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Pavel Rehak, Institute of Mathematics, Academy of Sciences CR, Zizkova 22, CZ-616 62 Brno, Czech Republic, e-mail: rehak@math.cas.cz
Abstract: The aim of this contribution is to study the role of the coefficient $r$ in the qualitative theory of the equation $(r(t)\Phi(y\del))\del+p(t)\Phi(y\sig)=0$, where $\Phi(u)=|u|^{\alpha-1}\sgn u$ 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.
Keywords: half-linear dynamic equation, time scale, transformation, comparison theorem, oscillation criteria
Classification (MSC 2010): 34C10, 34N05, 39A12, 39A13
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