The idea of [λ,μ]-almost convergence (briefly, F[λ,μ]-convergence) has been recently introduced and studied by Mohiuddine and Alotaibi (2014). In this paper first we define a norm on F[λ,μ] such that it is a Banach space and then we define and characterize those four-dimensional matrices which transform F[λ,μ]-convergence of double sequences x=(xjk) into F[λ,μ]-convergence. We also define a F[λ,μ]-core of x=(xjk) and determine a Tauberian condition for core inclusions and core equivalence.

1. Background, Notations, and Preliminaries

We begin by recalling the definition of convergence for double sequences which was introduced by Pringsheim [1]. A double sequence x=(xjk) is said to be convergent to L in the Pringsheim's sense (or P-convergent to L) if for given ϵ>0 there exists an integer N such that |xjk-L|<ϵ whenever j,k>N. We will write this as
(1)P-limj,k→∞xjk=L,
where j and k are tending to infinity independent of each other. We denote by CP the space of P-convergent sequences.

We say that a double sequence x=(xjk) is bounded if
(2)∥x∥=supj,k≥0|xjk|<∞.
Denote by L∞ the space of all bounded double sequences.

If a double sequence x=(xjk) in L∞ and x is also P-convergent to L, then we say that it is boundedly P-convergent to L (or, BP-convergent to L). We denote by CBP the space of all boundedly P-convergent double sequences. Note that CBP⊂L∞.

We remark that, in contrast to the case for single sequences, a P-convergent double sequence need not be bounded.

Let A=(apqjk:j,k=0,1,2,…) be a four-dimensional infinite matrix of real numbers for all p,q=0,1,2,… and S1 a space of double sequences. Let S2 be a double sequences space, converging with respect to a convergence rule ν∈{P,BP}. Define
(3)S1A,ν={x=(xjk):Ax=(Apq(x))-∑j,k∑kapqjkxjkexistsandAx∈S1hhhhhhhhhhhhh=ν-∑j,k∑kapqjkxjkexists,Ax∈S1}.
Then, we say that a four-dimensional matrix B maps the space S2 into the space S1 if S2⊂S1B,ν and is denoted by (S2,S1).

The idea of almost convergence of Lorentz [2] is narrowly connected with the limits of Banach (see [3]) as follows. A sequence x=(xj) in l∞ is almost convergent to L if all of its Banach limits are equal, where l∞ denotes the space of all bounded sequences. Mohiuddine [4] applies this concept to established some approximation theorems for sequence of positive linear operator. Móricz and Rhoades [5] extended the notion of almost convergence from single to double sequences as follows.

A double sequence x=(xj,k) of real numbers is said to be almost convergent to a number L if
(4)limp,q→∞supm,n>0|1pq∑j=mm+p-1∑k=nn+q-1xj,k-L|=0.
For more details on almost convergence for single and double sequences, one can refer to [6–13].

The two-dimensional analogue of Banach limit has been defined by Mursaleen and Mohiuddine [14] as follows. A linear functional L on L∞ is said to be Banach limit if it has the following properties:

L(x)≥0 if x≥0 (i.e., xjk≥0 for all j,k),

L(E)=1, where E=(ejk) with ejk=1 for all j,k,

L(S11x)=L(x)=L(S10x)=L(S01x),

where the shift operators S01, S10, and S11 are defined by
(5)S01x=(xj,k+1),S10x=(xj+1,k),S11x=(xj+1,k+1).
Denote by B the set of all Banach limits on L∞. Note that if (BL3) holds, then we may also write L(Sx)=L(x). A double sequence x=(xjk) is said to be almost convergent to a number L if L(x)=L for all L∈B.

Let λ=(λm:m=0,1,2,…) and μ=(μn:n=0,1,2,…) be two nondecreasing sequences of positive reals and each tending to ∞ such that λm+1≤λm+1, λ1=0, μn+1≤μn+1, μ1=0, and
(6)Imn(x)=1λmμn∑j∈Jm∑k∈Inxjk,
is called the double generalized de la Valée-Poussin mean, where Jm=[m-λm+1,m] and In=[n-μn+1,n]. We denote the set of all λ and μ type sequences by using the symbol Λ.

Quite recently, Mohiuddine and Alotaibi [15] presented a generalization of the notion of almost convergent double sequence with the help of de la Vallée-Poussin mean and called it [λ,μ]-almost convergent. In the same paper, they also defined and characterized some four-dimensional matrices. For more details on double sequences, four-dimensional matrices, and other related concepts, one can refer to [16–26].

A double sequence x=(xjk) of reals is said to be [λ,μ]-almost convergent (briefly, F[λ,μ]-convergent) [15] to some number L if x∈F[λ,μ], where
(7)F[λ,μ]={x=(xjk):P-limm,n→∞Ωmnst(x)=Lexists,hhhhhhhhhhhhhx=(xjk):P-limm,n→∞Ωmnst(x)uniformlyins,t;L=F[λ,μ]-limx},Ωmnst(x)=1λmμn∑j∈Jm∑k∈Inxj+s,k+t.
Denote by F[λ,μ] the space of all [λ,μ] almost convergent sequences (xj,k). Note that CBP⊂F[λ,μ]⊂L∞.

We remark that if we take λm=m and μn=n, then the notion of [λ,μ]-almost convergence coincides with the notion of almost convergence for double sequences due to Móricz and Rhoades [5].

We will assume throughout this paper that the limit of a double sequence means limit in the Pringsheim sense. We define the following matrix classes and establish interesting results.

Definition 1.

A four-dimensional matrix A=(apqjk) is said to be [λ,μ]-almost regular if Ax∈F[λ,μ] for all x=(xjk)∈CBP with F[λ,μ]-limAx=limx, and one denotes this by A∈(CBP,F[λ,μ])reg.

Definition 2.

A matrix A=(apqjk) is said to be of class (F[λ,μ],F[λ,μ]) if it maps every F[λ,μ]-convergent double sequence into F[λ,μ]-convergent double sequence; that is, Ax∈F[λ,μ] for all x=(xjk)∈F[λ,μ]. In addition, if F[λ,μ]-limAx=F[λ,μ]-limx, then A is F[λ,μ]-regular and, in symbol, one will write A∈(F[λ,μ],F[λ,μ])reg.

Now we define the norm on F[λ,μ] as follows.

Theorem 3.

F[λ,μ] is a Banach space normed by
(8)∥x∥=supm,n,s,t|Ωmnst(x)|.

Proof.

It can be easily verified that (8) defines a norm on F[λ,μ]. We show that F[λ,μ] is complete. Now, let (xb) be a Cauchy sequence in F[λ,μ]. Then for each j,k, (xjkb) is a Cauchy sequence in R. Therefore xjkb→xjk (say). Put x=(xjk); given ϵ there exists an integer N(ϵ)=N, say, such that, for each b,d>N,
(9)∥xb-xd∥<ϵ2.
Hence
(10)supm,n,s,t|Ωmnst(xb-xd)|<ϵ2;
then, for each m, n, s, t and b, d>N, we have
(11)|Ωmnst(xb-xd)|<ϵ2.
Taking limit d→∞, we have for b>N and for each of m,n,s,t(12)|Ωmnst(xb-x)|<ϵ2.
Now for fixed b, the above inequality holds. Since for fixed b, xb∈F[λ,μ], we get
(13)limm,n→∞Ωmnst(xb)=l
uniformly in s, t. For given ϵ>0, there exist positive integers m0, n0 such that
(14)|Ωmnst(xb)-l|<ϵ2,
for m≥m0, n≥n0 and for all s, t. Here m0, n0 are independent of s, t but depend upon ϵ. Now by using (12) and (14), we get
(15)|Ωmnst(x)-l|=|Ωmnst(x)-Ωmnst(xb)+Ωmnst(xb)-l|≤|Ωmnst(x)-Ωmnst(xb)|+|Ωmnst(xb)-l|<ϵ,
for m≥m0, n≥n0 and for all s, t. Hence x=(xjk)∈F[λ,μ] and F[λ,μ] is complete.

Now we characterize the matrix class (F[λ,μ],F[λ,μ]) as well as (F[λ,μ],F[λ,μ])reg. Let M[λ,μ] be the subspace of F[λ,μ] such that P-limm,n→∞Ωmnst(x)=0, uniformly in s,t; that is
(16)M[λ,μ]={x=(xjk)∈F[λ,μ]:P-limm,n→∞Ωmnst(x)=0,hx=(xjk)h∈F[λ,μ]lF[λ,μ]:P-limm,n→∞Ωmnst(x)uniformlyins,t}.
Note that every y∈F[λ,μ] can be written as
(17)y=x+lE,
where x∈M[λ,μ],l=P-limm,nΩmnst(y) uniformly in s,t, and E=(ejk) with ejk=1 for all j,k.

Theorem 4.

A matrix A=(apqjk)∈(F[λ,μ],F[λ,μ]) if and only if

∥A∥=supp,q∑j=0∞∑k=0∞|apqjk|<∞,

a=(∑j=0∞∑k=0∞apqjk)p,q=1∞∈F[λ,μ],

A(S-I)∈(L∞,F[λ,μ]),

where S is the shift operator.
Proof.

Necessity. Let A∈(F[λ,μ],F[λ,μ]). We know that CBP⊂F[λ,μ]⊂L∞, so we have A∈(CBP,L∞). Hence the necessity of (AC1) follows. Since ∈F[λ,μ], then AE∈F[λ,μ]. This is equivalent to
(18)(∑j=0∞∑k=0∞apqjk)p,q=1∞∈F[λ,μ];
that is, (AC2) holds. For each x=(xjk)∈L∞, we have Sx-x∈F[λ,μ] because
(19)L(Sx-x)=L(Sx)-L(x)=0
for all Banach limit L. Hence A(Sx-x)∈F[λ,μ]; that is, (AC3) holds.

Sufficiency. Let conditions (AC1)–(AC3) hold and y=(yjk)∈F[λ,μ]. Then
(20)y=x+lE,
where x=(xjk)∈M[λ,μ], l=P-limm,n→∞Ωmnst(y), uniformly in s,t and E=(ejk) with ejk=1 for all j,k. Taking A-transform in (20), we obtain
(21)Ay=Ax+lAE=Ax+l(∑j=0∞∑k=0∞apqjk)p,q=1∞.
If x=(xjk)∈L∞, then by (AC3) we have A(Sx-x)∈F[λ,μ]. Since by (AC1), A is bounded linear operator on L∞, we get AM[λ,μ]⊂F[λ,μ]. This yields Ax∈F[λ,μ]. Now from condition (AC2) and (21), Ay∈F[λ,μ]. Therefore A∈(F[λ,μ],F[λ,μ]).

Corollary 5.

A matrix A=(apqjk)∈(F[λ,μ],F[λ,μ])reg if and only if conditions (AC1) and (AC2) with F[λ,μ]-lima=1 and (AC3) hold.

3. Some Core Theorems

The core or Knopp core of a real number single sequence x is the closed interval [liminfx,limsupx] (see [27, 28]). In 1999, Patterson [29] extended the Knopp core from single sequences to double sequences and called it Pringsheim core (shortly, P-core) which is given by [P-liminfx,P-limsupx]. In the recent past, the M-core and σ-core for double sequences have been defined and studied by Mursaleen and Edely [30] and Mursaleen and Mohiuddine [31, 32], respectively, while the σ-core for single sequences is given by Mishra et al. [33]. In 2011, Kayaduman and Çakan [34] presented the concept of Cesáro core of double sequences.

We define the following sublinear functional on L∞:
(22)Γ(x)=limsupm,n→∞sups,t1λmμn∑j∈Jm∑k∈Inxj+s,k+t.
Then we define the F[λ,μ]-core of a real-valued bounded double sequence (xjk) to be the closed interval [-Γ(-x),Γ(x)].

Since every BP-convergent double sequence is F[λ,μ]-convergent, we have
(23)Γ(x)≤L(x),
where L(x)=P-limsupx, and hence it follows that F[λ,μ]-core{x}⊆P-core{x} for all x∈L∞.

Theorem 6.

For every x=(xjk)∈F[λ,μ],
(24)Γ(Ax)≤Γ(x)(orF[λ,μ]-core
{Ax}⊂F[λ,μ]-core
{x})
if and only if

where
(25)α(m,n,s,t,j,k)=1λmμn∑p∈Jm∑q∈Inap+s,q+t,j,k.Proof.

Necessity. Suppose that (24) holds for all x=(xjk)∈F[λ,μ]. One obtains
(26)-Γ(-x)≤-Γ(-Ax)≤Γ(Ax)≤Γ(x);
that is,
(27)F[λ,μ]-liminfx≤-Γ(-Ax)≤Γ(Ax)≤F[λ,μ]-limsupx.
If x=(xjk)∈F[λ,μ], then
(28)-Γ(-Ax)=Γ(Ax)=F[λ,μ]-limx;
that is,
(29)F[λ,μ]-lim(Ax)=F[λ,μ]-limx.
Therefore A is F[λ,μ]-regular. This yields the necessity of (CR1).

Now, with the help of Lemma 2.1 of [35], there is a double sequence x=(xjk)∈L∞ such that ∥x∥≤1 and
(30)limsupm,n→∞sups,t∑j=0∞∑k=0∞α(m,n,s,t,j,k)xjk=limsupm,n→∞sups,t∑j=0∞∑k=0∞|α(m,n,s,t,j,k)|.
If a double sequence x=(xjk) defined by
(31)xjk={1;ifj=k,0;otherwise,
then
(32)1=Γ′(Ax)=liminfm,n→∞sups,t∑j=0∞∑k=0∞|α(m,n,s,t,j,k)|≤Γ(Ax)≤Γ(x)≤∥x∥≤1,
where
(33)Γ′(x)=liminfm,n→∞sups,t1λmμn∑j=0p∑k=0qxj+s,k+t.
This yields the necessity of (CR2).

Sufficiency. We know that CBP⊂F[λ,μ]. Following the lines of Theorem 2 of [31] for translation mapping, one obtains
(34)Γ(Ax)≤L(x).
For any x′∈M[λ,μ], we have
(35)Γ(Ax+Ax′)≤L(x+x′).
Taking infimum over x′∈M[λ,μ], we obtain
(36)infx′∈M[λ,μ]Γ(Ax+Ax′)≤infx′∈M[λ,μ]limsupm,n→∞(xmn+xmn′)=w(x),say.
Thus
(37)sups,tlimsupm,n→∞Ωmnst(Ax)+infx′∈M[λ,μ]infs,tliminfm,n→∞Ωmnst(Ax′)≤w(x).
Since Ax′∈F[λ,μ], we can write
(38)Ax′=x¯+lE,
where x¯∈M[λ,μ],l=F[λ,μ]-limAx′(=F[λ,μ]-limx′, since A is F[λ,μ]-regular). Operating Ωmnst to (38), one obtains
(39)Ωmnst(Ax′)=Ωmnst(x¯)+Ωmnst(lE).
By [λ,μ]-almost regularity, we have
(40)liminfm,n→∞Ωmnst(Ax′)=limm,n→∞Ωmnst(x¯)+llimm,n→∞∑j=0∞∑k=0∞α(m,n,s,t,j,k).
From the definition of M[λ,μ], we get
(41)limm,n→∞Ωmnst(x¯)=0
uniformly in s,t. Also
(42)limm,n→∞∑j=0∞∑k=0∞α(m,n,s,t,j,k)=1.
Therefore we obtain from (40) that
(43)liminfm,n→∞Ωmnst(Ax′)=1
uniformly in s,t. Equations (37) and (43) give that
(44)Γ(Ax)+1≤w(x);
that is,
(45)Γ(Ax)≤w(x).
As w(x)=Γ(x), one obtains Γ(Ax)≤Γ(x).

Note that F[λ,μ]-core{x}⊆P-core{x}. This motivates us to prove the following result by adding a condition to get a more general result.

Theorem 7.

For x=(xjk)∈L∞, if
(46)lims,t(xst-xs+1,t+1)=0
holds, then P-core{x}⊆F[λ,μ]-core{x}.

Proof.

By the definition of P-core and F[λ,μ]-core, we have to show that L(x)≤Γ(x). Let Γ(x)=l. Then, for given ϵ>0, for all j,k,s,t and for large m,n it follows from the definition of Γ that
(47)1λmμn∑j∈Jm∑k∈Inxj+s,k+t<l+ϵ2.
We can write
(48)xst=xst-1λmμn∑j∈Jm∑k∈Inxj+s,k+t+1λmμn∑j∈Jm∑k∈Inxj+s,k+t≤|xst-1λmμn∑j∈Jm∑k∈Inxj+s,k+t|+l+ϵ2.
Since (46) holds, for given ϵ>0, we get that
(49)|xst-xj+s,k+t|<ϵ2,
for all j,k≥0. Thus we have
(50)|xst-1λmμn∑j∈Jm∑k∈Inxj+s,k+t|=1λmμn|λmμnxst-∑j∈Jm∑k∈Inxj+s,k+t|≤1λmμnλmμn|xst-xj+s,k+t|,j,k≥0.
Equation (49) yields
(51)|xst-1λmμn∑j∈Jm∑k∈Inxj+s,k+t|<ϵ2.
Taking limsups,t in (48) and using (51), one obtains L(x)≤l+ϵ. Since ϵ is arbitrary, we obtain L(x)≤Γ(x).

Corollary 8.

If (46) holds and x=(xjk) is F[λ,μ]-convergent, then x is convergent.

Finally, we define the concepts of [λ,μ]-almost uniformly positive and [λ,μ]-almost absolutely equivalent and establish a theorem related to these concepts.

Definition 9.

A matrix A=(apqjk) is said to be [λ,μ]-almost uniformly positive, denoted by F[λ,μ]-uniformly positive, if
(52)limm,n→∞sups,t∑j=0∞∑k=0∞1λmμn|∑p∈Jm∑q∈Inap+s,q+t,j,k|=1.

Definition 10.

Let A=(apqjk) and B=(bpqjk) be two F[λ,μ]-regular matrices and
(53)ypq=∑j=0∞∑k=0∞apqjkxjk,ypq′=∑j=0∞∑k=0∞bpqjkxjk.
Then A and B are said to be [λ,μ]-almost absolutely equivalent, denoted by F[λ,μ]-absolutely equivalent, on L∞ whenever F[λ,μ]-lim(ypq-ypq′)=0; that is, either (ypq) and (ypq′) both tend to the same F[λ,μ]-limit or neither of them tends to a F[λ,μ]-limit, but their difference tends to F[λ,μ]-limit zero.

Before proceeding further, first we state the following lemma which we will use to our next result.

Lemma 11.

For x,y∈L∞, if F[λ,μ]-lim|x-y|=0, then F[λ,μ]-core{x}=F[λ,μ]-core{y}.

Proof of the lemma is straightforward and thus omitted.

Theorem 12.

Let A=(apqjk) be a F[λ,μ]-regular matrix. Then, Γ(Ax)≤Γ(x) for all x=(xjk)∈L∞ if and only if there is a F[λ,μ]-regular matrix B=(bpqjk) such that B is F[λ,μ]-uniformly positive and F[λ,μ]-absolutely equivalent with A on L∞.

Proof.

Let there be a F[λ,μ]-regular matrix B such that B is F[λ,μ]-uniformly positive and F[λ,μ]-absolutely equivalent with A on L∞. Then, by (53) and F[λ,μ]-absolutely equivalent of A and B, we have
(54)F[λ,μ]-lim|ymn-ymn′|=limm,n→∞sups,t|∑j=0∞∑k=0∞1λmμnhhhhhhhhhhhhh×∑p∈Jm∑q∈In[ap+s,q+t,j,k-bp+s,q+t,j,k]xjk|≤∥x∥limm,n→∞sups,t∑j=0∞∑k=0∞1λmμnhhhhhhhhhhhhhhhhh×|∑p∈Jm∑q∈In[ap+s,q+t,j,k-bp+s,q+t,j,k]|=0,
uniformly in s,t. Now, by Lemma 11, F[λ,μ]-core{Ax}=F[λ,μ]-core{Bx} for all x∈L∞. By Theorem 6, we have Γ(Ax)≤Γ(x), since x is arbitrary.

Conversely, let Γ(Ax)≤γ(x) for all x∈L∞. Then by Theorem 6, A is F[λ,μ]-uniformly positive. Now we define a matrix B=(bpqjk) as
(55)bpqjk=12(apqjk+ap,q,j+1,k+1)
for all p, q, j, k∈N. Then it is easy to see that B is F[λ,μ]-regular since A is F[λ,μ]-regular, and
(56)F[λ,μ]-lim(Ax)=F[λ,μ]-lim(Bx).
Further
(57)limm,n→∞sups,t∑j=0∞∑k=0∞1λmμn|∑p∈Jm∑q∈Inbp+s,q+t,j,k|≤12[limm,n→∞sups,t∑j=0∞∑k=0∞1λmμn|∑p∈Jm∑q∈Inap+s,q+t,j,k|hhh+limm,n→∞sups,t∑j=0∞∑k=0∞1λmμnhhhhhhhhhhhhhh×|∑p∈Jm∑q∈Inap+s,q+t,j+1,k+1|].
Since B is F[λ,μ]-regular, we have by (57) that
(58)limm,n→∞sups,t∑j=0∞∑k=0∞1λmμn|∑p∈Jm∑q∈Inbp+s,q+t,j,k|=1.
Thus B is F[λ,μ]-uniformly positive. Further, it follows from (56) that A and B are F[λ,μ]-absolutely equivalent.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant no. (130-265-D1435). The authors, therefore, acknowledge with thanks DSR technical and financial support.

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