# Some Classes of Difference Sequence Spaces of Fuzzy Real Numbers Defined by Orlicz Function

Authors: Binod Chandra Tripathy; Stuti Borgohain
Journal: Advances in Fuzzy Systems
 (2011)

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## Abstract

We introduce the classes of generalized difference bounded, convergent, and null sequences of fuzzy real numbers defined by an Orlicz function. Some properties of these sequence spaces like solidness, symmetricity, and convergence-free are studied. We obtain some inclusion relations involving these sequence spaces.


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## Body

## 1. Introduction

The concept of fuzzy set theory was introduced by Zadeh in 1965. Later on sequences of fuzzy numbers have been discussed by Syau [1], Tripathy and Baruah [2], Tripathy and Borgohain [3], Tripathy and Dutta [4, 5], Tripathy and Sarma [6, 7], and many others.Kizmaz [8] defined the difference sequence spaces ℓ∞(Δ), c(Δ), and c0(Δ) of complex numbers as follows:
(1)Z(Δ)={x=(xk):(Δxk)∈Z},forZ=l∞,c,c0,whereΔx=(Δxk)=(xk-xk+1).The above spaces are Banach spaces, normed by(2)‖x‖Δ=|x1|+sup⁡k⁡|Δxk|.The idea of Kizmaz [8] was applied to introduce different type of difference sequence spaces and study their different properties by Et et al. [9], Tripathy et al. [10], Tripathy and Baruah [2], Tripathy and Borgohain [3], Tripathy and Esi [11], Tripathy et al. [12], Tripathy and Mahanta [13], and many others.Tripathy and Esi [11] introduced a new type of difference sequence spaces as follows. Let m∈N be fixed, then
(3)Z(Δm)={x=(xk):(Δmxk)∈Z},forZ=l∞,c,c0,whereΔmx=(Δmxk)=(xk-xk+m).The above sequence spaces are Banach spaces, normed by(4)‖x‖Δ=∑r=1m|xr|+sup⁡k⁡|Δmxk|.
Tripathy et al. [12] further generalized this notion and introduced the following. For m≥1 and n≥1,
(5)Z(Δmn)={x=(xk):(Δmnxk)∈Z},forZ=l∞,c,c0,whereΔmnxk=Δmn-1xk-Δmn-1xk+m,∀k∈N.This generalized difference has the following binomial representation:(6)Δmnxk=∑r=0n(-1)r(nr)xk+rm.An Orlicz function is a functionM:[0,∞)→[0,∞), which is continuous, nondecreasing, and convex with M(0)=0, M(x)>0, for x>0 and M(x)→∞, as x→∞ (one may refer to Korasnoselkii and Rutitsky [14]).An Orlicz functionM is said to satisfy Δ2-condition for all values of x, if there exists a constant K>0, such that M(Lx)≤KLM(x), for all x>0 and for L>1.Remark 1.
An Orlicz function satisfies the inequalityM(λx)≤λM(x), for all λ with 0<λ<1.Throughout the paperwF,ℓF,ℓ∞F represent the classes of all, absolutely summable,and boundedsequences of fuzzy real numbers, respectively.


## 2. Definitions and Background

LetC(Rn)={A⊂Rn:Aiscompactandconvex}. Then the space C(Rn) has linear structure induced by the operations A+B={a+b:a∈A,b∈B} and λA={λa:a∈A} for A, B∈C(Rn) and λ∈R.The Hausdorff distance betweenA and B of C(Rn) is defined as
(7)δ∞(A,B)=max⁡{sup⁡a∈A⁡inf⁡b∈B⁡‖a-b‖,sup⁡b∈B⁡inf⁡a∈A⁡‖a-b‖}.It is well known that(C(Rn),δ∞) is a complete metric space. A fuzzy real number on Rn is a function X:Rn→I(=[0,1]) associating each real number t∈Rn with its grade of membership X(t).A fuzzy real numberX is called convex if X(t)≥X(s)∧X(r)=min(X(s),X(r)), where s<t<r.If there existst0∈Rn 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 Rn.The class of allupper semi-continuous, normal, convex fuzzy real numbers is denoted by Rn(I). Let X∈Rn(I), then the α-level set Xα, for 0<α≤1, is defined by, Xα={t∈Rn:X(t)≥α} and is a nonempty compact convex subset of Rn. The 0-level set, that is, X0, is the closure of strong 0-cut, that is, X0=cl{t∈Rn:X(t)>0}. The absolute value of X∈R(I), that is, |X|, is defined by
(8)|X|(t)={max⁡{X(t),X(-t)},fort≥0,0,otherwise.Forr∈Rn, r̅∈Rn(I) is defined as,
(9)r̅(t)={1,fort=r,0,otherwise.The additive identity and multiplicative identity ofRn(I) are denoted by θ̅  and e̅, respectively, where θ=(0,0,…,0) and e=(1,1,…,1). The zero sequence of fuzzy real numbers is denoted by ϑ̅={θ̅,θ̅,…,θ̅,…}.The linear structure ofC(Rn) induces the addition X+Y and scalar multiplication λX, λ∈R in terms of α-level sets, by [X+Y]α=[X]α+[Y]α and [λX]α=λ[X]α, for each 0<α≤1, where
(10)λX(t)={0̅,forλ=0,X(λ-1t),otherwise.Define, for each1≤q<∞,
(11)dq(X,Y)=(∫01δ∞(Xα,Yα)qdα)1/q
and d∞:Rn(I)×Rn(I)→R such that d∞(X,Y)=sup0<α≤1δ∞(Xα,Yα), where δ∞ is the Hausdorff metric. Clearly, d∞(X,Y)=limq→∞dq(X,Y) with dq≤dr if q≤r. Moreover (Rn(I),d∞) is a complete, separable, and locally compact metric space.A sequenceX=(Xk) of fuzzy real numbers is said to converge to the fuzzy number X0, if for every ε>0, there exists k0∈N such that d∞(Xk,X0)<ε, for all k≥k0.A sequence spaceE is said to be solidif (Yn)∈E, whenever (Xn)∈E and |Yn|≤|Xn|, for all n∈N.LetX=(Xn) be a sequence, then S(X) denotes the set of all permutations of the elements of (Xn), that is, S(X)={(Xπ(n)):πisapermutationofN}. A sequence space E is said to be symmetric if S(X)⊂E for all X∈E.A sequence spaceE is said to be convergence-free if (Yn)∈E whenever (Xn)∈E and Xn=θ̅  implies Yn=θ̅.A sequence spaceE is said to be monotone if E contains the canonical preimages of all its step spaces.Lemma 2.
A class of sequencesE is solid which implies that E is monotone.Lindenstrauss and Tzafriri [15] used the notion of Orlicz function and introduced the sequence space:
(12)lM={(xk)∈w:∑k=1∞M(|xk|ρ)<∞,forsomeρ>0}.The spaceℓM with the norm,
(13)‖(xk)‖=inf⁡⁡{ρ>0:∑k=1∞M(|xk|ρ)≤1},
becomes a Banach space, which is called an Orlicz sequence space. The space ℓM is closely related to the space ℓp, which is an Orlicz sequence space with M(x)=xp, for 1≤p<∞.In the later stage different classes of Orlicz sequence spaces were introduced and investigated by Altin et al. [16], Et et al. [9], Tripathy et al. [10], Tripathy and Borgohain [3], Tripathy and Hazarika [17], Tripathy and Mahanta [13], Tripathy and Sarma [6, 7, 18], and many others.In this paper we introduce the following difference sequence spaces:(14)l∞F(M,Δmn)={(Xk)∈wF:sup⁡k⁡(d∞(ΔmnXk,θ̅)ρ)<∞,forsomeρ>0(d∞(ΔmnXk,θ̅ρ)}cF(M,Δmn)={(Xk)∈wF:lim⁡k→∞⁡(d∞(ΔmnXk,L)ρ)=0,forsomeρ>0,L∈Rn(I)(d∞(ΔmnXk,Lρ)}c0F(M,Δmn)={(Xk)∈wF:lim⁡k→∞⁡(d∞(ΔmnXk,θ̅)ρ)=0,forsomeρ>0(d∞(ΔmnXk,θ̅ρ)}.


## 3. Main Results

Theorem 3.
The classes of sequencesℓ∞F(M,Δmn), cF(M,Δmn), c0F(M,Δmn)  are complete metric spaces by the metric
(15)η(X,Y)=∑r=1mnd∞(Xr,Yr)+inf⁡⁡{ρ>0:sup⁡k⁡M(d∞(ΔmnXk,ΔmnYk)ρ)≤1},
for X, Y∈ℓ∞(M,Δmn)F, cF(M,Δmn), c0F(M,Δmn).Proof.
We establish the result for the class of sequencesℓ∞F(M,Δmn). The proof for the other cases will follow similarly. It can easily be verified that ℓ∞F(M,Δmn) is a metric space by the metric η defined above. Next we show that it is a complete metric space.
Let(X(i)) be a Cauchy sequence in ℓ∞F(M,Δmn) such that X(i)=(Xn(i))n=1∞. Let ε>0 be given. For a fixed x0>0, choose r>0 such that M(rx0/2)≥1. Then there exits a positive integer n0=n0(ε) such that
(16)η(X(i),X(j))<εrx0,∀i,j≥n0.
By the definition ofη, we have,
(17)∑r=1mnd∞(Xr(i),Xr(j))+inf⁡⁡{ρ>0:sup⁡k⁡M(d∞(ΔmnXr(i),ΔmnXr(j))ρ)≤1}≤ε,∀i,j≥n0,
which implies
(18)∑r=1mnd∞(Xr(i),Xr(j))<ε,∀i,j≥n0,⟹d∞(Xr(i),Xr(j))<ε,∀i,j≥n0,r=1,2,3,…,mn.
Hence(Xr(i)), for r=1,2,3,…,mn are Cauchy sequence in Rn(I) and hence are convergent in Rn(I), since Rn(I) is a complete metric space.
Let(19)lim⁡i→∞⁡Xr(i)=Xr,forr=1,2,3,…,mn.
Also,(20)sup⁡k⁡M(d∞(ΔmnXk(i),ΔmnXk(j))ρ)≤1,∀i,j≥n0,⟹M(d∞(ΔmnXk(i),ΔmnXk(j))η(X(i),X(j)))≤1≤M(rx02),∀i,j≥n0.
Since M is continuous, we get,
(21)d∞(ΔmnXk(i),ΔmnXk(j))≤rx02⋅η(X(i),X(j)),∀i,j≥n0,⟹d∞(ΔmnXk(i),ΔmnXk(j))<rx02⋅εrx0=ε2,∀i,j≥n0,⟹d∞(ΔmnXk(i),ΔmnXk(j))<ε2,∀i,j≥n0,
which implies (ΔmnXk(i)) is a Cauchy sequence in Rn(I) and so is convergent in Rn(I), since Rn(I) is complete metric space.
LetlimiΔmnXk(i)=Yk (say), in Rn(I), for each k∈N.
We have to prove that(22)lim⁡i⁡X(i)=X,X∈l∞F(M,Δmn).
Fork=1, we have, from (6) and (19),
(23)lim⁡i→∞⁡Xmn+1(i)=Xmn+1,form≥1,n≥1.
Proceeding in this way inductively, we get(24)lim⁡i→∞⁡Xk(i)=Xk,foreachk∈N.
Also,limiΔmnXk(i)=ΔmnXk, for each k∈N.
Next takingj→∞, keeping i fixed, and by the continuity of M, we have the following from (20):
(25)sup⁡k⁡M(d∞(ΔmnXk(i),ΔmnXk)ρ)≤1,forsomeρ>0.
Now on taking the infimum of suchρ’s, we get
(26)inf⁡{ρ>0:sup⁡k⁡M(d∞(ΔmnXk(i),ΔmnXk)ρ)≤1}<ε,∀i≥n0(by(2)).
Hence from (17) on taking limit as j→∞, we get
(27)∑r=1mnd∞(Xr(i),Xr)+inf⁡{ρ>0:sup⁡kM(d∞(ΔmnXk(i),ΔmnXk)ρ)≤1}<ε+ε=2ε,∀i≥n0,
which implies
(28)η(X(i),X)<2ε,∀i≥n0.
That is, limiX(i)=X.
Next we show thatX∈ℓ∞F(M,Δmn).
We know that(29)d∞(ΔmnXk,θ̅)≤d∞(ΔmnXk(i),ΔmnXk)+d∞(ΔmnXk(i),θ̅).
SinceM is continuous and nondecreasing, so we get
(30)sup⁡kM(d∞(ΔmnXk,θ̅)ρ)≤sup⁡k⁡M(d∞(ΔmnXk(i),ΔmnXk)ρ)⁡+sup⁡k⁡M(d∞(ΔmnXk(i),θ̅)ρ)<∞,
which implies X∈ℓ∞F(M,Δmn).
Henceℓ∞F(M,Δmn) is a complete metric space.
The other cases can be established similarly.
This completes the proof of the theorem.Result 1.
The classes of sequencesℓ∞F(M,∆mn), cF(M,∆mn), c0F(M,∆mn), are neither solid nor monotone in general.Proof.
The result follows from the following example.
Example 4.
Consider the sequence spaceℓ∞F(M,Δmn). Let m=2 and n=3. Let M(x)=|x|, for all x∈[0,∞).
Consider the sequence(Xk) defined by
(31)Xk(t)={1,fork∈N,t=(k,k,k,…),0,otherwise.
Then,(32)Δ23Xk(t)={1,fork∈N,t=(0,0,0,…),0,otherwise.
Then, we haved∞(Δ23Xk,θ̅)=0, for all k∈N.
Hence, we have(33)sup⁡k⁡M(d∞(Δ23Xk,θ̅)ρ)<∞,forsomeρ>0,
which implies (Xk)∈ℓ∞F(M,Δ23).
Consider the sequence(αk) of scalars defined by
(34)αk={1,fork=i2,i∈N,0,otherwise.
Fork=i2, we have
(35)αkXk(t)={1,t=(k,k,k,…),0,otherwise.
Fork≠i2, we have
(36)αkXk(t)={1,t=(0,0,0,…),0,otherwise,
which implies
(37)sup⁡k⁡M(d∞(Δ23Xk,θ̅)ρ)=∞,foreachρ>0.
Hence(αkXk)∉ℓ∞F(M,Δ23).
Thus,ℓ∞F(M,∆mn) is not solid in general.Similarly the other cases can be established. The classes of sequences are not monotone followed by Lemma2.Result 2.
The classes of sequencesℓ∞F(M,∆mn), cF(M,∆mn), and c0F(M,∆mn) are not symmetric in general.Proof.
The result follows from the following example.
Example 5.
Letm=2 and n=1. Let M(x)=x2, for all x∈[0,∞). Consider the sequence (Xk) defined by
(38)Xk(t)={1,fork∈N,t=(k,k,k,…),0,otherwise.
Then,(39)Δ2Xk(t)={1,fork∈N,t=(-2,-2,-2,…),0,otherwise.
Then,d∞(Δ2Xk,0̅)=1, for all k∈N, which shows (Xk)∈cF(M,Δ2)⊂ℓ∞F(M,Δ2).
Let(Yk) be a rearrangement of (Xk) such that (Yk)=(X1,X2,X4,X3,X9,X5,X16,X6,X25,…). Then we get d∞(Δ2Yk,0̅)≈k-(k-1)2≈k2, for all k∈N, which implies
(40)sup⁡k⁡M(d∞(Δ2Yk,0̅)ρ)=∞,foreachfixedρ>0.
Hence,(Yk)∉ℓ∞F(M,Δ2).
Thus the classes of sequencesℓ∞F(M,Δmn), cF(M,Δmn), and c0F(M,Δmn) are not symmetric in general.Note 1.
Form=n=0, the class of sequences ℓ∞F(M) and cF(M) are symmetric. For m≤1 and n≤1, the class of sequences c0F(M,Δmn) is symmetric.Proposition 6.
The classes of sequencesℓ∞F(M,Δmn), cF(M,Δmn), c0F(M,Δmn) are not convergence-free in general.Proof.
The result follows from the following example.
Example 7.
Letm=4 and n=1. Let M(x)=x3, for all x∈[0,∞). Consider the sequence (Xk) defined by
(41)Xk(t)={1,fork∈N,t=(1k,1k,1k,…),0,otherwise.
Then,
(42)Δ4Xk(t)={1,fork∈N,t=(4k(k+4),4k(k+4),4k(k+4),…),0,otherwise.
Hence we haved∞(Δ4Xk,θ̅)=4/k(k+4), which implies (Xn)∈c0F(M,Δ4)⊂cF(M,Δ4)⊂ℓ∞F(M,Δ4).
Consider the sequence(Yk) defined by
(43)Yk(t)={1,fork∈N,t=(k2,k2,k2,…),0,otherwise,
so that
(44)Δ4Yk(t)={1,fork∈N,t=(-(8k+16),-(8k+16),…),0,otherwise.
Thusd∞(Δ4Yk,θ̅)=8k+16, for all k∈N, which implies
(45)sup⁡k⁡M(d∞(Δ4Yk,θ̅)ρ)=∞,forsomeρ>0.
Thus(Yk)∉ℓ∞F(M,Δ4).
Hence the classes of sequencesℓ∞F(M,Δmn), cF(M,Δmn), c0F(M,Δmn) are not convergence-free in general.Theorem 8.
LetM, M1, and M2 be Orlicz functions satisfying Δ2-condition. Then, for Z=ℓ∞F, cF, and c0F, (i)
Z(M1,Δmn)⊆Z(M°M1,Δmn),(ii)
Z(M1,Δmn)∩Z(M2,Δmn)⊆Z(M1+M2,Δmn).Proof.
(i) Let(Xk)∈Z(M1,Δmn). Consider ε>0 and η>0 such that ε=M(η).
Then,(46)M1(d∞(ΔmnXk,L)ρ)<η,forsomeρ>0.
Let(47)Yk=M1(d∞(ΔmnXk,L)ρ),forsomeρ>0.
SinceM is continuous and non-decreasing, we get
(48)M(Yk)=M(M1(d∞(ΔmnXk,L)ρ))<M(η)=ε,forsomeρ>0,
which implies (Xk)∈Z(M°M1,Δmn).
This completes the proof.
(ii) Let(Xk)∈Z(M1,Δmn)∩Z(M2,Δmn).
Then,(49)M1(d∞(ΔmnXk,L)ρ)<ε,forsomeρ>0,M2(d∞(ΔmnXk,L)ρ)<ε,forsomeρ>0.
The proof follows from the equality(50)(M1+M2)(d∞(ΔmnXk,L)ρ)=M1(d∞(ΔmnXk,L)ρ)+M2(d∞(ΔmnXk,L)ρ),<ε+ε=2ε,forsomeρ>0,
which implies that (Xk)∈Z(M1+M2,Δmn).
This completes the proof.Proposition 9.
One hasZ(M,Δmi)⊂Z(M,Δmn), for 0≤i<n, for Z=ℓ∞F, cF, and c0F.Proof.
Let(Xk)∈ℓ∞F(M,Δmn-1). Then we have,
(51)sup⁡k≥1⁡M(d∞(Δmn-1Xk,θ̅)ρ)<∞.
Now we have(52)sup⁡k≥1⁡M(d∞(ΔmnXk,θ̅)ρ)=sup⁡k≥1⁡M(d∞(Δmn-1Xk-Δmn-1Xk+m,θ̅)ρ)≤12sup⁡k≥1⁡M(d∞(Δmn-1Xk,θ̅)ρ)+12sup⁡k≥1⁡M(d∞(Δmn-1Xk+m,θ̅)ρ)<∞.
Proceeding in this way, we haveZ(M,Δmi)⊂Z(M,Δmn), for 0≤i<n, for Z=ℓ∞F, cF, and c0F.
This completes the proof.



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