Implicit Error Bounds for Picard Iterations on Hilbert Spaces

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.

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Notes

  1. 1.

    Though it is called the Friedrichs angle, the notion goes back at least to Jordan[18, Eq. 60, pp. 122–130].

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Funding

D. Russell Luke was supported in part by German Israeli Foundation Grant G-1253-304.6 and Deutsche Forschungsgemeinschaft Research Training Grant RTG2088.

Nguyen H. Thao was supported by German Israeli Foundation Grant G-1253-304.6 and the European Research Council grant agreement no. 339681.

Matthew K. Tam was supported by Deutsche Forschungsgemeinschaft Research Training Grant 2088 and a Postdoctoral Fellowship from the Alexander von Humboldt Foundation.

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Correspondence to D. Russell Luke.

Additional information

This paper is dedicated to Professor Michel Théra on his 70th birthday.

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Luke, D.R., Thao, N.H. & Tam, M.K. Implicit Error Bounds for Picard Iterations on Hilbert Spaces. Vietnam J. Math. 46, 243–258 (2018). https://doi.org/10.1007/s10013-018-0279-x

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Keywords

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

Mathematics Subject Classification (2010)

  • 49J53
  • 65K10
  • 49M05
  • 49M27
  • 65K05
  • 90C30