Skip to main content
SpringerLink
Log in

Generalized convex spaces, L-spaces, and FC-spaces

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

Abstract

We show that FC-spaces due to Ding are particular types of L-spaces due to Ben-El-Mechaiekh et al., and hence particular types of G-convex spaces. Some counter-examples are given and related matters are also discussed.

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  3. Ding X.P.: Abstract convexity and generalizations of Himmelberg type fixed-point theorems. Comp. Math. Appl. 41, 497–504 (2001)

    Article  Google Scholar 

  4. Ding X.P.: Generalized L-KKM type theorems in L-convex spaces with applications. Comput. Math. Appl. 43, 1240–1256 (2002)

    Google Scholar 

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

    Article  Google Scholar 

  6. Ding X.P.: Generalized KKM type theorems in FC-spaces with applications (I). J. Glob. Optim. 36, 581–596 (2006)

    Article  Google Scholar 

  7. Ding X.P.: Continuous selection, collectively fixed points and system of coincidence theorems in product topological spaces. Acta Math. Sinica English Ser. 22, 1629–1638 (2006)

    Article  Google Scholar 

  8. Ding, X.P.: New generalized R-KKM type theorems in general topological spaces and applications. Acta Math. Sinica English Ser. (2006). doi:10.1007/s10114-005-0876-y

  9. Ding X.P.: Weak Pareto equilibria for generalized constrained multiobjective games in locally FC-spaces. Nonlinear Anal. 65, 538–545 (2006)

    Article  Google Scholar 

  10. Ding, X.P.: Collective fixed points and a system of coincidence theorems in product FC-spaces and applications. Nonlinear Anal. (2006). doi:10.1016/j.na.2006.03.043

  11. Ding, X.P. : Maximal elements of G KKM-majorized mappings in product FC-spaces and applications (II). Nonlinear Anal. (2006). doi:10.1016/j.na.2006.10.025

  12. Ding, X.P.: Generalized game and system of generalized vector quasi-equilibrium problems in locally FC-uniform spaces. Nonlinear Anal. (2006). doi:10.1016/j.na.2006.12.003

  13. Ding X.P.: Generalized KKM type theorems in FC-spaces with applications (II). J. Glob. Optim. 38, 367–385 (2007)

    Article  Google Scholar 

  14. Ding X.P.: Nonempty intersection theorems and generalized multi-objective games in product FC-spaces. J. Glob. Optim. 37, 63–73 (2007)

    Article  Google Scholar 

  15. Ding X.P.: Maximal elements of G KKM-majorized mappings in product FC-spaces and applications (I). Nonlinear Anal. 67, 963–973 (2007)

    Article  Google Scholar 

  16. Ding X.P., Ding T.M.: KKM type theorems and generalized vector equilibrium problems in noncompact FC-spaces. J. Math. Anal. Appl. 331, 1230–1245 (2007)

    Article  Google Scholar 

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

    Article  Google Scholar 

  18. Ding X.P., Park J.Y., Jung I.H.: Existence of solutions for nonlinear inequalities in G-convex spaces. Appl. Math. Lett. 15, 735–741 (2002)

    Article  Google Scholar 

  19. Ding, X.P., Wang, L.: Fixed points, minimax inequalities and equilibria of noncompact abstract economies in FC-spaces. Nonlinear Anal. (2007). doi:10.1016/j.na.2007.06.006

  20. Ding X.P., Yao J.C., Lin L.J.: Solutions of generalized vector quasi-equilibrium problems in locally G-convex spaces. J. Math. Anal. Appl. 298, 398–410 (2004)

    Article  Google Scholar 

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

    Article  Google Scholar 

  22. Fang M., Huang N.-J.: KKM type theorems with applications to generalized vector equilibrium problems in FC-spaces. Nonlinear Anal. 67, 809–817 (2007)

    Article  Google Scholar 

  23. Horvath C.D.: Contractibility and generalized convexity. J. Math. Anal. Appl. 156, 341–357 (1991)

    Article  Google Scholar 

  24. Huang J.: The matching theorems and coincidence theorems for generalized R-KKM mapping in topological spaces. J. Math. Anal. Appl. 312, 374–382 (2005)

    Article  Google Scholar 

  25. Knaster B., Kuratowski K., Mazurkiewicz S.: Ein Beweis des Fixpunktsatzes für n-Dimensionale Simplexe. Fund. Math. 14, 132–137 (1929)

    Google Scholar 

  26. Lu H., Tang D.: An intersection theorem in L-convex spaces and applications. J. Math. Anal. Appl. 312, 343–356 (2005)

    Article  Google Scholar 

  27. Park S.: Another five episodes related to generalized convex spaces. Nonlinear Funct. Anal. Appl. 3, 1–12 (1998)

    Google Scholar 

  28. Park S.: Ninety years of the Brouwer fixed point theorem. Vietnam J. Math. 27, 193–232 (1999)

    Google Scholar 

  29. Park S.: Elements of the KKM theory for generalized convex spaces. Korean J. Comput. Appl. Math. 7, 1–28 (2000)

    Google Scholar 

  30. Park S.: Remarks on topologies of generalized convex spaces. Nonlinear Func. Anal. Appl. 5, 67–79 (2000)

    Google Scholar 

  31. Park S.: On generalizations of the KKM principle on abstract convex spaces. Nonlinear Anal. Forum 11, 67–77 (2006)

    Google Scholar 

  32. Park S.: A unified fixed point theory in generalized convex spaces. Acta Math. Sinica English Ser. 23(8), 1509–1536 (2007)

    Article  Google Scholar 

  33. Park S.: Elements of the KKM theory on abstract convex spaces. J. Korean Math. Soc. 45(1), 1–27 (2008)

    Article  Google Scholar 

  34. Park S., Jeong K.S.: Fixed point and non-retract theorems – Classical circular tours. Taiwan. J. Math. 5, 97–108 (2001)

    Google Scholar 

  35. Park S., Kim H.: Admissible classes of multifunctions on generalized convex spaces. Proc. Coll. Nat. Sci. Seoul Nat. Univ. 18, 1–21 (1993)

    Google Scholar 

  36. Park S., Kim H.: Coincidence theorems on admissible maps on generalized convex spaces. J. Math. Anal. Appl. 197, 173–187 (1996)

    Article  Google Scholar 

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

    Article  Google Scholar 

  38. Park S., Lee W.: A unified approach to generalized KKM maps in generalized convex spaces. J. Nonlinear Convex Anal. 2, 157–166 (2001)

    Google Scholar 

  39. Sánchez M.C., Llinares J.-V., Subiza B.: A KKM-result and an application for binary and non-binary choice function. Econom. Th. 21, 185–193 (2003)

    Article  Google Scholar 

  40. Tang, G.-S, Zhang, Q.-B., Cheng, C.-Z.: W-G-F-KKM mapping, intersection theorems and minimax inequalities in FC-space. J. Math. Anal. Appl. (2007). doi:10.1016/j.jmaa.2007.01.040

  41. Zhang Q.-b., Cheng C.-z.: Some fixed-point theorems and minimax inequalities in FC-space. J. Math. Anal. Appl. 328, 1369–1377 (2007)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. The National Academy of Sciences, Seoul, 137-044, Korea

    Sehie Park

  2. Department of Mathematical Sciences, Seoul National University, Seoul, 151-747, Korea

    Sehie Park

Authors
  1. Sehie Park
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sehie Park.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Park, S. Generalized convex spaces, L-spaces, and FC-spaces. J Glob Optim 45, 203–210 (2009). https://doi.org/10.1007/s10898-008-9363-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-008-9363-1

Keywords

Mathematics Subject Classification (2000)

Search

Navigation