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 paper deals with the fixed-point findings under relational strict almost $(\varphi ,\psi )$-contraction. Our findings complement and strengthen prevailing results. In the course of the procedure, we also derive a related fixed point theorem for strict almost $(\varphi ,\psi )$-contraction. We provide several illustrative examples to support the validity of our outcomes. We also argue about the possibility of a unique solution of a nonlinear Fredholm integral equation via our outcomes.

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.