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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 Δ (t)) Δ +q(t)x σ (t)=0(*)

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 Δ stands for the Δ-derivative of x and x σ is the composition of x with the forward jump operator σ. Furthermore, p,q:𝕋 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.

34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
39A11Stability of difference equations (MSC2000)
39A12Discrete version of topics in analysis