Effective metastability for a method of alternating resolvents

TL;DR

This paper analyzes the strong convergence of a generalized alternating resolvents method for monotone operators, providing effective rates of metastability and asymptotic regularity using proof-theoretic techniques.

Contribution

It offers new quantitative convergence results for the method, removing the need for sequential weak compactness and applying proof mining techniques.

Findings

Established effective metastability rates.

Derived quasi-rates of asymptotic regularity.

Removed reliance on sequential weak compactness.

Abstract

A generalized method of alternating resolvents was introduced by Boikanyo and Moro{\c s}anu as a way to approximate common zeros of two maximal monotone operators. In this paper we analyse the strong convergence of this algorithm under two different sets of conditions. As a consequence we obtain effective rates of metastability (in the sense of Terence Tao) and quasi-rates of asymptotic regularity. Furthermore, we bypass the need for sequential weak compactness in the original proofs. Our quantitative results are obtained using proof-theoretical techniques in the context of the proof mining program.