Hartman-Wintner growth results for sublinear functional differential equations

TL;DR

This paper analyzes the growth rates of solutions to sublinear functional differential equations, establishing conditions under which solutions behave like those of an auxiliary ODE and providing sharp bounds.

Contribution

It introduces new growth rate results for scalar nonlinear functional differential equations with sublinear nonlinearities, extending Hartman-Wintner type theorems.

Findings

Solutions are asymptotic to auxiliary ODE solutions under certain growth conditions.

The growth rate of solutions depends on the nonlinearity's rate relative to a critical threshold.

Sharp bounds are derived for solutions when the nonlinearity grows more slowly than the critical rate.

Abstract

In this paper, we determine rates 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 a nonlinear function of the state. We assume the nonlinearity grows sublinearly at infinity, leading to subexponential growth in the solutions. Our main results show that the solutions of the functional differential equations are asymptotic to those of an auxiliary autonomous ordinary differential equation when the nonlinearity grows more slowly than a critical rate. If the nonlinearity grows more rapidly than this rate, the ODE dominates the FDE. If the nonlinearity tends to infinity at exactly this rate, the FDE and ODE grow at the same rate, modulo a constant non-unit factor. Finally, if the nonlinearity grows more slowly than the critical rate, then the ODE and FDE grow at the same rate asymptotically. We also prove a partial converse of the last result. In the case when the growth rate is slower than that of the ODE, we calculate sharp bounds on the solutions.