## 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

### Keywords:

oscillation; delay nonlinear dynamic equations; time scales
Full Text:

### References:

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