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On Krasnoselskii's Cone Fixed Point Theorem

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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Correspondence to Man Kam Kwong.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Kwong, M.K. On Krasnoselskii's Cone Fixed Point Theorem. Fixed Point Theory Appl 2008, 164537 (2008). https://doi.org/10.1155/2008/164537

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

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