A note on singular nonlinear boundary value problems

In this paper we study the existence and multiplicity of solutions of the following operator equation in Banach space E: u = lambdaAu, 0 < lambda < +infinity, u is an element of P \ {theta}, where lambda is a parameter and P a cone of Banach space E. Under certain conditions on the operator A we find lambda* such that the operator equation has at least two solutions for 0 < lambda < lambda*, at least one solution for lambda = lambda* and no solutions for lambda > lambda*. As an application, we investigate the existence and multiplicity of positive solutions of a singular second order boundary value problem. In addition, we briefly outline an application of our results which simplifies a previous theorem that appeared in the literature. (C) 2004 Elsevier Inc. All rights reserved.