Journal of Function Spaces

Journal of Function Spaces / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 724120 | 11 pages | 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 𝑋, we give some equivalent conditions that every element in 𝑋 is strongly unique of order 𝑝, bounded strongly unique of order 𝑝, and locally strongly unique of order 𝑝.

1. Notations and Definitions

Let 𝑋 be a Banach space and let 𝑋∗ be its dual space. Let us denote by 𝐵(𝑋) and 𝑆(𝑋) the closed unit ball and the unit sphere of 𝑋, and by 𝐵(𝑥,𝑟) the closed ball centered at 𝑥 of radius 𝑟>0. Let 𝑥∈𝑆(𝑋),𝐷(𝑥)={𝑓∈𝑆(𝑋∗)∶𝑓(𝑥)=1}. Let us denote by 𝑁𝐴(𝑋) the set of all norm-attaining functionals in 𝑋∗ and 𝑆0(𝑋∗)=𝑆(𝑋∗)∩𝑁𝐴(𝑋).

For a subset 𝐶⊂𝑋, the metric projection 𝑃𝐶∶𝑋→2𝐶 is defined by 𝑃𝐶(𝑥)={𝑦∈𝐶∶‖𝑥−𝑦‖=𝑑(𝑥,𝐶)}, where 𝑑(𝑥,𝐶)=inf{‖𝑥−𝑦‖∶𝑦∈𝐶},𝑥∈𝑋. If 𝑃𝐶(𝑥)≠∅ for each 𝑥∈𝑋, then 𝐶 is said to be proximinal. If 𝑃𝐶(𝑥) is at most a singleton for each 𝑥∈𝑋, then 𝐶 is said to be semi-Chebyshev. If 𝐶 is a simultaneously proximinal and semi-Chebyshev set, then 𝐶 is said to be a Chebyshev set.

Definition 1.1 (see [1]). One says that 𝑋 is strongly convex (resp., very convex/nearly strongly convex/nearly very convex) if for any 𝑥∈𝑆(𝑋) and {𝑥𝑛}⊂𝐵(𝑋) convergence 𝑥∗(𝑥𝑛)→1 as ğ‘›â†’âˆž for some 𝑥∗∈𝐷(𝑥) implies that 𝑥𝑛→𝑥 as ğ‘›â†’âˆž (resp., 𝑥𝑛𝑤−→𝑥 as ğ‘›â†’âˆž/{𝑥𝑛} is relatively compact/{𝑥𝑛} is weakly relatively compact).

Definition 1.2 (see [2]). One says that 𝑋 is midpoint locally uniformly rotund (MLUR) (resp., weakly midpoint locally uniformly rotund (WMLUR)) if any 𝑥0, 𝑥𝑛, 𝑦𝑛∈𝑆(𝑋), (𝑛=1,2,…), 𝑥𝑛+𝑦𝑛→2𝑥0 as ğ‘›â†’âˆž implies that 𝑥𝑛−𝑦𝑛→𝜃 as ğ‘›â†’âˆž (resp., 𝑥𝑛−𝑦𝑛𝑤−→𝜃 as ğ‘›â†’âˆž).

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 𝑋 is said to be very rotund if no 𝑥∗∈𝑆(𝑋∗) is simultaneously a norming element for some 𝑥∈𝑆(𝑋) and 𝑥∗∗∈𝑆(𝑋∗∗), where 𝑥≠𝑥∗∗. Z. H. Zhang and C. J. Zhang proved that very rotund space coincides with very convex space in [5]. In [3–5, 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 𝑋 is said to be ALUR (resp., WALUR) if for any 𝑥∈𝑆(𝑋), {𝑥𝑛}⊂𝐵(𝑋), and {𝑥𝑛}⊂𝐵(𝑋∗), the condition lim𝑚lim𝑛𝑥∗𝑚((𝑥𝑛+𝑥)/2)=1 implies 𝑥𝑛→𝑥 (resp., 𝑥𝑛𝑤−→𝑥). Many properties of these two classes of convexities were studied in [8–10] 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 {𝑧𝑛}⊂𝐶 is said to be minimizing for 𝑥∈𝑋⧵𝐶 if ‖𝑥−𝑧𝑛‖→𝑑(𝑥,𝐶) as ğ‘›â†’âˆž.

Definition 1.3 (see [10]). Let 𝐶 be a closed subset (resp., a weakly closed subset) of 𝑋 and 𝑥0∈𝑋⧵𝐶.(1)One says that 𝐶 is approximatively (resp., weakly approximatively) compact for 𝑥0 if every minimizing sequence {𝑧𝑛}⊂𝐶 for 𝑥0 has a convergent (resp., weakly convergent) subsequence.(2)One says that 𝐶 is strongly Chebyshev (resp., weakly strongly Chebyshev) for 𝑥0 if every minimizing sequence {𝑧𝑛}⊂𝐶 for 𝑥0 is convergent (resp., weakly convergent).

If 𝐶 is approximatively compact (resp., weakly approximatively compact/strongly Chebyshev/weakly strongly Chebyshev) for every 𝑥∈𝑋⧵𝐶, we say that 𝐶 is approximatively compact (resp., weakly approximatively compact/strongly Chebyshev/weakly strongly Chebyshev) in 𝑋.

Definition 1.4 (see [11]). Let 𝐺∈𝑋, 𝑥∈𝑋⧵𝐺, and 1≤𝑝<∞.(1)𝑔0∈𝑃𝐺(𝑥) is said to be strongly unique at 𝑥 if there exists a constant 𝑟𝑝>0 such that ‖‖‖𝑥−𝑔‖≥𝑥−𝑔0‖‖+𝑟𝑝‖‖𝑔−𝑔0‖‖,(1.1) for any 𝑔∈𝐺.(2)𝑔0∈𝐺 is said to be strongly unique of order 𝑝 at 𝑥 if there exists an 𝑟𝑝=𝑟𝑝(𝑥)>0 such that ‖𝑥−𝑔‖𝑝≥‖‖𝑥−𝑔0‖‖𝑝+𝑟𝑝‖‖𝑔−𝑔0‖‖𝑝,(1.2) for any 𝑔∈𝐺.(3)𝑔0∈𝐺 is said to be bounded strongly unique of order 𝑝 at 𝑥 if given any 𝑁>0, there exists an 𝑟𝑝,𝑁=𝑟𝑝,𝑁(𝑥) such that ‖‖‖𝑥−𝑔‖≥𝑥−𝑔0‖‖+𝑟𝑝,𝑁‖‖𝑔−𝑔0‖‖𝑝,(1.3) for any 𝑔∈𝐺,‖𝑔−𝑔0‖≤𝑁.(4)𝑔0∈𝐺 is said to be locally strongly unique of order 𝑝 at 𝑥 if there exist 𝑁>0 and 𝑟𝑝,𝑁=𝑟𝑝,𝑁(𝑥) such that ‖‖‖𝑥−𝑔‖≥𝑥−𝑔0‖‖+𝑟𝑝,𝑁‖‖𝑔−𝑔0‖‖𝑝,(1.4) for any 𝑔∈𝐺,‖𝑔−𝑔0‖≤𝑁.

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 𝑝 and the bounded strongly unique of order 𝑝, respectively. By [11], we know that the strongly unique of order 𝑝 and the bounded strongly unique of order 𝑝 all are generalizations of the strongly unique. The strongly unique of order 𝑝 and the bounded strongly unique of order 𝑝 imply the locally strongly unique of order 𝑝, but converse implied relation is not generally true. When 𝑝=1, the strongly unique of order 𝑝 and the bounded strongly unique of order 𝑝 all are strongly unique. The locally strongly unique of order 1 is not equivalent to the strongly unique.

Definition 1.5 (see [10]). For 𝑥∗∈𝑆(𝑋∗) and 𝑥∈𝑆(𝑋), let one define the following maps: 𝑥𝐷(𝑥)=∗𝑋∈𝑆∗∶𝑥∗(𝑥)=1,𝐷−1𝑥∗=𝑥∈𝑆(𝑋)∶𝑥∗.(𝑥)=1(1.5)𝐷 is called the duality map and 𝐷−1 is called the preduality map. Naturally, 𝐷−1 is defined only on 𝑆0(𝑋∗).

Definition 1.6 (see [10]). The preduality map 𝐷−1∶𝑆(𝑋∗)→𝑆(𝑋) is said to be upper semicontinuous (𝑛−𝜏) on 𝑆0(𝑋∗) if for any 𝑥∗∈𝑆0(𝑋∗) and any 𝜏 open set 𝑊 with 𝐷−1(𝑥∗)⊂𝑊, there exists 𝜀>0 such that 𝐷−1(𝑦∗)⊂𝑊 whenever ‖𝑥∗−𝑦∗‖<𝜀, where 𝜏 stands for norm or weak topology.

2. Convexity and Continuity of the Preduality Map

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

Lemma 2.1. Suppose that 𝜀>0, 𝑥0∈𝑆(𝑋), 𝑥∗0∈𝑆(𝑋∗), and 𝑥∗0(𝑥0)>1−𝜀, then there are 𝑥𝜀∈S(𝑋) and 𝑥∗𝜀∈𝐷(𝑥𝜀) such that ‖‖𝑥𝜀−𝑥0‖‖√<2‖‖𝑥𝜀,∗𝜀−𝑥∗0‖‖√<2𝜀.(2.1)

Theorem 2.2. Let 𝑋 be a Banach space. 𝑋 is strongly convex if and only if the preduality map 𝐷−1∶𝑆0(𝑋∗)→𝑆(𝑋) is singlevalued and continuous.

Proof. Necessity. Since strong convexity implies strict convexity, 𝐷−1 is singlevalued. Suppose that {𝑥∗𝑛}⊂𝑆0(𝑋∗), 𝑥∗∈𝑆0(𝑋∗) with 𝑥∗𝑛→𝑥∗, then there exist {𝑥𝑛}⊂𝑆(𝑋) and 𝑥∈𝑆(𝑋) such that 𝑥∗𝑛(𝑥𝑛)=1=𝑥∗(𝑥). We have that 𝑥∗𝐷−1𝑥∗𝑛=𝑥∗−𝑥∗𝑛+𝑥∗𝑛𝐷−1𝑥∗𝑛=𝑥∗−𝑥∗𝑛𝐷−1𝑥∗𝑛+𝑥∗𝑛𝐷−1𝑥∗𝑛.(2.2) Since ||𝑥∗−𝑥∗𝑛𝐷−1𝑥∗𝑛||≤‖‖𝑥∗𝑛−𝑥∗‖‖⟶0,asğ‘›âŸ¶âˆž,(2.3) and 𝑥∗𝑛(𝐷−1(𝑥∗𝑛))=𝑥∗𝑛(𝑥𝑛)=1, we have that 𝑥∗𝐷−1𝑥∗𝑛=𝑥∗𝑥𝑛⟶1,asğ‘›âŸ¶âˆž.(2.4) Since 𝑋 is strongly convex, we deduce that 𝑥𝑛→𝑥 as ğ‘›â†’âˆž, that is, 𝐷−1(𝑥∗𝑛)→𝐷−1(𝑥∗) as ğ‘›â†’âˆž. This means that 𝐷−1 is continuous.
Sufficiency. Let {𝑥𝑛}⊂𝑆(𝑋),𝑥∈𝑆(𝑋) with 𝑥∗(𝑥𝑛)→1 as ğ‘›â†’âˆž for some 𝑥∗∈𝐷(𝑥). Since 𝐷−1 is singlevalued, by Lemma 2.1, there exist {𝑦𝑛}⊂𝑆(𝑋) and {𝑦∗𝑛}⊂𝑆0(𝑋∗) such that 𝐷−1(𝑦∗𝑛)=𝑦𝑛 and ‖‖𝑦𝑛−𝑥𝑛‖‖‖‖𝑦⟶0,∗𝑛−𝑥∗‖‖⟶0,asğ‘›âŸ¶âˆž.(2.5) Since 𝐷−1 is continuous, we have that 𝐷−1(𝑦∗𝑛)→𝐷−1(𝑥∗) as ğ‘›â†’âˆž, that is, 𝑦𝑛→𝑥 as ğ‘›â†’âˆž. So 𝑥𝑛→𝑥 as ğ‘›â†’âˆž, which means that 𝑋 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 𝑋 be a Banach space. 𝑋 is very convex if and only if the preduality map 𝐷−1∶𝑆0(𝑋∗)→𝑆(𝑋) is singlevalued and weakly continuous.

Lemma 2.4. Let 𝐶 be a convex set of a strongly convex Banach space 𝑋. The following are equivalent:(1)𝐶 is proximinal;(2)𝐶 is weakly approximatively compact;(3)𝐶 is approximatively compact;(4)𝐶 is strongly Chebyshev.

Proof. We only need to prove (1)⇒(4).
Let 𝑥∈𝑋⧵𝐶, {𝑧𝑛}⊂𝐶 such that limğ‘›â†’âˆžâ€–â€–ğ‘¥âˆ’ğ‘§ğ‘›â€–â€–=𝑑(𝑥,𝐶).(2.6) In order to finish the proof, we will show that there exists some 𝑦0∈𝐶 such that ‖𝑦0−𝑧𝑛‖→0 as ğ‘›â†’âˆž.
Step 1. If 𝑥=0, then 𝑟=𝑑(0,𝐶)>0 and 𝐵(0,𝑟)∩𝐶=𝑃𝐶(0)≠∅,int𝐵(0,𝑟)∩𝐶=∅,(2.7) where 𝐵(0,𝑟)={𝑦∶‖𝑦‖≤𝑟}. By the separation theorem [2] and definition of norm, there exists an 𝑓∈𝑆(𝑋∗) such that ‖‖𝑦sup{𝑓(𝑦)∶𝑦∈𝐶}≤inf{𝑓(𝑦)∶𝑦∈𝐵(0,𝑟)}=−‖𝑓‖𝑟=−0‖‖,(2.8) for any 𝑦0∈𝑃𝐶(0). Hence, we have that −‖‖𝑦0‖‖𝑦≤𝑓0‖‖𝑦≤sup{𝑓(𝑦)∶𝑦∈𝐶}≤−0‖‖.(2.9) This shows that 𝑓∈𝐷(−𝑦0/‖𝑦0‖). From 𝑧𝑛∈𝐶, we get 𝑓(𝑧𝑛)≤𝑓(𝑦0). Combining it with the condition limğ‘›â†’âˆžâ€–0−𝑧𝑛‖=𝑑(0,𝐶), it follows that ‖‖0−𝑦0‖‖=𝑓0−𝑦0≤𝑓0−𝑧𝑛≤‖‖0−𝑧𝑛‖‖‖‖⟶𝑑(0,𝐶)=0−𝑦0‖‖,(2.10) as ğ‘›â†’âˆž. Hence, 𝑓(−𝑧𝑛/‖𝑧𝑛‖)→1 as ğ‘›â†’âˆž. Since 𝑋 is strongly convex, −𝑧𝑛/‖𝑧𝑛‖→−𝑦0/‖𝑦0‖ as ğ‘›â†’âˆž, that is, ‖𝑧𝑛−𝑦0‖→0 as ğ‘›â†’âˆž.Step 2. If 𝑥≠0, we set ğ¶î…ž=𝑥−𝐶. It is clear that ğ¶î…ž is proximinal and {𝑥−𝑧𝑛}âŠ‚ğ¶î…ž is a minimizing sequence for 0. By Step 1, there exists ğ‘¦î…ž0âˆˆğ¶î…ž, such that â€–ğ‘¦î…ž0−(𝑥−𝑧𝑛)‖→0 as ğ‘›â†’âˆž. This shows that ‖(ğ‘¥âˆ’ğ‘¦î…ž0)−𝑧𝑛‖→0 as ğ‘›â†’âˆž and ğ‘¥âˆ’ğ‘¦î…ž0=𝑦0∈𝐶.

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

Lemma 2.5. Let 𝐶 be a convex set of a nearly strongly convex Banach space 𝑋. The following are equivalent:(1)𝐶 is proximinal;(2)𝐶 is weakly approximatively compact;(3)𝐶 is approximatively compact.

Lemma 2.6. Let 𝑥∈𝑆(𝑋), 𝑓∈𝐷(𝑥), then hyperplane 𝐻={𝑦∈𝑋∶𝑓(𝑦)=1} is a proximinal convex subset in 𝑋.

Proof. Let ker𝑓={𝑦∈𝑋∶𝑓(𝑦)=0}.(2.11) We will prove that ker𝑓 is a proximinal convex subset in 𝑋. For any 𝑧∈𝑋, if 𝑑(𝑧,ker𝑓)=0, since ker𝑓 is a closed subspace, we know that 𝑧∈ker𝑓. Hence, 𝑧∈𝑃ker𝑓(𝑧). If 𝑑(𝑧,ker𝑓)>0, since 𝑋={𝛼𝑧}+ker𝑓, there exist 𝜆∈ℝ and 𝑦0∈ker𝑓 such that 𝑑(𝑧,ker𝑓)𝑥=𝜆𝑧+𝑦0. For any 𝑦∈𝑋, 𝑑(𝑦,ker𝑓)=|𝑓(𝑦)|, and 𝑓(𝑥)=1, we have that 𝑑(𝑥,ker𝑓)=1 and 𝑓(𝑑(𝑧,ker𝑓)𝑥)=𝑑(𝑑(𝑧,ker𝑓)𝑥,ker𝑓). Therefore, we have that 𝑑(𝑧,ker𝑓)=𝑑𝜆𝑧+𝑦0=||𝜆||𝑑,ker𝑓(𝑧,ker𝑓).(2.12) This means that |𝜆|=1. Hence, we have ‖‖𝑑(𝑧,ker𝑓)=‖𝑑(𝑧,ker𝑓)𝑥‖=𝜆𝑧+𝑦0‖‖=‖‖𝑧+𝜆𝑦0‖‖.(2.13) It follows that −𝜆𝑦0∈𝑃ker𝑓(𝑧), which means that ker𝑓 is proximinal set.
Furthermore, we will prove that hyperplane 𝐻={𝑦∈𝑋∶𝑓(𝑦)=1} is a proximinal convex subset in 𝑋. For 𝑥0∈𝐻, 𝐻=𝑥0+ker𝑓. For any 𝑧∈𝑋, 𝑑(𝑧,𝐻)=inf𝑦∈ker𝑓‖‖𝑥𝑧−0‖‖+𝑦=𝑑𝑧−𝑥0.,ker𝑓(2.14) Since ker𝑓 is a proximinal subset, there exists a 𝑦0∈ker𝑓 such that 𝑑𝑧−𝑥0=‖‖𝑥,ker𝑓𝑧−0+𝑦0‖‖.(2.15) Therefore, 𝑥0+𝑦0∈𝑃𝐻(𝑧), which means that 𝐻 is proximinal set in 𝑋.

Theorem 2.7. Let 𝑋 be a Banach space. 𝑋 is a nearly strongly convex if and only if the preduality map 𝐷−1∶𝑆0(𝑋∗)→𝑆(𝑋) is upper semicontinuous (𝑛−𝑛) on 𝑆0(𝑋∗) with norm compact images, where 𝑛 stands for norm topology.

Proof. Necessity. Arbitrarily take 𝑥∗∈𝑆0(𝑋∗). Then by Lemma 2.6, we know that ker𝑥∗=𝑌 is a proximinal convex subset in 𝑋. Since 𝑋 is nearly strongly convex, by Lemma 2.5, we know that 𝑌 is approximatively compact. Suppose that 𝐷−1 is not upper semicontinuous (𝑛−𝑛) at 𝑥∗, then for some open set 𝑊 in 𝑋 with 𝐷−1(𝑥∗)⊂𝑊, there exists {𝑥∗𝑛}⊂𝑆(𝑋∗) such that 𝑥∗𝑛→𝑥∗ as ğ‘›â†’âˆž and 𝐷−1(𝑥∗𝑛)̸⊆𝑊 for all 𝑛. Let 𝑧𝑛∈𝐷−1(𝑥∗𝑛)⧵𝑊. Fix 𝑥∈𝐷−1(𝑥∗). Let 𝑥𝑛=𝑥∗(𝑧𝑛)𝑥−𝑧𝑛, then {𝑥𝑛}⊂𝑌 is a minimizing sequence for 𝑥. Since 𝑌 is approximatively compact, {𝑥𝑛} has a convergent subsequence. So {𝑧𝑛} has convergent subsequence converging to 𝑧. Thus, 𝑧∈𝐷−1(𝑥∗)⊂𝑊, but 𝑧𝑛∈𝑋⧵𝑊 is closed, which is a contradiction. By the assumption, we easily know that the image of 𝐷−1 is compact.
Sufficiency. Let {𝑥𝑛}⊂𝑆(𝑋),𝑥∈𝑆(𝑋) with 𝑥∗(𝑥𝑛)→1 as ğ‘›â†’âˆž for some 𝑥∗∈𝐷(𝑥). By Lemma 2.1, there exist {𝑦𝑛}⊂𝑆(𝑋) and {𝑦∗𝑛}⊂𝑆(𝑋∗) such that 𝑦𝑛∈𝐷−1(𝑦∗𝑛) and ‖‖𝑦𝑛−𝑥𝑛‖‖‖‖𝑦⟶0,∗𝑛−𝑥∗‖‖⟶0,asğ‘›âŸ¶âˆž.(2.16) Since 𝐷−1(𝑥∗) is compact, 𝐷−1(𝑥∗) is proximinal. Let 𝑧𝑛∈𝐷−1(𝑥∗) such that ‖𝑦𝑛−𝑧𝑛‖=𝑑(𝑦𝑛,𝐷−1(𝑥∗)). In virtue of 𝐷−1 being upper semicontinuous (𝑛−𝑛) on 𝑆0(𝑋∗) and ‖𝑦∗𝑛−𝑥∗‖→0, for any 𝜀>0, there is 𝑛0 such that for all 𝑛≥𝑛0𝑑(𝑦𝑛,𝐷−1(𝑥∗))<𝜀,thatis,‖𝑦𝑛−𝑧𝑛‖<𝜀, which means that 𝑦𝑛−𝑧𝑛→𝜃 as ğ‘›â†’âˆž. Combining this with the compactness of 𝐷−1(𝑥∗),{𝑧𝑛} has convergent subsequence, and hence {𝑦𝑛} has convergent subsequence. By 𝑦𝑛−𝑥𝑛→𝜃 as ğ‘›â†’âˆž, {𝑥𝑛} has convergent subsequence, which means that 𝑋 is nearly strongly convex.

Theorem 2.8. Let 𝑋 be a Banach space. 𝑋 is nearly very convex if and only if the preduality map 𝐷−1∶𝑆0(𝑋∗)→𝑆(𝑋) is upper semicontinuous (𝑛−𝑤) on 𝑆0(𝑋∗) with weakly compact images.

Proof. In a similar way to the proof of Theorem 2.7, the necessity can be proved.
Sufficiency. Let {𝑥𝑛}⊂𝑆(𝑋) with 𝑥∗(𝑥𝑛)→1 as ğ‘›â†’âˆž for some 𝑥∗∈𝐷(𝑥). By Lemma 2.1, there are {𝑦𝑛}⊂𝑆(𝑋) and {𝑦∗𝑛}⊂𝑆(𝑋∗) such that 𝑦𝑛∈𝐷−1(𝑦∗𝑛) and ‖‖𝑦𝑛−𝑥𝑛‖‖‖‖𝑦⟶0,∗𝑛−𝑥∗‖‖⟶0asğ‘›âŸ¶âˆž.(2.17) Suppose that {𝑥𝑛} does not have weakly convergent subsequence. By (2.17), we know that {𝑦𝑛} does not have weakly convergent subsequence. Without loss of generality, we can assume that ||𝑓𝑦𝑛−𝑦𝑚||≥𝜀0(∀𝑛,𝑚∈𝑁,𝑛≠𝑚),(2.18) for some 𝑓∈𝑋∗⧵{𝜃} and 𝜀0>0. Since 𝐷−1(𝑥∗) is weakly compact, there exists {𝑧𝑖}𝑘𝑖=1⊂𝐷−1(𝑥∗), such that 𝐷−1(𝑥∗)⊂∪𝑘𝑖=1{𝑦∈𝑋∶|𝑓(𝑦−𝑧𝑖)|<𝜀0/2}. Combining upper semicontinuity (𝑛−𝑤) of 𝐷−1 with ‖𝑦∗𝑛−𝑥∗‖→0,{𝑦𝑛}𝑛≥𝑛0⊂∪𝑘𝑖=1{𝑦∈𝑋∶|𝑓(𝑦−𝑧𝑖)|<𝜀0/2} for some 𝑛0∈𝑁. So, there are subsequence {𝑦𝑛𝑘}𝑘≥1⊂{𝑦𝑛}𝑛≥𝑛0 and 𝑖0(1≤𝑖0≤𝑘) such that {𝑦𝑛𝑘}𝑘≥1⊂{𝑦∈𝑋∶|𝑓(𝑦−𝑧𝑖0)|<𝜀0/2}. Hence, ||𝑓𝑦𝑛𝑘−𝑦𝑛𝑙||≤||𝑓𝑦𝑛𝑘−𝑧𝑖0||+||𝑓𝑧𝑖0−𝑦𝑛𝑙||<𝜀02+𝜀02=𝜀0,(2.19) a contradiction with (2.18).

3. Convexity and Proximinality

Theorem 3.1. Let 𝐶 be a weakly approximatively compact subset of nearly strongly convex Banach space 𝑋, 𝑥∈𝑋⧵𝐶, and 𝑔0∈𝐶. If 𝑔0 is the unique element of best approximation of 𝑥, then the following are equivalent:(1)𝑔0 is strongly unique of order 𝑝 at 𝑥;(2)𝑔0 is bounded strongly unique of order 𝑝 at 𝑥;(3)𝑔0 is locally strongly unique of order 𝑝 at 𝑥.

Proof. (1)⇒(2). Let 𝑟𝑝>0 be such that ‖𝑥−𝑔‖𝑝≥‖‖𝑥−𝑔0‖‖𝑝+𝑟𝑝‖‖𝑔−𝑔0‖‖𝑝,(3.1) for any 𝑔∈𝐶. For any 𝑁>0 and each 𝑔∈𝐶,‖𝑔−𝑔0‖≤𝑁, since ‖‖‖𝑥−𝑔‖≤𝑥−𝑔0‖‖+‖‖𝑔−𝑔0‖‖≤‖‖𝑥−𝑔0‖‖+𝑁,(3.2) by Lagrange’s mean value theorem, ‖𝑥−𝑔‖𝑝−‖‖𝑥−𝑔0‖‖𝑝‖‖‖𝑥−𝑔‖−𝑥−𝑔0‖‖‖‖≤𝑝𝑥−𝑔0‖‖+𝑁𝑝−1.(3.3) Set 𝑟𝑝,𝑁=𝑟𝑝𝑝‖‖𝑥−𝑔0‖‖+𝑁1−𝑝.(3.4) Then ‖‖‖𝑥−𝑔‖−𝑥−𝑔0‖‖≥𝑟𝑝,𝑁‖‖𝑔−𝑔0‖‖𝑝.(3.5) Because 𝑟𝑝,𝑁 is independent of 𝑔,𝑔0 is bounded strongly unique of order 𝑝 at 𝑥.
(2)⇒(1). From limâ€–ğ‘”â€–â†’âˆžâ€–ğ‘¥âˆ’ğ‘”â€–ğ‘âˆ’â€–â€–ğ‘¥âˆ’ğ‘”0‖‖𝑝‖‖𝑔−𝑔0‖‖𝑝≥limâ€–ğ‘”â€–â†’âˆžâ€–â€–î‚¸î‚µ1−𝑥−𝑔0‖‖‖‖𝑔−𝑔0‖‖𝑝−‖‖𝑥−𝑔0‖‖‖‖𝑔−𝑔0‖‖𝑝=1,(3.6) we can take 𝑁>0 such that ‖𝑥−𝑔‖𝑝≥‖‖𝑥−𝑔0‖‖𝑝+12‖‖𝑔−𝑔0‖‖𝑝‖‖,∀𝑔∈𝐶,𝑔−𝑔0‖‖≥𝑁.(3.7) Again, by Lagrange’s mean value theorem, we have ‖𝑥−𝑔‖𝑝−‖‖𝑥−𝑔0‖‖𝑝‖‖‖𝑥−𝑔‖−𝑥−𝑔0‖‖‖‖≥𝑝𝑥−𝑔0‖‖𝑝−1.(3.8) Therefore, ‖𝑥−𝑔‖𝑝≥‖‖𝑥−𝑔0‖‖𝑝‖‖+𝑝𝑥−𝑔0‖‖𝑝−1⋅‖‖‖𝑥−𝑔‖−𝑥−𝑔0‖‖≥‖‖𝑥−𝑔0‖‖𝑝‖‖+𝑝𝑥−𝑔0‖‖𝑝−1⋅𝑟𝑝,𝑁‖‖𝑔−𝑔0‖‖𝑝,(3.9) for any 𝑔∈𝐶 and ‖𝑔−𝑔0‖≤𝑁. Set 𝑟𝑝=min{1/2,𝑝‖𝑥−𝑔0‖𝑝−1𝑟𝑝,𝑁}, then ‖𝑥−𝑔‖𝑝≥‖‖𝑥−𝑔0‖‖𝑝+𝑟𝑝‖‖𝑔−𝑔0‖‖𝑝,(3.10) for any 𝑔∈𝐶. That shows 𝑔0 is strongly unique of order 𝑝 at 𝑥.
Now, we only need to prove (3)⇒(2). Let 𝑁0>0,𝑟𝑝,𝑁0>0 such that ‖‖‖𝑥−𝑔‖≥𝑥−𝑔0‖‖+𝑟𝑝,𝑁0‖‖𝑔−𝑔0‖‖𝑝,(3.11) for all 𝑔∈𝐶 and ‖𝑔−𝑔0‖≤𝑁0.
If the condition (2) is not true, there exist 𝑁>𝑁0, 𝑔𝑛∈𝐶 with 𝑁0<‖𝑔𝑛−𝑔‖≤𝑁 such that ‖‖𝑥−𝑔𝑛‖‖<‖‖𝑥−𝑔0‖‖+1𝑛‖‖𝑔𝑛−𝑔0‖‖𝑝,(3.12) then, because {𝑔𝑛} is bounded, ‖‖𝑥−𝑔0‖‖≤lim𝑛‖‖𝑥−𝑔𝑛‖‖≤lim𝑛‖‖𝑥−𝑔0‖‖+1𝑛‖‖𝑔𝑛−𝑔0‖‖𝑝=‖‖𝑥−𝑔0‖‖,(3.13) that is, lim𝑛‖𝑥−𝑔𝑛‖=𝑑(𝑥,𝐶). This shows that {𝑔𝑛} is a minimizing sequence for 𝑥. Since 𝐶 is weakly approximatively compact, there is a weakly convergent subsequence of {𝑔𝑛}. Without loss of generality, we can assume that 𝑔𝑛𝑤−→𝑔0∈𝐶.(3.14) Based on weak lower semi-continuity of norm, we get that ‖‖𝑥−𝑔0‖‖≤lim𝑛‖‖𝑥−𝑔𝑛‖‖=‖‖𝑥−𝑔0‖‖.(3.15) Hence, 𝑔0∈𝑃𝐶(𝑥), consequently, 𝑔0=𝑔0. Take 𝑓∈𝐷((𝑥−𝑔0)/‖𝑥−𝑔0‖), then 𝑓𝑥−𝑔𝑛‖‖𝑥−𝑔𝑛‖‖⟶𝑓𝑥−𝑔0‖‖𝑥−𝑔0‖‖=1.(3.16) Since 𝑋 is nearly strongly convex, there exists {𝑥−𝑔𝑛𝑘}⊂{𝑥−𝑔𝑛} such that 𝑥−𝑔𝑛𝑘→𝑥−𝑔0, that is, 𝑔𝑛𝑘→𝑔0. It follows that ‖𝑔0−𝑔0‖≥𝑁0. This is a contradiction with the fact that 𝑔0=𝑔0.

Corollary 3.2. Let 𝐶 be a weakly approximatively compact subset of strongly convex Banach space 𝑋, 𝑥∈𝑋⧵𝐶, and 𝑔0∈𝑃𝐶(𝑥), then the following are equivalent:(1)𝑔0 is strongly unique of order 𝑝 at 𝑥;(2)𝑔0 is bounded strongly unique of order 𝑝 at 𝑥;(3)𝑔0 is locally strongly unique of order 𝑝 at 𝑥.

If 𝐶 is a convex set, we have the following result.

Theorem 3.3. Let 𝐶 be a convex subset of a nearly strongly convex Banach space 𝑋, 𝑥∈𝑋⧵𝐶, and 𝑔0∈𝐶. If 𝑔0 is the unique element of best approximation of 𝑥, then the following are equivalent:(1)𝑔0 is strongly unique of order 𝑝 at 𝑥;(2)𝑔0 is bounded strongly unique of order 𝑝 at 𝑥;(3)𝑔0 is locally strongly unique of order 𝑝 at 𝑥.

Proof. By the proof of Theorem 3.1, we have (1)⇔(2). Now, we only need to prove (3)⇒(2).
If the condition (2) is not true, there exist 𝑁>𝑁0,𝑔𝑛∈𝐶 with 𝑁0<‖𝑔𝑛−𝑔‖≤𝑁 such that ‖‖𝑥−𝑔𝑛‖‖<‖‖𝑥−𝑔0‖‖+1𝑛‖‖𝑔𝑛−𝑔0‖‖𝑝.(3.17) In the same way of the proof of (3)⇒(2) in Theorem 3.1, we can also prove that {𝑔𝑛} is a minimizing sequence for 𝑥. By Lemma 2.5, 𝐶 is approximatively compact. Hence, there exists a convergent subsequence of {𝑔𝑛}. Without loss of generality, we can assume that 𝑔𝑛⟶𝑔0∈𝐶.(3.18) Consequently, 𝑔0=𝑔0, but ‖𝑔0−𝑔0‖≥𝑁0, which is a contradiction.

Corollary 3.4. Let 𝐶 be a convex set of a strongly convex Banach space 𝑋. 𝑥∈𝑋⧵𝐶 and 𝑔0∈𝑃𝐶(𝑥). The following are equivalent:(1)𝑔0 is strongly unique of order 𝑝 at 𝑥;(2)𝑔0 is bounded strongly unique of order 𝑝 at 𝑥;(3)𝑔0 is locally strongly unique of order 𝑝 at 𝑥.

Acknowledgment

The authors very much appreciated the reviewer’s suggestions for the revision of this paper.

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