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On the oscillation for third-order nonlinear neutral delay dynamic equations on time scales. (English) Zbl 1377.34112

Summary: In this paper, we consider the third-order nonlinear neutral delay dynamic equations \[ \left( b(t)\left( \left( ((x(t)-p(t)x(\tau (t)))^\Delta)^{\alpha_1}\right)^\Delta \right)^{\alpha_2}\right)^\Delta +f(t,x(\delta (t)))=0 \] on a time scale \(\mathbb {T}\), where \(\alpha_i\) are quotients of positive odd integers, \(i=1,2\), \(|f(t,u)|\geq q(t)|u|\), \(b\), \(p\) 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.

MSC:

34N05 Dynamic equations on time scales or measure chains
34K11 Oscillation theory of functional-differential equations
34K40 Neutral functional-differential equations
34K25 Asymptotic theory of functional-differential equations
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