Bulletin of Mathematical Analysis and Applications ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 12 Issue 2 (2020), Pages 12-26.
REGULARLY IDEAL CONVERGENCE OF DOUBLE SEQUENCES IN FUZZY NORMED SPACES
ERD˙INC¸ D ¨UNDAR, MUHAMMED RECA˙I T ¨URKMEN AND N˙IMET PANCARO ˇGLU AKIN
Abstract. In this study, we introduce the notions of regularly (I2,I)-convergence, regularly (I∗2,I∗)-convergence, regularly (I2,I)-Cauchy and regularly (I∗2,I∗)- Cauchy double sequences in fuzzy normed linear spaces. Also, we establish some basic results related to these notions.
1. Introduction and Background
Throughout the paperNandRdenote the set of all positive integers and the set of all real numbers, respectively. The concept of convergence of a sequence of real numbers has been extended to statistical convergence independently by Fast [19]
and Schoenberg [37]. The idea of I-convergence was introduced by Kostyrko et al. [24] as a generalization of statistical convergence which is based on the structure of the idealI of subset of the set of natural numbers. Das et al. [5] introduced the concept of I-convergence of double sequences in a metric space and studied some properties of this type convergence. D¨undar [14] introduces the notions of regularly (I2,I)-convergence and (I2,I)-Cauchy double sequences of real valued functions.
The concept of ordinary convergence of a sequence of fuzzy numbers was firstly introduced by Matloka [27] and proved some basic theorems for sequences of fuzzy numbers. Nanda [30] studied the sequences of fuzzy numbers and showed that the set of all convergent sequences of fuzzy numbers form a complete metric space.
D¨undar and Talo [11,12] investigatedI2-convergence,I2∗-convergence andI2-Cauchy sequence of fuzzy numbers and D¨undar et al. [13] introduced regularly (I2,I)- convergence and regularly (I2,I)-Cauchy double sequences of fuzzy numbers. Haz- arika [21] studied the concepts of I-convergence, I∗-convergence and I-Cauchy sequence in a fuzzy normed linear space. Also, Hazarika and Kumar [22] defined the concepts ofI2-convergence,I2∗-convergence andI2-Cauchy sequence in a fuzzy normed linear space. D¨undar and T¨urkmen [15, 16] studied I2-convergence and I2-Cauchy double sequences in fuzzy normed spaces. A lot of developments have been made in this area after the works of [17, 23, 29, 35, 36, 39–42, 45].
2000Mathematics Subject Classification. 03E72, 40A05, 40A35, 40B05.
Key words and phrases. Double sequences; I-convergence; Regularly convergence; Fuzzy normed spaces.
c
2020 Universiteti i Prishtin¨es, Prishtin¨e, Kosov¨e.
Submitted April 9, 2020. Published April 28, 2020.
Communicated by Feyzi Basar.
12
Now, we recall the concept of ideal convergence, double sequence and fuzzy normed space and some basic definitions (see [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13–16, 18, 20–22, 24–28, 32–34, 38, 43, 44])
Fuzzy sets are considered with respect to a nonempty base setX of elements of interest. The essential idea is that each element x∈X is assigned a membership gradeu(x) taking values in [0,1], withu(x) = 0 corresponding to nonmembership, 0< u(x)<1 to partial membership, and u(x) = 1 to full membership. According to Zadeh [46], a fuzzy subset of X is a nonempty subset {(x, u(x)) : x ∈ X} of X×[0,1] for some functionu: X →[0,1]. The functionuitself is often used for the fuzzy set.
A fuzzy setuonRis called a fuzzy number if it has the following properties:
1. uis normal, that is, there exists anx0∈Rsuch thatu(x0) = 1;
2. uis fuzzy convex, that is, for x, y∈Rand 0 ≤λ≤1, u(λx+ (1−λ)y)≥ min[u(x), u(y)];
3. uis upper semicontinuous;
4. The set [u]0=cl{x∈R:u(x)>0}is compact.
LetL(R) be set of all fuzzy numbers. Ifu∈L(R) andu(t) = 0 fort <0,thenu is called a non-negative fuzzy number. We denote the set of all non-negative fuzzy numbers by L∗(R). We can say that u ∈ L∗(R) iff u−α ≥ 0 for each α ∈ [0,1]. Clearly we havee0∈L(R).Foru∈L(R),theαlevel set ofuis defined by
[u]α=
{x∈R:u(x)≥α}, if α∈(0,1]
cl{x∈R:u(x)>0}, if α= 0.
A partial ordering onL(R) is defined byuv ifu−α ≤v−α andu+α ≤vα+ for allα∈[0,1].
Some arithmetic operations forα-level sets are defined as follows:
u, v∈L(R) and [u]α= [u−α, u+α] and [v]α= [vα−, v+α], α∈(0,1].Then, [u⊕v]α= [u−α+v−α, u+α +v+α], [u v]α= [u−α−vα+, u+α−vα−], [uv]α= [u−α.vα−, u+α.v+α] and ˜1u
α=h
1 u+α, 1
u−α
i
,u−α >0.
Foru, v∈L(R),the supremum metric onL(R) defined as D(u, v) = sup
0≤α≤1
max
u−α−vα− ,
u+α −v+α .
It is known thatDis a metric onL(R) and (L(R), D) is a complete metric space.
A sequence x = (xk) of fuzzy numbers is said to be convergent to the fuzzy numberx0,if for everyε >0 there exists a positive integerk0such thatD(xk, x0)<
εfor k > k0 and a sequence x= (xk) of fuzzy numbers level-wise converges to x0
iff lim
k→∞[xk]α = [x0]−α and lim
k→∞[xk]α = [x0]+α, where [xk]α =h
(xk)−α,(xk)+αi and [x0]α=h
(x0)−α,(x0)+αi
, for everyα∈(0,1).
LetX be a vector space overR,k.k:X →L∗(R) and the mappingsL;R(respec- tively, left norm and right norm) : [0,1]×[0,1]→[0,1] be symetric, nondecreasing in both arguments and satisfyL(0,0) = 0 andR(1,1) = 1.
The quadruple (X,k.k, L, R) is called fuzzy normed linear space (briefly (X,k.k) FNS) andk.ka fuzzy norm if the following axioms are satisfied
(1) kxk=e0 iff x= 0,
(2) krxk=|r| kxk forx∈X, r∈R,
(3) For all x, y∈X
(a) kx+yk(s+t) ≥ L(kxk(s),kyk(t)), whenever s ≤ kxk−1 , t ≤ kyk−1 ands+t≤ kx+yk−1,
(b) kx+yk(s+t) ≤ R(kxk(s),kyk(t)), whenever s ≥ kxk−1 , t ≥ kyk−1 ands+t≥ kx+yk−1.
In the sequel we take L(p, q) = min(p, q) andR(p, q) = max(p, q) for allp, q ∈ [0,1]. So, we get triangle inequality askx+yk−α ≤ kxk−α +kyk−α andkx+yk+α ≤ kxk+α +kyk+α, for allα∈(0,1) andx, y∈X.Then, we say thatk.k−α andk.k+α are norms in the usual sense onX.
Let (X,k.kC) be an ordinary normed linear space. Then, a fuzzy normk.konX can be obtained by
kxk(t) =
0, if 0≤t≤akxkC or t≥bkxkC
t
(1−a)kxkC −1−aa , akxkC ≤t≤ kxkC
−t
(b−1)kxkC +b−1b , kxkC≤t≤bkxkC
(1)
wherekxkCis the ordinary norm of x(6=θ),0< a <1 and 1< b <∞.Forx=θ, definekxk=e0.Hence, (X,k.k) is a fuzzy normed linear space.
Let us consider the topological structure of anF N S (X,k.k).For anyε >0, α∈ [0,1] and x ∈ X, the (ε, α)− neighborhood of x is the setNx(ε, α) = {y ∈ X : kx−yk+α < ε}.Throughout the paper, we let (X,k.k) be an FNS.
A sequence (xn)∞n=1 in X is convergent to L ∈ X with respect to the fuzzy norm on X and we denote by xn
F N→ L or F N − lim
n→∞xn = L, provided that (D)− lim
n→∞kxn−Lk = e0; i.e., for every ε > 0 there is an N(ε) ∈ N such that D
kxn−Lk,e0
< ε,for all n≥N(ε).This means that for every ε >0 there is anN(ε)∈Nsuch that for alln≥N(ε), sup
α∈[0,1]
kxn−Lk+α =kxn−Lk+0 < ε.
If K ⊆ N, then Kn denotes the set {k ∈ K : k ≤ n} and |Kn| denotes the cardinality of Kn. The natural density of K is given by d(K) = lim
n→∞
1
n|Kn|, if it exists.
The number sequencex= (xk) is statistically convergent toLprovided that for every ε >0 we haved(K(ε)) = 0, whereK =K(ε) :={k∈N:|xk−L| ≥ε}. In this case, we writest−limx=L.
A double sequencex= (xmn)(m,n)∈N×Nof 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 shall write this as
m,n→∞lim xmn=L.
A double sequencex= (xmn) is said to be bounded if there exists a positive real number M such that|xmn|< M for allm, n∈N, that is,kxk∞= sup
m,n
|xmn|<∞.
We let the set of all bounded double sequences byL∞.
A double sequence (xmn) is said to be convergent to L ∈ X (in Pringsheim’s sense) with respect to the fuzzy norm onX if for everyε >0 there exist a number N = N(ε) such that D
kxmn−Lk,e0
< ε, for allm, n ≥ N. In this case, we writexmnF N→ L. This means that, for everyε >0 there exists a numberN=N(ε)
such that sup
α∈[0,1]
kxmn−Lk+α = kxmn−Lk+0 < ε, for all m, n ≥ N. In terms of neighborhoods, we have xmn
−→F N L provided that for any ε > 0, there exists a numberN =N(ε) such thatxmn∈ Nx(ε,0),wheneverm, n≥N.
LetK⊂N×N. LetKmn be the number of (j, k)∈K such thatj≤m,k≤n.
If the sequenceKmn
m.n has a limit in Pringsheim’s sense then we say thatK has double natural density and is denoted byd2(K) = lim
m,n→∞
Kmn m.n.
A double sequencex= (xmn) of real numbers is said to be statistically convergent to L∈Rif for anyε >0 we have d2(A(ε)) = 0, where A(ε) ={(m, n)∈N×N:
|xmn−L| ≥ε}.
LetX6=∅. A classI of subsets ofX is said to be an ideal in X provided:
(i)∅ ∈ I, (ii)A, B∈ I impliesA∪B∈ I, (iii)A∈ I,B⊂AimpliesB∈ I. I is called a nontrivial ideal if X6∈ I.
Let X 6= ∅. A non empty class F of subsets of X is said to be a filter in X provided:
(i)∅ 6∈ F, (ii)A, B∈ F impliesA∩B∈ F, (iii)A∈ F,A⊂B impliesB ∈ F.
LetI is a nontrivial ideal inX, thenF(I) ={M ⊂X : (∃A∈ I)(M =X\A)}
is a filter onX, called the filter associated withI.
A nontrivial idealI in X is called admissible if {x} ∈ I for eachx∈X.
Throughout the paper we takeI as an admissible ideal inN.
If we takeI=Id={A⊂N:d(A) = 0}, thenI=Id is a non-trivial admissible ideal of N and the ideal convergence coincides with statistical convergence with respect to the fuzzy norm onN.
An admissible ideal I ⊂ 2N is said to satisfy 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 andB=S∞
j=1Bj∈ I.
A nontrivial idealI2ofN×Nis called strongly admissible if{i} ×NandN× {i}
belong to I2 for each i ∈N. It is evident that a strongly admissible ideal is also admissible.
Let I20 = {A ⊂N×N : (∃m(A),(i, j)≥ m(A) ⇒ (i, j) 6∈ A)}. Then I20 is a nontrivial strongly admissible ideal and clearly an idealI2 is strongly admissible if and only ifI20⊂ I2.
Throughout the paper we takeI2 as a strongly admissible ideal inN×N. If we takeI2 =Id2 ={A⊂N×N:d2(A) = 0}, then I2 =Id2 is a nontrivial strongly admissible ideal ofN×Nand the ideal convergence coincides with statistical convergence with respect to the fuzzy norm onN.
We say that an admissible idealI2 ⊂2N×N satisfies the property (AP2), if for every countable family of mutually disjoint sets{A1, A2, ...}belonging toI2, there exists a countable family of sets{B1, B2, ...} such thatAj∆Bj ∈ I20, i.e., Aj∆Bj is included in the finite union of rows and columns in N×N for each j ∈N and B=S∞
j=1Bj∈ I2 (henceBj∈ I2 for eachj∈N).
A sequencex= (xm)m∈NinX is said to beI-convergent toL∈X with respect to fuzzy norm on X if for each ε >0, the set A(ε) =n
m∈N:kxm−Lk+0 ≥εo belongs toI.In this case, we writexm
−→FI LorFI − lim
m→∞xm=L. The element Lis called theI-limit of (xm) inX.
A sequence (xm) inX is said to beI∗ convergent toLinX with respect to the fuzzy norm onX if there exists a setM ∈ F(I),M ={m1< m2<· · · } ⊂Nsuch that lim
k→∞kxmk−Lk= 0. In this case, we writexm FI∗
−→LorFI∗− lim
m→∞xm=L.
A sequence (xm) inX is said to beI-Cauchy with respect to the fuzzy norm on X if for every ε > 0, there exists an integern =n(ε) in N such that {m ∈ N : kxm−xnk+0 ≥ε} ∈ I.
A double sequence x = (xmn) in X is said to be I2- convergent to L ∈ X with respect to fuzzy norm on X if for every ε > 0, A(ε) = {(m, n) ∈ N×N : kxmn−Lk+0 ≥ε} ∈ I2. In this case, we write xmn −→FI2 L or xmn → L(FI2) or FI2− lim
m,n→∞xmn=L .
A double sequencex= (xmn) inX is said to beI2∗-convergent to Lin X with respect to the fuzzy norm onX if there exists a setM ∈ F(I2), M ={m1<· · ·<
mk <· · · ;n1<· · ·< nl<· · · } ⊂N×Nsuch that lim
k,l→∞kxmknl−Lk.
A double sequencex= (xmn) in X is said to be I2-Cauchy with respect to the fuzzy norm onX if for each ε >0,there exists integerss=s(ε) andt=t(ε) such thatn
(m, n)∈N×N:kxmn−xstk+0 ≥εo
∈ I2.
A double sequence x = (xmn) in X is said to be I2∗-Cauchy double sequence with respect to fuzzy norm on X, if there exists a set M ∈ F(I2) (i.e., H = N×N\M ∈ I2) andk0=k0(ε) such that for everyε >0 and for (m, n),(s, t)∈M, kxmn−xstk+0 < ε,wheneverm, n, s, t > k0.In this case we write
m,n,s,t→∞lim kxmn−xstk+0 = 0.
Lemma 1.1. [15] Let(X,k.k)be a fuzzy normed space,(xmn)be a double sequence inX andL1∈X. Then,F P − lim
m,n→∞xmn=L1⇒FI2− lim
m,n→∞xmn=L1. Lemma 1.2. [21] Let (X,k.k) be a fuzzy normed space, x = (xmn) be a double sequence in X and L1 ∈ X. If x = (xmn) is I2∗-convergent to L1 then it is I2- convergent toL1.
Lemma 1.3. [21] LetI2⊂2N×Nbe a strongly admissible ideal with property (AP2), (X,k.k)be a fuzzy normed space,x= (xmn)be a double sequence inX andL1∈X.
If x= (xmn)isI2-convergent toL1 then it isI2∗-convergent toL1.
Lemma 1.4. [16] Let I2 be an admissible ideal of N×N. If a double sequence (xmn)in X is an FI2∗-Cauchy sequence, then it isFI2-Cauchy.
Lemma 1.5. [21] Let (X,k.k) be a fuzzy normed space, x = (xmn) be a double sequence inX. Ifx= (xmn)isI2-convergent, then it isI2-Cauchy sequence inX.
Lemma 1.6. [31] Let {Pi}∞i=1 be a countable collection of subsets ofN such that Pi∈ F(I)for each i, where F(I) is a filter associated with a strongly admissible idealI with the property(AP). Then, there exists a setP ⊂Nsuch thatP ∈ F(I) and the setP\Pi is finite for alli.
Lemma 1.7. [16] LetI2be an admissible ideal ofN×Nwith the property(AP2)and (xmn)be a double sequence in X. Then, the conceptsI2−Cauchy double sequence with respect to fuzzy norm on X and I2∗-Cauchy double sequence with respect to fuzzy norm on X coincide.
2. Main Results
In this section, we introduce the notions of regularly (I2,I)-convergence, regu- larly (I2∗,I∗)-convergence, regularly (I2,I)-Cauchy and regularly (I2∗,I∗)-Cauchy double sequences in fuzzy normed linear spaces. Also, we establish some basic results related to these notions.
Definition 2.1. A double sequence (xmn) inX is said to be regularly convergent with respect to fuzzy norm onX, if it is convergent in Pringsheim’s sense and the limits
F N− lim
m→∞xmn, (n∈N) andF N− lim
n→∞xmn, (m∈N),
exist for each fixedn∈Nand each fixedm∈N, respectively. Note that if (xmn) is regularly convergent toLinX, then the limits
F N− lim
n→∞ lim
m→∞xmn andF N− lim
m→∞ lim
n→∞xmn
exist and are equal toL. In this case we write F r− lim
m,n→∞xmn=L or xmn
−→F r L.
In terms of neighborhoods, we have xmn
−→F r L if for every ε >0, there exists an integer k = k0(ε) ∈ N such that xmn ∈ NL(ε,0), whenever m, n ≥k, xmn ∈ NL(ε,0), wheneverm≥kand for each fixedn∈Nandxmn∈ NL(ε,0), whenever n≥kand for each fixedm∈N.
Definition 2.2. A double sequence (xmn) in X is said to be regularly (I2,I)- convergent (F r(I2,I)-convergent) with respect to fuzzy norm onX, if it isFI2- convergent in Pringsheim’s sense and for eachε >0, the following statements hold:
{m∈N:kxmn−Lnk+0 ≥ε} ∈ I (2) for someLn ∈X and each fixedn∈Nand
{n∈N:kxmn−Kmk+0 ≥ε} ∈ I (3) for someKm∈X and each fixedm∈N.
If (xmn) isF r(I2,I)-convergent toL∈X, then the limits FI − lim
n→∞ lim
m→∞xmn andFI − lim
m→∞ lim
n→∞xmn exist and are equal toL. In this case we write
F r(I2,I)− lim
m,n→∞xmn=L or xmnF r(I−→2,I)L.
In terms of neighborhoods, we havexmn
F r(I2,I)
−→ Lif for everyε >0, {(m, n)∈N×N:xmn∈ N/ L(ε,0)} ∈ I2
and
{m∈N:xmn∈ N/ L(ε,0)} ∈ I and {n∈N:xmn∈ N/ L(ε,0)} ∈ I for each fixedn∈Nand each fixedm∈N, respectively.
A useful interpretation of the above definition is the following;
xmn
F r(I2,I)
−→ L⇔FI2− lim
m,n→∞kxmn−Lk+0 = 0,
FI − lim
m→∞kxmn−Lk+0 = 0, (for each fixedn∈N) and
FI − lim
n→∞kxmn−Lk+0 = 0, (for each fixedm∈N).
Note thatF r(I2,I)− lim
m,n→∞kxmn−Lk+0 = 0 implies that FI2− lim
m,n→∞kxmn−Lk−α =FI2− lim
m,n→∞kxmn−Lk+α = 0, FI − lim
m→∞kxmn−Lk−α =FI2− lim
m→∞kxmn−Lk+α = 0, (for each fixed n∈N) and
FI − lim
n→∞kxmn−Lk−α =FI2− lim
n→∞kxmn−Lk+α = 0, (for each fixedm∈N) for eachα∈[0,1], since
0≤ kxmn−Lk−α ≤ kxmn−Lk+α ≤ kxmn−Lk+0 , (for eachm, n∈N),
0≤ kxmn−Lk−α ≤ kxmn−Lk+α ≤ kxmn−Lk+0 , (for each m∈Nand fixedn∈N) and
0≤ kxmn−Lk−α ≤ kxmn−Lk+α ≤ kxmn−Lk+0 , (for each n∈Nand fixedm∈N) holds for eachα∈[0,1] .
Example 2.1. LetI =Id,I2=Id2, (Rm,k.k) be a FNS and (xkn)mk,n=1∈Rm be a fixed nonzero vector, where the fuzzy norm onRm is defined as in (1) such that kxkC =
m P
k=1 m
P
n=1
|xkn|2 1/2
.Now we define the double sequence (xkn) inRmas
xkn=
n, if k≤2
x, if n=k=j2, j∈Nand k≥3 θ, otherwise.
It is clear that for any ε satisfying 0< ε ≤bkxkC, where 1< b <∞. Then, for k≥3 we have
K(ε) ={(n, k)∈N×N:kxnk−θk+0 ≥ε}={(9,9),(16,16),· · · }, K1(ε) ={n∈N:kxnk−θk+0 ≥ε}={9,16,· · · },
for eachk∈Nand
K2(ε) ={k∈N:kxnk−θk+0 ≥ε}={9,16,· · · }
for each n ∈ N and so, d2(K(ε)) = 0, d(K1(ε)) = 0 and d(K2(ε)) = 0. If we choose ε > bkxkC thenK(ε) =∅,K1(ε) =∅andK2(ε) =∅and so,d2(K(ε)) = 0, d(K1(ε)) = 0 andd(K2(ε)) = 0. It is clear that (xkn) is I2-convergent to 0 but (xkn) is notF r(I2,I)-convergent in (Rm,k.k).
Theorem 2.1. If a double sequence (xmn) inX isF r-convergent, then (xmn)is F r(I2,I)-convergent.
Proof. Let (xmn) be any double sequence in X and suppose that (xmn) be F r- convergent. Then, (xmn) is convergent in Pringsheim’s sense and the limits
F N− lim
m→∞xmn, (n∈N) andF N− lim
n→∞xmn, (m∈N),
exist for each fixed n ∈ N and each fixed m ∈ N, respectively. By Lemma 1.1, (xmn) isI2-convergent. Also, for eachε >0 there existm=m0(ε) andn=n0(ε) such that for allm > m0
kxmn−Lnk+0 < ε,
for someLn and each fixedn∈Nand also, for alln > n0
kxmn−Kmk+0 < ε,
for someKmand each fixedm∈N. Then, sinceI is an admissible ideal so for each ε >0, we have
{m∈N:kxmn−Lnk+0 ≥ε} ⊂ {1,2, . . . , m0} ∈ I, {n∈N:kxmn−Kmk+0 ≥ε} ⊂ {1,2, . . . , n0} ∈ I.
Hence, (xmn) isF r(I2,I)-convergent inX.
The opposite of this theorem is not always true. Let’s see this with an example.
Example 2.2. LetI =Id,I2=Id2, (Rm,k.k) be a FNS and (xkn)mk,n=1∈Rm be a fixed nonzero vector, where the fuzzy norm onRm is defined as in (1) such that kxkC =
m P
k=1 m
P
n=1
|xkn|2 1/2
.Now we define a double sequence (xkn) inRm as
xkn=
x, if n, k=j3, j∈N θ, otherwise.
It is clear that for anyε satisfying 0 < ε≤bkxkC, where 1< b < ∞. Then, we have
K(ε) ={(n, k)∈N×N:kxnk−θk+0 ≥ε}={(1,1),(8,8),(27,27),· · · }, K1(ε) ={n∈N:kxnk−θk+0 ≥ε}={1,8,27,· · · },
for eachk∈Nand
K2(ε) ={k∈N:kxnk−θk+0 ≥ε}={1,8,27,· · · }
for each n ∈ N and so, d2(K(ε)) = 0, d(K1(ε)) = 0 and d(K2(ε)) = 0. If we choose ε > bkxkC thenK(ε) =∅,K1(ε) =∅andK2(ε) =∅and so,d2(K(ε)) = 0, d(K1(ε)) = 0 andd(K2(ε)) = 0. Hence, (xkn) isF r(I2,I)-convergent in (Rm,k.k).
But (xkn) is notF r-convergent in (Rm,k.k).
Definition 2.3. A double sequence (xmn) inXis said to beF r(I2∗,I∗)-convergent with respect to fuzzy norm on X, if there exist the setsM ∈ F(I2), M1 ∈ F(I) andM2∈ F(I) (i.e.,N×N\M ∈ I2,N\M1∈ I andN\M2∈ I) such that the limits
F N− lim
m,n→∞
(m,n)∈M
xmn, F N− lim
m→∞
m∈M1
xmn andF N− lim
n→∞
n∈M2
xmn
exist for each fixedn∈Nand each fixedm∈N, respectively.
Theorem 2.2. If a double sequence (xmn)in X isF r(I2∗,I∗)-convergent, then it isF r(I2,I)-convergent.
Proof. Let (xmn) inX beF r(I2∗,I∗)-convergent. Then, it isI2∗-convergent and so, by Lemma 1.2, it is I2-convergent. Also, there exist the setsM1, M2∈ F(I) such that
(∀ε >0) (∃m0=m0(ε)∈N) (∀m≥m0) (m∈M1) kxmn−Lnk+0 < ε, (n∈N) for someLn ∈X and
(∀ε >0) (∃n0=n0(ε)∈N) (∀n≥n0) (n∈M2) kxmn−Kmk+0 < ε, (m∈N) for someKm∈X. Hence, for eachε >0 we have
A(ε) = {m∈N:kxmn−Lnk+0 ≥ε} ⊂H1∪ {1,2, . . . , m0−1}, (n∈N), B(ε) = {n∈N:kxmn−Kmk+0 ≥ε} ⊂H2∪ {1,2, . . . , n0−1}, (m∈N), forH1, H2∈ I. SinceI is an admissible ideal we get
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 (xmn) is
F r(I2,I)-convergent inX.
Theorem 2.3. Let I2⊂2N×N be a strongly admissible ideal with property (AP2), I ⊂ 2N be an admissible ideal with property (AP). If a double sequence (xmn) is F r(I2,I)-convergent, then(xmn) isF r(I2∗,I∗)-convergent inX.
Proof. Let a double sequence (xmn) inX be F r(I2,I)-convergent. Then, (xmn) is I2-convergent and so (xmn) isI2∗-convergent by Lemma 1.3. Also, for each ε >0 we have
A(ε) ={m∈N:kxmn−Lnk+0 ≥ε} ∈ I for someLn ∈X and for each fixedn∈Nand
C(ε) ={n∈N:kxmn−Kmk+0 ≥ε} ∈ I for someKm∈X and for each fixedm∈N.
Now put
A1 = {m∈N:kxmn−Lnk+0 ≥1}, Ak =
m∈N: 1
k ≤ kxmn−Lnk+0 < 1 k−1
fork≥2, for some Ln ∈X and for each fixed n∈N. It is clear thatAi∩Aj =∅ for i6=j and Ai ∈ I for each i ∈N. By the property (AP) there is a countable family of sets{B1, B2, . . .}inI such thatAj4Bj is a finite set for eachj∈Nand B=S∞
j=1Bj∈ I. We prove that
F N− lim
m→∞
m∈M
xmn=Ln,
for someLn, each fixedn∈NandM =N\B∈ F(I). Letδ >0 be given. Choose k∈Nsuch that 1/k < δ. Then, we have
{m∈N:kxmn−Lnk+0 ≥δ} ⊂
k
[
j=1
Aj,
for someLn and each fixedn∈N.SinceAj4Bjis a finite set forj∈ {1,2, . . . , k}, there existsm0∈Nsuch that
[k
j=1
Bj
∩ {m:m≥m0}=[k
j=1
Aj
∩ {m:m≥m0}.
Ifm≥m0 andm6∈B then m6∈
k
[
j=1
Bj and som6∈
k
[
j=1
Aj.
Thus, we have kxmn−Lnk+0 < 1k < δ, for some Ln and each fixed n ∈N. This implies that
F N− lim
m→∞
m∈M
xmn=Ln. Hence, we have
FI∗− lim
m→∞xmn=Ln for someLn and each fixedn∈N.
Similarly, for the setC(ε) ={n∈N:kxmn−Kmk+0 ≥ε} ∈ I, we have FI∗− lim
n→∞xmn=Km
for someKmand each fixedm∈N. Hence, a double sequence (xmn) isF r(I2∗,I∗)-
convergent.
Now, we give the definitions of F r(I2,I)-Cauchy sequence and F r(I2∗,I∗)- Cauchy sequence.
Definition 2.4. A double sequence (xmn) in X is said to be regularly (I2,I)- Cauchy double sequence with respect to fuzzy norm onX(F r(I2,I)-Cauchy double sequence), if it isI2-Cauchy double sequence with respect to fuzzy norm onX and for each ε > 0 there exist kn = kn(ε) ∈ N and lm = lm(ε) ∈ N such that the following statements hold:
A1(ε) = {m∈N:kxmn−xknnk+0 ≥ε} ∈ I, (n∈N), A2(ε) = {n∈N:kxmn−xmlmk+0 ≥ε} ∈ I, (m∈N).
Theorem 2.4. If a double sequence (xmn) in X is F r(I2,I)-convergent, then (xmn)isF r(I2,I)-Cauchy double sequence.
Proof. Let (xmn) be aF r(I2,I)-convergent double sequence inX. Then, (xmn) is I2-convergent and by Lemma 1.5, it is I2-Cauchy double sequence. Also for each ε >0, we have
A1
ε 2
=n
m∈N:kxmn−Lnk+0 ≥ ε 2
o∈ I
for someLn and each fixedn∈Nand also A2ε
2
=n
n∈N:kxmn−Kmk+0 ≥ ε 2
o∈ I
for someKmand each fixedm∈N. SinceI is an admissible ideal, the sets Ac1ε
2
=n
m∈N:kxmn−Lnk+0 < ε 2 o
,(n∈N)
for someLn and Ac2ε
2
=n
n∈N:kxmn−Kmk+0 < ε 2 o
,(m∈N)
for some Km, are nonempty and belong to F(I). For kn ∈ Ac1(ε2), (n ∈ N and kn>0) we have
kxknn−Lnk+0 <ε 2, for someLn. Now, for eachε >0, we define the set
B1(ε) ={m∈N:kxmn−xknnk+0 ≥ε}, (n∈N),
where kn =kn(ε)∈ N. Let m ∈B1(ε). Sincek.k+0 is a norm in the usual sense, then forkn∈Ac1(ε2), (n∈Nandkn>0) we have
ε≤ kxmn−xknnk+0 ≤ kxmn−Lnk+0 +kxknn−Lnk+0
< kxmn−Lnk+0 +ε 2, for someLn. This shows that
ε
2 <kxmn−Lnk+0 and so m∈A1ε 2
. Hence, we haveB1(ε)⊂A1(ε2).
Similarly, for eachε >0 and forlm∈Ac2(ε2) (m∈Nandlm>0) we have kxmlm−Kmk+0 <ε
2, (m∈N) for someKm. Therefore, it can be seen that
B2(ε) ={m∈N:kxmlm−Kmk+0 ≥ε} ⊂A2(ε 2).
Hence, we have B1(ε), B2(ε) ∈ I. This shows that (xmn) is F r(I2,I)-Cauchy
double sequence.
Definition 2.5. A double sequence (xmn) is said to be regularly (I2∗,I∗)-Cauchy double sequence with respect to fuzzy norm on X (F r(I2∗,I∗)-Cauchy double se- quence), if there exist the sets M ∈ F(I2), M1 ∈ F(I) and M2 ∈ F(I) (i.e., N×N\M ∈ I2,N\M1∈ I andN\M2∈ I), for eachε >0 there existN =N(ε), s=s(ε),t=t(ε), (s, t)∈M,kn=kn(ε),lm=lm(ε)∈Nsuch that
kxmn−xstk+0 < ε, for (m, n),(s, t)∈M,
kxmn−xknnk+0 < ε, for eachm∈M1 and each fixedn∈N, kxmn−xmlmk+0 < ε, for eachn∈M2and each fixedm∈N, wheneverm, n, s, t, kn, lm≥N.
Theorem 2.5. If a double sequence(xmn) in X isF r(I2∗,I∗)-Cauchy double se- quence, then it isF r(I2,I)-Cauchy double sequence.
Proof. Since a double sequence (xmn) inX isF r(I2∗,I∗)-Cauchy double sequence, it isI2∗-Cauchy double sequence. We know thatI2∗-Cauchy double sequence implies I2-Cauchy double sequence by Lemma 1.4. Also, there exist the setsM1, M2∈ F(I) and for eachε >0 there exist kn =kn(ε)∈Nandlm=lm(ε)∈Nsuch that
kxmn−xknnk+0 < ε, for eachm∈M1and each fixedn∈N, kxmn−xmlmk+0 < ε, for eachn∈M2and each fixedm∈N,
forN =N(ε)∈Nandm, n, kn, lm≥N.
Therefore, forH1=N\M1∈ I andH2=N\M2∈ I we have
A1(ε) ={m∈N:kxmn−xknnk+0 ≥ε} ⊂H1∪ {1,2, . . . , N −1}, (n∈N) form∈M1and
A2(ε) ={n∈N:kxmn−xmlmk+0 ≥ε} ⊂H2∪ {1,2, . . . , N−1}, (m∈N) forn∈M2. SinceI is an admissible ideal,
H1∪ {1,2, . . . , N −1} ∈ I andH2∪ {1,2, . . . , N−1} ∈ I.
Hence, we haveA1(ε), A2(ε)∈ I and (xmn) isF r(I2,I)-Cauchy double sequence.
Theorem 2.6. Let I2⊂2N×N be a strongly admissible ideal with property (AP2), I ⊂2Nbe an admissible ideal with property (AP). If a double sequence(xmn)inX isF r(I2,I)-Cauchy double sequence, then it isF r(I2∗,I∗)-Cauchy double sequence.
Proof. Since F r(I2,I)-Cauchy double sequence, it is I2-Cauchy double sequence.
We know that I2-Cauchy double sequence implies I2∗-Cauchy double sequence by Lemma 1.7. Also, for every ε >0 there existkn =kn(ε)∈Nand lm=lm(ε)∈N such that the following statements hold:
A1(ε) = {m∈N:kxmn−xknnk+0 ≥ε} ∈ I, (n∈N), A2(ε) = {n∈N:kxmn−xmlmk+0 ≥ε} ∈ I, (m∈N).
Let
Pi =
m∈N:kxmn−xknink+0 <1 i
; (i= 1,2, . . .) and
Ri=
n∈N:kxmn−xmlmik+0 < 1 i
; (i= 1,2, . . .),
where kni = kn(1\i) and lmi = lm(1\i). It is clear that Pi, Ri ∈ F(I), (i = 1,2, . . .). Since I has the property (AP), then by Lemma 1.6 there exist the sets P, R⊂Nsuch thatP, R∈ F(I) andP\PiandR\Riare finite for alli. Now, firstly we show that for everyε >0,
kxmn−xknnk+0 < ε, for each m∈P and each fixedn∈N.
To prove this, letε >0 andj ∈Nsuch thatj >2/ε. Ifm∈P thenP\Pi is a finite set, so there existsk=k(j) such thatm∈Pj for allm, kn> k(j). Therefore,
kxmn−xknink+0 <1
j and kxknn−xknink+0 < 1 j
for allm, n, kn> k(j). Sincek.k+0 is a norm in the usual sense, then it follows that kxmn−xknnk+0 ≤ kxmn−xknink+0 +kxknn−xknink+0
< 1 j +1
j =2 j < ε
for all m, n, kn > k(j). Thus, for any ε > 0 there exists k = k(ε) such that for m, n, kn> k(ε)
kxmn−xknnk+0 < ε, for each m∈P and each fixedn∈N.
Similarly, we can show that for any ε > 0 there exists l = l(ε) such that for m, n, lm> l(ε)
kxmn−xmlmk+0 < ε, for each n∈R and each fixedm∈N.
This shows that the sequence (xmn) is anI2∗-Cauchy double sequence.
Acknowledgments. The authors would like to thank the anonymous referee for his/her comments that helped us improve this article.
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Erdinc¸ D¨undar
Department of Mathematics, Faculty of Science and Literature, Afyon Kocatepe Uni- versity, 03200, Afyonkarahisar, Turkey
E-mail address:[email protected]
Muhammed Recai T¨urkmen
Department of Mathematics, Faculty of Education, Afyon Kocatepe University, 03200, Afyonkarahisar, Turkey
E-mail address:[email protected]
Nimet Pancaroˇglu Akın
Department of Mathematics, Faculty of Education, Afyon Kocatepe University, 03200, Afyonkarahisar, Turkey
E-mail address:[email protected]
