Iteration methods for nonlinear problems

Abstract

Analogous methods have been used in practice, with apparent success, on nonlinear problems as well. For the most part, these have not been justified mathematically and this work is an attempt to fill this gap. In particular it is shown that the relaxation methods yield solutions to problems arising from the minimization of certain convex functions. In practice, these functions are obtained by approximating multiple integrals in a calculus of variations problem. It is shown that an approximate Plateau problem may be solved by a successive displacements method, or a method analogous to Liebmann's method. We at the same time obtain an extension of a free steering theorem for positive definite symmetric matrices given as Theorem 4 of [3], and results of Ostrowski [2].