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A fixed point theorem for discontinuous functions

  • HERINGS, P. Jean-Jacques
  • van der LAAN, Gerard
  • TALMAN, Dolf
  • YANG, Zaifu

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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Paper provided by Université catholique de Louvain, Center for Operations Research and Econometrics (CORE) in its series CORE Discussion Papers RP with number -2154.

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Handle: RePEc:cor:louvrp:-2154
Note: In : Operations Research Letters, 36, 89-93, 2008
Contact details of provider: Postal: Voie du Roman Pays 34, 1348 Louvain-la-Neuve (Belgium)
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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. repec:dgr:kubcen:198952 is not listed on IDEAS
  3. repec:cup:cbooks:9780521266550 is not listed on IDEAS
  4. Talman, A.J.J. & Dai, Y. & van der Laan, G. & Yamamoto, Y., 1991. "A simplicial algorithm for the nonlinear stationary point problem on an unbounded polyhedron," Other publications TiSEM d961dc7e-e203-4709-8a75-5, School of Economics and Management.
  5. Talman, A.J.J. & Yamamoto, Y., 1989. "A simplicial algorithm for stationary point problems on polytopes," Other publications TiSEM 0d6b2de0-17c0-4d5e-963f-5, School of Economics and Management.
  6. 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.
  7. repec:cup:cbooks:9780521319867 is not listed on IDEAS
  8. Drew Fudenberg & Jean Tirole, 1991. "Game Theory," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262061414, June.
  9. repec:ner:tilbur:urn:nbn:nl:ui:12-153111 is not listed on IDEAS
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