Tauberian conditions for double sequences which are statistically summable (C,1,1) in fuzzy number space

1 Introduction

The notion of statistical convergence for double sequences is presented by Mursaleen and Edely [24] and by Móricz [22] independently. Tauberian theorems for statistically convergent double sequence are given by many researchers [7, 31]. Also, the concept of statistically summability (C,1,1) is defined and necessary conditions under which statistical convergence follows from statistical summability (C,1,1) is given by Móricz [23]. Chan and Chen [8] extend the results given by Moricz to double sequence that are statistically summable by weighted mean.

The concept of fuzzy set is put forward by Lotfi A. Zadeh in [36] as an extension of the concept of classical set, and since then to today researchers have utilized the fuzzy sets to cope with problems occuring in many branches of mathematics [13–16, 37] as well as other areas of science. In connection with sequences, authors have defined many classes of sequences of fuzzy numbers and examined some properties of these classes. Different kinds of summability methods and related Tauberian theorems have been extended to fuzzy analysis [3, 33–35]. Besides various convergence methods for double sequences of fuzzy numbers have been introduced [12, 27]. In particular, the concept of statistical convergence has been extended to double sequences of fuzzy numbers recently [26]. In this paper, we define the notion of statical summability (C,1,1) for double sequence in E ⁿ . Also, we extend the results that were given by Móricz [23] to double sequence in E ⁿ .

2 Preliminaries

In this section, we introduce the basic notions of fuzzy numbers and refer to [4, 32] for more details.

Throughout this paper, ℝ is the real line and ℕ denotes the set of all natural numbers.

Let K c ( ℝ n ) denote the family of all nonempty compact convex subsets of ℝ n . If A , B ∈ K c ( ℝ n ) and k ∈ ℝ then the operations of addition and scalar multiplication are defined as A + B = { a + b : a ∈ A , b ∈ B } kA = { ka : a ∈ A } .

The Hausdorff metric on K c ( ℝ n ) is defined by d ( A , B ) = max { sup a ∈ A inf b ∈ B ∥ a - b ∥ , sup b ∈ B inf a ∈ A ∥ a - b ∥ } , where ∥.∥ denotes the usual Euclidean norm in ℝ n .

A fuzzy number is a mapping u : ℝ n → [ 0 , 1 ] which satisfies the following four conditions:

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

The set of all fuzzy numbers is denoted by E ⁿ and E ⁿ is called fuzzy number space. If u ∈ E ⁿ , then α-level set [u] _α of u, defined by [ u ] α : = { { x ∈ ℝ n : u ( x ) ≥ α } , ( 0 < α ≤ 1 ) , { x ∈ ℝ n : u ( x ) > 0 } ¯ , ( α = 0 ) , is a nonempty compact convex subset of ℝ n .

Let r ∈ ℝ n . We say that r ¯ is a crisp fuzzy number if r ¯ ( x ) : = { 1 , if x = r 0 , otherwise .

The operations addition and scalar multiplication on fuzzy numbers are defined by u + v = w ⇔ [ w ] α = [ u ] α + [ v ] α for all α ∈ [ 0 , 1 ] and [ ku ] α = k [ u ] α for all α ∈ [ 0 , 1 ] .

The operations addition and scalar multiplication on fuzzy numbers have the following properties.

Lemma 1. [28]

If 0 ¯ ∈ E n is neutral element with respect to +, i.e u + 0 ¯ = 0 ¯ + u = u , for all u ∈ E ⁿ .

The metric D on E ⁿ defined as follows: D ( u , v ) : = sup α ∈ [ 0 , 1 ] d ( [ u ] α , [ v ] α ) .

From [20], we have the following lemma.

Lemma 2. Let u, v, w, z ∈ E ⁿ and k ∈ ℝ .

(E ⁿ , D) is a complete metric space.

The concept of ordinary convergence of a double sequence of fuzzy numbers was firstly introduced by Savaş [25] and defined as follows:

A double sequence u = ( u mn ) m , n = 0 ∞ of fuzzy numbers is said to be convergent in the Pringsheim’s sense if for every ɛ > 0 there exists N ∈ ℕ such that D (u _mn , μ) < ɛ; whenever min(m, n) ≥ N and we denote by u _mn → μ.

The concept of statistical convergence was extended to double sequences of fuzzy numbers by Savaş and Mursaleen [26]. A sequence u = (u _mn ) of fuzzy numbers is said to be statistically convergent to a fuzzy number μ if the following equality holds for every ɛ > 0: lim M , N → ∞ 1 ( M + 1 ) ( N + 1 ) | { m ≤ M and n ≤ N : D ( u mn , μ ) ≥ ɛ } | = 0 where | · | denotes the cardinality of a given set. In this case we write u mn → s t μ or st - lim m , n → ∞ u mn = μ

Let u = (u _jk ) be a double sequences of fuzzy numbers. As usual, the (first) arithmetic mean σ _mn of a double sequence (u _jk ) is defined by

σ mn = 1 ( m + 1 ) ( n + 1 ) ∑ j = 0 m ∑ k = 0 n u jk whenever m , n = 0 , 1 , 2 …(1)

If st - lim m , n → ∞ σ mn = μ then, we say that (u _jk ) is statistically summable (C,1,1) to fuzzy number μ. We write u mn → s t μ ( C , 1 , 1 ) for the case that σ _mn → s t μ. If a double sequence (u _jk ) of fuzzy numbers is bounded, then u mn → s t μ ⇒ u mn → s t μ ( C , 1 , 1 ) .(2)

Indeed, D ( σ mn , μ ) = D ( 1 ( m + 1 ) ( n + 1 ) ∑ j = 0 m ∑ k = 0 n u jk , μ ) ≤ 1 ( m + 1 ) ( n + 1 ) ∑ j = 0 m ∑ k = 0 n D ( u jk , μ ) .

We know that double real sequence {D (u _jk , μ)} is bounded and D (u _jk , μ) → s t 0. So we have 1 ( m + 1 ) ( n + 1 ) ∑ j = 0 m ∑ k = 0 n D ( u jk , μ ) → s t 0 .

This implies that σ _mn → s t μ.

If sequence (u _jk ) of fuzzy numbers is not bounded, then the implication (2) can not be provided. The following example explains this suggestion.

Example 1. Define the double sequence (u _jk ) in E¹ for all x ∈ ℝ by u jk ( x ) = { x - k , for k ≤ x ≤ k + 1 - x + k + 2 , for k + 1 ≤ x ≤ k + 2 0 , otherwise . } , if j = 0 and for all k , 0 ¯ , otherwise

Since lim M , N → ∞ 1 ( M + 1 ) ( N + 1 ) | { j ≤ M and k ≤ N : D ( u jk , 0 ¯ ) ≥ ɛ } | ≤ lim M , N → ∞ N + 1 ( M + 1 ) ( N + 1 ) = 0 , (u _jk ) is statistically convergent to 0 ¯ . However, (u _jk ) is not statistically summable (C,1,1).

The converse of the implication (2) does not hold, in general.

Example 2. Define the double sequence (u _jk ) in E¹ by u jk = { μ , j is odd , for all k , ν , otherwise where for all x ∈ ℝ μ ( x ) = { - x + 1 , 0 ≤ x ≤ 1 , 0 , otherwise and ν ( x ) = { x + 1 , - 1 ≤ x ≤ 0 , 0 , otherwise .

Although (u _jk ) is statistically summable (C,1,1) to μ + ν 2 , (u _jk ) is not statistically convergent.

The purpose of this paper is to get Tauberian conditions under which u mn → s t μ ( C , 1 , 1 ) ⇒ u mn → s t μ .(3)

Firstly we need the following lemmas.

Lemma 3. Let u _mn → s t μ  (C, 1, 1). Then for every λ > 0, we have σ_{λ m ,λ n} → s t μ, σ_{m,λ n} → s t μ and σ_{λ m ,n} → s t μ where λ _n : = [λn].

The proof can be carried out in the same way in Lemma 2.3 in [8].

Lemma 4. Let (u _mn ) be a double sequence in E ⁿ with u _mn → s t μ  (C, 1, 1). Then, for every λ > 1,

st - lim m , n → ∞ 1 ( λ m - m ) ( λ n - n ) ∑ j = m + 1 λ m ∑ k = n + 1 λ n u jk = μ(4) and for every 0 < λ < 1,

st - lim m , n → ∞ 1 ( m - λ m ) ( n - λ n ) ∑ j = λ m + 1 m ∑ k = λ n + 1 n u jk = μ .(5)

Proof. Case λ > 1. If λ > 1 and m, n are large enough in the sense that λ _n > n and λ _m > m, then we have the following inequality by proceeding as in the proof of Lemma 3.8(i) in [10].

D ( 1 ( λ m - m ) ( λ n - n ) ∑ j = m + 1 λ m ∑ k = n + 1 λ n u jk , μ ) ≤ D ( 1 ( λ m - m ) ( λ n - n ) ∑ j = m + 1 λ m ∑ k = n + 1 λ n u jk , σ mn ) + D ( σ mn , μ ) ≤ ( λ n + 1 ) ( λ m + 1 ) ( λ n - n ) ( λ m - m ) D ( σ λ m , λ n , σ λ m , n ) + ( λ n + 1 ) ( λ m + 1 ) ( λ n - n ) ( λ m - m ) D ( σ mn , σ m , λ n ) + ( λ m + 1 ) ( λ m - m ) D ( σ λ m , n , σ mn ) + ( λ n + 1 ) ( λ n - n ) D ( σ m , λ n , σ mn ) + D ( σ mn , μ ) .(6)

Now (4) follows from inequality (6), Lemma 3, the statistical convergence of (σ _mn ) and the fact that for large enough m

λ m + 1 λ m - m < 2 λ λ - 1 .(7)

Case 0 < λ < 1. This time, we have the following inequality by proceeding as in the proof of Lemma 3.8(ii) in [10]: D ( 1 ( m - λ m ) ( n - λ n ) ∑ j = λ m + 1 m ∑ k = λ n + 1 n u jk , μ ) ≤ D ( 1 ( m - λ m ) ( n - λ n ) ∑ j = λ m + 1 m ∑ k = λ n + 1 n u jk , σ mn ) + D ( σ mn , μ ) ≤ ( λ n + 1 ) ( λ m + 1 ) ( n - λ n ) ( m - λ m ) D ( σ mn , σ m , λ n ) + ( λ n + 1 ) ( λ m + 1 ) ( m - λ m ) ( n - λ n ) D ( σ λ m , λ n , σ λ m , n ) + ( λ m + 1 ) ( m - λ m ) D ( σ mn , σ λ m , n ) + ( λ n + 1 ) ( n - λ n ) D ( σ mn , σ m , λ n ) + D ( σ mn , μ ) .(8)

Now (5) follows from inequality (8), Lemma 3, the statistical convergence of (σ _mn ) and the fact that for large enough m

λ m + 1 m - λ m < 2 λ 1 - λ .(9) □

Theorem 1. Let (u _mn ) be a double sequence in E ⁿ with u _mn → s t μ  (C, 1, 1). Then u _mn → s t μ if and only if one of the following two conditions holds: For every ɛ > 0, inf λ > 1 lim sup M , N → ∞ 1 ( M + 1 ) ( N + 1 ) | { m ≤ M and n ≤ N : D ( 1 ( λ m - m ) ( λ n - n ) ∑ j = m + 1 λ m ∑ k = n + 1 λ n u jk , u mn ) ≥ ɛ } | = 0(10)

or inf 0 < λ < 1 lim sup M , N → ∞ 1 ( M + 1 ) ( N + 1 ) | { m ≤ M and n ≤ N : D ( 1 ( m - λ m ) ( n - λ n ) ∑ j = λ m + 1 m ∑ k = λ n + 1 n u jk , u mn ) ≥ ɛ } | = 0(11)

Proof. Necessity. Let u _mn → s t μ  (C, 1, 1) and u _mn → s t μ. Then from Lemma 4, we have (10) for λ > 1 and (11) for 0 < λ < 1.

Sufficiency. Let u _mn → s t μ  (C, 1, 1) and (10) is satisfied. If we show that D ( u mn , σ mn ) → s t 0 ,(12) then u _mn → s t μ is proved. For λ > 1, we have D ( σ mn , u mn ) ≤ D ( 1 ( λ m - m ) ( λ n - n ) ∑ j = m + 1 λ m ∑ k = n + 1 λ n u jk , u mn ) + D ( 1 ( λ m - m ) ( λ n - n ) ∑ j = m + 1 λ m ∑ k = n + 1 λ n u jk , σ mn ) .

From the inequality (6) we obtain D ( σ mn , u mn ) ≤ ( λ n + 1 ) ( λ m + 1 ) ( λ n - n ) ( λ m - m ) D ( σ λ m , λ n , σ λ m , n ) + ( λ n + 1 ) ( λ m + 1 ) ( λ n - n ) ( λ m - m ) D ( σ mn , σ m , λ n ) + ( λ m + 1 ) ( λ m - m ) D ( σ λ m , n , σ mn ) + ( λ n + 1 ) ( λ n - n ) D ( σ m , λ n , σ mn ) + D ( 1 ( λ m - m ) ( λ n - n ) ∑ j = m + 1 λ m ∑ k = n + 1 λ n u jk , u mn ) .(13)

Hence, we have { m ≤ M and n ≤ N : D ( σ mn , u mn ) ≥ ɛ } ⊆ A MN ( ɛ ) ∪ B MN ( ɛ ) , where A MN ( ɛ ) = { m ≤ M and n ≤ N : D ( 1 ( λ m - m ) ( λ n - n ) ∑ j = m + 1 λ m ∑ k = n + 1 λ n u jk , u mn ) ≥ ɛ 2 }

and B MN ( ɛ ) = { m ≤ M and n ≤ N : ( λ n + 1 ) ( λ m + 1 ) ( λ n - n ) ( λ m - m ) D ( σ λ m , λ n , σ λ m , n ) + ( λ n + 1 ) ( λ m + 1 ) ( λ n - n ) ( λ m - m ) D ( σ mn , σ m , λ n ) + ( λ m + 1 ) ( λ m - m ) D ( σ λ m , n , σ mn ) + ( λ n + 1 ) ( λ n - n ) D ( σ m , λ n , σ mn ) ≥ ɛ 2 } .

For δ > 0, from (10) there exists some λ > 1 such that lim sup M , N → ∞ 1 ( M + 1 ) ( N + 1 ) | A MN ( ɛ ) | ≤ δ .

By Lemma 3, we have lim M , N → ∞ 1 ( M + 1 ) ( N + 1 ) | B MN ( ɛ ) | = 0 .

Therefore we get lim sup M , N → ∞ 1 ( M + 1 ) ( N + 1 ) | { m ≤ M and n ≤ N : D ( σ mn , u mn ) ≥ ɛ } | ≤ δ .

Since δ > 0 is arbitrary, it follows that for every ɛ > 0 lim M , N → ∞ 1 ( M + 1 ) ( N + 1 ) | { m ≤ M and n ≤ N : D ( σ mn , u mn ) ≥ ɛ } | = 0 .

This means that D (u _mn , σ _mn ) → s t 0.

For 0 < λ < 1, similarly from (11), (12) can be easily proved. □

A double sequence (u _mn ) in E ⁿ is said to be statistically slowly oscillating with respect to the first index if, for every ɛ > 0 inf λ > 1 lim sup M , N → ∞ 1 ( M + 1 ) ( N + 1 ) | { m ≤ M and n ≤ N : max m < j ≤ λ m D ( u jn , u mn ) ≥ ɛ } | = 0 .(14)

We say that (u _mn ) is statistically slowly oscillating in the strong sense with respect to the first index if (14) is satisfied with max m < j ≤ λ m n < k ≤ λ n D ( u jk , u mk ) instead of max m < j ≤ λ m D ( u jn , u mn ) .

The statistically slow oscillation property with respect to the second index is defined analogously. By Lemmas 1 and 2, we obtain D ( 1 ( λ m - m ) ( λ n - n ) ∑ j = m + 1 λ m ∑ k = n + 1 λ n u jk , u mn ) ≤ max m < j ≤ λ m n < k ≤ λ n D ( u jk , u mk ) + max n < k ≤ λ n D ( u mk , u mn ) .

Hence, if (u _mn ) is statistically slowly oscillating with respect to the second index and statistically slowly oscillating in the strong sense with respect to the first index, then (10) holds for all ɛ > 0. By Theorem 1, we obtain the following corollary.

corollary 1. For a double sequence (u _mn ) of fuzzy numbers implication (3) holds if (u _mn ) is statistically slowly oscillating with respect to both indices and, in addition, in the strong sense with respect to one of the indices.

Now we give two-sided Tauberian conditions of Hardy type for double sequences in E ⁿ . Consider jD ( u jn , u j - 1 , n ) ≤ H ( j , n > n 0 )(15) kD ( u mk , u m , k - 1 ) ≤ H ( m , k > n 0 ) .(16) where n₀ ≥ 1 and H > 0 are suitable constants. For λ > 1 and m, k > n₀ we have max m < j ≤ λ m n < k ≤ λ n D ( u jk , u mk ) ≤ max m < j ≤ λ m n < k ≤ λ n { ( ∑ l = m + 1 j 1 l ) ( sup m < l ≤ j lD ( u lk , u l - 1 , k ) ) } ≤ ∑ l = m + 1 λ m H l ≤ H log λ .

Therefore, if (15) holds, then (u _mn ) is statistically slowly oscillating in the strong sense with respect to the first index. Similarly, (16) implies the statistically slow oscillation property in the strong sense with respect to the second index. As a consequence of Corollary 1, we obtain the following result.

Corollary 2. Let (u _mn ) be a double sequence in E ⁿ with u _mn → s t μ  (C, 1, 1). If conditions (15) and (16) are satisfied, then u _mn → s t μ.

Now, we give an example which shows that any condition of our theorems can not be dropped.

Example 3. Define the double sequence (u _jk ) in E² by u jk = { μ , j is odd , for all k , 0 ¯ , otherwise where for all x = ( x 1 , x 2 ) ∈ ℝ 2 μ ( x ) = { 1 - x 1 2 - x 2 2 , x ∈ { ( x 1 , x 2 ) ∈ ℝ 2 : x 1 2 + x 2 2 ≤ 1 } , 0 , otherwise .

Obviously this sequence is not statistically convergent. However, (u _jk ) is statistically summable (C,1,1) to μ 2 . Since D (u _jk , u_j,k-1) =0, the condition (16) is satisfied. But for any fix k ∈ ℕ , D ( u jk , u j - 1 , k ) = D ( μ , 0 ¯ ) = 1 and we can not find a number H satisfying jD (u _jn , u_j-1,n) = j ≤ H for j, k > n₀. That is, the condition (15) is not satisfied. Consequently this example shows that both the conditions of Corollary 2 must be provided for statistical summable (C,1,1) double sequences in fuzzy number space to converge in E ⁿ .

Talo and Bayazit [30] proved the following corollary.

Corollary 3. Let (u _mn ) be a double sequence in E ⁿ with u _mn → s t μ. If conditions (15) and (16) are satisfied, then u _mn → μ.

As a result of this, if we consider Corollary 2 and 3 together, we have the following conclusion.

Corollary 4. Let (u _mn ) be a double sequence in E ⁿ with u _mn → s t μ  (C, 1, 1). If conditions (15) and (16) are satisfied, then u _mn → μ.

The Korovkin type approximation theory which deals with the problem of approximation of functions by the sequence of positive linear operators has important applications in different branches of mathematics. Recently, this theory is extended to fuzzy analysis. The approximation properties of sequences of positive linear operators defined on the space of continuous fuzzy number valued functions are given by Anastassia [1] (also see [2]). The Korovkin type approximation theorems for fuzzy-number-valued functions of two variables are first proved by Demirci and Karakuş [11]. In this chapter, we apply the notion of statically summable (C,1,1) of double sequences in E¹ to prove a fuzzy Korovkin-type approximation theorem for fuzzy positive linear operators.

K is a compact subset of ℝ 2 and we denote by C (K) the space of all continuous real functions on K. C (K) is a Banach spaces with the norm by ∥· ∥ ∥ h ∥ = sup ( x , y ) ∈ K | h ( x , y ) | .

E¹ is partially ordered and is not a linear space. If u ∈ E¹, then α-level set [u] _α of u is closed, bounded and non-empty interval and we can write [ u ] α : = [ u α - , u α + ] . The partial ordering relation on E¹ is defined as follows: u ⪯ v ⇔ [ u ] α ⪯ [ v ] α ⇔ u α - ≤ v α - and u α + ≤ v α + for all α ∈ [ 0 , 1 ] .

A fuzzy-number-valued function of two variable f : K → E¹ has the parametric representation [ f ( x , y ) ] α = [ f α - ( x , y ) , f α + ( x , y ) ] , for each (x, y) ∈ K and α ∈ [0, 1]. The set of all continuous fuzzy-number-valued-functions on K is denoted by C F ( K ) and C F ( K ) is a complete metric space with the metric D * ( f , g ) = sup ( x , y ) ∈ K D ( f ( x , y ) , g ( x , y ) ) = sup ( x , y ) ∈ K sup α ∈ [ 0 , 1 ] max { | f α - ( x , y ) - g α - ( x , y ) | , | f α + ( x , y ) - g α + ( x , y ) | } .

Now let L : C F ( K ) → C F ( K ) be an operator. Then L is said to be fuzzy linear if, for every λ 1 , λ 2 ∈ ℝ , f 1 , f 2 ∈ C F ( K ) and (x, y) ∈ K, L ( λ 1 f 1 + λ 2 f 2 ; x , y ) = λ 1 L ( f 1 ; x , y ) + λ 2 L ( f 2 ; x , y ) holds. Also L is called fuzzy positive linear operator if it is fuzzy linear and the condition L (f; x, y) ⪯ L (g; x, y) is satisfied for any f , g ∈ C F ( K ) and all (x, y) ∈ K with f (x, y) ⪯ g (x, y).

Theorem 2. Let { L mn } ( m , n ) ∈ ℕ 2 be a double sequence of fuzzy positive linear operators from C F ( K ) into itself. Assume that there exists a corresponding sequence { L ˜ mn } ( m , n ) ∈ ℕ 2 of positive linear operators from C (K) into itself with the property { L mn ( f ; x , y ) } α ± = L ˜ mn ( f α ± ; x , y )(17)for all ( x , y ) ∈ K , α ∈ [ 0 , 1 ] , ( m , n ) ∈ ℕ 2 and f ∈ C F ( K ) . Assume further that st - lim m , n → ∞ ∥ 1 ( m + 1 ) ( n + 1 ) ∑ j = 0 m ∑ k = 0 n L ˜ jk ( g i ) - g i ∥ = 0 , i = 0 , 1 , 2 , 3(18)

where g₀ (x,  y)   =  1, g₁ (x,  y)   =  x, g₂ (x, y)   =  y, g₃ (x, y) = x² + y².

Then, for all f ∈ C F ( K ) , we have st - lim m , n → ∞ D * ( 1 ( m + 1 ) ( n + 1 ) ∑ j = 0 m ∑ k = 0 n L jk ( f ) , f ) = 0 .(19)

Proof. By f ∈ C F ( K ) , there exist a M > 0 such that D ( f ( x , y ) , 0 ¯ ) ≤ M for all (x, y) ∈ K and for every ɛ > 0 there is a number δ > 0 such that D (f (s, t), f (x, y)) < ɛ for all (s, t) ∈ K satisfying ( s - x ) 2 + ( t - y ) 2 < δ . On the other hand, if ( s - x ) 2 + ( t - y ) 2 ≥ δ , then D ( f ( s , t ) , f ( x , y ) ) ≤ D ( f ( s , t ) , 0 ¯ ) + D ( f ( x , y ) , 0 ¯ ) ≤ 2 M δ 2 ( ( s - x ) 2 + ( t - y ) 2 )

Therefore, we obtain for (x, y) fixed, we obtain D ( f ( s , t ) , f ( x , y ) ) < ɛ + 2 M δ 2 ( ( s - x ) 2 + ( t - y ) 2 ) .

From definition of the metric D, for α ∈ [0, 1]

| f α ± ( s , t ) - f α ± ( x , y ) | < ɛ + 2 M δ 2 ( ( s - x ) 2 + ( t - y ) 2 ) .(20)

We know that L ˜ mn is linear and positive operator on C (K) and f α ± ∈ C ( K ) for α ∈ [0, 1]. From inequality (20), for α ∈ [0, 1] we see that | 1 ( m + 1 ) ( n + 1 ) ∑ j = 0 m ∑ k = 0 n L ˜ jk ( f α ± ; x , y ) - f α ± ( x , y ) | ≤ ɛ + ( ɛ + M + 2 M δ 2 ( A 2 + B 2 ) ) | 1 ( m + 1 ) ( n + 1 ) ∑ j = 0 m ∑ k = 0 n L ˜ jk ( g 0 ; x , y ) - g 0 ( x , y ) | + 2 M δ 2 | 1 ( m + 1 ) ( n + 1 ) ∑ j = 0 m ∑ k = 0 n L ˜ jk ( g 3 ; x , y ) - g 3 ( x , y ) | + 4 MA δ 2 | 1 ( m + 1 ) ( n + 1 ) ∑ j = 0 m ∑ k = 0 n L ˜ jk ( g 1 ; x , y ) - g 1 ( x , y ) | + 4 MB δ 2 | 1 ( m + 1 ) ( n + 1 ) ∑ j = 0 m ∑ k = 0 n L ˜ jk ( g 2 ; x , y ) - g 2 ( x , y ) |

where A = max {|x|}, B = max {|y|}, as in the proof of Theorem 3.3 in [5]. Taking supremum over (x, y) ∈ K, for α ∈ [0, 1] we obtain

∥ 1 ( m + 1 ) ( n + 1 ) ∑ j = 0 m ∑ k = 0 n L ˜ jk ( f α ± ) - f α ± ∥ ≤ ɛ + C ∑ i = 0 3 ∥ 1 ( m + 1 ) ( n + 1 ) ∑ j = 0 m ∑ k = 0 n L ˜ jk ( g i ) - g i ∥(21) where C = max { ( ɛ + M + 2 M δ 2 ( A 2 + B 2 ) ) , 2 M δ 2 , 4 MA δ 2 , 4 MB δ 2 } .

From definition of the metric D^* D * ( 1 ( m + 1 ) ( n + 1 ) ∑ j = 0 m ∑ k = 0 n L jk ( f ) , f ) = sup ( x , y ) ∈ K D ( 1 ( m + 1 ) ( n + 1 ) ∑ j = 0 m ∑ k = 0 n L jk ( f ; x , y ) , f ( x , y ) ) = sup x ∈ [ a , b ] sup α ∈ [ 0 , 1 ] max { | 1 ( m + 1 ) ( n + 1 ) ∑ j = 0 m ∑ k = 0 n L ˜ jk ( f α - ; x , y ) - f α - ( x , y ) | , | 1 ( m + 1 ) ( n + 1 ) ∑ j = 0 m ∑ k = 0 n L ˜ jk ( f α + ; x , y ) - f α + ( x , y ) | } = sup α ∈ [ 0 , 1 ] max { ∥ 1 ( m + 1 ) ( n + 1 ) ∑ j = 0 m ∑ k = 0 n L ˜ jk ( f α - ) - f α - ∥ , ∥ 1 ( m + 1 ) ( n + 1 ) ∑ j = 0 m ∑ k = 0 n L ˜ jk ( f α + ) - f α + ∥ }

Combining the above equality with (21), we have

D * ( 1 ( m + 1 ) ( n + 1 ) ∑ j = 0 m ∑ k = 0 n L jk ( f ) , f ) ≤ ɛ + C ∑ i = 0 3 ∥ 1 ( m + 1 ) ( n + 1 ) ∑ j = 0 m ∑ k = 0 n L ˜ jk ( g i ) - g i ∥ .(22)

Now, for a given ɛ′ > 0, choose ɛ > 0 such that 0 < ɛ < ɛ′ and also define the following sets: U : = { n ∈ ℕ : D * ( 1 ( m + 1 ) ( n + 1 ) ∑ j = 0 m ∑ k = 0 n L jk ( f ) , f ) ≥ ɛ ′ } , U i : = { n ∈ ℕ : ∥ 1 ( m + 1 ) ( n + 1 ) ∑ j = 0 m ∑ k = 0 n L ˜ jk ( g i ) - g i ∥ ≥ ɛ ′ - ɛ 4 C } i = 0 , 1 , 2 , 3 .

Then inequality (22) gives U ⊆ U₀ ∪ U₁ ∪ U₂ ∪ U₃ and so δ (U) ≤ δ (U₀) + δ (U₁) + δ (U₂) + δ (U₃). By considering this inequality and (18), we get (19). The proof is completed. □

Example 4. Consider the sequence of fuzzy Bernstein-type polynomials

B mn ( f ; x , y ) = ∑ j = 0 m ∑ k = 0 n f ( j m , k n ) ( m j ) x j ( 1 - x ) m - j ( n k ) y k ( 1 - y ) n - k(23) where (x, y) ∈ K = [0, 1] × [0, 1], f ∈ C F ( K ) , ( m , n ) ∈ ℕ 2 . By using these operators, define the following fuzzy positive linear operatorson C F ( K ) L mn ( f ; x , y ) = ( 1 + x mn ) B mn ( f ; x , y )(24) where x _mn = (-1) ⁿ . For all α ∈ [0, 1] we have { L mn ( f ; x , y ) } α ± = L ˜ mn ( f α ± ; x , y ) = ( 1 + x mn ) ∑ j = 0 m ∑ k = 0 n f α ± ( j m , k n ) ( m j ) x j ( 1 - x ) m - j ( n k ) y k ( 1 - y ) n - k .

Then, the sequence {L _mn } satisfies conditions of Theorem 2. Hence, we have st - lim m , n → ∞ D * ( 1 ( m + 1 ) ( n + 1 ) ∑ j = 0 m ∑ k = 0 n L jk ( f ) , f ) = 0 .

However, since (x _mn ) is neither convergent nor statistical convergent to 0 the double sequence ( ∥ L ˜ mn ( g i ) - g i ∥ ) is neither convergent nor statistically convergent to zero for i = 0, 1, 2, 3. So Theorem 2.2 in [11] and Theorem 2.1 in [11] do not work for our operators defined by (24).

In this paper, we have introduced statistical summability (C,1,1) of double sequences in fuzzy number space E ⁿ and given slowly oscillating type Tauberian conditions for statistically summable (C, 1, 1) double sequences of fuzzy numbers. Besides, we presented an application of the method to fuzzy Korovkin approximation theory. Many studies have been done on the concept of statistical summability methods of fuzzy numbers and on their applications recently, and in this way current paper is likely to help researchers who deals with the concept of statistical type weighted mean summability methods on fuzzy number space and with related applications.