Compared to the general metric space

themetric space endowed with a directed graph is more complex in space structure

and the directed graph itself does not have a linear structure

and it isrelatively difficult to explore the problems of the fixed points of themappings.In this paper

the convexstructure is introduced in the metric space endowed with a directed graph forthe first time

and the convex metric spaces endowed with a directed graph isdefined. Moreover

the convergencetheorem of the sequence which is generated by the Mann iterative algorithm ofGmonotone nonexpansive mappings is obtained.Furthermore

an example is given to illustrate that the sequencegenerated by Mann iteration does not necessarily converge to the fixed point ofGmonotone nonexpansive mapping.