## Interval oscillation criteria for second order super-half linear functional differential equations with delay and advanced arguments.(English)Zbl 1180.34070

Summary: Sufficient conditions are established for oscillation of second order super half linear equations containing both delay and advanced arguments of the form
$\big(\varphi_\alpha(k(t)x'(t))\big)'+ p(t) \varphi_\beta(x(\tau(t)))+ q(t) \varphi_\gamma(x(\sigma(t)))= e(t), \quad t\geq 0,$
where $$\varphi_\delta(u)= |u|^{\delta-1}u$$; $$\alpha>0$$, $$\beta\geq \alpha$$, and $$\gamma\geq \alpha$$ are real numbers; $$k,p,q,e,\tau,\sigma$$ are continuous real-valued functions; $$\tau(t)\leq t$$ and $$\sigma(t)\geq t$$ 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:

 34K11 Oscillation theory of functional-differential equations

### Keywords:

super-half linear; forced; delay; advanced; oscillation
Full Text:

### References:

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