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Lacunary Δ-statistical convergence in intuitionistic fuzzy n-normed space

Abstract

The concept of lacunary statistical convergence was introduced in intuitionistic fuzzy n-normed spaces in Sen and Debnath (Math. Comput. Model. 54:2978-2985, 2011). In this article, we introduce the notion of lacunary Δ-statistically convergent and lacunary Δ-statistically Cauchy sequences in an intuitionistic fuzzy n-normed space. Also, we give their properties using lacunary density and prove relation between these notions.

MSC:47H10, 54H25.

1 Introduction

Fuzzy set theory was introduced by Zadeh [1] in 1965. This theory has been applied not only in different branches of engineering such as in nonlinear dynamic systems [2], in the population dynamics [3], in the quantum physics [4], but also in many fields of mathematics such as in metric and topological spaces [57], in the theory of functions [8, 9], in the approximation theory [10]. 2-normed and n-normed linear spaces were initially introduced by Gähler [11, 12] and further studied by Kim and Cho [13], Malceski [14] and Gunawan and Mashadi [15]. Vijayabalaji and Narayanan [16] defined fuzzy n-normed linear space. After Saadati and Park [17] introduced the concept of intuitionistic fuzzy normed space, Vijayabalaji et al. [18] defined the notion of intuitionistic fuzzy n-normed space. The notion of statistical convergence was investigated by Steinhaus [19] and Fast [20]. Then a lot of authors applied this concept to probabilistic normed spaces [21, 22], random 2-normed spaces [23] and finally intuitionistic fuzzy normed spaces [24, 25]. Fridy and Orhan [26] introduced the idea of lacunary statistical convergence. Using this idea, Mursaleen and Mohiuddine [27], Sen and Debnath [28] investigated lacunary statistical convergence in intuitionistic fuzzy normed spaces and intuitionistic fuzzy n-normed spaces, respectively. The idea of difference sequences was introduced by Kızmaz [29] where Δx=(Δ x k )= x k x k + 1 . Başarır [30] introduced the Δ-statistical convergence of sequences. Bilgin [31] introduced the definition of lacunary strongly Δ-convergence of fuzzy numbers. Hazarika [32] gave the definition of lacunary generalized difference statistical convergence in random 2-normed spaces. Also, the generalized difference sequence spaces were studied by various authors [3335]. In this article, we shall introduce lacunary Δ-statistical convergence and lacunary Δ-statistically Cauchy sequences in IFnNLS.

2 Preliminaries, background and notation

In this section, we give the basic definitions.

Definition 2.1 ([27])

A binary operation :[0,1]×[0,1][0,1] is said to be a continuous t-norm if it satisfies the following conditions:

  1. (i)

    is associative and commutative,

  2. (ii)

    is continuous,

  3. (iii)

    a1=a for all a[0,1],

  4. (iv)

    abcd whenever ac and bd for each a,b,c,d[0,1].

Definition 2.2 ([27])

A binary operation :[0,1]×[0,1][0,1] is said to be a continuous t-conorm if it satisfies the following conditions:

  1. (i)

    is associative and commutative,

  2. (ii)

    is continuous,

  3. (iii)

    a0=a for all a[0,1],

  4. (iv)

    abcd whenever ac and bd for each a,b,c,d[0,1].

Definition 2.3 ([27])

Let nN and X be a real vector space of dimension dn (here we allow it to be infinite). A real-valued function ,, on X××X= X n satisfying the following four properties:

  1. (i)

    x 1 , x 2 ,, x n =0 if and only if x 1 , x 2 ,, x n are linearly dependent,

  2. (ii)

    x 1 , x 2 ,, x n are invariant under any permutation,

  3. (iii)

    x 1 , x 2 ,,α x n =|α| x 1 , x 2 ,, x n for any αR,

  4. (iv)

    x 1 , x 2 ,, x n 1 ,y+z x 1 , x 2 ,, x n 1 ,y+ x 1 , x 2 ,, x n 1 ,z,

is called an n-norm on X and the pair is called an n-normed space.

Definition 2.4 ([28])

An IFnNLS is the five-tuple (X,μ,υ,,) where X is a linear space over a field F, is a continuous t-norm, is a continuous t-conorm, μ, υ are fuzzy sets on X n ×(0,), μ denotes the degree of membership and υ denotes the degree of nonmembership of ( x 1 , x 2 ,, x n ,t) X n ×(0,) satisfying the following conditions for every ( x 1 , x 2 ,, x n ) X n and s,t>0:

  1. (i)

    μ( x 1 , x 2 ,, x n ,t)+υ( x 1 , x 2 ,, x n ,t)1,

  2. (ii)

    μ( x 1 , x 2 ,, x n ,t)>0,

  3. (iii)

    μ( x 1 , x 2 ,, x n ,t)=1 if and only if x 1 , x 2 ,, x n are linearly dependent,

  4. (iv)

    μ( x 1 , x 2 ,, x n ,t) is invariant under any permutation of x 1 , x 2 ,, x n ,

  5. (v)

    μ( x 1 , x 2 ,,c x n ,t)=μ( x 1 , x 2 ,, x n , t | c | ) for all c0, cF,

  6. (vi)

    μ( x 1 , x 2 ,, x n ,s)μ( x 1 , x 2 ,, x n ,t)μ( x 1 , x 2 ,, x n + x n ,s+t),

  7. (vii)

    μ( x 1 , x 2 ,, x n ,t):(0,)[0,1] is continuous in t,

  8. (viii)

    lim t μ( x 1 , x 2 ,, x n ,t)=1 and lim t 0 μ( x 1 , x 2 ,, x n ,t)=0,

  9. (ix)

    υ( x 1 , x 2 ,, x n ,t)<1,

  10. (x)

    υ( x 1 , x 2 ,, x n ,t)=0 if and only if x 1 , x 2 ,, x n are linearly dependent,

  11. (xi)

    υ( x 1 , x 2 ,, x n ,t) is invariant under any permutation of x 1 , x 2 ,, x n ,

  12. (xii)

    υ( x 1 , x 2 ,,c x n ,t)=υ( x 1 , x 2 ,, x n , t | c | ) for all c0, cF,

  13. (xiii)

    υ( x 1 , x 2 ,, x n ,s)υ( x 1 , x 2 ,, x n ,t)υ( x 1 , x 2 ,, x n + x n ,s+t)

  14. (xiv)

    υ( x 1 , x 2 ,, x n ,t):(0,)[0,1] is continuous in t,

  15. (xv)

    lim t υ( x 1 , x 2 ,, x n ,t)=0 and lim t 0 υ( x 1 , x 2 ,, x n ,t)=1.

Example 2.1 ([28])

Let (X,,,) be an n-normed linear space. Also let ab=ab and ab=min{a+b,1} for all a,b[0,1],

μ( x 1 , x 2 ,, x n ,t)= t t + x 1 , x 2 , , x n andυ( x 1 , x 2 ,, x n ,t)= x 1 , x 2 , , x n t + x 1 , x 2 , , x n .

Then (X,μ,υ,,) is an IFnNLS.

Definition 2.5 ([26])

A lacunary sequence is an increasing integer sequence θ={ k r } such that k 0 =0 and 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 k r k r 1 will be abbreviated as q r . Let KN. The number

δ θ (K)= lim r 1 h r |{k I r :kK}|

is said to be the θ-density of K, provided the limit exists.

Definition 2.6 ([28])

Let θ be a lacunary sequence. A sequence x={ x k } of numbers is said to be lacunary statistically convergent (or S θ -convergent) to the number L if for every ε>0, the set K(ε) has θ-density zero, where

K(ε)= { k N : | x k L | ε } .

In this case, we write S θ -limx=L.

3 Δ-Convergence and lacunary Δ-statistical convergence in IFnNLS

In this section, we define Δ-convergence and lacunary Δ-statistical convergence in intuitionistic fuzzy n-normed spaces.

Definition 3.1 Let (X,μ,υ,,) be an IFnNLS. A sequence x={ x k } in X is said to be Δ-convergent to LX with respect to the intuitionistic fuzzy n-norm ( μ , υ ) n if, for every ε>0, t>0 and y 1 , y 2 ,, y n 1 X, there exists k 0 N such that μ( y 1 , y 2 ,, y n 1 ,Δ x k L,t)>1ε and υ( y 1 , y 2 ,, y n 1 ,Δ x k L,t)<ε for all k k 0 , where kN and Δ x k =( x k x k + 1 ). It is denoted by ( μ , υ ) n -limΔx=L or Δ x k L as k.

Definition 3.2 Let (X,μ,υ,,) be an IFnNLS. A sequence x={ x k } in X is said to be lacunary Δ-statistically convergent or S θ (Δ)-convergent to LX with respect to the intuitionistic fuzzy n-norm ( μ , υ ) n provided that for every ε>0, t>0 and y 1 , y 2 ,, y n 1 X,

δ θ ( Δ ) ( { k N : μ ( y 1 , y 2 , , y n 1 , Δ x k L , t ) 1 ε or  υ ( y 1 , y 2 , , y n 1 , Δ x k L , t ) ε } ) = 0 ,

or, equivalently,

δ θ ( Δ ) ( { k N : μ ( y 1 , y 2 , , y n 1 , Δ x k L , t ) > 1 ε and  υ ( y 1 , y 2 , , y n 1 , Δ x k L , t ) < ε } ) = 1 .

It is denoted by S θ ( μ , υ ) n (Δ)-limx=L or x k L( S θ (Δ)). Using Definition 3.2 and properties of the θ-density, we can easily obtain the following lemma.

Lemma 3.1 Let (X,μ,υ,,) be an IFnNLS and θ be a lacunary sequence. Then, for every ε>0, t>0 and y 1 , y 2 ,, y n 1 X, the following statements are equivalent:

  1. (i)

    S θ ( μ , υ ) n (Δ)-limx=L,

  2. (ii)

    δ θ (Δ)({kN:μ( y 1 , y 2 ,, y n 1 ,Δ x k L,t)1ε})= δ θ (Δ)({kN:υ( y 1 , y 2 ,, y n 1 ,Δ x k L,t)ε})=0,

  3. (iii)

    δ θ (Δ)({kN:μ( y 1 , y 2 ,, y n 1 ,Δ x k L,t)>1ε and υ( y 1 , y 2 ,, y n 1 ,Δ x k L,t)<ε})=1,

  4. (iv)

    δ θ (Δ)({kN:μ( y 1 , y 2 ,, y n 1 ,Δ x k L,t)>1ε})= δ θ (Δ)({kN:υ( y 1 , y 2 ,, y n 1 ,Δ x k L,t)<ε})=1,

  5. (v)

    S θ -limμ( y 1 , y 2 ,, y n 1 ,Δ x k L,t)=1 and S θ -limυ( y 1 , y 2 ,, y n 1 ,Δ x k L,t)=0.

Proceeding exactly in a similar way as in [36], the following theorem can be proved.

Theorem 3.1 Let (X,μ,υ,,) be an IFnNLS and θ be a lacunary sequence. If a sequence x={ x k } in X is lacunary Δ-statistically convergent or S θ (Δ)-convergent to LX with respect to the intuitionistic fuzzy n-norm ( μ , υ ) n , S θ ( μ , υ ) n (Δ)-limx is unique.

Theorem 3.2 Let (X,μ,υ,,) be an IFnNLS and θ be a lacunary sequence. If ( μ , υ ) n -limΔx=L, then S θ ( μ , υ ) n (Δ)-limx=L.

Proof Let ( μ , υ ) n -limΔx=L. Then, for every ε>0, t>0 and y 1 , y 2 ,, y n 1 X, there exists k 0 N such that μ( y 1 , y 2 ,, y n 1 ,Δ x k L,t)>1ε and υ( y 1 , y 2 ,, y n 1 ,Δ x k L,t)<ε for all k k 0 . Hence the set

{ k N : μ ( y 1 , y 2 , , y n 1 , Δ x k L , t ) 1 ε or  υ ( y 1 , y 2 , , y n 1 , Δ x k L , t ) ε }

has a finite number of terms. Since every finite subset of has lacunary density zero,

δ θ ( Δ ) ( { k N : μ ( y 1 , y 2 , , y n 1 , Δ x k L , t ) 1 ε or  υ ( y 1 , y 2 , , y n 1 , Δ x k L , t ) ε } ) = 0 ,

that is, S θ ( μ , υ ) n (Δ)-limx=L.

It follows from the following example that the converse of Theorem 3.2 is not true in general.

Example 3.1 Consider X= R n with

x 1 , x 2 ,, x n =abs ( | x 11 x 1 n x n 1 x n n | ) ,

where x i =( x i 1 , x i 2 ,, x i n ) R n for each i=1,2,,n, and let ab=ab, ab=min{a+b,1} for all a,b[0,1]. Now, for all y 1 , y 2 ,, y n 1 ,x R n and t>0, μ( y 1 , y 2 ,, y n 1 ,x,t)= t t + y 1 , y 2 , , y n 1 , x and υ( y 1 , y 2 ,, y n 1 ,x,t)= y 1 , y 2 , , y n 1 , x t + y 1 , y 2 , , y n 1 , x . Then ( R n ,μ,υ,,) is an IFnNLS. Let I r and h r be as defined in Definition 2.5. Define a sequence x={ x k } whose terms are given by

x k ={ ( ( n [ h r ] + 1 ) ( n + [ h r ] ) 2 , 0 , , 0 ) R n if  1 k n [ h r ] , ( 1 2 k 2 + 1 2 k , 0 , , 0 ) R n if  n [ h r ] + 1 k n , ( 1 2 n 2 1 2 n , 0 , , 0 ) R n if  k > n

such that

Δ x k ={ ( k , 0 , , 0 ) N if  n [ h r ] + 1 k n , ( 0 , 0 , , 0 ) N otherwise .

For every 0<ε<1 and for any y 1 , y 2 ,, y n 1 X, t>0, let

K ( ε , t ) = { k I r : μ ( y 1 , y 2 , , y n 1 , Δ x k L , t ) 1 ε or  υ ( y 1 , y 2 , , y n 1 , Δ x k L , t ) ε } .

Now,

K ( ε , t ) = { k I r : y 1 , y 2 , , y n 1 , Δ x k ε t 1 ε > 0 } { k I r : Δ x k = ( k , 0 , , 0 ) R n } .

Thus we have 1 h r |{k I r :kK(ε,t)}| [ h r ] h r 0 as r. Hence S θ ( μ , υ ) n (Δ)-limx=0.

On the other hand, x={ x k } in X is not Δ-convergent to 0 with respect to the intuitionistic fuzzy n-norm since

μ ( y 1 , y 2 , , y n 1 , Δ x k , t ) = t t + y 1 , y 2 , , y n 1 , Δ x k = { t t + y 1 , y 2 , , y n 1 , Δ x k if  n [ h r ] + 1 k n , 1 , otherwise , 1

and

υ ( y 1 , y 2 , , y n 1 , Δ x k , t ) = y 1 , y 2 , , y n 1 , Δ x k t + y 1 , y 2 , , y n 1 , Δ x k = { y 1 , y 2 , , y n 1 , Δ x k t + y 1 , y 2 , , y n 1 , Δ x k if  n [ h r ] + 1 k n , 0 , otherwise 0 .

This completes the proof of the theorem. □

Theorem 3.3 Let (X,μ,υ,,) be an IFnNLS. Then S θ ( μ , υ ) n (Δ)-limx=L if and only if there exists an increasing sequence K={ k n } of the natural numbers such that δ θ (Δ)(K)=1 and ( μ , υ ) n - lim k K Δ x k =L.

Proof Necessity. Suppose that S θ ( μ , υ ) n (Δ)-limx=L. Then, for every y 1 , y 2 ,, y n 1 X, t>0 and j=1,2, ,

K ( j , t ) = { k N : μ ( y 1 , y 2 , , y n 1 , Δ x k L , t ) > 1 1 j and  υ ( y 1 , y 2 , , y n 1 , Δ x k L , t ) < 1 j } and M ( j , t ) = { k N : μ ( y 1 , y 2 , , y n 1 , Δ x k L , t ) 1 1 j or  υ ( y 1 , y 2 , , y n 1 , Δ x k L , t ) 1 j } .

Then δ θ (Δ)(M(j,t))=0 since

K(j,t)K(j+1,t)
(3.1)

and

δ θ (Δ) ( K ( j , t ) ) =1
(3.2)

for t>0 and j=1,2, . Now we have to show that for kK(j,t) suppose that for some kK(j,t), x={ x k } not Δ-convergent to L with respect to the intuitionistic fuzzy n-norm ( μ , υ ) n . Therefore there is α>0 and a positive integer k 0 such that μ( y 1 , y 2 ,, y n 1 ,Δ x k L,t)1α or υ( y 1 , y 2 ,, y n 1 ,Δ x k L,t)α for all k k 0 . Let α> 1 j and

K ( α , t ) = { k N : μ ( y 1 , y 2 , , y n 1 , Δ x k L , t ) > 1 α and  υ ( y 1 , y 2 , , y n 1 , Δ x k L , t ) < α } .

Then δ θ (Δ)(K(α,t))=0. Since α> 1 j , by (3.1) we have δ θ (Δ)(K(j,t))=0, which contradicts by equation (3.2).

Sufficiency. Suppose that there exists an increasing sequence K={ k n } of the natural numbers such that δ θ (Δ)(K)=1 and ( μ , υ ) n - lim k K Δ x k =L, i.e., for every y 1 , y 2 ,, y n 1 X, ε>0 and t>0, there exists n 0 N such that μ( y 1 , y 2 ,, y n 1 ,Δ x k L,t)>1ε and υ( y 1 , y 2 ,, y n 1 ,Δ x k L,t)<ε.

Let

M ( ε , t ) : = { k N : μ ( y 1 , y 2 , , y n 1 , Δ x k L , t ) 1 ε or  υ ( y 1 , y 2 , , y n 1 , Δ x k L , t ) ε } { k n 0 + 1 , k n 0 + 2 , }

and consequently δ θ (Δ)(M(ε,t))11=0. Hence S θ ( μ , υ ) n (Δ)-limx=L. This completes proof of the theorem. □

Theorem 3.4 Let (X,μ,υ,,) be an IFnNLS. Then S θ ( μ , υ ) n (Δ)-limx=L if and only if there exist a convergent sequence y={ y k } and a lacunary Δ-statistically null sequence z={ z k } with respect to the intuitionistic fuzzy n-norm ( μ , υ ) n such that ( μ , υ ) n -limy=L, Δx=y+Δz and δ θ (Δ)({kN:Δ z k =0})=1.

Proof Necessity. Suppose that S θ ( μ , υ ) n (Δ)-limx=L and

K ( j , t ) = { k N : μ ( y 1 , y 2 , , y n 1 , Δ x k L , t ) > 1 1 j and  υ ( y 1 , y 2 , , y n 1 , Δ x k L , t ) < 1 j } .

Using Theorem 3.3 for any y 1 , y 2 ,, y n 1 X, t>0 and jN, we can construct an increasing index sequence { r j } of the natural numbers such that r j K(j,t), δ θ (Δ)(K(j,t))=1, and so we can conclude that for all r> r j (jN),

1 h r | { k I r : μ ( y 1 , y 2 , , y n 1 , Δ x k L , t ) > 1 1 j and  υ ( y 1 , y 2 , , y n 1 , Δ x k L , t ) < 1 j } | > j 1 j .

We define y={ y k } and z={ z k } as follows. If 1<k< r 1 , we set y k =Δ x k and z k =0. Now suppose that j1 and r j <k r j + 1 . If kK(j,t), i.e., μ( y 1 , y 2 ,, y n 1 ,Δ x k L,t)>1 1 j and υ( y 1 , y 2 ,, y n 1 ,Δ x k L,t)< 1 j , we set y k =Δ x k and Δ z k =0. Otherwise y k =L and Δ z k =Δ x k L. Hence it is clear that Δx=y+Δz.

We claim that ( μ , υ ) n -limy=L. Let ε> 1 j . If kK(j,t) for all k> r j , μ( y 1 , y 2 ,, y n 1 , y k L,t)>1ε and υ( y 1 , y 2 ,, y n 1 , y k L,t)<ε. Since ε was arbitrary, we have proved the claim.

Next we claim that z={ z k } is a lacunary Δ-statistically null sequence with respect to the intuitionistic fuzzy n-norm ( μ , υ ) n , i.e., S θ ( μ , υ ) n (Δ)-limz=0. It suffices to see that δ θ (Δ)({kN:Δ z k =0})=1 to prove the claim. This follows from observing that

| { k I r : Δ z k = 0 } | | { k I r : μ ( y 1 , y 2 , , y n 1 , Δ z k , t ) > 1 ε  and  υ ( y 1 , y 2 , , y n 1 , Δ z k , t ) < ε } |

for any rN and ε>0.

We show that if δ>0 and jN such that 1 j <δ, then

1 h r |{k I r :Δ z k =0}|>1δ

for all r> r j . Recall from the construction that if kK(j,t), then Δ z k =0 for r j <k r j + 1 .

Now, for t>0 and sN, let

K ( s , t ) = { k N : μ ( y 1 , y 2 , , y n 1 , Δ x k L , t ) > 1 1 s and  υ ( y 1 , y 2 , , y n 1 , Δ x k L , t ) < 1 s } .

For s>j and r s <k r s + 1 by (3.2),

K ( s , t ) = { k N : μ ( y 1 , y 2 , , y n 1 , Δ x k L , t ) > 1 1 s and  υ ( y 1 , y 2 , , y n 1 , Δ x k L , t ) < 1 s } { k N : Δ z k = 0 } .

Consequently, if r s <k r s + 1 and s>j, then

1 h r | { k I r : Δ z k = 0 } | 1 h r | { k I r : μ ( y 1 , y 2 , , y n 1 , Δ x k L , t ) > 1 1 s and  υ ( y 1 , y 2 , , y n 1 , Δ x k L , t ) < 1 s } | > 1 1 s > 1 1 j > 1 δ .

Hence we get δ θ (Δ)({kN:Δ z k =0})=1, which establishes the claim.

Sufficiency. Let x, y and z be sequences such that ( μ , υ ) n -limy=L, Δx=y+Δz and δ θ (Δ)({kN:Δ z k =0})=1. Then, for any y 1 , y 2 ,, y n 1 X, ε>0 and t>0, we have

{ k N : μ ( y 1 , y 2 , , y n 1 , Δ x k L , t ) 1 ε  or  υ ( y 1 , y 2 , , y n 1 , Δ x k L , t ) ε } { k N : μ ( y 1 , y 2 , , y n 1 , y k L , t ) 1 ε  or  υ ( y 1 , y 2 , , y n 1 , y k L , t ) ε } { k N : Δ z k 0 } .

Therefore

δ θ ( Δ ) ( { k N : μ ( y 1 , y 2 , , y n 1 , Δ x k L , t ) 1 ε or  υ ( y 1 , y 2 , , y n 1 , Δ x k L , t ) ε } ) δ θ ( { k N : μ ( y 1 , y 2 , , y n 1 , y k L , t ) 1 ε  or  υ ( y 1 , y 2 , , y n 1 , y k L , t ) ε } ) + δ θ ( Δ ) ( { k N : Δ z k 0 } ) .

Since ( μ , υ ) n -limy=L, the set

{ k N : μ ( y 1 , y 2 , , y n 1 , y k L , t ) 1 ε  or  υ ( y 1 , y 2 , , y n 1 , y k L , t ) ε }

contains at most finitely many terms and thus

δ θ ( { k N : μ ( y 1 , y 2 , , y n 1 , y k L , t ) 1 ε  or  υ ( y 1 , y 2 , , y n 1 , y k L , t ) ε } ) .

Also by hypothesis, δ θ (Δ)({kN:Δ z k 0}). Hence,

δ θ ( Δ ) ( { k N : μ ( y 1 , y 2 , , y n 1 , Δ x k L , t ) 1 ε or  υ ( y 1 , y 2 , , y n 1 , Δ x k L , t ) ε } ) = 0 ,

and consequently S θ ( μ , υ ) n (Δ)-limx=L. □

4 Δ-Cauchy and lacunary Δ-statistically Cauchy sequences in IFnNLS

In this section, we introduce the notion of Cauchy sequences and lacunary statistically Cauchy sequences in IFnNLS.

Definition 4.1 Let (X,μ,υ,,) be an IFnNLS. A sequence x={ x k } in X is said to be Δ-Cauchy with respect to the intuitionistic fuzzy n-norm ( μ , υ ) n if, for every ε>0, t>0 and y 1 , y 2 ,, y n 1 X, there exists k 0 N such that μ( y 1 , y 2 ,, y n 1 ,Δ x k Δ x m ,t)>1ε and υ( y 1 , y 2 ,, y n 1 ,Δ x k Δ x m ,t)<ε for all k,m k 0 .

Definition 4.2 Let (X,μ,υ,,) be an IFnNLS. A sequence x={ x k } in X is said to be lacunary Δ-statistically Cauchy or S θ (Δ)-Cauchy with respect to the intuitionistic fuzzy n-norm ( μ , υ ) n if, for every ε>0, t>0 and y 1 , y 2 ,, y n 1 X, there exists a number mN satisfying

δ θ ( Δ ) ( { k N : μ ( y 1 , y 2 , , y n 1 , Δ x k Δ x m , t ) 1 ε or  υ ( y 1 , y 2 , , y n 1 , Δ x k Δ x m , t ) ε } ) = 0 .

Theorem 4.1 Let (X,μ,υ,,) be an IFnNLS. If a sequence x={ x k } in X is lacunary Δ-statistically convergent with respect to the intuitionistic fuzzy n-norm ( μ , υ ) n if and only if it is lacunary Δ-statistically Cauchy with respect to the intuitionistic fuzzy n-norm ( μ , υ ) n .

Proof Let x={ x k } be a lacunary Δ-statistically convergent sequence which converges to L. For a given ε>0, choose s>0 such that (1ε)(1ε)>1s and εε<s. Let

A ( ε , t ) = { k N : μ ( y 1 , y 2 , , y n 1 , Δ x k L , t / 2 ) 1 ε or  υ ( y 1 , y 2 , , y n 1 , Δ x k L , t / 2 ) ε } .

Then, for any t>0 and y 1 , y 2 ,, y n 1 X,

δ θ (Δ) ( A ( ε , t ) ) =0,
(4.1)

which implies that δ θ (Δ)( A c (ε,t))=1.

Let q A c (ε,t). Then

μ( y 1 , y 2 ,, y n 1 ,Δ x q L,t/2)>1ε

and

υ( y 1 , y 2 ,, y n 1 ,Δ x k L,t/2)<ε.

Now, let

B ( s , t ) = { k N : μ ( y 1 , y 2 , , y n 1 , Δ x k Δ x q , t ) 1 s or  υ ( y 1 , y 2 , , y n 1 , Δ x k Δ x q , t ) s } .

We need to show that B(s,t)A(ε,t). Let kB(s,t) A c (ε,t). Hence μ( y 1 , y 2 ,, y n 1 ,Δ x k Δ x q ,t)1s and μ( y 1 , y 2 ,, y n 1 ,Δ x k L,t/2)>1ε, in particular, μ( y 1 , y 2 ,, y n 1 ,Δ x q L,t/2)>1ε. Then

1 s μ ( y 1 , y 2 , , y n 1 , Δ x k Δ x q , t ) μ ( y 1 , y 2 , , y n 1 , Δ x k L , t / 2 ) μ ( y 1 , y 2 , , y n 1 , Δ x q L , t / 2 ) > ( 1 ε ) ( 1 ε ) > 1 s ,

which is not possible. On the other hand, υ( y 1 , y 2 ,, y n 1 ,Δ x k Δ x q ,t)s and υ( y 1 , y 2 ,, y n 1 ,Δ x k L,t/2)<ε, in particular, υ( y 1 , y 2 ,, y n 1 ,Δ x q L,t/2)<ε. Hence,

s υ ( y 1 , y 2 , , y n 1 , Δ x k Δ x q , t ) υ ( y 1 , y 2 , , y n 1 , Δ x k L , t / 2 ) υ ( y 1 , y 2 , , y n 1 , Δ x q L , t / 2 ) < ε ε < s ,

which is not possible. Hence B(s,t)A(ε,t) and by (4.1) δ θ (Δ)(B(ε,t))=0. This proves that x is lacunary Δ-statistically Cauchy with respect to the intuitionistic fuzzy n-norm ( μ , υ ) n .

Conversely, let x={ x k } be lacunary Δ-statistically Cauchy but not lacunary Δ-statistically convergent with respect to the intuitionistic fuzzy n-norm ( μ , υ ) n . For a given ε>0, choose s>0 such that (1ε)(1ε)>1s and εε<s. Since x is not lacunary Δ-convergent

μ ( y 1 , y 2 , , y n 1 , Δ x k Δ x m , t ) μ ( y 1 , y 2 , , y n 1 , Δ x k L , t / 2 ) μ ( y 1 , y 2 , , y n 1 , Δ x q L , t / 2 ) > ( 1 ε ) ( 1 ε ) > 1 s , υ ( y 1 , y 2 , , y n 1 , Δ x k Δ x m , t ) υ ( y 1 , y 2 , , y n 1 , Δ x k L , t / 2 ) υ ( y 1 , y 2 , , y n 1 , Δ x q L , t / 2 ) < ε ε < s .

Therefore δ θ (Δ)( E c (s,t))=0, where

B ( s , t ) = { k N : μ ( y 1 , y 2 , , y n 1 , Δ x k Δ x q , t ) 1 s or  υ ( y 1 , y 2 , , y n 1 , Δ x k Δ x q , t ) s }

and so δ θ (Δ)(E(s,t))=1, which is a contradiction, since x was lacunary Δ-statistically Cauchy with respect to the intuitionistic fuzzy n-norm ( μ , υ ) n . So, x must be lacunary Δ-statistically convergent with respect to the intuitionistic fuzzy n-norm ( μ , υ ) n . □

Corollary 4.1 Let (X,μ,υ,,) be an IFnNLS and θ be a lacunary sequence. Then, for any sequence x={ x k } in X, the following conditions are equivalent:

  1. (i)

    x is S θ (Δ)-convergent with respect to the intuitionistic fuzzy n-norm ( μ , υ ) n .

  2. (ii)

    x is S θ (Δ)-Cauchy with respect to the intuitionistic fuzzy n-norm ( μ , υ ) n .

  3. (iii)

    There exists an increasing sequence K={ k n } of the natural numbers such that δ θ (Δ)(K)=1 and the subsequence { x k n } is S θ (Δ)-Cauchy with respect to the intuitionistic fuzzy n-norm ( μ , υ ) n .

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Altundağ, S., Kamber, E. Lacunary Δ-statistical convergence in intuitionistic fuzzy n-normed space. J Inequal Appl 2014, 40 (2014). https://doi.org/10.1186/1029-242X-2014-40

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Keywords

  • statistical convergence
  • lacunary sequence
  • difference sequence
  • intuitionistic fuzzy n-normed space