2-Strict Convexity and Continuity of Set-Valued Metric Generalized Inverse in Banach Spaces

Abstract

Authors investigate the metric generalized inverses of linear operators in Banach spaces. Authors prove by the methods of geometry of Banach spaces that, if is approximately compact and is 2-strictly convex, then metric generalized inverses of bounded linear operators in are upper semicontinuous. Moreover, authors also give criteria for metric generalized inverses of bounded linear operators to be lower semicontinuous. Finally, a sufficient condition for set-valued mapping to be continuous mapping is given.

1. Introduction

Let be a real Banach space. Let and denote the unit sphere and the unit ball, respectively. By we denote the dual space of . Let , , and denote the set of natural numbers, reals, and nonnegative reals, respectively.

Let and . By we denote that is weakly convergent to . denotes closed hull of (weak closed hull) and dist denotes the distance of and . Let be a nonempty subset of . Then the set-valued mapping is called the metric projection operator from onto .

A subset of is said to be proximal if for all (see [1]). is said to be semi-Chebyshev if is at most a singleton for all . is said to be Chebyshev if it is proximal and semi-Chebyshev. It is well known that (see [1]) is reflexive if and only if each closed convex subset of is proximal and that is strictly convex if and only if each convex subset of is semi-Chebyshev.

Definition 1 (see [2]). A nonempty subset of is said to be approximatively compact if, for any and any satisfying , the sequence has a subsequence converging to an element in . is called approximatively compact if every nonempty closed convex subset of is approximatively compact.

Definition 2 (see [3]). Set-valued mapping is called upper semicontinuous at , if, for each norm open set with , there exists a norm neighborhood of such that for all in . is called lower continuous at , if, for any and any in with , there exists such that as . is called continuous at , if is upper semicontinuous and is lower continuous at .

Let us present the history of the approximative compactness and related notions. This notion has been introduced by Jefimow and Stechkin in [2] as a property of Banach spaces, which guarantees the existence of the best approximation element in a nonempty closed convex set for any . In 2007, Chen et al. (see [4]) proved that a nonempty closed convex of a midpoint locally uniformly rotund space is approximately compact if and only if is Chebyshev set and the metric projection operator is continuous. In 1972, Oshman (see [5]) proved that the metric projection operator is upper semicontinuous.

Definition 3 (see [6]). A Banach space is called nearly dentable space if, for any and any open set , we have and .

Definition 4 (see [7]). A Banach space is said to be -strictly convex if for any elements , if , then are linearly dependent.

Definition 5 (see [8]). A Banach space is said to be nearly strictly convex space if every convex subset of is relatively compact.

I. Singer defined in [7] the -strictly convex spaces and the dual notion (-smooth spaces) was introduced by Sullivan. In 1988, Skowski and Stachura [8] introduced the notion of nearly strict convexity of Banach spaces by means of the Kuratowski measure of noncompactness. It is well known that if is a -strictly convex space, then is a nearly strictly convex space. It is easy to see that, if is a nearly dentable space, then is a reflexive space. In 2011, Shang et al (see [6]) defined nearly dentable space and proved the following two results.

Theorem 6. A Banach space is approximatively compact if and only if (1)is a nearly dentable space,(2) is a nearly strictly convex space.

Theorem 7. Let be nearly a dentable space. Then for any closed convex set , the metric projection operator is upper semicontinuous.

Let be a linear bounded operator from into . Let , , and denote the domain, range, and null space of , respectively. If or , the operator equation is generally ill-posed. In applications, one usually looks for the best approximate solution (b.a.s.) to the equation (see [9]).

A point is called the best approximate solution to the operator equation , if where (see [9]).

Nashed and Votruba [9] introduced the concept of the (set-valued) metric generalized inverse as follows.

Definition 8. Let , be Banach spaces and let be a linear operator from to . The set-valued mapping defined by for any is called the (set-valued) metric generalized inverse of , where

During the last three decades, the linear generalized inverses of linear operators in Banach spaces and their applications have been investigated by many authors. In this paper, authors investigate the metric generalized inverses of linear operators in Banach spaces. Authors prove by the methods of geometry of Banach spaces that, if is approximatively compact and is 2-strictly convex space, then metric generalized inverse of a bounded linear operator is upper semicontinuous. Moreover, authors also give criteria for metric generalized inverses of bounded linear operators to be lower semicontinuous. Finally, authors give a sufficient condition for the set-valued mapping to be continuous mapping. The topic of this paper is related to the topic of [10–15].

2. Main Results

Theorem 9. Let and be nearly dentable Banach spaces and a closed subspace of . Then for any bounded linear operator , if is a closed subspace of and is a Chebyshev subspace of , then and , where(1) is a 2-strictly convex Banach space;(2)for any , there exist and such that the set-valued mapping satisfies the equality ;(3)the set-valued mapping is upper semicontinuous;(4)for any , the set-valued mapping is lower semicontinuous at if and only if the function is lower semicontinuous at .

In order to prove this theorem, we give a lemma.

Lemma 10. Let be a reflexive 2-strictly convex Banach space. Then for any closed convex set and , there exist and such that .

Proof. (a) We may assume without loss of generality that and . Hence, for any , , and , we have This implies that . Since is 2-strictly convex, we may assume that . By the Hahn-Banach theorem, there exists such that . Noticing that , , and , we have . Thus
(b) By , it is easy to see that . Since is a 2-strictly convex space, is a nearly convex space. Since is a nearly convex space and is closed convex set, is compact. Hence there exist and such that We may assume without loss of generality that is not a singleton. Moreover, for any , if , then . Otherwise, we have . Therefore, by the proof of (a), we have for any . Suppose that . Then Hence we have a contradiction. Hence . Similarly, we have . Thus . This means that, for any , we have . Hence we have the equation . This completes the proof.

Proof of Theorem 9. Consider that . Suppose that is not a 2-strictly convex Banach space. Then there exist , , and such that and , , and are linearly independent. Therefore, by the Hahn-Banach theorem, there exists such that . Noticing that , , and , we have . Pick . Define the subspace of . Since is a one-dimensional subspace of , we obtain that is a strictly convex Banach space. This implies that is a Chebyshev subspace of . Define the bounded linear operator Since is a bounded linear operator and is a Chebyshev subspace of , there exist and such that . Moreover, it is easy to see that This implies that , , . Hence there exist , , and such that Then Hence there exists such that Then This implies that , , and are linearly dependent, a contradiction. Hence we obtain that is a 2-strictly convex Banach space.
Consider that . (a) Since is a Chebyshev subspace of , we obtain that for any , is single-point set. Hence, for any , there exists such that . Moreover, by Lemma 10, there exist and such that . Thus
(b) By Theorem 7, the metric projector operator is upper semicontinuous. Since is a Chebyshev subspace, we obtain that is a single-valued operator. This means that the metric projector operator is continuous. Next we will prove that is upper semicontinuous; that is, for any , and any norm open set with , there exists a natural number such that whenever . Otherwise, there exists such that . Since the metric projector operator is continuous, we obtain that as . Noticing that , we have as . Since is a bounded linear operator, we obtain that is a closed subspace of . Put where and . It is easy to see that . Moreover, . In fact, suppose that . Then there exists such that . It is easy to see that . This implies that is not a Chebyshev subspace of , a contradiction. By , we obtain that is a Banach space. Moreover, it is easy to see that is a bounded linear operator and . This implies that the bounded linear operator is both injective and surjective. By the inverse operator theorem, is a bounded linear operator. Hence we have This means that as . Noticing that it is easy to see that and . Since , , and , we have as . We will derive a contradiction for each of the following two cases.
Case  1  . By (19), we have as . This implies that as . Thus as . By , we have . Moreover, by , we have as , which contradicts the equation .
Case  2  . By , we obtain that is a bounded sequence. Since the distance function is continuous, we have This implies that is a bounded sequence. Hence is a bounded sequence. Since is a bounded linear operator, we obtain that is a bounded sequence. By , we obtain that is a bounded sequence. Since is a nearly dentable Banach space, is reflexive. Hence, there exists a subsequence of such that . Since is a closed convex set, is a weakly closed convex set. Hence we obtain that . By and , we have . Noticing that , we have . Since , we have . By the Hahn-Banach theorem, there exists such that . Hence This implies that . By and , we have . Define By and , we have . Since , , and , we have , and .
Next we will prove that there exists a subsequence of such that as . Pick . Then for , we have . Since is a closed set, there exists such that dist for any . Noticing that , we have dist. Then dist. In fact, suppose that dist. Then for any , there exists such that dist. Hence, for any , there exists such that dist. We define the open set It is easy to see that and . Since is a nearly dentable space, we obtain that is a reflexive space. Hence is a weakly compact set and . By the separation theorem of locally convex space, there exists and such that Noticing that and , we have . This means that is impossible, a contradiction. Hence . Then there exists a subsequence of such that as . Then there exists a sequence such that as . Since is a 2-strictly convex space, we obtain that is a nearly strictly convex space. This means that is compact. Hence the sequence has a Cauchy subsequence. By , we have that has a Cauchy subsequence. Noticing that , we obtain that there exists a subsequence of such that as . By and , we have Moreover, we have , which contradicts the equation . Hence, for any bounded linear operator and , if is a closed subspace of and is Chebyshev subspace of , then there exist , such that the set-valued mapping is upper semicontinuous.
Consider that (1)(4). Let the function be lower semicontinuous at . Since is a 2-strictly convex Banach space, there exist and such that set-valued mapping . Hence . Let as . Then as . Otherwise, there exist and a subsequence of such that for any . We define the open set Since is a 2-strictly convex Banach space, by the implication (1)(2)+(3), the set-valued mapping is upper semicontinuous. Hence there exists a natural number such that whenever . Let . Then there exist and such that Then Noticing that and , we have . Moreover, we may assume without loss of generality that . Therefore, by (29), we have Then This implies that a contradiction. Then as . Hence there exists such that as . Similarly, there exists such that as . For any , there exists such that . Moreover, it is easy to see that . Hence, for any , we have This means that, if the function is lower semicontinuous at , then the set-valued mapping is lower semicontinuous at .
Let the set-valued mapping be lower semicontinuous at . Since is a 2-strictly convex Banach space, there exist and such that the set-valued mapping satisfies the equality . Hence we have . Let as and . Then . Since the set-valued mapping is lower semicontinuous at , there exist and such that and as . Since we have Hence the function is lower semicontinuous at . This completes the proof.

Theorem 11. Let and be nearly dentable Banach spaces and a closed subspace of . Then for any bounded linear operator , if is a closed subspace of , is a hyperplane of , and is a Chebyshev subspace of , then , where(1) is a 2-strictly convex Banach space;(2)for any , there exist and such that the set-valued mapping satisfies the equality ;(3)the set-valued mapping is continuous.

In order to prove the theorem, we first give a lemma.

Lemma 12. Let be a reflexive 2-strictly convex space and be a hyperplane of . Then the function is lower semicontinuous.

Proof. (a) We will prove that, if , , then . It is easy to see that and . Hence, for any , we have This implies that . Hence we have . Similarly, we have . By and , we have . This implies that .
(b) Let , , and . Next we will prove that . Pick . Then Moreover, we have In fact, noticing that and , we have for any . By we have This implies that By (a), there exists such that By (42) and (43), we have . Hence .
(c) We will prove that, if and , then , where and . Let and . Then Let . Therefore, by (44), we have This means that and . Hence we have . Then , . Otherwise, there exists such that . By , we have , a contradiction. This implies that .
(d) Suppose that there exists such that function is not lower semicontinuous at . Then there exists and such that and as . Moreover, we may assume without loss of generality that . Otherwise, let . Pick . By (a) and (b), there exists such that and as . Put Hence, for any , we have This implies that . Hence we have Let . Therefore, by (c) and , we have . Hence By and , we have , a contradiction. This completes the proof.

Proof of Theorem 11. By Theorem 9, we just need to prove that, for any , the function is lower semicontinuous on . Let as . Since is a nearly dentable Banach space and is a Chebyshev subspace of , by Theorem 7, we have as . Put where and . By the proof of Theorem 9, we obtain that is a bounded linear operator. Then Hence there exist and such that as . By Lemma 12, we obtain that , where . Noticing that and , we have . This completes the proof.

By Theorems 6 and 9, we have the following.

Theorem 13. Let and be approximatively compact Banach spaces and a closed subspace of . Then for any bounded linear operator , if is a closed subspace of and is a Chebyshev subspace of , then and , where(1) is a 2-strictly convex Banach space;(2)for any , there exist and such that the set-valued mapping satisfies the equality ;(3)the set-valued mapping is upper semicontinuous;(4)for any , the set-valued mapping is lower semicontinuous at if and only if the function is lower semicontinuous at .

By Theorems 6 and 11, we have the following.

Theorem 14. Let and be approximatively compact Banach spaces and a closed subspace of . Then for any bounded linear operator , if is a closed subspace of , is a hyperplane of , and is a Chebyshev subspace of , then , where(1) is a 2-strictly convex Banach space;(2)for any , there exist and such that the set-valued mapping satisfies the equality ;(3)the set-valued mapping is continuous.

Acknowledgments

This work is supported by the Fundamental Research Funds for the Central Universities, DL12BB36, and by Heilongjiang Provincial Department of Education Funds 12521070.