The Existence of Conjugate Points for Selfadjoint Differential Equations of Even Order

This paper presents sufficient conditions on the coefficents of <!-- MATH ${L_{2n}}y = \Sigma _{k = 0}^n{( - 1)^{n - k}}{({p_k}{y^{(n - k)}})^{(n - k)}}$ --> which insure that <!-- MATH ${L_{2n}}y = 0$ --> has conjugate points for all 0$">. The main theorem implies that <!-- MATH ${( - 1)^n}{y^{(2n)}} + py = 0$ --> has conjugate points for all 0$"> when <!-- MATH ${\smallint ^\infty }{x^\alpha }p(x)dx = - \infty$ --> for some <!-- MATH $\alpha < 2n - 1$ --> <img width="101" height="37" align="MIDDLE" border="0" src="images/img13.gif" alt="$ \alpha < 2n - 1$"> with no sign restrictions on .