Quantitative metastability of the Tikhonov-Mann iteration for countable families of mappings

TL;DR

This paper establishes quantitative rates of metastability for the Tikhonov-Mann iteration applied to countable families of mappings in CAT(0) spaces, extending previous convergence results to a broader setting.

Contribution

It provides the first explicit rates of metastability for this iteration in CAT(0) spaces, generalizing prior work on single mappings and nonexpansive families.

Findings

Derived explicit metastability rates in CAT(0) spaces.

Extended convergence analysis to countable families of mappings.

Connected iteration behavior with previous asymptotic regularity results.

Abstract

In this paper, we obtain rates of metastability for the Tikhonov-Mann iteration for countable families of mappings in CAT(0) spaces. This iteration was recently defined by the author in the setting of W-hyperbolic spaces as a generalization of the strongly convergent version of the Krasnoselskii-Mann iteration introduced by Bot and Meier for finding common fixed points of families of nonexpansive mappings in Hilbert spaces, and as an extension of the Tikhonov-Mann iteration for single mappings, for which Leustean and the author obtained rates of asymptotic regularity in W-hyperbolic spaces.