Fixed Point Theorems

Contents

Preface VII

Symbols used VIII

1 CONTRACTION MAPPINGS 1
1.1 Introduction 1
1.2. The contraction mapping theorem 2
1.3 The Cauchy-Lipschitz theorem 3
1.4 Implicit functions 5
1.5 Other applications 7

2 FIXED POINTS IN COMPACT CONVEX SETS 9
2.1 The fixed point property 9
2.2 Other proofs of Brouwer's theorem 12
2.3 Extensions to infinite-dimensional spaces 13
2.4 Kakutani's example 15


3 WHICH SETS HAVE THE FIXED POINT PROPERTY? 18
3.1 Compact contractible sets 18
3.2 Pathology 20

4 EXTENSIONS OF SCHAUDER'S THEOREM 25
4.1 Schauder's second theorem 25
4.2 Rothe's theorem 26
4.3 Continuation theorems 28
4.4 Krasnoselskii's theorem 31
4.5 Locally convex spaces 32

5 NON-EXPANSIVE MAPPINGS 35
5.1 Bounded convex sets 35
5.2 Various 38

6 EXISTENCE THEOREMS FOR DIFFERENTIAL EQUATIONS 41
6.1 Methods available 41
6.2 Ordinary D.E.s 43
6.3 Two-point boundary conditions 46
6.4 Periodic solutions 47
6.5 Partial D.E.s: use of a Green's function 49
6.6 The linearisation trick for partial D.E.s 50
6.7 The methods of Leray-Schauder and Schaefer 51

7 FIXED POINTS FOR FAMILIES OF MAPPINGS 53
7.1 Commuting mappings 53
7.2 Downward induction 57
7.3 Groups and semigroups of mappings 58

8 EXISTENCE OF INVARIANT MEANS 62
8.1 Almost periodic functions 6
8.2 Banach limits 63
8.3 Haar measure 65
8.4 Day's fixed point theorem 67

9 FIXED POINT THEOREMS FOR MANY-VALUED MAPPINGS 68
9.1 Kakutani's theorem 68
9.2 Generalisations 70
9.3 Theory of games 72

10 SOME NUMERICAL INVARIANTS 75
10.1 The rotation of a vector field 75
10.2 The degree for mappings of spheres 77
10.3 The degree for mappings of open sets 79
10.4 The index and Lefschetz number 83

11 FURTHER TOPICS 85

Bibliography 87

Index 93