The structure of fuzzy extended $ b $-metric spaces and some fixed-point theorems with applications

Yanrong An; Muhammad Aamir Ali; Jarunee Sirisin; Thanin Sitthiwirattham; Yanrong An; Muhammad Aamir Ali; Jarunee Sirisin; Thanin Sitthiwirattham

doi:10.3934/math.20251351

AIMS Mathematics

2025, Volume 10, Issue 12: 30785-30805. doi: 10.3934/math.20251351

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Research article

The structure of fuzzy extended $ b $-metric spaces and some fixed-point theorems with applications

1.
School of Management, Nanjing University of Posts and Telecommunications, Nanjing 210003, China
2.
School of Mathematics, Hohai University, Nanjing 210098, China
3.
Department of Mathematics, Faculty of Applied Science, King Mongkut's University of Technology North Bangkok, Bangkok 10800, Thailand
4.
Research Group for Fractional Calculus Theory and Applications, Science and Technology Research Institute, King Mongkut's University of Technology North Bangkok, Bangkok 10800, Thailand
5.
Mathematics Department, Faculty of Science and Technology, Suan Dusit University, Bangkok 10300, Thailand

Received: 26 September 2025 Revised: 17 December 2025 Accepted: 23 December 2025 Published: 30 December 2025
MSC : 35R13, 65M06, 65M12

This paper establishes a novel concept of fuzzy extended $ b $-metric space, which serves as an extension to both fuzzy metric spaces and extended $ b $-metric spaces. The introduction of fuzzy extended $ b $-metric spaces is motivated by the need to model uncertainty and vagueness in real-world problems, where classical metric spaces may fail to capture imprecise relationships. This framework extends traditional $ b $-metric spaces, enhancing their applicability in fuzzy environments and providing a foundation for advanced fixed-point results and applications. By utilizing the extended $ b $-comparison function, some contractive type fixed-point theorems are proved. As an application, we study the homeomorphism between the fuzzy extended $ b $-metric space and the extended $ b $-metric space. In addition, the existence and uniqueness of solutions to the Fredholm integral equation is presented.
- fuzzy extended $ b $-metric space,
- fixed point theorem,
- homeomorphism
Citation: Yanrong An, Muhammad Aamir Ali, Jarunee Sirisin, Thanin Sitthiwirattham. The structure of fuzzy extended $ b $-metric spaces and some fixed-point theorems with applications[J]. AIMS Mathematics, 2025, 10(12): 30785-30805. doi: 10.3934/math.20251351

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Abstract

This paper establishes a novel concept of fuzzy extended $ b $-metric space, which serves as an extension to both fuzzy metric spaces and extended $ b $-metric spaces. The introduction of fuzzy extended $ b $-metric spaces is motivated by the need to model uncertainty and vagueness in real-world problems, where classical metric spaces may fail to capture imprecise relationships. This framework extends traditional $ b $-metric spaces, enhancing their applicability in fuzzy environments and providing a foundation for advanced fixed-point results and applications. By utilizing the extended $ b $-comparison function, some contractive type fixed-point theorems are proved. As an application, we study the homeomorphism between the fuzzy extended $ b $-metric space and the extended $ b $-metric space. In addition, the existence and uniqueness of solutions to the Fredholm integral equation is presented.

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