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.

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482810_1_En_Print.indd

Pata, Vittorino

0009172

Springer Nature Switzerland AG

Springer Nature (Textbooks & Major Reference Works), Cham, 2019

Unitext, Cham, Switzerland, 2019

Switzerland, Switzerland

1, 2019-06-22

Jul 01, 2019

0

sm77222019

producers:
Acrobat Distiller 10.0.0 (Windows)

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Source title: Fixed Point Theorems and Applications (UNITEXT (116))

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

Preface 7
Introduction 8
Notation 9
Contents 14
About the Author 16
Part I Fixed Point Theorems 17
1 The Banach Contraction Principle 18
2 The Boyd-Wong Theorem 24
3 Further Extensions of the Contraction Principle 27
4 Weak Contractions 34
5 Contractions of ε-Type 40
6 Sequences of Maps and Fixed Points 48
7 Fixed Points of Non-expansive Maps 51
8 The Riesz Mean Ergodic Theorem 54
9 The Brouwer Fixed Point Theorem 58
10 The Schauder-Tychonoff Fixed Point Theorem 63
11 Further Consequences of the Schauder-Tychonoff Theorem 68
12 The Markov-Kakutani Theorem 73
13 The Kakutani-Ky Fan Theorem 75
Part II Some Applications 81
14 The Implicit Function Theorem 82
15 Location of Zeros 88
16 Ordinary Differential Equations in Banach Spaces 91
17 The Lax-Milgram Lemma 102
18 An Abstract Elliptic Problem 110
19 Semilinear Evolution Equations 114
20 An Abstract Parabolic Problem 122
21 The Invariant Subspace Problem 129
22 Measure Preserving Maps on Compact Hausdorff Spaces 133
23 Invariant Means on Semigroups 136
24 Haar Measures 140
25 Game Theory 148
Part III Some Problems 152
26 Problems 153
References 166
Index 169

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Erscheinungsdatum: 01.07.2019

2019-09-23