Almost automorphic solutions of semilinear evolution equations

We are concerned with the semilinear differential equation in a Banach space $\mathbb{X}$,

where $A$ generates an exponentially stable $C_0$-semigroup and $F(t,x): \mathbb{R}\times \mathbb{X}\to \mathbb{X}$is a function of the form $F(t,x)=P(t)Q(x)$. Under appropriate conditions on $P$ and $Q$, and using the Schauder fixed point theorem, we prove the existence of an almost automorphic mild solution to the above equation.