A new type of $ \mathcal{R} $-contraction and its best proximity points

Mustafa Aslantas; Hakan Sahin; Ishak Altun; Taif Hameed SAADOON SAADOON; Mustafa Aslantas; Hakan Sahin; Ishak Altun; Taif Hameed SAADOON SAADOON

doi:10.3934/math.2024474

AIMS Mathematics

2024, Volume 9, Issue 4: 9692-9704. doi: 10.3934/math.2024474

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Research article

A new type of $ \mathcal{R} $-contraction and its best proximity points

1.
Department of Mathematics, Faculty of Science, Çankırı Karatekin University, 18100 Çankırı, Turkey
2.
Department of Mathematics, Faculty of Engineering and Natural Sciences, Bursa Technical University, 16310 Bursa, Turkey
3.
Department of Mathematics, Faculty of Engineering and Natural Sciences, Kirikkale University, Yahsihan 71450, Kirikkale, Turkey

Received: 11 December 2023 Revised: 28 February 2024 Accepted: 04 March 2024 Published: 11 March 2024
MSC : Primary 54H25; Secondary 47H10, 14F35

In this paper, we aim to overcome the problem given by Abkar et al. [Abstr. Appl. Anal., 2013 (2013), 189567], and so to obtain real generalizations of fixed point results in the literature. In this direction, we introduce a new class of functions, which include $ \mathcal{R} $-functions. Thus, we present a new type of $ \mathcal{R} $ -contraction and weaken $ \mathcal{R} $-contractions that have often been studied recently. We also give a new definition of the $ P $-property. Hence, we obtain some best proximity point results, including fixed point results for the new kind of $ \mathcal{R} $-contractions. Then, we provide an example to show the effectiveness of our results. Finally, inspired by a nice and interesting technique, we investigate the existence of a best proximity point of the homotopic mappings with the help of our main result.
- $ \mathcal{R} $-contractions,
- homotopy,
- best proximity point,
- fixed point
Citation: Mustafa Aslantas, Hakan Sahin, Ishak Altun, Taif Hameed SAADOON SAADOON. A new type of $ \mathcal{R} $-contraction and its best proximity points[J]. AIMS Mathematics, 2024, 9(4): 9692-9704. doi: 10.3934/math.2024474

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Abstract

In this paper, we aim to overcome the problem given by Abkar et al. [Abstr. Appl. Anal., 2013 (2013), 189567], and so to obtain real generalizations of fixed point results in the literature. In this direction, we introduce a new class of functions, which include $ \mathcal{R} $-functions. Thus, we present a new type of $ \mathcal{R} $ -contraction and weaken $ \mathcal{R} $-contractions that have often been studied recently. We also give a new definition of the $ P $-property. Hence, we obtain some best proximity point results, including fixed point results for the new kind of $ \mathcal{R} $-contractions. Then, we provide an example to show the effectiveness of our results. Finally, inspired by a nice and interesting technique, we investigate the existence of a best proximity point of the homotopic mappings with the help of our main result.

References

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