Stability and oscillation of linear delay differential equations

TL;DR

This paper investigates the relationship between stability and oscillation in linear delay differential equations, providing sharp criteria and unifying classical results, with improvements on stability bounds for equations with measurable parameters.

Contribution

It offers new sharp results linking oscillation speed and stability, unifies classical stability and oscillation criteria, and generalizes the 3/2-stability criterion to measurable parameters.

Findings

Established sharp relations between oscillation speed and stability.

Unified classical results of Myshkis and Lillo.

Improved the 3/2-stability criterion for measurable parameters.

Abstract

There is a close connection between stability and oscillation of delay differential equations. For the first-order equation $$ x^{\prime}(t)+c(t)x(\tau(t))=0,~~t\geq 0, $$ where $c$ is locally integrable of any sign, $\tau(t)\leq t$ is Lebesgue measurable, $\lim_{t\rightarrow\infty}\tau(t)=\infty$, we obtain sharp results, relating the speed of oscillation and stability. We thus unify the classical results of Myshkis and Lillo. We also generalise the $3/2-$stability criterion to the case of measurable parameters, improving $1+1/e$ to the sharp $3/2$ constant.