Almost automorphic solutions of semilinear evolution equations

Abstract

We are concerned with the semilinear differential equation in a Banach space $\mathbb {X}$, \[ x’(t)=Ax(t)+F(t,x(t)),\;\ t\in \mathbb {R} ,\] 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.