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Oscillation criteria for second-order matrix dynamic equations on a time scale. (English) Zbl 1017.34030
The authors investigate oscillatory properties of the matrix Sturm-Liouville dynamic equation on a time scale \(\mathbb{T}\) (which is supposed to be unbounded from above) \[ [P(t)X^\Delta ]^\Delta + Q(t)X^\sigma =0, \tag{*} \] where \(P,Q\) are symmetric \(n\times n\)-matrices and \(P\) is positive definite. Basic facts of the time scale calculus can be found in the recent monograph of M. Bohner and A. Peterson [Dynamic equations on time scales. An introduction with applications. Basel: Birkhäuser (2001; Zbl 0978.39001)]. Recall that the time scale derivative \({}^\Delta \) reduces to the usual derivative \(\frac{d}{dt}\) if \(\mathbb{T}=\mathbb{R}\) and to the forward difference \(\Delta \) if \(\mathbb{T}=\mathbb{Z}\). The forward jump operator is defined by \(\sigma(t)=\inf\{s\in \mathbb{T}: s>t\}\) and the graininess by \(\mu(t)=\sigma(t)-t\). One of the main results of the paper reads as follows:
Suppose that for every \(t_0\in \mathbb{T}\) there exist \(t_0\leq a_0<b_0\) such that \(\mu(a_0)>0\), \(\mu(b_0)>0\) and \[ \lambda_{\max}\left(\int_{a_0}^{b_0} Q(t)\Delta t\right)\geq \frac{1}{\mu(a_0)}+\frac{1}{\mu(b_0)}, \] where \(\lambda_{\max}\) stands for the greatest eigenvalue of the matrix indicated. Then (*) with \(P(t)\equiv I\) is oscillatory.
The general oscillation criteria presented in the paper are illustrated by a number of examples and corollaries.

MSC:
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
39A10 Additive difference equations
34B24 Sturm-Liouville theory
34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)
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