Existence results and approximation methods for functional ordinary differential equations with singular diffusion-type differential operators

We derive sufficient conditions for the existence of extremal solutions for a second-order singular functional differential equation subject to initial data. The type of equations that we study here can be regarded as stationary and one-dimensional models for diffusion processes in which the diffusion coefficient is not a constant. We have also tried to relax the regularity assumptions as far as possible, in order to extend the applicability of our result. We present an example in which the extremal solutions are approximated.