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The fixed point property of Orlicz sequence spaces equipped with thep-Amemiya norm

Abstract

In this paper, the Opial modulus and the weakly convergent sequence coefficient ofOrlicz space l ϕ , p endowed with the p-Amemiya norm are calculated, the criteria for the uniformOpial property as well as for weakly uniform normal structure of l ϕ , p are presented. It is shown that the Orlicz sequence space equipped with thep-Amemiya norm has the fixed point property if and only if it isreflexive.

MSC: 47H10, 46E30, 46B20.

1 Introduction and preliminaries

The aim of this paper is to present criteria for some important geometric propertiesrelated to the metric fixed point theory in Orlicz sequence spaces.

The Opial property originates from the fixed point theorem proved by Opial in[1]. The uniform Opial property withrespect to the weak topology was defined in [2]and Opial modulus was introduced in [3]. It iswell known that the Opial property and normal structure of a Banach space Xplay an important role in metric fixed point theory for nonexpansive mappings, as wellas in the theory of differential and integral equations (see [1, 46]). The Opial property also plays an important role in the study ofweak convergence of iterates, random products of nonexpansive mappings and theasymptotic behavior of nonlinear semigroups [1, 79].Moreover, it can be introduced to the open unit ball of a complex Hilbert space,equipped with the hyperbolic metric, where it is useful in proving the existence offixed points of holomorphic self-mappings of X[8].

The coefficient WCS(X) was introduced by Bynum [10], who established their relations with normal structure andcalculated the value of WCS( l p ). A reflexive Banach space X withWCS(X)>1 has normalstructure and consequently it has the weakly fixed point property. This is probably oneof the Banach space constants which has been most widely studied, although withconsiderable confusion because there exist many equivalent definitions. Severaldifferent formulae for WCS(X) were found (see [4]), also see the work by Sims and Smyth [11].

The notion of p-Amemiya norm was introduced by Cui and Hudzik in [12], where they showed that the p-Amemiya norm Φ , p is equivalent to the Orlicz norm Φ aswell as to the Luxemburg norm Φ . They also illustrated thedescription of extreme points and strongly extreme points in Orlicz spaces equipped withthe p-Amemiya norm [12, 13]. In 2012, they presented the criteria for non-squareness,uniform non-squareness, and locally uniform non-squareness of these spaces[14]. Chen and Cui (see [15, 16]) gave the criteria forcomplex extreme points and complex strict convexity in Orlicz function spaces equippedwith the p-Amemiya norm, and for complex mid-point locally uniform rotundityand complex rotundity of Orlicz sequence spaces equipped with the p-Amemiyanorm.

The rest of the paper is organized as follows. In the first section, some basic notions,terminology and original results are reviewed, which will be used throughout the paper.In Section 2, the Opial modulus of Orlicz space l ϕ , p endowed with the p-Amemiya norm is calculated, and the criteria for the uniformOpial property of l ϕ , p are presented. The weakly convergent sequence coefficient is calculated inSection 3. Finally, the necessary and sufficient condition for fixed pointproperties to exist in l ϕ , p are given.

Let X be a Banach space. We denote by B(X) the unit ball ofX, by S(X) the unit sphere of X. Now we recall somenotions from fixed point theory.

A mapping T:CX defined on a subsetC of a Banach space X is said to be non-expansive ifTxTyxy for allx,yC.

We say that a Banach space X has the fixed point property if for every weaklycompact convex subset CX and forevery nonexpansive T:CC, T has a fixedpoint of C.

It is known that L 1 [0,1] does not have the fixed point property.

For any map Φ:R[0,],define

a Φ =max { u 0 : Φ ( u ) = 0 } , b Φ =max { u 0 : Φ ( u ) < } .

A map Φ is said to be an Orlicz function if Φ(0)=0, Φ is notidentically equal to zero, it is even and convex on the interval ( b Φ , b Φ ) and left-continuous at b Φ .

For every Orlicz function Φ, we define its complementary functionΨ:R[0,] by theformula

Ψ(v)=sup { u | v | Φ ( u ) : u 0 } .

And the convex modular by I Φ (x)= i = 1 Φ(x(i)) for anyx=(x(i)).

Definition 1.1[1719]

The Orlicz sequence space is defined as the set

l Φ = { x = ( x ( i ) ) : I Φ ( λ x ) <  for some  λ > 0 } .

The Luxemburg norm and the Orlicz norm are expressed as

x Φ =inf { λ > 0 : I Φ ( x λ ) 1 }

and

x Φ = inf k > 0 1 k ( 1 + I Φ ( k x ) ) ,

respectively. The Orlicz space equipped with the Luxemburg norm and the Orlicz norm isdenoted by l Φ and l Φ ,respectively.

For any 1p andu0,define

s p (u)= { ( 1 + u p ) 1 p for  1 p < , max { 1 , u } for  p =

and define s Φ , p (x)= s p I Φ (x) for all 1p. Note that thefunctions s p and s Φ , p are convex. Moreover, the function s p isincreasing on R + for1p<, but the function s isincreasing on the interval [1,) only.

Definition 1.2[12, 13]

Let 1p. For anyx=(x(i)), define the p-Amemiya norm by theformula

x Φ , p = inf k > 0 1 k s Φ , p (kx).

The Orlicz space equipped with the p-Amemiya norm will be denoted by l Φ , p .

It is known that x Φ , 1 = x Φ and x Φ , = x Φ . If 1<p< andx0,

1 2 x Φ x Φ x Φ , p 2 1 p x Φ < 2 1 p x Φ

(see [12]).

Let p + bethe right-hand side derivative of Φ on [0, b Φ ) and put p + ( b Φ )= lim u b Φ p + (u). Define the function α p : l Φ , p [1,] by

α p (x)= { I Φ p 1 ( x ) I Ψ ( p + ( | x | ) ) 1 , 1 p < , 1 , p = , I Φ ( x ) 1 , I Ψ ( p + ( | x | ) ) , p = , I Φ ( x ) > 1

and the functions k p : l Φ , p [0,), k p : l Φ , p [0,) by

k p ( x ) = inf { k 0 : α p ( k x ) 0 } ( with  inf ϕ = ) , k p ( x ) = inf { k 0 : α p ( k x ) 0 } .

It is obvious that k p (x) k p (x) for every 1p andx l Φ , p .

Set K p (x)={0<k<: k p (x)k k p (x)}.

Definition 1.3[17]

We say that an Orlicz function Φ satisfies the Δ 2 (0)-condition (Φ Δ 2 (0), for short) if there exist constantsK2 and u 0 >0such that

Φ(2u)KΦ(u)for every |u| u 0 .

For more details about Orlicz spaces, we refer to [13, 14, 18, 19].

Lemma 1.1[14]

Let1p.Then I Φ (x) x Φ , p forallx L Φ , p with x Φ , p 1.

Lemma 1.2[17]

If the Orlicz function Φ vanishes only at zeroandΦ Δ 2 (0), then the norm convergence and the modularconvergence are equivalent.

Lemma 1.3[12]

For every1pandeachx l Φ , p {0}, the following conditions hold.

  1. 1.

    If k p (x)= k p (x)=, K p (x)=ϕ, then

    x Φ , p = lim k 1 k ( 1 + I Φ p ( k x ) ) 1 p .
  2. 2.

    If k p (x)< k p (x)=, then the p-Amemiya norm x Φ , p is attained at everyk[ k p (x),).

  3. 3.

    If k P (x)<, then the p-Amemiya norm x Φ , p is attained at everyk[ k p (x), k p (x)].

Lemma 1.4 If the Orlicz function Φ vanishes only at zero,then l Φ , p isorder continuous if and only ifΦ Δ 2 (0).

Lemma 1.5 AssumeΦ Δ 2 (0), 1p<. Then,for anyL>0andε>0, thereexistsδ>0suchthat

I Φ (x)L, I Φ (y)δ | I Φ p ( x + y ) I Φ p ( x ) | <ε(x,y l Φ , p ).

Proof Let

h=sup { I Φ ( 2 x + 2 y ) : I Φ ( x ) L , I Φ ( y ) 1 } .

Then L<h<, sinceΦ Δ 2 (0). Without loss of generality, we assumeL>1 andε<1.Set β= ε h p .Since the modular convergence implies the norm convergence, so we can findδ>0 suchthat I Φ (y)δimplies y Φ , p min{ β 2 , ε 1 p β 1 1 p 2 }. Thus, applying theconvexity of Φ and Lemma 1.1, we have

I Φ p ( x + y ) = I Φ p ( ( 1 β ) x + β ( x + β 1 y ) ) ( 1 β ) I Φ p ( x ) + β I Φ p ( x + β 1 y ) ( 1 β ) I Φ p ( x ) + β 2 I Φ p ( 2 x ) + β 2 I Φ p ( 2 β 1 y ) ( 1 β ) I Φ p ( x ) + β 2 I Φ p ( 2 x ) + β 2 2 β 1 y Φ , p p I Φ p ( x ) + β h p 2 + 2 p 1 β p 1 y Φ , p p I Φ p ( x ) + ε .

Replacing x, y by x+y,−y, respectively, in the above inequalities, we have

I Φ p (x) I Φ p (x+y)+ε.

 □

Lemma 1.6[20]

l Φ , p isreflexive if and only ifΦ Δ 2 (0)andΨ Δ 2 (0).

Lemma 1.7 Let1p<. l Φ , p hasa subspace isomorphic to l 1 if andonly ifA>0,whereA= lim u 0 Φ ( u ) u .

Proof Since the function Φ ( u ) u isnondecreasing, lim u 0 Φ ( u ) u exists.

If A>0, by thecontinuity of Φ ( u ) u , thereexists A 1 >0such that

A|u|Φ(u) A 1 |u| ( | u | [ 0 , Φ 1 ( 1 ) ] ) .

Then Φ Δ 2 (0).

For any x l Φ , p ,we have I Φ ( x x Φ )=1, then | x ( i ) | x Φ [0, Φ 1 (1)] (i=1,2,),therefore,

A | x ( i ) | x Φ Φ ( | x ( i ) | x Φ ) A 1 | x ( i ) | x Φ ,

we have

A i = 1 | x ( i ) | x Φ i = 1 Φ ( | x ( i ) | x Φ ) A 1 i = 1 | x ( i ) | x Φ ,

then A x l 1 x Φ A 1 x l 1 .This yields that the l 1 norm and theLuxemburg norm are equivalent. Since the p-Amemiya norm and the Luxemburg normare equivalent, l Φ , p has a subspace isomorphic to l 1 . □

Therefore, if A>0, thespace l Φ , p has the Schur property, and it has the Opial property trivially because there is noweakly null sequence ( x n ) in S( l Φ ). The case when A>0 is notinteresting if we consider the Opial modulus and weakly convergent sequence coefficient.For this reason we will assume that A=0 in thefollowing whenever the Opial modulus and the weakly convergent sequence coefficient areconsidered.

2 Opial modulus for Orlicz sequence spaces

In this section we present some results on the Opial modulus. The obtained resultsextend the existing ones, which were presented by a number of papers studying thegeometry of Orlicz spaces endowed with the Luxemburg norm and the Orlicz norm,respectively. A formula for calculating the Opial modulus in Orlicz andMusielak-Orlicz spaces equipped with the Luxemburg or the Orlicz norm is found in[2126].

Definition 2.1[1]

We say that a Banach space X has the Opial property if for any weakly nullsequence ( x n ) in X and anyxX{0} there holds

lim inf n x n < lim inf n x+ x n .

Opial proved in [1] that l p (1<p<) has thisproperty, but L p [0,2π] does not have it ifp(1,),p2.

Definition 2.2[6]

We say that X has the uniform Opial property if for any ε>0 thereexists r>0 such thatfor any xXwith xεand any weakly null sequence ( x n ) in the unit sphere S(X) of X there holds

1+r lim inf n x n +x.

It is obvious that the uniform Opial property implies the Opial property.

Definition 2.3[2]

The Opial modulus of X is denoted by δ o and itis defined for ε(0;1] by the formula

δ o (ε)=inf { lim inf n x n + x : ( x n ) S ( X ) , x n 0  weakly , x = ε } .

It is easy to see that X has the uniform Opial property if and only if δ o (ε)>1 for anyε(0,1].

Theorem 2.1 If Φ is an Orlicz function,1p<, a Φ >0,then l Φ , p doesnot have the Opial property.

Proof Divide into a sequence ( N n ) of pairwise disjoint and infinite subsetsof such that inf n N n as n and define

x n = i N n + 1 a Φ e i ,x= i N 1 a Φ e i .

Then the sequence ( x n ) is weakly convergent to zero. For anyk>1, we have

I Φ (k x n )= I Φ (kx)= I Φ ( k ( x n + x ) ) =

and for any k(0,1],

1 k ( 1 + I Φ p ( k x n ) ) 1 p = 1 k ( 1 + I Φ p ( k x ) ) 1 p = 1 k ( 1 + I Φ p ( k ( x n + x ) ) ) 1 p = 1 k .

So x n Φ , p = x Φ , p = x n + x Φ , p =1.Therefore, l Φ , p does not have the Opial property. □

In the following we may assume that a Φ =0.

Theorem 2.2 Let Φ be an Orlicz functionsatisfying Δ 2 (0), 1p<. Then foranyε(0,1], we have

δ o ( ε ) = inf { c x y k > 0 : I Φ p ( k x c x y k ) + I Φ p ( k y c x y k ) = k p 1 , k > 1 , x Φ , p = 1 , y Φ , p = ε , x , y l Φ , p } .

Proof Set d(ε)=inf{ c x y k >0: I Φ p ( k x c x y k )+ I Φ p ( k y c x y k )= k p 1,k>1, x Φ , p =1, y Φ , p =ε,x,y l Φ , p }.

(1) For fixed k>1, consider the function

F(c)= I Φ p ( k x c ) + I Φ p ( k y c ) .

We have F is continuous on R + . Since

1= x Φ , p 1 k ( 1 + I Φ p ( k x ) ) 1 p ,

we have I Φ p (kx) k p 1for any k>0, thus

F(1)= I Φ p (kx)+ I Φ p (ky)> k p 1.

Moreover, lim c + F(c)=0, then thereexists unique c x y k (1,+) such that

F( c x y k )= I Φ p ( k x c x y k ) + I Φ p ( k y c x y k ) = k p 1,

which shows that the function d(ε) is well defined.

(2) Now, we will show that δ o (ε)d(ε) for any ε(0,1]. For any θ>0, there exist x 0 , y 0 X with x 0 =1, y 0 =ε and k 0 >1 such that d(ε)+θ> c x 0 y 0 k 0 . Put

x = ( y 0 ( 1 ) , 0 , y 0 ( 2 ) , 0 , y 0 ( 3 ) , 0 , y 0 ( 4 ) , 0 , ) , x n = i = 0 x 0 ( i + 1 ) e 2 n + 2 n i ( n = 1 , 2 , ) .

Then x n Φ , p = x 0 Φ , p =1, x Φ , p = y 0 Φ , p =ε,and x n (i)0 asn for any iN.

Since i = 1 Φ(x(i))<, there exists i 0 N such that i = i 0 + 1 Φ(x(i))<θ.

For fixed i 0 ,due to lim u 0 Φ ( u ) u =0,we have

lim l 0 1 l i = 1 i 0 Φ ( l x ( i ) ) = lim l 0 i = 1 i 0 Φ ( l x ( i ) ) l | x ( i ) | | x ( i ) | = i = 1 i 0 lim l 0 Φ ( l x ( i ) ) l | x ( i ) | | x ( i ) | = 0 .

So, for any nN and l(0,1) which is small enough, we get

I Φ ( l x n ) l = I Φ ( l x ) l = 1 l ( i = 1 i 0 Φ ( l x ( i ) ) + i = i 0 + 1 Φ ( l x ( i ) ) ) 1 l i = 1 i 0 Φ ( l x ( i ) ) + i = i 0 + 1 Φ ( x ( i ) ) < 2 θ .

Then lim l 0 I Φ ( l x n ) l =0.

Take any f l Ψ , q ,there exists λ>0satisfying I Ψ (λf)<. Since

inf(supp x n )as n,

as well as the Young inequality, we have

| f ( x n ) | =| i = 1 x n (i)f(i)| 1 l λ ( I Φ ( l x n ) + I Ψ ( λ f χ supp x n ) ) 0

(as n). Then we have proved that x n 0 weakly.

Since x and x n havedisjoint supports, we get

x n + x Φ , p d ( ε ) + θ 1 k 0 ( 1 + I Φ p ( k 0 ( x n + x ) d ( ε ) + θ ) ) 1 p = 1 k 0 ( 1 + I Φ p ( k 0 x n d ( ε ) + θ ) + I Φ p ( k 0 x d ( ε ) + θ ) ) 1 p 1 k 0 ( 1 + I Φ p ( k 0 x n c x 0 y 0 k 0 ) + I Φ p ( k 0 x c x 0 y 0 k 0 ) ) 1 p = 1 ,

hence x n + x Φ , p d(ε)+θ. By thearbitrariness of θ>0, wehave δ o (ε)d(ε).

(3) Assume that δ o (ε)<d(ε) for some ε(0,1]. Then there exists ε 0 >0 such that δ o (ε)d(ε)2 ε 0 , so there exist x l Φ , p and ( x n ) in S( l Φ , p ) such that x Φ , p =ε, x n 0 weakly and

lim n x n + x Φ , p <d(ε)2 ε 0 .

For fixed ε 0 4 ,there exists i 0 such that

i = i 0 + 1 x ( i ) e i Φ , p < ε 0 4 , i = 1 i 0 x n ( i ) e i Φ , p < ε 0 4

for nN large enough. Set

i = 1 i 0 x ( i ) e i Φ , p =a, i = i 0 + 1 x n ( i ) e i Φ , p = b n .

And define

y = ( ε x ( 1 ) a , ε x ( 2 ) a , , ε x ( i 0 ) a , 0 , 0 , ) , y n = ( 0 , 0 , 0 , x n ( i 0 + 1 ) b n , x n ( i 0 + 2 ) b n , , x n ( i 0 + m ) b n , ) ,

then y n Φ , p =1(for all nN), y Φ , p =ε, y n w 0.For n large enough, we have

y n + y Φ , p i = i 0 + 1 x n ( i ) b n e i x n Φ , p + i = 1 i 0 ε a x ( i ) e i x Φ , p + x n + x Φ , p i = 1 i 0 x n ( i ) e i Φ , p + | 1 b n | + i = i 0 + 1 x ( i ) e i Φ , p + | ε a | + d ( ε ) 2 ε 0 2 i = 1 i 0 x n ( i ) e i Φ , p + 2 i = i 0 + 1 x ( i ) e i Φ , p + d ( ε ) 2 ε 0 < d ( ε ) ε 0 .

Case 1. A={nN: K Φ ( y n + y d ( ε ) )ϕ} is infinite. For any nA, we take k n K Φ ( y n + y d ( ε ) ). Then

1 ε 0 d ( ε ) y n + y Φ , p d ( ε ) = 1 k n ( 1 + I Φ p ( k n ( y n + y ) d ( ε ) ) ) 1 p = 1 k n ( 1 + I Φ p ( k n y n d ( ε ) ) + I Φ p ( k n y d ( ε ) ) ) 1 p 1 k n ( 1 + I Φ p ( k n y n c y n y k n ) + I Φ p ( k n y c y n y k n ) ) 1 p = 1 .

This is a contradiction.

Case 2. B={nN: K Φ ( y n + y d ( ε ) )=ϕ} is infinite. For any nB,

1 ε 0 d ( ε ) y n + y Φ , p d ( ε ) = lim l 1 l ( 1 + I Φ p ( l ( y n + y ) d ( ε ) ) ) 1 p = lim l 1 l ( 1 + I Φ p ( l y n d ( ε ) ) + I Φ p ( l y d ( ε ) ) ) 1 p lim l 1 l ( 1 + I Φ p ( l y n c y n y l ) + I Φ p ( l y c y n y l ) ) 1 p = 1 ,

this is also a contradiction. The two cases have shown that δ o (ε)=d(ε). □

Remark The main result presented in this paper generalizes the existing result tothe p-Amemiya norm. In the case that p=1, the situationdegrades to the case of the classical Orlicz norm.

If Φ Δ 2 (0), then

δ o ( ε ) = inf { c x y k > 0 : I Φ ( k x c x y k ) + I Φ ( k y c x y k ) = k 1 , k > 1 , x Φ = 1 , y Φ = ε , x , y l Φ } .

In the following we will consider the uniform Opial property for the Orlicz sequenceequipped with the p-Amemiya norm.

Lemma 2.3[25]

Let X be a Köthe sequence space with the semi-Fatouproperty and without order continuity of the norm.Then X does not have the uniform Opial property. Infact, we even have that for anyε(0,1), δ o (ε)=1.

Theorem 2.4 Let1p<, l Φ , p hasthe uniform Opial property if and only ifΦ Δ 2 (0).

Proof If Φ Δ 2 (0) and l Φ , p does not have the uniform Opial property, then there exists some ε 0 (0,1] such that δ o ( ε 0 )=1. Therefore, for anysequence ( c n ) such that c n 1,there are two sequences ( x n ) and ( y n ) in l Φ , p with x n =1, y n = ε 0 and k n >1such that

I Φ p ( k n x n c n ) + I Φ p ( k n y n c n ) = k n p 1.

Since

1 c n = x n Φ , p c n 1 k n ( 1 + I Φ p ( k n x n c n ) ) 1 p , ε 0 c n = y n Φ , p c n 1 k n ( 1 + I Φ p ( k n y n c n ) ) 1 p ,

then

I Φ p ( k n x n c n ) ( k n c n ) p 1, I Φ p ( k n y n c n ) ( ε 0 k n c n ) p 1.

Thus we obtain ( k n c n ) p 1+ ( ε 0 k n c n ) p 1 k n p 1,then k n c n p 1 + ε 0 p ,which means that the sequence ( k n ) is bounded. Then

I Φ ( k n y n c n ) = k n p 1 I Φ p ( k n x n c n ) k n p 1 ( k n c n ) p +10.

But k n y n c n Φ , p = ε 0 k n c n > ε 0 c n ε 0 ,according to Lemma 1.2 and Φ Δ 2 (0), this is a contradiction.

If Φ Δ 2 (0), then l Φ , p does not have the order continuity property, by Lemma 2.3, the proof isfinished. □

3 Weakly convergent sequence coefficient for Orlicz sequence spaces

In this section, our main aim is to calculate the weakly convergent sequence coefficientfor an Orlicz sequence space and further discuss the fixed point property of thisspace.

Let X denote a reflexive infinite dimensional Banach space, without Schurproperty automatically. For each sequence ( x n ) in X, we define the asymptoticdiameter and asymptotic radius respectively by

dian a ( x n ) = lim sup k { x n x m : n , m k } , r a ( x n ) = inf { lim sup n x n y : y conv ¯ ( x n ) } .

The weakly convergent sequence coefficient concerned with normal structure is animportant geometric parameter. It was introduced by Bynum [10] as follows.

WCS(X) is the supremum of the set of all numbersM with the property that for each weakly convergent sequence( x n ) with asymptotic diameter A, thereis some y in the closed convex hull of the sequence such that

M lim sup n x n yA.

A sequence ( x n ) in X is said to be asymptoticequidistant if

dian a ( x n )= lim inf n { x i x j : i j , i , j n } .

In this paper, we use the following equivalent definition of WCS(X). This definition was introduced in [27], where it was proved that

WCS ( X ) = inf { dian a ( x n ) : ( x n )  is an asymptotic equidistant sequence in ( X )  and  x n 0  weakly } .

It is obvious that 1WCS(X)2 (see[10]). A Banach space X is saidto have weakly uniform normal structure provided WCS(X)>1. See[11] for further information about thiscoefficient.

For p1,WSC( l p )= 2 1 p and WCS( L p (Ω))=min{ 2 1 p , 2 1 1 p }. A formula for calculatingthe weakly convergent sequences of reflexive Orlicz and Musielak-Orlicz sequence spacesequipped with the Luxemburg or Amemiya norm is found in [21, 23], respectively.

Theorem 3.1 IfΦ Δ 2 (0), 1p<,then

WCS( l Φ , p )=inf { inf k > 1 { c x k : I Φ p ( k x c x k ) = k p 1 2 } : x S ( l Φ , p ) } .

Proof Let d=inf{ inf k > 1 { c x k : I Φ p ( k x c x k )= k p 1 2 }:xS( l Φ , p )}. For any ε>0, thereexists xS( l Φ , p ) such that

inf { inf k > 1 { c x k : I Φ p ( k x c x k ) = k p 1 2 } : x S ( l Φ , p ) } <d+ε.

Then there exist k>1 and c x k <d+εsuch that I Φ p ( k x c x k )= k p 1 2 .Define

x 1 = ( x ( 1 ) , 0 , x ( 2 ) , 0 , x ( 3 ) , 0 , x ( 4 ) , 0 , x ( 5 ) , 0 , x ( 6 ) , 0 , ) , x 2 = ( 0 , x ( 1 ) , 0 , 0 , 0 , x ( 2 ) , 0 , 0 , 0 , 0 , 0 , 0 , 0 , x ( 3 ) , 0 , 0 , 0 , ) , x 3 = ( 0 , 0 , 0 , x ( 1 ) , 0 , 0 , 0 , 0 , 0 , 0 , 0 , x ( 2 ) , 0 , 0 , 0 , 0 , 0 , 0 , 0 , ) , ,

where ( x n ) have pairwise disjoint supports. Then x n Φ , p = x Φ , p =1(nN) for all nm and allk>1.According to the same method as in the proof of Theorem 2.2, we have x n 0 weakly,

x n x m d + ε Φ , p 1 k ( 1 + I Φ p ( k x n x m d ε ) ) 1 p = 1 k ( 1 + 2 I Φ p ( k x d + ε ) ) 1 p < 1 k ( 1 + 2 I Φ p ( k x c x k ) ) 1 p = 1 k ( 1 + k p 1 ) 1 p = 1 .

Then x n x m Φ , p d+ε,so dian a ( x n )d+ε,since ε is arbitrary, we have WCS( l Φ , p )d.

On the other hand, let { x n } in S( l Φ , p ) be an arbitrary asymptoticequidistant sequence such that x n 0weakly.

Since Φ Δ 2 (0), then by Lemma 1.5 forε>0,there exists 0<δ<εsuch that

I Φ (x)1,k 1 ε and I Φ (y)δ| I Φ p ( k ( x + y ) d ) I Φ p ( k x d ) |<ε.

Let n 1 =1and pick m 1 such that j > m 1 Φ( x n 1 ( j ) ε d )<δ and choose n 2 > n 1 such that j m 1 Φ( x n 2 ( j ) ε d )<δ. By x n (i)0 asn for i=1,2, ,we can find m 2 with m 2 > m 1 such that j > m 2 Φ( x n 2 ( j ) ε d )<δ. And so on,by induction, we find the sequence { n i } and { m i } of natural numbers with n 1 < n 2 < , m 1 < m 2 <satisfying

j > m i Φ ( x n i ( j ) ε d ) <δ, j m i Φ ( x n i ( j ) ε d ) <δ.

Take k i j K Φ ( x n i x n j d ).

  1. (1)

    If k i j 1, then x n i x n j d Φ , p 1 k i j 1, so x n i x n j d.

  2. (2)

    If 1< k i j 1 ε , set n<m, then

    x n i x n j d Φ , p p 1 k i j p ( 1 + ( l m i Φ ( k i j x n i ( l ) d ) ) p ε + ( l > m i Φ ( k i j x n i ( l ) d ) ) p ε ) 1 k i j p ( 1 + I Φ p ( k i j x n i ( l ) d ) δ ε + I Φ p ( k i j x n i ( l ) d ) δ ε ) 1 k i j p ( 1 + I Φ p ( k i j x n i ( l ) c x n i k i j ) + I Φ p ( k i j x n i ( l ) c x n i k i j ) 4 ε ) = 1 k i j p ( 1 + 2 k i j p 1 2 4 ε ) > 1 4 ε .

Therefore x n i x n j Φ , p d.

(3) If k i j 1 ε , then

x n i x n j d Φ , p p > 1 k i j p I Φ p ( k i j x n i x n j d ) ε p I Φ p ( x n i x n j ε d ) ε p ( I Φ p ( x n i ε d ) + I Φ p ( x n j ε d ) 4 ε ) ε p ( I Φ p ( x n i ε c x n i 1 ε ) + I Φ p ( x n j ε c x n j 1 ε ) 4 ε ) = ε p ( ε p 1 4 ε ) = 1 ε p 4 ε p + 1 ,

hence we get x n i x n j Φ , p dagain. Consequently dian a ( x n )d. By thearbitrariness of { x n } in S( l Φ , p ), it follows thatWCS( l Φ , p )d. □

Theorem 3.2 Let1p<and a Φ =0. l Φ , p hasweakly uniform normal structure if and only ifΦ Δ 2 (0).

Proof Necessity. If Φ Δ 2 (0), by Theorem 3.1,

WCS( l Φ , p )=inf { inf k > 1 { c x k : I Φ p ( k x c x k ) = k p 1 2 } : x S ( l Φ , p ) } .

Assume WCS( l Φ , p )=1, then for any0<ε<1there exist xS( l Φ , p ) and k>1 satisfying c x k <1+ε.By x Φ , p =1and Φ Δ 2 (0), there exists δ>0 such that I Φ ( x 2 )δ. Hence,

1 = 1 k p ( 2 I Φ p ( k x c x k ) + 1 ) 1 k p ( 1 + I Φ p ( k x 1 + ε ) ) + 1 k p I Φ p ( k x 1 + ε ) x 1 + ε Φ , p p + I Φ p ( x 1 + ε ) ( 1 1 + ε ) p + I Φ p ( x 2 ) ( 1 1 + ε ) p + δ p 1 + δ p as  ε 0 .

This is a contradiction.

Sufficiency. If not, Φ Δ 2 (0), then for any ε>0 thereexists 0<u<εsuch that

Φ ( ( 1 + ε ) u ) > 1 ε Φ(u).

We can find mN such that

1Φ(2ε)<mΦ ( ( 1 + ε ) u ) 1.

Take c>0satisfying mΦ((1+ε)u)+Φ(c)=1. ThenΦ(c)<Φ(2ε). Set

Then x n x n Φ , p S( l Φ , p ) and { x n x n Φ , p } is an asymptoticequidistant sequence. And because lim sup l 0 I Φ ( l x n ) l =0,we have x n 0 weakly. Finally, x n Φ , p = x 1 Φ , p x 1 Φ , =1,then

1 1 + ε ( x n x n x m x m ) Φ , p p x n x m 1 + ε Φ , p p 1 + I Φ p ( x n x m 1 + ε ) = 1 + ( 2 m Φ ( u ) + 2 Φ ( c 1 + ε ) ) p 1 + ( 2 m ε Φ ( ( 1 + ε ) u ) + 2 Φ ( c ) ) p 1 + ( 2 ε + 2 Φ ( 2 ε ) ) p ,

which implies that

dian a ( x n )(1+ε) ( 1 + ( 2 ε + 2 Φ ( 2 ε ) ) p ) ,

so WCS( l Φ , p )=1. □

According to the above proof, we have the following.

Corollary 3.3 Let1p<, a Φ =0andΦ Δ 2 (0), thenWCS( l Φ , p )=1.

Remark In the case that p=1, the situationdegrades to the case of classical Orlicz norm.

  1. 1.

    If Φ Δ 2 (0), then WCS( l Φ )=1.

  2. 2.

    If Φ Δ 2 (0), then

    WCS ( l Φ ) =inf { inf k > 1 { c x k : I Φ ( k x c x k ) = k 1 2 } : x S ( l Φ ) } .

Corollary 3.4

WCS ( l p ) = { 1 2 p , 1 p < , 1 , p = .

Next, we discuss the fixed point property of l Φ , p .

Theorem 3.5Φ Δ 2 (0)and a Φ =0,1p,then l Φ , p containsan asymptotically isometric copy of c 0 .

Proof If Φ Δ 2 (0), then there exist the sequence{ u k }0 and{ n k }N such that

Φ ( u k ) 1 2 k + 1 p + 1 , 1 2 k + 1 p + 1 n k Φ ( u k ) 1 2 k + 1 p , Φ ( ( 1 + 1 k ) u k ) > 2 k + 1 p + 1 Φ ( u k ) for all  k N .

Set

Define P: c 0 l Φ , p by

Pt= n = 1 t n x n

for t=( t 1 , t 2 ,) c 0 .It is obvious that P is linear.

For any λ>0,since t n 0, then there exists j 0 N such thatλ| t i |<1 for alli j 0 ,hence

I Φ ( λ P t ) = i = 1 n i Φ ( λ t i u i ) = i = 1 j 0 n i Φ ( λ t i u i ) + i = j 0 + 1 n i Φ ( λ t i u i ) i = 1 j 0 n i Φ ( λ t i u i ) + i = j 0 + 1 1 2 1 p + i < i = 1 j 0 n i Φ ( λ t i u i ) + 1 < ,

which implies Pt l Φ , p .Moreover, for any t=( t 1 , t 2 ,) c 0 ,we have

I Φ ( P t t ) = i = 1 n i Φ ( t i t u i ) i = 1 1 2 1 p + i = 2 1 p ,

then P t Φ , p 2 1 p P t Φ , t .

On the other hand, for any λ(0,1), there exists j 1 N such that

( 1 ε j 1 ) | t j 1 | λ sup { ( 1 ε i ) | t i | : i N } >1,

where ε i = 1 i + 1 (iN), then

| t j 1 | λ sup { ( 1 ε i ) | t i | : i N } > 1 1 ε j 1 =1+ 1 j 1

and

I Φ ( P t λ sup { ( 1 ε i ) | t i | : i N } ) > n j 1 Φ ( ( 1 + 1 j 1 ) u j 1 ) > 2 j 1 + 1 p + 1 n j 1 Φ( u j 1 )>1.

Therefore,

P t Φ , p P t Φ , sup { ( 1 ε i ) | t i | : i N } ,

which implies that l Φ , p contains an asymptotically isometric copy of c 0 . □

Theorem 3.6 Let1p, a Φ =0,then l Φ , p hasthe fixed point property if and only if it is reflexive.

Proof Since a reflexive Banach space X with WCS(X)>1 has the fixedpoint property, we only need to prove the necessity.

Suppose Φ Δ 2 (0), then by Theorem 3.5, l Φ , p contains an asymptotically isometric copy of c 0 . Hence l Φ , p does not have the fixed point property.

Suppose Ψ Δ 2 (0), then there exists yS( l Ψ , q ) such that I Ψ (ly)= for any l>1, and for everysequence { ε n } decreasing to 0, thereexist 0= i 1 < i 2 < i 3 <such that

i = i n + 1 i n + 1 y ( i ) e i Ψ , q >1 ε n for all nN.

Set y n = i = i n + 1 i n + 1 y(i) e i ,then there exists x n S( l Φ , p ) such that y n , x n = y n Ψ , q .Hence, for any α=(α(n)) l 1 ,we have

n = 1 | α ( n ) | = n = 1 | α ( n ) | x n Φ , p n = 1 α ( n ) x n Φ , p n = 1 α ( n ) x n , n = 1 sign ( α ( n ) ) y n = n = 1 | α ( n ) | x n , y n ( 1 ε n ) n = 1 | α ( n ) | .

Hence l Φ , p contains an asymptotically isometric copy of l 1 . ByTheorem 2 of [28], l Φ , p does not have the fixed point property. □

Theorem 3.7 If a Φ >0,1p<,then l Φ , p doesnot have the fixed point property.

Proof If a Φ >0, forany nN, define , then

1 + I Φ ( x n ) = 1 , 1 k ( 1 + I Φ p ( k x n ) ) 1 p = 1 k > 1 ( k ( 0 , 1 ) ) , 1 k ( 1 + I Φ p ( k x n ) ) 1 p > 1 ( k > 1 ) .

Therefore, x n Φ , p =1for any nN.

Moreover, we can prove that n = 1 x n Φ , p =1.Hence, define P: l l Φ , p by

Py= n = 1 y n x n

for all y=( y 1 ,, y n ,) l . Thenthe operator P is obviously linear, since

y = sup n | y n | = sup n | y n | x n Φ , p n = 1 y n x n Φ , p = P y Φ , p sup n | y n | n = 1 x n Φ , p = sup n | y n | = y ,

so we have P y Φ , p = y , then P is an orderisometry of l ontoa closed subspace P( l ) of l Φ , p . □

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Acknowledgements

This work was supported by the Provincial Education Department Fund (12531185) andpartly supported by the National Natural Science Foundation of China (61203191).

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He, X., Cui, Y. & Hudzik, H. The fixed point property of Orlicz sequence spaces equipped with thep-Amemiya norm. Fixed Point Theory Appl 2013, 340 (2013). https://doi.org/10.1186/1687-1812-2013-340

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Keywords

  • Orlicz sequence space
  • Opial property
  • weakly convergent sequence coefficient
  • weakly uniform normal structure
  • p-Amemiya norm