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Some geometric properties of a new modular space defined by Zweier operator

Abstract

In this paper, we define the modular space Z σ (s,p) by using the Zweier operator and a modular. Then, we consider it equipped with the Luxemburg norm and also examine the uniform Opial property and property β. Finally, we show that this space has the fixed point property.

MSC:40A05, 46A45, 46B20.

Dedication

Dedicated to Professor Hari M Srivastava

1 Introduction

In literature, there are many papers about the geometrical properties of different sequence spaces such as [19]. Opial [10] introduced the Opial property and proved that the sequence spaces p (1<p<) have this property but L p [0,2π] (p2, 1<p<) does not have it. Franchetti [11] showed that any infinite dimensional Banach space has an equivalent norm that satisfies the Opial property. Later, Prus [12] introduced and investigated the uniform Opial property for Banach spaces. The Opial property is important because Banach spaces with this property have the weak fixed point property.

2 Definition and preliminaries

Let (X,) be a real Banach space and let S(X) (resp. B(X)) be the unit sphere (resp. the unit ball) of X. A Banach space X has the Opial property if for any weakly null sequence { x n } in X and any x in X{0}, the inequality lim n infx< lim n inf x n +x holds. We say that X has the uniform Opial property if for any ε>0 there exists r>0 such that for any xX with xε and any weakly null sequence { x n } in the unit sphere of X, the inequality 1+r lim n inf x n +x holds.

For a bounded set AX, the ball-measure of noncompactness was defined by β(A)=inf{ε>0:A can be covered by finitely many balls with diameter ε}. The function Δ defined by Δ(ε)=inf{1inf(x:xA):A is closed convex subset of B(X) with β(A)ε} is called the modulus of noncompact convexity. A Banach space X is said to have property (L), if lim ε 1 Δ(ε)=1. This property is an important concept in the fixed point theory and a Banach space X possesses property (L) if and only if it is reflexive and has the uniform Opial property.

A Banach space X is said to satisfy the weak fixed point property if every nonempty weakly compact convex subset C and every nonexpansive mapping T:CC(TxTyxy,x,yC) have a fixed point, that is, there exists xC such that T(x)=x. Property (L) and the fixed point property were also studied by Goebel and Kirk [13], Toledano et al. [14], Benavides [15], Benavides and Phothi [16]. A Banach space X is said to have property (H) if every weakly convergent sequence on the unit sphere is convergent in norm. Clarkson [17] introduced the uniform convexity, and it is known that the uniform convexity implies the reflexivity of Banach spaces. Huff [18] introduced the concept of nearly uniform convexity of Banach spaces. A Banach space X is called uniformly convex (UC) if for each ε>0, there is δ>0 such that for x,yS(X), the inequality xy>ε implies that 1 2 (x+y)<1δ. For any xB(X), the drop determined by x is the set D(x,B(X))=conv({x}B(X)). A Banach space X has the drop property (D) if for every closed set C disjoint with B(X), there exists an element xC such that D(x,B(X))C={x}. Rolewicz [19] showed that the Banach space X is reflexive if X has the drop property. Later, Montesinos [20] extended this result and proved that X has the drop property if and only if X is reflexive and has property (H). A sequence { x n } is said to be ε-separated sequence for some ε>0 if

sep( x n )=inf { x n x m : n m } >ε.

A Banach space X is called nearly uniformly convex (NUC) if for every ε>0, there exists δ(0,1) such that for every sequence ( x n )B(X) with sep( x n )>ε, we have conv( x n )((1δ)B(X)). Huff [18] proved that every (NUC) Banach space is reflexive and has property (H). A Banach space X has property (β) if and only if for each ε>0, there exists δ>0 such that for each element xB(X) and each sequence ( x n ) in B(X) with sep( x n )ε, there is an index k for which x + x k 2 <1δ.

For a real vector space X, a function ρ:X[0,] is called a modular if it satisfies the following conditions:

  1. (i)

    ρ(x)=0 if and only if x=0,

  2. (ii)

    ρ(αx)=ρ(x) for all scalar α with |α|=1,

  3. (iii)

    ρ(αx+βy)ρ(x)+ρ(y) for all x,yX and all α,β0 with α+β=1.

The modular ρ is called convex if

  1. (iv)

    ρ(αx+βy)αρ(x)+βρ(y) for all x,yX and all α,β0 with α+β=1.

For any modular ρ on X, the space

X ρ = { x X : ρ ( σ x ) <  for some  σ > 0 }

is called a modular space. In general, the modular is not subadditive and thus it does not behave as a norm or a distance. But we can associate the modular with an F-norm. A functional :X[0,] defines an F-norm if and only if

  1. (i)

    x=0x=0,

  2. (ii)

    αx=x whenever |α|=1,

  3. (iii)

    x+yx+y,

  4. (iv)

    if α n α and x n x0, then α n x n αx0.

F-norm defines a distance on X by d(x,y)=xy. If the linear metric space (X,d) is complete, then it is called an F-space. The modular space X ρ can be equipped with the following F-norm:

x=inf { α > 0 : ρ ( x α ) α } .

If the modular ρ is convex, then the equality x=inf{α>0:ρ( x α )1} defines a norm which is called the Luxemburg norm.

A modular ρ is said to satisfy the δ 2 -condition if for any ε>0, there exist constants K2, a>0 such that ρ(2u)Kρ(u)+ε for all u X ρ with ρ(u)a. If ρ provides the δ 2 -condition for any a>0 with K2 dependent on a, then ρ provides the strong δ 2 -condition (briefly ρ δ 2 s ).

Let us denote by 0 the space of all real sequences. The Cesàro sequence spaces

Ces p = { x 0 : n = 1 ( n 1 i = 1 n | x i | ) p < } ,1p<,

and

Ces = { x 0 : sup n n 1 i = 1 n | x i | < } ,

were introduced by Shiue [21]. Jagers [22] determined the Köthe duals of the sequence space Ces p (1<p<). It can be shown that the inclusion p Ces p is strict for 1<p< although it does not hold for p=1. Also, Suantai [23] defined the generalized Cesàro sequence space by

ces(p)= { x 0 : ρ ( λ x ) <  for some  λ > 0 } ,

where ρ(x)= n = 1 ( 1 n i = 1 n | x ( i ) | ) p n . If p=( p n ) is bounded, then

ces(p)= { x = ( x k ) : n = 1 ( n 1 i = 1 n | x ( i ) | ) p n < } .

The sequence space C(s,p) was defined by Bilgin [24] as follows:

C(s,p)= { x = ( x k ) : r = 0 ( 2 r r k s | x k | ) p r < , s 0 }

for p=( p r ) with inf p r >0, where r denotes a sum over the ranges 2 r k< 2 r + 1 . The special case of C(s,p) for s=0 is the space

Ces(p)= { x = ( x k ) : r = 0 ( 2 r r | x k | ) p r < }

which was introduced by Lim [25]. Also, the inclusion Ces(p)C(s,p) holds. A paranorm on C(s,p) is given by

ρ(x)= ( r = 0 ( 2 r r k s | x k | ) p r ) 1 / M

for M=max(1,H) and H=sup p r <.

The Z-transform of a sequence x=( x k ) is defined by ( Z x ) n = y n =α x n +(1α) x n 1 by using the Zweier operator

Z=( z n k )={ α , k = n , 1 α , k = n 1 , 0 , otherwise for n,kN and αF{0},

where is the field of all complex or real numbers. The Zweier operator was studied by Şengönül and Kayaduman [26].

Now we introduce a new modular sequence space Z σ (s,p) by

Z σ (s,p)= { x 0 : σ ( t x ) < ,  for some  t > 0 } ,

where σ(x)= r = 0 ( 2 r r k s | α x k + ( 1 α ) x k 1 | ) p r < and s0. If we take α=1, then Z σ (s,p)=C(s,p); if α=1 and s=0, then Z σ (s,p)=Ces(p). It can be easily seen that σ: Z σ (s,p)[0,] is a modular on Z σ (s,p). We define the Luxemburg norm on the sequence space Z σ (s,p) as follows:

x=inf { t > 0 : σ ( x t ) 1 } ,x Z σ (s,p).

It is easy to see that the space Z σ (s,p) is a Banach space with respect to the Luxemburg norm.

Throughout the paper, suppose that p=( p r ) is bounded with p r >1 for all rN and

for iN and x 0 . In addition, we will require the following inequalities:

| a k + b k | p k C ( | a k | p k + | b k | p k ) ,| a k + b k | t k | a k | t k +| b k | t k ,

where t k = p k M 1 and C=max{1, 2 H 1 } with H=sup p k .

3 Main results

Since p is reflexive and convex, (p)-type spaces have many useful applications, and it is natural to consider a geometric structure of these spaces. From this point of view, we generalized the space C(s,p) by using the Zweier operator and then obtained the equality Z σ (s,p)=Ces(p), that is, it was seen that the structure of the space Ces(p) was preserved. In this section, our goal is to investigate a geometric structure of the modular space Z σ (s,p) related to the fixed point theory. For this, we will examine property (β) and the uniform Opial property for Z σ (s,p). Finally, we will give some fixed point results. To do this, we need some results which are important in our opinion.

Lemma 3.1 [2]

If σ δ 2 s , then for any L>0 and ε>0, there exists δ>0 such that

|σ(u+v)σ(u)|<ε,

where u,v X σ with σ(u)L and σ(v)δ.

Lemma 3.2 [2]

If σ δ 2 s , convergence in norm and in modular are equivalent in X σ .

Lemma 3.3 [2]

If σ δ 2 s , then for any ε>0, there exists δ=δ(ε)>0 such that x1+δ implies σ(x)1+ε.

Now we give the following two lemmas without proof.

Lemma 3.4 If x L <1 for any x Z σ (s,p), then σ(x) x L .

Lemma 3.5 For any x Z σ (s,p), x L =1 if and only if σ(x)=1.

Lemma 3.6 If liminf p r >1, then for any x Z σ (s,p), there exist k 0 N and μ(0,1) such that

σ ( x k 2 ) 1 μ 2 σ ( x k )

for all kN with k k 0 , where and 2 r k< 2 r + 1 .

Proof Let kN be fixed. Then there exists r k N such that k I r k . Let γ be a real number 1<γliminf p r , and so there exists k 0 N such that γ< p r k for all k k 0 . Choose μ(0,1) such that ( 1 2 ) γ 1 μ 2 . Therefore, we have

σ ( x k 2 ) = r = 0 ( 2 r r k s | α x ( k ) + ( 1 α ) x ( k 1 ) 2 | ) p r = r = 0 ( 1 2 ) p r ( 2 r r k s | α x ( k ) + ( 1 α ) x ( k 1 ) | ) p r ( 1 2 ) γ r = 0 ( 2 r r k s | α x ( k ) + ( 1 α ) x ( k 1 ) | ) p r < 1 μ 2 σ ( x k )

for each x Z σ (s,p) and k k 0 . □

Lemma 3.7 If σ δ 2 s , then for any ε(0,1), there exists δ(0,1) such that σ(x)1ε implies x1δ.

Proof Suppose that lemma does not hold. So, there exist ε>0 and x n Z σ (s,p) such that σ( x n )<1ε and 1 2 x n 1. Take s n = 1 x n 1 , and so s n 0 as n. Let P=sup{σ(2 x n ):nN}. There exists D2 such that

σ(2u)Dσ(u)+1
(3.2)

for every u Z σ (s,p) with σ(u)<1, since σ δ 2 s . We have

σ(2 x n )Dσ( x n )+1<D+1

for all nN by (3.1). Therefore, 0<P< and from Lemma 3.5 we have

1 = σ ( x n x n ) = σ ( 2 s n x n + ( 1 s n ) x n ) s n σ ( 2 x n ) + ( 1 s n ) σ ( x n ) s n P + ( 1 ε ) ( 1 ε ) .

This is a contradiction. So, the proof is complete. □

Theorem 3.8 The space Z σ (s,p) has property (β).

Proof Let ε>0 and ( x n )B( Z σ (s,p)) with sep( x n )ε and xB( Z σ (s,p)). For each lN, we can find r k N such that 2 r k l< 2 r k + 1 . Let

Since for each iN, ( x n ( i ) ) i = 1 is bounded, by using the diagonal method, we can find a subsequence ( x n j ) of ( x n ) such that ( x n j (i)) converges for each iN with 1il. Therefore, there exists an increasing sequence of positive integers t l such that sep( ( x n j l ) j t l )ε. Thus, there exists a sequence of positive integers ( r l ) l = 1 with r 1 < r 2 < such that x r l l ε 2 for all lN. Since σ δ 2 s , there is η>0 such that

σ ( x r l l ) ηfor all lN
(3.4)

from Lemma 3.3. However, there exist k 0 N and μ(0,1) such that

σ ( v k 2 ) 1 μ 2 σ ( v k )
(3.5)

for all v Z σ (s,p) and k k 0 by Lemma 3.6. There exists δ>0 such that

σ(y)1 μ η 4 y1δ
(3.6)

for any y Z σ (s,p) by Lemma 3.7.

By Lemma 3.1, there exists δ 0 such that

|σ(u+v)σ(u)|< μ η 4 ,
(3.7)

where σ(u)1 and σ(v) δ 0 . Hence, we get that σ(x)1 since xB( Z σ (s,p)). Then there exists k k 0 such that σ( x k ) δ 0 . Let u= x r l l and v= x l . Then

σ ( u 2 ) <1andσ ( v 2 ) < δ 0 .

We obtain from (3.3) and (3.5) that

σ ( u + v 2 ) σ ( u 2 ) + μ η 4 1 μ 2 σ(u)+ μ η 4 .
(3.8)

Choose s i = r l i . By the inequalities (3.2), (3.3), (3.6) and the convexity of the function f(u)=|u | p r , we have

σ ( x + x s k 2 ) = r = 0 ( 2 r r k s | α ( x ( k ) + x s i ( k ) ) + ( 1 α ) ( x ( k 1 ) + x s i ( k 1 ) ) 2 | ) p r = r = 0 r k 1 ( 2 r r k s | α ( x ( k ) + x s i ( k ) ) + ( 1 α ) ( x ( k 1 ) + x s i ( k 1 ) ) 2 | ) p r + r = r k ( 2 r r k s | α ( x ( k ) + x s i ( k ) ) + ( 1 α ) ( x ( k 1 ) + x s i ( k 1 ) ) 2 | ) p r 1 2 r = 0 r k 1 ( 2 r r k s | α x ( k ) + ( 1 α ) x ( k 1 ) | ) p r + 1 2 r = 0 r k 1 ( 2 r r k s | α x s i ( k ) + ( 1 α ) x s i ( k 1 ) | ) p r + r = r k ( 2 r r k s | α x s i ( k ) + ( 1 α ) x s i ( k 1 ) 2 | ) p r + μ η 4 1 2 r = 0 r k 1 ( 2 r r k s | α x ( k ) + ( 1 α ) x ( k 1 ) | ) p r + 1 2 r = 0 r k 1 ( 2 r r k s | α x s i ( k ) + ( 1 α ) x s i ( k 1 ) | ) p r + 1 μ 2 r = r k ( 2 r r k s | α x s i ( k ) + ( 1 α ) x s i ( k 1 ) 2 | ) p r + μ η 4 1 2 r = 0 r k 1 ( 2 r r k s | α x ( k ) + ( 1 α ) x ( k 1 ) | ) p r + 1 2 r = 0 ( 2 r r k s | α x s i ( k ) + ( 1 α ) x s i ( k 1 ) | ) p r μ 2 r = r k ( 2 r r k s | α x s i ( k ) + ( 1 α ) x s i ( k 1 ) 2 | ) p r + μ η 4 1 2 + 1 2 μ η 2 + μ η 4 = 1 μ η 4 .

So, the inequality (3.4) implies that x + x s k 2 1δ. Consequently, the space Z σ (s,p) possesses property (β). □

Since property (β) implies NUC, NUC implies property (D) and property (D) implies reflexivity, we can give the following result from Theorem 3.8.

Corollary 3.9 The space Z σ (s,p) is nearly uniform convex, reflexive and also it has property (D).

Theorem 3.10 The space Z σ (s,p) has the uniform Opial property.

Proof Let ε>0 and x Z σ (s,p) be such that xε and ( x n ) be a weakly null sequence in S( Z σ (s,p)). By σ δ 2 s , there exists ζ(0,1) independent of x such that σ(x)>ζ by Lemma 3.2. Also since σ δ 2 s , by Lemma 3.1, there is ζ 1 (0,ζ) such that

|σ(y+z)σ(y)|< ζ 4
(3.10)

whenever σ(y)1 and σ(z) ζ 1 . Take r 0 N such that

r = r 0 + 1 ( 2 r r k s | α x ( k ) + ( 1 α ) x ( k 1 ) | ) p r < ζ 1 4 .
(3.11)

Hence, we have

ζ < r = 1 r 0 ( 2 r r k s | α x ( k ) + ( 1 α ) x ( k 1 ) | ) p r + r = r 0 + 1 ( 2 r r k s | α x ( k ) + ( 1 α ) x ( k 1 ) | ) p r r = 1 r 0 ( 2 r r k s | α x ( k ) + ( 1 α ) x ( k 1 ) | ) p r + ζ 1 4
(3.12)

and this implies that

r = 1 r 0 ( 2 r r k s | α x ( k ) + ( 1 α ) x ( k 1 ) | ) p r > ζ ζ 1 4 > ζ ζ 4 = 3 ζ 4 .
(3.13)

Since x n w 0, by the inequality (3.10), there exists r 0 N such that

r = 1 r 0 ( 2 r r k s | α ( x n ( k ) + x ( k ) ) + ( 1 α ) ( x n ( k 1 ) + x ( k 1 ) ) | ) p r > 3 ζ 4 .
(3.14)

Again, by x n w 0, there is r 1 > r 0 such that for all r> r 1

x n | r 0 <1 ( 1 ζ 4 ) 1 / M ,
(3.15)

where p r MN for all rN. Therefore, we obtain that

x n | N r 0 > ( 1 ζ 4 ) 1 / M
(3.16)

by the triangle inequality of the norm. It follows from the definition of the Luxemburg norm that

1 σ ( x n | N r 0 ( 1 ζ 4 ) 1 / M ) = r = r 0 + 1 ( 2 r r k s | α x n ( k ) + ( 1 α ) x n ( k 1 ) | ( 1 ζ 4 ) 1 / M ) p r ( 1 ( 1 ζ 4 ) 1 / M ) M r = r 0 + 1 ( 2 r r k s | α x n ( k ) + ( 1 α ) x n ( k 1 ) | ) p r
(3.17)

and this implies that

r = r 0 + 1 ( 2 r r k s | α x n ( k ) + ( 1 α ) x n ( k 1 ) | ) p r 1 ζ 4 .
(3.18)

By (3.7), (3.8), (3.11), (3.15) and since x n w 0 x n 0 (coordinatewise), we have for any r> r 1 that

σ ( x n + x ) = r = 1 r 0 ( 2 r r k s | α ( x n ( k ) + x ( k ) ) + ( 1 α ) ( x n ( k 1 ) + x ( k 1 ) ) | ) p r + r = r 0 + 1 ( 2 r r k s | α ( x n ( k ) + x ( k ) ) + ( 1 α ) ( x n ( k 1 ) + x ( k 1 ) ) | ) p r r = r 0 + 1 ( 2 r r k s | α ( x n ( k ) + x ( k ) ) + ( 1 α ) ( x n ( k 1 ) + x ( k 1 ) ) | ) p r ζ 4 + 3 ζ 4 3 ζ 4 + ( 1 ζ 4 ) ζ 4 = 1 + ζ 4 .

Since σ δ 2 s , it follows from Lemma 3.3 that there is τ depending on ζ only such that x n +x1+τ. □

Corollary 3.11 The space Z σ (s,p) has property (L) and the fixed point property.

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Et, M., Karakaş, M. & Çınar, M. Some geometric properties of a new modular space defined by Zweier operator. Fixed Point Theory Appl 2013, 165 (2013). https://doi.org/10.1186/1687-1812-2013-165

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