Almost Periodic Type Sequences And Their Applications To Differential Equations

Almost periodic type sequences are closely connected with almost periodic type functions and have important applications in discussing the solutions of differential and difference equations. Difference equations are arise as variational equations along the orbits of discrete dynamical systems and play a key role in various fields, such as numerical methods of differential equations, finite elements techniques, control theory, and computer science. The existence of some kinds of solutions to these equations and their applications to discrete dynamical systems and differential equations with piecewise constant arguments have been recently studied by some m- athmatical workers.In this paper, some results concerning asymptotically almost periodic type sequences are obtained, and the existence and uniqueness of asymptotically almost periodic functions solutions to two classes of differential equations with piecewise constant argument are discussed. The present paper is organized as follows:First, we show that the mean of asymptotically almost periodic sequence exists, and we investigate the relation of asymptotically almost periodic sequence with asy- mptotically almost periodic function, at the same time, like almost periodic matrix f- unction, we give the definition of asymptotically almost periodic matrix function, and a result concerning asymptotically almost periodic matrix function is given.Second, we discuss the existence of asymptotically almost periodic solution for a class of differential equation with piecewise constant argument by means of asymp- totically almost periodic sequence solutions of relevant difference equation and the unique decomposition of asymptotically almost periodic function. as well as, under the assumption of exponential dichotomy, we discuss the existence and uniqueness of asymptotically almost periodic function solutions for a class of differential equation with piecewise constant argument using contraction mapping principle, we discuss the existence and uniqueness of two classes of nonlinear differential equations with piecewise constant argument.