Singular non-linear two-point boundary value problems: Existence and uniqueness

A general approach is presented for proving existence and uniqueness of solutions to the singular boundary value problem y ''(x) + m/xy '(x) = f (x, y(x)), x is an element of (0, 1], y'(0) = 0, Ay(1) + By'(1) = C, A > 0, B, C >= 0. The proof is constructive in nature, and could be used for numerical generation of the solution. The only restriction placed on f (x, y) is that it not be a singular function of the independent variable x; singularities in y are easily avoided. Solutions are found in finite regions where partial derivative f/partial derivative y >= 0, using an integral equation whose Green's function contains an adjustable parameter that secures convergence of the Picard iterative sequence. Methods based on the theory are developed and applied to a set of problems that have appeared previously in published works. Published by Elsevier Ltd