Open Access

Remarks on Extensions of the Himmelberg Fixed Point Theorem

Fixed Point Theory and Applications20072007:016028

https://doi.org/10.1155/2007/16028

Received: 30 August 2007

Accepted: 16 November 2007

Published: 13 December 2007

Abstract

Recently, Jafari and Sehgal obtained an extension of the Himmelberg fixed point theorem based on the Kakutani fixed-point theorem. We give generalizations of the extension to almost convex sets instead of convex sets. We also give generalizations for a large class of better admissible multimaps instead of the Kakutani maps. Our arguments are based on the KKM principle and some of previous results due to the second author.

[12345678]

Authors’ Affiliations

(1)
Faculty of Business and Commerce, Keio University, Hiyoshi
(2)
The National Academy of Sciences, Seocho-gu
(3)
Department of Mathematical Sciences, College of Natural Science, Seoul National University

References

  1. Himmelberg CJ: Fixed points of compact multifunctions. Journal of Mathematical Analysis and Applications 1972, 38: 205–207. 10.1016/0022-247X(72)90128-XMathSciNetView ArticleGoogle Scholar
  2. Park S: Ninety years of the Brouwer fixed point theorem. Vietnam Journal of Mathematics 1999,27(3):187–222.MATHMathSciNetGoogle Scholar
  3. Jafari F, Sehgal VM: An extension to a theorem of Himmelberg. Journal of Mathematical Analysis and Applications 2007,327(1):298–301. 10.1016/j.jmaa.2006.04.048MATHMathSciNetView ArticleGoogle Scholar
  4. Park S: The Knaster-Kuratowski-Mazurkiewicz theorem and almost fixed points. Topological Methods in Nonlinear Analysis 2000,16(1):195–200.MATHMathSciNetGoogle Scholar
  5. Park S: Fixed point theorems for better admissible multimaps on almost convex sets. Journal of Mathematical Analysis and Applications 2007,329(1):690–702. 10.1016/j.jmaa.2006.06.066MATHMathSciNetView ArticleGoogle Scholar
  6. Park S: The KKM principle implies many fixed point theorems. Topology and Its Applications 2004,135(1–3):197–206.MATHMathSciNetView ArticleGoogle Scholar
  7. Park S, Tan DH: Remarks on Himmelberg-Idzik's fixed point theorem. Acta Mathematica Vietnamica 2000,25(3):285–289.MATHMathSciNetGoogle Scholar
  8. Ben-El-Mechaiekh H: Spaces and maps approximation and fixed points. Journal of Computational and Applied Mathematics 2000,113(1–2):283–308. 10.1016/S0377-0427(99)00262-9MATHMathSciNetView ArticleGoogle Scholar

Copyright

© H. Komiya and S. Park. 2007

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.