A Singular Perturbation Problem in Integrodifferential Equations

Consider the singular perturbation problem for εu(t; ε) + u(t; ε) = Au(t; ε) + ∫ t 0 K(t− s)Au(s; ε) ds+ f(t; ε) , where 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 when ε → 0, one has u(t; ε) → w(t) and u(t; ε) → w(t) in X uniformly on [0, T ] for any fixed T > 0. An application to viscoelasticity is given.