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

Abstract

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