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half-linear dynamic equation; time scale; transformation; comparison theorem; oscillation criteria
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 $\Phi (u)=|u|^{\alpha -1}\mathop{\rm sgn}u$ 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.
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