Asymptotic behavior of averaged and firmly nonexpansive mappings in geodesic spaces

TL;DR

This paper investigates the long-term behavior of averaged and firmly nonexpansive mappings in geodesic spaces, providing explicit bounds and rates of convergence for iterative algorithms used in nonlinear convex feasibility problems.

Contribution

It introduces a quantitative analysis of asymptotic behavior of Picard iterates for these mappings in geodesic spaces, extending previous results with effective bounds.

Findings

Explicit bounds on asymptotic regularity for firmly nonexpansive mappings

Effective rates of asymptotic regularity for algorithms in nonlinear convex feasibility

Generalization of known results to geodesic space setting

Abstract

We further study averaged and firmly nonexpansive mappings in the setting of geodesic spaces with a main focus on the asymptotic behavior of their Picard iterates. We use methods of proof mining to obtain an explicit quantitative version of a generalization to geodesic spaces of result on the asymptotic behavior of Picard iterates for firmly nonexpansive mappings proved by Reich and Shafrir. From this result we obtain effective uniform bounds on the asymptotic regularity for firmly nonexpansive mappings. Besides this, we derive effective rates of asymptotic regularity for sequences generated by two algorithms used in the study of the convex feasibility problem in a nonlinear setting.