Applications of Tauberian theorems for double sequences of fuzzy numbers via De La Vallée Poussin mean

1 Introduction and preliminaries

Tauberian theorem was initially proved by Tauber [34] in 1897. Since then several mathematicians studied this theorem from different aspects. Some of them are Estrada and Vindas [12], Schmidt [33], Dik [9], Edely and Mursaleen [11], Braha [5] and Maddox [19]. In [15] Fridy and Khan proved certain Tauberian theorems via statistical convergence. Subsequently, Mòricz and Stadtmüller [23] prove necessary and sufficient Tauberian conditions under which convergence follows from summability by weighted mean. Recently Raj et al. [30] established some Tauberian theorems for Orlicz spaces of double difference sequences of fuzzy numbers. In several fields of mathematics, Tauberian theorems serve many applications. Likewise in the analysis of differential operators, complex analysis, number theory and probability theory (see [1, 18]). Raj and Choudhary [31] studied some applications of Tauberian theorems for Cesàro Orlicz double sequences.

A double sequence x = (x _mn ) is an infinite array of elements x mn ∈ ℝ (or ℂ ) for all m , n ∈ ℕ . Bromwich [6] initiated the work on double sequences. Later on, the double sequences were studied by Moricz [21], Moricz and Rhoades [22], Başarir and Solankan [3], Tripathy et al. [39] and many others. Baliarsingh [2] studied some difference double sequence spaces of fractional order. In [26] Nayak and Baliarsingh explored some applications of double difference sequences. Recently, Choudhary and Raj [8] investigated some interesting results on the space of double difference sequences of fractional order. Ulusu and Gülle [41] worked on statistical convergence for double set sequences. Nuray et al. studied lacunary statistical convergence [27] and Cesàro summability [28] on double sequences of sets. A double sequence x = (x _mn ) is bounded if there exists a positive number k such that |x _mn | < k, for all m and n. Let c, c₀ and ℓ_∞ denote the space of convergent, null and bounded sequences respectively.

Initially, Kızmaz [17] introduced difference sequence spaces. The notion was further generalized by Et and Çolak [13]. Let r be a non-negative integer and w denotes the set of real or complex sequences. Then for Z = c, c₀ and ℓ_∞, we have the following sequence spaces Z ( Δ r ) = { x = ( x m ) ∈ w : ( Δ r x m ) ∈ Z } , where Δ ^r x = (Δ ^r x _m ) = (Δ^r-1x _m  - Δ^r-1x_m+1) and Δ⁰x _m  = x _m for all m ∈ ℕ , which is equivalent to the following binomial representation Δ r x m = ∑ v = 0 r ( - 1 ) v ( r v ) x m + v . Similarly, difference operator on double sequences can be defined as: Δ x mn = ( x mn - x m , n + 1 ) - ( x m + 1 , n - x m + 1 , n + 1 ) = x mn - x m , n + 1 - x m + 1 , n + x m + 1 , n + 1 , Δ 0 x mn = x mn and Δ r x mn = Δ r - 1 x mn - Δ r - 1 x m , n + 1 - Δ r - 1 x m + 1 , n + Δ r - 1 x m + 1 , n + 1 . Fuzzy set theory is an incredible tool for modelling uncertainty and vagueness in different issues emerging in the field of science and technology. This theory has wide range of applications in population dynamics, chaos control, computer programming and non-linear dynamical systems. In [42] Zadeh introduced the concept of a fuzzy number. For definitions and results on fuzzy numbers sequences one can refer to ([16, 40]). For applications in different fields, one can refer to [14] and [24]. Matloka [20] introduced the concept of usual convergence in fuzzy number sequences and Nanda [25] proved that the set of all convergent fuzzy number sequences form a complete metric space. Let F¹ denotes the space of fuzzy numbers. For x ∈ F¹, λ -level set [x]  _λ is defined by [ x ] λ = { { t ∈ ℝ : x ( t ) ≥ λ } , 0 < λ ≤ 1 , { t ∈ ℝ : x ( t ) > λ } ¯ , λ = 0 . { t ∈ ℝ : x ( t ) > λ } ¯ , λ = 0 . The set [x]  _λ is closed, bounded and non-empty interval for each λ ∈ [0, 1] which is defined by [x]  _λ  = [x^- (λ) , x⁺ (λ)].

Definition 1.1. [4] Let D : F 1 × F 1 → ℝ + , D ( x , y ) = sup λ ∈ [ 0 , 1 ] max { | x - ( λ ) - y - ( λ ) | , | x + ( λ ) - y + ( λ ) | } = sup λ ∈ [ 0 , 1 ] d ( [ x ] λ , [ y ] λ ) . Then D is called the Hausdorff distance between fuzzy numbers x and y.

Proposition 1.2. [4] Let x, y, z, t ∈ F¹ and k ∈ ℝ . Then the following hold:

(i) (F¹, D) is a complete metric space.

(ii) D (x + z, y + z) = D (x, y) .

(iii) D (kx, ky) = |k|D (x, y) .

(iv) D (x + y, z + t) ≤ D (x, z) + D (y, t) .

(v) | D ( x , 0 ¯ ) - D ( y , 0 ¯ ) | ≤ D ( x , y ) ≤ D ( x , 0 ¯ ) + D ( y , 0 ¯ ) .

Definition 1.3. A double sequence x = (x _mn ) of fuzzy numbers is a function x from ℕ × ℕ into the set F¹.

Definition 1.4. Let u = (u _kl ) be a double sequence of positive real numbers. A double sequence x = (x _mn ) of fuzzy numbers is said to be Δ u r - convergent to the fuzzy number L written as lim m , n u kl x mn = L if for every ɛ > 0 there exists a positive integer n₀ (ɛ) such that D (u _kl Δ ^r x _mn , L) < ɛ whenever m, n ≥ n₀. The number L is called the Δ u r - Pringsheim limit of x.

Definition 1.5. A double sequence (x _mn ) is said to be ( C , 1 , 1 ) Δ u r - summable to L if lim m , n → ∞ σ mn u kl , Δ r = L ,(1.1) where σ mn u kl , Δ r is a ( C , 1 , 1 ) Δ u r - means of (x _mn ) and σ mn u kl , Δ r = 1 ( m + 1 ) ( n + 1 ) ∑ k = 0 m ∑ l = 0 n u kl Δ r x kl for all non-negative integers m and n .

Definition 1.6. Let m and n be sufficiently large non-negative integers and for α _n the integer part of the product αn . Then the de la Vallée Poussin means of the double sequence (x _mn ) are defined by ϑ mn > , u kl , Δ r = 1 ( α m - m ) ( α n - n ) ∑ k = m + 1 α m ∑ l = n + 1 α n u kl Δ r x kl , if α > 1 and ϑ mn < , u kl , Δ r = 1 ( m - α m ) ( n - α n ) ∑ k = α m + 1 m ∑ l = α n + 1 n u kl Δ r x kl , if 0 < α < 1 .

Tripathy and Baruah [36] initiated the study of the concept of slowly oscillating sequences of fuzzy numbers.

Definition 1.7. A double sequence (x _mn ) is said to be Δ u r - slowly oscillating in the sense (1, 1) , if lim α ↓ 1 lim sup m , n → ∞ max m + 1 ≤ k ≤ α m n + 1 ≤ l ≤ α n D ( u kl Δ r x kl , u kl Δ r x mn ) = 0 . Identically, a double sequence (x _mn ) is slowly oscillating in the sense (1, 1) if for each ɛ > 0, there exists η₁ (ɛ) and α = (α _ɛ ) >1, such that D (u _kl Δ ^r x _kl , u _kl Δ ^r x _mn ) ≤ ɛ, whenever η₁ < m < k ≤ α _m and η₁ < n < l ≤ α _n  .

Definition 1.8. A double sequence (x _mn ) is said to satisfy the two-sided Δ u r - Hardy type Tauberian condition in the sense (1, 0) , if there exists a positive integer γ₁ and a constant G such that kD ( u kl Δ r x kn , u kl Δ r x k - 1 , n ) ≤ G , whenever k , n > γ 1 .

Definition 1.9. A double sequence (x _mn ) of fuzzy number is said to satisfy the two-sided Δ u r - Hardy type Tauberian condition in the sense (0, 1) , if there exists a positive integer δ₁ and a constant G₁ such that lD ( u kl Δ r x ml , u kl Δ r x m , l - 1 ) ≤ G 1 , whenever l , m > δ 1 .

Theorem 2.1. If x = (x _mn ) is bounded and Δ u r - convergent to L . Then x = (x _mn ) is ( C , 1 , 1 ) Δ u r - summable to a fuzzy number L .

Proof. Let x = (x _mn ) be a double sequence of fuzzy number convergent to L ∈ F¹ such that lim m , n → ∞ D ( u kl Δ r x mn , L ) = 0 . Now by using Proposition 1.2, we get D ( σ mn u kl , Δ r , L ) = D ( 1 ( m + 1 ) ( n + 1 ) ∑ k = 0 m ∑ l = 0 n u kl Δ r x kl , L ) ≤ 1 ( m + 1 ) ( n + 1 ) ∑ k = 0 m ∑ l = 0 n D ( u kl Δ r x kl , L ) . Since lim m , n → ∞ D ( u kl Δ r x kl , L ) = 0 and also a double sequence (x _mn ) is bounded. Hence, by using the regularity of (C, 1, 1) summability, we have lim m , n → ∞ D ( σ mn u kl , Δ r , L ) = 0 . □

The following example explains that the converse of Theorem 2.1 does not hold in general.

Example 2.2. Define the double sequence of fuzzy numbers x = (x _mn ) by u kl Δ r x mn = { τ ν , if m , n are even ; ς ν , otherwise , where τ ν ( s ) = { - s + 1 , if s ∈ [ 0 , 1 ] ; 0 , otherwise and ς ν ( s ) = { s + 1 , if s ∈ [ - 1 , 0 ] ; 0 , otherwise . Clearly, Δ u r x mn is not convergent. Now, we can check the endpoints of the λ-level set of u _kl Δ ^r x _mn . u kl Δ r x mn - ( λ ) = { 0 , if m , n are even ; λ - 1 , otherwise and u kl Δ r x mn + ( λ ) = { 1 - λ , if m , n are even ; 0 , otherwise . Then by the definition of ( C , 1 , 1 ) Δ u r mean, we have σ mn - u kl Δ r ( λ ) = { - 1 + λ 4 ( ( m + 2 ) n + 2 m ( n + 1 ) ( m + 1 ) ( n + 1 ) ) , if m , n are even ; - 1 + λ 4 ( 3 m + 2 m + 1 ) , if m is even and n is odd ; 3 ( - 1 + λ ) 4 , if m , n are odd ; - 1 + λ 4 ( 3 n + 2 n + 1 ) , if m is odd and n is even and σ mn + u kl Δ r ( λ ) = { 1 - λ 4 ( ( m + 2 ) ( n + 2 ) ( m + 1 ) ( n + 1 ) ) , if m , n are even ; 1 - λ 4 ( 3 m + 2 m + 1 ) , if m is even and n is odd ; 1 - λ 4 , if m , n are odd ; 1 - λ 4 ( n + 2 n + 1 ) , if m is odd and n is even . Hence, the sequences ( σ mn - u kl Δ r ( λ ) ) and ( σ mn + u kl Δ r ( λ ) ) converges to 3 ( - 1 + λ ) 4 and 1 - λ 4 as m, n→ ∞ respectively. Now by taking κ _ν  = (τ _ν  + 3ς _ν )/4, we get lim m , n → ∞ D ( σ mn u kl , Δ r , κ ν ) = 0 . Thus, (x _mn ) is ( C , 1 , 1 ) Δ u r - summable to κ _ν  . Therefore, we can conclude that a double sequence of fuzzy numbers (x _mn ) is ( C , 1 , 1 ) Δ u r - summable but not Δ u r - convergent to κ _ν .

Now, we show that if a double sequence of fuzzy numbers (x _mn ) is ( C , 1 , 1 ) Δ u r - summable then it is Δ u r - convergent with the help of certain Tauberian conditions. These types of results are known as Tauberian Theorems. Firstly, we shall give some lemmas that help to prove Tauberian Theorems.

Lemma 2.3. Let Δ ^r be a difference operator of order r . Then the followings hold:

(i) If α > 1, α _m  > m and α _n  > n, then D ( ϑ mn > , u kl , Δ r , σ mn u kl , Δ r ) ≤ ( α m + 1 ) ( α n + 1 ) ( α m - m ) ( α n - n ) D ( σ α m , α n u kl , Δ r , σ α m , n u kl , Δ r ) + ( α m + 1 ) ( α n + 1 ) ( α m - m ) ( α n - n ) D ( σ m , n u kl , Δ r , σ m , α n u kl , Δ r ) + α m + 1 α m - m D ( σ α m , n u kl , Δ r , σ m , n u kl , Δ r ) + α n + 1 α n - n D ( σ m , α n u kl , Δ r , σ m , n u kl , Δ r ) . (ii) If 0 < α < 1, α _m  < m and α _n  < n, then D ( ϑ mn < , u kl , Δ r , σ mn u kl , Δ r ) ≤ ( α m + 1 ) ( α n + 1 ) ( m - α m ) ( n - α n ) D ( σ α m , α n u kl , Δ r , σ α m , n u kl , Δ r ) + ( α m + 1 ) ( α n + 1 ) ( m - α m ) ( n - α n ) Δ r ) + α m + 1 m - α m D ( σ m , n u kl , Δ r , σ α m , n u kl , Δ r ) Δ r , σ m , α n u kl , Δ r ) .

Proof.  D ( ϑ mn > , u kl , Δ r , σ mn u kl , Δ r ) = D ( 1 ( α m - m ) ( α n - n ) ∑ k = m + 1 α m ∑ l = n + 1 α n u kl Δ r x kl + 1 ( α m - m ) ( α n - n ) ( ∑ k = 0 m ∑ l = n + 1 α n + ∑ k = m + 1 α m ∑ l = 0 n + ∑ k = 0 m ∑ l = 0 n ) u kl Δ r x kl , σ mn u kl , Δ r + 1 ( α m - m ) ( α n - n ) ( ∑ k = 0 m ∑ l = n + 1 α n + ∑ k = m + 1 α m ∑ l = 0 n + ∑ k = 0 m ∑ l = 0 n ) u kl Δ r x kl ) = D ( 1 ( α m - m ) ( α n - n ) ∑ k = 0 α m ∑ l = 0 α n u kl Δ r x kl + 1 ( α m - m ) ( α n - n ) ∑ k = 0 m ∑ l = 0 n u kl Δ r x kl , σ mn u kl , Δ r + 1 ( α m - m ) ( α n - n ) ∑ k = 0 m ∑ l = 0 α n u kl Δ r x kl + 1 ( α m - m ) ( α n - n ) ∑ k = 0 α m ∑ l = 0 n u kl Δ r x kl ) = D ( ( α m + 1 ) ( ( α n + 1 ) ( α m - m ) ( α n - n ) σ α m , α n u kl , Δ r + ( m + 1 ) ( n + 1 ) ( α m - m ) ( α n - n ) σ m , n u kl , Δ r , σ mn u kl , Δ r + ( m + 1 ) ( α n + 1 ) ( α m - m ) ( α n - n ) σ m , α n u kl , Δ r + ( α m + 1 ) ( n + 1 ) ( α m - m ) ( α n - n ) σ α m , n u kl , Δ r ) = D ( ( α m + 1 ) ( α n + 1 ) ( α m - m ) ( α n - n ) σ α m , α n u kl , Δ r + ( m + 1 ) ( n + 1 ) ( α m - m ) ( α n - n ) σ m , n u kl , Δ r + α n + 1 α n - n σ m , α n u kl , Δ r + α m + 1 α m - m σ α m , n u kl , Δ r , σ m , n u kl , Δ r + ( α m + 1 ) ( α n + 1 ) ( α m - m ) ( α n - n ) ( σ m , α n u kl , Δ r + σ α m , n u kl , Δ r ) ) ≤ ( α m + 1 ) ( α n + 1 ) ( α m - m ) ( α n - n ) D ( σ α m , α n u kl , Δ r , σ α m , n u kl , Δ r ) + D ( ( m + 1 ) ( n + 1 ) ( α m - m ) ( α n - n ) σ m , n u kl , Δ r + α n + 1 α n - n σ m , α n u kl , Δ r + α m + 1 α m - m σ α m , n u kl , Δ r + n + 1 α n - n σ m , n u kl , Δ r + m + 1 α m - m σ m , n u kl , Δ r + σ m , n u kl , Δ r , σ m , n u kl , Δ r + ( α m + 1 ) ( α n + 1 ) ( α m - m ) ( α n - n ) σ m , α n u kl , Δ r + n + 1 α n - n σ m , n u kl , Δ r + m + 1 α m - m σ m , n u kl , Δ r + σ m , n u kl , Δ r ) ≤ ( α m + 1 ) ( α n + 1 ) ( α m - m ) ( α n - n ) D ( σ α m , α n u kl , Δ r , σ α m , n u kl , Δ r ) + D ( ( m + 1 ) ( n + 1 ) ( α m - m ) ( α n - n ) σ m , n u kl , Δ r + α n + 1 α n - n σ m , α n u kl , Δ r + α m + 1 α m - m σ α m , n u kl , Δ r , ( α m + 1 ) ( α n + 1 ) ( α m - m ) ( α n - n ) σ m , α n u kl , Δ r + α n + 1 α n - n σ m , n u kl , Δ r + α m + 1 α m - m σ m , n u kl , Δ r ) ≤ ( α m + 1 ) ( α n + 1 ) ( α m - m ) ( α n - n ) D ( σ α m , α n u kl , Δ r , σ α m , n u kl , Δ r ) + ( α m + 1 ) ( α n + 1 ) ( α m - m ) ( α n - n ) D ( σ m , n u kl , Δ r , σ m , α n u kl , Δ r ) + α m + 1 α m - m D ( σ α m , n u kl , Δ r , σ m , n u kl , Δ r ) + α n + 1 α n - n D ( σ m , α n u kl , Δ r , σ m , n u kl , Δ r ) . (ii) D ( ϑ mn < , u kl , Δ r , σ mn u kl , Δ r ) = D ( 1 ( m - α m ) ( n - α n ) ∑ k = α m + 1 m ∑ l = α n + 1 n u kl Δ r x kl + 1 ( m - α m ) ( n - α n ) ( ∑ k = 0 α m ∑ l = α n + 1 n + ∑ k = α m + 1 m ∑ l = 0 α n + ∑ k = 0 α m ∑ l = 0 α n ) u kl Δ r x kl , σ mn u kl , Δ r + 1 ( m - α m ) ( n - α n ) ( ∑ k = 0 α m ∑ l = α n + 1 n + ∑ k = α m + 1 m ∑ l = 0 α n + ∑ k = 0 α m ∑ l = 0 α n ) u kl Δ r x kl ) = D ( 1 ( m - α m ) ( n - α n ) ∑ k = 0 m ∑ l = 0 n u kl Δ r x kl + 1 ( m - α m ) ( n - α n ) ∑ k = 0 α m ∑ l = 0 α n u kl Δ r x kl , σ mn u kl , Δ r + 1 ( m - α m ) ( n - α n ) ∑ k = 0 α m ∑ l = 0 n u kl Δ r x kl + 1 ( m - α m ) ( n - α n ) ∑ k = 0 m ∑ l = 0 α n u kl Δ r x kl ) = D ( ( m + 1 ) ( ( n + 1 ) ( m - α m ) ( n - α n ) σ m , n u kl , Δ r + ( α m + 1 ) ( α n + 1 ) ( m - α m ) ( n - α n ) σ α m , α n u kl , Δ r , σ mn u kl , Δ r + ( α m + 1 ) ( n + 1 ) ( m - α m ) ( n - α n ) σ α m , n u kl , Δ r + ( α m + 1 ) ( α n + 1 ) ( m - α m ) ( n - α n ) σ m , α n u kl , Δ r ) = D ( ( m + 1 ) ( n + 1 ) ( m - α m ) ( n - α n ) σ m , n u kl , Δ r + ( α m + 1 ) ( α n + 1 ) ( m - α m ) ( n - α n ) σ α m , α n u kl , Δ r , σ m , n u kl , Δ r + ( α m + 1 ) ( α n + 1 ) ( m - α m ) ( n - α n ) ( σ α m , n u kl , Δ r + σ m , α n u kl , Δ r ) + α n + 1 n - α n σ m , α n u kl , Δ r + α m + 1 m - α m σ α m , n u kl , Δ r ) ≤ ( α m + 1 ) ( α n + 1 ) ( m - α m ) ( n - α n ) D ( σ α m , α n u kl , Δ r , σ α m , n u kl , Δ r ) + D ( ( α m + 1 ) ( α n + 1 ) ( m - α m ) ( n - α n ) σ m , n u kl , Δ r + α n + 1 n - α n σ m , n u kl , Δ r + α m + 1 m - α m σ m , n u kl , Δ r + σ m , n u kl , Δ r , σ m , n u kl , Δ r + ( α m + 1 ) ( α n + 1 ) ( m - α m ) ( n - α n ) σ m , α n u kl , Δ r + α n + 1 n - α n σ m , α n u kl , Δ r + α m + 1 m - α m σ α m , n u kl , Δ r ) ≤ ( α m + 1 ) ( α n + 1 ) ( m - α m ) ( n - α n ) D ( σ α m , α n u kl , Δ r , σ α m , n u kl , Δ r ) + ( α m + 1 ) ( α n + 1 ) ( m - α m ) ( n - α n ) D ( σ m , n u kl , Δ r , σ m , α n u kl , Δ r ) + α m + 1 m - α m D ( σ m , n u kl , Δ r , σ α m , n u kl , Δ r ) + α n + 1 n - α n D ( σ m , n u kl , Δ r , σ m , α n u kl , Δ r ) . □ Lemma 2.4. If x = (x _mn ) is ( C , 1 , 1 ) Δ u r - summable to L, then lim m , n → ∞ ϑ mn > , u kl , Δ r = L(2.1)andlim m , n → ∞ ϑ mn < , u kl , Δ r = L .(2.2)

Proof. From Proposition 1.2, we have D ( ϑ mn > , u kl , Δ r , L ) = D ( ϑ mn > , u kl , Δ r + σ m , n u kl , Δ r , σ m , n u kl , Δ r + L ) ≤ D ( ϑ mn > , u kl , Δ r , σ m , n u kl , Δ r ) + D ( σ m , n u kl , Δ r , L ) .(2.3) Now, by using Lemma 2.3 (i) and (2.3) , we get

D ( ϑ mn > , u kl , Δ r , L ) ≤ ( α m + 1 ) ( α n + 1 ) ( α m - m ) ( α n - n ) D ( σ α m , α n u kl , Δ r , σ α m , n u kl , Δ r ) + ( α m + 1 ) ( α n + 1 ) ( α m - m ) ( α n - n ) D ( σ m , n u kl , Δ r , σ m , α n u kl , Δ r ) + α m + 1 α m - m D ( σ α m , n u kl , Δ r , σ m , n u kl , Δ r ) + α n + 1 α n - n D ( σ m , α n u kl , Δ r , σ m , n u kl , Δ r ) + D ( σ m , n u kl , Δ r , L ) . For all α > 1 and sufficiently large n, we have α α - 1 < α n + 1 α n - n < 2 α α - 1 .(2.4) The inequality (2.4) holds for α _m with sufficiently large m . Thus, (2.1) follows from (1.1) and (2.4) . Similarly, we can prove for (2.2) .□

Theorem 2.5. Let Δ ^r be a difference operator of order r . If x = (x _mn ) is ( C , 1 , 1 ) Δ u r - summable to L, then x = (x _mn ) is Δ u r - convergent to L if and only if either of the following holds:lim inf α ↓ 1 lim sup m , n → ∞ D ( ϑ mn > , u kl , Δ r , u kl Δ r x mn ) = 0 .(2.5) lim inf α ↑ 1 lim sup m , n → ∞ D ( ϑ mn < , u kl , Δ r , u kl Δ r x mn ) = 0 .(2.6)

Proof. Firstly, we prove the direct part of the theorem. Since x = (x _mn ) is ( C , 1 , 1 ) Δ u r - summable to a fuzzy number L and lim m , n → ∞ u kl Δ r x mn = L . Then, we have lim m , n → ∞ D ( u kl Δ r x mn , σ m , n u kl , Δ r ) = 0 .(2.7) Now, by using (2.7) , we have (2.5) and (2.6) follows by (2.1) and (2.2) respectively.

Conversely, consider that (2.5) is satisfied. Hence, for any given ɛ > 0, there exists α₀ > 1 such that lim sup m , n → ∞ D ( ϑ mn > , u kl , Δ r , u kl Δ r x mn ) ≤ ɛ(2.8) Also, we have D ( u kl Δ r x mn , L ) = D ( u kl Δ r x mn + ϑ mn > , u kl , Δ r , L + ϑ mn > , u kl , Δ r ) ≤ D ( u kl Δ r x mn , ϑ mn > , u kl , Δ r ) + D ( ϑ mn > , u kl , Δ r , L ) . Since x = (x _mn ) is ( C , 1 , 1 ) Δ u r - summable to L . Hence, by using (2.1) and (2.8) , we have lim sup m , n → ∞ D ( u kl Δ r x mn , L ) ≤ ɛ(2.9) and if (2.6) is satisfied, for any given ɛ > 0, there exist 0 < α₀ < 1 such that lim sup m , n → ∞ D ( ϑ mn < , u kl , Δ r , u kl Δ r x mn ) ≤ ɛ .(2.10) Also, D ( u kl Δ r x mn , L ) = D ( u kl Δ r x mn + ϑ mn < , u kl , Δ r , L + ϑ mn < , u kl , Δ r ) ≤ D ( u kl Δ r x mn , ϑ mn < , u kl , Δ r ) + D ( ϑ mn < , u kl , Δ r , L ) and x = (x _mn ) is ( C , 1 , 1 ) Δ u r - summable to L . Thus, by using (2.2) and (2.10) , we get lim sup m , n → ∞ D ( u kl Δ r x mn , L ) ≤ ɛ .

Theorem 2.6. If x = (x _mn ) is Δ u r - slowly oscillating in the sense (1, 1) and ( C , 1 , 1 ) Δ u r - summable to L, then x = (x _mn ) is Δ u r - convergent to L .

Proof. Consider that a double sequence x = (x _mn ) is Δ u r - slowly oscillating. Since, D ( ϑ mn > , u kl , Δ r , u kl Δ r x mn ) D ( 1 ( α m - m ) ( α n - n ) ∑ k = m + 1 α m ∑ l = n + 1 α n u kl Δ r x kl , u kl Δ r x mn ) 1 ( α m - m ) ( α n - n ) ∑ k = m + 1 α m ∑ l = n + 1 α n D ( u kl Δ r x kl , u kl Δ r x mn ) max m + 1 ≤ k ≤ α m n + 1 ≤ l ≤ α n D ( u kl Δ r x kl , u kl Δ r x mn ) . Hence, lim inf α ↓ 1 lim sup m , n → ∞ D ( ϑ mn > , u kl , Δ r , u kl Δ r x mn ) = 0 . Now, by Theorem 2.5, (x _mn ) is Δ u r - convergent to L .□