New directions in fixed point theory in $G$-metric spaces and applications to mappings contracting perimeters of triangles

TL;DR

This paper introduces new fixed point theorems in $G$-metric spaces, expanding the theoretical framework and providing applications to triangle perimeter contractions, addressing previous limitations in the field.

Contribution

The paper establishes novel Banach, Kannan, and Reich fixed point theorems in $G$-metric spaces, overcoming earlier methodological constraints.

Findings

New fixed point theorems in $G$-metric spaces

Applications to mappings contracting triangle perimeters

Clarification of limitations of previous approaches

Abstract

We are concerned with the study of fixed points for mappings $T: X\to X$, where $(X,G)$ is a $G$-metric space in the sense of Mustafa and Sims. After the publication of the paper [Journal of Nonlinear and Convex Analysis. 7(2) (2006) 289--297] by Mustafa and Sims, a great interest was devoted to the study of fixed points in $G$-metric spaces. In 2012, the first and third authors observed that several fixed point theorems established in $G$-metric spaces are immediate consequences of known fixed point theorems in standard metric spaces. This observation demotivated the investigation of fixed points in $G$-metric spaces. In this paper, we open new directions in fixed point theory in $G$-metric spaces. Namely, we establish new versions of the Banach, Kannan and Reich fixed point theorems in $G$-metric spaces. We point out that the approach used by the first and third authors [Fixed Point Theory Appl. 2012 (2012) 1--7] is inapplicable in the present study. We also provide some interesting applications related to mappings contracting perimeters of triangles.