Abstract

We study the singular perturbation problem{A formula is presented}for the integro-differential equation{A formula is presented}in a Banach space, when ε{lunate} → 0+. We assume that A is the generator of a strongly continuous cosine family. Then under some regularity conditions on the scalar-valued kernel K we show that problem (Eε{lunate}) has a unique solution uε{lunate}(t) for each small ε{lunate} > 0. Moreover uε{lunate}(t) converges to u(t) as ε{lunate} → 0+, the unique solution of equation (E). © 2005 Elsevier Inc. All rights reserved.