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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 $\Bbb T$ where $\gamma\ge 1$ is the quotient of odd positive integers, $a$ and $r$ are positive $rd$-continuous functions on $\Bbb T$, and the so-called delay function $\tau:\Bbb T\to\Bbb T$ satisfies $\tau(t)\le t$ for $t\in\Bbb T$ and $\lim_{t\to\infty}\tau(t)=\infty$ and $f\in C(\Bbb T\times \Bbb R,\Bbb R)$. 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 $\Bbb T=\Bbb R$ and $\Bbb T=\Bbb N$ involve and improve some oscillation results for third order delay differential and difference equations; when $\Bbb T=h\Bbb N$, $\Bbb T=q^{\Bbb N_0}$ and $\Bbb T=\Bbb N^2$ our oscillation results are essentially new. Some examples are given to illustrate the main results.

34K11Oscillation theory of functional-differential equations
39A10Additive difference equations
Full Text: DOI
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