Almost automorphic solutions of some differential equations in Banach spaces

Abstract

We discuss the conditions under which bounded solutions of the evolution equation x′(t)=Ax(t)+f(t) in a Banach space are almost automorphic whenever f(t) is almost automorphic and A generates a C0-group of strongly continuous operators. We also give a result for asymptotically almost automorphic solutions for the more general case of x′(t)=Ax(t)+f(t,x(t)).