Abstract
We establish strong convergence and Δ-convergence theorems of an iteration scheme associated to a pair of nonexpansive mappings on a nonlinear domain. In particular we prove that such a scheme converges to a common fixed point of both mappings. Our results are a generalization of well-known similar results in the linear setting. In particular, we avoid assumptions such as smoothness of the norm, necessary in the linear case. MSC:47H09, 46B20, 47H10, 47E10.
Highlights
Let C be a nonempty subset of a metric space (X, d) and T : C → C be a mapping
Mann and Ishikawa iterative procedures are well-defined in a vector space through its built-in convexity
Note that Mann iterative procedures were investigated in hyperbolic metric spaces [, ]
Summary
Let C be a nonempty subset of a metric space (X, d) and T : C → C be a mapping. Denote the set of fixed points of T by F(T). Throughout this paper we assume that if X is a uniformly convex hyperbolic space, for every s ≥ and ε > , there exists η(s, ε) > such that δ(r, ε) > η(s, ε) > for any r > s. Let {xn} be a bounded sequence in a metric space X and C be a nonempty subset.
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