Introduction

Abstract

The appearance of Brouwer’s fixed point theorem in 1912 and its generalizations have resulted in a great breakthrough of a number of scientific research areas. Brouwer’s theorem states that every continuous function from a compact and convex nonempty set into itself has a fixed point, i.e., an element which is mapped by the function into itself. However, the nonconstructive proofs of these fixed point theorems limited their further applications to real world problems. In 1967 Scarf gave the first elegant constructive proof of Brouwer’s fixed point theorem on the unit simplex. The unit simplex is the subset of the Euclidean space in which all components of every point are nonnegative and sum up to one. From then on a significant development in computing fixed points was initiated.