Fuzzy Entire Sequence Spaces

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International Journal of Mathematics and Mathematical Sciences Volume 2007, Article ID 58368,9pages

doi:10.1155/2007/58368

Research Article

Fuzzy Entire Sequence Spaces

J. Kavikumar, Azme Bin Khamis, and R. Kandasamy

Received 6 December 2006; Accepted 22 February 2007

Recommended by Laszlo Toth

We first investigate the notion of fuzzy entire sequence space with a suitable example. Also we deal with the properties of the space of fuzzy entire sequences. The concepts of subset and superset of the fuzzy entire sequence spaces are introduced and their properties are discussed.

Copyright © 2007 J. Kavikumar et al. 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.

1. Introduction

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2. Preliminaries

At first, we recall some definitions and results about fuzzy numbers. A fuzzy number space is a fuzzy set on the real axis, that is, denoteF(R)= {a|a:Rn_→_{[0, 1],}_a_{has the} following properties (a)–(d)}:

(a)ais normal, that is, there exists anx0∈Rn_{such that}_a₍_x0₎₌_1;

(b)ais convex, that is,a(λx+ (1−λ)y)≥min{a(x),a(y)}wheneverx,y∈Rn_and 0≤λ≤1;

(c)a(x) is upper semicontinuous;

(d) [a]0₌_cl_{_x_∈_Rn_:_a₍_x₎_>₀_}_{is a compact set.}

As obtained by Zadeh,ais convex if and only if each of itsα-level setsaα, whereaα= {x∈ Rn_:_a₍_x₎_≥_α_}_{for each}_α_∈_{(0, 1], is a nonempty compact convex subset of}_Rn_with com-pact support. Theα-level set of an upper semicontinuous convex normal fuzzy number is a closed interval [a−_λ,a+

λ], where the valuesa−λ = −∞anda+λ=+∞are admissible. Since eachx∈Rn_{can be considered as a fuzzy number}_a_{, defined by (2.1), the real number can} be embedded inF∗(R). A fuzzy numberais called nonnegative ifa(x)=0 for allx <0. The set of all nonnegative fuzzy numbers ofF∗(R) is denoted byF(R). Letk∈F(R) and k=λ∈[0,1]λ[k,k],

a(x)=

⎧ ⎨ ⎩

1 forx=k,k∈Rn_,

0 forx=k,k∈Rn_. (2.1)

For anya∈F(R),ais called fuzzy number andF(R) is called a fuzzy number space. For

a,b∈F(R), we definea≤bif and only if [a]_λ=[a−_λ,a+

λ]≤[b]λ=[b−λ,b+λ] and [a]λ≤[b]λ if and only ifa−_λ ≤b−_λ anda+

λ≤b+λ for anyλ∈[0, 1].

Theorem 2.1 (representation theorem). Fora∈F(R),

(1)a−_λ is a bounded left continuous nondecreasing function on (0, 1]; (2)a+_λ is a bounded left continuous nonincreasing function on (0, 1]; (3)a−_λ anda+

λ is right continuous atλ=0; (4)a−_λ ≤a+

λ.

Moreover, if the pair of functionsa(λ) andb(λ) satisfies (1)–(4), then there exists a uniquea∈F(R) such thataλ=[a(λ),b(λ)] for eachλ∈[0, 1].

DefineF(R)×F(R)→Rbyd(a,b)=sup₀_≤a≤₁δ∞(aα,bα) fora,b∈F(R) andδ∞(A,B)= max{sup_a∈Ainfb∈B a−b , supb∈Binfa∈A a−b }. It is known that (F(R),d) is a com-plete metric space.

Let0 and1 be defined by

0(x)=

⎧ ⎨

⎩1₀ for_forx_x=₌0,_0, 1(x)= ⎧ ⎨ ⎩

1 forx=1,

0 forx=1. (2.2)

The absolute value|a|ofa∈F(R) is defined by

|a|(t)= ⎧ ⎨ ⎩max

a(t),a(−t), t >0,

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Definition 2.2 [5]. A sequencex=(xk) of fuzzy numbers is said to be a Cauchy sequence to a fuzzy number (xm), written as limn→kxk=xm, if for everyε >0 there exists a positive integerN0such thatρ(xk,xm)< εfork,m > N0.

3. Fuzzy entire sequenceΓ(R)

Now we introduce the new fuzzy sequence space and we show that this is a complete metric space.

For each fixedn, define the fuzzy metric

ρ xn,yn

=

λ∈[0,1]

λ xn−1 − yn −

1 1/n_{, sup}

λ≤η≤1

xn−_η − yn−_η1/n∨ xn+_η− yn+_η1/n

.

(3.1)

Letx= {xn}andy= {yn}be sequences of fuzzy real numbers. Define their distance by

θ(x,y)

=

λ∈[0,1] λ

sup

(n) xn

− 1− yn

− 1

1/n_{, sup}

(n)

sup λ≤η≤1

xn

− η− yn

− η

1/n_∨ xn

+ η− yn

+ η

1/n .

(3.2)

Clearly, (F(R),ρ) is a complete metric space [6].

Definition 3.1. x= {xn}is called a fuzzy entire sequence if ρ(limn→∞xn)=0. In other words, givenε >0 there exists a positive integerNsuch thatρ(xn, 0)< εfor alln∈N. Let

Γ(R)={all fuzzy entire sequences}.

Example 3.2. Letan=1/n! forn=1, 2,... be a sequence fuzzy numbers. Now we have

a=(an)=(1/n!), wheren∈F(R).

ρ an,0

=

λ∈[0,1] λ

sup

₁

n! −

−0−, sup n

sup λ≤η≤1

₁

n! −

η−0

−1/n_∨1 n!

+

η−0 −1/n

=0.

(3.3)

This implies thata=(an)=(1/n!)∈Γ(R). Hence,a=(an) is a fuzzy entire sequence.

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Proof. It can be seen thatθis a metric forΓ(R). Let{x(i)_}_{be any fuzzy Cauchy sequence} inΓ(R). Then for everyε >0, there exist a positive integerNsuch that

ρ x(ni),y

(j) n ≤ λ∈[0,1] λ sup n _x(i)−

n1 −x

(j)−

n1 1/n , sup n sup λ≤η≤1

x(niη)−−x

(j)−

nη

1/n

∨x(niη)+−x

(j)+

nη

1/n

=θ x(i),x(j)< ε

5 fori,j > Nn.

(3.4)

This implies that{x(ni)}i=∞1is a Cauchy sequence inF(R) for each fixedn.

(F(R),ρ) is a complete metric space; hence the Cauchy sequence{xn(i)}∞i=1converges to

xn, that isρ(xn(i),xn)=0 asi→ ∞for each fixedn

=⇒lim i→∞

λ∈[0,1]

λx(ni1)−−x

− n1

1/n_{, sup}

λ≤η≤1

x(niη)−−x

− nη

1/n_∨ x(niη)+−x

+ nη

1/n₌₀ _∀ n =⇒lim i→∞ λ∈[0,1] λ sup

(n) _x(i)−

n1 −x

− n1

1/n_{, sup}

(n)

sup λ≤η≤1

x(i)− nη −x

− nη

1/n_∨ x(i)+

nη −x

+ nη

1/n₌₀ _∀ n

=⇒lim i→∞θ x

(i)_,_x₌_0, _where_x_∈ _x n.

(3.5)

Now we will show thatx∈Γ(R). In (3.4), letting j→ ∞, we getρ(x(ni),xn)< ε/5, since

{xn(i)}is a Cauchy sequence for eachn.

This implies thatρ(x(ni),x_k(i))< ε/5 forn,k∈Ni.

Similarly,ρ(x(nj),x(_kj))< ε/5 forn,k∈Nj. For each fixed j, putN=max{Nn,Ni,Nj}. Then for givenε >0, there existx(i)_,_x(j)_∈_F₍_R_{) in connection with (3.4) such that}

ρx(nj),x(_kj)

<ε 5, ρ

x(ni),x

(j) n

<ε

5 fori,j,k,n > N

=⇒ρx_k(i),x_k(j)≤ρx(i) n ,x

(j) n

+ρx(i)

n ,x (i) k

+ρx(nj),x(_kj) ≤ε 5+ ε 5+ ε 5= 3ε 5 fori,j,k≥N

=⇒x(i) n

∞

i=1is a fuzzy Cauchy sequence inF(R)

=⇒ by the completeness ofF(R),ρx(_ki),xk

≤3ε

5 for somexk∈F(R)

=⇒ρ xn,xk≤ρ

xn,x(ni)

+ρx(i)

n ,x (i) k

+ρx(_ki),xk ≤ ε 5+ ε 5+ 3ε 5 =ε for eachn,k≥N

=⇒xn

is a Cauchy sequence with respect toθ.

(3.6)

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Theorem 3.4. The space of the fuzzy entire sequences is a fuzzy linear space.

Proof. Letα,β∈Γ(R) and letα=(an) andβ=(bn), whereρlimn→∞an=0 andρlimn→∞bn

=0. To prove aα+bβ∈Γ(R) and aα+bβ=a(an) +b(bn). It is enough to prove that

ρlimn→∞(aan+bbn)=0. Sinceρlimn→∞an=0, givenε >0, there exists a positive integer N1such thatρ(an,0)< ε/|a|for alln≥N1. Sinceρlimn→∞bn=0, givenε >0, there exists a positive integerN2such thatρ(bn,0)< ε/|b|for alln≥N2.

LetN=max{N1,N2}. Then

ρlim

n→∞ aan+bbn

=ρaan+bbn,0=ρ aan,0+ρ bbn,0=aρan,0+bρbn,0

= aε |a|+

bε

|b|=ε+ε < ε ∀n≥Nρ aan+bbn,0

< ε.

(3.7)

This implies thatρlimn→∞(aan+bbn)=0. Therefore,aα+bβ∈Γ(R). Hence,Γ(R) is a

fuzzy linear space.

Theorem 3.5. The space of the fuzzy entire sequence is a fuzzy linear metric space.

Proof. Γ(R) is a fuzzy metric space if we define the metricρby

ρ(α,β)=

λ∈[0,1]

λ an−1− bn −

1 1/n_{, sup}

λ≤η≤1

an−_η − bn−_η1/n∨ an+_η− bn+_η1/n

,

(3.8)

whereα,β∈Γ(R) andα=(an) andβ=(bn). To prove thatΓ(R) is a fuzzy linear metric space, it is enough to proveα+β,kαwhereα,β∈Γ(R) andk∈R+_{. That}_α₊_β_{is fuzzy} continuous follows from the property|α+β| ≤ |α| ⊕ |β|. To prove thatkαis continuous; it is enough to prove thatαn→αinΓ(R) andkn→k⇒knα→kαfor eachα∈Γ(R). Case 1. Letαn→αinΓ(R). To prove thatknα→kαsinceαn→α, givenε >0, there exists Nsuch thatρ(αn,α)< ε/|k|nfor alln≥N. Now

ρ kαn,kα

≤

λ∈[0,1] λ

sup

(n) _k_α_n

1−kα

1/n_{, sup}

(n)

sup λ≤η≤1

kα−_nη−kα

−1/n_∨

kα+_nη−kα

+1/n_,

ρ kαn,kα

≤

λ∈[0,1] λ

sup

(n)

|k|1/n_α n1−α

1/n_{, sup}

(n)

sup λ≤η≤1

|k|1/n_α− nη−α

−1/n_{∨ |} k|1/n_α+

nη−α

+1/n_,

ρ kαn,kα

≤ |k|1/n_ρ_α n,α

≤ |k|1/n_ε/_|_k_|1/n_{< ε.}

(3.9)

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Case 2. Let kn→k. To prove knα for each α∈Γ(R), considerk=0. Let α=(an)∈

Γ(R), whereρlimn→∞an=0. Sinceρlimn→∞an=0, givenε >0, there existsN1such that

ρ(an,0) < εfor alln≥N1. Sincekn→0, we may suppose thatρ(kn,0)<1 for alln≥N1. From the fuzzy metric space [6], obviously we get|knα| ≤εforn≥N1. That is,knα→0 asn→ ∞. This proves thatΓ(R) is a fuzzy linear metric space.

Theorem 3.6. A fuzzy entire sequence space is separable.

Proof. LetC= {x1,x2,x3,...,xn, 0, 0, 0,...}be a countable subset of Γ(R) and xi∈Q⊂

Γ(R), whereQ={all fuzzy rational numbers}. Hence,Γ(R) is separable.

4. A subset ofΓ(R)

In fact,χis a subset ofΓ. For each fixedn, define the fuzzy metric

μ xn,yn=

λ∈[0,1] λ

_∠_n_x_n− 1−∠n yn

− 1

1/n_,

sup λ≤η≤1

∠n xn −

η−∠n yn −

η 1/n_∨_∠

n xn +

η−∠n yn +

η 1/n

. (4.1)

Letx= {xn}andy= {yn}be sequence of fuzzy real numbers. Define their distance by

d(x,y)

=

λ∈[0,1] λ

sup

(n)

_∠_n _x_n−

1−∠n yn −

1 1/n_{, sup}

(n)

sup λ≤η≤1

∠n xn −

η −∠n yn −

η 1/n

∨∠n xn+η−∠n yn+η 1/n

.

(4.2)

Definition 4.1. x= {xn} ∈χifμ(limn→∞xn)=0. In other words, givenε >0 there exists a positive integerNsuch thatμ(xn, 0)< εfor alln∈N. The set of all fuzzy subsets ofΓ(R) is denoted byχ(R). Note thatχ(R)⊂Γ(R).

Proposition 4.2. χ(R) is a proper subset ofΓ(R).

Proof. Let |xn| ≤∠n|xn|. This implies that |xn|1/n≤(∠n|xn|)1/n. Let (xn)∈χ(R)⇒ (∠n|xn|)1/n< ε⇒ |xn|1/n< εfor alln≥n0⇒(xn)∈Γ(R)⇒χ(R)⊂Γ(R).

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Proof

Step 1. (1/∠n)∈/ χ(R) But (1/∠n)∈Γ(R). This implies thatχ(R) is a proper subspace of

Γ(R).

Step 2. Suppose thata(p)_→_a_∈_χ₍_R₎_{⇒ |}_a(p)

0 −a0|, [∠n|a (p)

n −an|]1/n< εforp≥p0,

=⇒∠nan1/n≤

∠na(np)1/n+∠na(np)−an1/n <∠na(np)1/n+ (∠n)1/nε/n forp≥p0

=⇒∠nan 1/n

< ε1+kε, for suﬃciently largen.

(4.3)

From Stirling’s formula, whereε→0 asp→ ∞andkis a positive constant

=⇒μ ∠nan1/n,0

< ε ∀n≥n0=⇒a= an

∈χ(R). (4.4)

Theorem 4.4. A subsetχ(R) ofΓ(R) is a fuzzy complete.

Proof. Let{ap:p≥1} be a fuzzy Cauchy sequence inχ(R), where|∠na(np)|1/n→0 as n→ ∞, for eachp≥1.

Letε >0, there existsN=N(ε) such that

supa(0p)−a (m) 0 ,

∠na(np)−a(nm) 1/n

,n≥1< ε form,p≥N

=⇒a(0p)−a (m) 0 < ε,

∠na(np)−a(nm) 1/n

< ε form,p≥N (n=1, 2,...) (4.5)

⇒ {∠n|a(np)|}1/nand so{a(np)}is a fuzzy Cauchy sequence in the complex plane for each n≥1.

=⇒a(np)−→an asp−→ ∞ ∀n≥1=⇒∠na(np)−→∠nan asp−→ ∞. (4.6)

Now forn≥1,p≥1,|∠nan|1/n≤[∠n|an−a(np)|]1/n+{∠n|a(np)|}1/n.

Let ε >0, then there exists p0 such that [∠n|an(p)−an|]1/n< ε and so this holds in particular forp=p0.

Consequently,

_∠_na_n1/n

< ε+∠na(p0)

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Thus,

μ lim

n→∞∠nan 1/n_≤

ε, =⇒lim n→∞μ

∠nan 1/n₌

0, =⇒a= an

∈χ(R). (4.8)

Also (4.5) and (4.6),|a(0p)−a0|< ε; [∠n|a (p)

n −an|]1/n< εforp≥N(n=1, 2,...)⇒ |ap− a| →0 asp→ ∞ ⇒χ(R) is a fuzzy complete ofΓ(R).

Corollary 4.5. (χ(R),θ) is a complete fuzzy metric space.

Proof. FromProposition 4.2andTheorem 3.3, it follows thatχ(R) is a closed subspace of the complete fuzzy metric space ofΓ(R). Hence,χ(R) is complete fuzzy metric space.

5. A superset ofΓ(R)

In fact,Λis a superset ofΓ.

Definition 5.1. x= {xn} ∈Λis called a fuzzy analytic sequenceΛ, ifρ(xn,0)≤M, for all nand some constantM >0. The set of all fuzzy analytic sequences is denoted byΛ(R). Note thatΓ(R)⊂Λ(R).

Without proof, we state the following theorems.

Theorem 5.2. (Λ(R),θ) is a complete fuzzy metric space.

Proof. It is similar to the proof ofTheorem 3.3.

Theorem 5.3. Γ(R) is a closed subspace ofΛ(R).

Proof. It is similar to the proof ofTheorem 4.3.

6. Conclusion

In this paper, we try to introduce the concept of fuzzy entire sequence spaces and we deal with some topological properties also. Many known sequence spaces can be fuzzified. There is considerable scope for further research in this area like in matrix transformation.

Acknowledgements

The authors wish to express their thanks to thier Rector and the Director of Science Stud-ies Center, UTHM, Malaysia. The authors are grateful to the referee for his valuable sug-gestion.

References

[1] L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, pp. 338–353, 1965. [2] M. Matloka, “Sequences of fuzzy numbers,” BUSEFAL, vol. 28, pp. 28–37, 1986.

[3] S. Nanda, “On sequences of fuzzy numbers,” Fuzzy Sets and Systems, vol. 33, no. 1, pp. 123–126, 1989.

[4] V. Ganapathy Iyer, “On the space of integral functions. I,” The Journal of the Indian Mathematical

Society. New Series, vol. 12, pp. 13–30, 1948.

[5] F. Nuray and E. Savas¸, “Statistical convergence of sequences of fuzzy numbers,” Mathematica

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[6] P. Diamond and P. Kloeden, “Metric spaces of fuzzy sets,” Fuzzy Sets and Systems, vol. 35, no. 2, pp. 241–249, 1990.

J. Kavikumar: Pusat Pengajian Sains, Universiti Tun Hussein Onn Malaysia, Parit Raja, Batu Pahat 86400, Malaysia

Email address:[email protected]

Azme Bin Khamis: Pusat Pengajian Sains, Universiti Tun Hussein Onn Malaysia, Parit Raja, Batu Pahat 86400, Malaysia

Email address:[email protected]

R. Kandasamy: Pusat Pengajian Sains, Universiti Tun Hussein Onn Malaysia, Parit Raja, Batu Pahat 86400, Malaysia