Applied Mathematics and Mechanics

, Volume 28, Issue 1, pp 103–109 | Cite as

Weakly R-KKM mappings—intersection theorems and minimax inequalities in topological spaces

  • Deng Lei  (邓磊)Email author
  • Yang Ming-ge  (杨明歌)


In this paper, we introduce the concepts of weakly R-KKM mappings, R-convex and R-β-quasiconvex in general topological spaces without any convex structure. Relating to these, we obtain an extension to general topological spaces of Fan’s matching theorem, namely that Lemma 1.2 in this paper. On this basis, two intersection theorems are proved in topological spaces. By using intersection theorems, some minimax inequalities of Ky Fan type are also proved in topological spaces. Our results generalize and improve the corresponding results in the literature.

Key words

weakly R-KKM mapping R-convex R-β-quasiconvex generalized R-KKM mapping 

Chinese Library Classification


2000 Mathematics Subject Classification

47H04 54A05 54C10 54C60 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Knaster B, Kuratowski C, Mazurkiewicz S. Ein beweis des fixpunktsatzes für n-dimensionale simplexe[J]. Fund Math, 1929, 14(1):132–137.Google Scholar
  2. [2]
    Fan K. A generalization of Tychonoff’s fixed point theorem[J]. Math Ann, 1961, 142(3):305–310.CrossRefMathSciNetGoogle Scholar
  3. [3]
    Park S. Generalizations of Ky Fan’s matching theorems and their applications[J]. J Math Anal Appl, 1989, 141(1):164–176.CrossRefMathSciNetGoogle Scholar
  4. [4]
    Chang T H, Yen C L. KKM property and fixed point theorems[J]. J Math Anal Appl, 1996, 203(1):224–235.CrossRefMathSciNetGoogle Scholar
  5. [5]
    Lin L J, Ko C J, Park S. Coincidence theorems for set-valued maps with G-KKM property on generalized convex space[J]. Discuss Math Differential Incl, 1998, 18(1):69–85.MathSciNetGoogle Scholar
  6. [6]
    Balaj M. Weakly G-KKM mappings, G-KKM property, and minimax inequalities[J]. J Math Anal Appl, 2004, 294(1):237–245.CrossRefMathSciNetGoogle Scholar
  7. [7]
    Deng L, Xia X. Generalized R-KKM theorems in topological spaces and their applications[J]. J Math Anal Appl, 2003, 285(2):679–690.CrossRefMathSciNetGoogle Scholar
  8. [8]
    Park S, Kim H. Admissible classes of multifunctions on generalized convex spaces[J]. Proc Coll Natur Sci Seoul National University, 1993, 18(1):1–21.Google Scholar
  9. [9]
    Shih M H. Covering properties of convex sets[J]. Bull London Math Soc, 1986, 18(1):57–59.MathSciNetGoogle Scholar
  10. [10]
    Park S, Kim H. Foundations of the KKM theory on generalized convex spaces[J]. J Math Anal Appl, 1997, 209(2):551–571.CrossRefMathSciNetGoogle Scholar
  11. [11]
    Park S. Fixed point theorems in locally G-convex spaces[J]. Nonlinear Anal, 2002, 48(6):869–879.CrossRefMathSciNetGoogle Scholar
  12. [12]
    Aubin J P, Ekeland I. Applied nonlinear analysis[M]. New York: Wiley, 1984.Google Scholar

Copyright information

© Editorial Committee of Appl. Math. Mech. 2007

Authors and Affiliations

  • Deng Lei  (邓磊)
    • 1
    Email author
  • Yang Ming-ge  (杨明歌)
    • 1
    • 2
  1. 1.School of Mathematics and StatisticsSouthwest UniversityChongqingP. R. China
  2. 2.College of Mathematics ScienceLuoyang Normal UniversityLuoyangP. R. China

Personalised recommendations