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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 $$(p(t)x^\Delta(t))^\Delta+q(t)x^\sigma(t)=0\leqno(\ast)$$ on time scales ${\Bbb T}$. 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, $p,q:{\Bbb T}\rightarrow{\Bbb R}$ 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 {\it 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
Full Text:
References:
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