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Oscillation criteria for linear matrix Hamiltonian systems. (English) Zbl 0970.34025

The authors give some oscillation criteria for the linear matrix Hamiltonian system:

${U}^{\text{'}}=A\left(x\right)U+B\left(x\right)V,\phantom{\rule{4pt}{0ex}}{V}^{\text{'}}=C\left(x\right)U-{A}^{*}\left(x\right)V,\phantom{\rule{2.em}{0ex}}\left(*\right)$

where $A\left(x\right)$, $B\left(x\right)={B}^{*}\left(x\right)>0$ and $C\left(x\right)={C}^{*}\left(x\right)$ are real continuous $n×n$-matrix functions on the interval $\left[a,\infty \right)$.

The results given stand for extensions to the systems of the form (*), of the following oscillation criteria earlier derived by other authors: theorems 1 and 2 for the system (**) ${Y}^{\text{'}\text{'}}+Q\left(x\right)Y=0$ due to G. J. Etgen and J. F. Pawlowski [Pacific J. Math. 66, 99-110 (1976; Zbl 0355.34017)] and theorems 1-7 of L. H. Erbe, Q. Kong and S. Ruan [Proc. Am. Math. Soc. 117, No. 4, 957-962 (1993; Zbl 0777.34024)] for selfadjoint systems (***) ${\left(P\left(x\right){U}^{\text{'}}\right)}^{\text{'}}+Q\left(x\right)U=0$ as well as the theorem 1 for (**) given by F. Meng, J. Wang and Z. Zheng [Proc. Am. Math. Soc. 126, No. 2, 391-395 (1998; Zbl 0891.34037)]. Finally, the authors present a set of six examples illustrating the established theorems.

##### MSC:
 34C10 Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory 34A30 Linear ODE and systems, general