In this work, a novel Hilfer cotangent fractional derivative is presented. This derivative combines the characteristics of the Riemann-Liouville cotangent fractional derivative and the Caputo cotangent fractional derivative. The essential properties of the newly introduced derivative are discussed. By utilizing this derivative, a nonlinear fractional differential problem with a nonlocal initial condition is investigated, and its equivalence to a cotangent Volterra integral equation is demonstrated. The uniqueness and existence of solutions are established by employing fixed-point theorems. Additionally, two illustrative examples are provided to illustrate the obtained results.

This paper contains several novel definitions including neutrosophic $E_{\beta }$ metric space, neutrosophic quasi-$S_{\beta }$-metric space, neutrosophic pseudo-$S_{\beta }$-metric space, neutrosophic quasi-$E$-metric space and neutrosophic pseudo-$E_{\beta }$-metric space. Further, we present some generalized fixed point results with non-trivial examples and the decomposition theorem in the setting of the neutrosophic pseudo-$E_{\beta }$-metric space. Moreover, by using the main result, we examine the existence and uniqueness of the solution to an integral equation, a system of linear equations, and nonlinear fractional differential equations.

In this paper, we introduce a new class of contractions in normed spaces, referred to as generalized enriched Kannan contractions. These contractions expand the familiar enriched Kannan contractions to three-point versions, broadening the scope of Kannan contractions. These mappings are typically discontinuous, except at the fixed points, where they exhibit continuity, similar to enriched Kannan mappings. However, through suitable examples, we demonstrate that these two classes of mappings are distinct from one another. We present new results for generalized enriched Kannan contractions. Additionally, by incorporating conditions of continuity and asymptotic regularity, we extend the class of operators to which fixed-point methods can be applied. Additionally, we derive two more results for generalized enriched Kannan contractions in normed spaces, without the requirement that they be Banach spaces. Finally, we use our main result to demonstrate the existence of solutions for a boundary value problem involving a fractional differential equation.

In this manuscript, we investigated the coincidence points, best proximity points, and fixed-points results endowed with $ F $-contraction within the realm of suprametric spaces. The proximal point results obtained in this work show that our investigation is not purely theoretical; fundamental findings were supplemented with concrete examples that demonstrate their practical ramifications. Furthermore, this paper focuses on boundary value problems (BVPs) related to nonlinear fractional differential equations of order $ 2 < \varpi \leq 3 $. By cleverly translating the BVP into an integral equation, we obtained conditions that confirm the existence and uniqueness of fixed points under $ (\mathscr{F}_{\tau })_{F_{\mathscr{P}}} $-contraction. A relevant part of this work is the approximation of the Green's function, which is critical in proving the existence and uniqueness of solutions. Our work not only adds to the current body of knowledge but also provides strong approaches for dealing with hard mathematical problems in the field of fractional differential equations.

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.

This manuscript contains several new notions including intuitionistic fuzzy $N_b$ metric space, intuitionistic fuzzy quasi-$S_b$-metric space, intuitionistic fuzzy pseudo-$S_b$-metric space, intuitionistic fuzzy quasi-$N$-metric space and intuitionistic fuzzy pseudo $N_b$ fuzzy metric space. We prove decomposition theorem and fixed-point results in the setting of intuitionistic fuzzy pseudo $N_b$ fuzzy metric space. Further, we provide several non-trivial examples to show the validity of introduced notions and results. At the end, we solve an integral equation, system of linear equations and nonlinear fractional differential equations as applications.