Lacunary Statistical and Lacunary Strongly Convergence of Generalized Diﬀerence Sequences in Intuitionistic Fuzzy Normed Linear Spaces

: In this article we introduce the concepts of lacunary statistical con- vergence and lacunary strongly convergence of generalized diﬀerence sequences in intuitionistic fuzzy normed linear spaces and give their characterization. We obtain some inclusion relation relating to these concepts. Further some necessary and suf-ﬁcient conditions for equality of the sets of statistical convergence and lacunary sta- tistical convergence of generalized diﬀerence sequences have been established. The notion of strong Ces` a ro summability in intuitionistic fuzzy normed linear spaces has been introduced and studied. Also the concept of lacunary generalized diﬀerence statistically Cauchy sequence has been introduced and some results are established.


Introduction
Ever since the theory of fuzzy sets was introduced by Zadeh [38] in 1965, the potential of the introduced notion was realised by researchers and it has been applied in various branches of science like Statistics, Artificial Intelligence, Computer Programming, Operation Research, Quantum Physics, Pattern Recognition, Decision Making etc. Some of its application can be found in ( [8], [12], [14], [17], [19], [24]). An important development of the classical fuzzy sets theory is the theory of intuitionistic fuzzy sets(IFS) proposed by Atanassov [7]. IFS give us a very natural tool for modeling imprecision in real life situations and found applications in various areas of science and engineering. Generalizing the idea of ordinary normed linear 118 Mausumi Sen and Mikail Et space, Saadati and Park [28] introduced the notion of intuitionistic fuzzy normed linear space. There after the theory has emerged as an active area of research in many branches of Mathematics like approximation theory, stability of functional equations, summability theory etc. The idea of statistical convergence was first introduced by Steinhaus [34] and Fast [13] which was later on studied by many authors. Schoenberg [30] studied statistical convergence as a summability method and studied some properties of statistical convergence. Altin et al. [5] have studied statistical summability method (C, 1) for sequences of fuzzy real numbers. Karakus et al. [20] generalized the concept of statistical convergence on intuitionistic fuzzy normed spaces. Some works in this field can be found in( [1], [22], [23], [25], [32]). Generalizing the idea of statistical convergence, Fridy and Orhan [15] introduced the idea of lacunary statistical convergence. Some works in lacunary statistical convergence can be found in ( [2], [15], [16], [18], [26], [27], [29], [33], [36]).The idea of difference sequence was introduced by Kizmaz [21] and later on it was further investigated by different researchers in classical as well as fuzzy sequence spaces ( [3], [4], [6], [9], [10], [11], [35], [37]).
The aim of the present paper is to introduce the concepts of lacunary statistical convergence and lacunary strongly convergence of generalized difference sequences in intuitionistic fuzzy normed linear spaces (IFNLS) and obtain some important results on this concept. Also we have introduced the concept of lacunary generalized difference statistically Cauchy sequences and given some new characterizations of it.

Preliminaries
Throughout this paper R and N will denote the set of real numbers and the set of natural numbers respectively.
Et and Colak [11] introduced the notion of generalized difference sequences as follows : Definition 2.2. Let m be a non-negative integer, then the generalized difference operator ∆ m x k is defined as Using this concept, we can define ∆ m -convergent and ∆ m -Cauchy sequences in IFNLS as follows: Definition 2.4. Let (X, µ, ν, * , •) be an IFNLS. We say that a sequence x = {x k } in X is ∆ m -Cauchy with respect to the intuitionistic fuzzy norm (µ, ν) if, for every ε ∈ (0, 1) and t > 0, there exists k 0 ∈ N such that µ(∆ m x k − ∆ m x n , t) > 1 − ε and ν(∆ m x k − ∆ m x n , t) < ε for all k, n ≥ k 0 . Definition 2.5. Let (X, µ, ν, * , •) be an IFNLS. A sequence x = {x k } in X is said to be ∆ m -bounded with respect to the intuitionistic fuzzy norm (µ, ν) if, there exists ε ∈ (0, 1) and t > 0, such that µ(∆ m x k , t) > 1 − ε and ν(∆ m x k , t) < ε . Let ℓ (µ,ν) ∞ (∆ m ) denotes the set of all ∆ m -bounded sequences in IFNLS (X, µ, ν, * , •). Definition 2.6. A lacunary sequence is an increasing integer sequence θ = {k r } such that 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 kr−1 will be abbreviated as q r . Let K ⊆ N . The number is said to be the θ-density of K, provided the limit exists. Definition 2.7. Let θ be a lacunary sequence. A sequence x = {x k } of numbers is said to be lacunary statistically convergent (briefly S θ − convergent) to the number L if for every ε > 0, the set K(ε) has θ-density zero, where In this case we write S θ -lim x = L.

Lacunary ∆ m -statistical convergence in IFNLS
In this section we define lacunary generalized difference statistical convergence in IFNLS and obtain our main results.
Definition 3.1. Let (X, µ, ν, * , •) be an IFNLS and θ be a lacunary sequence. A sequence x = {x k } in X is said to be lacunary ∆ m -statistically convergent to L ∈ X with respect to the intuitionistic fuzzy norm (µ, ν) if, for every ε ∈ (0, 1) and t > 0, In this case we write S The following lemma can be easily obtained using Definition 3.1 and properties of the θ-density.
Theorem 3.4. Let (X, µ, ν, * , •) be an IFNLS and θ be a lacunary sequence. If The converse of Theorem 3.4 is not true in general which follows from the following example.
Define a sequence x = {x k } whose terms are given by , it is not convergent with respect to the intuitionistic fuzzy norm (µ, ν).
We state the following result without proof.
By Theorem 3.7, we get an increasing index sequence K = {k i } of the natural numbers such that δ θ (K) = 1 and (µ, ν) − lim ∆ m x ki = L. Consider the sequence y defined by Then y serves our purpose. Conversely suppose that x and y are be sequences such that (µ, ν)−lim ∆ m y k = L and δ θ ({k ∈ N : ∆ m x k = ∆ m y k }) = 1. Then for every ε ∈ (0, 1) and t > 0, we have Since (µ, ν) − lim ∆ m y k = L, so the set {k ∈ N : µ(∆ m y k − L, t) ≤ 1 − ε or ν(∆ m y k −L, t) ≥ ε} contains at most finitely many terms. Also by assumption, Define the sequences {y k } and {z k } as follows: and Then {y k } and {z k } serves our purpose. Conversely if such two sequences {y k } and {z k } exist with the required properties, then the result follows immediately from Theorem 3.4 and Lemma 3.6. ✷ Definition 3.10. Let (X, µ, ν, * , •) be an IFNLS. A sequence x = {x k } in X is said to be ∆ m -statistically convergent to L ∈ X with respect to the intuitionistic fuzzy norm (µ, ν) if, for every ε ∈ (0, 1) and t > 0 , lim n→∞ Let S(∆ m ) and S θ (∆ m ) denote the sets of all ∆ m -statistically and lacunary ∆ m -statistically convergent sequences respectively in an IFNLS (X, µ, ν, * , •).

Lacunary strongly ∆ m convergence in IFNLS
In this section we define lacunary strongly ∆ m convergence in IFNLS.
Consider r 1 = max(r 0 , n 0 ). Then we will get a l ∈ N such that Since ε is arbitrary, µ(L 1 − L 2 , t) = 1 for all t > 0 and so L 1 = L 2 . ✷ The following theorem can be proved using the standard techniques , so we state without proof.

Conclusion
In this paper, we have introduced the notion of lacunary ∆ m -statistically convergent and lacunary strongly ∆ m -convergent and strongly ∆ m -Cesàro summable sequences in IFNLS and proved several useful results for these notions. Also we have introduced and studied the concept of lacunary ∆ m -statistically Cauchy sequences in IFNLS. As every crisp norm can induce an intuitionistic fuzzy norm, the results obtained here are more general than the corresponding results for normed spaces.