Skip to content


  • Research Article
  • Open Access

Fixed point theorems in locally convex spaces—the Schauder mapping method

Fixed Point Theory and Applications20062006:57950

  • Received: 22 March 2005
  • Accepted: 6 September 2005
  • Published:


In the appendix to the book by F. F. Bonsal, Lectures on Some Fixed Point Theorems of Functional Analysis (Tata Institute, Bombay, 1962) a proof by Singbal of the Schauder-Tychonoff fixed point theorem, based on a locally convex variant of Schauder mapping method, is included. The aim of this note is to show that this method can be adapted to yield a proof of Kakutani fixed point theorem in the locally convex case. For the sake of completeness we include also the proof of Schauder-Tychonoff theorem based on this method. As applications, one proves a theorem of von Neumann and a minimax result in game theory.


  • Functional Analysis
  • Game Theory
  • Point Theorem
  • Mapping Method
  • Differential Geometry


Authors’ Affiliations

Faculty of Mathematics and Computer Science, Babeş-Bolyai University, Cluj-Napoca, 400084, Romania


  1. Benyamini Y, Lindenstrauss J: Geometric Nonlinear Functional Analysis. Vol. 1, American Mathematical Society Colloquium Publications. Volume 48. American Mathematical Society, Rhode Island; 2000.MATHGoogle Scholar
  2. Bohnenblust HF, Karlin S: On a theorem of Ville. In Contributions to the Theory of Games, Annals of Mathematics Studies, no. 24. Princeton University Press, New Jersey; 1950:155–160.Google Scholar
  3. Bonsal FF: Lectures on some Fixed Point Theorems and Functional Analysis, Notes by K. B. Vedak. Tata Institute of Fundamental Research, Bombay; 1962:iii+176.Google Scholar
  4. Dunford N, Schwartz JT: Linear Operators. I. General Theory, Pure and Applied Mathematics. Volume 7. Interscience, New York; 1958.MATHGoogle Scholar
  5. Edwards RE: Functional Analysis. Theory and Applications, Corrected reprint of the 1965 original. Dover, New York; 1995:xvi+783.Google Scholar
  6. Engelking R: General Topology, Sigma Series in Pure Mathematics. Volume 6. 2nd edition. Heldermann, Berlin; 1989.MATHGoogle Scholar
  7. Glicksberg IL: A further generalization of the Kakutani fixed point theorem, with application to Nash equilibrium points. Proceedings of the American Mathematical Society 1952,3(1):170–174.MathSciNetMATHGoogle Scholar
  8. Goebel K, Kirk WA: Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics. Volume 28. Cambridge University Press, Cambridge; 1990.View ArticleMATHGoogle Scholar
  9. Istrăţescu VI: Fixed Point Theory. An Introduction, Mathematics and Its Applications. Volume 7. D. Reidel, Dordrecht; 1981.Google Scholar
  10. Kakutani S: A generalization of Brouwer's fixed point theorem. Duke Mathematical Journal 1941, 8: 457–459. 10.1215/S0012-7094-41-00838-4MathSciNetView ArticleMATHGoogle Scholar
  11. Kantorovich LV, Akilov GP: Functional Analysis. 3rd edition. Nauka, Moscow; 1984. English translation of the 1959 edition: Macmillan, New York 1964MATHGoogle Scholar
  12. Khamsi MA, Kirk WA: An Introduction to Metric Spaces and Fixed Point Theory, Pure and Applied Mathematics (New York). Wiley-Interscience, New York; 2001.View ArticleGoogle Scholar
  13. Köthe G: Topological Vector Spaces. I., Translated from the German by D. J. H. Garling, Die Grundlehren der mathematischen Wissenschaften. Volume 159. Springer, New York; 1969.Google Scholar
  14. Lusternik LA, Sobolev VJ: Elements of Functional Analysis, International Monographs on Advanced Mathematics Physics. Hindustan, Delhi; Halsted Press [John Wiley & Sons], New York; 1974.MATHGoogle Scholar
  15. Nikaidô H: Convex Structures and Economic Theory, Mathematics in Science and Engineering. Volume 51. Academic Press, New York; 1968.MATHGoogle Scholar
  16. Schauder J: Der Fixpunktsatz in Funktionalräume. Studia Mathematica 1930, 2: 171–180.MATHGoogle Scholar
  17. Smart DR: Fixed Point Theorems, Cambridge Tracts in Mathematics, no. 66. Cambridge University Press, London; 1974.Google Scholar
  18. Tychonoff A: Ein Fixpunktsatz. Mathematische Annalen 1935,111(1):767–776. 10.1007/BF01472256MathSciNetView ArticleMATHGoogle Scholar
  19. von Neumann J: Über ein ökonomisches Gleichungssystem und eine Verallgemeinerung des Brouwerschen Fixpunktsatzes. Ergebnisse eines Mathematischen Kolloquiums 1937, 8: 73–83.MATHGoogle Scholar