  Let $(X,\|.\|)$ be a Banach space. Let $C$ be a nonempty, bounded, closed,
and convex subset of $X$ and $T: C \rightarrow C$ be a $G$-monotone
nonexpansive mapping. In this work, it is shown that the Mann iteration
sequence defined by $$x_{n+1} = t_n T(x_n) + (1-t_n)x_n, \, n = 1, 2, \cdots$$
can be proved the existence of a fixed point of $G$-monotone nonexpansive
mappings.
