Conjugate Points for Fourth Order Linear Differential Equations

Abstract

Let $r_{i_1 \,i_2 \cdots i_k } = \infty $ mean that no nontrivial solution of $y^{( n )} + \sum\nolimits_{i = 0}^{n - 1} {p_i } ( x )y^{( i )} = 0$ has an $i_1 - i_2 - \cdots - i_k $ distribution of zeros. The main result is the following theorem. THEOREM 1. If$n = 4$and$r_{121} = \infty $, then$r_{31} = r_{31} = \infty $and$\eta _k ( t )$, the k-th right conjugate point oft, is achieved by an extremal solution with double zeros attand$\eta _k ( t )$and only simple zeros in$( {t,\eta _k ( t )} )$. Further, an example is given showing that $r_{31} = r_{31} = \infty $ does not imply $r_{121} = \infty $. This example also shows that it is possible to have $r_{31} = r_{31} = \infty $ and still have extremal solutions for $\eta _k ( t )$ with more than two double zeros.