An Inmrrmtion~ Journal computers & mathematics PERGAMON Computers and Mathematics with Applications 44 (2002) 1467-1477 www.elsevier.com/locate/camwa Oscillation of Linear Hamiltonian Systems FANWEI MENG AND YUANGONG SUN Department of Mathematics, Qufu Normal University Qufu, Shandong, 273165, P.R. China (Received and accepted February 2001) Abstract--In this paper, sufficient conditions have been obtained for the oscillation of a class of linear Hamiltonian systems. (~) 2002 Elsevier Science Ltd. All rights reserved. Keywords--Oscillation, Hamiltonian systems. 1. INTRODUCTION In this paper, we are concerned with oscillation of solutions of linear Hamiltonian system X' = A(t)X + U(t)Y, Y' = C(t)X - A*(t)Y, t _> to, (1.1) where X(t), Y(t), A(t), B(t), C(t) are n × n real continuous matrix functions such that C(t) is symmetric and B(t) is symmetric and positive definite. By M*, we mean the transpose of the matrix M. For any two solutions Xl(t),Yl(t) and X2(t),Y2(t) of (1.1), the "Wronskian" X~(t)Y2(t)- Y~(t)X2(t) is a constant matrix. In particular, for any solution X(t), Y(t) of (1.1), X*(t)Y(t) - Y*(t)X(t) is a constant matrix. The solution X(t), Y(t) of (1.1) is said to be conjoined if X*(t)Y(t) - Y*(t)X(t) = O. (1.2) A conjoined solution X(t), Y(t) of (1.1) is said to be a conjoined basis of (1.1) if the rank of the 2n x n-matrix (X(t); Y(t)) is n. A conjoined basis X(t), Y(t) of (1.1) is said to be oscillatory on [0, co), if det X(t) has arbitrarily large zeros; otherwise, X(t), Y(t) is called nonoscillatory. System (1.1) is said to be oscillatory if every conjoined basis of (1.1) is oscillatory. The linear Hamiltonian system associated with (1.1) is given by x' = A(t)x + B(t)y, y' = C(t)x - A*(t)y, t > O. (1.3) If (X(t), Y(t)) is a solution of (1.1), then its columns are solutions of (1.3). The points to, tl C [0, +oc), to ~ tl, are said to be mutually conjugate relative to (1.3) if there exists a solution z(t) = This research was supported by the NSF of China and Shandong Province. 0898-1221/02/$ - see front matter (~) 2002 Elsevier Science Ltd. All rights reserved. Typeset by .AA/~S-2~X PII: S0898-1221 (02)00271-7