Generalized common best proximity point results in fuzzy multiplicative metric spaces

Umar Ishtiaq; Fahad Jahangeer; Doha A. Kattan; Manuel De la Sen; Umar Ishtiaq; Fahad Jahangeer; Doha A. Kattan; Manuel De la Sen

doi:10.3934/math.20231299

AIMS Mathematics

2023, Volume 8, Issue 11: 25454-25476. doi: 10.3934/math.20231299

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Research article

Generalized common best proximity point results in fuzzy multiplicative metric spaces

1.
Office of Research, Innovation and Commercialization, University of Management and Technology, Lahore 54770, Pakistan
2.
Department of Mathematics and Statistics, International Islamic University, Islamabad, Pakistan
3.
Department of Mathematics, Faculty of Sciences and Arts, King Abdulaziz University, Rabigh, Saudi Arabia
4.
Department of Electricity and Electronics, Institute of Research and Development of Processes, University of the Basque Country, Campus of Leioa, Leioa, Bizkaia, Spain

Received: 19 July 2023 Revised: 22 August 2023 Accepted: 23 August 2023 Published: 31 August 2023
MSC : 26E05, 26E25, 47H10

In this manuscript, we prove the existence and uniqueness of a common best proximity point for a pair of non-self mappings satisfying the iterative mappings in a complete fuzzy multiplicative metric space. We consider the pair of non-self mappings $ X:\mathcal{P}\rightarrow \mathcal{G} $ and $ Z:\mathcal{P }\rightarrow \mathcal{G} $ and the mappings do not necessarily have a common fixed-point. In a complete fuzzy multiplicative metric space, if $ \mathcal{\varphi } $ satisfy the condition $ \mathcal{\varphi } \left(b, Zb, \varsigma \right) = \mathcal{\varphi }\left(\mathcal{P}, \mathcal{ G}, \varsigma \right) = \mathcal{\varphi }\left(b, Xb, \varsigma \right) $, then $ b $ is a common best proximity point. Further, we obtain the common best proximity point for the real valued functions $ \mathcal{L}, \mathcal{M}:(0, 1]\rightarrow \mathbb{R} $ by using a generalized fuzzy multiplicative metric space in the setting of $ (\mathcal{L}, \mathcal{M}) $-iterative mappings. Furthermore, we utilize fuzzy multiplicative versions of the $ (\mathcal{L}, \mathcal{M}) $-proximal contraction, $ (\mathcal{L}, \mathcal{M}) $-interpolative Reich-Rus-Ciric type proximal contractions, $ (\mathcal{L}, \mathcal{M}) $-Kannan type proximal contraction and $ (\mathcal{L}, \mathcal{M}) $-interpolative Hardy-Rogers type proximal contraction to examine the common best proximity points in fuzzy multiplicative metric space. Moreover, we provide differential non-trivial examples to support our results.
- fuzzy multiplicative metric space,
- fixed-point,
- best proximity point theorems,
- contraction mappings,
- uniqueness
Citation: Umar Ishtiaq, Fahad Jahangeer, Doha A. Kattan, Manuel De la Sen. Generalized common best proximity point results in fuzzy multiplicative metric spaces[J]. AIMS Mathematics, 2023, 8(11): 25454-25476. doi: 10.3934/math.20231299

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Abstract

In this manuscript, we prove the existence and uniqueness of a common best proximity point for a pair of non-self mappings satisfying the iterative mappings in a complete fuzzy multiplicative metric space. We consider the pair of non-self mappings $ X:\mathcal{P}\rightarrow \mathcal{G} $ and $ Z:\mathcal{P }\rightarrow \mathcal{G} $ and the mappings do not necessarily have a common fixed-point. In a complete fuzzy multiplicative metric space, if $ \mathcal{\varphi } $ satisfy the condition $ \mathcal{\varphi } \left(b, Zb, \varsigma \right) = \mathcal{\varphi }\left(\mathcal{P}, \mathcal{ G}, \varsigma \right) = \mathcal{\varphi }\left(b, Xb, \varsigma \right) $, then $ b $ is a common best proximity point. Further, we obtain the common best proximity point for the real valued functions $ \mathcal{L}, \mathcal{M}:(0, 1]\rightarrow \mathbb{R} $ by using a generalized fuzzy multiplicative metric space in the setting of $ (\mathcal{L}, \mathcal{M}) $-iterative mappings. Furthermore, we utilize fuzzy multiplicative versions of the $ (\mathcal{L}, \mathcal{M}) $-proximal contraction, $ (\mathcal{L}, \mathcal{M}) $-interpolative Reich-Rus-Ciric type proximal contractions, $ (\mathcal{L}, \mathcal{M}) $-Kannan type proximal contraction and $ (\mathcal{L}, \mathcal{M}) $-interpolative Hardy-Rogers type proximal contraction to examine the common best proximity points in fuzzy multiplicative metric space. Moreover, we provide differential non-trivial examples to support our results.

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