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Zbl 1180.34070
Zafer, A.
Interval oscillation criteria for second order super-half linear functional differential equations with delay and advanced arguments. (English)
[J] Math. Nachr. 282, No. 9, 1334-1341 (2009). ISSN 0025-584X; ISSN 1522-2616/e

Summary: Sufficient conditions are established for oscillation of second order super half linear equations containing both delay and advanced arguments of the form $$\big(\varphi_\alpha(k(t)x'(t))\big)'+ p(t) \varphi_\beta(x(\tau(t)))+ q(t) \varphi_\gamma(x(\sigma(t)))= e(t), \quad t\ge 0,$$ where $\varphi_\delta(u)= |u|^{\delta-1}u$; $\alpha>0$, $\beta\ge \alpha$, and $\gamma\ge \alpha$ are real numbers; $k,p,q,e,\tau,\sigma$ are continuous real-valued functions; $\tau(t)\le t$ and $\sigma(t)\ge t$ with $\lim_{t\to\infty}\tau(t)=\infty$. The functions $p(t)$, $q(t)$, and $e(t)$ are allowed to change sign, provided that $p(t)$ and $q(t)$ are nonnegative on a sequence of intervals on which $e(t)$ alternates sign. As an illustrative example we show that every solution of $$\big(\varphi_\alpha(x'(t))\big)'+m_1\sin t\varphi_\beta(x(t-\pi/5))+ m_2\cos t\varphi_\gamma(x(t+\pi/20))= r_0\cos 2t$$ is oscillatory provided that either $m_1$ or $m_2$ or $r_0$ is sufficiently large.

MSC 2000:: *34K11 Oscillation theory of functional-differential equations

Keywords: super-half linear; forced; delay; advanced; oscillation

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