Asymptotically Almost Periodic Solutions For Differential Equations With Piecewise Constant Argument

As well known, one of the most fundamental and important problems in differential equation theory is the research on the properties of solutions for differential equations, especially, the existence of periodic type solutions and almost periodic type solutions have important theoretical and applied significance. In 1983, K.L.Cook and J.Wiener gave a survey of the status concerning the differential equations with piecewise constant argument. They have applications in certain biological models. These equations have the properties of continuous dynamical systems within intervals of unit length. Continuity of the 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 and have been applied in certain biological models and control theory.In this paper, some results concerning almost periodic type functions are given, then the existence of asymptotically almost periodic functions solutions to some classes of differential equations with piecewise constant argument are given, and the uniqueness of solutions are discussed. This paper is organized as follows:Firstly, we show that the space of asymptotically almost periodic functions is Banach space by the supremun norm and this space is a translation invariant C *-subalgebra of C ( R ); and the relation between the norm of asymptotically almost periodic function and the norm of it's almost periodic component is obtained.Secondly, the existence and uniqueness of asymptotically almost periodic functions solutions of a class first-order linear differential equations with piecewise contant argument and a class second-order linear equations with piecewise constant argument are studied by using the unique decomposition theorem of asymptotically almost periodic functions and asymptotically almost periodic sequence solutions of corresponding difference equations. Using contraction mapping principle, we discuss the existence and uniqueness of asymptotically almost periodic solutions of first-order nonlinear differential equations with piecewise constant argument.Thirdly, the existence of asymptotically almost periodic solutions of a class singularly perturbed differential equations with piecewise constant argument is studied. The existence of asymptotically almost periodic solutions of differential equations with piecewise constant argument is given. Then using the relevant results about differential equations and asymptotically almost periodic sequence solutions of corresponding difference equations, the conclusion is given.