Fixed point theorems in controlled rectangular metric spaces

TL;DR

This paper introduces controlled rectangular metric spaces, extending traditional metric spaces, and establishes fixed point theorems within this new framework, thereby generalizing and improving existing results in the field.

Contribution

It defines controlled rectangular metric spaces with a new inequality and proves fixed point theorems, extending and enhancing prior fixed point results.

Findings

Established fixed point theorems in controlled rectangular metric spaces.

Extended existing fixed point results to a broader class of spaces.

Provided an illustrative example supporting the theoretical results.

Abstract

In this paper, we introduce an extension of rectangular metric spaces called controlled rectangular metric spaces, by changing the rectangular inequality as follows: \begin{equation*} d(x, y)\leq\alpha(x, u)d(x, u)+\alpha(u, v)d(u, v)+\alpha(v, y)d(v, y), \end{equation*} for all distinct $x, y, u, v\in X$ with the function $\alpha: X\times X\rightarrow[1,\infty[$. We also establish some fixed point theorems for self-mappings defined on such spaces. Our main results extends and improves many results existing in the literature. Moreover, an illustrative example is presented to support the obtained results.