Quantitative translations for viscosity approximation methods in hyperbolic 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).

Abstract strongly convergent variants of the proximal point algorithm

“…Finally, Suzuki has shown in [49] that this sort of convergence theorems for Halpern iterations -even for families of mappings like in our case -directly yield convergence theorems for the corresponding viscosity iterations; this has been recently analyzed quantitatively by Kohlenbach and Pinto [30], and the results of that paper -specifically Lemma 3.4, Remark 3.5 and Theorem 3.11 -may be used to immediately derive from our results rates of metastability for the viscosity proximal point algorithm, thus further illustrating the modularity of proof mining approaches.…”