On Sturmian theory for second order systems

A generalization of the Sturm Comparison Theorem is given to nonselfadjoint second order linear systems. In addition, a theorem, involving the existence of a solution with strictly positive components, is proven. Counterexamples are given to show that the theorems are false without the stated assumptions. Introduction. Consider the vector differential equations (1) x" + A(t)x = 0 and (2) y" + B(t)y = O where A(t) = (a,,(t)) and B(t) -(b,(t)) are continuous n X n matrices. A number / is a conjugate point of a, a a,J(t) at some point t, t E [a, P] (where P is the first conjugate point of a relative to (1)), then the first conjugate point of a relative to (2) is less than the first conjugate point of a relative to (1). Later, in [4] it was shown that with the additional assumption that A(t) and B(t) be symmetric, the strict inequality b1_(t) > a,,(t) can be relaxed to hold only along the diagonal elements, i.e. b,,(t) > all(t). The purpose of our first theorem is to show that this result holds without the symmetry assumption on A(t) and B(t). Furthermore, we give a simple example to show that not only the comparison theorem established here but also those established in [2 and 4] are all false without the nonnegativity assumption on the elements of the coefficient matrices. This should eliminate any doubt about the importance of the nonnegativity assumption made in the earlier works [1-6]. In [1] it was shown that if (1) is disconjugate on [a, oc) and if A(t) is a symmetric matrix with nonnegative elements, which is irreducible at some point in [ a, oo), then (1) has a solution which vanishes at a and all of whose components are strictly positive on (a, oc). This result was later established [6] without the symmetry condition. In both proofs the existence of a nontrivial solution with nonnegative components followed independent of the irreducibility condition. In this paper, we Received by the editors June 21, 1982. 1980 Mathematics Suhject Classification. Primary 34A25; Secondary 34C10 Kev words anid phrases. Conjugate point, disconjugate, irreducible, symmetric, selfadjoint, solution. ")1983 American Mathematical Society 0002-9939/82/(XX)-0972/$02 ()(