Topological Methods in Nonlinear Analysis

Equivalent forms of the Brouwer fixed point theorem I

Adam Idzik, Władysław Kulpa, and Piotr Maćkowiak

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


In this paper we survey a set of Brouwer fixed point theorem equivalents. These equivalents are divided into four loops related to (1) the Borsuk retraction theorem, (2) the Himmelberg fixed point theorem, (3) the Gale lemma and (4) the Nash equilibrium theorem.

Article information

Topol. Methods Nonlinear Anal., Volume 44, Number 1 (2014), 263-276.

First available in Project Euclid: 11 April 2016

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Idzik, Adam; Kulpa, Władysław; Maćkowiak, Piotr. Equivalent forms of the Brouwer fixed point theorem I. Topol. Methods Nonlinear Anal. 44 (2014), no. 1, 263--276.

Export citation


  • C. Berge, Topological Spaces, Oliver & Boyd, 1963.
  • K. Border, Fixed Point Theorems with Applications to Economics and Game Theory, Cambridge University Press, 1985.
  • L. Brouwer, Beweis der Invarianz der Dimmensionenzahl, Math. Ann. 70 (1911), 161–165.
  • L. Brouwer, Über Abbildung von Mannigfaltigkeiten, Math. Ann. 71 (1912), 97–115.
  • J. Dugundji and A. Granas, Fixed Point Theory, Springer, 2003.
  • K. Fan, A Minimax Inequality and Applications, Inequalities Vol. 3 (O. Shisha, ed.), Academic Press, 1972, 103–113.
  • K. Fan, Some Properties of convex sets related to fixed point theorems, Math. Ann. 266 (1984), 519–537.
  • D. Gale, The law of supply and demand, Math. Scand. 3 (1955), 155–169.
  • C.J. Himmelberg, Fixed points of compact multifunctions, J. Math. Anal. App. 38 (1972), 205–207.
  • T. Ichiishi and A. Idzik, Equitable allocation of divisible goods, J. Math. Econom. 32 (1999), 389–400.
  • A. Idzik, Almost fixed point theorems, Proc. Amer. Math. Soc. 104 (1988), 779–784.
  • A. Idzik and M. van de Vel, Almost fixed point theorems I, Nonlinear Anal. 47 (2001), 619–625.
  • S. Kakutani, A generalization of Brouwer's fixed point theorem, Duke Math. 8 (1941), 457–459.
  • B. Knaster, K. Kuratowski and S. Mazurkiewicz, Ein Bewies des Fixpunktsatzes für $n$-dimensionale Simplexe, Fund. Math. 14 (1929), 132–137.
  • W. Kulpa, Topologia a ekonomia, Wydawnictwo Uniwersytetu Kardynała Stefana Wyszyńskiego, 2010.
  • W. Kulpa and A. Szymański, On Nash theorem, Acta Univ. Carolin. Math. Phys. 43(2) (2002), 55–67.
  • W. Kulpa and A. Szymański, Theorem on signatures, Acta Univ. Carolin. Math. Phys. 48 (2) (2007), 55–67.
  • K. Kuratowski and H. Steinhaus, Une application géometrique du théoreme de Brouwer sur les points invariants, Bull. de l'Academie Pol. Sci., Cl. III, 1 (1953), 83–86.
  • P. Maćkowiak, Some equivalents of Brouwer's fixed point theorem and the existence of economic equilibrium, Quantitative Methods in Economics 2012 (M. Matłoka, ed.), Wydawnictwo Uniwersytetu Ekonomicznego w Poznaniu, 2012, 164–171.
  • P. Maćkowiak, The existence of equilibrium in a simple exchange model, Fixed Point Theory Appl. 2013:104 doi:10.1186/1687-1812-2013-104 (2013), 1–11.
  • H. Nikaido and K. Isoda, Note on non-cooperative convex games, Pacific J. Math. 5 (Suppl. 1) (1955), 807–815.
  • H. Nikaido, Convex Structures and Economic Theory, Academic Press, 1968.
  • S. Park, Ninety Years of the Brouwer Fixed Point Theorem, Vietnam J. Math. 27 (3) (1999), 187–222.
  • Y. Takeuchi and T. Suzuki, An easily verifiable proof of the Brouwer fixed point theorem, Bull. Kyushu Inst. Technol. Pure Appl. Math. 59 (2012), 1–5.
  • J.P. Torres-Martinez, Fixed points as Nash equilibria, Fixed Point Theory Appl. 2006:36135 doi:10.1155/FPTA/2006/36135 (2006), 1–4.
  • H. Uzawa, The Walras existence theorem and Brouwer's fixed-point theorem, The Economic Studies Quarterly 13 (1) (1962), 59–62.
  • J. Zhao, Equivalence between the Existence Theorems on Nash Equilibrium, Core and Hybrid Equilibrium, Working Paper, Department of Economics, University of Saskatchewan, Saskatchewan, 2005.