Open Access

Generalized lacunary Δ m -statistically convergent sequences of fuzzy numbers using a modulus function

Journal of Inequalities and Applications20132013:559

https://doi.org/10.1186/1029-242X-2013-559

Received: 12 May 2013

Accepted: 16 October 2013

Published: 25 November 2013

Abstract

In this paper, we introduce the space of lacunary strongly Δ ( p ) m -summable sequences of fuzzy numbers and discuss relations between Δ m -statistically convergent sequences and lacunary Δ m -statistically convergent sequences of fuzzy numbers. We also study inclusion relations using different arbitrary lacunary sequences.

MSC:40A05, 40C05, 46A45.

Keywords

fuzzy number modulus function lacunary Δ m -statistical convergence lacunary refinement

1 Introduction

The concepts of fuzzy sets and fuzzy set operations were first introduced by Zadeh [1], and subsequently several authors have discussed various aspects of the theory and applications of fuzzy sets such as topological spaces, similarity relations and fuzzy orderings, fuzzy mathematical programming etc. Later on, various types of sequence spaces of fuzzy numbers have been constructed by several authors such as Matloka [2], Nanda [3], Nuray and Savaş [4], Mursaleen and Basarır [5], Malkowsky et al. [6], Tripathy and Chandra [7], Tripathy and Borgogain [8] and so on. Later on, the fuzzy sequence space got momentum after the introduction of new convergence methods and theories in the process as well as the requirement. Some of them are statistical convergence, lacunary statistical convergence etc.

Nuray and Savaş [4] introduced the idea of statistical convergence of fuzzy numbers and Nuray [9] introduced the related concept of convergence with the help of a lacunary sequence. Using their ideas, many authors such as Kwon and Shim [10], Bilgin [11], Altin et al. [12], Esi [13], Tripathy and Baruah [14, 15], Tripathy and Dutta [16, 17] and others constructed different types of sequence spaces.

2 Definitions and preliminaries

Definition 2.1 A fuzzy number X is a mapping X : R [ 0 , 1 ] associating each real number t with its grade of membership X ( t ) .

Definition 2.2 If there exists t R such that X ( t ) = 1 , then the fuzzy number X is called normal.

Definition 2.3 A fuzzy number X is said to be convex if X ( t ) X ( s ) X ( r ) = min { X ( s ) , X ( r ) } , where s < t < r .

Definition 2.4 A fuzzy number X is said to be upper semi-continuous if for each ε > 0 , X 1 ( [ 0 , a + ε ) ) for all a [ 0 , 1 ] is open in the usual topology of .

L ( R ) denotes the set of all upper semi-continuous, normal, convex fuzzy numbers such that [ X ] α = { t R : X ( t ) > 0 } ¯ is compact, where { t R : X ( t ) > 0 } ¯ denotes the closure of the set { t R : X ( t ) > 0 } in the usual topology of .

Definition 2.5 The set L ( R ) forms a linear space under addition and scalar multiplication in terms of α-level sets as defined below:
[ X + Y ] α = [ X ] α + [ Y ] α and [ λ X ] α = λ [ X ] α for each  0 α 1 ,
where X α is given as
X α = { t : X ( t ) α if  α ( 0 , 1 ] , t : X ( t ) > 0 if  α = 0 .

For each α [ 0 , 1 ] , the set X α is a closed, bounded and nonempty interval of .

Let D denote the set of all closed and bounded intervals A = [ a 1 , a 2 ] on the real line . For A , B D , ( D , d ) is a complete metric space where the metric d is defined as
d ( A , B ) = max { | a 1 b 1 | , | a 2 b 2 | }

for any A = [ a 1 , a 2 ] and B = [ b 1 , b 2 ] .

It is easy to verify that d ¯ : L ( R ) × L ( R ) R , defined by
d ¯ ( X , Y ) = sup 0 α 1 d ( X α , Y α ) ,

is a metric on L ( R ) .

Definition 2.6 Let X = ( X k ) be a sequence of fuzzy numbers. Then the sequence X = ( X k ) is said to be Δ m -convergent to the fuzzy number X 0 , denoted as lim k Δ m X k = X 0 , if for every ε > 0 , there exists a positive integer k 0 such that d ¯ ( Δ m X k , X 0 ) < ε for all k > k 0 .

Definition 2.7 The sequence X = ( ( X k ) k = 1 ) of fuzzy numbers is said to be Δ m -statistically convergent to the fuzzy number ξ if for every ε > 0 ,
lim n 1 n | { k n : d ¯ ( Δ m X k , ξ ) ε } | = 0 .

In this case, we write X k ξ ( S F ( Δ m ) ) and S F ( Δ m ) denotes the set of all Δ m -statistically convergent sequences of fuzzy numbers.

By a lacunary sequence we mean an increasing sequence of integers θ = ( k r ) such that k 0 = 0 and h r = k r k r 1 as r . Throughout this paper, the intervals determined by θ will be denoted by I r = ( k r 1 , k r ] and q r = k r k r 1 .

Definition 2.8 Let θ be a lacunary sequence. A lacunary refinement of θ = ( k r ) is a lacunary sequence θ = ( k r ) satisfying { k r } { k r } .

Definition 2.9 The sequence X = ( X k ) of fuzzy numbers is said to be lacunary Δ m -statistically convergent to the fuzzy number ξ if for every ε > 0 ,
lim r 1 h r | { k I r : d ¯ ( Δ m X k , ξ ) ε } | = 0 .

In this case, we write X k ξ ( S θ F ( Δ m ) ) , and S θ F ( Δ m ) denotes the set of all lacunary Δ m -statistically convergent sequences of fuzzy numbers.

Definition 2.10 A metric d ¯ on L ( R ) is said to be translation invariant if d ¯ ( X + Z , Y + Z ) = d ¯ ( X , Y ) for all X , Y , Z L ( R ) .

Lemma 2.1 (Mursaleen and Basarır [5])

If d ¯ is a translation invariant metric on L ( R ) , then
  1. (i)

    d ¯ ( X + Y , 0 ¯ ) d ¯ ( X , 0 ¯ ) + d ¯ ( Y , 0 ¯ ) ,

     
  2. (ii)

    d ¯ ( λ X , 0 ¯ ) | λ | d ¯ ( X , 0 ¯ ) , | λ | > 1 .

     

Lemma 2.2 (Maddox [18])

Let a k , b k for all k be sequences of complex numbers, and let ( p k ) be a bounded sequence of positive real numbers, then
| a k + b k | p k C ( | a k | p k + | b k | p k )
and
| λ | p k max ( 1 , | λ | H ) ,

where C = max ( 1 , 2 H 1 ) , H = sup p k and λ is any complex number.

Lemma 2.3 (Maddox [18])

Let a k 0 , b k 0 for all k be sequences of complex numbers and 1 p k sup p k < , then
( k | a k + b k | p k ) 1 M ( k | a k | p k ) 1 M + ( k | b k | p k ) 1 M ,

where M = max ( 1 , H ) , H = sup p k .

3 Main results

Now, we introduce the space N θ F ( Δ ( p ) m , f ) as follows:
N θ F ( Δ ( p ) m , f ) = { X = ( X k ) : X k L ( R )  such that  lim r 1 h r k I r ( f ( d ¯ ( Δ m X k , ξ ) ) ) p k = 0 } ,

where θ = ( k r ) is a lacunary sequence, f is any modulus function and p = ( p k ) is any sequence of strictly positive real numbers.

Now we define a lacunary strongly Δ p m -summable sequence of fuzzy numbers as follows.

The sequence X = ( X k ) is said to be lacunary strongly Δ ( p ) m -summable to the fuzzy number ξ L ( R ) if for every ε > 0 ,
lim r 1 h r k I r ( f ( d ¯ ( Δ m X k , ξ ) ) ) p k = 0 .

In this case, we write X k ξ ( N θ F ( Δ ( p ) m , f ) ) . Thus the class N θ F ( Δ ( p ) m , f ) denotes the set of all lacunary strongly Δ ( p ) m -summable sequences of fuzzy numbers.

For suitable choices of m, f and p k , some of the known sequence spaces, which are obtained from the space N θ F ( Δ ( p ) m , f ) , are as follows.
  1. (i)

    Let f ( x ) = x , m = 0 and p k p for all k N , the sequence space N θ F ( Δ ( p ) m , f ) represents the space N θ studied by Kwon and Shim [10].

     
  2. (ii)

    Let f ( x ) = x , m = 1 and p k 1 for all k N , the sequence space N θ F ( Δ ( p ) m , f ) denotes the space N θ ( Δ ) investigated by Bilgin [11].

     
  3. (iii)

    Let f ( x ) = x , the sequence space N θ F ( Δ ( p ) m , f ) denotes the space N θ ( Δ p m ) studied by Altin et al. [12].

     
  4. (iv)

    Let f ( x ) = x and p k 1 for all k N , the sequence space N θ F ( Δ ( p ) m , f ) reduces to the space N θ ( Δ m , F ) investigated by Esi [13].

     

Thus, the study of the sequence space N θ F ( Δ ( p ) m , f ) gives a unified approach to many of the earlier known spaces.

Theorem 3.1 Let ( p k ) be a bounded sequence of positive real numbers. Then the class N θ F ( Δ ( p ) m , f ) is a linear space over .

Proof Using Lemma 2.1, Lemma 2.2, the subadditivity property of a modulus function f and the result f ( λ x ) ( 1 + [ | λ | ] ) f ( x ) , it is easy to show that N θ F ( Δ ( p ) m , f ) is a linear space over the real field . □

Theorem 3.2 Let ( p k ) be a bounded sequence of positive real numbers such that inf p k > 0 . Then the sequence space N θ F ( Δ ( p ) m , f ) is a complete metric space with respect to the metric
h ( X , Y ) = i = 1 m f ( d ¯ ( X i , Y i ) ) + sup r ( 1 h r k I r ( f ( d ¯ ( Δ m X k , Δ m Y k ) ) ) p k ) 1 M ,

where M = max { 1 , sup p k } .

Proof It is easy to verify that h gives a metric on N θ F ( Δ ( p ) m , f ) . To show completeness, let us assume that ( X u ) is a Cauchy sequence in N θ F ( Δ ( p ) m , f ) , where X u = ( X k u ) k = 1 N θ F ( Δ ( p ) m , f ) for each u N . So, for given ε > 0 , there exists u 0 N such that
g ( X u , X v ) < ε for all  u , v u 0 ,
i.e.,
i = 1 m f ( d ¯ ( X i u , X i v ) ) + sup r ( 1 h r k I r ( f ( d ¯ ( Δ m X k u , Δ m X k v ) ) ) p k ) 1 M < ε for all  u , v u 0 .
This means that
i = 1 m f ( d ¯ ( X i u , X i v ) ) < ε for all  u , v u 0
(3.1)
as well as
1 h r k I r ( f ( d ¯ ( Δ m X k u , Δ m X k v ) ) ) p k < ε for all  u , v u 0  and for all  r .
(3.2)
Since f is a modulus function, so equation (3.1) gives d ¯ ( X i u , X i v ) < ε 1 for all u , v u 0 , for some ε 1 such that ε > ε 1 > 0 and for each i = 1 , 2 , , m , i.e.,
( X i u )  is a Cauchy sequence in  L ( R ) for each  i = 1 , 2 , , m .
(3.3)
Similarly, as f is a modulus function, so by choosing suitable ε 2 > 0 , equation (3.2) gives d ¯ ( Δ m X k u , Δ m X k v ) < ε 2 for all u , v u 0 and for each k, i.e.,
( Δ m X k u )  is a Cauchy sequence in  L ( R ) for each  k N .
(3.4)

Using equation (3.3) and equation (3.4), it can be easily shown that ( X k u ) is a Cauchy sequence in L ( R ) . But L ( R ) is complete, so the sequence ( X k u ) converges to X = ( X k ) in L ( R ) as u .

Keeping u fixed and letting v in equation (3.1) and equation (3.2), we get
i = 1 m f ( d ¯ ( X i u , X i ) ) < ε for all  u u 0
and
1 h r k I r ( f ( d ¯ ( Δ m X k u , Δ m X k ) ) ) p k < ε for all  u u 0  and for all  r ,
(3.5)
i.e.,
g ( X u , X ) < ε for all  u u 0 .

Next we show that the limit point X N θ F ( Δ ( p ) m , f ) , for which the proof is as follows.

Since X u = ( X k u ) N θ F ( Δ ( p ) m , f ) for u N , so for each u, there exist L u L ( R ) and r u N such that
1 h r k I r ( f ( d ¯ ( Δ m X k u , L u ) ) ) p k < ε for all  r r u .
(3.6)
Similarly, for each v, there exist L v L ( R ) and r v N such that
1 h r k I r ( f ( d ¯ ( Δ m X k v , L v ) ) ) p k < ε for all  r r v .
(3.7)
Now let u , v u 0 and r 0 = max ( r u , r v ) . Then from equation (3.2), equation (3.6), equation (3.7) and using Lemma 2.2, we have
1 h r k I r ( f ( d ¯ ( L u , L v ) ) ) p k C 1 h r k I r ( f ( d ¯ ( L u , Δ m X k u ) ) ) p k + C 1 h r k I r ( f ( d ¯ ( Δ m X k u , Δ m X k v ) ) ) p k + C 1 h r k I r ( f ( d ¯ ( Δ m X k v , L v ) ) ) p k < 3 C ε for all  u , v u 0  and for all  r r 0 .
(3.8)
Now using the fact that the modulus function is monotone and for a suitable choice of ε 3 > 0 , we get
d ¯ ( L u , L v ) < ε 3 for all  u , v u 0 ,
i.e., ( L u ) is a Cauchy sequence in L ( R ) . Since L ( R ) is complete, so there exists ξ L ( R ) such that L u ξ as u . Keeping u fixed and letting v in equation (3.8), we get
1 h r k I r ( f ( d ¯ ( L u , ξ ) ) ) p k < 3 C ε for  u u 0  and for  r r 0 .
(3.9)
Hence, from equation (3.5), equation (3.6) and equation (3.9), we have
1 h r k I r ( f ( d ¯ ( Δ m X k , ξ ) ) ) p k C 1 h r k I r ( f ( d ¯ ( Δ m X k , Δ m X k u 0 ) ) ) p k + C 1 h r k I r ( f ( d ¯ ( Δ m X k u 0 , L u 0 ) ) ) p k + C 1 h r k I r ( f ( d ¯ ( L u 0 , ξ ) ) ) p k < 2 C ε + 3 C 2 ε ε 1 for all  r r 0 ,

which implies 1 h r k I r ( f ( d ¯ ( Δ m X k , ξ ) ) ) p k 0 as r , i.e., X = ( X k ) N θ F ( Δ ( p ) m , f ) and hence the sequence space N θ F ( Δ ( p ) m , f ) is a complete metric space. □

Theorem 3.3 Let θ = ( k r ) be a lacunary sequence and X = ( X k ) be a sequence of fuzzy numbers. Then:
  1. (i)

    X k ξ ( N θ F ( Δ ( p ) m , f ) ) implies X k ξ ( S θ F ( Δ m ) ) .

     
  2. (ii)

    If f is a bounded modulus function and X k ξ ( S θ F ( Δ m ) ) , then X k ξ ( N θ F ( Δ ( p ) m , f ) ) .

     

Proof Easy, so omitted. □

Theorem 3.4 Let θ = ( k r ) be a lacunary sequence, and let X = ( X k ) be a sequence of fuzzy numbers. Then:
  1. (i)

    W F ( Δ m , f ) N θ F ( Δ m , f ) if and only if lim inf r q r > 1 .

     
  2. (ii)

    N θ F ( Δ m , f ) W F ( Δ m , f ) if and only if lim sup r q r < ,

     
where
W F ( Δ m , f ) = { ( X k ) : 1 n k = 1 n f ( d ¯ ( Δ m X k , ξ ) ) 0  as  n }
and
N θ F ( Δ m , f ) = { ( X k ) : 1 h r k I r f ( d ¯ ( Δ m X k , ξ ) ) 0  as  r } .

Proof (i) Assume that lim inf r q r > 1 . Then there exists δ > 0 such that q r 1 + δ for sufficiently large r.

Let X k ξ ( W F ( Δ m , f ) ) . Then
1 n k = 1 n f ( d ¯ ( Δ m X k , ξ ) ) 0 as  n .
Consider
1 h r k I r f ( d ¯ ( Δ m X k , ξ ) ) = 1 h r k = 1 k r f ( d ¯ ( Δ m X k , ξ ) ) 1 h r k = 1 k r 1 f ( d ¯ ( Δ m X k , ξ ) ) k r h r ( 1 k r k = 1 k r f ( d ¯ ( Δ m X k , ξ ) ) ) k r 1 h r ( 1 k r 1 k = 1 k r 1 f ( d ¯ ( Δ m X k , ξ ) ) ) .
Since h r = k r k r 1 , we have
k r h r = k r k r k r 1 = q r q r 1 = 1 + 1 q r 1 1 + 1 δ = δ + 1 δ
and
k r 1 h r = k r 1 k r k r 1 = 1 q r 1 1 δ .

Since X W F ( Δ m , f ) . So both the terms 1 k r k = 1 k r f ( d ¯ ( Δ m X k , ξ ) ) and 1 k r 1 k = 1 k r 1 f ( d ¯ ( Δ m X k , ξ ) ) converge to 0 as r and hence 1 h r k r I r f ( d ¯ ( Δ m X k , ξ ) ) converges to 0 as r , i.e., X N θ F ( Δ m , f ) and hence W F ( Δ m , f ) N θ F ( Δ m , f ) .

Conversely, let X k ξ ( W F ( Δ m , f ) ) imply X k ξ ( N θ F ( Δ m , f ) ) but lim inf r q r = 1 .

Since θ is a lacunary sequence, we can select a subsequence ( k r ( j ) ) of θ satisfying
k r ( j ) k r ( j ) 1 < 1 + 1 j and k r ( j ) 1 k r ( j 1 ) > j , where  r ( j ) r ( j 1 ) + 2 .
Define
Δ m X k = { 1 ¯ if  k I r ( j )  for some  j = 1 , 2 , , 0 ¯ otherwise .
Now we prove that the sequence X for which Δ m X k , which is defined as above, is strongly Δ m -summable but not lacunary strongly Δ m -summable. Since
1 h r ( j ) k I r ( j ) f ( d ¯ ( Δ m X k , 0 ¯ ) ) = f ( 1 ) for  r = r ( j )
and
1 h r k I r f ( d ¯ ( Δ m X k , 0 ¯ ) ) = 0 for  r r ( j ) ,
which implies X k 0 ¯ ( N θ F ( Δ m , f ) ) , but if n is sufficiently large integer and j is the unique integer with k r ( j ) 1 < n k r ( j + 1 ) 1 , then, since n implies j , we have
1 n k = 1 n f ( d ¯ ( Δ m X k , 0 ¯ ) ) k r ( j ) k r ( j ) 1 k r ( j ) 1 + h r ( j ) k r ( j ) 1 2 j 0 as  n ,

i.e., X k 0 ¯ ( W F ( Δ m , f ) ) . This shows that W F ( Δ m , f ) N θ F ( Δ m , f ) , which leads to a contradiction. Therefore it proves that lim inf q r > 1 .

(ii) (Sufficiency) Suppose lim sup r q r < . Then there exists a constant A > 0 such that q r < A for all r N . Let X k ξ ( N θ F ( Δ m , f ) ) . So lim r 1 h r k I r f ( d ¯ ( Δ m X k , ξ ) ) = 0 .

Let ε > 0 . Then we can find R > 0 and K > 0 such that
sup r R 1 h r r I r f ( d ¯ ( Δ m X k , ξ ) ) < ε
and
1 h r i I j f ( d ¯ ( Δ m X k , ξ ) ) < K for all  j = 1 , 2 , .
Then if n is any integer with k r 1 < n k r , where r > R , then we can write
1 n k = 1 n f ( d ¯ ( Δ m X k , ξ ) ) 1 k r 1 k = 1 k r f ( d ¯ ( Δ m X k , ξ ) ) = 1 k r 1 ( I 1 f ( d ¯ ( Δ m X k , ξ ) ) + I 2 f ( d ¯ ( Δ m X k , ξ ) ) + + I r f ( d ¯ ( Δ m X k , ξ ) ) ) , where  I r = ( k r 1 , k r ] = k 1 k r 1 1 h 1 I 1 f ( d ¯ ( Δ m X k , ξ ) ) + k 2 k 1 k r 1 1 h 2 I 2 f ( d ¯ ( Δ m X k , ξ ) ) + + k R k R 1 k r 1 1 h R I R f ( d ¯ ( Δ m X k , ξ ) ) + + k r k r 1 k r 1 1 h r I r f ( d ¯ ( Δ m X k , ξ ) ) k r k r 1 ( sup 1 i R 1 h i i f ( d ¯ ( Δ m X k , ξ ) ) ) + k r k R k r 1 ( sup i R 1 h i i f ( d ¯ ( Δ m X k , ξ ) ) ) < K k r k r 1 + ε ( q r k R k r 1 ) < K k r k r 1 + ε q r < K k r k r 1 + A ε ,

which implies 1 n k = 1 n f ( d ¯ ( Δ m X k , ξ ) ) 0 as n and it follows that X k ξ ( W F ( Δ m , f ) ) .

(Necessity) Suppose that X k ξ ( W θ F ( Δ m , f ) ) implies X k ξ ( N F ( Δ m , f ) ) but lim sup r q r = .

Since θ is any lacunary sequence, we can select a subsequence ( k r ( j ) ) of θ satisfying q r ( j ) > j and define a bounded difference sequence X = ( X k ) by
Δ m X k = { 1 ¯ if  k r ( j ) 1 < k 2 k r ( j ) 1  for some  j = 1 , 2 , , 0 ¯ otherwise .
Let ε > 0 be given. Then we have if r = r ( j ) , then
1 h r ( j ) I r ( j ) f ( d ¯ ( Δ m X k , 0 ¯ ) ) = 2 k r ( j ) 1 k r ( j ) 1 k r ( j ) k r ( j ) 1 f ( 1 ) = k r ( j ) 1 k r ( j ) k r ( j ) 1 f ( 1 ) < f ( 1 ) j 1 ,
and if r r ( j ) , then
1 h r ( j ) I r ( j ) f ( d ¯ ( Δ m X k , 0 ¯ ) ) = 0 ,

which implies that X k 0 ¯ ( N θ F ( Δ m , f ) ) .

For the above sequence and for k = 1 , 2 , , k r ( j ) ,
1 k r ( j ) k = 1 k r ( j ) f ( d ¯ ( Δ m X k , 1 ¯ ) ) 1 k r ( j ) ( k r ( j ) 2 k r ( j ) 1 ) f ( 1 ) = ( 1 2 q r ( j ) ) f ( 1 ) > ( 1 2 j ) f ( 1 ) ,
and for k = 1 , 2 , , 2 k r ( j ) 1 ,
1 2 k r ( j ) 1 k = 1 2 k r ( j ) 1 f ( d ¯ ( Δ m X k , 1 ¯ ) ) k r ( j ) 1 2 k r ( j ) 1 f ( 1 ) = f ( 1 ) 2 .

It proves that X = ( X k ) ( W F ( Δ m , f ) ) and hence the result follows immediately. □

Theorem 3.5 Let θ = ( k r ) be a lacunary sequence and X = ( X k ) be a sequence of fuzzy numbers. Then:
  1. (i)

    S θ F ( Δ m ) S F ( Δ m ) if and only if lim sup r q r < .

     
  2. (ii)

    S F ( Δ m ) S θ F ( Δ m ) if and only if lim inf r q r > 1 .

     

Proof The proof can be established following the technique used in Theorem 3.4. □

Theorem 3.6 If θ = ( k r ) is a lacunary refinement of θ = ( k r ) and X k ξ ( S θ F ( Δ m ) ) , then X k ξ ( S θ F ( Δ m ) ) .

Theorem 3.7 Let β = ( l i ) be a lacunary refinement of the lacunary sequence θ = ( k i ) . Let I i = ( k i 1 , k i ] and J i = ( l i 1 , l i ] , i = 1 , 2 , 3 ,  , be the corresponding intervals of θ and β, respectively, and h i = k i k i 1 and g i = l i l i 1 , respectively. If there exists δ > 0 such that
g j h i δ for every  J j I i ,

then X k ξ ( S θ F ( Δ m ) ) implies X k ξ ( S β F ( Δ m ) ) .

Proof Let ε > 0 be given, X k ξ ( S θ F ( Δ m ) ) and for every J j , we can find I i such that J j I i , then we have
1 g j | { k J j : d ¯ ( Δ m X k , ξ ) ε } | = h i g j 1 h i | { k J j : d ¯ ( Δ m X k , ξ ) ε } | h i g j 1 h i | { k I i : d ¯ ( Δ m X k , ξ ) ε } | ( as  J j I i ) 1 δ 1 h i | { k I i : d ¯ ( Δ m X k , ξ ) ε } | ,

which implies that X k ξ ( S β F ( Δ m ) ) . □

Theorem 3.8 Let β = ( l i ) and θ = ( k i ) be two lacunary sequences. Let I i = ( k i 1 , k i ] and J i = ( l i 1 , l i ] , i = 1 , 2 , 3 ,  , be the corresponding intervals of θ and β, respectively, and I i j = I i J j , i , j = 1 , 2 , 3 ,  , and h i = k i k i 1 and g i = l i l i 1 . If there exists δ > 0 such that
| I i j | h i δ for every  i , j = 1 , 2 , 3 ,  provided  I i , j ,

then X k ξ ( S θ F ( Δ m ) ) implies X k ξ ( S β F ( Δ m ) ) .

Proof Let α = β θ . Then α is a lacunary refinement of both the lacunary sequences β and θ. The interval sequence of α is { I i j = I j J j : I i j ϕ } .

Let ε > 0 be given, X k ξ ( S θ F ( Δ m ) ) , and for every I i , j , we can find I i such that I i j I i and I i j . Then we have
1 | I i j | | { k I i j : d ¯ ( Δ m X k , ξ ) ε } | = h i | I i j | 1 h i | { k I i j : d ¯ ( Δ m X k , ξ ) ε } | h i | I i j | 1 h i | { k I i : d ¯ ( Δ m X k , ξ ) ε } | ( as  I i j I i ) 1 δ 1 h i | { k I i : d ¯ ( Δ m X k , ξ ) ε } | ,

which implies that X k ξ ( S α F ( Δ m ) ) .

Since α is a lacunary refinement of β, so by Theorem 3.6 it follows that X k ξ ( S α F ( Δ m ) ) implies X k ξ ( S β F ( Δ m ) ) and hence the result follows. □

Theorem 3.9 Let β = ( l i ) and θ = ( k i ) be two lacunary sequences. Let I i = ( k i 1 , k i ] , J i = ( l i 1 , l i ] , i = 1 , 2 , 3 ,  , and I i j = I i J j , i , j = 1 , 2 , 3 ,  . If there exists δ > 0 such that
| I i j | ( h i + g j ) δ for every  i , j = 1 , 2 , 3 ,  provided  I i , j ,

then X k ξ ( S θ F ( Δ m ) ) if and only if X k ξ ( S β F ( Δ m ) ) .

Definition 3.1 Let β = ( l i ) and θ = ( k r ) be two lacunary sequences. Let J i = ( l i 1 , l i ] , I i = ( k i 1 , k i ] , i = 1 , 2 ,  . There exist a sequence X and a fuzzy number L such that X k ξ ( S β F ( Δ m ) ) and X k ξ ( S θ F ( Δ m ) ) if and only if there exist ( s r ) , ( t r ) N and δ > 0 which satisfy the following conditions:
  1. (i)

    J s r I t r ϕ ;

     
  2. (ii)

    lim r | J s r | | I t r | = ;

     
  3. (iii)

    | J s r I t r | | I t r | δ , r = 1 , 2 ,  .

     
Proof Let there exist a sequence X = ( X k ) and a fuzzy number ξ such that X k ξ ( S β F ( Δ m ) ) and X k ξ ( S θ F ( Δ m ) ) . Then we can find a subsequence ( t r ) N , ε > 0 and δ > 0 such that
( 1 | I t r | ) | { k I t r : d ¯ ( Δ m X k , ξ ) ε } | δ , r = 1 , 2 , .
For each t r , there exist a positive integer s r and a whole number u r such that
J s r + j I t r ϕ for every  j = 0 , 1 , 2 , , r = 1 , 2 , .
Then we can write
δ ( 1 | I t r | ) | { k I t r : d ¯ ( Δ m X k , ξ ) ε } | = ( 1 | I t r | ) | { k ( j = 0 n J s r + j ) I t r : d ¯ ( Δ m X k , ξ ) ε } | = ( 1 | I t r | ) j = 0 u r | { k J s r + j I t r : d ¯ ( Δ m X k , ξ ) ε } | δ ( 1 | I t r | ) j = 0 , u r | { k J s r + j I t r : d ¯ ( Δ m X k , ξ ) ε } | δ + ( 1 | I t r | ) 0 < j < u r | { k J s r + j I t r : d ¯ ( Δ m X k , ξ ) ε } | δ = ( 1 | I t r | ) j = 0 , u r | { k J s r + j I t r : d ¯ ( Δ m X k , ξ ) ε } | δ + ( 1 | I t r | ) 0 < j < u r | { k J s r + j : d ¯ ( Δ m X k , ξ ) ε } | δ = ( 1 | I t r | ) j = 0 , u r | { k J s r + j I t r : d ¯ ( Δ m X k , ξ ) ε } | δ + 0 < j < u r ( | J s r + j | | I t r | ) ( 1 | J s r + j | ) | { k J s r + j : d ¯ ( Δ m X k , ξ ) ε } | .
(3.10)
Since X k ξ ( S β F ( Δ m ) ) , so for sufficiently large s r + j , we have
( 1 | J s r + j | ) | { k J s r + j : d ¯ ( Δ m X k , ξ ) ε } | < δ 2 .
(3.11)
It can be seen that 0 < j < u r ( | J s r + j | | I t r | ) 1 . Hence, by using equation (3.11) in equation (3.10), we get
δ ( 1 | I t r | ) | { k I t r : d ¯ ( Δ m X k , ξ ) ε } | ( 1 | I t r | ) j = 0 , u r | { k J s r + j I t r : d ¯ ( Δ m X k , ξ ) ε } | + δ 2 .
This implies that ( 1 | I t r | ) j = 0 , u r | { k J s r + j I t r : d ¯ ( Δ m X k , ξ ) ε } | δ 2 , which implies that at least one of the following two inequalities holds:
( 1 | I t r | ) | { k J s r I t r : d ¯ ( Δ m X k , ξ ) ε } | δ 4
(3.12)
or
( 1 | I t r | ) | { k J s r + u r I t r : d ¯ ( Δ m X k , ξ ) ε } | δ 4 .
(3.13)

Suppose that equation (3.12) holds. Since | { k J s r I t r : d ¯ ( Δ m X k , ξ ) ε } | | J s r I t r | , so using equation (3.12), we conclude that δ 4 | J s r I t r | | I t r | , which proves (iii).

For such s r , t r chosen in the proof of (iii), from equation (3.12), we have
δ 4 ( 1 | I t r | ) | { k J s r I t r : d ¯ ( Δ m X k , ξ ) ε } | ( | J s r | | I t r | ) ( 1 | J s r | ) | { k J s r I t r : d ¯ ( Δ m X k , ξ ) ε } | .
(3.14)

Since ( 1 | J s r | ) | { k J s r I t r : d ¯ ( Δ m X k , ξ ) ε } | 0 as r , from equation (3.14) we have | J s r | | I t r | as r , which implies that condition (ii) is satisfied.

Conversely, let for the two lacunary sequences β = ( l i ) and θ = ( k i ) there exist sequences ( s r ) , ( t r ) N and δ > 0 which satisfy the above three conditions. Define
Δ m X k = { 1 ¯ if  k J s r I t r , 0 ¯ otherwise .

Then, for any 0 < ε < 1 , if j s r for any r = 1 , 2 , 3 ,  , 1 | J r | | { k J r : d ¯ ( Δ m X k , 0 ¯ ) ε } | = 0 and if j = s r for some r, 1 J s r | { k J s r : d ¯ ( Δ m X k , 0 ¯ ) ε } | = | J s r I t r | | J s r | | I t r | | J s r | 0 as r . Hence X k 0 ¯ ( S β F ( Δ m ) ) .

But ( 1 | I t r | ) | { k I t r : d ¯ ( Δ m X k , 0 ¯ ) ε } | = | J s r I t r | | I t r | δ for r = 1 , 2 ,  , which implies X k 0 ¯ ( S θ F ( Δ m ) ) . □

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Indian Institute of Technology

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© Srivastava and Mohanta; licensee Springer. 2013

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