Applications of deferred Cesàro statistical convergence of sequences of fuzzy numbers of order ( ξ,ω )

1 Introduction and preliminaries

The perception of statistical convergence was introduced by Fast [18] which was further studied by Schoenberg [32]. Later on, the idea of statistical convergence was investigated from the sequence spaces point of view and linked with summability theory by many mathematicians such as Salat [34], Fridy [19], Connor [11], Hazarika et al. [21] and Edely et al. [13]. The order of statistical convergence for a sequence of numbers was introduced by Gadjiev and Orhan [20]. The statistical convergence of order β was studied by Altinok [3]. Recently, Altinok and M. Et [4] generalized the results of [3] by introducing statistical convergence of order (β, γ) for sequences of fuzzy numbers. The statistical convergence and Cesàro summability in sequences of fuzzy numbers are very important topics in fuzzy mathematics. In recent years, there has been an increasing interest in focusing on summability methods of sequences of fuzzy numbers. İ. Çanak et al. [8] have established some Tauberian theorems for statistical Cesàro summability methods of sequences of fuzzy numbers. Statistical σ-convergence of double sequences was given by Mursaleen et al. [26]. The notion of statistical convergence is an effective tool to resolve many problems in Ergodic theory, Fuzzy set theory, Trigonometric series, Fourier analysis and number theory. Statistical convergence is also found in random graph theory (see [36]) in the sense that almost convergence, which is same as the statistical convergence, which means convergence with a probability of 1, whereas in usual statistical convergence the probability is not necessarily 1. The generalization of statistical convergence developed the study of strong integral summability and the structure of bounded continuous functions on locally compact spaces. For a detailed study on statistical convergence (see [33] and [35]).

The concept of fuzzy set theory was introduced by Zadeh [39]. Matloka [28] extended the study of Zadeh by introducing the notion of fuzzy sequences. The idea of fuzzy set corresponding to unexplained physical situations gives useful applications on many topics such as statistics, data processing and linguistics. Recently, the notion of fuzzy sequences through difference operator together with weighted mean has been defined by Mohiuddine et al. [27]. Later on, several authors have discussed various aspects of fuzzy set theory and its applications such as topological spaces, similarity relations, fuzzy ordering, fuzzy measures of fuzzy events and fuzzy mathematical programming. In [25] F. Maturo and Š. Hoškovà-Mayerovà used alternative operations to solve some problems of addition and multiplication between fuzzy numbers in fuzzy regression models. Sequences of fuzzy numbers have been studied by Altinok [7] and İ. Çanak et al. [9]. By using fuzzy numbers, Tauberian theorem for double sequences was presented by Z. Önder et al. in [30].

A fuzzy set is a mapping Z: ℝ → [ 0 , 1 ] which satisfies the following criteria:

(i) Z is normal, i.e. there exists an u ∈ ℝ such that Z (u) =1,

(ii) Z is fuzzy convex, i.e. Z [λm + (1 - λ) n] ≥ min {Z (m), Z (n)} for all m , n ∈ ℝ and for all λ ∈ [0, 1],

(iii) Z is upper semi-continuous, i.e. for every ɛ > 0, Z^-1 ([0, a + ɛ)) is open in usual topology of ℝ , where a ∈ [0, 1],

(iv) suppZ = cl { u ∈ ℝ : Z ( u ) > 0 } is compact.

For a fuzzy number Z, the α-level set is given as [ Z ] α = { { u ∈ ℝ : Z ( u ) ≥ α } , if 0 < α ≤ 1 supp Z , if α = 0 . It is clear that Z is a fuzzy number if and only if [Z]  ^α is a closed interval for each α ∈ (0, 1] and [Z] ¹ ≠ φ. Throughout the paper, L ( ℝ ) denotes the space of all fuzzy numbers.

For any two fuzzy numbers Z₁ and Z₂, the distance between Z₁ and Z₂ can be evaluated as: d ( Z 1 , Z 2 ) = sup 0 ≤ α ≤ 1 d M ( [ Z 1 ] α , [ Z 2 ] α ) .

The Hausdorff metric is represented by d _M and is defined as d M ( [ Z 1 ] α , [ Z 2 ] α ) = max { | Z 1 _ α - Z 2 _ α | , | Z 1 ¯ α - Z 2 ¯ α | } , where Z 1 ¯ α and Z 1 _ α are the upper and lower bounds of α-level cut.

The concept of difference sequence spaces was introduced by Kizmaz [23] and then it was further generalized by Altinok et al. [6], Et [15], Et and Colak [17]. This concept of Δ was also used by Hüseyin et al. in [10] to define Δ-quasi-slowly oscillating continuity. It is known that there are many applications of sequences and difference sequences of fuzzy numbers. For instance, sequences of numbers have practical applications in areas of science and engineering. Let Ω (F) be set of all sequence spaces, then the operator Δ : Ω (F) → Ω (F) is given by: Δ Z r = Z r - Z r + 1 . Let Z = (Z _r ) be a sequence of fuzzy numbers, then it is called Δ-convergent to fuzzy number Z₀ if there exists a positive integer r₀ such that d (ΔZ _r , Z₀) < ɛ, for every ɛ > 0, r > r₀. A sequence Z = (Z _r ) is Δ-bounded if the set { Δ Z r : r ∈ ℕ } of fuzzy numbers is bounded.

Suppose Z = (Z _r ) be a sequence of fuzzy numbers. Then (Z _r ) is said to be statistical convergent to fuzzy number Z₀ if for each ɛ > 0 lim n → ∞ 1 n | { r ≤ n : d ( Z r , Z 0 ) ≥ ɛ } | = 0 , where the vertical bars represent the cardinality of the enclosed set. In this case, we write S (F) - lim Z _r  = Z₀. For more details of fuzzy sequence spaces one may refer ([22, 38]).

As defined by Agnew [1], let (a _n )  and  (b _n ) be sequences of non-negative integers. Then the deferred Cesàro mean of sequences of real numbers is given by ( D a , b z ) n = 1 ( b n - a n ) ∑ r = a n + 1 b n z r , where the sequences (a _n ), (b _n ) satisfies a _n  < b _n and lim n → ∞ b n = + ∞ , ∀ n ∈ ℕ .

In [24] Küçükaslan and Yılmaztürk defined deferred density and deferred statistical convergence for sequences of real numbers. Let K _d  (n) be the representation of the set {r : a _n  < r ≤ b _n , r ∈ K}, where K is subset of ℕ . Then the deferred density of K is defined by δ d ( K ) = lim n → ∞ 1 ( b n - a n ) | K d ( n ) | , provided that the limit exists. Consider a mapping f : [0, ∞) → [0, ∞), then f is said to be modulus function if

(i) f (z) =0 iff z = 0,

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

(iii)f is increasing,

(iv) f is right continuous at 0.

From above definition it is clear that f is continuous everywhere on (0, ∞]. For more details (see [5, 37]).

H. Altinok studied Δ ^m -statistical convergence of order β for sequences of fuzzy numbers in [3]. Later on, in [4], Altinok introduced statistical convergence of order (β, γ) and strong p-Cesaro summability of order (β, γ) for sequences of fuzzy numbers.

Essentially motivated by the aforementioned investigations and outcomes defined above, our aim is to manifest some new results related to (Δ, F, f)-deferred Cesàro statistically convergence and strongly p-deferred Cesàro summability of order (ξ, ω). We extend the study of Altinok [4] by using (Δ, F, f)-deferred Cesàro statistical convergence with the help of parameters ξ and ω. Further, some of the results are also presented graphically with the help of MATLAB software. It is well known that boundedness and statistical convergence of a sequence of fuzzy numbers leads to strongly p-Cesàro summable but we have demonstrated that this is not valid in our case when ξ and ω are used.

2 Main results

Definition 2.1. A sequence Z = (Z _r ) of fuzzy numbers is said to be (Δ, F, f)-convergent if for ɛ > 0, there exists a positive integer r₀ such that f ( d ( Δ Z r , Z 0 ) ) < ɛ for r > r₀ and it is called (Δ, F, f)-bounded if the set { f ( Δ Z r ) : r ∈ ℕ } of fuzzy numbers is bounded.

Definition 2.2. Let Z = (Z _r ) be a sequence of fuzzy numbers, then it is said to be (Δ, F, f)-deferred Cesàro statistical convergent of order (ξ, ω) for 0 < ξ ≤ ω ≤ 1 (or S d ( ξ , ω ) ( Δ , F , f ) - convergent) to a fuzzy number Z₀ if for every ɛ > 0, S d ( ξ , ω ) ( Δ , F , f ) = lim n → ∞ 1 f ( b n - a n ) ξ × f ( | { a n < r ≤ b n : d ( Δ Z r , Z 0 ) ≥ ɛ } | ω ) = 0 where f is any modulus function. If S d ( ξ , ω ) ( Δ , F , f ) is convergent to Z₀, then it can be written as S d ( ξ , ω ) ( Δ , F , f ) - lim Z r = Z 0 .

Definition 2.3. Let 0 < ξ ≤ ω ≤ 1, 0< p < ∞ and Z = (Z _r ) be sequence of fuzzy numbers. A sequence Z = (Z _r ) is said to be strongly (Δ, F, f, p)-deferred Cesàro summable of order (ξ, ω) if there exists a fuzzy real number Z₀ such that lim n → ∞ 1 ( b n - a n ) ξ ( ∑ a n + 1 b n f [ d ( Δ Z r , Z 0 ) ] p ) ω = 0 , or we can write it as W d ( ξ , ω ) ( Δ , F , f ) - summable to Z₀.

Theorem 2.4. Every (Δ, F, f)-convergent sequence is S d ( ξ , ω ) ( Δ , F , f ) - convergent for 0 < ξ ≤ ω ≤ 1. But the converse is not true as shown in the following example.

Example 2.5. Define the sequence of fuzzy numbers as follows

If r ≠ n², Z r ( z ) = 0 ¯

and if r = n² Z r ( z ) = { 2 3 z - 4 3 , for 2 ≤ z ≤ 7 2 - 2 3 z + 10 3 , for 7 2 ≤ z ≤ 5 0 , otherwise , for n = 1, 2, 3, ⋯. Take modulus function f (z) = z. Then, we can find α-level set of sequence (Z _r ) and (ΔZ _r ) as follows [ Z r ] α = { [ 3 α 2 + 2 , - 3 α 2 + 5 ] , if r = n 2 [ 0 , 0 ] , if r ≠ n 2 and [ Δ Z r ] α = { [ 3 α 2 + 2 , - 3 α 2 + 5 ] , if r = n 2 [ 3 α 2 - 5 , - 3 α 2 - 2 ], if r + 1 = n 2 [ 0 , 0 ] , otherwise . Thus, we have lim n → ∞ 1 f ( b n - a n ) ξ f ( | { a n < r ≤ b n : d ( Δ Z r , 0 ¯ ) ≥ ɛ } | ω ) ≤ f ( ( b n - a n ) ω + 1 ) f ( ( b n - a n ) ξ ) = ( b n - a n ) ω + 1 ( b n - a n ) ξ . Then, Z = (Z _r ) is (Δ, F, f)-deferred Cesàro statistical convergent of order (ξ, ω) for ξ > ω 2 , but it is not convergent.

OPEN IN VIEWER

Fig. 1

Z = (Z _r ) is (Δ, F, f)-statistical convergent of order (ξ, ω), but not convergent.

Remark 2.6. S _d  (Δ, F, f)-convergent of order (ξ, ω) is well defined for ξ ≤ ω, but it is not well defined for ω < ξ. To show this, consider the sequence Z r ( z ) = { z + 10 , for - 10 ≤ z ≤ - 9 - z - 8 , for - 9 ≤ z ≤ - 8 0 , otherwise } = Z ′ if r is odd and Z r ( z ) = { z - 8 , for 8 ≤ z ≤ 9 - z + 10 , for 9 ≤ z ≤ 10 0 , otherwise } = Z ″ if r is even.

Now, the α-level set of (Z _r ) and (ΔZ _r ) are given as [ Z r ] α = { [ α - 10 , - α - 8 ] , if r is odd [ α + 8 , - α + 10 ] , if r is even and [ Δ Z r ] α = { 2 [ α - 10 , - α - 8 ] , if r is odd 2[ α + 8 , - α + 10 ] , if r is even . Then, we can write

1 f ( n ) ξ f ( | { a n < r ≤ b n : d ( Δ Z r , Z ′ ) ≥ ɛ } | ω ) ≤ f ( n 2 ) ω f ( n ξ ) , 1 f ( n ) ξ f ( | { a n < r ≤ b n : d ( Δ Z r , Z ″ ) ≥ ɛ } | ω ) ≤ f ( n 2 ) ω f ( n ξ ) and lim n → ∞ 1 f ( n ) ξ f ( | { a n < r ≤ b n : d ( Δ Z r , Z ′ ) ≥ ɛ } | ω ) = 0 , lim n → ∞ 1 f ( n ) ξ f ( | { a n < r ≤ b n : d ( Δ Z r , Z ″ ) ≥ ɛ } | ω ) = 0 for ξ > ω, b _n  = 7n, a _n  = 6n and ∀ ɛ > 0 using the property lim t → ∞ f ( t ) t > 0 . Here, [ Z ′ ] α = 2 [ α − 10 , − α − 8 ] and [ Z ″ ] α = 2 [ α + 8 , − α + 10 ] . Therefore, the sequence Z = (Z _r ) of fuzzy numbers is S d ( ξ , ω ) ( Δ , F , f ) - convergent to both Z′ and Z″ of order (ξ, ω), which is not possible.

Theorem 2.7. Let f be an unbounded modulus function and 0 < ξ ≤ ω ≤ 1.

(i) If S d ( ξ , ω ) ( Δ , F , f ) - lim Z r = Z 0 and k ∈ ℂ , then S d ( ξ , ω ) ( Δ , F , f ) - lim kZ r = k Z 0 .

(ii) If S d ( ξ , ω ) ( Δ , F , f ) - lim Y r = Y 0 and S d ( ξ , ω ) ( Δ , F , f ) - lim Z r = Z 0 , then S d ( ξ , ω ) ( Δ , F , f ) - lim ( Y r + Z r ) = Y 0 + Z 0 .

Proof. The Proof (i) follows from the inequality, 1 f ( b n - a n ) ξ f ( | { a n < r ≤ b n : d ( k Δ Z r , kZ 0 ) ≥ ɛ } | ω ) ≤ 1 f ( b n - a n ) ξ f ( | { a n < r ≤ b n : d ( Δ Z r , Z 0 ) ≥ ɛ | k | } | ω ) . Proof of (ii) follows from inequality 1 f ( b n - a n ) ξ f ( | { a n < r ≤ b n : d ( Δ Y r + Δ Z r , Y 0 + Z 0 ) ≥ ɛ } | ω )

≤ 1 f ( b n - a n ) ξ f ( | { a n < r ≤ b n : d ( Δ Y r , Y 0 ) + d ( Δ Z r , Z 0 ) ≥ ɛ } | ω ) ≤ 1 f ( b n - a n ) ξ f ( | { a n < r ≤ b n : d ( Δ Y r , Y 0 ) ≥ ɛ 2 } | ω ) + 1 f ( b n - a n ) ξ f ( | { a n < r ≤ b n : d ( Δ Z r , Z 0 ) ≥ ɛ 2 } | ω ) .

Remark 2.8. Let f be unbounded modulus function and 0 < ξ ≤ ω ≤ 1, we have S d ξ ( Δ , F , f ) ⊂ S d ( ξ , ω ) ( Δ , F , f ) . This means that if S d ξ ( Δ , F , f ) - lim Z r = Z 0 , then S d ( ξ , ω ) ( Δ , F , f ) - lim Z r = Z 0 but converse is not true.

For converse we give the example as shown below.

Example 2.9. Define the sequence of fuzzy numbers as follows

If r = n⁶

Z r ( z ) = { 2 3 z - 16 3 , for 8 ≤ z ≤ 19 2 - 2 3 z + 22 3 , for 19 2 ≤ z ≤ 11 0 , otherwise and if r ≠ n⁶ Z r ( z ) = { 2 3 z + 22 3 , for - 11 ≤ z ≤ - 19 2 - 2 3 z - 16 3 , for - 19 2 ≤ z ≤ - 8 0 , otherwise . Now, the α-level set of (Z _r ) and (ΔZ _r ) are given as [ Z r ] α = { [ 3 2 α + 8 , - 3 2 α + 11 ] , if r = n 6 [ 3 2 α - 11 , - 3 2 α - 8 ] , if r ≠ n 6 and [ Δ Z r ] α = { 2[ 3 2 α + 8 , - 3 2 α + 11 ] , if r = n 6 2[ 3 2 α - 11 , - 3 2 α - 8 ] , if r + 1 = n 6 [ 3 α - 3 , - 3 α + 3 ] , otherwise . After some routine verification, S d ( ξ , ω ) ( Δ , F , f ) - lim Z r = Z 0 , where [Z₀]  ^α  = [3α - 3, - 3α + 3] for ξ = 1 6 and ω = 5 8 , but not deferred Cesàro statistically convergent for ξ = 1 6 .

Another example for Remark 2.8 is given below.

Define the sequence of fuzzy numbers as follows

If r = n²

Z r ( z ) = { z - 3 , for 3 ≤ z ≤ 4 - z + 5 , for 4 ≤ z ≤ 5 0 , otherwise and if r ≠ n² Z r ( z ) = { z + 5 , for - 5 ≤ z ≤ - 4 - z - 3 , for - 4 ≤ z ≤ - 3 0 , otherwise . Now, the α-level set of (Z _r ) and (ΔZ _r ) are given as [ Z r ] α = { [ α + 3 , - α + 5 ] , if r = n 2 [ α - 5 , - α - 3 ] , if r ≠ n 2 and [ Δ Z r ] α = { [ 2 ( α + 3 , - α + 5 ) ] , if r = n 2 [ 2 ( α - 5 , - α - 3 ) ] , if r + 1 = n 2 [ 2 α - 2 , - 2 α + 2 ] , otherwise . Then it can easily seen that S d ( ξ , ω ) ( Δ , F , f ) - lim Z r = Z 0 , where [Z₀]  ^α  = [2α - 2, - 2α + 2] for ξ = 1 2 and ω = 1 3 , but not deferred Cesàro statistically convergent for ξ = 1 2 .

Theorem 2.10. If S d ( ξ 1 , ω 2 ) ( Δ , F , f ) - lim Z r = Z 0 , then S d ( ξ 2 , ω 1 ) ( Δ , F , f ) - lim Z r = Z 0 for 0 < ξ₁ ≤ ξ₂ ≤ ω₁ ≤ ω₂ ≤ 1, but converse need not true.

Proof. Suppose S d ( ξ 1 , ω 2 ) ( Δ , F , f ) - lim Z r = Z 0 . Let ω₁ ≤ ω₂ and ξ₁ ≤ ξ₂, then 1 f ( b n - a n ) ξ 2 f ( | { a n < r ≤ b n : d ( Δ Z r , Z 0 ) ≥ ɛ } | ω 1 ) ≤ 1 f ( b n - a n ) ξ 1 f ( | { a n < r ≤ b n : d ( Δ Z r , Z 0 ) ≥ ɛ } | ω 2 ) for every ɛ > 0, so S d ( ξ 2 , ω 1 ) ( Δ , F , f ) - lim Z r = Z 0 . To show inclusion is strict consider a sequence Z = (Z _r ) as follows

for r = n⁷

Z r ( z ) = { z - 14 , for 14 ≤ z ≤ 15 - z + 16 , for 15 ≤ z ≤ 16 0 , otherwise and for r ≠ n⁷ Z r ( z ) = { z - 17 , for 17 ≤ z ≤ 18 - z + 19 , for 18 ≤ z ≤ 19 0 , otherwise . Then, [ Z r ] α = { [ α + 14 , - α + 16 ] , if r = n 7 [ α + 17 , - α + 19 ] , if r ≠ n 7 and [ Δ Z r ] α = { [ 2 ( α - 5 2 ) , 2 ( - α - 1 2 ) ] , if r = n 7 [ 2 ( α + 1 2 ) , 2 ( - α + 5 2 ) ] , if r + 1 = n 7 [ 2 α - 2 , - 2 α + 2 ] , otherwise . For r > 1, b _n  = 2n, a _n  = n and by taking f (z) = z ^s where s ∈ (0, 1]. We have S d ( ξ 2 , ω 1 ) ( Δ , F , f ) - lim Z r = Z 0 for 1 35 < ξ 2 ≤ 1 5 and ω 1 = 1 5 , but S d ( ξ 1 , ω 2 ) ( Δ , F , f ) - lim Z r ≠ Z 0 for 0 < ξ 1 ≤ 1 35 and ω 2 = 1 5 .

Corollary 2.11. Let 0 < ξ₁ ≤ ξ₂ ≤ ω₁ ≤ ω₂ ≤ 1.

(i) If ξ₁ = ξ₂ and ω₁ = ω₂, then S d ( ξ 1 , ω 2 ) ( Δ , F , f ) = S d ( ξ 2 , ω 1 ) ( Δ , F , f ) . (ii) If ξ₁ = ξ₂ and ω₁ = ω₂ = 1, then S d ξ 1 ( Δ , F , f ) = S d ξ 2 ( Δ , F , f ) .

(iii) If ξ₂ = 1 and ω₁ = ω₂ = 1, then S d ξ 1 ( Δ , F , f ) ⊆ S d ( Δ , F , f ) .

(iv) If ω₂ = 1 then S d ξ 1 ( Δ , F , f ) ⊆ S d ( ξ 2 , ω 1 ) ( Δ , F , f ) .

Theorem 2.12. For 0 < ξ₁ ≤ ξ₂ ≤ ω₁ ≤ ω₂ ≤ 1. If Z r → Z 0 ( W d ( ξ 1 , ω 2 ) ( p , Δ , F , f ) ) , then Z r → Z 0 ( W d ( ξ 2 , ω 1 ) ( p , Δ , F , f ) ) but converse is not true.

Proof. Suppose that Z r → Z 0 ( W d ( ξ 1 , ω 2 ) ( p , Δ , F , f ) for 0 < ξ₁ ≤ ξ₂ ≤ ω₁ ≤ ω₂ ≤ 1 and 0 < p < ∞. Then 1 ( b n - a n ) ξ 2 ( ∑ a n + 1 b n f [ d ( Δ Z r , Z 0 ) ] p ) ω 1 ≤ 1 ( b n - a n ) ξ 1 ( ∑ a n + 1 b n f [ d ( Δ Z r , Z 0 ) ] p ) ω 2 . But converse is not true as given below.

If r ≠ n⁴, then Z r ( z ) = 0 ¯

and if r = n⁴, then

Z r ( z ) = { z - 4 , for 4 ≤ z ≤ 5 - z + 6 , for 5 ≤ z ≤ 6 0 , otherwise . Now, the α-level set of (Z _r ) and (ΔZ _r ) are given as [ Z r ] α = { [ α + 4 , - α + 6 ] , if r = n 4 [ 0 , 0 ] , if r ≠ n 4 and [ Δ Z r ] α = { [ α + 4 , - α + 6 ] , if r = n 4 [ α - 6 , - α - 4 ] , if r + 1 = n 4 [ 0 , 0 ] , otherwise . Then, we can write

lim n → ∞ 1 ( b n - a n ) ξ 2 ( ∑ a n + 1 b n f [ d ( Δ Z r , 0 ¯ ) ] p ) ω 1 ≤ 2 ( ( b n 4 - a n 4 ) ω 1 + 1 ) ( b n - a n ) ξ 2 f ( 6 ) and so Z r → Z 0 ( W d ( ξ 2 , ω 1 ) ( p , Δ , F , f ) for 1 12 < ξ 2 ≤ 1 3 , p = 1, f (z) = z and ω 1 = 1 3 , but since ( b n 4 - a n 4 ) ω 2 - 1 ( b n - a n ) ξ 1 f ( 6 ) ≤ 1 ( b n - a n ) ξ 1 ( ∑ a n + 1 b n f [ d ( Δ Z r , 0 ¯ ) ] p ) ω 2 and ( b n 4 - a n 4 ) ω 2 - 1 ( b n - a n ) ξ 1 f ( 6 ) → ∞ as n → ∞ . So, (Z _r ) is not strongly p-deferred Cesàro summable of order (ξ₁, ω₂) for 0 < ξ 1 ≤ 1 12 and ω 2 = 1 3 .

Corollary 2.13. Suppose 0 < ξ₁ ≤ ξ₂ ≤ ω₁ ≤ ω₂ ≤ 1, 0< p < ∞ and f be any unbounded modulus function. Then following results hold.

(i) If ξ₁ = ξ₂ and ω₁ = ω₂, then ( W d ( ξ 1 , ω 2 ) ( p , Δ , F , f ) ) = ( W d ( ξ 2 , ω 1 ) ( p , Δ , F , f ) ) ,

(ii) if ξ₁ = ξ₂ and ω₁ = ω₂ = 1, then ( W d ξ 1 ( p , Δ , F , f ) ) = ( W d ξ 2 ( p , Δ , F , f ) ) ,

(iii) if ( W d ξ 1 ( p , Δ , F , f ) ) ⊆ ( W d ( p , Δ , F , f ) ) , for each 0 < ξ₁ ≤ 1,

(iv) if ( W d ξ 1 ( p , Δ , F , f ) ) ⊆ ( W d ( ξ 2 , ω 1 ) ( p , Δ , F , f ) ) , for each 0 < ω₁ ≤ 1.

Theorem 2.14. Consider 0 < ξ₁ ≤ ξ₂ ≤ ω₁ ≤ ω₂ ≤ 1, 0< p < ∞ and f be any unbounded modulus function. If Z r → Z 0 ( W d ( ξ 1 , ω 2 ) ( p , Δ , F , f ) ) , then Z r → Z 0 ( S d ( ξ 2 , ω 1 ) ( p , Δ , F , f ) ) and inclusion is strict.

Proof. Suppose Z r → Z 0 ( W d ( ξ 1 , ω 2 ) ( p , Δ , F , f ) ) and for given ɛ > 0. So,

( ∑ a n + 1 b n f [ d ( Δ Z r - Z 0 ) ] p ) ω 2 ≥ ( ∑ a n + 1 b n f [ d ( Δ Z r , Z 0 ) ] p ) ω 1          ≥ ɛ ^{pω ₁} f (| {a _n  < r ≤ b _n  : d (ΔZ _r , Z₀) ≥ ɛ} | ^{ω ₁} )

and 1 ɛ p ω 1 ( b n - a n ) ξ 1 ( ∑ a n + 1 b n f [ d ( Δ Z r , Z 0 ) ] p ) ω 2 ≥ 1 ( b n - a n ) ξ 1 f ( | { a n < r ≤ b n : d ( Δ Z r , Z 0 ) ≥ ɛ } | ω 1 ) . To show strictness of relation, consider the sequence Z = (Z _r ) as defined in the Theorem 2.9. By taking f (z) = z, b _n  = 3n - 1 and a _n  = 2n - 1 we have lim n → ∞ 1 n ξ 2 | { k ≤ n : d ( Δ Z r , 0 ¯ ) ≥ ɛ } | ω 1 ≤ lim n → ∞ n ω 1 4 n ξ 2 = lim n → ∞ 1 n ξ 2 - ω 1 4 = 0 , when ω 1 = 1 4 and 1 16 < ξ 2 ≤ 1 4 but 1 n ξ 1 ( ∑ r = 1 n [ d ( Δ Z r , 0 ¯ ) ] p ) ω 2 → ∞ , when ξ ∈ ( 0 , 1 16 ] and ω 2 = 1 4 . Therefore, Z = (Z _r ) is not strongly p-deferred Cesàro summable of order (ξ₁, ω₂).

Remark 2.15. It is well known that if the sequence Z = (Z _r ) of fuzzy numbers is (Δ, F, f)-bounded and (Δ, F, f)-statistical convergent, then the sequence is strongly ( W d ( ξ , ω ) ( p , Δ , F , f ) ) -summable. But this is not possible for our case. Let us prove this by an example.

Example 2.16. Let f be modulus function such that f (z) = z, b _n  = 5n, a _n  = 4n.

For r ≠ n³, we have Z r ( z ) = { r 2 z + 1 , for - 2 r ≤ z ≤ 0 - r 2 z + 1 , for 0 ≤ z ≤ 2 r 0 , otherwise ∥

and for r = n³, the fuzzy sequence is defined as Z r ( z ) = { z - 8 , for 8 ≤ z ≤ 9 - z + 10 , for 9 ≤ z ≤ 10 0 , otherwise . ∥

Then for n = 1, 2, 3, ⋯, we find the respective α-level set of sequences (Z _r ) and (ΔZ _r ) as [ Z r ] α = { [ 2 α - 2 r , - 2 α + 2 r ] , for r ≠ n 3 [ α + 8 , - α + 10 ] , for r = n 3 and [ Δ Z r ] α = { [ 8 r + 1 - 2 + α ( 2 + r + 1 ) r + 1 , 10 r + 1 + 2 - α ( 2 + r + 1 ) r + 1 ] , for r = n 3 [ - 10 r - 2 + α ( 2 + r ) r , - 8 r + 2 - α ( 2 + r ) r ] , for r + 1 = n 3 [ - 2 r - 2 r + 1 + 2 α ( r + r + 1 ) r r + 1 , 2 r + 2 r + 1 - 2 α ( r + r + 1 ) r r + 1 ] , otherwise . Here, we see that (Z _r ) is (Δ, F, f)-bounded and statistical convergent to Z₀ = [0, 0] for ω = 1 2 , 1 6 < ξ ≤ 1 2 . But (Z _r ) is not strongly p-deferred Cesàro summable defined by modulus function. To show this, let A r = { r ≤ n : r = n 3 , n ∈ ℕ } and B r = { r ≤ n : r + 1 = n 3 , n ∈ ℕ } .

OPEN IN VIEWER

Fig. 2

The blue lines in the graph (i), (ii), (iii) represents for particular values of r, i.e. for r = 7, 2, 1, respectively whereas the red lines in the graph (i), (ii), (iii) shows when the value of r + 1 = n³ i.e. when r = 26, 63, ⋯, when neither s ≠ n³ nor s + 1 = n³ i.e. for 3, 4, ⋯, and when s = n³ i.e. for 8, 27, ⋯, respectively.

Now, we have

1 n ξ ( ∑ r = 1 n f ( [ d ( [ Δ Z r ] α , [ Z 0 ] α ) ] p ) ω ≥ 1 n ξ ( ∑ r ∈ A r f ( [ d ( [ Δ Z r ] α , [ Z 0 ] α ) ] p ) ) ω + 1 n ξ ( ∑ r ∈ B r f ( [ d ( [ Δ Z r ] α , [ Z 0 ] α ) ] p ) ) ω + 1 n ξ ( ∑ r ∉ A r r ∉ B r f ( [ d ( [ Δ Z r ] α , [ Z 0 ] α ) ] p ) ) ω = 1 n ξ ( ∑ r ∈ A r 10 + 2 r + 1 ) ω + 1 n ξ ( ∑ r ∈ B r 10 + 2 r + 1 ) ω + 1 n ξ ( ∑ r ∉ A r , r ∉ B r 2 r + 2 r + 1 ) ω > 1 n ξ ( n 3 ) ω . Thus, for p = 1 and f (z) = z, (Z _r ) is not strongly ( W d ( ξ , ω ) ( p , Δ , F , f ) ) -summable for 0 < ξ ≤ 1 6 , ω = 1 2 .

Over years, many authors have widely focused on statistically convergence of sequence of fuzzy numbers. Statistical convergence has many applications in different fields of mathematics such as in rough convergence, rough continuity and rough statistical convergence. Since the fuzzy approximation theory is the theoretical approach to solve numerical approximation in the field of mathematical analysis. Also approximation theory is applied in computer simulation. In [2] Anastassiou and Duman explore the application of statistical convergence of fuzzy numbers through approximation theory. As our work is concerned with statistical convergence of fuzzy numbers so we can expect applications of presented work in the field of approximation theory also. Our results are applicable on pointwise and uniform statistical convergence for sequence of functions. The work presented in this paper give a new direction to previous work done by several authors. Regarding the topic of rough statistical convergence, one can obtain the corresponding results of present paper by using Δ ^m -rough statistical convergence of order (ξ, ω).

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