Nonlinear Singular BVP of Limit Circle Type and the Presence of Reverse-Ordered Upper and Lower Solutions

Abstract

We consider the following class of nonlinear singular differential equation −(𝑝(𝑥)𝑦′(𝑥))′+𝑞(𝑥)𝑓(𝑥,𝑦(𝑥),𝑝(𝑥)𝑦′(𝑥))=0,0<𝑥<1 subject to the Neumann boundary condition 𝑦′(0)=𝑦′(1)=0. Conditions on 𝑝(𝑥) and 𝑞(𝑥) ensure that 𝑥=0 is a singular point of limit circle type. A simple approximation scheme which is iterative in nature is considered. The initial iterates are upper and lower solutions which can be ordered in one way (𝑣0≤𝑢0) or the other (𝑢0≤𝑣0).