Open Access

On Krasnoselskii's Cone Fixed Point Theorem

Fixed Point Theory and Applications20082008:164537

https://doi.org/10.1155/2008/164537

Received: 27 August 2007

Accepted: 5 March 2008

Published: 17 March 2008

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.

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Authors’ Affiliations

(1)
Department of Applied Mathematics, The Hong Kong Polytechnic University
(2)
Department of Mathematics, Statistics, and Computer Science, University of Illinois

Copyright

© Man Kam Kwong. 2008

This article is published under license to BioMed Central Ltd. 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.