Effective results on a fixed point algorithm for families of nonlinear mappings

TL;DR

This paper applies proof mining techniques to establish a uniform rate of asymptotic regularity for a fixed point algorithm targeting families of nonlinear mappings, enhancing understanding of convergence behavior.

Contribution

It introduces a novel use of proof mining to derive explicit convergence rates for a parallel fixed point algorithm in Hilbert spaces.

Findings

Established a uniform rate of asymptotic regularity for the algorithm.

Demonstrated the applicability of logical metatheorems to fixed point problems.

Provided a rigorous foundation for convergence analysis of nonlinear mappings.

Abstract

We use proof mining techniques to obtain a uniform rate of asymptotic regularity for the instance of the parallel algorithm used by L\'opez-Acedo and Xu to find common fixed points of finite families of $k$-strict pseudocontractive self-mappings of convex subsets of Hilbert spaces. We show that these results are guaranteed by a number of logical metatheorems for classical and semi-intuitionistic systems.