Geometric properties of Banach spaces and nonlinear iterations

Charles Chidume (auth.)1848821891, 9781848821897

Nonlinear functional analysis and applications is an area of study that has provided fascination for many mathematicians across the world. This monograph delves specifically into the topic of the geometric properties of Banach spaces and nonlinear iterations, a subject of extensive research over the past thirty years.

Chapters 1 to 5 develop materials on convexity and smoothness of Banach spaces, associated moduli and connections with duality maps. Key results obtained are summarized at the end of each chapter for easy reference. Chapters 6 to 23 deal with an in-depth, comprehensive and up-to-date coverage of the main ideas, concepts and results on iterative algorithms for the approximation of fixed points of nonlinear nonexpansive and pseudo-contractive-type mappings. This includes detailed workings on solutions of variational inequality problems, solutions of Hammerstein integral equations, and common fixed points (and common zeros) of families of nonlinear mappings.

Carefully referenced and full of recent, incisive findings and interesting open-questions, this volume will prove useful for graduate students of mathematical analysis and will be a key-read for mathematicians with an interest in applications of geometric properties of Banach spaces, as well as specialists in nonlinear operator theory.

Table of contents :
Front Matter….Pages i-xvii
Some Geometric Properties of Banach Spaces….Pages 1-9
Smooth Spaces….Pages 11-18
Duality Maps in Banach Spaces….Pages 19-28
Inequalities in Uniformly Convex Spaces….Pages 29-44
Inequalities in Uniformly Smooth Spaces….Pages 45-55
Iterative Method for Fixed Points of Nonexpansive Mappings….Pages 57-86
Hybrid Steepest Descent Method for Variational Inequalities….Pages 87-111
Iterative Methods for Zeros of Ф – Accretive-Type Operators….Pages 113-127
Iteration Processes for Zeros of Generalized Ф —Accretive Mappings….Pages 129-140
An Example; Mann Iteration for Strictly Pseudo-contractive Mappings….Pages 141-149
Approximation of Fixed Points of Lipschitz Pseudo-contractive Mappings….Pages 151-160
Generalized Lipschitz Accretive and Pseudo-contractive Mappings….Pages 161-167
Applications to Hammerstein Integral Equations….Pages 169-191
Iterative Methods for Some Generalizations of Nonexpansive Maps….Pages 193-204
Common Fixed Points for Finite Families of Nonexpansive Mappings….Pages 205-214
Common Fixed Points for Countable Families of Nonexpansive Mappings….Pages 215-229
Common Fixed Points for Families of Commuting Nonexpansive Mappings….Pages 231-242
Finite Families of Lipschitz Pseudo-contractive and Accretive Mappings….Pages 243-250
Generalized Lipschitz Pseudo-contractive and Accretive Mappings….Pages 251-256
Finite Families of Non-self Asymptotically Nonexpansive Mappings….Pages 257-270
Families of Total Asymptotically Nonexpansive Maps….Pages 271-282
Common Fixed Points for One-parameter Nonexpansive Semigroup….Pages 283-285
Single-valued Accretive Operators; Applications; Some Open Questions….Pages 287-299
Back Matter….Pages 301-332