Asymptotic and Pseudo Almost Periodicity of the Convolution Operator and Applications to Differential and Integral Equations

Abstract

We examine conditions which do ensure the asymptotic almost periodicity (respectively, pseudo almost periodicity) of the convolution function $f\ast h$ of $f$ with $h$ whenever $f$ is asymptotically almost periodic (respectively, pseudo almost periodic) and $h$ is a (Lebesgue) measurable function satisfying some additional assumptions. Next we make extensive use of those results to investigate on the asymptotically almost periodic (respectively, pseudo almost periodic) solutions to some differential, functional, and integral equations.