Advances in Difference Equations
Volume 2010 (2010), Article ID 478291, 31 pages
Research Article

On Regularly Varying and History-Dependent Convergence Rates of Solutions of a Volterra Equation with Infinite Memory

Edgeworth Centre for Financial Mathematics, School of Mathematical Sciences, Dublin City University, Dublin 9, Ireland

Received 30 December 2009; Accepted 28 February 2010

Academic Editor: Ağacik Zafer

Copyright © 2010 John A. D. Appleby. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We consider the rate of convergence to equilibrium of Volterra integrodifferential equations with infinite memory. We show that if the kernel of Volterra operator is regularly varying at infinity, and the initial history is regularly varying at minus infinity, then the rate of convergence to the equilibrium is regularly varying at infinity, and the exact pointwise rate of convergence can be determined in terms of the rate of decay of the kernel and the rate of growth of the initial history. The result is considered both for a linear Volterra integrodifferential equation as well as for the delay logistic equation from population biology.