Fixed Point Results of a New Family of Contractions in Metric Space Endowed with a Graph

Abstract

In this note, a general class of contractive inequality, namely, admissible hybrid $\left(H-\alpha -\varphi \right)$ contraction, is proposed in metric space endowed with a graph, and new criteria for which the mapping is a Picard operator are examined. The significance of this type of contraction is connected with the possibility that its inequality can be particularized in more than one way, depending on the provided constants. Relevant examples are designed to support the assumptions of our obtained notions and to show how they are different from the known ones. A corollary which reduces our obtained result to a recently published result in the literature is pointed out and discussed.