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Research | Open | Published:

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

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)1δ, where sep( x n )=inf{d( x n , x m ):nm}>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 k2 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)rδ [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 [1418], 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 if t n (x) 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 kN, 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 kN, 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 kN. 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 xB(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 NN such that ( x n r (k)) converges for each kN with 1kN, since ( x n ( k ) ) k = 1 is bounded for each kN. Therefore, there is t N N for each NN 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 NN. Then there exists κ>0 such that for all NN,

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 xB(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 kN, 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 <jδ 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 k2.

Proof Let ε>0 and ( x n ) B d (V[λ,p]) with sep( x n )ε. For each mN, 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 iN, 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 kN. 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 mN. 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 k2. 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,,n1) 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 mN. For 1in1, 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 (iN), 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 <rδ 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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Correspondence to Murat Karakaş.

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The authors declare that they have no competing interests.

Authors’ contributions

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)