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Fixed point and periodic point theorems

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Abstract

We introduce a weaker form of continuity which is a necessary and sufficient condition for the existence of fixed points. The obtained theorems exhibit interesting fixed point – eventual fixed point patterns. If we slightly weaken the conditions then the mappings admit periodic points besides fixed points and such mappings possess interesting combinations of fixed and periodic points. Our results are applicable to contractive type as well as non-expansive type mappings. Our theorems are independent of almost all the existing results for contractive type mappings. The last theorem of Sect. 2 is applicable to mappings having various geometric patterns as their domain and is perhaps the first result of its type that also opens up scope for the study of periodic points and periodic point structures. We also give an application of our theorem to obtain the solutions of a nonlinear Diophantine equation; and also show that various well-known fixed point theorems are not applicable in solving this equation.

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Acknowledgements

The authors are thankful to the referees for their useful suggestions for the improvement of the paper.

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Authors and Affiliations

  1. Nainital, India

    R. P. Pant

  2. Faculty of Sciences and Mathematics, University of Niš, Višegradska 33, 18000, Niš, Serbia

    Vladimir Rakočevič

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  1. R. P. Pant
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  2. Vladimir Rakočevič
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All the authors contributed equally.

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Correspondence to Vladimir Rakočevič.

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Pant, R.P., Rakočevič, V. Fixed point and periodic point theorems. Acta Sci. Math. (Szeged) 90, 175–192 (2024). https://doi.org/10.1007/s44146-024-00126-w

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