Regularly (𝓘₂,𝓘)-Convergence and Regularly (𝓘₂,𝓘)-Cauchy Double Sequences of Fuzzy Numbers

Abstract

In this paper, we introduce the notions of regularly (ℐ₂,ℐ), (ℐ^∗₂,ℐ^∗)-convergence and regularly (ℐ₂,ℐ), (ℐ^∗₂,ℐ^∗)-Cauchy double sequence of fuzzy numbers. Also, we study some properties of these concepts.

1. Introduction, Notations, and Definitions

The concept of ordinary convergence of a sequence of fuzzy numbers was firstly introduced by Matloka [1] and proved some basic theorems for sequences of fuzzy numbers. Nanda [2] studied the sequences of fuzzy numbers and showed that the set of all convergent sequences of fuzzy numbers forms a complete metric space. Recently, Nuray and Savaş [3] defined the concepts of statistical convergence and statistical Cauchy for sequence of fuzzy numbers. They proved that a sequence of fuzzy numbers, is statistically convergent if and only if it is statistically Cauchy. Nuray [4] introduced Lacunary statistical convergence of sequences of fuzzy numbers whereas Savaş [5] studied some equivalent alternative conditions for a sequence of fuzzy numbers to be statistically Cauchy. A lot of developments have been made in this area after the works of [6–8].

Throughout the paper ℕ and ℝ denote the set of all positive integers and the set of all real numbers, respectively. The idea of ℐ-convergence was introduced by Kostyrko et al. [9] as a generalization of statistical convergence which is based on the structure of the ideal ℐ of subset of the set ℕ. Nuray and Ruckle [10] independently introduced the same with another name generalized statistical convergence. Das et al. [11] introduced the concept of ℐ-convergence of double sequences in a metric space and studied some properties of this convergent sequences of this type. Balcerzak et al. [12] studied on statistical convergence and ideal convergence for sequences of functions. Komisarski [13] discussed the pointwise ℐ-convergence and ℐ-convergence in measure of sequences of functions. Mursaleen and Alotaibi investigated the notion of ideal convergence in [14] for random 2-normed space and construct some interesting examples. Mursaleen and Mohiuddine defined and studied the concept of ℐ-convergence, ℐ^∗-convergence, ℐ-limit points and ℐ-cluster points in probabilistic normed space, in [15]. Şahiner et al. introduced and investigated ℐ-convergence in 2-normed spaces, and also defined and examined some new sequence spaces using norm, in [16]. A lot of developments have been made in this area after the works of [17–21].

V. Kumar and K. Kumar studied the concepts of ℐ-convergence, ℐ^∗-convergence, and ℐ-Cauchy sequence for sequences of fuzzy numbers, in [22]. Mursaleen et al. studied the concept of ideal convergence and ideal Cauchy for double sequences in intuitionistic fuzzy normed spaces, in [23]. Recently, Dündar and Talo have introduced the concepts of ℐ₂-convergence, ℐ^∗₂-convergence for double sequences of fuzzy numbers and studied their some properties and relations, in [24]. Quite recently, Dündar and Talo have introduced the concepts of ℐ₂-Cauchy, ℐ^∗₂-Cauchy double sequences of fuzzy numbers, in [25].

In this paper, we introduce the notions of regularly (ℐ₂,ℐ), (ℐ^∗₂,ℐ^∗)-convergence and regularly (ℐ₂,ℐ), (ℐ^∗₂,ℐ^∗)-Cauchy double sequence of fuzzy numbers. Also, we study some properties of those sequences.

In a possible application we can say that if we choose the statistical convergence of a special case of the ideal convergence, the results in this paper can be obtained for statistical convergence.

Now, we recall the concept of fuzzy numbers, convergence, ideal convergence of the sequences and double sequences (see [9, 11, 21, 24–30]).

A double sequence 𝑥=(𝑥_𝑚𝑛)_{𝑚,𝑛∈ℕ} of real numbers is said to be convergent to 𝐿∈ℝ in Pringsheim’s sense, if for any 𝜀>0 there exists 𝑁_𝜀∈ℕ such that |𝑥_𝑚𝑛−𝐿|<𝜀 whenever 𝑚,𝑛>𝑁_𝜀. In this case we write 𝑃−lim_{𝑚,𝑛→∞}𝑥_𝑚𝑛=𝐿orlim_{𝑚,𝑛→∞}𝑥_𝑚𝑛=𝐿. (1)

Let 𝑋≠∅. A class ℐ of subsets of 𝑋 is said to be an ideal in 𝑋 provided that (i)∅∈ℐ,(ii)𝐴,𝐵∈ℐ implies 𝐴∪𝐵∈ℐ, (iii)𝐴∈ℐ, 𝐵⊂𝐴 implies 𝐵∈ℐ.

ℐ is called a nontrivial ideal if 𝑋∉ℐ.

Let 𝑋≠∅. A non-empty class ℱ of subsets of 𝑋 is said to be a filter in 𝑋 provided that (i)∅∉ℱ, (ii)𝐴,𝐵∈ℱ implies 𝐴∩𝐵∈ℱ, (iii)𝐴∈ℱ, 𝐴⊂𝐵 implies 𝐵∈ℱ.

If ℐ is a nontrivial ideal in 𝑋, 𝑋≠∅, then the class ℱ(ℐ)={𝑀⊂𝑋∶(∃𝐴∈ℐ)(𝑀=𝑋⧵𝐴)} (2) is called a filter on 𝑋 associated with ℐ.

A nontrivial ideal ℐ in 𝑋 is called admissible if {𝑥}∈ℐ for each 𝑥∈𝑋.

Throughout the paper we take ℐ as a nontrivial admissible ideal in ℕ.

Let ℐ⊂2^ℕ be a nontrivial ideal, and let (𝑋,𝜌) be a metric space. A sequence (𝑥_𝑛) of elements of 𝑋 is said to be ℐ-convergent to 𝐿∈𝑋, if for each 𝜀>0 we have 𝐴(𝜀)={𝑛∈ℕ∶𝜌(𝑥_𝑛,𝐿)≥𝜀}∈ℐ.

A sequence (𝑥_𝑛) of elements of 𝑋 is said to be ℐ^∗-convergent to 𝐿∈𝑋 if and only if there exists a set 𝑀∈ℱ(ℐ) (i.e., ℕ⧵𝑀∈ℐ), 𝑀={𝑚₁<𝑚₂<⋯<𝑚_𝑘<⋯} such that 𝜌(𝑥_𝑚𝑘,𝐿)→0,as𝑘→∞.

Throughout the paper we take ℐ₂ as a nontrivial admissible ideal in ℕ×ℕ.

A nontrivial ideal ℐ₂⊂2^ℕ×ℕ is called strongly admissible if {𝑖}×ℕ and ℕ×{𝑖} belong to ℐ₂ for each 𝑖∈ℕ. It is evident that a strongly admissible ideal is also admissible.

Let ℐ⁰₂={𝐴⊂ℕ×ℕ∶(∃𝑚(𝐴)∈ℕ)(𝑖,𝑗≥𝑚(𝐴)⇒(𝑖,𝑗)∉𝐴)}. Then ℐ⁰₂ is a nontrivial strongly admissible ideal and clearly an ideal ℐ₂ is strongly admissible if and only if ℐ⁰₂⊂ℐ₂.

Let (𝑋,𝜌) be a linear metric space, and let ℐ₂⊂2^ℕ×ℕ be a strongly admissible ideal. A double sequence 𝑥=(𝑥_𝑚𝑛) in 𝑋 is said to be ℐ₂-convergent to 𝐿∈𝑋, if for any 𝜀>0 we have 𝐴(𝜀)={(𝑚,𝑛)∈ℕ×ℕ∶𝜌(𝑥_𝑚𝑛,𝐿)≥𝜀}∈ℐ₂ and is written ℐ₂−lim_{𝑚,𝑛→∞}𝑥_𝑚𝑛=𝐿.

If ℐ₂⊂2^ℕ×ℕ is a strongly admissible ideal, then usual convergence implies ℐ₂-convergence.

Let (𝑋,𝜌) be a linear metric space, and let ℐ₂⊂2^ℕ×ℕ be a strongly admissible ideal. A double sequence 𝑥=(𝑥_𝑚𝑛) of elements of 𝑋 is said to be ℐ^∗₂-convergent to 𝐿∈𝑋, if and only if there exists a set 𝑀∈ℱ(ℐ₂) (i.e., ℕ×ℕ⧵𝑀∈ℐ₂) such that lim_{𝑚,𝑛→∞}𝑥_𝑚𝑛=𝐿, for (𝑚,𝑛)∈𝑀 and is written ℐ^∗₂−lim_{𝑚,𝑛→∞}𝑥_𝑚𝑛=𝐿.

Let (𝑋,𝜌) be a linear metric space, and let ℐ₂⊂2^ℕ×ℕ be a strongly admissible ideal. A double sequence 𝑥=(𝑥_𝑚𝑛) of elements of 𝑋 is said to be ℐ₂-Cauchy, if for every 𝜀>0 there exist 𝑠=𝑠(𝜀),𝑡=𝑡(𝜀)∈ℕ such that 𝐴(𝜀)={(𝑚,𝑛)∈ℕ×ℕ∶𝜌(𝑥_𝑚𝑛,𝑥_𝑠𝑡)≥𝜀}∈ℐ₂.

Let ℐ₂ be an ideal of ℕ×ℕ and ℐ be an ideal of ℕ, then a double sequence 𝑥=(𝑥_𝑚𝑛) in ℂ, which is the set of complex numbers, is said to be regularly (ℐ₂,ℐ)-convergent (r (ℐ₂,ℐ)-convergent), if it is ℐ₂-convergent in Pringsheim’s sense and for every 𝜀>0, the following statements hold: 𝑚∈ℕ∶𝑥_𝑚𝑛−𝐿_𝑛≥𝜀∈ℐ (3) for some 𝐿_𝑛∈ℂ, for each 𝑛∈ℕ and 𝑛∈ℕ∶𝑥_𝑚𝑛−𝐾_𝑚≥𝜀∈ℐ (4) for some 𝐾_𝑚∈ℂ, for each 𝑚∈ℕ.

We say that an admissible ideal ℐ⊂2^ℕ satisfies the property (AP), if for every countable family of mutually disjoint sets {𝐴₁,𝐴₂,…} belonging to ℐ, there exists a countable family of sets {𝐵₁,𝐵₂,…} such that 𝐴_𝑗△𝐵_𝑗 is a finite set for 𝑗∈ℕ and 𝐵=⋃^∞_𝑗=1𝐵_𝑗∈ℐ. (hence 𝐵_𝑗∈ℐ for each 𝑗∈ℕ).

We say that an admissible ideal ℐ₂⊂2^ℕ×ℕ satisfies the property (AP2), if for every countable family of mutually disjoint sets {𝐴₁,𝐴₂,…} belonging to ℐ₂, there exists a countable family of sets {𝐵₁,𝐵₂,…} such that 𝐴_𝑗△𝐵_𝑗∈ℐ⁰₂, that is, 𝐴_𝑗△𝐵_𝑗 is included in the finite union of rows and columns in ℕ×ℕ for each 𝑗∈ℕ and 𝐵=⋃^∞_𝑗=1𝐵_𝑗∈ℐ₂ (hence 𝐵_𝑗∈ℐ₂ for each 𝑗∈ℕ).

A fuzzy number is a fuzzy set on the real axis, that is, a mapping 𝑢∶ℝ→[0,1] which satisfies the following four conditions. (i)𝑢 is normal, that is, there exists an 𝑥₀∈ℝ such that 𝑢(𝑥₀)=1. (ii)𝑢 is fuzzy convex, that is, 𝑢[𝜆𝑥+(1−𝜆)𝑦]≥min{𝑢(𝑥),𝑢(𝑦)} for all 𝑥,𝑦∈ℝ and for all 𝜆∈[0,1]. (iii)𝑢 is upper semicontinuous. (iv)The set [𝑢]₀∶={𝑥∈ℝ∶𝑢(𝑥)>0} is compact, (cf. Zadeh [31]), where {𝑥∈ℝ∶𝑢(𝑥)>0} denotes the closure of the set {𝑥∈ℝ∶𝑢(𝑥)>0} in the usual topology of ℝ.

We denote the set of all fuzzy numbers on ℝ by 𝐸¹ and called it as the space of fuzzy numbers. 𝛼-level set [𝑢]_𝛼 of 𝑢∈𝐸¹ is defined by [𝑢]_𝛼{(𝑡)},(),∶=𝑡∈ℝ∶𝑥≥𝛼0<𝛼≤1{𝑡∈ℝ∶𝑥(𝑡)>𝛼},(𝛼=0). (5)

The set [𝑢]_𝛼 is closed, bounded, and nonempty interval for each 𝛼∈[0,1] which is defined by [𝑢]_𝛼∶=[𝑢⁻(𝛼),𝑢⁺(𝛼)]. ℝ can be embedded in 𝐸¹, since each 𝑟∈ℝ can be regarded as a fuzzy number 𝑟 defined by 𝑟(𝑥)(),().∶=1,𝑥=𝑟0,𝑥≠𝑟 (6)

Theorem 1 (see [27]). Let [𝑢]_𝛼=[𝑢⁻(𝛼),𝑢⁺(𝛼)] for 𝑢∈𝐸¹ and for each 𝛼∈[0,1]. Then the following statements hold.(i)𝑢⁻ is a bounded and nondecreasing left continuous function on (0,1]. (ii)𝑢⁺ is a bounded and nonincreasing left continuous function on (0,1]. (iii)The functions 𝑢⁻ and 𝑢⁺ are right continuous at the point 𝛼=0. (iv)𝑢⁻(1)≤𝑢⁺(1).
Conversely, if the pair of functions 𝛼 and 𝛽 satisfies the conditions (i)–(iv), then there exists a unique 𝑢∈𝐸¹ such that [𝑢]_𝛼∶=[𝑢⁻(𝛼),𝑢⁺(𝛼)] for each 𝛼∈[0,1]. The fuzzy number 𝑢 corresponding to the pair of functions 𝑢⁻ and 𝑢⁺ is defined by 𝑢∶ℝ→[0,1], 𝑢(𝑥)∶=sup{𝛼∶𝑢⁻(𝛼)≤𝑥≤𝑢⁺(𝛼)}.

Let 𝑢,𝑣,𝑤∈𝐸¹ and 𝑘∈ℝ. Then the operations addition, scalar multiplication, and product are defined on 𝐸¹ by 𝑢+𝑣=𝑤⟺[𝑤]_𝛼=[𝑢]_𝛼+[𝑣]_𝛼,∀𝛼∈[0,1]⟺𝑤⁻(𝛼)=𝑢⁻(𝛼)+𝑣⁻(𝛼),𝑤⁺(𝛼)=𝑢⁺(𝛼)+𝑣⁺(𝛼),[𝑘𝑢]_𝛼=𝑘[𝑢]_𝛼[],[𝑤],∀𝛼∈0,1𝑢𝑣=𝑤⟺_𝛼=[𝑢]_𝛼[𝑣]_𝛼,∀𝛼∈[0,1], (7) where it is immediate that 𝑤⁻(𝛼)=min𝑢⁻(𝛼)𝑣⁻(𝛼),𝑢⁻(𝛼)𝑣⁺(𝛼),𝑢⁺(𝛼)𝑣⁻(𝛼),𝑢⁺(𝛼)𝑣⁺(𝛼,𝑤₊)(𝛼)=max𝑢⁻(𝛼)𝑣⁻(𝛼),𝑢⁻(𝛼)𝑣⁺(𝛼),𝑢⁺(𝛼)𝑣⁻(𝛼),𝑢⁺(𝛼)𝑣⁺(𝛼), (8) for all 𝛼∈[0,1].

Let 𝑊 be the set of all closed bounded intervals 𝐴 of real numbers with endpoints 𝐴 and 𝐴, that is, 𝐴∶=[𝐴,𝐴]. Define the relation 𝑑 on 𝑊 by 𝑑(𝐴,𝐵)∶=max𝐴−𝐵,𝐴−𝐵. (9)

Then it can easily be observed that 𝑑 is a metric on 𝑊 and (𝑊,𝑑) is a complete metric space, (cf. Nanda [2]). Now, we may define the metric 𝐷 on 𝐸¹ by means of the Hausdorff metric 𝑑 as 𝐷(𝑢,𝑣)∶=sup_[]𝛼∈0,1𝑑[𝑢]_𝛼,[𝑣]_𝛼∶=sup_[]𝛼∈0,1max|𝑢⁻(𝛼)−𝑣⁻(𝛼)|,𝑢⁺(𝛼)−𝑣⁺(𝛼). (10)

One can see that 𝐷𝑢,0=sup_𝛼∈[0,1]max|𝑢⁻(𝛼)|,𝑢⁺(𝛼𝑢)=max|⁻(0)|,𝑢⁺(0). (11)

The partial ordering relation ⪯ on 𝐸¹ is defined as follows: 𝑢⪯𝑣⟺𝑢⁻(𝛼)≤𝑣⁻(𝛼),𝑢⁺(𝛼)≤𝑣⁺(𝛼)∀𝛼∈[0,1]. (12)

Now, we may give the following.

Proposition 2 (see [6]). Let 𝑢,𝑣,𝑤,𝑧∈𝐸¹ and 𝑘∈ℝ. Then, the following statements hold.(i)(𝐸¹,𝐷) is a complete metric space. (ii)𝐷(𝑘𝑢,𝑘𝑣)=|𝑘|𝐷(𝑢,𝑣). (iii)𝐷(𝑢+𝑣,𝑤+𝑣)=𝐷(𝑢,𝑤). (iv)𝐷(𝑢+𝑣,𝑤+𝑧)≤𝐷(𝑢,𝑤)+𝐷(𝑣,𝑧). (v)|𝐷(𝑢,0)−𝐷(𝑣,0)|≤𝐷(𝑢,𝑣)≤𝐷(𝑢,0)+𝐷(𝑣,0).

Following Matloka [1], we give some definitions concerning the sequences of fuzzy numbers which are needed in the text.

A sequence 𝑢=(𝑢_𝑘) of fuzzy numbers is a function 𝑢 from the set ℕ into the set 𝐸¹. The fuzzy number 𝑢_𝑘 denotes the value of the function at 𝑘∈ℕ and is called as the general term of the sequence. By 𝑤(𝐹), we denote the set of all sequences of fuzzy numbers

A sequence (𝑢_𝑛)∈𝑤(𝐹) is called convergent with limit 𝑢∈𝐸¹, if for every 𝜀>0 there exists an 𝑛₀=𝑛₀(𝜀)∈ℕ such that 𝐷(𝑢_𝑛,𝑢)<𝜀 for all 𝑛≥𝑛₀.

A double sequence 𝑢=(𝑢_𝑛𝑘) of fuzzy real numbers is defined by a function 𝑢 from the set ℕ×ℕ into the set 𝐸¹. The fuzzy number 𝑢_𝑛𝑘 denotes the value of the function at (𝑛,𝑘)∈ℕ×ℕ.

A double sequence 𝑢=(𝑢_𝑚𝑛) of fuzzy numbers is said to be convergent in the Pringsheim’s sense or P-convergent, if for every 𝜀>0 there exists 𝑘∈ℕ such that 𝐷(𝑢_𝑚𝑛,𝑢₀)<𝜀 for all 𝑚,𝑛≥𝑘 and is denoted by 𝑃−lim_{𝑚,𝑛→∞}𝑢_𝑚𝑛=𝑢₀. The fuzzy number 𝑢₀ is called the Pringsheim limit of 𝑢.

Let ℐ₂⊂2^ℕ×ℕ be a strongly admissible ideal. (i)A double sequence 𝑢=(𝑢_𝑚𝑛) of fuzzy numbers is said to be ℐ₂-convergent to a fuzzy number 𝑢₀ if for any 𝜀>0 we have 𝐴(𝜀)={(𝑚,𝑛)∈ℕ×ℕ∶𝐷(𝑢_𝑚𝑛,𝑢₀)≥𝜀}∈ℐ₂ and is written ℐ₂−lim_{𝑚,𝑛→∞}𝑢_𝑚𝑛=𝑢₀. (ii)A double sequence 𝑢=(𝑢_𝑚𝑛) of fuzzy numbers is said to be ℐ^∗₂-convergent to 𝑢₀∈𝐸¹ if there exists 𝑀∈ℱ(ℐ₂) (i.e., 𝐻=ℕ×ℕ⧵𝑀∈ℐ₂) such that lim_{𝑚,𝑛→∞(𝑚,𝑛)∈𝑀}𝑢_𝑚𝑛=𝑢₀ (13) and is written ℐ^∗₂−lim_{𝑚,𝑛→∞}𝑢_𝑚𝑛=𝑢₀.(iii)A double sequence 𝑢=(𝑢_𝑚𝑛) of fuzzy numbers is said to be ℐ₂-Cauchy, if for each 𝜀>0, there exist 𝑠=𝑠(𝜀),𝑡=𝑡(𝜀)∈ℕ such that 𝐴(𝜀)={(𝑚,𝑛)∈ℕ×ℕ∶𝐷(𝑢_𝑚𝑛,𝑢_𝑠𝑡)≥𝜀}∈ℐ₂.(iv)A double sequence 𝑢=(𝑢_𝑚𝑛) of fuzzy numbers is said to be ℐ^∗₂-Cauchy if there exists a set 𝑀∈ℱ(ℐ₂) (i.e., 𝐻=ℕ×ℕ⧵𝑀∈ℐ₂) such that for every 𝜀>0 and for (𝑚,𝑛),(𝑠,𝑡)∈𝑀, 𝑚,𝑛,𝑠,𝑡>𝑘₀=𝑘₀(𝜀), 𝐷(𝑢_𝑚𝑛,𝑢_𝑠𝑡)<𝜀, that is, lim_{(),()𝑚,𝑛,𝑠,𝑡→∞𝑚,𝑛𝑠,𝑡∈𝑀}𝐷𝑢_𝑚𝑛,𝑢_𝑠𝑡=0. (14)

Now, we begin with quoting the following five lemmas due to Dündar and Talo [24, 25] which are needed throughout the paper.

Lemma 3 (see [24, Theorem 3.3]). Let ℐ₂⊂2^ℕ×ℕ be a strongly admissible ideal, let 𝑢=(𝑢_𝑚𝑛) be a double sequence of fuzzy numbers, and let 𝑢₀ be a fuzzy number. Then, 𝑃−lim_{𝑚,𝑛→∞}𝑢_𝑚𝑛=𝑢₀ implies ℐ₂−lim_{𝑚,𝑛→∞}𝑢_𝑚𝑛=𝑢₀.

Lemma 4 (see [24, Theorem 4.2]). Let ℐ₂⊂2^ℕ×ℕ be a strongly admissible ideal, let 𝑢=(𝑢_𝑚𝑛) be a double sequence of fuzzy numbers, and let 𝑢₀∈𝐸¹. Then, ℐ^∗₂−lim_{𝑚,𝑛→∞}𝑢_𝑚𝑛=𝑢₀ implies ℐ₂−lim_{𝑚,𝑛→∞}𝑢_𝑚𝑛=𝑢₀.

Lemma 5 (see [24, Theorem 4.4]). Let ℐ₂⊂2^ℕ×ℕ be a strongly admissible ideal with property (AP2), let 𝑢=(𝑢_𝑚𝑛) be a double sequence of fuzzy numbers, and let 𝑢₀ be a fuzzy real number. Then, ℐ₂−lim_{𝑚,𝑛→∞}𝑢_𝑚𝑛=𝑢₀ implies ℐ^∗₂−lim_{𝑚,𝑛→∞}𝑢_𝑚𝑛=𝑢₀.

Lemma 6 (see [25, Theorem 3.2]). Let ℐ₂⊂2^ℕ×ℕ be a strongly admissible ideal. A double sequence 𝑢=(𝑢_𝑚𝑛) of fuzzy numbers is ℐ₂-convergent if and only if it is ℐ₂-Cauchy sequence.

Lemma 7 (see [25, Theorem 3.4]). Let ℐ₂⊂2^ℕ×ℕ be a strongly admissible ideal. If a double sequence 𝑢=(𝑢_𝑚𝑛) of fuzzy numbers is an ℐ^∗₂-Cauchy sequence, then it is ℐ₂-Cauchy.

2. Main Results

In this section, we study certain properties of regular convergence, regularly (ℐ₂,ℐ)-convergence and regularly (ℐ₂,ℐ)-Cauchy double sequences of fuzzy numbers.

Definition 8. A double sequence (𝑢_𝑚𝑛) of fuzzy numbers is said to be regularly convergent, if it is convergent in Pringsheim’s sense and the limits lim_𝑚→∞𝑢_𝑚𝑛,(𝑛∈ℕ),lim_𝑛→∞𝑢_𝑚𝑛,(𝑚∈ℕ), (15) exist for each fixed 𝑛∈ℕ and 𝑚∈ℕ, respectively. Note that if (𝑢_𝑚𝑛) is regularly convergent to a fuzzy number 𝑢₀, then the limits lim_𝑛→∞lim_𝑚→∞𝑢_𝑚𝑛,lim_𝑚→∞lim_𝑛→∞𝑢_𝑚𝑛 (16) exist and are equal to 𝑢₀. In this case we write 𝑟−lim_{𝑚,𝑛→∞}𝑢_𝑚𝑛=𝑢₀or𝑢^𝑟_𝑚𝑛⟶𝑢₀as𝑚,𝑛⟶∞. (17)

Definition 9. Let ℐ₂⊂2^ℕ×ℕ be a strongly admissible ideal, and let ℐ⊂2^ℕ be an admissible ideal. A double sequence (𝑢_𝑚𝑛) of fuzzy numbers is said to be regularly (ℐ₂,ℐ)-convergent (𝑟(ℐ₂,ℐ)-convergent), if it is ℐ₂-convergent in Pringsheim’s sense and for every 𝜀>0, the following statements hold: 𝑚∈ℕ∶𝐷𝑢_𝑚𝑛,𝑣_𝑛≥𝜀∈ℐ (18) for some fuzzy numbers 𝑣_𝑛, for each 𝑛∈ℕ and 𝑛∈ℕ∶𝐷𝑢_𝑚𝑛,𝑤_𝑚≥𝜀∈ℐ (19) for some fuzzy numbers 𝑤_𝑚, for each 𝑚∈ℕ.

In the case (𝑢_𝑚𝑛) is regularly (ℐ₂,ℐ)-convergent (𝑟(ℐ₂,ℐ)-convergent) to a fuzzy number 𝑢₀, then the limits ℐ−lim_𝑛→∞lim_𝑚→∞𝑢_𝑚𝑛 and ℐ−lim_𝑚→∞lim_𝑛→∞𝑢_𝑚𝑛 exist and are equal to 𝑢₀.

Theorem 10. Let ℐ₂ be a strongly admissible ideal of ℕ×ℕ, and let ℐ be an admissible ideal of ℕ. If a double sequence (𝑢_𝑚𝑛) of fuzzy numbers is regularly convergent, then it is 𝑟(ℐ₂,ℐ)-convergent.

Proof. Let (𝑢_𝑚𝑛) be regularly convergent. Then (𝑢_𝑚𝑛) is convergent in Pringsheim’s sense and the limits lim_𝑚→∞𝑢_𝑚𝑛(𝑛∈ℕ) and lim_𝑛→∞𝑢_𝑚𝑛(𝑚∈ℕ) exist. By Lemma 3, (𝑢_𝑚𝑛) is ℐ₂-convergent. Also, for 𝜀>0, there exist 𝑚=𝑚₀(𝜀) and 𝑛=𝑛₀(𝜀) such that 𝐷𝑢_𝑚𝑛,𝑣_𝑛<𝜀 (20) for some fuzzy numbers 𝑣_𝑛 and each fixed 𝑛∈ℕ for every 𝑚≥𝑚₀ and 𝐷𝑢_𝑚𝑛,𝑤_𝑚<𝜀 (21) for some fuzzy numbers 𝑤_𝑚 and each fixed 𝑚∈ℕ for every 𝑛≥𝑛₀. Then, since ℐ is admissible ideal so for 𝜀>0, we have 𝑚∈ℕ∶𝐷𝑢_𝑚𝑛,𝑣_𝑛≥𝜀=1,2,…,𝑚₀𝑢−1∈ℐ,𝑛∈ℕ∶𝐷_𝑚𝑛,𝑤_𝑚≥𝜀=1,2,…,𝑛₀−1∈ℐ. (22)
Hence, (𝑢_𝑚𝑛) is 𝑟(ℐ₂,ℐ)-convergent.

Definition 11. Let ℐ₂ be a strongly admissible ideal of ℕ×ℕ, and let ℐ be an admissible ideal of ℕ. A double sequence (𝑢_𝑚𝑛) of fuzzy numbers is said to be 𝑟(ℐ^∗₂,ℐ^∗)-convergent, if there exist the sets 𝑀∈ℱ(ℐ₂) (i.e., ℕ×ℕ⧵𝑀∈ℐ₂), 𝑀₁∈ℱ(ℐ), and 𝑀₂∈ℱ(ℐ) (i.e., ℕ⧵𝑀₁∈ℐ and ℕ⧵𝑀₂∈ℐ) such that the limits lim_{𝑚,𝑛→∞(𝑚,𝑛)∈𝑀}𝑢_𝑚𝑛,lim_{𝑚→∞𝑚∈𝑀1}𝑢_𝑚𝑛,lim_{𝑛→∞𝑛∈𝑀2}𝑢_𝑚𝑛 (23) exist for each fixed 𝑛∈ℕ and 𝑚∈ℕ, respectively.

Theorem 12. Let ℐ₂ be a strongly admissible ideal of ℕ×ℕ, and let ℐ be an admissible ideal of ℕ. If a double sequence (𝑢_𝑚𝑛) of fuzzy numbers is 𝑟(ℐ^∗₂,ℐ^∗)-convergent, then it is 𝑟(ℐ₂,ℐ)-convergent.

Proof. Let (𝑢_𝑚𝑛) be 𝑟(ℐ^∗₂,ℐ^∗)-convergent. Then, it is ℐ^∗₂-convergent and so, by Lemma 4, it is ℐ₂-convergent. Also, there exist the sets 𝑀₁,𝑀₂∈ℱ(ℐ) such that (∀𝜀>0)∃𝑚₀∈ℕ∀𝑚≥𝑚₀𝑚∈𝑀₁𝐷𝑢_𝑚𝑛,𝑣_𝑛<𝜀,(𝑛∈ℕ) (24) for some fuzzy numbers 𝑣_𝑛 and (∀𝜀>0)∃𝑛₀∈ℕ∀𝑛≥𝑛₀𝑛∈𝑀₂𝐷𝑢_𝑚𝑛,𝑤_𝑚<𝜀,(𝑚∈ℕ) (25) for some fuzzy numbers 𝑤_𝑚. Hence, we have 𝐴(𝜀)=𝑚∈ℕ∶𝐷𝑢_𝑚𝑛,𝑣_𝑛≥𝜀⊂𝐻₁∪1,2,…,𝑚₀,(),𝐵(𝜀)=𝑢−1𝑛∈ℕ𝑛∈ℕ∶𝐷_𝑚𝑛,𝑤_𝑚≥𝜀⊂𝐻₂∪1,2,…,𝑛₀,(),−1𝑚∈ℕ (26) for 𝐻₁,𝐻₂∈ℐ. Since ℐ is admissible ideal we get 𝐻₁∪1,2,…,𝑚₀𝐻−1∈ℐ,₂∪1,2,…,𝑛₀−1∈ℐ, (27) and therefore 𝐴(𝜀),𝐵(𝜀)∈ℐ. This shows that the double sequence (𝑢_𝑚𝑛) is 𝑟(ℐ₂,ℐ)-convergent.

Theorem 13. Let ℐ₂⊂2^ℕ×ℕ be a strongly admissible ideal with property (AP2), and let ℐ⊂2^ℕ be an admissible ideal with property (AP). If a double sequence (𝑢_𝑚𝑛) of fuzzy numbers is 𝑟(ℐ₂,ℐ)-convergent, then (𝑢_𝑚𝑛) is 𝑟(ℐ^∗₂,ℐ^∗)-convergent.

Proof. Let a double sequence (𝑢_𝑚𝑛) of fuzzy numbers be 𝑟(ℐ₂,ℐ)-convergent. Then (𝑢_𝑚𝑛) is ℐ₂-convergent and so (𝑢_𝑚𝑛) is ℐ^∗₂-convergent, by Lemma 5. Also, for every 𝜀>0 we have 𝐴(𝜀)=𝑚∈ℕ∶𝐷𝑢_𝑚𝑛,𝑣_𝑛≥𝜀∈ℐ (28) for some fuzzy numbers 𝑣_𝑛, for each 𝑛∈ℕ and 𝐶(𝜀)=𝑛∈ℕ∶𝐷𝑢_𝑚𝑛,𝑤_𝑚≥𝜀∈ℐ (29) for some fuzzy numbers 𝑤_𝑚, for each 𝑚∈ℕ.
Now put 𝐴₁=𝑚∈ℕ∶𝐷𝑢_𝑚𝑛,𝑣_𝑛,𝐴≥1_𝑘=1𝑚∈ℕ∶𝑘≤𝐷𝑢_𝑚𝑛,𝑣_𝑛<1𝑘−1 (30) for 𝑘≥2, for some fuzzy numbers 𝑣_𝑛 and for each 𝑛∈ℕ. It is clear that 𝐴_𝑖∩𝐴_𝑗=∅ for 𝑖≠𝑗 and 𝐴_𝑖∈ℐ for each 𝑖∈ℕ. By the property (𝐴𝑃) there is a countable family of sets {𝐵₁,𝐵₂,…} in ℐ such that 𝐴_𝑗△𝐵_𝑗 is a finite set for each 𝑗∈ℕ and 𝐵=⋃^∞_𝑗=1𝐵_𝑗∈ℐ.
We prove that lim_{𝑚→∞𝑚∈𝑀}𝑢_𝑚𝑛=𝑣_𝑛forsomefuzzynumbers𝑣_𝑛andforeach𝑛∈ℕ (31) for 𝑀=ℕ⧵𝐵∈ℱ(ℐ). Let 𝛿>0 be given. Choose 𝑘∈ℕ such that 1/𝑘<𝛿. Then, we have 𝑚∈ℕ∶𝐷𝑢_𝑚𝑛,𝑣_𝑛⊂≥𝛿_𝑘𝑗=1𝐴_𝑗forsomefuzzynumbers𝑣_𝑛andforeach𝑛∈ℕ. (32)
Since 𝐴_𝑗△𝐵_𝑗 is a finite set for 𝑗∈{1,2,…,𝑘}, there exists 𝑚₀∈ℕ such that ^𝑘_𝑗=1𝐵_𝑗∩𝑚∶𝑚≥𝑚₀=^𝑘_𝑗=1𝐴_𝑗∩𝑚∶𝑚≥𝑚₀. (33)
If 𝑚≥𝑚₀ and 𝑚∉𝐵 then 𝑚∉^𝑘_𝑗=1𝐵_𝑗andso𝑚∉^𝑘_𝑗=1𝐴_𝑗. (34)
Thus, we have 𝐷(𝑢_𝑚𝑛,𝑣_𝑛)<1/𝑘<𝛿 for some fuzzy numbers 𝑣_𝑛 and for each 𝑛∈ℕ. This implies that lim_{𝑚→∞𝑚∈𝑀}𝑢_𝑚𝑛(𝑥)=𝑣_𝑛. (35)
Hence, we have ℐ^∗−lim_𝑚→∞𝑢_𝑚𝑛=𝑣_𝑛 (36) for some fuzzy numbers 𝑣_𝑛 and for each 𝑛∈ℕ.
Similarly, for the set 𝐶(𝜀)={𝑛∈ℕ∶𝐷(𝑢_𝑚𝑛,𝑤_𝑚)≥𝜀}∈ℐ, we have ℐ^∗−lim_𝑛→∞𝑢_𝑚𝑛=𝑤_𝑚 (37) for some fuzzy numbers 𝑤_𝑚 and for each 𝑚∈ℕ. Hence, a double sequence (𝑢_𝑚𝑛) of fuzzy numbers is 𝑟(ℐ^∗₂,ℐ^∗)-convergent.

Now, we give the definitions of 𝑟(ℐ₂,ℐ)-Cauchy sequence and 𝑟(ℐ^∗₂,ℐ^∗)-Cauchy sequence.

Definition 14. Let ℐ₂ be a strongly admissible ideal of ℕ×ℕ, and let ℐ be an admissible ideal of ℕ. A double sequence (𝑢_𝑚𝑛) of fuzzy numbers is said to be regularly (ℐ₂,ℐ)-Cauchy (𝑟(ℐ₂,ℐ)-Cauchy), if it is ℐ₂-Cauchy in Pringsheim’s sense and for every 𝜀>0 there exist 𝑘_𝑛=𝑘_𝑛(𝜀)∈ℕ and 𝑙_𝑚=𝑙_𝑚(𝜀)∈ℕ such that the following statements hold: 𝐴₁(𝜀)=𝑚∈ℕ∶𝐷𝑢_𝑚𝑛,𝑢_𝑘𝑛𝑛(),𝐴≥𝜀∈ℐ,𝑛∈ℕ₂(𝜀)=𝑛∈ℕ∶𝐷𝑢_𝑚𝑛,𝑢_𝑚𝑙𝑚≥𝜀∈ℐ,(𝑚∈ℕ). (38)
A double sequence (𝑢_𝑚𝑛) is said to be regularly (ℐ^∗₂,ℐ^∗)-Cauchy (𝑟(ℐ^∗₂,ℐ^∗)-Cauchy), if there exist the sets 𝑀∈ℱ(ℐ₂), 𝑀₁∈ℱ(ℐ), and 𝑀₂∈ℱ(ℐ) (i.e., ℕ×ℕ⧵𝑀∈ℐ₂, ℕ⧵𝑀₁∈ℐ, and ℕ⧵𝑀₂∈ℐ) and for every 𝜀>0 there exist 𝑁=𝑁(𝜀), 𝑠=𝑠(𝜀), 𝑡=𝑡(𝜀), 𝑘_𝑛=𝑘_𝑛(𝜀), 𝑙_𝑚=𝑙_𝑚(𝜀)∈ℕ such that 𝐷𝑢_𝑚𝑛,𝑢_𝑠𝑡<𝜀for(),()𝐷𝑢𝑚,𝑛𝑠,𝑡∈𝑀,_𝑚𝑛,𝑢_𝑘𝑛𝑛<𝜀foreach𝑚∈𝑀₁andforeach𝐷𝑢𝑛∈ℕ,_𝑚𝑛,𝑢_𝑚𝑙𝑚<𝜀foreach𝑛∈𝑀₂andforeach𝑚∈ℕ, (39) whenever 𝑚,𝑛,𝑠,𝑡,𝑘_𝑛,𝑙_𝑚≥𝑁.

Theorem 15. Let ℐ₂ be a strongly admissible ideal of ℕ×ℕ, and let ℐ be an admissible ideal of ℕ. If a double sequence (𝑢_𝑚𝑛) of fuzzy numbers is 𝑟(ℐ^∗₂,ℐ^∗)-Cauchy, then it is 𝑟(ℐ₂,ℐ)-Cauchy.

Proof. Since a double sequence (𝑢_𝑚𝑛) of fuzzy numbers is 𝑟(ℐ^∗₂,ℐ^∗)-Cauchy, it is ℐ^∗₂-Cauchy. We know that ℐ^∗₂-Cauchy implies ℐ₂-Cauchy by Lemma 7. Also, since the double sequence (𝑢_𝑚𝑛) of fuzzy numbers is 𝑟(ℐ^∗₂,ℐ^∗)-Cauchy so there exist the sets 𝑀₁,𝑀₂∈ℱ(ℐ) and for every 𝜀>0 there exist 𝑘_𝑛=𝑘_𝑛(𝜀), 𝑙_𝑚=𝑙_𝑚(𝜀)∈ℕ such that 𝐷𝑢_𝑚𝑛,𝑢_𝑘𝑛𝑛<𝜀foreach𝑚∈𝑀₁andforeach𝐷𝑢𝑛∈ℕ,_𝑚𝑛,𝑢_𝑚𝑙𝑚<𝜀foreach𝑛∈𝑀₂andforeach𝑚∈ℕ (40)
for 𝑁=𝑁(𝜀)∈ℕ and 𝑚,𝑛,𝑘_𝑛,𝑙_𝑚≥𝑁. Therefore, for 𝐻₁=ℕ⧵𝑀₁,𝐻₂=ℕ⧵𝑀₂∈ℐ we have 𝐴₁(𝜀)=𝑚∈ℕ∶𝐷𝑢_𝑚𝑛,𝑢_𝑘𝑛𝑛≥𝜀⊂𝐻₁∪{1,2,…,𝑁−1},(𝑛∈ℕ) (41) for 𝑚∈𝑀₁ and 𝐴₂(𝜀)=𝑛∈ℕ∶𝐷𝑢_𝑚𝑛,𝑢_𝑚𝑙𝑚≥𝜀⊂𝐻₂∪{1,2,…,𝑁−1},(𝑚∈ℕ) (42) for 𝑛∈𝑀₂. Since ℐ is admissible ideal 𝐻₁∪{1,2,…,𝑁−1}∈ℐ,𝐻₂∪{1,2,…,𝑁−1}∈ℐ. (43) Hence, we have 𝐴₁(𝜀),𝐴₂(𝜀)∈ℐ and (𝑢_𝑚𝑛) is 𝑟(ℐ₂,ℐ)-Cauchy.

Theorem 16. Let ℐ₂ be a strongly admissible ideal of ℕ×ℕ, and let ℐ be an admissible ideal of ℕ. If a double sequence (𝑢_𝑚𝑛) of fuzzy numbers is 𝑟(ℐ₂,ℐ)-convergent, then (𝑢_𝑚𝑛) is 𝑟(ℐ₂,ℐ)-Cauchy sequence.

Proof. Let (𝑢_𝑚𝑛) be a 𝑟(ℐ₂,ℐ)-convergent double sequence of fuzzy numbers. Then (𝑢_𝑚𝑛) is ℐ₂-convergent, and by Lemma 6, it is ℐ₂-Cauchy sequence of fuzzy numbers. Also for every 𝜀>0, we have 𝐴₁𝜀2=𝑚∈ℕ∶𝐷𝑢_𝑚𝑛,𝑣_𝑛≥𝜀2∈ℐ (44) for some fuzzy numbers 𝑣_𝑛, for each 𝑛∈ℕ and 𝐴₂𝜀2=𝑛∈ℕ∶𝐷𝑢_𝑚𝑛,𝑤_𝑚≥𝜀2∈ℐ (45) for some fuzzy numbers 𝑤_𝑚, for each 𝑚∈ℕ. Since ℐ is admissible ideal, the sets 𝐴^𝑐₁𝜀2=𝑚∈ℕ∶𝐷𝑢_𝑚𝑛,𝑣_𝑛<𝜀2,(𝑛∈ℕ) (46) for some fuzzy numbers 𝑣_𝑛 and 𝐴^𝑐₂𝜀2=𝑛∈ℕ∶𝐷𝑢_𝑚𝑛,𝑤_𝑚<𝜀2,(𝑚∈ℕ) (47) for some fuzzy numbers 𝑤_𝑚, are nonempty and belong to ℱ(ℐ). For 𝑘_𝑛∈𝐴^𝑐₁(𝜀/2), (𝑛∈ℕ and 𝑘_𝑛>0) we have 𝐷(𝑢_𝑘𝑛𝑛,𝑣_𝑛)<𝜀/2 for some fuzzy numbers 𝑣_𝑛. Now, for 𝜀>0 we define the set 𝐵₁(𝜀)=𝑚∈ℕ∶𝐷𝑢_𝑚𝑛,𝑢_𝑘𝑛𝑛≥𝜀,(𝑛∈ℕ), (48) where 𝑘_𝑛=𝑘_𝑛(𝜀). Let 𝑚∈𝐵₁(𝜀). Then for 𝑘_𝑛∈𝐴^𝑐₁(𝜀/2), (𝑛∈ℕ and 𝑘_𝑛>0) we have 𝜀≤𝐷𝑢_𝑚𝑛,𝑢_𝑘𝑛𝑛≤𝐷𝑢_𝑚𝑛,𝑣_𝑛+𝐷𝑢_𝑘𝑛𝑛,𝑣_𝑛<𝐷𝑢_𝑚𝑛,𝑣_𝑛+𝜀2 (49) for some fuzzy numbers 𝑣_𝑛. This shows that 𝜀2<𝐷𝑢_𝑚𝑛,𝑣_𝑛andso𝑚∈𝐴₁𝜀2. (50)
Hence, we have 𝐵₁(𝜀)⊂𝐴₁(𝜀/2).
Similarly, for 𝑙_𝑚∈𝐴^𝑐₂(𝜀/2) (𝑚∈ℕ and 𝑙_𝑚>0) we have 𝐷𝑢_𝑚𝑙𝑚,𝑤_𝑚<𝜀2,(𝑚∈ℕ) (51) for some fuzzy numbers 𝑤_𝑚. Therefore, it can be seen that 𝐵₂(𝜀)=𝑚∈ℕ∶𝐷𝑢_𝑚𝑙𝑚,𝑤_𝑚≥𝜀⊂𝐴₂𝜀2. (52)
Hence, we have 𝐵₁(𝜀),𝐵₂(𝜀)∈ℐ. This shows that (𝑢_𝑚𝑛) is 𝑟(ℐ₂,ℐ)-Cauchy sequence of fuzzy numbers.

Acknowledgment

The authors would like to express their pleasure to the referees for making some useful remarks and helpful suggestions which improved the presentation of the paper.