The existence of conjugate points for selfadjoint differential equations of even order

Abstract:

This paper presents sufficient conditions on the coefficents of ${L_{2n}}y = \Sigma _{k = 0}^n{( - 1)^{n - k}}{({p_k}{y^{(n - k)}})^{(n - k)}}$ which insure that ${L_{2n}}y = 0$ has conjugate points $\eta (a)$ for all $a > 0$. The main theorem implies that ${( - 1)^n}{y^{(2n)}} + py = 0$ has conjugate points $\eta (a)$ for all $a > 0$ when ${\smallint ^\infty }{x^\alpha }p(x)dx = - \infty$ for some $\alpha < 2n - 1$ with no sign restrictions on $p(x)$.