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A role of the coefficient of the differential term in qualitative theory of half-linear equations. (English) Zbl 1224.34293
Summary: The aim of this contribution is to study the role of the coefficient $$r$$ in the qualitative theory of the equation $(r(t)\Phi (y^{\Delta }))^{\Delta } +p(t)\Phi (y^{\sigma })=0,$ where $$\Phi (u)=| u| ^{\alpha -1}\operatorname {sgn}u$$ with $$\alpha >1$$. We discuss sign and smoothness conditions posed on $$r$$, (non)availability of some transformations, and, mainly, we show how the behavior of $$r$$, along with the behavior of the graininess of the time scale, affect some comparison results and (non)oscillation criteria. At the same time, we provide a survey of recent results acquired by sophisticated modifications of the Riccati type technique, which are supplemented by some new observations.

##### MSC:
 34N05 Dynamic equations on time scales or measure chains 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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