# Relation-theoretic metrical fixed point results via w-distance with an application in nonlinear fractional differential equations

## TL;DR

This paper extends fixed point theory using w-distance in relation-structured metric spaces and applies these results to establish existence and uniqueness of solutions for nonlinear fractional differential equations.

## Contribution

It introduces a new relation-theoretic fixed point theorem using w-distance and applies it to fractional differential equations involving Caputo derivatives.

## Key findings

- Established a generalized Banach fixed point theorem in relation-structured metric spaces.
- Provided examples illustrating the new fixed point results.
- Applied the theory to prove existence and uniqueness of solutions for nonlinear fractional differential equations.

## Abstract

In this article, utilizing the concept of w-distance, we prove the celebrated Banach's fixed point theorem in metric spaces equipped with an arbitrary binary relation. Necessarily our findings unveil another direction of relation-theoretic metrical fixed point theory. Also, our paper consists of several non-trivial examples which signify the motivation for such investigations. Finally, our obtained results enable us to explore the existence and uniqueness of solutions of nonlinear fractional differential equations involving the Caputo fractional derivative.