Abstract and Applied Analysis

Volume 2010 (2010), Article ID 582475, 11 pages

http://dx.doi.org/10.1155/2010/582475

## Convergence of a Sequence of Sets in a Hadamard Space and the Shrinking Projection Method for a Real Hilbert Ball

Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo 152-8552, Japan

Received 27 August 2010; Accepted 18 December 2010

Academic Editor: Mitsuharu Ôtani

Copyright © 2010 Yasunori Kimura. 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

We propose a new concept of set convergence in a Hadamard space and obtain its equivalent condition by using the notion of metric projections. Applying this result, we also prove a convergence theorem for an iterative scheme by the shrinking projection method in a real Hilbert ball.

#### 1. Introduction

A Hadamard space is defined as a complete geodesic metric space satisfying the CAT(0) inequality for each pair of points in every triangle. Since this concept includes various important spaces, it has been widely studied by a large number of researchers. In 2004, Kirk [1] proved a fixed point theorem for a nonexpansive mapping defined on a subset of a Hadamard space, and, since then, the study of approximation theory for fixed points of nonlinear mappings has been rapidly developed. See [2–4] and references therein. Kirk and Panyanak [5] proposed a concept of convergence called -convergence, which was originally introduced by Lim [6]. This notion corresponds to usual weak convergence in Banach spaces, and they share many useful properties.

On the other hand, the notion of set convergence for a reflexive Banach space has also been investigated by many researchers. In this paper, we will focus on the Mosco convergence. The relationship between convergence of a sequence of closed convex sets and the corresponding sequence of projections plays an important role in this field [7–10]. In recent research, this concept is applied to convergence of an approximating scheme, which is called the shrinking projection method in Hilbert and Banach spaces; see [11, 12].

Motivated by these results, we propose a new concept of set convergence for a sequence of subsets in a Hadamard space, which follows the notion of Mosco convergence in a Banach space. We adopt -convergence for weak convergence in a Hadamard space. In the main result, we obtain an equivalent condition for this convergence by using the notion of metric projections. In the final section, applying our main result, we prove a convergence theorem for an iterative scheme by the shrinking projection method in a real Hilbert ball.

#### 2. Preliminaries

Let be a metric space with a metric . For a subset of , the closure of is denoted by . For , a mapping , where , is called a geodesic with endpoints if , , and for . If, for every , a geodesic with endpoints exists, then we call a geodesic metric space. Furthermore, if a geodesic is unique for each , then is said to be uniquely geodesic. To introduce some notations, we do not need to assume the uniqueness of geodesics. However, since CAT(0) spaces, which we mainly use in this paper, are always uniquely geodesic, we will assume that is uniquely geodesic in what follows.

Let be a uniquely geodesic metric space. For , the image of a geodesic with endpoints is called a geodesic segment joining and and is denoted by . A geodesic triangle with vertices is a union of geodesic segments , , and , and we denote it by . A comparison triangle in for is a triangle in the 2-dimensional Euclidean space with vertices such that , , and , where is the Euclidean norm on . A point is called a comparison point for if . If, for any and their comparison points , the inequality holds for all triangles in , then we call a CAT(0) space. This inequality is called the CAT(0) inequality. Hadamard spaces are defined as complete CAT(0) spaces.

The CAT(0) space has been investigated in various fields in mathematics, and a great deal of results have been obtained. For more details, see [13].

For and , there exists a unique point such that and . We denote it by . From the CAT(0) inequality, it is easy to see that for every and .

A subset of is said to be convex if, for every , a geodesic segment is included in . For a subset of , a convex hull of is defined as an intersection of all convex sets including , and we denote it by .

Let be a subset of . A mapping is said to be nonexpansive if holds for every . The set of all fixed points of is denoted by ; that is, . We know that is closed and convex if is nonexpansive. The following fixed point theorem for nonexpansive mappings on Hadamard spaces plays an important role in our results.

Theorem 2.1 (Kirk [1]). *Let be a bounded open subset of a Hadamard space and a nonexpansive mapping. Suppose that there exists such that every in the boundary of does not belong to . Then, has a fixed point in .*

Let be a nonempty closed convex subset of a Hadamard space . Then, for each , there exists a unique point such that . The mapping is called a metric projection onto and is denoted by . We know that is nonexpansive; see [13, pages 176-177].

Let be a bounded sequence in a metric space . For , let The asymptotic center of is a set of points satisfying that . It is known that the asymptotic center of consists of one point for every bounded sequence in a Hadamard space; see [3]. The following property of asymptotic centers is important for our results.

Theorem 2.2 (Dhompongsa et al. [3]). *Let be a closed convex subset of a Hadamard space and a bounded sequence in . Then, the asymptotic center of is included in .*

The notion of -convergence was firstly introduced by Lim [6] in a general metric space setting. Following [5], we apply it to Hadamard spaces. Let be a sequence in . We say that is -convergent to if is the unique asymptotic center of any subsequence of . We know that every bounded sequence in a Hadamard space has a -convergent subsequence; see [5, 14].

#### 3. Convergence of a Sequence of Sets

Let be a sequence of closed convex subsets of a Hadamard space . As an analogy of Mosco convergence in Banach spaces [15], we introduce a new concept of set convergence. First let us define subsets and of as follows: if and only if there exists such that converges to 0 and that for all . On the other hand, if and only if there exist a sequence and a subsequence of such that has an asymptotic center and that for all . If a subset of satisfies that , it is said that converges to in the sense of -Mosco, and we write . Since the inclusion is always true, to obtain is a limit of in the sense of -Mosco, it suffices to show that .

It is easy to show that, if every is convex, then so is . Moreover, we know that is always closed. Therefore, is closed and convex whenever is a sequence of closed convex subsets of .

The following lemma is essentially obtained in [5] as the Kadec-Klee property in CAT(0) spaces. We modify it to a suitable form for our purpose. For the sake of completeness, we give the proof.

Lemma 3.1. * Let be a Hadamard space and a sequence in . Suppose that is -convergent to and converges to for some . Then, converges to .*

*Proof. *Let be comparison triangles in for with an identical geodesic segment . Then, we have that , , and for all . We know that is bounded in . Let be an arbitrary subsequence of converging to . Then, by assumption, we have that
Let be a metric projection of onto a closed convex set . Since is continuous, we have that converges to . Let be a point corresponding to . Using the CAT(0) inequality, we have that
and hence . By the uniqueness of the asymptotic center of , we obtain that , and thus . Since
for every , it follows that
and thus . Tending , we obtain that . Since any convergent subsequence of a bounded sequence in has a limit , we have that converges to . Thus we have that as , and hence converges to .

Now we state the main theorem of this section. Using a sequence of metric projections corresponding to a sequence of closed convex subsets, we give a characterization of -Mosco convergence in a Hadamard space.

Theorem 3.2. * Let be a Hadamard space and a nonempty closed convex subset of . Then, for a sequence of nonempty closed convex subsets in , the following are equivalent: *(i)* converges to in the sense of -Mosco; *(ii)* converges to for every . *

*Proof. *We first show that (i) implies (ii). Fix , and let for . Since , there exists such that for all and that converges to . By the definition of metric projection, we have that for . Thus, tending , we have that
It also follows that is bounded. Let be an arbitrary subsequence of and an asymptotic center of . Then, for fixed , it holds that
for sufficiently large . Since the closed ball with the center and the radius is convex, by Theorem 2.2, we have that , and hence
On the other hand, since , we have that , and therefore we have that , which implies that . Since all subsequences of have the same asymptotic center , is -convergent to .

Let us show that . If it were not true, then there exists a subsequence of satisfying that . Let be an asymptotic center of . For , we have that for sufficiently large , where . Since the closed ball with the center and the radius is convex, we have , and hence . Since , we get that
a contradiction. Therefore, we obtain that
and thus converges to . Using Lemma 3.1, we have that converges to . Hence (ii) holds.

Next we suppose (ii) and show that (i) holds. By assumption, for , a sequence converges to . Since for all , we have that , and hence . Let . Then, there exist and such that for all and is an asymptotic center of . Since each is convex, from the definition of metric projection, it follows that
for and . Then, we have that
and thus
Tending , we get that
for every , and since converges to as , we have that
Since is an asymptotic center of , we have that
It follows that
and therefore , which implies that . Consequently we have that converges to in the sense of -Mosco, and hence (ii) holds.

Using the result in [8], we obtain the following characterization in a Hilbert space.

Theorem 3.3. * Let be a Hilbert space and a nonempty closed convex subset of . Let be a sequence of nonempty closed convex subsets in . Then, converges to in the sense of Mosco if and only if converges to in the sense of -Mosco.*

*Proof. *By [8, Theorems 4.1 and 4.2], converges to in the sense of Mosco if and only if converges strongly to for all . Therefore, using Theorem 3.2, we obtain the desired result.

This result shows that Mosco convergence in Hilbert spaces is an example of -Mosco convergence. Let us see other simple examples.

*Example 3.4. *Let be a sequence of nonempty closed convex subsets of a Hadamard space . Then, as a direct consequence of the definition, we obtain that
In particular, if is a decreasing sequence with respect to inclusion, then is -Mosco convergent to . Likewise, if is increasing, then the limit is .

*Example 3.5. *Let be a sequence of nonempty bounded closed convex subsets of a Hadamard space . If converges to a bounded closed convex subset with respect to the Hausdorff metric, then also converges to in the sense of -Mosco. The Hausdorff metric between nonempty bounded closed subsets of is defined by
where and for .

Let us prove this fact. For , we have that and since as , there exists a sequence converging to such that for all . It follows that , and hence .

To show , let . Then, there exists a subsequence of and a sequence whose asymptotic center is and for all . Let be arbitrary. Then, since as , there exists such that for every .

Let . Then, is closed and convex in . Indeed, for and , there exist such that and . Considering the comparison triangle of and using the CAT(0) inequality, we have that

In the same way, considering the comparison triangle of , we have that
Thus, we have that . Since is convex, we have that , and hence . This shows that is convex. It is obvious that is closed since the function is continuous.

Since for , using Theorem 2.2, we have that ; that is, . Since is arbitrary and is closed, we obtain that , and hence . Consequently we have that converges to in the sense of -Mosco.

#### 4. Shrinking Projection Method in a Real Hilbert Ball

As an example of Hadamard spaces, let us deal with a real Hilbert ball in this section. Let be the open unit ball of a complex Hilbert space with an inner product and induced norm . For an orthonormal basis of , let . Then, a real Hilbert ball is a metric space defined by and by for . It is known that a real Hilbert ball is an example of Hadamard spaces. One of the most important properties for our results in this section is that a half space is convex for any ; see [16, 17].

Theorem 4.1. *Let be a real Hilbert ball with the metric . Let be a family of nonexpansive mappings of into itself with a nonempty set of their common fixed points. Let be nonnegative real numbers in such that for each . For , generate an iterative sequence by , , and
**
for all . Then, is well defined and converges to .*

*Proof. *Since each is closed and convex, so is . For , it follows that
for all , and thus for every . Therefore, we have and is nonempty for . Further, is closed and convex by the property of a real Hilbert ball . Hence, the metric projection exists, and is well defined for all . Since is decreasing with respect to inclusion, as in Example 3.4, we have that converges to in the sense of -Mosco. By Theorem 3.2, we have that converges to . Since for all , we have that for all and . Fix arbitrarily, and let be a subsequence of converging to . Then, since , we have that
for , and, as , we obtain that ; that is, . Since is arbitrary, we have that , and therefore , which is the desired result.

Next, we consider the case of a single mapping. Motivated by [18], we obtain the following theorem. It shows that, without assuming the existence of fixed points, we may prove that the iterative sequence is well defined. Moreover, the boundedness of the sequence guarantees that the set of fixed points is nonempty.

Theorem 4.2. *Let be a real Hilbert ball and a nonexpansive mapping. Let be a nonnegative real sequence in such that . For , generate an iterative sequence by , , and
**
for all . Then, is well defined and the following are equivalent: *(i)* is nonempty;*(ii)* is convergent; *(iii)* is bounded; *(iv)* is nonempty. ** Moreover, in this case the limit of is .*

*Proof. *First we show that is well defined. Since is given and , is nonempty. Suppose that are nonempty. Then, and are defined. Let and . Then, since is nonempty, bounded, closed, and convex, there exists a metric projection . Since is nonexpansive, it follows that is also a nonexpansive mapping of into itself. Moreover, has a nonempty interior, and does not intersect the boundary of for every . Thus, by Theorem 2.1, there exists such that . Since for and is convex, it follows from the definition of the metric projection that
for . Thus, we have that
and thus
Tending , we have that , and hence for . It gives us that
for all , and hence . This shows that is nonempty and obviously it is closed and convex. Therefore, is defined. By induction, we obtain that is well defined.

Next, we show that (i)–(iv) are equivalent. We know from Theorem 4.1 for a single mapping that (i) implies (ii). We also have that converges to . It is trivial that (ii) implies (iii). Let us suppose that (iii) holds and show (iv). Since is bounded, there exists a subsequence which is -convergent to some . From the definition of subsequence, for any , there exists such that for all . Since is decreasing with respect to inclusion, we have for all . By Theorem 2.2, we have that for every , and hence (iv) holds. Lastly, we show that (iv) implies (i). Assume that is nonempty. By Theorem 3.2, converges to . Then, as in the proof of Theorem 4.1, we have that , and thus (i) holds. Consequently, these four conditions are all equivalent.

#### Acknowledgment

The author is supported by Grant-in-Aid for Scientific Research no. 22540175 from Japan Society for the Promotion of Science.

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