On the non-exponential decay to equilibrium of solutions of nonlinear scalar Volterra integro-differential equations

Abstract

We study the rate of decay of solutions of the scalar nonlinear Volterra equation \[ x'(t)=-f(x(t))+ \int_{0}^{t} k(t-s)g(x(s))\,ds,\quad x(0)=x_0 \] which satisfy $x(t)\to 0$ as $t\to\infty$. We suppose that $xg(x)>0$ for all $x\not=0$, and that $f$ and $g$ are continuous, continuously differentiable in some interval $(-\delta_1,\delta_1)$ and $f(0)=0$, $g(0)=0$. Also, $k$ is a continuous, positive, and integrable function, which is assumed to be subexponential in the sense that $k(t-s)/k(t)\to 1$ as $t\to\infty$ uniformly for $s$ in compact intervals. The principal result of the paper asserts that $x(t)$ cannot converge to $0$ as $t\to\infty$ faster than $k(t)$.