A homotopy-like simplicial method for approximating fixed points of an increasing mapping from R+ n into itself

A simple generalization of Tarski's fixed point theorem shows that, if f is an increasing mapping from R₊ ⁿ into itself and Δ(γ) = {x ∈ R₊ ⁿ | x ≤ f(x) + γ} is bounded for any given γ ∈ 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 triangulation of Rⁿ x [0, 1] with a mesh size of δ > 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*∥ ≤ δ. If the accuracy is not good enough, the mesh size δ of the triangulation can be refined and the method can be restarted from y*. Furthermore, by letting δ = 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. ©2009 IEEE.