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.
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.
Editors 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. Hilberts 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
CHAPTER 4. Brouwers Fixed Point Theorem
4.0. Introduction
4.1. The Fixed Point Property
4.2. Brouwers Fixed Point theorem. Equivalent Formulations
4.3. Robbins Complements of Brouwers Theorem
4.4. The Borsuk-Ulam Theorem
4.5. An Elementary Proof of Brouwers Theorem
4.6. Some Examples
4.7. Some Applications of Brouwers Fixed Point Theorem
4.8. The Computation of Fixed Points. Scarfs Theorem
CHAPTER 5. Schauders Fixed Point Theorem and Some Generalizations
5.0. Introduction
5.1. The Schauder Fixed Point Theorem
5.2. Darbos Generalization of Schauders Fixed Point Theorem
5.3. Krasnoselskiis， Rothes and Altmans Theorems
5.4. Browders and Fans Generalizations of Schauders and Tychonoffs Fixed Point Theorem
5.5. Some Applications
CHAPTER 6. Fixed Point Theorems jbr 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 amt Monotome 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. Markovs and Kakutanis Results
9.2. The RylI-Nardzewski Fixed 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 Topological Degree
12.3. Lerays Example
12.4. The Topological Degree for k-Set Contractions
12.5. The Uniqueness Problem for the Topological Degree
I2.6. The Computation of the Topological Degree
12.7. Some Applications of the Topological Degree
BIBLIOGRAPHY
INDEX
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