I 2 - λ ¯ -statistically convergence of double sequences in fuzzy normed spaces

1 Introduction

The notion of statistical convergence of sequences of numbers was introduced by Fast [12] and Schoenberg [57] independently. Later on, statistical convergence turned out to be one of the most active areas of research in summability theory after the works of Fridy [13] and Šalát [14]. Connor [8] have studied the concept statistical convergence in summability theory. Mursaleen and Edely [23] extended the above idea from single to double sequences of scalars and established relations between statistical convergence and strongly Cesàro summable double sequences. Tripathy [38] studied on statistical convergence of double sequences. A lot of useful developments of double sequences in summability methods can be found in Çakan and Altay [4], Altay and Başar [5].

The idea of λ-statistical convergence was introduced and studied by Mursaleen [24] as an extension of the [V, λ] summability of Leindler [25]. Many mathematicians such as Çolak [6], Et and Cinar [7] have studied on λ-statistical convergence.

The concept of ideal convergence plays a vital role not only in pure mathematics but also in other branches of science involving mathematics, especially in information theory, computer science, biological science, dynamical systems, geographic information systems, population modeling and motion planning in robotics.

Among various developments of the theory of fuzzy sets [42] a progressive development has been made to find the fuzzy analogues of the classical set theory.

In fact the fuzzy set theory has become an area of active research for the last 40 years. The notion of fuzzyness are using by many persons for Cybernetics, Artificial Intelligence, Expert System and Fuzzy control, Pattern recognition, Operation research, Decision making, Image analysis, Projectiles, Probability theory, Agriculture, Weather forecasting.

The fuzzy set theory has been used widely in many engineering applications, such as, in bifurcation of non-linear dynamical systems, in the control of chaos, in the non-linear operator, in population dynamics.

In many situations, we can’t determine the norm of a vector exactly and hence it seems that the concept of a fuzzy norm is more suitable than a crisp norm in these cases, namely, we can model the inexactness by fuzzy norm.

The idea of fuzzy norm was initiated by Katsaras [44] in studying fuzzy topological vector spaces. Also Alimohammady and Roohi [1] introduced compactness in fuzzy minimal spaces. Using the notion of fuzzy number, Felbin [15] put forward the concept of fuzzy norm on a linear space, which is based on the treatment of a fuzzy metric introduced by Kaleva and Seikkala [45]. Some topological properties of fuzzy normed linear spaces were found in [29, 43]. Other types of fuzzy normed linear spaces can be found in [2, 3].

The concept of ordinary convergence of a sequence of fuzzy numbers was firstly introduced by Matloka [19]. Nanda [28] studied the seqeuences of fuzzy numbers and showed that the set of all convergent sequences of fuzzy numbers form a complete metric space.

Nuray and Savaş [30] gave the notion of statistical convergence of fuzzy numbers. Savaş and Mursaleen [31] studied on statistical convergent double sequences of fuzzy numbers. See also [32 –35] for details.

Şençimen and Pehlivan [36] introduced the notions of statistically convergent sequence and statistically Cauchy sequence in a fuzzy normed linear space. Mohiuddine et al. [26] studied statistical convergence of double sequences in fuzzy normed spaces.

The idea of I -convergence was introduced by Kostyrko et al. [21] as a generalization of statistical convergence. Kostyrko et al. [22] studied the idea of I -convergence and extremal I -limit points.

Let X≠ ∅. A class of sets I of subsets is said to be an ideal in X provided (i) ∅ ∈ I , (ii) For each A , B ∈ I we have A ∪ B ∈ I , (iii) For each A ∈ I and each B ⊆ A we have B ∈ I . Let X≠ ∅. A class of sets F of subsets is said to be an ideal in X provided (i) ∅ ∉ F , (ii) For each A , B ∈ F we have A ∩ B ∈ F , (iii) For each A ∈ F and each A ⊆ B we have B ∈ F . If I is a nontrivial ideal in X, X≠ ∅, then the class F ( I ) = { M ⊂ N : ( ∃ A ∈ I ) ( M = X / A ) } is a filter on X, called the filter associated with I .

Recently, Das et al. [11] introduced new notions, namely I -statistical convergence and I -lacunary statistical convergence by using ideal.

I -convergence has been discussed in more general abstract spaces such as the fuzzy normed spaces [39], 2-normed linear spaces [17], n-normed linear spaces [18]. The notion of intuitionsitc fuzzy sets initially introduced by Anastassiou [16]. Kumar and Kumar [40] introduced I -convergence in intuitionsitc fuzzy normed space. Quite recently Hazarika [51] introduced the notion of ideal convergence in fuzzy normed space and proved some intresting results. Tripathy and Tripathy [53] introduced the concept of I -convergence and I -Cauchy of double sequences and proved some properties related to the solidity, symmetricity, completeness and denseness. Recently, Das et al. [9] introduced the concepts of I and I ∗ -convergence of double sequence in a metric space and studied some properties of this convergence. Savaş [55], Kumar [41] discussed the basic properties of I -convergence and I ∗ -convergence for double sequences. Gürdal et al. [20], Mursaleen et al. [27], studied about ideal convergence of double sequences. Dündar et al. [10] introduced I 2 -convergence of double sequences of fuzzy numbers. Since the study of convergence of a sequence in a fuzzy normed space is very important to fuzzy functional analysis, Hazarika and Kumar [51] introduced the notion of I -convergent and I -Cauchy in a fuzzy normed linear space and establish some basic results related to these notions. Hazarika and Kumar [52] gave the concept of I -convergence of double sequences in fuzzy normed spaces.

Savaş [56] presented λ-statistical convergence of fuzzy numbers. Double λ-statistical convergence was studied and examined by Savaş ([47, 48] and Savaş and Patterson ([49, 50]). Furthermore, Hazarika and Savaş [54] introduced the concept of (λ, μ)-statistically convergence of double sequences in n-normed spaces, gave some properties of the new concept. λ-statistical convergence in a fuzzy normed linear space was studied by Türkmen and Çınar [37].

2 Preliminaries

Fuzzy sets are considered with respect to a nonempty base set X of elements of interest. The essential idea is that each element x ∈ X is assigned a membership grade u (x) taking values in [0, 1], with u (x) =0 corresponding to nonmembership, 0 < u (x) <1 to partial membership, and u (x) =1 to full membership. According to Zadeh [42], a fuzzy subset of X is a nonempty subset {(x, u (x)) : x ∈ X} of X × [0, 1] for some function u : X → [0, 1]. The function u itself is often used for the fuzzy set.

A fuzzy set u on ℝ is called a fuzzy number if it has the following properties:

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

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

3. u is upper semicontinuous;

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

Let L ( ℝ ) be set of all fuzzy numbers. If u ∈ L ( ℝ ) and u (t) =0 for t < 0, then u is called a non-negative fuzzy number. We write L ∗ ( ℝ ) by the set of all nonnegative fuzzy numbers. We can say that L ∗ ( ℝ ) iff u α - ≥ 0 for each α ∈ [0, 1]. Clearly we have 0 ˜ ∈ L ( ℝ ) . For u ∈ L ( ℝ ) , the α level set of u is defined by [ u ] α = { { x ∈ ℝ : u ( x ) > α } , if α ∈ ( 0 , 1 ] suppu , if α = 0 A partial order on L ( ℝ ) is defined by uv if u α - v α - and u α + v α + for all α ∈ [0, 1].

Arithemetic operation for t ∈ ℝ , ⊕, ⊖, ⊙ and ø on L ( ℝ ) × L ( ℝ ) are defined by

( u ⊕ v ) ( t ) = sup s ∈ ℝ { u ( s ) Λ v ( t - s ) } ,

( u ⊖ v ) ( t ) = sup s ∈ ℝ { u ( s ) Λ v ( s - t ) } ,

( u ⊙ v ) ( t ) = sup s ∈ ℝ , s ≠ 0 { u ( s ) Λ v ( t / s ) } and

( u ø v ) ( t ) = sup s ∈ ℝ { u ( st ) Λ v ( s ) } .

For k ∈ ℝ + , ku is defined as ku (t) = u (t/k) and 0 u ( t ) = 0 ˜ , t ∈ ℝ .

Some arithmetic operations for α-level sets are defined as follows:

[ u ⊕ v ] α = [ u α - + v α - , u α + + v α + ] ,

[ u ⊖ v ] α = [ u α - - v α + , u α + - v α - ] ,

[ u ⊙ v ] α = [ u α - . v α - , u α + . v α + ] and

[ 1 ˜ ø u ] α = [ 1 u α + , 1 u α - ] , u α - > 0 .

For u , v ∈ L ( ℝ ) , the supremum metric on L ( ℝ ) defined as D ( u , v ) = sup 0 ≤ α ≤ 1 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 said to be convergent to the fuzzy number x ₀, if for every ɛ > 0 there exists a positive integer k ₀ such that D (x _k , x ₀) < ɛ for k > k ₀ and a sequence x = (x _k ) of fuzzy numbers converges to levelwise to x ₀ iff lim k → ∞ [ x k ] α = [ x 0 ] α - and lim k → ∞ [ x k ] α = [ x 0 ] α + , where [ x k ] α = [ ( x k ) α - , ( x k ) α + ] and [ x 0 ] α = [ ( x 0 ) α - , ( x 0 ) α + ] , for every α ∈ (0, 1).

Let X be a vector space over ℝ , | | . | | : X → L ∗ ( ℝ ) and the mappings L ; R (respevtively,left norm and right norm): [0, 1] × [0, 1] → [0, 1] be symetric, nondecreasing in both arguments and satisfy L (0, 0) =0 and R (1, 1) =1 .

The quadruple (X, || . ||, L, R) is called fuzzy normed linear space (briefly (X, || . ||) FNS) and || . || a fuzzy norm if the following axioms are satisfied

1. | | x | | = 0 ˜ iff x = 0 ;

2. ||rx|| = |r| ⊙ ||x|| for x ∈ X, r ∈ ℝ ,

3. For all x, y ∈ X

(a) ||x + y|| (s + t) ≥ L (||x|| (s) , ||y|| (t)), whenewer s ≤ | | x | | 1 - , t ≤ | | y | | 1 - and s + t ≤ | | x + y | | 1 - ,

(b) ||x + y|| (s + t) ≤ R (||x|| (s) , ||y|| (t)), whenewer s ≥ | | x | | 1 - , t ≤ | | y | | 1 - and s + t ≥ | | x + y | | 1 - .

Let (X, || . || _C ) be an ordinary normed linear space. Then, a fuzzy norm || . || on X can be obtained by

| | x | | ( t ) = { 0 , if 0 ≤ t ≤ a | | x | | C or t ≥ b | | x | | C t ( 1 - a ) | | x | | C - a 1 - a , a | | x | | C ≤ t ≤ | | x | | C - t ( b - 1 ) | | x | | C + b b - 1 , | | x | | C ≤ t ≤ b | | x | | C(2.1) where ||x|| _C is the ordinary norm of x (≠ 0), 0 < α < 1 and 1< b < ∞. For x = θ, define | | x | | = 0 ˜ . Hence, (X, || . ||) is a fuzzy normed linear space.

Let us consider be an FNS (X, || . ||). For any ɛ > 0, α ∈ [0, 1] and x ∈ X, the (ɛ, α)-neighborhood of x is the set N x ( ɛ , α ) = { y ∈ X : | | x - y | | α + < ɛ } .

Let (X, || . ||) be an FNS. A sequence ( x n ) n = 1 ∞ in X is convergent to x ∈ X with respect to the fuzzy norm on X and we denote by x n → FN x , provided that ( D ) - lim n → ∞ | | x n - x | | = 0 ˜ ; i.e., for every ɛ > 0 there is an N ( ɛ ) ∈ ℕ such that D ( | | x n - x | | , 0 ˜ ) < ɛ for all n ≥ N (ɛ). This means that for every ɛ > 0 there is an N ( ɛ ) ∈ ℕ such that for all n ≥ N (ɛ), sup α ∈ [ 0 , 1 ] | | x n - x | | α + = | | x n - x | | α - < ɛ .

Let (X, || . ||) be an FNS. A sequence ( x n ) n = 1 ∞ in X is statistically convergent to L ∈ X with respect to the fuzzy norm on X and we denote by x n → FS x , provided that for each ɛ > 0, we have δ ( { k ∈ ℕ : D ( | | x n - x | | , 0 ˜ ) ≥ ɛ } ) = 0 . This implies that for each ɛ > 0, the set K ( ɛ ) = { k ∈ ℕ : | | x n - x | | 0 + ≥ ɛ } has natural density zero; namely, for each ɛ > 0, | | x n - x | | 0 + < ɛ for almost all k.

A sequence x = (x _k ) in X is said to be I -statistically convergent to L in X or FS I -convergent to L with respect to fuzzy norm on X if for each ɛ, δ > 0 { n ∈ ℕ : 1 n | { k ≤ n : ( D | | x k - L | | , 0 ˜ ) ≥ ɛ } | ≥ δ } ∈ I . In this case, we write x k → FS I L or x k → L ( FS I ) . The class of all I -statistically convergent sequences in fuzzy normed space X will be denoted by FS I ( X ) .

A sequence x = (x _k ) in X is said to be I λ -statistically convergent or FS λ I -convergent to L ∈ X with respect to fuzzy norm on X if for each ɛ, δ > 0 { n ∈ ℕ : 1 λ n | { k ∈ I n : ( D | | x k - L | | , 0 ˜ ) ≥ ɛ } | ≥ δ } ∈ I . In this case, we write x k → FS λ I L or x k → L ( FS λ I ) or FS λ I - lim k → ∞ x k = L .

A double sequence x = (x _mn ) of real numbers is said to be bounded if there exists a positive real number M such that |x _mn | < M, for all m , n ∈ ℕ . That is, | | x | | ∞ = sup k , l | x k , l | < ∞ .

A double sequence x = (x _k,l ) of real numbers is said to be convergent to L ∈ ℝ in Pringsheim’s sense [46](shortly, p-convergent to L ∈ ℝ ), if for any ɛ > 0, there exists N ɛ ∈ ℕ such that |x _k,l  - L| < ɛ, whenever k, l > N _ɛ . In this case, we write lim k , l → ∞ x k , l = L .

We recall that a subset K of ℕ × ℕ is said to have natural density d (K) if d ( K ) = lim k , l → ∞ K ( k , l ) k . l , where K ( k , l ) = | { ( j , k ) ∈ ℕ × ℕ : j ≤ k , k ≤ l } | .

A double sequence x = (x _jk ) is said to be statistically convergent to the number L if for each ɛ > 0, the set {j, k : j≤ m and k ≤ n, |x _jk  - L| ≥ ɛ } has double natural density zero. In this case, we write st 2 - lim jk x jk = L .

Let (X, || . ||) be an FNS. Then a double sequence (x _jk ) is said to be convergent to x ∈ X with respect to the fuzzy norm on X if for each ɛ > 0 there exist a number N = N (ɛ) such that D ( | | x jk - x | | , 0 ˜ ) < ɛ , for all j , k ≥ N .

In this case, we write x jk → FN x . This means that for every ɛ > 0 there is an N ( ɛ ) ∈ ℕ such that sup α ∈ [ 0 , 1 ] | | x jk - x | | α + = | | x jk - x | | α - < ɛ , for all j, k ≥ N. In terms of neighnorhoods, we have x jk → FN x provided that for any ɛ > 0, there exists a number N = N (ɛ) such that x _jk ∈ N x ( ɛ , 0 ) , when every j, k ≥ N .

Let (X, || . ||) be an FNS. Then a double sequence (x _jk ) in X is statistically convergent to L ∈ X with respect to the fuzzy norm on X and we denote by x jk → FS x , provided that for each ɛ > 0, we have δ 2 ( { k ∈ ℕ : D ( | | x jk - x | | , 0 ˜ ) ≥ ɛ } ) = 0 . This implies that for each ɛ > 0, the set K ( ɛ ) = { ( j . k ) ∈ ℕ × ℕ : | | x jk - L | | 0 + ≥ ɛ } has natural density zero; namely, for each ɛ > 0, | | x jk - x | | 0 + < ɛ for almost all j, k. In this case, we write x jk → st 2 ( FN ) x .

A nontrivial ideal I 2 of ℕ × ℕ is called strongly admissible if { i } × ℕ and ℕ × { i } belong to I 2 for each i ∈ ℕ . It is evident that a strongly admissible ideal is admissible also.

Let (X, || . ||) be an FNS. A double sequence x = (x _k,l ) in X is said to be I 2 -convergent or F I 2 convergent to L ∈ X with respect to fuzzy norm on X if for each ɛ > 0, the set K ( ɛ ) = { ( k , l ) ∈ ℕ × ℕ : | | x jk - x | | 0 + ≥ ɛ } ∈ I 2 . In this case, we write x k , l → I 2 L . The element L is called the I 2 -limit of (x _k,l ) in X.

In terms of neighnorhoods, we have x k , l → I 2 L provided that for any ɛ > 0, { ( k , l ) ∈ ℕ × ℕ : x jk ∉ K L ( ɛ , 0 ) } ∈ I 2 . A useful interpretation of the above definition is the following; x k , l → I 2 L ⇔ I 2 - lim | | x k , l - L | | 0 + = 0 . Note that I 2 - lim | | x k , l - L | | 0 + = 0 implies that I 2 - lim | | x k , l - L | | α - = I 2 - lim | | x k , l - L | | α + = 0 , for each α ∈ [0, 1].

Throughout the paper, we consider λ ¯ r , s = λ r μ s be the collection of such sequences λ ¯ will be denoted by Δ ₂.

Let λ = (λ _r ) and μ = (μ _s ) be two non-decreasing sequences of positive real numbers, each tending to ∞ and such that λ _r+1 ≤ λ _r  + 1, λ ₁ = 1; μ _s+1 ≤ μ _s  + 1, μ ₁ = 1. Let I _r  = [r - λ _r  + 1, r], I _s  = [s - μ _s  + 1, s] and I _r,s  = I _r  × I _s .

For any set X ⊆ ℕ × ℕ , the number, δ λ ¯ ( X ) = P - lim r , s → ∞ 1 λ ¯ r , s | { ( k , l ) ∈ I r × I s : ( k , l ) ∈ X } | ; is said to be λ ¯ -density of the set X, provided the limit exists,where λ ¯ r , s = λ r μ s .

A double sequence x = (x _k,l ) in X is said to be double λ ¯ -statistically convergent or FS λ ¯ -convergent to L ∈ X with respect to fuzzy norm on X if for each ɛ > 0 lim r , s → ∞ 1 λ ¯ r , s | { ( k , l ) ∈ I r , s : D ( | | x k , l - L | | , 0 ˜ ) ≥ ɛ } | = 0 . In this case, we write x k , l → FS λ ¯ L or x k , l → L ( FS λ ¯ ) or FS λ ¯ - lim k , l → ∞ x k , l = L . This implies that for each ɛ > 0, the set K ( ɛ ) = { ( k . l ) ∈ I r , s : | | x k , l - L | | 0 + ≥ ɛ } has natural density zero; namely, for each ɛ > 0, | | x k , l - x | | 0 + < ɛ for almost all k, l. In terms of neighborhoods, we have x k , l → FS λ ¯ L if for each ɛ > 0, δ λ ¯ ( { ( k , l ) ∈ I r , s : x k , l ∉ N L ( ɛ , 0 ) } ) = 0 , that is, for each ɛ > 0, x k , l ∈ N L ( ɛ , 0 ) for almost all (k, l).

The set of all λ ¯ -statistically convergent sequence in fuzzy normed space is shown by FS λ ¯ , which is defined by FS λ ¯ ( X ) = { x = ( x k , l ) : for some L , FS λ ¯ - lim k , l → ∞ x kl = L } .

If λ ¯ r , s = rs , for all r, s, then the set of double λ ¯ -statistically convergent sequences reduces to the set of double statistically convergent sequences in fuzzy normed space.

In this section, we introduce the concepts of double I 2 -statistically convergence, double I 2 - λ ¯ -statistically convergence, double I 2 - [ V , λ ¯ ] FN -summability, double I 2 -[C, 1]  _FN -summability for double sequences in fuzzy normed spaces. Also, we investigate some properties and relationships between these concepts.

Definition 3.1. A double sequence x = (x _k,l ) in X is said to be double I 2 -statistically convergent or I 2 -FSt ₂ convergent to L ∈ X with respect to fuzzy norm on X if for each ɛ, δ > 0 { ( r , s ) ∈ ℕ × ℕ : 1 rs | { ( k , l ) , k ≤ r , l ≤ s : ( D | | x k , l - L | | , 0 ˜ ) ≥ ɛ } | ≥ δ } ∈ I 2 . In this case, we write x k , l → I 2 - FSt 2 L or x k , l → L ( I 2 - FSt 2 ) .

Definition 3.2. A double sequence x = (x _k,l ) in X is said to be double I 2 - λ ¯ -statistically convergent or I 2 - FS λ ¯ convergent to L ∈ X with respect to fuzzy norm on X if for each ɛ, δ > 0 { ( r , s ) ∈ ℕ × ℕ : 1 λ ¯ r , s | { ( k , l ) ∈ I r , s : D ( | | x k , l - L | | , 0 ˜ ) ≥ ɛ } | ≥ δ } ∈ I 2 .

In this case, we write x k , l → I 2 - FS λ ¯ L or x k , l → L ( I 2 - FS λ ¯ ) .

In terms of neighborhoods, we have x k , l → I 2 - FS λ ¯ L if for each ɛ > 0, { ( r , s ) ∈ ℕ × ℕ : x k , l ∉ N L ( ɛ , 0 ) } ∈ I 2 , that is, for each ɛ > 0, x k , l ∈ N L ( ɛ , 0 ) for almost all (k, l).

A useful interpretation of the above definition is the following; x k , l → I 2 - FS λ ¯ L ⇔ I 2 - FS λ ¯ - lim | | x k , l - L | | 0 + = 0 . Note that I 2 - FS λ ¯ - lim | | x k , l - L | | 0 + = 0 implies that I 2 - FS λ ¯ - lim | | x k , l - L | | α - = I 2 - FS λ ¯ - lim | | x k , l - L | | α + = 0 , for each α ∈ [0, 1], since 0 ≤ | | x k , l - L | | α - ≤ | | x k , l - L | | α + + | | x k , l - L | | 0 + , holds for every ( k , l ) ∈ ℕ × ℕ and for each α ∈ [0, 1].

The set of all I 2 - λ ¯ -statistically convergent double sequence with respect to fuzzy norm on X will be denoted by I 2 - FS λ ¯ = { x : for some L , I 2 - FS λ , μ - lim x = L } .

If λ _r  = r and μ _s  = s for all r, s then the space I 2 - λ ¯ -statistically convergence is reduced to I 2 -statistically convergence. in fuzzy normed space.

Remark 3.1. Note that if I 2 is the ideal I 2 0 = { A ⊂ ℕ × ℕ : ∃ m ( A ) ∈ ℕ such that i , j ≥ m ( A ) ⇒ ( i , j ) ∉ A } , then, I 2 0 is a strongly admissible ideal and clearly an ideal I 2 is strongly admissible if and only if I 2 0 ⊂ I 2 .

Example 3.1. Let ( ℝ m , | | . | | ) be a FNS and x = ( x 1 , x 2 , . . . , x m ) ∈ ℝ m be a fixed non-zero vector where the fuzzy norm on ℝ m is defined as in (2.1) such that | | x | | C = ∑ i = 1 m x i 2 and λ ¯ r , s = λ r μ s ∈ Δ 2 , I 2 be a strongly admissible ideal in ℕ × ℕ .

Now, we define a sequence x = (x _k,l ) in ℝ m as x k , l = { x , if k = t 2 , l = m 2 , θ , if k ≠ t 2 , l ≠ m 2 , where t , m ∈ ℕ . We see that for any ɛ satisfying 0 < ɛ ≤ b ( x 1 2 + x 2 2 + . . . + x m 2 ) 1 2 , then, we have K ( ɛ ) = { ( r , s ) ∈ ℕ : 1 λ ¯ r , s | ( k , l ) ∈ I r , s : | | x k , l - θ | | 0 + ≥ ɛ | ≥ δ } = { 1 , 4 , 9 , 16 , . . . } ∈ I 2 d ,

where I 2 d , is the ideal of sets which have zero density. Also, if we take I 2 d for I 2 , double I 2 - λ ¯ -statistical convergence coincides with double λ ¯ -statistical convergence for double sequences in fuzzy normed spaces.

If we choose ɛ > b ( x 1 2 + x 2 2 + . . . + x m 2 ) 1 2 , then we get K (ɛ) =∅ and hence δ ₂ (∅) = 0. Thus, ( x k , l ) → θ ( I 2 - F S λ ¯ ) but (x _k,l ) is not convergent since the set {1, 4, 9, 16, . . .} has infinitely many elements. □

We define the generalized double de la Vallėe Poussin mean by t r , s ( x ) = 1 λ ¯ r , s ∑ k ∈ I r ∑ l ∈ I s x k , l .

Definition 3.3. A double sequence x = (x _k,l ) in X is said to be I 2 - [ V , λ ¯ ] FN -summable to L ∈ X with respect to fuzzy norm on X if for each ɛ > 0 { ( r , s ) ∈ ℕ × ℕ : 1 λ ¯ r , s ∑ k , l ∈ I r , s ( D | | x k , l - L | | , 0 ˜ ) ≥ ɛ } ∈ I 2 . If λ _r,s  = r, s, then I 2 - [ V , λ ¯ ] FN -summability reduced to I 2 -[C, 1]  _FN -summability with respect to fuzzy norm on X. We write I 2 - [ C , 1 ] FN : = { ( r , s ) ∈ ℕ × ℕ : 1 rs ∑ k , l ∈ 1 , 1 r , s ( D | | x k , l - L | | , 0 ˜ ) ≥ ɛ } ∈ I 2 .

Example 3.2. Let ( ℝ , | | . | | ) be a FNS as defined in Example 3.1, I 2 be a strongly admissible ideal in ℕ × ℕ and λ = (λ _r ) and μ = (μ _s ) be two non-decreasing sequences of positive real numbers, each tending to ∞ and such that λ _r+1 ≤ λ _r  + 1, λ ₁ = 1; μ _s+1 ≤ μ _s  + 1, μ ₁ = 1 and ( x k , l ) = { 1 , if r - [ λ r ] + 1 ≤ k ≤ r and s - [ μ s ] + 1 ≤ l ≤ s ( r , s = 1 , 2 , . . . ) 0 , otherwise . be a double sequence in ℝ . In this case, we write x k , l → 0 ( I 2 - [ V , λ ¯ ] ) . In this case, we say that (x _k,l ) is strongly I 2 - λ ¯ -summable to 0 with respect to norm on ℝ .

Theorem 3.1. Let X be a fuzzy-normed space and λ ¯ ∈ Δ 2 . If (x _k,l ) is a double sequence in X such that I 2 - FS λ ¯ -lim x = L exists, then it is unique.

Proof. Suppose that there exists elements L ₁, L ₂ (L ₁ ≠ L ₂) in X such that I 2 - FS λ ¯ - lim x = L 1 ; I 2 - FS λ ¯ - lim x = L 2 . Since L ₁ ≠ L ₂, then L ₁ - L ₂ ≠ 0. Therefore | | L 1 - L 2 | | 0 + = 2 ɛ > 0 . Since | | . | | 0 + is a norm in the usual sense. Since I 2 - FS λ ¯ -lim x = L ₁ and I 2 - FS λ ¯ -lim x = L ₂ it follows that { ( r , s ) ∈ ℕ × ℕ : 1 λ ¯ r , s | ( k , l ) ∈ I r , s : | | x k , l - L 1 | | 0 + ≥ ɛ | ≥ δ } ∈ I 2 , { ( r , s ) ∈ ℕ × ℕ : 1 λ ¯ r , s | ( k , l ) ∈ I r , s : | | x k , l - L 2 | | 0 + ≥ ɛ | ≥ δ } ∈ I 2 . There are (k, l) ∈ I _r,s such that | | x k , l - L 1 | | 0 + < ɛ and | | x k , l - L 2 | | 0 + < ɛ . Further, for these k, l we have | | L 1 - L 2 | | 0 + ≤ | | x k , l - L 1 | | 0 + + | | x k , l - L 2 | | 0 + < 2 ɛ which is a contradiction. This completes the proof.

Theorem 3.2. Let (X, || . ||) be an FNS and λ ¯ ∈ Δ 2 . (x _kl ) and (y _k,l ) be double sequences in X such that x k , l → I 2 - FS λ ¯ L 1 and y k , l → I 2 - FS λ ¯ L 2 , where L ₁, L ₂ ∈ X. Then, we have the following;

(i) ( x k , l + y k , l ) → I 2 - FS λ ¯ L 1 + L 2 ,

(ii) β x k , l → I 2 - FS λ ¯ β L 1 , ( β ∈ ℝ ) ,

(iii) ( x k , l y k , l ) → I 2 - FS λ ¯ L 1 L 2 ,

(iv) | | x k , l | | → I 2 - FS λ ¯ | | L 1 | | .

Proof. (i) Assume that x k , l → I 2 - FS λ ¯ L 1 and y k , l → I 2 - FS λ ¯ L 2 . Since | | . | | 0 + is a norm in the usual sense, we get | | ( x k , l + y k , l ) - ( L 1 + L 2 ) | | 0 + ≤ | | x k , l - L 1 | | 0 + + | | y k , l - L 2 | | 0 +(3.1) for all (k, l) ∈ I _r,s . Now, let us write K ( ɛ , δ ) = { ( r , s ) ∈ ℕ × ℕ : 1 λ ¯ r , s | ( k , l ) ∈ I r , s : | | ( x k , l + y k , l ) - ( L 1 + L 2 ) | | 0 + ≥ ɛ | ≥ δ , K 1 ( ɛ 2 , δ ) = { ( r , s ) ∈ ℕ × ℕ : 1 λ ¯ r , s | ( k , l ) ∈ I r , s : | | x k , l - L 1 | | 0 + ≥ ɛ 2 | ≥ δ } and K 2 ( ɛ 2 , δ ) = { ( r , s ) ∈ ℕ × ℕ : 1 λ ¯ r , s | ( k , l ) ∈ I r , s : | | y k , l - L 2 | | 0 + ≥ ɛ 2 | ≥ δ } . From (3.1) that K ( ɛ , δ ) ⊆ K 1 ( ɛ 2 , δ ) ∪ K 2 ( ɛ 2 , δ ) . Bow, by assumption, we get K 1 ( ɛ 2 , δ ) , K 2 ( ɛ 2 , δ ) belongs to I 2 , so we have K 1 ( ɛ 2 , δ ) ∪ K 2 ( ɛ 2 , δ ) ∈ I 2 . This implies K ( ɛ , δ ) ∈ I 2 . This completes the proof.   (ii) Let β ∈ ℝ . If β = 0, we have nothing to prove, so we assume β ≠ 0. Let ɛ > 0 be given. Since | | . | | 0 + is a norm in the usual sense, we get | | β x k , l | | 0 + = | β | | | x k , l | | 0 + . Since x k , l → I 2 - FS λ ¯ L 1 therefore the set A ( ɛ ) = { ( r , s ) ∈ ℕ × ℕ : 1 λ ¯ r , s | ( k , l ) ∈ I r , s : | | x k , l - L 1 | | 0 + ≥ ɛ | ≥ δ } ∈ I 2 . Let A 1 ( ɛ ) = { ( r , s ) ∈ ℕ × ℕ : 1 λ ¯ r , s | ( k , l ) ∈ I r , s : | | β x k , l - β L 1 | | 0 + ≥ ɛ | ≥ δ } . We need to show that A ₁ (ɛ) contained in A (ɛ ₁) for some ɛ ₁ > 0. Let p, q ∈ A ₁ (ɛ), then ɛ ≤ | | β x p , q - β L 1 | | 0 + = | β | | | x p , q - L 1 | | 0 + . This implies that | | x p , q - L 1 | | 0 + = ɛ β = ɛ 1 (say). Therefore p, q ∈ A (ɛ ₁). Then we have A ₁ (ɛ) ⊂ A (ɛ ₁). By definition of I 2 , we get A 1 ( ɛ ) ∈ I 2 . This completes the proof.   (iii) Since x k , l → I 2 - FS λ ¯ L 1 , then we have the set A ( 1 ) = { ( r , s ) ∈ ℕ × ℕ : 1 λ r , s | ( k , l ) ∈ I r , s : | | x k , l - L 1 | | 0 + ≥ ɛ | < 1 } ∈ F ( I 2 ) . Now | | . | | 0 + being a norm in the usual sense, we get | | x k , l y k , l - L 1 L 2 | | 0 + ≤ | | x k , l | | 0 + | | y k , l - L 2 | | 0 + + | | x k , l - L 1 | | 0 + | | L 2 | | 0 + . For k ∈ A (1), we have | | x k , l | | 0 + < | | L 1 | | 0 + + 1 and it follows that | | x k , l y k , l - L 1 L 2 | | 0 + ≤ ( | | x k , l | | 0 + + 1 ) | | y k , l - L 2 | | 0 + + | | L 2 | | 0 + ( | | x k , l - L 1 | | 0 + ) .(3.2) Let ɛ > 0 be given. Choose η > 0 such that 0 < 2 η < ɛ | | x k , l | | 0 + + | | y k , l | | 0 + + 1 .(3.3) Since (X, || . ||) such that x k , l → I 2 - FS λ ¯ L 1 and y k , l → I 2 - FS λ ¯ L 2 therefore we set A 1 ( η ) = { ( r , s ) ∈ ℕ × ℕ : 1 λ ¯ r , s | { ( k , l ) ∈ I r , s : | | x k , l - L 1 | | 0 + ≥ ɛ } | < η } , A 2 ( η ) = { ( r , s ) ∈ ℕ × ℕ : 1 λ ¯ r , s | { ( k , l ) ∈ I r , s : | | y k , l - L 2 | | 0 + ≥ ɛ } | < η } . Thus A 1 ( η ) , A 2 ( η ) ∈ F ( I 2 ) . Obviously, A ( 1 ) ∩ A 1 ( η ) ∩ A 2 ( η ) ∈ F ( I 2 ) and for each k ∈ A (1) ∩ A ₁ (η) ∩ A ₂ (η), we have from (3.2) and (3.3) | | x k , l y k , l - L 1 L 2 | | 0 + < ɛ . This implies that { ( r , s ) ∈ ℕ × ℕ : 1 λ ¯ r , s | ( k , l ) ∈ I r , s : | | x k , l y k , l - L 1 L 2 | | 0 + ≥ ɛ | ≥ δ } ∈ I 2 , i.e. ( x k , l y k , l ) → I 2 - FS λ ¯ L 1 L 2 .   (iv) Since | | . | | α - and | | . | | α + are norms in the usual sense, we have the inequalities 0 ≤ | | | x k , l | | α - - | | L 1 | | α - | ≤ | | x k , l - L 1 | | α - and 0 ≤ | | | x k , l | | α + - | | L 1 | | α + | ≤ | | x k , l - L 1 | | α + for all α ∈ [0, 1] and (k, l) ∈ I _r,s . Hence, 0 ≤ max { | | | x k , l | | α - - | | L 1 | | α - | , | | | x k , l | | α + - | | L 1 | | α + | } ≤ | | x k , l - L 1 | | α + , for all α ∈ [ 0 , 1 ] and ( k , l ) ∈ I r , s . Taking supremum over α ∈ [0, 1], we get 0 ≤ D ( | | x k , l | | , | | L 1 | | ) ≤ | | x k , l - L 1 | | 0 + . Hence, we have | | x k , l | | → I 2 - FS λ ¯ | | L 1 | | .