Remarks on Caristi’s fixed point theorem

In this work, we give a characterization of the existence of minimal elements in partially ordered sets in terms of fixed points of multivalued maps. This characterization shows that the assumptions in Caristi’s fixed point theorem can, a priori, be weakened. Finally, we discuss Kirk’s problem on an...

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Bibliographic Details
Published in:Nonlinear analysis 2009-07, Vol.71 (1), p.227-231
Main Author: Khamsi, M.A.
Format: Article
Language:English
Subjects:
Contraction and nonexpansive mappings
Exact sciences and technology
Fixed point
Global analysis, analysis on manifolds
Mathematical analysis
Mathematics
Minimal element
Multivalued mappings
Order preserving mappings
Partially ordered sets
Sciences and techniques of general use
Subadditive functions
Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
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Summary:In this work, we give a characterization of the existence of minimal elements in partially ordered sets in terms of fixed points of multivalued maps. This characterization shows that the assumptions in Caristi’s fixed point theorem can, a priori, be weakened. Finally, we discuss Kirk’s problem on an extension of Caristi’s theorem and prove a new positive result which illustrates the weakening mentioned before.
ISSN:0362-546X
1873-5215
DOI:10.1016/j.na.2008.10.042