A Fixed Point Theorem for Discontinuous Functions

Abstract

In this paper we prove the following fixed point theorem. Consider a non-empty bounded polyhedron P and a function Æ : P → P such that for every x є P for which Æ (x) ≠ x there exists δ > 0 such that for all y, z є B (x, δ) ∩ P it holds that (Æ(y)-y)2 (Æ(z)-z) ≤ 0, where B (x, δ) is the ball in Rⁿ centered at x with radius δ . Then Æ has a fixed point, i.e., there exists a point x* є P satisfying Æ (x*) = x* . The condition allows for various discontinuities and irregularities of the function. In case f is a continuous function, the condition is automatically satisfied and thus the Brouwer fixed point theorem is implied by the result. We illustrate that a function that satisfies the condition is not necessarily upper or lower semi-continuous. A game-theoretic application is also discussed.mathematical economics and econometrics ;