Fixed Point Theory and Applications
Volume 2008 (2008), Article ID 164537, 18 pages
Research Article

On Krasnoselskii's Cone Fixed Point Theorem

Man Kam Kwong1,2

1Department of Applied Mathematics, The Hong Kong Polytechnic University, Hunghom, Hong Kong
2Department of Mathematics, Statistics, and Computer Science, University of Illinois, Chicago, IL 60607-7045, USA

Received 27 August 2007; Accepted 5 March 2008

Academic Editor: Jean Mawhin

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.


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.