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Some oscillation criteria are given for the second order matrix differential system Y" + Q(t)Y = 0, where Y and Q are n x n real continuous matrix functions with Q(t) symmetric, t E [to, cc). These results improve oscillation criteria recently discovered by Erbe, Kong and Ruan by using a generalized Riccati transformation V(t) = a(t){Y'(t)Y-1(t) + f(t)I}, where I is the n x n identity matrix, f E C1 is a given function on [to, oo) and a(t) = exp{-2 f t f (s) ds}. Consider the second order linear differential system (1) Y" + Q(t)Y = O, t C [tooo), where Y and Q are n x n real continuous matrix functions with Q(t) symmetric. A solution Y(t) of (1) is said to be a nontrivial solution if det Y(t) :$ 0 for at least one t E [to, oo), and a nontrivial solution Y(t) of (1) is said to be prepared if (2) Y*(t)YI(t) (y*(t))Iy(t) 0_ , t E [tooo), where for any matrix A, the transpose of A is denoted by A*. System (1) is said to be oscillatory on [to, oo) in case the determinant of every nontrivial prepared solution vanishes on [T, oo) for each T > to. For matrix system (1), many authors have given some important simple oscillation criteria (see [1], [2], [3], [6]). We particularly mention the results of Erbe, Kong and Ruan [3] who proved the following theorem. Erbe, Kong and Ruan's Theorem. Let H(t, s) and h(t, s) be continuous on D = {(t,s): t > s > to} such that H(t,t) 0 0 for t > to and H(t,s) > 0 for t > s > to. We assume further that the partial derivative &H(t, s) = Hs(t, s) is nonpositive and continuous for t > s > to and h(t, s) is defined by Hs(t, s) = -h(t, s)[H(t, s)]1/2, (t, s) E D. Finally, we assume that 3) lo sup A1 [Jt (H(t's)Q(s)h2(ts)I) dsl o (3) sup H(tj to) A1 k0 ,)%S 4 ' =0, Received by the editors May 25, 1996. 1991 Mathematics Subject Classification. Primary 34C10.

A note on Kamenev type theorems for second order matrix differential systems