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

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.

Table of Contents:
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.

Review :
From the reviews: “The aim of the present book is to give an introduction to this very active area of investigation. … the book is of great help for graduate and postgraduate students, as well as for researchers interested in fixed point theory, geometry of Banach spaces and numerical solution of various kinds of equations - operator differential equations, differential inclusions, variational inequalities.” (S. Cobzaş, Studia Universitatis Babeş-Bolyai. Mathematica, Vol. LIV (4), December, 2009) “The topic of this monograph falls within the area of nonlinear functional analysis. … The main purpose of this book is to expose in depth the most important results on iterative algorithms for approximation of fixed points or zeroes of the mappings mentioned above. … this book picks up the most important results in the area, its explanations are comprehensive and interesting and I think that this book will be useful for mathematicians interested in iterations for nonlinear operators defined in Banach spaces.” (Jesus Garcia-Falset, Mathematical Reviews, Issue 2010 f)