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=(λm) be a non-decreasing sequence of positive numbers tending to infinity such that λm + 1≤λm + 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.


Introduction
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],Salát [4], Ç akalli [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-Cech compactification of the natural numbers. Moreover, statistical convergence is closely related to the concept of convergence in probability [13]. In this case, we write S − lim x = or x k → (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][17][18][19][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: http://www.iaumath.com/content/6/1/62 Let λ = (λ m ) be a non-decreasing sequence of positive numbers tending to ∞ such that The collection of such sequence λ will be denoted by . 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 λ m m ≤ 1, δ(K) ≤ δ λ (K) for every K ⊆ N. Definition 1.2. [24] A sequence x = (x k ) is said to be λstatistically convergent or S λ -convergent to if for every ε > 0, the set {k ∈ I m : |x k − | ≥ ε} has λ-density of zero. In this case we write S λ − lim x = or x k → (S λ ) and k=0 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 x k → l as k → ∞, where is said to be -statistically convergent to if for every ε > 0 the set {k ∈ N : | x k − | ≥ 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][28][29][30][31]. Further details on generalization of statistical convergence can be found in [11,12,32,33].

Preliminaries
Definition 2.1. A function f : R → R + 0 is called a distribution function if it is a non-decreasing and left continuous with inf t∈R f (t) = 0 and sup t∈R f (t) = 1. By D + we denote the set of all distribution functions such that A triangle function τ is a binary operation on D + , which is commutative, associative, and τ (f , In [22], Gähler introduced the following concept of 2normed space.
A trivial example of an 2-normed space is X = R 2 , equipped with the Euclidean 2-norm ||x 1 , x 2 || E = the volume of the parallellogram spanned by the vectors x 1 , x 2 which may be given explicitly by the formula Recently, Goleţ [21] used the idea of 2-normed space to define the random 2-normed space.
In other words we can write the sequence (x k ) statistical converges to in random 2-normed space In this case, we write S R2N − lim x = and 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 (X, F, * ).
λ -convergent to some ∈ X with respect to F 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 ) λ- i.e., In this case, we write S R2N λ − lim x = and is called the λ (X) denotes the set of all statistical convergent sequences in random 2-normed space (X, F, * ).
In this paper, we define and study -statistical convergence in random 2-normed space which is quite a new and interesting idea to work with. We show that some properties of -statistical convergence of real numbers also hold for sequences in random 2-normed spaces. We find some relations of -statistical convergent sequences in random 2-normed spaces. Also, we find out the relation between -statistical convergent and -statistical Cauchy sequences in this spaces.

-statistical convergence in random 2-normed space
In this section, we define -statistical convergent sequence in random 2-normed (X, F, * ). 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 2normed space (X, F, * ) is said to be -convergent to ∈ X with respect to F if for each ε > 0, θ ∈ (0, 1) there exists a positive integer n 0 such that F( x k − , z; ε) > 1 − θ, whenever k ≥ n 0 and for non zero z ∈ X. In this case, we write F − lim k x k = , and is called the F -limit of x = (x k ).

Definition 3.2.
A sequence x = (x k ) in a random 2-normed space (X, F, * ) is said to be -Cauchy with respect to F if for each ε > 0, θ ∈ (0, 1) there exists a positive integer n 0 = n 0 (ε) such that F( x k − x s , z; ε) < 1 − θ, whenever k, s ≥ n 0 and for non zero z ∈ X.

Definition 3.4.
A sequence x = (x k ) in a random 2normed space (X, F, * ) is said to be -statistical Cauchy with respect to F 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

Definition 3.3 immediately implies the following Lemma.
Lemma 3.1. Let (X, F, * ) 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: Proof. Suppose that there exist elements 1 Let ε > 0 be given. Choose r > 0 such that Then, for any t > 0 and for non zero z ∈ X, we define Since S R2N − lim k→∞ x k = 1 and S R2N − lim k→∞ x k = 2 , we have δ (K 1 (r, t)) = 0 and δ (K 2 (r, t)) = 0 for all t > 0. Now let K(r, t) = K 1 (r, t) ∪ K 2 (r, t), then it is easy to observe that δ (K(r, t) It follows by (3.1) that Since ε > 0 was arbitrary, we get F( 1 − 2 , z; t) = 1 for all t > 0 and non zero z ∈ X. Hence 1 = 2 . The next theorem gives the algebraic characterization of -statistical convergence on random 2-normed spaces. The proof of the theorem is straightforward, thus omitted.
Proof. Let F − lim x k = . Then for every ε > 0, t > 0 and non zero z ∈ X, there is a positive integer n 0 such that has, at most, many finite terms. Also, since every finite subset of N has δ -density zero, consequently, we have δ (K(ε, t)) = 0. This shows that S R2N − lim x k = .
Nor for every 0 < ε < 1 and t > 0, we write Taking the limit m which approaches to ∞, we get This shows that x k → 0(S R2N (X)).
On the other hand, the sequence is not F -convergent to zero as Proof. Suppose first that S R2N −lim x k = . Then for any t > 0, r = 1, 2, 3, ... and non zero z ∈ X, let Since S R2N − lim x k = , it follows that δ (K(r, t)) = 0. Now for t > 0 and r = 1, 2, 3, ..., we observe that

A(r, t) ⊃ A(r + 1, t)
and δ (A(r, t) Now we have to show that for k ∈ A(r, t), F − lim x k = . Suppose that for k ∈ A(r, t), (x k ) is not convergent to with respect to F . Then, there exists some s > 0 such that Furthermore, A(r, t) ⊂ A(s, t) implies that δ (A(r, t)) = 0, which contradicts (3.2) as δ (A(r, t) Conversely, suppose that there exists a subset K ⊆ N such that δ (K) = 1 and F − lim x k = .
Combining Theorem 3.7 and Theorem 3.8 we get the following corollary. Corollary 3.9. Let (X, F, * ) 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 -statistically convergent. (b) x is -statistically Cauchy. (c) There exists a subset K ⊆ N such that δ (K) = 1 and F − lim x k = .

λ-statistical limit points and statistical cluster points on random-2-normed space
In this section, we will define the − statistical limit points and cluster point and we will give connection between this classes.
Definition 4.1. Let (X, F, * ) be a R2N−space. l ∈ X is called a − limit point of the sequence x = (x k ) with respect to F provided that there is a subsequence of x that − converges to l with respect to F. We will denote by L R2N (x) the set of all − limit points of the sequence x = (x k ).

Definition 4.2.
Let (X, F, * ) be a R2N−space. Then ξ ∈ X is called a − statistical limit point of the sequence x = (x k ) with respect to F provided that there is a subsequence {x k(j) } of x = (x k ) that − converges to l with respect to F and δ (K) = 0, where K = {k(j) ∈ N : j ∈ N}.
We will denote by R2N (x) the set of all − statistical limit points of the sequence x = (x k ).
We will denote by R2N (x) the set of all − statistical cluster of the sequence x = (x k ).

Conclusions
In this paper, we have define and studied the notion of -statistical convergence and -statistical Cauchy sequences in random 2-normed spaces , where λ = (λ m ) http://www.iaumath.com/content/6/1/62 be a non-decreasing sequence of positive numbers tending to infinity such that λ m+1 ≤ λ m + 1, λ 1 = 1 and we have proved some theorems. In last section we have given the definition of the − limit and cluster points and we have shown their relation between those classes.