Skip to main content

Minimax inequalities equivalent to the Fan-Knaster-Kuratowski-Mazurkiewicz theorems

Applied Mathematics and Optimization volume 28, pages 173–179 (1993)Cite this article

Abstract

The purpose of this note is to give further generalizations of the Ky Fan minimax inequality by relaxing the compactness and convexity of sets and the quasi-concavity of the functional and to show that our minimax inequalities are equivalent to the Fan-Knaster-Kuratowski-Mazurkiewicz (FKKM) theorem and a modified FKKM theorem given in this note.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

39,95 €

Price includes VAT (Finland)

References

  1. Allen G (1977) Variational inequalities, complementarity problems, and duality theorems. J Math Anal Appl 58:1–10

    Google Scholar 

  2. Aubin JP (1979) Mathematical Methods of Game and Economic Theory. North-Holland, Amsterdam

    Google Scholar 

  3. Aubin JP, Ekeland I (1984) Applied Nonlinear Analysis. Wiley, New York

    Google Scholar 

  4. Ben-El-Mechaickh H, Deguire P, Granas A (1982) Points fixes et coincidences pour les applications multivoques (applications de Ky Fan). C R Acad Sci Paris Sér I Math 295:337–340

    Google Scholar 

  5. Ben-El-Mechaickh H, Deguire P, Granas A (1982) Points fixes et coincidences pour les fonctions multivoques II (applications de type ϕ et ϕ*). C R Acad Sci Paris Sér I Math 295:381–384

    Google Scholar 

  6. Border KC (1985) Fixed Point Theorems with Applications to Economics and Game Theory. Cambridge University press, Cambridge

    Google Scholar 

  7. Fan K (1969) Extensions of two fixed point theorems of F. E. Browder. Math Z 112:234–240

    Google Scholar 

  8. Fan K (1972) A minimax inequality and application. In: Inequalities, vol 3 (Shisha O., ed). Academic Press, New York, pp 103–113

    Google Scholar 

  9. Fan K (1979) Fixed-point and related theorems for non-compact sets. In: Game Theory and Related Topics (Moeschlin O, Pallaschke D, eds). North-Holland, Amsterdam, pp 151–156

    Google Scholar 

  10. Fan K (1984) Some properties of convex sets related to fixed points theorems. Math Ann 266:519–537

    Google Scholar 

  11. Hartman PT, Stampacchia G (1966) On some nonlinear ecliptic differential functional equations. Acta Math 115:153–188

    Google Scholar 

  12. Knaster B, Kuratowski C, Mazurkiewicz S (1929) Ein Beweis des Fixpunktsatzes für n-demensionale Simpliexe. Fund Math 14:132–137

    Google Scholar 

  13. Lassonde M (1983) On the use of KKM multifonctions in fixed point theory and related topics. J Math Anal Appl 97:151–201

    Google Scholar 

  14. Mosco U (1976) Implicit variational problems and quasi-variational inequalities. Lecture Notes in Mathematics, vol 543. Springer-Verlag, Berlin, pp 83–156

    Google Scholar 

  15. Tarafdar E (1987) A fixed point theorem equivalent to the Fan-Knaster-Kuratowski-Mazurkiewicz theorem. J Math Anal Appl 128:475–479

    Google Scholar 

  16. Tian G (1991) Fixed points theorems for mappings with non-compact and non-convex domains. J Math Anal Appl 158:160–167

    Google Scholar 

  17. Tian G (1992) Existence of equilibrium in abstract economies with discontinuous payoffs and non-compact choice spaces. J Math Econom 21:379–388

    Google Scholar 

  18. Tian G, Zhou J (1991) Quasi-variational inequalities with non-compact sets. J Math Anal Appl 160:583–595

    Google Scholar 

  19. Zhou J, Chen G (1988) Diagonal convexity conditions for problems in convex analysis and quasi-variational inequalities. J Math Anal Appl 132:213–225

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Graduate School of Management, University of California, 92521, Riverside, CA, USA

    Y. Joseph Lin

  2. Department of Economics, Texas A&M University, 77843, College Station, TX, USA

    Guoqiang Tian

Authors
  1. Y. Joseph Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Guoqiang Tian
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by R. Triggiani

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Lin, Y.J., Tian, G. Minimax inequalities equivalent to the Fan-Knaster-Kuratowski-Mazurkiewicz theorems. Appl Math Optim 28, 173–179 (1993). https://doi.org/10.1007/BF01182980

Download citation

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01182980

Key words

  • The minimax inequality
  • Variational inequalities
  • The FKKM theorem
  • Noncompact and nonconvex sets
  • Equivalence

AMS classification

  • 49A29
  • 90C33
  • 90C50