Nonlinear singular BVP of limit circle type and the presence of reverse-ordered upper and lower solutions (Q763010)

Summary: We consider the following class of nonlinear singular differential equations \[ -(p(x)y'(x))' + q(x)f(x, y(x), p(x) y'(x)) = 0, ~0 < x < 1, \] 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 \((v_0 \leq u_0)\) or the other \((u_0 \leq v_0)\).