Computing Equilibria and Fixed Points pp 217-237 | Cite as

# The Computation of Antipodal Fixed Points

## Abstract

In this chapter we consider the problem of finding zero points of a continuous function *f* from the *n*-dimensional symmetric cube *C* ^{ n } into ℝ^{ n }.It will be shown in a constructive way that there exists a zero point of *f* in *C* ^{ n } if for every ∈ *bd*(*C* ^{ n }), *f* (−*x*) ≠ *α* *f* (−*x*) for all *α* > 0. It will be argued that this result is so strong that it implies several powerful fixed point theorems, including Borsuk-Ulam’s theorem and Brouwer’s fixed point theorem. We will introduce two algorithms to compute zero points of the continuous function. One is an integer labeling algorithm and the other a vector labeling algorithm. The integer labeling algorithm will lead to a constructive proof for Tucker’s theorem. The algorithm we introduce here is a modification of the algorithms developed by Todd and Wright [1980], Freund and Todd [1981], and van der Laan [1984] in the sense that the modified algorithm can start with an arbitrarily chosen point in *C* ^{ n }, whereas the existing methods can only start with the point 0^{ n }. The modified algorithm will be built upon a new triangulation of *C* ^{ n }, called the *AS*-triangulation.

## Keywords

Fixed Point Theorem Label Function Constructive Proof Vector Label Tucker Theorem## Preview

Unable to display preview. Download preview PDF.