Electron.n. J. Diff. Eqns. Vol. 1993(1993), No. 02, pp. 1-10.

### A singular perturbation problem in integrodifferential equations

James H. Liu

**Abstract:**

Consider the singular perturbation problem for

where
,
,
, and

in a Banach space X when
.
Here A is the generator of a strongly continuous cosine family
and a
strongly continuous semigroup, and
is a bounded linear operator for
. With some convergence conditions on initial data and
and smoothness conditions on
, we prove
that when
, one has
and
in X uniformly on
[0,T] for any fixed
.
An application to viscoelasticity is given.

Submitted June 14, 1993. Published September 16, 1993.

Math Subject Classification: 45D, 45J, 45N.

Key Words: Singular perturbation, Convergence in solutions and
derivatives.

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James H. Liu

Department of Mathematics,
James Madison University

Harrisonburg, VA 22807, USA

e-mail: liu@math.jmu.edu

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