Fixed Point Theory and Applications
Volume 2008 (2008), Article ID 164537, 18 pages
doi:10.1155/2008/164537
Research Article
On Krasnoselskii's Cone Fixed Point Theorem
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.
Abstract
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.