On increasing solutions of half-linear delay differential equations

TL;DR

This paper investigates the asymptotic behavior of increasing solutions to half-linear delay differential equations, establishing conditions for regular variation and deriving precise asymptotic formulas, with novel insights even in the linear case.

Contribution

It provides new conditions and asymptotic formulas for solutions of half-linear delay differential equations, including original results for the linear case and non-functional equations.

Findings

All eventually positive increasing solutions are regularly varying under certain conditions.

Derived precise asymptotic formulas for these solutions.

Identified significant differences between delayed and non-delayed cases for decreasing solutions.

Abstract

We establish conditions guaranteeing that all eventually positive increasing solutions of a half-linear delay differential equation are regularly varying and derive precise asymptotic formulae for them. The results here presented are new also in the linear case and some of the observations are original also for non-functional equations. A substantial difference between the delayed and non-delayed case for eventually positive decreasing solutions is pointed out.