Strong Convergence of Viscosity Approximation Methods for Nonexpansive Mappings in CAT(0) Spaces

Abstract

Viscosity approximation methods for nonexpansive mappings in CAT(0) spaces are studied. Consider a nonexpansive self-mapping 𝑇 of a closed convex subset 𝐶 of a CAT(0) space 𝑋. Suppose that the set Fix(𝑇) of fixed points of 𝑇 is nonempty. For a contraction 𝑓 on 𝐶 and 𝑡∈(0,1), let 𝑥_𝑡∈𝐶 be the unique fixed point of the contraction 𝑥↦𝑡𝑓(𝑥)⊕(1−𝑡)𝑇𝑥. We will show that if 𝑋 is a CAT(0) space satisfying some property, then {𝑥_𝑡} converge strongly to a fixed point of 𝑇 which solves some variational inequality. Consider also the iteration process {𝑥_𝑛}, where 𝑥₀∈𝐶 is arbitrary and 𝑥_𝑛+1=𝛼_𝑛𝑓(𝑥_𝑛)⊕(1−𝛼_𝑛)𝑇𝑥_𝑛 for 𝑛≥1, where {𝛼_𝑛}⊂(0,1). It is shown that under certain appropriate conditions on 𝛼_𝑛,{𝑥_𝑛} converge strongly to a fixed point of 𝑇 which solves some variational inequality.

1. CAT(0) Spaces

A metric space 𝑋 is a CAT(0) space if it is geodesically connected and if every geodesic triangle in 𝑋 is at least as thin as its comparison triangle in the Euclidean plane. The precise definition is given below. It is well known that any complete, simply connected Riemannian manifold having nonpositive sectional curvature is a CAT(0) space. Other examples include pre-Hilbert spaces [1], R-trees [2], Euclidean buildings [3], the complex Hilbert ball with a hyperbolic metric [4], and many others. For a thorough discussion of these spaces and of the fundamental role they play in geometry, we refer the reader to Bridson and Haefliger [1].

Fixed-point theory in CAT(0) spaces was first studied by Kirk (see [5, 6]). He showed that every nonexpansive (single-valued) mapping defined on a bounded closed convex subset of a complete CAT(0) space always has a fixed point. Since then, the fixed-point theory for single-valued and multivalued mappings in CAT(0) spaces has been rapidly developed, and many papers have appeared [2, 7–17].

The purpose of this paper is to study the iterative scheme defined as follows. Consider a nonexpansive self-mapping 𝑇 of a closed convex subset 𝐶 of a CAT(0) space 𝑋. Suppose that the set Fix(𝑇) of fixed points of 𝑇 is nonempty. For a contraction 𝑓 on 𝐶 and 𝑡∈(0,1), let 𝑥_𝑡∈𝐶 be the unique fixed point of the contraction 𝑥↦𝑡𝑓(𝑥)⊕(1−𝑡)𝑇𝑥. Consider the iteration process {𝑥_𝑛}, where 𝑥₀∈𝐶 is arbitrary and𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛⊕1−𝛼_𝑛𝑇𝑥_𝑛,(1.1) for 𝑛≥1, where {𝛼_𝑛}⊂(0,1). We show that {𝑥_𝑛} converge strongly to a fixed point of 𝑇 under certain appropriate conditions on 𝛼_𝑛, and the fixed point of 𝑇 solves some variational inequality.

Let (𝑋,𝑑) be a metric space. A geodesic path joining 𝑥∈𝑋 to 𝑦∈𝑋 (or, more briefly, a geodesic from 𝑥 to 𝑦) is a map 𝑐 from a closed interval [0,𝑙]⊂𝑅 to 𝑋 such that 𝑐(0)=𝑥,𝑐(𝑙)=𝑦, and 𝑑(𝑐(𝑡),𝑐(𝑡^))=|𝑡−𝑡^| for all 𝑡,𝑡∈[0,𝑙]. In particular, 𝑐 is an isometry and 𝑑(𝑥,𝑦)=𝑙. The image 𝛼 of 𝑐 is called a geodesic (or metric) segment joining 𝑥 and 𝑦. When it is unique, this geodesic segment is denoted by [𝑥,𝑦]. The space (𝑋,𝑑) is said to be a geodesic space if every two points of 𝑋 are joined by a geodesic, and 𝑋 is said to be uniquely geodesic if there is exactly one geodesic joining 𝑥 and 𝑦 for each 𝑥,𝑦∈𝑋. A subset 𝑌⊆𝑋 is said to be convex if 𝑌 includes every geodesic segment joining any two of its points.

A geodesic triangle ▵(𝑥₁,𝑥₂,𝑥₃) in a geodesicmetric space (𝑋,𝑑) consists of three points 𝑥₁,𝑥₂, and 𝑥₃ in 𝑋 (the vertices of ▵) and a geodesic segment between each pair of vertices (the edges of ▵). A comparison triangle for the geodesic triangle ▵(𝑥₁,𝑥₂,𝑥₃) in (𝑋,𝑑) is a triangle ▵(𝑥₁,𝑥₂,𝑥₃)∶=▵(𝑥₁,𝑥₂,𝑥₃) in the Euclidean plane 𝔼² such that 𝑑_𝔼²(𝑥_𝑖,𝑥_𝑗)=𝑑(𝑥_𝑖,𝑥_𝑗) for 𝑖,𝑗∈1,2,3.

A geodesic space is said to be a CAT(0) space if all geodesic triangles satisfy the following comparison axiom.

CAT(0): let ▵ be a geodesic triangle in 𝑋, and let ▵ be a comparison triangle for ▵. Then, ▵ is said to satisfy the CAT(0) inequality if for all 𝑥,𝑦∈▵ and all comparison points 𝑥,𝑦∈▵,𝑑(𝑥,𝑦)≤𝑑_𝔼²𝑥,𝑦.(1.2)

Let 𝑥,𝑦∈𝑋, and by Lemma 2.1(iv) of [18] for each 𝑡∈[0,1], there exists a unique point 𝑧∈[𝑥,𝑦] such that𝑑(𝑥,𝑧)=𝑡𝑑(𝑥,𝑦),𝑑(𝑦,𝑧)=(1−𝑡)𝑑(𝑥,𝑦).(1.3) From now on, we will use the notation (1−𝑡)𝑥⊕𝑡𝑦 for the unique point 𝑧 satisfying the above equation.

We now collect some elementary facts about CAT(0) spaces which will be used in the proofs of our main results.

Lemma 1.1. Let 𝑋 be a CAT(0) space. Then,(i)(see [18, Lemma 2.4]) for each 𝑥,𝑦,𝑧∈𝑋 and 𝑡∈[0,1], one has𝑑((1−𝑡)𝑥⊕𝑡𝑦,𝑧)≤(1−𝑡)𝑑(𝑥,𝑧)+𝑡𝑑(𝑦,𝑧),(1.4)(ii) (see [7]) for each 𝑥,𝑦,𝑧∈𝑋 and 𝑡,𝑠∈[0,1], one has𝑑((1−𝑡)𝑥⊕𝑡𝑦,(1−𝑠)𝑥⊕𝑠𝑦)≤|𝑡−𝑠|𝑑(𝑥,𝑦),(1.5)(iii) (see [6, Lemma 3]) for each 𝑥,𝑦,𝑧∈𝑋 and 𝑡∈[0,1], one has𝑑((1−𝑡)𝑧⊕𝑡𝑥,(1−𝑡)𝑧⊕𝑡𝑦)≤𝑡𝑑(𝑥,𝑦),(1.6)(iv) (see [18, Lemma 2.5]) for each 𝑥,𝑦,𝑧∈𝑋 and 𝑡∈[0,1], one has𝑑((1−𝑡)𝑥⊕𝑡𝑦,𝑧)²≤(1−𝑡)𝑑(𝑥,𝑧)²+𝑡𝑑(𝑦,𝑧)²−𝑡(1−𝑡)𝑑(𝑥,𝑦)².(1.7)

Let 𝑋 be a complete CAT(0) space, let {𝑥_𝑛} be a bounded sequence in a complete X, and for 𝑥∈𝑋, set𝑟𝑥,𝑥_𝑛=limsup_𝑛→∞𝑑𝑥,𝑥_𝑛.(1.8)

The asymptotic radius 𝑟({𝑥_𝑛}) of {𝑥_𝑛} is given by𝑟𝑥_𝑛=inf𝑟𝑥,𝑥_𝑛∶𝑥∈𝑋,(1.9)

and the asymptotic center 𝐴({𝑥_𝑛}) of {𝑥_𝑛} is the set𝐴𝑥_𝑛=𝑥∈𝑋∶𝑟𝑥,𝑥_𝑛=𝑟𝑥_𝑛.(1.10)

It is known (see, e.g., [11, Proposition 7]) that in a CAT(0) space, 𝐴({𝑥_𝑛}) consists of exactly one point.

A sequence {𝑥_𝑛} in 𝑋 is said to ▵-converge to 𝑥∈𝑋 if 𝑥 is the unique asymptotic center of {𝑢_𝑛} for every subsequence {𝑢_𝑛} of {𝑥_𝑛}. In this case, we write ▵-lim_𝑛𝑥_𝑛=𝑥 and call 𝑥 the ▵-limit of {𝑥_𝑛}.

Lemma 1.2. Assume that 𝑋 is a CAT(0) space. Then,(i) (see [14]) every bounded sequence in 𝑋 has a ▵-convergent subsequence,(ii) (see [14, Proposition 3.7]) if 𝐾 is a closed convex subset of 𝑋, and 𝑓∶𝐾→𝑋 is a nonexpansive mapping, then the conditions {𝑥_𝑛}▵-converge to 𝑥 and 𝑑(𝑥_𝑛,𝑓(𝑥_𝑛))→0 and imply 𝑥∈𝐾 and 𝑓(𝑥)=𝑥.

Lemma 1.3 (see [19, Proposition 3.5]). Assume that 𝑋 is a CAT(0) space, 𝐶 is a closed convex subset of 𝑋. Then the metric (nearest point) projection 𝑃_𝐶∶𝑋→𝐶,𝑃_𝐶(𝑥)∶=inf{𝑑(𝑥,𝑦);𝑦∈𝐶} is a nonexpansive mapping. one calls a CAT(0) space 𝑋 satisfying property 𝒫 if for 𝑥,𝑢,𝑦₁,𝑦₂∈𝑋, 𝑑𝑥,𝑃_[𝑥,𝑦1](𝑢)𝑑𝑥,𝑦₁≤𝑑𝑥,𝑃_[𝑥,𝑦2](𝑢)𝑑𝑥,𝑦₂+𝑑(𝑥,𝑢)𝑑𝑦₁,𝑦₂.(1.11)

Remark 1.4. The property 𝒫 in Hilbert space corresponds to the inequality ||⟨𝑢−𝑥,𝑦₁−𝑥⟩||≤||⟨𝑢−𝑥,𝑦₂−𝑥⟩||+‖𝑢−𝑥‖⋅‖‖𝑦₁−𝑦₂‖‖.(1.12)
Recall that a continuous linear functional 𝜇 on 𝑙^∞, the Banach space of bounded real sequences, is called a Banach limit if ‖𝜇‖=𝜇(1,1,…)=1 and 𝜇_𝑛(𝑎_𝑛)=𝜇_𝑛(𝑎_𝑛+1) for all {𝑎_𝑛}∈𝑙^∞.

Lemma 1.5 (see [20, Proposition 2]). Let (𝑎₁,𝑎₂,…)∈𝑙^∞ be such that 𝜇_𝑛(𝑎_𝑛)≤0 for all Banach limits 𝜇 and limsup_𝑛(𝑎_𝑛+1−𝑎_𝑛)≤0. Then limsup_𝑛𝑎_𝑛≤0.

Lemma 1.6 (see [21, Lemma 2.3]). Let {𝑠_𝑛} be a sequence of nonnegative real numbers, {𝛼_𝑛} a sequence of real numbers in [0,1] with ∑^∞𝑛=1𝛼_𝑛=∞, {𝑢_𝑛} a sequence of nonnegative real numbers with ∑^∞𝑛=1𝑢_𝑛<∞, and {𝑡_𝑛} a sequence of real numbers with limsup_𝑛𝑡_𝑛≤0. Suppose that 𝑠_𝑛+1=1−𝛼_𝑛𝑠_𝑛+𝛼_𝑛𝑡_𝑛+𝑢_𝑛,∀𝑛∈ℕ.(1.13) Then lim_𝑛→∞𝑠_𝑛=0.

2. Viscosity Iteration for a Single Mapping

In this section, we prove the main results of this paper.

Lemma 2.1. Let 𝐶 be a closed convex subset of a complete CAT(0) space 𝑋, and let 𝑇∶𝐶→𝐶 be a nonexpansive mapping. Let 𝑓 be a contraction on 𝐶 with coefficient 𝛼<1. For each 𝑡∈[0,1], the mapping 𝑆_𝑡∶𝐶→𝐶 defined by 𝑆_𝑡𝑥=𝑡𝑓(𝑥)⊕(1−𝑡)𝑇𝑥,for𝑥∈𝐶(2.1) has a unique fixed point 𝑥_𝑡∈𝐶, that is, 𝑥_𝑡=𝑡𝑓𝑥_𝑡⊕(1−𝑡)𝑇𝑥_𝑡.(2.2)

Proof. For 𝑥,𝑦∈𝐶, according to Lemma 1.1, we have the following: 𝑑𝑆_𝑡(𝑥),𝑆_𝑡(𝑦)=≤≤≤+𝑑𝑑𝑡(𝑑1𝑑(((𝑡𝑡−𝑓𝑓𝑓(𝑡𝑓𝑡(((𝑥𝑥𝑥((1)))𝑦,−⊕⊕)𝑓⊕𝛼((()𝑦11)()1𝑑−−)−+(𝑡𝑡𝑥))𝑡(,𝑇𝑇)1𝑦𝑥𝑥𝑇)−,,𝑥.𝑡𝑡,𝑡𝑓𝑓)𝑡𝑓𝑑((𝑥𝑦(())𝑦𝑇⊕⊕)𝑥⊕,𝑇((11𝑦()1−−−𝑡𝑡))𝑡𝑇𝑇)𝑦𝑦𝑇))𝑦)(2.3) This implies that 𝑆_𝑡 is a contraction mapping, and hence, the conclusion follows.

The following result is to prove that the net {𝑥_𝑡} converge strongly to a fixed point of 𝑇.

Theorem 2.2. Let 𝐶 be a closed convex subset of a complete CAT(0) space 𝑋 satisfying the property 𝒫, and let 𝑇∶𝐶→𝐶 be a nonexpansive mapping. Let 𝑓 be a contraction on 𝐶 with coefficient 𝛼<1. For each 𝑡∈[0,1], let {𝑥_𝑡} be given by 𝑥_𝑡=𝑡𝑓𝑥_𝑡⊕(1−𝑡)𝑇𝑥_𝑡.(2.4) Then one has lim_𝑡→0𝑥_𝑡=∶𝑥̃ and 𝑥̃=𝑃_Fix(𝑇)𝑓(𝑥̃).

Proof. We first show that {𝑥_𝑡} is bounded. Indeed choose a 𝑝∈Fix(𝑇), and using Lemma 1.1 and the nonexpansive of 𝑇, we derive that 𝑑𝑥_𝑡,𝑝≤=𝑑𝑡𝑑𝑡𝑓𝑓𝑥𝑥_𝑡𝑡⊕(1−𝑡)𝑇𝑥_𝑡,𝑝≤𝑡𝑑𝑓𝑥_𝑡,𝑝+(1−𝑡)𝑑𝑇𝑥_𝑡,𝑝,𝑝+(1−𝑡)𝑑𝑥_𝑡,𝑝.(2.5) It follows that 𝑑𝑥_𝑡,𝑝≤𝑑𝑓𝑥_𝑡,𝑝≤≤𝛼𝑑𝑑𝑓𝑥_𝑡𝑥_𝑡,𝑓(𝑝)+𝑑(𝑓(𝑝),𝑝),𝑝+𝑑(𝑓(𝑝),𝑝).(2.6) Hence, 𝑑𝑥_𝑡,𝑝≤11−𝛼𝑑(𝑓(𝑝),𝑝),(2.7) and {𝑥_𝑡} is bounded, so are {𝑇𝑥_𝑡} and {𝑓(𝑥_𝑡)}. As a result, we can get that lim_𝑡→0𝑑𝑥_𝑡,𝑇𝑥_𝑡==lliimm_{𝑡→0𝑡→0}𝑑𝑡𝑓𝑥_𝑡⊕(1−𝑡)𝑇𝑥_𝑡,𝑇𝑥_𝑡𝑡𝑑𝑓𝑥_𝑡,𝑇𝑥_𝑡=0.(2.8)
Assume that {𝑡_𝑛}⊆(0,1) is such that 𝑡_𝑛→0 as 𝑛→∞. Put 𝑥_𝑛∶=𝑥_𝑡𝑛. We will show that {𝑥_𝑛} contains a subsequence converging strongly to 𝑥̃, where 𝑥̃∈Fix(𝑇).
Since {𝑥_𝑛} is bounded, by Lemma 1.2(i),(ii), we may assume that {𝑥_𝑛}▵-converges to a point 𝑥̃, and 𝑥̃∈Fix(𝑇).
Next we will prove that {𝑥_𝑛} converge strongly to 𝑥̃.
Indeed, according to Lemma 1.1 and the property of 𝑇 and 𝑓, we can get that 𝑑²𝑥_𝑛,𝑥̃≤=𝑑𝑡²_𝑛𝑡_𝑛𝑓𝑥_𝑛⊕1−𝑡_𝑛𝑇𝑥_𝑛,𝑥̃≤𝑡_𝑛𝑑²𝑓𝑥_𝑛,𝑥̃+1−𝑡_𝑛𝑑²𝑇𝑥_𝑛,𝑥̃−𝑡_𝑛1−𝑡_𝑛𝑑²𝑓𝑥_𝑛,𝑇𝑥_𝑛𝑑²𝑓𝑥_𝑛,𝑥̃+1−𝑡_𝑛𝑑²𝑥_𝑛,𝑥̃−𝑡_𝑛1−𝑡_𝑛𝑑²𝑓𝑥_𝑛,𝑇𝑥_𝑛.(2.9) It follows that 𝑑²𝑥_𝑛,𝑥̃=≤𝑑𝑑²₂𝑓𝑥_𝑛,𝑥̃−1−𝑡_𝑛𝑑²𝑓𝑥_𝑛,𝑇𝑥_𝑛𝑓𝑥_𝑛,𝑥̃−𝑑²𝑓𝑥_𝑛,𝑇𝑥_𝑛+𝑡_𝑛𝑑²𝑓𝑥_𝑛,𝑇𝑥_𝑛.(2.10)
Since lim_𝑡→0𝑑(𝑥_𝑡,𝑇𝑥_𝑡)=0, we can get that limsup_𝑛→∞𝑑²𝑥_𝑛,𝑥̃≤limsup_𝑛→∞𝑑²𝑓𝑥_𝑛,𝑥̃−𝑑²𝑓𝑥_𝑛,𝑥_𝑛.(2.11) Let Δ(𝑥̃,𝑥_𝑛,𝑓(𝑥_𝑛)) be a comparison triangle for ▵(𝑥̃,𝑥_𝑛,𝑓(𝑥_𝑛)) in 𝔼². Then, 𝑑²𝑓𝑥_𝑛,𝑥̃−𝑑²𝑓𝑥_𝑛,𝑥_𝑛==𝑑²𝑓𝑥_𝑛,𝑥̃−𝑑²𝑓𝑥_𝑛,𝑥_𝑛=2𝑓𝑥_𝑛−𝑥̃,𝑓𝑥_𝑛−𝑥̃−𝑓𝑥_𝑛−𝑥̃_𝑛,𝑓𝑥_𝑛−𝑥̃_𝑛=2𝑓𝑥_𝑛−𝑥̃,𝑥_𝑛−𝑥̃−𝑥_𝑛−𝑥̃,𝑥_𝑛−𝑥̃=2𝑓𝑥_𝑛−𝑥̃,𝑥_𝑛−𝑥̃−𝑑²𝑥_𝑛,𝑥̃𝑓𝑥_𝑛−𝑥̃,𝑥_𝑛−𝑥̃−𝑑²𝑥_𝑛,𝑥̃.(2.12) Hence, limsup_𝑛→∞𝑑²𝑥_𝑛,𝑥̃≤limsup_𝑛→∞𝑓𝑥_𝑛−𝑥̃,𝑥_𝑛−𝑥̃.(2.13)
Let ▵(𝑥̃,𝑥_𝑛,𝑓(𝑥̃)) be a comparison triangle for ▵(𝑥̃,𝑥_𝑛,𝑓(𝑥̃)) in 𝔼². For each 𝑛, let 𝑢_𝑛 be the point of the segment [𝑥̃,𝑓(𝑥̃)] which is nearest to 𝑥_𝑛, and let 𝑢_𝑛 be the point of the segment [𝑥̃,𝑓(𝑥̃)] for which 𝑑(𝑢_𝑛,𝑥̃)=𝑑(𝑢_𝑛,𝑥̃).
By passing to subsequences again, we may suppose that {𝑢_𝑛} converges to 𝑢∈[𝑥̃,𝑓(𝑥̃)], {𝑢_𝑛} converges to 𝑢∈[𝑥̃,𝑓(𝑥̃)].
Since {𝑥_𝑛}▵-converges to a point 𝑥̃, we have 𝑟𝑥_𝑛==lliimm_𝑛𝑛sup𝑑𝑥,𝑥_𝑛≥lim_𝑛sup𝑑𝑥,𝑥_𝑛=lim_𝑛sup𝑑𝑢_𝑛,𝑥_𝑛≥lim_𝑛sup𝑑𝑢,𝑥_𝑛sup𝑑𝑢,𝑥_𝑛.(2.14) Thus, 𝑟(𝑢,{𝑥_𝑛})≤𝑟({𝑥_𝑛}). This implies that 𝑢=𝑥 by uniqueness of the asymptotic center. Hence, 𝑢=𝑥. That is to say, {𝑢_𝑛} converges to 𝑥̃, and {𝑢_𝑛} converges to 𝑥̃.
Moreover, since 𝑋 satisfies the property 𝒫, we can get that |||𝑓𝑥_𝑛−𝑥̃,𝑥_𝑛−𝑥̃|||=𝑑𝑥̃,𝑃_[𝑥̃_{,𝑓(𝑥𝑛)]}=𝑑𝑥̃,𝑃_[𝑥_𝑛⋅𝑑𝑥̃,𝑓𝑥_𝑛≤𝑑𝑥̃,𝑃_[𝑥̃_{,𝑓(𝑥𝑛)]}𝑥_𝑛⋅𝑑𝑥̃,𝑓𝑥_𝑛+𝑑𝑥̃,𝑥𝑥_𝑛̃_,𝑓(𝑥̃_)]𝑥_𝑛⋅𝑑(𝑥̃,𝑓(𝑥̃))=𝑑⋅𝑑𝑓𝑥_𝑛,𝑓(𝑥̃)𝑥̃,𝑃_[𝑥̃_,𝑓(𝑥̃_)]+𝑑𝑥̃,𝑥_𝑛𝑥_𝑛⋅𝑑𝑥̃,𝑓(𝑥̃)≤𝑑⋅𝑑𝑓𝑥_𝑛,𝑓(𝑥̃)𝑢_𝑛,𝑥̃𝑑𝑥̃,𝑓(𝑥̃)+𝛼𝑑²𝑥_𝑛,𝑥̃.(2.15) It follows that limsup_𝑛→∞𝑑²𝑥_𝑛,𝑥̃≤11−𝛼limsup^𝑛→∞𝑑𝑢_𝑛,𝑥̃𝑑𝑓(𝑥̃),𝑥̃.(2.16) Since {𝑢_𝑛} converges to 𝑥̃, we obtain that limsup_𝑛→∞𝑑²(𝑥_𝑛,𝑥̃)=0, that is, {𝑥_𝑛} converge strongly to 𝑥̃. Since {𝑡_𝑛}⊆(0,1) is such that 𝑡_𝑛→0 as 𝑛→∞ is arbitrarily selected, we can get that lim_𝑡→0𝑥_𝑡=𝑥̃.
Finally, we will prove that 𝑥̃ satisfy the equation 𝑥̃=𝑃_Fix(𝑇)𝑓(𝑥̃).
Indeed, for any 𝑦∈Fix(𝑇), 𝑑𝑥_𝑡,𝑦==𝑑𝑡𝑑𝑡𝑓𝑓𝑥𝑥_𝑡𝑡⊕(1−𝑡)𝑇𝑥_𝑡,𝑦≤𝑡𝑑𝑓𝑥_𝑡,𝑦+(1−𝑡)𝑑𝑇𝑥_𝑡,𝑦,𝑦+(1−𝑡)𝑑𝑥_𝑡,𝑦.(2.17) It follows that 𝑑𝑥_𝑡,𝑦≤𝑑𝑓𝑥_𝑡,𝑦.(2.18) Since lim_𝑡→0𝑥_𝑡=𝑥̃, we can get that 𝑑(𝑥̃,𝑦)≤𝑑(𝑓(𝑥̃),𝑦).(2.19)
Hence, 𝑑𝑑((𝑓𝑥̃,(𝑓𝑥̃)(,𝑥𝑦̃)))≥≥𝑑||𝑑(𝑓(𝑥(̃,𝑥𝑦̃)),−𝑦)𝑑.(𝑥̃,𝑓(𝑥̃))||,(2.20) That is to say, 𝑥̃=𝑃_Fix(𝑇)𝑓(𝑥̃).

Consider now the iteration process𝑥𝑥₀∈𝐶,_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛⊕1−𝛼_𝑛𝑇𝑥_𝑛,𝑛≥0,(2.21) where {𝛼_𝑛}⊆(0,1) satisfies(H1)𝛼_𝑛→0,(H2)∑^∞𝑛=0𝛼_𝑛=∞,(H3)either ∑^∞𝑛=0|𝛼_𝑛+1−𝛼_𝑛|<∞ or lim_𝑛→∞(𝛼_𝑛+1/𝛼_𝑛)=1.

Theorem 2.3. Let 𝑋 be a CAT(0) space satisfying the property 𝒫, 𝐶 a closed convex subset of 𝑋, 𝑇∶𝐶→𝐶 a nonexpansive mapping with Fix(𝑇)≠∅, and 𝑓∶𝐶→𝐶 a contraction with coefficient 𝛼<1. Let 𝑥₀∈𝐶, {𝑥_𝑛} be generated by 𝑥_𝑛+1=𝛼_𝑛𝑓(𝑥_𝑛)⊕(1−𝛼_𝑛)𝑇𝑥_𝑛,𝑛≥0. Then under the hypotheses (H1 )–(H3 ), 𝑥_𝑛→𝑥̃, where 𝑥̃=𝑃_Fix(𝑇)𝑓(𝑥̃).

Proof. We first show that the sequence {𝑥_𝑛} is bounded. Let 𝑝∈Fix(𝑇). Then, 𝑑𝑥_𝑛+1,𝑝=≤𝑑𝛼_𝑛𝛼_𝑛𝑓𝑥_𝑛⊕1−𝛼_𝑛𝑇𝑥_𝑛,𝑝≤𝛼_𝑛𝑑𝑓𝑥_𝑛,𝑝+1−𝛼_𝑛𝑑𝑇𝑥_𝑛,𝑝≤max𝑑𝑓𝑑𝑥𝑥_𝑛𝑛,𝑓(𝑝)+𝑑(𝑓(𝑝),𝑝)+1−𝛼_𝑛𝑑𝑥_𝑛,𝑝,𝑝,11−𝛼𝑑(𝑓(𝑝),𝑝).(2.22) By induction, we have 𝑑𝑥_𝑛,𝑝≤max𝑑𝑥₀,𝑝,11−𝛼𝑑(𝑓(𝑝),𝑝),(2.23) for all 𝑛∈ℕ. This implies that {𝑥_𝑛} is bounded and so is the sequence {𝑇𝑥_𝑛} and {𝑓(𝑥_𝑛)}.
We claim that 𝑑(𝑥_𝑛+1,𝑥_𝑛)→0. Indeed, we have 𝑑𝑥_𝑛+1,𝑥_𝑛≤=𝑑𝑑𝛼𝛼_𝑛𝑛𝑓𝑥_𝑛⊕1−𝛼_𝑛𝑇𝑥_𝑛,𝛼_𝑛−1𝑓𝑥_𝑛−1⊕1−𝛼_𝑛−1𝑇𝑥_𝑛−1+𝑑𝛼𝑓_𝑛𝑥_𝑛⊕1−𝛼_𝑛𝑇𝑥_𝑛,𝛼_𝑛𝑓𝑥_𝑛⊕1−𝛼_𝑛𝑇𝑥_𝑛−1+𝑑𝛼_𝑛𝑓𝑥_𝑛⊕1−𝛼_𝑛𝑇𝑥_𝑛−1,𝛼_𝑛𝑓𝑥_𝑛−1⊕1−𝛼_𝑛𝑇𝑥_𝑛−1≤1−𝛼_𝑛𝑓𝑥_𝑛−1⊕1−𝛼_𝑛𝑇𝑥_𝑛−1,𝛼_𝑛−1𝑓𝑥_𝑛−1⊕1−𝛼_𝑛−1𝑇𝑥_𝑛−1≤1−𝛼_𝑛𝑑𝑇𝑥_𝑛,𝑇𝑥_𝑛−1+𝛼_𝑛𝑑𝑓𝑥_𝑛,𝑓𝑥_𝑛−1+||𝛼_𝑛−𝛼_𝑛−1||𝑑𝑓𝑥_𝑛−1,𝑇𝑥_𝑛−1𝑑𝑥_𝑛,𝑥_𝑛−1+𝛼_𝑛𝛼𝑑𝑥_𝑛,𝑥_𝑛−1+||𝛼_𝑛−𝛼_𝑛−1||𝑑𝑓𝑥_𝑛−1,𝑇𝑥_𝑛−1.(2.24) By the conditions H2 and H3, we have 𝑑𝑥_𝑛+1,𝑥_𝑛⟶0.(2.25) Consequently, by the condition H1, 𝑑𝑥_𝑛,𝑇𝑥_𝑛≤=𝑑𝑑𝑥𝑥_𝑛𝑛,𝑥_𝑛+1+𝑑𝑥_𝑛+1,𝑇𝑥_𝑛=𝑑𝑥_𝑛,𝑥_𝑛+1+𝑑𝛼_𝑛𝑓𝑥_𝑛⊕1−𝛼_𝑛𝑇𝑥_𝑛,𝑇𝑥_𝑛,𝑥_𝑛+1+𝛼_𝑛𝑑𝑓𝑥_𝑛,𝑇𝑥_𝑛⟶0.(2.26)
Since {𝑥_𝑛} is bounded, we may assume that {𝑥_𝑛}▵-converges to a point 𝑥̂. By Lemma 1.2, we have 𝑥̂∈Fix(𝑇).
Next we will prove that {𝑥_𝑛} converge strongly to 𝑥̂ and 𝑥̂=𝑥̃. Indeed, according to Lemma 1.1 and the property of 𝑇 and 𝑓, we can get that 𝑑²𝑥_𝑛+1,𝑥̂=≤𝛼𝑑²_𝑛𝑡_𝑛𝑓𝑥_𝑛⊕1−𝑡_𝑛𝑇𝑥_𝑛,𝑥̂≤1𝑑−²𝛼𝑓_𝑛𝑥_𝑛,𝑥̂+1−𝛼_𝑛𝑑²𝑇𝑥_𝑛,𝑥̂−𝛼_𝑛1−𝛼_𝑛𝑑²𝑓𝑥_𝑛,𝑇𝑥_𝑛𝑑²𝑥_𝑛,𝑥̂+𝛼_𝑛𝑑²𝑓𝑥_𝑛,𝑥̂−1−𝛼_𝑛𝑑²𝑓𝑥_𝑛,𝑇_𝑛.(2.27) With a minor modification of the proof of the analogous statement in Theorem 2.2, we can get that 𝑑²𝑓𝑥_𝑛,𝑥̂−𝑑²𝑓𝑥_𝑛,𝑥_𝑛=≤22𝑑𝑓𝑥_𝑛−𝑥̂,𝑥_𝑛−𝑥̂−𝑑²𝑥_𝑛,𝑥̂𝑢_𝑛,𝑥̂𝑑𝑓(𝑥̂),𝑥̂+2𝛼𝑑²𝑥_𝑛,𝑥̂−𝑑²𝑥_𝑛,𝑥̂,(2.28) and 𝑑(𝑢_𝑛,𝑥̂)→0.
Thus, 𝑑²𝑥_𝑛+1,𝑥̂≤+1−1−𝛼_𝑛𝛼_𝑛𝑑²𝑥_𝑛,𝑥̂+𝛼_𝑛𝑑²𝑓𝑥_𝑛,𝑥̂−1−𝛼_𝑛𝑑²𝑓𝑥_𝑛,𝑥_𝑛≤1−2(1−𝑑𝛼²)𝛼𝑓_𝑛𝑥_𝑛,𝑥_𝑛−1−𝛼_𝑛𝑑²𝑇𝑥_𝑛,𝑥_𝑛+2(1−𝛼)𝛼_𝑛𝑑²𝑥_𝑛,𝑥̂(1−1𝛼)𝑑𝑢_𝑛,𝑥̂𝑑𝑓(𝑥̂),𝑥̂+1(1−𝛼)𝛽^𝑛,(2.29) where 𝛽_𝑛=(1−𝛼_𝑛)𝑑²(𝑓(𝑥_𝑛),𝑥_𝑛)−(1−𝛼_𝑛)𝑑²(𝑇𝑥_𝑛,𝑥_𝑛)]. Since 𝑑(𝑢_𝑛,𝑥̂)→0 and 𝑑(𝑥_𝑛,𝑇𝑥_𝑛)→0, we obtain that lim_𝑛sup1(1−𝛼)𝑑𝑢_𝑛,𝑥̂𝑑𝑓(𝑥̂),𝑥̂+1(1−𝛼)𝛽^𝑛≤0.(2.30)
According to Lemma 1.6, we can get 𝑑²(𝑥_𝑛,𝑥̂)→0.
Finally, we prove that 𝑥̂=𝑥̃.
Indeed, for any 𝑧∈Fix(𝑇), 𝑑²𝑥_𝑛+1,𝑧≤≤𝛼𝛼_𝑛𝑛𝑑²𝑧,𝑓𝑥_𝑛+1−𝛼_𝑛𝑑²𝑧,𝑇𝑥_𝑛−𝛼_𝑛1−𝛼_𝑛𝑑²𝑓𝑥_𝑛,𝑇𝑥_𝑛𝑑²𝑧,𝑓𝑥_𝑛+1−𝛼_𝑛𝑑²𝑧,𝑥_𝑛−𝛼_𝑛1−𝛼_𝑛𝑑²𝑓𝑥_𝑛,𝑇𝑥_𝑛.(2.31)
Let 𝜇 be a Banach limit. Then, 𝜇_𝑛𝑑²𝑥_𝑛+1,𝑧≤𝜇_𝑛𝑑²𝑧,𝑓𝑥_𝑛−𝜇_𝑛𝑑²𝑓𝑥_𝑛,𝑇𝑥_𝑛.(2.32)
Since 𝑥_𝑛→𝑥̂, we obtain that 𝑑²(𝑥̂,𝑧)≤𝑑²(𝑧,𝑓(𝑥̂))−𝑑²(𝑓(𝑥̂),𝑥̂).(2.33) It follows that 𝑑²(𝑓(𝑥̂),𝑥̂)≤𝑑²(𝑧,𝑓(𝑥̂)),(2.34) that is to say, 𝑥̂=𝑃_Fix(𝑇)𝑓(𝑥̂). Since 𝑃_Fix(𝑇)𝑓 is a contraction and 𝑥̃=𝑃_Fix(𝑇)𝑓(𝑥̃), we know that 𝑥̂=𝑥̃.

Acknowledgment

This paper was supported by NSFC Grant no. 11071279.