Singular perturbation for Volterra equations of convolution type

摘要

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)when ϵ → 0+, for the integrodifferential equation(E)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).