On Wardowski Type Results in the Framework of G-Metric Spaces

In 2012, D. Wardowski introduced the notion of F- contraction and proved a fixed point theorem that modifies the famous Banach result in a complete metric space. Since then, many extensions and generalizations of this new type mapping, in metric and generalized metric spaces, have appeared. Sometimes these investigations have produced interesting results, but there are also gaps. Recently, M. Kumar and S. Arora introduced a concept of generalized F-contraction of type S_G and M_G, in the G-metric spaces and established two fixed point results. In this chapter, we will explain how we improved their results because the evidence in them is questionable, for us the claims in it are open and interesting. The correction consists of the following:

using only the first property of mapping F;

reformulating the statement of their first theorem more precisely andcorrectingtheproof;

changing the structure of the set S_f;

using our two lemmas for proving that the sequence is Cauchy;

stating tree consequences of our improved theorem that represent Wardowski's results in concept of G-metric spaces;

analyzing and improving some parts of their two examples that support our obtained results.

We hope these notes will be useful to many researchers who deal with F-contractions within generalized G-metric spaces.