Oscillation of first order delay differential equations

Abstract

In this article, we establish some new criteria for the oscillation of the delay differential equation
$$
y^{\prime}(x)+q(x) y(\tau(x))=0, x \geq x_0, \tau(x)<x .
$$
For the case where
$$
\int_{\tau(x)}^x q(t) d t \geq \frac{1}{e} \text { and } \lim _{x \rightarrow \infty} \int_{\tau(x)}^x q(t) d t=\frac{1}{e} .
$$
An open problem by A. Elbert and I. P. Stavroulakis (1995, Proc. Amer. Math. Soc., 123, 1503-1510) is solved.