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Some geometric properties of the metric space V[λ,p]

Journal of Inequalities and Applicationsvolume 2013, Article number: 28 (2013) | Download Citation

Abstract

In this study, we consider the space V[λ,p] with an invariant metric. Then, we examine some geometric properties of the linear metric space V[λ,p] such as property (β), property (H) and k-NUC property.

MSC:40A05, 46A45, 46B20.

1 Introduction

Let X be a vector space over the scalar field of real numbers and d be an invariant metric on X. We denote B d (X) and S d (X) as follows:

Let (X,d) be a linear metric space and B d (X) (resp., S d (X)) be a closed unit ball (resp., the unit sphere) of X. A linear metric space (X,d) has property (β) if and only if for each r>0 and ε>0, there exists δ>0 such that for each element x∈ B d (0,r) and each sequence ( x n ) in B d (0,r) with sep( x n )≥ε, there is an index k for which d( x + x k 2 ,0)≤1−δ, where sep( x n )=inf{d( x n , x m ):n≠m}>ε [1]. If for each x∈ S d (0,r) and ( x n )⊂ S d (0,r), x n → w x implies x n →x, a linear metric space (X,d) is said to have property (H). Let k≥2 be an integer. A linear metric space (X,d) is said to be k-nearly uniform convex (k-NUC) if for every ε>0 and r>0, there exists δ>0 such that for any sequence ( x n )⊂ B d (0,r) with sep( x n )≥ε, there are s 1 , s 2 ,…, s k such that d( x s 1 + x s 2 + ⋯ + x s k k ,0)≤r−δ [2]. These properties have been studied by Mongkolkeha and Pumam [3], Sanhan and Suantai [4], Cui et al. [5] and Cui and Hudzik [6].

Ahuja et al. [7] introduced the notions of strict convexity and U.C.I (uniform convexity) in linear metric spaces which are generalizations of the corresponding concepts in linear normed spaces. Later, Sastry and Naidu [8] introduced the notions of U.C.II and U.C.III in linear metric spaces and showed that these three forms are not always equivalent. Further, Junde et al. [9, 10] showed that if a linear metric space is complete and U.C.I, then it is reflexive.

In summability theory, de la Vallée-Poussin mean was first used to define the (V,λ)-summability by Leindler [11]. (V,λ)-summable sequences have been studied by many authors including Et et al. [12, 13], Savas [14–18], Savas and Malkowsky [19] and Şimsek et al. [20, 21]. Let Λ=( λ k ) be a nondecreasing sequence of positive real numbers tending to infinity and let λ 1 =1 and λ k + 1 ≤ λ k +1. The generalized de la Vallée-Poussin mean is defined by t n (x)= 1 λ n ∑ k ∈ I n x k , where I n =[n− λ n +1,n] for n=1,2,… . A sequence x=( x k ) is said to be (V,λ)-summable to a number ℓ if t n (x)→ℓ as n→∞. If λ n =n, then (V,λ)-summability is reduced to Cesàro summability.

Let w be the space of all real sequences. Let p=( p k ) be a bounded sequence of positive real numbers. Şimşek et al. [20] defined the space V[λ,p] as follows:

V[λ,p]= { x = ( x k ) ∈ ω : ∑ k = 1 ∞ ( 1 λ k ∑ j ∈ I k | x j | ) p k < ∞ } .

If λ k =k, then V[λ,p]=ces(p) [22]. If λ k =k and p k =p for all k∈N, then V[λ,p]= ces p  [23]. Paranorm on V[λ,p] is given by

h(x)= ( ∑ k = 1 ∞ ( 1 λ k ∑ j ∈ I k | x j | ) p k ) 1 M ,

where M=max{1,H} and H=sup p k . If p k =p for all k∈N, the notation V p (λ) is used in place of V[λ,p] and the norm on V p (λ) is as follows:

∥ x ∥ V p ( λ ) = ( ∑ k = 1 ∞ ( 1 λ k ∑ j ∈ I k | x j | ) p ) 1 p .

ρ: V ρ [λ,p]→[0,∞], ρ(x)=( ∑ k = 1 ∞ ( 1 λ k ∑ j ∈ I k | x j | ) p k ) is a modular on V ρ [λ,p] and the Luxemburg norm on V ρ [λ,p] is defined by ∥ x ∥ L =inf{σ>0:ρ( x σ )≤1} for all x∈ V ρ [λ,p]. The Amemiya norm on the space V ρ [λ,p] can be similarly introduced as follows:

∥ x ∥ A = inf σ > 0 1 σ ( 1 + ρ ( σ x ) ) for all x∈ V ρ [λ,p].

2 Main results

In this part of the paper, our main purpose is to define a metric on V[λ,p] and show that V[λ,p] possesses property (β), property (H) and k-NUC property. Let p=( p k ) be a bounded sequence of real numbers with p k >1 for all k∈N. The mapping d(x,y)= ( ∑ k = 1 ∞ ( 1 λ k ∑ j ∈ I k | x ( j ) − y ( j ) | ) p k ) 1 / H is a metric on the space V[λ,p], where M=max(1,H=sup p k ) and m=inf p k since the function | t | p is convex for p>1. First, we will show that the space V[λ,p] has property (β) under the above metric. To do this, we need the following two lemmas. To prove these lemmas, we use the technique given in Sanhan and Mongkolkeha [1].

Lemma 2.1 Let y,z∈(V[λ,p],d). If β∈(0,1), then

( d ( y + z , 0 ) ) M ≤ ( d ( y , 0 ) ) M + 2 M β ( d ( y , 0 ) ) M + 2 M β M − 1 ( d ( z , 0 ) ) M .

Proof Let y,z∈(V[λ,p],d) and 0<β<1. Then

( d ( y + z , 0 ) ) M = ∑ k = 1 ∞ ( 1 λ k ∑ j ∈ I k | y ( j ) + z ( j ) | ) p k ≤ ∑ k = 1 ∞ ( ( 1 − β ) 1 λ k ∑ j ∈ I k | y ( j ) | + β 1 λ k ∑ j ∈ I k | y ( j ) + z ( j ) β | ) p k ≤ ( 1 − β ) ∑ k = 1 ∞ ( 1 λ k ∑ j ∈ I k | y ( j ) | ) p k + β ∑ k = 1 ∞ ( 1 λ k ∑ j ∈ I k | y ( j ) + z ( j ) β | ) p k ≤ ∑ k = 1 ∞ ( 1 λ k ∑ j ∈ I k | y ( j ) | ) p k + 2 M β ∑ k = 1 ∞ ( 1 λ k ∑ j ∈ I k | y ( j ) | ) p k + 2 M ∑ k = 1 ∞ ( 1 λ k ∑ j ∈ I k | z ( j ) β | ) p k ≤ ∑ k = 1 ∞ ( 1 λ k ∑ j ∈ I k | y ( j ) | ) p k + 2 M β ∑ k = 1 ∞ ( 1 λ k ∑ j ∈ I k | y ( j ) | ) p k + 2 M β M − 1 ∑ k = 1 ∞ ( 1 λ k ∑ j ∈ I k | z ( j ) | ) p k = ( d ( y , 0 ) ) M + 2 M β ( d ( y , 0 ) ) M + 2 M β M − 1 ( d ( z , 0 ) ) M .

 □

Lemma 2.2 Let y,z∈(V[λ,p],d). Then for any ε>0 and L>0, there exists δ>0 such that

| ( d ( y + z , 0 ) ) M − ( d ( y , 0 ) ) M |<ε,

where ( d ( y , 0 ) ) M ≤L and ( d ( z , 0 ) ) M ≤δ.

Proof Let ε>0 and L>0. For β= ε 2 M + 1 ( L + ε ) , we take δ= ε β M − 1 2 M + 1 . From Lemma 2.1, we have

( d ( y + z , 0 ) ) M ≤ ( d ( y , 0 ) ) M + 2 M β ( d ( y , 0 ) ) M + 2 M β M − 1 ( d ( z , 0 ) ) M ≤ ( d ( y , 0 ) ) M + 2 M β L + 2 M β M − 1 δ ≤ ( d ( y , 0 ) ) M + 2 M ε 2 M + 1 L L + ε + 2 M β M − 1 ε β M − 1 2 M + 1 ≤ ( d ( y , 0 ) ) M + ε 2 + ε 2 ≤ ( d ( y , 0 ) ) M + ε
(2.1)

and

( d ( y , 0 ) ) M ≤ ( d ( y + z , 0 ) ) M + 2 M β ( d ( y + z , 0 ) ) M + 2 M β M − 1 ( d ( − z , 0 ) ) M ≤ ( d ( y + z , 0 ) ) M + 2 M β ( ( d ( y , 0 ) ) M + ε ) + 2 M β M − 1 δ ≤ ( d ( y + z , 0 ) ) M + 2 M β ( L + ε ) + 2 M β M − 1 ε β M − 1 2 M + 1 = ( d ( y + z , 0 ) ) M + 2 M ε 2 M + 1 ( L + ε ) ( L + ε ) + ε 2 = ( d ( y + z , 0 ) ) M + ε 2 + ε 2 = ( d ( y + z , 0 ) ) M + ε .
(2.2)

From (2.1) and (2.2), we obtain that | ( d ( y + z , 0 ) ) M − ( d ( y , 0 ) ) M |<ε. □

Theorem 2.3 The space (V[λ,p],d) has property (β).

Proof Let ε>0 and ( x n )⊂B(V[λ,p],d) such that sep( x n )≥ε and x∈B(V[λ,p],d). We take y N =(0,0,…,0, ∑ k = 1 N y(k),y(N+1),y(N+2),…). By using the diagonal method, we can find a subsequence ( x n r ) of ( x n ) for each N∈N such that ( x n r (k)) converges for each k∈N with 1≤k≤N, since ( x n ( k ) ) k = 1 ∞ is bounded for each k∈N. Therefore, there is t N ∈N for each N∈N such that sep( ( x n N ) r > t N )≥ε. So, there is a sequence of positive integers ( t N ) N = 1 ∞ with t 1 < t 2 < t 3 ⋯ such that d( x t N N ,0)≥ ε 2 for all N∈N. Then there exists κ>0 such that for all N∈N,

∑ k = N ∞ ( 1 λ k ∑ j ∈ I k | x t N | ) p k ≥κ.
(2.3)

By Lemma 2.2, there exists δ 0 such that

| ( d ( y + z , 0 ) ) M − ( d ( y , 0 ) ) M |< κ 2 m ,
(2.4)

where ( d ( y , 0 ) ) M < j M and ( d ( z , 0 ) ) M ≤ δ 0 . There exists N 1 ∈N such that ( d ( x N 1 , 0 ) ) M ≤ δ 0 if x∈B(V[λ,p]) and ( d ( x , 0 ) ) M ≤ δ 0 . Let us take y= x t N 1 N 1 and z= x N 1 . Hence, we have

∑ k = N 1 ∞ ( 1 λ k ∑ j ∈ I k | x ( j ) + x t N 1 ( j ) 2 | ) p k ≤ ∑ k = N 1 ∞ ( 1 λ k ∑ j ∈ I k | x t N 1 ( j ) 2 | ) p k + κ 2 m .
(2.5)

From (2.3), (2.4), (2.5) and by using the convexity of the function f(t)= | t | p k for all k∈N, we obtain that

( d ( y + z 2 , 0 ) ) M = ∑ k = 1 ∞ ( 1 λ k ∑ j ∈ I k | x ( j ) + x t N 1 ( j ) 2 | ) p k = ∑ k = 1 N 1 − 1 ( 1 λ k ∑ j ∈ I k | x ( j ) + x t N 1 ( j ) 2 | ) p k + ∑ k = N 1 ∞ ( 1 λ k ∑ j ∈ I k | x ( j ) + x t N 1 ( j ) 2 | ) p k ≤ ∑ k = 1 N 1 − 1 ( 1 λ k ∑ j ∈ I k | x ( j ) + x t N 1 ( j ) 2 | ) p k + ∑ k = N 1 ∞ ( 1 λ k ∑ j ∈ I k | x t N 1 ( k ) 2 | ) p k + κ 2 m ≤ 1 2 ∑ k = 1 N 1 − 1 ( 1 λ k ∑ j ∈ I k | x ( j ) | ) p k + 1 2 ∑ k = 1 N 1 − 1 ( 1 λ k ∑ j ∈ I k | x t N 1 ( j ) | ) p k + 1 2 m ∑ k = N 1 ∞ ( 1 λ k ∑ j ∈ I k | x t N 1 ( j ) | ) p k + κ 2 m ≤ 1 2 ∑ k = 1 N 1 − 1 ( 1 λ k ∑ j ∈ I k | x ( j ) | ) p k + 1 2 ∑ k = 1 ∞ ( 1 λ k ∑ j ∈ I k | x t N 1 ( j ) | ) p k − 2 m − 2 2 m + 1 ∑ k = N 1 ∞ ( 1 λ k ∑ j ∈ I k | x t N 1 ( j ) | ) p k + κ 2 m < j M 2 + j M 2 − 2 m − 2 2 m + 1 κ + κ 2 m = j M − κ 2 .

Therefore, we have d( y + z 2 ,0)< ( j M − κ 2 ) 1 / M <j−δ whenever δ∈(0,j− ( j M − κ 2 ) 1 / M ). Consequently, the space (V[λ,p],d) possesses property (β). □

Now, we will show that the space (V[λ,p],d) has k-NUC property.

Theorem 2.4 The space V[λ,p] is k-NUC for any integer k≥2.

Proof Let ε>0 and ( x n )⊂ B d (V[λ,p]) with sep( x n )≥ε. For each m∈N, let

x n m = ( 0 , 0 , … , x n ( m ) , x n ( m + 1 ) , … ) .
(2.6)

Since the sequence ( x n ( i ) ) i = 1 ∞ is bounded for each i∈N, by using the diagonal method, we can find a subsequence ( x n l ) of ( x n ) such that ( x n l (k)) converges for each k∈N. Therefore, there is an increasing sequence t m with sep( ( x n l m ) l > t m )≥ε. Hence, there exists a sequence of positive integers ( r m ) m = 1 ∞ with r 1 < r 2 < r 3 <⋯ such that d( x r m m ,0)≥ ε 2 for all m∈N. Then there is ζ>0 such that

∑ k = m ∞ ( 1 λ k ∑ j ∈ I k | x r m | ) p k ≥ζ.
(2.7)

Let α>0 such that 1<α< lim k → ∞ inf p k . Let ε 1 = n α − 1 − 1 ( n − 1 ) n α ζ 2 for k≥2. From Lemma 2.2, there is a δ>0 such that

| ( d ( y + z , 0 ) ) M − ( d ( y , 0 ) ) M |< ε 1 ,
(2.8)

where ( d ( y , 0 ) ) M < r M and ( d ( z , 0 ) ) M ≤δ. Then there exist positive integers m i (i=1,2,…,n−1) with m 1 < m 2 <⋯< m n − 1 such that d( x i m i ,0)≤δ. Now, define m n = m n − 1 +1. Then we have d( x r m n m n ,0)≥ζ for all m∈N. For 1≤i≤n−1, let s i =i and s n = r m n . By using (2.6), (2.7), (2.8) and the convexity of the function f i (u)= | u | p i (i∈N), we obtain

Thus, we have d( x s 1 ( j ) + x s 2 ( j ) + ⋯ + x s n ( j ) n ,0)< ( r M − ( n α − 1 − 1 n α ) ζ 2 ) 1 / M <r−δ for δ∈(0,r− ( r M − ( n α − 1 − 1 n α ) ζ 2 ) 1 / M ). Hence, (V[λ,p],d) is k-NUC. □

Since k-NUC implies NUC and NUC implies property (H), by using the previous theorem, we can give the following result.

Corollary 2.5 The space (V[λ,p],d) has property (H).

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  1. Department of Mathematics, Muş Alparslan University, Muş, 49100, Turkey

    • Muhammed Çinar

  2. Department of Statistics, Bitlis Eren University, Bitlis, 13000, Turkey

    • Murat Karakaş

  3. Department of Mathematics, Firat University, Elazig, 23119, Turkey

    • Mikail Et

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Correspondence to Murat Karakaş.

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The authors declare that they have no competing interests.

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MC, MK and ME have contributed to all parts of the article. All authors read and approved the final manuscript.

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Keywords

  • linear metric space
  • Luxemburg norm
  • de la Vallée-Poussin mean
  • k-NUC property
  • property (H)