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

Authors: Erdinç Dündar; Özer Talo; Feyzi Başar
Journal: International Journal of Analysis
 (2013)

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## Abstract

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


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## Body

## 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 paperℕ 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 ofℐ-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, ℐ2*-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, ℐ2*-Cauchy double sequences of fuzzy numbers, in [25].In this paper, we introduce the notions of regularly (ℐ2,ℐ), (ℐ2*,ℐ*)-convergence and regularly (ℐ2,ℐ), (ℐ2*,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 sequencex=(xmn)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 Nε∈ℕ such that |xmn-L|<ε whenever m,n>Nε. In this case we write
(1)P-limm,n→∞xmn=Lorlimm,n→∞xmn=L.LetX≠∅. 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.LetX≠∅. 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.Ifℐ is a nontrivial ideal in X, X≠∅, then the class
(2)ℱ(ℐ)={M⊂X:(∃A∈ℐ)(M=X∖A)}
is called a filter on X associated with ℐ.A nontrivial idealℐ in X is called admissible if {x}∈ℐ for each x∈X.Throughout the paper we takeℐ as a nontrivial admissible ideal in ℕ.Letℐ⊂2ℕ be a nontrivial ideal, and let (X,ρ) be a metric space. A sequence (xn) 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(xn,L)≥ε}∈ℐ.A sequence(xn) 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<⋯<mk<⋯} such that ρ(xmk,L)→0,ask→∞.Throughout the paper we takeℐ2 as a nontrivial admissible ideal in ℕ×ℕ.A nontrivial idealℐ2⊂2ℕ×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.Letℐ20={A⊂ℕ×ℕ:(∃m(A)∈ℕ)(i,j≥m(A)⇒(i,j)∉A)}. Then ℐ20 is a nontrivial strongly admissible ideal and clearly an ideal ℐ2 is strongly admissible if and only if ℐ20⊂ℐ2.Let(X,ρ) be a linear metric space, and let ℐ2⊂2ℕ×ℕ be a strongly admissible ideal. A double sequence x=(xmn) 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(xmn,L)≥ε}∈ℐ2 and is written ℐ2-limm,n→∞xmn=L.Ifℐ2⊂2ℕ×ℕ is a strongly admissible ideal, then usual convergence implies ℐ2-convergence.Let(X,ρ) be a linear metric space, and let ℐ2⊂2ℕ×ℕ be a strongly admissible ideal. A double sequence x=(xmn) of elements of X is said to be ℐ2*-convergent to L∈X, if and only if there exists a set M∈ℱ(ℐ2) (i.e., ℕ×ℕ∖M∈ℐ2) such that limm,n→∞xmn=L, for (m,n)∈M and is written ℐ2*-limm,n→∞xmn=L.Let(X,ρ) be a linear metric space, and let ℐ2⊂2ℕ×ℕ be a strongly admissible ideal. A double sequence x=(xmn) 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(xmn,xst)≥ε}∈ℐ2.Letℐ2 be an ideal of ℕ×ℕ and ℐ be an ideal of ℕ, then a double sequence x=(xmn) 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:
(3){m∈ℕ:|xmn-Ln|≥ε}∈ℐ
for some Ln∈ℂ, for each n∈ℕ and
(4){n∈ℕ:|xmn-Km|≥ε}∈ℐ
for some Km∈ℂ, for each m∈ℕ.We say that an admissible idealℐ⊂2ℕ 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 Aj△Bj is a finite set for j∈ℕ and B=⋃j=1∞‍Bj∈ℐ. (hence Bj∈ℐ for each j∈ℕ).We say that an admissible idealℐ2⊂2ℕ×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 Aj△Bj∈ℐ20, that is, Aj△Bj is included in the finite union of rows and columns in ℕ×ℕ for each j∈ℕ and B=⋃j=1∞‍Bj∈ℐ2 (hence Bj∈ℐ2 for each j∈ℕ).A fuzzy number is a fuzzy set on the real axis, that is, a mappingu: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 onℝ by E1 and called it as the space of fuzzy numbers. α-level set [u]α of u∈E1 is defined by
(5)[u]α≔{{t∈ℝ:x(t)≥α},(0<α≤1),{t∈ℝ:x(t)>α}¯,(α=0).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
(6)r¯(x)≔{1,(x=r),0,(x≠r).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 functionsu- and u+ are right continuous at the point α=0.(iv)
u-(1)≤u+(1).
Conversely, if the pair of functionsα 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)}.Letu,v,w∈E1 and k∈ℝ. Then the operations addition, scalar multiplication, and product are defined on E1 by
(7)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,∀α∈[0,1],uv=w⟺[w]α=[u]α[v]α,∀α∈[0,1],
where it is immediate that
(8)w-(α)=min{u-(α)v-(α),u-(α)v+(α),u+(α)v-(α),u+(α)v+(α)},w+(α)=max{u-(α)v-(α),u-(α)v+(α),u+(α)v-(α),u+(α)v+(α)},
for all α∈[0,1].LetW 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
(9)d(A,B)≔max{|A_-B_|,|A¯-B¯|}.Then it can easily be observed thatd 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
(10)D(u,v)≔supα∈[0,1]d([u]α,[v]α)≔supα∈[0,1]max{|u-(α)-v-(α)|,|u+(α)-v+(α)|}.One can see that(11)D(u,0¯)=supα∈[0,1]max{|u-(α)|,|u+(α)|}=max{|u-(0)|,|u+(0)|}.The partial ordering relation⪯ on E1 is defined as follows:
(12)u⪯v⟺u-(α)≤v-(α),u+(α)≤v+(α)∀α∈[0,1].Now, we may give the following.Proposition 2 (see [6]).
Letu,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.Asequenceu=(uk) of fuzzy numbers is a function u from the set ℕ into the set E1. The fuzzy number uk 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 numbersA sequence(un)∈w(F) is called convergent with limit u∈E1, if for every ε>0 there exists an n0=n0(ε)∈ℕ such that D(un,u)<ε for all n≥n0.Adouble sequenceu=(unk) of fuzzy real numbers is defined by a function u from the set ℕ×ℕ into the set E1. The fuzzy number unk denotes the value of the function at (n,k)∈ℕ×ℕ.A double sequenceu=(umn) 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(umn,u0)<ε for all m,n≥k and is denoted by P-limm,n→∞umn=u0. The fuzzy number u0 is called the Pringsheim limit of u.Letℐ2⊂2ℕ×ℕ be a strongly admissible ideal. (i)
A double sequenceu=(umn) 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(umn,u0)≥ε}∈ℐ2 and is written ℐ2-limm,n→∞umn=u0.(ii)
A double sequenceu=(umn) of fuzzy numbers is said to be ℐ2*-convergent to u0∈E1 if there exists M∈ℱ(ℐ2) (i.e., H=ℕ×ℕ∖M∈ℐ2) such that
(13)limm,n→∞(m,n)∈Mumn=u0
and is written ℐ2*-limm,n→∞umn=u0.(iii)
A double sequenceu=(umn) 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(umn,ust)≥ε}∈ℐ2.(iv)
A double sequenceu=(umn) of fuzzy numbers is said to be ℐ2*-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(umn,ust)<ε, that is,
(14)limm,n,s,t→∞(m,n),(s,t)∈MD(umn,ust)=0.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]).
Letℐ2⊂2ℕ×ℕ be a strongly admissible ideal, let u=(umn) be a double sequence of fuzzy numbers, and let u0 be a fuzzy number. Then, P-limm,n→∞umn=u0 implies ℐ2-limm,n→∞umn=u0.Lemma 4 (see [24, Theorem 4.2]).
Letℐ2⊂2ℕ×ℕ be a strongly admissible ideal, let u=(umn) be a double sequence of fuzzy numbers, and let u0∈E1. Then, ℐ2*-limm,n→∞umn=u0 implies ℐ2-limm,n→∞umn=u0.Lemma 5 (see [24, Theorem 4.4]).
Letℐ2⊂2ℕ×ℕ be a strongly admissible ideal with property (AP2), let u=(umn) be a double sequence of fuzzy numbers, and let u0 be a fuzzy real number. Then, ℐ2-limm,n→∞umn=u0 implies ℐ2*-limm,n→∞umn=u0.Lemma 6 (see [25, Theorem 3.2]).
Letℐ2⊂2ℕ×ℕ be a strongly admissible ideal. A double sequence u=(umn) of fuzzy numbers is ℐ2-convergent if and only if it is ℐ2-Cauchy sequence.Lemma 7 (see [25, Theorem 3.4]).
Letℐ2⊂2ℕ×ℕ be a strongly admissible ideal. If a double sequence u=(umn) of fuzzy numbers is an ℐ2*-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(umn) of fuzzy numbers is said to be regularly convergent, if it is convergent in Pringsheim’s sense and the limits
(15)limm→∞umn,(n∈ℕ),limn→∞umn,(m∈ℕ),
exist for each fixed n∈ℕ and m∈ℕ, respectively. Note that if (umn) is regularly convergent to a fuzzy number u0, then the limits
(16)limn→∞limm→∞umn,limm→∞limn→∞umn
exist and are equal to u0. In this case we write
(17)r-limm,n→∞umn=u0orumn⟶ru0asm,n⟶∞.Definition 9.
Letℐ2⊂2ℕ×ℕ be a strongly admissible ideal, and let ℐ⊂2ℕ be an admissible ideal. A double sequence (umn) 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:
(18){m∈ℕ:D(umn,vn)≥ε}∈ℐ
for some fuzzy numbers vn, for each n∈ℕ and
(19){n∈ℕ:D(umn,wm)≥ε}∈ℐ
for some fuzzy numbers wm, for each m∈ℕ.In the case(umn) is regularly (ℐ2,ℐ)-convergent (r(ℐ2,ℐ)-convergent) to a fuzzy number u0, then the limits ℐ-limn→∞limm→∞umn and ℐ-limm→∞limn→∞umn exist and are equal to u0.Theorem 10.
Letℐ2 be a strongly admissible ideal of ℕ×ℕ, and let ℐ be an admissible ideal of ℕ. If a double sequence (umn) of fuzzy numbers is regularly convergent, then it is r(ℐ2,ℐ)-convergent.Proof.
Let(umn) be regularly convergent. Then (umn) is convergent in Pringsheim’s sense and the limits limm→∞umn(n∈ℕ) and limn→∞umn(m∈ℕ) exist. By Lemma 3, (umn) is ℐ2-convergent. Also, for ε>0, there exist m=m0(ε) and n=n0(ε) such that
(20)D(umn,vn)<ε
for some fuzzy numbers vn and each fixed n∈ℕ for every m≥m0 and
(21)D(umn,wm)<ε
for some fuzzy numbers wm and each fixed m∈ℕ for every n≥n0. Then, since ℐ is admissible ideal so for ε>0, we have
(22){m∈ℕ:D(umn,vn)≥ε}={1,2,…,m0-1}∈ℐ,{n∈ℕ:D(umn,wm)≥ε}={1,2,…,n0-1}∈ℐ.
Hence,(umn) is r(ℐ2,ℐ)-convergent.Definition 11.
Letℐ2 be a strongly admissible ideal of ℕ×ℕ, and let ℐ be an admissible ideal of ℕ. A double sequence (umn) of fuzzy numbers is said to be r(ℐ2*,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
(23)limm,n→∞(m,n)∈Mumn,limm→∞m∈M1umn,limn→∞n∈M2umn
exist for each fixed n∈ℕ and m∈ℕ, respectively.Theorem 12.
Letℐ2 be a strongly admissible ideal of ℕ×ℕ, and let ℐ be an admissible ideal of ℕ. If a double sequence (umn) of fuzzy numbers is r(ℐ2*,ℐ*)-convergent, then it is r(ℐ2,ℐ)-convergent.Proof.
Let(umn) be r(ℐ2*,ℐ*)-convergent. Then, it is ℐ2*-convergent and so, by Lemma 4, it is ℐ2-convergent. Also, there exist the sets M1,M2∈ℱ(ℐ) such that
(24)(∀ε>0)(∃m0∈ℕ)(∀m≥m0)(m∈M1)D(umn,vn)<ε,(n∈ℕ)
for some fuzzy numbers vn and
(25)(∀ε>0)(∃n0∈ℕ)(∀n≥n0)(n∈M2)D(umn,wm)<ε,(m∈ℕ)
for some fuzzy numbers wm. Hence, we have
(26)A(ε)={m∈ℕ:D(umn,vn)≥ε}⊂H1∪{1,2,…,m0-1},(n∈ℕ),B(ε)={n∈ℕ:D(umn,wm)≥ε}⊂H2∪{1,2,…,n0-1},(m∈ℕ),
for H1,H2∈ℐ. Since ℐ is admissible ideal we get
(27)H1∪{1,2,…,(m0-1)}∈ℐ,H2∪{1,2,…,n0-1}∈ℐ,
and therefore A(ε),B(ε)∈ℐ. This shows that the double sequence (umn) is r(ℐ2,ℐ)-convergent.Theorem 13.
Letℐ2⊂2ℕ×ℕ be a strongly admissible ideal with property (AP2), and let ℐ⊂2ℕ be an admissible ideal with property (AP). If a double sequence (umn) of fuzzy numbers is r(ℐ2,ℐ)-convergent, then (umn) is r(ℐ2*,ℐ*)-convergent.Proof.
Let a double sequence(umn) of fuzzy numbers be r(ℐ2,ℐ)-convergent. Then (umn) is ℐ2-convergent and so (umn) is ℐ2*-convergent, by Lemma 5. Also, for every ε>0 we have
(28)A(ε)={m∈ℕ:D(umn,vn)≥ε}∈ℐ
for some fuzzy numbers vn, for each n∈ℕ and
(29)C(ε)={n∈ℕ:D(umn,wm)≥ε}∈ℐ
for some fuzzy numbers wm, for each m∈ℕ.
Now put(30)A1={m∈ℕ:D(umn,vn)≥1},Ak={m∈ℕ:1k≤D(umn,vn)<1k-1}
for k≥2, for some fuzzy numbers vn and for each n∈ℕ. It is clear that Ai∩Aj=∅ for i≠j and Ai∈ℐ for each i∈ℕ. By the property (AP) there is a countable family of sets {B1,B2,…} in ℐ such that Aj△Bj is a finite set for each j∈ℕ and B=⋃j=1∞‍Bj∈ℐ.
We prove that(31)limm→∞m∈Mumn=vnfor  some  fuzzy  numbersvnand  for  eachn∈ℕ
for M=ℕ∖B∈ℱ(ℐ). Let δ>0 be given. Choose k∈ℕ such that 1/k<δ. Then, we have
(32){m∈ℕ:D(umn,vn)≥δ}⊂⋃j=1kAjfor  some  fuzzy  numbersvnand  for  eachn∈ℕ.
SinceAj△Bj is a finite set for j∈{1,2,…,k}, there exists m0∈ℕ such that
(33)(⋃j=1k‍Bj)∩{m:m≥m0}=(⋃j=1k‍Aj)∩{m:m≥m0}.
Ifm≥m0 and m∉B then
(34)m∉⋃j=1k‍Bjandsom∉⋃j=1k‍Aj.
Thus, we haveD(umn,vn)<1/k<δ for some fuzzy numbers vn and for each n∈ℕ. This implies that
(35)limm→∞m∈Mumn(x)=vn.
Hence, we have(36)ℐ*-limm→∞umn=vn
for some fuzzy numbers vn and for each n∈ℕ.
Similarly, for the setC(ε)={n∈ℕ:D(umn,wm)≥ε}∈ℐ, we have
(37)ℐ*-limn→∞umn=wm
for some fuzzy numbers wm and for each m∈ℕ. Hence, a double sequence (umn) of fuzzy numbers is r(ℐ2*,ℐ*)-convergent.Now, we give the definitions ofr(ℐ2,ℐ)-Cauchy sequence and r(ℐ2*,ℐ*)-Cauchy sequence.Definition 14.
Letℐ2 be a strongly admissible ideal of ℕ×ℕ, and let ℐ be an admissible ideal of ℕ. A double sequence (umn) 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 kn=kn(ε)∈ℕ and lm=lm(ε)∈ℕ such that the following statements hold:
(38)A1(ε)={m∈ℕ:D(umn,uknn)≥ε}∈ℐ,(n∈ℕ),A2(ε)={n∈ℕ:D(umn,umlm)≥ε}∈ℐ,(m∈ℕ).
A double sequence(umn) is said to be regularly (ℐ2*,ℐ*)-Cauchy (r(ℐ2*,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), kn=kn(ε), lm=lm(ε)∈ℕ such that
(39)D(umn,ust)<εfor(m,n),(s,t)∈M,D(umn,uknn)<εforeachm∈M1andforeachn∈ℕ,D(umn,umlm)<εforeachn∈M2andforeachm∈ℕ,
whenever m,n,s,t,kn,lm≥N.Theorem 15.
Letℐ2 be a strongly admissible ideal of ℕ×ℕ, and let ℐ be an admissible ideal of ℕ. If a double sequence (umn) of fuzzy numbers is r(ℐ2*,ℐ*)-Cauchy, then it is r(ℐ2,ℐ)-Cauchy.Proof.
Since a double sequence(umn) of fuzzy numbers is r(ℐ2*,ℐ*)-Cauchy, it is ℐ2*-Cauchy. We know that ℐ2*-Cauchy implies ℐ2-Cauchy by Lemma 7. Also, since the double sequence (umn) of fuzzy numbers is r(ℐ2*,ℐ*)-Cauchy so there exist the sets M1,M2∈ℱ(ℐ) and for every ε>0 there exist kn=kn(ε), lm=lm(ε)∈ℕ such that
(40)D(umn,uknn)<εforeachm∈M1andforeachn∈ℕ,D(umn,umlm)<εforeachn∈M2andforeachm∈ℕ
forN=N(ε)∈ℕ and m,n,kn,lm≥N. Therefore, for H1=ℕ∖M1,H2=ℕ∖M2∈ℐ we have
(41)A1(ε)={m∈ℕ:D(umn,uknn)≥ε}⊂H1∪{1,2,…,N-1},(n∈ℕ)
for m∈M1 and
(42)A2(ε)={n∈ℕ:D(umn,umlm)≥ε}⊂H2∪{1,2,…,N-1},(m∈ℕ)
for n∈M2. Since ℐ is admissible ideal
(43)H1∪{1,2,…,N-1}∈ℐ,H2∪{1,2,…,N-1}∈ℐ.
Hence, we have A1(ε),A2(ε)∈ℐ and (umn) is r(ℐ2,ℐ)-Cauchy.Theorem 16.
Letℐ2 be a strongly admissible ideal of ℕ×ℕ, and let ℐ be an admissible ideal of ℕ. If a double sequence (umn) of fuzzy numbers is r(ℐ2,ℐ)-convergent, then (umn) is r(ℐ2,ℐ)-Cauchy sequence.Proof.
Let(umn) be a r(ℐ2,ℐ)-convergent double sequence of fuzzy numbers. Then (umn) is ℐ2-convergent, and by Lemma 6, it is ℐ2-Cauchy sequence of fuzzy numbers. Also for every ε>0, we have
(44)A1(ε2)={m∈ℕ:D(umn,vn)≥ε2}∈ℐ
for some fuzzy numbers vn, for each n∈ℕ and
(45)A2(ε2)={n∈ℕ:D(umn,wm)≥ε2}∈ℐ
for some fuzzy numbers wm, for each m∈ℕ. Since ℐ is admissible ideal, the sets
(46)A1c(ε2)={m∈ℕ:D(umn,vn)<ε2},(n∈ℕ)
for some fuzzy numbers vn and
(47)A2c(ε2)={n∈ℕ:D(umn,wm)<ε2},(m∈ℕ)
for some fuzzy numbers wm, are nonempty and belong to ℱ(ℐ). For kn∈A1c(ε/2), (n∈ℕ and kn>0) we have D(uknn,vn)<ε/2 for some fuzzy numbers vn. Now, for ε>0 we define the set
(48)B1(ε)={m∈ℕ:D(umn,uknn)≥ε},(n∈ℕ),
where kn=kn(ε). Let m∈B1(ε). Then for kn∈A1c(ε/2), (n∈ℕ and kn>0) we have
(49)ε≤D(umn,uknn)≤D(umn,vn)+D(uknn,vn)<D(umn,vn)+ε2
for some fuzzy numbers vn. This shows that
(50)ε2<D(umn,vn)andsom∈A1(ε2).
Hence, we haveB1(ε)⊂A1(ε/2).
Similarly, forlm∈A2c(ε/2) (m∈ℕ and lm>0) we have
(51)D(umlm,wm)<ε2,(m∈ℕ)
for some fuzzy numbers wm. Therefore, it can be seen that
(52)B2(ε)={m∈ℕ:D(umlm,wm)≥ε}⊂A2(ε2).
Hence, we haveB1(ε),B2(ε)∈ℐ. This shows that (umn) is r(ℐ2,ℐ)-Cauchy sequence of fuzzy numbers.



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*Source: 749684-2013-04-03.xml*