Constructive existence theorems for problems of Thomas‐Fermi Type

Abstract

A minimal positive solution of the Thomas‐Fermi problem ẅ = λt−1/2 w3/2, w(0) = 1, w(1) = w(1) is shown to exist for each λ > 0. It is proved that all positive solutions, for a given value of λ, are strictly ordered and that the minimal positive solution wλ is a decreasing function of λ. Upper and lower analytic bounds for wλ are given and these bounds are shown to initiate sequences of Picard and Newton iterates which converge monotonically to wλ. A comparative analysis of the efficiency of the iteration schemes is presented. The methods used are of a general nature and can be applied to a variety of nonlinear boundary value problems of convex type [14].