ZHOU CHENG-HUA,GONG WAN-ZHONGAND ZHANG DAO-XIANG

(Department of Mathematics,Anhui Normal University,Wuhu,Anhui,241000)

Abstract: In this paper we give some characterizations of O-convexity of Banach spaces,and show the criteria for O-convexity in Orlicz-Bochner function space LM(μ,X)and Orlicz-Bochner sequence space lM(Xs)endowed with Orlicz norm.Moreover,we give a sufficient condition for the dual of such a space to have the fixed point property.

Key words:O-convexity,Orlicz norm,Orlicz-Bochner sequence space,Orlicz-Bochner function space

1 Introduction

It is well known that convexities and re fl exivity play an important role in Banach space theory.Since B-convexity(write(BC))was introduced by Beck[1],some revelent properties including uniform non-squareness(write(U-NS))and P-convexity(write(PC))were given by Brown[2],Giesy[3],James[4]and Kottman[5].From their achievements we know that

where(UC)denotes uniform convexity,(US)uniform smoothness,and(Rfx)re fl exivity.A natural and interesting question raised by Brown[6]:“Is there a B-convex space that is not P-convex?”Though Giesy[7]and James[8]provided answer to the above question,Naidu and Sastry[9]introduced a new geometric conception in a Banach space X named O-convexity(write(OC)):if there exists an ε＞ 0 and an n0∈ N+such that for every x(1),x(2),···,x(n0)∈ S(X)there holds

In[9],the authors showed

where(S-Rfx)denotes super-re fl exivity.They also proved thatnormed by‖(x,y)‖=is O-convex whenever X and Y are O-convex for 1≤p＜ ∞.In recent years,many works indicate that O-convexity is closely related to the fixed point property(see[10]and[11]).

For Orlicz-Bochner function space LM(μ,X)or Orlicz-Bochner sequence space lM(Xs)with Orlicz norm,a fundamental question is that whether or not a geometrical property lifts from X to LM(μ,X),or Xsto lM(Xs).The answer may often be guessed,but usually,the result exceed the guess and the proof is nontrivial.Various kinds of convexity for Lebesgue-Bochner space Lp(μ,X),Orlicz-Bochner function space LM(μ,X)and Orlicz-Bochner sequence space lM(Xs)were carried out by many authors(see[12]–[17],etc).

In this paper,a characterization of O-convexity of LM(μ,X)or lM(Xs)endowed with Orlicz norm is given.As a corollary of the main result we get that for 1＜p＜∞,the equi-O-convexity of{Xs}implies the O-convexity of lp(Xs),and the O-convexity of X implies the O-convexity of Lp(μ,X).Moreover,we show that the dual space of LM(μ,X)(or lM(Xs))has the fixed point property whenever LMand X are O-convex(or lMand Xsare equi-O-convex).

equipped with Orlicz norm

endowed with Orlicz norm.

In this paper,LMmeans LM(μ,R),and so for lM.For every Orlicz function M we de fine its complementary function N:R→[0,∞)by the formula

It is well known that for x∈lM(Xs)withthe equality

holds if and only if k ∈ K(x)=[k?,k??],where

Similarly for x ∈ LM(μ,X).

We say that an Orlicz function M satis fies condition δ2(or?2)if there exists u0＞ 0 and K＞2 such that

for every|u|≤u0(or|u|≥u0).In this case,we write M ∈δ2(or M ∈?2).If(1.1)holds for any u∈R,we write

For more properties about Orlicz space,Orlicz-Bochner space,O-convexity and any other convexities we refer to[3]–[22].

2 De finition and Lemmas

De finition 2.1Let Xsbe Banach spaces for s ∈ N+.Then we callare equi-O-convex if there exists an ε＞ 0 and an n0∈ N+such that for all s∈ N+and,there holds

The series(n0,ε)here is called the equi-O-constant of{Xs}.

Lemma 2.1A Banach space X is O-convex if and only if there exists an n0∈N+and a δ＞ 0 such that for any elements x1,x2,···,xn0∈ X{0},two indices i0,j0∈can be found such that

Without loss of generality,we may assume that

Then

It follows that

Therefore,

That is,

Sufficiency.It follows immediately from the de finition of O-convexity.

Lemma 2.2Let X be an O-convex Banach space.Then there exists an n0∈N+and a δ＞ 0 such that for any elements x1,x2,···,xn0∈ X{0}and k,l∈ R+,two indicescan be found such that

Proof. By Lemma 2.1,we get that there exists an n0∈ N+and a δ＞ 0 such that for any elements x1,x2,···,xn0∈ X{0}and k,l ∈ R+,two indices i0,j0∈ {1,2,···,n0}can be found such that

Hence

which finishes the proof.

Lemma 2.3[16],[22]For any Orlicz function M the following assertions hold true:

Lemma 2.4Suppose that M∈δ2,N∈δ2.Then for any l≥m＞0 and w＞0,there exists r=r(w,m,l)∈(0,1)such that for any equi-O-convex Banach spaces{Xs:s=1,2,···}with equi-O-constant(n0,ε),there holds

Suppose that M ∈ δ2,N ∈ δ2and{Xs:s=1,2,···}is equi-O-convex with equi-O-constant(n0,ε).Takeandsatisfying

Let h0=h0(s)be the index such that

holds true for all i,j ∈ {1,2,···,n0}.By Lemma 2.3,there exists a δ= δ(l,m,w)∈ (0,1)and a c=c(l,m,w)∈(0,1)such that the inequality

Moreover,by the convexity of M,we get for all i,j ∈ {1,2,···,n0}.Therefore,by inequalities(2.3)and(2.4),we have

Let i0,j0∈ {1,2,···,n0}be a pair of indices from Lemma 2.2.Then

which implies that

Hence

Using Lemma 2.2 and inequality(2.5),we obtain

By inequality(2.6)and the convexity of M one can get

Therefore,

Since M∈δ2,for any l0＞1,w＞0 there exists a kl0=kl0(a,w)＞1 such that

Therefore,using inequalities(2.7)and(2.8),we know that for any j?=i,Finally,denoting

we obtain the inequality

Corollary 2.1Let X be an O-convex Banach space and.Then there exists an r∈(0,1)and an n0∈N+such that

Lemma 2.5[20](a)If M ∈δ2,N ∈δ2,then the set{K(u):u∈S(lM)}is bounded and there exists a δ∈ (0,1)such that

3 Main Results

Theorem 3.1The following statements are equivalent:

(a)lM(Xs)is O-convex;

(b)lMis O-convex andare equi-O-convex;

(c)lMis re fl exive andare equi-O-convex;

(d)M ∈ δ2,N ∈ δ2,andare equi-O-convex.

Therefore,for any y1,y2,···,yn0∈ S(Xs),one has

which shows that{(Xs,‖ ·‖s)}are equi-O-convex.

Similarly,lMis O-convex.

(b)?(c).Every O-convex Banach spaces are re fl exive.Consequently,lMis re fl exive.

(c)?(d).lMis re fl exive if and only if M ∈ δ2and N ∈ δ2.

(d)?(a).Suppose that M ∈ δ2,N ∈ δ2andis equi-O-convex with equi-O-constant(n0,ε).Denote

By Lemma 2.5,we have 1+δ＜η＜l＜+∞for some δ∈(0,1).Takewhere(i=1,2,···,n0).Obviously,Let w=M?1(l?1).

Hence,by Lemma 2.4,there exists an r∈(0,1)such that the inequality

which implies the inequality

i.e.,

Therefore,

The inequality above shows that lM(Xs)is O-convex.

Corollary 3.1Let Xsbe Banach spaces,1＜p＜+∞.Then lp(Xs)is O-convex if and only if Xsare equi-O-convex.

Theorem 3.2The following statements are equivalent:

(a)LM(μ,X)is O-convex;

(b)Both LM(μ)andare O-convex;

(c)LM(μ)is re fl exive andis O-convex;

Proof. Here we only need to prove(d)?(a).

By Lemma 2.5,we have 1+δ＜η＜l＜+∞ for some δ∈(0,1).

For clarity,we divide the proof into two parts.

By Corollary 2.1,there exists an r∈(0,1)such that

holds true forμ-a.e.t∈ T.Then

which yields the inequality

Hence LM(μ,X)is O-convex.

II.Suppose thatμ? ＜ +∞,M ∈ ?2and N ∈ ?2.TakeDe fine

Since

we have

Noticing ki＞1,we know that

By Corollary 2.2,there exists an r=r(w)∈(0,1)such that

Therefore,LM(μ,X)is O-convex.

Corollary 3.2Let X be a Banach space,1＜ p＜ +∞.Then Lp(μ,X)is O-convex if and only if X is O-convex.

4 The Fixed Point Property in Orlicz-Bochner Spaces

Let C be a nonempty bounded,closed and convex subset of a Banach space X.A mapping T:C→C is called nonexpansive if

X is said to have the fixed point property(write(FPP))if every nonexpansive mapping has a fixed point.(FPP)has always been the focus of attention for its wide application.We have known that uniformly nonsequare Banach spaces has(FPP)(see[18]),certainly,for uniformly convex Banach spaces.A well-known open problem is that whether every re fl exive Banach space has(FPP).Fetter H.Nathansky[10]and Dowling[11]shown that X?has(FPP)when X is O-convex.In view of O-convexity implying re fl exivity,and re fl exivity implying Radon-Nikodym property,by the representation theorem for the dual of Orlicz-Bochner space(see[23]and[24]),we have

Theorem 4.1Letbe an O-convex Banach space,and eitherandwheneverμ?=+∞,or M∈?2and N∈?2whenever 0＜μ?＜∞.Then the Orlicz-Bochner function space L(N)(μ,X?)has(FPP),where L(N)(μ,X?)means that the linear space LN(μ,X?)equipped with the Luxemburg norm

Theorem 4.2Letbe equi-O-convex,M ∈ δ2and N ∈ δ2.Then the Orlicz-Bochner sequence spacehas(FPP),wheremeans that the linear spaceequipped with the Luxemburg norm.