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

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1 Bol. Soc. Paran. Mat. (3s.) v. 38 (2020): c SPM ISSN on line ISSN in press SPM: doi:0.5269/bspm.v38i.3484 Lacunary Statistical and Lacunary Strongly Convergence of Generalized Difference Sequences in Intuitionistic Fuzzy Normed Linear Spaces Mausumi Sen and Mikail Et abstract: In this article we introduce the concepts of lacunary statistical convergence 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 sufficient conditions for equality of the sets of statistical convergence and lacunary statistical convergence of generalized difference sequences have been established. The notion of strong Cesàro 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. Key Words: Statistical convergence, Lacunary sequence, Intuitionistic fuzzy normed linear space, Difference sequence, Cesàro convergence. Contents Introduction 7 2 Preliminaries 8 3 Lacunary m -statistical convergence in IFNLS 20 4 Lacunary strongly m convergence in IFNLS 24 5 Lacunary m -statistically Cauchy sequences in IFNLS 26 6 Conclusion 27. Introduction Eversincethe theoryoffuzzy setswasintroducedby Zadeh [38] in 965,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 Makingetc. Someofits applicationcan be found in ( [8], [2], [4], [7], [9], [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 200 Mathematics Subject Classification: 40A05, 46A99. Submitted January, 207. Published July 5, Typeset by B S P M style. c Soc. Paran. de Mat.

2 8 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 [3] 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, ) 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( [], [22], [23], [25], [32]). Generalizing the idea of statistical convergence, Fridy and Orhan [5] introduced the idea of lacunary statistical convergence. Some works in lacunary statistical convergence can be found in ( [2], [5], [6], [8], [26], [27], [29], [33], [36]).The idea of difference sequence was introduced by Kizmaz [2] and later on it was further investigated by different researchers in classical as well as fuzzy sequence spaces ( [3], [4], [6], [9], [0], [], [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. 2. Preliminaries Throughout this paper R and N will denote the set of real numbers and the set of natural numbers respectively. Using the definitions of continuous t-norm and continuous t-conorm found in [3], an IFNLS is defined as follows: Definition 2.. An intuitionistic fuzzy normed linear space (in short, IFNLS) is a five-tuple (X,µ,ν,, ), where X is a linear space, is a continuous t-norm, is a continuous t-conorm and µ, ν are fuzzy sets on X (0, ) satisfying the following conditions for every x,y X and s,t > 0: (i) µ(x,t)+ν(x,t), (ii) µ(x,t) > 0, (iii) µ(x,t) = if and only if x = 0, t (iv) µ(αx,t) = µ(x, α ) for each α 0, (v) µ(x,t) µ(y,s) µ(x+y,t+s), (vi) µ(x,t) : (0, ) [0,] is continuous in t, (vii)lim t µ(x,t) = and lim t 0 µ(x,t) = 0, (viii) ν(x,t) <, (ix) ν(x,t) = 0 if and only if x = 0,

3 Lacunary Strongly Convergence of Generalized Difference Sequences 9 t (x) ν(αx,t) = ν(x, α ) for each α 0, (xi) ν(x,t) ν(y,s) ν(x+y,t+s), (xii) ν(x,t) : (0, ) [0,] is continuous in t, (xiii) lim t ν(x,t) = 0 and lim t 0 ν(x,t) =. Et and Colak [] 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 m x k = m x k m x k+, where 0 x k = x k, for all k N. Using this concept, we can define m -convergent and m -Cauchy sequences in IFNLS as follows: Definition 2.3. Let (X,µ,ν,, ) be an IFNLS. A sequence x = {x k } in X is said to be m -convergent to L X witespect to the intuitionistic fuzzy norm (µ,ν) if, for every ε (0,) and t > 0, there exists k 0 N such that µ( m x k L,t) > ε and ν( m x k L,t) < ε for all k k 0. It is denoted by (µ,ν) lim m x k = L. Definition 2.4. Let (X,µ,ν,, ) be an IFNLS. We say that a sequence x = {x k } in X is m -Cauchy witespect to the intuitionistic fuzzy norm (µ,ν) if, for every ε (0,) and t > 0, there exists k 0 N such that µ( m x k m x n,t) > 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 witespect to the intuitionistic fuzzy norm (µ,ν) if, there exists ε (0,) and t > 0, such that µ( m x k,t) > 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 = {k r } such that = k r k r as r. The intervals determined by will be denoted by I r = (k r,k r ], and the ratio kr k r will be abbreviated as q r. Let K N. The number δ (K) = lim {k I r : k K} r is said to be the -density of K, provided the limit exists. Definition 2.7. Let be a lacunary sequence. A sequence x = {x k } of numbers is said to be lacunary statistically convergent (briefly S convergent) to the number L if for every ε > 0, the set K(ε) has -density zero, where In this case we write S -limx = L. K(ε) = {k I r : x k L ε}.

4 20 Mausumi Sen and Mikail Et 3. 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.. Let (X,µ,ν,, ) be an IFNLS and be a lacunary sequence. A sequence x = {x k } in X is said to be lacunary m -statistically convergent to L X witespect to the intuitionistic fuzzy norm (µ,ν) if, for every ε (0,) and t > 0, δ ({k N : µ( m x k L,t) ε or ν( m x k L,t) ε}) = 0. In this case we write S (µ,ν) lim m x k = L. The following lemma can be easily obtained using Definition 3. and properties of the -density. Lemma 3.2. Let (X,µ,ν,, ) be an IFNLS and be a lacunary sequence. Then for every ε (0,) and t > 0, the following statements are equivalent: (i) S (µ,ν) lim m x k = L, (ii) δ ({k N : µ( m x k L,t) ε}) = δ ({k N : ν( m x k L,t) ε}) = 0, (iii) δ ({k N : µ( m x k L,t) > ε and ν( m x k L,t) < ε}) =, (iv) δ ({k N : µ( m x k L,t) > ε}) = δ ({k N : ν( m x k L,t) < ε}) =, (v) S limµ( m x k L,t) = and S limν( m x k L,t) = 0. We formulate the following two preliminary results without proof.. Theorem 3.3. Let (X,µ,ν,, ) be an IFNLS and be a lacunary sequence. If x = {x k } is a sequence in X such that S (µ,ν) lim m x k exists, then the limit is unique. Theorem 3.4. Let (X,µ,ν,, ) be an IFNLS and be a lacunary sequence. If (µ,ν) lim m x k = L, then S (µ,ν) lim m x k = L. The converse of Theorem 3.4 is not true in general which follows from the following example. Example 3.5. Consider (R,. ), the space of real numbers with usual norm. Let a b = ab, a b = min{a+b,} for all a,b [0,]. For all t > 0 and x R, let us define µ(x,t) = t t+ x and ν(x,t) = x t+ x. Then (R,µ,ν,, ) is an IFNLS. Let = {k r } be a lacunary sequence. Define a sequence x = {x k } whose terms are given by { m k if n [ hr ]+ k n x k = 0 otherwise. For every ε (0,) and t > 0, let K r (ε,t) = {k I r : µ( m x k,t) ε or ν( m x k,t) ε}. Now K r (ε,t) = {k I r : m x k εt ε > 0} {k I r : m x k = k}. Thus

5 Lacunary Strongly Convergence of Generalized Difference Sequences 2 {k I r : k K r (ε,t)} [ ] 0 as r. Hence S (µ,ν) lim m x k = 0. However the sequence { m x k } is not convergent in (R,. ), it is not convergent witespect to the intuitionistic fuzzy norm (µ, ν). We state the following result without proof. Lemma 3.6. Let (X,µ,ν,, ) be an IFNLS. Then (i) If S (µ,ν) lim m x k = L and S (µ,ν) lim m y k = L 2, then S (µ,ν) lim m (x k + y k ) = L +L 2. (ii) If S (µ,ν) lim m x k = L and α R, then S (µ,ν) lim m (αx k ) = αl. The following result can easily be proved using standard techniques. Theorem 3.7. Let (X,µ,ν,, ) be an IFNLS. Then S (µ,ν) lim m x k = L if and only if there exists an increasing index sequence K = {k i } of natural numbers such that δ (K) = and (µ,ν) lim m x ki = L. Theorem 3.8. Let (X,µ,ν,, ) be an IFNLS. Then S (µ,ν) lim m x k = L if and only if there exists a sequence y = {y k } such that (µ,ν) lim m y k = L and δ ({k N : m x k = m y k }) =. Proof. Let S (µ,ν) lim m x k = L. By Theorem 3.7, we get an increasing index sequence K = {k i } of the natural numbers such that δ (K) = and (µ,ν) lim m x ki = L. Consider the sequence y defined by { m y k = m x k if k K L otherwise Then y serves our purpose. Converselysuppose that x and y are be sequencessuch that (µ,ν) lim m y k = L and δ ({k N : m x k = m y k }) =. Then for every ε (0,) and t > 0, we have {k N : µ( m x k L,t) ε or ν( m x k L,t) ε} {k N : µ( m y k L,t) ε or ν( m y k L,t) ε} {k N : m x k m y k }. Since (µ,ν) lim m y k = L, so the set {k N : µ( m y k L,t) ε or ν( m y k L,t) ε}containsatmostfinitely manyterms. Alsobyassumption, δ ({k N : m x k m y k }) = 0. Hence δ ({k N : µ( m x k L,t) ε or ν( m x k L,t) ε}) = 0. and so S (µ,ν) lim m x k = L. Theorem 3.9. Let (X,µ,ν,, ) be an IFNLS. Then S (µ,ν) lim m x k = L if and only if there exist sequences {y k } and {z k } in X such that m x k = m y k + m z k, for all k N where (µ,ν) lim m y k = L and S (µ,ν) lim m z k = 0

6 22 Mausumi Sen and Mikail Et Proof. Let S (µ,ν) lim m x k = L. By Theorem 3.7,there exists an increasing sequence K = {k i } ofnatural numbers such that δ (K) = and (µ,ν) lim m x ki = L. Define the sequences {y k } and {z k } as follows: { m y k = m x k, if k K L, otherwise. and m z k = { 0, if k K m x k L, otherwise. 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 3.0. Let (X,µ,ν,, ) be an IFNLS. A sequence x = {x k } in X is said to be m -statistically convergent to L X witespect to the intuitionistic fuzzy norm (µ,ν) if, for every ε (0,) and t > 0, lim n n {k n : µ( m x k L,t) ε or ν( m x k L,t) ε} = 0 and we write S (µ,ν) lim m x k = L. Let S( m ) and S ( m ) denote the sets of all m - statistically and lacunary m -statistically convergent sequences respectively in an IFNLS (X,µ,ν,, ). For x X,t > 0 and α (0,), the ball centered at x witadius α is defined by B Y (x,α,t) = {y X : µ(x y,t) > α and ν(x y,t) < α}. Theorem 3.. Let (X,µ,ν,, ) be an IFNLS. For any lacunary sequence, S ( m ) S( m ) if and only if limsup r q r <. Proof. If limsup r q r <, then there exists H > 0 such that q r < H for all r. Suppose that x S ( m ) and S (µ,ν) lim m x k = L. For t > 0 and λ (0,), let N r = {k I r : µ( m x k L,t) λ or ν( m x k L,t) λ}. Then for ε > 0, there exists r 0 N such that N r < ε for all r > r 0. (3.) Now let K = max{n r : r r 0 } and choose n such that k r < n k r. Then we have n {k n : µ( m x k L,t) λ or ν( m x k L,t) λ} k r {k k r : µ( m x k L,t) λ or ν( m x k L,t) λ} = k r {N +N 2 + +N r0 +N r0+ + +N r }

7 Lacunary Strongly Convergence of Generalized Difference Sequences 23 K k r r 0 + k r {0+ Nr Nr } r0k k r +εq r r0k k r +εh. To prove the converse suppose that limsup r q r =. Since is a lacunary sequence, we can select a subsequence {k r(j) } of such that q r(j) > j. Let ξ( 0) X. We define a sequence x = {x k } as follows: { m ξ if kr(j) < k 2k x k = r(j) for some j =,2,3,... 0 otherwise. Since ξ( 0), we can choose t > 0, λ (0,) such that ξ / B Y (0,λ,t). Now for j >, (j) {k k r(j) : µ( m x k,t) λ or ν( m x k,t) λ} < j. Thus x S ( m ). But x / S( m ). For 2k r(j) k 2k r(j) : µ( m x k,t) λ or ν( m x k,t) λ} > 2 and k r(j) {k k r(j) : µ( m x k ξ,t) λ or ν( m x k ξ,t) λ} 2 j, which is a contradiction. Theorem 3.2. Let (X,µ,ν,, ) be an IFNLS. For any lacunary sequence, S( m ) S ( m ) if and only if liminf r q r > Proof. Let liminf r q r >. Then there exists δ > 0 such that q r + δ for sufficiently large r, which implies that hr k r δ +δ. Let S (µ,ν) lim m x k = L. Then for each t > 0, λ (0,) and sufficiently large r, we have k r {k k r : µ( m x k L,t) λ or ν( m x k L,t) λ} δ +δ {k I r : µ( m x k L,t) λ or ν( m x k L,t) λ}. Thus S (µ,ν) lim m x k = L. To prove the converse suppose that liminf r q r =. Since is a lacunary sequence, we can select a subsequence {k r(j) } of = {k r } such that k r(j) < + k r(j) j and k r(j) > j where r(j) r(j )+2. k r(j ) Let ξ( 0) X. Consider the sequence x = {x k } defined by: { m ξ if k Ir(j) for some j =,2,3,... x k = 0 otherwise.

8 24 Mausumi Sen and Mikail Et We show that x S( m ). Let t > 0 and λ (0,). Choose t > 0 and λ (0,) such that B(0,λ,t ) B(0,λ,t) and ξ / B(0,λ,t ). Also for each n N we can find a positive integer j n such that k r(jn) < n k r(jn). Then for each n N, we have n {k n : µ( m x k,t) λ or ν( m x k,t) λ} k r(jn )+(jn k r(jn) 2 j n. Thus S (µ,ν) lim m x k = L. Next we show that x / S ( m ). Since ξ( 0), so we can choose t > 0 and λ (0,) such that ξ / B(0,λ,t). Thus lim j (j) {k I r(j) : µ( m x k,t) λ or ν( m x k,t) λ} = and for r r j, lim j {k I r : µ( m x k ξ,t) λ or ν( m x k ξ,t) λ} =. Hence x / S ( m ), a contradiction. Theorem 3. and Theorem 3.2 together give the following corollary. Corollary 3.3. Let (X,µ,ν,, ) be an IFNLS. For any lacunary sequence, S( m ) = S ( m ) if and only if < liminf r q r limsup r q r <. 4. Lacunary strongly m convergence in IFNLS In this section we define lacunary strongly m convergence in IFNLS. Definition 4.. Let (X,µ,ν,, ) be an IFNLS and be a lacunary sequence. A sequence x = {x k } in X is said to be lacunary strongly m - convergent to L X witespect to the intuitionistic fuzzy norm (µ,ν) if, for every ε (0,) and t > 0,there exist r 0 N such that h k I r µ( m x k L,t) > ε and ν( m x k L,t) < ε for all r r 0. In this case we write N (µ,ν) lim m x k = L. Definition 4.2. Let (X,µ,ν,, ) be an IFNLS and be a lacunary sequence. A sequence x = {x k } in X is strongly m -Cesàro summable to L X witespect to the intuitionistic fuzzy norm (µ,ν) if, for every ε (0,) and t > 0,there exists n 0 N such that n n k= µ( m x k L,t) > ε and n n k= ν( m x k L,t) < ε for all n n 0. In this case we write σ (µ,ν) lim m x k = L. Let N ( m ) and σ( m ) denote the sets of all lacunary strongly m - convergent sequences and strongly m -Cesàro summable sequences in the IFNLS (X,µ,ν,, ). Theorem 4.3. Let (X,µ,ν,, ) be an IFNLS and = (k r ) be a lacunary sequence. Then σ( m ) N ( m ) if liminf r q r >.

9 Lacunary Strongly Convergence of Generalized Difference Sequences 25 Proof. Let liminf r q r > and x = (x k ) σ( m ). Then there exists δ > 0 such that q r +δ for all r. Then µ( m x k L,t) = kr k= µ( m x k L,t) kr k= µ( m x k L,t) ] ] = kr k= µ( m x k L,t) kr k= µ( m x k L,t) [ k r kr [ Since = k r k r, kr +δ δ and kr k r kr Also the terms k r kr k= µ( m x k L,t) and k r kr k= µ( m x k L,t) both converges to 0. So, µ( m x k L,t). Similarly ν( m x k L,t) 0. So (x k ) N ( m ). Theorem 4.4. Let (X,µ,ν,, ) be an IFNLS and = (k r ) be a lacunary sequence. Then N ( m ) σ( m ) if liminf r q r =. Proof. Let liminf r q r = and x = (x k ) N ( m ). Then for t > 0, we have H r = µ( m x k L,t) and H r = ν( m x k L,t) 0 as r. Then for ε > 0, there exists r 0 N such that H r < +ε for all r r 0. Also we can find T > 0 such that H r < T and H r < T, r =,2,. Let n be an integer with k r < n k r. Then n n k= µ( m x k L,t) kr k r k= µ( m x k L,t) [ = k r i I µ( m x k L,t)+ + i I 2 µ( m x k L,t) ] k = sup r r0 H r0 r k r + hr 0 + k r H r hr k r H r < T kr 0 k r +(+ε) kr kr 0 k r. Since k r as n, it follows that n n k= µ( m x k L,t). Similarly we can show that n n k= ν( m x k L,t) 0. Hence x σ( m ). Theorem 4.5. If x = {x k } N ( m ) σ( m ), then N (µ,ν) lim m x k = σ (µ,ν) lim m x k. Proof. Let N (µ,ν) lim m x k = L and σ (µ,ν) lim m x k = L 2. Given ε > 0, choose λ (0,) such that ( λ) ( λ) > ε and λ λ < ε. Now for t > 0, there exists r 0 N such that δ ) > λ and ν( m x k L, t 2 k I r µ( m x k L, t 2 r r 0. Also, there exists n 0 N such that n n k= µ( m x k L 2, t 2 ) > λ and n n k= ν( m x k L 2, t 2 n n 0. ) < λ for all ) < λ for all

10 26 Mausumi Sen and Mikail Et Consider r = max(r 0,n 0 ). Then we will get a l N such that µ( m x l L, t 2 ) µ( m x k L, t 2 ) > λ and µ( m x l L 2, t 2 ) n n k= µ( m x k L 2, t 2 ) > λ. Therefore µ(l L 2,t) ( λ) ( λ) > ε. Since ε is arbitrary, µ(l L 2,t) = for all t > 0 and so L = L 2. The following theorem can be proved using the standard techniques, so we state without proof. Theorem 4.6. Let = (k r ) be a lacunary sequence and x = {x k } be a sequence in (X,µ,ν,, ). Then (i)n (µ,ν) lim m x k = L implies S (µ,ν) lim m x k = L (ii)x l (µ,ν) ( m ) and S (µ,ν) lim m x k = L implies N (µ,ν) lim m x k = L (iii)l (µ,ν) ( m ) S ( m ) = l (µ,ν) ( m ) N ( m ). 5. Lacunary m -statistically Cauchy sequences in IFNLS Definition 5.. Let (X,µ,ν,, ) be an IFNLS. A sequence x = {x k } in X is said to be lacunary m - statistically Cauchy witespect to the intuitionistic fuzzy norm (µ,ν) if there is a subsequence (x k (r)) I r for eac, (µ,ν) lim m x k (r) = L and for each ε (0,), t > 0, lim r {k I r : µ( m x k m x k (r),t) ε or ν( m x k m x k (r),t) ε} = 0. Theorem 5.2. The sequence x = {x k } in an IFNLS (X,µ,ν,, )is m - statistically convergent if and onlu if it is lacunary m -statistically Cauchy in X. Proof. Let S (µ,ν) lim m x k = L and for each n we write K n = {k N : µ( m x k L,t) > n and ν( m x k L,t) < n }. K Then K n+ K n for each n and lim n I r r =. So there exists p such that r > p and K Ir > 0 i.e. K I r. We next choose p 2 > p such that r p 2 implies K 2 I r. Then for eac satisfying p r p 2 we choose k (r) I r such that k (r) K I r. In general we choose p n+ > p n such that r p n+ implies k (r) K n I r. Thus k (r) I r for eac and µ( m x k (r) L,t) > n and ν( m x k (r) L,t) < n. Hence (µ,ν) lim m x k (r) = L. Using Theorem and 3.4 Lemma 3.6, it can be easily seen that lim r {k I r : µ( m x k m x k (r),t) ε or ν( m x k m x k (r),t) ε} = 0. Conversely suppose that {x k } is a lacunary m -statistically Cauchy sequence in X.For ε > 0 choose λ (0,) such that ( λ) ( λ) > ε and λ λ < ε. Then for any t > 0 define K µ, = {k N : µ( m x k m x k (r), t 2 > λ} and K µ,2 = {k N : µ( m x k (r) L, t 2 > λ} Let K µ = K µ, K µ,2. Then δ (K µ ) = and for k K µ,

11 Lacunary Strongly Convergence of Generalized Difference Sequences 27 µ( m x k L,t) µ( m x k m x k (r), t 2 ) µ( m x k (r) L, t 2 ) > ε Similarly if we define K ν, = {k N : ν( m x k m x k (r), t 2 ) < λ} and K ν,2 = {k N : ν( m x k (r) L, t 2 ) < λ} Then δ (K ν ) = δ (K ν, K ν,2 ) = and for k K ν, ν( m x k L,t) < ε. Therefore δ ({k N : µ( m x k L,t) > delta and ν( m x k L,t) < delta }) =. Hence x = {x k } is m - statistically convergent. Corollary 5.3. Any lacunary m - statistically convergent sequence has a m - convergent subsequence. 6. 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. References. Alghamdi, M. A., Alotaibi, A., Lohani, Q. M. D., Mursaleen, M., Statistical limit superior and limit inferior in intuitionistic fuzzy normed spaces. J. Ineq. Appl. 96, doi:0.86/ x (202). 2. Alotaibi, A., On lacunary statistical convergence of double sequences witespect to the intuitionistic fuzzy normed space. Int. J. Contemp. Math. Sci. 5, (200). 3. Altin,Y., Et, M., Çolak, R., Lacunary statistical and lacunary strongly convergence of generalized difference sequences of fuzzy numbers. Comput. Math. Appl.52(6-7), (2006). 4. Altin,Y., Et, M., Basarir, M., On some generalized difference sequences of fuzzy numbers. Kuwait J. Sci. Engrg. 34(A), -4 (2007). 5. Altin,Y., Mursaleen, M., Altinok,H., Statistical summability (C; ) for sequences of fuzzy real numbers and a Tauberian theorem. J. Intell. Fuzzy Systems. 2(6), (200). 6. Altinok, H., Mursaleen,M., -Statistically boundedness for sequences of fuzzy numbers. Taiwanese J. Math. 5(5), (20). 7. Atanassov, K. T., Intuitionistic fuzzy sets. Fuzzy Sets and Systems. 20, (986). 8. Barros, L. C., Bassanezi, R. C., Tonelli, P. A., Fuzzy modelling in population dynamics. Ecol. Model. 3 28, 27-3 (2000). 9. Colak,R., Altinok, H., Et,M., Generalized difference sequences of fuzzy numbers. Chaos, Solitons and Fractals 40(3), 06-7 (2009). 0. Colak, R., Altin, Y., Mursaleen, M., On some sets of difference sequences of fuzzy numbers. Soft Computing. 5, (20).. Et, M., Colak,R., On some generalized difference sequence spaces. Soochow J. Math. 2 (4), (995).

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13 Lacunary Strongly Convergence of Generalized Difference Sequences Tripathy, B.C., Dutta, A. J., Lacunary bounded variation sequence of fuzzy real numbers, Journal of Intelligent and Fuzzy Systems 24(), (203). 37. Tripathy, B.C., Dutta, H., On some lacunary difference sequence spaces defined by a sequence of Orlicz functions and q-lacunary n m -statistical convergence, Analele Stiintifice ale Universitatii Ovidius, Seria Matematica 20(), (202). 38. Zadeh,L. A., Fuzzy sets. Information and Control. 8, (965). M. Sen, Department of Mathematics, National Institute of Technology Silchar, Silchar 78800, India, Tel.: , Fax: address: senmausumi@gmail.com and M. Et, Department of Mathematics, Firat University 239; Elazig ; TURKEY. address: mikailet@yahoo.com