### Asymptotic behavior of solutions to half-linear $q$-difference equations.

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We study oscillatory properties of the second order half-linear difference equation $$\Delta ({r}_{k}|\Delta {y}_{k}{|}^{\alpha -2}\Delta {y}_{k})-{p}_{k}{\left|{y}_{k+1}\right|}^{\alpha -2}{y}_{k+1}=0,\phantom{\rule{1.0em}{0ex}}\alpha >1.\phantom{\rule{2.0em}{0ex}}\left(\mathrm{HL}\right)$$ It will be shown that the basic facts of oscillation theory for this equation are essentially the same as those for the linear equation $$\Delta \left({r}_{k}\Delta {y}_{k}\right)-{p}_{k}{y}_{k+1}=0.$$ We present here the Picone type identity, Reid Roundabout Theorem and Sturmian theory for equation (HL). Some oscillation criteria are also given.

The aim of this contribution is to study the role of the coefficient $r$ in the qualitative theory of the equation ${\left(r\left(t\right)\Phi \left({y}^{\Delta}\right)\right)}^{\Delta}+p\left(t\right)\Phi \left({y}^{\sigma}\right)=0$, where $\Phi \left(u\right)={\left|u\right|}^{\alpha -1}\mathrm{sgn}u$ 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...

The paper can be understood as a completion of the $q$-Karamata theory along with a related discussion on the asymptotic behavior of solutions to the linear $q$-difference equations. The $q$-Karamata theory was recently introduced as the theory of regularly varying like functions on the lattice ${q}^{{\mathbb{N}}_{0}}:=\{{q}^{k}:k\in {\mathbb{N}}_{0}\}$ with $q>1$. In addition to recalling the existing concepts of $q$-regular variation and $q$-rapid variation we introduce $q$-regularly bounded functions and prove many related properties. The $q$-Karamata theory is then...

We establish conditions which guarantee that the second order difference equation $${\Delta}^{2}{x}_{k}+{p}_{k}{x}_{k+1}=0$$ possesses a nontrivial solution with at least two generalized zero points in a given discrete interval

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