On the Continuity of Conjugate Point Functions

Abstract

With the kth conjugate point of t$\eta _k (t)$, defined for a given nth order linear differential equation as the minimum number r greater than t such that there exists a nontrivial solution of the differential equation having zeros at t and r and at least $n + k - 1$ zeros on the interval $[ t,r ]$, it is well known that $\eta _k (t)$ is a strictly increasing continuous function of t when $k = 1$; however, this question remains open when $k > 1$. For $k\geqq 1$, the main theorems of this paper give conditions which guarantee that $\eta _k (t)$ is strictly increasing and continuous and that $\eta _k (t)$ depends continuously on the coefficients in the given differential equation. The relationship between these results and previous results for fourth order differential equations is indicated.