Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-27T23:57:02.496Z Has data issue: false hasContentIssue false
  • English
  • Français

A PURELY METRIC PROOF OF THE CARISTI FIXED POINT THEOREM

Part of: Nonlinear operators and their properties

Published online by Cambridge University Press:  02 November 2016

WOJCIECH M. KOZLOWSKI
WOJCIECH M. KOZLOWSKI*
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia email [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we prove Caristi’s fixed point theorem using only purely metric techniques.

Keywords

fixed pointCaristi fixed point theoremnonexpansive mappingcontraction

MSC classification

Primary: 47H09: Contraction-type mappings, nonexpansive mappings, $A$-proper mappings, etc.
Secondary: 47H10: Fixed-point theorems
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 95 , Issue 2 , April 2017 , pp. 333 - 337
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

References

Brøndsted, A., ‘On a lemma of Bishop and Phelps’, Pacific J. Math. 55 (1974), 335–341.Google Scholar
Caristi, J., ‘Fixed point theorems for mappings satisfying inwardness conditions’, Trans. Amer. Math. Soc. 215 (1976), 241–251.Google Scholar
Cobzas, S., ‘Fixed points in metric and generalized metric spaces’, Preprint, 2015, arXiv:1508.05173v1.Google Scholar
Downing, D. and Kirk, W. A., ‘A generalization of Caristi’s theorem with applications to nonlinear mapping theory’, Pacific J. Math. 69(2) (1977), 339–346.Google Scholar
Du, W.-S., ‘On Caristi-type mappings without semicontinuity assumptions’, J. Fixed Point Theory Appl. 17(4) (2015), 733–752.CrossRefGoogle Scholar
Ekeland, I., ‘On the variational principle’, J. Math. Anal. Appl. 47 (1974), 324–353.CrossRefGoogle Scholar
Gamir, G. H., ‘Solovay’s axiom and functional analysis’, in: Proceedings of the Symposium on Functional Analysis (Istanbul, 1973), Publication of the Mathematical Research Institute, Istanbul, 1 (Mathematical Research Institute, Istanbul, 1974), 57–68.Google Scholar
Jachymski, J. R., ‘Caristi’s fixed point theorem and selections of set-valued contractions’, J. Math. Anal. Appl. 227 (1998), 55–67.Google Scholar
Khamsi, M. A., ‘Remarks on Caristi’s fixed point theorem’, Nonlinear Anal. 71(2) (2009), 227–231.Google Scholar
Khamsi, M. A., ‘Introduction to metric fixed point theory’, in: Topics in Fixed Point Theory (eds. Almezel, S., Ansari, Q. H. and Khamsi, M. A.) (Springer, New York–Heidelberg–Dordrecht–London, 2014), 1–32.Google Scholar
Khamsi, M. A. and Kozlowski, W. M., Fixed Point Theory in Modular Function Spaces (Birkhauser–Springer International, Basel, 2015).Google Scholar
Kirk, W. A., ‘Caristi’s fixed point theorem and metric convexity’, Colloq. Math. 36 (1976), 81–86.Google Scholar
Kirk, W. A., ‘Metric fixed point theory: a brief retrospective’, Fixed Point Theory Appl. 2015 (2015), article 215.Google Scholar
Oettli, W. and Thera, M., ‘Equivalents of Ekeland’s principle’, Bull. Aust. Math. Soc. 48 (1993), 385–392.CrossRefGoogle Scholar
van Rooij, A. C. M., ‘The axiom of choice in p-adic functional analysis’, in: p-Adic Functional Analysis (Laredo, 1990), Lecture Notes in Pure and Applied Mathematics, 137 (Dekker, New York, 1992), 151–156.Google Scholar