Geometric Properties of Banach Spaces and Nonlinear Iterations

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.

Self-contained, with detailed motivations, explanations and examples In-depth, comprehensive and up-to-date coverage Contains interesting, important and reasonable open problems Summaries of key inequalities and theorems for easy reference Includes supplementary material: sn.pub/extras

Klappentext

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.

Inhalt
Some Geometric Properties of Banach Spaces.- Smooth Spaces.- Duality Maps in Banach Spaces.- Inequalities in Uniformly Convex Spaces.- Inequalities in Uniformly Smooth Spaces.- Iterative Method for Fixed Points of Nonexpansive Mappings.- Hybrid Steepest Descent Method for Variational Inequalities.- Iterative Methods for Zeros of ? Accretive-Type Operators.- Iteration Processes for Zeros of Generalized ? Accretive Mappings.- An Example; Mann Iteration for Strictly Pseudo-contractive Mappings.- Approximation of Fixed Points of Lipschitz Pseudo-contractive Mappings.- Generalized Lipschitz Accretive and Pseudo-contractive Mappings.- Applications to Hammerstein Integral Equations.- Iterative Methods for Some Generalizations of Nonexpansive Maps.- Common Fixed Points for Finite Families of Nonexpansive Mappings.- Common Fixed Points for Countable Families of Nonexpansive Mappings.- Common Fixed Points for Families of Commuting Nonexpansive Mappings.- Finite Families of Lipschitz Pseudo-contractive and Accretive Mappings.- Generalized Lipschitz Pseudo-contractive and Accretive Mappings.- Finite Families of Non-self Asymptotically Nonexpansive Mappings.- Families of Total Asymptotically Nonexpansive Maps.- Common Fixed Points for One-parameter Nonexpansive Semigroup.- Single-valued Accretive Operators; Applications; Some Open Questions.