# On Coincidence Theorem in Intuitionistic Fuzzyb-Metric Spaces with Application

Authors: Shazia Kanwal; Akbar Azam; Faria Ahmed Shami
Journal: Journal of Function Spaces
 (2022)

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## Abstract

The concept of intuitionistic fuzzy b-metric spaces (shortly, IFbMS) has been introduced and studied to generalize both the notion of intuitionistic fuzzy metric spaces and fuzzyb-metric spaces. The existence of coincident point and common fixed point for two self-mappings has been established. In order to show the strength of these results, some interesting examples are established as well. Our results generalize many previous results existing in literature. Some nontrivial examples are furnished as well as an application is created to give the strength of our main result.


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## Body

## 1. Introduction

The advancement and the rich progress of fixed-point theorems in metric spaces (and in its various generalizations) have significant theoretic and useful applications. These developments in the last three decades were fabulous. Most of them used Banach’s contraction theorem [1] in their reference result. Many problems in engineering and research can be solved by confining nonlinear equations to similar fixed-point cases. A fixed point Fx=x can be proved for an operator sum Gx=0, where F is a self-mapping in some relevant discipline. Fixed-point theory has various key modes for addressing difficulties arising from a variety of mathematical inspection offshoots, such as split feasibility concerns, supporting problems, equilibrium problems, and matching and selection issues. The theory of fixed points is a fascinating and energising field of study. This idea has already been identified as an over-the-top attempt to pack nonlinear analysis into a small amount of time.In 1965, Zadeh [2] initiated the concept of FSs which is used to characterize/manipulate information and data having nonstatistical uncertainties. The objectives of the theory of FS are to offer rational and set hypothetical tools to deal with such problems where errors and degree of uncertainties are present. Later, the idea of intuitionistic-FSs was given by Atanassov [3] in 1986. This set theory not only defines the degree of membership but also degree of nonmembership which is a more generalized form of FS-theory. Many authors apply this concept in different fields of mathematics. Gulzar et al. (e.g., [4–6]) have applied this theory in groups and its characteristics. Akber [7] used this idea to define intuitionistic fuzzy mappings and obtained common fixed points for such types of mappings. Since the notion of distance function have a central part in approximation theory, therefore, FSs further have been applied to the classical notion of metric. In this direction, some authors [8–10] suggested the generalization of metric spaces to the fuzzy situations.In 1975, the concept of FMS was offered by Kramosil and Michalek [11], which was further developed by George and Veeramani [12] in 1994, in order to build a Hausdorff topology using fuzzy metric. Rehman and Aydi [13] constructed results in fuzzy cone metric space. Bakhtin [14] first proposed the concept of b-metric. b-metric spaces are a broader category than metric spaces. Later, Javed et al. [15] established results on orthogonal partial b-metric spaces. Nădăban [16] proposed the notion of fuzzy b-metric space in 2016, which is a generalization of FMSs. Javed et al. [17] worked on finding fixed-point results in fuzzy b-metric spaces. Shazia et al. [18] established fixed points for various contractions in fuzzy strong b-metric spaces. Park [19] defined IFMSs as a refinement of FMSs by using theory of intuitionistic-FSs and continuous t-norm and continuous t-conorm in the year 2004. Jungck [20] enhanced Banach’s theorem in 1976 by analyzing at coincidence and common fixed points of commuting mappings. Jungck [21] established the concept of compatible maps for a pair of self-mappings, as well as the existence of shared fixed points, in 1986. Turkoglu et al. [22] expanded Jungck’s common fixed-point theorems in IFMSs in a paper published in 2006. Jungck and Rhoades [23] introduced weakly compatible mappings in 2006. Weakly compatible mappings are more generic since any pair of compatible mappings is weakly compatible, but not the other way around. Altun et al. [24,25] recently established excellent results on the best proximity points in 2021.Grabiec [26] defined the completion of FMS in 1988. We proved the existence of the coincidence theorem and the common fixed-point theorem in IFbMS in this work. The structure of the paper is as follows.After the preliminaries, in Section3, the notion of IFbMSs has been defined, and this concept is clarified with the help of comprehensible examples. The conceptual definitions of convergent sequence, Cauchy sequence, and topology induced by an IFbMS are presented as well. In Section 4, we formulate and prove our main results regarding coincidence points and common fixed point of weakly compatible mappings in IFbMS and establish some nontrivial examples to justify the validity of our results. Section 5 consists of an application of our main result.


## 2. Preliminaries

For the reader’s convenience, some definitions and results are recalled.Definition 2.1.
(see [27]). Let ζ be an arbitrary nonempty set and s≥1 be a given real number. A function ϖ:ζ×ζ⟶[0,∞) is a b-metric on ζ if, for all ω,υ,z∈ζ, the following conditions are satisfied:b1ϖω,υ=0⇔ω=υb2ϖω,υ=ϖυ,ωb3ϖω,z≤sϖω,υ+ϖυ,z
The tripleζ,ϖ,s will be called b-metric space.Example 1.
(see [28]). The space ki0<i<1,(1)ki=ωj⊂R:∑j=1∞ωji<∞,with a function ϖ:ki×ki⟶R,(2)ϖω,υ=∑j=1∞ωj−υji1/i,is a b-metric space, where ω=ωjandυ=υj∈ki. By some calculation, we obtain(3)ϖω,z≤21/iϖω,υ+ϖυ,z.Here, s=21/i>1.Example 2.
(see [28]). The space Gi0<i<1, of all real functions ωt,t∈0,1 such that ∫01ωtidt<∞, is b-metric space if we take(4)ϖω,υ=∫01ωt−υti1/i,for each ω,υ∈Gi.Definition 2.2.
(see [2]). Consider a nonempty set ζ. A mapping from ζ to 0,1 is called the fuzzy set. If K is a FS and η∈ζ, then the function value Kη is called the grade of membership of η in K.Definition 2.3.
(see [3]). For a nonempty set ζ an intuitionistic-FS is defined as(5)A=ω∈ζ:μAω,νAω,where μA:ζ⟶0,1 is called the degree of membership and νA:ζ⟶0,1 is called the degree of nonmembership of every ω to the set A such that μAω+νAω∈0,1, for all ω∈ζ.Definition 2.4.
(see [29]). A binary operation ⊙:0,1×0,1⟶0,1 is known as continuous-t-norm if the following axioms are satisfied:(1)
Associativity and commutativity properties are satisfied by⊙(2)
⊙ is a continuous function(3)
λ⊙1=λ,∀λ∈0,1(4)
Ifλ≤k and ϵ≤l with λ,ϵ,k,l∈0,1, then λ⊙ϵ≤k⊙lExample.
(1)
λ⊙1ε=minλ,ε(2)
λ⊙2ε=λ.ε(3)
λ⊙3ε=maxλ+ε−1,0Definition 2.5.
(see [29]). A binary operation °:0,1×0,1⟶0,1 is known as continuous-t-conorm if the following axioms are satisfied:(1)
Associativity and commutativity properties are satisfied by°(2)
° is continuous function(3)
λ°0=λ,∀λ∈0,1(4)
λ°ϵ≤k°l, whenever λ≤k and ϵ≤l for all λ,ϵ,k,l∈0,1Example.
(1)
λ°1ε=minλ+ε,1(2)
λ°2ε=λ+ε−λε(3)
λ°3ε=maxλ,εDefinition 2.6.
(see [16]). Let ζ be a nonempty set. Let s∈Rsetofreals,s≥1, and ⊙ be a continuous-t-norm. A FS Φ on ζ×ζ×[0,∞) is called fuzzy b-metric if, for all ω,υ,z∈ζ, the following conditions hold:(bM1)Φω,υ,0=0(bM2)Φω,υ,ι=1,∀ι≥0⇔Sω=υ(bM3)Φω,υ,ι=Φυ,ω,ι,∀ι≥0(bM4)Φω,z,sι+θ≥Φω,υ,ι⊙Φυ,z,θ,∀ι,θ≥0(bM5)limι⟶∞Φω,υ,ι=1 and Φ(ω,υ,.):[0,∞)⟶[0,1] is left continuous
The quadrupleζ,Φ,⊙,s is known as fuzzy b-metric space.


## 3. Intuitionistic Fuzzyb-Metric Spaces and Coincidence Point Results

Definition 3.1.
A 6-tupleζ,Φ,φ,⊙,°,s is said to be IFbMS if ζ is an arbitrary set, s≥1 is a given real number, ⊙ is a continuous-t-norm, ° is a continuous-t-conorm, Φ and φ are FSs on ζ2×[0,∞) satisfying the following conditions. For all ω,υ,z∈ζ,(a)
Φω,υ,ι+φω,υ,ι≤1(b)
Φω,υ,0=0(c)
Φω,υ,ι=1,∀ι>0 iff ω=υ(d)
Φω,υ,ι=Φυ,ω,ι,∀ι>0(e)
Φω,z,sι+θ≥Φω,υ,ι⊙Φυ,z,θ,∀ι,θ>0(f)
Φ(ω,υ,.):[0,∞)⟶[0,1] is left continuous and limι⟶∞Φω,υ,ι=1(g)
φω,υ,0=1(h)
φω,υ,ι=0,∀ι>0 iff ω=υ(i)
φω,υ,ι=φυ,ω,ι,∀ι>0(j)
φω,z,sι+θ≤φω,υ,ι°φυ,z,θ,∀ι,θ>0(k)
limι⟶∞φω,υ,ι=0 and φ(ω,υ,.):[0,∞)⟶[0,1] is right continuous
Here,Φω,υ,ι and φω,υ,ι denote the degree of nearness and the degree of nonnearness between ω and υ with respect to t, respectively.Example 3.2.1.
Letζ,ϖ,s be a b-metric space and a=mina,b, a°b=maxa,b∀a,b∈0,1, and let Φϖ , φϖ be FSs on ζ2×[0,∞), defined as follows:(6)Φϖ(ω,υ,ι)=ιι+ϖ(ω,υ),ifι>0,0,ifι=0,φϖ(ω,υ,ι)=ϖ(ω,υ)ι+ϖ(ω,υ),ifι>0,1,ifι=0.
We check only axiomse and j of Definition 3.1 because verification of the other conditions is standard. Let ω,υ,z∈ζ and ι,u>0.
Without restraining the generality, we assume that(7)Φϖω,υ,ι≤Φϖυ,z,u,φϖω,υ,ι≥φϖυ,z,u.
Thus,(8)ιι+ϖω,υ≤uu+ϖυ,z,ϖω,υι+ϖω,υ≥ϖυ,zu+ϖυ,z,i.e.,(9)ιϖυ,z≤uϖω,υ.
On the contrary,(10)Φϖω,z,sι+u=sι+usι+u+ϖω,z≥sι+usι+u+sϖω,υ+ϖυ,z=ι+uι+u+ϖω,υ+ϖυ,z.
Also,(11)φϖω,z,sι+u=ϖω,zsι+u+ϖω,z≤sϖω,υ+ϖυ,zsι+u+sϖω,υ+ϖυ,z=ϖω,υ+ϖυ,zι+u+ϖω,υ+ϖυ,z,ϖω,υ+ϖυ,zι+u+ϖω,υ+ϖυ,z≤ϖω,υι+ϖω,υ.
Hence, we will obtain that(12)Φϖω,z,sι+u≥Φϖω,υ,ι=Φϖω,υ,ι⊙Φϖυ,z,u,φϖω,z,sι+u≤φϖω,υ,ι=φϖω,υ,ι°φϖυ,z,u,which had to be verified. We remark that(13)ι+uι+u+ϖω,υ+ϖυ,z≥ιι+ϖω,υ⇔ι2+uι+ιϖω,υ+uϖω,υ≥ι2+uι+ιϖω,υ+ιϖυ,z⇔uϖω,υ≥ιϖυ,z,which is true.
Also,(14)ϖω,υ+ϖυ,zι+u+ϖω,υ+ϖυ,z≤ϖω,υι+ϖω,υ⇔ιϖω,υ+ιϖυ,z+ϖω,υϖυ,z+ϖω,υ2≤ιϖω,υ+uϖω,υ+ϖx,υϖυ,z+ϖω,υ2⇔ιϖυ,z≤uϖω,υ,which is true.
Hence,ζ,Φϖ,φϖ,⊙,°,s is (standard) IFbMS.Example 3.2.2.
Letζ,ϖ,s be a b-metric space and a=mina,b and a°b=maxa,b∀a,b∈0,1, and let Φ and φ be FSs on ζ2×[0,∞), defined as follows:(15)Φω,υ,t=expϖω,υt−1,ift>0,0,ift=0,φϖω,υ,t=expϖω,υ/t−1expϖω,υ/tift>0,1,ift=0.
Then,ζ,Φ,φ,⊙,°,s is an IFbMS.Definition 3.3.
Lets≥1 be a given real number. A function f:R⟶R will be called s-nondecreasing if t<u implies that ft≤fsu and f is called s-nonincreasing if t<u implies that ft≥fsu.Proposition 3.4.
In an IFbMSζ,Φ,φ,⊙,°,s, Φ(ω,υ,.):[0,∞)⟶[0,1] is s-nondecreasing and φ(ω,υ,.):[0,∞)⟶[0,1] is s-nonincreasing, for all ω,υ∈ζ.Proof.
For0<t<u, we have(16)Φ(ω,υ,su)=Φ(ω,υ,s(u−t+t)≥Φω,ω,u−t⊙Φω,υ,t=1⊙Φω,υ,t=Φω,υ,t.
Also,(17)φ(ω,υ,su)=φ(ω,υ,s(u−t+t)≤φω,ω,u−t°φω,υ,t=0°φω,υ,t=φω,υ,t.Definition 3.5.
Letζ,Φ,φ,⊙,°,s be an IFbMS.(a)
Any sequenceωn in ζ is said to be convergent if there exists ω∈ζ such that limn⟶∞Φωn,ω,t=1 and limn⟶∞φωn,ω,t=0,∀t>0. ω is called the limit of the sequence ωn, and it is written as limn⟶∞ωn=ω, or ωn⟶ω.(b)
Any sequenceωn in ζ,Φ,⊙,°,s is said to be a Cauchy sequence if, for every ϵ in 0,1, there is n0∈φ such that Φωn,ωm,t>1−ϵandφωn,ωm,t<ϵ, for all m,n≥n0 and t>0.(c)
ζ is said to be complete if every Cauchy sequence in ζ is convergent in ζ.
In 2012, Tirado [30] proved that (standard) FMS is complete. It can be easily checked that (standard) IFbMS is also complete.Definition 3.6.
Letζ,Φ,φ,⊙,°,s be an IFbMS. An open ball Bω,r,t with center ω∈ζ and radius r, 0<r<1, and t>0 is defined as Bω,r,t=υ∈ζ:Φω,υ,t>1−r,φω,υ,t<r.Definition 3.7.
Letζ,Φ,φ,⊙,°,s be an IFbMS and A be a subset of ζ. A is said to be open if, for each ω∈A, there is an open ball B ω,r,t contained in A.
Result: letζ,Φ,φ,⊙,°,s be an IFbMS. Define τΦ,φ as τΦ,φ=A∈Pζ:ω∈Aiff∃ι>0andr∈0,1:Bω,r,ι⊂A, then τΦ,φ is a topology on ζ, where Pζ is the power set of ζ.


## 4. Coincidence and Common Fixed-Point theorems

This section concerns with the constructing and proving of coincidence theorem and common fixed-point theorem in IFbMS. Many useful results existing in literature are presented here as corollaries of our results.Definition 4.1.
Letζ be a nonempty set and Π,σ:ζ⟶ζ be two mappings on ζ.(i)
A pointω∈ζ is called a coincidence point of Π and σ if Πω=σω(ii)
A pointυ∈ζ is called point of coincidence of Π and σ if there exists ω∈ζ such that υ=Πω=σω(iii)
A pointz∈ζ is known as common fixed point of Π and σ if z=Πz=σzDefinition 4.2.
Two self-mapsΠ,σ:ζ⟶ζ are said to be weakly compatible if Πσω=σΠω when Πω=σω.Theorem 4.1.
Letζ be a nonempty set and Y,Φ,φ,⊙,°,s be an IFbMS and Π,σ:ζ⟶Y be mappings satisfying the following conditions:(1)
σζ⊆Πζ;\(2)
There isk, 0≤k<1, such that, for all ω,υ∈ζ,(18)Φσω,συ,kt≥ΦΠω,Πυ,t,Πφσω,συ,kt≤φΠω,Πυ,t.
IfΠζ or σζ is complete, then there exists a point z∈ζ such that Πz=σz. Moreover, Π and σ have a unique point of coincidence.Proof.
Letω0∈ζ. By (1), we can find ω1∈ζ such that Πω1=σω0.
Fork=0,(19)Φ(σ(ω0),σ(ω1),0t)≥Φ((Π(ω0),Π(ω1),t),φ(σ(ω0),σ(ω1),0t)≤φ((Π(ω0),Π(ω1),t),⇒Φσω0,σω1,0t=1,φσω0,σω1,0t=0.
Hence,(20)σω0=σω1⇒Πω1=σω1.
This implies thatω1 is the coincidence point of Π and σ.
Fork≠0, by induction, we can define a sequence ωn in ζ such that Πωn=σωn−1:(21)ΦΠωn,Πωn+1,t=Φσωn−1,σωn,t≥ΦΠωn−1,Πωn,t/k≥…≥ΦΠω0,Πω1,t/kn.
Clearly,1≥ΦΠωn,Πωn+1,t≥ΦΠω0,Πω1,t/kn⟶1, when n⟶∞.
Thus,
Limn⟶∞ΦΠωn,Πωn+1,t=1.
And(22)φΠωn,Πωn+1,t=φσωn−1,σωn,t≤φΠωn−1,Πωn,t/k≤…≤φΠω0,Πω1,t/kn.
Clearly,0≤φΠωn,Πωn+1,t≤φΠω0,Πω1,t/kn⟶0, when n⟶∞.
Thus,Limn⟶∞φΠωn,Πωn+1,t=0.
Letτnt=ΦΠωn,Πωn+1,t and μnt=φΠωn,Πωn+1,t, for all n∈N∪0, t>0.
Clearly,limn⟶∞τnt=1 and limn⟶∞μnt=0.
To show thatΠωn is a Cauchy sequence, suppose it is not; then, there exists 0<ϵ<1 and two sequences pη and qη such that, for every η∈N∪0, t>0, pη>qη≥η, ΦΠωpη,Πωqη,t≤1−ε, and φΠωpη,Πωqη,t≥ε.
Then,ΦΠωpη−1,Πωqη−1,t>1−ε, ΦΠωpη−1,Πωqη,t>1−ε, and φΠωpη−1,Πωqη−1,t<ε, φΠωpη−1,Πωqη,t<ε.
Now,(23)1−ε≥ΦΠωpη,Πωqη,t≥ΦΠωpη−1,Πωpη,t/2s⊙ΦΠωpη−1,Πωqη,t/2s>τpη−1t/2s⊙1−ε,ε≤φΠωpη,Πωqη,t≤φΠωpη−1,Πωpη,t/2s°φΠωpη−1,Πωqη,t/2s<μpη−1t/2s°ε.
Sinceτpη−1t/2s⟶1 as η⟶∞ and μpη−1t/2s⟶0 as η⟶∞ for every t, supposing that η⟶∞, we have 1−ε≥ΦΠωpη,Πωqη,t>1−ε, ε≤φΠωpη,Πωqη,t<ε.
Clearly, this leads to the contradiction.
Hence,Πωn is a Cauchy sequence in Πζ.Case I.
: suppose thatΠζ is complete; then, there exists a point υ∈Πζ such that limn⟶∞Πωn=υ.
This implies that there existsz∈ζ such that υ=Πz.
Now,(24)ΦΠz,σz,t≥ΦΠz,Πωn,t/2s⊙ΦΠωn,σz,t/2s=ΦΠz,Πωn,t/2s⊙Φσωn−1,σz,t/2s≥ΦΠz,Πωn,t/2s⊙ΦΠωn−1,Πz,t/2sk≥1⊙1=1,asn⟶∞,φΠz,σz,t≤φΠz,Πωn,t/2s∘φΠωn,σz,t/2s=φΠz,Πωn,t/2s∘φσωn−1,σz,t/2s≤φΠz,Πωn,t/2s∘φΠωn−1,Πz,t/2sk≤0∘0=0,asn⟶∞.
By (c) and (h) of Definition3.1, it follows that Πz=σz.
Hence,z is a coincidence point and υ is the point of coincidence of Π and σ.Case II
: suppose thatσζ is complete; then, there exists a point υ∈σζ such that limn⟶∞Πωn=υ.
However,σζ⊆Πζ; this implies that υ∈Πζ, so there exists z∈ζ such that υ=Πz.
Next onward, proof is the same as in case I.
Now, we show that the point of coincidence ofΠ and σ is unique.
Letυ1 be another point of coincidence of Π and σ. Then, υ1=Πz1=σz1 for some z1 in ζ:(25)1≥Φυ,υ1,t=Φσz,σz1,t≥ΦΠz,Πz1,t/k=Φυ,υ1,t/k≥…≥Φυ,υ1,t/kn.
Also,(26)0≤φυ,υ1,t=φσz,σz1,t≤φΠz,Πz1,t/k=φυ,υ1,t/k≤…≤φυ,υ1,t/kn.
Thus, byΠ and k of Definition 3.1, limn⟶∞Φυ,υ1,t/kn=1 and limn⟶∞φυ,υ1,t/kn=0.
It follows that1≥Φυ,υ1,t≥1and0≤φυ,υ1,t≤0, which implies that υ=υ1 by c and h of Definition 3.1. This completes the proof.
Note: the uniqueness of the coincidence point will be sure when Π or σ is one-one.
The following result gives common fixed point ofΠ and σ with the assumption of weakly compatibility.Theorem 4.2.
Letζ,Φ,φ,⊙,°,s be a complete IFbMS and Π,σ:ζ⟶ζ be mappings satisfying the following conditions:(1)
σζ⊆Πζ.(2)
There isk,0≤k<1, such that, for all ω,υ∈ζ,(27)Φσω,συ,kt≥ΦΠω,Πυ,tφσω,συ,kt≤φΠω,Πυ,t.(3)
Π and σ are weakly compatible.
Then,Π and σ have a unique-common fixed point in ζ.Proof.
By the above theorem, there is a unique point of coincidence ofΠ and σ in ζ. That is, we can get z,υ in ζ such that υ=Πz=σz.
Sinceυ=Πz and Π and σ are weakly compatible, so συ=σΠz=Πσz=Πυ.
Letu=Πυ=συ; then, u is a point of coincidence of Π and σ. Since the point of coincidence is unique, this implies that u=υ⇒υ=Πυ=συ.
Hence,υ is unique-common fixed point of Π and σ. This completes the proof.Corollary 1.
Letζ,Φ,φ,⊙,° be a complete IFMS and Π,σ:ζ⟶ζ be mappings satisfying the following conditions:(1)
σζ⊆Πζ.(2)
There isk,0≤k<1, such that, for all ω,υ∈ζ,(28)Φσω,συ,kt≥ΦΠω,Πυ,t,φσω,συ,kt≤φΠω,Πυ,t.(3)
Π and σ are weakly compatible.Then,Π and σ have unique common fixed point in ζ.Proof.
By puttings=1 in Theorem 4.2, we get the required result.Corollary 2.
Letζ,Φ,⊙ be a complete fuzzy b-metric space and Π,σ:ζ⟶ζ be mappings satisfying the following conditions:(1)
σζ⊆Πζ.(2)
There existk∈[0,1) such that ∀ω,υ∈ζ,(29)Φσω,συ,kt≥ΦΠω,Πυ,t.(3)
Π and σ are weakly compatible.
Then,Π and σ have unique-common fixed point in ζ.Proof.
By puttingφ=O (i.e., φ is a zero function) in Theorem 4.2, we get the required result.Corollary 3 (see [31]).
Letζ,Φ,⊙ be a complete FMS and Π,σ:ζ⟶ζ be mappings satisfying the following conditions:(1)
σζ⊆Πζ.(2)
There existk∈[0,1) such that ∀ω,υ∈ζ,(30)Φσω,συ,kt≥ΦΠω,Πυ,t.(3)
Π and σ are weakly compatible.
Then,Π and σ have unique-common fixed point in ζ.Proof.
By puttingφ=O (i.e., φ is a zero function) and s=1 in Theorem 4.2, we get the required result.Example.
Letζ=R and Π:ζ⟶ζ be a self-map on ζ defined as Πω=3ω,∀ω∈ζ.
DefineΦ,φ:ζ2×[0,∞)⟶[0,1] as(31)Φ(ω,υ,t)=tt+|ω−υ|,ift>0,0,ift=0,φ(ω,υ,t)=ω−υt+|ω−υ|,ift>0,1,ift=0.
By [30] and Example 3.2.1, it is clear that ζ,Φ,φ,⊙,°,s is complete IFbMS, where a=mina,b, a°b=maxa,b, and ∀a,b∈0,1. Note that conventional Banach’s contraction principle fails to find the fixed point of Π as Π is not a contraction.
Now, defineσ:ζ⟶ζ as σω=2ω,∀ω∈ζ.
It is evident thatσζ⊆Πζ and Π and σ are weakly compatible. Then,(32)Φσω,συ,2t/3=2t/32t/3+|2ω−2υ|=t/3t/3+|ω−υ|≥tt+|3ω−3υ|,=ΦΠω,Πυ,tφσω,συ,2t/3=2ω−2υ2t/3+|2ω−2υ|=ω−υt/3+|ω−υ|≤3ω−3υt+|3ω−3υ|=φΠω,Πυ,t.
Thus, all the conditions of Theorem4.2 are satisfied for k=2/3; hence, Π and σ have a unique-common fixed point: 0=Π0=σ0.


## 5. Application

Now, as an application of coincidence theorem, we give the following theorem.Theorem.
LetF,G:R×I⟶R and f:R⟶R be continuous mappings such that(33)Gω,u=Fω,u+fω,where I=u∈R:a≤u≤b,a,b∈R.
LetCI be the collection of all continuous functions defined from I into R. Suppose that, for each ω∈CI, there exists υ∈CI, such that fυu=Gωu,u and fω:ω∈CI is complete. If there exists a number k∈[0,1) such that, for all ω1,ω2∈CI and u∈I,(34)|Gω1u,u−Gω2u,u|≤k|fω1u−fω2u|,then the equation,(35)Fω,u=0,defines a continuous function ω in terms of u.Proof.
Letζ=Y=CI.
DefineΦ,φ:ζ2×[0,∞)⟶[0,1] as(36)Φϖ(ω,υ,t)=tt+maxu∈I|ω(u)−υ(u)|,ift>0,0,ift=0,φϖ(ω,υ,t)=maxu∈I|ω(u)−υ(u)|t+maxu∈I|ω(u)−υ(u)|,ift>0,1,ift=0.
Define mappingσ:ζ⟶ζ as follows:(37)σωu=Gωu,u.
Then, by assumption,fζ=fω:ω∈ζ is complete. Let ω⊙∈σζ; then, ω⊙=σω for ω∈ζ and ω⊙u=σωu=Gωu,u.
By assumptions, there existsυ∈ζ such that σωu=Gωu,u=fυu.
Hence,σζ⊆fζ.
Since(38)σωu−συu=|Gωu,u−Gυu,u|≤k|fωu−fυu|≤kmaxu∈I|fωu−fυu|,it further implies that(39)maxu∈I|σωu−συu|≤kmaxu∈I|fωu−fυu|⇒maxu∈I|σωu−συu|kt≤maxu∈I|fωu−fυu|t⇒ktmaxu∈I|σωu−συu|≥tmaxu∈I|fωu−fυu|⇒ktkt+maxu∈I|σωu−συu|≥tt+maxu∈I|fωu−fyu|⇒Φσω,συ,kt≥Φfω,fυ,t.
Also, inequality (39) implies that(40)maxu∈I|σωu−συu|kt≤maxu∈I|fωu−fυu|t⇒maxu∈I|σωu−συu|kt+maxu∈I|σωu−συu|≤maxu∈I|fωu−fυu|t+maxu∈I|fωu−fυu|⇒φσω,συ,kt≤φfω,fυ,t.
Hence, all the conditions of theorem (4.1) are satisfied to obtain a continuous functionz:I⟶R such that σz=fz. Then,(41)Gzu,u−fzu=0,where z will be a solution of the equation Fz,u=0.Remark.
If we consider an implicit formFω,u=10ω5u−1+u, then, by the assumptions Gω,u=10ω5u−1+u+90ω5 and fωu=90ω5 in Theorem 4.3, we can easily obtain the explicit representation as ω=5u/101−u.
For a nontrivial example, consider the implicit equation,(42)u+sin8ω5u−ω5=0,in the space C−1/9,1/9. Let(43)Fω,u=u+sin8ω5u−ω5,fω=5ω5−5,where F:R×−1/9,1/9⟶R and f:R⟶R. Then, let Gω,u=u+sin8ω5u+4ω5−5. Define σ:C−1/9,1/9⟶C−1/9,1/9 as(44)σωu=Gωu,u=u+sin8ω5uu+4ω5u−5.
Here,fω=5ω5−5 implies that fR=R.
Now,(45)σω1−σω2|=|Gω1,u−Gω2,u=|u+sin8ω15u+4ω15−5−u−sin8ω25u−4ω25+5|≤|sin8ω15u−sin8ω25u+4ω15−4ω25|≤|sin8ω15u−sin8ω25u|+4|ω15−ω25|≤8|u‖ω15−ω25|+4|ω15−ω25|≤4445|5ω15−5−5ω25+5|.
Hence, all the conditions of Theorem 4.3 are satisfied. To apply Theorem4.1, choose an initial guess ω0u=0; then,(46)σω0u=Gω0u,u=u−5=fω1u=5ω15−5.
This implies thatω1u=5u/5.
So,(47)σω1u=Gω1u,u=u+sin8ω15u+4ω15−5=u+sin8u25+4u5−5,fω2=u+sin8u25+4u5−5,5ω25u=u+sin8u25+4u5,⇒ω2u=sin8u2/5+9u/555.
Now,(48)σω2u=Gω2u,u=u+sin8ω25u+4ω25−5,fω3=u+sin8usin8u2/5+9u2/55+4sin8u2/5+9u/55−5,⇒ω3=u+sin8usin8u2/5+9u2/5/5+4sin8u2/5+9u/5/555,is an approximation of the explicit form of Fω,u.
It is worthwhile to point out here that the application given in the above remark is not found in the literature even as an application of the following corollary of Theorem4.1 regarding metric spaces.Corollary.
Letζ be a nonempty set and Y,ϖ be a b-metric space and f,σ:ζ⟶Y be mappings satisfying the following conditions:(1)
σζ⊆fζ.(2)
There existk∈[0,1) such that ∀ω,υ∈ζ:(49)ϖσω,συ≤kϖfω,fυ.
Iffζ or σζ is complete, then there exist a point z∈ζ such that fz=σz. Moreover, f and σ have a unique point of coincidence.


## 6. Conclusion

Metric spaces play a vital role in functional analysis and its related concepts. Modern and latest developments are due to fuzzy theory, fuzzy logic, and its vast applications in almost all fields of research. This motivated us to define metric-type spaces in fuzzy version. We called this IFbMSs. Moreover, interesting nontrivial examples are created as well. These spaces generalize fuzzyb-metric spaces and IFMSs which are already generalized forms of classic metric spaces. Since fixed-point techniques have a lot of wonderful applications in science and technology, so in this research article, we intended to put our efforts in obtaining coincidence points and common fixed points in IFbMS. In such a way, many useful, present, and conventional results are presented as consequences of our results. Furthermore, as an application we have proved an implicit function theorem with the help of our main result. This activity will definitely motivate researchers to do further work in these spaces and fixed-point theory.



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*Source: 5616824-2022-03-31.xml*