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Journal of Applied Mathematics
VolumeΒ 2012Β (2012), Article IDΒ 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: YonghongΒ Yao

Copyright Β© 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 𝐢 of a CAT(0) space 𝑋. Suppose that the set Fix(𝑇) of fixed points of 𝑇 is nonempty. For a contraction 𝑓 on 𝐢 and π‘‘βˆˆ(0,1), let π‘₯π‘‘βˆˆπΆ be the unique fixed point of the contraction π‘₯↦𝑑𝑓(π‘₯)βŠ•(1βˆ’π‘‘)𝑇π‘₯. We will show that if 𝑋 is a CAT(0) space satisfying some property, then {π‘₯𝑑} converge strongly to a fixed point of 𝑇 which solves some variational inequality. Consider also the iteration process {π‘₯𝑛}, where π‘₯0∈𝐢 is arbitrary and π‘₯𝑛+1=𝛼𝑛𝑓(π‘₯𝑛)βŠ•(1βˆ’π›Όπ‘›)𝑇π‘₯𝑛 for 𝑛β‰₯1, where {𝛼𝑛}βŠ‚(0,1). It is shown that under certain appropriate conditions on 𝛼𝑛,{π‘₯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 𝐢 of a CAT(0) space 𝑋. Suppose that the set Fix(𝑇) of fixed points of 𝑇 is nonempty. For a contraction 𝑓 on 𝐢 and π‘‘βˆˆ(0,1), let π‘₯π‘‘βˆˆπΆ be the unique fixed point of the contraction π‘₯↦𝑑𝑓(π‘₯)βŠ•(1βˆ’π‘‘)𝑇π‘₯. Consider the iteration process {π‘₯𝑛}, where π‘₯0∈𝐢 is arbitrary andπ‘₯𝑛+1=𝛼𝑛𝑓π‘₯π‘›ξ€ΈβŠ•ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘‡π‘₯𝑛,(1.1) for 𝑛β‰₯1, where {𝛼𝑛}βŠ‚(0,1). We show that {π‘₯𝑛} 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 π‘₯βˆˆπ‘‹ to π‘¦βˆˆπ‘‹ (or, more briefly, a geodesic from π‘₯ to 𝑦) is a map 𝑐 from a closed interval [0,𝑙]βŠ‚π‘… to 𝑋 such that 𝑐(0)=π‘₯,𝑐(𝑙)=𝑦, and 𝑑(𝑐(𝑑),𝑐(π‘‘ξ…ž))=|π‘‘βˆ’π‘‘ξ…ž| for all 𝑑,π‘‘βˆˆ[0,𝑙]. In particular, 𝑐 is an isometry and 𝑑(π‘₯,𝑦)=𝑙. The image 𝛼 of 𝑐 is called a geodesic (or metric) segment joining π‘₯ and 𝑦. When it is unique, this geodesic segment is denoted by [π‘₯,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 π‘₯ and 𝑦 for each π‘₯,π‘¦βˆˆπ‘‹. A subset π‘ŒβŠ†π‘‹ is said to be convex if π‘Œ includes every geodesic segment joining any two of its points.

A geodesic triangle β–΅(π‘₯1,π‘₯2,π‘₯3) in a geodesicmetric space (𝑋,𝑑) consists of three points π‘₯1,π‘₯2, and π‘₯3 in 𝑋 (the vertices of β–΅) and a geodesic segment between each pair of vertices (the edges of β–΅). A comparison triangle for the geodesic triangle β–΅(π‘₯1,π‘₯2,π‘₯3) in (𝑋,𝑑) is a triangle β–΅(π‘₯1,π‘₯2,π‘₯3)∢=β–΅(π‘₯1,π‘₯2,π‘₯3) in the Euclidean plane 𝔼2 such that 𝑑𝔼2(π‘₯𝑖,π‘₯𝑗)=𝑑(π‘₯𝑖,π‘₯𝑗) for 𝑖,π‘—βˆˆ1,2,3.

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

CAT(0): let β–΅ be a geodesic triangle in 𝑋, and let β–΅ be a comparison triangle for β–΅. Then, β–΅ is said to satisfy the CAT(0) inequality if for all π‘₯,π‘¦βˆˆβ–΅ and all comparison points π‘₯,π‘¦βˆˆβ–΅,𝑑(π‘₯,𝑦)≀𝑑𝔼2ξ€·π‘₯,𝑦.(1.2)

Let π‘₯,π‘¦βˆˆπ‘‹, and by Lemma 2.1(iv) of [18] for each π‘‘βˆˆ[0,1], there exists a unique point π‘§βˆˆ[π‘₯,𝑦] such that𝑑(π‘₯,𝑧)=𝑑𝑑(π‘₯,𝑦),𝑑(𝑦,𝑧)=(1βˆ’π‘‘)𝑑(π‘₯,𝑦).(1.3) From now on, we will use the notation (1βˆ’π‘‘)π‘₯βŠ•π‘‘π‘¦ 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 π‘₯,𝑦,π‘§βˆˆπ‘‹ and π‘‘βˆˆ[0,1], one has𝑑((1βˆ’π‘‘)π‘₯βŠ•π‘‘π‘¦,𝑧)≀(1βˆ’π‘‘)𝑑(π‘₯,𝑧)+𝑑𝑑(𝑦,𝑧),(1.4)(ii) (see [7]) for each π‘₯,𝑦,π‘§βˆˆπ‘‹ and 𝑑,π‘ βˆˆ[0,1], one has𝑑((1βˆ’π‘‘)π‘₯βŠ•π‘‘π‘¦,(1βˆ’π‘ )π‘₯βŠ•π‘ π‘¦)≀|π‘‘βˆ’π‘ |𝑑(π‘₯,𝑦),(1.5)(iii) (see [6, Lemma 3]) for each π‘₯,𝑦,π‘§βˆˆπ‘‹ and π‘‘βˆˆ[0,1], one has𝑑((1βˆ’π‘‘)π‘§βŠ•π‘‘π‘₯,(1βˆ’π‘‘)π‘§βŠ•π‘‘π‘¦)≀𝑑𝑑(π‘₯,𝑦),(1.6)(iv) (see [18, Lemma 2.5]) for each π‘₯,𝑦,π‘§βˆˆπ‘‹ and π‘‘βˆˆ[0,1], one has𝑑((1βˆ’π‘‘)π‘₯βŠ•π‘‘π‘¦,𝑧)2≀(1βˆ’π‘‘)𝑑(π‘₯,𝑧)2+𝑑𝑑(𝑦,𝑧)2βˆ’π‘‘(1βˆ’π‘‘)𝑑(π‘₯,𝑦)2.(1.7)

Let 𝑋 be a complete CAT(0) space, let {π‘₯𝑛} be a bounded sequence in a complete X, and for π‘₯βˆˆπ‘‹, setπ‘Ÿξ€·ξ€½π‘₯π‘₯,𝑛=limsupπ‘›β†’βˆžπ‘‘ξ€·π‘₯,π‘₯𝑛.(1.8)

The asymptotic radius π‘Ÿ({π‘₯𝑛}) of {π‘₯𝑛} is given byπ‘Ÿπ‘₯ξ€·ξ€½π‘›ξ€½π‘Ÿξ€·ξ€½π‘₯ξ€Ύξ€Έ=infπ‘₯,π‘›ξ€Ύξ€Ύξ€ΈβˆΆπ‘₯βˆˆπ‘‹,(1.9)

and the asymptotic center 𝐴({π‘₯𝑛}) of {π‘₯𝑛} is the set𝐴π‘₯𝑛=ξ€½ξ€·ξ€½π‘₯ξ€Ύξ€Έπ‘₯βˆˆπ‘‹βˆΆπ‘Ÿπ‘₯,𝑛π‘₯ξ€Ύξ€Έ=π‘Ÿξ€·ξ€½π‘›ξ€Ύξ€Έξ€Ύ.(1.10)

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

A sequence {π‘₯𝑛} in 𝑋 is said to β–΅-converge to π‘₯βˆˆπ‘‹ if π‘₯ is the unique asymptotic center of {𝑒𝑛} for every subsequence {𝑒𝑛} of {π‘₯𝑛}. In this case, we write β–΅-lim𝑛π‘₯𝑛=π‘₯ and call π‘₯ the β–΅-limit of {π‘₯𝑛}.

Lemma 1.2. Assume that 𝑋 is a CAT(0) space. Then,(i) (see [14]) every bounded sequence in 𝑋 has a β–΅-convergent subsequence,(ii) (see [14, Proposition 3.7]) if 𝐾 is a closed convex subset of 𝑋, and π‘“βˆΆπΎβ†’π‘‹ is a nonexpansive mapping, then the conditions {π‘₯𝑛}β–΅-converge to π‘₯ and 𝑑(π‘₯𝑛,𝑓(π‘₯𝑛))β†’0 and imply π‘₯∈𝐾 and 𝑓(π‘₯)=π‘₯.

Lemma 1.3 (see [19, Proposition 3.5]). Assume that 𝑋 is a CAT(0) space, 𝐢 is a closed convex subset of 𝑋. Then the metric (nearest point) projection π‘ƒπΆβˆΆπ‘‹β†’πΆ,𝑃𝐢(π‘₯)∢=inf{𝑑(π‘₯,𝑦);π‘¦βˆˆπΆ} is a nonexpansive mapping. one calls a CAT(0) space 𝑋 satisfying property 𝒫 if for π‘₯,𝑒,𝑦1,𝑦2βˆˆπ‘‹, 𝑑π‘₯,𝑃[π‘₯,𝑦1]𝑑(𝑒)π‘₯,𝑦1≀𝑑π‘₯,𝑃[π‘₯,𝑦2]𝑑(𝑒)π‘₯,𝑦2𝑦+𝑑(π‘₯,𝑒)𝑑1,𝑦2ξ€Έ.(1.11)

Remark 1.4. The property 𝒫 in Hilbert space corresponds to the inequality ||βŸ¨π‘’βˆ’π‘₯,𝑦1||≀||βˆ’π‘₯βŸ©βŸ¨π‘’βˆ’π‘₯,𝑦2||β€–β€–π‘¦βˆ’π‘₯⟩+β€–π‘’βˆ’π‘₯β€–β‹…1βˆ’π‘¦2β€–β€–.(1.12)
Recall that a continuous linear functional πœ‡ on π‘™βˆž, the Banach space of bounded real sequences, is called a Banach limit if β€–πœ‡β€–=πœ‡(1,1,…)=1 and πœ‡π‘›(π‘Žπ‘›)=πœ‡π‘›(π‘Žπ‘›+1) for all {π‘Žπ‘›}βˆˆπ‘™βˆž.

Lemma 1.5 (see [20, Proposition 2]). Let (π‘Ž1,π‘Ž2,…)βˆˆπ‘™βˆž be such that πœ‡π‘›(π‘Žπ‘›)≀0 for all Banach limits πœ‡ and limsup𝑛(π‘Žπ‘›+1βˆ’π‘Žπ‘›)≀0. Then limsupπ‘›π‘Žπ‘›β‰€0.

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 βˆ‘βˆžπ‘›=1𝛼𝑛=∞, {𝑒𝑛} a sequence of nonnegative real numbers with βˆ‘βˆžπ‘›=1𝑒𝑛<∞, and {𝑑𝑛} a sequence of real numbers with limsup𝑛𝑑𝑛≀0. Suppose that 𝑠𝑛+1=ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘ π‘›+𝛼𝑛𝑑𝑛+𝑒𝑛,βˆ€π‘›βˆˆβ„•.(1.13) Then limπ‘›β†’βˆžπ‘ π‘›=0.

2. Viscosity Iteration for a Single Mapping

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

Lemma 2.1. Let 𝐢 be a closed convex subset of a complete CAT(0) space 𝑋, and let π‘‡βˆΆπΆβ†’πΆ be a nonexpansive mapping. Let 𝑓 be a contraction on 𝐢 with coefficient 𝛼<1. For each π‘‘βˆˆ[0,1], the mapping π‘†π‘‘βˆΆπΆβ†’πΆ defined by 𝑆𝑑π‘₯=𝑑𝑓(π‘₯)βŠ•(1βˆ’π‘‘)𝑇π‘₯,forπ‘₯∈𝐢(2.1) has a unique fixed point π‘₯π‘‘βˆˆπΆ, that is, π‘₯𝑑π‘₯=π‘‘π‘“π‘‘ξ€ΈβŠ•(1βˆ’π‘‘)𝑇π‘₯𝑑.(2.2)

Proof. For π‘₯,π‘¦βˆˆπΆ, according to Lemma 1.1, we have the following: 𝑑𝑆𝑑(π‘₯),𝑆𝑑(𝑦)=𝑑(𝑑𝑓(π‘₯)βŠ•(1βˆ’π‘‘)𝑇π‘₯,𝑑𝑓(𝑦)βŠ•(1βˆ’π‘‘)𝑇𝑦)≀𝑑(𝑑𝑓(π‘₯)βŠ•(1βˆ’π‘‘)𝑇π‘₯,𝑑𝑓(π‘₯)βŠ•(1βˆ’π‘‘)𝑇𝑦)+𝑑(𝑑𝑓(𝑦)βŠ•(1βˆ’π‘‘)𝑇π‘₯,𝑑𝑓(𝑦)βŠ•(1βˆ’π‘‘)𝑇𝑦)≀𝑑𝑑(𝑓(π‘₯),𝑓(𝑦))+(1βˆ’π‘‘)𝑑(𝑇π‘₯,𝑇𝑦)≀(1βˆ’π‘‘(1βˆ’π›Ό))𝑑(π‘₯,𝑦).(2.3) This implies that 𝑆𝑑 is a contraction mapping, and hence, the conclusion follows.

The following result is to prove that the net {π‘₯𝑑} converge strongly to a fixed point of 𝑇.

Theorem 2.2. Let 𝐢 be a closed convex subset of a complete CAT(0) space 𝑋 satisfying the property 𝒫, and let π‘‡βˆΆπΆβ†’πΆ be a nonexpansive mapping. Let 𝑓 be a contraction on 𝐢 with coefficient 𝛼<1. For each π‘‘βˆˆ[0,1], let {π‘₯𝑑} be given by π‘₯𝑑π‘₯=π‘‘π‘“π‘‘ξ€ΈβŠ•(1βˆ’π‘‘)𝑇π‘₯𝑑.(2.4) Then one has lim𝑑→0π‘₯𝑑=βˆΆΜƒπ‘₯ and Μƒπ‘₯=𝑃Fix(𝑇)𝑓(Μƒπ‘₯).

Proof. We first show that {π‘₯𝑑} is bounded. Indeed choose a π‘βˆˆFix(𝑇), and using Lemma 1.1 and the nonexpansive of 𝑇, we derive that 𝑑π‘₯𝑑π‘₯,𝑝=π‘‘π‘‘π‘“π‘‘ξ€ΈβŠ•(1βˆ’π‘‘)𝑇π‘₯𝑑𝑓π‘₯,𝑝≀𝑑𝑑𝑑,𝑝+(1βˆ’π‘‘)𝑑𝑇π‘₯𝑑𝑓π‘₯,𝑝≀𝑑𝑑𝑑+ξ€·π‘₯,𝑝(1βˆ’π‘‘)𝑑𝑑.,𝑝(2.5) It follows that 𝑑π‘₯𝑑𝑓π‘₯,𝑝≀𝑑𝑑𝑓π‘₯,𝑝≀𝑑𝑑π‘₯,𝑓(𝑝)+𝑑(𝑓(𝑝),𝑝)≀𝛼𝑑𝑑,𝑝+𝑑(𝑓(𝑝),𝑝).(2.6) Hence, 𝑑π‘₯𝑑≀1,𝑝1βˆ’π›Όπ‘‘(𝑓(𝑝),𝑝),(2.7) and {π‘₯𝑑} is bounded, so are {𝑇π‘₯𝑑} and {𝑓(π‘₯𝑑)}. As a result, we can get that lim𝑑→0𝑑π‘₯𝑑,𝑇π‘₯𝑑=lim𝑑→0𝑑π‘₯π‘‘π‘“π‘‘ξ€ΈβŠ•(1βˆ’π‘‘)𝑇π‘₯𝑑,𝑇π‘₯𝑑=lim𝑑→0𝑓π‘₯𝑑𝑑𝑑,𝑇π‘₯𝑑=0.(2.8)
Assume that {𝑑𝑛}βŠ†(0,1) is such that 𝑑𝑛→0 as π‘›β†’βˆž. Put π‘₯π‘›βˆΆ=π‘₯𝑑𝑛. We will show that {π‘₯𝑛} contains a subsequence converging strongly to Μƒπ‘₯, where Μƒπ‘₯∈Fix(𝑇).
Since {π‘₯𝑛} is bounded, by Lemma 1.2(i),(ii), we may assume that {π‘₯𝑛}β–΅-converges to a point Μƒπ‘₯, and Μƒπ‘₯∈Fix(𝑇).
Next we will prove that {π‘₯𝑛} converge strongly to Μƒπ‘₯.
Indeed, according to Lemma 1.1 and the property of 𝑇 and 𝑓, we can get that 𝑑2ξ€·π‘₯𝑛,Μƒπ‘₯=𝑑2𝑑𝑛𝑓π‘₯π‘›ξ€ΈβŠ•ξ€·1βˆ’π‘‘π‘›ξ€Έπ‘‡π‘₯𝑛,Μƒπ‘₯≀𝑑𝑛𝑑2𝑓π‘₯𝑛+ξ€·,Μƒπ‘₯1βˆ’π‘‘π‘›ξ€Έπ‘‘2𝑇π‘₯𝑛,Μƒπ‘₯βˆ’π‘‘π‘›ξ€·1βˆ’π‘‘π‘›ξ€Έπ‘‘2𝑓π‘₯𝑛,𝑇π‘₯𝑛≀𝑑𝑛𝑑2𝑓π‘₯𝑛+ξ€·,Μƒπ‘₯1βˆ’π‘‘π‘›ξ€Έπ‘‘2ξ€·π‘₯𝑛,Μƒπ‘₯βˆ’π‘‘π‘›ξ€·1βˆ’π‘‘π‘›ξ€Έπ‘‘2𝑓π‘₯𝑛,𝑇π‘₯𝑛.(2.9) It follows that 𝑑2ξ€·π‘₯𝑛,Μƒπ‘₯≀𝑑2𝑓π‘₯π‘›ξ€Έξ€Έβˆ’ξ€·,Μƒπ‘₯1βˆ’π‘‘π‘›ξ€Έπ‘‘2𝑓π‘₯𝑛,𝑇π‘₯𝑛=𝑑2𝑓π‘₯𝑛,Μƒπ‘₯βˆ’π‘‘2𝑓π‘₯𝑛,𝑇π‘₯𝑛+𝑑𝑛𝑑2𝑓π‘₯𝑛,𝑇π‘₯𝑛.(2.10)
Since lim𝑑→0𝑑(π‘₯𝑑,𝑇π‘₯𝑑)=0, we can get that limsupπ‘›β†’βˆžπ‘‘2ξ€·π‘₯𝑛,Μƒπ‘₯≀limsupπ‘›β†’βˆžπ‘‘2𝑓π‘₯𝑛,Μƒπ‘₯βˆ’π‘‘2𝑓π‘₯𝑛,π‘₯𝑛.(2.11) Let Ξ”(Μƒπ‘₯,π‘₯𝑛,𝑓(π‘₯𝑛)) be a comparison triangle for β–΅(Μƒπ‘₯,π‘₯𝑛,𝑓(π‘₯𝑛)) in 𝔼2. Then, 𝑑2𝑓π‘₯𝑛,Μƒπ‘₯βˆ’π‘‘2𝑓π‘₯𝑛,π‘₯𝑛=𝑑2𝑓π‘₯𝑛,̃π‘₯βˆ’π‘‘2𝑓π‘₯𝑛,π‘₯𝑛=𝑓π‘₯π‘›ξ€Έβˆ’Μƒπ‘₯,𝑓π‘₯π‘›ξ€Έβˆ’ξ‚­βˆ’ξ‚¬Μƒπ‘₯𝑓π‘₯π‘›ξ€Έβˆ’Μƒπ‘₯𝑛,𝑓π‘₯π‘›ξ€Έβˆ’Μƒπ‘₯𝑛=2𝑓π‘₯π‘›ξ€Έβˆ’Μƒπ‘₯,π‘₯π‘›βˆ’ξ‚­βˆ’ξ‚¬Μƒπ‘₯π‘₯π‘›βˆ’Μƒπ‘₯,π‘₯π‘›βˆ’ξ‚­ξ‚¬Μƒπ‘₯=2𝑓π‘₯π‘›ξ€Έβˆ’Μƒπ‘₯,π‘₯π‘›βˆ’ξ‚­Μƒπ‘₯βˆ’π‘‘2ξ‚€π‘₯𝑛,̃π‘₯=2𝑓π‘₯π‘›ξ€Έβˆ’Μƒπ‘₯,π‘₯π‘›βˆ’ξ‚­Μƒπ‘₯βˆ’π‘‘2ξ€·π‘₯𝑛.,Μƒπ‘₯(2.12) Hence, limsupπ‘›β†’βˆžπ‘‘2ξ€·π‘₯𝑛,Μƒπ‘₯≀limsupπ‘›β†’βˆžξ‚¬π‘“ξ€·π‘₯π‘›ξ€Έβˆ’Μƒπ‘₯,π‘₯π‘›βˆ’ξ‚­.Μƒπ‘₯(2.13)
Let β–΅(Μƒπ‘₯,π‘₯𝑛,𝑓(Μƒπ‘₯)) be a comparison triangle for β–΅(Μƒπ‘₯,π‘₯𝑛,𝑓(Μƒπ‘₯)) in 𝔼2. For each 𝑛, let 𝑒𝑛 be the point of the segment [Μƒπ‘₯,𝑓(Μƒπ‘₯)] which is nearest to π‘₯𝑛, and let 𝑒𝑛 be the point of the segment [Μƒπ‘₯,𝑓(Μƒπ‘₯)] for which 𝑑(𝑒𝑛,Μƒπ‘₯)=𝑑(𝑒𝑛,Μƒπ‘₯).
By passing to subsequences again, we may suppose that {𝑒𝑛} converges to π‘’βˆˆ[Μƒπ‘₯,𝑓(Μƒπ‘₯)], {𝑒𝑛} converges to π‘’βˆˆ[Μƒπ‘₯,𝑓(Μƒπ‘₯)].
Since {π‘₯𝑛}β–΅-converges to a point Μƒπ‘₯, we have π‘Ÿπ‘₯𝑛=lim𝑛sup𝑑π‘₯,π‘₯𝑛=lim𝑛sup𝑑π‘₯,π‘₯𝑛β‰₯lim𝑛sup𝑑𝑒𝑛,π‘₯𝑛=lim𝑛sup𝑑𝑒,π‘₯𝑛β‰₯lim𝑛sup𝑑𝑒,π‘₯𝑛.(2.14) Thus, π‘Ÿ(𝑒,{π‘₯𝑛})β‰€π‘Ÿ({π‘₯𝑛}). This implies that 𝑒=π‘₯ by uniqueness of the asymptotic center. Hence, 𝑒=π‘₯. That is to say, {𝑒𝑛} converges to Μƒπ‘₯, and {𝑒𝑛} converges to Μƒπ‘₯.
Moreover, since 𝑋 satisfies the property 𝒫, we can get that |||𝑓π‘₯π‘›ξ€Έβˆ’Μƒπ‘₯,π‘₯π‘›βˆ’ξ‚­|||ξ‚€Μƒπ‘₯=𝑑̃π‘₯,𝑃[Μƒπ‘₯,𝑓(π‘₯𝑛)]ξ€·π‘₯𝑛⋅𝑑̃π‘₯,𝑓π‘₯𝑛=𝑑̃π‘₯,𝑃[Μƒπ‘₯,𝑓(π‘₯𝑛)]ξ€·π‘₯𝑛π‘₯⋅𝑑̃π‘₯,𝑓𝑛≀𝑑̃π‘₯,𝑃[Μƒπ‘₯,𝑓(Μƒπ‘₯)]ξ€·π‘₯𝑛⋅𝑑(Μƒπ‘₯,𝑓(Μƒπ‘₯))+𝑑̃π‘₯,π‘₯𝑛𝑓π‘₯⋅𝑑𝑛,𝑓(Μƒπ‘₯)=𝑑̃π‘₯,𝑃[Μƒπ‘₯,𝑓(Μƒπ‘₯)]ξ€·π‘₯𝑛⋅𝑑̃π‘₯,𝑓(Μƒπ‘₯)+𝑑̃π‘₯,π‘₯𝑛𝑓π‘₯⋅𝑑𝑛,𝑓(Μƒπ‘₯)≀𝑑𝑒𝑛,𝑑̃π‘₯Μƒπ‘₯,𝑓(Μƒπ‘₯)+𝛼𝑑2ξ€·π‘₯𝑛.,Μƒπ‘₯(2.15) It follows that limsupπ‘›β†’βˆžπ‘‘2ξ€·π‘₯𝑛≀1,Μƒπ‘₯1βˆ’π›Όlimsupπ‘›β†’βˆžπ‘‘ξ‚€π‘’π‘›,𝑑̃π‘₯𝑓(Μƒπ‘₯),.Μƒπ‘₯(2.16) Since {𝑒𝑛} converges to Μƒπ‘₯, we obtain that limsupπ‘›β†’βˆžπ‘‘2(π‘₯𝑛,Μƒπ‘₯)=0, that is, {π‘₯𝑛} converge strongly to Μƒπ‘₯. Since {𝑑𝑛}βŠ†(0,1) is such that 𝑑𝑛→0 as π‘›β†’βˆž is arbitrarily selected, we can get that lim𝑑→0π‘₯𝑑=Μƒπ‘₯.
Finally, we will prove that Μƒπ‘₯ satisfy the equation Μƒπ‘₯=𝑃Fix(𝑇)𝑓(Μƒπ‘₯).
Indeed, for any π‘¦βˆˆFix(𝑇), 𝑑π‘₯𝑑π‘₯,𝑦=π‘‘π‘‘π‘“π‘‘ξ€ΈβŠ•(1βˆ’π‘‘)𝑇π‘₯𝑑𝑓π‘₯,𝑦=𝑑𝑑𝑑,𝑦+(1βˆ’π‘‘)𝑑𝑇π‘₯𝑑𝑓π‘₯,𝑦≀𝑑𝑑𝑑+ξ€·π‘₯,𝑦(1βˆ’π‘‘)𝑑𝑑.,𝑦(2.17) It follows that 𝑑π‘₯𝑑𝑓π‘₯,𝑦≀𝑑𝑑,𝑦.(2.18) Since lim𝑑→0π‘₯𝑑=Μƒπ‘₯, we can get that 𝑑(Μƒπ‘₯,𝑦)≀𝑑(𝑓(Μƒπ‘₯),𝑦).(2.19)
Hence, ||||,𝑑(𝑓(Μƒπ‘₯),𝑦)β‰₯𝑑(Μƒπ‘₯,𝑦)βˆ’π‘‘(Μƒπ‘₯,𝑓(Μƒπ‘₯))𝑑(Μƒπ‘₯,𝑓(Μƒπ‘₯))β‰₯𝑑(𝑓(Μƒπ‘₯),𝑦).(2.20) That is to say, Μƒπ‘₯=𝑃Fix(𝑇)𝑓(Μƒπ‘₯).

Consider now the iteration processπ‘₯0π‘₯∈𝐢,𝑛+1=𝛼𝑛𝑓π‘₯π‘›ξ€ΈβŠ•ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘‡π‘₯𝑛,𝑛β‰₯0,(2.21) where {𝛼𝑛}βŠ†(0,1) satisfies(H1)𝛼𝑛→0,(H2)βˆ‘βˆžπ‘›=0𝛼𝑛=∞,(H3)either βˆ‘βˆžπ‘›=0|𝛼𝑛+1βˆ’π›Όπ‘›|<∞ or limπ‘›β†’βˆž(𝛼𝑛+1/𝛼𝑛)=1.

Theorem 2.3. Let 𝑋 be a CAT(0) space satisfying the property 𝒫, 𝐢 a closed convex subset of 𝑋, π‘‡βˆΆπΆβ†’πΆ a nonexpansive mapping with Fix(𝑇)β‰ βˆ…, and π‘“βˆΆπΆβ†’πΆ a contraction with coefficient 𝛼<1. Let π‘₯0∈𝐢, {π‘₯𝑛} be generated by π‘₯𝑛+1=𝛼𝑛𝑓(π‘₯𝑛)βŠ•(1βˆ’π›Όπ‘›)𝑇π‘₯𝑛,𝑛β‰₯0. Then under the hypotheses (H1 )–(H3 ), π‘₯𝑛→̃π‘₯, where Μƒπ‘₯=𝑃Fix(𝑇)𝑓(Μƒπ‘₯).

Proof. We first show that the sequence {π‘₯𝑛} is bounded. Let π‘βˆˆFix(𝑇). Then, 𝑑π‘₯𝑛+1𝛼,𝑝=𝑑𝑛𝑓π‘₯π‘›ξ€ΈβŠ•ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘‡π‘₯𝑛,𝑝≀𝛼𝑛𝑑𝑓π‘₯𝑛+ξ€·,𝑝1βˆ’π›Όπ‘›ξ€Έπ‘‘ξ€·π‘‡π‘₯𝑛,𝑝≀𝛼𝑛𝑑𝑓π‘₯𝑛+ξ€·,𝑓(𝑝)+𝑑(𝑓(𝑝),𝑝)1βˆ’π›Όπ‘›ξ€Έπ‘‘ξ€·π‘₯𝑛𝑑π‘₯,𝑝≀max𝑛,1,𝑝.1βˆ’π›Όπ‘‘(𝑓(𝑝),𝑝)(2.22) By induction, we have 𝑑π‘₯𝑛𝑑π‘₯,𝑝≀max0ξ€Έ,1,𝑝1βˆ’π›Όπ‘‘(𝑓(𝑝),𝑝),(2.23) for all π‘›βˆˆβ„•. This implies that {π‘₯𝑛} is bounded and so is the sequence {𝑇π‘₯𝑛} and {𝑓(π‘₯𝑛)}.
We claim that 𝑑(π‘₯𝑛+1,π‘₯𝑛)β†’0. Indeed, we have 𝑑π‘₯𝑛+1,π‘₯𝑛𝛼=𝑑𝑛𝑓π‘₯π‘›ξ€ΈβŠ•ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘‡π‘₯𝑛,π›Όπ‘›βˆ’1𝑓π‘₯π‘›βˆ’1ξ€ΈβŠ•ξ€·1βˆ’π›Όπ‘›βˆ’1𝑇π‘₯π‘›βˆ’1𝛼≀𝑑𝑛𝑓π‘₯π‘›ξ€ΈβŠ•ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘‡π‘₯𝑛,𝛼𝑛𝑓π‘₯π‘›ξ€ΈβŠ•ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘‡π‘₯π‘›βˆ’1𝛼+𝑑𝑛𝑓π‘₯π‘›ξ€ΈβŠ•ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘‡π‘₯π‘›βˆ’1,𝛼𝑛𝑓π‘₯π‘›βˆ’1ξ€ΈβŠ•ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘‡π‘₯π‘›βˆ’1𝛼+𝑑𝑛𝑓π‘₯π‘›βˆ’1ξ€ΈβŠ•ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘‡π‘₯π‘›βˆ’1,π›Όπ‘›βˆ’1𝑓π‘₯π‘›βˆ’1ξ€ΈβŠ•ξ€·1βˆ’π›Όπ‘›βˆ’1𝑇π‘₯π‘›βˆ’1≀1βˆ’π›Όπ‘›ξ€Έπ‘‘ξ€·π‘‡π‘₯𝑛,𝑇π‘₯π‘›βˆ’1ξ€Έ+𝛼𝑛𝑑𝑓π‘₯𝑛π‘₯,π‘“π‘›βˆ’1+||π›Όξ€Έξ€Έπ‘›βˆ’π›Όπ‘›βˆ’1||𝑑𝑓π‘₯π‘›βˆ’1ξ€Έ,𝑇π‘₯π‘›βˆ’1≀1βˆ’π›Όπ‘›ξ€Έπ‘‘ξ€·π‘₯𝑛,π‘₯π‘›βˆ’1ξ€Έ+𝛼𝑛π‘₯𝛼𝑑𝑛,π‘₯π‘›βˆ’1ξ€Έ+||π›Όπ‘›βˆ’π›Όπ‘›βˆ’1||𝑑𝑓π‘₯π‘›βˆ’1ξ€Έ,𝑇π‘₯π‘›βˆ’1ξ€Έ.(2.24) By the conditions H2 and H3, we have 𝑑π‘₯𝑛+1,π‘₯π‘›ξ€ΈβŸΆ0.(2.25) Consequently, by the condition H1, 𝑑π‘₯𝑛,𝑇π‘₯𝑛π‘₯≀𝑑𝑛,π‘₯𝑛+1ξ€Έξ€·π‘₯+𝑑𝑛+1,𝑇π‘₯𝑛π‘₯=𝑑𝑛,π‘₯𝑛+1𝛼+𝑑𝑛𝑓π‘₯π‘›ξ€ΈβŠ•ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘‡π‘₯𝑛,𝑇π‘₯𝑛π‘₯=𝑑𝑛,π‘₯𝑛+1ξ€Έ+𝛼𝑛𝑑𝑓π‘₯𝑛,𝑇π‘₯π‘›ξ€ΈβŸΆ0.(2.26)
Since {π‘₯𝑛} is bounded, we may assume that {π‘₯𝑛}β–΅-converges to a point Μ‚π‘₯. By Lemma 1.2, we have Μ‚π‘₯∈Fix(𝑇).
Next we will prove that {π‘₯𝑛} converge strongly to Μ‚π‘₯ and Μ‚π‘₯=Μƒπ‘₯. Indeed, according to Lemma 1.1 and the property of 𝑇 and 𝑓, we can get that 𝑑2ξ€·π‘₯𝑛+1ξ€Έ,Μ‚π‘₯=𝑑2𝑑𝑛𝑓π‘₯π‘›ξ€ΈβŠ•ξ€·1βˆ’π‘‘π‘›ξ€Έπ‘‡π‘₯𝑛,Μ‚π‘₯≀𝛼𝑛𝑑2𝑓π‘₯𝑛+ξ€·,Μ‚π‘₯1βˆ’π›Όπ‘›ξ€Έπ‘‘2𝑇π‘₯𝑛,Μ‚π‘₯βˆ’π›Όπ‘›ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘‘2𝑓π‘₯𝑛,𝑇π‘₯𝑛≀1βˆ’π›Όπ‘›ξ€Έπ‘‘2ξ€·π‘₯𝑛,Μ‚π‘₯+𝛼𝑛𝑑2𝑓π‘₯π‘›ξ€Έξ€Έβˆ’ξ€·,Μ‚π‘₯1βˆ’π›Όπ‘›ξ€Έπ‘‘2𝑓π‘₯𝑛,𝑇𝑛.ξ€Έξ€»(2.27) With a minor modification of the proof of the analogous statement in Theorem 2.2, we can get that 𝑑2𝑓π‘₯𝑛,Μ‚π‘₯βˆ’π‘‘2𝑓π‘₯𝑛,π‘₯𝑛=2𝑓π‘₯π‘›ξ€Έβˆ’Μ‚π‘₯,π‘₯π‘›βˆ’ξ‚­Μ‚π‘₯βˆ’π‘‘2ξ€·π‘₯𝑛,Μ‚π‘₯≀2𝑑𝑒𝑛,𝑑̂π‘₯𝑓(Μ‚π‘₯),̂π‘₯+2𝛼𝑑2ξ€·π‘₯𝑛,Μ‚π‘₯βˆ’π‘‘2ξ€·π‘₯𝑛,,Μ‚π‘₯(2.28) and 𝑑(𝑒𝑛,Μ‚π‘₯)β†’0.
Thus, 𝑑2ξ€·π‘₯𝑛+1≀,Μ‚π‘₯1βˆ’π›Όπ‘›ξ€Έπ‘‘2ξ€·π‘₯𝑛,Μ‚π‘₯+𝛼𝑛𝑑2𝑓π‘₯π‘›ξ€Έξ€Έβˆ’ξ€·,Μ‚π‘₯1βˆ’π›Όπ‘›ξ€Έπ‘‘2𝑓π‘₯𝑛,π‘₯𝑛+ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘‘2𝑓π‘₯𝑛,π‘₯π‘›ξ€Έβˆ’ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘‘2𝑇π‘₯𝑛,π‘₯𝑛≀1βˆ’2(1βˆ’π›Ό)𝛼𝑛𝑑2ξ€·π‘₯𝑛,Μ‚π‘₯+2(1βˆ’π›Ό)𝛼𝑛1𝑑(1βˆ’π›Ό)𝑒𝑛,𝑑̂π‘₯𝑓(Μ‚π‘₯),+1Μ‚π‘₯𝛽(1βˆ’π›Ό)𝑛,(2.29) where 𝛽𝑛=(1βˆ’π›Όπ‘›)𝑑2(𝑓(π‘₯𝑛),π‘₯𝑛)βˆ’(1βˆ’π›Όπ‘›)𝑑2(𝑇π‘₯𝑛,π‘₯𝑛)]. Since 𝑑(𝑒𝑛,Μ‚π‘₯)β†’0 and 𝑑(π‘₯𝑛,𝑇π‘₯𝑛)β†’0, we obtain that lim𝑛1sup𝑑(1βˆ’π›Ό)𝑒𝑛,𝑑̂π‘₯𝑓(Μ‚π‘₯),+1Μ‚π‘₯𝛽(1βˆ’π›Ό)𝑛≀0.(2.30)
According to Lemma 1.6, we can get 𝑑2(π‘₯𝑛,Μ‚π‘₯)β†’0.
Finally, we prove that Μ‚π‘₯=Μƒπ‘₯.
Indeed, for any π‘§βˆˆFix(𝑇), 𝑑2ξ€·π‘₯𝑛+1ξ€Έ,𝑧≀𝛼𝑛𝑑2ξ€·ξ€·π‘₯𝑧,𝑓𝑛+ξ€·ξ€Έξ€Έ1βˆ’π›Όπ‘›ξ€Έπ‘‘2𝑧,𝑇π‘₯π‘›ξ€Έβˆ’π›Όπ‘›ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘‘2𝑓π‘₯𝑛,𝑇π‘₯𝑛≀𝛼𝑛𝑑2ξ€·ξ€·π‘₯𝑧,𝑓𝑛+ξ€·ξ€Έξ€Έ1βˆ’π›Όπ‘›ξ€Έπ‘‘2𝑧,π‘₯π‘›ξ€Έβˆ’π›Όπ‘›ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘‘2𝑓π‘₯𝑛,𝑇π‘₯𝑛.(2.31)
Let πœ‡ be a Banach limit. Then, πœ‡π‘›π‘‘2ξ€·π‘₯𝑛+1ξ€Έ,π‘§β‰€πœ‡π‘›π‘‘2ξ€·ξ€·π‘₯𝑧,π‘“π‘›ξ€Έξ€Έβˆ’πœ‡π‘›π‘‘2𝑓π‘₯𝑛,𝑇π‘₯𝑛.(2.32)
Since π‘₯𝑛→̂π‘₯, we obtain that 𝑑2(Μ‚π‘₯,𝑧)≀𝑑2(𝑧,𝑓(Μ‚π‘₯))βˆ’π‘‘2(𝑓(Μ‚π‘₯),Μ‚π‘₯).(2.33) It follows that 𝑑2(𝑓(Μ‚π‘₯),Μ‚π‘₯)≀𝑑2(𝑧,𝑓(Μ‚π‘₯)),(2.34) that is to say, Μ‚π‘₯=𝑃Fix(𝑇)𝑓(Μ‚π‘₯). Since 𝑃Fix(𝑇)𝑓 is a contraction and Μƒπ‘₯=𝑃Fix(𝑇)𝑓(Μƒπ‘₯), we know that Μ‚π‘₯=Μƒπ‘₯.

Acknowledgment

This paper was supported by NSFC Grant no. 11071279.

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