OSCILLATION CRITERIA FOR FIRST-ORDER DELAY EQUATIONS

Abstract

This paper is concerned with the oscillatory behaviour of first-order delay differential equations of the form \begin{equation} x^{\prime}(t) + p(t)x(\tau(t)) = 0, \quad t \geqslant t_0, \end{equation} where $p, \tau \in C([t_0,\infty),{\rm I\!R}^+),{\rm I\!R}^+ = [0,\infty),\tau(t)$ is non-decreasing, $\tau(t) < t$ for $t \geqslant t_0$ and $\lim_{t\rightarrow\infty}\tau(t) = \infty$. Let the numbers $k$ and $L$ be defined by \[ k = \mathop{{\lim\inf}}_{{t\rightarrow\infty}} \int\nolimits^t_{\tau(t)} p(s)ds \quad\hbox{and}\quad L = \mathop{\lim\sup}_{{t\rightarrow\infty}} \int\nolimits^t_{\tau(t)} p(s)ds. \]

It is proved here that when $L < 1$ and $0 < k \leqslant 1/e$ all solutions of equation (1) oscillate in several cases in which the condition \[ L > \frac{\ln\lambda_1 - 1 + \sqrt{5 - 2\lambda_1 + 2k\lambda_1}}{\lambda_1} \] holds, where $\lambda_1$ is the smaller root of the equation $\lambda = e^{k\lambda}$.