Oscillation of linear Hamiltonian systems. (Q1416357)

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 \textit{G. J. Butler, L. H. Erbe} and \textit{A. B. Mingarelli} [Trans. Am. Math. Soc. 303, 263--282 (1987; Zbl 0648.34031)].