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

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

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Article ID: iaor20171901
Volume: 67
Issue: 3
Start Page Number: 595
End Page Number: 620
Publication Date: Jul 2017
Journal: Computational Optimization and Applications
Authors: ,
Keywords: approximation, Hilbert space, fixed point theory
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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