Asymptotically almost automorphic functions of order n and applications to dynamic equations on time scales

Abstract

We introduce the concept of asymptotically almost automorphic $\R ^m$-value functions of order $n\in\N^*$ on time scales. Then we investigate properties of such functions and we apply this notion to study the initial value problem associated to the semilinear equation
$x^\Delta(t)=A(t)x(t)+f(t,x(t)),\hbox{ } x(t_0)=x_0$, $t\in\T_+$ defined one time scales. Under suitable conditions on the data of the problem, we establish the existence and the uniqueness of solution asymptotically almost automorphic of order one to this problem. The result is achieved by means of the Banach Fixed-Point Theorem.