Skip to main content
SpringerLink
Log in

Generalized KKM Type Theorems in FC-Spaces with Applications (I)

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

The class KKM(X,Y) (resp., s-KKM(X,Y,Z)) of set-valued mappings with KKM (resp., s-KKM) property is introduced in FC-spaces without any convexity structure. Some generalized KKM (resp., s-KKM) type theorems are proved in FC-spaces under much weak assumptions. As applications, some new section theorems and coincidence theorems are established in FC-spaces. These theorems generalize many known results in literature. The further applications of these results will be given in a follow-up paper.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Knaster H., Kuratowski C., Mazurkiewicz S. (1929). Ein Beweis des Fixpunktsatzes für n-dimensionale simplexe. Fundam. Math. 14:132–137

    Google Scholar 

  2. Fan Ky. (1961). A generalization of Tychonoff’s fixed point theorem. Math. Ann. 142:305–310

    Article  Google Scholar 

  3. Ding X.P. (2002). KKM type theorems and coincidence theorems on K-convex spaces. Acta. Math. Sci. 22(3):419–426

    Google Scholar 

  4. Ding X.P. (2002). Generalized G-KKM theorems in generalized convex spaces and their applications. J. Math. Anal. Appl. 266(1):21–37

    Article  Google Scholar 

  5. Ding X.P. (2002). Generalized L-KKM type theorems in L-convex spaces with applications. Comput. Math. Appl. 43:1249–1256

    Article  Google Scholar 

  6. Ding X.P. (2004). KKM type theorems, minimax inequalities and saddle point theorems in generalized convex spaces. Acta. Math. Sin. (in Chinese) 47(4):711–722

    Google Scholar 

  7. Ding X.P. (2005). Generalized R-KKM type theorems in topological spaces and applications. J. Sichuan Norm. Univ. (NS) 28(5):505–513

    Google Scholar 

  8. Ding X.P., Liou Y.C., Yao J.C. (2005). Generalized R-KKM type theorems in topological spaces with applications. Appl. Math. Lett. 18(12):1345–1350

    Article  Google Scholar 

  9. Deng L., Xia X. (2003). Generalized R-KKM theorems in topolgical space and their applications. J. Math. Anal. Appl. 285:679–690

    Article  Google Scholar 

  10. Chang T.H., Yen C.L. (1996). KKM property and fixed point theorems. J. Math. Anal. Appl. 203:224–235

    Article  Google Scholar 

  11. Park S., Singh S.P., Watson B. (1994). Some fixed point theorems for composites of acyclic maps. Proc. Am. Math. Soc. 121:1151–1158

    Article  Google Scholar 

  12. Park S., Kim H. (1997). Foundations of the KKM theory on generalized convex spaces. J. Math. Anal. Appl. 209(2):551–571

    Article  Google Scholar 

  13. Ben-El-Mechaiekh H., Deguire P. (1992). Approachability and fixed point for non-convex set-valued maps. J. Math. Anal. Appl. 170:477–500

    Article  Google Scholar 

  14. Lin L.J., Ansari Q.H., Wu J.Y. (2003). Geometric properties and coincidence theorems with applications to generalized vector equilibrium problems. J. Optim. Theory Appl. 117(1):121–137

    Article  Google Scholar 

  15. Lin L.J., Wan W.P. (2004). KKM type theorems and coincidence theorems with applications to the existence of equilibria. J. Optim. Theory Appl. 123(1):105–122

    Article  Google Scholar 

  16. Lin L.J., Chen H.L. (2005). The study of KKM theorems with applications to vector equilibrium problems and implicit vector variational inequalities problems. J. Global Optim. 32:135–157

    Article  Google Scholar 

  17. Ding X.P. (1996). Coincidence theorems and equilibria of generalized games. Indian J. Pure Appl. Math. 27(11):1057–1071

    Google Scholar 

  18. Lin L.J. (2003). System of coincidence theorems with applications. J. Math. Anal. Appl. 285(2):408–418

    Article  Google Scholar 

  19. Ding X.P. (1999). Coincidence theorems in topological spaces and their applications. Appl. Math. Lett. 12:99–105

    Article  Google Scholar 

  20. Ding X.P. (2005). Maximal element theorems in product FC-spaces and generalized games. J. Math. Anal. Appl. 305(1):29–42

    Article  Google Scholar 

  21. Horvath C.D. (1991). Contractibility and general convexity. J. Math. Anal. Appl. 156(2):341–357

    Article  Google Scholar 

  22. Ben-El-Mechaiekh H., Chebbi S., Flornzano M., Llinares J.V. (1998). Abstract convexity and fixed points. J. Math. Anal. Appl. 222:138–150

    Article  Google Scholar 

  23. Chang T.H., Huang Y.Y., Jeng J.C., Kuo K.W. (1999). On S-KKM property and related topies. J. Math. Anal. Appl. 229:212–227

    Article  Google Scholar 

  24. Lin L.J. (2001). Applications of a fixed theorem in G-convex space. Nonlinear Anal. 46:601–608

    Article  Google Scholar 

  25. Aubin J.P., Ekeland I. (1984). Applied nonlinear analysis. Wiley, New York

    Google Scholar 

  26. Ha C.W. (1980). Minimax and fixed-point theorems. Math. Annal. 248:73–77

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Mathematics and Software Science, Sichuan Normal University, Chengdu, Sichuan, 610066, P. R. China

    Xie Ping Ding

Authors
  1. Xie Ping Ding
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xie Ping Ding.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ding, X.P. Generalized KKM Type Theorems in FC-Spaces with Applications (I). J Glob Optim 36, 581–596 (2006). https://doi.org/10.1007/s10898-006-9028-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-006-9028-x

Keywords

Search

Navigation