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

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.