Fixed Point Theorems and Applications

Description

This book addresses fixed point theory, a fascinating and far-reaching field with applications in several areas of mathematics. The content is divided into two main parts. The first, which is more theoretical, develops the main abstract theorems on the existence and uniqueness of fixed points of maps. In turn, the second part focuses on applications, covering a large variety of significant results ranging from ordinary differential equations in Banach spaces, to partial differential equations, operator theory, functional analysis, measure theory, and game theory. A final section containing 50 problems, many of which include helpful hints, rounds out the coverage. Intended for Master’s and PhD students in Mathematics or, more generally, mathematically oriented subjects, the book is designed to be largely self-contained, although some mathematical background is needed: readers should be familiar with measure theory, Banach and Hilbert spaces, locally convex topological vector spaces and, in general, with linear functional analysis. Front Matter ....Pages i-xvii Front Matter ....Pages 1-1 The Banach Contraction Principle (Vittorino Pata)....Pages 3-8 The Boyd-Wong Theorem (Vittorino Pata)....Pages 9-11 Further Extensions of the Contraction Principle (Vittorino Pata)....Pages 13-19 Weak Contractions (Vittorino Pata)....Pages 21-26 Contractions of \(\varepsilon \)-Type (Vittorino Pata)....Pages 27-34 Sequences of Maps and Fixed Points (Vittorino Pata)....Pages 35-37 Fixed Points of Non-expansive Maps (Vittorino Pata)....Pages 39-41 The Riesz Mean Ergodic Theorem (Vittorino Pata)....Pages 43-46 The Brouwer Fixed Point Theorem (Vittorino Pata)....Pages 47-51 The Schauder-Tychonoff Fixed Point Theorem (Vittorino Pata)....Pages 53-57 Further Consequences of the Schauder-Tychonoff Theorem (Vittorino Pata)....Pages 59-63 The Markov-Kakutani Theorem (Vittorino Pata)....Pages 65-66 The Kakutani-Ky Fan Theorem (Vittorino Pata)....Pages 67-72 Front Matter ....Pages 2-73 The Implicit Function Theorem (Vittorino Pata)....Pages 75-80 Location of Zeros (Vittorino Pata)....Pages 81-83 Ordinary Differential Equations in Banach Spaces (Vittorino Pata)....Pages 85-95 The Lax-Milgram Lemma (Vittorino Pata)....Pages 97-104 An Abstract Elliptic Problem (Vittorino Pata)....Pages 105-108 Semilinear Evolution Equations (Vittorino Pata)....Pages 109-116 An Abstract Parabolic Problem (Vittorino Pata)....Pages 117-123 The Invariant Subspace Problem (Vittorino Pata)....Pages 125-128 Measure Preserving Maps on Compact Hausdorff Spaces (Vittorino Pata)....Pages 129-131 Invariant Means on Semigroups (Vittorino Pata)....Pages 133-136 Haar Measures (Vittorino Pata)....Pages 137-144 Game Theory (Vittorino Pata)....Pages 145-148 Front Matter ....Pages 149-149 Problems (Vittorino Pata)....Pages 151-163 Back Matter ....Pages 165-171