Fixed point theory for Lipschitzian-type mappings with applications

373 Pages·2009·2.57 MB·English

by  D. R. Sahu

Fixed Point Theory for Lipschitzian-type
Mappings with Applications
Topological Fixed Point Theory and Its
Applications
VOLUME 6
Fixed Point Theory for
Lipschitzian-type Mappings with
Applications
by
Ravi P. Agarwal
Florida Institute of Technology
Melbourne, FL, USA
Donal O’Regan
National University of Ireland
Galway, Ireland
and
D.R. Sahu
Banaras Hindu University
Varanasi, India
123
RaviP.Agarwal DonalO’Regan
DepartmentofMathematicalSciences InstituteofMathematics
FloridaInsituteofTechnology UniversityCollegeGalway
Melbourne,FL32901USA NationalUniversityofIreland
agarwal@fit.edu Galway,Ireland
[email protected]
D.R.Sahu
DepartmentofMathematics
FacultyofScience
BanarasHinduUniversity
Varanasi,India
[email protected]
ISBN978-0-387-75817-6 e-ISBN978-0-387-75818-3
DOI10.1007/978-0-387-75818-3
SpringerDordrechtHeidelbergLondonNewYork
LibraryofCongressControlNumber: 2009927662
AMSSubjectClassifications(2000):47H09,47H10
(cid:2)c 2009SpringerScience+BusinessMedia,LLC.
All rights reserved. This work may not be translated or copied in whole or in part without the written
permission of the publisher (Springer Science+Business Media LLC, 233 Spring Street, New York, NY
10013,U.S.A.),exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanalysis.Useinconnection
withanyformofinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar
ordissimilarmethodologynowknownorhereafterdevelopedisforbidden.
Theuseinthispublicationoftradenames,trademarks,servicemarksandsimilarterms,eveniftheyare
notidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubjectto
proprietaryrights.
Printedonacid-freepaper.
SpringerispartofSpringerScience+BusinessMedia(www.springer.com)
Dedicated to our daughters
Sheba Agarwal
Lorna Emily O’Regan
Gargi Sahu
Preface
Over the past few decades, fixed point theory of Lipschitzian and non-
Lipschitzian mappings has been developed into an important field of study in
bothpureandappliedmathematics. Themainpurposeofthisbookistopresent
manyofthebasictechniquesandresultsofthistheory. Ofcourse,notallaspects
of this theory could be included in this exposition.
The book contains eight chapters. The first chapter is devoted to some
of the basic results of nonlinear functional analysis. The final section in this
chapter deals with the classic results of fixed point theory. Our goal is to
study nonlinear problems in Banach spaces. We remark here that it is hard to
study these without the geometric properties of Banach spaces. As a result in
Chapter 2, we discuss elements of convexity and smoothness of Banach spaces
and properties of duality mappings. This chapter also includes many interest-
ingresultsrelatedtoBanachlimits,metricprojectionmappings,andretraction
mappings. In Chapter 3, we consider normal structure coefficient, weak normal
structure coefficient, and related coefficients. This includes the most recent
workintheliterature. Ourtreatment ofthemain subjectinthebookbeginsin
Chapter 4. In this chapter, we consider the problem of existence of fixed points
of Lipschitzian and non-Lipschitzian mappings in metric spaces. Chapter 5
is devoted to problems of existence of fixed points of nonexpansive, asympto-
tically nonexpansive, pseudocontractive mappings in Banach spaces. Most of
the results are discussed in infinite-dimensional Banach spaces. The theory
of iteration processes for computing fixed points of nonexpansive, asympto-
tically nonexpansive, pseudocontractive mappings is developed in Chapter 6.
In Chapter 7, we prove strong convergence theorems for nonexpansive, pseudo-
contractive, and asymptotically pseudocontractive mappings in Banach spaces.
Finally in Chapter 8, we discuss several applicable problems arising in different
fields.
Eachchapterinthisbookcontainsabriefintroductiontodescribethetopic
thatiscovered. Also,anexercisesectionisincludedineachchapter. Becausethe
book is self-contained, the book should be of interest to graduate students and
mathematicians interested in learning fundamental theorems about the theory
of Lipschitzian and non-Lipschitzian mappings and fixed points.
We wish to express our deepest appreciation to Professors Y. Alber,
T. D. Benavides, J. S. Jung, W. A. Kirk, S. Prus, S. Reich, B. E. Rhoades,
B. K. Sharma, W. Takahashi, H. K. Xu, and J. C. Yao for their encouragement
and personal support.
viii Preface
FinallytheauthorsareverygratefultoMs.V.Damle,Ms.B.Marciaandthe
staff from the Springer Publishers for their indefatigable cooperation, patience
and understanding.
R.P. Agarwal
Donal O’Regan
D.R. Sahu
Contents
Preface vii
1 Fundamentals 1
1.1 Topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Normed spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Dense set and separable space . . . . . . . . . . . . . . . . . . . . 20
1.4 Linear operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.5 Space of bounded linear operators . . . . . . . . . . . . . . . . . 25
1.6 Hahn-Banach theorem and applications . . . . . . . . . . . . . . 28
1.7 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.8 Reflexivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.9 Weak topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1.10 Continuity of mappings . . . . . . . . . . . . . . . . . . . . . . . 43
2 Convexity, Smoothness, and Duality Mappings 49
2.1 Strict convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.2 Uniform convexity . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.3 Modulus of convexity . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.4 Duality mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
2.5 Convex functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
2.6 Smoothness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
2.7 Modulus of smoothness . . . . . . . . . . . . . . . . . . . . . . . 94
2.8 Uniform smoothness . . . . . . . . . . . . . . . . . . . . . . . . . 98
2.9 Banach limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
2.10 Metric projection and retraction mappings . . . . . . . . . . . . . 115
3 Geometric Coefficients of Banach Spaces 127
3.1 Asymptotic centers and asymptotic radius . . . . . . . . . . . . . 127
3.2 The Opial and uniform Opial conditions . . . . . . . . . . . . . . 136
3.3 Normal structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
3.4 Normal structure coefficient . . . . . . . . . . . . . . . . . . . . . 153
3.5 Weak normal structure coefficient . . . . . . . . . . . . . . . . . . 162
3.6 Maluta constant . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
3.7 GGLD property. . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
x Contents
4 Existence Theorems in Metric Spaces 175
4.1 Contraction mappings and their generalizations . . . . . . . . . . 175
4.2 Multivalued mappings . . . . . . . . . . . . . . . . . . . . . . . . 188
4.3 Convexity structure and fixed points . . . . . . . . . . . . . . . . 197
4.4 Normal structure coefficient and fixed points . . . . . . . . . . . 201
4.5 Lifschitz’s coefficient and fixed points. . . . . . . . . . . . . . . . 206
5 Existence Theorems in Banach Spaces 211
5.1 Non-self contraction mappings . . . . . . . . . . . . . . . . . . . 211
5.2 Nonexpansive mappings . . . . . . . . . . . . . . . . . . . . . . . 222
5.3 Multivalued nonexpansive mappings . . . . . . . . . . . . . . . . 237
5.4 Asymptotically nonexpansive mappings . . . . . . . . . . . . . . 243
5.5 Uniformly L-Lipschitzian mappings . . . . . . . . . . . . . . . . . 250
5.6 Non-Lipschitzian mappings . . . . . . . . . . . . . . . . . . . . . 259
5.7 Pseudocontractive mappings . . . . . . . . . . . . . . . . . . . . . 264
6 Approximation of Fixed Points 279
6.1 Basic properties and lemmas . . . . . . . . . . . . . . . . . . . . 279
6.2 Convergence of successive iterates . . . . . . . . . . . . . . . . . . 286
6.3 Mann iteration process . . . . . . . . . . . . . . . . . . . . . . . . 288
6.4 Nonexpansive and quasi-nonexpansive mappings . . . . . . . . . 292
6.5 The modified Mann iteration process . . . . . . . . . . . . . . . . 300
6.6 The Ishikawa iteration process . . . . . . . . . . . . . . . . . . . 303
6.7 The S-iteration process. . . . . . . . . . . . . . . . . . . . . . . . 307
7 Strong Convergence Theorems 315
7.1 Convergence of approximants of self-mappings. . . . . . . . . . . 315
7.2 Convergence of approximants of non-self mappings . . . . . . . . 324
7.3 Convergence of Halpern iteration process . . . . . . . . . . . . . 327
8 Applications of Fixed Point Theorems 333
8.1 Attractors of the IFS . . . . . . . . . . . . . . . . . . . . . . . . . 333
8.2 Best approximation theory. . . . . . . . . . . . . . . . . . . . . . 335
8.3 Solutions of operator equations . . . . . . . . . . . . . . . . . . . 336
8.4 Differential and integral equations . . . . . . . . . . . . . . . . . 339
8.5 Variational inequality . . . . . . . . . . . . . . . . . . . . . . . . 341
8.6 Variational inclusion problem . . . . . . . . . . . . . . . . . . . . 343
Appendix A 349
A.1 Basic inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
A.2 Partially ordered set . . . . . . . . . . . . . . . . . . . . . . . . . 350
A.3 Ultrapowers of Banach spaces . . . . . . . . . . . . . . . . . . . . 350
Bibliography 353
Index 365