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Topological Methods in Nonlinear Analysis

Existence of a fixed point and stability results for contractive mappings on metric spaces with graphs
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Existence of a fixed point and stability results for contractive mappings on metric spaces with graphs

Authors

  • Simeon Reich https://orcid.org/0000-0003-0780-1559
  • Alexander J. Zaslavski

DOI:

https://doi.org/10.12775/TMNA.2023.045

Keywords

Complete metric space, contractive mapping, fixed point, graph

Abstract

It is known that a strict contraction on complete metric spaces with graphs possesses a fixed point. In the present paper we show that this property holds for mappings on complete metric spaces with graphs which are merely contractive. We also obtain some stability results.

References

S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math. 3 (1922), 133–181.

F.S. de Blasi and J. Myjak, Sur la convergence des approximations successives pour les contractions non linéaires dans un espace de Banach, C.R. Acad. Sci. Paris 283 (1976), 185–187.

K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge University Press, Cambridge, 1990.

K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, New York and Basel, 1984.

G. Gwóźdź-Lukawska and J. Jachymski, IFS on a metric space with a graph structure and extensions of the Kelisky-Rivlin theorem, J. Math. Anal. Appl. 356 (2009), 453–463.

J. Jachymski, The contraction principle for mappings on a metric space with a graph, Proc. Amer. Math. Soc. 136 (2008), 1359–1373.

W.A. Kirk, Contraction Mappings and Extensions, Handbook of Metric Fixed Point Theory, Kluwer, Dordrecht, 2001, pp. 1–34.

A. Nicolae, D. O’Regan and A. Petruşel, Fixed point theorems for singlevalued and multivalued generalized contractions in metric spaces endowed with a graph, Georgian Math. J. 18 (2011), 307–327.

A. Petruşel, G. Petruşel and J.-C. Yao, Multi-valued graph contraction principle with applications, Optimization 69 (2020), 1541–1556.

A. Petruşel, G. Petruşel and J.-C. Yao, Graph contractions in vector-valued metric spaces and applications, Optimization 70 (2021), 763–775.

E. Rakotch, A note on contractive mappings, Proc. Amer. Math. Soc. 13 (1962), 459–465.

S. Reich and I. Shafrir, Nonexpansive iterations in hyperbolic spaces, Nonlinear Analysis 15 (1990), 537–558.

S. Reich and A.J. Zaslavski, Genericity in Nonlinear Analysis, Developments in Mathematics,vol. 34, Springer, New York, 2014.

S. Reich and A.J. Zaslavski, Contractive mappings on metric spaces with graphs, Mathematics 9 (2021), 2774, DOI: 10.3390/math9212774.

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Published

2024-03-03

How to Cite

1.
REICH, Simeon and ZASLAVSKI, Alexander J. Existence of a fixed point and stability results for contractive mappings on metric spaces with graphs. Topological Methods in Nonlinear Analysis. Online. 3 March 2024. Vol. 63, no. 1, pp. 233 - 244. [Accessed 13 December 2025]. DOI 10.12775/TMNA.2023.045.
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Issue

Vol 63, No 1 (March 2024)

Section

Articles

License

Copyright (c) 2024 Simeon Reich, Alexander J. Zaslavski

Creative Commons License

This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.

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