Fixed Point Theory for Lipschitzian-Type Mappings with Applications

By: Ravi P. Agarwal, Donal O'Regan, D. R. Sahu

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Hardcover | 27 May 2009

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380 Pages

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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.

Industry Reviews

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)

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Preface	p. vii
Fundamentals	p. 1
Topological spaces	p. 1
Normed spaces	p. 8
Dense set and separable space	p. 20
Linear operators	p. 22
Space of bounded linear operators	p. 25
Hahn-Banach theorem and applications	p. 28
Compactness	p. 32
Reflexivity	p. 34
Weak topologies	p. 36
Continuity of mappings	p. 43
Convexity, Smoothness, and Duality Mappings	p. 49
Strict convexity	p. 49
Uniform convexity	p. 53
Modulus of convexity	p. 58
Duality mappings	p. 67
Convex functions	p. 79
Smoothness	p. 91
Modulus of smoothness	p. 94
Uniform smoothness	p. 98
Banach limit	p. 106
Metric projection and retraction mappings	p. 115
Geometric Coefficients of Banach Spaces	p. 127
Asymptotic centers and asymptotic radius	p. 127
The Opial and uniform Opial conditions	p. 136
Normal structure	p. 146
Normal structure coefficient	p. 153
Weak normal structure coefficient	p. 162
Maluta constant	p. 165
GGLD property	p. 172
Existence Theorems in Metric Spaces	p. 175
Contraction mappings and their generalizations	p. 175
Multivalued mappings	p. 188
Convexity structure and fixed points	p. 197
Normal structure coefficient and fixed points	p. 201
Lifschitz's coefficient and fixed points	p. 206
Existence Theorems in Banach Spaces	p. 211
Non-self contraction mappings	p. 211
Nonexpansive mappings	p. 222
Multivalued nonexpansive mappings	p. 237
Asymptotically nonexpansive mappings	p. 243
Uniformly L-Lipschitzian mappings	p. 250
Non-Lipschitzian mappings	p. 259
Pseudocontractive mappings	p. 264
Approximation of Fixed Points	p. 279
Basic properties and lemmas	p. 279
Convergence of successive iterates	p. 286
Mann iteration process	p. 288
Nonexpansive and quasi-nonexpansive mappings	p. 292
The modified Mann iteration process	p. 300
The Ishikawa iteration process	p. 303
The S-iteration process	p. 307
Strong Convergence Theorems	p. 315
Convergence of approximants of self-mappings	p. 315
Convergence of approximants of non-self mappings	p. 324
Convergence of Halpern iteration process	p. 327
Applications of Fixed Point Theorems	p. 333
Attractors of the IFS	p. 333
Best approximation theory	p. 335
Solutions of operator equations	p. 336
Differential and integral equations	p. 339
Variational inequality	p. 341
Variational inclusion problem	p. 343
	p. 349
Basic inequalities	p. 349
Partially ordered set	p. 350
Ultrapowers of Banach spaces	p. 350
Bibliography	p. 353
Index	p. 365
Table of Contents provided by Ingram. All Rights Reserved.

Fixed Point Theory for Lipschitzian-Type Mappings with Applications

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