Almost Periodic Functions and Differential Equations

✍ Scribed by B. M. Levitan, V. V. Zhikov

• Publisher: Cambridge University Press
• Year: 1983
• Tongue: English
• Leaves: 224
• Category: Library

✦ Synopsis

Text: English, Russian (translation)

✦ Table of Contents

Cover
Half-title
Title
Copyright
Contents
Preface
Chapter 1. Almost periodic functions in metric spaces
1 Definition and elementary properties of almost periodic functions
2 Bochner's criterion
3 The connection with stable dynamical systems
4 Recurrence
5 A theorem of A. A. Markov
6 Some simple properties of trajectories
Comments and references to the literature
Chapter 2. Harmonic analysis of almost periodic functions
1 Prerequisites about Fourier–Stieltjes integrals
2 Proof of the approximation theorem
3 The mean-value theorem; the Bohr transformation; Fourier series; the uniqueness theorem
4 Bochner–Fejer polynomials
5 Almost periodic functions with values in a Hilbert space; Parseval's relation
6 The almost periodic functions of Stepanov
Comments and references to the literature
Chapter 3. Arithmetic properties of almost periods
1 Kronecker's theorem
2 The connection between the Fourier exponents of a function and its almost periods
3 Limit-periodic functions
4 Theorem of the argument for continuous numerical complex-valued almost periodic functions
Comments and references to the literature
Chapter 4. Generalisation of the uniqueness theorem (N-almost periodic functions)
1 Introductory remarks, definition and simplest properties of N-almost periodic functions
2 Fourier series, the approximation theorem, and the uniqueness theorem
Comments and references to the literature
Chapter 5. Weakly almost periodic functions
1 Definition and elementary properties of weakly almost periodic functions
2 Harmonic analysis of weakly almost periodic functions
3 Criteria for almost periodicity
Comments and references to the literature
Chapter 6. A theorem concerning the integral and certain questions of harmonic analysis
1 The Bohl–Bohr–Amerio theorem
2 Further theorems concerning the integral
3 Information from harmonic analysis
4 A spectral condition for almost periodicity
5 Harmonic analysis of bounded solutions of linear equations
Comments and references to the literature
Chapter 7. Stability in the sense of Lyapunov and almost periodicity
Notation
1 The separation properties
2 A lemma about separation
3 Corollaries of the separation lemma
4 Corollaries of the separation lemma (continued)
5 A theorem about almost periodic trajectories
6 Proof of the theorem about a zero-dimensional fibre
7 Statement of the principle of the stationary point
8 Realisation of the principle of the stationary point when the dimension m<=3
9 Realisation of the principle of the stationary point under monotonicity conditions
Comments and references to the literature
Chapter 8. Favard theory
1 Introduction
2 Weak almost periodicity (the case of a uniformly convex space)
3 Certain auxiliary questions
4 Weak almost periodicity (the general case)
5 Problems of compactness and almost periodicity
6 Weakening of the stability conditions
7 On solvability in the Besicovitch class
Comments and references to the literature
Chapter 9. The method of monotonic operators
1 General properties of monotonic operators
2 Solvability of the Cauchy problem for an evolution equation
3 The evolution equation on the entire line: questions of the boundedness and the compactness of solutions
4 Almost periodic solutions of the evolution equation
Comments and references to the literature
Chapter 10. Linear equations in a Banach space (questions of admissibility and dichotomy)
Notation
1 Preliminary results
2 The connection between regularity and the exponential dichotomy on the whole line
3 Theorems on regularity
4 Examples
Comments and references to the literature
Chapter 11. The averaging principle on the whole line for parabolic equations
1 Bogolyubov's lemma
2 Some properties of parabolic operators
3 The linear problem about averaging
4 A non-linear equation
5 The Navier–Stokes equation
6 The problem on the whole space
Comments and references to the literature
Bibliography
Additional references
Index
Back Cover