# Some Double Sequence Spaces of Fuzzy Real Numbers of Paranormed Type

Authors: Bipul Sarma
Journal: Journal of Mathematics
 (2013)

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## Abstract

We study different properties of convergent, null, and bounded double sequence spaces of fuzzy real numbers like completeness, solidness, sequence algebra, symmetricity, convergence-free, and so forth. We prove some inclusion results too.


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## Body

## 1. Introduction

Throughout the paper, a double sequence is denoted by〈Xnk〉, a double infinite array of elements Xnk, where each Xnk  is a fuzzy real number.The initial work on double sequences is found in Bromwich [1]. Later on, it was studied by Hardy [2], Móricz [3], Tripathy [4], Basarir and Sonalcan [5], and many others. Hardy [2] introduced the notion of regular convergence for double sequences.The concept of paranormed sequences was studied by Nakano [6] and Simons [7] at the initial stage. Later on, it was studied by many others.After the introduction of fuzzy real numbers, different classes of sequences of fuzzy real numbers were introduced and studied by Tripathy and Nanda [8], Choudhary and Tripathy [9], Tripathy et al. [10–13], Tripathy and Dutta [14–16], Tripathy and Borgogain [17],  Tripathy and Das [18], and many others.LetD denote the set of all closed and bounded intervals X=[a1,a2] on R, the real line. For  X, Y∈D, we define
(1)d(X,Y)=max(|a1-b1|,|a2-b2|),
where  X=[a1,a2] and  Y=[b1,b2]. It is known that (D,d)  is a complete metric space.Afuzzy real numberX  is a fuzzy set on  R, that is, a mapping  X:R→I(=[0,1]) associating each real number  t  with its grade of membership  X(t).Theα-level set [X]α of the fuzzy real number X, for 0<α≤1, is defined as  [X]α={t∈R:X(t)≥α}.The set of all upper semicontinuous, normal, and convex fuzzy real numbers is denoted byR(I), and throughout the paper, by a fuzzy real number, we mean that the number belongs to  R(I).LetX,Y∈R(I), and let the α-level sets be [X]α=[a1α,a2α],[Y]α=[b1α,b2α],andα∈[0,1]; the product of X and Y is defined by
(2)[X⊗Y]α=[mini,j∈{1,2}aiα·bjα,maxi,j∈{1,2}aiα·bjα].


## 2. Definitions and Preliminaries

Afuzzy real numberX is called convex if X(t)≥X(s)∧X(r)=min(X(s),X(t)), where  s<t<r.If there existst0∈R such that X(t0)=1, then the fuzzy real number X  is called normal.A fuzzy real numberX  is said to be upper semicontinuous if, for each ε>0, X-1([0,a+ε)), for all  a∈I,  is open in the usual topology of  R.The setR of all real numbers can be embedded in R(I). For r∈R, r¯∈R(I) is defined by
(3)r¯(t)={1,fort=r,0,fort≠r.Theabsolutevalue, |X| of  X∈R(I), is defined by (see, e.g., [19])
(4)|X|(t)=max{X(t),X(-t)}ift≥0,|X|(t)=0ift<0.A fuzzy real numberX is called nonnegative if X(t)=0, for all t<0. The set of all nonnegative fuzzy real numbers is denoted by R*(I).Letd¯:R(I)×R(I)→R  be defined by
(5)d¯(X,Y)=sup0≤α≤1d([X]α,[Y]α).Thend¯ defines a metric on R(I).The additive identity and multiplicative identity inR(I) are denoted by 0¯and1¯, respectively.A  sequence(Xk) of fuzzy real numbers is said to beconvergent to the fuzzy real number L  if, for every  ε>0, there exists  n0∈N such that d¯(Xk,L)<ε, for all  k≥n0.A sequence of fuzzy numbers(fn) converges to a fuzzy number f if both limn→∞[fn]α-=[f]α- and limn→∞[fn]α+=[f]α+ hold for every α∈(0,1] [20].A sequence(xn) of generalized fuzzy numbers converges weakly to a generalized fuzzy number x (and we write xn→wx) if distribution functions (xnl) converge weakly to xl and (xnr) converge weakly to xr [21].A double sequence(Xnk) of fuzzy real numbers is said to beconvergent in Pringsheim’s senseto the fuzzy real number  L  if, for every  ε>0, there exists  n0,  k0∈N such that d¯(Xnk,L)<ε, for all  n≥n0, k≥k0.A double sequence(Xnk) of fuzzy real numbers is said to beregularly convergentif it converges in Pringsheim’s sense, and the following limits exist:
(6)limnd¯(Xnk,Lk)=0,forsomeLk∈R(I),foreachk∈N,limkd¯(Xnk,Jn)=0,forsomeJn∈R(I),foreachn∈N.A fuzzy real number sequence(Xk) is said to be boundedif supk|Xk|≤μ, for some μ∈R*(I).Forr∈R and  X∈R(I),  we define
(7)rX(t)=X(r-1t)ifr≠0,rX(t)=0¯ifr=0.Throughout the paper2wF, (2ℓ∞)F, 2cF, (2c0)F,  2cFR, and (2c0R)F denote the classes of  all, bounded,  convergent in Pringsheim’s sense,null in Pringsheim’s sense,regularly convergent, and regularly null fuzzy real number sequences, respectively.A double sequence spaceEF is said to be solid (ornormal) if 〈Ynk〉∈EF, whenever |Ynk|≤|Xnk|, for all n, k∈N,  for some 〈Xnk〉∈EF.LetK={(ni,ki):i∈N;n1<n2<n3<⋯andk1<k2<k3<⋯}⊆N×N,  and let EF  be a double sequence space. A K-step space of EF is a sequence space
(8)λKE={〈Xniki〉∈2wF:〈Xnk〉∈EF}.Acanonical preimage of a sequence 〈Xniki〉∈EF  is a sequence 〈Ynk〉 defined as follows:
(9)Ynk={Xnkif(n,k)∈K,0,otherwise.Acanonical preimage of a step spaceλKE is a set of canonical preimages of all elements in λKE.A double sequence spaceEF  is said to be monotone if EF  contains the canonical preimage of all its step spaces.From the above definitions, we have the following remark.Remark 1.
A sequence spaceEF  is solid ⇒EF  is monotone.
A double sequence spaceEF is said to be symmetric if (Xπ(n,k))∈EF, whenever (Xnk)∈EF, where  π  is a permutation of  N×N.
A double sequence spaceEF is said to be sequence algebra if (Xnk⊗Ynk)∈EF, whenever  (Xnk), (Ynk)∈EF.
A double sequence spaceEF is said to be convergence-free if  (Ynk)∈EF,  whenever  (Xnk)∈EF,andXnk=0¯   implies that Ynk=0¯.
Sequences of fuzzy real numbers relative to the paranormed sequence spaces were studied by Choudhary and Tripathy [9].
In this paper, we introduce the following sequence spaces of fuzzy real numbers.
Letp=〈pnk〉 be a sequence of positive real numbers
(10)(2ℓ∞)F(p)={supn,k{d¯(Xnk,0¯)}pnk<∞〈Xnk〉∈2wF:supn,k{d¯(Xnk,0¯)}pnksupn,k{d¯(Xnk,0¯)}pnk<∞},2cF(p)={〈Xnk〉∈2wF:limn,k{d¯(Xnk,L)}pnk=0,〈Xnk〉∈2wF:limn,k{d¯(Xnk,L)}pnkforsomeL∈R(I)}.
ForL=0¯, we get the class (2cF)0(p).
Also a fuzzy sequence〈Xnk〉∈2cFR(p) if  〈Xnk〉∈2cF(p), and the following limits exist:
(11)limn{d¯(Xnk,Lk)}pnk=0,forsomeLk∈R(I),foreachk∈N,limk{d¯(Xnk,Jn)}pnk=0,forsomeJn∈R(I),foreachn∈N.
For the class of sequences(2cFR)0(p), L=Lk=Jn=0.
We definemF(p)=2cFR(p)∩(2ℓ∞)F(p), (m0)F(p)=(2cFR)0(p)∩(2ℓ∞)F(p).


## 3. Main  Results

Theorem 2.
Let〈pnk〉 be bounded. Then, the classes of sequences (2ℓ∞)F(p), 2cFR(p),  (2cFR)0(p), mF(p), and (m0)F(p) are complete metric spaces with respect to the metric defined by
(12)f(X,Y)=supn,k{d¯(Xnk,Ynk)}pnk/H,whereH=max(1,suppnk).Proof.
We prove the result for2ℓ∞(p). Let  〈Xnki〉 be a Cauchy sequence in (2ℓ∞)F(p). Then, for a given ε>0, there exists m0∈N such that
(13)f(Xnki,Xnkj)<ε<εpnk/H,∀i,j≥m0,⇒d¯(Xnki,Xnkj)<ε,∀i,j≥m0⇒〈Xnkj〉j=1∞isaCauchysequenceoffuzzyrealnumberforeachn,k∈N.
SinceR(I) is complete, there exist fuzzy numbers  Xnk such that  limj→∞Xnkj=Xnk, for each n, k∈N.
Takingj→∞ in (13), we have
(14)f(Xnki,Xnk)<ε.
Using the triangular inequality(15)f(〈Xnk〉,0¯)≤f(〈Xnk〉,〈Xnkj〉)+f(〈Xnkj〉,0¯),
we have 〈Xnk〉∈(2ℓ∞)F(p).  Hence, (2ℓ∞)F(p) is complete.Property 1.
The space(2ℓ∞)F(p) is symmetric, but the spaces 2cF(p), 2cFR(p), (2cF)0(p), (2cFR)0(p), mF(p), and (m0)F(p) are not symmetric.Proof.
Obviously the space(2ℓ∞)F(p) is symmetric. For the other spaces, consider the following example.
Example  3. Consider the sequence space 2cF(p). Let  p1k=2,  for all k∈N and pnk=3, otherwise. Let the sequence 〈Xnk〉 be defined by
(16)X1k=1¯∀k∈N,
and  for n>1,
(17)Xnk(t)={t+2,for-2≤t≤-1,-t,for-1≤t≤0,0,otherwise.
Let〈Ynk〉 be a rearrangement of  〈Xnk〉 defined by
(18)Ynn=1¯,
and for n≠k,
(19)Ynk(t)={t+2,for-2≤t≤-1,-t,for-1≤t≤0,0,otherwise.
Then,〈Xnk〉∈2cF(p), but 〈Ynk〉∉2cF(p). Hence, 2cF(p) is not symmetric. Similarly, it can be established that the other spaces are also not symmetric.Theorem 4.
The spaces
(
2
ℓ
∞
)
F
(
p
), (2c0)F(p), (2c0R)F(p),  and(m0)F(p)are solid.Proof.
Consider the sequence space(2ℓ∞)F(p). Let 〈Xnk〉∈(2ℓ∞)F(p), and let 〈Ynk〉 be such that  d¯(Ynk,0¯)≤d¯(Xnk,0¯).
The result follows from the inequality(20){d¯(Ynk,0¯)}pnk≤{d¯(Xnk,0¯)}pnk.
Hence, the space(2ℓ∞)F(p) is solid. Similarly, the other spaces are also solid.Property 2.
The spaces2cF(p), (2cR)F(p), and mF(p) are not monotone and hence are not solid.Proof.
The result follows from the following example.
Example 5.  Consider the sequence space 2cF(p). Let  pnk=3  for  n+k even and pnk=2, otherwise. Let J={(n,k):n+kiseven}⊆N×N. Let  〈Xnk〉  be defined by the following:
for alln, k∈N,
(21)Xnk(t)={t+3,for-3≤t≤-2,nt(3n-1)-1+3n(3n-1)-1,for-2≤t≤-1+n-1,0,otherwise.
Then,  〈Xnk〉∈2cF(p). Let 〈Ynk〉 be the canonical preimage of  〈Xnk〉J for the subsequence J of N. Then,
(22)Ynk={Xnk,for(n,k)∈J,0¯,otherwise.
Then,  〈Ynk〉∉2cF(p). Thus, 2cF(p) is not monotone. Similarly, the other spaces are also not monotone. Hence, the spaces 2cF(p), (2cR)F(p), and mF(p) are not solid.Property 3.
The spaces(2ℓ∞)F(p), 2cF(p), (2c0)F(p), (2cR)F(p), (2c0R)F(p), mF(p),  and  (m0)F(p)   are not convergence-free.The result follows from the following example.Example 6.
Consider the sequence space2cF(p). Let  p1k=1, for all k∈N,  pnk=3, otherwise. Consider the sequence 〈Xnk〉 defined by
(23)X1k=0¯,
and for other values,
(24)Xnk(t)={t+2,for-2≤t≤-1,-nt(n+1)-1+(n+1)-1,for-1≤t≤n-1,0,otherwise.
Let the sequence〈Ynk〉 be defined by
(25)Y1k=0¯,
and for other values,
(26)Ynk(t)={1,for0≤t≤1,(n-1)(n-1)-1,for1≤t≤n,0,otherwise.
Then, 〈Xnk〉∈2cF(p), but 〈Ynk〉∉2cF(p). Hence, the space 2cF(p) is not convergence-free. Similarly, the other spaces are also not convergence-free.Theorem 7.
Z
(
p
)
⊆
(
2
ℓ
∞
)
F
(
p
),  for Z=(2cR)F, (2c0R)F,  mF, (m0)F. The inclusions are strict.Proof.
Since convergent sequences are bounded, the proof is clear.Theorem 8.
Let0<qij≤pij<∞,  for all  i,j∈N.  Then,  Z(p)⊆Z(q)   for  Z=2cF,  (2cR)F, (2cF)0, (2c0R)F,  mF, (m0)F.Proof.
Consider the sequence spaces2cF(p) and 2cF(q). Let 〈Xnk〉∈2cF(p).
Then,{d¯(Xnk,L)}pnk<ε, for all n≥n0,k≥k0.
The result follows from the inequality{d¯(Xnk,L)}qnk≤{d¯(Xnk,L)}pnk.Theorem 9.
The spaces
(
2
ℓ
∞
)
F
(
p
), 2cF(p), (2c0)F(p), (2cR)F(p), (2c0R)F(p), mF(p),   and(m0)F(p)   are sequence algebras.Proof.
Consider the sequence space(2c0)F(p). Let 〈Xnk〉, 〈Ynk〉∈(2c0)F(p). Then, the result follows immediately from the inequality
(27){d¯(XnkYnk,0¯)}pnk≤{d¯(Xnk,0¯)}pnk{d¯(Ynk,0¯)}pnk.



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*Source: 627047-2013-02-13.xml*