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Analytical Meir–Keeler type contraction mappings and equivalent characterizations

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

The aim of this paper is to obtain a fixed point theorem which gives a new solution to the Rhoades’ problem on the existence of contractive mappings that admit discontinuity at the fixed point; and it is the first Meir–Keeler type solution of this problem. We prove that our theorem characterizes the completeness of the metric space. We also give the structure of complete subspaces of the real line in which contractive mappings do not admit discontinuity at the fixed point and, thus, in the setting of the real line we completely resolve the Rhoades’ question.

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Acknowledgements

The first author is thankful to Professor M. C. Joshi, Kumaun University, Nainital, for his valuable suggestions and encouragement during the course of these investigations. This work was supported by Thammasat University Research Unit in Fixed Points and Optimization.

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Authors and Affiliations

  1. Department of Mathematics, D. S. B. Campus, Kumaun University, Nainital, India

    Abhijit Pant

  2. Graphic Era Hill University, Nainital, India

    Rajendra Prasad Pant

  3. Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University Rangsit Center, Pathum Thani, 12120, Thailand

    Wutiphol Sintunavarat

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  1. Abhijit Pant
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  2. Rajendra Prasad Pant
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  3. Wutiphol Sintunavarat
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Correspondence to Wutiphol Sintunavarat.

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Pant, A., Pant, R.P. & Sintunavarat, W. Analytical Meir–Keeler type contraction mappings and equivalent characterizations. RACSAM 115, 37 (2021). https://doi.org/10.1007/s13398-020-00939-8

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