Geometric Properties of Banach Spaces and Nonlinear Iterations (Lecture Notes in Mathematics Book 1965)

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. 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 The contents of this monograph fall within the general area of nonlinear functional analysis and applications. We focus on an important topic within this area: geometric properties of Banach spaces and nonlinear iterations, a topic of intensive research e?orts, especially within the past 30 years, or so. In this theory, some geometric properties of Banach spaces play a crucial role. In the ?rst part of the monograph, we expose these geometric properties most of which are well known. As is well known, among all in?nite dim- sional Banach spaces, Hilbert spaces have the nicest geometric properties. The availability of the inner product, the fact that the proximity map or nearest point map of a real Hilbert space H onto a closed convex subset K of H is Lipschitzian with constant 1, and the following two identities 2 2 2 ||x y|| =||x|| 2 x,y ||y|| , (?) 2 2 2 2 ||?x (1??)y|| = ?||x|| (1??)||y|| ??(1??)||x?y|| , (??) which hold for all x,y? H, are some of the geometric properties that char- terize inner product spaces and also make certain problems posed in Hilbert spaces more manageable than those in general Banach spaces. However, as has been rightly observed by M. Hazewinkel, “... many, and probably most, mathematical objects and models do not naturally live in Hilbert spaces”. Consequently,toextendsomeoftheHilbertspacetechniquestomoregeneral Banach spaces, analogues of the identities (?) and (??) have to be developed. Annotation This monograph focuses on geometric properties of Banach spaces and nonlinear iterations. The first half of the monograph (Chapters 1 to 5) develops materials on convexity and smoothness of Banach spaces, associated moduli and connections with duality maps. Key results obtained in each chapter are summarized at the end of the chapter for easy reference. The second half (Chapters 6 to 23) deals with an in-depth, comprehensive and up-to-date coverage of the main ideas, concepts and most important results on iterative algorithms for the approximation of fixed points of nonlinear nonexpansive and pseudo-contractive-type mappings.As a flourishing area of research for numerous mathematicians, there has been an explosion of research papers on these topics. This self-contained volume will be useful for graduate students of mathematical analysis, as well as being a vital text for mathematicians interested in learning about the subject and for specialists in nonlinear operator theory