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 = λAu, 0<λ<+∞, u ∈ P \ {θ}, where λ is a parameter and P a cone of Banach space E. Under certain conditions on the operator A we find λ∗ such that the operator equation has at least two solutions for 0 < λ<λ∗, at least one solution for λ = λ∗ and no solutions for λ >λ∗. 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.  2004 Elsevier Inc. All rights reserved.