This book is intended as an introduction to fixed point theory and its applications. The topics treated range from fairly standard results (such as the Principle of Contraction Mapping, Brouwer's and Schauder's fixed point theorems) to the frontier of what is known, but we have not tried to achieve maximal generality in all possible directions. We hope that the references quoted may be useful for this purpose.

Editor's Preface

Foreword

CHAPTER 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

CHAPTER 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 Principle

CHAPTER 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 Theorem

CHAPTER 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

CHAPTER 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

CHAPTER 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

CHAPTER 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

CHAPTER 9. Families of Mappings and Fixed Points

 9.0. Introduction

 9.1. Markov's and Kakutani's Results

 9.2. The Ryll-Nardzewski Fixe'd~Point Theorem 

 9.3. Fixed Points for Families of Nonexpansive Mappings 

 9.4. lnvariant Means on Semigroups and Fixed Point for Families of Mappings 

CHAPTER 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

CHAPTER 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

CHAPTER 12. The Topological Degree

 12.0. Introduction

 12.1. The Topological Degree in Finite-Dimensional Spaces

 12.2. The Leray-Schauder Topo!ogical 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

BIBLIOGRAPHY

INDEX