Open Access

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

Journal of Inequalities and Applications20142014:40

https://doi.org/10.1186/1029-242X-2014-40

Received: 8 October 2013

Accepted: 26 December 2013

Published: 24 January 2014

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.

Keywords

statistical convergencelacunary sequencedifference sequenceintuitionistic fuzzy n-normed space

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 Δ x = ( 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)

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

     
  4. (iv)

    a b c d whenever a c and b d 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)

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

     
  4. (iv)

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

     

Definition 2.3 ([27])

Let n N and X be a real vector space of dimension d n (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 α R ,

     
  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 c 0 , c F ,

     
  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 c 0 , c F ,

     
  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 a b = a b and a b = 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 and υ ( 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 K N . The number
δ θ ( K ) = lim r 1 h r | { k I r : k K } |

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 - lim x = 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 L X 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 ) > 1 ε and υ ( y 1 , y 2 , , y n 1 , Δ x k L , t ) < ε for all k k 0 , where k N and Δ x k = ( x k x k + 1 ) . It is denoted by ( μ , υ ) n - lim Δ x = 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 L X 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 ) - lim x = 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 ) - lim x = L ,

     
  2. (ii)

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

     
  3. (iii)

    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 ,

     
  4. (iv)

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

     
  5. (v)

    S θ - lim μ ( y 1 , y 2 , , y n 1 , Δ x k L , t ) = 1 and S θ - lim υ ( 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 L X with respect to the intuitionistic fuzzy n-norm ( μ , υ ) n , S θ ( μ , υ ) n ( Δ ) - lim x is unique.

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

Proof Let ( μ , υ ) n - lim Δ x = 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 ) > 1 ε 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 ) - lim x = 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 = a b s ( | 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 a b = a b , a b = 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 : k K ( ε , t ) } | [ h r ] h r 0 as r . Hence S θ ( μ , υ ) n ( Δ ) - lim x = 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 ) - lim x = 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 ) - lim x = 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 k K ( j , t ) suppose that for some k K ( 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 ) 1 α 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 ) > 1 ε 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 ) ) 1 1 = 0 . Hence S θ ( μ , υ ) n ( Δ ) - lim x = L . This completes proof of the theorem. □

Theorem 3.4 Let ( X , μ , υ , , ) be an IFnNLS. Then S θ ( μ , υ ) n ( Δ ) - lim x = 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 - lim y = L , Δ x = y + Δ z and δ θ ( Δ ) ( { k N : Δ z k = 0 } ) = 1 .

Proof Necessity. Suppose that S θ ( μ , υ ) n ( Δ ) - lim x = 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 j N , 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 ( j N ),
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 j 1 and r j < k r j + 1 . If k K ( 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 Δ x = y + Δ z .

We claim that ( μ , υ ) n - lim y = L . Let ε > 1 j . If k K ( j , t ) for all k > r j , μ ( y 1 , y 2 , , y n 1 , y k L , t ) > 1 ε 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 ) - lim z = 0 . It suffices to see that δ θ ( Δ ) ( { k N : 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 r N and ε > 0 .

We show that if δ > 0 and j N such that 1 j < δ , then
1 h r | { k I r : Δ z k = 0 } | > 1 δ

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

Now, for t > 0 and s N , 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 ) ( { k N : Δ z k = 0 } ) = 1 , which establishes the claim.

Sufficiency. Let x, y and z be sequences such that ( μ , υ ) n - lim y = L , Δ x = y + Δ z and δ θ ( Δ ) ( { k N : 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 - lim y = 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 ) ( { k N : 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 ) - lim x = 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 ) > 1 ε 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 m N 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 ( 1 ε ) ( 1 ε ) > 1 s and ε ε < 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 ) > 1 ε
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 k B ( s , t ) A c ( ε , t ) . Hence μ ( y 1 , y 2 , , y n 1 , Δ x k Δ x q , t ) 1 s and μ ( y 1 , y 2 , , y n 1 , Δ x k L , t / 2 ) > 1 ε , in particular, μ ( y 1 , y 2 , , y n 1 , Δ x q L , t / 2 ) > 1 ε . 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 ( 1 ε ) ( 1 ε ) > 1 s and ε ε < 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 .

     

Declarations

Acknowledgements

The authors would like to thank the referees for their careful reading of the manuscript and for their suggestions.

Authors’ Affiliations

(1)
Department of Mathematics, Sakarya University

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© Altundağ and Kamber; licensee Springer. 2014

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