Application of a Fixed Point Search Algorithm to Nonlinear Boundary Value Problems Having Several Solutions

Abstract

A fixed point search algorithm is outlined and utilized to approximate the solutions of finite dimensional analogues of quasilinear elliptic and ordinary differential equation boundary value problems having several solutions. Two kinds of finite dimensional analogues are considered: finite differences and finite orthogonal expansions. In either case the finite dimensional analogue may be recast as a finite dimensional fixed point problem. If a mapping has several fixed points, the problem of approximating them by iterative methods is often severely complicated by such difficulties as locating appropriate starting points and magnetic properties of the fixed points. The search type algorithm used here is generally impervious to all such complications. Several boundary value problems having a finite number of isolated solutions are treated numerically and it is found that for sufficiently fine meshes, the search algorithm applied to the finite difference analogue yields approximations to all of the solutions. On the other hand, the search algorithm applied to the finite Fourier expansion analogue has not always yielded approximations to all of the solutions of a boundary value problem.