Almost automorphic solutions for differential equations with piecewise constant argument in a Banach space

Using spectral theory we obtain sufficient conditions for the almost automorphy of bounded solutions to differential equations with piecewise constant argument of the form x′(t)=A(t)x([t])+f(t),t∈Rx′(t)=A(t)x([t])+f(t),t∈R, where A(t) is an almost automorphy operator, f(t) is an XX-valued almost automorphic function and XX is a finite dimensional Banach space.