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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,Y ' =C(t)X-A * (t)Y,tt 0 ,

where X(t),Y(t),A(t),B(t)=B * (t)>0 and C(t)=C * (t) are n×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 Φ(t) for the linear equation v ' =A(t)v, they show that R(t)=Φ * (t)W(t)Φ(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 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)].

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