Strict pseudocontractions and demicontractions, their properties and applications

TL;DR

This paper explores properties of strict pseudocontractions and demicontractions in Hilbert spaces, establishing conditions for their combinations and introducing a generalized relaxation, with applications to convergence of iterative algorithms.

Contribution

It provides new conditions for combining these operators and introduces a generalized relaxation method, extending the applicability of iterative algorithms.

Findings

Conditions for convex combinations and compositions to remain strict pseudocontractions or demicontractions.

Introduction of a generalized relaxation of demicontraction composition.

Numerical comparisons showing the proposed methods outperform the Douglas-Rachford algorithm.

Abstract

We give properties of strict pseudocontractions and demicontractions defined on a Hilbert space, which constitute wide classes of operators that arise in iterative methods for solving fixed point problems. In particular, we give necessary and sufficient conditions under which a convex combination and composition of strict pseudocontractions as well as demicontractions that share a common fixed point is again a strict pseudocontraction or a demicontraction, respectively. Moreover, we introduce a generalized relaxation of composition of demicontraction and give its properties. We apply these properties to prove the weak convergence of a class of algorithms that is wider than the Douglas-Rachford algorithm and projected Landweber algorithms. We have also presented two numerical examples, where we compare the behavior of the presented methods with the Douglas-Rachford method.