Abstract
We show a necessary and sufficient condition for the existence of metric projection on a class of half-space in Banach space. Two representations of metric projections and are given, respectively, where stands for dual half-space of in dual space . By these representations, a series of continuity results of the metric projections and are given. We also provide the characterization that a metric projection is a linear bounded operator.
1. Introduction
The metric projection in Banach space is an enduring question for study or discussion. It has been used in many areas of mathematics such as the theories of optimization and approximation, fixed point theory, nonlinear programming, and variational inequalities. On continuity of the metric projection, many mathematicians, for example, Nevesenko [1], Oshman [2], Wang [3], Fang and Wang [4], and Zhang and Shi [5] have done profound research. In practical application, giving the representations of metric projection is very necessary. Generally speaking, this is very difficult. In recent years, Wang and Yu [6] gave a representation of single-valued metric projection on a class of hyperplanes in reflexive, smooth, and strictly convex Banach space . Song and Cao [7] gave a representation of metric projection on a class of half-space in the reflexive, smooth, and strictly convex Banach space . Wang [8] and Ni [9] extended the result of Wang and Yu to general Banach space, respectively. Wang [8] also discussed continuity of the metric projection on the hyperplane in Banach space.
In this paper, let be a Banach space and let be the dual of . Let and be the unit sphere and unit ball of , respectively. Let , let , let , let , let , let , and let . It is easily proved that , , and for all . For , the metric projection is defined by , where . Obviously, is a set-valued mapping. If for each , then is said to be proximinal. It is well known that is single-valued when is strictly convex and is proximal.
Cabrera and Sadarangani [10] introduced the geometrical properties of Banach spaces as follows.
A Banach space is called nearly strictly convex (resp., weakly nearly strictly convex) whenever, for any , the set is compact (resp., weakly compact). A Banach space is called nearly smooth (resp., weakly nearly smooth) whenever, for any , the set is compact (resp., weakly compact).
The metric projection is said to be norm-norm (resp., norm-weakly) upper semicontinuous if, for all in and for all norm (resp., weakly) open set , there exists a norm neighborhood of such that .
In this paper, firstly, we established a necessary and sufficient condition for the existence of metric projection on a class of half-space in Banach space. Secondly, we give two representations of the metric projections and by using a different method from the literatures [5–9]. Thirdly, by these representations, we prove that if is weakly nearly strictly convex (resp., weakly nearly smooth), then metric projection (resp., ) is norm-weakly upper semicontinuous. Finally, the characterization of the metric projection from to a subspace to be a linear bounded operator is given. We extend the corresponding results in [5–9].
2. The Representations of the Metric Projection on Two Classes of Half-Spaces in Banach Spaces
Lemma 1. Let be a Banach space and let ; then for all .
Proof. Firstly, suppose that . Let . For any , since
we deduce that
On the other hand, for any , there exists in such that . Set . Then
Consequently, and
It follows that
By arbitrariness of , we deduce that
This means that
Secondly, for and , since
from (7), we may obtain that
Remark 2. For given and , by Lemma 1, we have that for any .
Theorem 3. Let be a Banach space, let , and let ; then for any if and only if .
Proof. On necessity: take ; then there exists a . Set ; by Lemma 1, we have that
Hence, .
On the other hand,
This shows that , that is, and .
On sufficiency: take such that . We discuss that in two cases.
Case 1. If , then .
Case 2. If , since
then we have that . By Lemma 1,
It follows that .
Theorem 4. Let be a Banach space, let , let attain its norm on , and let . Then
Proof. Take . We discuss that in two cases.
Case 1. If , then .
Case 2. If , we arbitrarily take . Let . Similar to the proof of Theorem 3, we may obtain that . Therefore,
On the other hand, we arbitrarily take . Let ; similar to the proof of Theorem 3, we may obtain that . Therefore,
that is,
By Case 1 and Case 2, we have
for any .
By the similar proof to that in Lemma 1, we can obtain the following result.
Lemma 5. Let be a Banach space, let , and let . Then for any .
By a similar proof to that in Theorem 4, we can also prove the following result according to Lemma 5.
Theorem 6. Let be a Banach space, let , and let . Then for any .
3. Continuity of the Metric Projection on the Two Classes of Half-Spaces in Banach Spaces
Theorem 7. Let , let attain its norm on , and let . If is weakly nearly strictly convex, then the metric projection is norm-weakly upper semicontinuous.
Proof. Let , , and let as . Our proof will be divided into two cases.
Case 1. Suppose that . Since is a closed set, . Clearly, .
Case 2. Suppose that .
If there are an infinite number of for which , then we can choose a subsequence with . Therefore, as .
If there are an infinite number of for which , without loss of generality, we may assume that . Taking , by Theorem 4, we have
We assume that , where . Since is weakly nearly strictly convex, we know that has a weakly convergent subsequence with as . Consequently,
Noting and , we know that . Therefore,
where . This shows that .
Now, we will show that is norm-weakly upper semicontinuous at . Otherwise, there exist a weakly open set and a sequence with as , but for all . Taking , , similar to previous arguments, we can observe the fact that there exists a subsequence of such that as and . This means that there exists for some large enough, which is a contradiction.
Similar to the proof of Theorem 8, we may prove the following theorem.
Theorem 8. Let be a Banach space. (1)Let , let attain its norm on , and let . If is nearly strictly convex, then the metric projection is norm-norm upper semicontinuous.(2)Let and let . If is weakly nearly smooth, then the metric projection is norm-weakly upper semicontinuous.(3)Let and let . If is nearly smooth, then the metric projection is norm-norm upper semicontinuous.
Lemma 9 (see [11]). Let be a proximal subspace. Then for any , one has the decomposition where and If is a Chebyshev subspace, the decomposition is unique, and
Lemma 10. Let be a strictly convex Banach space and let be a proximal subspace. Then, for any , one has
Proof. Let , for any , we have that . Consider By the definition of , we obtain . Since is strictly convex, we know that is single-valued, and hence we have .
Similar to the proof Theorem in [6], we can prove the following result by Lemmas 9 and 10.
Lemma 11. Let be a strictly convex Banach space and let be a proximal subspace. is single-valued operator from into , and is a metric projection from into . Then if and only if the following conditions are satisfied:(1); (2).
Theorem 12. Let be a strictly convex Banach space and let be a proximal subspace. Then the metric projection is a linear bounded operator if and only if is a linear subspace.
Proof. On necessity: let be a linear operator. Since is strictly convex and is proximal, then is single valued. By Lemma 11, for any , , , then
and hence . This shows that is a linear subspace.
On sufficiency: let be a linear subspace and let be a metric projection; since is strictly convex, by Lemma 11, is also a linear subspace. For any , we have that
By Lemma 11, we have that
It follows that . Note that is homogeneous; we obtain that is a linear operator. In addition, for any , since , we have that
This shows that is a bounded operator.
Conflict of Interests
The authors declare that they have no conflict of interests.
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China (Grant no. 11271248) and Scientific Research Foundation of Shanghai University of Engineering Science (Grant nos. A-0501-12-43, nhky-2012-13).