Abstract
A general viscosity iterative method for a finite family of generalized asymptotically quasi-nonexpansive mappings in a convex metric space is introduced. Special cases of the new iterative method are the viscosity iterative method of Chang et al. (Appl. Math. Comput. 212:51-59, 2009), an analogue of the viscosity iterative method of Fukhar-ud-din et al. (J. Nonlinear Convex Anal. 16:47-58, 2015) and an extension of the multistep iterative method of Yildirim and Ozdemir (Arab. J. Sci. Eng. 36:393-403, 2011). Our results generalize and extend the corresponding known results in uniformly convex Banach spaces and $\operatorname{CAT}(0)$ spaces simultaneously.
Highlights
Introduction and preliminaries LetC be a nonempty subset of a metric space X and T : C → C be a mapping
The class of generalized asymptotically quasi-nonexpansive mappings includes the class of asymptotically quasi-nonexpansive mappings
2 Convergence in convex metric spaces we prove some results for the viscosity iterative method ( . ) to converge to a common fixed point of a finite family of generalized asymptotically quasi-nonexpansive mappings in a convex metric space
Summary
Introduction and preliminaries LetC be a nonempty subset of a metric space X and T : C → C be a mapping. ), we define {xn} as follows: x ∈ C, xn+ = W f (xn), Snxn, αn and call it a general viscosity iterative method in a convex metric space. ) to a common fixed point of a finite family of generalized asymptotically quasi-nonexpansive mappings on a convex metric space; (ii) prove strong convergence and -convergence results for the iterative method ) to a common fixed point of a finite family of generalized asymptotically quasi-nonexpansive mappings on a uniformly convex metric space.
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