A homotopy-like simplicial method for approximating fixed points of an increasing mapping from R&lt;sup&gt;n&lt;/sup&gt;&lt;inf&gt;&amp;#x002B;&lt;/inf&gt; into itself

Chuangyin Dang

doi:10.1109/icca.2009.5410219

Abstract

A simple generalization of Tarski's fixed point theorem shows that, if f is an increasing mapping from R n + into itself and ¿(¿) = {x ¿ R n + x ¿ f(x) + ¿ is bounded for any given ¿ ¿ R n + , there is a point x* in R n + such that f(x*) = x*, which is a fixed point of f and has many applications in economic analysis. However, it remains a challenging problem to approximate fixed points of such a mapping. To overcome this difficulty, we develop a homotopy-like simplicial method in this paper by applying a discrete increasing mapping, an integer labeling rule and a tri-angulation of R n X [0,1] with a mesh size of ¿ > 0. The method consists of two phases, one of which forms an (n + 1)-dimensional pivoting procedure and the other an n-dimensional pivoting procedure. Starting from an arbitrary point of R n + X {0}, the method interchanges from one phase to the other, if necessary, and follows a finite simplicial path that leads to an approximate fixed point y* satisfying that ||f(y* - y*|| ¿ ¿. If the accuracy is not good enough, the mesh size ¿ of the tri-angulation can be refined and the method can be restarted from y*. Furthermore, by letting (5=1 and the starting point be a point of Z n + x {0}, the method can be applied to compute fixed points of an increasing mapping from Z n + into itself.

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