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Fixed Point Theory for Lipschitzian-type Mappings with Applications : Topological Fixed Point Theory and Its Applications - Ravi P. Agarwal

Fixed Point Theory for Lipschitzian-type Mappings with Applications

Topological Fixed Point Theory and Its Applications


Published: 1st June 2009
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In recent years, the fixed point theory of Lipschitzian-type mappings has rapidly grown into an important field of study in both pure and applied mathematics. It has become one of the most essential tools in nonlinear functional analysis.

This self-contained book provides the first systematic presentation of Lipschitzian-type mappings in metric and Banach spaces. The first chapter covers some basic properties of metric and Banach spaces. Geometric considerations of underlying spaces play a prominent role in developing and understanding the theory. The next two chapters provide background in terms of convexity, smoothness and geometric coefficients of Banach spaces including duality mappings and metric projection mappings. This is followed by results on existence of fixed points, approximation of fixed points by iterative methods and strong convergence theorems. The final chapter explores several applicable problems arising in related fields.

This book can be used as a textbook and as a reference for graduate students, researchers and applied mathematicians working in nonlinear functional analysis, operator theory, approximations by iteration theory, convexity and related geometric topics, and best approximation theory.

From the reviews:

"The present book explains many of the basic techniques and ... the classical results of fixed point theory, and normal structure properties. ... Exercises are included in each chapter. As such, it is a self-contained book that can be used in a course for graduate students." (Srinivasa Swaminathan, Zentralblatt MATH, Vol. 1176, 2010)

"This book provides a presentation of fixed point theory for Lipschitzian type mappings in metric and Banach spaces. ... An exercise section is included at the end of each chapter, containing interesting and well chosen material in order to cover topics complementing the main body of the text. ... It is worthwhile to point out that a beginner in this area is certainly well served with this text ... . A book including all ... topics together for sure should be welcomed for graduate students." (Enrique Llorens-Fuster, Mathematical Reviews, Issue 2010 e)

Prefacep. vii
Fundamentalsp. 1
Topological spacesp. 1
Normed spacesp. 8
Dense set and separable spacep. 20
Linear operatorsp. 22
Space of bounded linear operatorsp. 25
Hahn-Banach theorem and applicationsp. 28
Compactnessp. 32
Reflexivityp. 34
Weak topologiesp. 36
Continuity of mappingsp. 43
Convexity, Smoothness, and Duality Mappingsp. 49
Strict convexityp. 49
Uniform convexityp. 53
Modulus of convexityp. 58
Duality mappingsp. 67
Convex functionsp. 79
Smoothnessp. 91
Modulus of smoothnessp. 94
Uniform smoothnessp. 98
Banach limitp. 106
Metric projection and retraction mappingsp. 115
Geometric Coefficients of Banach Spacesp. 127
Asymptotic centers and asymptotic radiusp. 127
The Opial and uniform Opial conditionsp. 136
Normal structurep. 146
Normal structure coefficientp. 153
Weak normal structure coefficientp. 162
Maluta constantp. 165
GGLD propertyp. 172
Existence Theorems in Metric Spacesp. 175
Contraction mappings and their generalizationsp. 175
Multivalued mappingsp. 188
Convexity structure and fixed pointsp. 197
Normal structure coefficient and fixed pointsp. 201
Lifschitz's coefficient and fixed pointsp. 206
Existence Theorems in Banach Spacesp. 211
Non-self contraction mappingsp. 211
Nonexpansive mappingsp. 222
Multivalued nonexpansive mappingsp. 237
Asymptotically nonexpansive mappingsp. 243
Uniformly L-Lipschitzian mappingsp. 250
Non-Lipschitzian mappingsp. 259
Pseudocontractive mappingsp. 264
Approximation of Fixed Pointsp. 279
Basic properties and lemmasp. 279
Convergence of successive iteratesp. 286
Mann iteration processp. 288
Nonexpansive and quasi-nonexpansive mappingsp. 292
The modified Mann iteration processp. 300
The Ishikawa iteration processp. 303
The S-iteration processp. 307
Strong Convergence Theoremsp. 315
Convergence of approximants of self-mappingsp. 315
Convergence of approximants of non-self mappingsp. 324
Convergence of Halpern iteration processp. 327
Applications of Fixed Point Theoremsp. 333
Attractors of the IFSp. 333
Best approximation theoryp. 335
Solutions of operator equationsp. 336
Differential and integral equationsp. 339
Variational inequalityp. 341
Variational inclusion problemp. 343
p. 349
Basic inequalitiesp. 349
Partially ordered setp. 350
Ultrapowers of Banach spacesp. 350
Bibliographyp. 353
Indexp. 365
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9780387758176
ISBN-10: 0387758178
Series: Topological Fixed Point Theory and Its Applications
Audience: Tertiary; University or College
Format: Hardcover
Language: English
Number Of Pages: 368
Published: 1st June 2009
Publisher: Springer-Verlag New York Inc.
Country of Publication: US
Dimensions (cm): 23.5 x 15.5  x 3.18
Weight (kg): 0.71