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

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.

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]). Let be a strictly convex and reflexive real Banach space satisfying the Kadec-Klee property. Let be a sequence of nonempty closed convex sets and a nonempty closed convex subset such that converges to in the sense of Mosco. Then converges strongly to for any , where is the metric projection of onto a nonempty closed convex subset of .

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 -Mosco convergence in complete geodesic spaces using a notion of asymptotic centre instead of weak convergence.

Theorem 2 (Kimura [2]). Let be a complete CAT(0) space. Let be a sequence of nonempty closed convex sets and a nonempty closed convex subset such that converges to in the sense of -Mosco. Then converges to for any , where is the metric projection of onto a nonempty closed convex subset of .

Moreover, Kimura and Satô [3] introduced -Mosco convergence in complete CAT() spaces and obtain the following result:

Theorem 3 (Kimura and Satô [3]). Let be a complete admissible CAT() space for . Let be a sequence of nonempty closed convex sets and a nonempty closed convex subset such that converges to in the sense of -Mosco. Then converges to for any , where is the metric projection of onto a nonempty closed convex subset of .

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 function is said to be a gauge if is strictly increasing, continuous, and . We know that if is a real sequence of such that for some gauge function , then .

Let be a real Banach space. Then the following propositions are equivalent: (i) is uniformly convex(ii)For , , and , it holds that whenever(iii)For any , there exists a convex gauge function such that

for any , where and for

For more details about the properties of uniformly convex real Banach spaces, see [4].

Let be a metric space and let . A geodesic path from to is a mapping such that , and for any . Let . If for any such that , a geodesic path from to exists, then we say that is -geodesic. Moreover, if such a geodesic path is unique for each pair of points, then is said to be -uniquely geodesic. In a -uniquely geodesic space, the image of a geodesic path from to is called a geodesic segment joining and and it is denoted by . For and , there exists a unique point such that and . We denote it by . A geodesic triangle with vertices is the union of geodesic segments , and . We denote it by .

To define a CAT() space, we use the following notation called a model space. For , the two-dimensional model space is the two-dimensional Euclidean space with the metric induced from the Euclidean norm. For , is the two-dimensional sphere whose metric is a length of a minimal great arc joining each two points. For , is the two-dimensional hyperbolic space with the metric defined by a usual hyperbolic distance.

The diameter of is denoted by , defined by

We know that is a -uniquely geodesic space for each .

Let . For in a geodesic space satisfying that , there exist points such that , , . We call the triangle having vertices , and in a comparison triangle of . Notice that it is unique up to an isometry of . For a specific choice of comparison triangles, we denote it by . A point is called a comparison point for if .

Let and a -geodesic space. If for any with , for any , and for their comparison points , the CAT() inequality holds, then we call a CAT() space. It is well known that any CAT() space is also a CAT() space whenever . Therefore, a CAT() space is a CAT(0) space for any .

Let be a CAT() space for . If for any , then we say that is admissible. A CAT() space is always admissible when .

A subset of a -uniquely geodesic space is said to be convex if for every and . For a subset of -uniquely geodesic space , a closed convex hull of is defined as the intersection of all closed convex sets including S, and we denote it by clco S.

Let be a complete admissible CAT() space for and a nonempty closed convex subset of . Then for , there exists a unique point such that

We call such a mapping defined by , the metric projection of onto .

Let be a CAT(0) space. From the CAT(0) inequality, it is easy to see that for every and .

The following lemma shows that a CAT(0) space has a similar property to the uniform convexity of Banach spaces.

Lemma 4. Let be a CAT(0) space. For , a point and two sequences , it holds that whenever

Proof. For , and , if then which implies Letting , we have This is the desired result.

Let be a metric space. For a point and a nonempty subset , the distance between them is defined by .

Let be a metric space. A function is said to be lower semicontinuous if for and , whenever . Moreover, a function is said to be proper if there exists a point such that and the domain of defined by

Let be a uniquely geodesic space. A function is said to be convex if for and , satisfies

3. Uniform Convexity of a Complete Geodesic Space

In the following, we always suppose that for any in geodesic space , a geodesic joining to is unique.

A geodesic space is said to satisfy the condition (D) if: for and .

We introduce two new concepts of uniform convexity on a geodesic space.

Let be a geodesic space. is said to be sequentially uniformly convex if satisfies the condition (D) and, for , a point and two sequences , it holds that whenever

Let be a geodesic space. is said to be uniformly convex if for any there exists a convex gauge function such that for any , where with and .

Uniformly convex real Banach spaces and CAT(0) spaces are uniformly convex in this sense.

Theorem 5. Let be a uniformly convex geodesic space. Then, is sequentially uniformly convex.

Proof. Let and let . Then there exists a convex gauge function , satisfying that for any and hence That is, satisfies the condition (D).
Moreover, for , and , if then there exists such that for any . From uniform convexity of , there exists a convex gauge function , satisfying that It follows that Therefore, we obtain and hence , that is, is sequentially uniformly convex.

Theorem 6. Let be a sequentially uniformly convex geodesic space. For and with , if, then .

Proof. For and with , we suppose that . If , then, since we have . From the sequential uniform convexity of , we have . This is a contradiction. Therefore, we have . This is the desired result.

Theorem 7. Let be a sequentially uniformly convex complete geodesic space and let be a nonempty closed convex subset of . Then, for , there exists a unique point such that .

Proof. For , let . Then, for , we can take a sequence such that Then, we have . Suppose that is not a Cauchy sequence. That is, there exists such that for any , there exist such that . In this way, we take two subsequences . Then, and we have Hence, from the sequential uniform convexity of , we have This is a contradiction and thus is a Cauchy sequence. Since is complete and is closed, there exists such that . Therefore, we have Next, we show the uniqueness of . Suppose that satisfy and . Then, from Theorem 6, we have This is a contradiction. Therefore, for , there exists a unique point such that .

Let be a sequentially uniformly convex complete uniquely geodesic space and let be a nonempty closed convex subset of . Then for , there exists a unique point such that

We call such a mapping defined by , the metric projection of onto .

Theorem 8. Let be a sequentially uniformly convex complete geodesic space and a sequence of nonempty bounded closed convex subsets which is decreasing with respect to inclusion, that is, for any . Then, is nonempty.

Proof. Since is nonempty bounded closed convex subset for , for , we can take a sequence by , where is the metric projection of onto a nonempty closed convex subset of . Then is a bounded increasing real sequence and hence has a limit . That is, we have First, we show that converges to some point . If , then, since , we have as . Hence, we may suppose that . Suppose that is not a Cauchy sequence. That is, there exists such that for any , there exist such that . Without loss of generality, we can suppose that . In this way, we take two subsequences . Then, Since , we have and thus From the sequential uniform convexity of , we have . This is a contradiction. Therefore, is a Cauchy sequence and thus there exists such that .
We show that . For , for and thus . Therefore, and it completes the proof.

4. -Convergence

Let be a metric space and a bounded sequence. An asymptotic centre of is defined by

Lemma 9. Let be a sequentially uniformly convex complete geodesic space and let be a bounded sequence. Then, there exists a point such that That is, is nonempty. Moreover, is bounded, closed and convex.

Proof. Let and define by for any . Then, for any , is nonempty and bounded. If and , then Letting , we have and hence . Moreover, for and , we have It implies . Therefore, is closed and convex for any . Moreover, is decreasing with respect to inclusion. Hence, we have and thus Also, we know that is closed and convex.

If we suppose uniform convexity for , we can prove that an asymptotic centre is a singleton.

Theorem 10. Let be a uniformly convex complete geodesic space and let be a bounded sequence of . Then, is a singleton.

Proof. Let . If with , then, since is bounded, there exists such that and for all . Then, there exists a convex gauge function , and we have and hence Since , we have and thus This is a contradiction. Therefore, is a singleton.

Let be a bounded sequence and . We say -converges to a-limit if is the unique asymptotic centre of any subsequences of , and we denote it by .

Let be a geodesic space. is said to satisfy the condition (C), if any nonempty closed convex subset is -closed, that is, if and , then .

CAT(0) spaces satisfy the condition (C). Let be a uniformly convex real Banach space. Then, the following propositions are equivalent: (i)For any bounded sequence , -converges to if and only if converges weakly to (ii) satisfies the condition (C)(iii) satisfies Opial’s condition

See [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]). Let be a uniformly convex complete geodesic space. Then, for any bounded sequence of has a -convergent subsequence.

Corollary 12. Let be a uniformly convex complete geodesic space satisfying the condition (C) and let be a nonempty bounded closed convex subset of . Then, for any sequence in has a -convergent subsequence and its -limit belongs to .

Theorem 13 (Bačák [6], He, Fang, Lopez and Li [7]). Let be a uniformly convex complete geodesic space satisfying the condition (C). Let be a proper lower semicontinuous function and a sequence such that . Then,

Corollary 14 (-lower semicontinuity of the distance function). Let be a uniformly convex complete geodesic space satisfying the condition (C) and let be a point. Let be a sequence such that . Then,

Lemma 15. Let be a uniformly convex complete geodesic space and let be a sequence such that -converges to . Then,

Proof. Since is bounded, so is . For any subsequence of , if , then, we have Therefore, since , we have . Hence, is the unique asymptotic centre of any subsequence of and it completes the proof.

Theorem 16 (-Kadec-Klee property). Let be a uniformly convex complete geodesic space satisfying the condition (C) and a sequence such that and for some . Then, .

Proof. Let be a sequence such that for any . Since , From Lemma 15, we have . Then, since is -lower semicontinuous and , we have and thus . Therefore, we obtain If , then . If , from sequential uniform convexity of , we obtain . Since for any , we have .

5. Convergence of a Sequence of Sets

Let be a sequence of nonempty closed convex subsets of a uniformly convex complete geodesic space . -Mosco convergence is defined by using a notion of asymptotic centre by Kimura [2]. First, we define subsets and of as follows: if and only if there exists such that and for all ; if and only if there exist a bounded sequence and a subsequence of such that and for all . If a subset of satisfies that , we say that converges to in the sense of -Mosco.

Here, we introduce a new concept of the set-convergence. We define subsets of as follows: if and only if there exists a bounded sequence and a subsequence of such that and for all .

Since a convergent sequence is a -convergent sequence, the inclusion is always true. If a subset of satisfies that , we say that converges to in the sense of -Mosco. Furthermore, the following inclusion holds:

Therefore, if converges to in the sense of -Mosco, then converges to in the sense of -Mosco.

Lemma 17. Let be a uniformly convex complete geodesic space satisfying the condition (C) and a sequence of nonempty closed convex sets which is decreasing with respect to inclusion, that is, for any . if is nonempty, then converges to in the sense of -Mosco.

Proof. We show that . If , then for any . Let be a sequence of such that for any . Then, and for any . Therefore, we obtain and hence .
Next, we show that . If , then there exists a sequence and such that for any and . For any , if , then . Therefore, from the condition (C) of , we have . Since is arbitrarily, we obtain . Hence, we have .
Since is always true, we have That is, converges to in the sense of -Mosco.

Theorem 18. Let be a uniformly convex complete geodesic space satisfying the condition (C). Let be a sequence of nonempty closed convex sets and a nonempty closed convex subset of . If converges to in the sense of -Mosco, then converges to for any , where is the metric projection of onto a nonempty closed convex subset of .

Proof. Fix arbitrarily. Since , there exists a sequence such that and for every . Since for any and is bounded, is also bounded. Moreover, letting , we have Here, we take a subsequence of arbitrarily. Since is bounded, there exists a subsequence of and such that Let for any . Then, since for any there exists such that for any , it holds that Let . Then we have for any . Since is closed and convex, from the condition (C) of , we obtain . Therefore, we have Letting , we obtain Since , we have and thus Hence, we obtain . Since is -lower semicontinuous and , we have and hence Therefore, . Since and , from the -Kadec-Klee property of , we obtain . Since for any subsequence of , there exists a subsequence of such that , we have .

Let be a uniformly convex complete geodesic space. If has the condition (C), we can show that convergence of a sequence of metric projections from -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 (C).

Theorem 19. Let be a sequentially uniformly convex complete geodesic space and a sequence of nonempty closed convex sets which is decreasing with respect to inclusion, that is, for any . Suppose that is nonempty. Then, converges to for any , where is the metric projection of onto a nonempty closed convex subset of .

Proof. Fix arbitrarily. Since , if , then for any , we have and hence is bounded. Furthermore, since for any , we have and thus is increasing. Therefore, has a limit First, we show that converges to some point . Since if