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Oscillation criteria for delay equations

Authors:

Institution:

Department of Mathematics, Boston University, Boston, Massachusetts 02215 ; Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece

I. P. Stavroulakis ; Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece

Abstract:

This paper is concerned with the oscillatory behavior of first-order delay differential equations of the form

$\displaystyle x^{\prime}(t)+p(t)x({\tau}(t))=0, \quad t\geq t_{0},$

(1)

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

and

be defined by

$\begin{displaymath}k=\liminf_{t{\rightarrow}{\infty}} \int_{\tau (t)}^{t}p(s)ds ... ... L=\limsup_{t{\rightarrow}{\infty}} \int_{\tau (t)}^{t}p(s)ds. \end{displaymath}$

It is proved here that when

and $0<k \leq \frac{1}{e}$ all solutions of Eq. (1) oscillate in several cases in which the condition

$\begin{displaymath}L>2k+\frac{2}{{\lambda}_{1}}-1 \end{displaymath}$

holds, where ${\lambda _1}$ is the smaller root of the equation $\lambda =e^{k \lambda}$ .

Keywords:

Oscillation delay differential equations

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