Abstract
A complete CAT(0) space X is said to have the nice projection property (property $\mathcal{N}$ for short) if its metric projection onto a geodesic segment preserves points on each geodesic segment, that is, for any geodesic segment L in X and $x,y\in X$ , $m\in[x,y]$ implies $P_{L}(m)\in[P_{L}(x), P_{L}(y)]$ , where $P_{L}$ denotes the metric projection from X onto L. In this paper, we prove a strong convergence theorem of a two-step viscosity iteration method for nonexpansive mappings in CAT(0) spaces without the condition on the property $\mathcal{N}$ . Our result gives an affirmative answer to a problem raised by Piatek (Numer. Funct. Anal. Optim. 34:1245-1264, 2013).
Highlights
A mapping T on a metric space (X, ρ) is said to be a contraction if there exists a constant k ∈ [, ) such that ρ T(x), T(y) ≤ kρ(x, y) for all x, y ∈ X. ( )If ( ) is valid when k =, T is called nonexpansive
Let C be a nonempty, closed, and convex subset of a Hilbert space H and T : C → C be a nonexpansive mapping with Fix(T) = ∅, the following scheme is known as the viscosity iteration method: x = u ∈ C arbitrarily chosen, xn+
Theorem A ([ ], Theorem . ) Let C be a nonempty, closed, and convex subset of a complete CAT( ) space X, T : C → C be a nonexpansive mapping with Fix(T) = ∅, and f : C → C be a contraction with k ∈ [, )
Summary
A mapping T on a metric space (X, ρ) is said to be a contraction if there exists a constant k ∈ [ , ) such that ρ T(x), T(y) ≤ kρ(x, y) for all x, y ∈ X. The first extension of Moudafi’s result to the so-called CAT( ) space was proved by Shi and Chen [ ] They assumed that the space (X, ρ) must satisfy the property P, i.e., for x, u, y , y ∈ X, one has ρ(x, m )ρ(x, y ) ≤ ρ(x, m )ρ(x, y ) + ρ(x, u)ρ(y , y ), where m and m are the unique nearest points of u on the segments [x, y ] and [x, y ], respectively. ) Let C be a nonempty, closed, and convex subset of a complete CAT( ) space X, T : C → C be a nonexpansive mapping with Fix(T) = ∅, and f : C → C be a contraction with k ∈ [ , ).
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