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Kamenev-type and interval oscillation criteria for second-order linear differential equations on a measure chain. (English) Zbl 1056.34050

This paper is concerned with oscillation theory for selfadjoint second-order scalar dynamic equations of the form

${\left(p\left(t\right){x}^{{\Delta }}\left(t\right)\right)}^{{\Delta }}+q\left(t\right){x}^{\sigma }\left(t\right)=0\phantom{\rule{2.em}{0ex}}\left(*\right)$

on time scales $𝕋$. Here, a time scale is an arbitrary nonempty closed subset of the real numbers, denoted as measure chain by the authors, ${x}^{{\Delta }}$ stands for the ${\Delta }$-derivative of $x$ and ${x}^{\sigma }$ is the composition of $x$ with the forward jump operator $\sigma$. Furthermore, $p,q:𝕋\to ℝ$ are assumed to be rd-continuous.

The authors provide Kamenev-type and interval oscillation criteria for such linear dynamic equations on time scales. These criteria generalize corresponding theorems for ODEs by Ch. G. Philos [Arch. Math. 53, 482–492 (1989; Zbl 0661.34030)] or the second author [J. Math. Anal. Appl. 229, 258–270 (1999; Zbl 0924.34026)], respectively, and are new for difference equations in particular. The paper closes with four examples illustrating the obtained results, two of them for difference equations and one on a time scale with unbounded graininess.

##### MSC:
 34C10 Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory 39A11 Stability of difference equations (MSC2000) 39A12 Discrete version of topics in analysis