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)+mxy′(x)=f(x,y(x)),x∈(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)f(x,y) is that it not be a singular function of the independent variable xx; singularities in yy are easily avoided. Solutions are found in finite regions where ∂f/∂y⩾0∂f/∂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.