Opis: Geometric Properties of Banach Spaces and Nonlinear Iterations - Charles Chidume

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. 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. Cobzas, Studia Universitatis Babes-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)1 Geometric properties. -1.1 Introduction. -1.2 Uniformly convex spaces. -1.3 Strictly convex Banach spaces. -1.4 The modulus of convexity. -1.5 Uniform convexity, strict convexity and reflexivity. -1.6 Historical remarks. -2 Smooth Spaces. -2.1 Introduction. -2.2 The modulus of smoothness. -2.3 Duality between spaces. -2.4 Historical remarks. -3 Duality Maps in Banach Spaces. -3.1 Motivation. -3.2 Duality maps of some concrete spaces. -3.3 Historical remarks. -4 Inequalities in Uniformly Convex Spaces. -4.1 Introduction. -4.2 Basic notions of convex analysis. -4.3 p-uniformly convex spaces. -4.4 Uniformly convex spaces. -4.5 Historical remarks. -5 Inequalities in Uniformly Smooth Spaces. -5.1 Definitions and basic theorems. -5.2 q-uniformly smooth spaces. -5.3 Uniformly smooth spaces. -5.4 Characterization of some real Banach spaces by the duality map. -5.4.1 Duality maps on uniformly smooth spaces. -5.4.2 Duality maps on spaces with uniformly Gateaux differentiable norms. -6 Iterative Method for Fixed Points of Nonexpansive Mappings. -6.1 Introduction. -6.2 Asymptotic regularity. -6.3 Uniform asymptotic regularity. -6.4 Strong convergence. -6.5 Weak convergence. -6.6 Some examples. -6.7 Halpern-type iteration method. -6.7.1 Convergence theorems. -6.7.2 The case of non-self mappings. -6.8 Historical remarks. -7 Hybrid Steepest Descent Method for Variational Inequalities. -7.1 Introduction. -7.2 Preliminaries. -7.3 Convergence Theorems. -7.4 Further Convergence Theorems. -7.4.1 Convergence Theorems. -7.5 The case of Lp spaces, 1 2. -7.6 Historical remarks. 8 Iterative Methods for Zeros of F -Accretive-Type Operators. -8.1 Introduction and preliminaries. -8.2 Some remarks on accretive operators. -8.3 Lipschitz strongly accretive maps. -8.4 Generalized F -accretive self-maps. -8.5 Generalized F -accretive non-self maps. -8.6 Historical remarks. -9 Iteration Processes for Zeros of Generalized F -Accretive Mappings. -9.1 Introduction. -9.2 Uniformly continuous generalized F -hemi-contractive maps. -9.3 Generalized Lipschitz, generalized F -quasi-accretive mappings. -9.4 Historical remarks. -10 An Example; Mann Iteration for Strictly Pseudo-contractive Mappings. -10.1 Introduction and a convergence theorem. -10.2 An example. -10.3 Mann iteration for a class of Lipschitz pseudo-contractive maps. -10.4 Historical remarks. -11 Approximation of Fixed Points of Lipschitz Pseudo-contractive Mappings. -11.1 Lipschitz pseudo-contractions. -11.2 Remarks. -12 Generalized Lipschitz Accretive and Pseudo-contractive Mappings. -12.1 Introduction. -12.2 Convergence theorems. -12.3 Some applications. -12.4 Historical remarks. -13 Applications to Hammerstein Integral Equations. -13.1 Introduction. -13.2 Solution of Hammerstein equations. -13.2.1 Convergence theorems for Lipschitz maps. -13.2.2 Convergence theorems for bounded maps. -13.2.3 Explicit algorithms. -13.3 Convergence theorems with explicit algorithms. -13.3.1 Some useful lemmas. -13.3.2 Convergence theorems with coupled schemes for the case of Lipschitz maps. -13.3.3 Convergence in Lp spaces, 1 2: . -13.4 Coupled scheme for the case of bounded operators. -13.4.1 Convergence theorems. -13.4.2 Convergence for bounded operators in Lp spaces, 1 2:. -13.4.3 Convergence theorems for generalized Lipschitz maps. -13.5 Remarks and open questions. -13.6 Exercise. -13.7 Historical remarks. -14 Iterative Methods for Some Generalizations of Nonexpansive Maps. -14.1 Introduction. -14.2 Iteration methods for asymptotically nonexpansive mappings. -14.2.1 Modified Mann process. -14.2.2 Iteration method of Schu. -14.2.3 Halpern-type process. -14.3 Asymptotically quasi-nonexpansive mappings. -14.4 Historical remarks. -14.5 Exercises. -15 Common Fixed Points for Finite Families of Nonexpansive Mappings. -15.1 Introduction. -15.2 Convergence theorems for a family of nonexpansive mappings. -15.3 Non-self mappings. -16 Common Fixed Points for Countable Families of Nonexpansive Mappings. -16.1 Introduction. -16.2 Path Convergence theorems. -16.3 Path Convergence in uniformly convex real Banach spaces. -16.4 Iterative convergence in uniformly convex real Banach spaces. -16.5 Non-self mappings. -16.6 Historical remarks. -17 Common Fixed Points for Families of Commuting Nonexpansive Mappings. -17.1 Introduction. -17.2 Three commuting nonexpansive mappings. -17.3 Common fixed points for family of commuting nonexpansive mappings. -17.4 Convergence theorems for infinite family of commuting nonexpansive mappings. -17.5 Historical remarks. -18 Finite Families of Lipschitz Pseudo-contractive and Accretive Mappings. -18.1 Introduction. -18.2 Convergence theorems. -18.3 Finite families of Lipschitz accretive operators. -18.4 Some applications. -18.5 Historical remarks. -19 Generalized Lipschitz Pseudo-contractive and Accretive Mappings. -19.1 Introduction. -19.2 Generalized Lipschitz pseudo-contractive mappings. -19.3 Generalized Lipschitz accretive operators. -19.4 Some applications. -19.5 Historical remarks. -20 Finite Families of Non-self Asymptotically Nonexpansive Mappings. -20.1 Introduction. -20.2 Preliminaries. -20.3 Strong convergence theorems. -20.4 Weak convergence theorems. -20.5 The case for nonexpansive mappings. -20.6 Historical remarks. -21 Families of Total Asymptotically Nonexpansive Maps. -21.1 Introduction. -21.2 Convergence theorems. -21.2.1 Necessary and sufficient conditions for convergence in real Banach spaces. -21.2.2 Convergence theorem in real uniformly convex Banach spaces. -21.3 The case of non-self maps. -21.4 Historical remarks. -22 Common Fixed Points for One-parameter Nonexpansive Semigroup. -22.1 Introduction. -22.2 Existence theorems. -22.3 Convergence theorems. -22.4 Historical remarks. -23 Single-valued Accretive Operators; Applications; Some Open Questions. -23.1 Introduction. -23.2 Lower semi-continuous accretive operators are single-valued. -23.3 An application to variational inequalities. -23.4 General comments on some fixed point theorems. -23.5 Examples of accretive operators. -23.6 Examples of nonexpansive retracts. -23.7 Some questions of interest. -23.8 Further reading. -References. -Index.

Szczegóły: Geometric Properties of Banach Spaces and Nonlinear Iterations - Charles Chidume

Tytuł: Geometric Properties of Banach Spaces and Nonlinear Iterations
Autor: Charles Chidume
Producent: Springer Verlag
ISBN: 9781848821897
Rok produkcji: 2009
Ilość stron: 352
Oprawa: Miękka
Waga: 0.54 kg

Recenzje: Geometric Properties of Banach Spaces and Nonlinear Iterations - Charles Chidume