Singular perturbations of integro-differential equations

We study the singular perturbation problem(Eϵ)ϵ2uϵ″(t)+uϵ′(t)=Auϵ(t)+(K∗Auϵ)(t)+fϵ(t),t⩾0,ϵ>0for the integro-differential equation(E)w′(t)=Aw(t)+(K∗Aw)(t)+f(t),t⩾0,in a Banach space, when ϵ → 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) has a unique solution uϵ(t) for each small ϵ > 0. Moreover uϵ(t) converges to u(t) as ϵ → 0+, the unique solution of equation (E).