Fixed Point Theory An Introduction

Approach your problems from the right It isn't that they can't see the solution. It end and begin with the answers. Then, is that they can't see the problem. one day, perhaps you will find the final G. K. Chesterton, The Scandal of Father question. Brown 'The Point of a Pin'. 'The Hermit Clad in Crane Feathers' in R. Van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of mono­ graphs and textbooks on increasingly specialized topics. However, the 'tree' of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces.

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 PointTheorems 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 Theoremsfor 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.