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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<sub>m</sub>-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.

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The <inline-formula> <tex-math notation="LaTeX">$G_M$ </tex-math></inline-formula>-Contraction Principle for Mappings on an <inline-formula> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula>-Metric Spaces Endowed With a Graph and Fixed Point Theorems

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