| Summary: | This article determines the rate of growth to infinity of scalar
autonomous nonlinear functional and Volterra differential equations.
In these equations, the right-hand side is a positive continuous
linear functional of f(x). We assume f grows sublinearly,
leading to subexponential growth in the solutions. The main results
show that the solution of the functional differential equations are
asymptotic to that of an auxiliary autonomous ordinary differential
equation with right-hand side proportional to f. This happens provided f
grows more slowly than l(x)=x/log(x). The linear-logarithmic growth rate
is also shown to be critical: if f grows more rapidly than l,
the ODE dominates the FDE; if f is asymptotic to a constant multiple of l,
the FDE and ODE grow at the same rate, modulo a constant non-unit factor;
if f grows more slowly than l, the ODE and FDE grow at exactly the same rate.
A partial converse of the last result is also proven. In the case when the
growth rate is slower than that of the ODE, sharp bounds on the growth rate
are determined. The Volterra and finite memory equations can have differing
asymptotic behaviour and we explore the source of these differences.
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