Subexponential Growth Rates in Functional Differential Equations

TL;DR

This paper analyzes the growth rates of solutions to scalar nonlinear functional differential equations with finite delay, establishing conditions under which solutions grow faster than polynomial but slower than exponential, with a focus on sublinear growth functions.

Contribution

It introduces a detailed asymptotic analysis of solutions based on the growth rate of the nonlinear functional, identifying a critical growth rate at $x/ ext{log} x$ that determines the solution's asymptotic behavior.

Findings

Solutions grow faster than polynomial but slower than exponential.

The growth rate of solutions is asymptotic to an auxiliary ODE under certain conditions.

The growth rate $x/ ext{log} x$ is critical for the solution's asymptotic behavior.

Abstract

This paper determines the rate of growth to infinity of a scalar autonomous nonlinear functional differential equation with finite delay, where the right hand side is a positive continuous linear functional of $f(x)$. We assume $f$ grows sublinearly, and is such that solutions should exhibit growth faster than polynomial, but slower than exponential. Under some technical conditions on $f$, it is shown that the solution of the functional differential equation is asymptotic to that of an auxiliary autonomous ordinary differential equation with righthand side proportional to $f$ (with the constant of proportionality equal to the mass of the finite measure associated with the linear functional), provided $f$ grows more slowly than $l(x)=x/\log x$. This 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.