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

Author

Listed:
  • Christian Kanzow

    (University of Würzburg)

  • Yekini Shehu

    (University of Nigeria
    University of Würzburg)

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.

Suggested Citation

  • Christian Kanzow & Yekini Shehu, 2017. "Generalized Krasnoselskii–Mann-type iterations for nonexpansive mappings in Hilbert spaces," Computational Optimization and Applications, Springer, vol. 67(3), pages 595-620, July.
    • Handle: RePEc:spr:coopap:v:67:y:2017:i:3:d:10.1007_s10589-017-9902-0
    DOI: 10.1007/s10589-017-9902-0
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    Cited by:

    1. Yunda Dong, 2021. "Weak convergence of an extended splitting method for monotone inclusions," Journal of Global Optimization, Springer, vol. 79(1), pages 257-277, January.
    2. Ke Guo & Deren Han, 2018. "A note on the Douglas–Rachford splitting method for optimization problems involving hypoconvex functions," Journal of Global Optimization, Springer, vol. 72(3), pages 431-441, November.

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