Existence of common fixed points of generalized set-contractions on a $b$-metric space with a directed graph

This paper deals with a type of fixed point obtained from a family of generalized graph contractions that we adapt in the framework of a $b$-metric space. We state and prove many new results in the underlying spaces. We provide a few applications of our results through some examples in order to support and illustrate the main results. To the best of our knowledge, the results derived herein improve and generalizes many results in the cited references.