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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 965751, 11 pages
http://dx.doi.org/10.1155/2012/965751
Research Article

Iterative Algorithm and Δ-Convergence Theorems for Total Asymptotically Nonexpansive Mappings in CAT(0) Spaces

1Department of Mathematics, Yibin University, Yibin, Sichuan 644007, China
2College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China
3Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong

Received 9 July 2012; Accepted 9 August 2012

Academic Editor: Yongfu Su

Copyright © 2012 J. F. Tang et al. 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

The main purpose of this paper is first to introduce the concept of total asymptotically nonexpansive mappings and to prove a Δ-convergence theorem for finding a common fixed point of the total asymptotically nonexpansive mappings and the asymptotically nonexpansive mappings. The demiclosed principle for this kind of mappings in CAT(0) space is also proved in the paper. Our results extend and improve many results in the literature.

1. Introduction

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. Fixed point theory in a CAT(0) space was first studied by Kirk [1, 2]. He showed that every nonexpansive 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 various mappings in CAT(0) space has been developed rapidly and many papers have appeared [310]. On the other hand, Browder [11] introduced the demiclosed principle which states that if 𝑋 is a uniformly convex Banach space, 𝐶 is a nonempty closed convex subset of 𝑋, and if 𝑇𝐶𝐶 is nonexpansive mapping, then 𝐼𝑇 is demiclosed at each 𝑦𝑋, that is, for any sequence {𝑥𝑛} in 𝐶 conditions 𝑥𝑛𝑥 weakly and (𝐼𝑇)𝑥𝑛𝑦 strongly imply that (𝐼𝑇)𝑥=𝑦 (where 𝐼 is the identity mapping of 𝑋). Xu [12] proved the demiclosed principle for asymptotically nonexpansive mappings in the setting of a uniformly convex Banach space. Nanjaras and Panyanak [13] proved the demiclosed principle for asymptotically nonexpansive mappings in CAT(0) space and obtained a Δ-convergence theorem for the Krasnosel’skii-Mann iteration.

Motivated and inspired by the researches going on in this direction, especially inspired by Nanjaras and Panyanak, and so forth [13], the purpose of this paper is to introduce a general mapping, namely, total asymptotically nonexpansive mapping and to prove its demiclosed principle in CAT(0) space. As a consequence, we construct a hierarchical iterative algorithm to study the fixed point of the total asymptotically nonexpansive mappings and obtain a Δ-convergence theorem.

2. Preliminaries and Lemmas

Let (𝑋,𝑑) be a metric space and 𝑥,𝑦𝑋 with 𝑑(𝑥,𝑦)=𝑙. A geodesic path from 𝑥 to 𝑦 is a isometry 𝑐[0,𝑙]𝑋 such that 𝑐(0)=𝑥, 𝑐(𝑙)=𝑦. The image of a geodesic path is called geodesic segment. A space (𝑋,𝑑) is a (uniquely) geodesic space if every two points of 𝑋 are joined by only one geodesic segment. A geodesic triangle Δ(𝑥1,𝑥2,𝑥3) in a geodesic metric space (𝑋,𝑑) consists of three points 𝑥1,𝑥2,𝑥3 in 𝑋 (the vertices of Δ) and a geodesic segment between each pair of vertices (the edges of Δ). A comparison triangle for the geodesic triangle Δ(𝑥1,𝑥2,𝑥3) in (𝑋,𝑑) is a triangle Δ(𝑥1,𝑥2,𝑥3)=Δ(𝑥1,𝑥2,𝑥3) in the Euclidean space 2 such that 𝑑2(𝑥𝑖,𝑥𝑗)=𝑑(𝑥𝑖,𝑥𝑗) for 𝑖,𝑗{1,2,3}.

A geodesic space is said to be a CAT(0) space if for each geodesic triangle Δ(𝑥1,𝑥2,𝑥3) in 𝑋 and its comparison triangle Δ=Δ(𝑥1,𝑥2,𝑥3) in 2, the CAT(0) inequality 𝑑(𝑥,𝑦)𝑑𝔼2𝑥,𝑦.(2.1) is satisfied for all 𝑥,𝑦Δ and 𝑥,𝑦Δ.

In this paper, we write (1𝑡)𝑥𝑡𝑦 for the unique point 𝑧 in the geodesic segment joining from 𝑥 to 𝑦 such that 𝑑(𝑥,𝑧)=𝑡𝑑(𝑥,𝑦),𝑑(𝑦,𝑧)=(1𝑡)𝑑(𝑥,𝑦).(2.2) We also denote by [𝑥,𝑦] the geodesic segment joining from 𝑥 to 𝑦, that is, [𝑥,𝑦]={(1𝑡)𝑥𝑡𝑦𝑡[0,1]}.

A subset 𝐶 of a CAT(0) space 𝑋 is said to be convex if [𝑥,𝑦]𝐶 for all 𝑥,𝑦𝐶.

Lemma 2.1 (see [14]). A geodesic space 𝑋 is a CAT(0) space, if and only if the following inequality 𝑑((1𝑡)𝑥𝑡𝑦,𝑧)2(1𝑡)𝑑(𝑥,𝑧)2+𝑡𝑑(𝑦,𝑧)2𝑡(1𝑡)𝑑(𝑥,𝑦)2(2.3) is satisfied for all 𝑥,𝑦,𝑧𝑋 and 𝑡[0,1]. In particular, if 𝑥,𝑦,𝑧 are points in a CAT(0) space and 𝑡[0,1], then 𝑑((1𝑡)𝑥𝑡𝑦,𝑧)(1𝑡)𝑑(𝑥,𝑧)+𝑡𝑑(𝑦,𝑧).(2.4)

Let {𝑥𝑛} be a bounded sequence in a CAT(0) space 𝑋. For 𝑥𝑋, one sets 𝑟𝑥𝑥,𝑛=limsup𝑛𝑑𝑥,𝑥𝑛.(2.5) The asymptotic radius 𝑟({𝑥𝑛}) of {𝑥𝑛} is given by 𝑟𝑥𝑛=inf𝑥𝑋𝑟𝑥𝑥,𝑛,(2.6) the asymptotic radius 𝑟𝐶({𝑥𝑛}) of {𝑥𝑛} with respect to 𝐶𝑋 is given by 𝑟𝐶𝑥𝑛=inf𝑥𝐶r𝑥𝑥,𝑛,(2.7) the asymptotic center 𝐴({𝑥𝑛}) of {𝑥𝑛} is the set 𝐴𝑥𝑛=𝑥𝑥𝑋𝑟𝑥,𝑛𝑥=𝑟𝑛,(2.8) the asymptotic center 𝐴𝐶({𝑥𝑛}) of {𝑥𝑛} with respect to 𝐶𝑋 is the set 𝐴𝐶𝑥𝑛=𝑥𝑥𝐶𝑟𝑥,𝑛=𝑟𝐶𝑥𝑛.(2.9)

Recall that a bounded sequence {𝑥𝑛} in 𝑋 is said to be regular if 𝑟({𝑥𝑛})=𝑟({𝑢𝑛}) for every subsequence {𝑢𝑛} of {𝑥𝑛}.

Proposition 2.2 (see [15]). If {𝑥𝑛} is a bounded sequence in a complete CAT(0) space 𝑋 and 𝐶 is a closed convex subset of 𝑋, then(1)there exists a unique point 𝑢𝐶 such that 𝑟𝑥𝑢,𝑛=inf𝑥𝐶𝑟𝑥𝑥,𝑛;(2.10)(2)𝐴({𝑥𝑛}) and 𝐴𝐶({𝑥𝑛}) are both singleton.

Lemma 2.3 2.3 (see [16]). If 𝐶 is a closed convex subset of a complete CAT(0) space 𝑋 and if {𝑥𝑛} is a bounded sequence in 𝐶, then the asymptotic center of {𝑥𝑛} is in 𝐶.

Definition 2.4 (see [17]). A sequence {𝑥𝑛} in a CAT(0) space 𝑋 is said to Δ-converge to 𝑥𝑋 if 𝑥 is the unique asymptotic center of {𝑢𝑛} for every subsequence {𝑢𝑛} of {𝑥𝑛}. In this case one writes Δlim𝑛𝑥𝑛=𝑥 and call 𝑥 the Δ-limit of {𝑥𝑛}.

Lemma 2.5 (see [17]). Every bounded sequence in a complete CAT(0) space always has a Δ-convergent subsequence.

Let {𝑥𝑛} be a bounded sequence in a CAT(0) space 𝑋 and let 𝐶 be a closed convex subset of 𝑋 which contains {𝑥𝑛}. We denote the notation 𝑥𝑛𝑤iΦ(𝑤)=inf𝑥𝐶Φ(𝑥),(2.11) where Φ(𝑥)=limsup𝑛𝑑(𝑥𝑛,𝑥).

Now one gives a connection between the “” convergence and Δ-convergence.

Proposition 2.6 2.6 (see [13]). Let {𝑥𝑛} be a bounded sequence in a CAT(0) space 𝑋 and let 𝐶 be a closed convex subset of 𝑋 which contains {𝑥𝑛}. Then(1)Δlim𝑛𝑥𝑛=𝑥 implies that {𝑥𝑛}𝑥;(2){𝑥𝑛}𝑥 and {𝑥𝑛} is regular imply that Δlim𝑛𝑥𝑛=𝑥;

Let 𝐶 be a closed subset of a metric space (𝑋,𝑑). Recall that a mapping 𝑇𝐶𝐶 is said to be nonexpansive if 𝑑(𝑇𝑥,𝑇𝑦)𝑑(𝑥,𝑦),𝑥,𝑦𝑋.(2.12)𝑇 is said to be asymptotically nonexpansive if there is a sequence {𝑘𝑛}[1,+) with lim𝑛𝑘𝑛=1 such that 𝑑(𝑇𝑛𝑥,𝑇𝑛𝑦)𝑘𝑛𝑑(𝑥,𝑦),𝑛1,𝑥,𝑦𝑋.(2.13)

𝑇 is said to be closed if, for any sequence {𝑥𝑛}𝐶 with 𝑑(𝑥𝑛,𝑥)0 and 𝑑(𝑇𝑥𝑛,𝑦)0, then 𝑇𝑥=𝑦.

𝑇 is called 𝐿-uniformly Lipschitzian, if there exists a constant 𝐿>0 such that 𝑑(𝑇𝑛𝑥,𝑇𝑛𝑦)𝐿𝑑(𝑥,𝑦),𝑥,𝑦𝐶,𝑛1.(2.14)

Definition 2.7. Let (𝑋,𝑑) be a metric space and let 𝐶 be a closed subset of 𝑋. A mapping 𝑇𝐶𝐶 is said to be ({𝑣𝑛},{𝜇𝑛},𝜁)total asymptotically nonexpansive if there exist nonnegative real sequences {𝑣𝑛},{𝜇𝑛} with 𝑣𝑛0, 𝜇𝑛0(𝑛) and a strictly increasing continuous function 𝜁[0,+)[0,+) with 𝜁(0)=0 such that 𝑑(𝑇𝑛𝑥,𝑇𝑛𝑦)𝑑(𝑥,𝑦)+𝑣𝑛𝜁(𝑑(𝑥,𝑦))+𝜇𝑛,𝑛1,𝑥,𝑦𝐶.(2.15)

Remark 2.8. (1) It is obvious that If 𝑇 is uniformly Lipschitzian, then 𝑇 is closed.
(2) From the definitions, it is to know that, each nonexpansive mapping is a asymptotically nonexpansive mapping with sequence {𝑘𝑛=1}, and each asymptotically nonexpansive mapping is a total asymptotically nonexpansive mapping with 𝑣𝑛=𝑘𝑛1, 𝜇𝑛=0, for all 𝑛1, and 𝜁(𝑡)=𝑡, 𝑡0.

Lemma 2.9 (demiclosed principle for total asymptotically nonexpansive mappings). Let 𝐶 be a closed and convex subset of a complete CAT(0) space 𝑋 and let 𝑇𝐶𝐶 be a 𝐿-uniformly Lipschitzian and ({𝑣𝑛},{𝜇𝑛},𝜁)total asymptotically nonexpansive mapping. Let {𝑥𝑛} be a bounded sequence in 𝐶 such that lim𝑛𝑑(𝑥𝑛,𝑇𝑥𝑛)=0 and 𝑥𝑛𝑤. Then 𝑇𝑤=𝑤.

Proof. By the definition, 𝑥𝑛𝑤 if and only if 𝐴𝐶({𝑥𝑛})={𝑤}. By Lemma 2.3, we have 𝐴({𝑥𝑛})={𝑤}.
Since lim𝑛𝑑(𝑥𝑛,𝑇𝑥𝑛)=0, by induction we can prove that lim𝑛𝑑𝑥𝑛,𝑇𝑚𝑥𝑛=0,𝑚1.(2.16)
In fact, it is obvious that, the conclusion is true for 𝑚=1. Suppose the conclusion holds for 𝑚1, now we prove that the conclusion is also true for 𝑚+1. In fact, since 𝑇 is a 𝐿-uniformly Lipschitzian mapping, we have 𝑑𝑥𝑛,𝑇𝑚+1𝑥𝑛𝑥𝑑𝑛,𝑇𝑥𝑛+𝑑𝑇𝑥𝑛,𝑇𝑇𝑚𝑥𝑛𝑥𝑑𝑛,𝑇𝑥𝑛𝑥+𝐿𝑑𝑛,𝑇𝑚𝑥𝑛(0as𝑛).(2.17) Equation (2.16) is proved. Hence for each 𝑥𝐶 and 𝑚1 from (2.16) we have Φ(𝑥)=limsup𝑛𝑑𝑥𝑛,𝑥=limsup𝑛𝑑𝑇𝑚𝑥𝑛.,𝑥(2.18) In (2.18) taking 𝑥=𝑇𝑚𝑤,𝑚1, we have Φ(𝑇𝑚𝑤)=limsup𝑛𝑑𝑇𝑚𝑥𝑛,𝑇𝑚𝑤limsup𝑛𝑑𝑥𝑛,𝑤+𝑣𝑚𝜁𝑑𝑥𝑛,𝑤+𝜇𝑚.(2.19) Let 𝑚 and taking superior limit on the both sides, it gets that limsup𝑚Φ(𝑇𝑚𝑤)Φ(w).(2.20) Furthermore, for any 𝑛,𝑚1 it follows from inequality (2.3) with 𝑡=1/2 that 𝑑2𝑥𝑛,𝑤𝑇𝑚𝑤212𝑑2𝑥𝑛+1,𝑤2𝑑2𝑥𝑛,𝑇𝑚𝑤14𝑑2(𝑤,𝑇𝑚𝑤).(2.21) Let 𝑛 and taking superior limit on the both sides of the above inequality, for any 𝑚1 we get Φ𝑤𝑇𝑚𝑤2212Φ(𝑤)2+12Φ(𝑇𝑚𝑤)214𝑑(𝑤,𝑇𝑚𝑤)2.(2.22) Since 𝐴({𝑥𝑛})={𝑤}, we have Φ(𝑤)2Φ𝑤𝑇𝑚𝑤2212Φ(𝑤)2+12Φ(𝑇𝑚𝑤)214𝑑(𝑤,𝑇𝑚𝑤)2,𝑚1,(2.23) which implies that 𝑑2(𝑤,𝑇𝑚𝑤)2Φ(𝑇𝑚𝑤)22Φ(𝑤)2.(2.24) By (2.20) and (2.24), we have lim𝑚𝑑(𝑤,𝑇𝑚𝑤)=0. This implies that lim𝑚𝑑(𝑤,𝑇𝑚+1𝑤)=0. Since 𝑇 is uniformly Lipschitzian, 𝑇 is uniformly continuous. Hence we have 𝑇𝑤=𝑤. This completes the proof of Lemma 2.9.

The following proposition can be obtained from Lemma 2.9 immediately which is a generalization of Kirk and Panyanak [17] and Nanjaras and Panyanak [13].

Proposition 2.10. Let 𝐶 be a closed and convex subset of a complete CAT(0) space 𝑋 and let 𝑇𝐶𝐶 be an asymptotically nonexpansive mapping. Let {𝑥𝑛} be a bounded sequence in 𝐶 such that lim𝑛𝑑(𝑥𝑛,𝑇𝑥𝑛)=0 and Δlim𝑛𝑥𝑛=𝑤. Then 𝑇(𝑤)=𝑤.

Definition 2.11 (see [18]). Let 𝑋 be a CAT(0) space then 𝑋 is uniformly convex, that is, for any given 𝑟>0,𝜖(0,2] and 𝜆[0,1], there exists a 𝜂(𝑟,𝜖)=𝜖2/8 such that, for all 𝑥,𝑦,𝑧𝑋, 𝜖𝑑(𝑥,𝑧)𝑟𝑑(𝑦,𝑧)𝑟𝑑(𝑥,𝑦)𝜖𝑟𝑑((1𝜆)𝑥𝜆𝑦,𝑧)12𝜆(1𝜆)28𝑟,(2.25) where the function 𝜂(0,)×(0,2](0,1] is called the modulus of uniform convexity of CAT(0).

Lemma 2.12 (see [14]). If {𝑥𝑛} is a bounded sequence in a complete CAT(0) space with 𝐴({𝑥𝑛})={𝑥}, {𝑢𝑛} is a subsequence of {𝑥𝑛} with 𝐴({𝑢𝑛})={𝑢}, and the sequence {𝑑(𝑥𝑛,𝑢)} converges, then 𝑥=𝑢.

Lemma 2.13. Let {𝑎𝑛},{𝑏𝑛}, and {𝛿𝑛} be sequences of nonnegative real numbers satisfying the inequality 𝑎𝑛+11+𝛿𝑛𝑎𝑛+𝑏𝑛.(2.26) If Σ𝑛=1𝛿𝑛< and Σ𝑛=1𝑏𝑛<, then {𝑎𝑛} is bounded and lim𝑛𝑎𝑛 exists.

Lemma 2.14 (see [13]). Let 𝑋 be a CAT(0) space, 𝑥𝑋 be a given point and {𝑡𝑛} be a sequence in [𝑏,𝑐] with 𝑏,𝑐(0,1) and 0<𝑏(1𝑐)1/2. Let {𝑥𝑛} and {𝑦𝑛} be any sequences in 𝑋 such that limsup𝑛𝑑𝑥𝑛,𝑥𝑟,limsup𝑛𝑑𝑦𝑛,𝑥𝑟,lim𝑛𝑑1𝑡𝑛𝑥𝑛𝑡𝑛𝑦𝑛,𝑥=𝑟,(2.27) for some 𝑟0. Then lim𝑛𝑑𝑥𝑛,𝑦𝑛=0.(2.28)

3. Main Results

In this section, we will prove our main theorem.

Theorem 3.1. Let 𝐶 be a nonempty bounded closed and convex subset of a complete CAT(0) space 𝑋. Let 𝑆𝐶𝐶 be a asymptotically nonexpansive mapping with sequence {𝑘𝑛}[1,), 𝑘𝑛1 and 𝑇𝐶𝐶 be a uniformly L-Lipschitzian and ({𝑣𝑛},{𝜇𝑛},𝜁)total asymptotically nonexpansive mapping such that =𝐹(𝑆)𝐹(𝑇). From arbitrary 𝑥1𝐶, defined the sequence {𝑥𝑛} as follows: 𝑦𝑛=𝛼𝑛𝑆𝑛𝑥𝑛1𝛼𝑛𝑥𝑛,𝑥𝑛+1=𝛽𝑛𝑇𝑛𝑦𝑛1𝛽𝑛𝑥𝑛(3.1) for all 𝑛1, where {𝛽𝑛} is a sequence in (0, 1). If the following conditions are satisfied:(i)Σ𝑛=1𝑣𝑛<;Σ𝑛=1𝜇𝑛<;Σ𝑛=1(𝑘𝑛1)<;(ii)there exists a constant 𝑀>0 such that 𝜁(𝑟)𝑀𝑟, 𝑟0;(iii)there exist constants 𝑏,𝑐(0,1) with 0<𝑏(1𝑐)1/2 such that {𝛼𝑛}[𝑏,𝑐];(iv)Σ𝑛=1sup{𝑑(𝑧,𝑆𝑛z)𝑧𝐵}< for each bounded subset 𝐵 of 𝐶.Then the sequence {𝑥𝑛}Δ-converges to a fixed point of .

Proof. We divide the proof of Theorem 3.1 into four steps. (I) First we prove that for each 𝑝 the following limit exists lim𝑛𝑑𝑥𝑛.,𝑝(3.2)In fact, for each 𝑝, we have 𝑑𝑦𝑛𝛼,𝑝=𝑑𝑛𝑆𝑛𝑥𝑛1𝛼𝑛𝑥𝑛,𝑝𝛼𝑛𝑑𝑆𝑛𝑥𝑛+,𝑝1𝛼𝑛𝑑𝑥𝑛,𝑝=𝛼𝑛𝑑𝑆𝑛𝑥𝑛,𝑆𝑛𝑝+1𝛼𝑛𝑑𝑥𝑛,𝑝𝛼𝑛𝑘𝑛𝑑𝑥𝑛+,𝑝1𝛼𝑛𝑑𝑥𝑛=,𝑝1+𝛼𝑛𝑘𝑛𝑑𝑥1𝑛,𝑑𝑥,𝑝𝑛+1𝛽,𝑝=𝑑𝑛𝑇𝑛𝑦𝑛1𝛽𝑛𝑥𝑛,𝑝𝛽𝑛𝑑𝑇𝑛𝑦𝑛+,𝑝1𝛽𝑛𝑑𝑥𝑛,𝑝=𝛽𝑛𝑑𝑇𝑛𝑦𝑛,𝑇𝑛𝑝+1𝛽𝑛𝑑𝑥𝑛,𝑝𝛽𝑛𝑑𝑦𝑛,𝑝+𝑣𝑛𝜁𝑑𝑦𝑛,𝑝+𝜇𝑛+1𝛽𝑛𝑑𝑥𝑛,𝑝𝛽𝑛𝑑𝑦𝑛,𝑝+𝑣𝑛𝑀𝑑𝑦𝑛,𝑝+𝜇𝑛+1𝛽𝑛𝑑𝑥𝑛𝑥,𝑝𝑑𝑛,𝑝+𝛽𝑛𝛼𝑛𝑘𝑛𝑑𝑥1𝑛,𝑝+𝛽𝑛𝑣𝑛𝑀1+𝛼𝑛𝑘𝑛𝑑𝑥1𝑛,𝑝+𝜇𝑛𝑘1+𝑛1+𝑣𝑛𝑀1+𝛼𝑛𝑘𝑛𝑑𝑥1𝑛,𝑝+𝜇𝑛.(3.3) It follows from Lemma 2.13 that {𝑑(𝑥𝑛,𝑝)} is bounded and lim𝑛𝑑(𝑥𝑛,𝑝) exists. Without loss of generality, we can assume lim𝑛𝑑(𝑥𝑛,𝑝)=𝑐0.  (II) Next we prove that lim𝑛𝑑𝑥𝑛,𝑇𝑥𝑛=0.(3.4) In fact, since 𝑑𝑇𝑛𝑦𝑛𝑇,𝑝=𝑑𝑛𝑦𝑛,𝑇𝑛𝑝𝑦𝑑𝑛,𝑝+𝑣𝑛𝜁𝑑𝑦𝑛,𝑝+𝜇𝑛1+𝑣𝑛𝑀𝑑𝑦𝑛,𝑝+𝜇𝑛1+𝑣𝑛𝑀1+𝛼𝑛𝑘𝑛𝑑𝑥1𝑛,𝑝+𝜇𝑛(3.5) for all 𝑛 and 𝑝, we have limsup𝑛𝑑𝑇𝑛𝑦𝑛,𝑝𝑐.(3.6) On the other hand, since lim𝑛𝑑𝛽𝑛𝑇𝑛𝑦𝑛1𝛽𝑛𝑥𝑛,𝑝=lim𝑛𝑑𝑥𝑛+1,𝑝=𝑐,(3.7) by Lemma 2.14, we have lim𝑛𝑑𝑇𝑛𝑦𝑛,𝑥𝑛=0.(3.8)
From condition (iv), we have 𝑑𝑥𝑛,𝑦𝑛𝑥=𝑑𝑛,1𝛼𝑛𝑥𝑛𝛼𝑛𝑆𝑛𝑥𝑛𝛼𝑛𝑑𝑥𝑛,𝑆𝑛𝑥𝑛0(𝑛).(3.9) Hence from (3.8) and (3.9) we have that 𝑑𝑥𝑛,𝑇𝑛𝑥𝑛𝑥𝑑𝑛,𝑇𝑛𝑦𝑛𝑇+𝑑𝑛𝑦𝑛,𝑇𝑛𝑥𝑛𝑥𝑑𝑛,𝑇𝑛𝑦𝑛𝑦+𝐿𝑑𝑛,𝑥𝑛0(𝑛).(3.10) By (3.9) and (3.10) it gets that 𝑑𝑥𝑛+1,𝑇𝑛𝑥𝑛=𝑑1𝛽𝑛𝑥𝑛𝛽𝑛𝑇𝑛𝑦𝑛,𝑇𝑛𝑥𝑛1𝛽𝑛𝑑𝑥𝑛,𝑇𝑛𝑥𝑛+𝛽𝑛𝑑𝑇𝑛𝑦𝑛,𝑇𝑛𝑥𝑛1𝛽𝑛𝑑𝑥𝑛,𝑇𝑛𝑥𝑛+𝛽𝑛𝑦𝐿𝑑𝑛,𝑥𝑛0(𝑛).(3.11) Hence from (3.10) and (3.11) we have that 𝑑𝑥𝑛,𝑥𝑛+10(𝑛).(3.12) Again since 𝑇 is uniformly 𝐿-Lipschitzian, from (3.10) and (3.12) we have that 𝑑𝑥𝑛,𝑇𝑥𝑛𝑥𝑑𝑛,𝑥𝑛+1𝑥+𝑑𝑛+1,𝑇𝑛+1𝑥𝑛+1𝑇+𝑑𝑛+1𝑥𝑛+1,𝑇𝑛+1𝑥𝑛𝑇+𝑑𝑛+1𝑥𝑛,𝑇𝑥𝑛𝑥(𝐿+1)𝑑𝑛,𝑥𝑛+1𝑥+𝑑𝑛+1,𝑇𝑛+1𝑥𝑛+1𝑇+𝐿𝑑𝑛𝑥𝑛,𝑥𝑛0(𝑛).(3.13) Equation (3.4) is proved. (III) Now we prove that 𝑤𝜔𝑥𝑛=𝑢𝑛𝑥𝑛𝐴𝑢𝑛(3.14)and 𝑤𝜔(𝑥𝑛) consists exactly of one point.
In fact, let 𝑢𝑤𝜔(𝑥𝑛), then there exists a subsequence {𝑢𝑛} of {𝑥𝑛} such that 𝐴({𝑢𝑛})={𝑢}. By Lemmas 2.5 and 2.3, there exists a subsequence {𝜈𝑛} of {𝑢𝑛} such that Δlim𝑛𝜈𝑛=𝜈𝐶. By Lemma 2.9, we have 𝜈𝐹(𝑇). By Lemma 2.12, 𝑢=𝜈. This shows that 𝑤𝜔(𝑥𝑛).
Let {𝑢𝑛} be a subsequence of {𝑥𝑛} with 𝐴({𝑢𝑛})={𝑢} and let 𝐴({𝑥𝑛})={𝑥}. Since 𝑢𝑤𝜔(𝑥𝑛) and {𝑑(𝑥𝑛,𝑢)} converges, by Lemma 2.12, we have 𝑥=𝑢. This shows that 𝑤𝜔(𝑥𝑛) consists of exactly one point.  (IV) Finally we prove {𝑥𝑛}Δ-converges to a point of .
In fact, it follows from (3.2) that {𝑑(𝑥𝑛,𝑝)} is convergent for each 𝑝. By (3.4) lim𝑛𝑑(𝑥𝑛,𝑇𝑥𝑛)=0. By (3.14) 𝑤𝜔(𝑥𝑛) and 𝑤𝜔(𝑥𝑛) consists of exactly one point. This shows that {𝑥𝑛}Δ-converges to a point of .
This completes the proof of Theorem 3.1.

Conflict of Interests

The authors declare that they have no competing interests.

Authors’ Contribution

All the authors contributed equally to the writing of the present article. And they also read and approved the final paper.

Acknowledgments

The authors would like to express their thanks to the referees for their helpful suggestions and comments. This study was supported by the Scientific Research Fund of Sichuan Provincial Education Department (12ZB346).

References

  1. W. A. Kirk, “Geodesic geometry and fixed point theory,” in Seminar of Mathematical Analysis: Proceedings, Universities of Malaga and Seville (Spain), September 2002–February 2003, vol. 64 of Colección Abierta, pp. 195–225, Seville University Publications, Seville, Spain, 2003. View at Zentralblatt MATH
  2. W. A. Kirk, “Geodesic geometry and fixed point theory II,” in International Conference on Fixed Point Theory and Applications, pp. 113–142, Yokohama Publications, Yokohama, Japan, 2004. View at Zentralblatt MATH
  3. 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, vol. 70, no. 12, pp. 4268–4273, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. S. Dhompongsa, W. A. Kirk, and B. Sims, “Fixed points of uniformly Lipschitzian mappings,” Nonlinear Analysis, vol. 65, no. 4, pp. 762–772, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. N. Hussain and M. A. Khamsi, “On asymptotic pointwise contractions in metric spaces,” Nonlinear Analysis, vol. 71, no. 10, pp. 4423–4429, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. A. Kaewcharoen and W. A. Kirk, “Proximinality in geodesic spaces,” Abstract and Applied Analysis, vol. 2006, Article ID 43591, 10 pages, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. 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
  8. 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
  9. N. Shahzad, “Fixed point results for multimaps in CAT(0) spaces,” Topology and Its Applications, vol. 156, no. 5, pp. 997–1001, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. N. Shahzad, “Invariant approximations in CAT(0) spaces,” Nonlinear Analysis, vol. 70, no. 12, pp. 4338–4340, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. F. E. Browder, “Semicontractive and semiaccretive nonlinear mappings in Banach spaces,” Bulletin of the American Mathematical Society, vol. 74, pp. 660–665, 1968. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. H. K. Xu, “Existence and convergence for fixed points of mappings of asymptotically nonexpansive type,” Nonlinear Analysis, vol. 16, no. 12, pp. 1139–1146, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. B. Nanjaras and B. Panyanak, “Demiclosed principle for asymptotically nonexpansive mappings in CAT(0) spaces,” Fixed Point Theory and Applications, vol. 2010, Article ID 268780, 14 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. 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
  15. S. Dhompongsa, W. A. Kirk, and B. Sims, “Fixed points of uniformly Lipschitzian mappings,” Nonlinear Analysis, vol. 65, no. 4, pp. 762–772, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. 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
  17. W. A. Kirk and B. Panyanak, “A concept of convergence in geodesic spaces,” Nonlinear Analysis, vol. 68, no. 12, pp. 3689–3696, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. 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