The theory of almost periodic functions was first developed by the Danish mathematician H. Bohr during 1925-1926. Then Bohr's work was substantially extended by S. Bochner, H. Weyl, A. Besicovitch, J. Favard, J. von Neumann, V. V. Stepanov, N. N. Bogolyubov, and oth ers. Generalization of the classical theory of almost periodic functions has been taken in several directions. One direction is the broader study of functions of almost periodic type. Related this is the study of ergodic ity. It shows that the ergodicity plays an important part in the theories of function spectrum, semigroup of bounded linear operators, and dynamical systems. The purpose of this book is to develop a theory of almost pe riodic type functions and ergodicity with applications-in particular, to our interest-in the theory of differential equations, functional differen tial equations and abstract evolution equations. The author selects these topics because there have been many (excellent) books on almost periodic functions and relatively, few books on almost periodic type and ergodicity. The author also wishes to reflect new results in the book during recent years. The book consists of four chapters. In the first chapter, we present a basic theory of four almost periodic type functions. Section 1. 1 is about almost periodic functions. To make the reader easily learn the almost periodicity, we first discuss it in scalar case. After studying a classical theory for this case, we generalize it to finite dimensional vector-valued case, and finally, to Banach-valued (including Hilbert-valued) situation.
1 Almost periodic type functions.- 1.1.- 1.1.1 Numerical almost periodic functions.- 1.1.2 Uniform almost periodic functions.- 1.1.3 Vector-valued almost periodic functions.- 1.2 Asymptotically almost periodic functions.- 1.3 Weakly almost periodic functions.- 1.3.1 Vector-valued weakly almost periodic functions.- 1.3.2 Ergodic theorem.- 1.3.3 Invariant mean and mean convolution.- 1.3.4 Fourier series of WAV(?, H).- 1.3.5 Uniformly weakly almost periodic functions.- 1.4 Approximate theorem and applications.- 1.4.1 Numerical approximate theorem.- 1.4.2 Vector-valued approximate theorem.- 1.4.3 Unique decomposition theorem.- 1.5 Pseudo almost periodic functions.- 1.5.1 Pseudo almost periodic functions.- 1.5.2 Generalized pseudo almost periodic functions.- 1.6 Converse problems of Fourier expansions.- 1.7 Almost periodic type sequences.- 1.7.1 Almost periodic sequences.- 1.7.2 Other almost periodic type sequences.- 2 Almost periodic type differential equations.- 2.1 Linear differential equations.- 2.1.1 Ordinary differential equations.- 2.1.2 Abstract differential equations.- 2.1.3 Integration of almost periodic type functions.- 2.2 Partial differential equations.- 2.2.1 Dirichlet Problems.- 2.2.2 Parabolic equations.- 2.2.3 Second-order equations with gradient operators.- 2.3 Means, introversion and nonlinear equations.- 2.3.1 General theory of means and introversions.- 2.3.2 Applications to (weakly) almost periodic functions.- 2.3.3 Nonlinear differential equations.- 2.3.4 Implications of almost periodic type solutions.- 2.4 Regularity and exponential dichotomy.- 2.4.1 General theory of regularity.- 2.4.2 Stability of regularity.- 2.4.3 Almost periodic type solutions.- 2.5 Equations with piecewise constant argument.- 2.5.1 Exponential dichotomy for difference equations.- 2.5.2 Equations with piecewise constant argument.- 2.5.3 Almost periodic difference equations.- 2.6 Equations with unbounded forcing term.- 2.7 Almost periodic structural stability.- 2.7.1 Topological equivalence and structural stability.- 2.7.2 Exponential dichotomy and structural stability.- 3 Ergodicity and abstract differential equations.- 3.1 Ergodicity and regularity.- 3.1.1 Ergodicity and regularity.- 3.1.2 Solutions of almost periodic type equations.- 3.2 Ergodicity and nonlinear equations.- 3.3 Semigroup of operators and applications.- 3.3.1 Semigroup of operators.- 3.3.2 Almost periodic type solutions.- 3.4 Delay differential equations.- 3.4.1 Introduction of delay differential equations.- 3.4.2 Linear autonomous equations.- 3.4.3 Linear nonautonomous equations.- 3.5 Spectrum of functions.- 3.6 Abstract Cauchy Problems.- 3.6.1 Harmonic analysis of solutions.- 3.6.2 Asymptotic behavior of solutions.- 3.6.3 Mild solutions.- 3.6.4 Weakly almost periodic solutions.- 4 Ergodicity and averaging methods.- 4.1 Ergodicity and its properties.- 4.2 Quantitative theory.- 4.2.1 Introduction.- 4.2.2 Quantitative theory of averaging methods.- 4.2.3 Example and comments.- 4.3 Perturbations of noncritical linear systems.- 4.4 Qualitative theory of averaging methods.- 4.4.1 Almost periodic type solutions of nonlinear equations.- 4.4.2 Some examples.- 4.5 Averaging methods for functional equations.- 4.5.1 Averaging for functional differential equations.- 4.5.2 Averaging for delay difference equations.- Notations.