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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 $𝕋$ (which is supposed to be unbounded from above)

${\left[P\left(t\right){X}^{{\Delta }}\right]}^{{\Delta }}+Q\left(t\right){X}^{\sigma }=0,\phantom{\rule{2.em}{0ex}}\left(*\right)$

where $P,Q$ are symmetric $n×n$-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 $𝕋=ℝ$ and to the forward difference ${\Delta }$ if $𝕋=ℤ$. The forward jump operator is defined by $\sigma \left(t\right)=inf\left\{s\in 𝕋:s>t\right\}$ and the graininess by $\mu \left(t\right)=\sigma \left(t\right)-t$. One of the main results of the paper reads as follows:

Suppose that for every ${t}_{0}\in 𝕋$ there exist ${t}_{0}\le {a}_{0}<{b}_{0}$ such that $\mu \left({a}_{0}\right)>0$, $\mu \left({b}_{0}\right)>0$ and

${\lambda }_{max}\left({\int }_{{a}_{0}}^{{b}_{0}}Q\left(t\right){\Delta }t\right)\ge \frac{1}{\mu \left({a}_{0}\right)}+\frac{1}{\mu \left({b}_{0}\right)},$

where ${\lambda }_{max}$ stands for the greatest eigenvalue of the matrix indicated. Then (*) with $P\left(t\right)\equiv I$ is oscillatory.

The general oscillation criteria presented in the paper are illustrated by a number of examples and corollaries.

##### MSC:
 34C10 Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory 39A10 Additive difference equations 34B24 Sturm-Liouville theory 34B30 Special ODE (Mathieu, Hill, Bessel, etc.)
##### Keywords:
measure chains; time scales; Riccati equation; oscillation