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Abstract and Applied Analysis
Volume 2010, Article ID 582475, 11 pages
http://dx.doi.org/10.1155/2010/582475
Research Article

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 [24] and references therein. Kirk and Panyanak [5] proposed a concept of convergence called Δ-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 [710]. 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 Δ-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 𝑋 be a metric space with a metric 𝑑. For a subset 𝐴 of 𝑋, the closure of 𝐴 is denoted by cl𝐴. For 𝑥,𝑦𝑋, a mapping 𝑐[0,𝑙]𝑋, where 𝑙0, is called a geodesic with endpoints 𝑥,𝑦 if 𝑐(0)=𝑥, 𝑐(𝑙)=𝑦, and 𝑑(𝑐(𝑡),𝑐(𝑠))=|𝑡𝑠| for 𝑡,𝑠[0,𝑙]. If, for every 𝑥,𝑦𝑋, a geodesic with endpoints 𝑥,𝑦 exists, then we call 𝑋 a geodesic metric space. Furthermore, if a geodesic is unique for each 𝑥,𝑦𝑋, then 𝑋 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 𝑋 is uniquely geodesic in what follows.

Let 𝑋 be a uniquely geodesic metric space. For 𝑥,𝑦𝑋, the image of a geodesic 𝑐 with endpoints 𝑥,𝑦 is called a geodesic segment joining 𝑥 and 𝑦 and is denoted by [𝑥,𝑦]. A geodesic triangle with vertices 𝑥,𝑦,𝑧𝑋 is a union of geodesic segments [𝑥,𝑦], [𝑦,𝑧], and [𝑧,𝑥], and we denote it by Δ(𝑥,𝑦,𝑧). A comparison triangle Δ(𝑥,𝑦,𝑧) in 𝔼2 for Δ(𝑥,𝑦,𝑧) is a triangle in the 2-dimensional Euclidean space 𝔼2 with vertices 𝑥,𝑦,𝑧𝔼2 such that 𝑑(𝑥,𝑦)=|𝑥𝑦|𝔼2, 𝑑(𝑦,𝑧)=|𝑦𝑧|𝔼2, and 𝑑(𝑧,𝑥)=|𝑧𝑥|𝔼2, where ||𝔼2 is the Euclidean norm on 𝔼2. A point 𝑝[𝑥,𝑦] is called a comparison point for 𝑝[𝑥,𝑦] if 𝑑(𝑥,𝑝)=|𝑥𝑝|𝔼2. If, for any 𝑝,𝑞Δ(𝑥,𝑦,𝑧) and their comparison points 𝑝,𝑞Δ(𝑥,𝑦,𝑧), the inequality||𝑑(𝑝,𝑞)𝑝𝑞||𝔼2(2.1) holds for all triangles in 𝑋, then we call 𝑋 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 𝑥,𝑦𝑋 and 𝑡[0,1], there exists a unique point 𝑧[𝑥,𝑦] such that 𝑑(𝑥,𝑧)=(1𝑡)𝑑(𝑥,𝑦) and 𝑑(𝑧,𝑦)=𝑡𝑑(𝑥,𝑦). We denote it by 𝑡𝑥(1𝑡)𝑦. From the CAT(0) inequality, it is easy to see that𝑑(𝑧,𝑡𝑥(1𝑡)𝑦)2𝑡𝑑(𝑧,𝑥)2+(1𝑡)𝑑(𝑧,𝑦)2𝑡(1𝑡)𝑑(𝑥,𝑦)2(2.2) for every 𝑥,𝑦,𝑧𝑋 and 𝑡[0,1].

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

Let 𝑌 be a subset of 𝑋. A mapping 𝑆𝑌𝑋 is said to be nonexpansive if 𝑑(𝑆𝑥,𝑆𝑦)𝑑(𝑥,𝑦) holds for every 𝑥,𝑦𝑌. The set of all fixed points of 𝑆 is denoted by 𝐹(𝑆); that is, 𝐹(𝑆)={𝑧𝑌𝑆𝑧=𝑧}. We know that 𝐹(𝑆) is closed and convex if 𝑆 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 𝑈 be a bounded open subset of a Hadamard space 𝑋 and 𝑆cl𝑈𝑋 a nonexpansive mapping. Suppose that there exists 𝑝𝑈 such that every 𝑥 in the boundary of 𝑈 does not belong to [𝑝,𝑆𝑥]{𝑆𝑥}. Then, 𝑆 has a fixed point in cl𝑈.

Let 𝐶 be a nonempty closed convex subset of a Hadamard space 𝑋. Then, for each 𝑥𝑋, there exists a unique point 𝑦𝑥𝐶 such that 𝑑(𝑥,𝑦𝑥)=inf𝑦𝐶𝑑(𝑥,𝑦). The mapping 𝑥𝑦𝑥 is called a metric projection onto 𝐶 and is denoted by 𝑃𝐶. We know that 𝑃𝐶 is nonexpansive; see [13, pages 176-177].

Let {𝑥𝑛} be a bounded sequence in a metric space 𝑋. For 𝑥𝑋, let𝑟𝑥𝑥,𝑛=limsup𝑛𝑑𝑥,𝑥𝑛,𝑟𝑥𝑛=inf𝑥𝑋𝑟𝑥𝑥,𝑛.(2.3) The asymptotic center of {𝑥𝑛} is a set of points 𝑥𝑋 satisfying that 𝑟(𝑥,{𝑥𝑛})=𝑟({𝑥𝑛}). It is known that the asymptotic center of {𝑥𝑛} consists of one point for every bounded sequence {𝑥𝑛} 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 𝐶 be a closed convex subset of a Hadamard space 𝑋 and {𝑥𝑛} a bounded sequence in 𝐶. Then, the asymptotic center of {𝑥𝑛} is included in 𝐶.

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

3. Convergence of a Sequence of Sets

Let {𝐶𝑛} be a sequence of closed convex subsets of a Hadamard space 𝑋. As an analogy of Mosco convergence in Banach spaces [15], we introduce a new concept of set convergence. First let us define subsets 𝑑-Li𝑛𝐶𝑛 and Δ-Ls𝑛𝐶𝑛 of 𝑋 as follows: 𝑥𝑑-Li𝑛𝐶𝑛 if and only if there exists {𝑥𝑛}𝑋 such that {𝑑(𝑥𝑛,𝑥)} converges to 0 and that 𝑥𝑛𝐶𝑛 for all 𝑛. On the other hand, 𝑦Δ-Ls𝑛𝐶𝑛 if and only if there exist a sequence {𝑦𝑖}𝑋 and a subsequence {𝑛𝑖} of such that {𝑦𝑖} has an asymptotic center {𝑦} and that 𝑦𝑖𝐶𝑛𝑖 for all 𝑖. If a subset 𝐶0 of 𝑋 satisfies that 𝐶0=𝑑-Li𝑛𝐶𝑛=Δ-Ls𝑛𝐶𝑛, it is said that {𝐶𝑛} converges to 𝐶0 in the sense of Δ-Mosco, and we write 𝐶0=ΔM-lim𝑛𝐶𝑛. Since the inclusion 𝑑-Li𝑛𝐶𝑛Δ-Ls𝑛𝐶𝑛 is always true, to obtain 𝐶0 is a limit of {𝐶𝑛} in the sense of Δ-Mosco, it suffices to show that Δ-Ls𝑛𝐶𝑛𝐶0𝑑-Li𝑛𝐶𝑛.

It is easy to show that, if every 𝐶𝑛 is convex, then so is 𝑑-Li𝑛𝐶𝑛. Moreover, we know that 𝑑-Li𝑛𝐶𝑛 is always closed. Therefore, ΔM-lim𝑛𝐶𝑛 is closed and convex whenever {𝐶𝑛} is a sequence of closed convex subsets of 𝑋.

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 𝑋 be a Hadamard space and {𝑥𝑛} a sequence in 𝑋. Suppose that {𝑥𝑛} is Δ-convergent to 𝑥𝑋 and {𝑑(𝑥𝑛,𝑝)} converges to 𝑑(𝑥,𝑝) for some 𝑝𝑋. Then, {𝑥𝑛} converges to 𝑥.

Proof. Let {Δ(𝑥,𝑝,𝑥𝑛)} be comparison triangles in 𝔼2 for 𝑛 with an identical geodesic segment [𝑝,𝑥]. Then, we have that |𝑥𝑝|𝔼2=𝑑(𝑥,𝑝), |𝑥𝑛𝑝|𝔼2=𝑑(𝑥𝑛,𝑝), and |𝑥𝑛𝑥|𝔼2=𝑑(𝑥𝑛,𝑥) for all 𝑛. We know that {𝑥𝑛} is bounded in 𝔼2. Let {𝑥𝑛𝑖} be an arbitrary subsequence of {𝑥𝑛} converging to 𝑦𝔼2. Then, by assumption, we have that ||𝑦𝑝||𝔼2=lim𝑖||𝑥𝑛𝑖𝑝||𝔼2=lim𝑖𝑑𝑥𝑛𝑖||,𝑝=𝑑(𝑥,𝑝)=𝑥𝑝||𝔼2.(3.1) Let 𝑃=𝑃[𝑝,𝑥] be a metric projection of 𝔼2 onto a closed convex set [𝑝,𝑥]. Since 𝑃 is continuous, we have that {𝑃𝑥𝑛𝑖} converges to 𝑃𝑦𝔼2. Let 𝑧[𝑝,𝑥]𝑋 be a point corresponding to 𝑧=𝑃𝑦[𝑝,𝑥]𝔼2. Using the CAT(0) inequality, we have that 𝑟𝑥𝑛𝑖=limsup𝑖𝑑𝑥,𝑥𝑛𝑖=limsup𝑖||𝑥𝑥𝑛𝑖||𝔼2limsup𝑖||𝑃𝑥𝑛𝑖𝑥𝑛𝑖||𝔼2=limsup𝑖||𝑧𝑥𝑛𝑖||𝔼2limsup𝑖𝑑𝑧,𝑥𝑛𝑖,(3.2) and hence 𝑟(𝑧,{𝑥𝑛𝑖})𝑟({𝑥𝑛𝑖}). By the uniqueness of the asymptotic center of {𝑥𝑛𝑖}, we obtain that 𝑧=𝑥, and thus 𝑧=𝑥. Since ||𝑥𝑦||𝔼2=||𝑧𝑦||𝔼2=||𝑃𝑦𝑦||𝔼2||(1𝑡)𝑥+𝑡𝑝𝑦||𝔼2(3.3) for every 𝑡]0,1[, it follows that ||𝑥𝑦||2𝔼2||(1𝑡)𝑥+𝑡𝑝𝑦||2𝔼2||=(1𝑡)𝑥𝑦||2𝔼2||+𝑡𝑝𝑦||2𝔼2||𝑡(1𝑡)𝑥𝑝||2𝔼2||=(1𝑡)𝑥𝑦||2𝔼2+𝑡2||𝑝𝑥||2𝔼2,(3.4) and thus |𝑥𝑦|2𝔼2𝑡|𝑝𝑥|2𝔼2. Tending 𝑡0, we obtain that 𝑥=𝑦. Since any convergent subsequence {𝑥𝑛𝑖} of a bounded sequence {𝑥𝑛} in 𝔼2 has a limit 𝑥, we have that {𝑥𝑛} converges to 𝑥. Thus we have that 𝑑(𝑥𝑛,𝑥)=|𝑥𝑛𝑥|𝔼20 as 𝑛, and hence {𝑥𝑛} converges to 𝑥𝑋.

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 Δ-Mosco convergence in a Hadamard space.

Theorem 3.2. Let 𝑋 be a Hadamard space and 𝐶0 a nonempty closed convex subset of 𝑋. Then, for a sequence {𝐶𝑛} of nonempty closed convex subsets in 𝑋, the following are equivalent: (i){𝐶𝑛} converges to 𝐶0 in the sense of Δ-Mosco; (ii){𝑃𝐶𝑛𝑥} converges to 𝑃𝐶0𝑥𝑋 for every 𝑥𝑋.

Proof. We first show that (i) implies (ii). Fix 𝑥𝑋, and let 𝑝𝑛=𝑃𝐶𝑛𝑥 for 𝑛. Since 𝑃𝐶0𝑥𝐶0=𝑑-Li𝑛𝐶𝑛, there exists {𝑦𝑛}𝑋 such that 𝑦𝑛𝐶𝑛 for all 𝑛 and that {𝑦𝑛} converges to 𝑃𝐶0𝑥. By the definition of metric projection, we have that 𝑑(𝑥,𝑝𝑛)𝑑(𝑥,𝑦𝑛) for 𝑛. Thus, tending 𝑛, we have that limsup𝑛𝑑𝑥,𝑝𝑛lim𝑛𝑑𝑥,𝑦𝑛=𝑑𝑥,𝑃𝐶0𝑥.(3.5) It also follows that {𝑝𝑛} is bounded. Let {𝑝𝑛𝑖} be an arbitrary subsequence of {𝑝𝑛} and 𝑝0 an asymptotic center of {𝑝𝑛𝑖}. Then, for fixed 𝜖>0, it holds that 𝑑𝑥,𝑝𝑛i𝑑𝑥,𝑃𝐶0𝑥+𝜖(3.6) for sufficiently large 𝑖. Since the closed ball with the center 𝑥 and the radius 𝑑(𝑥,𝑃𝐶0𝑥)+𝜖 is convex, by Theorem 2.2, we have that 𝑑(𝑥,𝑝0)𝑑(𝑥,𝑃𝐶0𝑥)+𝜖, and hence 𝑑𝑥,𝑝0𝑑𝑥,𝑃𝐶0𝑥.(3.7) On the other hand, since 𝑝0Δ-Ls𝑛𝐶𝑛=𝐶0, we have that 𝑑(𝑥,𝑃𝐶0𝑥)𝑑(𝑥,𝑝0), and therefore we have that 𝑑(𝑥,𝑃𝐶0𝑥)=𝑑(𝑥,𝑝0), which implies that 𝑝0=𝑃𝐶0𝑥. Since all subsequences of {𝑝𝑛} have the same asymptotic center 𝑃𝐶0𝑥, {𝑝𝑛} is Δ-convergent to 𝑃𝐶0𝑥.
Let us show that liminf𝑛𝑑(𝑥,𝑝𝑛)𝑑(𝑥,𝑃𝐶0𝑥). If it were not true, then there exists a subsequence {𝑝𝑛𝑖} of {𝑝𝑛} satisfying that liminf𝑛𝑑(𝑥,𝑝𝑛)=lim𝑖𝑑(𝑥,𝑝𝑛𝑖)<𝑑(𝑥,𝑃𝐶0𝑥). Let 𝑝𝑋 be an asymptotic center of {𝑝𝑛𝑖}. For 𝜖>0, we have that 𝑑(𝑥,𝑝𝑛𝑖)𝛿+𝜖 for sufficiently large 𝑖, where 𝛿=lim𝑖𝑑(𝑥,𝑝𝑛𝑖). Since the closed ball with the center 𝑥 and the radius 𝛿+𝜖 is convex, we have 𝑑(𝑥,𝑝)𝛿+𝜖, and hence 𝑑(𝑥,𝑝)𝛿=lim𝑖𝑑(𝑥,𝑝𝑛𝑖). Since 𝑝Δ-Ls𝑛𝐶𝑛=C0, we get that 𝑑𝑥,𝑃𝐶0𝑥>lim𝑖𝑑𝑥,𝑝𝑛𝑖𝑑(𝑥,𝑝)𝑑𝑥,𝑃𝐶0𝑥,(3.8) a contradiction. Therefore, we obtain that𝑑𝑥,𝑃𝐶0𝑥liminf𝑛𝑑𝑥,𝑝𝑛limsup𝑛𝑑𝑥,𝑝𝑛𝑑𝑥,𝑃𝐶0𝑥,(3.9) and thus {𝑑(𝑥,𝑝𝑛)} converges to 𝑑(𝑥,𝑃𝐶0𝑥). Using Lemma 3.1, we have that {𝑝𝑛} converges to 𝑃𝐶0𝑥. Hence (ii) holds.
Next we suppose (ii) and show that (i) holds. By assumption, for 𝑦𝐶0, a sequence {𝑃𝐶𝑛𝑦} converges to 𝑃𝐶0𝑦=𝑦. Since 𝑃𝐶𝑛𝑦𝐶𝑛 for all 𝑛, we have that 𝑦𝑑-Li𝑛𝐶𝑛, and hence 𝐶0𝑑-Li𝑛𝐶𝑛. Let 𝑧Δ-Ls𝑛𝐶𝑛. Then, there exist {𝑧𝑖}𝑋 and {𝑛𝑖} such that 𝑧𝑖𝐶𝑛𝑖 for all 𝑖 and 𝑧 is an asymptotic center of {𝑧𝑖}. Since each 𝐶𝑛𝑖 is convex, from the definition of metric projection, it follows that 𝑑𝑧,𝑃𝐶𝑛𝑖𝑧𝑑𝑧,(1𝑡)𝑃𝐶𝑛𝑖𝑧𝑡𝑧𝑖(3.10) for 𝑡]0,1[ and 𝑖. Then, we have that𝑑𝑧,𝑃𝐶𝑛𝑖𝑧2𝑑𝑧,(1𝑡)𝑃𝐶𝑛𝑖𝑧𝑡𝑧𝑖2(1𝑡)𝑑𝑧,𝑃𝐶𝑛𝑖𝑧2+𝑡𝑑𝑧,𝑧𝑖2𝑃𝑡(1𝑡)𝑑𝐶𝑛𝑖𝑧,𝑧𝑖2,(3.11) and thus 𝑑𝑧,𝑃𝐶𝑛𝑖𝑧2𝑃+(1𝑡)𝑑𝐶𝑛𝑖𝑧,𝑧𝑖2𝑑𝑧,𝑧𝑖2.(3.12) Tending 𝑡0, we get that 𝑑𝑧,𝑃𝐶𝑛𝑖𝑧2𝑃+𝑑𝐶𝑛𝑖𝑧,𝑧𝑖2𝑑𝑧,𝑧𝑖2(3.13) for every 𝑖, and since {𝑃𝐶𝑛𝑖𝑧} converges to 𝑃𝐶0𝑧 as 𝑖, we have that 𝑑𝑧,𝑃𝐶0𝑧2+limsup𝑖𝑑𝑃C0𝑧,𝑧𝑖2limsup𝑖𝑑𝑧,𝑧𝑖2.(3.14) Since 𝑧 is an asymptotic center of {𝑧𝑖}, we have that limsup𝑖𝑑𝑧,𝑧𝑖𝑧=𝑟𝑧,𝑖𝑧=𝑟𝑖𝑃𝑟𝐶0𝑧𝑧,𝑖=limsup𝑖𝑑𝑃𝐶0𝑧,𝑧𝑖.(3.15) It follows that 𝑑𝑧,𝑃𝐶0𝑧2limsup𝑖𝑑𝑧,𝑧𝑖2limsup𝑖𝑑𝑃𝐶0𝑧,𝑧𝑖20,(3.16) and therefore 𝑧=𝑃𝐶0𝑧𝐶0, which implies that Δ-Ls𝑛𝐶𝑛𝐶0. Consequently we have that {𝐶𝑛} converges to 𝐶0 in the sense of Δ-Mosco, and hence (ii) holds.

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

Theorem 3.3. Let 𝐻 be a Hilbert space and 𝐶0 a nonempty closed convex subset of 𝐻. Let {𝐶𝑛} be a sequence of nonempty closed convex subsets in 𝐻. Then, {𝐶𝑛} converges to 𝐶0 in the sense of Mosco if and only if {𝐶𝑛} converges to 𝐶0 in the sense of Δ-Mosco.

Proof. By [8, Theorems 4.1 and 4.2], {𝐶𝑛} converges to 𝐶0 in the sense of Mosco if and only if {𝑃𝐶𝑛𝑥} converges strongly to 𝑃𝐶0𝑥 for all 𝑥𝐻. Therefore, using Theorem 3.2, we obtain the desired result.

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

Example 3.4. Let {𝐶𝑛} be a sequence of nonempty closed convex subsets of a Hadamard space 𝑋. Then, as a direct consequence of the definition, we obtain that cl𝑚=1𝑛=𝑚𝐶𝑛𝑑-Li𝑛𝐶𝑛Δ-Ls𝑛𝐶𝑛𝑚=1clco𝑛=𝑚𝐶𝑛.(3.17) In particular, if {𝐶𝑛} is a decreasing sequence with respect to inclusion, then {𝐶𝑛} is Δ-Mosco convergent to 𝑛=1𝐶𝑛. Likewise, if {𝐶𝑛} is increasing, then the limit is cl𝑛=1𝐶𝑛.

Example 3.5. Let {𝐶𝑛} be a sequence of nonempty bounded closed convex subsets of a Hadamard space 𝑋. If {𝐶𝑛} converges to a bounded closed convex subset 𝐶0𝑋 with respect to the Hausdorff metric, then {𝐶𝑛} also converges to 𝐶0 in the sense of Δ-Mosco. The Hausdorff metric between nonempty bounded closed subsets 𝐴,𝐵 of 𝑋 is defined by (𝐴,𝐵)=max{𝑒(𝐴,𝐵),𝑒(𝐵,𝐴)},(3.18) where 𝑒(𝐴,𝐵)=sup𝑥𝐴𝑑(𝑥,𝐵) and 𝑑(𝑥,𝐵)=inf𝑦𝐵𝑑(𝑥,𝑦) for 𝑥𝑋.
Let us prove this fact. For 𝑥𝐶0, we have that 𝑑(𝑥,𝐶𝑛)𝑒(𝐶0,𝐶𝑛)(𝐶0,𝐶𝑛) and since (𝐶𝑛,𝐶0)0 as 𝑛, there exists a sequence {𝑥𝑛}𝑋 converging to 𝑥 such that 𝑥𝑛𝐶𝑛 for all 𝑛. It follows that 𝑥𝑑-Li𝑛𝐶𝑛, and hence 𝐶0𝑑-Li𝑛𝐶𝑛.
To show Δ-Ls𝑛𝐶𝑛𝐶0, let 𝑥Δ-Ls𝑛𝐶𝑛. Then, there exists a subsequence {𝑛𝑖} of and a sequence {𝑥𝑖}𝑋 whose asymptotic center is 𝑥 and 𝑥𝑖𝐶𝑛𝑖 for all 𝑖. Let 𝜖>0 be arbitrary. Then, since 𝑑(𝑥𝑖,𝐶0)𝑒(𝐶𝑛𝑖,𝐶0)(𝐶𝑛𝑖,𝐶0)0 as 𝑖, there exists 𝑖0 such that 𝑑(𝑥𝑖,𝐶0)<𝜖 for every 𝑖𝑖0.
Let 𝐷𝜖={𝑦𝑋𝑑(𝑦,𝐶0)𝜖}. Then, 𝐷𝜖 is closed and convex in 𝑋. Indeed, for 𝑦1,𝑦2𝐷𝜖 and 𝑡]0,1[, there exist 𝑧1,𝑧2𝐶0 such that 𝑑(𝑦1,𝑧1)<𝜖 and 𝑑(𝑦2,𝑧2)<𝜖. Considering the comparison triangle of (𝑦1,𝑧1,𝑧2) and using the CAT(0) inequality, we have that 𝑑𝑡𝑦1(1𝑡)𝑧2,𝑡𝑧1(1𝑡)𝑧2𝑦𝑡𝑑1,𝑧1.(3.19)
In the same way, considering the comparison triangle of (𝑦1,𝑦2,𝑧2), we have that 𝑑𝑡𝑦1(1𝑡)𝑦2,𝑡𝑦1(1𝑡)𝑧2𝑦(1𝑡)𝑑2,𝑧2.(3.20) Thus, we have that 𝑑(𝑡𝑦1(1𝑡)𝑦2,𝑡𝑧1(1𝑡)𝑧2)(1𝑡)𝑑(𝑦2,𝑧2)+𝑡𝑑(𝑦1,𝑧1)𝜖. Since 𝐶0 is convex, we have that 𝑡𝑧1(1𝑡)𝑧2𝐶0, and hence 𝑡𝑦1(1𝑡)𝑦2𝐷𝜖. This shows that 𝐷𝜖 is convex. It is obvious that 𝐷𝜖 is closed since the function 𝑑(,𝐶0) is continuous.
Since 𝑥𝑖𝐷𝜖 for 𝑖𝑖0, using Theorem 2.2, we have that 𝑥𝐷𝜖; that is, 𝑑(𝑥,𝐶0)𝜖. Since 𝜖 is arbitrary and 𝐶0 is closed, we obtain that 𝑥𝐶0, and hence Δ-Ls𝑛𝐶𝑛𝐶0. Consequently we have that {𝐶𝑛} converges to 𝐶0 in the sense of Δ-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 𝐵𝐻 be the open unit ball of a complex Hilbert space 𝐻 with an inner product , and induced norm . For an orthonormal basis {𝑒𝑖𝑖𝐼} of 𝐻, let 𝐻𝑅={𝑧𝐻Im𝑧,𝑒𝑖=0𝑖𝐼}. Then, a real Hilbert ball (𝐵,𝜌) is a metric space defined by 𝐵=𝐵𝐻𝐻𝑅 and 𝜌𝐵×𝐵 by𝜌(𝑥,𝑦)=arctanh11𝑥21𝑦2||||1𝑥,𝑦2(4.1) for 𝑥,𝑦𝐵. 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 𝐶={𝑧𝐵𝜌(𝑧,𝑦)𝜌(𝑧,𝑥)} is convex for any 𝑥,𝑦𝐵; see [16, 17].

Theorem 4.1. Let 𝐵 be a real Hilbert ball with the metric 𝜌. Let {𝑇𝑖𝑖𝐼} be a family of nonexpansive mappings of 𝐵 into itself with a nonempty set 𝐹 of their common fixed points. Let {𝛼𝑛(𝑖)𝑖𝐼,𝑛} be nonnegative real numbers in [0,1] such that liminf𝑛𝛼𝑛(𝑖)<1 for each 𝑖𝐼. For 𝑥𝐵, generate an iterative sequence {𝑥𝑛} by 𝑥1=𝑥, 𝐶0=𝐵, and 𝑦𝑛(𝑖)=𝛼𝑛(𝑖)𝑥𝑛1𝛼𝑛𝑇(𝑖)𝑖𝑥𝑛𝐶,foreach𝑖I,𝑛=𝑧𝐵sup𝑖𝐼𝜌𝑧,𝑦𝑛(𝑖)𝜌𝑧,𝑥𝑛𝐶𝑛1,𝑥𝑛+1=𝑃𝐶𝑛𝑥(4.2) for all 𝑛. Then, {𝑥𝑛} is well defined and converges to 𝑃𝐹𝑥𝐵.

Proof. Since each 𝐹(𝑇𝑖) is closed and convex, so is 𝐹=𝑖𝐼𝐹(𝑇𝑖). For 𝑧𝐹, it follows that 𝜌𝑧,𝑦𝑛(𝑖)2𝛼𝑛(𝑖)𝜌𝑧,𝑥𝑛2+1𝛼𝑛𝜌(𝑖)𝑧,𝑇𝑖𝑥𝑛2𝛼𝑛(𝑖)1𝛼𝑛𝜌𝑥(𝑖)𝑛,𝑇𝑖𝑥𝑛2𝛼𝑛(𝑖)𝜌𝑧,𝑥𝑛2+1𝛼𝑛𝜌𝑇(𝑖)𝑖𝑧,𝑇𝑖𝑥𝑛2𝜌𝑧,𝑥𝑛2(4.3) for all 𝑖𝐼, and thus sup𝑖𝐼𝜌(𝑧,𝑦𝑛(𝑖))𝜌(𝑧,𝑥𝑛) for every 𝑛. Therefore, we have 𝐹𝐶𝑛 and 𝐶𝑛 is nonempty for 𝑛. Further, 𝐶𝑛 is closed and convex by the property of a real Hilbert ball 𝐵. Hence, the metric projection 𝑃𝐶𝑛 exists, and 𝑥𝑛 is well defined for all 𝑛. Since {𝐶𝑛} is decreasing with respect to inclusion, as in Example 3.4, we have that {𝐶𝑛} converges to 𝐶=𝑛=1𝐶𝑛 in the sense of Δ-Mosco. By Theorem 3.2, we have that {𝑥𝑛} converges to 𝑥0=𝑃𝐶𝑥. Since 𝑥0𝐶𝑛 for all 𝑛, we have that 𝜌(𝑥0,𝑦𝑛(𝑖))𝜌(𝑥0,𝑥𝑛) for all 𝑛 and 𝑖𝐼. Fix 𝑖𝐼 arbitrarily, and let {𝛼𝑛𝑘(𝑖)} be a subsequence of {𝛼𝑛(𝑖)} converging to 𝛼0(𝑖)[0,1[. Then, since 𝜌(𝑥𝑛,𝑦𝑛(𝑖))=(1𝛼𝑛(𝑖))𝜌(𝑥𝑛,𝑇𝑖𝑥𝑛), we have that 𝜌𝑥0,𝑇𝑖𝑥0𝑥𝜌0,𝑥𝑛𝑘𝑥+𝜌𝑛𝑘,𝑇𝑖𝑥𝑛𝑘𝑇+𝜌𝑖𝑥𝑛𝑘,𝑇𝑖𝑥0𝑥2𝜌0,𝑥𝑛𝑘+11𝛼𝑛𝑘(𝜌𝑥𝑖)𝑛𝑘,𝑦𝑛𝑘𝑥(𝑖)2𝜌0,𝑥𝑛𝑘+11𝛼𝑛𝑘𝜌𝑥(𝑖)𝑛𝑘,𝑥0𝑥+𝜌0,𝑦𝑛𝑘1(𝑖)21+1𝛼𝑛𝑘𝜌𝑥(𝑖)0,𝑥𝑛𝑘(4.4) for 𝑘, and, as 𝑘, we obtain that 𝑥0=𝑇𝑖𝑥0; that is, 𝑥0𝐹(𝑇𝑖). Since 𝑖𝐼 is arbitrary, we have that 𝑃𝐶𝑥=𝑥0𝐹𝐶, and therefore 𝑥0=𝑃𝐹𝑥, 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 𝐵 be a real Hilbert ball and 𝑇𝐵𝐵 a nonexpansive mapping. Let {𝛼𝑛} be a nonnegative real sequence in [0,1] such that liminf𝑛𝛼𝑛<1. For 𝑥𝐵, generate an iterative sequence {𝑥𝑛} by 𝑥1=𝑥, 𝐶0=𝐵, and 𝑦𝑛=𝛼𝑛𝑥𝑛1𝛼𝑛𝑇𝑥𝑛𝐶,foreach𝑖I,𝑛=𝑧𝐵𝜌𝑧,𝑦𝑛𝜌𝑧,𝑥𝑛𝐶𝑛1,𝑥𝑛+1=𝑃𝐶𝑛𝑥(4.5) for all 𝑛. Then, {𝑥𝑛} is well defined and the following are equivalent: (i)𝐹(𝑇) is nonempty;(ii){𝑥𝑛} is convergent; (iii){𝑥𝑛} is bounded; (iv)𝑛=1𝐶𝑛 is nonempty. Moreover, in this case the limit of {𝑥𝑛} is 𝑃𝐹(𝑇)𝑥=𝑃𝑛=1𝐶𝑛𝑥.

Proof. First we show that {𝑥𝑛} is well defined. Since 𝑥1 is given and 𝑦1𝐶1, 𝐶1 is nonempty. Suppose that 𝐶1,𝐶2,,𝐶𝑛1 are nonempty. Then, 𝑥1,𝑥2,,𝑥𝑛 and 𝑦1,𝑦2,,𝑦𝑛 are defined. Let 𝑟=max1𝑘𝑛𝜌(𝑥,𝑇𝑥𝑘) and 𝐷={𝑤𝐵𝜌(𝑥,𝑤)𝑟}. Then, since 𝐷 is nonempty, bounded, closed, and convex, there exists a metric projection 𝑃𝐷𝐵𝐷. Since 𝑃𝐷 is nonexpansive, it follows that 𝑃𝐷𝑇𝐷 is also a nonexpansive mapping of 𝐷 into itself. Moreover, 𝐷 has a nonempty interior, and [𝑥,𝑣]{𝑣} does not intersect the boundary of 𝐷 for every 𝑣𝐷. Thus, by Theorem 2.1, there exists 𝑢𝐷 such that 𝑢=𝑃𝐷𝑇𝑢. Since 𝑇𝑥𝑘𝐷 for 𝑘=1,2,,𝑛 and 𝐷 is convex, it follows from the definition of the metric projection that 𝜌(𝑇𝑢,𝑢)=𝜌𝑇𝑢,𝑃𝐷𝑇𝑢𝜌𝑇𝑢,(1𝑡)𝑢𝑡𝑇𝑥𝑘(4.6) for 𝑡]0,1[. Thus, we have that 𝜌(𝑇𝑢,𝑢)2𝜌𝑇𝑢,(1𝑡)𝑢𝑡𝑇𝑥𝑘2(1𝑡)𝜌(𝑇𝑢,𝑢)2+𝑡𝜌𝑇𝑢,𝑇𝑥𝑘2𝑡(1𝑡)𝜌𝑢,𝑇𝑥𝑘2,(4.7) and thus 𝜌(𝑇𝑢,𝑢)2𝜌𝑇𝑢,𝑇𝑥𝑘2(1𝑡)𝜌𝑢,𝑇𝑥𝑘2𝜌𝑢,𝑥𝑘2(1𝑡)𝜌𝑢,𝑇𝑥𝑘2.(4.8) Tending 𝑡0, we have that 0𝜌(𝑇𝑢,𝑢)2𝜌(𝑢,𝑥𝑘)2𝜌(𝑢,𝑇𝑥𝑘)2, and hence 𝜌(𝑢,𝑇𝑥𝑘)𝜌(𝑢,𝑥𝑘) for 𝑘=1,2,,𝑛. It gives us that 𝜌𝑢,𝑦𝑘2=𝜌𝑢,𝛼𝑘𝑥𝑘1𝛼𝑘𝑇𝑥𝑘2𝛼𝑘𝜌𝑢,𝑥𝑘2+1𝛼𝑘𝜌𝑢,𝑇𝑥𝑘2𝜌𝑢,𝑥𝑘2(4.9) for all 𝑘=1,2,,𝑛, and hence 𝑢𝐶𝑛. This shows that 𝐶𝑛 is nonempty and obviously it is closed and convex. Therefore, 𝑥𝑛+1=𝑃𝐶𝑛𝑥 is defined. By induction, we obtain that {𝑥𝑛} 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 {𝑥𝑛} converges to 𝑃𝐹(𝑇)𝑥=𝑃𝑛=1𝐶𝑛𝑥. It is trivial that (ii) implies (iii). Let us suppose that (iii) holds and show (iv). Since {𝑥𝑛} is bounded, there exists a subsequence {𝑥𝑛𝑘} which is Δ-convergent to some 𝑥0𝐵. From the definition of subsequence, for any 𝑛, there exists 𝑘0 such that 𝑛𝑘>𝑛 for all 𝑘𝑘0. Since {𝐶𝑛} is decreasing with respect to inclusion, we have 𝑥𝑛𝑘𝐶𝑛𝑘1𝐶𝑛 for all 𝑘𝑘0. By Theorem 2.2, we have that 𝑥0𝐶𝑛 for every 𝑛, and hence (iv) holds. Lastly, we show that (iv) implies (i). Assume that 𝐶=𝑛=1𝐶𝑛 is nonempty. By Theorem 3.2, {𝑥𝑛} converges to 𝑃𝐶𝑥. Then, as in the proof of Theorem 4.1, we have that 𝑃𝐶𝑥𝐹(𝑇), 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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