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.
University Grants CommissionF./2015-16/NFO-2015-17-OBC-ASS-36722/(SA-III/Website)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 mapping v:R→[0,1], which satisfies the following four conditions:
v is normal; i.e., there exists an x0∈R such that vx0=1.
v is fuzzy convex; i.e., vλx+1-λy≥min{vx,vy} for all x,y∈Randforallλ∈0,1.
v is upper semicontinuous.
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 on R 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 set vλ 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]).
Let x,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=minx-λy-λ,x-λy+λ,x+λy-λ,x+λy+λand(5)z+λ=maxx-λy-λ,x-λy+λ,x+λy-λ,x+λy+λfor all λ∈0,1.
Definition 3 (Talo and Başar [20]).
Let W 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=maxA1-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∈0,1dxλ,yλ=supλ∈0,1maxx-λ-y-λ,x+λ-y+λ
The partial ordering relation on E1 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∈E1 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∈0,1maxx-λ,x+λ=maxx-0,x+0
Lemma 5 (Bede and Gal [21]).
Letx,y,z∈E1 andk∈R. Then
E1,D is a complete metric space.
Dkx,ky=kDx,y.
Dx+y,z+y=Dx,z.
Dx+y,z+u≤Dx,z+Dy,u.
D(x,0-)-D(y,0-)≤Dx,y≤Dx,0-+Dy,0-.
Lemma 6 (Talo and Başar [20]).
The following statements hold:
Dxy,0-≤Dx,0-Dy,0-forallx,y∈E1.
If xk→x,ask→∞thenDxk,0-→Dx,0-ask→∞.
By WFwe 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 sequence xk∈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∈NDxk,0-=supk∈Nsupλ∈0,1maxxk-λ,xk+λ<∞.
Definition 8.
Consider the sequence of fuzzy numbers xk∈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 sets l∞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:supkDxk,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 from 0 to ∞; for example,(12)∑kxkmeans that(13)∑k=0∞xk.
Definition 9 (Talo and Başar [20]).
Let xk∈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 sequence sn is bounded then we say that the series (21)∑kxkof fuzzy numbers is bounded.
Definition 10 (Talo and Başar [20]).
Let μF 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:
If q=(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 any G>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 of CespF 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 space Cesp,qnF and determine the Köthe-Toeplitz dual and give some related matrix transformations.
2. Complete Metric Structure
We equip Cesp,qnF with a metric and show that this set is complete with respect to the metric defined in the following theorem.
Theorem 12.
Cesp,qnF 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 that d∗x,y=0⇔x=y and d∗x,y=d∗y,x.
Now we prove the triangle inequality.
Suppose x=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.
Suppose x=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 that x=xk∈Cesp,qnF.
We have that d∗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 DualDefinition 13 (Talo and Başar [20]).
The Köthe-Toeplitz dual or the α-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.
If 1<p<∞ and 1/p+1/t=1, then(39)Cesp,qnFα=a=ak:∑r=0∞Q2rmaxrDak,0-qktG-t<∞forsomeintegerG>1.
Proof.
Let 1<p<∞ and 1/p+1/t=1.
We define (40)μtF=a=ak:∑r=0∞Q2rmaxrDak,0-qktG-t<∞forsomeintegerG>1.We want to show that Cesp,qnFα=μtF.
Let x=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∞maxrDak,0-qk∑rqkDxk,0-=∑r=0∞Q2rmaxrDak,0-qk1Q2r∑rqkDxk,0-≤G∑r=0∞Q2rmaxrDak,0-qktG-t+1Q2r∑rqkDxk,0-p=G∑r=0∞Q2rmaxrDak,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∞Q2rmaxrDak,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+1-1Q2rmaxrDak,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=maxrDak,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+1-1Q2rmaxrDak,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-1maxrDak,0-qkt-1γ+2-tMγ-1pNow(51)∑r=nγnγ+1-1Q2rt-1pmaxrDak,0-qkt-1pγ+2tpMγ-p=γ+2-2Mγ-1∑r=nγnγ+1-1Q2rmaxrDak,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 obtain Cesp,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]).
Let μ1F,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∈(μ1F:μ2F) 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=maxrDank,0-qkwhere, for each n∈N, the maximum is taken with respect to k∈2r,2r+1.
Theorem 16.
Let A=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 integer G>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∞maxrDank,0-qk∑rqkDxk,0-=∑r=0∞Q2rmaxrDank,0-qk1Q2r∑rqkDxk,0-Using inequality (26), we obtain(57)DAnx,0-≤G∑r=0∞Q2rmaxrDank,0-qktG-t+1Q2r∑rqkDxk,0-p=G∑r=0∞Q2rmaxrDank,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 if A∈Cesp,qnF:l∞F then the condition (55) holds or not.
Theorem 17.
Let A=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-k
Proof.
Suppose conditions (55) and (58) hold. Then(59)∑r=0∞Q2rmaxrDα-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 each x=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)D∑k=2m0∞ankxk,∑k=2m0∞α-kxk≤∑k=2m0∞Dankxk,α-kxk≤∑k=2m0∞Dankxk,0-+∑k=2m0∞Dα-kxk,0-≤∑k=2m0∞Dank,0-Dxk,0-+∑k=2m0∞Dα-k,0-Dxk,0-≤∑r=m0∞Q2rmaxrDank,0-qk1Q2r∑rqkDxk,0-+∑r=m0∞Q2rmaxrDα-k,0-qk1Q2r∑rqkDxk,0-≤∑r=m0∞Q2rmaxrDank,0-qkG-tGt1Q2r∑rqkDxk,0-+∑r=m0∞Q2rmaxrDα-k,0-qkG-tGt1Q2r∑rqkDxk,0-=Gt∑r=m0∞Q2rmaxrDank,0-qktG-t1/t∑r=m0∞1Q2r∑rqkDxk,0-p1/p+Gt∑r=m0∞Q2rmaxrDα-k,0-qktG-t1/t∑r=m0∞1Q2r∑rqkDxk,0-p1/p≤Gt∑r=m0∞Q2rmaxrDank,0-qktG-t1/t+Gt∑r=m0∞Q2rmaxrDα-k,0-qktG-t1/tε≤GUG1/t+GUG1/tε=2GUG1/tε.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.
Let A=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.
Data Availability
The paper is theoretical in nature and all necessary references are included in the References with proper citation within the main text.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors would like to express their gratitude to the University Grants Commission, New Delhi, India, for offering Fellowship to the author J. Gogoi via Award Letter no F./2015-16/NFO-2015-17-OBC-ASS-36722/(SA-III/Website).
ZadehL. A.Fuzzy sets19658338353MR0219427MatlokaH.Sequences of fuzzy numbers1986282837NandaS.On sequences of fuzzy numbers198933112312610.1016/0165-0114(89)90222-4MR1021128Zbl0707.540032-s2.0-0000437763NurayF.SavaşE.Statistical convergence of sequences of fuzzy numbers1995453269273MR1361821DiamondP.KloedenP.Metric spaces of fuzzy sets199035224124910.1016/0165-0114(90)90197-EMR1050604Zbl0704.540062-s2.0-0000944768EsiA.On some new paranormed sequence spaces of fuzzy numbers defined by Orlicz functions and statistical convergence2006114379388KalevaO.On the convergence of fuzzy sets1985171536510.1016/0165-0114(85)90006-5MR808463Zbl0584.54004NandaS.198374Trieste, ItalyICTPMR866486DuboisD.PradeH.Operations on fuzzy numbers197896613626MR0491199Zbl0383.9404510.1080/00207727808941724AltinokH.ÇolakR.AltinY.On the class of λ-statistically convergent difference sequences of fuzzy numbers2012166102910342-s2.0-8486100351210.1007/s00500-011-0800-6Zbl1266.40003StojakovicM.StojakovicZ.Addition and series of fuzzy sets199683341346StojakovićM.StojakovićZ.Series of fuzzy sets2009160213115312710.1016/j.fss.2008.12.013MR2567096Zbl1183.030542-s2.0-69249242843MursaleenM.SrivastavaH. M.SharmaS. K.Generalized statistically convergent sequences of fuzzy numbers20163031511151810.3233/IFS-1518582-s2.0-84961233640SubrahmanyamP. V.Cesàro summability of fuzzy real numbers19997159168MR1735002TaloÖ.ÇakanC.On the Cesàro convergence of sequences of fuzzy numbers201225676681AltinY.MursaleenM.AltinokH.Statistical summability (C, 1) for sequences of fuzzy real numbers and a Tauberian theorem2010216379384YavuzE.Comparison theorems for summability methods of sequences of fuzzy numbers20181810.3233/JIFS-162050YavuzE.Euler summability method of sequences of fuzzy numbers and a Tauberian theorem201610GoetschelJ.VoxmanW.Elementary fuzzy calculus1986181314310.1016/0165-0114(86)90026-6MR825618Zbl0626.260142-s2.0-46149135272TaloÖ.BaşarF.Determination of duals of classical sets of sequences of fuzzy numbers and related matrix transformations20095871773310.1016/j.camwa.2009.05.002MR2554016BedeB.GalS. G.Almost periodic fuzzy-number-valued functions2004147338540310.1016/j.fss.2003.08.004MR2100833Zbl1053.420152-s2.0-4544258343KhanF. M.RahmanM. F.Infinite matrices and Cesàro sequence spaces199723311MaddoxI. J.Continuous and Köthe-Toeplitz dual of certain sequence spaces196965431435LimK. P.Matrix transformation in the Cesàro sequence spaces197414221227MR0374752ShiueJ. S.On the Cesàro sequence spaces197011925MR0262737LeibowitzG. M.A note on the Cesàro sequence spaces19712151157KhanF. M.KhanM. A.The sequence space ces(p, s) and related matrix transformations19912195104KhanF. M.KhanM. A.Matrix transformations between Cesàro sequence spaces1994256641645JohnsonP. D.MohapatraR. N.Density of finitely non-zero sequences in some sequence spaces1979243253262RahmanM. F.KarimA. B. M.Dual spaces of generalized Cesàro sequence space and related matrix mapping2016444450