On Abstract Nonlinear Integro-Dynamic Equations in Time Scale

In this paper, we investigate the existence of the asymptotically almost automorphic solution of the following type of abstract nonlinear integro-dynamic equation \begin{eqnarray*} y^Δ(s) &=&Ay(s)+$\mathcal{F}$\left(s,y(s),\int\limits_{t_0}^{s}{$\mathcal{H}$(s,τ,y(τ))}Δτ\right),~ s\in$\mathbb{T}$^k, y(0)&=&y_0 \end{eqnarray*} in the Banach space of continuous function on a time scale $\mathbb{T}$. We apply the Krasnoselskii fixed point theorem to show the existence of an almost automorphic solution of the above dynamic equation.