Topological methods for some boundary value problems

We establish existence of solutions for a finite difference approximation to y″ = f(x,y,y′) on [0, 1], subject to nonlinear two-point Sturm-Liouville boundary conditions of the form giy′(i), y′(i)) = 0, i = 0, 1, assuming f satisfies one-sided growth bounds with respect to y′.