A Fixed Point Theorem for Discontinuous Functions
AbstractIn 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.
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Bibliographic InfoPaper provided by Maastricht : METEOR, Maastricht Research School of Economics of Technology and Organization in its series Research Memoranda with number 011.
Date of creation: 2005
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Other versions of this item:
- Herings, P.J.J. & Laan, G. van der & Talman, A.J.J. & Yang, Z.F., 2008. "A fixed point theorem for discontinuous functions," Open Access publications from Tilburg University urn:nbn:nl:ui:12-357915, Tilburg University.
- HERINGS, P. Jean-Jacques & van der LAAN, Gerard & TALMAN, Dolf & YANG, Zaifu, . "A fixed point theorem for discontinuous functions," CORE Discussion Papers RP -2154, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
- Jean-Jacques Herings & Gerard van der Laan & Dolf Talman & Zaifu Yang, 2004. "A Fixed Point Theorem for Discontinuous Functions," Tinbergen Institute Discussion Papers 05-004/1, Tinbergen Institute.
- Herings, P.J.J. & Laan, G. van der & Talman, A.J.J. & Yang, Z.F., 2005. "A Fixed Point Theorem for Discontinuous Functions," Discussion Paper 2005-4, Tilburg University, Center for Economic Research.
- Herings,Jean-Jacques & Laan,Gerard,van der & Talman,Dolf & Yang,Zaifu, 2005. "A Fixed Point Theorem for Discontinuous Functions," Research Memorandum 011, Maastricht University, Maastricht Research School of Economics of Technology and Organization (METEOR).
- C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
- C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques
This paper has been announced in the following NEP Reports:
- NEP-ALL-2005-04-09 (All new papers)
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