# Relation-theoretic metrical coincidence and common fixed point theorems under nonlinear contractions

## TL;DR

This paper establishes new fixed point theorems in metric spaces with binary relations using nonlinear contractions, extending and unifying several existing results in the literature.

## Contribution

It introduces generalized fixed point results under nonlinear contractions on metric spaces with arbitrary binary relations, broadening previous theorems and providing sharper conditions.

## Key findings

- Extended fixed point theorems under nonlinear contractions.
- Unified multiple known results into a broader framework.
- Provided examples demonstrating the improvements over existing theorems.

## Abstract

In this paper, we prove coincidence and common fixed points results under nonlinear contractions on a metric space equipped with an arbitrary binary relation. Our results extend, generalize, modify and unify several known results especially those are contained in Berzig [J. Fixed Point Theory Appl. 12, 221-238 (2012))] and Alam and Imdad [To appear in Filomat (arXiv:1603.09159 (2016))]. Interestingly, a corollary to one of our main results under symmetric closure of a binary relation remains a sharpened version of a theorem due to Berzig. Finally, we use examples to highlight the accomplished improvements in the results of this paper.