Fixed point results for a new mapping related to mean nonexpansive mappings

TL;DR

This paper explores fixed point properties of a new mapping related to mean nonexpansive mappings, specifically analyzing the less-studied composition $T igl( extstyle{rac{eta}{eta+1}}I + rac{1}{eta+1}Tigr)$ in Banach spaces.

Contribution

It introduces and investigates fixed point properties of a novel composition of mean nonexpansive mappings, expanding understanding of their behavior in Banach spaces.

Findings

Established fixed point results for the new mapping.

Compared properties of the new mapping with existing related mappings.

Discussed relationships between different compositions of mean nonexpansive mappings.

Abstract

Mean nonexpansive mappings were first introduced in 2007 by Goebel and Japon Pineda and advances have been made by several authors toward understanding their fixed point properties in various contexts. For any given $(\alpha_1, \alpha_2)$-nonexpansive mapping $T$ of a Banach space, many of the positive results have been derived from properties of the mapping $T_\alpha = \alpha_1 T + \alpha_2T^2= (\alpha_1I + \alpha_2T)\circ T$ which is nonexpansive. However, the related mapping $T \circ (\alpha_1I + \alpha_2T)$ has not yet been studied. In this paper, we investigate some fixed point properties of this new mapping and discuss relationships between $(\alpha_1I + \alpha_2T)\circ T$ and $T\circ(\alpha_1I + \alpha_2T)$.