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Oscillation and asymptotic behavior of a third-order nonlinear dynamic equation. (English) Zbl 1145.34329
Summary: We establish some new oscillation criteria for the third-order nonlinear dynamic equation
\[ (c(t)((a(t)x^\Delta)^\gamma)^\Delta +f(t,x(t))=0,\quad t\in[a,\infty)_{\mathbb T} \]
on time scales, where \(\gamma\geq 1\) is a quotient of odd integers. Our results not only unify the oscillation theory for third-order nonlinear differential and difference equations but also are new for the \(q\)-difference equations and can be applied on different time scales. The results improve some of the main results in the literature in the case when \(f = 1\).

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
39A13 Difference equations, scaling (\(q\)-differences)
39A10 Additive difference equations