Asymptotically typed solutions to a semilinear integral equation

Abstract

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