Template-type: ReDif-Paper 1.0
Author-Name: Herings Jean-Jacques
Author-Name: Laan Gerard van der
Author-Name: Talman Dolf
Author-Name: Yang Zaifu
Author-workplace-name: METEOR
Title: 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 ƒ : P → P  such that for every x є P for which ƒ (x) ≠ x  there exists δ > 0 such that for all y, z  є B (x, δ) ∩ P it holds that (ƒ(y)-y)2 (ƒ(z)-z) ≤ 0, where B (x, δ) is the ball in Rⁿ  centered at x with radius δ . Then ƒ has a fixed point, i.e., there exists a point x* є P satisfying ƒ (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.
Keywords: mathematical economics and econometrics ; 
Series: Research Memoranda
Creation-Date: 2005
Number: 011
File-URL: http://digitalarchive.maastrichtuniversity.nl/fedora/objects/guid:9ca0d19d-35b2-46fe-b0fa-660aa118cf07/datastreams/ASSET1/content
File-Format: application/pdf
File-Size: 134456
Handle: RePEc:unm:umamet:2005011