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.)

Zbl 0978.39001
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References:

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