**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.

Communicated by T. A. Burton.
| Received on 2003-08-01 Appeared on 2004-08-31 |

**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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