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Oscillation of third order nonlinear delay dynamic equations on time scales. (English) Zbl 1175.34086

Summary: This paper gives oscillation criteria for the third order nonlinear delay dynamic equation
\[ \left(a(t)\left\{\left[r(t)x^\Delta(t)\right]^\Delta\right\}^\gamma\right)^\Delta+f(t,x(\tau(t)))=0 \]
on a time scale \(\mathbb T\) where \(\gamma\geq 1\) is the quotient of odd positive integers, \(a\) and \(r\) are positive \(rd\)-continuous functions on \(\mathbb T\), and the so-called delay function \(\tau:\mathbb T\to\mathbb T\) satisfies \(\tau(t)\leq t\) for \(t\in\mathbb T\) and \(\lim_{t\to\infty}\tau(t)=\infty\) and \(f\in C(\mathbb T\times \mathbb R,\mathbb R)\). Our results are new for third order delay dynamic equations and extend many known results for oscillation of third order dynamic equation. These results in the special cases when \(\mathbb T=\mathbb R\) and \(\mathbb T=\mathbb N\) involve and improve some oscillation results for third order delay differential and difference equations; when \(\mathbb T=h\mathbb N\), \(\mathbb T=q^{\mathbb N_0}\) and \(\mathbb T=\mathbb N^2\) our oscillation results are essentially new. Some examples are given to illustrate the main results.

MSC:

34K11 Oscillation theory of functional-differential equations
39A10 Additive difference equations
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