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Cyclic uniform Lipschitzian mappings and proximal uniform normal structure

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Abstract

For \(x\in A\cup B,\) define \(d_n=d\left( T^nx,T^{n-1}x\right) , n\ge 1\) where A and B are subsets of a metric space and T is a cyclic map on \(A\cup B\). In this paper, we introduce a new class of mappings called cyclic uniform Lipschitzian mappings for which \(\{d_n\}\) is not necessarily a non-increasing sequence and therein prove the existence of a best proximity pair. We also introduce a notion called proximal uniform normal structure and using the same we prove the existence of a best proximity pair for such mappings. Some open problems in this direction are also discussed.

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Acknowledgements

The authors sincerely thank to the reviewers for their valuable comments and constructive suggestions for the improvement of the manuscript.

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Authors and Affiliations

  1. Department of Mathemtics, IIT Ropar, Rupnagar, Punjab, 140 001, India

    Abhik Digar & G. Sankara Raju Kosuru

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  1. Abhik Digar
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  2. G. Sankara Raju Kosuru
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Correspondence to Abhik Digar.

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Communicated by Patrick Dowling.

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Digar, A., Kosuru, G.S.R. Cyclic uniform Lipschitzian mappings and proximal uniform normal structure. Ann. Funct. Anal. 13, 5 (2022). https://doi.org/10.1007/s43034-021-00152-7

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  • DOI: https://doi.org/10.1007/s43034-021-00152-7

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