On deferred statistical convergence of order β of sequences of fuzzy numbers

1 Introduction, definitions and preliminaries

In the first edition of his monograph puplished in Warsaw in 1935, Zygmund [43] mentioned the idea of statistical convergence and after then the notion was introduced by Steinhaus [36] and Fast [19] and later on reintroduced by Schoenberg [34]. Recently many mathematicians such as Salat [33], Fridy [20] and Connor [10] have linked the concept with summability theory. Nowadays statistical convergence have been studied by Altın et al. [2], Altınok et al. [3, 4], Çınar et al. [8], Et et al. [15–18], Işıket al. [22–25], Mursaleen [31], Nuray [32], Şengül and Et [35] and many others. As a generalization of statistical convergence, statistical convergence of order was generalized by Gadjiev and Orhan [21] and then Çolak [9] generalized the statistical convergence by ordering the interval [0, 1] and defined the statistical convergence of order β and strong p-Cesáro summability of order β for β ∈ (0, 1] and p ∈ ℝ + . Altinok et al. [4] introduced the concepts of statistical convergence of order β and strong p-Cesáro summability of order β for sequences of fuzzy numbers, where β ∈ (0, 1] and p ∈ ℝ + .

The purpose of this paper is to generalize the study of Altinok et al. [4] so as to fill up the existing gaps in the summability theory of fuzzy numbers. For a detailed account of many more interesting investigations concerning spaces of fuzzy numbers, one may refer to [5–7, 39–42].

The deferred Cesáro mean of sequences of real numbers was introduced by Agnew [1] such as: ( D p , q x ) n = 1 q ( n ) - p ( n ) ∑ p ( n ) + 1 q ( n ) x k ,(1) where {p (n)} and {q (n)} are sequences of non-negative integers satisfying p ( n ) < q ( n ) and lim n → ∞ q ( n ) = + ∞ .

The concepts of deferred density and deferred statistical convergence for real sequences were given by Küçükaslan and Yılmaztürk [27, 37] such as:

Let K be a subset of ℕ and denote the set {k: p (n)< k ≤ q (n), k ∈ K } by K _d (n). Then deferred density of K is defined by δ d ( K ) = lim n → ∞ 1 q ( n ) - p ( n ) | K d ( n ) | ,(2) provided the limit exists .

The vertical bars in (2) represents the cardinality of the set K _d (n). If q (n) = n and p (n) = 0, then deferred density coincides natural density of K.

Let {p (n)} and {q (n)} be two sequences as above and 0 < β ≤ 1 be given. We define deferredβ-density of subset K of ℕ by δ d β ( K ) = lim n → ∞ 1 ( q ( n ) - p ( n ) ) β | K d ( n ) | , provided the limit exists .

Deferred β-density δ d β ( K ) reduces to natural density δ (K) in the special cases β = 1 and q (n) = n, p (n) = 0. It can be clearly seen that every finite subset of ℕ has zero β-deferred density. Besides, it does not need to hold δ d β ( K c ) = 1 - δ d β ( K ) for 0 < β < 1 in general. It can be shown that δ d γ ( K ) ≤ δ d β ( K ) for β, γ ∈ (0, 1] such that β ≤ γ.

A fuzzy set u on ℝ is called a fuzzy number if it has 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 semi-continuous;

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

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 .

If u ∈ L ( ℝ ) , then u is called a fuzzy number.

Let u , v ∈ L ( ℝ ) , k ∈ ℝ and the α-level sets of fuzzy numbers u and v be [ u ] α = [ u _ α , u ¯ α ] and [ v ] α = [ v _ α , v ¯ α ] , α ∈ (0, 1]. Then, a partial ordering " ≤ " in L ( ℝ ) is defined by u ≤ v ⇔ u _ α ≤ v _ α and u ¯ α ≤ v ¯ α for all α ∈ (0, 1]. Some arithmetic operations for α-level sets are defined as follows: [ u + v ] a = [ u _ α + v _ α , u ¯ α + v ¯ α ] [ u - v ] a = [ u _ α - v ¯ α , u ¯ α - v _ α ] [ ku ] α = { [ k u _ α , k u ¯ α ] , if k ≥ 0 [ k u ¯ α , k u _ α ] , otherwise .

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 as 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 (for detail see [11, 38]).

By ℓ_∞ (F) and c (F) we denote the set of all bounded sequences and all convergent sequences of fuzzy numbers, respectively.

In this section, we define and examine the concepts of deferred statistical convergence of order β and strong r-deferred Cesáro summability of order β for sequences of fuzzy numbers, where β is any real number such that 0 < β ≤ 1.

Definition 1. Let {p (n)} and {q (n)} be two sequences as above and β ∈ (0, 1] be given. A sequence X = (X _k ) of fuzzy numbers is called to be deferred statistically convergent of order β if lim n → ∞ 1 ( q ( n ) - p ( n ) ) β | { p ( n ) < k ≤ q ( n ) } : d ( X k , X 0 ) ≥ ɛ } | = 0 .

In this case we write S d β - lim X k = X 0 . By S d β ( F ) we denote the set of all deferred statistically convergent sequences of fuzzy numbers of order β. If q (n) = n and p (n) = 0, then deferred statistical convergence of order β coincides statistical convergence of order β of sequences of fuzzy numbers which was introduced by Altinoket al. [4], and also in the special cases q (n) = n, p (n) = 0 and β = 1 deferred statistical convergence of order β coincides usual statistical convergence of sequences of fuzzy numbers.

Remark. Let {p (n)} and {q (n)} be sequences of natural numbers as above and β be any real number such that 0 < β ≤ 1. Let us define X ∈ [ S d ] β ⇔ lim n → ∞ 1 [ q ( n ) ] β - [ p ( n ) ] β | { [ p ( n ) ] β < k ≤ [ q ( n ) ] β : d ( X k , X 0 ) ≥ ɛ } | = 0 .

Since [q (n)] ^β - [p (n)] ^β ≤ (q (n) - p (n)) ^β we have [ S d ] β ⊂ S d β for β ∈ (0, 1], where { [p (n)] ^β } = { [p (1)] ^β , [p (2)] ^β ,… } and { [q (n)] ^β } = { [q (1)] ^β , [q (2)] ^β ,… }.

The set S d β ( F ) contains some deferred statistically convergent sequences of order β for β ∈ (0, 1]. For this consider a sequence X = (X _k ) defined by X k ( x ) = { x - 4 , for 4 ≤ x ≤ 5 - x + 6 , for 5 ≤ x ≤ 6 0 , otherwise } if k = m 3 ( m = 1 , 2 , 3 , . . . ) x - 1 , for 1 ≤ x ≤ 2 - x + 3 , for 2 ≤ x ≤ 3 0 , otherwise } : = X 0 if k ≠ m 3 . Then, we calculate α-level set of this sequence as follows: [ X k ] α = { [ 4 + α , 6 - α ] , if k = m 3 [ 1 + α , 3 - α ] , if k ≠ m 3 and so we have for q (n) = n and p (n) = 0 lim n → ∞ 1 n β | { k ≤ n : d ( [ X k ] α , [ X 0 ] α ) ≥ ɛ } | ≤ lim n → ∞ n 3 n β .

Thus, X = (X _k ) is statistically convergent of order β, to fuzzy number X₀, where [X₀] ^α = [1 + α, 3 - α] for β ∈ ( 1 3 , 1 ] .

Theorem 1. Let β be any real number such that 0 < β ≤ 1. Every statistically convergent sequence of fuzzy numbers is deferred statistically convergent of order β, but the converse is not true.

Proof. The first part of proof is easy. To show the strictness of the inclusion c ( F ) ⊆ S d β ( F ) , let us define a sequence of fuzzy numbes X = (X _k ) by X k ( x ) = { x - 5 , if 5 ≤ x ≤ 6 - x + 7 , if 6 ≤ x ≤ 7 0 , otherwise } , if k = n 3 0 ¯ , if k ≠ n 3 , then we can easily calculate α-level set of X such as [ X k ] α = { [ α + 5 , 7 - α ] , if k = n 3 [ 0 , 0 ] , if k ≠ n 3 . So we have 1 ( q ( n ) - p ( n ) ) β | { p ( n ) < k ≤ q ( n ) : d ( X k , 0 ¯ ) ≥ ɛ } | ≤ q ( n ) 3 - p ( n ) 3 + 1 ( q ( n ) - p ( n ) ) β and so X = (X _k ) is deferred statistically convergent of order β with S d β - lim X k = 0 ¯ for β > 1 3 , but it is not statistically convergent (See Fig. 1). OPEN IN VIEWER

The deferred statistical convergence of order β is well defined for 0 < β ≤ 1, but it is not well defined for β > 1 in general. For this X = (X _k ) be defined as follows: X k ( x ) = { x + 5 , - x - 3 , 0 - 5 ≤ x ≤ - 4 - 4 ≤ x ≤ - 3 otherwise } : = X 0 , if k is odd x - 3 , - x + 5 , 0 3 ≤ x ≤ 4 4 ≤ x ≤ 5 otherwise } : = X 0 ′ , if k is even then we calculate α-level set as: [ X k ] α = { [ α - 5 , - 3 - α ] : = [ X 0 ] α , if k is odd [ α + 3 , 5 - α ] : = [ X 0 ′ ] α , if k is even . Then both lim n → ∞ 1 n β | { p ( n ) < k ≤ q ( n ) : d ( X k , X 0 ) ≥ ɛ } | ≤ lim n → ∞ n 2 n β = 0 and lim n → ∞ 1 n β | { p ( n ) < k ≤ q ( n ) : d ( X k , X 0 ′ ) ≥ ɛ } | ≤ lim n → ∞ n + 1 2 n β = 0 for β > 1 and q (n) = 4n, p (n) = 3n. So we have S d β - lim X k = X 0 and S d β - lim X k = X 0 ′ , but this is impossible.

Definition 2. Let {p (n)} and {q (n)} be two sequences as above, β and r be two positive real numbers. A sequence X = (X _k ) is said to be strongly r-deferred Cesáro summable of order β if lim n → ∞ 1 ( q ( n ) - p ( n ) ) β ∑ p ( n ) + 1 q ( n ) [ d ( X k , X 0 ) ] r = 0 .

By W d β [ r ] we denote the set of all strongly r-deferred Cesáro summable sequences of order β. If q (n) = n, p (n) = 0, then strong r-deferred Cesáro summability of order β coincides strong r-Cesáro summability of order β of sequences of fuzzy numbers which was introduced by Altinok et al. [4], and also in the special cases q (n) = n, p (n) = 0 and β = 1 strong r-deferred Cesáro summability of order β coincides strong r-Cesáro summability of order β of sequences of fuzzy numbers.

The set W d β [ r ] contains some strongly r-deferred Cesáro summable sequences of order β for β ∈ (0, 1]. For this, consider the sequence X = (X _k ) of fuzzy numbers as follows: X k ( x ) = { x - 3 , for 3 ≤ x ≤ 4 - x + 5 , for 4 ≤ x ≤ 5 0 , otherwise } if k = m 2 ( m = 1 , 2 , 3 , . . . ) x , for 0 ≤ x ≤ 1 - x + 2 , for 1 ≤ x ≤ 2 0 , otherwise } : = X 0 if k ≠ m 2

Then, the α-level set of sequence (X _k ) is [ X k ] α = { [ 3 + α , 5 - α ] , if k = m 2 [ α , 2 - α ] , if k ≠ m 2 .

Then, it is easy to see that 1 n β ∑ k = 1 n [ d ( [ X k ] α , [ X 0 ] α ) ] r ≤ 3 n n β , for r = 1 and q (n) = n, p (n) = 0. So X = (X _k ) is strongly r-deferred Cesáro summable of order β, to fuzzy number X₀ for β ∈ ( 1 2 , 1 ] , where [X₀] ^α = [α, 2 - α].

The following result can be established using standard techniques, so we state the result without proof.

Theorem 2. Let X = (X _k ), Y = (Y _k ) be sequences of fuzzy numbers.

Let 0 < β ≤ 1, then

(i) If S d β - lim X k = X and c ∈ ℝ then S d β - lim cX k = cX ,

(ii) If S d β - lim X k = X and S d β - lim Y k = Y , then S d β - lim ( X k + Y k ) = X + Y ,

Let β and r be two positive real numbers, then

(iii) If W d β [ r ] - lim X k = X and c ∈ ℝ , then W d β [ r ] - lim cX k = cX ,

(iv) If W d β [ r ] - lim X k = X and W d β [ r ] - lim Y k = Y , then W d β [ r ] - lim ( X k + Y k ) = X + Y .

Theorem 3. Let β, γ ∈ (0, 1] such that β < γ. If a sequence X = (X _k ) is deferred statistically convergent of order β, then it is deferred statistically convergent of order γ, i.e. S d β ( F ) ⊆ S d γ ( F ) and the inclusion is strict.

Proof. The first part of proof is easy.

Now, we show that the inclusion S d β ( F ) ⊆ S d γ ( F ) is strict. For this, consider a sequence X = (X _k ) of fuzzy numbers as follows: X k ( x ) = { x , If 0 ≤ x ≤ 1 - x + 2 , If 1 ≤ x ≤ 2 0 , otherwise } , If k = n 3 0 , If k ≠ n 3 .

Then, the α-level set of sequence (X _k ) is [ X k ] α = { [ α , 2 - α ] , If k = n 3 [ 0 , 0 ] , If k ≠ n 3 . It can be easily seen that X = (X _k ) is deferred statistically convergent of order γ for γ ∈ ( 1 3 , 1 ] , but not deferred statistically convergent of order β for β ∈ ( 0 , 1 3 ] and q (n) = 3n - 1, p (n) = 2n - 1. (See Fig. 2) OPEN IN VIEWER

Now as a result of Theorem 3 we have the following.

Corollary 1. The inclusion S d β ( F ) ⊆ S d ( F ) is strict for β ∈ (0, 1].

Theorem 4. Let 0 < β ≤ γ ≤ 1 and r be a positive real number. If X = (X _k ) is strongly r-deferred Cesáro summable of order β to X₀, then it is strongly r-deferred Cesáro summable of order γ to X₀, i.e. W d β [ r ] ⊂ W d γ [ r ] and the inclusion is strict.

Proof. The first part of proof is easy.

To show the inclusion is strict consider a sequence X = (X _k ) of fuzzy numbers as follows: X k ( x ) = { x - 1 , If 1 ≤ x ≤ 2 - x + 3 , If 1 ≤ x ≤ 3 0 , otherwise } , If k = n 2 0 , If k ≠ n 2 .

Then, the α-level set of sequence (X _k ) is [ X k ] α = { [ α + 1 , 3 - α ] , If k = n 2 [ 0 , 0 ] , If k ≠ n 2 .

Then we have 1 ( q ( n ) - p ( n ) ) γ ∑ p ( n ) + 1 q ( n ) [ d ( X k , 0 ¯ ) ] r = 1 ( q ( n ) - p ( n ) ) γ ∑ p ( n ) + 1 q ( n ) [ d ( X k , 0 ¯ ) ] r ≤ q ( n ) - p ( n ) + 1 ( q ( n ) - p ( n ) ) γ → 0 for 1 2 < γ ≤ 1 and so we have X ∈ W d γ [ r ] , but since q ( n ) - p ( n ) - 1 ( q ( n ) - p ( n ) ) β ≤ 1 ( q ( n ) - p ( n ) ) β ∑ p ( n ) + 1 q ( n ) [ d ( X k , 0 ¯ ) ] r → ∞ X ∉ W d β [ r ] for 0 < α < 1 2 .

Theorem 4 yields the following corollary.

Corollary 2. The inclusion W d β [ r ] ⊂ W d [ r ] is strict for β ∈ (0, 1].

Theorem 5. Let 0 < β ≤ γ ≤ 1 and r be a positive real number. If X = (X _k ) is strongly r-Cesáro summable of order β, then it is statistically convergent of order γ, i.e. W d β [ r ] ⊆ S d γ ( F ) .

Proof.Omitted.

Even if X = (X _k ) is a bounded sequence of fuzzy numbers, the converse of Theorem 5 doesn’t hold, in general. To show this we must find a sequence of fuzzy numbers that bounded and deferred statistically convergent of order β, but need not to be strongly r-deferred Cesáro summable of order γ. To show this let p (n) =0 and q (n) = n for all n ∈ ℕ and X = (X _k ) be defined as follows: X k ( x ) = { 2 k - 2 k k + 1 x k - k + 1 , If 1 k + 1 ≤ x ≤ k + k + 1 2 k k + 1 2 k k + 1 x - 2 k + 1 k - k + 1 , If k + k + 1 2 k k + 1 ≤ x ≤ 1 k 0 , otherwise } , If k ≠ m 3 ( m = 1 , 2 , 3 , . . . ) x - 1 , If 1 ≤ x ≤ 2 - x + 3 , If 2 ≤ x ≤ 3 0 , otherwise } , If k = m 3

Then, the α-level set of sequence (X _k ) is [ X k ] α = { [ - α ( k - k + 1 ) + 2 k 2 k k + 1 , α ( k - k + 1 ) + 2 k + 1 2 k k + 1 ] , if k ≠ m 3 [ 1 + α , 3 - α ] , if k = m 3 . It can be shown that X bounded and deferred statistically convergent of order β for β ∈ ( 1 3 , 1 ] . First of all, recall that the inequality ∑ k = 1 n 1 k > n is satisfied for n ≥ 2. Define H _n ={ p (n) < k ≤ q (n): k ≠ m³, m = 1, 2, 3,… } and take r = 1. Since ∑ p ( n ) + 1 q ( n ) [ d ( [ X k ] α , 0 ¯ ) ] r = ∑ k = 1 n [ d ( [ X k ] α , 0 ¯ ) ] = ∑ k ∈ H n [ d ( [ X k ] α , 0 ¯ ) ] + ∑ k ∉ H n [ d ( [ X k ] α , 0 ¯ ) ] = ∑ k ∈ H n 1 k + ∑ k ∉ H n 3 > ∑ k ∈ H n 1 k + ∑ k ∉ H n 1 k = ∑ k = 1 n 1 k > n . we have 1 ( q ( n ) - p ( n ) ) β ∑ p ( n ) + 1 q ( n ) [ d ( [ X k ] α , 0 ¯ ) ] r = 1 n β ∑ k = 1 n [ d ( [ X k ] α , 0 ¯ ) ] > 1 n β n = 1 n β - 1 2 → ∞ as n → ∞ , for β ∈ ( 0 , 1 2 ) . So x ∉ W d β [ r ] for β ∈ ( 0 , 1 2 ) and therefore ( X k ) ∈ S d β - W d β [ r ] for β ∈ ( 1 3 , 1 2 ) . (See Fig. 3) OPEN IN VIEWER

The following result is easily derivable from Theorem 5.

Corollary 3. The inclusion W d β [ r ] ⊆ S d ( F ) holds for β ∈ (0, 1]

Theorem 6. Let {p (n)}, { q (n) } be given as above, β and γ be two real numbers such that 0 < β ≤ γ ≤ 1 and suppose that lim n ( q ( n ) - p ( n ) ) γ n β > 0 . If a sequence of fuzzy numbers is deferred statistically convergent of order β then it is deferred statistically convergent of order γ.

Proof.Let X = (X _k ) be statistically convergent of order β and lim n ( q ( n ) - p ( n ) ) γ n β > 0 . For a given ɛ > 0, we have { k ≤ n : d ( X k , X 0 ) ≥ ɛ } ⊇ { p ( n ) < k ≤ q ( n ) : d ( X k , X 0 ) ≥ ɛ } , and so 1 n β | { k ≤ n : d ( X k , X 0 ) ≥ ɛ } | ≥ ( q ( n ) - p ( n ) ) γ n β 1 ( q ( n ) - p ( n ) ) γ | { p ( n ) < k ≤ q ( n ) : d ( X k , X 0 ) ≥ ɛ } | .

Taking limit as n→ ∞ and using the fact that lim n ( q ( n ) - p ( n ) ) γ n β > 0 , we get X = (X _k ) is deferred statistically convergent of order γ.

Theorem 6 yields the following corollary.

Corollary 4. Let {p (n)}, { q (n) } be given as above, β be a real number such that 0 < β ≤ 1 and suppose that lim n q ( n ) - p ( n ) n β > 0 . If a sequence of fuzzy numbers is deferred statistically convergent of order β then it is deferred statistically convergent.

Theorem 7. Let {p (n)}, { q (n) } be given as above, β and γ be two real numbers such that 0 < β ≤ γ ≤ 1 and suppose that the sequence ( p ( n ) q ( n ) - p ( n ) ) is bounded. If a sequence of fuzzy numbers is strong r-Cesáro convergent of order β then it is strong r-deferred Cesáro convergent of order γ.

Proof.Since the sequence ( p ( n ) q ( n ) - p ( n ) ) is bounded there exists a number K such that p ( n ) q ( n ) - p ( n ) ≤ K . Let us assume that the sequence X = (X _k ) of fuzzy numbers strong r-Cesáro convergent of order β, then we have 1 ( q ( n ) - p ( n ) ) γ ∑ p ( n ) + 1 q ( n ) [ d ( X k , X 0 ) ] r ≤ 1 ( q ( n ) - p ( n ) ) β [ ∑ k = 1 q ( n ) [ d ( X k , X 0 ) ] r + ∑ k = 1 p ( n ) [ d ( X k , X 0 ) ] r ] = [ p ( n ) ] β ( q ( n ) - p ( n ) ) β 1 [ p ( n ) ] β ∑ k = 1 p ( n ) [ d ( X k , X 0 ) ] r + ( q ( n ) q ( n ) - p ( n ) ) β 1 [ q ( n ) ] β ∑ k = 1 q ( n ) [ d ( X k , X 0 ) ] r = K β 1 [ p ( n ) ] β ∑ k = 1 p ( n ) [ d ( X k , X 0 ) ] r + ( 1 + K β ) 1 [ q ( n ) ] β ∑ k = 1 q ( n ) [ d ( X k , X 0 ) ] r . This yields the proof.

The following result is easily derivable from Theorem 7.

Corollary 5. Under the conditions of Theorem 7 if a sequence of fuzzy numbers is strong r-Cesáro convergent of order β then it is strong r-deferred Cesáro convergent.

In the following theorem, by changing the conditions on the sequences {p (n)} and {q (n)} we give the same relation with Theorem 7.

Theorem 8. Let {p (n)}, { q (n) } be given as above, β and γ be two real numbers such that 0 < β ≤ γ ≤ 1 and suppose tha the sequence ( q ( n ) + p ( n ) q ( n ) - p ( n ) ) is bounded, If a sequence of fuzzy numbers is strong r-Cesáro convergent of order β then it is strong r-deferred Cesáro convergent of order γ.

Proof.Proof follows from the following inequality and Theorem 7 q ( n ) + p ( n ) q ( n ) - p ( n ) = ( q ( n ) - p ( n ) ) + 2 p ( n ) q ( n ) - p ( n ) = 1 + 2 p ( n ) q ( n ) - p ( n ) .

The sequences of fuzzy numbers were introduced and studied by Matloka [30] and Altinok et al. [4] introduced the concepts of statistical convergence of order β and strong p-Cesáro summability of order β for sequences of fuzzy numbers. Now in this paper we study the concepts of deferred statistical convergence of order β and strong r-Cesáro summability of order β for sequences of fuzzy numbers, where β is any real number such that 0 < β ≤ 1. To do this we introduce some of fairly wide classes of sequences of fuzzy numbers using two sequences of non-negative integers {p (n)} and {q (n)} satisfying p ( n ) < q ( n ) and lim n → ∞ q ( n ) = + ∞ .

Statistical convergence has many applications in mathematics such as: rough convergence and rough continuity. The theory of soft rough sets and soft rough hemirings studied by Zhan et al. [39–42] and Ma et al. [29]. Future time, rough statistical convergence of order α can be studied. This is an open problem to work for researchers.