Pseudo-almost automorphic mild solutions to semilinear integral equations in a Banach space

Abstract

In this paper, we consider the existence of pseudo-almost automorphic solutions of the semilinear integral equation x ( t ) = ∫ − ∞ t a ( t − s ) [ A x ( s ) + f ( s , x ( s ) ) ] d s , t ∈ R in a Banach space X , where a ∈ L 1 ( R + ) , A is the generator of an integral resolvent family of linear bounded operators defined on the Banach space X , and f : R × X → X is a pseudo-almost automorphic function. The main results are proved by using integral resolvent families combined with the theory of fixed points.