Stability and oscillation of linear delay differential equations

Abstract

There is a close connection between stability and oscillation of delay differential equations. For the first-order equation $$x^{\prime }(t)+c(t)x(\tau (t))=0,t\ge 0,$$ where $c$ is locally integrable of any sign, $\tau (t)\le t$ is Lebesgue measurable, ${\lim }_{t\to \infty }\tau (t)=\infty$, 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.

22 pages, one figure, submitted to Journal of Differential Equations on June 20, 2019

Cited by in corpus (1)

• Exponential stability for a system of second and first order delay differential equations