TAILIEUCHUNG - Báo cáo hóa học: "Research Article On Krasnoselskii’s Cone Fixed Point Theorem"

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 On Krasnoselskii’s Cone Fixed Point Theorem | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2008 Article ID 164537 18 pages doi 2008 164537 Research Article On Krasnoselskii s Cone Fixed Point Theorem Man Kam Kwong1 2 1 Department of Applied Mathematics The Hong Kong Polytechnic University Hunghom Hong Kong 2 Department of Mathematics Statistics and Computer Science University of Illinois Chicago IL 60607-7045 USA Correspondence should be addressed to Man Kam Kwong mkkwong@ Received 27 August 2007 Accepted 5 March 2008 Recommended by Jean Mawhin In recent years the Krasnoselskii fixed point theorem for cone maps and its many generalizations have been successfully applied to establish the existence of multiple solutions in the study of boundary value problems of various types. In the first part of this paper we revisit the Krasnoselskii theorem in a more topological perspective and show that it can be deduced in an elementary way from the classical Brouwer-Schauder theorem. This viewpoint also leads to a topology-theoretic generalization of the theorem. In the second part of the paper we extend the cone theorem in a different direction using the notion of retraction and show that a stronger form of the often cited Leggett-Williams theorem is a special case of this extension. Copyright 2008 Man Kam Kwong. 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 The classical Brouwer-Schauder fixed point theorem is an undeniably important tool in the study of the existence of solutions to mathematical problems . see 1-3 . In recent years another fixed point theorem due to Krasnoselskii 4 5 and its generalizations have been successfully applied to obtain existence results for multiple positive solutions of various types of boundary value problems notably in the case of ordinary differential equations .

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