Two concepts—one of statistical convergence and the other of de la Vallée-Poussin mean—play an important role in recent research on summability theory. In this work we define a new type of summability methods and statistical completeness involving the ideas of de la Vallée-Poussin mean and statistical convergence in the framework of probabilistic normed spaces.
1. Introduction, Definitions, and Preliminaries
Fast [1] presented the following definition of statistical convergence for sequences of real numbers. Let K⊆ℕ, the set of natural numbers, and Kn={k≤n:k∈K}. The natural density of K is defined by δ(K)=limnn-1|Kn| if the limit exists, where |Kn| denotes the cardinality of Kn.
The sequence x=(xj) is said to be statistically convergent to the number ℓ if for every ϵ>0 the set Kɛ:={k∈ℕ:|xk-ℓ|≥ɛ} has natural density zero; that is, for each ϵ>0,
(1)limn1n|{j≤n:|xj-ℓ|≥ϵ}|=0.
Note that every convergent sequence is statistically convergent to the same limit, but its converse need not be true.
In 1985, Fridy [2] has defined the notion of statistically Cauchy sequence and proved that it is equivalent to statistical convergence and since then a large amount of work has appeared. Various extensions, generalizations, variants, and applications have been given by several authors so far, for example, [3–8] and references therein. In the recent past, Mursaleen [9] presented a generalization of statistical convergence by using de la Vallée-Poussin mean which is known λ-statistical convergence and further studied by Çolak and Bektas [10, 11]. For more details related to this concept we refer to [12–18].
Let λ=(λn) be a nondecreasing sequence of positive numbers tending to ∞ such that
(2)λn+1≤λn+1,λ1=0.
The generalized de la Vallée-Poussin mean is defined by
(3)tn(x)=:1λn∑j∈Inxj,
where In=[n-λn+1,n].
A sequence x=(xj) is said to be (V,λ)-summable to a number ℓ if
(4)tn(x)⟶ℓasn⟶∞.
In this case ℓ is called λ-limit of x.
Let K⊆ℕ be a set of positive integers; then
(5)δλ(K)=limn1λn|{n-λn+1≤j≤n:j∈K}|
is said to be λ-density of K.
In case λn=n, λ-density reduces to the natural density. Also, since (λn/n)≤1, δ(K)≤δλ(K) for every K⊆ℕ.
The number sequence x=(xj) is said to be λ-statistically convergent to the number ℓ if, for each ϵ>0, δλ(Kϵ)=0, where Kϵ={j∈ℕ:|xj-ℓ|>ϵ}; that is,
(6)limn1λn|{j∈In:|xj-ℓ|>ϵ}|=0.
In this case we write Sλ-limjxj=ℓ and we denote the set of all λ-statistically convergent sequences by Sλ.
A distribution function is an element of Δ+, where Δ+={f:ℝ→[0,1];f is left-continuous, nondecreasing, f(0)=0 and f(+∞)=1} and the subset D+⊆Δ+ is the set D+={f∈Δ+;l-f(+∞)=1}. Here l-f(+∞) denotes the left limit of the function f at the point x. The space Δ+ is partially ordered by the usual pointwise ordering of functions; that is, f≤g if and only if f(x)≤g(x) for all x∈ℝ.
A triangle function is a binary operation on Δ+, namely, a function τ:Δ+×Δ+→Δ+ that is associative, commutative nondecreasing and which has e as unit; that is, for all f,g,h∈Δ+, we have
τ(τ(f,g),h)=τ(f,τ(g,h)),
τ(f,g)=τ(g,f),
τ(f,h)=τ(g,h) whenever f≤g,
τ(f,e)=f.
Here e is the d.f. defined by
(7)e(x)={0ifx≤0;1ifx>0.
We remark that the set Δ as well as its subsets can be partially ordered by the usual pointwise order: in this order, e is the maximal element in Δ+.
There are two definitions of probabilistic normed space, the original one by Šerstnev [19] who used the idea of Menger [20] to define such space and the other one by Alsina et al. [21].
According to Šerstnev, a probabilistic normed space (for short, PN-space) is a triple (X,ν,τ), where X is a real linear space, τ is a triangle function, and ν is the probabilistic norm; that is, ν is a map from X into Δ+ that satisfies the following conditions:
νx=e if and only if x=θ, where θ is the null vector of X,
ναx(t)=νx(t/|α|) for all t>0, α∈ℝ with α≠0, and x∈X,
νx+y≥τ(νx,νy) whenever x,y∈X.
Here νx(t) denotes the value of νx at t∈ℝ.
In this paper, using the notions of statistical convergence and de la Vallée-Poussin mean, we define and study a new type of summability methods in the setting of probabilistic normed spaces. We also introduce a new type of statistical completeness through de la Vallée-Poussin mean in this framework.
2. Statistical Summability through de la Vallée-Poussin Mean
Here we introduce the notions of λ-summable and statistically λ-summable in PN-space and give some of its properties. We will assume throughout this paper that (X,ℱ,τ) is a probabilistic normed space.
Definition 1.
A sequence x=(xk) is said to be λ-summable in PN-space (X,ν,τ) or simply (λ)ν-summable to ξ if for each ɛ>0, θ∈(0,1) there exists a positive integer j0 such that νtj(x)-ξ(ɛ)>1-θ for all j≥j0. In this case one writes ν(λ)-limxk=ξ and ξ is called the ν(λ)-limit of the sequence x=(xk).
Definition 2.
A sequence x=(xk) is said to be statistically λ-summable in (X,ν,τ) or simply S(λ)ν-summable to ξ if for each ɛ>0, θ∈(0,1) the set Kɛ(λ)={j∈ℕ:νtj(x)-ξ(ɛ)≤1-θ} has natural density zero (briefly, δ(Kɛ(λ))=0); that is,
(8)limn1n|{j≤n:νtj(x)-ξ(ɛ)≤1-θ}|=0.
In this case one writes ν(Sλ)-limxk=ξ, and ξ is called the ν(Sλ)-limit of x. One may write (8) in the alternative form as
(9)limn1n|{j≤n:νtj(x)-ξ(ɛ)>1-θ}|=1.
Theorem 3.
If a sequence x=(xk) is statistically λ-summable in PN-space, that is, ν(Sλ)-lim
xk=ξ exists, then it is unique.
Proof.
Suppose that there exist two elements ξ1,ξ2∈X with ξ1≠ξ2 such that ν(Sλ)-limxk=ξ1 and ν(Sλ)-limxk=ξ2. Let ϵ>0 be given. Choose q>0 such that
(10)τ((1-q),(1-q))>1-ϵ.
Then, for any t>0, we define
(11)Mq′(λ)={j∈ℕ:νtj(x)-ξ1(t)≤1-q},Mq′′(λ)={j∈ℕ:νtj(x)-ξ2(t)≤1-q}.
Since ν(Sλ)-limx=ξ1 implies δ(Mq′(λ))=0 and, similarly, we have δ(Mq′′(λ))=0. Now, let Mq(λ)=Mq′(λ)∩Mq′′(λ). It follows that δ(Mq(λ))=0 and hence the complement Mqc(λ) is nonempty set and δ(Mqc(λ))=1. Now, if k∈ℕ∖Mq(λ), then
(12)νξ1-ξ2(t)≥τ(νtj(x)-ξ1(t2),νtj(x)-ξ2(t2))>τ((1-q),(1-q))>1-ϵ.
Since ϵ>0 was arbitrary, we get νξ1-ξ2(t)=1 for all t>0. Hence ξ1=ξ2. This means that ν(Sλ)-limit is unique.
The following theorem gives the algebraic properties of statistically λ-summable sequences in PN-spaces.
Theorem 4.
Let x=(xk) and y=(yk) be two sequences. If ν(Sλ)-lim
xk=ξ1 and ν(Sλ)-lim
yk=ξ2, then
ν(Sλ)-lim
(xk±yk)=ξ1±ξ2,
ν(Sλ)-lim
αxk=αξ, α(≠0)∈ℝ.
Proof of the theorem is straightforward and so omitted.
Theorem 5.
If a sequence x=(xk) is λ-summable to ξ in PN-space; then it is statistically λ-summable to the same limit.
Proof.
Let ν(λ)-limxk=ξ. Then for every ϵ>0 and t>0, there is a positive integer j0 such that
(13)νtj(x)-ξ(t)>1-ϵ
for all j≥j0. Since the set
(14)Kϵ(λ):={j∈ℕ:νtj(x)-ξ(t)≤1-ϵ}
is contained in {1,2,3,…,j0-1}. As we know, every finite subset of ℕ has natural density zero; that is, δ(Kϵ(λ))=0. Hence, a sequence x=(xk) is S(λ)ν-summable to ξ.
Remark 6.
The converse of the above theorem is not true in general, which is verified by the following example.
Example 7.
Let (ℝ,|·|) denote the space of all real numbers with the usual norm and τ(a,b)=ab for all a,b∈[0,1]. Let νx(t)=t/(t+|x|) for all x∈X and t>0. In this case, we observe that (ℝ,ν,τ) is a PN-space. Define a sequence x=(xk) by
(15)tj(x)={j;ifj=n2,n∈ℕ0;otherwise.
For ϵ>0 and t>0, write
(16)Kϵ(λ)={j∈ℕ:νtj(x)(t)≤1-ϵ}.
It is easy to see that
(17)νtj(x)(t)=tt+|tj(x)|={tt+j,forj=n2,n∈ℕ;1,otherwise;
hence
(18)limνtj(x)(t)={0,forifj=n2,n∈ℕ;1,otherwise.
Therefore, the sequence (xk) is not (λ)ν-summable. But the set Kϵ(λ) has natural density zero since Kϵ(λ)⊂{1,4,9,16,…}. From here, we conclude that the converse of Theorem 5 need not be true.
Theorem 8.
A sequence x=(xk) is statistically λ-summable in PN-space to ξ if and only if there exists a subset K={k1<k2<⋯<kn<⋯}⊆ℕ such that δ(K)=1 and ν(λ)-lim
xkn=ξ.
Proof.
Suppose that there exists a subset K={k1<k2<⋯<kn<⋯}⊆ℕ such that δθ(K)=1 and ν(λ)-limxkn=ξ. Then there exists a positive integer N∈ℕ such that for n≥N(19)νtn(x)-ξ(t)>1-ϵ.
Put Kϵ(λ)={n∈ℕ:νtn(x)-ξ(t)≤1-ϵ} and K′={kN+1,kN+2,…}. Then δ(K′)=1 and Kϵ(λ)⊆ℕ-K′ which implies that δ(Kϵ(λ))=0. Hence x=(xk) is statistically λ-summable to ξ in PN-space.
Conversely, let sequence x=(xk) is statistically λ-summable to ξ. For q=1,2,3,… and t>0, write
(20)Kq(λ)={j∈ℕ:νtkj(x)-ξ(t)≥1-1q},Mq(λ)={j∈ℕ:νtkj(x)-ξ(t)<1-1q}.
Then δ(Kq(λ))=0 and
(21)M1(λ)⊃M2(λ)⊃⋯Mi(λ)⊃Mi+1(λ)⊃⋯,(22)δ(Mq(λ))=1,q=1,2,….
Now we have to show that for j∈Mq(λ), x=(xkj) is (λ)ν-summable to ξ. Suppose that x=(xkj) is not (λ)ν-summable to ξ. Therefore there is ϵ>0 such that νtkj-ξ(t)≥ϵ for infinitely many terms. Let
(23)Mϵ(λ)={j∈ℕ:νtkj-ξ(t)<ϵ},
and ϵ>1/q with q=1,2,3…. Then
(24)δ(Mϵ(λ))=0,
and, by (21), Mq(λ)⊂Mϵ(λ). Hence δ(Mq(λ))=0, which contradicts (22) and therefore x=(xkj) is (λ)ν-summable to ξ.
Similarly we can prove the following dual statement.
Theorem 9.
A sequence x=(xk) is λ-statistically summable in PN-space to ξ if and only if there exists a subset K={k1<k2<⋯<kn<⋯}⊆ℕ such that δλ(K)=1 and ν-lim
xkn=ξ.
3. Statistically Complete through de la Vallée-Poussin Mean
In this section, we define the notions of statistically λ-Cauchy and statistically λ-complete with respect to probabilistic normed space and prove related results.
Definition 10.
A sequence x=(xk) is said to be statistically λ-Cauchy in (X,ν,τ) or simply S(λ)ν-Cauchy if, for every ϵ>0 and θ∈(0,1), there exists a number N=N(ϵ) such that, for all j,h≥N, the set Sϵ(λ)={j∈ℕ:νtj(x)-th(x)(ϵ)≤1-θ} has natural density zero (briefly, δ(Sϵ(λ))=0); that is,
(25)limn1n|{j≤n:νtj(x)-th(x)(ϵ)≤1-θ}|=0.
Theorem 11.
A sequence x=(xk) is statistically λ-summable in PN-space; then it is statistically λ-Cauchy.
Proof.
Suppose that ν(Sλ)-limxk=ℓ. Let ϵ>0 be a given number and choose q>0 such that
(26)τ((1-q),(1-q))>1-ϵ.
Then, for t>0, we have δ(Aq(λ))=0, where Aq(λ)={j∈ℕ:νtj(x)-ℓ(t/2)≤1-q} which implies that
(27)δ(Aqc(λ))=δ({j∈ℕ:νtj(x)-ℓ(t2)>1-q})=1.
Let m∈Aqc(λ). Then νtm(x)-ℓ(t/2)>1-q.
Now, let
(28)Bϵ(λ)={j∈ℕ:νtj(x)-tm(x)(t)≤1-ϵ}.
We need to show that Bϵ(λ)⊂Aq(λ). Let j∈Bϵ(λ). Then νtj(x)-tm(x)(t)≤1-ϵ and hence νtj(x)-ℓ(t/2)≤1-q; that is, j∈Aq(λ). Otherwise, if νtj(x)-ℓ(t/2)>1-q, then
(29)1-ϵ≥νtj(x)-tm(x)(t)≥τ(νtj(x)-ℓ(t2),νtm(x)-ℓ(t2))>τ((1-q),(1-q))>1-ϵ,
which is not possible. Therefore Bϵ(λ)⊂Aq(λ) and hence a sequence x is statistically λ-Cauchy in PN-space.
Definition 12.
Let (X,ν,τ) be a PN-space. Then,
PN-space is said to be complete if every Cauchy sequence is convergent in (X,ν,τ);
PN-space is said to be statistically λ-complete or simply S(λ)ν-complete if every statistically λ-Cauchy sequence in (X,ν,τ) is statistically λ-summable.
Theorem 13.
Every probabilistic normed space (X,ν,τ) is statistically λ-complete but not complete in general.
Proof.
Suppose that x=(xk) is statistically λ-Cauchy in PN-space but not statistically λ-summable. Then there exists M∈ℕ such that
(30)δ(Eϵ(λ))=δ({j∈ℕ:νtj(x)-tm(x)(t)≤1-ϵ})=0,δ(Fϵ(λ))=δ({j∈ℕ:νtj(x)-ℓ(t2)>1-ϵ})=0.
This implies that δ(Fϵc(λ))=1. Since
(31)νtj(x)-tm(x)(t)≥2νtj(x)-ℓ(t2)>1-ϵ,
if νtj(x)-ℓ(t/2)>(1-ϵ)/2, then δ(Eϵc(λ))=0; that is, δ(Eϵ(λ))=1, which leads to a contradiction, since x=(xk) was statistically λ-Cauchy. Hence x=(xk) must be statistically λ-summable in PN-space.
A probabilistic normed space is not complete in general; we verify this by the following example.
Example 14. 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 sequence (1/(n+1)) is Cauchy with respect to (X,ν,τ) but not convergent with respect to the present PN-space.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The author thanks the referees for their comments.
FastH.Sur la convergence statistique19512241244MR0048548ZBL0044.33605FridyJ. A.On statistical convergence198554301313MR816582ZBL0588.40001MursaleenM.MohiuddineS. A.On ideal convergence in probabilistic normed spaces201262149622-s2.0-8485526558710.2478/s12175-011-0071-9MR2886653ZBL1274.40034MursaleenM.AlotaibiA.Statistical summability and approximation by de la Vallée-Poussin mean20112433203242-s2.0-7865000901610.1016/j.aml.2010.10.014MR2741037ZBL1216.40003TripathyB. C.On genralized difference paranormed statistically convergent sequences20043556556632-s2.0-3142742717MR2071731ZBL1073.46004TripathyB. C.DuttaH.On some lacunary difference sequence spaces defined by a sequence of Orlicz functions and Δmn-statistical convergence2012201417430MR2928432TripathyB. C.BaruahA.EtM.GungorM.On almost statistical convergence of new type of generalized difference sequence of fuzzy numbers2012362147155MR2959937ZBL1278.40002TripathyB. C.BaruahA.Lacunary statically convergent and lacunary strongly convergent generalized difference sequences of fuzzy real numbers20105045655742-s2.0-7865092466910.5666/KMJ.2010.50.4.565MR2747754ZBL1229.40007Mursaleenλ-statistical convergence2000501111115MR1764349ZBL0953.40002ÇolakR.BektasÇ. A.λ-statistical convergence of order α201131395395910.1016/S0252-9602(11)60288-9MR2830535ZBL1240.40016KarakusS.Statistical convergence on probabilistic normed space2007121123ÇakalliH.Lacunary statistical convergence in topological groups1995262113119MR1318105ZBL0835.43006ÇakalliH.KhanM. K.Summability in topological spaces20112433483522-s2.0-7864999165810.1016/j.aml.2010.10.021MR2741044ZBL1216.40009MohiuddineS. A.AiyubM.Lacunary statistical convergence in random 2-normed spaces201263581585MR2950573MohiuddineS. A.AlotaibiA.MursaleenM.Statistical convergence of double sequences in locally solid Riesz spaces201220129719729MR2975305ZBL1262.4000510.1155/2012/719729MursaleenM.KarakayaV.ErtürkM.GürsoyF.Weighted statistical convergence and its application to Korovkin type approximation theorem2012218189132913710.1016/j.amc.2012.02.068MR2923012ZBL1262.40004SavasE.MohiuddineS. A.λ̅-statistically convergent double sequences in probabilistic normed spaces20126219910810.2478/s12175-011-0075-5MR2886657ZBL1274.40016TripathyB. C.SenM.NathS.I-convergence in probabilistic n-normed space20121661021102710.1007/s00500-011-0799-8ZBL1264.40006ŠerstnevA. N.Random normed spaces: problems of completeness19621224320MR0174032MengerK.Statistical metrics19422812535537MR000757610.1073/pnas.28.12.535ZBL0063.03886AlsinaC.SchweizerB.SklarA.On the definition of a probabilistic normed space1993461-291982-s2.0-000273687910.1007/BF01834000MR1220724ZBL0792.46062