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**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.

\[ \big(\varphi_\alpha(k(t)x'(t))\big)'+ p(t) \varphi_\beta(x(\tau(t)))+ q(t) \varphi_\gamma(x(\sigma(t)))= e(t), \quad t\geq 0, \]

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 |

Full Text:
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