Memory dependent growth in sublinear Volterra differential equations

Abstract

We investigate memory dependent asymptotic growth in scalar Volterra equations with sublinear nonlinearity. In order to obtain precise results we extensively utilize the powerful theory of regular variation. 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 upon the memory of the system. Finally, we employ a fixed point argument for determining 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.