Stability and oscillation of linear delay differential equations

There is a close connection between stability and oscillation of delay differential equations. For the first-order equation $x^{\prime}(t)+c(t)x(τ(t))=0,~~t\geq 0,$ where $c$ is locally integrable of any sign, $τ(t)\leq t$ is Lebesgue measurable, $\lim_{t\rightarrow\infty}τ(t)=\infty$, 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.

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