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Interval criteria for oscillation of linear Hamiltonian systems. (English) Zbl 1085.34521

The authors consider the linear matrix Hamiltonian system

${X}^{\text{'}}=A\left(t\right)X+B\left(t\right)Y,\phantom{\rule{1.em}{0ex}}{Y}^{\text{'}}=C\left(t\right)X-{A}^{*}\left(t\right)Y,\phantom{\rule{1.em}{0ex}}t\ge {t}_{0},$

where $X\left(t\right),Y\left(t\right),A\left(t\right),B\left(t\right)={B}^{*}\left(t\right)>0$ and $C\left(t\right)={C}^{*}\left(t\right)$ are $n×n$-matrices whose entries real-valued continuous functions. By employing the substitution $W\left(t\right)=a\left(t\right)\left[Y\left(t\right){X}^{-1}\left(t\right)+f\left(t\right){B}^{-1}\left(t\right)\right]$ and a fundamental matrix ${\Phi }\left(t\right)$ for the linear equation ${v}^{\text{'}}=A\left(t\right)v$, they show that $R\left(t\right)={{\Phi }}^{*}\left(t\right)W\left(t\right){\Phi }\left(t\right)$ solves a matrix Riccati equation. Based on this Riccati equation and the $H$-function averaging method, they establish some new interval oscillation criteria for the system above. Among earlier published papers on the subject are Q. Kong [Differ. Equ. Dyn. Systems, 8, 99-110 (2000; Zbl 0993.34034)]; Q. G. Yang [Ann. Pol. Math., 79, 185-198 (2002; Zbl 1118.34315)] and Q.-R. Wang [J. Math. Anal. Appl., 276 373–395 (2002; Zbl 1022.34032)].

##### MSC:
 34C10 Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory 37J99 Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems 34A30 Linear ODE and systems, general
##### Keywords:
Interval criteria; Oscillation; Hamiltonian systems