The purpose of this paper is to define some new types of summability methods for double sequences involving the ideas of de la Vallée-Poussin mean in the framework of probabilistic normed spaces and establish some interesting results.

1. Introduction and Preliminaries

Throughout the paper, the symbols ℕ and ℝ will denote the set of all natural and real numbers, respectively. The notion of convergence for double sequence was introduced by Pringsheim [1]: we say that a double sequence x=(xj,k)j,k∈ℕ of reals is convergent to L in Pringsheim’s sense (briefly, (P) convergent) provided that given ϵ>0 there exists a positive integer N such that |xj,k-L|<ϵ whenever j,k≥N.

The idea of statistical convergence is a generalization of convergence of real sequences which was first presented by Fast [2] and Steinhaus [3], independently. Some of its basic properties and interesting concepts, especially, the notion of statistically Cauchy sequence, were proved by Schoenberg [4], Šalát [5], and Fridy [6]. See, for instance, [7–16] and references therein. Mursaleen and Edely [17] introduced the two-dimensional analogue of natural (or asymptotic) density as follows: let A⊆ℕ×ℕ and A(h,l)={j≤h,k≤l:(j,k)∈A}, where h,l∈ℕ. Then
(1)δ-2(A)=(P)limsuph,l→∞|A(h,l)|hl,δ_2(A)=(P)liminfh,l→∞|A(h,l)|hl
are called the upper and lower asymptotic densities of a two-dimensional set A, respectively, where the vertical bars stand for cardinality of the enclosed set. If δ-2(A)=δ_2(A), then
(2)δ2(A)=(P)limh,l→∞|A(h,l)|hl
is called the double natural density of the set A. In the same paper, using the notion of double natural density, they extended the idea of statistical convergence from single to double sequences (for recent work, see [18–23]).

The double sequence x=(xj,k) is statistically convergent to the number L if, for each ϵ>0, the set {(j,k),j≤h,k≤l:|xj,k-L|≥ϵ} has double natural density zero. We denote this by S-limx=L (or xj,k→L(S)).

Mursaleen initiated the notion of λ-statistical convergence (single sequences) with the help of de la Vallée-Poussin mean, in [24]. For detail of λ-statistical convergence, one can be referred to [25–31] and many others. In [32], Mursaleen et al. presented the notion of (λ,μ)-statistical convergence and (λ,μ)-statistically bounded for double sequences and showed that (λ,μ)-statistically bounded double sequences are (λ,μ)-statistical convergence if and only if (λ,μ)-statistical limit infimum of x=(xj,k) is equal to (λ,μ)-statistical limit supremum of x (also see [33]).

Suppose that λ=(λm) and μ=(μn) are two nondecreasing sequences of positive real numbers such that
(3)λm+1≤λm+1,λ1=0,μn+1≤μn+1,μ1=0
and each tends to infinity.

Recall that (λ,μ)-density of the set K⊆ℕ×ℕ is given by
(4)δλ,μ(K)=(P)limm,n1λmμn×|{m-λm+1≤j≤m,n-μn+1≤k≤n:(j,k)∈K}|
provided that the limit exists.

We remark, that, for λm=m and μn=n, the above density reduces to the double natural density.

The generalized double de la Vallée-Poussin mean is defined as
(5)tm,n(x)=1λmμn∑j∈Jm∑k∈Inxj,k,
where Jm=[m-λm+1,m] and In=[n-μn+1,n].

We say that x=(xj,k) is (λ,μ)-statistically convergent to the number L if, for every ϵ>0,
(6)(P)limm,n1λmμn|{j∈Jm,k∈In:|xj,k-L|≥ϵ}|=0.
We denote this by Sλ,μ-limx=L.

The symbol Δ+ will denote the set of all distribution functions (d.f.) f:ℝ→[0,1] which are nondecreasing, left continuous on ℝ, equal to zero on [-∞,0], and such that f(+∞)=1. The space Δ+ is partially ordered by the usual pointwise ordering of functions.

A triangular norm (or a t-norm) [34] is a binary operation τ:[0,1]×[0,1]→[0,1] which satisfies the following conditions. For all h1,h2,h3∈[0,1]

τ(τ(h1,h2),h3)=τ(h1,τ(h2,h3)),

τ(h1,h2)=τ(h2,h1),

τ(h1,h3)≤τ(h2,h3) whenever h1≤h2,

τ(h1,1)=h1.

In the literature, we have two definitions of probabilistic normed space or, briefly, PN-space; the original one is given by Šerstnev [35] in 1962 who used the concept of Menger [36] to define such space and the other one by Alsina et al. [37] (for more details, see [38–40]).

According to Šerstnev [35], a probabilistic normed space is a triple (X,ν,τ), where X is a real linear space, ν is the probabilistic norm, that is, ν is a function from X into Δ+, for x∈X, the d.f. ν(x) is denoted by νx, νx(t) which is the value of νx at t∈ℝ, and τ is a t-norm that satisfies the following conditions:

νx(0)=0;

νx(t)=1 for all t>0 if and only if x=0;

ναx(t)=νx(t/|α|) for all t>0, α∈ℝ with α≠0 and x∈X;

νx+y(t1+t2)≥τ(νx(t1),νy(t2)) for all x,y∈X and t1,t2∈ℝ+={x∈ℝ:x≥0}.

2. Main Results

We define the notions of (λ,μ)-summable, statistically (λ,μ)-summable, statistically (λ,μ)-Cauchy, and statistically (λ,μ)-complete for double sequences with respect to PN-space and establish some interesting results.

Definition 1.

A double sequence x=(xj,k) is said to be (λ,μ)-summable in (X,ν,τ) (or, shortly, ν(λ,μ)-summable) to L if for each ϵ>0, θ∈(0,1) there exists N∈ℕ such that νtm,n(x)-L(ϵ)>1-θ for all m,n≥N. In this case, one writes ν(λ,μ)-limx=L.

Definition 2.

A double sequence x=(xj,k) is said to be statistically(λ,μ)-summable in (X,ν,τ) (or, shortly, ν(Sλ,μ)-summable) to L if δ2(Kλ,μ)=0, where Kλ,μ={(m,n)∈ℕ×ℕ:νtm,n(x)-L(ϵ)≤1-θ}; that is, if, for each ϵ>0, θ∈(0,1),
(7)(P)limh,l1hl|{m≤h,n≤l:νtm,n(x)-L(ϵ)≤1-θ}|=0
or equivalently
(8)(P)limh,l1hl|{m≤h,n≤l:νtm,n(x)-L(ϵ)>1-θ}|=1.
In this case, we write ν(Sλ,μ)-limx=L, and L is called the ν(Sλ,μ)-limit of x.

Definition 3.

A double sequence x=(xj,k) is said to be statistically (λ,μ)-Cauchy in (X,ν,τ) (or, shortly, ν(Sλ,μ)-Cauchy) if, for every ϵ>0 and θ∈(0,1), there exist M,N∈ℕ such that, for all m,p≥M, n,q≥M, the set Sϵ(λ,μ)={(m,n)∈ℕ×ℕ:νtm,n(x)-tp,q(x)(ϵ)≤1-θ} has double natural density zero; that is,
(9)(P)limh,l1hl|{m≤h,n≤l:νtm,n(x)-tp,q(x)(ϵ)≤1-θ}|=0.

Theorem 4.

If a double sequence x=(xj,k) is statistically (λ,μ)-summable in (X,ν,τ), that is, ν(Sλ,μ)-lim x=L exists, then ν(Sλ,μ)-limit of (xj,k) is unique.

Proof.

Assume that ν(Sλ,μ)-lim x=L1 and ν(Sλ,μ)-lim x=L2. We have to prove that L1≠L2. For given ϵ>0, choose q>0 such that
(10)τ((1-q),(1-q))>1-ϵ.
Then, for any t>0, we define
(11)Mq′(λ,μ)={(m,n)∈ℕ×ℕ:νtm,n(x)-L1(t)≤1-q},Mq′′(λ,μ)={(m,n)∈ℕ×ℕ:νtm,n(x)-L2(t)≤1-q}.
Since ν(Sλ,μ)-lim x=L1 implies δ2(Mq′(λ,μ))=0 and similarly we have δ2(Mq′′(λ,μ))=0. Now, let Mq(λ,μ)=Mq′(λ,μ)∩Mq′′(λ,μ). It follows that δ2(Mq(λ,μ))=0 and hence the complement Mqc(λ,μ) is nonempty set and δ2(Mqc(λ,μ))=1. Now, if (m,n)∈ℕ×ℕ∖Mq(λ,μ), then
(12)νL1-L2(t)≥τ(νtm,n(x)-L1(t2),νtm,n(x)-L2(t2))>τ((1-q),(1-q))>1-ϵ.
Since ϵ>0 was arbitrary, we obtain νL1-L2(t)=1 for all t>0. Hence L1=L2. This means that ν(Sλ,μ)-limit is unique.

Theorem 5.

If a double sequence x=(xj,k) is ν(λ,μ)-summable to L, then it is ν(Sλ,μ)-summable to the same limit.

Proof.

Let us consider that ν(λ,μ)-lim x=L. For every ϵ>0 and t>0, there exists a positive integer N such that
(13)νtm,n(x)-L(t)>1-ϵ
holds for all m,n≥N. Since
(14)Kϵ(λ,μ)≔{(m,n)∈ℕ×ℕ:νtm,n(x)-L(t)≤1-ϵ}
is contained in ℕ×ℕ, hence δ2(Kϵ(λ,μ))=0; that is, x=(xj,k) is ν(Sλ,μ)-summable to L.

Example 6.

This example proves that the converse of Theorem 5 need not be true. We denote by (ℝ,|·|) the set of all real numbers with the usual norm and τ(a,b)=ab for all a,b∈[0,1]. Assume that νx(t)=t/(t+|x|) for all x∈X and all t>0. Here, we observe that (ℝ,ν,τ) is a PN-space. The double sequence x=(xj,k) is defined by
(15)tm,n(x)={mn;ifm,n=w2,w∈ℕ0;otherwise.
For ϵ>0 and t>0, write
(16)Kϵ(λ,μ)={(m,n)∈ℕ×ℕ:νtm,n(x)(t)≤1-ϵ}.
It is easy to see that
(17)νtm,n(x)(t)=tt+|tm,n(x)|={tt+mn,form,n=w2,w∈ℕ;1,otherwise;
and hence
(18)limνtm,n(x)(t)={0,forifm,n=w2,w∈ℕ;1,otherwise.
We see that the sequence (xj,k) is not (λ,μ)-summable in (ℝ,ν,τ). But the set Kϵ(λ,μ) has double natural density zero since Kϵ(λ,μ)⊂{(1,1),(4,4),(9,9),(16,16),…}. From here, we conclude that the converse of Theorem 5 need not be true.

Theorem 7.

A double sequence x=(xj,k) is ν(Sλ,μ)-summable to L if and only if there exists a subset K={(jm,kn):j1<j2<⋯<jm<⋯;k1<k2<⋯<kn<⋯}⊆ℕ×ℕ such that δ2(K)=1 and ν(λ,μ)-lim xjm,kn=L.

Proof.

Assume that there exists a subset K = {(jm,kn):j1<j2<⋯<jm<⋯; k1 < k2 < ⋯ < kn < ⋯}⊆ℕ×ℕ such that δ2(K)=1 and ν(λ,μ)-lim xjm,kn=L. Then there exists N∈ℕ such that
(19)νtm,n(x)-L(t)>1-ϵ
holds for all m,n>N. Put Kϵ(λ,μ)={(m,n)∈ℕ×ℕ:νtjm,kn(x)-ξ(t)≤1-ϵ} and K′ = {(jN+1,kN+1), (jN+2,kN+2),…}. Then δ2(K′) = 1 and Kϵ(λ,μ)⊆ℕ-K′ which implies that δ2(Kϵ(λ,μ))=0. Hence x=(xj,k) is statistically (λ,μ)-summable to L in PN-space.

Conversely, suppose that x=(xj,k) is ν(Sλ,μ)-summable to L. For q=1,2,3,… and t>0, write
(20)Kq(λ,μ)={(m,n)∈ℕ×ℕ:νtjm,kn(x)-L(t)≤1-1q},Mq(λ,μ)={(m,n)∈ℕ×ℕ:νtjm,kn(x)-L(t)>1q}.
Then δ2(Kq(λ,μ))=0 and
(21)M1(λ,μ)⊃M2(λ,μ)⊃⋯Mi(λ,μ)⊃Mi+1(λ,μ)⊃⋯,(22)δ2(Mq(λ,μ))=1,q=1,2,⋯.

Now, we have to show that, for (m,n)∈Mq(λ,μ), x=(xjm,kn) is ν(λ,μ)-summable to L. Suppose that x=(xjm,kn) is not ν(λ,μ)-summable to L. Therefore, there is ϵ>0 such that νtjm,kn-L(t)≤ϵ for infinitely many terms. Let
(23)Mϵ(λ,μ)={(m,n)∈ℕ×ℕ:νtjm,kn-ξ(t)>ϵ},
and ϵ>1/q with q=1,2,3,…. Then
(24)δ(Mϵ(λ,μ))=0,
and by (21), Mq(λ,μ)⊂Mϵ(λ,μ). Hence δ(Mq(λ,μ))=0, which contradicts (22) and therefore x=(xjm,kn) is ν(λ,μ)-summable to L.

Theorem 8.

If a double sequence x=(xj,k) is statistically (λ,μ)-summable in PN-space, then it is statistically (λ,μ)-Cauchy.

Proof.

Suppose that ν(Sλ,μ)-lim x=L. Let ϵ>0 be a given number so that we choose q>0 such that
(25)τ((1-q),(1-q))>1-ϵ.
Then, for t>0, we have
(26)δ2(Aq(λ,μ))=0,
where Aq(λ,μ)={(m,n)∈ℕ×ℕ:νtm,n(x)-L(t/2)≤1-q} which implies that
(27)δ2(Aqc(λ,μ))=δ2({(m,n)∈ℕ×ℕ:νtm,n(x)-L(t2)>1-q})=1.
Let (f,g)∈Aqc(λ,μ). Then νtf,g(x)-L(t/2)>1-q.

Now, let
(28)Bϵ(λ,μ)={(m,n)∈ℕ×ℕ:νtm,n(x)-tf,g(x)(t)≤1-ϵ}.
We need to show that Bϵ(λ,μ)⊂Aq(λ,μ). Let (m,n)∈Bϵ(λ,μ)∖Aq(λ,μ). Then νtm,n(x)-tf,g(x)(t)≤1-ϵ, νtm,n(x)-L(t/2)>1-q, and in particular νtf,g(x)-L(t/2)>1-q. Then
(29)1-ϵ≥νtm,n(x)-tf,g(x)(t)≥τ(νtm,n(x)-L(t2),νtf,g(x)-L(t2))>τ((1-q),(1-q))>1-ϵ,
which is not possible. Hence Bϵ(λ,μ)⊂Aq(λ,μ). Therefore, by (26) δ2(Bϵ(λ,μ))=0. Hence, x is statistically (λ,μ)-Cauchy in PN-space.

Definition 9.

Let (X,ν,τ) be a PN-space. Then,

PN-space is said to be complete if every Cauchy double sequence is P-convergent in (X,ν,τ);

PN-space is said to be statistically (λ,μ)-complete (or, shortly, ν(Sλ,μ)-complete) if every statistically (λ,μ)-Cauchy sequence in PN-space is statistically (λ,μ)-summable.

Theorem 10.

Every probabilistic normed space (X,ν,τ) is ν(Sλ,μ)-complete but not complete in general.

Proof.

Suppose that x=(xj,k) is ν(Sλ,μ)-Cauchy but not ν(Sλ,μ)-summable. Then there exist M,N∈ℕ such that, for all m,p≥M, n,q≥M, the set Eϵ(λ,μ)={(m,n)∈ℕ×ℕ:νtm,n(x)-tp,q(x)(t)≤1-ϵ} has double natural density zero; that is, δ2(Eϵ(λ,μ))=0 and
(30)δ2(Fϵ(λ,μ))=δ2({(m,n)∈ℕ×ℕ:νtm,n(x)-L(t2)>1-ϵ})=0.
This implies that δ2(Fϵc(λ,μ))=1, since
(31)νtm,n(x)-tp,q(x)(t)≥2νtm,n(x)-L(t2)>1-ϵ,
if νtm,n(x)-L(t/2)>(1-ϵ)/2. Therefore δ2(Eϵc(λ,μ))=0; that is, δ2(Eϵ(λ,μ))=1, which leads to a contradiction, since x=(xj,k) was ν(Sλ,μ)-Cauchy. Hence x=(xj,k) must be ν(Sλ,μ)-summable.

To see that a probabilistic normed space is not complete in general, we have the following example.

Example 11.

Let X=(0,1] and νx(t)=t/(t+|x|) for t>0. Then (X,ν,τ) is a probabilistic normed space but not complete, since the double sequence (1/mn) is Cauchy with respect to (X,ν,τ) but not P-convergent with respect to the present PN-space.

Conflict of Interests

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

Acknowledgments

This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant no. (303/130/1433). The authors, therefore, acknowledge with thanks DSR technical and financial support.

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