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volume 7, issue 5, article 161, 2006.

Received 21 September, 2006;

accepted 13 October, 2006.

Communicated by:S.S. Dragomir

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Journal of Inequalities in Pure and Applied Mathematics

SOME ESTIMATES ON THE WEAKLY CONVERGENT SEQUENCE COEFFICIENT IN BANACH SPACES

FENGHUI WANG AND HUANHUAN CUI

Department of Mathematics Luoyang Normal University Luoyang 471022, China.

EMail:[email protected]

c

2000Victoria University ISSN (electronic): 1443-5756 241-06

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Some Estimates on the Weakly Convergent Sequence Coefficient in Banach Spaces Fenghui Wang and Huanhuan Cui

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J. Ineq. Pure and Appl. Math. 7(5) Art. 161, 2006

http://jipam.vu.edu.au

Abstract

In this paper, we study the weakly convergent sequence coefficient and obtain its estimates for some parameters in Banach spaces, which give some sufficient conditions for a Banach space to have normal structure.

2000 Mathematics Subject Classification:46B20.

Key words: Weakly convergent sequence coefficient; James constant; Von Neumann-Jordan constant; Modulus of smoothness.

The authors would like to thank the referee for his helpful suggestions.

1. Introduction

A Banach space X said to have (weak) normal structure provided for every (weakly compact) closed bounded convex subset C ofX withdiam(C) > 0, contains a nondiametral point, i.e., there existsx₀ ∈Csuch thatsup{kx−x₀k: x∈C}<diam(C).It is clear that normal structure and weak normal structure coincides whenXis reflexive.

The weakly convergent sequence coefficientW CS(X), a measure of weak normal structure, was introduced by Bynum in [3] as the following.

Definition 1.1. The weakly convergent sequence coefficient ofXis defined by (1.1) W CS(X)

= inf

diama({xn})

r_a({x_n}) :{x_n}is a weakly convergent sequence

, where diama({xn}) = lim sup_k→∞{kxn−xmk : n, m≥ k}is the asymptotic diameter of{x_n}andr_a({x_n}) = inf{lim sup_n→∞kx_n−yk:y ∈co({x¯ _n})is the asymptotic radius of{x_n}.

One of the equivalent forms ofW CS(X)is W CS(X) = inf

n,m,n6=mlim kx_n−x_mk:x_n→^w 0,kx_nk= 1

and lim

n,m,n6=mkx_n−x_mkexists

.

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Obviously,1≤W CS(X)≤2,and it is well known thatW CS(X)>1implies thatXhas a weak normal structure.

The constantR(a, X), which is a generalized García-Falset coefficient [10], was introduced by Domínguez [7] as: For a given real numbera >0,

(1.2) R(a, X) = supn

lim inf

n→∞ kx+x_nko ,

where the supremum is taken over allx∈ Xwithkxk ≤aand all weakly null sequences{x_n} ⊆B_X such that

(1.3) lim

n,m,n6=mkx_n−x_mk ≤1.

We shall assume throughout this paper that BX and SX to denote the unit ball and unit sphere ofX, respectively. x_n→^w xstands for weak convergence of sequence{x_n}inXto a pointxinX.

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2. Main Results

The James constant, or the nonsquare constant, was introduced by Gao and Lau in [8] as

J(X) = sup{kx+yk ∧ kx−yk:x, y ∈S_X}

= sup{kx+yk ∧ kx−yk:x, y ∈B_X}.

A relation between the constant R(1, X)and the James constant J(X)can be found in [6,12]:

R(1, X)≤J(X).

We now state an inequality between the James constantJ(X)and the weakly convergent sequence coefficientW CS(X).

Theorem 2.1. LetX be a Banach space with the James constantJ(X).Then

(2.1) W CS(X)≥ J(X) + 1

(J(X))² .

Proof. IfJ(X) = 2, it suffices to note thatW CS(X)≥1.Thus our estimate is a trivial one.

If J(X) < 2, thenX is reflexive. Let {xn} be a weakly null sequence in S_X. Assume thatd = limn,m,n6=mkx_n−x_mkexists and consider a normalized functional sequence {x^∗_n}such thatx^∗_n(x_n) = 1. Note that the reflexivity ofX guarantees, by passing through the subsequence, that there existsx^∗ ∈X^∗ such thatx^∗_n →^w x^∗.Let0< <1and chooseN large enough so that|x^∗(x_N)|< /2 and

d− <kxN −xmk< d+

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for allm > N.Note that

n,m,n6=mlim

x_n−x_m d+

≤1 and

x_N d+

≤1.

Then by the definition ofR(1, X), we can choose M > N large enough such that

x_N +x_M d+

≤R(1, X) +≤J(X) +, |(x^∗_M −x^∗)(x_N)|< /2, and|x^∗_N(x_M)|< .Hence

|x^∗_M(xN)| ≤ |(x^∗_M −x^∗)(xN))|+|x^∗(xN)|< . Putα=J(X),

x= x_N −x_M

d+ , and y= x_N +x_M (d+)(α+). It follows thatkxk ≤1,kyk ≤1,and also that

kx+yk= 1 (d+)(α+)

(α+ 1 +)x_N −(α−1 +)x_M

≥ 1

(d+)(α+) (α+ 1 +)x^∗_N(x_N)−(α−1 +)x^∗_N(x_M)

≥ α+ 1− (d+)(α+),

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kx−yk= 1 (d+)(α+)

(α+ 1 +)xM −(α−1 +)xN

≥ 1

(d+)(α+) (α+ 1 +)x^∗_M(x_M)−(α−1 +)x^∗_M(x_N)

≥ α+ 1− (d+)(α+).

Thus, from the definition of the James constant,

J(X)≥ α+ 1−

(d+)(α+) = J(X) + 1− (d+)(J(X) +). Letting→0,we get

d ≥ J(X) + 1 (J(X))² .

Since the sequence{x_n}is arbitrary, we get the inequality (2.1).

As an application of Theorem2.1, we can obtain a sufficient condition for X to have normal structure in terms of the James constant.

Corollary 2.2 ([4, Theorem 2.1]). Let X be a Banach space with J(X) <

(1 +√

5)/2.ThenX has normal structure.

The modulus of smoothness [14] ofXis the functionρ_X(τ)defined by ρ_X(τ) = sup

kx+τ yk+kx−τ yk

2 −1 :x, y ∈S_X

.

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It is readily seen that for anyx, y ∈S_X,

kx±yk ≤ kx±τ yk+ (1−τ) (0< τ ≤1), which implies thatJ(X)≤ρ_X(τ) + 2−τ.

In [2], Baronti et al. introduced a constantA₂(X), which is defined by A₂(X) = ρ_X(1) + 1 = sup

kx+yk+kx−yk

2 :x, y ∈S_X

. It is worth noting thatA₂(X) = A₂(X^∗).

We now state an inequality between the modulus of smoothnessρ_X(τ)and the weakly convergent sequence coefficientW CS(X).

Theorem 2.3. LetXbe a Banach space with the modulus of smoothnessρX(τ).

Then for any0< τ ≤1,

(2.2) W CS(X)≥ ρ_X(τ) + 2

(ρ_X(τ) + 1)(ρ_X(τ)−τ + 2). Proof. Let0< τ ≤1.IfρX(τ) =τ, it suffices to note that

ρ_X(τ) + 2

(ρ_X(τ) + 1)(ρ_X(τ)−τ+ 2) = τ + 2

2(τ + 1) ≤1.

Thus our estimate is a trivial one.

IfρX(τ)< τ, thenXis reflexive. Let{xn}be a weakly null sequence inSX. Assume that d= limn, m, n6=mkx_n−x_mkexists and consider a normalized

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functional sequence {x^∗_n} such that x^∗_n(x_n) = 1. Note that the reflexivity of X guarantees that there exists x^∗ ∈ X^∗ such that x^∗_n →^w x^∗. Let > 0 and x_M, x_N, xandyselected as in Theorem2.1. Similarly, we get

kx±τ yk ≥ α(τ) +τ− (d+)(α(τ) +),

whereα(τ) = ρ_X(τ) + 2−τ.Then by the definition ofρ_X(τ), we obtain ρ_X(τ)≥ α(τ) +τ −

(d+)(α(τ) +)−1.

Letting→0,

ρ_X(τ) + 1≥ α(τ) +τ

dα(τ) = ρ_X(τ) + 2 d(ρ_X(τ)−τ + 2), which gives that

d≥ ρX(τ) + 2

(ρ_X(τ) + 1)(ρ_X(τ)−τ + 2).

Since the sequence{x_n}is arbitrary, we get the inequality (2.2).

It is known that ifρ_X(τ)< τ /2for someτ >0,thenXhas normal structure (see [9]). Using Theorem2.3, We can improve this result in the following form:

Corollary 2.4. LetX be a Banach space with ρ_X(τ)< τ −2 +√

τ² + 4 2

for some τ ∈ (0,1].Then X has normal structure. In particular, if A₂(X) <

(1 +√

5)/2,thenX and its dualX^∗ have normal structure.

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In connection with a famous work of Jordan-von Neumann concerning inner products, the Jordan-von Neumann constant CNJ(X)of X was introduced by Clarkson (cf. [1,11]) as

CNJ(X) = sup

kx+yk²+kx−yk²

2(kxk²+kyk²) :x, y ∈Xand not both zero

. A relationship betweenJ(X)andC_NJ(X)is found in ([11] Theorem 3):J(X)≤ p2CNJ(X).

In [5], Dhompongsa et al. proved the following inequality (2.3). We now re- state this inequality without the ultra product technique and the factC_NJ(X) = C_NJ(X^∗).

Theorem 2.5 ([5] Theorem 3.8). LetXbe a Banach space with the von Neumann- Jordan constantC_NJ(X).Then

(2.3) (W CS(X))² ≥ 2C_NJ(X) + 1

2(C_NJ(X))² .

Proof. If C_NJ(X) = 2, it suffices to note that W CS(X) ≥ 1. Thus our estimates is a trivial one.

IfC_NJ(X)< 2, thenX is reflexive. Let{x_n}be a weakly null sequence in SX. Assume thatd = limn,m,n6=mkxn−xmkexists and consider a normalized functional sequence {x^∗_n}such thatx^∗_n(x_n) = 1. Note that the reflexivity ofX gurantees that there esistsx^∗ ∈X^∗such thatx^∗_n→^w x^∗.Let >0and chooseN large enough so that|x^∗(xN)|< /2and

d− <kxN −xmk< d+

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for allm > N.Note that

n,m,n6=mlim

xn−xm

d+

≤1 and

xN

d+

≤1.

Then by the definition ofR(1, X), we can choose M > N large enough such that

x_N −x_M d+

≤R(1, X) +≤p

2CNJ(X) +, |(x^∗_M −x^∗)(xN)|< /2, and|x^∗_N(x_M)|< .Hence

|x^∗_M(x_N)|<|(x^∗_M −x^∗)(x_N))|+|x^∗(x_N)|< . Putα =p

2C_NJ(X), x =α²(x_N −x_M), y =x_N +x_M.It follows thatkxk ≤ α²(d+),kyk ≤(α+)(d+), and also that

kx+yk=k(α²+ 1)x_N −(α²−1)x_Mk

≥(α²+ 1)x^∗_N(x_N)−(α²−1)x^∗_N(x_M)

≥α²+ 1−3,

kx−yk=k(α²+ 1)x_M −(α²−1)x_Nk

≥(α²+ 1)x^∗_M(x_M)−(α²−1)x^∗_M(x_N)

≥α²+ 1−3.

Thus, from the definition of the von Neumann-Jordan constant, CNJ(X)≥ 2(α²+ 1−3)²

2(α⁴(d+)²+ (α+)²(d+)²)

= 1

(d+)² · (α²+ 1−3)² α⁴+ (α+)² .

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Sinceis arbitrary andα =p

2C_NJ(X), we get

CNJ(X)≥ 1 d²

1 + 1

α²

= 2C_NJ(X) + 1 d²·2C_NJ(X), which implies that

d² ≥ 2C_NJ(X) + 1 2(C_NJ(X))² .

Since the sequence{xn}is arbitrary, we obtain the inequality (2.3).

Using Theorem2.5, we can get a sufficient condition forX to have normal structure in terms of the von Neumann-Jordan constant.

Corollary 2.6 ([6, Theorem 3.16], [13, Theorem 2]). LetXbe a Banach space withC_NJ(X)<(1 +√

3)/2.ThenX and its dualX^∗ have normal structure.

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References

[1] J.A. CLARKSON, The von Neumann-Jordan constant for the Lebesgue space, Ann. of Math., 38 (1937) 114–115.

[2] M. BARONTI, E. CASINI AND P.L. PAPINI, Triangles inscribed in a semicircle, in Minkowski planes, and in normed spaces, J. Math. Anal.

Appl., 252 (2000), 124–146.

[3] W.L. BYNUM, Normal structure coefficients for Banach spaces, Pacific.

J. Math., 86 (1980), 427–436.

[4] S. DHOMPONGSA, A. KAEWKHAO AND S. TASENA, On a general- ized James constant, J. Math. Anal. Appl., 285 (2003), 419–435.

[5] S. DHOMPONGSA, T. DOMINGUEZ BENAVIDES, A.

KAEWCHAROEN, A. KAEWKHAO AND B. PANYANAK, The

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[6] S. DHOMPONGSAANDA. KAEWKHAO, A note on properties that im- ply the fixed point property, Abstr. Appl. Anal., 2006 (2006), Article ID 34959.

[7] T. DOMINGUEZ BENAVIDES, A geometrical coefficient impling the fixed point prpperty and stability results, Houston J. Math., 22 (1996) 835–

849.

[8] J. GAO AND K. S. LAU, On two classes Banach spaces with uniform normal structure, Studia Math., 99 (1991), 41–56.

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[9] J. GAO, Normal structure and smoothness in Banach spaces, J. Nonlinear Functional Anal. Appl., 10 (2005), 103–115.

[10] J. GARCIA-FALSET, Stability and fixed point for nonexpansive map- pings, Houston J. Math., 20 (1994) 495–506.

[11] M. KATO, L. MALIGRANDA AND Y. TAKAHASHI, On James and Jordan-von Neumann constants and normal structure coefficient of Banach spaces, Studia Math., 114 (2001), 275–295.

[12] E.M. MAZCUÑÁN-NAVARRO, Geometry of Banach spaces in metric fixed point theory, Ph.D. thesis, University of Valencia, Valencia, 2003.

[13] S. SAEJUNG, On James and von Neumann-Jordan constants and suffi- cient conditions for the fixed point property, J. Math. Anal. Appl., 323 (2006), 1018–1024.

[14] J. LINDENSTRAUSS, On the modulus of smoothness and divergent series in Banach spaces, Michigan Math. J., 10 (1963), 241–252.