Almost Periodic Functions and Differential Equations

Preface ix

1. Almost periodic functions in metric spaces 1
1.1. Definition and elementary properties of almost periodic functions 1
1.2. Bochner\'s criterion 4
1.3. The connection with stable dynamical systems 8
1.4. Recurrence 9
1.5. A theorem of A. A. Markov 10
1.6. Some simple properties of trajectories 11
Comments and references to the literature 12

2. Harmonic analysis of almost periodic functions 14
2.1. Prerequisites about Fourier-Stieltjes integrals 14
2.2. Proof of the approximation theorem 17
2.3. The mean-value theorem; the Bohr transformation; Fourier series; the uniqueness theorem 21
2.4. Bochner-Fejer polynomials 25
2.5. Almost periodic functions with values in a Hilbert space; Parseval\'s relation 31
2.6. The almost periodic functions of Stepanov 33
Comments and references to the literature 36

3. Arithmetic properties of almost periods 37
3.1. Kronecker\'s theorem 37
3.2. The connection between the Fourier exponents of a function and its almost periods 40
3.3. Limit-periodic functions 45
3.4. Theorem of the argument for continuous numerical complex-valued almost periodic functions 48
Comments and references to the literature 51

4. Generalisation of the uniqueness theorem ($N$-almost periodic functions) 53
4.1. Introductory remarks, definition and simplest properties of N-almost periodic functions 53
4.2. Fourier series, the approximation theorem, and the uniqueness theorem 59
Comments and references to the literature 62

5. Weakly almost periodic functions 64
5.1. Definition and elementary properties of weakly almost periodic functions 64
5.2. Harmonic analysis of weakly almost periodic functions 68
5.3. Criteria for almost periodicity 70
Comments and references to the literature 76

6. A theorem concerning the integral and certain questions of harmonic analysis 77
6.1. The Bohl-Bohr-Amerio theorem 77
6.2. Further theorems concerning the integral 81
6.3. Information from harmonic analysis 87
6.4. A spectral condition for almost periodicity 91
6.5. Harmonic analysis of bounded solutions of linear equations 92
Comments and references to the literature 96

7. Stability in the sense of Lyapunov and almost periodicity 98
Notation 98
7.1. The separation properties 98
7.2. A lemma about separation 101
7.3. Corollaries of the separation lemma 105
7.4. Corollaries of the separation lemma (continued) 107
7.5. A theorem about almost periodic trajectories 109
7.6. Proof of the theorem about a zero-dimensional fibre 113
7.7. Statement of the principle of the stationary point 116
7.8. Realisation of the principle of the stationary point when the dimension $m\\leq3$ 117
7.9. Realisation of the principle of the stationary point under monotonicity conditions 121
Comments and references to the literature 123

8. Favard theory 124
8.1. Introduction 124
8.2. Weak almost periodicity (the case of a uniformly convex space) 127
8.3. Certain auxiliary questions 130
8.4. Weak almost periodicity (the general case) 134
8.5. Problems of compactness and almost periodicity 135
8.6. Weakening of the stability conditions 140
8.7. On solvability in the Besicovitch class 142
Comments and references to the literature 147

9. The method of monotonic operators 149
9.1. General properties of monotonic operators 149
9.2. Solvability of the Cauchy problem for an evolution equation 153
9.3. The evolution equation on the entire line: questions of the boundedness and the compactness of solutions 157
9.4. Almost periodic solutions of the evolution equation 161
Comments and references to the literature 165

10. Linear equations in a Banach space (questions of admissibility and dichotomy) 166
Notation 166
10.1. Preliminary results 166
10.2. The connection between regularity and the exponential dichotomy on the whole line 170
10.3. Theorems on regularity 172
10.4. Examples 176
Comments and references to the literature 181

11. The averaging principle on the whole line for parabolic equations 182
11.1. Bogolyubov\'s lemma 182
11.2. Some properties of parabolic operators 183
11.3. The linear problem about averaging 186
11.4. A non-linear equation 189
11.5. The Navier-Stokes equation 193
11.6. The problem on the whole space 195
Comments and references to the literature 199

Bibliography 200

Index 208