Growth Rates of Sublinear Functional and Volterra Differential Equations

Abstract

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. µ 2 (ds). (1.2)

Memory dependent growth in sublinear Volterra differential equations

“…The authors have previously obtained this conclusion for sublinear equations of the form (1.3) without regular variation but with lim t→∞ M (t) = M ∈ (0, ∞). Therefore Theorem 4 can be thought of as a continuous extension of our previous results for (1.1) with sublinear nonlinearities and finite measures (see [6] for further details).…”

“…[10,Theorem 7.2.3]). In [6] we also show that if lim t→∞ M (t) = ∞, then lim t→∞ F (x(t))/t = ∞. This result suggests that allowing the total measure to be infinite makes the long run dynamics more sensitive to the memory but that comparison with a non-autonomous ordinary differential equation may be necessary in this case.…”

“…(1.4) As with the unperturbed equation, it is useful to consider an integral form of (1. In [6], with µ a finite measure, we demonstrate that when f is sublinear and asymptotically increasing, the solution of (1.1) obeys lim t→∞ F (x(t))/t = [0,∞) µ(ds) < ∞, where…”

On the admissibility of unboundedness properties of forced deterministic and stochastic sublinear Volterra summation equations

“…If we still assume that k ∈ ℓ 1 (N), f is sublinear in the sense of (2.2), and f : (0, ∞) → (0, ∞) and k is non-negative, then all solutions of (2.13) with positive initial condition grow to infinity at a rate determined by f . A continuous analogue of (2.13) with these positivity assumptions is considered in [8]. Some existing economic models take the form of (2.5) or are closely related to it.…”