A fixed point theorem for discontinuous functions
type: Item
instance of: scholarly article
title: A fixed point theorem for discontinuous functions
zbMATH Open document ID: 1135.91317
author: P. Jean-Jacques Herings
author: Zaifu Yang
author: Gerard van der Laan
author: Dolf Talman
published in: Operations Research Letters
publication date: 2008-05-29T00:00:00Z
review text: Let \(P\) be an non-empty polytope in the \(n\)-dimensional Euclidean space \(\mathbb{R}^n\) and \(f:P\to P\) be a function. A function \(f\) is locally gross direction preserving if for each \(x\in P\) for which \(f(x)\neq x\), there exists \(\delta>0\) such that for every \(y,z\in B(x,\delta)\cap P\), the function \(f\) satisfies \((f(y)-y)^T(f(z)-z)\geq 0\). This condition allows for various kinds od discontinuities and irregularities of the function \(f\). The main result of the present paper is the following theorem: Let \(P\) be an non-empty polytope in \(\mathbb{R}^n\) and the function \(f:P\to P\) satisfy the locally gross direction preserving property. Then \(f\) has a fixed point. Some applications of the result in the area of non-cooperative game theory are given.
reviewed by: Jarosław Górnicki
Mathematics Subject Classification ID: 91A12
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 65K10
Mathematics Subject Classification ID: 91A10
zbMATH DE Number: 5280662
zbMATH Keywords: fixed point
zbMATH Keywords: simplicial subdivision
zbMATH Keywords: discontinuity
zbMATH Keywords: discontinuous games
MaRDI profile type: Publication
OpenAlex ID: W2137534004
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DOI: 10.1016/J.ORL.2007.03.008
description: scientific article; zbMATH DE number 5280662