Interval criteria for oscillation of certain matrix differential systems (Q1856986)

Interval criteria for the oscillation of the second-order matrix differential system of the form \[ [P(t) Y']'+ Q(t) Y= 0,\tag{\(*\)} \] where \(t\geq t_0\) and \(Y(t)\), \(P(t)\) and \(Q(t)\) are real \(n\times n\)-matrix functions with \(P(t)\), \(Q(t)\) symmetric on \([t_0, \infty)\), \(P(t)\) positive definite a.e. on \([t_0,\infty)\) and \(P^{-1}\), \(Q\in L_{\text{loc}}([t_0, \infty),M)\), \(M\) being the real linear space of all real \(n\times n\)-matrices. These criteria seem to improve some recent oscillation results due to \textit{R. Zhuang} [Acta Math. Sin. 44, No. 6, 1037--1044 (2001; Zbl 1027.34042)]. The proposed criteria are based only on the behavior of system \((*)\) (or \(P\) and \(Q\)) and only on a sequence of subintervals of \([t_0,\infty)\) rather than on the whole interval \([t_0,\infty)\) as in other works [see \textit{Q. Kong}, Differ. Equ. Dyn. Syst. 8, No. 2, 99--110 (2000; Zbl 0993.34034); \textit{Q. Wang}, Arch. Math. 76, No. 5, 385--390 (2001; Zbl 0989.34024) and see also the related paper by \textit{F. Meng}, \textit{J. Wang} and \textit{Z. Zheng}, Proc. Am. Math. Soc. 126, No. 2, 391--395 (1998; Zbl 0891.34037)]. The generalized Riccati technique and an averaging technique are employed in the paper under review and the results are presented in the form of a high degree of generality. Several examples illustrating the above oscillation criteria are finally given.