Lacunary statistical convergence defined by an Orlicz function in sequences of fuzzy numbers

1 Introduction

In order to generalize the concept of convergence of real sequences, the notion of statistical convergence was introduced by Fast [14] and Schoenberg [31], independently. Over the years and under different names statistical convergence has been discussed in the theory of Fourier analysis, Ergodic theory and Number theory. Later on it was further investigated from the sequence space point of view and linked with summability theory by Fridy [15], Šalát [30], Connor [9], topological groups by Çakallı [6], function spaces by Caserta et al. [7]. Fridy and Orhan [17] introduced the concept of lacunary statistical convergence. Some works on lacunary statistical convergence can be found in [10, 33]. Çolak [8] generalized the statistical convergence by ordering the interval [0, 1] and defined the statistical convergence of order α for α ∈ (0, 1] . For an extensive view on this subject we refer [3, 12].

Matloka [21] defined the notion of fuzzy sequence and introduced bounded and convergent sequences of fuzzy real numbers and studied their some properties. Subsequently, there has been increasing interest in the study of fuzzy numbers (see [22, 27–29]). Nuray and Savaş [26] defined the notion of statistical convergence for sequences of fuzzy numbers. The rough set theory and rough convergence which is generalization of classical convergence for sequences of real and fuzzy numbers also studied by Aytar [5], Akçay and Aytar [2], Zhan et al. [36].

One of the topics studied in summability theory is the concept of difference sequences. The difference sequence spaces were defined by Kızmaz [20], who studied the spaces l_∞ (Δ), c (Δ) and c₀ (Δ) and have been studied many authors for real and fuzzy cases (see [3, 24]). However, there are many applications of the sequences and difference sequences of numbers (complex and fuzzy numbers). For instance, sequences of numbers have interesting and practical uses in many areas of science and engineering, including acoustics. One of these applications was given on a paper of Kawamura et al. [18]. In that paper, Kawamura et al. [18] showed that the earthquake ground motions have very simple conditioned fuzzy set rules with non-fuzzy parameters of the first and second order differences ΔX _i and Δ²X _i defined by membership functions μ’s in order to predict earthquakes waves.

2 Definitions and preliminaries

In this section, we recall some basic definitions and notations that we are going to use in this paper.

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 [16].

An Orlicz function is a function M : [0, ∞) → [0, ∞), which is continuous, non-decreasing and convex with M (0) =0, M (x) >0 for x > 0 and M (x)→ ∞ as x→ ∞. It is well known that if M is a convex function and M (0) = 0, then M (λx) ≤ λM (x) for all λ with 0 < λ < 1 . Recently, Orlicz functions have been studied by Altın et al. [1], Mursaleen et al. [23], Savaş and Rhoades [32] and many others.

Now, we give one of the frequently used inequalities as follows:

Let p = (p _k ) be a sequence of positive real numbers with 0 < p _k  ≤ sup p _k  = H, and let D = max(1, 2^H-1) . Then for a k , b k ∈ ℂ for all k ∈ ℕ , we have | a k + b k | p k ≤ D { | a k | p k + | b k | p k } .

A fuzzy set u on ℝ is called a fuzzy number if it has the following properties:

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

It is clear that u is a fuzzy number if and only if [u]  ^α is a closed interval for each α ∈ [0, 1] and [u] ¹ ≠ ∅ . We denote space of all fuzzy numbers by L ( ℝ ) .

In order to calculate the distance between two fuzzy numbers u and v, we use the metric d ( u , v ) = sup 0 ≤ α ≤ 1 d H ( [ u ] α , [ v ] α ) where d _H is the Hausdorff metric defined by d H ( [ u ] α , [ v ] α ) = max { | u _ α - v _ α | , | u ¯ α - v ¯ α | } .

It is known that d is a metric on L ( ℝ ) , and ( L ( ℝ ) , d ) is a complete metric space.

A sequence X = (X _k ) of fuzzy numbers is a function X : ℕ → L ( ℝ ) . Let X = (X _k ) be a sequence of fuzzy numbers. The sequence X = (X _k ) of fuzzy numbers is said to be bounded if the set { X k : k ∈ ℕ } of fuzzy numbers is bounded and convergent to the fuzzy number X₀, written as lim k X k = X₀, if for every ɛ > 0 there exists a positive integer k₀ such that d (X _k , X₀) < ɛ for k > k₀ . Let ℓ_∞ (F) and c (F) denote the set of all bounded sequences and all convergent sequences of fuzzy numbers, respectively [21].

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 the 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].

In this study, we extend the notion of lacunary statistical convergence of order β with respect to an Orlicz function for sequences of fuzzy numbers using generalized difference operator Δ ^m such that Δ ^m X _k  = Δ^m-1X _k  - Δ^m-1X_k+1 for (m = 1, 2, 3, . . .) and give some relation theorems so as to fill up the existing gaps in the theory of lacunary statistical convergence of fuzzy numbers.

3 Main results

In this section at first, we recall the notion S θ β ( F , Δ m ) -statistical convergence defined by Altınok and Yağdıran [3]. After then, we define the strong N θ β ( p , F , Δ m , M ) -summability and give some inclusion relations among them.

Definition 3.1. [3] 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 Δ ^m -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 ) .

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 a sequence X = (X _k ) of fuzzy numbers such that X k ( t ) = { t - 1 , for 1 ≤ t ≤ 3 - t + 5 , for 3 ≤ t ≤ 5 0 , otherwise } : = X ′ , if k is odd t - 5 , for 5 ≤ t ≤ 7 - t + 9 , for 7 ≤ t ≤ 9 0 , otherwise } : = X ′ ′ , if k is even

We can find the α-level sets of sequences (X _k ) and (Δ ^m X _k ) after some arithmetic operations as follows, respectively [ X k ] α = { [ α + 1 , 5 - α ] , if k is odd [ α + 5 , 9 - α ] , if k is even and [ Δ m X k ] α = { [ 2 m ( α - 4 ) , - 2 m α ] : = X ′ , if k is odd [ 2 m α , 2 m ( 4 - α ) ] : = X ′ ′ , if k is even , ( 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 - k 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 the sequence X = (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.

Definition 3.2. Let M be an Orlicz function, X = (X _k ) be a sequence of fuzzy numbers, p = (p _k ) be a sequence of strictly positive real numbers with 0 < p _k  ≤ sup p _k  = H, β ∈ (0, 1] be any real number, Δ ^m be a generalized difference operator and θ ={ k _r  } be a lacunary sequence. Now we define N θ β ( p , F , Δ m , M ) = { X = ( X k ) : lim r → ∞ 1 h r β ∑ k ∈ I r [ M ( d ( Δ m X k , X 0 ) ρ ) ] p k = 0 , for some L and ρ > 0 } .

If X ∈ N θ β ( p , F , Δ m , M ) , then we say that X is strongly N θ β ( p , F , Δ m , M ) -summable with respect to the Orlicz function M . In the special case M (X) = X, we shall write N θ β ( p , F , Δ m , M ) - summability reduces to the strong N θ β ( p , F , Δ m ) summability defined in [3].

Theorem 3.3. Let θ ={ k _r  } and θ^∣ ={ s _r  } be two lacunary sequence. If

( i ) lim r → ∞ inf h r β l r γ > 0(3.1) then S θ ∣ γ ( F , Δ m ) ⊆ S θ β ( F , Δ m ) and if ( ii ) lim r → ∞ l r h r γ = 1(3.2) then S θ β ( F , Δ m ) ⊆ S θ ∣ γ ( F , Δ m ) for each r ∈ ℕ , where I _r  ⊂ J _r ve 0 < β ≤ γ ≤ 1 .

Proof. (i) Suppose that I _r  ⊂ J _r and (3.1) holds for each r ∈ ℕ . Then we have for ɛ > 0 { k ∈ J r : d ( Δ m X k , X 0 ) ≥ ɛ } ⊇ { k ∈ I r : d ( Δ m X k , X 0 ) ≥ ɛ } and 1 l r γ | { k ∈ J r : d ( Δ m X k , X 0 ) ≥ ɛ } | ≥ h r β l r γ 1 h r β | { k ∈ I r : d ( Δ m X k , X 0 ) ≥ ɛ } | for 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 . If we take limit in the last inequality as r→ ∞ and use (3.1), then we get S θ ∣ γ ( F , Δ m ) ⊆ S θ β ( F , Δ m ) .

(ii) Let X = ( X k ) ∈ S θ β ( F , Δ m ) and (3.2) holds. Then we can write 1 l r γ | { k ∈ J r : d ( Δ m X k , X 0 ) ≥ ɛ } | = 1 l r γ | { s r - 1 < k ≤ k r - 1 : d ( Δ m X k , X 0 ) ≥ ɛ } | + 1 l r γ | { k r < k ≤ s r : d ( Δ m X k , X 0 ) ≥ ɛ } | + 1 l r γ | { k r - 1 < k ≤ k r : d ( Δ m X k , X 0 ) ≥ ɛ } | ≤ k r - 1 - s r - 1 l r γ + s r - k r l r γ + 1 l r γ | { k ∈ I r : d ( Δ m X k , X 0 ) ≥ ɛ } | ≤ l r - h r γ h r γ + 1 h r γ | { k ∈ I r : d ( Δ m X k , X 0 ) ≥ ɛ } | ≤ ( l r h r γ - 1 ) + 1 h r γ | { k ∈ I r : d ( Δ m X k , X 0 ) ≥ ɛ } | for ɛ > 0 since I _r  ⊂ J _r . On the other hand we have S θ β ( F , Δ m ) ⊆ S θ ∣ γ ( F , Δ m ) since lim r → ∞ l r h r γ = 1 and X = ( X k ) ∈ S θ β ( F , Δ m ).

We can obtain following two results from Theorem 3.3.

Corollary 3.4. Let θ ={ k _r  } and θ^∣ ={ s _r  } be two lacunary sequence and I _r  ⊂ J _r for β ∈ (0, 1] and each r ∈ ℕ. If (3.1) holds, then we can write

S θ ∣ β ( F , Δ m ) ⊆ S θ β ( F , Δ m ) ,

Corollary 3.5. Let θ ={ k _r  } and θ^∣ ={ s _r  } be two lacunary sequence and I _r  ⊂ J _r for β ∈ (0, 1] and each r ∈ ℕ. If (3.2) holds, then we can write

S θ β ( F , Δ m ) ⊆ S θ ∣ β ( F , Δ m ) ,

Theorem 3.6. Let 0 < β ≤ 1 and X = (X _k ), Y = (Y _k ) be sequences of fuzzy numbers, Then

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

Proof. Omitted.

Theorem 3.7. Let θ ={ k _r  } and θ^∣ ={ s _r  } be two lacunary sequence. Then

If (3.1) holds, then N θ ∣ γ ( p , F , Δ m , M ) ⊂ N θ β ( p , F , Δ m , M )

Proof. (i) Suppose that I _r  ⊂ J _r for all r ∈ ℕ and (3.1) holds. Then we have 1 l r γ ∑ k ∈ I r [ M ( d ( Δ m X k , X 0 ) ρ ) ] p k > h r β l r γ 1 h r β ∑ k ∈ I r [ M ( d ( Δ m X k , X 0 ) ρ ) ] p k since N θ ∣ γ ( p , F , Δ m , M ) ⊂ N θ β ( p , F , Δ m , M ).

(ii) Suppose that X = (X _k ) ∈ l_∞ (F, Δ ^m ) and (3.2) holds. Since X = (X _k ) is a Δ ^m -bounded sequence, there is a real number T > 0 such that d (Δ ^m X _k , X₀) ≤ T for all k ∈ ℕ . On the other hand we can write h _r  ≤ l _r since I _r  ⊂ J _r for all r ∈ ℕ and also p _k  = p for all k ∈ ℕ 1 l r γ ∑ k ∈ J r [ M ( d ( Δ m X k , X 0 ) ρ ) ] p = 1 l r γ ∑ k ∈ J r - I r [ M ( d ( Δ m X k , X 0 ) ρ ) ] p + 1 l r γ ∑ k ∈ I r [ M ( d ( Δ m X k , X 0 ) ρ ) ] p ≤ ( l r - h r l r γ ) T p + 1 l r γ ∑ k ∈ I r [ M ( d ( Δ m X k , X 0 ) ρ ) ] p ≤ ( l r - h r γ h r γ ) T p + 1 h r γ ∑ k ∈ I r [ M ( d ( Δ m X k , X 0 ) ρ ) ] p ≤ ( l r - h r γ h r γ ) T p + 1 h r β ∑ k ∈ I r [ M ( d ( Δ m X k , X 0 ) ρ ) ] p .

Hence we conclude that l ∞ ( F , Δ m ) ∩ N θ β ( p , F , Δ m , M ) ⊂ N θ ∣ γ ( p , F , Δ m , M ).

Corollary 3.8. Let θ ={ k _r  } and θ^∣ ={ s _r  } be two lacunary sequence and I _r  ⊂ J _r for β ∈ (0, 1] and all r ∈ ℕ. If (3.1) holds, then

N θ ∣ β ( p , F , Δ m , M ) ⊆ N θ β ( p , F , Δ m , M ) ,

Corollary 3.9. Let θ ={ k _r  } and θ^∣ ={ s _r  } be two lacunary sequence and I _r  ⊂ J _r for β ∈ (0, 1] and all r ∈ ℕ. If (3.2) holds, then

l ∞ ( F , Δ m ) ∩ N θ β ( p , F , Δ m , M ) ⊆ N θ ∣ β ( p , F , Δ m , M ) ,

Theorem 3.10. Let 0 < β ≤ γ ≤ 1, p = (p _k ) be a sequence of strictly positive real numbers with 0 < p _k  ≤ sup p _k  = H, and I _r  ⊂ J _r for all r ∈ ℕ.

If a sequence X = (X _k ) is strongly N θ ∣ γ ( p , F , Δ m , M ) -summable to X₀, then it is S θ β ( F , Δ m ) -statistically convergent to X₀ in case (3.1) holds

Proof. (i) We can write 1 l r γ ∑ k ∈ J r [ M ( d ( Δ m X k , X 0 ) ρ ) ] p k = 1 l r γ ∑ k ∈ J r d ( Δ m X k , X 0 ) ≥ ϵ [ M ( d ( Δ m X k , X 0 ) ρ ) ] p k + 1 l r γ ∑ k ∈ J r d ( Δ m X k , X 0 ) < ϵ [ M ( d ( Δ m X k , X 0 ) ρ ) ] p k ≥ 1 l r γ ∑ k ∈ I r d ( Δ m X k % , X 0 ) ≥ ϵ [ M ( d ( Δ m X k , X 0 ) ρ ) ] p k + 1 l r γ ∑ k ∈ I r d ( Δ m X k , X 0 ) < ϵ [ M ( d ( Δ m X k , X 0 ) ρ ) ] p k ≥ 1 l r γ ∑ k ∈ I r d ( Δ m X k , X 0 ) ≥ ϵ [ M ( d ( Δ m X k , X 0 ) ρ ) ] p k and hence ⇒ 1 l r γ ∑ k ∈ J r [ M ( d ( Δ m X k , X 0 ) ρ ) ] p k ≥ 1 l r γ ∑ k ∈ I r d ( Δ m X k % , X 0 ) ≥ ϵ [ M ( d ( Δ m X k , X 0 ) ρ ) ] p k ≥ 1 l r γ ∑ k ∈ I r d ( Δ m X k , X 0 ) ≥ ϵ [ M ( ε ρ ) ] p k ≥ 1 l r γ ∑ k ∈ I r d ( Δ m X k , X 0 ) ≥ ϵ min { [ M ( ε 1 ) ] inf p k , [ M ( ε 1 ) ] sup p k } , ε 1 = ε ρ ≥ 1 l r γ | { k ∈ I r : d ( Δ m X k , X 0 ) ≥ ε } | . min { [ M ( ε 1 ) ] inf p k , [ M ( ε 1 ) ] sup p k } ≥ h r β l r γ 1 h r β | { k ∈ I r : d ( Δ m X k , X 0 ) ≥ ε } | . min { [ M ( ε 1 ) ] inf p k , [ M ( ε 1 ) ] sup p k % }

From above inequalities we can say that X = (X _k ) is S θ β ( F , Δ m ) -statistically convergent to X₀ since (3.1) holds.

(ii) Proof is similar to operations in one of (i).

Corollary 3.11. Let θ ={ k _r  } and θ^∣ ={ s _r  } be two lacunary sequence and I _r  ⊂ J _r for β ∈ (0, 1] and all r ∈ ℕ. If (3.1) holds, then

N θ ∣ β ( p , F , Δ m , M ) ⊆ S θ β ( F , Δ m ) ,

Corollary 3.12. Let θ ={ k _r  } and θ^∣ ={ s _r  } be two lacunary sequence and I _r  ⊂ J _r for β ∈ (0, 1] and all r ∈ ℕ. If (3.2) holds, then

l ∞ ( F , Δ m ) ∩ S θ β ( F , Δ m ) ⊆ N θ ∣ β ( p , F , Δ m , M ) ,

Theorem 3.13. If lim p _k  > 0 and X = (X _k ) is strongly N θ β ( p , F , Δ m , M ) -summable to X′ with respect to the Orlicz function M, then N θ β ( p , F , Δ m , M ) - lim X k is unique.

Proof. Let lim p _k  = s > 0, N θ β ( p , F , Δ m , M ) - lim X k = X 0 and N θ β ( p , F , Δ m , M ) - lim X k = X 0 ′ . Then we write lim r → ∞ 1 h r β ∑ k ∈ I r [ M ( d ( Δ m X k , X 0 ) ρ 1 ) ] p k = 0 , ρ 1 > 0 and lim r → ∞ 1 h r β ∑ k ∈ I r [ M ( d ( Δ m X k , X 0 ′ ) ρ 2 ) ] p k = 0 , ρ 2 > 0

Take ρ = max(2ρ₁, 2ρ₂) . Since M is non-decreasing and convex we can write 1 h r β ∑ k ∈ I r [ M ( d ( X 0 , X 0 ′ ) ρ ) ] p k ≤ D h r β ∑ k ∈ I r 1 2 p k ( [ M ( d ( Δ m X k , X 0 ) ρ 1 ) ] p k + [ M ( d ( Δ m X k , X 0 ′ ) ρ 2 ) ] p k ) ≤ D h r β ∑ k ∈ I r [ M ( d ( Δ m X k , X 0 ) ρ 1 ) ] p k + D h r β ∑ k ∈ I r [ M ( d ( Δ m X k , X 0 ′ ) ρ 2 ) ] p k → 0 , where sup p _k  = H and D = max(1, 2^H-1) . From above inequalities, we have lim r → ∞ 1 h r β ∑ k ∈ I r [ M ( d ( X 0 , X 0 ′ ) ρ ) ] p k = 0 and so lim k → ∞ 1 h r β ∑ k ∈ I r [ M ( d ( X 0 , X 0 ′ ) ρ ) ] p k = [ M ( d ( X 0 , X 0 ′ ) ρ ) ] s since lim p _k  = s . It means that X 0 = X 0 ′ and limit is unique.