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Almost automorphic solutions to nonautonomous semilinear evolution equations in Banach spaces. (English) Zbl 1192.43005
Summary: This paper is concerned with almost automorphy of the solutions to a nonautonomous semilinear evolution equation ${u}^{\text{'}}\left(t\right)=A\left(t\right)u\left(t\right)+f\left(t,u\left(t\right)\right)$ in a Banach space with a Stepanov-like almost automorphic nonlinear term. We establish a composition theorem for Stepanov-like almost automorphic functions. Furthermore, we obtain some existence and uniqueness theorems for almost automorphic solutions to the nonautonomous evolution equation, by means of the evolution family and the exponential dichotomy. Some results in this paper are new even if $A\left(t\right)$ is time independent.
##### MSC:
 43A60 Almost periodic functions on groups, etc.; almost automorphic functions 34G20 Nonlinear ODE in abstract spaces