Quantitative results for Bruck iterations of demicontinuous pseudocontractions

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.