Common fixed point of set valued graph Aφ-contraction pair and generalized φ-weak G-contraction on metric space endowed with a graph

In this paper, we define the notion of graph A(phi)-contraction pair and generalized phi-weak G contraction on subsets of a metric space involving a graph. Using such contractions, the existence as well as uniqueness of common fixed point for set valued mappings with set valued domain involving a directed graph have been examined. Suitable examples are presented to validate the non-triviality the results. We particularly generalize and extend the results due to Zhang and Song [Fixed point theory for generalized yo-weak contractions. Appl. Math. Lett., 22:75-78, 2009].