Zeitschrift f¨ ur Analysis und ihre Anwendungen c  European Mathematical Society Journal for Analysis and its Applications Volume 25 (2006), 327–340 Asymptotic and Pseudo Almost Periodicity of the Convolution Operator and Applications to Differential and Integral Equations Dariusz Bugajewski, Toka Diagana, and Cr´ epin M. Mahop Abstract. We examine conditions which do ensure the asymptotic almost periodicity (respectively, pseudo almost periodicity) of the convolution function f * 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. Keywords. Almost periodic function, asymptotically almost periodic function, Ba- nach fixed-point principle, convolution operator, differential equation, integral equa- tion, functional equation, pseudo almost periodic function, Zima’s fixed-point theorem Mathematics Subject Classification (2000). Primary 44A35, secondary 42A85, 42A75 1. Introduction Given two functions f,h : R → R, the convolution function, if it exists, of f with h denoted f ∗ h is defined by (f ∗ h)(t) :=  +∞ −∞ f (σ)h(t − σ)dσ, ∀t ∈ R. (1) Several properties of the convolution operation ∗ can be found in most good books in functional analysis. Among others, setting u = t − σ in Eq. (1) it is routine to show that the convolution operation is commutative, i.e., f ∗ h = h ∗ f . D. Bugajewski: Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87, 61-614 Pozna´ n, Poland; ddbb@amu.edu.pl T. Diagana: Department of Mathematics, Howard University, 2441 6th Street N.W., Washington, D.C. 20059, USA; tdiagana@howard.edu C. M. Mahop: Department of Mathematics, Howard University, 2441 6th Street N.W., Washington, D.C. 20059, USA; cmahop@howard.edu