Coincidence point and fixed point theorems for a new type of G-contraction multivalued mappings on a metric space endowedwith a graph

In this paper, a new type of G-contraction multivalued mappings in a metric space endowed with a directed graph is introduced and studied. This type of mappings is more general than that of Mizoguchi and Takahashi (J. Math. Anal. Appl. 141: 177-188, 1989), Berinde and Berinde (J. Math. Anal. Appl. 326: 772-782, 2007), Du (Topol. Appl. 159: 49-56, 2012), and Sultana and Vetrivel (J. Math. Anal. Appl. 417: 336-344, 2014). A fixed point and coincidence point theorem for this type of mappings is established. Some examples illustrating our main results are also given. The main results obtained in this paper extend and generalize those in (Tiammee and Suantai in Fixed Point Theory Appl. 2014: 70, 2014) and many well-known results in the literature.