A sharp oscillation criterion for a linear delay differential equation

It is well-known that for the oscillation of all solutions of the linear delay differential equation x'(t) + p(t)x(t - tau) = 0, t >= to, with p is an element of C([t(0), infinity),R+) and tau > 0 it is necessary that B := lim sup(t ->infinity) A(t) >= 1/e, where A(t) := integral(t)(t-tau) p(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 > 1/e 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. (C) 2019 Elsevier Ltd. All rights reserved.