Generalized spaces of double sequences for Orlicz functions and bounded regular matrices over n normed spaces

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R E S E A R C H

Open Access

Generalized spaces of double sequences for

Orlicz functions and bounded-regular

matrices over

n

-normed spaces

Syed Abdul Mohiuddine

1*

_{, Kuldip Raj}

2

_{and Abdullah Alotaibi}

1

*_{Correspondence:} [email protected] 1_{Department of Mathematics,} Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

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

Abstract

The aim of this paper is to introduce some generalized spaces of double sequences with the help of the Musielak-Orlicz functionM= (Mjk) and four-dimensional bounded-regular (shortly,RH-regular) matricesA= (anmjk) overn-normed spaces. Some topological properties and inclusion relations between these spaces are investigated.

MSC: 40A05; 40D25

Keywords: double sequence; Orlicz function; diﬀerence sequence; paranormed space; overn-normed spaces; bounded-regular matrices

1 Introduction, notations, and preliminaries

The concept of -normed spaces was first introduced by Gähler [] in the mid-s, while that ofn-normed spaces one can find in Misiak []. Since then, many others have studied this concept and obtained various results; see Gunawan [, ] and Gunawan and Mashadi []. Letn∈NandXbe a linear space over the field of real numbersRof dimen-siond, 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∈R, and

() x+x,x, . . . ,xn ≤ x,x, . . . ,xn+x,x, . . . ,xn

is called ann-norm onX, and the pair (X,·, . . . ,·) is called an-normed space over the ﬁeldR.

For example, we may takeX=Rn_{being equipped with the}_n-norm_x

,x, . . . ,xnE=

the volume of then-dimensional parallelepiped spanned by the vectorsx,x, . . . ,xnwhich

may be given explicitly by the formula

x,x, . . . ,xnE=det(xij),

wherexi= (xi,xi, . . . ,xin)∈Rnfor eachi= , , . . . ,n. Let (X,·, . . . ,·) be ann-normed

space of dimensiond≥n≥ and{a,a, . . . ,an}be a linearly independent set inX. Then

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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.

If every Cauchy sequence inXconverges to someL∈X, then Xis said to be complete with respect to then-norm. A completen-normed space is calledn-Banach space.

Of the definitions of convergence commonly employed for double series, only that due to Pringsheim permits a series to converge conditionally. Therefore, in spite of any disad-vantages which it may possess, this definition is better adapted than others to the study of many problems in double sequences and series. Chief among the reasons why the theory of double sequences, under the Pringsheim definition of convergence, presents difficulties not encountered in the theory of simple sequences is the fact that a double sequence{xij}

may converge withoutxijbeing a bounded function ofiandj. Thus it is not surprising

that many authors in dealing with the convergence of double sequences should have re-stricted themselves to the class of bounded sequences or, in dealing with the summability of double series, to the class of series for which the function whose limit is the sum of the series is a bounded function ofiandj. Without such a restriction, peculiar things may sometimes happen; for example, a double power series may converge with partial sum {Sij}unbounded at a place exterior to its associated circles of convergence. Nevertheless

there are problems in the theory of double sequences and series where this restriction of boundedness as it has been applied is considerably more stringent than need be. In [], Hardy introduced the concept of regular convergence for double sequences. Some im-portant work on double sequences has also been done by Bromwich []. Later on, it was studied by various authors,e.g.Móricz [], Móricz and Rhoades [], Başarır and Sonalcan [], Mursaleen and Mohiuddine [, ], and many others. Mursaleen [] has deﬁned and characterized the notion of almost strong regularity of four-dimensional matrices and applied these matrices to establish a core theorem (also see []). Altay and Başar [] have recently introduced the double sequence spacesBS,BS(t),CSp,CSbp,CSr, and

BV consisting of all double series whose sequence of partial sums are in the spacesMu,

Mu(t),Cp,Cbp,Cr, andLu, respectively. Başar and Sever [] extended the well known

spaceqfrom single sequence to double sequences, denoted byLq, and established its

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The notion of diﬀerence sequence spaces was introduced by Kızmaz [], who studied the diﬀerence sequence spacesl∞(),c(), andc(). The notion was further generalized by Et and Çolak [] by introducing the spacesl∞(r),c(r), andc(r).

Letwbe the space of all complex or real sequencesx= (xk) and letrandsbe two

non-negative integers. Then forZ=l∞,c,c, we have the following sequence spaces:

Zr_s=x= (xk)∈w:

r_sxk

∈Z,

wherer

sx= (rsxk) = (rs–xk–sr–xk+) andxk=xkfor allk∈N, which is equivalent

to the following binomial representation:

r_sxk= r

v= (–)v

r v xk+sv.

We remark that fors=  andr=s= , we obtain the sequence spaces which were intro-duced and studied by Et and Çolak [] and Kızmaz [], respectively. For more details as regards sequence spaces, see [–] and references therein.

AnOrlicz function M: [,∞)→[,∞) is a continuous, nondecreasing, and convex such thatM() = ,M(x) >  forx>  andM(x)→ ∞asx→ ∞. If convexity of the Orlicz func-tion is replaced byM(x+y)≤M(x) +M(y), then this function is calledmodulus function. Lindenstrauss and Tzafriri [] used the idea of Orlicz function to deﬁne the sequence space

M=

x= (xk)∈w:

∞

k= M

_|

xk| ρ

<∞, for someρ> 

,

known as an Orlicz sequence space. The spaceMis a Banach space with the norm

x=inf

ρ>  : ∞

k= M

_|

xk| ρ

≤

.

Also it was shown in [] that every Orlicz sequence spaceMcontains a subspace

isomor-phic top(p≥). An Orlicz functionMcan always be represented in the integral form

M(x) =

x



η(t)dt,

whereηis known as the kernel ofM, is a right diﬀerentiable fort≥,η() = ,η(t) > ,η

is nondecreasing andη(t)→ ∞ast→ ∞.

A sequence M= (Mk) of Orlicz functions is said to beMusielak-Orlicz function(see

[, ]). A sequenceN= (Nk) is deﬁned by

Nk(v) =sup

|v|u–Mk(u) :u≥

, k= , , . . .

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hMare deﬁned as follows:

t_M=x∈w:I_M(cx) <∞for somec> ,

hM=x∈w:IM(cx) <∞for allc> ,

whereIMis a convex modular deﬁned by

I_M(x) = ∞

k=

Mk(xk), x= (xk)∈tM.

We considert_Mequipped with the Luxemburg norm

x=inf

k>  :IM

x k

≤

or equipped with the Orlicz norm

x=inf

 k

 +I_M(kx):k> 

.

A Musielak-Orlicz function M= (Mk) is said to satisfy the-conditionif there exist constantsa,K> , and a sequencec= (ck)k∞=∈l+ (the positive cone ofl) such that the inequality

Mk(u)≤KMk(u) +ck

holds for allk∈Nandu∈R+_{, whenever}_M

k(u)≤a.

A double sequencex= (xjk) is said to beboundedifx(∞,)=supj,k|xjk|<∞. We denote

byl

∞, the space of all bounded double sequences.

By the convergence of double sequencex= (xjk) we mean the convergence in the

Pring-sheim sensei.e.a double sequencex= (xjk) is said toconvergeto the limitLin

Pring-sheim sense (denoted byP-limx=L) provided that given>  there existsn∈Nsuch that|xjk–L|<wheneverj,k>n(see []). We shall write more brieﬂy asP-convergent.

If, in addition,x∈l

∞, thenxis said to beboundedly P-convergenttoL. We shall denote the space of all bounded convergent double sequences (or, boundedlyP-convergent) byc

∞. LetS⊆N×Nand let>  be given. ByχS(x;), we denote the characteristic function of

the setS(x;) ={(j,k)∈N×N:|xjk| ≥}.

LetA= (anmjk) be a four-dimensional inﬁnite matrix of scalars. For allm,n∈N, where

N:=N∪ {}, the sum

ynm=

∞,∞

j,k=, anmjkxjk

is called theA-meansof the double sequence (xjk). A double sequence (xjk) is said to be

A-summableto the limitLif theA-means exist for allm,nin the sense of Pringsheim’s convergence:

P- lim

p,q→∞

p,q

j,k=,

anmjkxjk=ynm and P- lim

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A four-dimensional matrix Ais said to bebounded-regular (orRH-regular) if every boundedP-convergent sequence isA-summable to the same limit and theA-means are also bounded.

The following is a four-dimensional analog of the well-known Silverman-Toeplitz theo-rem [].

Theorem .(Robison [] and Hamilton []) The four-dimensional matrix A is

RH-regular if and only if

(RH) P-limn,manmjk= for eachjandk,

(RH) P-limn,m

∞,∞

j,k=,|anmjk|= ,

(RH) P-limn,m

_∞

j=|anmjk|= for eachk,

(RH) P-limn,m

∞

k=|anmjk|= for eachj,

(RH) ∞,∞j,k=,|anmjk|<∞for alln,m∈N.

2 Some spaces of double sequences overn-normed spaces

Recently, Yurdakadim and Tas [] defined the spaces of double sequences for RH-regular four-dimensional matrices and Orlicz functions and also established some interesting re-sults. Quite recently, Mohiuddineet al.[] defined and studied some paranormed double difference sequence spaces for four-dimensional bounded-regular matrices and Musielak-Orlicz functions.

Recall that a linear topological spaceXover the real ﬁeldR(the set of real numbers) is said to be aparanormed spaceif there is a subadditive functiong:X→Rsuch that g(θ) = ,g(x) =g(–x) and scalar multiplication is continuous,i.e.,|αn–α| → andg(xn–

x)→ implyg(αnxn–αx)→ for allα’s inRand allx’s inX, whereθ is the zero vector

in the linear spaceX.

The linear spacesl∞(p),c(p),c(p) were deﬁned by Maddox [] (also, see Simons []). Let (X,·, . . . ,·) be an-normed space andw(n–X) denotes the space ofX-valued se-quences. LetM= (Mjk) be a Musielak-Orlicz function, that is,Mis a sequence of

Or-licz functions and let A= (anmjk) be a nonnegative four-dimensional bounded-regular

matrix. Then we deﬁne the following double diﬀerence sequence spaces overn-normed spaces:

W_A,M,u,r_s,p,·, . . . ,·

=

x= (xjk)∈w(n–X) :P-lim n,m

 nm

∞,∞

j,k=, anmjk

Mjk

ujkrsxjk

ρ ,z, . . . ,zn–

pjk

=  for someρ> 

and

W_A,_M_,_u,r

s,p,·, . . . ,·

=

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P-lim

n,m

 nm

∞,∞

j,k=, anmjk

Mjk

ujkrsxjk–L

ρ ,z, . . . ,zn–

pjk

=  for someρ>  andL∈C

,

where p= (pjk) is a double sequence of real numbers such that pjk >  for j, k and

sup_j_,_kpjk =H <∞, and u = (ujk) is a double sequence of strictly positive real

num-bers.

We obtain the following sequence spaces from the above sequence spaces:W

(A,M,u,

r

s,p,·, . . . ,·) andW(A,M,u,rs,p,·, . . . ,·) by giving particular values toM, p, u,

andA.

(i) IfM(x) =x, then we writeW_(A,u,r

s,p,·, . . . ,·)andW(A,u,rs,p,·, . . . ,·)

instead ofW

(A,M,u,rs,p,·, . . . ,·)andW(A,M,u,rs,p,·, . . . ,·),

respectively.

(ii) Ifp= (pjk) = for allj,k, then we writeW(A,M,u,rs,·, . . . ,·)andW(A,M,u, r_s,·, . . . ,·)instead ofW_(A,M,u,_sr,p,·, . . . ,·)andW(A,M,u,r_s,p, ·, . . . ,·), respectively.

(iii) Ifu= (ujk) = for allj,k, then we writeW(A,M,rs,p,·, . . . ,·)andW(A,M, r

s,p,·, . . . ,·)instead ofW(A,M,u,rs,p,·, . . . ,·)andW(A,M,u,rs,p,

·, . . . ,·), respectively.

(iv) IfA= (C, , ), then we writeW

(M,u,rs,p,·, . . . ,·)andW(M,u,rs,p,

·, . . . ,·)instead ofW(A,M,u,rs,p,·, . . . ,·)andW(A,M,u,rs,p,·, . . . ,·),

respectively, where(C, , )denotes thenmth Cesàro mean of double sequence(xjk).

(v) IfA= (C, , )andM(x) =x, then we writeW

(u,rs,p,·, . . . ,·)andW(u,rs,p,

·, . . . ,·)instead ofW

(A,M,u,rs,p,·, . . . ,·)andW(A,M,u,rs,p,·, . . . ,·),

respectively.

Throughout the paper, we shall use the following inequality: Let (ajk) and (bjk) be two

double sequences. Then

|ajk+bjk|pjk≤K

|ajk|pjk+|bjk|pjk

, (.)

whereK=max(, H–_{) and}_sup

j,kpjk=H(see []).

3 Main results

Theorem . Let M= (Mjk)be a Musielak-Orlicz function, A= (anmjk) be a

nonneg-ative four-dimensional RH-regular matrix, p= (pjk) be a bounded sequence of

posi-tive real numbers and u= (ujk) be a sequence of strictly positive real numbers. Then

W_(A,M,u,r_s,p,·, . . . ,·) and W(A,M,u,r_s,p,·, . . . ,·) are linear spaces over the ﬁeldRof reals.

Proof Supposex= (xjk) andy= (yjk)∈W(A,M,u,rs,p,·, . . . ,·) andα,β ∈R. Then

there exist positive numbersρ,ρsuch that

lim

n,m

 nm

∞,∞

j,k=, anmjk

Mjk

ujkrsxjk

ρ

,z, . . . ,zn–

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and

lim

n,m

 nm

∞,∞

j,k=, anmjk

Mjk

ujkrsyjk

ρ

,z, . . . ,zn–

pjk=  for someρ> .

Letρ=max(|α|ρ, |β|ρ). SinceM= (Mjk) is a nondecreasing and convex so by using

inequality (.), we have

lim

n,m

 nm

∞,∞

j,k=, anmjk

Mjk

ujkrs(αxjk+βyjk) ρ

,z, . . . ,zn–

pjk

≤Klim

n,m

 nm

∞,∞

j,k=,  pjkanmjk

Mjk

ujkrsxjk

ρ

,z, . . . ,zn–

pjk

+Klim

n,m

 nm

∞,∞

j,k=,  pjkanmjk

Mjk

ujkrsyjk

ρ

,z, . . . ,zn–

pjk

≤Klim

n,m

 nm

∞,∞

j,k=, anmjk

Mjk

ujkrsxjk

ρ

,z, . . . ,zn–

pjk

+Klim

n,m

 nm

∞,∞

j,k=, anmjk

Mjk

ujkrsyjk

ρ

,z, . . . ,zn–

pjk

= .

Thusαx+βy∈W_(A,M,u,r_s,p,·, . . . ,·). This proves thatW_(A,M,u,r_s,p,·, . . . ,·) is a linear space. Similarly we can prove thatW(A,M,u,r_s,p,·, . . . ,·) is also a linear

space.

Theorem . LetM= (Mjk)be a Musielak-Orlicz function, A= (anmjk) be a

W

(A,M,u,rs,p,·, . . . ,·)and W(A,M,u,rs,p,·, . . . ,·)are paranormed spaces with

the paranorm

g(x) =inf

(ρ)pjkM _:_lim n,m

 nm

∞,∞

j,k=, anmjk

Mjk

ujkrsxjk

ρ ,z, . . . ,zn–

pjk

 M

≤for someρ> 

,

where <pjk≤suppjk=H<∞and M=max(,H).

Proof (i) Clearlyg(x)≥ forx= (xjk)∈W(A,M,u,rs,p,·, . . . ,·). SinceMjk() = , we

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(iii) Letx= (xjk),y= (yjk)∈W(A,M,u,rs,p,·, . . . ,·) there exist positive numbersρ andρsuch that

lim

n,m

 nm

∞,∞

j,k=, anmjk

Mjk

ujkrsxjk

ρ

,z, . . . ,zn–

pjk≤

and

lim

n,m

 nm

∞,∞

j,k=, anmjk

Mjk

ujkrsyjk

ρ

,z, . . . ,zn–

pjk≤.

Letρ=ρ+ρ. Then by using Minkowski’s inequality, we have

lim

n,m

 nm

∞,∞

j,k=, anmjk

Mjk

ujkrs(xjk+yjk)

ρ ,z, . . . ,zn–

pjk

=lim

n,m

 nm

∞,∞

j,k=, anmjk

Mjk

ujkrs(xjk+yjk) ρ+ρ

,z, . . . ,zn–

pjk

=lim

n,m

 nm

∞,∞

j,k=, anmjk

Mjk

ujkrsxjk

ρ+ρ

,z, . . . ,zn–

pjk

+lim

n,m

 nm

∞,∞

j,k=, anmjk

Mjk

ujkrsyjk

ρ+ρ

,z, . . . ,zn–

pjk

≤

ρ

ρ+ρ

lim

n,m

 nm

∞,∞

j,k=, anmjk

Mjk

ujkrsxjk

ρ

,z, . . . ,zn–

pjk

+

ρ

ρ+ρ

lim

n,m

 nm

∞,∞

j,k=, anmjk

Mjk

ujkrsyjk

ρ

,z, . . . ,zn–

pjk

≤,

and thus

g(x+y)

=inf

(ρ)pjkM :_lim n,m

 nm

∞,∞

j,k=, anmjk

Mjk

ujkrs(xjk+yjk)

ρ ,z, . . . ,zn–

pjk  M ≤ ≤inf

(ρ)pjkM _:_lim n,m

 nm

∞,∞

j,k=, anmjk

Mjk

ujkrsxjk

ρ

,z, . . . ,zn–

pjk  M ≤ +inf

(ρ)

pjk M _:_lim

n,m

 nm

∞,∞

j,k=, anmjk

Mjk

ujkrsyjk

ρ

,z, . . . ,zn–

pjk  M ≤ .

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Finally, we prove that the scalar multiplication is continuous. Letλbe any complex num-ber. By deﬁnition,

g(λx) =inf

(ρ)pjkM _:_lim n,m

 nm

∞,∞

j,k=, anmjk

Mjk

ujkrsλxjk

ρ ,z, . . . ,zn–

pjk

 M

≤

=inf

|λ|t

pjk M _:_lim

n,m

 nm

∞,∞

j,k=, anmjk

Mjk

ujkrsxjk

t ,z, . . . ,zn–

pjk

 M

≤

,

wheret=_|ρ_λ_|> . Since|λ|pjk≤_max(,|_λ|suppjk_{), we have}

g(λx)≤max,|λ|suppjk

×inf

tpjkM _:_lim n,m

 nm

∞,∞

j,k=, anmjk

Mjk

ujkrsxjk

t ,z, . . . ,zn–

pjk

 M

≤

.

So, the fact that the scalar multiplication is continuous follows from the above inequality. This completes the proof of the theorem.

Theorem . LetM= (Mjk)be a Musielak-Orlicz function, A= (anmjk) be a

W

(A,M,u,rs,p,·, . . . ,·)and W(A,M,u,rs,p,·, . . . ,·)are complete topological

lin-ear spaces.

Proof Let (xq_jk) be a Cauchy sequence inW_(A,M,u,_sr,p,·, . . . ,·), that is,g(xq–xt)→ asq,t→ ∞. Then we have

 nm

∞,∞

j,k=, anmjk

Mjk

ujkrsx

q

jk–ujkrsxtjk

ρ ,z, . . . ,zn–

pjk→.

Thus for each ﬁxedjandkasq,t→ ∞, sinceA= (anmjk) is nonnegative, we are granted

that

Mjk

ujkrsx

q

jk–ujkrsxtjk

ρ ,z, . . . ,zn–

→

and by continuity ofM= (Mjk), (xqjk) is a Cauchy sequence inRfor each ﬁxedjandk.

SinceRis complete ast→ ∞, we havexq_jk→xjk for each (j,k). For> , there exists a

natural numberNsuch that

lim

n,m

 nm

∞,∞

j,k=,q,t>N

anmjk

Mjk

ujkrsx

q

jk–ujkrsxtjk

ρ ,z, . . . ,zn–

pjk< for allm,n.

Since for any ﬁxed natural numberM, we have

lim

n,m

 nm

∞,∞

j,k≤Mq,t>N

anmjk

Mjk

ujkrsx

q

jk–ujkrsxtjk

ρ ,z, . . . ,zn–

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and by lettingt→ ∞in the above expression we obtain

lim

n,m

 nm

∞,∞

j,k≤Mq>N

anmjk

Mjk

ujkrsx

q

jk–ujkrsxjk

ρ ,z, . . . ,zn–

pjk<.

SinceMis arbitrary, by lettingM→ ∞we obtain

lim

n,m

 nm

∞,∞

j,k=, anmjk

Mjk

ujkrsx

q

jk–ujkrsxjk

ρ ,z, . . . ,zn–

pjk< for allm,n.

Thusg(xq_–_x)_→_{ as}_q_{→ ∞}_{. This proves that}_W

(A,M,u,rs,p,·, . . . ,·) is a complete

topological linear space.

Now we shall show that W(A,M,u,r_s,p,·, . . . ,·) is a complete topological linear space. For this, since (xq_{) is also a sequence in}_W_(A,_M_,_u,r

s,p,·, . . . ,·) by deﬁnition of

W_(A,_M_,_u,r

s,p,·, . . . ,·), for eachqthere existsLqwith

 nm

∞,∞

j,k=, anmjk

Mjk

ujkrsx

q

jk–ujkrsLq

ρ ,z, . . . ,zn–

pjk→ asm,n→ ∞,

whence, from the fact that sup_nm_nm ∞,∞_j_,_k_=,anmjk <∞ and from the deﬁnition of a

Musielak-Orlicz function, we haveMjk(

r_sLq–r_sL

ρ ,z, . . . ,zn–)→ asq→ ∞and soL q

converges toL. Thus

 nm

∞,∞

j,k=, anmjk

Mjk

ujkrsxjk–L

ρ ,z, . . . ,zn–

pjk→ asm,n→ ∞.

Hencex∈W_(A,_M_,_u,r

s,p,·, . . . ,·) and this completes the proof.

Theorem . Let M = (Mjk) be a Musielak-Orlicz function which satisﬁes the

-condition.Then W_(A,_u,r

s,p,·, . . . ,·)⊆W(A,M,u,rs,p,·, . . . ,·).

Proof Letx= (xk)∈W(A,u,rs,p,·, . . . ,·), that is,

lim

n,m

 nm

j,k

anmjk

ujkrsxjk–L

ρ ,z, . . . ,zn–

pjk= .

Let >  and choose δ with  <δ <  such that Mjk(t) < for ≤t≤δ. Writeyjk =

(ujk

r sxjk–L

ρ ,z, . . . ,zn–) and consider

lim

n,m

 nm

j,k

anmjk

Mjk(yjk)pjk

=lim

n,m

 nm

j,k:|yjk|≤δ

anmjk

Mjk(yjk)pjk

+lim

n,m

 nm

j,k:|yjk|>δ

anmjk

Mjk(yjk)pjk

=lim

n,m

 nm

j,k:|yjk|≤δ

anmjk+lim n,m

 nm

j,k:|yjk|>δ

anmjk

Mjk(yjk)pjk

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Foryjk>δ, we use the fact thatyjk< yjk

δ <  + yjk

δ . Hence

Mjk(yjk) <Mjk

 +yjk

δ

<Mjk()  +

 Mjk

yjk

δ

.

SinceMsatisﬁes the-condition, we have

Mjk(yjk) <K

yjk

δMjk() +K

yjk

δMjk() =K

yjk δ Mjk(),

and hence

lim

n,m

 nm

j,k:|yjk|>δ

anmjk

Mjk(yjk)pjk

≤KMjk

δ ()limn,m

 nm

j,k

anmjk

ujkrsxjk–L

ρ ,z, . . . ,zn–

pjk.

Since A is RH-regular and x∈W_(A,_u,r

s,p,·, . . . ,·), we get x∈W(A,M,u,rs,p,

·, . . . ,·).

Theorem . LetM= (Mjk)be a Musielak-Orlicz function and let A= (anmjk)be a

non-negative four-dimensional RH-regular matrix.Suppose thatβ=lim_t_→∞Mjk(t)

t <∞.Then

WA,u,r_s,p,·, . . . ,·=WA,M,u,r_s,p,·, . . . ,·.

Proof In order to prove thatW_(A,_u,r

s,p,·, . . . ,·) =W(A,M,u,rs,p,·, . . . ,·). It is

suﬃcient to show that W(A,M,u,r_s,p,·, . . . ,·)⊂W(A,u,r_s,p,·, . . . ,·). Now, let

β> . By deﬁnition ofβ, we haveMjk(t)≥βtfor allt≥. Sinceβ> , we havet≤_βMjk(t)

for allt≥. Letx= (xjk)∈W(A,M,u,rs,p,·, . . . ,·). Thus, we have

lim

n,m

 nm

∞,∞

j,k=, anmjk

ujkrsxjk–L

ρ ,z, . . . ,zn–

pjk

≤ 

βlimn,m

 nm

∞,∞

j,k=, anmjk

Mjk

ujkrsxjk–L

ρ ,z, . . . ,zn–

pjk,

which implies thatx= (xjk)∈W(A,u,rs,p,·, . . . ,·). This completes the proof.

Theorem .

(i) Let <infpjk<pjk≤.Then

WA,M,u,r_s,p,·, . . . ,·⊆WA,M,u,r_s,·, . . . ,·.

(ii) Let≤pjk≤suppjk<∞.Then

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Proof (i) Letx= (xjk)∈W(A,M,u,sr,p,·, . . . ,·). Then since  <infpjk <pjk≤, we

obtain the following:

lim

n,m

 nm

∞,∞

j,k=, anmjk

Mjk

ujkrsxjk–L

ρ ,z, . . . ,zn–

≤lim

n,m

 nm

∞,∞

j,k=, anmjk

Mjk

ujkrsxjk–L

ρ ,z, . . . ,zn–

pjk.

Thusx= (xjk)∈W(A,M,u,rs,·, . . . ,·).

(ii) Letpjk≥ for eachjandkandsuppjk<∞. Letx= (xjk)∈W(A,M,u,rs,·, . . . ,·).

Then for each  <<  there exists a positive integerNsuch that

lim

n,m

 nm

∞,∞

j,k=, anmjk

Mjk

ujkrsxjk–L

ρ ,z, . . . ,zn–

≤<  for allm,n≥N.

This implies that

lim

n,m

 nm

∞,∞

j,k=, anmjk

Mjk

ujkrsxjk–L

ρ ,z, . . . ,zn–

pjk

≤lim

n,m

 nm

∞,∞

j,k=, anmjk

Mjk

ujkrsxjk–L

ρ ,z, . . . ,zn–

.

Thereforex= (xjk)∈W(A,M,u,rs,p,·, . . . ,·). This completes the proof.

Lemma . Let G be an ideal in l

∞and let x= (xjk)∈l∞ .Then x is in the closure of G in l

∞if and only ifχS(x;)∈G for all> .

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

Lemma . LetM= (Mjk)be a Musielak-Orlicz function which satisﬁes the-condition

and let A= (anmjk)be a nonnegative four-dimensional RH-regular matrix.Then W(A,M, u,r_s,p,·, . . . ,·)∩l

∞is an ideal in l∞.

Proof Letx∈W_(A,M,u,r_s,p,·, . . . ,·)∩l_∞ andy∈l_∞. We need to show thatxy∈ W_(A,M,u,r

s,p,·, . . . ,·)∩l∞ . Sincey∈l∞, there existsT>  such thaty<T. In

this case|xjkyjk|<T|xjk|for allj,k. SinceMis nondecreasing and satisﬁes-condition,

we have

Mjk

ujkrs(xjkyjk)

ρ ,z, . . . ,zn–

pjk<

Mjk

T

ujkrsxjk

ρ ,z, . . . ,zn–

pjk

≤T(T)

Mjk

ujkrsxjk

ρ ,z, . . . ,zn–

pjk,

for allj,k, andT> . Thereforelimn,m_nm

j,kanmjk[Mjk(

ujkrs(xjkyjk)

ρ ,z, . . . ,zn–) pjk_{] = .}

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Lemma . If A is a nonnegative four-dimensional RH-regular matrix,then W

(A,u,rs,

p,·, . . . ,·)∩l

∞is a closed ideal in l∞.

Proof We haveW

(A,M,u,rs,p,·, . . . ,·)∩l∞ ⊂l∞ and it is clear thatW(A,M,u,rs,

p,·, . . . ,·)∩l

∞ = . For x,y∈W(A,M,u,rs,p,·, . . . ,·)∩l∞ , we get |xjk +yjk| <

|xjk|+|yjk|. Now, we have

Mjk

ujkrs(xjk+yjk)

ρ ,z, . . . ,zn–

pjk ≤ Mjk ujkrsxjk

ρ ,z, . . . ,zn–

+ujk

r syjk

ρ ,z, . . . ,zn–

pjk <  Mjk

ujk

r sxjk

ρ ,z, . . . ,zn–

pjk+ 

Mjk

ujk

r syjk

ρ ,z, . . . ,zn–

pjk < K Mjk ujkrsxjk

ρ ,z, . . . ,zn–

pjk+ K

Mjk

ujkrsyjk

ρ ,z, . . . ,zn–

pjk

by the-condition and the convexity ofM. Since

lim

n,m

 nm

j,k

anmjk

Mjk

ujkrs(xjk+yjk)

ρ ,z, . . . ,zn–

pjk

≤ Klimn,m

 nm

j,k

anmjk

Mjk

ujkrsxjk

ρ ,z, . . . ,zn–

pjk

+ Klimn,m

 nm

j,k

anmjk

Mjk

ujkrsyjk

ρ ,z, . . . ,zn–

pjk,

whereK=max{K,K}, sox+y,x–y∈W

(A,M,u,rs,p,·, . . . ,·)∩l∞.

Letx∈W

(A,M,u,rs,p,·, . . . ,·)∩l∞andy∈l∞. Thus, there exists a positive integer K, so that for everyj,k, we have|xjkyjk| ≤K|xjk|. Therefore

Mjk

ujkrs(xjkyjk)

ρ ,z, . . . ,zn–

pjk≤

Mjk

Kujk

r sxjk

ρ ,z, . . . ,zn–

pjk ≤T Mjk ujkrsxjk

ρ ,z, . . . ,zn–

pjk,

and so

lim

n,m

 nm

j,k

anmjk

Mjk

ujkrs(xjkyjk)

ρ ,z, . . . ,zn–

pjk

≤Tlim

n,m

 nm

j,k

anmjk

Mjk

ujkrsxjk

ρ ,z, . . . ,zn–

pjk.

Hencexy∈W

(A,M,u,rs,p,·, . . . ,·)∩l∞. So W(A,M,u,rs,p,·, . . . ,·)∩l∞ is an ideal inl

∞for a Musielak-Orlicz function which satisﬁes the-condition.

Now, we have to show that W_(A,M,u,r_s,p,·, . . . ,·) ∩ l_∞ is closed. Let x ∈ W_(A,M,u,r

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such thatxcd_→_x_∈_l

∞. For every>  there existsN()∈Nsuch that for allc,d>N(), |xcd_–_x_|_<_{. Now, for}_{> , we have}

lim

n,m

 nm

j,k

anmjk

Fjk

ujkrsxjk

ρ ,z, . . . ,zn–

pjk

=lim

n,m

 nm

j,k

anmjk

Mjk

ujkrsxjk–ujksrxcdjk +ujkrsxcdjk

ρ ,z, . . . ,zn–

pjk

≤lim

n,m

 nm

j,k

anmjk

Mjk

ujkrsxjk–ujkrsxcdjk

ρ ,z, . . . ,zn–

+ujk

r sxcdjk

ρ ,z, . . . ,zn–

pjk

≤ limn,m

 nm

j,k

anmjk

Mjk

ujk

r

sxjk–ujkrsxcdjk

ρ ,z, . . . ,zn–

pjk

+ limn,m

 nm

j,k

anmjk

Mjk

ujk

r sxcdjk

ρ ,z, . . . ,zn–

pjk

≤

KMjk()limn,m

 nm

j,k

anmjk

+ Klimn,m

 nm

j,k

anmjk

Mjk

ujkrsxcdjk

ρ ,z, . . . ,zn–

pjk.

Sincexcd_∈_W

(A,M,u,rs,p,·, . . . ,·)∩l∞andAis RH-regular, we get

lim

n,m

 nm

j,k

anmjk

Mjk

ujkrsxjk

ρ ,z, . . . ,zn–

pjk= ,

sox∈W

(A,M,u,rs,p,·, . . . ,·)∩l∞. This completes the proof.

Theorem . Let x= (xjk)be a bounded sequence,M= (Mjk)be a Musielak-Orlicz

func-tion which satisﬁes the-condition and A be a nonnegative four-dimensional RH-regular matrix.Then W(A,M,u,_sr,p,·, . . . ,·)∩l_∞ =W(A,u,r_s,p,·, . . . ,·)∩l_∞.

Proof Without loss of generality we may takeL=  and establish

W_A,M,u,_sr,p,·, . . . ,·∩l_∞ =W_A,u,r_s,p,·, . . . ,·∩l_∞.

Since W

(A,u,rs,p,·, . . . ,·)⊆ W(A,M,u,rs,p,·, . . . ,·), therefore W(A,u,rs,p,

·, . . . ,·)∩l

∞⊆W(A,M,u,rs,p,·, . . . ,·)∩l∞. We need to show thatW(A,M,u,rs,

p,·, . . . ,·)∩l

∞⊆W(A,u,rs,p,·, . . . ,·)∩l∞. Notice that ifS⊂N×N, then

lim

n,m

 nm

j,k

anmjk

Mjk

χS(j,k)

pjk

=Mjk()lim n,m

 nm

j,k

anmjk

χS(j,k)

pjk

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for alln,m. Observe thatχS(j,k)∈W(A,u,rs,p,·, . . . ,·)∩l∞wheneverx∈W(A,M, u,r

s,p,·, . . . ,·)∩l∞by Lemma . and Lemma ., so

W_A,M,u,_sr,p,·, . . . ,·∩l_∞ ⊆W_A,u,r_s,p,·, . . . ,·∩l_∞.

The proof is complete.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors contributed equally and signiﬁcantly in writing this paper. The 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_{School of Mathematics, Shri Mata Vaishno Devi University, Katra, Jammu and Kashmir 182320, India.}

Acknowledgements

The authors gratefully acknowledge the ﬁnancial support from King Abdulaziz University, Jeddah, Saudi Arabia.

Received: 20 April 2014 Accepted: 5 August 2014 Published: 2 September 2014 References

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doi:10.1186/1029-242X-2014-332