Hyperbolic singular perturbations for integrodifferential equations

We study the convergence of solutions ofε2u′′(t;ε)+u′(t;ε)=(ε2A+B)u(t;ε)+∫0tK(t−s)(ε2A+B)u(s;ε)ds+f(t;ε),t⩾0,u(0;ε)=u0(ε),u′(0;ε)=u1(ε),to solutions ofw′(t)=Bw(t)+∫0tK(t−s)Bw(s)ds+f(t),t⩾0,w(0)=w0,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.