λ-statistical convergence in n-normed spaces

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λ-statistical convergence in n-normed spaces

Bipan Hazarika^1∗ and Ekrem Sava¸s²

Abstract

In this paper, we introduce the concept ofλ-statistical convergence in n-normed spaces. Some inclusion relations between the sets of statistically convergent andλ-statistically convergent sequences are established.

We find its relations to statistical convergence, (C,1)-summability and strong (V, λ)-summability inn-normed spaces.

1 Introduction

The notion of statistical convergence was introduced by Fast [8] and Schoen- berg [28] independently. Over the years and under different names statistical convergence has been discussed in the theory of Fourier analysis, ergodic theory and number theory. Later on it was further investigated from various points of view. For example, statistical convergence has been investigated in summability theory by ( Fridy [10], ˘Sal´at [26]), topological groups (C¸ akalli [1], [2]), topological spaces (Di Maio and Ko˘cinac[20]), function spaces (Caserta and Ko˘cinac [3], Caserta, Di Maio and Ko˘cinac [4]), locally convex spaces (Maddox[19]), measure theory (Cheng et al., [5], Connor and Swardson [6], Millar[21]) , fuzzy mathematics (Nuray and Sava¸s [24], Sava¸s [27]). In the re- cent years, generalization of statistical convergence have appeared in the study of strong integral summability and the structure of ideals of bounded continu- ous functions [6]. Mursaleen [23], introduced the λ-statistical convergence for real sequences. In this article, we consider only sequences of real numbers, so

Key Words: Statistical convergence;λ-statistical convergence;n-norm.

2010 Mathematics Subject Classification: 40A05; 40B50; 46A19; 46A45.

Received: November 2012 Revised: March 2013 Accepted: May 2013

∗Corresponding author

141

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that ”a sequence” means ”a sequence of real numbers”.

The notion of statistical convergence depends on the (natural or asymp- totic) density of subsets ofN.A subset of N is said to have natural density δ(E) if

δ(E) = lim

n→∞

1 n

n

X

k=1

χ_E(k) exists.

Definition 1.1. A sequencex= (xk) is said to bestatistically convergent to`if for every ε >0

δ({k∈N:|xk−`| ≥ε}) = 0.

In this case, we writeS−limx=`or (xk)−→^S `andS denotes the set of all statistically convergent sequences.

Letλ= (λm) be a non-decreasing sequence of positive numbers tending to

∞such that

λm+1≤λm+ 1, λ1= 1.

The collection of such sequencesλwill be denoted by ∆.

The generalized de la Vall´ee-Poussin mean is defined by tm(x) = 1

λm

X

k∈Im

xk,

whereI_m= [m−λ_m+ 1, m].

Definition 1.2.[17] A sequencex= (xk) is said to be (V, λ)-summable to a number`if

tm(x)→`, as m→ ∞.

If λm = m, then (V, λ)-summability reduces to (C,1)-summability. We write

[C, λ] = (

x= (xk) :∃`∈R, lim

m→∞

1 m

m

X

k=1

|xk−`|= 0 )

and

[V, λ] = (

x= (xk) :∃`∈R, lim

m→∞

1 λm

X

k∈Im

|xk−`|= 0 )

for the sets of sequencesx= (xk) which arestrongly Ces`aro summable (see [9]) andstrongly(V, λ)-summableto`, i.e. (xk)^[C,1]−→`and (xk)^[V,λ]−→`,respectively.

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Definition 1.3. [23] A sequence x = (xk) is said to be λ-satistically convergent orSλ-convergent to` if for everyε >0

m→∞lim 1

λ_m|{k∈Im:|xk−`| ≥ε}|= 0.

In this case we writeS_λ−limx=`or (x_k)−→^S^λ ` and S_λ={x= (x_k) :∃`∈R, S_λ−limx=`}.

It is clear that ifλ_m=m, thenS_λ is same asS.

The concept of 2-normed space was initially introduced by G¨ahler[12], in the mid of 1960’s, while that of n-normed spaces can be found in Misiak [22].

Since then, many others authors have studied this concept and obtained various results (see, for instance, Gunawan[14] ,G¨ahler[11], Gunawan and Mashadi ([13], [15]), Lewandowska[18], Dutta [7]).

2 Definitions and Preliminaries

Let n be a non negative integer and X be a real vector space of dimension d ≥n (d may be infinite). A real-valued function ||., ..., .|| from Xⁿ into R satisfying the following conditions:

(1)||x1, x₂, ..., x_n||= 0 if and only ifx₁, x₂, ..., x_n are linearly dependent, (2)||x₁, x₂, ..., x_n|| is invariant under permutation,

(3)||αx₁, x₂, ..., x_n||=|α|||x₁, x₂, ..., x_n||,for anyα∈R, (4)||x+x, x₂, ..., x_n|| ≤ ||x, x₂, ..., x_n||+||x, x₂, ..., x_n||

is called ann-norm onXand the pair (X,||., ..., .||) is called ann-normed space.

A trivial example of an n-normed space is X = Rⁿ, equipped with the Euclidean n-norm||x1, x2, ..., xn||E= the volume of the n-dimensional paral- lelepiped spanned by the vectorsx1, x2, ..., xnwhich may be given expicitly by the formula

||x1, x2, ..., xn||E=|det(xij)|=abs(det(< xi, xj >)), where xi= (xi1, xi2, ..., xin)∈Rⁿ for eachi= 1,2,3..., n.

Let (X,||., ..., .||) be an n-normed space of dimension d ≥ n ≥ 2 and {a1, a2, ..., an}be a linearly independent set inX.Then the function||., ..., .||_∞

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fromXⁿ⁻¹into Rdefined by

||x1, x2, ..., x_n−1||_∞= max

1≤i≤n{||x1, x2, ..., x_n−1, ai||}

defines an (n−1)-norm onX with respect to{a1, a2, ...an}and this is known as the derived (n−1)-norm (for details see [13]).

The standard n-norm on a real inner product space of dimensiond≥nis as follows:

||x₁, x₂, ..., x_n||_S = [det(< x_i, x_j>)]¹²,

where<, > denotes the inner product onX.If we takeX =Rⁿ then this n- norm is exactly the same as the Euclideann-norm||x1, x₂, ..., x_n||Ementioned earlier. Forn= 1 thisn-norm is the usual norm ||x1||=√

< x₁, x₁>(for further details see [13]).

Definition 2.1. A sequence (x_k) in an n-normed space (X,||., ..., .||) is said to beconvergent to `∈X with respect to the n-norm if for each ε >0 there exists an positive integern₀ such that||x_k−`, z₁, z₂, ..., z_n−1||< ε,for allk≥n0 and for everyz1, z2, ..., z_n−1∈X.

Definition 2.2. A sequence (xk) in ann-normed space (X,||., ...., .||) is said to be Cauchy with respect to the n-norm if for eachε > 0 there exists a positive integern0=n0(ε) such that ||xk−xm, z1, z2, ..., z_n−1||< ε, for all k, m≥n0 and for everyz1, z2, ..., z_n−1∈X.

If every Cauchy sequence in X converges to some ` ∈X, then X is said to becomplete with respect to the n-norm. Any completen-normed space is said to be ann-Banach space.

Definition 2.3. A sequence (xk) in ann-normed space (X,||., ...., .||) is said to be statistically-convergent to some`∈X with respect to the n-norm if for each ε >0 the set {k ∈N : ||xk−`, z1, z2, ..., zn−1|| ≥ε} has natural density zero, for everyz1, z2, ..., zn−1∈X.

In other words the sequence (xk) statistical converges to ` an n-normed spaceX if

m→∞lim 1

m|{k∈N:||xk−`, z1, z2, ..., z_n−1|| ≥ε}|= 0,

for each z1, z2, ..., z_n−1 ∈ X. Let S^nN(X) denotes the set of all statistically convergent sequences inn-normed spaceX.

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Recently, G¨urdal and Pehlivan [16] studied statistical convergence in 2- normed spaces. B.S. Reddy [25] extended this idea to n-normed space and studied some properties.

In the present paper we studyλ-statistical convergence inn-normed spaces.

We show that some properties of λ-statistical convergence of real numbers also hold for sequences inn-normed spaces. We find some relations related to statistical convergent, λ-statistical convergent sequences, (C,1)-summability and strong (V, λ)-summability inn-normed spaces.

3 λ-statistical convergent sequences in n-normed space X

In this section we defineλ-statistically convergent sequences inn-normed linear spaceX. Also, we obtained some basic properties of this notion inn-normed spaces.

Definition 3.1. A sequencex= (xk) in ann-normed space (X,||., ..., .||) is said to beλ-satistically convergent orSλ-convergent to `∈X with respect to then-norm if for everyε >0

m→∞lim 1

λ_m|{k∈Im:||xk−`, z1, z2, ..., z_n−1|| ≥ε}|= 0, for eachz1, z2, ..., zn−1∈X.In this case we writeS_λ^nN−limx=`or (xk)^S

nN

−→λ ` and

S_λ^nN(X) ={x= (xk) :∃`∈R, S_λ^nN−limx=`}.

LetS^nN_λ (X) denotes the set of allλ-statistically convergent sequences in the n-normed spaceX.

Definition 3.2. A sequencex= (x_k) in ann-normed space (X,||., ..., .||) is said to be (V, λ)-summable to`∈X with respect to then-norm if

tm(x)→`, as m→ ∞.

Ifλm=m,then (V, λ)-summability reduces to (C,1)-summability with respect to then-norm. We write

[C, λ]^nN(X) = (

x= (xk) :∃`∈R, lim

m→∞

1 m

m

X

k=1

||xk−`, z1, ..., z_n−1||= 0 )

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and [V, λ]^nN(X) =

(

x= (xk) :∃ `∈R, lim

m→∞

1 λ_m

X

k∈Im

||xk−`, z1, ..., z_n−1||= 0 )

for the sets of X-valued sequences x = (x_k) which are strongly Ces`aro summable andstrongly (V, λ)-summable to`with respect to then-norm, i.e., (x_k)^[C,1]

nN

−→ `and (x_k)^[V,λ]

nN

−→ `, respectively.

Theorem 3.1. Let X be an n−normed space and λ= (λn)∈∆. If (xk) is a sequence in X such that S^nN_λ −limxk =`exists, then it is unique.

Proof. Suppose that there exist elements`1, `2 (`16=`2) in X such that S^nN_λ − lim

k→∞x_k=`₁;S_λ^nN− lim

k→∞x_k =`₂.

Since`₁6=`₂,then`₁−`₂6= 0,so there existz₁, z₂, ..., z_n−1∈X such that

`₁−`₂ andz₁, z₂, ..., z_n−1are linearly independent. Therefore,

||`1−`2, z1, z2, ...z_n−1||= 2ε >0.

SinceS_λ^nN−limk→∞xk =`1andS_λ^nN−limk→∞xk =`2it follows that

m→∞lim 1 λm

|{k∈Im:||xk−`1, z1, z1, ...zn−1|| ≥ε}|= 0 and

m→∞lim 1

λ_m|{k∈Im:||xk−`2, z1, z2, ...z_n−1|| ≥ε}|= 0.

There isk∈I_m such that

||xk−`1, z1, z1, ...z_n−1||< εand||xk−`2, z1, z1, ...z_n−1||< ε.

Further, for thisk we have

||`1−`2, z1, z1, ...z_n−1|| ≤ ||xk−`1, z1, z1, ...z_n−1||+||xk−`2, z1, z1, ...z_n−1||<2ε which is a contradiction. This completes the proof.

The next theorem gives the algebraic characterization of λ-statistical convergence onn-normed spaces.

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Theorem 3.2. LetX be ann-normed space,λ= (λn)∈∆, x= (xk)and y= (yk)be two sequences inX.

(a)If S_λ^nN−limk→∞xk=`and c(6= 0)∈R,then S_λ^nN−limk→∞cxk=c`.

(b) If S_λ^nN −limk→∞xk = `1 and S_λ^nN −limk→∞yk = `2, then S_λ^nN − limk→∞(xk+yk) =`1+`2.

Proof of the theorem is straightforward, thus omitted.

Theorem 3.3. S_λ^nN(X)∩`_∞(X)is a closed subset of`_∞(X), if X is an n-Banach space.

Proof. Suppose that (xⁱ)_i∈N, xⁱ = (xⁱ_k)_k∈N, is a convergent sequence in S_λ^nN(X)∩`_∞(X) converging to x = (xk) ∈ `_∞(X). We need to prove that x ∈ S_λ^nN(X)∩`_∞(X). Assume that (xⁱ_k)k

S_λ^nN

→ `i, for all i ∈ N. Take a positive decreasing convergent sequence (εi)_i∈N,where εi= ₂^εi,for a given ε > 0. Clearly (εi)_i∈N converges to 0. Choose a positive integer i such that

||x−xⁱ, z1, z2, ..., z_n−1||_∞<^ε₄ⁱ, for everyz1, z2, ..., z_n−1∈X.Then we have

m→∞lim 1 λm

|{k∈Im:||xⁱ_k−`i, z1, z2, ..., z_n−1|| ≥ εi

4}|= 0 and

m→∞lim 1 λm

|{k∈Im:||xⁱ⁺¹_k −`i+1, z1, z2, ..., zn−1|| ≥ ε_i+1 4 }|= 0.

Since, 1 λm

nk∈I_m : ||xⁱ_k−`_i, z₁, z₂, ..., z_n−1|| ≥ ε_i

4 ∨

||xⁱ⁺¹_k −`i+1, z1, z2, ..., z_n−1|| ≥ εi+1

4 o

<1 and form∈N

n

k∈Im:||xⁱ_k−`i, z1, z2, ..., z_n−1|| ≥ εi

4 o ∩ nk∈I_m:||xⁱ⁺¹_k −`_i+1, z₁, z₂, ..., z_n−1|| ≥ ε_i+1

4 o

is infinite. Hence there must exists ak∈I_mfor which we have simultaneously,

||xⁱ_k−`i, z1, z2, ..., z_n−1||<εi

4 and ||xⁱ⁺¹_k −`i+1, z1, z2, ..., z_n−1||<εi+1

4 .

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Then it follows that

||`i−`i+1, z1, z2, ..., zn−1||

≤ ||`i−xⁱ_k, z₁, z₂, ...z_n−1||+||xⁱ_k−xⁱ⁺¹_k , z₁, z₂, ...z_n−1||+

+||xⁱ⁺¹_k −`_i+1, z₁, z₂, ...z_n−1||

≤ ||xⁱ_k−`i, z1, z2, ...z_n−1||+||xⁱ⁺¹_k −`i+1, z1, z2, ...z_n−1||

+||x−xⁱ, z1, z2, ...z_n−1||_∞+||x−xⁱ⁺¹, z1, z2, ...z_n−1||_∞

< ε_i 4 +ε_i+1

4 +ε_i 4 +ε_i+1

4 < εi.

This implies that (`_i) is a Cauchy sequence inX and there is an element`∈X such that`i→`asi→ ∞. We need to prove that (xk)^S

nN

−→λ `.

For anyε >0, choosei∈N such thatεi< ^ε₄,

||xk−xⁱ_k, z1, z2, ..., z_n−1||_∞< ε

4,||`i−`, z1, z2, ..., z_n−1||< ε 4.

Then 1

λm

|{k∈I_m:||x_k−`, z₁, z₂, ..., z_n−1|| ≥ε}|

≤ 1 λm

|{k∈Im:||xⁱ_k−`i, z1, z2, ..., z_n−1||

+||xk−xⁱ_k, z1, z2, ..., z_n−1||_∞+||`i−`, z1, z2, ..., z_n−1|| ≥ε}|

≤ 1 λm

|{k∈Im:||xⁱ_k−`i, z1, z2, ..., zn−1||+ε 4 +ε

4 ≥ε}|

≤ 1 λm

|{k∈I_m:||xⁱ_k−`_i, z₁, z₂, ..., z_n−1|| ≥ ε

2}| →0 as m→ ∞.

This gives that (x_k)^S

nN

−→λ `,which completes the proof.

Theorem 3.4. LetX be ann-normed space and letλ= (λm)∈∆.Then (i) (x_k)^[V,λ]

nN

−→ `⇒(x_k)^S

nN

−→λ `,

(ii) [V, λ]^nN(X)is a proper subset of S_λ^nN(X), (iii)x∈`_∞(X)and(xk)^S

nN

−→λ `,then(xk)^[V,λ]

nN

−→ `and hence (xk)^[C,1]

nN

−→ `, provided x= (xk)is not eventually constant,

(iv)S_λ^nN(X)∩`_∞(X) = [V, λ]^nN(X)∩`_∞(X).

Proof. (i) Ifε >0 and (xk)^[V,λ]

nN

−→ `,we can write

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X

k∈I_m

||xk−`, z1, z2, ..., z_n−1|| ≥

≥ X

k∈Im,||xk−`,z1,z₂,...,zn−1||≥ε

||xk−`, z1, z2, ..., zn−1|| ≥

≥ε|{k∈Im:||xk−`, z1, z2, ...zn−1|| ≥ε}|

and so

1 ελm

X

k∈Im

||x_k−`, z₁, z₂, ..., z_n−1|| ≥ 1

λ_m|{k∈Im:||xk−`, z1, z2, ..., z_n−1|| ≥ε}|.

This proves the result.

(ii) In order to establish that the inclusion [V, λ]^nN(X) ⊂ S_λ^nN(X) is proper, we define a sequencex= (xk) by

x_k=

k, ifm−[√

λ_m] + 1≤k≤m;

0, otherwise.

Thenx /∈`_∞and for every ε >0(0< ε <1), 1

λm

X

k∈Im

||x_k−0, z₁, z₂, ..., z_n−1|| ≤ [√ λ_m] λm

→0, as m→ ∞,

i.e. (x_k)^S

nN

−→λ 0. On the other hand, 1

λm

| {k∈I_m:||xk−0, z₁, z₂, ..., z_n−1|| ≥ε} | → ∞, as m→ ∞, i.e. (x_k) does not converge to 0 in [V, λ]^nN(X).

(iii) Suppose that (xk)^S

nN

−→λ ` and (xk)∈`∞(X). Then there exists aM >0 such that||xk−`, z1, z2, ...zn−1|| ≤M for allk∈N.Givenε >0,we have

1 λm

X

k∈Im

||xk−`, z1, z2, ..., z_n−1||= 1

λm

X

k∈Im,||xk−`,z1,z₂,...,zn−1||≥^ε₂

||xk−`, z1, z2, ..., zn−1||

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+ 1 λm

X

k∈Im,||xk−`,z1,z₂,...,zn−1||<^ε₂

||xk−`, z1, z2, ..., zn−1||

≤ M λm

|{k∈I_m:||xk−`, z₁, z₂, ..., z_n−1|| ≥ ε 2}|+ε

2, This shows that (x_k)^[V,λ]

nN

−→ `.

Again, we have 1 m

m

X

k=1

||xk−`, z1, z2, ..., zn−1||

≤ 1 m

m−λm

X

k=1

||xk−`, z1, z2, ..., zn−1||+ 1 m

X

k∈Im

||xk−`, z1, z2, ..., zn−1||

≤ 1 λ_m

m−λm

X

k=1

||xk−`, z1, z2, ..., z_n−1||+ 1 λ_m

X

k∈Im

||xk−`, z1, z2, ..., z_n−1||

≤ 2 λm

X

k∈Im

||xk−`, z1, z2, ..., zn−1||.

Hence (xk)^[C,1]

nN

−→ `, because (xk)^[V,λ]

nN

−→ `.

(iv) This is an immediate consequence of (i), (ii) and (iii).

Theorem 3.5. Let X be ann-normed space and let λ= (λm)∈∆.Then S^nN(X)⊂S_λ^nN(X)if and only if lim infmλm

m >0.

Proof. Suppose first that lim infmλm

m >0.Then a givenε >0,we have 1

m|{k≤m:||x_k−`, z₁, z₂, ..., z_n−1|| ≥ε}| ≥ 1

m|{k∈Im:||xk−`, z1, z2, ..., z_n−1|| ≥ε}| ≥ λm

m. 1 λm

|{k∈Im:||xk−`, z1, z2, ..., z_n−1|| ≥ε}|.

It follows that (x_k)^S

nN

−→`⇒(x_k)^S

nN

−→λ `. HenceS^nN(X)⊂S_λ^nN(X).

Conversely, suppose that lim inf_m^λ_m^m = 0. Then we can select a subsequence (m(j))^∞_j=1 such that

λ_m(j) m(j) < 1

j.

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We define a sequence x= (xk) as follows:

xk=

1, ifk∈I_m(j), j= 1,2,3, ...;

0, otherwise.

Thenxis statistically convergent, sox∈S^nN(X).Butx /∈[V, λ]^nN(X).

Theorem 3.4(iii) implies thatx /∈S_λ^nN(X).This completes the proof.

Theorem 3.6. Let X be ann-normed space and let λ= (λm)∈∆ such that limmλm

m = 1.Then S_λ^nN(X)⊂S^nN(X).

Proof. Since lim_m^λ_m^m = 1, then forε >0,we observe that 1

m|{k≤m:||xk−`, z1, z2, ..., zn−1|| ≥ε}|

≤ 1

m|{k≤m−λ_m:||xk−`, z₁, z₂, ..., z_n−1|| ≥ε}|

+1

m|{k∈Im:||xk−`, z1, z2, ..., z_n−1|| ≥ε}|

≤ m−λm

m + 1

m|{k∈Im:||xk−`, z1, z2, ..., z_n−1|| ≥ε}|

= m−λm

m +λm

m 1

λ_m|{k∈Im:||xk−`, z1, z2, ..., z_n−1|| ≥ε}|.

This implies that (xk) is statistically convergent, if (xk) isλ-statistically convergent. HenceS_λ^nN(X)⊂S^nN(X).

Remark: We do not know whether the condition lim_m^λ_m^m = 1 in the Theorem 3.6 is necessary and leave it as an open problem.

Acknowledgements: The authors thank the referees for their comments which improved the presentation of the paper.

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1Department of Mathematics, Rajiv Gandhi University,

Rono Hills, Doimukh-791112, Arunachal Pradesh, India E-mail: bh [email protected]

2Department of Mathematics, Istanbul Ticaret University, Uskudar-Istanbul, Turkey E-mail: [email protected]

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