Almost automorphic solutions of nonautonomous evolution equations

In this paper, we establish some new theorems about the existence of almost automorphic solutions to nonautonomous evolution equations u′(t)=A(t)u(t)+f(t)u′(t)=A(t)u(t)+f(t) and u′(t)=A(t)u(t)+f(t,u(t))u′(t)=A(t)u(t)+f(t,u(t)) in Banach spaces. As we will see, our results allow for a more general A(t)A(t) to some extent. An example is also given to illustrate our results. In addition, by means of an example, we show that one cannot ensure the existence of almost automorphic solutions to u′(t)=A(t)u(t)+f(t)u′(t)=A(t)u(t)+f(t) even if the evolution family U(t,s)U(t,s) generated by A(t)A(t) is exponentially stable and f∈AA(X)f∈AA(X).