Almost Periodic Type Solutions For Semilinear And Nonlinear Differential Equations

One of the most fundamental and important problems in differential equation theory is the solutions'properties of differential equations, the existence of periodic and almost periodic type solutions for systems has more important significance in theory and value in application. The theory of almost periodic functions and the theory of asymptotically almost periodic, weakly almost periodic and pseudo almost periodic functions who are derivated by the theory of almost periodic functions construct the theory of almost periodic type functions. The theory of almost periodic type functions plays an important role in differential equations, integral equations and difference equations, the asymptotically almost periodic and pseudo almost periodic functions to some classes of semilinear and nonlinear differential equations are mainly applied in this paper, the existence of asymptotically almost periodic solutions and pseudo almost periodic solutions to some classes of semilinear and nonlinear differential equations is discussed.Two aspects of the job are mainly did in this paper: the frist, the composition promble of the asymptotically almost periodic functions is obtained through the other method. The second, the existence of almost periodic type solutions on some classes of semilinear and nonlinear differential equations is discussed. The specifical content as follows:Firstly, the composition promble of the asymptotically almost periodic functions has been proved in some papers by the properties of asymptotically almost periodic functions, the result is proved by the definition of the asymptotically almost periodic functions in this paper.Secondly, the existence of mild asymptotically almost periodic solution on one kind of semilinear differential equations is discussd by the properties of asymp-totically almost periodic functions and the contraction mapping principle. The existence and uniqueness of the other kind of asymptotically almost periodic functions are discussed by the properties of asymptotically almost periodic functions and the theorem of Banach fixed-point. The necessary and sufficient condition of the existence and uniqueness of asymptotically almost periodic solution to a kind of nonlinear differential equations is discussed according the equivalence theorem of homeomorphism. The existence and uniqueness of almost periodic type solution to the other kind of nonlinear differential equations are discussed by using the properties of asymptotically almost periodic type functions, exponential trichotomy and fixed-point theorem.