Mann Iteration Process for Monotone Nonexpansive Mappings with a Graph

TL;DR

This paper investigates the Mann iteration process for G-monotone nonexpansive mappings in Banach spaces, establishing conditions under which fixed points exist for such mappings.

Contribution

It extends fixed point theory by proving the existence of fixed points for G-monotone nonexpansive mappings using Mann iteration in Banach spaces.

Findings

Fixed points exist for G-monotone nonexpansive mappings under certain conditions.

The Mann iteration converges to a fixed point for these mappings.

The results generalize previous fixed point theorems in Banach spaces.

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.