  This paper considers the growth rates of positive solutions of scalar
nonlinear functional and Volterra differential equations. The equations are
assumed to be autonomous (or asymptotically so), and the nonlinear dependence
grows less rapidly than any linear function. We impose extra regularity
properties on a function asymptotic to this nonlinear function, rather than on
the nonlinearity itself. The main result of the paper demonstrates that the
growth rate of the solution can be found by determining the rate of growth of a
trivial functional differential equation (FDE) with the same nonlinearity and
all its associated measure concentrated at zero; the trivial FDE is nothing
other than an autonomous nonlinear ODE. We also supply direct asymptotic
information about the solution of the FDE under additional conditions on the
nonlinearity, and exploit the theory of regular variation to sharpen and extend
the results.
