This manuscript contains several fixed point results for ${\alpha }_{\Gamma }$-F-fuzzy contractive mappings in the framework of orthogonal fuzzy metric spaces. The symmetric property guarantees that the distance function is consistent and does not favour any one direction in orthogonal fuzzy metric spaces. No matter how the points are arranged, it enables a fair assessment of the separations between all of them. In fixed point results, the symmetry condition is preserved for several types of contractive self-mappings. Moreover, we provide several non-trivial examples to show the validity of our main results. Furthermore, we solve non-linear fractional differential equations, the Atangana–Baleanu fractional integral operator and Fredholm integral equations by utilizing our main results.

The symmetry of fuzzy metric spaces has benefits for flexibility, ambiguity tolerance, resilience, compatibility, and applicability. They provide a more comprehensive description of similarity and offer a solid framework for working with ambiguous and imprecise data. We give fuzzy versions of some celebrated iterative mappings. Further, we provide different concrete conditions on the real valued functions $\mathcal{J},\mathcal{S}:(0,1]\to \mathbb{R}$ for the existence of the best proximity point of generalized fuzzy $(\mathcal{J},\mathcal{S})$-iterative mappings in the setting of fuzzy metric space. Furthermore, we utilize fuzzy versions of $\left(\mathcal{J},\mathcal{S}\right)$-proximal contraction, $\left(\mathcal{J},\mathcal{S}\right)$-interpolative Reich–Rus–Ciric-type proximal contractions, $\left(\mathcal{J},\mathcal{S}\right)$-Kannan type proximal contraction and $\left(\mathcal{J},\mathcal{S}\right)$-interpolative Hardy Roger’s type proximal contraction to examine the common best proximity points in fuzzy metric space. Also, we establish several non-trivial examples and an application to support our results.

In this manuscript, we present several types of interpolative proximal contraction mappings including Reich–Rus–Ciric-type interpolative-type contractions and Kannan-type interpolative-type contractions in the setting of bipolar metric spaces. Further, taking into account the aforementioned mappings, we prove best proximity point results. These results are an extension and generalization of existing ones in the literature. Furthermore, we provide several nontrivial examples, an application to find the solution of an integral equation, and a nonlinear fractional differential equation to show the validity of the main results.

In this paper, we present the concept of $(\mathrm{{\rm Y}},\mathrm{\Omega })$-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 $(\mathrm{{\rm Y}},\mathrm{\Omega })$-fuzzy bipolar proximal contraction, $(\mathrm{{\rm Y}},\mathrm{\Omega })$-Reich–Rus–Ciric type proximal contraction, $(\mathrm{{\rm Y}},\mathrm{\Omega })$-Kannan type proximal contraction and $(\mathrm{{\rm Y}},\mathrm{\Omega })$-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.

In this paper, we present Proinov-type fixed point theorems in the setting of bi-polar metric spaces and fuzzy bi-polar metric spaces. Fuzzy bi-polar metric spaces with symmetric property extend classical metric spaces to address dual structures and uncertainty, ensuring consistency and balance. We provide different concrete conditions on the real-valued functions $\Omega ,\Pi :\left(0,\infty \right)\to \mathbb{R}$ for the existence of fixed points via the $(\Omega ,\Pi )$-contraction in bi-polar metric spaces. Further, we define real-valued functions $\Omega ,\Pi :(0,1]\to \mathbb{R}$ to obtain fixed point theorems in fuzzy bi-polar metric spaces. We apply $\left(\Omega ,\Pi \right)$ fuzzy bi-polar version of a Banach fixed point theorem to show the existence of solutions. Furthermore, we provide some non-trivial examples to show the validity of our results. In the end, we find the existence and uniqueness of a solution of integral equations and boundary value problem used in chemical sciences by applying main results.

In this work, we establish the conditions for ensuring the existence and uniqueness of common best proximity points for non-self-mappings defined on the general metric spaces. A unified theoretical framework is formulated to cover a broad class of contraction mappings. We describe the required conditions on the real-valued functions (ℵ, Φ) : [0,∞) → R and verify that these secure the existence of common best proximity points for (ℵ, Φ)−interpolative contractions in complete metric spaces. The study further extends this concept by examining multiple forms of interpolative proximal-type contractions, such as proximal, Ćirić —Reich—Rus, Kannan, and Hardy-Rogers variants, through the use of the auxiliary functions (ℵ, Φ). Several illustrated examples are included to demonstrate the applicability of our findings. Finally, we conclude with an application involving a nonlinear fractional differential equation, showing that it fully satisfies the assumption of our main result.