Subexponential Growth Rates in Functional Differential Equations
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 linearlogarithmic 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 nonunit factor.
 Publication:

arXiv eprints
 Pub Date:
 September 2014
 arXiv:
 arXiv:1409.4201
 Bibcode:
 2014arXiv1409.4201A
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 34K25;
 34C11
 EPrint:
 10 pages