Oscillation theorems for systems of linear differential equations

Abstract

where A = A(x) is a continuous n x n matrix on an x-interval R, and y is an ndimensional column vector. We shall assume that the elements of A are real, and we shall consider only real solution vectors of (1.1). This is not an essential restriction since, in the complex case, (1.1) can be replaced by an equivalent real system with a 2n x 2n coefficient matrix. We shall say that a nontrivial solution vector y = (Yi, . . ., Yn) of (1.1) is oscillatory on R if each of its components takes the value zero at some point of R, i.e., Yk(Xk)=O, Xk c R, k= 1, . . ., n. The system (1.1) itself will be said to be oscillatory if it possesses at least one oscillatory solution vector. If there is no such solution vector, i.e., if every nontrivial solution vector has a component which does not vanish on R, the system will be said to be nonoscillatory on R. In a recent paper by B. Schwarz [16], systems with the latter property are called disconjugate , rather than nonoscillatory , and a word of justification for this change of terminology is in order. The term disconjugate, as introduced by Wintner [21], refers to the absence of a conjugate point in the sense of Jacobi, and thus originally applied only to selfadjoint equations and systems [3], [4], [14], [15], [19], [20]. However, this concept generalizes in a natural way to general nth order differential equations [1], [8], [9], [10], [13], [17], [18] and thus also to systems which are equivalent to such equations. In all these cases, the right conjugate point rj(xo) of x0 (-q(x0)>x x) is a continuous function of xo, and the left conjugate point of -q(xo) coincides with xo [17], [18]. In the case of a system which can be reduced to an nth order equation, rj(xo) can be defined in the following way: there exists a solution vector of (1.1) such that every component of y vanishes either at xo or at -q(xo), and -q(xo) is the smallest number with this property. It can then be shown that -q(xo) = inf b, where b is such that the system is oscillatory in [xo, b) [9], [13], [17]. In the case of a general system (1.1), the conjugate point may be defined in the same way, but it is in general not true that -q(xo) = 6(xo), where 6(xo) = inf b, and