Weighted pseudo almost automorphic mild solutions to semilinear integral equations with $S^{p}$-weighted pseudo almost automorphic coefficients

Abstract

In this paper, we consider the existence of weighted pseudo almost automorphic solutions of the semilinear integral equation $x(t)= \int_{-\infty}^{t}a(t-s)[Ax(s) + f(s,x(s))]ds, t\in\mathbb{R}$ in a Banach space $\mathbb{X}$, where $a\in L^{1}(\mathbb{R}_{+})$, $A$ is the generator of an integral resolvent family of linear bounded operators defined on the Banach space $\mathbb{X}$, and $f : \mathbb{R}\times\mathbb{X} \rightarrow \mathbb{X}$ is a weighted pseudo almost automorphic function. The main results are proved by using integral resolvent families, suitable composition theorems combined with the theory of fixed points.