Nonlinear functional analysis (lecture notes)

Preface......Page 4
Differentiation and integration in Banach spaces......Page 8
Contraction principles......Page 12
Ordinary differential equations......Page 15
Introduction......Page 18
Definition of the mapping degree and the determinant formula......Page 20
Extension of the determinant formula......Page 24
The Brouwer fixed-point theorem......Page 31
Kakutani's fixed-point theorem and applications to game theory......Page 32
Further properties of the degree......Page 36
The Jordan curve theorem......Page 38
The mapping degree on finite dimensional Banach spaces......Page 40
Compact operators......Page 41
The Leray--Schauder mapping degree......Page 42
The Leray--Schauder principle and the Schauder fixed-point theorem......Page 44
Applications to integral and differential equations......Page 46
Introduction and motivation......Page 50
An insert on Sobolev spaces......Page 51
Existence and uniqueness of solutions......Page 57
Monotone operators......Page 60
The nonlinear Lax--Milgram theorem......Page 62
The main theorem of monotone operators......Page 64
Bibliography......Page 68
Index......Page 70