A Fixed Point Theorem for Discontinuous Functions

P. Jean-Jacques Herings (), Gerard van der Laan, Dolf Talman () and Zaifu Yang
Additional contact information
Zaifu Yang: METEOR

No 11, Research Memoranda from Maastricht : METEOR, Maastricht Research School of Economics of Technology and Organization

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 (search for similar items in EconPapers)
Date: 2005
View list of references View citations in EconPapers

Downloads: (external link)
http://edocs.ub.unimaas.nl/loader/file.asp?id=1046 (application/pdf)

Related works:
Working Paper: A fixed point theorem for discontinuous functions (2005)
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: http://EconPapers.repec.org/RePEc:dgr:umamet:2005011

Access Statistics for this paper

More papers in Research Memoranda from Maastricht : METEOR, Maastricht Research School of Economics of Technology and Organization
Series data maintained by Willy Villevoye ().