This book presents recent methods of study on the asymptotic behavior of solutions of abstract differential equations such as stability, exponential dichotomy, periodicity, almost periodicity, and almost automorphy of solutions. The chosen methods are described in a way that is suitable to those who have some experience with ordinary differential equations. The book is intended for graduate students and researchers in the related areas. Contents: Stability and Exponential Dichotomy; Almost Periodic Solutions; Almost Automorphic Solutions; Nonlinear Equations.
Author(s): James H. Liu, Gaston M. N'Guerekata, Nguyen Van Minh
Edition: First Edition
Year: 2008
Language: English
Pages: 219
Contents......Page 8
Preface......Page 6
1.1.1 Banach Spaces......Page 12
1.1.2 Linear Operators......Page 13
1.1.3 Spectral Theory of Linear (Closed) Operators......Page 14
1.1.3.1 Several Properties of Resolvents......Page 16
1.2.1 Definition and Basic Properties......Page 18
1.2.2 Compact Semigroups and Analytic Strongly Continuous Semigroups......Page 23
1.2.3 Spectral Mapping Theorems......Page 25
1.2.4 Commuting Operators......Page 28
1.3.2 Spectrum of a Bounded Function......Page 30
1.3.3 Uniform Spectrum of a Bounded Function......Page 35
1.3.4.1 De nition and basic properties......Page 37
1.3.5 Sprectrum of an Almost Periodic Function......Page 40
1.3.6 A Spectral Criterion for Almost Periodicity of a Function......Page 41
1.3.7 Almost Automorphic Functions......Page 42
2.1 Perron Theorem......Page 50
2.2 Evolution Semigroups and Perron Theorem......Page 58
2.3.1 Exponential Stability......Page 62
2.3.2 Strong Stability......Page 65
2.4.1 Further Reading Guide......Page 69
2.4.2 Comments......Page 70
3.1.1 An Example......Page 72
3.1.2 Evolution Semigroups......Page 74
3.1.3 The Finite Dimensional Case......Page 75
3.1.4 The Infinite Demensional Case......Page 76
3.1.5.1 Invariant functions spaces of evolution semigroups......Page 79
3.1.5.2 Monodromy operators......Page 81
3.1.5.3 Unique solvability of the inhomogeneous equations in P(1)......Page 84
3.1.5.4 Unique solvability in AP(X) and exponential dichotomy......Page 85
3.1.5.5 Unique solvability of the inhomogeneous equations in M(f)......Page 87
3.1.5.6 Unique solvability of nonlinearly perturbed equations......Page 88
3.1.5.7 Example 1......Page 89
3.1.5.8 Example 2......Page 91
3.2.1 Invariant Function Spaces......Page 92
3.2.2 Differential Operator d/dt – A and Notions of Admissibility......Page 94
3.2.3 Admissibility for Abstract Ordinary Differential Equations......Page 97
3.2.4 Higher Order Differential Equations......Page 100
3.2.5 Abstract Functional Differential Equations......Page 107
3.2.6 Examples and Applications......Page 109
3.3 Decomposition Theorem......Page 114
3.3.1 Spectral Decomposition......Page 118
3.3.2 Spectral Criteria For Almost Periodic Solutions......Page 125
3.4.1 Further Reading Guide......Page 129
3.4.2 Comments......Page 130
4.1 The Inhomogeneous Linear Equation......Page 132
4.2 Method of Invariant Subspaces and Almost Automorphic Solutions of Second-Order Differential Equations......Page 138
4.3 Existence of Almost Automorphic Solutions to Semilinear Differential Equations......Page 142
4.4 Method of Sums of Commuting Operators and Almost Automorphic Functions......Page 146
4.5 Almost Automorphic Solutions of Second Order Evolution Equations......Page 150
4.5.1.2 Mild Solutions and Weak solutions......Page 151
4.5.2 Operators A......Page 152
4.5.3 Nonlinear Equations......Page 156
4.6 The Equations x'=f(t,x)......Page 157
4.7 Comments and Further Reading Guide......Page 162
5.1.1 Nonlinear Equations Without Delay......Page 164
5.1.2 Nonlinear Equations With Finite Delay......Page 173
5.1.3 Nonlinear Equations With Infinite Delay......Page 177
5.1.4 Non-Densely Defined Equations......Page 191
5.2.1 Evolution Semigroups......Page 194
5.2.2.1 Almost periodic solutions of di erential equations without delay......Page 197
5.2.2.2 Almost periodic solutions of di erential equations with delays......Page 198
5.3.1 Further Reading Guide......Page 201
5.3.2 Comments......Page 202
A.1 Lipschitz Operators......Page 204
A.2 Fixed Point Theorems......Page 206
A.3 Invariant Subspaces......Page 208
A.4 Semilinear Evolution Equations......Page 209
Bibliography......Page 212
Index......Page 218
