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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
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  1. Shoven,John B. & Whalley,John, 1992. "Applying General Equilibrium," Cambridge Books, Cambridge University Press, number 9780521319867.
  2. 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, Tilburg University, School of Economics and Management.
  3. 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.
  4. P. Jean-Jacques Herings, 2000. "Two simple proofs of the feasibility of the linear tracing procedure," Economic Theory, Springer, vol. 15(2), pages 485-490.
  5. Gérard Debreu (ed.), 1996. "General Equilibrium Theory," Books, Edward Elgar, volume 0, number 548, March.
  6. Drew Fudenberg & Jean Tirole, 1991. "Game Theory," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262061414, June.
  7. Talman, A.J.J. & Yamamoto, Y., 1989. "A simplicial algorithm for stationary point problems on polytopes," Other publications TiSEM 0d6b2de0-17c0-4d5e-963f-5, Tilburg University, School of Economics and Management.
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