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Interval criteria for oscillation of linear Hamiltonian systems. (English) Zbl 1085.34521
The authors consider the linear matrix Hamiltonian system $$X'=A(t)X+B(t)Y,\quad Y'=C(t)X-A^*(t)Y,\quad t\ge t_0,$$ where $X(t), Y(t), A(t), B(t)=B^*(t)>0$ and $C(t)=C^*(t)$ are $n\times n$-matrices whose entries real-valued continuous functions. By employing the substitution $W(t)=a(t)[Y(t)X^{-1}(t)+f(t)B^{-1}(t)]$ and a fundamental matrix $\Phi(t)$ for the linear equation $v'=A(t)v$, they show that $R(t)=\Phi^*(t)W(t)\Phi(t)$ 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 {\it Q. Kong} [Differ. Equ. Dyn. Systems, 8, 99-110 (2000; Zbl 0993.34034)]; {\it Q. G. Yang} [Ann. Pol. Math., 79, 185-198 (2002; Zbl 1118.34315)] and {\it Q.-R. Wang} [J. Math. Anal. Appl., 276 373--395 (2002; Zbl 1022.34032)].

34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
37J99Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
34A30Linear ODE and systems, general
Full Text: DOI
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