This article is comprised of outcomes on fixed points for almost Matkowski contractions via locally $\mathcal{J} $-transitive relations. Our outcomes sharpen, unify, enrich and improve many fixed point theorems of the existing literatures. Several examples are furnished to demonstrate the credibility of our results. By implementing our outcomes, we ascertain a unique solution for certain boundary value problem.

In this research, a few metrical fixed-point outcomes consisting of an almost nonlinear Pant-type contraction employing a locally finitely transitive relation have been established. The findings of our research extrapolate, unify, develop, and improve a number of previously mentioned results. In the present investigation, we formulate a fixed-point finding for almost nonlinear Pant-type contractions in abstract metric space. To assist our study, we formulate numerous examples to illustrate our outcomes. Using our findings, we describe the existence and uniqueness of solutions to a nonlinear Fredholm integral equation.

This article aims to adopt some notions for mapping $\mathfrak{f}:{\mathfrak{X}}^k\to \mathfrak{X}$, (where integer k is positive) and to prove the nonlinear-Prešić-type results on metric spaces employing a $\mathfrak{f}$-reflexive and locally finitely $\mathfrak{f}$-transitive binary relation (not necessarily partial order). The outcomes proven herewith are extended and generalized to several fixed point findings of literature. Lastly, examples are provided to support the applicability of these outcomes.

In this paper, we explore specific fixed-point outcomes under a Mukherjea-Type nonlinear description of almost contractions in a metric space with a locally ℏ-transitive binary relation. Numerous previous insights are expanded, developed, improved, and consolidated in the outcomes reported herein. To demonstrate the reliability of our results, a couple of instances are supplied. Our outcomes are deployed to evaluate the accuracy of the unique solution of a boundary value problem.

The objective of this paper is to propose some fixed-point findings under a relational contraction of Pant type employing a pair of auxiliary functions and through a generalized class of transitive binary relations. Our outcomes extend, sharpen, modify and enrich many existing findings. To facilitate our research, we create a few instances that convey our findings. Through the use of our outcomes, we demonstrate the existence and uniqueness of solutions for a nonlinear integral equation.

This manuscript seeks to study couple fixed-point findings over a relational extended contraction through a locally, finitely transitive binary relation and a pair of generalized altering distance functions. Various prior discoveries are expanded, refined, modified, compiled, and enhanced by our findings. We craft some instances for demonstrating the practical relevance of our findings. We embark on our research in an attempt to recognize a unique solution to a boundary value problem connected with a second-order ordinary differential equation.