AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 908676 10.1155/2014/908676 908676 Research Article The Representations and Continuity of the Metric Projections on Two Classes of Half-Spaces in Banach Spaces Zhang Zihou Liu Chunyan Ezzinbi Khalil College of Fundamental Studies Shanghai University of Engineering Science Shanghai 201620 China sues.edu.cn 2014 1722014 2014 20 10 2013 31 12 2013 17 2 2014 2014 Copyright © 2014 Zihou Zhang and Chunyan Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We show a necessary and sufficient condition for the existence of metric projection on a class of half-space Kx0*,c={xX:x*(x)c} in Banach space. Two representations of metric projections PKx0*,c and PKx0,c are given, respectively, where Kx0,c stands for dual half-space of Kx0*,c in dual space X*. By these representations, a series of continuity results of the metric projections PKx0*,c and PKx0,c 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 , Oshman , Wang , Fang and Wang , and Zhang and Shi  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  gave a representation of single-valued metric projection on a class of hyperplanes Hx0*,c={xX:x0*(x)=c} in reflexive, smooth, and strictly convex Banach space X. Song and Cao  gave a representation of metric projection on a class of half-space Kx0*,c in the reflexive, smooth, and strictly convex Banach space X. Wang  and Ni  extended the result of Wang and Yu to general Banach space, respectively. Wang  also discussed continuity of the metric projection on the hyperplane Hx0*,c in Banach space.

In this paper, let X be a Banach space and let X* be the dual of X. Let S(X) and B(X) be the unit sphere and unit ball of X, respectively. Let x0*X*{θ},  let c, let Kx0*,c={xX:x0*(x)c}, let D-1(x0*)={xX:x0*(x)=x0*x=x0*2=x2}, let x0X{θ}, let Kx0,c={x*X*:x*(x0)c}, and let D(x0)={x*X*:x*(x0)=x*x0=x*2=x02}. It is easily proved that D(αx)=αD(x), D-1(αx*)=αD-1(x*), and for all α. For MX, the metric projection PM:X2M is defined by PM(x)={yM:x-y=d(x,M)}, where d(x,M)=inf{x-y:yM}. Obviously, PM is a set-valued mapping. If PM(x) for each xX, then M is said to be proximinal. It is well known that PM is single-valued when X is strictly convex and M is proximal.

Cabrera and Sadarangani  introduced the geometrical properties of Banach spaces as follows.

A Banach space X is called nearly strictly convex (resp., weakly nearly strictly convex) whenever, for any x*S(X*), the set D-1(x*) is compact (resp., weakly compact). A Banach space X is called nearly smooth (resp., weakly nearly smooth) whenever, for any xS(X), the set D(x) is compact (resp., weakly compact).

The metric projection PM is said to be norm-norm (resp., norm-weakly) upper semicontinuous if, for all x in X and for all norm (resp., weakly) open set WPM(x), there exists a norm neighborhood U of x such that PM(U)W.

In this paper, firstly, we established a necessary and sufficient condition for the existence of metric projection on a class of half-space Kx0*,c in Banach space. Secondly, we give two representations of the metric projections PKx0*,c and PKx0,c by using a different method from the literatures . Thirdly, by these representations, we prove that if X is weakly nearly strictly convex (resp., weakly nearly smooth), then metric projection PKx0*,c(resp., PKx0,c) is norm-weakly upper semicontinuous. Finally, the characterization of the metric projection PM from X to a subspace M to be a linear bounded operator is given. We extend the corresponding results in .

2. The Representations of the Metric Projection on Two Classes of Half-Spaces in Banach Spaces Lemma 1.

Let X be a Banach space and let x0*X*{θ}; then d(x,Kx0*,c)=|x0*(x)-c|/x0* for all xXKx0*,c.

Proof.

Firstly, suppose that x0*=1. Let xX/Kx0*,c. For any yKx0*,c, since (1)x0*(x-y)x0*(x)-c>0, we deduce that (2)d(x,Kx0*,c)|x0*(x)-c|.

On the other hand, for any ε>0(ε<1/4), there exists uε in S(X) such that 1-ε<x0*(uε)1. Set yε=x-(1+2ε)(x0*(x)-c)uε. Then (3)x0*(yε)=x0*(x)-(1+2ε)(x0*(x)-c)x0*(uε)<x0*(x)-(1+2ε)(x0*(x)-c)(1-ε)=x0*(x)-(1+ε-2ε2)(x0*(x)-c)x0*(x)-(x0*(x)-c)=c. Consequently, yεKx0*,c and (4)x-yε=(1+2ε)|x0*(x)-c|. It follows that (5)d(x,Kx0*,c)(1+2ε)|x0*(x)-c|. By arbitrariness of ε, we deduce that (6)d(x,Kx0*,c)|x0*(x)-c|. This means that (7)d(x,Kx0*,c)=|x0*(x)-c|. Secondly, for x*X*θ and x*1, since (8)Kx0*,c={xX:x0*(x)c}={xX:x0*x0*(x)cx0*}, from (7), we may obtain that (9)d(x,Kx0*,c)=|x0*(x)-c|x0*,xXKx0*,c.

Remark 2.

For given x0*X*{θ} and  c, by Lemma 1, we have that (10)d(x,Kx0*,c)=d(x,Hx0*,c), for any xXKx0*,c.

Theorem 3.

Let X be a Banach space, let x0*X*θ, and let c; then PKx0*,c(x) for any xX if and only if D-1(x0*).

Proof.

On necessity: take xXKx0*,c; then there exists a yPKx0*,c(x). Set u=(x0*2/(x0*(x)-c))(x-y); by Lemma 1, we have that (11)u=x0*2|x0*(x)-c|x-y=x0*2|x0*(x)-c||x0*(x)-c|x0*=x0*. Hence, x0*(u)x0*u=x0*2.

On the other hand, (12)x0*(u)=x0*2x0*(x)-c(x0*(x)-x0*(y))x0*2x0*(x)-c(x0*(x)-c)=x0*2. This shows that x0*(u)=x0*2=u2, that is, uD-1(x0*) and D-1(x0*).

On sufficiency: take xS(X) such that x0*(x)=x0*x=x0*2=x2. We discuss that in two cases.

Case  1. If xKx0*,c, then xPKx0*,c(x).

Case  2. If xKx0*,c, since (13)x0*(x-x0*(x)-cx0*2x0)=x0*(x)-(x0*(x)-c)=c;then we have that x-((x0*(x)-c)/x0*2)x0Kx0*,c. By Lemma 1, (14)x-(x-x0*(x)-cx0*2x0)=x0*(x)-cx0*=d(x,Kx0*). It follows that x-((x0*(x)-c)/x0*2)x0PKx0*,c(x).

Theorem 4.

Let X be a Banach space, let x0*X*{θ}, let x0* attain its norm on S(X), and let c. Then (15)PKx0*,c(x)=x-max{0,x0*(x)-cx0*2}D-1(x0*).

Proof.

Take xX. We discuss that in two cases.

Case  1. If xKx0*,c, then PKx0*,c(x)={x}.

Case  2. If xKx0*,c, we arbitrarily take x0D-1(x0*). Let y=x-((x0*(x)-c)/x0*2)x0. Similar to the proof of Theorem 3, we may obtain that yPKx0*,c(x). Therefore, (16)x-x0*(x)-cx0*2D-1(x0*)PKx0*,c(x).

On the other hand, we arbitrarily take yPKx0*,c(x). Let u=(x0*2/(x0*(x)-c))(x-y); similar to the proof of Theorem 3, we may obtain that uD-1(x0*). Therefore, (17)y=x-x0*(x)-cx0*2ux-x0*(x)-cx0*2D-1(x0*), that is, (18)PKx0*,c(x)x-x0*(x)-cx0*2D-1(x0*).

By Case 1 and Case 2, we have (19)PKx0*,c(x)=x-max{0,x0*(x)-cx0*2}D-1(x0*), for any xX.

By the similar proof to that in Lemma 1, we can obtain the following result.

Lemma 5.

Let X be a Banach space, let x0X{θ}, and let  c. Then (20)d(x*,Kx0,c)=|x*(x0)-c|x0, for any x*X*Kx0,c.

By a similar proof to that in Theorem 4, we can also prove the following result according to Lemma 5.

Theorem 6.

Let X be a Banach space, let x0X{θ}, and let c. Then (21)PKx0,c(x*)=x*-max{0,x*(x0)-cx02}D(x0), for any x*X*.

3. Continuity of the Metric Projection on the Two Classes of Half-Spaces in Banach Spaces Theorem 7.

Let x0*X{θ},  let x0* attain its norm on S(X), and let c. If X is weakly nearly strictly convex, then the metric projection PKx0*,c is norm-weakly upper semicontinuous.

Proof.

Let x,  xnX, and let xnx as n. Our proof will be divided into two cases.

Case  1. Suppose that {xn}Kx0*,c. Since Kx0*,c is a closed set, xKx0*,c. Clearly, PKx0*,c(xn)=xnx=PKx0*,c(x).

Case  2. Suppose that {xn}Kx0*,c.

If there are an infinite number of n for which xnKx0*,c, then we can choose a subsequence {xnk}{xn} with {xnk}Kx0*,c. Therefore, PKx0*,c(xnk)=xnkx=PKx0*,c(x) as k.

If there are an infinite number of n for which xnKx0*,c, without loss of generality, we may assume that {xn}XKx0*,c. Taking ynPKx0*,c(xn), by Theorem 4, we have (22)PKx0*,c(xn)=xn-x0*(xn)-cx02D-1(x0*).

We assume that yn=xn-((x0*(xn)-c)/x0*2)zn, where znD-1(x0*). Since X is weakly nearly strictly convex, we know that {zn} has a weakly convergent subsequence {znk} with znkwz as k. Consequently, (23)ynk=xnk-x0*(xnk)-cx0*znkwx-x0*(x)-cx0*z.

Noting x0*(z)=limkx0*(znk)=limkx0*2=znk2 and zlimk¯znk,  we know that x0*(z)x0*·z. Therefore, (24)x0*(z)=x0*·z=x0*2=z2, where zD-1(x0*). This shows that ynkwx-((x0*(x)-c)/x0*)zPKx0*,c(x).

Now, we will show that PKx0*,c is norm-weakly upper semicontinuous at x. Otherwise, there exist a weakly open set W0PKx0*,c(x) and a sequence {xm} with xmx as m, but PKx0*,c(xm)W0 for all m. Taking ymPKx0*,c(xm)W0,  m=1,2,, similar to previous arguments, we can observe the fact that there exists a subsequence {ymk} of {ym} such that ymkwy as k and yPKx0*,c(x). This means that there exists ymkW0 for some k large enough, which is a contradiction.

Similar to the proof of Theorem 8, we may prove the following theorem.

Theorem 8.

Let X be a Banach space.

Let x0*X*{θ},  let x0* attain its norm on S(X), and let c. If X is nearly strictly convex, then the metric projection PKx0*,c is norm-norm upper semicontinuous.

Let x0X{θ}  and let c. If X is weakly nearly smooth, then the metric projection PKx0,c is norm-weakly upper semicontinuous.

Let x0X{θ}  and let c. If X is nearly smooth, then the metric projection PKx0,c is norm-norm upper semicontinuous.

Lemma 9 (see [<xref ref-type="bibr" rid="B12">11</xref>]).

Let M be a proximal subspace. Then for any xX, one has the decomposition (25)x=x1+x2,x1PM(x),x2D-1(M), where M={x*X*:x*(x)=0,xM} and (26)D-1(M)={xX:D(x)M}. If M is a Chebyshev subspace, the decomposition is unique, and (27)x=PM(x)+x2,x2D-1(M).

Lemma 10.

Let X be a strictly convex Banach space and let M be a proximal subspace. Then, for any xX, one has (28)PM(x+y)=PM(x)+y,yM.

Proof.

Let yM, for any zM, we have that w=z-yM. Consider (29)PM(x)+y-(x+y)=PM(x)-xw-x=(w+y)-(x+y)=z-(x+y). By the definition of PM, we obtain PM(x)+yPM(x+y). Since X is strictly convex, we know that PM is single-valued, and hence we have PM(x+y)=PM(x)+y.

Similar to the proof Theorem 2.1(1) in , we can prove the following result by Lemmas 9 and 10.

Lemma 11.

Let X be a strictly convex Banach space and let M be a proximal subspace. P is single-valued operator from X into M, and PM is a metric projection from X into M. Then P=PM if and only if the following conditions are satisfied:

P-1(θ)=D-1(M);

P(x+y)=P(x)+y,for  allyM.

Theorem 12.

Let X be a strictly convex Banach space and let M be a proximal subspace. Then the metric projection PM is a linear bounded operator if and only if D-1(M) is a linear subspace.

Proof.

On necessity: let PM be a linear operator. Since X is strictly convex and M is proximal, then PM is single valued. By Lemma 11(1), for any x,yD-1(M)=PM-1(θ),  α,  β, then (30)PM(αx+βy)=αPM(x)+βPM(y)=0, and hence αx+βyPM-1(θ)=D-1(M). This shows that D-1(M) is a linear subspace.

On sufficiency: let D-1(M) be a linear subspace and let PM be a metric projection; since X is strictly convex, by Lemma 11(1), PM-1(θ) is also a linear subspace. For any x,yX,x-PM(x),y-PM(y){x-PM(x):xX}, we have that (31)(x+y)-(PM(x)+PM(y))=(x-PM(x))+(y-PM(y)){z-PM(z):zX}=PM-1(θ). By Lemma 11(2), we have that (32)0=PM((x+y)-(PM(x)+PM(y)))=PM(x+y)-(PM(x)+PM(y)). It follows that PM(x+y)=PM(x)+PM(y). Note that PM is homogeneous; we obtain that PM is a linear operator. In addition, for any xX, since θM, we have that (33)PM(x)=PM(x)-x+xPM(x)-x+xθ-x+x=2x. This shows that PM 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).

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