The $G_M$ -Contraction Principle for Mappings on an $M$ -Metric Spaces Endowed With a Graph and Fixed Point Theorems

Fixed point theory is a very important tool in mathematics and applied sciences. Latterly, many application examples have been presented for communication network and computer science fields. The proposed schema can be considered as a theoretical foundation for such a type of applications. In this paper, we introduce the notion of the $G_{m}$ -contraction to generalize and extend the notion of $G$ -contraction. We investigate the existence and uniqueness of the fixed point for such contractions in $M$ -metric space endowed with a graph. Our results extend and generalize various results in the existing literature, in particular the results of Jachymski. Some examples are included, which illustrate the results proved herein.