Oscillation criteria for delay equations.

This paper is concerned with the oscillatory behavior of first-order delay differential equations of the form \begin{eqnarray} x^{\prime}(t)+p(t)x({\tau}(t))=0, \quad t\geq t_{0}, \end{eqnarray} where $p, {\tau} \in C([t_{0}, \infty), \mathbb{R}^+), \mathbb{R}^+=[0, \infty), \tau (t)$ is non-decreasing, $\tau (t) <t$ for $t \geq t_{0}$ and $\lim_{t{\rightarrow}{\infty}} \tau (t) = \infty$. Let the numbers $k$ and $L$ be defined by \[ k=\liminf_{t{\rightarrow}{\infty}} \int_{\tau (t)}^{t}p(s)ds \quad \mbox{and} \quad L=\limsup_{t{\rightarrow}{\infty}} \int_{\tau (t)}^{t}p(s)ds. \] It is proved here that when $L<1$ and $0<k \leq \frac{1}{e}$ all solutions of Eq. (1) oscillate in several cases in which the condition \[ L>2k+\frac{2}{{\lambda}_{1}}-1 \] holds, where ${\lambda _1}$ is the smaller root of the equation $\lambda =e^{k \lambda}$. [ABSTRACT FROM AUTHOR]