Singular Perturbations of Integrodifferential Equations in Banach Space

Let ε > 0 and consider ε2u′′(t; ε) + u′(t; ε) = Au(t; ε) + ∫ t 0 K(t− s)Au(s; ε)ds+ f(t; ε), t ≥ 0, u(0; ε) = u0(ε), u ′(0; ε) = u1(ε), and w′(t) = Aw(t) + ∫ t 0 K(t− s)Aw(s)ds+ f(t), t ≥ 0, w(0) = w0, in a Banach space X when ε → 0. Here A is the generator of a strongly continuous cosine family and a strongly continuous semigroup, and K(t) is a bounded linear operator for t ≥ 0. With some convergence conditions on initial data and f(t; ε) and smoothness conditions on K(·), we prove that if ε → 0, then u(t; ε) → w(t) in X uniformly for t ∈ [0, T ] for any fixed T > 0. We will apply this to an equation in viscoelasticity.