Double sequence spaces over n normed spaces defined by a sequence of Orlicz functions

R E S E A R C H

Open Access

Double sequence spaces over

n

-normed

spaces defined by a sequence of Orlicz

functions

Abdullah Alotaibi

_{, Mohammad Mursaleen}

_{and Sunil K Sharma}

*_{Correspondence:}

[email protected]

2_{Department of Mathematics,}

Aligarh Muslim University, Aligarh, 202002, India

Full list of author information is available at the end of the article

Abstract

In the present paper we introduce double sequence spacem2_(M,A,

_φ

MSC: 40A05; 46A45

Keywords: double sequence spaces; paranormed space; Orlicz function;n-normed

space

1 Introduction and preliminaries

The initial works on double sequences is found in Bromwich []. Later on, it was studied by Hardy [], Moricz [], Moricz and Rhoades [], Başarır and Sonalcan [] and many oth-ers. Hardy [] introduced the notion of regular convergence for double sequences. Quite recently, Zeltser [] in her PhD thesis has essentially studied both the theory of topologi-cal double sequence spaces and the theory of summability of double sequences. Mursaleen and Edely [] have recently introduced the statistical convergence which was further stud-ied in locally solid Riesz spaces []. Nextly, Mursaleen [] and Mursaleen and Savas [] have defined the almost regularity and almost strong regularity of matrices for double se-quences and applied these matrices to establish core theorems and introduced theM-core for double sequences and determined those four dimensional matrices transforming ev-ery bounded double sequencesx= (xk,l) into one whose core is a subset of theM-core ofx.

More recently, Altay and Başar [] have defined the spacesBS,BS(t),CSp,CSbp,CSrand

BVof double sequences consisting of all double series whose sequence of partial sums are in the spacesMu,Mu(t),Cp,Cbp,CrandLu, respectively and also examined some

prop-erties of these sequence spaces and determined theα-duals of the spacesBS,BV,CSbp

and theβ(v)-duals of the spacesCSbpandCSrof double series. Recently Başar and Sever

[] have introduced the Banach spaceLqof double sequences corresponding to the well

known spaceqof single sequences and examined some properties of the spaceLq. Now,

recently Raj and Sharma [] have introduced entire double sequence spaces. By the con-vergence of a double sequence we mean the concon-vergence in the Pringsheim sensei.e.a double sequencex= (xk,l) has Pringsheim limitL(denoted byP-limx=L) provided that

given>  there existsn∈Nsuch that|xk,l–L|<wheneverk,l>n, see []. The double

sequencex= (xk,l) is bounded if there exists a positive numberMsuch that|xk,l|<Mfor

allkandl.

Throughout this paper,NandCdenote the set of positive integers and complex num-bers, respectively. A complex double sequence is a function xfromN×NintoCand briefly denoted by{xk,l}. If for all> , there isn∈Nsuch that|xk,l–a|<wherek>n

andl>n, then a double sequence{xk,l}is said to be convergent toa∈C. A real double

se-quence{xk,l}is non-decreasing, ifxk,l≤xp,qfor (k,l) < (p,q). A double series is infinite sum

_∞

k,l=xk,l and its convergence implies the convergence of partial sums sequence{Sn,m},

whereSn,m= m

k=

n

l=xk,l(see []). For recent development on double sequences, we

refer to [–] and [–].

A double sequence spaceEis said to be solid if{xk,lyk,l} ∈Efor all double sequences {yk,l}of scalars such that|yk,l|<  for allk,l∈Nwhenever{xk,l} ∈E.

Letx={xk,l}be a double sequence. A setS(x) is defined by

S(x) ={Xπ(k),π(k)}:πandπare permutation ofN

.

IfS(x)⊆Efor allx∈E, thenEis said to be symmetric. Now letPsbe a family of subsetsσ

having at most elementssinN. AlsoPs,tdenotes the class of subsetsσ=σ×σinN×N such that the element numbers ofσandσare at mostsandt, respectively. Besides{φk,l}

is taken as a non-decreasing double sequence of the positive real numbers such that

kφk+,l≤(k+ )φk,l, lφk,l+≤(l+ )φk,l.

An Orlicz functionM: [,∞)→[,∞) is a continuous, non-decreasing, and convex func-tion such thatM() = ,M(x) >  forx>  andM(x)→ ∞asx→ ∞.

Lindenstrauss and Tzafriri [] used the idea of Orlicz function to define the following sequence space:

M=

x∈w: ∞

k=

M _|

xk|

ρ

<∞ ,

which is called an Orlicz sequence space. AlsoMis a Banach space with the norm

x=inf

ρ>  : ∞

k=

M _|

xk|

ρ

≤ .

Also, it was shown that every Orlicz sequence spaceMcontains a subspace isomorphic

top(p≥). The-condition is equivalent toM(Lx)≤LM(x), for allLwith  <L< .

An Orlicz functionMcan always be represented in the following integral form:

M(x) =

x



η(t)dt,

For further reading on Orlicz spaces, we refer to [–].

LetXbe a linear metric space. A functionp:X→Ris called a paranorm if

() p(x)≥for allx∈X, () p(–x) =p(x)for allx∈X,

() p(x+y)≤p(x) +p(y)for allx,y∈X,

() if(λn)is a sequence of scalars withλn→λasn→ ∞and(xn)is a sequence of vectors withp(xn–x)→asn→ ∞, thenp(λnxn–λx)→asn→ ∞.

A paranormpfor whichp(x) =  impliesx=  is called a total paranorm and the pair (X,p) is called a total paranormed space. It is well known that the metric of any linear metric space is given by some total paranorm (see [], Theorem .., p.).

The concept of -normed spaces was initially developed by Gähler [] in the mid-s, while that ofn-normed spaces one can see in Misiak []. Since then, many others have studied this concept and obtained various results; see Gunawan [, ] and Gunawan and Mashadi [] and references therein. Letn∈NandXbe a linear space over the field K, whereKis the field of real or complex numbers of dimensiond, whered≥n≥. A real valued function·, . . . ,·onXnsatisfying the following four conditions:

() x,x, . . . ,xn= if and only ifx,x, . . . ,xnare linearly dependent inX; () x,x, . . . ,xnis invariant under permutation;

() αx,x, . . . ,xn=|α|x,x, . . . ,xnfor anyα∈K, and () x+x,x, . . . ,xn ≤ x,x, . . . ,xn+x,x, . . . ,xn

is called a n-norm onX, and the pair (X,·, . . . ,·) is called an-normed space over the field K. For example, we may take X=Rn _{being equipped with the Euclideann-norm} x,x, . . . ,xnE, the volume of then-dimensional parallelepiped spanned by the vectors x,x, . . . ,xnwhich may be given explicitly by the formula

x,x, . . . ,xnE=det(xij),

where xi= (xi,xi, . . . ,xin)∈Rn_{for eachi= , , . . . ,n. Let (X,·, . . . ,·) be a n-normed}

space of dimensiond≥n≥ and{a,a, . . . ,an}be linearly independent set inX. Then the function·, . . . ,·∞onXn–defined by

x,x, . . . ,xn–∞=maxx,x, . . . ,xn–,ai:i= , , . . . ,n

defines an (n– )-norm onXwith respect to{a,a, . . . ,an}.

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

lim

k→∞xk–L,z, . . . ,zn–=  for everyz, . . . ,zn–∈X.

A sequence (xk) in an-normed space (X,·, . . . ,·) is said to be Cauchy if

lim k→∞ p→∞

xk–xp,z, . . . ,zn–=  for everyz, . . . ,zn–∈X.

The spacem(φ) was introduced by Sargent []:

m(φ) =

x= (xk)∈w:xm(φ)= sup

s≥,σ∈Ps



φs

k∈σ

|xk|<∞

,

which was further studied in [, ] and []. Recently, Duyar and Oˇgur [] introduced the sequence spacem_{(M,A,φ,p) and studied some of its properties.}

LetA= (aijkl) be an infinite double matrix of complex numbers,M= (Mk,l) be a

se-quence of Orlicz functions, andp= (pk,l) be a bounded double sequence of positive real

numbers. In the present paper we define the following sequence space:

mM,A,φ,p,·, . . . ,·

=

x= (xk,l)∈w(X) :sup



φs,t

i∈σ

j∈σ

∞

k,l=

Mk,l

Aij_ρ(x),z, . . . ,zn–

pij:

(s,t)≥(, ),σ×σ∈Ps,t <∞for someρ>  ,

whereA(x) = (Aij(x)) ifAij(x) =∞_k,l=aijklxk,lconverges for each (i,j)∈N×N.

Ifp= (pij) = , we have

mM,A,φ,·, . . . ,· =

x= (xk,l)∈w(X) :sup



φs,t

i∈σ

j∈σ

∞

k,l=

Mk,l

Aij(x)

ρ ,z, . . . ,zn–

:

(s,t)≥(, ),σ×σ∈Ps,t <∞for someρ>  .

The following inequality will be used throughout the paper:

|a+b|pij≤_{max, }H–|_a|pij₊|_b|pij_{, (.)}

wherea,b∈CandH=sup{pij: (i,j)∈N×N}.

We examine some topological properties ofm_{(M,A,φ,p,·, . . . ,·) and establish some} inclusion relations.

2 Main results

Theorem . LetM= (Mk,l)be a sequence of Orlicz functions and p= (pk,l)be a bounded sequence of positive real numbers,then the space m_{(M,A,φ,p,·, . . . ,·)is linear space}

over the field of complex numberC.

Proof Letx= (xk,l),y= (yk,l)∈m(M,A,φ,p,·, . . . ,·) andα,β∈C. Then there exist

pos-itive numbersρandρsuch that

sup



φs,t

i∈σ

j∈σ

∞

k,l=

Mk,l

Aij_ρ(_x),z, . . . ,zn–

and

sup



φs,t

i∈σ

j∈σ

∞

k,l=

Mk,l

Aij(y)

ρ

,z, . . . ,zn–

pij: (s,t)≥(, ),σ×σ∈Ps,t <∞.

Letρ=max(|α|ρ, |β|ρ). SinceMis a non-decreasing and convex function, we have

i∈σ

j∈σ

∞

k,l=

Mk,l

Aij(αx+βy)

ρ

,z, . . . ,zn–

pij

≤

i∈σ

j∈σ

∞

k,l=

Mk,l

αaijklxk,l+βaijklyk,l

ρ

,z, . . . ,zn–

pij

≤

i∈σ

j∈σ

∞

k,l=

Mk,l

αaijklxk,l

|α|ρ

,z, . . . ,zn–

+βaijklyk,l |β|ρ

,z, . . . ,zn–

pij

=

i∈σ

j∈σ

∞

k,l=

Mk,l

aijklxk,l

ρ

,z, . . . ,zn–

+aijklyk,l ρ

,z, . . . ,zn–

pij

≤max, H–

i∈σ

j∈σ

∞

k,l=

Mk,l

aijklxk,l

ρ

,z, . . . ,zn–

pij

+

i∈σ

j∈σ

∞

k,l=

Mk,l

aijklyk,l

ρ

,z, . . . ,zn–

pij

.

Thus, we have

sup



φs,t

i∈σ

j∈σ

∞

k,l=

Mk,l

Aij(α_ρx_+βy),z, . . . ,zn–

pij:

(s,t)≥(, ),σ×σ∈Ps,t

≤max, H–

sup



φs,t

i∈σ

j∈σ

∞

k,l=

Mk,l

aijklxk,l

ρ

,z, . . . ,zn–

pij:

(s,t)≥(, ),σ×σ∈Ps,t

+sup



φs,t

i∈σ

j∈σ

Mk,l

aijklyk,l

ρ

,z, . . . ,zn–

pij:

(s,t)≥(, ),σ×σ∈Ps,t .

This proves that αx+βy∈m(M,A,φ,p,·, . . . ,·). Hence m_{(M,A,φ,p,·, . . . ,·) is a} linear space. This completes the proof of the theorem.

space with the paranorm defined by

g(x) =inf

ρpqrH _:

sup



φs,t

i∈σ

j∈σ

∞

k,l=

Mk,l

Aij_ρ(x),z, . . . ,zn–

pij:

(s,t)≥(, ),σ×σ∈Ps,t ≤ /H

,q∈N,r∈N .

Proof It is clear thatg(x) =g(–x) andg(x) =  ifx= . Then there exist positive numbers

ρandρsuch that

sup



φs,t

i∈σ

j∈σ

∞

k,l=

Mk,l

Aij(x)

ρ

,z, . . . ,zn–

pij: (s,t)≥(, ),σ×σ∈Ps,t < 

and

sup



φs,t

i∈σ

j∈σ

∞

k,l=

Mk,l

Aij_ρ(_y),z, . . . ,zn–

pij: (s,t)≥(, ),σ×σ∈Ps,t < .

Then, by using Minkowski’s inequality, we have

sup



φs,t

i∈σ

j∈σ

∞

k,l=

Mk,l

Aij(x+y)

ρ+ρ

,z, . . . ,zn–

pij:

(s,t)≥(, ),σ×σ∈Ps,t

≤sup



φs,t

i∈σ

j∈σ

∞

k,l=

Mk,l

Aij(x)

ρ+ρ

,z, . . . ,zn–

+ Aij(y)

ρ+ρ

,z, . . . ,zn–

pij

: (s,t)≥(, ),σ×σ∈Ps,t

≤

ρ

ρ+ρ

h



φs,t

i∈σ

j∈σ

∞

k,l=

Mk,l

Aij(x)

ρ

,z, . . . ,zn–

pij:

(s,t)≥(, ),σ×σ∈Ps,t

+

ρ

ρ+ρ

h



φs,t

i∈σ

j∈σ

∞

k,l=

Mk,l

Aij(y)

ρ

,z, . . . ,zn–

pij:

(s,t)≥(, ),σ×σ∈Ps,t ,

whereh=infpij. This shows thatg(x+y)≤g(x) +g(y). Using this triangle inequality we can write

Thus we have

gλnxn–λnx

=inf ρ pqr H n : sup 

φs,t

i∈σ

j∈σ

∞

k,l=

Mk,l

Aij(λnx_ρn_n–λnx),z, . . . ,zn–

pij:

(s,t)≥(, ),σσ∈Ps,t ≤ _/H

≤,q∈N,r∈N

=inf

ρnpqr/H:

sup



φs,t

i∈σ

j∈σ

∞

k,l=

Mk,l

Aij_(ρ(_nx_/|n_λ–n|x₎),z, . . . ,zn– pij:

(s,t)≥(, ),σ×σ∈Ps,t ≤ _/H

≤,q∈N,r∈N

=inf

λnρn pqr/H

:

sup



φs,t

i∈σ

j∈σ

∞

k,l=

Mk,l

Aij(x_ρn_n–x),z, . . . ,zn–

pij:

(s,t)≥(, ),σ×σ∈Ps,t

/H

≤,q∈N,r∈N

≤maxλnh/H,λn×inf

λnρn

pqr/H

:

sup



φs,t

i∈σ

j∈σ

∞

k,l=

Mk,l

Aij(x_ρn_n–x),z, . . . ,zn–

pij:

(s,t)≥(, ),σ×σ∈Ps,t

/H

≤,q∈N,r∈N

=maxλnh/H,λn·gxn–x.

Thus

gλnx–λx

=inf

ρnpqr/H:

sup



φs,t

i∈σ

j∈σ

∞

k,l=

Mk,l

Aij((λ_ρn_n–λ)x),z, . . . ,zn–

pij:

(s,t)≥(, ),σ×σ∈Ps,t

/H

≤,q∈N,r∈N

=inf

ρnpqr/H:

sup



φs,t

i∈σ

j∈σ

∞

k,l=

Mk,l

Aij(x)

ρn/|λn–λ|

,z, . . . ,zn–

pij:

(s,t)≥(, ),σ×σ∈Ps,t

/H

≤,q∈N,r∈N

=inf

λn–λρn pqr/H

:

sup



φs,t

i∈σ

j∈σ

∞

k,l=

Mk,l

Aij_ρ(_nx),z, . . . ,zn–

(s,t)≥(, ),σ×σ∈Ps,t

/H

≤,q∈N,r∈N

≤maxλn–λh/H,λn–λg(x).

Hence g(λnxn _{–λx)→  where λ}n _{→ λ and x}n _{→ x as n → ∞. This proves that} m(M,A,φ,p,·, . . . ,·) is a paranormed space with the paranorm defined byg. This com-pletes the proof of the theorem.

Theorem . Let φ and ψ be two double sequences then m(M,A,φ,p,·, . . . ,·)⊆

m_{(M,A,ψ,p,·, . . . ,·)if and only ifsup}

(s,t)≥(,)(φs,t/ψs,t) <∞.

Proof LetK =sup_{(s,t)≥(,)}(φs,t/ψs,t) <∞. Then φs,t ≤K·ψs,t for all (s,t)≥(, ). If x= {xk,l} ∈m(M,A,φ,p,·, . . . ,·), then

sup



φs,t

i∈σ

j∈σ

∞

k,l=

Mk,l

Aij_ρ(x),z, . . . ,zn–

pij: (s,t)≥(, ),σ×σ∈Ps,t

<∞ for someρ> . Thus

sup



Kψs,t

i∈σ

j∈σ

∞

k,l=

Mk,l

Aij(x)

ρ ,z, . . . ,zn–

pij: (s,t)≥(, ),σ×σ∈Ps,t

<∞ for someρ> ,

and hencex={xk,l} ∈m(M,A,ψ,p,·, . . . ,·). This shows that mM,A,φ,p,·, . . . ,·⊆mM,A,ψ,p,·, . . . ,·.

Conversely, let m(M,A,φ,p,·, . . . ,·)⊆m(M,A,ψ,p,·, . . . ,·) and αs,t = φψss,,tt for all

(s,t)≥(, ), and supposesup_{(s,t)≥(,)}αs,t=∞. Then there exists a subsequence{αsi,ti}of

{αs,t}such thatlimi→∞αsi,ti=∞. Ifx={xk,l} ∈m(M,A,φ,p,·, . . . ,·), then we have

sup



ψs,t

i∈σ

j∈σ

∞

k,l=

M_k∞_,lAij(x)

ρ ,z, . . . ,zn–

pij: (s,t)≥(, ),σ×σ∈Ps,t

=sup

αs,t



φs,t

i∈σ

j∈σ

∞

k,l=

M∞_k,lAij(x)

ρ ,z, . . . ,zn–

pij:

(s,t)≥(, ),σ×σ∈Ps,t

≥sup

m≥αsm,tm

sup



φsm,tm

i∈σ

j∈σ

∞

k,l=

M∞_k,lAij(x)

ρ ,z, . . . ,zn–

pij:

This is a contradiction asx={xk,l}∈/m(M,A,φ,p,·, . . . ,·). This completes the proof of

the theorem.

Corollary . Let φ and ψ be two double sequences then m_{(M,A,φ,p,·, . . . ,·) =}

m_{(M,A,ψ,p,·, . . . ,·)if and only ifsup}

(s,t)≥(,)αs,t<∞andsup(s,t)≥(,)αs–,t <∞.

Proof It is easy to prove so we omit the details.

Theorem . LetM= (Mk,l),M = (M_k,l)andM = (M_{k ,l})be sequences of Orlicz func-tions satisfying-condition.Then

(i) m_{(M,φ,·, . . . ,·)⊆m}_{(M◦M,φ,·, . . . ,·),}

(ii) m(M,A,φ,p,·, . . . ,·)∩m(M ,A,φ,p,·, . . . ,·)⊆

m_{(M +M ,A,φ,p,·, . . . ,·).}

Proof (i) Letx={xk,l} ∈m(M,A,φ,p,·, . . . ,·). Then there existsρ>  such that

sup



φs,t

i∈σ

j∈σ

∞

k,l=

Mk,l

Aij_ρ(x),z, . . . ,zn–

pij: (s,t)≥(, ),σ×σ∈Ps,t <∞.

By the continuity ofM, we can take a numberδwith  <δ<  such thatMk,l(t) <,

when-ever ≤t<δ, for arbitrary  << . Now let

yi,j=

Ai,j(x)

ρ ,z, . . . ,zn–

.

Thus we have

i∈σ

j∈σ

∞

k,l=

Mk,l(yi,j)pi,j=

yi,j≤δ

∞

k,l=

Mk,l(yi,j)pi,j+

yi,j>δ

∞

k,l=

Mk,l(yi,j)pi,j.

By the properties of the Orlicz function we have

yi,j≤∞

∞

k,l=

Mk,l(yi,j)pi,j≤max

,Mk,l()H yi,j≤∞

(yi,j)pi,j.

Again, we have

Mk,l(yi,j) <Mk,l

 +yi,j

δ

< 

Mk,l() +  Mk,l

yi,j

δ

foryi,j>δ. IfMsatisfies the-condition, then we have

Mk,l(yi,j) <

 T

yi,j

δ Mk,l() +

 T

yi,j

δ Mk,l(),

and so

yi,j>δ

Mk,l(yi,j)pi,j≤max

,

T

δMk,l()

H

Hence, we have

sup



φs,t

i∈σ

j∈σ

∞

k,l=

Mk,l

M_k,lAij(x)

ρ ,z, . . . ,zn–

pij:

(s,t)≥(, ),σ×σ∈Ps,t

<∞

≤max,Mk,l()H

sup



φs,t

i∈σ

j∈σ

∞

k,l=

Mk,l(yi,j)pij: (s,t)≥(, ),σ×σ∈Ps,t

<∞+max

,

T

δMk,l()

H

×sup



φs,t

i∈σ

j∈σ

∞

k,l=

Mk,l(yi,j)pij: (s,t)≥(, ),σ×σ∈Ps,t <∞.

Thus, we have x={xk,l} ∈m(M◦ M,φ,·, . . . ,·) and hence m(M,φ,·, . . . ,·)⊆ m_{(M◦M,φ,·, . . . ,·).}

(ii) Letx={xk,l} ∈m(M,A,φ,p,·, . . . ,·)∩m(M ,A,φ,p,·, . . . ,·). Then there exists

aρ>  such that

sup



φs,t

i∈σ

j∈σ

∞

k,l=

M_k,lAi,j(x)

ρ ,z, . . . ,zn–

pij: (s,t)≥(, ),σ×σ∈Ps,t <∞

and

sup



φs,t

i∈σ

j∈σ

∞

k,l=

M_{k ,l}Ai,j(x)

ρ ,z, . . . ,zn–

pij: (s,t)≥(, ),σ×σ∈Ps,t <∞.

By the inequality, we have

i∈σ

j∈σ

∞

k,l=

M_k,l+M _k,lAi,j(x)

ρ ,z, . . . ,zn–

pij

≤max, H–

i∈σ

j∈σ

∞

k,l=

M_k,lAi,j(x)

ρ ,z, . . . ,zn–

pij

+max, H–

i∈σ

j∈σ

∞

k,l=

M _k,lAi,j(x)

ρ ,z, . . . ,zn–

pij.

Hence

mM,A,φ,p,·, . . . ,·∩mM ,A,φ,p,·, . . . ,·

⊆mM +M ,A,φ,p,·, . . . ,·.

Theorem . The sequence space m_{(M,φ,p,·, . . . ,·)is solid.}

Proof Letα={αk,l} be a double sequence of scalars such that|αk,l| ≤ andy={yk,l} ∈ m_{(M,φ,p,·, . . . ,·). Then we have}

sup



φs,t

k∈σ

l∈σ

Mk,l

αk,lxk,l

ρ ,z, . . . ,zn–

pij: (s,t)≥(, ),σ×σ∈Ps,t

≤sup



φs,t

k∈σ

l∈σ

Mk,l

αk,lyk,l

ρ ,z, . . . ,zn–

pij: (s,t)≥(, ),σ×σ∈Ps,t

≤sup



φs,t

k∈σ

l∈σ

Mk,l

yk,l

ρ ,z, . . . ,zn–

pij: (s,t)≥(, ),σ×σ∈Ps,t

.

This implies that{αk,lyk,l} ∈m(M,φ,p,·, . . . ,·). This proves that the spacem(M,φ,p, ·, . . . ,·) is a solid.

Corollary . The sequence space m_{(M,φ,p,·, . . . ,·)is monotone.}

Proof It is trivial so we omit the details.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Author details

1_{Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia.} 2_{Department of Mathematics, Aligarh Muslim University, Aligarh, 202002, India.}3_{Department of Mathematics, Model}

Institute of Engineering and Technology, Kot Bhalwal, JK 181122, India.

Received: 21 February 2014 Accepted: 9 May 2014 Published:29 May 2014 References

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