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

A simple generalization of Tarski´s fixed point theorem shows that, if f is an increasing mapping from Rⁿ ₊ into itself and ¿(¿) = {x ¿ Rⁿ ₊ x ¿ f(x) + ¿ is bounded for any given ¿ ¿ Rⁿ ₊, there is a point x* in Rⁿ ₊ 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 Rⁿ 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 Rⁿ ₊ 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 Zⁿ ₊ x {0}, the method can be applied to compute fixed points of an increasing mapping from Zⁿ ₊ into itself.