On modified Halpern and Tikhonov-Mann iterations

TL;DR

This paper establishes the equivalence and convergence properties of modified Halpern and Tikhonov-Mann iterations in geodesic spaces, providing new proofs, generalizations to CAT(0) spaces, and quantitative convergence rates.

Contribution

It demonstrates the equivalence of two iterative schemes in geodesic spaces, extends convergence results to CAT(0) spaces, and derives explicit convergence rates.

Findings

Proves asymptotic regularity and strong convergence equivalence.

Extends convergence results from Hilbert to CAT(0) spaces.

Provides explicit $O(1/n)$ convergence rates for specific iterations.

Abstract

We show that the asymptotic regularity and the strong convergence of the modified Halpern iteration due to T.-H. Kim and H.-K. Xu and studied further by A. Cuntavenapit and B. Panyanak and the Tikhonov-Mann iteration introduced by H. Cheval and L. Leu\c{s}tean as a generalization of an iteration due to Y. Yao et al. that has recently been studied by Bo\c{t} et al. can be reduced to each other in general geodesic settings. This, in particular, gives a new proof of the convergence result in Bo\c{t} et al. together with a generalization from Hilbert to CAT(0) spaces. Moreover, quantitative rates of asymptotic regularity and metastability due to K. Schade and U. Kohlenbach can be adapted and transformed into rates for the Tikhonov-Mann iteration corresponding to recent quantitative results on the latter of H. Cheval, L. Leu\c{s}tean and B. Dinis, P. Pinto respectively. A transformation in the converse direction is also possible. We also obtain rates of asymptotic regularity of order $O(1/n)$ for both the modified Halpern (and so in particular for the Halpern iteration) and the Tikhonov-Mann iteration in a general geodesic setting for a special choice of scalars.