# On a NewI-Convergent Double-Sequence Space

Authors: Vakeel A. Khan; Nazneen Khan
Journal: International Journal of Analysis
 (2013)

---

## Abstract

The sequence spaceBVσ was introduced and studied by Mursaleen (1983). In this article we introduce the sequence space 2BVσI and study some of its properties and inclusion relations.


---

## Body

## 1. Introduction and Preliminaries

Letℕ, ℝ, and ℂ  be the sets of all natural, real, and complex numbers, respectively. We write
(1)ω={x=(xk):xk∈ℂ},showing the space of all real or complex sequences.Definition 1.
A double sequence of complex numbers is defined as a functionx:ℕ×ℕ→ℂ. We denote a double sequence as (xij) where the two subscripts run through the sequence of natural numbers independent of each other [1]. A number a∈ℂ is called a double limit of a double sequence (xij) if for every ϵ>0 there exists some N=N(ϵ)∈ℕ such that
(2)|(xij)-a|<ϵ,∀i,j≥N,(see  [2]).Letl∞ and c denote the Banach spaces of bounded and convergent sequences, respectively, with norm ∥x∥∞=supk|xk|. Let v denote the space of sequences of bounded variation; that is,
(3)v={x=(xk):∑k=0∞‍|xk-xk-1|<∞,x-1=0},
where vis a Banach space normed by
(4)∥x∥=∑k=0∞‍|xk-xk-1|,(see  [3]).Definition 2.
Letσ be a mapping of the set of the positive integers into itself having no finite orbits. A continuous linear functional ϕ on l∞ is said to be an invariant mean or σ-mean if and only if(i)
ϕ(x)≥0 when the sequence x=(xk) has xk≥0 for all k;(ii)
ϕ(e)=1, where e={1,1,1,…};(iii)
ϕ(xσ(n))=ϕ(x) for all x∈l∞.In caseσ is the translation mapping n→n+1, a σ-mean is often called a Banach limit (see [4]), and Vσ, the set of bounded sequences all of whose invariant means are equal, is the set of almost convergent sequences (see [5]).Ifx=(xk), then Tx=(Txk)=(xσ(k)). Then it can be shown that
(5)Vσ={∑m=1∞‍x=(xk):∑m=1∞‍tm,k(x)=Luniformlyink,∑m=1∞‍L=σ-limx},
where m≥0, k>0. Consider
(6)tm,k(x)=xk+xσ(k)+⋯+xσm(k)m+1,t-1,k=0,
where σm(k) denote the mth iterate of σ(k) at k. The special case of (5) in which σ(n)=n+1 was given by Lorentz [5, Theorem 1], and that the general result can be proved in a similar way. It is familiar that a Banach limit extends the limit functional on c.Theorem 3.
Aσ-mean extends the limit functional on c in the sense that ϕ(x)=limx for all x∈c if and only if σ has no finite orbits; that is to say, if and only if, for all k≥0, j≥1, (see [3])
(7)σj(k)≠k.
Put
(8)ϕm,k(x)=tm,k(x)-tm-1,k(x),
assuming that t-1,k=0. A straight forward calculation shows (see [6]) that
(9)ϕm,k(x)={1m(m+1)∑j=1mJ(xσj(k)-xσj-1(k))(m≥1),xk,(m=0).For any sequencex, y, and scalar λ, we have
(10)ϕm,k(x+y)=ϕm,k(x)+ϕm,k(y),ϕm,k(λx)=λϕm,k(x).Definition 4.
A sequencex∈l∞ is of σ-bounded variation if and only if (i)
∑k=0∞|ϕm,k(x)| converges uniformly in n;(ii)
limm→∞tm,k(x), which must exist, should take the same value for all k.We denote byBVσ, the space of all sequences of σ-bounded variation (see [7]):
(11)BVσ={x∈l∞:∑m‍|ϕm,k(x)|<∞,uniformlyinn}.Theorem 5.
BV


σ is a Banach space normed by
(12)∥x∥=supk∑m=0∞‍|ϕm,k(x)|, (see [8]).Subsequently, invariant means have been studied by Ahmad and Mursaleen [9], Mursaleen et al. [3, 6, 8, 10–14], Raimi [15], Schaefer  [16], Savas and Rhoades [17], Vakeel et al. [18–20], and many others [21–23]. For the first time, I-convergence was studied by Kostyrko et al. [24]. Later on, it was studied by Šalát et al. [25, 26], Tripathy and Hazarika [27], Ebadullah et al. [18–20, 28], and Vakeel et al. [1, 29].Definition 6 (see [30, 31]).
LetX be a nonempty set. Then, a family of sets I⊆2X (2X denoting the power set of X) is said to be an ideal in X if(i)
∅∈I;(ii)
I is additive; that is, A,B∈I⇒A∪B∈I;(iii)
I is hereditary that is, A∈I, B⊆A⇒B∈I;An IdealI⊆2X is called nontrivial if I≠2X. A non-trivial ideal I⊆2X is called admissible if {{x}:x∈X}⊆I.A non-trivial idealI is maximal if there cannot exist any non-trivial ideal J≠I containing I as a subset.For each idealI, there is a filter £(I) corresponding to I. That is,
(13)£(I)={K⊆N:Kc∈I},whereKc=N-K.Definition 7 (see [24, 31, 32]).
A double sequence(xij)∈ω is said to be I-convergent to a number L if for every ϵ>0,
(14){i,j∈ℕ:|xij-L|≥ϵ}∈I.
In this case, we write I-limxij=L.Definition 8 (see [2]).
A double sequence(xij)∈ω is said to be I-null if L=0.  In this case, we write
(15)I-limxij=0.Definition 9.
A double sequence(xij)∈ω is said to be I-cauchy if for every ϵ>0 there exist numbers m=m(ϵ), n=n(ϵ) such that
(16){i,j∈ℕ:|xij-xmn|≥ϵ}∈I.Definition 10.
A double sequence(xij)∈ω is said to be I-bounded if there exists M>0 such that
(17){i,j∈ℕ:|xij|>M}.Definition 11.
A double-sequence spaceE is said to be solid or normal if (xij)∈E implies (αijxij)∈E for all sequence of scalars (αij) with |αij|<1 for all i,j∈ℕ.Definition 12 (see [24, 33]).
A nonempty family of sets£(I)⊆2X is said to be filter on X if and only if(i)
Φ∉£(I);(ii)
forA,B∈£(I), we have A∩B∈£(I);(iii)
for eachA∈£(I) and A⊆B implies B∈£(I).Definition 13.
LetX be a linear space. A function g:X→R is called a paranorm, if for all x,y,z∈X,(i)
g(x)=0 if x=θ;(ii)
g(-x)=g(x);(iii)
g(x+y)≤g(x)+g(y);(iv)
if(λn) is a sequence of scalars with λn→λ(n→∞)  and xn,a∈X with xn→a(n→∞), in the sense that g(xn-a)→0(n→∞), in the sense that g(λnxn-λa)→0(n→∞).The concept of paranorm is closely related to that of linear metric spaces. It is a generalization of that of absolute value (see [34, 35]).


## 2. Main Results

In this paper, we introduce the sequence space(18)BVσI2∶={x=(xij)∈ω:{i,j∈ℕ:|ϕmnij(x)-L|≥ϵ}∈IforsomeL∈ℂ{(xij)}}.Theorem 14.

2


BV


σ
I is a linear space.Proof.
Let(xij),(yij)∈BVσI2 and α, β be two scalars in ℂ. Then for a given ϵ>0, we have
(19){i,j∈ℕ:|ϕmn,ij(x)-L1|≥ϵ2}∈I,forsomeL1∈ℂ,{i,j∈ℕ:|ϕmn,ij(y)-L2|≥ϵ2}∈I,forsomeL2∈ℂ.
Now let,
(20)A1={i,j∈ℕ:|ϕmn,ij(x)-L1|<ϵ2}∈I,forsomeL1∈ℂ,A2={i,j∈ℕ:|ϕmn,ij(y)-L2|<ϵ2}∈I,forsomeL2∈ℂ
be such that A1c,A2c∈I. Now consider
(21)|ϕmn,ij(αx+βy)-(αL1+βL2)|=|ϕmn,ij(αx)+ϕmn,ij(βy)-αL1-βL2|=|ϕmn,ij(αx)-αL1+ϕmn,ij(βy)-βL2|≤|ϕmn,ij(αx)-αL1|+|ϕmn,ij(βy)-βL2|=|α||ϕmn,ij(x)-L1|+|β||ϕmn,ij(y)-L2|≤|α|ϵ2+|β|ϵ2=(|α|+|β|)ϵ2≤ϵ′(say),
this implies that the sequence space
(22)A3={i,j∈ℕ:|ϕmn,ij(αx+βy)-(αL1+βL2)|<ϵ′}∈I,forsomeL1,L2∈ℂ.
Hence, (αx+βy)∈BVσI2. Therefore, BVσI2 is a linear space.Theorem 15.
The space2BVσI is a paranormed space, paranormed by
(23)g(xij)=supij|ϕmn,ij(x)|.Proof.
Forx=(xij)=0, g(xij)=0 is trivial.
Forx=(xij)≠0, g(xij)≠0, we have(i)
g(x)=supij|ϕmn,ij(x)|≥ 0 for all x∈BVσI2.(ii)
g(-x)=supij|ϕmn,ij(-x)|=supij|-ϕmn,ij(x)|=supij|ϕmn,ij(x)|=g(x) for all x∈BVσI2.(iii)
g(x+y)=supij|ϕmn,ij(x+y)|≤supij|ϕmn,ij(x)|+supij|ϕmn,ij(y)|=g(x)+g(y).(iv)
Let(λij) be a sequence of scalars with λij→λ(ij→∞)  and (x)∈BVσI2 such that
(24)ϕmn,ij(x)→L(ij→∞),in the sense that
(25)g(ϕmn,ij(x)-L)→0(ij→∞).
Therefore,
(26)g(λijϕmn,ij(x)-λL)≤g(λijϕmn,ij(x))-g(λL)=λijg(ϕmn,ij(x))-λg(L)→0as(ij→∞).
Hence,   2BVσI is a paranormed space.Theorem 16.

2


BV


σ
I is a closed subspace of l∞I2.Proof.
Let(xij(pq)) be a cauchy sequence in BVσI2 such that x(pq)→x. We show that x∈BVσI2. Since (xij(pq))∈BVσI2, then there exists apq such that
(27){i,j∈ℕ:|ϕmn,ij(x(pq))-apq|≥ϵ}∈I.
We need to show that(i)
(apq) converges to a.(ii)
IfU={i,j∈ℕ:|xij-a|<ϵ}, then Uc∈I.
Since(xij(pq)) is a cauchy sequence in BVσI2, then for a given ϵ>0, there exists k0∈ℕ such that
(28)supij|ϕmn,ij(xij(pq))-ϕmn,ij(xij(rs))|<ϵ3,∀p,q,r,s≥k0.
For a givenϵ>0, we have
(29)Bpqrs={i,j∈ℕ:|ϕmn,ij(xij(pq))-ϕmn,ij(xij(rs))|<ϵ3},Bpq={i,j∈ℕ:|ϕmn,ij(xij(pq))-apq|<ϵ3},Brs={i,j∈ℕ:|ϕmn,ij(xij(rs))-ars|<ϵ3}.
Then Bpqrsc,Bpqc, and Brsc∈I. Let Bc=Bpqrsc∩Bpqc∩Brsc,where B={i,j∈ℕ:|apq-ars|<ϵ}. Then Bc∈I. We choose k0∈Bc, then for each p,q,r,s≥k0, we have
(30){i,j∈ℕ:|apq-ars|<ϵ}⊇{i,j∈ℕ:|ϕmn,ij(xij(pq))-apq|<ϵ3}∩{i,j∈ℕ:|ϕmn,ij(xij(pq))-ϕmn,ij(xij(rs))|<ϵ3}∩{i,j∈ℕ:|ϕmn,ij(xij(rs))-ars|<ϵ3}.
Then (apq) is a cauchy sequence of scalars in ℂ, so there exists a scalar a∈ℂ  such that (apq)→a, as p,q→∞.
For the next step, let0<δ<1 be given. Then, we show that if
(31)U={i,j∈ℕ:|ϕmn,ij(x)-a|<δ},
then Uc∈I. Since ϕmn,ij(x(pq))→ϕmn,ij(x), then there exists p0,q0∈ℕ such that
(32)P={i,j∈ℕ:|ϕmn,ij(xij(p0q0))-ϕmn,ij(x)|<δ3}
which implies that Pc∈I. The number p0,  q0 can be so chosen that together with (32), we have
(33)Q={i,j∈ℕ:|ap0q0-a|<δ3}
such that Qc∈I. Since {i,j∈ℕ:|ϕmn,ij(xij(p0q0))-ap0q0|≥δ}∈I, then we have a subset S of ℕ such that Sc∈I, where
(34)S={i,j∈ℕ:|ϕmn,ij(xij(p0q0))-ap0q0|<δ3}.
Let Uc=Pc∩Qc∩Sc, where U={i,j∈ℕ:|ϕmn,ij(x)-a|<δ}.
Therefore, for eachi,j∈Uc, we have
(35){i,j∈ℕ:|ϕmn,ij(x)-a|<δ}⊇{i,j∈ℕ:|ϕmn,ij(xij(p0q0))-ϕmn,ij(x)|<δ3}∩{i,j∈ℕ:|ϕmn,ij(xij(p0q0))-ap0q0|<δ3}∩{i,j∈ℕ:|ap0q0-a|<δ3}.
Hence, the result BVσI2⊂l∞I2 follows.Theorem 17.
The space2BVσI is nowhere dense subsets of l∞I2.Proof.
Proof of the result follows from the previous theorem.Theorem 18.
The space2BVσI is solid and monotone.Proof.
Let(xij)∈BVσI2 and αij be a sequence of scalars with |αij|≤1 for all i,j∈ℕ. Then, we have
(36)|αijϕmn,ij(x)|≤|αij||ϕmn,ij(x)|≤|ϕmn,ij(x)|,∀i,j∈ℕ.
The space   2BVσI is solid follows from the following inclusion relation:
(37){i,j∈ℕ:|ϕmn,ij(x)|≥ϵ}⊇{i,j∈ℕ:|αijϕmn,ij(x)|≥ϵ}.
Also a sequence space is solid implies monotone. Hence, the space  2BVσIis monotone.Theorem 19.

2
c
0
I
⊂

2


BV


σ
I
⊂
l
∞
I
2 and the inclusions are proper.Proof.
Letx=(xij)∈c0I2. Then, we have {i,j∈ℕ:|xij|≥ϵ}∈I. Since  2c0⊂BVσ2, x=(xij)∈BVσI2 implies
(38){i,j∈ℕ:|ϕmn,ij(x)|≥ϵ}∈I.
Now let,(39)A1={i,j∈ℕ:|xij|<ϵ},A2={i,j∈ℕ:|ϕmn,ij(x)|<ϵ}
be such that A1c,A2c∈I. As l∞={x=(xij):supij|xij|<∞}, taking supremum over i,j we get A1c⊂A2c. Hence,  2c0I⊂2BVσI⊂2l∞I.
Next we show that the inclusion is proper
(i) First for2c0I⊂BVσI2. Consider x∈BVσI2, then by the definition
(40)BVσI2∶={x=(xij)∈ω:{i,j∈ℕ:|ϕmnij(x)-L|≥ϵ}∈IforsomeL∈ℂ{(xij)}},
we have
(41)ϕmn,ij(x)=tmn,ij(x)-t(m-1)(n-1),ij(x),
where
(42)tmn,ij(x)=xij+xσ(ij)+xσ2(ij)+⋯+xσmn(ij)mn.
Therefore,
(43)tmn,ij(x)-t(m-1)(n-1),ij(x)=xij+xσ(ij)+xσ2(ij)+⋯+xσmn(ij)mn-xij+xσ(ij)+xσ2(ij)+⋯+xσ(m-1)(n-1)(ij)(m-1)(n-1)=(m-1)(n-1)(xij+xσ(ij)+⋯+xσmn(ij))mn(m-1)(n-1)-mn(xij+xσ(ij)+⋯+xσ(m-1)(n-1)(ij))mn(m-1)(n-1).
On solving, we get
(44)ϕmn,ij(x)=mnxσmn(ij)mn(m-1)(n-1)+(1-m-n)(xij+xσ(ij)+xσ2(ij)+⋯+xσmn(ij))mn(m-1)(n-1).
As σ is a translation map, that is, σ(n)=n+1, we have
(45)ϕmn,ij(x)=mnx(i+m)(j+n)mn(m-1)(n-1)+(1-m-n)(xij+x(i+1)(j+1)+⋯+x(i+m)(j+n))mn(m-1)(n-1).
Taking limi,j→∞, we have
(46)lim(i,j)→∞ϕmn,ij(x)=lim(i,j)→∞[(eemnx(i+m)(j+n)+(1-m-n)×(xij+x(i+1)(j+1)+⋯+x(i+m)(j+n))ee)×(mn(m-1)(n-1))-1ee],L(mn(m-1)(n-1))=lim(i,j)→∞[mnx(i+m)(j+n)+(1-m-n)mnx(i+m)(j+n)×(xij+x(i+1)(j+1)+⋯+x(i+m)(j+n))].
Since m,n,L≠0, therefore limi,j→∞ϕmn,ij(x)≠0 which implies that x∉(2c0I).  Hence, we get that the inclusion is proper.
(ii) Second for2BVσI⊂l∞I2.
The result of this part follows from the proof of Theorem18.Theorem 20.

c
I
2
⊂


BV


σ
I
2
⊂
l
∞
I
2 and the inclusions are proper.Proof.
Letx=(xij)∈cI2. Then, we have
(47){i,j∈ℕ:|xij-L|≥ϵ}∈I.
Since c2⊂BVσ2⊂l∞2, which implies x=(xij)∈BVσI2 implies
(48){i,j∈ℕ:|ϕmn,ij(x)-L|≥ϵ}∈I.
Now let,(49)B1={i,j∈ℕ:|ϕi,j-L|<ϵ},B2={i,j∈ℕ:|ϕmn,ij(x)-L|<ϵ}
be such that B1c,B2c∈I. As
(50)2l∞={x=(xij):supij|xij|<∞},
taking limsup over i,j, we get B1c⊂B2c. Hence, cI2⊂BVσI2⊂l∞I2.
Next, we show that the inclusion is proper
(i) First forcI2⊂BVσI2. We show that cI2⊊BVσI2.
Letx=(xij)∈BVσI2, then by the definition
(51)2BVσI∶={x=(xij)∈ω:{i,j∈ℕ:|ϕmnij(x)-L|≥ϵ}∈I{i,j∈ℕ:|ϕmnij(x)-L|≥ϵ}forsomeL∈ℂ}.
We have, |ϕmnij(x)-L|≥ϵ. We say that the I-lim(ϕmnij(x))=L.
Now considering the case when|ϕmnij(x)-L|<ϵ, then
(52)|tmn,ij(x)-t(m-1)(n-1),ij(x)-L|<ϵ,
when m,n=0, then we have ϕmnij(x)=tij(x)=xij. Therefore we get,
(53)|xij-L|<ϵ∀i,j∈ℕ.
Hence, x∉cI2={i,j∈ℕ:|xij-L|≥ϵ}∈I. Hence, the inclusion is proper.
(ii) Second forBVσI2⊂l∞I2.
The result follows from the proof of Theorem18.



---

*Source: 126163-2013-03-12.xml*