Viscosity approximation methods for asymptotically nonexpansive mapping in CAT(0) spaces (Q2017689)

In the paper under review, the authors prove the strong convergence of the following version of Moudafi's viscosity approximation method \[ y_n = \alpha_nf(y_n) \oplus(1-\alpha_n)T^ny_n \] to \(x^*=P_{F(T)}f(x^*)\) in CAT(0) spaces, where \(T\) is an asymptotically nonexpansive mapping and \(f\) is a contraction. The characterization theorem of \textit{H. Dehghan} and \textit{J. Rooin} [``A characterization of metric projection in CAT(0) spaces'', in: Proceedings of the international conference on functional equations, geometric functions and applications (ICFGA 2012), Payame Noor University, Tabriz, Iran. 41--43 (2012), \url{arXiv:1311.4174}] guarantees that \(x^*=P_{F(T)}f(x^*)\) is the solution of the variational inequality \[ \langle \overrightarrow{x^*f(x^*)}, \overrightarrow{xx^*}\rangle\text{ fo rall }x\in F(T). \] Reviewer's remark. On page 3, line -10, the authors wrongly mentioned that ``Goebel and Kirk's fixed point theorem for asymptotically nonexpansive mappings in Banach spaces guarantees the existence of fixed points in CAT(0) spaces.'' Yet, this error has been corrected in the main results by inserting it in the hypothesis.