Geodesic
A role of the coefficient of the differential term in qualitative theory of half-linear equations
Mathematica Bohemica, Tome 135 (2010) no. 2, pp. 151-162
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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.
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.
DOI : 10.21136/MB.2010.140692
Classification : 34C10, 34N05, 39A12, 39A13
Keywords: half-linear dynamic equation; time scale; transformation; comparison theorem; oscillation criteria 
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Řehák, Pavel. A role of the coefficient of the differential term in qualitative theory of half-linear equations. Mathematica Bohemica, Tome 135 (2010) no. 2, pp. 151-162. doi: 10.21136/MB.2010.140692

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