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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 $\bbfT$ (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 {\it M. Bohner} and {\it 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 $\bbfT=\bbfR$ and to the forward difference $\Delta $ if $\bbfT=\bbfZ$. The forward jump operator is defined by $\sigma(t)=\inf\{s\in \bbfT: s>t\}$ and the graininess by $\mu(t)=\sigma(t)-t$. One of the main results of the paper reads as follows: Suppose that for every $t_0\in \bbfT$ 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.

34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
39A10Additive difference equations
34B24Sturm-Liouville theory
34B30Special ODE (Mathieu, Hill, Bessel, etc.)
Full Text: DOI
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