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

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 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 [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 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,𝑦∈𝑋, a mapping 𝑐∶[0,𝑙]→𝑋, where 𝑙≥0, is called a geodesic with endpoints 𝑥,𝑦 if 𝑐(0)=𝑥, 𝑐(𝑙)=𝑦, and 𝑑(𝑐(𝑡),𝑐(𝑠))=|𝑡−𝑠| for 𝑡,𝑠∈[0,𝑙]. If, for every 𝑥,𝑦∈𝑋, a geodesic with endpoints 𝑥,𝑦 exists, then we call 𝑋 a geodesic metric space. Furthermore, if a geodesic is unique for each 𝑥,𝑦∈𝑋, 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,𝑦∈𝑋, 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,𝑧∈𝑋 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,𝑧∈𝔼² such that 𝑑(𝑥,𝑦)=|𝑥−𝑦|_𝔼², 𝑑(𝑦,𝑧)=|𝑦−𝑧|_𝔼², and 𝑑(𝑧,𝑥)=|𝑧−𝑥|_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)=|𝑥−𝑝|_𝔼². If, for any 𝑝,𝑞∈Δ(𝑥,𝑦,𝑧) and their comparison points 𝑝,𝑞∈Δ(𝑥,𝑦,𝑧), the inequality𝑑(𝑝,𝑞)≤||𝑝−𝑞||_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,𝑦∈𝑋 and 𝑡∈[0,1], there exists a unique point 𝑧∈[𝑥,𝑦] such that 𝑑(𝑥,𝑧)=(1−𝑡)𝑑(𝑥,𝑦) and 𝑑(𝑧,𝑦)=𝑡𝑑(𝑥,𝑦). We denote it by 𝑡𝑥⊕(1−𝑡)𝑦. From the CAT(0) inequality, it is easy to see that𝑑(𝑧,𝑡𝑥⊕(1−𝑡)𝑦)²≤𝑡𝑑(𝑧,𝑥)²+(1−𝑡)𝑑(𝑧,𝑦)²−𝑡(1−𝑡)𝑑(𝑥,𝑦)²(2.2) for every 𝑥,𝑦,𝑧∈𝑋 and 𝑡∈[0,1].

A subset 𝐶 of 𝑋 is said to be convex if, for every 𝑥,𝑦∈𝐶, 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 𝑆∶𝑌→𝑋 is said to be nonexpansive if 𝑑(𝑆𝑥,𝑆𝑦)≤𝑑(𝑥,𝑦) holds for every 𝑥,𝑦∈𝑌. The set of all fixed points of 𝑆 is denoted by 𝐹(𝑆); that is, 𝐹(𝑆)={𝑧∈𝑌∶𝑆𝑧=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 𝑆∶cl𝑈→𝑋 a nonexpansive mapping. Suppose that there exists 𝑝∈𝑈 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 𝑥∈𝑋, there exists a unique point 𝑦_𝑥∈𝐶 such that 𝑑(𝑥,𝑦_𝑥)=inf_𝑦∈𝐶𝑑(𝑥,𝑦). The mapping 𝑥↦𝑦_𝑥 is called a metric projection onto 𝐶 and is denoted by 𝑃_𝐶. We know that 𝑃_𝐶 is nonexpansive; see [13, pages 176-177].

Let {𝑥_𝑛} be a bounded sequence in a metric space 𝑋. For 𝑥∈𝑋, let𝑟𝑟𝑥,𝑥𝑥_𝑛𝑛=limsup_𝑛→∞𝑑𝑥,𝑥_𝑛,=inf_𝑥∈𝑋𝑟𝑥,𝑥_𝑛.(2.3) The asymptotic center of {𝑥_𝑛} is a set of points 𝑥∈𝑋 satisfying that 𝑟(𝑥,{𝑥_𝑛})=𝑟({𝑥_𝑛}). It is known that the asymptotic center of {𝑥_𝑛} consists of one point for every bounded sequence {𝑥_𝑛} 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 {𝑥_𝑛} a bounded sequence in 𝐶. Then, the asymptotic center of {𝑥_𝑛} 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 {𝑥_𝑛} be a sequence in 𝑋. We say that {𝑥_𝑛} is Δ-convergent to 𝑥∈𝑋 if 𝑥 is the unique asymptotic center of any subsequence of {𝑥_𝑛}. We know that every bounded sequence {𝑥_𝑛} in a Hadamard space 𝑋 has a Δ-convergent subsequence; see [5, 14].

3. Convergence of a Sequence of Sets

Let {𝐶_𝑛} 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-Li_𝑛𝐶_𝑛 and Δ-Ls_𝑛𝐶_𝑛 of 𝑋 as follows: 𝑥∈𝑑-Li_𝑛𝐶_𝑛 if and only if there exists {𝑥_𝑛}⊂𝑋 such that {𝑑(𝑥_𝑛,𝑥)} converges to 0 and that 𝑥_𝑛∈𝐶_𝑛 for all 𝑛∈ℕ. On the other hand, 𝑦∈Δ-Ls_𝑛𝐶_𝑛 if and only if there exist a sequence {𝑦_𝑖}⊂𝑋 and a subsequence {𝑛_𝑖} of ℕ such that {𝑦_𝑖} has an asymptotic center {𝑦} and that 𝑦_𝑖∈𝐶_𝑛𝑖 for all 𝑖∈ℕ. If a subset 𝐶₀ of 𝑋 satisfies that 𝐶₀=𝑑-Li_𝑛𝐶_𝑛=Δ-Ls_𝑛𝐶_𝑛, it is said that {𝐶_𝑛} converges to 𝐶₀ in the sense of Δ-Mosco, and we write 𝐶₀=ΔM-lim_𝑛→∞𝐶_𝑛. Since the inclusion 𝑑-Li_𝑛𝐶_𝑛⊂Δ-Ls_𝑛𝐶_𝑛 is always true, to obtain 𝐶₀ is a limit of {𝐶_𝑛} in the sense of Δ-Mosco, it suffices to show that Δ-Ls_𝑛𝐶_𝑛⊂𝐶₀⊂𝑑-Li_𝑛𝐶_𝑛.

It is easy to show that, if every 𝐶_𝑛 is convex, then so is 𝑑-Li_𝑛𝐶_𝑛. Moreover, we know that 𝑑-Li_𝑛𝐶_𝑛 is always closed. Therefore, ΔM-lim_𝑛→∞𝐶_𝑛 is closed and convex whenever {𝐶_𝑛} 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 {𝑥_𝑛} a sequence in 𝑋. Suppose that {𝑥_𝑛} is Δ-convergent to 𝑥∈𝑋 and {𝑑(𝑥_𝑛,𝑝)} converges to 𝑑(𝑥,𝑝) for some 𝑝∈𝑋. Then, {𝑥_𝑛} converges to 𝑥.

Proof. Let {Δ(𝑥,𝑝,𝑥_𝑛)} be comparison triangles in 𝔼² for 𝑛∈ℕ with an identical geodesic segment [𝑝,𝑥]. Then, we have that |𝑥−𝑝|_𝔼²=𝑑(𝑥,𝑝), |𝑥_𝑛−𝑝|_𝔼2=𝑑(𝑥_𝑛,𝑝), and |𝑥_𝑛−𝑥|_𝔼2=𝑑(𝑥_𝑛,𝑥) for all 𝑛∈ℕ. We know that {𝑥_𝑛} is bounded in 𝔼². Let {𝑥_𝑛𝑖} be an arbitrary subsequence of {𝑥_𝑛} converging to 𝑦∈𝔼². Then, by assumption, we have that ||𝑦−𝑝||_𝔼2=lim_𝑖→∞||𝑥_𝑛𝑖−𝑝||_𝔼2=lim_𝑖→∞𝑑𝑥_𝑛𝑖,𝑝=𝑑(𝑥,𝑝)=||𝑥−𝑝||_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 {𝑃𝑥_𝑛𝑖} converges to 𝑃𝑦∈𝔼². Let 𝑧∈[𝑝,𝑥]⊂𝑋 be a point corresponding to 𝑧=𝑃𝑦∈[𝑝,𝑥]⊂𝔼². Using the CAT(0) inequality, we have that 𝑟𝑥_𝑛𝑖=limsup_𝑖→∞𝑑𝑥,𝑥_𝑛𝑖=limsup_𝑖→∞||𝑥−𝑥_𝑛𝑖||_𝔼2≥limsup_𝑖→∞||𝑃𝑥_𝑛𝑖−𝑥_𝑛𝑖||_𝔼2=limsup_𝑖→∞||𝑧−𝑥_𝑛𝑖||_𝔼2≥limsup_𝑖→∞𝑑𝑧,𝑥_𝑛𝑖,(3.2) and hence 𝑟(𝑧,{𝑥_𝑛𝑖})≤𝑟({𝑥_𝑛𝑖}). By the uniqueness of the asymptotic center of {𝑥_𝑛𝑖}, we obtain that 𝑧=𝑥, and thus 𝑧=𝑥. Since ||𝑥−𝑦||_𝔼2=||𝑧−𝑦||_𝔼2=||𝑃𝑦−𝑦||_𝔼2≤||(1−𝑡)𝑥+𝑡𝑝−𝑦||_𝔼2(3.3) for every 𝑡∈]0,1[⊂ℝ, it follows that ||𝑥−𝑦||^2𝔼₂≤||(1−𝑡)𝑥+𝑡𝑝−𝑦||^2𝔼₂=(1−𝑡)||𝑥−𝑦||^2𝔼₂+𝑡||𝑝−𝑦||^2𝔼₂−𝑡(1−𝑡)||𝑥−𝑝||^2𝔼₂=(1−𝑡)||𝑥−𝑦||^2𝔼₂+𝑡²||𝑝−𝑥||^2𝔼₂,(3.4) and thus |𝑥−𝑦|^2𝔼₂≤𝑡|𝑝−𝑥|^2𝔼₂. Tending 𝑡↓0, we obtain that 𝑥=𝑦. Since any convergent subsequence {𝑥_𝑛𝑖} of a bounded sequence {𝑥_𝑛} in 𝔼² has a limit 𝑥, we have that {𝑥_𝑛} converges to 𝑥. Thus we have that 𝑑(𝑥_𝑛,𝑥)=|𝑥_𝑛−𝑥|_𝔼2→0 as 𝑛→∞, and hence {𝑥_𝑛} converges to 𝑥∈𝑋.

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 {𝐶_𝑛} of nonempty closed convex subsets in 𝑋, the following are equivalent: (i){𝐶_𝑛} converges to 𝐶₀ in the sense of Δ-Mosco; (ii){𝑃_𝐶𝑛𝑥} converges to 𝑃_𝐶0𝑥∈𝑋 for every 𝑥∈𝑋.

Proof. We first show that (i) implies (ii). Fix 𝑥∈𝑋, and let 𝑝_𝑛=𝑃_𝐶𝑛𝑥 for 𝑛∈ℕ. Since 𝑃_𝐶0𝑥∈𝐶₀=𝑑-Li_𝑛𝐶_𝑛, there exists {𝑦_𝑛}⊂𝑋 such that 𝑦_𝑛∈𝐶_𝑛 for all 𝑛∈ℕ and that {𝑦_𝑛} converges to 𝑃_𝐶0𝑥. By the definition of metric projection, we have that 𝑑(𝑥,𝑝_𝑛)≤𝑑(𝑥,𝑦_𝑛) for 𝑛∈ℕ. Thus, tending 𝑛→∞, we have that limsup_𝑛→∞𝑑𝑥,𝑝_𝑛≤lim_𝑛→∞𝑑𝑥,𝑦_𝑛=𝑑𝑥,𝑃_𝐶0𝑥.(3.5) It also follows that {𝑝_𝑛} is bounded. Let {𝑝_𝑛𝑖} be an arbitrary subsequence of {𝑝_𝑛} and 𝑝₀ an asymptotic center of {𝑝_𝑛𝑖}. Then, for fixed 𝜖>0, it holds that 𝑑𝑥,𝑝_𝑛i≤𝑑𝑥,𝑃_𝐶0𝑥+𝜖(3.6) for sufficiently large 𝑖∈ℕ. Since the closed ball with the center 𝑥 and the radius 𝑑(𝑥,𝑃_𝐶0𝑥)+𝜖 is convex, by Theorem 2.2, we have that 𝑑(𝑥,𝑝₀)≤𝑑(𝑥,𝑃_𝐶0𝑥)+𝜖, and hence 𝑑𝑥,𝑝₀≤𝑑𝑥,𝑃_𝐶0𝑥.(3.7) On the other hand, since 𝑝₀∈Δ-Ls_𝑛𝐶_𝑛=𝐶₀, we have that 𝑑(𝑥,𝑃_𝐶0𝑥)≤𝑑(𝑥,𝑝₀), and therefore we have that 𝑑(𝑥,𝑃_𝐶0𝑥)=𝑑(𝑥,𝑝₀), which implies that 𝑝₀=𝑃_𝐶0𝑥. Since all subsequences of {𝑝_𝑛} have the same asymptotic center 𝑃_𝐶0𝑥, {𝑝_𝑛} is Δ-convergent to 𝑃_𝐶0𝑥.
Let us show that liminf_𝑛→∞𝑑(𝑥,𝑝_𝑛)≥𝑑(𝑥,𝑃_𝐶0𝑥). If it were not true, then there exists a subsequence {𝑝_𝑛𝑖} of {𝑝_𝑛} satisfying that liminf_𝑛→∞𝑑(𝑥,𝑝_𝑛)=lim_𝑖→∞𝑑(𝑥,𝑝_𝑛𝑖)<𝑑(𝑥,𝑃_𝐶0𝑥). Let 𝑝∈𝑋 be an asymptotic center of {𝑝_𝑛𝑖}. For 𝜖>0, we have that 𝑑(𝑥,𝑝_𝑛𝑖)≤𝛿+𝜖 for sufficiently large 𝑖∈ℕ, where 𝛿=lim_𝑖→∞𝑑(𝑥,𝑝_𝑛𝑖). 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=lim_𝑖→∞𝑑(𝑥,𝑝_𝑛𝑖). Since 𝑝∈Δ-Ls_𝑛𝐶_𝑛=C₀, we get that 𝑑𝑥,𝑃_𝐶0𝑥>lim_𝑖→∞𝑑𝑥,𝑝_𝑛𝑖≥𝑑(𝑥,𝑝)≥𝑑𝑥,𝑃_𝐶0𝑥,(3.8) a contradiction. Therefore, we obtain that𝑑𝑥,𝑃_𝐶0𝑥≤liminf_𝑛→∞𝑑𝑥,𝑝_𝑛≤limsup_𝑛→∞𝑑𝑥,𝑝_𝑛≤𝑑𝑥,𝑃_𝐶0𝑥,(3.9) and thus {𝑑(𝑥,𝑝_𝑛)} converges to 𝑑(𝑥,𝑃_𝐶0𝑥). Using Lemma 3.1, we have that {𝑝_𝑛} converges to 𝑃_𝐶0𝑥. Hence (ii) holds.
Next we suppose (ii) and show that (i) holds. By assumption, for 𝑦∈𝐶₀, a sequence {𝑃_𝐶𝑛𝑦} converges to 𝑃_𝐶0𝑦=𝑦. Since 𝑃_𝐶𝑛𝑦∈𝐶_𝑛 for all 𝑛∈ℕ, we have that 𝑦∈𝑑-Li_𝑛𝐶_𝑛, and hence 𝐶₀⊂𝑑-Li_𝑛𝐶_𝑛. Let 𝑧∈Δ-Ls_𝑛𝐶_𝑛. Then, there exist {𝑧_𝑖}⊂𝑋 and {𝑛_𝑖}⊂ℕ such that 𝑧_𝑖∈𝐶_𝑛𝑖 for all 𝑖∈ℕ and 𝑧 is an asymptotic center of {𝑧_𝑖}. Since each 𝐶_𝑛𝑖 is convex, from the definition of metric projection, it follows that 𝑑𝑧,𝑃_𝐶𝑛𝑖𝑧≤𝑑𝑧,(1−𝑡)𝑃_𝐶𝑛𝑖𝑧⊕𝑡𝑧_𝑖(3.10) for 𝑡∈]0,1[ and 𝑖∈ℕ. Then, we have that𝑑𝑧,𝑃_𝐶𝑛𝑖𝑧²≤𝑑𝑧,(1−𝑡)𝑃_𝐶𝑛𝑖𝑧⊕𝑡𝑧_𝑖²≤(1−𝑡)𝑑𝑧,𝑃_𝐶𝑛𝑖𝑧²+𝑡𝑑𝑧,𝑧_𝑖²−𝑡(1−𝑡)𝑑𝑃_𝐶𝑛𝑖𝑧,𝑧_𝑖²,(3.11) and thus 𝑑𝑧,𝑃_𝐶𝑛𝑖𝑧²+(1−𝑡)𝑑𝑃_𝐶𝑛𝑖𝑧,𝑧_𝑖²≤𝑑𝑧,𝑧_𝑖².(3.12) Tending 𝑡↓0, we get that 𝑑𝑧,𝑃_𝐶𝑛𝑖𝑧²+𝑑𝑃_𝐶𝑛𝑖𝑧,𝑧_𝑖²≤𝑑𝑧,𝑧_𝑖²(3.13) for every 𝑖∈ℕ, and since {𝑃_𝐶𝑛𝑖𝑧} converges to 𝑃_𝐶0𝑧 as 𝑖→∞, we have that 𝑑𝑧,𝑃_𝐶0𝑧²+limsup_𝑖→∞𝑑𝑃_C0𝑧,𝑧_𝑖²≤limsup_𝑖→∞𝑑𝑧,𝑧_𝑖².(3.14) Since 𝑧 is an asymptotic center of {𝑧_𝑖}, we have that limsup_𝑖→∞𝑑𝑧,𝑧_𝑖=≤𝑟𝑟𝑃𝑧,_𝐶0𝑧_𝑖=𝑟𝑧_𝑖=limsup_𝑖→∞𝑧,𝑧_𝑖𝑑𝑃_𝐶0𝑧,𝑧_𝑖.(3.15) It follows that 𝑑𝑧,𝑃_𝐶0𝑧²≤limsup_𝑖→∞𝑑𝑧,𝑧_𝑖²−limsup_𝑖→∞𝑑𝑃_𝐶0𝑧,𝑧_𝑖²≤0,(3.16) and therefore 𝑧=𝑃_𝐶0𝑧∈𝐶₀, which implies that Δ-Ls_𝑛𝐶_𝑛⊂𝐶₀. Consequently we have that {𝐶_𝑛} 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 {𝐶_𝑛} be a sequence of nonempty closed convex subsets in 𝐻. Then, {𝐶_𝑛} converges to 𝐶₀ in the sense of Mosco if and only if {𝐶_𝑛} converges to 𝐶₀ in the sense of Δ-Mosco.

Proof. By [8, Theorems 4.1 and 4.2], {𝐶_𝑛} converges to 𝐶₀ in the sense of Mosco if and only if {𝑃_𝐶𝑛𝑥} converges strongly to 𝑃_𝐶0𝑥 for all 𝑥∈𝐻. 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 {𝐶_𝑛} be a sequence of nonempty closed convex subsets of a Hadamard space 𝑋. Then, as a direct consequence of the definition, we obtain that cl^∞_∞𝑚=1_𝑛=𝑚𝐶_𝑛⊂𝑑-Li_𝑛𝐶_𝑛⊂Δ-Ls_𝑛𝐶_𝑛⊂^∞_𝑚=1clco^∞_𝑛=𝑚𝐶_𝑛.(3.17) In particular, if {𝐶_𝑛} is a decreasing sequence with respect to inclusion, then {𝐶_𝑛} is Δ-Mosco convergent to ⋂^∞𝑛=1𝐶_𝑛. Likewise, if {𝐶_𝑛} is increasing, then the limit is cl⋃^∞𝑛=1𝐶_𝑛.

Example 3.5. Let {𝐶_𝑛} be a sequence of nonempty bounded closed convex subsets of a Hadamard space 𝑋. If {𝐶_𝑛} converges to a bounded closed convex subset 𝐶₀⊂𝑋 with respect to the Hausdorff metric, then {𝐶_𝑛} also converges to 𝐶₀ in the sense of Δ-Mosco. The Hausdorff metric ℎ between nonempty bounded closed subsets 𝐴,𝐵 of 𝑋 is defined by ℎ(𝐴,𝐵)=max{𝑒(𝐴,𝐵),𝑒(𝐵,𝐴)},(3.18) where 𝑒(𝐴,𝐵)=sup_𝑥∈𝐴𝑑(𝑥,𝐵) and 𝑑(𝑥,𝐵)=inf_𝑦∈𝐵𝑑(𝑥,𝑦) for 𝑥∈𝑋.
Let us prove this fact. For 𝑥∈𝐶₀, we have that 𝑑(𝑥,𝐶_𝑛)≤𝑒(𝐶₀,𝐶_𝑛)≤ℎ(𝐶₀,𝐶_𝑛) and since ℎ(𝐶_𝑛,𝐶₀)→0 as 𝑛→∞, there exists a sequence {𝑥_𝑛}⊂𝑋 converging to 𝑥 such that 𝑥_𝑛∈𝐶_𝑛 for all 𝑛∈ℕ. It follows that 𝑥∈𝑑-Li_𝑛𝐶_𝑛, and hence 𝐶₀⊂𝑑-Li_𝑛𝐶_𝑛.
To show Δ-Ls_𝑛𝐶_𝑛⊂𝐶₀, let 𝑥∈Δ-Ls_𝑛𝐶_𝑛. Then, there exists a subsequence {𝑛_𝑖} of ℕ and a sequence {𝑥_𝑖}⊂𝑋 whose asymptotic center is 𝑥 and 𝑥_𝑖∈𝐶_𝑛𝑖 for all 𝑖∈ℕ. Let 𝜖>0 be arbitrary. Then, since 𝑑(𝑥_𝑖,𝐶₀)≤𝑒(𝐶_𝑛𝑖,𝐶₀)≤ℎ(𝐶_𝑛𝑖,𝐶₀)→0 as 𝑖→∞, there exists 𝑖₀∈ℕ such that 𝑑(𝑥_𝑖,𝐶₀)<𝜖 for every 𝑖≥𝑖₀.
Let 𝐷_𝜖={𝑦∈𝑋∶𝑑(𝑦,𝐶₀)≤𝜖}. Then, 𝐷_𝜖 is closed and convex in 𝑋. Indeed, for 𝑦₁,𝑦₂∈𝐷_𝜖 and 𝑡∈]0,1[, there exist 𝑧₁,𝑧₂∈𝐶₀ 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 𝑑𝑡𝑦₁⊕(1−𝑡)𝑧₂,𝑡𝑧₁⊕(1−𝑡)𝑧₂≤𝑡𝑑𝑦₁,𝑧₁.(3.19)
In the same way, considering the comparison triangle of (𝑦₁,𝑦₂,𝑧₂), we have that 𝑑𝑡𝑦₁⊕(1−𝑡)𝑦₂,𝑡𝑦₁⊕(1−𝑡)𝑧₂≤(1−𝑡)𝑑𝑦₂,𝑧₂.(3.20) Thus, we have that 𝑑(𝑡𝑦₁⊕(1−𝑡)𝑦₂,𝑡𝑧₁⊕(1−𝑡)𝑧₂)≤(1−𝑡)𝑑(𝑦₂,𝑧₂)+𝑡𝑑(𝑦₁,𝑧₁)≤𝜖. Since 𝐶₀ is convex, we have that 𝑡𝑧₁⊕(1−𝑡)𝑧₂∈𝐶₀, and hence 𝑡𝑦₁⊕(1−𝑡)𝑦₂∈𝐷_𝜖. This shows that 𝐷_𝜖 is convex. It is obvious that 𝐷_𝜖 is closed since the function 𝑑(⋅,𝐶₀) is continuous.
Since 𝑥_𝑖∈𝐷_𝜖 for 𝑖≥𝑖₀, using Theorem 2.2, we have that 𝑥∈𝐷_𝜖; that is, 𝑑(𝑥,𝐶₀)≤𝜖. Since 𝜖 is arbitrary and 𝐶₀ is closed, we obtain that 𝑥∈𝐶₀, and hence Δ-Ls_𝑛𝐶_𝑛⊂𝐶₀. Consequently we have that {𝐶_𝑛} 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 𝐵_𝐻 be the open unit ball of a complex Hilbert space 𝐻 with an inner product ⟨⋅,⋅⟩ and induced norm ‖⋅‖. For an orthonormal basis {𝑒_𝑖∶𝑖∈𝐼} of 𝐻, let 𝐻_𝑅={𝑧∈𝐻∶Im⟨𝑧,𝑒_𝑖⟩=0∀𝑖∈𝐼}. Then, a real Hilbert ball (𝐵,𝜌) is a metric space defined by 𝐵=𝐵_𝐻∩𝐻_𝑅 and 𝜌∶𝐵×𝐵→ℝ by𝜌(𝑥,𝑦)=arctanh⎷1−1−‖𝑥‖²1−‖𝑦‖²1−||⟨𝑥,𝑦⟩||²(4.1) for 𝑥,𝑦∈𝐵. 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={𝑧∈𝐵∶𝜌(𝑧,𝑦)≤𝜌(𝑧,𝑥)} is convex for any 𝑥,𝑦∈𝐵; see [16, 17].

Theorem 4.1. Let 𝐵 be a real Hilbert ball with the metric 𝜌. Let {𝑇_𝑖∶𝑖∈𝐼} be a family of nonexpansive mappings of 𝐵 into itself with a nonempty set 𝐹 of their common fixed points. Let {𝛼_𝑛(𝑖)∶𝑖∈𝐼,𝑛∈ℕ} be nonnegative real numbers in [0,1] such that liminf_𝑛→∞𝛼_𝑛(𝑖)<1 for each 𝑖∈𝐼. For 𝑥∈𝐵, generate an iterative sequence {𝑥_𝑛} by 𝑥₁=𝑥, 𝐶₀=𝐵, and 𝐶𝑦_𝑛(𝑖)=𝛼_𝑛(𝑖)𝑥_𝑛⊕1−𝛼_𝑛(𝑖)𝑇_𝑖𝑥_𝑛,foreach𝑖∈I,_𝑛=𝑧∈𝐵∶sup_𝑖∈𝐼𝑥𝜌𝑧_𝑛+1,𝑦_𝑛(𝑖)≤𝜌𝑧,𝑥_𝑛∩𝐶_𝑛−1,=𝑃_𝐶𝑛𝑥(4.2) for all 𝑛∈ℕ. Then, {𝑥_𝑛} is well defined and converges to 𝑃_𝐹𝑥∈𝐵.

Proof. Since each 𝐹(𝑇_𝑖) is closed and convex, so is 𝐹=⋂_𝑖∈𝐼𝐹(𝑇_𝑖). For 𝑧∈𝐹, it follows that 𝜌𝑧,𝑦_𝑛(𝑖)²≤𝛼_𝑛(𝑖)𝜌𝑧,𝑥_𝑛²+1−𝛼_𝑛(𝑖)𝜌𝑧,𝑇_𝑖𝑥_𝑛²−𝛼_𝑛(𝑖)1−𝛼_𝑛(𝑖)𝜌𝑥_𝑛,𝑇_𝑖𝑥_𝑛²≤𝛼_𝑛(𝑖)𝜌𝑧,𝑥_𝑛²+1−𝛼_𝑛(𝑖)𝜌𝑇_𝑖𝑧,𝑇_𝑖𝑥_𝑛²≤𝜌𝑧,𝑥_𝑛²(4.3) for all 𝑖∈𝐼, and thus sup_𝑖∈𝐼𝜌(𝑧,𝑦_𝑛(𝑖))≤𝜌(𝑧,𝑥_𝑛) for every 𝑛∈ℕ. Therefore, we have 𝐹⊂𝐶_𝑛 and 𝐶_𝑛 is nonempty for 𝑛∈ℕ. Further, 𝐶_𝑛 is closed and convex by the property of a real Hilbert ball 𝐵. Hence, the metric projection 𝑃_𝐶𝑛 exists, and 𝑥_𝑛 is well defined for all 𝑛∈ℕ. Since {𝐶_𝑛} is decreasing with respect to inclusion, as in Example 3.4, we have that {𝐶_𝑛} converges to 𝐶=⋂^∞𝑛=1𝐶_𝑛 in the sense of Δ-Mosco. By Theorem 3.2, we have that {𝑥_𝑛} converges to 𝑥₀=𝑃_𝐶𝑥. Since 𝑥₀∈𝐶_𝑛 for all 𝑛∈ℕ, we have that 𝜌(𝑥₀,𝑦_𝑛(𝑖))≤𝜌(𝑥₀,𝑥_𝑛) for all 𝑛∈ℕ and 𝑖∈𝐼. Fix 𝑖∈𝐼 arbitrarily, and let {𝛼_𝑛𝑘(𝑖)} be a subsequence of {𝛼_𝑛(𝑖)} converging to 𝛼₀(𝑖)∈[0,1[. Then, since 𝜌(𝑥_𝑛,𝑦_𝑛(𝑖))=(1−𝛼_𝑛(𝑖))𝜌(𝑥_𝑛,𝑇_𝑖𝑥_𝑛), we have that 𝜌𝑥₀,𝑇_𝑖𝑥₀≤≤2𝜌𝜌𝑥𝑥₀,₀𝑥_𝑛𝑘+𝜌𝑥_𝑛𝑘,𝑇_𝑖𝑥_𝑛𝑘+𝜌𝑇_𝑖𝑥_𝑛𝑘,𝑇_𝑖𝑥₀,𝑥_𝑛𝑘+11−𝛼_𝑛𝑘(𝑖)𝜌𝑥^𝑛𝑘≤2𝜌𝑥₀,𝑦_𝑛𝑘(𝑖),𝑥_𝑛𝑘+11−𝛼_𝑛𝑘(𝑖)𝜌𝑥^𝑛𝑘≤21+1,𝑥₀+𝜌𝑥₀,𝑦_𝑛𝑘(𝑖)1−𝛼_𝑛𝑘(𝑖)𝜌𝑥⁰,𝑥_𝑛𝑘(4.4) for 𝑘∈ℕ, and, as 𝑘→∞, we obtain that 𝑥₀=𝑇_𝑖𝑥₀; that is, 𝑥₀∈𝐹(𝑇_𝑖). Since 𝑖∈𝐼 is arbitrary, we have that 𝑃_𝐶𝑥=𝑥₀∈𝐹⊂𝐶, and therefore 𝑥₀=𝑃_𝐹𝑥, 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 𝑇∶𝐵→𝐵 a nonexpansive mapping. Let {𝛼_𝑛} be a nonnegative real sequence in [0,1] such that liminf_𝑛→∞𝛼_𝑛<1. For 𝑥∈𝐵, generate an iterative sequence {𝑥_𝑛} by 𝑥₁=𝑥, 𝐶₀=𝐵, and 𝐶𝑦_𝑛=𝛼_𝑛𝑥_𝑛⊕1−𝛼_𝑛𝑇𝑥_𝑛,foreach𝑖∈I,_𝑛=𝑧∈𝐵∶𝜌𝑥𝑧,_𝑛+1𝑦_𝑛≤𝜌𝑧,𝑥_𝑛∩𝐶_𝑛−1,=𝑃_𝐶𝑛𝑥(4.5) for all 𝑛∈ℕ. Then, {𝑥_𝑛} is well defined and the following are equivalent: (i)𝐹(𝑇) is nonempty;(ii){𝑥_𝑛} is convergent; (iii){𝑥_𝑛} is bounded; (iv)⋂^∞𝑛=1𝐶_𝑛 is nonempty. Moreover, in this case the limit of {𝑥_𝑛} is 𝑃_𝐹(𝑇)𝑥=𝑃_⋂^∞𝑛=1_𝐶𝑛𝑥.

Proof. First we show that {𝑥_𝑛} is well defined. Since 𝑥₁ is given and 𝑦₁∈𝐶₁, 𝐶₁ is nonempty. Suppose that 𝐶₁,𝐶₂,…,𝐶_𝑛−1 are nonempty. Then, 𝑥₁,𝑥₂,…,𝑥_𝑛 and 𝑦₁,𝑦₂,…,𝑦_𝑛 are defined. Let 𝑟=max_{1≤𝑘≤𝑛}𝜌(𝑥,𝑇𝑥_𝑘) and 𝐷={𝑤∈𝐵∶𝜌(𝑥,𝑤)≤𝑟}. Then, since 𝐷 is nonempty, bounded, closed, and convex, there exists a metric projection 𝑃_𝐷∶𝐵→𝐷. Since 𝑃_𝐷 is nonexpansive, it follows that 𝑃_𝐷𝑇∣_𝐷 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 𝑣∈𝐷. Thus, by Theorem 2.1, there exists 𝑢∈𝐷 such that 𝑢=𝑃_𝐷𝑇𝑢. Since 𝑇𝑥_𝑘∈𝐷 for 𝑘=1,2,…,𝑛 and 𝐷 is convex, it follows from the definition of the metric projection that 𝜌(𝑇𝑢,𝑢)=𝜌𝑇𝑢,𝑃_𝐷𝑇𝑢≤𝜌𝑇𝑢,(1−𝑡)𝑢⊕𝑡𝑇𝑥_𝑘(4.6) for 𝑡∈]0,1[. Thus, we have that 𝜌(𝑇𝑢,𝑢)²≤𝜌𝑇𝑢,(1−𝑡)𝑢⊕𝑡𝑇𝑥_𝑘²≤(1−𝑡)𝜌(𝑇𝑢,𝑢)²+𝑡𝜌𝑇𝑢,𝑇𝑥_𝑘²−𝑡(1−𝑡)𝜌𝑢,𝑇𝑥_𝑘²,(4.7) and thus 𝜌(𝑇𝑢,𝑢)²≤𝜌𝑇𝑢,𝑇𝑥_𝑘²−(1−𝑡)𝜌𝑢,𝑇𝑥_𝑘²≤𝜌𝑢,𝑥_𝑘²−(1−𝑡)𝜌𝑢,𝑇𝑥_𝑘².(4.8) Tending 𝑡↓0, we have that 0≤𝜌(𝑇𝑢,𝑢)²≤𝜌(𝑢,𝑥_𝑘)²−𝜌(𝑢,𝑇𝑥_𝑘)², and hence 𝜌(𝑢,𝑇𝑥_𝑘)≤𝜌(𝑢,𝑥_𝑘) for 𝑘=1,2,…,𝑛. It gives us that 𝜌𝑢,𝑦_𝑘²=𝜌𝑢,𝛼_𝑘𝑥_𝑘⊕1−𝛼_𝑘𝑇𝑥_𝑘²≤𝛼_𝑘𝜌𝑢,𝑥_𝑘²+1−𝛼_𝑘𝜌𝑢,𝑇𝑥_𝑘²≤𝜌𝑢,𝑥_𝑘²(4.9) for all 𝑘=1,2,…,𝑛, and hence 𝑢∈𝐶_𝑛. This shows that 𝐶_𝑛 is nonempty and obviously it is closed and convex. Therefore, 𝑥_𝑛+1=𝑃_𝐶𝑛𝑥 is defined. By induction, we obtain that {𝑥_𝑛} 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 {𝑥_𝑛} converges to 𝑃_𝐹(𝑇)𝑥=𝑃_⋂^∞𝑛=1_𝐶𝑛𝑥. It is trivial that (ii) implies (iii). Let us suppose that (iii) holds and show (iv). Since {𝑥_𝑛} is bounded, there exists a subsequence {𝑥_𝑛𝑘} which is Δ-convergent to some 𝑥₀∈𝐵. From the definition of subsequence, for any 𝑛∈ℕ, there exists 𝑘₀∈ℕ such that 𝑛_𝑘>𝑛 for all 𝑘≥𝑘₀. Since {𝐶_𝑛} is decreasing with respect to inclusion, we have 𝑥_𝑛𝑘⊂𝐶_𝑛𝑘−1⊂𝐶_𝑛 for all 𝑘≥𝑘₀. By Theorem 2.2, we have that 𝑥₀∈𝐶_𝑛 for every 𝑛∈ℕ, and hence (iv) holds. Lastly, we show that (iv) implies (i). Assume that 𝐶=⋂^∞𝑛=1𝐶_𝑛 is nonempty. By Theorem 3.2, {𝑥_𝑛} converges to 𝑃_𝐶𝑥. Then, as in the proof of Theorem 4.1, we have that 𝑃_𝐶𝑥∈𝐹(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.