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On Discontinuity Problem at Fixed Point

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Abstract

We obtain some fixed point theorems under contractive conditions which admit discontinuity at the fixed point. Our results subsume all the known results of similar type, provide new answers to the question of continuity of contractive mappings at their fixed points, and also extend some recent results. Furthermore, we consider the fixed-circle problem on metric spaces. We obtain new fixed-circle results as the generalizations of some known fixed-circle theorems. Some applications of obtained results to discontinuous activation functions are also given.

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Acknowledgements

The authors would like to thank the anonymous referee for his/her comments that helped us improve this article. The second and third authors were supported by Balıkesir University Research Grant No: 2018/019.

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Authors and Affiliations

  1. Department of Mathematics, Kumaun University, Nainital, India

    R. P. Pant

  2. Department of Mathematics, Balıkesir University, 10145, Balıkesir, Turkey

    Nihal Yilmaz Özgür & Nihal Taş

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  1. R. P. Pant
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  2. Nihal Yilmaz Özgür
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  3. Nihal Taş
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Correspondence to Nihal Yilmaz Özgür.

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Rosihan M. Ali.

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Pant, R.P., Özgür, N.Y. & Taş, N. On Discontinuity Problem at Fixed Point. Bull. Malays. Math. Sci. Soc. 43, 499–517 (2020). https://doi.org/10.1007/s40840-018-0698-6

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  • DOI: https://doi.org/10.1007/s40840-018-0698-6

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