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Journal of Function Spaces and Applications
VolumeΒ 2012Β (2012), Article IDΒ 724120, 11 pages
http://dx.doi.org/10.1155/2012/724120
Research Article

Convexity and Proximinality in Banach Space

College of Fundamental Studies, Shanghai University of Engineering Science, Shanghai 201620, China

Received 9 March 2011; Accepted 9 November 2011

Academic Editor: AnnaΒ Kaminska

Copyright Β© 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.

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 π‘‹βˆ— be its dual space. Let us denote by 𝐡(𝑋) and 𝑆(𝑋) the closed unit ball and the unit sphere of 𝑋, and by 𝐡(π‘₯,π‘Ÿ) the closed ball centered at π‘₯ of radius π‘Ÿ>0. Let π‘₯βˆˆπ‘†(𝑋),𝐷(π‘₯)={π‘“βˆˆπ‘†(π‘‹βˆ—)βˆΆπ‘“(π‘₯)=1}. Let us denote by 𝑁𝐴(𝑋) the set of all norm-attaining functionals in π‘‹βˆ— and 𝑆0(π‘‹βˆ—)=𝑆(π‘‹βˆ—)βˆ©π‘π΄(𝑋).

For a subset πΆβŠ‚π‘‹, the metric projection π‘ƒπΆβˆΆπ‘‹β†’2𝐢 is defined by 𝑃𝐢(π‘₯)={π‘¦βˆˆπΆβˆΆβ€–π‘₯βˆ’π‘¦β€–=𝑑(π‘₯,𝐢)}, where 𝑑(π‘₯,𝐢)=inf{β€–π‘₯βˆ’π‘¦β€–βˆΆπ‘¦βˆˆπΆ},π‘₯βˆˆπ‘‹. If 𝑃𝐢(π‘₯)β‰ βˆ… for each π‘₯βˆˆπ‘‹, then 𝐢 is said to be proximinal. If 𝑃𝐢(π‘₯) is at most a singleton for each π‘₯βˆˆπ‘‹, then 𝐢 is said to be semi-Chebyshev. If 𝐢 is a simultaneously proximinal and semi-Chebyshev set, then 𝐢 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 π‘₯βˆˆπ‘†(𝑋) and {π‘₯𝑛}βŠ‚π΅(𝑋) convergence π‘₯βˆ—(π‘₯𝑛)β†’1 as π‘›β†’βˆž for some π‘₯βˆ—βˆˆπ·(π‘₯) implies that π‘₯𝑛→π‘₯ as π‘›β†’βˆž (resp., π‘₯π‘›π‘€βˆ’β†’π‘₯ as π‘›β†’βˆž/{π‘₯𝑛} is relatively compact/{π‘₯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 π‘₯0, π‘₯𝑛, π‘¦π‘›βˆˆπ‘†(𝑋), (𝑛=1,2,…), π‘₯𝑛+𝑦𝑛→2π‘₯0 as π‘›β†’βˆž implies that π‘₯π‘›βˆ’π‘¦π‘›β†’πœƒ as π‘›β†’βˆž (resp., π‘₯π‘›βˆ’π‘¦π‘›π‘€βˆ’β†’πœƒ as π‘›β†’βˆž).

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 π‘₯βˆ—βˆˆπ‘†(π‘‹βˆ—) is simultaneously a norming element for some π‘₯βˆˆπ‘†(𝑋) and π‘₯βˆ—βˆ—βˆˆπ‘†(π‘‹βˆ—βˆ—), where π‘₯β‰ π‘₯βˆ—βˆ—. 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 π‘₯βˆˆπ‘†(𝑋), {π‘₯𝑛}βŠ‚π΅(𝑋), and {π‘₯𝑛}βŠ‚π΅(π‘‹βˆ—), the condition limπ‘šlim𝑛π‘₯βˆ—π‘š((π‘₯𝑛+π‘₯)/2)=1 implies π‘₯𝑛→π‘₯ (resp., π‘₯π‘›π‘€βˆ’β†’π‘₯). 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}βŠ‚πΆ is said to be minimizing for π‘₯βˆˆπ‘‹β§΅πΆ if β€–π‘₯βˆ’π‘§π‘›β€–β†’π‘‘(π‘₯,𝐢) as π‘›β†’βˆž.

Definition 1.3 (see [10]). Let 𝐢 be a closed subset (resp., a weakly closed subset) of 𝑋 and π‘₯0βˆˆπ‘‹β§΅πΆ.(1)One says that 𝐢 is approximatively (resp., weakly approximatively) compact for π‘₯0 if every minimizing sequence {𝑧𝑛}βŠ‚πΆ for π‘₯0 has a convergent (resp., weakly convergent) subsequence.(2)One says that 𝐢 is strongly Chebyshev (resp., weakly strongly Chebyshev) for π‘₯0 if every minimizing sequence {𝑧𝑛}βŠ‚πΆ for π‘₯0 is convergent (resp., weakly convergent).

If 𝐢 is approximatively compact (resp., weakly approximatively compact/strongly Chebyshev/weakly strongly Chebyshev) for every π‘₯βˆˆπ‘‹β§΅πΆ, we say that 𝐢 is approximatively compact (resp., weakly approximatively compact/strongly Chebyshev/weakly strongly Chebyshev) in 𝑋.

Definition 1.4 (see [11]). Let πΊβˆˆπ‘‹, π‘₯βˆˆπ‘‹β§΅πΊ, and 1≀𝑝<∞.(1)𝑔0βˆˆπ‘ƒπΊ(π‘₯) is said to be strongly unique at π‘₯ if there exists a constant π‘Ÿπ‘>0 such that β€–β€–β€–π‘₯βˆ’π‘”β€–β‰₯π‘₯βˆ’π‘”0β€–β€–+π‘Ÿπ‘β€–β€–π‘”βˆ’π‘”0β€–β€–,(1.1) for any π‘”βˆˆπΊ.(2)𝑔0∈𝐺 is said to be strongly unique of order 𝑝 at π‘₯ if there exists an π‘Ÿπ‘=π‘Ÿπ‘(π‘₯)>0 such that β€–π‘₯βˆ’π‘”β€–π‘β‰₯β€–β€–π‘₯βˆ’π‘”0‖‖𝑝+π‘Ÿπ‘β€–β€–π‘”βˆ’π‘”0‖‖𝑝,(1.2) for any π‘”βˆˆπΊ.(3)𝑔0∈𝐺 is said to be bounded strongly unique of order 𝑝 at π‘₯ if given any 𝑁>0, there exists an π‘Ÿπ‘,𝑁=π‘Ÿπ‘,𝑁(π‘₯) such that β€–β€–β€–π‘₯βˆ’π‘”β€–β‰₯π‘₯βˆ’π‘”0β€–β€–+π‘Ÿπ‘,π‘β€–β€–π‘”βˆ’π‘”0‖‖𝑝,(1.3) for any π‘”βˆˆπΊ,β€–π‘”βˆ’π‘”0‖≀𝑁.(4)𝑔0∈𝐺 is said to be locally strongly unique of order 𝑝 at π‘₯ if there exist 𝑁>0 and π‘Ÿπ‘,𝑁=π‘Ÿπ‘,𝑁(π‘₯) such that β€–β€–β€–π‘₯βˆ’π‘”β€–β‰₯π‘₯βˆ’π‘”0β€–β€–+π‘Ÿπ‘,π‘β€–β€–π‘”βˆ’π‘”0‖‖𝑝,(1.4) for any π‘”βˆˆπΊ,β€–π‘”βˆ’π‘”0‖≀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 π‘₯βˆ—βˆˆπ‘†(π‘‹βˆ—) and π‘₯βˆˆπ‘†(𝑋), let one define the following maps: ξ€½π‘₯𝐷(π‘₯)=βˆ—ξ€·π‘‹βˆˆπ‘†βˆ—ξ€ΈβˆΆπ‘₯βˆ—ξ€Ύ(π‘₯)=1,π·βˆ’1ξ€·π‘₯βˆ—ξ€Έ=ξ€½π‘₯βˆˆπ‘†(𝑋)∢π‘₯βˆ—ξ€Ύ.(π‘₯)=1(1.5)𝐷 is called the duality map and π·βˆ’1 is called the preduality map. Naturally, π·βˆ’1 is defined only on 𝑆0(π‘‹βˆ—).

Definition 1.6 (see [10]). The preduality map π·βˆ’1βˆΆπ‘†(π‘‹βˆ—)→𝑆(𝑋) is said to be upper semicontinuous (π‘›βˆ’πœ) on 𝑆0(π‘‹βˆ—) if for any π‘₯βˆ—βˆˆπ‘†0(π‘‹βˆ—) and any 𝜏 open set π‘Š with π·βˆ’1(π‘₯βˆ—)βŠ‚π‘Š, there exists πœ€>0 such that π·βˆ’1(π‘¦βˆ—)βŠ‚π‘Š whenever β€–π‘₯βˆ—βˆ’π‘¦βˆ—β€–<πœ€, 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 πœ€>0, π‘₯0βˆˆπ‘†(𝑋), π‘₯βˆ—0βˆˆπ‘†(π‘‹βˆ—), and π‘₯βˆ—0(π‘₯0)>1βˆ’πœ€, then there are π‘₯πœ€βˆˆS(𝑋) and π‘₯βˆ—πœ€βˆˆπ·(π‘₯πœ€) such that β€–β€–π‘₯πœ€βˆ’π‘₯0β€–β€–βˆš<2β€–β€–π‘₯πœ€,βˆ—πœ€βˆ’π‘₯βˆ—0β€–β€–βˆš<2πœ€.(2.1)

Theorem 2.2. Let 𝑋 be a Banach space. 𝑋 is strongly convex if and only if the preduality map π·βˆ’1βˆΆπ‘†0(π‘‹βˆ—)→𝑆(𝑋) is singlevalued and continuous.

Proof. Necessity. Since strong convexity implies strict convexity, π·βˆ’1 is singlevalued. Suppose that {π‘₯βˆ—π‘›}βŠ‚π‘†0(π‘‹βˆ—), π‘₯βˆ—βˆˆπ‘†0(π‘‹βˆ—) with π‘₯βˆ—π‘›β†’π‘₯βˆ—, then there exist {π‘₯𝑛}βŠ‚π‘†(𝑋) and π‘₯βˆˆπ‘†(𝑋) such that π‘₯βˆ—π‘›(π‘₯𝑛)=1=π‘₯βˆ—(π‘₯). We have that π‘₯βˆ—ξ€·π·βˆ’1ξ€·π‘₯βˆ—π‘›=ξ€·π‘₯ξ€Έξ€Έβˆ—βˆ’π‘₯βˆ—π‘›+π‘₯βˆ—π‘›π·ξ€Έξ€·βˆ’1ξ€·π‘₯βˆ—π‘›=ξ€·π‘₯ξ€Έξ€Έβˆ—βˆ’π‘₯βˆ—π‘›π·ξ€Έξ€·βˆ’1ξ€·π‘₯βˆ—π‘›ξ€Έξ€Έ+π‘₯βˆ—π‘›ξ€·π·βˆ’1ξ€·π‘₯βˆ—π‘›.ξ€Έξ€Έ(2.2) Since ||ξ€·π‘₯βˆ—βˆ’π‘₯βˆ—π‘›ξ€Έπ·βˆ’1ξ€·π‘₯βˆ—π‘›ξ€Έ||≀‖‖π‘₯βˆ—π‘›βˆ’π‘₯βˆ—β€–β€–βŸΆ0,asπ‘›βŸΆβˆž,(2.3) and π‘₯βˆ—π‘›(π·βˆ’1(π‘₯βˆ—π‘›))=π‘₯βˆ—π‘›(π‘₯𝑛)=1, we have that π‘₯βˆ—ξ€·π·βˆ’1ξ€·π‘₯βˆ—π‘›ξ€Έξ€Έ=π‘₯βˆ—ξ€·π‘₯π‘›ξ€ΈβŸΆ1,asπ‘›βŸΆβˆž.(2.4) Since 𝑋 is strongly convex, we deduce that π‘₯𝑛→π‘₯ as π‘›β†’βˆž, that is, π·βˆ’1(π‘₯βˆ—π‘›)β†’π·βˆ’1(π‘₯βˆ—) as π‘›β†’βˆž. This means that π·βˆ’1 is continuous.
Sufficiency. Let {π‘₯𝑛}βŠ‚π‘†(𝑋),π‘₯βˆˆπ‘†(𝑋) with π‘₯βˆ—(π‘₯𝑛)β†’1 as π‘›β†’βˆž for some π‘₯βˆ—βˆˆπ·(π‘₯). Since π·βˆ’1 is singlevalued, by Lemma 2.1, there exist {𝑦𝑛}βŠ‚π‘†(𝑋) and {π‘¦βˆ—π‘›}βŠ‚π‘†0(π‘‹βˆ—) such that π·βˆ’1(π‘¦βˆ—π‘›)=𝑦𝑛 and β€–β€–π‘¦π‘›βˆ’π‘₯π‘›β€–β€–β€–β€–π‘¦βŸΆ0,βˆ—π‘›βˆ’π‘₯βˆ—β€–β€–βŸΆ0,asπ‘›βŸΆβˆž.(2.5) Since π·βˆ’1 is continuous, we have that π·βˆ’1(π‘¦βˆ—π‘›)β†’π·βˆ’1(π‘₯βˆ—) as π‘›β†’βˆž, that is, 𝑦𝑛→π‘₯ as π‘›β†’βˆž. So π‘₯𝑛→π‘₯ as π‘›β†’βˆž, 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 π·βˆ’1βˆΆπ‘†0(π‘‹βˆ—)→𝑆(𝑋) is singlevalued and weakly continuous.

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

Proof. We only need to prove (1)β‡’(4).
Let π‘₯βˆˆπ‘‹β§΅πΆ, {𝑧𝑛}βŠ‚πΆ such that limπ‘›β†’βˆžβ€–β€–π‘₯βˆ’π‘§π‘›β€–β€–=𝑑(π‘₯,𝐢).(2.6) In order to finish the proof, we will show that there exists some 𝑦0∈𝐢 such that ‖𝑦0βˆ’π‘§π‘›β€–β†’0 as π‘›β†’βˆž.
Step 1. If π‘₯=0, then π‘Ÿ=𝑑(0,𝐢)>0 and 𝐡(0,π‘Ÿ)∩𝐢=𝑃𝐢(0)β‰ βˆ…,int𝐡(0,π‘Ÿ)∩𝐢=βˆ…,(2.7) where 𝐡(0,π‘Ÿ)={π‘¦βˆΆβ€–π‘¦β€–β‰€π‘Ÿ}. By the separation theorem [2] and definition of norm, there exists an π‘“βˆˆπ‘†(π‘‹βˆ—) such that ‖‖𝑦sup{𝑓(𝑦)βˆΆπ‘¦βˆˆπΆ}≀inf{𝑓(𝑦)βˆΆπ‘¦βˆˆπ΅(0,π‘Ÿ)}=βˆ’β€–π‘“β€–π‘Ÿ=βˆ’0β€–β€–,(2.8) for any 𝑦0βˆˆπ‘ƒπΆ(0). Hence, we have that βˆ’β€–β€–π‘¦0‖‖𝑦≀𝑓0‖‖𝑦≀sup{𝑓(𝑦)βˆΆπ‘¦βˆˆπΆ}β‰€βˆ’0β€–β€–.(2.9) This shows that π‘“βˆˆπ·(βˆ’π‘¦0/‖𝑦0β€–). From π‘§π‘›βˆˆπΆ, we get 𝑓(𝑧𝑛)≀𝑓(𝑦0). Combining it with the condition limπ‘›β†’βˆžβ€–0βˆ’π‘§π‘›β€–=𝑑(0,𝐢), it follows that β€–β€–0βˆ’π‘¦0β€–β€–ξ€·=𝑓0βˆ’π‘¦0≀𝑓0βˆ’π‘§π‘›ξ€Έβ‰€β€–β€–0βˆ’π‘§π‘›β€–β€–β€–β€–βŸΆπ‘‘(0,𝐢)=0βˆ’π‘¦0β€–β€–,(2.10) as π‘›β†’βˆž. Hence, 𝑓(βˆ’π‘§π‘›/‖𝑧𝑛‖)β†’1 as π‘›β†’βˆž. Since 𝑋 is strongly convex, βˆ’π‘§π‘›/β€–π‘§π‘›β€–β†’βˆ’π‘¦0/‖𝑦0β€– as π‘›β†’βˆž, that is, β€–π‘§π‘›βˆ’π‘¦0β€–β†’0 as π‘›β†’βˆž.Step 2. If π‘₯β‰ 0, we set πΆξ…ž=π‘₯βˆ’πΆ. It is clear that πΆξ…ž is proximinal and {π‘₯βˆ’π‘§π‘›}βŠ‚πΆξ…ž is a minimizing sequence for 0. By Step 1, there exists π‘¦ξ…ž0βˆˆπΆξ…ž, such that β€–π‘¦ξ…ž0βˆ’(π‘₯βˆ’π‘§π‘›)β€–β†’0 as π‘›β†’βˆž. This shows that β€–(π‘₯βˆ’π‘¦ξ…ž0)βˆ’π‘§π‘›β€–β†’0 as π‘›β†’βˆž and π‘₯βˆ’π‘¦ξ…ž0=𝑦0∈𝐢.

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

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

Lemma 2.6. Let π‘₯βˆˆπ‘†(𝑋), π‘“βˆˆπ·(π‘₯), then hyperplane 𝐻={π‘¦βˆˆπ‘‹βˆΆπ‘“(𝑦)=1} is a proximinal convex subset in 𝑋.

Proof. Let ker𝑓={π‘¦βˆˆπ‘‹βˆΆπ‘“(𝑦)=0}.(2.11) We will prove that ker𝑓 is a proximinal convex subset in 𝑋. For any π‘§βˆˆπ‘‹, if 𝑑(𝑧,ker𝑓)=0, since ker𝑓 is a closed subspace, we know that π‘§βˆˆker𝑓. Hence, π‘§βˆˆπ‘ƒker𝑓(𝑧). If 𝑑(𝑧,ker𝑓)>0, since 𝑋={𝛼𝑧}+ker𝑓, there exist πœ†βˆˆβ„ and 𝑦0∈ker𝑓 such that 𝑑(𝑧,ker𝑓)π‘₯=πœ†π‘§+𝑦0. For any π‘¦βˆˆπ‘‹, 𝑑(𝑦,ker𝑓)=|𝑓(𝑦)|, and 𝑓(π‘₯)=1, we have that 𝑑(π‘₯,ker𝑓)=1 and 𝑓(𝑑(𝑧,ker𝑓)π‘₯)=𝑑(𝑑(𝑧,ker𝑓)π‘₯,ker𝑓). Therefore, we have that 𝑑(𝑧,ker𝑓)=π‘‘πœ†π‘§+𝑦0ξ€Έ=||πœ†||𝑑,ker𝑓(𝑧,ker𝑓).(2.12) This means that |πœ†|=1. Hence, we have ‖‖𝑑(𝑧,ker𝑓)=‖𝑑(𝑧,ker𝑓)π‘₯β€–=πœ†π‘§+𝑦0β€–β€–=‖‖𝑧+πœ†π‘¦0β€–β€–.(2.13) It follows that βˆ’πœ†π‘¦0βˆˆπ‘ƒker𝑓(𝑧), which means that ker𝑓 is proximinal set.
Furthermore, we will prove that hyperplane 𝐻={π‘¦βˆˆπ‘‹βˆΆπ‘“(𝑦)=1} is a proximinal convex subset in 𝑋. For π‘₯0∈𝐻, 𝐻=π‘₯0+ker𝑓. For any π‘§βˆˆπ‘‹, 𝑑(𝑧,𝐻)=infπ‘¦βˆˆker𝑓‖‖π‘₯π‘§βˆ’0ξ€Έβ€–β€–ξ€·+𝑦=π‘‘π‘§βˆ’π‘₯0ξ€Έ.,ker𝑓(2.14) Since ker𝑓 is a proximinal subset, there exists a 𝑦0∈ker𝑓 such that π‘‘ξ€·π‘§βˆ’π‘₯0ξ€Έ=β€–β€–ξ€·π‘₯,kerπ‘“π‘§βˆ’0+𝑦0ξ€Έβ€–β€–.(2.15) Therefore, π‘₯0+𝑦0βˆˆπ‘ƒπ»(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 π·βˆ’1βˆΆπ‘†0(π‘‹βˆ—)→𝑆(𝑋) is upper semicontinuous (π‘›βˆ’π‘›) on 𝑆0(π‘‹βˆ—) with norm compact images, where 𝑛 stands for norm topology.

Proof. Necessity. Arbitrarily take π‘₯βˆ—βˆˆπ‘†0(π‘‹βˆ—). Then by Lemma 2.6, we know that kerπ‘₯βˆ—=π‘Œ is a proximinal convex subset in 𝑋. Since 𝑋 is nearly strongly convex, by Lemma 2.5, we know that π‘Œ is approximatively compact. Suppose that π·βˆ’1 is not upper semicontinuous (π‘›βˆ’π‘›) at π‘₯βˆ—, then for some open set π‘Š in 𝑋 with π·βˆ’1(π‘₯βˆ—)βŠ‚π‘Š, there exists {π‘₯βˆ—π‘›}βŠ‚π‘†(π‘‹βˆ—) such that π‘₯βˆ—π‘›β†’π‘₯βˆ— as π‘›β†’βˆž and π·βˆ’1(π‘₯βˆ—π‘›)ΜΈβŠ†π‘Š for all 𝑛. Let π‘§π‘›βˆˆπ·βˆ’1(π‘₯βˆ—π‘›)β§΅π‘Š. Fix π‘₯βˆˆπ·βˆ’1(π‘₯βˆ—). Let π‘₯𝑛=π‘₯βˆ—(𝑧𝑛)π‘₯βˆ’π‘§π‘›, then {π‘₯𝑛}βŠ‚π‘Œ is a minimizing sequence for π‘₯. Since π‘Œ is approximatively compact, {π‘₯𝑛} has a convergent subsequence. So {𝑧𝑛} has convergent subsequence converging to 𝑧. Thus, π‘§βˆˆπ·βˆ’1(π‘₯βˆ—)βŠ‚π‘Š, but π‘§π‘›βˆˆπ‘‹β§΅π‘Š is closed, which is a contradiction. By the assumption, we easily know that the image of π·βˆ’1 is compact.
Sufficiency. Let {π‘₯𝑛}βŠ‚π‘†(𝑋),π‘₯βˆˆπ‘†(𝑋) with π‘₯βˆ—(π‘₯𝑛)β†’1 as π‘›β†’βˆž for some π‘₯βˆ—βˆˆπ·(π‘₯). By Lemma 2.1, there exist {𝑦𝑛}βŠ‚π‘†(𝑋) and {π‘¦βˆ—π‘›}βŠ‚π‘†(π‘‹βˆ—) such that π‘¦π‘›βˆˆπ·βˆ’1(π‘¦βˆ—π‘›) and β€–β€–π‘¦π‘›βˆ’π‘₯π‘›β€–β€–β€–β€–π‘¦βŸΆ0,βˆ—π‘›βˆ’π‘₯βˆ—β€–β€–βŸΆ0,asπ‘›βŸΆβˆž.(2.16) Since π·βˆ’1(π‘₯βˆ—) is compact, π·βˆ’1(π‘₯βˆ—) is proximinal. Let π‘§π‘›βˆˆπ·βˆ’1(π‘₯βˆ—) such that β€–π‘¦π‘›βˆ’π‘§π‘›β€–=𝑑(𝑦𝑛,π·βˆ’1(π‘₯βˆ—)). In virtue of π·βˆ’1 being upper semicontinuous (π‘›βˆ’π‘›) on 𝑆0(π‘‹βˆ—) and β€–π‘¦βˆ—π‘›βˆ’π‘₯βˆ—β€–β†’0, for any πœ€>0, there is 𝑛0 such that for all 𝑛β‰₯𝑛0𝑑(𝑦𝑛,π·βˆ’1(π‘₯βˆ—))<πœ€,thatis,β€–π‘¦π‘›βˆ’π‘§π‘›β€–<πœ€, which means that π‘¦π‘›βˆ’π‘§π‘›β†’πœƒ as π‘›β†’βˆž. Combining this with the compactness of π·βˆ’1(π‘₯βˆ—),{𝑧𝑛} has convergent subsequence, and hence {𝑦𝑛} has convergent subsequence. By π‘¦π‘›βˆ’π‘₯π‘›β†’πœƒ as π‘›β†’βˆž, {π‘₯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 π·βˆ’1βˆΆπ‘†0(π‘‹βˆ—)→𝑆(𝑋) is upper semicontinuous (π‘›βˆ’π‘€) on 𝑆0(π‘‹βˆ—) with weakly compact images.

Proof. In a similar way to the proof of Theorem 2.7, the necessity can be proved.
Sufficiency. Let {π‘₯𝑛}βŠ‚π‘†(𝑋) with π‘₯βˆ—(π‘₯𝑛)β†’1 as π‘›β†’βˆž for some π‘₯βˆ—βˆˆπ·(π‘₯). By Lemma 2.1, there are {𝑦𝑛}βŠ‚π‘†(𝑋) and {π‘¦βˆ—π‘›}βŠ‚π‘†(π‘‹βˆ—) such that π‘¦π‘›βˆˆπ·βˆ’1(π‘¦βˆ—π‘›) and β€–β€–π‘¦π‘›βˆ’π‘₯π‘›β€–β€–β€–β€–π‘¦βŸΆ0,βˆ—π‘›βˆ’π‘₯βˆ—β€–β€–βŸΆ0asπ‘›βŸΆβˆž.(2.17) Suppose that {π‘₯𝑛} 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 ||π‘“ξ€·π‘¦π‘›βˆ’π‘¦π‘šξ€Έ||β‰₯πœ€0(βˆ€π‘›,π‘šβˆˆπ‘,π‘›β‰ π‘š),(2.18) for some π‘“βˆˆπ‘‹βˆ—β§΅{πœƒ} and πœ€0>0. Since π·βˆ’1(π‘₯βˆ—) is weakly compact, there exists {𝑧𝑖}π‘˜π‘–=1βŠ‚π·βˆ’1(π‘₯βˆ—), such that π·βˆ’1(π‘₯βˆ—)βŠ‚βˆͺπ‘˜π‘–=1{π‘¦βˆˆπ‘‹βˆΆ|𝑓(π‘¦βˆ’π‘§π‘–)|<πœ€0/2}. Combining upper semicontinuity (π‘›βˆ’π‘€) of π·βˆ’1 with β€–π‘¦βˆ—π‘›βˆ’π‘₯βˆ—β€–β†’0,{𝑦𝑛}𝑛β‰₯𝑛0βŠ‚βˆͺπ‘˜π‘–=1{π‘¦βˆˆπ‘‹βˆΆ|𝑓(π‘¦βˆ’π‘§π‘–)|<πœ€0/2} for some 𝑛0βˆˆπ‘. So, there are subsequence {π‘¦π‘›π‘˜}π‘˜β‰₯1βŠ‚{𝑦𝑛}𝑛β‰₯𝑛0 and 𝑖0(1≀𝑖0β‰€π‘˜) such that {π‘¦π‘›π‘˜}π‘˜β‰₯1βŠ‚{π‘¦βˆˆπ‘‹βˆΆ|𝑓(π‘¦βˆ’π‘§π‘–0)|<πœ€0/2}. Hence, ||π‘“ξ€·π‘¦π‘›π‘˜βˆ’π‘¦π‘›π‘™ξ€Έ||≀||π‘“ξ€·π‘¦π‘›π‘˜βˆ’π‘§π‘–0ξ€Έ||+||𝑓𝑧𝑖0βˆ’π‘¦π‘›π‘™ξ€Έ||<πœ€02+πœ€02=πœ€0,(2.19) a contradiction with (2.18).

3. Convexity and Proximinality

Theorem 3.1. Let 𝐢 be a weakly approximatively compact subset of nearly strongly convex Banach space 𝑋, π‘₯βˆˆπ‘‹β§΅πΆ, and 𝑔0∈𝐢. If 𝑔0 is the unique element of best approximation of π‘₯, then the following are equivalent:(1)𝑔0 is strongly unique of order 𝑝 at π‘₯;(2)𝑔0 is bounded strongly unique of order 𝑝 at π‘₯;(3)𝑔0 is locally strongly unique of order 𝑝 at π‘₯.

Proof. (1)β‡’(2). Let π‘Ÿπ‘>0 be such that β€–π‘₯βˆ’π‘”β€–π‘β‰₯β€–β€–π‘₯βˆ’π‘”0‖‖𝑝+π‘Ÿπ‘β€–β€–π‘”βˆ’π‘”0‖‖𝑝,(3.1) for any π‘”βˆˆπΆ. For any 𝑁>0 and each π‘”βˆˆπΆ,β€–π‘”βˆ’π‘”0‖≀𝑁, since β€–β€–β€–π‘₯βˆ’π‘”β€–β‰€π‘₯βˆ’π‘”0β€–β€–+β€–β€–π‘”βˆ’π‘”0‖‖≀‖‖π‘₯βˆ’π‘”0β€–β€–+𝑁,(3.2) by Lagrange’s mean value theorem, β€–π‘₯βˆ’π‘”β€–π‘βˆ’β€–β€–π‘₯βˆ’π‘”0‖‖𝑝‖‖‖π‘₯βˆ’π‘”β€–βˆ’π‘₯βˆ’π‘”0‖‖‖‖≀𝑝π‘₯βˆ’π‘”0β€–β€–ξ€Έ+π‘π‘βˆ’1.(3.3) Set π‘Ÿπ‘,𝑁=π‘Ÿπ‘π‘ξ€·β€–β€–π‘₯βˆ’π‘”0β€–β€–ξ€Έ+𝑁1βˆ’π‘.(3.4) Then β€–β€–β€–π‘₯βˆ’π‘”β€–βˆ’π‘₯βˆ’π‘”0β€–β€–β‰₯π‘Ÿπ‘,π‘β€–β€–π‘”βˆ’π‘”0‖‖𝑝.(3.5) Because π‘Ÿπ‘,𝑁 is independent of 𝑔,𝑔0 is bounded strongly unique of order 𝑝 at π‘₯.
(2)β‡’(1). From limβ€–π‘”β€–β†’βˆžβ€–π‘₯βˆ’π‘”β€–π‘βˆ’β€–β€–π‘₯βˆ’π‘”0β€–β€–π‘β€–β€–π‘”βˆ’π‘”0‖‖𝑝β‰₯limβ€–π‘”β€–β†’βˆžβ€–β€–ξ‚Έξ‚΅1βˆ’π‘₯βˆ’π‘”0β€–β€–β€–β€–π‘”βˆ’π‘”0β€–β€–ξ‚Άπ‘βˆ’ξ‚΅β€–β€–π‘₯βˆ’π‘”0β€–β€–β€–β€–π‘”βˆ’π‘”0‖‖𝑝=1,(3.6) we can take 𝑁>0 such that β€–π‘₯βˆ’π‘”β€–π‘β‰₯β€–β€–π‘₯βˆ’π‘”0‖‖𝑝+12β€–β€–π‘”βˆ’π‘”0‖‖𝑝‖‖,βˆ€π‘”βˆˆπΆ,π‘”βˆ’π‘”0β€–β€–β‰₯𝑁.(3.7) Again, by Lagrange’s mean value theorem, we have β€–π‘₯βˆ’π‘”β€–π‘βˆ’β€–β€–π‘₯βˆ’π‘”0‖‖𝑝‖‖‖π‘₯βˆ’π‘”β€–βˆ’π‘₯βˆ’π‘”0β€–β€–β€–β€–β‰₯𝑝π‘₯βˆ’π‘”0β€–β€–π‘βˆ’1.(3.8) Therefore, β€–π‘₯βˆ’π‘”β€–π‘β‰₯β€–β€–π‘₯βˆ’π‘”0‖‖𝑝‖‖+𝑝π‘₯βˆ’π‘”0β€–β€–π‘βˆ’1β‹…ξ€Ίβ€–β€–β€–π‘₯βˆ’π‘”β€–βˆ’π‘₯βˆ’π‘”0β€–β€–ξ€»β‰₯β€–β€–π‘₯βˆ’π‘”0‖‖𝑝‖‖+𝑝π‘₯βˆ’π‘”0β€–β€–π‘βˆ’1β‹…π‘Ÿπ‘,π‘β€–β€–π‘”βˆ’π‘”0‖‖𝑝,(3.9) for any π‘”βˆˆπΆ and β€–π‘”βˆ’π‘”0‖≀𝑁. Set π‘Ÿπ‘=min{1/2,𝑝‖π‘₯βˆ’π‘”0β€–π‘βˆ’1π‘Ÿπ‘,𝑁}, then β€–π‘₯βˆ’π‘”β€–π‘β‰₯β€–β€–π‘₯βˆ’π‘”0‖‖𝑝+π‘Ÿπ‘β€–β€–π‘”βˆ’π‘”0‖‖𝑝,(3.10) for any π‘”βˆˆπΆ. That shows 𝑔0 is strongly unique of order 𝑝 at π‘₯.
Now, we only need to prove (3)β‡’(2). Let 𝑁0>0,π‘Ÿπ‘,𝑁0>0 such that β€–β€–β€–π‘₯βˆ’π‘”β€–β‰₯π‘₯βˆ’π‘”0β€–β€–+π‘Ÿπ‘,𝑁0β€–β€–π‘”βˆ’π‘”0‖‖𝑝,(3.11) for all π‘”βˆˆπΆ and β€–π‘”βˆ’π‘”0‖≀𝑁0.
If the condition (2) is not true, there exist 𝑁>𝑁0, π‘”π‘›βˆˆπΆ with 𝑁0<β€–π‘”π‘›βˆ’π‘”β€–β‰€π‘ such that β€–β€–π‘₯βˆ’π‘”π‘›β€–β€–<β€–β€–π‘₯βˆ’π‘”0β€–β€–+1π‘›β€–β€–π‘”π‘›βˆ’π‘”0‖‖𝑝,(3.12) then, because {𝑔𝑛} is bounded, β€–β€–π‘₯βˆ’π‘”0‖‖≀lim𝑛‖‖π‘₯βˆ’π‘”π‘›β€–β€–β‰€lim𝑛‖‖π‘₯βˆ’π‘”0β€–β€–+1π‘›β€–β€–π‘”π‘›βˆ’π‘”0‖‖𝑝=β€–β€–π‘₯βˆ’π‘”0β€–β€–,(3.13) that is, lim𝑛‖π‘₯βˆ’π‘”π‘›β€–=𝑑(π‘₯,𝐢). This shows that {𝑔𝑛} is a minimizing sequence for π‘₯. Since 𝐢 is weakly approximatively compact, there is a weakly convergent subsequence of {𝑔𝑛}. Without loss of generality, we can assume that π‘”π‘›π‘€βˆ’β†’π‘”0∈𝐢.(3.14) Based on weak lower semi-continuity of norm, we get that β€–β€–π‘₯βˆ’π‘”0‖‖≀lim𝑛‖‖π‘₯βˆ’π‘”π‘›β€–β€–=β€–β€–π‘₯βˆ’π‘”0β€–β€–.(3.15) Hence, 𝑔0βˆˆπ‘ƒπΆ(π‘₯), consequently, 𝑔0=𝑔0. Take π‘“βˆˆπ·((π‘₯βˆ’π‘”0)/β€–π‘₯βˆ’π‘”0β€–), then 𝑓π‘₯βˆ’π‘”π‘›β€–β€–π‘₯βˆ’π‘”π‘›β€–β€–ξ‚Άξ‚΅βŸΆπ‘“π‘₯βˆ’π‘”0β€–β€–π‘₯βˆ’π‘”0β€–β€–ξ‚Ά=1.(3.16) Since 𝑋 is nearly strongly convex, there exists {π‘₯βˆ’π‘”π‘›π‘˜}βŠ‚{π‘₯βˆ’π‘”π‘›} such that π‘₯βˆ’π‘”π‘›π‘˜β†’π‘₯βˆ’π‘”0, that is, π‘”π‘›π‘˜β†’π‘”0. It follows that ‖𝑔0βˆ’π‘”0β€–β‰₯𝑁0. This is a contradiction with the fact that 𝑔0=𝑔0.

Corollary 3.2. Let 𝐢 be a weakly approximatively compact subset of strongly convex Banach space 𝑋, π‘₯βˆˆπ‘‹β§΅πΆ, and 𝑔0βˆˆπ‘ƒπΆ(π‘₯), then the following are equivalent:(1)𝑔0 is strongly unique of order 𝑝 at π‘₯;(2)𝑔0 is bounded strongly unique of order 𝑝 at π‘₯;(3)𝑔0 is locally strongly unique of order 𝑝 at π‘₯.

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

Theorem 3.3. Let 𝐢 be a convex subset of a nearly strongly convex Banach space 𝑋, π‘₯βˆˆπ‘‹β§΅πΆ, and 𝑔0∈𝐢. If 𝑔0 is the unique element of best approximation of π‘₯, then the following are equivalent:(1)𝑔0 is strongly unique of order 𝑝 at π‘₯;(2)𝑔0 is bounded strongly unique of order 𝑝 at π‘₯;(3)𝑔0 is locally strongly unique of order 𝑝 at π‘₯.

Proof. By the proof of Theorem 3.1, we have (1)⇔(2). Now, we only need to prove (3)β‡’(2).
If the condition (2) is not true, there exist 𝑁>𝑁0,π‘”π‘›βˆˆπΆ with 𝑁0<β€–π‘”π‘›βˆ’π‘”β€–β‰€π‘ such that β€–β€–π‘₯βˆ’π‘”π‘›β€–β€–<β€–β€–π‘₯βˆ’π‘”0β€–β€–+1π‘›β€–β€–π‘”π‘›βˆ’π‘”0‖‖𝑝.(3.17) In the same way of the proof of (3)β‡’(2) in Theorem 3.1, we can also prove that {𝑔𝑛} is a minimizing sequence for π‘₯. By Lemma 2.5, 𝐢 is approximatively compact. Hence, there exists a convergent subsequence of {𝑔𝑛}. Without loss of generality, we can assume that π‘”π‘›βŸΆπ‘”0∈𝐢.(3.18) Consequently, 𝑔0=𝑔0, but ‖𝑔0βˆ’π‘”0β€–β‰₯𝑁0, which is a contradiction.

Corollary 3.4. Let 𝐢 be a convex set of a strongly convex Banach space 𝑋. π‘₯βˆˆπ‘‹β§΅πΆ and 𝑔0βˆˆπ‘ƒπΆ(π‘₯). The following are equivalent:(1)𝑔0 is strongly unique of order 𝑝 at π‘₯;(2)𝑔0 is bounded strongly unique of order 𝑝 at π‘₯;(3)𝑔0 is locally strongly unique of order 𝑝 at π‘₯.

Acknowledgment

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

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