Quantitative results for Bruck iterations of demicontinuous pseudocontractions

TL;DR

This paper provides a detailed quantitative analysis of Bruck's iteration scheme for demicontinuous pseudocontractions in Hilbert spaces, including rates of metastability and asymptotic regularity, extending previous proof mining results.

Contribution

It introduces new quantitative bounds for Bruck's iteration scheme, generalizing earlier work from Lipschitzian to demicontinuous pseudocontractions in Hilbert spaces.

Findings

Established a rate of metastability for Bruck's iteration scheme.

Derived a metastable version of asymptotic regularity under uniform continuity.

Provided comprehensive quantitative analysis of the iteration scheme.

Abstract

Our first result is a rate of metastability in the sense of Tao for Bruck's iteration scheme for demicontinuous pseudocontractions in Hilbert space, extracted from Bruck's original proof. This result generalizes earlier work in the ongoing program of proof mining from Lipschitzian to demicontinuous pseudocontractions. Our second main result is a metastable version of asymptotic regularity under the additional assumption that the underlying operator is norm-to-norm uniformly continuous on bounded subsets. These results (and their intermediate versions given in this paper) provide a thorough quantitative analysis of Bruck's iteration scheme for pseudocontractions in Hilbert space.