# Some Coincidence and Common Fixed Point Results in Fuzzy Metric Space with an Application to Differential Equations

Authors: Saif Ur Rehman; Iqra Shamas; Naeem Jan; Abdu Gumaei; Mabrook Al-Rakhami
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, we study some coincidence point and common fixed point theorems in fuzzy metric spaces by using three-self-mappings. We prove the uniqueness of some coincidence point and common fixed point results by using the weak compatibility of three-self-mappings. In support of our results, we present some illustrative examples for the validation of our work. Our results extend and improve many results given in the literature. In addition, we present an application of fuzzy differential equations to support our work.


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## Body

## 1. Introduction

Zadeh [1] introduced the concept of fuzzy sets which is defined as “a set contracted from a function having a domain is a nonempty set Ω and range in 0,1 is called a fuzzy set, that is if G:Ω⟶0,1.” In 1975, Kramosil and Michalek [2] introduced the notion of fuzzy metric (FM) space, and they compared the concept of fuzzy metric with the statistical metric space and proved that both the conceptions are equivalent in some cases. Later on, George and Veeramani [3] modify the concept of Kramosil and Michalek [2] and proved that every metric induces a fuzzy metric. They proved some basic properties and Baire’s theorem for fuzzy metric spaces. In 1988, Grabiec [4] used the concept of Kramosil and Michalek [2] and proved fixed point (FP) theorems of “Banach and Edelstein contraction mapping theorems on complete and compact FM spaces, respectively.” Gregori and Sapena [5], Imdad and Ali [6], Mihet [7, 8], Bari and Vetro [9], and Som [10] proved some FP and common fixed point (CFP) theorems in FM spaces. Aliouche et al. [11] and Rao et al. [12] established some related FP in FM spaces.Hadzic and Pap [13] established some multivalued FP results in probabilistic metric spaces with an application in FM spaces. Later on, Kiany and Amini-Haradi [14] obtained some FP and end-point theorems for set-valued contractive type mappings in FM spaces. In [15], Beg et al. proved some FP results for self-mappings satisfying an implicit relation in a complete FM space. Rolden et al. [16] established some new FP theorems in FM spaces, while in [17], Jeli et al. presented some results by using cyclic (ψ,ϕ)-contractions in Kaleva-Seikkala’s type FM spaces. Later on, Li et al. [18] proved some strong coupled FP theorems in complete FM spaces with an integral type of application. The concept of rational type fuzzy-contraction is given by Rehman et al. [19]. They proved some unique FP theorems with the application of nonlinear integral in FM spaces. Shamas et al. [20] proved some unique FP results in FM spaces with an application to Fredholm integral equations. Recently, Jabeen et al. [21] presented the concept of weakly compatible self-mappings in fuzzy cone metric spaces, and they proved some coincidence point and CFP theorems in the said space with integral type application. Some more coincidence points, coupled coincidence points, and CFP findings in deferent types of metric spaces can be found in (e.g., see [22–35] the references therein).In this paper, we establish some unique coincidence points and CFP theorems in FM space by using the concept of Gregori and Sapena [5] and Jabeen et al. [21]. We establish some generalized fuzzy-contraction results for weakly compatible three self-mappings in FM spaces without the assumption that the “fuzzy contractive sequences are Cauchy.” We present some illustrative examples and an application of fuzzy differential equation to support our work. By using this concept, researchers can prove more coincidence points and CFP results for different contractive type mappings in FM spaces with the application of integral operators.The layout of this paper is as follows: Section2 consists of preliminary concepts. While Section 3 deals with the main results of this paper in which we shall prove unique coincidence points and CFP theorems by using weakly-compatible three self-mappings in FM spaces with some illustrative examples. In Section 4, we establish an application of fuzzy differential equations to support our main work. While in the last section, that is, Section 5 is the conclusion part of our paper.


## 2. Preliminaries

In this section, we recall some basic definitions related to our main work such as continuoust-norm, FM space, Cauchy sequence, fuzzy-contraction, and weak-compatible mappings. The concept of continuous t-norm is given by Schweizer and Sklar [36].Definition 1 (see [36]).
An operation∗:0,1×0,1⟶0,1 is known as a continuous t-norm if it fulfils the following axioms;
(1)
∗ is associative, commutative, and continuous.(2)
1∗ρ1=ρ1 and ρ1∗ρ2≤ρ3∗ρ4, whenever ρ1≤ρ3 and ρ2≤ρ4, for each ρ1,ρ2,ρ3,ρ4∈0,1.
The basic continuoust-norms are (see; [36]): the minimum, the product, and the Lukasiewicz t-norms are defined respectively as follows;
(1)ρ1∗ρ2=minρ1,ρ2,ρ1∗ρ2=ρ1ρ2andρ1∗ρ2=maxρ1+ρ2−1,0.Definition 2 (see [3]).
A 3-tupleΩ,MF,∗ is said to be a FM space if Ω is an arbitrary set, ∗ is a continuous t-norm, and MF is a fuzzy set on Ω×Ω×0,∞ satisfying the following;
(i)
MFξ,η,t>0 and MFξ,η,t=1⇔ξ=η.(ii)
MFξ,η,t=MFη,ξ,t.(iii)
MFξ,ζ,t∗MFζ,η,s≤MFξ,η,t+s.(iv)
MFξ,η,·:0,∞⟶0,1 is continuousfor all ξ,ζ,η∈Ω and t,s>0.Definition 3 (see [3, 5]).
LetΩ,MF,∗ be a FM space, ξ∈Ω and ξj be a sequence in Ω. Then
(i)
ξj is said to converge to ξ if for t>0 and ε∈0,1,∃j1∈N where (N represent the set of natural numbers) such that MFξj,ξ,t>1−ε,∀j≥j1. We represent this by limj⟶∞ξj=ξ or ξj⟶ξ as j⟶∞.(ii)
wi is said to be a Cauchy sequence, if for t>0 and ε∈0,1,,∃j1∈N such that MFξj,ξm,t>1−ε,∀j,m≥j1.(iii)
Ω,MF,∗ is complete, if every Cauchy sequence is convergent in Ω.(iv)
ξj is known as a fuzzy-contractive, if there is 0<β<1 so that
(2)1MFξj,ξj+1,t−1≤β1MFξi−1,ξi,t−1fort>0,andj≥1.Lemma 4 (see [3]).
MFξ,η,∗ is nondecreasing ∀ξ,η∈Ω.Lemma 5 (see [3]).
LetΩ,MF,∗ be a FM space and let a sequence ξj in Ω converges to a point ξ∈Ω iff MFξj,ξ,t⟶1, as j⟶∞, for t>0.Definition 6 (see [9]).
LetΩ,MF,∗ be a FM space. The FM MF is triangular if
(3)1MFξ,η,t−1≤1MFξ,ζ,t−1+1MFζ,η,t−1,for all ξ,ζ,η∈Ω and t>0.
Note: easily one can prove that a FMMF is triangular if we define a mapping MF:Ω×Ω×intC⟶0,1 by MFξ,η,t=t/t+ξ−η,forallξ,η∈Ω,andt>0.Lemma 7 (see [3]).
LetΩ,MF,∗ be a FM space. Let ξ∈Ω and ξj be a sequence in Ω. Then ξj⟶ξ iff limj⟶∞MFξj,ξ,t=1, for t>0.Definition 8 (see [5]).
LetΩ,MF,∗ be a FM space and G:Ω⟶Ω. Then, G is known as a fuzzy-contraction, if there is 0<β<1 so that
(4)1MFGξ,Gη,t−1≤β1MFξ,η,t−1,for all ξ,η∈Ω and t>0.Definition 9 (see [37]).
LetG and ℓ be two self-mappings on a nonempty set Ω (i.e., G,ℓ:Ω⟶Ω). If ∃ξ∈Ω and ξ=Gω=ℓω for some ω∈Ω. Then ω is called a coincidence point of G and ℓ, and ξ is called a point of coincidence of the mappings G and ℓ. The mappings G and ℓ are said to be weakly-compatible if they commute at their coincidence point, i.e., Aω=ℓω for some ω∈Ω, then Aℓω=ℓAω.Proposition 10 (see [37]).
LetG and ℓ be weakly-compatible self-mappings on a nonempty set Ω. If G and ℓ have a unique point of coincidence such that ξ=Aω=ℓω, then ξ is known as the unique CFP of G and ℓ.


## 3. Main Results

In this section, we present some unique coincidence points and CFP theorems in FM spaces by using weakly-compatible three self-mappings with illustrative examples.Theorem 11.
LetΩ,MF,∗ be a FM space in which a FM MF is triangular. Let G,H,ℓ:Ω⟶Ω be three self-mappings that satisfy
(5)1MFGξ,Hη,t−1≤β11MFℓξ,ℓη,t−1+β21MFℓξ,Gξ,t−1+1MFℓη,Hη,t−1+1MFℓη,Gξ,t−1+1MFℓξ,Hη,t−1+β3max1MFℓξ,Gξ,t−1,1MFℓη,Hη,t−1,1MFℓξ,Hη,t−1,1MFℓη,Gξ,t−1,for all ξ,η∈Ω,t>0, and β1,β2,β3∈0,1 with β1+4β2+2β3<1. If GΩ∪HΩ⊂ℓΩ and ℓΩ is a complete subspace of Ω. Then, G,H, and ℓ have a unique point of coincidence. In addition, if G,ℓ and H,ℓ are weakly-compatible, then, G,H, and ℓ have a unique CFP.Proof.
Letξ0∈Ω be arbitrary and by using the hypothesis GΩ∪HΩ⊂ℓΩ. Now, we choose a sequence ξj such that
(6)ℓξ2j+1=Gξ2jandℓξ2j+2=Hξ2j+1forallj≥0.
Now by the view of (5) and (6), for t>0, we have
(7)1MFℓξ2j+1,ℓξ2j+2,t−1=1MFGξ2j,Hξ2j+1,t−1≤β11MFℓξ2j,ℓξ2j+1,t−1+β21MFℓξ2j,Gξ2j,t−1+1MFℓξ2j+1,Hξ2j+1,t−1+1MFℓξ2j+1,Gξ2j,t−1+1MFℓξ2j,Hξ2j+1,t−1+β3max1MFℓξ2j,Gξ2j,t−1,1MFℓξ2j+1,Hξ2j+1,t−1,1MFℓξ2j+1,Gξ2j,t−1,1MFℓξ2j,Hξ2j+1,t−1,≤β11MFℓξ2j,ℓξ2j+1,t−1+β21MFℓξ2j,ℓξ2j+1,t−1+1MFℓξ2j+1,ℓξ2j+2,t−1+1MFℓξ2j,ℓξ2j+1,t−1+1MFℓξ2j+1,ℓξ2j+2,t−1+β3max1MFℓξ2j,ℓξ2j+1,t−1,1MFℓξ2j+1,ℓξ2j+2,t−1,1MFℓξ2j,ℓξ2j+2,t−1.
After simplification, fort>0, we obtain
(8)1MFℓξ2j+1,ℓξ2j+2,t−1≤β1+2β21−2β21MFℓξ2j,ℓξ2j+1,t−1+β31−2β2max1MFℓξ2j,ℓξ2j+1,t−1,1MFℓξ2j+1,ℓξ2j+2,t−1,1MFℓξ2j,ℓξ2j+2,t−1.
Now three possibilities arise:(i)
If1/MFℓξ2j,ℓξ2j+1,t−1 is a maximum term in (8) for t>0, then after simplification, we get that
(9)1MFℓξ2j+1,ℓξ2j+2,t−1≤β1+2β2+β31−2β21MFℓξ2j,ℓξ2j+1,t−1fort>0.(ii)
If1/MFℓξ2j+1,ℓξ2j+2,t−1 is a maximum term in (8) for t>0, then after simplification, we get that
(10)1MFℓξ2j+1,ℓξ2j+2,t−1≤β1+2β21−2β2−β31MFℓξ2j,ℓξ2j+1,t−1fort>0.(iii)
If1/MFℓξ2j,ℓξ2j+2,t−1 is a maximum term in (8) for t>0, then after simplification, we get that
(11)1MFℓξ2j+1,ℓξ2j+2,t−1≤β1+2β2+β31−2β2−β31MFℓξ2j,ℓξ2j+1,t−1fort>0.
Assume thatλ≔maxβ1+2β2+β3/1−2β2,β1+2β2/1−2β2−β3,β1+2β2+β3/1−2β2−β3<1, then from (9), (10), and (11), we get that
(12)1MFℓξ2j+1,ℓξ2j+2,t−1≤λ1MFℓξ2j,ℓξ2j+1,t−1fort>0.
Similarly, again from (5) and (6), for t>0,
(13)1MFℓξ2j+2,ℓξ2j+3,t−1=1MjGξ2j+2,Hξ2j+1,t−1≤β11MFℓξ2j+2,ℓξ2j+1,t−1+β21MFℓξ2j+2,Gξ2j+2,t−1+1MFℓξ2j+1,Hξ2j+1,t−1+1MFℓξ2j+1,Gξ2j+2,t−1+1MFℓξ2j+2,Hξ2j+1,t−1+β3max1MFℓξ2j+2,Gξ2j+2,t−1,1MFℓξ2j+1,Hξ2j+1,t−1,1MFℓξ2j+1,Gξ2j+2,t−1,1MFℓξ2j+2,Hξ2j+1,t−1≤β11MFℓξ2j+2,ℓξ2j+1,t−1+β21MFℓξ2j+2,ℓξ2j+3,t−1+1MFℓξ2j+1,ℓξ2j+2,t−1+1MFℓξ2j+1,ℓξ2j+2,t−1+1MFℓξ2j+2,ℓξ2j+3,t−1+β3max1MFℓξ2j+2,ℓξ2j+3,t−1,1MFℓξ2j+1,ℓξ2j+2,t−1,1MFℓξ2j+1,ℓξ2j+3,t−1.
After simplification, fort>0, we obtain
(14)1MFℓξ2j+2,ℓξ2j+3,t−1≤β1+2β21−2β21MFℓξ2j+1,ℓξ2j+2,t−1+β31−2β2max1MFℓξ2j+2,ℓξ2j+3,t−1,1MFℓξ2j+1,ℓξ2j+2,t−1,1MFℓξ2j+1,ℓξ2j+3,t−1.
Now three possibilities arise,(i)
If1/MFℓξ2j+2,ℓξ2j+3,t−1 is a maximum term in (14) for t>0, then after simplification, we get that
(15)1MFℓξ2j+2,ℓξ2j+3,t−1≤β1+2β21−2β2−β31MFℓξ2j+1,ℓξ2j+2,t−1fort>0.(ii)
If1/MFℓξ2j+1,ℓξ2j+2,t−1 is a maximum term in (14) for t>0, then after simplification, we get that
(16)1MFℓξ2j+2,ℓξ2j+3,t−1≤β1+2β2+β31−2β21MFℓξ2j+1,ℓξ2j+2,t−1fort>0.(iii)
If1/MFℓξ2j+1,ℓξ2j+3,t−1 is a maximum term in (14) for t>0, then after simplification, we get that
(17)1MFℓξ2j+2,ℓξ2j+3,t−1≤β1+2β2+β31−2β2−β31MFℓξ2j+1,ℓξ2j+2,t−1fort>0.
Sinceλ=maxβ1+2β2+β3/1−2β2,β1+2β2/1−2β2−β3,β1+2β2+β3/1−2β2−β3<1, as defined in (12), then from (15), (16), and (17), we get that
(18)1MFℓξ2j+2,ℓξ2j+3,t−1≤λ1MFℓξ2j+1,ℓξ2j+2,t−1fort>0.
Now, from (12) and (18), and by induction, for t>0, we have
(19)1MFℓξ2j+2,ℓξ2j+3,t−1≤λ1MFℓξ2j+1,ℓξ2j+2,t−1≤λ21MFℓξ2j,ℓξ2j+1,t−1≤⋯≤λ2j+21MFℓξ0,ℓξ1,t−1⟶0,asj⟶∞.
Hence, from the above, we get that the sequenceℓξjj≥0 is a fuzzy-contractive in Ω, therefore,
(20)limj⟶∞MFℓξj,ℓξj+1,t=1fort>0.
Next, we have to prove that the sequenceℓξjj≥0 is a Cauchy sequence in Ω,MF,∗. Then, from Definition 2 and by using (20), for m>j where m=j+n for n=1,2,⋯ and t>0, we have that
(21)MFℓξj,ℓξm,t=MFℓξj,ℓξj+n,t≥MFℓξj,ℓξj+1,tn∗MFℓξj+1,ℓξj+2,tn∗⋯∗MFℓξj+n−1,ℓξj+n,tn⟶1∗1∗⋯∗1=1,asj⟶∞.
This implies that(22)limj,m⟶∞MFℓξj,ℓξm,t=1fort>0.
Hence, proved thatℓξj is a Cauchy sequence. Since, ℓΩ is a complete subspace of Ω,MF,∗, then there exits ω,κ∈Ω such that ℓξj⟶κ=ℓω as j⟶∞, therefore,
(23)limj⟶∞MFℓξj,κ,t=limj⟶∞MFℓξj,ℓω,t=1fort>0.
Now, we have to show thatℓω=Gω=Hω. First, we estimate that MFℓω,Gω,t=MFκ,Gω,t for t>0. Since, MF is triangular, therefore,
(24)1MFℓω,Gω,t−1≤1MFℓω,ℓξ2j+2,t−1+1MFℓξ2j+2,Gω,t−1fort>0.
Now by using (5), (20), and (23), for t>0, we have
(25)1MFℓξ2j+2,Gω,t−1=1MFGω,Hξ2j+1,t−1≤β11MFℓω,ℓξ2j+1,t−1+β21MFℓω,Gω,t−1+1MFℓξ2j+1,Hξ2j+1,t−1+1MFℓξ2j+1,Gω,t−1+1MFℓω,Hξ2j+1,t−1+β3max1MFℓω,Gω,t−1,1MFℓξ2j+1,Hξ2j+1,t−1,1MFℓξ2j+1,Gω,t−1,1MFℓω,Hξ2j+1,t−1=β11MFℓω,ℓξ2j+1,t−1+β21MFℓω,Gω,t−1+1MFℓξ2j+1,ℓξ2j+2,t−1+1MFℓξ2j+1,Gω,t−1+1MFℓω,ℓξ2j+2,t−1+β3max1MFℓω,Gω,t−1,1MFℓξ2j+1,ℓξ2j+2,t−1,1MFℓξ2j+1,Gω,t−1,1MFℓω,ℓξ2j+2,t−1⟶2β2+β31MFℓω,Gω,t−1asj⟶∞.
Thus,(26)limj⟶∞sup1MFℓξ2j+2,Gω,t−1≤2β2+β31MFℓω,Gω,t−1fort>0.
The above (26) is together with (23) and (24), we get that
(27)1MFℓω,Gω,t−1≤2β2+β31MFℓω,Gω,t−1⇒1−2β2−β31MFℓω,Gω,t−1≤0fort>0,
which is a contradiction, as1−2β2−β3≠0, therefore, MFℓω,Gω,t=1⇒ℓω=Gω, for t>0. Hence, ℓω=Gω=κ. Similarly, again by using the triangular property of MF,
(28)1MFℓω,Hω,t−1≤1MFℓω,ℓξ2j+1,t−1+1MFℓξ2j+1,Hω,t−1fort>0.
Again by using (5), (20), and (23), similar to the above simplification, we get
(29)limj⟶∞sup1MFℓξ2j+1,Hω,t−1≤2β2+β31MFℓω,Hω,t−1fort>0.
The above (29) is together with (23) and (28), we get that
(30)1MFℓω,Hω,t−1≤2β2+β31MFℓω,Hω,t−1⇒1−2β2−β31MFℓω,Hω,t−1≤0fort>0,which is a contradiction, as 1−2β2−β3≠0, therefore, MFℓω,Hω,t=1⇒ℓω=Hω, for t>0. Hence, ℓω=Hω=κ. It follows that ω is a coincidence point, and κ is a common point of coincidence point for the self-mappings ℓ,G, and H, that is,
(31)ℓω=Gω=Hω=κ.
Next, we prove that the point of coincidence point of the three self-mappings is unique. Let∃ another point κ1∈Ω such that
(32)κ1=ℓω1=Gω1=Hω1forsomeω1∈Ω.
Now by using (5), for t>0,
(33)1MFκ,κ1,t−1=1MFGω,Hω1,t−1≤β11MFℓω,ℓω1,t−1+β21MFℓω,Gω,t−1+1MFℓω1,Hω1,t−1+1MFℓω1,Gω,t−1+1Mmℓω,Hω1,t−1+β3max1MFℓω,Gω,t−1,1MFℓω1,Hω1,t−1,1MFℓω1,Gω,t−1,1Mmℓω,Hω1,t−1=β1+2β2+β31MFκ,κ1,t−1.
After simplification, we get that(34)1MFκ,κ1,t−1≤β1+2β2+β31MFκ,κ1,t−1,⇒1−β1−2β2−β31MFκ,κ1,t−1≤0fort>0,which is a contradiction. Hence, MFκ,κ1,t=1⇒κ=κ1 for t>0. Further, by using the weak compatibility of the pairs G,ℓ and H,ℓ and by Proposition 10, it follows that the three self-mappings G,H, and ℓ have a unique CFP, that is, ℓκ=Gκ=Hκ=κ.Corollary 12.
LetΩ,MF,∗ be a FM space in which a FM MF is triangular. Let G,H,ℓ:Ω⟶Ω be three self-mappings that satisfy
(35)1MFGξ,Hη,t−1≤β11MFℓξ,ℓη,t−1+β21MFℓξ,Gξ,t−1+1MFℓη,Hη,t−1+1MFℓη,Gξ,t−1+1MFℓξ,Hη,t−1,for all ξ,η∈Ω,t>0, and β1,β2∈0,1 with β1+4β2<1. If GΩ∪HΩ⊂ℓΩ and ℓΩ is a complete subspace of Ω. Then G,H, and ℓ have a unique point of coincidence. In addition, if G,ℓ and H,ℓ are weakly-compatible, then, G,H, and ℓ have a unique CFP.Corollary 13.
LetΩ,MF,∗ be a FM space in which a FM MF is triangular. Let G,H,ℓ:Ω⟶Ω be three self-mappings that satisfy
(36)1MFGξ,Hη,t−1≤β11MFℓξ,ℓη,t−1+β3max1MFℓξ,Gξ,t−1,1MFℓη,Hη,t−1,1MFℓξ,Hη,t−1,1MFℓη,Gξ,t−1,,for all ξ,η∈Ω,t>0, and β1,β3∈0,1 with β1+2β3<1. If GΩ∪HΩ⊂ℓΩ and ℓΩ is a complete subspace of Ω. Then, G,H, and ℓ have a unique point of coincidence. In addition, if G,ℓ and H,ℓ are weakly-compatible, then G,H, and ℓ have a unique CFP.If we use identity map instead of the mappingℓ, i.e., ℓ=I, in Theorem 11, we get the following corollary.Corollary 14.
LetΩ,MF,∗ be a complete FM space in which a FM MF is triangular. Let G,H:Ω⟶Ω be a pair of self-mappings that satisfy
(37)1MFGξ,Hη,t−1≤β11MFξ,η,t−1+β21MFξ,Gξ,t−1+1MFη,Hη,t−1+1MFη,Gξ,t−1+1MFξ,Hη,t−1+β3max1MFξ,Gξ,t−1,1MFη,Hη,t−1,1MFξ,Hη,t−1,1MFη,Gξ,t−1,,for all ξ,η∈Ω,t>0, and β1,β2,β3∈0,1 with β1+4β2+2β3<1. Then, the mappings G and H have a unique CFP.Example 15.
LetΩ=0,1 and ∗ is a continuous t-norm. Let a FM MF:Ω×Ω×0,∞⟶0,1 be defined by
(38)MFξ,η,t=t/t+mξ,η,wheremξ,η=ξ−η,∀ξ,η∈Ω,andt>0.
Then easily one can verify thatMF is triangular FM space Ω,MF,∗. Now we define the mappings G,H,ℓ:Ω⟶Ω by Gξ=Hξ=2ξ/27 and ℓξ=ξ/3 for all ξ∈Ω and satisfying the hypothesis GΩ∪HΩ⊂ℓΩ. By using the constants β1=2/9,β2=1/6, and β3=1/20 in (5), then the three self-mappings satisfying the inequality (5) of Theorem 11. For verification, we present the following calculation for the defined mappings with constant values used in the inequality (5), for t>0, we have
(39)1MFGξ,Hη,t−1=Gξ−Hηt=2ξ−η27t≤13ξ+η81t≤2ξ−η27t+7x+y81t+1540tmax7ξ,7η,9η−2ξ,9ξ−2η≤2ξ−η27t+16t14x+y27+120tmax7ξ27,7η27,9η−2ξ27,9ξ−2η27≤29tξ3−η3+16tξ3−2ξ27+η3−2η27+η3−2ξ27+ξ3−2η27+120tmaxξ3−2ξ27,η3−2η27,η3−2ξ27,ξ3−2η27=β11MFℓξ,ℓη,t−1+β21MFℓξ,Gξ,t−1+1MFℓη,Hη,t−1+1MFℓη,Gξ,t−1+1MFℓξ,Hη,t−1+β3max1MFℓξ,Gξ,t−1,1MFℓη,Hη,t−1,1MFℓξ,Hη,t−1,1MFℓη,Gξ,t−1.
Hence, proved that all the conditions of Theorem11 are satisfied with β1=2/9,β2=1/6, and β3=1/20, where β1+4β2+2β3=2/9+41/6+21/20=89/90<1 and the three-self-mappings G, H, and ℓ have a unique coincidence point and CFP, that is, 0∈Ω.Theorem 16.
LetΩ,MF,∗ be a FM space in which a FM MF is triangular. Let G,H,ℓ:Ω⟶Ω be three self-mappings that satisfy
(40)1MFGξ,Hη,t−1≤β11MFℓξ,ℓη,t−1+β21MFℓξ,Gξ,t−1+β31MFℓη,Hη,t−1+β41MFℓη,Gξ,t−1+β51MFℓξ,Hη,t−1+β61ℵG,H,ℓ,ξ,η,t−1,where
(41)1ℵG,H,ℓ,ξ,η,t−1=min1MFℓξ,Gξ,t−1,1MFℓη,Hη,t−1,1MFℓξ,Hη,t−1,1MFℓη,Gξ,t−1,,for all ξ,η∈Ω,t>0, and β1,β2,β3,β4,β5,β6∈0,1 with β1+β2+β3+β4+β5+β6<1 and β2=β3 or β4=β5. If GΩ∪HΩ⊂ℓΩ and ℓΩ is a complete subspace of Ω. Then G,H, and ℓ have a unique point of coincidence. In addition, if G,ℓ and H,ℓ are weakly-compatible, then G,H, and ℓ have a unique CFP.Proof.
Letξ0∈Ω be arbitrary and by using the hypothesis GΩ∪HΩ⊂ℓΩ. Now, we choose a sequence ξj such that
(42)ℓξ2j+1=Gξ2jandℓξ2j+2=Hξ2j+1forj≥0.
Now, from (40), for t>0, we have
(43)1MFℓξ2j+1,ℓξ2j+2,t−1=1MFGξ2j,Hξ2j+1,t−1≤β11MFℓξ2j,ℓξ2j+1,t−1+β21MFℓξ2j,Gξ2j,t−1+β31MFℓξ2j+1,Hξ2j+1,t−1+β41MFℓξ2j+1,Gξ2j,t−1+β51MFℓξ2j,Hξ2j+1,t−1+β61ℵG,H,ℓ,ξ2j,ξ2j+1,t−1=β11MFℓξ2j,ℓξ2j+1,t−1+β21MFℓξ2j,ℓξ2j+1,t−1+β31MFℓξ2j+1,ℓξ2j+2,t−1+β51MFℓξ2j,ℓξ2j+2,t−1+β61ℵG,H,ℓ,ξ2j,ξ2j+1,t−1≤β1+β2+β51MFℓξ2j,ℓξ2j+1,t−1+β3+β51Mℓξ2j+1,ℓξ2j+2,t−1+β61ℵG,H,ℓ,ξ2j,ξ2j+1,t−1,where
(44)1ℵG,H,ℓ,ξ2j,ξ2j+1,t−1=min1MFℓξ2j,Gξ2j,t−1,1MFℓξ2j+1,Hξ2j+1,t−1,1MFℓξ2j+1,Gξ2j,t−1,1MFℓξ2j,Hξ2j+1,t−1=min1MFℓξ2j,ℓξ2j+1,t−1,1MFℓξ2j+1,ℓξ2j+2,t−1,0,1MFℓξ2j,ℓξ2j+2,t−1=0,
Hence, we get that(45)1MFℓξ2j+1,ℓξ2j+2,t−1≤A1MFℓξ2j,ℓξ2j+1,t−1fort>0,where A=β1+β2+β5/1−β3−β5. Similarly, again by the view of (40), for t>0,
(46)1MFℓξ2j+2,ℓξ2j+3,t−1=1MFGξ2j+2,Hξ2j+1,t−1≤β11MFℓξ2j+2,ℓξ2j+1,t−1+β21MFℓξ2j+2,Gξ2j+2,t−1+β31MFℓξ2j+1,Hξ2j+1,t−1+β41MFℓξ2j+1,Gξ2j+2,t−1+β51MFℓξ2j+2,Hξ2j+1,t−1+β61ℵG,H,ℓ,ξ2j+2,ξ2j+1,t−1=β11MFℓξ2j+2,ℓξ2j+1,t−1+β21MFℓξ2j+2,ℓξ2j+3,t−1+β31MFℓξ2j+1,ℓξ2j+2,t−1+β41MFℓξ2j+1,ℓξ2j+3,t−1+β61ℵG,H,ℓ,ξ2j+2,ξ2j+1,t−1≤β1+β3+β41MFℓξ2j,ℓξ2j+1,t−1+β2+β41Mℓξ2j+1,ℓξ2j+2,t−1+β61ℵG,H,ℓ,ξ2j+2,ξ2j+1,t−1,where
(47)1ℵG,H,ℓ,ξ2j+2,ξ2j+1,t−1=min1MFℓξ2j+2,Gξ2j+2,t−1,1MFℓξ2j+1,Hξ2j+1,t−1,1MFℓξ2j+1,Gξ2j+2,t−1,1MFℓξ2j+2,Hξ2j+1,t−1=min1MFℓξ2j+2,ℓξ2j+3,t−1,1MFℓξ2j+1,ℓξ2j+2,t−1,1MFℓξ2j+1,ℓξ2j+3,t−1,0=0.
Hence, we get that(48)1MFℓξ2j+2,ℓξ2j+3,t−1≤B1MFℓξ2j+1,ℓξ2j+2,t−1fort>0,where B=β1+β3+β4/1−β2−β4. Now from (45) and (48),
(49)1MFℓξ2j+2,ℓξ2j+3,t−1≤B·A1MFℓξ2j,ℓξ2j+1,t−1fort>0.
Now by the view of (45), (48), (49), and by induction, for t>0, we obtain that
(50)1MFℓξ2j+1,ℓξ2j+2,t−1≤A1MFℓξ2j,ℓξ2j+1,t−1≤A·B1MFℓξ2j−1,ℓξ2j,t−1≤A·B·A1MFℓξ2j−2,ℓξ2j−1,t−1≤⋯≤AB·Aj1MFℓξ0,ℓξ1,t−1,(51)1Mℓξ2j+2,ℓξ2j+3,t−1≤B1Mℓξ2j+1,ℓξ2j+2,t−1≤⋯≤B·Aj+11Mℓξ0,ℓξ1,t−1.
Since,A=β1+β2+β5/1−β3−β5 and B=β1+β3+β4/1−β2−β4. Now, if β2=β3,
(52)A·B=β1+β2+β51−β3−β5·β1+β3+β41−β2−β4=β1+β2+β51−β2−β4·β1+β3+β41−β3−β5<1·1=1,
In second case, ifβ4=β5,
(53)A·B=β1+β2+β51−β3−β5·β1+β3+β41−β2−β4<1·1=1.
Hence, from (50), (52), and (53), we get that ℓξjj≥0 is a fuzzy-contractive sequence, therefore,
(54)limj⟶∞MFℓξj,ℓξj+1,t=1fort>0.
Next, we have to prove that a sequenceℓξjj≥0 is a Cauchy sequence in Ω,MF,∗. Then, from Definition 2 and by using (54), for m>j where m=j+n for n=1,2,⋯ and t>0, we have that
(55)MFℓξj,ℓξm,t=MFℓξj,ℓξj+n,t≥MFℓξj,ℓξj+1,tn∗MFℓξj+1,ℓξj+2,tn∗⋯∗MFℓξj+n−1,ℓξj+n,tn⟶1∗1∗⋯∗1=1,asj⟶∞.
This implies that(56)limj,m⟶∞MFℓξj,ℓξm,t=1fort>0.
Hence, proved thatℓξj is a Cauchy sequence. Since, ℓΩ is a complete subspace of Ω,MF,∗, then there exits ω,κ∈Ω such that ℓξj⟶κ=ℓω as j⟶∞, therefore,
(57)limj⟶∞MFℓξj,κ,t=limj⟶∞MFℓξj,ℓω,t=1fort>0.
Now we have to show thatℓω=Gω=Hω. First, we estimate that MFℓω,Gω,t=MFκ,Gω,t for t>0. Since, MF is triangular, therefore,
(58)1MFℓω,Gω,t−1≤1MFℓω,ℓξ2j+2,t−1+1MFℓξ2j+2,Gω,t−1fort>0.
Now by using (40), (54), and (57), for t>0, we have
(59)1MFℓξ2j+2,Gω,t−1=1MFGω,Hξ2j+1,t−1≤β11MFℓω,ℓξ2j+1,t−1+β21MFℓω,Gω,t−1+β31MFℓξ2j+1,Hξ2j+1,t−1+β41MFℓξ2j+1,Gω,t−1+β51MFℓω,Hξ2j+1,t−1+β61ℵG,H,ℓ,ω,ξ2j+1,t−1=β11MFℓω,ℓξ2j+1,t−1+β21MFℓω,Gω,t−1+β31MFℓξ2j+1,ℓξ2j+2,t−1+β41MFℓξ2j+1,Gω,t−1+β51MFℓω,ℓξ2j+2,t−1+β61ℵG,H,ℓ,ω,ξ2j+1,t−1⟶β2+β41MFℓω,Gω,t−1+β61ℵG,H,ℓ,ω,ξ2j+1,t−1,asj⟶∞,where
(60)1ℵG,H,ℓ,ω,ξ2j+1,t−1=min1MFℓω,Gω,t−1,1MFℓξ2j+1,Hξ2j+1,t−1,1MFℓω,Hξ2j+1,t−1,1MFℓξ2j+1,Gω,t−1=min1MFℓω,Gω,t−1,1MFℓξ2j+1,ℓξ2j+2,t−1,1MFℓω,ℓξ2j+2,t−1,1MFℓξ2j+1,Gω,t−1⟶min0,1MFℓω,Gω,t−1=0,asj⟶∞.
Thus,(61)limj⟶∞sup1MFℓξ2j+2,Gω,t−1≤β2+β41MFℓω,Gω,t−1fort>0.
The above (61) is together with (57) and (58), we get that
(62)1MFℓω,Gω,t−1≤β2+β41MFℓω,Gω,t−1⇒1−β2−β41MFℓω,Gω,t−1≤0fort>0,
which is a contradiction, as1−β2−β4≠0, therefore, MFℓω,Gω,t=1⇒ℓω=Gω, for t>0. Hence, ℓω=Gω=κ.
Similarly, again by using the triangular property ofMF,
(63)1MFℓω,Hω,t−1≤1MFℓω,ℓξ2j+1,t−1+1MFℓξ2j+1,Hω,t−1fort>0.
Again by using (40), (54), and (57), similar as above simplification, we get
(64)limj⟶∞sup1MFℓξ2j+1,Hω,t−1≤β3+β51MFℓω,Hω,t−1fort>0.
The above (64) is together with (57) and (63), we get that
(65)1MFℓω,Hω,t−1≤β3+β51MFℓω,Hω,t−1⇒1−β3−β51MFℓω,Hω,t−1≤0fort>0,which is a contradiction, as 1−β3−β5≠0, therefore, MFℓω,Hω,t=1⇒ℓω=Hω, for t>0. Hence, ℓω=Hω=κ. It follows that ω is a coincidence point and κ is a common point of coincidence point for the self-mappings ℓ,G, and H, that is,
(66)ℓω=Gω=Hω=κ.
Next, we prove that the point of coincidence point of the three self-mappings is unique. Let∃ another point κ1∈Ω such that
(67)κ1=ℓω1=Gω1=Hω1forsomeω1∈Ω.
Now by using (40), for t>0,
(68)1MFκ,κ1,t−1=1MFGω,Hω1,t−1≤β11MFℓω,ℓω1,t−1+β21MFℓω,Gω,t−1+β31MFℓω1,Hω1,t−1+β41MFℓω1,Gω,t−1+β51MFℓω,Hω1,t−1+β61ℵG,H,ℓ,ω,ω1,t−1=β1+β4+β51MFκ,κ1,t−1+β61ℵG,H,ℓ,ω,ω1,t−1,where
(69)1ℵG,H,ℓ,ω,ω1,t−1=min1MFℓω,Gω,t−1,1MFℓω1,Hω1,t−1,1MFℓω,Hω1,t−1,1MFℓω1,Gω,t−1=min1MFκ,κ1,t−1,0=0.
Hence, from the above, we get that(70)1MFκ,κ1,t−1≤β1+β4+β51MFκ,κ1,t−1,⟹1−β1−β4−β51MFκ,κ1,t−1≤0fort>0,which is a contradiction. Hence MFκ,κ1,t=1⇒κ=κ1 for t>0. Further, by using the weak compatibility of the pairs G,ℓ and H,ℓ and by Proposition 10, it follows that the three self-mappings G,H, and ℓ have a unique CFP, that is, ℓκ=Gκ=Hκ=κ.Corollary 17.
LetΩ,MF,∗ be a FM space in which a FM MF is triangular. Let G,H,ℓ:Ω⟶Ω be three self-mappings that satisfy
(71)1MFGξ,Hη,t−1≤β11MFℓξ,ℓη,t−1+β21MFℓξ,Gξ,t−1+β31MFℓη,Hη,t−1+β61ℵG,H,ℓ,ξ,η,t−1,where
(72)1ℵG,H,ℓ,ξ,η,t−1=min1MFℓξ,Gξ,t−1,1MFℓη,Hη,t−1,1MFℓξ,Hη,t−1,1MFℓη,Gξ,t−1,,
for allξ,η∈Ω,t>0, and β1,β2,β3,β6∈0,1 with β1+β2+β3<1. If GΩ∪HΩ⊂ℓΩ and ℓΩ is a complete subspace of Ω. Then G,H, and ℓ have a unique point of coincidence. In addition, if G,ℓ and H,ℓ are weakly-compatible, then G,H, and ℓ have a unique CFP.Corollary 18.
LetΩ,MF,∗ be a FM space in which a FM MF is triangular. Let G,H,ℓ:Ω⟶Ω be three self-mappings that satisfy
(73)1MFGξ,Hη,t−1≤β11MFℓξ,ℓη,t−1+β41MFℓη,Gξ,t−1+β51MFℓξ,Hη,t−1+β61ℵG,H,ℓ,ξ,η,t−1,where
(74)1ℵG,H,ℓ,ξ,η,t−1=min1MFℓξ,Gξ,t−1,1MFℓη,Hη,t−1,1MFℓξ,Hη,t−1,1MFℓη,Gξ,t−1,for all ξ,η∈Ω,t>0, and β1,β4,β5,β6∈0,1 with β1+β4+β5<1. If GΩ∪HΩ⊂ℓΩ and ℓΩ is a complete subspace of Ω. Then G,H, and ℓ have a unique point of coincidence. In addition, if G,ℓ and H,ℓ are weakly-compatible, then G,H, and ℓ have a unique CFP.If we use identity map instead of the mappingℓ, i.e., ℓ=I, in Theorem 16, we get the following corollary.Corollary 19.
LetΩ,MF,∗ be a complete FM space in which a FM MF is triangular. Let G,H:Ω⟶Ω be a pair of self-mappings and satisfies
(75)1MFGξ,Hη,t−1≤β11MFξ,η,t−1+β21MFξ,Gξ,t−1+β31MFη,Hη,t−1+β41MFη,Gξ,t−1+β51MFξ,Hη,t−1+β61ℵG,H,I,ξ,η,t−1,where
(76)1ℵG,H,I,ξ,η,t−1=min1MFξ,Gξ,t−1,1MFη,Hη,t−1,1MFξ,Hη,t−1,1MFη,Gξ,t−1,for all ξ,η∈Ω,t>0, and β1,β2,β3,β4,β5,β6∈0,1 with β1+β2+β3+2β4+2β5<1. Then G and H have a unique CFP.Example 20.
From Example15, let we define the three-self-mappings G,H,ℓ:Ω⟶Ω by: as for each ξ∈Ω,
(77)Gξ=Hξ=3102ξ7+35,ifξ≠0,0,ifξ=0.
And(78)ℓξ=2ξ7+35,ifξ≠0,0,ifξ=0.
Since,GΩ∪HΩ⊂ℓΩ, then, we have
(79)1MFGξ,Hη,t−1=Gξ−Hηt=3102ξ7−2η7=3101MFℓξ,ℓη,t−1forallξ,η∈Ωandt>0,which shows that the three self-mappings are weakly-compatible fuzzy-contractive in a complete FM space Ω,MF,∗. Thus, all the conditions of Theorem 16 are satisfied with β1=3/10,β2=β3=1/5,β4=β5=2/15, and β6=0, and the three self-mappings have a unique coincidence point and CFP in Ω, that is, 0.


## 4. Application

This section deals with the application of the fuzzy differential equations (FDEs) to support our work. From the book of Lakshmikantham and Mohapatra [38], we have the following FDEs.LetF be the space of all fuzzy subsets ξ of R where ξ:R⟶0,1. Let the boundary value problem (BVP) is
(80)ξ′′τ=Pτ,ξτ,ξ′τ,τ∈I=a,b,ξτ1=ξ1,ξτ2=ξ2,τ1,τ2∈I=a,b,where P:I×F×F⟶F is a continuous function. This problem is equivalent to the integral equation
(81)ξτ=∫τ1τ2Qτ,vPv,ξv,ξ′vdv+ℏτ,where Green’s function Q is given by
(82)Qτ,v=τ2−τv−τ1τ2−τ1,τ1≤v≤τ≤τ2,τ2−vτ−τ1τ2−τ1,τ1≤τ≤v≤τ2.Andℏτ satisfies ℏ′′=0,ℏτ1=ξ1,ℏτ2=ξ2. Here, we recall some properties of Qτ,v, that are
(83)∫τ1τ2Qτ,vdv≤τ2−τ128,and
(84)∫τ1τ2Qττ,vdv≤τ2−τ12.LetΩ=Ω1I,F, ∗ is a continuous t-norm, and a fuzzy metric MF:Ω×Ω×0,∞⟶0,1 be defined as
(85)MFξ,η,t=tt+mξ,ηwheremξ,η=ξ−η,∀ξ,η∈Ω, and t>0. Then easily one can prove that MF is triangular and Ω,MF,∗ is a complete FM space.Now, we prove the existing result for the above BVP by using Theorem11.Theorem 21.
Suppose thatP1,P2:I×F×F⟶F and let there exist α,β∈0,1 with α≤β such that for all ξ,η∈Ω1I,F, satisfies
(86)P1τ,ξτ,ξ′τ−P2τ,ητ,η′τ≤α∣ξτ−ητ∣+βξ′τ−η′τ.
Let there existsk∈0,1 such that
(87)mξτ,ητ=mξ,η≤kDG,H,ℓ,ξ,η,∀ξ,η∈Ω,
where(88)DG,H,ℓ,ξ,η=ξ−η,ξ−Aξ−ℏ+η−Bη−ℏ+η−Aξ−ℏ+ξ−Bη−ℏmaxξ−Aξ−ℏ,η−Bη−ℏ,η−Aξ−ℏ,ξ−Bη−ℏ,
Then the integral equations(89)ξτ=∫τ1τ2Qτ,vP1v,ξv,ξ′vdv+ℏτ,τ∈I,and
(90)ητ=∫τ1τ2Qτ,vP2v,ηv,η′vdv+ℏv,τ∈I,have a unique common solution in Ω1τ1,τ2,F.Proof.
Suppose thatΩ=τ1,τ2,F with metric
(91)mξ,η=maxτ1≤τ≤τ2αξτ−ητ+βξ′τ−η′τ.
The spaceΩ,m is a complete metric space. Now, we define the operators G,H,ℓ:Ω⟶Ω as
(92)Gξ=Aξ+ℏ,Hη=Bη+ℏ,ℓξ=ξ,andℓη=η,where
(93)Aξτ=∫τ1τ2Qτ,vP1v,ξv,ξ′vdv,τ∈I,and
(94)Bηv=∫τ1τ2Qτ,vP2v,ηv,η′vdv,τ∈I,where P1,P2∈ΩI×F×F,F,ξ,η∈Ω1I,F, and ℏ∈ΩI,F. Now by the properties of Qτ,v, and from (91), (92) and by using the hypothesis, we have
(95)Gξτ−Hητ≤∫τ1τ2Qτ,vP1v,ξv,ξ′v−P2v,ηv,η′vdv≤mξ,η∫τ1τ2Qτ,vdv≤τ2−τ128mξ,η≤mξ,η8.
And(96)Gξ′τ−Hη′τ≤∫τ1τ2Qττ,vP1v,ξv,ξ′v−P2v,ηv,η′vdv≤mξ,η∫τ1τ2Qττ,vdv≤τ2−τ12mξ,η≤mξ,η2.
Now, from the above and by view of (86), and (91), we have that
(97)mGξ,Hη=maxτ1≤τ≤τ2αGξτ−Hητ+βGξ′τ−Hη′τ≤αmξ,η8+βmξ,η2≤58βmξ,η.
Now, from (87), we have that
(98)mGξ,Hη≤58βmξ,η≤Y∗DG,H,ℓ,ξ,η,∀ξ,η∈Ω,where Y∗=5/8βk<1. Now we are in the position to apply Theorem 11 to get that G,H, and ℓ have a unique CFP u∗∈Ω, i.e., u∗ is a solution of the BVP. We have the following two main cases:
(a)
IfDG,H,ℓ,ξ,η=ξ−η in (88), then from (85) and (98), for t>0, we have(99)1MrGξ,Hη,t−1=mGξ,Hηt≤Y∗DG,H,ℓ,ξ,ηt=Y∗ξ−ηt=Y∗1Mrξ,η,t−1,for all ξ,η∈Ω. Hence, the operators G,H, and ℓ satisfy all the conditions of Theorem 11 with Y∗=β1 and β2=β3=0 in (5). Thus, the operators G,H, and ℓ have a unique CFP u∗∈Ω, i.e., u∗ is a solution of the BVP (80).
(b)
IfDG,H,ℓ,ξ,η=ξ−Aξ−ℏ+η−Bη−ℏ+η−Aξ−ℏ+ξ−Bη−ℏ in (88), then from (85) and (98), for t>0, we have(100)1MrGξ,Hη,t−1=mGξ,Hηt≤Y∗DG,H,ℓ,ξ,ηt=Y∗ξ−Aξ−ℏ+η−Bη−ℏ+η−Aξ−ℏ+ξ−Bη−ℏt=Y∗ξ−Gξt+η−Hηt+η−Gξt+ξ−Hηt=Y∗1MFℓξ,Gξ,t−1+1MFℓη,Hη,t−1+1MFℓη,Gξ,t−1+1MFℓξ,Hη,t−1,for all ξ,η∈Ω. Hence, the operators G,H, and ℓ satisfy all the conditions of Theorem 11 with Y∗=β2 and β1=β3=0 in (5). Thus, the operators G,H, and ℓ have a unique CFP u∗∈Ω, i.e., u∗ is a solution of the BVP (80).
(c)
IfDG,H,ℓ,ξ,η=maxξ−Aξ−ℏ,η−Bη−ℏ,η−Aξ−ℏ,ξ−Bη−ℏ in (88), then, we may have the following four subcases.
(i)
Ifξ−Aξ−ℏ is the maximum term in ξ−Aξ−ℏ,η−Bη−ℏ,η−Aξ−ℏ,ξ−Bη−ℏ. Then, DG,H,ℓ,ξ,η=ξ−Aξ−ℏ in (88). Now from (85) and (98), for t>0, we have(101)1MrGξ,Hη,t−1=mGξ,Hηt≤Y∗DG,H,ℓ,ξ,ηt=Y∗ξ−Aξ−ℏt=Y∗ξ−Gξt=Y∗1Mrξ,Gξ,t−1,forallξ,η∈Ω.(ii)
Ifη−Bη−ℏ is the maximum term in ξ−Aξ−ℏ,η−Bη−ℏ,η−Aξ−ℏ,ξ−Bη−ℏ. Then, DG,H,ℓ,ξ,η=η−Bη−ℏ in (88). Now from (85) and (98), for t>0, we have(102)1MrGξ,Hη,t−1=mGξ,Hηt≤Y∗DG,H,ℓ,ξ,ηt=Y∗η−Bη−ℏt=Y∗η−Hηt=Y∗1Mrη,Hη,t−1,forallξ,η∈Ω.(iii)
Ifη−Aξ−ℏ is the maximum term in ξ−Aξ−ℏ,η−Bη−ℏ,η−Aξ−ℏ,ξ−Bη−ℏ. Then, DG,H,ℓ,ξ,η=η−Aξ−ℏ in (88). Now from (85) and (98), for t>0, we have(103)1MrGξ,Hη,t−1=mGξ,Hηt≤Y∗DG,H,ℓ,ξ,ηt=Y∗η−Aξ−ℏt=Y∗η−Gξt=Y∗1Mrη,Gξ,t−1,forallξ,η∈Ω.(iv)
Ifξ−Bη−ℏ is the maximum term in ξ−Aξ−ℏ,η−Bη−ℏ,η−Aξ−ℏ,ξ−Bη−ℏ. Then, DG,H,ℓ,ξ,η=ξ−Bη−ℏ in (88). Now from (98) and (98), for t>0, we have(104)1MrGξ,Hη,t−1=mGξ,Hηt≤Y∗DG,H,ℓ,ξ,ηt=Y∗ξ−Bη−ℏt=Y∗ξ−Bη−ℏt=Y∗1Mrξ,Hη,t−1,forallξ,η∈Ω.for all ξ,η∈Ω. Hence, from (i)-(vi), the operators G,H, and ℓ satisfy all the conditions of Theorem 11 with Y∗=β3 and β1=β2=0 in (5). Thus, the operators G,H, and ℓ have a unique CFP u∗∈Ω, i.e., u∗ is a solution of the BVP (80)


## 5. Conclusion

In this paper, we proved some generalized unique coincidence points and CFP theorems for weakly-compatible three self-mappings in FM spaces without the assumption that the “fuzzy contractive sequences are Cauchy.” The “triangular property of FM” is used as a basic tool throughout the complete paper to get the existence of unique coincidence points and CFP results in FM spaces. In support of our main work, we presented two illustrative examples, that is, Examples15 and 20. In addition, we established an application of fuzzy differential equations to ensure the existence of a unique common solution to support our work. By using this concept, one can contribute different contractive types of FP, CFP, and coincidence points result in FM spaces with different types of applications.



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*Source: 9411993-2021-10-29.xml*