Fixed Point Theory: An Introduction

Fixed Point Theory: An Introduction
ISBN-10
9027712247
ISBN-13
9789027712240
Category
Mathematics
Pages
488
Language
English
Published
2001-11-30
Publisher
Springer
Author
Vasile I. Istrățescu

Description

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 Spaces.- 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.- 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 Contractive Type.- 3.5. Approximate Iteration.- 3.6. A Converse of the Contraction Principle.- 3.7. Some Applications of the Contraction Principle.- 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. Scarf's Theorem.- 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 Applications.- 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 Points.- 7. Sequences of Mappings and Fixed Points.- 7.0. Introduction.- 7.1. Convergence of Fixed Points for Contractions or Related Mappings.- 7.2. Sequences of Mappings and Measures of Noncompactness.- 8. Duality Mappings and Monotone Operators.- 8.0. Introduction.- 8.1. Duality Mappings.- 8.2. Monotone Mappings and Classes of Nonexpansive Mappings.- 8.3. Some Surjectivity Theorems on Real Banach Spaces.- 8.4. Some Surjectivity Theorems in Complex Banach Spaces.- 8.5. Some Surjectivity Theorems in Locally Convex Spaces.- 8.6. Duality Mappings and Monotonicity for Set-Valued Mappings.- 8.7. Some Applications.- 9. Families of Mappings and Fixed Points.- 9.0. Introduction.- 9.1. Markov's and Kakutani's Results.- 9.2. The Ryll-Nardzewski Fixed Point Theorem.- 9.3. Fixed Points for Families of Nonexpansive Mappings.- 9.4. Invariant Means on Semigroups and Fixed Point for Families of Mappings.- 10. Fixed Points and Set-Valued Mappings.- 10.0 Introduction.- 10.1 The Pompeiu-Hausdorff Metric.- 10.2. Continuity for Set-Valued Mappings.- 10.3. Fixed Point Theorems for Some Classes of Set-valued Mappings.- 10.4. Set-Valued Contraction Mappings.- 10.5. Sequences of Set-Valued Mappings and Fixed Points.- 11. Fixed Point Theorems for Mappings on PM-Spaces.- 11.0. Introduction.- 11.1. PM-Spaces.- 11.2. Contraction Mappings in PM-Spaces.- 11.3. Probabilistic Measures of Noncompactness.- 11.4. Sequences of Mappings and Fixed Points.- 12. The Topological Degree.- 12.0. Introduction.- 12.1. The Topological Degree in Finite-Dimensional Spaces.- 12.2. The Leray-Schauder Topological Degree.- 12.3. Leray's Example.- 12.4. The Topological Degree for k-Set Contractions.- 12.5. The Uniqueness Problem for the Topological Degree.- 12.6. The Computation of the Topological Degree.- 12.7. Some Applications of the Topological Degree.

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