Journal of Function Spaces

Journal of Function Spaces / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 724120 | https://doi.org/10.1155/2012/724120

Z. H. Zhang, C. Y. Liu, "Convexity and Proximinality in Banach Space", Journal of Function Spaces, vol. 2012, Article ID 724120, 11 pages, 2012. https://doi.org/10.1155/2012/724120

Convexity and Proximinality in Banach Space

Academic Editor: Anna Kaminska
Received09 Mar 2011
Accepted09 Nov 2011
Published04 Jan 2012

Abstract

By the continuity of preduality map, we give some necessary and sufficient conditions of the strongly convex and very convex spaces, respectively. Using nearly strong convexity of ๐‘‹, we give some equivalent conditions that every element in ๐‘‹ is strongly unique of order ๐‘, bounded strongly unique of order ๐‘, and locally strongly unique of order ๐‘.

1. Notations and Definitions

Let ๐‘‹ be a Banach space and let ๐‘‹โˆ— be its dual space. Let us denote by ๐ต(๐‘‹) and ๐‘†(๐‘‹) the closed unit ball and the unit sphere of ๐‘‹, and by ๐ต(๐‘ฅ,๐‘Ÿ) the closed ball centered at ๐‘ฅ of radius ๐‘Ÿ>0. Let ๐‘ฅโˆˆ๐‘†(๐‘‹),๐ท(๐‘ฅ)={๐‘“โˆˆ๐‘†(๐‘‹โˆ—)โˆถ๐‘“(๐‘ฅ)=1}. Let us denote by ๐‘๐ด(๐‘‹) the set of all norm-attaining functionals in ๐‘‹โˆ— and ๐‘†0(๐‘‹โˆ—)=๐‘†(๐‘‹โˆ—)โˆฉ๐‘๐ด(๐‘‹).

For a subset ๐ถโŠ‚๐‘‹, the metric projection ๐‘ƒ๐ถโˆถ๐‘‹โ†’2๐ถ is defined by ๐‘ƒ๐ถ(๐‘ฅ)={๐‘ฆโˆˆ๐ถโˆถโ€–๐‘ฅโˆ’๐‘ฆโ€–=๐‘‘(๐‘ฅ,๐ถ)}, where ๐‘‘(๐‘ฅ,๐ถ)=inf{โ€–๐‘ฅโˆ’๐‘ฆโ€–โˆถ๐‘ฆโˆˆ๐ถ},๐‘ฅโˆˆ๐‘‹. If ๐‘ƒ๐ถ(๐‘ฅ)โ‰ โˆ… for each ๐‘ฅโˆˆ๐‘‹, then ๐ถ is said to be proximinal. If ๐‘ƒ๐ถ(๐‘ฅ) is at most a singleton for each ๐‘ฅโˆˆ๐‘‹, then ๐ถ is said to be semi-Chebyshev. If ๐ถ is a simultaneously proximinal and semi-Chebyshev set, then ๐ถ is said to be a Chebyshev set.

Definition 1.1 (see [1]). One says that ๐‘‹ is strongly convex (resp., very convex/nearly strongly convex/nearly very convex) if for any ๐‘ฅโˆˆ๐‘†(๐‘‹) and {๐‘ฅ๐‘›}โŠ‚๐ต(๐‘‹) convergence ๐‘ฅโˆ—(๐‘ฅ๐‘›)โ†’1 as ๐‘›โ†’โˆž for some ๐‘ฅโˆ—โˆˆ๐ท(๐‘ฅ) implies that ๐‘ฅ๐‘›โ†’๐‘ฅ as ๐‘›โ†’โˆž (resp., ๐‘ฅ๐‘›๐‘คโˆ’โ†’๐‘ฅ as ๐‘›โ†’โˆž/{๐‘ฅ๐‘›} is relatively compact/{๐‘ฅ๐‘›} is weakly relatively compact).

Definition 1.2 (see [2]). One says that ๐‘‹ is midpoint locally uniformly rotund (MLUR) (resp., weakly midpoint locally uniformly rotund (WMLUR)) if any ๐‘ฅ0, ๐‘ฅ๐‘›, ๐‘ฆ๐‘›โˆˆ๐‘†(๐‘‹), (๐‘›=1,2,โ€ฆ), ๐‘ฅ๐‘›+๐‘ฆ๐‘›โ†’2๐‘ฅ0 as ๐‘›โ†’โˆž implies that ๐‘ฅ๐‘›โˆ’๐‘ฆ๐‘›โ†’๐œƒ as ๐‘›โ†’โˆž (resp., ๐‘ฅ๐‘›โˆ’๐‘ฆ๐‘›๐‘คโˆ’โ†’๐œƒ as ๐‘›โ†’โˆž).

Wu and Li defined strong convexity in [3], and Wang and Zhang in [4] defined very convexity, nearly strong convexity, and nearly very convexity which are two generalizations of locally uniformly rotund (LUR) and weakly locally uniformly rotund (WLUR) spaces. By [3, 5], we have the following relations:

724120.fig.001

Sullivan defined very rotund space in [6]. A Banach space ๐‘‹ is said to be very rotund if no ๐‘ฅโˆ—โˆˆ๐‘†(๐‘‹โˆ—) is simultaneously a norming element for some ๐‘ฅโˆˆ๐‘†(๐‘‹) and ๐‘ฅโˆ—โˆ—โˆˆ๐‘†(๐‘‹โˆ—โˆ—), where ๐‘ฅโ‰ ๐‘ฅโˆ—โˆ—. Z. H. Zhang and C. J. Zhang proved that very rotund space coincides with very convex space in [5]. In [3โ€“5, 7], many results of these four classes of convexities were proved. In particular, Zhang and Shi proved that they have important applications in approximation theory in [1]. In [8], Bandyapadhyay et al. also proposed two generalizations of locally uniformly rotund space, which are called almost locally uniformly rotund space and weakly almost locally uniformly rotund space. A Banach space ๐‘‹ is said to be ALUR (resp., WALUR) if for any ๐‘ฅโˆˆ๐‘†(๐‘‹), {๐‘ฅ๐‘›}โŠ‚๐ต(๐‘‹), and {๐‘ฅ๐‘›}โŠ‚๐ต(๐‘‹โˆ—), the condition lim๐‘šlim๐‘›๐‘ฅโˆ—๐‘š((๐‘ฅ๐‘›+๐‘ฅ)/2)=1 implies ๐‘ฅ๐‘›โ†’๐‘ฅ (resp., ๐‘ฅ๐‘›๐‘คโˆ’โ†’๐‘ฅ). Many properties of these two classes of convexities were studied in [8โ€“10] too. Recently, we proved that almost locally uniformly rotund space is equivalent to strongly convex space and that weakly almost locally uniformly rotund space is equivalent to very convex space [7]. Thus, we unified the results of the studies about the strongly convex space (resp., very convex space) and the almost locally uniform rotundity (resp., weakly almost locally uniform rotundity). This shows that these convexities have important effects on and applications in geometry of Banach space and approximation theory.

A sequence {๐‘ง๐‘›}โŠ‚๐ถ is said to be minimizing for ๐‘ฅโˆˆ๐‘‹โงต๐ถ if โ€–๐‘ฅโˆ’๐‘ง๐‘›โ€–โ†’๐‘‘(๐‘ฅ,๐ถ) as ๐‘›โ†’โˆž.

Definition 1.3 (see [10]). Let ๐ถ be a closed subset (resp., a weakly closed subset) of ๐‘‹ and ๐‘ฅ0โˆˆ๐‘‹โงต๐ถ.(1)One says that ๐ถ is approximatively (resp., weakly approximatively) compact for ๐‘ฅ0 if every minimizing sequence {๐‘ง๐‘›}โŠ‚๐ถ for ๐‘ฅ0 has a convergent (resp., weakly convergent) subsequence.(2)One says that ๐ถ is strongly Chebyshev (resp., weakly strongly Chebyshev) for ๐‘ฅ0 if every minimizing sequence {๐‘ง๐‘›}โŠ‚๐ถ for ๐‘ฅ0 is convergent (resp., weakly convergent).

If ๐ถ is approximatively compact (resp., weakly approximatively compact/strongly Chebyshev/weakly strongly Chebyshev) for every ๐‘ฅโˆˆ๐‘‹โงต๐ถ, we say that ๐ถ is approximatively compact (resp., weakly approximatively compact/strongly Chebyshev/weakly strongly Chebyshev) in ๐‘‹.

Definition 1.4 (see [11]). Let ๐บโˆˆ๐‘‹, ๐‘ฅโˆˆ๐‘‹โงต๐บ, and 1โ‰ค๐‘<โˆž.(1)๐‘”0โˆˆ๐‘ƒ๐บ(๐‘ฅ) is said to be strongly unique at ๐‘ฅ if there exists a constant ๐‘Ÿ๐‘>0 such that โ€–โ€–โ€–๐‘ฅโˆ’๐‘”โ€–โ‰ฅ๐‘ฅโˆ’๐‘”0โ€–โ€–+๐‘Ÿ๐‘โ€–โ€–๐‘”โˆ’๐‘”0โ€–โ€–,(1.1) for any ๐‘”โˆˆ๐บ.(2)๐‘”0โˆˆ๐บ is said to be strongly unique of order ๐‘ at ๐‘ฅ if there exists an ๐‘Ÿ๐‘=๐‘Ÿ๐‘(๐‘ฅ)>0 such that โ€–๐‘ฅโˆ’๐‘”โ€–๐‘โ‰ฅโ€–โ€–๐‘ฅโˆ’๐‘”0โ€–โ€–๐‘+๐‘Ÿ๐‘โ€–โ€–๐‘”โˆ’๐‘”0โ€–โ€–๐‘,(1.2) for any ๐‘”โˆˆ๐บ.(3)๐‘”0โˆˆ๐บ is said to be bounded strongly unique of order ๐‘ at ๐‘ฅ if given any ๐‘>0, there exists an ๐‘Ÿ๐‘,๐‘=๐‘Ÿ๐‘,๐‘(๐‘ฅ) such that โ€–โ€–โ€–๐‘ฅโˆ’๐‘”โ€–โ‰ฅ๐‘ฅโˆ’๐‘”0โ€–โ€–+๐‘Ÿ๐‘,๐‘โ€–โ€–๐‘”โˆ’๐‘”0โ€–โ€–๐‘,(1.3) for any ๐‘”โˆˆ๐บ,โ€–๐‘”โˆ’๐‘”0โ€–โ‰ค๐‘.(4)๐‘”0โˆˆ๐บ is said to be locally strongly unique of order ๐‘ at ๐‘ฅ if there exist ๐‘>0 and ๐‘Ÿ๐‘,๐‘=๐‘Ÿ๐‘,๐‘(๐‘ฅ) such that โ€–โ€–โ€–๐‘ฅโˆ’๐‘”โ€–โ‰ฅ๐‘ฅโˆ’๐‘”0โ€–โ€–+๐‘Ÿ๐‘,๐‘โ€–โ€–๐‘”โˆ’๐‘”0โ€–โ€–๐‘,(1.4) for any ๐‘”โˆˆ๐บ,โ€–๐‘”โˆ’๐‘”0โ€–โ‰ค๐‘.

In order to study the uniqueness of best approximation in nonlinear approximation theory, Wulbert [12] defined the strong uniqueness of best approximation. Smarzewski [13] and Schmidt [14] defined the strongly unique of order ๐‘ and the bounded strongly unique of order ๐‘, respectively. By [11], we know that the strongly unique of order ๐‘ and the bounded strongly unique of order ๐‘ all are generalizations of the strongly unique. The strongly unique of order ๐‘ and the bounded strongly unique of order ๐‘ imply the locally strongly unique of order ๐‘, but converse implied relation is not generally true. When ๐‘=1, the strongly unique of order ๐‘ and the bounded strongly unique of order ๐‘ all are strongly unique. The locally strongly unique of order 1 is not equivalent to the strongly unique.

Definition 1.5 (see [10]). For ๐‘ฅโˆ—โˆˆ๐‘†(๐‘‹โˆ—) and ๐‘ฅโˆˆ๐‘†(๐‘‹), let one define the following maps: ๎€ฝ๐‘ฅ๐ท(๐‘ฅ)=โˆ—๎€ท๐‘‹โˆˆ๐‘†โˆ—๎€ธโˆถ๐‘ฅโˆ—๎€พ(๐‘ฅ)=1,๐ทโˆ’1๎€ท๐‘ฅโˆ—๎€ธ=๎€ฝ๐‘ฅโˆˆ๐‘†(๐‘‹)โˆถ๐‘ฅโˆ—๎€พ.(๐‘ฅ)=1(1.5)๐ท is called the duality map and ๐ทโˆ’1 is called the preduality map. Naturally, ๐ทโˆ’1 is defined only on ๐‘†0(๐‘‹โˆ—).

Definition 1.6 (see [10]). The preduality map ๐ทโˆ’1โˆถ๐‘†(๐‘‹โˆ—)โ†’๐‘†(๐‘‹) is said to be upper semicontinuous (๐‘›โˆ’๐œ) on ๐‘†0(๐‘‹โˆ—) if for any ๐‘ฅโˆ—โˆˆ๐‘†0(๐‘‹โˆ—) and any ๐œ open set ๐‘Š with ๐ทโˆ’1(๐‘ฅโˆ—)โŠ‚๐‘Š, there exists ๐œ€>0 such that ๐ทโˆ’1(๐‘ฆโˆ—)โŠ‚๐‘Š whenever โ€–๐‘ฅโˆ—โˆ’๐‘ฆโˆ—โ€–<๐œ€, where ๐œ stands for norm or weak topology.

2. Convexity and Continuity of the Preduality Map

Using the Bronsted-Rockafeller Theorem (see [15, Theoremโ€‰โ€‰3.18, page 51]), we can prove the following lemma.

Lemma 2.1. Suppose that ๐œ€>0, ๐‘ฅ0โˆˆ๐‘†(๐‘‹), ๐‘ฅโˆ—0โˆˆ๐‘†(๐‘‹โˆ—), and ๐‘ฅโˆ—0(๐‘ฅ0)>1โˆ’๐œ€, then there are ๐‘ฅ๐œ€โˆˆS(๐‘‹) and ๐‘ฅโˆ—๐œ€โˆˆ๐ท(๐‘ฅ๐œ€) such that โ€–โ€–๐‘ฅ๐œ€โˆ’๐‘ฅ0โ€–โ€–โˆš<2โ€–โ€–๐‘ฅ๐œ€,โˆ—๐œ€โˆ’๐‘ฅโˆ—0โ€–โ€–โˆš<2๐œ€.(2.1)

Theorem 2.2. Let ๐‘‹ be a Banach space. ๐‘‹ is strongly convex if and only if the preduality map ๐ทโˆ’1โˆถ๐‘†0(๐‘‹โˆ—)โ†’๐‘†(๐‘‹) is singlevalued and continuous.

Proof. Necessity. Since strong convexity implies strict convexity, ๐ทโˆ’1 is singlevalued. Suppose that {๐‘ฅโˆ—๐‘›}โŠ‚๐‘†0(๐‘‹โˆ—), ๐‘ฅโˆ—โˆˆ๐‘†0(๐‘‹โˆ—) with ๐‘ฅโˆ—๐‘›โ†’๐‘ฅโˆ—, then there exist {๐‘ฅ๐‘›}โŠ‚๐‘†(๐‘‹) and ๐‘ฅโˆˆ๐‘†(๐‘‹) such that ๐‘ฅโˆ—๐‘›(๐‘ฅ๐‘›)=1=๐‘ฅโˆ—(๐‘ฅ). We have that ๐‘ฅโˆ—๎€ท๐ทโˆ’1๎€ท๐‘ฅโˆ—๐‘›=๎€ท๐‘ฅ๎€ธ๎€ธโˆ—โˆ’๐‘ฅโˆ—๐‘›+๐‘ฅโˆ—๐‘›๐ท๎€ธ๎€ทโˆ’1๎€ท๐‘ฅโˆ—๐‘›=๎€ท๐‘ฅ๎€ธ๎€ธโˆ—โˆ’๐‘ฅโˆ—๐‘›๐ท๎€ธ๎€ทโˆ’1๎€ท๐‘ฅโˆ—๐‘›๎€ธ๎€ธ+๐‘ฅโˆ—๐‘›๎€ท๐ทโˆ’1๎€ท๐‘ฅโˆ—๐‘›.๎€ธ๎€ธ(2.2) Since ||๎€ท๐‘ฅโˆ—โˆ’๐‘ฅโˆ—๐‘›๎€ธ๐ทโˆ’1๎€ท๐‘ฅโˆ—๐‘›๎€ธ||โ‰คโ€–โ€–๐‘ฅโˆ—๐‘›โˆ’๐‘ฅโˆ—โ€–โ€–โŸถ0,as๐‘›โŸถโˆž,(2.3) and ๐‘ฅโˆ—๐‘›(๐ทโˆ’1(๐‘ฅโˆ—๐‘›))=๐‘ฅโˆ—๐‘›(๐‘ฅ๐‘›)=1, we have that ๐‘ฅโˆ—๎€ท๐ทโˆ’1๎€ท๐‘ฅโˆ—๐‘›๎€ธ๎€ธ=๐‘ฅโˆ—๎€ท๐‘ฅ๐‘›๎€ธโŸถ1,as๐‘›โŸถโˆž.(2.4) Since ๐‘‹ is strongly convex, we deduce that ๐‘ฅ๐‘›โ†’๐‘ฅ as ๐‘›โ†’โˆž, that is, ๐ทโˆ’1(๐‘ฅโˆ—๐‘›)โ†’๐ทโˆ’1(๐‘ฅโˆ—) as ๐‘›โ†’โˆž. This means that ๐ทโˆ’1 is continuous.
Sufficiency. Let {๐‘ฅ๐‘›}โŠ‚๐‘†(๐‘‹),๐‘ฅโˆˆ๐‘†(๐‘‹) with ๐‘ฅโˆ—(๐‘ฅ๐‘›)โ†’1 as ๐‘›โ†’โˆž for some ๐‘ฅโˆ—โˆˆ๐ท(๐‘ฅ). Since ๐ทโˆ’1 is singlevalued, by Lemma 2.1, there exist {๐‘ฆ๐‘›}โŠ‚๐‘†(๐‘‹) and {๐‘ฆโˆ—๐‘›}โŠ‚๐‘†0(๐‘‹โˆ—) such that ๐ทโˆ’1(๐‘ฆโˆ—๐‘›)=๐‘ฆ๐‘› and โ€–โ€–๐‘ฆ๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–โ€–โ€–๐‘ฆโŸถ0,โˆ—๐‘›โˆ’๐‘ฅโˆ—โ€–โ€–โŸถ0,as๐‘›โŸถโˆž.(2.5) Since ๐ทโˆ’1 is continuous, we have that ๐ทโˆ’1(๐‘ฆโˆ—๐‘›)โ†’๐ทโˆ’1(๐‘ฅโˆ—) as ๐‘›โ†’โˆž, that is, ๐‘ฆ๐‘›โ†’๐‘ฅ as ๐‘›โ†’โˆž. So ๐‘ฅ๐‘›โ†’๐‘ฅ as ๐‘›โ†’โˆž, which means that ๐‘‹ is strongly convex.

Using Lemma 2.1, in a similar way to prove Theorem 2.2, we can prove the following result.

Theorem 2.3. Let ๐‘‹ be a Banach space. ๐‘‹ is very convex if and only if the preduality map ๐ทโˆ’1โˆถ๐‘†0(๐‘‹โˆ—)โ†’๐‘†(๐‘‹) is singlevalued and weakly continuous.

Lemma 2.4. Let ๐ถ be a convex set of a strongly convex Banach space ๐‘‹. The following are equivalent:(1)๐ถ is proximinal;(2)๐ถ is weakly approximatively compact;(3)๐ถ is approximatively compact;(4)๐ถ is strongly Chebyshev.

Proof. We only need to prove (1)โ‡’(4).
Let ๐‘ฅโˆˆ๐‘‹โงต๐ถ, {๐‘ง๐‘›}โŠ‚๐ถ such that lim๐‘›โ†’โˆžโ€–โ€–๐‘ฅโˆ’๐‘ง๐‘›โ€–โ€–=๐‘‘(๐‘ฅ,๐ถ).(2.6) In order to finish the proof, we will show that there exists some ๐‘ฆ0โˆˆ๐ถ such that โ€–๐‘ฆ0โˆ’๐‘ง๐‘›โ€–โ†’0 as ๐‘›โ†’โˆž.
Step 1. If ๐‘ฅ=0, then ๐‘Ÿ=๐‘‘(0,๐ถ)>0 and ๐ต(0,๐‘Ÿ)โˆฉ๐ถ=๐‘ƒ๐ถ(0)โ‰ โˆ…,int๐ต(0,๐‘Ÿ)โˆฉ๐ถ=โˆ…,(2.7) where ๐ต(0,๐‘Ÿ)={๐‘ฆโˆถโ€–๐‘ฆโ€–โ‰ค๐‘Ÿ}. By the separation theorem [2] and definition of norm, there exists an ๐‘“โˆˆ๐‘†(๐‘‹โˆ—) such that โ€–โ€–๐‘ฆsup{๐‘“(๐‘ฆ)โˆถ๐‘ฆโˆˆ๐ถ}โ‰คinf{๐‘“(๐‘ฆ)โˆถ๐‘ฆโˆˆ๐ต(0,๐‘Ÿ)}=โˆ’โ€–๐‘“โ€–๐‘Ÿ=โˆ’0โ€–โ€–,(2.8) for any ๐‘ฆ0โˆˆ๐‘ƒ๐ถ(0). Hence, we have that โˆ’โ€–โ€–๐‘ฆ0โ€–โ€–๎€ท๐‘ฆโ‰ค๐‘“0๎€ธโ€–โ€–๐‘ฆโ‰คsup{๐‘“(๐‘ฆ)โˆถ๐‘ฆโˆˆ๐ถ}โ‰คโˆ’0โ€–โ€–.(2.9) This shows that ๐‘“โˆˆ๐ท(โˆ’๐‘ฆ0/โ€–๐‘ฆ0โ€–). From ๐‘ง๐‘›โˆˆ๐ถ, we get ๐‘“(๐‘ง๐‘›)โ‰ค๐‘“(๐‘ฆ0). Combining it with the condition lim๐‘›โ†’โˆžโ€–0โˆ’๐‘ง๐‘›โ€–=๐‘‘(0,๐ถ), it follows that โ€–โ€–0โˆ’๐‘ฆ0โ€–โ€–๎€ท=๐‘“0โˆ’๐‘ฆ0๎€ธ๎€ทโ‰ค๐‘“0โˆ’๐‘ง๐‘›๎€ธโ‰คโ€–โ€–0โˆ’๐‘ง๐‘›โ€–โ€–โ€–โ€–โŸถ๐‘‘(0,๐ถ)=0โˆ’๐‘ฆ0โ€–โ€–,(2.10) as ๐‘›โ†’โˆž. Hence, ๐‘“(โˆ’๐‘ง๐‘›/โ€–๐‘ง๐‘›โ€–)โ†’1 as ๐‘›โ†’โˆž. Since ๐‘‹ is strongly convex, โˆ’๐‘ง๐‘›/โ€–๐‘ง๐‘›โ€–โ†’โˆ’๐‘ฆ0/โ€–๐‘ฆ0โ€– as ๐‘›โ†’โˆž, that is, โ€–๐‘ง๐‘›โˆ’๐‘ฆ0โ€–โ†’0 as ๐‘›โ†’โˆž.Step 2. If ๐‘ฅโ‰ 0, we set ๐ถ๎…ž=๐‘ฅโˆ’๐ถ. It is clear that ๐ถ๎…ž is proximinal and {๐‘ฅโˆ’๐‘ง๐‘›}โŠ‚๐ถ๎…ž is a minimizing sequence for 0. By Step 1, there exists ๐‘ฆ๎…ž0โˆˆ๐ถ๎…ž, such that โ€–๐‘ฆ๎…ž0โˆ’(๐‘ฅโˆ’๐‘ง๐‘›)โ€–โ†’0 as ๐‘›โ†’โˆž. This shows that โ€–(๐‘ฅโˆ’๐‘ฆ๎…ž0)โˆ’๐‘ง๐‘›โ€–โ†’0 as ๐‘›โ†’โˆž and ๐‘ฅโˆ’๐‘ฆ๎…ž0=๐‘ฆ0โˆˆ๐ถ.

Similarly to the proof of Lemma 2.4, we can prove the following result.

Lemma 2.5. Let ๐ถ be a convex set of a nearly strongly convex Banach space ๐‘‹. The following are equivalent:(1)๐ถ is proximinal;(2)๐ถ is weakly approximatively compact;(3)๐ถ is approximatively compact.

Lemma 2.6. Let ๐‘ฅโˆˆ๐‘†(๐‘‹), ๐‘“โˆˆ๐ท(๐‘ฅ), then hyperplane ๐ป={๐‘ฆโˆˆ๐‘‹โˆถ๐‘“(๐‘ฆ)=1} is a proximinal convex subset in ๐‘‹.

Proof. Let ker๐‘“={๐‘ฆโˆˆ๐‘‹โˆถ๐‘“(๐‘ฆ)=0}.(2.11) We will prove that ker๐‘“ is a proximinal convex subset in ๐‘‹. For any ๐‘งโˆˆ๐‘‹, if ๐‘‘(๐‘ง,ker๐‘“)=0, since ker๐‘“ is a closed subspace, we know that ๐‘งโˆˆker๐‘“. Hence, ๐‘งโˆˆ๐‘ƒker๐‘“(๐‘ง). If ๐‘‘(๐‘ง,ker๐‘“)>0, since ๐‘‹={๐›ผ๐‘ง}+ker๐‘“, there exist ๐œ†โˆˆโ„ and ๐‘ฆ0โˆˆker๐‘“ such that ๐‘‘(๐‘ง,ker๐‘“)๐‘ฅ=๐œ†๐‘ง+๐‘ฆ0. For any ๐‘ฆโˆˆ๐‘‹, ๐‘‘(๐‘ฆ,ker๐‘“)=|๐‘“(๐‘ฆ)|, and ๐‘“(๐‘ฅ)=1, we have that ๐‘‘(๐‘ฅ,ker๐‘“)=1 and ๐‘“(๐‘‘(๐‘ง,ker๐‘“)๐‘ฅ)=๐‘‘(๐‘‘(๐‘ง,ker๐‘“)๐‘ฅ,ker๐‘“). Therefore, we have that ๐‘‘๎€ท(๐‘ง,ker๐‘“)=๐‘‘๐œ†๐‘ง+๐‘ฆ0๎€ธ=||๐œ†||๐‘‘,ker๐‘“(๐‘ง,ker๐‘“).(2.12) This means that |๐œ†|=1. Hence, we have โ€–โ€–๐‘‘(๐‘ง,ker๐‘“)=โ€–๐‘‘(๐‘ง,ker๐‘“)๐‘ฅโ€–=๐œ†๐‘ง+๐‘ฆ0โ€–โ€–=โ€–โ€–๐‘ง+๐œ†๐‘ฆ0โ€–โ€–.(2.13) It follows that โˆ’๐œ†๐‘ฆ0โˆˆ๐‘ƒker๐‘“(๐‘ง), which means that ker๐‘“ is proximinal set.
Furthermore, we will prove that hyperplane ๐ป={๐‘ฆโˆˆ๐‘‹โˆถ๐‘“(๐‘ฆ)=1} is a proximinal convex subset in ๐‘‹. For ๐‘ฅ0โˆˆ๐ป, ๐ป=๐‘ฅ0+ker๐‘“. For any ๐‘งโˆˆ๐‘‹, ๐‘‘(๐‘ง,๐ป)=inf๐‘ฆโˆˆker๐‘“โ€–โ€–๎€ท๐‘ฅ๐‘งโˆ’0๎€ธโ€–โ€–๎€ท+๐‘ฆ=๐‘‘๐‘งโˆ’๐‘ฅ0๎€ธ.,ker๐‘“(2.14) Since ker๐‘“ is a proximinal subset, there exists a ๐‘ฆ0โˆˆker๐‘“ such that ๐‘‘๎€ท๐‘งโˆ’๐‘ฅ0๎€ธ=โ€–โ€–๎€ท๐‘ฅ,ker๐‘“๐‘งโˆ’0+๐‘ฆ0๎€ธโ€–โ€–.(2.15) Therefore, ๐‘ฅ0+๐‘ฆ0โˆˆ๐‘ƒ๐ป(๐‘ง), which means that ๐ป is proximinal set in ๐‘‹.

Theorem 2.7. Let ๐‘‹ be a Banach space. ๐‘‹ is a nearly strongly convex if and only if the preduality map ๐ทโˆ’1โˆถ๐‘†0(๐‘‹โˆ—)โ†’๐‘†(๐‘‹) is upper semicontinuous (๐‘›โˆ’๐‘›) on ๐‘†0(๐‘‹โˆ—) with norm compact images, where ๐‘› stands for norm topology.

Proof. Necessity. Arbitrarily take ๐‘ฅโˆ—โˆˆ๐‘†0(๐‘‹โˆ—). Then by Lemma 2.6, we know that ker๐‘ฅโˆ—=๐‘Œ is a proximinal convex subset in ๐‘‹. Since ๐‘‹ is nearly strongly convex, by Lemma 2.5, we know that ๐‘Œ is approximatively compact. Suppose that ๐ทโˆ’1 is not upper semicontinuous (๐‘›โˆ’๐‘›) at ๐‘ฅโˆ—, then for some open set ๐‘Š in ๐‘‹ with ๐ทโˆ’1(๐‘ฅโˆ—)โŠ‚๐‘Š, there exists {๐‘ฅโˆ—๐‘›}โŠ‚๐‘†(๐‘‹โˆ—) such that ๐‘ฅโˆ—๐‘›โ†’๐‘ฅโˆ— as ๐‘›โ†’โˆž and ๐ทโˆ’1(๐‘ฅโˆ—๐‘›)ฬธโІ๐‘Š for all ๐‘›. Let ๐‘ง๐‘›โˆˆ๐ทโˆ’1(๐‘ฅโˆ—๐‘›)โงต๐‘Š. Fix ๐‘ฅโˆˆ๐ทโˆ’1(๐‘ฅโˆ—). Let ๐‘ฅ๐‘›=๐‘ฅโˆ—(๐‘ง๐‘›)๐‘ฅโˆ’๐‘ง๐‘›, then {๐‘ฅ๐‘›}โŠ‚๐‘Œ is a minimizing sequence for ๐‘ฅ. Since ๐‘Œ is approximatively compact, {๐‘ฅ๐‘›} has a convergent subsequence. So {๐‘ง๐‘›} has convergent subsequence converging to ๐‘ง. Thus, ๐‘งโˆˆ๐ทโˆ’1(๐‘ฅโˆ—)โŠ‚๐‘Š, but ๐‘ง๐‘›โˆˆ๐‘‹โงต๐‘Š is closed, which is a contradiction. By the assumption, we easily know that the image of ๐ทโˆ’1 is compact.
Sufficiency. Let {๐‘ฅ๐‘›}โŠ‚๐‘†(๐‘‹),๐‘ฅโˆˆ๐‘†(๐‘‹) with ๐‘ฅโˆ—(๐‘ฅ๐‘›)โ†’1 as ๐‘›โ†’โˆž for some ๐‘ฅโˆ—โˆˆ๐ท(๐‘ฅ). By Lemma 2.1, there exist {๐‘ฆ๐‘›}โŠ‚๐‘†(๐‘‹) and {๐‘ฆโˆ—๐‘›}โŠ‚๐‘†(๐‘‹โˆ—) such that ๐‘ฆ๐‘›โˆˆ๐ทโˆ’1(๐‘ฆโˆ—๐‘›) and โ€–โ€–๐‘ฆ๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–โ€–โ€–๐‘ฆโŸถ0,โˆ—๐‘›โˆ’๐‘ฅโˆ—โ€–โ€–โŸถ0,as๐‘›โŸถโˆž.(2.16) Since ๐ทโˆ’1(๐‘ฅโˆ—) is compact, ๐ทโˆ’1(๐‘ฅโˆ—) is proximinal. Let ๐‘ง๐‘›โˆˆ๐ทโˆ’1(๐‘ฅโˆ—) such that โ€–๐‘ฆ๐‘›โˆ’๐‘ง๐‘›โ€–=๐‘‘(๐‘ฆ๐‘›,๐ทโˆ’1(๐‘ฅโˆ—)). In virtue of ๐ทโˆ’1 being upper semicontinuous (๐‘›โˆ’๐‘›) on ๐‘†0(๐‘‹โˆ—) and โ€–๐‘ฆโˆ—๐‘›โˆ’๐‘ฅโˆ—โ€–โ†’0, for any ๐œ€>0, there is ๐‘›0 such that for all ๐‘›โ‰ฅ๐‘›0๐‘‘(๐‘ฆ๐‘›,๐ทโˆ’1(๐‘ฅโˆ—))<๐œ€,thatis,โ€–๐‘ฆ๐‘›โˆ’๐‘ง๐‘›โ€–<๐œ€, which means that ๐‘ฆ๐‘›โˆ’๐‘ง๐‘›โ†’๐œƒ as ๐‘›โ†’โˆž. Combining this with the compactness of ๐ทโˆ’1(๐‘ฅโˆ—),{๐‘ง๐‘›} has convergent subsequence, and hence {๐‘ฆ๐‘›} has convergent subsequence. By ๐‘ฆ๐‘›โˆ’๐‘ฅ๐‘›โ†’๐œƒ as ๐‘›โ†’โˆž, {๐‘ฅ๐‘›} has convergent subsequence, which means that ๐‘‹ is nearly strongly convex.

Theorem 2.8. Let ๐‘‹ be a Banach space. ๐‘‹ is nearly very convex if and only if the preduality map ๐ทโˆ’1โˆถ๐‘†0(๐‘‹โˆ—)โ†’๐‘†(๐‘‹) is upper semicontinuous (๐‘›โˆ’๐‘ค) on ๐‘†0(๐‘‹โˆ—) with weakly compact images.

Proof. In a similar way to the proof of Theorem 2.7, the necessity can be proved.
Sufficiency. Let {๐‘ฅ๐‘›}โŠ‚๐‘†(๐‘‹) with ๐‘ฅโˆ—(๐‘ฅ๐‘›)โ†’1 as ๐‘›โ†’โˆž for some ๐‘ฅโˆ—โˆˆ๐ท(๐‘ฅ). By Lemma 2.1, there are {๐‘ฆ๐‘›}โŠ‚๐‘†(๐‘‹) and {๐‘ฆโˆ—๐‘›}โŠ‚๐‘†(๐‘‹โˆ—) such that ๐‘ฆ๐‘›โˆˆ๐ทโˆ’1(๐‘ฆโˆ—๐‘›) and โ€–โ€–๐‘ฆ๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–โ€–โ€–๐‘ฆโŸถ0,โˆ—๐‘›โˆ’๐‘ฅโˆ—โ€–โ€–โŸถ0as๐‘›โŸถโˆž.(2.17) Suppose that {๐‘ฅ๐‘›} does not have weakly convergent subsequence. By (2.17), we know that {๐‘ฆ๐‘›} does not have weakly convergent subsequence. Without loss of generality, we can assume that ||๐‘“๎€ท๐‘ฆ๐‘›โˆ’๐‘ฆ๐‘š๎€ธ||โ‰ฅ๐œ€0(โˆ€๐‘›,๐‘šโˆˆ๐‘,๐‘›โ‰ ๐‘š),(2.18) for some ๐‘“โˆˆ๐‘‹โˆ—โงต{๐œƒ} and ๐œ€0>0. Since ๐ทโˆ’1(๐‘ฅโˆ—) is weakly compact, there exists {๐‘ง๐‘–}๐‘˜๐‘–=1โŠ‚๐ทโˆ’1(๐‘ฅโˆ—), such that ๐ทโˆ’1(๐‘ฅโˆ—)โŠ‚โˆช๐‘˜๐‘–=1{๐‘ฆโˆˆ๐‘‹โˆถ|๐‘“(๐‘ฆโˆ’๐‘ง๐‘–)|<๐œ€0/2}. Combining upper semicontinuity (๐‘›โˆ’๐‘ค) of ๐ทโˆ’1 with โ€–๐‘ฆโˆ—๐‘›โˆ’๐‘ฅโˆ—โ€–โ†’0,{๐‘ฆ๐‘›}๐‘›โ‰ฅ๐‘›0โŠ‚โˆช๐‘˜๐‘–=1{๐‘ฆโˆˆ๐‘‹โˆถ|๐‘“(๐‘ฆโˆ’๐‘ง๐‘–)|<๐œ€0/2} for some ๐‘›0โˆˆ๐‘. So, there are subsequence {๐‘ฆ๐‘›๐‘˜}๐‘˜โ‰ฅ1โŠ‚{๐‘ฆ๐‘›}๐‘›โ‰ฅ๐‘›0 and ๐‘–0(1โ‰ค๐‘–0โ‰ค๐‘˜) such that {๐‘ฆ๐‘›๐‘˜}๐‘˜โ‰ฅ1โŠ‚{๐‘ฆโˆˆ๐‘‹โˆถ|๐‘“(๐‘ฆโˆ’๐‘ง๐‘–0)|<๐œ€0/2}. Hence, ||๐‘“๎€ท๐‘ฆ๐‘›๐‘˜โˆ’๐‘ฆ๐‘›๐‘™๎€ธ||โ‰ค||๐‘“๎€ท๐‘ฆ๐‘›๐‘˜โˆ’๐‘ง๐‘–0๎€ธ||+||๐‘“๎€ท๐‘ง๐‘–0โˆ’๐‘ฆ๐‘›๐‘™๎€ธ||<๐œ€02+๐œ€02=๐œ€0,(2.19) a contradiction with (2.18).

3. Convexity and Proximinality

Theorem 3.1. Let ๐ถ be a weakly approximatively compact subset of nearly strongly convex Banach space ๐‘‹, ๐‘ฅโˆˆ๐‘‹โงต๐ถ, and ๐‘”0โˆˆ๐ถ. If ๐‘”0 is the unique element of best approximation of ๐‘ฅ, then the following are equivalent:(1)๐‘”0 is strongly unique of order ๐‘ at ๐‘ฅ;(2)๐‘”0 is bounded strongly unique of order ๐‘ at ๐‘ฅ;(3)๐‘”0 is locally strongly unique of order ๐‘ at ๐‘ฅ.

Proof. (1)โ‡’(2). Let ๐‘Ÿ๐‘>0 be such that โ€–๐‘ฅโˆ’๐‘”โ€–๐‘โ‰ฅโ€–โ€–๐‘ฅโˆ’๐‘”0โ€–โ€–๐‘+๐‘Ÿ๐‘โ€–โ€–๐‘”โˆ’๐‘”0โ€–โ€–๐‘,(3.1) for any ๐‘”โˆˆ๐ถ. For any ๐‘>0 and each ๐‘”โˆˆ๐ถ,โ€–๐‘”โˆ’๐‘”0โ€–โ‰ค๐‘, since โ€–โ€–โ€–๐‘ฅโˆ’๐‘”โ€–โ‰ค๐‘ฅโˆ’๐‘”0โ€–โ€–+โ€–โ€–๐‘”โˆ’๐‘”0โ€–โ€–โ‰คโ€–โ€–๐‘ฅโˆ’๐‘”0โ€–โ€–+๐‘,(3.2) by Lagrangeโ€™s mean value theorem, โ€–๐‘ฅโˆ’๐‘”โ€–๐‘โˆ’โ€–โ€–๐‘ฅโˆ’๐‘”0โ€–โ€–๐‘โ€–โ€–โ€–๐‘ฅโˆ’๐‘”โ€–โˆ’๐‘ฅโˆ’๐‘”0โ€–โ€–๎€ทโ€–โ€–โ‰ค๐‘๐‘ฅโˆ’๐‘”0โ€–โ€–๎€ธ+๐‘๐‘โˆ’1.(3.3) Set ๐‘Ÿ๐‘,๐‘=๐‘Ÿ๐‘๐‘๎€ทโ€–โ€–๐‘ฅโˆ’๐‘”0โ€–โ€–๎€ธ+๐‘1โˆ’๐‘.(3.4) Then โ€–โ€–โ€–๐‘ฅโˆ’๐‘”โ€–โˆ’๐‘ฅโˆ’๐‘”0โ€–โ€–โ‰ฅ๐‘Ÿ๐‘,๐‘โ€–โ€–๐‘”โˆ’๐‘”0โ€–โ€–๐‘.(3.5) Because ๐‘Ÿ๐‘,๐‘ is independent of ๐‘”,๐‘”0 is bounded strongly unique of order ๐‘ at ๐‘ฅ.
(2)โ‡’(1). From limโ€–๐‘”โ€–โ†’โˆžโ€–๐‘ฅโˆ’๐‘”โ€–๐‘โˆ’โ€–โ€–๐‘ฅโˆ’๐‘”0โ€–โ€–๐‘โ€–โ€–๐‘”โˆ’๐‘”0โ€–โ€–๐‘โ‰ฅlimโ€–๐‘”โ€–โ†’โˆžโ€–โ€–๎‚ธ๎‚ต1โˆ’๐‘ฅโˆ’๐‘”0โ€–โ€–โ€–โ€–๐‘”โˆ’๐‘”0โ€–โ€–๎‚ถ๐‘โˆ’๎‚ตโ€–โ€–๐‘ฅโˆ’๐‘”0โ€–โ€–โ€–โ€–๐‘”โˆ’๐‘”0โ€–โ€–๎‚ถ๐‘๎‚น=1,(3.6) we can take ๐‘>0 such that โ€–๐‘ฅโˆ’๐‘”โ€–๐‘โ‰ฅโ€–โ€–๐‘ฅโˆ’๐‘”0โ€–โ€–๐‘+12โ€–โ€–๐‘”โˆ’๐‘”0โ€–โ€–๐‘โ€–โ€–,โˆ€๐‘”โˆˆ๐ถ,๐‘”โˆ’๐‘”0โ€–โ€–โ‰ฅ๐‘.(3.7) Again, by Lagrangeโ€™s mean value theorem, we have โ€–๐‘ฅโˆ’๐‘”โ€–๐‘โˆ’โ€–โ€–๐‘ฅโˆ’๐‘”0โ€–โ€–๐‘โ€–โ€–โ€–๐‘ฅโˆ’๐‘”โ€–โˆ’๐‘ฅโˆ’๐‘”0โ€–โ€–โ€–โ€–โ‰ฅ๐‘๐‘ฅโˆ’๐‘”0โ€–โ€–๐‘โˆ’1.(3.8) Therefore, โ€–๐‘ฅโˆ’๐‘”โ€–๐‘โ‰ฅโ€–โ€–๐‘ฅโˆ’๐‘”0โ€–โ€–๐‘โ€–โ€–+๐‘๐‘ฅโˆ’๐‘”0โ€–โ€–๐‘โˆ’1โ‹…๎€บโ€–โ€–โ€–๐‘ฅโˆ’๐‘”โ€–โˆ’๐‘ฅโˆ’๐‘”0โ€–โ€–๎€ปโ‰ฅโ€–โ€–๐‘ฅโˆ’๐‘”0โ€–โ€–๐‘โ€–โ€–+๐‘๐‘ฅโˆ’๐‘”0โ€–โ€–๐‘โˆ’1โ‹…๐‘Ÿ๐‘,๐‘โ€–โ€–๐‘”โˆ’๐‘”0โ€–โ€–๐‘,(3.9) for any ๐‘”โˆˆ๐ถ and โ€–๐‘”โˆ’๐‘”0โ€–โ‰ค๐‘. Set ๐‘Ÿ๐‘=min{1/2,๐‘โ€–๐‘ฅโˆ’๐‘”0โ€–๐‘โˆ’1๐‘Ÿ๐‘,๐‘}, then โ€–๐‘ฅโˆ’๐‘”โ€–๐‘โ‰ฅโ€–โ€–๐‘ฅโˆ’๐‘”0โ€–โ€–๐‘+๐‘Ÿ๐‘โ€–โ€–๐‘”โˆ’๐‘”0โ€–โ€–๐‘,(3.10) for any ๐‘”โˆˆ๐ถ. That shows ๐‘”0 is strongly unique of order ๐‘ at ๐‘ฅ.
Now, we only need to prove (3)โ‡’(2). Let ๐‘0>0,๐‘Ÿ๐‘,๐‘0>0 such that โ€–โ€–โ€–๐‘ฅโˆ’๐‘”โ€–โ‰ฅ๐‘ฅโˆ’๐‘”0โ€–โ€–+๐‘Ÿ๐‘,๐‘0โ€–โ€–๐‘”โˆ’๐‘”0โ€–โ€–๐‘,(3.11) for all ๐‘”โˆˆ๐ถ and โ€–๐‘”โˆ’๐‘”0โ€–โ‰ค๐‘0.
If the condition (2) is not true, there exist ๐‘>๐‘0, ๐‘”๐‘›โˆˆ๐ถ with ๐‘0<โ€–๐‘”๐‘›โˆ’๐‘”โ€–โ‰ค๐‘ such that โ€–โ€–๐‘ฅโˆ’๐‘”๐‘›โ€–โ€–<โ€–โ€–๐‘ฅโˆ’๐‘”0โ€–โ€–+1๐‘›โ€–โ€–๐‘”๐‘›โˆ’๐‘”0โ€–โ€–๐‘,(3.12) then, because {๐‘”๐‘›} is bounded, โ€–โ€–๐‘ฅโˆ’๐‘”0โ€–โ€–โ‰คlim๐‘›โ€–โ€–๐‘ฅโˆ’๐‘”๐‘›โ€–โ€–โ‰คlim๐‘›๎‚ƒโ€–โ€–๐‘ฅโˆ’๐‘”0โ€–โ€–+1๐‘›โ€–โ€–๐‘”๐‘›โˆ’๐‘”0โ€–โ€–๐‘๎‚„=โ€–โ€–๐‘ฅโˆ’๐‘”0โ€–โ€–,(3.13) that is, lim๐‘›โ€–๐‘ฅโˆ’๐‘”๐‘›โ€–=๐‘‘(๐‘ฅ,๐ถ). This shows that {๐‘”๐‘›} is a minimizing sequence for ๐‘ฅ. Since ๐ถ is weakly approximatively compact, there is a weakly convergent subsequence of {๐‘”๐‘›}. Without loss of generality, we can assume that ๐‘”๐‘›๐‘คโˆ’โ†’๐‘”0โˆˆ๐ถ.(3.14) Based on weak lower semi-continuity of norm, we get that โ€–โ€–๐‘ฅโˆ’๐‘”0โ€–โ€–โ‰คlim๐‘›โ€–โ€–๐‘ฅโˆ’๐‘”๐‘›โ€–โ€–=โ€–โ€–๐‘ฅโˆ’๐‘”0โ€–โ€–.(3.15) Hence, ๐‘”0โˆˆ๐‘ƒ๐ถ(๐‘ฅ), consequently, ๐‘”0=๐‘”0. Take ๐‘“โˆˆ๐ท((๐‘ฅโˆ’๐‘”0)/โ€–๐‘ฅโˆ’๐‘”0โ€–), then ๐‘“๎‚ต๐‘ฅโˆ’๐‘”๐‘›โ€–โ€–๐‘ฅโˆ’๐‘”๐‘›โ€–โ€–๎‚ถ๎‚ตโŸถ๐‘“๐‘ฅโˆ’๐‘”0โ€–โ€–๐‘ฅโˆ’๐‘”0โ€–โ€–๎‚ถ=1.(3.16) Since ๐‘‹ is nearly strongly convex, there exists {๐‘ฅโˆ’๐‘”๐‘›๐‘˜}โŠ‚{๐‘ฅโˆ’๐‘”๐‘›} such that ๐‘ฅโˆ’๐‘”๐‘›๐‘˜โ†’๐‘ฅโˆ’๐‘”0, that is, ๐‘”๐‘›๐‘˜โ†’๐‘”0. It follows that โ€–๐‘”0โˆ’๐‘”0โ€–โ‰ฅ๐‘0. This is a contradiction with the fact that ๐‘”0=๐‘”0.

Corollary 3.2. Let ๐ถ be a weakly approximatively compact subset of strongly convex Banach space ๐‘‹, ๐‘ฅโˆˆ๐‘‹โงต๐ถ, and ๐‘”0โˆˆ๐‘ƒ๐ถ(๐‘ฅ), then the following are equivalent:(1)๐‘”0 is strongly unique of order ๐‘ at ๐‘ฅ;(2)๐‘”0 is bounded strongly unique of order ๐‘ at ๐‘ฅ;(3)๐‘”0 is locally strongly unique of order ๐‘ at ๐‘ฅ.

If ๐ถ is a convex set, we have the following result.

Theorem 3.3. Let ๐ถ be a convex subset of a nearly strongly convex Banach space ๐‘‹, ๐‘ฅโˆˆ๐‘‹โงต๐ถ, and ๐‘”0โˆˆ๐ถ. If ๐‘”0 is the unique element of best approximation of ๐‘ฅ, then the following are equivalent:(1)๐‘”0 is strongly unique of order ๐‘ at ๐‘ฅ;(2)๐‘”0 is bounded strongly unique of order ๐‘ at ๐‘ฅ;(3)๐‘”0 is locally strongly unique of order ๐‘ at ๐‘ฅ.

Proof. By the proof of Theorem 3.1, we have (1)โ‡”(2). Now, we only need to prove (3)โ‡’(2).
If the condition (2) is not true, there exist ๐‘>๐‘0,๐‘”๐‘›โˆˆ๐ถ with ๐‘0<โ€–๐‘”๐‘›โˆ’๐‘”โ€–โ‰ค๐‘ such that โ€–โ€–๐‘ฅโˆ’๐‘”๐‘›โ€–โ€–<โ€–โ€–๐‘ฅโˆ’๐‘”0โ€–โ€–+1๐‘›โ€–โ€–๐‘”๐‘›โˆ’๐‘”0โ€–โ€–๐‘.(3.17) In the same way of the proof of (3)โ‡’(2) in Theorem 3.1, we can also prove that {๐‘”๐‘›} is a minimizing sequence for ๐‘ฅ. By Lemma 2.5, ๐ถ is approximatively compact. Hence, there exists a convergent subsequence of {๐‘”๐‘›}. Without loss of generality, we can assume that ๐‘”๐‘›โŸถ๐‘”0โˆˆ๐ถ.(3.18) Consequently, ๐‘”0=๐‘”0, but โ€–๐‘”0โˆ’๐‘”0โ€–โ‰ฅ๐‘0, which is a contradiction.

Corollary 3.4. Let ๐ถ be a convex set of a strongly convex Banach space ๐‘‹. ๐‘ฅโˆˆ๐‘‹โงต๐ถ and ๐‘”0โˆˆ๐‘ƒ๐ถ(๐‘ฅ). The following are equivalent:(1)๐‘”0 is strongly unique of order ๐‘ at ๐‘ฅ;(2)๐‘”0 is bounded strongly unique of order ๐‘ at ๐‘ฅ;(3)๐‘”0 is locally strongly unique of order ๐‘ at ๐‘ฅ.

Acknowledgment

The authors very much appreciated the reviewerโ€™s suggestions for the revision of this paper.

References

  1. Z. H. Zhang and Z. R. Shi, โ€œConvexities and approximative compactness and continuity of metric projection in Banach spaces,โ€ Journal of Approximation Theory, vol. 161, no. 2, pp. 802โ€“812, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  2. J. Diestel, Geometry of Banach Spaces—Selected Topics, vol. 485 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1975.
  3. C. X. Wu and Y. J. Li, โ€œStrong convexity in Banach spaces,โ€ Chines Journal of Mathematics, vol. 13, no. 1, pp. 105โ€“108, 1993. View at: Google Scholar | Zentralblatt MATH
  4. J. H. Wang and Z. H. Zhang, โ€œCharacterizations of the CK property,โ€ Acta Mathematica Scientia. Series A, vol. 17, no. 3, pp. 280โ€“284, 1997. View at: Google Scholar | Zentralblatt MATH
  5. Z. H. Zhang and C. J. Zhang, โ€œOn very rotund Banach space,โ€ Applied Mathematics and Mechanics, vol. 21, no. 8, pp. 965โ€“970, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  6. F. Sullivan, โ€œGeometrical peoperties determined by the higher duals of a Banach space,โ€ Illinois Journal of Mathematics, vol. 21, no. 2, pp. 315โ€“331, 1977. View at: Google Scholar
  7. Z. H. Zhang and C. Y. Liu, โ€œSome generalizations of locally and weakly locally uniformly convex space,โ€ Nonlinear Analysis, vol. 74, no. 12, pp. 3896โ€“3902, 2011. View at: Publisher Site | Google Scholar
  8. P. Bandyopadhyay, D. Huang, B.-L. Lin, and S. L. Troyanski, โ€œSome generalizations of locally uniform rotundity,โ€ Journal of Mathematical Analysis and Applications, vol. 252, no. 2, pp. 906โ€“916, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  9. P. Bandyopadhyay, D. Huang, and B.-L. Lin, โ€œRotund points, nested sequence of balls and smoothness in Banach spaces,โ€ Commentationes Mathematicae. Prace Matematyczne, vol. 44, no. 2, pp. 163โ€“186, 2004. View at: Google Scholar | Zentralblatt MATH
  10. P. Bandyopadhyay, Y. J. Li, B.-L. Lin, and D. Narayana, โ€œProximinality in Banach spaces,โ€ Journal of Mathematical Analysis and Applications, vol. 341, no. 1, pp. 309โ€“317, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  11. S. Y. Xu, C. Li, and W. S. Yang, The Nonlinear Approximation Theory in Banach Spaces[M], Science Publication, Beijing, China, 1998.
  12. D. E. Wulbert, โ€œUniqueness and differential characterization of approximations from manifolds of functions,โ€ American Journal of Mathematics, vol. 93, pp. 350โ€“366, 1971. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  13. R. Smarzewski, โ€œStrongly unique best approximation in Banach spaces,โ€ Journal of Approximation Theory, vol. 47, no. 3, pp. 184โ€“194, 1986. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  14. D. Schmidt, โ€œStrong unicity and Lipschitz conditions of order 1/2 for monotone approximation,โ€ Journal of Approximation Theory, vol. 27, no. 4, pp. 346โ€“354, 1979. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  15. R. R. Phelps, Convex Functions, Monotone Operators and Differentiability, vol. 1364 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1989.

Copyright © 2012 Z. H. Zhang and C. Y. 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.

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