Equivalent forms of the Brouwer fixed point theorem II

Adam Idzik, Władysław Kulpa, Piotr Maćkowiak

DOI: http://dx.doi.org/10.12775/TMNA.2020.036

Abstract


Equivalents of the Brouwer fixed point theorem are proved. They involve formulations either for the standard simplex or for the cube. Characterizations of continuous functions defined on the standard simplex are also presented. The famous Steinhaus chessboard theorem is generalized.

Keywords


Brouwer fixed point theorem; Steinhaus chessboard problem

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References


R.B. Bapat, A constructive proof of a permutation-based generalization of Sperner’s lemma, Math. Prog. 44 (1989), 113–120.

R.B. Bapat, Sperner’s lemma with multiple labels, Modeling, Computation and Optimization (S.K. Neogy, A.K. Das and R.B. Bapat, eds.), World Scientific, 2009, pp. 257–262.

R. Engelking, Theory of Dimensions: Finite and Infinite, Heldermann, Lemgo, 1995.

D. Gale, The game of hex and the Brouwer fixed-point theorem, Amer. Math. Monthly 86 (1979), 818–827.

D. Gale, Equilibrium in a discrete exchange economy with money, Internat. J. Game Theory 13 (1984), 61–64.

T. Ichiishi and A. Idzik, Equitable allocation of divisible goods, J. Math. Econom. 32 (1999), 389–400.

A. Idzik, W. Kulpa and P. Maćkowiak, Equivalent forms of the Brouwer fixed point theorem I, Topol. Methods Nonlinear Anal. 44 (2014), 263–276.

W. Kulpa, The indexed open covering theorem, Acta Univ. Carolin. Math. Phys. 34, (1993), 75–82.

W. Kulpa, Convexity and the Brouwer fixed point theorem, Topology Proc. 22 (1999), 211–235.

W. Kulpa, L. Socha and M. Turzański, Steinhaus chessboard theorem, Acta Univ. Carolin. Math. Phys. 41 (2000), 47–50.

W. Kulpa, A. Szymański and M. Turzański, Function and colorful extensions of the KKM theorem, Topol. Methods Nonlinear Anal. 56 (2020), No. 1, 313–324.

K. Kuratowski, Topology, vol. II , Academic Press, New York, 1968.

S. Park, Ninety years of the Brouwer fixed point theorem, Vietnam J. Math. 27 (1999), 187–222.

S. Park and K.S. Jeong, A proof of the Sperner lemma from the Brouwer fixed point theorem, Nonlinear Anal. Forum 8 (2003), 65–67.

H. Scarf, The computation of equilibrium prices: an exposition, Handbook of Mathematical Economics, vol. 2 (K.J. Arrow and M.D. Intriligator, eds.), North-Holland, 1982, pp. 1006–1061.

H. Steinhaus, Mathematical Snapshots, 3rd edition, Oxford University Press, Oxford, 1983.

P. Tkacz and M. Turzański, An n-dimensional version of Steinhaus’ chessboard theorem, Topology Appl. 155 (2008), 354–361.

P. Tkacz and M. Turzański, The Bolzano–Poincaré type theorems, Int. J. Math. Math. Sci. 2011 (2011), 1–9.

Z. Yang, Computing Equilibria and Fixed Points, Kluwer, Boston, 1999.


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