Asymptotically Almost Periodic Solutions For Two Classes Of Differential Equations

As well known, one of the most fundamental and important problems in differential equations theory is the research on the properties of solutions to differential equations, among these the existence of periodic solutions and almost periodic type solutions for systems has theoretical and applied significance. In this paper, some results concerning asymptotically almost periodic function are obtained, and the existence of asymptotically almost periodic solutions to two classes of differential equations is discussed. The present paper is organized as follows:First, since asymptotically almost periodic functions are extensions of almost periodic functions, some equivalent definitions and fundamental properties of almost periodic functions are generalized to asymptotically almost periodic functions and some properties of asymptotically almost periodic functions are presented, so that asymptotically almost periodic functions can be applied in differential and integral equation theory.Second, the conditions for the existence of unique asymptotically almost periodic solution to a class of linear differential equations are presented by the ergodicity of functions and exponential dichotomy of equations as well as unique decomposition of asymptotically almost periodic functions.In 1984, K.L.Cook and J.Wiener gave a survey of the status concerning the differential equations with piecewise constant argument. These equations have the structure of continuous dynamical systems within intervals of unit length. The continuity of a solution at a point joining any two consecutive intervals implies recursion relations for the values of the solution at such points. Therefore, they combine the properties of differential equations and difference equations. In the rest part of the present paper, the existence of asymptotically almost periodic solutions to two differential equations with piecewise constant argument is studied; the existence and uniqueness of these equations are discussed based on asymptotically almost periodic sequence solutions to difference equations.In conclusion, some of the results in this paper improve and generalize the well-known results, others are new.