# Uniform Convexity and Convergence of a Sequence of Sets in a Complete Geodesic Space

Authors: Yasunori Kimura; Shuta Sudo
Journal: Abstract and Applied Analysis
 (2022)

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## Abstract

In this paper, we first introduce two new notions of uniform convexity on a geodesic space, and we prove their properties. Moreover, we reintroduce a concept of the set-convergence in complete geodesic spaces, and we prove a relation between the metric projections and the convergence of a sequence of sets.


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## Body

## 1. Introduction

There are a lot of works dealing with the relation between convergence of a sequence of sets and convergence of a sequence of projections corresponding to it. In particular, the following theorem on a reflexive and strictly convex real Banach space is one of the important results.Theorem 1 (Tsukada [1]).
LetE be a strictly convex and reflexive real Banach space satisfying the Kadec-Klee property. Let Cn⊂2E be a sequence of nonempty closed convex sets and C0⊂E a nonempty closed convex subset such that Cn converges to C0 in the sense of Mosco. Then PCnx converges strongly to PC0x∈E for any x∈X, where PK:E⟶K is the metric projection of E onto a nonempty closed convex subset K of X.Since a uniformly convex real Banach space is strictly convex, reflexive, and satisfying the Kadec-Klee property, this theorem is true in uniformly convex real Banach spaces. Moreover, since a real Hilbert space is a uniformly convex real Banach space, it is also true in real Hilbert spaces.On the other hand, we know that a Hadamard space is another generalization of Hilbert spaces. It is defined as a complete metric space having a particular convexity structure and it also has various useful properties that Hilbert spaces have. Kimura [2] introduced Δ-Mosco convergence in complete geodesic spaces using a notion of asymptotic centre instead of weak convergence.Theorem 2 (Kimura [2]).
LetX be a complete CAT(0) space. Let Cn⊂2X be a sequence of nonempty closed convex sets and C0⊂X a nonempty closed convex subset such that Cn converges to C0 in the sense of Δ-Mosco. Then PCnx converges to PC0x∈X for any x∈X, where PK:X⟶K is the metric projection of X onto a nonempty closed convex subset K of X.Moreover, Kimura and Satô [3] introduced Δκ-Mosco convergence in complete CAT(κ) spaces and obtain the following result:Theorem 3 (Kimura and Satô [3]).
LetX be a complete admissible CAT(κ) space for κ>0. Let Cn⊂2X be a sequence of nonempty closed convex sets and C0⊂X a nonempty closed convex subset such that Cn converges to C0 in the sense of Δκ-Mosco. Then PCnx converges to PC0x∈X for any x∈X, where PK:X⟶K is the metric projection of X onto a nonempty closed convex subset K of X.In this work, we introduce a new concept of the set-convergence and we obtain a similar result as above under the assumptions that both uniformly convex real Banach spaces and complete CAT(0) spaces have.


## 2. Preliminaries

A functiong:0,∞⟶0,∞ is said to be a gauge if g is strictly increasing, continuous, and g0=0. We know that if αn is a real sequence of 0,∞ such that gαn⟶0 for some gauge function g, then αn⟶0.LetE be a real Banach space. Then the following propositions are equivalent:
(i)
E is uniformly convex(ii)
Forr>0, z∈E, and xn,yn⊂E, it holds that limn⟶∞xn−yn=0 whenever(1)limn⟶∞xn−z=limn⟶∞yn−z=limn⟶∞12xn+12yn−z=r(iii)
For anyK>0, there exists a convex gauge function g¯K:0,∞⟶0,∞ such that(2)tx+1−ty2≤tx2+1−ty2−t1−tg¯Kx−yfor anyt∈0,1, where x≤K and y≤K for x,y∈EFor more details about the properties of uniformly convex real Banach spaces, see [4].LetX,d be a metric space and let x,y∈X. A geodesic path from x to y is a mapping γ:0,dx,y⟶X such that γ0=x, γdx,y=y and dγs,γt=s−t for any s,t∈0,dx,y. Let D∈0,∞. If for any x,y∈X such that dx,y<D, a geodesic path from x to y exists, then we say that X is D-geodesic. Moreover, if such a geodesic path is unique for each pair of points, then X is said to be D-uniquely geodesic. In a D-uniquely geodesic space, the image of a geodesic path from x to y is called a geodesic segment joining x and y and it is denoted by x,y. For x,y∈X and t∈0,1, there exists a unique point z∈x,y such that dx,z=1−tdx,y and dy,z=tdx,y. We denote it by z=tx⊕1−ty. A geodesic triangle with vertices x,y,z∈X is the union of geodesic segments x,y, y,z and z,x. We denote it by Δx,y,z.To define a CAT(κ) space, we use the following notation called a model space. For κ=0, the two-dimensional model space Mκ2=M02 is the two-dimensional Euclidean space E2 with the metric induced from the Euclidean norm. For κ>0, Mκ2 is the two-dimensional sphere 1/κS2 whose metric is a length of a minimal great arc joining each two points. For κ<0, Mκ2 is the two-dimensional hyperbolic space 1/−κℍ2 with the metric defined by a usual hyperbolic distance.The diameter ofMκ2 is denoted by Dκ, defined by
(3)Dκ=∞κ≤0,π/κκ>0.We know thatMκ2 is a Dκ-uniquely geodesic space for each κ∈ℝ.Letκ∈ℝ. For Δx,y,z in a geodesic space X satisfying that dx,y+dy,z+dz,x<2Dκ, there exist points x¯,y¯,z¯∈Mκ2 such that dx,y=dMκ2x¯,y¯, dy,z=dMκ2y¯,z¯, dz,x=dMκ2z¯,x¯. We call the triangle having vertices x¯, y¯ and z¯ in Mκ2 a comparison triangle of Δx,y,z. Notice that it is unique up to an isometry of Mκ2. For a specific choice of comparison triangles, we denote it by Δ¯x¯,y¯,z¯. A point p¯∈x¯,y¯ is called a comparison point for p∈x,y if dx,p=dMκ2x¯,p¯.Letκ∈ℝ and X a Dκ-geodesic space. If for any x,y,z∈X with dx,y+dy,z+dz,x<2Dκ, for any p,q∈Δx,y,z, and for their comparison points p¯,q¯∈Δ¯x¯,y¯,z¯, the CAT(κ) inequality
(4)dp,q≤dMκ2p¯,q¯holds, then we call X a CAT(κ) space. It is well known that any CAT(κ) space is also a CAT(κ′) space whenever κ<κ′. Therefore, a CAT(κ) space is a CAT(0) space for any κ<0.LetX be a CAT(κ) space for κ∈ℝ. If
(5)du,v<Dκ2for any u,v∈X, then we say that X is admissible. A CAT(κ) space is always admissible when κ≤0.A subsetC of a D-uniquely geodesic space X is said to be convex if tx⊕1−ty∈C for every x,y∈C and t∈0,1. For a subset S of D-uniquely geodesic space X, a closed convex hull of S is defined as the intersection of all closed convex sets including S, and we denote it by clco S.LetX be a complete admissible CAT(κ) space for κ∈ℝ and C a nonempty closed convex subset of X. Then for x∈X, there exists a unique point yx∈C such that
(6)dx,yx=dx,C.We call such a mapping defined byPCx=yx, the metric projection of X onto C.LetX be a CAT(0) space. From the CAT(0) inequality, it is easy to see that
(7)dtx⊕1−ty,z2≤tdx,z2+1−tdy,z2−t1−tdx,y2for every x,y,z∈X and t∈0,1.The following lemma shows that a CAT(0) space has a similar property to the uniform convexity of Banach spaces.Lemma 4.
LetX be a CAT(0) space. For r>0, a point z∈X and two sequences xn,yn⊂X, it holds that limn⟶∞dxn,yn=0 whenever
(8)limn⟶∞dxn,z=limn⟶∞dyn,z=limn⟶∞d12xn⊕12yn,z=r.Proof.
Forxn,yn⊂X, z∈X and r∈0,∞, if
(9)limn⟶∞dxn,z=limn⟶∞dyn,z=limn⟶∞d12xn⊕12yn,z=r,then
(10)d12xn⊕12yn,z2≤12dxn,z2+12dyn,z2−14dxn,yn2,which implies
(11)dxn,yn2≤2dxn,z2+2dyn,z2−4d12xn⊕12yn,z2.
Lettingn⟶∞, we have
(12)limn⟶∞dxn,yn=0.
This is the desired result.LetX be a metric space. For a point x∈X and a nonempty subset C⊂X, the distance between them is defined by dx,C=infy∈Cdx,y.LetX be a metric space. A function f:X⟶−∞,∞ is said to be lower semicontinuous if for x0∈X and xn⊂X,
(13)fx0≤liminfn⟶∞fxnwhenever xn⟶x0. Moreover, a function f:X⟶−∞,∞ is said to be proper if there exists a point z0∈X such that fz0<∞ and the domain off defined by
(14)domf=x∈X∣fx<∞.LetX be a uniquely geodesic space. A function f:X⟶−∞,∞ is said to be convex if for x,y∈X and t∈0,1, f satisfies
(15)ftx⊕1−ty≤tfx+1−tfy.


## 3. Uniform Convexity of a Complete Geodesic Space

In the following, we always suppose that for anyx,y in geodesic space X, a geodesic joining x to y is unique.A geodesic spaceX is said to satisfy the condition (D) if:
(16)dtx⊕1−ty,z2≤tdx,z2+1−tdy,z2for x,y,z∈X and t∈0,1.We introduce two new concepts of uniform convexity on a geodesic space.LetX be a geodesic space. X is said to be sequentially uniformly convex if X satisfies the condition (D) and, for r>0, a point z∈X and two sequences xn,yn⊂X, it holds that
(17)limn⟶∞dxn,yn=0whenever
(18)limn⟶∞dxn,z=limn⟶∞dyn,z=limn⟶∞d12xn⊕12yn,z=r.LetX be a geodesic space. X is said to be uniformly convex if for any K>0 there exists a convex gauge function g¯K:0,∞⟶0,∞ such that for any t∈0,1,
(19)dtx⊕1−ty,z2≤tdx,z2+1−tdy,z2−t1−tg¯Kdx,y,where x,y,z∈X with dx,z≤K and dy,z≤K.Uniformly convex real Banach spaces and CAT(0) spaces are uniformly convex in this sense.Theorem 5.
LetX be a uniformly convex geodesic space. Then, X is sequentially uniformly convex.Proof.
Letx,y,z∈X and let K=maxdx,z,dy,z<∞. Then there exists a convex gauge function g¯K:0,∞⟶0,∞, satisfying that
(20)dtx⊕1−ty,z2≤tdx,z2+1−tdy,z2−t1−tg¯Kdx,yfor any t∈0,1 and hence
(21)dtx⊕1−ty,z2≤tdx,z2+1−tdy,z2.
That is,X satisfies the condition (D).
Moreover, forr>0, z∈X and xn,yn⊂X, if
(22)limn⟶∞dxn,z=limn⟶∞dyn,z=limn⟶∞d12xn⊕12yn,z=r,then there exists n0∈ℕ such that
(23)dxn,z≤r+1,dyn,z≤r+1for any n≥n0. From uniform convexity of X, there exists a convex gauge function g¯r+1, satisfying that
(24)d12xn⊕12yn,z2≤12dxn,z2+12dyn,z2−14g¯r+1dxn,yn.
It follows that(25)0≤g¯r+1dxn,yn≤2dxn,z2+2dyn,z2−4d12xn⊕12yn,z2⟶2r2+2r2−4r2=0.
Therefore, we obtaing¯r+1dxn,yn⟶0 and hence limn⟶∞dxn,yn=0, that is, X is sequentially uniformly convex.Theorem 6.
LetX be a sequentially uniformly convex geodesic space. For r>0 and x,y,z∈X with x≠y, ifdx,z=dy,z=r, then d1/2x⊕1/2y,z<r.Proof.
Forr>0 and x,y,z∈X with x≠y, we suppose that dx,z=dy,z=r. If r≤d1/2x⊕1/2y,z, then, since
(26)r2≤d12x⊕12y,z2≤12dx,z2+12dy,z2=r2,we have d1/2x⊕1/2y,z=r. From the sequential uniform convexity of X, we have dx,y=0. This is a contradiction. Therefore, we have d1/2x⊕1/2y,z<r. This is the desired result.Theorem 7.
LetX be a sequentially uniformly convex complete geodesic space and let C be a nonempty closed convex subset of X. Then, for x∈X, there exists a unique point y0∈C such that dx,y0=dx,C.Proof.
Forx∈X, let d=dx,C. Then, for n∈ℕ, we can take a sequence yn⊂C such that
(27)d≤dx,yn≤d+1n.
Then, we havedx,yn⟶d. Suppose that yn is not a Cauchy sequence. That is, there exists ε>0 such that for any i∈ℕ, there exist mi,ni≥i such that dymi,yni≥ε. In this way, we take two subsequences ymi,yni⊂yn. Then,
(28)limi⟶∞dx,ymi=limi⟶∞dx,yni=dand we have
(29)d2≤dx,12ymi⊕12yni2≤12dx,ymi2+12dx,yni2⟶d2.
Hence, from the sequential uniform convexity ofX, we have
(30)limi⟶∞dymi,yni=0.
This is a contradiction and thusyn is a Cauchy sequence. Since X is complete and C is closed, there exists y0∈C such that yn⟶y0. Therefore, we have
(31)dx,y0=limn⟶∞dx,yn=d=dx,C.
Next, we show the uniqueness ofy0. Suppose that y0,z0∈C satisfy y0≠z0 and dx,y0=dx,z0=dx,C. Then, from Theorem 6, we have
(32)dx,12y0⊕12z0<dx,C.
This is a contradiction. Therefore, forx∈X, there exists a unique point y0∈C such that dx,y0=dx,C.LetX be a sequentially uniformly convex complete uniquely geodesic space and let C be a nonempty closed convex subset of X. Then for x∈X, there exists a unique point yx∈C such that
(33)dx,yx=dx,C.We call such a mappingPC:X⟶C defined by PCx=yx, the metric projection of X onto C.Theorem 8.
LetX be a sequentially uniformly convex complete geodesic space and Cn⊂2X a sequence of nonempty bounded closed convex subsets which is decreasing with respect to inclusion, that is, Cn+1⊂Cn for any n∈ℕ. Then, ⋂n=1∞Cn is nonempty.Proof.
SinceCn is nonempty bounded closed convex subset for n∈ℕ, for x∈X, we can take a sequence xn⊂X by xn=PCnx, where PK is the metric projection of X onto a nonempty closed convex subset K of X. Then dx,xn is a bounded increasing real sequence and hence dx,xn has a limit c∈0,∞. That is, we have
(34)limn⟶∞dx,xn=c.
First, we show thatxn converges to some point x0∈X. If c=0, then, since dx,PCnx⟶0, we have xn⟶x0 as x0=x. Hence, we may suppose that c>0. Suppose that xn is not a Cauchy sequence. That is, there exists ε>0 such that for any i∈ℕ, there exist mi,ni≥i such that dxmi,xni≥ε. Without loss of generality, we can suppose that mi≥ni. In this way, we take two subsequences xmi,xni⊂xn. Then,
(35)limi⟶∞dx,xmi=limi⟶∞dx,xni=c.
Sincexmi,xni∈Cni, we have
(36)dx,xni=dx,PCnix≤dx,12xmi⊕12xni2≤12dx,xmi2+12dx,xni2and thus
(37)limi⟶∞dx,12xmi⊕12xni=c.
From the sequential uniform convexity ofX, we have limi⟶∞dxmi,xni=0. This is a contradiction. Therefore, xn is a Cauchy sequence and thus there exists x0∈X such that xn⟶x0.
We show thatx0∈⋂n=1∞Cn. For n0∈ℕ, xn∈Cn0 for n≥n0 and thus x0∈Cn0. Therefore, x0∈⋂n=1∞Cn and it completes the proof.


## 4.Δ-Convergence

LetX be a metric space and xn⊂X a bounded sequence. An asymptotic centreACxn of xn is defined by
(38)ACxn=u∈X∣limsupn⟶∞du,xn=infx∈Xlimsupn⟶∞dx,xn.Lemma 9.
LetX be a sequentially uniformly convex complete geodesic space and let xn⊂X be a bounded sequence. Then, there exists a point u∈X such that
(39)limsupn⟶∞du,xn=infx∈Xlimsupn⟶∞dx,xn.
That is,ACxn is nonempty. Moreover, ACxn is bounded, closed and convex.Proof.
LetM=infx∈Xlimsupn⟶∞dx,xn and define Ck⊂2X by
(40)Ck=u∈X∣M≤limsupn⟶∞du,xn≤M+1kfor any k∈ℕ. Then, for any k∈ℕ, Ck is nonempty and bounded. If um⊂Ck and um⟶u0, then
(41)M≤limsupn⟶∞du0,xn≤du0,um+limsupn⟶∞dum,xn≤du0,um+M+1k.
Lettingm⟶∞, we have
(42)M≤limsupn⟶∞du0,xn≤M+1kand hence u0∈Ck. Moreover, for u,v∈Ck and t∈0,1, we have
(43)M2≤limsupn⟶∞dtu⊕1−tv,xn2≤tlimsupn⟶∞du,xn2+1−tlimsupn⟶∞dv,xn2≤M+1k2.
It impliestu⊕1−tv∈Ck. Therefore, Ck is closed and convex for any k∈ℕ. Moreover, Ck is decreasing with respect to inclusion. Hence, we have
(44)⋂k=1∞Ck=⋂k=1∞u∈X∣M≤limsupn⟶∞du,xn≤M+1k≠∅and thus
(45)ACxn=⋂k=1∞u∈X∣M≤limsupn⟶∞du,xn≤M+1k≠∅.
Also, we know thatACxn is closed and convex.If we suppose uniform convexity forX, we can prove that an asymptotic centre is a singleton.Theorem 10.
LetX be a uniformly convex complete geodesic space and let xn be a bounded sequence of X. Then, ACxn is a singleton.Proof.
LetM=infx∈Xlimsupn⟶∞dx,xn. If u,v∈ACxn with u≠v, then, since xn⊂X is bounded, there exists K>0 such that du,xn≤K and dv,xn≤K for all n∈ℕ. Then, there exists a convex gauge function g¯K, and we have
(46)d12u⊕12v,xn2≤12du,xn2+12dv,xn2−14g¯Kdu,vand hence
(47)limsupn⟶∞d12u⊕12v,xn2≤12limsupn⟶∞du,xn2+12limsupn⟶∞dv,xn2−14g¯Kdu,v≤M2−14g¯Kdu,v.
Sincedu,v≠0, we have g¯Kdu,v>0 and thus
(48)limsupn⟶∞d12u⊕12v,xn2<M2.
This is a contradiction. Therefore,ACxn is a singleton.Letxn⊂X be a bounded sequence and x0∈X. We say xnΔ-converges to aΔ-limit x0 if x0 is the unique asymptotic centre of any subsequences of xn, and we denote it by xn⇀Δx0.LetX be a geodesic space. X is said to satisfy the condition (ΔC), if any nonempty closed convex subset C⊂X is Δ-closed, that is, if xn⊂C and xn⇀Δx0, then x0∈C.CAT(0) spaces satisfy the condition (ΔC). Let E be a uniformly convex real Banach space. Then, the following propositions are equivalent:
(i)
For any bounded sequencexn⊂E, xnΔ-converges to x0∈E if and only if xn converges weakly to x0∈E(ii)
E satisfies the condition (ΔC)(iii)
E satisfies Opial’s conditionSee [5] for details.The following two theorems can be proved by the same method as the corresponding results in [6–8].Theorem 11 (Bačák [6], Kirk and Panyanak [8]).
LetX be a uniformly convex complete geodesic space. Then, for any bounded sequence of X has a Δ-convergent subsequence.Corollary 12.
LetX be a uniformly convex complete geodesic space satisfying the condition (ΔC) and let C be a nonempty bounded closed convex subset of X. Then, for any sequence in C has a Δ-convergent subsequence and its Δ-limit belongs to C.Theorem 13 (Bačák [6], He, Fang, Lopez and Li [7]).
LetX be a uniformly convex complete geodesic space satisfying the condition (ΔC). Let f:X⟶−∞,∞ be a proper lower semicontinuous function and xn⊂X a sequence such that xn⇀Δx0∈domf. Then,
(49)fx0≤liminfn⟶∞fxn.Corollary 14 (Δ-lower semicontinuity of the distance function).
LetX be a uniformly convex complete geodesic space satisfying the condition (ΔC) and let z∈X be a point. Let xn⊂X be a sequence such that xn⇀Δx0. Then,
(50)dx0,z≤liminfn⟶∞dxn,z.Lemma 15.
LetX be a uniformly convex complete geodesic space and let xn⊂X be a sequence such that Δ-converges to x0∈X. Then,
(51)12xn⊕12x0⇀Δx0.Proof.
Sincexn is bounded, so is 1/2xn⊕1/2x0. For any subsequence 1/2xni⊕1/2x0 of 1/2xn⊕1/2x0, if y0=AC1/2xni⊕1/2x0, then, we have
(52)limsupi⟶∞dy0,xni≤limsupi⟶∞dy0,12xni⊕12x0+limsupi⟶∞d12xni⊕12x0,xni≤limsupi⟶∞dx0,12xni⊕12x0+limsupi⟶∞d12xni⊕12x0,xni≤12limsupi⟶∞dx0,xni+12limsupi⟶∞dx0,xni=limsupi⟶∞dx0,xni.
Therefore, sinceACxni=x0, we have y0=x0. Hence, x0 is the unique asymptotic centre of any subsequence of 1/2xn⊕1/2x0 and it completes the proof.Theorem 16 (Δ-Kadec-Klee property).
LetX be a uniformly convex complete geodesic space satisfying the condition (ΔC) and xn⊂X a sequence such that xn⇀Δx0 and dxn,p⟶dx0,p for some p∈X. Then, xn⟶x0.Proof.
Letyn⊂X be a sequence such that yn=x0 for any n∈ℕ. Since xn⇀Δx0, From Lemma 15, we have 1/2xn⊕1/2x0⇀Δx0. Then, since d·,p is Δ-lower semicontinuous and dxn,p⟶dx0,p, we have
(53)dx0,p2≤liminfn⟶∞d12xn⊕12x0,p2≤limsupn⟶∞d12xn⊕12x0,p2≤12limsupn⟶∞dxn,p2+12limsupn⟶∞dx0,p2=dx0,p2and thus limn⟶∞d1/2xn⊕1/2x0,p=dx0,p. Therefore, we obtain
(54)limn⟶∞dxn,p=limn⟶∞dyn,p=limn⟶∞d12xn⊕12yn,p=dx0,p.
Ifdx0,p=0, then xn⟶p=x0. If dx0,p>0, from sequential uniform convexity of X, we obtain limn⟶∞dxn,yn=0. Since yn=x0 for any n∈ℕ, we have xn⟶x0.


## 5. Convergence of a Sequence of Sets

LetCn⊂2X be a sequence of nonempty closed convex subsets of a uniformly convex complete geodesic space X. Δ-Mosco convergence is defined by using a notion of asymptotic centre by Kimura [2]. First, we define subsets d‐LinCn and Δ‐LsnCn of X as follows: x∈d‐LinCn if and only if there exists xn⊂X such that xn⟶x and xn∈Cn for all n∈ℕ; y∈Δ‐LsnCn if and only if there exist a bounded sequence yi⊂X and a subsequence ni of ℕ such that y=ACyi and yi∈Cni for all i∈ℕ. If a subset C0 of X satisfies that C0=d‐LinCn=Δ‐LsnCn, we say that Cn converges to C0 in the sense of Δ-Mosco.Here, we introduce a new concept of the set-convergence. We define subsetsΔ¯‐LsnCn of X as follows: y∈Δ¯‐LsnCn if and only if there exists a bounded sequence yi⊂X and a subsequence ni of ℕ such that yi⇀Δy and yi∈Cni for all i∈ℕ.Since a convergent sequence is aΔ-convergent sequence, the inclusion
(55)d‐LinCn⊂Δ¯‐LsnCnis always true. If a subset C0 of X satisfies that C0=d‐LinCn=Δ¯‐LsnCn, we say that Cn converges to C0 in the sense of Δ¯-Mosco. Furthermore, the following inclusion holds:
(56)d‐LinCn⊂Δ¯‐LsnCn⊂Δ‐LsnCn.Therefore, ifCn converges to C0 in the sense of Δ-Mosco, then Cn converges to C0 in the sense of Δ¯-Mosco.Lemma 17.
LetX be a uniformly convex complete geodesic space satisfying the condition (ΔC) and Cn⊂2X a sequence of nonempty closed convex sets which is decreasing with respect to inclusion, that is, Cn+1⊂Cnfor any n∈ℕ. ifC0=⋂n=1∞Cn is nonempty, then Cn converges to C0 in the sense of Δ¯-Mosco.Proof.
We show thatΔ¯‐LsnCn⊂C0⊂d‐LinCn. If x∈C0=⋂n=1∞Cn, then x∈Cn for any n∈ℕ. Let xn be a sequence of X such that xn=x for any n∈ℕ. Then, xn⟶x and xn∈Cn for any n∈ℕ. Therefore, we obtain x∈d‐LinCn and hence C0⊂d‐LinCn.
Next, we show thatΔ¯‐LsnCn⊂C0. If y∈Δ¯‐LsnCn, then there exists a sequence yi⊂X and ni⊂ℕ such that yi∈Cni for any i∈ℕ and yi⇀Δy. For any i0∈ℕ, if i≥i0, then yi⊂Cni0. Therefore, from the condition (ΔC) of X, we have yi⇀Δy∈Cni0. Since i0∈ℕ is arbitrarily, we obtain y∈⋂n=1∞Cn=C0. Hence, we have Δ¯‐LsnCn⊂C0⊂d‐LinCn.
Sinced‐LinCn⊂Δ¯‐LsnCn is always true, we have
(57)Δ¯‐LsnCn=C0=d‐LinCn.
That is,Cn converges to C0 in the sense of Δ¯-Mosco.Theorem 18.
LetX be a uniformly convex complete geodesic space satisfying the condition (ΔC). Let Cn⊂2X be a sequence of nonempty closed convex sets and C0 a nonempty closed convex subset of X. If Cn converges to C0 in the sense of Δ¯-Mosco, then PCnx converges to PC0x∈X for any x∈X, where PK:X⟶K is the metric projection of X onto a nonempty closed convex subset K of X.Proof.
Fixx∈X arbitrarily. Since PC0x∈C0=d‐LinCn, there exists a sequence xn⊂X such that xn⟶PC0x and xn∈Cn for every n∈ℕ. Since dx,PCnx≤dx,xn for any n∈ℕ and xn is bounded, PCnx is also bounded. Moreover, letting n⟶∞, we have
(58)limsupn⟶∞dx,PCnx≤dx,PC0x.
Here, we take a subsequencePCnix of PCnx arbitrarily. Since PCnix is bounded, there exists a subsequence PCnijx of PCnix and p0∈X such that
(59)PCnijx⇀Δp0.
Letpj=PCnijx for any j∈ℕ. Then, since
(60)limsupj⟶∞dx,pj≤dx,PC0x,for any ε>0 there exists j0∈ℕ such that for any j≥j0, it holds that
(61)dx,pj≤dx,PC0x+ε.
LetD=y∈X∣dx,y≤dx,PC0x+ε. Then we have pj∈D for any j≥j0. Since D is closed and convex, from the condition (ΔC) of X, we obtain pj⇀Δp0∈D. Therefore, we have
(62)dx,p0≤dx,PC0x+ε.
Lettingε↘0, we obtain
(63)dx,p0≤dx,PC0x.
Sincep0∈Δ¯‐LsnCn=C0, we have dx,PC0x≤dx,p0 and thus
(64)dx,PC0x=dx,p0.
Hence, we obtainpj⇀ΔPC0x. Since dx,· is Δ-lower semicontinuous and pj⇀ΔPC0x, we have
(65)dx,PC0x≤liminfj⟶∞dx,pjand hence
(66)dx,PC0x≤liminfj⟶∞dx,pj≤limsupj⟶∞dx,pj≤limsupn⟶∞dx,PCnx≤dx,PC0x.
Therefore,dx,pj⟶dx,PC0x. Since pj⇀ΔPC0x and dx,pj⟶dx,PC0x, from the Δ-Kadec-Klee property of X, we obtain pj⟶PC0x. Since for any subsequence PCnix of PCnx, there exists a subsequence PCnijx of PCnix such that PCnijx⟶PC0x, we have PCnx⟶PC0x.LetX be a uniformly convex complete geodesic space. If X has the condition (ΔC), we can show that convergence of a sequence of metric projections from Δ¯-Mosco convergence. If a sequence of sets which is decreasing with respect to inclusion, then we can show it on sequentially uniformly convex complete geodesic spaces without the condition (ΔC).Theorem 19.
LetX be a sequentially uniformly convex complete geodesic space and Cn⊂2X a sequence of nonempty closed convex sets which is decreasing with respect to inclusion, that is, Cn+1⊂Cn for any n∈ℕ. Suppose that C0=⋂n=1∞Cn is nonempty. Then, PCnx converges to PC0x∈X for any x∈X, where PK:X⟶K is the metric projection of X onto a nonempty closed convex subset K of X.Proof.
Fixx∈X arbitrarily. Since C0=⋂n=1∞Cn≠∅, if p∈C0, then for any n∈ℕ, we have
(67)dx,PCnx≤dx,pand hence dx,PCnx is bounded. Furthermore, since Cn+1⊂Cn for any n∈ℕ, we have
(68)dx,PCnx≤dx,PCn+1xand thus dx,PCnx is increasing. Therefore, dx,PCnx has a limit
(69)c=limn⟶∞dx,PCnx.
First, we show thatPCnx converges to some point x0∈X. Since PCnx⟶x if c=0, we may assume that c>0. Suppose that PCnx is not a Cauchy sequence. That is, there exist ε>0 such that for any i∈ℕ, there exist mi,ni≥i such that dPCmix,PCnix≥ε. Without loss of generality, we can suppose that mi≥ni. In this way, we take two sequences PCmix,PCnix⊂PCnx. Then, PCmix and PCnix satisfies
(70)limi⟶∞dx,PCmix=limi⟶∞dx,PCnix=c.
SincePCmix,PCnix∈Cni, we obtain
(71)dx,PCnix2≤dx,12PCmix⊕12PCnix2≤12dx,PCmix2+12dx,PCnix2.
Lettingi⟶∞, we obtain
(72)limi⟶∞dx,12PCmix⊕12PCnix=c.
From sequential uniform convexity ofX, we have
(73)limi⟶∞dPCmix,PCnix=0.
This is contradictory to(74)dPCmix,PCnix≥ε>0and hence PCnx is a Cauchy sequence. Therefore, since X is complete, there exists a point x0∈X such that PCnx⟶x0.
Next, we show thatx0∈C0. For any n0∈ℕ, since PCnx⊂Cn0 if n≥n0 and Cn0 is closed, we have x0∈Cn0 and thus x0∈⋂n=1∞Cn.
Finally, we show thatx0=PC0x. For any n∈ℕ, form the property of the metric projection, we have
(75)dx,PCnx≤dx,PC0x≤dx,x0.
Lettingn⟶∞, we obtain
(76)dx,PC0x=dx,x0and hence x0=PC0x.
Consequently, we havePCnx⟶PC0x for any x∈X.


## 6. Convergence of a Sequence of Sets in CAT(κ) Spaces

In this section, we will consider CAT(κ) spaces for κ∈ℝ.LetX be a metric space and let xn be a bounded sequence of X. The asymptotic radiusrxn of xn defined by
(77)rxn=infx∈Xlimsupn⟶∞dx,xn.LetCn⊂2X be a sequence of nonempty closed convex subsets of a complete admissible CAT(κ) space for κ∈ℝ. Δκ-Mosco convergence is defined by Kimura and Satô [3]. First, we define subsets d‐LinCn and Δκ‐LsnCn of X as follows: d‐LinCn is the same as in the case of uniformly convex complete geodesic spaces, that is, x∈d‐LinCn if and only if there exists xn⊂X such that xn⟶x and xn∈Cn for all n∈ℕ; y∈Δκ‐LsnCn if and only if there exist a sequence yi⊂X and a subsequence ni of ℕ such that ryi<Dκ/2, y=ACyi and yi∈Cni for all i∈ℕ. If a subset C0 of X satisfies that C0=d‐LinCn=Δκ‐LsnCn, it is said that Cn converges to C0 in the sense of Δκ-Mosco. The notion of Δκ-Mosco convergence coincides with Δ-Mosco convergence in complete CAT(κ) spaces for κ≤0.We define a subsetΔ¯κ‐LsnCn of X as follows: y∈Δ¯κ‐LsnCn if and only if there exist a sequence yi⊂X and a subsequence ni of ℕ such that ryi<Dκ/2, yi⇀Δy and yi∈Cni for all i∈ℕ. If κ≤0, then Δ¯‐LsnCn=Δ¯κ‐LsnCn.Since a convergent sequence is aΔ-convergent sequence, the inclusion
(78)d‐LinCn⊂Δ¯κ‐LsnCnis always true. If a subset C0 of X satisfies that C0=d‐LinCn=Δ¯κ‐LsnCn, it is said that Cn converges to C0 in the sense of Δ¯κ-Mosco. Furthermore, the following inclusion holds:
(79)d‐LinCn⊂Δ¯κ‐LsnCn⊂Δκ‐LsnCn.Therefore, ifCn converges to C0 in the sense of Δκ-Mosco, then Cn converges to C0 in the sense of Δ¯κ-Mosco.Δ¯κ-Mosco convergence coincides with Δ¯-Mosco convergence in complete CAT(κ) spaces for κ≤0.Since complete CAT(κ) space is uniformly convex and satisfies the condition (ΔC) for any κ≤0, we have the following theorem:Theorem 20.
LetX be a complete CAT(κ) space for κ≤0. Let Cn⊂2X be a sequence of nonempty closed convex sets and C0 a nonempty closed convex subset of X. If Cn is converges to C0 in the sense of Δ¯-Mosco, then PCnx converges to PC0x∈X for any x∈X, where PK:X⟶K is the metric projection of X onto a nonempty closed convex subset K of X.We also have the following result [2].Theorem 21 (Kimura [2]).
LetX be a complete CAT(κ) space for κ≤0 and PK:X⟶K is the metric projection of X onto a nonempty closed convex subset K of X. Let Cn⊂2X be a sequence of nonempty closed convex sets and C0 a nonempty closed convex subset of X. If PCnx converges to PC0x∈X for any x∈X, then Cn converges to C0 in the sense of Δ-Mosco.LetX be a complete CAT(κ) space κ≤0 and PK:X⟶K is the metric projection of X onto a nonempty closed convex subset K of X. Let Cn⊂2X be a sequence of nonempty closed convex sets and C0 a nonempty closed convex subset of X. Then, from Theorem 19 and Theorem 21, the following propositions are equivalent:
(i)
Cn converges to C0 in the sense of Δ-Mosco(ii)
Cn converges to C0 in the sense of Δ¯-Mosco(iii)
PCnx converges to PC0x for any x∈XThat is, the notion ofΔκ-Mosco convergence coincides with Δ¯κ-Mosco convergence in complete CAT(κ) spaces for κ≤0.Since most of the results for CAT(κ) spaces for κ>0 are easily deduced from that for CAT(1) spaces, in the following, we focus on CAT(1) spaces.We first see the following known results for CAT(1) spaces.Theorem 22 (Espínola and Fernádez-León [9]).
LetX be a complete CAT(1) space. If a sequence xn⊂X satisfies rxn<π/2, then there exists a Δ-convergent subsequence of xn.Theorem 23 (Espínola and Fernádez-León [9]).
LetX be a complete admissible CAT(1) space and let xn⊂X be a sequence satisfies rxn<π/2. Then,
(80)ACxn⊂⋂k=0∞clcoxk,xk+1,xk+2,⋯.Theorem 24 (He, Fang, Lopez, and Li [7]).
LetX be a complete CAT(1) space and let xn⊂X be a sequence such that xn⇀Δx0∈X. Then for any p∈X with limsupn⟶∞dp,xn<π/2, the following inequality holds:
(81)dp,x0≤liminfn⟶∞dp,xn.Theorem 25 (Kimura and Satô [3]).
LetX be a complete admissible CAT(1) space and let xn⊂X be a sequence. Suppose that xn⇀Δx0∈X and dxn,p⟶dx0,p. Then xn⟶x0.Using these results, we obtain the relation betweenΔ¯1-Mosco convergence of sets and corresponding sequence of metric projections.Theorem 26.
LetX be a complete admissible CAT(1) space. Let Cn⊂2X be a sequence of nonempty closed convex sets and C0 a nonempty closed convex subset of X. If Cn converges to C0 in the sense of Δ¯1-Mosco, then PCnx converges to PC0x∈X for any x∈X, where PK:X⟶K is the metric projection of X onto a nonempty closed convex subset K of X.Proof.
Fixx∈X arbitrarily. Since PC0x∈C0=d‐LinCn, there exists a sequence xn⊂X such that xn⟶PC0x. Since dx,PCnx≤dx,xn for any n∈ℕ. Moreover, letting n⟶∞, we obtain
(82)limsupn⟶∞dx,PCnx≤dx,PC0x<π2.
Here, we take a subsequencePCnix of PCnx arbitrarily. From Theorem 22, there exists a subsequence PCnijx of PCnix and p0∈X such that
(83)PCnijx⇀Δp0.
Letpj=PCnijx for any j∈ℕ. Then, since
(84)limsupj⟶∞dx,pj≤dx,PC0x<π2,for any ε∈0,π/2‐dx,PC0x there exists j0∈ℕ such that for any j≥j0, it holds that
(85)dx,pj≤dx,PC0x+ε<π2.
LetD=y∈X∣dx,y≤dx,PC0x+ε. Then we have pj∈D for any j≥j0. Since D is closed and convex, from Theorem 23, we have pj⇀Δp0∈D. Therefore, we have
(86)dx,p0≤dx,PC0x+ε.
Lettingε↘0, we obtain
(87)dx,p0≤dx,PC0x.
Sincep0∈Δ¯1‐LsnCn=C0, we have dx,PC0x≤dx,p0 and thus
(88)dx,PC0x=dx,p0.
Hence, we obtainpj⇀ΔPC0x. From Theorem 24, we have
(89)dx,PC0x≤liminfj⟶∞dx,pjand hence
(90)dx,PC0x≤liminfj⟶∞dx,pj≤limsupj⟶∞dx,pj≤limsupn⟶∞dx,PCnx≤dx,PC0x.
Therefore,dx,pj⟶dx,PC0x. Since pj⇀ΔPC0x and dx,pj⟶dx,PC0x, from Theorem 25, we have pj⟶PC0x. Since for any subsequence PCnix of PCnx, there exists a subsequence PCnijx of PCnix such that PCnijx⟶PC0x, we have PCnx⟶PC0x.We also have the following result [3].Theorem 27 (Kimura and Satô [3]).
LetX be a complete admissible CAT(1) space and PK:X⟶K is the metric projection of X onto a nonempty closed convex subset K of X. Let Cn⊂2X be a sequence of nonempty closed convex sets and C0 a nonempty closed convex subset of X. If PCnx converges to PC0x∈X for any x∈X, then Cn converges to C0 in the sense of Δκ-Mosco.LetX be a complete admissible CAT(1) space and PK:X⟶K is the metric projection of X onto a nonempty closed convex subset K of X. Let Cn⊂2X be a sequence of nonempty closed convex sets and C0 a nonempty closed convex subset of X. Then, from Theorem 26 and Theorem 27, the following propositions are equivalent:
(i)
Cn converges to C0 in the sense of Δ1-Mosco(ii)
Cn converges to C0 in the sense of Δ¯1-Mosco(iii)
PCnx converges to PC0x for any x∈XThat is,Δ1-Mosco convergence coincides with Δ¯1-Mosco convergence in complete admissible CAT(1) spaces.For the case whereκ>0, by using standard modification, we can obtain the same result. That is, Δκ-Mosco convergence coincides with Δ¯κ-Mosco convergence in complete admissible CAT(κ) spaces for κ>0.Consequently,Δκ-Mosco convergence coincides with Δ¯κ-Mosco convergence for every κ∈ℝ and we can replace the definition. That is, if a subset C0 of a complete admissible CAT(κ) space X for κ∈ℝ satisfies that
(91)C0=d‐LinCn=Δ¯κ‐LsnCn,then we can say that Cn converges to C0 in the sense of Δκ-Mosco.



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