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 ÃÂ : P â P such that for every x Ñ P for which ÃÂ (x) â x there exists Î´ > 0 such that for all y, z Ñ B (x, Î´) â© P it holds that (ÃÂ(y)-y)2 (ÃÂ(z)-z) â¤ 0, where B (x, Î´) is the ball in Râ¿ centered at x with radius Î´ . Then ÃÂ has a fixed point, i.e., there exists a point x* Ñ P satisfying ÃÂ (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 Tilburg University in its series Open Access publications from Tilburg University with number urn:nbn:nl:ui:12-357915.
Date of creation: 2008
Date of revision:
Publication status: Published in Operations Research Letters (2008) v.36, p.89-93
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Web page: http://www.tilburguniversity.edu/
Other versions of this item:
- 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).
- 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, 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).
- 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
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