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Zbl 0951.34045
Kon, M.; Sficas, Y.G.; Stavroulakis, I.P.
Oscillation criteria for delay equations.
(English)
[J] Proc. Am. Math. Soc. 128, No.10, 2989-2997 (2000). ISSN 0002-9939; ISSN 1088-6826/e

Summary: This paper is concerned with the oscillatory behavior of first-order delay differential equations of the form $$x'(t)+p(t)x({\tau}(t))=0, \quad t\geq t_{0},\tag 1$$ with $p, {\tau} \in C([t_{0}, \infty), \bbfR^+)$, $\bbfR^+=[0, \infty), \tau(t)$ is nondecreasing, $\tau(t) <t$ for $t \geq t_{0}$ and $\lim_{t{\rightarrow}{\infty}} \tau(t) = \infty$. Let the numbers $k$ and $L$ be defined by $$k=\liminf_{t\to\infty} \int_{\tau(t)}^{t}p(s)ds \text{ and }L=\limsup_{t{\rightarrow}{\infty}} \int_{\tau(t)}^{t}p(s)ds.$$ It is proved that when $L<1$ and $0<k \leq \frac{1}{e}$ all solutions to (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 $\lambda =e^{k \lambda}$.
MSC 2000:
*34K11 Oscillation theory of functional-differential equations
34C10 Qualitative theory of oscillations of ODE: Zeros, etc.

Keywords: oscillation; delay differential equations

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