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 Fixed Point Theory for Admissible Type Maps with Applications | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2009 Article ID 439176 22 pages doi 2009 439176 Research Article Fixed Point Theory for Admissible Type Maps with Applications Ravi P. Agarwal1 and Donal O Regan2 1 Department of Mathematical Sciences Florida Institute of Technology Melbourne FL 32901 USA 2 Department of Mathematics National University of Ireland Galway Ireland Correspondence should be addressed to Ravi P. Agarwal agarwal@ Received 8 December 2008 Accepted 18 June 2009 Recommended by Marlene Frigon We present new Leray-Schauder alternatives Krasnoselskii and Lefschetz fixed point theory for multivalued maps between Frechet spaces. As an application we show that our results are directly applicable to establish the existence of integral equations over infinite intervals. Copyright 2009 R. P. Agarwal and D. O Regan. 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 this paper assuming a natural sequentially compact condition we establish new fixed point theorems for Urysohn type maps between Fréchet spaces. In Section 2 we present new Leray-Schauder alternatives Krasnoselskii and Lefschetz fixed point theory for admissible type maps. The proofs rely on fixed point theory in Banach spaces and viewing a Frechet space as the projective limit of a sequence of Banach spaces. Our theory is partly motivated by a variety of authors in the literature see 1-6 and the references therein . Existence in Section 2 is based on a Leray-Schauder alternative for Kakutani maps see 4 5 7 for the history of this result which we state here for the convenience of the reader. Theorem . Let B be a Banach space U an open subset of B and 0 e U. Suppose T U CK B is an upper semicontinuous compact or countably condensing map here CK B denotes the family of .