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Comparison and oscillatory behavior for certain second order nonlinear dynamic equations. (English) Zbl 1219.34115

The authors consider the second order nonlinear dynamic equation

${\left(a{\left({x}^{{\Delta }}\right)}^{\alpha }\right)}^{{\Delta }}\left(t\right)+q\left(t\right){x}^{\beta }\left(t\right)=0$

on an arbitrary time scale $𝕋$, where $\alpha$ and $\beta$ are ratios of positive odd integers, $a$ and $q$ are positive rd-continuous functions on $𝕋$. They establish comparison results with the inequality

${\left(a{\left({x}^{{\Delta }}\right)}^{\alpha }\right)}^{{\Delta }}\left(t\right)+q\left(t\right){x}^{\beta }\left(t\right)\le 0$

which are applied to neutral equations. A necessary and sufficient condition is obtained for the oscillation property of second order equations on time scales.

##### MSC:
 34N05 Dynamic equations on time scales or measure chains 34C10 Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
##### References:
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