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A fixed point theorem for discontinuous functions

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Author Info

  • Herings, P.J.J.

    (Tilburg University)

  • Laan, G. van der
  • Talman, A.J.J.

    (Tilburg University)

  • Yang, Z.F.

    (Tilburg University)

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.

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Bibliographic Info

Paper provided by Tilburg University in its series Open Access publications from Tilburg University with number urn:nbn:nl:ui:12-357915.

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Date of creation: 2008
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Publication status: Published in Operations Research Letters (2008) v.36, p.89-93
Handle: RePEc:ner:tilbur:urn:nbn:nl:ui:12-357915

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Web page: http://www.tilburguniversity.edu/

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  1. Philip J. Reny, 1999. "On the Existence of Pure and Mixed Strategy Nash Equilibria in Discontinuous Games," Econometrica, Econometric Society, vol. 67(5), pages 1029-1056, September.
  2. Shoven,John B. & Whalley,John, 1992. "Applying General Equilibrium," Cambridge Books, Cambridge University Press, number 9780521319867, October.
  3. Herings, P.J.J., 1997. "Two Simple Proofs of the Feasibility of the Linear Tracing Procedure," Discussion Paper 1997-77, Tilburg University, Center for Economic Research.
  4. Talman, A.J.J. & Yamamoto, Y., 1989. "A simplicial algorithm for stationary point problems on polytopes," Open Access publications from Tilburg University urn:nbn:nl:ui:12-153111, Tilburg University.
  5. Carolyn Pitchik, 1981. "Equilibria of a Two-Person Non-Zero Sum Noisy Game of Timing," Cowles Foundation Discussion Papers 579, Cowles Foundation for Research in Economics, Yale University.
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Cited by:
  1. repec:hal:cesptp:halshs-00265464 is not listed on IDEAS
  2. Philippe Bich, 2006. "Some fixed point theorems for discontinuous mappings," Cahiers de la Maison des Sciences Economiques b06066, Université Panthéon-Sorbonne (Paris 1).

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