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<p>The Krasnosel’skiĭ-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 <i>weakly</i> to a fixed point provided the parameter sequence satisfies a divergent-series condition. In this paper, we show that <i>strong</i> convergence holds provided the underlying nonexpansive mapping is <i>linear</i>. 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.</p>

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Baillon-Bruck-Reich Revisited: Divergent-Series Parameters and Strong Convergence in the Linear Case

  • 期刊论文

  • 发表日期：

    2026年4月16日

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  • Sedi Bartz,
  • Heinz H. Bauschke,
  • Yuan Gao

摘要

The Krasnosel’skiĭ-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 weakly to a fixed point provided the parameter sequence satisfies a divergent-series condition. In this paper, we show that strong convergence holds provided the underlying nonexpansive mapping is 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.