On the Lacunary Statistical Convergence of Triple Sequences in Fuzzy Metric Spaces

. In this research paper, we analyze the lacunary statistical convergence and lacunary statistical Cauchy concepts of triple sequence in fuzzy metric space. We also introduce the concept of triple lacunary statistical completeness and prove some basic properties.


Introduction
In recent years, the fuzzy theory has emerged as the most active area of research in many branches of mathematics and engineering. In 1965, Zadeh [34] first introduced the fuzzy set theory and since then a large number of research papers have appeared by using the concept of fuzzy set (numbers) and fuzzification of many classical theories has also been made. One of the main problems in the theory of fuzzy topological spaces is to obtain an appropriate and consistent notion of a fuzzy metric space. Many authors have defined several concepts of fuzzy metric space in different ways [2,8,17,18]. In [8,9], George and Veeramani introduced and studied the concept of fuzzy metric space (in short FMS) with the help of continuous t-norms. Recently, several convergences in fuzzy metric spaces, as p-convergence, s-convergence, st-convergence and std-convergence, were studied by Gregori et al [12,13,14,15].
In this research paper, we analyze the lacunary statistical convergence and lacunary statistical Cauchy concepts of triple sequence in fuzzy metric space. We also introduce the concept of lacunary statistical completeness which would provide a more general framework to study the completeness of fuzzy metric spaces.

Preliminaries
We recall the following basic concepts from [1,8,11,18,31]. They will be needed in the course of the paper. By N and R, we mean the set of all natural and real numbers, respectively. Definition 2. The 3-tuple (X, M, * ) is said to be a fuzzy metric space if X is a nonempty set, * is a continuous t-norm and M is a fuzzy set on X 2 × (0, ∞) satisfying the following conditions for all x, y, z ∈ X and s, t > 0 : If (X, M, * ) is a fuzzy metric space, then we will call (M, * ), or simply M , a fuzzy metric on X.
Let (X, d) be a metric space. Denote by a.b the usual multiplication for all a, b ∈ [0, 1] , and let M d be the fuzzy set defined on X 2 × (0, ∞) by .
) is a fuzzy metric on X called standard fuzzy metric space [8]. with centre x ∈ X and radius r, 0 < r < 1, t > 0 as Let (X, M, * ) be a fuzzy metric space. Define Then τ M is a topology on X. George and Veeramani [8] proved that {B M (x, r, t) : x ∈ X, t > 0, r ∈ (0, 1)} forms a base of a topology τ M in X.
The next characterization of convergent sequences was gave in [8] .
Proposition 1. Let (X, M, * ) be a fuzzy metric space and τ M be the topology induced by the fuzzy metric. Then for a sequence {x n } in X, x n converges to x in X if and only if M (x n , x, t) tends to 1 as n tends to ∞, for t > 0.
It is easy to see that {x n } converges to x 0 which is equivalent to the following : For each r ∈ (0, 1) and each t > 0, there exists n 0 ∈ N such that M (x n , x 0 , t) > 1 − r for all n > n 0 . Definition 5. A fuzzy metric space is said to be complete if every Cauchy sequence is convergent.

Triple Lacunary statistical convergence in FMS
Li et al. [19] introduced the concepts of statistical convergence in fuzzy metric spaces. In this section, we present the notion of lacunary statistically convergent and lacunary statistically Cauchy of triple sequences in fuzzy metric spaces. We now recall that the concept of statistical convergence for triple sequences was presented by Şahiner, Gürdal and Düden [25] as follows: A triple sequence (x jkl ) is said to be convergent to L in Pringsheim's sense if for every ε > 0, there exists n 0 (ε) ∈ N such that |x jkl − L| < ε whenever j, k, l ≥ n 0 .
We start this section with the following definition.
Definition 7. Let (X, M, * ) be a fuzzy metric space. A triple sequence {x jkl } in X is said to be convergent to L ∈ X if, for each r ∈ (0, 1) and each t > 0, there exists n 0 ∈ N such that M (x jkl , L, t) > 1 − r whenever j > n 0 , k > n 0 , l > n 0 .
In this case we write x jkl → L (S (sts 3 )) . In such a case we say that {x jkl } is sts 3 -convergent to L (or, {x jkl } sts 3 -converges to L). It is obvious that x, y ∈ R and every t > 0, consider In this case observe that (X, M, * ) is a fuzzy metric space. Now define a sequence x = {x jkl } whose terms are given by It is immediate to see that {x jkl } is not convergent to 0, but {x jkl } is sts 3 -convergent to 0.
Theorem 2. Let (X, M, * ) be a fuzzy metric space and θ 3 = θ r,s,t be a lacunary triple sequence. If triple sequence {x jkl } in X is convergent to L, then {x jkl } is lacunary statistical convergent to L. But converse need not be true.
Proof. Suppose that {x jkl } is convergent to L. Let r ∈ (0, 1) and t > 0. Then there exists n 0 ∈ N such that M (x jkl , L, t) > 1 − r for all j > n 0 , k > n 0 , l > n 0 . Hence that is x jkl → L (S θ3 (sts)) . We complete the proof.
So {x jkl } converges to L.
The following example shows that the converse of the above theorem need not be true.
for all x, y ∈ X and t > 0. Then (X, M, * ) is a fuzzy metric space and triple sequence {x jkl } in (X, M, * ) is sts θ3 -Cauchy, but it is not sts θ3 -convergent.
Let α ∈ (0, 1) and t > 0. Then there exists p, q, r ∈ N such that for all j > p, k > q, l > r. Hence Corollary 1. If a triple sequence in a fuzzy metric space is Cauchy, then it is sts θ3 -Cauchy. However, the converse is false.
Now using a similar technique in the proof of Theorem 3, one can get the following result at once.
Theorem 5. Let (X, M, * ) be a fuzzy metric space, and let x = {x jkl } be a triple sequence whose terms are in X. Then, for θ 3 = θ r,s,t lacunary triple sequence, the following conditions are equivalent:.

Concluding remarks
In this paper, we have dealt with the notion of the lacunary statistical convergence and lacunary statistical Cauchy of triple sequence in fuzzy metric space.
However, further investigation in these aspects is required. We also introduce the concept of triple lacunary statistical completeness and prove some of the basic properties.

Compliance with ethical standards
Conflict of interest The authors declare that they have no conflict of interest.