Asymptotically Almost Periodic Mild Solutions And Asymptotically Almost Automorphic Mild Solutions Of Differential Equations

Periodic movement is a perfect movement which has many good characteristics. The nature as a whole is inferred basing its local nature known by us. But it is too simple to be used in many questions. Many movements in nature can not be described by a single periodic movement, so almost periodic movement is necessary. Almost periodic movement is called almost periodic function on math. In order to keep up with practical problems, and expand the application, many mathematics workers, one after one, put forward the asymptotically almost periodic function, pseudo almost periodic function, remote almost periodic functions and almost automorphic function, asymptotically almost automorphic function, pseudo almost automorphic function and other related concepts.The application in differential equation is a very important part of periodic type functions theoretical research and almost automorphic type functions theoretical research. In this paper the main work is to apply the theory of asymptotically almost periodic functions and asymptotically almost automorphic functions theory on two kinds of neutral differential equations, and to discuss how the correlative mild solutions of two kinds of differential equations is not only existence but also uniqueness.The results of this paper are divided into two parts:In the first part, the existence and uniqueness of asymptotically almost periodic mild solution for a class of neutral differential equation is discussed by fixed point theorem and some knowledge on C0-semigroup.In the second part, it is researched that the existence and uniqueness of asymptotically almost automorphic mild solutions for another class of neutral differential equation by using the convolution of the exponential dichotomy and fixed point theorem.