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Strong convergence of a general iteration scheme in $CAT\left(0\right)$ spaces. (English) Zbl 1202.47076
Summary: We introduce and study strong convergence of a general iteration scheme for a finite family of asymptotically quasi-nonexpansive maps in convex metric spaces and $CAT\left(0\right)$ spaces. The new iteration scheme includes modified Mann and Ishikawa iterations, the three-step iteration scheme of B.-L. Xu and M. A. Noor [J. Math. Anal. Appl. 267, No. 2, 444–453 (2002; Zbl 1011.47039)] and the scheme of A. R. Khan, A. A. Domlo and H. Fukhar-Ud-Din [J. Math. Anal. Appl. 341, No. 1, 1–11 (2008; Zbl 1137.47053)] as special cases in Banach spaces. Our results are refinements and generalizations of several recent results from the current literature.
##### MSC:
 47J25 Iterative procedures (nonlinear operator equations) 54H25 Fixed-point and coincidence theorems in topological spaces 47H09 Mappings defined by “shrinking” properties