# Fixed-Point Results Related tob-Intuitionistic Fuzzy Metric Space

Authors: Mohamed Altanji; Annamalai Santhi; Vediyappan Govindan; Shyam Sundar Santra; Samad Noeiaghdam
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this study, we contribute to the terminological debate about various fixed-point results’ use of the term intuitionistic fuzzyb-metric space in defining the structure based on fuzzy sets. As a predominant result, we give an adequate condition for a sequence to be Cauchy in the intuitionistic fuzzy b-metric space. Subsequently, we simplify the proofs of manifold fixed-point theorems in the intuitionistic fuzzy b-metric spaces under the prominent contraction conditions. Also, we give a satisfactory condition for a solution to Cauchy in the intuitionistic fuzzy b-metric spaces.


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## Body

## 1. Introduction

Zadeh [1] was very striking with fuzzy logic in which the truth values of variables may be any real number in the interval [0,1]. In this real world, we all are surrounded by the problems of uncertainty. To counter this problem of uncertainty, Zadeh establishes fuzzy logic. Moreover, fixed-point theory can be explored in fuzzy metric space (briefly, FMS) in several different ways. Fuzzy generalization of Banach contraction principle [2] expressed fuzzy mapping as a notion and manifested a theorem for a fuzzy contraction on a fixed point in linear metric spaces.Fuzzy metric space was pioneered by Kramosil and Michalek [3] by referring to the concept of the fuzzy set. Subsequently, Grabiec [4] interpreted the concept of completeness in FMS and extended the Banach contraction principle to G-complete FMS. Then, later on, George and Veeramani [5] also played a vital role in the theory of FMS and amended the idea of Cauchy sequence which was established by Grabiec and, meantime, amended the notion of a FMS which was introduced by Kramosil and came up with the new idea of Hausdorff topology on FMS.On a different note, Atanassov [6] generalized the FS and brought up the idea of IFS. Using this concept, Park [7] discovered the concept of IFMS in 2006. Saadati Park [8] introduced the related results. Furthermore, we refer the readers to [9–16]. In metric space, still there are enough scopes. Bakhtin [17] and Czerwik [18] revealed a weaker condition of metric space to generalize the Banach contraction principle [2]. They labeled it as b-MS. To explore more about these spaces, see [14, 19–21]. The b-metric and FMSs are related, which we can see in [22]. The idea of a fuzzy b-MS was established in [23]. Using this idea, we proved a very useful lemma by using the extension of t-norm and t-conorm setting that the sequence ðy is a Cauchy sequence. This paper comprises various fixed-point results in the b-IFMS. After the criticism presented in [24] on the adoption of the term “intuitionistic” in his original algebraic structures, a defense of his position is given by Atanassov in [25] by an “a posteriori” argument. Indeed, in [24], besides the above discussed argument about the use of “intuitionistic” term attributed to an algebraic negation which does not satisfy the great part of the accepted principles of intuitionistic logic, it is stressed that, in a paper of 1984 [26], the term “intuitionistic fuzzy set theory by Takeuti and Titani is an absolute legitimate approach, in the scope of intuitionistic logic, but it has nothing to do with Atanassov’s intuitionistic fuzzy sets.”We show a novel utilization of IFMS in a really difficult space of dynamic (for example, professional decision). An illustration of vocation assurance will be introduced, accepting there is a dataset (for example, a portrayal of a bunch of subjects and a bunch of vocations). We will portray the condition of understudies knowing the after effects of their presentation. The difficult portrayal utilizes the idea of IFS that makes it conceivable to deliver two significant realities. To begin with, the upsides of each subject presentation change for every understudy. Second, in a professional assurance dataset depicting vocation for various understudies, it ought to be considered that, for various understudies focusing on a similar vocation, the upsides of a similar subject exhibition can be extraordinary. We utilize the standardized Euclidean metric strategy surrendered to gauge the metric between every understudy and each profession. The littlest acquired worth calls attention to a legitimate vocation assurance dependent on scholarly execution.


## 2. Preliminaries

Through this paper, we considerℕ be a set of all natural numbers and k be a positive real number.Definition 1 (see [17]).
A functiond:ℨ×ℨ⟶0,∞ is b-metric if, for every b,e,w∈ℨ,(a)
db,e=0⇔b=e(b)
db,e=de,b(c)
db,w≤sdb,e+de,w
The ordered pairℨ,d is a b-MS. It is key to note that b-MS are not metrizable, especially b-metric might not be a continuous function of its variable.Definition 2 (see [27]).
A binary operation in such a wayT:0,12⟶0,1 is said to be a continuous triangular norm (t-norm) if(a)
Tp,b=Tb,p and Tp,b,r=Tp,b,r(b)
T is continuous(c)
Tp,1=p,∀p∈0,1(d)
Tp,b≤Tr,S whenever p≤r&b≤S,forallp,b,r,S∈0,1Definition 3 (see [28]).
LetT be a t-norm. Then, Ty:0,12⟶0,1,y∈ℕ, is defined by(1)T1k=Tk,k,Ty+1k=TTyk,k,y∈ℕ,k∈0,1.
Let(2)sup0<k<1Tk,k=1.
Then, t-normT is of H-type with the assuming the functions family Tmkm=1∞ which is equicontinuous at k=1, where(3)T1k=k,Tm+1k=TTmk,m=1,2,3,…,andk∈0,1.
The t-normTminð,v=minð,v is an example of H-type.
Each t-normT and t-conorm S can be drawn out to an n-ary operation taking ð1,ð2,…,ðy∈0,1y (see [27]):(4)Ti=11ði=ð1,Ti=1y=TTi=1yði,ðy=Tð1,ð2,…,ðy.Definition 4.
Tmin,TL, and TP can be extended in n-ary in the following ways:(5)Tminð1,ð2,…,ðy=minð1,ð2,…,ðyTLð1,ð2,…,ðy=max∑i=y∞ði−y−1,0TPð1,ð2,…,ðy=∏i=1yði.Definition 5 (see [29], t-conorm).
“A binary operationS:0,1×0,1⟶0,1 is continuous t-conorm if(a)
Sa,q=Sq,a&Sa,q,c=Sa,q,c(b)
T is continuous(c)
Sa,0=a,∀a∈0,1(d)
Sa,q≤Sc,s whenever a≤c&q≤s,∀a,q,c,s∈0,1Definition 6.
A t-conorm,Sy:0,1⟶0,1,y∈ℕ is defined by(6)S1k=Sk,k,Sy+1k=SSyk,k,y∈ℕ,k∈0,1.
Let(7)inf0<k<1Sk,k=1.
Then, t-conormS is of H-type with the assuming the functions’ family Smkm=1∞ which is equicontinuous at k=0, where(8)S1k=k,Sm+1k=SSmk,m=1,2,3,…,andk∈0,1.
The t-conormSmaxð,v=maxð,v is an example of H-type.
The t-conorm can be stretched out by associativity in an n-ary operation taking, forð1,ð2,…,ðy∈0,1y, the values(9)Si=11ði=SSi=1yði,ðy=Sð1,ð2,…,ðy.Example 1.
The extensions of the t-conorm in n-ary areSmin,SL,andSP are the following:(10)Smaxð1,ð2,…,ðy=maxð1,ð2,…,ðySLð1,ð2,…,ðy=min∑i=y∞ði,1SPð1,ð2,…,ðy=∑i=y∞ði−∏i=1yði.
For anyðyy∈ℕ in [0,1], the t-norm and t-conorm can be stretched out to a countable illimitable operation, that is,(11)T∞i=1ði=limn⟶∞Tyi=1ði,S∞i=1ði=limn⟶∞Syi=1ði.
The sequenceTyi=1ði is increasing, and Syi=1ði is decreasing and has an upper bound and lower bound, respectively, so the limit exists.
In this fixed-point theory, see [30, 31], it is very interesting to discuss the classes of t-norms T and the classes of t-conorm S and ðy in [0, 1] such that limn⟶∞ðy=1 and(12)T∞i=1ði=limn⟶∞Tyi=1ðy+i=1,(13)S∞i=1ði=limn⟶∞Syi=1ðy+i=0.Proposition 1.
Letðyy∈ℕ in [0, 1] be limn⟶∞ðy=1 and T be a t-norm of H-type. Then,(14)T∞i=1ði=limn⟶∞Tyi=1ðy+i=1.Proposition 2.
Letðyy∈ℕ in [0, 1] ne limn⟶∞ðy=0 and S be a t-conorm of H-type. Then,(15)S∞i=1ði=limn⟶∞Syi=1ðy+i=0.Definition 7 (see [23]).
A 3-tupleℨ,ℳ,T is called a b-FMS. If ∀ð,v,z∈ℨ,s,k>0,(a)
ℳð,v,k>0(b)
ℳð,v,k=1,∀k>0 iff ð=v(c)
ℳð,v,k=ℳv,ð,k(d)
ℳð,v,k+s≥Tℳð,a,k/b,ℳa,v,s/b(e)
ℳð,v,⋅:0,∞⟶0,1 (left continuous)where ℨ is a nonempty set, T is continuous t-norm, and ℳ is a FS on ℨ2×0,∞.Example 2 (see [9]).
Letℳð,v,k=e−ð−vp, where p>1. Then, ℳ is a b-metric with b=2p−1.Definition 8.
(see [7, 8]). A 5-tuple ℨ,ℳ,N,T,S is called intuitionistic fuzzy rectangular metric space. If, for every ð,v,z∈ℨ and s,k>0,(a)
ℳð,v,k+Nð,v,k≤1(b)
ℳð,v,0=0(c)
ℳð,v,k=1,∀k>0 iff ð=v(d)
ℳð,v,k=ℳv,ð,k(e)
ℳð,v,k+s≥Tℳð,a,k/b,ℳa,v,s/b(f)
ℳð,v,⋅:0,∞⟶0,1 (left continuous)(g)
limk⟶∞ℳð,v,k=1,∀ð,v∈ℨ(h)
Nð,v,0=1(i)
Nð,v,k=0,∀k>0 iff ð=v(j)
Nð,v,k+s≥SNð,a,k/b,ℳa,v,s/b(k)
Nð,v,⋅:0,∞⟶0,1 (right continuous)(l)
limk⟶∞Nð,v,k=0,∀ð,v∈ℨwhere ℨ is a random set, T is continuous t-norm, S is continuous t-conorm, and ℳandN are FSs on X2×0,∞.Example 3.
Letℳð,v,k=e−dð,v/k,Na,e,k=1−e−da,e/k, where p>1. Then, ℳ and N are b-metric with b=2p−1.Definition 9 (see [9]).
A functionζ:ℜ⟶ℜ is called b-nondecreasing if ð>bv implies ζð≥ζð,∀ð,v∈ℜ.Definition 10.
A functionζ:ℜ⟶ℜ is called b-nonincreasing if ð<bv implies ζð≤ζv,∀ð,v∈ℜ.Lemma 1 (see [23, 32]).
Letℳð,v,T be a b-FMS. Then, ℳð,v,k is b-increasing with respect to t,∀ð,v∈ℨ.Lemma 2.
LetNð,v,S be an b-IFMS. Then, Nð,v,k is b-decreasing with respect to t,∀ð,v∈ℨ.Definition 11 (see [23, 32]).
Letℨ,ℳ,N,T,S be an b-IFMS. For k>0, the open ball ℬv,r,k with centre ð∈ℨ and radius 0<r<1 are defined as(16)ℬð,r,k=v∈ℨ:ℳð,v,k>1−r,ℬð,r,k=v∈ℨ:Nð,v,k<1−r.
A sequenceðy(a)
Converges toð if ℳðy,ð,k⟶1 as n⟶∞, for each k>0. In this case, we write limn⟶∞ðy=ð.(b)
Is a Cauchy sequence if∀0<ε<1&k>0,∃y0∈ℕ in such a way that ℳðy,ðm,k>1−ε,∀y,m≥y0.(c)
Converges toð if Nðy,ð,k⟶0 as y⟶∞, for each k>0. In this case, we write limn⟶∞ðy=ð.(d)
Is a Cauchy sequence if∀0<ε<1&k>0,∃y0∈ℕ in such a way that Nðy,ðm,k<1−ε,∀y,m≥y0.Definition 12.
The intuitionistic fuzzy b-MSℨ,ℳ,N,T,S is complete if and only if every Cauchy sequence is a convergent sequence.Lemma 3.
In an b-IFMSℨ,ℳ,N,T,S, we have a sequence ðy in ℨ converges to ð; then, it is definitely a Cauchy sequence and ð is unique.In a b-IFMS, we possess the successive proposition.Proposition 3.
Letℨ,ℳ,N,T,S be a b-IFMS and ðy converges to x. Then,(17)ℳð,v,kb≤limn⟶∞supℳðy,v,k≤ℳð,v,bk,ℳð,v,kb≤limn⟶∞infℳðy,v,k≤ℳð,v,bk,Nð,v,bk≤limn⟶∞supNðy,v,k≤ℳð,v,kb,Nð,v,bk≤limn⟶∞infNðy,v,k≤ℳð,v,kb.Remark 1.
A b-IFM is not continuous generally.Example 4.
Letℨ=0,∞, ℳð,v,k=e−dð,v/k, and Na,e,k=1−e−da,e/k, and(18)dð,v=0,ifð=v2ð−v,ifð,v∈0,112ð−v,otherwise.
Then,ℨ,ℳ,N,T,S be a b-IFMS with b=6.


## 3. Main Results

Furthermore, we will use a b-IFMS in terms of definition with additional condition,limk⟶∞ℳð,v,k=1.Lemma 4.
Letðy in a b-IFMS ℨ,ℳ,N,T,S be(19)ℳðy,ðy+1,k≥ℳðy−1,ðy,kμ,y∈ℕ,k>0,Nðy,ðy+1,k≤Nðy−1,ðy,kμ,y∈ℕ,k>0,and there exist ð0,ð1∈ℨ and v∈0,1 such that(20)limi⟶∞Ti=1∞ð1,ð1,kvi=1,limi⟶∞Si=1∞ð1,ð1,kvi=0,k>0.
Then,ðy is a Cauchy sequence.Proof.
Letω∈0,1. Then, the sum ∑i=1∞ωi is convergent, and then, there exists y0∈ℕ such that ∑i=1∞ωi<1, for every n>y0. Let n>m>y0. Since ℳ is b-increasing by Definition 8(e) and N is b-decreasing by Definition 8(j), for every k>0, we obtain(21)ℳðy,ðy+m,k≥ℳðy,ðy+m,k∑i=yy+m−1ωib≥Tℳðy,ðy+1,kωyb2,ℳðy+1,ðy+m,k∑i=y+1y+m−1ωib2≥Tℳðy,ðy+1,kωyb2,Tℳðy+1,ðy+2,kωy+1b3,…,ℳðy+m−1,ðy+m,kωy+m−1bm.
By (19), it implies ℳðy,ðy+1,k≥ℳð0,ð1,k/μy,y∈ℕ,k>0:(22)Nðy,ðy+m,k≤Nðy,ðy+m,k∑i=yy+m−1ωib≤SNðy,ðy+1,kωyb2,Nðy+1,ðy+m,k∑i=y+1y+m−1ωib2≥SNðy,ðy+1,kωyb2,SNðy+1,ðy+2,kωy+1b3,…,Nðy+m−1,ðy+m,kωy+m−1bm.
By (19), it implies Nðy,ðy+1,k≤Nð0,ð1,k/μy,y∈ℕ,k>0.
Sincen>m and b>1, we have(23)ℳðy,ðy+m,k≥Tℳð0,ð1,kωyb2μy,Tℳð0,ð1,kωy+1b3μy+1,…,ℳð0,ð1,kωy+m−1bm+1μy+m−1,…≥Ti=yy+m−1ℳð0,ð1,kωibi−y+2μi≥Ti=yy+m−1ℳð0,ð1,kωibiμi≥Ti=yy+m−1ℳð0,ð1,kvi,Nðy,ðy+m,k≥SNð0,ð1,kωyb2μy,SNð0,ð1,kωy+1b3μy+1,…,Nð0,ð1,kωy+m−1bm+1μy+m−1,…≥Si=yy+m−1Nð0,ð1,kωibi−y+2μi,Nðy,ðy+m,k≥Si=yy+m−1Nð0,ð1,kωibiμi≥Si=yy+m−1Nð0,ð1,kvi,where v=bμ/ω is v∈0,1. By equation (20), it implies ðy is a Cauchy sequence.Corollary 1.
Letðy be a b-IFMS ℨ,ℳ,N,T,S, where T and S are H-type. If there exists μ∈0,1/b such that(24)ℳðy,ðy+1,k≥ℳðy−1,ðy,kμ,y∈ℕ,k>0,Nðy,ðy+1,k≤Nðy−1,ðy,kμ,y∈ℕ,k>0,holds, then ðy is a Cauchy sequence.Lemma 5.
If there existsμ∈0,1 and ð,v∈ℨ such that(25)ℳð,v,k≥ℳðy−1,ðy,kμ,y∈ℕ,k>0,Nð,v,k≤Nðy−1,ðy,kμ,y∈ℕ,k>0,holds, then ð=v.Proof.
Condition (25) implies that(26)ℳð,v,k≥ℳð,v,kμ,y∈ℕ,k>0≥limn⟶∞ℳð,v,kμy,y∈ℕ,k>0,Nð,v,k≤Nðy−1,ðy,kμ,y∈ℕ,k>0≤limn⟶∞Nðy−1,ðy,kμ,y∈ℕ,k>0.
By definition of b-IFMS, we haveð=v.Theorem 1.
Letℨ,ℳ,N,T,S be a b-IFMS, which is complete and Q:ℨ⟶ℨ. Suppose there exists μ∈0,1/b such that(27)ℳQð,Qv,k≥ℳð,v,kμ,ð,v∈ℨ,k>0,NQð,Qv,k≥Nð,v,kμ,ð,v∈ℨ,k>0,holds. Furthermore, there exists ð0∈ℨ and v∈0,1 such that(28)limn⟶∞Ti=y∞ℳð0,Qð0,kvi=1,limn⟶∞Si=y∞Nð0,Qð0,kvi=0,holds. Then, Q has a unique fixed point in ℨ.Proof.
Letð0∈ℨ and ðy+1=Qðy,y∈ℕ. If we take ð=ðy and y=ðy−1 in (27), then we have(29)ℳðy,ðy+1,k≥ℳðy−1,ðy,kμ,y∈ℕ,k>0,Nðy,ðy+1,k≤Nðy−1,ðy,kμ,y∈ℕ,k>0.
By Lemma4, it implies ðy is Cauchy sequence. Since ℨ,ℳ,N,T,S is complete, then limn⟶∞ðy=ð,ð∈ℨ. Therefore,(30)limn⟶∞ℳð,ðy,k=1,limn⟶∞Nð,ðy,k=0,and(31)ℳQð,ð,k≥TℳQð,ðy,k2b,ℳðy,ð,k2b≥Tℳð,ðy−1,k2bμ,ℳðy,ð,k2b,NQð,ð,k≥TNQð,ðy,k2b,Nðy,ð,k2b≥TNð,ðy−1,k2bμ,Nðy,ð,k2b,∀k>0.
By (27), as n⟶∞, we have(32)ℳQð,ð,k≥T1,1=1,NQð,ð,k≤S0,0=0.
Let us assume thatx and y are fixed points for Q. By (24), we obtain(33)ℳð,v,k=ℳQð,Qð,k≥ℳð,v,kμ,Nð,v,k=ℳQð,Qð,k≤ℳð,v,kμ.
Using Lemma5, we have ð=v.Example 5.
Assumeℨ=0,1. Using Example 3 for p=2 implies ℨ,ℳ,N,T,S is a b-IFMS with b=2 and b-IFM. We define(34)ℳð,v,k=e−ð−v2/kandNð,v,k=1−e−x−y2/k.
LetQv=kv,k<2/2,ð∈ℨ. Then,(35)ℳQð,Qv,k=e−k2ð−v2/k≥e−μð−v2/k=ℳð,v,kb,ð,v∈ℨ,t>0,NQð,Qv,k=1−ℳQð,Qv,k=1−e−k2ð−v2/k≤1−e−μð−v2/k=ℳð,v,kb,ð,v∈ℨ,t>0,for 1/b>μ>k2; hence, equation (27) will be satisfied, and we can say that Q possess a fixed point in ℨ which is unique.Theorem 2.
Letℨ,ℳ,N,T,S be a b-IFMS and Q:ℨ⟶ℨ. Assume that μ∈0,1/b such that(36)ℳQð,Qv,k≥minℳð,v,kμ,ℳQð,ð,kμ,ℳQv,v,kμ,NQð,Qv,k≥maxNð,v,kμ,NQð,ð,kμ,NQv,v,kμ,∀ð,v∈ℨ,k>0,and there exists ð0∈ℨ,v∈0,1 such that(37)limn⟶∞Ti=y∞ℳð0,Qð0,kvi=1,limn⟶∞Si=y∞Nð0,Qð0,kvi=0,∀k>0.
Then,Q has a unique fixed point in ℨ.Proof.
Letð0∈ℨ,ðy+1=Qðy,y∈ℕ. By (36), with ð=ðy and v=ðy−1, for every y∈ℕ,k>0, we have(38)ℳðy+1,ðy,k≥minℳðy,ðy−1,kμ,ℳðy+1,ðy,kμ,ℳðy,ðy−1,kμ≥minℳðy,ðy−1,kμ,ℳðy+1,ðy,kμ,Nðy+1,ðy,k≤minNðy,ðy−1,kμ,Nðy+1,ðy,kμ,Nðy,ðy−1,kμ≤minNðy,ðy−1,kμ,Nðy+1,ðy,kμ.
Ify∈ℕ,k>0, then(39)ℳðy+1,ðy,k≥ℳðy+1,ðy,kμ,Nðy+1,ðy,k≤Nðy+1,ðy,kμ.
Lemma5 implies ðy=ðy+1,y∈ℕ. Therefore,(40)ℳðy+1,ðy,k≥ℳðy,ðy−1,kμ,Nðy+1,ðy,k≤Nðy,ðy−1,kμ,and by lemma, we have ðy is a Cauchy sequence. Hence, limn⟶∞ðy=v,ð∈ℨ:(41)limn⟶∞ℳð,ðy,k=1,k>0,limn⟶∞Nð,ðy,k=0,k>0.
Let us proveð is a fixed point for Q. Let ω1∈μb,1 and ω2=1−ω1. By (36), we have(42)ℳQð,ð,k≥TℳQð,Qðy,tω1b,ℳðy+1,ð,tω2b≥Tminℳð,ðy,tω1bμ,ℳð,Qð,tω1bμ,ℳðy+1,ð,tω2bμ,ℳðy+1,ð,tω2b,NQð,ð,k≤SNQð,Qðy,tω1b,Nðy+1,ð,tω2b≤SmaxNð,ðy,tω1bμ,Nð,Qð,tω1bμ,Nðy+1,ð,tω2bμ,Nðy+1,ð,tω2b.
Takingn⟶∞ and using (41), we have(43)ℳQð,ð,k≥Tmin1,ℳð,Qð,tω1bμ,1,1=Tℳð,Qð,tω1bμ,1=ℳð,Qð,kv,k>0,NQð,ð,k≤Smax0,Nð,Qð,tω1bμ,1,0=SNð,Qð,tω1bμ,0=Nð,Qð,kv,k>0,where v=bμ/ω1∈0,1. Therefore, we obtain(44)ℳQð,ð,k≥ℳð,Qð,kv,k>0,NQð,ð,k≥Nð,Qð,kv,k>0.
Lemma5 implies Qð=ð.
Suppose thatð and v are fixed points for Q (i.e., Qð=ð&Qv=v). By (36), we obtain(45)ℳQð,Qv,k≥minℳð,v,kμ,ℳð,Qð,kμ,ℳy,Qv,kμ=minℳð,v,kμ,1,1=ℳð,v,kμ=ℳQð,Qv,kμ,NQð,Qv,k≤maxNð,v,kμ,Nð,Qð,kμ,Ny,Qv,kμ=maxNð,v,kμ,1,1=Nð,v,kμ=NQð,Qv,kμ,fork>0.
By Lemma5, it follows that Qð=Qv, which implies that ð=v.Example 6.
LetX=0,2,ℳð,v,k=e−ð−v2/k, and Nð,v,k=1−e−ð−v2/k. Then, ð,ℳ,N,T,S is b-IFMS which is complete with b=2. Let(46)Qx=2−ð,ifð∈0,1,1,ifð∈1,2.Case (i): ifð,v∈1,2, then ℳQð,Qv,k=1andNQð,Qv,k=0,k>0, and conditions (36) will be trivially satisfied.Case (ii): ifð∈1,2,v∈0,1, then μ∈1/4,1/2; we have(47)ℳQð,Qv,k=e−1−v2/k≥e−4μ1−v2/k=ℳQv,v,kλ,k>0,NQð,Qv,k=1−ℳQð,Qv,k=1−e−1−v2/k≤1−e−4μ1−v2/k=NQv,v,kλ,k>0.Case (iii): forμ∈1/4,1/2, we have(48)ℳQð,Qv,k=ℳQv,v,kλ,k>0,NQð,Qv,k=NQv,v,kλ,k>0.Case (iv): ifð,v∈0,1, then μ∈1/4,1/2; we have(49)ℳQð,Qv,k≥e−ð−v2/k≥e−1−v2/k≥e−4μ1−v2/k=ℳQv,v,kμ,NQð,Qv,k=1−ℳQð,Qv,k=1−e−1−v2/k≤1−e−4μ1−v2/k=NQv,y,kμ,ℳQð,Qv,k≥ℳQv,v,kμ,NQð,Qv,k≤NQv,v,kμ.
So, condition (36) is satisfied ∀ð,v∈ð,k>0, and by Theorem 2, Q possess a unique fixed point.Theorem 3.
Letð,ℳ,N,T,S be a complete b-IFMS and Q:X⟶X. If(50)ℳQð,Qv,k≥minℳð,v,kμ,ℳQð,ð,kμ,ℳQv,v,kμ,ℳQð,v,2tμ,ℳð,fv,kμ,NQð,Qv,k≥minNð,v,kμ,NQð,ð,kμ,NQv,v,kμ,NQð,v,2tμ,ℳð,fv,kμ,for μ0,1/b3, then Q possess a unique fixed point in v.Proof.
Letð0∈ℨ and ðy+1=Qðy,y∈ℕ. By (37), with ð=ðy and v=vy, using Definition 8(d) and T=Tmin, we have(51)ℳðy+1,ðy,k≥minℳðy,ðy−1,kμ,ℳðy+1,ðy,kμ,ℳðy,ðy−1,kμ,minℳðy+1,ðy,kbμ,ℳðy,ðy−1,kbμ,ℳðy,ðy,kμ≥minℳðy,ðy−1,kbμ,ℳðy+1,ðy,kbμ,Nðy+1,ðy,k≤maxNðy,ðy−1,kμ,Nðy+1,ðy,kμ,Nðy,ðy−1,kμmaxNðy+1,ðy,kbμ,Nðy,ðy−1,kbμ,Nðy,ðy,kμ≤maxNðy,ðy−1,kbμ,ℳðy+1,ðy,kbμ.
Proceeding as in proof of Theorem2, Lemma 5, and Corollary 1, it follows that(52)ℳðy+1,ðy,k≥ℳðy,ðy−1,kbμ,Nðy+1,ðy,k≤Nðy,ðy−1,kbμ,and ðy is a Cauchy sequence. So, there exists ð∈ℨ such that limn⟶∞ðy=ð and(53)limn⟶∞ℳð,ðy,k=1,k>0,limn⟶∞Nð,ðy,k=0,k>0.
Letω1∈μb3,1 and ω2=1−ω1. By (50) and Definition 8(e), for T=Tmin, we have(54)ℳQð,ð,k≥minℳQðy,Qðy−1,tω1b,ℳQðy,ð,tω2b≥minminℳð,ðy,tω2bμ,ℳð,Qð,tω2bμ,ℳðy,ðy+1,tω2bμ,minℳQð,ð,tω1b3μ,ℳð,ðy,tω2b3μ,ℳð,ðy+1,tω1bμ,ℳðy+1,ð,tω2b,NQð,ð,k≤maxNQðy,Qðy−1,tω1b,NQðy,ð,tω2b≤maxmaxNð,ðy,tω2bμ,Nð,Qð,tω2bμ,Nðy,ðy+1,tω2bμ,maxNQð,ð,tω1b3μ,Nð,ðy,tω2b3μ,Nð,ðy+1,tω1bμ,Nðy+1,ð,tω2b,for all y∈ℕ and k>0. Taking n⟶∞ and using (41), we obtain(55)ℳQð,ð,k≥minmin1,ℳð,Qð,tω1bμ,1,minℳQð,ð,tω2b3μ,1,1=ℳQð,ð,tω1b3μ,k>0,NQð,ð,k≤maxmax1,Nð,Qð,tω1bμ,1,maxNQð,ð,tω2b3μ,1,1=NQð,ð,tω1b3μ,k>0,and by Lemma 5, with v=b2μ/ω1∈0,1, it follows that Qð=ð.
By condition (50), ð=Qð and v=Qv; we have(56)ℳQð,Qv,k≥minℳð,v,kμ,ℳQð,ð,kμ,ℳQv,v,kμ,minℳQð,ð,kbμ,ℳð,v,kbμ,ℳð,Qv,kμ=minℳð,v,kμ,1,1,min,ℳð,v,kbμ,ℳð,v,kμ=ℳð,v,kbμ=ℳQð,Qv,kbμ,NQð,Qv,k≤maxNð,v,kμ,NQð,ð,kμ,NQv,v,kμ,maxNQð,ð,kbμ,Nð,v,kbμ,Nð,Qv,kμ=maxNð,v,kμ,1,1,max,Nð,v,kbμ,Nð,v,kμ=Nð,v,kbμ=NQð,Qv,kbμ,and by Lemma 3, it follows that ð=v.Theorem 4.
Letð,ℳ,N,T,S,T≥Tp, be a b-IFMS which is complete and Q:ℨ⟶ℨ, for some μ∈0,1/b2, such that(57)ℳQð,Qv,k≥minℳð,v,kμ,ℳQð,ð,kμ,ℳQv,v,kμ,ℳQð,v,2tμ,ℳð,fv,kμ,NQð,Qv,k≤maxNð,v,kμ,NQð,ð,kμ,NQv,v,kμ,NQð,v,2tμ,Nð,fv,kμ,and there exists ð0∈ℨ,v∈0,1; we have(58)limn⟶∞Ti=y∞ℳð0,Qð0,kvi=1,limn⟶∞Si=y∞Nð0,Qð0,kvi=0.Then,Q possess a unique fixed point in ℨ.Proof.
Letð0∈ℨ&ðy+1=Qðy,y∈ℕ. Taking ð=ðy and v=vy in condition (57), and T=Tp, we have(59)ℳðy+1,ðy,k≥minℳðy,ðy−1,kμ,ℳðy+1,ðy,kμ,ℳðy,ðy−1,kμ,ℳðy+1,ðy,kbμ,ℳðy,ðy−1,kbμ,ℳðy,ðy,kbμ,Nðy+1,ðy,k≤maxNðy,ðy−1,kμ,Nðy+1,ðy,kμ,Nðy,ðy−1,kμ,Nðy+1,ðy,kbμ,Nðy,ðy−1,kbμ,Nðy,ðy,kbμ.
Sinceℳðy+1,ðy,k is b-nondecreasing and a.b≥mina,b; Nðy+1,ðy,k is b-nonincreasing and a.b≤maxa,b.
Therefore,(60)ℳðy+1,ðy,k≥minℳðy+1,ðy,kbμ,ℳðy,ðy−1,kbμ,Nðy+1,ðy,k≤maxNðy+1,ðy,kbμ,Nðy,ðy−1,kbμ,for y∈ℕ,k>0. By lemma,(61)ℳðy+1,ðy,k≥ℳðy,ðy−1,kbμ,Nðy+1,ðy,k≥Nðy,ðy−1,kbμ,and ðy is a Cauchy sequence. Since ð,ℳ,N,T,S is a complete b-IFMS, then limn⟶∞ðy=v,ð∈ℨ and(62)limn⟶∞ℳð,ðy,k=1,limn⟶∞Nð,ðy,k=0,ω1∈b3μ,1 and ω1∈b3μ,1. Also and for T≥Tp, we have(63)ℳQð,ð,k≥TℳQð,Qðy,tω1b,ℳQðy,Qð,tω2b≥Tminℳð,ðy,tω1bμ,ℳð,Qð,tω1bμ,ℳðy,ðy+1,tω1bμℳQð,ðy,tω1b3μ,ℳð,ðy,tω1b3μ,ℳð,ðy+1,tω1bμ,ℳðy+1,ð,tω2b≥Tminℳð,ðy,tω1bμ,ℳð,Qð,tω1bμ,ℳðy,ðy+1,tω1bμ,minℳQð,ð,tω1b3μ,ℳð,ðy,tω1b3μ,ℳðy,ðy+1,tω1bμ,ℳðy+1,ð,tω2b,NQð,ð,k≤SNQð,Qðy,tω1b,NQðy,Qð,tω2b≤SmaxNð,ðy,tω1bμ,Nð,Qð,tω1bμ,Nðy,ðy+1,tω1bμNQð,ðy,tω1b3μ,Nð,ðy,tω1b3μ,Nð,ðy+1,tω1bμ,Nðy+1,ð,tω2b≤SminNð,ðy,tω1bμ,Nð,Qð,tω1bμ,Nðy,ðy+1,tω1bμ,maxNQð,ð,tω1b3μ,Nð,ðy,tω1b3μ,Nðy,ðy+1,tω1bμ,Nðy+1,ð,tω2b,for every y∈ℕ and k>0. Taking n⟶∞ and using(64)ℳQð,ð,k≥Tmin1,ℳð,Qð,tω1bμ,1,minℳQð,ð,tω1b3μ,1,1,1≥ℳQð,ð,tω1b3μ,k>0,NQð,ð,k≤Smax0,Nð,Qð,tω1bμ,0,maxNQð,ð,tω1b3μ,0,0,0≤NQð,ð,tω1b3μ,k>0,and by Lemma 5 with v=b3μ/ω1∈0,1, we have Qð=ð and(65)ℳQð,Qv,k≥Tℳð,v,kμ,ℳQð,ð,kμ,ℳQv,v,kμ,ℳQð,ð,kbμ,ℳð,v,kbμ,ℳð,Qv,kμ≥Tℳð,v,kμ,1,1,min1,ℳð,v,kbμ,ℳð,v,kbμ=ℳQð,Qv,kbμ,k>0,NQð,Qv,k≤SNð,v,kμ,NQð,ð,kμ,NQv,v,kμ,ℳQð,ð,kbμ,Nð,v,kbμ,Nð,Qv,kμ≤SNð,v,kμ,1,1,max1,Nð,v,kbμ,Nð,v,kbμ=NQð,Qv,kbμ,k>0.
Thus, by Lemma5, we have ð=v.Example 7.
Letℨ=0,1,3,ℳð,v,k=e−ð,v2/k,Nð,v,k=1−e−ð−v2/k, or Nð,v,k=e−ð−v2/t−ð−v2, T=Tp,S=Sp. Then, ℨ,ℳ,N,T,S is a complete b-IFM with b=2.
DefineQ:ℨ⟶ℨ as f0=f1=0,f3=0.
We observe that ifð=v or ð,v∈0,1, then ℳQð,Qv,k=1andNQð,Qv,k=0,k>0, and (57) is satisfied.
Ifx=1 and y=3, then μ∈1/9,1/4 and(66)ℳfð,fv,k=e−1/k≥mine−4μ/k,1,e−9μ/k,e−μ/k,1,Nfð,fv,k=e−1/t−1≤maxe−4μ/t−4μ,1,e−9μ/t−9μ,e−μ/2t−4μ,e−μ/t−μ.
In the same way, if we takeð=3 and v=1 and for μ∈1/9,1/4, condition (57) is satisfied, for all ð,v∈ℨ,k>0, and Q possess a unique fixed in ℨ.


## 4. Applications ofb-Intuitionistic Fuzzy Metric Space

The substance of giving satisfactory data to understudies to legitimate vocation decision cannot be overemphasized. This is a principle on the grounds that the various issues of absence of legitimate vocation control looked by understudies are of extraordinary result on their profession decision and effectiveness. Accordingly, it is practical that understudies be given adequate data on vocation assurance or decision to upgrade satisfactory arranging, arrangement, and capability. Among the vocation deciding elements such as scholarly execution, interest, and character make-up; the first-referenced is by all accounts’ abrogating. We use b-IFMS as a device since it joins the enrollment degree (i.e., the marks of the questions answered by the student), the nonparticipation degree (i.e., the marks of the questions the student failed), and the dithering degree (which is the mark allocated to the questions the student do not attempt).


## 5. Conclusions

The authors introduced and discussed several notions of intuitionistic fuzzy b-metric space from different points of view with a suitable notion for the intuitionistic fuzzy metric of a given intuitionistic fuzzy b-metric space. In particular, we explore several properties of the intuitionistic fuzzy b-metric space. We have presented the b-IFMS and identified with fixed-point results about career determination which is of incredible importance since it gives precise and appropriate professional decision dependent on scholastic execution. We give a satisfactory condition for an arrangement to Cauchy in the b-IFMS. Accordingly, we work on the verifications of complex fixed-point hypotheses. Profession decision is a fragile independent direction issue since it has a reverberatory impact on productivity, and capability is appropriately dealt with. In the proposed application, we utilized standardized Euclidean distance to compute the distance of every understudy from each career regarding the subjects, to acquire outcomes. We use b-IFMS as an instrument since it fuses the membership degree, the nonmembership degree, and the hesitation degree.



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