Linear Differential Equations and Function Spaces (eBook)

Front Cover 1
Linear Differential Equations and Function Spaces 4
Copyright Page 5
Contents 16
Preface 8
PART I 22
Chapter 1. Geometry of Banach spaces 24
10. Introduction 24
11. Angles, splittings, and dihedra 28
12. Coupled spaces 34
13. The class of subspaces of a Banach space 39
14. Hilbert space 47
15. Notes to Chapter 1 52
Chapter 2. Function spaces 54
20. Introduction 54
21. N-spaces 62
22. F-spaces 67
23. F-spaces 78
24. Spaces of continuous functions 97
25. Notes to Chapter 2 104
Chapter 3. Linear differential equations 105
30. Introduction 105
31. Solutions 107
32. Associate equations in coupled spaces 110
33. D-solutions of homogeneous equations 113
34. Notes to Chapter 3 118
PART II 120
Chapter 4. Dichotomies 122
40. Introduction 122
41. Ordinary dichotomies 123
42. Exponential dichotomies 131
43. Dichotomies for associate equations 138
44. Finite-dimensional space 141
45. Notes to Chapter 4 143
Chapter 5. Admissibility and related concepts 145
50. Introduction 145
51. Admissibility 147
52. (B, D)-manifolds 159
53. (B, D)-manifolds, admissibility, and the associate equations 170
54. (B, D)-subspaces and the associate equations 176
55. Finite-dimensional space 181
56. Notes to Chapter 5 183
Chapter 6. Admissibility and dichotomies 186
60. Introduction 186
61. The fundamental inequalities 188
62. Predichotomy behavior of the solutions of the homogeneous equation 191
63. Admissibility, (B, D)-subspaces, and dichotomies: the general case 200
64. Admissibility, (B, D)-subspaces, and dichotomies: the equation with A e M(X) 209
65. Examples and comments 213
66. Behavior of the solutions of the associate homogeneous equation 232
67. Notes to Chapter 6 242
Chapter 7. Dependence on A 244
70. Introduction 244
71. Admissibility classes and (B, D)-subspaces 245
72. Dichotomy classes 258
73. Connection in dichotomy classes: Banach spaces 266
74. Connection in dichotomy classes: Hilbert space 272
75. Notes to Chapter 7 290
Chapter 8. Equations on R 292
80. Introduction 292
81. (B, D)-dihedra and admissibility 294
82. Double dichotomies. Connections with admissibility and (B, D)-dihedra 300
83. Associate equations 314
84. Dependence on A 317
PART III 330
Chapter 9. Ljapunov’s method 332
90. Introduction 332
91. Ljapunov functions 337
92. Exponential dichotomies 341
93. Ordinary dichotomies 348
94. Notes to Chapter 9 353
Chapter 10. Equations with almost periodic A 354
100. Introduction 354
101. The condition Xo*a = {0} 359
102. Exponential dichotomies 362
103. Reflexive and finite-dimensional spaces 364
104. Notes to Chapter 10 366
Chapter 11. Equations with periodic A 369
110. Introduction 369
111. Floquet representation 372
112. Periodic equations and periodic solutions 375
113. The solutions of the homogeneous equation 379
114. Individual periodic equations 390
Chapter 12. Higher-order equations 394
120. Introduction 394
121. The (m + 1)st-order equation 397
122. Admissibility and (B, D)-manifolds 402
123. The main theorems 407
References 414
Index Author and subject 420
Notation 423