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 manuscript deals with proving certain metrical fixed-point findings for a class of generalized contractions involving a couple $ (\psi, \varphi) $ of test functions employing a local class of transitive relation. The outcomes investigated herein refine, modify, unify, and sharpen various existing outcomes. To attest the accuracy of our outcomes, we provide a few examples. By means of our outcomes, we impart an explanation of the reality of a solution to a boundary value problem.

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.