ON k-NEARLY UNIFORM CONVEX PROPERTY IN GENERALIZED CESÀRO SEQUENCE SPACES

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ON k-NEARLY UNIFORM CONVEX PROPERTY IN GENERALIZED CESÀRO SEQUENCE SPACES

WINATE SANHAN and SUTHEP SUANTAI Received 19 January 2003

We define a generalized Cesàro sequence space ces(p), wherep=(pk)is a bounded sequence of positive real numbers, and consider it equipped with the Luxemburg norm. The main purpose of this paper is to show that ces(p)isk-nearly uniform convex (k-NUC) fork≥2 when lim_n→∞infpn>1. Moreover, we also obtain that the Cesàro sequence space ces_p(where 1< p <∞)isk-NUC,kR, NUC, and has a drop property.

2000 Mathematics Subject Classification: 46B20, 46B45.

1. Introduction. Let(X,·)be a real Banach space and letB(X)andS(X) be the closed unit ball and the unit sphere ofX, respectively. For any sub- setAofX, we denote by conv(A)(resp., conv(A)) the convex hull (resp., the closed convex hull) of Clarkson [1] who introduced the concept of uniform convexity, and it is known that uniform convexity implies reflexivity of Banach spaces. There are different uniform geometric properties which have been de- fined between the uniform convexity and the reflexivity of Banach spaces. Huff [6] introduced the nearly uniform convexity of Banach spaces. He has proved that the class of nearly uniformly convexifiable spaces is strictly between su- perreflexive and reflexive Banach spaces.

A Banach spaceXis calleduniformly convex(UC) if for each >0, there is δ >0 such that forx,y∈S(X), the inequalityx−y> implies that

1

2(x+y)

<1−δ. (1.1)

For anyx∈B(X), thedropdetermined byxis the set D

x,B(X)

=conv

{x}∪B(X)

. (1.2)

Rolewicz [12], basing on Daneš drop theorem [4], introduced the notion of drop property for Banach spaces.

A Banach spaceXhas thedrop property(D) if for every closed setCdisjoint withB(X), there exists an elementx∈Csuch that

D

x,B(X)

∩C= {x}. (1.3)

A Banach spaceXis said to have theKadec-Klee property(orproperty(H)) if every weakly convergent sequence on the unit sphere is convergent in norm.

In [13], Rolewicz proved that if the Banach spaceXhas the drop property, thenXis reflexive. Montesinos [11] extended this result by showing thatXhas the drop property if and only ifXis reflexive and has the property (H).

Recall that a sequence{xn} ⊂Xis said to be-separated sequencefor some >0 if

sep xn

=infxn−xm:n=m

> . (1.4)

A Banach spaceXis said to benearly uniformly convex (NUC) if for every >0, there existsδ∈(0,1)such that for every sequence(xn)⊆B(X) with sep(xn) > , we have

conv xn

∩

(1−δ)B(X)

= ∅. (1.5)

Huff [6] proved that every NUC Banach space is reflexive and it has property (H).

Kutzarova [7] has definedk-nearly uniformly convex Banach spaces. Letk≥ 2 be an integer. A Banach spaceXis said to bek-nearly uniformly convex(k- NUC) if for any >0, there existsδ >0 such that for any sequence(xn)⊂B(X) with sep(xn)≥, there aren1,n2,...,nk∈Nsuch that

xn1+xn2+xn3+···+xnk

k

<1−δ. (1.6)

Clearly,k-NUC Banach spaces are NUC but the opposite implication does not hold in general (see [7]).

Fan and Glicksberg [5] have introduced fullyk-convex Banach spaces. A Ba- nach spaceXis said to befullyk-rotund(kR) if for every sequence(xn)⊂B(X), xn1+xn2+ ··· +xn_k →kasn1,n2,...,nk→ ∞implies that(xn)is conver- gent.

It is well known that UC implieskRandkRimplies(k+1)R, andkRspaces are reflexive and rotund, and it is easy to see thatk-NUC implieskR.

Denote byNandRthe set of all natural and real numbers, respectively.

LetXbe a real vector space. A functional:X→[0,∞]is called amodular if it satisfies the following conditions:

(i) (x)=0 if and only ifx=0;

(ii) (αx)=(x)for all scalarαwith|α| =1;

(iii) (αx+βy)≤(x)+(y)for allx,y∈Xand allα,β≥0 withα+β=1.

The modularis calledconvexif

(iv) (αx+βy)≤ α(x)+β(y) for allx,y ∈ X and allα,β≥ 0 with α+β=1.

k ...

Ifis a modular inX, we define

X=

x∈X: lim

λ→0⁺(λx)=0

, X^∗=

x∈X:(λx) <∞for someλ >0 .

(1.7)

It is clear thatX⊆X^∗. Ifis a convex modular, forx∈X, we define

x =inf

λ >0 :x λ ≤1

. (1.8)

Orlicz [10] proved that ifis a convex modular onX, thenX=X^∗and · is a norm onXfor whichXis a Banach space. The norm·, defined as in (1.8), is called the Luxemburg norm.

A modularis said to satisfy theδ2-condition(∈δ2)if for any >0, there exist constantsK≥2 anda >0 such that

(2u)≤K(u)+ (1.9)

for allu∈Xwith(u)≤a.

Ifsatisfies theδ2-condition for anya >0 withK≥2 dependent ona, we say thatsatisfies the strongδ2-condition (∈δ^s2).

The following known results are very important for our consideration.

Theorem1.1. If∈δ^s2, then for anyL >0andε >0, there existsδ >0such that

(u+v)−(u)< ε (1.10)

wheneveru,v∈Xwith(u)≤Land(v)≤δ. Proof. See [2, Lemma 2.1].

Theorem 1.2. (1) If ∈δ^s2, then for anyx ∈X, x =1 if and only if (x)=1.

(2) If ∈δ2, then for any sequence (xn) in X, xn →0 if and only if (xn)→0.

Proof. See [2, Corollary 2.2 and Lemma 2.3].

Theorem1.3. If∈δ^s2, then for any∈(0,1), there existsδ∈(0,1)such that(x)≤1−impliesx ≤1−δ.

Proof. Suppose that the theorem does not hold, then there exist >0 and xn∈Xsuch that(xn) <1−and 1/2≤ xn 1. Letan=1/xn−1. Then an→0 asn→ ∞. LetL=sup{(2xn); n∈N}. Since∈δ^s2, there existsK≥2

such that

(2u)≤K(u)+1 (1.11)

for everyu∈Xwith(u) <1.

By (1.11), we have(2xn)≤K(xn)+1< K+1 for alln∈N. Hence, 0< L <

∞. ByTheorem 1.2(1), we have

1= xn

xn

=

2anxn+ 1−an

xn

≤an 2xn

+ 1−an

xn

≤anL+(1−)→1−,

(1.12)

which is a contradiction.

Letl⁰be the space of all real sequences. For 1< p <∞, the Cesàro sequence space (ces_p) is defined by

ces_p=



x∈l⁰: ∞ n=1



1 n

n i=1

x(i)



p

<∞



 (1.13)

equipped with the norm

x =



^∞

n=1



1 n

n i=1

x(i)



p



1/p

. (1.14)

This space was first introduced by Shiue [14], which is useful in the theory of Matrix operator and others (see [8, 9]). Some geometric properties of the Cesàro sequence space ces_p were studied by many authors. It is known that (ces_p,·)is locally uniformly rotund (LUR) and has property (H) (see [9]). Cui and Meng [3] proved that(ces_p,·)has property(β).

Letp=(pn)be a sequences of positive real numbers withpn≥1 for all n∈N. The generalized Cesàro sequence space ces(p)is defined by

ces(p)=

x∈l⁰:ρ(λx) <∞for someλ >0

, (1.15)

where

ρ(x)= ∞ n=1



1 n

n i=1

x(i)



pn

(1.16)

is a convex modular on ces(p).

We consider ces(p)equipped with the Luxemburg norm:

x =inf

ε >0 :ρx ε ≤1

. (1.17)

k ...

Whenpn=qfor alln∈N, we see that ces(p)=ces_qand the Luxemburg norm on ces(p)given in (1.17) is equal to the norm·given in (1.14). In this paper, we show that ces(p)equipped with the Luxemburg norm isk-NUC fork≥2, so it iskRand (NUC).

Throughout this paper, we assume thatp=(pn)is bounded with

n→∞liminfpn>1 (1.18)

and thatM=sup_npn. 2. Main results

Proposition2.1. Forx∈ces(p), the modularρonces(p)satisfies the fol- lowing properties:

(1) if0< a <1, thena^Mρ(x/a)≤ρ(x)andρ(ax)≤aρ(x), (2) ifa≥1, thenρ(x)≤a^Mρ(x/a),

(3) ifa≥1, thenρ(x)≤aρ(x)≤ρ(ax).

Proof. All assertions are clearly obtained by the definition and convexity ofρ.

Proposition2.2. For anyx∈ces(p), (1) ifx ≤1, thenρ(x)≤ x, (2) ifx>1, thenρ(x)≥ x, (3) x =1if and only ifρ(x)=1.

Proof. (1) Suppose thatx ≤1. Ifx=0, thenρ(x)= x =0. Suppose x=0. By the definition of·, there is a sequence(n)withn↓ xsuch that ρ(x/n)≤1. This implies thatρ(x/x)≤1. ByProposition 2.1(1), we have

ρ(x)=ρ

x·x

x ≤ xρ x

x ≤ x. (2.1)

(2) Suppose that x> 1. Then for ∈(0,(x−1)/x), we have (1− )x>1. ByProposition 2.1(1), we have

1< ρ x

(1−)x ≤ ρ(x)

(1−)x, (2.2)

so that(1−)x< ρ(x). By taking→0, we haveρ(x)≥ x. (3) It follows fromTheorem 1.2(1) becauseρ∈δ^s2.

Proposition2.3. For anyL >0andε >0, there existsδ >0such that ρ(u+v)−ρ(u)< ε (2.3) wheneveru,v∈ces(p)withρ(u)≤Landρ(v)≤δ.

Proof. Sincep=(pn)is bounded, it is easy to see thatρ∈δ^s2. Hence, the proposition is obtained directly fromTheorem 1.1.

Proposition2.4. For every sequence(xn)∈ces(p),xn →0if and only ifρ(xn)→0.

Proof. It follows directly fromTheorem 1.2(2) becauseρ∈δ^s2.

Theorem2.5. For anyx∈ces(p)and∈(0,1), there existsδ∈(0,1)such thatρ(x)≤1−impliesx ≤1−δ.

Proof. Sinceρ∈δ^s2, the theorem is obtained directly fromTheorem 1.3.

Theorem2.6. The spaceces(p)isk-NUC for any integerk≥2.

Proof. Let >0 and(xn)⊂B(ces(p))with sep(xn)≥. For eachm∈N, let

x_n^m=



0,0,...,0

m−1

,xn(m),xn(m+1),...



. (2.4)

Since for eachi∈N, (xn(i))^∞_n=1 is bounded, by using the diagonal method, we have that for eachm∈N, we can find a subsequence(xn_j)of(xn)such that(xnj(i))converges for eachi∈N, 1≤i≤m. Therefore, there exists an in- creasing sequence of positive integer(tm)such that sep((x^m_nj)j>tm)≥. Hence, there is a sequence of positive integers(rm)^∞_m=1withr1< r2< r3<··· such thatx_r^m_m ≥/2 for allm∈N. Then byProposition 2.4, we may assume that there existsη >0 such that

ρ x_r^m_m

≥η ∀m∈N. (2.5)

Letα >0 be such that 1< α <lim_n→∞infpn. For fixed integerk≥2, let1= ((k^α−1−1)/(k−1)k^α)(η/2). Then byProposition 2.3, there is aδ >0 such that ρ(u+v)−ρ(u)< 1 (2.6) wheneverρ(u)≤1 andρ(v)≤δ.

Since byProposition 2.2(1)ρ(xn)≤1 for alln∈N, there exist positive inte- gersmi(i=1,2,...,k−1)withm1< m2<···< mk−1such thatρ(x_i^mi)≤δ andα≤pjfor allj≥mk−1. Definemk=mk−1+1. By (2.5), we haveρ(x^mr_mk^k)≥ η. Letsi=ifor 1≤i≤k−1 andsk=rm_k.

Then in virtue of (2.5), (2.6), and convexity of functionfi(u)= |u|^pi (i∈N), we have

ρ

xs1+xs2+···+xsk

k

= ∞ n=1



1 n

n i=1

xs1(i)+xs2(i)+···+xsk(i) k





pn

k ...

=

m1

n=1



1 n

n i=1

xs1(i)+···+xs_k(i) k





pn

+ ∞ n=m1+1



1 n

n i=1

xs1(i)+xs2(i)+···+xs_k(i) k





pn

≤

m1

n=1



1 n

n i=1

xs1(i)+···+xsk(i) k





pn

+ ∞ n=m1+1



1 n

n i=1

xs2(i)+xs3(i)+···+xs_k(i) k





pn

+1

≤

m1

n=1

1 k

k j=1



1 n

n i=1

xsj(i)



pn

+

m2

n=m1+1



1 n

n i=1

xs2(i)+xs3(i)+···+xsk(i) k





pn

+ ∞ n=m2+1



1 n

n i=1

xs2(i)+xs3(i)+···+xsk(i) k





pn

+1

≤

m1

n=1

1 k

k j=1



1 n

n i=1

xs_j(i)



pn

+

m2

n=m1+1



1 n

n i=1

xs2(i)+xs3(i)+···+xs_k(i) k





pn

+ ∞ n=m2+1



1 n

n i=1

xs3(i)+xs4(i)+···+xs_k(i) k





pn

+21

≤

m1

n=1

1 k

k j=1



1 n

n i=1

xsj(i)



pn

+

m2

n=m1+1

1 k

k j=2



1 n

n i=1

xsj(i)



pn

+

m3

n=m2+1

1 k

k j=3



1 n

n i=1

xsj(i)



pn

+···+

mk

n=m_k−1+1

1 k

k j=k−1



1 n

n i=1

xs_j(i)



pn

+ ∞ n=m_k+1



1 n

n i=1

xs_k(i)

k





pn

+(k−1)1

≤ρ xs1

+···+ρ xs_k−1

k +1

k

mk n=1



1 n

n i=1

xs_k(i)



pn

+ ∞ n=m_k+1



1 n

n i=1

xs_k(i)

k





pn

+(k−1)1

≤k−1 k +1

k

m_k

n=1



1 n

n i=1

xs_k(i)



pn

+ 1 k^α

∞ n=m_k+1



1 n

n i=1

xs_k(i)



pn

+(k−1)1

≤1−1 k+1

k



1−

∞ n=mk+1



1 n

n i=1

xsk(i)



pn



+ 1 k^α

∞ n=mk+1



1 n

n i=1

xsk(i)



pn

+(k−1)1

≤1+(k−1)1−

k^α−1−1 k^α

∞ n=mk+1



1 n

n i=1

xsk(i)



pn

≤1+(k−1)1−k^α−1−1 k^α η

=1−

k^α−1−1 k^α

η 2 .

(2.7)

ByTheorem 2.5, there existγ >0 such that(xs1+xs2+···+xsk)/k<1−γ. Therefore, ces(p)isk-NUC.

Sincek-NUC implieskRandkRimpliesRand reflexivity holds, andk-NUC implies NUC and NUC implies property (H) and reflexivity holds, byTheorem 2.6, the following results are obtained.

Corollary2.7. The spaceces(p)iskR, NUC, and has a drop property.

Corollary2.8. For1< p <∞, the spaceces_pisk-NUC.

Corollary2.9. For1< p <∞, the spaceces_piskRand NUC.

Corollary2.10. For1< p <∞, the spaceces_phas the drop property.

Acknowledgments. Suthep Suantai would like to thank the Thailand Re- search Fund for the financial support and the referee for pointing out the work of Cui and Hudzik [2]. Winate Sanhan was supported by The Royal Golden Jubilee Project.

k ...

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Winate Sanhan: Department of Mathematics, Faculty of Science, Chiang Mai Univer- sity, Chiang Mai 50200, Thailand

E-mail address:[email protected]

Suthep Suantai: Department of Mathematics, Faculty of Science, Chiang Mai Univer- sity, Chiang Mai 50200, Thailand

E-mail address:[email protected]