Lacunary statistical boundedness of order β for sequences of fuzzy numbers

1 Introduction, Definitions and Preliminaries

The concept of statistical convergence which is a powerful mathematical tool for studying the convergence problems of numerical problems through the concept of density was introduced by Steinhaus [32] and Fast [17] and later reintroduced by Schoenberg [30] independently. Several authors have discussed various aspects of the theory and applications of statistical convergence such as Fourier analysis theory, Ergodic theory, Number theory, Measure theory, Trigonometric series, Turnpike theory and Banach spaces. Later on it was further investigated from the sequence spaces point of view and linked with summability theory by Altinok and Et [4], Connor [11], Et et al. ([13, 16]), Fridy [19], Fridy and Orhan [19] and Orhan [20], Mursaleen [25] and many others.

By a lacunary sequence, we mean an increasing integer sequence θ = (k _r ) such that 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 the ratio k r k r - 1 will be abbreviated by q _r . Recently, lacunary sequences have been discussed by various mathematicians ([5, 18, 31]).

The concepts of fuzzy sets and basic operations on a fuzzy set were introduced by Zadeh [34] as an extension of the concept of classical set. Fuzzy sets plays an important role in various fields like Possibility theory, Linguistic and numerical modeling, decision making in complex phenomena which are difficult to describe. Really, the theory of fuzzy set has become an active area of research in science and engineering, recently. This idea has been used not only in many engineering applications, such as, in the computer programming, population dynamics, in quantum physics, but also in various branches of mathematics, such as, in the theory of metric and topological spaces [1] and in studies of convergence of sequences of functions ([9, 33]).

The theory of sequence of fuzzy numbers was introduced by Matloka [24], where he proved some basic theorems related to sequences of fuzzy numbers. Later, the notion of statistical convergence for sequences of fuzzy numbers was defined and studied by Nuray and Savaş [26]. Since then, there has been increasing interest in the studies of fuzzy numbers and statistical convergence of fuzzy sequences (see [6],[27–29]).

Çolak [10] defined the concepts of statistical convergence of order β and strong p-Cesàro summability of order β for classical sequences and later, Altınok et al [3] generalized these concepts for sequences of fuzzy numbers.

The statistical boundedness which is generalization of boundedness for sequences of fuzzy numbers was defined by Aytar and Pehlivan [6] and generalized by Altinok and Mursaleen [2] using a difference operator Δ such that ΔX _k = X _k - X_k+1. Aytar et al [7] defined the statistical limit superior and limit inferior of statistically bounded sequences of fuzzy numbers and studied some of their properties. Altinok and Et [4] studied the concept of λ-statistical boundedness of order β of sequences of fuzzy numbers using a nondecreasing sequence λ = (λ _n ) of positive real numbers such that λ_n+1 ≤ λ _n + 1, λ₁ = 1, λ _n → ∞ as n→ ∞ and also examined the solidity, monotonicity and symmetricity of the sequence class S λ β B ( F ) .

For the real number sequences, the concept of lacunary statistical boundedness of order β was given by Et et al [15]. Let θ = (k _r ) be a lacunary sequence and 0 < β ≤ 1 be given. The lacunaryβ-density of subset 𝔼 of ℕ defined by δ θ β ( 𝔼 ) = lim n 1 h r β | { k r - 1 < k ≤ k r : k ∈ 𝔼 } | , provided the limit exists . Lacunary β-density δ θ β ( 𝔼 ) reduces to natural density δ ( 𝔼 ) in the special cases β = 1 and θ = (2 ^r ).

If X = (X _k ) is a sequence of fuzzy numbers such that (X _k ) satisfies property p (k) for all k except a set of lacunary β-density zero, then we say that (X _k ) satisfies p (k) for “”lacunary almost all k according to β” and we abbreviate this by "a. a. k _r (β)".

In this study, we are interested in lacunary statistical boundedness of order β and examine some inclusion relations between statistical boundedness and lacunary statistical boundedness of order β for sequences of fuzzy numbers.

Now, we give the basic notions which will be used throughout the paper.

A fuzzy number is a fuzzy set which satisfies the following conditions on ℝ :

u is normal, that is, there exists an x 0 ∈ ℝ such that u (x₀) =1 ;

α-level set [u] ^α of a fuzzy number u is defined by [ u ] α = { [ c ] cc { x ∈ ℝ : u ( x ) ≥ α } , if α ∈ ( 0 , 1 ] * supp u , if α = 0 . Above four conditions guarantee that interval [u] ^α is non-empty, closed and bounded for each α ∈ [0, 1], which is defined by [ u ] α = [ u _ α , u ¯ α ] .

Let u , v ∈ L ( ℝ ) , k ∈ ℝ and intervals [ u ] α = [ u _ α , u ¯ α ] and [ v ] α = [ v _ α , v ¯ α ] denote the α-level sets of fuzzy numbers u and v for α ∈ [0, 1]. Generally, we use the metric d ( u , v ) = sup 0 ≤ α ≤ 1 d H ( [ u ] α , [ v ] α ) between fuzzy numbers u and v, where d _H is the Hausdorff metric defined by d H ( [ u ] α , [ v ] α ) = max { | u _ α - v _ α | , | u ¯ α - v ¯ α | } . Diamond and Kloeden [12] showed that d satisfies metric space axioms on L ( ℝ ) , and metric space ( L ( ℝ ) , d ) is complete.

For any α ∈ [0, 1], the order relation on L ( ℝ ) is defined by u ≤ v if and only if u _ α ≤ u ¯ α and v _ α ≤ v ¯ α . Let u and v be two fuzzy numbers. If neither u ≤ v nor v ≤ u, then fuzzy numbers u and v are called to be incomparable and in this case, we prefer to use the notation u≁v ([12, 22]).

Let (X _k ) be a sequence of fuzzy numbers. The sequence X = (X _k ) is said to be convergent to the fuzzy number X₀ if for every ɛ > 0 and k > k₀, there exists a number k 0 ∈ ℕ such that d (X _k , X₀) < ɛ, written as lim k → ∞ X k = X₀. On the other hand, for each k ∈ ℕ , if there exist two fuzzy numbers u and v such that u ≤ X _k ≤ v, then the sequence (X _k ) is said to be bounded [24].

Let E (F) be a fuzzy sequence space. Then,

Normal (or solid), if d ( Y k , 0 ¯ ) ≤ d ( X k , 0 ¯ ) for all k ∈ ℕ implies (Y _k ) ∈ E (F), whenever (X _k ) ∈ E (F),

It is well known that; If E (F) is solid, then E (F) is monotone.

2 Main Results

Definition 1. Let θ = (k _r ) be a lacunary sequence, X = (X _k ) be a sequence of fuzzy numbers and β ∈ (0, 1] be a real number. A sequence X = (X _k ) is said to be lacunary statistically Cauchy sequence of order β provided that for every ɛ > 0 there exists a number N (= N (ɛ)) such that d ( X k , X N ) < ɛ for a . a . k r ( β ) i.e. lim r ← ∞ 1 h r β | { k ∈ I r : d ( X k , X N ) ≥ ɛ } | = 0 .

Definition 2. Let θ = (k _r ) be a lacunary sequence, β ∈ (0, 1] and X = (X _k ) be a sequence of fuzzy numbers. A sequence (X _k ) is said to be lacunary statistically bounded above of order β if there exists a fuzzy number L such that lim r ← ∞ 1 h r β | { k ∈ I r : X k > L } ∪ { k ∈ I r : X k ≁ L } | = 0 , and a sequence (X _k ) is said to be lacunary statistically bounded below of order β if there exists a fuzzy number L such that lim r ← ∞ 1 h r β | { k ∈ I r : X k < L } ∪ { k ∈ I r : X k ≁ L } | = 0 , where I _r = (k_r-1, k _r ] and h β = ( h r β ) = ( h 1 β , h 2 β , . . . , h r β , . . . ) .

Since the set L ( ℝ ) is partially ordered set, it must be considered the incomparable elements in L ( ℝ ) . Therefore we have added the elements of the set { k ∈ ℕ : X k ≁ L } to the set { k ∈ ℕ : X k > L } .

If a sequence X = (X _k ) of fuzzy numbers is both lacunary statistically bounded above of order β and lacunary statistically bounded below of order β, then it is called lacunary statistically bounded of order β.

The set of all lacunary statistically bounded sequences of order β will be denoted by S θ β ( F , b ) . For θ = (2 ^r ) we shall write S ^β (F, b) instead of S θ β ( F , b ) and in the special cases β = 1 and θ = (2 ^r ) we shall write S (F, b) instead of S θ β ( F , b ) . For β = 1 we obtain the set S _θ (F, b) of all lacunary statistically bounded sequences of fuzzy numbers.

Now, we give an example related to above definition as follows:

Example 3. Consider the sequence X = (X _k ) of fuzzy numbers defined by X k ( x ) = { 2 x - ( 2 k - 1 ) , for x ∈ [ k - 1 2 , k ] - 2 x + ( 2 k + 1 ) , for x ∈ [ k , k + 1 2 ] 0 , otherwise } , if k = n 3 ( n = 1 , 2 , . . . ) 2 x - 1 , for x ∈ [ 1 2 , 1 ] - 2 x + 3 , for x ∈ [ 1 , 3 2 ] 0 , otherwise : = L , otherwise α-level set of this sequence is [ X k ] α = { [ c ] cc [ α - 1 2 + k , k + 1 - α 2 ] , if k = n 3 [ α + 1 2 , 3 - α 2 ] : = L α , otherwise . Hence, we can write for β > 1 3 δ β ( { k ∈ I r : X k > L } ∪ { k ∈ I r : X k ≁ L } ) = δ β ( { 8 , 27 , 81 , . . . } ∪ { ∅ } ) = 0 and δ β ( { k ∈ I r : X k < L } ∪ { k ∈ I r : X k ≁ L } ) = δ β ( { ∅ } ) = 0 Finally, the sequence X = (X _k ) is lacunary statistically bounded of order β, but not bounded for lacunary sequence θ = (2 ^r ).

In view of the existing standard techniques and routine operations, proofs of the following three results omitted.

Proposition 4. Every convergent sequence of fuzzy numbers is lacunary statistically bounded, but the converse is not true.

Proposition 5. Let X = (X _k ) be a sequence of fuzzy numbers and 0 < β ≤ 1 be given. Then, every lacunary statistically convergent sequence of order β is lacunary statistically Cauchy sequence of order β.

Proposition 6. Let 0 < β ≤ 1 be given. Then, every lacunary statistically Cauchy sequence of order β of fuzzy numbers is lacunary statistically bounded of order β.

Theorem 7. Let 0 < β ≤ 1 be given. Then, every bounded sequence of fuzzy numbers is lacunary statistically bounded of order β, but the converse is not true.

Proof. First part of proof is easy. For second part, define a sequence X = (X _k ) of fuzzy numbers as follows: X k ( x ) = { x - k + 1 , for k - 1 ≤ x ≤ k - x + k + 1 , for k ≤ x ≤ k + 1 0 , otherwise } , if k = n 2 ( n = 1 , 2 , . . . ) x - 2 , for 2 ≤ x ≤ 3 - x + 4 , for 3 ≤ x ≤ 4 0 , otherwise } : = L , if k ≠ n 2 and α-level set is [ X k ] α = { [ α + k - 1 , - α + k + 1 ] , if k = n 2 [ α + 2 , 4 - α ] : = L α , if k ≠ n 2 . Hence, we can write δ β ( { k ∈ I r : X k > L } ∪ { k ∈ I r : X k ≁ L } ) = δ β ( { 4 , 9 , 16 , . . . } ∪ { ∅ } ) = 0 and δ β ( { k ∈ I r : X k < L } ∪ { k ∈ I r : X k ≁ L } ) = δ β ( { ∅ } ) = 0 for β > 1 2 . Then, it can be seen that X = (X _k ) is lacunary statistically bounded of order β, but not bounded for θ = (2 ^r ) (see Fig. 1).

OPEN IN VIEWER

Corollary 8. If a sequence of fuzzy numbers is bounded, then it is lacunary statistically bounded, but the converse does not hold.

Theorem 9. Let 0 < β ≤ 1 be given. Every lacunary statistically convergent sequence of fuzzy numbers of order β is lacunary statistically bounded of order β, but the converse is not true.

Proof. The proof of first part follows from Proposition 5 and Proposition 6. To show the strictness of the inclusion, let θ = (2 ^r ) be given and the sequence X = (X _k ) defined by X k ( x ) = { x - 1 , for 1 ≤ x ≤ 2 - x + 3 , for 2 ≤ x ≤ 3 0 , otherwise } : = l 1 , if k is odd x - 4 , for 4 ≤ x ≤ 5 - x + 6 , for 5 ≤ x ≤ 6 0 , otherwise } : = l 2 , if k is even and the α-level set is [ X k ] α = { [ α + 1 , 3 - α ] : = l 1 , if k is odd [ α + 4 , 6 - α ] : = l 2 , if k is even Then X ∈ S θ β ( F , b ) , but X ∉ S θ β ( F ) in the special cases β = 1 and θ = (2 ^r ). (See Fig. 2)

OPEN IN VIEWER

The following result is a consequence of Theorem 9.

Corollary 10. If a sequence of fuzzy numbers is lacunary statistically convergent, then it is lacunary statistically bounded, but the converse does nothold.

Theorem 11. i) S θ β ( F , b ) is not symmetric,

ii) S θ β ( F , b ) is normal and hence monotone,

Proof. i) Consider the following sequence of fuzzy numbers: X k ( x ) = { x - k , for k ≤ x ≤ k + 1 - x + k + 2 , for k + 1 ≤ x ≤ k + 2 0 , otherwise } , if k = n 2 0 ¯ , if k ≠ n 2 After some arithmetic operations related to membership function of (X _k ), we can find the α-level sets of sequence (X _k ) as [ X k ] α = { [ α + k , - α + k + 2 ] , if k = n 2 [ 0 , 0 ] , if k ≠ n 2 This sequence belongs to the set S θ β ( F , b ) for β > 1 2 and θ = (2 ^r ). Let Y = (Y _k ) be a rearrangement of (X _k ), which is defined as follows: ( Y k ) = ( X 1 , X 2 , X 4 , X 3 , X 9 , X 5 , X 16 , X 6 , X 25 , X 7 , X 36 , X 8 , X 49 , X 10 , . . . ) . Clearly for any fuzzy number L, we can write δ θ β ( { k ∈ I r : Y k > L } ∪ { k ∈ I r : Y k ≁ L } ) ≠ 0 in the special case β = 1 and θ = (2 ^r ), so ( Y k ) ∉ S θ β ( F , b ) .

ii) Let X = ( X k ) ∈ S θ β ( F , b ) and Y = (Y _k ) be a sequence such that d ( Y k , 0 ¯ ) ≤ d ( X k , 0 ¯ ) for all k ∈ ℕ . Since X ∈ S θ β ( F , b ) there exists a fuzzy number L such that δ θ β ( { k ∈ I r : X k > L } ∪ { k ∈ I r : X k ≁ L } ) = 0 . Clearly ( Y k ) ∈ S θ β ( F , b ) as { k ∈ I r : Y k > L } ∪ { k ∈ I r : Y k ≁ L } ⊂ { k ∈ I r : X k > L } ∪ { k ∈ I r : X k ≁ L } . Hence S θ β ( F , b ) is normal and so it is monotone since every normal space is monotone.

Theorem 12. If 0 < β ≤ γ ≤ 1 then S θ β ( F , b ) ⊆ S θ γ ( F , b ) and the inclusion is strict.

Proof. It is enough to show the strictness of the inclusion since the inclusion part of proof is easy. For this, let θ be given and the sequence X = (X _k ) defined by X k ( x ) = { [ c ] cc [ c ] cc x - [ h r ] + 1 , for [ h r ] - 1 ≤ x ≤ [ h r ] - x + [ h r ] + 1 , for [ h r ] ≤ x ≤ [ h r ] + 1 0 , otherwise } , k = 1 , 2 , 3 , . . . , [ h r ] 0 ¯ , otherwise Then X ∈ S θ γ ( F , b ) for 1 2 < γ ≤ 1 but X ∉ S θ β ( F , b ) for 0 < β ≤ 1 2 .

Theorem 12 yields the following corollary.

Corollary 13. If a sequence of fuzzy numbers is lacunary statistically bounded of order β, then it is lacunary statistically bounded.

Theorem 14. Let 0 < β ≤ 1 and θ = (k _r ) be a lacunary sequence. If lim inf r q r > 1 , then S β ( F , b ) ⊂ S θ β ( F , b ) .

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, which implies that 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 ) ∈ S ^β (F, b), then for a fuzzy number L and sufficiently large r, we have 1 k r β | { k ≤ k r : X k > L } ∪ { k ≤ k r : X k ≁ L } | ≥ 1 k r β | { k ∈ I r : X k > L } ∪ { k ∈ I r : X k ≁ L } | ≥ δ β ( 1 + δ ) β 1 h r β | { k ∈ I r : X k > L } ∪ { k ∈ I r : X k ≁ L } | , this proves the proof.

Theorem 15. Let 0 < β ≤ 1, X = (X _k ) be a sequence of fuzzy numbers and θ = (k _r ) be a lacunary sequence. If lim sup r q r < ∞ , then S θ β ( F , b ) ⊂ S ( F , b ) .

Proof. Omitted.

Remark 16. Although every subsequence of a bounded sequences of fuzzy numbers is bounded, this is no longer true for subsequences of lacunary statistically bounded of order β sequences of fuzzy numbers. For this we can give the following example.

Example 17. Let θ = (k _r ) be a lacunary sequence, with lim inf r q r > 1 and define a sequence X = (X _k ) of fuzzy numbers as follow:

X k ( x ) = { x - k + 1 , for k - 1 ≤ x ≤ k - x + k + 1 , for k ≤ x ≤ k + 1 0 , otherwise } , if k = n 2 0 ¯ , if k ≠ n 2 We can calculate the α-level set of this sequence as [ X k ] α = { [ α + k - 1 , k + 1 - α ] , if k = n 2 [ 0 , 0 ] , if k ≠ n 2 Then X = ( X k ) ∈ S β ( F , b ) ⊂ S θ β ( F , b ) since lim inf r q r > 1 , that is, X = (X _k ) is lacunary statistically bounded for β > 1 2 . On the other hand, define a fuzzy subsequence (Y _k ) of (X _k ) by Y k ( x ) = { x - k 2 + 1 , for k 2 - 1 ≤ x ≤ k 2 - x + k 2 + 1 , for k 2 ≤ x ≤ k 2 + 1 0 , otherwise . The α-level set of subsequence (Y _k ) is [Y _k ] ^α = [α + k² - 1, k² + 1 - α]. It is easy to see that subsequence (Y _k ) is not lacunary statistically bounded.

Theorem 18. Let 0 < β ≤ 1, X = (X _k ) be a sequence of fuzzy numbers and θ = (k _r ) be a lacunary sequence. If lim r → ∞ inf h r β k r > 0 1(1) then S ( F , b ) ⊂ S θ β ( F , b ) .

Proof. For a fuzzy number L, we have { k ≤ k r : X k > L } ∪ { k ≤ k r : X k ≁ L } ⊃ { k ∈ I r : X k > L } ∪ { k ∈ I r : X k ≁ L } . Therefore, 1 k r | { k ≤ k r : X k > L } ∪ { k ≤ k r : X k ≁ L } | ≥ 1 k r | { k ∈ I r : X k > L } ∪ { k ∈ I r : X k ≁ L } | = h r β k r 1 h r β | { k ∈ I r : X k > L } ∪ { k ∈ I r : X k ≁ L } | . Taking limit as r→ ∞ and using (1), we get X k → S ( F , b ) ⇒ X k → S θ β ( F , b ) .

Let θ = ( k r ) r ∈ ℕ and θ ′ = ( s r ) r ∈ ℕ be two lacunary sequences such that I _r ⊆ J _r for all r ∈ ℕ . Suppose also that the parameters β and γ are fixed real numbers such that 0 < β ≤ γ ≤ 1. We shall now give general inclusion relation for different choices of θ.

Theorem 19. Let X = (X _k ) be a sequence of fuzzy numbers, θ = (k _r ) and θ′ = (s _r ) be two lacunary sequences such that I _r ⊂ J _r for all r ∈ ℕ and β and γ be such that 0 < β ≤ γ ≤ 1,

(i) If lim r → ∞ inf h r β ℓ r γ > 0(2) then

(ii) If lim r → ∞ ℓ r h r γ = 1(3) then S θ β ( F , b ) ⊆ S θ ′ γ ( F , b ) .

Proof. (i) Suppose that I _r ⊂ J _r for all r ∈ ℕ and let (2) be satisfied. For a fuzzy number L we have { k ∈ J r : X k > L } ∪ { k ∈ J r : X k ≁ L } ⊇ { k ∈ I r : X k > L } ∪ { k ∈ I r : X k ≁ L } and so 1 ℓ r γ | { k ∈ J r : X k > L } ∪ { k ∈ J r : X k ≁ L } | ≥ h r β ℓ r γ 1 h r β | { k ∈ I r : X k > L } ∪ { k ∈ I r : X k ≁ L } | for all r ∈ ℕ , where I _r = (k_r-1, k _r ], J _r = (s_r-1, s _r ], h _r = k _r - k_r-1, ℓ _r = s _r - s_r-1. Now taking the limit as r→ ∞ in the last inequality and using (2) we get

(ii) Let X = ( X k ) ∈ S θ β ( F , b ) and (3) be satisfied. Since I _r ⊂ J _r , for a fuzzy number L we may write 1 ℓ r γ | { k ∈ J r : X k > L } ∪ { k ∈ J r : X k ≁ L } | [ c ] c = 1 ℓ r γ | { s r - 1 < k ≤ k r - 1 : X k > L } ∪ { s r - 1 < k ≤ k r - 1 : X k ≁ L } | + 1 ℓ r γ | { k r < k ≤ s r : X k > L } ∪ { k r < k ≤ s r : X k ≁ L } | + 1 ℓ r γ | { k r - 1 < k ≤ k r : X k > L } ∪ { k r - 1 < k ≤ k r : X k ≁ L } | ≤ k r - 1 - s r - 1 ℓ r γ + s r - k r ℓ r γ + 1 ℓ r γ | { k ∈ I r : X k > L } ∪ { k ∈ I r : X k ≁ L } | = ℓ r - h r ℓ r γ + 1 ℓ r γ | { k ∈ I r : X k > L } ∪ { k ∈ I r : X k ≁ L } | ≤ ℓ r - h r γ h r γ + 1 h r γ | { k ∈ I r : X k > L } ∪ { k ∈ I r : X k ≁ L } | ≤ ( ℓ r h r γ - 1 ) + 1 h r β | { k ∈ I r : X k > L } ∪ { k ∈ I r : X k ≁ L } | for all r ∈ ℕ . Since lim r → ∞ ℓ r h r β = 1 by (3) the first term and since X ∈ S θ β ( F , b ) the second term of right hand side of above inequality tend to 0 as r→ ∞. This implies that S θ β ( F , b ) ⊆ S θ ′ γ ( F , b ) .

The following results are derivable easily from Theorem 19.

Corollary 20. Let X = (X _k ) be a sequence of fuzzy numbers, and let θ = (k _r ) and θ′ = (s _r ) be two lacunary sequences such that I _r ⊆ J _r for all r ∈ ℕ .

If the condition (2) is satisfied, then

(i) S θ β ( F , b ) ⊆ S θ ′ β ( F , b ) for each β ∈ (0, 1],

(ii) S θ β ( F , b ) ⊆ S θ ′ β ( F , b ) for each β ∈ (0, 1],

(iii) S θ β ( F , b ) ⊆ S θ ′ β ( F , b )

Furthermore, if the condition (3) is satisfied, then

(i) S θ β ( F , b ) ⊆ S θ ′ β ( F , b ) for each β ∈ (0, 1],

(ii) S θ β ( F , b ) ⊆ S θ ′ ( F , b ) for each β ∈ (0, 1],

(iii) S _θ (F, b)⊆S_θ′ (F, b).

The notion of statistical convergence and statistical boundedness for sequences of fuzzy numbers defined by Nuray and Savaş [26] and Aytar and Pehlivan [6], respectively. For β ∈ (0, 1], Çolak [10] generalized the definition of statistical convergence for real sequences and introduced the statistical convergence of order β. Et et al [15] defined the lacunary statistical boundedness of order β for real sequences using a lacunary sequence θ = (k _r ) such that h _r = (k _r - k_r-1)→ ∞ as r → ∞. In this study we generalized the study of Et et al [15] for sequences of fuzzy numbers and so defined the lacunary statistical boundedness of order β. Furthermore, we gave some inclusion theorems relation to statistical boundedness and lacunary statistical boundedness, and examined the some properties like symmetricity, normality and monotonicity.

Statistical convergence has several generalizations and applications in different fields of mathematics such as: rough convergence, rough continuity and rough statistical convergence. The theory of soft rough sets and soft rough hemirings studied by Zhan et al. ([35–38]) and Ma et al [23]. Regarding this topic, the concept of rough statistical boundedness for sequence of fuzzy numbers can be studied. This is an open problem to work for researchers.