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New Kamenev type oscillation criteria for linear matrix Hamiltonian systems. (English) Zbl 1060.34018
Summary: Some new Kamenev-type criteria are obtained for the oscillation of the linear matrix Hamiltonian system \[ X'=A(t)X+ B(t)Y,\quad Y'= C(t)X- A^*(t)Y, \] under the hypothesis: \(A(t)\), \(B(t)= B^*(t)> 0\) and \(C(t)= C^*(t)\) are real continuous \(n\times n\)-matrix functions on the interval \([t_0,\infty)\), \(t_0>-\infty\). Our results are different from most known ones in the sense that they are given in the form of \(\lim_{t\to\infty}\sup g[\cdot]>\text{const.}\), rather than in the form of \(\lim_{t\to\infty}\sup \lambda_1[\cdot]= \infty\), where \(g\) is a positive linear functional on the linear space of \(n\times n\)-matrices with real entries. Consequently, our results improve some previous results to some extent, which can be seen by the examples given at the end of this paper.

MSC:
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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