Generalized difference sequence spaces associated with a multiplier sequence on a real n normed space

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R E S E A R C H

Open Access

Generalized difference sequence spaces

associated with a multiplier sequence on a

real

n

-normed space

¸Sükran Konca

*

_{and Metin Ba¸sarır}

*_{Correspondence:}

[email protected] Department of Mathematics, Sakarya University, Sakarya, 54187, Turkey

Abstract

The purpose of this paper is to introduce new sequence spaces associated with a multiplier sequence by using an inﬁnite matrix, an Orlicz function and a generalized

B-diﬀerence operator on a realn-normed space. Some topological properties of these spaces are examined. We also deﬁne a new concept, which will be called

(Bμ)n_-statistical_A_{-convergence, and establish some inclusion connections between} the sequence spaceW(A,Bμ,p,·,. . .,·) and the set of all (Bμ)n_{-statistically}

A-convergent sequences.

MSC: Primary 40A05; secondary 40B50; 46A19; 46A45

Keywords: statistical convergence; multiplier sequence; generalized diﬀerence operator; inﬁnite matrix;n-norm

1 Introduction

Letw,l∞,candcbe the linear spaces of all, bounded, convergent and null sequences

x= (xk) for allk∈N, respectively.

LetXandYbe two subsets ofw. By (X,Y), we denote the class of all matrices ofAsuch thatAm(x) =

∞

k=amkxkconverges for eachm∈N, the set of all natural numbers, and the

sequenceAx= (Am(x))m∞=∈Yfor allx∈X.

LetA= (amk) be an inﬁnite matrix of complex numbers. ThenAis said to be regular if

and only if it satisﬁes the following well-known Silverman-Toeplitz conditions: () sup_m∞_k_=|amk|<∞,

() lim_m_→∞amk= for eachk∈N,

() limm→∞∞k=amk= .

The idea of statistical convergence was given by Zygmund [] in . The concept of statistical convergence was introduced by Fast [] and Schoenberg [] independently for the real sequences. Later on, it was further investigated from a sequence point of view and linked with the summability theory by Fridy [] and many others. The natural density of a subsetEofNis denoted by

δ(E) = lim

m→∞



m{k∈E:k≤m},

where the vertical bar denotes the cardinality of the enclosed set.

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Spaces of strongly summable sequences were studied by Kuttner [], Maddox [] and others. The class of sequences that are strongly Cesaro summable with respect to a mod-ulus was introduced by Maddox [] as an extension of the definition of strongly Cesaro summable sequences. Connor [] has further extended this definition to a definition of strongA-summability with respect to a modulus, whereA= (amk) is a non-negative

regu-lar matrix, and established some connections between strongA-summability with respect to a modulus andA-statistical convergence.

Assume now thatAis a non-negative regular summability matrix. Then a sequencex= (xk) is said to beA-statistically convergent to a numberLifδA(K) =limm→∞

∞ k=amk×

χK(k) =  or, equivalently,limm→∞k∈Kamk=  for everyε> , whereK={k∈N:|xk–

L| ≥ε}andχK(k) is the characteristic function ofK. We denote this limit bystA-limx=L

[] (see also [, , ]).

ForA=C, the Cesaro matrix,A-statistical convergence reduces to statistical

conver-gence (see [, ]). TakingA=I, the identity matrix,A-statistical convergence coincides with ordinary convergence. We note that ifA= (amk) is a regular summability matrix for

whichlimmmaxk|amk|= , thenA-statistical convergence is stronger than usual

conver-gence []. It should be also noted that the concept ofA-statistical convergence may also be given in normed spaces [].

The notion of diﬀerence sequence space was introduced by Kızmaz []. It was further generalized by Et and Çolak [] as follows:Z(μ_{) =}_{_x_{= (}_x

k)∈w: (μxk)∈Z}forZ=

l∞,candc, whereμis a non-negative integer,μxk=μ–xk–μ–xk+,xk=xkfor

allk∈Nor equivalent to the following binomial representation:

μxk=

μ

v=

(–)v

μ

v

xk+v.

These sequence spaces were generalized by Et and Başarır [] taking Z=l∞(p),c(p)

andc(p).

Dutta [] introduced the following diﬀerence sequence spaces using a new diﬀerence operator:Z((η)) ={x= (xk)∈w:(η)x∈Z}forZ=l∞,candc, where(η)x= ((η)xk) =

(xk–xk–η) for allk,η∈N.

In [], Dutta introduced the sequence spacesc(·,·,₍μ_η₎,p),c(·,·,μ₍η),p),l∞(·,·,

μ₍_η₎,p), m(·,·,μ₍_η₎,p) and m(·,·,n(η),p), where η, μ∈ N and

μ

(η)x= (

μ

(η)xk) =

(μ₍_η–₎ xk–μ(η–) xk–η) and₍η)xk=xkfor allk,η∈N, which is equivalent to the following

binomial representation:

μ₍_η₎xk=

μ

v=

(–)v

μ

v

xk–ηv.

The diﬀerence sequence spaces have been studied by several authors, [–]. Başar and Altay [] introduced the generalized diﬀerence matrixB= (bmk) for allk,m∈N, which

is a generalization of()-diﬀerence operator, by

bmk=

⎧ ⎪ ⎨ ⎪ ⎩

r, k=m, s, k=m– ,

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Başarır and Kayıkçı [] deﬁned the matrixBμ_{= (}_bμ

mk) which reduced the diﬀerence

ma-trixμ_()in caser= ,s= –. The generalizedBμ_{-diﬀerence operator is equivalent to the} following binomial representation:

Bμx=Bμ(xk) =

μ

v=

μ

v

rμ–vsvxk–v.

Related articles can be found in [–].

The concept of -normed space was initially introduced by Gähler [] in the mid of s, while that ofn-normed spaces can be found in Misiak []. Since then, many others have used these concepts and obtained various results; see, for instance, Gunawan [], Gunawan and Mashadi [], Gunawanet al.[] (see also [–]).

2 Deﬁnitions and preliminaries

Letnbe a non-negative integer and letXbe a real vector space of dimensiond≥n≥. A real-valued function·, . . . ,·onXn_{satisﬁes the following conditions:}

() x, . . . ,xn= if and only ifx, . . . ,xnare linearly dependent,

() x, . . . ,xnis invariant under permutation,

() αx, . . . ,xn–,xn=|α|x, . . . ,xn–,xnfor anyα∈R,

() x, . . . ,xn–,y+z ≤ x, . . . ,xn–,y+x, . . . ,xn–,z.

Then it is called ann-norm onXand the pair (X,·, . . . ,·) is called ann-normed space. A trivial example of an n-normed space isX=Rn_{equipped with the following Euclidean}_n

-norm:x, . . . ,xnE=|det(xij)|, wherexi= (xi, . . . ,xin)∈Rnfor eachi= , . . . ,n. The stan-dardn-norm onX, whereXis a real inner product space of dimensiond≥n, is deﬁned as

x, . . . ,xnS:=

x,x · · · x,xn

..

. . .. ...

xn,x · · · xn,xn

 

,

where ·,· denotes the inner product onX. IfX=Rn_{, then this}_n_{-norm is exactly the}

same as the Euclidean n-normx, . . . ,xnE as mentioned earlier. Notice that forn= ,

then-norm above is the usual normxS=x,x



 _{which gives the length of}_x__{, while}

forn= , it deﬁnes the standard -normx,xS= (xS.xS–x,x )



 _which

rep-resents the area of the parallelogram spanned by x and x. Further, if X =R, then x,x,xS=x,x,xErepresents the volume of the parallelograms spanned byx,x

andx. In generalx, . . . ,xnSrepresents the volume of then-dimensional parallelepiped

spanned byx, . . . ,xninX.

A sequence (xk) in an n-normed space (X,·, . . . ,·) is said to converge to someL∈

X in the n-norm if for each ε>  there exists a positive integer n=n(ε) such that xk–L,z, . . . ,zn–<εfor allk≥nand for everyz, . . . ,zn–∈X[].

An Orlicz function is a function M: [,∞) →[,∞) which is continuous, non-decreasing and convex withM() = ,M(x) >  forx>  andM(x)→ ∞asx→ ∞. It is well known that ifMis a convex function, thenM(αx)≤αM(x) with  <α< .

Let= (k) be a sequence of nonzero scalars. Then, for a sequence spaceE, the

multi-plier sequence spaceE, associated with the multiplier sequence, is deﬁned as

E=

x= (xk)∈w: (kxk)∈E

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The following well-known inequality will be used throughout the paper. Let p= (pk)

be any sequence of positive real numbers with  <h=inf_kpk≤pk ≤supkpk=H,D= max{, H–_}_{. Then we have, for all}_a

k,bk∈Cand for allk∈N,

|ak+bk|pk≤D

|ak|pk₊|_bk|pk_, _(.)

and fora∈C,|a|pk≤_max{|_a|h_,|_a|H}_.

In this paper, we introduce some new sequence spaces on a realn-normed space by using an inﬁnite matrix, an Orlicz function and a generalizedBμ-diﬀerence operator. Further, we examine some topological properties of these sequence spaces. We also introduce a new concept which will be called (Bμ)n-statisticalA-convergence in ann-normed space.

3 Main results

In this section, we give some new sequence spaces on a realn-normed space and investi-gate some topological properties of these spaces. We also give some inclusion relations.

Let A= (amk) be an inﬁnite matrix of non-negative real numbers, let p= (pk) be a

bounded sequence of positive real numbers for all k ∈N, and let = (k) be a

se-quence of nonzero scalars. Further, letMbe an Orlicz function and (X,·, . . . ,·) be an n-normed space. We denote the space of allX-valued sequence spaces byw(n–X) and x= (xk)∈w(n–X) byx= (xk) for brevity. We deﬁne the following sequence spaces for

every nonzeroz,z, . . . ,zn–∈Xand for someρ> :

WA,Bμ,M,p,·, . . . ,·

=

x= (xk) : lim

m→∞

∞

k=

amk

MB

μ xk–L

ρ ,z, . . . ,zn–

pk

= 

for someL∈X

,

W

A,Bμ,M,p,·, . . . ,·

=

x= (xk) : lim

m→∞

∞

k=

amk

MB

μ xk

ρ ,z, . . . ,zn–

pk

= 

,

W∞A,Bμ,M,p,·, . . . ,·

=

x= (xk) :sup m

∞

k=

amk

MB

μ xk

ρ ,z, . . . ,zn–

pk<∞

,

where and throughout the paperBμxk=

μ

v=

μ

v

rμ–v_sv_x

k–vk–vandμ,k∈N. If we

con-sider some special cases of the spaces above, the following are obtained:

() If we takeμ= , then the spaces above are reduced toW(A,,M,p,·, . . . ,·), W(A,,M,p,·, . . . ,·),W∞(A,,M,p,·, . . . ,·), respectively.

() If we taker= ,s= –, then we get the spacesW(A,μ,M,p,·, . . . ,·), W(A,μ,M,p,·, . . . ,·),W∞(A,

μ

,M,p,·, . . . ,·).

() IfM(x) =x, then the spaces above are denoted byW(A,Bμ,p,·, . . . ,·), W(A,Bμ,p,·, . . . ,·),W∞(A,B

μ

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() Ifpk= for allk∈Nand= (k) = (, , , . . .), then the spaces above are reduced

to the sequence spacesW(A,Bμ_,_M_,_·_{, . . . ,}_·₎_,_W

(A,Bμ,M,·, . . . ,·),

W∞(A,Bμ,M,·, . . . ,·), respectively.

() IfM(x) =xandpk= for allk∈N, then the spaces above are denoted by

W(A,Bμ,·, . . . ,·),W(A,Bμ,·, . . . ,·),W∞(A,B μ

,·, . . . ,·), respectively. () If we takeA=C,i.e., the Cesaro matrix, then the spaces above are reduced to the

spacesW(Bμ,M,p,·, . . . ,·),W(Bμ,M,p,·, . . . ,·),W∞(B μ

,M,p,·, . . . ,·). () If we takeA= (amk)is de la Vallee Poussin mean,i.e.,

amk=



λm, k∈Im= [m–λm+ ,m],

, otherwise, (.)

whereλmis a non-decreasing sequence of positive numbers tending to∞and

λm+≤λm+ ,λ= , then the spaces above are denoted by

W(λ,Bμ,M,p,·, . . . ,·),W(λ,Bμ,M,p,·, . . . ,·),W∞(λ,B μ

,M,p,·, . . . ,·). () By a lacunary sequenceθ= (km),m= , , . . ., wherek= , we mean an increasing

sequence of non-negative integers withhm= (km–km–)→ ∞asm→ ∞. The

intervals determined byθare denoted byIm= (km–,km]. Let

amk=



hm, km–<k≤km,

, otherwise. (.)

Then we obtain the spacesW(θ,Bμ,M,p,·, . . . ,·),W(θ,Bμ,M,p,·, . . . ,·)and W∞(θ,Bμ,M,p,·, . . . ,·), respectively.

() If we takeA=I, whereIis an identity matrix andpk= for allk∈N, then the

spaces above are reduced to the sequence spacesc(Bμ,M,·, . . . ,·), c(Bμ,M,·, . . . ,·)andl∞(B

μ

,M,·, . . . ,·), respectively.

() If we takeA=I, whereIis an identity matrix,M(x) =xandpk= for allk∈N,

then we denote the spaces above by the sequence spacesc(Bμ,·, . . . ,·), c(Bμ,·, . . . ,·)andl∞(B

μ

,·, . . . ,·).

Theorem . W(A,Bμ,M,p,·, . . . ,·), W(A,Bμ,M,p,·, . . . ,·) and W∞(A,B μ ,M,p,

·, . . . ,·)are linear spaces.

Proof We consider onlyW(A,Bμ,M,p,·, . . . ,·). Others can be treated similarly. Letx,y∈ W(A,Bμ,M,p,·, . . . ,·) andα,βbe scalars. Suppose thatx→Landy→L. Then there

exists|α|ρ+|β|ρ>  such that

∞

k=

amk

MB

μ

(αxk+βyk) – (αL+βL) |α|ρ+|β|ρ

,z, . . . ,zn–

pk

≤ ∞

k=

amk

M

_|

α|ρ |α|ρ+|β|ρ

Bμxk–L

ρ

,z, . . . ,zn–

+ |β|ρ

|α|ρ+|β|ρ

Bμyk–L

ρ

,z, . . . ,zn–Bμ

pk

≤ ∞

k=

amk

|α|ρ |α|ρ+|β|ρ

MB μ xk–L

ρ

,z, . . . ,zn–

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+ |β|ρ

|α|ρ+|β|ρ

MB μ yk–L

ρ

,z, . . . ,zn–

pk

≤D

∞

k=

amk

MB

μ xk–L

ρ

,z, . . . ,zn–

pk

+D

∞

k=

amk

MB

μ yk–L

ρ

,z, . . . ,zn–

pk,

which leads us, by taking limit asm→ ∞, to the fact that we get (αx+βy)∈W(A,Bμ,M,p,

·, . . . ,·).

Theorem . For any two sequences p= (pk)and q= (qk)of positive real numbers and for

any two n-norms·, . . . ,·,·, . . . ,·on X,the following holds:Z(A,Bμ,M,p,·, . . . ,·)∩

Z(A,Bμ,M,q,·, . . . ,·)=∅,where Z=W,Wand W∞.

Proof Since the zero element belongs to each of the above classes of sequences, thus the

intersection is non-empty.

Theorem . Let A= (amk)be a non-negative matrix,and let p= (pk)be a bounded

se-quence of positive real numbers.Then,for any ﬁxed m∈N,the sequence space W∞(A,Bμ, M,p,·, . . . ,·)is a paranormed space for every nonzero z, . . . ,zn–∈X and for someρ> 

with respect to the paranorm deﬁned by

gm(x) =inf

ρpmH _: _∞

k=

amk

MB

μ xk

ρ ,z, . . . ,zn–

pk



H <∞

.

Proof Thatgm(θ) =  andgm(–x) =gm(x) are easy to prove. So, we omit them. Let us take

x= (xk) andy= (yk) inW∞(A,Bμ,M,p,·, . . . ,·). Let

A(x) =

ρ>  :

∞

k=

amk

MB

μ xk

ρ ,z, . . . ,zn–

pk<∞

,

A(y) =

ρ>  :

∞

k=

amk

MB

μ yk

ρ ,z, . . . ,zn–

pk<∞

for every nonzeroz, . . . ,zn–∈X. Letρ∈A(x) andρ∈A(y), then we have

_∞

k=

amk

MB

μ

(xk+yk)

(ρ+ρ)

,z, . . . ,zn–

pkH

<∞

by using Minkowski’s inequality forp= (pk) > . Thus,

gm(x+y) =inf

(ρ+ρ)

pm

H _:_ρ_∈_A₍_x_),_ρ_∈_A₍_y₎

≤infρ

pm

H _:_ρ_∈_A₍_x₎₊_inf_ρ_pmH _:_ρ_∈_A₍_y₎

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We also getgm(x+y)≤gm(x) +gm(y) for  <pk≤ by using (.). Hence, we complete the

proof of this condition of the paranorm. Finally, we show that the scalar multiplication is continuous. Wheneverα→ andxis ﬁxed implygm(αx)→. Also, wheneverx→θ

andαis any number implygm(αx)→. By using the deﬁnition of the paranorm, for every

nonzeroz, . . . ,zn–∈X, we have

gm(αx) =inf

ρpmH _: _∞

k=

amk

MB

μ (αxk)

ρ ,z, . . . ,zn–

pkH

<∞

.

Then

gm(αx) =inf

(α)pmH _:

_∞

k=

amk

MB

μ xk

,z, . . . ,zn–

pkH

<∞

,

where=ρ_α. Since|α|pk≤_max{|_α|h_,|_α|H}_{, therefore}|_α|pkH ≤₍_max{|_α|h_,|_α|H}₎H_{. Then the} required proof follows from the following inequality

gm(αx)≤

max|α|h,|α|H



H

·inf

pmH _: _∞

k=

amk

MB

μ xk

,z, . . . ,zn–

pkH

<∞

=max|α|h,|α|H



H_g

m(x).

Theorem . Let M,M,Mbe Orlicz functions.Then the following hold:

() Let <h≤pk≤.ThenZ(A,Bμ,M,p,·, . . . ,·)⊆Z(A,B μ

,M,·, . . . ,·),where Z=W,W.

() Let <pk≤H<∞.ThenZ(A,Bμ,M,·, . . . ,·)⊆Z(A,B μ

,M,p,·, . . . ,·),where Z=W,W.

() W(A,Bμ,M,p,·, . . . ,·)∩W(A,Bμ,M,p,·, . . . ,·)⊆

W(A,Bμ,M+M,p,·, . . . ,·).

Proof () We give the proof for the sequence spaceW(A,Bμ,M,p,·, . . . ,·) only. The other can be proved by a similar argument. Let (xk)∈W(A,Bμ,M,p,·, . . . ,·) and  < h≤pk≤, then

∞

k=

amk

MB

μ xk

ρ ,z, . . . ,zn–

≤

∞

k=

amk

MB

μ xk

ρ ,z, . . . ,zn–

pk

.

Hence, we have the result by taking the limit asm→ ∞. This completes the proof. () Let  <pk≤H<∞and (xk)∈W(A,Bμ,M,·, . . . ,·). Then, for each  <ε< , there exists a positive integerMsuch that

∞

k=

amk

MB

μ xk

ρ ,z, . . . ,zn–

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for allm>M. This implies that ∞ k= amk MB

μ xk

ρ ,z, . . . ,zn–

pk ≤ ∞ k= amk MB

μ xk

ρ ,z, . . . ,zn–

.

Hence we have the result.

() Letx= (xk)∈W(A,Bμ,M,p,·, . . . ,·)∩W(A,Bμ,M,p,·, . . . ,·). Then, by the

following inequality, the result follows

∞

k=

amk

(M+M)

Bμxk

ρ ,z, . . . ,zn–

pk ≤D ∞ k= amk M Bμxk

ρ ,z, . . . ,zn–

pk +D ∞ k= amk M Bμxk

ρ ,z, . . . ,zn–

pk.

If we take the limit asm→ ∞, then we get (xk)∈W(A,Bμ,M+M,p,·, . . . ,·). This

completes the proof.

Theorem . Z(A,Bμ–,M,p,·, . . . ,·)⊂ Z(A,B μ

,M,p,·, . . . ,·) and the inclusion is strict for μ≥ . In general, Z(A,Bj,M,p,·, . . . ,·)⊂ Z(A,Bμ,M,p,·, . . . ,·) for j = , , , . . . ,μ– and the inclusions are strict,where Z=W,Wand W∞.

Proof We give the proof forW(A,Bμ–,M,p,·, . . . ,·) only. The others can be proved by a similar argument. Letx= (xk) be any element in the spaceW(A,Bμ–,M,p,·, . . . ,·), then there existsρ=|r|ρ+|s|ρ>  such that

lim m→∞ ∞ k= amk MB

μ–

xk

ρ ,z, . . . ,zn–

pk

= .

SinceMis non-decreasing and convex, it follows that

∞

k=

amk

M B

μ xk |r|ρ+|s|ρ

,z, . . . ,zn–

pk = ∞ k= amk MrB

μ–

xk+sB μ–

xk–

|r|ρ+|s|ρ

,z, . . . ,zn–

pk ≤D ∞ k= amk MB

μ–

xk

ρ

,z, . . . ,zn–

pk +D ∞ k= amk MB

μ–

xk–

ρ

,z, . . . ,zn–

pk.

The result holds by taking the limit asm→ ∞.

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Example . LetM(x) =x,pk=  for allk∈N,= (k) = (, , . . .),A=C,i.e., the Cesaro

matrix,r= ,s= –, whereBμxk=

μ

v=

μ

v

rμ–v_sv_x

k–vk–vfor allr,s∈R–{}. Consider

the sequencex= (xk) = (kμ–). Then x= (xk) belongs toW(Bμ,M,p,·, . . . ,·) but does

not belong toW(Bμ–,M,p,·, . . . ,·).

Theorem . Let A= (amk)be a non-negative regular matrix and p= (pk)be such that

 <h≤pk≤H<∞.Then

l∞Bμ,M,·, . . . ,·

⊆W∞A,Bμ,M,p,·, . . . ,·

.

Proof Let l∞(Bμ,M,·, . . . ,·). Then there exists T >  such that [M( Bμxk

ρ ,z, . . . , zn–)]≤Tfor allk∈Nand for every nonzeroz, . . . ,zn–∈X. SinceA= (amk) is a

non-negative regular matrix, we have the following inequality by () of Silverman-Toeplitz conditions:

sup m

∞

k=

amk

MB

μ xk

ρ ,z, . . . ,zn–

pk≤maxT_h,T_Hsup m

∞

k=

amk<∞.

Hencel∞(Bμ,M,·, . . . ,·)⊆W∞(A,B μ

,M,p,·, . . . ,·).

4 (Bμ)n_{-statistically}_A_{-convergent sequences}

In this section we introduce and study a new concept of (Bμ)n-statisticalA-convergence in ann-normed space as follows.

Deﬁnition . Let (X,·, . . . ,·) be ann-normed space and letA= (amk) be a non-negative

regular matrix. A real sequencex= (xk) is said to be (Bμ)n-statisticallyA-convergent to a numberLifδ_A₍_Bμ

)n(K) =limm→∞

∞

k=amkχK(k) =  or, equivalently,limm→∞

k∈Kamk=

 for eachε>  and for every nonzeroz, . . . ,zn–∈X, whereK={k∈N:Bμxk–L,z, . . . ,

zn– ≥ε}andχKis the characteristic function ofK.

In this case, we write (Bμ)nstat-A-limx=L. S(A(B μ

)n) denotes the set of all (B μ )n -statisticallyA-convergent sequences.

If we consider some special cases of the matrix, then we have the following: () IfA=C, the Cesaro matrix, then the deﬁnition reduces to(Bμ)n-statistical

convergence.

() IfA= (amk)is de la Vallee Poussin mean, which is given by(.), then the deﬁnition

reduces to(Bμ)n-statisticalλ-convergence.

() If we takeA= (amk)as in(.), then the deﬁnition reduces to(Bμ)n-statistical lacunary convergence.

Theorem . Let p= (pk)be a sequence of non-negative bounded real numbers such that infkpk> .Then W(A,Bμ,p,·, . . . ,·)⊂S(A(B

μ )n).

Proof Assume that x = (xk) ∈ W(A,B

μ

,p,·, . . . ,·). So, we have for every nonzero z, . . . ,zn–∈X

lim

m→∞

∞

k=

amkBμxk–L,z, . . . ,zn– pk

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Letε>  andK={k∈N:Bμxk–L,z, . . . ,zn– ≥ε}. We obtain the following:

∞

k=

=

k∈K

amkBμxk–L,z, . . . ,zn–pk+

k∈/K

amkBμxk–L,z, . . . ,zn–pk

≥minεh,εH

k∈K

amk.

If we take the limit asm→ ∞, then we getx∈S(A(Bμ)n). This completes the proof.

Theorem . Let p= (pk)be a sequence of non-negative bounded real numbers such that infkpk> .Then

l∞Bμ,·, . . . ,·

∩SABμ n

⊂WA,Bμ,p,·, . . . ,·

.

Proof Suppose thatx= (xk)∈l∞(Bμ,·, . . . ,·)∩S(A(B μ

)n). Then there exists an integer T such thatBμxk–L,z, . . . ,zn– ≤Tfor allk>  and for every nonzeroz, . . . ,zn–∈X,

andlim_m_→∞_k_∈_Kamk= , whereK={k∈N:B

μ

xk–L,z, . . . ,zn– ≥ε}. Then we can

write

∞

k=

amkBμxk–L,z, . . . ,zn–pk

=

k∈/K

amkB

μ

xk–L,z, . . . ,zn– pk

+

k∈K

amkB

μ

xk–L,z, . . . ,zn– pk

<maxεh,εH

k∈/K

amk+max

Th,TH

k∈K

amk.

SinceA= (amk) is a non-negative regular matrix, then we have

 = lim

m→∞

∞

k=

amk

= lim

m→∞

k∈/K

amk+lim

m→∞

k∈K

amk.

Hence,limm→∞

k∈/Kamk= . Thus

lim

m→∞

∞

k=

<ε lim

m→∞

k∈/K

amk+T lim

m→∞

k∈K

amk

<ε,

wheremax{εh_,_εH_}₌_ε_and_max_{_Th_,_TH_}₌_T_.

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Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally in the preparation of this article. Both the authors read and approved the ﬁnal manuscript.

Acknowledgements

This paper is supported by Sakarya University BAPK Project No: 2012-50-02-032.

Received: 21 February 2013 Accepted: 9 July 2013 Published: 23 July 2013 References

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