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Short note on oscillation of matrix Hamiltonian systems. (English) Zbl 1062.34032

The authors study the linear matrix system

${U}^{\text{'}}=A\left(x\right)U+B\left(x\right)V,\phantom{\rule{1.em}{0ex}}{V}^{\text{'}}=C\left(x\right)U+\left(\mu \left(x\right)I-{A}^{*}\left(x\right)\right)V,\phantom{\rule{2.em}{0ex}}\left(1\right)$

where $A\left(x\right)$, $B\left(x\right)$, $C\left(x\right)$ are continuous real-valued $n×n$-matrix functions on the interval $J=\left[a,\infty \right)$ and $\mu \left(x\right)$ is a real-valued continuous function on $J$. The matrices $B$ and $C$ are supposed to be symmetric and $B$ is supposed to be either positive definite or negative definite. The authors transform the system by a transformation preserving oscillatory properties into a Hamiltonian system and use an oscillation criterion due to I. Kumari and S. Umamaheswaram [J. Differ. Equations 165, No.1, 174-198 (2000; Zbl 0970.34025)] to derive a new oscillation criterion for system (1).

##### MSC:
 34C10 Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory 34A30 Linear ODE and systems, general
##### Keywords:
oscillation; Hamiltonian system; linear system