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

One of the applicable concepts in metric fixed point theory is the notion of hybrid functional equations. In the same vein, the role of graphs in computational sciences and nonlinear functional analysis is currently well known. However, as duly revealed from the available literature, we understand that hybrid fixed point notions in metric space endowed with graph have not been well considered. In this note, therefore, a general class of contractive inequality, namely admissible hybrid (H-α-ϕ)-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 some recently announced results in the literature is pointed out and discussed.