  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.
