Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Remarks on Extensions of the Himmelberg Fixed Point Theorem | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2007 Article ID 16028 5 pages doi 2007 16028 Research Article Remarks on Extensions of the Himmelberg Fixed Point Theorem Hidetoshi Komiya and Sehie Park Received 30 August 2007 Accepted 16 November 2007 Recommended by Anthony To-Ming Lau 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 B 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. Copyright 2007 H. Komiya and S. 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. 1. Introduction In 1972 Himmelberg 1 derived the following from the Kakutani fixed point theorem. Theorem . Let T be a nonvoid convex subset of a separated locally convex space L. Let F T - T be a . multimap such that F x is closed and convex for all x e T and F T is contained in some compact subset C of T. Then F has a fixed point. Recall that Theorem is usually called the Himmelberg fixed point theorem and is a common generalization of historically well-known fixed point theorems due to Brouwer Schauder Tychonoff Kakutani Bohnenblust and Karlin Fan Glicksberg and Hukuhara see 2 . Recall also that the multimap F is usually called a Kakutani map. Recently Jafari and Sehgal 3 obtained an extension of Theorem based on the Kakutani fixed point theorem. Our aim in this paper is to give generalizations of the extension to almost convex sets instead of convex sets. We also give generalizations for a large class B of better admissible multimaps instead of the Kakutani maps. Our
