Lacunary Generalized Difference Statistical Convergence in Random 2-normed Spaces

Recently in [22], Mursaleen introduced the concept of statistical convergence in random 2-normed spaces. In this paper, we define and study the notion of lacunary ∆ n-statistical convergence and lacunary ∆ n-statistical Cauchy sequences in random 2-normed spaces using la-cunary density and prove some interesting theorems.


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 modelling, and motion planning in robotics.
The notion of statistical convergence was introduced by Fast [7] and Schoenberg [27] independently.A lot of developments have been made in this areas after the works of Salát [26], Fridy [8] and Miller [21].Over the years and under different names statistical convergence has been discussed in the theory of Fourier analysis, ergodic theory and number theory.Fridy and Orhan [9] introduced the concept of lacunary statistical convergence.In [23], Mursaleen and Mohiuddine introduced the concept of lacunary statistical convergence with respect to the intuitionistic fuzzy normed space.Some work on lacunary statistical convergence can be found in [2], [10], [19], [25].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, (see [3]).
The probabilistic metric space was introduced by Menger [20] which is an interesting and important generalization of the notion of a metric space.Karakus [17] studied the concept of statistical convergence in probabilistic normed spaces.The theory of probabilistic normed(or metric) spaces was initiated and developed in [1], [28], [29], [30], [31] and further it was extended to random/probabilistic 2-normed spaces by Golet ¸[13] using the concept of 2-norm which is defined by Gähler [11], and Gürdal and Pehlivan [15] studied statistical convergence in 2-Banach spaces.
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 where the vertical bars denote the cardinality of the enclosed set.
A single sequence x = (x k ) is said to be statistically convergent to if for every ε > 0 In this case we write S-lim x = or x k → (S) (see [7], [8]).
where n ∈ N and ), and also this generalized difference notion has the following binomial representation:

!
x k+i for all k ∈ N.

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 f (0 In [11],Gähler introduced the following concept of 2-normed spaces.Definition 2.2.Let X be a linear space of dimension d > 1 (d may be infinite).A real-valued function ||., .||from X 2 into R satisfying the following conditions: (1) ||x 1 , x 2 || = 0 if and only if x 1 , x 2 are linearly dependent, (2) is called an 2-norm on X and the pair (X, ||., .||) is called an 2-normed space.
Remark 2.1.Every 2-normed space (X, ||., .||)can be made a random 2-normed space in a natural way, by setting (i)F(x, y; t) = H 0 (t − ||x, y||), for every x, y ∈ X, t > 0 and a for each ε > 0, η ∈ (0, 1) and non zero z ∈ X there exists an positive integer In this case we write F −lim k x k = , and is called the F-limit of x = (x k ).Definition 2.5.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, η ∈ (0, 1) and non zero z ∈ X, there exists a positive integer In [14], Gürdal and Pehlivan studied statistical convergence in 2-normed spaces and in 2-Banach spaces in [15].In fact, Mursaleen [22] studied the concept of statistical convergence of sequences in random 2-normed spaces, in [24], Mohiuddine and Aiyub introduced the concept of lacunary statistical convergence in random 2-normed space.Recently in [4], Esi and Özdemir introduced and studied the concept of generalized ∆ m -statistical convergence of sequences in probabilistic normed spaces and in [5] Esi and Özdemir introduced and studied the concept of lacunary statistical convergence in random n-normed space.Definition 2.6.[22] A sequence x = (x k ) in a random 2-normed space (X, F, * ) is said to be statistical-convergent or S R2N -convergent to some ∈ X with respect to F if for each ε > 0, η ∈ (0, 1) and non zero z ∈ X such that δ ({k In other words we can write the sequence (x k ) statistical converges to in random 2-normed space (X, F, * ) if or equivalently 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, * ).
In this paper we define and study lacunary ∆ n -statistical convergence in random 2-normed space which is quite a new and interesting idea to work with.We show that some properties of lacunary ∆ n -statistical convergence of real numbers also hold for sequences in random 2-normed spaces.We find some relations related to lacunary ∆ n -statistical convergent sequences in random 2-normed spaces.Also we find out the relation between lacunary ∆ n -statistical convergent and lacunary ∆ n -statistical Cauchy sequences in this spaces.

Lacunary ∆ n -statistical convergence in random 2-normed spaces
In this section we define lacunary ∆ n -statistical convergent sequence in random 2-normed (X, F, * ).Also we obtained some basic properties of this notion in random 2-normed spaces.
Definition 3.1.By a lacunary sequence θ = (k r ), where k 0 = 0 , we shall mean an increasing sequence of non-negative integers with h r : k r − k r−1 → ∞ as r → ∞.The intervals determined by θ will be denoted by I r = (k r−1 , k r ] and the ratio kr k r−1 will be defined by q r .Let θ be a lacunary sequence and Let θ be a lacunary sequence.A sequence x = (x k ) is said to be S θ -convergent to provided that for each ε > 0, the set has θ-density zero.In this case we write S θ -lim x = or x k → (S θ ) (for details see [9], [10]).
To examine the above notion we will provide some examples.
Example 3.1.Let θ = (2 r − 1) and K = {i 2 : i ∈ N}.Then we have We define the ∆ n -convergence in random 2-normed spaces as follows: Definition 3.3.A sequence x = (x k ) in a random 2-normed space (X, F, * ) is said to be ∆ n -convergent to ∈ X with respect to F if for each ε > 0, η ∈ (0, 1) and non zero z ∈ X, there exists an positive integer n 0 = n 0 (ε, z) such that F(∆ n x k − , z; ε) > 1 − η, whenever k ≥ n 0 .In this case we write F − lim k ∆ n x k = , and is called the Definition 3.4.A sequence x = (x k ) in a random 2-normed space (X, F, * ) is said to be ∆ n -Cauchy with respect to F if for each ε > 0, η ∈ (0, 1) and non zero z ∈ X, there exists a positive integer In [16] Hazarika and Savas and [24] Mohiuddine and Aiyub independently introduced lacunary satistically convergence in random 2-normed spaces as follows.
(X) denotes the set of all lacunary statistical convergent sequences in random 2-normed space (X, F, * ).Definition 3.6.( [16], [24]) A sequence x = (x k ) in a random 2-normed space (X, F, * ) is said to be lacunary statistical Cauchy with respect to F if for each ε > 0, η ∈ (0, 1) and non zero z ∈ X there exists a positive integer n = n(ε, z) such that Now, we define the lacunary ∆ n -statistically convergence in random 2-normed spaces.Definition 3.7.A sequence x = (x k ) in a random 2-normed space (X, F, * ) is said to be lacunary ∆ n -satistically convergent or S θ(∆ n ) -convergent to ∈ X with respect to F if for every ε > 0, η ∈ (0, 1) and non zero z ∈ X such that or equivalently In this case we write and Let S R2N θ(∆ n ) denotes the set of all lacunary ∆ n -statistical convergent sequences in random 2-normed space (X, F, * ).Definition 3.8.A sequence x = (x k ) in a random 2-normed space (X, F, * ) is said to be lacunary ∆ n -statistical Cauchy with respect to F if for every ε > 0, η ∈ (0, 1) and non zero z ∈ X there exists a positive integer m = m(ε, z) such that for all k, s ≥ m 3.7, 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, η ∈ (0, 1) and for non zero z ∈ X, then the following statements are equivalent. (i) Proof.Suppose that there exist elements 1 , 2 ( 1 6 = 2 ) in X such that Let ε > 0 be given.Choose a > 0 such that Then, for any t > 0 and for non zero z ∈ X we define Nor for every 0 < ε < 1 and t > 0, write We see that Therefore we get This shows that S R2N θ(∆ n ) − lim x k = 0. On the other hand the sequence is not F-convergent to zero as Theorem 3.7.Let (X, F, * ) be a random 2-normed space.If x = (x k ) be a sequence in X, then S R2N θ(∆ n ) − lim x k = if and only if there exists a subset Then for any t > 0, a = 1, 2, 3, ... and non zero z ∈ X, let Now for t > 0 and a = 1, 2, 3, ..., we observe that Now we have to show that, for k ∈ A(a, t), F ∆ n − lim x k = .Suppose that for k ∈ A(a, t), (x k ) not convergent to with respect to F. Then there exists some s > 0 such that
Conversely, suppose that there exists a subset K ⊆ N such that δ θ(∆ n ) (K) = 1 and F ∆ n − lim x k = .
Then for every ε > 0, t > 0 and non zero z ∈ X, we can find out a positive integer m = m(ε, z) such that Finally, we establish the Cauchy convergence criteria in random 2normed spaces.
Theorem 3.8.Let (X, F, * ) be a random 2-normed space.Then a sequence (x k ) in X is lacunary ∆ n -statistically convergent if and only if it is lacunary ∆ n -statistically Cauchy.
Proof.Let (x k ) be a lacunary ∆ n -statistically convergent sequence in X.We assume that S R2N θ(∆ n ) − lim x k = .Let ε > 0 be given.Choose a > 0 such that (3.1) is satisfied.For t > 0 and for non zero z ∈ X define
Conversely, suppose (x k ) is lacunary ∆ n -statistically Cauchy but not lacunary ∆ n -statistically convergent.Then there exists positive integer p and for non zero z ∈ X such that For a > 0 such that (3.1) is satisfied and we take 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 lacunary ∆ n -statistically convergent.(b) x is lacunary ∆ n -statistically Cauchy.(c) there exists a subset K ⊆ N such that δ θ(∆ n ) (K) = 1 and F ∆ n −lim x k = .
Now, we introduce the completeness of random 2-normed spaces.
As a consequence of the Theorem 3.8, for n = 0, we define the following definition in random 2-normed spaces.Definition 3.10.A random 2-normed space (X, F, * ) is said to be S θcomplete if every S θ -Cauchy sequence is S θ -convergent in (X, F, * ).Theorem 3.10.Let θ be a lacunary sequence.Then every random 2normed space (X, F, * ) is S θ -complete but not complete in general.
Proof.First part of the proof of the theorem follows from the Theorem 3.8, for n = 0.
To see that random 2-normed space (X, F, * ) is not complete in general, we consider the following example: Cauchy with respect to F but not convergent with respect to the present F.This completes the proof of the theorem.

Acknowledgements
The author expresses his heartfelt gratitude to the anonymous reviewer for such excellent comments and suggestions which have enormously enhanced abelian monoid with unit one and c * d ≥ a * b if 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, H 0 ) = f for every f ∈ D + .