Singular perturbation for Volterra equations of convolution type

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 problemequation(Eϵ)ϵ2uϵ″(t)+uϵ′(t)=Auϵ(t)+(K∗Auϵ)(t)+fϵ(t),(t⩾0)(ϵ>0),when ϵ → 0+, for the integrodifferential equationequation(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) has a unique solution uϵ(t) for each small ϵ > 0. Moreover uϵ(t) converges as ϵ → 0+, to the unique solution u(t) of equation (E).