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Some Topological Properties of Intuitionistic Fuzzy Normed Spaces

By Vakeel Ahmad Khan, Hira Fatima and Mobeen Ahmad

Submitted: October 15th 2018Reviewed: November 14th 2018Published: April 25th 2019

DOI: 10.5772/intechopen.82528

Downloaded: 103

Abstract

In 1986, Atanassov introduced the concept of intuitionistic fuzzy set theory which is based on the extensions of definitions of fuzzy set theory given by Zadeh. This theory provides a variable model to elaborate uncertainty and vagueness involved in decision making problems. In this chapter, we concentrate our study on the ideal convergence of sequence spaces with respect to intuitionistic fuzzy norm and discussed their topological and algebraic properties.

Keywords

  • ideal
  • intuitionistic fuzzy normed spaces
  • Orlicz function
  • compact operator
  • I-convergence

1. Introduction

In recent years, the fuzzy theory has emerged as the most active area of research in many branches of mathematics, computer and engineering [1]. After the excellent work of Zadeh [2], a large number of research work have been done on fuzzy set theory and its applications as well as fuzzy analogues of the classical theories. It has a wide number of applications in various fields such as population dynamics [3], nonlinear dynamical system [4], chaos control [5], computer programming [6], etc. In 2006, Saadati and Park [7] introduced the concept of intuitionistic fuzzy normed spaces after that the concept of statistical convergence in intuitionistic fuzzy normed space was studied for single sequence in [8]. The study of intuitionistic fuzzy topological spaces [9], intuitionistic fuzzy 2-normed space [10] and intuitionistic fuzzy Zweier ideal convergent sequence spaces [11] are the latest developments in fuzzy topology.

First, let us recall some notions, basic definitions and concepts which are used in sequel.

Definition 1.1. (See Ref. [7]). The five-tuple Xμνis said to be an intuitionistic fuzzy normed space (for short, IFNS) if Xis a vector space, is a continuous t-norm, is a continuous t-conorm, and μand νare fuzzy sets on X×0satisfying the following conditions for every x,yXand s,t>0:

(a) μxt+νxt1,

(b) μxt>0,

(c) μxt=1if and only if x=0,

(d) μαxt=μxtαfor each α0,

(e) μxtμysμx+yt+s,

(f) μx.:001is continuous,

(g) limtμxt=1and limt0μxt=0,

(h) νxt<1,

(i) νxt=0if and only if x=0,

(j) ναxt=νxtαfor each α0,

(k) νxtνysνx+yt+s,

(l) νx.:001is continuous,

(m) limtνxt=0and limt0νxt=1.

In this case μνis called an intuitionistic fuzzy norm.

Example 1.1. Let X.be a normed space. Denote ab=aband ab=mina+b1for all a,b01and let μ0and ν0be fuzzy sets on X×0defined as follows:

μ0xt=tt+x,andν0xt=xt+x

for all t+. Then Xμνis an intuitionistic fuzzy normed space.

Definition 1.2. Let Xμνbe an IFNS. Then a sequence x=xkis said to be convergent to LXwith respect to the intuitionistic fuzzy norm μνif, for every ε>0and t>0, there exists k0such that μxkLt>1εand νxkLt<εfor all kk0. In this case we write μν-limx=L.

In 1951, the concept of statistical convergence was introduced by Steinhaus [12] and Fast [13] in their papers “Sur la convergence ordinaire et la convergence asymptotique” and “Sur la convergence statistique,” respectively. Later on, in 1959, Schoenberg [14] reintroduced this concept. It is a very useful functional tool for studying the convergence of numerical problems through the concept of density. The concept of ideal convergence, which is a generalization of statistical convergence, was introduced by Kostyrko et al. [15] and it is based on the ideal Ias a subsets of the set of positive integers and further studied in [16, 17, 18, 19, 20].

Let Xbe a non-empty set then a family I2Xis said to be an ideal in Xif I, Iis additive, i.e., for all A,BIABIand Iis hereditary, i.e., for all AI,BABI. A non empty family of sets F2Xis said to be a filter on Xif for all A,BFimplies ABFand for all AFwith ABimplies BF. An ideal I2Xis said to be nontrivial if I2X, this non trivial ideal is said to be admissible if Ix:xXand is said to be maximal if there cannot exist any nontrivial ideal JIcontaining Ias a subset. For each ideal I, there is a filter FIcalled as filter associate with ideal I, that is (see [15]),

FI=KX:KcI,whereKc=X\K.E1

A sequence x=xkωis said to be I-convergent [21, 22] to a number Lif for every ε>0, we have k:xkLεI.In this case, we write Ilimxk=L.

2. IF-ideal convergent sequence spaces using compact operator

This section consists of some double sequence spaces with respect to intuitionistic fuzzy normed space and study the fuzzy topology on the said spaces. First we recall some basic definitions on compact operator.

Definition 2.1. (See [23]). Let Xand Ybe two normed linear spaces and T:DTYbe a linear operator, where DX.Then, the operator Tis said to be bounded, if there exists a positive real ksuch that

Txkx,forallxDT.

The set of all bounded linear operators BXY[24] is a normed linear spaces normed by

T=supxX,x=1Tx

and BXYis a Banach space if Yis a Banach space.

Definition 2.2. (See [23]). Let Xand Ybe two normed linear spaces. An operator T:XYis said to be a compact linear operator (or completely continuous linear operator), if

(i) Tis linear,

(ii) Tmaps every bounded sequence xkin Xon to a sequence Txkin Ywhich has a convergent subsequence.

The set of all compact linear operators CXYis a closed subspace of BXYand CXYis Banach space, if Yis a Banach space.

In 2015, Khan et al. [11] introduced the following sequence spaces:

ZμνI=xkω:k:μxk/Lt1εorνxk/LtεI,
Z0μνI=xkω:k:μxk/t1εorνxk/tεI.

Motivated by this, we introduce the following sequence spaces with the help of compact operator in intuitionistic fuzzy normed spaces:

MμνIT={xk:{k:μTxkLt1εorνTxkLtε}I}E2
M0μνIT={xk:{k:μTxkt1εorνTxktε}I}.E3

Here, we also define an open ball with center xand radius rwith respect to tas follows:

BxrtT={yk:{k:μTxkTykt1εorνTxkTyktε}I}.E4

Now, we are ready to state and prove our main results. This theorem is based on the linearity of new define sequence spaces which is stated as follows.

Theorem 2.1. The sequence spaces MμνITand M0μνITare linear spaces.

Proof. Let x=xk,y=ykMμνITand α,βbe scalars. Then for a given ε>0, we have the sets:

P1=k:μTxkL1t2α1εorνTxkL1t2αεI;
P2=k:μTykL2t2β1εorνTykL2t2βεI.

This implies

P1c=k:μTxkL1t2α>1εorνTxkL1t2α<εFI;
P2c=k:μTykL2t2β>1εorν(TykL2t2β<εFI.

Now, we define the set P3=P1P2, so that P3I. It shows that P3cis a non-empty set in FI. We shall show that for each xk,ykMμνIT.

P3c{k:μαTxk+βTykαL1+βL2t>1εorναTxk+βTykαL1+βL2t<ε}.

Let mP3c, in this case

μTxmL1t2α>1εorνTxmL1t2α<ε

and

μTymL2t2β>1εorνTymL2t2β<ε.

Thus, we have

μαTxm+βTymαL1+βL2tμαTxmαL1t2μβTxmβL2t2=μTxmL1t2αμTxmL2t2β>1ε1ε=1ε.

and

ναTxm+βTymαL1+βL2tναTxmαL1t2νβTxmβL2t2=μTxmL1t2αμTxmL2t2β<εε=ε.

This implies that

P3c{k:μαTxk+βTykαL1+βL2t>1εorναTxk+βTykαL1+βL2t<ε.

Therefore, the sequence space MμνITis a linear space.

Similarly, we can proof for the other space. □

In the following theorems, we discussed the convergence problem in the said sequence spaces. For this, firstly we have to discuss about the topology of this space. Define

τμνIT={AMμνIT:foreachxAthereexistst>0andr01suchthatBxrtTA}.

Then τμνITis a topology on MμνIT.

Theorem 2.2. Let MμνITis an IFNS and τμνITis a topology on MμνIT. Then a sequence xkMμνIT,xkxif and only if μTxkTxt1and νTxkTxt0as k.

Proof. Fix t0>0. Suppose xkx. Then for r01, there exists n0such that xkBxrtTfor all kn0. So, we have

Bxrt0T=k:μTxkTxt1rorνTxkTxt0rI,

such that Bxcrt0TFI. Then 1μTxkTxt0<rand νTxkTxt0<r. Hence μTxkTxt01and νTxkTxt00as k.

Conversely, if for each t>0,μTxkTxt1and νTxkTxt0as k, then for r01,there exists n0, such that 1μTxkTxt<rand νTxkTxt<r, for all kn0. It shows that μTxkTxt>1rand νTxkTxt<rfor all kn0.Therefore xkBxcrtTfor all kn0and hence xkx.

There are some facts that arise in connection with the convergence of sequences in these spaces. Let us proceed to the next theorem on Ideal convergence of sequences in these new define spaces.

Theorem 2.3. A sequence x=xkMμνITis I-convergent if and only if for every ε>0and t>0there exists a number N=Nxεtsuch that

N:μTxNLt2>1εorνTxNLt2<εFI.

Proof. Suppose that Iμνlimx=Land let t>0. For a given ε>0, choose s>0such that 1ε1ε>1sand εε<s.Then for each xMμνIT,

R=k:μTxkLt21εorνTxkLt2εI,

which implies that

Rc=k:μTxkLt2>1εorνTxkLt2<εFI.

Conversely, let us choose NRc. Then

μTxNLt2>1εorνTxNLt2<ε.

Now, we want to show that there exists a number N=Nxεtsuch that

k:μTxkTxNt1sorνTxkTxNtsI.

For this, we define for each xMμνIT

S=k:μTxkTxNt1sorνTxkTxNtsI.

So, we have to show that SR. Let us suppose that SR, then there exists nSand nR. Therefore, we have

μTxnTxNt1sorμTxnLt2>1ε.

In particular μTxNLt2>1ε.Therefore, we have

1sμTxnTxNtμTxnLt2μTxNLt21ε1ε>1s,

which is not possible. On the other hand

νTxnTxNtsorνTxnLt2<ε.

In particular νTxNLt2<ε.So, we have

sνTxnTxNtνTxnLt2νTxNLt2εε<s,

which is not possible. Hence SR. RIwhich implies SI.□

3. IF-ideal convergent sequence spaces using Orlicz function

In this section, we have discussed the ideal convergence of sequences in Intuitionistic fuzzy I-convergent sequence spaces defined by compact operator and Orlicz function. We shall now define the concept of Orlicz function, which is basic definition in our work.

Definition 3.1. An Orlicz function is a function F:00,which is continuous, non-decreasing and convex with F0=0,Fx>0for x>0and Fxas x. If the convexity of Orlicz function Fis replaced by Fx+yFx+Fy, then this function is called modulus function.

Remark 3.1. If Fis an Orlicz function, then FλxλFxfor all λwith 0<λ<1.

In 2009, Mohiuddine and Lohani [18] introduced the concept of statistical convergence in intuitionistic fuzzy normed spaces in their paper published in Chaos, Solitons and Fractals. This motivated us to introduced some sequence spaces defined by compact operator and Orlicz function which are as follows:

MμνITF=xk:{k:FμTxkLtρ1εorFνTxkLtρε}I;E5
M0μνITF=xk:{k:FμTxktρ1εorFνTxktρε}I.E6

We also define an open ball with center xand radius rwith respect to tas follows:

BxrtTF=yk:{k:FμTxkTyktρ1εorFνTxkTyktρε}I.E7

We shall now consider some theorems of these sequence spaces and invite the reader to verify the linearity of these sequence spaces.

Theorem 3.1. Every open ball BxrtTFis an open set in MμνITF.

Proof. Let BxrtTFbe an open ball with center xand radius rwith respect to t. That is

BxrtTF=y=yk:k:FμTxkTyktρ1rorFνTxkTyktρrI.

Let yBxcrtTF, then FμTxkTyktρ>1rand

FνTxkTyktρ<r.Since FμTxkTyktρ>1r,there exists t00tsuch that FμTxkTykt0ρ>1rand FνTxkTykt0ρ<r.

Putting r0=FμTxkTykt0ρ, so we have r0>1r, there exists s01such that r0>1s>1r. For r0>1s, we have r1,r201such that r0r1>1sand 1r01r0s.Putting r3=maxr1r2. Now we consider a ball Byc1r3tt0TF. And we prove that

Byc1r3tt0TFBxcrtTF.

Let z=zkByc1r3tt0TF, then FμTykTzktt0ρ>r3and FνTykTzktt0ρ<1r3. Therefore, we have

FμTxkTzktρFμTxkTykt0ρFμTykTzktt0ρr0r3r0r11s1r

and

FνTxkTzktρFνTxkTykt0ρFνTykTzktt0ρ1r01r31r01r2sr.

Thus zBxcrtTFand hence, we get

Byc1r3tt0TFBxcrtTF.

Remark 3.2. MμνITFis an IFNS.

Define

τμνITF={AMμνITF:foreachxAthereexistst>0andr01suchthatBxrtTFA}.

Then τμνITFis a topology on MμνITF.

In the above result we can easily verify that the open sets in these spaces are open ball in the same spaces. This theorem itself will have various applications in our future work.

Theorem 3.2. The topology τμνITFon M0μνITFis first countable.

Proof. Bx1n1nTF:n=1,2,3is a local base at x, the topology τμνITFon M0μνITFis first countable.□

Theorem 3.3. MμνITFand M0μνITFare Hausdorff spaces.

Proof. Let x,yMμνITFsuch that xy. Then 0<FμTxTytρ<1and 0<FνTxTytρ<1.

Putting r1=FμTxTytρ, r2=FνTxTytρand r=maxr11r2.For each r0r1there exists r3and r4such that r3r4r0and 1r31r41r0.

Putting r5=maxr31r4and consider the open balls Bx1r5t2and By1r5t2. Then clearly Bxc1r5t2Byc1r5t2=ϕ. For if there exists zBxc1r5t2Byc1r5t2, then

r1=FμTxTytρμTxTzt2ρFμTzTyt2ρr5r5r3r3r0>r1

and

r2=FνTxTytρFνTxTzt2ρFνTzTyt2ρ1r51r51r41r41r0<r2

which is a contradiction. Hence, MμνITFis Hausdorff. Similarly the proof follows for M0μνITF.□

4. Conclusion

The concept of defining intuitionistic fuzzy ideal convergent sequence spaces as it generalized the fuzzy set theory and give quite useful and interesting applications in many areas of mathematics and engineering. This chapter give brief introduction to intuitionistic fuzzy normed spaces with some basic definitions of convergence applicable on it. We have also summarized different types of sequence spaces with the help of ideal, Orlicz function and compact operator. At the end of this chapter some theorems and remarks based on these new defined sequence spaces are discussed for proper understanding.

Conflict of interest

The authors declare that they have no competing interests.

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Vakeel Ahmad Khan, Hira Fatima and Mobeen Ahmad (April 25th 2019). Some Topological Properties of Intuitionistic Fuzzy Normed Spaces [Online First], IntechOpen, DOI: 10.5772/intechopen.82528. Available from:

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