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Solution to the Rhoades’ problem under minimal metric structure

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Abstract

An open problem proposed by Rhoades (Contemp Math 72:233–245, 1988) is the following, “Is there a contractive condition that guarantees a fixed point’s existence but does not require the mapping to be continuous at that point?” In this paper, we generalize a result of Bisht (J Fixed Point Theory Appl 25:11, 2023), which allows us to find a new solution to this open problem. Furthermore, we have validated the result generated in the article by producing several examples.

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Acknowledgements

The authors thank Editor-in-Chief/Area Editors and Referee(s) for their valuable comments and suggestions, which were very much useful to improve the paper significantly. This work was initiated while the second author (Dr. Gopal) was working as visiting Professor at the Department of Mathematics, University of Jean Spain during February 20 to July 14, 2023. He thanks administration of Guru Ghasidas Vishawavidyala for granting extraordinary leave for the mentioned period. He also thank University of Jean for their kind hospitality and support.

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No funding was received to assist with the preparation of this manuscript.

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Author notes

  1. Dhananjay Gopal, Juan Martinez Moreno and Shailesh Kumar Srivastava have contributed equally to this work.

Authors and Affiliations

  1. Department of Mathematics, Sardar Vallabhbhai National Institute of Technology, Surat, 395007, India

    Jayesh Savaliya & Shailesh Kumar Srivastava

  2. Department of Mathematics, Guru Ghasidas Vishwavidyalaya, Bilaspur, 495009, India

    Dhananjay Gopal

  3. Department of Mathematics, University of Jaen, Jaen, 23071, Spain

    Juan Martinez Moreno

Authors
  1. Jayesh Savaliya
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  2. Dhananjay Gopal
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  3. Juan Martinez Moreno
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  4. Shailesh Kumar Srivastava
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Correspondence to Jayesh Savaliya.

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Communicated by Simeon Reich.

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Savaliya, J., Gopal, D., Moreno, J.M. et al. Solution to the Rhoades’ problem under minimal metric structure. J Anal (2024). https://doi.org/10.1007/s41478-024-00722-7

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  • DOI: https://doi.org/10.1007/s41478-024-00722-7

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