A fixed point theorem for discontinuous functions
In 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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|Note:||In : Operations Research Letters, 36, 89-93, 2008|
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- Drew Fudenberg & Jean Tirole, 1991. "Game Theory," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262061414, June.
- 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.
- repec:cup:cbooks:9780521266550 is not listed on IDEAS
- repec:ner:tilbur:urn:nbn:nl:ui:12-153111 is not listed on IDEAS
- P. Jean-Jacques Herings, 2000.
"Two simple proofs of the feasibility of the linear tracing procedure,"
Springer, vol. 15(2), pages 485-490.
- 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.
- repec:cup:cbooks:9780521319867 is not listed on IDEAS
- Dai, Y. & van der Laan, G. & Talman, A.J.J. & Yamamoto, Y., 1989.
"A simplicial algorithm for the nonlinear stationary point problem on an unbounded polyhedron,"
1989-52, Tilburg University, Center for Economic Research.
- 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.
- repec:dgr:kubcen:198952 is not listed on IDEAS
- 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.
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