Abstract
In this paper, we show the existence of function which is not asymptotically periodic, but which is asymptotically periodic in the Stepanov sense. We give sufficient conditions for the existence and uniqueness of asymptotically omega periodic solutions for a nonautonomous differential equation with piecewise constant argument in a Banach space when omega is an integer. This is done using the Banach fixed point Theorem. An example involving the heat operator is discussed as an illustration of the theory.