Abstract and Applied Analysis

Volume 2012 (2012), Article ID 965751, 11 pages

http://dx.doi.org/10.1155/2012/965751

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

^{1}Department of Mathematics, Yibin University, Yibin, Sichuan 644007, China^{2}College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China^{3}Department 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 -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 is a CAT(0) space if it is geodesically connected and if every geodesic triangle in 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 [3–10]. On the other hand, Browder [11] introduced the demiclosed principle which states that if is a uniformly convex Banach space, is a nonempty closed convex subset of , and if is nonexpansive mapping, then is demiclosed at each , that is, for any sequence in conditions weakly and strongly imply that (where is the identity mapping of ). 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 -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 -convergence theorem.

#### 2. Preliminaries and Lemmas

Let be a metric space and with . A geodesic path from to is a isometry such that , . The image of a geodesic path is called geodesic segment. A space is a (uniquely) geodesic space if every two points of are joined by only one geodesic segment. A geodesic triangle in a geodesic metric space consists of three points in (the vertices of ) and a geodesic segment between each pair of vertices (the edges of ). A comparison triangle for the geodesic triangle in is a triangle in the Euclidean space such that for .

A geodesic space is said to be a CAT(0) space if for each geodesic triangle in and its comparison triangle in , the CAT(0) inequality is satisfied for all and .

In this paper, we write for the unique point in the geodesic segment joining from to such that We also denote by the geodesic segment joining from to , that is, .

A subset of a CAT(0) space is said to be convex if for all .

Lemma 2.1 (see [14]). * A geodesic space is a CAT(0) space, if and only if the following inequality
**
is satisfied for all and . In particular, if are points in a CAT(0) space and , then
*

Let be a bounded sequence in a CAT(0) space . For , one sets
The *asymptotic radius * of is given by
the *asymptotic radius * of with respect to is given by
the *asymptotic center * of is the set
the *asymptotic center * of with respect to is the set

Recall that a bounded sequence in is said to be *regular* if for every subsequence of .

Proposition 2.2 (see [15]). * If is a bounded sequence in a complete CAT(0) space and is a closed convex subset of , then*(1)*there exists a unique point such that
*(2)* and are both singleton.*

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

*Definition 2.4 (see [17]). * A sequence in a CAT(0) space is said to -converge to if is the unique asymptotic center of for every subsequence of . In this case one writes and call the -limit of .

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

Let be a bounded sequence in a CAT(0) space and let be a closed convex subset of which contains . We denote the notation where .

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

Proposition 2.6 2.6 (see [13]). *Let be a bounded sequence in a CAT(0) space and let be a closed convex subset of which contains . Then*(1)* implies that ;*(2)* and is regular imply that ;*

Let be a closed subset of a metric space . Recall that a mapping is said to be * nonexpansive* if
is said to be * asymptotically nonexpansive* if there is a sequence with such that

is said to be * closed* if, for any sequence with and , then .

is called *uniformly Lipschitzian*, if there exists a constant such that

*Definition 2.7. * Let be a metric space and let be a closed subset of . A mapping is said to be total asymptotically nonexpansive if there exist nonnegative real sequences with , and a strictly increasing continuous function with such that

*Remark 2.8. *(1) It is obvious that If is uniformly Lipschitzian, then is closed.

(2) From the definitions, it is to know that, each nonexpansive mapping is a asymptotically nonexpansive mapping with sequence , and each asymptotically nonexpansive mapping is a total asymptotically nonexpansive mapping with , , for all , and , .

Lemma 2.9 (demiclosed principle for total asymptotically nonexpansive mappings). *Let be a closed and convex subset of a complete CAT(0) space and let be a uniformly Lipschitzian and total asymptotically nonexpansive mapping. Let be a bounded sequence in such that and . Then .*

*Proof. * By the definition, if and only if . By Lemma 2.3, we have .

Since , by induction we can prove that

In fact, it is obvious that, the conclusion is true for . Suppose the conclusion holds for , now we prove that the conclusion is also true for . In fact, since is a uniformly Lipschitzian mapping, we have
Equation (2.16) is proved. Hence for each and from (2.16) we have
In (2.18) taking , we have
Let and taking superior limit on the both sides, it gets that
Furthermore, for any it follows from inequality (2.3) with that
Let and taking superior limit on the both sides of the above inequality, for any we get
Since , we have
which implies that
By (2.20) and (2.24), we have . This implies that . Since is uniformly Lipschitzian, is uniformly continuous. Hence we have . 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 be a closed and convex subset of a complete CAT(0) space and let be an asymptotically nonexpansive mapping. Let be a bounded sequence in such that and . Then .*

*Definition 2.11 (see [18]). *Let be a CAT(0) space then is uniformly convex, that is, for any given and , there exists a such that, for all ,
where the function is called the modulus of uniform convexity of CAT(0).

Lemma 2.12 (see [14]). * If is a bounded sequence in a complete CAT(0) space with , is a subsequence of with , and the sequence converges, then .*

Lemma 2.13. * Let , and be sequences of nonnegative real numbers satisfying the inequality
**
If and , then is bounded and exists.*

Lemma 2.14 (see [13]). *Let be a CAT(0) space, be a given point and be a sequence in with and . Let and be any sequences in such that
**
for some . Then
*

#### 3. Main Results

In this section, we will prove our main theorem.

Theorem 3.1. * Let be a nonempty bounded closed and convex subset of a complete CAT(0) space . Let be a asymptotically nonexpansive mapping with sequence , and be a uniformly L-Lipschitzian and total asymptotically nonexpansive mapping such that . From arbitrary , defined the sequence as follows:
**
for all , where is a sequence in (0, 1). If the following conditions are satisfied:*(i)*;*(ii)*there exists a constant such that , ;*(iii)*there exist constants with such that ;*(iv)* for each bounded subset of .**Then the sequence -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 the following limit exists
In fact, for each , we have
It follows from Lemma 2.13 that is bounded and exists. Without loss of generality, we can assume .** (II) Next we prove that
In fact, since
for all and , we have
On the other hand, since
by Lemma 2.14, we have

From condition (iv), we have
Hence from (3.8) and (3.9) we have that
By (3.9) and (3.10) it gets that
Hence from (3.10) and (3.11) we have that
Again since is uniformly -Lipschitzian, from (3.10) and (3.12) we have that
Equation (3.4) is proved.** (III) Now we prove that
and consists exactly of one point.

In fact, let , then there exists a subsequence of such that . By Lemmas 2.5 and 2.3, there exists a subsequence of such that . By Lemma 2.9, we have . By Lemma 2.12, . This shows that .

Let be a subsequence of with and let . Since and converges, by Lemma 2.12, we have . This shows that consists of exactly one point.** (IV) Finally we prove -converges to a point of .

In fact, it follows from (3.2) that is convergent for each . By (3.4) . By (3.14) and consists of exactly one point. This shows that -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

- 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 Google Scholar · View at Zentralblatt MATH - 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 Google Scholar · View at Zentralblatt MATH - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - S. Dhompongsa and B. Panyanak, “On $\mathrm{\Delta}$-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 - 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 - 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 Google Scholar · View at Zentralblatt MATH - 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 - 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