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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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