Mann Iteration Process for Monotone Nonexpansive Mappings with a Graph

Abstract

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.