Statistical Convergence of order β for ( λ,μ ) double 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 [18]. Schoenberg [44] gave some basic properties of statistical convergence. For more details about statistical convergence one can refer to Connor [15], Fridy [19], Šalát [38], Gadjiev and Orhan [20] introduced the definition of order of statistical convergence for positive linear operator and after that generalization of statistical convergence was introduced by Çolak [12] under name of statistical convergence of order α. Over the years and under different names statistical convergence has been discussed in the theory of Fourier analysis, Ergodic theory and Number theory. 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-Cech compactification of the natural numbers. Moreover, statistical convergence is closely related to the concept of convergence in probability. Aizpuru et al. [1] defined the f-density of the subset A of ℕ by using an unbounded modulus function. After that, Bhardwaj and Dhawan [10] introduced f-statistical convergence of order α and strong Cesaro summability of order α with respect to a modulus function f- for real sequences. For a detailed account of many more interesting investigations concerning statistical convergence of order β, one may refer to ([4, 50]).

Mursaleen and Edely [32] introduced the definition of double statistical convergence of number sequences using double natural density of positive integers. Besides this topic was studied by many authors ([11, 48]).

Recently, the fuzzy theory has emerged as the most active area of research in many branches of mathematics and engineering. Zadeh [51] in 1965 introduced the concept of fuzzy set, which is defined with the help of grades of membership. Zadeh [51] put forward the concept of fuzzy sets as a formal mathematical system to model human reasoning and decision making processes in uncertain environments. Zadeh’s [51] study attracted many researchers in different fields of science and found numerous applications ranging from engineering to mathematics.This theory has been developed and influenced in many areas of application.

The idea of bounded and convergent sequences of fuzzy numbers initially discussed by Matloka [25] where it was shown that every convergent sequence is bounded. Nanda [34] proved that the spaces of bounded and convergent sequences of fuzzy numbers are complete metric spaces. After that, Nuray and Savaş [36] defined the concept of statistical convergence for sequences of fuzzy numbers. For some further works in this direction we refer to ([2, 49]) Savaş [41] introduced the double sequences of fuzzy numbers. Savaş and Mursaleen [43] studied the statistical convergence for double sequences of fuzzy numbers and others ([8, 46]).

Mursaleen [31] introduced λ- statistical convergence as an extension of (V, λ)- summability of Leindler [24] with the help of a non-decreasing sequence λ = (λ _n ) of positive numbers tending to ∞ with λ_n+1 ≤ λ _n +1, λ₁ = 1 . The generalized de la Vallee-Poussion mean is defined by t n ( x ) = 1 λ n ∑ k ∈ I n x k , where I _n  = [n - λ _n  + 1, n] .Meenaskshi et.al. also studied these concepts (λ, μ) statistical convergence for double sequences [26]. Later Işık and Altın [21] introduced the concepts of f_λ,μ-statistical convergence for double sequences of order α ˜ . Savaş [42] introduced and discussed the concepts of λ- statistically convergence of fuzzy numbers. The theory of λ- statistical convergence for fuzzy sequences and their properties has been studied extensively by various authors (see [5–7]). This paper organizes as follows: In Section 2, we give the basic notions which will be used throughout the paper. In Section 3, we define the concepts of φ_λ,μ-double statistically convergence of order β in fuzzy sequences and strongly λ- double Cesaro summable of order β for sequences of fuzzy numbers. Also we give some inclusion theorems. Finally in the Section 4, we mention content of paper and studies future time.

2 Definitions and preliminaries

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

Definition 2.1. A fuzzy number is fuzzy set u : ℝ → [0, 1] with the following properties:

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

ii) u is fuzzy convex, that is, for x , y ∈ ℝ and 0≤ λ ≤ 1, u (λx + (1 - λ) y) ≥ min [u (x) , u (y)] ;

iii) u is upper semicontinuous;

iv) supp u = cl { x ∈ ℝ : u ( x ) > 0 } , or denoted by [u] ⁰, is compact, then it is called a fuzzy number.

Definition 2.2. α-level set [u]  ^α of a fuzzy number u is defined by [ u ] α = { { x ∈ ℝ : u ( x ) ≥ α } , if α ∈ ( 0 , 1 ] supp u , if α = 0 . 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 ) . Definition 2.3. 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 )

Definition 2.4. Let u = [ u _ α , u ¯ α ] and v = [ v _ α , v ¯ α ] be two fuzzy numbers. Then, the Hausdorff metric is 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 ) .

Definition 2.5. A sequence X = (X _k ) of fuzzy numbers is called bounded if and only if the set { X k : k ∈ ℕ } of fuzzy numbers consisting of the terms the sequence X _k is a bounded set.

Definition 2.6. A sequence X = (X _k ) of fuzzy numbers is called convergent with limit X 0 ∈ L ( ℝ ) , if and only if for every ɛ > 0 there exists a positive integer k₀ such that d (X _k , X₀) < ɛ for all k > k₀ .

Let s (F) , ℓ_∞ (F) and c (F) denote the set of all sequences, all bounded sequences and all convergent sequences of fuzzy numbers, respectively [25].

Definition 2.7. Let φ be real-valued function defined on [0, ∞) satisfying the following:

i) φ (t) = 0 iff t = 0,

ii) φ (t + u) ≤ φ (t) + φ (u) for t, u ≥ 0,

iii) φ is increasing,

iv) φ is right-continuous at 0, see [35].

Such a function is called a modulus function; some examples of modulus functions are t ^p , (0 < p ≤ 1) , log(1 + t) . A modulus function can be bounded or unbounded.

For an extensive view on this subject we refer ([39, 45]).

Throughout the paper, we will take β instead of (a, b) and γ instead of (c, d) for a, b, c, d ∈ (0, 1] as follows [13]:

β ≺ γ ⇔ a < c and b < d , β ⪯ γ ⇔ a ≤ c and b ≤ d , β ≃ γ ⇔ a = c and b = d , β ∈ ( 0 , 1 ] ⇔ a , b ∈ ( 0 , 1 ] , γ ∈ ( 0 , 1 ] ⇔ c , d ∈ ( 0 , 1 ] , β ≃ 1 ⇔ a = b = 1 β ≻ 1 ⇔ a > 1 , b > 1 .

Let λ = (λ _n ) and μ = (μ _m ) be two non-decreasing sequences of positive real numbers tending to ∞ with λ_n+1 ≤ λ _n  + 1, λ₁ = 0 ; μ_n+1 ≤ μ _n  + 1, μ₁ = 0 and β ∈ (0, 1] be given.

Let K ⊆ ℕ × ℕ be two dimensional set of positive integers and φ be an unbounded modulus function.

Definition 2.8. Then, δ β f 2 ( λ , μ ) - double density of K is defined as δ β f 2 ( K ) = lim n , m → ∞ 1 φ ( λ n s μ m t ) φ ( | { ( k , ℓ ) ∈ I n × I m : ( i , j ) ∈ K } | ) , if the limit exists , see [ 21 ] .

3 Main results

In this section we give the concept of φ_λ,μ-double statistically convergence of order β in fuzzy sequences.

Definition 3.1. Let λ = (λ _n ) and μ = (μ _m ) be two non-decreasing sequences of positive real numbers as above and β ∈ (0, 1] be given and X = (X_kℓ) be a double sequence of fuzzy numbers. If for every ɛ > 0, lim n , m → ∞ 1 φ ( λ n a μ m b ) φ ( | { ( j , k ) ∈ I n × I m : d ( X k ℓ , X 0 ) ≥ ɛ } | ) = 0 , a double sequence X = (X_kℓ) of fuzzy numbers is said to be φ_λ,μ-double statistically convergent of order β to the fuzzy number X₀ . In this case we write S 2 β ( F , λ , μ , φ ) - lim X k ℓ = X 0 or X k ℓ → X 0 ( S 2 β ( F , λ , μ , φ ) ) where φ is an unbounded modulus function. By S 2 β ( F , λ , μ , φ ) , we shall denote the set of all double sequences of fuzzy numbers which are φ_λ,μ-statistically convergent of order β . Where I _n  = [n - λ _n  + 1, n] and I _m  = [m - μ _m  + 1, m] .

It is easy to see that every convergent sequence is φ_λ,μ-double statistically convergent of order β, but converse does not hold as following example:

Example 3.2. Take modulus function φ (x) = x ^p for 0 < p ≤ 1 and λ _n  = n,μ _m  = m . Consider the sequence X k ℓ ( x ) = { x ( k ℓ ) 3 + 1 , for x ∈ [ - ( k ℓ ) 3 , 0 ] - x ( k ℓ ) 3 + 1 , for x ∈ [ 0 , - ( k ℓ ) 3 ] 0 , otherwise } if k = n 3 , ℓ = m 3 x - 4 , for x ∈ [ 4 , 5 ] - x + 6 , for x ∈ [ 5 , 6 ] 0 , otherwise } otherwise Then, we calculate the α-level set of sequences (X_kℓ) as follows [ X k ℓ ] α = { [ ( k ℓ ) 3 ( α - 1 ) , - ( k ℓ ) 3 ( α - 1 ) ] , if k = n 3 , ℓ = m 3 [ α + 4 , 6 - α ] , otherwise . Hence (X_kℓ) is φ_λ,μ-double statistically convergent of order β, for β ∈ ( 1 3 , 1 ] to fuzzy number X₀, where [X₀]  ^α  = [α + 4, 6 - α] , but not convergent.

Theorem 3.3. Let β ∈ (0, 1] and X = (X_kℓ) , Y = (Y_kℓ) be two double sequences of fuzzy numbers. Then(i) X k ℓ → X 0 ( S 2 β ( F , λ , μ , φ ) ) and ν ∈ ℂ implies ( ν X k ℓ ) → ν X 0 ( S 2 β ( F , λ , μ , φ ) ) , (ii) X k ℓ → X 0 ( S 2 β ( F , λ , μ , φ ) ) and Y k ℓ → Y 0 ( S 2 β ( F , λ , μ , φ ) ) implies ( X k ℓ + Y k ℓ ) → ( X 0 + Y 0 ) ( S 2 β ( F , λ , μ , φ ) ) .

Proof. (i) Proof follows from the Minkoski inequality in [23],

lim n , m → ∞ 1 φ ( λ n a μ m b ) φ ( | { ( k , ℓ ) ∈ I n × I m : d ( ν X k ℓ , ν X 0 ) ≥ ɛ } | ) ≤ 1 φ ( λ n a μ m b ) φ ( | { ( k , ℓ ) ∈ I n × I m : d ( X k ℓ , X 0 ) ≥ ɛ | ν | } | ) (ii) It follows from the inequality 1 φ ( λ n a μ m b ) φ ( | { ( k , ℓ ) ∈ I n × I m : d ( X k ℓ + Y k ℓ , X 0 + Y 0 ) ≥ ɛ } | ) ≤ 1 φ ( λ n a μ m b ) φ ( | { ( k , ℓ ) ∈ I n × I m : d ( X k ℓ , X 0 ) ≥ ɛ 2 } | ) + 1 φ ( λ n a μ m b ) φ ( | { ( k , ℓ ) ∈ I n × I m : d ( Y k ℓ , Y 0 ) ≥ ɛ 2 } | )

Theorem 3.4. Let φ be unbounded modulus function, X = (X_kℓ) be a double sequences of fuzzy numbers and β, γ ∈ (0, 1]. Then S 2 β ( F , λ , μ , φ ) ⊂ S 2 γ ( F , λ , μ , φ ) and the inclusion is strict.

Proof. It can be easily shown the inclusion by using the fact that φ is increasing for β ⪯ γ. Now, we show that the inclusion is strict. For this, consider fuzzy sequence X = (X_kℓ) defined by

X k ℓ ( x ) = { x - k ℓ + 1 , if k ℓ - 1 ≤ x ≤ k ℓ - x + k ℓ + 1 , if k ℓ ≤ x ≤ k ℓ + 1 0 , otherwise } , if k = n 2 , ℓ = m 2 x - 2 , if 2 ≤ x ≤ 3 - x + 4 , if 3 ≤ x ≤ 4 0 , otherwise } , otherwise and take modulus function φ (x) = x ^p , 0 < p ≤ 1. We can find the α-level set of sequence (X_kℓ) as follows: [ X k ℓ ] α = { [ k ℓ + α - 1 , k ℓ + 1 - α ] , if k = n 2 , ℓ = m 2 [ α + 2 , 4 - α ] , otherwise . Then, the fuzzy double sequence (X_kℓ) is φ-double statistically convergent of order γ for 1 2 < γ ≤ 1 , but not φ-double statistically convergent of order β for β ∈ ( 0 , 1 2 ] .

Corollary 3.5. Let X = (X_kℓ) be a fuzzy double sequence, φ be an unbounded modulus function and β ∈ (0, 1]. Then S 2 β ( F , λ , μ , φ ) ⊂ S 2 ( F , λ , μ , φ ) and the inclusion is strict, also the limits of sequence X = (X_kℓ) of fuzzy numbers are same.

Corollary 3.6. Let X = (X_kℓ) be a fuzzy double sequence, φ be an unbounded modulus function and β, γ ∈ (0, 1] . Then

i) S 2 β ( F , λ , μ , φ ) = S 2 γ ( F , λ , μ , φ ) if and only if β ≃ γ,

ii) S 2 β ( F , λ , μ , φ ) = S 2 ( F , λ , μ , φ ) if and only if γ ≃ 1,

iii) S 2 β ( F , λ , μ , φ ) ⊆ S 2 γ ( F , λ , μ , φ ) ⊆ S 2 ( F , λ , μ, φ) for β ⪯ γ .

Definition 3.7. Let X = (X_kℓ) be a double sequence of fuzzy numbers, φ be an unbounded modulus function and β ∈ (0, 1]. If there is a fuzzy number X₀ such that lim n , m → ∞ 1 λ n a μ m b ∑ k ∈ I n ∑ ℓ ∈ I m φ ( d ( X k ℓ , X 0 ) ) = 0 , a double sequence X = (X_kℓ) of fuzzy numbers is said to be strongly λ- double Cesaro summable of order β to a fuzzy number X₀ . By w 2 β ( F , λ , μ , φ ) , we shall denote the set of all double sequences of fuzzy numbers which are strongly λ- double Cesaro summable of order β.

Theorem 3.8. Let X = (X_kℓ) be a double sequence of fuzzy numbers, φ be an unbounded modulus function and 0 < β ⪯ γ ≤ 1. Then w 2 β ( F , λ , μ , φ ) ⊂ w 2 γ ( F , λ , μ , φ ) and also the limits are same.

Proof. It is easy to show the inclusion relation. For strictness of inclusion, let φ be a modulus function and consider the fuzzy double sequence X = (X_kℓ) defined by X k ℓ ( x ) = { x 2 , - x 2 + 2 , 0 0 ≤ x ≤ 2 2 ≤ x ≤ 4 otherwise } , if k = n 3 , ℓ = m 3 0 ¯ , otherwise We can write 1 λ n c μ m d ∑ k ∈ I n ∑ ℓ ∈ I m φ ( d ( X k ℓ , 0 ¯ ) ) ≤ 4 λ n 3 μ m 3 λ n c μ m d φ ( 4 ) for every n , m ∈ ℕ . So, we have ( X k ℓ ) ∈ w 2 γ ( F , λ , μ , φ ) since the right side tends to zero for γ > 1 3 as m, n → ∞ . On the other hand, we obtain

1 λ n a μ m b ∑ k ∈ I n ∑ ℓ ∈ I m φ ( d ( X k ℓ , 0 ¯ ) ) ≥ 4 λ n 3 μ m 3 - 4 λ n c μ m d φ ( 4 ) for every n ∈ ℕ . Hence we have ( X k ℓ ) ∉ w 2 β ( F , λ , μ , φ ) for 0 < β ≤ 1 3 as n, m → ∞ .

Corollary 3.9. Let φ be an unbounded modulus function and β, γ ∈ (0, 1] , then

i) w 2 β ( F , λ , μ , φ ) = w 2 γ ( F , λ , μ , φ ) if and only if β ≃ γ,

ii) w 2 β ( F , λ , μ , φ ) = w 2 ( F , λ , μ , φ ) for β ∈ (0, 1] and γ ≃ 1 .

Theorem 3.10. Let X = (X_kℓ) be a double sequence of fuzzy numbers and 0 < β ⪯ γ ≤ 1. Also, φ be an unbounded modulus function such that φ (xy) ≥ cφ (x) φ (y) for some positive constant c and for all x, y ≥ 0 and lim t → ∞ φ ( t ) t > 0 . Then w 2 β ( F , λ , μ , φ ) ⊂ S 2 γ ( F , λ , μ , φ ) .

Proof. Take any double sequence X = (X_kℓ) of fuzzy numbers and ɛ > 0 . Let H _nm  = { (k, ℓ) , k ≤ n, ℓ ≤ m : d (X_kℓ, X₀) ≥ ɛ } . From the definition of modulus function, we can write ∑ k ∈ I n ∑ ℓ ∈ I m φ ( d ( X k ℓ , X 0 ) ) = ∑ k ∈ H nm ∑ ℓ ∈ H nm φ ( d ( X k ℓ , X 0 ) ) + ∑ k ∉ H nm ∑ ℓ ∉ H nm φ ( d ( X k ℓ , X 0 ) ) ≥ ∑ k ∈ H nm ∑ ℓ ∈ H nm φ ( d ( X k ℓ , X 0 ) ) ≥ φ ( ∑ k ∈ H nm ∑ ℓ ∈ H nm φ ( d ( X k ℓ , X 0 ) ) ) ≥ φ ( | { ( k , ℓ ) , k ≤ n , ℓ ≤ m : d ( X k ℓ , X 0 ) ≥ ɛ } | ɛ ) ≥ c φ ( | { ( k , ℓ ) , k ≤ n , ℓ ≤ m : d ( X k ℓ , X 0 ) ≥ ɛ } | ) φ ( ɛ ) and 1 λ n a μ m b ∑ k ∈ I n ∑ ℓ ∈ I m φ ( d ( X k ℓ , X 0 ) ) ≥ 1 λ n a μ m b c φ ( | { ( k , ℓ ) , k ≤ n , ℓ ≤ m : d ( X k ℓ , X 0 ) ≥ ɛ } | ) φ ( ɛ ) ≥ 1 λ n c μ m d φ ( λ n c μ m d ) c φ ( | { ( k , ℓ ) , k ≤ n , ℓ ≤ m : d ( X k ℓ , X 0 ) ≥ ɛ } | ) φ ( ɛ ) φ ( λ n c μ m d ) . Hence, we have X ∈ S 2 γ ( F , λ , μ , φ ) using the fact that lim t → ∞ φ ( t ) t > 0 and X ∈ w 2 β ( F , λ , μ , φ ) . So, the proof is completed.

Corollary 3.11. Let φ be an unbounded modulus function and β, γ ∈ (0, 1] , then w 2 β ( F , λ , μ , φ ) = S 2 γ ( F , λ , μ , φ ) if and only if β ≃ γ.

4 Conclusion

Meenaskshi et al. [26] studied the concept of (λ, μ)-statistical convergence for double sequences. Later Işık and Altın [21] introduced the concepts of f_λ,μ-statistical convergence for double sequences of order α ˜ . Now in this paper we generalized the study of Meenaskshi et.al. [26] grading the interval [0, 1] by helping a value β, defined the sequence classes S 2 β ( F , λ , μ , φ ) and w 2 β ( F , λ , μ , φ ) which are φ_λ,μ-statistically convergent of order β and strongly λ- double Cesaro summable of order β, and gave some inclusion relations between them. Statistical convergence has several generalizations and applications in different fields of mathematics such as: rough convergence, rough continuity, rough statistical convergence. Regarding this topic, the concept of rough convergence of order β for sequences of fuzzy or real numbers can be studied future time.