Some new results on bounded solutions to a semilinear integro-differential equation in Banach spaces

Abstract

In this article, by a concept of Stepanov type $\mu$-pseudo almost automorphic functions developed recently, we investigate some new existence results on bounded solutions to a semilinear integro-differential equation in Banach spaces. We first establish a new composition theorem of such functions, and then we prove the main results via ergodicity and composition theorems of Stepanov type $\mu$-pseudo almost automorphic functions combined with theories of uniformly exponentially stable and strongly continuous family of operators. These bounded solutions can cover (weighted) pseudo almost automorphic solutions with a Stepanov type forcing term as special cases.