Implicit error bounds for Picard iterations on Hilbert spaces

D.R. Luke, Thao Nguyen, M.K. Tam

Research output: Contribution to journalArticleScientificpeer-review

1 Citation (Scopus)

Abstract

We investigate the role of error bounds, or metric subregularity, in the convergence of Picard iterations of nonexpansive maps in Hilbert spaces. Our main results show, on one hand, that the existence of an error bound is sufficient for strong convergence and, on the other hand, that an error bound exists on bounded sets for nonexpansive mappings possessing a fixed point whenever the space is finite dimensional. In the Hilbert space setting, we show that a monotonicity property of the distances of the Picard iterations is all that is needed to guarantee the existence of an error bound. The same monotonicity assumption turns out also to guarantee that the distance of Picard iterates to the fixed point set converges to zero. Our results provide a quantitative characterization of strong convergence as well as new criteria for when strong, as opposed to just weak, convergence holds.

Original languageEnglish
Pages (from-to)243-258
JournalVietnam Journal of Mathematics
Volume46
Issue number2
DOIs
Publication statusPublished - 2018

Keywords

  • Averaged operators
  • Error bounds
  • Fixed points
  • Metric regularity
  • Metric subregularity
  • Nonexpansiveness
  • Picard iteration
  • Strong convergence

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