A sharp oscillation criterion for a linear delay differential equation

Abstract

It is well-known that for the oscillation of all solutions of the linear delay differential equation x′(t)+p(t)x(t−τ)=0,t≥t0, with p∈C([t0,∞),R+) and τ>0 it is necessary that B≔lim supt→∞A(t)≥1e,whereA(t)≔∫t−τtp(s)ds. Our main result shows that if the function A is slowly varying at infinity (in additive form), then under mild additional assumptions B>1e implies the oscillation of all solutions of the above linear delay differential equation. The applicability of the obtained results and the importance of the slowly varying assumption on A are illustrated by examples.