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Generalized Krasnoselskii–Mann-type iterations for nonexpansive mappings in Hilbert spaces

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Abstract

The Krasnoselskii–Mann iteration plays an important role in the approximation of fixed points of nonexpansive operators; it is known to be weakly convergent in the infinite dimensional setting. In this present paper, we provide a new inexact Krasnoselskii–Mann iteration and prove weak convergence under certain accuracy criteria on the error resulting from the inexactness. We also show strong convergence for a modified inexact Krasnoselskii–Mann iteration under suitable assumptions. The convergence results generalize existing ones from the literature. Applications are given to the Douglas–Rachford splitting method, the Fermat–Weber location problem as well as the alternating projection method by John von Neumann.

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Author notes

  1. Yekini Shehu

    Present address: Institute of Mathematics, University of Würzburg, Campus Hubland Nord, Emil-Fischer-Str. 30, 97074, Würzburg, Germany

Authors and Affiliations

  1. Institute of Mathematics, University of Würzburg, Campus Hubland Nord, Emil-Fischer-Str. 30, 97074, Würzburg, Germany

    Christian Kanzow

  2. Department of Mathematics, University of Nigeria, Nsukka, Nigeria

    Yekini Shehu

Authors
  1. Christian Kanzow
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  2. Yekini Shehu
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Correspondence to Christian Kanzow.

Additional information

Yekini Shehu: The research of this author is supported by the Alexander von Humboldt-Foundation.

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Kanzow, C., Shehu, Y. Generalized Krasnoselskii–Mann-type iterations for nonexpansive mappings in Hilbert spaces. Comput Optim Appl 67, 595–620 (2017). https://doi.org/10.1007/s10589-017-9902-0

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