A note on strong convergence to common fixed points of nonexpansive mappings in Hilbert spaces

TL;DR

This paper explores the strong convergence properties of algorithms for finding common fixed points of nonexpansive mappings in Hilbert spaces, linking various iterative methods and extending existing theorems.

Contribution

It establishes new connections between ${ m T}_C$-class algorithms, CQ Algorithm, and shrinking projection methods, extending convergence results for nonexpansive mappings.

Findings

Strong convergence of algorithms is related to coherent ${ m T}_C$-class sequences.

Examples demonstrate applicability to nonexpansive finite sets and semigroups.

Extensions of existing theorems on fixed point convergence.

Abstract

The aim of this paper is to investigate the links between ${\cal T}_C$-class algorithms, CQ Algorithm and shrinking projection methods. We show that strong convergence of these algorithms are related to coherent ${\cal T}_C$-class sequences of mapping. Some examples dealing with nonexpansive finite set of mappings and nonexpansive semigroups are given. They extend some existing theorems.