Asymptotic behavior of mild solutions to semilinear integro differential equations

Abstract

We deal in this paper with $\rho$-pseudo almost periodic and $\rho$-pseudo almost automorphic mild solutions to the semilinear integro-differential equation $u'(t)=Au(t)+\int_{-\infty}^{t}a(t-s)Au(s)ds+f(t,Bu(t)).\;\;t\in \R,$ where $A$ generates a uniformly integrable semigroup of operators in a Banach space $X$, and $B\in \mathcal{L}(X)$. We give an example of non uniqueness of the decomposition of $\rho$-pseudo almost automorphic functions. We also discuss some properties of weakly almost automorphic functions. These results extend and complement some recent ones on this topic.