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

Authors: Yasunori Kimura
Journal: Abstract and Applied Analysis
 (2010)

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## 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.


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## Body

## 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 adoptΔ-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

LetX 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.LetX 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 inequality(2.1)d(p,q)≤|p¯-q¯|E2
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].Forx,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 that(2.2)d(z,tx⊕(1-t)y)2≤td(z,x)2+(1-t)d(z,y)2-t(1-t)d(x,y)2
for every x,y,z∈X and t∈[0,1].A subsetC 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.LetY 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]).
LetU 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.LetC be a nonempty closed convex subset of a Hadamard space X. Then, for each x∈X, there exists a unique point yx∈C such that d(x,yx)=infy∈Cd(x,y). The mapping x↦yx 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 x∈X, let(2.3)r(x,{xn})=limsup⁡n→∞d(x,xn),r({xn})=inf⁡x∈Xr(x,{xn}).
The asymptotic center of {xn} is a set of points x∈X 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]).
LetC 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 ofΔ-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 x∈X 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: x∈d-LinCn if and only if there exists {xn}⊂X such that {d(xn,x)} converges to 0 and that xn∈Cn for all n∈ℕ. On the other hand, y∈Δ-LsnCn if and only if there exist a sequence {yi}⊂X and a subsequence {ni} of ℕ such that {yi} has an asymptotic center {y} and that yi∈Cni for all i∈ℕ. 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-limn→∞Cn. Since the inclusion d-LinCn⊂Δ-LsnCn is always true, to obtain C0 is a limit of {Cn} in the sense of Δ-Mosco, it suffices to show that Δ-LsnCn⊂C0⊂d-LinCn.It is easy to show that, if everyCn is convex, then so is d-LinCn. Moreover, we know that d-LinCn is always closed. Therefore, ΔM-limn→∞Cn 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.
LetX be a Hadamard space and {xn} a sequence in X. Suppose that {xn} is Δ-convergent to x∈X and {d(xn,p)} converges to d(x,p) for some p∈X. Then, {xn} 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(xn,p), and |x¯n-x|𝔼2=d(xn,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
(3.1)|y¯-p¯|E2=lim⁡i→∞|x¯ni-p¯|E2=lim⁡i→∞d(xni,p)=d(x,p)=|x¯-p¯|E2.
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
(3.2)r({xni})=limsup⁡i→∞d(x,xni)=limsup⁡i→∞|x¯-x¯ni|E2≥limsup⁡i→∞|Px¯ni-x¯ni|E2=limsup⁡i→∞|z¯-x¯ni|E2≥limsup⁡i→∞d(z,xni),
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
(3.3)|x¯-y¯|E2=|z¯-y¯|E2=|Py¯-y¯|E2≤|(1-t)x¯+tp¯-y¯|E2
for every t∈]0,1[⊂ℝ, it follows that
(3.4)|x¯-y¯|E22≤|(1-t)x¯+tp¯-y¯|E22=(1-t)|x¯-y¯|E22+t|p¯-y¯|E22-t(1-t)|x¯-p¯|E22=(1-t)|x¯-y¯|E22+t2|p¯-x¯|E22,
and thus |x¯-y¯|𝔼22≤t|p¯-x¯|𝔼22. 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(xn,x)=|x¯n-x¯|𝔼2→0 as n→∞, and hence {xn} 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 ofΔ-Mosco convergence in a Hadamard space.Theorem 3.2.
LetX 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 PC0x∈X for every x∈X.Proof.
We first show that (i) implies (ii). Fixx∈X, and let pn=PCnx for n∈ℕ. Since PC0x∈C0=d-LinCn, there exists {yn}⊂X such that yn∈Cn for all n∈ℕ and that {yn} converges to PC0x. By the definition of metric projection, we have that d(x,pn)≤d(x,yn) for n∈ℕ. Thus, tending n→∞, we have that
(3.5)limsup⁡n→∞d(x,pn)≤lim⁡n→∞d(x,yn)=d(x,PC0x).
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
(3.6)d(x,pni)≤d(x,PC0x)+ϵ
for sufficiently large i∈ℕ. 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
(3.7)d(x,p0)≤d(x,PC0x).
On the other hand, since p0∈Δ-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 thatliminfn→∞d(x,pn)≥d(x,PC0x). If it were not true, then there exists a subsequence {pni} of {pn} satisfying that liminfn→∞d(x,pn)=limi→∞d(x,pni)<d(x,PC0x). Let p∈X be an asymptotic center of {pni}. For ϵ>0, we have that d(x,pni)≤δ+ϵ for sufficiently large i∈ℕ, where δ=limi→∞d(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=limi→∞d(x,pni). Since p∈Δ-LsnCn=C0, we get that
(3.8)d(x,PC0x)>lim⁡i→∞d(x,pni)≥d(x,p)≥d(x,PC0x),
a contradiction. Therefore, we obtain that(3.9)d(x,PC0x)≤liminf⁡n→∞d(x,pn)≤limsup⁡n→∞d(x,pn)≤d(x,PC0x),
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, fory∈C0, a sequence {PCny} converges to PC0y=y. Since PCny∈Cn for all n∈ℕ, we have that y∈d-LinCn, and hence C0⊂d-LinCn. Let z∈Δ-LsnCn. Then, there exist {zi}⊂X and {ni}⊂ℕ such that zi∈Cni for all i∈ℕ and z is an asymptotic center of {zi}. Since each Cni is convex, from the definition of metric projection, it follows that
(3.10)d(z,PCniz)≤d(z,(1-t)PCniz⊕tzi)
for t∈]0,1[ and i∈ℕ. Then, we have that(3.11)d(z,PCniz)2≤d(z,(1-t)PCniz⊕tzi)2≤(1-t)d(z,PCniz)2+td(z,zi)2-t(1-t)d(PCniz,zi)2,
and thus
(3.12)d(z,PCniz)2+(1-t)d(PCniz,zi)2≤d(z,zi)2.
Tending t↓0, we get that
(3.13)d(z,PCniz)2+d(PCniz,zi)2≤d(z,zi)2
for every i∈ℕ, and since {PCniz} converges to PC0z as i→∞, we have that
(3.14)d(z,PC0z)2+limsup⁡i→∞d(PC0z,zi)2≤limsup⁡i→∞d(z,zi)2.
Since z is an asymptotic center of {zi}, we have that
(3.15)limsup⁡i→∞d(z,zi)=r(z,{zi})=r({zi})≤r(PC0z,{zi})=limsup⁡i→∞d(PC0z,zi).
It follows that
(3.16)d(z,PC0z)2≤limsup⁡i→∞d(z,zi)2-limsup⁡i→∞d(PC0z,zi)2≤0,
and therefore z=PC0z∈C0, which implies that Δ-LsnCn⊂C0. 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.
LetH 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 x∈H. Therefore, using Theorem 3.2, we obtain the desired result.This result shows that Mosco convergence in Hilbert spaces is an example ofΔ-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
(3.17)cl⁡⋃m=1∞⋂n=m∞Cn⊂d-LinCn⊂Δ-LsnCn⊂⋂m=1∞cl co⋃n=m∞Cn.
In particular, if {Cn} is a decreasing sequence with respect to inclusion, then {Cn} is Δ-Mosco convergent to ⋂n=1∞Cn. Likewise, if {Cn} is increasing, then the limit is cl⋃n=1∞Cn.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 C0⊂X with respect to the Hausdorff metric, then {Cn} 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
(3.18)h(A,B)=max⁡{e(A,B),e(B,A)},
where e(A,B)=supx∈Ad(x,B) and d(x,B)=infy∈Bd(x,y) for x∈X.
Let us prove this fact. Forx∈C0, we have that d(x,Cn)≤e(C0,Cn)≤h(C0,Cn) and since h(Cn,C0)→0 as n→∞, there exists a sequence {xn}⊂X converging to x such that xn∈Cn for all n∈ℕ. It follows that x∈d-LinCn, and hence C0⊂d-LinCn.
To showΔ-LsnCn⊂C0, let x∈Δ-LsnCn. Then, there exists a subsequence {ni} of ℕ and a sequence {xi}⊂X whose asymptotic center is x and xi∈Cni for all i∈ℕ. Let ϵ>0 be arbitrary. Then, since d(xi,C0)≤e(Cni,C0)≤h(Cni,C0)→0 as i→∞, there exists i0∈ℕ such that d(xi,C0)<ϵ for every i≥i0.
LetDϵ={y∈X:d(y,C0)≤ϵ}. Then, Dϵ is closed and convex in X. Indeed, for y1,y2∈Dϵ 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
(3.19)d(ty1⊕(1-t)z2,tz1⊕(1-t)z2)≤td(y1,z1).
In the same way, considering the comparison triangle of(y1,y2,z2), we have that
(3.20)d(ty1⊕(1-t)y2,ty1⊕(1-t)z2)≤(1-t)d(y2,z2).
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∈Dϵ. This shows that Dϵ is convex. It is obvious that Dϵ is closed since the function d(·,C0) is continuous.
Sincexi∈Dϵ for i≥i0, using Theorem 2.2, we have that x∈Dϵ; that is, d(x,C0)≤ϵ. Since ϵ is arbitrary and C0 is closed, we obtain that x∈C0, and hence Δ-LsnCn⊂C0. 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. LetBH be the open unit ball of a complex Hilbert space H with an inner product 〈·,·〉 and induced norm ∥·∥. For an orthonormal basis {ei:i∈I} of H, let HR={z∈H:Im〈z,ei〉=0∀i∈I}. Then, a real Hilbert ball (B,ρ) is a metric space defined by B=BH∩HR and ρ:B×B→ℝ by(4.1)ρ(x,y)=arctanh1-(1-‖x‖2)(1-‖y‖2)1-|〈x,y〉|2
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.
LetB be a real Hilbert ball with the metric ρ. Let {Ti:i∈I} be a family of nonexpansive mappings of B into itself with a nonempty set F of their common fixed points. Let {αn(i):i∈I,n∈ℕ} be nonnegative real numbers in [0,1] such that liminfn→∞αn(i)<1 for each i∈I. For x∈B, generate an iterative sequence {xn} by x1=x, C0=B, and
(4.2)yn(i)=αn(i)xn⊕(1-αn(i))Tixn,foreachi∈I,Cn={z∈B:sup⁡i∈Iρ(z,yn(i))≤ρ(z,xn)}∩Cn-1,xn+1=PCnx
for all n∈ℕ. Then, {xn} is well defined and converges to PFx∈B.Proof.
Since eachF(Ti) is closed and convex, so is F=⋂i∈IF(Ti). For z∈F, it follows that
(4.3)ρ(z,yn(i))2≤αn(i)ρ(z,xn)2+(1-αn(i))ρ(z,Tixn)2-αn(i)(1-αn(i))ρ(xn,Tixn)2≤αn(i)ρ(z,xn)2+(1-αn(i))ρ(Tiz,Tixn)2≤ρ(z,xn)2
for all i∈I, and thus supi∈Iρ(z,yn(i))≤ρ(z,xn) for every n∈ℕ. Therefore, we have F⊂Cn and Cn is nonempty for n∈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∈ℕ. Since {Cn} is decreasing with respect to inclusion, as in Example 3.4, we have that {Cn} converges to C=⋂n=1∞Cn in the sense of Δ-Mosco. By Theorem 3.2, we have that {xn} converges to x0=PCx. Since x0∈Cn for all n∈ℕ, we have that ρ(x0,yn(i))≤ρ(x0,xn) for all n∈ℕ and i∈I. Fix i∈I arbitrarily, and let {αnk(i)} be a subsequence of {αn(i)} converging to α0(i)∈[0,1[. Then, since ρ(xn,yn(i))=(1-αn(i))ρ(xn,Tixn), we have that
(4.4)ρ(x0,Tix0)≤ρ(x0,xnk)+ρ(xnk,Tixnk)+ρ(Tixnk,Tix0)≤2ρ(x0,xnk)+11-αnk(i)ρ(xnk,ynk(i))≤2ρ(x0,xnk)+11-αnk(i)(ρ(xnk,x0)+ρ(x0,ynk(i)))≤2(1+11-αnk(i))ρ(x0,xnk)
for k∈ℕ, and, as k→∞, we obtain that x0=Tix0; that is, x0∈F(Ti). Since i∈I is arbitrary, we have that PCx=x0∈F⊂C, 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.
LetB be a real Hilbert ball and T:B→B a nonexpansive mapping. Let {αn} be a nonnegative real sequence in [0,1] such that liminfn→∞αn<1. For x∈B, generate an iterative sequence {xn} by x1=x, C0=B, and
(4.5)yn=αnxn⊕(1-αn)Txn,foreachi∈I,Cn={z∈B:ρ(z,yn)≤ρ(z,xn)}∩Cn-1,xn+1=PCnx
for all n∈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=1∞Cn is nonempty. Moreover, in this case the limit of {xn} is PF(T)x=P⋂n=1∞Cnx.Proof.
First we show that{xn} is well defined. Since x1 is given and y1∈C1, C1 is nonempty. Suppose that C1,C2,…,Cn-1 are nonempty. Then, x1,x2,…,xn and y1,y2,…,yn are defined. Let r=max1≤k≤nρ(x,Txk) and D={w∈B:ρ(x,w)≤r}. Then, since D is nonempty, bounded, closed, and convex, there exists a metric projection PD:B→D. Since PD is nonexpansive, it follows that PDT∣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=PDTu. Since Txk∈D for k=1,2,…,n and D is convex, it follows from the definition of the metric projection that
(4.6)ρ(Tu,u)=ρ(Tu,PDTu)≤ρ(Tu,(1-t)u⊕tTxk)
for t∈]0,1[. Thus, we have that
(4.7)ρ(Tu,u)2≤ρ(Tu,(1-t)u⊕tTxk)2≤(1-t)ρ(Tu,u)2+tρ(Tu,Txk)2-t(1-t)ρ(u,Txk)2,
and thus
(4.8)ρ(Tu,u)2≤ρ(Tu,Txk)2-(1-t)ρ(u,Txk)2≤ρ(u,xk)2-(1-t)ρ(u,Txk)2.
Tending t↓0, we have that 0≤ρ(Tu,u)2≤ρ(u,xk)2-ρ(u,Txk)2, and hence ρ(u,Txk)≤ρ(u,xk) for k=1,2,…,n. It gives us that
(4.9)ρ(u,yk)2=ρ(u,αkxk⊕(1-αk)Txk)2≤αkρ(u,xk)2+(1-αk)ρ(u,Txk)2≤ρ(u,xk)2
for all k=1,2,…,n, and hence u∈Cn. 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 Theorem4.1 for a single mapping that (i) implies (ii). We also have that {xn} converges to PF(T)x=P⋂n=1∞Cnx. 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 x0∈B. From the definition of subsequence, for any n∈ℕ, there exists k0∈ℕ such that nk>n for all k≥k0. Since {Cn} is decreasing with respect to inclusion, we have xnk⊂Cnk-1⊂Cn for all k≥k0. By Theorem 2.2, we have that x0∈Cn for every n∈ℕ, and hence (iv) holds. Lastly, we show that (iv) implies (i). Assume that C=⋂n=1∞Cn is nonempty. By Theorem 3.2, {xn} converges to PCx. Then, as in the proof of Theorem 4.1, we have that PCx∈F(T), and thus (i) holds. Consequently, these four conditions are all equivalent.



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*Source: 582475-2011-01-19.xml*