Abstract

Under the assumption that A is the generator of a twice integrated cosine family and K is a scalar valued kernel, we solve the singular perturbation problem(Eε{lunate}){Mathematical expression}when ε{lunate} → 0+, for the integrodifferential equation(E)w′ (t) = Aw (t) + (K * Aw) (t) + f (t), (t ≥ 0),on a Banach space. If the kernel K verifies some regularity conditions, then we show that problem (Eε{lunate}) has a unique solution uε{lunate}(t) for each small ε{lunate} > 0. Moreover uε{lunate}(t) converges as ε{lunate} → 0+, to the unique solution u(t) of equation (E). © 2006 Elsevier Inc. All rights reserved.