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