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 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.