Multiple positive solutions of conjugate boundary value problems with singularities

The existence of one or several positive solutions for some nth order differential equations with conjugate boundary conditions is obtained. The approach is to employ the well-known results on the existence of positive solutions for Hammerstein integral equations obtained recently by the author. This avoids utilizing the theory of fixed point index for compact maps defined on cones directly. New properties on the kernels corresponding to the boundary value problems are provided and then employed to prove new properties of nonzero positive solutions for these boundary value problems and new inequalities on functions satisfying the conjugate boundary conditions. Our results improve and generalize many recent results. (C) 2002 Elsevier Inc. All rights reserved.