Open access peer-reviewed chapter

A Study of Fuzzy Sequence Spaces

Written By

Vakeel A. Khan, Mobeen Ahmad and Masood Alam

Reviewed: 25 September 2020 Published: 12 January 2022

DOI: 10.5772/intechopen.94202

Chapter metrics overview

196 Chapter Downloads

View Full Metrics


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.


  • 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=kr such that k0=0 and hr=krkr1 as r. The intervals are determined by θ and 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.

Definition 2.1. A mapping F:Rλ=01 is a real fuzzy number on the set R associating real number s with its grade of membership Fs. Let C denote the set of all closed and bounded intervals F=f1f2 on the real line R. For G=g1g2 in C, one define FG if and only if f1f2 and g1g2. Determine a metric ρ on C by


It can be easily seen that is a partial order on C and Cρ is a complete metric space. The absolute value F of FRλ is defined by


Suppose ρ¯:Rλ×RλR be determined as


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

Definition 2.2. A family of subsets J of the power set PN of the natural number N is known as an ideal if and only if the following conditions are satisfied [8]

  1. J,

  2. for every A1,A2J one obtain A1A2J,

  3. for every A1J and every A2A1 one obtain A2J.

An ideal J is known as non– trivial if JPN and non– trivial ideal is said to be an admissible if Jn:nN.

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

Definition 2.3. A family of subsets H of the power set PN of the natural number N is known as filter in if and only if following condition are satisfied [8].

  1. H,

  2. for every A1,A2H one have A1A1H,

  3. for every A1H and A2A1 one have A2H.

Remark 1. Filter associated with the ideal J is defined by the family of sets


Definition 2.4. A sequence Fk of fuzzy real numbers is known as J convergent to fuzzy real numbers F0 if for each ε>0, the set [9].


Definition 2.5. A sequence Fk is known as J null if there exists a fuzzy real numbers 0¯ such that for each ε>0 [9],


Definition 2.6. A sequence Fk of fuzzy real numbers is known as J-Cauchy if there exists a subsequence Flε of Fk in such a way that for every ε>0 [13],


Definition 2.7. A sequence Fk is known as J bounded if there exists a fuzzy real numbers M>0 so that, the set [9].


Definition 2.8. Suppose K=kiN:k1<k2<N and E be a sequence space. A K– step space of E is a sequence space [12].


The canonical pre-image of a sequence xkiλKE is a sequence ykω defined as follows:


y is in canonical pre-image of ΛKE if y is canonical pre-image of some element xΛKE.

Definition 2.9. A sequence space E is known as monotone, if it is contains the canonical pre-images of it is step space [12].

That is, if for all infinite KN and xkE the sequence αkxk, where αk=1 for kK and αk=0 otherwise, belongs to E.

Definition 2.10. [12] A sequence space E is known as convergent free, if xkE whenever ykE and yk=0 implies that xk=0 for all kN.

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

Definition 2.11. Suppose U and V are normed spaces. An operator T:UV is known as compact linear operator if [12].

  1. T is linear

  2. T maps every bounded sequence xk in U onto a sequence Txk in V which 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 ωF the class of all sequences of fuzzy real numbers and J be an admissible ideal of the subset of the natural numbers N.


Theorem 3.1. The sequence spaces FSJθT,FS0JθT and FSJT are linear spaces.

Proof. Suppose α and β be scalars, and assume that F=Fk, G=gkFSJθT. Since Fk,GkFSJθT. Then for a given ε>0, there exists F1,F2 in such a manner that




Now, let


be such that A1c,A2cJ. Therefore, the set


Thus, the sets on right hand side of (2) belong to HJ. Therefore A3c belongs to J. Therefore, αFk+βGkFSJθT. Hence FSJθT is linear space.

In similar manner, one can easily prove that FS0JθT and FSJT are linear spaces. □

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


Example 3.2. Suppose J=Jf=BN:Bisfinite. Jf is an admissible ideal in N and FSJfT=FSθT.

Theorem 3.2. The spaces FSJθT and FS0JθT are not convergent free.

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

Example 3.3. Suppose J=Jδ and TFk=Fk.

Consider the sequence FkFS0JθTFSJθT as:

For ki2,iN


For k=i2,iN,Fks=0¯. Therefore


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


For k=i2,iN,Fks=0¯. Therefore


It can be easily seen that GkFS0JθTFSJθT.

Hence FSJθT and FS0JθT are not convergence free.

Theorem 3.3. The sequence F=FkFSθT is J convergent if and only if for every ε>0, there exists NεN in such a way that


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


Fix an NεBε. Then, we have


Which holds for all NεBε. Hence


On Contrary, assume that


That is


Then, the set


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


That is


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


With the property that




Then there exists a ξIk where kN in such a way that ξ=JlimTFk. Therefore, the result holds. □

Theorem 3.4.The inclusionsFS0JθTFSJθTFSJθThold.

Proof. Let F=FkFSJθT. Then there exists a number F0R such that


That is, the set




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


Then, it proves that FS0JθTFSJθT. Hence FS0JθTFSJθTFSJθT. □

Theorem 3.5. The space FSJθT is neither normal nor monotone if J is not maximal ideal.

Example 3.4. Suppose fuzzy number


Then GksFSJθT. Applying Lemma 1, there exists a subset K of N in such a way that KJ and N\KJ. Determine a sequence G=Gk as


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

Theorem 3.6. The sequence space FS0JθT is solid and monotone.

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


Suppose that sequence of scalars αk with the property αk1kN. Therefore


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




Therefore, αkFkFS0JθT. Then FS0JθT is solid and monotone by lemma 2.1. □

Theorem 3.7. The sequence space FSJθT is closed subspace of FSθT.

Proof. Suppose Fkq be a Cauchy sequence in FSJθT. Then, FkqF in FST as q. Since FkqFSJθT, then for each ε>0 there exists aq such that converges to a.


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


For a given ε>0,






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


Assume n0Bc. Then, for every q,sn0 it has


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


Since FkqF, there exists q0N so that


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


Which implies that QcJ. Since


Then it has a subset S such that ScJ, where


Suppose that Uc=PcQcSc. Then, for every kUc it has


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 FSJθT is closed subspace of FSθT.

Example 3.5. Suppose that sequence of fuzzy number determine by


Hence, ρ¯Fk0¯=supρFk0¯. Therefore FkFSJθT and L=0¯. So, it can be easily seen that FSJθT is 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 FSJθT is 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.


  1. 1. Esi A, Mehmet Acikgoz, On almost λ-statistical convergence of fuzzy numbers, Acta Scientiarum.Technology, 2014, vol. 36(1), 129-133.
  2. 2. Alaba BA, Norahun WZ, Fuzzy Ideals and Fuzzy Filters of Pseudocomplemented Semilattices, Advances in Fuzzy Systems, 2019, Vol. (2019), 1-13.
  3. 3. Hazarika B, Fuzzy real valued lacunary I-convergent sequences, Applied Mathematics Letters, 2012, 25(3):466-470.
  4. 4. Nuray F, Savas E, Statistical convergence of sequences of fuzzy numbers, Mathematica Slovaca, 1995, 45(3):269-273.
  5. 5. Fast H, Sur la convergence statistique Colloquium Math., 1951, (2), 241-244.
  6. 6. Zadeh LA, Fuzzy sets. Information and control, 1965, 8(3):338-353.
  7. 7. Matloka M, Sequences of fuzzy numbers, Busefal, 1986, 28(1):28-37.
  8. 8. Kostyrko P, Wilczynski W, Salat T, I-convergence, Real Analysis Ex-change, 2000, 26(2):669-686.
  9. 9. Nanda S, On sequences of fuzzy numbers, Fuzzy sets and systems, 1989, 33(1):123-126.
  10. 10. Khan VA, Ahmad M, Fatima F, Khna MF, On some results in intuitionistic fuzzy ideal convergence double sequence spaces, Advances in Difference Equations, 2019, vol.(2019): 1-10.
  11. 11. Khan VA, Kara EE, Altaf H, Ahmad M, Intuitionistic fuzzy I-convergent Fibonacci difference sequence spaces, Journal of Inequalities and Applications, 2019, Vol.(2019)(1), 1-7
  12. 12. Khan VA, Rababah RKA, Esi A, Sameera Ameen Ali Abdullah, and Kamal Mohammed Ali Suleiman Alshlool, Some new spaces of ideal convergent double sequences by using compact operator, Journal of Applied Sciences, 2017, 17(9):467-474.
  13. 13. Kumar V, Kumar K, On the ideal convergence of sequences of fuzzy numbers, Information Sciences, 2008, 178(24):4670-4678.

Written By

Vakeel A. Khan, Mobeen Ahmad and Masood Alam

Reviewed: 25 September 2020 Published: 12 January 2022