不动点理论导论
2009-5
世界图书出版公司
伊斯特拉泰斯库
466
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This book is intended as an introduction to fixed point theory and itsapplications. The topics treated range from fairly standard results （such asthe Principle of Contraction Mapping， Brouwers and Schauders fixedpoint theorems） to the frontier of what is known， but we have not tried toachieve maximal generality in all possible directions. We hope that thereferences quoted may be useful for this purpose. The point of view adopted in this book is that of functional analysis; forthe readers more interested in the algebraic topological point of view wehave added some references at the end of the book. A knowledge offunctional analysis is not a prerequisite， although a knowledge of anintroductory course in functional analysis would be profitable. However，the book contains two introductory chapters， one on general topology andanother on Banach and Hilbert spaces. As a special feature of these chapterswe note the study of measures of noncompactness; first in the case of metricspaces， and second in the case of Banach spaces. Chapter 3 contains a detailed account of the Contraction Principle，perhaps the best known fixed point theorem. Many generalizations of theContraction Principle are also included. We note here the connectionbetween ideas from projective geometry and contractive mappings. Afterpresenting some ways to compute the fixed points for contractivemappings， we discuss several applications in various areas. Chapter 4 presents Brouwers fixed point theorem， perhaps the mostimportant fixed point theorem. After some historical notes concerningopinions about Brouwers proof- which have been influential for the futureof the fixed point theory （Alexander and Birkhoff and Kellogg）-wepresent many proofs of this theorem of Brouwer， of interest to differentcategories of readers. Thus we present an elementary one， which requiresonly elementary properties of polynomials and continuous functions;another uses differential forms; still another uses differential topology; andone relies on combinatorial topology. These different proofs may be used indifferent ways to compute the fixed points for mappings. In this connection，some algorithms for the computation of fixed points are given.
This book is intended as an introduction to fixed point theory and itsapplications. The topics treated range from fairly standard results （such asthe Principle of Contraction Mapping， Brouwers and Schauders fixedpoint theorems） to the frontier of what is known， but we have not tried toachieve maximal generality in all possible directions. We hope that thereferences quoted may be useful for this purpose. The point of view adopted in this book is that of functional analysis; forthe readers more interested in the algebraic topological point of view wehave added some references at the end of the book. A knowledge offunctional analysis is not a prerequisite， although a knowledge of anintroductory course in functional analysis would be profitable. However，the book contains two introductory chapters， one on general topology andanother on Banach and Hilbert spaces.
Editor's PrefaceForewordCHAPTER 1. Topological Spaces and Topological Linear Spaces 1.1. Metric Spaces 1.2. Compactness in Metric Spaces. Measures of Noncompactness 1.3. Baire Category Theorem 1.4. Topological Spaces 1.5. Linear Topological Spaces. Locally Convex SpacesCHAPTER 2. Hilbert spaces and Banach spaces 2.1. Normed Spaces. Banach Spaces 2.2. Hilbert Spaces 2.3. Convergence in X, X* and L(X) 2.4. The Adjoint of an Operator 2.5. Classes of Banach Spaces 2.6. Measures of Noncompactness in Banach Spaces 2.7. Classes of Special Operators on Banach Spaces CHAPTER 3. The Contraction Principle 3.0. Introduction 3.1. The Principle of Contraction Mapping in Complete Metric Spaces 3.2. Linear Operators and Contraction Mappings 3.3. Some Generalizations of the Contraction Mappings 3.4. Hilbert's Projective Metric and Mappings of ContractiveType 3.5. Approximate Iteration 3.6, A Converse of the Contraction Principle 3.7. Some Applications of the Contraction PrincipleCHAPTER 4. Brouwer's Fixed Point Theorem 4.0. Introduction 4.1. The Fixed Point Property 4.2. Brouwer's Fixed Point theorem. Equivalent Formulations 4.3. Robbins' Complements of Brouwer's Theorem 4.4. The Borsuk-Ulam Theorem 4.5. An Elementary Proof of Brouwer's Theorem 4.6. Some Examples 4.7. Some Applications of Brouwer's Fixed Point Theorem 4.8. The Computation of Fixed Points. Scarfs TheoremCHAPTER 5. Schauder's Fixed Point Theorem and Some Generalizations 5.0. Introduction 5.1. The Schauder Fixed Point Theorem 5.2. Darbo's Generalization of Schauder's Fixed Point Theorem 5.3. Krasnoselskii's, Rothe's and Altman's Theorems 5.4. Browder's and Fan's Generalizations of Schauder's and Tychonoff's Fixed Point Theorem 5.5. Some ApplicationsCHAPTER 6. Fixed Point Theorems for Nonexpansive Mappings and Related Classes of Mappings 6.0. Introduction 6.1. Nonexpansive Mappings 6.2. The Extension of Nonexpansive Mappings 6.3. Some General Properties of Nonexpansive Mappings 6.4. Nonexpansive Mappings on Some Classes of Banach Spaces 6.5. Convergence of Iterations of Nonexpansive Mappings 6.6. Classes of Mappings Related to Nonexpansive Mappings 6.7. Computation of Fixed Points for Classes of Nonexpansive Mappings 6.8. A Simple Example of a Nonexpansive Mapping on a Rotund Space Without Fixed PointsCHAPTER 7.Sequences of Mappings and Fixed PointsCHAPTER 8.Duality Masppings and Monotone OperatorsCHAPTER 9.Families of Mappings and Fixed PointsCHAPTER 10.Fixed Points and Set-Valued MappingsCHAPTER 11.Fixed Point Theorems for Mappings on PM-SpacesCHAPTER 12.The Topological DegreeBIBLIOGRAHYINDEX
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