Asymptotically almost automorphic solutions of dynamic equations on time scales

Abstract

In the present work, we introduce the concept of asymptotically almost automorphic functions on time scales and study their main properties. We study nonautonomous dynamic equations on time scales given by $x^{\Delta} (t) = A(t) x(t) + f(t)$ and $x^{\Delta} (t) = A(t) x(t) + f(t, x(t))$, $t \in \mathbb T$, where $\mathbb T$ is an invariant under translations time scale and $A \in \mathcal{R}(\mathbb T, \mathbb R^{n \times n})$. We give new criteria ensuring the existence of an asymptotically almost automorphic solution for both equations.