Local Uniform Kadec-Klee Property (LUKK) and Modulus of (LUKK)

Hindawi Journal of Function Spaces Volume 2020, Article ID 8191878, 5 pages https://doi.org/10.1155/2020/8191878

Research Article Local Uniform Kadec-Klee Property (LUKK) and Modulus of (LUKK)

Yunan Cui 1 and Xiaoxia Wang 2

1Department of Mathematics, Harbin University of Science and Technology, Harbin 150080, China 2Faculty of Mathematics and Computer Engineering, Ordos Institution of Applied Technology, Ordos 017000, China

Correspondence should be addressed to Xiaoxia Wang; [email protected]

Received 9 February 2020; Revised 19 May 2020; Accepted 22 June 2020; Published 7 July 2020

Academic Editor: Tomonari Suzuki

Copyright © 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 ðLUKKÞ and local near uniform convexity ðLNUCÞ, L uniformly Kadec-Klee (UKK), (H) are investigated in Banach space. Furthermore, the modulus PXðεÞ of (LUKK) and the L L L modulus ΔXðεÞ of ðLNUCÞ are introduced and the relationship of size between PXðεÞ and ΔXðεÞ is also investigated in Banach L p space. Finally, several formulas for PXðεÞ are calculated in classical Banach space l .

1. Introduction where

Let ðX, k·kÞ be a Banach space, X∗ be the dual space of X.By μ A = inf ε >0: A can be covered with a f inite number of UðXÞ and SðXÞ, we denote the closed unit ball and the unit ðÞ f sphere of Banach space X, respectively. By coðAÞ and co ðAÞ, sets of radii smaller than εg, we denote the convex hull and closed convex hull of the set w A, respectively. xn ⟶ x as n ⟶ ∞ denotes fxng is weakly αðÞA = inf fε >0: A can be covered by f initely many sets x n ⟶ ∞ converges to as . with diameter < εg: It is well known that the condition equivalent to near 2 uniform convexity (NUC) was independently formulated in ð Þ [1] (see also [2]). Recall that a notion of noncompactness with Hausdorﬀ measure and Kuratowski measure of a set b The functions ΔXðεÞ and ΔXðεÞ are called the moduli (see [3]). Let A be a bounded subset of X. Fix ε ∈ ½0, 1 for ﬀ A ⊂ U X μ A ≥ ε of noncompact convexity with Hausdor measure and all convex closed sets ð Þ with ð Þ , we put Kuratowski measure of X, respectively. It is clear that X is ðNUCÞ if and only if ΔXðεÞ >0 for every ε ∈ ð0, 1. Other properties of ΔXðεÞ were investigated in [4–7]. ΔXðÞε = inffg 1 − dðÞθ, A : A ⊂ UXðÞ, A = coðÞ A , μðÞA ≥ ε , Recall also that a function b ΔXðÞε = inffg 1 − dðÞθ, A : A ⊂ UXðÞ, A = coðÞ A , αðÞA ≥ ε , ð1Þ ε1ðÞX = sup fgε : ΔXðÞε =0 : ð3Þ 2 Journal of Function Spaces

J. Banas’ proved that if ε1ðXÞ <1, then X is reflexive and and scalars λ1, ⋯, λN ≥ 0 with ∑ λn =1such that ∥∑ λnxn∥≤ has normal structure (see [3]). 1 − δ. The modulus of ðUKKÞ was introduced in [8] by J. R. We say that a Banach space X has local near uniform Partington that is PX : ½0, 1 ⟶ ½0, 1 convexity property (LNUC) if for every ε >0and x ∈ UðXÞ there exists δ = δðx, εÞ >0such that for any sequence fxng ⊂ U X sep x ≥ ε PXðÞε = inff 1−∥x∥ : ∃x ∈ UXðÞand ð Þ with ð nÞ then w ð4Þ fgxn ⊂ UXðÞs:t:xn ⟶ x and sepðÞ xn ≥ εg, coðÞfg xn, x ∩ BδðÞ0 ≠∅: ð5Þ where sepðxnÞ = inf f∥xn − xm∥ : m ≠ ng. He proved that X i.e., has ðUKKÞ property if and only if PXðεÞ >0 whenever ε ∈ 0, 1 P ½ . He also proved that the function X is nonde- ∥λ y +1− λ x∥ < δwhere y ∈ co x , 6 creasing on [0,1]. 0 ðÞ0 ðÞfgn ð Þ There are many recent papers concerning the Kadec-Klee property, such as Kadec¨CKlee property and fixed points and for some λ0 ∈ ð0, 1Þ and y ∈ coðxnÞ (see [16]). fi Dual Kadec-Klee property and xed points studied by Jean w Saint Raymond (see [9, 10]). In a recent paper [11], Maciej Lemma 1. Let X be a Banach Space, fxng ⊂ X and xn ⟶ x0. Ciesielski, Paweł Kolwicz, and Ryszard Płuciennik were Then, interested in local approach to Kadec¨CKlee property in ∞ ÀÁ symmetric function spaces. Moreover, normal structure fgx0 = ∩ n=1co fg xk k≥n : ð7Þ ∗ and moduli of ðUKKÞ, ðNUCÞ, and ðUKKÞ in Banach spaces have been deeply investigated by Satit Saejung 3. Materials and Methods and Ji Gao. The new kind of Banach spaces: ðsemi − UKKÞ, semi − NUC semi − UKK ð Þ, modulus of ð Þ, and modulus of In this paper, we take Kutazrova and Bor-Luh Lin’s approach semi − NUC ð Þ are introduced in terms of this u-separation to localize the ðUKKÞ property and obtain the ðLUKKÞ measure in their paper.(see [12]). property. By using the same localized method, we localize Since Banach space is more extensive than Hilbert space, the modulus ðPXðεÞÞ of ðUKKÞ which introduced by Par- ffi ff it is quite di cult to describe its geometry structure. An e ec- tington and the modulus of noncompact convexity with tive method is to introduce new geometric properties for Hausdorff measure ðΔXðεÞÞ and obtain two new moduli Banach space and to define an appropriate function, usually L L PXðεÞ and ΔXðεÞ; then, we study the relationship of size called a modulus or a geometric constant. Because the range L L between P ðεÞ and Δ ðεÞ in Banach space by using the of values of these geometric constants directly determines the X X Corollary of Hahn-Banach Theorem and the weak lower existence of some geometric properties; therefore, many semi continuity of norm. scholars are interested to calculate the modules and constants fi of some speci c spaces. The starting point of the present 4. Results and Discussion paper is the observation that the ðUKKÞ property can be localized. We call this new property named the local uniform We begin this section by formulating some definitions. Kadec-Klee property (LUKK), and we observe that it lies H UKK strictly between ð Þ and ð Þ properties. By using the Definition 2. A Banach space X is said to have local uniform P ε same localized method, we localize the modulus ð Xð ÞÞ of Kadec− Klee property (LUKK), for every ε >0and x ∈ SðXÞ UKK w ð Þ introduced by J. R. Partington and the modulus of there exists δ >0such that if ∥xn∥≤1, xn ⟶ x0, sepðxnÞ ≥ ε noncompact convexity with Hausdorff measure ðΔXðεÞÞ ∥x − x∥≥ε L L and n then obtain two new moduli PXðεÞ and ΔXðεÞ, and we observe that ΔL ε ≥ PL ε Xð Þ Xð Þ. coðÞ x0, x ∩ BδðÞ0 ≠∅where BδðÞ0 = fgx : ∥x∥≤δ , ð8Þ

2. Preliminaries i.e., Before starting with our results, we need to recall some ∥λ x +1− λ x∥ < δ, 9 notions and a lemma in [13–15]. 0 0 ðÞ0 ð Þ We say that a Banach space X has ðHÞ property if for any w x ∈ SðXÞ, fxng ⊂ X, limn⟶∞∥xn∥ = ∥x∥ and xn ⟶ x, then for some λ0 ∈ ð0, 1Þ. limn→∞∥xn − x∥ =0. We say that a Banach space X has Kadec− Klee (UKK) Definition 3. Let X be a Banach Space. For every ε >0 and property if for every ε >0 there exists 0<δ <1 such that x ∈ SðXÞ x ⊂ U X x ⟶w x sep x ≥ ε x ∈ B 0 f ng ð Þ, n and ð nÞ then δð Þ n B 0 = x : ∥x∥≤δ w where δð Þ f g. PL ε = inf 1−∥co x, x ∥ : ∣x ∥≤1, x ⟶ x , X XðÞ ðÞ0 n n 0 We say that a Banach space has near uniform convexity o ð10Þ property (NUC) if for every ε >0there exists 0<δ = δðεÞ <1 sepðÞ xn ≥ ε,∥xn − x∥≥ε , such that fxng ⊂ UðXÞ with sepðxnÞ ≥ ε then there is a N ≥ 1 Journal of Function Spaces 3 is said to be the modulus of ðLUKKÞ property or local Proof. Suppose that X does not have ðLUKKÞ property. Partington’s coefficient. Then, there exists ε0 >0, x0 ∈ SðXÞ and fyng ⊂ UðXÞ with se w pðynÞ ≥ 2ε0, yn ⟶ y0 with Definition 4. For every ε >0and x ∈ SðXÞ, we put co x , y ∩ B 0 = ∅: 15 L ðÞfg0 0 1−1/nðÞ ð Þ ΔXðÞε = inff 1 − dðÞθ, coðÞ x, coðÞ xn : ∥xn∥≤1, ð11Þ αðÞcoðÞ x, coðÞ xn ≥ ε,∥xn − x∥≥εg, Since X has ðLNUCÞ property, for ε0 >0 and x0 ∈ SðXÞ mentioned above, there exists δ = δðx0, ε0Þ >0 such that where coðx0, coðfynÞgÞ ∩ B1−δð0Þ ≠∅, which means for some λ0 ∈ ′ ½0, 1 and y ∈ coðfyngÞ, we have αðÞA = inf fε >0: A can be covered by f initely many sets with diameter < εg ′ λ0x0 +1ðÞ− λ0 y ≤ 1 − δ, ð16Þ ð12Þ

L ΔXðεÞ is said to be the modulus of ðLNUCÞ with Kura- from (15) it follows that for any λ ∈ ½0, 1 we have towski measure. 1 Corollary 5. X LUKK λx +1− λ y >1− ⟶ 1, n ⟶ ∞: 17 If a Banach space has ð Þ property, then kk0 ðÞ0 n ð Þ X has ðHÞ property.

X Proof. We prove the contrapositive. Suppose does not y = y′ H x ⊂ S X Case (i) if 0 , then (16) contradicts with (17). have ð Þ property, then there exists f ng ð Þ and ′ w w Case (ii) if y0 ≠ y , since yn ⟶ y0, then by Lemma 1, we x0 ∈ SðXÞ such that although xn ⟶ x0 as n ⟶ ∞,we y = ∩ ∞ co y y′ ∈ co y still have xn½x0 what means there exists ε0 >0 and nk > n have f 0g n=1 ðf kgk≥nÞ, since ðf ngÞ then for n ∈ N ∥x − x∥≥ε any x ∈ SðXÞ and λ ∈ ½0, 1. Let for any such that nk 0 this implies that sep x ≥ ε ð nk Þ 0 holds. YxðÞδ = fgy ∈ BXðÞ:1− δ < kkλx +1ðÞ− λ y <1 , X LUKK ε >0 x ∈ S X has ð Þ property, for 0 and 0 ð Þ men- ψλ = λx +1− λ y , tioned above, there exists δ ∈ ð0, 1Þ such that ðÞ kkðÞ ð18Þ L = fgλ ∈ ½0, 1 : ψλðÞ≤ 1 − δ , coðÞ x0, x0 ∩ BδðÞ0 ≠∅, ð13Þ λ′ = inf fgλ ∈ ½0, 1 : ψλðÞ≤ 1 − δ , i.e., It is obvious that ψð0Þ ≤ ψð1Þ =1, L ≠∅ , and ψðλ′Þ = 1 − δ x0 ∩ BδðÞ0 ≠∅, ð14Þ . From (16), we get ′ y ∈Yx δ : ð19Þ this shows ∥x0∥≤δ <1, a contradiction. Thus, the assumption 0 ðÞ does not hold. ′ ′ ′ The following conclusion follows from the deﬁnitions of For another facts, for λ ∈ ½0, 1, let z0 = λ y + ð1 − λ Þx0; ðLUKKÞ and ðUKKÞ. then, we have Corollary 6. If Banach space X has ðUKKÞ property, then X ∥λz +1− λ x ∥ = ∥λλ′y′ +1− λλ′ x ∥ LUKK 0 ðÞ0 0 has ð Þ property. ð20Þ = ψλλ′ >1− δ, It follows from previous Corollaries, we conclude the following Corollary. i.e., Corollary 7. For every Banach space X, the implication UKK ⟹ LUKK ⟹ H ′ ′ ′ ð Þ ð Þ ð Þ holds. ∥λλ y +1− λλ x0∥ >1− δ: ð21Þ

We are now ready to prove the main theorems of this y′ ∈ Y δ paper. Thus, x0 ð Þ which contradicts with (19); therefore, the assumption is not true. Theorem 8. If a Banach space X is ðLNUCÞ, then X has L L ðLUKKÞ property. Theorem 9. For every Banach space X, we have PXðεÞ ≥ ΔXðεÞ. 4 Journal of Function Spaces

w Proof. Fix ε ∈ ½0, 1 and take an arbitrary sequence fxng ⊂ then yn ⟶ y0. By the weak lower semicontinuity of w UðXÞ with sepðxnÞ ≥ ε, xn ⟶ x0 as n ⟶ ∞, αðxnÞ ≥ ε. norm function, we get For every x ∈ SðXÞ and some λ ∈ ð0, 1Þ, we let ∥y ∥≤inf lim∥y ∥: ð32Þ 0 n n yn = λx +1ðÞ− λ xn, ð22Þ y = λx +1− λ x : y ⊆ y K ∈ N 0 ðÞ0 Then, there exists a subsequence f nk g f ng and ∥y ∥≤∥y ∥ k > K such that 0 nk for all . Hence, By the corollary of Hahn-Banach theorem, there exists no ∗ p p p f0 ∈ SðX Þ such that f0ðynÞ = ∥yn∥. Picking η >0 be small ∥y0∥ ≤ min ∥yn ∥ ,∥yn ∥ : i, j > k enough and considering the following set i j p p p p ∣∥yn ∥ +∥yn ∥ ∣−∣∥yn ∥ −∥yn ∥ ∣ = i j i j ð33Þ DfðÞ0, η = fgy ∈ UXðÞ: f0ðÞy ≥∥y0∥−η : ð23Þ 2 p 2−∥yn − yn ∥ It is obvious that the set Dðf , ηÞ is closed, convex, and ≤ i j : 0 2 d θ, Df, η ≥∥y ∥−η: ð24Þ ðÞðÞ0 0 Since w Since xn ⟶ x0, then ∥y − y ∥ = λ∥x − x ∥≥λε, 34 ni nj ni nj ð Þ

f0ðÞyn ⟶ f0ðÞy0 = ∥y0∥: ð25Þ then we get

Then, there exists n0 ∈ N such that p p ∥yn − yn ∥ λε ∥y ∥p ≤ 1 − i j ≤ 1 − ðÞ, f y ≥∥y ∥−η when n ≥ n , 26 0 2 2 0ðÞn 0 0 ð Þ λε p 1/p ∥y ∥≤ 1 − ðÞ , 35 Y = y : n ≥ n 0 ð Þ this implies that the set n0 f n 0g is a subset of 2 D f , η ð 0 Þ. Then, we get λpεp 1/p À ÀÁ 1−∥y0∥≥1 − 1 − , d θ, co Y ≥∥y ∥−η: 27 2 n0 0 ð Þ thus Since αðYn Þ ≥ ε, then 0 ÀÁÀÁ ÀÁÀÁ λpεp 1/p L L α co Yn ≥ εand Δ ε ≤ 1 − d θ, co Yn : ð28Þ Plp ðÞε ≥ 1 − 1 − where 1 < p < ∞, ð36Þ 0 XðÞ 0 2 from (27) it follows that it follows that ÀÁÀÁ 1 − d θ, co Yn ≤ 1−∥y0∥+η: ð29Þ p 1/p 0 L ε P p ε ≥ 1 − 1 − where 1 < p < ∞: 37 l ðÞ 2 ð Þ Consequently, ÀÁÀÁ L Theorem 11. If X is a reﬂexive Banach space, then for any ΔXðÞε ≤ 1 − d θ, co Yn ≤ 1−∥y0∥+η, L L 0 ε ∈ ½0, 2, we have Δ ðεÞ ≥ P ðε/2Þ. ð30Þ X X L 1−∥y0∥≥ΔXðÞε − η: Proof. Fix ε ∈ ½0, 2. Take fxng ⊂ UðXÞ with αðxnÞ ≥ ε then α co x ≥ ε L L ð ð nÞÞ ; here, we let Thus, PXðεÞ ≥ ΔXðεÞ − η. Since η >0 is small enough, PL ε ≥ ΔL ε then we get Xð Þ Xð Þ and the proof is complete. yn = λx +1ðÞ− λ xn, ð38Þ

p Theorem 10. For Banach space l ð1 < p<∞Þ, we have for every x ∈ SðXÞ and λ ∈ ½0, 1. Thus, there exists fzng ⊂ L p 1/p co y sep z ≥ ε/2 ﬂ X Plp ðεÞ ≥ 1 − ð1 − ðε/2Þ Þ . ð nÞ such that ð nÞ . By the re exivity of , there z ⊂ z ⊂ co y z ∈ U X exists subsequence f nk g f ng ð nÞ and 0 ð Þ ε >0 x ∈ S lp λ ∈ 0, 1 x ⊂ z ⟶w z Proof. For every , ð Þ and ð Þ, let f ng such that nk 0. It is obvious that p w Uðl Þ such that xn ⟶ x0, sepðxnÞ ≥ ε and ∥xn − x∥≥ε. Let L ε ∥coðÞ z0, x ∥≤1 − PX , y = λx +1− λ x , 2 n ðÞn ð39Þ ð31Þ L ε 1 − PX ≥ sup fg∥αz0 +1ðÞ− α x∥ : α ∈ ðÞ0, 1 : y0 = λx +1ðÞ− λ x0, 2 Journal of Function Spaces 5

And consequently, [2] K. Goebel and W. A. Kirk, Topics in metric fixed point theory, Cambridge University Press, Cambridge, 1990. ÀÁ L ε ∥z0∥≤1 − PX and z0 ∈ co zn , ð40Þ [3] J. Banas and K. Goebel, Measures of noncompactness in Banach 2 k spaces, Lecture Notes in Pure and Application Mathematical, Marcel Dekker, New York, 1980. this implies that [4] J. Banas, “On modulus of noncompact convexity and its properties,” Canadian Mathematical Bulletin, vol. 30, no. 2, d θ, co co x, x = d θ, co y ≤ d θ, co z ðÞðÞðÞn ðÞðÞn ðÞðÞn pp. 186–192, 1987. ÀÁÀÁ ε L [5] J. Banas, “Compactness conditions in the geometric theory of ≤ d θ, co zn ≤∥z0∥≤1 − PX : k 2 Banach spaces,” Nonlinear Analysis, vol. 16, no. 7-8, pp. 669– ð41Þ 682, 1991. [6] S. Prus, “Banach spaces with the uniform Opial property,” From (41), it follows that Nonlinear Analysis, vol. 18, no. 8, pp. 697–704, 1992. “ ε [7] S. Prus, On the modulus of noncompact convexity of a Banach L space,” Archiv der Mathematik, vol. 63, no. 5, pp. 441–448, 1 − dðÞθ, coðÞ coðÞ x, xn ≥ PX : ð42Þ 2 1994. “ ” L L [8] J. R. Partington, On nearly uniformly convex Banach spaces, Thus, ΔXðεÞ ≥ PXðε/2Þ for any ε ∈ ½0, 2, and the proof Mathematical Proceedings of the Cambridge Philosophical Soci- is complete. ety, vol. 93, no. 1, pp. 127–129, 1983. [9] J. Saint Raymond, “Kadec-Klee property and fixed points,” 5. Conclusions Journal of Functional Analysis, vol. 266, no. 8, pp. 5429– 5438, 2014. LUKK In this paper, we introduce a new geometric property ð Þ [10] J. Saint Raymond, “Dual Kadec-Klee property and fixed that lies between two classical geometric properties (UKK) ” L L points, Journal of Functional Analysis, vol. 272, no. 9, and (H). Moreover, two new moduli PXðεÞ and ΔXðεÞ for pp. 3825–3844, 2017. LUKK LNUC ( )andð Þ are introduced in Banach spaces; these [11] M. Ciesielski, P. Kolwicz, and R. Płuciennik, “Local approach fi new notions introduced in our paper play a very signi cant to Kadec-Klee properties in symmetric function spaces,” Jour- role in some recent trends of the geometric theory of Banach nal of Mathematical Analysis and Applications, vol. 426, no. 2, spaces. Furthermore, we give some further facts concerning pp. 700–726, 2015. the implication between ðLUKKÞ and ðLNUCÞ. Moreover, [12] S. Saejung and J. Gao, “Normal structure and moduli of UKK, L L the relationship of size between the moduli PXðεÞ and ΔXðεÞ NUC, and UKK∗ in Banach spaces,” Applied Mathematics Let- L – is discussed in Banach spaces, and PXðεÞ is calculated in clas- ters, vol. 25, no. 10, pp. 1548 1553, 2012. sical Banach spaces lp meanwhile. We believe that these intro- [13] R. Huff, “Banach spaces which are nearly uniformly convex,” duced concepts will be useful and can be used to further solve Rocky Mountain Journal of Mathematics, vol. 10, no. 4, – the problems of accurately reflecting the shape and geometric pp. 743 750, 1980. structure of the unit sphere in Banach space. [14] B.-L. Lin and W. Y. Zhang, “Some geometric properties related to uniformly convexity of Banach spaces, function spaces,” Data Availability Lecture Notes in Pure and Applied Mathematics, vol. 136, pp. 281–291, 1992. No data were used to support this study. [15] S. Rolewicz, “On Δ-uniform convexity and drop property,” Studia Mathematica, vol. 87, no. 2, pp. 181–191, 1987. Conflicts of Interest [16] D. N. Kutzarova and B.-L. Lin, “Locally k-nearly uniformly convex Banach spaces,” Mathematica Balkanica, vol. 8, The authors declare that there is no conflict of interest pp. 203–210, 1994. 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). References

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