Oscillation criteria for delay equations

Abstract:

This paper is concerned with the oscillatory behavior of first-order delay differential equations of the form \begin{eqnarray} x’ (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 $\lambda =e^{k \lambda }$.