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# A Study of Fuzzy Sequence Spaces

By Vakeel A. Khan, Mobeen Ahmad and Masood Alam

Reviewed: September 25th 2020Published: January 12th 2022

DOI: 10.5772/intechopen.94202

## Abstract

The purpose of this chapter is to introduce and study some new ideal convergence sequence spaces FSJθT, FS0JθT and FS∞JθT on a fuzzy real number F defined by a compact operator T. We investigate algebraic properties like linearity, solidness and monotinicity with some important examples. Further, we also analyze closedness of the subspace and inclusion relations on the said spaces.

### Keywords

• Ideal
• J{convergence
• J{Cauchy
• Fuzzy number
• Lacunary sequence
• Compact operator

## 1. Introduction

The concepts of fuzzy sets were initiated by Zadeh [1], since then it has become an active area of researchers. Matloka [2] initiated the notion of ordinary convergence of a sequence of fuzzy real numbers and studied convergent and bounded sequences of fuzzy numbers and some of their properties, and proved that every convergent sequence of fuzzy numbers is bounded. Nanda [3] investigated some basic properties for these sequences and showed that the set of all convergent sequences of fuzzy real numbers form a complete metric space. Alaba and Norahun [4] studied fuzzy Ideals and fuzzy filters of pseudocomplemented semilattices Moreover, Nuray and Savas [5] extended the notion of convergence of the sequence of fuzzy real numbers to the notion of statistical convergence.

Fast [6] introduced the theory of statistical convergence. After that, and under different names, statistical convergence has been discussed in the ergodic theory, Fourier analysis and number theory. Furthermore, it was examined from the sequence space point of view and linked with summability theory. Esi and Acikgoz [7] examined almost λ-statistical convergence of fuzzy numbers. Kostyrko et al. [8] introduced ideal Jconvergence which is based on the natural density of the subsets of positive integers. Kumar and Kumar [9] extended the theory of ideal convergence to apply to sequences of fuzzy numbers. Khan et al. [10, 11, 12] studied the notion of J-convergence in intuitionistic fuzzy normed spaces. Subsequently, Hazarika [3] studied the concept of lacunary ideal convergent sequence of fuzzy real numbers. Where a lacunary sequence is an increasing integer sequence θ=krsuch that k0=0and hr=krkr1as r. The intervals are determined by θand defined by Ir=kr1kr.

We outline the present work as follows. In Section 2, we recall some basic definitions related to the fuzzy number, ideal convergent, monotonic sequence and compact operator. In Section 3, we introduce the spaces of fuzzy valued lacunary ideal convergence of sequence with the help of a compact operator and prove our maim results. In Section 4, we state the conclusion of this chapter.

## 2. Preliminaries

In this section, we recall some basic notion, definitions and lemma that are required for the following sections.

Definition2.1. A mapping F:Rλ=01is a real fuzzy number on the set Rassociating real number swith its grade of membership Fs. Let Cdenote the set of all closed and bounded intervals F=f1f2on the real line R. For G=g1g2in C, one define FGif and only if f1f2and g1g2. Determine a metric ρon Cby

ρFG=maxf1g1f2g2.E1

It can be easily seen that is a partial order on Cand Cρis a complete metric space. The absolute value Fof FRλis defined by

Fs=maxFsFs,ifs>00,ifs<0.

Suppose ρ¯:Rλ×RλRbe determined as

ρ¯FG=supρFG.

Hence, ρ¯defines a metric on Rλ. The multiplicative and additive identity in Rλare denoted by 1¯and 0¯, respectively.

Definition2.2. A family of subsets Jof the power set PNof the natural number Nis known as an ideal if and only if the following conditions are satisfied [8]

1. J,

2. for every A1,A2Jone obtain A1A2J,

3. for every A1Jand every A2A1one obtain A2J.

An ideal Jis known as non– trivial if JPNand non– trivial ideal is said to be an admissible if Jn:nN.

Lemma 1.If idealJis maximal, then for everyANwe have eitherAJorN\AJ[8].

Definition2.3. A family of subsets Hof the power set PNof the natural number Nis known as filter in if and only if following condition are satisfied [8].

1. H,

2. for every A1,A2Hone have A1A1H,

3. for every A1Hand A2A1one have A2H.

Remark1. Filter associated with the ideal Jis defined by the family of sets

HJ=KN:AJ:K=N\A

Definition2.4. A sequence Fkof fuzzy real numbers is known as Jconvergent to fuzzy real numbers F0if for each ε>0, the set [9].

kN:ρ¯FkF0εJ.

Definition2.5. A sequence Fkis known as Jnull if there exists a fuzzy real numbers 0¯such that for each ε>0[9],

kN:ρ¯Fk0¯εJ.

Definition2.6. A sequence Fkof fuzzy real numbers is known as J-Cauchy if there exists a subsequence Flεof Fkin such a way that for every ε>0[13],

kN:ρ¯FkFlεεJ.

Definition2.7. A sequence Fkis known as Jbounded if there exists a fuzzy real numbers M>0so that, the set [9].

kN:ρ¯Fk0¯>MJ.

Definition2.8. Suppose K=kiN:k1<k2<Nand Ebe a sequence space. A K– step space of Eis a sequence space [12].

ΛKE=xkiω:xkE.

The canonical pre-image of a sequence xkiλKEis a sequence ykωdefined as follows:

yk=xk,ifnK0,otherwise.

yis in canonical pre-image of ΛKEif yis canonical pre-image of some element xΛKE.

Definition2.9. A sequence space Eis known as monotone, if it is contains the canonical pre-images of it is step space [12].

That is, if for all infinite KNand xkEthe sequence αkxk, where αk=1for kKand αk=0otherwise, belongs to E.

Definition2.10. [12] A sequence space Eis known as convergent free, if xkEwhenever ykEand yk=0implies that xk=0for all kN.

Lemma 2.1.Every solid space is monotone [12].

Definition2.11. Suppose Uand Vare normed spaces. An operator T:UVis known as compact linear operator if [12].

1. Tis linear

2. Tmaps every bounded sequence xkin Uonto a sequence Txkin Vwhich has a convergent subsequence.

## 3. Main results

In this section, we introduce the spaces of fuzzy valued lacunary ideal convergence of sequence with the help of a compact operator and investigate some topological and algebraic properties on these spaces. We denote ωFthe class of all sequences of fuzzy real numbers and Jbe an admissible ideal of the subset of the natural numbers N.

FSJθT=F=FkωF:{rN:hr1kJrρ¯(TFkF0)ε}JforsomeF0Rλ,
FS0JθT=F=FkωF:{rN:hr1kJrρ¯(TFk0¯)ε}J,
FSJθT=F=FkωF:M>0:{rN:hr1kJrρ¯(TFk0¯)M}J,
FSθT=F=FkωF:suprhr1kJrρ¯(TFk0¯)<.

Theorem 3.1.The sequence spaces FSJθT,FS0JθTand FSJTare linear spaces.

Proof.Suppose αand βbe scalars, and assume that F=Fk, G=gkFSJθT. Since Fk,GkFSJθT. Then for a given ε>0, there exists F1,F2in such a manner that

rN:hr1kJrρ¯(TFkF1)ε2J.

and

rN:hr1kJrρ¯(TFkF2)ε2J.

Now, let

A1=rN:hr1kJrρ¯(TFkF1)<t2αHJ.
A2=rN:hr1kJrρ¯(TFkF2)<t2βHJ.

be such that A1c,A2cJ. Therefore, the set

A3=rN:hr1kJrρ¯(TαFk+βGkαL1+βL2)<εrN:hr1kJrρ¯(TFkL1)<ε2αrN:hr1kJrρ¯(TFkL1)<ε2β.E2

Thus, the sets on right hand side of (2) belong to HJ. Therefore A3cbelongs to J. Therefore, αFk+βGkFSJθT. Hence FSJθTis linear space.

In similar manner, one can easily prove that FS0JθTand FSJTare linear spaces. □

Example 3.1.Suppose J=Jρ=BN:ρB=0, where ρBdenotes the asymptotic density of B. In this case FSJρT=FSθT, where

FSθT=FkωF:ρrN:hr1kJrρ¯TFkF0ε=0forsomeF0Rγ.

Example 3.2.Suppose J=Jf=BN:Bisfinite. Jfis an admissible ideal in Nand FSJfT=FSθT.

Theorem 3.2.The spaces FSJθTand FS0JθTare not convergent free.

Proof.For the proof of the theorem, we consider the subsequent example. □

Example 3.3.Suppose J=Jδand TFk=Fk.

Consider the sequence FkFS0JθTFSJθTas:

For ki2,iN

Fks=1,if0sk10,otherwise.

For k=i2,iN,Fks=0¯. Therefore

Fk=00,ifk2=i0k1,ki2.

Hence, Fk0¯ask. Thus FkFS0JθTFSJθT. Let Gkbe sequence in such a way that, for k=i2,iN,Fks=0¯. Therefore, one obtain

Gks=1,if0sk0,otherwise.

For k=i2,iN,Fks=0¯. Therefore

Gk=00,ifk2=i0k,ki2.

It can be easily seen that GkFS0JθTFSJθT.

Hence FSJθTand FS0JθTare not convergence free.

Theorem 3.3.The sequence F=FkFSθTis Jconvergent if and only if for every ε>0, there exists NεNin such a way that

rN:hr1kJrρ¯(TFkTFNε)<εHJ.E3

Proof.Suppose that F=FkFSθTLet F0=JlimF. Then for every ε>0, the set

Bε=rN:hr1kJrρ¯(TFkF0)<ε2HJ.

Fix an NεBε. Then, we have

ρ¯TFkTFNερ¯TFkF0
+ρ¯TFNεF0<ε2+ε2=ε.

Which holds for all NεBε. Hence

rN:hr1kJrρ¯(TFkTFNε)<εHJ.

On Contrary, assume that

rN:hr1kJrρ¯(TFkTFNε)<εHJ.

That is

rN:hr1kJrρ¯(TFkTFNε)<εHJforallε>0.

Then, the set

Cε=kN:TFk[TFNεεTFNε+ε]HJforallε>0.

Let Iε=TFNεεTFNε+ε. If we fix an ε>0, then we have CεHJas well as Cε2HJ. Hence Cε2CεHJ. This implies that I=IεIε2. That is

kN:TFkIHJ.

That is

diamIdiamIε.

Where the diam of Idenotes the length of interval I. Continuing in this way, by induction, we get the sequence of closed intervals.

Iε=I0I1Ik

With the property that

diamIk12diamIk1fork=2,3,4

and

kN:TFkIkHJfork=2,3,4.

Then there exists a ξIkwhere kNin such a way that ξ=JlimTFk. Therefore, the result holds. □

Theorem 3.4.The inclusionsFS0JθTFSJθTFSJθThold.

Proof.Let F=FkFSJθT. Then there exists a number F0Rsuch that

Jlimkρ¯TFkF0=0.

That is, the set

rN:hr1kJrρ¯(TFkF0)εJ.

Here

hr1kJrρ¯TFk0¯=hr1kIrρ¯TFkF0+F0hr1kIrρ¯TFkF0+hr1kJrρ¯TFkF0E4

On the both sides, taking the supremum over kof the above equation, we obtain FkFSJθT. Therefore inclusion holds.

rN:hr1kJrρ¯(TFkpap)εJ.

Then, it proves that FS0JθTFSJθT. Hence FS0JθTFSJθTFSJθT. □

Theorem 3.5.The space FSJθTis neither normal nor monotone if Jis not maximal ideal.

Example 3.4.Suppose fuzzy number

Gks=1+s2,if1s13s2,if1s30,otherwise.

Then GksFSJθT. Applying Lemma 1, there exists a subset Kof Nin such a way that KJand N\KJ. Determine a sequence G=Gkas

Gk=Fk,kK0,otherwise.

Therefore Gkbelong to the canonical pre- image of the K- step spaces of FSJθT. But GkFSJθT. Hence FSJθTis not monotone. Hence, by Lemma (2.1) FSJθTis not normal.

Theorem 3.6.The sequence space FS0JθTis solid and monotone.

Proof.Suppose F=FkFS0JθT, then for ε>0, the set

rN:hr1kJrρ¯(TFkF0)εJ.E5

Suppose that sequence of scalars αkwith the property αk1kN. Therefore

ρ¯TαkFkF0=ρ¯αkTFkF0αkρ¯TFkF0forallkN.E6

Hence, from the Eq. (5) and above inequality, one obtain

rN:hr1kJrρ¯(TαkFkF0)ε
rN:hr1kJrρ¯(TFkF0)εJ.

Implies

rN:hr1kJrρ¯(TαkFkF0)εJ.

Therefore, αkFkFS0JθT. Then FS0JθTis solid and monotone by lemma 2.1. □

Theorem 3.7.The sequence space FSJθTis closed subspace of FSθT.

Proof.Suppose Fkqbe a Cauchy sequence in FSJθT. Then, FkqFin FSTas q. Since FkqFSJθT, then for each ε>0there exists aqsuch that converges to a.

JlimF=a.

Since Fkqbe a Cauchy sequence in FSIθT. Then for each ε>0there exists n0N, such that

supkρ¯TFkqTFks<ε3forallq,sn0.

For a given ε>0,

Bq,s=rN:hr1kJrρ¯(TFkqTFks)<ε3.

Now,

Bq=rN:hr1kJrρ¯(TFkpaq)<ε3

and

Bs=rN:hr1kIrρ¯(TFksas)<ε3.

Then Bq,sc,Bqc,BscJ. Let Bc=Bq,scBqcBsc. Where

B=rN:hr1kJrρ¯aqas<εthenBcJ.

Assume n0Bc. Then, for every q,sn0it has

ρ¯aqasρ¯TFkqaq+ρ¯TFksas+ρ¯TFkqTFks<ε3+ε3+ε3=ε.

Hence, aqis a Cauchy sequence of scalars in , so there exists scalar ain such a way that aqaas q. For this step, let 0<α<1be given. Therefore it proved that whenever

U=rN:hr1kJrρ¯(TFa)<αthenUcJ.E7

Since FkqF, there exists q0Nso that

P=rN:hr1kJrρ¯(TFq0TF)<α3.

implies that PcJ. The number, qocan be chosen together with Eq. (7), it have

Q=rN:hr1kJrρ¯aq0a<α3.

Which implies that QcJ. Since

rN:hr1kJrρ¯(TFkq0aq0)α3J.

Then it has a subset Ssuch that ScJ, where

S=rN:hr1kJrρ¯(TFkq0aq0)<α3.

Suppose that Uc=PcQcSc. Then, for every kUcit has

rN:hr1kJrρ¯TFa<αrN:hr1kIrρ¯(TFq0F)<α3rN:hr1kJrρ¯aq0a<α3rN:hr1kJrρ¯(TFkq0aq0)<α3.E8

The right hand side of Eq. (8) belongs to HJ. Hence, the sets on the left hand side of Eq. (8) belong to HJ. Therefore its complement belongs to J. Thus, Jlimρ¯aq0a=0. □

In the following example to prove that FSJθTis closed subspace of FSθT.

Example 3.5.Suppose that sequence of fuzzy number determine by

Fks=211+s,fors11213s,fors130,otherwise.

Hence, ρ¯Fk0¯=supρFk0¯. Therefore FkFSJθTand L=0¯. So, it can be easily seen that FSJθTis closed subspace of FSθT.

## 4. Conclusion

The spaces of fuzzy valued lacunary ideal convergence of sequence with the help of a compact operator and investigate algebraic and topological properties together with some examples on the given spaces. We proved that the new introduced sequence spaces are linear. Some spaces are convergent free and we also proved that space FSJθTis closed subspace of FSθT. These new spaces and results provide new tools to help the authors for further research and to solve the engineering problems.

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Vakeel A. Khan, Mobeen Ahmad and Masood Alam (January 12th 2022). A Study of Fuzzy Sequence Spaces, Fuzzy Systems - Theory and Applications, Constantin Volosencu, IntechOpen, DOI: 10.5772/intechopen.94202. Available from:

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