Asymptotically Almost Periodic Solution And Almost Automorphic Mild Solution On Two Classes Of Differential Equations

As well known, the existence problems on all kinds of solutions for differential equations are aroused more and more attention by mathematical workers all the time. It becomes one important study direction of the field for differential equation. Among these, the research for almost periodic type solutions and almost automorphic type solutions of differential equation is more wide than that for periodic solutions.The existence and uniqueness of asymptotically almost periodic solution and almost automorphic mild solution of two classes differential equations are discussed in this paper. The contents of this paper are divied into four parts. The source of the topic in this paper is introduced in the first part, Meanwhile, the research status and development tend of almost periodic functions, asymptotically almost periodic functions and almost automorphic functions are also introduced in this part. The research contents of this dissertation are introduced at the end.The concepts and the related basic properties of almost periodic functions, asymptotically almost periodic functions and almost automorphic functions are introduced in the second part. In addition,the knowledge of C0-semigroup and the dominated convergence theorem of Lebesgue are also introduced in this part. To provide theoretical basis for asymptotically almost periodic solution and almost automorphic mild solution of differential equations.The existence and uniqueness of asymptotically almost periodic solution for a class of third-order differential equations are discussed by by using the Banach fixed point theorem and the related basic properties of almost periodic function and asymptotically almost periodic function. The content of this part is one of the main results of this dissertation.The existence and uniqueness of almost automorphic mild solution for a class of differential equations,are discussed by by using the fixed point theorem and the related theoretical knowledge of almost automorphic function, and the related knowledge of C0-semigroup and the dominated convergence theorem of Lebesgue. The content of this part is the other main result of this dissertation.On the basis of some literature for the research of almost periodic solutions for two classes of differential equations, the existence and uniqueness of asymptotically almost periodic solution and almost automorphic mild solution on them are investigated. So these contents of research will be used more widely.