In this article, we prove fixed point results for a Meir–Keeler type contraction due to orthogonal M-metric spaces. The results of the paper improve and extend some recent developments in fixed point theory. The extension is assured by the concluding remarks and followed by the main theorem. Finally, an application of the main theorem is established in proving theorems for some integral equations and integral-type contractive conditions.

This paper presents the development of fuzzy multiplicative metric spaces, an extended framework that combines the principles of multiplicative metric spaces with fuzzy logic to better address uncertainty and imprecision inherent in many real-world problems. By replacing additive distance measures with multiplicative ones, this approach proves particularly effective in contexts where relative variations or proportional relationships are more meaningful than absolute differences. Within this generalized setting, we establish a series of fixed-point theorems of Banach, Kannan, and Chatterjee types, along with related corollaries, each supported by concrete examples. The theoretical results are further validated through their application to the solution of nonlinear integral equations, demonstrating the versatility and applicability of the proposed framework across mathematical analysis and applied disciplines.

In the present research article, the notion of hesitant fuzzy mapping, contraction hesitant fuzzy mapping, generalized contraction of hesitant fuzzy mapping are introduced in setting of extended b-metric structure, which in particular case reduced to b-metric space. The newly introduced concepts are used to prove hesitant fuzzy fixed point theorems. The main results have extended and unified recent results in literature. The results are validated with the help of examples. Finally, as an application, the results are applied to solve integral equation of fredholm type.

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 \((\beta{-}\theta)\)-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.

The present study introduces the notion of compatibility in partially ordered G-metric spaces and uses this perception to establish a coupled coincidence point result. Our effort extend the recent work of Choudhary and Maity [B. S. Choudhary, P. Maity, Coupled fixed point results in generalized metric spaces, Mathematical and Computer Modelling 54 (2011) 73-79]. The example demonstrates that our main result is an actual improvement over the results which are generalized