Hyperbolic singular perturbations for integrodifferential equations

Abstract

We study the convergence of solutions of ε 2u ′′(t;ε)+u ′(t;ε)=(ε 2A+B)u(t;ε)+∫ 0 tK(t−s)(ε 2A+B)u(s;ε) ds +f(t;ε), t⩾0, u(0;ε)=u 0(ε), u ′(0;ε)=u 1(ε), to solutions of w ′(t)=Bw(t)+∫ 0 tK(t−s)Bw(s) ds+f(t), t⩾0, w(0)=w 0, when ε→0. Here A and B are linear unbounded operators in a Banach space X, K( t) is a linear bounded operator for each t⩾0 in X, and f( t; ε) and f( t) are given X-valued functions. Our result extends the studies in Fattorini [J. Diff. Eq. 70 (1987) 1] for equations without the integral term and in Liu [Proc. Am. Math. Soc. 122 (1994) 791] for parabolic singular perturbation problems.