# Duals and Matrix Classes Involving Cesàro Type Classes of Sequences of Fuzzy Numbers

Authors: Hemen Dutta; Jyotishmaan Gogoi
Journal: Advances in Fuzzy Systems
 (2018)

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## Abstract

We first define Cesàro type classes of sequences of fuzzy numbers and equip the set with a complete metric. Then we compute the Köthe-Toeplitz dual and characterize some related matrix classes involving such classes of sequences of fuzzy numbers.


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## Body

## 1. Introduction

In 1965, Zadeh [1] introduced the concept of fuzzy sets and fuzzy set operations as an extension of the classical notion of the set theory. Later on several authors have discussed different aspects of the theory of fuzzy sets and applied it in various areas of science and engineering such as fuzzy topological spaces, similarity relations and fuzzy orderings, fuzzy possibility theory, fuzzy measures of fuzzy events, and fuzzy mathematical programming. Nowadays, fuzzy set theory is used as a powerful mathematical tool in solving complex real life problems which yields a notion of uncertainty and vagueness. Matloka [2] introduced bounded and convergent sequences of fuzzy numbers and studied their properties. In [3], Nanda studied sequences of fuzzy numbers and proved that every Cauchy sequence of fuzzy numbers is convergent. Since then, different classes of sequences of fuzzy numbers were introduced and studied by various authors. For the works on convergence of fuzzy sequences and series, we refer to Nuray and Savaş [4], Diamond and Kloeden [5], Matloka [2], Esi [6], Kaleva [7], Nanda [8],[3], Dubois and Prade [9], Altınok, Çolak, and Altın [10], Stojaković and Stojaković [11],[12], and Mursaleen, Srivastava, and Sharma [13]. In [14], Subrahmanyam defined the Cesàro summability of sequences of fuzzy numbers and proved some related Tauberian theorems. Some interesting results related to Cesàro summability method of sequences of fuzzy numbers and the Tauberian conditions which guarantee the convergence of summable sequences of fuzzy numbers can be found in Subrahmanyam [14], Talo and Çakan [15], Altın, Mursaleen, and Altınok [16], and Yavuz [17],[18].Definition 1 (Goetschel and Voxman [19]).
A fuzzy number is a fuzzy set on the real axis, i.e., a mappingv:R→[0,1], which satisfies the following four conditions:(i)
v is normal; i.e., there exists an x0∈R such that vx0=1.(ii)
v is fuzzy convex; i.e., vλx+1-λy≥min⁡{vx,vy} for all x,y∈Randforallλ∈0,1.(iii)
v is upper semicontinuous.(iv)
The set[v]0={x∈R:v(x)>0}- is compact, where {x∈R:v(x)>0}- denotes the closure of the set {x∈R:v(x)>0} in the usual topology of R.
We denote the set of all fuzzy numbers onR by E1 and called it the space of fuzzy numbers. λ-level set [v]λ of v∈E1 is defined by vλ=t∈R:vt≥λ,0<λ≤1,{t∈R:v(t)>λ,}-,λ=0.
The setvλ is a closed, bounded, and nonempty interval for each λ∈[0,1] which is defined by vλ=v-λ,vλ.  R can be embedded in E1, since each r∈R can be regarded as a fuzzy number r- defined as(1)r-t=1,t=r,0,t≠r.Definition 2 (Talo and Başar [20]).
Letx,y,z∈E1 and k∈R. Then the operations addition, scalar multiplication, and product are defined on E1 by(2)x+y=z⇔zλ=xλ+yλ⟺z-λ=x-λ+y-λ,z+λ=x+λ+y+λkxλ=kxλ∀λ∈0,1and (3)xy=z⇔zλ=xλyλ∀λ∈0,1,where it is immediate that(4)z-λ=min⁡x-λy-λ,x-λy+λ,x+λy-λ,x+λy+λand(5)z+λ=max⁡x-λy-λ,x-λy+λ,x+λy-λ,x+λy+λfor all λ∈0,1.Definition 3 (Talo and Başar [20]).
LetW be the set of all closed bounded intervals A of real numbers such that A=[A1,A2]. Define the relation d on W as follows:(6)dA,B=max⁡A1-B1,A2,B2,where B=[B1,B2]∈W. Then W,d is a complete metric space (see Diamond and Kloeden [5], Nanda [8]). Talo and Başar [20] defined the metric D on E1 by means of Hausdorff metric d as(7)Dx,y=supλ∈0,1⁡dxλ,yλ=supλ∈0,1⁡max⁡x-λ-y-λ,x+λ-y+λ
The partial ordering relation onE1 is defined as follows:(8)x≼y⇔xλ≼yλ⇔x-λ≤y-λandx+λ≤y+λ∀λ∈0,1.Definition 4 (Talo and Başar [20]).
x ∈ E 1 is a nonnegative fuzzy number if and only if xx0=0 for all x0<0. It is immediate that x≽0- if u is a nonnegative fuzzy number.
One can see that(9)Dx,0-=supλ∈0,1⁡maxx-λ,x+λ=maxx-0,x+0Lemma 5 (Bede and Gal [21]).
L e t  x , y , z ∈ E 1 andk∈R. Then (i)
E1,D is a complete metric space.(ii)
Dkx,ky=kDx,y.(iii)
Dx+y,z+y=Dx,z.(iv)
Dx+y,z+u≤Dx,z+Dy,u.(v)
D(x,0-)-D(y,0-)≤Dx,y≤Dx,0-+Dy,0-.Lemma 6 (Talo and Başar [20]).
The following statements hold:(i)
Dxy,0-≤Dx,0-Dy,0-forallx,y∈E1.(ii)
Ifxk→x,ask→∞thenDxk,0-→Dx,0-ask→∞.ByWFwe denote the set of all single sequences of fuzzy numbers on R.Matloka [2] introduced bounded and convergent sequences of fuzzy numbers and studied their properties. We now quote the following definitions given by Talo and Başar [20] which we will use in a later part of this paper.Definition 7.
A sequence of fuzzy numbers(xk) is said to be bounded if the set of fuzzy numbers consisting of the terms of the sequence xk is a bounded set. That is to say that a sequence xk∈WF is said to be bounded if and only if there exist two fuzzy numbers m and M such that m≼xk≼M for all k∈N. This means that m-(λ)≤xk-(λ)≤M-(λ) and m+(λ)≤xk+(λ)≤M+(λ) for all λ∈0,1.
The fact that the boundedness of the sequencexk∈WF is equivalent to the uniform boundedness of the functions xk-(λ) and xk+(λ) on 0,1. Therefore, one can say that the boundedness of the sequence xk∈WFis equivalent to the fact that (10)supk∈N⁡Dxk,0-=supk∈N⁡supλ∈0,1⁡max⁡xk-λ,xk+λ<∞.Definition 8.
Consider the sequence of fuzzy numbersxk∈WF. If for every ε>0 there exists n0=n0ε∈N and l∈E1suchthatDxk,l<ε for all k>n0, then we say that the sequence is said to be convergent to the limit l and write(11)limk→∞⁡Dxk,l=0,
and we have the setsl∞F,CF,C0F consisting of the bounded, convergent, and convergent to 0- sequences of fuzzy numbers (Talo and Başar [20]) as follows:l∞F=xk∈WF:supk⁡Dxk,0-<∞.CF=xk∈WF:∃l∈E1suchthatlimk→∞⁡D(xk,l)=0.C0F=xk∈WF:limk→∞⁡D(xk,0-)=0.
Throughout the text, the summations without limits run from0 to ∞; for example,(12)∑kxkmeans that(13)∑k=0∞xk.Definition 9 (Talo and Başar [20]).
Letxk∈WF. Then the expression(14)∑kxkis called a series corresponding to the sequence xk of fuzzy number. We denote (15)sn=∑k=1nxk∀n∈N.
If the sequence (sn) converges to a fuzzy number x, then we say that the series (16)∑kxkconverges to x and write (17)∑kxk=x,which implies as n→∞ that(18)∑k=0nxk-λ→x-λand∑k=0nxk+λ→x+λ,uniformly in λ∈0,1. Conversely, if the fuzzy numbers xk=xk-λ,xk+λ:λ∈0,1, (19)∑kxk-λ=x-λ,and (20)∑kxk+λ=x+λconverge uniformly in λ∈0,1, then x={(x-λ,x+λ:λ∈[0,1]} defines a fuzzy number such that x=∑k=0∞xk.
Otherwise, we say the series of fuzzy numbers diverges. Additionally, if the sequencesn is bounded then we say that the series (21)∑kxkof fuzzy numbers is bounded.Definition 10 (Talo and Başar [20]).
LetμF be a space of convergent sequences of fuzzy numbers. The sum of a series(22)∑kxkwith respect to this rule is defined by(23)limn→∞⁡∑kxk.Definition 11.
Following Khan and Rahman [22], we define the Cesàro sequence space Cesp,qnF as follows:
Ifq=(qn) is a positive sequence of real numbers, then, for 1<p<∞,(24)Cesp,qnF=x=xk∈WF:∑r=0∞1Q2r∑rqkDxk,0-p<∞where Q2r=q2r+q2r+1+⋯+q2r+1-1 and ∑r denotes summation over the range 2r≤k<2r+1. If qn=1 for all n∈N, then Cesp,qnF reduces to CespF defined by(25)CespF=x=xk∈WF:∑r=0∞12r∑rDxk,0-p<∞.Following Maddox [23], throughout the paper we use the following inequality.
For anyG>0 and a,b∈E1 we have(26)Dab,0-≤Da,0-Db,0-≤GDa,0-tG-t+Db,0-p,where p>1 and 1/p+1/t=1.
The classical analogy ofCespF was introduced and studied by Lim [24].
The classical Cesàro sequence space and its algebraic dual and related matrix transformations were introduced and studied by various authors like Shiue [25], Leibowitz [26], Lim [24], Khan and Khan [27],[28], Khan and Rahman [22], Johnson and Mohapatra [29], Rahman and Karim [30], etc.
The main purpose of this paper is to define and study the Cesàro sequence spaceCesp,qnF and determine the Köthe-Toeplitz dual and give some related matrix transformations.


## 2. Complete Metric Structure

We equipCesp,qnF with a metric and show that this set is complete with respect to the metric defined in the following theorem.Theorem 12.
C e s p , q n F is complete with the metric d∗ defined by(27)d∗x,y=∑r=0∞1Q2r∑rqkDxk,ykp1/p,where x=xk,y=yk∈Cesp,qnF.Proof.
We first show that(Cesp,qnF,d∗) is a metric space.
It is obvious thatd∗x,y=0⇔x=y and d∗x,y=d∗y,x.
Now we prove the triangle inequality.
Supposex=xk,y=yk,z=zk∈Cesp,qnF.(28)d∗x,z=∑r=0∞1Q2r∑rqkDxk,zkp1/p≤∑r=0∞1Q2r∑rqkDxk,yk+Dyk,zkp1/p≤∑r=0∞1Q2r∑rqkDxk,yk+1Q2r∑rqkDyk,zkp1/pThen, using Minkowski’s inequality,(29)d∗x,z≤∑r=0∞1Q2r∑rqkDxk,ykp1/p+∑r=0∞1Q2r∑rqkDyk,zkp1/p=d∗x,y+d∗y,zNext, to show that Cesp,qnF is complete under d∗, let us consider that (xi) is a Cauchy sequence in Cesp,qnF. Then, for given ε>0, there exists k0>0 such that(30)d∗xi,xj<ε∀i,j≥k0⇒∑r=0∞1Q2r∑rqkDxki,xkjp1/p<ε,which implies that Dxki,xkj<ε∀i,j>k0; that is, xki is a Cauchy sequence in E1. So xki converges to a limit, say xk∈E1; i.e., limi→∞⁡xki=xk,∀k∈N.
Supposex=xk. For given ε>0, there exists k0>0 such that, for any t∈N,(31)∑r=0t1Q2r∑rqkDxki,xkjp1/p≤d∗xi,xj<ε∀i,j≥k0Letting j→∞, we obtain(32)∑r=0t1Q2r∑rqkDxki,xkp1/p<ε∀i≥k0Since t is arbitrary, letting t→∞, we obtain(33)∑r=0∞1Q2r∑rqkDxki,xkp1/p<ε∀i≥k0which implies that d∗xi,x<ε,∀i≥k0.
Next we show thatx=xk∈Cesp,qnF.
We have thatd∗xi,0- is bounded in Cesp,qnF; i.e., there exists K>0 such that d∗x,0-≤K. Now, for any t∈N,(34)∑r=0t1Q2r∑rqkDxki,0-p1/p≤d∗xi,0-≤K∀i≥k0Now,(35)∑r=0t1Q2r∑rqkDxk,0-p1/p≤∑r=0t1Q2r∑rqkDxk,xki+Dxki,0-p1/p≤∑r=0t1Q2r∑rqkDxk,xkip1/p+∑r=0t1Q2r∑rqkDxki,0-p1/p.Letting t→∞, we obtain(36)∑r=0∞1Q2r∑rqkDxk,0-p1/p≤ε+K<∞.This implies that x=xk∈Cesp,qnF.
This step completes the proof.


## 3. Computation of the Köthe-Toeplitz Dual

Definition 13 (Talo and Başar [20]).
The Köthe-Toeplitz dual or theα-dual of a set μF⊂WF, denoted by {μF}α, is defined as follows:(37)μFα≔xk∈WF:xkyk∈l1F∀yk∈μFwhere l1F denotes the absolutely summable sequences of fuzzy numbers defined as follows:(38)l1F=xk∈WF:∑kDxk,0-<∞.We now give the following theorem by which the Köthe-Toeplitz dual Cesp,qnFα of Cesp,qnF will be determined.Theorem 14.
If1<p<∞ and 1/p+1/t=1, then(39)Cesp,qnFα=a=ak:∑r=0∞Q2rmaxr⁡Dak,0-qktG-t<∞forsomeintegerG>1.Proof.
Let1<p<∞ and 1/p+1/t=1.
We define(40)μtF=a=ak:∑r=0∞Q2rmaxr⁡Dak,0-qktG-t<∞forsomeintegerG>1.We want to show that Cesp,qnFα=μtF.
Letx=xk∈Cesp,qnF and a=ak∈μtF. Then, using inequality (26) together with Lemma 6, we get(41)∑k=1∞Dakxk,0-=∑r=0∞∑rDakxk,0-≤∑r=0∞∑rDak,0-Dxk,0-=∑r=0∞∑r1qkDak,0-qkDxk,0-≤∑r=0∞maxr⁡Dak,0-qk∑rqkDxk,0-=∑r=0∞Q2rmaxr⁡Dak,0-qk1Q2r∑rqkDxk,0-≤G∑r=0∞Q2rmaxr⁡Dak,0-qktG-t+1Q2r∑rqkDxk,0-p=G∑r=0∞Q2rmaxr⁡Dak,0-qktG-t+∑r=0∞1Q2r∑rqkDxk,0-p<∞,which implies that a=ak∈Cesp,qnFα. Thus μtF⊆Cesp,qnFα.
Conversely, suppose that(42)∑k=1∞Dakxk,0-<∞for all x=xk∈Cesp,qnF but a∉μtF. Then, for every G>1,(43)∑r=0∞Q2rmaxr⁡Dak,0-qktG-t=∞.So, following Khan and Rahman [22], we can define a sequence 0=n0<n1<n2<… such that γ=0,1,2,… and we have(44)Mγ=∑r=nγnγ+1-1Q2rmaxr⁡Dak,0-qktγ+2-t/p>1.Now we define a sequence x=xk as follows:(45)xNr=Q2rtANrt1DaNr,0-γ+2-tMγ-1fornγ≤r≤nγ+1-1,γ=0,1,2,…,and xk=0-∀k≠Nr, where Nr is such that(46)ANr/DaNr,0-<L,forsomeL>0and(47)ANr=maxr⁡Dak,0-qk,where the maximum is taken with respect to k in 2r,2r+1.
Therefore,(48)∑k=2nγ2nγ+1-1Dakxk,0-=∑r=nγnγ+1-1Q2rtANrtγ+2-tMγ-1=γ+2-1Mγ-1∑r=nγnγ+1-1Q2rmaxr⁡Dak,0-qktγ+2-t/p=γ+2-1Mγ-1Mγ=γ+2-1.It follows that (49)∑k=1∞Dakxk,0-=∑γ=0∞γ+2-1is divergent. Moreover(50)∑r=nγnγ+1-11Q2r∑rqkDxk,0-p≤∑r=nγnγ+1-1LQ2rt-1maxr⁡Dak,0-qkt-1γ+2-tMγ-1pNow(51)∑r=nγnγ+1-1Q2rt-1pmaxr⁡Dak,0-qkt-1pγ+2tpMγ-p=γ+2-2Mγ-1∑r=nγnγ+1-1Q2rmaxr⁡Dak,0-qktγ+2-t/pMγ1-pγ+21-p=γ+2-2Mγ-1MγMγ1-pγ+21-p=γ+2-2Mγp/tγ+2p/t≤γ+2-2.Thus, (52)∑r=0∞1Q2r∑rqkDxk,0-p≤Lp∑r=0∞γ+2-2<∞.So, x=xk∈Cesp,qnF, which is a contradiction to our assumption. Hence a∈μtF.
That is,Cesp,qnFα⊆μtF.
Then, combining the two results, we obtainCesp,qnFα=μtF.
This step completes the proof.


## 4. Characterization of Matrix Classes

An infinite matrix is one of the most general linear operators between two sequence spaces. The study of theory of matrix transformations has always been of great interest to mathematicians in the study of sequence spaces, which is motivated by special results in summability theory.Definition 15 (Talo and Başar [20]).
Letμ1F,μ2F⊂WF and A=(ank) be any two-dimensional infinite matrix of fuzzy numbers. Then we say that A defines a mapping from μ1F into μ2F and denote it by A:μ1F→μ2F if, for every sequence x=xk∈μ1F, the A-transform of x, Ax={(Ax)n}, given by(53)Axn=∑kankxkexists for each each n∈N and is in μ2F.
A ∈ ( μ 1 F : μ 2 F ) if and only if the series on the right hand side of (53) converges for each n∈N and every x=xk∈μ1F and we have Ax={(Ax)n}n∈N. A sequence of fuzzy numbers x=xk is said to be A-summable to α if Ax converges to α which is called the A-limit of x. Also by A∈(μ1F:μ2F;P) we denote that A preserves the limit; that is, A-limit of x is equal to the limit of x for all x=xk∈μ1F.
We write(54)Arn=maxr⁡Dank,0-qkwhere, for each n∈N, the maximum is taken with respect to k∈2r,2r+1.Theorem 16.
LetA=ank be an infinite matrix of fuzzy numbers and 1<p<∞. Then A∈Cesp,qnF:l∞F if there exists an integer G>1 such that UG<∞ where(55)UG=supn⁡∑r=0∞Q2rArntG-tand 1/p+1/t=1.Proof.
Suppose there exists an integerG>1 such that UG<∞. Let x=xk∈Cesp,qnF. Then(56)DAnx,0-=D∑k=1∞ankxk,0-≤∑k=1∞Dankxk,0-=∑r=0∞∑rDankxk,0-≤∑r=0∞∑rDank,0-Dxk,0-=∑r=0∞∑r1qkDank,0-qkDxk,0-≤∑r=0∞maxr⁡Dank,0-qk∑rqkDxk,0-=∑r=0∞Q2rmaxr⁡Dank,0-qk1Q2r∑rqkDxk,0-Using inequality (26), we obtain(57)DAnx,0-≤G∑r=0∞Q2rmaxr⁡Dank,0-qktG-t+1Q2r∑rqkDxk,0-p=G∑r=0∞Q2rmaxr⁡Dank,0-qktG-t+∑r=0∞1Q2r∑rqkDxk,0-p=G∑r=0∞Q2rArntG-t+∑r=0∞1Q2r∑rqkDxk,0-p<∞.Therefore, A∈Cesp,qnF:l∞F.
The necessity of the above theorem is still open; i.e., we do not know ifA∈Cesp,qnF:l∞F then the condition (55) holds or not.Theorem 17.
LetA=ank be an infinite matrix of fuzzy numbers and 1<p<∞. Then A∈Cesp,qnF:CF if (55) holds and, for fixed k∈N, we have(58)limn→∞⁡ank=α-kProof.
Suppose conditions (55) and (58) hold. Then(59)∑r=0∞Q2rmaxr⁡Dα-k,0-qktG-t≤UG<∞.From (59), using similar argument as in Theorem 14, it is easy to verify that the series(60)∑k=1∞Dα-kxk,0-<∞.This implies that (61)D∑k=1∞α-kxk,0-≤∑k=1∞Dα-kxk,0-<∞.Thus, (62)∑k=1∞α-kxkexists.
Now, for eachx=xk∈Cesp,qnF and ε>0, we can choose an integer m0≥1 such that(63)dm0∗x,0-=∑r=m0∞1Q2r∑rqkDxk,0-p<εP.Then (64)
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.It follows that(65)D∑k=1∞ankxk,∑k=1∞α-kxk→0,as n→∞. This shows that A∈Cesp,qnF:CF which proves the theorem.Corollary 18.
LetA=ank be an infinite matrix of fuzzy numbers and 1<p<∞. Then A∈Cesp,qnF:C0F if (55) holds and, for fixed k∈N, we have(66)limn→∞⁡ank=0-.Proof.
The proof is obvious.



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*Source: 8636121-2018-08-08.xml*