On a conjecture for oscillation of second-order ordinary differential systems

We present here some results pertaining to the oscillatory behavior at infinity of the vector differential equation y"+Q(t)y=O, tE[O,a4), where Q(t) is a real continuous n x n symmetric matrix function. It has been conjectured (cf., e.g. [6]) that the criterion lim Ai{f Q(s) ds =oo where AI(-) denotes the maximum eigenvalue of the matrix concerned, implies oscillation. We show that this is so under the tacit assumption lim inf t-' trt Q(s) ds} > 00 where tr(*) represents the trace of the matrix under consideration. 1. We will be mainly concerned with the differential equation Y" + Q(t)y = 0, t E J, (1.1) for a vectory where Q(t) = Q*(t) is a real n x n symmetric matrix continuous on some interval J. Points a #1,8 in J will be called (mutually) conjugate relative to (1.1) provided there exists a solution y(t) = 0 of (1.1) which vanishes at a and fi. The equation (1.1) will be called disconjugate on [a, b] if there are no conjugate points therein, i.e., if every nontrivial solution vanishes at most once in [a, b]. When J = [0, oo) it will be termed oscillatory at oo if for every a > 0 there exists b > a such that (1.1) fails to be disconjugate on [a, b]. It will be nonoscillatory otherwise. Associated with (1.1) is the matrix differential system Y"+Q(t)Y=O, t E J, (1.2) for a matrix Y where Q is as in (1.1). A solution Y(t) of (1.2) is said to be nontrivial if det Y(t) 7# 0 for at least one t E J. A simple differentiation shows that whenever Y is a solution of (1.2) Y*(t) Y'(t) Y*'(t) Y(t) = K (1.3) for t E J where K is a constant matrix. A nontrivial solution Y will be called prepared or self-conjugate (cf. [4]) if K = 0 in (1.3), i.e., Y* Y' = (Y* Y')*. Received by the editors August 5, 1980. 1980 Mathematics Subject Classification. Primary 34A30, 34C10.