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The weakest contractive conditions for Edelstein’s mappings to have a fixed point in complete metric spaces

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Abstract

In this paper, we give the answer to the following problem: Let (X, d) be a complete metric space and let T be a mapping on X satisfying \(d(Tx, Ty) < d(x, y)\) for any \(x, y \in X\) with \(x \ne y\). Then what are the weakest additional assumptions to imply the same conclusion as in the Banach contraction principle?

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Acknowledgements

The author is very grateful to the referee for his/her careful reading.

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Conflict of interest The author declares that he has no competing interests.

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Authors and Affiliations

  1. Department of Basic Sciences, Faculty of Engineering, Kyushu Institute of Technology, Tobata, Kitakyushu, 804-8550, Japan

    Tomonari Suzuki

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  1. Tomonari Suzuki
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Correspondence to Tomonari Suzuki.

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The author is supported in part by JSPS KAKENHI Grant Number 16K05207 from Japan Society for the Promotion of Science.

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Suzuki, T. The weakest contractive conditions for Edelstein’s mappings to have a fixed point in complete metric spaces. J. Fixed Point Theory Appl. 19, 2361–2368 (2017). https://doi.org/10.1007/s11784-017-0430-4

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