Open Access

Coincidence and fixed point theorems for functions in -KKM class on generalized convex spaces

  • Tian-Yuan Kuo1Email author,
  • Young-Ye Huang2,
  • Jyh-Chung Jeng3 and
  • Chen-Yuh Shih4
Fixed Point Theory and Applications20062006:72184

https://doi.org/10.1155/FPTA/2006/72184

Received: 25 October 2004

Accepted: 1 September 2005

Published: 26 February 2006

Abstract

We establish a coincidence theorem in -KKM class by means of the basic defining property for multifunctions in -KKM. Based on this coincidence theorem, we deduce some useful corollaries and investigate the fixed point problem on uniform spaces.

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Authors’ Affiliations

(1)
Fooyin University
(2)
Center for General Education, Southern Taiwan University of Technology
(3)
Nan-Jeon Institute of Technology
(4)
Department of Mathmatics, Cheng Kung University

References

  1. Browder FE: The fixed point theory of multi-valued mappings in topological vector spaces. Mathematische Annalen 1968, 177: 283–301. 10.1007/BF01350721MathSciNetView ArticleMATHGoogle Scholar
  2. Browder FE: Coincidence theorems, minimax theorems, and variational inequalities. In Conference in Modern Analysis and Probability (New Haven, Conn, 1982), Contemp. Math.. Volume 26. American Mathematical Society, Rhode Island; 1984:67–80.View ArticleGoogle Scholar
  3. Chang T-H, Huang Y-Y, Jeng J-C, Kuo K-H: On -KKM property and related topics. Journal of Mathematical Analysis and Applications 1999,229(1):212–227. 10.1006/jmaa.1998.6154MathSciNetView ArticleMATHGoogle Scholar
  4. Chang T-H, Yen C-L: KKM property and fixed point theorems. Journal of Mathematical Analysis and Applications 1996,203(1):224–235. 10.1006/jmaa.1996.0376MathSciNetView ArticleMATHGoogle Scholar
  5. Fan K: A generalization of Tychonoff's fixed point theorem. Mathematische Annalen 1960/1961, 142: 305–310.MathSciNetView ArticleMATHGoogle Scholar
  6. Granas A, Liu FC: Coincidences for set-valued maps and minimax inequalities. Journal de Mathématiques Pures et Appliquées. Neuvième Série(9) 1986,65(2):119–148.MathSciNetMATHGoogle Scholar
  7. Horvath CD: Contractibility and generalized convexity. Journal of Mathematical Analysis and Applications 1991,156(2):341–357. 10.1016/0022-247X(91)90402-LMathSciNetView ArticleMATHGoogle Scholar
  8. Kelley JL: General Topology. D. Van Nostrand, Toronto; 1955:xiv+298.Google Scholar
  9. Lassonde M: On the use of KKM multifunctions in fixed point theory and related topics. Journal of Mathematical Analysis and Applications 1983,97(1):151–201. 10.1016/0022-247X(83)90244-5MathSciNetView ArticleMATHGoogle Scholar
  10. Lin L-J, Park S: On some generalized quasi-equilibrium problems. Journal of Mathematical Analysis and Applications 1998,224(2):167–181. 10.1006/jmaa.1998.5964MathSciNetView ArticleMATHGoogle Scholar
  11. Park S: Foundations of the KKM theory via coincidences of composites of upper semicontinuous maps. Journal of the Korean Mathematical Society 1994,31(3):493–519.MathSciNetMATHGoogle Scholar
  12. Park S, Kim H: Coincidence theorems for admissible multifunctions on generalized convex spaces. Journal of Mathematical Analysis and Applications 1996,197(1):173–187. 10.1006/jmaa.1996.0014MathSciNetView ArticleMATHGoogle Scholar
  13. Park S, Kim H: Foundations of the KKM theory on generalized convex spaces. Journal of Mathematical Analysis and Applications 1997,209(2):551–571. 10.1006/jmaa.1997.5388MathSciNetView ArticleMATHGoogle Scholar
  14. Watson PJ: Coincidences and fixed points in locally G-convex spaces. Bulletin of the Australian Mathematical Society 1999,59(2):297–304. 10.1017/S0004972700032901MathSciNetView ArticleMATHGoogle Scholar

Copyright

© Tian-Yuan Kuo et al. 2006

This article is published under license to BioMed Central Ltd. 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.