On Λ-statistical convergence in random 2-normed space

Recently, Mursaleen introduced the concepts of statistical convergence in random 2-normed spaces. In this paper, we define and study the notion of Λ-statistical convergence and Λ-statistical Cauchy sequences in random 2-normed spaces , where λ=(λ_m) be a non-decreasing sequence of positive numbers tending to infinity such that λ_{m + 1}≤λ_m + 1,λ₁=1 and prove some theorems. In last section we will give the definition of the Λ− limit and cluster points and we will show their relation between those classes.

The concept of statistical convergence play a vital role not only in pure mathematics but also in other branches of science involving mathematics, especially in information theory, computer science, biological science, dynamical systems, geographic information systems, population modeling, and motion planning in robotics.

The notion of statistical convergence was introduced by Fast [1] and Schoenberg [2] independently. Over the years and under different names, statistical convergence has been discussed in the theory of Fourier analysis, ergodic theory, and number theory. Later on it was further investigated by Fridy [3], S̆alát [4], Çakalli [5], Maio and Kocinac [6], Miller [7], Maddox [8], Leindler [9], Mursaleen and Alotaibi [10], Mursaleen and Edely [11], Mursaleen and Edely [12], and many others. In the recent years, generalization of statistical convergence have appeared in the study of strong integral summability and the structure of ideals of bounded continuous functions on Stone-C̆ech compactification of the natural numbers. Moreover, statistical convergence is closely related to the concept of convergence in probability [13].

The notion of statistical convergence depends on the density of subsets of N. A subset of N is said to have density δ(E) if

Definition 1.1

A sequence x=(x_k) is said to be statistically convergent to ℓ if for every ε>0

In this case, we write S−limx=ℓor x_k→ℓ(S) and S denotes the set of all statistically convergent sequences.

The probabilistic metric space was introduced by Menger [14] which is an interesting and important generalization of the notion of a metric space. Karakus [15] studied the concept of statistical convergence in probabilistic normed spaces. The theory of probabilistic normed spaces was initiated and developed in [16-20] and it was further extended to random/probabilistic 2-normed spaces by Goleţ [21] using the concept of 2-norm which is defined by Gähler [22], and Gürdal and Pehlivan [23] studied statistical convergence in 2-Banach spaces.

Mursaleen [24], introduced the λ-statistical convergence for real sequences as follows:

Let λ=(λ_m) be a non-decreasing sequence of positive numbers tending to ∞such that

The collection of such sequence λwill be denoted by Δ.

Let K⊆N be a set of positive integers. Then

is said to be λ-density of K. In case λ_m=m, then λ-density reduces to natural density, so S_λ is the same as S. Also, since for every K⊆N.

Definition 1.2

[24] A sequence x=(x_k) is said to be λstatistically convergent or S_λconvergent to ℓif for every ε>0, the set {k∈I_m:|x_k−ℓ|≥ε} has λ-density of zero. In this case we write S_λ−limx=ℓor x_k→ℓ(S_λ) and

Definition 1.3

Let be a strictly increasing sequence of positive real numbers tending to the infinity which is

Mursaleen and Noman [25] introduced the notion of μ−convergent sequences as follows: A sequence x=(x_k) is said to be μ−convergent to the number l if Λ x_k→las where

A sequence x=(x_k) is said to be Λ statistically convergent to ℓ if for every ε>0 the set {k∈N:|Λ x_k−ℓ|≥ε} has natural density zero, i.e.,

The existing literature on statistical convergence and its generalizations appears to have been restricted to real or complex sequences, but in recent years these ideas have been also extended to the sequences in fuzzy normed [26] and intutionistic fuzzy normed spaces [27-31]. Further details on generalization of statistical convergence can be found in [11,12,32,33].

Preliminaries

Definition 2.1

A function is called a distribution function if it is a non-decreasing and left continuous with inf_t∈Rf(t)=0 and sup_t∈Rf(t)=1. By D ⁺ we denote the set of all distribution functions such that f(0)=0. If , then H_a∈D ⁺ , where

It is obvious that H₀≥ffor all f∈D ⁺ .

A t-norm is a continuous mapping ∗:[0,1]×[0,1]→[0,1] such that ([0,1],∗) is abelian monoid with unit one and c∗d≥a∗bif c≥a and d≥b for all a,b,c∈[0,1]. A triangle function τ is a binary operation on D ⁺ , which is commutative, associative, and τ(f,H₀)=ffor every f∈D ⁺ .

In [22], Gähler introduced the following concept of 2-normed space.

Definition 2.2

Let X be a real vector space of dimension d>1 (d may be infinite). A real-valued function ||.,.|| from X²into R satisfying the following conditions:

(1) ||x₁,x₂||=0 if and only if x₁,x₂are linearly dependent,

(2) ||x₁,x₂|| is invariant under permutation,

(3) ||αx₁,x₂||=|α|||x₁,x₂||, for any α∈R, and

is called an 2-norm on X, and the pair (X,||.,.||) is called an 2-normed space.

A trivial example of an 2-normed space is X=R², equipped with the Euclidean 2-norm ||x₁,x₂||_E= the volume of the parallellogram spanned by the vectors x₁,x₂ which may be given explicitly by the formula

where x_i=(x_i1,x_i2)∈R² for each i=1,2.

Recently, Goleţ [21] used the idea of 2-normed space to define the random 2-normed space.

Definition 2.3

Let X be a linear space of dimension d>1 (d may be infinite), τa triangle, and Then is called a probabilistic 2-norm and a probabilistic 2-normed space if the following conditions are satisfied:

1. if x and y are linearly dependent, where denotes the value of at t∈R,

2. if x and y are linearly independent,

3. for all x,y∈X,

4. for every t>0,α≠0 and x,y∈X,

5. whenever x,y,z∈X.

If (P2N₅) is replaced by for all x,y,z∈Xand

then is called a random 2-normed space (for short, R2NS).

Remark 2.1

Every 2-normed space (X,||.,.||) can be made a random 2-normed space in a natural way, by setting

for every x,y∈X,t>0 and a∗b=min{a,b},a,b∈[0,1];

for every x,y∈X,t>0 and a∗b=ab,a,b∈[0,1].

In [34], Gürdal and Pehlivan studied statistical convergence in 2-normed spaces and in 2-Banach spaces in [23]. In fact, Mursaleen [35] studied the concept of statistical convergence of sequences in random 2-normed space. Recently in [36], Esi and Özdemir introduced and studied the concept of generalized Δ^m-statistical convergence of sequences in probabilistic normed space. In [37], Hazarika introduced the generalized statistical convergence in random 2-normed spaces.

Definition 2.4

[35] A sequence x=(x_k) in a random 2-normed space is said to be statistical convergent or S^R2Nconvergent to some ℓ∈Xwith respect to if for each ε>0,θ∈(0,1) and for non zero, z∈X, such that

In other words we can write the sequence (x_k)statistical converges to ℓ in random 2-normed space if

or equivalently

i.e.,

In this case, we write S^R2N−limx=ℓand ℓ is called the S^R2N−limit of x. Let S^R2N(X) denotes the set of all statistical convergent sequences in random 2-normed space .

Definition 2.5

[37] A sequence x=(x_k) in a random 2-normed space is said to be λstatistical convergent or convergent to some ℓ∈X with respect to if for each ε>0,θ∈(0,1) and for non zero z∈Xsuch that

In other words, we can write the sequence (x_k)λ-statistical converges to ℓ in random 2-normed space if

or equivalently,

i.e.,

In this case, we write and ℓ is called the of x. Let denotes the set of all statistical convergent sequences in random 2-normed space

In this paper, we define and study Λ-statistical convergence in random 2-normed space which is quite a new and interesting idea to work with. We show that some properties of Λ-statistical convergence of real numbers also hold for sequences in random 2-normed spaces. We find some relations of Λ-statistical convergent sequences in random 2-normed spaces. Also, we find out the relation between Λ-statistical convergent and Λ-statistical Cauchy sequences in this spaces.

Λ-statistical convergence in random 2-normed space

In this section, we define Λ-statistical convergent sequence in random 2-normed Also, we obtained some basic properties of this notion in random 2-normed space.

Definition 3.1

A sequence x=(x_k) in a random 2-normed space is said to be Λ-convergent to ℓ∈Xwith respect to if for each ε>0,θ∈(0,1) there exists a positive integer n₀such that whenever k≥n₀and for non zero z∈X. In this case, we write and ℓis called the -limit of x=(x_k).

Definition 3.2

A sequence x=(x_k) in a random 2-normed space is said to be Λ-Cauchy with respect to if for each ε>0,θ∈(0,1) there exists a positive integer n₀=n₀(ε) such that whenever k,s≥n₀and for non zero z∈X.

Definition 3.3

A sequence x=(x_k) in a random 2-normed space is said to be Λ-statistically convergent or S _Λ -convergent to ℓ∈X with respect to if for every ε>0,θ∈(0,1) and for non zero z∈Xsuch that

In other ways, we can write

or equivalently,

i.e.,

In this case, we write or and

Let denotes the set of all Λ-statistical convergent sequences in random 2-normed space

Definition 3.4

A sequence x=(x_k) in a random 2-normed space is said to be Λ-statistical Cauchy with respect to if for every ε>0,θ∈(0,1) and for non zero z∈X, there exists a positive integer n=n(ε) such that for all k,s≥n

or equivalently,

Definition 3.3 immediately implies the following Lemma.

Lemma 3.1

Let be a random 2-normed space. If x=(x_k) is a sequence in X, then for every ε>0,θ∈(0,1) and for non zero z∈X, then the following statements are equivalent:

(i) SΛ−lim_k→∞x_k=ℓ.

(ii)

(iii)

(iv)

Theorem 3.2

Let be a random 2-normed space. If x=(x_k) is a sequence in X such that exists, then it is unique.

Proof

Suppose that there exist elements ℓ₁,ℓ₂(ℓ₁≠ℓ₂) in X such that

□

Let ε>0 be given. Choose r>0 such that

Then, for any t>0 and for non zero z∈X, we define

Since and , we have

δ _Λ (K₁(r,t))=0 and δ _Λ (K₂(r,t))=0 for all t>0.

Now let K(r,t)=K₁(r,t)∪K₂(r,t), then it is easy to observe that δ _Λ (K(r,t))=0. But we have δ _Λ (K^c(r,t))=1.

Now if k∈K^c(r,t), then we have

It follows by (3.1) that

Since ε>0 was arbitrary, we get for all t>0 and non zero z∈X. Hence ℓ₁=ℓ₂. The next theorem gives the algebraic characterization of Λ-statistical convergence on random 2-normed spaces.

Theorem 3.3

Let be a random 2-normed space, and x=(x_k) and y=(y_k) be two sequences in X:

(a) If and c(≠0)∈R, then

(b) If S _Λ −limx_k=ℓ₁and then

The proof of the theorem is straightforward, thus omitted.

Theorem 3.4

Let be a random 2-normed space. If x=(x_k) be a sequence in X such that , then .

Proof

Let . Then for every ε>0,t>0 and non zero z∈X, there is a positive integer n₀such that

for all k≥n₀. Since the set

has, at most, many finite terms. Also, since every finite subset of N has δ _Λ -density zero, consequently, we have δ _Λ (K(ε,t))=0. This shows that □

Remark 3.5

The converse of the above theorem is not true in general. It follows from the following example.

Example 3.6

Let X=R², with the 2-norm ||x,z||=|x₁z₂−x₂z₁|,x=(x₁,x₂),z=(z₁,z₂), and a×b=ab for all a,b∈[0,1]. Let for all x,z∈X,z₂≠0, and t>0. Now we define a sequence x=(x_k) by

Nor for every 0<ε<1 and t>0, we write

so we get

Taking the limit m which approaches to ∞, we get

This shows that

On the other hand, the sequence is not -convergent to zero as

Theorem 3.7

Let be a random 2-normed space. If x=(x_k) be a sequence in X, then if and only if there exists a subset K⊆N, such that δ _Λ (K)=1 and

Proof

Suppose first that Then for any t>0,r=1,2,3,… and non zero z∈X, let

and

□

Since , it follows that

Now for t>0 and r=1,2,3,…, we observe that

and

Now we have to show that for Suppose that for k∈A(r,t),(x_k) is not convergent to ℓwith respect to Then, there exists some s>0 such that

for infinitely many terms x_k. Let

and

Then we have

Furthermore, A(r,t)⊂A(s,t) implies that δ _Λ (A(r,t))=0, which contradicts (3.2) as δ _Λ (A(r,t))=1. Hence, .

Conversely, suppose that there exists a subset K⊆Nsuch that δ _Λ (K)=1 and .

Then for every ε>0,t>0 and non zero z∈X, we can find out a positive integer n such that

for all k≥n. If we take

then it is easy to see that

and consequently,

Hence, .

Finally, we establish the Cauchy convergence criteria in random 2-normed spaces.

Theorem 3.8

Let be a random 2-normed space. Then a sequence (x_k) in X is Λ-statistically convergent if and only if it is Λ-statistically Cauchy.

Proof

Let (x_k) be a Λ-statistically convergent sequence in X. We assume that Let ε>0 be given. Choose r>0 such that (3.1) is satisfied. For t>0 and for non zero z∈X, we define

and

□

Since , it follows that δ _Λ (A(r,t))=0 and consequently, δ _Λ (A^c(r,t))=1. Let p∈A^c(r,t). Then

If we take

then to prove the result, it is sufficient to prove that B(ε,t)⊆A(r,t). Let n∈B(ε,t), then for non zero z∈X

If , then we have and therefore n∈A(r,t). As otherwise i.e., if then by (3.1), (3.3), and (3.4) we get

which is not possible. Thus, B(ε,t)⊂A(r,t). Since δ _Λ (A(r,t))=0, it follows that δ _Λ (B(ε,t))=0. This shows that (x_k) is Λ-statistically Cauchy. Conversely, suppose (x_k) is Λ-statistically Cauchy but not Λ-statistically convergent. Then there exists positive integer p and for non zero z∈X such that if we take

and

then

and consequently,

Since

if , then we have

i.e., δ _Λ (A^c(ε,t))=0, which contradicts (3.5) as δ _Λ (A^c(ε,t))=1. Hence, (x_k) is Λ-statistically convergent.

Combining Theorem 3.7 and Theorem 3.8 we get the following corollary.

Corollary 3.9

Let be a random 2-normed space and and x=(x_k) be a sequence in X. Then the following statements are equivalent:

(a) x is Λ-statistically convergent.

(b) x is Λ-statistically Cauchy.

(c) There exists a subset K⊆Nsuch that δ _Λ (K)=1 and

λ-statistical limit points and statistical cluster points on random-2-normed space

In this section, we will define the Λ− statistical limit points and cluster point and we will give connection between this classes.

Definition 4.1

Let be a R2N−space. l∈Xis called a Λ− limit point of the sequence x=(x_k) with respect to provided that there is a subsequence of x that Λ− converges to l with respect to .

We will denote by the set of all Λ− limit points of the sequence x=(x_k).

Definition 4.2

Let be a R2N−space. Then ξ∈Xis called a Λ− statistical limit point of the sequence x=(x_k) with respect to provided that there is a subsequence {x_k(j)} of x=(x_k) that Λ− converges to l with respect to and δ _Λ (K)≠0, where .

We will denote by the set of all Λ− statistical limit points of the sequence x=(x_k).

Definition 4.3

Let be a R2N−space. Then ϑ∈X is called a Λ− statistical cluster point of the sequence x=(x_k) with respect to provided that for every ε>0,λ∈(0,1), and z∈X∖{0},

We will denote by the set of all Λ− statistical cluster of the sequence x=(x_k).

Theorem 4.1

Let be a R2N−space. For any x∈X,

Theorem 4.2

Let be a R2N−space. For any x∈X,

Theorem 4.3

Let be a R2N−space. For a sequence x=(x_k), if then we get .

In this paper, we have define and studied the notion of Λ-statistical convergence and Λ-statistical Cauchy sequences in random 2-normed spaces , where λ=(λ_m) be a non-decreasing sequence of positive numbers tending to infinity such that λ_{m + 1}≤λ_m + 1,λ₁=1 and we have proved some theorems. In last section we have given the definition of the Λ− limit and cluster points and we have shown their relation between those classes.

The authors declare that they have no competing interests.

AE has introduce and studied the concept λ− statistical convergence in random 2-normed space. NB has introduce and studied the concept of λ-statistical limit points and statistical cluster points on random-2-normed space. Both authors read and approved the final manuscript.

1. Fast, H: Sur la convergence statistique. Colloq. Math. 2, 241–244 (1951)

2. Schoenberg, IJ: The integrability of certain functions and related summability methods. Amer. Math. Monthly. 66, 361–375 (1959). Publisher Full Text

3. Fridy, JA: On statistical convergence. Analysis. 5, 301–313 (1985)

4. S̆alát, T: On statistically convergent sequences of real numbers. Math. Slovaca. 30, 139–150 (1980)

5. Çakalli, H: Funct. Anal. Approx. Comput. 1(2), 19–24,MR2662887 (2009)

6. Maio, GD, Kocinac, LDR: Statistical convergence in topology. Topology Appl. 156, 28–45 (2008). Publisher Full Text

7. Miller, HI: A measure theoretical subsequence characterization of statistical convergence. Trans. Amer. Math. Soc. 347(5), 1811–1819 (1995). Publisher Full Text

8. Maddox, IJ: Statistical convergence in a locally convex space. Math. Proc. Cambridge Philos. Soc. 104(1), 141–145 (1988). Publisher Full Text

9. Leindler, L: Uber die de la Vallee-Pousinsche Summierbarkeit allgenmeiner Othogonalreihen. Acta. Math. Acad. Sci. Hungar. 16, 375–387 (1965). Publisher Full Text

10. Mursaleen, M, Alotaibi, A: Statistical summability and approximation by de la Vallee-Pousin mean. Appl. Math. Lett. 24, 320–324 (2011). Publisher Full Text

11. Mursaleen, M, Edely, OHH: Statistical convergence of double sequences. J. Math. Anal. Appl. 288, 223–231 (2003). Publisher Full Text

12. Mursaleen, M, Edely, OHH: On the invariant mean and statistical convergence. Appl. Math. Lett. 22, 1700–1704 (2009). Publisher Full Text

13. Connor, J, Swardson, MA: Measures and ideals of C∗(X). Ann. N.Y. Acad. Sci. 704, 80–91 (1993). Publisher Full Text

14. Menger, K: Statistical metrics. Proc. Natl. Acad. Sci. USA. 28, 535–537 (1942). PubMed Abstract | Publisher Full Text | PubMed Central Full Text

15. Karakus, S: Statistical convergence on probabilistic normed spaces. Math. Commun. 12, 11–23 (2007)

16. Alsina, C, Schweizer, B, Sklar, A: Continuity properties of probabilistic norms. J. Math. Anal. Appl. 208, 446–452 (1997). Publisher Full Text

17. Schweizer, B, Sklar, A: Statistical metric spaces. Pacific J. Math. 10, 313–334 (1960). Publisher Full Text

18. Schweizer, B, Sklar, A: Probabilistic Metric Spaces, Dover, New York (1983)

19. Sempi, C: A short and partial history of probabilistic normed spaces. Mediterr. J. Math. 3, 283–300 (2006). Publisher Full Text

20. Serstnev, AN: On the notion of a random normed space. Dokl. Akad Nauk SSSR. 149, 280–283 (1963)

21. Goleţ, I: On probabilistic 2-normed spaces. Novi Sad. J. Math. 35, 95–102 (2006)

22. Gähler, S: 2-metrische Raume and ihre topologische Struktur. Math. Nachr. 26, 115–148 (1963). Publisher Full Text

23. Gürdal, M, Pehlivan, S: The statistical convergence in 2-Banach spaces. Thai. J. Math. 2(1), 107–113 (2004)

24. Mursaleen, M: λ-statistical convergence. Mathematica Slovaca. 50(1), 111–115 (200)

25. Mursaleen, M, Noman, AK: On the spaces of λ−convergent and bounded sequences. Thai. J. Math. 8(2), 311–329 (2010)

26. Sencimen, C, Pehlivan, S: Statistical convergence in fuzzy normed linear spaces. Fuzzy Set. Syst. 159, 361–370 (2008). Publisher Full Text

27. Karakus, S, Demirci, K, Duman, O: Statistical convergence on intuitionistic fuzzy normed spaces. Chaos Soliton Fract. 35, 763–769 (2008). Publisher Full Text

28. Mohiuddine, SA, Danish, Lohani Q M.: On generalized statistical convergence in intuitionistic fuzzy normed space. Chaos Soliton Fract. 42, 1731–1737 (2009). Publisher Full Text

29. Mursaleen, M, Mohiuddine, SA: Statistical convergence of double sequences in intuitionistic fuzzy normed spaces. Chaos Soliton Fract. 41, 2414–2421 (2009). Publisher Full Text

30. Mursaleen, M, Mohiuddine, SA: On lacunary statistical convergence with respect to the intuitionistic fuzzy normed space. J. Comput. Math. 233(2), 142–149 (2009). Publisher Full Text

31. Kumar, V, Mursaleen, M: On (λ,μ)-statistical convergence of double sequences on intuitionistic fuzzy normed spaces. Filomat. 25(2), 109–120 (2011). Publisher Full Text

32. Mursaleen, M, Çakan, C, Mohiuddine, SA, Savas, E: Generalized statistical convergence and statistical core of double sequences. Acta. Math. Sinica. 26(11), 2131–2144 (2010). Publisher Full Text

33. Mursaleen, M, Edely, OHH: Generalized statistical convergence. Inf. Sci. 162, 287–294 (2004). Publisher Full Text

34. Gürdal, M, Pehlivan, S: Statistical convergence in 2-normed spaces. South. Asian Bull. Math. 33, 257–264 (2009)

35. Mursaleen, M: On statistical convergence in random 2-normed spaces. Acta Sci. Math.(Szeged). 76(1-2), 101–109 (2010)

36. Esi, A, Özdemir, MK: Generalized Δm-statistical convergence in probabilistic normed space. J. Comput. Anal. Appl. 13(5), 923–932 (2011)

37. Hazarika, B: On generalized statistical convergence in random 2-normed spaces. In: Scientia Magna. 8, 58–97