A new technique for coincidence point theory in metric spaces endowed with graph

Coincidence theory is a generalization of fixed point theory. There are many researchs combining fixed point theory and graph theory. In this paper, a new type of multi-valued mapping and g-l-graph preserving is proposed to prove a coincidence point theorem on complete metric spaces endowed with a directed graph. Supported examples of there main theorems are also introduced. Main results are sufficiently conditions for finding a vertex in the directed graph such that itself image of the defined surjective mapping is contained in the defined multivalued mapping. The proposed theorem can be apply use to obtain the similar result in a matrix space endowed with a partial order set.