μ-Pseudo Almost Automorphic Solutions To Some Nonlinear Evolution Equations

Some properties of μ-pseudo almost automorphic functions and related functions are discussed in this thesis. Applications of these properties to evolution equations are also presented. The theory of this work generalizes the classical results on pseudo almost automorphic functions. And in this thesis, we investigate many important results on the new theory of μ-pseudo almost automorphic functions. This thesis consists of six parts.In Chapter1, some research backgrounds and main results of the theory of almost automorphic function will be presented. Meanwhile, we also introduce some recently important research results about almost automorphic functions.Chapter2is on preliminaries of this thesis. We introduce some notations and basic results needed in the sequel. It mainly includes some definitions and basic properties about almost automorphic functions,μ-pseudo almost automorphic functions,μ-pseudo almost automorphic functions of class p and μ-pseudo almost automorphic functions of class oo, which form a basis for exploring further applications of these functions to evolution equations. Moreover, we recall some useful knowledge about integral resolvent families, intermediate spaces and infinite delay spaces.In Chapter3, we consider the existence of μ-pseudo almost automorphic mild solutions to a semi-linear fractional differential equation: We establish composition theorem at first, and then apply sectorial operator theories and fixed point theorems to obtain our main results.In Chapter4, we are mainly concerned with the existence of μ-pseudo almost auto-morphic mild solutions to the following differential equations: We prove our main results by the interpolation theory and fixed point methods.In Chapter5, we deal with the existence of μ-pseudo almost automorphic mild solu-tions to the class of semi-linear integral equations in the abstract form: Combing with fixed point technique, the main results are established by integral resolvent families.In the last chapter, we investigate the existence of μ-pseudo almost automorphic mild solutions to the class of non-autonomous first-order neutral functional-differential equations with infinite delay:For that, the first task there consisted of introducing some new spaces called μ-pseudo almost automorphic functions of class p, μ-pseudo almost automorphic functions of class∞, which are heavily linked with the presence of infinite delay. Next, some composition results of μ-pseudo almost almost automorphic functions of class p were established. At last, we make use of the previous results to investigate the neutral system.