The goal of this paper is to study the existence of a mild solution and controllability for a class of neutral stochastic differential equations (SDEs) involving the $\mathrm{\Psi }$-Hilfer fractional derivatives, a generalization of the well-known Riemann–Liouville fractional derivative using almost sectorial operators. Sufficient conditions for controllability are established using the notion of measure of noncompactness (MNC) and the Mönch fixed-point theorem. An example is given to illustrate the abstract findings.

In this article, we prove certain fixed point findings for a nonlinear almost contraction map of the Pant type in a metric space endowed with a binary relation. The outcomes presented here expand, develop, enhance, and consolidate several existing findings. With a view to arguing for our findings, we furnish a few examples. As an application of our outcomes, we discuss the existence of a unique solution of a boundary value problem of the first order.

This paper concludes a few fixed-point outcomes involving almost Matkowski contraction-inequality of Pant type in a relational metric space. The findings established here enhance, expand, consolidate and develop several noted outcomes. In order to argue for our investigations, we construct some illustrative examples. We exploit our outcomes to analyze the availability of a (unique) positive solution to certain singular fractional differential equations.

This article consists of some new fixed point theorems verifying relation-theoretic strict almost nonlinear contractions that are preserved by a locally finitely transitive relation. Our results improve certain well-known results in the literature. To confirm the reliability of our results, some examples are delivered. We carry out our findings to solve a nonlinear Fredholm integral equation.

This study presents a novel and efficient iterative approach to approximating the fixed points of contraction mappings in Banach spaces, specifically approximating the solutions of nonlinear fractional differential equations of the Caputo type. We establish two theorems proving the stability and convergence of the proposed method, supported by numerical examples and graphical comparisons, which indicate a faster convergence rate compared to existing methods, including those by Agarwal, Gursoy, Thakur, Ali and Ali, and $D^{\ast \ast }$. Additionally, a data dependence result for approximate operators using the proposed method is provided. This approach is applied to achieve the solutions for Caputo-type fractional differential equations with boundary conditions, demonstrating the efficacy of the method in practical applications.

This article presents a few fixed-point results under Matkowski-type functional contractive mapping using locally $\mathbf{J}$-transitive binary relations. Our results strengthen, enhance, and consolidate numerous existent fixed-point results. To argue for the efficacy of our results, several illustrated examples are supplied. With the help of our findings, we deal with the existence and uniqueness theorems pertaining to the solution of a variety of singular fractional differential equations.