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Erschienen in:
11.04.2016 | Original Research
On the oscillation for third-order nonlinear neutral delay dynamic equations on time scales
Erschienen in: Journal of Applied Mathematics and Computing | Ausgabe 1-2/2017
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Abstract
In this paper, we consider the third-order nonlinear neutral delay dynamic equations on a time scale \(\mathbb {T}\), where \(\alpha _i\) are quotients of positive odd integers, \(i=1\), 2, \(|f(t,u)|\ge q(t)|u|\), \(b,\ p\) and q are real-valued positive rd-continuous functions defined on \(\mathbb {T}\). By using the Riccati transformation technique and integral averaging technique, some new sufficient conditions which ensure that every solution oscillates or tends to zero are established. Our results are new for third-order nonlinear neutral delay dynamic equations and extend many known results for oscillation of third order dynamic equations. Some examples are given here to illustrate our main results.
$$\begin{aligned} \left( b(t)\left( \left( ((x(t)-p(t)x(\tau (t)))^\Delta )^{\alpha _1}\right) ^\Delta \right) ^{\alpha _2}\right) ^\Delta +f(t,x(\delta (t)))=0 \end{aligned}$$