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Zbl 0938.34062
Jaroš, J.; Stavroulakis, I.P.
Oscillation tests for delay equations.
(English)
[J] Rocky Mt. J. Math. 29, No.1, 197-207 (1999). ISSN 0035-7596

The authors study the oscillatory behavior of delay equations of the form $$x'(t)+ p(t)x(\tau(t))= 0,\quad t\ge T,\tag 1$$ with $p,\tau\in C([T,\infty),[0,\infty))$, $\tau(t)$ is decreasing, $\tau(t)<t$ for $t\ge T$, $\lim_{t\to\infty} \tau(t)= \infty$. Let $$k= \liminf_{t\to\infty} \int^t_{\tau(t)} p(s) ds,\quad L= \limsup_{t\to\infty} \int^t_{\tau(t)} p(s) ds.$$ It is proved that when $L<1$ and $0< k\le 1/e$, all solutions to equation (1) oscillate if $$L> {\ln\lambda_1+ 1\over \lambda_1}- {1-k- \sqrt{1- 2k- k^2}\over 2},$$ where $\lambda_1$ is the smaller root of the equation $\lambda= e^{k\lambda}$.
[Aleksandra Rodkina (Jamaica)]
MSC 2000:
*34K11 Oscillation theory of functional-differential equations
34C10 Qualitative theory of oscillations of ODE: Zeros, etc.

Keywords: oscillation; delay; advanced differential equations

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