Existence and Uniqueness Results for Fuzzy Bipolar Metric Spaces

Ishtiaq, Umar; Jahangeer, Fahad; Garayev, Mubariz; Popa, Ioan-Lucian

doi:10.3390/sym17020180

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Existence and Uniqueness Results for Fuzzy Bipolar Metric Spaces

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Office of Research, Innovation and Commercialization, University of Management and Technology, Lahore 54770, Pakistan

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Department of Mathematics and Statistics, International Islamic University Islamabad, Islamabad 44000, Pakistan

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Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

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Department of Computing, Mathematics and Electronics, “1 Decembrie 1918” University of Alba Iulia, 510009 Alba Iulia, Romania

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Faculty of Mathematics and Computer Science, Transilvania University of Brasov, Iuliu Maniu Street 50, 500091 Brasov, Romania

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Authors to whom correspondence should be addressed.

Symmetry 2025, 17(2), 180; https://doi.org/10.3390/sym17020180

Submission received: 20 December 2024 / Revised: 18 January 2025 / Accepted: 22 January 2025 / Published: 24 January 2025

(This article belongs to the Special Issue Nonlinear Analysis and Applications, Geometry of Banach Spaces and Symmetry II)

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Abstract

In this paper, we present the concept of (Υ,Ω)-iterativemappings in the setting of fuzzy bipolar metric space. The symmetric property in fuzzy bipolar metric spaces guarantees that the distance between any two elements remains invariant under permutation, ensuring consistency and uniformity in measurement regardless of the order in which the elements are considered. Furthermore, we prove several best proximity point results by utilizing (Υ,Ω)-fuzzy bipolar proximal contraction, (Υ,Ω)-Reich–Rus–Ciric type proximal contraction, (Υ,Ω)-Kannan type proximal contraction and (Υ,Ω)-Hardy–Rogers type contraction. Furthermore, we provide some non-trivial examples to show the comparison with the existing results in the literature. At the end, we present an application to find the existence and uniqueness of a solution of an integral equation by applying the main result.

Keywords: fuzzy bipolar metric space; iterative mappings; proximal contractions; integral equations

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MDPI and ACS Style

Ishtiaq, U.; Jahangeer, F.; Garayev, M.; Popa, I.-L. Existence and Uniqueness Results for Fuzzy Bipolar Metric Spaces. Symmetry 2025, 17, 180. https://doi.org/10.3390/sym17020180

AMA Style

Ishtiaq U, Jahangeer F, Garayev M, Popa I-L. Existence and Uniqueness Results for Fuzzy Bipolar Metric Spaces. Symmetry. 2025; 17(2):180. https://doi.org/10.3390/sym17020180

Chicago/Turabian Style

Ishtiaq, Umar, Fahad Jahangeer, Mubariz Garayev, and Ioan-Lucian Popa. 2025. "Existence and Uniqueness Results for Fuzzy Bipolar Metric Spaces" Symmetry 17, no. 2: 180. https://doi.org/10.3390/sym17020180

APA Style

Ishtiaq, U., Jahangeer, F., Garayev, M., & Popa, I.-L. (2025). Existence and Uniqueness Results for Fuzzy Bipolar Metric Spaces. Symmetry, 17(2), 180. https://doi.org/10.3390/sym17020180

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Existence and Uniqueness Results for Fuzzy Bipolar Metric Spaces

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