μ-pseudo Almost Automorphic Solutions To A Non-autonomous Evolution Equation

The notion of almost automorphic functions is an important generalization of almostperiodicity, which paves the road to many applications to diferential equations. Thecomposition theorems and basic properties of almost automorphic functions and pseudoalmost automorphic functions, form a basis for exploring further applications to variousevolution equations. In the last decades, the existence of mild solutions of pseudo almostautomorphic functions has been extensively applied, combined with suitable fixed pointtheorems.This thesis is concerned with some properties of μ-pseudo almost automorphic func-tions with their applications to the following nonautonomous semilinear evolution equa-tion:y (t)=A(t)y(t)+f(t, y(t h)), t∈R.where h≥0is a fixed constant, and {A(t)}t∈Rsatisfies some conditions.The whole paper is divided into four chapters.As the introduction, in Chapter1, we present the backgrounds of the theory of almostautomorphic functions and main aims of this thesis.Chapter2is preliminaries, some definitions and lemmas about μ-pseudo almost au-tomorphic functions are briefly described. Moreover, we introduce briefly the definitionsand related notations of the nonautonomous semilinear evolution equation, intermediatespace theory, which are important for investigating further applications of these functionsto the above evolution equation.In Chapter3, we are mainly concerned with the existence of μ-pseudo almost auto-morphic mild solutions of the above equation. Combining with fixed point technique, themain results are established by evolution family theory, the Leray-Shauder Alternativetheorem.In Chapter4, we are mainly concerned with the existence of μ-pseudo almost automor-phic mild solutions of the above equation. We obtain our main results by the interpolationtheory and fixed point methods.