Weighted pseudo almost automorphic solutions to nonautonomous semilinear evolution equations with delay and $S^{p}$-weighted pseudo almost automorphic coefficients

Abstract

By some new properties of Stepanov-like weighted pseudo almost automorphic functions established by Chang, Zhang and N'Guerekata recently, we shall deal with weighted pseudo almost automorphic solutions to the nonautonomous semilinear evolution equations with a constant delay: $u'(t)=A(t)u(t)+f(t,u(t-h))$, $ t\in\mathbb{R}$ in a Banach space $\mathbb{X}$, where $A(t),t\in\mathbb{R}$ generates an exponentially stable evolution family $\{U(t,s)\}$ and $f \colon\mathbb{R}\times\mathbb{X} \rightarrow \mathbb{X}$ is a $\S^{p}$-weighted pseudo almost automorphic function satisfying some suitable conditions. We obtain our main results by the Leray-Shauder Alternative theorem.