Geometric Properties of Banach Spaces and Nonlinear Iterations
(Sprache: Englisch)
This text is a comprehensive and up-to-date look at geometric properties of Banach spaces and nonlinear iterations. Recent years have seen an explosion of research and papers in this field, and this text incorporates all the new information.
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This text is a comprehensive and up-to-date look at geometric properties of Banach spaces and nonlinear iterations. Recent years have seen an explosion of research and papers in this field, and this text incorporates all the new information.
Klappentext zu „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.
Inhaltsverzeichnis zu „Geometric Properties of Banach Spaces and Nonlinear Iterations “
1 Geometric properties1.1 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 < p
... mehr
< 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 < p < 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 < p < 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
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 < p < 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 < p < 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
... weniger
Bibliographische Angaben
- Autor: Charles Chidume
- 2009, XVII, 326 Seiten, Maße: 15,5 x 23,5 cm, Kartoniert (TB), Englisch
- Verlag: Springer, Berlin
- ISBN-10: 1848821891
- ISBN-13: 9781848821897
- Erscheinungsdatum: 27.03.2009
Sprache:
Englisch
Rezension zu „Geometric Properties of Banach Spaces and Nonlinear Iterations “
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)
Pressezitat
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)
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