The link between integral equations and higher order ODEs

We study a class of partial integro-differential equations defined on a spatially extended domain that arise in the modeling of neuronal networks. In previous studies the Fourier transform was used to derive associated fourth order ordinary differential equations when the solution of the time independent integral equation is a homoclinic orbit. This gives rise to the question of whether solutions of the ODE whose Fourier transform is not well defined are also solutions of the time independent integral equation. We address this question and show that any solution of the ODE that satisfies a fairly relaxed growth condition is also a solution of the integral equation. Applications to two specific examples are given.  2003 Elsevier Inc. All rights reserved.