Baillon-Bruck-Reich revisited: divergent-series parameters and strong convergence in the linear case

TL;DR

This paper proves that the Krasnoselskii-Mann iteration converges strongly to a fixed point when applied to linear nonexpansive mappings, improving previous weak convergence results.

Contribution

It establishes strong convergence for linear nonexpansive mappings, extending classical results by relaxing parameter conditions.

Findings

Strong convergence under linearity assumption

Improvement over classical weak convergence results

Relaxation of parameter sequence conditions

Abstract

The Krasnoselskii-Mann iteration is an important algorithm in optimization and variational analysis for finding fixed points of nonexpansive mappings. In the general case, it produces a sequence converging \emph{weakly} to a fixed point provided the parameter sequence satisfies a divergent-series condition. In this paper, we show that \emph{strong} convergence holds provided the underlying nonexpansive mapping is \emph{linear}. This improves on a celebrated result by Baillon, Bruck, and Reich from 1978, where the parameter sequence was assumed to be constant as well as on recent work where the parameters were bounded away from $0$ and $1$.