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

J. A. D. Appleby, CMDE, School of Mathematical Sciences, Dublin City University, Dublin 9, Ireland
D. W. Reynolds, CMDE, School of Mathematical Sciences, Dublin City University, Dublin 9, Ireland

E. J. Qualitative Theory of Diff. Equ., Proc. 7'th Coll. Qualitative Theory of Diff. Equ., No. 3. (2004), pp. 1-14.

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 $k(t)$.

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