An extension of Brouwer's fixed point theorem allowing discontinuities
type: Item
instance of: scholarly article
title: An extension of Brouwer's fixed point theorem allowing discontinuities
zbMATH Open document ID: 1074.47028
author: Philippe Bich
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2004-08-20T00:00:00Z
review text: Brouwer's fixed point theorem states that every continuous mapping \(f: \mathcal{B} \rightarrow \mathcal{B},\) where \(\mathcal{B}\) is a closed ball of \(\mathbb{R}^n,\) has a fixed point. This paper extends this famous theorem to the situation where discontinuities of the single-valued mapping \(f\) are permitted. In economics, a class of problems that takes the incompleteness of financial markets into account has recently emerged. The utility of the results in this paper is that it may be applied to explain discontinuites in economic situations arising from markets that are not complete. In particular, the authors consider an equilibrium existence problem in economics that involves a fixed point of mappings \(f(x,\text{span}~ V(x)),\) where \(V(x)\) is a matrix specifying the returns on financial assets.
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 47N10
Mathematics Subject Classification ID: 91B50
zbMATH DE Number: 2093950
zbMATH Keywords: Brouwer's fixed point theorem
zbMATH Keywords: discontinuities
zbMATH Keywords: incompleteness of financial markets
zbMATH Keywords: equilibrium existence
MaRDI profile type: Publication
full work available at URL: j.crma.2004.03.001
OpenAlex ID: W2571335432
cites work: Q5588318
cites work: Equilibrium in incomplete markets. I: A basic model of generic existence
cites work: A geometric approach to a class of equilibrium existence theorems
cites work: Handbook of mathematical economics. Volume IV
cites work: Characteristic Classes. (AM-76)
DOI: 10.1016/J.CRMA.2004.03.001
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description: scientific article; zbMATH DE number 2093950