Korean J. Math. Vol. 32 No. 4 (2024) pp.759-768
DOI: https://doi.org/10.11568/kjm.2024.32.4.759

A note on best proximity points for $F$-contractive non-self mappings

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Published: Dec 30, 2024
Keywords:
Best proximity point, fixed point, $F$-contraction, $F$-proximal contraction of the first kind, $F$-proximal contraction of the second kind
Mathematics Subject Classification:
47H10, 54H25

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Sumit Som
Mathematics Division, School of Advanced Sciences and Languages (SASL), VIT Bhopal University, Kothrikalan, Sehore, Madhya Pradesh-466114, India.

Abstract

In the year 2012, Wardowski [Wardowski, D., Fixed Point Theory Appl., 94 (2012), 6pp] introduced the notion of $F$-contraction mapping and presented a fixed point result on complete metric space which generalized the Banach contraction principle. Then, in the year 2014, Omidvari et al. [Omidvari, M., Vaezpour, S.M., Saadati, R., Miskolc Math Notes, 15 (2014), 615-623] considered the concept of $F$-contraction non-self mappings and presented a best proximity point theorem for this class of mappings to generalize the fixed point theorem of Wardowski. In this note, we show that the existence of best proximity point for $F$-contraction non-self mappings follow from the Wardowski's fixed point theorem. Also, in this note, we provide a new version of [15, Theorem 2] where instead of considering the continuity of $F$-proximal contraction of the first kind, we use the concept of $p$-property. We apply Wardowski's fixed point theorem to prove [15, Theorem 2]. In the last part, we also prove a best proximity point result regarding $F$-proximal contraction of the second kind where we drop some conditions.


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