On increasing solutions of half-linear delay differential equations
Serena Matucci, Pavel Řehák
TL;DR
This work analyzes the asymptotic behavior of all eventually positive increasing solutions to the half-linear delay differential equation $$(r(t)\Phi(y'))'=p(t)\Phi(y(\tau(t)))$$ with $\Phi(u)=|u|^{\alpha-1}\operatorname{sgn}u$ and $\alpha>1$. Employing the theory of regular variation and the De Haan class $\Pi$, the authors show that all such solutions are regularly varying (or slowly varying) and derive explicit asymptotic formulas that depend on integrals of $G(t)=\left(\frac{tp(t)}{r(t)}\right)^{\beta-1}$, where $\beta$ is the conjugate of $\alpha$. The results cover a range of regimes, including the critical case $\delta=-1$, and extend to coefficients not in standard regular variation via tau-transformations; they also highlight notable differences between delayed and non-delayed equations, especially for decreasing solutions. The unified framework yields precise growth/decay rates for increasing solutions and provides by-products such as transformed representations and generalized regularly varying functions, enhancing understanding of functional differential equations in the regular variation setting.
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.