Almost automorphic solutions of neutral functional differential equations

In this article, we prove the existence and uniqueness of almost automorphic solutions to the non-autonomous evolution equation $$ frac{d}{dt}(u(t)-F_1(t,B_1u(t)))=A(t)(u(t)-F_1(t,Bu(t)))+F_2(t,u(t),B_2u(t)), quad tin mathbb{R} $$ where $A(t)$ generates a hyperbolic evolution family $U(t,s)$ (not necessarily periodic) in a Banach space, and $B_1,B_2$ are bounded linear operators. The results are obtained by means of fixed point methods.