Rates of asymptotic regularity for the alternating Halpern-Mann iteration

TL;DR

This paper extends quantitative asymptotic regularity results for the alternating Halpern-Mann iteration to $UCW$-hyperbolic spaces, providing new insights even for uniformly convex normed spaces and computing explicit rates in specific spaces.

Contribution

It generalizes existing results to broader spaces and calculates explicit linear and quadratic rates of asymptotic regularity for particular parameter choices.

Findings

Established new asymptotic regularity results in $UCW$-hyperbolic spaces.

Computed linear rates of asymptotic regularity in $W$-hyperbolic spaces.

Derived quadratic rates of asymptotic regularity in CAT(0) spaces.

Abstract

In this paper we extend to $UCW$-hyperbolic spaces the quantitative asymptotic regularity results for the alternating Halpern-Mann iteration obtained by Dinis and the second author for CAT(0) spaces. These results are new even for uniformly convex normed spaces. Furthermore, for a particular choice of the parameter sequences, we compute linear rates of asymptotic regularity in $W$-hyperbolic spaces and quadratic rates of $T$- and $U$-asymptotic regularity in CAT(0) spaces.