The first positive root of the fundamental solution is an optimal oscillation bound for linear delay differential equations

We consider the nonautonomous linear delay differential equationx′(t)+p(t)x(t−1)=0,t≥t0, where p:[t0,∞)→R is a continuous function such that p(t)≥a>1/e for t≥t0. It is shown that the smallest positive constant ω=ω(a) such that every solution has at least one zero in [t0−1,t0+ω] coincides with the first positive root ζ(a) of the fundamental solution of the equation with constant coefficienty′(t)+ay(t−1)=0. A detailed analysis of the function ζ:(1/e,∞)→(1,∞) is given with the emphasis on the asymptotic description of ζ(a) as a→1/e+. As corollaries new oscillation criteria are obtained which improve some previous results in the literature. With appropriate modifications the method applies to a more general class of linear differential equations with time-dependent delay.