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Δ-Convergence of the Modified Mann Iteration in Complete CAT(0) Spaces

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Abstract

In this paper, we prove Δ-convergence of the sequence given by the modified inexact Mann iteration

$$x_{n+1} = \alpha_{n} S_{n}Py_{n} \oplus (1- \alpha_{n})T_{n}Py_{n}, \quad d(y_{n}, x_{n}) \leq e_{n}, \quad x_{0} \in C, $$

to a common fixed point of the families (T n ) and (S n ) of nonexpansive self-mappings on a closed and convex subset C of a complete CAT(0) space X, where (α n ) ⊂ [0, 1], (e n ) ⊂ ℝ+ and P is the nearest point projection on C. The strong convergence is also discussed in the exact case with certain conditions. This extends a result of Mainge.

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Acknowledgments

The authors are grateful to the referees for their valuable comments and suggestions.

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Authors and Affiliations

  1. Department of Mathematics, University of Zanjan, P.O. Box 45195-313, Zanjan, Iran

    Hadi Khatibzadeh & Sajad Ranjbar

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  1. Hadi Khatibzadeh
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  2. Sajad Ranjbar
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Correspondence to Hadi Khatibzadeh.

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Khatibzadeh, H., Ranjbar, S. Δ-Convergence of the Modified Mann Iteration in Complete CAT(0) Spaces. Vietnam J. Math. 44, 307–313 (2016). https://doi.org/10.1007/s10013-015-0134-2

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  • DOI: https://doi.org/10.1007/s10013-015-0134-2

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