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Oscillation of linear Hamiltonian systems. (English) Zbl 1047.34030
The authors investigate oscillatory properties of the linear Hamiltonian system $$ x'=A(t)x+B(t)u,\quad u'=C(t)x-A^T(t)u, \tag{*} $$ where $A,B,C$ are $n\times n$-matrices with continuous entries for $t\in [t_0,\infty)$, $B,C$ are symmetric and the matrix $B$ is supposed to be positive definite. Using a transformation which preserves the oscillatory nature of transformed systems, system (*) is transformed into the “diagonal off” system $y'=\widetilde B(t)z$, $z'=-\widetilde C(t)y$ with $\widetilde B$ positive definite, hence this system is equivalent to the second-order vector-matrix Sturm-Liouville differential equation $$ (\widetilde B^{-1}(t)y')'+\widetilde C(t)y=0. \tag{**} $$ The main result of the paper is formulated under the assumption that the matrix $\widetilde B(t)-I$ is nonnegative definite ($I$ denotes the identity matrix). Under this assumption, system (**) is a Sturmian majorant of the system $y''+\widetilde C(t)y=0$. So, oscillation of the last system implies oscillation of (**) and hence, in turn, also of (*). From this point of view, the results of the paper are very close to those presented in the paper of {\it G. J. Butler, L. H. Erbe} and {\it A. B. Mingarelli} [Trans. Am. Math. Soc. 303, 263--282 (1987; Zbl 0648.34031)].

34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
34A30Linear ODE and systems, general
Full Text: DOI
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