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

Abstract

In this paper, we define the notion of graph A_φ-contraction pair and generalized φ-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 has been examined. Example to support the non-triviality of our results is also provided. Our results generalize and extend some existing results in the literature, especially those of Zhang and Song [Fixed point theory for generalized φ-weak contractions. Appl. Math. Lett., 22:75-78, 2009].