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Note on KKM maps and applications

Abstract

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.

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References

  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)90076-5

    Article  MathSciNet  MATH  Google Scholar 

  3. Browder FE: The fixed point theory of multi-valued mappings in topological vector spaces. Mathematische Annalen 1968, 177: 283–301. 10.1007/BF01350721

    Article  MathSciNet  MATH  Google 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-N

    Article  MathSciNet  MATH  Google 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.38.2.121

    Article  MathSciNet  MATH  Google Scholar 

  7. Fan K: Sur un théorème minimax. Comptes Rendus Academie des Sciences Paris 1964, 259: 3925–3928.

    MATH  Google Scholar 

  8. Fan K: Applications of a theorem concerning sets with convex sections. Mathematische Annalen 1966, 163: 189–203. 10.1007/BF02052284

    Article  MathSciNet  MATH  Google Scholar 

  9. Fan K: Extensions of two fixed point theorems of F. E. Browder. Mathematische Zeitschrift 1969, 112: 234–240. 10.1007/BF01110225

    Article  MathSciNet  MATH  Google 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.

    Book  MATH  Google 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-L

    Article  MathSciNet  MATH  Google Scholar 

  13. Knaster B, Kuratowski C, Mazurkiewicz S: Ein Beweis des Fixpunktsatzes für -dimensionale simplexe. Fundamenta Mathematicae 1929, 14: 132–137.

    MATH  Google 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.1996.0014

    Article  MathSciNet  MATH  Google 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.

    Book  MATH  Google Scholar 

  16. Sion M: On general minimax theorems. Pacific Journal of Mathematics 1958, 8: 171–176.

    Article  MathSciNet  MATH  Google 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, 1027–1034

    MathSciNet  MATH  Google Scholar 

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Correspondence to YJ Cho.

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Chen, Y., Cho, Y., Kim, J. et al. Note on KKM maps and applications. Fixed Point Theory Appl 2006, 53286 (2006). https://doi.org/10.1155/FPTA/2006/53286

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