Table of Contents Author Guidelines Submit a Manuscript
International Journal of Mathematics and Mathematical Sciences
Volume 2008 (2008), Article ID 423596, 10 pages
http://dx.doi.org/10.1155/2008/423596
Research Article

Remarks on Weakly KKM Maps in Abstract Convex Spaces

1Natural Sciences Division, The National Academy of Sciences, Seoul 137-044, South Korea
2Department of Mathematics, Seoul National University, Seoul 151-747, South Korea

Received 26 September 2007; Accepted 7 January 2008

Academic Editor: Petru Jebelean

Copyright © 2008 Sehie Park. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. S. Park and H. Kim, “Admissible classes of multifunctions on generalized convex spaces,” Proceedings of College of Natural Sciences, Seoul National University, vol. 18, pp. 1–21, 1993. View at Google Scholar
  2. H. Ben-El-Mechaiekh, S. Chebbi, M. Florenzano, and J.-V. Llinares, “Abstract convexity and fixed points,” Journal of Mathematical Analysis and Applications, vol. 222, no. 1, pp. 138–150, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. J. Huang, “The matching theorems and coincidence theorems for generalized R-KKM mapping in topological spaces,” Journal of Mathematical Analysis and Applications, vol. 312, no. 1, pp. 374–382, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. X. P. Ding, “Maximal element theorems in product FC-spaces and generalized games,” Journal of Mathematical Analysis and Applications, vol. 305, no. 1, pp. 29–42, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. X. P. Ding, “Generalized KKM type theorems in FC-spaces with applications. I,” Journal of Global Optimization, vol. 36, no. 4, pp. 581–596, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. S. Park, “Various subclasses of abstract convex spaces for the KKM theory,” Proceedings of the National Institute of Mathematical Science, vol. 2, no. 4, pp. 35–47, 2007. View at Google Scholar
  7. B. Knaster, K. Kuratowski, and S. Mazurkiewicz, “Ein Beweis des fixpunktsatzes für n-dimensionale simplexe,” Fundamenta Mathematicae, vol. 14, pp. 132–137, 1929. View at Google Scholar · View at Zentralblatt MATH
  8. S. Park, “On generalizations of the KKM principle on abstract convex spaces,” Nonlinear Analysis Forum, vol. 11, no. 1, pp. 67–77, 2006. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. S. Park, “Elements of the KKM theory on abstract convex spaces,” Journal of the Korean Mathematical Society, vol. 45, no. 1, pp. 1–27, 2008. View at Google Scholar
  10. S. Park, “Fundamental theory of the KKM spaces,” Fixed Point Theory and Applications. In press.
  11. S. Park, “Equilibrium existence theorems in KKM spaces,” 2007, Nonlinear Analysis: Theory, Methods & Applications. In press. View at Publisher · View at Google Scholar
  12. F.-C. Liu, “On a form of KKM principle and SupInfSup inequalities of von Neumann and of Ky Fan type,” Journal of Mathematical Analysis and Applications, vol. 155, no. 2, pp. 420–436, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. M. Balaj, “Weakly G-KKM mappings, G-KKM property, and minimax inequalities,” Journal of Mathematical Analysis and Applications, vol. 294, no. 1, pp. 237–245, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. G.-S. Tang, Q.-B. Zhang, and C.-Z. Cheng, “W-G-F-KKM mapping, intersection theorems and minimax inequalities in FC-space,” Journal of Mathematical Analysis and Applications, vol. 334, no. 2, pp. 1481–1491, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. S. Park, “Examples of 𝒦𝒞-maps and 𝒦𝒪-maps on abstract convex spaces,” Soochow Journal of Mathematics, vol. 33, no. 3, pp. 477–486, 2007. View at Google Scholar · View at MathSciNet
  16. S. Park, “Coincidence, almost fixed point, and minimax theorems on generalized convex spaces,” Journal of Nonlinear and Convex Analysis, vol. 4, no. 1, pp. 151–164, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. B. P. Sortan, “Introduction to Axiomatic Theory of Convexity,” Kishineff, New York, NY, USA, 1984. View at Google Scholar
  18. S. Park, “Ninety years of the Brouwer fixed point theorem,” Vietnam Journal of Mathematics, vol. 27, no. 3, pp. 187–222, 1999. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. S. Park, “Elements of the KKM theory for generalized convex spaces,” The Korean Journal of Computational & Applied Mathematics, vol. 7, no. 1, pp. 1–28, 2000. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. S. Park, “Remarks on topologies of generalized convex spaces,” Nonlinear Functional Analysis and Applications, vol. 5, no. 2, pp. 67–79, 2000. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. S. Park and W. Lee, “A unified approach to generalized KKM maps in generalized convex spaces,” Journal of Nonlinear and Convex Analysis, vol. 2, no. 2, pp. 157–166, 2001. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. H. Komiya, “Convexity on a topological space,” Fundamenta Mathematicae, vol. 111, no. 2, pp. 107–113, 1981. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. M. Lassonde, “On the use of KKM multifunctions in fixed point theory and related topics,” Journal of Mathematical Analysis and Applications, vol. 97, no. 1, pp. 151–201, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. C. D. Horvath, “Contractibility and generalized convexity,” Journal of Mathematical Analysis and Applications, vol. 156, no. 2, pp. 341–357, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. C. D. Horvath, “Extension and selection theorems in topological spaces with a generalized convexity structure,” Annales de la Faculté des Sciences de Toulouse. Mathématiques, vol. 2, no. 2, pp. 253–269, 1993. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. X. P. Ding and T.M. Ding, “KKM type theorems and generalized vector equilibrium problems in noncompact FC-spaces,” Journal of Mathematical Analysis and Applications, vol. 331, no. 2, pp. 1230–1245, 2006, (French). View at Publisher · View at Google Scholar · View at MathSciNet
  27. Q.-B. Zhang and C.-Z. Cheng, “Some fixed-point theorems and minimax inequalities in FC-space,” Journal of Mathematical Analysis and Applications, vol. 328, no. 2, pp. 1369–1377, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet