Common fixed point theorems under an implicit contractive condition on metric spaces endowed with an arbitrary binary relation and an application

TL;DR

This paper develops new fixed point theorems in metric spaces with arbitrary relations under implicit contractions, unifying many known conditions and applying results to solve integral equations.

Contribution

It introduces a general implicit contractive condition for fixed points in metric spaces with arbitrary relations, covering many existing results and providing new insights.

Findings

Established generalized fixed point theorems under implicit conditions

Provided an example demonstrating the broad applicability of the results

Applied the theorems to prove existence of solutions to integral equations

Abstract

The aim of this paper is to establish some metrical coincidence and common fixed point theorems with an arbitrary relation under an implicit contractive condition which is general enough to cover a multitude of well known contraction conditions in one go besides yielding several new ones. We also provide an example to demonstrate the generality of our results over several well known corresponding results of the existing literature. Finally, we utilize our results to prove an existence theorem for ensuring the solution of an integral equation.