Quantitative translations for viscosity approximation methods in hyperbolic spaces

TL;DR

This paper develops quantitative convergence rates for viscosity approximation methods in hyperbolic spaces and Banach spaces, using proof-theoretic techniques to extend and analyze classical iterative algorithms.

Contribution

It introduces a systematic way to derive explicit convergence rates for viscosity iterations from known rates of classical sequences in hyperbolic spaces.

Findings

Convergence of Browder-type sequences implies convergence of viscosity versions.

Transformation of rates from classical to viscosity iterations is possible.

Uniform moduli of uniqueness lead to Cauchy rates in Banach spaces.

Abstract

In the setting of hyperbolic spaces, we show that the convergence of Browder-type sequences and Halpern iterations respectively entail the convergence of their viscosity version with a Rakotch map. We also show that the convergence of a hybrid viscosity version of the Krasnoselskii-Mann iteration follows from the convergence of the Browder type sequence. Our results follow from proof-theoretic techniques (proof mining). From an analysis of theorems due to T. Suzuki, we extract a transformation of rates for the original Browder type and Halpern iterations into rates for the corresponding viscosity versions. We show that these transformations can be applied to earlier quantitative studies of these iterations. From an analysis of a theorem due to H.-K. Xu, N. Altwaijry and S. Chebbi, we obtain similar results. Finally, in uniformly convex Banach spaces we study a strong notion of accretive operator due to Brezis and Sibony and extract an uniform modulus of uniqueness for the property of being a zero point. In this context, we show that it is possible to obtain Cauchy rates for the Browder type and the Halpern iterations (and hence also for their viscosity versions).