Memory Dependent Growth in Sublinear Volterra Differential Equations

TL;DR

This paper studies how memory effects influence the long-term growth of solutions in sublinear Volterra equations, revealing explicit dependencies on the memory kernel and effects of perturbations.

Contribution

It introduces a novel analysis of memory-dependent growth in Volterra equations using regular variation and fixed point methods, extending understanding of solution behavior under perturbations.

Findings

Growth rate depends explicitly on the memory kernel.

Strong memory effects lead to solutions tracking the system's memory.

Perturbed equations exhibit asymptotic tracking of the forcing term.

Abstract

We investigate memory dependent asymptotic growth in scalar Volterra equations with sublinear nonlinearity. To obtain precise results we utilise the powerful theory of regular variation extensively. By computing the growth rate in terms of a related ordinary differential equation we show that when the memory effect is so strong that the kernel tends to infinity, the growth rate of solutions depends explicitly on the memory of the system. Finally, we employ a fixed point argument to determine analogous results for a perturbed Volterra equation and show that, for a sufficiently large perturbation, the solution tracks the perturbation asymptotically, even when the forcing term is potentially highly non-monotone.