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Zbl 0922.34060
Philos, Ch.G.; Sficas, Y.G.
An oscillation criterion for first-order linear delay differential equations.
(English)
[J] Can. Math. Bull. 41, No. 2, 207-213 (1998). ISSN 0008-4395; ISSN 1496-4287/e

The authors consider the oscillation behavior of the delay differential equation $$x'(t)+p(t)x (t-\tau(t))=0$$ with $p,\tau\in C([0, \infty), [0,\infty))$, $t-\tau(t)$ is increasing, and $\lim_{t\to\infty}(t-\tau (t))= \infty$. It is shown that the equation is oscillatory if $M+{L^2\over 2(1-L)}+ {L^2\over 2}\lambda_0 >1$ where $$L=\lim \inf_{t\to \infty} \int^t_{t-\tau (t)}p(s)ds, \quad M=\lim \sup_{t\to \infty} \int^t_{t-\tau (t)}p(s)ds,$$ and $\lambda_0$ is the smaller real root of the equation $\lambda= e^{L\lambda}$. This result allows $$\lim\inf_{t\to\infty} \int^t_{t-\tau (t)}p(s) ds\le 1/e.$$
[Bingtuan Li (Tempe)]
MSC 2000:
*34K11 Oscillation theory of functional-differential equations
34C15 Nonlinear oscillations of solutions of ODE

Keywords: oscillation; first-order linear delay differential equations

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