Fixed Point Theorems in the Fuzzy Supermetric Spaces and Application

This article introduces a new mathematical concept that extends the notion of a supermetric space, termed a fuzzy supermetric space. An illustrative example is provided to validate the proposed concept, and a class of contractive mappings is developed and employed to establish the existence and uniqueness of fixed-point theorems within the framework of fuzzy supermetric spaces. These results generalize both classical and recent findings as special cases within a broader, more inclusive setting, offering a more suitable framework for modeling and solving real-world problems. In addition, the concept of \((β{-}θ)\)-contractive mappings is examined in this context. Finally, the theoretical results are applied to demonstrate the existence of a solution to the Fredholm integral equation.