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