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Oscillation criteria for half-linear dynamic equations on time scales. (English) Zbl 1156.34022

The author considers the second-order half-linear dynamic equation

${\left(r\left(t\right){\left({x}^{{\Delta }}\left(t\right)\right)}^{\gamma }\right)}^{{\Delta }}+p\left(t\right){x}^{\gamma }\left(t\right)=0\phantom{\rule{2.em}{0ex}}\left(1\right)$

on an arbitrary time scale $𝕋$ ($sup𝕋=\infty$), where $\gamma$ is the quotient of odd positive integers, $r\left(t\right)$ and $p\left(t\right)$ are positive rd-continuous functions on $𝕋$. Main results of the paper are sufficient conditions for every solution of (1) to be oscillatory. As the author remarks, when $𝕋=ℝ$, the obtained results improve several results known for differential equations and when $𝕋=ℕ$, then they improve some results known for second order difference equations.

##### MSC:
 34C10 Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory 39A10 Additive difference equations
##### References:
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