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    This book provides a comprehensive theory of almost periodic type functions with a large number of the applications to differential equations，functional equations and evolution equations. In addition，it also presents a basic theory on ergodicity and its applications in the theory of function spectrum，semi group of bounded linear operators and dynamical systems。It reflects new establishment of recent years in the field。This monograph is self-contained，the only prerequisite being a basic knowledge of functional analysis and ordinary differential equations。It is written for the mathematicians who wish to learn about the subject and is of interest to the specialists in the areas of abstract harmonic analysis，functional analysis，differential (functional) equations，dynamical system and ergodicity。It is also suitable as a textbook for graduates in Mathematical Analysis。

Preface
Chapter 1 Almost periodic type functions
　1.1 Almost periodic functions
　　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 WAP(R,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
Chapter 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
　2.5 Equations with piecewise constant argument
　　2.5.1 Exponential dichotomy for difference equations
　　5.2.2 Equations with piecewise constant argument
　　5.2.3 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
Chapter 3 Ergodicity and abstract difference 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 difference equations
　　3.4.1 Introduction of delay difference 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
Chapter 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
　　4.4.1 Almost periodic type solutions of nonlinear equations
　　4.4.2 Some examples
　4.5 Averaging methods for functional equations
　　4.5.1 Averaging methods for functional differential equations
　　4.5.2 Averaging methods for delay difference equations
Bibliography
Notations
Index