Convexity and Proximinality in Banach Space

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 𝑃_𝐶∶𝑋→2^𝐶 is defined by 𝑃_𝐶(𝑥)={𝑦∈𝐶∶‖𝑥−𝑦‖=𝑑(𝑥,𝐶)}, where 𝑑(𝑥,𝐶)=inf{‖𝑥−𝑦‖∶𝑦∈𝐶},𝑥∈𝑋. If 𝑃_𝐶(𝑥)≠∅ for each 𝑥∈𝑋, then 𝐶 is said to be proximinal. If 𝑃_𝐶(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 {𝑥_𝑛}⊂𝐵(𝑋) convergence 𝑥^∗(𝑥_𝑛)→1 as 𝑛→∞ for some 𝑥^∗∈𝐷(𝑥) implies that 𝑥_𝑛→𝑥 as 𝑛→∞ (resp., 𝑥−→_𝑛𝑤𝑥 as 𝑛→∞/{𝑥_𝑛} is relatively compact/{𝑥_𝑛} 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, 𝑥_𝑛, 𝑦_𝑛∈𝑆(𝑋), (𝑛=1,2,…), 𝑥_𝑛+𝑦_𝑛→2𝑥₀ as 𝑛→∞ implies that 𝑥_𝑛−𝑦_𝑛→𝜃 as 𝑛→∞ (resp., 𝑥_𝑛−𝑦−→_𝑛𝑤𝜃 as 𝑛→∞).

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:

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), {𝑥_𝑛}⊂𝐵(𝑋), and {𝑥_𝑛}⊂𝐵(𝑋^∗), the condition lim_𝑚lim_𝑛𝑥^∗𝑚((𝑥_𝑛+𝑥)/2)=1 implies 𝑥_𝑛→𝑥 (resp., 𝑥−→_𝑛𝑤𝑥). 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 {𝑧_𝑛}⊂𝐶 is said to be minimizing for 𝑥∈𝑋⧵𝐶 if ‖𝑥−𝑧_𝑛‖→𝑑(𝑥,𝐶) as 𝑛→∞.

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 {𝑧_𝑛}⊂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 {𝑧_𝑛}⊂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∈𝑃_𝐺(𝑥) is said to be strongly unique at 𝑥 if there exists a constant 𝑟_𝑝>0 such that ‖𝑥−𝑔‖≥‖‖𝑥−𝑔₀‖‖+𝑟_𝑝‖‖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 𝑟_𝑝=𝑟_𝑝(𝑥)>0 such that ‖𝑥−𝑔‖^𝑝≥‖‖𝑥−𝑔₀‖‖^𝑝+𝑟_𝑝‖‖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 𝑟_𝑝,𝑁=𝑟_𝑝,𝑁(𝑥) such that ‖𝑥−𝑔‖≥‖‖𝑥−𝑔₀‖‖+𝑟_𝑝,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 𝑟_𝑝,𝑁=𝑟_𝑝,𝑁(𝑥) such that ‖𝑥−𝑔‖≥‖‖𝑥−𝑔₀‖‖+𝑟_𝑝,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: 𝐷(𝑥)=𝑥^∗∈𝑆𝑋^∗∶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 𝑥_𝜀∈S(𝑋) and 𝑥^∗𝜀∈𝐷(𝑥_𝜀) such that ‖‖𝑥_𝜀−𝑥₀‖‖<2√𝜀,‖‖𝑥^∗𝜀−𝑥^∗0‖‖<2√𝜀.(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 {𝑥_𝑛}⊂𝑆(𝑋) and 𝑥∈𝑆(𝑋) such that 𝑥^∗𝑛(𝑥_𝑛)=1=𝑥^∗(𝑥). We have that 𝑥^∗𝐷⁻¹𝑥^∗𝑛=𝑥^∗−𝑥^∗𝑛+𝑥^∗𝑛𝐷⁻¹𝑥^∗𝑛=𝑥^∗−𝑥^∗𝑛𝐷⁻¹𝑥^∗𝑛+𝑥^∗𝑛𝐷⁻¹𝑥^∗𝑛.(2.2) Since ||𝑥^∗−𝑥^∗𝑛𝐷⁻¹𝑥^∗𝑛||≤‖‖𝑥^∗𝑛−𝑥^∗‖‖⟶0,as𝑛⟶∞,(2.3) and 𝑥^∗𝑛(𝐷⁻¹(𝑥^∗𝑛))=𝑥^∗𝑛(𝑥_𝑛)=1, we have that 𝑥^∗𝐷⁻¹𝑥^∗𝑛=𝑥^∗𝑥_𝑛⟶1,as𝑛⟶∞.(2.4) Since 𝑋 is strongly convex, we deduce that 𝑥_𝑛→𝑥 as 𝑛→∞, that is, 𝐷⁻¹(𝑥^∗𝑛)→𝐷⁻¹(𝑥^∗) as 𝑛→∞. This means that 𝐷⁻¹ is continuous.
Sufficiency. Let {𝑥_𝑛}⊂𝑆(𝑋),𝑥∈𝑆(𝑋) with 𝑥^∗(𝑥_𝑛)→1 as 𝑛→∞ for some 𝑥^∗∈𝐷(𝑥). Since 𝐷⁻¹ is singlevalued, by Lemma 2.1, there exist {𝑦_𝑛}⊂𝑆(𝑋) and {𝑦^∗𝑛}⊂𝑆₀(𝑋^∗) such that 𝐷⁻¹(𝑦^∗𝑛)=𝑦_𝑛 and ‖‖𝑦_𝑛−𝑥_𝑛‖‖⟶0,‖‖𝑦^∗𝑛−𝑥^∗‖‖⟶0,as𝑛⟶∞.(2.5) Since 𝐷⁻¹ is continuous, we have that 𝐷⁻¹(𝑦^∗𝑛)→𝐷⁻¹(𝑥^∗) as 𝑛→∞, that is, 𝑦_𝑛→𝑥 as 𝑛→∞. So 𝑥_𝑛→𝑥 as 𝑛→∞, 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, {𝑧_𝑛}⊂𝐶 such that lim_𝑛→∞‖‖𝑥−𝑧_𝑛‖‖=𝑑(𝑥,𝐶).(2.6) In order to finish the proof, we will show that there exists some 𝑦₀∈𝐶 such that ‖𝑦₀−𝑧_𝑛‖→0 as 𝑛→∞.
Step 1. If 𝑥=0, then 𝑟=𝑑(0,𝐶)>0 and 𝐵(0,𝑟)∩𝐶=𝑃_𝐶(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 sup{𝑓(𝑦)∶𝑦∈𝐶}≤inf{𝑓(𝑦)∶𝑦∈𝐵(0,𝑟)}=−‖𝑓‖𝑟=−‖‖𝑦₀‖‖,(2.8) for any 𝑦₀∈𝑃_𝐶(0). Hence, we have that −‖‖𝑦₀‖‖≤𝑓𝑦₀≤sup{𝑓(𝑦)∶𝑦∈𝐶}≤−‖‖𝑦₀‖‖.(2.9) This shows that 𝑓∈𝐷(−𝑦₀/‖𝑦₀‖). From 𝑧_𝑛∈𝐶, we get 𝑓(𝑧_𝑛)≤𝑓(𝑦₀). Combining it with the condition lim_𝑛→∞‖0−𝑧_𝑛‖=𝑑(0,𝐶), it follows that ‖‖0−𝑦₀‖‖=𝑓0−𝑦₀≤𝑓0−𝑧_𝑛≤‖‖0−𝑧_𝑛‖‖⟶𝑑(0,𝐶)=‖‖0−𝑦₀‖‖,(2.10) as 𝑛→∞. Hence, 𝑓(−𝑧_𝑛/‖𝑧_𝑛‖)→1 as 𝑛→∞. Since 𝑋 is strongly convex, −𝑧_𝑛/‖𝑧_𝑛‖→−𝑦₀/‖𝑦₀‖ as 𝑛→∞, that is, ‖𝑧_𝑛−𝑦₀‖→0 as 𝑛→∞.Step 2. If 𝑥≠0, we set 𝐶^=𝑥−𝐶. It is clear that 𝐶^ is proximinal and {𝑥−𝑧_𝑛}⊂𝐶^ is a minimizing sequence for 0. By Step 1, there exists 𝑦^0∈𝐶^, such that ‖𝑦^0−(𝑥−𝑧_𝑛)‖→0 as 𝑛→∞. This shows that ‖(𝑥−𝑦^0)−𝑧_𝑛‖→0 as 𝑛→∞ and 𝑥−𝑦^0=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∈𝑃_ker𝑓(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)=𝑑𝜆𝑧+𝑦₀,ker𝑓=||𝜆||𝑑(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∈𝑃_ker𝑓(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)=inf_{𝑦∈ker𝑓}‖‖𝑧−𝑥₀+𝑦‖‖=𝑑𝑧−𝑥₀,ker𝑓.(2.14) Since ker𝑓 is a proximinal subset, there exists a 𝑦₀∈ker𝑓 such that 𝑑𝑧−𝑥₀,ker𝑓=‖‖𝑧−𝑥₀+𝑦₀‖‖.(2.15) Therefore, 𝑥₀+𝑦₀∈𝑃_𝐻(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 𝑛→∞ and 𝐷⁻¹(𝑥^∗𝑛)⊈𝑊 for all 𝑛. Let 𝑧_𝑛∈𝐷⁻¹(𝑥^∗𝑛)⧵𝑊. Fix 𝑥∈𝐷⁻¹(𝑥^∗). Let 𝑥_𝑛=𝑥^∗(𝑧_𝑛)𝑥−𝑧_𝑛, then {𝑥_𝑛}⊂𝑌 is a minimizing sequence for 𝑥. Since 𝑌 is approximatively compact, {𝑥_𝑛} has a convergent subsequence. So {𝑧_𝑛} has convergent subsequence converging to 𝑧. Thus, 𝑧∈𝐷⁻¹(𝑥^∗)⊂𝑊, but 𝑧_𝑛∈𝑋⧵𝑊 is closed, which is a contradiction. By the assumption, we easily know that the image of 𝐷⁻¹ is compact.
Sufficiency. Let {𝑥_𝑛}⊂𝑆(𝑋),𝑥∈𝑆(𝑋) with 𝑥^∗(𝑥_𝑛)→1 as 𝑛→∞ for some 𝑥^∗∈𝐷(𝑥). By Lemma 2.1, there exist {𝑦_𝑛}⊂𝑆(𝑋) and {𝑦^∗𝑛}⊂𝑆(𝑋^∗) such that 𝑦_𝑛∈𝐷⁻¹(𝑦^∗𝑛) and ‖‖𝑦_𝑛−𝑥_𝑛‖‖⟶0,‖‖𝑦^∗𝑛−𝑥^∗‖‖⟶0,as𝑛⟶∞.(2.16) Since 𝐷⁻¹(𝑥^∗) is compact, 𝐷⁻¹(𝑥^∗) is proximinal. Let 𝑧_𝑛∈𝐷⁻¹(𝑥^∗) such that ‖𝑦_𝑛−𝑧_𝑛‖=𝑑(𝑦_𝑛,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(𝑦_𝑛,𝐷⁻¹(𝑥^∗))<𝜀,thatis,‖𝑦_𝑛−𝑧_𝑛‖<𝜀, which means that 𝑦_𝑛−𝑧_𝑛→𝜃 as 𝑛→∞. Combining this with the compactness of 𝐷⁻¹(𝑥^∗),{𝑧_𝑛} has convergent subsequence, and hence {𝑦_𝑛} has convergent subsequence. By 𝑦_𝑛−𝑥_𝑛→𝜃 as 𝑛→∞, {𝑥_𝑛} 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 {𝑥_𝑛}⊂𝑆(𝑋) with 𝑥^∗(𝑥_𝑛)→1 as 𝑛→∞ for some 𝑥^∗∈𝐷(𝑥). By Lemma 2.1, there are {𝑦_𝑛}⊂𝑆(𝑋) and {𝑦^∗𝑛}⊂𝑆(𝑋^∗) such that 𝑦_𝑛∈𝐷⁻¹(𝑦^∗𝑛) and ‖‖𝑦_𝑛−𝑥_𝑛‖‖⟶0,‖‖𝑦^∗𝑛−𝑥^∗‖‖⟶0as𝑛⟶∞.(2.17) Suppose that {𝑥_𝑛} does not have weakly convergent subsequence. By (2.17), we know that {𝑦_𝑛} does not have weakly convergent subsequence. Without loss of generality, we can assume that ||𝑓𝑦_𝑛−𝑦_𝑚||≥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 {𝑧_𝑖}^𝑘𝑖=1⊂𝐷⁻¹(𝑥^∗), such that 𝐷⁻¹(𝑥^∗)⊂∪^𝑘𝑖=1{𝑦∈𝑋∶|𝑓(𝑦−𝑧_𝑖)|<𝜀₀/2}. Combining upper semicontinuity (𝑛−𝑤) of 𝐷⁻¹ with ‖𝑦^∗𝑛−𝑥^∗‖→0,{𝑦_𝑛}_𝑛≥𝑛0⊂∪^𝑘𝑖=1{𝑦∈𝑋∶|𝑓(𝑦−𝑧_𝑖)|<𝜀₀/2} for some 𝑛₀∈𝑁. So, there are subsequence {𝑦_𝑛𝑘}_𝑘≥1⊂{𝑦_𝑛}_𝑛≥𝑛0 and 𝑖₀(1≤𝑖₀≤𝑘) such that {𝑦_𝑛𝑘}_𝑘≥1⊂{𝑦∈𝑋∶|𝑓(𝑦−𝑧_𝑖0)|<𝜀₀/2}. Hence, ||𝑓𝑦_𝑛𝑘−𝑦_𝑛𝑙||≤||𝑓𝑦_𝑛𝑘−𝑧_𝑖0||+||𝑓𝑧_𝑖0−𝑦_𝑛𝑙||<𝜀⁰2+𝜀2⁰=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 𝑟_𝑝>0 be such that ‖𝑥−𝑔‖^𝑝≥‖‖𝑥−𝑔₀‖‖^𝑝+𝑟_𝑝‖‖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‖‖≤𝑝‖‖𝑥−𝑔⁰‖‖+𝑁^𝑝−1.(3.3) Set 𝑟_𝑝,𝑁=𝑟^𝑝𝑝‖‖𝑥−𝑔⁰‖‖+𝑁^1−𝑝.(3.4) Then ‖𝑥−𝑔‖−‖‖𝑥−𝑔₀‖‖≥𝑟_𝑝,𝑁‖‖𝑔−𝑔₀‖‖^𝑝.(3.5) Because 𝑟_𝑝,𝑁 is independent of 𝑔,𝑔₀ is bounded strongly unique of order 𝑝 at 𝑥.
(2)⇒(1). From lim_{‖𝑔‖→∞}‖𝑥−𝑔‖^𝑝−‖‖𝑥−𝑔₀‖‖^𝑝‖‖𝑔−𝑔₀‖‖^𝑝≥lim_{‖𝑔‖→∞}1−‖‖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‖‖≥𝑝‖‖𝑥−𝑔⁰‖‖^𝑝−1.(3.8) Therefore, ‖𝑥−𝑔‖^𝑝≥≥‖‖‖‖𝑥𝑥−−𝑔𝑔₀₀‖‖^𝑝+𝑝‖‖𝑥−𝑔₀‖‖^𝑝−1⋅‖𝑥−𝑔‖−‖‖𝑥−𝑔₀‖‖‖‖^𝑝+𝑝‖‖𝑥−𝑔₀‖‖^𝑝−1⋅𝑟_𝑝,𝑁‖‖𝑔−𝑔₀‖‖^𝑝,(3.9) for any 𝑔∈𝐶 and ‖𝑔−𝑔₀‖≤𝑁. Set 𝑟_𝑝=min{1/2,𝑝‖𝑥−𝑔₀‖^𝑝−1𝑟_𝑝,𝑁}, then ‖𝑥−𝑔‖^𝑝≥‖‖𝑥−𝑔₀‖‖^𝑝+𝑟_𝑝‖‖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,𝑟_𝑝,𝑁0>0 such that ‖𝑥−𝑔‖≥‖‖𝑥−𝑔₀‖‖+𝑟_𝑝,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, 𝑔_𝑛∈𝐶 with 𝑁₀<‖𝑔_𝑛−𝑔‖≤𝑁 such that ‖‖𝑥−𝑔_𝑛‖‖<‖‖𝑥−𝑔₀‖‖+1𝑛‖‖𝑔^𝑛−𝑔₀‖‖^𝑝,(3.12) then, because {𝑔_𝑛} is bounded, ‖‖𝑥−𝑔₀‖‖≤lim_𝑛‖‖𝑥−𝑔_𝑛‖‖≤lim_𝑛‖‖𝑥−𝑔⁰‖‖+1𝑛‖‖𝑔^𝑛−𝑔₀‖‖^𝑝=‖‖𝑥−𝑔⁰‖‖,(3.13) that is, lim_𝑛‖𝑥−𝑔_𝑛‖=𝑑(𝑥,𝐶). This shows that {𝑔_𝑛} is a minimizing sequence for 𝑥. Since 𝐶 is weakly approximatively compact, there is a weakly convergent subsequence of {𝑔_𝑛}. Without loss of generality, we can assume that 𝑔−→_𝑛𝑤𝑔₀∈𝐶.(3.14) Based on weak lower semi-continuity of norm, we get that ‖‖𝑥−𝑔₀‖‖≤lim_𝑛‖‖𝑥−𝑔_𝑛‖‖=‖‖𝑥−𝑔₀‖‖.(3.15) Hence, 𝑔₀∈𝑃_𝐶(𝑥), consequently, 𝑔₀=𝑔₀. Take 𝑓∈𝐷((𝑥−𝑔₀)/‖𝑥−𝑔₀‖), then 𝑓𝑥−𝑔^𝑛‖‖𝑥−𝑔_𝑛‖‖⟶𝑓𝑥−𝑔₀‖‖𝑥−𝑔₀‖‖=1.(3.16) Since 𝑋 is nearly strongly convex, there exists {𝑥−𝑔_𝑛𝑘}⊂{𝑥−𝑔_𝑛} such that 𝑥−𝑔_𝑛𝑘→𝑥−𝑔₀, that is, 𝑔_𝑛𝑘→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∈𝑃_𝐶(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,𝑔_𝑛∈𝐶 with 𝑁₀<‖𝑔_𝑛−𝑔‖≤𝑁 such that ‖‖𝑥−𝑔_𝑛‖‖<‖‖𝑥−𝑔₀‖‖+1𝑛‖‖𝑔^𝑛−𝑔₀‖‖^𝑝.(3.17) In the same way of the proof of (3)⇒(2) in Theorem 3.1, we can also prove that {𝑔_𝑛} is a minimizing sequence for 𝑥. By Lemma 2.5, 𝐶 is approximatively compact. Hence, there exists a convergent subsequence of {𝑔_𝑛}. Without loss of generality, we can assume that 𝑔_𝑛⟶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∈𝑃_𝐶(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.