The purpose of this paper is to generalize the concept of almost convergence for double sequence through the notion of de la Vallée-Poussin mean for double sequences. We also define and characterize the generalized regularly almost conservative and almost coercive four-dimensional matrices. Further, we characterize the infinite matrices which transform the sequence belonging to the space of absolutely convergent double series into the space of generalized almost convergence.

1. Introduction and Preliminaries

Let l∞ be the Banach space of real bounded sequences x=xn with the usual norm ∥x∥=sup|xn|. There exist continuous linear functionals on l∞ called Banach limits [1]. It is well known that any Banach limit of x lies between liminfx and limsupx. The idea of almost convergence of Lorentz [2] is narrowly connected with the limits of S. Banach; that is, a sequence xn∈l∞ is almost convergent to l if all of its Banach limits are equal. As an application of almost convergence, Mohiuddine [3] obtained some approximation theorems for sequence of positive linear operator through this notion. For double sequence, the notion of almost convergence was first introduced by Móricz and Rhoades [4]. The authors of [5] introduced the notion of Banach limit for double sequence and characterized the spaces of almost and strong almost convergence for double sequences through some sublinear functionals. For more details on these concepts, one can refer to [6–12].

We say that a double sequence x=(xj,k:j,k=0,1,2,…) of real or complex numbers is bounded if
(1)∥x∥=supj,k|xj,k|<∞,
denoted by ℒ∞, the space of all bounded sequence (xj,k).

A double sequence x=(xj,k) of reals is called convergent to some number L in Pringsheim’s sense (briefly, P-convergent to L) [13] if for every ϵ>0 there exists N∈ℕ such that |xj,k-L|<ϵ whenever j,k≥N, where ℕ≔{1,2,3,…}.

If a double sequence x=(xj,k) in ℒ∞ and x is also P-convergent to L, then we say that it is boundedly P-convergent to L (briefly, BP-convergent to L).

A double sequence x=(xjk) is said to converge regularly to L (briefly, R-convergent to L) if x is converges in Pringsheim’s sense, and the limits xj≔limkxjk(j∈ℕ) and xk≔limjxjk(k∈ℕ) exist. Note that in this case the limits limjlimkxjk and limklimjxjk exist and are equal to the P-limit of x.

Throughout this paper, by 𝒞P, 𝒞BP, and 𝒞R, we denote the space of all P-convergent, BP-convergent, and R-convergent double sequences, respectively. Also, the linear space of all continuous linear functionals on 𝒞R is denoted by 𝒞R′.

Let B=(bp,q,j,k: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,R}. Define
(2)S1B,ν={-∑j,kbp,q,j,kxj,kexistsandBx∈S1}x=(xj,k):Bx=(Bp,q(x))=ν-∑j,kbp,q,j,kxj,kexistsandBx∈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).

Móricz and Rhoades [4] extended the notion of almost convergence from single to double sequence and characterized some matrix classes involving this concept. A double sequence x=(xj,k) of real numbers is said to be almost convergent to a number L if
(3)limp,q→∞supm,n>0|1pq∑j=mm+p-1∑k=nn+q-1xj,k-L|=0.

For more details on double sequences and 4-dimensional matrices, one can refer to [14–20].

Using the notion of almost convergence for single sequence, King [21] introduced a slightly more general class of matrices than the conservative and regular matrices, that is, almost conservative and almost regular matrices, and presented its characterization. In [22], Schaefer presented some interesting characterization for almost convergence. The Steinhaus-type theorem for the concepts of almost regular and almost coercive matrices was proved by Başar and Solak [23]. In this paper, we generalize the concept of almost convergence for double sequences with the help of double generalized de la Vallée-Poussin mean and called it (Λ)almost convergence. Using this concept, we define the notions of regularly (Λ)almost conservative and (Λ)almost coercive four-dimensional matrices and obtain their necessity and sufficient conditions. Further, we introduce the space ℒ1 of all absolutely convergent double series and characterize the matrix class (ℒ1,ℱΛ), where ℱΛ denotes the space of (Λ)almost convergence for double sequences.

2. Main ResultsDefinition 1.

Let λ=(λm:m=0,1,2,…) and μ=(μn:n=0,1,2,…) be two nondecreasing sequences of positive reals with each tending to ∞ such that λm+1≤λm+1, λ1=0, μn+1≤μn+1, μ1=0, and
(4)ℑm,n(x)=1λmμn∑j∈Jm∑k∈Inxj,k
is called the double generalized de la Vallée-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.

Definition 2.

A double sequence x=(xj,k) of reals is said to be (Λ)almost convergent (briefly, ℱΛ-convergent) to some number L if x∈ℱλ,μ, where
(5)ℱΛ={x=(xjk):P-limm,n→∞Ωm,n,s,t(x)=Lexists,uniformlyins,t;L=ℱΛ-limx{x=(xjk):P-limm,n→∞Ωm,n,s,t(x)},Ωm,n,s,t(x)=1λmμn∑j∈Jm∑k∈Inxj+s,k+t,
denoted by ℱΛ, the space of all (Λ)almost convergent sequences (xj,k). Note that 𝒞BP⊂ℱΛ⊂ℒ∞.

Remark 3.

If we take λm=m and μn=n, then the notion of (Λ)almost convergence reduced to almost convergence due to Móricz and Rhoades [4].

Definition 4.

A four-dimensional matrix B=(bp,q,j,k) is said to be regularly (Λ)almost conservative if it maps every R-convergent double sequence into ℱΛ-convergent double sequence; that is, B∈(𝒞R,ℱΛ). In addition, if ℱΛ-limAx=R-limx, then B is regularly (Λ)almost regular.

Definition 5.

A matrix B=(bp,q,j,k) is said to be (Λ)almost coercive if it maps every BP-convergent double sequence (xj,k) into ℱΛ-convergent double sequence, briefly, a matrix B in (𝒞BP,ℱΛ).

Theorem 6.

A matrix B=(bp,q,j,k) is regularly (Λ)almost conservative if and only if

∥B∥=supp,q∑j,k|bp,q,j,k|<∞,

limm,n→∞α(m,n,s,t,j,k)=ujk, for each j, k (uniformly in s, t),

limm,n→∞∑k|α(m,n,s,t,j,k)|=uj0, for each j (uniformly in s, t),

limm,n→∞∑j|α(m,n,s,t,j,k)|=u0k, for each k (uniformly in s, t),

limm,n→∞∑j,kα(m,n,s,t,j,k)=u, (uniformly in s, t),

where
(6)β(m,n,s,t,j,k)=1λmμn∑p∈Jm∑q∈Inbp+s,q+t,j,k.
In this case, the ℱΛ-limit of Bx is
(7)ℓu+∑j=0∞(ℓj-ℓ)uj0+∑k=0∞(hk-ℓ)u0k+∑j=0∞∑k=0∞(xj,k-ℓj-hk-ℓ)ujk,
where ℓ=R-limx.
Proof.

Necessity. Suppose that B is regularly (Λ)almost conservative matrix. Fix s, t∈ℤ, the set of integers. Let
(8)Ωm,n,s,t(x)=1λmμn∑p∈Jm∑q∈Imρp+s,q+t(x),
where
(9)ρp,q(x)=∑j=0∞∑k=0∞bp,q,j,kxj,k.
It is clear that
(10)ρp,q∈𝒞R′,p,q=0,1,2,….
Hence Ωm,n,s,t∈𝒞R′ for m, n∈ℕ. Since B is regularly (Λ)almost conservative, we have
(11)P-limm,n→∞Ωm,n,s,t(x)=Ω(x)(say),
uniformly in s, t. It follows that (Ωm,n,s,t(x)) is bounded for x∈𝒞R and fixed s, t. Hence, ∥Ωm,n,s,t(x)∥ is bounded by the uniform boundedness principle.

For each i, v∈ℤ+, define the sequence y=y(m,n,s,t) by
(12)yj,k={sgn(∑p∈Jm∑q∈Inbp+s,q+t,j,k)if0≤j≤i,0≤k≤v,0,ifv<k,i<j.
Then, a double sequence y∈𝒞R, ∥y∥=1, and
(13)|Ωm,n,s,t(y)|=1λmμn∑j=0i∑k=0v|∑p∈Jm∑q∈Inbp+s,q+t,j,k|.
Hence
(14)|Ωm,n,s,t(y)|≤∥Ωm,n,s,t∥∥y∥=∥Ωm,n,s,t∥.
Therefore
(15)1λmμn∑j=0∞∑k=0∞|∑p∈Jm∑q∈Inbp+s,q+t,j,k|≤∥Ωm,n,s,t∥,
so that condition (CR1) follows.

The sequences F(b,c)=(fj,k(b,c)), F(b)=(fj,k(b)), G(c)=(gj,k(c)), and G=(gj,k) are defined by
(16)fj,k(q,r)={1,if(j,k)=(b,c),0,otherwise,fj,k(b)={1,ifj=b,0,otherwise,gj,k(c)={1,ifk=c,0,otherwise,gj,k=1,∀j,k.
Since F(j,k), F(j), G(k), G∈𝒞R, the P-limit of Ωm,n,s,t(F(j,k)), Ωm,n,s,t(F(j)), Ωm,n,s,t(G(k)), and Ωm,n,s,t(G) must exist, uniformly in s, t. Hence, the conditions (CR2)–(CR5) must hold, respectively.

Sufficiency. Suppose that the conditions (CR1)–(CR5) hold and a double sequence x=(xjk)∈𝒞R. Fix s, t∈ℤ. Then
(17)Ωm,n,s,t(x)=1λmμn∑j=0∞∑k=0∞∑p∈Jm∑q∈Inbp+s,q+t,j,kxj,k,|Ωm,n,s,t(x)|≤1λmμn∑j=0∞∑k=0∞|∑p∈Jm∑q∈Inbp+s,q+t,j,k|∥xj,k∥.
Therefore, by (CR1), we have |Ωm,n,s,t(x)|≤Cs,t∥x∥, where Cs,t is a constant independent of m, n. Hence Ωm,n,s,t∈𝒞R′ and the sequence (∥Ωm,n,s,t∥) is bounded for each s,t∈ℤ+. It follow from the conditions (CR2), (CR3), (CR4), and (CR5) that the P-limit of Ωm,n,s,t(F(j,k)), Ωm,n,s,t(F(j)), Ωm,n,s,t(G(k)), and Ωm,n,s,t(G) exist for all j, k, s, and t. Since {G, F(j), G(k) and F(j,k)} is a fundamental set in 𝒞R (see [24]), it follows that
(18)limm,n→∞Ωm,n,s,t(x)=Ωs,t(x)
exists and Ωs,t∈𝒞R′. Therefore, Ωs,t has the form
(19)Ωs,t(x)=ℓΩs,t(G)+∑j=0∞(ℓj-ℓ)Ωs,t(F(j))+∑k=0∞(hk-ℓ)Ωs,t(G(k))+∑j=0∞∑k=0∞(xj,k-ℓj-hk+ℓ)Ωs,t(F(j,k)).
But Ωs,t(F(j,k))=ujk, Ωs,t(F(j))=uj0, Ωs,t(G(k))=u0k, and Ωs,t(G)=u by the conditions (CR2)–(CR5), respectively. Hence
(20)limm,n→∞Ωm,n,s,t(x)=Ω(x)
exists for each x∈𝒞R and s, t=0,1,2,… with
(21)Ω(x)=ℓu+∑j=0∞(ℓj-ℓ)uj0+∑k=0∞(hk-ℓ)u0k+∑j=0∞∑k=0∞(xj,k-ℓj-hk-ℓ)ujk.
Since Ωm,n,s,t(x)∈𝒞R′ for each m, n, s, and t, it has the form
(22)Ωm,n,s,t(x)=ℓΩm,n,s,t(G)+∑j=0∞(ℓj-ℓ)Ωm,n,s,t(F(j))+∑k=0∞(hk-ℓ)Ωm,n,s,t(G(k))+∑j=0∞∑k=0∞(xj,k-ℓj-hk-ℓ)Ωm,n,s,t(F(j,k)).
It is easy to see from (21) and (22) that the convergence of (Ωm,n,s,t(x)) to Ω(x) is uniform in s, t, since Ωm,n,s,t(G)→u, Ωm,n,s,t(F(j))→uj0, Ωm,n,s,t(G(k))→u0k, and Ωm,n,s,t(F(j,k))→ujk(m,n→∞) uniformly in s, t. Therefore, B is regularly (Λ)almost conservative.

Let us recall the following lemma, which is proved by Mursaleen and Mohiuddine [25].

Lemma 7.

Let A(s,t)=(am,n,j,k(s,t)), s, t=0,1,2,…, be a sequence of infinite matrices such that

∥A(s,t)∥<H<+∞ for all s, t; and

for each j, klimm,nam,n,j,k(s,t)=0 uniformly in s, t.

Then
(23)limm,n∑j∑kam,n,j,k(s,t)xj,k=0uniformlyins,tforeachx∈ℒ∞
if and only if
(24)limm,n∑j∑k|am,n,j,k(s,t)|=0uniformlyins,t.Theorem 8.

A matrix B=(bp,q,j,k) is (Λ)almost coercive if and only if

∥B∥=supp,q∑j,k|bp,q,j,k|<∞,

limm,n→∞α(m,n,s,t,j,k)=ujk, for each j, k (uniformly in s, t),

limm,n→∞∑j=1∞∑k=1∞(1/λmμn)|∑p∈Jm∑q∈Inbp+s,q+t,j,k-ujk|=0, uniformly in s, t.

In this case, the ℱΛ-limit of Bx is ∑j=1∞∑k=1∞ujkxj,k for every (xj,k)∈ℒ∞.
Proof.

Sufficiency. Assume that conditions (AC1)–(AC3) hold. For any positive integers J, K(25)∑j=1J∑k=1K|ujk|=∑j=1J∑k=1Klimm,n→∞1λmμn|∑p∈Jm∑q∈Inbp+s,q+t,j,k|=limm,n→∞1λmμn∑j=1J∑k=1K|∑p∈Jm∑q∈Inbp+s,q+t,j,k|≤limsupm,n1λmμn∑p∈Jm∑q∈In∑j=1∞∑k=1∞|bp+s,q+t,j,k|≤∥B∥.
This shows that ∑j=1∞∑k=1∞|ujk| converges and that ∑j=1∞∑k=1∞ujkxj,k is defined for every double sequence x=(xj,k)∈ℒ∞.

Let (xj,k) be any arbitrary bounded double sequence. For every positive integers m, n(26)∥∑j=1∞∑k=1∞(1λmμn∑p∈Jm∑q∈Inbp+s,q+t,j,k-ujk)xj,k∥=∥∑j=1∞∑k=1∞[1λmμn∑p∈Jm∑q∈In[bp+s,q+t,j,k-ujk]]xj,k∥≤sups,t[|∑j=1∞∑k=1∞[1λmμn∑p∈Jm∑q∈In[bp+s,q+t,j,k-ujk]]xj,k|]≤∥x∥sups,t[1λmμn∑j=1∞∑k=1∞|∑p∈Jm∑q∈In[bp+s,q+t,j,k-ujk]|].
Letting p, q→∞ and using condition (AC3), we get
(27)1λmμn∑j=1∞∑k=1∞∑p∈Jm∑q∈Inbp+s,q+t,j,kxj,k⟶∑j=1∞∑k=1∞ujkxj,k.
Hence, Bx∈ℱΛ with ℱΛ-limBx=∑j=1∞∑k=1∞ujkxj,k.

Necessity. Let B be (Λ)almost coercive matrix. This implies that a four-dimensional matrix B is (Λ)almost conservative; then we have conditions (AC1) and (AC2) from Theorem 6. Now we have to show that (AC3) holds.

Suppose that, for some s, t, we have
(28)limsupm,n1λmμn∑j=1∞∑k=1∞|∑p∈Jm∑q∈In[bp+s,q+t,j,k-ujk]|=N>0.
Since ∥B∥ is finite, therefore N is also finite. We observe that since ∑j=1∞∑k=1∞|ujk|<+∞ and B is (Λ)almost coercive, the matrix A=(ap,q,j,k), where ap,q,j,k=bp,q,j,k-ujk is also (Λ)almost coercive matrix. By an argument similar to that of Theorem 2.1 in [26] for single sequences, one can find x∈ℒ∞ for which Ax∉ℱΛ. This contradiction implies the necessity of (AC3).

Now, we use Lemma 7 to show that this convergence is uniform in s, t. Let
(29)hm,n,j,k(s,t)=1λmμn∑p∈Jm∑q∈In[bp,q,j,k-ujk]
and let H(s,t) be the matrix (hm,n,j,k(s,t)). It is easy to see that ∥H(s,t)∥≤2∥B∥ for every s, t; and from condition (AC2)
(30)limm,nhm,n,j,k(s,t)=0foreachj,k,uniformlyins,t.
For any x∈ℒ∞,
(31)limm,n∑j=1∞∑k=1∞hm,n,j,k(s,t)xj,k=ℱΛ-limBx-∑j=1∞∑k=1∞ujkxj,k
and the limit exists uniformly in s, t, since Bx∈ℱΛ. Moreover, this limit is zero since
(32)|∑j=1∞∑k=1∞hm,n,j,k(s,t)xj,k|≤∥x∥∑j=1∞∑k=1∞|∑p∈Jm∑q∈In[bp,q,j,k-ujk]|λmμn.
Hence
(33)limm,n∑j=1∞∑k=1∞|hm,m,j,k(s,t)|=0uniformlyins,t.
This shows that matrix B=(bp,q,j,k) satisfies condition (AC3).

In the following theorem, we characterize the four-dimensional matrices of type (ℒ1,ℱΛ), where
(34)ℒ1={x=(xj,k):∑j=0∞∑k=0∞|xj,k|<∞},
the space of all absolutely convergent double series.

Theorem 9.

A matrix B∈(ℒ1,ℱΛ) if and only if it satisfies the following conditions:

supm,n,s,t,j,k|(1/λmμn)∑p∈Jm∑q∈Inbp+s,q+t,j,k|<∞,

and the condition (CR2) of Theorem 6 holds.
Proof.

Sufficiency. Suppose that conditions (i) and (CR2) hold. For any double sequence x=(xj,k)∈ℒ1, we see that
(35)limm,n,→∞1λmμn∑j=1∞∑k=1∞∑p∈Jm∑q∈Inbp+s,q+t,j,kxj,k=∑j=1∞∑k=1∞ujkxj,k,
uniformly in s, t and it also converges absolutely. Furthermore,
(36)1λmμn∑j=1∞∑k=1∞∑p∈Jm∑q∈Inbp+s,q+t,j,kxj,k
converges absolutely for each m, n, s, and t. Given ϵ>0, there exist j∘=j∘(ϵ) and k∘=k∘(ϵ) such that
(37)∑j>j∘∑k>k∘|xj,k|<ϵ.
By the condition (CR2), we can find m∘, n∘∈ℕ such that
(38)|∑j≤j∘∑k≤k∘[1λmμn∑p∈Jm∑q∈Inbp+s,q+t,j,k-ujk]xj,k|<∞,
for all m>m∘ and n>n∘, uniformly in s, t. Now, by using the conditions (37), (38), and (CR2), we get
(39)|∑j=1∞∑k=1∞[1λmμn∑p∈Jm∑q∈Inbp+s,q+t,j,k-ujk]xj,k|≤|∑j≤j∘∑k≤k∘[1λmμn∑p∈Jm∑q∈Inbp+s,q+t,j,k-ujk]xj,k|+∑j>j∘∑k>k∘|1λmμn∑p∈Jm∑q∈Inbp+s,q+t,j,k-ujk||xj,k|,
for all m>m∘, n>n∘ and uniformly in s, t. Hence (37) holds.

Necessity. Suppose that B∈(ℒ1,ℱΛ). The condition (CR2) follows from the fact that E∈ℒ1, where E=(e(j,k)) with e(j,k)=1 for all j, k. To verify the condition (i), we define a continuous linear functional Lm,n,s,t(x) on ℒ1 by
(40)Lm,n,s,t(x)=1λmμn∑j=1∞∑k=1∞∑p∈Jm∑q∈Inbp+s,q+t,j,kxj,k.
Now
(41)|Lm,n,s,t(x)|≤supj,k|1λmμn∑p∈Jm∑q∈Inbp+s,q+t,j,k|∥x∥1
and hence
(42)∥Lm,n,s,t∥≤supj,k|1λmμn∑p∈Jm∑q∈Inbp+s,q+t,j,k|.
For any fixed j,k∈ℕ, we define a double sequence x=(xi,ℓ) by
(43)xi,ℓ={sgn(1λmμn∑p∈Jm∑q∈Inbp+s,q+t,j,k),for(i,ℓ)=(j,k),0,for(i,ℓ)≠(j,k).
Then ∥x∥1=1, and
(44)|Lm,n,s,t(x)|=|1λmμn∑p∈Jm∑q∈Inbp+s,q+t,j,kxj,k|=|1λmμn∑p∈Jm∑q∈Inbp+s,q+t,j,k|∥x∥1,
so that
(45)∥Lm,n,s,t∥≥supj,k|1λmμn∑p∈Jm∑q∈Inbp+s,q+t,j,k|.
It follows from (42) and (45) that
(46)∥Lm,n,s,t∥=supj,k|1λmμn∑p∈Jm∑q∈Inbp+s,q+t,j,k|.
Since B∈(ℒ1,ℱΛ), we have
(47)supm,n,s,t|Lm,n,s,t(x)|=supm,n,s,t|1λmμn∑j=1∞∑k=1∞∑p∈Jm∑q∈Inbp+s,q+t,j,kxj,k|<∞.
Hence, by the uniform boundedness principle, we obtain
(48)supm,n,s,t∥Lm,n,s,t(x)∥=supm,n,s,t,j,k|1λmμn∑p∈Jm∑q∈Inbp+s,q+t,j,k|<∞.

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. (130-100-D1434). The authors, therefore, acknowledge with thanks DSR technical and financial support.

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