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

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.