Research On Almost Periodic Problems Of Lebesgue Spaces With Variable Exponents For Several Classes Of Differential Equation Models

The concept of almost periodic function was first proposed by Danish mathematician Bohr from 1924 to 1926.It is well known that the classical Bohr function is continuous.Since then,Stepanov,Besicovitch et al.have defined a more extensive space of almost periodic functions successively.These generalized almost periodic functions are only defined in measurable space and can be discontinuous.The space of almost periodic functions defined on Lebesgue Spaces of variable exponentials is a further extension of the space of generalized almost periodic functions.In this paper,we focus on the existence of almost periodic solutions in the Lebesgue Spaces with variable exponents for several kinds of differential equation models,including fuzzy cel-lular neural networks with Clifford value,abstract semilinear equations and non-autonomous differential equations.The first chapter is the introduction of this paper,introduces the research background and overview of the almost periodic function,the Lebesgue Spaces with variable exponents and the significance of this paper.In chapter 2,we study Stepanov almost periodic solutions in Lebesgue Spaces of variable exponents of Clifford valued fuzzy cellular neural networks.Based on Lebesgue space theory of variable exponential,some new composition theorems of Stepanov almost periodic functions in the Lebesgue space with variable exponential are obtained.By using the fixed point theorem and a new inequality technique,the existence,uniqueness and global exponential stability of S^p（x）-almost periodic solutions for Clifford valued neural network are obtained.In chapter 3,we first introduce the concept of Besicovitch almost automorphic function in the Lebesgue Spaces with variable exponents,prove the completeness of this space,and obtain some new composition theorems.Finally,we discuss the existence and uniqueness of B^p（x）--almost automorphic solutions for a class of abstract semilinear equations.In chapter 4,based on chapter 3,the existence and uniqueness of B^p（x）--almost periodic solutions for a class of non-autonomous differential equations are studied by using B^p（x）--almost periodic theory.Finally,we give an example on Lebesgue Spaces with variable exponents to show the correctness and validity of the theory.