Existence of an asymptotically almost periodic solution for a fractional semilinear problem

In this research, we consider the fractional semilinear problem in a sequentially compact Banach space $X$: $x^{\alpha}(t)=A(t)x(t)+f(t,x(t))$, $t\in \mathbb R^{+} $, with the initial condition $x(0)=x_{0}$, $ x_{0} \in X $, where $A$ is the generator of an evolution system $({U(t,s)})_{t\leq s \leq {0}}$ and $f$ is a given function satisfying some assumptions. We study this fractional semilinear integro-differential equation and examine when it has an asymptotically almost periodic solution.