Regularity of the Product of Two Relaxed Cutters with Relaxation Parameters Beyond Two

TL;DR

This paper investigates the regularity properties of the product of two relaxed cutters with relaxation parameters beyond two, demonstrating that under certain conditions, the product inherits regularity and ensuring convergence of related algorithms.

Contribution

It extends the analysis of relaxed cutters to cases with relaxation parameters greater than two, showing inheritance of regularity and convergence properties.

Findings

Product inherits regularity under specific conditions

Convergence of algorithms using such products is established

Results apply to weak, norm, and linear convergence

Abstract

We study the product of two relaxed cutters having a common fixed point. We assume that one of the relaxation parameters is greater than two so that the corresponding relaxed cutter is no longer quasi-nonexpansive, but rather demicontractive. We show that if both of the operators are (weakly/linearly) regular, then under certain conditions, the resulting product inherits the same type of regularity. We then apply these results to proving convergence in the weak, norm and linear sense of algorithms that employ such products.