JFSJournal of Function Spaces2314-88882314-8896Hindawi10.1155/2020/81918788191878Research ArticleLocal Uniform Kadec-Klee Property (LUKK) and Modulus of (LUKK)https://orcid.org/0000-0001-6756-2233CuiYunan1https://orcid.org/0000-0002-4903-9390WangXiaoxia2SuzukiTomonari1Department of MathematicsHarbin University of Science and TechnologyHarbin 150080Chinahrbust.edu.cn2Faculty of Mathematics and Computer EngineeringOrdos Institution of Applied TechnologyOrdos 017000China202077202020200902202019052020220620207720202020Copyright © 2020 Yunan Cui and Xiaoxia Wang.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.

A new geometry property and two new moduli are introduced in Banach space. First, the concept of local uniform Kadec-Klee property (LUKK) is introduced and the implication relationships between LUKK and local near uniform convexity LNUC, uniformly Kadec-Klee (UKK), (H) are investigated in Banach space. Furthermore, the modulus PXLε of (LUKK) and the modulus ΔXLε of LNUC are introduced and the relationship of size between PXLε and ΔXLε is also investigated in Banach space. Finally, several formulas for PXLε are calculated in classical Banach space lp.

The Science Research Project of Ordos Institution of Applied TechnologyKYYB2017014The Science Research Project of Inner Mongolia Autonomous RegionNJZY18253The National Science Foundation of China11871181
1. Introduction

Let X,· be a Banach space, X be the dual space of X. By UX and SX, we denote the closed unit ball and the unit sphere of Banach space X, respectively. By coA and co¯A, we denote the convex hull and closed convex hull of the set A, respectively. xnwx as n denotes xn is weakly converges to x as n.

It is well known that the condition equivalent to near uniform convexity (NUC) was independently formulated in  (see also ). Recall that a notion of noncompactness with Hausdorff measure and Kuratowski measure of a set (see ). Let A be a bounded subset of X. Fix ε0,1 for all convex closed sets AUX with μAε, we put (1)ΔXε=inf1dθ,A: AUX,A=coA,μAε,Δ^Xε=inf1dθ,A: AUX,A=coA,αAε,where (2)μA=infε>0:Acanbecoveredwithafinitenumberofsetsofradiismallerthanε,αA=infε>0:Acanbecoveredbyfinitelymanysetswithdiameter<ε.

The functions ΔXε and Δ^Xε are called the moduli of noncompact convexity with Hausdorff measure and Kuratowski measure of X, respectively. It is clear that X is NUC if and only if ΔXε>0 for every ε0,1. Other properties of ΔXε were investigated in .

Recall also that a function (3)ε1X=supε:ΔXε=0.

J. Banas’ proved that if ε1X<1, then X is reflexive and has normal structure (see ).

The modulus of UKK was introduced in  by J. R. Partington that is PX:0,10,1(4)PXε=inf1x:xUXandxnUXs.t.xnwxandsepxnε,where sepxn=infxnxm:mn. He proved that X has UKK property if and only if PXε>0 whenever ε0,1. He also proved that the function PX is nondecreasing on [0,1].

There are many recent papers concerning the Kadec-Klee property, such as Kadec¨CKlee property and fixed points and Dual Kadec-Klee property and fixed points studied by Jean Saint Raymond (see [9, 10]). In a recent paper , Maciej Ciesielski, Paweł Kolwicz, and Ryszard Płuciennik were interested in local approach to Kadec¨CKlee property in symmetric function spaces. Moreover, normal structure and moduli of UKK, NUC, and UKK in Banach spaces have been deeply investigated by Satit Saejung and Ji Gao. The new kind of Banach spaces: semiUKK, semiNUC, modulus of semiUKK, and modulus of semiNUC are introduced in terms of this u-separation measure in their paper.(see ).

Since Banach space is more extensive than Hilbert space, it is quite difficult to describe its geometry structure. An effective method is to introduce new geometric properties for Banach space and to define an appropriate function, usually called a modulus or a geometric constant. Because the range of values of these geometric constants directly determines the existence of some geometric properties; therefore, many scholars are interested to calculate the modules and constants of some specific spaces. The starting point of the present paper is the observation that the UKK property can be localized. We call this new property named the local uniform Kadec-Klee property (LUKK), and we observe that it lies strictly between H and UKK properties. By using the same localized method, we localize the modulus PXε of UKK introduced by J. R. Partington and the modulus of noncompact convexity with Hausdorff measure ΔXε obtain two new moduli PXLε and ΔXLε, and we observe that ΔXLεPXLε.

2. Preliminaries

Before starting with our results, we need to recall some notions and a lemma in .

We say that a Banach space X has H property if for any xSX, xnX, limnxn=x and xnwx, then limnxnx=0.

We say that a Banach space X has KadecKlee (UKK) property if for every ε>0 there exists 0<δ<1 such that xnUX, xnwx and sepxnε then xBδ0 where Bδ0=x:xδ.

We say that a Banach space X has near uniform convexity property (NUC) if for every ε>0 there exists 0<δ=δε<1 such that xnUX with sepxnε then there is a N1 and scalars λ1,,λN0 with λn=1 such that λnxn1δ.

We say that a Banach space X has local near uniform convexity property (LNUC) if for every ε>0 and xUX there exists δ=δx,ε>0 such that for any sequence xnUX with sepxnε then (5)coxn,xBδ0.

i.e., (6)λ0y+1λ0x<δwhereycoxn,for some λ00,1 and ycoxn (see ).

Lemma 1.

Let X be a Banach Space, xnX and xnwx0. Then, (7)x0=n=1co¯xkkn.

3. Materials and Methods

In this paper, we take Kutazrova and Bor-Luh Lin’s approach to localize the UKK property and obtain the LUKK property. By using the same localized method, we localize the modulus PXε of UKK which introduced by Partington and the modulus of noncompact convexity with Hausdorff measure ΔXε and obtain two new moduli PXLε and ΔXLε; then, we study the relationship of size between PXLε and ΔXLε in Banach space by using the Corollary of Hahn-Banach Theorem and the weak lower semi continuity of norm.

4. Results and Discussion

We begin this section by formulating some definitions.

Definition 2.

A Banach space X is said to have local uniform KadecKlee property (LUKK), for every ε>0 and xSX there exists δ>0 such that if xn1, xnwx0, sepxnε and xnxε then (8)cox0,xBδ0whereBδ0=x:xδ,

i.e., (9)λ0x0+1λ0x<δ,for some λ00,1.

Definition 3.

Let X be a Banach Space. For every ε>0 and xSX(10)PXLε=inf1cox,x0:xn1,xnwx0,sepxnε,xnxε,is said to be the modulus of LUKK property or local Partington’s coefficient.

Definition 4.

For every ε>0 and xSX, we put (11)ΔXLε=inf1dθ,cox,coxn: xn1,αcox,coxnε,xnxε,where (12)αA=infε>0:Acanbecoveredbyfinitelymanysetswithdiameter<ε

ΔXLε is said to be the modulus of LNUC with Kuratowski measure.

Corollary 5.

If a Banach space X has LUKK property, then X has H property.

Proof.

We prove the contrapositive. Suppose X does not have H property, then there exists xnSX and x0SX such that although xnwx0 as n, we still have xn½x0 what means there exists ε0>0 and nk>n for any nN such that xnkxε0 this implies that sepxnkε0 holds.

X has LUKK property, for ε0>0 and x0SX mentioned above, there exists δ0,1 such that (13)cox0,x0Bδ0,

i.e., (14)x0Bδ0,this shows x0δ<1, a contradiction. Thus, the assumption does not hold.

The following conclusion follows from the definitions of LUKK and UKK.

Corollary 6.

If Banach space X has UKK property, then X has LUKK property.

It follows from previous Corollaries, we conclude the following Corollary.

Corollary 7.

For every Banach space X, the implication UKKLUKKH holds.

We are now ready to prove the main theorems of this paper.

Theorem 8.

If a Banach space X is LNUC, then X has LUKK property.

Proof.

Suppose that X does not have LUKK property. Then, there exists ε0>0, x0SX and ynUX with sepyn2ε0, ynwy0 with (15)cox0,y0B11/n0=.

Since X has LNUC property, for ε0>0 and x0SX mentioned above, there exists δ=δx0,ε0>0 such that cox0,coynB1δ0, which means for some λ00,1 and ycoyn, we have (16)λ0x0+1λ0y1δ,from (15) it follows that for any λ0,1 we have (17)λx0+1λy0>11n1,n.

Case (i) if y0=y, then (16) contradicts with (17).

Case (ii) if y0y, since ynwy0, then by Lemma 1, we have y0=n=1co¯ykkn, since ycoyn then for any xSX and λ0,1. Let (18)Yxδ=yBX: 1δ<λx+1λy<1,ψλ=λx+1λy,L=λ0,1: ψλ1δ,λ=infλ0,1: ψλ1δ,

It is obvious that ψ0ψ1=1, L , and ψλ=1δ. From (16), we get (19)yYx0δ.

For another facts, for λ0,1, let z0=λy+1λx0; then, we have (20)λz0+1λx0=λλy+1λλx0=ψλλ>1δ,

i.e., (21)λλy+1λλx0>1δ.

Thus, yYx0δ which contradicts with (19); therefore, the assumption is not true.

Theorem 9.

For every Banach space X, we have PXLεΔXLε.

Proof.

Fix ε0,1 and take an arbitrary sequence xnUX with sepxnε, xnwx0 as n, αxnε. For every xSX and some λ0,1, we let (22)yn=λx+1λxn,y0=λx+1λx0.

By the corollary of Hahn-Banach theorem, there exists f0SX such that f0yn=yn. Picking η>0 be small enough and considering the following set (23)Df0,η=yUX: f0yy0η.

It is obvious that the set Df0,η is closed, convex, and (24)dθ,Df0,ηy0η.

Since xnwx0, then (25)f0ynf0y0=y0.

Then, there exists n0N such that (26)f0yny0ηwhennn0,this implies that the set Yn0=yn:nn0 is a subset of Df0,η. Then, we get (27)dθ,coYn0y0η.

Since αYn0ε, then (28)αcoYn0εandΔXLε1dθ,coYn0.from (27) it follows that (29)1dθ,coYn01y0+η.

Consequently, (30)ΔXLε1dθ,coYn01y0+η,1y0ΔXLεη.

Thus, PXLεΔXLεη. Since η>0 is small enough, then we get PXLεΔXLε and the proof is complete.

Theorem 10.

For Banach space lp1<p<, we have PlpLε11ε/2p1/p.

Proof.

For every ε>0, xSlp and λ0,1, let xnUlp such that xnwx0, sepxnε and xnxε. Let (31)yn=λx+1λxn,y0=λx+1λx0,then ynwy0. By the weak lower semicontinuity of norm function, we get (32)y0inflimnyn.

Then, there exists a subsequence ynkyn and KN such that y0ynk for all k>K. Hence, (33)y0pminynip,ynjp:i,j>k=ynip+ynjpynipynjp22yniynjp2.

Since (34)yniynj=λxnixnjλε,then we get (35)y0p1yniynjp21λεp2,y01λεp21/p,1y011λpεp21/p,thus (36)PlpLε11λpεp21/pwhere1<p<,it follows that (37)PlpLε11εp21/pwhere1<p<.

Theorem 11.

If X is a reflexive Banach space, then for any ε0,2, we have ΔXLεPXLε/2.

Proof.

Fix ε0,2. Take xnUX with αxnε then αcoxnε; here, we let (38)yn=λx+1λxn,for every xSX and λ0,1. Thus, there exists zncoyn such that sepznε/2. By the reflexivity of X, there exists subsequence znkzncoyn and z0UX such that znkwz0. It is obvious that (39)coz0,x1PXLε2,1PXLε2supαz0+1αx:α0,1.

And consequently, (40)z01PXLε2andz0coznk,this implies that (41)dθ,cocox,xn=dθ,coyndθ,cozndθ,coznkz01PXLε2.

From (41), it follows that (42)1dθ,cocox,xnPXLε2.

Thus, ΔXLεPXLε/2 for any ε0,2, and the proof is complete.

5. Conclusions

In this paper, we introduce a new geometric property LUKK that lies between two classical geometric properties (UKK) and (H). Moreover, two new moduli PXLε and ΔXLε for (LUKK) and LNUC are introduced in Banach spaces; these new notions introduced in our paper play a very significant role in some recent trends of the geometric theory of Banach spaces. Furthermore, we give some further facts concerning the implication between LUKK and LNUC. Moreover, the relationship of size between the moduli PXLε and ΔXLε is discussed in Banach spaces, and PXLε is calculated in classical Banach spaces lp meanwhile. We believe that these introduced concepts will be useful and can be used to further solve the problems of accurately reflecting the shape and geometric structure of the unit sphere in Banach space.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there is no conflict of interest regarding the publication of this paper.

Acknowledgments

The authors are grateful to the referee for comments which improved the paper. This paper is supported by “The National Science Foundation of China” (11871181); “The Science Research Project of Inner Mongolia Autonomous Region” (NJZY18253); “The Science Research Project of Ordos Institution of Applied Technology” (KYYB2017014).

GoebelK.SekowskiT.The modulus of noncompact convexityAnnales Universitatis Mariae Curie-Sklodowska, sectio AA1984384148GoebelK.KirkW. A.Topics in metric fixed point theory1990CambridgeCambridge University PressBanasJ.GoebelK.Measures of noncompactness in Banach spaces1980New YorkLecture Notes in Pure and Application Mathematical, Marcel DekkerBanasJ.On modulus of noncompact convexity and its propertiesCanadian Mathematical Bulletin198730218619210.4153/CMB-1987-027-8BanasJ.Compactness conditions in the geometric theory of Banach spacesNonlinear Analysis1991167-866968210.1016/0362-546X(91)90174-Y2-s2.0-0001316402PrusS.Banach spaces with the uniform Opial propertyNonlinear Analysis199218869770410.1016/0362-546X(92)90165-B2-s2.0-0000198390PrusS.On the modulus of noncompact convexity of a Banach spaceArchiv der Mathematik199463544144810.1007/BF011966752-s2.0-0039873420PartingtonJ. R.On nearly uniformly convex Banach spacesMathematical Proceedings of the Cambridge Philosophical Society198393112712910.1017/S03050041000603942-s2.0-84976197113Saint RaymondJ.Kadec-Klee property and fixed pointsJournal of Functional Analysis201426685429543810.1016/j.jfa.2013.11.0232-s2.0-84897652171Saint RaymondJ.Dual Kadec-Klee property and fixed pointsJournal of Functional Analysis201727293825384410.1016/j.jfa.2016.12.0162-s2.0-85009247892CiesielskiM.KolwiczP.PłuciennikR.Local approach to Kadec-Klee properties in symmetric function spacesJournal of Mathematical Analysis and Applications2015426270072610.1016/j.jmaa.2015.01.0642-s2.0-84923602924SaejungS.GaoJ.Normal structure and moduli of UKK, NUC, and UKK in Banach spacesApplied Mathematics Letters201225101548155310.1016/j.aml.2012.01.0132-s2.0-84862999561HuffR.Banach spaces which are nearly uniformly convexRocky Mountain Journal of Mathematics198010474375010.1216/RMJ-1980-10-4-7432-s2.0-84880065824LinB.-L.ZhangW. Y.Some geometric properties related to uniformly convexity of Banach spaces, function spacesLecture Notes in Pure and Applied Mathematics1992136281291RolewiczS.On ∆-uniform convexity and drop propertyStudia Mathematica198787218119110.4064/sm-87-2-181-191KutzarovaD. N.LinB.-L.Locally k-nearly uniformly convex Banach spacesMathematica Balkanica19948203210