Skip to content


  • Research Article
  • Open Access

Note on KKM maps and applications

Fixed Point Theory and Applications20062006:53286

  • Received: 6 March 2005
  • Accepted: 11 August 2005
  • Published:


We apply the KKM technique to study fixed point theory, minimax inequality and coincidence theorem. Some new results on Fan-Browder fixed point theorem, Fan's minimax theorem and coincidence theorem are obtained.


  • Point Theorem
  • Differential Geometry
  • Fixed Point Theorem
  • Point Theory
  • Computational Biology


Authors’ Affiliations

Department of Mathematics, Foshan University, Foshan, Guangdong, 528000, China
Department of Mathematics Education and the RINS, College of Education, Gyeongsang National University, Chinju, 660-701, Korea
Department of Mathematics Education, College of Education, Kyungnam University, Masan, 631-701, Korea
Department of Mathematics, Kyungsung University, Pusan, 608-735, Korea


  1. Aubin J-P, Ekeland I: Applied Nonlinear Analysis, Pure and Applied Mathematics (New York). John Wiley & Sons, New York; 1984:xi+518.Google Scholar
  2. Bardaro C, Ceppitelli R: Some further generalizations of Knaster-Kuratowski-Mazurkiewicz theorem and minimax inequalities. Journal of Mathematical Analysis and Applications 1988,132(2):484–490. 10.1016/0022-247X(88)90076-5MathSciNetView ArticleMATHGoogle Scholar
  3. Browder FE: The fixed point theory of multi-valued mappings in topological vector spaces. Mathematische Annalen 1968, 177: 283–301. 10.1007/BF01350721MathSciNetView ArticleMATHGoogle Scholar
  4. Chang S-S, Zhang Y: Generalized KKM theorem and variational inequalities. Journal of Mathematical Analysis and Applications 1991,159(1):208–223. 10.1016/0022-247X(91)90231-NMathSciNetView ArticleMATHGoogle Scholar
  5. Dugundji J, Granas A: Fixed Point Theory. I, Monografie Matematyczne. Volume 61. Państwowe Wydawnictwo Naukowe (PWN), Warsaw; 1982:209.Google Scholar
  6. Fan K: Fixed-point and minimax theorems in locally convex topological linear spaces. Proceedings of the National Academy of Sciences of the United States of America 1952, 38: 121–126. 10.1073/pnas.38.2.121MathSciNetView ArticleMATHGoogle Scholar
  7. Fan K: Sur un théorème minimax. Comptes Rendus Academie des Sciences Paris 1964, 259: 3925–3928.MATHGoogle Scholar
  8. Fan K: Applications of a theorem concerning sets with convex sections. Mathematische Annalen 1966, 163: 189–203. 10.1007/BF02052284MathSciNetView ArticleMATHGoogle Scholar
  9. Fan K: Extensions of two fixed point theorems of F. E. Browder. Mathematische Zeitschrift 1969, 112: 234–240. 10.1007/BF01110225MathSciNetView ArticleMATHGoogle Scholar
  10. Granas A: KKM-maps and their applications to nonlinear problems. In The Scottish Book. Edited by: Mauldin RD. Birkhäuser, Massachusetts; 1981:45–61.Google Scholar
  11. Granas A, Dugundji J: Fixed Point Theory, Springer Monographs in Mathematics. Springer, New York; 2003:xvi+690.View ArticleMATHGoogle Scholar
  12. Horvath CD: Contractibility and generalized convexity. Journal of Mathematical Analysis and Applications 1991,156(2):341–357. 10.1016/0022-247X(91)90402-LMathSciNetView ArticleMATHGoogle Scholar
  13. Knaster B, Kuratowski C, Mazurkiewicz S: Ein Beweis des Fixpunktsatzes für -dimensionale simplexe. Fundamenta Mathematicae 1929, 14: 132–137.MATHGoogle Scholar
  14. Park S, Kim H: Coincidence theorems for admissible multifunctions on generalized convex spaces. Journal of Mathematical Analysis and Applications 1996,197(1):173–187. 10.1006/jmaa.1996.0014MathSciNetView ArticleMATHGoogle Scholar
  15. Singh S, Watson B, Srivastava P: Fixed Point Theory and Best Approximation: The KKM-Map Principle, Mathematics and Its Applications. Volume 424. Kluwer Academic, Dordrecht; 1997:x+220.View ArticleMATHGoogle Scholar
  16. Sion M: On general minimax theorems. Pacific Journal of Mathematics 1958, 8: 171–176.MathSciNetView ArticleMATHGoogle Scholar
  17. Yuan GX-Z: KKM Theory and Applications in Nonlinear Analysis, Monographs and Textbooks in Pure and Applied Mathematics. Volume 218. Marcel Dekker, New York; 1999:xiv+621.Google Scholar
  18. Zhang SS, Yang GS: Some further generalizations of Ky Fan's minimax inequality and its applications to variational inequalities. Applied Mathematics and Mechanics. Yingyong Shuxue he Lixue 1990,11(11):961–968. Applied Mathematics and Mechanics (English ed.) 11 (1990), no. 11, 1027–1034MathSciNetMATHGoogle Scholar


© Y. Q. Chen et al. 2006

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.