Differential Equations in Abstract Spaces

Front Cover......Page 1
Differential Equations in Abstract Spaces......Page 4
Copyright Page......Page 5
Contents......Page 6
Preface......Page 10
Acknowledgments......Page 12
1.0 Introduction......Page 14
1.1 Abstract Functions......Page 15
1.2 The Mean Value Theorem......Page 16
1.3 The Riemann Integral Abstract Functions......Page 17
1.4 Abstract Lebesgue Integrals......Page 21
1.5 The Abstract Stieltjes Integral......Page 23
1.6 Gateaux and Fréchet Differentials......Page 25
1.7 Notes......Page 33
2.0 Introduction......Page 34
2.1 Strongly Continuous Semigroups of Operators......Page 36
2.2 The Infinitesimal Generator......Page 39
2.3 The Hille–Yosida–Phillips Theorem......Page 45
2.4 Linear Autonomous Funtional Differential Equations......Page 55
2.5 Analytic Semigroups......Page 61
2.6 Notes......Page 67
3.0 Introduction......Page 68
3.1 Definitions and Hypotheses......Page 69
3.2 Statements of the Main Theorems and Some Heuristic Arguments......Page 73
3.3 Properties of the Semigroup {exo[–tA(t)]}......Page 75
3.4 Existence of a Fundamental Solution......Page 87
3.5 Uniqueness of the Fundamental Solution......Page 92
3.6 Solution of the Abstract Cauchy Problem......Page 97
3.7 Differentiability of Solutions......Page 101
3.8 Asymptotic Behavior......Page 104
3.9 Notes......Page 106
4.0 Introduction......Page 107
4.1 Lower Bounds, Uniqueness, and Convexity (Special Results)......Page 108
4.2 Lower Bounds, Uniqueness, and Convexity (General Results)......Page 119
4.3 Approximate Solutions, Bounds, and Uniqueness......Page 131
4.4 Application to Parabolic Equations......Page 135
4.5 Notes......Page 138
5.0 Introduction......Page 139
5.1 Counterexamples......Page 140
5.2 Existence and Uniqueness......Page 146
5.3 Nonlinear Variation of Constants Formula......Page 154
5.4 Stability and Asymptotic Behavior......Page 165
5.5 Chaplygin’s Method......Page 170
5.6 Global Existence and Asymptotic Equilibrium......Page 174
5.7 Lyapunov Functions and Stability Criteria......Page 180
5.8 Notes......Page 184
6.0 Introduction......Page 185
6.1 Nonlinear Semigroups and Differential Equations......Page 186
6.2 Functional Differential Equations in Banach Spaces......Page 198
6.3 Second-Order Evolution Equations......Page 204
6.4 Notes......Page 212
Appendixes......Page 213
Bibliography......Page 224
Index......Page 228