JMATH Journal of Mathematics 2314-4785 2314-4629 Hindawi Publishing Corporation 627047 10.1155/2013/627047 627047 Research Article Some Double Sequence Spaces of Fuzzy Real Numbers of Paranormed Type Sarma Bipul D'Urso Pierpaolo 1 Department of Mathematics Madhab Choudhury College-Gauhati University Barpeta Assam 781301 India gauhati.ac.in 2013 13 2 2013 2013 07 11 2012 03 01 2013 16 01 2013 2013 Copyright © 2013 Bipul Sarma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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 . Later on, it was studied by Hardy , Móricz , Tripathy , Basarir and Sonalcan , and many others. Hardy  introduced the notion of regular convergence for double sequences.

The concept of paranormed sequences was studied by Nakano  and Simons  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 , Choudhary and Tripathy , Tripathy et al. , Tripathy and Dutta , Tripathy and Borgogain , Tripathy and Das , and many others.

Let D denote the set of all closed and bounded intervals X=[a1,a2] on R, the real line. For X, YD, 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.

A fuzzy real number X is a fuzzy set on R, that is, a mapping X:RI(=[0,1]) associating each real number t with its grade of membership X(t).

The α-level set [X]α of the fuzzy real number X, for 0<α1, is defined as [X]α={tR:X(t)α}.

The set of all upper semicontinuous, normal, and convex fuzzy real numbers is denoted by R(I), and throughout the paper, by a fuzzy real number, we mean that the number belongs to R(I).

Let X,YR(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)[XY]α=[mini,j{1,2}aiα·bjα,maxi,j{1,2}aiα·bjα].

2. Definitions and Preliminaries

A fuzzy real number X is called convex if X(t)X(s)X(r)=min(X(s),X(t)), where s<t<r.

If there exists t0R such that X(t0)=1, then the fuzzy real number X is called normal.

A fuzzy real number X is said to be upper semicontinuous if, for each ε>0, X-1([0,a+ε)), for all aI, is open in the usual topology of R.

The set R of all real numbers can be embedded in R(I). For rR, r¯R(I) is defined by (3)r¯(t)={1,fort=r,0,fortr.

The absolute value, |X| of XR(I), is defined by (see, e.g., ) (4)|X|(t)=max{X(t),X(-t)}ift  0,|X|(t)=0ift<0.

A fuzzy real number X is called nonnegative if X(t)=0, for all t<0. The set of all nonnegative fuzzy real numbers is denoted by R*(I).

Let d¯:R(I)×R(I)R be defined by (5)d¯(X,Y)=sup0α1d([X]α,[Y]α).

Then d¯ defines a metric on R(I).

The additive identity and multiplicative identity in R(I) are denoted by 0¯    and    1¯, respectively.

A sequence (Xk) of fuzzy real numbers is said to be convergent to the fuzzy real number L if, for every ε>0, there exists n0N such that d¯(Xk,L)<ε, for all kn0.

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] .

A sequence (xn) of generalized fuzzy numbers converges weakly to a generalized fuzzy number x (and we write xnwx) if distribution functions (xnl) converge weakly to xl and (xnr) converge weakly to xr .

A double sequence (Xnk) of fuzzy real numbers is said to be convergent in Pringsheim’s sense to the fuzzy real number L if, for every ε>0, there exists n0,  k0N such that d¯(Xnk,L)<ε, for all nn0, kk0.

A double sequence (Xnk) of fuzzy real numbers is said to be regularly convergent if it converges in Pringsheim’s sense, and the following limits exist: (6)limnd¯(Xnk,Lk)=0,for  some  LkR(I),for  each  kN,limkd¯(Xnk,Jn)=0,for  some  JnR(I),for  each  nN.

A fuzzy real number sequence (Xk) is said to be bounded if supk|Xk|μ, for some μR*(I).

For rR and XR(I), we define (7)rX(t)=X(r-1t)ifr0,rX(t)=0¯ifr=0.

Throughout the paper 2wF, (2)F, 2cF, (2c0)F,  2cFR, and (2c0R)F denote the classes of all, bounded, convergent in Pringsheims sense, null in Pringsheim’s sense, regularly convergent, and regularly null fuzzy real number sequences, respectively.

A double sequence space EF is said to be solid (or normal) if YnkEF, whenever |Ynk||Xnk|, for all n, kN, for some XnkEF.

Let K={(ni,ki):iN;n1<n2<n3<  and    k1<k2<k3<}N×N, and let EF be a double sequence space. A K-step space of EF is a sequence space (8)λKE={Xniki2wF:XnkEF}.

A canonical preimage of a sequence XnikiEF is a sequence Ynk defined as follows: (9)Ynk={Xnkif(n,k)K,0,otherwise.

A canonical preimage of a step space λKE is a set of canonical preimages of all elements in λKE.

A double sequence space EF 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 space EF is solid EF is monotone.

A double sequence space EF is said to be symmetric if (Xπ(n,k))EF, whenever (Xnk)EF, where π is a permutation of N×N.

A double sequence space EF is said to be sequence algebra if (XnkYnk)EF, whenever (Xnk), (Ynk)  EF.

A double sequence space EF is said to be convergence-free if (Ynk)EF, whenever (Xnk)EF, and Xnk=0¯ implies that Ynk=0¯.

Sequences of fuzzy real numbers relative to the paranormed sequence spaces were studied by Choudhary and Tripathy .

In this paper, we introduce the following sequence spaces of fuzzy real numbers.

Let p=pnk be a sequence of positive real numbers (10)(2)F(p)={supn,k{d¯(Xnk,0¯)}pnk<Xnk2wF:supn,k{d¯(Xnk,0¯)}pnksupn,k{d¯(Xnk,0¯)}pnk<},2cF(p)={Xnk2wF:limn,k{d¯(Xnk,L)}pnk=0,Xnk2wF:limn,k{d¯(Xnk,L)}pnkfor  someLR(I)}.

For L=0¯, we get the class (2cF)0(p).

Also a fuzzy sequence Xnk2cFR(p) if Xnk2cF(p), and the following limits exist: (11)limn{d¯(Xnk,Lk)}pnk=0,  for  some  LkR(I),for  each  kN,limk{d¯(Xnk,Jn)}pnk=0,  for  some  JnR(I),for  each  nN.

For the class of sequences (2cFR)0(p), L=Lk=Jn=0.

We define mF(p)=2cFR(p)(2)F(p), (m0)F(p)=(2cFR)0(p)(2)F(p).

3. Main Results Theorem 2.

Let pnk be bounded. Then, the classes of sequences (2)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,where  H=max(1,suppnk).

Proof.

We prove the result for 2(p). Let Xnki be a Cauchy sequence in (2)F(p). Then, for a given ε>0, there exists m0N such that (13)f(Xnki,Xnkj)<ε<εpnk/H,i,jm0,d¯(Xnki,Xnkj)<ε,    i,jm0Xnkjj=1is  a  Cauchy  sequence  of  fuzzyreal  number  for  each  n,kN.

Since R(I) is complete, there exist fuzzy numbers Xnk such that limjXnkj=Xnk, for each n, kN.

Taking j 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(2)F(p). Hence, (2)F(p) is complete.

Property 1.

The space (2)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 (2)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 kN and pnk=3, otherwise. Let the sequence Xnk be defined by (16)X1k=1¯kN, and for n>1, (17)Xnk(t)={t+2,for-2t-1,-t,for-1t0,0,otherwise.

Let Ynk be a rearrangement of Xnk defined by (18)Ynn=1¯, and for nk, (19)Ynk(t)={t+2,for-2t-1,-t,for-1t0,0,otherwise.

Then, Xnk2cF(p), but Ynk2cF(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 ) F ( p ) , (2c0)F(p), (2c0R)F(p),  and (m0)F(p) are solid.

Proof.

Consider the sequence space (2)F(p). Let Xnk(2)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 (2)F(p) is solid. Similarly, the other spaces are also solid.

Property 2.

The spaces 2cF(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+kis  even}N×N. Let Xnk be defined by the following:

for all n, kN, (21)Xnk(t)={t+3,for-3t-2,nt(3n-1)-1+3n(3n-1)-1,for-2t-1+n-1,0,otherwise. Then, Xnk2cF(p). Let Ynk be the canonical preimage of XnkJ for the subsequence J of N. Then, (22)Ynk={Xnk,for(n,k)J,0¯,otherwise. Then, Ynk2cF(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 (2)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 space 2cF(p). Let p1k=1, for all kN, pnk=3, otherwise. Consider the sequence Xnk defined by (23)X1k=0¯, and for other values, (24)Xnk(t)={t+2,for-2t-1,-nt(n+1)-1+(n+1)-1,for-1tn-1,0,otherwise.

Let the sequence Ynk be defined by (25)Y1k=0¯, and for other values, (26)Ynk(t)={1,for0t1,(n-1)(n-1)-1,for1tn,0,otherwise. Then, Xnk2cF(p), but Ynk2cF(p). Hence, the space 2cF(p) is not convergence-free. Similarly, the other spaces are also not convergence-free.

Theorem 7.

Z ( p ) ( 2 ) 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.

Let 0<qijpij<,  for all i,jN. Then, Z(p)Z(q)   for  Z=2cF,  (2cR)F, (2cF)0, (2c0R)F,  mF, (m0)F.

Proof.

Consider the sequence spaces 2cF(p) and 2cF(q). Let Xnk2cF(p).

Then, {d¯(Xnk,L)}pnk<ε, for all nn0,kk0.

The result follows from the inequality {d¯(Xnk,L)}qnk{d¯(Xnk,L)}pnk.

Theorem 9.

The spaces ( 2 ) 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.

Acknowledgment

The author’s work is supported by UGC Project no. F. 5-294/2009-10 (MRP/NERO).

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