegenbauer-Humbert polynomial sequence with parameters C 0 and y 0 , which is defined by (7) with coefficient polynomials p(x) = 2x/C 0 and q(x) = −y 0 /C 0 and initial conditions a0 (x) = 1/C 0 and a1 (x) = 2x/(C 0 )2 , then we have the following 0 0 transfer formula from {Pn1,y,C (x)}n≥0 to {Pn1,y ,C (x)}n≥0 : 0 0 Pn1,y ,C (x)

C n+2 = C0

s ±

y0 yCC 0

!n Pn1,y,C

 √  x yC ±√ 0 0 . yC 0

0

(34)

In particular, every polynomials sequence {Pn1,y ,C (x)} defined by (7) can be transfered to the Chebyshev polynomial sequence of the second kind by using the formula

716

16

T. X. He

0 0 Pn1,y ,C (x)

1 = 0 C

r ±

y0 C0

!n

  x Un ± √ 0 0 . yC

(35)

Proof. We first modify the explicit formula of the polynomial sequences defined by linear recurrence relation (32) of order 2. If α(x) 6= β(x), the first formula in (32) can be written as

an (x) =

a1 (x)((α(x))n − (β(x))n ) − a0 (x)α(x)β(x)((α(x))n−1 − (β(x))n−1 ) . α(x) − β(x)

Noting that −α(x)β(x) = α(x)(α(x) − p(x)) = β(x)(β(x) − p(x)), we may further write the above expression of an (x) as an (x) =

1 [a1 (x)((α(x))n − (β(x))n ) + a0 (x)α(x)(α(x) − p(x)) α(x) − β(x)  ×(α(x))n−1 − a0 (x)β(x)(β(x) − p(x))(β(x))n−1

a0 (x)((α(x))n+1 − (β(x))n+1 ) + (a1 (x) − a0 (x)p(x))((α(x))n − (β(x))n ) . α(x) − β(x) (36) p p Denote r(x) = x + x2 − Cy and s(x) = x − x2 − Cy. To find a transfer formula between expressions (9) and (36), we set =

α(x) :=

r(x) k(x)

and β(x) :=

s(x) k(x)

(37)

for a nonzero real or complex valued function k(x), which are two roots of t2 − p(x)t − q(x) = 0. Thus, adding and multiplying two equations of (37) side by side, we obtain 2x k(x) yC α(x)β(x) = −q(x) = . (k(x))2 α(x) + β(x) = p(x) =

The above system implies s k(x) = ±

Cy , −q(x)

717

characterization of some orthogonal polynomials

17

and at p(x)k(x) p(x) x= =± 2 2

s

yC , −q(x)

r(x) and s(x) give expressions of α(x) and β(x) as q q     p(x) yC yC s ± r ± p(x) 2 2 −q(x) −q(x) q q , and β(x) = . α(x) = yC yC ± −q(x) ± −q(x)

(38)

It is clear that α(x) and β(x) satisfy α(x) + β(x) = p(x) and α(x)β(x) = −q(x). p We first consider the case of k(x) = −yC/q(x). Substituting the corresponding (38) with positive sign into (36), we have an (x) a0 (x)(rn+1 (x) − sn+1 (x)) + k(x)(a1 (x) − a0 (x)p(x))(rn (x) − sn (x)) = k n (x)(r(x) − s(x)) s !n   −q(x) k(x)p(x) n+2 1,y,C = a0 (x)C Pn yC 2 s !n−1   −q(x) k(x)p(x) 1,y,C n+1 +(a1 (x) − a0 (x)p(x))C Pn−1 yC 2 s s !n ! −q(x) yC p(x) = a0 (x)C n+2 Pn1,y,C yC 2 −q(x) s s !n−1 ! −q(X) p(x) yC 1,y,C +(a1 (x) − a0 (x)p(x))C n+1 Pn−1 . yC 2 −q(x) (39) p Similarly, for k(x) = − −yC/q(x), we have s

!n

s ! p(x) yC n+2 1,y,C an (x) = a0 (x)C − Pn − 2 −q(x) s s !n−1 ! −q(x) p(x) yC 1,y,C n+1 +(a1 (x) − a0 (x)p(x))C − Pn−1 − . yC 2 −q(x) −q(x) yC

(40)

718

18

T. X. He Therefore, an (x) defined by (31) can be presented as s !n ! −q(x) yC p(x) an (x) = a0 (x)C n+2 ± Pn1,y,C ± yC 2 −q(x) s s !n−1 ! −q(X) p(x) yC 1,y,C n+1 +(a1 (x) − a0 (x)p(x))C ± Pn−1 ± , yC 2 −q(x) s

(41) where {Pn1,y,c } is the sequence of any generalized Gegenbauer-Humbert polynomials with λ = 1. In particular, an (x) can be expressed in terms of {Pn1,1,1 = Un }, the sequence of Chebyshev polynomials of the second kind: ! p(x) ± p 2 −q(x) n−1  p Un−1 +(a1 (x) − a0 (x)p(x)) ± −q(x)

 p n an (x) = a0 (x) ± −q(x) Un

p(x) ± p 2 −q(x)

! , (42)

which is a special case of (41) for (y, C) = (1, 1). 0 0 If an (x) = Pn1,y C (x) defined by (7) with coefficient polynomials p(x) = 2x/C 0 and q(x) = −y 0 /C 0 and initial conditions a0 (x) = 1/C 0 and a1 (x) = 2x/(C 0 )2 , then a1 (x) − a0 (x)p(x) = 0 and (41) and (42) are reduced to (34) and (35), respectively.

From Theorem 4.2, we immediately have transfer formulas Pn+1 (x) = (±i)n Un (∓xi) ,   xi n Fn+1 (x) = (±i) Un ∓ , 2  x  Bn (x) = (±1)n Un ± +1 , 2   √ n x Φn+1 (x) = ± 2 Un ± √ , 2 2   √
n 1 x Dn (x, a) = ± a Un ± √ . 2 2 a

719

characterization of some orthogonal polynomials

19

Remark 2 It is obvious that when both y and C are integers, the corresponding generalized Gegenbauer-Humbert polynomials have integer coefficients. Formulas (34) can be used to transfer between the generalized Gegenbauer-Humbert polynomials with integer coefficients and the generalized Gegenbauer-Humbert polynomials with non-integer coefficients. For instance, the last transfer formula shown above presents the Dickson polynomial of the second kind with real coefficients in terms of the Chebyshev polynomials of the second kind. If yC > 0, from Theorem 2.1 we know that {Pn1,y,C (x)} is an orthogonal polynomial sequence. Let w(x) and S = [a, b] be the density function and its support interval of {Pn1,y,C (x)}. We now use Theorem 4.2 to find the density function and its support interval of {Pn1,y,C (g(x))}, where g(x) is a one-to-one and differentiable function.

Theorem 4.3 Let {Pn1,y,C (x)} be a polynomial sequence defined by (7), and let g(x) be a one-to-one and differential function. Then sequence {Pn1,y,C (g(x))} is an orthogonal polynomial sequence associated with the density function

p w(x) = g 0 (x) 1 − (g(x))2 /(yC) √ √ with support interval between g −1 (− yC) and g −1 ( yC), where g −1 (x) is the composition inverse of g(x), i.e., (g −1 ◦ g)(x) = (g ◦ g −1 )(x) = x. Furthermore,

Z

√ g −1 ( yC)

√ g −1 (− yC)

√ (g(x))2 π yC  y n 1− dx = δn,m , yC 2C 2 C (43)

s 1,y,C Pn1,y,C (x)Pm (x)g 0 (x)

where δn,m is the Kronecker symbol. In particular, if g(x) = x, then {Pn1,y,C p (x)} is an orthogonal polynomial sequence with respect to density function 1 − x2 /(yC) over support interval  √ √  − yC, yC , and {Pn1,y,C (x)} satisfies (43) when g(x) = g −1 (x) = x.

1,y,C Proof. Let us consider inner product hPn1,y,C (x), Pm (x)i√1−x2 /(yC) over √ √ [− yC, yC], in which the transfer formula (35) will be applied:

720

20

T. X. He

√

√ − yC Z √yC

= = =

s

yC

x2 dx yC    r n+m  s x x 1 x2 y √ √ ± ± ± U U 1 − dx n m √ 2 C yC yC yC − yC C  r n+m Z 1 p p y 1 2 yCdx ± U (x)U (x) 1 − x n m C2 C −1 √   yC y n π δn,m , C2 C 2

Z

1,y,C P 1,y,C (x)Pm (x) 1 −

where the rightmost integral yields (π/2)δn,m due to the orthogonality of {Un (x)} (see, for examples, [24] by Mason and Handscomb and [25] by Rivlin). Hence, using a transformation we obtain √ g −1 ( yC)

Z

=

s 1,y,C P 1,y,C (x)Pm (x)g 0 (x)

√ g −1 (− yC) s Z √yC 1,y,C P 1,y,C (x)Pm (x) 1 √ − yC

√

=

−

1−

(g(x))2 dx yC

x2 dx yC

yC  y n π δn,m . C2 C 2

Corollary 4.4 Let {Pn1,C,C (x)}, C 6= 0, be a polynomial sequence defined by (7) with λ = 1, and let g(x) be a one-to-one and differential function. Then sequence {Pn1,C,C (g(x))} is an orthogonal polynomial sequence satisfying recurrence relation (10) associated with the density function w(x) =

g 0 (x) p 2 C − (g(x))2 |C|

with support interval between g −1 (−|C|) and g −1 (|C|), where g −1 (x) is the composition inverse of g(x), i.e., (g −1 ◦ g)(x) = (g ◦ g −1 )(x) = x. Furthermore, Z

g −1 (|C|)

g −1 (−|C|)

1,C,C Pn1,C,C (x)Pm (x)

g 0 (x) p 2 π|C| C − (g(x))2 dx = δn,m , |C| 2C 2

(44)

721

characterization of some orthogonal polynomials

21

where δn,m is the Kronecker symbol. In particular, if g(x) = x, then {Pn1,C,C p(x)} is an orthogonal polynomial sequence with respect to density function 1 − x2 /C 2 over support interval [−|C|, |C|], and {Pn1,C,C (x)} satisfies (44) when g(x) = g −1 (x) = x. Example 5 From Theorem 4.3, Morgan-Voyc polynomial sequence {Bn (x) =
1,1,1 x P 2 + 1 } is orthogonal with respect to the density function w(x) = √n −4x − x2 /4 with support [−4, 0]. The sequence of Fermat polynomials of 1,2,1 the first kind, {Φn (x) = Pn−1 (x/2)}, is orthogonal with respect to the den√ √ √ √ 2 sity function w(x) = 8 − x /(4 2) with support [−2 2, 2 2]. Dickson polynomials {Dn (x, a) = Pn1,2a,2 (x)} of the second kind when √ are orthogonal √ a > 0 with respect to the density function w(x) = 4a − x2 /(2 a) over the √ √ support interval [−2 a, 2 a]. In addition, we have √

0

−4x − x2 π dx = δn,m , 4 2 −4 √ Z 2 √2 8 − x2 √ dx = π2n−(1/2) δn,m , Φ (x)Φ (x) n m √ 4 2 −2 2 √ Z 2 √a 4a − x2 π √ Dn (x, a)Dm (x, a) dx = an+1/2 δn,m . √ 4 2 a −2 a Z

Bn (x)Bm (x)

References [1] R. Askey and M. Ismail, Recurrence Relations, Continued Fractions and Orthogonal Polynomials, Mem. AMS, Vol. 49, Num. 300, AMS, Providence, Rhode Island, 1984. [2] J. Dombrowski and P. Nevai, Orthogonal polynomials, measures and recurrence relations. SIAM J. Math. Anal. 17 (1986), no. 3, 752–759. [3] C. F. Dunkl and Y. Xu, Orthogonal Polynomials of Several Variables, Encyclopedia of Mathematics and Its Applications 81, Cambridge University Press, Cambridge, UK, 2001. [4] G. Szeg¨o, Orthogonal polynomials. Fourth edition. American Mathematical Society, Colloquium Publications, Vol. XXIII. American Mathematical Society, Providence, R.I., 1975. [5] T. S. Chihara, 45 years of orthogonal polynomials: a view from the wings, J. Comp. Appl. Math., 133 (2001), 13-21.

722

22

T. X. He [6] L. Comtet, Advanced Combinatorics, Reidel, Dordrecht, 1974. [7] H. W. Gould, Inverse series relations and other expansions involving Humbert polynomials, Duke Math. J. 32 (1965), 697–711. [8] T. X. He, L. C. Hsu, P. J.-S. Shiue, A symbolic operator approach to several summation formulas for power series II, Discrete Math. 308 (2008), no. 16, 3427–3440. [9] T. Koshy, Fibonacci and Lucas numbers with applications, Pure and Applied Mathematics (New York), Wiley-Interscience, New York, 2001.

[10] R. Lidl, G. L. Mullen, and G. Turnwald, Dickson polynomials. Pitman Monographs and Surveys in Pure and Applied Mathematics, 65. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. [11] L. C. Hsu, On Stirling-type pairs and extended Gegenbauer-HumbertFibonacci polynomials. Applications of Fibonacci numbers, Vol. 5 (St. Andrews, 1992), 367–377, Kluwer Acad. Publ., Dordrecht, 1993. [12] T. X. He and P. J.-S. Shiue, On sequences of numbers and polynomials defined by linear recurrence relations of order 2, International Journal of Mathematics and Mathematical Sciences, Volume 2009 (2009), Article ID 709386. [13] A. F. Horadam and J. M. Mahon, Pell and Pell- Lucas polynomials. Fibonacci Quart. 23 (1985), no. 1, 7–20. [14] A. F. Horadam, Chebyshev and Fermat polynomials for diagonal functions. Fibonacci Quart. 17 (1979), no. 4, 328–333. [15] S. Roman, The Umbral Calculus, Acad. Press., New York, 1984. [16] S. Roman and G.-C. Rota, The Umbral Calculus, Adv. Math., 1978, 95-188. [17] T. X. He, L. C. Hsu, P. J.-S. Shiue, The Sheffer Group and the Riordan Group, Discrete Appl. Math., (155) 2007, 1895-1909. [18] T. X. He, L. C. Hsu, P. J.-S. Shiue, Multivariate Expansion Associated with Sheffer-type Polynomials and Operators, Bull. Inst. Math. Acad. Sin. (N.S.) 1 (2006), no. 4, 451–473.

723

characterization of some orthogonal polynomials

23

[19] J. Meixner, Orthogonale Polynomsysteme mit einer besonderen Gestalt der erzeugende Funktion, J. London Math. Soc., 9 (1934), 6-13. [20] L. C. Hsu, Computational Combinatorics (Chinese), First edition, Shanghai Scientific & Techincal Publishers, Shanghai, 1983. [21] G. Strang, Linear algebra and its applications. Second edition. Academic Press (Harcourt Brace Jovanovich, Publishers), New YorkLondon, 1980. [22] H. S. Wilf, Generatingfunctionology, Academic Press, New York, 1990. [23] S. J. Miller and R. Takloo-Bighash, An Invitation to Modern Number Theory, Princeton University Press, Princeton and Oxford, 2006. [24] J. C. Mason and D. C. Handscomb, Chebyshev Polynomials, Chapman & Hall/CRC, Boca Raton, FL, 2003. [25] T. J. Rivlin, Chebyshev polynomial: from approximation theory to algebra and number theory, Second edition, John Wiley, NJ, 1990.

JOURNAL 724 OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.4, 724-729, 2011, COPYRIGHT 2011 EUDOXUS PRESS, LLC

On the stability of quadratic double centralizers on Banach algebras

M. Eshaghi Gordji and A. Bodaghi Department of Mathematics, Semnan University, P. O. Box 35195-363, Semnan, Iran e-mail: [email protected] Department of Mathematics, Azad University, Garmsar Branch, Garmsar, Iran e-mail: abasalt [email protected]

Abstract. In this paper, we investigate the generalized Hyers–Ulam–Rassias stability of the system of functional equations   f (x + y) + f (x − y) = 2f (x) + 2f (y), .   f (xy) = f (x)y, on Banach algebras.

1. Introduction Speaking of the stability of a functional equation we follow the question of S. Ulam: “when is it true that the solution of an equation differing slightly from a given one, must of necessity be close to the solution of the given equation?” (see [39], p. 63) As the words “differing slightly” and “be close” may have various meanings, different kinds of stability can be dealt with. The first answer to Ulam’s question, concerning the Cauchy equation was given by D.H. Hyers [8]. Thus we speak about the Hyers–Ulam stability. In 1950, Hyers’s theorem was generalized T. Aoki [2] for additive mappings and independently, in 1978, Th.M. Rassias [30] provided a generalization of Hyers’s theorem, which allows the Cauchy difference to be unbounded. This new concept is known as Hyers–Ulam–Rassias stability. The functional equation f (x1 + x2 ) + f (x1 − x2 ) = 2f (x1 ) + 2f (x2 )

(1.1)

is related to symmetric bi–additive function and is called a quadratic functional equation and every solution of the quadratic equation (1.1) is said to be a quadratic function. It is well known that a function f between two real vector spaces is quadratic if and only if there exists a unique symmetric bi–additive function B such that f (x1 ) = B(x1 , x1 ) for all x1 0 0

2000 Mathematics Subject Classification: 39B82, 39B52, 46H25. Keywords: Stability; Quadratic mapping; Double centralizer

725

M. Eshaghi-Gordji and A. Bodaghi

2

where 1 (f (x1 + x2 ) − f (x1 − x2 )) 4 (see [1,19]). Skof [38] proved Hyers–Ulam stability problem for quadratic functional equation (1.1) for a class of functions f : A −→ B, where A is normed space and B is a Banach space. In 1992, Czerwik [8] proved the Hyers–Ulam–Rassias stability of the equation (1.1). It may happen that all approximate solutions are in fact exact solutions. Then we speak about superstability. To get acquainted with the theory of the stability of functional equations we refer to [4,5,10,11,17,29,31,32] and [41]. Let A be a Banach algebra and let X be a Banach A−module. A quadratic mapping L : A → X is said to be quadratic left centralizer if L(ab) = L(a)b for all a, b ∈ A. Similarly, a quadratic mapping R : A → X that R(ab) = aR(b) for all a, b ∈ A is called a right centralizer from A into X. A quadratic double centralizer from A into X is a pair (L, R), where L is a quadratic left centralizer, R is a quadratic right centralizer and aL(b) = R(a)b for all a, b ∈ A. Example. Let A be a Banach algebra. Set   0 A A T :=  0 0 A  , 0 0 0 B(x1 , x2 ) =

then T is a Banach algebra by the following norm:   0 a b k  0 0 c  k = kak + kbk + kck 0 0 0 So

(a, b, c ∈ A).

 0 A∗ A∗ 0 A∗  , T∗ = 0 0 0 0 following norm  f g 0 h  k = M ax{kf k, kgk, khk} 0 0 

is the dual of T by the  0 k 0 0 Let the left module define as follows.  0 f h 0 0 0 0

(f, g, h ∈ A∗ ).

action of T on T ∗ be trivial and let the right module action of T on T ∗  g 0 h  0 0 0

a 0 0

  b 0 c , 0 0 0

∗

for all f, g, h ∈ A , a, b, c, x, y, z ∈ A. Then T T ∗ . We define D : T  0 D( 0 0

−→ T ∗ by   a b 0 0 c ) =  0 0 0 0

k 0 0

 y z i = f (ax) + g(by) + h(cz), 0  0 k is a Banach T -module. Let  0 0 0 0 x 0 0

∗

 g 0 h  0 0 0

0 0 0

 ac 0  0

(a, b, c ∈ A).

Then we can show that (D, D) is a quadratic double centralizer from T into T ∗ .

 g h ∈ 0

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On the stability of quadratic double centralizers on Banach algebras

3

2. Main results Moslehian, Rahbarnia and Sahoo [11] established the stability of double centralizers in the framework of Banach algebras. In this paper, we prove the same results for quadratic double centralizers. Theorem 2.1. Let A be a Banach algebra.Suppose f : A −→ A is a mapping with f (0) = 0. If there exist mapping g : A −→ A with g(0) = 0 and functions χj : A4 −→ [0, ∞)(j = 1, 2) and σ : A2 −→ [0, ∞) such that χ fj (a, b, c, d) :=

∞ X 1 χ (kn a, kn b, kn c, kn d) < ∞ (1) 2n j k n=0

1 σ(akn a, kn b) = 0 (2) k3n k f (ka + b + tcd) + f (ka − b + tcd) − 2k2 f (a) − 2f (b) − 2tf (c)d k≤ χ1 (a, b, c, d) (3) lim

n−→∞

k g(ka + b + tcd) + g(ka − b + tcd) − 2k2 g(a) − 2g(b) − 2tcg(d) k≤ χ2 (a, b, c, d) (4) k af (b) − g(a)b k≤ σ(a, b) (5), for all a, b, c, d ∈ A and t ∈ R such that 0 < t < 1. Then there exists a unique quadratic double centralizer (L, R) such that for a ∈ A 1 k f (a) − L(a) k≤ 2 χ f1 (a, 0, 0, 0) 2k 1 k g(a) − R(a) k≤ 2 χ f2 (a, 0, 0, 0). 2k Proof. Setting b = c = d = 0 in (3), we have k 2f (ka) − 2k2 f (a) k≤ χ(a, 0, 0, 0)

,

for all a ∈ A. Thus

1 1 f (ka) − f (a) k≤ 2 χ(a, 0, 0, 0) , k2 2k for all a ∈ A. Hence for all nonnegative integer p, q with q > p and all a ∈ A, we have k

k

q−1 X 1 1 1 p q f (k a) − f (k a) k≤ χ(kn a, 0, 0, 0) 2n+2 k2p k2q 2k n=p

(6),

1 It follows from (1) and (6) that sequence { k2n f (kn a)} is cauchy for all a ∈ A. By comn 1 pleteness of A, the sequence { k2n f (k a)} converges. we define the mapping L : A −→ A by 1 L(a) := lim 2n f (kn a) n→∞ k for all a ∈ A. By (2.10). Putting c = d = 0 in (3) we get k L(ka + b) − L(ka − b) − 2k2 L(a) − 2L(b) k 1 = limn−→∞ k2n k f (kn+1 a + K n b) − L(kn+1 a − K n b) − 2k2 f (kn a) − 2L(kn b) k 1 ≤ limn→∞ k2n χ(kn a, kn b) = 0. The last equality and inequality follow from (1) and (3). So the mapping L is quadratic. Putting a = b = 0, t = k1n and replacing c and d by kn c and kn d, respectively, we have

k 2f (kn cd) − 2f (kn c)d k≤ χ1 (0, 0, kn c, kn d). Hence by dividing the both sides of the above inequality by k2n , we get k

f (kn cd) f (kn c) 1 − d k≤ 2n χ1 (0, 0, kn c, kn d). 2n k k2n 2k

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M. Eshaghi-Gordji and A. Bodaghi

4

Taking limit as n −→ ∞, we conclude L(cd) = L(c)d. It follows from (6) with p = 0 and passing the limit q −→ ∞, we get 1 k f (a) − L(a) k≤ 2n χ1 (a, 0, 0, 0). 2k Let L0 : A −→ A be a map that satisfied k f (a) − L(a) k≤ 2k12n χ1 (a, 0, 0, 0). Since L and L0 are quadratic, for each a ∈ A, we have 1 k L(a) − L0 (a) k = 2n k L(kn a) − L0 (kn a) k k 1 ≤ 2n (k L(kn a) − f (kn a) k + k L0 (kn a) − f (kn a) k) k 1 e(kn a, 0, 0, 0). ≤ 2n+2 χ k Which tend to zero as n −→ ∞, for all a ∈ A, so we have L(a) = L0 (a), for all a ∈ A. A similar argument for χ2 and g show that there exist a unique quadratic function right n a) multiplier R defined by R(A) := limn→∞ g(k .Replacing a and b by kn a and kn b in (5) k2n 3n and dividing by k , we get f (kn b) g(kn a) 1 − b k≤ 2n σ(kn a, kn b). k2n k2n 2k Passing to the limit as n −→ ∞, we obtain aL(b) = R(a)b for a, b ∈ A. ka



Corollary 2.2. Let A be a Banach algebra and p < 2 and  be a positive real numbers.Let f, g : A −→ A be a mappings satisfy f (0) = g(0) = 0. If there is function σ : A2 −→ [0, ∞) such that 1 lim σ(kn a, kn b) = 0, n−→∞ k 3n k f (ka+b+tcd)+f (ka−b+tcd)−2k2 f (a)−2f (b)−2tf (c)d k≤ (k a kp + k b kp + k c kp + k d kp ), k g(ka+b+tcd)+g(ka−b+tcd)−2k2 g(a)−2g(b)−2tcg(d) k≤ (k a kp + k b kp + k c kp + k d kp ), k af (b) − g(a)b k≤ (k a kp + k b kp ), for all a, b, c, d ∈ A and t ∈ R such that 0 < t < 1. Then there exist a unique quadratic double centralizer (L, R) such that for a ∈ A  k a kp , 8 − 2p+1  k a kp k g(a) − R(a) k≤ . 8 − 2p+1 k f (a) − L(a) k≤

Proof. For j = 1, 2 we put χj (a, b, c, d) = (k a kp + k b kp + k c kp + k d kp ) and σ(a, b) = (k a kp + k b kp ) in Theorem 2.1.  Theorem 2.3. Let A be a Banach algebra.Suppose f : A −→ A is a mapping with f (0) = 0. If there exist mapping g : A −→ A with g(0) = 0 and functions χj : A4 −→ [0, ∞)(j = 1, 2) and σ : A2 −→ [0, ∞) such that χ fj (a, b, c, d) :=

∞ X

k2n χj (k−n a, k−n b, k−n c, k−n d) < ∞

n=0

lim k3n σ(k−n a, k−n b) = 0

n−→∞

k f (ka + b + tcd) + f (ka − b + tcd) − 2k2 f (a) − 2f (b) − 2tf (c)d k≤ χ1 (a, b, c, d) k g(ka + b + tcd) + g(ka − b + tcd) − 2k2 g(a) − 2g(b) − 2tcg(d) k≤ χ2 (a, b, c, d)

728

On the stability of quadratic double centralizers on Banach algebras

5

k af (b) − g(a)b k≤ σ(a, b), for all a, b, c, d ∈ A and t ∈ R such that 0 < t < 1. Then there exist a unique quadratic double centralizer (L, R) such that for a ∈ A 1 a χ f1 ( , 0, 0, 0) 2 k a 1 f2 ( , 0, 0, 0). k g(a) − R(a) k≤ χ 2 k k f (a) − L(a) k≤

Proof. Using the same method as in Theorem 2.1.



Corollary 2.4. Let A be a Banach algebra and p > 2 and  be a positive real numbers.Let f, g : A −→ A be a mappings satisfy f (0) = g(0) = 0. If there is function σ : A2 −→ [0, ∞) such that lim k3n σ(k−n a, k−n b) = 0, n−→∞

k f (ka+b+tcd)+f (ka−b+tcd)−2k2 f (a)−2f (b)−2tf (c)d k≤ (k a kp + k b kp + k c kp + k d kp ), k g(ka+b+tcd)+g(ka−b+tcd)−2k2 g(a)−2g(b)−2tcg(d) k≤ (k a kp + k b kp + k c kp + k d kp ), k af (b) − g(a)b k≤ (k a kp + k b kp ), for all a, b, c, d ∈ A and t ∈ R such that 0 < t < 1. Then there exist a unique quadratic double centralizer (L, R) such that for a ∈ A k f (a) − L(a) k≤

 k a kp , 2p+1 − 8

k g(a) − R(a) k≤

 k a kp . 2p+1 − 8

Proof. For j = 1, 2 we put χj (a, b, c, d) = (k a kp + k b kp + k c kp + k d kp ) and σ(a, b) = (k a kp + k b kp ) in Theorem 2.3. 

References [1] J. Aczel, J. Dhombres, Functional Equations in Several Variables, Cambridge Univ. Press, 1989. [2] T. Aoki, On the stability of the linear transformation in Banach spaces, J.Math.Soc.Japan2 (1950) 64–66. [3] Y. Benyamini, J. Lindenstrauss,Geometric Nonlinear Functional Analysis, vol. 1, Colloq. Publ. vol. 48, Amer. Math. Soc., Providence, RI, 2000. [4] L. C˘ adariu, V. Radu, Fixed points and the stability of quadratic functional equations, Analele Universitatii de Vest din Timisoara, vol. 41, no. 1, pp. 25-48, 2003. [5] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992) 59–64. [6] M. Eshaghi Gordji, H. Khodaei, C. Park, A fixed point approach to the Cauchy-Rassias stability of general Jensen type quadratic-quadratic mappings, To appear in Bull. Korean Math. Soc. [7] M. Eshaghi Gordji, H. Khodaei, Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi–Banach spaces,Nonlinear Analysis: Theory, Methods & Applications, Volume 71, Issue 11, 2009, Pages 5629–5643. [8] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. 27 (1941) 222–224.

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[9] S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, 2001. [10] Pl. Kannappan, Quadratic functional equation and inner product spaces, Results Math. 27 (1995) 368–372. [11] M. S. Moslehian, F. Rahbarnia and P. K. Sahoo, Approximate double centralizers are exact double centralizers, Bol. Soc. Mat. Mexicana, 13, (2007), 111–122. [12] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978) 297–300. ˇ [13] Th. M. Rassias, P. Semrl, On the Hyers-Ulam stability of linear mappings, J.Math. Anal. Appl. 173 (1993) 325-338. [14] Th.M. Rassias, K. Shibata,, Variational problem of some quadratic functions in complex analysis, J. Math. Anal. Appl. 228 (1998) 234-253. [15] F. Skof, Propriet locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano, 53 (1983) 113-129. [16] S. M. Ulam, Problems in Modern Mathematics, Chapter VI, science Editions., Wiley, New York, 1964.

JOURNAL 730 OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.4, 730-733, 2011, COPYRIGHT 2011 EUDOXUS PRESS, LLC

Generalized multi–valued contraction mappings M. Eshaghi Gordji, H. Baghani, H. Khodaei and M. Ramezani Department of Mathematics, Semnan University, P. O. Box 35195-363, Semnan, Iran e-mail: [email protected], [email protected],[email protected],[email protected] Abstract. In this paper, the famous Nadler’s fixed point theorem [S.B. Nadler Jr., Multivalued contraction mappings, Pacific J. Math. 30 (1969) 475-487] is extended to the case of generalized multi-valued mappings.

1. Introduction Let (X, d) be a metric space. CB(X) denotes the collection of all nonempty closed bounded subsets of X. For A, B ∈ CB(X) and x ∈ X, define distd (x, A) := inf{d(x, a); a ∈ A} and Hd (A, B) := max{sup distd (a, B), sup distd (b, A)}. a∈A

b∈B

It is easy to see that Hd is a metric on CB(X). Hd is called the Hausdorff metric induced by d. Let k ∈ N, and M0 := X, H0 := d, for each i ∈ {1, 2, ..., k}, put Mi := CB(Mi−1 ) and Hi := HHi−1 . One can show that (Mi , Hi ) is a complete metric space for all i ∈ {1, 2, ..., k}, whenever (X, d) is a complete metric space (see for example Lemma 8.1.4, of [4]). Every mapping T from X into Mk is called generalized multi-valued mapping. Definition 1.1. Let k ∈ N. An element x ∈ X is said to be a fixed point of a genralized multi–valued mapping T : X → Mk , if there exist A1 ∈ M1 , A2 ∈ M2 , · · · , Ak−2 ∈ Mk−2 and Ak−1 ∈ Mk−1 such that x ∈ A1 ∈ A2 ∈ · · · ∈ Ak−1 ∈ T (x). In 1969, Nadler [3] extended the Banach contraction principle [1] to set– valued mappings. Theorem 1.2. (See Nadler [3].) Let (X, d) be a complete metric space and let T be a mapping from X into CB(X). Assume that there exists r ∈ [0, 1) such that Hd (T x, T y) ≤ rd(x, y) for all x, y ∈ X. Then there exists z ∈ X such that z ∈ T (z). 0 0

2000 Mathematics Subject Classification: 54H25. Keywords: Hausdorff metric; Multi-valued contraction; Nadler’s fixed point theorem

MULTI-VALUED CONTRACTION MAPPINGS

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M. Eshaghi Gordji et al.

In this paper, we give a generalization of Nadler’s fixed point theorem. 2. Main result The following lemma has important role in the proving of the main theorem. Lemma 2.1. (See Nadler [3].) Let (X, d) be a metric space and let A, B ∈ CB(X). Then for each a ∈ A and  > 0 there exists b ∈ B such that

d(a, b) ≤ Hd (A, B) + . Now, we proceed the main result of the paper, which can be regarded as an extension of Nadler’s fixed point theorem. Theorem 2.2. Let k ∈ N and let (X, d) be a complete metric space. Let T be a mapping from X into Mk . Assume that there exists l ∈ [0, 1) such that Hk (T x, T y) ≤ ld(x, y) for all x, y ∈ X. Then T has a fixed point. Proof. If l = 0, then the proof is clear. Now, assume l ∈ (0, 1), x0 ∈ X, and

∈ T x0 . A11 ∈ A21 ∈ · · · ∈ Ak−1 1

By putting  := l in above lemma, it follows that  k−1 k ∈ T x1 ; Hk−1 (Ak−1 ∃Ak−1  1 , A2 ) ≤ H (T x0 , T x1 ) + l, 2       k−2 k−1 k−2 k−1 (Ak−1 , Ak−1 ) + l,   Hk−2 (Ak−2  ∃A2 ∈ A2 ; 1 , A2 ) ≤ H 1 2    . .    .     ∃A12 ∈ A22 ; H1 (A11 , A12 ) ≤ H2 (A21 , A22 ) + l,       ∃x2 ∈ A12 ; d(x1 , x2 ) ≤ H1 (A21 , A12 ) + l. Similarly by putting  := l2 in Lemma 2.1, we get

731

732

MULTI-VALUED CONTRACTION MAPPINGS

3

 k−1 k 2 ∃Ak−1 ∈ T x2 ; Hk−1 (Ak−1  3 2 , A3 ) ≤ H (T x1 , T x2 ) + l ,       k−2 k−1 (Ak−1 , Ak−1 ) + l2 ,   ∃Ak−2 ∈ Ak−1 Hk−2 (Ak−2  3 3 ; 2 , A3 ) ≤ H 2 3    . .    .     ∃A13 ∈ A23 ; H1 (A12 , A13 ) ≤ H2 (A22 , A23 ) + l2 ,       ∃x3 ∈ A13 ; d(x2 , x3 ) ≤ H1 (A12 , A13 ) + l2 . Hence, we have d(x2 , x3 ) ≤ Hk (T x1 , T x2 ) + kl2 ≤ ld(x1 , x2 ) + kl2 . Using the same reason as above, it can be shown by induction that there exist sequences {xn }n∈N in X and {Ain }n∈N in Mi , i = 1, 2, · · · , k − 1, such that  i i i i+1 n H (An , An+1 ) ≤ Hi+1 (Ai+1  n , An+1 ) + l , i = 1, 2, · · · , k − 2,     k−1 k n Hk−1 (Ak−1 n , An+1 ) ≤ H (T xn−1 , T xn ) + l ,      d(xn , xn+1 ) ≤ H1 (A1n , A1n+1 ) + ln . Hence, d(xn , xn+1 ) ≤ Hk (T xn−1 , T xn ) + kln ≤ ld(xn−1 , xn ) + kln

(1)

for all n ∈ N. It can be conclude that d(xn , xn+1 ) ≤ ln d(x0 , x1 ) + nkln for all n ∈ N. Therefore ∞ ∞ ∞ X X X d(xn , xn+1 ) ≤ ln d(x0 , x1 ) + k nln < ∞. n=1

n=1

n=1

It follows that {xn }n∈N is a Cauchy sequence. Since X is complete, then there exists x∗ ∈ X such that limn→∞ xn = x∗ . Similarly we can show that Hi (Ain , Ain+1 ) ≤ ln−1 Hi (Ai1 , Ai2 ) + (n − 1)kln for all n ∈ N, i = 1, 2, · · · , k − 1. Therefore {Ain }n∈N is a Cauchy sequence in Mi , for i = 1, 2, · · · , k − 1. On the other hand, we know that (Mi , Hi ) is a

MULTI-VALUED CONTRACTION MAPPINGS

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M. Eshaghi Gordji et al.

4

complete metric space for each i ∈ {1, 2, · · · , k − 1}. Hence, there exist A∗1 ∈ M1 , A∗2 ∈ M2 , · · · , A∗k−2 ∈ Mk−2 and A∗k−1 ∈ Mk−1 such that limn→∞ Ain = A∗i , i = 1, · · · , k − 1. Since xn → x∗ , and T is a generalized multi–value contraction mapping, then ∗ k ∗ lim distHk−1 (Ak−1 n , T x ) ≤ lim H (T xn , T x ) = 0.

n→∞

n→∞

It follows that ∗ k−1 ∗ 0 = lim distHk−1 (Ak−1 n , T x ) = distHk−1 ( lim An , T x ) n→∞

n→∞

= Hence,

A∗k−1

∈

T x∗

=

T x∗ .

distHk−1 (A∗k−1 , T x∗ ).

Similarly we can show that

x∗ ∈ A∗1 ∈ A∗2 · · · ∈ A∗k−1 and this completes the proof.



Many fixed point theorems have been proved by various authors as generalizations to the Nadler’s theorem. One such generalization is due to Mizogochi and Takahashi [2] (see also [5]). Theorem 2.3. (See Mizogochi and Takahashi [2].) Let (X, d) be a complete metric space and let T be a mapping from X into CB(X). Let α be a mapping from [0, ∞) into [0, 1) such that lim supr→t+ α(r) < 1 for all t ∈ [0, ∞) and let Hd (T x, T y) ≤ α(d(x, y))d(x, y) for all x, y ∈ X. Then T has a fixed point. The question arises here is whether Mizoguchi–Takahashi’s fixed point theorem can be extended similar to generalized multi–valued mappings or not? References [1] S. Banach, Sur les operations dans onsembles abstraits et leurs application aux equations integrales, Fund. Math. 3 (1922) 133-181. [2] N. Mizogochi and W. Takahashi, Fixed point theorem for multi-valued mappings on complete metric spaces, J. Math. Anal. Appl. 141 (1989) 177-188. [3] N.B. Nadler Jr., Multi-valued contraction mappings, Pacific J. Math. 30 (1969) 475-488. [4] I.A. Rus, Generalized Contractions and Applications, Cluj Univercity Press, ClujNappa, 2001. [5] T. Suzuki, Mizoguchi–Takahashi ’s fixed point theorem is a real generalization of Nadler ’s, J. Math. Anal. Appl. 340 (2008) 752-755.

JOURNAL 734 OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.4, 734-742, 2011, COPYRIGHT 2011 EUDOXUS PRESS, LLC

Nearly higher derivations in unital C ∗ −algebras 1

M. Eshaghi Gordji, 2 R. Farokhzad Rostami and 3 S. A. R. Hosseinioun

1

Department of Mathematics, Semnan University, P. O. Box 35195-363, Semnan, Iran 2,3 Department of Mathematics, Shahid Beheshti University, Tehran, Iran e-mail: [email protected], [email protected],[email protected] Abstract. In this paper we investigate the Hyers–Ulam–Rassias stability of higher derivations and higher Jordan derivation in unital C ∗ −algebras.

1. Introduction A classical question in the theory of functional equations is that “when is it true that a mapping which approximately satisfies a functional equation E must be somehow close to an exact solution of E.” Such a problem was formulated by S.M. Ulam [22] in 1940 and solved in the next year for the Cauchy functional equation by D.H. Hyers [8]. It gave rise to the stability theory for functional equations. During the last decades several stability problems of functional equations have been investigated by many mathematicians. A large list of references concerning the stability of functional equations can be found in [3, 9, 10, 14, 19, 20, 21]. It seems that approximate derivations was first investigated by K.-W. Jun and D.-W. Park [12]. Recently, the stability of derivations have been investigated by some authors; see [1, 4, 5, 6, 7, 12, 16, 17] and references therein. Jun and Lee [13] proved the following: Let X and Y be Banach spaces. Denote by ˜ y) = P∞ 3−n φ(3n x, 3n y) < ∞ φ : X − {0} × Y − {0} → [0, ∞) a function such that φ(x, n=0 for all x, y ∈ X − {0}. Suppose that f : X −→ Y is a mapping satisfying x+y ) − f (x) − f (y)k ≤ φ(x, y) k2f ( 2 for all x, y ∈ X − {0}. Then there exists a unique additive mapping T : X −→ Y such that kf (x) − f (0) − T (x)k ≤

1 ˜ ˜ (φ(x, −x) + φ(−x, 3x)) 3

for all x ∈ X − {0}. B. E. Johnson (Theorem 7.2 of [11]) investigated almost algebra ∗−homomorphisms between Banach ∗−algebras: Suppose that U and B are Banach ∗−algebras which satisfy the conditions of (Theorem 3.1 of [11]). Then for each positive  and K there is a positive δ such that if T ∈ L(U, B) with kT k < K, kT ∨ k < δ and kT (x∗ )∗ − T (x)k < δkxk(x ∈ U ), then there is a ∗−homomorphism T 0 : U −→ B with kT − T 0 k < . Here L(U, B) is the space of bounded linear maps from U into B, and T ∨ (x, y) = T (xy)−T (x)T (y)(x, y ∈ U ). See [11] for details. Throughout this paper, let A be a unital C ∗ −algebra. Let U (A) be the set of unitary elements in A, Asa := {x ∈ A|x = x∗ }, and I1 (Asa ) = {v ∈ Asa |kvk = 1, v ∈ Inv(A)}. 0

Keywords: Hyers–Ulam–Rassias stability; higher derivation; Jordan higher derivation.

735

2

M. Eshaghi Gordji, R. Farokhzad Rostami and S. A. R. Hosseinioun

In this paper, we investigate the Hyers–Ulam–Rassias stability of ∗−higher derivations and ∗−higher Jordan derivations in unital C ∗ −algebras. A linear mapping d : A → A is said to be a derivation if d(xy) = d(x)y + xd(y) holds for all x, y ∈ A. Let N be the set of natural numbers. For m ∈ N ∪ {0} = N0 , a sequence H = {h0 , h1 , ..., hm } (resp. H = {h0 , h1 , ..., hn , ...}) of linear mappings from A into B is called a higher derivation of rank m (resp. infinite rank) from A into B if X hn (xy) = hi (x)hj (y) i+j=n

holds for each n ∈ {0, 1, ..., m} (resp. n ∈ N0 ) and all x, y ∈ A. The higher derivation H from A into B is said to be onto if h0 : A → B is onto. The higher derivation H on A is called be strong if h0 is an identity mapping on A. Of course, a higher derivation of rank 0 from A into B (resp. a strong higher derivation of rank 1 on A) is a homomorphism (resp. a derivation). So a higher derivation is a generalization of both a homomorphism and a derivation. Note that a unital C ∗ −algebra is of real rank zero, if the set of invertible self–adjoint elements is dense in the set of self–adjoint elements (see [2]). 2. Main Results We start our work with the following theorem to investigate the generalized Hyers–Ulam– Rassias stability of ∗−higher derivations in unital C ∗ −algebras. Theorem 2.1. Suppose that F = {f0 , f1 , ..., fn , ...} is a sequence of mappings from A into A such that fn (0) = 0 for each n ∈ N0 . If there exists a function φ : A5 → [0, ∞) such that ˜ y, z, u, w) := φ(x,

∞ X

3−n φ(3n x, 3n y, 3n z, 3n u, 3n w) < ∞

(2.1)

n=0

for all x, y, z, u, w ∈ A and for each n ∈ N0 , µx + µy + wz )−µfn (x)−µfn (y)−wfn (z)−fn (w)z+fn (u∗ )−fn (u)∗ k ≤ φ(x, y, z, u, w), 2 (2.2) for all µ ∈ T and all x, y, z ∈ A, u, w ∈ (U (A) ∪ {0}). Then there exists a unique ∗−higher derivation H = {h0 , h1 , ..., hn , ...} of any rank from A into A such that for each n ∈ N0 , k2fn (

kfn (x) − hn (x)k ≤

1 ˜ ˜ (φ(x, −x, 0, 0, 0) + φ(−x, 3x, 0, 0, 0)) 3

(2.3)

for all x ∈ A − {0}. Proof. Put w = u = 0, µ = 1 in (2.2), it follows from Theorem 1 of [13] that there exists a unique additive mapping hn : A → A satisfies (2.3). This mapping is given by hn (x) = lim m

fn (3m x) 3m

for all x ∈ A. By the same reasoning as the proof of Theorem 1 of [18], hn is C−linear and ∗−preserving. Then we have hn (3m x) = 3m hn (x)

(2.4)

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Nearly higher derivations in unital C ∗ −algebras

3

for all x ∈ A and for each n ∈ N0 . Next, we need to show that the sequence H = {h0 , h1 , ..., hn , ...} satisfies the identity X hn (xy) = hi (x)hj (y) i+j=n

for each n ∈ N0 and all x, y ∈ A. It follows from (2.1) that X fi (3m u)fj (3m z) X fn (3m u3m z) − [ k khn (uz) − hi (u)hj (z)k = lim k m m 9 9m i+j=n i+j=n ≤ lim m

1 1 φ(0, 0, 3m z, 3m u, 0) ≤ lim m φ(0, 0, 3m z, 3m u, 0) m 3 9m

=0 for all u ∈ U (A), all z ∈ A and for each n ∈ N0 . This means that X hn (uz) = hi (u)hj (z)

(2.5)

i+j=n

for all u ∈ U (A), all z ∈ A and for each n ∈ N0 . Let x ∈P A. By Theorem 4.1.7 of [15], x is a finite linear combination of unitary elements, i.e., x = m k=1 ck uk (ck ∈ C, uk ∈ U (A)), it follows from (2.5) that hn (xy) = hn (

m X

ck uk y) =

k=1

=

m X

ck (

=

X

ck hn (uk y)

k=1

X

hi (uk )hj (y))

i+j=n

k=1

=

m X

[hi (

m X

ck uk )hj (y)]

i+j=n

k=1

X

hi (x)hj (y)

i+j=n

for all y ∈ A and for each n ∈ N0 . This means that H is a higher derivation. And this completes the proof of theorem.  Corollary 2.2. Let p ∈ (0, 1), θ ∈ [1, ∞) be real numbers. Suppose that F = {f0 , f1 , ..., fn , ...} is a sequence of mappings from A into A such that fn (0) = 0 for each n ∈ N0 . Suppose that for each n ∈ N0 , µx + µy + wz ) − µfn (x) − µfn (y) − wfn (z) − fn (w)z + fn (u∗ ) − fn (u)∗ k k2fn ( 2 ≤ θ(kxkp + kykp + kzkp + kwkp + kukp ) for all µ ∈ T and all x, y, z, u, w ∈ A. Then there exists a unique ∗− higher derivation H = {h0 , h1 , ..., hn , ...} of any rank from A into A such that for each n ∈ N0 , kfn (x) − hn (x)k ≤

3 + 3p θkxkp 3 − 3p

for all x ∈ A − {0}. Proof. Setting φ(x, y, z, u, w) := θ(kxkp + kykp + kzkp + kukp + kwkp ) all x, y, z, u, w ∈ A. Then by Theorem 2.1, we get the desired result. 

737

4

M. Eshaghi Gordji, R. Farokhzad Rostami and S. A. R. Hosseinioun

Theorem 2.3. Let A be a C ∗ −algebra of real rank zero. Suppose that F = {f0 , f1 , ..., fn , ...} is a sequence of bounded mappings from A into A such that fn (0) = 0 for each n ∈ N0 . Suppose that there exists a function φ : A5 → [0, ∞) such that ˜ y, z, u, w) := φ(x,

∞ X

3−n φ(3n x, 3n y, 3n z, 3n u, 3n w) < ∞

n=0

for all x, y, z, u, w ∈ A and for each n ∈ N0 µx + µy + wz )−µfn (x)−µfn (y)−wfn (z)−fn (w)z+fn (u∗ )−fn (u)∗ k ≤ φ(x, y, z, u, w), k2fn ( 2 (2.6) for all µ ∈ T, x, y, z ∈ A and all u, w ∈ (I1 (Asa ) ∪ {0}). Then there exists a unique continuous ∗−higher derivation H = {h0 , h1 , ..., hn , ...} of any rank from A into A such that for each n ∈ N0 , kfn (x) − hn (x)k ≤

1 ˜ ˜ (φ(x, −x, 0, 0, 0) + φ(−x, 3x, 0, 0, 0)) 3

(2.7)

for all x ∈ A − {0}. Proof. By the same reasoning as the proof of Theorem 2.1, there exists a unique sequence H = {h0 , h1 , ..., hn , ...} of involutive C-linear mappings of any rank from A into A satisfying (2.7). It follows from (2.6) that X X fi (3m u)fj (3m z) fn (3m u3m z) khn (uz) − hi (u)hj (z)k = lim k − [ k m m 9 9m i+j=n i+j=n ≤ lim m

1 1 φ(0, 0, 3m z, 3m u, 0) ≤ lim m φ(0, 0, 3m z, 3m u, 0) m 3 9m

=0

(2.8)

for all u ∈ I1 (Asa ), all z ∈ A and for each n ∈ N0 . On the other hand A is real rank zero. On can easily show that I1 (Asa ) is dense in {x ∈ Asa : kxk = 1}. Let v ∈ {x ∈ Asa : kxk = 1}. Then there exists a sequence {uk } in I1 (Asa ) such that limk uk = v. Since fn is bounded it is easy to see that hn is bounded and then hn is continuous for each n ∈ N0 , it follows from (2.8) that X hn (vz) = lim hn (uk z) = lim hi (uk )hj (z) k

k

=

X

i+j=n

hi (v)hj (z)

(2.9)

i+j=n

for all z ∈ A and for each n ∈ N0 . Now, let x ∈ A. Then we have x = x1 + ix2 , where ∗ ∗ x1 := x+x and x2 = x−x are self-adjoint. 2 2i First consider x1 = 0, x2 6= 0. Since T is C−linear, it follows from (2.9) that x2 hn (xz) = hn (ix2 z) = hn (ikx2 k z) kx2 k X x2 x2 = ikx2 khn ( z) = ikx2 k hi ( )hj (z) kx2 k kx 2k i+j=n X X = hi (ix2 )hj (z) = hi (x)hj (z) i+j=n

for all z ∈ A and for each n ∈ N0 . If x2 = 0, x1 6= 0, then by (2.9), we have

i+j=n

738

Nearly higher derivations in unital C ∗ −algebras

5

x1 z) kx1 k X x1 x1 = kx1 khn ( z) = kx1 k hi ( )hj (z) kx1 k kx 1k i+j=n X = hi (x1 )hj (z)

hn (xz) = hn (x1 z) = hn (kx1 k

i+j=n

for all z ∈ A and each n ∈ N0 . Finally, consider the case that x1 6= 0, x2 6= 0. Then it follows from (2.9) that x2 x1 z) + hn (ikx2 k z) kx1 k kx2 k x1 x2 = kx1 khn ( z) + ikx2 khn ( z) kx1 k kx2 k X X x1 x2 = kx1 k hi ( )hj (z) + ikx2 k hi ( )hj (z) kx kx 1k 2k i+j=n i+j=n X X = hi (x1 )hj (z) + hi (ix2 )hj (z)

hn (xz) = hn (x1 z + ix2 z) = hn (kx1 k

i+j=n

=

X

i+j=n

hi (x)hj (z)

i+j=n

for all z ∈ A and each n ∈ N0 . This means that H = {h0 , h1 , ..., hn , ...} is a higher derivation.  Corollary 2.4. Let A be a C ∗ −algebra of rank zero. Let p ∈ (0, 1), θ ∈ [1, ∞) be real numbers. Suppose that for each n ∈ N0 , k2fn (

µx + µy + wz ) − µfn (x) − µfn (y) − wfn (z) − fn (w)z + fn (u∗ ) − fn (u)∗ k 2 ≤ θ(kxkp + kykp + kzkp + kwkp + kukp )

for all µ ∈ T and all x, y, z ∈ A and all u, w ∈ I1 (Asa ) Then there exists a unique ∗− higher derivation H = {h0 , h1 , ..., hn , ...} of any rank from A into A such that for each n ∈ N0 , kfn (x) − hn (x)k ≤

3 + 3p θkxkp 3 − 3p

for all x ∈ A − {0}. Proof. Setting φ(x, y, z, w, u) := θ(kxkp + kykp + kzkp + kukp + kwkp ) all x, y, z, w, u ∈ A. Then by Theorem 2.3 we get the desired result.  Note that a sequence H = {h0 , h1 , ..., hn , ...} of linear mappings from A into A is a Jordan higher derivation if X hn (ab + ba) = [hi (a)hj (b) + hi (b)hj (a)] i+j=n

for all a, b ∈ A and for each n ∈ N0 . Now, we investigate the generalized Hyers–Ulam–Rassias stability of Jordan ∗−higher derivations in unital C ∗ −algebras.

739

6

M. Eshaghi Gordji, R. Farokhzad Rostami and S. A. R. Hosseinioun

Theorem 2.5. Suppose that F = {f0 , f1 , ..., fn , ...} is a sequence of mappings from A into A such that fn (0) = 0 for each n ∈ N0 . If there exists a function φ : A5 → [0, ∞) such that ∞ X ˜ y, z, u, w) := φ(x, 3−n φ(3n x, 3n y, 3n z, 3n u, 3n w) < ∞ n=0

for all x, y, z, u, w ∈ A and that µx + µy + wz + zw ) − µfn (x) − µfn (y) − wfn (z) − fn (w)z − zfn (w) − fn (z)w+ k2fn ( 2 fn (u∗ ) − fn (u)∗ k ≤ φ(x, y, z, u, w), (2.10) for all µ ∈ T and all x, y, z ∈ A, u, w ∈ (U (A) ∪ {0}). Then there exists a unique Jordan ∗−higher derivation H = {h0 , h1 , ..., hn , ...} of any rank from A into A such that for each n ∈ N0 , 1 ˜ ˜ kfn (x) − hn (x)k ≤ (φ(x, −x, 0, 0, 0) + φ(−x, 3x, 0, 0, 0)) (2.11) 3 for all x ∈ A − {0}. Proof. By the same reasoning as the proof of Theorem 2.1, the mapping hn : A → A given by fn (3m x) hn (x) = lim m 3m for all x ∈ A, and each n ∈ N0 is C−linear and ∗−preserving. hn satisfies (2.11). It follows from (2.10) that X khn (uz + zu) − [hi (u)hj (z) + hi (z)hj (u)]k i+j=n

X fi (3m u)fj (3m z) fn (3m u3m z + 3m z3m u) fi (3m z)fj (3m u) = lim k − [ + k m m m 9 9 9m i+j=n ≤ lim m

1 1 φ(0, 0, 3m z, 3m u, 0) ≤ lim m φ(0, 0, 3m z, 3m u, 0) m 3 9m

=0 for all u ∈ U (A), all z ∈ A and for each n ∈ N0 . This means that X hn (uz + zu) = [hi (u)hj (z) + hi (z)hj (u)]

(2.12)

i+j=n

for all u ∈ U (A), all z ∈ A and for each n ∈ N0 . Let x ∈P A. By Theorem 4.1.7 of [15], x is a finite linear combination of unitary elements, i.e., x = m k=1 ck uk (ck ∈ C, uk ∈ U (A)), it follows from (2.12) that m m m m X X X X hn (xy + yx) = hn ( ck uk y) + hn ( ck yuk ) = ck hn (uk y) + ck hn (yuk ) k=1

=

m X

ck (

X

[hi (

i+j=n

=

X

m X

k=1

k=1

hi (uk )hj (y)) +

i+j=n

k=1

=

k=1

X

ck uk )hj (y)] +

m X

ck (

X

k=1

hi (y)hj (uk ))

k=1

i+j=n

X

[hi (y)hj (

i+j=n

m X

ck uk )]

k=1

[hi (x)hj (y) + hi (y)hj (x)]

i+j=n

for all y ∈ A and for each n ∈ N0 . This means that H is a Jordan higher derivation.



740

Nearly higher derivations in unital C ∗ −algebras

7

Corollary 2.6. Let p ∈ (0, 1), θ ∈ [0, ∞) be real numbers. Suppose that F = {f0 , f1 , ..., fn , ...} is a sequence of mappings from A into A such that fn (0) = 0 for each n ∈ N0 . Suppose that µx + µy + wz + zw ) − µfn (x) − µfn (y) − wfn (z) − fn (w)z − zfn (w) − fn (z)w+ k2fn ( 2 fn (u∗ ) − fn (u)∗ k ≤ θ(kxkp + kykp + kzkp + kwkp + kukp ) for all µ ∈ T and all x, y, z, u, w ∈ A and for each n ∈ N0 . Then there exists a unique Jordan ∗−higher derivation H = {h0 , h1 , ..., hn , ...} of any rank from A into A such that for each n ∈ N0 , 3 + 3p kfn (x) − hn (x)k ≤ θkxkp 3 − 3p for all x ∈ A − {0}. Proof. Setting φ(x, y, z, u, w) := θ(kxkp + kykp + kzkp + kukp + kwkp ) all x, y, z, u, w ∈ A. Then by Theorem 2.5 we get the desired result.  Theorem 2.7. Let A be a C ∗ −algebra of real rank zero. Suppose that F = {f0 , f1 , ..., fn , ...} is a sequence of mappings from A into A such that fn (0) = 0 for each n ∈ N0 . Suppose that there exists a function φ : A5 → [0, ∞) such that ˜ y, z, u, w) := φ(x,

∞ X

3−n φ(3n x, 3n y, 3n z, 3n u, 3n w) < ∞

n=0

for all x, y, z, u, w ∈ A and for each n ∈ N0 , µx + µy + wz k2fn ( )−µfn (x)−µfn (y)−wfn (z)−fn (w)z+fn (u∗ )−fn (u)∗ k ≤ φ(x, y, z, u, w), 2 (2.13) for all µ ∈ T, x, y, z ∈ A and all u, w ∈ (I1 (Asa ) ∪ {0}). Then there exists a unique continuous Jordan ∗−higher derivation H = {h0 , h1 , ..., hn , ...} of any rank from A into A such that for each n ∈ N0 , 1 ˜ ˜ kfn (x) − hn (x)k ≤ (φ(x, −x, 0, 0, 0) + φ(−x, 3x, 0, 0, 0)) (2.14) 3 for all x ∈ A. Proof. By the same reasoning as the proof of Theorem 2.1, there exists a unique H = {h0 , h1 , ..., hn , ...} of involutive C−linear mappings satisfying (2.14). It follows from (2.13) that X khn (uz + zu) − [hi (u)hj (z) + hi (z)hj (u)]k i+j=n

X fi (3m u)fj (3m z) fn (3m u3m z + 3m z3m u) fi (3m z)fj (3m u) = lim k − [ + k m m m 9 9 9m i+j=n ≤ lim m

1 1 φ(0, 0, 3m z, 3m u, 0) ≤ lim m φ(0, 0, 3m z, 3m u, 0) m 3 9m

=0

(2.15)

for all u ∈ I1 (Asa ), all z ∈ A and for each n ∈ N0 . Let v ∈ {x ∈ Asa : kxk = 1}. Then there exists a sequence {uk } in I1 (Asa ) such that limk uk = v. Since hn is continuous for each n ∈ N0 , it follows from (2.15) that X hn (vz + zv) = lim hn (uk z + zuk ) = lim [hi (uk )hj (z) + hi (z)hj (uk )] k

k

i+j=n

741

8

M. Eshaghi Gordji, R. Farokhzad Rostami and S. A. R. Hosseinioun =

X

hi (v)hj (z) + hi (z)hj (v)

(2.16)

i+j=n

for all z ∈ A and for each n ∈ N0 . Now, let x ∈ A. Then we have x = x1 + ix2 , where ∗ ∗ and x2 = x−x are self-adjoint. x1 := x+x 2 2i First consider x1 = 0, x2 6= 0. Since T is C−linear, it follows from (2.16) that x2 x2 x2 x2 hn (xz + zx) = hn (ix2 z + izx2 ) = hn (ikx2 k z + iz ) = ikx2 khn ( z+z ) kx2 k kx2 k kx2 k kx2 k X x2 x2 )hj (z) + hi (z)hj ( )] [hi ( = ikx2 k kx kx 2k 2k i+j=n X = hi (ix2 )hj (z) + hi (z)hj (ix2 ) i+j=n

=

X

hi (x)hj (z) + hi (z)hj (x)

i+j=n

for all z ∈ A and for each n ∈ N0 . If x2 = 0, x1 6= 0, then by (2.16), we have x1 x1 x1 x1 hn (xz + xz) = hn (x1 z + zx1 ) = hn (kx1 k z+z ) = kx1 khn ( z + z( )) kx1 k kx1 k kx1 k kx1 k X x1 x1 [hi ( )hj (z) + hi (z)hj ( )] = kx1 k kx kx 1k 1k i+j=n X = hi (x1 )hj (z) + hi (z)hj (x1 ) i+j=n

for all z ∈ A and each n ∈ N0 . Finally, consider the case that x1 6= 0, x2 6= 0. We can show that hn (xz + zx) =

X

hi (x)hj (z) + hi (z)hj (x)

i+j=n

for all z ∈ A and each n ∈ N0 . This means that H = {h0 , h1 , ..., hn , ...} is a Jordan higher derivation.  Corollary 2.8. Let A be a C ∗ −algebra of rank zero. Let p ∈ (0, 1), θ ∈ [1, ∞) be real numbers. Suppose that F = {f0 , f1 , ..., fn , ...} is a sequence of mappings from A into A such that fn (0) = 0 for each n ∈ N0 . Suppose that for each n ∈ N0 , k2fn (

µx + µy + wz ) − µfn (x) − µfn (y) − wfn (z) − fn (w)z + fn (u∗ ) − fn (u)∗ k 2 ≤ θ(kxkp + kykp + kzkp + kukp + kwkp )

for all µ ∈ T, x, y, z ∈ A and all u, w ∈ (I1 (Asa ) ∪ {0}). Then there exists a unique Jordan ∗−higher derivation H = {h0 , h1 , ..., hn , ...} such that kfn (x) − hn (x)k ≤

3 + 3p θkxkp 3 − 3p

for all x ∈ A − {0} and each n ∈ N0 . Proof. Setting φ(x, y, z, u, w) := θ(kxkp + kykp + kzkp + kukp + kwkp ) all x, y, z, u, w ∈ A. Then by Theorem 2.7 we get the desired result. 

742

Nearly higher derivations in unital C ∗ −algebras

9

References [1] R. Badora, On approximate derivations, Math. Inequal. Appl. 9 (2006), no. 1, 167–173. [2] L. Brown and G. Pedersen, C ∗ −algebras of real rank zero, J. Funct. Anal. 99 (1991) 131–149. [3] S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, River Edge, NJ, 2002. [4] M. Eshaghi Gordji and M. S. Moslehian, A trick for investigation of approximate derivations, To appear. [5] M. Eshaghi Gordji, N. Ghobadipour and J. M. Rassias, Generalized Hyers-Ulam Stability of Generalized N,K-Derivations, Abstract and Applied Analysis Volume 2009, Article ID 437931, 9 pages. [6] M. Eshaghi Gordji, T. Karimi and S. Kaboli Gharetapeh, Approximately n-Jordan Homomorphisms on Banach Algebras, Journal of Inequalities and Applications Volume 2009, Article ID 870843, 8 pages. [7] M. Eshaghi Gordji, M. B. Ghaemi, S. Kaboli Gharetapeh, S. Shams and A. Ebadian, On the stability of J ∗ −derivations, Journal of Geometry and Physics, 2009, doi:10.1016/j.geomphys.2009.11.004. [8] D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222–224. [9] D.H. Hyers and Th.M. Rassias, Approximate homomorphisms, Aequationes Math. 44 (1992), no. 2-3, 125–153. [10] D.H. Hyers, G. Isac and Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkh¨ auser, Basel, 1998. [11] B.E. Johnson, Approximately multiplicative maps between Banach algebras, J. London Math. Soc. (2) 37 (1988) 294-316. [12] K.-W. Jun and D.-W. Park, Almost derivations on the Banach algebra C n [0, 1], Bull. Korean Math. Soc. 33 (1996), no. 3, 359–366. [13] K. Jun, Y. Lee, A generalization of the Hyers-Ulam-Rassias stability of Jensens equation, J. Math. Anal. Appl. 238 (1999) 305-315. [14] S.-M. Jung, Hyers–Ulam–Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, 2001. [15] R. V. Kadison, J. R. Ringrose, Fundamentals of the Theory of Operator Algebras, Elementary Theory, Academic Press, New York, 1983. [16] C. Park, Linear derivations on Banach algebras, Nonlinear Funct. Anal. Appl. 9 (2004), no. 3, 359–368. [17] C. Park, Linear ∗−Derivations on C ∗ −Algebras, Tamsui Oxford Journal of Mathematical Sciences 23(2) (2007) 155–171. [18] C. Park, D.-H. Boo and J.-S. An, Homomorphisms between C ∗ -algebras and linear derivations on C ∗ -algebras, J. Math. Anal. Appl. 337 (2008), no. 2, 1415–1424. [19] C. Park, W. Park, On the Jensens equation in Banach modules, Taiwanese J. Math. 6 (2002) 523-531. [20] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978) 297–300. [21] Th.M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math. 62 (2000), no. 1, 23–130. [22] S. M. Ulam, Problems in Modern Mathematics, Chapter VI, science ed. Wiley, New York, 1940.

JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.4, 743-755, 2011, COPYRIGHT 2011 EUDOXUS PRESS, 743 LLC

Numerical solution of fuzzy differential equations by hybrid predictor-corrector method Cheng-Fu Yang∗ Department of Mathematics of Hexi University, Zhangye Gansu,734000, P. R. China

Abstract In this paper, a hybrid predictor-corrector method is proposed to solve the ”fuzzy initial value problem”. The method is obtained by combining an explicit three-step method and an implicit trapezoidal rule calculating integration. Convergence of the proposed methods are also presented in detail. These methods are illustrated by solving some examples. Keywords: Fuzzy differential equations; Fuzzy Cauchy problem; Adams-Bashforth method; Trapezoidal rule; Hybrid predictor-corrector method

1

Introduction

The fuzzy differential equations is a very important topic from the theoretical point of view [1,2] as well as the applied point of view [3]. The use of fuzzy differential equations is a natural way to model dynamical systems under possibilistic uncertainty. The concept of differential equations in a fuzzy environment was formulated by Kaleva [4]. The last few years, several authors have produced a wide range of results in both the theoretical and applied fields. A variety of exact, approximate, and purely numerical methods are available to find the solution of a fuzzy initial value problem(FIVP). For instance, Abbasbandy et al.[5] introduced the Taylor method, Allahviranloo et al.[6,7]introduced the predictor-corrector method. In this paper, a hybrid predictor-corrector method is introduced. ∗

E-mail: [email protected] (C.F.Yang)

1

744

YANG: FUZZY DIFFERENTIAL EQUATIONS

2

Preliminaries 2.1. Notations and definitions

A tilde is placed over a symbol to denote a fuzzy set so α˜ 1 , f˜(t), . . . Definition 2.1[6]. A fuzzy number u in parametric form is a pair (u(α), u(α)) of function u(α), u(α), 0 ≤ α ≤ 1, which satisfies the following requirements: 1. u(α) is a bounded monotonic increasing left continuous function, 2. u(α) is a bounded monotonic decreasing left continuous function, 3. u(α) ≤ u(α), 0 ≤ α ≤ 1. Let E be the set of all upper semicontinuous normal convex fuzzy numbers with bounded α-level intervals. Meaning if v ∈ E then the α-level set [v]α = {s|v(s) ≥ α},

0 < α ≤ 1,

is a closed bounded interval which is denoted by [v]α = [vα , vα ]. Let I be a real interval. A mapping y : I → E is called a fuzzy process and its α-level set is denoted by [y(t)]α = [yα (t), yα (t)],

t ∈ I, α ∈ (0, 1].

Triangular fuzzy numbers are those fuzzy sets in E which are characterized by an ordered triple (xl , xc , xr ) ∈ R3 with xl ≤ xc ≤ xr such that [U]0 = [xl , xr ] and [U]1 = {xc } then [U]α = [xc − (1 − α)(xc − xl ), xc + (1 − α)(xr − xc )] for any α ∈ I. Definition 2.2[6]. The supremum metric d∞ on E is defined by d∞ (U, V) = sup{dH ([U]α , [V]α ) : α ∈ I}, and (E, d∞ ) is a complete metric space. Definition 2.3[6]. A mapping F : T → E is Hukuhara differentiable at t0 ∈ T ⊆ R if for some h0 > 0 the Hukuhara differences F(t0 + ∆t) ∼h F(t0 ),

F(t0 ) ∼h F(t0 − ∆t),

exist in E for all 0 < ∆t < h0 and if there exists a F 0 (t0 ) ∈ E such that lim d∞ ((F(t0 + ∆t) ∼h F(t0 ))/∆t, F 0 (t0 )) = 0

∆t→0+

and lim d∞ ((F(t0 ) ∼h F(t0 − ∆t))/∆t, F 0 (t0 )) = 0

∆t→0+

2

YANG: FUZZY DIFFERENTIAL EQUATIONS

745

the fuzzy set F 0 (t0 ) is called the Hukuhara derivative of F at t0 . Recall that U ∼h V = W ∈ E are defined on level sets, where [U]α ∼h [V]α = [W]α for all α ∈ I. By consideration of definition of the metric d∞ , all the level set mappings [F(.)]α are Hukuhara differentiable at t0 with Hukuhara derivatives [F 0 (t0 )]a for each α ∈ I when F : T → E is Hukuhara differentiable at t0 with Hukuhara derivative [F 0 (t0 )]α . Definition 2.4[6]. The fuzzy integral Z b y(t)dt, 0 ≤ a ≤ b ≤ 1, a

is defined by Z b Z b Z b α α [ y(t)dt] = [ y (t)dt, yα (t)dt], a

a

(2.1)

a

provided the Lebesgue integrals on the right exist. α Theorem 2.1[8]. Let f : R → E be a function and f (t) = ( f α (t), f (t)), for each α ∈ [0, 1]. α

If f is Hukuhara differentiable , then f α (t), f (t) are differentiable functions and α

[ f (t)]0 = ([ f α (t)]0 , [ f (t)]0 ) Theorem 2.2[8]. For t0 ∈ R, the fuzzy differential equation

(2.2)

y0 (t) = f (t, y(t)), y(t0 ) = y0 ∈ E, where f : R × E → E is supposed to be continuous, is equivalent to the integral equation Z t y(t) = y0 + f (s, y(s))ds, ∀t ∈ [t0 , t1 ] (2.3) t0

on some interval (t0 , t1 ) ⊂ R, under the Hukuhara derivate. 2.2. A fuzzy Cauchy problem Consider the first-order fuzzy differential equation y0 (t) = f (t, y(t)), where y is a fuzzy function of t, f (t, y) is a fuzzy function of crisp variable t and fuzzy variable y, and y0 (t) is Hukuhara fuzzy derivative of y. If an initial value y˜ (t0 ) = α˜ 0 is given, a fuzzy Cauchy problem of first order will be obtained as follows: ( 0 y (t) = f (t, y(t)), t0 ≤ t ≤ T, (2.4) y˜ (t0 ) = α˜ 0 . Sufficient conditions for the existence of a unique solution to Eq. (2.4) are: • Continuity of f , • Lipschitz condition d∞ ( f (t, x), f (t, y)) ≤ Ld∞ (x, y), L > 0. 3

746

YANG: FUZZY DIFFERENTIAL EQUATIONS

3

Adams-Bashforth method

Let fuzzy initial value problem be y0 (t) = f (t, y(t)), and the fuzzy initial values are y˜ (ti−1 ), y˜ (ti ), y˜ (ti+1 ), i.e. f˜(ti−1 , y(ti−1 )), f˜(ti , y(ti )), f˜(ti+1 , y(ti+1 )), which are triangular fuzzy numbers. Allahviranloo et al.[6] gave Adams-Bashforth three-step method as follows:  α   yα (ti+2 ) = yα (ti+1 ) + 12h [5 f α (ti−1 , y(ti−1 )) − 16 f (ti , y(ti )) + 23 f α (ti+1 , y(ti+1 ))],         α α  α α α h y (t  i+2 ) = y (ti+1 ) + 12 [5 f (ti−1 , y(ti−1 )) − 16 f (ti , y(ti )) + 23 f (ti+1 , y(ti+1 ))],           yα (ti−1 ) = α0 , yα (ti ) = α1 , yα (ti+1 ) = α2 , yα (ti−1 ) = α3 , yα (ti ) = α4 , yα (ti+1 ) = α5 .

4

The trapezoidal rule α

To calculate the integrals of f α (t, y(t)) and f (t, y(t)), we apply the trapezoidal rule [9,10]. The interval [a, b] is partitioned by equally spaced points: a = t0 ≤ t1 ≤ . . . ≤ tN = b, h = b−a = ti+1 − ti , 1 ≤ i ≤ N − 1. N Define sN (α) = h[

f α (a, y(a)) + f α (b, y(b)) 2

+

N−1 X

f α (ti , y(ti ))],

i=1

α

α

f (a, y(a)) + f (b, y(b)) X α sN (α) = h[ f (ti , y(ti ))]. + 2 i=1 N−1

For arbitrary fixed α [9,10] we have Z b lim sN (α) = F(α) = f α (t, y(t))dt, N→∞

Z lim sN (α) = F(α) =

N→∞

(4.1)

a

b

α

f (t, y(t))dt.

(4.2)

a

Theorem 4.1[10]. If f (t) is continuous , the convergence of sN (α), sN (α) to F(α), F(α), respectively, is uniform in α. From (2.1),(2.3), (4.1) and (4.2), the following results will be obtained:

4

YANG: FUZZY DIFFERENTIAL EQUATIONS

747

R ti+2 α  α α  y (t ) = y (t ) + f (t, y(t))dt  i+2 i−1  ti−1         f α (ti−1 ,y(ti−1 ))+ f α (ti+2 ,y(ti+2 ))  α   ≈ y (t ) + h[ + f α (ti , y(ti )) + f α (ti+1 , y(ti+1 ))],  i−1 2        R ti+2 α  α α (4.3)  y (t ) = y (t ) + f (t, y(t))dt i+2 i−1  ti−1        α α  α α  f (ti−1 ,y(ti−1 ))+ f (ti+2 ,y(ti+2 )) α   ≈ y (t ) + h[ + f (ti , y(ti )) + f (ti+1 , y(ti+1 ))],  i−1  2         yα (ti−1 ) = α0 , yα (ti ) = α1 , yα (ti+1 ) = α2 , yα (ti−1 ) = α3 , yα (ti ) = α4 , yα (ti+1 ) = α5 . where h =

5

ti+2 −ti−1 . 3

Hybrid predictor-corrector method

The following algorithm is based on Adams-Bashforth three-step method as a predictor and trapezoidal rule as a corrector. Algorithm (Hybrid predictor-corrector method). To approximate the solution of following fuzzy initial value problem  0  y (t) = f (t, y(t)), t0 ≤ t ≤ T,          α y (t0 ) = α0 , yα (t1 ) = α1 , yα (t2 ) = α2 ,           yα (t0 ) = α3 , yα (t1 ) = α4 , yα (t2 ) = α5 . positive integer N is chosen Step 1. Let h =

T −t0 N

,

wα (t0 ) = α0 , wα (t1 ) = α1 , wα (t2 ) = α2 , wα (t0 ) = α3 , wα (t1 ) = α4 , wα (t2 ) = α5 . Step 2. Let i = 1. Step 3. Let w(0) (ti+2 ) = (w(0)α (ti+2 ), w(0)α (ti+2 )) i.e.  (0)α α  w (ti+2 ) = wα (ti+1 ) + 12h [5 f α (ti−1 , w(ti−1 )) − 16 f (ti , w(ti )) + 23 f α (ti+1 , w(ti+1 ))],          w(0)α (ti+2 ) = wα (ti+1 ) + h [5 f α (ti−1 , w(ti−1 )) − 16 f α (ti , w(ti )) + 23 f α (ti+1 , w(ti+1 ))], 12 5

748

YANG: FUZZY DIFFERENTIAL EQUATIONS

Step 4. Let ti+2 = t0 + (i + 2)h. Step 5. Let  f α (ti−1 ,w(ti−1 ))+ f α (ti+2 ,w(0) (ti+2 ))  α α   w (t ) = w (t ) + h[ + f α (ti , w(ti )) + f α (ti+1 , w(ti+1 ))], i+2 i−1  2      α α    wα (ti+2 ) = wα (ti−1 ) + h[ f (ti−1 ,w(ti−1 ))+ f (ti+2 ,w(0) (ti+2 )) + f α (ti , w(ti )) + f α (ti+1 , w(ti+1 ))], 2 Step 6. i = i + 1. Step 7. if i ≤ N − 2 go to step 3. Step 8. algorithm will be completed and (wα (T ), wα (T )) approximates real value of α (Y α (T ), Y (T )).

6 Convergence To integrate the system given in Eq. (2.4) from t0 to a prefixed T > t0 , the interval [t0 , T ] will be replaced by a set of discrete equally spaced grid points t0 < t1 < t2 < . . . < tN = T which the exact solution (Y(t, α), Y(t, α)) is approximated by some (y(t, α), y(t, α)). The exact and approximate solutions at tn , 0 ≤ n ≤ N are denoted by Yn (α) = (Y n (α), Y n (α)), and yn (α) = (y (α), yn (α)), respectively. The grid points which the n solution is calculated are tn = t0 + nh, h = (T − t0 )/N, 1 ≤ n ≤ N. From Eq.(4.3), the polygon curves n o y(t, h, α) = [t0 , y (α)], [t1 , y (α)], · · · , [tN , y (α)] , 0

1

N

 y(t, h, α) = [t0 , y0 (α)], [t1 , y1 (α)], · · · , [tN , yN (α)] , are the approximates of Y(t, α) and Y(t, α), respectively, over the interval t0 ≤ t ≤ tN . The following lemma will be applied to show convergence of these approximates, i.e. lim y(t, h, α) = Y(t, α), h→0

lim y(t, h, α) = Y(t, α). h→0

N Lemma 6.1. Let a sequence of numbers {wn }n=0 satisfy:

|wn+2 | ≤ A |wn−1 | + B |wn | + C |wn+1 | + D,

1≤n≤ N−2

for some given positive constants A, B, C and D. Then

6

YANG: FUZZY DIFFERENTIAL EQUATIONS

749

|wn+2 | ≤ β |w0 | + γ |w1 | + δ |w2 | + (C n−1 + C n−1 + · · · + C + 1)D +[(n − 2)C n−3 + (n − 3)C n−4 + · · · + 2C + 1]BD +[(n − 3)C n−4 + (n − 4)C n−5 + · · · + 2C + 1]AD +[ζ1C n−5 + ζ2C n−6 + · · · + ζn−5C + ζn−4 ]B2 D +[λ1C n−6 + λ2C n−7 + · · · + λn−6C + λn−5 ]ABD +[µ1C n−7 + µ2C n−8 + · · · + µn−7C + µn−6 ]A2 D + · · · , where β, γ, δ, ζ s , λt , µk are constants for all s, t and k. The proof, by using mathematical induction is straightforward. Theorem 6.1. For arbitrary fixed α : 0 ≤ α ≤ 1, the trapezoidal rule approximates of Eq. (4.3) converge to the exact solutions Y(t, a), Y(t, α) for Y, Y ∈ C 3 [t0 , T ]. Proof. It is sufficient to show lim y (α) = Y(T, α), h→0 N

lim yN (α) = Y(T, α). h→0

By using exact value the following results will be obtained:   Y n+2 (α) = Y n−1 (α) + 2h f (tn−1 , Y n−1 (α)) + h2 f (tn+2 , Y n+2 (α)) + h f (tn , Y n (α))         3    +h f (tn+1 , Y n+1 (α)) − h4 Y 000 (ξn ),          Y n+2 (α) = Y n−1 (α) + 2h f (tn−1 , Y n−1 (α)) + h2 f (tn+2 , Y n+2 (α)) + h f (tn , Y n (α))         000 3    +h f (tn+1 , Y n+1 (α)) − h4 Y (ζn ),     where tn−1 ≤ ξn , ζn ≤ tn+2 . Consequently Y n+2 (α) − y

n+2

(α) = Y n−1 (α) − y

n−1

(α) + h2 ( f (tn−1 , Y n−1 (α)) − f (tn−1 , y

+ h2 ( f (tn+2 , Y n+2 (α)) − f (tn+2 , y

(α)))

n+2

(α))) + h( f (tn , Y n (α)) − f (tn , y (α)))

n+1

(α))) − 41 h3 Y 000 (ξn ),

+h( f (tn+1 , Y n+1 (α)) − f (tn+1 , y 7

n−1

n

750

YANG: FUZZY DIFFERENTIAL EQUATIONS

Y n+2 (α) − yn+2 (α) = Y n−1 (α) − yn−1 (α) + h2 ( f (tn−1 , Y n−1 (α)) − f (tn−1 , yn−1 (α))) + h2 ( f (tn+2 , Y n+2 (α)) − f (tn+2 , yn+2 (α))) + h( f (tn , Y n (α)) − f (tn , yn (α))) 000

+h( f (tn+1 , Y n+1 (α)) − f (tn+1 , yn+1 (α))) − 14 h3 Y (ζn ). Denote wn = Y n (α) − y (α), vn = Y n (α) − yn (α). Then n

2 + hL1 2hL3 2hL4 h3 M, |wn−1 | + |wn | + |wn+1 | + 2 − hL2 2 − hL2 2 − hL2 4 − 2hL2

|wn+2 | ≤

2 + hL5 2hL7 2hL8 h3 M. |vn−1 | + |vn | + |vn+1 | + 2 − hL6 2 − hL6 2 − hL6 4 − 2hL6 000 Where M = maxt0 ≤t≤T Y 000 (t, α) and M = maxt0 ≤t≤T Y (t, α) . |vn+2 | ≤

Set L = max{L1 , L2 , L3 , L4 , L5 , L6 , L7 , L8 } < 2h , then |wn+2 | ≤

2 + hL 2hL 2hL h3 M, |wn−1 | + |wn | + |wn+1 | + 2 − hL 2 − hL 2 − hL 4 − 2hL

|vn+2 | ≤

2 + hL 2hL 2hL h3 M. |vn−1 | + |vn | + |vn+1 | + 2 − hL 2 − hL 2 − hL 4 − 2hL

are obtained, so by Lemma (6.1), since w0 = v0 = 0, w1 = v1 = 0 and w2 = v2 = 0 we have: |wn+2 | ≤

2hL n ( 2−hL ) −1 2hL 2−hL −1

3

h 2hL n−3 2hL n−4 2hL × 4−2hL M+[(n−2)( 2−hL ) +(n−3)( 2−hL ) +· · ·+2× 2−hL +1]

2hL n−4 2hL n−5 +[(n − 3)( 2−hL ) + (n − 4)( 2−hL ) + ··· + 2 × 2hL n−5 2hL n−6 +[ζ1 ( 2−hL ) + ζ2 ( 2−hL ) + · · · + ζn−5 ×

|vn+2 | ≤

2hL 2−hL

2hL 2−hL

+ 1]

2+hL 2−hL

×

3 2hL × h M 2−hL 4−2hL

h3 M 4−2hL

h3 M 4−2hL

2hL 2 + ζn−4 ]( 2−hL ) ×

2hL n−6 2hL n−7 +[λ1 ( 2−hL ) + λ2 ( 2−hL ) + · · · + λn−6 ×

2hL 2−hL

2+hL 2hL + λn−5 ] 2−hL × ( 2−hL ) ×

2hL n−7 2hL n−8 +[µ1 ( 2−hL ) + µ2 ( 2−hL ) + · · · + µn−7 ×

2hL 2−hL

2+hL 2 + µn−6 ]( 2−hL ) ×

2hL n ( 2−hL ) −1 2hL 2−hL −1

h3 M 4−2hL

3

h 2hL n−3 2hL n−4 2hL × 4−2hL M+[(n−2)( 2−hL ) +(n−3)( 2−hL ) +· · ·+2× 2−hL +1]

2hL n−5 2hL n−4 ) + (n − 4)( 2−hL ) + ··· + 2 × +[(n − 3)( 2−hL

8

2hL 2−hL

+ 1]

2+hL 2−hL

h3 M 4−2hL

×

+ ...,

3 2hL × h M 2−hL 4−2hL

h3 M 4−2hL

YANG: FUZZY DIFFERENTIAL EQUATIONS

2hL n−5 2hL n−6 ) + ζ2 ( 2−hL ) + · · · + ζn−5 × +[ζ1 ( 2−hL

2hL 2−hL

2hL 2 + ζn−4 ]( 2−hL ) ×

751

h3 M 4−2hL

2hL n−6 2hL n−7 +[λ1 ( 2−hL ) + λ2 ( 2−hL ) + · · · + λn−6 ×

2hL 2−hL

2+hL 2hL + λn−5 ] 2−hL × ( 2−hL ) ×

2hL n−7 2hL n−8 +[µ1 ( 2−hL ) + µ2 ( 2−hL ) + · · · + µn−7 ×

2hL 2−hL

2+hL 2 + µn−6 ]( 2−hL ) ×

h3 M 4−2hL

h3 M 4−2hL

+ ...

If h → 0 then wn+2 → 0, vn+2 → 0, which concludes the proof.

7

Example Example 7.1[7,11]. Consider the initial value problem

y0 (t) = −y(t) + t + 1, y(0) = (0.96 + 0.04α, 1.01 − 0.01α), y(0.01) = (0.01 + (0.985 + 0.015α)e−0.01 − (1 − α)0.025e0.01 , 0.01 + (0.985 + 0.015α)e−0.01 + (1 − α)0.025e0.01 ), y(0.02) = (0.02 + (0.985 + 0.015α)e−0.02 − (1 − α)0.025e0.02 , 0.02 + (0.985 + 0.015α)e−0.02 + (1 − α)0.025e0.02 ). The exact solution at t = 0.1 is given by Y(0.1, α) = (0.1 + (0.985 + 0.015α)e−0.1 − (1 − α)0.025e0.1 , 0.1 + (0.985 + 0.015α)e−0.1 + (1 − α)0.025e0.1 ). By using the hybrid predictor-corrector method with N = 10 the following results are obtained:

9

752

YANG: FUZZY DIFFERENTIAL EQUATIONS

α 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

y 0.96362436409891 0.96774560164242 0.97186683918593 0.97598807672944 0.98010931427295 0.98423055181646 0.98835178935997 0.99247302690348 0.99659426444699 1.00071550199050 1.00483673953401

Y 0.96363558381353 0.96775576723577 0.97187595065802 0.97599613408026 0.98011631750250 0.98423650092474 0.98835668434699 0.99247686776923 0.99659705119147 1.00071723461372 1.00483741803596

Error 1.121971461737203e-005 1.016559335087486e-005 9.111472084599726e-006 8.057350818102549e-006 7.003229551716395e-006 5.949108285108196e-006 4.894987018722041e-006 3.840865752113842e-006 2.786744485727688e-006 1.732623219119489e-006 6.785019528443570e-007

α 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

y 1.01889394829001 1.01748822741251 1.01608250653501 1.01467678565751 1.01327106478000 1.01186534390250 1.01045962302500 1.00905390214750 1.00764818127000 1.00624246039250 1.00483673951500

Y 1.01889412971731 1.01748845854918 1.01608278738104 1.01467711621291 1.01327144504477 1.01186577387664 1.01046010270850 1.00905443154037 1.00764876037223 1.00624308920409 1.00483741803596

Error 1.814273025146918e-007 2.311366684715210e-007 2.808460342063057e-007 3.305554001631350e-007 3.802647658979197e-007 4.299741318547490e-007 4.796834973674891e-007 5.293928631022737e-007 5.791022290591030e-007 6.288115947938877e-007 6.785209607507170e-007

The results of Example 7.1 are shown in Fig. 1. Example 7.2[7,11]. Consider the initial value problem y0 (t) = −y(t), y(0) = (0.96 + 0.04α, 1.01 − 0.01α), y(0.01) = ((0.985+0.015α)e−0.01 −(1−α)0.025e0.01 , (0.985+0.015α)e−0.01 +(1−α)0.025e0.01 ), y(0.02) = ((0.985+0.015α)e−0.02 −(1−α)0.025e0.02 , (0.985+0.015α)e−0.02 +(1−α)0.025e0.02 ). The exact solution at t = 0.1 is given by 10

YANG: FUZZY DIFFERENTIAL EQUATIONS

753

Y(0.1, α) = ((0.985+0.015α)e−0.1 −(1−α)0.025e0.1 , (0.985+0.015α)e−0.1 +(1−α)0.025e0.1 ). By using the hybrid predictor-corrector method with N = 10 the following results are obtained: α 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

y 0.86362436409891 0.86774560164242 0.87186683918593 0.87598807672944 0.88010931427295 0.88423055181646 0.88835178935997 0.89247302690348 0.89659426444699 0.90071550199050 0.90483673953401

Y 0.86363558381353 0.86775576723577 0.87187595065802 0.87599613408026 0.88011631750250 0.88423650092474 0.88835668434699 0.89247686776923 0.89659705119147 0.90071723461372 0.90483741803596

Error 1.121971461726101e-005 1.016559335076384e-005 9.111472084377681e-006 8.057350817991527e-006 7.003229551605372e-006 5.949108284997173e-006 4.894987018499997e-006 3.840865752113842e-006 2.786744485727688e-006 1.732623219230511e-006 6.785019527333347e-007

α 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

y 0.91889394829001 0.91748822741251 0.91608250653501 0.91467678565751 0.91327106478000 0.91186534390250 0.91045962302500 0.90905390214750 0.90764818127000 0.90624246039250 0.90483673951500

Y 0.91889412971731 0.91748845854918 0.91608278738104 0.91467711621291 0.91327144504477 0.91186577387664 0.91046010270850 0.90905443154037 0.90764876037223 0.90624308920409 0.90483741803596

Error 1.814273025146918e-007 2.311366684715210e-007 2.808460340952834e-007 3.305553999410904e-007 3.802647660089420e-007 4.299741316327044e-007 4.796834974785114e-007 5.293928632132960e-007 5.791022289480807e-007 6.288115947938877e-007 6.785209606396947e-007

The results of Example 7.2 are shown in Fig. 2.

8

Conclusion

In this paper, The Adams-Bashforth method and the trapezoidal rule are combined, and a hybrid predictor-corrector three-step method is given. The correctness of the proposed method is shown by some examples. It is worth mentioning that the hybrid predictor11

754

YANG: FUZZY DIFFERENTIAL EQUATIONS

corrector three-step method can naturally be generalized to hybrid predictor-corrector mstep methods.

References [1] M. Ma, M. Friedman, A. Kandel, Numerical Solutions of fuzzy differential equations, Fuzzy Sets and Systems 105(1999)133-138. [2] M.T. Mizukoshi, L.C. Barros, Y. Chalco-Cano, H.Rom`an-Flores, R.C. Bassanezi, Fuzzy differential equations and the extension principle, Information Sciences 177(2007)3627-3635. [3] S. Abbasbandy, T. Allahviranloo, Oscar Lopez-Pouso, Juan J. Nieto, Numerical methods for fuzzy differential inclusions, Journal of Computer and Mathematics With Applications 48(2004)1633-1641. [4] O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems 24(1987)301-317. [5] S. Abbasbandy, T. Allahviranloo, Numerical solutions of fuzzy differential equations by Taylor method, Journal of Computational Methods in Applied 393 Mathematics 2(2002)113124. [6] T. Allahviranloo, N. Ahmady, E. Ahmady, Numerical solution of fuzzy differential equations by predictor-corrector method, Information Sciences 177(2007)1633-1647. [7] T. Allahviranloo, S.Abbasbandy, N. Ahmady, E. Ahmady, Improved predictor-corrector method for solving fuzzy initial value problems, Information Sciences 179 (2009) 945-955. [8] T. Allahviranloo, N.A. Kiani, M. Barkhordari, Toward the existence and uniqueness of solutions of second-order 3 fuzzy differential equations, Information Sciences 179 (2009)12071215. [9] M. Friedman, A. Kandel, Fundamentals of Computer Numerical Analysis, CRC Press, Boca Raton, FL, 1994, pp. 307-351. [10] Menahem Friedman, Ming Ma , Abraham Kandel, Numerical solutions of fuzzy differential and integral equations, Fuzzy Sets and Systems 106 (1999) 35-48. [11] B. Bede, Note on numerical solutions of fuzzy differential equations by predictor-corrector method, Information Sciences 178(2008)1917-1922.

12

YANG: FUZZY DIFFERENTIAL EQUATIONS

1

755

Hybrid Predictor−Corrector Real Value

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 0.96

0.97

0.98

0.99

1

1.01

1.02

1.03

Fig.1 The results of Example 7.1

1

Hybrid Predictor−Corrector Real Value

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 0.86

0.87

0.88

0.89

0.9

Fig.2. The results of Example 7.2.

13

0.91

0.92

JOURNAL 756 OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.4, 756-770, 2011, COPYRIGHT 2011 EUDOXUS PRESS, LLC

On a Construction of Positive Linear Operators for Approximation of Continuous Functions in the Weighted Spaces T¨ ulin Co¸skun Department of Mathematics, Zonguldak Karaelmas University 67100 Zonguldak, Turkey e-mail: tulin [email protected] Abstract The modificated form of Gadjiyev-Ibragimov operators and study of the approximation of continuous functions by these operators in the weighted norm with polynomial weight are given. Key Words: positive linear operators, Korovkin theorems, Gadjiev-Ibragimov operators, weighted spaces.

1

Introduction and premilinaries

The sequence of linear positive operators ½ ν ¾ £ ∂ ¤ ν (−αn ψn (0))ν Ln (f, x) = f( 2 ) K (x, t, u) · n u=αn ψn (t);t=0 n ψn (0) ∂uν ν! ν=0 ∞ X

(1)

were introduced by Ibrahimov-Gadjiev∗ under some conditions on functions Kn (x, t, u), ψn (t) and the sequence (αn )n∈IN , which will be noted below. Some new properties of the operators (1) were established in different papers, and we refer to [2], [6], [7], [11] ,[15], [16]. Radatz and Wood in [16] have given an approximation of derivatives of functions in a certain class by the derivatives of the operators (1). Note that all of these papers, including the Ibrahimov-Gadjiev’s paper [13], are devoted to the case of finite interval [0, A] (see also [12], [14].) The aim of this paper is to solve the problem of weighted approximation of continuous and unbounded functions defined on semiaxis by modificated operators (1), which will be defined below. This modification may be applied also to different other known operators, for example to Baskakov operators [3]. Note that (1) give a more general construction than the Baskakov operators [3] (see [14] and especially [1] where may be found different modifications of Baskakov operators). We will use the weighted Korovkin’s type theorems established in [8], [9]. Note that some generalizations of these theorems were proved in [4], [5]. ∗ Gadziev=Gadjiev, see [10]) 2000 Mathematics Subject Classification: Primary 41A36; Secondary 47A58.

1

COSKUN: POSITIVE LINEAR OPERATORS

757

Let ρ(x) be increasing continuous function on [0, ∞), ρ(0) = 1 and Bρ be the set of all functions f defined on the semi axis satisfying the conditions |f (x)| ≤ Mf · ρ(x) with some constant Mf , depending only on f . By Cρ , we denote the subspace of all continuous functions belonging to Bρ . Also let Cρ0 be the subspace of all functions f in Cρ , for which (x) is finite. lim fρ(x) x→∞

From the general results obtained [8] and [9] and devoted to approximation of continuous functions in the weighted space Bρ we give here the special theorem in the case ρ(x) = 1 + x2p , where p is a natural number. In this case we will denote the spaces Cρ and Bρ 0 respectively as C2p and B2p and the space Cρ0 as C2p . Moreover for a sequence of positive numbers (an ) such that lim an = ∞ we denote by n→∞

B2p [0, an ] the subspace B2p with the norm kf k2p,[0,an ] =

sup 0≤x≤an

|f (x)| . 1 + x2p

Theorem 1.1 ([8], [9]) Let the sequence of linear positive operators (Ln )n∈IN , acting from C2p to B2p satisfy the conditions lim kLn (tν , x) − xν kρ = 0,

n→∞

ν = 0, p, 2p,

0 , where ρ(x) = 1 + x2p . Then for any function f ∈ C2p

lim kLn f − f kρ = 0

n→∞

0 such that and there exists a function f ∗ ∈ C2p \ C2p

lim sup kLn f ∗ − f ∗ kρ ≥ 1. n→∞

For linear positive operators, acting from C2p to B2p,[0,an ] , Theorem 1.1 gives the following result (see also [10]). Theorem 1.2 The conditions lim kLn (tν , x) − xν k2p,[0,an ] = 0,

n→∞

ν = 0, p, 2p

(2)

imply lim kLn f − f k2p,[0,an ] = 0

n→∞

for any function f ∈ Cρ0 . Now let us determine the modificated operators (1). Let (γn )n∈IN be the sequence of positive numbers, which has finite or infinite limit and let (Kn (x, t, u))n∈IN be a sequence of functions of three variables x, t, u, where x, t ∈ [0, γn ] and u ≥ 0, satisfying the following conditions: 2

758

COSKUN: POSITIVE LINEAR OPERATORS

(i) for each x, t ∈ [0, γn ] and for each n ∈ IN, Kn is entire analytic function with respect to variable u. (ii) Kn (x, 0, 0) = 1 for any x ∈ [0, γn ] (n ∈ IN). ½ ¾ £ ∂ν ¤ (iii) (−1)ν ∂u ≥ 0, for ν = 0, 1, 2, · · · , ν Kn (x, t, u) u=u1 ;t=0 (iv) There exist a number m(n) ∈ IN with the property lim

n→∞

n ∈ IN and x ∈ [0, γn ]. m(n) n

= 1 such that

· ν−1 ¸ ∂ν ∂ Kn (x, t, u) = −nx Km(n) (x, t, u) (1 + βν,n (u)) ∂uν ∂uν−1 for all x ∈ [0, γn ] and n, ν ∈ IN, where in u = 0, n → ∞ uniformly in ν.

βν,n (0) convergences to zero for

Moreover, let (ϕn (t))n∈IN and (ψn (t))n∈IN are sequences of functions from C[0, ∞) such that ϕn (0) = 0, ψn (t) > 0, for all t and lim

n→∞

1 = 0. n2 ψn (0)

(3)

Also let (αn )n∈IN denote a sequence of positive numbers satisfying the conditions αn = 1. (4) n→∞ n We call the operators (1) under the above conditions the modificated Gadjiev-Ibrahimov operators. Note that in the case of lim γn = A, m(n) = n + m, βν,n (u) = 0 we obtain Gadjievlim

n→∞

Ibrahimov operators, defined in (1). Since K(x, t, u) is entire functions respectively the variable u, we can write for any u1 ∈ IR the following Taylor expansion Kn (x, t, u) =

∞ X ∂ ν Kn (x, t, u) ¯¯ (u − u1 )ν ¯ ∂uν ν! u=u1 ν=0

(5)

Replacing u = ϕn (t), u1 = αn ψn (t) and then t = 0, where (αn ) is the sequence defined in (4), ∞ X (−αn ψn (0))ν ∂ ν Kn (x, t, u) ¯¯ Kn (x, 0, 0) = ¯ ν ∂u ν! u=αn ψn (t),t=0 ν=0 is obtained by the condition ϕn (0) = 0. Taking into account that Kn (x, 0, 0) = 1 by (ii) and denoting for simplicity ∂ ν Kn (x, t, u) ¯¯ = Kn(ν) (x, 0, αn ψn (0)) ¯ ∂uν u=αn ψn (t),t=0 3

COSKUN: POSITIVE LINEAR OPERATORS

759

we obtain the equality ∞ X

Kn(ν) (x, 0, αn ψn (0))

ν=0

(−αn ψn (0))ν =1 ν!

which means that Ln (1, x) = 1 for any x ∈ [0, γn ]. From the equality (1) we obtain ∞ ∂Kn (x, t, u) X ∂ ν Kn (x, t, u) ¯¯ (u − u1 )ν−1 = ν ¯ ∂u ∂uν ν! u=u1 ν=0

and

∞ ∂ 2 Kn (x, t, u) X ∂ ν Kn (x, t, u) ¯¯ (u − u1 )ν−2 = ν(ν − 1) . ¯ ∂u2 ∂uν ν! u=u1 ν=0

Therefore, for u = ϕn (t), u1 = αn ψn (t) and t = 0, Kn0 (x, 0, 0) =

∞ X

Kn(ν) (x, 0, αn ψn (0))ν

ν=0

Kn00 (x, 0, 0) =

∞ X

(−αn ψn (0))ν−1 ν!

Kn(ν) (x, 0, αn ψn (0))ν(ν − 1)

ν=0

(6)

(−αn ψn (0))ν−2 . ν!

Applying the conditions (iv) to the left hands of these equalities ∞ X

Kn(ν) (x, 0, αn ψn (0))ν

ν=0

∞ X

Kn(ν) (x, 0, αn ψn (0))ν(ν − 1)

ν=0

(−αn ψn (0))ν−1 = −nx[1 + β1,n (0)] ν!

(7)

(−αn ψn (0))ν−2 = nm(n)x2 [1 + β2,n (0)][1 + β1,n (0)] (8) ν!

are obtained. From (1) it is easy to see also that for any natural number r ∞ ∂ r Kn (x, t, u) X ∂ ν+r Kn (x, t, u) ¯¯ = (u − u1 )ν ¯ r ∂ur ∂u u=u1 ν=0

and from this Kn(r) (x, 0, 0)

=

∞ X

Kn(ν+r) (x, 0, αn ψn (0))

ν=0

4

(−αn ψn (0))ν . ν!

(9)

760

COSKUN: POSITIVE LINEAR OPERATORS

2

The main results

We shall give some properties of modificated operators (1). Lemma 2.1 For any natural k Ln (tk , x) = d(k, n)xk +

k X

(n)

gj,k (x)qj,k

(10)

j=1 (n)

where qj,k −→ 0 as n −→ ∞ uniformly on k, j and gj,k ∈ Ck [0, ∞) lim d(k, n) = 1.

(11)

n→∞

Proof. It’s clear that the equation (10) is true for k = 0. Consider (10) for k = 1 we obtain Ln (t, x) =

∞ X ν=0

= =

− −

ν n2 ψ

αn n2

n (0)

∞ X

Kn(ν) (x, 0, αn ψn (0))

Kn(ν) (x, 0, αn ψn (0))

ν=1

(−αn ψn (0))ν ν!

(−αn ψn (0))ν−1 (ν − 1)!

αn ∂Kn (x, 0, 0) n2 ∂u

by the formula (6). Since by (7) ∂Kn (x, 0, 0) = −nx[1 + β1,n (0)], ∂u we obtain Ln (t, x) = Denoting

αn n

= d(1, n),

αn n β1,n (0)

αn αn x+ xβ1,n (0). n n

(n)

= q1,1 , ψ1,1 (x) = x,

lim d(1, n) = 1,

n→∞

(n) lim q n→∞ 1,1

=0

can be seen and therefore for k = 1 the equation (10) is true. Suppose that (10) holds for any k ≤ m and using the induction method we will prove (10) for k = m + 1. Obviously we can found the numbers a1 , a2 , · · · , am such that the following equality holds m

(

ν ν(ν − 1) · · · (ν − m) X ν 1 )m+1 = + ( 2 )j aj 2 . 2 2 m+1 n ψn (0) (n ψn (0)) n ψn (0) (n ψn (0))m+1−j j=1

5

COSKUN: POSITIVE LINEAR OPERATORS

761

Therefore, by the conditions (ii) and (iv), we have Ln (tm+1 , x) = (

∞ X 1 ν(ν − 1) · · · (ν − m) (ν) m+1 ) Kn (x, 0, αn ψn (0))(−αn ψn (0))ν n2 ψn (0) ν! ν=0

+

m X j=1

aj (

1 n2 ψ

n (0)

)m−j+1 Ln (tj , x)

(12)

If the first term in right hand side is considered, it is equal to 1 (n2 ψn (0))m+1

·

∞ X (−αn ψn (0))ν (ν) K (x, 0, αn ψn (0)) (ν − m − 1)! n ν=m+1

∞ −αn m+1 X (−αn ψn (0))ν (ν+m+1) (x, 0, αn ψn (0)) ) Kn n2 ν! ν=0 αn = ( 2 )m+1 Kn(m+1) (x, 0, 0) n αn m+1 m+1 = ( 2) x nm(n)m(m(n)) · · · m(m(· · · (m(n))) n ·(1 + βm+1,n (0))(1 + βm,m(n) (0)) · · · (1 + β0,q(n) (0))

= (

where q(n) = m(m(· · · (m(n))). Denoting h(m + 1, n)

= nm(n)m(m(n)) · · · q(n)(1 + βm+1,n (0)) · (1 + βm,m(n) (0)) αn · · · (1 + β0,q(n) (0))( 2 )m+1 n

we see that by condition (iv) and the equation (4) lim h(m + 1, n) = 1.

(13)

n→∞

Therefore, (12) has the form Ln (tm+1 , x)

= h(m + 1, n)xm+1 +

m X

aj

1 Ln (tj , x) (n2 ψn (0))m−j+1

aj

X 1 (n) gi,j (x)qi,j ] [d(j, n)xj + 2 m−j+1 (n ψn (0)) i=1

j=1

= h(m + 1, n)xm+1 +

m X

j

j=1 (n)

(n)

where gi,j (x) ∈ Cj (0, ∞) and lim qi,j = 0. Denoting xj = g0,j (x) and d(j, n) = q0,j we n→∞ can write Ln (tm+1 , x) = h(m + 1, n)xm+1 +

m X j=1

6

j X aj (n) gi,j (x)qi,j (n2 ψn (0))m−j+1 i=0

762

COSKUN: POSITIVE LINEAR OPERATORS

(n)

where lim q0,j = 1. n→∞

This formula show that the coeficient of every function gi,j (x) in right hand side tends to zero as n → ∞ uniformly in i and j, because for any i, j (n)

qi,j lim = 0. n→∞ (n2 ψn (0))m−j+1 (n)

Moreover, by the assumption for any i ≤ m, gi,j (x) ∈ Cm+1 [0, ∞), the proof could be said to have been completed. 2 Lemma 2.2 The operators Ln given by (1) acts from C2p [0, ∞) to B2p [0, γn ], for any natural p. Proof. Let f ∈ C2p [0, ∞). Then |Ln (f, x)| ≤ Mf Ln (1 + t2p , x) = Mf [1 + Ln (t2p , x)] and therefore sup 0≤x≤γn

|Ln (f, x)| Ln (t2p , x) ≤ [1 + sup ]. 2p 2p 1+x 0≤x≤γn 1 + x

By the Lemma 1.3 the sup in right hand side is bounded and therefore Ln f ∈ B2p [0, γn ].

2

Theorem 2.3 Let f ∈ C2p [0, ∞) and let wa+1 (f, δ) be the modulus of continuity of f on the finite interval [0, a + 1], a > 0. Then for a sequence of the positive operators (1) the inequality kLn (f, x) − f (x)kC[0,a] ≤ ³ ´ ≤ K(f, a) wa+1 (f, δn ) + δn2p (14) holds, where K(f, a) is the positive constant depending on the interval [0, a] and the function f , and n o 1 1 p δn = max |d(2p, n) − 2d(p, n) + 1| 2p + 2p n2 ψn (0)

(15)

which the numbers d(ν, n) tending to 1 as n → ∞ by the Lemma 2.1. Proof. Since lim γn = ∞, we can take a sufficiently large n, that [0, a] ⊂ [0, γn ]. n→∞

Obviously that for x ∈ [0, a] and t ∈ (0, γn ] we can divide the positive semi-axis in two point sets E1 = {(x, t) : x ∈ [0, a], t > a + 1} E2 = {(x, t) : x ∈ [0, a], t ≤ a + 1}.

7

COSKUN: POSITIVE LINEAR OPERATORS

763

Let (x, t) ∈ E1 . In this case tp − xp > 1 and we easily obtain the inequality |f (t) − f (x)| ≤ 3Mf (1 + ap )2 (tp − xp )2 .

(16)

If (x, t) ∈ E2 , we have the inequality ³ |t − x| ´ |f (t) − f (x)| ≤ wa+1 (f, |t − x|) ≤ wa+1 (f, δn ) 1 + δn

(17)

with the some positive δn . (16) and (17) give the inequality ³ |t − x| ´ |f (t) − f (x)| ≤ 3Mf (1 + ap )2 (tp − xp )2 + wa+1 (f, δn ) 1 + δn

(18)

for any t ≥ 0 and x ∈ [0, a]. Applying Ln to both sides of (18) and using H¨older inequality, we obtain Ln (|f (t) − f (x)|, x) ≤ 3Mf (1 + ap )2 Ln ((tp − xp )2 , x)+ ³ ´ p 1 2p wa+1 (f, δn ) 1 + Ln ((t − x)2p , x) δn

(19)

On the other hand, by using the monotonity of the positive operators and the inequality (t − x)2p ≤ (tp − xp )2 , we get Ln (|f (t) − f (x)|, x) ≤ 3Mf (1 + ap )2 Ln ((tp − xp )2 , x)+ ³ ´ p 1 2p wa+1 (f, δn ) 1 + Ln ((tp − xp )2 , x) . δn

(20)

Now, we calculate Ln ((tp − xp )2 , x) in (19). From (8), we obtain Ln ((tp − xp )2 , x)

= Ln ((t2p , x) − 2xp Ln (tp , x) + x2p Ln (1, x) ³ g1,2p (x) g2,2p (x) gp−1,2p (x) = d(2p, n)x2p + 2 + + ... + 2 n ψn (0) (n2 ψn (0))2 (n ψn (0))p−1 gp,2p (x) g2p,2p (x) ´ + 2 + ... + 2 − p (n ψn (0)) (n ψn (0))2p ³ g2,p (x) gp,p (x) ´ 2p g1,p (x) + 2 + . . . + +x ≤ −2xp d(p, n)xp + 2 n ψn (0) (n ψn (0))2 (n2 ψn (0))p 1 Since for a large n, 2 is less than 1 by (3) we can write n ψn (0) ≤ (d(2p, n) − 2d(p, n) + 1)x2p +

8

764

COSKUN: POSITIVE LINEAR OPERATORS

nX o X 1 p |g (x) − 2x g (x)| + |g (x)| . k,2p k,p k,2p n2 ψn (0) Hence

p

2p

k=1

k=p

³ ´ sup Ln (tp − xp )2 , x ≤ |d(2p, n) − 2d(p, n) + 1|a2p +

0≤x≤a

³X ´ X 1 p |g (x) − 2x g (x)| + |g (x)| . k,2p k,p k,2p n2 ψn (0) p

2p

k=1

k=p

Therefore, 1 ³ ³ ´ ´ 2p 1 1 p sup Ln (tp − xp )2 , x) ≤ Cf (a) |d(2p, n) − 2d(p, n) + 1| 2p + 2p n2 ψn (0) 0≤x≤a

is obtained. Then choosing δn as in (15) we can see that with some positive constant Cf (a) ³ ´ Ln (tp − xp )2 , x ≤ Cf (a)δn2p . Using these results in (20), one concludes that Ln (|f (t) − f (x)|, x) ≤ Cf (a)(1 + ap )2 δn2p + 2wa+1 (f, δn ) and, since Ln (1, x) = 1, we obtain kLn (f, x) − f (x)kC[0,a] ≤ Cf (a)(1 + ap )2 δn2p + 2wa+1 (f, δn ) ´ ³ ≤ K(f, a) δn2p + wa+1 (f, δn )

(21)

2

which gives the proof. Corollary 2.4 For sufficiently large n the inequality kLn f − f kC[0,a] ≤ K1 (f, a)wa+1 (f, δn ) holds.

Proof. Since δn → 0 as n → ∞ for large n, δn2p < δn and by the properties of modulus of continuity 2 wa+1 (f, δn ) δn ≤ wa+1 (f, 1) Denoting

³ K1 (f, a) =

´ 2 + 1 K(f, a) wa+1 (f, 1)

2

we obtain the desired result from (21). 9

COSKUN: POSITIVE LINEAR OPERATORS

765

0 Theorem 2.5 For any natural p and any function f ∈ C2p (0, ∞)

lim

sup

n→∞ 0≤x≤γn

|Ln (f, x) − f (x)| = 0. 1 + x2p

Proof. . According to the Theorem 1.2 given above, we can verify only the conditions (2). For ν = 0, (2) is obvious. We will prove (2) only for ν = p since for ν = 2p the proof is analogous. Using (10), we can write Ln (tp − xp ) = [d(p, n) − 1]xp +

p X

(n)

gj,p (x)qj,p

j=1

where by the Lemma 2.1 lim (d(p, n) − 1) = 0,

n→∞

(n) lim q n→∞ j,p

=0

and the functions gj,p belongs to Cp (0, ∞) and therefore to C2p (0, ∞) too. Therefore kLn (tp , x) − xp k2p,[0,γn ] ≤ |d(p, n) − 1| +

p X

(n)

kgj,p k2p qj,p

j=1

2

which gives the proof.

3

Approximation the first derivatives

Now we consider the operators (1) in the space of functions f belonging to the Lipschitz space LipM α, that is for any x, y ∈ [0, ∞) |f (x) − f (y)| ≤ M |x − y|α ,

0 ≤ α < 1.

We assume that the function Kn (x, t, u) in addition to the condition (i)-(iv) satisfies also the condition (v) for any fixed t and u the function Kn (x, t, u) is continuously differentiable with respect to variable x ∈ (0, γn ] and for any ν x

∂ (ν) K (x, 0, u1 ) = νKn(ν) (x, 0, u1 ) + u1 Kn(ν+1) (x, 0, u1 ). ∂x n

Lemma 3.1 Let the function Kn (x, t, u) satisfy the conditions (i)-(v) and the function f ∈ LipM α on [0, ∞). Then the operator Ln (f, x) defined in (1) has the first derivative d dx Ln (f, x) and ∞ h i X ν+1 ν d (ν) f( 2 Ln (f, x) = nαn ψn (0) ) − f( 2 ) Km(n) (x, 0, αn ψn (0)) dx n ψ (0) n ψ (0) n n ν=0

10

766

COSKUN: POSITIVE LINEAR OPERATORS

(−αn ψn (0))ν £ 1 + βν,n (0)]. ν!

(22)

Proof. . Firstly we note by the properties (iv) all derivatives of function Kn (x, t, u) are equal zero in the point x = 0. We shall show that for a fixed n the series in right hand side of (22) is absolutely convergence on [0, γn ]. Realy, for each function f ∈ LipM α and each x ∈ [0, γn ] ∞ ¯ X ¯ ¯f (

¯ ν (−αn ψn (0))ν ¯ (ν) ) K (x, 0, α ψ (0)) ¯ n n n n2 ψn (0) ν! ν=0 ∞ ¯ ¯ X¯ ν (−αn ψn (0))ν ¯α ≤ M − x¯ Kn(ν) (x, 0, αn ψn (0)) + M |f (x)| ¯ 2 n ψn (0) ν! ν=0 ≤ M

∞ ³ ³X ν=0

´2 ν (−αn ψn (0))ν ´ α2 (ν) − x K (x, 0, α ψ (0)) +M |f (x)| n n n n2 ψn (0) ν!

and the first term in right hand side may be easy calculated. Using the properties (v) we obtain d Ln (f, x) dx

=

∞ X

f(

ν=0 ∞ X

−

ν n2 ψn (0)

f(

ν=0 ∞

=

ν (−αn ψn (0))ν ) Kn(ν) (x, 0, αn ψn (0)) x ν!

ν (−αn ψn (0)) (ν+1) (−αn ψn (0))ν ) K (x, 0, α ψ (0)) n n n n2 ψn (0) x ν!

i 1 Xh ν+1 ν (−αn ψn (0))ν+1 f( 2 ) − f( 2 ) Kn(ν+1) (x, 0, αn ψn (0)) . x ν=0 n ψn (0) n ψn (0) ν!

Applying the properties (iv), the formula (22) is obtained. Theorem 3.2 Let Kn (x, t, u) satisfy the conditions (i)-(v), f is differentiable function on [0, ∞) and f 0 ∈ LipM α, then |L0n (f, x) − f 0 (x)| = 0. n→∞ x∈[0,γ ] 1 + xα n lim

sup

Proof. By mean value theorem we can write f(

ν ν+θ 1 ν+1 ) − f( 2 ) = f 0( 2 ) 2 n ψn (0) n ψn (0) n ψn (0) n (0)

n2 ψ

where 0 < θ < 1. Using this equality in (22), taking into account that lim

n→∞

αn = 1, n

lim

n→∞

11

1 =0 n2 ψn (0)

COSKUN: POSITIVE LINEAR OPERATORS

and

∞ X

(ν)

Km(n) (x, 0, αn ψn (0))

ν=0

(−αn ψn (0))ν =1 ν!

and denoting (ν)

Pν,n (x) = Km(n) (x, 0, αn ψn (0)) we obtain L0n (f, x) =

767

(−αn ψn (0))ν ν!

∞ αn X 0 ν + θ f( 2 )Pν,n (x)[1 + βν,n (0)]. n ν=0 n ψn (0)

Since lim βν,n (0) = 0 uniformly on ν, there exits a sequence εn → 0 as n → ∞ such n→∞ that for all ν |βν,n (0)| ≤ εn . Therefore L0n (f, x) − f 0 (x) =

∞ i αn Xh 0 ν + θ f( 2 ) − f 0 (x) n ν=0 n ψn (0)

·Pν,n (x)[1 + βν,n (0)] + [

+f 0 (x)

∞ X

∞ X αn − 1]f 0 (x) Pν,n (x)[1 + βν,n (0)] n ν=0

βν,n (0)Pν,n (x) = I1 + I2 + I3 .

(23)

ν=0

Obviously |I2 + I3 | ≤ |

αn − 1| · |f 0 (x)|(1 + 2εn ). n

Since f 0 ∈ LipM α we obtain |f 0 (x)| ≤ |f 0 (0)| + M xα ≤ M1 (1 + xα ). Therefore |I2 + I3 | ≤ |

αn − 1|(1 + 2εn )M1 (1 + xα ). n

By H¨older inequality |I1 |

≤ M

∞ ¯ ¯α X αn ¯ ν+θ ¯ (1 + εn ) − x¯ Pν,n (x) ¯ 2 n n ψ (0) n ν=0

≤ M

∞ ³ ³X ´2 ´ α2 ν+θ αn (1 + εn ) − x P (x) ν,n n n2 ψn (0) ν=0

Using the properties (iv) it is easy to see that Ln (t, x) =

αn m(n) x(1 + εn ) n n 12

(24)

768

COSKUN: POSITIVE LINEAR OPERATORS

h α i m(n) m(m(n)) 2 αn m(n) 1 n Ln (t2 , x) = ( )2 x + x (1 + ε0n ), n n n n n n2 ψn (0) where (εn ) and (ε0n ) tends to zero as n → ∞. Therefore |I1 |

∞ ∞ iα X αn hX³ 1 + ν ´2 1+ν P (x) − 2x Pν,n (x) + x2 ν,n 2 2 n ν=0 n ψn (0) n ψn (0) ν=0 h iα 1 αn 2 2x 2 2 = M + L (t, x) + L (t , x) − − 2xL (t, x) + x n n n n n4 ψn2 (0) n2 ψn (0) n2 ψn (0) h αn 1 2 αn m(n) αn m(n) m(m(n)) 2 = M + x + ( )2 x n n4 ψn2 (0) n2 ψn (0) n n n n n iα αn m(n) 1 1 2 αn m(n) 2 2 0 + x − 2x − 2x + x + xε + x ε n n n n n2 ψn (0) n2 ψn (0) n n

≤ M

From this equality we obtain |I1 | ≤ M

iα αn h an x + bn x2 + cn n

(25)

where an bn and cn tends to zero as n → ∞. Therefore, from (24), (25) and (23) sup 0≤x≤γn

|L0n (f, x) − f 0 (x)| 1 + xα

tends to zero as n → ∞ which gives the proof.

2

In conclusion we note that the theorems 2.3, 2.5 and 3.2 may be applied to different sequences of positive linear operators which are special case of operators (1). For example, these theorems may be applied to the classical Bernstein polynomials, to BernsteinChlodowsky polynomials, to Sz´asz operators [17] and Baskakov [3] operators in the form of Mastroiani [1]. Moreover, in the case of Kn (x, t, u) =

1 (1 + t + u)n

we obtain the modificied Baskakov operators Vn (f, x) =

¶ ∞ µ ´ν X 1 ν n+ν−1 ³ x f( 2 ) n (1 + αn ψn (0)x) ν=0 1 + αn ψn (0)x n ψn (0) ν

for which m(n) = n + 1 and βν,n (u) ≡ 0.

13

COSKUN: POSITIVE LINEAR OPERATORS

769

References [1] F. Altomare and M. Campiti, Korovkin-type Approximation Theory and its Applications, Walter de Gruyter, Berlin. New York, 1994. [2] A. Aral, Approximation by Ibragimov-Gadjiyev operators in polynomial weighted space, Proc. of IMM of NAS of Azerbaijan, Vol.XIX, 35–44, (2003). [3] V.A. Baskakov, On a sequence of linear positive operators, In ”Research in modern constructive functions theory”, Moscow, 1961. [4] T. Co¸skun, Some properties of linear positive operators on the spaces of weight functions, Commun. Fac. Sci. Univ. Ank. Series, A1, V. 47, 175-181, (1998). [5] T. Co¸skun, Weighted approximation of continuous functions by sequences of linear positive operators, Proc. Indian Acad. Sci. (Math. Sci.), Vol.10, No. 4, 357-362, (2000). [6] T. Co¸skun, Weighted approximation of unbounded continuous functions by sequence of linear positive operators, Indian J. Pure Appl. Math., 34(3), 477-485, (2003). [7] O. Do˘gru, On a certain family linear positive operators, Tr. J. Math., 21, 387-399, (1997). [8] A.D. Gadzhiev, The convergence problem for a sequence of positive linear operators on unbounded sets and theorems analogous to that of P.P. Korovkin, English translated Sov. Math. Dokl., Vol 15, No 5, (1974). [9] A.D. Gadzhiev, On P.P Korovkin type theorems, Mathem. Zametki, Vol 20, No 5, (Engl. Transl., Math. Notes 20, 995-998, (1976).) [10] A.D. Gadjiev, I. Efendiev and E. Ibikli, Generalized Bernstein-Chlodowsky polynomials, Rocky Mount. J. Math., V. 28, No 4, 1267-1277, (1998). [11] A.D. Gadjiev and N. Ispir, On a sequence of linear positive operators in weighted spaces, Proc. of IMM of Azerbaijan AS, Vol. XI(XIX), 45-56, (1999). [12] H. Hacısaliho˘glu and A.D. Gadjiev, On Convergence of the Sequences of Linear Positive Operators (in Turkish), Ankara, 1995. [13] I. I. Ibragimov and A. D. Gadjiev, On a certain Family of linear positive operators, Soviet Math. Dokl., English Trans., 11, 1092-1095,(1970). [14] P.P. Korovkin, Linear operators and approximation theory, Delhi, 1960. [15] T. Po˘gany, Some Korovkin-type theorem for stocastic processes, Teor. Imovi Stat., No 61, 145-151, (1999). Trans. in theory Prob. Math. Stat. N61, 153–159, (2000).

14

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COSKUN: POSITIVE LINEAR OPERATORS

[16] P. Radatz and B. Wood, Approximating Derivativer of Functions Unbounded on the Positive Axis with Lineare Operators, Rev. Roum. Math. Pures et Appl., Bucarest, Tome XXIII, No 5, 771-781, (1978). [17] O. Sz´asz, Generalization of Bernstein’s polynomials to the infinite interval, J. Res. Nat. Bur. Standards 45, 239-245, (1950).

15

JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.4, 771-775, 2011, COPYRIGHT 2011 EUDOXUS PRESS, 771 LLC

THE PRE-SCHWARZIAN DERIVATIVE AND NONLINEAR INTEGRAL TRANSFORMS R. AGHALARY AND A. EBADIAN

Abstract. By making use of extension of Becker’s univalence criterion we obtain the univalent condition for families of integral operators. A number of new univalent conditions would follow upon specializing the parameters in our main result.

1. Introduction and preliminaries Let H denote the class of all analytic functions in the open unit disc D = {z : |z| < 1} and A denote the class of function f ∈ H normalized by f (0) = 0 = f 0 (0) − 1. Also let S denote the class of all univalent functions in A. For a constant o < λ ≤ 1, consider the class U (λ) defined by U (λ) = {f ∈ A : |f 0 (z)(z/f (z))2 | < λ, z ∈ D}. It is known [10] that U (λ) ⊂ S for 0 < λ ≤ 1. In particular, the Bieberbach theorem yields that |a2 | = |f 00 (0)/2| < 2 for f ∈ U (1). Set Uσ (λ) = {f ∈ U (λ) : |f 00 (0)| ≤ 2σ}, for σ ≥ 0. Furthermore, for some real m with 0 < m ≤ 2 we define a subclass P (m) of A consisting of all functions f (z) which satisfy z 00 ( ) (z ∈ U ). f (z) ≤ m Singh [9] has shown that P (2m) ⊂ S for 0 < m ≤ 1. Set Pσ (m) = {f ∈ P (m) : |f 00 (0)| ≤ 2σ}, for σ ≥ 0. For gi ∈ A(i = 1, 2, ..., n) and α ∈ C, we define an integral operator by 1   n(α−1)+1 Z z α−1 α−1 . (1.1) Gn,α (z) = [n(α − 1) + 1] [g1 (t)] ...[gn (t)] dt 0

Also for fi ∈ A(i = 1, 2, ..., n) and α, β ∈ C, we define an integral operator )1/β ( Z n z Y 1 f (t) i (1.2) Fα,β (z) = β tβ−1 ( ) α dt . t 0 i=1 For some recent investigations of univalent conditions involving the families of operators defined by (1.1) and (1.2) see [1], [2], [3], [4], [5], [8]. In the present paper by using other methods we obtain new univalent conditions for operators defined by (1.1) and (1.2). The following familiar result is of fundamental importance in our investigation. 2000 Mathematics Subject Classification. Primary 30C45; Secondary 30C80. Key words and phrases. Univalence, pre-Schwarzian, nonlinear integral operator. 1

772

2

AGHALARY, EBADIAN

Theorem 1.1. ([6,7]) Let β ∈ C, 0. If f ∈ A satisfies 1 − |z|2γ zf 00 (z) (z ∈ D), f 0 (z) ≤ 1 γ then the integral operator 1/β f (t)dt ,

z

 Z Lβ (z) = β

t

β−1 0

0

is in f ∈ S. 2. Main Results We now start our first result begin with the following: Theorem 2.1. Let 21 < λ ≤ 1 and µ, σ be non-negative numbers with µ = σ+λ ≤ 1. Suppose that each of the functions gi ∈ A (i = 1, 2, ...n) belongs to Uσ (λ). Also let |α − 1| ≤

µ2 √ , 2n(λ + µ)(1 − 1 − µ)2

then the function Gn,α defined by (1.1) belongs to S. Proof. We begin by setting Z f (z) =

n z Y

0 j=1

(

gj (t) α−1 ) dt, t

so that, obviously, f 0 (z) =

(2.1)

n Y gj (z) α−1 ( ) . z j=1

Taking a logarithmic differentiation from (2.1) and multiplying both sides by z we obtain  n  X zgj0 (z) zf 00 (z) (2.2) = (α − 1) −1 . f 0 (z) gj (z) j=1 Let gj (z) = z + a2 z 2 + ... be in Uσ (λ). Since z 2 gj0 (z) = 1 + (a3 + 3a22 )z 2 + ..., gj2 (z) we can write (2.3)

z 2 gj0 (z) = 1 + λz 2 ωj (z), gj2 (z)

(j = 1, 2, ..., n)

where ωj (z) is an analytic function in D with |ωj (z)| ≤ 1. If we set hj (z) = gj1(z) − z1 , then we see that hj is analytic in D and hj (0) = −a2 . Using the identity h0j (z) = −

gj0 (z) 1 + = −λωj (z), gj2 (z) z 2

(j = 1, 2, ..., n)

we get the representation (2.4)

z = 1 − a2 z − λz 2 gj (z)

Z

1

ωj (tz)dt, 0

(j = 1, 2, ..., n)

773

THE PRE-SCHWARZIAN DERIVATIVE

3

of gj (z). Now from (2.3) and (2.4) we get R1 zgj0 (z) λz 2 ωj (z) + a2 z + λz 2 0 ωj (tz)dt −1= . R1 gj (z) 1 − a2 z + λz 2 0 ωj (tz)dt This implies that 0 zgj (z) |z|(2λ + |a2 |) gj (z) − 1 ≤ 1 − (|a2 | + λ)|z| . Since |a2 | + λ < σ + λ = µ ≤ 1 we obtain 0 zgj (z) |z|(λ + µ) ≤ (2.5) − 1 . gj (z) 1 − µ|z| Now from (2.2) and (2.5) we find that 00   zf (z) ≤ 2(1 − |z|)|α − 1| |z|(λ + µ) (2.6) 2(1 − |z|) 0 f (z) 1 − µ|z| Let us define k(x) =

(1 − x)x , 1 − µx

(0 ≤ x < 1)

then it is easy to see that k(x) attains it’s maximum value on [0, 1) at x = So we have √ (1 − 1 − µ)2 k(x) ≤ , µ2 and so

√ 1− 1−µ . µ

00 √ zf (z) (1 − 1 − µ)2 ≤ 2|α − 1|n(λ + µ) . (z ∈ D) 2(1 − |z|) 0 f (z) µ2 Using conditions mentioned in the hypothesis of Theorem 2.1 one can observe that 00 zf (z) 1 ≤ 1. 0 such that if a function f : G1 −→ G2 satisfies the inequality d(f (xy), f (x)f (y)) < δ for all x, y ∈ G1, then there exists a homomorphism T : G1 → G2 such that d(f (x), T (x)) <  for all x ∈ G1 ? As mentioned above, when this problem has a solution, we say that the homomorphisms from G1 to G2 are stable. In 1941, D. H. Hyers [14] gave a partial solution of U lam, s problem for the case of approximate additive mappings under the assumption that G1 and G2 are Banach spaces. In 1950, T. Aoki [25] was the second author to treat this problem for additive mappings (see also [16]). In 1978, Th. M. Rassias [17] generalized the theorem of Hyers by considering the stability problem with unbounded Cauchy differences. This phenomenon of stability that was introduced by Th. M. Rassias [17] is called the Hyers–Ulam–Rassias stability. According to Th. M. Rassias theorem: Theorem 1.1. Let f : E −→ E 0 be a mapping from a norm vector space E into a Banach space E 0 subject to the inequality kf (x + y) − f (x) − f (y)k ≤ (kxkp + kykp ) for all x, y ∈ E, where  and p are constants with  > 0 and p < 1. Then there exists a unique additive mapping T : E −→ E 0 such that 2 kf (x) − T (x)k ≤ kxkp 2 − 2p for all x ∈ E. If p < 0 then inequality (1.3) holds for all x, y 6= 0, and (1.4) for x 6= 0. Also, if the function t 7→ f (tx) from R into E 0 is continuous for each fixed x ∈ E, then T is linear. During the last decades several stability problems of functional equations have been investigated by many mathematicians. A large list of references concerning the stability of functional equations can be found in [18–29]. C. Park [30] has contributed works to the stability problem of ternary homomorphisms and ternary derivations (see also [31]). Recently, C˘ adariu and Radu applied the fixed point method to the investigation of the functional equations. (see also [32–38]). In this paper, we will adopt the fixed point alternative of C˘ adariu and Radu to prove the generalized Hyers–Ulam–Rassias stability of ternary derivations on ternary Banach algebras associated with the following functional equation x − 2y + z x + y − 2z x+y+z ) + µf ( ) + µf ( ) = f (µx) . µf ( 3 3 3 Throughout this paper, assume that (A, [ ]A ) is a ternary Banach algebra and X is a ternary Banach A−module. 2. Main Results Before proceeding to the main results, we will state the following theorem. Theorem 2.1. (the alternative of fixed point [32]). 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 m x, T m+1 x) = ∞ for all m ≥ 0, or other exists a natural number m0 such that

778

The fixed point alternative and the stability of ternary derivations ? ? ? ?

3

d(T m x, T m+1 x) < ∞ for all m ≥ m0 ; the sequence {T m x} is convergent to a fixed point y ∗ of T ; y ∗ is the unique fixed point of T in the set Λ = {y ∈ Ω : d(T m0 x, y) < ∞}; 1 d(y, T y) for all y ∈ Λ. d(y, y ∗ ) ≤ 1−L

We start our work with the following theorem which establishes the generalized Hyers– Ulam–Rassias stability of ternary derivations. Theorem 2.2. Let f : A → X be a mapping for which there exists a function φ : A6 → [0, ∞) such that x+y+z x − 2y + z x + y − 2z ) + µf ( ) + µf ( ) − f (µx) 3 3 3 +f ([abc]A ) − [f (a)bc]X − [af (b)c]X − [abf (c)]X kX ≤ φ(x, y, z, a, b, c), kµf (

(2.1)

for all µ ∈ T and all x, y, z, a, b, c ∈ A. If there exists an L < 1 such that x y z a a c φ(x, y, z, a, b, c) ≤ 3Lφ( , , , , , ) 3 3 3 3 3 3 for all x, y, z, a, b, c ∈ A, then there exists a unique ternary derivation D : A → X such that kf (x) − D(x)kB ≤

L φ(x, 0, 0, 0, 0, 0) 1−L

(2.2)

for all x ∈ A. Proof. It follows from x y z a b c φ(x, y, z, a, b, c) ≤ 3Lφ( , , , , , ) 3 3 3 3 3 3 that limj 3−j φ(3j x, 3j y, 3i z, 3i a, 3i b, 3i c) = 0 for all x, y, z, a, b, c ∈ A. Put µ = 1, y = z = a = b = c = 0 in (2.1) to obtain x k3f ( ) − f (x)kB ≤ φ(x, 0, 0, 0, 0, 0) 3 for all x ∈ A. Hence, 1 1 k f (3x) − f (x)kB ≤ φ(3x, 0, 0, 0, 0, 0) ≤ Lφ(x, 0, 0, 0, 0, 0) 3 3

(2.3)

(2.4)

(2.5)

for all x ∈ A. Consider the set X 0 := {g | g : A → B} and introduce the generalized metric on X 0 : d(h, g) := inf {C ∈ R+ : kg(x) − h(x)kB ≤ Cφ(x, 0, 0, 0, 0, 0)∀x ∈ A}. It is easy to show that (X 0 , d) is complete. Now we define the linear mapping J : X 0 → X 0 by 1 J(h)(x) = h(3x) 3 for all x ∈ A. By Theorem 3.1 of [32], d(J(g), J(h)) ≤ Ld(g, h) 0

for all g, h ∈ X . It follows from (2.5) that d(f, J(f )) ≤ L.

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4

By Theorem 1.2, J has a unique fixed point in the set X1 := {h ∈ X 0 : d(f, h) < ∞}. Let D be the fixed point of J. D is the unique mapping with D(3x) = 3D(x) for all x ∈ A satisfying there exists C ∈ (0, ∞) such that kD(x) − f (x)kB ≤ Cφ(x, 0, 0, 0, 0, 0) for all x ∈ A. On the other hand we have limn d(J n (f ), D) = 0. It follows that limn for all x ∈ A. It follows from d(f, D) ≤

1 f (3n x) = D(x) 3n 1 d(f, J(f )), 1−L

d(f, D) ≤

(2.6)

that

L . 1−L

This implies the inequality (2.2). It follows from (2.1), (2.3) and (2.6) that x+y+z x − 2y + z x + y − 2z ) + D( ) + D( ) − D(x)kX 3 3 3 1 = limn n kf (3n−1 (x + y + z)) + f (3n−1 (x − 2y + z)) + f (3n−1 (x + y − 2z)) − f (3n x)kX 3 1 ≤ limn n φ(3n x, 3n y, 3n z, 3n a, 3n b, 3n c) = 0 3 for all x, y, z ∈ A. So kD(

D(

x − 2y + z x + y − 2z x+y+z ) + D( ) + D( ) = D(x) 3 3 3

for all x, y, z ∈ A. Put w = x+y+z , t = x−2y+z and s = x+y−2z in above equation, we get 3 3 3 D(w + t + s) = D(w) + D(t) + D(s) for all w, t, s ∈ A. Hence, D is Cauchy additive. By putting y = z = x, a = b = c = 0 in (2.1), we have kµf (x) − f (µx)kX ≤ φ(x, x, x, 0, 0, 0) for all x ∈ A. It follows that 1 1 kD(µx)−µD(x)kX = limn n kf (µ3n x)−µf (3n x)kX ≤ limn n φ(3n x, 3n x, 3n x, 3n a, 3n b, 3n c) = 0 3 3 for all µ ∈ T, and all x ∈ A. One can show that the mapping D : A → B is C−linear. It follows from (2.1) that kD([xyz]A ) − [D(x)yz]X − [xD(y)z]X − [xyD(z)]X kX 1 1 = limn k n D([3n x3n y3n z]A ) − n ([D(3n x)3n y3n z]X 27 27 1 + [3n xD(3n y)3n z]X + [3n x3n yD(3n z)]X )kX ≤ limn n φ(0, 0, 0, 3n x, 3n y, 3n z) 27 1 ≤ limn n φ(0, 0, 0, 3n x, 3n y, 3n z) 3 =0 for all x, y, z ∈ A. So D([xyz]A ) = [D(x)yz]X + [xD(y)z]X + [xyD(z)]X for all x, y, z ∈ A. Hence, D : A → X is a ternary derivation satisfying (2.2), as desired. 

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The fixed point alternative and the stability of ternary derivations

5

We prove the following Hyers–Ulam–Rassias stability problem for ternary derivations on ternary Banach algebras. Corollary 2.3. Let p ∈ (0, 1), θ ∈ [0, ∞) be real numbers. Suppose f : A → X satisfies kµf (

x+y+z x − 2y + z x + y − 2z ) + µf ( ) + µf ( ) − f (µx)kX ≤ θ(kxkpA + kykpA + kzkpA ), 3 3 3 kf ([abc]A ) − [f (a)bc]X − [af (b)c]X − [abf (c)]X kX , ≤ θ(kakpA + kbkpA + kckpA ),

for all µ ∈ T and all a, b, c, x, y, z ∈ A. Then there exists a unique ternary derivation D : A → X such that 2p θ kxkpA kf (x) − D(x)kB ≤ 2 − 2p for all x ∈ A. Proof. Setting φ(x, y, z, a, b, c) := θ(kxkpA +kykpA +kzkpA +kakpA +kbkpA +kckpA ) all x, y, z, a, b, c ∈ A. Then by L = 2p−1 , we get the desired result.  Theorem 2.4. Let f : A → X be a mapping for which there exists a function φ : A6 → [0, ∞) satisfying (2.1). If there exists an L < 1 such that φ(x, y, z, a, b, c) ≤ 31 Lφ(3x, 3y, 3z, 3a, 3b, 3c) for all x, y, z, a, b, c ∈ A, then there exists a unique ternary derivation D : A → X such that kf (x) − D(x)kX ≤

L φ(x, 0, 0, 0, 0, 0) 3 − 3L

(2.7)

for all x ∈ A. Proof. It follows from (2.4) that x L x k3f ( ) − f (x)kX ≤ φ( , 0, 0, 0, 0, 0) ≤ φ(x, 0, 0, 0, 0, 0) 3 3 3 0 for all x ∈ A. We consider the linear mapping J : X → X 0 such that x J(h)(x) = 3h( ) 3 for all x ∈ A. It follows from (2.9) that

(2.8)

L . 3 By Theorem 2.1, J has a unique fixed point in the set X1 := {h ∈ X 0 : d(f, h) < ∞}. Let D be the fixed point of J, that is, D(3x) = 3D(x) d(f, J(f )) ≤

for all x ∈ A satisfying there exists C ∈ (0, ∞) such that kD(x) − f (x)kX ≤ Cφ(x, 0, 0, 0, 0, 0) n

for all x ∈ A. We have d(J (f ), D) → 0 as n → 0. This implies the equality x limn 3n f ( n ) = D(x) 3 for all x ∈ A. It follows from d(f, D) ≤

1 d(f, J(f )), 1−L

d(f, D) ≤

(2.9)

that

L , 3 − 3L

which implies the inequality (2.7). The rest of the proof is similar to the proof of Theorem 2.2. 

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6

Corollary 2.5. Let p ∈ (3, ∞), θ ∈ [0, ∞) be real numbers. Suppose f : A → X satisfies kµf (

x+y+z x − 2y + z x + y − 2z ) + µf ( ) + µf ( ) − f (µx)kX ≤ θ(kxkpA + kykpA + kzkpA ), 3 3 3 kf ([abc]A ) − [f (a)bc]X − [af (b)c]X − [abf (c)]X kX , ≤ θ(kakpA + kbkpA + kckpA ),

for all µ ∈ T and all a, b, c, x, y, z ∈ A. Then there exists a unique ternary derivation D : A → X such that kf (x) − D(x)kX ≤

θ kxkpA 3p − 3

for all x ∈ A. Proof. Setting φ(x, y, z, a, b, c) := θ(kxkpA + kykpA + kzkpA + kakpA + kbkpA + kckpA ) for all x, y, z, a, b, c ∈ A. Then by L = 31−p , we get the desired result.  Now we investigate the generalized Hyers–Ulam–Rassias stability of Jordan ternary derivations. Theorem 2.6. Let f : A → X be a mapping for which there exists a function φ : A4 → [0, ∞) such that x+y+z x − 2y + z x + y − 2z ) + µf ( ) + µf ( ) − f (µx) 3 3 3 +f ([aaa]A ) − [f (a)aa]X − [af (a)a]X − [aaf (a)]X kX ≤ φ(x, y, z, a), kµf (

(2.10)

for all µ ∈ T and all x, y, z, a, b, c ∈ A. If there exists an L < 1 such that x y z a φ(x, y, z, a) ≤ 3Lφ( , , , ) 3 3 3 3 for all x, y, z, a ∈ A, then there exists a unique Jordan ternary derivation D : A → X such that L φ(x, 0, 0, 0) (2.11) kf (x) − D(x)kX ≤ 1−L for all x ∈ A. Proof. By the same reasoning as the proof of Theorem 2.2, there exists a unique involutive C−linear mapping D : A → X satisfying (2.11). The mapping D is given by D(x) = limn

1 f (3n x) 3n

for all x ∈ A. The relation (2.10) follows that kD([xxx]A ) − [D(x)xx]X − [xD(x)x]X − [xxD(x)]X kX 1 1 = limn k n D([3n x3n x3n x]A ) − n ([D(3n x)3n x3n x]X 27 27 1 + [3n xD(3n x)3n x]X + [3n x3n xD(3n x)]X )kX ≤ limn n φ(0, 0, 0, 3n x) 27 1 ≤ limn n φ(0, 0, 0, 3n x) 3 =0 for all x ∈ A. So D([xxx]A ) = [D(x)xx]X + [xD(x)x]X + [xxD(x)]X

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The fixed point alternative and the stability of ternary derivations

7

for all x ∈ A. Hence, D : A → X is a Jordan ternary derivation satisfying (2.11), as desired.  We prove the following Hyers–Ulam–Rassias stability problem for Jordan ternary derivations on ternary Banach algebras. Corollary 2.7. Let p ∈ (0, 1), θ ∈ [0, ∞) be real numbers. Suppose f : A → B satisfies x+y+z x − 2y + z x + y − 2z kµf ( ) + µf ( ) + µf ( ) − f (µx)kB ≤ θ(kxkpA + kykpA + kzkpA ), 3 3 3 kf ([xxx]A ) − [f (x)xx]X − [xf (x)x]X − [xxf (x)]X kX ≤ 3θ(kxkpA ) for all µ ∈ T, and all x, y, z ∈ A. Then there exists a unique Jordan ternary derivation D : A → X such that 2p θ kf (x) − D(x)kX ≤ kxkpA 2 − 2p for all x ∈ A. Proof. Setting φ(x, y, z, a) := θ(kxkpA + kykpA + kzkpA + kakpA ) all x, y, z, a ∈ A. Then by L = 2p−1 , we get the desired result.  Theorem 2.8. Let f : A → X be a mapping for which there exists a function φ : A4 → [0, ∞) satisfying (2.10). If there exists an L < 1 such that φ(x, y, z, a) ≤ 31 Lφ(3x, 3y, 3z, 3a) for all x, y, z, a ∈ A, then there exists a unique Jordan ternary derivation D : A → X such that L φ(x, 0, 0, 0) kf (x) − D(x)kX ≤ 3 − 3L for all x ∈ A. Proof. The proof is similar to the proofs of Theorems 2.4 and 2.6.



Corollary 2.9. Let p ∈ (3, ∞), θ ∈ [0, ∞) be real numbers. Suppose f : A → X satisfies x+y+z x − 2y + z x + y − 2z kµf ( ) + µf ( ) + µf ( ) − f (µx)kB ≤ θ(kxkpA + kykpA + kzkpA ), 3 3 3 kf ([xxx]A ) − [f (x)xx]X − [xf (x)x]X − [xxf (x)]X kX ≤ 3θ(kxkpA ) for all µ ∈ T, and all x, y, z ∈ A. Then there exists a unique Jordan ternary derivation D : A → X such that θ kf (x) − D(x)kX ≤ p kxkpA 3 −3 for all x ∈ A. Proof. Setting φ(x, y, z, a) := θ(kxkpA + kykpA + kzkpA + kakpA ) all x, y, z ∈ A in above theorem. Then by L = 31−p , we get the desired result. 

References [1] A. Cayley, On the 34 concomitants of the ternary cubic, Am. J. Math. 4, 1 (1881). [2] M. Kapranov, I. M. Gelfand and A. Zelevinskii, Discrimininants, Resultants and Multidimensional Determinants, Birkhauser, Berlin, 1994. [3] V. Abramov, R. Kerner and B. Le Roy, Hypersymmetry a Z3 graded generalization of supersymmetry, J. Math. Phys. 38, 1650 (1997). [4] F. Bagarello and G. Morchio, Dynamics of mean-field spin models from basic results in abstract differential equations, J. Stat. Phys. 66 (1992) 849-866. MR1151983 (93c:82034) [5] N. Bazunova, A. Borowiec and R. Kerner, Universal differential calculus on ternary algebras, Lett. Matt. Phys. 67 (2004), no. 3, 195-206.

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[6] R. Haag and D. Kastler, An algebraic approach to quantum field theory, J. Math. Phys. 5 (1964) 848-861. MR0165864 [7] R. Kerner, Ternary algebraic structures and their applications in physics, Univ. P. M. Curie preprint, Paris (2000), http://arxiv.org/list/math-ph/0011. [8] R. Kerner, The cubic chessboard: Geometry and physics, Class. Quantum Grav. 14, A203 (1997). [9] G. L. Sewell, Quantum Mechanics and its Emergent Macrophysics, Princeton Univ. Press, Princeton, NJ, 2002. MR1919619 (2004b:82001) [10] L. Takhtajan, On foundation of the generalized Nambu mechanics, Comm. Math. Phys. 160 (1994), no. 2, 295-315. [11] L. Vainerman and R. Kerner, On special classes of n-algebras, J. Math. Phys. 37 (1996), no. 5, 2553-2565. [12] H. Zettl, A characterization of ternary rings of operators, Adv. Math. 48 (1983) 117-143. MR0700979 (84h:46093) [13] S. M. Ulam, Problems in Modern Mathematics,Chapter VI, science ed. Wiley, New York, 1940. [14] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. 27 (1941) 222-224. [15] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan. 2(1950), 64-66. [16] D. G. Bourgin, Approximately isometric and multiplicative transformations on continuous function rings,Duke Math. J. 16, (1949). 385–397. [17] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978) 297-300. [18] M. Bavand Savadkouhi, M. Eshaghi Gordji, N. Ghobadipour and J. M. Rassias, Approximate ternary Jordan derivations on Banach ternary algebras, Journal of Mathematical Phisics, 50, 042303 (2009). [19] J. M. Rassias, Solution of a problem of Ulam, J. Approx. Theory 57 (1989), no. 3, 268-273. [20] J. M. Rassias and H. M. Kim, Approximate homomorphisms and derivations between C ∗ -ternary algebras.J. Math. Phys. 49 (2008), no. 6, 063507, 10 pp. 46Lxx (39B82) [21] P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984) 76-86. MR0758860 (86d:39016) [22] S. Czerwik, Stability of functional equations of Ulam-Hyers-Rassias type, Hadronic Press, 2003. [23] Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14(1991) 431-434. [24] D. H. Hyers, G. Isac and Th. M. Rassias, Stability of functional Equations in Several Variables, Birkhauser, Boston, Basel, Berlin, 1998. [25] Th. M. Rassias, On the stability of functional equations and a problem of Ulam,Acta Math. Appl. 62 (2000) 23-130. MR1778016 (2001j:39042) [26] Th. M. Rassias, On the stability of functional equations in Banach spaces,J. Math. Anal. Appl. 251 (2000) 264-284. MR1790409 (2003b:39036) [27] Th. M. Rassias, The problem of S.M.Ulam for approximately multiplicative mappings,J. Math. Anal. Appl. 246(2)(2000),352-378. [28] G. Isac and Th. M. Rassias, On the Hyers-Ulam stability of -additive mappings, J. Approx. Theorey 72 (1993), 131-137.

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[29] M. Eshaghi Gordji, H. Khodaei, Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi–Banach spaces, Nonlinear Analysis. (2009), article in press. [30] C. Park, Isomorphisms between C*-ternary algebras, J. Math. Anal. Appl. 327 (2007), 101-115. [31] C. Park, Homomorphisms between Poisson JC*-algebras, Bull. Braz. Math. Soc. 36 (2005) 79-97. MR2132832 (2005m:39047) [32] L. C˘ adariu and V. Radu, On the stability of the Cauchy functional equation: a fixed point approach, Grazer Mathematische Berichte 346 (2004), 43–52. [33] V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), 91–96. [34] I.A. Rus, Principles and Applications of Fixed Point Theory, Ed. Dacia, Cluj-Napoca, 1979 (in Romanian). [35] L. C˘ adariu, V. Radu, The fixed points method for the stability of some functional equations, Carpathian Journal of Mathematics 23 (2007), 63–72. [36] L. C˘ adariu, V. Radu, Fixed points and the stability of quadratic functional equations, Analele Universitatii de Vest din Timisoara 41 (2003), 25–48. [37] L. C˘ adariu and V. Radu, Fixed points and the stability of Jensen’s functional equation, J. Inequal. Pure Appl. Math. 4 (2003), Art. ID 4. [38] S. Rolewicz, Metric Linear Spaces,PWN-Polish Sci. Publ./Reidel, Warszawa/Dordrecht, 1984.

J.COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.4, 785-795, 2011, COPYRIGHT 2011 EUDOXUS 785 PRESS

NEW ASYMPTOTIC EXPANSIONS OF THE GAMMA FUNCTION AND IMPROVEMENTS OF STIRLING’S TYPE FORMULAS ´ AND NEVEN ELEZOVIC ´ TOMISLAV BURIC Abstract. New asymptotic expansions of the gamma function are derived and some new accurate approximations for the factorial function are given.

1. Introduction The classical Stirling approximation for the factorial function  n √ n n! ≈ 2πn e is a shortening of the following two well known asymptotic expansions [1, 8, 9], Laplace expansion:  n   √ 1 1 139 571 n + 1+ n! ∼ 2πn − − + . . . (1.1) e 12n 288n2 51840n3 2488320n4 and Stirling series

 n   √ 1 1 n 1 − exp + + ... . n! ∼ 2πn e 12n 360n3 1260n5

(1.2)

There are lots of variations of such formulas, see an overview at [10]. Let us mention some of the formulas given there, which will be improved in this paper. The first one is the Karatsuba-Ramanujan formula [7]  n  √ 1 1 1 11 79 n 6 n! ≈ 2πn + 1+ + − + . (1.3) e 2n 8n2 240n3 1920n4 26880n5 We shall explain the choice of the index 6 in this formula and using our methods, the following two even better formulas will be derived:  n  √ 1 1 2 1 1 149 n 12 n! ∼ 2πn 1+ + 2 + + + + + . . .. e n 2n 15n3 120n4 840n5 25200n6 (1.4)  n  √ 2 2 19 8 16 127 n 24 1+ + 2 + + + + + . . .. (1.5) n! ∼ 2πn e n n 15n3 15n4 105n5 3150n6 The next on the improvement list is the Nemes formula [10]: n  n  √ 1 1 239 n 1+ + + (1.6) n! ≈ 2πn e 12n2 1440n4 362880n6 Version May 25, 2010.

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BURIC, ELEZOVIC ASYMPTOTIC EXPANSION OF THE GAMMA FUNCTION

2

and the Gosper formula

   n  √ 1 1 23 n 1+ . n+ − n! ≈ 2π e 6 144n2 6480n3

(1.7)

The formula of Wehmeier [10] will be also improved. This formula is rediscovered by Batir and Mortici using other methods in recent papers [2, 11]:  n  √ 1 1 31 139 9871 n + n! ≈ 2πn 1+ − − + . (1.8) e 6n 72n2 6480n3 155520n4 6531840n5 The following one by Batir [2] is similar to Karatsuba-Ramanujan expansion:  n  √ 1 1 2 31 n 4 + 1+ − − . (1.9) n! ≈ 2πn e 3n 18n2 405n3 9720n4 All of these formulas have been considered separately and the computation of each term is a tedious job. An attempt to derive more general expansion using the same technique was given in [3]. Using new method given in [4] and [5] these results can be generalized and an efficient algorithm for calculating such and more general expansions can be established. Moreover, better approximations for the gamma and factorial function are obtained. Numerical results and applications will be presented in the final section.

2. Asymptotic expansion of the gamma function The logarithm of gamma function has asymptotic expansion in terms of the Bernoulli polynomials [8, p. 32]: log Γ(x + t) ∼ (x + t − 12 ) log x − x +

1 2

log(2π) +

∞  (−1)n+1 Bn+1 (t) −n x . (2.1) n(n + 1) n=1

We shall now generalize formulas (1.3), (1.8) and (1.9). It will be proved that these formulas follow from the general expression which depends on one parameter m. The result is given in the next theorem. Theorem 2.1. It holds log Γ(x + t) ∼ (x + t −

1 2 ) log x

−x+

1 2

∞  1 −n log log(2π) + Pn (t)x m n=0

(2.2)

where polynomials Pn (t) are defined by: P0 (t) = 1 m  (−1)k+1 Bk+1 (t) Pn−k (t), n k+1 n

Pn (t) =

n≥1

(2.3)

k=1

and Bk (t) stands for the Bernoulli polynomials. The manipulations with asymptotic series in the proof of this theorem and in the sequell are justified by properties of asymptotic power series, see e.g. [6, §1.6].

BURIC, ELEZOVIC

787

ASYMPTOTIC EXPANSION OF THE GAMMA FUNCTION

3

Proof. Differentiating (2.2) we get 1 1 ψ(x + t) ∼ log x + (t − 12 ) + x m



∞ 

−1  Pn x−n

n=0

∞ 

(−n)Pn x−n−1

n=1

It can be written in a way  ∞   ∞  1 −n 1 ψ(x + t) − log x − (t − 2 ) ∼ m Pn x (−n)Pn x−n−1 x n=0 n=1 Since (see [8, p. 33]) ψ(x + t) ∼ log x −

∞  (−1)n+1 Bn+1 (t) −n−1 x n+1 n=0

we have ψ(x + t) − log x − (t − 12 ) Therefore m



∞ 

 −n

Pn x

n=0

∞  1 (−1)n Bn+1 (t) −n−1 ∼ x . x n=1 n+1

∞  (−1)k Bk+1 (t)

k=1

k+1

−k−1

x

∞ 

∼−

(2.4)

nPn x−n−1 ,

n=1

wherefrom it follows: −nPn = m

n  (−1)k Bk+1 (t) k=1

k+1

Pn−k . 

Statement of the theorem follows. The first few polynomials Pn are: P0 = 1 m (1 − 6t + 6t2 ) P1 = 12 m (m − 24t − 12mt + 72t2 + 48mt2 − 48t3 − 72mt3 + 36mt4 ) P2 = 288 m (−144 + 5m2 − 360mt − 90m2 t + 4320t2 + 3240mt2+ P3 = 51840 + 630m2 t2 − 8640t3 − 9360mt3 − 2160m2t3 + 4320t4 + 10800mt4+

(2.5)

+ 3780m2t4 − 4320mt5 − 3240m2 t5 + 1080m2t6 ) The further calculation is a tedious job, but for each m coefficients of the expansion can be easily derived using (2.3). Let us derive the consequence of this formula in approximations of the factorial function. For t = 1 and x = n we have Γ(x + t) = n! and Bn (1) = (−1)n Bn , where Bn are the Bernoulli numbers. Corollary 2.2. It holds

1/m  n  ∞ √ n −k Pk n , n! ∼ 2πn e k=0

(2.6)

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BURIC, ELEZOVIC ASYMPTOTIC EXPANSION OF THE GAMMA FUNCTION

4

where (Pn ) is a sequence defined by P0 = 1, m Pn = n

(n+1)/2



k=1

B2k Pn−2k+1 , 2k

(2.7)

n ≥ 1.

Hence, for each m it holds   n   √ 1 m2 m 1 m 1 n m3 · + 5 2· 2+ 1+ 2 − 3 2 n! ∼ 2πn 7 4 e 2 ·3 n 2 ·3 n 2 ·3 2 · 3 · 5 n3    1/m m2 1 m4 + − + . . . , (2.8) 211 · 35 2 5 · 3 3 · 5 n4 One can see that m = 6 is a good choice if the goal is to obtain a simple formula, because of the values in denominators of this expression. But better choice is m = 12 or m = 24. This expansions are given in (1.4) and (1.5) and numerical calculations will be presented in the final section. The choice t = 12 and x = n + 12 leads to even better numerical results: Corollary 2.3. It holds n! ∼

√



n+ 2π e

1 2

n+ 12  ∞

1/m Pk (n +

1 −k 2)

,

(2.9)

k=0

where (Pn ) is a sequence defined by P0 = 1 Pn =

m n

(n+1)/2



k=1

(2−2k+1 − 1)B2k Pn−2k+1 , 2k

n≥1

(2.10)

As an example, we shall give three formulas for the choice of m = 6, m = 12 and m = 24:  1  √ n + 12 n+ 2 6 1 1 23 + n! ∼ 2π 1− + + ... e 4(n + 12 ) 32(n + 12 )2 1920(n + 12 )3 (2.11) 1  n+ 2 √ n + 12 1 1 1 12 + 1− + + . . . (2.12) n! ∼ 2π e 2(n + 12 ) 8(n + 12 )2 120(n + 12 )3  1  √ n + 12 n+ 2 24 1 1 13 n! ∼ 2π 1− + − + ... (2.13) 1 1 2 e n+ 2 2(n + 2 ) 120(n + 12 )3 

3. New very accurate formula of Stirling type In the formula (1.3), (1.4), (1.5) and (1.9), i.e. in the formula (2.2), one can see that the better results are obtained for bigger values of the parameter m. Therefore, it is natural to assume that the approximation with m supstituted by variable x will be better for big values of x. This is indeed the case.

BURIC, ELEZOVIC

789

ASYMPTOTIC EXPANSION OF THE GAMMA FUNCTION

5

Theorem 3.1. The following asymptotic expression is valid   ∞ 1 1 −n 1 1 + log Γ(x + t) ∼ (x + t − 2 ) log x − x + 2 log(2π) + (3.1) Pn (t)x 12x x n=0 where P0 (t) = 1 and 1  (−1)k kBk+2 (t) Pn−k (t), n (k + 1)(k + 2) n

Pn (t) =

n ≥ 1.

(3.2)

k=1

Proof. By comparing the following two asymptotic expansions: ∞

 (−1)k+1 Bk+1 (t) 1 1 1 Γ(x + t) ∼ (x + t − ) log x − x + log(2π) + + x−k , 2 2 12x k(k + 1) k=2   ∞ 1 1 1 1 + log Pn (t)x−n Γ(x + t) ∼ (x + t − ) log x − x + log(2π) + 2 2 12x x n=0 we can write log

 ∞

Pn (t)x−n

n=0

 ∼

∞  (−1)k Bk+2 (t) −k x (k + 1)(k + 2)

k=1

By differentiating it follows  ∞ ∞ ∞    (−1)k k Bk+2 (t) −n−1 −n −k−1 x . nPn (t)x ∼ Pn (t)x (k + 1)(k + 2) n=1 n=0 k=1

Hence, nPn (t) =

n  (−1)k k Bk+2 (t) k=1

(k + 1)(k + 2)

Pn−k (t) 

and the statement follows.

Corollary 3.2. The following asymptotic expansion for the factorial function holds:

∞ 1/n  n  √ n n! ∼ 2πn e1/12n P2m n−2m , (3.3) e m=0 where (Pm ) is a sequence defined by P0 = 1 and 1  k B2k+2 P2m−2k . 2m (k + 1)(2k + 1) m

P2m =

(3.4)

k=1

Therefore, it holds   n 1/n √ 1 1447 1170727 n 1/12n 1− e + − +... . (3.5) n! ∼ 2πn e 360n2 1814400n4 1959552000n6 Proof. The statements follows from Theorem 3.1 by taking x = n, t = 1 and using the fact that Bn (1) = (−1)n Bn . It is easy to see that it holds P1 = 0, and since B2k+1 = 0 for k ≥ 1, all odd members of the sequence (Pn ) vanish.  The approximation of the factorial function given in the formula above is very accurate. In the last section some numerical calculations which will show the quality of this approximation will be given.

790

BURIC, ELEZOVIC ASYMPTOTIC EXPANSION OF THE GAMMA FUNCTION

6

Using the same technique as in Theorem 3.1, a generalization of Nemes formula (1.6) can be derived: Theorem 3.3. The following asymptotic expression is valid   ∞ Γ(x + t) ∼ (x + t − 12 ) log x − x + 12 log(2π) + x log Pn (t)x−n

(3.6)

n=0

where P0 (t) = 1 and 1  (−1)k Bk (t) Pn−k (t), n k−1 n

Pn (t) =

n ≥ 1.

(3.7)

k=2

The general form of the Nemes expansion can be written: Corollary 3.4. The following asymptotic expansion for the factorial function holds: n  n  ∞ √ n −2m P2m n , (3.8) n! ∼ 2πn e m=0 where (Pm ) is a sequence defined by P0 = 1 and 1  B2k P2m−2k , 2m 2k − 1 m

P2m =

m≥1

(3.9)

k=1

As before, the “n and a half” formula easily follows: Corollary 3.5. It holds n+ 12  1 ∞  √ n + 12 n+ 2  1 −2m P2m (n + 2 ) , n! ∼ 2π e

(3.10)

k=0

where (Pm ) is a sequence defined by P0 = 1 and 1  (2−2k+1 − 1)B2k P2m−2k , 2m 2k − 1 m

P2m =

m≥1

(3.11)

k=1

4. Gosper formula and its generalizations We shall now discuss the formula (1.7). It can also be embedded into more general one-parameter formula like examples before. Theorem 4.1. The following asymptotic expansion is valid for every real ω ∞    ω −n 1 1 1 log Γ(x+t) ∼ 2 log(2π)+(x+t− 2 ) log x−x+ 2 log 1+ +log Pn (t, ω)x x n=0 (4.1) where polynomials Pn (t) are defined by P0 (t) = 1 and   n 1 Bk+1 (t) ω k − Pn−k (t, ω), Pn (t, ω) = (−1)k+1 n≥1 (4.2) n k+1 2 k=1

BURIC, ELEZOVIC

791

ASYMPTOTIC EXPANSION OF THE GAMMA FUNCTION

7

Proof. The following asymptotic expansion of the logarithm of gamma function is well known, see [8, p. 32]: log Γ(x + t) ∼

1 2

log(2π) + (x + t − 12 ) log x − x +

∞ 

Cn (t)x−n .

n=1

Therefore, we can start with the asymptotic expansion of the following type: ∞    ω −n 1 1 1 +log Pn (t, ω)x log Γ(x+t) ∼ 2 log(2π)+(x+t− 2 ) log x−x+ 2 log 1+ x n=0 since coefficients Pn (t, ω) will compensate the influence of the term 12 log(1 + ωx ). Now we have ω −1  ∞ ∞ − 2   1 1 −k −n−1 1 x . Pk (t, ω)x (−n)Pn (t, ω)x ψ(x+t)−log x−(t− 2 ) ∼ · + x 2 1+ ω n=1 k=0 x Since it holds ω ∞ ∞ k k 1 ω  1 − x2 kω k ω · (−1) = (−1) = − 2 1+ ω 2x2 xk 2 xk+1 k=0 k=1 x and using the asymptotic expansion (2.4), we can write ∞  ∞   (−1)k+1 Bk+1 (t) x−k−1 Pk (t, ω)x−k k+1 k=0 k=1   ∞ ∞  (−1)k ω k −k−1 x = + nPn (t, ω)x−n−1 2 n=1 k=1



and the statement of the theorem follows.

Remark 1. Between all possible choices of the parameter ω in (4.1), the known one ω = 16 is the best one. Namely, in this case and if t = 1, for the first coefficient P1 we have P1 = 12 B2 − 12 ω = 0 and we deduce Gosper formula (1.7), as stated in the following corollary. Corollary 4.2. The following formula is valid  ∞  n  √ 1  n −m n! ∼ 2π n+ Pm n e 6 m=0 where Pm =

 m  (−1)k 1  Bk+1 + Pm−k m k+1 2 · 6k

(4.3)

(4.4)

k=2

and Bk are the Bernoulli numbers. In recent paper [12], Nemes improved formula (1.7) and presented the following expansion for the factorial function:     ∞ 1 Gk x −x Γ(x + 1) ∼ x e 2π x + 6 (x + 14 )k k=0

792

BURIC, ELEZOVIC ASYMPTOTIC EXPANSION OF THE GAMMA FUNCTION

8

but the calculation of the coefficients is cumbersome:  k  k−1    −j  Gj −1/2 ak−j = , 6j j k − j 4k−j j=0 j=0 where ak are coefficients from the Laplace expansion (1.1) ∞  Γ(x + 1) an √ ∼ . xx e−x 2πx n=0 xn

We shall show that this expansion can be improved to be more natural such that calculation of the coefficients can be significantly simplified. It is sufficient to substitute x in (4.1) with n + 14 and take t = 34 . Then n + 16 from (1.7) will be equal 1 . Hence, the result of the theorem to n + 14 + ω wherefrom it follows that ω = − 12 can be stated as the following corollary. Corollary 4.3. The following approximation for the factorial function is valid       ∞ n + 14 n −1/4 1 Pk (4.5) e 2π n + n! ∼ e 6 (n + 14 )k k=0

where P0 = 1 and

 n 1 1  (−1)k+1 Bk+1 ( 34 ) + . Pn = n k+1 2 · 12k

(4.6)

k=1

Let us mention that the value of the Bernoulli polynomials Bn ( 34 ) can be calculated from the well known formula [1, p. 806]

 Bn ( 34 ) = (−1)n+1 2−n (1 − 21−2n )Bn + n 4−n En−1 where Bn are Bernoulli and En Euler numbers. The first few values in the expansion (4.5) are P1 =

1 , 32

P2 =

185 , 18432

P3 =

14647 , 26542080

P4 = −

3929497 3397386240

5. Applications and numerical results In this section we shall summarize our results and apply them to improve some known approximations. Some numerical calculations for factorial function will be presented. Let us start with expansion (2.2). As we mentioned before, the formulas by Batir and Mortici can be improved. Taking t = 1 and m = 2 we improve (1.8) and with m = 4 we improve (1.9):  n  √ 1 1 31 139 n + 1+ n! ∼ 2πn − − + e 6n 72n2 6480n3 155520n4 1/2 (5.1) 324179 9871 + + ... + 6531840n4 1175731200n6

BURIC, ELEZOVIC ASYMPTOTIC EXPANSION OF THE GAMMA FUNCTION

and n! ∼

√

793 9

 n  1 1 2 31 n + 1+ 2πn − − + e 3n 18n2 405n3 9720n4

(5.2) 1/4 19349 529 + + ... + 204120n5 18370800n6 For simplicity, only one more term in the formulas will be presented. Of course, using general expansions from previous sections one can derive as many coefficients as necessary. Secondly, by taking m = 6 Karatsuba-Ramanujan formula (1.3) is improved and new better formulas (1.4) and (1.5) for m = 12 and m = 24 are obtained. For the convenience of the reader, we shall recall these formulas:  n  √ 1 1 1 11 79 3539 n 6 + 2+ n! ∼ 2πn 1+ − + + + ... 3 4 5 e 2n 8n 240n 1920n 26880n 1612800n6  n  √ 1 1 2 1 1 149 n 12 n! ∼ 2πn 1+ + 2 + + + + + ... e n 2n 15n3 120n4 840n5 25200n6  n  √ 2 2 19 8 16 127 n 24 n! ∼ 2πn 1+ + 2 + + + + + ... 3 4 5 e n n 15n 15n 105n 3150n6 Note that m = 1 gives well known Laplace formula (1.1). Now we shall compare numerical precision of these formulas. Formula m=1 m=2 m=4 m=6 m=12 m=24 m=1 m=2 m=4 m=6 m=12 m=24 m=1 m=2 m=4 m=6 m=12 m=24

n (1) (2) (3) (4) (5) (6) (7) (8) 100 6.5 8.6 11.7 13.1 16.2 17.2 20.4 21.1 100 6.2 8.6 11.4 13.1 15.9 17.2 20.0 21.1 100 5.9 8.9 11.1 13.2 15.6 17.3 19.7 21.1 100 5.7 9.2 11.0 13.3 15.4 17.3 19.5 21.1 100 5.4 8.0 11.2 14.0 15.3 17.6 19.3 21.3 100 5.1 7.3 9.7 12.2 14.8 17.4 19.3 22.7 1000 8.5 11.6 15.6 18.1 22.2 24.2 28.3 30.1 1000 8.2 11.6 15.4 18.1 21.9 24.2 28.0 30.1 1000 7.9 11.9 15.1 18.2 21.6 24.3 27.7 30.1 1000 7.7 12.2 15.0 18.3 21.4 24.3 27.5 30.1 1000 7.4 11.0 15.2 19.0 21.3 24.6 27.3 30.3 1000 7.1 10.3 13.7 17.2 20.8 24.4 27.3 32.1 10000 10.5 14.6 19.6 23.1 28.2 31.2 36.3 39.1 10000 10.2 14.6 19.3 23.1 27.9 31.2 36.0 39.1 10000 9.9 14.9 19.1 23.2 27.6 31.3 35.7 39.1 10000 9.7 15.2 19.0 23.3 27.4 31.3 35.5 39.1 10000 9.4 14.0 19.2 24.0 27.3 31.6 35.3 39.3 10000 9.1 13.3 17.7 22.2 26.8 31.4 35.3 41.1

The previous table displays the number of exact decimal digits (EDD) of the formulas for some values of n. Exact decimal digits are defined by:    formula(n)  EDD(n) = − log10 1 −  n!

794

BURIC, ELEZOVIC ASYMPTOTIC EXPANSION OF THE GAMMA FUNCTION

10

The (i)-th column is the EDD of the given approximation using the series up to the i-th order term. Calculations are done with Mathematica. One can see that m = 1 (Laplace expansion) is the best choice for odd order terms, and as we take more even order terms our new approximations for bigger m give better numerical results. Even better results can be obtained with “n and half” formulas (from expression (2.9)). Recall that for m = 6, 12, 24 we have:  1 n + 12 n+ 2 6 1 1 23 n! ∼ 2π + 1− + + ... e 4(n + 12 ) 32(n + 12 )2 1920(n + 12 )3  1  √ n + 12 n+ 2 12 1 1 1 + n! ∼ 2π 1− + + ... 1 1 2 e 2(n + 2 ) 8(n + 2 ) 120(n + 12 )3  1  √ n + 12 n+ 2 24 1 1 13 n! ∼ 2π 1− + − + ... e n + 12 2(n + 12 )2 120(n + 12 )3 √



Precision is given in the following table (notations are same as before). For bigger m better approximations than in previous table are achieved. Formula n (1) (2) (3) (4) m=6 100 6.3 8.7 11.2 13.2 m=12 100 6.0 9.2 11.0 13.3 m=24 100 5.7 8.3 11.2 14.0 m=6 1000 8.3 11.7 15.2 18.2 m=12 1000 8.0 12.2 15.0 18.3 m=24 1000 7.7 11.3 15.2 19.0 m=6 10000 10.3 14.7 19.2 23.2 m=12 10000 10.0 15.2 19.0 23.3 m=24 10000 9.7 14.3 19.2 24.0

(5) (6) 15.7 17.3 15.4 17.3 15.3 17.7 21.7 24.3 21.4 24.3 21.3 24.6 27.7 31.3 27.4 31.3 27.3 31.6

Finally, we shall present numerical quality of our new very accurate approximation formula (3.3) (if m = n). Let us recall this formula:   n 1/n √ 1 1447 1170727 n 1/12n 1− e + − + ... n! ∼ 2πn e 360n2 1814400n4 1959552000n6 Because of its special form, we shall put it in a separate table: Formula n (2) (4) (6) (8) New 100 13.1 17.2 21.1 24.7 New 1000 18.1 24.2 30.1 35.7 New 10000 23.1 31.2 39.1 46.7 Acknowledgment. The research of the authors was supported in part by the Croatian Ministry of Science, Education and Sports, under the project 036-11708891054.

BURIC, ELEZOVIC ASYMPTOTIC EXPANSION OF THE GAMMA FUNCTION

795 11

References [1] M. Abramowitz and I. A. Stegun (Eds), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series 55, 9th printing, Washington, 1970. [2] N. Batir, Very accurate approximations for the factorial function, J. Math. Inequal., (to appear) [3] J. Bukac, T. Buri´c and N. Elezovi´ c, Stirling’s formula revisited via some new and classical inequalities, submitted. [4] T. Buri´c and N. Elezovi´ c, Bernoulli polynomials and asymptotic expansions of the quotient of gamma functions, submitted. [5] T. Buri´c and N. Elezovi´c, New asymptotic expansions of the quotient of gamma functions, submitted [6] A. Erd´elyi, Asymptotic expansions, Dover Publications, New York, 1956. [7] E. A. Karatsuba, On the asymptotic representation of the Euler gamma function by Ramanujan, J. Comp. Appl. Math. 135 (2001), 225–240. [8] Y.L. Luke, The Special Functions and Their Approximations, Vol. I, Academic Press, New York, 1969. [9] Y.L. Luke, Mathematical Functions and Their Approximations, Vol. I, Academic Press, New York, 1975. [10] P. Luschny, Approximation formulas for the factorial function n!, http://www.luschny.de/math/factorial/approx/SimpleCases.html [11] C. Mortici, Sharp inequalities related to Gosper’s formula, C. R. Acad. Sci. Paris, Ser. I, 348 (2010), 137–140. [12] G. Nemes, More accurate approximations for the gamma function, preprint ´, University of Zagreb, Faculty of Electrical Engineering and ComTomislav Buric puting, Unska 3, 10000 Zagreb, Croatia, E-mail address: [email protected] ´, University of Zagreb, Faculty of Electrical Engineering and ComNeven Elezovic puting, Unska 3, 10000 Zagreb, Croatia, E-mail address: [email protected]

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL.13, NO.4, 2011

A Common Fixed Point Theorem of a Gregus Type on Convex Cone Metric Spaces, Thabet Abdeljawad and Erdal Karapinar,……………………………………….....………………….609 A Gregus type common fixed point theorem of set-valued mappings in cone metric spaces, T. Abdeljawad, P.P. Murthy and K. Tas,…………………………………………………………622 The Existence of Weak Solution for Degenerate ∑ Δpi(x) -Equation, Ravi P. Agarwal, M.B. Ghaeme and S. Saiedinezhad,………………………………………………………………….629 q-Euler Numbers and Polynomials Associated With Multiple q-Zeta Functions, Taekyun Kim, Lee-Chae Jang and Byungje Lee,………………………………………………………………642 Numerical solution of Volterra integro-differential equations, S. Karimi Vanani and A. Aminataei,………………………………………………………………………………………654 Sandwich Theorems Associated With New Multiplier Transformations, Adriana Catas,……...663 The rate of convergence of the q-analogue of Favard-Szász type operators, Çigdem Atakut and Ibrahim Büyükyazici,…………………………………………………………………………....673 Multilateral Generating Functions for Classes of Polynomials Involving Multivariable Laguerre Polynomials, Mehmet Ali Özarslan and Cem Kaanoglu,…………………………………….....683 On a new class Wp,q of Segal algebras, Birsen Sagir, Cenap Duyar,…………………………...692 Characterizations of Orthogonal Generalized Gegenbauer-Humbert Polynomials and Orthogonal Sheffer-type Polynomials, Tian-Xiao He,……………………………………………………....701 On the stability of quadratic double centralizers on Banach algebras, M. Eshaghi Gordji and A. Bodaghi,………………………………………………………………………………………...724 Generalized multi–valued contraction mappings, M. Eshaghi Gordji, H. Baghani, H. Khodaei and M. Ramezani,………………………………………………………………………………730 Nearly higher derivations in unital C*−algebras, M. Eshaghi Gordji, R. Farokhzad Rostami and S.A.R. Hosseinioun,…………………………………………………………………………….734 Numerical solution of fuzzy differential equations by hybrid predictor-corrector method, ChengFu Yang,………………………………………………………………………………………...743

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL.13, NO.4, 2011 (continues)

On a Construction of Positive Linear Operators for Approximation of Continuous Functions in the Weighted Spaces, Tulin Coskun,…………………………………………………………..756 The Pre-Schwarzian Derivative and Nonlinear Integral Transforms, R. Aghalary and A. Ebadian,………………………………………………………………………………………...771 The fixed point alternative and the stability of ternary derivations, A. Ebadian and Sh. Najafzadeh,……………………………………………………………………………………. 776 New Asymptotic Expansions of the Gamma Function and Improvements of Stirling’s Type Formulas, Tomislav Buric and Neven Elezovic,…………………………………………….....785  

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Volume 13, Number 5 ISSN:1521-1398 PRINT,1572-9206 ONLINE

July 2011

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(seven 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. Webmaster:Ray Clapsadle 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 $470,Electronic $300,Print and Electronic $500.Individual:Print $150,Electronic $100,Print &Electronic $200.For any other part of the world add $50 more to the above prices for Print.No credit card payments. Copyright©2011 by Eudoxus Press,LLCAll 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 20) Hrushikesh N.Mhaskar 1) George A. Anastassiou Department of Mathematical Sciences Department Of Mathematics California State University The University of Memphis Los Angeles,CA 90032 Memphis,TN 38152,U.S.A 626-914-7002 Tel.901-678-3144 e-mail: [email protected] e-mail: [email protected] Approximation Theory,Real Analysis, Orthogonal Polynomials, Approximation Theory,Splines, Wavelets, Neural Wavelets, Neural Networks Networks,Probability, Inequalities. 21) M.Zuhair Nashed Department Of Mathematics 2) J. Marshall Ash University of Central Florida Department of Mathematics PO Box 161364 De Paul University Orlando, FL 32816-1364 2219 North Kenmore Ave. e-mail: [email protected] Chicago,IL 60614-3504 Inverse and Ill-Posed problems, 773-325-4216 Numerical Functional Analysis, e-mail: [email protected] Integral Equations,Optimization, Real and Harmonic Analysis Signal Analysis 3) Mark J.Balas 22) Mubenga N.Nkashama Department Head and Professor Department OF Mathematics Electrical and Computer Engineering University of Alabama at Dept. Birmingham College of Engineering Birmingham,AL 35294-1170 University of Wyoming 205-934-2154 1000 E. University Ave. e-mail: [email protected] Laramie, WY 82071 Ordinary Differential Equations, 307-766-5599 Partial Differential Equations e-mail: [email protected] Control Theory,Nonlinear Systems, 23) Charles E.M.Pearce Neural Networks,Ordinary and Applied Mathematics Department Partial Differential Equations, University of Adelaide Functional Analysis and Operator Adelaide 5005, Australia Theory e-mail: [email protected] 4) Drumi D.Bainov Stochastic Processes, Probability Department of Mathematics Theory, Harmonic Analysis, Medical University of Sofia Measure Theory, P.O.Box 45,1504 Sofia,Bulgaria Special Functions, Inequalities e-mail: [email protected] 24) Josip E. Pecaric e-mail:[email protected] Differential Equations/Inequalities Faculty of Textile Technology University of Zagreb Pierottijeva 6,10000 5) Carlo Bardaro Zagreb,Croatia Dipartimento di Matematica e e-mail: [email protected] Informatica Inequalities,Convexity Universita di Perugia Via Vanvitelli 1

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JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.5, 809-821, 2011, COPYRIGHT 2011 EUDOXUS PRESS, 809 LLC

A STUDY OF FUNCTIONAL PROPERTIES OF WEIGHTED LIPSCHITZ-LORENTZ SPACES WITH THEIR COMPACT EMBEDDINGS I·LKER ERYILMAZ Abstract. Weighted Lipschitz-Lorentz spaces are de…ned and some basic properties are showed. The author also showed the necessary conditions for compact embeddings of these spaces.

1. Introduction and preliminaries

Let G be a metrizable locally compact abelian group with Haar measure and Lip ( ; p) and lip( ; p) denote the Lipschitz spaces de…ned on G for 1 p < 1. Quek and Yap in [15],Feichtinger in [6],[7] proved a series of results concerning about Lipschitz spaces and multipliers from L1 (G) to the Lipschitz spaces. In ^ particular, it was shown that the spaces (L1 (G) ; lip( ; p)) ; lip( ; p); Lip( ; p) and ^ (L1 (G) ; Lip( ; p)) are isometrically isomorphic where lip( ; p) denotes the relative completion of lip( ; p) in [15]. It was also showed that L1 (G) lip( ; p) = lip( ; p) = L1 (G) Lip( ; p) where is usual convolution operation. Let w be a Beurling’s weight function, i.e. a measurable, locally bounded function on G, satisfying w (x) 1; w (x + y) w (x) w (y) for all x; y 2 G. For any two weight functions w1 and w2 , we write w1 w2 if there exists C > 0 such that w1 (x) Cw2 (x) for all x 2 G. We write w1 w2 if and only if w1 w2 and w2 w1 .We will de…ne weighted Lipschitz-Lorentz spaces and then show that the properties mentioned above hold for weighted Lorentz spaces which is denoted by L (p; q; wd ) (G). In [2], Blozinski de…ned the convolution operators and proved some convolution theorems for Lorentz spaces. Duyar and Gürkanl¬ generalized these properties to the weighted Lorentz spaces in [5]. For the convenience of the reader, we now review brie‡y what we need from the theory of L (p; q; wd ) (G) spaces. Lorentz spaces over weighted measure spaces L (p; q; wd ) (G) are de…ned and discussed in [5],[8]. Instead of Haar measure ; let us take the measure as wd . Let f be complex-valued, measurable and de…ned on the measure space (G; wd ). Then the distribution function of f is R w (x) d (x) ; y 0 f;w (y) = w fx 2 G : jf (x)j > yg = fx2G: jf (x)j>yg

2000 Mathematics Subject Classi…cation. , 46E30, 26A16, 14E25. Key words and phrases. weighted Lorentz, Lipschitz, Compact Embedding. 1

810

I·LKER ERYILM AZ

2

found. The nonnegative rearrangement of f is given by fw (t) = inf fy > 0 :

where we assume that inf (0; 1) is given by

f;w

(y)

t g = sup fy > 0 :

= 1 and sup

f;w

(y) > t g ; t

0

= 0. Also the average function of f on

1R t f (s) ds: t 0 w Note that f;w ( ) ; fw ( ) and fw ( ) are nonincreasing and right continuous functions. The weighted Lorentz space L (p; q; wd ) (G) is the collection of all the functions f such that kf kp;q;w < 1, where fw (t) =

R q q 1 p p 0 t

kf kp;q;w =

1

q

[fw (t)] dt

1 p

supt fw (t) t>0

1 q

; 0 < p; q < 1

:

; 0 0 and n; m " > 0 such that (2.6)

kfn

)

kLy fN

fN kp;q;w

" 4 (2 +


0: Since w is bounded, there exists a constant C 0 such that w (x) C for all x 2 G. 1. Since f 2 lipw ( ; p; q), there exists " > 0 such that sup

kLy f

f kp;q;w


0 and for all y 2 G; 0 < jyj is used, we get for the same " > 0 sup

kLy (f


0 such that sup

kLy f

f kp;q;w

< sup

" 1 p

w (x)

jyj

for all y 2 G; 0 < jyj we have


0; for all y 2 G; 0 < jyj

kLy (Lx f )

(Lx f )kp;q;w

jyj


0 such that

(f

en )kp;q;w

jyj

0 such that kf k( ;p;q;w2 ) c kf k( ;p;q;w1 ) for all f 2 lipw1 ( ; p; q): 2. lipw ( ; p; q1 ) lipw ( ; p; q2 ) if 0 < q1 q2 1. Proposition 10. Let w1 and w2 be weight functions on G and 0 < p; q < 1.Then w1 w2 if and only if lipw2 ( ; p; q) lipw1 ( ; p; q). Proposition 11. Let p = 1 and 0 < q 1 . Then for any f 2 lipw ( ; 1; q), the function x ! kLx f k( ;p;q;w) is equivalent to the weight function w, i.e. there exists c1 (f ) ; c2 (f ) > 0 such that c1 w (x)

kLx f k(

;p;q;w)

c2 w (x) :

Proposition 12. Let w1 and w2 be weight functions on G and 0 < q 1. Then the embedding i : lipw1 ( ; 1; q) ,! lipw2 ( ; 1; q) is continuous if and only if w2 w1 . Proposition 13. Let 1 q p < 1. Then for any f 2 lipwp ( ; p; q1 ), the function x ! kLx f k( ;p;q;wp ) is equivalent to the weight function w, i.e. there exists c1 (f ) ; c2 (f ) > 0 such that c1 w (x)

kLx f k(

;p;q;wp )

c2 w (x) :

Proposition 14. Let w1 and w2 be weight functions on G and 1 q p < 1. Then the embedding i : lipw1p ( ; p; q) ,! lipw2p ( ; p; q) is continuous if and only if w2 w1 . Remark 4. Propositions 8-14 are also valid for Lipw ( ; p; q); k k(

;p;q;w)

spaces.

After this point, we will work on G = Rd with Lebesgue measure dx. We denote by Cc Rd the space of complex-valued, continuous functions with compact support. We will bene…t from the techiques which are used in [10]. Lemma 3. Let (fn )n2N be a sequence in lipw ( ; p; q). If (fn )n2N converges to zero in lipw ( ; p; q) then (fn )n2N also converges to zero in the vague topology, i.e. for n!1 R f (x) k (x) dx ! 0; Rd n for all k 2 Cc Rd .

Proof. Let k 2 Cc Rd . We write R (3.1) f (x) k (x) dx Rd n

kkkp0 ;q0 kfn kp;q kkkp0 ;q0 kfn k(

kkkp0 ;q0 kfn kp;q;w

;p;q;w)

818

I·LKER ERYILM AZ

10

for 1 (3.2)

p; q < 1 and R f (x) k (x) dx Rd n

kkk1 kfn k1;q kkk1 kfn k(

kkk1 kfn k1;q;w

;1;q;w)

for p = 1 and 0 < q 1 by Hölder’s inequality where p1 + p10 = 1; 1q + q10 = 1. Hence the sequence (fn )n2N converges to zero in the vague topology by (3.1) and (3.2). Proposition 15. Let w1 , w2 be Beurling weight functions on Rd ; w1 (x) ! 1 2 (x) d as x ! 1 and 0 < q 1. If w2 w1 and w w1 (x) doesn’t tend to zero in R for x ! 1 then the embedding of the space lipw1 ( ; 1; q) into L (1; q; w2 dx) Rd is never compact. Proof. Since w2 w1 and 0 < q 1, there exists a constant C > 0 such that w2 (x) Cw1 (x). This implies that lipw1 ( ; 1; q) L (1; q; w2 dx) Rd by proposiw2 (x) tion 12. Let (tn )n2N be a sequence in Rd such that tn ! 1 as n ! 1. Since w 1 (x)

2 (x) doesn’t tend to zero in Rd as x ! 1, there exists > 0 such that w >0 w1 (x) for x ! 1. For the proof of the embedding of the space lipw1 ( ; 1; q) into L (1; q; w2 dx) Rd is never compact, let us take any …xed f 2 lipw1 ( ; 1; q) and de…ne a sequence (fn )n2N , where

fn = w1 1 (tn ) Ltn f

(3.3)

which is bounded in lipw1 ( ; 1; q). Indeed by proposition 11, we have kfn k(

;1;q;w1 )

=

w1 1 (tn ) Ltn (f )

( ;1;q;w1 )

1

w1 (tn ) w1 (tn ) kf k(

= w1 1 (tn ) kLtn (f )k(

;1;q;w1 )

= kf k(

;1;q;w1 )

;1;q;w1 )

:

Now we will prove that there wouldn’t exist a subsequence of (fn )n2N which is convergent in L (1; q; w2 dx) Rd . The sequence in the above certainly converges in the vague topology. Indeed for all k 2 Cc Rd , we get R R (3.4) Rd fn (x) k (x) dx = w 1 (tn ) Ltn f (x) k (x) dx Rd 1 =

R 1 d Lt f (x) k (x) dx w1 (tn ) R n 1 1 kkk1 kf k1;q;w1 kkk1 kf k( w1 (tn ) w1 (tn )

;1;q;w1 )

Since right hand side of (3.4) tends to zero for n ! 1, we have R f (x) k (x) dx ! 0: Rd n

Finally,

R

f Rd n

(x) k (x) dx

kkk1 kfn k1

kkk1 kfn k1;q;w2

kkk1 kfn k1;q;w1

kkk1 kfn k(

;1;q;w1 )

by this inequality, the only possible limit of (fn )n2N in L (1; q; w2 dx) Rd is zero. It is known that the function x ! kLx f k1;q;w2 is equivalent to the weight function w2 , i.e. there exists c1 (f ) ; c2 (f ) > 0 such that c1 w2 (x)

kLx f k1;q;w2

c2 w2 (x)

819

W EIGHTED LIPSCHITZ-LORENTZ SPACES

for all x 2 Rd . Therefore (3.5)

= w1 1 (tn ) kLtn (f )k1;q;w2

kfn k1;q;w2

is found. Since

w2 (tn ) w1 (tn )

11

c1 w1 1 (tn ) w2 (tn )

> 0 for all tn , by using (3.5) we write kfn k1;q;w2

c1 w1 1 (tn ) w2 (tn )

c1 :

This means that it is not possible to …nd norm convergent subsequence of (fn )n2N in L (1; q; w2 dx) Rd . So the embedding lipw1 ( ; 1; q) ,! L (1; q; w2 dx) Rd is never compact. Proposition 16. Let w1 , w2 be Beurling weight functions on Rd ; w1 (x) ! 1 as w2 (x) x ! 1 and 0 < q 1. If w2 w1 and w doesn’t tend to zero in Rd for x ! 1 1 (x) then the embeddings of the spaces lipw1 ( ; 1; q) into lipw2 ( ; 1; q) and lipw1 ( ; 1; q) into Lipw2 ( ; 1; q) are never compact. Proof. Let us assume that w2 w1 . Then there is a constant C > 0 such that w2 (x) C1 w1 (x) . By proposition 10, this implies that lipw1 ( ; 1; q) lipw2 ( ; 1; q) and the unit function i from lipw1 ( ; 1; q) into lipw2 ( ; 1; q) is continw2 (x) doesn’t tend to zero in Rd for x ! 1 and (fn )n2N be uous. Now assume that w 1 (x) a bounded sequence in lipw1 ( ; 1; q) . If any subsequence of (fn )n2N is convergent in lipw2 ( ; 1; q), then this subsequence is also convergent in L (1; q; w2 dx) Rd . However this is not possible by the preceding proposition, since the embedding of the space lipw1 ( ; 1; q) ,! L (1; q; w2 dx) Rd is never compact. One can use the same method to show the non-compactness of the embedding lipw1 ( ; 1; q) ,! Lipw2 ( ; 1; q). Proposition 17. Let w1 , w2 be Beurling weight functions on Rd ; w1 (x) ! 1 as w2 (x) doesn’t tend to zero in Rd x ! 1 and 1 q p < 1. If w2 w1 and w 1 (x) for x ! 1 then the embedding of the space lipw1p ( ; p; q) into L (p; q; w2p dx) Rd is never compact. Proof. Firstly we assume that . Since w2 w1 and 1 q p < 1, there exists a constant C > 0 such that w2 (x) Cw1 (x). This implies that lipw1p ( ; p; q) L (p; q; w2p dx) Rd by proposition 14. Let (tn )n2N be a sequence in Rd such that w2 (x) doesn’t tend to zero in Rd as x ! 1, there exists tn ! 1 as n ! 1. Since w 1 (x) 2 (x) > 0 such that w > 0 for x ! 1. For the proof of the embedding of the w1 (x) space lipw1p ( ; p; q) into L (p; q; w2p dx) Rd is never compact, let us take any …xed f 2 lipw1p ( ; p; q) and de…ne a sequence (fn )n2N , where

fn = w1 1 (tn ) Ltn f

(3.6)

which is bounded in lipw1p ( ; p; q). Indeed, we have kfn k(

;p;q;w1p )

=

w1 1 (tn ) Ltn (f ) (

;p;q;w1p )

w1 1 (tn ) w1 (tn ) kf k(

= w1 1 (tn ) kLtn (f )k(

;p;q;w1p )

= kf k(

;p;q;w1p )

;p;q;w1p )

:

Now we will prove that there wouldn’t exist a subsequence of (fn )n2N which is convergent in L (p; q; w2p dx) Rd . The sequence in the above certainly converges in

820

I·LKER ERYILM AZ

12

the vague topology. Indeed for all k 2 Cc Rd , we get R R f (x) k (x) dx = (3.7) w 1 (tn ) Ltn f (x) k (x) dx Rd n Rd 1 R 1 = d Lt f (x) k (x) dx w1 (tn ) R n 1 kkkp0 ;q0 kfn kp;q;wp 1 w1 (tn ) 1 kkkp0 ;q0 kfn k( ;p;q;wp ) : 1 w1 (tn ) Since right hand side of (3.7) tends to zero for n ! 1, we have R f (x) k (x) dx ! 0: Rd n Finally the inequality R f (x) k (x) dx Rd n

kkkp0 kfn kp

kkkp0 kfn kp;wp 2

kkkp0 kfn kp;q;wp 2

kkk1 kfn k(

L (p; q; w2p dx)

;1;q;w1 ) d

says that, the only possible limit of (fn ) in R is zero. It is known that the function x ! kLx f kp;q;wp is equivalent to the weight function w2 , 2 i.e. there exists c1 (f ) ; c2 (f ) > 0 such that c1 w2 (x) for all x 2 Rd . Therefore (3.8)

is found. Since

2

c2 w2 (x)

= w1 1 (tn ) kLtn (f )kp;q;wp

kfn kp;q;wp 2

w2 (tn ) w1 (tn )

kLx f kp;q;wp

2

c1 w1 1 (tn ) w2 (tn )

> 0 for all tn , by using (3.8) we write kfn kp;q;wp 2

c1 w1 1 (tn ) w2 (tn )

c1 :

This means that it is not possible to …nd norm convergent subsequence of (fn )n2N in L (p; q; w2p dx) Rd . Therefore the embedding lipw1p ( ; p; q) ,! L (p; q; w2p dx) Rd is never compact. Proposition 18. Let w1 , w2 be Beurling weight functions on Rd ; w1 (x) ! 1 as w2 (x) doesn’t tend to zero in Rd x ! 1 and 1 q p < 1. If w2 w1 and w 1 (x) for x ! 1 then the embeddings of the spaces lipw1p ( ; p; q) into lipw2p ( ; p; q) and lipw1p ( ; p; q) into Lipw2p ( ; p; q) are never compact. Proof. Let us assume that w2 w1 . Then there is a constant C > 0 such that w2 (x) Cw1 (x) . By proposition 14, this implies that lipw1p ( ; p; q) lipw2p ( ; p; q) and the unit function i from lipw1p ( ; p; q) into lipw2p ( ; p; q) is contin2 (x) d uous. Now assume that w w1 (x) doesn’t tend to zero in R for x ! 1 and (fn )n2N be a bounded sequence in lipw1p ( ; p; q). If any subsequence of (fn )n2N is convergent in lipw2p ( ; p; q), then this subsequence is also convergent in L (p; q; w2p dx) Rd . However this is not possible by the preceding proposition, since the embedding of the space lipw1p ( ; p; q) ,! L (p; q; w2p dx) Rd is never compact. One can use the same method to show the non-compactness of the embedding lipw1p ( ; p; q) ,! Lipw2p ( ; p; q).

821

W EIGHTED LIPSCHITZ-LORENTZ SPACES

13

References [1] Bloom,W.R., Multipliers of Lipschitz spaces on zero dimensional groups. Math. Z., 176(1981) 485–488. [2] Blozinski,A.P., On a convolution theorem for L(p; q) spaces, Trans.Amer.Math.Soc. 164 (1972), 255-264. [3] Blozinski,A.P., Convolution of L(p; q) functions, Proc. of the .Amer. Math. Soc. 32-1(1972), 237-240. [4] Chen,Y.K. and Lai,H.C., Multipliers of Lorentz spaces, Hokkaido Math. J. 4 (1975), 247-260 [5] Duyar,C. and Gürkanl¬,A.T., Multipliers and Relative completion in weighted Lorentz spaces, Acta Math. Sci. 23 (2003), 467-476. [6] Feichtinger,H.G., Multipliers from L1 (G) to a homogeneous Banach space, J.Math. Anal. Appl. 61 (1977), 341-356. [7] Feichtinger,H.G., Konvolutoren von L1 (G) nach Lipschitz-R¨ aumen, Anz. d. österr. Akad. Wiss., 6 (1979), 148–153. [8] Ferreyra,E.V., Weighted Lorentz norm inequalities for integral operators, Stud.Math. 96 (1990), 125-134. [9] Gürkanl¬, A.T., On the inclusion of some Lorentz spaces, J. Math. Kyoto Univ., 44-2 (2004), 441-450. [10] Gürkanl¬, A.T., Compact Embeddings of the Spaces Apw;! Rd , Taiwanese Journal of Math., 12-7 (2008), 1757-1767. [11] Hewitt,E. and Ross,K.A., Abstract Harmonic Analysis,Vol.1, Springer-Verlag, Berlin, 1963. [12] Hunt,R.A., On L(p; q) spaces, L’enseignement Mathematique, TXII-4 (1966), 249-276. [13] O’Neil,R., Convolution operators and L(p; q) spaces, Duke Math. Journal 30 (1963), 129-142. [14] Onneweer,C.W., Generalized Lipschitz spaces and Herz spaces on certain totally disconnected groups. In Martingale theory in harmonic analysis and Banach spaces, Proc. NSF-CBMS Conf., Cleveland/Ohio 1981, Lect. Notes Math. 939 (1982), 106-121. [15] Quek,T.S. and Yap,L.Y.H., Multipliers from L1 (G) to a Lipschitz space, J.Math. Anal. Appl. 69 (1979), 531-539. [16] Rie¤el,M., Induced Banach representations of Banach Algebras and locally compact groups, J. Func. Anal.1 (1967), 443-491. [17] Saeki,S.and Thome,E.L., Lorentz spaces as L1 -modules and multipliers, Hokkaido Math. J. 23 (1994), 55-92 .

Address: OndokuzMayis University Faculty of Science and Arts Department of Math. 55139 Kurupelit-SAMSUN TURKEY E-mail address : [email protected]

JOURNAL 822 OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.5, 822-829, 2011, COPYRIGHT 2011 EUDOXUS PRESS, LLC

Nikol’ski˘ı-type inequality with doubling weights ∗ †

Feilong Cao1

Shaobo Lin2

1. Institute of Metrology and Computational Sciences, China Jiliang University, Hangzhou 310018, Zhejiang Province, P R China 2. Institute for Information and System Sciences, Xi’an Jiaotong University, Xi’an 710049, Shannxi Province, P R China

Abstract There have been some polynomial inequalities with doubling weights such as Bernstein inequality, Marcinkiewicz inequality, etc. This paper deals with weighted Nikol’ski˘ı-type inequality. By using the modulus of smoothness and the best polynomial approximation as tools, a Nikol’ski˘ı-type inequality with doubling weights for trigonometric polynomials is established. MSC(2000): 41A17 Keywords: Nikol’ski˘ı inequality; Doubling weight; Modulus of smoothness.

1

Introduction

For the sake of brevity, we denote by C a positive constant which may be different in different occasion. Let Pn and Tn denote the set of all algebraic and trigonometric polynomials of degree at most n with real coefficients, respectively. And denote by Lp (a, b) the space of real valued and p-integrable functions on (a, b) endowed with the norms ∥f ∥∞ := ∥f ∥L∞ (a,b) := ess sup |f (x)| x∈(a,b)

and ∥f ∥p := ∥f ∥Lp (a,b) :=

{∫

a

∗

b

}1/p |f (x)| dx < ∞, p

0 < p < ∞.

The research was supported by the National Natural Science Foundation of China(Nos.

60873206, 60874029) and the Natural Science Foundation of Zhejiang Province of China (No. Y7080235). † Corresponding author: Feilong Cao, E-mail: [email protected]

1

CAO, LIN: NIKOLSKII INEQUALITY

823

An integrable periodic weight function W is called doubling weight if W satisfies so-called doubling condition W (2I) ≤ LW (I),

(1)

for all intervals I, where L is a constant independent of I, 2I is the interval twice the length of I and with midpoint at the midpoint of I, and ∫ W (I) = W (u)du. I

In other words, W has the doubling property if the measure of a twice enlarged interval is less than a constant times the measure of the original interval (here by “measure” we mean W (I)). Many of the weights that appear in analysis satisfy this doubling condition; in particular, all weights of the form W (t) = h(t)

k ∏

|t − xj |γj ,

γj > −1,

t ∈ [−π, π),

j=1

where h is a positive measurable function bounded away from zero and infinity (socalled generalized Jacobi weights). The constant L in (1) will be referred to as the doubling constant. The doubling condition is closely related to Muckenhoupt’s Ap conditions, which play an important role in harmonic analysis and weighted inequalities. In fact, all Ap weights satisfy the A∞ condition which is defined as follows: for every α > 0 there is a β > 0 such that W (E) ≥ βW (I) for an interval I and any measurable set E ⊂ I with |E| ≥ α|I|, here, and in what follows, |E| denotes the Lebesgue measure of the set E. It is easy to see that the doubling condition is exactly this A∞ condition stated for interval E’s rather than measurable ones (see the Lemma 2.1 and Lemma 5.1 of [6]). Thus, the A∞ property implies the doubling condition, and they are very close, indeed. However, they are not identical, for there are nonzero doubling weights that vanish on a set of positive measure (see [9], Chap. 1, Sec. 8.8) and this cannot happen for an A∞ weight (see also [3]). Recent years several researches have focused on the inequalities of the algebraic and trigonometric polynomials with doubling weight W (see also [1], [2], [5], [6], [7]). Various important, weighted, algebraic and trigonometric polynomial inequalities such as Bernstein, Marcinkiewicz, Schur, Remez, etc., have been proved for 1 ≤ p ≤ ∞ by Mastroianni and Totik in [6] under minimal assumption on the weights. They stated that in most cases this minimal assumption is the doubling condition. 2

824

CAO, LIN: NIKOLSKII INEQUALITY

Sometimes, however, as for the Remez and Nikol’ski˘ı inequalities, one needs the slightly stronger A∞ condition. The classical Nikol’ski˘ı inequality says that if 1 ≤ p < q ≤ ∞, then there is a constant C such that for all trigonometric polynomials of degree at most n we have 1

∥Tn ∥q ≤ Cn p

− 1q

∥Tn ∥p ,

Tn ∈ Tn .

In [6], Mastroianni and Totik proved Nikol’ski˘ı-type inequality with A∞ weights (see Theorem 5.5 of [6]). In this paper we will establish a full analogue for doubling weights. In fact, we will prove the following theorem. Theorem 1. Let W (x) be a doubling weight. Then for all 1 ≤ p < q < ∞ and Tn ∈ Tn there is a constant C depending only on q such that (∫ π )1 (∫ π )1 q p p 1 1 − p q ≤ Cn p q |Tn (x)| W (x) q dx . |Tn (x)| W (x)dx −π

2

−π

(2)

Proof of Theorem 1

To prove Theorem 1, we need the following lemmas. The first lemma is a Hardy-type inequality, its proof is a simple modification for the proof of Hardy inequality (see P. 24 of [4]). Lemma 1. Let θ > 0, 1 ≤ q < ∞, 0 ≤ a < b ≤ ∞. Then for each positive measurable function ϕ on R+ , there holds ]q ∫ ∫ b[ ∫ b ]q dt 1 b[ θ ds dt θ t ϕ(t) ≤ q . t ϕ(s) s t θ a t a t

(3)

The next lemma was proved by Milman [8]. Lemma 2. Let f ∈ Lp (0, 1) (1 ≤ p < ∞), and f ∗ be the decreasing rearrangement of f . Then ω(f ∗ , t)Lp (0,1) ≤ 7ω(f, t)Lp (0,1) ,

(4)

where ω(f, t)Lp (0,1) is the modulus of smoothness of functions f defined by ω(f, t)p := ω(f, t)Lp (a,b) := sup

00 if there exists K j Lp (Sn−1 ) such that If ∥Iρ (f ) − f ∥p = o(φ(ρ)), then Iρ (f ) = f ; ∥Iρ (f ) − f ∥p = O(φ(ρ)) if and only if f ∈ K;

(i) (ii)

then Iρ is said to be saturated on Lp (Sn−1 ) with order O(φ(ρ)) and K is called its saturation class.

3

Some Lemmas

In this section, we show some lemmas as the preparation for the proof of the main results. Lemma 3.1 For any f ∈ Lp (Sn−1 ) and any positive integers k, r, u, s, we have, (i) ⊕u Jk,s (f ) is a spherical polynomial of degree no more than s(k − 1), which e r (⊕u Jk,s (f )) ∈ Lp (Sn−1 ); implies ∆ (ii)

(iii) (iv)

∥ ⊕u Jk,s (f )∥p ≤ 2u ∥f ∥p ; e r (⊕u Jk,s (f ))∥p ≤ Ck 2r ∥f ∥p ; ∥∆

e r g ∈ Lp (Sn−1 ), then ∥∆ e r (⊕u Jk,s (g))∥p ≤ 2u ∥∆ e r g∥p . If ∆

Proof. (i) Since Dk,s (θ) is an even trigonometric polynomial of degree no more than k(s − 1), then Jk,s (f ) is a spherical polynomial of degree no more than s(k − 1). Thus we can prove by induction that ⊕u Jk,s (f ) is a spherical polynomial of degree no more than s(k − 1). (ii) Using the contraction of translation operator that (see for instance [13]) ∥Sθ (f )∥p ≤ ∥f ∥p ,

0 1. Then there exists positive a ≤ k < mN +1 . Take sv ≥ 0 such that km−v−1 < sv ≤ km−v ,

Proof. First we take m > 1, such that ln integer N , such that mN v = 0, 1, . . . , N as well as

τs v ≤ τj Set sN +1 = 1, bm = σk ≤

( as )p

≤ k

0

k

−p

N ∑

≤ mp

ln m a < 1. Clearly bm → 1, as m → ∞. Then ln m

σs0 + τs0 v+1

(a

v=0

N +1 ∑

(km−v−1 < j ≤ km−v ).

m

sv )

(

σs v −

( m) ln a − ln m pv

v=0

= Cm k −bm p

p

k ∑

(

asv+1 sv

)p

σsv+1



)

N +1 ∑

τsv ≤ Cm k −bm p 

+ τs0 ≤ m

p

N ( ∑ m )−p(v+1) v=0

∑

v=0 km−v−1 0 is a sequence of operators on Lp (Sn−1 ), and there exists series {λρ (j)}∞ j=1 with respect to ρ, such that Iρ (f )(x) =

∞ ∑

λρ (j)Yj (f )(x)

j=0

for every f ∈ Lp (Sn−1 ). If for any j = 0, 1, 2, . . . , there exists φ(ρ) → 0+ (ρ → ρ0 ) such that 1 − λρ (j) = τj ̸= 0, lim ρ→ρ0 φ(ρ) then {Iρ }ρ>0 is saturated on Lp (Sn−1 ) with the order O(φ(ρ)) and the collection of all constants is the invariant class for {Iρ }ρ>0 on Lp (Sn−1 ).  7

WANG, CAO: JACKSON OPERATORS ON THE SPHERE

4

837

Main Results and Their Proof

In this section, we shall state and prove the main results, that is, the lower and upper bounds as well as the saturation order for Boolean sums of Jackson operators on Lp (Sn−1 ). Theorem 4.1 Let 2s ≥ n, and let {⊕r Jk,s }∞ k=1 be the sequence of Boolean sums of Jackson operators defined above. Then for any positive integers k and r as well as e r g ∈ Lp (Sn−1 ), we have sufficiently smoothing g ∈ Lp (Sn−1 ), 1 ≤ p ≤ ∞ such that ∆ e r g∥p , ∥ ⊕r Jk,s (g) − g∥p ≤ C1 k −2r ∥∆

(4.13)

∥ ⊕r Jk,s (f ) − f ∥p ≤ C2 ω 2r (f, k −1 )p ,

(4.14)

therefore, for any f ∈ Lp (Sn−1 ), we have

where C1 and C2 are constants independent of f and k. Proof. By Definition 2.1, we have ⊕r Jk,s (g)(x) − g(x) = −(I − Jk,s )r (g)(x). Now we prove (4.13) by induction. For r = 1, Sθ (g)(x) − g(x) =

∫

θ

sin−2λ τ

0

∫

0

τ

e sin2λ uSu (∆g)(x)dudτ

(see [10]) implies (explained below)

∫ π

2λ

∥Jk,s (g) − g∥p = Dk,s (θ) (Sθ (g)(·) − g(·)) sin θdθ

0

∫

π

∫

p

θ

∫

τ



e ≤ Dk,s (θ) sin2λ θ sin−2λ τ sin2λ u Su (∆g)

dudτ dθ p 0 0 0 ) } (∫ π { ∫ θ ∫ τ −2 −2λ −2λ 2 2λ e p ≤ sup θ sin τ sin udu dτ θ Dk,s (θ) sin θdθ ∥∆g∥ θ>0 −2

≤ Ck

0

0

0

e p, ∥∆g∥

(4.15)

where the Minkowski inequality is used in the first inequality, the second one by (3.10) and the third one is deduced from Lemma 3.2. Assume that for any fixed positive integer u,

Then

e u g∥p . ∥ ⊕u Jk,s (g) − g∥p ≤ Ck −2u ∥∆

e u Jk,s (g) − g)∥p ∥ ⊕u+1 Jk,s (g) − g∥p = ∥(Jk,s − I)(⊕u Jk,s (g) − g)∥p ≤ Ck −2 ∥∆(⊕ e − ∆g∥ e p ≤ Ck −2u−2 ∥∆ e u+1 g∥p , = Ck −2 ∥ ⊕u Jk,s (∆g) 8

838

WANG, CAO: JACKSON OPERATORS ON THE SPHERE

where the first inequality is by (4.15), the second one by (3.11), the last by induction assumption. Therefore, (4.13) holds. Using (2.1) and noticing that ⊕u Jk,s is a linear operator, we obtain (4.14). This completes the proof of the theorem.  Next, we establish an inverse inequality of strong type for ⊕r Jk,s on Lp (Sn−1 ). Theorem 4.2 For positive r ≥ 1 and f ∈ Lp (Sn−1 ), 1 ≤ p ≤ ∞, there exists a constant C independent of f and k such that ω 2r (f, k −1 )p ≤ C max ∥ ⊕r Jv,s (f ) − f ∥p .

(4.16)

v≥k

Proof. We first establish a Steckin-Marchaud type inequality, that is, for f ∈ Lp (Sn−1 ), ω 2r (f, k −1 )p ≤ Cm k −2bm r

k ∑

v 2bm r−1 ∥ ⊕r Jv,s (f ) − f ∥p ,

v=1

where 0 < bm < 1, bm → 1, as m → ∞. Set e r (⊕r Jv,s (f )) ∥p , τv = ∥ ⊕r Jv,s (f ) − f ∥p , σv = v −2r ∥∆

v ≥ 1.

Using Lemma 3.1, we have

e r (⊕r Jk,s (⊕r Jv,s (f ))) ∥p + k −2r ∥∆ e r (⊕r Jk,s (⊕r Jv,s (f ) − f )) ∥p σk ≤ k −2r ∥∆ ( v )2r ( ) e r (⊕r Jv,s (f )) ∥p + C ∥ ⊕r Jv,s (f ) − f ∥p ≤ 2r v −2r ∥∆ k ( √ )2r 2v = σv + Cτv . k ∑ By Lemma 3.3, we have σk ≤ Cm k −2bm r kv=1 v 2bm r−1 τv for some large enough m. That is, e r (⊕u Jk,s (f )) ∥p ≤ Cm k −2bm r k −2r ∥∆

k ∑

v 2bm r−1 ∥ ⊕r Jv,s (f ) − f ∥p .

v=1

k ≤ k0 ≤ k, such that 2 k ∥ ⊕r Jk0 ,s (f ) − f ∥p ≤ ∥ ⊕r Jv,s (f ) − f ∥p , ≤ v ≤ k. 2

For k ≥ 1, there exists a positive integer k0 ,

Thus e k −2r )p ≤ ∥ ⊕r Jk ,s (f ) − f ∥p + k −2r ∥∆ e r (⊕r Jk ,s (f )) ∥p K2r (f, ∆, 0 0 ∑ 2r−1 r 2r −2r v ∥ ⊕ Jv,s (f ) − f ∥p ≤ 2 k k ≤v≤k 2

+Cm k ≤ Cm k

−2bm r

k ∑

v 2bm r−1 ∥ ⊕r Jv,s (f ) − f ∥p

v=1 k ∑ −2bm r 2bm r−1

v

v=1

9

∥ ⊕r Jv,s (f ) − f ∥p .

WANG, CAO: JACKSON OPERATORS ON THE SPHERE

839

From (2.1) it follows that 2r

ω (f, k

−1

) p ≤ Cm k

−2bm r

k ∑

v 2bm r−1 ∥ ⊕u Jv,s (f ) − f ∥p .

v=1

To finish our proof, we need the following inequalities. 1 max v 2r ∥ ⊕r Jv,s (f ) − f ∥p k 2r 1≤v≤k 1 2r+ 14 ∥ ⊕r Jv,s (f ) − f ∥p . 1 max v 2r+ 4 1≤v≤k k

ω 2r (f, k −1 )p ≈ ≈

(4.17)

In the first place, we prove the former inequality of (4.17) (explained below). 2r

ω (f, k

−1

)p ≤ C1 k

−2bm r

k ∑

v 2bm r−1 ∥ ⊕r Jv,s (f ) − f ∥p

v=1

≤ C1 ≤ ≤

(

k

−2bm r

k ∑

v

−2(1−bm )r−1

)

v=1 C2 k max v 2r ∥ ⊕r Jv,s (f ) − 1≤v≤k C3 k −2r max v 2r ω 2r (f, v −1 )p 1≤v≤k −2r

≤ C4

(

k

−2r

max v

2r

1≤v≤k

max v 2r ∥ ⊕r Jv,s (f ) − f ∥p

1≤v≤k

f ∥p

( )2r ) k ω 2r (f, k −1 )p ≤ C4 ω 2r (f, k −1 )p , v

where the fourth inequality is deduced by Theorem 4.1 and the fifth is by (2.2). Thus 1 ω 2r (f, k −1 )p ≈ 2r max v 2r ∥ ⊕r Jv,s (f ) − f ∥p . k 1≤v≤k In the same way, we have ω 2r (f, k −1 )p ≤ C1 k −2bm r

k ∑

v 2bm r−1 ∥ ⊕r Jv,s (f ) − f ∥p

v=1

≤ C1

(

k −2bm r

k ∑

v

−2(1−bm )r− 41 −1

v=1

≤ C5 k

−2r− 41

max v

2r+ 14

1≤v≤k

)

1

max v 2r+ 4 ∥ ⊕r Jv,s (f ) − f ∥p

1≤v≤k

∥ ⊕r Jv,s (f ) − f ∥p

≤ C6 k −2r− 4 max v 2r+ 4 ω 2r (f, v −1 )p 1≤v≤k ( ( )2r ) 1 1 k ≤ C6 k −2r− 4 max v 2r+ 4 ω 2r (f, k −1 )p ≤ C7 ω 2r (f, k −1 )p , 1≤v≤k v 1

1

that is, ω 2r (f, k −1 )p ≈

1 k

2r+ 14

1

max v 2r+ 4 ∥ ⊕r Jv,s (f ) − f ∥p . v≥k

10

840

WANG, CAO: JACKSON OPERATORS ON THE SPHERE

Therefore ω 2r (f, k −1 )p ≈ ≈

1 max v 2r ∥ ⊕r Jv,s (f ) − f ∥p k 2r v≥k 1 2r+ 14 ∥ ⊕r Jv,s (f ) − f ∥p . 1 max v 2r+ 4 v≥k k

(4.18)

Now we can complete the proof of (4.16). Clearly, there exists 1 ≤ k1 ≤ k such that 2r+ 41

k1

1

∥ ⊕r Jk1 ,s (f ) − f ∥p = max v 2r+ 4 ∥ ⊕r Jv,s (f ) − f ∥p . 1≤v≤k

Then it is deduced from (4.18) that 1 max v 2r ∥ ⊕r Jv,s (f ) − f ∥p k 2r 1≤v≤k 1 1 ≤ C8 max v 2r+ 4 ∥ ⊕r Jv,s (f ) − f ∥p 2r+ 14 1≤v≤k k

k −2r k12r ∥ ⊕r Jk1 ,s (f ) − f ∥p ≤

2r+ 14

= C8 k −2r− 4 k1 1

∥ ⊕r Jk1 ,s (f ) − f ∥p .

This implies k1 ≈ k. Applying (4.18) again implies ω 2r (f, k −1 )p ≤ C5 = C5

1 k

2r+ 14

1 k

2r+ 14

1

max v 2r+ 4 ∥ ⊕r Jv,s (f ) − f ∥p

1≤v≤k

2r+ 14

(k1

∥ ⊕r Jk1 ,s (f ) − f ∥p ) ≤ C5 max ∥ ⊕r Jv,s (f ) − f ∥p . k1 ≤v≤k

Noticing that k1 ≈ k, we may rewrite the above inequality as ω 2r (f, k −1 )p ≤ C max ∥ ⊕r Jv,s (f ) − f ∥p . v≥k

This completes the proof of Theorem 4.2.



p n−1 ) with order k −2r and the collecTheorem 4.3 {⊕r Jk,s }∞ k=1 are saturated on L (S tion of constants is their invariant class.

Proof. We first prove for j = 0, 1, 2, . . . , 1 − 1 ξk (j) k→∞ 1 − 1 ξk (1) lim

=

j(j + 2λ) . 2λ + 1

(4.19)

In fact, for any 0 < δ < π, it follows from (3.12) that ∫ π ∫ π ( )3 θ Dk,s (θ) sin2λ θ dθ ≤ Dk,s (θ) sin2λ θ dθ δ δ δ ∫ π −3 ≤ δ θ 3 Dk,s (θ) sin2λ θ dθ ≤ Cδ,s k −3 . 0

For v = 1, 2, . . . , we have, using (3.12) again, ) ( ∫ π ∫ π θ Gλ1 (cos θ) 2λ 1 sin θ dθ ≈ Dk,s (θ) sin2 sin2λ θ dθ 1 − ξk (1) = Dk,s (θ) 1 − λ 2 G1 (1) 0 0 ≈ k −2 .

11

(4.20)

WANG, CAO: JACKSON OPERATORS ON THE SPHERE

841

We deduce from (2.9) that for any ϵ > 0, there exists δ > 0, for 0 < θ < δ, such that ( ) 1 − Pjn (cos θ) − j(j + 2λ) (1 − P1n (cos θ)) ≤ ϵ (1 − P1n (cos θ)) . 2λ + 1 Then it follows that ( ) ( ) 1 − 1 ξk (j) − j(j + 2λ) 1 − 1 ξk (1) 2λ + 1 ∫ π ( ) = Dk,s (θ) 1 − Pjn (cos θ) sin2λ θ dθ 0 ∫ π j(j + 2λ) n 2λ − Dk,s (θ) (1 − P1 (cos θ)) sin θ dθ 2λ + 1 ∫ π0 ( ) ( ) ( ) j(j + 2λ) n n 2λ = Dk,s (θ) 1 − Pj (cos θ) − 1 − P1 (cos θ) sin θ dθ 2λ + 1 0 ( ) ∫ δ ∫ π j(j + 2λ) n 2λ 2λ ≤ Dk,s (θ) ϵ (1 − P1 (cos θ)) sin θ dθ + 2 Dk,s (θ) sin θ 1 + dθ 2λ + 1 0 δ ≤ Cϵ k −2 + Cδ,s k −3 . So, (4.19) holds. By Lemma 3.4, ⊕ Jk,s (f ) − f = r

∞ ∑

(1 −r ξk,s (j)) Yj (f )

j=0

and for j = 1, 2, . . . , (∫ r 1 − ξk,s (j) =

0

γ

Dk,s (θ) 1 − (

Pjn (cos θ)

)

2λ

sin

θ dθ

)r

( )r = 1 − 1 ξk,s (j) .

Combining with (4.19) and (4.20), we have, 1 − r ξk,s (j) = lim k→∞ 1 − r ξk,s (1)

(

j(j + 2λ) 2λ

)r

̸= 0

and 1 − r ξk,s (1) ≈ k −2r . Using of Lemma 3.5, we finish the proof of Theorem 4.3.  We obtain the following corollary from Theorem 4.1, Theorem 4.2 and Theorem 4.3. Corollary 4.1 For positive integers r and s, 2s ≥ n, 0 < α ≤ 2r, f ∈ Lp (Sn−1 ), 1 ≤ p ≤ ∞, and the sequence of Boolean sums of Jackson operators {⊕r Jk,s }∞ k=1 given by (2.4), the following statements are equivalent. (i) ∥ ⊕r Jk,s (f ) − f ∥p = O(k −α ) (k → ∞); (ii) ω 2r (f, δ)p = O(δ α ) (δ → 0). 

References [1] H. Berens, P. L. Butzer, S. Pawelke, Limitierungsverfahren von reihen mehrdimensionaler kugelfunktionen und deren saturationsverhalten, Publ. Res. Inst. Math. Sci. Ser. A, 4 (2) (1968), 201-268. 12

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WANG, CAO: JACKSON OPERATORS ON THE SPHERE

[2] P. L. Butzer, H. Johnen, Lipschitz spaces on compact manifolds, J. Funct. Anal., 7 (1971), 242-266. [3] P. L. Butzer, R. J. Nessel, W. Trebels, On summation processes of Fourier expansions in Banach spaces II. Saturation theorems, Tˆohoku Math. J., 24 (4) (1972), 551-569. [4] Z. Ditzian, K. G. Ivanov, Strong converse inequalities, J. D’Analyse Math´ematique, 61 (1993), 61-111. [5] Z. Ditzian, Jackson-type inequality on the sphere, Acta Math. Hungar., 102 (1-2) (2004), 1-35. [6] L. Li, R. Yang, Approximation by Jackson polynomials on the sphere, Journal of Beijing Normal University (Natural Science), 27 (1) (1991), 1-12 (in Chinese). [7] I. P. Lizorkin, S. M. Nikol’skiˇı, A theorem concerning approximation on the sphere, Anal. Math., 9 (1983), 207-221. [8] C. M¨ uller, Spherical harmonics, Lecture Notes in Mathematics, 17, Springer, Berlin, 1966. [9] S. M. Nikol’skiˇı, I. P. Lizorkin, Approximation by spherical functions, Proc. Steklov Inst. Math., 173 (1987), 195-203. ¨ [10] S. Pawelke, Uber die approximationsordnung bei kugelfunktionen und algebraischen polynomen, Tˆohoku Math. J., 24 (3) (1972), 473-486. [11] K. V. Rustamov, On equivalence of different moduli of smoothness on the sphere, Proc. Stekelov Inst. Math., 204 (3) (1994), 235-260. [12] E. M. Stein, G. Weiss, An introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, New Jersey, 1971. [13] K. Wang, L. Li, Harmonic analysis and approximation on the unit sphere, Science Press, Beijing, 2006. [14] E. van Wickeren, Steckin-Mauchaud type inequalities in connection with Bernstein polynomials, Constr. Approx., 2 (1986), 331-337.

13

JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.5, 843-849, 2011, COPYRIGHT 2011 EUDOXUS PRESS, 843 LLC

On The Absolute Riesz Summability W. T. Sulaiman Department of Computer Engineering College of Engineering University of Mosul, Iraq. [email protected]

Abstract. A new result concerning absolute Riesz summability factors of infinite series improved and generalized a known reslt due to Leindler [5] has been obtained. 2000 Mathematics Subject Classification : 40D15, 40F05. Key words : Absolute summability, Riesz summability, infinite series.

1. Introduction Let T be a lower triangular matrix, (s n ) a sequence, then n

Tn := ∑ t nv s v .

A series

∑a

(1.1)

v =0

n

is said to be summable T k , k ≥ 1, if ∞

∑n

k −1

n =1

∆Tn −1

k

< ∞.

(1.2)

Given any lower triangular matrix T one can associate the matrices T and Tˆ , with entries defined by n

t nv = ∑ t ni , n, i = 0,1, 2...,

tˆnv = t nv − t n −1,v

i =v

respectively. With s n = ∑i =0 ai λi , n

n

t n = ∑ t nv sv = v =0

Yn := t n − t n −1 =

n

v

v =0

i =0

∑ tnv ∑ ai λi =

n

n −1

i =0

i =0

n

n

i =0

v =i n

∑ ai λi ∑ tnv =

∑ t ni ai λi − ∑ t n−1,i ai λi =

∑ tˆ i =0

ni

n

∑t i =0

ni

a i λi ,

a i λi .

(1.3)

as t n −1, n = 0.

(1.4)

We also have tˆnn = t nn .

( pn )

is assumed to be a positive sequence of numbers such that Pn = p0 + p1 + ... + p n → ∞, as n → ∞,

. A positive sequence (a n ) is said to be almost increasing if there exists a positive increasing sequence (bn ) and two positive constants A and B such that (see[1]), A bn ≤ a n ≤ B bn . It is easy to verify that a sequence (a n ) is almost increasing if and only if it is quasi increasing, that is, if there exists a constant K = K (a n ) ≥ 1 such that

844

SULAIMAN: RIESZ SUMMABILITY

Ka n ≥ a m ≥ 0 ,

(1.5)

holds for all n ≥ m . The series

∑a

n

is said to be summable N , p n k , k ≥ 1, if k −1

⎛ Pn ⎞ ⎜⎜ ⎟⎟ ∑ n =1 ⎝ p n ⎠ ∞

t n − t n −1

k

< ∞,

(1.6)

.

(1.7)

where

tn =

1 Pn

n

∑p s

v v

v =0

Here we give the following definition A series

∑a

n

is said to be T , pn k , T = (tnv ), k ≥ 1, if ⎛ Pn ⎞ ⎜⎜ ⎟⎟ ∑ n =1 ⎝ p n ⎠ ∞

k −1

Yn − Yn −1 < ∞, k

(1.8)

where n

Yn = ∑ tˆnv a v .

(1.9)

v =0

In the special case when t nv = p v / Pn , T , pn k summability reduces to

N , pn k summability. Leindler [5] established the following result Theorem 1.1. Let λ n → 0. Suppose there exists a positive quasi increasing sequence ( X n ) such that ∞

∑X n =1

n

∆λ n < ∞ ,

(1.10)

∞

1 k t n = Ο ( X m ), n =1 n pn k t n = Ο ( X m ), Pn

X m* := ∑ m

∑ n =1

m

1

∑n n =1

(1.11) (1.12)

λn < ∞ ,

(1.13)

∆ ∆λ n < ∞ ,

(1.14)

and ∞

∑ nX then the series

∑a λ n

n =1

n

* n

is summable N , p n , k ≥ 1. k

2. Main Result. We prove the following

2

SULAIMAN: RIESZ SUMMABILITY

845

Theorem 2.1. Let λ n → 0. Suppose there exists a positive quasi increasing sequence ( X n ) such that ∞

1

∑n X n =1

λn < ∞ ,

n

⎛ Pn ⎞ ⎜⎜ ⎟⎟ ∑ n =v +1 ⎝ pn ⎠ ∞

k −1 k

tˆnv < ∞ ,

n −1

(2.2)

= Ο ( t nn ),

∑ ∆ tˆ

v nv

v =1

(2.1)

(2.3)

t nn = Ο ( p n / Pn ), ∞

= Ο ( t vv ),

∑ ∆ tˆ

v nv

n = v +1

(2.4) (2.5)

k

1 tn = Ο ( X m ), ∑ k −1 n =1 n X n m

m

∑ n =1

(2.6)

k

pn t n = Ο ( X m ), Pn X nk −1

⎛ Pn ⎞ ⎜⎜ ⎟⎟ ∑ n = v +1 ⎝ p n ⎠ ∞

k −1

t nn

k −1

(2.7)

∆ v tˆnv = Ο ( t vv ),

(2.8) ∞

∑X n =1

n

∆λ n < ∞ ,

(2.9)

and ∞

∑n X n =1

n

then the series

∆ ∆λ n < ∞ ,

∑a λ n

n

(2.10)

is summable T , p n k , k ≥ 1.

Remark 1. In comparing theorems (1.1) and (2.1), although condition (2.1) is stronger than condition (1.13), two improvements are achieved 1. Conditions (2.6) and (2.7) are weaker than conditions (1.11) and (1.12) respectively (Lemma 2.2). 2. Summability T , pn k is more general than summability N , pn k . Remark 2. Via the proof of theorem 1.1, there are powers lost through k −1 estimations, as an example λ n has been estimated as Ο(1) since λ n = Ο (1),

while theorem 2.1 with its conditions has no such cases.

Lemma 2.2. Conditions (2.6) and (2.7) are weaker than conditions (1.11) and (1.12) respectively. Proof. If condition (1.12) is satisfied, then

3

846

SULAIMAN: RIESZ SUMMABILITY

k

⎛ 1 ⎞ m pn k pn t n tn = Ο (X m ) , = Ο ⎜⎜ k −1 ⎟⎟ ∑ ∑ k −1 P X = 1 n =1 Pn X n n n ⎝ 1 ⎠ while if condition (2.7) is holds, then k m m pn k pn t n tn = ∑ X nk −1 ∑ k −1 P P X n =1 n =1 n n n m −1 n m p p k k = ∑ (∆X nk −1 )∑ v tv + X mk −1 ∑ n t n n =1 v =1 Pv n =1 Pn m

m −1

= Ο (1) ∑ X n ∆X nk −1 + Ο ( X m ) X mk −1 n =1

m −1

(

)

( )

= Ο ( X m −1 ) ∑ X nk+−11 − X nk + Ο X mk n =1 k −1 m

( ) + Ο (X ) = Ο (X ) ≠ Ο ( X ). = Ο X m −1 X k m

The rest follows by putting

k m

m

p n = 1, Pn = n .

Lemma 2.3. The conditions

(i) λ n → 0, (ii) ( X n ) is positive quasi increasing sequence, in addition (2.9) and (2.10) implies the following X n λn < ∞ ,

(2.11)

n X n ∆λ n < ∞ .

(2.12)

For the proof see [5].

Proof of Theorem 2.1. We have n n ⎛ tˆ λ ⎞ Tn := ∑ tˆnv av λv = ∑ vav ⎜⎜ nv v ⎟⎟ v =1 v =1 ⎝ v ⎠ n −1 ⎞t λ ⎛ v ⎞ ⎛ tˆ λ ⎞ ⎛ n = ∑ ⎜ ∑ ra r ⎟ ∆ v ⎜⎜ nv v ⎟⎟ + ⎜ ∑ vav ⎟ nn n v =1 ⎝ r =1 ⎠ ⎝ v ⎠ ⎝ v =1 ⎠ n n −1 ⎛ tˆ λ 1 (∆ v tˆnv λv + tˆn,v+1∆λv )⎞⎟⎟ + n + 1 t n t nn λn = ∑ (v + 1) t v ⎜⎜ nv v + n v =1 ⎠ ⎝ v(v + 1) v + 1 n −1 n −1 n −1 1 n +1 = ∑ t v tˆnv λv + ∑ t v ∆tˆnv λv + ∑ t v tˆn ,v +1 ∆λv + t n t nn λ n n v =1 v =1 v =1 v = Tn1 + Tn 2 + Tn 3 + Tn 4 .

In order to prove the theorem, by Minkowski's inequality, it is sufficient to show that

4

SULAIMAN: RIESZ SUMMABILITY

∞

∑ (P n =1

/ pn )

n

k −1

Tnr

847

< ∞ , r =1,2,3,4.

k

Making use of Holder's inequality, ⎛ Pn ⎞ ⎜⎜ ⎟⎟ ∑ n =1 ⎝ p n ⎠ m

k −1

Tn1

k

⎛P ⎞ = ∑ ⎜⎜ n ⎟⎟ n =1 ⎝ p n ⎠

k −1

⎛P ⎞ ≤ ∑ ⎜⎜ n ⎟⎟ n =1 ⎝ p n ⎠

k −1

m

m

n −1

1 ˆ t v t nv λv ∑ v =1 v

k 1 ⎛ n −1 1 ⎞ k ˆ t t λ X v λv ⎟ ∑ nv v ⎜∑ k −1 v v =1 vX v ⎝ v =1 v ⎠ n −1

m

1 k = Ο (1) ∑ k −1 t v λv v =1 vX v tv

m

= Ο (1) ∑ v =1

k

⎛ Pn ⎞ ⎜⎜ ⎟⎟ ∑ n = v +1 ⎝ p n ⎠ m

k −1

k −1

tˆnv

k

k

λv

vX vk −1

k m ⎛ v tr k ⎞ tv ⎟ ⎜ = Ο (1) ∑ ∑ k −1 ∆ λv + Ο (1) λ m ∑ k −1 ⎟ ⎜ v =1 r =1 rX r v =1 vX v ⎠ ⎝ m −1

m

= Ο (1) ∑ X v ∆λv + Ο (1) X m λ m v =1

= Ο (1) , in view of (2.1), (2.2), (2.6), (2.9) and (2.11).

⎛ Pn ⎞ ⎜⎜ ⎟⎟ ∑ n =1 ⎝ p n ⎠ m

k −1

Tn 2

k

⎛P ⎞ = ∑ ⎜⎜ n ⎟⎟ n =1 ⎝ p n ⎠ m

⎛P ⎞ ≤ ∑ ⎜⎜ n ⎟⎟ n =1 ⎝ p n ⎠ m

k −1

∑

t v ∆ v tˆnv λv

v =1

k −1

n −1

∑ v =1

⎛P ⎞ = Ο (1) ∑ ⎜⎜ n ⎟⎟ n =1 ⎝ p n ⎠ m

m

k

n −1

t v ∆ v tˆnv λv k

k −1

= Ο (1) ∑ t v λv k

k

v =1 m

k −1

t nn

n −1

∑

t v ∆ v tˆnv λv k

v =1

⎛ Pn ⎜⎜ ∑ n = v +1 ⎝ p n m

⎛ n −1 ˆ ⎞ ⎜ ∑ ∆ v t nv ⎟ ⎝ v =1 ⎠

k

⎞ ⎟⎟ ⎠

k −1

k

k −1

t nn

k −1

∆ v tˆnv

= Ο (1) ∑ t v λv t vv k

k

v =1 m

= Ο (1) ∑ v =1 m

= Ο (1) ∑ v =1

k

pv t v λv ( X v λv Pv X vk −1

)

k −1

k

pv t v λv Pv X vk −1

k m ⎛ v p tr k ⎞ pv t v r ⎟ ⎜ ∆ λ + Ο (1) λm ∑ = Ο (1) ∑ ∑ k −1 k −1 ⎟ v ⎜ v =1 r =1 Pr X r v =1 Pv X v ⎠ ⎝ m −1

5

848

SULAIMAN: RIESZ SUMMABILITY

m

= Ο (1) ∑ X v ∆λv + Ο (1) X m λ m v =1

= Ο (1) , in view of (2.3), (2.7), (2.8), (2.9), and (2.11),

⎛ Pn ⎞ ⎜⎜ ⎟⎟ ∑ n =1 ⎝ p n ⎠ m

k −1

Tn 3

k

⎛P ⎞ = ∑ ⎜⎜ n ⎟⎟ n =1 ⎝ p n ⎠

k −1

⎛P ⎞ ≤ ∑ ⎜⎜ n ⎟⎟ n =1 ⎝ p n ⎠

k −1

m

m

= Ο (1) ∑ v =1

∑ t tˆ

v n ,v +1

v =1

n −1

∑

∆λv k

k t v tˆn ,v +1k

⎛ n −1 ⎞ ∆λv ⎜ ∑ X v ∆λv ⎟ ⎝ v =1 ⎠

X vk −1

v =1

vX vk −1

v ∆λv

m −1

= Ο (1) ∑ ∆(v ∆λv v =1

)∑ v

r =1

tr

k

rX rk −1

(

m

k −1

k

tv

m

k

n −1

+ Ο (1) m ∆λ m

m

tv

∑ vX v =1

k k −1 v

)

= Ο (1) ∑ ∆λv + (v + 1) ∆ ∆λv X v + Ο (1) m ∆λ m X m v =1 m

m

v =1

v =1

= Ο (1) ∑ X v ∆λv + Ο (1) ∑ v X v ∆ ∆λv + Ο (1) m ∆λ m X m = Ο (1) , in view of (2.9), (2.10), and (2.12).

⎛ Pn ⎞ ⎜⎜ ⎟⎟ ∑ n =1 ⎝ p n ⎠ m

k −1

Tn 5

k

⎛P ⎞ = ∑ ⎜⎜ n ⎟⎟ n =1 ⎝ p n ⎠ m

k −1

n +1 t n t nn λ n n

⎛P ⎞ ≤ Ο (1) ∑ ⎜⎜ n ⎟⎟ n =1 ⎝ p n ⎠ m

= Ο (1)

m

∑t n =1

k

k −1

t n t nn λ n k

k

k

t n λ n , in view of (2.4) k

nn

k

= Ο (1) , as in the case of Tn 2 . This completes the proof of the theorem .

3. Application The following result is an improvement of theorem 1.1. Corollary 3.1. Let λ n → 0. Suppose there exists a positive quasi increasing sequence ( X n ) such that such with (λ n ) they are satisfying (2.1), (2.6)-(2.7) and

(2.9)-(2.10), then the series

∑a λ n

n

is summable N , p n k , k ≥ 1.

6

SULAIMAN: RIESZ SUMMABILITY

Proof. The proof follows from theorem 2.1 by putting T ≡ ( N , p n ) , that is by putting p p P p p t nv = v , tˆnv = n v −1 , ∆ v tˆnv = − n v . Pn Pn Pn −1 Pn Pn −1

References [1] L. S. Aljancic and D. Arandelovic, 0-regularly varying functions, Publ. Inst. Math., 22 (1977), 5-22. [2] R. P. Boas Jr., Quasi positive sequence and trigonometric series, Proc. London Math. Soc., 14 (1965), 38-46. [3] H. Bor, A note on two sunnability methods, Proc. Amer. Math. Soc., 98 (1986), 81-84. [4] H. Bor, An application of almost increasing and δ − quasi-monotone sequences, J. Inequal. Pure and Appl. Math. 1 (2) (2000), Art. 18. [5] L. Leindler, On the absolute Riesz summability factors, J. Inequal. Pure and Appl. Math., 5 (2) (2004), Art. 29.

7

849

JOURNAL 850 OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.5, 850-856, 2011, COPYRIGHT 2011 EUDOXUS PRESS, LLC

On Necessary and Sufficient Conditions for Inclusion Relations for Absolute Summability W. T. Sulaiman Department of Computer Engineering, College of Engineering, University of Mosul, Iraq.

Abstract. We obtain the necessary and sufficient conditions for U k summability of ∑ a n to imply T s summability of

∑a λ n

n

, 1 < k ≤ s < ∞ . Other results are also obtained.

2000 Mathematics subject Classification : 40F05, 40D25, 40G99. Key words : Absolute summability, weighted mean matrix, Holder's inequality.

1. Introduction Let T be a lower triangular matrix, (s n ) be the sequence of the n-th partial sums

of the series ∑ a n , then, we define

n

Tn := ∑ t nv s v .

(1) A series

∑a

v =0

n

is said to be summable T k , k ≥ 1, if ∞

∑n

(2)

k −1

n =1

∆Tn −1

k

< ∞.

Given any lower triangular matrix T one can associate the matrices T and Tˆ , with entries defined by n

t nv = ∑ t ni , n, i = 0,1, 2...,

tˆnv = t nv − t n −1,v

i =v

respectively. With s n = ∑i =0 ai λi , we define and derive n

n

(3)

v =0

(4)

n

v

v =0

i =0

n

n

n

i =0

v =i n

i =0

t n = ∑ t nv sv = ∑ t nv ∑ ai λi = ∑ ai λi ∑ t nv = ∑ t ni ai λi . Yn := t n − t n −1 =

n −1

n

∑t i =0

ni

a i λi − ∑ t n −1,i a i λi = i =0

∑ tˆ i =0

ni

a i λi ,

as t n −1, n = 0.

We call T a triangle if T is lower triangular and t nn ≠ 0 for all n. A triangle A is called factorable if its nonzero entries a mn can be written in the form bm c n for each m and n . We also assume that U = (u ij ) is a triangle .

SULAIMAN: ABSOLUTE SUMMABILITY

851

If we assumed that T and U to be factorable, then t nv = a n bv and u nv = c n d v , and therefore we have n

n

r =v

r =v n −1

t nv = ∑ t nr = an ∑ br := an Bv ,

(5)

tˆnv = t nv − t n −1,v = a n Bv − a n −1 ∑ br = a n Bv − a n −1 Bv + a n −1bn −1

(6)

r =v

= a n Bv − a n −1 Bv = − ∆a n −1 Bv , as a n −1bn −1 = t n −1,n −1 = 0 . Also, we have dv , Bn = bn and Dn = d n . Dv Dv +1 are assumed to be positive sequences of numbers such that Pn = p 0 + p1 + ... + p n → ∞, as n → ∞,

∆Bv = bv , ∆Dv = d v , ∆Dv−1 = −

( p n ), (q n )

Qn = q 0 + q1 + ... + q n → ∞, The series

∑a

n

as n → ∞.

is said to be summable R, p n k , k ≥ 1, if ∞

∑n

k −1

n =1

∆z n −1

k

< ∞,

where n

z n = ∑ pi si . i =0

In their paper, Rhoades and Savas [2] have used the notation N , p n k to denote

the R, p n

summability. They proved the following result .

k

Theorem 1.1. Let

( pn )

and (q n ) be positive sequences, 1 < k ≤ s < ∞ . Then N , pn

k

⇒ N , qn

s

iff n1 / k −1 / s

(7)

q n Pn = Ο (1) , p n Qn 1/ k*

k ⎛ m q v Pv 1 ⎞⎟ ⎜ (8) = Ο (1) , Qv − ⎜⎜ ∑ pv v ⎟⎟ v =1 ⎝ ⎠ * where k denotes the conjugate index of k, i.e., 1 / k + 1 / k * = 1. s ⎛ ∞ ⎛ ⎞ ⎜ ⎜ n1−1 / s q n ⎞⎟ ⎟ ⎜ ⎜ n∑ Qn Qn −1 ⎟⎠ ⎟ ⎝ =m ⎝ ⎠

1/ s

*

The aim of this paper is to present the following generalization

2. Results The following is our main result

2

852

SULAIMAN: ABSOLUTE SUMMABILITY

Theorem 2.1. Let 1 < k ≤ s < ∞ , (λ n ) be a sequence of constants. Let T and U be factorable triangles with bounded entries, t nv = a n bv , u nv = c n d v . Then the implication ∑ a n summable U k ⇒ ∑ a n λn summable T s

holds iff the following holds ⎛ ∆a b λ ⎞ n1 / k −1 / s ⎜⎜ n −1 n n ⎟⎟ = Ο (1) , ⎝ ∆c n −1 d n ⎠ k* ⎞ 1/ s ⎛ m s ⎞ ⎛ ⎞ d B b 1 ⎛ ∞ 1−1 / s ⎟ ⎜ v ⎜⎜ − (10) ∆a n −1 ⎟ ⎜ ∑ Bv λv + v ∆λv + v λv +1 ⎟⎟ ⎟ ⎜∑ n Dv +1 Dv +1 ⎝ n=m ⎠ ⎜ v =1 Dv ⎝ Dv Dv +1 ⎠ ⎟⎠ ⎝ = Ο (1) . Proof. Let (u n ) and (t n ) denote the nth terms of the U and T transforms of

(9)

n

∑ a j and j =0

n

∑a λ j =0

j

j

respectively . Then X n := u n − u n −1 =

n

∑ uˆ

v =0 n

n

nv

a v = − ∆c n −1 ∑ Dv a v , v =0 n

Yn := y n − y n −1 = ∑ tˆnv a v λv = − ∆a n −1 ∑ Bv a v λv . v =0

v =0

By Abel’s transformation, we have n

Yn = − ∆a n −1 ∑ Dv a v v =1

Bv λ v Dv

n −1 ⎛ v ⎞ ⎛B λ = − ∆a n −1 ∑ ⎜ ∑ Dr a r ⎟ ∆ v ⎜⎜ v v v =1 ⎝ r =1 ⎠ ⎝ Dv

Xv ⎛ ⎜⎜ ∆ Dv−1 v =1 ∆c v −1 ⎝

n −1

= ∆a n −1 ∑

⎞ ⎞B λ ⎛ n ⎟⎟ − ∆a n −1 ⎜ ∑ Dv a v ⎟ n n ⎠ Dn ⎝ v =1 ⎠ ⎞ ⎛ ∆a B λ ∆B v B Bv λ v + v ∆(λv ) + λv +1 ⎟⎟ + ⎜⎜ n −1 n n Dv +1 Dv +1 ⎠ ⎝ ∆c n −1 Dn

( )

Xv ⎛ dv ⎜⎜ − Dv Dv +1 v =1 ∆c v −1 ⎝

n −1

= ∆a n −1 ∑

⎞ ⎟⎟ X n ⎠ ⎞ ⎛ ∆a b λ ⎞ b 1 Bv λ v + Bv ∆λv + v λv +1 ⎟⎟ + ⎜⎜ n −1 n n ⎟⎟ X n Dv +1 Dv +1 ⎠ ⎝ ∆c n −1 d n ⎠

⎞ ⎛ ∆a b λ dv b 1 ⎛ 1 ⎜⎜ − Bv λ v + Bv ∆λv + v λv +1 ⎟⎟ X v + ⎜⎜ n −1 n n Dv Dv +1 Dv +1 Dv +1 v =1 ∆c v −1 ⎝ ⎠ ⎝ ∆c n −1 d n

n −1

= ∆a n −1 ∑

Define X n* = n1−1 / k X n ,

Yn* = n1−1 / s Yn ,

then, we have

3

⎞ ⎟⎟ X n ⎠

SULAIMAN: ABSOLUTE SUMMABILITY

853

⎞ dv b 1 ⎛ 1 ⎜⎜ − Bv λ v + Bv ∆λv + v λv +1 ⎟⎟ X v Dv Dv +1 Dv +1 Dv +1 v =1 ∆c v −1 ⎝ ⎠

n −1

Yn* = n1−1 / s ∆a n −1 ∑

⎛ ∆a b λ + n1 / k −1 / s ⎜⎜ n −1 n n ⎝ ∆c n −1 d n

⎞ ⎟⎟ X n . ⎠

Therefore Yn* =

(11)

n

∑a v =1

a nv

Then U

k

nv

X v* , where

⎧ 1−1 / s ⎞ dv b 1 ⎛ 1 ⎜⎜ − Bv λ v + Bv ∆λv + v λv +1 ⎟⎟ , 1 ≤ v < n ∆a n −1 ⎪n Dv +1 Dv +1 ∆c n −1 ⎝ Dv Dv +1 ⎠ ⎪ ⎪ ⎛ ∆a b λ ⎞ ⎪ v=n = ⎨n1 / k −1 / s ⎜⎜ n −1 n n ⎟⎟ , ⎝ ∆c n −1 d n ⎠ ⎪ ⎪0 , v > n. ⎪ ⎪⎩ of

∑a

n

implies

∑X

* k n

T

s

of

0) ,

(3)

¡ ¢ was estimated in several ways. In this paper we study, for x > −1, the sequence Cn (x) n∈N given as Cn (x) :=

n µ X k=1

¶ ³ 1 1 ´ − ln 1 + , x+k x+k

(4)

¡ ¢ en (x) and its alternating version C , n∈N en (x) := C

n X

µ (−1)

k+1

k=1

¶ ³ 1 ´ 1 − ln 1 + . x+k x+k

(5)

The sequence (4) is closely related to (3); we have Cn (x) = γn (x + 1) + ln

x+n x+n+1

(x > −1, n ∈ N) .

We shall show that both sequences converge and define the Euler-constant functions C(x) := e en (x). The rate of convergence will be estimated in both cases. lim Cn (x) and C(x) := lim C

n→∞

n→∞

Either of the two sequences could be analyzed using the Euler-Maclaurin summation formula which is the subject of the next section.

2

Preliminaries

Concerning the convergence of infinite series, we have two theorems at our disposal [9], where the main role play the sequence of weighted Bernoulli 1-periodic functions Wn (x) and the sequence of weighted Bernoulli polynomials Vn (x) defined inductively as follows: (i)

V0 (x) = 1, for x ∈ R

(ii)

Vn0 (x) = Vn−1 (x), for n ≥ 1 and x ∈ R

(iii)

Vn (0) := Vn (1),

for n ≥ 2 ,

2

(6)

LAMPRET: GENERALIZED EULER CONSTANT

859

and, for n ∈ {0, 1}, (i)

W0 (x) = 1 for x ∈ R

(ii)

W1 (0) := 0

(iii)

W1 (x) := x − 12 ,

if 0 < x < 1

(iv)

W1 (x + 1) := W1 (x),

for x ∈ R .

(i)

Wn (x) := Vn (x),

if 0 ≤ x ≤ 1

(ii)

Wn (x + 1) := Wn (x),

for x ∈ R .

(7)

and, for n ≥ 2, (8)

Since the sequence of Bernoulli polynomials Bn (x) is unambiguously inductively defined as (i)

B0 (x) = 1, for x ∈ R

(ii)

Bn0 (x) = n Bn−1 (x), for n ≥ 1 and x ∈ R

(iii)

Bn (0) := Bn (1),

for n ≥ 2 ,

we have Vn (x) ≡

Bn (x) n!

(n ≥ 0, x ∈ R).

(9)

Thus, referring to [1, 23.1.13], we estimate |V2q (x)| < |V2q (0)|

(q ≥ 1, 0 < x < 1).

(10)

and, considering (9) and [1, 23.1.15], also 1 2 2 < (−1)q+1 V2q (0) < · 2q −q (2π) 1−2·4 (2π)2q

(q ≥ 1).

(11)

Additionally, according to (6)–(9) and [1, 23.1.18], we have W2q (x) =

∞ 2(−1)q−1 X cos(2kπx) (2π)2q k 2q

(q ≥ 1, x ∈ R).

(12)

k=1

Hence, W2q (−x) = W2q (x)

(q ≥ 1, x ∈ R).

(13)

Bernoulli coefficients are defined as Bk := Bk (0) ≡ (k!)Vk (0) 3

(k ≥ 0)

(14)

860

LAMPRET: GENERALIZED EULER CONSTANT

and obey the following relations [1, items 23.1.3 and 23.1.19] B1 = −

1 2

and

B2j+1 = 0

(j ≥ 1).

(15)

The estimate £ ¤ ¡ ¢ 0 < (−1)q−1 · V2q (0) − W2q (x) ≤ 2 1 − 4−q |V2q (0)|
0. (2q)!

(17)

Hence, (−1)q−1 Dq (x) > 0

(q ≥ 1, 0 < x < 1).

(18)

Moreover, considering (6)(iii) we have Dq (0) = Dq (1). Thus, Dq (x) attain its absolute extreme on the interval [0, 1] at its stationary point in the open interval (0, 1). But, thanks to (6)(ii), (9) and [1, 23.1.14] this stationary point coincides with

1 2,

the only zero of B2q−1 (x) within the interval

(0, 1). Hence, using (17) and (9), ¯ ¯ ¡ ¢ max |Dq (x)| = ¯Dq ( 21 )¯ = 2 1 − 4−q |V2q (0)| .

0≤x≤1

This, together with (18) and (11), proves (16). Lemma 1. [9, 3.14] (infinite series convergence criterion) Let m and p be positive integers, 0 ≤ ω ≤ 1, and let f ∈ C p [1, ∞) fulfils the following two conditions: (a)

R ∞ ¯ (p) ¯ ¯f (x)¯ dx < ∞, 1

(b) Finite λk := Then

R∞ 1

lim

n∈N, n7→∞

f (k) (n) exists for every integer k, 0 ≤ k ≤ p − 1.

Wp (ω − x) f (p) (x) dx converges absolutely and the next two propositions hold: 4

LAMPRET: GENERALIZED EULER CONSTANT

(1)

P∞

k=1 f (k

+ ω) converges.

(2) If the series ∞ X

f (k + ω) =

P∞

k=1 f (k

m−1 X

⇐⇒ Finite

exists.

+ ω) converges, then λ0 = 0 and, Z

∞

f (x) dx +

f (k + ω) + m

k=1

k=1

Rn lim f (x) dx n∈N, n7→∞ 1

861

p X

£ ¤ Vj (ω) λj−1 − f (j−1) (m) + ρp (m, ω),

(19)

j=1

where the remainder,

Z

∞

ρp (m, ω) = −

Wp (ω − x) f (p) (x) dx,

(20)

m

is roughly estimated as Z

∞

|ρp (m, ω)| ≤ max |Vp (x)| · 0≤x≤1

¯ ¯ ¯ (p) ¯ ¯f (x)¯ dx.

(21)

m

Setting p = 2q and ω = 0 into Lemma 1, and considering (13)–(15), we obtain the next lemma: Lemma 2. Let m and q be positive integers, and f ∈ C 2q [1, ∞) fulfils the next three conditions: (a) (b) (c)

R ∞ ¯ (2q) ¯ ¯f (x)¯ dx < ∞, 1 f (n) = 0,

lim

f (2j−1) (n) = 0, for every 1 ≤ j ≤ q − 1.

n∈N, n7→∞

Then (1)

lim

n∈N, n7→∞

R∞ 1

W2q (x) f (2q) (x) dx converges absolutely and we have:

P∞

k=1 f (k)

converges.

(2) If the series

P∞

k=1 f (k)

∞ X

⇐⇒ Finite

Rn

n∈N, n7→∞ 1

f (x) dx exists.

converges, then1

f (k) =

k=1

lim

m−1 X

Z

q−1

∞

f (k) +

f (x) dx + m

k=1

Z

∞

+

f (m) X B2i (2i−1) − f (m) 2 (2i)! i=1

[V2q (0) − W2q (x)] f (2q) (x) dx.

(22)

m

We need also a formula for summation the alternating series. This is the Euler-Boole formula revisited recently in [3] and [9]. 1 By

definition

P0

i=1

xi = 0.

5

862

LAMPRET: GENERALIZED EULER CONSTANT

Lemma 3. [9, 3.16] (alternating infinite series convergence criterion) Let m and p be positive integers and let f ∈ C p [1, ∞) obeys the following three conditions: (a) (b)

R ∞ ¯ (p) ¯ ¯f (x)¯ dx < ∞, 1 lim

n∈N, n7→∞

f (2n) = 0,

(c) Finite λ∗∗ 2i−1 :=

lim

n∈N, n7→∞

f (2i−1) (2n) exists for every integer2 i, 1 ≤ i ≤ b p2 c.

Then, R∞ R∞ P∞ k+1 f (k) converges, the integrals 1 Wp (x) f (p) (2x) dx and 1 Wp (x − 21 ) f (p) (2x) dx k=1 (−1) converge absolutely, and the equality ∞ X

(−1)k+1 f (k) =

k=1

m X £ ¤ f (2m) f (2j − 1) − f (2j) + 2 j=1 bp/2c

−

X ¡

(23)

¢ £ ¤ (2i−1) 4i − 1 V2i (0) λ∗∗ (2m) + ρ∗∗ 2i−1 − f p (m),

i=1

holds, where the remainder Z p ρ∗∗ p (m) = (−2)

∞

m

h

i Wp (x) − Wp (x − 21 ) f (p) (2x) dx

is estimated as ¯ ∗∗ ¯ ¯ρp (m)¯ ≤ 2p · max |Vp (x)| · 0≤x≤1

Z

∞

¯ ¯ ¯ (p) ¯ ¯f (t)¯ dt.

(24)

(25)

2m

Putting p = 2q + 1 into Lemma 3, referring to (13)–(15) and to [1, 23.1.8, 23.1.14], we get the lemma below. Lemma 4. [9, 3.16] Let m and q be positive integers and f ∈ C 2q+1 [1, ∞) obeys the following three conditions: (a) (b) (c)

R ∞ ¯ (2q+1) ¯ ¯f (x)¯ dx < ∞, 1 lim

f (2n) = 0,

lim

f (2i−1) (2n) exists for every integer i, 1 ≤ i ≤ q.

n∈N, n7→∞

n∈N, n7→∞

2 The

symbol bxc means the integer part or floor of x.

6

LAMPRET: GENERALIZED EULER CONSTANT

863

Then, R∞ R∞ P∞ k+1 f (k) converges, the integrals 1 Wp (x) f (p) (2x) dx and 1 Wp (x − 21 ) f (p) (2x) dx k=1 (−1) converge absolutely, and the equality ∞ X

(−1)k+1 f (k) =

k=1

m X £ ¤ f (2m) f (2j − 1) − f (2j) + 2 j=1

+

q X ¡

¢ B2i (2i−1) em (q) f (2m) + E (2i)!

4i − 1

i=1

(26)

holds true with the estimate ¯ ¯ ¯e ¯ ¯Em (q)¯
−1 and its sum C(x) can be expressed in the form C(x) = Dm (q, x) + Em (q, x),

(29)

where m and q are any positive integers (parameters), Dm (q, x) =

m−1 X k=1

−

+

1 1 x+m + − ln x + k 2(x + m) x+1

µ ¶ µ ¶x+m+1 1 1 1 ln 1 + − 1 − ln 1 − 2 x+m x+m+1 q−1 X i=1

· ¸ 2i − 1 1 1 B2i + − , (2i − 1)(2i) (x + m)2i (x + m + 1)2i−1 (x + m)2i−1

7

(30)

864

LAMPRET: GENERALIZED EULER CONSTANT

the sequence m 7→ (−1)q−1 Em (q, x) is strictly decreasing, and the following estimates hold ½ ¾ |B2q | 1 1 (−1) Em (q, x) > · max , 2(x + m + 2) (x + m + 2)2q q(x + m + 1)2q µ ¶2q √ πq q >2 , x + m + 2 π e(x + m + 2) µ ¶ √ µ ¶2q πq 1 − 4−q q (−1)q−1 Em (q, x) < 4 e1/(24q) · . 1 − 2 · 4−q x + m π e(x + m) q−1

(31)

(32)

Proof. The series (28) originates from the function fx : (−x, ∞) → R with x > −1, fx (t) :=

µ ¶ 1 1 1 − ln 1 + ≡ + ln(x + t) − ln(x + t + 1), x+t x+t x+t

(33)

satisfying all the conditions of Lemma 2. It has the derivatives fx(k) (t) and the integral

k

¡

≡ (−1) (k − 1) !

Z

∞

¢

µ

1 1 k − + k k (x + t + 1) (x + t) (x + t)k+1

µ fx (t) dt = −1 − ln 1 −

m

1 x+m+1

¶ (34)

¶x+m+1 > 0.

(35)

Introducing in (34) the new variable τ = x + t, and using the functions Fk (τ ) ≡ τ −k together with the first order Taylor’s formula, we obtain (k)

(−1)k fx (t) (1) = Fk (τ + 1) − Fk (τ ) − Fk (τ ) (k − 1)! 1 (2) k(k + 1) = Fk (τ + Θ) = , 2 2(τ + Θ)k+2 for some Θ ∈ (0, 1). Hence, we get the estimate3 (k)

k(k + 1) (−1)k fx (t) k(k + 1) < < k+2 2(x + t + 1) (k − 1)! 2(x + t)k+2

(k ≥ 1) .

(36)

In addition, with the same substitution τ = x + t > 0 in (34) and applying Bernoulli’s inequality, 3 Thus,

(1)

fx (t) < 0, consequently fx (t) > 0, because lim fx (t) = 0. x→∞

8

LAMPRET: GENERALIZED EULER CONSTANT

865

we have (k)

(−1)k fx (t) k 1 1 = k+1 − k + (k − 1)! τ τ (τ + 1)k · ¸ ¡ τ ¢k ´ 1 k ³ = k − 1 − τ +1 τ τ · ¸ ¡ ¢k ´ 1 k ³ 1 = k − 1 − 1 − τ +1 τ τ · ¸ ¡ ¢´ 1 k ³ k > k − 1 − 1 − τ +1 τ τ k = k+1 τ (τ + 1) =

k (x + t)k+1 (x + t + 1)

(k ≥ 1) .

(37)

From the above considerations, using Lemma 2, we obtain C(x) =

m−1 X k=1

+

µ ¶x+m+1 1 1 fx (k) + − 1 − ln 1 − 2(x + m) x+m+1

q−1 X i=1

Z

· ¸ B2i 2i − 1 1 1 + − (2i − 1)(2i) (x + m)2i (x + m + 1)2i−1 (x + m)2i−1

∞

+ m

[V2q (0) − W2q (t)] fx(2q) (t) dt.

(38)

Here we have, thanks to the telescoping of logarithms, m−1 X k=1

fx (k) ≡

m−1 X k=1

x+m 1 − ln . x+k x+1

This identity together with (38) confirms (30) when the last summand, the integral, in (38) is considered as the error term Em (q, x). Moreover, considering (36) and (16), and Stirling’s factorial

9

866

LAMPRET: GENERALIZED EULER CONSTANT

formula [1, 6.1.38], Z q−1

0 < (−1)

q−1

∞

Em (q, x) := (−1)

m




− fx(2q−1) (m + 1)

|B2q | · max 2(x + m + 2)

867

´

½

1 1 , (x + m + 2)2q q(x + m + 1)2q

¾

[1,23.1.15] |B2q | (2q)! 1 > 2 · 2q+1 2(x + m + 2) (2π)2q 2(x + m + 2)2q+1 µ ¶2q p [1,6.1.38] 2q 1 1 > 2π · 2q · · e (2π)2q (x + m + 2)2q+1 µ ¶2q √ πq q =2 . x + m + 2 π e(x + m + 2)

≥

¥

Corollary 1.1. The generalized-Euler-constant function C(x) is infinitely differentiable, strictly decreasing and strictly convex on the interval (−1, ∞). Proof. According to (33) we have the symmetry fx (t) ≡ ft (x). Thus, d d fx (t) ≡ fx (t) dx dt

and

Consequently, considering (36), the series C(x) ≡

d2 d2 fx (t) ≡ fx (t). 2 dx d t2 P∞ k=1

fx (k) can be differentiated term by term

as many times as we please. In particular, using (36), we have C (1) (x) ≡

∞ X

fx(1) (k) < 0

and

C (2) (x) ≡

k=1

∞ X

fx(2) (k) > 0.

k=1

The following corollary is a direct consequence of Theorem 1. Corollary 1.2. For x > −1 and any integer m ≥ 1 there holds the estimate ∗ Dm (x) := Dm (5, x) +

0.0375 0.0758 ∗∗ < C(x) < Dm (x) := Dm (5, x) + , (x + m + 2)11 (x + m)11

11

¥

868

LAMPRET: GENERALIZED EULER CONSTANT

where Dm (5, x) =

m−1 X k=1

− + + − +

1 x+m − ln x+k x+1

µ ¶ µ ¶x+m+1 1 1 1 ln 1 + − 1 − ln 1 − 2 x+m x+m+1 µ ¶ 1 5 1 1 + + 12 x + m x + m + 1 (x + m)2 µ ¶ 1 1 1 3 − − 360 (x + m)3 (x + m + 1)3 (x + m)4 µ ¶ 1 1 1 5 − + 1260 (x + m)5 (x + m + 1)5 (x + m)6 µ ¶ 1 1 1 7 − − . 1680 (x + m)7 (x + m + 1)7 (x + m)8

The graph of the function x 7→ C(x) is illustrated in Figure1 where the graphs of its lower and upper bounds, x 7→ D2∗ (5, x) and x 7→ D2∗∗ (5, x), respectively, are drawn. On the left there is seen practical coincidence of both graphs. 1.4 1.2 1.0 0.8 0.6 0.4 0.2 -1

D*2 HxL D** 2 HxL 0.5776 0.5772

CH0L=0.57721

0.5768 1

2

3

4

-0.0010 -0.0005

0.0005

... x 0.0010

Figure 1: On the left and on the right are depicted the graphs of the functions D2∗ (5, x) and D2∗∗ (5, x) bounding the function C(x).

Corollary 1.3. The rough estimate C ∗ (x) :=

1 1 1 1 < C(x) < C ∗∗ (x) := + + , 4(x + 1) 2(x + 1) x + 2 5(x + 1)3

holds true: on the left for x ∈ (−1, − 34 ) and on the right for x ≥ 0. Consequently, lim C(x) = ∞

x↓−1

and

12

lim C(x) = 0.

x→∞

LAMPRET: GENERALIZED EULER CONSTANT

869

Proof. Referring to Theorem 1, we have · ¸ ³ 1 1 1 ´ C(x) > D1 (1, x) = − ln 1 + . 2 x+1 x+1

(39)

1 Moreover, for x ∈ (−1, − 43 ), the number y := x+1 ≥ 4. But, for y ≥ 4 we have 1 + y4 ≥ 2, ¡ ¢ ¡ ¢ ¡ ¢2 ¡ ¢ consequently 1 + y2 1 + y4 ≥ 1 + y. Thus, exp y2 > 1 + y2 + 12 y2 = 1 + y2 1 + y4 > 1 + y or y 2

> ln(1 + y). Therefore, y 2

y − ln(1 + y) >

(y ≥ 4).

(40)

The relations (39) and (40) verify the left inequality in the corollary. Moreover, according to Theorem 1, we have also 0.16 (x + 1)3 ³ − 1 − ln 1 −

C(x) < D1 (1, x) +
−1 − x+2 x+2

(x ≥ 0).

(42)

Thus, we can confirm the right inequality in the corollary by appealing to (41) and (42).

¥

Figure2 shows the graph of the function C(x) together with the graphs (dashed lines) of its bounds C ∗ (x) and C ∗∗ (x) from Corollary 1.3. 30 25 20 15 10 5

0.7 0.6 0.5 0.4 0.3 0.2 0.1 -0.95

-0.90

-0.85

-0.80

2

-0.75

4

6

8

10

Figure 2: The graph of the function C(x) together with its bounds (dashed lines); C ∗ (x), on the left, and C ∗∗ (x), on the right.

13

870

LAMPRET: GENERALIZED EULER CONSTANT

4

An alternating generalized-Euler-constant function

The rate of convergence of the sequence in (5) is given in the following theorem. Theorem 2. The alternating series ∞ X

µ (−1)k+1

k=1

¶ ³ 1 1 ´ − ln 1 + x+k x+k

(43)

e is convergent for any x > −1, its sum C(x) can be expressed in the form e e m (q, x) + E em (q, x), C(x) =D

(44)

where m and q are any positive integers, µ ¶¸ m · X 1 1 e m (q, x) = D + ln 1 − (x + 2j − 1)(x + 2j) (x + 2j)2 j=1 µ ¶ 1 1 1 + ln 1 − 2x + 4m 2 x + 2m + 1 ¡ ¢ · ¸ q X 4i − 1 B2i 2i − 1 1 1 − + − (2i − 1)(2i) (x + 2m)2i (x + 2m + 1)2i−1 (x + 2m)2i−1 i=1

+

em (q, x) is estimated as and the error term E p µ ¶2q+1 ¯ ¯ 2π(2q + 1) 2(2q + 1) ¯e ¯ . ¯Em (q, x)¯ < 2(4q − 1)(x + 2m) π e(x + 2m)

(45)

(46)

Proof. The series in question also originates from the function fx appearing in the proof of Theorem 1. Since, fx satisfies all the conditions of Lemma 4 the theorem can be verified easily using the relations (33), (34) and (36). For example Z ∞¯ ¯ ¯ ¯ 42q+1 ¯e ¯ ¯ (2q+1) ¯ f (t) ¯Em (q, x)¯ < q ¯ ¯ dt x (4 − 1)(2π)2q+1 2m =
0,

k=1

and k 7→

(2) (fx (k)

are strictly increasing and

decreasing, respectively.

¥

Theorem 2 also has the following direct consequence. Corollary 2.2. For x > −1 and integer m ≥ 1 we have the estimate ∗ em e m (5, x) − D (x) := D

where e m (5, x) = D

m X j=1

+ − + − +

1 4

135 135 ∗∗ e em e m (5, x) + < C(x) 0.

Thus, the function ϕ(x) is decreasing on the interval [−1, ∞). Consequently, ¢ ¡ ϕ(x) ≥ ϕ − 32 > −0.9, Additionally, for x ∈ (−1, − 32 ), the variable y := ey = 1 + y +

for 1 x+1

x ∈ (−1, − 23 ].

(49)

≥ 3. However, for y ≥ 3, we estimate

y3 y2 ³ y´ y2 + + ... > 1+ ≥ y2 , 2 6 2 3

i.e. y > 2 ln y or ln y < y2 . Hence 1 y 1 + ln(x + 1) = y − ln y > = , x+1 2 2(x + 1)

for

x ∈ (−1, − 23 ] .

(50)

Using (46), we estimate ¯ ¯ ¯e ¯ ¯E1 (1, x)¯
−1.

Now, appealing to (44) and (48)–(51), we obtain 1 0.251 − 0.45 − 2(x + 1) (x + 2)4 0.25 1 − 0.5 − > 2(x + 1) (x + 2)4 µ ¶ 1 1 1 −x 1 = −1 − = − , 2 x+1 4(x + 2)4 2(x + 1) 4(x + 2)4

e C(x) >

16

(51)

LAMPRET: GENERALIZED EULER CONSTANT

873

for x ∈ (−1, − 23 ]. Similarly, using (44), (47) and (51), we estimate also 1 1 1 0.251 + + + (x + 1)(x + 2) 2(x + 2) 4(x + 2) (x + 2)4 1 1 0.251 − + = x + 1 4(x + 2) (x + 2)4 1 1 < + , for x > −1. x + 1 4(x + 2)4

e C(x)
0, cn > 0, bn , dn ∈ R and nondegenerate distributions G(x) and L(y) such that F n (an x + bn ) → G(x),

(1 − F (cn y + dn ))n → 1 − L(y)

or equivalently n (1 − F (un (x))) → − log G(x),

nF (vn (y)) → − log (1 − L(y)) ,

(1.1)

where un (x) = an x+bn and vn (y) = cn y+dn . It is easy to prove that the normalized partial maxima and minima of independent and identically distributed (i.i.d.) random variables are asymptotically independent if (1.1) holds, G(x) is an extreme value distribution of the normalized maxima and L(y) is an extreme value distribution of the normalized minima, see Leadbetter et al. (1983) for details. For the weakly dependent case, Leadbetter (1974) obtained the limiting distribution of the maxima with an extremal index 0 < θ ≤ 1 when D(un ) holds. Davis (1979) showed that the joint limiting behavior of the normalized maximum and minimum is the same as that of the i.i.d. case if the considered random sequence satisfies some weak dependence conditions like those provided by Leadbetter (1974). Recently, Mladenovi´c and Piterbarg (2006) considered the asymptotic distributions of maxima for complete and incomplete samples of stationary random sequences. The stochastic model is: Let (Xn ) be a strictly stationary random sequence with the marginal distribution function F (x) = P {X1 ≤ x} and suppose that some of the random variables X1 , X2 , · · · can be observed and the others not. Let Sn = ε1 + ε2 + · · · + εn denote the number of observed random variables from the 1

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set {X1 , X2 , · · · , Xn }, where εk is the indicator of the event that random variable Xk is observed. For convenience, define Mn = max{X1 , · · · , Xn }, mn = min{X1 , · · · , Xn },   max{Xj : εj = 1, k + 1 ≤ j ≤ n}, if Sn ≥ 1; f Mn =  inf{t|F (t) > 0}, if Sn = 0 and m en =

  min{Xj : εj = 1, k + 1 ≤ j ≤ n}, if Sn ≥ 1; 

inf{t|F (t) > 0},

if Sn = 0.

In this note, we are interested in the joint limiting distribution of the normalized random vector fn , m (M e n , Mn , mn ) under some weak dependence conditions similar to those provided by Davis fn , Mn ) and (m (1979) and Mladenovi´c and Piterbarg (2006). The main results show that (M e n , mn ) are asymptotically independent. To formulate our main results, we state the conditions of weak dependence of (Xn ). Definition 1.1. Let (Xn ) be a strictly stationary random sequence and denote Nn = {1, 2, · · · , n}. Then condition D(un (x1 ), vn (y1 ), un (x2 ), vn (y2 )) is satisfied, if for any n and all A1 , A2 , B1 , B2 ⊂ Nn , such that A1 ∩ A2 = ∅, B1 ∩ B2 = ∅ and b − a ≥ l, where a ∈ A1 ∪ A2 and b ∈ B1 ∪ B2 , the following inequalities hold:   \ \ 0 0  {γin < Xj ≤ δin } ∩ {γin < Xj ≤ δin } P j∈A1 ∪B1 j∈A2 ∪B2   \ \ 0 0 −P  {γin < Xj ≤ δin } ∩ {γin < Xj ≤ δin } j∈A1

j∈A2

 ×P 

\

{γin < Xj ≤ δin } ∩

j∈B1

\ j∈B2

0

{γin

 0 < Xj ≤ δin } ≤ αn,ln

for i = 1, 2, 3, where γ1n = −∞, δ1n = un (x2 ), γ2n = vn (y2 ), δ2n = ∞, γ3n = γ2n , δ3n = δ1n , 0 0 0 0 0 0 0 0 γ1n = −∞, δ1n = un (x1 ), γ2n = vn (y1 ), δ2n = ∞, γ3n = γ2n , δ3n = δ1n and αn,ln → 0 as n → ∞ for some sequence ln → ∞ with ln /n → 0. The condition corresponding to i = 1 is D(un (x1 ), vn (y1 ), un (x2 ), vn (y2 )), a natural extension of D(un ) provided by Leadbetter (1974). The condition corresponding to i = 2 is D0 (un (x), vn (y)), due to Davis (1979), which may eliminate the phenomena of clustering for the considered random sequence, see Leadbetter et al. (1983). Definition 1.2. Let (Xn ) be a strictly stationary random sequence. For real sequences (un (x)) and (vn (y)), the condition D0 (un (x), vn (y)) is satisfied if lim sup n n→∞

[n/k] h

X

P (X1 > un (x), Xj+1 > un (x)) + P (X1 > un (x), Xj+1 ≤ vn (y))

j=1

i +P (X1 ≤ vn (y), Xj+1 > un (x)) + P (X1 ≤ vn (y), Xj+1 ≤ vn (y)) = o(1) as k → ∞. 2

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2

877

Main results

In this section we state our main results, i.e. the limiting distributions of the normalized random fn , m vector (M e n , Mn , mn ). Related proofs are deferred to Section 3. Theorem 2.1. Let (Xn ) be a strictly stationary random sequence. Suppose (1.1) holds for un (x), vn (y). Suppose too both the condition D0 (un (x), vn (y)) and the condition D (un (x1 ), vn (y1 ), un (x2 ), vn (y2 )) hold for x2 < x1 , y2 > y1 . Furthermore, suppose that the indicator random variables (εn ) P

are independent of (Xn ) and Sn /n → p ∈ [0, 1] as n → ∞. Then we have   fn ≤ un (x2 ), vn (y1 ) < mn ≤ Mn ≤ un (x1 ) lim P vn (y2 ) < m en ≤ M n→∞

= Gp (x2 )(1 − L(y2 ))p G1−p (x1 )(1 − L(y1 ))1−p . Based on Theorem 2.1 and the continuity of G(x) and L(y), we have the following result. Theorem 2.2. Under the conditions of Theorem 2.1, we have lim P (m e n > vn (y2 ), mn > vn (y1 )) = (1 − L(y2 ))p (1 − L(y1 ))1−p

n→∞

for y1 < y2 and   fn ≤ un (x2 ), Mn ≤ un (x1 ) = Gp (x2 )G1−p (x1 ) lim P M

n→∞

fn , Mn ) are asymptotically independent. for x2 < x1 . This means that (m e n , mn ) and (M The joint limiting distribution of the maxima and minima of observed random variables can be obtained using Theorem 2.1. Corollary 2.1. Under the conditions of Theorem 2.1, we have   fn ≤ un (x) = Gp (x)(1 − L(y))p . lim P vn (y) < m en ≤ M n→∞

To end this section, we give an example satisfying the conditions in Theorem 2.1. Let (Xn ) be a standard stationary Gaussian sequence with correlation rn = EX1 Xn+1 satisfying rn log n → 0,

as

n → ∞,

or n

1X |rk |(log k) exp(γ|rk | log k) → 0 n

as

n→∞

k=1

for some γ > 2. Set 1 an = (2 log n)−1/2 , bn = (2 log n)1/2 − (2 log n)−1/2 (log log n + log 4π), 2 and let un (x) = an x+bn , vn (y) = −an y −bn for x, y ∈ R. Then by Lemma 11.1.2, Lemma 4.5.3 and Lemma 4.4.1 in Leadbetter et al. (1983), we can prove that both the condition D0 (un (x), vn (y)) and the condition D(un (x1 ), vn (y1 ), un (x2 ), vn (y2 )) hold. 3

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Proofs

Before proving the main results, we need some lemmas. Lemma 3.1. Suppose that the equality (1.1) holds. Then lim n{1 − p[F (un (x2 )) − F (vn (y2 ))] − (1 − p)[F (un (x1 )) − F (vn (y1 ))]}

n→∞

= −p[log G(x2 ) + log(1 − L(y2 ))] − (1 − p)[log G(x1 ) + log(1 − L(y1 ))] for x2 < x1 and y2 > y1 . Proof. The result follows by elementary calculus. Let I1 , I2 , · · · , Ik be subsets of Nn = {1, 2, · · · , n}, such that |b − a| ≥ l for all a ∈ Is , b ∈ It , where s 6= t. Let the indicator random variables (εn ) be independent of (Xn ). Define M (Is ) = max{Xj : j ∈ Is },

f(Is ) = max{Xj : j ∈ Is , εj = 1}. M

By using the induction method, we have the following result. Lemma 3.2. Assume that (Xn ) satisfies the condition D(un (x1 ), vn (y1 ), un (x2 ), vn (y2 )) for x2 < x1 , y2 > y1 . Then we have ! k \ f(Is ) ≤ un (x2 ), vn (y1 ) < m(Is ) ≤ M (Is ) ≤ un (x1 )} {vn (y2 ) < m(I e s) ≤ M P s=1 ! k Y f(Is ) ≤ un (x2 ), vn (y1 ) < m(Is ) ≤ M (Is ) ≤ un (x1 ) − P vn (y2 ) < m(I e s) ≤ M s=1

≤ (k − 1)αn,ln . Now let k be a fixed positive integer, m = [n/k] and define Ks = {j : (s − 1)m + 1 ≤ j ≤ sm}. We have Lemma 3.3. Let (Xn ) be a strictly stationary random sequence such that both (1.1) and the condition D(un (x1 ), vn (y1 ), un (x2 ), vn (y2 )) hold. Then fn ≤ un (x2 ), vn (y1 ) < mn ≤ Mn ≤ un (x1 )) P (vn (y2 ) < m en ≤ M

lim

n→∞

−

k Y

! f(Ks ) ≤ un (x2 ), vn (y1 ) < m(Ks ) ≤ M (Ks ) ≤ un (x1 )) P (vn (y2 ) < m(K e s) ≤ M

= 0.

s=1

Proof. Define Nn = {1, 2, · · · , n} for any positive integer n. Let k be a fixed positive integer and m = [n/k]. For large n we can choose a positive integer l such that k < l < m. Let Nmk = (I1 ∪ J1 ) ∪ (I2 ∪ J2 ) ∪ · · · ∪ (Ik ∪ Jk ), where Is = {(s − 1)m + 1, · · · , sm − l} and Js = {sm − l + 1, · · · , sm} for s = 1, 2, · · · , k. Since mk ≤ n < (m + 1)k < mk + l, we get |Nn \Nmk | < k < l. Define sets Ik+1 and Jk+1 as Ik+1 = {mk − m + l + 1, · · · , mk − 1, mk}, Jk+1 = {mk + 1, · · · , mk + l − 1, mk + l}. 4

PENG ET AL: JOINT LIMITING DISTRIBUTIONS...

879

Then |Ik+1 | = m − l and |Jk+1 | = l. The set Ik+1 is a subset of Nmk and the set |Jk+1 | contains the set Nn \Nmk . The maxima (minima) on the sets I1 , I2 , · · · , Ik are weakly dependent, and the small intervals J1 , J2 , · · · , Jk , Jk+1 can be essentially neglected. For simplicity, define f(I) ≤ un (x2 ), vn (y1 ) < m(I) ≤ M (I) ≤ un (x1 )} A(I) = {vn (y2 ) < m(I) e ≤M for any consecutive interval I ⊂ Nn . Let ∆ = P (A(Nn )) −

k Y

P (A(Is ∪ Js )) =: ∆1 + ∆2 + ∆3 ,

s=1

where k \

∆1 = P

! A(Is )

− P (A(Nn )) ,

∆2 = P

s=1

k \

! A(Is )

s=1

−

k Y

P (A(Is ))

s=1

and ∆3 =

k Y

P (A(Is )) −

s=1

k Y

P (A(Is ∪ Js )).

s=1

By using Lemma 3.2, we have |∆2 | ≤ (k − 1)αn,ln → 0 as n → ∞. By arguments similar to those in the proofs of Lemma 3.1 in Davis (1979) and Lemma 4.3 in Mladenovi´c and Piterbarg (2006), we can get ∆i → 0 for i = 1, 3. The proof is complete. Proof of Theorem 2.1. Let k be a fixed positive integer and m = [n/k]. For x2 < x1 and y2 > y1 , we only consider 0 < G(x2 ) < 1, 0 < L(y1 ) < 1. Denote Ks = {j : (s − 1)m + 1 ≤ j ≤ sm}, Bs = {j : j ∈ Ks , εj = 1}, Asj

s ∈ {1, 2, · · · , k};

s ∈ {1, 2, · · · , k};

= {X(s−1)m+j > un (x2 )} ∪ {X(s−1)m+j ≤ vn (y2 )},

j ∈ {1, 2, · · · , m}.

By elementary calculus (cf. Davis (1979) and Leadbetter et al. (1983)), for fixed s ∈ {1, 2, · · · , k} we have 1−

m X

h P (Ssm − S(s−1)m = t) t(1 − F (un (x2 )) + F (vn (y2 )))

t=0

i +(m − t)(1 − F (un (x1 )) + F (vn (y1 )))  f(Ks ) ≤ un (x2 ), vn (y1 ) < m(Ks ) ≤ M (Ks ) ≤ un (x1 ) ) ≤ P ( vn (y2 ) < m(K e s) ≤ M m h X ≤ 1− P (Ssm − S(s−1)m = t) t(1 − F (un (x2 )) + F (vn (y2 ))) t=0 m i X +(m − t)(1 − F (un (x1 )) + F (vn (y1 ))) + m P (As1 Asj ). j=2

5

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PENG ET AL: JOINT LIMITING DISTRIBUTIONS...

By arguments similar to those in the proof of Theorem 3.2 in Mladenovi´c and Piterbarg (2006) and by using the condition D0 (un (x), vn (y)), Lemma 3.1 and Lemma 3.3, we obtain 

H(x1 , x2 , y1 , y2 ) 1+ k

k

fn ≤ un (x2 ), vn (y1 ) < mn ≤ Mn ≤ un (x1 )) ≤ lim inf P (vn (y2 ) < m en ≤ M n→∞

fn ≤ un (x2 ), vn (y1 ) < mn ≤ Mn ≤ un (x1 )) ≤ lim sup P (vn (y2 ) < m en ≤ M n→∞

 ≤

k H(x1 , x2 , y1 , y2 ) 1+ + T0 (k) . k

Finally, we obtain the desired result by letting k → ∞.

References [1] Davis, R. A. (1979). Maxima and minima of stationary sequences. Annals of Probability, 3, 453–460. [2] Leadbetter, M. R. (1974). On extreme values in stationary sequences. Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete, 28, 289–303. [3] Leadbetter, M. R., Lindgren, G. and Rootz´en, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer Verlag, New York. [4] Mladenovi´c, P. and Piterbarg, V. (2006). On asymptotic distribution of maxima of complete and incomplete samples from stationary sequences. Stochastic Processes and Their Applications, 116, 1977–1991.

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JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.5, 881-885, 2011, COPYRIGHT 2011 EUDOXUS PRESS, 881 LLC

TOPOLOGICAL SPACES AND SOFT SETS XUN GE, ZHAOWEN LI, AND YING GE

Abstract. In this paper, we establish some relations between topology and soft set theory. Our work is to give some characterizations of trivial (resp. discrete, T0 -, T1 -, TD - and R0 -) topological spaces by the null soft set, the absolute soft set, the identical soft set and so on. Our results may be useful in the application research of classical topology.

1. Introduction The concept of soft set theory was initiated by Molodtsov in [6]. As a mathematical tool for dealing with uncertainties, soft set theory has a rich potential for applications in several directions. In the past years, this theory had aroused us interesting and concerning and had been applied to many branches of mathematics, information science and computer science (see [1, 5, 6, 7], for example). In mathematical applications, the following is an interesting question (see [6], for example). Question 1.1. How to characterize topological spaces by soft sets? In this paper, we make some investigations for the above question. We give some characterizations of trivial (resp. discrete, T0 -, T1 -, TD - and R0 -) topological spaces by the null soft set, the absolute soft set, the identical soft set and so on. Throughout this paper, all spaces mean topological spaces. Let X be a space and A ⊂ X. The derived set and the closure of A is denoted by Ad and T A respectively. For x ∈ X, Bx denotes the open neighborhood base at x and Bx = {W : W ∈ Bx }. In this paper, both U and E are the set X, where U denotes an initial universe set and E denotes the set of all possible parameters under consideration with respect to U . We assume that each set of parameters is a subset of E. 2. Definitions and Remarks At first, we give some correlative concepts for topological space. Definitions of trivial space, discrete space, T0 -space, T1 -space, the derived set and the closure of a subset are well-known (see [4], for example). We give definitions of TD -space and R0 -space. Definition 2.1 ([2, 3]). Let X be a space. (1) X is called a TD -space if for each xT∈ X, there are an open subset W and a closed subset K of X such that {x} = W K. 1991 Mathematics Subject Classification. 54A05, 54D10, 54H99, 03E47. Key words and phrases. trivial space, discrete space, Ti -space (i=0,1,d), R0 -space, soft set. This project is supported by the National Natural Science Foundation of China (No. 10971185,10771056). 1

882

2

XUN GE, ZHAOWEN LI, AND YING GE

(2) X is called a R0 -space if {x} ⊂ W for each x ∈ X and each open neighborhood W of x. Remark 2.2. It is well-known that the following are hold. (1) T1 -space =⇒ TD -space. (2) T1 -space ⇐⇒ T0 - and R0 -space. Now we give some concepts for correlative soft sets. Definition 2.3 ([6]). Let A be a set of parameters. A pair (F, A) is called a soft set over U if F is a mapping given by F : A −→ P(U ), where P(U ) is the family of all subsets of U . Definition 2.4 ([5]). Let (F1 , A1 ) and (F2 , A2 ) be two soft sets. (1) (F1 , A1 ) and (F2 , A2 ) are called to be equivalent denoted by (F1 , A1 ) = (F2 , A2 ), if A1 = A2 and F1 (x) = F2 (x) for each x ∈ A1 . (2) (F1 , A1 ) is called a refinement of (F2 , A2 ) denoted by (F1 , A1 ) ≺ (F2 , A2 ), if A1 ⊂ A2 and F1 (x) ⊂ F2 (x) for each x ∈ A1 . Definition 2.5 ([5]). Let (F, A) be a soft set. (1) (F, A) is called a null soft set denoted by Φ, if F (x) = ∅ for each x ∈ A. e if F (x) = U for each x ∈ A. (2) (F, A) is called an absolute soft set denoted by A, Remark 2.6. In the notation Φ of a null soft set, the set of parameters for Φ is not clear, which will result in some confusions. For example, let (F1 , A1 ) and (F2 , A2 ) be two soft sets, where A1 6= A2 , F1 (x) = ∅ for each x ∈ A1 and F2 (x) = ∅ for each x ∈ A2 . Then (F1 , A1 ) 6= (F2 , A2 ) because A1 6= A2 . However, both (F1 , A1 ) and (F2 , A2 ) are null soft sets, and so (F1 , A1 ) = Φ = (F2 , A2 ). This is a contradiction. Remark 2.7. In this paper, we use the following notations for null soft set and absolute soft set. (1) (Φ, A) denotes a null soft set with a set A of parameters, i.e., Φ(x) = ∅ for each x ∈ A. (2) (Ω, A) denotes an absolute soft set with a set A of parameters, i.e., Ω(x) = U for each x ∈ A. T Definition 2.8 ([5]). Let (F1 , A1 ) and (F2 , A2 ) be two soft sets. If A1 A2 6= ∅ T T and F1 (x) = F2 (x) for each x ∈ A1 A2 , then the intersection (F1 , A1 ) e (F2 , A2 ) of T (F1 , A1 ) and (F2 , A2 ) is T defined as a soft set (H, A1 A2 ), where H(x) = F1 (x) = F2 (x) for each x ∈ A1 A2 . We give the following for our investigations. Notation 2.9. Let A be a set of parameters. (1) (I, A) denotes an identical soft set, where I(x) = {x} for each x ∈ A. (2) (N, A) denotes a neighborhood soft set, where N (x) = Bx for each x ∈ A. (3) (D, A) denotes a derived soft set, where D(x) = {x}d for each x ∈ A. (4) (C, A) denotes a closure soft set, where C(x) = {x} for each x ∈ A. (5) (D¬ , A) denotes a derived-complement soft set, where D¬ (x) = (X − {x})d for each x ∈ A. (6) (C ¬ , A) denotes a closure-complement soft set, where C ¬ (x) = X − {x} for each x ∈ A. (7) (D2 , A) denotes a bi-derived soft set, where D2 (x) = ({x}d )d for each x ∈ A.

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TOPOLOGICAL SPACES AND SOFT SETS

3

(8) (CD, A) denotes a closure-derived soft set, where CD(x) = {x}d for each x ∈ A. Definition 2.10. Let (F, A) be a soft set. (1) (F, A) is called discernible if F (x) 6= F (y) for all x, y ∈ A and x 6= y. (2) (F, A) is called partitioned if {F (x) : x ∈ X 0 } is a partition of X for some X 0 ⊂ X. (3) (F, A) is called closed if F (x) is closed in X for each x ∈ X. 3. The Main Results Theorem 3.1. The following are equivalent for a space X. (1) X is a trivial space. (2) (N, X) = (Ω, X). Proof. (1) =⇒ (2): Let X be a trivial space. For each x ∈ X, Bx = {X}, and hence Bx = X. So N (x) = Bx = X = U = Ω(x). Consequently, (N, X) = (Ω, X). (2) =⇒ (1): If X is not a trivial space, then there is an open subset W of X such that ∅ = 6 W 6= X. Choose x ∈ W , then N (x) = Bx ⊂ W 6= X = U = Ω(x). So (N, X) 6= (Ω, X).  Theorem 3.2. The following are equivalent for a space X. (1) X is a discrete space. T (2) (I, X) e (D¬ , X) = (Φ, X). T (3) (I, X) e (C ¬ , X) = (Φ, X). T Proof. (1) =⇒ (2): Let X be a discrete space. Put T (I, X) e (D¬ , X)T= (F, X). For each x ∈ X, (X −{x})d = ∅, and hence F (x) = I(x) D¬ (x) = {x} (X −{x})d = T T {x} ∅ = ∅ = Φ(x), So (F, X) = (Φ, X). It follows that (I, X) e (D¬ , X) = (Φ, X). T T (2) =⇒ (3): LetT(I, X) e (D¬ , X) = (Φ, X). Put (I, X) e (C ¬ , X) = (F, X). For T each x ∈ X, {x} (X − {x})d = I(x) D¬ (x) = Φ(x) = ∅, and hence F (x) = T S T T T I(x) SC ¬ (x)T= {x} X − {x}S= {x} ((X − {x}) (X − {x})d ) = ({x} (X − {x})) ({x} (X − {x})d ) = ∅ ∅ = ∅ = Φ(x). So (F, X) = (Φ, X). It follows that T (I, X) e (C ¬ , X) = (Φ, X). T T (3) =⇒ (1): Let (I, X) e (C ¬ , X) = (Φ, X). For each x ∈ X, {x} X − {x} = T ¬ I(x) C (x) = Φ(x) = ∅, So x 6∈ X − {x}, and hence X − {x} ⊂ X − {x}. It follows that X − {x} = X − {x}. So X − {x} is a closed subset of X, hence {x} is an open subset of X. Consequently, X is a discrete space.  The following three lemmas are known. Lemma 3.3 ([3]). The following are equivalent for a space X. (1) X is an R0 -space. T (2) For all x, y ∈ X, {x} = {y} or {x} {y} = ∅. Lemma 3.4 ([4]). The following are equivalent for a space X. (1) X is a T0 -space. (2) {x} = 6 {y} for all x, y ∈ X and x 6= y. Lemma 3.5 ([4]). The following are equivalent for a space X. (1) X is a T1 -space. (2) {x} = {x} for each x ∈ X.

884

4

XUN GE, ZHAOWEN LI, AND YING GE

(3) Bx = {x} for each x ∈ X. (4) {x}d = ∅ for each x ∈ X. The following three theorems are obtained immediately by Lemma 3.3, Lemma 3.4 and Lemma 3.5, respectively. Theorem 3.6. The following are equivalent for a space X. (1) X is an R0 -space. (2) (C, X) is partitioned. Theorem 3.7. The following are equivalent for a space X. (1) X is a T0 -space. (2) (C, X) is discernible. Theorem 3.8. The following are equivalent for a space X. (1) X is a T1 -space. (2) (I, X) = (C, X). (2) (I, X) = (N, X). (2) (D, X) = (Φ, X). By Remark 2.2(2), we have the following corollary. Corollary 3.9. The following are equivalent for a space X. (1) (C, X) = (I, X). (2) (C, X) is discernible and discernible. Lemma 3.10. Let T X be a space and x ∈ X. Then the following are equivalent. (1) {x} = W K for some open subset W and closed subset K of X. T (2) {x} = W {x} for some open subset W of X. (3) x 6∈ {x}d . (4) {x} = 6 {x}d . (5) x 6∈ ({x}d )d . (6) ({x}d )d ⊂ {x}d . (7) {x}d is a closed subset of X. Proof. (1) =⇒ (2): Assume that there are an open subset W and a closed subset T T K of X such that {x} = W K. Since x ∈ K, {x} ⊂ K. So {x} ⊂ W {x} ⊂ T T W K = {x}. It follows that {x} = W {x}. (2) =⇒ (1): It is clear. (2) =⇒ (3): Assume that there are an open subset W of X such that {x} = S T T S T T T W {x}, i.e., {x} = (W {x}d ) (W {x}) = (W {x}d ) {x}. Then W {x}d ⊂ T {x}. Note that x 6∈ {x}d . So W {x}d = ∅. It follows that x 6∈ {x}d (3) =⇒ (4): Assume that x 6∈ {x}d . Note that x ∈ {x}. It follows that {x} = 6 {x}d . (4) =⇒ (5): Assume that {x} = 6 {x}d . Since {x}d ⊂ {x}, {x}d ⊂ {x} = {x}. S S Thus {x} 6⊂ {x}d . Note that {x} = {x} {x}d and {x}d = {x}d ({x}d )d . So x 6∈ ({x}d )d . (5) =⇒ (6): Assume that x 6∈ ({x}d )d . Let y ∈ ({x}d )d , then y 6= x. We only need to prove that y ∈ {x}. Whenever Wy is an open neighborhood of y. It T T suffices to prove T that Wy {x} = 6 ∅. Since y ∈ ({x}d )d ⊂ {x}d , Wy {x}d 6= ∅. d d Choose T z ∈ Wy {x} . Then z ∈ Wy and T z ∈ {x} . Since Wy is open in X, Wy ({x} − {z}) 6= ∅. It follows that Wy {x} = 6 ∅.

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TOPOLOGICAL SPACES AND SOFT SETS

5

S (6) =⇒ (7): Assume that ({x}d )d ⊂ {x}d . Then {x}d = ({x}d )d {x}d = {x}d . So {x}d is a closed subset of X. (7) =⇒ (2): Assume that {x}d is a closed subset of X. Put W = X − {x}d , then W is an open subset of X. Note that x 6∈ {x}d , so x ∈ W . It follows that T S T S T S T  W {x} = W ({x} {x}d ) = (W {x}) (W {x}d ) = {x} ∅ = {x}. We have the following lemma by Lemma 3.10. Lemma 3.11. The following are equivalent for a space X. (1) X is a TD -space. T (2) For each x ∈ X, {x} = W {x} for some open subset W of X. (3) x 6∈ {x}d for each x ∈ X. (4) {x} = 6 {x}d for each x ∈ X. (5) x 6∈ ({x}d )d for each x ∈ X. (6) ({x}d )d ⊂ {x}d for each x ∈ X. (7) {x}d is a closed subset of X for each x ∈ X. The following theorem is obtained immediately by Lemma 3.11. Theorem 3.12. The following are equivalent for a space X. (1) X is a TD -space. T (2) (I, X) e (CD, X) = (Φ, X). (3) (C, X) 6= (CD, X). T (4) (I, X) e (D2 , X) = (Φ, X). (5) (D2 , X) ≺ (D, X). (6) (D, X) is closed. We do not known whether “W ” in Lemma 3.11(2) can be replaced by “Bx ”. More precisely, we have the following question. T Question 3.13. Let X be a space. Is X TD -space if (I, X) = (N, X) e (C, X)? References 1. H. Aktas and N. Cagman, Soft sets and soft groups, Info. Sci., 177(2007), 2726-2735. 2. C.E. Aull and W.J. Thron, Separation axioms between T0 and T1 , Indagations Math., 24(1962), 26-37. 3. A.S. Davis, Indexed systems of neighborhoods, Amer. Math. Monthly, 68(1961), 886-893. 4. R. Engelking, General Topology, Sigma Series in Pure Mathematics 6, Heldermann, Berlin, revised ed., 1989. 5. P.K. Maji, R. Biswas and A.R. Roy, Soft set theory, Comput. Math. Appl., 45(2003), 555-562. 6. D. Molodtsov, Soft set theory-First results, Comput. Math. Appl., 37(1999), 19-31. 7. Y. Zou and Z. Xiao, Data analysis approaches of soft sets under incomplete information, Knowledge-Based Systems, 21(2008), 941-945. Zhangjiagang College, Jiangsu University of Science and Technology, Zhangjiagang 215600, P. R. China E-mail address: [email protected] College of Mathematics and Computer Science, Guangxi University for Nationalities, Nanning 530006, P. R. China E-mail address: [email protected] School of Mathematical Science, Soochow University, Suzhou 215006, P. R. China E-mail address: [email protected]

JOURNAL 886 OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.5, 886-891, 2011, COPYRIGHT 2011 EUDOXUS PRESS, LLC

A new application of quasi-monotone and almost increasing sequences ¨ H¨ useyin BOR and H.S OZARSLAN Department of Mathematics, Erciyes University , 38039 Kayseri, Turkey E-mail:[email protected] ; [email protected]

Abstract In this paper, a general theorem dealing with | C, α, γ, β; δ |k summability factors has been proved. This theorem also includes several known results.

1

Introduction

A positive sequence (bn ) is said to be almost increasing if there exists a positive increasing sequence cn and two positive constants A and B such that Acn ≤ bn ≤ Bcn (see [1]). Obviously every increasing sequence is almost increasing but the converse need not be n

true as can be seen from the example bn = ne(−1) . A sequence (bn ) of positive numbers is said to be δ-quasi-monotone, if bn → 0, bn > 0 ultimately and ∆bn ≥ −δn , where (δn ) is a sequence of positive numbers (see [2]). Let

P

an be a given infinite series with partial sums

α,β (sn ). We denote by uα,β aro means of order (α, β), with α + β > −1, n and tn the n-th Ces`

of the sequence (sn ) and (nan ), respectively, i.e., (see [8]) uα,β n =

1

n X

Anα+β

v=0

β Aα−1 n−v Av sv

(1)

AMS 2010 Subject Classifications: 40D15, 40F05, 40G05 , 40G99. Key words. Quasi-monotone sequences , almost increasing sequences, summability factors.

1

BOR, OZARSLAN: ...ALMOST INCREASING SEQUENCES

tα,β n =

1

n X

Aα+β n

v=1

887

β Aα−1 n−v Av vav ,

(2)

where Aα+β = O(nα+β ), n The series

P

α + β > −1,

Aα+β = 1 and 0

Aα+β −n = 0 f or

n > 0.

(3)

an is said to be summable | C, α, γ, β; δ |k , k ≥ 1, δ ≥ 0, α + β > −1 and γ

is a real number, if (see [7]) ∞ X

k nγ(δk+k−1)−k | tα,β n | < ∞.

(4)

n=1

If we take γ = 1 and δ = 0 , then | C, α, γ, β; δ |k summability reduces to | C, α, β |k summability (see [9]). If we take β = 0 , then we have | C, α, γ; δ |k summability (see [12]). Also if we take γ = 1 and β = 0 , then we get | C, α; δ |k summability (see [11]). Furthermore if we take γ = 1 , β = 0 and δ = 0 , then we get | C, α |k summability (see [10]). Bor and Seyhan [5] have proved the following theorem dealing with | C, α, γ; δ |k summability factors. Theorem A.Let (Xn ) be an almost increasing sequence such that | ∆Xn |= O( Xnn ) and λn → 0 as

n → ∞. Suppose that there exists a sequence of numbers (Bn ) such that it

is δ-quasi-monotone with

P

nXn δn < ∞,

P

Bn Xn is convergent and | ∆λn |≤ Bn for all

n. If the sequence (θnα ) defined by ; θnα =| tαn |,

α=1

(5)

θnα = max | tαv |, 0 < α < 1

(6)

1≤v≤n

satisfies the condition m X

nγ(δk+k−1)−k (θnα )k = O(Xm ) as

m → ∞,

(7)

n=1

then the series

P

an λn is summable | C, α, γ; δ |k , k ≥ 1, 0 ≤ δ < α ≤ 1 and γ is a real

number such that (α + 1)k − γ(δk + k − 1) > 1. 2

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BOR, OZARSLAN: ...ALMOST INCREASING SEQUENCES

2. The aim of this paper is to prove the following general theorem . Theorem . Let (Xn ) be an almost increasing sequence such that | ∆Xn |= O( Xnn ) and λn → 0 as

n → ∞. Suppose that there exists a sequence of numbers (Bn ) such that it

is δ-quasi-monotone with

P

nXn δn < ∞,

P

Bn Xn is convergent and | ∆λn |≤ Bn for all

n. If the sequence (θnα,β ) defined by ; θnα,β =| tα,β n |,

α = 1 , β > −1

θnα,β = max | tα,β |, 0 < α < 1 , β > −1 v 1≤v≤n

(8) (9)

satisfies the condition m X

nγ(δk+k−1)−k (θnα,β )k = O(Xm ) as m → ∞,

(10)

n=1

then the series

P

an λn is summable | C, α, γ, β; δ |k , k ≥ 1, 0 ≤ δ < α ≤ 1 and γ is a real

number such that (α + β + 1)k − γ(δk + k − 1) > 1. We need the following lemmas for the proof of our theorem. Lemma 1 [6]). If 0 < α ≤ 1, β > −1 and 1 ≤ v ≤ n, then |

v X

β Aα−1 n−p Ap ap |≤ max | 1≤m≤v

p=0

m X

β Aα−1 m−p Ap ap | .

(11)

p=0

Lemma 2 ([3]). Under the conditions regarding (λn ) and (Xn ) of the Theorem, we have | λn | Xn = O(1) as n → ∞.

(12)

Lemma 3 ([4]). Under the conditions pertaining to (Xn ) and (Bn ) of the Theorem, we have that nBn Xn = O(1)

(13)

nXn | ∆Bn |< ∞.

(14)

∞ X n=1

3.

Proof of the theorem. Let (Tnα,β ) be the n-th (C, α, β) mean of the sequence

(nan λn ). Then, by means of (2) we have Tnα,β =

1

n X

Aα+β n

v=1

3

β Aα−1 n−v Av vav λv .

BOR, OZARSLAN: ...ALMOST INCREASING SEQUENCES

889

First, applying Abel’s transformation and then using Lemma 1 , we have that

| Tnα,β | ≤ ≤

n−1 X

1

Tnα,β =

Aα+β n

1

∆λv

v=1

n−1 X

Aα+β n

v=1

1

n−1 X

Aα+β n v=1

v X

β Aα−1 n−p Ap pap +

p=1

| ∆λv ||

v X

n λn X

Aα+β n

β Aα−1 n−p Ap pap | +

p=1

β Aα−1 n−v Av vav ,

v=1

| λn |

n X

|

Aα+β n

β Aα−1 n−v Av vav |

v=1

α,β α,β Aαv Aβv θvα,β | ∆λv | + | λn | θnα,β = Tn,1 + Tn,2 ,

say.

Since α,β k α,β k α,β k α,β | ), | + | Tn,2 | ≤ 2k (| Tn,1 + Tn,2 | Tn,1

in order to complete the proof of the theorem, by (4), it is sufficient to show that ∞ X

α,β k nγ(δk+k−1)−k | Tn,r | < ∞,

f or

r = 1, 2.

n=1

Whenever k > 1, we can apply H¨older’s inequality with indices k and k 0 , where

1 k

+ k10 = 1,

we get that m+1 X

α,β k | ≤ nγ(δk+k−1)−k | Tn,1

m+1 X

nγ(δk+k−1)−k |

n=2

n=2

= O(1) ×

(n−1 X

1

n−1 X

Aα+β n

v=1

Aαv Aβv θvα,β ∆λv |k

m+1 X

1

n=2

n(α+β+1)k−γ(δk+k−1)

(n−1 X

)

v

αk βk

v

Bv (θvα,β )k

v=1

)k−1

Bv

v=1

= O(1) = O(1) = O(1) = O(1)

m X v=1 m X v=1 m X

v (α+β)k Bv (θvα,β )k v (α+β)k Bv (θvα,β )k

m+1 X

1 (α+β+1)k−γ(δk+k−1) n n=v+1

Z ∞

dx

v

x(α+β+1)k−γ(δk+k−1)

vBv v γ(δk+k−1)−k (θvα,β )k

v=1 m−1 X

∆(vBv )

v=1

v X p=1

4

pγ(δk+k−1)−k (θpα,β )k

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BOR, OZARSLAN: ...ALMOST INCREASING SEQUENCES

+ O(1)mBm

m X

v γ(δk+k−1)−k (θvα,β )k

v=1

= O(1) = O(1)

m−1 X

|∆ (vBv )| Xv + O(1)mBm Xm

v=1 m−1 X

v |∆Bv | Xv + O(1)

v=1

m−1 X

Bv Xv + O(1)mBm Xm

v=1

= O(1) as

m → ∞,

in view of hypotheses of the theorem and Lemma 3. Similarly , we have that m X

α,β k nγ(δk+k−1)−k | Tn,2 | = O(1)

n=1

= O(1)

m X

|λn | nγ(δk+k−1)−k (θnα,β )k

n=1 m−1 X

∆ (|λn |)

n=1

n X

v γ(δk+k−1)−k (θvα,β )k

v=1

+ O(1) |λm |

m X

v γ(δk+k−1)−k (θvα,β )k

v=1

= O(1) = O(1)

m−1 X n=1 m−1 X

|∆λn | Xn + O(1) |λm | Xm Bn Xn + O(1) |λm | Xm = O(1) as

m → ∞,

n=1

by virtue of hypotheses of the Theorem and Lemma 2. Therefore , by (6) we get that ∞ X

α k nγ(βk+k−1)−k | Tn,r | < ∞ f or

r = 1, 2.

n=1

This completes the proof of the theorem. If we take β = 0 and γ = 1, then we get a result for | C, α; δ |k summability (see [3]) . If we take β = 0 ), then we get Theorem A . Also, if we take γ = 1 , then we have a new result for | C, α, β; δ |k . Furthermore, if we take γ = 1, β = 0 , α = 1 and δ = 0 , then we obtain a result for | C, 1 |k summability factors.

References [1] S. Aljancic and D. Arandelovic, O-regularly varying functions, Publ. Inst. Math., 22 (1977), 5-22. 5

BOR, OZARSLAN: ...ALMOST INCREASING SEQUENCES

891

[2] R.P. Boas, Quasi-positive sequences and trigonometric series, Proc. London Math. Soc. Ser. A, 14 (1965), 38-46. ¨ [3] H. Bor and H.S. Ozarslan, A note on absolute summability factors, Adv. Stud. Contemp. Math. (Kyungshang), 6 (2003), 1-11. [4] H. Bor and L. Leindler , A note on δ-quasi monotone and almost increasing sequences, Math. Inequal . Appl. 8 (2005), 129-134 . ¨ [5] H. Bor and H.S. Ozarslan, On the quasi-monotone and almost increasing sequences, J. Math. Inequal., 1 (2007), 529-534. [6] H. Bor, On a new application of quasi power increasing sequences , Proc. Estonian Acad. Sci. Phys. Math., 57 (2008), 205-209. [7] H. Bor, On the generalized absolute Ces`aro summability , Pac . J . Appl. Math., 2 (3) (2009), 35-40. [8] D. Borwein, Theorems on some methods of summability, Quart. J. Math., Oxford, Ser.9 (1958), 310-316. [9] G. Das, A Tauberian theorem for absolute summability, Proc. Camb. Phil. Soc., 67 (1970), 321-326. [10] T.M. Flett, On an extension of absolute summability and some theorems of Littlewood and Paley, Proc. London Math. Soc., 7 (1957), 113-141. [11] T.M. Flett, Some more theorems concerning the absolute summability of Fourier series, Proc. London Math. Soc., 8 (1958), 357-387. [12] A.N. Tuncer, On generalized absolute Ces`aro summability factors, Ann. Polon. Math., 78 (2002), 25-29.

6

JOURNAL 892 OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.5, 892-898, 2011, COPYRIGHT 2011 EUDOXUS PRESS, LLC

A note on a Tauberian theorem for (A, i) limitable method II ˙ Ibrahim C ¸ anak∗ March 6, 2010

Abstract In this paper which is a sequel to one by C ¸ anak and Albayrak [Int. J. Pure Appl. Math. 35 (3) (2007), 421–424], we obtain weaker Tauberian type conditions under which boundedness and subsequential convergence of (un ) follows from its (A, i) limitability.

Keywords: Tauberian theorem, general control modulo, (A, i) limitability, moderately oscillating sequence, slowly oscillating sequence.

1

Introduction

Using the concept of the generator sequence [7] and a corollary to Karamata’s main theorem [9], C ¸ anak [4] proved the generalized Littlewood Tauberian theorem that states if (un ) is Abel limitable to s and slowly oscillating, then (un ) converges to s. C ¸ anak and Totur [3] obtained a Tauberian condition in terms of the general control modulo for the Abel limitable sequence (un ) to be convergent. (m)

Theorem 1.1 [3] Let (un ) be Abel limitable to s. If (wn (u)) is (C,1) slowly oscillating, then (un ) converges to s. We note that the generalized Littlewood Tauberian theorem is a corollary to Theorem 1.1. Replacing the Abel limitability of (un ) by (A, i) limitability of (un ) in Theorem 1.1 we proved the following theorem in [1]. ∗ Department

˙ of Mathematics, Ege University, Izmir, Turkey, 35100, [email protected]

1

893

2

A note on a Tauberian theorem for (A, i) limitable method II (m)

Theorem 1.2 Let un → s (A, i) for any integer i ≥ 1. If (wn (u)) is (C,1) slowly oscillating, then (un ) converges to s. The purpose of this paper is to replace the Tauberian condition in Theorem 1.2 by weaker Tauberian type conditions to recover boundedness and subsequential convergence of (un ) from its (A, i) limitability.

2

Definitions and Notations

Throughout this paper, the symbols un = o(1) and un = O(1) mean that un → 0 as n → ∞ and (un ) is bounded, respectively. Let u = (un ) be a sequence of real numbers. Pn (i−1) (i) 1 (u) for i ≥ 1 and Define the repeated arithmetic means of (un ) by σn (u) = n+1 k=0 σk (i)

(0)

σn (u) = un . A sequence (un ) is said to be (H, i) limitable to s if limn→∞ σn (u) exists and is equal to s. The (H, 1) limitable method is the same as the (C, 1) limitable method. P∞ (i) A sequence (un ) is said to be (A, i) limitable [13] to s if limx→1− (1 − x) n=0 σn (u)xn exists and is equal to s. In this case, we write un → s (A, i). If i = 0, then (A, i) limitability reduces to Abel limitability. It is well-known that un → s (A, 0) implies un → s (A, i) for each integer i ≥ 1. The converse is not necessarily true in general. For example, in the case i = 1, the sequence (un ) which is the Taylor coefficients of the function f defined by f (x) = sin((1 − x)−1 ) on 0 < x < 1 is not Abel limitable, but (un ) is (A, 1) limitable. A sequence (un ) is said to be slowly oscillating in the sense of Stanojevi´c [12] if lim lim sup

λ→1+

max

n→∞ n≤k≤[λn]

|uk − un | = 0.

Denote the class of all slowly oscillating sequences by S. We say that (un ) is (C, 1) slowly (1)

oscillating if (σn (u)) is slowly oscillating. Note that every null sequence is slowly oscillating. Stanojevi´c’s definition of slow oscillation is more suitable in proofs than those of Landau [10] and Schmidt [11]. An equivalent definition of slow oscillation of a sequence is given in (0)

terms of generator sequence (Vn (∆u)) of (un ) by Dik [7] and he proved that a sequence (0)

(un ) is slowly oscillating if and only if (Vn (∆u)) is slowly oscillating and bounded. Stanojevi´c [12] extended the class S and introduced the following definition. A sequence (un ) is said to be moderately oscillating if for λ > 1, lim sup

max

n→∞ n≤k≤[λn]

|uk − un | < ∞.

894

˙ ˙ C IBRAH IM ¸ ANAK

3

Denote the class of all moderately oscillating sequences by M. We say that (un ) is (C, 1) (1)

moderately oscillating if (σn (u)) is moderately oscillating. It is straightforward to show that S ⊂ M. A sequence (un ) converges subsequentially [5] if there exists a finite interval I(u) such that all of the accumulation points of I(u) are in I(u) and every point of I(u) is an accumulation point of I(u). Equivalently, for every r ∈ I(u) there exists a subsequence (un(r) ) of (un ) such that limn(r) un(r) = r. It is clear that subsequential convergence implies boundedness of the sequence. However, the converse is not true. Namely, there are bounded sequences that do not converge subsequentially. For instance, ((−1)n ) is a bounded sequence, but not subsequentially convergent. A number of classical and neoclassical Tauberian-like conditions were introduced in [6] to retrieve subsequential convergence of (un ) out of its boundedness. Since it is difficult to recover convergence of (un ) out of its Abel limitability and Tauberian conditions weaker than those such as Hardy-Littlewood [8], Landau [10] and Schmidt [11], Stanojevi´c [14] introduced the concept of the general control modulo of the oscillatory behavior of integer order m ≥ 1 of a real sequence (un ), defined by inductively, for all nonnegative integers n, ωn(m) (u) = ωn(m−1) (u) − σn(1) (ω (m−1) (u))

(1)

(0)

where ωn (u) = n∆un . C ¸ anak and Totur [2] expressed (1) in a more explicit form in terms of a generator sequence. We define (n∆)m un = (n∆)m−1 ((n∆)un ) = n∆((n∆)m−1 un ) for each integer m ≥ 1 and each nonnegative integer n, where (n∆)0 un = un and (n∆)1 un = n∆un . (m)

(m−1)

(∆u) for each integer m ≥ 1, where Pn (0) (m) (1) 1 Vn (∆u) = σn (V (m−1) (∆u)) and Vn (∆u) = n+1 k=0 k∆uk . It is proved in [2] that ωn (u) = (n∆)m Vn

3

Lemmas

We need the following lemmas to prove the results in the next section. Lemma 3.1 [2] For each integer m ≥ 1 and for all nonnegative integers n, µ ¶ m−1 X (m) j m−1 ωn (u) = (−1) n∆Vn(j) (∆u), j j=0 µ where

¶

m−1 j

=

(m − 1)(m − 2)...(m − j) . j!

895

4

A note on a Tauberian theorem for (A, i) limitable method II Lemma 3.2 [5] Let (un ) be a bounded sequence. If ∆un = un − un−1 = o(1), then (un )

is subsequentially convergent. Lemma 3.3 [14] Let (un ) be Ces` aro limitable to s. If (un ) is slowly oscillating, then (un ) converges to s.

4

Results (m)

Theorem 4.1 Let un → s (A, i) for some integer i ≥ 1. If (wn (u)) is (C, 1) moderately oscillating, then (un ) is bounded. (2)

Proof Assume that σn (w(m) (u)) = an for some a = (an ) ∈ S. Noticing that σn(2) (w(m) (u)) = wn(m) (σ (2) (u)), we have , for every positive integer n, wn(m) (σ (2) (u)) = an and then wn(m) (σ (j) (u)) = σn(j−2) (a) for j = 2, 3, ..., i. From the equivalent definition of a sequence in S, it follows that (σn(j−2) (a)) ∈ S for j = 2, 3, ..., i. Since un → s (A, i), we have un → s (H, i) by Theorem 1.1. By the fact that every (C, 1) limitable sequence is Abel limitable, we have un → s (A, i − 1). (m)

(i−3)

Since (wn (σ (i−1) (u)) = (σn

(2)

(a)) ∈ S and un → s (A, i − 1), we obtain that un → s

(H, i − 1). By the same reasoning as in obtaining (2), we have un → s (A, i − 2). Continuing in this vein, we obtain that un → s (A, 1).

896

˙ ˙ C IBRAH IM ¸ ANAK

5

(1)

Since σn (w(m) (σ (1) (u))) = an for some a = (an ) ∈ S, we have un → s (C, 1) by Theorem 1.1. It follows by the fact that every (C, 1) limitable sequence is Abel limitable (m)

that un → s (A, 0). The assumption that (wn (u)) is (C, 1) moderately oscillating implies (1)

that σn (w(m+1) (u)) = O(1) as n → ∞. It follows by m+1 X

µ k

(−1)

k=0

¶ m+1 Vn(k) (∆u) = O(1), n → ∞ k

that (un ) is bounded. Now, we introduce some Tauberian-like conditions to retrieve subsequential convergence of (un ) out of boundedness of the sequence. We end this section with the following results. (m)

Theorem 4.2 Let un → s (A, i) for some integer i ≥ 1. If (wn (u)) is (C, 1) moderately (0)

oscillating and the sequence (∆Vn (∆u)) is slowly oscillating, then (un ) is subsequentially convergent. Proof We have by the previous theorem that (un ) is bounded. Taking the backward difference of both sides of the equality un − σn(1) (u) = Vn(0) (∆u) we have (0)

∆un =

Vn (∆u) + ∆Vn(0) (∆u). n

(3)

(0)

Since (∆Vn (∆u)) ∈ S and (0)

Vn (∆u) → 0, n → ∞, n

(4)

∆Vn(0) (∆u) = o(1), n → ∞.

(5)

we have, by Lemma 3.3,

Taking (3), (4) and (5) into account, we obtain ∆un = o(1), n → ∞. By Lemma 3.2, it follows that (un ) is subsequentially convergent. We have the following corollaries for Theorem 4.2.

897

6

A note on a Tauberian theorem for (A, i) limitable method II (m)

Corollary 4.3 Let un → s (A, i) for some integer i ≥ 1. If (wn (u)) is (C, 1) moderately oscillating and the sequence (∆un ) is slowly oscillating, then (un ) is subsequentially convergent. Proof Taking the backward difference of both sides of the equality un − σn(1) (u) = Vn(0) (∆u) we have (0)

∆un =

Vn (∆u) + ∆Vn(0) (∆u). n

(6)

(0)

Since (un ) is bounded by the previous theorem, (Vn (∆u)) is bounded. It follows from (6) (0)

that (∆Vn (∆u)) is slowly oscillating. This completes the proof by Theorem 4.2. (m)

Corollary 4.4 Let un → s (A, i) for some integer i ≥ 1. If (wn (u)) is (C, 1) moderately oscillating and the sequence (un ) is slowly oscillating, then (un ) is subsequentially convergent. Proof By the fact that slow oscillation of (un ) implies that ∆un = o(1), we have the proof of Corollary 4.4.

References ˙ C [1] I. ¸ anak, M. Albayrak, A note on a Tauberian theorem for (A, i) limitable method, Int. J. Pure Appl. Math., 35 (3) , 421–424, (2007). ˙ C ¨ Totur, A Tauberian theorem with a generalized one-sided condition, Ab[2] I. ¸ anak, U. stract and Applied Analysis, Vol. 2007, 12 pp. (2007). ˙ C ¨ Totur, A note on Tauberian theorems for regularly generated sequences, [3] I. ¸ anak, U. Tamkang Journal of Mathematics, 39 (2), 187–191, (2008). ˙ C [4] I. ¸ anak, A short proof of the generalized Littlewood Tauberian theorem, submitted for publication. [5] F. Dik, Tauberian theorems for convergence and subsequential convergence with moderately oscillatory behavior, Math. Morav., 5, 19–56, (2001).

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˙ ˙ C IBRAH IM ¸ ANAK

7

˙ C [6] F. Dik, M. Dik, I. ¸ anak, Applications of subsequential Tauberian theory to classical Tauberian theory, Appl. Math. Lett., 20 (8), 946–950, (2007). [7] M. Dik, Tauberian theorems for sequences with moderately oscillatory control modulo, Math. Morav., 5, 57–94, (2001). [8] G. H. Hardy, J. E. Littlewood, Tauberian theorems concerning power series and Dirichlet’s series whose coefficients are positive, Proc. London Math. Soc., 13 (2), 174–191, (1914). [9] J. Karamata,

¨ Uber die Hardy-Littlewoodschen umkehrungen des Abelschen

stetigkeitssatzes, Math. Z., 32, 319–320, (1930). ¨ [10] E. Landau, Uber die Bedeutung einiger neuen Grenzwerts¨atze der Herren Hardy und Axer, Prace Mat. -Fiz., 21, 97–177, (1910). ¨ [11] R. Schmidt, Uber divergente Folgen und lineare Mittelbildungen, Math. Z., 22, 89–152, (1925). ˇ V. Stanojevi´c, Analysis of Divergence: Control and Management of Divergent Pro[12] C. ˙ C cess, Graduate Research Seminar Lecture Notes, edited by I. ¸ anak, University of Missouri-Rolla, Missouri, 1998. ˇ V. Stanojevi´c, I. ˙ C [13] C. ¸ anak, V. B. Stanojevi´c, Tauberian theorems for generalized Abelian summability methods, in Analysis of divergence. Control and management of ˇ V. Stanojevi´c, eds.), Birkh¨auser, Boston, divergent processes (William O. Bray and C. MA, 1999, pp. 13–26. ˇ V. Stanojevi´c, Analysis of Divergence: Applications to the Tauberian Theory, Grad[14] C. uate Research Seminar, University of Missouri-Rolla, Missouri, 1999.

JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.5, 899-906, 2011, COPYRIGHT 2011 EUDOXUS PRESS, 899 LLC

STABILITY OF LINEAR NON-AUTONOMOUS SYSTEMS YAOWALUCK KHONGTHAM Department of Mathematics, Faculty of Science,Maejo University, Chiang Mai 50290, Thailand. Abstract. In this paper, we study the exponential stability of linear non-autonomous systems with multiple delays. Using Lyapunov-like function, we find sufficient conditions for the exponential stability in terms of the solution of a Riccati differential equation. Our results are illustrated with numerical examples. Keywords: Exponential stability; time-delay; Lyapunov function; Riccati equation. AMS Subject Classification: 34K20

1. Introduction The topic of Lyapunov stability of linear systems has been an interesting research area in the past decades. An integral part of the stability analysis of differential equations is the existence of inherent time delays. Time delays are frequently encountered in many physical and chemical processes as well as in the models of hereditary systems, Lotka-Volterra systems, control of the growth of global economy, control of epidemics, etc. Therefore, the stability problem of time-delay systems has been received considerable attention from many researchers (see; e.g. [5, 6, 10, 11, 12, 14] and references therein). One of the extended stability properties is the concept of the α-stability, which relates to the exponential stability with a convergent rate α > 0. Namely, a retarded system x˙ = f (t, x(t), x(t − h)),

t ≥ 0,

t ∈ [−h, 0],

x(t) = φ(t),

is α-stable, with α > 0, if there is a function ξ(.) such that for each φ(.), the solution x(t, φ) of the system satisfies kx(t, φ)k ≤ ξ(kφk)e−αt ,

∀t ≥ 0,

where kφk = max{kφ(t)k : t ∈ [−h, 0]}. This implies that for α > 0, the system can be made exponentially stable with the convergent rate α. It is well known that there are many different methods to study the stability problem of time-delay linear autonomous systems. The widely used method is the approach of Lyapunov functions with Razumikhin techniques and the asymptotic stability conditions are presented in terms of the solution of either linear matrix inequalities or Riccati equations [2, 7, 8]. By using both the time-domain and the frequency-domain techniques, the paper [15] derived sufficient conditions for the asymptotic stability Email address: [email protected]. 1

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Y. KHONGTHAM

of a linear autonomous system with multiple time delays of the form x(t) ˙ = A0 x(t) +

m X

Ai x(t − hi ),

i=1

x(t) = φ(t),

t ≥ 0,

(1.1)

t ∈ [−h, 0],

where Ai are given constant matrices, h = max{hi : i = 1, 2, . . . , m}. These conditions depend only on the eigenvalues of A0 and the norm values of Ai of the system. For studying the α-stability problem, based on the asymptotic stability of the linear undelayed part, i.e. A0 is a Hurwitz matrix, the papers [13, 14] proposed sufficient conditions for the α-stability of system (1.1) in terms of the solution of a scalar inequality involving the eigenvalues, the matrix measures and the spectral radius of the system matrices. It is worth noticing that although the approach used in these papers allows us to derive the less conservative stability conditions, but it can not be applied to non-autonomous delay systems. The reason is that, the assumption A0 (t) to be a Hurwitz matrix for each t ≥ 0, i.e. Re λ(A(t)) < 0, for each t, does not implies the exponential stability of the linear non-autonomous system x˙ = A0 (t)x. It is the purpose of this paper to search sufficient conditions for the α-stability of non-autonomous delay systems. Using the Lyapunov-like function method, we develop the results obtained in [3, 14] to the non-autonomous systems with multiple delays. Do not using any Lyapunov stability theorem, we establish sufficient conditions for the α-stability of system (2.1), which are given in terms of the solution of a Riccati differential equation (RDE). These conditions do not involve any stability property of the system matrix A0 (t). Although the problem of solving of RDEs is in general still not easy, various effective approaches for finding the solutions of RDEs can be found in [1, 4, 9, 16]. The paper is organized as follows. Section 2 presents notations, mathematical definitions and an auxiliary lemma used in the next section. The sufficient conditions for the α-stability are presented in Section 3. Numerical examples illustrated the obtained result are also given in Section 3. The paper ends with cited references. 2. Preliminaries The following notations will be used for the remaining this paper. R+ denotes the set of all real non-negative numbers; Rn denotes the n-dimensional space with the scalar product h., .i and the vector norm k.k; Rn×r denotes the space of all matrices of dimension (n × r). AT denotes the transpose of the vector/matrix A; a matrix A is symmetric if A = AT ; I denotes the identity matrix; λ(A) denotes the set of all eigenvalues of A; λmax (A) = max{Re λ : λ ∈ λ(A)}; kAk denotes the spectral norm of the matrix defined by q kAk = λmax (AT A); η(A) denotes the matrix measure of the matrix A given by 1 η(A) = λmax (A + AT ). 2 C([a, b], Rn ) denotes the set of all Rn -valued continuous functions on [a, b]; Matrix A is called semi-positive definite (A ≥ 0) if hAx, xi ≥ 0, for all x ∈ Rn ; A is positive definite (A > 0) if hAx, xi > 0 for all x 6= 0;

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STABILITY OF LINEAR NON-AUTONOMOUS SYSTEMS

3

In the sequel, sometimes for the sake of brevity, we will omit the arguments of matrix-valued functions, if it does not cause any confusion. Let us consider the following linear non-autonomous system with multiple delays x(t) ˙ = A0 (t)x(t) +

m X

Ai (t)x(t − hi ),

t ≥ 0,

i=1

(2.1)

t ∈ [−h, 0],

x(t) = φ(t),

where h = max{hi : i = 1, 2, . . . , m}, Ai (t), i = 0, 1, . . . , m, are given matrix functions and φ(t) ∈ C([−h, 0], Rn ). Definition. The system (2.1) is said to be α-stable, if there is a function ξ(.) : R+ → R+ such that for each φ(t) ∈ C([−h, 0], Rn ), the solution x(t, φ) of the system satisfies kx(t, φ)k ≤ ξ(kφk)e−αt , ∀t ∈ R+ . The following well-known lemma, which is derived from completing the square, will be used in the proof of our main result. Lemma 2.1. Assume that S ∈ Rn×n is a symmetric positive definite matrix. Then for every P, Q ∈ Rn×n , hP x, xi + 2hQy, xi − hSy, yi ≤ h(P + QS −1 QT )x, xi,

∀x, y ∈ Rn .

3. Main results Consider the linear non-autonomous delay system (2.1), where the matrix functions Ai (t), i = 0, 1, . . . , m, are continuous on R+ . Let us set A0,α (t) = A0 (t) + αI,

Ai,α (t) = eαhi Ai (t), i = 1, 2, . . . , m.

Theorem 3.1. The linear non-autonomous system (2.1) is α-stable if there is a symmetric semi-positive definite matrix P (t), t ∈ R+ such that P˙ (t) + AT0,α (t)[P (t) + I] + [P (t) + I]A0,α (t) +

m X

[P (t) + I]Ai,α (t)ATi,α (t)[P (t) + I] + mI = 0.

(3.1)

i=1

Proof. Let P (t) ≥ 0, t ∈ R+ be a solution of the RDE (3.1). We take the following change of the state variable y(t) = eαt x(t),

t ∈ R+ ,

then the linear delay system (2.1) is transformed to the delay system y(t) ˙ = A0,α (t)y(t) +

m X

Ai,α (t)y(t − hi ),

i=1 αt

y(t) = e φ(t),

t ∈ [−h, 0],

Consider the following time-varying Lyapunov-like function m Z t X 2 V (t, y(t)) = hP (t)y(t), y(t)i + ky(t)k + ky(s)k2 ds. i=1

t−hi

(3.2)

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4

Y. KHONGTHAM

Taking the derivative of V (.) in t along the solution of y(t) of system (3.2) and using the RDE (3.1), we have V˙ (t, y(t)) = hP˙ (t)y(t), y(t)i + 2hP (t)y(t), ˙ y(t)i + 2hy(t), ˙ y(t)i + mky(t)k2 −

m X

ky(t − hi )k2 ,

i=1

= hP˙ (t)y(t), y(t)i + 2hP (t)A0,α (t)y(t), y(t)i + 2

m X

hP (t)Ai,α (t)y(t − hi ), y(t)i

i=1

+ 2hA0,α (t)y(t), y(t)i + 2

m X

hAi,α (t)y(t − hi ), y(t)i

i=1

+ mky(t)k2 −

m X

ky(t − hi )k2 ,

i=1

= hP˙ (t)y(t), y(t)i + 2h(P (t) + I)A0,α (t)y(t), y(t)i +2

m X

h(P (t) + I)Ai,α (t)y(t − hi ), y(t)i + mky(t)k2 −

i=1

=−

m X

+2

i=1

m X

h[P (t) + I]Ai,α (t)y(t − hi ), y(t)i −

i=1 m X

ky(t − hi )k2 ,

h[P (t) + I]Ai,α (t)ATi,α (t)[P (t) + I]y(t), y(t)i

i=1 m X

=

m X

hy(t − hi ), y(t − hi )i

i=1

{−h[P (t) + I]Ai,α (t)ATi,α (t)[P (t) + I]y(t), y(t)i

i=1

+ 2h[P (t) + I]Ai,α (t)y(t − hi ), y(t)i − hy(t − hi ), y(t − hi )i}. (3.3) Applying Lemma 2.1 to the above equality, we have V˙ (t, y(t)) ≤ 0,

∀t ∈ R+ .

Integrating both sides of this inequality from 0 to t, we find ∀t ∈ R+ ,

V (t, y(t)) − V (0, y(0)) ≤ 0, and hence 2

hP (t)y(t), y(t)i + ky(t)k +

m Z X

t

ky(s)k2 ds

t−hi

i=1 m Z 0 X

≤ hP0 y(0), y(0)i + ky(0)k2 +

i=1

ky(s)k2 ds,

−hi

where P0 = P (0) ≥ 0 is any initial condition. Since Z t hP (t)y, yi ≥ 0, ky(s)k2 ds ≥ 0, t−hi

Z

0

−hi

ky(s)k2 ds ≤ kφk

Z

0

−hi

eαs ds =

1 (1 − e−αhi )kφk, α

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STABILITY OF LINEAR NON-AUTONOMOUS SYSTEMS

5

it follows that m

1X ky(t)k ≤ hP0 y(0), y(0)i + ky(0)k + (1 − e−αhi )kφk. α i=1 2

2

Therefore, the solution y(t, φ) of the system (3.2) is bounded. Returning to the solution x(t, φ) of system (2.1) and noting that ky(0)k = kx(0)k = φ(0) ≤ kφk, we have kx(t, φ)k ≤ ξ(kφk)e−αt for all t ∈ R+ , where m

ξ(kφk) := {kP0 kkφk2 + kφk2 +

1 1X (1 − e−αhi )kφk} 2 . α i=1

This implies system (2.1) begin α-stable and completes the proof.



Remark. Note that the existence of a semi-positive definite matrix solution P (t) of RDE (3.1) guarantees the boundedness of the solution of transformed system (3.2), and hence the exponential stability of the linear non-autonomous delay system (2.1). Also, the stability of A(t) is not assumed. Example 3.2. Consider the following linear non-autonomous delay system in R2 : x˙ = A0 (t)x + A1 (t)x(t − 0.5) + A2 (t)x(t − 1),

t ∈ R+ ,

with any initial function φ(t) ∈ C([−1, 0], R2 ) and     −0.5 e a1 (t) 0√ a0 (t) 0 , A0 (t) = , A1 (t) = 0 −7.5 0 e−0.5 3  −1  e a1 (t) 0√ A2 (t) = , 0 e−1 3 where

7e−9t − 5 1 , a1 (t) = √ . −9t 2(1 + e ) 2(1 + e−9t ) We have h1 = 0.5, h2 = 1, m = 2 and the matrix A0 (t) is not asymptotically stable, since Re λ(A(0)) = 0.5 > 0. Taking α = 1, we have     a1 (t) √0 a0 (t) + 1 0 A0,α (t) = , A1,α (t) = A2,α (t) = . 0 −6.5 0 3 a0 (t) =

The solution of RDE (3.1) is  P (t) =

e−9t 0

 0 ≥ 0, 1

∀t ∈ R+ .

Therefore, the system is 1-stable. For the autonomous delay systems, we have the following α-stability condition as a consequence. Corollary 3.3. The linear delay system (2.1), where Ai are constant matrices, is α-stable if there is a symmetric semi-positive definite matrix P ∈ Rn×n , which is a solution of the algebraic Riccati equation m X AT0,α [P + I] + [P + I]A0,α + [P + I]Ai,α ATi,α [P + I] + mI = 0. (3.4) i=1

904

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Y. KHONGTHAM

Example 3.4. Consider the linear autonomous delay system x(t) ˙ = A0 x(t) + A1 x(t − 2) + A2 x(t − 4), with any initial function φ(t) ∈ C([−4, 0], R2 ) and   −1  17  0 e 0 −6 , A1 = A0 = , 4 0 e−1 −3.5 3

t ∈ R+ , 

A2 =

e−2 0

0 e−2

 .

In this case, we have m = 2, h1 = 2, h2 = 4. Taking α = 0.5, we find     7 1 0 −3 0 , A1,α (t) = A2,α (t) = , A0,α (t) = 4 0 1 −3 3 and the solution of algebraic Riccati equation (3.4) is   1 −1 P = ≥ 0. −1 1 Therefore, the system is 0.5-stable. Remark. Note that we can estimate the value of V (t, y) as follows. Since 2(P + I)A0,α = AT0 P + P A0 + A0 + AT0 + 2α(P + I), from (3.3) it follows that V˙ (t, y(t)) = h[P˙ (t) + AT (t)P (t) + P (t)A0 (t) + mI]y(t), y(t)i 0

+ h[A0 (t) + AT0 (t)]y(t), y(t)i + 2αh(P (t) + I)y(t), y(t)i m n o X + 2h[P (t) + I]Ai,α (t)y(t − hi ), y(t)i − ky(t − hi )k2 . i=1

Using Lemma 2.1, we have m n o X 2h[P + I]Ai,α y(t − hi ), y(t)i − ky(t − hi )k2 i=1

≤

m X

h[P + I]Ai,α ATi,α [P + I]y(t), y(t)i.

i=1

On the other hand, since m X h[P (t) + I]Ai,α (t)ATi,α (t)[P (t) + I]y(t), y(t)i ≤ mkP (t) + Ik2 e2αh kA(t)k2 ky(t)k2 , i=1

with h = max{h1 , h2 , . . . , hm }, kA(t)k2 = max{kA1 (t)k2 , kA2 (t)k2 , . . . , kAm (t)k2 }, we obtain V˙ (t, y(t)) ≤ h[P˙ (t) + AT0 (t)P (t) + P (t)A0 (t) + mI]y(t), y(t)i h i + 2η(A0 (t)) + 2αkP (t) + Ik + mkP (t) + Ik2 e2αh kA(t)k2 ky(t)k2 . Therefore, the α-stability condition of Theorem 3.1 can be given in terms of the solution of the following Lyapunov equation, which does not involve α: P˙ (t) + AT0 (t)P (t) + P (t)A0 (t) + mI = 0. (3.5) In this case, if we assume that P (t), Ai (t) are bounded on R+ and η(A0 ) := sup η(A0 (t)) < +∞, t∈R+

(3.6)

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STABILITY OF LINEAR NON-AUTONOMOUS SYSTEMS

7

then the rate of convergence α > 0 can be defined as a solution of the scalar inequality m η(A0 ) + αkPI k + e2αh kPI k2 kAk2 ≤ 0, (3.7) 2 where PI = sup kP (t) + Ik, kAk2 = sup kA(t)k2 . t∈R+

t∈R+

Therefore, we have the following α-stability condition. Corollary 3.5. The linear delay system (2.1), where Ai (t) are constant matrices, is α-stable if there is a symmetric semi-positive definite P of the algebraic Lyapunov equation AT0 P + P A0 + mI = 0. (3.8) In this case, the convergent rate α > 0 is the solution of the scalar inequality m (3.9) η(A0 ) + αkPI k + kPI k2 e2αh kAk2 ≤ 0, 2 where PI = P + I, kAk2 = max{kAi k2 , i = 1, 2, . . . , m}. Example 3.6. Consider the linear autonomous delay system x(t) ˙ = A0 x(t) + A1 x(t − 0.5) + A2 x(t − 1),

t ∈ R+ ,

with any initial function φ(t) ∈ C([−1, 0], R2 ) and     −2 0.5 1/3 0 A0 = , A1 = A2 = e−0.4 . −1 −4 0 1/3 We have m = 2, h1 = 0.5, h2 = 1 and √ η(A0 ) = −3 + 0.5 4.25,

e−0.8 . 9 The solution of the algebraic Lyapunov equation (3.8) is   0.5 0 P = , 0 0.25 kAk2 =

and then kP + Ik = 1.5. The rate of convergence α = 0.4 satisfies the condition (3.9). Then, by Corollary 3.5 the system is 0.4-stable. References [1] Abou-Kandil H., G. Freiling, V.Ionescu and G. Jank, Matrix Riccati Equations in Control and Systems Theory, Basel, Birkhauser, 2003. [2] Boyd S., El. Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix and Control Theory. SIAM Studies in Appl. Math., SIAM PA, vol. 15, (1994). [3] D.Q. Cao, P. He and K. Zhang, Exponential stability criteria of uncertain systems with multiple time delays. J. Math. Anal. Appl., 283, 362-374 (2003). [4] J.S. Gibson, Riccati equations and numerical approximations. SIAM J. Contr. Optim., 21, 95-139 (1983). [5] J.D. Jefferey, Stability for time-varying linear dynamic systems on time scales. J. Comp. Appl. Math., 176, 381-410 (2005). [6] J. Hale and Stokes A., Conditions for the stability of non-autonomous differential equations. J. Math. Anal. Appl., 3, 50-69 (1961). [7] V.L. Kharitonov, Lyapunov-Krasovskii functionals for scalar time delay equations. Systems & Control Letters, 51, 133-149 (2004). [8] V.B. Kolmanovskii and V.R. Nosov, Stability of Functional Differential Equations. Academic Press, London, 1986.

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[9] A.J. Laub, Schur techniques for solving Riccati equations, IEEE Trans. AC, 24, 913-921 (1979). [10] V.N. Phat, Constrained Control Problems of Discrete Processes. World Scientific, Singapore, 1996. [11] V.N. Phat and N.M. Linh, Exponential stability of nonlinear time-varying differential equations and applications. Elect. J. of Diff. Equations, , N-34 , 1-14 (2001). [12] V.N. Phat, New stabilization criteria for linear time-varying systems with state delay and normed bounded uncertainties. IEEE Trans. AC, 12, 2095-2098 (2002). [13] Y.J Sun, J.G. Hsieh and Y.C. Hsieh, Exponential stability criterion for uncertain retarded systems with multiple time-varying delays. J. Math. Anal. Appl., 201, 430-446 (1996). [14] Y.J Sun and J.G. Hsieh, On α-stability criteria of nonlinear systems with multiple delays. J. Franklin Inst., 335B, 695-705 (1998). [15] B. Xu, Stability criteria for linear systems with uncertain delays. J. Math. Anal. Appl., 284, 455-470 (2003). [16] R. William Thomas, Riccati Differential Equations, Academic Press, New York, 1972.

JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.5, 907-922, 2011, COPYRIGHT 2011 EUDOXUS PRESS, 907 LLC

Moment Functional and Orthogonal Trigonometric Polynomials of Semi-Integer Degree G. V. Milovanovi´ca , A. S. Cvetkovi´cb , and M. P. Stani´cc a

c

Megatrend University, Faculty of Computer Sciences, Bulevar Umetnosti 29, 11070 Novi Beograd, Serbia, [email protected] b University of Niˇs, Faculty of Science and Mathematics, Viˇsegradska 33, 18000 Niˇs, Serbia, [email protected] University of Kragujevac, Faculty of Science, Radoja Domanovi´ca 12, 34000 Kragujevac, Serbia, [email protected]

In this paper we consider algebraic properties of orthogonal trigonometric polynomials of semi-integer degree. We investigate the theory of orthogonality with respect to a general linear functional, which maps the space of trigonometric polynomials to the real numbers. Under certain conditions imposed on the linear functional, we prove the existence of a sequence of orthogonal trigonometric polynomials of semi-integer degree, which satisfies three-term recurrence relations when it is treated in a suitable matrix settings. Key Words. trigonometric polynomial; semi-integer degree; orthogonality; moment functional; recurrence relation. 2000 Mathematics Subject Classification. 42A05; 42C05; 42C15

1

Introduction

The first results on orthogonal trigonometric polynomials of semi-integer degree were given in 1959 by Abram Haimovich Turetzkii (see [7]). They are connected with quadrature rules with an even maximal trigonometric degree of exactness in the case of an odd number of nodes. A trigonometric polynomial of semi-integer degree n + 21 is a trigonometric function of the following form n · X ν=0

¸ ³ ³ 1´ 1´ cν cos ν + x + dν sin ν + x , 2 2

(1)

where cν , dν ∈ R, |cn | + |dn | 6= 0. The coefficients cn and dn are called the leading coefficients. Let us denote by Tn , n ∈ N0 , the linear space of all trigonometric polynomials of degree less than or equal to n, i.e., the linear span of the following 1

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MILOVANOVIC ET AL: MOMENT FUNCTIONAL...

1/2

set {1, cos x, sin x, . . . , cos nx, sin nx}, by Tn , n ∈ N0 , the linear space of all trigonometric polynomials of semi-integer degree less than or equal to n + 21 , i.e., the linear span of {cos(k + 12 )x, sin(k + 12 )x : k = 0, 1, . . . , n}, and by T and T 1/2 the set of all trigonometric polynomials and the set of trigonometric polynomials of semi-integer degree, respectively. For an integrable and nonnegative weight function w(x) on the interval [0, 2π), vanishing there only on a set of a measure zero, and a given set xν , ν = 0, 1, . . . , 2n, of distinct points in [0, 2π), Turetzkii in [7] considered an interpolatory quadrature rule of the form Z

2π

t(x)w(x) dx = 0

2n X

wν t(xν ),

t ∈ Tn .

(2)

ν=0

Such a quadrature rule can be obtained from the trigonometric interpolation polynomial (cf. [2], [4]). A simple generalization dealing with a translation of the interval [0, 2π) was given in [5]. Thus, the mentioned problem can be considered on any interval whose length is equal to 2π, i.e., on any interval of the form [L, L + 2π), L ∈ R. Definition 1. A quadrature rule of the form Z

L+2π

f (x)w(x) dx = L

n X

wν f (xν ) + Rn (f ),

ν=0

where L ∈ R, L ≤ x0 < x1 < · · · < xn < L + 2π, has a trigonometric degree of exactness equal to d if Rn (f ) = 0 for all f ∈ Td and there exists g ∈ Td+1 such that Rn (g) 6= 0. Turetzkii tried to increase the trigonometric degree of exactness of a quadrature rule (2) in such a way that he did not specify in advance the nodes xν , ν = 0, 1, . . . , 2n. His approach was a simulation of the development of Gaussian quadrature rules for algebraic polynomials. He proved that the trigonometric degree of exactness of the quadrature rule (2) is 2n if and only if the nodes xν (∈ [0, 2π)), ν = 0, 1, . . . , 2n, are zeros of a trigonometric polynomial of semiinteger degree n + 12 which is orthogonal on [0, 2π) with respect to the weight function w(x) to every trigonometric polynomial of a semi-integer degree less than or equal to n − 12 . It is said that such a quadrature rule is of Gaussian type, because it has the maximal trigonometric degree of exactness. The trigonometric polynomial of semi-integer degree An+ 21 , which is orthogonal on [0, 2π) with respect to a weight function w(x) to every trigonometric polynomial of a semi-integer degree less than or equal to n − 12 , with given leading coefficients cn and dn , is uniquely determined (see [7, §3]) and it has in [0, 2π) exactly 2n + 1 distinct simple zeros (see [7, Theorem 3]). Orthogonal trigonometric polynomials of semi-integer degree were studied in detail in [5], where two choices of the leading coefficients for such orthogonal systems were considered: cn = 1, dn = 0 and cn = 0, dn = 1. It was proved that 2

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such orthogonal systems satisfied some five–term recurrence relations. Also, a numerical method for constructing Gaussian type quadratures based on the five–term recurrence relations was presented. For some special weight functions the explicit formulas for the five-term recursion coefficients were obtained in [6]. A concept of orthogonality in the space T 1/2 can be considered more generally. Namely, it is known that orthogonal algebraic polynomials can be defined with respect to a moment functional (see [1], [4]). In this paper we consider orthogonal trigonometric polynomials of semi-integer degree with respect to a linear functional defined on the vector space T. The paper is organized as follows. The second section gives a general concept of trigonometric polynomials of semi-integer degree which are orthogonal with respect to a given linear functional. It is also an introduction to a very suitable matrix notation for this purpose. The third section establishes the existence of three-term recurrence relations. Also, the corresponding Christofell-Darboux formulas are proved.

2

Orthogonality with respect to a moment functional

S Definition 2. Let m0 be a real number, {mC n }, {mn }, n ∈ N, two sequences of real numbers, and let L be a linear functional defined on the vector space T by

L[1] = m0 ,

L[cos nx] = mC n,

L[sin nx] = mSn ,

n ∈ N.

Then L is called the moment functional determined by m0 and by the sequences S {mC n }, {mn }. For a 2 × 2 type matrix [tij ], whose entries are trigonometric polynomials, for the brevity we denote by L[[tij ]] the following 2 × 2 type matrix [L[tij ]]. For each k ∈ N0 let denote by xk the column vector h ³ ³ 1´ 1 ´ iT xk = cos k + x sin k + x . 2 2 For k, j ∈ N0 , let define matrices mk,j by mk,j = L[xk (xj )T ].

(3)

By definition, mk,j is a matrix of type 2 × 2 and its elements are linear combiS nations of the moments m0 , {mC n }, {mn }, n ∈ N. For each n ∈ N0 , the matrices mk,j , k, j = 0, 1, . . . , n, are used to define the so-called moment matrix Mn = [mk,j ]nk,j=0 .

(4)

We denote its determinant by ∆n , i.e., ∆n = det Mn . Lemma 1. The moment matrix Mn , n ∈ N0 , is a symmetric matrix. 3

(5)

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Proof. It is easy to see that all of the matrices mk,k , k = 0, 1, . . . , n, are symmetric. Since £¡ ¢T ¤ L[xj (xk )T ] = L xk (xj )T , it follows that mj,k = mTk,j , k, j = 0, 1, . . . , n, i.e., the moment matrix Mn is symmetric. If L is a moment functional and An+ 12 (x) is a trigonometric polynomial 1/2

of semi-integer degree n + 12 such that L[An+ 12 t] = 0 for every t ∈ Tn−1 , then An+ 21 is an orthogonal trigonometric polynomial of semi-integer degree n + 12 with respect to the moment functional L. One can start with the © ª 1/2 basis cos(k + 21 )x, sin(k + 12 )x : k = 0, 1, . . . , n of Tn and use the GrammSchmidt orthogonalization method to generate a new basis, whose elements are 1/2 mutually orthogonal with respect to L. It is obvious that in any basis of Tn , for all k = 0, 1, . . . , n we have two linearly independent trigonometric polynomials of the same semi-integer degree k + 21 . The orthogonal trigonometric polynomials of semi-integer degree with respect to a suitable weight function w on [0, 2π), considered in [7], [5] and [6], are orthogonal trigonometric polynomials of semi-integer degree with respect to the linear functional Lw , defined by Z 2π Lw [t] := t(x) w(x) dx, t ∈ T. (6) 0

It is required that the orthogonal trigonometric polynomial of semi-integer de1/2 gree n + 12 is orthogonal to every element of Tn−1 . As a matter of fact, the orthogonality is considered only in terms of trigonometric polynomials of different semi-integer degrees, i.e., trigonometric polynomials of the same semiinteger degree have to be orthogonal to all trigonometric polynomials of lower semi-integer degrees, but they may not be orthogonal among themselves. So, we follow this idea when we define orthogonal trigonometric polynomials of semi-integer degree with respect to a moment functional. Let us denote by h iT (1) (2) Ak (x) = Ak+ 1 (x) Ak+ 1 (x) , 2

k ∈ N0 ,

2

the vector whose elements are two linearly independent trigonometric polynomials of semi-integer degree k + 21 . We use the following notation n o (1) (2) (1) (2) S{A0 (x), . . . , An (x)} = A 1 (x), A 1 (x), . . . , An+ 1 (x), An+ 1 (x) , n ∈ N0 , 2

2

2

2

for the set consisting of components of the vectors Ak (x), k = 0, 1, . . . , n. We may also call Ak (x) a trigonometric polynomial of semi-integer degree k + 21 . By 0 we denote the zero vector [0 0]T , as well as the 2 × 2 type zero matrix, which will be clear from the context, and finally, by I and Ib we denote

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the identity matrix of type 2 × 2 and the matrix " # 0 −1 Ib = , 1 0 respectively. Definition 3. Let L be a moment functional. A sequence of trigonometric polynomials of semi-integer degree {An (x)}+∞ n=0 is said to be orthogonal with respect to L if the following conditions are satisfied: L[xk ATn ] = 0,

k < n;

L[xn ATn ] = Kn ,

(7)

where Kn , n ∈ N0 , is an invertible 2 × 2 type matrix. Notice that in Definition 3 is indirectly assumed that L permits the existence +∞ of such an orthogonal sequence {An (x)}n=0 . Lemma 2. Let L be a moment functional and {Ak }+∞ k=0 be a sequence of orthogonal trigonometric polynomials of semi-integer degree with respect to L. Then 1/2 the set S{A0 , A1 , . . . , An } forms a basis for Tn , n ∈ N0 . 1/2

Proof. Since dim(Tn ) = 2n + 2 and the set S{A0 , A1 , . . . , An } has 2n + 2 elements, only we need to prove is a linear independence. Let consider the sum aT0 A0 + aT1 A1 + · · · + aTn An , where ak = [a1k a2k ]T , ajk ∈ R, k = 0, 1, . . . , n, j = 1, 2. Multiplying the previous sum from the right hand side by the (xk )T and applying L, due to orthogonality, it follows from aT0 A0 +aT1 A1 +· · ·+aTn An = 0 that aTk KkT = 0. Since Kk is invertible, it follows that ak = 0. Therefore, S{A0 , A1 , . . . , An } is a linearly independent system, which forms a basis for 1/2 Tn . Orthogonal trigonometric polynomials of semi-integer degree An , n ∈ N0 , can be written as An = Cn,n xn + Cn,n−1 xn−1 + · · · + Cn,0 x0 ,

(8)

where Cn,k , k = 0, 1, . . . , n, are 2 × 2 type real matrices. The matrix Cn,n is called the leading coefficient of An . Lemma 3. Let L be a moment functional and An , n ∈ N0 , be an orthogonal trigonometric polynomial of semi-integer degree n + 21 with respect to L. Then the leading coefficient Cn,n is an invertible matrix. 0 Proof. According to Lemma 2, there exists a matrix Cn,n such that 0 xn = Cn,n An + Bn−1 , 1/2

where Bn−1 is a vector whose components belong to Tn−1 . Comparing coeffi0 cients of xn , we obtain Cn,n Cn,n = I, which implies that Cn,n is invertible. 5

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If {An } is a sequence of orthogonal trigonometric polynomials of semi-integer e n = C −1 An yields degree and Cn,n denotes the leading coefficient of An , then A n,n e n } of orthogonal trigonometric polynomithe corresponding monic sequence {A als of semi-integer degree. Namely, we have e T ] = L[xk AT (C −1 )T ] = 0, L[xk A n n n,n and

k = 0, 1, . . . , n − 1,

e T ] = L[xn AT (C −1 )T ] = Kn (C −1 )T , L[xn A n n n,n n,n

−1 T and Kn (Cn,n ) is invertible matrix by Lemma 3. For an orthogonal system of trigonometric polynomials of semi-integer degree {An } with respect to a moment functional L, let us denote by µn , n ∈ N0 , the following matrix µn = L[An ATn ]. (9)

It is obvious that the matrix µn , n ∈ N0 , given by (9) is symmetric. Lemma 4. Let L be a moment functional and An , n ∈ N0 , be an orthogonal trigonometric polynomial of semi-integer degree n + 21 with respect to L. Then the matrix µn , n ∈ N0 , given by (9) is invertible. Proof. Since µn = L[An ATn ] = Cn,n L[xn ATn ] = Cn,n Kn , it is invertible according to Lemma 3. Theorem 1. Let L be a moment functional and An , n ∈ N0 , an orthogonal trigonometric polynomial of semi-integer degree n + 21 with respect to L. Then An is uniquely determined by the matrix Kn . Proof. Suppose contrary that there exist An and A0n , both satisfying the orthog0 denote the leading onality conditions (7) with the same Kn . Let Cn,n and Cn,n 0 coefficients of An and An , respectively. Since the system S{A0 , A1 , . . . , An } 1/2 forms a basis of Tn , the elements of A0n can be written in terms of that basis. So, there exist 2 × 2 type matrices Ck , k = 0, 1, . . . , n, such that A0n = Cn An + Cn−1 An−1 + · · · + C0 A0 . Multiplying the both hand sides of the above equation from the right by ATk , k = 0, 1, . . . , n−1, and applying the moment functional L, we get that Ck L[Ak ATk ] = 0, k = 0, 1, . . . , n − 1, by orthogonality. According to Lemma 4, it follows that Ck = 0 for all k = 0, 1, . . . , n − 1, i.e., A0n = Cn An . Comparing the 0 0 −1 leading coefficients leads to Cn,n = Cn Cn,n , i.e., Cn = Cn,n Cn,n and An = 0−1 0 Cn,n Cn,n An . By using (7) we obtain 0−1 T 0−1 T Kn = L[xn ATn ] = L[xn A0T n ](Cn,n Cn,n ) = Kn (Cn,n Cn,n ) , 0−1 0 which implies that Cn,n Cn,n = I. Thus, Cn,n = Cn,n and An = A0n .

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Theorem 2. Let L be a moment functional. A system of orthogonal trigonometric polynomials of semi-integer degree with respect to the moment functional L exists if and only if ∆n 6= 0, n ∈ N0 . Proof. Using the expanded form (8) of An and the matrices mk,j defined by (3), we get L[xk ATn ]

= L[xk (Cn,n xn + Cn,n−1 xn−1 + · · · + Cn,0 x0 )T ] T T T = mk,n Cn,n + mk,n−1 Cn,n−1 + · · · + mk,0 Cn,0 .

It is easy to see that the orthogonality conditions (7) are equivalent to the following system of linear equations 

T Cn,0 .. .

  Mn  T  Cn,n−1 T Cn,n

 0   ..    .  , =   0  Kn 



(10)

where Mn is the moment matrix, defined by (4). Let us suppose that a system of orthogonal trigonometric polynomials of semi-integer degree with respect to the moment functional L exists. For each Kn it is unique by Theorem 1. Hence, the system of equations (10) has a unique solution, which implies that ∆n 6= 0. Let us now suppose that ∆n 6= 0. Then for each invertible matrix Kn the systemPof equations (10) has a unique solution (Cn,0 , . . . , Cn,n ). Let denote n An = k=0 Cn,k xk . The system (10) is equivalent to the following L[xk ATn ] = 0,

k = 0, 1, . . . , n − 1;

L[xn ATn ] = Kn ,

i.e., orthogonal trigonometric polynomials of semi-integer degree with respect to the moment functional L exist. Definition 4. A moment functional L is said to be regular if ∆n 6= 0 for all n ∈ N0 . Definition 5. A moment functional L is said to be positive definite if for all t ∈ T 1/2 , t 6= 0, the following inequality L[t2 ] > 0 holds. Theorem 3. If a moment functional L is positive definite, then ∆n > 0 for all n ∈ N0 . Proof. Let us assume that L is positive definite. Let v be an eigenvector of the moment matrix Mn , corresponding to an eigenvalue λ. a trigonometric PFor n polynomial of semi-integer degree n+ 12 defined by t(x) = k=0 vkT xk , it follows that vT Mn v = L[t2 ] > 0. On the other hand, vT Mn v = λkvk2 , which implies that λ > 0. Therefore, all eigenvalues are positive and then ∆n = det Mn > 0.

7

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According to Theorems 2 and 3 we have the following result: Corollary 1. For a positive definite moment functional L, there exists a system of orthogonal trigonometric polynomials of semi-integer degree with respect to L, i.e., every positive definite moment functional is regular. Definition 6. Let L be a positive definite moment functional. A system of trigonometric polynomials of semi-integer degree {A∗n (x)}+∞ n=0 is said to be orthonormal with respect to L if the following conditions are satisfied L[A∗m (A∗n )T ] = δm,n I,

m, n ∈ N0 ,

(11)

where δm,n is Kronecker delta function. Lemma 5. Let L be a regular moment functional and let {An } be a system of orthogonal trigonometric polynomials of semi-integer degree with respect to L. Then L is a positive definite moment functional if and only if all of the matrices µn , n ∈ N0 , given by (9), are positive definite. Proof. If L is a positive definite moment functional, then for any nonzero vector a with real entries, t(x) = aT An (x) is a nonzero trigonometric polynomial of semi-integer degree n + 21 by Lemma 2. Therefore, aT µn a = L(t2 ) > 0, which means that µn is a positive definite matrix. Let us now suppose that all of the matrices µn , n ∈ N0 , given by (9), are positive definite. According to Lemma 2, every nonzero trigonometric polynomial P of semi-integer degree n + 12 can be represented in the following form n t(x) = k=0 tk (x), where tk (x) = aTk Ak (x), k = 0, 1, . . . , n, and an differs from the zero vector. Because of orthogonality we get L[t2 ] =

n X

L[t2k ] =

k=0

n X

aTk µk ak ,

k=0

which is positive since all of the matrices µn , n ∈ N0 , are positive definite. Theorem 4. If L is a positive definite moment functional, then there exists a system of orthonormal trigonometric polynomials of semi-integer degree {A∗n (x)} with respect to L. Proof. Let {An } be a system of orthogonal trigonometric polynomials of semiinteger degree with respect to L, and let µn , n ∈ N0 , be the matrix defined by (9). According to Lemma 5, the matrix µn is positive definite. Let ν n be the positive definite square root of µn , i.e., the unique positive definite matrix such that µn = ν n ν n (see [8]). Since µn is a symmetric matrix, the matrix ν n is also symmetric. Let define A∗n (x) = ν −1 n An (x). Then we have T −1 −1 −1 L[A∗n (A∗n )T ] = ν −1 n L[An An ]ν n = ν n µn ν n = I,

which proves the assertion.

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Remark 1. It is easy to see that a system of orthonormal trigonometric polynomials of semi-integer degree with respect to a positive definite moment functional L is not unique. As a matter of fact, if {A∗n } is an orthonormal system, then for any orthogonal 2 × 2 type matrix On , {On A∗n } is also an orthonormal sysb∗ tem with respect to the same moment functional L. Moreover, if A∗n and A n are two vectors of orthonormal trigonometric polynomials of semi-integer degree b ∗ differ with respect to a positive definite moment functional L, then A∗n and A n by multiplication by an orthogonal 2 × 2 type matrix.

3

Three-term recurrence relations

It is well known that orthogonal algebraic polynomials satisfy the three-term recurrence relation (see [1], [3], [4]). Such a recurrence relation is one of the most important piece of information for the constructive and computational use of orthogonal polynomials. Knowledge of the recursion coefficients allows the zeros of orthogonal polynomials to be computed as eigenvalues of a symmetric tridiagonal matrix, and with them the Gaussian quadrature rule, and also allows an efficient evaluation of expansions in orthogonal polynomials. For the orthogonal trigonometric polynomials of semi-integer degree with respect to a regular moment functional L, there exists three-term recurrence relations in a vector-matrix form. Actually, two kinds of recurrence relations exist, the first one with cosine function, and the second one with sine function.

3.1

Three-term recurrence relation with cosine function

Theorem 5. Let L be a regular moment functional and {An } be a system of orthogonal trigonometric polynomials of semi-integer degree with respect to L. Then, C C 2 cos xAn = γ C n An+1 + αn An + β n An−1 ,

n = 0, 1, . . . ;

A−1 = 0,

(12)

C C C where β C 0 is arbitrary 2 × 2 type matrix and αn , β n and γ n are 2 × 2 type matrices given by −1 T γC n = L[2 cos xAn An+1 ]µn+1 ,

αC n βC n

= =

L[2 cos xAn ATn ]µ−1 n , C T −1 µn (γ n−1 ) µn−1 , n

n ∈ N0 ,

(13)

n ∈ N0 , ∈ N.

Proof. Since the components of 2 cos xAn are trigonometric polynomials of semiinteger degree n + 1 + 21 , they can be represented as a linear combination of orthogonal trigonometric polynomials of semi-integer degree at most n + 1 + 12 by Lemma 2. Therefore, in a vector notation, there exist 2 × 2 type matrices Ck , k = 0, 1, . . . , n + 1, such that 2 cos xAn = Cn+1 An+1 + Cn An + · · · + C0 A0 .

9

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Multiplying the both hand sides of the previous equation by ATk , k = 0, 1, . . . , n − 2, from the right and applying the moment functional L, due to orthogonality we obtain Ck µk = L[2 cos xAn ATk ] = 0, which implies that Ck = 0, since µk is an invertible matrix. Therefore, the three-term recurrence relation (12) holds. Let us now multiply the both hand sides of the equation (12) by ATn from the right and apply the moment functional L. Due to orthogonality we obtain T C L[2 cos xAn ATn ] = αC n L[An An ] = αn µn ,

which yields the expression for αC n . In the similar way, multiplying the equation (12) by ATn+1 from the right and applying the moment functional L, because of orthogonality, we obtain the expression for γ C n. To finish the proof, we write the recurrence relation (12) with n + 1 instead of n, transpose the written equation, multiply by An from the left and apply L. Then, we obtain L[2 cos xAn ATn+1 ] = µn β C n+1 , T i.e., γ C n µn+1 = µn β n+1 . Changing n by n−1, it is easy to get what is stated.

Remark 2. Although the matrix coefficient β C 0 in (12) can be chosen arbitrarily, since it multiplies A−1 = 0, it is convenient for later purposes to define β C 0 = µ0 . C Lemma 6. All of the matrices γ C n , n ∈ N0 , and β n , n ∈ N, in (12) are invertible.

Proof. Writing An and An+1 in the recurrence relation (12) in the expanded forms (8) and comparing the highest coefficients at both hand sides it follows that Cn,n = γ n Cn+1,n+1 . Since the matrices Cn,n and Cn+1,n+1 are invertible by Lemma 3, the matrix γ n , is also invertible for all n ∈ N0 . The assertion for the matrix β n , n ∈ N, follows from the last equation in (13) and Lemma 4. We proved in Theorem 4 the existence of an orthonormal sequence {A∗n } of trigonometric polynomials of semi-integer degree for a positive definite moment functional L. For such a case, the recurrence relation can be considered, too. The steps in proof are the same as in Theorem 5 with µn = I, n ∈ N0 . Theorem 6. Let L be a positive definite moment functional and {A∗n } be a system of orthonormal trigonometric polynomials of semi-integer degree with respect to L. Then, ∗C T ∗ ∗ ∗C ∗ 2 cos xA∗n = β ∗C n+1 An+1 +αn An +(β n ) An−1 , n = 0, 1, . . . ; A−1 = 0, (14) ∗C ∗C where β ∗C 0 is arbitrary 2×2 type matrix and αn and β n are 2×2 type matrices given by ∗ ∗ T α∗C n = L[2 cos xAn (An ) ],

∗ ∗ T β ∗C n = L[2 cos xAn−1 (An ) ],

n ∈ N0 .

(15)

Remark 3. It is easy to see that each α∗C n , n ∈ N0 , is symmetric and all of , n ∈ N, are invertible. the matrices β ∗C n 10

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3.2

917

Three-term recurrence relation with sine function

Let {An } be a system of orthogonal trigonometric polynomials of semi-integer degree with respect to a regular moment functional L. Since the components of 2 sin xAn are trigonometric polynomials of semiinteger degree n + 1 + 12 , they can also be represented as a linear combination of orthogonal trigonometric polynomials of semi-integer degree at most n + 1 + 12 by Lemma 2. A consequence of this fact is that one can consider the following representation 2 sin xAn = Sn+1 An+1 + Sn An + · · · + S0 A0 , for some matrices Sk , k = 0, 1, . . . , n + 1. By using the above equation, in analogous way as in the proof of Theorem 5, the following result can be proved. Theorem 7. Let L be a regular moment functional and {An } be a system of orthogonal trigonometric polynomials of semi-integer degree with respect to L. Then, 2 sin xAn = γ Sn An+1 + αSn An + β Sn An−1 ,

n = 0, 1, . . . ;

A−1 = 0,

(16)

where αSn , β Sn and γ Sn are 2 × 2 type matrices given by γ Sn = L[2 sin xAn ATn+1 ]µ−1 n+1 , αSn β Sn

= =

L[2 sin xAn ATn ]µ−1 n , T −1 S µn (γ n−1 ) µn−1 , n

n ∈ N0 ,

(17)

n ∈ N0 , ∈ N.

The matrix coefficient β S0 can be chosen arbitrarily, but we define it as = µ0 . Analogously as in Lemma 6, we can prove that all of the matrices γ Sn , n ∈ N0 , and β Sn , n ∈ N, are invertible. With µn = I, n ∈ N0 , the following result can be easily proved. βC 0

Theorem 8. Let {A∗n } be a system of orthonormal trigonometric polynomials of semi-integer degree with respect to a positive definite moment functional L. Then we have the following three-term recurrence relation with sine function, ∗S T ∗ ∗ ∗S ∗ 2 sin xA∗n = β ∗S n+1 An+1 +αn An +(β n ) An−1 , n = 0, 1, . . . ; A−1 = 0, (18) ∗S where α∗S n and β n are 2 × 2 type matrices given by ∗ ∗ T α∗S n = L[2 sin xAn (An ) ],

∗ ∗ T β ∗S n = L[2 sin xAn−1 (An ) ],

n ∈ N0 .

(19)

The matrix coefficient β ∗S 0 can be chosen arbitrarily. ∗S Also, each α∗S n , n ∈ N0 , is symmetric and all of the matrices β n , n ∈ N, are invertible.

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3.3

Monic orthogonal trigonometric polynomials of semiinteger degree

In the sequel, by {An (x)} we will denote the sequence of the monic orthogonal trigonometric polynomials of semi-integer degree with respect to a regular moment functional L. Thus, An (x) is a vector of two trigonometric polynomials of semi-integer degree, such that the first one is with the leading cosine function, and the second one with the leading sine function. We use the following quite natural notation  C  An+ 1 (x) 2 , An (x) =  ASn+ 1 (x) 2

where

AC (x) n+ 21

AC n+ 21 (x)

and

ASn+ 1 (x) 2

have the following expanded forms

¸ n−1 ³ ³ ³ X· 1´ 1´ 1´ (n) (n) = cos n + x+ cν cos ν + x + dν sin ν + x , (20) 2 2 2 ν=0

¸ n−1 ³ ³ ³ X· 1´ 1´ 1´ ASn+ 1 (x) = sin n + x+ x + gν(n) sin ν + x , (21) fν(n) cos ν + 2 2 2 2 ν=0 (n)

(n)

(n)

(n)

for some real coefficients cν , dν , fν and gν , ν = 0, 1, . . . , n − 1. For a monic system of orthogonal trigonometric polynomials of semi-integer degree {An } with respect to a regular moment functional L, the matrix γ C n in (12) is the identity matrix I (see proof of Lemma 6), hence, recurrence relation (12) has the following form C 2 cos xAn = An+1 + αC n An + β n An−1 ,

n = 0, 1, . . . ;

A−1 = 0.

(22)

C −1 Here, we have β C n = µn µn−1 , n ∈ N, β 0 = µ0 . When the monic orthogonal trigonometric polynomials of semi-integer degree are in question, the situation with the recurrence relation with sine function is something different from the case with cosine function. The reason for this lies in the following simple equality " # " # cos(k + 21 )x sin(k + 1 + 12 )x − sin(k − 12 )x 2 sin x = . sin(k + 21 )x − cos(k + 1 + 21 )x + cos(k − 21 )x

In order to obtain the recurrence relation with sine function for the monic orthogonal trigonometric polynomials of semi-integer degree we only need to see the following equalities: # " # " cos(k + 12 )x sin(k + 21 )x b = , Ib2 = −I. I 1 1 − cos(k + 2 )x sin(k + 2 )x

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For the monic system of orthogonal trigonometric polynomials of semiinteger degree {An } with respect to a regular moment functional L, the recurrence relation (16) has the following form b n+1 + αSn An + β Sn An−1 , 2 sin xAn = −IA

n = 0, 1, . . . ;

A−1 = 0.

(23)

b T = I, b from the last equation in (17) and (23) we get here β Sn = Since (−I) −1 b µn Iµn−1 , n ∈ N.

3.4

Orthonormal trigonometric polynomials of semi-integer degree

If L is a positive definite moment functional, then by {A∗n (x)} we will denote the sequence of the orthonormal trigonometric polynomials of semi-integer degree with respect to L, given by A∗n (x) = ν −1 n An (x), where the matrix ν n is the positive square root of the matrix µn , n ∈ N0 . As it was said, {An (x)} is a sequence of the monic trigonometric polynomials of semi-integer degree with ∗S respect to L. Then, the recursion coefficients β ∗C n and β n are given as follows β ∗C n

β ∗S n

=

T −1 L[2 cos xA∗n−1 (A∗n )T ] = ν −1 n−1 L[2 cos xAn−1 An ]ν n

=

T −1 ν −1 n−1 L[An An ]ν n

=

T −1 L[2 sin xA∗n−1 (A∗n )T ] = ν −1 n−1 L[2 sin xAn−1 An ]ν n −1 b −1 b b n ATn ]ν −1 ν −1 L[−IA Iµn ν −1 Iν n . n = −ν n = −ν

=

=

−1 ν −1 n−1 µn ν n

n−1

=

(24)

ν −1 n−1 ν n ;

n−1

(25)

n−1

∗S Some simple properties of the recursion coefficients matrices α∗C n , αn , n ∈ ∗C ∗S N0 , and β n , β n , n ∈ N, of the recurrence relations (14) and (18) are given in Subsections 3.1 and 3.2. The following result gives some connections between these coefficients.

Theorem 9. Let {A∗n (x)} be the sequence of the orthonormal trigonometric polynomials of semi-integer degree with respect to a positive definite moment functional L, satisfying the three-term recurrence relations (14) and (18). Then the recursion coefficients matrices satisfy the following commutativity conditions: ∗S ∗S ∗C β ∗C k β k+1 = β k β k+1 ∗S ∗S ∗C ∗S ∗S ∗C ∗C β ∗C k+1 αk+1 + αk β k+1 = αk β k+1 + β k+1 αk+1 ∗S T ∗C ∗C ∗S T ∗S (β ∗C k ) β k + αk αk + β k+1 (β k+1 ) ∗C T ∗S ∗S ∗C T ∗C = (β ∗S k ) β k + αk αk + β k+1 (β k+1 ) , ∗S for k ≥ 0, where β ∗C 0 = β 0 = 0.

13

(26)

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MILOVANOVIC ET AL: MOMENT FUNCTIONAL...

Proof. Using the recurrence relations (14) and (18), there are two different ways of calculating the matrices L[4 cos x sin xA∗k (A∗k+2 )T ], L[4 cos x sin xA∗k (A∗k+1 )T ], L[4 cos x sin xA∗k (A∗k )T ], which lead to the desired commutativity equalities. Thus, using recurrence relations and the fact that {A∗n (x)} is orthonormal with respect to L, we have L[4 cos x sin xA∗k (A∗k+2 )T ] = L[2 cos xA∗k 2 sin x(A∗k+2 )T ] h¡ ¢ ∗C T ∗ ∗ ∗C ∗ = L β ∗C k+1 Ak+1 + αk Ak + (β k ) Ak−1 × ¡ ¢i T ∗ T ∗S T ∗ T ∗S × (A∗k+3 )T (β ∗S k+3 ) + (Ak+2 ) (αk+2 ) + (Ak+1 ) β k+2 ∗C ∗S ∗ ∗ T ∗S = L[β ∗C k+1 Ak+1 (Ak+1 ) β k+2 ] = β k+1 β k+2 ,

and, analogously, ∗C L[4 cos x sin xA∗k (A∗k+2 )T ] = L[2 sin xA∗k 2 cos x(A∗k+2 )T ] = β ∗S k+1 β k+2 ,

which leads to the first equation in (26). Further, from L[4 cos x sin xA∗k (A∗k+1 )T ] = L[2 cos xA∗k 2 sin x(A∗k+1 )T ] h¡ ¢ ∗C T ∗ ∗ ∗C ∗ = L β ∗C k+1 Ak+1 + αk Ak + (β k ) Ak−1 × ¡ ¢i T ∗ T ∗S T ∗ T ∗S × (A∗k+2 )T (β ∗S k+2 ) + (Ak+1 ) (αk+1 ) + (Ak ) β k+1 ∗S ∗C ∗S = β ∗C k+1 αk+1 + αk β k+1 ,

and L[4 cos x sin xA∗k (A∗k+1 )T ] =

L[2 sin xA∗k 2 cos x(A∗k+1 )T ]

∗C ∗S ∗C = β ∗S k+1 αk+1 + αk β k+1 ,

we obtain the second equation in (26). Notice, that we here use the fact that ∗S matrices α∗C k and αk are symmetric. Finally, from L[4 cos x sin xA∗k (A∗k )T ] = L[2 cos xA∗k 2 sin x(A∗k )T ] h¡ ¢ ∗C T ∗ ∗ ∗C ∗ = L β ∗C k+1 Ak+1 + αk Ak + (β k ) Ak−1 × ¢i ¡ T ∗S T ∗ T ∗S T ∗ × (A∗k+1 )T (β ∗S k+1 ) + (Ak ) (αk ) + (Ak−1 ) β k ∗C T ∗S ∗S T ∗C ∗S = β ∗C k+1 (β k+1 ) + αk αk + (β k ) β k ,

and L[4 cos x sin xA∗k (A∗k )T ] = =

L[2 sin xA∗k 2 cos x(A∗k )T ] ∗S T ∗C ∗C T ∗S ∗C β ∗S k+1 (β k+1 ) + αk αk + (β k ) β k ,

we get the third equation in (26). 14

MILOVANOVIC ET AL: MOMENT FUNCTIONAL...

3.5

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Christoffel-Darboux formulas

As a direct corollary of the three-term recurrence relation for algebraic orthogonal polynomials the Christoffel-Darboux formula can be proved (see [1], [3], [4]). According to the fact that we proved three-term recurrence relations for orthogonal trigonometric polynomials of semi-integer degree, a similar formula can be expected in this trigonometric case. Actually, since we have two recurrence relation, we have two Christoffel-Darboux formulas. Theorem 10 (Christoffel-Darboux formulas). Let {A∗n } be a sequence of orthonormal trigonometric polynomials of semi-integer degree with respect to a positive definite linear functional. Then, for all x, y ∈ R, and for all nonnegative integers n, the following formula n X

(A∗k (x))T A∗k (y) =

∗C T ∗ ∗ T ∗ ∗ (β ∗C n+1 An+1 (x)) An (y) − (An (x)) (β n+1 An+1 (y)) , 2(cos x − cos y)

(A∗k (x))T A∗k (y) =

∗S T ∗ ∗ ∗ T ∗ (β ∗S n+1 An+1 (x)) An (y) − (An (x)) (β n+1 An+1 (y)) 2(sin x − sin y)

k=0 n X k=0

hold. Proof. Put σ−1 = 0 and ¡ ¢T ∗ ¡ ¢ ∗ ∗ σk = β ∗C Ak (y) − (A∗k (x))T β ∗C k+1 Ak+1 (x) k+1 Ak+1 (y) ,

k = 0, 1, . . . , n.

By using the three-term recurrence relation (14), we get ³

σk

´T ∗C T ∗ ∗ 2 cos xA∗k (x) − α∗C A∗k (y) k Ak (x) − (β k ) Ak−1 (x) ´ ¡ ¢T ³ ∗C T ∗ ∗ − A∗k (x) 2 cos yA∗k (y) − α∗C k Ak (y) − (β k ) Ak−1 (y) ¢T ¢T ¡ ∗C T ¢ ∗ = 2(cos x − cos y)(A∗k (x) A∗k (y) − (A∗k (x) (αk ) − α∗C Ak (y) k ³¡ ´ ¢T ∗C ∗ T ∗C T ∗ ∗ ∗ − Ak−1 (x) β k Ak (y) − (Ak (x)) (β k ) Ak−1 (y) .

=

Since the all of the matrices α∗C are symmetric (see Remark 3), the second k term on the right hand side of the previous expression of σk is equal to zero. The third term of the same expression can be written as follows ¡

A∗k (x)

¡ ∗ ¢T ∗C ∗ T ∗ β k Ak (y) (β ∗C k ) Ak−1 (y) − Ak−1 (x) ´T ³ ¡ ¢T ∗ ∗ = β ∗C A∗k−1 (y) − A∗k−1 (x) β ∗C k Ak (x) k Ak (y) = σk−1 .

¢T

Therefore, we have σk = 2(cos x − cos y)(A∗k (x) 15

¢T

A∗k (y) + σk−1 ,

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MILOVANOVIC ET AL: MOMENT FUNCTIONAL...

i.e.,

¢T (cos x − cos y)(A∗k (x) A∗k (y) = σk − σk−1 .

Summing the previous equality for all k = 0, 1, . . . , n, we get the first formula. In the same way, by using the three-term recurrence relation (18), the second formula can be proved. Acknowledgments. The authors were supported in part by the Serbian Ministry of Science and Technological Developments (Project: Orthogonal Systems and Applications, grant number #144004G)

References [1] T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978. [2] R.A. DeVore, G.G. Lorentz, Constructive Approximation, Springer-Verlag, Berlin Heildeberg, 1993. [3] W. Gautschi, Orthogonal Polynomials: Computation and Approximation, Clarendon Press, Oxford, 2004. [4] G. Mastroianni, G.V. Milovanovi´c, Interpolation Processes - Basic Theory and Applications, Springer Monographs in Mathematics, Springer - Verlag, Berlin - Heidelberg, 2008. [5] G.V. Milovanovi´c, A.S. Cvetkovi´c, and M.P. Stani´c, Trigonometric orthogonal systems and quadrature formulae, Comput. Math. Appl. 56 (11), 2915– 2931 (2008). [6] G.V. Milovanovi´c, A.S. Cvetkovi´c, and M.P. Stani´c, Explicit formulas for five-term recurrence coefficients of orthogonal trigonometric polynomials of semi-integer degree, Appl. Math. Comput. 198, 559–573 (2008). [7] A.H. Turetzkii, On quadrature formulae that are exact for trigonometric polynomials, East J. Approx. 11, 337–359 (2005) (translation in English from Uchenye Zapiski, Vypusk 1(149), Seria math. Theory of Functions, Collection of papers, Izdatel’stvo Belgosuniversiteta imeni V.I. Lenina, Minsk, 31–54 (1959)) [8] E.W. Weisstein, Positive Definite Matrix, From MathWorld–A Wolfram Web Resource http://mathworld.wolfram.com/PositiveDefiniteMatrix.html

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JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.5, 923-932, 2011, COPYRIGHT 2011 EUDOXUS PRESS, 923 LLC

Generalized ∆m−Statistical Convergence in Probabilistic Normed Space ¨ *Ayhan ESI and **M. Kemal Ozdemir *Adiyaman University, Science and Arts Faculty, Department of Mathematics, TR-02040, Adiyaman, Turkey e-mail: [email protected] **Inonu University, Science and Arts Faculty, Department of Mathematics, TR-44280, Malatya, Turkey e-mail: [email protected] March 19, 2010 Abstract λ In this paper we define the concepts of S∆ m −statistical convergence λ and S∆ m −statistically Cauchy in probabilistic normed space and give some results. The main purpose of this paper is to generalize the results on statistical convergence in probabilistic normed space given by Karaku¸s [10] and Alotaibi [1] earlier. 2000 Mathematics Subject Classification: 40A05, 60B99. Keywords and Phrases: Statistical convergence, Difference sequence, t-norm, Probabilictic normed space.

1

Introduction and Background

An interesting and important generalization of the notion of metric space was introduced by Menger [12] under the name of statistical metric, which is now called probabilistic metric space. The notion of a probabilistic metric space corresponds to the situations when we do not know exactly the distance between two points, we know only probabilities of possible values of this distance. The theory of probabilistic metric space was developed by numerous authors, as it can be realized upon consulting the list of references in [4], as well as those in [15, 16]. Probabilistic normed spaces (briefly, PN-spaces) are linear spaces in which the norm of each vector is an appropriate probability distribution function rather than a number. Such spaces were introduced by Serstnev in 1963 [17]. In [2], Alsina et al. gave a new definition of PN-spaces which includes Serstnev’s a special case and leads naturally to the identification of the principle class

1

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ESI, OZDEMIR: STATISTICAL CONVERGENCE

of PN-spaces, the Menger spaces. An important family of probabilistic metric spaces are probabilistic normed spaces. The theory of probabilistic normed spaces is important as a generalization of deterministic results of linear normed space. It seems therefore reasonable to think if the concept of statistical convergence can be extended to probabilistic normed spaces and in that case enquire how the basic properties are affected. But basic properties do not hold on probabilistic normed spaces. The problem is that the triangle function in such spaces. In this paper we extend the concept of ∆m −statistical convergence to probabilistic normed spaces and observe that some basic properties are also preserved on probabilistic normed spaces. Since the study of convergence in PN-spaces is fundamental to probabilistic functional analysis, we feel that the concepts of ∆m −statistical convergence and ∆m −statistical Cauchy in a PN-space would provide a more general framework for the subject.

2

Preliminaries

Now we recall some notations and definitions used in paper. Definition 1 ([2]) A function f : R → R+ o is called a distribution function if it is non-decreasing and left continuous with inf t∈R f (t) = 0 and supt∈R f (t) = 1. We will denote the set of all distribution functions by D. Definition 2 ([2]) A triangular norm, briefly t-norm, is a binary operation on [0, 1] which is continuous, commutative, associative, non-decreasing and has 1 as neutral element, that is, it is the continuous mapping > : [0, 1] × [0, 1] → [0, 1] such that for all a, b, c ∈ [0, 1] : (1) a >1 = a, (2) a >b = b >a, (3) c >d ≥ a >b if c ≥ a and d ≥ b, (4) (a > b) >c = a > (b > c). Example 3 The > operations a >b = max {a + b − 1, 0}, a >b = a.b and a >b = min {a, b} on [0, 1] are t-norms. Definition 4 ([15, 16]) A triple (X, N, >) is called a probabilistic normed space or shortly PN-space if X is a real vector space, N is a mapping from X into D (for x ∈ X, the distribution function N (x) is denoted by Nx and Nx (t) is the value of Nx at t ∈ R) and > is a t-norm satisfying the following conditions: (PN-1) Nx (0) = 0, (PN-2) Nx (t) = 1 for all t > 0 if and only if x = 0, ( ) t for all α ∈ R\ {0}, (PN-3) Nαx (t) = Nx |α| (PN-4) Nx+y (s + t) ≥ Nx (s) > Ny (t) for all x, y ∈ X and s, t ∈ R+ o .

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925

Example 5 Suppose that (X, ∥.∥) is a normed space µ ∈ D with µ (0) = 0 and µ ̸= h, where { 0 , t≤0 h (t) = . 1 , t>0 Define

{ Nx (t) =

(h (t) ) µ

t ||x||

,

x=0

,

x ̸= 0

where x ∈ X, t ∈ R. Then (X, N, >) is a PN-space. For example if we define the functions µ and ν on R by { { 0 , x≤0 0 , x≤0 −1 µ (x) = , ν (x) = x , x>0 e x , x>0 1+x then we obtain the following well-known > norms: { { h (t) ) h (t) , x = 0 ( −∥x∥ Nx (t) = , Mx (t) = t , x = ̸ 0 e t t+∥x∥

,

x=0

,

x ̸= 0

.

We recall the concepts of convergence and Cauchy sequences in a probabilistic normed space. Definition 6 ([1]) Let (X, N, >) is a PN-space. Then a sequence x = (xk ) is said to be convergent to l ∈ X with respect to the probabilistic norm N if, for every ε > 0 and θ ∈ (0, 1) , there exists a positive integer ko such that Nxk −l (ε) > 1 − θ whenever k ≥ ko . It is denoted by N − lim x = L N

or xk → L as k → ∞. Definition 7 ([1]) Let (X, N, >) is a PN-space. Then a sequence x = (xk ) is called a Cauchy sequence with respect to the probabilistic norm N if, for every ε > 0 and θ ∈ (0, 1) , there exists a positive integer ko such that Nxk −xl (ε) > 1 − θ for all k, l ≥ ko . Definition 8 ([1]) Let (X, N, >) is a PN-space. Then a sequence x = (xk ) is said to be bounded in X, if there is a r ∈ R such that Nxk (r) > N 1−θ, where θ ∈ (0, 1). We denote by l∞ the space of all bounded sequences in PN space.

3

∆m -Statistical Convergence on PN-spaces

Definition 9 ([8]) Let K be a subset of N, the set of natural numbers. Then the asymptotic density of K, denoted by δ (K), is defined as δ (K) = lim n

1 |{k ≤ n : k ∈ K}| , n

where the vertical bars denote the cardinality of the enclosed set. Definition 10 ([13]) Let λ = (λn ) be a non-decreasing sequence of positive numbers tending to infinity such that λn+1 ≤ λn + 1, λ1 = 0. Let K ⊂ N. The number δλ (K) = lim n

1 |{n − λn + 1 ≤ k ≤ n : k ∈ K}| λn

3

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ESI, OZDEMIR: STATISTICAL CONVERGENCE

is said to be the λ−density of the set K. If λn = n for all n ∈ N then every λ−density is reduced to asymptotic density. The idea of statistical convergence was first introduced by Steinhaus in 1951 [18] and then studied by various authors, e.g. Salat [14], Fridy [9], Et and Nuray [6], Connor [3], Esi [5] and many others and in normed space by Kolk [11]. Recently Karakus [10] and Alotaibi [1] have studied the concept of statistical convergence in probabilistic normed spaces. Definition 11 ([7]) A number sequence x = (xk ) is said to be statistically convergent to the number l if for each ε > 0, the set K (ε) = {k ≤ n : |xk − l| ≥ ε} has asymptotic density zero, i.e., lim n

1 |{k ≤ n : |xk − l| ≥ ε}| = 0. n

In this case we write st − lim x = l. Definition 12 ([6]) A number sequence x = (xk ) is said to be λ−statistically convergent to the number l if for each ε > 0, lim n

1 |{k ≤ n : |∆m xk − l| ≥ ε}| = 0 n

where m ∈ N, ∆0 x = (xk ), ∆x = (xk − xk+1 ), ∆m x = (∆m xk ) = (∆m−1 xk − ∆m−1 xk+1 ) and ( ) m ∑ m ∆ m xk = (−1)v xk+v . v v=0 In this case we say that the sequence x = (xk ) is ∆m −statistically convergent to l. Definition 13 ([13]) A number sequence x = (xk ) is said to be λ−statistically convergent to the number l if for each ε > 0, the set K (ε) = {n − λn + 1 ≤ k ≤ n : |xk − l| ≥ ε} has λ−density zero, i.e., lim n

1 |{n − λn + 1 ≤ k ≤ n : |xk − l| ≥ ε}| = 0. λn

In this case we write stλ − lim x = l. Definition 14 ([10]) Let (X, N, >) is a PN-space. Then a sequence x = (xk ) is said to be statistically convergent to l ∈ X with respect to the probabilistic norm N provided that for every ε > 0 and θ ∈ (0, 1) δ ({k ∈ N : Nxk −l (ε) ≤ 1 − θ }) = 0, or equivalently lim n

1 |{k ≤ n : Nxk −l (ε) ≤ 1 − θ }| = 0. n

In this case we write stN − lim x = l.

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ESI, OZDEMIR: STATISTICAL CONVERGENCE

927

Definition 15 ([1]) Let (X, N, >) is a PN-space. Then a sequence x = (xk ) is said to be λ−statistically convergent to l ∈ X with respect to the probabilistic norm N provided that for every ε > 0 and θ ∈ (0, 1) δ ({n − λn + 1 ≤ k ≤ n : Nxk −l (ε) ≤ 1 − θ }) = 0, or equivalently lim n

1 |{n − λn + 1 ≤ k ≤ n : Nxk −l (ε) ≤ 1 − θ }| = 0. λn

In this case we write stλ (P N ) − lim x = l. We are now ready to obtain our main results. Definition 16 Let (X, N, >) is a PN-space. Then a sequence x = (xk ) is said to be ∆m −convergent to l ∈ X with respect to the probabilistic norm N provided that for every ε > 0 and θ ∈ (0, 1) there is a positive integer ko such that N∆m xk −l (ε) > 1 − θ whenever k ≥ ko . In this case we write N m

∆ N∆m − lim x = l or xk → l.

Definition 17 Let (X, N, >) is a PN-space. Then a sequence x = (xk ) is said to be ∆m λ −statistically convergent to l ∈ X with respect to the probabilistic norm N provided that for every ε > 0 and θ ∈ (0, 1) δ∆m ({n − λn + 1 ≤ k ≤ n : N∆m xk −l (ε) ≤ 1 − θ }) = 0, λ

or equivalently 1 |{n − λn + 1 ≤ k ≤ n : N∆m xk −l (ε) ≤ 1 − θ }| = 0. λn ( ) N In this case we write st∆m (P N ) − lim x = l or xk → l S∆m . λ λ lim n

Definition 18 Let (X, N, >) is a PN-space. Then a sequence x = (xk ) is said to be ∆m λ −statistically Cauchy in X if for every ε > 0 and θ ∈ (0, 1), there exists a number T = T (ε) such that δ∆m ({n − λn + 1 ≤ k ≤ n : N∆m xk −∆m xT (ε) ≤ 1 − θ }) = 0. λ

If λn = n for all n ∈ N in the Definition 16 and Definition 17 then we obtain: Definition 19 Let (X, N, >) is a PN-space. Then a sequence x = (xk ) is said to be ∆m −statistically convergent to l ∈ X with respect to the probabilistic norm N provided that for every ε > 0 and θ ∈ (0, 1) δ∆m ({k ≤ n : N∆m xk −l (ε) ≤ 1 − θ }) = 0, or equivalently lim n

1 |{k ≤ n : N∆m xk −l (ε) ≤ 1 − θ }| = 0. n N

In this case we write st∆m (P N ) − lim x = l. In this case we write xk → l (S∆m ).

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Definition 20 Let (X, N, >) is a PN-space. Then a sequence x = (xk ) is said to be ∆m −statistically Cauchy in X if for every ε > 0 and θ ∈ (0, 1), there exists a number T = T (ε) such that δ∆m ({k ≤ n : N∆m xk −∆m xT (ε) ≤ 1 − θ }) = 0. If λn = n for all n ∈ N and m = 0, then we obtain ordinary statistical convergence and statistical Cauchy in a PN-space which were defined by Karakus [10]. If we take m = 0, then we obtain λ−statistical convergence and λ−statistical Cauchy in a PN-space which were defined by Alotaibi [1]. Theorem 21 If a sequence x = (xk ) is ∆m λ −statistically convergent in PN-space X then st∆m (P N ) − lim x is unique. λ Proof. Suppose that st∆m (P N ) − lim x = lı and st∆m (P N ) − lim x = λ λ l2 , l1 ̸= l2 . Let ε > 0 and θ ∈ (0, 1). Choose γ ∈ (0, 1) such that (1 − γ) > (1 − γ) ≥ 1 − θ. Then we define the following sets as KN,1 (γ, ε) = {n − λn + 1 ≤ k ≤ n : N∆m xk −l1 (ε) ≤ 1 − θ } , KN,2 (γ, ε) = {n − λn + 1 ≤ k ≤ n : N∆m xk −l2 (ε) ≤ 1 − θ } . Since st∆m (P N ) − lim x = l1 and st∆m (P N ) − lim x = l2 we have λ λ δ∆m {KN,1 (γ, ε)} = 0 and we have δ∆m {KN,2 (γ, ε)} = 0, for all ε > 0, λ λ respectively. Now let KN,3 (γ, ε) = KN,1 (γ, ε) ∩ KN,2 (γ, ε) . Then we observe that δ∆m {KN,3 (γ, ε) } = 0 λ

which implies δ∆m {NKN,3 (γ, ε) } = 1. λ

If k ∈ NKN,3 (γ, ε), we have Nl1 −l2 (ε) = N(l1 −∆m xk )+(∆m xk −l2 ) (ε) ≥ Nl1 −∆m xk

(ε)

> N∆m xk −l2

2 > (1 − γ) > (1 − γ) ≥ 1 − θ .

(ε) 2

Since θ was arbitrary, we get Nl1 −l2 (ε) = 1 for all ε > 0 which gives l1 = l2 . Hence st∆m (P N ) − lim x is unique. λ Theorem 22 Let (X, N, >) is a PN-space. If N∆m − lim x = l then st∆m (P N ) − lim x = l. But converse does not hold. λ Proof. Let st∆m (P N ) − lim x = l. Then for every θ ∈ (0, 1) and ε > 0, there is a number ko such that N∆m xk −l (ε) > 1 − θ for all k ≥ ko . This guaranties that the set {n − λn + 1 ≤ k ≤ n : N∆m xk −l (ε) ≤ 1 − θ } has at most finitely many terms. Since every finite subset of the natural numbers has density zero, we can see that δ∆m ({n − λn + 1 ≤ k ≤ n : N∆m xk −l (ε) ≤ 1 − θ }) = 0. λ

For converse, we consider the following example:

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ESI, OZDEMIR: STATISTICAL CONVERGENCE

929

Example 23 Define a sequence x = (xk ) by { k , for n − λn + 1 ≤ k ≤ n ∆m xk = . 0 , otherwise Let ε > 0 and θ ∈ (0, 1) Kn (θ, ε) = {n − λn + 1 ≤ k ≤ n : N∆m xk (ε) ≤ 1 − θ } . Then δ∆m (Kn (θ, ε)) → 0 as n → ∞ this implies that st∆m (P N ) − lim x = 0, but it is obvious that N∆m − λ lim x ̸= 0. Theorem 24 Let (X, N, >) is a PN-space and x = (xk ) be a sequence. Then st∆m (P N ) − lim x = l if and only if there exists a subset K = λ {k1 < k2 < ...} ⊂ N such that δ∆m (K) = 1 and N∆m − limn xkn = l. λ

Proof. Suppose that st∆m (P N ) − lim x = l. Then, for any ε > 0 and λ s ∈ N, let { } 1 K (s, ε) = n − λn + 1 ≤ k ≤ n : N∆m xk −l (ε) ≤ 1 − s and

{ T (s, ε) =

n − λn + 1 ≤ k ≤ n : N

∆m xk −l

1 (ε) > 1 − s

} .

Then δ∆m (K (s, ε)) = 0 and λ

T (1, ε) ⊃ T (2, ε) ⊃ T (3, ε) ⊃ ... ⊃ T (j, ε) ⊃ T (j + 1, ε) ⊃ ...

(1)

δ∆m (T (s, ε)) = 1, s = 1, 2, 3, ...

(2)

and λ

Now we have to show that for k ∈ T (s, ε), x = (xk ) is xk

N∆m

→ l.

N∆m

Suppose that xk 9 l. Therefore there is a θ > 0 such that the set {n − λn + 1 ≤ k ≤ n : N∆m xk −l (ε) ≤ 1 − θ } has infinitely many terms. Let T (θ, ε) = {n − λn + 1 ≤ k ≤ n : N∆m xk −l (ε) > 1 − θ } and θ >

1 (s = 1, 2, 3, ...) . s

Then δ∆m (T (θ, ε)) = 0.

(3)

λ

By (1) we have T (s, ε) ⊂ T (θ, ε) . N m

∆ l that Hence δ∆m (T (s, ε)) = 0 which contradicts (2). Therefore xk → λ is N∆m − lim xk = l.

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ESI, OZDEMIR: STATISTICAL CONVERGENCE

Conversely, suppose that there exists a subset K = {k1 < k2 < ...} ⊂ N N m

∆ such that δ∆m (K) = 1 and xk → l. Then there exists ko ∈ N such that λ for every θ ∈ (0, 1) and ε > 0

N∆m xk −l (ε) > 1 − θ for all k ≥ ko . Now T (θ, ε) = {n − λn + 1 ≤ k ≤ n : N∆m xk −l (ε) ≤ 1 − θ } ⊂ N \ {kko +1 , kko +2 , kko +3 , ...} . Therefore δ∆m (T (θ, ε)) ≤ 1 − 1 = 0. λ

Hence st∆m (P N ) − lim x = l. λ Theorem 25 Let (X, N, >) is a PN-space. Then ∆m λ −statistically convergent if and only if it is ∆m λ −statistically Cauchy. Proof. Let x = (xk ) be ∆m λ −statistically convergent to l in PN-space X, i.e., st∆m (P N ) − lim x = l. Then for every ε > 0 and θ ∈ (0, 1), we have λ δ∆m ({n − λn + 1 ≤ k ≤ n : N∆m xk −l (ε) ≤ 1 − θ }) = 0. λ

Choose a number T = T (ε) such that N∆m xT −l (ε) ≤ 1 − θ. Now let A1 (θ, ε) = {n − λn + 1 ≤ k ≤ n : N∆m xk −∆m xT (ε) ≤ 1 − θ } , A2 (θ, ε) = {n − λn + 1 ≤ k ≤ n : N∆m xk −l (ε) ≤ 1 − θ } and A3 (θ, ε) = {n − λn + 1 ≤ T ≤ n : N∆m xT −l (ε) ≤ 1 − θ } . Then A1 (θ, ε) ⊂ A2 (θ, ε) ∪ A3 (θ, ε) and therefore δ∆m (A1 (θ, ε)) ≤ δ∆m (A2 (θ, ε)) + δ∆m (A3 (θ, ε)) . λ

λ

λ

∆m λ −statistically Cauchy. x = (xk ) is ∆m λ −statistically

Hence x = (xk ) is Conversely, let Cauchy but not ∆m λ −statistically convergent. Then there exists a natural number T = T (ε) such that the set A1 (θ, ε) has natural denstiy zero. Hence the set A4 (θ, ε) = {n − λn + 1 ≤ k ≤ n : N∆m xk −∆m xT (ε) < 1 − θ } has natural density 1, that is, δ∆m (A4 (θ, ε)) = 1. In particular, we can λ write N∆m xk −∆m xT (ε) ≤ 2N∆m xk −l (ε) < ε (4) if N∆m xk −l (ε) < 2ε . Since x = (xk ) is not ∆m λ −statistically convergent, the set A2 (θ, ε) has natural density 1, i.e., δ∆m ({n − λn + 1 ≤ k ≤ n : N∆m xk −l (ε) > 1 − θ }) = 0. λ

Therefore by (4), we have δ∆m ({n − λn + 1 ≤ k ≤ n : N∆m xk −∆m xT (ε) > 1 − θ }) = 0, λ

8

ESI, OZDEMIR: STATISTICAL CONVERGENCE

i.e., the set A1 (θ, ε) has natural density 1 which is contradiction. Hence x = (xk ) is ∆m λ −statistically convergent. We now show that ∆m λ −statistically convergence on PN-spaces has some arithmetical properties similar to properties of the usual convergence on R. Lemma 26 Let (X, N, >) is a PN-space. The following statements are true: (1) If st∆m (P N )−lim x = l1 and st∆m (P N )−lim y = l2 , then st∆m (P N )− λ λ λ lim (x + y) = l1 + l2 , (2) If st∆m (P N ) − lim x = l and β ∈ R, then st∆m (P N ) − lim βx = βl, λ λ (3) If st∆m (P N )−lim x = l1 and st∆m (P N )−lim y = l2 , then st∆m (P N )− λ λ λ lim (x − y) = l1 − l2 . Proof. It is similar to the Lemma 2 of Karakus [10], so we omit it. Conclusion 27 The idea of probabilistic norm is very useful to deal with the convergence problems of sequences of real numbers. The main purpose of this paper is to more generalize the results on statistical convergence proved by Karaku¸s [10] and Alotaibi [1]. We have introduced a more wider class of ∆m λ −statistically convergent sequences in a PN-space to deal with the sequences which are not covered in [10] and [1].

9

931

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ESI, OZDEMIR: STATISTICAL CONVERGENCE

References [1] A. Alotaibi, Generalized statistical convergence in probabilistic normed spaces, The Open Mathematics Journal 1(2008), 82-88. [2] C. Alsina, B. Schweizer and A. Sklar, On the definition of a probabilistic normed space, Aequationes Math.46(1993), 91-98. [3] J. S. Connor, The statistical and strong p-Cesaro convergence of sequences, Analysis, 8(1988), 47-63. [4] G. Constantin and I. Istratescu, Elements of probabilistic analysis with applications, Vol.36 of Mathematics and Its Applications (East European Series), Kluwer Academic Publishers, Dordrecht, The Netherlands, 1989. [5] A. Esi, The A-statistical and strongly (A-p)-Cesaro convergence of sequences, Pure and Appl. Mathematica Sciences, Vol:XLIII, No:12(1996), 89-93. [6] M. Et and F. Nuray, ∆m −Statistical convergence, Indian J. Pure Appl. Math., 32(6)(2001), 961-969. [7] H. Fast, Sur la convergence statistique, Colloq. Math., 2(1995), 241244. [8] A. R. Freedmann, J. J. Sember and M. Raphael, Some Cesaro type summability spaces, Proc. London Math. Soc., 37(1987), 508-520. [9] J. A. Fridy, On statistical convergence, Analysis, 5(1985), 301-313. [10] S. Karakus, Statistical convergence on probabilistic normed space, Math. Commun., 12(2007), 11-23. [11] E. Kolk, Statistically convergent sequences in normed spaces; Reports of convergence, “Methods of algebra and analysis”, Tartu, (1988), 63-66 (in Russian). [12] K. Menger, Statistical metrics, Proc. Natl. Acad. Sci. USA, 28(12)(1942), 535-537. [13] M. Mursaleen, λ−Statistical convergence, Math. Slovaca, 50(2000), 111-115. [14] T. Salat, On statistically convergent sequences of real numbers, Math. Slovaca, 30(1980), 139-150. [15] B. Schweizer and A. Sklar, Statistical metric spaces, Pacific J. Math., 10(1960), 313-334. [16] B. Schweizer and A. Sklar, Probabilistic metric spaces, NorthHolland Series in Probability and Applied Mathematics, NorthHolland, New York, NY,USA, 1983. [17] A. N. Serstnev, On the notion of a random normed space, Dokl. Akad. Nauk SSSR 149(1963), 280-283. [18] H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math., 2(1951), 73-74.

10

JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.5, 933-952, 2011, COPYRIGHT 2011 EUDOXUS PRESS, 933 LLC

Solving Several Complex Variables Equations Numerically Louis Paquette D´ epartement de Math´ ematiques et Informatique Universit´ e du Qu´ ebec a ` Trois-Rivi` eres C.P. 500, Trois-Rivi` eres, Qu´ ebec Canada, G9A 5H7 Email: [email protected] December 4, 2009 Abstract This paper addresses the problem of solving a system of nonlinear equations using complex numbers. To this end, second order Taylor expansions of the functions of the equations are used to minimize the sum of the modula of the errors in the equations. This gives a single multivariate equation which is not differentiable in complex space, but is differentiable when it is transposed to real space. Using complex number’s properties we demonstrate the existence of negative eigenvalues for the Hessian, in the neighborhood of a null Gradient. A sufficient condition for these negative eigenvalues to exist is: when the search vector is not a solution and the Gradient is null, then the bilinear form associated with second order derivatives is full rank. This result leads to an algorithm which avoids first order local minima of the total error in the equations. Convergence to a solution at an accumulation point is then demonstrated. Indeed, with complex numbers, negative curvatures are available when needed.

Keywords. Systems of nonlinear equations, Second Order Approximation, Global convergence, Complex Numbers, Numerical Algorithm, Negative Eigenvalues, Physics. 1

934

PAQUETTE: SEVERAL COMPLEX VARIABLES EQUATIONS

1

Introduction

In a previous paper [13], we have shown that the smaller root of a second order Taylor expansion of a function f (z) with complex numbers could always reduce the error of an estimate of a root. Convergence to a solution was obtained at an accumulation point. This approach was motivated by the Minimum Modulus theorem of complex analysis. This paper addresses the problem of finding a solution to a system of nonlinear equations which use complex numbers and it extends the approach of [13] to several complex numbers functions. With a system of equations, the motivation to use complex numbers is the solvability of a system of polynomial equations (see [5, 18]). Let present the notation for the problem. We require the functions of the equations to have Taylor expansions (see [16]). These functions are denoted fk (u), k = 1, ..., n, where u is a vector of complex numbers. The problem is to find a solution u such that: fk (u) = 0, k = 1, ..., n

(1)

u = (z1 , z2 , ..., zn )

(2)

This problem is equivalent to solving the following single equation (|z| is the modulus of z): n X

|fk (u)|2 = 0

(3)

k=1

Using a second order Taylor expansion of each function fk (u), the equation (3) has a complex space representation and a real space representation that uses 2n real variables. Combining properties of both representations, we demonstrate that, if u is not a solution, then we can always find a vector v such that : n X

|fk (v)|2
0 such that for every k, k > N , the matrices Q(uk ) have a negative eigenvalue denoted ek such that ek ≤ 12 e∗ < 0. Proof. Since the elements of the matrix Q are continuous, we have: lim ht2∗ Q(uk ) h2∗ = ht2∗ Q(u∗ ) h2∗ = e∗

k→∞

(30)

Define d = −e2 ∗ > 0. Therefore, there exist an integer N > 0, such that for all k > N , we have: |ht2∗ Q(uk ) h2∗ − ht2∗ Q(u∗ ) h2∗ | = |ht2∗ Q(uk ) h2∗ − e∗ | < d This means: 9

(31)

942

PAQUETTE: SEVERAL COMPLEX VARIABLES EQUATIONS

e∗ N , the matrices Q(uk ) have a negative eigenvalue ek ≤ e2∗ < 0 This demonstrates the Theorem. ht2∗ Q(uk ) h2∗ < d + e∗ =

The next Corollary demonstrates ||g(u)|| can be reduced in all cases. Corollary 4.4. Assume g(u) 6= 0. Assume the regularity condition is true (Definition 4.1). It is always possible to reduce ||g(u)|| in equation (12). t

Proof. If g 0 (u) 6= 0, then defining h = −g 0 (u) , we have g 0 (u)h = −||g 0 (u)||2 < 0, and it is the dominant term of equation (12) so we can reduce ||g(u)||. If g 0 (u) = 0, by Theorem 4.2 there is a negative eigenvalue of the matrix Q. Define h to be the eigenvector associated to the most negative eigenvalue of Q. Then we have that, in equation (12), the dominant terms are the second order terms which sum is negative. Therefore, in all case there exist a λ ∈ (0, 1] and a vector h such that ||g(u + λ h)|| < ||g(u)|| (see [10] for example).

5

An Algorithm for the Minimization of ||g(u)||2

Let first describe what needs to occur to solve equations numerically. On a computer, g 0 (u) is rarely 0 due to floating point errors, but it can be very small. On the other hand, an extremelly small g 0 (u) is still the dominant term in equation (12). That means: to bypass first order local minima (that is, g 0 (u) = 0) with equation (12) it is advisable to select properly the sign of h, when using negative eigenvalues, as explained in the next paragraph. Assume h is an eigenvector associated to the smallest negative eigenvalue of Q. Then the sum of second order terms in equation (12) is negative. Also, observe that ht2 Q h2 = (−h2 ) Q (−h2 ). That means we can also make negative the first order term in equation (12): if Re(g 0 (u)h) > 0 then invert the sign of h so that we have Re(g 0 (u)h) < 0 (bilinear forms are insensitive to sign). Robust Nonlinear Equations Algorithm

10

PAQUETTE: SEVERAL COMPLEX VARIABLES EQUATIONS

To demonstate convergence of the algorithm, we have to handle many cases, for generality. 1. Choose an arbitrary initial value for u. 2. When it is possible, define hN , hG , hE as follows: • Define hE to be the eigenvector of Q corresponding to the smallest negative eigenvalue of Q. If Re(g 0 (u)hE ) > 0 then invert the sign of hE so that we have Re(g 0 (u)hE ) < 0. • Define hN with Newton method using equation (26). t

• Define hG = −g 0 (u) . Note that this is called ”Gradient Method” or ”steepest descent” in real space algorithms since we have g 0 (u)hG = −||g 0 (u)||2 . 3. For each possible search vectors hN , hG , hE , perform a line search on λ ∈ (0, 1] using equation (12), and determine the smallest value of ||g(u+ λ h)||. Replace u by the best value u + λ h of the best search vector. Also, the line search is assumed to be exact (see Brent Method in [15]) and eigenvectors are normalized. 4. If ||g(u)|| is sufficiently small, stop. Otherwise, goto 2. Let comment the behaviour of the algorithm in practice. First, observe that at a solution u, that is, g(u) = 0, we must also have g 0 (u) = 0 and the matrix Q must be semi-positive definite, since otherwhise we could decrease ||g(u)||, which is impossible. Newton method is very effective sufficiently near a solution. Far away from a solution and at a quasi-local minimum, the Gradient Method might be usefull, in the case the smallest eigenvalue is not sufficiently negative. Finally, in the case g(u) 6= 0 and ||g 0 (u)|| is extremely small, eventually the matrix Q will have an effective negative eigenvalue, which allows to bypass local minima g 0 (u) = 0 (see Corollary 4.4).

6

Numerical Experiments

The first problem illustrates the case where g 0 (u) = 0. Let define 2 polynomial equations with random coefficients of the form: 11

943

944

PAQUETTE: SEVERAL COMPLEX VARIABLES EQUATIONS

Iteration ||g 0 (u)||

Method

Min. Eigenvalue

1 2 3 4 5 6

hE hN hN hN hN hN

-6844.8881207667 -4866.9101895053 13844.2960649869 18963.4537234783 18682.9583949293 18675.4013157256

0.00000001 6768.82970131 7589.418620428 740.5130745734 17.8679919882 0.0116490365 0.00000001

Error 161.9289967856 139.5100488482 38.9670588752 5.3091793595 0.1302209352 0.0000849397 0

Table 1: Example with polynomial equations. At the first iteration, we have g 0 (u) = 0, and only the Negative Eigenvalue Method is usable (the best method is hE).

p(x, y) = ax3 + bx2 y + cxy + dx2 + ey 3 + f y 2 + gx + hy + k = 0

(33)

The constants ”a, b,..., k” are random numbers, ajusted to make g 0 (u0 ) = 0. First, we find an initial vector u0 = (x0 , y0 ) such that the first line of the Jacobian is null. Also, we select the coefficient k of the second equation such that the equation is satisfied. Then, we have g 0 (u0 ) = 0. In Table 1, the only method available at the first iteration is the negative eigenvalue vector which allows to overcome the case g 0 (u0 ) = 0. The other iterations converge to a solution using Newton Method. Note that at a solution, negative eigenvalues disappear. The next example is a 3 equations problem defined as follows. The 3 variables are denoted u, v, w. We use several polynomials of the following type: a u2 + b v 2 + c w2 + d. These polynomials are denoted: pk (u, v, w) and qk (u, v, w), k = 1, 2, 3

(34)

Also, equations are defined as follows: fk (u, v, w) = pk (u, v, w) + sin(qk (u, v, w)) , k=1,2,3

12

(35)

PAQUETTE: SEVERAL COMPLEX VARIABLES EQUATIONS

Iteration ||g 0 (u)||

Method

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

hG hG hE hN hN hN hN hN hN hN hN hN hE hN hN hN hN hN hN hN

103301750.791628 40869.7938703847 1386.3838747446 2177.058115818 681.4418474771 1468.5123798148 2500.8032714471 2599.2958570336 1761.4695413557 2603.8917652289 1244.5790333155 1730.8170262823 1730.6150214975 1246.8647374114 2043.5794525719 2085.9658698154 315.6152263362 34.6975381965 1.8750188675 0.0048059319 0.000000032

Min. Eigenvalue

Error 4940.5007323635 -17753011.692712 121.9630033916 -25073.9462039774 63.1294232234 -9311.6062252306 55.3218863859 -807.314995412 50.6522683543 -512.280632562 50.4820296894 -549.5316670355 43.2462993432 -684.1680363503 39.5810959338 -308.3897167604 37.2992915339 -288.0966684209 36.8782650103 -408.7169111235 35.8862917189 -176.1049593285 29.9117083161 -287.0042885545 20.5019629678 -18.9056412314 18.6468392907 59.3686869061 16.6820787436 -191.8102552131 12.10827678 226.6209060283 4.0260689135 939.1277773846 0.6902699112 803.8304208882 0.0330157275 782.0564970093 0.0000839078 781.0729000976 0.0000000006

Table 2: Example with 3 equations of the type: p(u,v,w)+sin(q(u,v,w)). In this case, the best method is most frequently the Newton Method (hN).

13

945

946

PAQUETTE: SEVERAL COMPLEX VARIABLES EQUATIONS

The coefficients a, b,c,d for the polynomials in the equations are random complex numbers and the initial vector is chosen so that the initial error is large. The Table 2 shows the behavior of the algorithm with the second problem. Observe that all methods are used (the best method is indicated). Also, negative eigenvalues exist most of the time, but not always. Finally, negative eigenvalues disappear near a solution, as it should.

7

Convergence of the Algorithm

Recall that the algorithm always reduces the error ||g(u)||, by Corollary 4.4, if g(u) 6= 0. It requires functions to have a Taylor expansion and the regularity condition must be true (Definition 4.1). Also, observe that a solution might not exist: ef (u) has no solution. Theorem 7.1 (Convergence of the Proposed Algorithm). Assume the regularity condition is true (Definition 4.1). Assume also that the vectors u in the algorithm creates an accumulation point u∗ . Then, u∗ is a solution to equation (1). Proof. The proof proceeds by contraction, that is, we assume u∗ is not a solution, i.e. g(u∗ ) 6= 0. Let denote A the set of vectors u used in the algorithm. Since there is an accumulation point u∗ , then there is a subset B of A such that the vectors in B converge to u∗ . Let observe that the algorithm can be applied at u∗ . In the next paragraphs, we create a convergent sequence of vectors u in the algorithm and we define a direction vector hk as follows: if g 0 (u∗ ) 6= 0 then hk = hGk , and if g 0 (u∗ ) = 0 then hk = hEk . That is, we select only direction vectors of the Gradient Method or only direction vectors of the Negative Eigenvalue Method. Case g 0 (u∗ ) 6= 0 If g 0 (u∗ ) 6= 0, then we can apply the Gradient Method at u∗ , so that h∗ = hG∗ (see Corollary 4.4). Applying the line search we obtain λ∗ such that ||g(u∗ + λ∗ h∗ )|| < ||g(u∗ )||. In this case let denote the vectors in the set B by 14

PAQUETTE: SEVERAL COMPLEX VARIABLES EQUATIONS

947

uk and the corresponding Gradient Method vectors hGk by hk . Since g 0 (u) is a continuous function, we have: lim uk = u∗

(36)

lim hk = h∗

(37)

||g(u∗ + λ∗ h∗ )|| < ||g(u∗ )||

(38)

k→∞

k→∞

Case g 0 (u∗ ) = 0 In this case, we will use the vectors hE (u) in the algorithm. The real space representation of hE (u) will be denoted h2E (u). Since g 0 (u∗ ) = 0 and the vectors u in the set B converge to u∗ , we can apply Theorem 4.3 and its notation: for u sufficiently close to u∗ , we have: e∗ 0

(44)

Also, define the function e(α) as follows, where α is a short name to represent all the elements in vectors u, h and λ and ”(u, h, λ)” denotes also the vector α: e(α) = e(u, h, λ) = ||g(u + λ h)||

(45)

The function e(α) is continuous, with respect to u, h and λ or α, since all its elements are continuous. Also define ωj as follows (we want ωj to converge to λ∗ ), where j0 is the smallest positive integer such that λ∗ − j10 ∈ (0, 1] : 1 ωj = λ∗ − , j = j0 , ..., ∞ j

(46)

Note that ωj ∈ (0, 1] and we have ... lim ωj = λ∗

j→∞

16

(47)

PAQUETTE: SEVERAL COMPLEX VARIABLES EQUATIONS

949

Now, define αj , j = j0 , ..., ∞ and β as follows: αj = (uj , hj , ωj )

(48)

β = (u∗ , h∗ , λ∗ )

(49)

lim αj = β

(50)

lim e(αj ) = e(β) = ||g(w)|| < ||g(u∗ )||

(51)

We have: j→∞

j→∞

Therefore, by continuity, ∃N such that ∀j > N we have: | e(αj ) − e(β)| < d/2

(52)

That means ... ||g(u∗ )| − ||g(w)|| = 2 ||g(u∗ )|| + ||g(w)|| < ||g(u∗ )|| 2

e(αj ) < e(β) + d/2 = ||g(w)|| +

e(αj ) = ||g(uj + ωj hj )|| < ||g(u∗ )||

(53)

(54)

The last equation (54) is a contradiction for the following reason. First, we have ||g(uj )|| > ||g(uj+1 )|| > ||g(uj+2 )|| > ... > ||g(u∗ )||. Also, we have found ωj ∈ (0, 1] for which ||g(uj + ωj hj )|| < ||g(u∗ )|| which contredicts the fact that we have selected the best method in the algorithm, when we have encountered the vector uj . Since the case g 0 (u∗ ) = 0 is impossible and the case g 0 (u∗ ) 6= 0 is impossible we must have g(u∗ ) = 0, that is, u∗ is a solution to equation (1).

17

950

PAQUETTE: SEVERAL COMPLEX VARIABLES EQUATIONS

Corollary 7.2. Assume the regularity condition is true (Definition 4.1). The algorithm converges to a solution, or, ||uj || goes at infinity, always decreasing ||g(uj )|| without creating an accumulation point. Proof. If the algorithm creates an accumulation point, then by Theorem 7.1 we have that u∗ is a solution to equations (1). Otherwise the vector norms ||uj || goes at infinity, always decreasing ||g(uj )|| without creating an accumulation point. Decreasing ||g(uj )|| follows from Corollary 4.4.

8

Conclusion

The paper has presented a new approach for solving numerically a set of nonlinear equations that uses complex numbers. The proposed algorithm requires only the sufficient condition of Definition 4.1. With this condition, there is no such thing as a local minimum of ||g(u)|| , if u is not a solution. This follows from the existence of negative eigenvalues when g 0 (u) = 0. Convergence results were also obtained. Using real numbers implies, invariably, more restrictive conditions. To conclude, the algorithm that was presented aims at describing the use of the theoretical results of this paper. As mention in the introduction, in the case of large scale problems, the approaches of recent papers such as ”trust-region” (see [2, 4, 6, 11]) should be adapted to the use of complex numbers. Finally, an example of question worth investigating might be: in the case there exists a solution, is it possible to find a connected path to that solution? Another question might be: in the case of multivariate polynomial equations, does the algorithm always find a solution?

References [1] S. Abbasbandy, Y. Tan, S.J. Liao, Newton-homotopy analysis method for nonlinear equations, Applied Mathematics and Computation 188 (2007), p. 1794-1800. [2] P.-A. Absil, R. Mahony, and B. Andrews, Convergence of the iterates of descent methods for analytic cost functions, SIAM J. OPTIM. Vol. 16, No. 2, pp. 531-547 18

PAQUETTE: SEVERAL COMPLEX VARIABLES EQUATIONS

[3] R. L. Burden and J. D. Faires, Numerical Analysis, sixth edition, Brooks/Cole Publishing Company, 1997. [4] Andrew r. Conn, Katya Scheinberg, and Luis n. Vicente, Global convergence of general derivative-free trust-region algorithms to first- and second-order critical points, SIAM J. OPTIM., Vol. 20, No. 1, pp. 387415 [5] D. Cox, J. Little, D. O’Shea, Ideals, Varieties, and Algorithms, Springer, (2007). [6] J. E. Dennis, jr., Mahmoud el-Alem, and Karen Williamson, A trustregion approach to nonlinear systems of equalities and inequalities, SIAM J. OPTIM., Vol. 9, No. 2, pp. 291-315 [7] Crina Grosan and Ajith Abraham, A New Approach for solving Nonlinear Equations Systems, IEEE Transactions on Systems, Man, and Cybernetics-Part A: Systems and Humans, Vol. 38, no. 3, (2008). [8] K. Hoffman and R. Kunze, Linear Algebra, Prentice-Hall, (1961). [9] D. Li and M. Fukushima, A globally and superlinearly convergent GaussNewton-based BFGS method for symmetric nonlinear equations, SIAM J. Numer. Anal., Vol. 37, No. 1, pp. 152-172. [10] David Luenberger, Introduction to Linear and Nonlinear Programming, Addison-Wesley, 1973, p.330. [11] Alberto Olivares and Javier M. Moguerza, Improving directions of negative curvature in an efficient manner, Ann. Oper. Res. (2009) 166: 183-201 [12] Ben Noble, Applied Linear Algebra, Prentice Hall, 1969. [13] Louis Paquette, The Smaller Root Principle for Finding Roots of a Complex Number Function, to appear in Journal of Computational Analysis and Applications. [14] L. Pennisi, Elements of Complex Variables, Holt, Rinehart anWinston ed., 1963. [15] William H. Press et al., Numerical Recipes in C, Cambridge University Press, 1992. 19

951

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PAQUETTE: SEVERAL COMPLEX VARIABLES EQUATIONS

[16] Volker Scheidemann, Introduction to Complex Analysis in Several Variables, ed. Birkhuser Basel, 1 edition (2005) [17] R. B. Schnabel and P. Frank, Tensor methods for nonlinear equations, SIAM J. Numer. Anal. 21 (1984), pp. 814-843. [18] Bernd Sturmfels, Solving Systems of Polynomial Equations, Lectures Notes, Department of Mathematics, Berkeley, (2002). [19] Yunong Zhang and W.E. Leithead, Exploiting Hessian matrix and trustregion algorithm in hyperparameters estimation of Gaussian process, Applied Mathematics and Computation, Volume 171, Issue 2, Pages 12641281, (2005)

20

JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.5, 953-962, 2011, COPYRIGHT 2011 EUDOXUS PRESS, 953 LLC

Approximation by polynomials and rational functions in Orlicz spaces. Sadulla Z. Jafarov Department of Mathematics, Faculty of Art and Sciences, Pamukkale University, 20017, Denizli, Turkey, e-mail: [email protected] Abstract. In this work some direct theorems of approximation theory in the SmirnovOrlicz classes, defined in the domains with a Dini- smooth boundary, are proved. Keywords. Direct theorem, Cauchy singular integral, polynomial and rational approximation , Dini-smooth curve, Orlicz space, Smirnov-Orlicz class, modulus of smoothness. 1. Introduction, auxiliary and main results. Let Γ be a rectifiable Jordan curve in the complex plane C and let G : = intΓ , G− := extΓ. Without loss of generality we suppose that 0 ∈ G. Further let T : = {w ∈ C : |w| = 1} , D : = intT , D− : = extT . We denote by w = ϕ1 (z) and w = ϕ2 (z) the conformal mappings of G and G− onto D− normalized by the conditions ϕ1 (0) = 0 ,

lim z · ϕ1 (z) > 0

z→∞

and

ϕ2 (z) >0 z respectively and let z = ψ1 (w) and z = ψ2 (w) be the inverse mappings of w = ϕ1 (z) and w = ϕ2 (z). Let ϕ also be a continuous function on T . Its modulus of continuity is defined by ϕ2 (∞) = ∞,

lim

z→∞

w (t, ϕ) : = sup {|ϕ (t1 ) − ϕ (t2 )| : t1 , t2 ∈ T, |t1 − t2 | ≤ t} , The function ϕ is called Dini – continuous if Z a w (t, ϕ) dt < ∞ , t 0

t ≥ 0.

a > 0.

A curve Γ is called Dini – smooth [ 26 , p.48 ] if it has a parametrization Γ : h (τ ) , such that h0 (τ ) is Dini – continuous and h0 (τ ) 6= 0. If Γ is Dini – smooth, then [ 30 ] 0 0 0 < c1 < ψ2 (w) < c2 < ∞, 0 < c3 < φ2 (z) < c4 < ∞

τ ∈T

(1)

where the constants c1 , c2 , and c3 , c4 are independent of |w| ≥ 1 and z ∈ G− , respectively. Let Γ be a rectifiable Jordan curve and consider a function f ∈ L1 (Γ). Then the functions f + and f − defined by Z 1 f (ς) f + (z) : = dς , for z ∈ G, 2πi Γ ς − z Z 1 f (ς) − f (z) : = dς , for z ∈ G− 2πi Γ ς − z are analytic in G and G− respectively and f − (∞) = 0. The Cauchy singular integral of f at a point z ∈ Γ is defined by Z 1 f (ς) SΓ (f ) (z) : = lim dς ε→0 2πi Γ|Γ(z,ε) ς − z

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Sadulla Z. Jafarov

if the limit exists. The linear operator SΓ : f → SΓ f is called the Cauchy singular operator. If one of the functions f + or f − has the non-tangential limits almost everywhere(a.e.) on Γ, then SΓ f (z) exist a.e. on Γ and also the other one has non-tangential limits a.e. on Γ. Conversely, if SΓ f (z) exist a.e. on Γ, then both functions f + and f − have non-tangential limits a.e. on Γ. In both cases, the formulae 1 f + (z) = SΓ f (z) + f (z) , 2

1 f − (z) = SΓ f (z) − f (z) 2

(2)

and hence f = f+ − f−

(3)

hold almost everywhere on Γ (cf. [7 , p.431]). Let Γr be the image of the circle {w ∈ C : |w| = r , 0 < r < 1} under some conformal mapping of D onto G. By Ep (G) we denote the class of analytic functions f in G which satisfy Z p

|f (z)| |dz| < ∞

sup 0 0. The space LM (Γ) becomes a Banach space with the norm Z  kf kLM (Γ) : = sup |f (z) g (z)| |dz| : g ∈ LN (Γ) , ρ (g; N ) ≤ 1 , Γ

where

Z N [|g (z)|] |dz| .

ρ (g, N ) : = Γ

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The norm k · kLM (Γ) is called Orlicz norm and the Banach space LM (Γ) is called Orlicz space. Every function in LM (Γ) is integrable on Γ [27, p.50], i.e. LM (Γ) ⊂ L1 (Γ) . An N -function M satisfies the ∆2 -condition if lim sup

x→∞

M (2x) < ∞. M (x)

The Orlicz space LM (Γ) is reflexive if and only if the N -function M and its complementary function N both satisfy the ∆2 -condition [27, p.113]. Let Γr be the image of the circle {w ∈ C : |w| = r , 0 < r < 1} under some conformal mapping of D onto G and let M be an N -function. By EM (G) we denote the class of analytic functions f in G which satisfy Z M (|f (z)|) |dz| < ∞ Γr

uniformly r. The class EM (G) is called the Smirnov-Orlicz class. If M (x) = M (x, p) : = xp (1 < p < ∞), then the Smirnov-Orlicz class EM (G) coincides with the usual Smirnov class Ep (G). As was noted in [17 ], every function of class EM (G) has a.e. non-tangential boundary values and the boundary function belongs to LM (Γ). The class EM (G− ) can be defined similarly. The following hold:

EM (G) ⊂ E1 (G) , EM G− ⊂ E1 G− . The direct and inverse problems of approximation theory in Smirnov-Orlicz classes were investigated by several researchers(see, for example, [ 4 ], [ 5 ], [9], [10] and [17] ) For ς ∈ Γ we define the point ςh , ς−h ∈ Γ by
ςh : = ψ2 ϕ2 (ς) eih ,
ς−h : = ψ2 ϕ2 (ς) e−ih , h ∈ [0, 2π] . (2)

We define the modulus of continuity wM (f, ·) for f ∈ LM (Γ) as: (2)

wM (f, δ) : = sup kf (ςh ) + f (ς−h ) − 2f (ς)kLM (Γ) ,

δ≥0

Let Γ be a Dini-smooth curve and let f0 : = f ◦ ψ2 , f1 : = f ◦ ψ1 for f ∈ EM (Γ). Then from (1) we get f0 ∈ LM (T ) and f1 ∈ LM (T ) for f ∈ LM (Γ). We use c, c1 , c2 , c3 , ... to denote constants (which may, in general, differ in different relations) depending only on numbers that are not important for the questions of interest. For a > 0 and b > 0 , we will use the expression a b (order inequality) if a ≤ cb. The expression a b means that a  b and b  a simultaneously. In this work we use the approximation properties of the Faber polynomials some direct theorems for polynomial and rational approximation of functions in certain subclasses of Smirnov-Orlicz and Orlicz classes, defined in the domains with a Dini-smooth boundary, are proved. In Orlicz class, approximation problem of the function connected to defined parameters is examined. Boundedness of singular operator SΓ plays an important role in approximation theory. The boundedness problem of this operator was solved by David [13] in Lebesgue space Lp (Γ). Karlovich [19] proved the boundedness of the operator SΓ in Orlicz space by using David theorem and Boyd interpolation theorem. In the proof of the main results boundedness of singular integral in Orlicz spase is used The main results of this work can be given as follows :

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Theorem 1. Let G be a finite simply connected domain with the Dini-smooth boundary Γ, and let EM (G) be a reflexive Smirnov-Orlicz space on G. Then for every f ∈ EM (G) and any natural number n there exist an algebraic polynomial Pn (·, f ) of degree at most n such that   1 (2) kf − Pn (·, f )kLM (Γ) ≤ c5 wM f, n with some constant c5 > 0 independent of n. Theorem 2. Let Γ be the Dini-smooth curve and f (z) ∈ LM (Γ). Then for the function FAB (z) = A f (z) + B f (z) we obtain 1. If A = B, then for any n ∈ N there exists an algebraic polynomial Pn (z) of degree at most n such that   1 (2) kf − Pn kLM (Γ) ≤ c6 [2A] wM f, . n 2. If A 6= B, then for any n ∈ N there exists rational function Rn (z) of degree at most n such that      1 1 (2) (2) + (B − A) wM f, . kF − Rn kLM (Γ) ≤ c7 (A + B) wM f, n n 2. Proofs of main results Proof of Theorem 1. Let f ∈ LM (Γ). Then by LM (Γ) ⊂ L1 (Γ) we get f ∈ L1 (Γ). Since Γ is a Dini-smooth curve, we have f ◦ ψ2 ∈ L1 (T ). Hence we can associate formal series n n X X bk f (ψ2 (w)) ∼ ak wk + (4) wk+1 k=0

k=1

We know [29 , pp. 52, 255] that n

0

X φk (z) ψ2 (w) = , ψ2 (w) − z wk+1

z ∈ G , w ∈ D−

k=0

where ϕk (z) are the Faber polynomials of degree k with respect to z for the continuum G with the integral representation [29 , pp.35, 255] ϕk (z) =

1 2πi

0

Z |w|=R

wk ψ2 (w) dw , ψ2 (w) − z

z ∈ G , R > 1.

(5)

The detailed information about the Faber polynomials and their approximation properties can be found in the monographs [28], [29]. Let Kn (h) =

h X

(n)

λk eikh

k=−n

be an even, nonnegative trigonometric polynomial satisfying the conditions Z π 1 Kn (h) dh = 1, 2π −π Z π |Kn (h)| dh ≤ c8 ,

(6)

(7)

−π

Z

π

k

|h| |Kn (h)| dh ≤ c9 (k) (n + 1) −π

−k

,

(8)

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5

for every natural number n and with some constants c8 > 0 , c9 (k) > 0 (for example, the Jackson kernel)
4 3 sin nh 2 Jn (h) : =
4 n (2n2 + 1) sin h2 satisfies the above cited conditions, see [14 , p.203-204]. On the other hand, using (6) − (8) we get (see [11]). π

Z

−π



1 |t| + n

k

|Kn (h)| dh ≤ c10 (k) n−k ,

(n = 1, 2, 3, . . .) .

Consider the integral I (h, z) : =

1 2πi

Z

f (ς−h ) + f (ςh ) dς , z ∈ G. (9) ς −z Γ
eit and taking into account the relations (4) and (5)

Using the change of variables ς = ψ2 we obtain



 0 it
it Z π  f ψ2 ei(t−h) + f ψ2 ei(t+h) ψ2 e e 1 dt I (h, z) = 2πi −π ψ2 (eit ) − z =

h X

  ak ϕk (z) e−ikh + eikh .

k=0

Since I (h, z) ∈ L1 [−π, π] and Kn (h) is of bounded variation, by the generalized Parseval identity [6, p.225-228], we get 1 2πi

Z

π

I (h, z) dh = −π

n  X

(h)

(h)

µk + λk



ak ϕk (z) .

k=0

Then by (6) and (9) we have 1 4π 2 i

Z

π

n

Z Kn (h) dh

−π

Γ

 X  (h) f (ς−h ) + f (ςh ) (h) dς = µk + λk ak ϕk (z) . ς −z k=0

Hence we see that 1 Pn (z , f ) : = 4π 2 i 0

Z

π

Z

Fh (ς) dς , ς − z0

Kn (h) dh 0

Γ

z 0 ∈ G,

is an algebraic polynomial of degree n, where Fh (ς) = f (ς−h ) + f (ςh ) . It is clear that 1 f (z ) = 2πi 0

Z Γ

f (ς) dς , ς − z0

z 0 ∈ G.

By (2) a.e. for z ∈ Γ we obtain f (z) = or

1 f (z) = 2π

Z 0

π

1 f (z) + SΓ f (z) 2

1 Kn (h) f (z) dh + π

Z

π

Kn (h) SΓ f (z) dh. 0

(10)

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Sadulla Z. Jafarov

By (1) and (6) a.e. for z ∈ Γ we can write   Z π 1 1 Pn (z, f ) = Kn (h) Fh (z) + SΓ (Fh ) (z) dh. 2π 0 2

(11)

Then taking the limit z 0 → z ∈ Γ along all non-tangential paths inside Γ and using (10) and (11) we have Z π Z 1 π 1 Kn (h) f (z) dh + Kn (h) [SΓ (f ) (z)] dh f (z) − Pn (z, f ) = 2π 0 π 0   Z π 1 1 Kn (h) Fh (z) + SΓ (Fh ) (z) dh − (12) 2π 0 2 Z π Z π 1 1 = Kn (h) [2f (z) − Fh (z)] dh + Kn (h) [SΓ (2f − Fh ) (z)] dh 4π 0 2π 0 for almost all z ∈ Γ. Taking the supremum over all function g ∈ LN (Γ) with ρ (g, N ) ≤ 1 in the relation (12), we obtain Z kf − Pn (·, f )kLM (Γ) = sup |f (z) − Pn (z, f )| |g (z)| |dz| Γ

Z π Z 1 Kn (h) [2f (z) − Fh (z)] dh |g (z)| |dz| ≤ sup Γ 4π 0 Z Z π 1 + sup Kn (h) [SΓ (2f − Fh ) (z)] dh |g (z)| |dz| Γ 2π 0  Z  Z π 1 ≤ sup Kn (h) (|2f (z) − Fh (z)|) dh |g (z)| |dz| 4π 0 Γ  Z π Z  1 + sup Kn (h) (|SΓ (2f − Fh ) (z)|) dh |g (z)| |dz| . 2π 0 Γ Here by Fubini’s theorem and boundedness of singular integral we get   Z π Z 1 kf − Pn (·, f )kLM (Γ) ≤ Kn (h) sup [|2f (z) − Fh (z)|] |g (z)| |dz| dh 4π 0 Γ   Z π Z 1 + Kn (h) sup [|SΓ (2f − Fh ) (z)|] |g (z)| |dz| dh 2π 0 Γ Z π Z π h i h i 1 1 ≤ Kn (h) k2f − Fh kLM (Γ) dh + Kn (h) kSΓ (2f − Fh ) (z)kLM (Γ) dh 4π 0 2π 0 Z π h i ≤ c10 Kn (h) kFh − 2f kLM (Γ) + kFh − 2f kLM (Γ) dh. 0

(2)

Using the definition of wM (δ, f ) and features (6), (8) kernel Kn (h) consequently we obtain, Z π (2) kf − Pn (·, f )kLM (Γ) ≤ c11 Kn (h) wM (f, h) dh (13) 0

 Z π   1 1 (2) (2) ≤ c12 wM f, Kn (h) (nh + 1) dh ≤ c13 wM f, . n n 0 The theorem 1 is proved. Now, we investigate the approximation problem of the function FAB (z) = A f (z) + B f (z)

APPROXIMATION...ORLICZ SPACES

959

7

in the Dini- smooth curves. In particular, in approximation with the rational function and polynomials the function FAB (z), the role of the constants A and B is studied. Similar problems in the integral metric were studied in the work [25]. Proof of Theorem 2. Let Γ be a Dini-smooth curve and f ∈ LM (Γ). Since LM (Γ) ⊂ L1 (Γ) we obtain f ∈ L1 (Γ). Hence Cauchy’s singular integral SΓ (f ) (z) exists a.e. on Γ. Then according to the Privalov’s theorem [ 7 , p.431] Cauchy’s type integrals f + (z) and f − (z) have non-tangential limits a.e. on Γ. Then a.e. on Γ we have     FAB (t) = A f (t) + B f˜ (t) = A f + (t) − f − (t) + B f + (t) − f − (t) = (A + B) f + (t) + (B − A) f − (t)

(14) +

According to the above equality, for the approximation of function FAB (t), functions f (t) and f − (t) which are analytic inside and outside of the curve Γ, respectively, are sufficient to approximate. Using the property of the kernel Kn (h), the function f + can be written in the following form: Z Z π 1 1 π Kn (h) SΓ f (t) dh + 2Kn (h) f (t) dh. f + (t) = π 0 4π 0 By Theorem 1 we obtain  

+

1 (2)

f − Pn ≤ c13 wM f, . LM (Γ) n Then we get

(A + B) f + − (A + B) Pn L

M (Γ)

(2)

≤ c14 (A + B) wM

 f,

1 n

 .

(15)

Now, we investigate the approximate of the function f − (z). For this purpose, the plane z is mapped to the plane z 0 by the mapping z = z10 . Then the
curve Γ maps to the curve Γ0 and the function f (z) maps to the function f1 (z 0 ) = f z10 . We obtain f − (z) = f1+ (z 0 ) where the function f1+ is analytic at interior points of the Dini-smooth curve Γ0 . Then by the relation (13) there exist an algebraic polynomial P˜n (z 0 ) such that  

1

+ 0 (2) 0 ˜ ≤ c15 wM f1 , . (16)

f1 (z ) − Pn (z ) n LM (Γ) Later f1+ (t0 ) = f − (t) and since the point z = 0 is an interior point of the curve Γ, we have

 

− 1

+ 0 0 ˜ ˜

 f (t) − Pn . (17)

f1 (t ) − Pn (t ) t LM (Γ) LM (Γ0 ) We define the modulus of continuity in the following form: (2)

w eM (f, δ) = sup kf (e ςh ) − f (e ς−h ) − 2f (ς)kLM (Γ) |h|≤δ


ih

where ςeh = ψe ϕ e (ς) e the function ϕ e (ϕ e (0) = 0) maps the domain G conformally to the + e D , the function ψ is the inverse function of the ϕ e and

 0

 0 

(2) w eM (f1 , δ) = sup f1 ςeh − f1 ςe−h − 2f (ς) |h|≤δ

LM (Γ0 )


0 where ςeh = ψe2 ϕ e2 (ς) eih , the function ϕ e2 maps outside of the curve Γ0 conformally to the 0 outside of the unit circle j0 and ψe2 is the inverse function of the ϕ e2 . Using the proof scheme in [ 24 ] we get easily (2) (2) w eM (f1 , δ) = w eM (f, δ) (18)

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Sadulla Z. Jafarov

Thus by (16), (17) and (18) we obtain

   

−

1 (2)

f (t) − Pen 1 ≤ c16 w eM f, .

t LM (Γ) n

(19)

Using by (19) we can write

   

1 (2)

(B − A) f − (t) − (B − A) Pen 1 ≤ c17 (B − A) w eM f, .

t LM (Γ) n

(20)

With the use of (3), (15) and (20) we obtain

  



FAB (t) − (A + B) Pn (t) + (B − A) Pen 1 ≤ f + (t) (A + B) − (A + B) Pn (t) L (Γ)

M t LM (Γ)

     

1 1 1 (2) (2) − e

+ (B − A) f (t) − (B − A) Pn ≤ c18 (A + B) wM f, +c19 (B − A) w eM f, t LM (Γ) n n Setting Rn (t) = (A + B) Pn (t) + (B − A) Pen

  1 , t

we have kFAB (t) − Rn (t)kLM (Γ) ≤ c18 (A +

(2) B) wM



1 f, n

 + c19 (B −

(2) A) w ˜M

  1 f, . n

Now we study the following cases: 1. Let A = B; then kFAB (t) − 2A Pn (t)kLM (Γ) ≤

(2) 2c18 A wM



1 f, n



and in this case we have polynomial approximation of the function FAB (t). Specifically, if A = 21 , B = 12 we find 1 1 FAB (t) = f (t) + fe(t) = f + (t) 2 2 and thus we have  

+

1 (2)

f (t) − Pn (t) . ≤ c18 wM f, LM (Γ) n 2. Let A = B; then kFAB (z) − Rn (z)kLM (Γ) ≤ c18 (A +

(2) B) wM

    1 1 (2) f, + c19 (B − A) w eM f, n n

Specifically, if A = 1 , B = 0 we obtain FAB (z) = f (z) , Rn (z) = Pn (z) + Pen

  1 = Rn1 (z) . z

(21)

Then by (21) we have kf (z) − Rn1 (z)kLM (Γ) ≤

(2) c18 wM



1 f, n

 +

(2) c19 w eM



1 f, n

 .

Let A = − 12 and B = 12 . Thus 1 1 FAB (z) = − f (z) + fe(z) = f − (z) 2 2

(22)

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9

and by (22) we reach

−

f (z) − Rn (z) 2 L

M (Γ)

where Rn2 (z) = Pen

(2)

≤ c19 w eM

 f,

1 n

 ,

  1 . z

Thus the proof of Theorem 2 is completed. The author is grateful to Prof. Dzh.I. Mamedkhanov and Prof. D. M.Israfilov for their very useful discussions on the paper.

References [1] S.Ja. Al’per, Approximation in the Mean of Analytic functions of Class Ep , Gousudarst Izdat. Fiz.-Mat. Lit., Moscow, 1960, 273-236 (In Russian). [2] J.E. Andersson, On the degree of polynomial approximation in E p (D), J.Approximation Theory 19 (1977), 61-68. [3] M.I. Andrasko, On the approximation in the mean of analytic functions in regions with smoothboundaries , Problems in mathematical physics and function theory , Izdat. Akad. Nauk Ukrain. RSR, 1, p.3. Kiev, 1963. [4] R.Akgun and D.M.Israfilov, Approximation and moduli of fractional orders in SimirnovOrlicz classes, Clasnik matematiki, Vol. 43(63) (2008), 121-136. [5] R.Akgun and D.M.Israfilov, Polynomial approximation in weighted Simirnov-Orlicz space, Proc. A. Razmadze Math. Inst. 139(2005), 89-92. [6] N. K. Bary, A Treatise on Trigonometric Series, Volume I, Pergamon Press, Oxford, (1964). [7] G. M. Goluzin, Geometric theory of functions of a complex variable, Trasl. Math. Monogr. 2G, R. I., AMS, providence, 1969. [8] D. M. Galan, Approximation in the mean of regular fuctions of class E 1 in regions with smooth boundaries, Depovidi Akad. Nauk. Ukrain. RSR: Ser A, p. 673, 1967. [9] A. Guven and D. M. Israfilov, Polynomial approximation in Smirnov – Orlicz classes, Comput, Methods Funct. Theory 2-2(2002), 509-517. [10] A. Guven, D.M. Israfilov, Rational approximation in Orlicz spaces on Carleson curves, Bull. Belg. Math. Soc. 12(2005), 223-224. [11] V. K. Dzyadık, Introduction to the Theory of Uniform Approximation of Functions by Polynomials, Nauka, Moskow, 1977. [12] P.L. Duren, Theory of H p Spaces, Academic Pres, 1970, 258 p. [13] G. David, Operateurs Integrauxs Singuliers sur certainesd Courbes du plan Complexe, Ann. Sci. Ecole. Norm. Sur., 17(1984), 157-189. [14] R. A. Devore and G. G.Lorentz, Constructive Approximation, Springer – verlag, 1993. [15] I.I. Ibragimov and Dzh.I. Mamedkhanov, Constructive characterization of a certain class of functions, Sov. Math. Dokl. 16(1976), 820-823.

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[16] D.M. Israfilov, Approximation properties of generalized Faber series in an integral metric, Izv. Akad. Nauk. Az.SSR, Ser.Fiz-Tekh.Math.Nauk 2 (1987), 10-14 (in Russian). [17] V. Kokilashvili, On analytic functions of Smirnov-Orlicz class, Studia Mathematica, 31(1968), 43-59. [18] V. Kokilashvili, A direct theorem on mean approximation of analytic functions by polynomials, Sov. Math. Dokl. 10(1969), 411-414. [19] A. Yu. Karlovich, Algebras of singular integral operators with piecewise continuous coefficients on reflexive Orlicz spaces, Math . Nachr. 178 (1996), 187- 222. [20] A. Yu. Karlovich, Singular integral operators with regulated coefficients in reflexive Orlicz Spaces, Siberian Math. J., 38(1997), 253-266. [21] A. Yu. Karlovich, Singular integral operators with PC coefficients in reflexive rearrangement invariant spaces, Integ. Eq. and Oper. Th., 32(1998), 436-481. [22] A. Yu. Karlovich, Algebras of Singular integral operators with PC Coefficients in reflexive rearrangement invariant spaces with Muckenhoupt weights, J. Operator Theory, 47 (2002), 303-323. [23] M.A Krasnosel’skii and Ya. B. Rutitskii, Convex Functions and Orlicz spaces, Noordhoff, 1961. [24] Dzh. I. Mamedkhanov, Approximation in complex lane and simgular operators with a Cauchy kernel, Dissertation Doct. Phys-math. nauk. The University of Tblisi, 1984 (in Russian). [25] Mokhamad Ali, The problem of Approximation Theory on Complex plane, Avtorev. Diss. cand. Phys.-math. nauk, The University of Baku, 1990 (in Russian). [26] Ch. Pommerenke, Boundary behavior of conformal maps, Springer – Verlag, 1992. [27] M.M. Rao and Z.D. Ren, Theory of Orlicz spaces, Marcel Dekker, New York, 1991. [28] V. I. Smirnov and N. A. Lebedev, Functions of a Complex Variable Constructive Theory, Masschusetts Institute of Technology, Cambridge, 1968. [29] P.K. Suetin, Series of Faber polynomials, Goron and Breach, 1. Reading, 1998. [30] S.E. Warschawskii, ber das ranverhalten der Ableitung der Abildungsfunktion bei konformer Abbildung, Math. Z., 35(1932), 321.

JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.5, 963-970, 2011, COPYRIGHT 2011 EUDOXUS PRESS, 963 LLC

Fixed Points of Mappings Satisfying a New Condition in Cone Metric Spaces

Duran Turkoglu, Muhib Abuloha Department of Mathematics, Faculty of Science and Arts, Gazi University, 06500, Ankara-Turkey. [email protected]. Department of Mathematics, Institute of Science and Technology, Gazi University, 06500, Ankara-Turkey. [email protected].

Abstract In this paper we proved some …xed points of mappings satisfying a new condition ´ c in [4] are recovered. in cone metric spaces, where some of the main results of Ciri´

Key words: Fixed point, cone metric space, minihedral cone, strongly minihedral cone, cone metrically convex.

1

Introduction

Cone metric spaces were introduced by Huang and Zhang in [5]. The authors there described convergence in cone metric spaces and introduced completeness. Then they proved some …xed point theorems of contractive mappings on cone metric spaces. Some de…nitions and topological concepts were generalized in [2] and they proved there that every cone metric space is a topological space, they also generalized the concept of diametrically contractive mappings and proved some …xed point theorems in cone metric spaces. Furthermore, cone metric spaces were studied by many authors (see [1,3,6–11,13,14]). In this paper we proved some …xed points of mappings satisfying a new con´ c in [4] are dition in cone metric spaces, where some of the main results of Ciri´ recovered. Preprint submitted to Elsevier Science

11 May 2010

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2

Preliminaries

De…nition 1 Let E be a real Banach space and P a subset of E: Then, P is called a cone if and only if P1) P is closed, non empty and P 6= f0g P2) a; b 2 R a; b P3) x 2 P and

0; x; y 2 P ) ax + by 2 P x 2 P ) x = 0:

Given a cone P E, we de…ne a partial ordering with respect to P by x y if and only if y x 2 P: We write x < y to indicate that x y but x 6= y; while x > 0; there is N such that for all n > N; d(xn ; x) > 0; there is N such that for all n; m > N; d(xm ; xn ) 0 is arbitrary and a and h are such that min fKhdc (T x; T y); [dc (x; T x) + dc (y; T y)]g

a < 2 and h > (2r + a)

(2)

m(x; y) = min fdc (x; T y); dc (y; T x)g :

(3)

0 and

Then T has at least one …xed point in F:

PROOF. Let x0 2 X be arbitrary. Since F is convex, for each x; y 2 X and any 2 (0; 1) ; there exists z 2 F such that dc (x; z) = dc (x; y) and dc (z; y) = (1 )dc (x; y): Thus, we may choose x1 2 F such that dc (x0 ; x1 ) =

1 dc (x0 ; T x0 ) r+1

and dc (x1 ; T x0 ) =

r dc (x0 ; T x0 ): r+1

Or, and dc (x1 ; T x0 ) = rdc (x0 ; x1 )

dc (x0 ; T x0 ) = (r + 1)dc (x0 ; x1 ) 3

966

TURKOGLU, ABULOHA: FIXED POINTS ...CONE METRIC SPACES

Similarly, we may choose x2 2 F such that and dc (x2 ; T x1 ) = rdc (x1 ; x2 ):

dc (x1 ; T x1 ) = (r + 1)dc (x1 ; x2 )

Continuing in this process we can choose a sequence fxn g in F such that, dc (xn 1 ; T xn 1 ) = (r + 1)dc (xn 1 ; xn ) dc (xn ; T xn 1 ) = rdc (xn 1 ; xn ):

(4)

Now, we show that fxn g is a Cauchy sequence. By setting x = xn 1 ;

y = xn :

From (1) we get min fKhdc (T xn 1 ; T xn ); [dc (xn 1 ; T xn 1 ) + dc (xn ; T xn )]g 1 (2r + a) max dc (xn 1 ; xn ); dc (xn ; T xn 1 ) : r

(5) (6)

Using (4), we have min fKhdc (T xn 1 ; T xn ); (r + 1) [dc (xn 1 ; xn ) + dc (xn ; xn+1 )]g (2r + a) max fdc (xn 1 ; xn ); dc (xn 1 ; xn )g = (2r + a)dc (xn 1 ; xn ):

(7)

From (7). (I) we get, Khdc (T xn 1 ; T xn ) by triangle inequality we have,

d(xn ; T xn )

d(xn ; T xn ) d(xn ; T xn 1 )

(2r + a)dc (xn 1 ; xn )

d(xn ; T xn 1 ) + d(T xn 1 ; T xn ) d(T xn 1 ; T xn )

(8)

(9)

by normality, kd(xn ; T xn ) d(xn ; T xn 1 )k jkd(xn ; T xn )k kd(xn ; T xn 1 )kj jdc (xn ; T xn ) dc (xn ; T xn 1 )j

K kd(T xn 1 ; T xn )k K kd(T xn 1 ; T xn )k Kdc (T xn 1 ; T xn ):

(10)

By using (4), we get j(r + 1) dc (xn ; xn+1 )

rdc (xn 1 ; xn )j 4

Kdc (T xn 1 ; T xn ) :

(11)

TURKOGLU, ABULOHA: FIXED POINTS ...CONE METRIC SPACES

967

Case (i), if (r + 1) dc (xn ; xn+1 )

(12)

rdc (xn 1 ; xn )

then from (8) and (11) we have, (r + 1) dc (xn ; xn+1 ) rdc (xn 1 ; xn ) h (r + 1) dc (xn ; xn+1 ) hrdc (xn 1 ; xn )

and hence dc (xn ; xn+1 )

Kdc (T xn 1 ; T xn ) hKdc (T xn 1 ; T xn ) (2r + a)dc (xn 1 ; xn )

(2r + a) + hr dc (xn 1 ; xn ): h (r + 1)

(13)

Case (ii), if (r + 1) dc (xn ; xn+1 ) < rdc (xn 1 ; xn ); then this inequality implies dc (xn ; xn+1 )
0 is arbitrary and a and h satisfy (2) and m (x; y) = min fdc (x; T y) ; dc (y; Sx)g : Then S and T have a common …xed point. 6

(22)

TURKOGLU, ABULOHA: FIXED POINTS ...CONE METRIC SPACES

969

PROOF. Fix x0 2 X as in the proof of Theorem 8, we can choose a sequence fxn g in F such that 1 dc (x2n ; Sx2n ) ; r+1 r dc (x2n ; Sx2n ) dc (x2n+1 ; Sx2n ) = r+1 dc (x2n ; x2n+1 ) =

(23)

and 1 dc (x2n+1 ; T x2n+1 ) ; r+1 r dc (x2n+2 ; T x2n+1 ) = dc (x2n+1 ; T x2n+1 ) r+1 dc (x2n+1 ; x2n+2 ) =

(24)

n = 0; 1; 2; :::. Proceeding as in the proof of Theorem 8, replacing x = x2n+2 and y = x2n+1 in the condition (21), by (23) and (24), we obtain dc (x2n+2 ; x2n+3 )

dc (x2n+1 ; x2n+2 )

(25)

where is de…ned as in (16), so it can be shown that fxn g is a Cauchy sequence and u = lim xn and so Su = u and T u = u: The proof is complete. n!1

Remark 10 It is obvious that a contraction mapping also satis…es (1) with any a; r and h with a + 2r > h: In this case, assumption of cone metrically convexity of a space is unnecessary.

Acknowledgements

The authors would like to thank the referee for his/her useful comments. The research has been supported by The Scienti…c and Technological Research Council of Turkey (TUBITAK-Turkey).

References

[1] D. Ili´c, V. Rako´cevi´c, Common Fixed Points for Maps on Cone Metric Space. J. Math. Anal. Appl. 341: 876-882 (2008). [2] D. Turkoglu, M. Abuloha, Cone Metric Spaces and Fixed Point Theorems in Diametrically Contractive Mappings. Acta Mathematica Sinica, English Series (to appear).

7

970

TURKOGLU, ABULOHA: FIXED POINTS ...CONE METRIC SPACES

[3] R. Raja, S. M. Vaezpour, Some Extensions of Banach‘s Contraction Principle in Complete Cone Metric Spaces. Fixed Point Theory and Applications, Vol. 2008,doi: 10.1155/2008/768294. ´ c, Fixed Points of Mappings Satisfying a New Condition, Proc. Nat. [4] Lj. B. Ciri´ Acad. Sci. India Sect. A Phys. Sci. no. 3,73 (2003). [5] L.G. Huang, X. Zhang, Cone Metric Spaces and Fixed Point Theorems of Contractive Mappings. J. Math. Anal. Appl., 332, 1468-1476, 2007. [6] M. Abbas, G. Jungck, Common Fixed Point Results for Non Commuting Mappings Without Continuity in Cone Metric Spaces. J. Math. Anal. Appl., 341(1), 416-420 (2008). [7] P. Vetro, Common Fixed Points in Cone Metric Spaces. Rend. Circ. Mat. Palermo. 56: 464-468 (2007). [8] Sh. Rezapour, R. Hamlbarani, Some Notes on the Paper "Cone Metric Spaces and Fixed Point Theorems of Contractive Mappings". J. Math. Anal. Appl. 345, 719-724 (2008). [9] D. Ili´c, V. Rako´cevi´c, Quasi-Contraction on Cone Metric space. Applied Mathematics Letters, 22, 728-731 (2009). [10] M. Abbas, B.E.Rhoades, Fixed and Periodic Point Results in Cone metric Spaces. Applied Mathematics Letters, 22(4), 511-515 (2009). [11] D. Wardowski, Endpoint and Fixed Points of set-valued Contractions in Cone Metric Spaces, Nonlinear Analysis, 71, 512-516 (2009). [12] K. Deimling, Nonlinear Functional Analysis, Springer-Verlage, 1985. [13] Sh. Rezapour, Best Approximations in Cone Metric Spaces, Mathematica Moravica, Vol.11, 85-88, (2007). [14] C. Di Bari, P. Vetro, ' Pairs and Common Fixed Points in Cone Metric Spaces, Rend. Circ. Mat. Palermo, 57, 279-285, (2008).

8

JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.5, 971-976, 2011, COPYRIGHT 2011 EUDOXUS PRESS, 971 LLC

Distribution of the roots of the second kind Bernoulli polynomials C. S. Ryoo Department of Mathematics, Hannam University, Daejeon 306-791, Korea

Abstract : In [8], we introduced the second kind Bernoulli numbers and polynomials in the complex plane. In this paper, we investigate the zeros of the second kind Bernoulli polynomials Bn (x). Key words : Bernoulli numbers, Bernoulli polynomials, the second kind Bernoulli numbers and polynomials 1. Introduction Many mathematicians have studied Bernoulli numbers and polynomials, Euler numbers and polynomials(see [1-10]). These numbers and polynomials are used not only in complex analysis and mathematical physics, but also in p-adic analysis and other areas. The purpose of this paper is to obtain interesting properties of the second kind Bernoulli numbers and polynomials. In order to study the second kind Bernoulli numbers Bn and polynomials Bn (x), we must understand the structure of the second kind Bernoulli numbers Bn and polynomials Bn (x). Therefore, using computer, a realistic study for the second kind Bernoulli numbers Bn and polynomials Bn (x) is very interesting. It is the aim of this paper to observe an interesting phenomenon of ‘scattering’ of the zeros of the second kind Bernoulli polynomials Bn (x) in complex plane. The outline of this paper is as follows. We introduce the second kind Bernoulli numbers Bn and polynomials Bn (x). We give some properties of these numbers Bn and polynomials Bn (x). In Section 2, we describe the beautiful zeros of the second kind Bernoulli polynomials Bn (x) using a numerical investigation. Finally, we investigate the roots of the second kind Bernoulli polynomials Bn (x). We make some conclusions and discussions for further research in Section 3. Throughout this paper, we always make use of the following notations: R denotes the set of real numbers, C denotes the set of complex numbers, Zp denotes the ring of p-adic rational integers, Qp denotes the field of p-adic rational numbers, and Cp denotes the completion of algebraic closure of Qp . Let νp be the normalized exponential valuation of Cp with |p|p = p−νp (p) = p−1 . When one talks of q-extension, q is considered in many ways such as an indeterminate, a complex number q ∈ C, or p-adic number q ∈ Cp . If q ∈ C one normally assume that |q| < 1. If q ∈ Cp , we normally assume 1 that |q − 1|p < p− p−1 so that q x = exp(x log q) for |x|p ≤ 1. For g ∈ U D(Zp ) = {g|g : Zp → Cp is uniformly differentiable function}, the p-adic q-integral was defined by [2] 

N

p −1 1  Iq (g) = g(x)dμq (x) = lim g(x)q x , (cf. [1, 2, 3, 5, 7, 8]). N →∞ [pN ] Zp x=0

The bosonic integral was considered from a physical point of view to the bosonic limit q → 1, as follows:  pN −1 1  I1 (g) = lim Iq (g) = g(x)dμ1 (x) = lim N g(x), cf. [1, 2, 3, 5]) (1.1). q→1 N →∞ p Zp x=0

972

RYOO: SECOND KIND BERNOULLI POLYNOMIALS

By (1.1), we easily see that I1 (g1 ) = I1 (g) + g  (0), cf. [1, 2, 3, 4, 5, 8, 9, 10], dg(x)  . dx x=0 (2x+1)t In (1.2), if we take g(x) = e , then we obtain  e(2x+1)t dμ1 (x) =

(1.2)

where g1 (x) = g(x + 1) and g  (0) =

Zp

for |t| ≤ p

1 − p−1

2tet , −1

e2t

. Let us define the second kind Bernoulli numbers and polynomials as follows:  ∞  tn (1.3) e(2x+1)t dμ1 (x) = Bn , n! Zp n=0 

(2y+1+x)t

e

Zp

dμ1 (y) =

∞ 

Bn (x)

n=0

tn . n!

(1.4)

By (1.3) and (1.4), we obtain the following Witt’s formula. Theorem 1 (Witt formula).

 Zp

 Zp

(2x + 1)n dμ1 (x) = Bn ,

(x + 2y + 1)n dμ1 (y) = Bn (x).

The second kind Bernoulli polynomials Bn (x) of degree n in x, are defined by means of the following generate function: F (x, t) =

∞ tn 2tet xt  e . = B (x) n e2t − 1 n! n=0

(1.5)

In [8], we studied the second kind Bernoulli numbers Bn and polynomials Bn (x) and investigated their properties. The following elementary properties of the second kind Bernoulli numbers Bn and polynomials Bn (x) are readily derived form (1.5)( see, for details, [8]). We, therefore, choose to omit details involved. Theorem 2 (Distribution relation). For any positive integer m, we obtain n−1

Bn (x) = m

m−1 

 Bn

i=0

2i + x + 1 − m m

 for n ≥ 0.

Theorem 3 (Addition theorem). The second kind Bernoulli polynomials Bn (x) satisfies the following relation: l    l Bn (x)y l−n . Bl (x + y) = n n=0 Theorem 4. For n ∈ N, we have Bn (x) = (−1)n Bn (−x). Theorem 5 (Difference equation). For any positive integer n, we have Bn (x + 2) − Bn (x) = 2n(x + 1)n−1 .

2

RYOO: SECOND KIND BERNOULLI POLYNOMIALS

973

2. Zeros of the second kind Bernoulli polynomials Bn (x) In this section, we display the shapes of the second kind Bernoulli polynomials Bn (x) and we investigate the zeros of the second Bernoulli polynomials Bn (x). For n = 1, · · · , 10, we can draw a plot of the second kind Bernoulli polynomials Bn (x), respectively. This shows the ten plots combined into one. We display the shape of Bn (x), −5 ≤ x ≤ 5 (Figure 1).

300

200

100 Bn x 0

-100

-200 -4

-2

0 x

2

4

Figure 1: Curve of Bn (x)

We investigate the beautiful zeros of the second kind Bernoulli polynomials Bn (x) by using a computer. We plot the zeros of the second kind Bernoulli polynomials Bn (x) for n = 20, 30, 40, 50 and x ∈ C (Figure 2). In Figure 2 (top-left), we choose n = 20. In Figure 2 (top-right), we choose n = 30. In Figure 2 (bottom-left), we choose n = 40. In Figure 2 (bottom-right), we choose n = 50. Stacks of zeros of Bn (x) for 1 ≤ n ≤ 50 from a 3-D structure are presented (Figure 3). Our numerical results for approximate solutions of real zeros of Bn (x) are displayed (Tables 1, 2). Table 1. Numbers of real and complex zeros of Bn (x) degree n

real zeros

complex zeros

1

1

0

2

2

0

3

3

0

4

4

0

5

5

0

6

2

4

7

3

4

8

4

4

9

5

4

10

6

4

11

7

4

We observe a remarkably regular structure of the complex roots of the second kind Bernoulli polynomials Bn (x). We hope to verify a remarkably regular structure of the complex roots of the second kind Bernoulli polynomials Bn (x) (Table 1). Next, we calculated an approximate solution

3

974

RYOO: SECOND KIND BERNOULLI POLYNOMIALS

Imx

15

15

10

10

5

5

Imx

0

0

-5

-5

-10

-10

-10

-5

0

5

10

15

-10

-5

Rex

Imx

15

15

10

10

5

5

Imx

0

-5

-10

-10

-5

0

5

10

15

5

10

15

0

-5

-10

0

Rex

5

10

15

-10

-5

Rex

0

Rex

Figure 2: Zeros of Bn (x) for n = 20, 30, 40, 50 satisfying Bn (x), x ∈ R. The results are given in Table 2. Table 2. Approximate solutions of Bn (x) = 0, x ∈ R degree n

x

1

0.0000 −0.57735027,

2

−1.00000000,

3 4 5

−1.31540744, −1.5275252,

−1.0000000,

7

−1.494431,

8

10

−1.898212, −2.147943,

−1.0000000

−1.499849,

4

1.0000000,

1.5275252

1.0000000

0.5012392

0.000,

−0.50031057,

1.31540744

0.5049186

0.000,

−0.5012392,

1.00000000

0.51932962,

0.000,

−0.5049186,

6

9

0.0000,

−0.51932962,

−1.0000000,

0.57735027

1.494431

1.0000000,

0.50031057,

1.898212 1.499849,

2.147943

RYOO: SECOND KIND BERNOULLI POLYNOMIALS

Imx

975

10 5

0 -5 -10

40

n 20

0 -5 0 Rex

5

Figure 3: Stacks of zeros of Bn (x), 1 ≤ n ≤ 50

Figure 4 shows the distribution of real zeros of Bn (x) for 1 ≤ n ≤ 50.

40

n 20

0 -6 -5 -4 -3 -2 -1

1

0

2

3

4

5

6

Rex Figure 4: Plot of real zeros of Bn (x), 1 ≤ n ≤ 50

3. Directions for Further Research Finally, we shall consider the more general problems. Prove that Bn (x), x ∈ C, has Re(x) = 0 reflection symmetry in addition to the usual Im(x) = 0 reflection symmetry analytic complex functions(see Figures 2, 3). The obvious corollary is that the zeros of Bn (x) will also inherit these symmetries. If Bn (x0 ) = 0, then Bn (x∗0 ) = 0 (3.1) ∗ denotes complex conjugation (see Figure 2). Prove that Bn (x) = 0 has n distinct solutions. If B2n+1 (x) has Re(x) = 0 and Im(x) = 0 reflection symmetries, and 2n + 1 non-degenerate zeros, 5

976

RYOO: SECOND KIND BERNOULLI POLYNOMIALS

then 2n of the distinct zeros will satisfy (3.1). If the remaining one zero is to satisfy (3.1) too, it must reflect into itself, and therefore it must lie at 0 (see Figure 4), the center of the structure of the zeros, ie., Bn (0) = 0 ∀ odd n. By Theorem 4 and Theorem 5, we obtain Bn (1) = Bn (−1) = 0 ∀ odd n ≥ 3. Find all the real zeros of Bn (x) = 0. Prove that Bn (x) = 0 has n distinct solutions. Find the numbers of complex zeros CBn (x) of Bn (x), Im(x) = 0. Since n is the degree of the polynomial Bn (x), the number of real zeros RBn (x) lying on the real plane Im(x) = 0 is then RBn (x) = n − CBn (x) , where CBn (x) denotes complex zeros. See Table 1 for tabulated values of RBn (x) and CBn (x) . Find the equation of envelope curves bounding the real zeros lying on the plane. For related topics the interested reader is referred to [3, 5, 6, 7, 8, 9]. Acknowledgment This paper has been supported by the 2011 Hannam University Research Fund. REFERENCES 1. Carlitz, L. (1948) q-Bernoulli numbers and polynomials, Duke Math. J., v. 15, pp. 987-1000. 2. Kim, T. (2002). q-Volkenborn integration, Russ. J. Math. Phys., v. 9, pp. 288-299. 3. Kim, T., Ryoo, C.S., Jang, L.C., Rim, S.H.,(2005). Exploring the q-Riemann Zeta function and q-Bernoulli polynomials, Discrete Dynamics in Nature and Society, v.2005(2), pp.171-181. 4. Kim, M-S., Son, J-W.,(2007). Analytic properties of the q-volkenborn integral on the ring of p-adic integres, Bull. Korean Math. Soc., v.44, pp.1-12. 5. Ryoo,C.S., Kim, T., Agarwal, R.P. (2005). The structure of the zeros of the generalized Bernoulli polynomials, Neural Parallel Sci. Comput., v.13, pp. 371-379. 6. Ryoo, C.S., Rim, S.H. (2008). On the analogue of Bernoulli polynomials, Journal of Computational Analysis and Applications, v.10, pp. 163-172. 7. Ryoo, C.S., Kim, T., Agarwal, R.P. (2006). A numerical investigation of the roots of qpolynomials, Inter. J. Comput. Math., v.83, pp. 223-234. 8. Ryoo, C.S. (2010). A note on the second kind Bernoulli polynomials, to appear. 9. Ryoo, C.S. (2010). Calculating zeros of the second kind Euler polynomials, Journal of Computational Analysis and Applications, v.12, pp. 828-833. 10. Simesk, Y., Kurt, V., Kim, D. (2007). New approach to the complete sum of products of the twisted (h, q)-Bernoulli numbers and polynomials, J. Nonlinear Math. Phys., v.14, pp. 44-56.

6

JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.5, 977-985, 2011, COPYRIGHT 2011 EUDOXUS PRESS, 977 LLC

A numerical investigation on the structure of the roots of q-Genocchi polynomials C.S. Ryoo Department of Mathematics, Hannam University, Daejeon 306-791, Korea Abstract Recently several authors studied the q-extension of Genocchi numbers and polynomials(see [2-9]). In this paper we construct the q-Genocchi numbers Gn,q and polynomials Gn,q (x). We also observe the behavior of complex roots of q-Genocchi polynomials, using numerical investigation. By means of numerical experiments, we demonstrate a remarkably regular structure of the complex roots of q-Genocchi polynomials. Finally, we give a table for the solutions of the q-Genocchi polynomials. 2000 Mathematics Subject Classification : 11S80, 11B68 Key words : Genocchi numbers, Genocchi polynomials, q-Genocchi numbers, q-Genocchi polynomials

1. Introduction In the 21st century, the computing environment would make more and more rapid progress. Using computer, a realistic study for new analogs of Genocchi numbers and polynomials is very interesting. It is the aim of this paper to observe an interesting phenomenon of ‘scattering’ of the zeros of the q-Genocchi polynomials Gn,q (x). The outline of this paper is as follows. In Section 2, we study the the q-Genocchi polynomials Gn,q (x). In Section 3, we describe the beautiful zeros of the q-Genocchi polynomials Gn,q (x) using a numerical investigation. Also we display distribution and structure of the zeros of the the q-Genocchi polynomials Gn,q (x) by using computer. By using the results of our paper the readers can observe the regular behaviour of the roots of the q-Genocchi polynomials Gn,q (x). Finally, we carried out computer experiments for doing demonstrate a remarkably regular structure of the complex roots of q-Genocchi polynomials Gn,q (x). Throughout this paper we use the following notations. By Zp we denote the ring of p-adic rational integers. Q denotes the field of rational numbers, Qp denotes the field of p-adic rational numbers, C denotes the complex number field, and Cp denotes the completion of algebraic closure of Qp . Let νp be the normalized exponential valuation of Cp with |p|p = p−νp (p) = p−1 . When one talks of q-extension, q is considered in many ways such as an indeterminate, a complex number q ∈ C, or p-adic number q ∈ Cp . If q ∈ C 1 one normally assume that |q| < 1. If q ∈ Cp , we normally assume that |q − 1|p < p− p−1 so that q x = exp(x log q) for |x|p ≤ 1. We say that f is uniformly differentiable function at a point a ∈ Zp and denote this property by g ∈ U D(Zp ), if the difference quotients Fg (x, y) =

g(x) − g(y) x−y

have a limit l = g  (a) as (x, y) → (a, a). For g ∈ U D(Zp ), the p-adic q-integral was defined by [3,4,5]   1 Iq (g) = g(x)dμq (x) = lim g(x)q x . N N →∞ [p ] Zp N 0≤x 1, n

with the usual convention about replacing (Gq ) by Gn,q in the binomial expansion. Proof. From (2.3), we obtain ∞ ∞ n   tn 2t n t = = = eGq t G (G ) n,q q qet + 1 n=0 n! n=0 n!

which yields 2t = (qet + 1)eGq t = qe(Gq +1)t + eGq t . Using Taylor expansion of exponential function, we have 2t =

∞ 

n

n

{q (Gq + 1) + (Gq ) }

n=0 ∞ 

+

n

tn 0 0 1 1 = q (Gq + 1) + (Gq ) + q (Gq + 1) + (Gq ) n!

n

{q (Gq + 1) + (Gq ) }

n=2

tn . n!

The result follows by comparing the coefficients.  Since

  l ∞ ∞ ∞    2t tl tn  m tm tn l−n tl−n (x+y)t e = Gl,q (x + y) = t = Gn,q (x) y Gn,q (x) y l! qe + 1 n! m=0 m! n! (l − n)! n=0 l=0 l=0 n=0     ∞ l   l tl Gn,q (x)y l−n , = n l! n=0

∞ 

l=0

we have the following theorem. Theorem 4. q-Genocchi polynomials Gn,q (x) satisfies the following relation: Gl,q (x + y) =

l    l n=0

n

Gn,q (x)y l−n .

It is easy to see that 

a+x m−1 2t tn 1  2mt xt a a e = e m Gn,q (x) = t (−1) q m mt n! qe + 1 m q e + 1 n=0 a=0   m−1 ∞   a + x (mt)n 1 a a = (−1) q Gn,qm m a=0 m n! n=0   n  ∞ m−1   t a+x n−1 a a . = m (−1) q Gn,qm m n! n=0 a=0 ∞ 

Hence we have the below theorem. Theorem 5. For any positive integer m(=odd), we obtain n−1

Gn,q (x) = m

m−1 

 i i

(−1) q G

i=0

4

n,q m

i+x m

 for n ≥ 0.

 (mt)

RYOO: q-GENOCCHI POLYNOMIALS

981

From (2.3), we can derive the following equality:  n    ∞ ∞    n Gk+1,q 2 tn tn xt n−k e = x . En,q (x) = t n! qe + 1 n! k k+1 n=0 n=0 k=0

Hence, we give relation between Gn,q and En,q (x), nth q-Euler polynomials as follows: Theorem 6. For any positive integer n, we obtain En,q (x) =

n    n Gn+1,q

k

k=0

k+1

xn−k .

3. Distribution and Structure of the zeros In this section, we investigate the zeros of the q-Genocchi polynomials Gn,q (x) by using computer. We plot the zeros of Gn,q (x), q = 1/2, x ∈ C. (Figures 1, 2, 3, and 4). In Figure 1, we choose 8 4 6 4 2 2

Imx 0

Imx 0 -2

-2 -4 -6

-4 -4

-2

0

4

2

-6

-4

-2

Rex

Figure 1: Zeros of Gn,q (x)

8

6

6

4

4

2

2

Imx 0

Imx 0

-2

-2

-4

-4

-6

-6

-4

-2

0

2

4

6

8

6

8

Figure 2: Zeros of Gn,q (x)

8

-6

0

Rex

2

4

6

8

Rex

-6

-4

-2

0

2

4

Rex

Figure 3: Zeros of Gn,q (x)

Figure 4: Zeros of Gn,q (x)

5

982

RYOO: q-GENOCCHI POLYNOMIALS

n = 15 and q = 1/2. In Figure 2, we choosen = 20 and q = 1/2. In Figure 3, we choose n = 25 and q = 1/2. In Figure 4, we choosen = 30 and q = 1/2. Next, we plot the zeros of Gn,q (x), x ∈ C for q = 1/10, 1/100, 1/1000, 1/10000. (Figures 5, 6, 7, and 8). In Figure 5, we choose n = 30 and q = 1/10. In Figure 6, we choosen = 30 and q = 1/100. 6

6

4

4

2

2

Imx 0

Imx 0

-2

-2

-4

-4

-6

-6 -6

-4

-2

0

2

4

6

-6

-4

-2

Rex

0

2

4

6

Rex

Figure 5: Zeros of Gn,q (x)

Figure 6: Zeros of Gn,q (x)

6

6

4

4

2

2

Imx 0

Imx 0

-2

-2

-4

-4

-6

-6 -6

-4

-2

0

2

4

6

Rex

-6

-4

-2

0

2

4

6

Rex

Figure 7: Zeros of Gn,q (x)

Figure 8: Zeros of Gn,q (x)

In Figure 7, we choose n = 30 and q = 1/1000. In Figure 8, we choose n = 30 and q = 1/10000. In Figures 1-9, Gn,q (x), x ∈ C, has Re(x) = 1/2 reflection symmetry. This translates to the following open problem: Prove or disprove: Gn,q (x), x ∈ C, has Re(x) = 1/2 reflection symmetry. Our numerical results for numbers of real and complex zeros of Gn,q (x) are displayed in Table 1.

6

RYOO: q-GENOCCHI POLYNOMIALS

983

5

Imx 0 -5 30

20

n 10

0 -2 0 2

Rex

4

Figure 9: Stacks of zeros Gn,q (x), q = 1/2, for 1 ≤ n ≤ 30

Table 1. Numbers of real and complex zeros of Gn,q (x) q = 1/3

q = 1/2

degree n

real zeros

complex zeros

real zeros

complex zeros

2

1

0

1

0

3

2

0

2

0

4

3

0

3

0

5

2

2

2

2

6

3

2

3

2

7

4

2

4

2

8

3

4

3

4

9

4

4

4

4

10

3

6

3

6

11

4

6

4

6

Plot of real zeros of Gn,q (x) for 1 ≤ n ≤ 30, q = 1/10, 8/10 structure are presented (Figure 10). We shall consider the more general open problem. In general,how many roots does Gn,q (x) have ? Prove or disprove: Gn,q (x) has n − 1 distinct solutions. Find the numbers of complex zeros CGn,q (x) of Gn,q (x), Im(x) = 0. Prove or give a counterexample: Conjecture: Since n is the degree of the polynomial Gn,q (x), the number of real zeros RGn,q (x) lying on the real plane Im(x) = 0 is then RGn,q (x) = n − 1 − CGn,q (x) , where CGn,q (x) denotes complex zeros. See Table 1 for tabulated values of RGn,q (x) and CGn,q (x) . Find the equation of envelope curves bounding the real zeros lying on the plane, and the equation of a trajectory curve running through the complex zeros on any one of the arcs. We plot the Gn,q (x), respectively (Figures 1-10). These figures give mathematicians an unbounded capacity to create visual mathematical investigations of the behavior of the roots of the Gn,q (x). Moreover, it is possible to create a new mathematical ideas and analyze them in ways that generally are not possible by hand. The author has no doubt that investigation along this line will lead to a new approach employing numerical method in the field of research of the q-Genocchi 7

984

RYOO: q-GENOCCHI POLYNOMIALS

30

30

20

20

n

n

10

10

0 -2

-1

0

1

2

3

0

4

-3

-2

-1

0

Rex

1

2

3

4

Rex

Figure 10: Real zeros of Gn,q (x) for q = 1/10, 8/10, 1 ≤ n ≤ 30

polynomials Gn,q (x) to appear in mathematics and physics. For related topics the interested reader is referred to [6], [7], [8]. We calculated an approximate solution satisfying Gn,q (x), x ∈ C. The results are given in Table 2 and Table 3. Table 2. Approximate solutions of Gn,q (x) = 0, q = 1/2 degree n

x

2

0.33333

3 4 5

−0.13807, −0.42060, −0.4243 − 0.0718i,

0.8047

0.22004,

1.2006

−0.4243 + 0.0718i,

0.6547,

1.5273 6

−0.6454 − 0.3244i,

−0.6454 + 0.3244i,

1.0854, 7

−0.7648 − 0.4958i, 0.5160,

8

0.08542,

1.7866

−0.7648 + 0.4958i, 1.528,

1.958

−0.4719,

RYOO: q-GENOCCHI POLYNOMIALS

985

Table 3. Approximate solutions of Gn,q (x) = 0, q = 1/3 degree n

x

2

0.25000 −0.18301,

3

−0.39606,

4 5

−0.43794 − 0.21689i,

0.6830

0.07357,

1.0725

−0.43794 + 0.21689i,

0.4631

1.4128 6

−0.5860 − 0.3815i,

−0.5860 + 0.3815i,

0.8580, 7

−0.6620 − 0.5900i, 0.25042,

−0.14190,

1.7060

−0.6620 + 0.5900i, 1.2506,

−0.6265,

1.9495

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(20100008344) REFERENCES 1. Carlitz, L. (1948) q-Bernoulli numbers and polynomials, Duke Math. J., v. 15, pp. 987-1000. 2. Cenkci, M., Can, M., Kurt, V.(2004). q-adic interpolation functions and kummer type congruence for q-twisted and q-generalized twisted Euler numbers, Advan. Stud. Contemp. Math., v. 9, pp. 203-216. 3. Kim, T.(2008). Note on q-Genocchi numbers and polynomials , Advan. Stud. Contemp. Math., v. 17, pp. 9-15. 4. Kim, T.(2007). On the q-extension of Euler and Genocchi numbers, J. Math. Anal. Appl., v.326, pp. 1458-1465. 5. Kim, T., Jang, L.C., Pak, H-K.(2001). A note on q-Euler and Genocchi numbers , Proc. Japan Acad. , v.77 A, pp. 139-141. 6. Kim, T., Ryoo, C.S., Jang, L.C., Rim, S.H.,(2005). Exploring the q-Riemann Zeta function and q-Bernoulli polynomials , Discrete Dynamics in Nature and Society, v.2005(2), pp.171-181. 7. Ryoo, C.S. (2005). A numerical investigation on the zeros of the Genocchi polynomials, J. Appl. Math. and Computing, v.22(1-2), pp. 125-132. 8. Ryoo, C.S. (2008). Calculating zeros of the twisted Genocchi polynomials, Advan. Stud. Contemp. Math., v.17, pp. 147-159.

9

JOURNAL 986 OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.5, 986-992, 2011, COPYRIGHT 2011 EUDOXUS PRESS, LLC

A note on the second kind Genocchi polynomials C. S. Ryoo Department of Mathematics, Hannam University, Daejeon 306-791, Korea

Abstract : In this paper, we introduce the second kind Genocchi numbers and polynomials in the complex plane. We also give some properties of these numbers and polynomials. Finally, we investigate the zeros of the second kind Genocchi polynomials Gn (x). Key words : Genocchi numbers, Genocchi polynomials, the second kind Genocchi numbers and polynomials 1. Introduction In recent years, many mathematicians have investigated Bernoulli numbers and polynomials, Euler numbers and polynomials, Genocchi numbers and polynomials. These numbers and polynomials are used not only in complex analysis and mathematical physics, but also in p-adic analysis and other areas. In this paper, we introduce the second kind Genocchi numbers Gn and polynomials Gn (x). In order to study the second kind Genocchi numbers Gn and polynomials Gn (x), we must understand the structure of the second kind Genocchi numbers Gn and polynomials Gn (x). Therefore, using computer, a realistic study for the second kind Genocchi numbers Gn and polynomials Gn (x) is very interesting. It is the aim of this paper to observe an interesting phenomenon of ‘scattering’ of the zeros of the second kind Genocchi polynomials Gn (x) in complex plane. The outline of this paper is as follows. We introduce the second kind Genocchi numbers Gn and polynomials Gn (x). We give some properties of these numbers Gn and polynomials Gn (x). In Section 2, we describe the beautiful zeros of the the second kind Genocchi polynomials Gn (x) using a numerical investigation. Finally, we investigate the roots of the second kind Genocchi polynomials Gn (x). In complex number field C, the second kind Genocchi numbers Gn are defined by the generating function: ∞  π 2t tn , (|t| < ), (1) G(t) = t = G n e + e−t n! 2 n=0 where we use the technique method notation by replacing Gn by Gn (n ≥ 0) symbolically. From (1), we have ∞  tn 2t = Gn = eGt t −t e +e n! n=0 which yields 2t = e(G+1)t + e(G−1)t . Using Taylor expansion of exponential function, we obtain 2t =

∞ 

((G + 1)n + (G − 1)n )

n=0

tn . n!

By comparing the coefficients, we have  (G + 1) + (G − 1) = n

n

2, 0,

if n = 1, if n =  1.

RYOO: SECOND KIND GENOCCHI POLYNOMIALS

987

We obtain the first value of the second kind Genocchi numbers Gn : G0 = 0,

G1 = 1,

G8 = 0,

G9 = 12465,

G3 = −3,

G2 = 0,

G17 = 329655706465,

G10 = 0,

G4 = 0,

G5 = 25,

G11 = −555731,

G19 = −45692713833379

G7 = −427,

G6 = 0,

G13 = 35135945,

G15 = −2990414715,

G21 = 7777794952988025, · · ·

In general, it satisfies G2 = G4 = G6 = · · · = 0. We consider the second kind Genocchi polynomials Gn (x) as follows: G(x, t) =

∞  2t tn xt e = Gn (x) . t −t e +e n! n=0

(2)

By the above definition, we obtain ∞ 

Gl (x)

l=0

∞ ∞  2t tl tn  m tm xt = t e = G x n l! e + e−t n! m=0 m! n=0    l    ∞ l ∞     l tl tn l−n tl−n Gn xl−n = . Gn x = n n! (l − n)! l! n=0 n=0 l=0

l=0

l

By using comparing coefficients

t , we have l! Gn (x) =

n    n k=0

k

Gk xn−k .

In the special case x = 0, we define Gn (0) = Gn . From (2), we have ∞  n=0

En (x)

∞ ∞  tn ext  = F (t, x) = 2 2t (−1)n e(2n+1+x)t = (−1)n e(2n+1)t n! t n=0 n=0  ∞  ∞  ∞ m−1 m    tl t G t m+1 = = ext Gm xl m! m + 1 m! l! m=0 m=0 l=0     ∞ n ∞ n       tn n Gm+1 n−m tn Gm+1 n−m n! x = x . = m+1 n!(n − m)! n! n=0 m=0 m m + 1 n! n=0 m=0

Therefore, we give relation between the second kind Euler polynomials En (x) (see [1]) and the second kind Genocchi numbers Gn . Theorem 1. For any positive integer n, we have n    n Gk+1 n−k x En (x) = , k k+1 k=0

where En (x) denotes the second kind Euler polynomials cf. [1]. Let m be odd. We compute the following sum ∞  n=0

Gn (x)

m−1  2t tn 2temt (2a+x+1−m)t xt = t e e = (−1)a 2mt −t n! e +e e +1 a=0 



2a + x + 1 − m mt 1 2(mt) a m = (−1) mt e −mt m a=0 e +e   m−1 ∞  2a + x + 1 − m (mt)n 1  a = (−1) Gn m a=0 m n! n=0   n  ∞ m−1   t 2a + x + 1 − m . = mn−1 (−1)a Gn m n! n=0 a=0 m−1 

2

(3)

988

RYOO: SECOND KIND GENOCCHI POLYNOMIALS

Comparing the coefficient of

tn on both sides of (3), we obtain the following multiplication theorem. n!

Theorem 2. For any positive integer m(=odd), we obtain Gn (x) = mn−1

m−1 

 (−1)i Gn

i=0

2i + x + 1 − m m

 for n ≥ 0.

Since ∞ 

Gl (x + y)

l=0

∞ ∞  2t tl tn  m tm (x+y)t = t e = G (x) y n l! e + e−t n! m=0 m! n=0    l    ∞ l ∞     l tl tn l−n tl−n l−n Gn (x)y = , Gn (x) y = n n! (l − n)! l! n=0 n=0 l=0

l=0

we have the following addition theorem. Theorem 3. The second kind Genocchi polynomials Gn (x) satisfies the following relation: Gl (x + y) =

l    l n=0

Because

n

Gn (x)y l−n .

∞  tn d ∂ G(t, x) = tG(t, x) = Gn (x) , ∂x dx n! n=0

it follows the important relation d Gn (x) = nGn−1 (x). dx We have the integral formula as follows:  b 1 Gn−1 (x)dx = [Gn (b) − Gn (a)] . n a Since ∞ 

Gn (−x)

n=0

−2t (−t)n = G(−x, −t) = −t e(−x)(−t) n! e + et =

∞  −2t tn xt e = −G(x, t) = −Gn (x) , t −t e +e n! n=0

we obtain the following theorem. Theorem 4. For n ∈ N, we have Gn (x) = (−1)n−1 Gn (−x).

From (4), we have ∞  n=0

(Gn (x + 2) + Gn (x))

∞ ∞   tn = 2t (−1)n e(2n+1+x+2)t + 2t (−1)n e(2n+1+x+2)t n! n=0 n=0 (x+1)t

= 2te

= 2t

∞ 

n=0

3

nt

(x + 1)

n

n!

=

∞ 

n=0

n−1 t

2n(x + 1)

n

n!

.

(4)

RYOO: SECOND KIND GENOCCHI POLYNOMIALS

989

tn on both sides of (4), we get the following theorem. n! Theorem 5. For any positive integer n, we have

Comparing the coefficient of

Gn (x + 2) + Gn (x) = 2n(x + 1)n−1 . By using computer, the second kind Genocchi polynomials En (x) can be determined explicitly. A few of them are G1 (x) = 1,

G2 (x) = 2x,

G5 (x) = 5x4 − 30x2 + 25,

G3 (x) = 3x2 − 3,

G4 (x) = 4x3 − 12x,

G6 (x) = 6x5 − 60x3 + 150x,

G8 (x) = 8x7 − 168x5 + 1400x3 − 3416x,

G7 (x) = 7x6 − 105x4 + 525x2 − 427,

G9 (x) = 9x8 − 252x6 + 3150x4 − 15372x2 + 12465,

G10 (x) = 10x9 − 360x7 + 6300x5 − 51240x3 + 124650x, G11 (x) = 11x10 − 495x8 + 11550x6 − 140910x4 + 685575x2 − 555731 ···

2. Zeros of the second kind Genocchi polynomials Gn (x) In this section, we display the shapes of the second kind Genocchi polynomials Gn (x) and we investigate the zeros of the second Genocchi polynomials Gn (x). For n = 1, · · · , 10, we can draw a plot of the second kind Genocchi polynomials Gn (x), respectively. This shows the ten plots combined into one. We display the shape of Gn (x), −6 ≤ x ≤ 6 (Figure 1).

30000

20000

10000

Gn x 0

-10000

-20000 -6

-4

-2

0

2

4

6

x

Figure 1: Curve of Gn (x)

We investigate the beautiful zeros of the second kind Genocchi polynomials Gn (x) by using a computer. We plot the zeros of the second kind Genocchi polynomials Gn (x) for n = 10, 20, 30, 40 and x ∈ C (Figure 2). In Figure 2 (top-left), we choose n = 10. In Figure 2 (top-right), we choose n = 20. In Figure 2 (bottom-left), we choose n = 30. In Figure 2 (bottom-right), we choose n = 40. Stacks of zeros of Gn (x) for 1 ≤ n ≤ 50 from a 3-D structure are presented (Figure 3). Our numerical results for approximate solutions of real zeros of Gn (x) are displayed (Tables 1, 2).

4

990

RYOO: SECOND KIND GENOCCHI POLYNOMIALS

Imx

20

20

15

15

10

10

5

5

Imx

0

0

-5

-5

-10

-10

-15

-15

-15

-5

-10

5

0

10

15

20

-15

-10

-5

Rex

Imx

20

20

15

15

10

10

5

5

Imx

0

-5

-10

-10

-15

-15

-5

-10

5

0

5

10

15

20

5

10

15

20

0

-5

-15

0

Rex

10

15

20

-15

-10

-5

Rex

0

Rex

Figure 2: Zeros of Gn (x) for n = 10, 20, 30, 40

Table 1. Numbers of real and complex zeros of Gn (x) degree n

real zeros

complex zeros

2

1

0

3

2

0

4

3

0

5

4

0

6

5

0

7

2

4

8

3

4

9

4

4

10

5

4

11

6

4

We observe a remarkably regular structure of the complex roots of the second kind Genocchi polynomials Gn (x). We hope to verify a remarkably regular structure of the complex roots of the second kind Genocchi polynomials Gn (x) (Table 1). Next, we calculated an approximate solution

5

RYOO: SECOND KIND GENOCCHI POLYNOMIALS

Imx

991

20 10

0 -10 -20

40

n 20

0 -10 0

Rex

10

Figure 3: Stacks of zeros of Gn (x), 1 ≤ n ≤ 50

satisfying Gn (x), x ∈ R. The results are given in Table 2. Table 2. Approximate solutions of Gn (x) = 0, x ∈ R degree n

x

2

0.0000 −1.0000,

3

−1.7321,

4

−2.236,

5 6

−2.236,

−2.236,

−2.000,

10 11

−3.000, −3.000, −4.000,

−2.000

−3.000,

2.236,

2.236

2.000

−1.000

0.000,

1.000,

2.236

1.000

0.000,

−1.000,

1.7321

1.000,

0.000,

−1.000,

8 9

0.0000,

−1.000,

7

1.0000

− 2.000

12.000,

1.000,

3.000

3.000,

4.000

Figure 4 shows the distribution of real zeros of Gn (x) for 1 ≤ n ≤ 35. 3. Directions for Further Research Finally, we shall consider the more general problems. Prove that Gn (x), x ∈ C, has Re(x) = 0 reflection symmetry in addition to the usual Im(x) = 0 reflection symmetry analytic complex 6

992

RYOO: SECOND KIND GENOCCHI POLYNOMIALS

30

20

n

10

-10-9-8-7 -6-5-4-3 -2-1 0 1 2 3 4 5 6

0 8 9 10

Rex Figure 4: Plot of real zeros of Gn (x), 1 ≤ n ≤ 35

functions(see Figures 2, 3). The obvious corollary is that the zeros of Gn (x) will also inherit these symmetries. If Gn (x0 ) = 0, then Gn (x∗0 ) = 0

(7)

∗ denotes complex conjugation (see Figure 2). Prove that Gn (x) = 0 has n − 1 distinct solutions. If G2n (x) has Re(x) = 0 and Im(x) = 0 reflection symmetries, and 2n − 1 non-degenerate zeros, then 2n − 1 of the distinct zeros will satisfy (7). If the remaining one zero is to satisfy (7) too, it must reflect into itself, and therefore it must lie at 0 (see Figure 4), the center of the structure of the zeros, ie., Gn (0) = 0 ∀ even n. By Theorem 4 and Theorem 5, we obtain Gn (1) = Gn (−1) = 0 ∀ odd n ≥ 3. Find all the real zeros of Gn (x) = 0. Prove that Gn (x) = 0 has n − 1 distinct solutions. Find the numbers of complex zeros CGn (x) of Gn (x), Im(x) = 0. Since n − 1 is the degree of the polynomial Gn (x), the number of real zeros RGn (x) lying on the real plane Im(x) = 0 is then RGn (x) = (n − 1) − CGn (x) , where CGn (x) denotes complex zeros. See Table 1 for tabulated values of RGn (x) and CGn (x) . Find the equation of envelope curves bounding the real zeros lying on the plane. For related topics the interested reader is referred to [1, 2, 3, 4]. Acknowledgment This paper has been supported by the 2011 Hannam University Research Fund. REFERENCES 1. Ryoo, C.S. (2010). Calculating zeros of the second kind Euler polynomials, Journal of Computational Analysis and Applications, v.12, pp. 828-833. 2. Ryoo, C.S. (2008). Calculating zeros of the twisted Genocchi polynomials, Advanced Studies in Contemporary Mathematics, v.17, pp. 147-159. 3. Ryoo, C.S. (2008). A numerical computation on the structure of the roots of q-extension of Genocchii polynomials, Applied Mathematics Letters, v.21, pp. 348-354. 4. Ryoo, C.S., Kim, T., Agarwal, R.P. (2006). A numerical investigation of the roots of qpolynomials, Inter. J. Comput. Math., v.83, pp. 223-234.

7

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL.13, NO.5, 2011

A Study of Functional Properties of Weighted Lipschitz-Lorentz Spaces with Their Compact Embeddings, Ilker Eryilmaz,……………………………………….....………………………..809 Nikol'skii-type inequality with doubling weights, Feilong Cao and Shaobo Lin,……………..822 Approximation by Boolean Sums of Jackson Operators on the Sphere, Yuguang Wang and Feilong Cao,…………………………………………………………………………………….830 On The Absolute Riesz Summability, W. T. Sulaiman,………………………………………..843 On Necessary and Sufficient Conditions for Inclusion Relations for Absolute Summability, W. T. Sulaiman,…………………………………………………………………………………….850 Two generalized-Euler-constant functions estimated accurately, Vito Lampret,……………...857 Joint limiting distributions of maxima and minima for complete and incomplete samples from weakly dependent stationary sequences, Peng Zuoxiang, Weng Zhichao and Saralees Nadarajah,………………………………………………………………………………………875 Topological Spaces and Soft Sets, Xun Ge, Zhaowen Li and Ying Ge,…………………….....881 A new application of quasi-monotone and almost increasing sequences, Huseyin Bor and H.S. Ozarslan,………………………………………………………………………………………..886 A note on a Tauberian theorem for (A, i) limitable method II, Ibrahim Canak,…………….....892 Stability of Linear Non-Autonomous Systems, Yaowaluck Khongtham,…………………......899 Moment Functional and Orthogonal Trigonometric Polynomials of Semi-Integer Degree, G.V. Milovanovic, A.S. Cvetkovic and M.P. Stanic,………………………………………………..907 Generalized Δm-Statistical Convergence in Probabilistic Normed Space, Ayhan Esi and M. Kemal Ozdemir,………………………………………………………………………………..923 Solving Several Complex Variables Equations Numerically, Louis Paquette,………………..933 Approximation by polynomials and rational functions in Orlicz spaces, Sadulla Z. Jafarov,........................................................................................................................................953 Fixed Points of Mappings Satisfying a New Condition in Cone Metric Spaces, Duran Turkoglu and Muhib Abuloha,…………………………………………………………………………....963

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL.13, NO.5, 2011 (continues)

Distribution of the roots of the second kind Bernoulli polynomials, C.S. Ryoo,…………971 A numerical investigation on the structure of the roots of q-Genocchi polynomials, C.S. Ryoo,………………………………………………………………………………………977 A note on the second kind Genocchi polynomials, C.S. Ryoo,……………………………986  

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Volume 13, Number 6 ISSN:1521-1398 PRINT,1572-9206 ONLINE

October 2011

Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC

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Editorial Board Associate Editors 20) Hrushikesh N.Mhaskar 1) George A. Anastassiou Department of Mathematical Sciences Department Of Mathematics California State University The University of Memphis Los Angeles,CA 90032 Memphis,TN 38152,U.S.A 626-914-7002 Tel.901-678-3144 e-mail: [email protected] e-mail: [email protected] Approximation Theory,Real Analysis, Orthogonal Polynomials, Approximation Theory,Splines, Wavelets, Neural Wavelets, Neural Networks Networks,Probability, Inequalities. 21) M.Zuhair Nashed Department Of Mathematics 2) J. Marshall Ash University of Central Florida Department of Mathematics PO Box 161364 De Paul University Orlando, FL 32816-1364 2219 North Kenmore Ave. e-mail: [email protected] Chicago,IL 60614-3504 Inverse and Ill-Posed problems, 773-325-4216 Numerical Functional Analysis, e-mail: [email protected] Integral Equations,Optimization, Real and Harmonic Analysis Signal Analysis 3) Mark J.Balas 22) Mubenga N.Nkashama Department Head and Professor Department OF Mathematics Electrical and Computer Engineering University of Alabama at Dept. Birmingham College of Engineering Birmingham,AL 35294-1170 University of Wyoming 205-934-2154 1000 E. University Ave. e-mail: [email protected] Laramie, WY 82071 Ordinary Differential Equations, 307-766-5599 Partial Differential Equations e-mail: [email protected] Control Theory,Nonlinear Systems, 23) Charles E.M.Pearce Neural Networks,Ordinary and Applied Mathematics Department Partial Differential Equations, University of Adelaide Functional Analysis and Operator Adelaide 5005, Australia Theory e-mail: [email protected] 4) Drumi D.Bainov Stochastic Processes, Probability Department of Mathematics Theory, Harmonic Analysis, Medical University of Sofia Measure Theory, P.O.Box 45,1504 Sofia,Bulgaria Special Functions, Inequalities e-mail: [email protected] 24) Josip E. Pecaric e-mail:[email protected] Differential Equations/Inequalities Faculty of Textile Technology University of Zagreb Pierottijeva 6,10000 5) Carlo Bardaro Zagreb,Croatia Dipartimento di Matematica e e-mail: [email protected] Informatica Inequalities,Convexity Universita di Perugia Via Vanvitelli 1

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Germany 011-49-203-379-3542 e-mail: [email protected] Approximation Theory, Computer Aided Geometric Design 16) 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 17) Christian Houdre School of Mathematics Georgia Institute of Technology Atlanta,Georgia 30332 404-894-4398 e-mail: [email protected] Probability,Mathematical Statistics,Wavelets 18) V. Lakshmikantham Department of Mathematical Sciences Florida Institute of Technology Melbourne, FL 32901 e-mail: [email protected] Ordinary and Partial Differential Equations, Hybrid Systems, Nonlinear Analysis 19) 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 36) Ahmed I. Zayed Department Of Mathematical Sciences DePaul University 2320 N. Kenmore Ave. Chicago, IL 60614-3250 773-325-7808 e-mail: [email protected]

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1001

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JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.6, 1005-1011, 2011, COPYRIGHT 2011 EUDOXUS PRESS, 1005 LLC

A numerical computation of the roots of q-Bernoulli polynomials C. S. Ryoo Department of Mathematics, Hannam University, Daejeon 306-791, Korea

Abstract Recently several authors studied the q-extension of Bernoulli numbers and polynomials(see [1-6]). In this paper we construct the q-Bernoulli numbers Bn,q and polynomials Bn,q (x). We also observe the behavior of complex roots of q-Bernoulli polynomials, using numerical investigation. By means of numerical experiments, we demonstrate a remarkably regular structure of the complex roots of q-Bernoulli polynomials. Finally, we give a table for the solutions of the q-Bernoulli polynomials. Key words - Bernoulli numbers, Bernoulli polynomials, q-Bernoulli numbers, q-Bernoulli polynomials, p-adic q-integrals 1. Introduction Many mathematicians have studied Bernoulli numbers and Bernoulli polynomials. Bernoulli polynomials posses many interesting properties and arising in many areas of mathematics and physics. We introduce the q-Bernoulli numbers and polynomials. In the 21st century, the computing environment would make more and more rapid progress. Using computer, a realistic study for new analogs of Bernoulli numbers and polynomials is very interesting. It is the aim of this paper to observe an interesting phenomenon of ‘scattering’ of the zeros of the q-Bernoulli polynomials Bn,q (x). The outline of this paper is as follows. In Section 2, we study the the q-Bernoulli polynomials Bn,q (x). In Section 3, we describe the beautiful zeros of the q-Bernoulli polynomials Bn,q (x) using a numerical investigation. Also we display distribution and structure of the zeros of the the q-Bernoulli polynomials Bn,q (x) by using computer. By using the results of our paper the readers can observe the regular behaviour of the roots of the q-Bernoulli polynomials Bn,q (x). Finally, we carried out computer experiments for doing demonstrate a remarkably regular structure of the complex roots of q-Bernoulli polynomials Bn,q (x). First, we introduce the Bernoulli numbers and Bernoulli polynomials. The Bernoulli numbers Bn are defined by the generating function: F (t) =

∞ X t tn = Bn , (|t| < π), cf. [1,2,3] t e − 1 n=0 n!

(1.1)

where we use the technique method notation by replacing B n by Bn (n ≥ 0) symbolically. We consider the Bernoulli polynomials Bn (x) as follows: F (x, t) = Note that Bn (x) =

Pn k=0

∞ X t tn xt e = Bn (x) . t e −1 n! n=0

¡n¢ n−k . In the special case x = 0, we define Bn (0) = Bn . k Bk x

(1.2)

1006

RYOO: q-BERNOULLI POLYNOMIALS

2. The q-Bernoulli numbers and polynomials In this section, we define the q-Bernoulli numbers Bn,q and polynomials Bn,q (x) and investigate their properties. Let q be a complex number with |q| < 1. By the meaning of (1.1) and (1.2), let us define the q-Bernoulli numbers Bn,q and polynomials Bn,q (x) as follows: Fq (t) =

Fq (x, t) =

∞ X t tn = B , n,q qet − 1 n=0 n!

∞ X t tn xt e = Bn,q (x) . t qe − 1 n! n=0

(2.1)

(2.2)

We have the following remark. Remark. Note that (1) Bn,q (0) = Bn,q , (2) If q → 1, then Bn,q (x) = Bn (x), Bn,q = Bn , (3) If q → 1, then Fq (x, t) = F (x, t), Fq (t) = F (t). Here is the list of the first q-Bernoulli numbers Bn,q . 1 2q , B2,q = − , −1 + q (−1 + q)2 3q 6q 2 4q 24q 2 24q 3 = + , B = − + − ,··· , 4,q (−1 + q)2 (−1 + q)3 (−1 + q)2 (−1 + q)3 (−1 + q)4

B1,q = B3,q

By the above definition, we obtain ∞ X

Bl,q (x)

l=0

∞ ∞ X t tn X m tm tl = t ext = Bn,q x l! qe − 1 n! m=0 m! n=0 Ã ! ∞ l X X tn tl−n = Bn,q xl−n n! (l − n)! l=0 n=0 Ã ! µ ¶ ∞ l X X l tl = . Bn,q xl−n l! n n=0 l=0

By using comparing coefficients

tl , we have the following theorem. l!

Theorem 1. For any positive integer n, we have n µ ¶ X n Bn,q (x) = Bk,q xn−k . k k=0

Because

∞ X d tn ∂ Fq (x, t) = tFq (x, t) = Bn,q (x) , ∂x dx n! n=0

it follows the important relation

d Bn,q (x) = nBn−1,q (x). dx We have the integral formula as follows: Z b 1 Bn−1,q (x)dx = (Bn,q (b) − Bn,q (a)). n a

2

RYOO: q-BERNOULLI POLYNOMIALS

1007

Here is the list of the first q-Bernoulli Polynomials Gn,q (x). 1 2q 2x , B2,q = − + , 2 −1 + q (−1 + q) −1 + q 3q 6q 2 6qx 3x2 = + − + , 2 3 2 (−1 + q) (−1 + q) (−1 + q) −1 + q 4q 24q 2 24q 3 12qx 24q 2 x 12qx2 4x3 =− + − − + − + , 2 3 4 2 3 2 (−1 + q) (−1 + q) (−1 + q) (−1 + q) (−1 + q) (−1 + q) −1 + q

B1,q = B3,q B4,q

··· . From (2.1), we obtain ∞ ∞ n X X t tn n t = B = (Bq ) = eBq t n,q t qe − 1 n=0 n! n=0 n!

which yields t = (qet − 1)eBq t = qe(Bq +1)t − eBq t . Using Taylor expansion of exponential function, we have t=

∞ X

n

n

{q (Bq + 1) − (Bq ) }

n=0

tn . n!

By comparing the coefficients, we have the following theorem. Theorem 2. The q-Bernoulli numbers Bn,q are defined respectively by ( 1, if n = 1, n q (Bq + 1) − Bn,q = 0, if n > 1, n

with the usual convention about replacing (Bq ) by Bn,q in the binomial expansion. Since

∞ X

Bl,q (x + y)

l=0

∞ ∞ X tl t tn X m tm = t e(x+y)t = Bn,q (x) y l! qe − 1 n! m=0 m! n=0 ! Ã ∞ l X X tl−n tn = Bn,q (x) y l−n n! (l − n)! l=0 n=0 ! Ã ∞ l µ ¶ X X l tl l−n = Bn,q (x)y , n l! n=0 l=0

we have the following theorem. Theorem 3. q-Bernoulli polynomials Bn,q (x) satisfies the following relation: l µ ¶ X l Bl,q (x + y) = Bn,q (x)y l−n . n n=0 It is easy to see that ∞ X n=0

n

Bn,q (x)

t 1 t = t ext = n! qe − 1 m

m−1 X a=0

qa

µ

mt q m emt

+1

a+x e m

! (mt)

¶

m−1 ∞ 1 X aX a + x (mt)n q Bn,qm m a=0 n=0 m n! Ã ¶! n µ ∞ m−1 X X t a + x . = mn−1 q a Bn,qm m n! n=0 a=0

=

3

1008

RYOO: q-BERNOULLI POLYNOMIALS

Hence we have the below theorem. Theorem 4. For any positive integer m, we obtain Bn,q (x) = mn−1

m−1 X

µ q i Bn,qm

i=0

i+x m

¶ , for n ≥ 0.

From (2.2), we can derive the following equality: ∞ X

∞ X tn tn = (qBn,q (x + 1) − Bn,q (x)) nxn−1 . n! n=0 n! n=0

Hence, we obtain the following difference equation. Theorem 5. For any positive integer n, we obtain qBn,q (x + 1) − Bn,q (x) = nxn−1 .

3. Distribution and Structure of the zeros

In this section, we investigate the zeros of the q-Bernoulli polynomials Bn,q (x) by using computer. Our numerical results for numbers of real and complex zeros of Gn,q (x) are displayed in Table1. Table 1. Numbers of real and complex zeros of Bn,q (x) n

q = 1/3 real zeros complex zeros

q = 1/5 real zeros complex zeros

2

1

0

1

0

3

0

2

0

2

4

1

2

1

2

5

0

4

0

4

6

1

4

1

4

7

0

6

0

6

8

1

6

1

6

9

0

8

0

8

10

1

9

1

9

11

0

10

0

10

We plot the zeros of Bn,q (x), q = 1/3, 1/5, 1/7, 1/9x ∈ C. (Figures 1, 2, 3, and 4). In Figures 14, Bn,q (x), x ∈ C, has Re(x) = 0 reflection symmetry. This translates to the following open problem: Prove or disprove: Bn,q (x), x ∈ C, has Re(x) = 0 reflection symmetry. We shall consider the more general open problem. In general,how many roots does Bn,q (x) have ? Prove or disprove: Bn,q (x) has n − 1 distinct solutions. Find the numbers of complex zeros CBn,q (x) of Bn,q (x), Im(x) 6= 0. Prove or give a counterexample: Conjecture: Since n is the degree of the polynomial Bn,q (x), the number of real zeros RBn,q (x) lying on the real plane Im(x) = 0 is then RBn,q (x) = n − 1 − CBn,q (x) , where CBn,q (x) denotes complex zeros. See Table 1 for tabulated values of RBn,q (x) and CBn,q (x) .

4

RYOO: q-BERNOULLI POLYNOMIALS

ImHxL

10

10

7.5

7.5

5

5

2.5

2.5

ImHxL

0

0

-2.5

-2.5

-5

-5

-7.5

-7.5

-5

0

5

-5

10

ReHxL

10

7.5

7.5

5

5

2.5

2.5

ImHxL

0

10

0

-2.5

-2.5

-5

-5

-7.5

-7.5

0

5

Figure 2: Zeros of B20,1/5 (x)

10

-5

0

ReHxL

Figure 1: Zeros of B20,1/3 (x)

ImHxL

1009

5

ReHxL

Figure 3: Zeros of B20,1/7 (x)

10

-5

0

5

10

ReHxL

Figure 4: Zeros of B20,1/9 (x)

Find the equation of envelope curves bounding the real zeros lying on the plane, and the equation of a trajectory curve running through the complex zeros on any one of the arcs. Stacks of zeros of Bn,q (x) for q = 1/3, 1 ≤ n ≤ 30 from a 3-D structure are presented (Figure 5). We calculated an approximate solution satisfying Bn,q (x), x ∈ C. The results are given in Table 2.

5

1010

RYOO: q-BERNOULLI POLYNOMIALS

10

ImHxL 0 -10 30

20

n 10

0 0 10

ReHxL

Figure 5: Stacks of zeros of Bn,q (x), 1 ≤ n ≤ 30

Table 2. Approximate solutions of Bn,q (x) = 0, x ∈ R n

q = 1/3

q = 1/5

2

−0.5000

−0.25000

4

−1.0800

−0.6706

6

−1.62683145

−1.049708007

8

−2.16108834

−1.41664879

10

−2.6889094

−1.77828154

12

−3.2128087

−2.1368066

14

−3.734055

−2.4932728

16

−4.253382

−2.848272

18

−4.77125

−3.202173

20

−5.28798

−3.55522

This shows the distribution of real zeros of Bn,q (x) for 1 ≤ n ≤ 30(Figure 6). We want to

ReHxL -8

-7

-6

-5

-4

-3

-2

-1

0

30

20

n 10

0

Figure 6: Plot of real zeros of Bn,q (x), q = 1/3, 1 ≤ n ≤ 30 6

find a formula that best fits a given set of data points. The least squares method is used to fit a

RYOO: q-BERNOULLI POLYNOMIALS

1011

polynomials or a set of functions to a given set of data points. Using the least squares method, we can find a and b such that x = a + bn is the least squares fit to the data given in Table 2. The graph of the data points is shown in Figure 6. We obtain x = 0.215291 − 0.273788n. That is, the result is the best linear combination of the function 1 and n. The real zero x ∼ −∞ asymptotically as n → ∞. These figures give mathematicians an unbounded capacity to create visual mathematical investigations of the behavior of the roots of the Bn,q (x). Moreover, it is possible to create a new mathematical ideas and analyze them in ways that generally are not possible by hand. The author has no doubt that investigation along this line will lead to a new approach employing numerical method in the field of research of the q-Bernoulli polynomials Bn,q (x) to appear in mathematics and physics. For related topics the interested reader is referred to [2], [3], [4], [5], [7].

REFERENCES 1. Carlitz, L. (1948) q-Bernoulli numbers and polynomials, Duke Math. J., v. 15, pp. 987-1000. 2. Kim, T., Ryoo, C.S., Jang, L.C., Rim, S.H.,(2005). Exploring the q-Riemann Zeta function and q-Bernoulli polynomials , Discrete Dynamics in Nature and Society, v.2005(2), pp.171-181. 3. Ryoo, C.S., Kim, T., Agarwal, R.P.(2005). Exploring the multiple Changhee q-Bernoulli polynomials, Inter. J. Comput. Math., v.82(4), pp. 483-493. 4. Ryoo,C.S., Kim, T., Agarwal, R.P. (2005). The structure of the zeros of the generalized Bernoulli polynomials, Neural Parallel Sci. Comput., v.13, pp. 371-379. 5. Ryoo, C.S., Rim, S.H. (2008). On the analogue of Bernoulli polynomials, Journal of Computational Analysis and Applications, v.10, pp. 163-172. 6. Ryoo, C.S. (2008). A numerical computation on the structure of the roots of q-extension of Genocchi polynomials, Applied Mathematics Letters, v.21, pp. 348-354. 7. Ryoo, C.S. (2008). Calculating zeros of the twisted Genocchi polynomials, Advan. Stud. Contemp. Math., v.17, pp. 147-159.

7

JOURNAL 1012 OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.6, 1012-1018, 2011, COPYRIGHT 2011 EUDOXUS PRESS, LLC

A note on the q-Hurwitz Euler zeta functions C. S. Ryoo Department of Mathematics, Hannam University, Daejeon 306-791, Korea

Abstract In [3], we introduced the second kind q-Euler numbers En and polynomials En (x). By using En and En (x), the second kind q-Euler zeta function ζq (s) and q- Hurwitz Euler zeta functions ζq (s, x) are defined. It is the aim of this paper to observe an interesting phenomenon of ’scattering’ of the zeros of ζq (s, x)) in complex plane. Finally, we investigate the roots of q- Hurwitz Euler zeta functions ζq (s, x). Key words - q-Euler polynomials, q-Euler zeta function, q-Hurwitz Euler zeta, the second kind q-Euler numbers, the second kind q-Euler polynomials 1. Introduction Several mathematicians have studied q-Euler numbers, q-Euler polynomials, the second kind qEuler numbers, and the second kind q-Euler polynomials (see [1,2,3,4]). q-Euler numbers, q-Euler polynomials, the second kind q-Euler numbers, and the second kind q-Euler polynomials numbers posses many interesting properties and arising in many areas of mathematics and physics. In [3], we observed the behavior of complex roots of the second kind q-Euler polynomials En,q (x), using numerical investigation. By means of numerical experiments, we demonstrated a remarkably regular structure of the complex roots of the second kind q-Euler polynomials En,q (x). In this paper, we introduce the second kind q-Euler zeta function ζq (s) and q-Hurwitz Euler zeta functions ζq (s, x). In order to study the second kind q-Euler zeta function ζq (s) and q-Hurwitz Euler zeta functions ζq (s, x), we must understand the structure of the second kind q-Euler zeta function ζq (s) and qHurwitz Euler zeta functions ζq (s, x). Therefore, using computer, a realistic study for the second kind q-Euler zeta function ζq (s) and q-Hurwitz Euler zeta functions ζq (s, x) is very interesting. It is the aim of this paper to observe an interesting phenomenon of ‘scattering’ of the zeros of the q-Hurwitz Euler zeta functions ζq (s, x) in complex plane. Throughout this paper, we always make use of the following notations: N = {1, 2, 3, · · · } denotes the set of natural numbers, R denotes the set of real numbers, and C denotes the set of complex numbers First, we introduce the second kind q-Euler numbers En,q and polynomials En,q (x). The second kind q-Euler numbers En,q are defined by the generating function: Fq (t) =

∞ X 2et tn = E . n,q qe2t + 1 n=0 n!

(1)

We consider the second kind q-Euler polynomials En,q (x) as follows: Fq (x, t) =

∞ X 2et tn xt e = En,q (x) . 2t qe + 1 n! n=0

(2)

The following elementary properties of the second kind q-Euler numbers En,q and polynomials En,q (x) are readily derived form (1) and (2)( see, for details, [3]). We, therefore, choose to omit details involved.

RYOO: q-HURWITZ EULER ZETA FUNCTIONS

1013

Proposition 1.1. For any positive integer n, we have En,q (x) =

n µ ¶ X n k=0

k

Ek,q xn−k .

Proposition 1.2 (Integral formula). Z

b

En−1,q (x)dx = a

1 (En,q (b) − En,q (a)). n

Proposition 1.3 (Addition theorem). En,q (x + y) =

n µ ¶ X n Ek,q (x)y n−k . k

k=0

Proposition 1.4 (Difference equation). qEn,q (x + 2) + En,q (x) = 2(x + 1)n . Proposition 1.5. For n ∈ N, we have En,q (x) = (−1)n q −1 En,q−1 (−x). The outline of this paper is as follows. In Section 2, we introduce the second kind q-Euler zeta function ζq (s) and q-Hurwitz Euler zeta functions ζq (s, x). We derive the existence of a specific interpolation function which interpolate the second kind q-Euler numbers En,q and the second kind q-Euler polynomials En,q (x) at negative integer. In section 3, we describe the beautiful zeros of qHurwitz Euler zeta functions ζq (s, x) using a numerical investigation. Finally, we investigate the roots of the q-Hurwitz Euler zeta functions ζq (s, x).

2. The second kind q-Euler zeta function By using the second kind q-Euler numbers and polynomials, the second kind q-Euler zeta function and q-Hurwitz Euler zeta functions are defined. These functions interpolate the second kind q-Euler numbers and q-Euler polynomials, respectively. From (1), we note that ¯ ∞ X ¯ dk ¯ F (t) = 2 (−1)n q n (2n + 1)k , (k ∈ N). q ¯ dtk t=0 n=0 By using the above equation, we are now ready to define the second kind q-Euler zeta functions. Definition 2.1. Let s ∈ C. ζq (s) = 2

∞ X (−1)n q n . (2n + 1)s n=1

(3)

Note that ζq (s) is a meromorphic function on C. Relation between ζq (s) and Ek,q is given by the following theorem. Theorem 2.2. For k ∈ N, we have ζq (−k) = Ek,q .

2

1014

RYOO: q-HURWITZ EULER ZETA FUNCTIONS

Observe that ζq (s) function interpolates Ek,q numbers at non-negative integers. By using (2), we note that ¯ ∞ X ¯ dk ¯ F (t, x) = 2 (−1)n q n (2n + x + 1)k , (k ∈ N), (4) q ¯ dtk t=0 n=0 and

µ

d dt

¶k à X ∞

tn En,q (x) n! n=0

!¯ ¯ ¯ ¯ ¯

= Ek,q (x), for k ∈ N.

(5)

t=0

By (4) and (5), we are now ready to define the q-Hurwitz Euler zeta functions. Definition 2.3. Let s ∈ C. ζq (s, x) = 2

∞ X

(−1)n q n . (2n + x + 1)s n=0

Note that ζq (s, x) is a meromorphic function on C. Relation between ζq (s, x) and Ek,q (x) is given by the following theorem. Theorem 2.4. For k ∈ N, we have ζq (−k, x) = Ek,q (x). Observe that ζq (−k, x) function interpolates Ek,q (x) numbers at non-negative integers.

3. Zeros of the q-Hurwitz Euler zeta functions

In this section, we show a plot of ζ1/2 (s, x), −2 ≤ s ≤ 1, −1 ≤ x ≤ 1 (Fig. 1). For k = 1, · · · , 10, we can draw a plot of the ζq (k, x), respectively. This shows the ten plots combined into one. We display the shape of ζ1/2 (−k, x), −1/2 ≤ x ≤ 1/2 for any positive integer k (Fig. 2).

Ζ q Hs,xL

4 2 1.5

2 0 1

-2

0.5

-1 0 s

x

0 1 2 -0.5

Figure 1: Plot of ζq (s, x) The plot above shows ζ1/2 (s, x) for real s and x, with the zero contour indicated in black (Fig. 3). Next, we investigate the zeros of ζ1/9 (−k, x), k = 15, 20, 25, 30, x ∈ C (Fig. 4). Stacks of zeros of ζ1/9 (−k, x) for 1 ≤ k ≤ 30 from a 3-D structure are presented. (Fig. 5). Our numerical results for numbers of real and complex zeros of ζ1/9 (−k, x), x ∈ C are displayed (Table 1).

3

RYOO: q-HURWITZ EULER ZETA FUNCTIONS

1015

100

80

Ζ

q Hk,xL

60

40

20

0 -0.4

0 x

-0.2

0.2

0.4

Figure 2: Curve of ζq (k, x) Ζ q Hs,xL 2

1.5

x

1

0.5

0

-0.5 -2

-1

1

0 s

2

Figure 3: Zero contour ζq (s, x)

Table 1. Numbers of real and complex zeros of ζq (−k, x) k

real zeros

complex zeros

1

1

0

2

2

0

3

1

2

4

2

2

5

3

2

6

4

2

7

3

4

8

4

4

9

5

4

10

6

4

11

5

6

12

6

6

13

3

10

14

4

10

4

1016

RYOO: q-HURWITZ EULER ZETA FUNCTIONS

ImHxL

15

15

10

10

5

5

ImHxL

0

0

-5

-5

-10

-10

-10

-5

0

5

10

15

-10

-5

ReHxL

ImHxL

15

15

10

10

5

5

ImHxL

0

-5

-10

-10

-5

0

5

10

15

5

10

15

0

-5

-10

0

ReHxL

5

10

15

-10

-5

ReHxL

0

ReHxL

Figure 4: Zeros of ζq (−k, x), k = 15, 20, 25, 30

Table 2. Approximate solutions of ζq (−k, x) = 0, x ∈ R k

x

1

−0.800000000

2

−1.400000000,

3 4 5 6

0.4424213586 −0.93592197, −1.89533053,

−2.0189980,

8

−1.10844760,

1.064067194

−0.330925657,

0.2792761594, 7

−0.2000000000

−1.7570598 2.209536560

0.891552395,

−2.3025522, 1.503274016,

1.653905928

2.729984490

−0.496739048 3.213706760

5

RYOO: q-HURWITZ EULER ZETA FUNCTIONS

ImHxL

1017

20 10

0 -10 -20

40

n 20

0 -10 0

ReHxL

10

Figure 5: Stacks of zeros of ζq (−k, x)

Figure 6 show the distribution of real zeros of ζ1/9 (−k, x) for 1 ≤ k ≤ 30. 30

20

n 10

0 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9

ReHxL Figure 6: Plot of real zeros of ζq (−k, x), 1 ≤ k ≤ 30 We observe a remarkably regular structure of the complex roots of the ζq (−k, x). We hope to verify a remarkably regular structure of the complex roots of the ζq (−k, x) (Table 1). Next, we calculated an approximate solution satisfying ζq (−k, x) = 0, x ∈ R. The results are given in Table 2. Finally, we shall consider the more general problems. Prove that ζq (−k, x)) = 0 has n distinct solutions. Find the numbers of complex zeros Cζq (−k,x) of ζq (−k, x), Im(x) 6= 0. The number of real zeros Rζq (−k,x) lying on the real plane Im(x) = 0 is then Rζq (−k,x) = n − Cζq (−k,x) , where Cζq (−k,x) denotes complex zeros. See Table 1 for tabulated values of Rζq (−k,x) and Cζq (−k,x) . We prove that ζq (−k, x), x ∈ C, has Re(x) = 0 reflection symmetry in addition to the usual Im(x) = 0 reflection symmetry analytic complex functions(Figs. 4-5). For related topics the interested reader is referred to [2, 3, 4, 5, 6, 7, 8, 9]. Acknowledgment This paper has been supported by the 2011 Hannam University Research Fund.

6

1018

RYOO: q-HURWITZ EULER ZETA FUNCTIONS

REFERENCES 1. Kim, T., Jang, L. C., Park, H. K.(2001). A note on q-Euler and Genocchi numbers , Proc. Japan Acad., v.77 A , pp. 139-141. 2. Ryoo, C.S. (2010)Calculating zeros of the second kind Euler polynomials, Journal of Computational Analysis and Applications, v.12 , pp. 828-833. 3. Ryoo, C.S. A note on the second kind q-Euler polynomials, Journal of Computational Analysis and Applications, submitted. 4. Ryoo, C.S. (2008). A numerical computation on the structure of the roots of q-extension of Genocchi polynomials, Applied Mathematics Letters, v.21, pp. 348-354. 5. Ryoo, C.S. (2008). Calculating zeros of the twisted Genocchi polynomials, Advan. Stud. Contemp. Math., v.17, pp. 147-159. 6. Ryoo, C.S., Kim, T. (2004). Beautiful zeros of q-Euler polynomials of order k, Proceeding of the Jangjeon Mathematical Society, v.7, pp. 63-79. 7. Ryoo, C.S., Kim, T. (2008). On the real roots of the q-Hurwitz zeta-function, Proceeding of the Jangjeon Mathematical Society, v.11, pp. 205-214. 8. Ryoo, C.S., Kim, T., Agarwal, R.P. (2006). A numerical investigation of the roots of qpolynomials, Inter. J. Comput. Math., v.83, pp. 223-234. 9. Ryoo, C.S., Song, H., Argawal, R.P. (2004). On the real roots of the Changhee-Barnes’ qBernoulli polynomials, Advan. Stud. Contemp. Math., v.9, pp. 153-163. 10. Ryoo, C.S., Kim, T., Agarwal, R.P. (2005). Distribution of the roots of the Euler-Barnes’ type q-Euler polynomials , Neural, Parallel & Scientific Computations, v.13, pp. 381-392.

7

JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.6, 1019-1024, 2011, COPYRIGHT 2011 EUDOXUS PRESS, 1019 LLC

On the alternating sums of powers of consecutive odd integers C. S. Ryoo Department of Mathematics, Hannam University, Daejeon 306-791, Korea Abstract : By applying the symmetry of the fermionic p-adic q-integral on Zp , which is defined Kim [2], we give recurrence identities the second kind Euler polynomials and the alternating sums of powers of consecutive odd integers. Key words : the second kind Euler numbers, the second kind Euler polynomials, alternating sums 1. Introduction Euler numbers and polynomials were studied by many authors (see for details [1], [2], [3], [4], [5], [6]). Euler numbers and polynomials posses many interesting properties and arising in many areas of mathematics and physics. In [6], we studied the second kind Euler numbers En and polynomials En (x). By using computer, we observed an interesting phenomenon of ‘scattering’ of the zeros of the second kind Euler polynomials En (x) in complex plane. We also carried out computer experiments for doing demonstrate a remarkably regular structure of the complex roots of the second kind Euler polynomials En (x). Throughout this paper, we always make use of the following notations: N = {1, 2, 3, · · · } denotes the set of natural numbers, R denotes the set of real numbers, C denotes the set of complex numbers, Zp denotes the ring of p-adic rational integers, Qp denotes the field of p-adic rational numbers, and Cp denotes the completion of algebraic closure of Qp . Let νp be the normalized exponential valuation of Cp with |p|p = p−νp (p) = p−1 . When one talks of q-extension, q is considered in many ways such as an indeterminate, a complex number q ∈ C, or p-adic number q ∈ Cp . If q ∈ C one normally assume that |q| < 1. If q ∈ Cp , we normally 1 assume that |q − 1|p < p− p−1 so that q x = exp(x log q) for |x|p ≤ 1. [x]q = [x : q] =

1 − qx , cf. [1, 2, 3, 5] . 1−q

Hence, limq→1 [x] = x for any x with |x|p ≤ 1 in the present p-adic case. Let d be a fixed integer and let p be a fixed prime number. For any positive integer N , we set [ (a + dpZp ), X = lim(Z/dpN Z), X ∗ = ←− N

0 lim inf pα (xβ − x), ∀y 6= x.

Any given map T : M −→ X is said to be demiclosed at 0 if for each net {xβ } in M converging weakly to x and {T xβ } converging strongly to 0, we have T x = 0. A mapping T : X −→ X is said to be demicompact if each bounded net {xβ } in X such that {(I − T )(xβ )} converges, has a convergent subnet.

Let C be a nonempty convex subset of X and q ∈ C. A self mapping T on

4

GOUDARZI, VAEZPOUR: ...LOCALLY CONVEX SPACES

C is said to be affine if

T (λx + (1 − λ)y) = λT (x) + (1 − λ)T (y),

for all x, y ∈ C and λ ∈ (0, 1). Also T is called to be affine with respect to q if

T (λx + (1 − λ)q) = λT (x) + (1 − λ)T (q),

for all x ∈ C and λ ∈ (0, 1). There is an example of an affine mapping with respect to a point which is not affine(see[17]). Throughout this paper we denote by F (T ), the set of fixed points of a map T : X −→ X. Also we need the following theorem due to Hadzic. Theorem 2.1 [3] Let X be a Banach space and S, T : X −→ X be continuous mappings and let η be a family of self mappings A : X −→ X such that i) A(X) ⊆ S(X) ∩ T (X), ∀A ∈ η. ii) A commutes with S and T , ∀A ∈ η. iii) k Ax − By k≤ q k Sx − T y k, for any x, y ∈ X and for all A, B ∈ η where 0 ≤ q < 1. Then for any A ∈ η, S, T and A have a unique common fixed point in X.

3

Main results

In this section at first we prove a lemma which will be needed in the proof of our main theorem. Lemma 3.1 : Let η be a family of self mappings on a τ -bounded subset M of a Hausdorff locally convex space (X, τ ).

5

1043

1044

GOUDARZI, VAEZPOUR: ...LOCALLY CONVEX SPACES

i) Suppose that for self mappings S, T : M −→ M

pα (Ax − By) ≤ qpα (Sx − T y), for all x, y ∈ M, pα ∈ A∗ (τ ), A, B ∈ η and 0 ≤ q < 1. Then

kAx − BykB ≤ qkSx − T ykB .

ii) If (X, τ ) is complete then (XB , k.kB ) is complete. iii) If S, T : X −→ X are continuous then S,T are continuous in (XB , k.kB ). Proof : i) For any x, y ∈ M , by hypothesis we have

pα (Ax − By) ≤ qpα (Sx − T y), for all pα ∈ A∗ (τ ), A, B ∈ η and 0 ≤ q < 1. Then we have

sup pα ( α

Ax − By Sx − T y ) ≤ q sup pα ( ), λα λα α

and so k Ax − By kB ≤ q k Sx − T y kB . The other parts can be similarly proved.  Theorem 3.2 : Let (X, τ ) be a Hausdorff locally convex space and C be a subset of X. Suppose that S, T : C −→ C be self mappings and η be a family of self mappings A : C −→ C such that A(C) ⊆ S(C) ∩ T (C), A commutes with S and T , C is p-starshaped with p ∈ F (T ) ∩ F (S), T and S are affine with respect to p and for each A, B ∈ η we have

pα (Ax − By) ≤ pα (Sx − T y), ∀pα ∈ A∗ (τ ), x 6= y ∈ C. 6

(1)

GOUDARZI, VAEZPOUR: ...LOCALLY CONVEX SPACES

1045

Then for each A ∈ η, F (T ) ∩ F (S) ∩ F (A) 6= ∅ provided one of the following conditions hold. i) C is weakly compact, T and S are continuous in the weak topology and T − A and S − A are demiclosed at 0. ii) T and S are compact, C is τ -bounded sequentially compact, S is non expansive and A and T are S-non expansive. iii) (X, τ ) is an Opial space, C is weakly compact, A is continuous in the weak topology and T and S are A-non expansive for each A ∈ η. Proof : Choose the sequence {kn } such that 0 < kn < 1 and kn −→ 1 as n −→ ∞. For n ≥ 1, for each A ∈ η and x ∈ C we define

An x = kn Ax + (1 − kn )p.

(2)

Since C is p-starshaped with p ∈ F (T ), T is affine with respect p and A(C) ⊆ T (C), we have An x = kn Ax + (1 − kn )p = kn Ax + (1 − kn )T p ∈ T (C). Hence An (C) ⊆ T (C), for n ≥ 1. Also A commutes with T and then An T x = kn AT x + (1 − kn )p = kn T Ax + (1 − kn )T p = T (kn Ax + (1 − kn )p) = T An x, for each x ∈ C. Thus An and T are commutative for each n ≥ 1. Similarly we can prove that An and S are commutative. Also we have An (C) ⊆ S(C). Therefore An (C) ⊆ T (C)∩S(C) and each An commutes with S and T . Moreover

7

1046

GOUDARZI, VAEZPOUR: ...LOCALLY CONVEX SPACES

for each An and Bn constructed with A and B in η we have pα (An x − Bn y) = kn pα (Ax − By) ≤ kn pα (Sx − T y), for each x, y ∈ C. Hence by Lemma 3.1 we obtain

kAn x − Bn ykB ≤ kn kSx − T ykB ,

for each x, y ∈ C. Now, i) If C is τ -weakly compact, since X is Hausdorff space then C is complete in the weak topology. On the other hand if T and S are continuous in the weak topology then they are continuous in (XB , k.kB ) and C is complete in (XB , k.kB ) by Lemma 3.1. Hence the conditions of theorem 2.1 hold for η = {An }∞ n=1 , S and T . So we have F (T ) ∩ F (S) ∩ F (An ) = {xn }, for each n ≥ 1. C is τ -weakly compact. Then there exists a subsequence {xm } of {xn } such that xm −→ x in weak topology. By Lemma 3.1 xm −→ x in (XB , k.kB ). By continuity of T we have x ∈ F (T ). Also from (2) we have

(T − A)xm = (1 −

1 )(xm − p). km

The sequence {xm } is τ -bounded and also km −→ 1 as m −→ ∞. So (T − A)xm −→ 0 strongly as m −→ ∞. The demicloseness of T − A implies that 0 = (T − A)x and hence T x = Ax. The similar argument shows that Ax = Sx, where S − A is demiclosed and hence F (T ) ∩ F (S) ∩ F (A) 6= ∅. Since A ∈ η is arbitrary the proof of i) is complete. ii) Since C is τ -bounded then {xn } is τ -bounded. If T is compact then {T xn } has a subsequent {T xnj } such that T xnj −→ x∗ ∈ C. But xni = Tni xni = kni T xni +(1−kni )p. As i −→ ∞ and using the continuity of T we get x∗ = T x∗ .

8

GOUDARZI, VAEZPOUR: ...LOCALLY CONVEX SPACES

By the similar way we can prove that x∗ = Sx∗ . On the other hand by (1) we have pα (Ax∗ − Axni ) ≤ pα (Sx∗ − T xni ). If we get i −→ ∞, we have Ax∗ = x∗ . So F (T ) ∩ F (S) ∩ F (A) 6= ∅ and the proof is complete. iii) If we can show that (T − A) and (S − A) are demiclosed at 0 for each A ∈ η, the result will follow from i). Let {xα } be a net in C, {xβ } be its subsequence such that xβ −→ x in the weak topology and (T − A)xβ −→ 0. By weakly compactness of C, x ∈ C and weakly continuity of A implies that Axβ −→ Ax in weak topology. But T is A-non expansive. Then for each pα ∈ A∗ (τ ),

pα (T xβ − T x) ≤ pα (Axβ − Ax).

As (Axβ − T xβ ) −→ 0, for each pα ,

lim inf pα (Axβ − Ax) ≥ lim inf pα (T xβ − T x) = lim inf pα (Axβ − T x). β

β

β

Since X satisfies the Opial’s condition and Axβ −→ Ax in weak topology, so Ax = T x and (T − A)x = 0. By the same way (S − A)x = 0.  Remark 3.2 : In Theorem 3.2(i) if we replace the condition weakly compact by sequentially weakly complete for C, the result still holds(see the details of Proof).

The following corollary and its results which has been proved in [1](Theorem 2.2), are the special cases of the above Theorem. Corollary 3.3 Let D be a closed subset of a normed linear space X. I and T self maps of D with T (D) ⊆ I(D) and q ∈ F (I). If D is q-starshaped , closure

9

1047

1048

GOUDARZI, VAEZPOUR: ...LOCALLY CONVEX SPACES

of T (D) is compact, I is continuous and linear, I and T are commuting and T is I-non expansive. Then I and T have a common fixed point. As applications of the previous theorem we prove some results in best approximation theory. Theorem 3.4 Suppose that T, S be self mappings on the Hausdorff locally convex space (X, τ ), C ⊆ X and η be a family of mappings A : X −→ X, invariant on C such that A(∂C) ⊆ C. Let x0 ∈ F (T ) ∩ F (S) ∩ F (A). D = Pc (x0 ) is non empty and p-starshaped with p ∈ F (S) ∩ F (T ) , T and S are affine with respect to p, T (D) = S(D) = D, each A ∈ η commutes with T and S and satisfies the following condition

pα (Ax − By) ≤ pα (Sx − T y), ∀pα ∈ A∗ (τ ), A, B ∈ η,

for all x, y ∈ D ∪ {x0 }. Then D ∩ F (T ) ∩ F (S) ∩ F (A) 6= ∅, for all A ∈ η provided one of the following conditions hold. i) D is weakly compact, S and T are continuous in the weak topology on D, (T − A) and (S − A) are demiclosed at 0. ii) D is τ -bounded and sequentially compact, T and S are compact, S is non expansive and A and T are S-non expansive. iii) (X, τ ) is an Opial space, D is weakly compact, for each A ∈ η, A is weakly continuous on D and T and S are A-non expansive. Proof : Suppose that y ∈ D. Then y ∈ ∂(C). Since A(∂C) ⊆ C we imply that Ay ∈ C. On the other hand T x0 = Sx0 = Ax0 . Then for each pα ∈ A∗ (τ ) and

10

GOUDARZI, VAEZPOUR: ...LOCALLY CONVEX SPACES

A, B ∈ η, pα (Ay − x0 ) = pα (Ay − Bx0 ) ≤ pα (Sy − T x0 ) = pα (Sy − x0 ) = dpα (x0 , C). Then Ay ∈ D. Consequently for each A ∈ η, A(D) ⊆ T (D) = D = S(D). Now by Theorem 3.2 the proof is complete.  Corollary 3.5 ([10], Theorem 1.1) : Let X be a normed space and T : X −→ X be a linear and non expansive operator. Let M be a T-invariant subset of X and x0 ∈ F (T ). If D, the set of best approximations of x0 in M , is nonempty compact and convex, then there exists a y in D which is also a fixed point of T. Corollary 3.6 ([10], Theorem 1.3) : Let X be normed space, T : X −→ X be a non expansive mapping, M be a T-invariant subset of X and x0 ∈ F (T ). If D, the set of best approximations of x0 in M is nonempty compact and starshaped, then there exists a best approximation of x0 in M which is also a fixed point of T. Corollary 3.7 ([10], Theorem 1.4) : Let I and T be self maps of a normed space X with x0 ∈ F (I) ∩ F (T ), M ⊆ X with T : ∂M −→ M and p ∈ F (I). If D, the set of best approximations of x0 in M is nonempty compact and p-starshaped, I(D) = D, I is continuous and linear on D, I and T are commutating on D and T is I-non expansive on D ∪ {x0 }, then I and T have a common fixed point in D. Corollary 3.8 ([12], Theorem 3) : Let T, I : X −→ X be operators on the normed linear space X, C be a T-invariant subset of X such that T (∂C) ⊆ C and x ∈ F (T ) ∩ F (I). Further I and T are non expansive on D, the set of best approximations of x in C. Let I be linear and continuous on D and I and T are 11

1049

1050

GOUDARZI, VAEZPOUR: ...LOCALLY CONVEX SPACES

commutating on D. If D is nonempty, compact and starshaped with respect to a point q ∈ F (I) and if I(D) = D, then D ∩ F (T ) ∩ F (I) 6= ∅. The following corollary and its 8 corollaries also will be the results of theorem 3.4. Corollary 3.9 ([13], Theorem 2.1) : Let E be a Hausdorff locally convex space and T : E −→ E be p-non expansive, for each continuous seminorm p on E. Let C be T-invariant set and y ∈ F (T ). Assume that for each continuous seminorm p on E, the set D of best C-approximants to y with respect to p is nonempty, sequentially complete, bounded and starshaped. Furthermore assume either of the following holds : i) (I − T )(D) is closed. ii) T is demicompact. Then T has a fixed point which is a best approximation to y in D. S S (u) = (u) = {x ∈ M : Sx ∈ PM (u)} and DM For M ⊆ X if we define CM S PM (u) ∩ CM (u) (see [1]), we can present the following theorem.

Theorem 3.10 Let (X, τ ) be a Hausdorff locally convex space and S and T be self mappings on X. Suppose that M be a subset of X such that T (∂M ) ⊆ M , S (u) 6= ∅ and qS(∂M ) ⊆ M and u ∈ F (T ) ∩ F (S). Suppose that D = DM

starshaped with q ∈ F (T ) ∩ F (S). S is non expansive on PM (u) ∪ {u}, T is S-non expansive on D ∪ {u} and both are affine with respect to q. Let η be a family of self mappings A : X −→ X which commute with S and T and

pα (Ax − By) ≤ pα (Sx − T y), ∀pα ∈ A∗ (τ ), x, y ∈ D ∪ {u}.

Then F (T )∩F (S)∩F (A) 6= ∅ provided S(D) = D = T (D), D is weakly compact, (T − A) and (S − A) are demiclosed at 0 and A(D) ⊆ D.

12

GOUDARZI, VAEZPOUR: ...LOCALLY CONVEX SPACES

1051

Proof : Suppose that y ∈ D. Since T (D) = D we have T y ∈ D. According to the definition of D, y ∈ ∂M and by the hypothesis of T (∂M ) ⊆ M we implies that T y ∈ M . Since T is S-non expansive then for each pα ∈ A∗ (τ ),

pα (T y − u) = pα (T y − T u) ≤ pα (Sy − u).

(3)

Since T y ∈ M and Sy ∈ PM (u), from (3) we infer that T y ∈ PM (u). On the other hand S is non expansive on PM (u) ∪ {u} so for each pα ∈ A∗ (τ ) we have pα (ST y − u) = pα (ST y − Su) ≤ pα (T y − u) = pα (T y − T u) ≤ pα (Sy − Su) = pα (Sy − u). S Thus ST y ∈ PM (u). So T y ∈ CM (u) and hence T y ∈ D. Therefore S and T

are self mappings on D. Now the condition (1) of Theorem 3.2 holds and hence the proof is complete. 

References [1] M. A. Al-Thagafi, Common fixed points and best approximation, J. Approx. Theory, 85(1996), 318-323. [2] B. Brosowski, Fix punktsatze in der approximations theorie, Mathematica(Cluj)11(1994), 165-220. [3] O. Hadzic, Common fixed point theorem for family of mappings in complete metric spaces, Mathematica Japonica, 29(1984), 127-134.

13

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[4] T. L. Hicks and M. D. Humpheries, A note on fixed point theorems, J. Approx. Theory, 34(1982), 221-225. [5] N. Hussain and A. R. Khan, Common fixed point results in best approximation theory, Appl. Math. Letters, 16(2003). [6] G. Jungck and S. Sessa, Fixed point theorem in best approximation theory, Math. Japonica, 42(1995), 249-252. [7] G. Kothe, Topological vector spaces-I, Springer-Verlag, New York(1969). [8] G. Meinardus, Invarianze bie linearen approximationen, Arch. Rational Mech. Anal., 14(1963), 301-303. [9] H. K. Nashine, An application of a fixed point theorem to best approximation for generalized affine mapping, Mathematical Proccedings of the Royal Irish Academy, 107(2007), 131-136. [10] H. K. Nashine, Best approximation result in locally convex spaces, Kyungpook Math J., 46(2006), 389-397. [11] S. A. Sahab and M. S. Khan, A result in best approximation theory, J. Approx. Theory, 55(1988), 349-351. [12] S. A. Sahab, M. S. Khan and S. Sessa, A result in best approximation theory, J. Approx. Thoey, 55(1998), 349-351. [13] B. N. Sahney, K. L. Singh and H. M. Whitfield, Best approximations in locally convex spaces, J. Approx. Theory, 38(1983), 182-187. [14] S. P. Singh, An approximation of a fixed point theorem to approximation theory, J. Approx. Thoey, 25(1979), 89-90. [15] P. V. Subrahmanyam, An application of a fixed point theorem to best approximations, J. Approx. Thoey, 20(1977), 165-172. 14

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[16] E. Tarafdar, Some fixed point theorems on locally convex linear topological spaces, Bull. Austral. Math. Soc., 13(1975), 241-254. [17] P. Vijayaraju and M. Marudai, Some results on common fixed points and best approximations, Indian Journal of Mathematics, 40(2004), 233-244.

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JOURNAL 1054 OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.6, 1054-1066, 2011, COPYRIGHT 2011 EUDOXUS PRESS, LLC

The Coupled Viscous Burgers Equations with Fractional-Time Derivative by the Homotopy Perturbation Method M. Ghoreishi§ , A.I.B.Md. Ismail∗, and A. Rashid† February 27, 2010

Abstract In this paper, the Homotopy Perturbation Method (HPM) which is an approximate analytical method has been used to solve the nonlinear coupled viscous Burgers equations with fractional-time derivative. With the HPM, the solution can be obtained in the form of a convergent power series with terms that can be easily computed. The method does not need linearization and provides an analytical solution by utilizing only the initial condition. We also show that the HPM can generate highly accurate analytical solutions. Keywords: Coupled viscous Burgers equations, Homotopy perturbation method, Time-Fractional derivative

1

Introduction

The coupled viscous Burgers equations are defined by the following nonlinear partial differential equations: ut − uxx + ηuux + θ(uv)x = 0,

x ∈ Γ,

t ∈ [0, T ],

(1)

vt − vxx + ξvvx + µ(uv)x = 0,

x ∈ Γ,

t ∈ [0, T ],

(2)

with initial conditions u(x, 0) = f (x), v(x, 0) = g(x),

x ∈ Γ, x ∈ Γ,

(3) (4)

t ∈ [0, T ], t ∈ [0, T ],

(5) (6)

and boundary conditions are u(−L, t) = u(L, t), v(−L, t) = v(L, t),

where Γ = [−L, L], η and ξ are real constants, θ and µ are arbitrary constants which depend on parameters such as the P´ eclet number, the Stokes velocity of particles due to gravity and the Brownian diffusivity [3]. The coupled system was first derived by Esipov and it represents an idealised model of sedimentation of scaled volume concentration of two kinds of particles in fluid suspensions under the effect of gravity [8]. Khater [9] used the Chebyshev Spectral collocation numerical method to solve coupled viscous Burgers equations and obtained approximate solutions. Rashid and Ismail [14] applied a Fourier Pesudo-spectral method to solve the system of nonlinear coupled viscous Burgers equations. They compared the obtained results with the Chebyshev spectral collocation method in [9]. They also showed that Fourier pesudospectral method is more accurate and has better convergence rate than the Chebyshev spectral collocation method to solve the coupled viscous Burgers equation. Kaya [8] used the approximate analytical Adomian ∗§ School

of Mathematical Science, Universiti Sains Malaysia, 11800, Penang, Malaysia. of Mathematics, Gomal University, Dera Ismail Khan, Pakistan, E-Mail:

† Department

1

rashid− [email protected].

GHOREISHI ET AL: COUPLED VISCOUS BURGERS EQUATIONS

1055

Decomposition Method (ADM) for solving coupled viscous Burgers equations homogeneous and nonhomogeneous) for η = ξ = 1 and obtained the solution in the form of a convergent power series with less computational work. Abdou and Soliman [1] used the approximate analytical Variational Iteration Method (VIM) for solving Burgers and coupled Burgers equations. Abdou and Soliman [1] showed that VIM is more effective than ADM. In recent years, fractional differential equations have attracted a great deal of attention. They are used in many areas of science and engineering such as in electromagnetic theory, traffic flow, electrochemistry as well as materials and details are discussed in [2, 7, 13, 15]. The coupled viscous Burgers equations with fractional-time of derivative are expressed as the following system of nonlinear partial differential equations: Dtα u − uxx + ηuux + θ(uv)x = 0, Dtα v − vxx + ξvvx + µ(uv)x = 0,

x ∈ Γ, t ∈ [0, T ], x ∈ Γ, t ∈ [0, T ],

(7) (8)

v(x, 0) = g(x),

(9)

with initial conditions u(x, 0) = f (x),

x ∈ Γ,

x ∈ Γ,

where 0 < α ≤ 1. If α = 1, the system of partial differential equations (1.7)-(1.8) reduce to the system of partial differential equations (1.1)-(1.2). The Homotopy Perturbation Method (HPM) was developed by He and is described in [4, 5, 6]. This method is applied for solving many types of nonlinear partial differential equations. The HPM method, as ADM and VIM, does not involve discretization of the variables and hence is free from rounding off errors and, further, does not require large computer memory or time. No linearization of nonlinear terms are required. In this method, the solution is obtained in the form of an infinite series, which converges rapidly to highly accurate solutions. Hence, it is also an approximate analytical method. This method uses a technique from topology called the homotopy technique in which a homotopy is constructed with an embedding parameter p ∈ [0, 1], which is considered as a ”small parameter” for the perturbation. The layout of the paper is as follows. Section 2 discusses some background on fractional calculus while section 3, constructs the homotopy to the coupled viscous Burgers equations. In section 4, we use the HPM for solving three examples of nonlinear coupled viscous Burgers equation with fractional-time derivative. Finally, in section 5, we give the conclusions of this study.

2

Basic Definition

The following definitions from [16] are used in this paper. Details on fractional derivatives can be found in [10,11,12,13].

2.1

Definition

The (left-sided) Caputo fractional derivative of a function f (x) is defined to be Z x 1 (x − t)m−α−1 f (m) (t)dt D∗α f (x) = Dα−m Dm f (x) = Γ(m − α) 0

(10)

m for m − 1 < α ≤ m, m ∈ N , x > 0, f ∈ C−1 .

Note that a real function f (x), x > 0, is in the space Cµ , µ ∈ R if there exists a real number p(> µ) such that f (x) = xp f1 (x), where f1 (x) ∈ C[0, ∞). Further, it is said to be in the space Cµm if f (m) ∈ Cµ , m ∈ N.

2.2

Definition

The Reimann-Liouville fractional integral operator of order α ≥ 0 , for f ∈ Cµ ,µ ≥ −1 is defined as follows Z x 1 −α f (t)(x − t)α−1 dt (11) D f (x) = Γ(α) 0 2

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GHOREISHI ET AL: COUPLED VISCOUS BURGERS EQUATIONS

D0 f (x) = f (x)

For f ∈ Cµ , µ ≥ −1 , α , β ≥ 0 and γ > −1, the following properties of D−α are true D−α D−β f (x) = D−(α+β) f (x) D−α D−β f (x) = D−β D−α f (x) D−α (xγ ) =

2.3

Γ(γ + 1) xα+γ Γ(α + γ + 1)

Definition

For the smallest integer m that exceeds α, the Caputo time-fractional derivative operator of order α > 0 is defined to be  Rt m 1 m−α−1 ∂ u(x,τ )  m − 1 < α < m,  Γ(m−α) 0 (t − τ ) ∂τ m dτ, α ∂ u(x, t) α = D∗t u(x, t) =  ∂tα  ∂ m u(x,t) , α = m ∈ N. ∂tm

3

HPM for coupled viscous Burgers equations

In this section, the HPM is applied on the nonlinear coupled viscous Burgers equations with fractional derivative (1.1) and (1.2). We rewrite Eqs. (1.1) and (1.2) as Dtα u = uxx − ηuux − θ(uv)x ,

x ∈ [−L, L],

t ∈ [0, T ],

(12)

Dtα v

x ∈ [−L, L],

t ∈ [0, T ],

(13)

= vxx − ξvvx − µ(uv)x ,

where Dtα stands for the fractional derivative. We will use the Homotopy perturbation technique. For this purpose, we first construct a homotopy w(r, p) : Γ × [0, 1] −→ R, w0 (r, p) : Γ × [0, 1] −→ R, which satisfies H(w, p) = (1 − p)(Dtα w − Dtα u0 ) + p(Dtα w − wxx + ηwwx + θ(ww0 )x ) = 0, 0 + ξw0 wx0 + µ(ww0 )x ) = 0, H(w0 , p) = (1 − p)(Dtα w0 − Dtα v0 ) + p(Dtα w0 − wxx

where p ∈ [0, 1] is an embedding parameter whilst u0 and v0 are the initial approximations of (3.1) and (3.2). We obtain Dtα w − Dtα u0 + pDtα u0 + p(−wxx + ηwwx + θ(ww0 )x ) = 0, (14) and 0 Dtα w0 − Dtα v0 + pDtα v0 + p(−wxx + ξw0 wx0 + µ(ww0 )x ) = 0,

If p = 0, we have H(w, 0) = Dtα w − Dtα u0 = 0, H(w0 , 0) = Dtα w0 − Dtα v0 = 0, and if p = 1, then H(w, 1) = Dtα w − wxx + ηwwx + θ(ww0 )x = 0, 0 H(w0 , 1) = Dtα w0 − wxx + ξw0 wx0 + µ(ww0 )x = 0.

3

(15)

GHOREISHI ET AL: COUPLED VISCOUS BURGERS EQUATIONS

1057

We assume that the solutions of (3.3) and (3.4) can be expressed as an infinite series in the following form w= w0 =

∞ X i=0 ∞ X

wi pi ,

(16)

wi0 pi .

(17)

i=0

By substituting the series (3.5) into (3.3) and the series (3.6) into (3.4), we have " ∞ ! ∞ ! ∞ ∞ X X X X pi Dtα wi − Dtα u0 + pDtα u0 + p − (wi )xx pi + η wi pi (wi )x pi i=0

i=0

+θ

i=0 ∞ X

i=0

wi p

i

i=0

∞ X

wi0 pi

i=0

! #

(18)

= 0,

x

and ∞ X

p

i

Dtα wi0

−

Dtα v0

+

pDtα v0

"

+p −

i=0

∞ X

(wi0 )xx pi

+ξ

i=0

∞ X

wi0 pi

i=0

+µ

∞ X

wi pi

i=0

!

∞ X

∞ X

(wi0 )x pi

i=0

wi0 pi

i=0

! #

!

(19)

= 0.

x

Substituting p = 1 into (3.5) and (3.6) will give in the approximate solutions of (3.1) and (3.2) as follows u = lim w = w0 + w1 + w2 + · · · ,

(20)

v = lim w0 = w00 + w10 + w20 + · · · ,

(21)

p→1

p→1

The convergence of the infinite series in equation (3.9) and (3.10) has been discussed by He in [6].

4

Numerical Results

In this section, the HPM will be demonstrated on three examples of nonlinear coupled viscous Burgers equations with fractional-time derivative. For our computation and illustration, let the expression ψm (x, t) =

m−1 X

uk (x, t),

(22)

k=0

denote the m-term HPM approximation to u(x, t). We compare the approximate analytical solution obtained using HPM for our nonlinear coupled viscous Burgers equation with the exact solution (where it exists). We define Em (x, t) to be the absolute error between the exact solution and m-term approximate HPM solution ψm (x, t). It is defined as follows Em (x, t) = |u(x, t) − ψm (x, t)|.

(23)

Example 1: To evaluate the performance of the Homotopy Perturbation Method (HPM) for solving coupled viscous Burgers equation with fractional-time derivative, we set η = ξ = −2 , θ = µ = 1 and L = π. The system of equations then takes the following form [8]: Dtα u = uxx + 2uux − (uv)x , Dtα v = vxx + 2vvx − (uv)x ,

4

−π ≤ x ≤ π, −π ≤ x ≤ π,

t > 0, t > 0,

(24) (25)

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GHOREISHI ET AL: COUPLED VISCOUS BURGERS EQUATIONS

with the initial conditions

u(x, 0) = sin x, v(x, 0) = sin x.

(26) (27)

It can be verified that the exact solutions are u(x, t) = e−t sin x and v(x, t) = e−t sin x. Kaya [8] used the Adomian Decomposition Method (ADM) for solving (4.3) and (4.4) and obtained the approximate solution. According to the HPM, we construct a homotopy as follows Dtα w − Dtα u0 = p(wxx + 2wwx − (ww0 )x − Dtα u0 ), Dtα w0

−

Dtα v0

=

0 p(wxx

+ 2w

0

wx0

0

− (ww )x −

Dtα v0 ).

(28) (29)

By substituting the infinite series (3.5) and (3.6) into (4.5) and (4.6), respectively, and comparing the coefficients of terms corresponding to p, we obtain the following Dtα w0 − Dtα u0 = 0,

(30)

Dtα w00 − Dtα u0 = 0.

(31)

Thus, the first terms of the infinite series (3.5) and (3.6) are obtained as follows w0 (x, t) = u0 (x, t) = f (x),

(32)

w00 (x, t) = v0 (x, t) = g(x).

(33)

Hence, we have the following for other components of the infinite series (3.5) and (3.6) Dtα w1 = (w0 )xx + 2(w0 )(w0 )x − (w0 w00 )x − Dtα u0 ,

(34)

Dtα w10 = (w00 )xx + 2(w00 )(w00 )x − (w0 w00 )x − Dtα v0 .

(35)

α

−α

Applying the operator D , the inverse of D in the equation associated with (4.13) and (4.14) and using the initial conditions for (4.3) and (4.4) gives w1 (x, t) = f1

tα , Γ(α + 1)

w10 (x, t) = g1

tα , Γ(α + 1)

where f1 = fxx + 2f fx − (f g)x , g1 = gxx + 2ggx − (f g)x . For the other terms, we obtain w2 (x, t) = f2

t2α , Γ(2α + 1)

w20 (x, t) = g2

t2α , Γ(2α + 1)

where f2 = (f1 )xx + 2[f (f1 )x + fx f1 ] − (f g1 + gf1 )x , g2 = (g1 )xx + 2[g(g1 )x + gx g1 ] − (f g1 + gf1 )x , and w3 (x, t) = f3

t3α , Γ(3α + 1)

w30 (x, t) = g3

t3α , Γ(3α + 1)

5

GHOREISHI ET AL: COUPLED VISCOUS BURGERS EQUATIONS

where

1059

    Γ(2α + 1) Γ(2α + 1) + f2 fx − f g2 + f1 g1 2 + f2 g , f3 = (f2 )xx + 2 f (f2 )x + f1 (f1 )x 2 Γ (α + 1) Γ (α + 1) x     Γ(2α + 1) Γ(2α + 1) + g2 gx − f g2 + f1 g1 2 + f2 g . g3 = (g2 )xx + 2 g(g2 )x + g1 (g1 )x 2 Γ (α + 1) Γ (α + 1) x and so on. The solution of coupled viscous Burgers equation with fractional-time derivative (4.3) and (4.4) is then obtained as an infinite series of the form u(x, t) = f (x) + v(x, t) = g(x) +

∞ X

k=1 ∞ X

k=1

fk

tkα , Γ(kα + 1)

(36)

gk

tkα . Γ(kα + 1)

(37)

Table 1 shows the absolute error E4 for u and v and various variables x and t in the case of α = 1. From the table, it can be seen that for all test points, the HPM provides a very accurate approximation for nonlinear coupled viscous Burgers equation. Table 1: Absolute error E4 for variables x and t t/x

0.1

0.1

4.0779×10−7

3.6403×10−6

1.1

3.5259×10−6

2.1

1.6984×10−7

3.1

0.3 0.5

3.1769×10−5 2.3600×10−4

2.8360×10−4 2.1068×10−3

2.7469×10−4 2.0406×10−3

1.3231×10−5 9.8296×10−5

0.7

8.7373×10−4

7.7998×10−3

7.5547×10−3

3.6391×10−4

Figure 1 and 2 shows comparison between analytical solution obtained for α = 1 with four terms using HPM and the exact solution. It can clearly be seen that the HPM is very accurate for this example. Figure 3 shows analytical solution obtained for α = 0.7 with four terms using HPM. u 0.4

0.2

-3

-2

1

-1

2

3

X

-0.2

-0.4

Figure 1: Comparing the solution of coupled viscous Burgers equation with fractional-time derivative for α = 1 (u and v are same )by HPM with four terms and the exact solution at t = 0.5 of example 1. In the graph, solid line corresponds to the exact solution and the dotted line corresponds to the HPM solution.

6

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GHOREISHI ET AL: COUPLED VISCOUS BURGERS EQUATIONS

u 0.6 0.4 0.2

-3

-2

1

-1

2

3

X

-0.2

-0.4

-0.6

Figure 2: Comparing the solution of coupled viscous Burgers equation with fractional-time derivative for α = 1 (u and v are same )by HPM with four terms and the exact solution at t = 0.7 of example 1. In the graph, solid line corresponds to the exact solution and the dotted line corresponds to the HPM solution.

u

0.4

0.2

-3

-2

1

-1

2

3

X

-0.2

-0.4

Figure 3: The solution of coupled viscous Burgers equation with fractional-time derivative for α = 0.7 at t = 0.5 (u and v are same )by HPM with four terms of example 1

7

GHOREISHI ET AL: COUPLED VISCOUS BURGERS EQUATIONS

1061

Dtα u = uxx − 2uux − θ(uv)x , Dtα v = vxx − 2vvx − µ(uv)x ,

(38) (39)

Example 2: We now consider the coupled viscous Burgers equation with η = ξ = 2 and L = 10. The system of equations (3.1) and (3.2) now takes the form: −10 ≤ x ≤ 10, t > 0, −10 ≤ x ≤ 10, t > 0,

subject to initial conditions   2θ − 1 tanh Ax, −10 ≤ x ≤ 10, u(x, 0) = a0 − 2A 4θµ − 1     2θ − 1 2µ − 1 − 2A tanh Ax, −10 ≤ x ≤ 10. v(x, 0) = a0 2θ − 1 4θµ − 1 The exact solutions of this problem as given in [9, 14] are   2θ − 1 u(x, t) = a0 − 2A tanh(A(x − 2At)), −10 ≤ x ≤ 10 , t > 0, 4θµ − 1     2µ − 1 2θ − 1 − 2A tanh(A(x − 2At)), −10 ≤ x ≤ 10 , t > 0, v(x, t) = a0 2θ − 1 4θµ − 1

(40) (41)

(42) (43)

a0 (4θµ − 1) and a0 ,θ and µ are arbitrary constants. In the table 2, we list absolute error E4 2(2θ − 1) for various values of x,t,θ and µ. where A =

Table 2: Absolute error E4 for variables x and t x 0.5

t 0.5

θ

E4 (u)

E4 (v)

0.30

3.8901×10−5

8.2721×10−6

µ

0.1

0.5 1.0

0.5 1.0

0.3 0.1

0.03 0.30

5.1636×10−5 7.6282×10−5

8.9130×10−6 1.4451×10−5

1.0

1.0

0.3

0.03

1.0121×10−4

1.9177×10−5

Figures 4 and 5 show the comparison between the analytical solution obtained for α = 1 with four terms for u and v, respectively, using HPM and the exact solution. It is clear that the HPM is accurate for this example. Figure 6 shows analytical solution for v and α = 0.4 with four terms using HPM. u

0.0515 0.0510 0.0505 0.0500 0.0495 0.0490

-1.0

0.5

-0.5

1.0

X

Figure 4: Comparison of the solution u of coupled viscous Burgers equation with fractional-time derivative for α = 1,θ = 0.1,µ = 0.3 and a0 = 0.05 by HPM with four terms and analytical solution at t = 0.9 and x ∈ [−1, 1] of example 2. In the graph, solid line corresponds to the exact solution and the dotted line corresponds to the HPM solution.

8

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GHOREISHI ET AL: COUPLED VISCOUS BURGERS EQUATIONS

u 0.065 0.060 0.055 0.050 0.045 0.040

-10

5

-5

10

X

Figure 5: Comparing the solution v of coupled viscous Burgers equation with fractional-time derivative for α = 1,θ = 0.1,µ = 0.3 and a0 = 0.05 by HPM with four terms and analytical solution at t = 0.9 and x ∈ [−10, 10] of example 2. In the graph, solid line corresponds to the exact solution and the dotted line corresponds to the HPM solution.

u 0.040 0.035 0.030 0.025 0.020 0.015

-10

5

-5

10

X

Figure 6: The solution of coupled viscous Burgers equation with fractional-time derivative for α = 0.4 at t = 0.9( function v ) by HPM with four terms of example 2

9

GHOREISHI ET AL: COUPLED VISCOUS BURGERS EQUATIONS

1063

Example 3: We finally consider the following non-homogeneous and nonlinear coupled viscous Burgers equation (3.1) and (3.2) with η = ξ = θ = µ = 1 and L = 10 Dtα u = uxx − uux − (uv)x + x2 − 2t + 2x3 t2 + t2 , Dtα v = vxx − vvx − (uv)x +

(44)

2

1 2t t − − 3 + t2 , x x3 x

(45)

with the initial conditions u(x, 0) = 0,

(46)

v(x, 0) = 0,

(47)

and the exact solutions are given in [8] u(x, t) = x2 t,

v(x, t) =

t . x

Kaya [8] used the ADM to solve of (4.23) and (4.24). According to the HPM, the homotopy to this problem is constructed as follows Dtα w − Dtα u0 = p(wxx − wwx − (ww0 )x − Dtα u0 ), 0 Dtα w0 − Dtα v0 = p(wxx − w0 wx0 − (ww0 )x − Dtα v0 ).

(48) (49)

By substituting the infinite series (3.5) and (3.6) into (4.27) and (4.28), respectively, and comparing the coefficients of terms corresponding to p, we obtain Dtα w0 − Dtα u0 = x2 − 2t + 2x3 t2 + t2 , Dtα w00 − Dtα u0 =

1 2t t2 − 3 − 3 + t2 . x x x

Thus, we have w0 (x, t) = u(x, 0) + D−α (x2 − 2t + 2x3 t2 + t2 ), 2t t2 1 − 3 − 3 + t2 ). x x x Hence, the following zeroth components are obtained as w00 (x, t) = v(x, 0) + D−α (

w0 (x, t) =

x2 tα 2 2 − tα+1 + (2x3 + 1) tα+2 , Γ(α + 1) Γ(α + 2) Γ(α + 3)

tα 2 1 1 2 − tα+1 + (1 − 3 ) tα+2 . xΓ(α + 1) x3 Γ(2 + α) x Γ(3 + α) Similarly one can obtain the other components as w00 (x, t) =

w1 (x, t) = 2 w10 (x, t) =

t2α Γ(2α + 1) 1 − 2x 2 t3α − 12x2 t2α+2 + · · · , Γ(2α + 1) Γ (α + 1)Γ(3α + 1) Γ(α + 1)Γ(2α + 3)

t2α 24 t1+2α 1 t3α 8 Γ(2 + 2α) 2 − 5 + 3 2 − 5 × 3 x Γ(2α + 1) x Γ(2 + 2α) x Γ (α + 1)Γ(3α + 1) x Γ(1 + α)Γ(2 + α) t1+3α + ··· . Γ(2 + 3α)

The solution of coupled viscous Burgers equation with fractional-time derivative (4.23) and (4.24) is then obtained as an infinite series of the form u(x, t) = v(x, t) =

∞ X

k=0 ∞ X k=0

10

wk (x, t),

(50)

wk0 (x, t).

(51)

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GHOREISHI ET AL: COUPLED VISCOUS BURGERS EQUATIONS

Tables 3 and 4 show respectively the absolute error E4 for u and v for various variables x and t in the case of α = 1. From these table, it is seen that at all the test points, the HPM provides a very accurate approximation for nonlinear non-homogeneous coupled viscous Burgers equation with fractional-time derivative. Figures 7 and 8 show the absolute error between solution obtained using HPM with three terms and the exact solution at t = 0.01 and x ∈ (−10, 10) for u and v, respectively. In Figures 9 and 10 we plotted the solution obtained using HPM with three terms at t = 0.01 and α = 0.7 for u(x, t) and v(x, t), respectively. Table 3: Absolute error E4 for variables x and t for u(x, t) t/x

t=0.1

t=0.01

t=0.001

2

4.88994×10−7

6.38157×10−13

1.14480×10−18

4 6

1.57361×10−6 3.79539×10−5

2.28398×10−15 3.13102×10−14

6.42134×10−19 6.44246×10−19

8

2.46189×10−4

5.38195×10−14

4.26469×10−19

10

9.98390×10−4

5.54708×10−13

5.13291×10−18

Table 4: Absolute error E4 for variables x and t for v(x, t) t/x

t=0.1

t=0.01

t=0.001

2 4

7.61068×10−6 7.12370×10−8

6.66619×10−11 1.38848×10−13

6.57348×10−18 1.31809×10−18

6

1.79108×10−7

4.33657×10−15

4.78548×10−20

8 10

4.39415×10−7 8.82034×10−7

7.57382×10−16 8.17177×10−16

1.70906×10−20 5.02395×10−21

e

3. ´ 10-12 2.5 ´ 10-12 2. ´ 10-12 1.5 ´ 10-12 1. ´ 10-12 5. ´ 10-13

-10

5

-5

10

x

Figure 7: Absolute error between HPM solution and the exact solution at t = 0.01 and x ∈ (−10, 10) for u(x, t)

11

GHOREISHI ET AL: COUPLED VISCOUS BURGERS EQUATIONS

1065

e 5. ´ 10-10

4. ´ 10-10

3. ´ 10-10

2. ´ 10-10

1. ´ 10-10

-10

5

-5

10

x

Figure 8: Absolute error between HPM solution and the exact solution at t = 0.01 and x ∈ (−10, 10) for v(x, t)

u

-10

5

-5

10

x

-5

-10

Figure 9: The solution obtained using HPM with three terms at t = 0.01, x ∈ (−10, 10) and α = 0.7 for u(x, t)

v

0.04

0.02

-10

5

-5

10

x

-0.02

-0.04

Figure 10: The solution obtained using HPM with three terms at t = 0.01, x ∈ (−10, 10) and α = 0.7 for v(x, t)

12

1066

5

GHOREISHI ET AL: COUPLED VISCOUS BURGERS EQUATIONS

Conclusion

In this paper, we discussed the solution of coupled viscous Burgers equation with fractional-time derivative using HPM. The results obtained show good accuracy of HPM method. This method was tested on three examples and it was shown (via figures and tables) that this method is highly accurate and converges rapidly for the case of α = 1. We have solved the viscous Burgers equation for three particular cases when α < 1.

References [1] M. A. Abdou, A. A. Soliman, Variational iteration method for solving Burgers and coupled Burgers equations, J. Comput. Appl. Math, 181 (2),245-251(2005). [2] V. Daftardar-Gejji, H. Jafari, Adomian decomposition: a tool for solving a system of fractional differential equations, J. Math. Anal. Appl. 301,508-518 (2005) . [3] S. E. Esipov, Coupled Burgers equations: A model of polydispersive sedimentation, Phys. Rev. E, 52,3711-3718(1995). [4] J.H. He, A coupling method of a homotopy technique and a perturbation technique for non-linear problems, Internat. J. Non-Linear Mech. 35 (1),37-43(2000). [5] J.H. He, Homotopy perturbation method: A new nonlinear analytical technique, J. Appl. Math. Comput. 135 (1),73-79(2003). [6] J.H. He, Homotopy perturbation technique, J. Comput. Methods. Appl. Mech. Engrg. 178,257262(1999). [7] H. Jafari, V. Daftardar-Gejji, Solving a system of nonlinear fractional differential equations using Adomian decomposition, J. Comput. Appl. Math. 196,644-651(2006). [8] D. Kaya, An explicit solution of coupled viscous Burgers equations by the decomposition method, International Journal of Mathematics and Mathematical Sciences , 27 (11),675-680(2001). [9] A. H. Khater, R. S. Temsah, M. M. Hassan, A Chebyshev spectral collocation method for solving Burgers-type equations, J. Comput. Appl. Math, 222 (2),333-350(2008). [10] Y. Luchko, R. Gorenflo, The initial value problem for some fractional differential equations with the Caputo derivative, Preprint series A0898, Fachbreich Mathematik und Informatik, Freic Universitat Berlin, 1998. [11] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, New York, 1993. [12] K.B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York, 1974. [13] I. Podlubny, Fractional Differential Equations, Academic Press, SanDiego, 1999. [14] A. Rashid, A.I.B.Md.Ismail, The Fourier pseudo-spectral method for solving coupled viscous Burgers equation, J. Comput. Method. Appl. Math, 9 (4),1-9(2009). [15] G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integral and Derivative: Theory and Applications , Gordon and Breach, Yverdon, 1993. [16] M. Shateri, D.D. Ganji, Solitary wave solutions for a time-fraction generalized Hirota-Satsuma coupled KdV equation by a new analytical technique, International Jornal of Differential equations, 20 (10) (2010), Article ID 954674, 11 pages.

13

JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.6, 1067-1074, 2011, COPYRIGHT 2011 EUDOXUS PRESS, 1067 LLC

Stability and boundedness of solutions of non-autonomous differential equations of second order Cemil Tunç Department of Mathematics, Faculty of Arts and Sciences Yüzüncü Yıl University, 65080, Van - Turkey E-mail: [email protected]

Abstract: This paper establishes some new sufficient conditions to ensure the stability and boundedness of solutions of certain second order non-autonomous and non-linear differential equations. By defining two appropriate Liapunov functions, we obtain three new results on the subject and give some examples to illustrate the theoretical analysis in this work. By this work, as well as two new results, we also extend and improve an important result from the literature. Keywords: Non-autonomous; non-linear; differential equation; second order; stability; boundedness, uniform boundedness. AMS (MOS) Subject Classification: 34K20.

1. Introduction In 1992, Qian [12] considered the following second order nonlinear differential equation

x ′′ + { f ( x ) + g ( x ) x ′}x ′ + h( x ) = e(t ),

(1)

where f , g and h are continuous functions on ℜ = ( −∞ , ∞ ) and e(t ) is a continuous function on

ℜ + = [0, ∞). Let x

a( x ) = exp(∫ g (u )du) 0

and x

b( x) = ∫ a(u ) f (u )du. 0

Subject the above suppositions, Eq. (1) can be transformed to the following system

x′ =

1 {c( y ) − b( x)}, a ( x)

y ′ = − a ( x ){h ( x ) − e(t )},

(2)

where a is a positive and continuous function on ℜ, b, c and h are continuous functions on ℜ, and

e is a continuous function on ℜ + . Utilizing the preceding acceptations, Qian [12] proved the following theorem Theorem A (Qian [12, Theorem 1]). Assume that (i ) there exists a positive number m such that

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TUNC: NON-AUTONOMOUS D.E.

2

y

a ( x) c( y ) ≤ ∫ c( s )ds + m =C ( y ) + m for any x, y; 0 x

(ii )

h( x)b( x) = h( x) ∫ a (u ) f (u )du ≥ 0 for all x; 0 y

(iii ) C ( y ) = ∫ c( s )ds > 0 for y ≠ 0 0

and y

lim sup C ( y ) = lim sup ∫ c( s )ds = ∞; y →∞

y →∞

0

t

(iv ) E (t ) = ∫ e( s ) ds, E (∞ ) < ∞, 0 x

(v ) H ( x) = ∫ a 2 ( s )h( s )ds > 0 for x ≠ 0. 0

Then every solution of (2) is bounded under each of the following hypotheses: x

(vi ) lim sup H ( x) = lim sup ∫ a 2 ( s )h( s )ds = ∞; x → +∞

x →∞

0 x

(vii ) lim sup H ( x ) = lim sup ∫ a 2 ( s )h( s )ds = ∞ x →∞

x →∞

0

and x

lim sup(−b( x)) = lim sup(− ∫ a (u ) f (u )du ) = ∞;

x →−∞

x →−∞

0 x

(viii ) lim sup H ( x) = lim sup ∫ a 2 ( s)h( s )ds = ∞ x → −∞

x →−∞

0

and x

lim sup b( x) = lim sup ∫ a (u ) f (u )du = ∞; x →∞

x →∞

0 x

(ix ) lim sup(signx)b( x) = lim sup(signx) ∫ a (u ) f (u )du = ∞. x →∞

x →∞

0

By the above theorem, Qian [12] extended and improved some important boundedness results obtained in the literature (see Antosiewicz [2], Burton and Townsend [3], Čžan [7], Graef [8] and Omari and Zanolin [11]). For some recent works on the boundedness and convergence of solutions of nonlinear differential equations of second order, we also refer readers to the papers of Constantin [6], Jin [9] and Zhou [17], which are generalizations of the results in [12] but in a different direction from the one in the present paper. In this paper, we consider the following second order non-autonomous and non-linear differential equation (3) x ′′ + a (t ){ f ( x, x ′) + g ( x, x ′) x ′}x ′ + b(t ) h( x ) = e(t , x, x ′), in which a, b, f , g , h and e are continuous functions in their respective domains, and the derivative b ′(t ) exists and is continuous. We here investigate the stability of the null solution of Eq. (3) when e(t , x, x ′) = 0, and establish two results on the boundedness and uniform-boundedness of the solutions of Eq. (3) when e(t , x, x ′) ≠ 0. It is worth mentioning that, with respect to the observations

3 1069

TUNC: NON-AUTONOMOUS D.E.

in the literature, it is not found any research on the stability and boundedness of solutions of Eq. (3). To prove our main results we introduce two Liapunov functions. The work in this paper is especially motivated by Qian [12], and by the recent works of Constantin [6], Jin [9], Tunç ([13], [14]), C. Tunç and E. Tunç [15] and Zhou [17]. It should be noted that Eq. (3) and the assumptions will be established here are different from that in the papers mentioned above and that in the literature. Instead of Eq. (3), we consider the system x′ = y, (4) y ′ = − a (t ){ f ( x, y ) + g ( x, y ) y} y − b(t ) h( x ) + e(t , x, y ), which was obtained from Eq. (3). Consider a system of differential equations

dx = F (t , x), dt

(5)

where x is an n − vector. Suppose that F (t , x ) is continuous in (t , x ) on I × D, where I denotes the interval 0 ≤ t < ∞ and D is a connected open set in ℜ n . It is also assumed without loss of generality that F (t ,0) = 0 and D is a domain such that x < H , H > 0. Theorem 1. (Yoshizawa [16].) Suppose that there exists a Liapunov function V (t , x ) defined on

0 ≤ t < ∞, x ≥ R, where R may be large, which satisfies the following conditions; (i ) a( x ) ≤ V (t , x) ≤ b( x ), where a (r ) ∈ CI , a (r ) → ∞ as r → ∞ and b( r ) ∈ CI

(CI

denotes the families of continuous increasing functions),

(ii ) V& (t , x) ≤ 0. Then, the solutions of (5) are uniform-bounded. Theorem 2. (Yoshizawa [16].) Suppose that there exists a Liapunov function V (t , x ) defined on 0 ≤ t < ∞, x < H which satisfies the following conditions;

(i ) V (t ,0) ≡ 0, (ii ) a( x ) ≤ V (t , x), where a (r ) ∈ CIP , (CIP denotes the families of continuous increasing and positive definite functions),

(iii ) V& (t , x) ≤ 0. Then, the solution x (t ) ≡ 0 of (5) is stable. Theorem 3. (Yoshizawa [16].) If condition (ii) of Theorem 2 is replaced by

a( x ) ≤ V (t , x) ≤ b( x ), where a (r ) ∈ CIP and b (r ) ∈ CIP (CIP denotes the families of continuous increasing and positive definite functions). Then, the solution x (t ) ≡ 0 of (5) is uniform-stable.

2. Description of Problems The first main problem of this paper is the following theorem Theorem 4. In addition to the basic assumptions imposed on the functions a, b, f , g , h and

p , we assume that there exist positive constants b0 and δ

such that the following assumptions

hold:

a (t ) > 0, b0 ≥ b(t ) ≥ 1 for all t ∈ ℜ + , ℜ + = [0, ∞ ), 2 a (t )b(t ){ f ( x, y ) + g ( x, y ) y} + b ′(t ) ≥ δ for all t ∈ ℜ + and x, y ∈ ℜ,

1070

4

TUNC: NON-AUTONOMOUS D.E.

x

xh( x ) > 0 for all x ≠ 0 and ∫ h( s)ds → +∞ as x → ∞, 0 t

e(t , x, y) ≤ p(t ) , exp(∫ p( s) ds) < ∞. 0

Then all solutions of Eq. (3) are uniform-bounded. Proof. Define Liapunov function t

x

0

0

V (t , x, y ) = exp(−2∫ p( s ) ds){∫ h( s )ds +

1 y 2 + 1}. 2b(t )

Then, we have ∞

x

0

0

exp(−2∫ p( s ) ds){∫ h( s )ds +

x

1 1 y 2 + 1} ≤ V (t , x, y ) ≤ ∫ h( s )ds + y2 +1 2b(t ) 2 b ( t ) 0

t

∫

by exp( p ( s) ds) < ∞. 0

Hence, utilizing the assumption b0 ≥ b(t ) ≥ 1, it follows that ∞

x

exp(−2∫ p( s ) ds){∫ h( s )ds + 0

0

x

1 2 1 y + 1} ≤ V (t , x, y ) ≤ ∫ h( s )ds + y 2 + 1. 2b0 2 0

x

Now, in view of the fact

∫ h(s)ds → +∞ as

x → ∞, we arrive at condition (i) of Theorem 1 holds.

0

The time derivative of Liapunov function V (t , x, y ) along (4) gives that t

x

1 d y 2 + 1} V (t , x, y ) = − 2 p(t ) exp(−2∫ p( s) ds) {∫ h( s )ds + 2 b ( t ) dt 0 0 t  a(t ) b′(t )  − exp(−2∫ p( s) ds)  { f ( x, y ) + g ( x, y ) y} + 2  y 2 2b (t )   b(t ) 0 t

+ exp(−2∫ p( s) ds) 0

ye(t , x, y ) b(t )

t

x

∫

∫

= − 2 p (t ) exp(−2 p ( s ) ds) { h( s )ds + 0

−

0

1 y 2 + 1} 2b(t )

t

1 2b 2 (t )

exp(−2∫ p( s) ds) {2a(t )b(t ){ f ( x, y ) + g ( x, y ) y} + b′(t )} y 2 0 t

+ exp(−2∫ p( s ) ds) 0

ye(t , x, y ) . b(t )

By utilizing the assumptions of Theorem 4 and the inequality y < 1 + y 2 , it follows that t

x

1 d y 2 + 1} V (t , x, y ) ≤ − 2 p(t ) exp(−2∫ p( s) ds) {∫ h( s )ds + 2 b ( t ) dt 0 0 −

1 2b 2 (t )

t

exp(−2∫ p( s) ds) {2a(t )b(t ){ f ( x, y ) + g ( x, y ) y} + b′(t )} y 2 0

5 1071

TUNC: NON-AUTONOMOUS D.E.

t

+ exp(−2∫ p( s ) ds) 0

y e(t , x, y ) b(t )

t

≤ − 2 p(t ) exp(−2∫ p( s) ds) { 0 t

+ exp(−2∫ p( s ) ds) 0

t

1 δ exp(−2∫ p( s) ds) y 2 y 2 + 1} − 2 2b(t ) 2b (t ) 0

(1 + y 2 ) p(t ) b(t )

t

≤ − 2 p(t ) exp(−2∫ p( s) ds ) − 0

δ 2b 2 (t )

t

exp(−2∫ p( s ) ds) y 2 ≤ 0. 0

Then, by Theorem 1, the solutions of Eq. (3) are uniform-bounded. Example 1. As a special case of Eq. (3), we consider the following second order non-autonomous and non-linear differential equation

1   x ′′ + {2 + exp(−t 2 )} 2 + x ′ + (2 − e − t ){2 x ′ + x ′ exp(−1 − x 2 − x ′ 2 )} 2 2  1 + x + x′   1   + 2x 3 = exp (6) . 2 2 2   1 + t + x + x′  This equation can be stated as the system

x ′ = y,

 1 y ′ = − {2 + exp(−t 2 )} 2 + 2 1+ x + y2    1 . − 2x 3 = exp 2 2 2  1+ t + x + y 

  y − (2 − e − t ){2 y + y exp(−1 − x 2 − y 2 )} 

Hence, it follows the following

a(t ) = 2 + exp(−t 2 ) ≥ 2 > 0, t ≥ 0, b(t ) = 2 − e − t , t ≥ 0, 2 ≥ b(t ) = 2 − e −t ≥ 1, 2 a (t )b(t ) ≥ 4, b ′(t ) = e − t , 1 ≥ b ′(t ) = e −t ≥ 0, 1 f ( x, y ) = 2 + , g ( x, y ) y = 2 + exp(−1 − x 2 − y 2 ), 2 2 1+ x + y 1 + 2 + exp(−1 − x 2 − y 2 ) ≥ 4, f ( x, y ) + g ( x, y ) y = 2 + 2 2 1+ x + y 2 a (t )b(t ){ f ( x, y ) + g ( x, y ) y} + b ′(t ) ≥ 16 ≥ δ , x

x

0

0

h( x) = x 3 , x 4 = xh( x) > 0, ( x ≠ 0), ∫ h( s )ds = ∫ s 3 ds =

x4 → +∞ as x → ∞, 4

  1 1   ≤ exp e(t , x, y) = exp = p (t ) , 2 2 2  2  1+ t  1 + t + x + y  ∞ ∞ 1  π exp( ∫ p ( s ) ds) = exp ∫ ds  = exp( ) < ∞. 2 2 0  0 1+ s  The above discussion shows that all the assumptions of Theorem 4 hold. Thus, we conclude that all solutions of Eq. (6) are uniform-bounded. For the case e(t , x, y ) ≡ 0 in Eq. (3), the second main problem of this paper is the following theorem

1072

6

TUNC: NON-AUTONOMOUS D.E.

Theorem 5. In addition to the basic assumptions imposed on the functions a, b, f , g and h, we assume that there exist positive constants b0 , δ and α such that the following assumptions hold:

a (t ) > 0, b0 ≥ b(t ) > 0 for all t ∈ ℜ + , ℜ + = [0, ∞ ), 2 a (t )b(t ){ f ( x, y ) + g ( x, y ) y} + b ′(t ) ≥ δ for all x, y ∈ ℜ, h( x ) ≥ α for all x ∈ ℜ, ( x ≠ 0). h(0) = 0, x Then the null solution of Eq. (3) is stable. Proof. Define Liapunov function x

V0 (t , x, y ) = ∫ h( s )ds + 0

1 y 2. 2b(t )

Evidently, we have V0 (t ,0,0) = 0. The assumptions b0 ≥ b(t ) > 0 and

h( x) ≥ α imply that x

x

h( s ) 1 sds + y 2 ≥ D1 ( x 2 + y 2 ), s 2 b ( t ) 0

V0 (t , x, y ) = ∫ where D1 =

1 min{α , b0−1 }. 2

A straightforward calculation for the time derivative of the function V0 (t , x, y ) along (4) yields that

d a(t ) b ′(t ) V0 (t , x, y ) = − { f ( x, y ) + g ( x, y ) y} y 2 − 2 y 2 b(t ) dt 2b (t ) 1 = − 2 {2a (t )b(t ){ f ( x, y ) + g ( x, y ) y} + b ′(t )} y 2 . 2b (t ) The assumption

2a (t )b(t ){ f ( x, y ) + g ( x, y ) y} + b ′(t ) ≥ δ leads that

d δ V0 (t , x, y ) ≤ − 2 y 2 ≤ 0. dt 2b (t ) By the above discussion, we conclude that the null solution of Eq. (3) is stable (see also Theorem 2). Example 2. Consider the following second order non-autonomous and non-linear differential equation

1   x ′′ + (3 + e −t ) 2 + x ′ + (3 − e − t ){3x ′ + x ′ exp(−1 − x 2 − x ′ 2 )} 2 2  ′ 1 + x + x   + 2 x + sin x = 0. (7) The associated system of Eq. (7) is the following

x ′ = y,

  1  y − (3 − e − t ){3 y + y exp(−1 − x 2 − y 2 )} y ′ = − (3 + e −t ) 2 + 2 2  1+ x + y   − 2 x − sin x. Evidently,

a(t ) = 3 + e −t ≥ 3 > 0, t ≥ 0, b(t ) = 3 − e − t , t ≥ 0, 3 ≥ b(t ) = 3 − e −t ≥ 2 > 0, 2a(t )b(t ) ≥ 12, b ′(t ) = e −t ≥ 0, f ( x, y ) + g ( x, y ) y ≥ 6, 2a (t )b(t ){ f ( x, y ) + g ( x, y ) y} + b ′(t ) ≥ 72 ≥ δ ,

7 1073

TUNC: NON-AUTONOMOUS D.E.

h( x) = 2 x + sin x, h(0) = 0, sin x h( x ) h( x) , ( x ≠ 0), ≥ 1 = α. = 2+ x x x Clearly, the above discussion reveals that all the assumptions of Theorem 5 hold. Hence, we arrive at the null solution of Eq. (7) is stable. For the case e(t , x, y ) ≠ 0 in Eq. (3), the third and last main problem of this paper is the following theorem Theorem 6. In addition to the assumptions of Theorem 5, except h(0) = 0 and b0 ≥ b(t ) > 0, we assume that

b0 ≥ b(t ) ≥ 1, e(t , x, y ) ≤ q(t ), 1

where q ∈ L (0, ∞), L1 (0, ∞ ) is space of Lebesgue integrable functions. Then, there exists a finite positive constant K such that every solution x (t ) of Eq. (3) satisfies

x(t ) ≤ K , x ′(t ) ≤ K for all t ≥ t 0 . Proof. For the case e(t , x, y ) ≠ 0, under the assumptions of Theorem 6, the time derivative of the function V0 (t , x, y ) can be disposed as the following: d 1 V0 (t , x, y ) ≤ − δy 2 + ye(t , x, y ). dt b(t ) Hence, plainly, we pursue that

d 1 1 V0 (t , x, y ) ≤ y e(t , x, y ) ≤ (1 + y 2 ) q (t ) dt b(t ) b(t ) 1 ≤ {1 + D1−1V (t , x, y )} q (t ) ≤ q (t ) + D1−1 q(t )V (t , x, y ). b(t ) Integrating the last inequality from 0 to t , using the assumption q ∈ L1 (0, ∞) and Gronwall-ReidBellman inequality (see Ahmad and Rama Mohana Rao [1]), for a positive constant K1 we can obtain

V0 (t , x, y ) ≤ K 1 . In view of the above discussion, one can arrive at

x 2 + y 2 ≤ D1−1V0 (t , x, y ) ≤ K , where K = K 1 D1−1 . Hence, we conclude that

x(t ) ≤ K , y (t ) ≤ K , for all t ≥ t 0 . The proof of Theorem 6 is now complete. Example 3. Consider the equation

1   x ′′ + (3 + e −t ) 2 + x ′ + (2 − e − t ){2 x ′ + x ′ exp(−1 − x 2 − x ′ 2 )} 2 2  1 + x + x′   2 + 2 x + sin x = . 2 1 + t + x 2 + x′ 2

(8)

Clearly, ∞

e(t , x, y ) =

∞

2 2 2 ds = π < ∞. = q (t ), ∫ q ( s )ds = ∫ ≤ 2 2 2 2 2 1+ t 1+ t + x + y 0 0 1+ s

In view of the discussion in Example 2 and the above, it follows that all the assumptions of Theorem 6 holds. Hence, we conclude that all solutions of Eq. (8) are bounded. Remark 1. Theorem 4 raises a new result in the literature on the uniform boundedness of solutions of nonlinear differential equations of second order.

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TUNC: NON-AUTONOMOUS D.E.

8

Remark 2. It is obvious that Eq. (3) considered in the present paper generalizes that investigated by Qian [12], and the assumptions established here are different from those in Qian [12, Theorem 1]. Another possible generalization was proposed in Zhou [17], where a nonlinear growth suitable for global existence (see Constantin [4, 5]) was introduced. This alternative generalization was further investigated in the paper by Yin [10], and opens up the possibility of consideration in combination with the present approach. Further, the assumptions of Theorem 4 can also be easily applied to a general second order nonlinear differential equation, Eq. (3), than that considered in Antosiewicz [2], Burton and Townsend [3], Čžan [7], Graef [8], Omari and Zanolin [11] and Qian [12]. That is to say the following: (r1 ) Our equation, Eq. (3), includes and improves the equation discussed by Antosiewicz [2], Burton and Townsend [3], Čžan [7], Graef [8], Omari and Zanolin [11] and Qian [12]. (r2 ) When we compare the assumptions of Theorem 4 with that of Theorem A (proved by Qian [12, Theorem1]), it is clear that our assumptions have a very simple form. Also the applicability of our assumptions can be very easily confirmed. Remark 3. Theorem 5 and 6 give some additional new results to that of Qian [12, Theorem 1]). The procedures used in the proofs of Theorem 4, 5 and 6 are very clear and comprehensible. References [1] Ahmad, S.; Rama Mohana Rao, M., Theory of ordinary differential equations. With applications in biology and engineering. Affiliated East-West Press Pvt. Ltd., New Delhi, 1999. [2] Antosiewicz, H. A., On non-linear differential equations of the second order with integrable forcing term. J. London Math. Soc. 30, (1955), 64-67. [3] Burton, T. A.; Townsend, C. G., On the generalized Liénard equation with forcing function. J. Differential Equations 4 (1968), 620-633. [4] Constantin, A., Solutions globales d'équations différentielles perturbées. (French) [Global solutions of perturbed differential equations] C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 11, 1319-1322. [5] Constantin, A., Global existence of solutions for perturbed differential equations. Ann. Mat. Pura Appl. (4) 168 (1995), 237-299. [6] Constantin, A., A note on a second-order nonlinear differential system. Glasg. Math. J. 42 (2000), no. 2, 195-199. [7] Čžan, Pan-gin’, On stability with arbitrary initial perturbations of the solutions of a system of two differential equations. (Chinese) Acta Math. Sinica 9 (1959), 442-445. [8] Graef, John R., On the generalized Liénard equation with negative damping. J. Differential Equations 12 (1972), 34-62. [9] Jin, Z., Boundedness and convergence of solutions of a second-order nonlinear differential system. J. Math. Anal. Appl. 256 (2001), no. 2, 360-374. [10] Yin, Z. Global existence and boundedness of solutions to a second order nonlinear differential system. Studia Sci. Math. Hungar. 41 (2004), no. 4, 365-378. [11] Omari, P.; Zanolin, F.On the existence of periodic solutions of forced Liénard differential equations. Nonlinear Anal. 11 (1987), no. 2, 275-284. [12] Qian, C., Boundedness and asymptotic behaviour of solutions of a second-order nonlinear system. Bull. London Math. Soc. 24 (1992), no. 3, 281-288. [13] Tunç, C., Some stability and boundedness results to nonlinear differential equations of Liénard type with finite delay. J. Comput. Anal. Appl. 11(4), (2009), 711-727. [14] Tunç, C., Some new stability and boundedness results of solutions of Liénard type equations with deviating argument. Nonlinear Anal., Hybrid Syst. 4, No. 1, 85-91 (2010). [15] Tunç, C. and Tunç, E., On the asymptotic behavior of solutions of certain second-order differential equations. J. Franklin Inst., Engineering and Applied Mathematics 344 (5), (2007), 391398. [16] Yoshizawa, T., Asymptotic behavior of solutions of a system of differential equations. Contributions to Differential Equations 1 (1963), 371-387. [17] Zhou, J., Necessary and sufficient conditions for boundedness and convergence of a second-order nonlinear differential system. (Chinese) Acta Math. Sinica (Chin. Ser.) 43 (2000), no. 3, 415-420.

JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.6, 1075-1080, 2011, COPYRIGHT 2011 EUDOXUS 1075 PRESS, LLC

WEAKLY COMPACT COMPOSITION OPERATORS ON BERGMAN SPACES OF THE UPPER HALF PLANE ZHI JIE JIANG Abstract. Let Π = {z ∈ C : Imz > 0} denote the upper half plane in the complex plane C. In this paper we prove that there are no weakly compact composition operators on Bergman spaces Ap (Π) for all p (1 ≤ p < ∞).

1. Introduction Let Π = {z ∈ C : Imz > 0} be the upper half plane in the complex plane C and let H(Π) be the space of all analytic functions on Π. Let dA(z) be the area measure on Π. For 1 ≤ p < ∞ the Bergman space Ap (Π) consists of all f ∈ H(Π) such that Z p kf kAp (Π) = |f (z)|p dA(z) < ∞. Π

The Bergman space Ap (Π) with the norm k · kAp (Π) is a Banach space. Let ϕ : Π → Π be an analytic self-map of Π. For f ∈ H(Π), the composition operator Cϕ is defined by Cϕ f (z) = f (ϕ(z)),

z ∈ Π.

For some information of Bergman spaces or weighted Bergman spaces of the upper half plane and some operators on them see, e.g., [4, 6, 8, 9, 10, 13, 14]. During the past few decades, composition operators have been studied extensively on spaces of analytic functions on the unit disk D or the unit ball B. For some recent results see [7, 11, 12, 13, 16, 17]. As a consequence of the Littlewood’s subordination theorem it is well known that every composition operator is bounded on Hardy spaces and weighted Bergman spaces of the open unit disk D. However, if we consider the Hardy space, or the Bergman space of the upper half plane, the situation is entirely different. There do exist unbounded composition operators on these spaces. Moreover, Matache [11] proved that there didn’t exist compact composition operators on Hardy spaces of the upper half plane. Shapiro and Smith [14] also showed that there were no compact composition operators on Bergman spaces of the upper half plane. Once boundedness and compactness have been established, the next natural question one can ask about any composition operator on Bergman space of the upper half plane is: Is it weakly compact? Let X and Y be Banach spaces, L : X → Y be a bounded linear operator. Recall that L : X → Y is weakly compact if it maps bounded sets into relatively weakly compact sets. For some results in this topic see [2] and [5]. Since the Bergman 2000 Mathematics Subject Classification. Primary 47B38; Secondary 47B33, 47B37. Key words and phrases. Composition operator, Bergman space, weakly compact operator. This work was supported by the Scientific Research Fund of School of Science SUSE. 1

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2

ZHI JIE JIANG

space Ap (Π) (1 < p < ∞) is reflexive, the compactness of composition operator on Ap (Π) is equivalent to the weak compactness. Thus, by the results in [14], for the case 1 < p < ∞ we know that there is no weakly compact composition operator on Ap (Π). However, since the space A1 (Π) is not reflexive, this leads us to wonder the question is: Is the compactness of composition operator on A1 (Π) equivalent to the weak compactness? In this paper, we are going to investigate this question and give an affirmative answer. Throughout this paper, constants are denoted by C, they are positive and may differ from one occurrence to the other. The notation A ³ B means that there is a positive constant C such that A/C ≤ B ≤ CA. 2. Auxiliary results In this section we prove several auxiliary results which will be used in the sequel. In order to deal with the weak compactness of composition operator, we need the pseudo-hyperbolic metric on Π. Recall that for w, z ∈ Π the pseudo-hyperbolic metric on Π is defined by |z − w| . ρ(z, w) = |z − w| For 0 < r < 1 and w = x + iy ∈ Π, let D(w, r) = {z ∈ C : ρ(z, w) < r} denote the pseudo-hyperbolic metric disk with center w and radius r. It is easy to see that 2ry 1+r 2 z ∈ D(w, r) if and only if z ∈ B((x, 1−r 2 y), 1−r 2 ), where B(w, r) is the Euclidean disk. For pseudo-hyperbolic metric on Π and some related information, see [8]. Lemma 2.1 ([8, Lemma 4.4]) For 0 < r < 1, there is a sequence (zn )n∈N in Π such that ∪∞ n=1 D(zn , r) = Π and there is a natural number N such that each z ∈ Π belongs to at most N of the sets D(zn , r). Lemma 2.2 Suppose p ≥ 1 and 0 < r < 1, then for any positive Borel measure µ on Π the following conditions are equivalent. (a) There is a constant C1 > 0 such that Z Z |f (z)|p dµ(z) ≤ C1 |f (z)|p dA(z) for all f ∈ Ap (Π); Π

Π

(b) There exists a constant C2 such that µ(D(z, r)) ≤ C2 (Imz)2 for all z ∈ Π. Proof. Assume that condition (a) holds. For w ∈ Π, setting 1 fw (z) = , z ∈ Π, (z − w)4/p we have Z π . |fw (z)|p dA(z) = 4(Imw)2 Π By an easy calculation, we get Z Z Z C1 π p p = C |f (z)| dA(z) ≥ |f (z)| dµ(z) ≥ |fw (z)|p dµ(z) 1 w w 4(Imw)2 Π Π D(w,r) ≥ inf{|fw (z)| : z ∈ D(w, r)}µ(D(w, r)) =

(1 − r)4 µ(D(w, r)). 16(Imw)4

Then µ(D(w, r)) ≤ C2 (Imw)2 and this shows that condition (b) holds.

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WEAKLY COMPACT COMPOSITION OPERATORS

3

Next assume that condition (b) holds. For each f ∈ Ap (Π), by Lemma 2.1 and |D(z, r)| ³ (Imz)2 , we have Z ∞ Z X |f (z)|p dµ(z) ≤ |f (z)|p dµ(z) Π

≤

n=1 ∞ X

D(zn ,r)

sup{|f (z)|p : z ∈ D(zn , r)}µ(D(zn , r))

n=1 ∞ X

Z µ(D(zn , r)) ≤C |f (z)|p dA(z) 2r+1 |D(z , r)| n D(z , ) n 3 n=1 n µ(D(z , r)) oZ n ≤ CN sup :z∈Π |f (z)|p dA(z) |D(zn , r)| Π Z p ≤ C1 |f (z)| dA(z), Π

from which condition (a) holds. If the positive Borel measure µ on Π satisfies the conditions in Lemma 2.2, we say that µ is a Carleson measure for Ap (Π). Note that for each f ∈ Ap (Π) we have Z Z p p kCϕ f kAp (Π) = |f (ϕ(z))| dA(z) = |f (z)|p dA ◦ ϕ−1 (z), Π

Π

−1

and denote A ◦ ϕ by µϕ . Then by Lemma 2.2, it is easy to show that the operator Cϕ : Ap (Π) → Ap (Π) is bounded if and only if µϕ is a Carleson measure for Ap (Π). As the proof of Lemma 2.2, we can prove the following lemma, which is the little oh version of Lemma 2.2. Lemma 2.3 Suppose p ≥ 1 and 0 < r < 1, then for any positive Borel measure µ on Π the following conditions are equivalent. (a) The identity map is compact from Ap (Π) to Lp (Π, dµ); (b) The limit µ(D(z, r)) lim = 0. Imz→0 (Imz)2 If the positive Borel measure µ on Π satisfies the conditions in Lemma 2.3, we say that µ is a vanishing Carleson measure for Ap (Π). Lemma 2.4 ([2, Proposition 1]) Let X, Y , Z be Banach spaces and let T : X → Y , S : X → Z be bounded operators such that kSxk ≤ kT xk. Suppose that there are two linear topologies τ1 on X and τ2 on Y such that T is τ1 −τ2 continuous, (BX , τ1 ) is metrizable and compact and the weak topology of Y is finer than τ2 . If T is weakly compact, then so is S. 3. Main results Here we formulate and prove the main results of this paper. Theorem 3.1 Suppose that the operator Cϕ : A1 (Π) → A1 (Π) is bounded, then the following statements are equivalent: (i) Cϕ : A1 (Π) → A1 (Π) is compact; (ii) Cϕ : A1 (Π) → A1 (Π) is weakly compact;

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4

(iii)

ZHI JIE JIANG

lim

Imz→0

µϕ (D(z,r)) (Imz)2

= 0.

Proof. It is obvious that (i) implies (ii). Now we want to prove that (ii) implies (iii). Assume that the operator Cϕ : A1 (Π) → A1 (Π) is weakly compact. Let τ1 the topology of uniform convergence on compact subsets of Π, τ2 the topology of the pointwise convergence, X = Y = A1 (Π), Z = L1 (Π, µϕ ) and S : A1 (Π) → L1 (Π, µϕ ) given by f 7→ f . Then by Lemma 2.4, it follows that S : A1 (Π) → L1 (Π, µϕ ) is weakly compact. Suppose to the contrary that (iii) is not true. Then there exists ε0 > 0 and (zn )n∈N ⊂ Π with Imzn → 0 such that µϕ (D(zn , r)) ≥ ε0 (Imz)2 . For each n ∈ N, defining 1 fn (z) = , (z − z n )4 we have fn ∈ A1 (Π) and kfn kA1 (Π) = π/4(Imzn )2 . Taking gn = fn /kfn kA1 (Π) , n ∈ N, we need prove that for each subsequence (gnk )k∈N of (gn )n∈N the sequence (Sgnk )k∈N is not weakly convergent in L1 (Π, µϕ ). By [1, p.137], it will enough to show that (Sgnk )k∈N is not uniformly integrable, i.e., there exists ε > 0 such that Rfor every η > 0 there is a measurable subset A of Π such that µϕ (A) ≤ η and A |gnk |dµϕ ≥ ε. Take ε = ε0 and fix an arbitrary η. Since µϕ is a Carleson measure, there is a constant C > 0 such that µϕ (D(z, r)) ≤ C(Imz)2 for all z ∈ Π. Since Imzn → 0 as n → ∞, we can choose k ∈ N such that µϕ (D(znk , r)) ≤ η. On the other hand, for z ∈ D(znk , r), we have |fnk (z)| ≥ (Imznk )−4 . Thus, we get Z (Imznk )−4 (Imznk )−4 µϕ (D(znk , r)) ≥ (Imznk )2 ε0 |gnk (z)|dµϕ (z) ≥ kfnk kA1 (Π) kfnk kA1 (Π) D(zn ,r) k

=

4 ε0 . π

Next, we prove that (iii) implies (i). Assume that (fn )n∈N is a bounded sequence by a positive constant M in A1 (Π) and fn → 0 uniformly on every compact subset of Π as n → ∞. It is enough to show kCϕ fn kA1 (Π) → 0 as n → ∞. Fix a sequence (zn )n∈N in Lemma 2.1. Since |D(zn , r)| ³ (Imzn )2 and condition (iii) holds, then lim

n→∞

µϕ (D(zn , r)) = 0, |D(zn , r)|

as n → ∞. Given ε > 0, there is a positive integer N0 such that µϕ (D(zn , r)) < ε, n→∞ |D(zn , r)| lim

(n ≥ N0 ).

By the proof of Lemma 2.2, there is a constant C > 0 such that Z ∞ Z ∞ X X µ(D(zn , r)) |fk (z)|dµϕ (z) ≤ C |fk (z)|dA(z) |D(zn , r)| D(zn , 2r+1 3 ) n=1 n=N0 D(zn ,r) Z ≤ εC |fk (z)|dA(z) D(zn , 2r+1 3 )

Z

≤ εCN

|fk (z)|dA(z) Π

≤ εCN M.

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WEAKLY COMPACT COMPOSITION OPERATORS

5

for all k ≥ 1. Since lim

k→∞

NX 0 −1 Z n=1

|fk (z)|dµϕ (z) = 0

D(zn ,r)

by uniform convergence, we have Z 0 −1 Z ³ NX lim sup |fk |dµϕ ≤ lim sup k→∞

Π

k→∞

n=1

|fk |dµϕ +

D(zn ,r)

∞ Z X n=N0

´ |fk |dµϕ

D(zn ,r)

≤ εCN M. Because ε is arbitrary, we have

Z

lim

k→∞

|fk (z|dµϕ (z) = 0, Π

and hence Cϕ : A1 (Π) → A1 (Π) is compact. As the proof of Theorem 3.1, the following theorem can be proved similarly. We omit the detail here. Theorem 3.2 Let p > 1 and the operator Cϕ : Ap (Π) → Ap (Π) be bounded. Then the following statements are equivalent: (i) Cϕ : Ap (Π) → Ap (Π) is compact; (ii) Cϕ : Ap (Π) → Ap (Π) is weakly compact; µ (D(z,r)) (iii) lim ϕ(Imz)2 = 0. Imz→0

Remark. By [14], we know that there are no compact composition operators on Ap (Π) for all p (1 ≤ p < ∞). Since Ap (Π) (1 < p < ∞) is a reflexive Banach space, the compactness of composition operator on Ap (Π) is equivalent to the weak compactness. Although the space A1 (Π) is not reflexive, we prove that the compactness of composition operator on A1 (Π) is also equivalent to the weak compactness in Theorem 3.1. Thus, in this paper we obtain the following important fact: There are no weakly compact composition operators on Ap (Π) for all p (1 ≤ p < ∞). References [1] S. Banach, Linear operator, Chelsea, New York, 1932. [2] M. D. Contress, H. Diaz, Weighted composition operators on Hardy spaces, J. Math. Anal. Appl. 263 (2001), 224-233. [3] S. Diaz, Weak compactness in L1 (X, µ), Proc. Amer. Math. Soc. 124 (1996), 2685-2693. [4] P. Duren, E. A. Gallardo-Guti´ errez, and A.Montes-Rodr´ıguez, A Paley -Wiener theorem for Bergman spaces with application to invariant subspaces, Bull. London Math. Soc. 39 (2007), 459-466. [5] Z. J. Jiang, Weighted composition operator on Hardy space H p (BN ), Advances in Mathematics. 37 (6) (2008), 749-754 (In Chinese). [6] Z. J. Jiang, Composition operators from Bergman spaces to some spaces of analytic functions on the upper half-plane, (to appear). [7] Z. J. Jiang, G. F. Cao, Composition operator on Bergman-Orlicz space, Journal of Inequalities and Applications, Volume 2009, Article ID 832686, 14 pages, (2009). [8] S. H. Kang, J. Y. Kim, Toeplitz operators on the weighted Bergman spaces of the half-plane, Bull. Korean Math. Soc. 37 (3) (2000), 437-450. [9] S. H. Kang, J. Y. Kim, The radial derivatives on weighted Bergman spaces, Commun. Korean Math. Soc. 18 (2) (2003), 243-249. [10] S. H. Kang, Berezin transforms and Toeplitz operators on the weighted Bergman spaces of the half-plane, Bull. Korean Math. Soc. 44 (2) (2007), 281-290.

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[11] V. Matache, Composition operators on Hardy spaces of a half-plane, Proc. Amer. Math. Soc. 127 (5) (1999), 1483-1491. [12] S. Li and S. Stevi´ c, Products of Volterra type operator and composition operator from H ∞ and Bloch spaces to Zygmund spaces, Journal of Mathematical Analysis and Applications. vol. 345 (1) (2008), 40-52. [13] M. Nishio, M. Yamada, Carleson type measures on parabolic Bergman spaces, J. Math. Soc. Japan. 58 (1) (2006), 83-96. [14] J. H. Shapiro, W. Smith, Hardy spaces that support no compact composition operators, J. Functional Analysis. 205 (2003), 62-89. [15] S. D. Sharma, A. K. Sharma, and S. Ahmed, Composition operators between Hardy and Bloch-type spaces of the upper half-plane, Bull. Korean Math. Soc. 43 (3) (2007), 475-482. [16] S. Stevi´ c, Composition operators between H ∞ and the α-Bloch spaces on the polydisc, Z. Anal. Anwend. 25 (2006), 457–466. [17] S. Stevi´ c, Composition operators from the Hardy Space to the Zygmund-type space on the upper half-plane, Abstract andApplied Analysis, Volume 2009, Article ID 161528, 8 pages, (2009). Zhi Jie Jiang, School of Science, Sichuan University of Science and Engineering, Zigong, Sichuan, 643000, P. R. China E-mail address: [email protected]

JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.6, 1081-1087, 2011, COPYRIGHT 2011 EUDOXUS 1081 PRESS, LLC

Estimate for Heinz Inequality in the Small Dilatation of Harmonic Quasiconformal Mappings ∗ Jian-Feng Zhu1 1

Xiao-Ming Zeng2

School of M athematical Sciences, Huaqiao U niversity, Quanzhou, 362021, China. E-Mail: [email protected] 2

Department of M athematics, Xiamen U niversity, Xiamen, 361005, China. E-Mail: [email protected]

Abstract. Let w(z) be a harmonic K−quasiconformal self-mapping of the unit disk satisfying w(0) then Heinz inequality can be improved as ª © = 0, 27 , where MK is a strictly decreasing func|wz (0)|2 + |wz¯(0)|2 ≥ max MK , 4π 2 tion of K. Moreover, if K ∈ [1, 1.03], the previous corresponding results in [3] are improved. Keywords. Harmonic mapping; Quasiconformal mapping; Harmonic quasiinvariant measure; Heinz inequality; Schwarz lemma. MR(2000) Subject Classification. 30C62, 30F15, 30C20.

1. Introduction Suppose w(z) = U (x, y) + iV (x, y) is a complex function defined in the unit 2 2 disk D = {z : |z| < 1}. If w(z) is continuous in D and if 4w = ∂∂xw2 + ∂∂yw2 = 0 at every point of D, then w(z) is said to be harmonic in D. Let p(r, x − ϕ) =

1 − r2 , 2π(1 − 2r cos(x − ϕ) + r2 )

denote the poisson kernel, then every bounded harmonic function w defined in D has the following representation Z2π p(r, x − ϕ)f (eix )dx,

w(z) = P [f ](z) =

(1.1)

0

where z = reiϕ ∈ D and f is a bounded integrable function defined on the unit circle ∂D. Since D is a simply connected region, w(z) can also be presented as w(z) = h(z) + g(z), here h(z) and g(z) are both analytic function in D. ∗ This work is supported by Huaqiao University Science Foundation under Grant 08HZR19 and FuJian Provincial Natural Science Foundation of China under Grant 2008J0195.

1

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ZHU, ZENG: HEINZ INEQUALITY

In 1952, E. Heinz [1] proved the following theorem. Theorem A. Let w(z) be a one-to-one harmonic mapping of the unit disk onto itself, normalized by w(0) = 0, then |wz (0)|2 + |wz¯(0)|2 ≥ c,

(1.2)

for some absolute constant c > 0. 4 Subsequently, in 1959, E. Heinz [2] proved the constant c = 1 − 2π 3 + π . He 27 also conjected the sharp value c = 4π2 , and this was finally verified by R. Hall [3] in 1982. Theorem B. Let w(z) = h(z) + g(z) =

∞ P n=1

an z n +

∞ P n=1

bn z n be an univalent

harmonic mapping of the unit disk onto itself, then its coefficients satisfy the inequality 27 |a1 |2 + |b1 |2 ≥ . (1.3) 4π 2 27 The lower bound 4π 2 is best possible. Assuming w(z) = P [f ](z) is an univalent harmonic function defined on D, with boundary function f (eit ) = eiγ(t) . w(z) is called a harmonic K−quasiconformal mapping, if there exist a constant k, satisfy ¯ ¯ ¯ wz¯ ¯ k = ess sup ¯¯ ¯¯ < 1, z∈D wz 1+k where K = 1−k . If we additionally assume that w(z) is a K− quasiconfoemal mapping, then the lower bound in (1.3) is not accurate. In fact, by improving schwarz lemma of harmonic quasiconformal mapping, D. Partyka and K. Sakan proved the following Theorem C in [4] which shown that the constant c in (1.2) can attain 1.

Theorem C. Given K ≥ 1, let w(z) be a harmonic K−quasiconformal self-mapping of D satisfying w(0) = 0, then the inequality ½ ¾ ½ ¾ 1 1 4 2 2 2 |∂z w(z)| + |∂z¯w(z)| ≥ 1 + 2 max ,L , (1.4) 2 K π2 K holds for every z ∈ D. Here LK =

2 π

√ Φ1/K (1/ 2)2

R 0

dt √ √ ΦK ( t)Φ1/K ( 1−t)

is a strictly decreasing function

of K, satisfying lim LK = L1 = 1, lim LK = 0, and function ΦK (s) := K→1 K→∞ ´ ³ , 0 < s < 1. Moreover, L has the following lipschz continuity. µ−1 µ(s) K K Lemma C. For every K ≥ 1, LK is a strictly decreasing function of K ≥ 1, such that |LK2 − LK1 | ≤ L|K2 − K1 |, K1 , K2 ≥ 1, (1.5) 2

ZHU, ZENG: HEINZ INEQUALITY

where the constant L =

4 π (1

1083

+ 65 ln 2).

The above result in (1.4) can be seen as an extended ©form on ª Heinz inequality © ª under harmonic quasiconformal mappings. Set PK = 21 1 + K12 max π42 , L2K , if K = 1, then the mapping w(z) coincides with a rotation, so the left side in (1.4) equals 1, also the right side equals P1 = 1. Thus the lower bound in (1.4) is sharp. However, according to (1.5), we have LK ≥ 1 − L(K − 1). Only when 1 ≤ K ≤ 1 + 1−2/π = 1.0062, then LK > π2 . This shows that Partyk and Skan L improved Heinz inequality [5] only when K ∈ [1, 1.0062]. In this paper, by using the quasi-invariance of harmonic measure and the representation of boundary values in harmonic quasiconformal mappings, a new accurate lower bound MK on (1.3) has been founded in Theorem 3.1. Further27 more, when K nears to 1, MK also be compared with 4π 2 and PK in Theorem 3.2.

2. Auxiliary results The following two auxiliary results play the key role in the proofs of our main Theorems. Lemma 2.1. Let w(z) = P [f ](z) be a harmonic quasiconformal self-mapping of D, with boundary function f (eit ) = eiγ(t) . For every z1 = ei(s+t) , z2 = ei(s−t) ∈ ∂D, set θ = γ(s + t) − γ(s − t), then f (z1 ) = eiθ f (z2 ) and the inequality θ 210−10K sin2K (t) ≤ sin2 ≤ 210−10/K sin2/K (t) 2 holds for every 0 ≤ s, t ≤ 2π. Proof. According to the quasi-invariance of harmonic measure [see 4, (1.9)], for every 0 ≤ s, t ≤ 2π we have θ t t Φ1/K (cos ) ≤ cos ≤ ΦK (cos ). 2 4 2 √ Since Φ2K (x) + Φ21/K ( 1 − x2 ) = 1, for every 0 ≤ x ≤ 1, we obtain t θ t Φ1/K (sin ) ≤ sin ≤ ΦK (sin ). 2 4 2

(2.1)

(2.2)

By the H¨ ubner inequality [cf.4, (1.22), (1.38)], we have s1/K ≤ ΦK (s) ≤ 41−1/K s1/K and 41−K sK ≤ Φ1/K (s) ≤ sK . Now, applying (2.1), (2.2) and the above two inequalities, we have 210−10K sin2K (t) ≤ sin2

θ ≤ 210−10/K sin2/K (t). 2

Lemma 2.2. Assume α > 0, set Zπ/4 cos(2x) sinα (x)dx, Aα = 0

3

(2.3)

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ZHU, ZENG: HEINZ INEQUALITY

Zπ/3 Bα = cos(2x) sinα (x)dx,

(2.4)

π/4

then " √ # √ 1 1 2 α απ 2 α ≤ ( ) − ( ) , 1 + α/2 2 2 8(1 + α) 2 √ √ 2 α 3 1 ≤ ( ) ( − ). 2 4 2

Aα Bα

Proof. Applying integration by parts we know that

Aα

=

Zπ/4 cos(2x) sinα (x)dx 0

=

√ Zπ/4 1 2 α ( ) −α sinα x cos2 xdx 2 2 0

√

=

1 2 α α ( ) − 2 2 2 Ã

hence Aα =

1 1+α/2

√

1 2 α 2( 2 )

−

Zπ/4 Zπ/4 α α sin xdx − sinα x cos(2x)dx, 2 0

α 2

0

π/4 R

! sinα xdx . Since

0

√ 2 2 π x

≤ sin x ≤ x, for

every x ∈ [0,³π/4], then ´ h √ √ √ 1+α 1 2 α α 2 2 α (π/4) 2 α 1 1 1 Aα ≤ 1+α/2 = 1+α/2 2( 2 ) − 2 ( π ) 1+α 2( 2 ) − Applying integration by substitution we have

Bα

=

Zπ/3 cos(2x) sinα (x)dx π/4 π/12 Z

sin(2x) sinα (x + π/4)dx

= − 0

√

π/12 Z 2 α ≤ −( ) sin(2x)dx 2 0 √ √ 2 α 3 1 ) ( − ). = ( 2 4 2

4

i √ απ 2 α 8(1+α) ( 2 ) .

ZHU, ZENG: HEINZ INEQUALITY

1085

3. Main results Theorem 3.1. Let w(z) = P [f ](z) be a harmonic K− quasiconformal selfmapping onto D satisfying w(0) = 0, then the following inequality |wz (0)|2 + |wz¯(0)|2 ≥ max{MK ,

27 } 4π 2

(3.1)

n √ 3 10−10/K holds. Here MK = 3 π 3 − 16 A2/K + 43 210−10K B2K + π 42 satisfies lim MK = M1 = 1.

√

3 1+1/K 3 10−10/K ( 4 ) 2 2 2+2/K

K→1

Proof. Let w(z) = P [f ](z) = h(z) + g(z), where h(z) = ∞ P n=1

∞ P n=1

an z n , g(z) =

bn z n , its boundary function f (eit ) = eiγ(t) . An application of Parserval’s

relation [cf. 6, P67 ] leads to the expression 1 2π

Z2π i[γ(s+t)−γ(s−t)]

e

ds =

∞ X

(|an |2 + |bn |2 )e2int

n=1

0

for arbitrary t ∈ R. Taking real parts, we arrive at the formula 1 − 2J(t) =

∞ X

(|an |2 + |bn |2 ) cos(2nt),

(3.2)

n=1

where 1 J(t) = 2π

µ

Z2π sin2 0

γ(s + t) − γ(s − t) 2

¶ ds.

Since w(z) is a harmonic quasiconformal mapping, by Lemma 2.1 we have 210−10K sin2K (t) ≤ J(t) ≤ 210−10/K sin2/K (t). Let  cos2 t − cos2 ( π3 + t), 0 ≤ t ≤ π6      π π cos2 t − cos2 ( 2π M (t) = 3 − t), 6 ≤ t ≤ 3      0 , π3 ≤ t ≤ π2 applying (3.2), then 8 π

√ ∞ Zπ/2 ¡ ¢ 3 3X 1 2 2 M (t)(1 − 2J(t))dt = |a1 | + |b1 | − |a3n |2 + |b3n |2 π n=1 9n2 − 1 0

≤ |a1 |2 + |b1 |2 . Since 8 π

√ Zπ/2 3 3 M (t)dt = , π 0

5

(3.3)

(3.4)

o ,

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ZHU, ZENG: HEINZ INEQUALITY

and # √ Zπ/2 Zπ/3" 3 3 M (t)J(t)dt = cos(2t) + sin t cos t J(t)dt 4 2 0

=

≤

3 4

0

Zπ/4 0

3 cos(2t)J(t)dt + 4 Zπ/4

cos(2t) sin 0

√

=

0

π/4

3 (10−10/K) ·2 4

3 (10−10/K) ·2 2

+

√ Zπ/3 Zπ/3 3 cos(2t)J(t)dt + sin t cos tJ(t)dt 2

2/K

3 tdt + · 2(10−10K) 4

Zπ/3 cos(2t) sin2K tdt

π/4

Zπ/3 cos t sin(2/K+1) tdt 0

√ 3 3 10−10/K 3 10−10/K (3/4)1+1/K ·2 · A2/K + · 210−10K · B2K + ·2 · , 4 4 2 2 + 2/K

by (3.3) and (3.4), we have Zπ/2 Zπ/2 8 16 |wz (0)| + |wz¯(0)| ≥ M (t)dt − M (t)J(t)dt π π 0 0 " # √ √ 3 3 16 3 10−10/K 3 10−10K 3 10−10/K (3/4)1+1/K ≥ − 2 A2/K + 2 B2K + 2 π π 4 4 2 2 + 2/K 2

2

= MK . (3.5) 27 According to (1.3) and (3.5), we have |wz (0)| + |wz¯(0)| ≥ max{MK , 4π 2 }. √ 1 π 5 3 π 1 Particularly, if K → 1, A2 = 4 − 16 , B2 = 32 − 48 − 4 , hence M1 = √ √ 16 3 3 π 3 3 π − π ( 16 − 16 ) = 1. Here, the lower bound 1 is best possible. 2

2

Theorem 3.2. Let MK as the definition of the above, then MK is a con27 tinuous decreasing function of K. Moreover, when 1 ≤ K ≤ 1.03, MK > 4π 2. Proof. Applying B2K < 0, we can easily obtain that 210−10/K A2/K , 210−10K B2K 1+1/K

are all strictly increasing functions of K, hence by (3.5) and 210−10/K (3/4) 2+2/K we obtain that MK is strictly decreasing of K. √ i h function √ √ √ 1 1 2 α απ According to Lemma 2.2, Aα ≤ 1+α/2 2 ( 2 ) − 8(1+α) ( 22 )α , Bα ≤ ( 22 )α ( 43 − 1 2 ).

Hence

MK

=

" # √ √ 3 10−10K 3 10−10/K (3/4)1+1/K 3 3 16 3 10−10/K − 2 A2/K + 2 B2K + 2 π π 4 4 2 2 + 2/K 6

ZHU, ZENG: HEINZ INEQUALITY

1087

√ µ ¶ 3 3 16 3 10−10/K 1 π(1/2)1/K − { 2 (1/2)1+1/K − π π 4 1 + 1/K 4(K + 2) √ √ 3 10−10K 3 3 10−10/K (3/4)1+1/K + 2 (1/2)K ( − 1/2) + 2 } 4 4 2 2 + 2/K = NK . ≥

27 By direct calculating we obtain that if 1 ≤ K ≤ 1.03, then MK ≥ NK > 4π 2. Finally, we compare our result (3.1) with (1.4) by the following table.

K PK ≥ MK ≥

1.03 0.3937 0.6877

1.0062 0.4026 0.8859

1.0029 0.6868 0.9118

1 1 1

From the above table we can see that MK is more closer to 1 than PK when K closes to 1. This shows that our result is better than (1.4) at z = 0.

References [1] E. Heinz, u ¨ber die L¨ osungen der Minimalfl¨ achengleichung, Nachr. Akad. wiss. G¨ ottir. Math. -Phys. kl. 51-56 (1952). [2] E. Heinz, On one-to-one harmonic mappings, pacific J. math. 9, 101-105 (1959). [3] P. R. Hall, On an inequality of E. Heina, J. Analyse Math. 42, 185-198 (1982/83). [4] D. Partyka and K. Sakan, On an asymptotically sharp variant of heinz’s inequality, Ann. Acad. Sci. Fenn. Math. 30, 167-182 (2005). [5] E. Heinz, On one-to-one harmonic mappings, pacific J. math. 62, 1-16 (1988). [6] P. Duren, Harmonic mappings in the plane, Cambridge university press 2004. [7] P. Duren, Theory of H p Spaces, Academic Press, New York, 32-52, 1970. [8] J. Clunie and T. Sheil-Small , Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A. I. Math. 9, 3-25 (1984). [9] G. Choquet, Sur une type de transformation analytique g´en´eralisant la representation conforme et d´efinie au moyen function harmoniques, Bull. Sci. Math. 69(2), 156-165 (1945). [10] H. Lewy, On the non-vanishing of the Jocobian in certain one-to-one mappings, Bull. Am. Math. Soc. 42, 689-692 (1932). [11] M. Pavlovic, Boundary correspondence under harmonic quasiconformal homeomorphisma of the unit disk, Ann. Sci.Fenn. Math. 27, 365-372 (2002).

7

JOURNAL1088 OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.6, 1088-1096, 2011, COPYRIGHT 2011 EUDOXUS PRESS, LLC

Superstability of homomorphisms and derivations on C ∗ −algebras: A fixed point approach 1 1,2

3

S. Kaboli Gharetapeh, 2 M. Aghaei and 3 T. Karimi

Department of Mathematics, Payame Noor University, Mashhad Branch, Mashhad, Iran e-mail: [email protected]

Department of Mathematics, Payame Noor University, Fariman Branch, Fariman, Iran e-mail: karimi [email protected]

Abstract. Let A be a unital C ∗ −algebras, B be a Banach algebra and let X be a Banach A−module. By using fixed pint methods, we prove that: i) Every almost linear mapping h : A −→ B which satisfies h(2n uy) = h(2n u)h(y) for all u ∈ A+ , all y ∈ A, and all n = 0, 1, 2, ..., is a homomorphism. ii) Every almost linear continuous mapping d : A −→ X is a derivation when d(2n uy) = d(2n u)y + 2n ud(y) holds for all u ∈ A+ , all y ∈ A, and all n = 0, 1, 2, ... .

1. Introduction S. M. Ulam [46], in 1940 proposed the following problem: ”Given a group G1 , a metric group (G2 , d) and a positive number , does there exist a δ > 0 such that if a function f : G1 −→ G2 satisfies the inequality d(f (xy), f (x)f (y)) < δ for all x, y ∈ G1, then there exists a homomorphism T : G1 → G2 such that d(f (x), T (x)) <  for all x ∈ G1 ?”. This problem solved in the next year for the Cauchy functional equation on Banach spaces by D.H. Hyers [30]. It gave rise to the stability theory for functional equations. Subsequently, various approaches to the problem have been introduced by several authors. There are cases in which each ‘approximate function’ is actually a ‘true function’. In such cases, we call the equation is superstable. In 1978, Th. M. Rassias [41] formulated and proved the following theorem, which implies Hyers’ Theorem as a special case. Suppose that E and F are real normed spaces with F a complete normed space, f : E → F is a mapping such that for each fixed x ∈ E the mapping t 7−→ f (tx) is continuous on R, and let there exist ε > 0 and p ∈ [0, 1) such that kf (x + y) − f (x) − f (y)k ≤ ε(kxkp + kykp )

(1.1)

for all x, y ∈ E. Then there exists a unique linear mapping T : E → F such that kf (x) − T (x)k ≤

2ε kxkp 2 − 2p

(1.2)

for all x ∈ E. The case of the existence of a unique additive mapping had been obtained by T. Aoki [1]. Th.M. Rassias [41] was the first to prove that there exists a unique linear mapping T satisfying (1.2). 0 0

2000 Mathematics Subject Classification. Primary 39B52; Secondary 39B82; 46HXX. Keywords: Alternative fixed point; homomorphism, derivation.

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2

In 1990, Th.M. Rassias [42] during the 27th International Symposium on Functional Equations asked the question whether such a theorem can also be proved for p ≥ 1. In 1991, Gajda [28] gave an affirmative solution to this question for p > 1 by following the same approach as in Rassias’ paper [41]. It was proved by Gajda [28], as well as by Th.M. Rassias ˇ and Semrl [44] that one cannot prove a Rassias type theorem when p = 1. In 1994, P. G˘ avruta [29] provided a generalization of Rassias’ theorem in which he replaced the bound ε(kxkp + kykp ) in ([41]) by a general control function ϕ(x, y). The paper of Th.M. Rassias [41] has provided a lot of influence in the development of what we now call the generalized Hyers– Ulam stability of functional equations. During the last decades several stability problems for various functional equations have been investigated by many mathematicians; we refer the reader to the monographs [5, 31, 32, 33, 35, 43]. (See also [2], [7]– [11], [16]– [26], [27], [36],[37] and [38]). We recall a fundamental result in fixed point theory (see [40, 45] for more details). Theorem 1.1. (The alternative of fixed point [3]). 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 m x, T m+1 x) = ∞ for all m ≥ 0, or other exists a natural number m0 such that d(T m x, T m+1 x) < ∞ for all m ≥ m0 ; the sequence {T m x} is convergent to a fixed point y ∗ of T ; y ∗ is the unique fixed point of T in the set Λ = {y ∈ Ω : d(T m0 x, y) < ∞}; d(y, y ∗ ) ≤

1 d(y, T y) 1−L

for all y ∈ Λ.

Throughout this paper, let A be a unital C ∗ −algebra with unit e, B a unital Banach algebra, and let X be a Banach A−module and let G(A) be the set of invertible elements in A. B.E. Johnson (Theorem 7.2 of [34]) investigated almost algebra ∗−homomorphisms between Banach ∗−algebras. Recently, C. Park, D.-H. Boo and J.-S. An [39] investigated almost homomorphisms between unital C ∗ −algebras. In this paper, by using the fixed point methods, we prove that every almost linear mapping h : A −→ B is a homomorphism when h(2n uy) = h(2n u)h(y) holds for all u ∈ A+ , all y ∈ A, and all n = 0, 1, 2, ... , and every almost linear continuous mapping d : A −→ X is a derivation when d(2n uy) = d(2n u)y + 2n ud(y) holds for all u ∈ A+ , all y ∈ A, and all n = 0, 1, 2, ... . In the other words, we prove the superstability of homomorphisms and derivations on unital C ∗ −algebras by fixed point methods. 2. Main results We start our work with the following theorem which investigate almost homomorphisms on unital C ∗ −algebras. Theorem 2.1. Let f : A → B be a mapping such that f (0) = 0 and that f (2n uy) = f (2n u)f (y)

(2.1)

for all u ∈ A+ , all y ∈ A, and all n = 0, 1, 2, ... . Let φ : A2 → [0, ∞) be a function such that

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Superstability of homomorphisms and derivations on C ∗ −algebras...

3

x+y x−y ) + µf ( ) − f (µx)k ≤ φ(x, y) (2.2) 2 2 for all µ ∈ T and all x, y ∈ A. Suppose that there exists an L < 1 such that φ(x, y) ≤ n 2Lφ( x2 , y2 ) for all x, y ∈ A. If limn f (22n e) ∈ G(B), then the mapping f : A → B is a homomorphism. kµf (

Proof. It follows from φ(x, y) ≤ 2Lφ( x2 , y2 ) that lim 2−j φ(2j x, 2j y) = 0

(2.3)

x k2f ( ) − f (x)k ≤ φ(x, 0) 2

(2.4)

1 1 k f (2x) − f (x)k ≤ φ(2x, 0) ≤ Lφ(x, 0) 2 2

(2.5)

j

for all x, y ∈ A. Put µ = 1, y = 0 in (2.2) to obtain

for all x ∈ A. Hence,

for all x ∈ A. Consider the set X := {g | g : A → B} and introduce the generalized metric on X: d(h, g) := inf {C ∈ R+ : kg(x) − h(x)k ≤ Cφ(x, 0)∀x ∈ A}. It is easy to show that (X, d) is complete. Now we define the linear mapping J : X → X by J(h)(x) =

1 h(2x) 2

for all x ∈ A. By Theorem 3.1 of [3], d(J(g), J(h)) ≤ Ld(g, h) for all g, h ∈ X. It follows from (2.5) that d(f, J(f )) ≤ L. By Theorem 1.1, J has a unique fixed point in the set X1 := {h ∈ X : d(f, h) < ∞}. Let H be the fixed point of J. H is the unique mapping with H(2x) = 2H(x) for all x ∈ A satisfying there exists C ∈ (0, ∞) such that kH(x) − f (x)k ≤ Cφ(x, 0) for all x ∈ A. On the other hand we have limn d(J n (f ), D) = 0. It follows that 1 f (2n x) = H(x) 2n for all x ∈ A. It follows from (2.2), (2.3) and (2.6) that x−y x+y ) + H( ) − H(x)k kH( 2 2 1 = lim n kf (2n−1 (x + y)) + f (2n−1 (x − y)) − f (2n x)k n 2 1 ≤ lim n φ(2n x, 2n y) n 2 =0 lim n

(2.6)

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S. Kaboli Gharetapeh, M. Aghaei and T. Karimi

4

for all x, y ∈ A. So x+y x−y ) + H( ) = H(x) 2 2 , t = x−y in above equation, we get for all x, y ∈ A. Put z = x+y 2 2 H(

H(z) + H(t) = H(z + t)

(2.7)

for all z, t ∈ A. Hence, H is Cauchy additive. By putting y = x in (2.2), we have kµf (

2x ) − f (µx)k ≤ φ(x, x) 2

for all x ∈ A. It follows that 1 1 kf (2µ2m x) − 2µf (2m x)k ≤ lim m φ(2m x, 2m x) = 0 m 2 2m for all µ ∈ T, and all x ∈ A. One can show that the mapping H : A → B is C−linear. Now, we show that H is a homomorphism. To this end, let u ∈ A+ , y ∈ A. Then by linearity of H and (2.1), we have kH(2µx) − 2µH(x)k = lim m

H(uy) = lim n

f (2n uy) f (2n u) = lim[ f (y)] = H(u)f (y) n 2n 2n

(2.9)

for all u ∈ A+ , all y ∈ A. Since H is additive, then by (2.9), we have 2n H(uy) = H(u(2n y)) = H(u)f (2n y) for all u ∈ A+ and all y ∈ A. Hence, H(uy) = lim[H(u) n

f (2n y) ] = H(u)H(y) 2n

(2.10)

for all u ∈ A+ and all y ∈ A. P Now, let x ∈ A. Then there exist u1 , u2 , u3 , u4 ∈ A+ such that x = 4j=1 uj . It follows from (2.10) that 4 4 X X H(xy) = H( uj y) = H(uj y) j=1

=

j=1

4 X

4 X (H(uj )H(y))

j=1

j=1

(H(uj y)) =

4 X

= H(

uj )H(y)

j=1

= H(x)H(y) for all y ∈ A. This means that H is a homomorphism. On the other hand, we have H(e) = lim n

f (2n e) ∈ G(B). 2n

Hence, it follows from (2.9) and (2.10) that H(e)H(y) = H(ye) = H(e)f (y) for all y ∈ A. Since H(e) is invertible, then H(y) = f (y) for all y ∈ A. This means that f is a homomorphism, and the proof of theorem is complete. 

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Superstability of homomorphisms and derivations on C ∗ −algebras...

5

Corollary 2.2. Let p ∈ (0, 1), θ ∈ [1, ∞) be real numbers. Let f : A → B be a mapping such that f (0) = 0 and that f (2n uy) = f (2n u)f (y) for all u ∈ A+ , all y ∈ A, and all n = 0, 1, 2, .... Suppose that kµf (

x+y x−y ) + µf ( ) − f (µx)k ≤ θ(kxkp + kykp ) 2 2

for all µ ∈ T and all x, y ∈ A. If limn homomorphism.

f (2n e) 2n

∈ G(B), then the mapping f : A → B is a

Proof. It follows from Theorem 2.1, by putting φ(x, y) := θ(kxkp + kykp ) all x, y ∈ A and L = 2p−1 .  Corollary 2.3. Let p ∈ (0, 21 ), θ ∈ [1, ∞) be real numbers. Let f : A → B be a mapping such that f (0) = 0 and that f (2n uy) = f (2n u)f (y) for all u ∈ A+ , all y ∈ A, and all n = 0, 1, 2, .... Suppose that kµf (

x+y x−y ) + µf ( ) − f (µx)k ≤ θ(kxkp kykp ) 2 2

for all µ ∈ T and all x, y ∈ A. If limn homomorphism.

f (2n e) 2n

∈ G(B), then the mapping f : A → B is a

Proof. It follows from Theorem 2.1, by putting φ(x, y) := θ(kxkp kykp ) all x, y ∈ A and L = 22p−1 .  Now, we use Theorem 2.1 to investigate the derivations on unital C ∗ −algebras. Theorem 2.4. Let A be a C ∗ −algebra. Let f : A → X be a mapping such that f (0) = 0 and that f (2n uy) = f (2n u)y + (2n u)f (y)

(2.11)

+

for all u ∈ A , all y ∈ A, and all n = 0, 1, 2, ... . Suppose there exists a function φ : A2 → [0, ∞) such that x−y x+y ) + µf ( ) − f (µx)k ≤ φ(x, y) (2.12) 2 2 for all µ ∈ T and all x, y ∈ A. If there exists an L < 1 such that φ(x, y) ≤ 2Lφ( x2 , y2 ) for all n x, y ∈ A. Also, if limn f (22n e) = 0, then the mapping f : A → X is a derivation. kµf (

Proof. It is easy to show that X ⊕1 A is a Banach algebra equipped with the following `1 -norm k(x, a)k = kxk + kak

(a ∈ A, x ∈ X)

and the product (x1 , a1 )(x2 , a2 ) = (x1 · a2 + a1 · x2 , a1 a2 )

(a1 , a2 ∈ A, x1 , x2 ∈ X) .

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S. Kaboli Gharetapeh, M. Aghaei and T. Karimi

6

The algebra X ⊕1 A is called a module extension Banach algebra (see [12]– [15]). Now, we define the mapping ϕf : A → X ⊕1 A by a 7→ (f (a), a). It follows from (2.11) that ϕf (2n uy) = (f (2n uy), 2n uy) = (2n uf (y) + f (2n u)y, 2n uy) = (f (2n u), 2n u)(f (y), y) = ϕf (2n u)ϕf (y) for all u ∈ A+ , all y ∈ A, and all n = 0, 1, 2, ... . On the other hand, by (2.12), we get x−y x+y ) + µϕf ( ) − ϕf (µx)k kµϕf ( 2 2 x+y x+y x−y x−y = kµ(f ( ), ) + µ(f ( ), ) − (f (µx), µx)k 2 2 2 2 x+y x−y = kµf ( ) + µf ( ) − f (µx)k 2 2 ≤ φ(x, y) for all µ ∈ T and all x, y ∈ A. It follows from assumption that ϕf (2n e) (f (2n e), 2n e) = lim = (0, e) = 1X⊕1 A ∈ G(X ⊕1 A). n n 2 2n It follows from Theorem 2.1 that ϕf is a homomorphism. Hence, we have lim n

k(f (xy), xy) = ϕf (xy) = varphif (x)ϕf (y) = (f (x), x)(f (y), y) = (f (x)y + xf (y), xy) for all x, y ∈ A. So we conclude that f (xy) = f (x)y + xf (y) for all x, y ∈ A. This implies that f is a derivation.  Corollary 2.5. Let p ∈ (0, 1), θ ∈ [1, ∞) be real numbers. Let f : A → X be a mapping such that f (0) = 0 and that f (2n uy) = f (2n u)y + (2n u)f (y) for all u ∈ A+ , all y ∈ A, and all n = 0, 1, 2, .... Suppose t that x−y x+y ) + µf ( ) − f (µx)k ≤ θ(kxkp + kykp ) 2 2 n for all µ ∈ T and all x, y ∈ A. If limn f (22n e) = 0, then the mapping f : A → X is a derivation. kµf (

Proof. It follows from Theorem 2.4, by putting φ(x, y) := θ(kxkp + kykp ) all x, y ∈ A and L = 2p−1 .  Corollary 2.6. Let p ∈ (0, 21 ), θ ∈ [1, ∞) be real numbers. Let f : A → X be a mapping such that f (0) = 0 and that f (2n uy) = f (2n u)y + 2n uf (y) for all u ∈ A+ , all y ∈ A, and all n = 0, 1, 2, .... Suppose that kµf (

x+y x−y ) + µf ( ) − f (µx)k ≤ θ(kxkp kykp ) 2 2

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Superstability of homomorphisms and derivations on C ∗ −algebras... for all µ ∈ T and all x, y ∈ A. If limn derivation.

f (2n e) 2n

7

= 0, then the mapping f : A → X is a

Proof. It follows from Theorem 2.4, by putting φ(x, y) := θ(kxkp kykp ) all x, y ∈ A and L = 22p−1 . 

References [1] T. Aoki, On the stability of the linear transformationin Banach spaces, J. Math. Soc. Japan 2 (1950) 64-66. [2] M. Bavand Savadkouhi, Gordji, M. Eshaghi, J. M. Rassias, and N. Ghobadipour, Approximate ternary Jordan derivations on Banach ternary algebras, J. Math. Phys. 50 (2009), no. 4, 042303, 9 pp. [3] L. C˘ adariu and V. Radu, On the stability of the Cauchy functional equation: a fixed point approach, Grazer Mathematische Berichte 346 (2004), 43–52. [4] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59–64. [5] S. Czerwik, Functional equations and inequalities in several variables, World Scientific, New Jersey, London, Singapore, Hong Kong, 2002. [6] P.W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76–86. [7] M. Eshaghi Gordji, Stability of an additive-quadratic functional equation of two variables in F -spaces, J. Nonlinear Sci. Appl. 2 (2009), no. 4, 251–259. [8] M. Eshaghi Gordji and M. B. Savadkouhi, Stability of mixed type cubic and quartic functional equations in random normed spaces, J. Inequal. Appl. 2009, Art. ID 527462, 9 pp. [9] M. Eshaghi Gordji, M. Bavand Savadkouhi and C. Park, Quadratic-quartic functional equations in RN-spaces, J. Inequal. Appl. 2009, Art. ID 868423, 14 pp. [10] M. Eshaghi Gordji, S. Abbaszadeh and C. Park, On the stability of a generalized quadratic and quartic type functional equation in quasi-Banach spaces, J. Inequal. Appl. 2009, Art. ID 153084, 26 pp. [11] M. Eshaghi Gordji and M. Bavand Savadkouhi, Approximation of generalized homomorphisms in quasi-Banach algebras, An. Stiint. Univ. “Ovidius” Constanta Ser. Mat. 17 (2009), no. 2, 203–213. [12] M. Eshaghi Gordji and M. Filali, Arens regularity of module actions, Studia Math. 181 (2007), no. 3, 237–254. [13] M. Eshaghi Gordji and M. Filali, Weak amenability of the second dual of a Banach algebra,. Studia Math. 182 (2007), no. 3, 205–213. [14] M. Eshaghi Gordji, F. Habibian and B. Hayati, Ideal amenability of module extensions of Banach algebras, Arch. Math. (Brno) 43 (2007), no. 3, 177–184. [15] M. Eshaghi Gordji, F. Habibian and A. Rejali, Ideal amenability of module extension Banach algebras Int. J. Contemp. Math. Sci. 2 (2007), no. 5-8, 213–219. [16] M. Eshaghi Gordji, J. M. Rassias and M. Bavand Savadkouhi, Approximation of the quadratic and cubic functional equations in RN-spaces, Eur. J. Pure Appl. Math. 2 (2009), no. 4, 494–507. [17] M. Eshaghi Gordji, S. Zolfaghari, J. M. Rassias and M. B. Savadkouhi, Solution and stability of a mixed type cubic and quartic functional equation in quasi-Banach spaces, Abstr. Appl. Anal. 2009, Art. ID 417473, 14 pp. [18] M. Eshaghi Gordji and M. Bavand Savadkouhi, On approximate cubic homomorphisms, Adv. Difference Equ. 2009, Art. ID 618463, 11 pp.

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[19] M. Eshaghi Gordji, A. Ebadian and S. Zolfaghari, Stability of a functional equation deriving from cubic and quartic functions, Abstr. Appl. Anal. 2008, Art. ID 801904, 17 pp. 39B82 [20] M. Eshaghi Gordji, T. Karimi and S. Kaboli Gharetapeh, Approximately n-Jordan homomorphisms on Banach algebras, J. Inequal. Appl. 2009, Art. ID 870843, 8 pp. [21] M. Eshaghi Gordji, M. B. Ghaemi, H. Majani and C. Park, Generalized UlamHyers Stability of Jensen Functional Equation in Serstnev PN Spaces, Journal of Inequalities and Applications, vol. 2010, Article ID 868193, 14 pages, 2010. doi:10.1155/2010/868193. [22] M. Eshaghi Gordji and H. Khodaei, On the generalized Hyers-Ulam-Rassias stability of quadratic functional equations, Abstr. Appl. Anal. 2009, Art. ID 923476, 11 pp. [23] M. Eshaghi Gordji, J. M. Rassias and N. Ghobadipour, Generalized Hyers-Ulam stability of generalized (N, K)-derivations, Abstr. Appl. Anal. 2009, Art. ID 437931, 8 pp. [24] M. Eshaghi Gordji and H. Khodaei, Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spaces, Nonlinear Anal. (TMA), 71 (2009), no. 11, 5629–5643. [25] M. Eshaghi Gordji, S. Kaboli, S. Zolfaghari and J. M. Rassias, Solution and stability of a mixed type additive, quadratic, and cubic functional equation, Adv. Difference Equ. 2009, Art. ID 826130, 17 pp. [26] M. Eshaghi,M. S. Moslehian, S. Kaboli and S. Zolfaghari, Stability of a mixed type additive, quadratic, cubic and quartic functional equation, Nonlinear Analysis and Variational Problems Publisher Springer New York DOI 10.1007/978-1-4419-0158-3 Copyright 2010 ISBN 978-1-4419-0178-1 (Print) 978-1-4419-0158-3 Part 1, pp. 65-80. [27] R. Farokhzad and S. A. R. Hosseinioun, Perturbations of Jordan higher derivations in Banach ternary algebras: An alternative fixed point approach, Internat. J. Nonlinear Anal. Appl. 1 (2010), 42–53. [28] Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), 431–434. [29] P. G˘ avruta, A generalization of the Hyers–Ulam–Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436. [30] D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222–224. [31] D.H. Hyers, G. Isac and Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkh¨ auser, Basel, 1998. [32] G. Isac and Th. M. Rassias, On the Hyers–Ulam stability of ψ-additive mappings, J. Approx. Theory 72 (1993), 131–137. [33] G. Isac and Th.M. Rassias, Functional inequalities for approximately additive mappings, Stability of mappings of Hyers-Ulam type, 117–125, Hadronic Press Collect. Orig. Artic., Hadronic Press, Palm Harbor, FL, 1994. [34] B.E. Johnson, Approximately multiplicative maps between Banach algebras, J. London Math. Soc. (2) 37 (1988) 294-316. [35] S.-M. Jung, Hyers–Ulam–Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, Florida, 2001. [36] H. Khodaei and Th.M. Rassias, Approximately generalized additive functions in several variables, Int. J. Nonlinear Anal. Appl. 1 (2010), 22–41. [37] C. Park, On the stability of the orthogonally quartic functional equation, Bull. Iranian Math. Soc. 31 (2005), no. 1, 63–70. [38] W.-G. Park and J.-H. Bae, On a bi-quadratic functional equation and its stability, Nonlinear Anal. 62 (2005), no. 4, 643–654.

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[39] C. Park, D.-H. Boo and J.-S. An, Homomorphisms between C ∗ −algebras and linear derivations on C ∗ −algebras, J. Math. Anal. Appl. 337 (2008), no. 2, 1415–1424. [40] V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), 91–96. [41] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [42] Th.M. Rassias, Problem 16 ; 2, Report of the 27th International Symp.on Functional Equations, Aequationes Math. 39 (1990), 292–293; 309. [43] Th.M. Rassias (ed.), Functional equations, inequalities and applications, Kluwer Academic Publishers, Dordrecht, Boston and London, 2003. ˇ [44] Th.M. Rassias and P. Semrl, On the behaviour of mappings which do not satisfy Hyers– Ulam stability, Proc. Amer. Math. Soc. 114 (1992), 989–993. [45] I. A. Rus, Principles and Applications of Fixed Point Theory, Ed. Dacia, Cluj-Napoca, 1979 (in Romanian). [46] S.M. Ulam, Problems in Modern Mathematics, Chapter VI, Science Editions, Wiley, New York, 1964.

JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.6, 1097-1105, 2011, COPYRIGHT 2011 EUDOXUS PRESS, 1097 LLC

STABILITY OF TERNARY QUADRATIC DERIVATION ON TERNARY BANACH ALGEBRAS S. Shagholi, M. E. Gordji, M. Bavand Savadkouhi Department of Mathematics, Semnan University, P. O. Box 35195-363, Semnan, Iran Abstract. Let A be a ternary Banach algebra with norm k.kA and B be a ternary Banach algebra with norm k.kB . A mapping D : A → B is called a ternary quadratic derivation if D be a quadratic function, 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, we investigate ternary quadratic derivation on ternary Banach algebras, associated with the following functional equation f (x + y) + f (x − y) = 2f (x) + 2f (y). Moreover, we prove the generalized Hyers-Ulam-Rassias stability of ternary quadratic derivations on ternary Banach algebras.

1. Introduction The study of stability problems for functional equations is related to a question of Ulam [52] concerning the stability of group homomorphisms and affirmatively answered for Banach spaces by Hyers [38]. Subsequently, the result of Hyers was generalized by Aoki [3] for additive mappings and by Th. M. Rassias [48] for linear mappings by considering an unbounded Cauchy difference. The paper of Th. M. Rassias has provided a lot of influence in the development of what we now call a generalized Hyers–Ulam stability of functional equations. We refer the interested readers for more information on such problems to the papers [7],[37],[40] and [47]. In 1991, Z. Gajda [25] answered the question for the case p > 1, which was rased by Rassias. This new concept is known as Hyers–Ulam–Rassias stability of functional equations (see [26] and [39]). The functional equation f (x + y) + f (x − y) = 2f (x) + 2f (y)

(1.1)

is related to symmetric bi-additive function. It is natural that this equation is called a quadratic functional equation. In particular, every solution of the quadratic equation (1.1) is said to be a quadratic function. It is well known that a function f between real vector spaces is quadratic if and only if there exists a unique symmetric bi-additive function B such that f (x) = B(x, x) for all x (see [2] and [41]). The bi-additive function B is given by B(x, y) =

1 (f (x + y) − f (x − y)) 4

Hyers–Ulam–Rassias stability problem for the quadratic functional equation (1.1) was proved by Skof for functions f : A −→ B, where A is normed space and B Banach space (see [50]). Cholewa [6] noticed that the Theorem 0

2000 Mathematics Subject Classification: ———-. Keywords: Generalized Hyers–Ulam–Rassias stability; Quadratic functional equation; ternary Banach algebra; ternary quadratic derivation. 0 E-mail: [email protected],maj [email protected],[email protected] 0

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of Skof is still true if relevant domain A is replaced an abelian group. In the paper [7] , Czerwik proved the Hyers–Ulam–Rassias stability of the equation (1.1). Grabiec [36] has generalized these result mentioned above. For more detailed definitions of such terminologies, we can refer to [8]–[17] A nonempty set G with a ternary operation [., ., .] : G3 → G is called a ternary groupoid and is denoted by (G, [., ., .]). The ternary groupoid (G, [., ., .]) is called commutative if [x1 , x2 , x3 ] = [xδ(1) , xδ(2) , xδ(3) ] for all x1 , x2 , x3 ∈ G and all permutations δ of {1, 2, 3}. If a binary operation ◦ is defined on G such that [x, y, z] = (x◦y)◦z for all x, y, z ∈ G, then we say that [., ., .] is derived from ◦. We say that (G, [., ., .]) is a ternary semigroup if the operation [., ., .] is associative, i.e., if [[x, y, z], u, v] = [x, [y, z, u], v] = [x, y, [z, u, v]] holds for all x, y, z, u, v ∈ G (see Ref. [5]). Definition 1.1. A mapping D : A → B is called a ternary quadratic derivation on ternary Banach algebras if (1) D be a quadratic function, (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. Recently, M. Bavand Savadkouhi, M. Eshaghi Gordji, J. M. Rassias and N. Ghobadipour in [4], proved Approximate ternary Jordan derivations on Banach ternary algebras. For more detailed definitions of such terminologies, we can refer to [1], [8]–[24], [27]–[35], [42]–[46] and [51]. The main purpose in this paper is to offer the Generalized Hyers–Ulam-Rassias stability of ternary quadratic derivations on ternary Banach algebras associated with the following functional equation: f (x + y) + f (x − y) = 2f (x) + 2f (y).

2. Stability of ternary quadratic derivations We investigate ternary quadratic derivations on ternary Banach algebras. Let A be a ternary Banach algebra with norm k.kA and B be a ternary Banach algebra with norm k.kB . Theorem 2.1. Let f : A → B be a mapping for which there exists a function φ : A × A × A → [0, ∞) such that ∞ X 1 φ(2j x, 2j y, 2j z) < ∞ j 4 j=0

(2.1)

kf (x + y) + f (x − y) − 2f (x) − 2f (y)kB ≤ φ(x, y, 0),

(2.2)

kf ([x, y, z]) − [f (x), y 2 , z 2 ] − [x2 , f (y), z 2 ] − [x2 , y 2 , f (z)])kB ≤ φ(x, y, z),

(2.3)

for all x, y, z ∈ A. Then there exists a unique ternary quadratic derivation D : A → B such that 1˜ kf (x) − D(x)kB ≤ φ(x, x, 0), 4 for all x ∈ A. Here, ∞ X 1 ˜ y, z) := φ(x, φ(2j x, 2j y, 2j z), j 4 j=0 for all x, y, z ∈ A.

(2.4)

(2.5)

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STABILITY OF TERNARY QUADRATIC DERIVATION...

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Proof. By 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 f (2x) φ(x, x, 0) − f (x)kB ≤ , (2.6) 4 4 for all x ∈ A. Now we use the Rassias’ method on inequality (2.6) ([25]). One can use induction on n to show that k

k

n−1 1 X φ(2j x, 2j x, 0) f (2n x) − f (x)k ≤ , B 22n 4 j=0 4j

(2.7)

for all x ∈ A and all non-negative integers n. Hence, n+m−1 f (2n+m x) f (2m x) 1 X φ(2j x, 2j x, 0) k 2(n+m) − kB ≤ , 22m 4 j=m 4j 2

(2.8)

for all non-negative integers n and m with n ≥ m and all x ∈ A. It follows from the convergence (2.1) that the n

x) sequence { f (2 22n } is Cauchy. Due to the completeness of B, this sequence is convergent. So one can define the

mapping D : A → B by Set f (2n x) n→∞ 22n n n for all x ∈ A. Replacing x, y by 2 x, 2 y, respectively, in (2.2) and multiply both sides of (2.2) by

(2.9)

D(x) := lim

1 22n ,

we get

kD(x + y) + D(x − y) − 2D(x) − 2D(y)kB 1 kf (2n (x + y)) + f (2n (x − y)) − 2f (2n x) − 2f (2n y)kB 22n φ(2n x, 2n y, 0) =0 ≤ lim n→∞ 22n for all x, y ∈ A and all non-negative integers n. So = lim

n→∞

D(x + y) + D(x − y) = 2D(x) + 2D(y),

(2.10)

for all x, y ∈ A. Moreover, it follows from (2.7) and (2.9) that kf (x) − D(x)kB ≤

1˜ φ(x, x, 0) 4

for all x ∈ A. It follows from (2.3) we get 1 43n n n n n n 2 n 2 n 2 n kf ([2 x, 2 y, 2 z]) − [f (2 x), (2 y) , (2 z) ] − [(2 x) , f (2 y), (2 z) ] − [(2n x)2 , (2n y)2 , f (2n z)]kB kD([x, y, z]) − [D(x), y 2 , z 2 ] − [x2 , D(y), z 2 ] − [x2 , y 2 , D(z)]kB ≤ lim

n→∞ n 2

φ(2n x, 2n y, 2n z) n→∞ 43n for all x, y, z ∈ A. So ≤ lim

(2.11)

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. Now, let D0 : A → B be another ternary quadratic derivation satisfying (2.4). Then we have kD(x) − D0 (x)kB =

1 kD(2n x) − D0 (2n x)kB 22n

1 (kD(2n x) − f (2n x)kB + kf (2n x) − D0 (2n x)kB ) 22n 2 ≤ 2n φ(2n x, 2n x, 0) 2 ≤

(2.12)

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S. Shagholi, M. E. Gordji, M. Bavand Savadkouhi

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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 → B is a unique ternary quadratic derivation satisfying (2.4).



Theorem 2.2. Let f : A → B be a mapping for which there exists a function φ : A × A × A → [0, ∞) such that ∞ X

43j φ(

j=0

x y z , , ) 2. Proof. Assume p < 2. One can use induction on n to show that k

n−1 f (2n x) θ X j(p−2) − f (x)k ≤ 2 kxkp , B 22n 2 j=0

(2.30)

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for all x ∈ A and all non-negative integers n. Hence, k

n+m−1 f (2(n+m) x) f (2m x) θ X j(p−2) − k ≤ 2 kxkp , B 22m 2 j=m 22(n+m)

(2.31) n

x) for all non-negative integers n and m with n ≥ m and all x ∈ A. It follows from p < 2 that the sequence { f (2 22n }

is Cauchy. Due to the completeness of B, this sequence is convergent. So one can define the mapping D : A → B by Set f (2n x) n→∞ 22n n n for all x ∈ A. Replacing x, y by 2 x), 2 y, respectively, in (2.27) and multiply both sides of (2.27) by D(x) := lim

(2.32) 1 22n ,

we get

kD(x + y) + D(x − y) − 2D(x) − 2D(y)kB f (2n x + 2n y) f (2n x − 2n y) f (2n x) f (2n y) + − 2 − 2 kB n→∞ 22n 22n 22n 22n ≤ lim 2n(p−2) θ(kxkp + kykp )  0

= lim k n→∞

for all x, y ∈ A and all non-negative integers n. So D(x + y) + D(x − y) = 2D(x) + 2D(y), for all x, y ∈ A. Moreover, it follows from (2.30) and (2.32) that ∞

kf (x) − D(x)kB ≤

θ X j(p−2) 2 kxkp 2 j=0

for all x ∈ A. It follows from (2.27) we get 1 43n n n n n n 2 n 2 n 2 n kf ([2 x, 2 y, 2 z]) − [f (2 x), (2 y) , (2 z) ] − [(2 x) , f (2 y), (2 z) ] − [(2n x)2 , (2n y)2 , f (2n z)]kB

kD([x, y, z]) − [D(x), y 2 , z 2 ] − [x2 , D(y), z 2 ] − [x2 , y 2 , D(z)]kB ≤ lim

n→∞ n 2

≤ lim 2n(p−6) θ(kxkp + kykp + kzkp )

(2.33)

n→∞

for all x, y, z ∈ A. So n → ∞ in (2.33) and by p < 2, we have 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. Now, let D0 : A → B be another ternary quadratic derivation satisfying (2.28). Then we have kD(x) − D0 (x)kB =

1 kD(2n x) − D0 (2n x)kB 22n

1 (kD(2n x) − f (2n x)kB + kf (2n x) − D0 (2n x)kB ) 22n ∞ ∞ X X ≤ 2(n+j)(p−2) θkxkp ≤ 2j(p−2) θkxkp ≤

j=0

j=n

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 → B is a unique ternary quadratic derivation satisfying (2.28). Similarly, one obtains the result for the case p > 2. The following corollary is the Hyers–Ulam stability of the functional equation (1.1).



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7

Corollary 2.4. Assume  be nonnegative real numbers, and let f : A → B be a mapping such that kf (x + y) + f (x − y) − 2f (x) − 2f (y)kB ≤ ,

(2.34)

kf ([x, y, z]) − [f (x), y 2 , z 2 ] − [x2 , f (y), z 2 ] − [x2 , y 2 , f (z)])kB ≤ ,

(2.35)

for all x, y, z ∈ A. Then there exists a unique ternary quadratic derivation D : A → B such that kf (x) − D(x)kB ≤ ,

(2.36)

holds for all x ∈ X. Proof. In Corollary 2.3, putting p := 0 and  := θ2 , we obtain the conclusion of the corollary.



References [1] S. Abbaszadeh, Intuitionistic fuzzy stability of a quadratic and quartic functional equation, Int. J. Nonlinear Anal. Appl. 1 (2010),2, 100–124. [2] J. Aczel and J. Dhombres, Functional equations in several variables, Cambridge Univ. Press, 1989. [3] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan. 2 (1950), 64–66. [4] M. Bavand Savadkouhi, M. Eshaghi Gordji, J. M. Rassias and N. Ghobadipour, Approximate ternary Jordan derivations on Banach ternary algebras, J. Math. Phys, 50 (2009), 9 pages. [5] N. Bazunova, A. Borowiec and R. Kerner, Universal differential calculus on ternary algebras, Lett. Math. Phys, 67, (2004). [6] P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76–86. [7] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59–64. [8] A. Ebadian, A. Najati, M. E. Gordji, On approximate additive–quartic and quadratic–cubic functional equations in two variables on abelian groups, Results Math., DOI 10.1007/s00025-010-0018-4 (2010). [9] M. Eshaghi Gordji, Stability of a functional equation deriving from quartic and additive functions, Bull. Korean Math. Soc., Vol. 47, No.3, 2010, 491–502. [10] M. Eshaghi Gordji, Stability of an additive-quadratic functional equation of two variables in F–spaces , Journal of Nonlinear Sciences and Applications, Vol 2, No 4,(2009) pp.251–259. [11] M. Eshaghi Gordji, S. Abbaszadeh and C. Park, On the stability of generalized mixed type quadratic and quartic functional equation in quasi-Banach spaces, J. Ineq. Appl., Volume 2009 (2009), Article ID 153084, 26 pages. [12] M. Eshaghi Gordji, M. Bavand Savadkouhi, Approximation of generalized homomorphisms in quasi-Banach algebras , Analele Univ. Ovidius Constata, Math series, Vol. 17(2), 2009, 203–214. [13] M. Eshaghi Gordji, M. Bavand Savadkouhi and C. Park, Quadratic-quartic functional equations in RN-spaces , J. Ineq. Appl., Volume 2009 (2009), Article ID 868423, 14 pages. [14] M. Eshaghi Gordji, M. Bavand Savadkouhi, N. Ghobadipour, A. Ebadian and C. Park, Approximate ternary Jordan homomorphisms on Banach ternary algebras, Abs. Appl. Anal., 2010, Art. ID:467525. [15] M. Eshaghi Gordji, M. Bavand Savadkouhi and M. Bidkham, Stability of a mixed type additive and quadratic functional equation in non-Archimedean spaces, Journal of Computational Analysis and Applications, VOL.12, No.2, 2010, 454–462. [16] M. Eshaghi Gordji and A. Bodaghi, On the stability of quadratic double centralizers on Banach, , To appear in Journal of Computational Analysis and Applications. [17] M. Eshaghi Gordji, M. B. Ghaemi, S. Kaboli Gharetapeh, S. Shams, A. Ebadian, On the stability of J ∗ derivations, Journal of Geometry and Physics 60(3) (2010), 454–459. [18] M. Eshaghi Gordji, M. B. Ghaemi, H. Majani, Generalized Hyers-Ulam-Rassias Theorem in Menger Probabilistic Normed Spaces, Discrete Dynamics in Nature and Society, 2010, Article ID 162371, 11 pages.

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[19] M. Eshaghi Gordji, M. B. Ghaemi, H. Majani, C. Park, Generalized Ulam-Hyers Stability of Jensen Functional Equation in erstnev PN Spaces, J. Ineq. Appl., 2010, Article ID 868193, 14 pages. [20] M. Eshaghi Gordji, M. Ghanifard, H. Khodaei and C. Park, A fixed point approach to the random stability of a functional equation driving from quartic and quadratic mappings, Discrete Dynamics in Nature and Society, 2010, Article ID: 670542. [21] M. Eshaghi Gordji, N. Ghobadipour, Stability of (α, β, γ)−derivations on Lie C ∗ −algebras, To appear in International Journal of Geometric Methods in Modern Physics (IJGMMP). [22] M. Eshaghi Gordji and M. B. Savadkouhi, Stability of cubic and quartic functional equations in nonArchimedean spaces, Acta Appl. Math. 110 (2010) 1321-1329. [23] M. Eshaghi Gordji and M. B. Savadkouhi, Stability of a mixed type cubic and quartic functional equations in random normed spaces , J. Ineq. Appl., Volume 2009 (2009), Article ID 527462, 9 pages. [24] M. Eshaghi Gordji and M. B. Savadkouhi, Stability of a mixed type cubicquartic functional equation in non-Archimedean spaces, Appl. Math. Lett. 23, No.10, (2010), 1198-1202. [25] Z. Gajda, On stability of additive mappings, Internat. J. Math. Sci. 14 (1991), 431–434. [26] P. Gˇavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl, 184 (1994), 431–436. [27] M. E. Gordji, H. Khodaei, Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi–Banach spaces, Nonlinear Analysis–TMA 71 (2009) 5629–5643. [28] M. E. Gordji, H. Khodaei and R. Khodabakhsh, General quartic–cubic–quadratic functional equation in non–Archimedean normed spaces, U.P.B. Sci. Bull., Series A, Vol. 72, Iss. 3, 2010, 69-84. [29] M. E. Gordji, H. Khodaei, On the Generalized Hyers–Ulam–Rassias Stability of Quadratic Functional Equations, Abs. Appl. Anal., Volume 2009, Article ID 923476, 11 pages. [30] M. E. Gordji, S. Kaboli Gharetapeh, J.M. Rassias and S. Zolfaghari, Solution and stability of a mixed type additive, quadratic and cubic functional equation, Advances in difference equations, Volume 2009, Article ID 826130, 17 pages, doi:10.1155/2009/826130. [31] M. E. Gordji, S. Kaboli Gharetapeh, T. Karimi , E. Rashidi and M. Aghaei, Ternary Jordan derivations on C ∗ −ternary algebras, Journal of Computational Analysis and Applications, VOL.12, No.2, 2010, 463–470. [32] M. E. Gordji and M. S. Moslehian, A trick for investigation of approximate derivations, Math. Commun. 15 (2010), no. 1, 99-105. [33] M. E. Gordji and A. Najati, Approximately J ∗ -homomorphisms: A fixed point approach, Journal of Geometry and Physics 60 (2010), 809–814. [34] M. E. Gordji, M. Ramezani, A. Ebadian and C. Park, Quadratic double centralizers and quadratic multipliers, Advances in Difference Equations, (in press). [35] M. E. Gordji, J.M. Rassias, N. Ghobadipour, Generalized Hyers–Ulam stability of the generalized (n, k)– derivations, Abs. Appl. Anal., Volume 2009, Article ID 437931, 8 pages. [36] A. Grabiec, The generalized Hyers-Ulam stability of a class of functional equations, Publ. Math. Debrecen. 48 (1996), 217–235. [37] D. H. Hyers, G. Isac and Th. M. Rassias, Stability of functional equations in several variables, Birkha˘er, Basel, (1998). [38] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci, 27 (1941), 222–224. [39] G. Isac and Th. M. Rassias, On the Hyers–Ulam stability of ψ-additive mappings,J. Approx. Theory, 72 (1993), 131–137. [40] S. Jung, HyersUlamRassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, (2001). [41] Pl. Kannappan, Quadratic functional equation and inner product spaces,Results Math, 27 (1995), 368–372. [42] H. Khodaei and M. Kamyar, Fuzzy approximately additive mappings, Int. J. Nonlinear Anal. Appl. 1 (2010),2, 44–53.

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[43] H. Khodaei and Th. M. Rassias, Approximately generalized additive functions in several variables, Int. J. Nonlinear Anal. Appl. 1 (2010),1, 22–41. [44] C. Park and M. Eshaghi Gordji, Comment on Approximate ternary Jordan derivations on Banach ternary algebras [Bavand Savadkouhi et al. J. Math. Phys. 50, 042303 (2009)], J. Math. Phys. 51, 044102 (2010) (7 pages). [45] C. Park and A. Najati, Generalized additive functional inequalities in Banach algebras, Int. J. Nonlinear Anal. Appl. 1 (2010),2, 54–62. [46] C. Park and Th.M. Rassias, Isomorphisms in unital C ∗ -algebras, Int. J. Nonlinear Anal. Appl. 1 (2010),2, 1–10. [47] Th. M. Rassias, Functional Equations, Inequalities and Applications, Kluwer Academic Publishers, Dordrecht, Boston and London, (2003). [48] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [49] S. Rolewicz, Metric Linear Spaces, PWN-Polish Sci. Publ., Reidel, Dor-drecht, (1984). [50] F. Skof, Propriet locali e approssimazione di operatori,Rend. Sem. Mat. Fis. Milano, 53 (1983), 113-129. [51] S. Shakeri, R. Saadati and C. Park, Stability of the quadratic functional equation in non-Archimedean L−fuzzy normed spaces, Int. J. Nonlinear Anal. Appl. 1 (2010),2, 72–83. [52] S. M. Ulam, Problems in modern mathematics, Chapter VI, science ed., Wiley, New York, (1940).

JOURNAL 1106 OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.6, 1106-1114, 2011, COPYRIGHT 2011 EUDOXUS PRESS, LLC

NEARLY TERNARY CUBIC HOMOMORPHISM IN TERNARY ´ FRECHET ALGEBRAS S. Shagholi, M. E. Gordji, M. Bavand Savadkouhi Department of Mathematics, Semnan University, P. O. Box 35195-363, Semnan, Iran Abstract. Let A, B be two ternary algebras. A mapping H : A → B is called a ternary cubic homomorphism if H is a cubic function, which satisfies: H([x, y, z]) = [H(x), H(y), H(z)] for all x, y, z ∈ A. In this paper, we investigate ternary cubic homomorphisms on ternary Fr´echet algebras.

1. Introduction The stability problem of functional equations originated from a question of Ulam [39] in 1940, concerning the stability of group homomorphisms. In 1941, D. H. Hyers [30] gave a first affirmative answer to the question of Ulam for Banach spaces. In 1978, Th. M. Rassias [38] provided a generalization of Hyers’ Theorem which allows the Cauchy difference to be unbounded. In 1991, Z. Gajda [20] answered the question for the case p > 1, which was rased by Rassias. This new concept is known as Hyers–Ulam–Rassias stability of functional equations (see [1],[2], [21],[22]–[28],[29],[31],[33]–[37] ). Jun and Kim [32] introduced the following functional equation f (2x + y) + f (2x − y) = 2f (x + y) + 2f (x − y) + 12f (x)

(1.1)

and they established the general solution and the generalized Hyers–Ulam–Rassias stability for the functional equation (1.1). The function f (x) = x3 satisfies the functional equation (1.1), which is thus called a cubic functional equation. Every solution of the cubic functional equation is said to be a cubic function. Jun and Kim proved that a function f between real vector spaces X and Y is a solution of (1.1) if and only if there exists a unique function C : X × X × X → Y such that f (x) = C(x, x, x) for all x ∈ X, and C is symmetric for each fixed one variable and is additive for fixed two variables. For more detailed definitions of such terminologies, we can refer to [4],[6],[7],[10]–[14],[16] and [18]. Definition 1.1. A mapping H : A → B is called a ternary cubic homomorphism between ternary algebras A, B if (1) H is a cubic function, (2) H([x, y, z]) = [H(x), H(y), H(z)], for all x, y, z ∈ A. 0

2000 Mathematics Subject Classification:46K05;39B82;47B47. Keywords: Generalized Hyers–Ulam–Rassias stability; Cubic functional equation; Fr´echet spaces; ternary cubic homomorphism. 0 E-mail: [email protected],maj [email protected],[email protected] 0

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S. Shagholi, M. E. Gordji, M. Bavand Savadkouhi

2

Recently, M. Eshaghi Gordji and M. Bavand Savadkouhi, in [9], investigated approximate cubic homomorphisms on Banach algebras. For more detailed definitions of such terminologies, we can refer to [3],[8],[15],[17] and [19].

Definition 1.2. A topological vector space X is a Fr´echet space if it satisfies the following three properties: (1) it is complete as a uniform space, (2) it is locally convex, (3) its topology can be induced by a translation invariant metric, i.e. a metric d : X × X → R such that d(x, y) = d(x + a, y + a) for all a, x, y ∈ X. For more detailed definitions of such terminologies, we can refer to [5]. Note that a ternary algebra is called ternary Fr´echet algebra is it is a Fr´echet space with a metric d.

2. Stability of ternary cubic homomorphisms we investigate the ternary cubic homomorphisms in ternary Fr´echet algebras. Theorem 2.1. Let A and B be two Fr´echet algebras by metrics d1 and d2 , respectively. Let f : A → B be a mapping for which there exist a function φ : A × A × A → [0, ∞) such that ∞ X 1 φ(2j x, 2j y, 2j z) < ∞ 3j 2 j=0

(2.1)

d2 (f (2x + y) + f (2x − y), 2f (x + y) + 2f (x − y) + 12f (x)) ≤ φ(x, y, 0),

(2.2)

d2 (f ([x, y, z]), [f (x), f (y), f (z)]) ≤ φ(x, y, z),

(2.3)

for all x, y, z ∈ A. Then there exists a unique ternary cubic homomorphism H : A → B such that d2 (f (x), H(x)) ≤

1 ˜ φ(x, 0, 0), 16

(2.4)

for all x ∈ A. Here, ˜ y, z) := φ(x,

∞ X 1 φ(2j x, 2j y, 2j z), 3j 2 j=0

(2.5)

for all x, y, z ∈ A. Proof. By putting x = y = 0 in (2.2), we get f (0) = 0. If we putting y = 0 in (2.2), we get d2 (2f (2x), 16f (x)) ≤ φ(x, 0, 0), for all x ∈ A. Now multiply both sides of (2.6) by d2 (

1 16 ,

(2.6)

we get

f (2x) φ(x, 0, 0) , , f (x)) ≤ 3 2 24

(2.7)

for all x ∈ A. Now we use the Rassias’ method on inequality (2.7) ([20]). One can use induction on n to show that d2 (

n−1 f (2n x) 1 X φ(2j x, 0, 0) , f (x)) ≤ , 23n 16 j=0 23j

(2.8)

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3

for all x ∈ A and all non-negative integers n. Hence, n+m−1 f (2n+m x) f (2m x) 1 X φ(2j x, 0, 0) d2 ( 3(n+m) , )≤ , 23m 16 j=m 23j 2

(2.9)

for all non-negative integers n and m with n ≥ m and all x ∈ A. It follows from the convergence (2.1) that the n

x) sequence { f (2 23n } is Cauchy. Due to the completeness of B, this sequence is convergent. So one can define the

mapping H : A → B by Set f (2n x) n→∞ 23n n n for all x ∈ A. Replacing x, y by 2 x, 2 y, respectively, in (2.2) and multiply both sides of (2.2) by H(x) := lim

(2.10) 1 23n ,

we get

d2 (H(2x + y) + H(2x − y), 2H(x + y) + 2H(x − y) + 12H(x)) 1 d2 (f (2n (2x + y)) + f (2n (2x − y)), 2f (2n (x + y)) + 2f (2n (x − y)) + 12f (2n x)) 23n φ(2n x, 2n y, 0) ≤ lim n→∞ 23n for all x, y ∈ A and all non-negative integers n. Taking the limit as n → ∞ we obtain = lim

n→∞

H(2x + y) + H(2x − y) = 2H(x + y) + 2H(x − y) + 12H(x),

(2.11)

for all x, y ∈ A. Moreover, it follows from (2.8) and (2.10) that d2 (f (x), D(x)) ≤

1 ˜ φ(x, 0, 0) 16

for all x ∈ A. It follows from (2.3) that 1 d2 (f ([2n x, 2n y, 2n z]), [f (2n x), f (2n y), f (2n z)]) n→∞ 29n φ(2n x, 2n y, 2n z) ≤ lim n→∞ 29n

d2 (H([x, y, z]), [H(x), H(y), H(z)]) = lim

(2.12)

for all x, y, z ∈ A. So H([x, y, z]) = [H(x), H(y), H(z)],

(2.13)

for all x, y, z ∈ A. Now, let H 0 : A → B be another ternary cubic homomorphism satisfying (2.4). Then we have d2 (H(x), H 0 (x)) =

1 d2 (H(2n x), H 0 (2n x)) 23n

1 (d2 (H(2n x), f (2n x)) + d2 (f (2n x), H 0 (2n x))) 23n 1 ˜ n ≤ φ(2 x, 0, 0) 8.23n which tends to zero as n → ∞ for all x ∈ A. So we can conclude that H(x) = H 0 (x) for all x ∈ A. This proves the ≤

uniqueness of H. Thus, the mapping H : A → B is a unique ternary cubic homomorphism satisfying (2.4).



Theorem 2.2. Let A and B be two Fr´echet algebras by metrics d1 and d2 , respectively. Let f : A → B be a mapping for which there exist a function φ : A × A × A → [0, ∞) such that ∞ X j=0

29j φ(

x y z , , ) 3.



Corollary 2.6. Let A and B will be two ternary Banach algebras. Suppose p ≥ 0 be given with p 6= 3. Assume θ be nonnegative real numbers, and let f : A → B be a mapping such that kf (2x + y) + f (2x − y) − 2f (x + y) − 2f (x − y) − 12f (x)k ≤ θ(kxkp + kykp ),

(2.45)

kf ([x, y, z]) − [f (x), f (y), f (z)]k ≤ θ(kxkp + kykp + kzkp ),

(2.46)

for all x, y, z ∈ A. Then there exists a unique ternary cubic homomorphism H : A → B such that θ kxkp , kf (x) − H(x)k ≤ 2(8 − 2p )

(2.47)

holds for all x ∈ X, where p < 3, or the inequality kf (x) − H(x)k ≤

θ kxkp , 2(2p − 8)

(2.48)

holds for all x ∈ A, where p > 3. Proof. In Theorem 2.5, by putting d1 (x, y) = kx − yk and d2 (x, y) = kx − yk for all x, y ∈ A, we obtain the conclusion of the Theorem.



The following corollary is the Hyers–Ulam stability of the functional equation (1.2). Corollary 2.7. Let A and B will be two ternary Banach algebras. Assume  be nonnegative real numbers, and let f : A → B be a mapping such that kf (2x + y) + f (2x − y) − 2f (x + y) − 2f (x − y) − 12f (x)k ≤ ,

(2.49)

kf ([x, y, z]) − [f (x), f (y), f (z)]k ≤ ,

(2.50)

for all x, y, z ∈ A. Then there exists a unique ternary cubic homomorphism H : A → B such that kf (x) − H(x)k ≤ ,

(2.51)

1113

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S. Shagholi, M. E. Gordji, M. Bavand Savadkouhi

holds for all x ∈ X. Proof. In Corollary 2.6, by putting p := 0 and  := θ2 , we obtain the conclusion of the corollary.



References [1] S. Abbaszadeh, Intuitionistic fuzzy stability of a quadratic and quartic functional equation, Int. J. Nonlinear Anal. Appl. 1 (2010),2, 100–124. [2] J. Aczel and J. Dhombres, Functional equations in several variables, Cambridge Univ. Press, 1989. [3] M. Bavand Savadkouhi, M. E. Gordji, J. M. Rassias and N. Ghobadipour, Approximate ternary Jordan derivations on Banach ternary algebras, J. Math. Phys. 50, 042303 (2009), 9 pages. [4] A. Ebadian, A. Najati and M. Eshaghi Gordji, On approximate additive–quartic and quadratic–cubic functional equations in two variables on abelian groups, Results Math., DOI 10.1007/s00025-010-0018-4 (2010). [5] M. Eshaghi Gordji, Stability of an additive-quadratic functional equation of two variables in F–spaces , Journal of Nonlinear Sciences and Applications, Vol 2, No 4,(2009) pp. 251–259. [6] M. Eshaghi Gordji and M. Bavand Savadkouhi, Stability of cubic and quartic functional equations in nonArchimedean spaces, Acta Appl. Math. 110 (2010), 1321-1329. [7] M. Eshaghi Gordji and M. Bavand Savadkouhi, Stability of a mixed type cubic-quartic functional equation in non-Archimedean spaces, Appl. Math. Lett. 23, No.10, (2010), 1198–1202. [8] M. Eshaghi Gordji, M. Bavand Savadkouhi, Approximation of generalized homomorphisms in quasi-Banach algebras, Analele Univ. Ovidius Constata, Math series, Vol. 17(2), (2009), 203–214. [9] M. Eshaghi Gordji and M. Bavand Savadkouhi, On approximate cubic homomorphisms, Advances in difference equations, Volume (2009), Article ID 618463, 11 pages ,doi:10.1155/2009/618463. [10] M. Eshaghi Gordji, M. Bavand Savadkouhi, J. M. Rassias and S. Zolfaghari, Solution and stability of a mixed type cubic and quartic functional equation in quasi-Banach spaces, Abs. Appl. Anal., Volume 2009, Article ID 417473, 14 pages doi:10.1155/2009/417473. [11] M. Eshaghi Gordji and M. Bavand Savadkouhi, Stability of a mixed type cubic and quartic functional equations in random normed spaces , J. Ineq. Appl., Volume 2009 (2009), Article ID 527462, 9 pages. [12] M. Eshaghi Gordji, A. Ebadian and S. Zolfaghari, Stability of a functional equation deriving from cubic and quartic functions, Abs. Appl. Anal., Volume (2008), Article ID 801904, 17 pages. [13] M. Eshaghi Gordji, S. Kaboli Gharetapeh, J.M. Rassias and S. Zolfaghari, Solution and stability of a mixed type additive, quadratic and cubic functional equation, Advances in difference equations, Volume 2009, Article ID 826130, 17 pages, doi:10.1155/2009/826130. [14] M. Eshaghi Gordji, S. Kaboli-Gharetapeh, C. Park and S. Zolfaghri, Stability of an additive-cubic-quartic functional equation, Advances in Difference Equations, (2009), Article ID 395693, 20 pages. [15] M. Eshaghi Gordji, T. Karimi and S. Kaboli Gharetapeh, Approximately n–Jordan homomorphisms on Banach algebras, J. Ineq. Appl. Volume 2009, Article ID 870843, 8 pages. [16] M. Eshaghi Gordji and H. Khodaei, Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi–Banach spaces, Nonlinear Analysis–TMA 71 (2009), 5629–5643. [17] M. Eshaghi Gordji and A. Najati, Approximately J ∗ -homomorphisms: A fixed point approach, Journal of Geometry and Physics, 60 (2010), 809–814. [18] M. Eshaghi Gordji, S. Zolfaghari , J. M. Rassias and M. Bavand Savadkouhi, Solution and Stability of a Mixed type Cubic and Quartic functional equation in Quasi–Banach spaces, Abs. Appl. Anal., Volume (2009), Art. ID 417473, 1–14. [19] R. Farokhzad and S. A. R. Hosseinioun, Perturbations of Jordan higher derivations in Banach ternary algebras: An alternative fixed point approach, Int. J. Nonlinear Anal. Appl. 1 (2010),1, 42–53. [20] Z. Gajda, On stability of additive mappings, Internat. J. Math. Sci. 14 (1991), 431–434. [21] P. Gˇavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl, 184 (1994), 431–436.

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[22] P. Gˇ avruta and L. Gˇ avruta, A new method for the generalized Hyers–Ulam–Rassias stability, Int. J. Nonlinear Anal. Appl, 1 (2010), 2, 11–18. [23] M. E. Gordji, H. Khodaei and R. Khodabakhsh, General quartic–cubic–quadratic functional equation in non–Archimedean normed spaces, U.P.B. Sci. Bull., Series A, Vol. 72, Iss. 3, 2010, 69-84. [24] M. E. Gordji, H. Khodaei, On the Generalized Hyers–Ulam–Rassias Stability of Quadratic Functional Equations, Abs. Appl. Anal., Volume 2009, Article ID 923476, 11 pages. [25] M. E. Gordji, S. Kaboli Gharetapeh, J.M. Rassias and S. Zolfaghari, Solution and stability of a mixed type additive, quadratic and cubic functional equation, Advances in difference equations, Volume 2009, Article ID 826130, 17 pages, doi:10.1155/2009/826130. [26] M. E. Gordji, S. Kaboli Gharetapeh, T. Karimi , E. Rashidi and M. Aghaei, Ternary Jordan derivations on C ∗ −ternary algebras, Journal of Computational Analysis and Applications, VOL.12, No.2, 2010, 463–470. [27] M. E. Gordji, M. Ramezani, A. Ebadian and C. Park, Quadratic double centralizers and quadratic multipliers, Advances in Difference Equations, (in press). [28] M. E. Gordji, J.M. Rassias, N. Ghobadipour, Generalized Hyers–Ulam stability of the generalized (n, k)– derivations, Abs. Appl. Anal., Volume 2009, Article ID 437931, 8 pages. [29] D. H. Hyers, G. Isac and Th. M. Rassias, Stability of functional equations in several variables, Birkha˘er, Basel, (1998). [30] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci, 27 (1941), 222–224. [31] G. Isac and Th. M. Rassias, On the Hyers–Ulam stability of ψ-additive mappings,J. Approx. Theory, 72 (1993), 131–137. [32] K. W. Jun and and H. M. Kim, The generalized Hyers–Ulam–Russias stability of a cubic functional equation, J. Math. Anal. Appl. 274 (2002), no. 2, 267–278. [33] H. Khodaei and M. Kamyar, Fuzzy approximately additive mappings, Int. J. Nonlinear Anal. Appl. 1 (2010), 2, 44–53. [34] H. Khodaei and Th. M. Rassias, Approximately generalized additive functions in several variables, Int. J. Nonlinear Anal. Appl. 1 (2010), 1, 22–41. [35] C. Park and M. E. Gordji, Comment on Approximate ternary Jordan derivations on Banach ternary algebras [ Bavand Savadkouhi et al. J. Math. Phys. 50, 042303 (2009)], J. Math. Phys. 51, 044102 (2010) (7 pages). [36] C. Park and A. Najati, Generalized additive functional inequalities in Banach algebras, Int. J. Nonlinear Anal. Appl. 1 (2010),2, 54–62. [37] C. Park and Th.M. Rassias, Isomorphisms in unital C ∗ -algebras, Int. J. Nonlinear Anal. Appl. 1 (2010),2, 1–10. [38] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [39] S. M. Ulam, Problems in modern mathematics, Chapter VI, science ed., Wiley, New York, (1940).

JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.6, 1115-1122, 2011, COPYRIGHT 2011 EUDOXUS PRESS, 1115 LLC

26.02.2010 - submitted ,

ON COMMON FIXED POINT THEOREMS WITHOUT COMMUTING CONDITIONS IN TVS-CONE METRIC SPACES ˘ ¨ ERDAL KARAPINAR, UGUR YUKSEL,

Abstract. In this manuscript, some common fixed point theorems without any commuting conditions investigated in TVS-valued metric spaces.

1. Introduction and Preliminaries Topological vector space valued metric space, namely TVS-cone metric spaces (in short TVS-CMS), is a generalization of the notion of the metric space that is introduced by Du [13]. TVS-cone metric spaces is also a generalization of the notion of the cone metric spaces (in short CMS) that is obtained by replacing real numbers with an ordered real Banach space in the definition of metric (see e.g. [21, 18, 14]). Lately, many results on fixed point theorems have been extended to cone metric spaces (see e.g.[14],[19],[23],[5],[16],[3],[6], [22]). Recently, Du [13] showed that Banach contraction principles in usual metric spaces and in TVS-CMS are equivalent. The author also deduce that generalization of some fixed point theorems (e.g. Kannan type [15], Chatterjea type[9], given in [1, 2, 19]) on usual metric space to TVS-cone metric space can be established easily. In this manuscript, some common fixed point theorems that require no commuting conditions (see [8]) are generalized from real-valued metric space to Banachvalued metric spaces. The method used in [13] is not sufficient to get such simple establishment in the main result of this paper. Throughout this paper, (E, S) stands for real Hausdorff locally convex topological vector space (t.v.s.) with S its generating system of seminorms. A non-empty subset P of E is called cone if P + P ⊂ P , λP ⊂ P for λ ≥ 0 and P ∩ (−P ) = {0}. For a given cone P , one can define a partial ordering (denoted by ≤ or ≤P ) with respect to P by x ≤ y if and only if y − x ∈ P . The notation x < y indicates that x ≤ y and x 6= y while x 0). Definition 1. (See [10], [12], [13]) For c ∈ intP , the nonlinear scalarization function φc : E → R is defined by φc (y) = inf{t ∈ R : y ∈ tc − P }, for all y ∈ E. 2000 Mathematics Subject Classification. 47H10,54H25. Key words and phrases. Common fixed point, non-commuting mapping, TVS-valued metric spaces. Submitted February 26, 2010. 1

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¨ E. KARAPINAR, U. YUKSEL ,

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Lemma 2. (See [10], [12], [13]) For each t ∈ R and y ∈ E, the following are satisfied: (i) φc (y) ≤ t ⇔ y ∈ tc − P , (ii) φc (y) > t ⇔ y ∈ / tc − P , (iii) φc (y) ≥ t ⇔ y ∈ / tc − intP , (iv) φc (y) < t ⇔ y ∈ tc − intP , (v) φc (y) is positively homogeneous and continuous on E, (vi) if y1 ∈ y2 + P , then φc (y2 ) ≤ φc (y1 ), (vii) φc (y1 + y2 ) ≤ φc (y1 ) + φc (y2 ), for all y1 , y2 ∈ E. Definition 3. Let X be a non-empty set and E, as usual, be a Hausdorff locally convex topological space. Suppose a vector-valued function p : X × X → E satisfies: (M 1) 0 ≤ p(x, y) for all x, y ∈ X, (M 2) p(x, y) = 0 if and only if x = y, (M 3) p(x, y) = p(y, x) for all x, y ∈ X (M 4) p(x, y) ≤ p(x, z) + p(z, y), for all x, y, z ∈ X. Then, p is called TVS-cone metric on X, and the pair (X, p) is called a TVS-cone metric space (in short, TVS-CMS). Note that in [14], the authors considered E as a real Banach space in the definition of TVS-CMS. Thus, a cone metric space (in short, CMS) in the sense of Huang and Zhang [14] is a special case of TVS-CMS. Lemma 4. (See [13]) Let (X, p) be a TVS-CMS. Then, dp : X × X → [0, ∞) defined by dp = φc ◦ p is a metric. Remark 5. Since a cone metric space (X, p) in the sense of Huang and Zhang [14], is a special case of TVS-CMS, then dp : X × X → [0, ∞) defined by dp = φc ◦ d is also a metric. Definition 6. (See [13]) Let (X, p) be a TVS-CMS, x ∈ X and {xn }∞ n=1 a sequence in X. (i) {xn }∞ n=1 TVS-cone converges to x ∈ X whenever for every 0 > 0 find δ > 0 and ρ ∈ S such that q(b) < δ implies b n0 and so p(xn , x) n0 . Therefore xn → x in (X, p). (ii) The proof is similar to that in (i).  Lemma 13. (see [4]) Let (X, p) be a TVS-cone metric space over a normal cone of a locally convex space (E, S), where S is the family of seminorms defining the locally convex topology. Let {xn } and {yn } be two sequences in X and xn → x, yn → y. Then p(xn , yn ) → p(x, y) in (E, S).

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Proof. Let ε > 0 and q ∈ S be given. Choose c ∈ E with c >> 0 such that ε q(c) < . Since xn → x and yn → y, one can find an n0 such that for all n > n0 , 6 p(xn , x) m. Then by (2.9) and triangular inequality, one can obtain p(yn , ym ) ≤ p(yn , yn−1 ) + p(yn−1 , yn+1 ) + · · · + p(ym−1 , ym ) ≤ rn p(y0 , y1 ) + rn−1 p(y0 , y1 ) + · · · + rm p(y0 , y1 ) rm ≤ 1−r p(y0 , y1 ).

(2.10)

To show {yn } is a Cauchy sequence take 0 0 rm and q ∈ S such that q(b) < δ ⇒ b 0, then limr→0+ φ(r) = 0. Taking n → ∞ in (2.7), we have lim F (xn , yn ) = F (x, y). n→∞

Noting g(xn+1 ) = F (xn , yn ) and (2.6), we get F (x, y) = g(x). Similarly, we have F (y, x) = g(y). Thus (x, y) is a coupled coincidence point of F and g. Remark 2.1. In Theorem 2.1, F need not commute with g and it is not necessary to be continuous for g. Therefore Theorem 2.1 extends and improves Theorem 2.1 in [14]. Corollary 2.1Suppose that F has the mixed g− comparable property and assume there exists a k ∈ [0, 1) with d(F (x, y), F (u, v)) ≤

k [d(g(x), g(u)) + d(g(y), g(v))] 2

(2.8)

for all x, u, y, v ∈ X for which g(x), g(u) are comparable and g(y), g(v) are comparable. Also suppose g(X) = X and X has the following properties: (i) if a comparable sequence xn → x, then xn , x are comparable for all n; (ii) if a comparable sequence yn → y, then y, yn are comparable for all n; (iii)if {zn } ⊂ X is a comparable sequence, then for any m and l, zm , zl are comparable. If there exist x0 , y0 ∈ X such that g(x0 ), F (x0 , y0 ) are comparable and F (y0 , x0 ), g(y0 ) are comparable, then there exist x, y ∈ X such that F (x, y) = g(x) and F (y, x) = g(y), that is, F and g have a coupled coincidence point. 4

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Proof. Taking φ(t) = k · t with k ∈ [0, 1) in Theorem 2.1, we obtain Corollary 2.1. Recall that (X, ≤) is a partially ordered set and d is a metric on X such that (X, d) is a complete metric space. Further, we endow the product space X × X with the following partial order: for (x, y), (u, v) ∈ X × X, (u, v) ≤ (x, y) ⇔ x ≥ u, y ≤ v. Theorem 2.2. In addition to the hypotheses of Theorem 2.1, suppose that for every (x, y), (y ∗ , x∗ ) ∈ X × X, there exists (u, v) ∈ X × X such that (F (u, v), F (v, u)) is comparable to (F (x, y), F (y, x)) and (F (x∗ , y ∗ ), F (y ∗ , x∗ )), and F is commutable with g, that is, g(F (x, y)) = F (g(x), g(y)) for all x, y ∈ X. Then F and g have a unique coupled common fixed point, that is, there exists a unique (x, y) ∈ X × X such that x = g(x) = F (x, y), andy = g(y) = F (y, x). Proof. Taking a proof similar to that of Theorem 2.2 in [14] and following from Theorem 2.1, we can obtain Theorem 2.2 easily, so we omit its proof. In the rest of this section, we consider the existence and uniqueness of a solution for the equation F (x, x) = g(x). Recall that F : X × X → X has the mixed monotone property if F is monotone non-decreasing in its first argument and is monotone non-increasing in its second argument, that is, for any x, y ∈ X, x1 , x2 ∈ X; x1 ≤ x2 ⇒ F (x1 , y) ≤ F (x2 , y) and y1 , y2 ∈ X; y1 ≤ y2 ⇒ F (x, y1 ) ≥ F (x, y2 ). Theorem 2.3. Let F : X ×X → X be a mapping having the mixed monotone property on X and g : D ⊂ X → X satisfy g(D) = X and the following condition (C): for x, y ∈ D, g(x) ≤ g(y) implies x ≤ y. Assume that there is a function φ : [0, +∞) → [0, +∞) with φ(t) < t and limr→t+ < t for each t > 0 such that ( ) d(g(x), g(u)) + d(g(y), g(v)) d(F (x, y), F (u, v)) ≤ φ , ∀x ≥ u, y ≤ v. (2.9) 2 Assume also that X has the following properties: (i) if a non-decreasing sequence xn → x, then xn ≤ x, ∀n; (ii) if a non-increasing sequence yn → y, then y ≤ yn , ∀n; Moreover, if there exist x0 , y0 ∈ D such that g(x0 ) ≤ F (x0 , y0 ) ≤ F (y0 , x0 ) ≤ g(y0 ). Then, there exists a unique x∗ ∈ K such that F (x∗ , x∗ ) = g(x∗ ), where K = {x ∈ D : g(x0 ) ≤ g(x) ≤ g(y0 )}.

5

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Proof. Similarly as in the proof of Theorem 2.1 in [14], we obtain two sequences {xn }, {yn } in D such that, for each n, g(xn+1 ) = F (xn , yn ), g(yn+1 ) = F (yn , xn )

(2.10)

g(x0 ) ≤ g(x1 ) ≤ · · · ≤ g(xn ) ≤ · · · ≤ g(yn ) ≤ · · · ≤ g(y1 ) ≤ g(y0 ),

(2.11)

and

which together with the condition (C) implies that x0 ≤ x1 ≤ · · · ≤ xn ≤ · · · ≤ yn ≤ · · · ≤ y1 ≤ y0 .

(2.12)

Moreover, {g(xn )} and {g(yn )} are Cauchy sequences in X. Since X is complete and g(D) = X, there exists x, y ∈ D such that lim g(xn ) = g(x) and

n→∞

lim g(yn ) = g(y).

n→∞

(2.13)

It follows from (2.11) and hypothesis (i), (ii) that, for any n, g(xn ) ≤ g(x) and g(y) ≤ g(yn ).

(2.14)

xn ≤ x, y ≤ yn

(2.15)

Thus

as g satisfies the condition (C). On the other hand, from (2.9) and (2.12), we have also d(g(xn ), g(yn )) = d(F (xn−1 , yn−1 ), F (yn−1 , xn−1 )) ≤ φ(d(g(xn−1 ), g(yn−1 ))) ≤ · · · ≤ φn (d(g(x0 ), g(y0 ))). It is known that φ(t) < t and limr→t+ φ(r) < t implies limn→∞ φn (t) = 0 for each t > 0. Hence, noting (2.13), we obtain that g(x) = g(y).

(2.16)

Now, we prove that g is an injection. Indeed, if g(z1 ) = g(z2 ) for z1 , z2 ∈ D, then g(z1 ) ≤ g(z2 ). The condition (C) yields z1 ≤ z2 . Similarly, it holds that z2 ≤ z1 . So z1 = z2 , i.e. g is an injection. It follows from (2.16) that x = y. Set x∗ = x = y, then for each n, it holds that g(xn ) ≤ g(x∗ ) ≤ g(yn ), that is x∗ ∈ K, and xn ≤ x∗ ≤ yn . Moreover, limn→∞ g(xn ) = g(x∗ ) and limn→∞ g(yn ) = g(x∗ ).

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Next we prove that F (x∗ , x∗ ) = g(x∗ ). In fact, from (2.9), we obtain, for each n ≥ 0, that d(F (x∗ , x∗ ), g(x∗ )) ≤ d(F (x∗ , x∗ ), g(xn )) + d(g(xn ), g(x∗ )) = d(F (x∗ , x∗ ), F (xn−1 , yn−1 )) + d(g(xn ), g(x∗ )) ( ) d(g(x∗ ), g(xn−1 )) + d(g(x∗ ), g(yn−1 )) ≤ φ + d(g(xn ), g(x∗ )). 2 Since φ(t) < t and limr→t+ φ(r) < t for each t > 0, then limr→0+ φ(r) = 0. Letting n → ∞ in the above inequality and noting (2.13), we get F (x∗ , x∗ ) = g(x∗ ), that is x∗ is a coincidence point of F and g. Suppose that there exists another element y ∗ ∈ K such that F (y ∗ , y ∗ ) = g(y ∗ ), then similarly as in the proof of g(xn ) ≤ g(x∗ ) ≤ g(yn ), one can show that g(xn ) ≤ g(y ∗ ) ≤ g(yn ), ∀n ≥ 0. Hence, xn ≤ y ∗ ≤ yn , ∀n ≥ 0. Now using (2.9), we have ) ( d(g(xn ), g(y ∗ )) + d(g(y ∗ ), g(yn )) ∗ ∗ ∗ . d(g(xn+1 ), g(y )) = d(F (xn , yn ), F (y , y )) ≤ φ 2 (2.17) Similarly, we can get ( ) d(g(yn ), g(y ∗ )) + d(g(y ∗ ), g(xn )) ∗ ∗ ∗ d(g(yn+1 ), g(y )) = d(F (yn , xn ), F (y , y )) ≤ φ . 2 (2.18) Combining (2.17) and (2.18), we obtain that ( ) d(g(xn+1 ), g(y ∗ )) + d(g(yn+1 ), g(y ∗ )) d(g(yn ), g(y ∗ )) + d(g(y ∗ ), g(xn )) ≤φ . (2.19) 2 2 Since limr→0+ φ(r) = 0 and limn→∞ g(xn ) = g(x∗ ) and limn→∞ g(yn ) = g(x∗ ), taking n → ∞ in (2.19), we get d(g(x∗ ), g(y ∗ )) + d(g(x∗ ), g(y ∗ )) = 0.

(2.20)

Thus g(x∗ ) = g(y ∗ ) and x∗ = y ∗ as g is an injection. This makes end to the proof. Corollary 2.2. Let F : X × X → X be a mapping having the mixed monotone property on X and g : D ⊂ X → X satisfy g(D) = X and the condition (C). Assume that there is a k ∈ [0, 1) with d(F (x, y), F (u, v)) ≤

k [d(g(x), g(u)) + d(g(y), g(v))], ∀x ≥ u, y ≤ v. 2 7

(2.21)

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Assume also that X has the following properties: (i) if a non-decreasing sequence xn → x, then xn ≤ x, ∀n; (ii) if a non-increasing sequence yn → y, then y ≤ yn , ∀n; Moreover, if there exist x0 , y0 ∈ D such that g(x0 ) ≤ F (x0 , y0 ) ≤ F (y0 , x0 ) ≤ g(y0 ). Then, there exists a unique x∗ ∈ K such that F (x∗ , x∗ ) = g(x∗ ), where K = {x ∈ D : g(x0 ) ≤ g(x) ≤ g(y0 )}.

Proof. Taking φ(t) = k · t with k ∈ [0, 1) in Theorem 2.3, we obtain Corollary 2.2. Remark 2.2. Set D = X and g = I in Theorem 2.3, we obtain the unique fixed point of F , where I : X → X is the identity mapping on X.

3

Applications

Let X = [0, +∞), then (X, d) is a complete metric space under the usual metric d(x, y) = |x − y| on R, where R denotes the set of real number. Endowed X with the usual order as x ≤ y ⇔ x − y ≤ 0, then (X, ≤) is a partially ordered set. Suppose now that F (x, y) = x2 + y21+1 and g(x) = 4x2 for any x, y ∈ X, then F : X × X → X is a mixed monotone mapping and g : X → X satisfies g(X) = X and the condition (C). Let u0 = 0, v0 = 1, then g(u0 ) = 0 ≤ F (u0 , v0 ) =

1 ≤ F (v0 , u0 ) = 2 ≤ g(v0 ) = 4. 2

Now for any x, y, u, v ∈ X, x ≥ u, y ≤ v, we have 1 1 2 − u − | y2 + 1 v2 + 1 2 2 v − y 2 2 ≤ |x − u | + 2 (y + 1)(v 2 + 1)

|F (x, y) − F (u, v)| = |x2 +

≤

1 [|g(x) − g(u)| + |g(v) − g(y)| ]. 4

Taking k = 12 , it follows from Corollary 2.2 that there is a unique x∗ in [0, 1] such that F (x∗ , x∗ ) = g(x∗ ).

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References [1] S. Banach, Sur les op´ erations dans les ensembles abstraits et leur application aux e´quations intgrales, Fund. Math. 3, 133-181 (1922) . [2] A. Meir, E. Keeler, A theorem on contraction mappings, J. Math. Anal. Appl. 28, 326-329 (1969) . [3] D. R. Smart, Fixed Point Theorems, Cambridge University Press, Cambridge, 1974. [4] E. Zeidler, Nonlinear Functional Analysis and Its Applications I: Fixed Point Theorems, Springer-Verlag, Berlin, 1986. [5] R. P. Agarwal, M. Meehan, D. O’Regan, Fixed Point Theory and Applications, Cambridge University Press, 2001. [6] A. Branciari, A fixed point theorem for mappings satisfying a general contractive condition of integral type, IJMMS 29, 531-536 (2002). [7] J. Dugundji, A. Granas, Fixed Point Theory, Springer-Verlag, 2003. [8] J. J. Nieto, R. Rodriguez-Lopez, Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta Math. Sinica, Engl. Ser. 23, 2205-2212(2007). [9] R. P. Agarwal, M.A. El-Gebeily, D. O’Regan, Generalized contractions in partially ordered metric spaces, Appl. Anal. 87, 1-8(2008) . [10] A. C. M. Ran, M. C. B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132, 1435-1443(2004). [11] J. Harjani, K. Sadarangani, Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations, Nonlinear Anal. 72, 1188-1197 (2010). [12] J. J. Nieto, R. L. Pouso, R. Rodriguez-Lopez, Fixed point theorems in ordered abstract spaces, Proc. Amer. Math. Soc. 135, 2505-2517 (2007). [13] T. Gnana Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65, 1379-1393 (2006). ´ c, Coupled fixed point theorems for nonlinear contractions in [14] V. Lakshmikantham, L. Ciri´ partially ordered metric spaces, Nonlinear Anal. 70, 4341-4349(2009).

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JOURNAL1132 OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.6, 1132-1142, 2011, COPYRIGHT 2011 EUDOXUS PRESS, LLC

The relaxed Newton-like method for a nonsymmetric algebraic Riccati equation∗ Jian-Lei Lia†, Ting-Zhu Huanga , Zhi-Jiang Zhangb a

School of Mathematics and Sciences, University of Electronic

Science and Technology of China, Chengdu, Sichuan, 611731, PR China. b

College of Minsheng, Henan University, Kaifeng, Henan, 475004, PR China

Abstract The non-symmetric algebraic Riccati equation arising in transport theory can be rewritten as a vector equation and the minimal positive solution of the non-symmetric algebraic Riccati equation can be obtained by solving the vector equation. In this paper, based on the relaxation technique, we propose a relaxed Newton-like method containing a relaxation parameter for solving the vector equation. Some convergence results are presented. The convergence analysis shows that sequence of vectors generated by the relaxed Newton-like method is monotonically increasing and converges to the minimal positive solution of the vector equation. Finally, numerical experiments are reported. Key words: Non-symmetric algebraic Riccati equation; M -matrix; Transport theory; Minimal positive solution; Relaxed Newton-like method. AMSC(2000): 15A24, 65F10, 82C70

1

Introduction

In transport theory, sometimes, it should solve the following integral equation µ ¶ µ ¶µ ¶ Z Z 1 1 1 1 X(ω, ν) 1 1 X(µ, ω) + X(µ, ν) = c 1 + dω 1+ dω , µ+α ν−α 2 −α ω + α 2 α ω−α (1.1) where 0 < c ≤ 1, 0 ≤ α < 1, and X(µ, ν) is a real-valued scattering function on the domain Ω = (µ, ν)|µ, ν) ∈ [−α, 1] × [α, 1], see [1, 27]. By discretizing the equation ∗ †

This research was supported by NSFC (60973015). Corresponding author: E-mail: [email protected].

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(1.1), the following nonsymmetric algebraic Riccati equation (NARE) (see [15–17] and the references cited therein) can be obtained: XCX − XE − AX + B = 0,

(1.2)

where A, B, C, E ∈ Rn×n have the following special form: A = ∆ − eq T , B = eeT , C = qq T , E = D − qeT .

(1.3)

Here and in the following, e = (1, 1, ..., 1)T , q = (q1 , q2 , ..., qn )T with qi = ci /2ωi ,  1    ∆ = diag(δ1 , δ2 , ..., δn ) with δi = cω (1 + α) , i (1.4) 1   D = diag(d , d , ..., d ) with d = ,  1 2 n i cωi (1 − α) and 0 < ωn < ... < ω2 < ω1 < 1,

n X

ci = 1, ci > 0, i = 1, 2, ..., n.

(1.5)

i=1

The form of the Riccati equation (1.2) also arises in Markov models [30] and in nuclear physics [8, 15], and it has many positive solutions in the componentwise sense. There have been a lot of studies about algebraic properties [25,27] and iterative methods for the nonnegative solution of the nonsymmetric algebraic Riccati equations (1.2), including the basic fixed-point iterations [5, 9, 12, 16, 28], the Schur method [19, 29], the Matrix Sign Function method [10, 26], the doubling algorithm [11] and the alternately linearized implicit iteration method [2], and so on; see related references therein. The existence of positive solutions of (1.2) has been shown in [15] and [17], but only the minimal positive solution is physically meaningful. So it is important to develop some effective and efficient procedures to compute the minimal positive solution of Equation (1.2). Recently, Lu [23] has shown that the matrix equation (1.2) is equivalent to a vector equation and has developed a simple and efficient iterative procedure to compute the minimal positive solution of (1.2). The fixed-point iteration methods were further studied in [1, 3] for solving the vector equation. In [1], Bai et. al. proposed two nonlinear splitting iteration methods: the nonlinear block Jacobi and the nonlinear block Gauss-Seidel iteration methods. In [3] Bao et. al. proposed a modified simple iteration method for solving the vector equation. Furthermore, the convergence rates of various fixed-point iterations [1, 3, 23] were determined and compared in [13]. The Newton method has been presented and analyzed by Lu for solving the vector equation in [24]. It has been shown that the Newton method for the vector equation is more simple and efficient than using the corresponding Newton method [12] directly for the original Riccati equation (1.2). And other variable Newton methods for the vector equation were further studied in [21, 22]. 2

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Based on the relaxation or extrapolation technique [1, 6, 7, 14, 20, 31], in this paper, we further study the Newton method and propose a relaxed Newton-like method to solve the vector equation. The convergence analysis shows that the sequence of vectors generated by the relaxed Newton-like method is monotonically increasing and converges to the minimal positive solution of the vector equation, which can be used to obtain the minimal positive solution of the original Riccati equation. For convenience, firstly, we give some definitions and notations. For any matrices A = [ai,j ] and B = [bi,j ] ∈ Rm×n , we write A ≥ B(A > B) if ai,j ≥ bi,j (ai,j > bi,j ) holds for all i, j. The Hadamard product of A and B is defined by A ◦ B = [ai,j · bi,j ]. I denotes the identity matrix with appropriate dimension. A real square matrix A is called a Z-matrix if all its off-diagonal elements are non-positive. Any Z-matrix A can be written as sI − B with B > 0. uT denotes the transpose of a vector u. The following Lemma will be used later. Lemma 1.1 [4] For a Z-matrix A, the following statements are equivalent: (1) A is a nonsingular M -matrix; (2) A is nonsingular and A−1 ≥ 0; (3) Av > 0 for some vector v ≥ 0. The rest of the paper is organized as follows. In Section 2, we review the Newton method and some useful results, and present the relaxed Newton-like method. Some convergence results are given in Section 3. Section 4 and 5 give numerical experiments and conclusions, respectively.

2

The relaxed Newton-like method method

It has been shown in [23, 24] that the solution of (1.1) must have the following form: X = T ◦ (uv T ) = (uv T ) ◦ T, where T = [ti,j ] = [1/(δi + dj )] and u, v are two vectors, which satisfy the vector equations: ( u = u ◦ (P v) + e, (2.1) v = v ◦ (P˜ u) + e, where P = [pi,j ] = [qj /(δi + dj )], P˜ = [˜ pi,j ] = [qj /(δj + di )]. Define w = [uT , v T ]T . The equation (2.1) can be rewritten equivalently as f (w) = w − w ◦ Pw − e = 0, where

· P=

0 P P˜ 0 3

¸ .

(2.2)

LI ET AL: RELAXED NEWTON-LIKE METHOD

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The minimal positive solution of (1.2) can be obtained via computing the minimal positive solution of the vector equation (2.2). The Newton method presented by Lu in [24] for the vector equation (2.2) is the following: wk+1 = wk − f 0 (wk )−1 f (wk ), k = 0, 1, 2... (2.3) where for any w ∈ R2n , the Jacobian matrix f 0 (w) of f (w) is given by · ¸ G1 (v) H1 (u) 0 f (w) = I2n − G(w), with G(w) = H2 (v) G2 (u)

(2.4)

where G1 (v) = diag(P v), G2 (u) = diag(P˜ u), H1 (u) = [u ◦ p1 , u ◦ p2 , ..., u ◦ pn ] and H2 (v) = [v ◦ p˜1 , v ◦ p˜2 , ..., v ◦ p˜n ]. For i = 1, 2, ..., n, pi and p˜i are the ith column of P and P˜ , respectively. Obviously, when w > 0, G(w) ≥ 0 and f 0 (w) is a Z-matrix. In [22], Lin et. al applied the modified Newton method presented in [18] for solving the vector equation (2.2), the modified Newton method is given in [22] as follows: Algorithm 2.1 (The modified Newton method) For k = 0, 1, 2, ..., ( w˜k = wk + f 0 (wk )−1 f (wk ), (2.5) wk+1 = w ˜k − f 0 (wk )−1 f (w˜k ). The numerical experiments has been shown in [22] that the modified Newton method has a better convergence than the Newton method [24]. Based on the relaxation or extrapolation technique [1, 6, 7, 14, 20, 31], here, we introduce a relaxation or extrapolation parameter λ, and give the following relaxed Newton-like method: Algorithm 2.2 (The relaxed Newton-like method) For k = 0, 1, 2, ... and appropriate real parameter |λ| ≤ 1, the relaxed Newton-like method is defined as follows:  0 −1   w¯k = wk + f (wk ) f (wk ), w˜k = (1 − λ)wk + λw¯k , (2.6)   0 −1 wk+1 = w ˜k − f (wk ) f (w˜k ).

Remark 2.1 In fact, the relaxed Newton-like method contains three Newton type methods [21, 22, 24]. When λ = −1, the relaxed Newton-like method becomes the Newton-Shamanskii method [21]. When λ = 0, the relaxed Newton-like method becomes the Newton method [24]. When λ = 1, the relaxed Newton-like method becomes the modified Newton method [22]. Before we give the convergence analysis of the relaxed Newton-like method, let us now state some results which are indispensable for our subsequent discussions.

4

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Lemma 2.1 [24] For any vectors w+ , w ∈ R2n , we have 1 f (w+ ) = f (w) + f 0 (w)(w+ − w) + f 00 (w)(w+ − w, w+ − w). 2

(2.7)

In particular, if w+ = w∗ , the minimal positive solution of (2.2), then 1 0 = f (w) + f 0 (w)(w∗ − w) + f 00 (w)(w∗ − w, w∗ − w). 2

(2.8)

Furthermore, for any y > 0 or y < 0, f 00 (w)y 2 < 0

(2.9)

1 f (w) = f 0 (w)(w − w∗ ) − f 00 (w)(w − w∗ )2 , 2

(2.10)

1 f 0 (w)(w − w∗ ) = f (w) + f 00 (w)(w − w∗ )2 . 2

(2.11)

and f 00 (w)y 2 is independent of w. By (2.8), we have

and

Lemma 2.2 [24] If 0 ≤ w < w∗ and f (w) < 0, then f 0 (w) is a nonsingular M -matrix.

3

Convergence analysis of the relaxed Newton-like method

Now, we analyse the convergence of the relaxed Newton-like method (2.6). Theorem 3.1 Given a vector wk ∈ R2n . wk+1 are obtained by the relaxed Newton-like method (2.6). If 0 ≤ wk < w∗ and f (wk ) < 0, then wk < wk+1 < w∗ and f (wk+1 ) < 0, moreover, f 0 (wk+1 ) is a nonsingular M -matrix. Proof. Since wk < w∗ and f (wk ) < 0, by Lemma 2.2, we can easily obtain that 0 f (wk ) is a nonsingular M -matrix. By Lemma 1.1, we have f 0 (wk )−1 ≥ 0. By Eq. (2.6), we obtain that w˜k = wk + λf 0 (wk )−1 f (wk ), wk+1 = wk + f 0 (wk )−1 [λf (wk ) − f (w˜k )]. Let ek = wk − w∗ , hk = f 0 (wk )−1 f (wk ), rk = f 0 (wk )−1 [λf (wk ) − f (w˜k )], then ek < hk < 0, w˜k = wk + λhk , wk+1 = wk + rk 5

(3.1)

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It follows from (2.7) that f (w˜k ) = f (wk + λhk ) 1 = f (wk ) + λf (wk ) + f 00 (wk )(λhk )2 2 1 00 = (λ + 1)f (wk ) + f (wk )(λhk )2 . 2

(3.2)

By Eq. (3.2), we have 1 λf (wk ) − f (w˜k ) = −f (wk ) − f 00 (wk )(λhk )2 > 0, 2 it means that rk > 0, i.e., wk+1 > wk . By Eqs. (2.9), (2.11), (3.1) and (3.2), we have the following error vectors equation ek+1 = ek + rk 1 = f 0 (wk )−1 [f (wk ) + f 00 (wk )e2k ] + f 0 (wk )−1 [λf (wk ) − f (w˜k )] 2 1 = f 0 (wk )−1 [(λ + 1)f (wk ) − f (w˜k ) + f 00 (wk )e2k ] 2 1 0 = f (wk )−1 f 00 (wk )[e2k − (λhk )2 ] 2 1 ≤ f 0 (wk )−1 f 00 (wk )[e2k − h2k ] < 0. 2 Hence, wk+1 < w∗ . By Eqs. (2.7), (2.11), (3.1) and (3.2), we have f (wk+1 ) = f (wk + rk ) 1 = f (wk ) + f 0 (wk )rk + f 00 (wk )rk2 2 1 = (λ + 1)f (wk ) − f (w˜k ) + f 00 (wk )rk2 2 1 00 1 = f (wk )rk2 − f 00 (wk )(λhk )2 2 2 1 00 ≤ f (wk )[rk2 − h2k ]. 2 It is easy to know that rk + hk = f 0 (wk )−1 [(λ + 1)f (wk ) − f (w˜k )] 1 = f 0 (wk )−1 [− f 00 (wk )(λhk )2 ] > 0, 2 6

(3.3)

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LI ET AL: RELAXED NEWTON-LIKE METHOD

which means that rk2 − h2k > 0. Therefore, f (wk+1 ) < 0. By Lemma 2.2, it can be concluded that f 0 (wk+1 ) is a nonsingular M -matrix. The proof of the theorem is completed. ¤ Remark 3.1 In fact, there always exists the initial vector wk such that 0 ≤ wk < w∗ and f (wk ) < 0, for example wk = 0. The sequence of vectors generated by the relaxed Newton-like method is monotonically increasing and converges to the minimal positive solution of the vector equation. Moreover, the range of the parameter λ may be larger, r 2 e2 the relaxation parameter λ can be chosen such that it satisfies λ2 ≤ min{ hk2 , hk2 }. The k k numerical experiments also show this point.

4

Numerical experiments

In this section, we give numerical experiments to illustrate the effectiveness of the relaxed Newton-like method presented in Section 3. In order to show numerically the feasibility and effectiveness of the relaxed Newton-like method, we list the number of iteration steps (denoted as IT), the CPU time in seconds (denoted as CPU), and relative residual error (denoted as ERR) for different parameter λ. The residual error is defined by ¾ ½ kuk+1 − uk k2 kvk+1 − vk k2 , , ERR = max kuk+1 k2 kvk+1 k2 where k · k2 is the 2-norm for a vector. All the experiments are run in MATLAB 7.0 on a personal computer with Intel(R) Pentium(R) D 3.00GHz CPU and 1 GB memory, and all iterations are terminated once the current iterate satisfies ERR ≤ n · eps, where eps = 1 × 10−16 . In the test example, the constants ci and wi , i = 1, 2, ...n, are given by the numerical quadrature formula on the interval [0, 1], which are obtained by dividing [0, 1] into n4 subintervals of equal length and applying a Gauss-Legendre quadrature with 4 nodes to each subinterval; see the Example 5.2 in [12]. We test several different values (c, α). In Table 1, we list ITs, CPUs and ERRs for the relaxed Newton-like method with different relaxation parameter λ. Figure 1 and Figure 2 describe the CPU time with different λ and different n. From these Tables and Figures, we can see that the relaxed Newton-like method is effective with appropriate relaxation parameter λ. Especially, when (c, α) is very close to (0, 1), the relaxed Newton-like method has a better convergence results.

5

Conclusion

In this paper, based on the relaxation or extrapolation technique, we have proposed a relaxed Newton-like method for solving the minimal positive solution of the non7

LI ET AL: RELAXED NEWTON-LIKE METHOD

1139

Table 1: Numerical results different pairs of (c, α) and λ n=256, (c, α) (0.9999, 0.0001) (0.999, 0.001) 8 7 0.4370 0.4060 5.9786e-15 1.8973e-15 9 8 0.5000 0.4530 3.1694e-15 1.5532e-15 12 10 0.6570 0.5940 1.3377e-15 1.4118e-15 9 8 0.4840 0.4680 1.4923e-15 1.6885e-15 8 7 0.4380 0.4060 2.8412e-15 1.4318e-15

λ -1.4

-1

0

1

1.4

IT CPU ERR IT CPU ERR IT CPU ERR IT CPU ERR IT CPU ERR

n=512, (c, α) (0.999, 0.001) (0.99, 0.01) 7 6 2.8910 2.5000 1.7734e-15 1.7586e-15 8 6 3.3120 2.2970 1.7769e-15 2.4157e-15 10 9 4.0630 3.5780 2.2549e-15 1.5605e-15 8 6 3.1090 2.3290 1.5937e-15 2.4313e-15 7 6 2.7030 2.9060 1.7675e-15 1.7058e-15

CPU time with different(c, α),n=256

CPU time with different(c, α),n=128 0.11

0.7

0.1

0.65 0.6 CPU time (seconds)

CPU time (seconds)

0.09

0.08

0.07

0.06

(c, α)=(0.9999,0.0001) (c, α)=(0.999,0.001) (c, α)=(0.99,0.01)

0.05

0.04 −2

−1.5

−1

−0.5

0 λ

0.5

0.55 0.5 0.45 (c, α)=(0.9999,0.0001) (c, α)=(0.999,0.001) (c, α)=(0.99,0.01)

0.4 0.35

1

1.5

2

−2

−1.5

−1

−0.5

Figure 1: CPU time for different (c, α) and λ.

8

0 λ

0.5

1

1.5

2

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CPU time with different(c, α),n=512

CPU time with different(c, α),n=1024

4.5

28 26

4

CPU time (seconds)

CPU time (seconds)

24 3.5

3

2.5

1.5 −2

−1.5

−1

−0.5

0 λ

0.5

20 18 16

(c, α)=(0.999,0.001) (c, α)=(0.99,0.01) (c, α)=(0.9,0.1)

2

22

(c, α)=(0.999,0.001) (c, α)=(0.99,0.01) (c, α)=(0.9,0.1)

14

1

1.5

12 −2

2

−1.5

−1

−0.5

0 λ

0.5

1

1.5

2

Figure 2: CPU time for different (c, α) and λ. symmetric algebraic Riccati equation arising in transport theory and have given the convergence analysis. The convergence analysis shows that the iteration sequence generated by the relaxed Newton-like method is monotonically increasing and converges to the minimal positive solution of the vector equation. Numerical experiments show that the relaxed Newton-like method is feasible and effective for the nonsymmetric algebraic Riccati equation, especially, when (c, α) is very close to (0, 1). Since the relaxed Newton-like method containing a relaxation parameter λ, hence, the determination of the optimum value of the relaxation parameter λ such that the relaxed Newton-like method has a better convergence rate needs further to be studied.

References [1] Z. Z. Bai, Y. H. Gao and L. Z. Lu. Fast iterative schemes for nonsymmetric algebraic raccati equations arising from transport theory. SIAM J.Sci.Comput., 30 (2) (2008), pp. 804-818. [2] Z. Z. Bai, X. X. Guo and S. F. Xu. Alternately linearized implicit iteration methods for the minimal nonnegative solutions of the nonsymmetric algebraic Riccati equations. Numer. Linear Algebra Appl., 13 (8) (2006), pp. 655-674. [3] L. Bao, Y. Q. Lin and Y. M. Wei. A modified simple iterative method for nonsymmetric algebraic Riccati equations arising in transport theory. Appl. Math. Comput., 181 (2006), pp. 1499-1504. [4] A. Berman, R. J. Plemmons. Nonnegative Matrices in the Mathematical Sciences. SIAM, Philadelphia, PA, 1994. [5] D. A. Bini, B. Iannazzo and F. Poloni. A fast Newton’s method for a nonsymmetric algebraic Riccati equations. SIAM J. Matrix Anal. Appl., 30 (2008), pp. 276-290.

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[6] Z. H. Cao. A convergence theorem on an extrapolated iterative method and its applications. Appl. Numer. Math., 27 (1998), pp. 203-209. [7] F. Chen, Y. L. Jiang. A generalization of the inexact parameterized Uzawa methods for saddle point problems. Appl. Math. Comput., 206 (2008), pp. 765-771. [8] B. D. Ganapol. An investigating of a simple transport model. Transport Theory Statist. Phys., 21 (1992), pp. 1-37. [9] C. H. Guo. Nonsymmetric algebraic Riccati equations and Wiener-Hopf factorization for M-matrices. SIAM J. Matrix Anal. Appl., 23 (1), (2001), pp. 225-242. [10] X. X. Guo and Z. Z. Bai. On the minimal nonnegative solution of nonsymmetric algebraic Riccati equation. J. Comput. Math., 23 (2005), pp. 305-320. [11] C. H. Guo, B. Iannazzo and B. Meini. On the doubling algorithm for a (shifted) nonsymmetric algebraic Riccati equation. SIAM J. Matrix Anal. Appl., 29 (2007), pp. 1083-1100. [12] C. H. Guo and A. J. Laub. On the iterative solution of a class of nonsymmetric algebraicRiccati equations. SIAM J. Matrix Anal. Appl., 22 (2)(2000), pp. 376-391. [13] C. H. Guo and W. W. Lin. Convergence rates of some iterative methods for nonsymmetric algebraic Riccati equations arising in transport theory. Linear Algebra Appl., 432 (2010), pp. 283-291. [14] A. Hadjidimos, A. Yeyios. The principle of extrapolation in connection with the accelerated overrelaxation method. Linear Algebra Appl., 30 (1980), pp. 115-128. [15] J. Juang. Existence of algebraic matrix Riccati equations arising in transport theory. Linear Algebra Appl., 230 (1995), pp. 89-100. [16] J. Juang and I. D. Chen. Iterative solution for a certain class of algebraic matrix Riccati equations arising in transport theory. Transport Theory Statist. Phys., 22 (1993), pp. 65-80. [17] J. Juang and W. W. Lin. Nonsymmetric algebraic Riccati equations and Hamiltonianlike matrices. SIAM J. Matrix Anal. Appl., 20 (1) (1999), pp. 228-243. [18] J. S. Kou, Y. T. Li and X. H. Wang. A modification of Newton method with third-order convergence. Appl. Math. Comput., 181 (2006), pp. 1106-1111. [19] A. J. Laub. A Schur method for solving algebraic Riccati equations. IEEE Transactions on automatic control., 24 (1979), pp. 913-921. [20] C. J. Li, Z. Li, X. Y. Nie, D. J. Evans. Generalized AOR method for the augmented system. Inter. J. Comput Math., 81 (2004), pp. 495-504. [21] Y. Q. Lin and L. Bao. Convergence analysis of the Newton-Shamanskii method for a nonsymmetric algebraic Riccati equations. Numer. Linear Algebra Appl., 15 (2008), pp. 535-546.

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[22] Y. Q. Lin, L. Bao and Y. M. Wei. A modified Newton method for solving non-symmetric algebraic Raccati equations arising in transport theory. IMA J. Numer. Anal., 28 (2008), pp. 215-224. [23] L. Z. Lu. Solution form and simple iteration of a nonsymmetric algebraic Riccati equation arising in transport theory. SIAM J. Matrix Anal. Appl., 26 (3) (2005), pp. 679-685. [24] L. Z. Lu. Newton iterations for a non-symmetric algebraic Riccati equation. Numer. Linear Algebra Appl., 12 (2005), pp. 191-200. [25] L. Z. Lu and M. K. Ng. Effects of a parameter on a nonsymmetric algebraic Riccati equation. Appl. Math. Comput., 172 (2006), pp. 753-761. [26] L. Z. Lu and C. E. M. Pearce On the mstrix-sign-function method for solving algebraic Riccati equations. Appl. Math. Comput., 86 (1997), pp. 157-170. [27] V. Mehrmann and H. G. Xu. Explicit solutions for a Riccati equation from transport theory. SIAM J. Matrix Anal. Appl., 30 (4) (2008), pp. 1339-1357. [28] N. J. Nigham and C. H. Guo. Iterative solution of a nonsymmetric algebraic Riccati equation. SIAM J. Matrix Anal. Appl., 29 (2) (2007), pp. 396-412. [29] C. Paige and C. V. Loan. A Schur decomposition for Hamitonian matrices. Linear Algebra Appl., 41 (1981), pp. 11-32. [30] L. C. G. Rogers. Fluid models in queueing theory and Wiener-Hopf factorization of Markov Chains. Ann. Appl. Probab., 4 (1994), pp. 390-413. [31] Y. Z. Song, L. Wang. On the semiconvergence of extrapolated iterative methods for singular linear systems. Appl. Numer. Math., 44 (2003), pp. 401-413.

11

JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.6, 1143-1156, 2011, COPYRIGHT 2011 EUDOXUS 1143 PRESS, LLC

The W 1,1 -Seminorm Estimate for the Discrete Derivative Green’s Function for the 5D Poisson Equation∗ Jinghong Liu1 , Qiding Zhu2

(1 Department of Fundamental Courses, Ningbo Institute of Technology, Zhejiang University, Ningbo 315100, China) E-mail : [email protected]

(2 College of Mathematics and Computer Science, Hunan Normal University, Changsha 410081, China) E-mail : [email protected]

Abstract In this paper, we first introduce the definitions of the regularized derivative Green’s function ∂Z,` G∗Z and the discrete derivative Green’s function ∂Z,` GhZ for the Poisson equation in five dimensions. Then the W 1,1 seminorm estimate for ∂Z,` G∗Z is derived. Finally, applying the triangle inequality and the error estimate of the finite element approximation, we 7 obtain the W 1,1 -seminorm estimate with order O(| ln h| 5 ) for ∂Z,` GhZ . Keywords: regularized derivative Green’s function; discrete derivative Green’s function; 5D Poisson equation; finite element; W 1,1 -seminorm estimate

1. INTRODUCTION AND PRELIMINARIES It is well known that estimates for the Green’s function play very important roles in the study of the superconvergence (especially, pointwise superconvergence) of the finite element method (see [1–8]). For one- and two-dimensional ∗

Supported by the Natural Science Foundation of Zhejiang Province (No. Y6090131), and the Natural Science Foundation of Ningbo City (No. 2010A610101).

1

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elliptic problems, one have obtained many optimal estimates for the Green’s function (see [1]). For three-dimensional elliptic problems, the W 2,1 -seminorm 2 optimal estimate with order O(| ln h| 3 ) for the discrete Green’s function and 4 the W 1,1 -seminorm optimal estimate with order O(| ln h| 3 ) for the discrete derivative Green’s function were derived too (see [5, 9]). Recently, we studied four-dimensional elliptic equations, and obtained the estimate for the fourdimensional discrete derivative Green’s function. It might seem that in current practical applications of the finite element method there is no need for simplicial higher-dimensional elements. Nevertheless, it is well known that for example in areas like financial mathematics, particle physics, statistical physics and general relativity, higher-dimensional PDEs need to be solved (see [10]). Therefore, it is meaningful to study the estimates for the Green’s function in four and up space dimensions. In this paper, we will derive the W 1,1 -seminorm estimate 7 with order O(| ln h| 5 ) for the discrete derivative Green’s function for the fivedimensional Poisson equation, which may be used in the study of the pointwise superconvergence of the derivative for the finite element method. In this paper, we shall use the symbol C to denote a generic constant, which is independent from the discretization parameter h and which may not be the same in each occurrence and also use the standard notations for the Sobolev spaces and their norms. We consider the following problem: Lu ≡ −∆u = f in Ω,

u = 0 on ∂Ω,

(1.1)

where Ω ⊂ R5 is a bounded polytopic domain and ∂i u = ∂xi u(x1 , x2 , x3 , x4 , x5 ), i = 1, 2, 3, 4, 5 stand for usual partial derivatives. Let {T h } be a regular family ¯ (see [10]). Denote the space of continuous piecewise of uniform partitions of Ω linear polynomials with respect to this kind of partitions by S h (Ω) and let S0h = S h (Ω)∩H01 (Ω). Discretizing the above equation using S0h as approximating space means finding uh ∈ S0h such that a(uh , v) = (f , v) for all v ∈ S0h , where Z

a(uh , v) ≡

Ω

and

∇uh · ∇v dx1 dx2 dx3 dx4 dx5 ,

Z

(f , v) ≡

Ω

f v dx1 dx2 dx3 dx4 dx5 .

2

LIU, ZHU: ...5D POISSON EQUATION

1145

This gives the Galerkin orthogonality relation a(u − uh , v) = 0 ∀ v ∈ S0h .

(1.2)

For every Z ∈ Ω, we define the discrete derivative δ function ∂Z,` δZh ∈ S0h and the L2 -projection Ph u ∈ S0h such that (see [1]) (v, ∂Z,` δZh ) = ∂` v(Z) ∀ v ∈ S0h ,

(1.3)

(u − Ph u, v) = 0 ∀ v ∈ S0h ,

(1.4)

where ` ∈ R5 and |`| = 1. ∂` v(Z) stands for the onesided directional derivative v(Z + ∆Z) − v(Z) , ∆Z = |∆Z|`. |∆Z|→0 |∆Z|

∂` v(Z) = lim

Remark 1. Since ∆Z = |∆Z|`, that is, ∆Z is of the same direction as `. Thus, provided that the direction ` is given, the above limit exists, and hence that no matter what direction is given, ∂` v(Z) is well defined. Let ∂Z,` G∗Z ∈ H 2 (Ω)∩H01 (Ω) be the solution of the elliptic problem L∂Z,` G∗Z = ∂Z,` δZh . We may call ∂Z,` G∗Z the regularized derivative Green’s function, and denote by ∂Z,` GhZ (the so-called discrete derivative Green’s function) the finite element approximation to ∂Z,` G∗Z . Thus, a(∂Z,` G∗Z − ∂Z,` GhZ , v) = 0 ∀ v ∈ S0h .

(1.5)

The main result of this work is the following estimate: ¯ ¯ ¯ ¯ ¯∂Z,` GhZ ¯

7

1, 1

≤ C| ln h| 5 .

2. THE W 1,1 -SEMINORM ESTIMATE FOR THE FIVE-DIMENSIONAL DISCRETE DERIVATIVE GREEN’S FUNCTION ∂Z,` GhZ To derive the estimate for the discrete derivative Green’s function, we introduce the weight function defined by ³

¯ 2 + θ2 φ ≡ φ(X) = |X − X|

´− 5 2

¯ ∀ X ∈ Ω,

(2.1)

¯ ∈ Ω ¯ is a fixed point, θ = γh, and γ ∈ [5, +∞) is a suitable real where X number. 3

1146

LIU, ZHU: ...5D POISSON EQUATION

For every α ∈ R, we give the following notations: µZ ¶1 m ¯ X ¯¯ X 2 β ¯2 n α n 2 |∇ v| = ¯D v ¯ , |∇ v|φα , Ω = φ |∇ v| dX , kvk2m, φα , Ω = |∇n v|2φα , Ω , n

2

Ω

|β|=n

n=0

where β = (β1 , β2 , β3 , β4 , β5 ), |β| = β1 + β2 + β3 + β4 + β5 , and βi ≥ 0, i = 1, 2, 3, 4, 5 are integers. In particular, for the case of m = 0, we write µZ

kvkφα , Ω =

¶1 α

Ω

2

φ |v| dX

2

.

We assume that there exists an q0 (1 < q0 ≤ ∞) such that L : W 2, q (Ω) ∩ W01, q (Ω) −→ Lq (Ω) (1 < q < q0 ) is a homeomorphism (see [1]). It means that for all v ∈ W 2, q (Ω) ∩ W01, q (Ω), we have the so-called a priori estimate: kvk2, q, Ω ≤ C(q)kLvk0, q, Ω ,

(2.2)

where C(q) denotes a positive constant only depending on q. Next, we give some lemmas used in the proofs of our main results. Lemma 2.1. For φ the weight function defined by (2.1), we have the following estimates: n |∇n φα | ≤ C(α, n)φα+ 5 , α ∈ R, n = 1, 2. (2.3) Proof. From (2.1),

Thus,

5 ¯ 2 + θ2 ). ln φα = − α ln(|X − X| 2 ¯ ∇φα X −X = −5α ¯ 2 + θ2 . φα |X − X|

(2.4)

Further, ¯ ¯ ¯ ∇φα ¯ ¯ |X − X| ¯ ¯ ¯ 2 + θ2 )− 21 . ≤ 5|α|(|X − X| ¯ α ¯ = 5|α| 2 2 ¯ ¯ φ ¯ |X − X| + θ

It follows that 1

1

|∇φα | ≤ 5|α|φα+ 5 = C(α)φα+ 5 , which shows that the result (2.3) holds when n = 1. 4

(2.5)

LIU, ZHU: ...5D POISSON EQUATION

1147

When n = 2, the operator ∇2 is the Hessian matrix of second derivatives. ¯ = (¯ We set X = (x1 , x2 , x3 , x4 , x5 ), X x1 , x¯2 , x¯3 , x¯4 , x¯5 ). From (2.4), we have 2

∂i φα = −5α(xi − x¯i )φα+ 5 , i = 1, ..., 5. Hence, 2

4

∂i2 φα = −5αφα+ 5 + 5α(5α + 2)(xi − x¯i )2 φα+ 5 , i = 1, ..., 5, 4

∂j ∂k φα = 5α(5α + 2)(xj − x¯j )(xk − x¯k )φα+ 5 , j < k.

(2.6) (2.7)

Combining (2.6)–(2.7) yields |∇2 φα |2 =

5 X 5 X

4

|∂j ∂k φα |2 ≤ [25α2 (5α + 2)2 + 50α2 |5α + 2| + 25α2 ]φ2α+ 5 ,

j=1 k=j

namely, 2

|∇2 φα | ≤ C(α)φα+ 5 , which shows that the result (2.3) holds when n = 2. 2 Lemma 2.2. For φ the weight function defined by (2.1), we have the following estimates: Z φα dX ≤ C(α − 1)−1 θ−5(α−1) ∀ α > 1, (2.8) Ω

and

Z Ω

φ dX ≤ C(k)| ln θ|, θ ≤ k < 1.

(2.9)

¯ D) = {X ∈ R5 : |X − X| ¯ ≤ D}. When Proof. Set D = diam(Ω) and B(X, α > 1, we have Z

Z

Z α

Ω

φ (X)dX ≤

α

¯ B(X,D)

Z

≤ C

φ (X)dX ≤ C

r4 dr

D

5

(r2 + θ2 ) 2 α

0

dr5 ≤ C(α − 1)−1 θ−5(α−1) . 5 5 α (r + θ )

∞

0

The proof of the result (2.8) is completed. In addition, when α = 1, Z Ω

Z

φ(X)dX ≤ C

Z

r4 dr

D

5

≤C

D

dr5 r5 + θ5

0 (r2 + θ2 ) 2 µ ¶ 1 ≤ C | ln θ| + ln(D5 + θ5 ) 5 Ã ! ln(1 + D5 ) ≤ C 1+ | ln θ|, 5| ln k| 0

5

1148

LIU, ZHU: ...5D POISSON EQUATION

where we used the fact that θ5 < 1 in the last inequality. Thus, Z Ω

φ(X)dX ≤ C(k)| ln θ|, θ ≤ k < 1,

which is the result (2.9). 2 With the similar arguments in [5], we may derive Lemma 2.3. For ∂Z,` δZh the discrete derivative δ function defined by (1.3), we have the following estimates: |∂Z,` δZh (X)| ≤ Ch−6 e−Ch

−1 |X−Z|

,

(2.10)

¯ and C is independent of h, X, and Z. where X, Z ∈ Ω, Lemma 2.4. For ∂Z,` δZh the discrete derivative δ function defined by (1.3), and for all α > 0, we have ° ° 5α−7 ° h° (2.11) °∂Z,` δZ ° −α ≤ Ch 2 , φ

° 3 ° ° −5 h° °φ ∂Z,` δZ °

0,

5 3

≤ C.

(2.12)

Proof. From (2.10), ° ° ° h °2 °∂Z,` δZ ° −α φ

≤ C ≤ C

Z ³ Ω Z ∞ 0

|X − Z|2 + θ2 ³

r2 + θ2

´ 5α 2

´ 5α 2

h−12 e−Ch

h−12 e−Ch

−1 r

−1 |X−Z|

dX

r4 dr,

Set h−1 r = t, then Z ° ° ° h °2 °∂Z,` δZ ° −α ≤ Ch5α−7 φ

0

∞

³

t2 + γ 2

´ 5α 2

e−Ct t4 dt ≤ Ch5α−7 .

The proof of (2.11) is completed. Similarly, the result (2.12) may be proved. 2 ¯ and 1 ≤ q ≤ Lemma 2.5. For Ph w the L2 -projection of w ∈ W 1, q (Ω) ∩ C(Ω), ∞, we have the following stability estimate: kPh wk1, q ≤ Ckwk1, q .

(2.13)

Proof. Denote by Πu ∈ S0h the interpolant of w, and then by the triangle inequality, the interpolation error estimate and an inverse inequality, we have kw − Ph wk1, q ≤ Ckw − Πwk1, q + Ch−1 kΠw − Ph wk0, q ≤ Ckwk1, q + Ch−1 kΠw − wk0, q + Ch−1 kw − Ph wk0, q ≤ Ckwk1, q + Ch−1 kw − Ph wk0, q . (2.14) 6

LIU, ZHU: ...5D POISSON EQUATION

1149

Similar to the proof of Lemma 2.5 in [5], we may derive kPh wk0, q ≤ C t kwk0, q ,

(2.15)

¯ ¯ ¯ ¯ where t = ¯1 − 2q ¯.

For an arbitrary v ∈ S0h , replacing w with w − v in (2.15) gives kPh w − vk0, q ≤ C t kw − vk0, q ,

(2.16)

where we used the fact that Ph v = v for all v ∈ S0h . Thus, applying the triangle inequality and (2.16) results in kw − Ph wk0, q ≤ kw − vk0, q + kPh w − vk0, q ≤ (1 + C t )kw − vk0, q .

(2.17)

Taking v = Πw in (2.17), and using the interpolation error estimate, we have kw − Ph wk0, q ≤ Ckw − Πwk0, q ≤ Chkwk1, q .

(2.18)

Combined with (2.14) yields kw − Ph wk1, q ≤ Ckwk1, q .

(2.19)

The result (2.13) follows from (2.19) and the triangle inequality immediately.2 Lemma 2.6. Let w ∈ W 1, 5 (Ω), then we have 4

kwk0, q ≤ Cq 5 kwk1, 5 ,

(2.20)

where q >> 1. Remark 2. This lemma may be found in [2]. Lemma 2.7. For ∂Z,` G∗Z the regularized derivative Green’s function, we have the following W 1, 1 -seminorm estimate: 7

|∂Z,` G∗Z |1, 1 ≤ C |ln h| 5 .

(2.21)

Proof. Set g = ∂Z,` G∗Z , and thus, |g|21, 1

Z

≤

Ω

φ dX · k∇gk2φ−1 .

(2.22)

Furthermore, k∇gk2φ−1 ≤ a(g , φ−1 g) + C kgk2φ− 53

≤ (∂Z,` δZh , φ−1 g) + C kgk2φ− 35

≤ k∂Z,` δZh k2 − 7 + C φ

7

5

kgk2φ− 53

.

(2.23)

1150

LIU, ZHU: ...5D POISSON EQUATION

Nevertheless, from an inverse estimate, the stability estimate (2.13), a priori estimate (2.2), (2.20), the Sobolev Embedding Theorem [11], and the Poincar´e inequality, 3

kgk2φ− 35

= (φ− 5 g , g) = a(w , g) = (∂Z,` δZh , w) = ∂` Ph w(Z) 5

5

≤ |Ph w|1, ∞ ≤ Ch− q |Ph w|1, q ≤ Ch− q |w|1, q 5 4 5 4 3 ≤ Ch− q q 5 kwk2, 5 ≤ Ch− q q 5 kφ− 5 gk0, 5 5 4 3 ≤ Ch− q q 5 kφ− 5 gk2, 5 5

4

(2.24)

3

3

≤ Ch− q q 5 kL(φ− 5 g)k0, 5 , 3

3

where Lw = φ− 5 g in Ω, and w|∂Ω = 0. Taking q = | ln h| in (2.24) yields 4

3

kgk2φ− 35 ≤ C |ln h| 5 kL(φ− 5 g)k0, 5 .

(2.25)

3

Further, ³

3

3

2

1

kL(φ− 5 g)k0, 5 ≤ C kφ− 5 Lgk0, 5 + kφ− 5 ∇gk0, 5 + kφ− 5 gk0, 5 ³

3

3

3

3

2

1

´ ´

3

= C kφ− 5 ∂Z,` δZh k0, 5 + kφ− 5 ∇gk0, 5 + kφ− 5 gk0, 5 . 3

3

(2.26)

3

Obviously, − 25

kφ

− 51

kφ

µZ

∇gk0, 5 = 3

Ω

µZ

gk0, 5 = 3

− 23

φ

− 13

Ω

φ

¶3

5 3

5

|∇g| dX ¶3

5 3

|g| dX

5

µZ

≤

Ω

¶

φ dX

µZ

≤

Ω

¶

φ dX

1 10

1 10

µZ Ω

µZ

− 35

Ω

¶1 −1

φ

2

φ |∇g| dX |g|2 dX

.

(2.27)

¶1 2

2

.

(2.28)

From (2.9), (2.12), and (2.25)–(2.28), kgk2φ− 35

4

9

4

9

9

≤ C |ln h| 5 + C |ln h| 10 k∇gkφ−1 + C |ln h| 10 kgkφ− 35

9

≤ C |ln h| 5 + C |ln h| 10 k∇gkφ−1 + εkgk2 − 3 + C(ε) |ln h| 5 . φ

Taking ε =

1 2

(2.29)

5

in (2.29) gives 9

9

kgk2φ− 53 ≤ C |ln h| 5 + C |ln h| 10 k∇gkφ−1 . Combined with (2.23) yields k∇gk2φ−1 ≤ k∂Z,` δZh k2 − 7 + C kgk2φ− 35 φ

5

9

9

≤ k∂Z,` δZh k2 − 7 + C |ln h| 5 + C |ln h| 10 k∇gkφ−1 ≤

φ 5 h 2 k∂Z,` δZ k − 7 φ 5

9 5

(2.30) 9 5

+ C |ln h| + ηk∇gk2φ−1 + C(η) |ln h| . 8

LIU, ZHU: ...5D POISSON EQUATION

Taking η =

1 2

1151

in (2.30) and applying (2.11), we obtain 9

k∇gk2φ−1 ≤ C |ln h| 5 .

(2.31)

Finally, the result (2.21) follows from (2.9), (2.22), and (2.31) immediately. 2 Lemma 2.8. For Πu ∈ S0h the interpolant of u, there hold the following interpolation error estimates: |u − Πu|n, p, e ≤ Ch2−n |u|2, p, e ∀ e ∈ T h , ¯

¯

|∇n (u − Πu)|φα , e ≤ Ch2−n ¯¯∇2 u¯¯ ∗h

µ

φα , e

(2.32)

∀ e ∈ T h,

(2.33)

¶

|v|φ2α+ 25 −β + |∇v|φ2α−β ∀ v ∈ S0h , (2.34) θ where α, β ∈ R, C ∗ = C ∗ (α, β), n = 0, 1, 2, s = 0, 1, θ ≥ 5|α|h, 1 ≤ p ≤ ∞, C is independent of α, h and u. Remark 2. The proof of Lemma 2.8 is almost the same as that of Lemma 4 in [1] (see p.110 in [1]). Lemma 2.9. For ∂Z,` G∗Z and ∂Z,` GhZ , the regularized derivative Green’s function and the discrete derivative Green’s function, respectively, we have the following estimate: ¯ ¯ 13 ¯ ¯ ≤ C| ln h| 10 . (2.35) ¯∂Z,` G∗Z − ∂Z,` GhZ ¯ s

α

α

|∇ (φ v − Π(φ v))|φ−β ≤ C

1, 1

Proof. For simplicity, we write g = ∂Z,` G∗Z , and gh = ∂Z,` GhZ . Obviously, |g −

gh |21, 1

Z

≤

Ω

φ dX · |g − gh |21, φ−1 .

(2.36)

Moreover, similar to the proof of (2.43) in [5], we have ¯ ¯

¯2 ¯

|g − gh |21, φ−1 ≤ Ch2 ¯∇2 g ¯

φ−1

+ Cˆ kg − gh k2φ− 35 .

(2.37)

Further, 3

kg − gh k2φ− 35

= (φ− 5 (g − gh ), g − gh ) = a(w, g − gh ) = a(w − Πw, g − gh ) ≤ ε |g − gh |21, φ−1 + C(ε) |w − Πw|21, φ . − 35

where Lw = φ (2.33) yields

(2.38) (g − gh ). Applying the weighted interpolation error estimate ¯ ¯

¯2 ¯

kg − gh k2φ− 35 ≤ ε |g − gh |21, φ−1 + C(ε)h2 ¯∇2 w¯ . φ

9

(2.39)

1152

LIU, ZHU: ...5D POISSON EQUATION

From the H¨older inequality and (2.8), µZ ¶1 ¯ ¯ ¯ ¯2 α ¯ ¯ 2 ¯2 ¯ α ¯∇ w ¯ ≤ φ dX ¯∇ 2 w ¯ φ

0,

Ω

5 3+µ

where 1 < α < +∞. Taking α =

2α α−1

¯ ¯ ¯ 2 ¯2 ¯∇ w ¯

5(α−1) α

≤ Cθ−

0,

2α α−1

,

(0 < µ < 1) yields

¯ ¯ ¯ ¯2 ¯ 2 ¯2 ¯ ¯ ¯∇ w ¯ ≤ Cθ −2+µ ¯∇2 w ¯ φ

0,

.

10 2−µ

In addition, from a priori estimate (2.2), and the Sobolev Embedding Theorem [11], we have 2

2

|∇2 w|φ ≤ Cθ−2+µ |∇2 w|0,

¯

¯2

≤ Cθ−2+µ ¯¯φ− 5 (g − gh )¯¯

10 2−µ

3

10

0,

2−µ ° 3 °2 ° °2 3 ° ° ° ° ≤ Cθ−2+µ °φ− 5 (g − gh )° 10 ≤ Cθ−2+µ °L(φ− 5 (g − gh ))° 10 2, 6−µ 0, 6−µ Ã ° 3 °2 ° 2 °2 ≤ Cθ−2+µ °°φ− 5 L(g − gh )°° 10 + °°φ− 5 ∇(g − gh )°° 10 0, 6−µ 0, 6−µ ! ° 1 °2 ° ° + °φ− 5 (g − gh )° 10 .

0,

6−µ

(2.40) It is easy to prove ° 3 °2 ° −5 ° °φ L(g − gh )°

0,

° °

°2 °

3

= °φ− 5 ∂Z,` δZh °

10 6−µ

10 6−µ

0,

≤ Ch−µ .

(2.41)

Moreover, ° 2 °2 ° −5 ° °φ ∇(g − gh )°

0,

µZ 10 6−µ

=

Ω

−4 6−µ

φ

µZ

≤

≤ Cθ namely,

1 1−µ

φ

Ω −µ

° 2 °2 ° −5 ° °φ ∇(g − gh )°

0,

|∇(g − gh )|

10 6−µ

¶ 6−µ

dX

5

¶ 1−µ Z

dX

5

Ω

φ−1 |∇(g − gh )|2 dX

|g − gh |21, φ−1 ,

10 6−µ

≤ Cθ−µ |g − gh |21, φ−1 .

Whereas, °2 ° 1 ° ° −5 °φ (g − gh )°

0,

µZ 10 6−µ

=

Ω

φ

µZ

≤

φ

Ω −µ

≤ Cθ

−2 6−µ

1 1−µ

|g − gh |

¶ 1−µ Z

dX

kg −

10

10 6−µ

5

gh k2φ− 35

Ω

,

¶ 6−µ 5

dX 3

φ− 5 |g − gh |2 dX

(2.42)

LIU, ZHU: ...5D POISSON EQUATION

namely,

° 1 °2 ° −5 ° °φ (g − gh )°

0,

10 6−µ

1153

≤ Cθ−µ kg − gh k2φ− 35 .

(2.43)

Combining (2.39)–(2.43) yields kg −

gh k2φ− 35

≤ ε |g −

gh |21, φ−1

µ

+ C(ε)γ

−1

1 + |g −

gh |21, φ−1

+ kg −

gh k2φ− 53

¶

.

(2.44) ˆ Taking an ε ∈ (0, 1) such that 0 < 4εC < 1, and Choosing γ ∈ [5, +∞) such that 0 < C(ε)γ −1 < min(ε, 12 ), we then have kg − gh k2φ− 35 ≤ 4ε |g − gh |21, φ−1 + 1.

(2.45)

Combining (2.37) and (2.45) yields ¯ ¯

¯2 ¯

|g − gh |21, φ−1 ≤ Ch2 ¯∇2 g ¯

φ−1

+ C.

(2.46)

In addition, ¯ ¯ ¯ 2 ¯2 ¯∇ g ¯ −1 φ

Z

=

¯ ¯

¯2 ¯

φ−1 ¯∇2 g ¯ dX =

Z ³

1

¯ ¯

¯ ´2 ¯

φ− 2 ¯∇2 g ¯

dX

Ω µZ ¯ ¶ Z ¯ Z ¯ ¯ ¯ ¯2 ¯ 2 − 12 ¯2 ¯ ¯ ¯ 2 − 12 ¯2 − 21 ≤ C ¯∇ (φ g)¯ dX + ¯g∇ φ ¯ dX + ¯∇φ · ∇g ¯ dX Ω Ω µ°Ω ¶ °2 1 ° ° ≤ C °∇2 (φ− 2 g)° + kgk2φ− 15 + |g|21, φ− 35 0 µ° ¶ °2 1 ° ° 2 2 −2 ≤ C °L(φ g)° + kgkφ− 15 + |g|1, φ− 53 0 ¶ µ Ω

≤ C kLgk2φ−1 + kgk2φ− 15 + |g|21, φ− 35

µ° ¶ ° ° 2 h °2 − 35 ≤ C °∂Z,` δZ ° −1 + kgkφ− 15 + a(g, φ g) φ µ° ¶ °2 ° ° 2 h h − 35 = C °∂Z,` δZ ° −1 + kgkφ− 15 + (∂Z,` δZ , φ g) φ ° °2 ° ° ≤ C °∂Z,` δZh ° −1 + C kgk2 − 51 , φ

φ

namely,

° ° ¯ ¯ ° ¯ 2 ¯2 2 h °2 ° −1 + C kgk − 51 . ¯∇ g ¯ −1 ≤ C °∂Z,` δZ φ

φ

11

φ

(2.47)

1154

LIU, ZHU: ...5D POISSON EQUATION

We will now estimate kgk2φ− 15 . Since 1

kgk2φ− 51

= (φ− 5 g , g) = a(v , g) = (∂Z,` δZh , v) = ∂` Ph v(Z) 5

5

≤ |Ph v|1, ∞ ≤ Ch− q |Ph v|1, q ≤ Ch− q |v|1, q 5 4 5 4 1 ≤ Ch− q q 5 kvk2, 5 ≤ Ch− q q 5 kφ− 5 gk0, 5 5 4 1 ≤ Ch− q q 5 kφ− 5 gk2, 5 5

4

(2.48)

3

1

≤ Ch− q q 5 kL(φ− 5 g)k0, 5 , 3

1

where Lv = φ− 5 g in Ω, and v|∂Ω = 0. Thus, taking q = | ln h| in (2.48) yields 4

1

kgk2φ− 15 ≤ C |ln h| 5 kL(φ− 5 g)k0, 5 .

(2.49)

3

Further, ³

1

1

1

kL(φ− 5 g)k0, 5 ≤ C kφ− 5 Lgk0, 5 + k∇gk0, 5 + kφ 5 gk0, 5 3

³

3

1

3

1

´ ´

3

= C kφ− 5 ∂Z,` δZh k0, 5 + k∇gk0, 5 + kφ 5 gk0, 5 . 3

3

(2.50)

3

Similar to the proof of (2.11), we easily obtain 1

kφ− 5 ∂Z,` δZh k0, 5 ≤ Ch−2 .

(2.51)

3

Moreover, from a priori estimate (2.2), and the Sobolev Embedding Theorem [11], we have k∇gk0, 5 ≤ Ckgk2, 5 ≤ Ck∂Z,` δZh k0, 5 ≤ Ch−2 . 3

4

(2.52)

4

1

As for kφ 5 gk0, 5 , we get 3

1 5

µZ

kφ gk0, = 5 3

Ω

1 3

5 3

φ |g| dX

¶3 5

µZ

≤

¶ 3

Ω

φ dX

1 10

µZ

− 15

Ω

φ

¶1 2

|g| dX

2

≤ Ch−1 kgkφ− 15 . (2.53)

From (2.49)–(2.53), kgk2φ− 15

4

4

4

8

≤ Ch−2 |ln h| 5 + Ch−1 |ln h| 5 kgkφ− 15 ≤ Ch−2 |ln h| 5 + Ch−2 |ln h| 5 +

1 kgk2φ− 15 , 2

namely, 8

kgk2φ− 15 ≤ Ch−2 |ln h| 5 . 12

(2.54)

LIU, ZHU: ...5D POISSON EQUATION

1155

From (2.11), (2.47), and (2.54), ¯ ¯ 8 ¯ 2 ¯2 ¯∇ g ¯ −1 ≤ Ch−2 |ln h| 5 . φ

(2.55)

Finally, the result (2.35) follows from (2.9), (2.36), (2.46), and (2.55). 2 Combining (2.21) and (2.35), and applying the triangle inequality, we get the following result. Theorem 2.1. For ∂Z,` GhZ ∈ S0h the discrete derivative Green’s function, we have the following estimate: ¯ ¯ ¯ ¯ ¯∂Z,` GhZ ¯

7

1, 1

≤ C| ln h| 5 .

References 1. Q. D. Zhu and Q. Lin, Superconvergence theory of the finite element methods (in Chinese), Hunan Science and Technology Press, Changsha, China, 1989. 2. C. M. Chen and Y. Q. Huang, High accuracy theory of finite element methods (in Chinese), Hunan Science and Technology Press, Changsha, China, 1995. 3. C. M. Chen, Construction theory of superconvergence of finite elements (in Chinese), Hunan Science and Technology Press, Changsha, China, 2001. 4. G. Goodsell, Pointwise superconvergence of the gradient for the linear tetrahedral element, Numer. Meth. Part. Differ. Equ. 10 (1994), 651-666. 5. J. H. Liu, B. Jia, and Q. D. Zhu, An estimate for the three-dimensional discrete Green’s function and applications, J. Math. Anal. Appl. 370 (2010), 350-363. 6. J. H. Liu and Q. D. Zhu, Uniform superapproximation of the derivative of tetrahedral quadratic finite element approximation, J. Comput. Math. 23 (2005), 75-82. 7. J. H. Liu and Q. D. Zhu, Maximum-norm superapproximation of the gradient for the trilinear block finite element, Numer. Meth. Part. Differ. Equ. 23 (2007), 1501-1508. 8. J. H. Liu and Q. D. Zhu, Pointwise supercloseness of tensor-product block finite elements, Numer. Meth. Part. Differ. Equ. 25 (2009), 990-1008.

13

1156

LIU, ZHU: ...5D POISSON EQUATION

9. J. H. Liu and Q. D. Zhu, The estimate for the W 1,1 -seminorm of discrete derivative Green’s function in three dimensions (in Chinese), J. Hunan Univ. Arts Sci. 16 (2004), 1-3. 10. J. H. Brandts and M. Kˇr´ıˇzek, Gradient superconvergence on uniform simplicial partitions of polytopes, IMA. J. Numer. Anal. 23 (2003), 489–505. 11. R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.

14

JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.6, 1157-1170, 2011, COPYRIGHT 2011 EUDOXUS 1157 PRESS, LLC

Existence of Extremal Solutions for Impulsive Fuzzy Differential Equations with Periodic Boundary Value in n-dimensional Fuzzy Vector Space Young Chel Kwun∗, Jeong Soon Kim Department of Mathematics, Dong-A University, Pusan 604-714, South Korea [email protected](Y.C. Kwun), [email protected](J.S. Kim), Jin Han Park† Department of Applied Mathematics, Pukyong National University, Pusan 608-737, South Korea [email protected] August 20, 2010

Abstract In this paper, we study the existence of extremal solutions for impulsive fuzzy differential equations with periodic boundary value in ndimensional fuzzy vector space (EN )n . Our result is an extension of the result of Rodr´ıguez-L´ opez [Fuzzy Sets and Systems 159 (2008) 1384] to n-dimensional fuzzy vector space.

1

Introduction

The concept of fuzzy set was initiated by Zadeh [8] via membership function in 1965. Many authors have studied the fuzzy equations. Fuzzy differential equations are a field of increasing interest, due to their applicability to the analysis of phenomena where imprecision is inherent. In [2], Rodr´ıguez-L´ opez studied the existence and approximation of extremal solutions for fuzzy differential equation by using monotone iterative technique in one dimensional fuzzy vector space E 1 . Nieto and Rodr´ıguez-L´opez [6] studied ∗ This

study was supported by research funds from Dong-A University. author: [email protected]

† Corresponding

1

1158

KWUN ET AL

Existence of Extremal Solutions for Impulsive Fuzzy Differential Equations

2

existence of extremal solutions for quadratic fuzzy equations. Rodr´ıguez-L´opez [1] proved the existence of solutions for impulsive fuzzy differential equations with periodic boundary value using monotone method in one-dimensional fuzzy space. In this paper, we study the existence of extremal solutions for impulsive fuzzy differential equations with periodic boundary value in n-dimensional fuzzy vector space (EN )n . First we consider the following equation:  dx (t) i  = fi (t, xi (t)), t ∈ [0, T ], dt (1) xi (0) = xi (T ),  xi (t+ k ) = Ik (xi (tk )), t 6= tk , k = 1, 2, · · · , m, i = 1, 2, · · · , n, i where T > 0, J = [0, T ], 0 = t0 < t1 < · · · < tm < tm+1 = T,, EN is the set of j i all upper semi-continuously convex fuzzy numbers on R with EN 6= EN (i 6= j), i i fi : [0, T ] × EN → EN is nonlinear regular continuous fuzzy function, Ik ∈ i i C(EN , EN ) are bounded functions.

2

Preliminaries

In this section, we give basic definitions, terminologies, notations and Lemmas which are most relevant to our investigated and are needed in later sections. All undefined concepts and notions used here are standard. A fuzzy set of Rn is a function u : Rn → [0, 1]. For each fuzzy set u, we denote by [u]α = {x ∈ Rn : u(x) ≥ α} for any α ∈ [0, 1], its α-level set. Let u, v be fuzzy sets of Rn . It is well known that [u]α = [v]α for each α ∈ [0, 1] implies u = v. Let E n denote the collection of all fuzzy sets of Rn that satisfies the following conditions: (1) u is normal, i.e., there exists an x0 ∈ Rn such that u(x0 ) = 1; (2) u is fuzzy convex, i.e., u(λx + (1 − λ)y) ≥ min{u(x), u(y)} for any x, y ∈ Rn , 0 ≤ λ ≤ 1; (3) u(x) is upper semi-continuous, i.e., u(x0 ) ≥ limk→∞ u(xk ) for any xk ∈ Rn (k = 0, 1, 2, · · ·), xk → x0 ; (4) [u]0 is compact. We call u ∈ E n a n-dimension fuzzy number. Wang et al. [4] defined n-dimensional fuzzy vector space and investigated its properties. For any ui ∈ E, i = 1, 2, · · · , n, we call the ordered one-dimension fuzzy number class u1 , u2 , · · · , un (i.e., the Cartesian product of one-dimension fuzzy number u1 , u2 , · · · , un ) a n-dimension fuzzy vector, denote it as (u1 , u2 , · · · , un ), and call the collection of all n-dimension fuzzy vectors (i.e., the Cartesian prodz }| { uct E × E × · · · × E) n-dimensional fuzzy vector space, and denote it as (E)n . Definition 2.1Q[4] If u ∈ E n , and [u]α is a hyperrectangle, i.e., [u]α can be n α α α α α α α represented by i=1 [uα il , uir ], i.e., [u1l , u1r ] × [u2l , u2r ] × · · · × [unl , unr ] for every α α α α α ∈ [0, 1], where uil , uir ∈ R with uil ≤ uir when α ∈ (0, 1], i = 1, 2, · · · , n, then

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we call u a fuzzy n-cell number. We denote the collection of all fuzzy n-cell numbers by L(E n ). Qn α Theorem 2.2 [4] For any u ∈ L(E n ) with [u]α = i=1 [uα il , uir ] (α ∈ [0, 1]), n α α there exists a unique (u1 , u2 , · · · , un ) ∈ (E) such that [ui ] = [uα il , uir ] (i = 1, 2, · · · , n and α ∈ [0, 1]). α Conversely, for any (u1 , u2 , · · · , un ) ∈ (E)n with [ui ]α = [uα il , uir ] (i = n 1, 2, · · · , n and α ∈ [0, 1]), there exists a unique u ∈ L(E ) such that [u]α = Q n α α i=1 [uil , uir ] (α ∈ [0, 1]). Note 2.3 Theorem 2.1 indicates that fuzzy n-cell numbers and n-dimension fuzzy vectors can represent each other, so L(E n ) and (E)n may be regarded as identity. If (u1 , u2 , · · · , un ) ∈ (E)n is the unique n-dimension fuzzy vector determined by u ∈ L(E n ), then we denote u = (u1 , u2 , · · · , un ). i n 1 2 n i Let (EN ) = EN × EN × · · · × EN , EN (i = 1, 2, · · · , n) is fuzzy subset of R. i n n Then (EN ) ⊆ (E) . i n ) (i = 1, 2, · · · , n) Definition 2.4 Let u, v ∈ (EN

dε ([u]α , [v]α ) = dH

n Y i=1

=

n X

[ui ]α ,

n Y

[vi ]α



i=1

(dH ([ui ]α , [vi ]α ))2

1/2

i=1

where dH is the Hausdorff distance. i n Definition 2.5 The complete metric d∞ on (EN ) is defined by

d∞ (u(t), v(t)) = sup dε ([u(t)]α , [v(t)]α ) 0 0 there is a δ > 0 such that if α, β ∈ [0, 1], |α − β| < δ implies dH ([x(t)]α , [x(t)]β ) < ε. i n ) ) be compact and {{xn } : x ∈ B} ⊂ Theorem 2.13 [7] Let B ⊂ C([0, T ], (EN B. If {{xn } : x ∈ B} is bounded and equicontinuous, then any sequence in B has a uniformly convergent subsequence.

Instead of the equation (1), we consider the problem:  dx (t) i  = M xi (t) + Fi (t, xi (t)), t ∈ [0, T ], dt i , xi (0) = xi (T ) ∈ EN  + xi (tk ) = Ik (xi (tk )), t 6= tk , k = 1, 2, · · · , m, i = 1, 2, · · · , n,

(2)

where real number M, M > 0, T > 0, J = [0, T ], 0 = t0 < t1 < · · · < tm < i tm+1 = T , EN is the set of all upper semi-continuously convex fuzzy numbers on j i i i (i 6= j), Fi (t, xi (t)) = fi (t, xi (t)) − M xi (t), Ik ∈ C(EN , EN ) R with EN 6= EN are bounded functions. To define a solution for the impulsive fuzzy differential equations, we consider the following n space : i i Ωi = xi : J → EN : (xi )k ∈ C(Jk , EN ), Jk = (tk , tk+1 ], and there exist o + − + − xi (0 ), xi (T ), xi (tk ), with xi (tk ) = xi (tk ), (k = 1, 2, · · · , m, i = 1, 2, · · · , n) , n i i Θi = xi ∈ C(J, EN ) : (xi )0k ∈ C 1 (Jk , EN ), Jk = (tk , tk+1 ], and there exist o 0 − x0i (0+ ), x0i (T − ), x0i (t+ ), x (t ) (k = 1, 2, · · · , m, i = 1, 2, · · · , n) , i k k T T 1 0 i 0 i Let QnT ], EN ), Θi = Θi C ([0, T ], EN ), i = 1, 2, · · · , n, and QnΩi = Ωi C([0, Ω0 = i=1 Ω0i , Θ0 = i=1 Θ0i .

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Assume the following: i n (H1) F (t, x) ≤n F (t, y) for t ∈ J, x, y ∈ (EN ) , a(t) ≤n x ≤n y ≤n b(t).

(H2) Given there exist δ1 > 0 such that, for α, β ∈ [0, 1], with |α − β| < δ1 , i = 1, 2, · · · , n,   ε dH [ain (0)]α , [ain (0)]β ≤ √ . 3c n (H3) Given there exist δ2 > 0 such that, for α, β ∈ [0, 1], with |α − β| < δ2 , i = 1, 2, · · · , n,   ε dH [Fi (t, xi (t))]α , [Fi (t, xi (t))]β ≤ √ . 3c nt (H4) Given there exist δ3 > 0 such that, for α, β ∈ [0, 1], with |α − β| < δ3 , i = 1, 2, · · · , n, dH

h X

iβ  iα h X Ik (ai(n−1) (tk )) ≤ Ik (ai(n−1) (tk )) , 0 0 may be arbitrarily small, we have f (z) ≤ max{f (x), f (y)} − βλ(1 − λ)kη(x, y)k2 , which is a contradicition to the inequality (3.6). Thus, f is a strongly prequasi-invex function for the same η on K. Theorem 3.2. Let K ⊆ Rn be an open invex set with respect to vector-valued map η : Rn ×Rn −→ Rn . If f : K −→ R is lower semicontinuous on K and Condition C and D are satisfied, then, f is a strongly prequasi-invex function for the same η on K if and only if there exist a constant β > 0 and α ∈ (0, 1) such that, for all x, y ∈ K, f (y + αη(x, y)) ≤ max{f (x), f (y)} − βα(1 − α)kη(x, y)k2 . Proof . The necessity is obvious from Definition 8. We only prove the sufficiency. Let A = {λ ∈ [ 0, 1] | f (y + λη(x, y)) ≤ max{f (x), f (y)} − βλ(1 − λ)kη(x, y)k2 , ∃β > 0, ∀x, y ∈ K} From Lemma 3.1, A is dense in [ 0, 1], Thus, for any λ ∈ (0, 1), there exists λn ∈ (0, 1)(n = 1, 2, · · ·) such that λn ∈ A, λn −→ λ. Thus, for any x, y ∈ K, we have f (y + λn η(x, y)) ≤ max{f (x), f (y)} − βλn (1 − λn )kη(x, y)k2 . 7

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Since f is a lower semicontinuous function, we have f (y + λη(x, y)) ≤

lim f (y + λn η(x, y))

n−→∞

≤ max{f (x), f (y)} − βλ(1 − λ)kη(x, y)k2 ,

∀x, y ∈ K.

Hence f is a strongly prequasi-invex function for the same α and η on K. Theorem 3.3. If η : Rn × Rn −→ Rn is a vector-valued map and K is an invex set with respect to η, and f : K −→ R be a differentiable function on K. Then f is a strongly prequasi-invex function for the same η on K if and only if f is a strongly quasi-invex function for the same η on K. Proof. We first prove the only if part. Suppose that f is a strongly prequasi- invex function for the same η on K, then there exists a constant β > 0 such that for any x, y ∈ K, the following inequality hold f (y + λη(x, y)) ≤ max{f (x), f (y)} − βλ(1 − λ)kη(x, y)k2 ,

∀λ ∈ [ 0, 1].

(3.8)

Let x, y ∈ K be such that f (x) ≤ f (y).

(3.9)

We need to show that there exists a constant β > 0 η(x, y)T ∇f (y) + βkη(x, y)k2 ≤ 0. Because of f is a differentiable function on K, thus, from (3.8) and (3.9), we get f (y + λη(x, y)) − f (y) ≤ −β(1 − λ)kη(x, y)k2 . λ Letting λ −→ 0 in the above inequality, we have η(x, y)T ∇f (y) + βkη(x, y)k2 ≤ 0. Hence f is a strongly quasi-invex function for the same η on K. Now we prove the if part. Suppose that f is a strongly quasi-invex function for the same η on K. Then there exists a constant β > 0, for any x, y ∈ K such that f (x) ≤ f (y)

(3.10)

η(x, y)T ∇f (y) + βkη(x, y)k2 ≤ 0.

(3.11)

implies By f is a differentiable function on K, thus, (3.11) becomes f (y + λη(x, y)) − f (y) + βkη(x, y)k2 ≤ 0. λ

(3.12)

Integrating the (3.10) and (3.12), the following inequality hold f (y + λη(x, y)) ≤ f (y) − βλkη(x, y)k2 =

max{f (x), f (y)} − βλkη(x, y)k2

< max{f (x), f (y)} − βλ(1 − λ)kη(x, y)k2 ,

∀λ ∈ [ 0, 1].

Hence f is a strongly prequasi-invex function for the same η on K. Theorem 3.4. Let K ⊆ Rn be an open invex set with respect to vector-valued map η : Rn ×Rn −→ 8

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Rn and η satisfy Condition C, and f : K −→ R be a differentiable function on K. If f is a strongly prequasi-invex function for the same η on K, then ∇f is a strongly quasi-monotone on K. Proof. Suppose that f is a strongly prequasi-invex function with respect to η. Let β > 0 and x, y ∈ K be such that η(y, x)T ∇f (x) + βkη(y, x)k2 > 0. (3.13) We need to show that η(x, y)T ∇f (y) ≤ 0. Because of f is a strongly prequasi-invex function with respect to η on K, then, from Theorem 3.3, f is a strongly quasi-invex function for the same η on K. Thus, from (3.13), we have f (y) > f (x).

(3.14)

Again from the strong quasi-invexity of f with respect to η, inequality (3.14) implies that η(x, y)T ∇f (y) ≤ −βkη(x, y)k2 ≤ 0. Hence ∇f is a strongly quasi-monotone on K. Theorem 3.5. Let K ⊆ Rn be an invex set with respect to vector-valued map η : Rn × Rn −→ Rn and η satisfy Condition C, for any x, y ∈ K with x 6= y, η(x, y) 6= 0. If f : K −→ R satisfy Condition D, then f is a strongly prequasi-invex function for the same η on K if and only if for any x, y ∈ K, the function Φ(λ) = f (y + λη(x, y)) is a strongly quasi-convex function in the interval [ 0, 1]. Proof. We first prove the only if part. Suppose that f is a strongly prequasi-invex with respect to η. From the definition of strong prequasi-invexity, there exists a constant β > 0 such that for any x, y ∈ K, we have f (y + λη(x, y)) ≤ max{f (x), f (y)} − βλ(1 − λ)kη(x, y)k2 ,

∀λ ∈ [ 0, 1].

(3.15)

For any α1 , α2 ∈ [ 0, 1], ∀λ ∈ [ 0, 1]. If α1 = α2 . By f is a strongly prequasi-invex with respect to η, then, we have Φ(λα1 + (1 − λ)α2 )

=

Φ(α2 )

= f (y + α2 η(x, y)) =

max{Φ(α1 ), Φ(α2 )} − βλ(1 − λ)kα1 − α2 k2 .

Hence, Φ(λ) is a strongly quasi-convex function in the interval [ 0, 1]. If α1 6= α2 . Without loss of generality, suppose that α1 > α2 . By α1 , α2 ∈ [ 0, 1], thus, α2 6= 1 and 1 −α2 0 < α1−α ≤ 1, We have 2 Φ(λα1 + (1 − λ)α2 )

= f (y + (α2 + λ(α1 − α2 ))η(x, y)) = f (y + α2 η(x, y) + λ(α1 − α2 )η(x, y)).

(3.16)

From Condition C, we have η(y + α1 η(x, y), y + α2 η(x, y))

= η(y + α2 η(x, y) + (α1 − α2 )η(x, y), y + α2 η(x, y)) = η(y + α2 η(x, y) + =

α1 −α2 1−α2 η(x, y

=

(α1 − α2 )η(x, y).

α1 −α2 1−α2 η(x, y

+ α2 η(x, y)), y + α2 η(x, y))

+ α2 η(x, y)) (3.17)

9

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From (3.15)–(3.17), we obtain Φ(λα1 + (1 − λ)α2 )

= f (y + (α2 + λ(α1 − α2 ))η(x, y)) = f (y + α2 η(x, y) + λη(y + α1 η(x, y), y + α2 η(x, y))) ≤ max{f (y + α1 η(x, y)), f (y + α2 η(x, y))} −βλ(1 − λ)kη(y + α1 η(x, y), y + α2 η(x, y))k2 =

max{Φ(α1 ), Φ(α2 )} − βλ(1 − λ)kα1 − α2 k2 · kη(x, y)k2

≤ max{Φ(α1 ), Φ(α2 )} − β 0 λ(1 − λ)kα1 − α2 k2 . Taking β 0 = min {βkη(x, y)k2 }. ∀x,y∈K

If x 6= y, because of for any x, y ∈ K with x 6= y, η(x, y) 6= 0, then, β 0 > 0. Hence, Φ(λ) is a strongly quasi-convex function in the interval [ 0, 1]. If x = y, then f (x) = f (y). Since η satify Condition C, thus, from x = y, we can deduce that η(x, y) = 0. i.e., f (y + λη(x, y))

= f (y) =

Φ(0)

= λΦ(0) + (1 − λ)Φ(0) − βλ(1 − λ)k0 − 0k2 . Hence, Φ(λ) is a strongly quasi-convex function in the interval [ 0, 1]. Now we prove the if part. Suppose that Φ(λ) is strongly quasi-convex function in the interval [ 0, 1], then, there exists a constant β > 0 such that for any α1 , α2 ∈ [ 0, 1], we have Φ(λα1 + (1 − λ)α2 ) ≤ max{Φ(α1 ), Φ(α2 )} − βλ(1 − λ)kα1 − α2 k2 ,

∀λ ∈ [ 0, 1].

If x 6= y. Because of for any x, y ∈ K with x 6= y, η(x, y) 6= 0. From the strong quasi-convexity of Φ(λ) in the interval [ 0, 1], and f satisfy Condition D, then, thus, exists a constant β > 0, ∀ λ ∈ [ 0, 1], for any x, y ∈ K, we obtain f (y + λη(x, y))

= Φ(λ) =

Φ(λ · 1 + (1 − λ) · 0)

≤ max{Φ(1), Φ(0)} − βλ(1 − λ) · k1 − 0k2 ≤ max{f (y + η(x, y), f (y)} − β 0 λ(1 − λ)kη(x, y)k2 ≤ max{f (x), f (y)} − β 0 λ(1 − λ)kη(x, y)k2 . β where take β 0 = min { kη(x,y)k 2 }. Hence, f is a strongly prequasi-invex function for the same η ∀x,y∈K

on K. If x = y, then, f (x) = f (y). Since η satify Condition C, thus, from x = y, we can deduce that η(x, y) = 0. i.e., exists a constant β > 0 such that for any x, y ∈ K with x = y, the following inequality hold f (y + λη(x, y)) ≤ max{f (x), f (y)} − βλ(1 − λ)kη(x, y)k2 ,

∀λ ∈ [ 0, 1].

Hence, f is a strongly prequasi-invex function for the same η on K. This completes the proof. Theorem 3.6. Let K ⊆ Rn be an invex set with respect to vector-valued map η : Rn × Rn −→ Rn 10

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1307

and η satisfy Condition C, for any x, y ∈ K with x 6= y, η(x, y) 6= 0. If f : K −→ R is a strictly prequasi-invex function for the same η on K, then for any x, y ∈ K, the function Φ(λ) = f (y + λη(x, y)) is a strictly quasi-convex function in the interval ( 0, 1). Proof. Suppose that f is a strictly prequasi-invex with respect to η. From the definition of strict prequasi-invexity, for any x, y ∈ K, with x 6= y, we have f (y + λη(x, y)) < max{f (x), f (y)},

∀λ ∈ ( 0, 1).

For any α1 , α2 ∈ ( 0, 1), ∀λ ∈ ( 0, 1). If α1 6= α2 . Without loss of generality, suppose that α1 > α2 , then α2 6= 1 and 0 < Thus,We have Φ(λα1 + (1 − λ)α2 )

(3.18) α1 −α2 1−α2

= f (y + (α2 + λ(α1 − α2 ))η(x, y)) = f (y + α2 η(x, y) + λ(α1 − α2 )η(x, y)).

≤ 1.

(3.19)

From Condition C, using the same way as in theorems 3.5, we have η(y + α1 η(x, y), y + α2 η(x, y)) = (α1 − α2 )η(x, y).

(3.20)

By for any x, y ∈ K with x, y ∈ K with x 6= y, η(x, y) 6= 0, and α1 6= α2 , then, y + α1 η(x, y) 6= y + α2 η(x, y), thus,From (3.18)–(3.20), we obtain Φ(λα1 + (1 − λ)α2 )

= f (y + (α2 + λ(α1 − α2 ))η(x, y)) = f (y + α2 η(x, y) + λη(y + α1 η(x, y), y + α2 η(x, y))) < max{f (y + α1 η(x, y)), f (y + α2 η(x, y))} =

max{Φ(α1 ), Φ(α2 )}.

Hence, Φ(λ) is a strictly quasi-convex function in the interval ( 0, 1). Theorem 3.7. Let K ⊆ Rn be an invex set with respect to vector-valued map η : Rn × Rn −→ Rn and η satisfy Condition C. If f : K −→ R is a semistrictly prequasi-invex function for the same η on K, then for any x, y ∈ K, the function φ(λ) = f (y + λη(x, y)) is semistrictly quasi-convex function in the interval ( 0, 1). Proof. Suppose that f is semistrictly prequasi-invex with respect to η. From the definition of semistrict prequasi-invexity, for any x, y ∈ K, with f (x) 6= f (y), we have f (y + λη(x, y)) < max{f (x), f (y)},

∀λ ∈ ( 0, 1).

(3.21)

For any α1 , α2 ∈ ( 0, 1), ∀λ ∈ ( 0, 1). If Φ(α1 ) 6= Φ(α2 ), then,f (y +α1 η(x, y)) 6= f (y +α2 η(x, y)) and α1 6= α2 . Without loss of generality, 1 −α2 suppose that α1 > α2 , then α2 6= 1 and 0 < α1−α ≤ 1. Thus, we have 2 Φ(λα1 + (1 − λ)α2 )

= f (y + (α2 + λ(α1 − α2 ))η(x, y)) = f (y + α2 η(x, y) + λ(α1 − α2 )η(x, y)). 11

(3.22)

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From Condition C, we have η(y + α1 η(x, y), y + α2 η(x, y)) = (α1 − α2 )η(x, y).

(3.23)

From f (y + α1 η(x, y)) 6= f (y + α2 η(x, y)), (3.21)–(3.23), we obtain Φ(λα1 + (1 − λ)α2 )

= f (y + (α2 + λ(α1 − α2 ))η(x, y)) = f (y + α2 η(x, y) + λη(y + α1 η(x, y), y + α2 η(x, y))) < max{f (y + α1 η(x, y)), f (y + α2 η(x, y))} =

max{Φ(α1 ), Φ(α2 )}.

Hence, Φ(λ) is semistrictly quasi-convex function in the interval ( 0, 1).

References [1] T. Weir, B. Mond, Pre-invex functions in multiple objective optimization, J. Math. Anal. Appl. 136(1988) 29–38. [2] X.M. Yang, D. Li, Semistrictly preinvex functions, J. Math. Anal. Appl. 258(2001) 287–308. [3] X.M. Yang, D. Li, On properties of preinvex functions, J. Math. Anal. Appl. 256(2001) 229– 241. [4] X.M. Yang, Semistrictly convex functions, Opsearch, 31(1994) 15–27. [5] X.M. Yang, X.Q. Yang , D. Li, Characterizations and Applications of Prequasi-Invex Functions, J. Optim. Theory Appl. 110(2001) 647–668. [6] W.M. Tang, Q. Liu, X.M. Yang, The Sufficiency and Necessity Conditions of Strongly Prequasi-invex Functions, J. Or Transactions.11(2007) 21–30. [7] O. L. Mangasarian, Nonlinear Programming[M], McGraw-Hill, New York, 1969. [8] S.R. Mohan, S.K. Neogy, On invex set and preinvex functions, J. Math. Anal. Appl. 189(1995) 901–908. [9] X.M. Yang, X.Q. Yang, K.L.Teo., Criteria for generalized invex monotonicities, European J. Oper. Res. 164(2005) 115–119. [10] M.A. Noor, K.I. Noor, Some characterizations of strongly preinvex functions, J. Math. Anal. Appl. 316(2006) 697–706. [11] S. Karamardian, S. Schaible, Seven kinds of Monotone Maps, J. Optim. Theory Appl. 66(1990) 37–46.

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JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.7, 1309-1318, 2011, COPYRIGHT 2011 EUDOXUS 1309 PRESS, LLC

L p(p > 1) Convergence results for Particle Filtering ∗ Yuhua Guo

Xiaoyuan Yang

Department of Mathematics, Beihang University, LMIB of the Ministry of Education, Beijing 100191, China Corresponding author:

[email protected]

Abstract

In this paper, we prove L p -convergence for p > 1 of the particle filter for a class of unbounded functions. Furthermore, it can be shown that the approximation solution converges in probability to the true optimal estimate for the case 1 < p ≤ 2 and the approximation solution converges almost surely to the true optimal estimate for the case p > 2. In addition, some numerical experiments are presented to illustrate the main convergence results. Key words Bayesian estimation, Conditional expectation, Particle filter, Probability density function.

1

Introduction

Many problems in science require estimation of the state of a system. Bayesian approach provides a rigorous general framework for dynamic state estimation problems. The general solution of the state estimation problem is given by the Bayesian recursive relations which generate conditional probability density functions (pdf’s) of the unknown state of the system. The exact closed-form solution of the Bayesian recursive relations has been known only for linear Gaussian systems [1] and several special cases. However in many real-world applications, these linearity and Gaussianity assumptions are not valid and one needs to use numerical methods. The most widely used approach is the extended Kalman filter (EKF)[2], which is based on the Taylor series expansion of the state. However, the EKF is prone to divergence due to poor representations of the nonlinear functions. The problem of divergence of the EKF has been resolved by the unscented Kalman filter (UKF)[3, 4, 5], which uses deterministic sets of points in the space of the state variable to obtain more accurate approximations to the mean and covariance than the EKF. Unfortunately, the UKF has the shortcoming that it can’t be applied to general non-Gaussian distributions. Particle filter is an effective solution for handling nonlinear/non-Gaussian dynamic problems. Over the past years, this method has been applied in many different fields including computer vision, signal processing, and so on[8, 9]. Since particle filter has become the most common and useful method in approximating the optimal nonlinear filtering problem, it is necessary to consider the approximation solution converges to the true optimal estimate as the number of particles tends to infinity. The particles are interacting, thus, classical convergence results on Monte Carlo methods, based on independent and identically distribution (i.i.d.) assumptions, do not apply. An extensive treatment of the currently existing convergence results can be found in [10, 11]. However, these results only treated bounded test functions. From a practitioner viewpoint, it seems unsatisfactory to have to assume the test functions bounded to obtain some convergence results. The excellent paper [12] extended the existing convergence results to a class of unbounded test functions and proved L4 -convergence of the modified particle filter. Moreover, the existing L4 results were sufficient for almost sure convergence. Compared with the sampling ∗

This research was supported by the National Key Basic Research Program (973) of China under grant 2009CB724001.

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importance resampling (SIR) particle filter, the modified particle filter added a step to check whether the normalizing constant greater than a threshold γt . Hence the modified particle filter algorithm had random cost per iteration. However, this modification was motivated from the mathematics in the proof and leaded to an improved result in practice. Furthermore, it was related to the degeneracy of the particle weights[11]. In this paper, we prove some important inequalities based on the conditional expectation and obtain L p -convergence for p > 1 of the particle filter under reasonable conditions for a class of unbounded functions. In addition, numerical experiments are also provided. The rest of this paper is organized as follows. In Section 2, the notations we need are introduced and a brief description of a particle filter algorithm is given. In Section 3, some relevant inequalities are proved. Section 4 investigates the convergence results for 1 < p ≤ 2. In section 5 we present the case p > 2. Section 6 provides some simulation results. Finally, our contribution is summarized in Section 7.

2

Preliminaries

Before we proceed, we introduce some notations and represent the particle filter algorithms used throughout the paper. We begin with a complete probability space (Ω, C, P) on which the process X = {Xt } denotes the evolution of the hidden state of a dynamic system, and Y = {Yt } describes the observation process of the same system. The system process {Xt } is assumed to be a Markov process with state transition kernel K(dxt |xt−1 ) and the observation process is defined by Yt = ht (Xt ) + wt where wt is an i.i.d. noise process. The prior initial state distribution, denoted by τ0 (dx0 ). The transition from τt−1|t−1 to τt|t is defined using the Bayes recursion as follows: Z µ(yt |xt )τt|t−1 (dxt ) τt−1|t−1 (dxt ) → τt|t−1 (dxt ) = . (2.1) τt−1|t−1 (dxt−1 )K(dxt |xt−1 ) → τt|t (dxt ) = R µ(y |x )τ (dx ) Rn x t t t|t−1 t n x R Then (τt|t−1 , ψ) = (τt−1|t−1 , τψ) → (τt|t , ψ) =

(τt|t−1 , ψµ) (τt|t−1 , µ)

(2.2)

Hence, E(ψ(xt )|y1:t ) = (τt|t , ψ). Obviously, (τt|t−1 , µ) > 0 by the definition of the conditional expectation. Let the filter state estimate xˆt = E(xt |Y1:t ) be the mean of the conditional distribution τt|t (dxt ) = P(Xt ∈ dxt |Y1:t = y1:t ) and xˆtN be an approximation of xˆt = E(xt |Y1:t ). In the following, the particle filter, as it was introduced in [7], will be referred to as the standard particle filter. For a thorough introduction to the standard particle filter, see [11] and [12]. We define a new set of weights β i according to β i = (β i1 , β i2 , . . . , β iN )

(2.3)

satisfying β ij ≥ 0,

N X

β ij = 1,

j=1

N X

β ij = 1.

(2.4)

i=1

Note that if β ij = 1 for j = i, and β ij = 0 for j , i, then we can obtain the standard particle filter algorithm. The extended particle filter proceeds as follows. Algorithm 2.1 The extended particle filter algorithm[12] N ∼ τ (dx ). 1) Initialize the particles, {x0i }i=1 0 0 2) Predict the particles by drawing independent samples according to x˜ti ∼ 1, . . . , N. 2

N P j=1

j

β ij K(dxt |xt−1 ), i =

GUO, YANG: PARTICLE FILTERING

1311

N , ωi = µ(y | x˜ i ), i = 1, . . . , N, and normalize ω Compute the importance weights {ωit }i=1 ˜ it = t t t N P j ωit / ωt .

3)

j=1

N (dx ), i = 1, . . . , N. 4) Resample, xti ∼ τ˜ t|t t

It is clear that the optimal filter recursion requires that (τt|t−1 , µ) > 0.

(2.5)

In the step 3 of Algorithm 2.1 we have used: N (τt|t−1 , µ) ≈ (˜τt|t−1 , µ) =

N 1 X µ(yt | x˜ti ). N i=1

(2.6)

Then the filter algorithm has to be modified to ensure that (2.5) is fulfilled. The modification is to choose a threshold γt so that N 1 X (τt|t−1 , µ) = µ(yt | x˜ti ) ≥ γt > 0. N i=1

(2.7)

The detail of the modified particle filter can be found in [12]. Thus, the modified algorithm can be described as in Algorithm 2.2. Algorithm 2.2 The Modified Particle Filter N ∼ τ (dx ) 1) Initialize the particles, {x0i }i=1 0 0 2) Predict the particles by drawing independent samples according to x¯ti ∼ 1, . . . , N. 3) If (1/N)

N P i=1

N P j=1

j

β ij K(dxt |xt−1 ), i =

µ(yt | x¯ti ) ≥ γt , proceed to step 4 otherwise return to step 2.

N , and normalize Rename x˜ti = x¯ti , i = 1, . . . , N and compute the importance weights {ωit }i=1 N P j ω ˜ it = ωit / ωt .

4)

j=1

N (dx ) = 5) Resample, xti ∼ τ˜ t|t t

N P i=1

ω ˜ it δ x˜ti (dxt ), i = 1, . . . , N.

f) Set t:=t+1, and repeat from step 2. As we will show in Section 4, the modified algorithm will not run into an infinite loop in step 2-3 if γt is chosen small enough.

3

Some important inequalities

In this section we present some inequalities with respect to the conditional expectation. This inequalities play an important role in establishing the main convergence results. We can get Lemma 3.1 by the Marcinkiewicz-Zygmund (M-Z) inequality. It is easy to obtain the following Lemmas 3.2-3.3. For the sake of brevity we omit their proof. Lemma 3.1 For 1 < p ≤ 2, let {ζi , i = 1, · · · , n} be conditionally independent random variables given σ- algebra F such that E(ζi |F ) = 0, E(|ζi | p |F ) < ∞. Then there exists a constant B p depending only on p, such that E(|

n X

p

ζi | |F ) ≤ B p E (

i=1

n X i=1

3

p
|ζi |2 ) 2 |F .

(3.1)

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GUO, YANG: PARTICLE FILTERING

Lemma 3.2 For p > 2, let {ζi , i = 1, · · · , n} be conditionally independent random variables given σalgebra F such that E(ζi |F ) = 0, E(|ζi | p |F ) < ∞, then E(|

n X

n n X X p p ζi | |F ) ≤ K p max{ E(|ζi | |F ), ( E(|ζi |2 |F )) 2 }, p

i=1

i=1

(3.2)

i=1

where K p is a constant that depends only on p. Lemma 3.3 Let 1 < p ≤ 2. Suppose that {ζi , i = 1, · · · , n} are conditionally independent random variables given σ-algebra F with E(ζi |F ) = 0, such that E(|ζi | p |F ) < ∞, for all i. Then E|

n X

p

ζi | |F ) ≤ 2

p

n X

i=1


E(|ζi | p |F .

(3.3)

i=1

Lemma 3.4 Let 1 < p ≤ 2. Suppose that {ζi , i = 1, · · · , n} are conditionally independent random variables given σ-algebra F with E(ζi |F ) = 0, such that E(|ζi | p |F ) < ∞, for all i. Then 2p 1X p
E| ζi | |F ≤ p−1 max E(|ζi | p |F ). 1≤i≤n n i=1 n n

(3.4)

Proof: Using Lemma 3.3, it is clear that X
1X p
ζi ζi | |F ≤ 2 p E | | p |F n i=1 n i=1 n

E| =

n

n 2p X 2p p max E(|ζi | p |F ). |F ) ≤ (E|ζ | i n p i=1 n p−1 1≤i≤n

This completes the proof of the Lemma. Lemma 3.5 Let p > 2. Suppose that {ζi , i = 1, · · · , n} are conditionally independent random variables given σ-algebra F with E(ζi |F ) = 0, such that E(|ζi | p |F ) < ∞, for all i. Then E(|

n Kp 1X p ζi | |F ) ≤ p max E(|ζi | p |F ). n i=1 n 2 1≤i≤n

(3.5)

Proof: By Lemma 3.2, we have n n n X p X 1X p 1 p E(| ζi | |F ) ≤ p K p max E(|ζi | |F ), ( E(|ζi |2 |F )) 2 . n i=1 n i=1 i=1

(3.6)

Notice that [

n X

p

n P i=1

p

2

p

1≤i≤n

i=1

and

p

p

E(|ζi |2 |F )] 2 ≤ n 2 max [(E(|ζi |2· 2 |F )) p ] 2 = n 2 max E(|ζi | p |F ) 1≤i≤n

(3.7)

E(|ζi | p |F ) ≤ n × max E(|ζi | p |F ), then 1≤i≤n

1 1 1X p ζi | |F ) ≤ K p max{ p max E(|ζi | p |F ), p−1 max E(|ζi | p |F )}. E(| 1≤i≤n 1≤i≤n n i=1 n n2 n

p

Since p > 2, then 1/n 2 > 1/n p−1 . Thus the assertion follows. 4

(3.8)

GUO, YANG: PARTICLE FILTERING

4

1313

L p -convergence for an arbitrary 1 < p ≤ 2

In this section, we focus on establishing L p -convergence results for 1 < p ≤ 2 of unbounded test functions arising from Algorithm 2.2. Let us list the following assumptions needed for the proof of Theorem 4.1 and 5.1. A0 : For given y1:s , s = 1, . . . , t, (τ s|s−1 , µ) > 0; and the constant γ s used in Algorithm 2.2 satisfies 0 < γ s < (τ s|s−1 , µ), s = 1, . . . , t.

(4.9)

A1 : g(y s |x s ) < ∞; K(x s |x s−1 ) < ∞ for given y1:s , s = 1, . . . , t. A2 : Let p > 1. The function ψ(·) satisfies sup xs |ψ(x s )| p µ(y s |x s ) < A(y1:s ) for given y1:s , s = 1, . . . , t. Note that A1 and A2 imply that the conditional moment for any p > 1 of ψ is bounded, that is R Z |ψ(x)| p µ(y s |x)τ s|s−1 (dx) p |ψ(x)| τ s|s (dx) = (τ s|s−1 , µ) R Cy1:s τ s|s−1 (dx) ≤ < ∞. (τ s|s−1 , µ) Furthermore, the set of all bounded functions is a subset of the class of functions specified in A2. Let p Lt (g) be the class of functions ψ satisfying A2, where g satisfies A1. Using the above assumptions, we have the following theorem. Theorem 4.1 Let 1 < p ≤ 2, under assumptions A0, A1 and A2 , there exists At|t independent of N such p that for any ψ ∈ Lt (g), we have p

N E|(τt|t , ψ) − (τt|t , ψ)| p ≤ At|t

k ψ kt,p N p−1

,

(4.10)

1

where k ψ kt,p = max{1, (τ s|s , |ψ| p ) p , s = 0, 1, . . . , t} and τNs|s is generated by Algorithm 2.2. Proof: Since most of the computations are similar to those made in the proof of Theorem 6.1 of [12], the proof will be only sketched. N be independent random variables with the same distribution τ (dx ). Applying Initialization: Let {x0i }i=1 0 0 lemma 3.4, we obtain X 1 E| (ψ(x0i ) − E(ψ(x0i )))| p p N i=1 N

E|(τ0N , ψ) − (τ0 , ψ)| p =

p

≤

k ψ k0,p 22p p , A k ψ k . 0|0 0,p N p−1 N p−1

(4.11)

Similarly E|(τ0N , |ψ| p ) − (τ0 , |ψ| p )| ≤ 2E|ψ(x0i )| p .

(4.12)

p

(4.13)

Then E|(τ0N , |ψ| p )| ≤ 3E|τ(x0i )| p , H0|0 k ψ k0,p . p

Prediction : Let us assume that for any ψ ∈ Lt (µ) p

N E|(τt−1|t−1 , ψ)

p

− (τt−1|t−1 , ψ)| ≤ At−1|t−1

k ψ kt−1,p N p−1

(4.14)

and p

N E|(τt−1|t−1 , |ψ| p )| ≤ Ht−1|t−1 k ψ kt−1,p

5

(4.15)

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GUO, YANG: PARTICLE FILTERING

i }N . Notice holds, where At−1|t−1 > 0 and Ht−1|t−1 > 0. Let Gt−1 denote the σ-algebra generated by {xt−1 i=1 that N (˜τt|t−1 , ψ) − (τt|t−1 , ψ) , Γ1 + Γ2 + Γ3

where N Γ1 , (˜τt|t−1 , ψ) −

N 1 X E[ψ( x˜ti )|Gt−1 ] N i=1

N N 1 X 1 X N,βi i Γ2 , E[ψ( x˜t )|Gt−1 ] − (τ , Kψ) N i=1 N i=1 t−1|t−1

Γ3 , N,β

i = and τt−1|t−1

N P j=1

N 1 X N,βi (τ , Kψ) − (τt|t−1 , ψ) N i=1 t−1|t−1

β ij δ x j (dxt−1 ). Let us now introduce the notation t−1

N,β

i i , Kψ), E[ψ( x¯t−1 )|Gt−1 ] = (τt−1|t−1

(4.16)

N,β

i where x¯ti be drawn from the distribution (τt−1|t−1 , K) as in step 2 of Algorithm 2.2. By (4.16) and Chebyshev inequality, we obtain

p

N P[(τt−1|t−1 , Kµ)

p

k µ kt−1,p At−1|t−1 k K k p k µ kt−1,p < γt ] ≤ · , A . · γ t |γt − (τt|t−1 , µ)| p N p−1 N p−1

(4.17)

By Lemma 3.3 and (4.16), it is clear that p

E|Γ1 |

p

p

k ψ kt−1,p 22p k K k p Ht−1|t−1 k ψ kt−1,p · , A · . Γ 1 1 − t N p−1 N p−1

≤

Using (4.16), we have that p2

p

E|Γ2 | p ≤

≤

p 2 γt

2 2 p Aγ2t k µ kt−1,p

(1 − t ) p N

p(p−1) 2

N (τt−1|t−1 , Kψ p )

p2

p

p

2 2 p A k µ kt−1,p k K k k ψ kt−1,p k ψ kt−1,p · , A k K k · . Γ 2 (1 − t ) p N p−1 N p−1

Besides, p

p

p

E|Γ3 | ≤ At−1|t−1 · k K k ·

k ψ kt−1,p N p−1

p

, AΓ3

k ψ kt−1,p N p−1

.

Thus, N E|(˜τt|t−1 , ψ)

− (τt|t−1 , ψ)|

p

1 p

1 p

1 p

≤ (AΓ1 + [AΓ2 k K k] + AΓ3 ) , A˜ t|t−1

k

p ψ kt−1,p . N p−1

p

p

k ψ kt−1,p N p−1

Since N E E (˜τt|t−1 , |ψ| p ) −

N

1 X N,βi p (τt−1|t−1 , K|ψ| p )|Gt ≤ 2 k K k p Ht−1|t−1 k ψ kt−1,p . N i=1

6

(4.18)

GUO, YANG: PARTICLE FILTERING

1315

by applying (4.15), it follows that p p N E|(˜τt|t−1 , |ψ| p ) − (˜τt|t−1 , |ψ| p )| ≤ (3Ht−1|t−1 + 1) k K k p k ψ kt−1,p , H˜ t|t−1 k ψ kt−1,p .

U pdate : Clearly, N (˜τt|t , ψ)

− (τt|t , ψ) =

N , µψ) (˜τt|t−1

=

N , µψ) (˜τt|t−1 N , µ) (˜τt|t−1

−

(τt|t−1 , µψ) (τt|t−1 , µ)

(˜τt|t−1 , µψ) (˜τt|t−1 , µψ) (τt|t−1 , µψ) + − . (τt|t−1 , µ) (τt|t−1 , µ) (τt|t−1 , µ)

(4.19)

(˜τt|t−1 , µψ) k µψ k N |≤ |(τt|t−1 , µ) − (˜τt|t−1 , µ)|. (τt|t−1 , µ) γt (τt|t−1 , µ)

(4.20)

N , µ) (˜τt|t−1

−

where |

N , µψ) (˜τt|t−1 N , µ) (˜τt|t−1

−

Here, γt is the threshold used in step 3 of the Algorithm 2.2. Thus, 1

N E|(˜τt|t , ψ)

− (τt|t , ψ)|

p

≤ [

p k µ k (k µψ k +γt ) A˜ t|t−1

γt (τt|t−1 , µ)

p

]

p

k ψ kt−1,p N p−1

p

, A˜ t|t

k ψ kt−1,p N p−1

.

(4.21)

Similarly, k µψ p k ·2 k µ k H˜ t|t−1 max{k µ k, 1} p + k ψ kt−1,p . γt (τt|t−1 , µ) (τt|t−1 , µ)

N N E|(˜τt|t , |ψ| p ) − (τt|t , |ψ| p )| ≤

(4.22)

Since k ψ k s,p ≥ 1 is increasing with respect to s, we have N E|(˜τt|t , |ψ| p )| ≤ 3max

 k µψ p k ·2 k µ k H˜ t|t−1 max{k µ k, 1} p p , , 1 · k ψ kt,p , H˜ t|t k ψ kt,p . γt (τt|t−1 , µ) (τt|t−1 , µ)

(4.23)

Resampling : It is easy to see that N N N N (τt|t , ψ) − (τt|t , ψ) = (τt|t , ψ) − (˜τt|t , ψ) + (˜τt|t , ψ) − (τt|t , ψ).

(4.24)

Then, we have 1

N E[|(τt|t , ψ) − (τt|t , ψ)| p ] p

1

N ≤ E[|(τt|t , ψ) − (˜τt|t , ψ)| p ] p 1

N + E[|(˜τt|t , ψ) − (τt|t , ψ)| p ] p .

(4.25)

N . Notice that Let Ct be the σ-field generated by { x˜t(i) }i=1 N N E[(τt|t , ψ)|Ct ] = (˜τt|t , ψ)

(4.26)

and p

N E[|(τt|t , ψ) − (˜τt|t , ψ)| p |Ct ] ≤ 2 p B p H˜ t|t

k ψ kt,p

(4.27)

N p−1

giving 1

N E p |(τt|t , ψ) − (τt|t , ψ)| p

1

1

≤ ([2 p B p H˜ t|t ] p + A˜ t|tp ) 1

, At|tp 7

k ψ kt,p N

p−1 p

.

k ψ kt,p N

p−1 p

(4.28)

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GUO, YANG: PARTICLE FILTERING

Thus, p

N E|(τt|t , ψ)

− (τt|t , ψ)|

p

≤ At|t

k ψ kt,p N p−1

.

Furthermore, by (4.25) we have p

N E|(τt|t , |ψ| p ) − (τt|t , |ψ| p )| ≤ (3H˜ t|t + 1) k ψ kt,p .

(4.29)

Hence p

p

N E|(τt|t , |ψ| p )| ≤ (3H˜ t|t + 2) k ψ kt,p , Ht|t k ψ kt,p .

(4.30)

Consequently, we obtain the assertion of the theorem. By Chebyshev inequality, we have the immediate consequence below. p

N , ψ) converges in pr. to Corollary 4.1.1 Let 1 < p ≤ 2, if A1 and A2 hold, then for any ψ ∈ Lt (ρ), (τt|t N , ψ) − (τ , ψ)| > } = 0. (τt|t , ψ). That is, for every  > 0, we have lim P{|(τt|t t|t N→∞

5

L p -convergence for an arbitrary p > 2

This section presents the L p -convergence result for p > 2 of the particle filter. Similar to the proof of the Theorem 4.1, we have the following Theorem 5.1. Theorem 5.1 Under assumptions A0, A1 and A2 , there exists A0t|t independent of N such that for any p ψ ∈ Lt (g) with p > 2, p

N E|(τt|t , ψ)

p

− (τt|t , ψ)| ≤

A0t|t

k ψ kt,p p

N2

,

(5.31)

1

where k ψ kt,p = max{1, (τ s|s , |ψ| p ) p , s = 0, 1, . . . , t} and τNs|s is generated by Algorithm 2.2. Consequently, by Chebyshev inequality and the Borel-Cantelli lemma [13], we have a corollary as follows. p

Corollary 5.1.1 Let p > 2, if A1 and A2 hold, then for any ψ ∈ Lt (ρ) N lim (τt|t , ψ) = (τt|t , ψ), almost surely.

N→∞

6

(5.32)

Numerical Illustration

In this section we provide simulation results obtained for the problem described in the previous Theorems. The nonlinear process model and measurement model are as follows [12]. xt+1 = yt =

xt 25xt + + 8cos(1.2t) + vt , vt ∼ N(0, 10). 2 1 + xt2 xt2 + wt , wt ∼ N(0, 1). 20

The initial state was x0 ∼ N(0, 5) and γt = 10−4 . We used 300 time instants and 500 simulations.

8

GUO, YANG: PARTICLE FILTERING

1317

Numerical Illustration 6.1

1800 1600 1400 1200 1000 800 600 400 200 0 400

300 200 100 350

300

250

200

150

100

50

0 t

N

Fig.1. The figure shows the second moment convergence.

Numerical Illustration 6.2

6

x 10 3.5 3 2.5 2 1.5 1 0.5

300 200

0 400

350

100 300

250

200

150 N

100

50

0 t

Fig.2. The figure shows the fourth moment convergence.

In figure 1 and figure 2, the vertical axis was denoted by the error xt − xˆt , where xˆt was computed from Algorithm 2.2. As expected, the errors in performance were decreased as the number of particles N was increased.

7 Conclusion The main contribution of this paper is that we have proved the L p -convergence for p > 1 of the modified particle filter for a class of unbounded test functions. Apparently, the L4 -convergence is a special case. In addition, some numerical results have been shown to illustrate the convergence results. The case of p = 1 will be further discussed in future. 9

1318

8

GUO, YANG: PARTICLE FILTERING

Acknowledgement

The authors gratefully thanks the anonymous reviewers for their careful reading of earlier version of this work.

References [1] B. D. O. Anderson and J. B. Moore, Optimal filtering, Englewood Cliffs, NJ: Prentice-Hall, 1979. [2] A. Gelb, J. F. Kasper, R. A. Nash, C. F. Price, A. A. Sutherland, Applied optimal estimation. Cambridge, MA: MIT Press, 1974. [3] S. J. Julier, J. K. Uhlmann, A general method for approximation nonlinear transformations of probability distributions, Technical Report, RRG, Department of Engineering Science, University of Oxford, 1996. [4] S. J. Julier, J. K. Uhlmann, Unscented filtering and nonlinear estimation, IEEE. Proceedings, vol. 92, pp. 401-422, 2004. [5] J. H. Kotecha, P. M. Djuri´c, Gaussian sum particle filtering, IEEE Trans. Signal Process., vol. 51, no. 10, pp. 2602-2612, Oct. 2003. J. Carpenter, P. Clifford, and P. Fearnhead, An improved particle filter for nonlinear problems, in Proc. Inst. Elect. Eng. Radar Sonar Navigat., vol. 146, pp. 2-7, 1999. [6] A. Doucet, N. de Freitas, and N. Gordon, Eds., Sequential Monte Carlo methods in practice. New York: Springer Verlag, 2001. [7] N. J. Gordon, D. J. Salmond, A. F. M. Smith, Novel approach to nonlinear and non-Gaussian Bayesian state estimation, Proc. Inst. Elect. Eng., Rader S ignal Process., vol. 140, pp. 107113, 1993. [8] C. S. Manohar, D.Roy, Monte Carlo filters for identification of nonlinear structural dynamical systems. Sadhana Acad. Proc. Eng. Sci. 31, pp. 399-427, 2006. [9] S. J. Ghosh, C. S. Manohar, D. Roy, A sequential importance sampling filter with a new proposal distribution for state and parameter estimation of nonlinear dynamical systems. Proc. R. Soc. Lond. Ser. A 464, pp. 25-47, 2007. [10] T. Bando, T. Shibata, K. Doya and S.Ishii, Switching particle filters for efficient visual tracking, Elsevier B.V., vol. 54, pp: 873-884, 2006. [11] D. Crisan and A. Doucet, A survey of convergence results on particle filtering methods for practitioners, IEEE Trans. Signal Process., vol. 50, no. 3, pp. 736-746, Mar. 2002. [12] X. L. Hu, T. B. Schon, ¨ and L. Ljung, A basic convergence result for particle filtering, IEEE Trans. On Signal Processing, vol. 56, pp. 1337-1347, 2008. [13] K. L. Chung, A course in probability theory, 2nd ed. New York: Academic, Probability and Mathematical Statistics. vol. 21, 1974. [14] H. Rosenthal, On the subspaces of l p (p > 2) spanned by sequences of independent random variables, I sraelJ.Math., vol. 8, pp. 273-303, 1970.

10

JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.7, 1319-1328, 2011, COPYRIGHT 2011 EUDOXUS 1319 PRESS, LLC

A Derivative-Free Method for Nonlinear System of Equations with Global Convergence∗ Jianguo Zhang†, Yun-Hai Xiao†,‡ and Dangzhen Zhou†

Abstract In this paper, we propose a simple derivative-free method for solving large-scale nonlinear system of equations. It comes from a simple sufficient descent method [17] for unconstrained optimization problems. A remarkable property of the proposed method is that it can solve nonlinear system without requiring Jacobian matrix information. It is also suitable to large-scale equations due to its lower storage requirement. Under appropriate conditions, we show that the method with nonmonotone derivative-free line search is globally convergent. Preliminary numerical results show that the proposed method is promising.

Key words. nonlinear system, sufficient descent direction, derivative-free method, nonmonotone line search, global convergence AMS subject classifications. 90C25 65H10

1.

Introduction

In this paper, we concerned with the development of a derivative-free method for solving the nonlinear system of equations F (x) = 0, x ∈ Rn , (1.1) where F : Rn → Rn is a continuous and monotone mapping. Many methods for this problem fall into the Newton and quasi-Newton category (e.g. [2, 8, 12, 13, 14]). A general quasi-Newton method for (1.1) generates a sequence of iterates {xk } by letting xk+1 = xk + αk dk , where αk is a steplength, and dk is a solution of the system of linear equations: Bk d + F (xk ) = 0.

(1.2)

If matrix Bk is replace by the Jacobian of F at xk , and αk ≡ 1, this method reduces to the well-known Newton method. The main advantage of a quasi-Newton method is its local superlinear convergence rate without requiring computation of Jacobians [10]. Despite the quasi-Newton method enjoy some attractive properties, still it need to solve a linear system of equations at each iteration using the Jacobian matrix or an approximation of it. For this reason, this method is quite efficient for nonlinear system with relatively small dimensions, but unattractive for largescale case. In this paper, we are interested in the case where n is large, or the Jacobian of F is not available, or requires a prohibitive amount of storage. In this case, to solve the problem, the derivativefree methods are welcome. ∗ The

work was supported by Chinese NSF granted 10761001. of Applied Mathematics, College of Mathematics and Information Science, Henan university, Kaifeng, 475004, P. R. China. (Email: [email protected], [email protected]) ‡ Current Address: Department of Mathematics, Nanjing University, Nanjing, 210093, P. R. China. † Institute

1

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To solve problems (1.1) with high dimensions, the derivative-free or matrix free methods have received much attentions in these years. The first work is due to Cruz, Mrt´ınez, and Raydan [6], in which they extended the two-pint stepsize method [1] to solve (1.1), and developed the Spectral Algorithm for Nonlinear Equations (SANE). SANE firstly uses the residual ±F (xk ), in a systematic way, as a search direction. The first trial point at each iteration is xk − σk F (xk ), where σk is a spectral coefficient. Global convergence is guaranteed by means of a variation of the nonmonotone strategy of Grippo, Lampariello and Lucidi (GLL) [9]. Moreover, to improve the performance of SANE, Cruz, Mart´ınez, and Raydan [7] develped a so-called derivative-free algorithm DF-SANE. DF-SANE uses the same direction and the same initial steplength as SANE. However, a derivative-free line search which based on LF condition [12] and GLL condition [9] is used to guardee the global convergence. DF-SANE is truly a derivative-free method, and numerical results show that the line search benefits to its performance. Recently, to avoid the drawbacks of GLL line search, Cheng and Li [4] appealed the nonmonotone line search of Zhang & Hager to take the place of GLL one in DF-SANE, and developed a new derivative-free spectral residual method N-DF-SANE. The reported numerical experiments on a large set of test problems show that N-DF-SANE is very promising. As it is well-known, conjugate gradient method, due to its simplicity and low storage, it is very suit for solving large-scale unconstrained minimization problems. However, the work of conjugate gradient methods for solving large-scale nonlinear equations is relatively fewer. To best of our knowledge, [5] is the only regular paper we can exceed. The generated directions of the method in [5] can be regarded as a convex combination of two sufficient descent conjugate gradient method in [16, 3] for unconstrained optimization. There, they were applied in a different environment. In this paper, we also develop an efficient derivative-free algorithm for solving (1.1). Our work can be considered as a further reaserch of the simple sufficient descent method of Zhang, Xiao, and Zhou [17] in unconstrained optimization. Here, we extend it to solving nonlinear equations with some modifications. An attractive feature of this algorithm is that the Jacobian of F is fully not used. Moreover, preliminary numerical results indicate that the proposed method performs well. The reminder of this paper is organized as follows. In the next section, we aims to construct our new algorithm with a derivative-free line search. In section 3, we establish the global convergence result under some appreciate conditions . Finally, we do some experiments by using some large-scale nonlinear equations to show the efficiency of the method. Throughout the paper, J(x) denotes the Jacobian matrix of F computed at x, N is the set of natural numbers. Symbol k · k denotes the Eucilidean norm of vectors.

2.

Algorithm

In this section, we firstly focus on the unconstrained optimization problem min f (x) : x ∈ Rn ,

(2.1)

where f : Rn − R is a continuously differentiable function, and its gradient at point xk is denoted by g(xk ), or gk for the sake of simplicity. The simple sufficient descent method (SSD) of Xiao et al. [17] is to generate a sequence {xk } such that xk+1 = xk + αk dk ,

k = 0, 1, . . .

where the steplength αk is determined by a line search, and the search direction dk is generated by ( −g0 , ³ if k = 0, ´ T (2.2) dk = gk gk −gk + I − kgk k2 gk−1 , if k > 0, 2

ZHANG ET AL: NONLINEAR SYSTEM OF EQUATIONS

1321

where I is an identify matrix. Obviously, the generated direction is a linear combination of gradient at current at previous step. It is easy to see that dk satisfies gkT dk = −kgk k2 ,

(2.3)

and this property is independent of any line search which is used. Consequently, (2.3) shows that vector dk is a sufficient descent direction of f at xk . The reported numerical results show that the SSD method works well. As an attempt, the better performance and the simple iterative form of SDD motivated us to extend it to solve the nonlinear system of equations (1.1). Similar with Eq. (2.2), we define the search directions as ( −F0 , if k = 0, ³ ´ dk = (2.4) Fk FkT −Fk + λk I − kFk k2 Fk−1 , if k > 0, where λk is a scalar and trends to zero as k → ∞. Clearly, the definition of dk does’t need the Jacobian information of F . Moreover, the following equality always holds without requiring any line search: dTk Fk = −kFk k2 . We know that, many numerical methods for solving (2.1) are mainly to solve the nonlinear system of equations g(x) = 0, and using kg(x)k2 as a merit function to globalize the process. Therefore, a natural connection for solving (1.1) is to apply the same technique but now forcing F (x) = 0 and using f (x) =

1 kF (x)k2 , 2

(2.5)

as a merit function. From now on, we abbreviate f (xk ) as fk for the sake of simplicity. The main purpose of this paper is to use the direction dk defined in (2.4) to solve (1.1). Obviously, the search direction dk may not be a descent direction for (2.5), hence we replace dk with −dk in this case. In order to get the global convergence of the proposed method, we consider the following derivative-free nonmonotone line search. That is find αk satisfies f (xk + αk dk ) ≤ Ck + ²k − γαk2 f (xk ) where γ ∈ (0, 1), C0 = f0 , the given positive sequence {²k } satisfies Qk+1 = ηk Qk + 1, Ck+1 =

(2.6)

P∞ k=0

< ∞, and Ck is updated by

ηk Qk (Ck + ²k ) + fk+1 , Qk+1

with Q0 = 1 and ηk ∈ [0, 1]. The nonmonotone line search is originated from Li & Fukushima [12], and Zhang & Hager [15], which has been proved very efficient when its applied in unconstrained optimization [11] and nonlinear system [4]. Now we are ready to state the steps of the derivative-free SSD method for nonlinear system of equations as follows. Algorithm 2.1. (DF-SSD algorithm) Step 0. Given an initial point x0 ∈ Rn , constants ² > 0, 0 ≤ ηmin ≤ ηmax ≤ 1, 0 < ρmin < ρmax < 1, β > 0, and γ > 0. Choose a positive sequence {²k } satisfies ∞ X

²k ≤ ².

k=0

3

(2.7)

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Set C0 = f0 , Q0 = 1, and k = 0. Step 1. If kF (xk )k = 0, then stop. Step 2. Compute dk by (2.4). Set α+ = β and α− = β. Step 3. Nonmonotone line search. If 2 f (xk + α+ dk ) ≤ Ck + ²k − γα+ f (xk ),

(2.8)

then set αk = α+ , xk+1 = xk + αk dk . Else if 2 f (xk − α− dk ) ≤ Ck + ²k − γα− f (xk ),

(2.9)

then set αk = α− , xk+1 = xk − α− dk . Else choose α+new ∈ [ρmin α+ , ρmax α+ ], α−new ∈ [ρmin α− , ρmax α− ]. Replace α+ = α+new , α− = α−new and go to Step 3. End if Step 4. Choose ηk ∈ [ηmin , ηmax ] and compute Qk+1 = ηk Qk + 1, Ck+1 =

ηk Qk Ck + fk+1 , Qk+1

(2.10)

Step 5. Let k = k + 1. Go to Step 1. Let Ak be the average function value for k, i.e., k

Ak =

1 X fi . k + 1 i=0

(2.11)

The following result shows that for any choice of ηk , Ck lies between fk and Ak . The proof of this lemma can be found in [4, Lemma 2.2]. Lemma 2.1. The iterates generated by Algorithm 2.1 satisfy fk ≤ Ck ≤ Ak for all k. Moreover, the sequence {Ck } satisfies Ck ≤ Ck−1 + ²k−1 .

3.

Convergence analysis

This section is devoted to establish the global convergence of Algorithm 2.1. The following assumption is very important to analyze the global convergence result. Assumption 3.1. The level set Ω = {x : f (x) ≤ f (x0 ) + ²} is bounded, where ² is defined in (2.7). Assumption 3.2. In some neighborhood N of Ω, the mapping F (x) is Lipschitz continuous, i.e., there exists a constant L > 0 such that kF (x) − F (y)k ≤ Lkx − yk, ∀ x, y ∈ N .

(3.1)

The assumptions show that there exists a positive constant γ¯ such that kg(x)k ≤ γ¯ ,

∀ x ∈ Ω.

From Step 4 in Algorithm 2.1, we know that f (xk+1 ) ≤ Ck + ²k . 4

(3.2)

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From Lemma 2.1, we have f (xk+1 ) ≤ Ck−1 + ²k−1 + ²k ≤ ··· ≤ f (x0 ) +

∞ X

²k

k=0

≤ f (x0 ) + ².

(3.3)

Consequently, combining with Assumption 3.1, (3.3) shows that the generated sequence {xk } is contained in Ω. The following lemma comes from [4, Lemma 3.2], we list here without proof for conversance. Lemma 3.1. Let the sequence {xk } be generated by Algorithm 2.1. Then there exists an infinite index set K ⊂ N such that lim αk2 f (xk ) = 0. (3.4) k∈K

Moreover, if ηmax < 1 then lim αk2 f (xk ) = 0.

k→∞

(3.5)

Using the preceding lemmas, we will prove that at every limit x∗ of the subsequence {xk }K one necessarily has that F (x∗ )T J(x∗ )F (x∗ ) = 0. In other words the gradient of kF (x)k2 at x∗ is orthogonal to the residual F (x∗ ). Theorem 3.1. Suppose that Assumptions 3.1-3.2 hold. Let the sequence {xk } is generated by Algorithm 2.1 and ηmax < 1. Then we have lim inf kF (xk )k = 0, (3.6) k→∞

∗

or every limit point x of {xk } satisfies F (x∗ )T J(x∗ )F (x∗ ) = 0.

(3.7)

Proof. Let x∗ be any limit point of {xk } and let K1 ⊂ N be an infinite sequence of indices such that limk∈K1 xk = x∗ . By (3.5), we have limk∈K1 αk kFk k = 0. Therefore, the proof can be divided into two cases. Case I: If lim supk∈K αk 6= 0, then there exists an infinite set K2 ⊂ K1 such that αk is bounded away from zero for all k ∈ K2 . By (3.5), we have limk∈K2 kF (xk )k = 0. Since F is continuous and limk∈K2 xk = x∗ . This implies that limk∈K2 kF (x∗ )k = 0, thus (3.6) holds. Case II: If lim αk = 0, (3.8) k∈K1

then there exists an index k0 ∈ K1 such that αk < 1 for all k > k0 with k ∈ K1 . From (3.8), we can suppose that, at iteration k, the number of inner loops in Step 4 is denoted by mk . Let αk+ and αk− be the values of α+ and α− , respectively, in the last unsuccessful steplength. By the choice of α+new and α−new in Step 3 of the Algorithm 2.1, we have that k αk ≥ ρm min

for all k > k0 , k ∈ K1 . By (3.8), we have limk∈K1 mk = ∞. From the choice of α+new and α−new , we have k −1 αk+ ≤ ρm max , and k −1 αk− ≤ ρm max .

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Since ρmax < 1 and lim mk = ∞, we can get k∈K1

lim αk+ = lim αk− = 0.

k∈K1

k∈K1

By the line search rule, we have for all k ∈ K1 with k ≥ k0 , 2 f (xk + α+ dk ) > Ck + ²k − γα+ f (xk ),

(3.9)

2 f (xk + α− dk ) > Ck + ²k − γα− f (xk ),

(3.10)

and By using Lemma 2.1, we have 2 f (xk + α+ dk ) > fk − γα+ f (xk ),

and 2 f (xk + α− dk ) > fk − γα− f (xk ),

Because (3.3) implies that fk ≤ f0 + ²k = C, then we have 2 f (xk + α+ dk ) − fk > −γCα+ ,

(3.11)

2 f (xk + α− dk ) − fk > −γCα− ,

(3.12)

kF (xk + α+ dk )k2 − kFk k2 > −2Cα+ . α+

(3.13)

and From (3.11), we obtain that

By the mean value theorem and (3.13), there exists a ξk ∈ (0, 1) such that hJ(xk + ξk α+ dk )T F (xk + ξk α+ dk ), dk i > −2Cα+ . Therefore, we get from (2.4) that hJ(xk + ξk α+ dk )T F (xk + ξk α+ dk ), Fk − λk Hk Fk−1 i < −2Cα+ .

(3.14)

Notice that λk → 0 as k → ∞, and take limits in (3.14), we obtain that F (x∗ )T J(x∗ )F (x∗ ) ≤ 0. Using (3.12) and proceeding an analogous way, we obtain F (x∗ )T J(x∗ )F (x∗ ) ≥ 0. The last two inequalities imply (3.7). If the mapping F (x) is strict and admits a solution, its solution must be unique, which can be stated as the following corollary. Corollary 3.1. Suppose that Assumptions 3.1-3.2 hold. Let the sequence {xk } is generated by the algorithm 2.1 and the mapping g(x) is strict. Then the every bounded subsequence {xk } converges to the unique solution of (1.1).

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4.

1325

Numerical experiments

The main work of this section is to report the performance of the algorithm DF-SSD and compare it with DF-SANE. The DF-SSD code was written in Fortran77 and in double precision arithmetic. All runs are performed on a PC (CPU 1.6 GHz, 256 memory) with Window XP operation system. Our experiments are performed on a set of 44 nonlinear equations from [7]. We test each problems twice with different size. Altogether, we tested 88 problems. We performed DF-SSD with the following parameters: ηk = 0.85, ρmin = 0.1, ρmax = 0.5, γ = 10−4 (x0 )k and ²k = kF (1+k)2 for all k. We choose the initial stepsize β in a various way. Specifically, we let β=

sTk sk , sTk yk

where sk = xk+1 − xk and yk = Fk+1 − Fk . Moreover, if β ∈ / [10−10 , 1010 ], we replace β by   1, β= kF (x )k−1 ,  5 k 10 ,

if kF (xk )k > 1, if 1 ≥ kF (xk )k ≥ 10−5 , if kF (xk )k < 10−5 .

We implemented the DF-SANE algorithm with the same parameters as Cruz, Martinez and Raydan [7]. That is: nexp = 2, σmin = 10−10 , σmax = 1010 , σ0 = 1, τmin = 0.1, τmax = 0.5, γ = 10−4 , M = 10, (x0 )k and ηk kF (1+k)2 for all k ∈ N . Both in DF-SANE and DF-SSD, we stop the process when kF (xk )k kF (x0 )k √ ≤ ea + er √ , n n

(4.1)

where ea = 10−5 and er = 10−4 . The numerical results of the algorithms DF-SANE and DF-SDD are listed in the following table. The columns have the following meanings: No.: Dim: Iter: Nf: Time:

number of the test problem; dimension of the test problem; number of iterations; number of function evaluations; CPU time in seconds.

The symbol ”-” indicates the related algorithm which is failed on this problem. Observing Table 4.1, we see that algorithms DF-SSD and DF-SANE work successfully almost on all these test problems. Moreover, in many cases, the number of iterations, the number of function evaluations and CPU times of both algorithm are identical. In summary, we observed the following: • DF-SSD and DF-SANE failed to reached a stationary point based on the stopping criteria (4.1) on 4 and 3 problems, respectively; • 16 problems where DF-SSD was superior to DF-SANE in Iter; • 17 problems where DF-SANE was superior to DF-SSD in Iter; • 13 problems where DF-SSD was superior to DF-SANE in Nf; • 19 problems where DF-SSD was superior to DF-SANE in Nf. From the above numerical analysis, we conclude that our proposed method provided a valid approach for solving large-scale nonlinear system of equations. Specifically, for some specific problems, the enhancement of DF-SSD on limited test problems is still noticeable. Moreover, preliminary experimental comparisons also indicate that algorithm DF-SSD is competitive with the well-known method DF-SANE.

7

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Table 4.1 Test Results of DF-SSD and DF-SANE NO. 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33

Dim 1000 10000 1000 5000 1000 10000 1000 10000 50 100 100 10000 100 500 1000 10000 100 1000 1000 10000 1000 5000 1000 10000 100 1000 1000 10000 1000 10000 1000 5000 1000 10000 50 100 1000 10000 100 1000 399 9999 1000 10000 500 1000 100 500 100 500 1000 10000 50 100 100 1000 100 1000 99 9999 1000 5000 500 1000 1000 5000

Iter 5 2 40 37 14 28 27 27 551 851 3 3 233 1 1 6 6 2 2 13237 55 29 3 4 12 12 5 5 14 17 7 7 19 5 5 30 102 5 5 1 1 2 2 13 2 3 1 1 10 11 1 1 1 1 12 12 6 6 6 6 7 4

DF-SSD Nf Time 5 0.0000 2 0.0156 54 0.0938 55 0.1875 21 0.0156 72 0.5156 99 0.0781 99 0.3750 3009 0.2344 5723 0.3281 3 0.0000 3 0.0000 286 0.0313 1 0.0000 1 0.0000 6 0.0000 6 0.8750 12 0.0156 12 0.1875 121451 68.8281 186 0.3281 121 1.6250 7 0.0000 8 0.0156 18 0.0156 20 0.1250 5 0.0469 5 0.0781 16 0.0156 17 0.0781 9 0.0000 9 0.1094 21 0.0000 5 0.0156 5 0.0462 42 0.0000 208 0.2188 7 0.0000 7 0.1094 2 0.0000 2 0.0000 18 0.0000 20 0.0000 22 0.0000 6 0.0000 9 0.0000 1 0.0000 1 0.0000 10 0.1250 11 0.2500 1 0.0000 1 0.0000 5 0.0000 5 0.0000 21 0.0000 21 0.1875 6 0.0156 6 0.0781 7 0.0000 7 0.0156 21 0.0313 16 0.0469

8

Iter 5 2 210 339 14 96 188 2144 3 3 23 23 1 1 6 6 2 2 17 17 34 15 3 4 12 12 5 5 14 17 7 7 19 5 5 40 49 5 5 1 1 2 2 27 59 2 3 1 1 10 11 1 1 1 1 11 11 6 6 6 6 45 4

DF-SANE Nf Time 5 0.0000 2 0.0156 218 0.1562 345 1.2500 21 0.0156 200 1.1875 792 1.5312 9734 0.5000 3 0.0000 3 0.0000 29 0.0000 29 0.0156 1 0.0000 1 0.0000 6 0.0000 6 0.9375 12 0.0156 12 0.0469 49 0.0469 49 0.1875 75 0.0938 33 0.4062 7 0.0000 8 0.0156 18 0.0156 20 0.0469 5 0.0156 5 0.0781 16 0.0156 17 0.0313 9 0.0000 9 0.0625 21 0.0000 5 0.0156 5 0.0469 42 0.0000 57 0.0156 7 0.0000 7 0.0469 2 0.0000 2 0.0000 18 0.0000 20 0.0000 34 0.0156 99 0.0156 6 0.0000 9 0.0000 1 0.0000 1 0.0156 10 0.0469 11 0.2656 1 0.0000 1 0.0000 5 0.0000 5 0.0000 16 0.0000 16 0.0781 6 0.0313 6 0.0781 7 0.0156 7 0.0000 62 0.0469 16 0.0469

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Table 4.1 Continued· · ·

34 34 35 35 36 36 37 37 38 38 39 39 40 40 41 41 42 42 43 43 44 44

Dim 1000 5000 1000 5000 100 500 1000 5000 1000 5000 1000 5000 1000 5000 500 1000 1000 5000 100 500 1000 5000

Iter 35 12 21 22 24 70 30 30 22 22 17 17 1 1 7 3 19 19 111 708 4 3

DF-SSD Nf Time 73 0.0313 18 0.0313 33 0.0156 32 0.0781 34 0.0000 182 0.0469 48 0.0156 48 0.0938 36 0.0156 36 0.1094 29 0.0156 29 0.0625 1 0.0000 1 0.0156 9 0.0000 3 0.0000 27 0.0000 27 0.0625 173 0.0000 1692 0.5469 4 0.0156 3 0.0313

Iter 28 12 21 38 52 45 21 21 25 25 14 14 1 1 7 3 44 44 111 270 4 3

DF-SANE Nf Time 38 0.0156 18 0.0313 27 0.0000 48 0.0781 62 0.0000 57 0.0156 27 0.0000 27 0.0469 30 0.0156 30 0.0781 20 0.0156 20 0.0156 1 0.0000 1 0.0000 9 0.0000 3 0.0000 52 0.0156 52 0.0781 145 0.0156 436 0.0625 4 0.0156 3 0.0313

References [1] J. Barzilai and J.M. Borwein, Two point step size gradient method, IMA J. Numer. Anal., 8, 141-148 (1988). [2] C. G. Broyden, A class of methods for solving nonlinear simultaneous equations, Math. Comput., 19, 577-593 (1965). [3] W. Cheng, A two term PRP based descent Method, Numer. Func. Anal. Optim., 28, 1217-1230 (2007). [4] W. Cheng and D.H. Li, A derivative-free nonmonotone line search and its application to the spectral residual method, IMA J. Numer. Anal., 29, 814-825 (1009). [5] W. Cheng, Y. Xiao, and Q. Hu, A family of derivative-free conjugate gradient methods for large-scale nonlinear systems of equations, J. Comput. Appl. Math., 224, 11-19 (2009). [6] W. La Cruz, J.M. Mrt´ınez, and M. Raydan, Nonmonotone spectral methods for large-scale nonlinear systems, Optim. Methods Softw., 18, 583-599 (2003). [7] W. La Cruz, J.M. Mrt´ınez, and M. Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations, Math. Comput., 75, 1429-1448 (2006). [8] M. Gasparo, A nonmonotone hybrid method for nonlinear systems, Optim. Methods Softw., 13, 79-94 (2000). [9] L. Grippo, F. Lampariello, and S. Lucidi, A nonmonotone line search technique for Newtons method, SIAM J. Numer. Anal., 23, 707-716 (1986). [10] G.Z. Gu, D.H. Li, L. Qi, and S.Z. Zhou, Descent directions of quasi-Newton methods for symmetric nonlinear equations, SIAM J. Numer. Anal., 40, 1763-1774 (2002). [11] W.W. Hager and H. Zhang, A new conjugate gradient method with guaranteed descent and an efficient line search, SIAM J. Optim., 16, 170-192 (2005). 9

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[12] D.H. Li and M. Fukushima, A derivative-free line search and globla convergence of Broyden-like method for nonlinear equations, Optim. Methods Softw., 13 , 583-599 (2000). [13] D.H. Li and M. Fukushima, A globally and superlinearly convergent Gauss-Newton-based BFGS method for symmetric nonlinear equations, SIAM J. Numer. Anal., 37, 152- 172 (1999). [14] J.M. Mart´ınez, A family of quasi-Newton methods for nonlinear equations with direct secant updates of matrix factorizations, SIAM J. Numer. Anal., 27, 1034-1049 (1990). [15] H. Zhang and W. W. Hager, A nonmonotone line search technique and its application to unconstrained optimization, SIAM J. Optim., 14, 1043-1056 (2004). [16] L. Zhang, W. Zhou, and D.H. Li, A descent modified Polak-Ribi`ere-Polyak conjugate gradient method and its global convergence, IMA J. Numer. Anal., 26, 629-640 (2006). [17] M.L. Zhang, Y.H. Xiao, and D. Zhou, A simple sufficient descent method for unconstrained optimization, working paper.

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JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.7, 1329-1334, 2011, COPYRIGHT 2011 EUDOXUS 1329 PRESS, LLC

NOTE ON p-ADIC q-EULER MEASURE YOUNG-HEE KIM, LEE-CHAE JANG, TAEKYUN KIM, BYUNGJE LEE, AND SEOG-HOON RIM

Abstract In [1], Carlitz defined q-Bernoulli numbers as follows : ( 1 if k = 1, β0,q = 1, q(qβ + 1)k − βk,q = 0 if k > 1, with the usual convention of replacing β k by βk,q . In [13], Kolbitz constructed a padic Carlitz’s q-Bernoulli measure for studying the q-extension of p-adic L-function. In [4, 8], T. Kim considered the Carlitz’s type q-Euler numbers and polynomials. In this paper, we consider Nasybullin’s type p-adic q-measure and we derive a Carlitz’s type q-Euler measure on Zp from the Nasybullin’s type p-adic q-measure. Key words and phrases : q-Euler numbers and polynomials, p-adic q-Euler measure 1. Introduction Throughout this paper, the symbol Z, Q, Zp , Qp , and Cp denote the ring of integers, the field of rational numbers, the ring of p-adic rational integers, the field of p-adic rational numbers, and the completion of algebraic closure of Qp , respectively. Let N be the set of natural numbers and Z+ = N ∪ {0}. Let νp be the normalized exponential valuation of Cp such that |p|p = p−νp (p) = p1 . When one talks of qextension, q is variously considered as an indeterminate, a complex number q ∈ C or a p-adic number q ∈ Cp . If q ∈ Cp , we normally assume |1 − q|p < 1. Now we use the notation of q-number : 1 − qx , (see [1-13]). [x]q = [x : q] = 1−q Note that lim [x]q = x for any x with |x|p ≤ 1 in the present p-adic case. q→1

Let p be a fixed odd prime. For f ∈ N with f ≡ 1 (mod 2), let f = lcm(f, p ) be the least common multiple of f and p. For f ∈ N, we set X = Xf

=

X ∗ = Xf∗

=

a + f pN Zp

lim (Z/f pN Z), X1 = Zp , ← − N

∪ 0 0, with the usual convention of replacing E i by Ei,q . These numbers are represented by p-adic q-invariant integral on Zp as follows : Z Z [x]nq dµ−q (x) = [x]nq dµ−q (x) = En,q , for n ∈ N. (3) Xf

Zp

The q-Euler polynomials are defined by n   X n En,q (x) = [x]n−l q lx El,q , q l

(see [4]).

l=0

From (2) and (3), we can easily derive Z En,q (x) = [x + t]nq dµ−q (t) Zp n   q lx [2]q X n (−1)l = n l 1 + q l+1 (1 − q) l=0 Z n   X n n−l lx = [x]q q [t]lq dµ−q (t) l Zp l=0

=

(q x E + [x]q )n .

(4)

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1331

3

By (4), we can easily derive the following distribution relation for the Carlitz’s type q-Euler polynomials as follows :

En,q (x) = [p]nq

p−1 [2]q X x+a (−q)a En,qp ( ), [2]qp a=0 p

for n ∈ Z+ .

(5)

Many authors have been interested in p-adic measures (cf. [13-14]). In [13], Kolbitz constructed p-adic Carlitz’s q-Bernoulli measure. In this paper, we consider Nasybullin’s type p-adic q-measure and we derive a Carlitz’s type q-Euler measure on Zp from the Nasybullin’s type p-adic q-measure. 2. p-adic q-measure First, we consider the Nasybullin’s type p-adic q-measure. Theorem 1. Let Rq be a K-valued function defined on Q(f ) with the following properties : (i) there exist two constants A, B ∈ K such that

p−1 X k=0

Rq p (

x+k )(−q)k = ARq (x) + BRq1/p (px), p

and (ii) Rq (x + 1) = Rq (x), for any number x ∈ Q(f ) . Suppose that ρ is a root of the cusp equation y 2 = Ay + Bp. Then there exists a K(ρ)-measure µ on Xf∗ such that

µ(Ia,n ) = ρ−n (−q)a Rqpn f (

a pn f

) + Bρ−(n+1) (−q)a Rqpn−1 f (

for any interval Ia,n .

Proof. It is sufficient to show that

p−1 X k=0

µ(Ia+pn f k,n+1 ) = µ(Ia,n ).

a pn−1 f

),

1332

KIM ET AL: p-ADIC q-EULER MEASURE

4

From the definition, we note that p−1 X

µ(Ia+pn f k,n+1 )

k=0

= ρ−(n+1)

p−1 X

Rqpn+1 f (

k=0 −(n+2)

+Bρ

p−1 X

n a + pn f k )(−q)a+p f k n+1 p f

Rqpn f (

k=0

= ρ−(n+1) (−q)a

p−1 X

n a + pn f k )(−q)a+p f k n p f

R(qpn f )p (

a pn f

k=0

+Bρ−(n+2) (−q)a

p−1 X

Rqpn f (

k=0

+k p a pn f

n

)(−q)p

fk

n

+ k)(−q)p

fk

a a ) + ρ−(n+1) (−q)a BRqpn−1 f ( n−1 ) pn f p f a −(n+2) a +Bρ (−q) pRqpn f ( n ) p f a a = ρ−(n+2) (−q)a (ρA + pB)Rqpn f ( n ) + ρ−(n+1) (−q)a BRqpn−1 f ( n−1 ) p f p f a a −(n+1) a −n a (−q) Rqpn−1 f ( n−1 ) = ρ (−q) Rqpn f ( n ) + Bρ p f p f = µ(Ia,n ). =ρ

−(n+1)

a

(−q) ARqpn f (

Thus, we have µ(Ia,n ) =

X

µ(Ib,n+1 ).

b (mod pn+1 f ) b≡a (mod pn f )

This proves our assertion, because any open-closed subset is a disjoint union of some finite intervals.  Now we derive a Carlitz’s type q-Euler measure on Zp from the Nasybullin’s type p-adic q-measure. Let Em,q (x) be the m-th Kim’s Carlitz q-Euler polynomials and let Em,q hxi be the m-th Carlitz’s type q-Euler functions, that is, for 0 ≤ x < 1, Em,q hxi = Em,q (x). Note that lim Em,q hxi = Em hxi is the m-th Euler function. For any positive integer q→1

m with m ≡ 1 (mod 2) and k ∈ Z+ , we have Ek,q (x) = [m]kq

m−1 [2]q X x+i (−q)i Ek,qm ( ). [2]qm i=0 m

In the special case m = p, Ek,q (x) = [p]kq

p−1 x+i [2]q X (−q)i Ek,qp ( ). [2]qp i=0 p

(6)

KIM ET AL: p-ADIC q-EULER MEASURE

1333

5

Thus, we note that Em,q hxi satisfies the properties of Theorem 1 with constants A = [p]−m q

[2]qp , B = 0. [2]q

It is easy to show that ρ 6= 0 is equal to [p]−m q simply to ρ2 = [p]−m q measure.

[2]qp [2]q

[2]qp [2]q

, as ρ2 = Aρ + Bp reduces

ρ. By Theorem 1, we obtain the following p-adic q-Euler

Theorem 2. (p-adic Carlitz type q-Euler measure) For m ∈ Z+ , let the function µm = µm:q be defined on Ia,n as follows: [2]q a (−q)a Em,qf pn ( n ). [2]qpn f p f

µm (Ia,n ) = [f pn ]m q Then µm is a Qp (q)-measure on Xf∗ .

Let χ be a primitive Dirichlet character modulo f . Then the n-th generalized Carlitz’s type q-Euler numbers attached to χ are defined as Z En,χ,q = [x]nq dµ−q (x) X

=

[f ]nq

f −1 a [2]q X χ(a) (−q)a En,qf ( ). [2]qf a=0 f

We can compute the p-adic q-l-function of Kim by the following p-adic q-MellinMazur transform with respect to µm . Z χ(a)dµm (a) L(µm , χ) = X∗ f

=

X

lim

n→∞

χ(a)µm (Ia,n ).

a (mod pn f ) a∈Z, (a, p)=1

Since the character χ is constant on the interval Ia,0 , it follows that X χ(a)µm (Ia,0 ) L(µm , χ) = a (mod f ) a∈Z, (a, p)=1

=

X

χ(a)[f ]m q

a (mod f ) a∈Z, (a, p)=1

= Em,χ,q − χ(p)

a [2]q (−q)a Em,qf ( ) [2]qf f

[2]q m [p] Em,χ,qp , [2]qp q

where Em,χ,q is the m-th Carlitz’s type q-Euler number attached to χ. References [1] L. Carlitz, q-Bernoulli numbers and polynomials, Duke Math. J. 15 (1948), 987–1000. [2] T. Kim, On p-adic interpolating function for q-Euler numbers and its dericatives, J. Math. Anal. Appl. 339 (2008), 598–608.

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6

[3] T. Kim, On p-adic q-l-functions and sums of powers, J. Math. Anal. Appl. 329 (2007), 1472–1481. [4] T. Kim, On the q-extension of Euler and Genocchi numbers, J. Math. Anal. Appl. 326 (2007), 1458–1465. [5] T. Kim, Note on the Euler q-zeta functions, J. Number Theory 129 (2009), no. 7, 1798–1804. [6] T. Kim, Note on the q-Euler numbers of higher order, Adv. Stud. Contemp. Math. 19 (2009), 25–29. [7] T. Kim, A note on the generalized q-Euler numbers, Proc. Jangjeon Math. Soc. 12 (2009), no. 1, 45-50. [8] T. Kim, q-Euler numbers and polynomials associated with p-adic q-integrals, J. Nonlinear Math. Phys. 14 (2007), no. 1, 15–27. [9] T. Kim, On the multiple q-Genocchi and Euler Numbers, Russ. J. Math. Phys. 15 (2008), no. 4, 481–486. [10] T. Kim, Some identities on the q-Euler polynomials of higher order and qStirling numbers by the fermionic p-adic integral on Zp , Russ. J. Math. Phys. 16 (2009), 484-491. [11] T. Kim, q-Volkenborn integration, Russ. J. Math. Phys. 9 (2002), no. 3, 288299. [12] Y.-H. Kim, W. Kim, L.-C. Jang, On the q-extension of Apostol-Euler numbers and polynomials, Abstr. Appl. Anal. 2008 (2008), Article ID 296159, 10 pages. [13] N. Koblitz, On Carlitz’s q-Bernoulli numbers, J. Number Theory 14 (1982), no. 3, 332–339. [14] S.-H. Rim, T. Kim, A note on p-adic Euler measure on Zp , Russ. J. Math. Phys. 13 (2006), no. 3, 358–361. Young-Hee Kim. Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea, E-mail address: [email protected] Lee-Chae Jang. Department of Mathematics and Computer Science, KonKuk University, Chungju 380-701, Republic of Korea E-mail address: [email protected] Taekyun Kim. Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea, E-mail address: [email protected] Byungje Lee. Department of Wireless Communications Engineering, Kwangwoon University, Seoul 139-701, Republic of Korea, E-mail address: bj− [email protected] Seog-Hoon Rim. Department of Mathematics Education, Kyungpook National University, Taegu 702-701, Republic of Korea, E-mail address: [email protected]

JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.7, 1335-1341, 2011, COPYRIGHT 2011 EUDOXUS 1335 PRESS, LLC

Wolfe type second-order symmetric duality in multiobjective programming over cones ∗ Yu Chen† Zhi-ming Luo School of Information, Hunan University of Commerce, 410205, Changsha, P.R. China‡

Abstract In this paper, a pair of Wolfe type second-order symmetric duality in multiobjective programming over arbitrary cones is fomulated. The weak, strong and converse duality theorems are also established for these programs by using η-invexity assumptions. Our results generalize these existing dual formulations which were discussed by the authors in [8, 10, 11, 20]. Key words. Multiobjective programming, symmetric duality, cones, η-invexity. MR(2000)Subject Classification: 49N15,90C30

1.

Introduction

The concept of symmetric dual were first introduced by Dorn [1] in 1960 for quadratic programming. Subsequently, Dantzig [2] and Mond [3] in 1965 extended his results to general nonlinear programs for convex\concavity functions. Later on, another pair of symmetric dual nonlinear programs under weaker convexity assumptions were presented by Mond et al. [6] in 1981. Weir et al. [8] in 1988 as well as Gulati et al.[10] in 1997 proved multiobjective symmetric duality results. In 1998, Chandra and Kumar [11] studied Mond-Weir type symmetric duality with cone constraints. His results were extended by Khurana [20] in 2005 to the case which the objective function has been optimized with respect to a closed convex cone. Mangasarian [4] in 1975 introduced the concept of second-order duality for nonlinear programs. He has also indicated a possible computational advantage of the second-order dual ∗

This work was supported by the Scientific Research Fund of Hunan Provincial Education Department of China (09C565) † Corresponding author. E-mail address: [email protected] ‡ Corresponding address.

1

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CHEN, LUO: MULTIOBJECTIVE PROGRAMMING OVER CONES

over the first order dual. This motivated several authors [5, 9, 12, 13, 14, 15, 17, 18, 19, 21, 22] in this field. In 2005, Yang et al. [19, 21] studied second-order multiobjective symmetric dual programs and established the duality results under F-convexity assumptions. Gulati et al. [22] in 2008 studied Wolfe and Mond-Weir type second-order symmetric duality over arbitrary cones under η-bonvexity\η-pseudobonvexity assumptions. In this paper, we consider Wolfe type second-order multiobjective symmetric dual programs over arbitrary cones and prove weak, strong, converse duality results under η-invexity assumptions. Our results generalize the work in [8, 10, 11, 20].

2.

Preliminaries

We consider the following multiobjective programming problem: K − minimize s.t.

f (x) −g(x) ∈ Q, x ∈ S,

where S ⊆ Rn+m is open, f : S → Rk , g : S → Rm , K and Q are closed convex pointed cones with nonempty interiors in Rk and Rm , respectively. Let X 0 = {x ∈ S : −g(x) ∈ Q} be the set of all feasible solution for (P) and f be differentiable on S. Definition 2.1 [20]A point x ∈ X 0 is an efficient solution of (P) if there exists no x ∈ X 0 such that f (x) − f (x) ∈ K\{0}. Definition 2.2 [7]The function f is invex at u ∈ S with respect to η : S × S → Rn if for any x ∈ S, f (x) − f (u) ≥ η(x, u)T ∇f (u). Definition 2.3 [16]Let C be a closed convex cone in Rn with nonempty interiors. The positive polar cone C ∗ of C is defined by C ∗ = {z ∈ Rn : xT z ≥ 0 f or all x ∈ C}.

3.

Wolfe type symmetric duality

We consider the following pair of Wolfe type second-order multiobjective symmetric dual problem and establish weak, strong and converse duality theorems. Primal(MP): K − minimize s.t.

f (x, y) − y T (∇y (λT f )(x, y) + ∇yy (ω T g)(x, y)p)e −(∇y (λT f )(x, y) + ∇yy (ω T g)(x, y)p) ∈ C2∗ , λ ∈ intK ∗ , (x, y) ∈ C1 × C2 , e ∈ K, λT e = 1, 2

CHEN, LUO: MULTIOBJECTIVE PROGRAMMING OVER CONES

1337

Dual(MD): K − maximize s.t.

f (u, v) − uT (∇x (λT f )(u, v) + ∇xx (ω T g)(u, v)r)e (∇x (λT f )(u, v) + ∇xx (ω T g)(u, v)r) ∈ C1∗ , λ ∈ intK ∗ , (u, v) ∈ C1 × C2 , e ∈ K, λT e = 1,

where (i)e = (1, 1, · · · , 1)T ∈ Rk , (ii)S1 ∈ Rn , S2 ∈ Rm are open sets, (iii)f : S1 × S2 → Rk is a twice differentiable function of x and y, (iv)g : S1 × S2 → Rq is a twice differentiable function of x and y, (v)λ ∈ Rk , ω ∈ Rq , p ∈ Rm , r ∈ Rn , (vi)for i = 1, 2, Ci ∈ Si is a closed convex cone with nonempty interior and Ci∗ is its positive polar cone. Theorem 3.1 (Weak duality) Let (x, y, λ, ω, p) be feasible for (MP) and (u, v, λ, ω, r) be feasible for (MD). Suppose that (i) (λT f )(·, v) be η1 -invex at u with respect to η1 for fixed v; (ii) −(λT f )(x, ·) be η2 -invex in the second variable at y; (iii) η1 (x, u) + u ∈ C1 for all x ∈ C1 ; (iv) η2 (v, y) +y∈ C2 for all v ∈ C2   rT 0 ∇xx (ω T g)(u, v) 0 η1 (x, u) (v) ≤0 . 0 pT 0 −∇yy (ω T g)(x, y) η2 (v, y) Then f (u, v) − uT (∇x (λT f )(u, v) + ∇xx (ω T g)(u, v)r)e − f (x, y) + y T (∇y (λT f )(x, y) + ∇yy (ω T g)(x, y)p)e ∈ / K\0. Proof. Suppose,to the contrary, that f (u, v) − uT (∇x (λT f )(u, v) + ∇xx (ω T g)(u, v)r)e − f (x, y) + y T (∇y (λT f )(x, y) + ∇yy (ω T g)(x, y)p)e ∈ K\0 Since λ ∈ intK ∗ , we obtain λT {−f (u, v) + uT (∇x (λT f )(u, v) + ∇xx (ω T g)(u, v)r)e + f (x, y) − y T (∇y (λT f )(x, y) + ∇yy (ω T g)(x, y)p)e} < 0.

(3.1)

In view of λT e = 1, one gets −λT f (u, v) + uT (∇x (λT f )(u, v) + ∇xx (ω T g)(u, v)r) + λT f (x, y) − y T (∇y (λT f )(x, y) + ∇yy (ω T g)(x, y)p) < 0.

(3.2)

By η1 -invexity of λT f (·, v), η2 -invexity of −λT f (x, ·) and hypothesis (v), we have λT f (x, v) − λT f (u, v) ≥ η1T (x, u){∇x (λT f )(u, v) + ∇xx (ω T g)(u, v)r}, 3

(3.3)

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CHEN, LUO: MULTIOBJECTIVE PROGRAMMING OVER CONES

λT f (x, y) − λT f (x, v) ≥ −η2T (v, y){∇y (λT f )(x, y) + ∇yy (ω T g)(x, y)p},

(3.4)

The first constraint in (MD) and hypothesis (iii) implies that η1T (x, u){∇x (λT f )(u, v) + ∇xx (ω T g)(u, v)r} ≥ −uT {∇x (λT f )(u, v) + ∇xx (ω T g)(u, v)r}. (3.5) Similarly, by hypothesis (iv) and the first constraint in (MP), −η2T (v, y){∇y (λT f )(x, y) + ∇yy (ω T g)(x, y)p} ≥ y T {∇y (λT f )(x, y) + ∇yy (ω T g)(x, y)p}. (3.6) Finally, the above four inequalities (3.3)(3.4)(3.5)(3.6) yield −λT f (u, v) + uT (∇x (λT f )(u, v) + ∇xx (ω T g)(u, v)r)e + λT f (x, y) − y T (∇y (λT f )(x, y) + ∇yy (ω T g)(x, y)p)e ≥ 0, which contradicts (3.2). Theorem 3.2 (Strong Duality)Suppose that (x, y, λ, ω, p) be an efficient solution for (MP). Let (i) ∇yy (ω T g)(x, y) be nonsingular, (ii) the set {∇y fi (x, y), i = 1, · · · , k} be linearly independent, and (iii) ∇yy (ω T g)(x, y)p ∈ / span{∇y f1 (x, y), · · · , ∇y fk (x, y)}\{0}. T T T (iv) x ∇x (λ f )(x, y) = y T ∇y (λ f )(x, y). Then, (x, y, ω, r = 0) is feasible for (MD)λ , and the objective function values of (MP) and (MD)λ are equal. Also, if the hypotheses of a weak duality theorem are satisfied for all feasible solutions of (MP )λ and (MD)λ , then (x, y, ω, r = 0) is an efficient solution for (MD)λ . Proof. Since (x, y, λ, ω, p) be an efficient solution for (MP), by using the Fritz John type necessary optimality conditions established by Suneja et al. in 2002 (See Lemma 1 in [16]), there exist α ∈ K ∗ , β ∈ C2 , such that the following conditions are satisfied at (x, y, λ, ω, p): T

(x − x)T [∇x f α + (∇yx (λ f ) + ∇x (∇yy (ω T g)p))(β − αT ey)] ≥ 0, f or all x ∈ C1 ,

(3.7)

T

(y − y)T {∇y f (α − αT eλ) + [∇yy (λ f ) + ∇y (∇yy (ω T g)p)](β − αT ey) − αT e∇yy (ω T g)p} ≥ 0, (3.8) m for all y ∈ R , (3.9) [(β − αT ey)T ∇y f ](λ − λ) ≥ 0, f or all λ ∈ intK ∗ , (β − αT ey)T (∇ω (∇yy (ω T g)p)) = 0,

(3.10)

(β − αT ey)T ∇yy (ω T g) = 0,

(3.11)

T

β T (∇y (λ f ) + ∇yy (ω T g)p) = 0,

(3.12)

(α, β) 6= 0.

(3.13)

4

CHEN, LUO: MULTIOBJECTIVE PROGRAMMING OVER CONES

1339

(3.8) and (3.9) yield the equations T

∇y f (α − αT eα) + [∇yy (λ f ) + ∇y (∇yy (ω T g)p)](β − αT ey) − αT e∇yy (ω T g)p = 0

(3.14)

and (β − αT ey)T ∇y f = 0.

(3.15)

By hypothesis (i) and (3.11), we have β = αT ey.

(3.16)

Now, we claim that αT e 6= 0. Indeed, if αT e = 0, then β = 0 from (3.16). Therefore, from (3.14), we get (∇y f )α = 0, which by hypothesis (ii) give α = 0, and contradicts (α, β) 6= 0. Substituting (3.16) into (3.14), we have ∇y f (α − αT eλ) = αT e∇yy (ω T g)p. Using hypothesis(iii), the above relation implies αT e∇yy (ω T g)p = 0, which in view of hypothesis(i) yields p = 0. Thus ∇y f (α − αT eλ) = 0. By hypothesis (ii), one gets α = αT eλ. Further, the above equation, (3.7)and (3.16) imply T

(x − x)T ∇x (λ f ) ≥ 0 f or all x ∈ C1 . Let x ∈ C1 , then x + x ∈ C1 and the above inequality implies T

xT ∇x (λ f ) ≥ 0 f or all x ∈ C1 . T

Therefore ∇x (λ f ) ∈ C1∗ . Hence (x, y, ω, r = 0) satisfies the constraints of (MD)λ , that is, it is feasible for the dual problem (MD)λ . Moreover, (MP) and (MD)λ have equal objective function value from hypothesis (iv). Now, suppose (x, y, ω, r = 0) is not an efficient solution for (MD)λ , then there exists a feasible solution (u, v, ω, r) for (MD)λ , such that f (u, v) − uT (∇x (λT f )(u, v) + ∇xx (ω T g)(u, v)r)e − f (x, y) + y T (∇y (λT f )(x, y) + ∇yy (ω T g)(x, y)p)e ∈ K\0, which contradicts weak duality. Hence (x, y, ω, r = 0) is an efficient solution for (MD)λ .

5

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CHEN, LUO: MULTIOBJECTIVE PROGRAMMING OVER CONES

Theorem 3.3 (Converse Duality)Suppose that (u, v, λ, ω, r) be an efficient solution for (MD). Let (i) ∇yy (ω T g)(u, v) be nonsingular, (ii) the set {∇y fi (u, v), i = 1, · · · , k} be linearly independent, and (iii) ∇yy (ω T g)(u, v)r ∈ / span{∇y f1 (u, v), · · · , ∇y fk (u, v)}\{0}. T T T (iv) u ∇x (λ f )(u, v) = v T ∇y (λ f )(u, v). Then, (u, v, ω, p = 0) is feasible for (MP )λ , and the objective function values of (MD) and (MP )λ are equal. Also, if the hypotheses of a weak duality theorem are satisfied for all feasible solutions of (MP )λ and (MD)λ , then (u, v, ω, p = 0) is an efficient solution for (MP )λ . Proof. Follows on the lines of Theorem 3.2.

References [1] W.S.Dorn, A symmetric dual theorem for quadratic programs, Journal of Operations Society of Japan 2 (1960) 93-97. [2] G.B.Dantzig, E.Eisenberg, R.W.Cottle, Symmetric dual nonlinear programs, Pacific Journal of Mathematics 15 (1965) 809-812. [3] B.Mond, A symmetric dual theorem for nonlinear programs, Quarterly of Applied Mathematics 23 (1965) 265-269. [4] O.L.Mangasarian, Second- and higher duality in nonlinear programming, Journal of Mathematical Analysis and Applications 51 (1975) 607-620. [5] B.Mond, Second-order duality for nonlinear programs, Opsearch 11 (1974) 90-99. [6] B.Mond, T.Weir, Generalized concavity and duality, in: S.Schaible, W.T.Ziemba(Eds.), Generalized Concavity in Optimization and Economics, Academic Press, New York, 1981, pp.263280. [7]

A.Ben-Istael, B.Mond, What is invexity Journal of Australion Mathematical Societty 28 (1986) 1-9.

[8] T.Weir, B.Mond, Symmetric and self duality in multiple objective programming, Asian Pacific Journal of Operational Research 5 (1988) 124-133. [9] M.A.Hanson, Second-order invexity and duality in mathematical programming, Opsearch 30 (1993) 313-320. [10] T.R.Gulati, I.Husain, A.Ahmed, Multiobjective symmetric duality with invexity Bulletin of Australion Mathematical Society 56 (1997) 25-36.

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[11] S.Chandra, V.Kumar, A note on pseudo-invexity and symmetric duality European Journal of Operational Research 105 (1998) 626-629. [12] G.Devi, Symmetric duality for nonlinear programming problem involving η-bonvex functons, European Journal of Operational Research 104 (1998) 615-621. [13] D.S.Kim, Y.B.Yun, H.Kuk, Second-order symmetric and self duality in multiple objective programming, Applied Mathematics Letter 10 (1997) 17-22. [14] S.H.Hou, X.M.Yang, On second-order symmetric duality in nondifferentiable programming, Journal of Mathematical Analysis and Applications 255 (2001) 491-498. [15] X.M.Yang, S.H.Hou, Second-order symmetric duality in multiobjective programming, Applied Mathematics Letter 14 (2001) 587-592. [16] S.K.Suneja, S.Aggarwal, S.Davar, Multiobjective symmetric duality involving cones, European Journal of Operational Research 141 (2002) 471-479. [17] S.K.Suneja, C.S.Lalitha, Seema Khurana, Second-order symmetric duality in multiobjective programming, European Journal of Operational Research 144 (2003) 492-500. [18] I.Ahmad, Z.Husian, Nondifferentiable second-order symmetric duality in multiobjective programming, Applied Mathematics Letter 18 (2005) 721-728. [19] X.M.Yang, X.Q.Yang, K.L.Teo, S.H.Hou, Multiobjective second-order symmetric duality with F-convexity European Journal of Operational Research 165 (2005) 585-591. [20] S.Khurana, Symmetric duality in multiobjective programming involving generalized coneinvex functions, European Journal of Operational Research 165 (2005) 592-597. [21] X.M.Yang, X.Q.Yang, K.L.Teo, S.H.Hou, Second-order symmetric duality in nondifferentiable multiobjective programming with F-convexity, European Journal of Operational Research 164 (2005) 406-416. [22]

T.R.Gulati, S.K.Gupta, I.Ahmad, Second-order symmetric duality with cone constraints, Journal of Computional and Applied Mathematics 220 (2008) 347-354.

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JOURNAL 1342 OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL. 13, NO.7, 1342-1353, 2011, COPYRIGHT 2011 EUDOXUS PRESS, LLC

A MULTIVARIATE SPECTRAL APPROACH FOR LARGE SCALE NONLINEAR SYSTEM ZHENSHENG YU1 AND

1

JINHONG YU2

College of Science, University of Shanghai for Science and Technology, Shanghai, 200093, P.R.China email: [email protected]

2

College of Foreign Language, University of Shanghai for Science and Technology, Shanghai, 200093, P.R.China email: [email protected]

Abstract: In this paper, we present a multivariate spectral approach for the system of nonlinear equations. Combined with some quasi Newton property, the multivariate spectral approach allows an individual adaptive stepsize along each coordinate direction. Based on the nonmonotone line search, we establish the global convergence of the proposed method. The numerical comparison with the classical spectral approach show the efficiency of the proposed approach. Keywords: Nonlinear equations, Multivariate spectral approach, Nonmonotone line search, Global convergence, Large scale problems

1. INTRODUCTION

In this paper, we consider the nonlinear system of nonlinear equations: finding a vector 𝑥∗ ∈ 𝑅𝑛 such that: 𝐹 (𝑥∗ ) = 0,

(1)

where 𝐹 : ℝ𝑛 → ℝ𝑛 is a continuous differentiable function. Such problem has many important applications and arises from engineering, management and economy. Different methods have been developed for (1). The 1

1343

2

A Multivariate Spectral Approach for Large Scale Nonlinear System

most popular schemes are based on Newton’s method, or quasi-Newton methods [3, 7, 8, 11, 13, 14]. The advantage of these methods is that they converge rapidly from any sufficiently good initial guess. The main drawback of these methods, for large scale problems, is that they need to solve a linear system of equations at each iteration using the Jacobian matrix or an approximation of it. Recently, La.Cruz and Raydan [6] extended the spectral gradient method for unconstrained optimization[1, 2, 12] to problem (1) and introduced a spectral approach(SANE) for problem (1). The SANE uses systematic way ±𝐹 (𝑥) as search directions, global convergence is guaranteed by means of a variation of nonmonotone line search strategy of Grippo, Lampariello and S.Lucidi [9]. Since it does not use the Jacobian matrix at each iteration, it is suitable for large scale problem, and some related variation algorithms have been developed [4, 5]. More recently, by replacing the classical spectral stepsize with a matrix, Han, Yu and Guan [10] defined a new iterative scheme and propose a multivariate spectral gradient method for unconstrained optimization. The new method is finitely convergent for positive definite quadratics and globally convergent for general function. In this paper, we aim to extend the multivariate spectral algorithm to nonlinear equations (1). Combined with some quasi Newton property, the multivariate spectral method allows an individual adaptive stepsize along each coordinate direction. By using the nonmonotone line search [9], we establish the global convergence of the proposed method. The remainder of our paper is organized as follows. In Section 2, we introduce the multivariate spectral algorithm and discuss the global convergence. Section 3, we present our numerical results. The conclusion is presented in Section 4. Throughout this work ∥ ⋅ ∥ denotes the 2-norm of vectors and matrices. For a given 𝑥 ∈ ℝ𝑛 , we use 𝑥𝑘 to denote the 𝑘𝑡ℎ iteration point, 𝑥𝑇 denote its transpose 𝑥𝑖𝑘 denote its 𝑖𝑡ℎ sub-variable for 𝑖 = 1, 2, ⋅ ⋅ ⋅ , 𝑛.

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3

2. ALGORITHM AND CONVERGENCE

In this section, we describe our multivariate spectral algorithm for solving problem (1). In what follows, we first introduce the spectral approach for nonlinear systems problem: 𝐹 (𝑥) = 0,

(2)

the iterations are defined as 𝑥𝑘+1 = 𝑥𝑘 −

1 𝐹 (𝑥𝑘 ), 𝛼𝑘

(3)

where the scalar 𝛼𝑘 is given by 𝛼𝑘 =

𝑠𝑇𝑘−1 𝑦𝑘−1 , 𝑠𝑇𝑘−1 𝑠𝑘−1

(4)

with 𝑠𝑘−1 = 𝑥𝑘 − 𝑥𝑘−1 , 𝑦𝑘−1 = 𝐹 (𝑥𝑘 ) − 𝐹 (𝑥𝑘−1 ). Define 𝑓 (𝑥) = ∥𝐹 (𝑥)∥2 and let 𝐽(𝑥𝑘 ) be the Jacobian of 𝐹 at the vector 𝑥𝑘 . Since in general the symmetric part of 𝐽(𝑥𝑘 ) is not positive define, then 𝐹 (𝑥𝑘 )𝑇 𝐽(𝑥𝑘 )𝐹 (𝑥𝑘 ) could be positive, negative or even zero, 𝑑𝑘 = −𝐹 (𝑥𝑘 ) is not necessarily a descent direction for the function 𝑓 (𝑥). To overcome this difficulty, La.Curz and M.Raydan [6] introduced 𝑑𝑘 = ±𝐹 (𝑥𝑘 ) as the search direction. By replacing 𝛼𝑘 with a matrix, the multivariate spectral approach generate a iteration sequence by the following iteration: Denote the 𝑖𝑡ℎ diagonal element of 𝛼𝑘 by 𝜆𝑖𝑘 and let 𝑑𝑖𝑎𝑔{𝜆1𝑘 , 𝜆2𝑘 , . . . , 𝜆𝑛𝑘 } be generated by minimizing ∥𝑑𝑖𝑎𝑔{𝜆1 , 𝜆2 , . . . , 𝜆𝑛 }𝑠𝑘−1 − 𝑦𝑘−1 ∥

(5)

with respect to {𝜆𝑖 }𝑛𝑖=1 , then the multivariate spectral approach is defined as: 𝑥𝑘+1 = 𝑥𝑘 ± 𝑑𝑖𝑎𝑔{1/𝜆1𝑘 , 1/𝜆2𝑘 , . . . , 1/𝜆𝑛𝑘 }𝐹𝑘 .

(6)

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A Multivariate Spectral Approach for Large Scale Nonlinear System

Based on Eqs.(5) (6), we describe the multivariate spectral method for problem (1) as follows:

Algorithm 2.1(MSANE) Step 0. Given 𝑥0 ∈ 𝑅𝑛 , 𝛼0 ∈ 𝑅𝑛×𝑛 , an positive integer 𝑀 ≥ 1, 𝛾 ∈ (0, 1), 𝛿 > 0, 0 < 𝜎1 < 𝜎2 < 1, 𝜀 > 0. Set 𝑘 = 0. Step 1. If ∥𝐹𝑘 ∥ < 𝜀, stop. Step 2. If ∣𝐹𝑘𝑇 𝐽𝑘 𝐹𝑘 ∣/𝐹𝑘𝑇 𝐹𝑘 < 𝜀, stop. Step 3. Compute 𝑠𝑔𝑛𝑘 = 𝑠𝑔𝑛(𝐹𝑘𝑇 𝐽𝑘 𝐹𝑘 ), set 𝑑𝑘 = −𝑠𝑔𝑛𝑘 𝐹𝑘 . Step 4. (a) If 𝑘 = 0, set 𝑥𝑘+1 = 𝑥𝑘 − 𝛼𝑘 𝑑𝑘 , go to Step 8. 𝑖 𝑖 (b) If 𝑦𝑘−1 /𝑠𝑖𝑘−1 > 0, set 𝜆𝑖𝑘 = 𝑦𝑘−1 /𝑠𝑖𝑘−1 ; otherwise, set 𝜆𝑖𝑘 = 𝑠𝑇𝑘−1 𝑦𝑘−1 /𝑠𝑇𝑘−1 𝑠𝑘−1 , 𝑖 =

1, 2, ⋅ ⋅ ⋅ , 𝑛. (c) If 𝜆𝑖𝑘 < 𝜀 or 𝜆𝑖𝑘 > 1/𝜀, Set 𝜆𝑖𝑘 = 𝛿, 𝑖 = 1, 2, ⋅ ⋅ ⋅ , 𝑛. Step 5. Define 𝛼𝑘 = 𝑑𝑖𝑎𝑔{1/𝜆1𝑘 , 1/𝜆2𝑘 , . . . , 1/𝜆𝑛𝑘 } and 𝑧𝑘 = 𝛼𝑘 𝑑𝑘 . Step 6. Set 𝜏 = 1, If 𝑓 (𝑥𝑘 + 𝜏 𝑧𝑘 ) ≤

max

𝑗∈[0,min[𝑘,𝑀 −1]]

𝑓 (𝑥𝑘−𝑗 ) + 𝛾𝜏 𝐹𝑘𝑇 𝐽𝑘 𝑧𝑘

(7)

then set 𝜏𝑘 = 𝜏, 𝑥𝑘+1 = 𝑥𝑘 + 𝜏𝑘 𝑧𝑘 , 𝑠𝑘 = 𝑥𝑘+1 − 𝑥𝑘 , 𝑦𝑘 = 𝐹𝑘+1 − 𝐹𝑘 and go to Step 8. Step 7. If (7) does not hold, then define 𝜏𝑛𝑒𝑤 ∈ [𝜎1 𝜏𝑘 , 𝜎2 𝜏𝑘 ], set 𝜏𝑘 = 𝜏𝑛𝑒𝑤 and go to Step 6. Step 8. Set k:=k+1, go to Step 1. The following Lemma shows that in most cases the parameter 𝜆𝑖𝑘 is positive.

Lemma 1. Let 𝜆𝑖𝑘 be generated by Step (4), then 𝜆𝑖𝑘 > 0 when one of the following cases holds. 𝑇 (i) 𝐹𝑘−1 𝐹𝑘 < 0; 𝑇 (i) 𝐹𝑘−1 𝐹𝑘 > 0 and ∥𝐹𝑘 ∥ < ∥𝐹𝑘−1 ∥.

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Zhensheng Yu Jinhong Yu

5

𝑖 𝑖 𝑖 Proof. By Step 4(b), if 𝑦𝑘−1 /𝑠𝑖𝑘−1 > 0, then 𝜆𝑖𝑘 = 𝑦𝑘−1 /𝑠𝑖𝑘−1 > 0, if 𝑦𝑘−1 /𝑠𝑖𝑘−1 ≤

0, then 𝜆𝑖𝑘 =

𝑠𝑇 𝑘−1 𝑦𝑘−1 , 𝑠𝑇 𝑘−1 𝑠𝑘−1

and therefore

𝑇 𝑇 𝑇 𝑠𝑔𝑛(𝜆𝑖𝑘 ) = −𝑠𝑔𝑛(𝐹𝑘−1 𝑦𝑘−1 /𝐹𝑘−1 𝐹𝑘−1 ) = −𝑠𝑔𝑛(𝐹𝑘−1 𝑦𝑘−1 ).

If (i) holds, then 𝑇 𝑇 𝑇 𝐹𝑘−1 𝑦𝑘−1 = 𝐹𝑘−1 𝐹𝑘 − 𝐹𝑘−1 𝐹𝑘−1 < 0,

hence 𝜆𝑖𝑘 > 0. If (ii) holds, then by Cauchy-Schwarz inequality, we obtain 𝑇 𝑇 0 < 𝐹𝑘−1 𝐹𝑘 ≤ ∥𝐹𝑘−1 ∥∥𝐹𝑘 ∥ < ∥𝐹𝑘−1 ∥2 = 𝐹𝑘−1 𝐹𝑘−1 ,

and so 𝑇 𝑇 𝑇 𝐹𝑘−1 𝑦𝑘−1 = 𝐹𝑘−1 𝐹𝑘 − 𝐹𝑘−1 𝐹𝑘−1 < 0,

hence 𝜆𝑖𝑘 > 0. Notice that the only case in which 𝜆𝑖𝑘 could be negative, and as a consequence 𝑇 we would have to choose 𝛿 > 0 in Step 4, is when 𝐹𝑘−1 𝐹𝑘 > 0 and ∥𝐹𝑘 ∥ ≥

∥𝐹𝑘−1 ∥, i.e., no descent is observed in the merit function. Hence we know that the matrix 𝛼𝑘 defined by Step 5 is positive define. To analyze the global convergence of the algorithm, we make the following assumptions: Assumption A (i) The level set ℒ = {𝑥 ∈ 𝑅𝑛 ∣𝑓 (𝑥) < 𝑓 (𝑥0 )} is bounded. (ii) 𝐹 (𝑥) is continuously differentiable ℒ. (iii) 𝐽(𝑥) is nonsingular for all 𝑥 ∈ ℒ. The following Lemma gives the property of 𝑧𝑘 defined by Step 5. Lemma 2. Under Assumption A, if {𝑥𝑘 } is generated by Algorithm 2.1, then there exist positive constants 𝑐1 , 𝑐2 and 𝑐3 such that

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A Multivariate Spectral Approach for Large Scale Nonlinear System

∥𝑧𝑘 ∥ ≤ 𝑐1 ∥∇𝑓 (𝑥𝑘 )∥,

(8)

∥∇𝑓 (𝑥𝑘 )∥ ≤ 𝑐2 ∥𝑧𝑘 ∥,

(9)

𝐹𝑘𝑇 𝐽𝑘 𝑧𝑘 ≤ −𝑐3 ∥∇𝑓 (𝑥𝑘 )∥2 .

(10)

and

Proof. Let 𝑀1 and 𝑀2 be positive constants such that ∥𝐽(𝑥)−1 ∥ ≤ 𝑀1 and ∥𝐽(𝑥)∥ ≤ 𝑀2 for all 𝑥 ∈ Ω. Since min(𝜀, 1/𝛿) < 𝜆𝑖𝑘 < max(1/𝜀, 1/𝛿) and 𝛼𝑘 = 𝑑𝑖𝑎𝑔{1/𝜆1𝑘 , 1/𝜆2𝑘 , . . . , 1/𝜆𝑛𝑘 }. By ∥𝑧𝑘 ∥ = ∥𝛼𝑘 𝑑𝑘 ∥ = ∥𝛼𝑘 𝐹𝑘 ∥ and 𝐹𝑘 = 1 −𝑇 2 𝐽𝑘 ∇𝑓 (𝑥𝑘 ),

we have

∥𝑧𝑘 ∥ ≤ i.e., 𝑐1 =

𝑀1 min(𝜀,1/𝛿)

1 −1 𝑀1 ∥𝐽𝑘 ∥∥𝛼𝑘 ∥∥∇𝑓 (𝑥𝑘 )∥ ≤ ∥∇𝑓 (𝑥𝑘 )∥, 2 min(𝜀, 1/𝛿) > 0.

On the other hand, since ∇𝑓 (𝑥𝑘 ) = 2𝐽𝑘𝑇 𝐹𝑘 and 𝐹𝑘 = −𝑠𝑔𝑛𝑘 𝛼𝑘−1 𝑧𝑘 , we have ∥∇𝑓 (𝑥𝑘 )∥ ≤ 2∥𝐽𝑘 ∥∥𝐹𝑘 ∥ ≤ 2∥𝐽𝑘 ∥∥𝛼𝑘−1 ∥∥𝑧𝑘 ∥ ≤ 2𝑀2 max(1/𝜀, 1/𝛿)∥𝑧𝑘 ∥, i.e., 𝑐2 = 2𝑀2 max(1/𝜀, 1/𝛿).

Finally, for all 𝑘 > 0, we have from Step 3 that ∣𝐹𝑘𝑇 𝐽𝑘 𝐹𝑘 ∣ ≥ 𝜀∥𝐹𝑘 ∥2 , which means ∥𝛼𝑘−1 ∥∣𝐹𝑘𝑇 𝐽𝑘 𝑧𝑘 ∣ ≥ 𝜀∥𝐹𝑘 ∥2 .

(11)

Hence from Step 5, we have

𝐹𝑘𝑇 𝐽𝑘 𝑧𝑘 = 𝐹𝑘𝑇 𝐽𝑘 (−𝑠𝑔𝑛𝑘 )𝛼𝑘 𝐹𝑘 =

⎧  ⎨ −𝐹 𝑇 𝐽𝑘 𝛼𝑘 𝐹𝑘 𝐹 𝑇 𝐽𝑘 𝐹𝑘 > 0, 𝑘 𝑘  ⎩ 𝐹𝑘𝑇 𝐽𝑘 𝛼𝑘 𝐹𝑘 𝐹𝑘𝑇 𝐽𝑘 𝐹𝑘 < 0,

(12)

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Zhensheng Yu Jinhong Yu

7

which implies that 𝐹𝑘𝑇 𝐽𝑘 𝑧𝑘 < 0, hence by (11) and (8), we have 𝐹𝑘𝑇 𝐽𝑘 𝑧𝑘 ≤

−𝜀 −𝜀 2 ∥𝛼−1 𝑧𝑘 ∥2 ≤ −𝜀 min(𝜀, 1/𝛿)∥𝑧𝑘 ∥2 ≤ −𝑐3 ∇𝑓 (𝑥𝑘 )∥2 , −1 ∥𝐹𝑘 ∥ ≤ ∥𝛼𝑘 ∥ ∥𝛼𝑘−1 ∥ 𝑘

where 𝑐3 = −𝜀 min(𝜀, 1/𝛿)𝑐−2 2 > 0. By Lemma 2, the convergence theorem in [9], and similar to proof of Theorem 3.4 [6], we can easily obtain the main convergence theorem as follows:

Theorem 1. Under Assumption A, Algorithm 2.1 either terminates at a finite iteration 𝑗 where 𝐹𝑗 = 0 or ∣𝐹𝑗𝑇 𝐽𝑗 𝐹𝑗 ∣ < 𝜀∥𝐹𝑗 ∥2 , or it generates a sequence {𝑥𝑘 } such that lim ∥𝐹𝑘 ∥ = 0.

𝑘→∞

3. NUMERICAL TESTS

In this section, we implemented Algorithm 2.1 in Matlab 7.0. We compare the performance of the MSANE algorithm, on a set of large-scale test problems, with the SANE algorithm in [6]. The problems used in our tests are well-known large functions which chosen from [6], Table I lists the problems and the starting points 𝑥0 . In both algorithms, we use parameters: 𝛾 = 0.01, 𝜀 = 10−5 , 𝜎1 = 0.1, 𝜎2 = 0.9, 𝛼0 = 𝐼, 𝑀 = 5,

𝛿=

⎧     ⎨

1

∥𝐹𝑘 ∥     ⎩ 10−5

𝑖𝑓 ∥𝐹𝑘 ∥ > 1, 𝑖𝑓 10−5 ≤ ∥𝐹𝑘 ∥ ≤ 1, 𝑖𝑓 ∥𝐹𝑘 ∥ < 10−5 .

The numerical results are shown in Tables II and III. We report the problem number and the dimension of the problem (No(n)), the number of iterations (IT), the number of function evaluations (F), Time denotes the CPU time (in second) used when the iteration is stopped, ∥𝐹 (𝑥∗ )∥ denotes the final function

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A Multivariate Spectral Approach for Large Scale Nonlinear System

value. The numerical tests show that our algorithm works quiet well for those test problems, and the method is comparable with the method [6]. Table I: Tests problems Problem

Name

Start point

1

Exponential function 1

(𝑛/(𝑛 − 1), 𝑛/(𝑛 − 1), ⋅ ⋅ ⋅ , 𝑛/(𝑛 − 1))𝑇

2

Exponential function 2

(𝑛/𝑛2 , 1/𝑛2 , ⋅ ⋅ ⋅ , 1/𝑛2 )𝑇

3

Exponential function 3

(1/4𝑛2 , 2/4𝑛2 , ⋅ ⋅ ⋅ , 𝑛/4𝑛2 )𝑇

4

Extended Rosenbrock function

(5, 1, 5, 1, ⋅ ⋅ ⋅ , 5, 1)𝑇

5

Chandrasekhars H-equation

(1, 1, ⋅ ⋅ ⋅ , 1)𝑇

6

Logarithmic function

(1, 1, ⋅ ⋅ ⋅ , 1)𝑇

7

Broyden Tridiagonal function

(−1, −1, ⋅ ⋅ ⋅ , −1)𝑇

8

Trigexp function

(0, 0, ⋅ ⋅ ⋅ , 0)𝑇

9

Function 15

(−1, −1 ⋅ ⋅ ⋅ , −1)𝑇

10

Strictly convex function 1

(1/𝑛, 2/𝑛, ⋅ ⋅ ⋅ , 1)𝑇

11

Strictly convex function 2

(1, 1, ⋅ ⋅ ⋅ , 1)𝑇

12

Function 18

(0, 0, ⋅ ⋅ ⋅ , 0)𝑇

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Zhensheng Yu Jinhong Yu

9

Table II: Numerical Results for MSANE No(n)

Iter

F

Time

∥𝐹 (𝑥∗ )∥

No(n)

Iter

F

Time

∥𝐹 (𝑥∗ )∥

1(1000)

8

8

1.000

4.3258e-06

7(500)

35

37

2.899

9.8167e-06

1(5000)

7

7

16.592

5.0582e-06

7(1000)

42

42

12.716

3.8522e-06

1(10000)

7

7

0

5.0582e-06

7(2000)

35

39

22.924

7.6898e-06

7(5000)

32

32

112.384

6.1711e-06

2(200)

3

7

0.074

5.7747e-06

8(100)

19

22

0.420

4.0179e-06

2(500)

2

2

0.321

9.3255e-06

8(500)

21

21

1.952

6.7657e-06

2(1000)

4

6

0.760

1.9317e-07

8(1000)

17

17

3.867

5.4889e-06

2(2000)

3

59

13.042

7.2816e-06

8(2000)

13

13

8.064

5.1012e-06

8(5000)

12

12

40.029

9.9579e-06

3(100)

6

6

0.254

5.1855e-06

9(100)

65

83

1.693

5.7020e-06

3(200)

5

5

0.216

9.1314e-06

9(500)

59

73

4.155

6.7910e-06

3(1000)

3

3

0.528

4.0507e-06

9(1000)

82

106

16.938

5.8209e-06

3(2000)

3

3

2.055

4.8157e-06

3(5000)

2

2

0.216

6.8302e-06

4(1000)

14

15

4.396

1.2990e-06

10(1000)

8

8

0.930

9.1993e-07

4(2000)

11

11

0.254

5.8340e-06

10(5000)

8

8

19.980

2.0372e-06

4(5000)

9

9

47.328

8.4091e-06 10(10000)

8

8

38.003

2.5746e-06

5(100)

17

18

4.331

2.8086e-06

11(100)

12

43

0.220

4.3625e-06

5(200)

27

27

40.469

3.7744e-06

11(500)

30

605

7.540

4.5363e-09

5(500)

24

36

579.850

5.2465e-06

11(1000)

48

1207

44.325

1.6228e-10

6(1000)

6

6

1.402

2.5837e-07

12(99)

6

6

0.211

4.8803e-09

6(2000)

6

6

2.261

3.5822e-07

12(399)

6

6

0.417

1.4567e-08

6(5000)

6

6

12.823

5.5970e-07

12(999)

6

6

1.320

2.3049e-08

6(10000)

6

6

59.147

7.8840e-07

12(3999)

6

6

13.176

4.6116e-08

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A Multivariate Spectral Approach for Large Scale Nonlinear System

Table III: Numerical Results for SANE No(n)

Iter

F

Time

∥𝐹 (𝑥∗ )∥

No(n)

Iter

F

Time

∥𝐹 (𝑥∗ )∥

1(1000)

14

17

1.438

7.9822e-06

7(500)

23

24

0.962

4.5644e-06

1(5000)

11

13

24.174

3.6738e-06

7(1000)

23

24

2.395

5.5515e-06

1(10000)

9

12

81.503

8.7411e-06

7(2000)

25

25

9.389

4.5721e-06

7(5000)

26

26

53.275

6.1626e-06

2(200)

22

33

0.537

6.1833e-06

8(100)

10

13

0.383

5.5431e-06

2(500)

22

32

0.914

9.7121e-06

8(500)

9

12

0.710

3.3164e-06

2(1000)

14

23

1.767

9.2286e-06

8(1000)

8

11

1.501

8.1520e-06

2(2000)

13

23

5.802

8.2819e-06

8(2000)

8

11

3.639

4.0945e-06

8(5000)

7

10

18.773

9.6326e-06

3(100)

6

6

0.373

5.1554e-06

9(100)

48

62

1.047

3.0012e-06

3(200)

5

5

0.453

6.5307e-06

9(500)

49

59

2.765

9.5874e-06

3(1000)

3

3

0.993

9.6314e-06

9(1000)

41

48

5.856

4.1449e-06

3(2000)

3

3

1.163

4.8157e-06

3(5000)

2

2

4.724

4.4575e-06

4(1000)

25

44

4.888

6.8517e-06

10(1000)

9

9

1.060

0

4(2000)

52

109

18.007

1.039e-06

10(5000)

9

9

20.360

0

4(5000)

11

13

26.001

1.0407e-06 10(10000)

9

9

90.440

0

5(100)

8

8

2.088

1.1053e-06

11(100)

76

128

1.575

9.4372e-06

5(200)

8

8

12.795

1.5671e-06

11(500)

1149

5161

84.610

9.9310e-06

5(500)

8

8

148.729

2.4796e-06

11(1000)

2378

13122

582.349

9.9813e-6

6(1000)

6

6

0.697

2.5837e-07

12(99)

5

6

0.180

7.2559e-09

6(2000)

6

6

2.643

3.5822e-07

12(399)

5

6

0.280

1.4567e-08

6(5000)

6

6

20.510

5.5970e-07

12(999)

5

6

0.836

2.3049e-08

6(10000)

6

7

41.657

7.8840e-07

12(3999)

5

6

10.820

4.6116e-08

1352

Zhensheng Yu Jinhong Yu

11

4. CONCLUSION In this paper, we extend the multivariate spectral approach to the nonlinear equations problems, compared with the classical spectral approach, our method allows an individual adaptive stepsize along each coordinate direction. The numerical tests for a set of large scale problems shows that our algorithm works quiet well. How to improve the algorithm to obtain the local linear convergence rate deserves further study, we leave it as the future work.

ACKNOWLEDGMENTS

This work is supported by Innovation Program of Shanghai Municipal Education Commission(No.10YZ99) and Shanghai Leading Academic Discipline Project (S.30501).

REFERENCES [1] J. Barzilai, J.M. Borwein, Two p stepsize gradient methods, IMA J. Numer. Anal, 8, 141-148 (1988). [2] E.G. Birgin, J.M. Martinez and M. Raydan, Spectral Projected gradient methods, Encyclopedia of Optimization (C.A. Floudas and P.M. Pardalos (Eds.), Springer, 2009, pp. 3652C3659. [3] E.G. Birgin, N. K. Krejic and J.M. Mart´ınez, Globally convergent inexact quasi-Newton methods for solving nonlinear systems, Numer. Algorithms, 32, 249-260 (2003). [4] W.Y. Cheng, D.H. Li, A derivative-free nonmonotone line search and its application to the spectral residual method, IMA J. Numer. Anal, 29, 814-825 (2009). [5] W.La Cruz, J.M. Mart´ınez, M. Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations, Math. Comput, 75, 1429-1448 (2006).

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A Multivariate Spectral Approach for Large Scale Nonlinear System

[6] W. La Cruz, M. Raydan, Nonmonotone spectral methods for large-scale nonlinear systems, Optim. Methods Softw, 18, 583-599 (2003). [7] J.E. Dennis and R.B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall, Englewood Cliffs, NJ(1983). [8] M. Gasparo, A nonmonotone hybrid method for nonlinear systems, Optim. Meth. Soft, 13, 79-94(2002). [9] L. Grippo, F. Lampariello and S. Lucidi, A nonmonotone line search technique for Newton’s method, SIAM J. Numer. Anal, 23, 707-716 (1986). [10] L.Han, G.H.Yu, L.T.Guan, Multivariate spectral gradient method for unconstrained optimization, Appl. Math. Comput, 201, 621-630 (2008). [11] D.H. Li and M. Fukushima, A derivative-free line search and global convergence of Broyden-like method for nonlinear equations, Optim. Meth. Soft, 13, 181-201 (2000). [12] M. Raydan, The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem, SIAM J. Optim, 7, 26-33 (1997). [13] M. Gomez-Ruggiero, J.M. Mart´ınez and A. Moretti, Comparing algorithms for solving sparse nonlinear systems of equations, SIAM J. Sci. Comput, 23, 459-483 (1992). [14] M.V. Solodov, B.F. Svaiter, A globally convergent inexact Newton method for systems of monotone equations, In: Fukushima M, Qi L (eds)Reformulation: piecewise smooth, semi-smooth and smoothing methods, Kluwer, Holanda, (1998), pp.355-369.

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL.13, NO.7, 2011

Some Properties on the p-Adic Invariant Integral on Ζp Associated with Genocchi and Bernoulli Polynomials, Taekyun Kim, Lee-Chae Jang and Young-Hee Kim,…………………………1201 On the q-Extension of the Twisted Generalized Euler Numbers and Polynomials Attached to χ, Y.H. Kim, B. Lee and T. Kim,………………………………………………………………1208 Symmetric p-Adic Invariant Integral on Ζp for the Generalized Twisted Euler Polynomials, L.C. Jang, T. Kim and B. Lee,…………………………………………………………………….1214 Algorithm for a New System of Variational Inclusions with B-monotone Operators in Banach Spaces, Jiu-ping Xu, Xue-ping Luo and Nan-jing Huang,…………………………………...1223 A Study of Uniquely Remotal Sets, R. Khalil and M. Sababheh,…………………………....1233 On The Symmetric Properties for the Generalized Genocchi Polynomials, Seog-Hoon Rim, EunJung Moon, Jeong-Hee Jin and Sun-Jung Lee,……………………………………………....1240 A shift-splitting Jacobi-gradient algorithm for Lyapunov matrix equations arising from control theory, Sheng-Kun Li and Ting-Zhu Huang,………………………………………………...1246 A New Proof of the Symmetric Properties for the Twisted Generalized Euler Polynomials of Higher Order, Seog-Hoon Rim, Young-Hee Kim, Byungje Lee and Taekyun Kim,………..1258 On the Global Convergence for a Variant of SQP Method, Si-chun Wang,………………....1263 Note on ω-limit set of a graph map, Taixiang Sun, Hongjian Xi and Hailan Liang,………....1268 A generalized mixed additive-cubic functional equation, Tian Zhou Xu, John Michael Rassias and Wan Xin Xu,……………………………………………………………………………...1273 Fast Gaussian elimination with pivoting for solving linear systems of rational Krylov matrices, Xiang Wang and Linzhang Lu,……………………………………………………………….1283 An Identity of Symmetry For The Generalized Euler Polynomials, Taekyun Kim,……….....1292 On properties of Strongly Prequasi-invex Functions, Wan Mei Tang,…………………….....1297 Lp (p > 1) Convergence results for Particle Filtering, Yuhua Guo and Xiaoyuan Yang,…......1309 A Derivative-Free Method for Nonlinear System of Equations with Global Convergence, Jianguo Zhang, Yun-Hai Xiao and Dangzhen Zhou…………………………………………………...1319

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL.13, NO.7, 2011 (continues)

Note on p-Adic q-Euler Measure, Young-Hee Kim, Lee-Chae Jang, Taekyun Kim, Byungje Lee and Seog-Hoon Rim,…………………………………………………………………………..1329 Wolfe type second-order symmetric duality in multiobjective programming over cones, Yu Chen and Zhi-Ming Luo,…………………………………………………………………………….1335 A Multivariate Spectral Approach for Large Scale Nonlinear System, Zhensheng Yu and Jinhong Yu,……………………………………………………………………………………………..1342