Lacunary statistical convergence of order β in difference sequences of fuzzy numbers

1 Introduction

The idea of statistical convergence was given by Zygmund [36] in the first edition of his monograph published in Warsaw in 1935. The concept of statistical convergence was introduced by Steinhaus [32] and Fast [19] and later reintroduced by Schoenberg [31] independently. Over the years and under different names statistical convergence has been discussed in the theory of Fourier analysis, ergodic theory, number theory, measure theory, trigonometric series, turnpike theory and Banach spaces. Later on it was further investigated from the sequence space point of view and linked with summability theory by Connor [15], Et [16], Fridy [21], Mursaleen [26] and many others. In recent years, generalizations of statistical convergence have appeared in the study of strong integral summability and the structure of ideals of bounded continuous functions on locally compact spaces. Statistical convergence and its generalizations are also connected with subsets of the Stone-Čech compactification of the natural numbers. Moreover, statistical convergence is closely related to the concept of convergence in probability.

The theory of sequences of fuzzy numbers was first introduced by Matloka [28]. Matloka [28] introduced bounded and convergent sequences of fuzzy numbers, studied some of their properties and showed that every convergent sequence of fuzzy numbers is bounded.

The order of statistical convergence of a sequence of real numbers was given by Gadjiev and Orhan [23] and later on statistical convergence of order α and strong p-Cesàro summability of order α was studied by Çolak [12]. Recently, Altınok et al. [4] introduced the concepts of statistical convergence of order β and strong p-Cesàro summability of order β for sequences of fuzzy numbers.

Altin et al. [2], Çanak ([7–10]) and Önder et al. [11] examined the Tauberian theorems for summability methods of sequences of fuzzy numbers.

Freedman et al. [20] introduced some Cesàro-type summability spaces using lacunary sequences and later Fridy and Orhan [22] defined the concepts of lacunary statistical convergence for real number sequences. Lacunary sequences of fuzzy numbers and real (or complex) numbers were studied by Gokhan et al. [24], Mursaleen and Mohuiddin [27], Tripathy and Baruah [35] and many others. Lacunary statistical convergence of order α in real number sequences was defined by Şengül and Et [33]. Altinok et al. [5] introduced and examined the class of sequence bv _θ  (Δ, F) using a lacunary sequence θ and the difference operator Δ in sequences of fuzzy numbers, and study some of its properties like solidity and symmetricity.

The purpose of this paper is to generalize the study of Şengül and Et [33] so as to fill up the existing gaps in the theory of lacunary statistical convergence for sequences of fuzzy numbers.

This paper organizes as follows: In Section 1, we give the basic notions related to fuzzy numbers and a brief information about statistical convergence and difference sequences. In Section 2, we define the sets of lacunary statistically convergent sequences of order β and strongly summable lacunary statistically convergent sequences of order β for sequences of fuzzy numbers using generalized difference operator Δ ^m and give some inclusion relations between them. In the last section, we obtain some results using a modulus function f .

Now, we recall some basic concepts which we shall use throughout the paper.

By a lacunary sequence we mean an increasing integer sequence θ = (r) such that k₀ = 0 and h _r  = (r - k_r-1)→ ∞ as r → ∞ . Throught this paper 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 by q _r [20].

The difference spaces ℓ_∞ (Δ), c (Δ) and c₀ (Δ), consisting of all real valued sequences x = (x _k ) such that Δx = Δ¹x = (x _k  - x_k+1) in the sequence spaces ℓ_∞, c and c₀, were defined by Kızmaz [25]. The idea of difference sequences was generalized by Et [17], Et and Çolak [18], Altinok [3], Çolak et al. ([13, 14]) and many others.

A fuzzy number is a mapping X : ℝ ⟶ [ 0 , 1 ] which satisfies the following properties:

i) X is normal, i.e., there exists an t 0 ∈ ℝ such that X (t₀) =1 ;

ii) X is fuzzy convex, i.e., for all λ ∈ [0, 1] and all s , t ∈ ℝ and 0≤ λ ≤ 1, X (λs + (1 - λ) t) ≥ min [X (s) , X (t)] ;

iii) X is upper semicontinuous on real numbers set;

iv) The set { t ∈ ℝ : X ( t ) > 0 } ¯ , denoted by [X] ⁰, is compact, where { t ∈ ℝ : X ( t ) > 0 } ¯ is closure of { t ∈ ℝ : X ( t ) > 0 } with usual topology of ℝ.

L ( ℝ ) denotes the set of all fuzzy numbers on real numbers set ℝ and is said to be a fuzzy number space.

For α ∈ (0, 1], the α-level set [X]  ^α of fuzzy number X is defined by. [ X ] α = { [ c ] cc { t ∈ ℝ : X ( t ) ≥ α } , for α ∈ ( 0 , 1 ] { t ∈ ℝ : X ( t ) > α } ¯ , for α = 0

The aritmetic operations for α-level sets are defined as follows:

For X , Y ∈ L ( ℝ ) , now and what follows we will denote α-level sets by [ X ] α = [ u 1 α , v 1 α ] ,[ Y ] α = [ u 2 α , v 2 α ] ,α ∈ [0, 1] . Then we have [ X + Y ] α = [ u 1 α + u 2 α , v 1 α + v 2 α ] [ X - Y ] α = [ u 1 α - v 2 α , v 1 α - u 2 α ] [ X . Y ] α = [ min i , j ∈ [ 1 , 2 ] u i α v j α , max i , j ∈ [ 1 , 2 ] u i α v j α ] .

A sequence X = (X _k ) of fuzzy numbers is a function X from the set ℕ of all natural numbers into L ( ℝ ) [28].

Let β ∈ (0, 1] and X = (X _k ) be a sequence of fuzzy numbers. Then the sequence X = (X _k ) of fuzzy numbers is said to be statistically convergent of order β, to fuzzy number X₀ if for every ɛ > 0, lim n → ∞ 1 n β | { k ≤ n : d ( X k , X 0 ) ≥ ɛ } | = 0 , where the vertical bars indicate the number of elements in the enclosed set. In this case we write S ^β  (F) - lim X _k  = X₀ . We denote the set of all statistically convergent sequences of order β by S ^β  (F) [4].

Let w (F) be the set of all sequences of fuzzy numbers. The operator Δ ^m  : w (F) → w (F) is defined by ( Δ 0 X ) k = X k , ( Δ 1 X ) k = Δ 1 X k = X k - X k + 1 , ( Δ m X ) k = Δ 1 ( Δ m - 1 X ) k , ( m ≥ 2 ) , for all k ∈ ℕ .

2 Main results

In this section, we define the spaces S θ β ( F , Δ m ) and N θ β ( p , F , Δ m ) and examine some inclusion relations between them and space w p β ( F , Δ m ) .

Definition 2.1. Let θ = (k _r ) be a lacunary sequence, X = (X _k ) be a sequence of fuzzy numbers and β ∈ (0, 1] be given. The sequence X = (X _k ) ∈ w (F) is said to be S θ β ( F , Δ m ) -statistically convergent (or lacunary statistically convergent sequence of order β) if there is a fuzzy number X₀ such that lim r → ∞ 1 h r β | { k ∈ I r : d ( Δ m X k , X 0 ) ≥ ɛ } | = 0 , where I _r  = (k_r-1, k _r ] and h r β denote the βth power (h _r )  ^β of h _r , that is h β = ( h r β ) = ( h 1 β , h 2 β , . . . , h r β , . . . ) . In this case we write S θ β ( F , Δ m ) - lim X k = X 0. The set of all S θ β ( F , Δ m ) -statistically convergent sequences will be denoted by S θ β ( F , Δ m ). For θ =

(2 ^r ) , we shall write S ^β  (F, Δ ^m ) instead of S θ β ( F , Δ m ) and we shall write S (F, Δ ^m ) instead of S θ β ( F , Δ m ) in the special case β = 1 and θ = (2 ^r ) .

Lemma 2.2. Let θ = (k _r ) be a lacunary sequence and K ⊂ ℕ . Then S θ γ ( K ) ≤ S θ β ( K ) for 0 < β ≤ γ ≤ 1.

Proof. Let0 < β ≤ γ ≤ 1. We can write 1 h r γ | { k r - 1 < k ≤ k r : k ∈ K } | ≤ 1 h r β | { k r - 1 < k ≤ k r : k ∈ K } | since h r β ≤ h r γ. Hence S θ γ ( K ) ≤S θ β ( K ).

Let β ∈ (0, 1] . If a sequence of fuzzy numbers is lacunary statistically convergent of order β, then it is lacunary statistically convergent. Really, we obtain 1 h r β | { k ∈ I r : d ( Δ m X k , X 0 ) ≥ ɛ } | ≥ 1 h r | { k ∈ I r : d ( Δ m X k , X 0 ) ≥ ɛ } | since h r β ≤ h r for β ∈ (0, 1] . Taking limit as r→ ∞ we get X k → X 0 ( S θ β , Δ m ) and so X _k  → X₀ (S _θ , Δ ^m ) .

Note that the lacunary statistical convergence of order β is well defined for β ∈ (0, 1], but not well defined for β > 1. To show this, consider the sequence X = (X _k ) of fuzzy numbers following: X k ( t ) = { [ c ] cc [ c ] cc t , for 0 ≤ t ≤ 1 - t + 2 , for 1 ≤ t ≤ 2 0 , otherwise } : = X ′ , [ c ] c if k = 2 r ( r = 1 , 2 , 3 , . . . ) [ c ] cc t - 3 , for 3 ≤ t ≤ 4 - t + 5 , for 4 ≤ t ≤ 5 0 , otherwise } : = X ′ ′ , if k ≠ 2 r

We can find the α-level sets of sequences (X _k ) and (Δ ^m X _k ) after some arithmetic operations as follows [ X k ] α = { [ c ] cc [ α , 2 - α ] , if k = 2 r [ 3 + α , 5 - α ] , if k ≠ 2 r . and [ Δ m X k ] α = { [ c ] cc [ 2 m α - 2 m - 1 . 5 , - 2 m α - 2 m - 1 ] : = X ′ , if k = 2 r [ 2 m α + 2 m - 1 , - 2 m α + 2 m - 1 . 5 ] : = X ′ ′ , if k ≠ 2 r , ( m = 1 , 2 , . . . ) then we can write lim r → ∞ 1 h r β | { k ∈ I r : d ( Δ m X k , X ′ ) ≥ ɛ } | ≤ lim r → ∞ k r - r - 1 2 h r β = lim r → ∞ h r 2 h r β = 0 and lim r → ∞ 1 h r β | { k ∈ I r : d ( Δ m X k , X ′ ′ ) ≥ ɛ } | ≤ lim r → ∞ k r - k r - 1 2 h r β = lim r → ∞ h r 2 h r β = 0 for β > 1 . Hence sequence (X _k ) is Δ ^m -lacunary statistically convergent of order β, both to X′ and X^′′, i.e., S θ β ( F , Δ m ) - lim X k = X′ and S θ β ( F , Δ m ) - lim X k = X^′′ . But this is impossible.

Theorem 2.3. Let 0 < β ≤ 1 and X = (X _k ) , Y = (Y _k ) be sequences of fuzzy numbers, then

(i) If S θ β ( F , Δ m ) - lim X k = X 0 and c ∈ ℂ , then S θ β ( F , Δ m ) - lim ( cx k ) = cx 0 ,

(ii) If S θ β ( F , Δ m ) - lim X k = X 0 and S θ β ( F , Δ m )- lim Y _k  = Y₀, then S θ β ( F , Δ m ) - lim ( X k + Y k ) = X 0 + Y 0 .

Proof. (i) It is clear for the case c = 0 . Suppose that c ≠ 0, then the proof of (i) follows from 1 h r β | { k ∈ I r : d ( c Δ m X k , cX 0 ) ≥ ɛ } | = 1 h r β | { k ∈ I r : d ( Δ m X k , X 0 ) ≥ ɛ | c | } | and that of (ii) follows from 1 h r β | { k ∈ I r : d ( Δ m ( X k + Y k ) , ( X 0 + Y 0 ) ) ≥ ɛ } | ≤ 1 h r β | { k ∈ I r : d ( Δ m X k , X 0 ) ≥ ɛ 2 } | + 1 h r β | { k ∈ I r : d ( Δ m Y k , Y 0 ) ≥ ɛ 2 } | .

Definition 2.4. Let X = (X _k ) be a sequence of fuzzy numbers, θ = (k _r ) be a lacunary sequence, β ∈ (0, 1] be any real number and let p be a positive real number. A sequence X of fuzzy numbers is said to be strongly N θ β ( p , F , Δ m ) -summable (or strongly N _θ  (p, F, Δ ^m )-summable of order β) if there is a fuzzy number X₀ such that lim r → ∞ 1 h r β ∑ k ∈ I r d ( Δ m X k , X 0 ) p = 0 .

In this case we write N θ β ( p , F , Δ m ) - lim X k = X 0. The strong N θ β ( p , F , Δ m ) -summability reduces to the strong N _θ  (p, F, Δ ^m )-summability for β = 1. The set of all strongly N θ β ( p , F , Δ m ) -summable sequences will be denoted by N θ β ( p , F , Δ m ).

If we take θ = (2 ^r ) in space N θ β ( p , F , Δ m ) , then we obtain strongly p-Cesaro summable sequences set of order β following w p β ( F , Δ m ) = { X = ( X k ) : ∃ X 0 , lim n 1 n β ∑ k = 1 n [ d ( Δ m X k , X 0 ) ] p = 0 }

This set will be denoted by w p , 0 β in case of X 0 = 0 ¯ [3].

Theorem 2.5. If 0 < β < γ ≤ 1 then S θ β ( F , Δ m ) ⊂ S θ γ ( F , Δ m ) and the inclusion is strict.

Proof. Proof can be obtained from the inequality 1 h r γ | { k ∈ I r : d ( Δ m X k , X 0 ) ≥ ɛ } | ≤ 1 h r β | { k ∈ I r : d ( Δ m X k , X 0 ) ≥ ɛ } | .

Taking θ = (2 ^r ) we show the strictness of the inclusion S θ β ( F , Δ m ) ⊂ S θ γ ( F , Δ m ) for a special case. For this, consider the sequence X = (X _k ) of fuzzy numbers defined by X k ( t ) = { [ c ] ll [ c ] ll 2 t - ( 2 k - 1 ) , for k - 1 2 ≤ t ≤ k - 2 t + ( 2 k + 1 ) , for k ≤ t ≤ k + 1 2 0 , otherwise } , if k = n 3 [ c ] ll 2 t - 1 , for 1 2 ≤ t ≤ 1 - 2 t + 3 , for 1 ≤ t ≤ 3 2 0 , otherwise } , if k ≠ n 3

After calculating α-level sets of sequences (X _k ) and (Δ ^m X _k ) , we obtain S θ γ ( F , Δ m ) - lim X k = [ 2 m - 1 ( α - 1 ) , 2 m - 1 ( 1 - α ) ] , i.e. X ∈ S θ γ ( F , Δ m ) for 1 3 < γ ≤ 1 , but X ∉ S θ β ( F , Δ m ) for 0 < β ≤ 1 3 . This implies that the inclusion S θ β ( F , Δ m ) ⊂ S θ γ ( F , Δ m ) is strict for β, γ ∈ (0, 1] such that β ∈ ( 0 , 1 3 ] and γ ∈ ( 1 3 , 1 ] .

Corollary 2.6. If a sequence of fuzzy numbers is S θ β ( F , Δ m ) -statistically convergent to fuzzy number X₀, then it is S _θ  (F, Δ ^m )-statistically convergent to fuzzy number X₀ .

Theorem 2.7. Let β and γ be fixed real numbers such that 0 < β ≤ γ ≤ 1, X = (X _k ) be a sequence of fuzzy numbers and θ = (k _r ) be a lacunary sequence. For 0 < p < ∞ , N θ β ( p , F , Δ m ) ⊂ S θ γ ( F , Δ m ) and the inclusion is strict for some β’s and γ’s.

Proof. For any sequence X = (X _k ) of fuzzy numbers and ɛ > 0, we have ∑ k ∈ I r [ d ( Δ m X k , X 0 ) ] p = ∑ K ∈ I r d ( △ m X L , X 0 ) ≥ ε [ d ( △ m X k , X 0 ) ] p + ∑ K ∈ I r d ( △ m X L , X 0 ) ≥ ε [ d ( △ m X k , X 0 ) ] p ≥ ∑ K ∈ I r d ( △ m X L , X 0 ) ≥ ε [ d ( △ m X k , X 0 ) ] p ≥ | { k ∈ I r : d ( Δ m X k , X 0 ) ≥ ɛ } | ɛ p and so that 1 h r β ∑ k ∈ I r [ d ( Δ m X k , X 0 ) ] p ≥ 1 h r β | { k ∈ I r : d ( Δ m X k , X 0 ) ≥ ɛ } | ɛ p ≥ 1 h r γ | { k ∈ I r : d ( Δ m X k , X 0 ) ≥ ɛ } | ɛ p .

It follows that if X = (X _k ) is strongly N θ β ( p ,F, Δ ^m )-summable to fuzzy number X₀, then it is S θ γ ( F , Δ m ) -statistically convergent to X₀.

Taking β = γ, θ = (2 ^r ) and p = 1 we show the strictness of the inclusion N θ β ( p , F , Δ m ) ⊂ S θ γ ( F , Δ m ) for a special case. For this we can select a sequence of fuzzy numbers as follows: X k ( t ) = { [ c ] cc [ c ] cc t k + 1 , for - k ≤ t ≤ 0 - t k + 1 , for 0 ≤ t ≤ k 0 , otherwise } , [ c ] c if k = n 2 ( n = 1 , 2 , . . . ) [ c ] cc t 2 - 1 , for 2 ≤ t ≤ 4 - t 2 + 3 , for 4 ≤ t ≤ 6 0 , otherwise } : = X 0 if k ≠ n 2

After calculating α-level sets of sequences (X _k ) and (Δ ^m X _k ) , we see that (X _k ) is Δ ^m -statistically convergent of order γ to X₀ for 1 2 < β ≤ 1 , where [X₀]  ^α  = [2^m+1 (α - 1) , 2 ^m  (1 - α)] , but it is not strongly N θ β ( p , F , Δ m )-summable to X₀ . Therefore N θ β ( p , F , Δ m ) ⊂ S θ β ( F , Δ m ) for 1 2 < β ≤ 1 .

Corollary 2.8. Let X = (X _k ) be a sequence of fuzzy numbers, β ∈ (0, 1] and p is a positive real number. Then

i) If X ∈ N θ β ( p , F , Δ m ) , then X ∈ S θ β ( F , Δ m ) and the limits are same.

ii) If X ∈ N θ β ( p , F , Δ m ) , then X ∈ S _θ  (F, Δ ^m ) and the limits are same.

Theorem 2.9. Let X = (X _k ) be a sequence of fuzzy numbers, β ∈ (0, 1] and θ = (k _r ) be a lacunary sequence. If lim inf r q r > 1 , then S β ( F , Δ m ) ⊂ S θ β ( F , Δ m ) .

Proof. Suppose that lim inf r q r > 1 , then there exists a number δ > 0 such that q _r  ≥ 1 + δ for sufficiently large r and so h r k r ≥ δ 1 + δ ⇒ ( h r k r ) β ≥ ( δ 1 + δ ) β ⇒ 1 k r β ≥ δ β ( 1 + δ ) β 1 h r β .

If X _k  → X₀ (S ^β  (F, Δ ^m )) , then we prove the suffıciency from following inequalities for every ɛ > 0 and for sufficiently large r 1 k r β | { k ≤ k r : d ( Δ m X k , X 0 ) ≥ ɛ } | ≥ 1 k r β | { k ∈ I r : d ( Δ m X k , X 0 ) ≥ ɛ } | ≥ δ β ( 1 + δ ) β 1 h r β | { k ∈ I r : d ( Δ m X k , X 0 ) ≥ ɛ } | ;

Theorem 2.10 If lim r → ∞ inf h r β k r > 0 , then S (F, Δ ^m )⊂ S θ β ( F , Δ m ) .

Proof. For a given ɛ > 0, we can write { k ≤ k r : d ( Δ m X k , X 0 ) ≥ ɛ } ⊃ { k ∈ I r : d ( Δ m X k , X 0 ) ≥ ɛ } .

From here we obtain 1 k r | { k ≤ k r : d ( Δ m X k , X 0 ) ≥ ɛ } | ≥ 1 k r | { k ∈ I r : d ( Δ m X k , X 0 ) ≥ ɛ } | = h r β k r 1 h r β | { k ∈ I r : d ( Δ m X k , X 0 ) ≥ ɛ } | .

Taking limit according to r and using lim r → ∞inf h r β k r > 0 , we get X k → X 0 ( S ( , F , Δ m ) ) ⇒ X k → X 0 ( S θ β , ( F , Δ m ) ) .

Theorem 2.11. Let X = (X _k ) be a sequence of fuzzy numbers, 0 < β ≤ 1, and θ = (k _r ) be a lacunary sequence. If lim sup r q r < ∞ , then S θ β ( F , Δ m ) ⊂ S ( F , Δ m ) .

Proof. Omitted.

Theorem 2.12. Let 0 < β < γ ≤ 1, X = (X _k ) be a sequence of fuzzy numbers and p > 0, then there exists inclusion N θ β ( p , F , Δ m ) ⊂ N θ γ ( p , F , Δ m ) and this is strict.

Proof. Let X = ( X k ) ∈ N θ β ( p , F , Δ m ). Then given β and γ such that 0 < β < γ ≤ 1 and a p > 0, we can write inequality 1 h r γ ∑ k ∈ I r [ d ( Δ m X k , X 0 ) ] p ≤ 1 h r β ∑ k ∈ I r [ d ( Δ m X k , X 0 ) ] p and this gives that N θ β ( p , F , Δ m ) ⊂ N θ γ ( p , F , Δ m ) .

To show that the inclusion is strict take lacunary sequence θ = (2 ^r ) and p = 1 . Consider the sequence X = (X _k ) of fuzzy numbers as follows X k ( t ) = { [ c ] cc [ c ] cc t - 2 , for 2 ≤ t ≤ 3 - t + 4 , for 3 ≤ t ≤ 4 0 , otherwise } , [ c ] c if k = n 3 ( n = 1 , 2 , . . . ) [ c ] cc t 3 - 5 3 , for 5 ≤ t ≤ 8 - t 3 + 11 3 , for 8 ≤ t ≤ 11 0 , otherwise } : = X 0 if k ≠ n 3

After calculating α-level sets of sequences (X _k ) and (Δ ^m X _k ) , we obtain X ∈ N θ γ ( p , F , Δ m ) since (X _k ) is strongly N θ β ( p , F , Δ m )-summable to X₀ for 1 3 < γ ≤ 1 , where [X₀]  ^α  = [2 ^m  (3α - 3) , 2 ^m  (3 - 3α)] , but X ∉ N θ β ( p , F , Δ m ) for 0 < β ≤ 1 3 , i.e. it is not strongly N θ β ( p , F , Δ m )-summable to fuzzy number X₀ . So, this shows that inclusion is strict. We show the limits of sequence Δ ^m X _k in Fig. 1.

Corollary 2.13. Let 0 < β ≤ γ ≤ 1, X = (X _k ) be a sequence of fuzzy numbers and p > 0. Then

(i) If β = γ, then N θ β ( p , F , Δ m ) = N θ γ ( p , F , Δ m ) ,

(ii) For each β ∈ (0, 1] and 0 < p < ∞ , N θ β ( p , F , Δ m ) ⊆ N θ ( p , F , Δ m ) .

Theorem 2.14. Let X = (X _k ) be a sequence of fuzzy numbers and θ = (k _r ) be a lacunary sequence. If lim inf r q r > 1 , then w p β ( F , Δ m ) ⊆ N θ β ( p , F , Δ m ) for 0 < β ≤ 1 and 0 < p < ∞ .

Proof.If lim inf r q r > 1 , then there exists a number δ > 0 such that q _r  ≥ 1 + δ for all r ≥ 1 . For 0 < β ≤ 1 and X ∈ w p , 0 β we can write τ r β = 1 h r β ∑ i = 1 k r d ( Δ m X i , 0 ¯ ) p - 1 h r β ∑ i = 1 k r - 1 d ( Δ m X i , 0 ¯ ) p = k r β h r β ( 1 k r β ∑ i = 1 k r d ( Δ m X i , 0 ¯ ) p ) - k r - 1 β h r β ( 1 k r - 1 β ∑ i = 1 k r - 1 d ( Δ m X i , 0 ¯ ) p ) and so k r β h r β ≤ ( 1 + δ ) β δ β and k r - 1 β h r β ≤ 1 δ β for h _r  = k _r  - k_r-1. It can be seen that both of terms1 k r β ∑ i = 1 k r d ( Δ m X i , 0 ¯ ) p and 1 k r - 1 β ∑ i = 1 k r - 1 d ( Δ mX i , 0 ¯ ) p converge to 0 and so τ r β does. Hence X ∈ N θ , 0 β ( p , F , Δ m ), that is w p β ( F , Δ m ) ⊆ N θ β ( p , F , Δ m ).

Theorem 2.15. If lim sup r k r k r - 1 β < ∞, then N _θ  (p, F, Δ m ) ⊆ w p β ( F , Δ m ) for 0 < β ≤ 1 .

Proof.If lim sup r k r k r - 1 β < ∞ , then there exists a number M > 0 such that k r k r - 1 β < M for all r ≥ 1 . Let X ∈ N_θ,0 (p, F, Δ ^m ) and ɛ > 0 . Then we can find numbers R > 0 and K > 0 such that sup i ≥ R τ i < ɛ and τ _i  < K for i = 1, 2, 3, . . .. Therefore we can write 1 t β ∑ i = 1 t d ( Δ m X i , 0 ¯ ) p [ c ] c ≤ 1 k r - 1 β ∑ i = 1 k r - 1 d ( Δ m X i , 0 ¯ ) p [ c ] c = 1 k r - 1 β ( ∑ I 1 d ( Δ m X i , 0 ¯ ) p + ∑ I 2 d ( Δ m X i , 0 ¯ ) p + . . . + ∑ I r d ( Δ m X i , 0 ¯ ) p ) [ c ] c = k 1 k r - 1 β τ 1 + k 2 - k 1 k r - 1 β τ 2 + . . . + k R - k R - 1 k r - 1 β τ R + k R + 1 - k R k r - 1 β τ R + 1 + . . . + kr - k r - 1 k r - 1 β τ r [ c ] c ≤ ( sup i ≥ 1 τ i ) k R k r - 1 β + ( sup i ≥ R τ i ) k r - k R k r - 1 β [ c ] c < K k R k r - 1 β + ɛ M where k_r-1 < t ≤ k _r and r > R . We get 1 t β ∑ i = 1 td ( Δ m X i , 0 ¯ ) p → 0 since t→ ∞ for k_r-1 → ∞ , hence X ∈ w p , 0 β ( F , Δ m ) .

Theorem 2.16. Let X = (X _k ) be a sequence of fuzzy numbers, θ = (k _r ) be a lacunary sequence and β ∈ (0, 1]. If X ∈ w β ( F , Δ m ) ∩ N θ β ( p , F , Δ m ) and lim sup r k r k r - 1 β < ∞ , then the limits of (X _k ) are same.

Proof. Suppose that w p β ( F , Δ m ) - lim X k = X ′ ,N θ , p β ( p , F , Δ m ) - lim X k = X ′ ′ and X′ ≠ X^′′. We get N θ , 0 ( p , F , Δ m ) ⊆ w p , 0 β ( F , Δ m ) since lim sup r k r k r - 1 β < ∞ from Theorem 2.15. Because of d (Δ ^m X _i , X^′′) ∈ N_θ,0 (p, F, Δ ^m ), so d ( Δ m X i , X ′ ′ ) ∈ w p , 0 β ( F , Δ m ). Therefore for p = 1 1 t β ∑ i = 1 t d ( Δ m X i , X ′ ′ ) → 0

In the following inequality, both of terms at left converges to 0, but this is a contradiction and so X′ = X^′′ . 1 t β ∑ i = 1 t d ( Δ m X i , X ′ ′ ) + 1 t β ∑ i = 1 t d ( Δ m X i , X ′ ) ≥ 1 t β d ( X ′ , X ′ ′ ) > 0

3 Results related to modulus function

We give relation between sets S θ β ( F , Δ m ) and w p β [ θ , f , F , Δ m ] according to modulus function f.

We first quote the definition of a modulus function:

The concept of modulus function was formally introduced by Nakano [29]. A mapping f : [0, ∞) → [0, ∞) is said to be a modulus if

i) f (x) = 0 iff x = 0,

ii) f (x + y) ≤ f (x) + f (y) for x, y ≥ 0,

iii) f is increasing,

iv) f is right-continuous at 0 .

The continuity of f everywhere on [0, ∞) follows from above definition. A modulus function can be bounded or unbounded. For example f (x) = x ^p , (0 < p ≤ 1) is bounded and f ( x ) = x 1 + x is bounded. The concept of modulus function for sequences of fuzzy numbers was first investigated by Sarma [30]. Later on it was explotied by Talo and Başar [34].

Definition 3.1. Let f be a modulus function, X = (X _k ) be a sequence of fuzzy numbers and p = (p _k ) be a sequence of positive real numbers and β ∈ (0, 1]. We define space w p β ( θ , f , F , Δ m ) as follows w p β ( θ , f , F , Δ m ) = { x = ( x k ) : lim r → ∞ 1 h r β ∑ k ∈ I r [ d ( Δ m X k , X 0 ) ] p k = 0 , for some X 0 }

We shall write N θ β ( p , F , Δ m ) instead of w p β ( θ , f , F , Δ m ) in the special cases p _k  = 1 and f (x) = x for all k ∈ ℕ. We consider that p = (p _k ) is bounded and 0 < h = inf k p k ≤ p k ≤ sup k p k = H < ∞ in the following theorems.

Theorem 3.2. Let β, γ ∈ (0, 1] , β ≤ γ, f be a modulus function, X = (X _k ) be a sequence of fuzzy numbers and θ = (k _r ) be a lacunary sequence. Then, w p β ( θ , f , F , Δ m ) ⊂ S θ γ ( F , Δ m ) .

Proof. Takex ∈ w p β ( θ , f , F , Δ m ), ɛ > 0 and let ∑₁ and ∑₂ be summations on k ∈ I _r , d (Δ ^m X _k , X₀) ≥ ɛ and k ∈ I _r , d (Δ ^m X _k , X₀) < ɛ, respectively. Since h r β < h r γ for each r, we get 1 h r β ∑ k ∈ I r [ f ( d ( Δ m X k , X 0 ) ) ] p k = 1 h r β [ ∑ 1 [ f ( d ( Δ m X k , X 0 ) ) ] p k + ∑ 2 [ f ( d ( Δ m X k , X 0 ) ) ] p k ] ≥ 1 h r γ [ ∑ 1 [ f ( d ( Δ m X k , X 0 ) ) ] p k + ∑ 2 [ f ( d ( Δ m X k , X 0 ) ) ] p k ] ≥ 1 h r γ ∑ 1 [ f ( ɛ ) ] p k ≥ 1 h r γ ∑ 1 min ( [ f ( ɛ ) ] h , [ f ( ɛ ) ] H ) ≥ 1 h r γ | { k ∈ I r : d ( Δ m X k , X 0 ) ≥ ɛ } | min ( [ f ( ɛ ) ] h , [ f ( ɛ ) ] H ) Left side of above inequality converges to 0 as r→ ∞ because of X ∈ w p β ( θ , f , F , Δ m ), so right one converges to zero and hence X ∈ S θ γ ( F , Δ m ).

Theorem 3.3. Let f be a bounded modulus function, X = (X _k ) be a sequence of fuzzy numbers and θ = (k _r ) be a lacunary sequence. If lim r → ∞ h r h r β = 1 , then S θ β ( F , Δ m ) ⊂ w p β ( θ , f , F , Δ m ).

Proof. Suppose that f is bounded and X ∈ S θ β ( F , Δ m ). Given ɛ > 0 . There is an integer K such that (fX) ≤ K since f is bounded. Now we can write 1 h r β ∑ k ∈ I r [ f ( d ( Δ m X k , X 0 ) ) ] p k = 1 h r β ∑ 1 [ f ( d ( Δ m X k , X 0 ) ) ] p k + 1 h r β ∑ 2 [ f ( d ( Δ m X k , X 0 ) ) ] p k ≤ 1 h r β ∑ 1 max ( K h , K H ) + 1 h r β ∑ 2 [ f ( ɛ ) ] p k ≤ max ( K h , K H ) 1 h r β | { k ∈ I r : f ( d ( Δ m X k , X 0 ) ) ≥ ɛ } | + h r h r β max ( f ( ɛ ) h , f ( ɛ ) H ) .