 
Summary: JOURNAL OF INTEGRAL EQUATIONS
AND APPLICATIONS
Volume 14, Number 2, Summer 2002
ON THE NONEXPONENTIAL CONVERGENCE
OF ASYMPTOTICALLY STABLE SOLUTIONS
OF LINEAR SCALAR VOLTERRA
INTEGRODIFFERENTIAL EQUATIONS
JOHN A.D. APPLEBY AND DAVID W. REYNOLDS
ABSTRACT. We study the stability of the scalar linear
Volterra equation
x (t) = ax(t) +
t
0
k(t  s)x(s) ds, x(0) = x0
under the assumption that all solutions satisfy x(t) 0 as
t . It is shown that if k is a continuously differentiable,
positive, integrable function which is subexponential in the
sense that k (t)/k(t) 0 as t , then x(t) cannot converge
to 0 as t faster than k(t).
1. Introduction. In this note we consider the asymptotic stability
