# StatisticalΛ-Convergence in Probabilistic Normed Spaces

Authors: M. Aldhaifallah; K. S. Nisar; H. M. Srivastava; M. Mursaleen
Journal: Journal of Function Spaces
 (2017)

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## Abstract

The main objective of the study was to understand the notion of Λ-convergence and to study the notion of probabilistic normed(PN) spaces. The study has also aimed to define the statistical Λ-convergence and statistical Λ-Cauchy in PN-spaces. The concepts of these approaches have been defined by some examples, which have demonstrated the concepts of statistical Λ-convergence and statistical Λ-Cauchy in PN-spaces. Previous studies have also been used to understand similar terminologies and notations for the extraction of outcomes.


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## Body

## 1. Introduction

The notion of statistical metric spaces [1–3], called probabilistic metric spaces, was introduced by Menger [4]; it is an important generalization of metric spaces. The concept of probabilistic normed (PN) spaces [5] is a key generalization of the concept of normed spaces. The idea of statistical convergence for sequences of real numbers was introduced by Fast [3] and Steinhaus [6] individualistically. This concept has been studied by many researchers in different set-ups such as in normed linear spaces, in probabilistic normed spaces, in intuitionist fuzzy normed spaces, and in locally convex spaces. The theory of probabilistic metric spaces was studied by several authors (see, for details, [2, 7–10]). Karakus [11] studied the concept of statistical convergence in PN-spaces. It was subsequently carried out in the works due to [12–14]. To define the concept of Λ-convergence, statistical Λ-convergence, and statistical Λ-Cauchy in PN-spaces, the subsequent definitions are needed. The terminology and notations used in this paper are standard as in the recent works [1, 4, 15, 16].Definition 1.
LetM be the subset of set of N. The natural density, δM, is characterized by(1)δM=limn→∞⁡1nm≦n:m∈M,where · is the cardinality of the enclosed set. A number sequence y=(ym) is said to be statistically convergent to the number L if, for each ϵ>0, the set(2)Mϵ=m≦n:ym-L>ϵhas natural density zero; that is,(3)limn→∞⁡1nm≦n:ym-L>ϵ=0and it can be written as stat lim⁡y=L.Definition 2.
A functionh:R→R+0 is a distribution function if it is nondecreasing and left-continuous with(4)inf⁡hu=0u∈R,sup⁡hu=0u∈R.Here D+ is the set of all distribution functions such that h(0)=0. If a∈R+0, then Ha∈D+, where(5)Hau=0u≦a1u>a,clearly for all h∈D+.
At-norm is a continuous mapping ∗:[0,1]×[0,1]→[0,1] such that ([0,1],∗) is an Abelian monoid with unit one and m∗n=k∗l if m=k and n=l for all k,l,m,n∈[0,1].Definition 3.
LetY be a linear space of dimension ≥ 1. Suppose also that ∗ is a t-norm and Y→D+. Then G is said to be a probabilistic norm and (Y;G,⁎) is referred to as a probabilistic normed space if it satisfies the following:(i)
Gy0=0 if y and z are linearly dependent, where Gy;z,t is the value of Gy;z at t∈R.(ii)
Gyt=1 for all t>0 if and only if y=0.(iii)
Gρyt=Gyt/ρ.(iv)
Gy+zs+t=Gys∗Gzt for all y,z∈Y and s,t∈R0.Recently, in [17–19], some λ-sequence spaces were introduced and studied. In this paper, let λ=λjj=0∞ be a strictly increasing sequence of positive real numbers tending to infinity; that is,(6)limj→∞⁡λj⟶∞,0<λ0<λ1<⋯<λj<⋯.In this case a sequence y=ymm=0∞ is λ-convergent to the number L∈C, which is known as λ-limit of y, if Λn(y)→L as n→∞, where(7)Λmy=1λm∑j=0mλj-λj-1yjm∈N.Hence it says that y=ymm=0∞ is λ-convergent to the number L if and only if the sequence Λn(y) is convergent to L. Here (and in the sequel), it will take the convention that any term with a negative subscript is equal to zero; for example, λ-1=0 and  y-1=0.
The sets of all Λ-bounded, Λ-convergent, and Λ-null sequencesl∞Λ,  cΛ, and c0Λ, respectively, are defined as follows:(8)l∞Λ=y=ymm=1∞:Λmymsup<∞,cΛ=y=ymm=1∞:limn→∞⁡Λmyexists,c0Λ=y=ymm=1∞:limm→∞⁡Λmy=0.In the present investigation, it is proposed to systematically study the idea of Λ-convergence in PN-spaces. In particular, statistical Λ-convergence and statistical Λ-Cauchy were investigated in PN-spaces and give some illustrative examples to demonstrate these concepts.Definition 4.
Let(Y,G,∗) be a PN-space. A sequence y=(ym) is called convergent in (Y;G,∗) or, simply, G-convergent to ζ if, for every ϵ>0 and ∅∈(0,1), there exists a positive integer m0∋Gym-ζ(ϵ)>1-∅ whenever m⊂m0 and it is written as(9)Glimm→∞⁡ym=ζand ζ is G limit of y=(ym).Definition 5.
Let(Y;G,∗) be a PN-space. A sequence y⊂(ym) is called statistically convergent in (Y;G,∗) or, simply, G  (stat)-convergent to ζ if, for ϵ>0 and ∅∈(0,1),(10)δm∈N:Gym-ϛϵ≦1-∅=0,or equivalently(11)δm∈N:Gym-ϛϵ>∅=1and this case is stated by(12)Gstatlim⁡y=ζand ζ is G(stat) limit of y.Definition 6.
Let(Y;G,∗) be a PN-space. A sequence y=(ym) is known to be statistically Cauchy in (Y;G,∗) or, simply, G (stat)-Cauchy if, for ϵ>0, there exists a number N=N(ϵ)∋,(13)δm∈N:Gyn-ykϵ≦1-∅=0,∀n,k≥N.In addition to the above definitions, the following definitions are given.Definition 7.
A sequencey=ymm=1∞ is called statistically Λ-convergent to the number L if, for each ϵ>0, the set given by(14)Mϵ=m≦n:Λmy-L≧ϵhas asymptotic density zero; that is,(15)limn→∞⁡1nm≦n:Λmy-L≧ϵ=0.Definition 8.
A sequencey=ymm=1∞ is known to be statistically Λ-Cauchy sequence if, for every ϵ>0, there exists a number N=N(ϵ)∋,(16)limn→∞⁡1nm≦nΛmy-ΛNy≧ϵ=0.Based on the previous definitions, the concept of Λ-convergence, statistical Λ-convergence, and statistical Λ-Cauchy in PN-spaces is defined.Definition 9.
Let(Y;G,∗) be a PN-space. A sequence y=(ym) is said to be convergent in (Y;G,∗) or, simply, GΛ-convergent to ζ if, for ϵ>0 and ∅∈(0,1), there exists a positive integer m0∋GΛy-ζ(ϵ)>1-∅ whenever m=m0. In this case GΛlim⁡ym=ζ can be written and ζ is called GΛ limit of the sequence y=(ym).Definition 10.
Let(Y;G,∗) be a PN-space. A sequence y=(ym) is known to be statistically Λ-convergent in (Y;G,∗) or, simply, GΛ(stat)-convergent if, for every ϵ>0 and ∅∈(0,1),(17)δm∈N:GΛmy-ϛϵ≦1-∅=0.Or,(18)δm∈N:GΛmy-ϛϵ≦1-∅=1;that is, (19)limn→∞⁡1nm∈N:GΛmy-ϛϵ≦1-∅=0;since(20)GΛstatlim⁡y=ϛand ϛ is called GΛ(stat) limit of the sequence y.Definition 11.
Let(Y;G,∗) be a PN-space. A sequence y=(ym) is called statistically Λ-Cauchy in (Y;G,∗) or, simply, GΛ(stat)-Cauchy to ζ if, for ϵ>0, there exists a number N=N(ϵ)∋,(21)δm∈N:GΛny-Λkyϵ≦1-∅=0,∀n,k≥N.


## 2. Main Results

By making use of the definitions given in the preceding section, it is proposed here to systematically investigate the notion of statistical Λ-convergence and statistical Λ-Cauchy in PN-spaces and apply our findings to the problem of approximating positive linear operators.Theorem 12.
Let(Y;G;∗) be a PN-space. If a sequence y=(ym) is GΛ(stat)-convergent, then the GΛ(stat) limit is unique.Proof.
Suppose that(22)GΛstatlim⁡Y=ϛ1,GΛstatlim⁡Y=ϛ2.For a given ∅>0, let ρ∈(0,1)∋,(23)1-ρ∗1-ρ>1-∅.Then, for any ϵ>0,(24)MG,1ρ,ϵ=m∈N:GΛmy-ϛ1ϵ≦1-ρ,MG,1ρ,ϵ=m∈N:GΛmy-ϛ2ϵ≦1-ρ;since(25)GΛstatlim⁡Y=ϛ1δMG,1ρ,ϵ=0for ϵ>0. Furthermore, by using(26)GΛstatlim⁡ym=ϛ2,it has(27)δMG,2ρ,ϵ=0for all ϵ>0.
Let(28)MGρ,ϵ=MG,1ρ,ϵ∩MG,2ρ,ϵ.Clearly,(29)δMGρ,ϵ=0,which implies that(30)δN∖MGρ,ϵ=1.For n∈N∖MG(ρ,ϵ), by using (2) and (3), (31)Gϛ1-ϛ2ϵ≧G∧my-ϛ1ϵ2∗G∧my-ϛ2ϵ2≧1-ρ∗1-ρ>1-∅.Since ∅>0 is arbitrary, by using (5),(32)Gϛ1-ϛ2ϵ=1∀ϵ>0. This implies ζ1=ζ2. Hence, GΛ (stat) limit is unique. This establishes Theorem 12.Theorem 13.
Let(Y;G;∗) be a PN-space. If(33)G∧lim⁡y=ϛ,then(34)G∧statlim⁡y=ϛ,but the converse is not necessarily true in general.Proof.
By hypothesis, for every∅∈(0,1) and ϵ>0, there exists a positive integer m0 such that(35)G∧my-ϛϵ>1-∅whenever m≧m0. This guarantees that the set(36)m∈z+:G∧my-ϛϵ≦1-∅has at most finitely many terms. As every finite subset of the set N of positive integers has density zero, (37)δm∈z+:GΛmy-ϛϵ≦1-∅=0,which establishes Theorem 13.Example 14.
This example would show that the converse of the assertion in Theorem13 needs not be true in general. Let (R,·) be the space of real numbers with the usual norm. Let(38)a∗b=ab,G∧myu=uu+Λmyu,where u=0. Here, it is noted that (R,G,∗) is a probabilistic normed space. If it takes a sequence Λm(y) whose terms are(39)Λmy≔1m=k2;k∈z+0otherwisethen, ∀∅∈(0,1) and for any ϵ>0, let(40)Mm0∅,ϵ≔m≦m0:GΛmyϵ≦1-∅.Since(41)Mm0∅,ϵ=m≦m0:uu+Λmy≦1-∅=m≦m0:Λmy≧∅u1-∅>0=m≦m0:Λmy=1=m≦m0:m=k,k∈z+,it gets(42)1m0Mm0∅,ϵ≦1m0m≦m0:m=k2,k∈z+≦m0m0which implies that(43)limm0→∞⁡1m0Mm0∅,ϵ=0.Hence, by Definition 7,(44)GΛystatlim⁡y=0.Nevertheless, as the sequence (Λm(y)) shown in (39) is not convergent in the space (R,·), by Remark 1 of [11], it is clear that the sequence (Λm(y)) is not convergent with respect to the probabilistic norm.Theorem 15.
Let(Y;G;∗) be a PN-space. If(45)GΛstatlim⁡y=ϛ1,GΛstatlim⁡z=ϛ2then(46)iGΛstatlim⁡y±z=ϛ1±ϛ2,iiGΛstatlim⁡σy=σϛ1σ∈R.Proof.
(i) Let(47)GΛstatlim⁡y=ϛ1,GΛstatlim⁡y=ϛ2,ϵ>0.Also let ∅∈(0,1). Choose ρ∈(0,1) such that(48)1-ρ∗1-ρ>1-∅.Then,(49)MG,1ρ,ϵ=m0∈N:GΛmy-ϛ1ϵ≦1-ρ,MG,2ρ,ϵ=m0∈N:GΛmy-ϛ2ϵ≦1-ρ.Since(50)GΛstatlim⁡y=ϛ1it has(51)δMG,1ρ,ϵ=0for all ϵ>0. Furthermore, by using(52)GΛstatlim⁡z=ϛ2get(53)δMG,2ρ,ϵ=0∀ϵ>0. Let(54)MGρ,ϵ=MG,1ρ,ϵ∩MG,2ρ,ϵ.Then(55)δMG,1ρ,ϵ=0,which implies (56)δN∖MGρ,ϵ=1.If(57)m0=N∖MGρ,ϵ,then (58)GΛmy-ϛ1ϵ+Λmz-ϛ2ϵ≧GΛmy-ϛ1ϵ2+GΛmy-ϵ2+GΛmz-ϛ2ϵ2>1-ρ∗1-ρ>1-∅.This shows that(59)δm0∈N:GΛmy-ϛ1+Λmz-ϛ2ϵ≦1-∅=0.So (60)GΛstatlim⁡y±z=ϛ1±ϛ2.
(ii) Let(61)GΛstatlim⁡y=ϛand suppose that ∅∈(0,1) and ϵ>0. Firstly, consider σ=0. Then, (62)G0Λmy-0·ϛϵ=Gϵ=1>1-∅.So (63)G0·Λmy=0.Now let σ∈R  σ≠0. Since(64)GΛmstatlim⁡y=ϛ,it follows from Theorem 13 that(65)GΛstatlim⁡y=ϛ.If (66)MGρ,ϵ≔m0∈N:GΛmy-ϛϵ≦1-∅,then (67)δMGρ,ϵ=0∀ϵ>0. In this case, (68)δN∖MGρ,ϵ=1.If m0∈N∖MGρ,ϵ, then (69)GΛmy-ρϛϵ=GΛmy-ϛϵσ≧GΛmy-ϛϵ∗G0ϵσ-ϵ=GΛmy-ϛϵ∗1=GΛmy-ϛϵ=1-∅for σ∈R (σ≠0).
This demonstrates that(70)δm0∈N:GσΛmy-σϛϵ≦1-∅=0,so(71)GΛstatlim⁡σy=σϛ,thereby completing the proof of Theorem 15.Theorem 16.
Let(Y;G;⁎) be a PN-space. Then(72)GΛstatlim⁡y=ϛ,if and only if there exists a set(73)m=m1<m2<m3<⋯<mk<⋯⊆Nwith(74)δM=1such that(75)Glim⁡Λmny=ϛ.Proof.
In order to prove the necessity part, first assume that(76)GΛstatlim⁡y=ϛ.Now, for every ϵ>0 and j∈N, let(77)Mj,ϵ:m∈N:GΛmy-ϛϵ≦1-1j,Kj,ϵ≔m∈N:GΛmy-ϛϵ>1-1j.Then(78)δMj,ϵ=0,(79)K1,ϵ⊃K2,ϵ⊃⋯⊃KI,ϵ⊃Ki+1,ϵ⊃⋯It has(80)δKj,ϵ=1j∈N.Now we prove that, for m∈K(j,ϵ), the sequence (ym) is GΛstat-convergent to ζ. Suppose, on the contrary, that the sequence (ym) is not GΛstat-convergent to ζ. Therefore, there exits ρ>0 such that the set(81)m∈N:GΛmy-ϛϵ≦1-ρhas infinitely many terms. Let(82)Kρ,ϵ≔m∈N:GΛmy-ϛϵ>1-ρ,ρ>1jj∈N.Then(83)δKp,ϵ=0,so that, by using (79), (84)Kj,ϵ⊂Kρ,ϵ.Hence(85)δKρ,ϵ=0,which contradicts with (80). Consequently, the sequence (ym) is GΛstat-convergent to ζ.
Next, to prove the sufficiency part, it is assumed that there exists a subset(86)M=m1<m2<m3<⋯<mk<⋯⊆N,δM=1,Glimn→∞⁡Λmny=ϛ.Then, for every ρ∈(0,1) and ϵ>0, (87)GΛmy-ϛϵ>1-ρm∈N.Now(88)Kρ,ϵ≔m∈N:GΛmy-ϛϵ≦1-ρ⊆N-MN+1,MN+2,….Therefore,(89)δKp,ϵ≦1-1=0.Hence(90)GΛstatlim⁡y=ϛ.Theorem 17.
A sequencey=(ym) in a PN-space (Y;G;∗) is GΛstat-convergent if and only if it is GΛ (stat)-Cauchy.Proof.
Let the sequencey be a GΛ (stat)-convergent to ζ in PN-space; that is,(91)GΛstatlim⁡y=ζ.Then, ∀ϵ>0 and ρ∈(0,1), (92)δm∈N:GΛmy-ζϵ≦1-ρ=0.Select a number N=N(ϵ), (93)GΛmy-ζϵ≦1-ρ.Now let(94)Aρ,ϵ=m∈N:GΛMy-ΛNyϵ≦1-ρ,Bρ,ϵ=m∈N:GΛMy-ζϵ≦1-ρCρ,ϵ=m=N∈N:GΛNy-ζϵ≦1-ρ.Then(95)Aρ,ϵ⊆Bρ,ϵ∪Cρ,ϵ.Therefore, (96)δAρ,ϵ≦δBρ,ϵ+δCρ,ϵ.Hence, y is statistically Λ-Cauchy.
Conversely, lety be a statistically Λ-Cauchy sequence, but not statistically Λ-convergent. There exists N such that the set A  (ρ,ϵ) has natural density zero. Therefore, the set(97)Eρ,ϵ=M∈N:GΛMy-ΛNyϵ>1-ρhas natural density 1; that is,(98)Eρ,ϵ=1.Particularly, it can be expressed as (99)GΛMy-ΛNyϵ≦2GΛMy-ζ<ϵif(100)GΛMy-ζ<ϵ2.Since y is not statistically Λ-convergent, the set (ρ,ϵ) has natural density 1; that is, (101)δM∈N:GΛMy-ζϵ>1-ρ=0.Hence, by (10), we get(102)δM∈N:GΛMy-ΛNyϵ>1-ρ=0.This is the contradiction that the set (ρ,ϵ) has natural density 1. Therefore, the sequence y is statistically Λ-convergent.


## 3. Conclusion

This paper has used the notion of Λ-convergence and studied it in the context of probabilistic normed (PN) spaces. Statistical Λ-convergence and statistical Λ-Cauchy were defined in PN-spaces and gave some illustrative examples to demonstrate these concepts. Λ-convergence can be used to study the Korovkin type approximation theorems in PN-spaces (see, for example, [1] and the references cited therein).



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