ON THE NON-EXPONENTIAL
CONVERGENCE OF ASYMPTOTICALLY
STABLE SOLUTIONS
OF LINEAR SCALAR VOLTERRA
INTEGRO-DIFFERENTIAL EQUATIONS

JOHN A.D. APPLEBY AND DAVID W. REYNOLDS

Abstract:

We study the stability of the scalar linear Volterra equation

$$x^\prime(t)=-ax(t)+ \int_{0}^{t} k(t-s)x(s)\,ds,\quad x(0)=x_0$$

under the assumption that all solutions satisfy $x(t)\to 0$ as $t\to\infty$. It is shown that if $k$ is a continuously differentiable, positive, integrable function which is subexponential in the sense that $k^\prime(t)/k(t)\to 0$ as $t\to\infty$, then $x(t)$ cannot converge to $0$ as $t\to\infty$ faster than $k(t)$.