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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 421050, 11 pages
http://dx.doi.org/10.1155/2012/421050
Research Article

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

Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China

Received 30 March 2012; Accepted 27 April 2012

Academic Editor: Yonghong Yao

Copyright © 2012 Luo Yi Shi and Ru Dong Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 𝑥𝑡𝐶 be the unique fixed point of the contraction 𝑥𝑡𝑓(𝑥)(1𝑡)𝑇𝑥. We will show that if 𝑋 is a CAT(0) space satisfying some property, then {𝑥𝑡} converge strongly to a fixed point of 𝑇 which solves some variational inequality. Consider also the iteration process {𝑥𝑛}, where 𝑥0𝐶 is arbitrary and 𝑥𝑛+1=𝛼𝑛𝑓(𝑥𝑛)(1𝛼𝑛)𝑇𝑥𝑛 for 𝑛1, where {𝛼𝑛}(0,1). It is shown that under certain appropriate conditions on 𝛼𝑛,{𝑥𝑛} 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, 717].

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 𝑥𝑡𝐶 be the unique fixed point of the contraction 𝑥𝑡𝑓(𝑥)(1𝑡)𝑇𝑥. Consider the iteration process {𝑥𝑛}, where 𝑥0𝐶 is arbitrary and𝑥𝑛+1=𝛼𝑛𝑓𝑥𝑛1𝛼𝑛𝑇𝑥𝑛,(1.1) for 𝑛1, where {𝛼𝑛}(0,1). We show that {𝑥𝑛} converge strongly to a fixed point of 𝑇 under certain appropriate conditions on 𝛼𝑛, and the fixed point of 𝑇 solves some variational inequality.

Let (𝑋,𝑑) be a metric space. A geodesic path joining 𝑥𝑋 to 𝑦𝑋 (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))=|𝑡𝑡| 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,𝑦𝑋. A subset 𝑌𝑋 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 𝑑𝔼2(𝑥𝑖,𝑥𝑗)=𝑑(𝑥𝑖,𝑥𝑗) for 𝑖,𝑗1,2,3.

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)𝑑𝔼2𝑥,𝑦.(1.2)

Let 𝑥,𝑦𝑋, 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)=(1𝑡)𝑑(𝑥,𝑦).(1.3) From now on, we will use the notation (1𝑡)𝑥𝑡𝑦 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,𝑧𝑋 and 𝑡[0,1], one has𝑑((1𝑡)𝑥𝑡𝑦,𝑧)(1𝑡)𝑑(𝑥,𝑧)+𝑡𝑑(𝑦,𝑧),(1.4)(ii) (see [7]) for each 𝑥,𝑦,𝑧𝑋 and 𝑡,𝑠[0,1], one has𝑑((1𝑡)𝑥𝑡𝑦,(1𝑠)𝑥𝑠𝑦)|𝑡𝑠|𝑑(𝑥,𝑦),(1.5)(iii) (see [6, Lemma 3]) for each 𝑥,𝑦,𝑧𝑋 and 𝑡[0,1], one has𝑑((1𝑡)𝑧𝑡𝑥,(1𝑡)𝑧𝑡𝑦)𝑡𝑑(𝑥,𝑦),(1.6)(iv) (see [18, Lemma 2.5]) for each 𝑥,𝑦,𝑧𝑋 and 𝑡[0,1], one has𝑑((1𝑡)𝑥𝑡𝑦,𝑧)2(1𝑡)𝑑(𝑥,𝑧)2+𝑡𝑑(𝑦,𝑧)2𝑡(1𝑡)𝑑(𝑥,𝑦)2.(1.7)

Let 𝑋 be a complete CAT(0) space, let {𝑥𝑛} be a bounded sequence in a complete X, and for 𝑥𝑋, set𝑟𝑥𝑥,𝑛=limsup𝑛𝑑𝑥,𝑥𝑛.(1.8)

The asymptotic radius 𝑟({𝑥𝑛}) of {𝑥𝑛} is given by𝑟𝑥𝑛𝑟𝑥=inf𝑥,𝑛𝑥𝑋,(1.9)

and the asymptotic center 𝐴({𝑥𝑛}) of {𝑥𝑛} is the set𝐴𝑥𝑛=𝑥𝑥𝑋𝑟𝑥,𝑛𝑥=𝑟𝑛.(1.10)

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

A sequence {𝑥𝑛} in 𝑋 is said to -converge to 𝑥𝑋 if 𝑥 is the unique asymptotic center of {𝑢𝑛} for every subsequence {𝑢𝑛} of {𝑥𝑛}. In this case, we write -lim𝑛𝑥𝑛=𝑥 and call 𝑥 the -limit of {𝑥𝑛}.

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 𝑓𝐾𝑋 is a nonexpansive mapping, then the conditions {𝑥𝑛}-converge to 𝑥 and 𝑑(𝑥𝑛,𝑓(𝑥𝑛))0 and imply 𝑥𝐾 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 𝑃𝐶𝑋𝐶,𝑃𝐶(𝑥)=inf{𝑑(𝑥,𝑦);𝑦𝐶} is a nonexpansive mapping. one calls a CAT(0) space 𝑋 satisfying property 𝒫 if for 𝑥,𝑢,𝑦1,𝑦2𝑋, 𝑑𝑥,𝑃[𝑥,𝑦1]𝑑(𝑢)𝑥,𝑦1𝑑𝑥,𝑃[𝑥,𝑦2]𝑑(𝑢)𝑥,𝑦2𝑦+𝑑(𝑥,𝑢)𝑑1,𝑦2.(1.11)

Remark 1.4. The property 𝒫 in Hilbert space corresponds to the inequality ||𝑢𝑥,𝑦1||||𝑥𝑢𝑥,𝑦2||𝑦𝑥+𝑢𝑥1𝑦2.(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 𝜇𝑛(𝑎𝑛)=𝜇𝑛(𝑎𝑛+1) for all {𝑎𝑛}𝑙.

Lemma 1.5 (see [20, Proposition 2]). Let (𝑎1,𝑎2,)𝑙 be such that 𝜇𝑛(𝑎𝑛)0 for all Banach limits 𝜇 and limsup𝑛(𝑎𝑛+1𝑎𝑛)0. Then limsup𝑛𝑎𝑛0.

Lemma 1.6 (see [21, Lemma 2.3]). Let {𝑠𝑛} be a sequence of nonnegative real numbers, {𝛼𝑛} a sequence of real numbers in [0,1] with 𝑛=1𝛼𝑛=, {𝑢𝑛} a sequence of nonnegative real numbers with 𝑛=1𝑢𝑛<, and {𝑡𝑛} a sequence of real numbers with limsup𝑛𝑡𝑛0. Suppose that 𝑠𝑛+1=1𝛼𝑛𝑠𝑛+𝛼𝑛𝑡𝑛+𝑢𝑛,𝑛.(1.13) Then lim𝑛𝑠𝑛=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 𝑇𝐶𝐶 be a nonexpansive mapping. Let 𝑓 be a contraction on 𝐶 with coefficient 𝛼<1. For each 𝑡[0,1], the mapping 𝑆𝑡𝐶𝐶 defined by 𝑆𝑡𝑥=𝑡𝑓(𝑥)(1𝑡)𝑇𝑥,for𝑥𝐶(2.1) has a unique fixed point 𝑥𝑡𝐶, that is, 𝑥𝑡𝑥=𝑡𝑓𝑡(1𝑡)𝑇𝑥𝑡.(2.2)

Proof. For 𝑥,𝑦𝐶, according to Lemma 1.1, we have the following: 𝑑𝑆𝑡(𝑥),𝑆𝑡(𝑦)=𝑑(𝑡𝑓(𝑥)(1𝑡)𝑇𝑥,𝑡𝑓(𝑦)(1𝑡)𝑇𝑦)𝑑(𝑡𝑓(𝑥)(1𝑡)𝑇𝑥,𝑡𝑓(𝑥)(1𝑡)𝑇𝑦)+𝑑(𝑡𝑓(𝑦)(1𝑡)𝑇𝑥,𝑡𝑓(𝑦)(1𝑡)𝑇𝑦)𝑡𝑑(𝑓(𝑥),𝑓(𝑦))+(1𝑡)𝑑(𝑇𝑥,𝑇𝑦)(1𝑡(1𝛼))𝑑(𝑥,𝑦).(2.3) This implies that 𝑆𝑡 is a contraction mapping, and hence, the conclusion follows.

The following result is to prove that the net {𝑥𝑡} 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 𝑇𝐶𝐶 be a nonexpansive mapping. Let 𝑓 be a contraction on 𝐶 with coefficient 𝛼<1. For each 𝑡[0,1], let {𝑥𝑡} be given by 𝑥𝑡𝑥=𝑡𝑓𝑡(1𝑡)𝑇𝑥𝑡.(2.4) Then one has lim𝑡0𝑥𝑡=̃𝑥 and ̃𝑥=𝑃Fix(𝑇)𝑓(̃𝑥).

Proof. We first show that {𝑥𝑡} is bounded. Indeed choose a 𝑝Fix(𝑇), and using Lemma 1.1 and the nonexpansive of 𝑇, we derive that 𝑑𝑥𝑡𝑥,𝑝=𝑑𝑡𝑓𝑡(1𝑡)𝑇𝑥𝑡𝑓𝑥,𝑝𝑡𝑑𝑡,𝑝+(1𝑡)𝑑𝑇𝑥𝑡𝑓𝑥,𝑝𝑡𝑑𝑡+𝑥,𝑝(1𝑡)𝑑𝑡.,𝑝(2.5) It follows that 𝑑𝑥𝑡𝑓𝑥,𝑝𝑑𝑡𝑓𝑥,𝑝𝑑𝑡𝑥,𝑓(𝑝)+𝑑(𝑓(𝑝),𝑝)𝛼𝑑𝑡,𝑝+𝑑(𝑓(𝑝),𝑝).(2.6) Hence, 𝑑𝑥𝑡1,𝑝1𝛼𝑑(𝑓(𝑝),𝑝),(2.7) and {𝑥𝑡} is bounded, so are {𝑇𝑥𝑡} and {𝑓(𝑥𝑡)}. As a result, we can get that lim𝑡0𝑑𝑥𝑡,𝑇𝑥𝑡=lim𝑡0𝑑𝑥𝑡𝑓𝑡(1𝑡)𝑇𝑥𝑡,𝑇𝑥𝑡=lim𝑡0𝑓𝑥𝑡𝑑𝑡,𝑇𝑥𝑡=0.(2.8)
Assume that {𝑡𝑛}(0,1) is such that 𝑡𝑛0 as 𝑛. Put 𝑥𝑛=𝑥𝑡𝑛. We will show that {𝑥𝑛} contains a subsequence converging strongly to ̃𝑥, where ̃𝑥Fix(𝑇).
Since {𝑥𝑛} is bounded, by Lemma 1.2(i),(ii), we may assume that {𝑥𝑛}-converges to a point ̃𝑥, and ̃𝑥Fix(𝑇).
Next we will prove that {𝑥𝑛} converge strongly to ̃𝑥.
Indeed, according to Lemma 1.1 and the property of 𝑇 and 𝑓, we can get that 𝑑2𝑥𝑛,̃𝑥=𝑑2𝑡𝑛𝑓𝑥𝑛1𝑡𝑛𝑇𝑥𝑛,̃𝑥𝑡𝑛𝑑2𝑓𝑥𝑛+,̃𝑥1𝑡𝑛𝑑2𝑇𝑥𝑛,̃𝑥𝑡𝑛1𝑡𝑛𝑑2𝑓𝑥𝑛,𝑇𝑥𝑛𝑡𝑛𝑑2𝑓𝑥𝑛+,̃𝑥1𝑡𝑛𝑑2𝑥𝑛,̃𝑥𝑡𝑛1𝑡𝑛𝑑2𝑓𝑥𝑛,𝑇𝑥𝑛.(2.9) It follows that 𝑑2𝑥𝑛,̃𝑥𝑑2𝑓𝑥𝑛,̃𝑥1𝑡𝑛𝑑2𝑓𝑥𝑛,𝑇𝑥𝑛=𝑑2𝑓𝑥𝑛,̃𝑥𝑑2𝑓𝑥𝑛,𝑇𝑥𝑛+𝑡𝑛𝑑2𝑓𝑥𝑛,𝑇𝑥𝑛.(2.10)
Since lim𝑡0𝑑(𝑥𝑡,𝑇𝑥𝑡)=0, we can get that limsup𝑛𝑑2𝑥𝑛,̃𝑥limsup𝑛𝑑2𝑓𝑥𝑛,̃𝑥𝑑2𝑓𝑥𝑛,𝑥𝑛.(2.11) Let Δ(̃𝑥,𝑥𝑛,𝑓(𝑥𝑛)) be a comparison triangle for (̃𝑥,𝑥𝑛,𝑓(𝑥𝑛)) in 𝔼2. Then, 𝑑2𝑓𝑥𝑛,̃𝑥𝑑2𝑓𝑥𝑛,𝑥𝑛=𝑑2𝑓𝑥𝑛,̃𝑥𝑑2𝑓𝑥𝑛,𝑥𝑛=𝑓𝑥𝑛̃𝑥,𝑓𝑥𝑛̃𝑥𝑓𝑥𝑛̃𝑥𝑛,𝑓𝑥𝑛̃𝑥𝑛=2𝑓𝑥𝑛̃𝑥,𝑥𝑛̃𝑥𝑥𝑛̃𝑥,𝑥𝑛̃𝑥=2𝑓𝑥𝑛̃𝑥,𝑥𝑛̃𝑥𝑑2𝑥𝑛,̃𝑥=2𝑓𝑥𝑛̃𝑥,𝑥𝑛̃𝑥𝑑2𝑥𝑛.,̃𝑥(2.12) Hence, limsup𝑛𝑑2𝑥𝑛,̃𝑥limsup𝑛𝑓𝑥𝑛̃𝑥,𝑥𝑛.̃𝑥(2.13)
Let (̃𝑥,𝑥𝑛,𝑓(̃𝑥)) be a comparison triangle for (̃𝑥,𝑥𝑛,𝑓(̃𝑥)) in 𝔼2. For each 𝑛, let 𝑢𝑛 be the point of the segment [̃𝑥,𝑓(̃𝑥)] which is nearest to 𝑥𝑛, and let 𝑢𝑛 be the point of the segment [̃𝑥,𝑓(̃𝑥)] for which 𝑑(𝑢𝑛,̃𝑥)=𝑑(𝑢𝑛,̃𝑥).
By passing to subsequences again, we may suppose that {𝑢𝑛} converges to 𝑢[̃𝑥,𝑓(̃𝑥)], {𝑢𝑛} converges to 𝑢[̃𝑥,𝑓(̃𝑥)].
Since {𝑥𝑛}-converges to a point ̃𝑥, we have 𝑟𝑥𝑛=lim𝑛sup𝑑𝑥,𝑥𝑛=lim𝑛sup𝑑𝑥,𝑥𝑛lim𝑛sup𝑑𝑢𝑛,𝑥𝑛=lim𝑛sup𝑑𝑢,𝑥𝑛lim𝑛sup𝑑𝑢,𝑥𝑛.(2.14) Thus, 𝑟(𝑢,{𝑥𝑛})𝑟({𝑥𝑛}). This implies that 𝑢=𝑥 by uniqueness of the asymptotic center. Hence, 𝑢=𝑥. That is to say, {𝑢𝑛} converges to ̃𝑥, and {𝑢𝑛} converges to ̃𝑥.
Moreover, since 𝑋 satisfies the property 𝒫, we can get that |||𝑓𝑥𝑛̃𝑥,𝑥𝑛|||̃𝑥=𝑑̃𝑥,𝑃[̃𝑥,𝑓(𝑥𝑛)]𝑥𝑛𝑑̃𝑥,𝑓𝑥𝑛=𝑑̃𝑥,𝑃[̃𝑥,𝑓(𝑥𝑛)]𝑥𝑛𝑥𝑑̃𝑥,𝑓𝑛𝑑̃𝑥,𝑃[̃𝑥,𝑓(̃𝑥)]𝑥𝑛𝑑(̃𝑥,𝑓(̃𝑥))+𝑑̃𝑥,𝑥𝑛𝑓𝑥𝑑𝑛,𝑓(̃𝑥)=𝑑̃𝑥,𝑃[̃𝑥,𝑓(̃𝑥)]𝑥𝑛𝑑̃𝑥,𝑓(̃𝑥)+𝑑̃𝑥,𝑥𝑛𝑓𝑥𝑑𝑛,𝑓(̃𝑥)𝑑𝑢𝑛,𝑑̃𝑥̃𝑥,𝑓(̃𝑥)+𝛼𝑑2𝑥𝑛.,̃𝑥(2.15) It follows that limsup𝑛𝑑2𝑥𝑛1,̃𝑥1𝛼limsup𝑛𝑑𝑢𝑛,𝑑̃𝑥𝑓(̃𝑥),.̃𝑥(2.16) Since {𝑢𝑛} converges to ̃𝑥, we obtain that limsup𝑛𝑑2(𝑥𝑛,̃𝑥)=0, that is, {𝑥𝑛} converge strongly to ̃𝑥. Since {𝑡𝑛}(0,1) is such that 𝑡𝑛0 as 𝑛 is arbitrarily selected, we can get that lim𝑡0𝑥𝑡=̃𝑥.
Finally, we will prove that ̃𝑥 satisfy the equation ̃𝑥=𝑃Fix(𝑇)𝑓(̃𝑥).
Indeed, for any 𝑦Fix(𝑇), 𝑑𝑥𝑡𝑥,𝑦=𝑑𝑡𝑓𝑡(1𝑡)𝑇𝑥𝑡𝑓𝑥,𝑦=𝑡𝑑𝑡,𝑦+(1𝑡)𝑑𝑇𝑥𝑡𝑓𝑥,𝑦𝑡𝑑𝑡+𝑥,𝑦(1𝑡)𝑑𝑡.,𝑦(2.17) It follows that 𝑑𝑥𝑡𝑓𝑥,𝑦𝑑𝑡,𝑦.(2.18) Since lim𝑡0𝑥𝑡=̃𝑥, we can get that 𝑑(̃𝑥,𝑦)𝑑(𝑓(̃𝑥),𝑦).(2.19)
Hence, ||||,𝑑(𝑓(̃𝑥),𝑦)𝑑(̃𝑥,𝑦)𝑑(̃𝑥,𝑓(̃𝑥))𝑑(̃𝑥,𝑓(̃𝑥))𝑑(𝑓(̃𝑥),𝑦).(2.20) That is to say, ̃𝑥=𝑃Fix(𝑇)𝑓(̃𝑥).

Consider now the iteration process𝑥0𝑥𝐶,𝑛+1=𝛼𝑛𝑓𝑥𝑛1𝛼𝑛𝑇𝑥𝑛,𝑛0,(2.21) where {𝛼𝑛}(0,1) satisfies(H1)𝛼𝑛0,(H2)𝑛=0𝛼𝑛=,(H3)either 𝑛=0|𝛼𝑛+1𝛼𝑛|< or lim𝑛(𝛼𝑛+1/𝛼𝑛)=1.

Theorem 2.3. Let 𝑋 be a CAT(0) space satisfying the property 𝒫, 𝐶 a closed convex subset of 𝑋, 𝑇𝐶𝐶 a nonexpansive mapping with Fix(𝑇), and 𝑓𝐶𝐶 a contraction with coefficient 𝛼<1. Let 𝑥0𝐶, {𝑥𝑛} be generated by 𝑥𝑛+1=𝛼𝑛𝑓(𝑥𝑛)(1𝛼𝑛)𝑇𝑥𝑛,𝑛0. Then under the hypotheses (H1 )–(H3 ), 𝑥𝑛̃𝑥, where ̃𝑥=𝑃Fix(𝑇)𝑓(̃𝑥).

Proof. We first show that the sequence {𝑥𝑛} is bounded. Let 𝑝Fix(𝑇). Then, 𝑑𝑥𝑛+1𝛼,𝑝=𝑑𝑛𝑓𝑥𝑛1𝛼𝑛𝑇𝑥𝑛,𝑝𝛼𝑛𝑑𝑓𝑥𝑛+,𝑝1𝛼𝑛𝑑𝑇𝑥𝑛,𝑝𝛼𝑛𝑑𝑓𝑥𝑛+,𝑓(𝑝)+𝑑(𝑓(𝑝),𝑝)1𝛼𝑛𝑑𝑥𝑛𝑑𝑥,𝑝max𝑛,1,𝑝.1𝛼𝑑(𝑓(𝑝),𝑝)(2.22) By induction, we have 𝑑𝑥𝑛𝑑𝑥,𝑝max0,1,𝑝1𝛼𝑑(𝑓(𝑝),𝑝),(2.23) for all 𝑛. This implies that {𝑥𝑛} is bounded and so is the sequence {𝑇𝑥𝑛} and {𝑓(𝑥𝑛)}.
We claim that 𝑑(𝑥𝑛+1,𝑥𝑛)0. Indeed, we have 𝑑𝑥𝑛+1,𝑥𝑛𝛼=𝑑𝑛𝑓𝑥𝑛1𝛼𝑛𝑇𝑥𝑛,𝛼𝑛1𝑓𝑥𝑛11𝛼𝑛1𝑇𝑥𝑛1𝛼𝑑𝑛𝑓𝑥𝑛1𝛼𝑛𝑇𝑥𝑛,𝛼𝑛𝑓𝑥𝑛1𝛼𝑛𝑇𝑥𝑛1𝛼+𝑑𝑛𝑓𝑥𝑛1𝛼𝑛𝑇𝑥𝑛1,𝛼𝑛𝑓𝑥𝑛11𝛼𝑛𝑇𝑥𝑛1𝛼+𝑑𝑛𝑓𝑥𝑛11𝛼𝑛𝑇𝑥𝑛1,𝛼𝑛1𝑓𝑥𝑛11𝛼𝑛1𝑇𝑥𝑛11𝛼𝑛𝑑𝑇𝑥𝑛,𝑇𝑥𝑛1+𝛼𝑛𝑑𝑓𝑥𝑛𝑥,𝑓𝑛1+||𝛼𝑛𝛼𝑛1||𝑑𝑓𝑥𝑛1,𝑇𝑥𝑛11𝛼𝑛𝑑𝑥𝑛,𝑥𝑛1+𝛼𝑛𝑥𝛼𝑑𝑛,𝑥𝑛1+||𝛼𝑛𝛼𝑛1||𝑑𝑓𝑥𝑛1,𝑇𝑥𝑛1.(2.24) By the conditions H2 and H3, we have 𝑑𝑥𝑛+1,𝑥𝑛0.(2.25) Consequently, by the condition H1, 𝑑𝑥𝑛,𝑇𝑥𝑛𝑥𝑑𝑛,𝑥𝑛+1𝑥+𝑑𝑛+1,𝑇𝑥𝑛𝑥=𝑑𝑛,𝑥𝑛+1𝛼+𝑑𝑛𝑓𝑥𝑛1𝛼𝑛𝑇𝑥𝑛,𝑇𝑥𝑛𝑥=𝑑𝑛,𝑥𝑛+1+𝛼𝑛𝑑𝑓𝑥𝑛,𝑇𝑥𝑛0.(2.26)
Since {𝑥𝑛} is bounded, we may assume that {𝑥𝑛}-converges to a point ̂𝑥. By Lemma 1.2, we have ̂𝑥Fix(𝑇).
Next we will prove that {𝑥𝑛} converge strongly to ̂𝑥 and ̂𝑥=̃𝑥. Indeed, according to Lemma 1.1 and the property of 𝑇 and 𝑓, we can get that 𝑑2𝑥𝑛+1,̂𝑥=𝑑2𝑡𝑛𝑓𝑥𝑛1𝑡𝑛𝑇𝑥𝑛,̂𝑥𝛼𝑛𝑑2𝑓𝑥𝑛+,̂𝑥1𝛼𝑛𝑑2𝑇𝑥𝑛,̂𝑥𝛼𝑛1𝛼𝑛𝑑2𝑓𝑥𝑛,𝑇𝑥𝑛1𝛼𝑛𝑑2𝑥𝑛,̂𝑥+𝛼𝑛𝑑2𝑓𝑥𝑛,̂𝑥1𝛼𝑛𝑑2𝑓𝑥𝑛,𝑇𝑛.(2.27) With a minor modification of the proof of the analogous statement in Theorem 2.2, we can get that 𝑑2𝑓𝑥𝑛,̂𝑥𝑑2𝑓𝑥𝑛,𝑥𝑛=2𝑓𝑥𝑛̂𝑥,𝑥𝑛̂𝑥𝑑2𝑥𝑛,̂𝑥2𝑑𝑢𝑛,𝑑̂𝑥𝑓(̂𝑥),̂𝑥+2𝛼𝑑2𝑥𝑛,̂𝑥𝑑2𝑥𝑛,,̂𝑥(2.28) and 𝑑(𝑢𝑛,̂𝑥)0.
Thus, 𝑑2𝑥𝑛+1,̂𝑥1𝛼𝑛𝑑2𝑥𝑛,̂𝑥+𝛼𝑛𝑑2𝑓𝑥𝑛,̂𝑥1𝛼𝑛𝑑2𝑓𝑥𝑛,𝑥𝑛+1𝛼𝑛𝑑2𝑓𝑥𝑛,𝑥𝑛1𝛼𝑛𝑑2𝑇𝑥𝑛,𝑥𝑛12(1𝛼)𝛼𝑛𝑑2𝑥𝑛,̂𝑥+2(1𝛼)𝛼𝑛1𝑑(1𝛼)𝑢𝑛,𝑑̂𝑥𝑓(̂𝑥),+1̂𝑥𝛽(1𝛼)𝑛,(2.29) where 𝛽𝑛=(1𝛼𝑛)𝑑2(𝑓(𝑥𝑛),𝑥𝑛)(1𝛼𝑛)𝑑2(𝑇𝑥𝑛,𝑥𝑛)]. Since 𝑑(𝑢𝑛,̂𝑥)0 and 𝑑(𝑥𝑛,𝑇𝑥𝑛)0, we obtain that lim𝑛1sup𝑑(1𝛼)𝑢𝑛,𝑑̂𝑥𝑓(̂𝑥),+1̂𝑥𝛽(1𝛼)𝑛0.(2.30)
According to Lemma 1.6, we can get 𝑑2(𝑥𝑛,̂𝑥)0.
Finally, we prove that ̂𝑥=̃𝑥.
Indeed, for any 𝑧Fix(𝑇), 𝑑2𝑥𝑛+1,𝑧𝛼𝑛𝑑2𝑥𝑧,𝑓𝑛+1𝛼𝑛𝑑2𝑧,𝑇𝑥𝑛𝛼𝑛1𝛼𝑛𝑑2𝑓𝑥𝑛,𝑇𝑥𝑛𝛼𝑛𝑑2𝑥𝑧,𝑓𝑛+1𝛼𝑛𝑑2𝑧,𝑥𝑛𝛼𝑛1𝛼𝑛𝑑2𝑓𝑥𝑛,𝑇𝑥𝑛.(2.31)
Let 𝜇 be a Banach limit. Then, 𝜇𝑛𝑑2𝑥𝑛+1,𝑧𝜇𝑛𝑑2𝑥𝑧,𝑓𝑛𝜇𝑛𝑑2𝑓𝑥𝑛,𝑇𝑥𝑛.(2.32)
Since 𝑥𝑛̂𝑥, we obtain that 𝑑2(̂𝑥,𝑧)𝑑2(𝑧,𝑓(̂𝑥))𝑑2(𝑓(̂𝑥),̂𝑥).(2.33) It follows that 𝑑2(𝑓(̂𝑥),̂𝑥)𝑑2(𝑧,𝑓(̂𝑥)),(2.34) that is to say, ̂𝑥=𝑃Fix(𝑇)𝑓(̂𝑥). Since 𝑃Fix(𝑇)𝑓 is a contraction and ̃𝑥=𝑃Fix(𝑇)𝑓(̃𝑥), we know that ̂𝑥=̃𝑥.

Acknowledgment

This paper was supported by NSFC Grant no. 11071279.

References

  1. M. R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Grundlehren der Mathematischen Wissenschaften, Springer, Berlin, Germany, 1999.
  2. W. A. Kirk, “Fixed point theorems in CAT(0) spaces and -trees,” Fixed Point Theory and Applications, no. 4, pp. 309–316, 2004. View at Publisher · View at Google Scholar
  3. K. S. Brown, Buildings, Springer, New York, NY, USA, 1989.
  4. K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, vol. 83 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1984.
  5. W. A. Kirk, “Geodesic geometry and fixed point theory,” in Seminar of Mathematical Analysis (Malaga/Seville, 2002/2003), vol. 64 of Colecc. Abierta, pp. 195–225, University of Seville, Secretary Publication, Seville, Spain, 2003. View at Zentralblatt MATH
  6. W. A. Kirk, “Geodesic geometry and fixed point theory. II,” in International Conference on Fixed Point Theory and Applications, pp. 113–142, Yokohama, Japan, Yokohama, 2004. View at Zentralblatt MATH
  7. P. Chaoha and A. Phon-on, “A note on fixed point sets in CAT(0) spaces,” Journal of Mathematical Analysis and Applications, vol. 320, no. 2, pp. 983–987, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. S. Dhompongsa, W. Fupinwong, and A. Kaewkhao, “Common fixed points of a nonexpansive semigroup and a convergence theorem for Mann iterations in geodesic metric spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 12, pp. 4268–4273, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. S. Dhompongsa, A. Kaewkhao, and B. Panyanak, “Lim's theorems for multivalued mappings in CAT(0) spaces,” Journal of Mathematical Analysis and Applications, vol. 312, no. 2, pp. 478–487, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. S. Dhompongsa, W. A. Kirk, and B. Panyanak, “Nonexpansive set-valued mappings in metric and Banach spaces,” Journal of Nonlinear and Convex Analysis, vol. 8, no. 1, pp. 35–45, 2007. View at Zentralblatt MATH
  11. S. Dhompongsa, W. A. Kirk, and B. Sims, “Fixed points of uniformly Lipschitzian mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 65, no. 4, pp. 762–772, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. S. Dhompongsa and B. Panyanak, “On Δ-convergence theorems in CAT(0) spaces,” Computers & Mathematics with Applications, vol. 56, no. 10, pp. 2572–2579, 2008. View at Publisher · View at Google Scholar
  13. K. Fujiwara, K. Nagano, and T. Shioya, “Fixed point sets of parabolic isometries of CAT(0)-spaces,” Commentarii Mathematici Helvetici, vol. 81, no. 2, pp. 305–335, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. W. A. Kirk and B. Panyanak, “A concept of convergence in geodesic spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 12, pp. 3689–3696, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. L. Leustean, “A quadratic rate of asymptotic regularity for CAT(0)-spaces,” Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 386–399, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. N. Shahzad, “Invariant approximations in CAT(0) spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 12, pp. 4338–4340, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. N. Shahzad and J. Markin, “Invariant approximations for commuting mappings in CAT(0) and hyperconvex spaces,” Journal of Mathematical Analysis and Applications, vol. 337, no. 2, pp. 1457–1464, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. S. Dhompongsa and B. Panyanak, “On δ-convergence theorems in CAT(0) spaces,” Computers & Mathematics with Applications, vol. 56, no. 10, pp. 2572–2579, 2008. View at Publisher · View at Google Scholar
  19. R. Espínola and A. Fernández-León, “CAT(K)-spaces, weak convergence and fixed points,” Journal of Mathematical Analysis and Applications, vol. 353, no. 1, pp. 410–427, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. N. Shioji and W. Takahashi, “Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces,” Proceedings of the American Mathematical Society, vol. 125, no. 12, pp. 3641–3645, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. K. Aoyama, Y. Kimura, W. Takahashi, and M. Toyoda, “Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 8, pp. 2350–2360, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH