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A necessary and sufficient condition for oscillation of the Sturm-Liouville dynamic equation on time scales. (English) Zbl 1009.34033

This is a very nice paper about oscillatory properties of the second-order dynamic equation

${\left(r\left(t\right){x}^{{\Delta }}\right)}^{{\Delta }}+c\left(t\right){x}^{\sigma }=0\phantom{\rule{2.em}{0ex}}\left(1\right)$

on an arbitrary time scale $𝕋$. Equation (1) contains as special cases the well known second-order Sturm-Liouville differential $\left(𝕋=ℝ\right)$ and difference $\left(𝕋=ℤ\right)$ equations. A necessary and sufficient condition for the oscillation of equation (1) is established by transforming equation (1) into a scalar trigonometric system (other terminology is a self-reciprocal system). The classification of (1) to be oscillatory/nonoscillatory makes sense, since the time scales Sturmian separation theorem holds for equation (1). The main tool for the proof is the time scales trigonometric transformation. This transformation preserves the oscillatory behavior of transformed systems and generalizes the corresponding continuous-time $\left(𝕋=ℝ\right)$ and discrete-time $\left(𝕋=ℤ\right)$ trigonometric transformations (the latter one obtained by M. Bohner and the first author [J. Differ. equations 163, No. 1, 113-129 (2000; Zbl 0956.39011)]). A further oscillation criterion for equation (1) is obtained via the Riccati technique.

This paper will be useful for researchers interested in (non)oscillatory behavior of differential, difference, and/or dynamic equations.

##### MSC:
 34C10 Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory 34B24 Sturm-Liouville theory 39A10 Additive difference equations 39A11 Stability of difference equations (MSC2000) 39A12 Discrete version of topics in analysis
##### Keywords:
time scale; oscillation; trigonometric system; Riccati equation