Geometric Properties of Banach Spaces and Nonlinear Iterations

Abstract

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. -9.2Uniformly 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 Po