# On Multivalued Fuzzy Contractions in Extendedb-Metric Spaces

Authors: Nayab Alamgir; Quanita Kiran; Hassen Aydi; Yaé Ulrich Gaba
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we establish a Hausdorff metric over the family of nonempty closed subsets of an extendedb-metric space. Thereafter, we introduce the concept of multivalued fuzzy contraction mappings and prove related α-fuzzy fixed point theorems in the context of extended b-metric spaces that generalize Nadler’s fixed point theorem as well as many preexisting results in the literature. Further, we establish α-fuzzy fixed point theorems for Ćirić type fuzzy contraction mappings as a generalization of previous results. Moreover, we give some examples to support the obtained results.


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## Body

## 1. Introduction

In 1928, Von Neumann [1] introduced the concept of fixed points for multivalued mappings because of its applications in several branches of mathematics. The development of the geometric fixed point theory for multivalued mappings was initiated by the work of Nadler [2]. He used the Pompeiu–Hausdorff metric to prove the multivalued contraction principle over the collection of nonempty closed and bounded subsets of a metric space. After that, several researchers have studied and generalized Nadler’s contraction principle in many directions.In 1965, Zadeh [3] initiated the concept of fuzzy set theory. After that, several authors extended the Banach contraction principle for single and multivalued mappings in the context of fuzzy sets. In 1981, Heilpern [4] proved a fixed point theorem for fuzzy contraction mappings as a generalization of Nadler’s contraction principle. Consequently, several authors studied and generalized fuzzy fixed point theorems in many directions (see [5–11]). In 2015, Phiangsungnoen and Kumam [12] established the concept of multivalued fuzzy contraction mappings in b-metric spaces and proved a related α-fuzzy fixed point theorem. In [13], Anita extended α-fuzzy fixed point theorems involving Ćirić type fuzzy contraction mappings.Meanwhile, Kamran in [14] introduced the concept of an extended b-metric, as a generalization of a b-metric, and proved fixed point results on such space. Thereafter, many researchers have studied and generalized fixed point results for single and multivalued mappings (see [15–24]).In this paper, we establish a Hausdorff metric over the family of nonempty closed subsets of an extendedb-metric space. After that, we introduce the concept of multivalued fuzzy contraction mappings and prove α-fuzzy fixed point theorems for such mappings in the context of extended b-metric spaces that generalize many preexisting results in the literature. To justify our results, we give some examples. In the last section, we further establish the fact that, by utilizing these concepts, we can also derive results for multivalued mappings. Throughout this paper, we will denote by CLBU the collection of nonempty closed and bounded subsets of U and by CLDU the collection of nonempty closed subsets of U.Definition 1.
(see [14]). Let U be a nonempty set with θ:U×U⟶1,∞. Then, a mapping dθ:U×U⟶0,∞ is called an extended b-metric, if for all μ,ν,ω∈U, it satisfies the following:(1)
dθμ,ν=0iffμ=ν(2)
dθμ,ν=dθν,μ(3)
dθμ,ω≤θμ,ωdθμ,ν+dθν,ωClearly, everyb-metric space is an extended b-metric space with θμ,ν=s≥1.Example 1.
(see [14]). Let U=0,∞. Define dθ:U×U⟶0,∞ by(1)dθμ,ν=μ−ν2,for all μ,ν∈U. Thus, dθ is an extended b-metric on U, where θ:U×U⟶1,∞ is defined as θμ,ν=μ+ν+1.Samreen et al. in [25] established the concept of an extended b-comparison function as an extension of a b-comparison function and generalized the concept of α-ψ-contraction mappings in the framework of extended b-metric spaces.Definition 2.
(see [25]). Let U,dθ be an extended b-metric space. Then, a function φ:0,∞⟶0,∞ is called an extended b-comparison function if it is increasing, and there exists a mapping f:D⊂U⟶U such that, for some μ0∈D, Oμ0⊂D, ∑n=0∞φnt∏i=1nθμi,μm converges for all t∈0,∞ and for every m∈ℕ. Here, μn=fnμ0 for n=1,2,… and Oμ0 is an orbit at a point μ0∈U. We say that φ is an extended b-comparison function for f at μ0. It is known that, for each extended b-comparison function φ, we have φt<t for all t>0 and φ0=0 for t=0.We denote byΨθ the collection of all extended b-comparison functions. If we put θμi,μm=s in Definition 2, we get ∑n=0∞φnt∏i=1nsn<∞, which is a b-comparison function. We denote by Ψb the collection of all b-comparison functions.In [17], Subashi and Gjini introduced the concept of a Pompeiu–Hausdorff metric on the collection of all compact subsets of extended b-metric spaces. On the other hand, Subashi in [18] initiated Pompeiu–Hausdorff metric on the collection of all nonempty closed and bounded subsets of extended b-metric spaces.Next, recall definitions of fuzzy sets, fuzzy mappings,α-fuzzy fixed point, Ćirić type contraction, and related fixed point theorem in b-metric spaces from [12, 13, 26].LetU,dθ be a b-metric space. Then a fuzzy set A in U is characterized by a membership function(2)fA:U⟶0,1,which assigns every member of U a membership grade in A. Denote by FU the collection of all fuzzy sets in U. Let us take A∈FU and α∈0,1. The α-level set of A is denoted by Aα and is defined as follows:(3)Aα=μ∈U:Aμ≥α,α∈0,1,A0=μ∈U:Aμ>0¯,where A¯ denotes the closure of A. Clearly, Aα and A0 are subsets of U.ForA,B∈FU, a fuzzy set A is said to be more accurate than a fuzzy set B and is denoted by A⊂B, if and only if fAμ≤fBμ for each μ∈U. Now, for μ∈U, A,B∈FU, α∈0,1, and Aα,Bα∈CLDU, define(4)ραμ,Aα=infdμ,a:a∈Aα,ραAα,Bα=infda,b:a∈Aα,b∈Bα,ρAα,Bα=supαραAα,Bα.Then Hausdorff fuzzyb-metric is denoted by HAα,Bα and is defined as follows:(5)HAα,Bα=maxsupa∈Aαda,Bα,supb∈Bαdb,Aα.Remark 1.
(see [12]). The function H:CLDU×CLDU⟶FU is a generalized Hausdorff fuzzy b-metric induced by d and is defined as follows:(6)HAα,Bα=maxsupa∈Aαda,Bα,supb∈Bαdb,Aα,if the maximum exists;∞,otherwise,where Aα,Bα∈CLDU.Definition 3.
(see [12]). Let U be a nonempty set and V be a b-metric space. Then(1)
A mappingℜ:U⟶FV is called a fuzzy mapping(2)
For a fuzzy mappingℜ:U⟶FU, an element ξ∈U is called a α fuzzy fixed point of ℜ, if ξ∈ℜξαTheorem 1 (see [12]).
LetU,d be a complete b-metric space with s≥1. Let ℜ:U⟶FU and α∈0,1 such that, for each μ∈U, ℜμα is a nonempty closed subset of U and φ∈Ψb such that(7)Hℜμα,ℜνα≤φdμ,ν,for all μ,ν∈U. Then, ℜ has an α-fuzzy fixed point.Definition 4.
(see [26]). A self-mapping ℜ:U⟶U on a metric space U,d is called a Ćirić type contraction if and only if for all μ,ν∈U, there exists h<1 such that(8)dℜμ,ℜν≤hmaxdμ,ν,dμ,ℜμ,dν,ℜν,dμ,ℜν+dν,ℜμ2.Theorem 2 (see [13]).
LetU,d be a complete b-metric space with s≥1. Let ℜ:U⟶FU and α∈0,1 such that, for each μ∈U, ℜμα is a nonempty closed subset of U and φ∈Ψb such that(9)Hℜμα,ℜνα≤φMμ,ν,where(10)Mμ,ν=maxdμ,ν,dμ,ℜμα,dν,ℜνα,dμ,ℜνα+dν,ℜμα2s,for all μ,ν∈U. Then, ℜ has an α-fuzzy fixed point.Remark 2.
IfMμ,ν=dμ,ν in Theorem 2, we get Theorem 1. Hence, Theorem 2 is an extension of Theorem 1.


## 2. Main Results

ForA⊂U, θμ,A=infa∈Aθμ,a and dθμ,A=infa∈Adθμ,a. The following lemma is essential in the sequel.Lemma 1.
LetU,dθ be an extended b-metric space. Then(11)dθμ,A≤θμ,Adθμ,ν+θν,Adθν,A,for all μ,ν∈U and a∈A⊂U, where θμ,A=infa∈Aθμ,a.Proof.
SinceU,dθ is an extended b-metric space, therefore by using the triangle inequality one writes(12)dθμ,a≤θμ,adθμ,ν+dθν,a,for allμ,ν,a∈U.
This implies that(13)dθμ,a≤θμ,adθμ,ν+θμ,adθν,a,for all μ,ν,a∈U. By taking infimum over A in equation (13), we obtain(14)infa∈Adθμ,a≤infa∈Aθμ,ainfa∈Adθμ,ν+dθν,a=infa∈Aθμ,adθμ,ν+infa∈Aθμ,ainfa∈Adθν,a.
Sincedθμ,A=infa∈Adθμ,a and θμ,A=infa∈Aθμ,a, thus from equation (14), we have(15)dθμ,A≤θμ,Adθμ,ν+θν,Adθν,A.Lemma 2.
Letμkk=0n⊂U. Then(16)dθμ0,μn≤θμ0,μndθμ0,μ1+θμ0,μnθμ1,μndθμ1,μ2+⋯+θμ0,μnθμ1,μn+⋯+θμn−1,μndθμn−1,μn.Proof.
From triangle inequality, we have(17)dθμ0,μn≤θμ0,μndθμ0,μ1+dθμ1,μn.
This implies that(18)dθμ0,μn≤θμ0,μndθμ0,μ1+θμ0,μndθμ1,μn.
Again, by the triangle inequality,(19)dθμ0,μn≤θμ0,μndθμ0,μ1+θμ0,μnθμ1,μndθμ1,μ2+dθμ2,μn,dθμ0,μn≤θμ0,μndθμ0,μ1+θμ0,μnθμ1,μndθμ1,μ2+θμ0,μnθμ1,μndθμ2,μn.
By continuing in this fashion, we have(20)dθμ0,μn≤θμ0,μndθμ0,μ1+θμ0,μnθμ1,μndθμ1,μ2+⋯+θμ0,μnθμ1,μn+⋯+θμn−1,μndθμn−1,μn.Theorem 3.
Letμn be a sequence in an extended b-metric space with the property that, for all n, dθμn,μn+1<ε/kn and limn⟶∞θμn,μn+1k<1, where k≥1 is a real constant. Then, μn is a Cauchy sequence.Proof.
Letε>0 and choose a positive integer N. Then, from the triangle inequality for all m>n≥N, we have(21)dθμn,μm≤θμn,μmdθμn,μn+1+θμn,μmθμn+1,μmdθμn+1,μn+2+⋯+θμn,μmθμn+1,μm+⋯+θμm−1,μmdθμm−1,μm,dθμn,μm≤θμn,μmεkn+θμn,μmθμn+1,μmεkn+1+⋯+θμn,μmθμn+1,μm+⋯+θμm−1,μmεkm−1,dθμn,μm≤θμ1,μmθμ2,μm+⋯+θμn,μmεkn+θμ1,μmθμ2,μm+⋯+θμn+1,μmεkn+1+⋯+θμ1,μmθμ2,μm+⋯+θμn,μmθμn+1,μm+⋯+θμm−1,μmεkm−1.
Sincelimn,m⟶∞θμn+1,μmk<1, the series ∑j=1∞ε/kn∏i=1jθμi,μm converges by the ratio test for each m∈ℕ. Let(22)S=∑p=1∞εkp∏q=1pθμq,μm,Sn=∑p=1nεkp∏q=1pθμq,μm.
Thus, form>n, the above inequality implies(23)dθμn,μm≤Sm−1−Sn.
Let us taken⟶∞. Hence, we conclude that μn is a Cauchy sequence.Now we will introduce the Pompeiu–Hausdorff metric.Definition 5.
Consider nonempty subsetsA,B of extended b-metric space U,dϕ, and we define(24)HϕA,B=maxsupa∈Adθa,B,supb∈Bdθb,A,if the maximum exists;∞,otherwise.By following the same procedure as Lemma2 of [27], we state the following lemma.Lemma 3.
For allA,B,C⊂U, we have(25)HθA,C≤maxsupa∈Aθa,C,supc∈Cθc,AHθA,B+maxsupa∈Aθa,C,supc∈Cθc,AHθB,C.Same as Theorem 2.1 of [27], we have the following theorem.Theorem 4.
LetU,dθ is an extended b-metric space, then the function Hθ:CLDU×CLDU⟶0,∞ is a generalized extended b-metric space in CLDU.Definition 6.
a
∈
A
¯, where A¯ is the closure of a set A⊂U, if and only if there exists a sequence an in A such that a=limn⟶∞an.Denote forε>0 and A⊂U(26)Bε,A=μ∈U:dθμ,A≤ε.Lemma 4.
Ifμ∈Bε,A¯, then dθμ,A≤θμ,Aε, where(27)θμ,A=infa∈Aθμ,a.Proof.
Letμ∈Bε,A¯, then there exists a sequence μn in Bε,A, where n=1,2,3,… such that μ=limn⟶∞μn. Hence, by Lemma 1,(28)dθμ,A≤θμ,Adθμ,μn+θμ,Adθun,A≤θμ,Aε.
This implies that(29)dθμ,A≤θμ,Aε,which proves the lemma.Definition 7.
The upper topological limit of a sequenceAkk=1∞ in the extended b-metric space U is denoted by lt¯Ak which is determined by(30)a∈lt¯Ak,ifflimk⟶∞infdθa,Ak=0.FollowingTheorem 2.2 of [27], we have the following.Theorem 5.
A pointa∈lt¯Ak if and only if there exists a subsequence ank⊂A such that limk⟶∞ank=a and ank∈Ank, for k=1,2,3,…AsTheorem 2.3 of [27], we have the following theorem.Theorem 6.
L
=
lt
¯
A
k is closed.Similar toCorollary 2.1 of [27], we state the following.Corollary 1.
(31)
lt
¯
A
k
=
∩
k
=
1
∞
∪
n
=
0
∞
A
k
+
n
¯
.Again, as Corollary2.2 of [27], we have the following.Corollary 2.
(32)
lim
k
⟶
∞
A
k
=
lt
¯
A
k
¯
=
lt
¯
A
k
¯
.By applying the same procedure asTheorem 2.4 of [27], we state the following.Theorem 7.
Let us consider a complete extendedb-metric space U,dθ with limn,m⟶∞θμn,μmk<1 for all μm,μn∈U, where k≥1. Then, CLDU,Hθ is complete.Definition 8.
LetU,dθ be an extended b-metric space with θ:U×U⟶1,∞. Then, a fuzzy set Aθ in U is characterized by a membership function(33)fAθ:U⟶0,1,which assigns every member of U a membership grade in Aθ.We denote byFθU the collection of all fuzzy sets in U. Let us take Aθ∈FθU and α∈0,1. The α-level set of Aθ is denoted by Aθα and is defined as follows:(34)Aθα=μ∈U:Aθμ≥α,α∈0,1,Aθ0=μ∈U:Aθμ>0¯,where B¯ denotes the closure of B. Clearly, Aθα and Aθ0 are subsets of the extended b-metric space U. For Aθ,ℬθ∈FθU, a fuzzy set Aθ is said to be more accurate than a fuzzy set ℬθ and is denoted by Aθ⊂ℬθ, if and only if fAθμ≤fℬθμ for each μ∈U. Now, for μ∈U, Aθ,ℬθ∈FθU, α∈0,1, and Aθα,ℬθα∈CLDU, define(35)ραμ,Aθα=infdθμ,ℬθ:a∈Aθα,ραAθα,ℬθα=infdθa,b:a∈Aθα,b∈ℬθα,ρAθα,ℬθα=supαραAθα,ℬθα.Remark 3.
From Theorem4, the function Hθ:CLDU×CLDU⟶0,∞ defined by(36)HθAθα,ℬθα=maxsupa∈Aθαdθa,ℬθα,supb∈ℬθαdθb,Aθα,if the maximum exists;∞,otherwiseis a Hausdorff extended fuzzy b-metric on CLDU.Theorem 8.
LetU,dθ be a complete extended b-metric space with θ:U×U⟶1,∞. Let ℜ:U⟶FU be a fuzzy mapping and α:U⟶0,1 such that, for each μ∈U, ℜμα is a nonempty closed subset of U and φ∈Ψθ such that(37)Hθℜμα,ℜνα≤φdθμ,ν,for all μ,ν∈U. Then, ℜ has an α-fuzzy fixed point.Proof.
Let us take an arbitrary pointμ0∈U. Suppose there exists μ1∈ℜμ0α. As ℜμ0α is a nonempty closed subset of U, thus clearly if μ0=μ1 and μ1∈ℜμ1α, we get μ1 as an α-fuzzy fixed point of T and the proof is complete. Hence, throughout our proof, we will assume μ0≠μ1 and μ1∉ℜμ1α. Thus, dθμ1,ℜμ1α>0. From the definition of Hausdorff metric, equation (37), and φ∈Ψθ, we have(38)0<dθμ1,ℜμ1α≤Hθℜμ0α,ℜμ1α≤φdθμ0,μ1<φrdθμ0,μ1,where r>1 is a real number. Suppose there exists μ2∈ℜμ1α with μ1≠μ2 such that(39)0<dθμ1,μ2≤dθμ1,ℜμ1α≤Hθℜμ0α,ℜμ1α≤φdθμ0,μ1<φrdθμ0,μ1.
Asℜμ2α is a nonempty closed subset of U, therefore we assume μ2∉ℜμ2α, so dθμ2,ℜμ2α>0. From the definition of Hausdorff metric, equation (37), and φ∈Ψθ, we have(40)0<dθμ2,ℜμ2α≤Hθℜμ1α,ℜμ2α≤φdθμ1,μ2<φ2rdθμ0,μ1.
Suppose there existsμ3∈ℜμ2α with μ2≠μ3 such that(41)0<dθμ2,μ3≤φdθμ1,μ2<φ2rdθμ0,μ1.
Asℜμ3α is a nonempty closed subset of U, so we assume μ3∉ℜμ3α, so dθμ3,ℜμ3α>0. From the definition of Hausdorff metric, equation (37), and φ∈Ψθ, we have(42)0<dθμ3,ℜμ3α≤Hθℜμ2α,ℜμ3α≤φdθμ2,μ3<φ3rdθμ0,μ1.
By induction, we can construct a sequenceμnn=0∞ in U such that μn∉ℜμnα, μn+1∈ℜμnα, and(43)0<dθμn,ℜμnα≤Hθℜμn−1α,ℜμnα≤φdθμn−1,μn<φnrdθμ0,μ1,for all n∈ℕ. From the triangle inequality, for all m>n, we have(44)dθμn,μm≤θμn,μmdθμn,μn+1+θμn,μmθμn+1,μmdθμn+1,μn+2+⋯+θμn,μmθμn+1,μm+⋯+θμm−1,μmdθμm−1,μm≤θμn,μmφnrdθμ0,μ1+θμn,μmθμn+1,μmφn+1rdθμ0,μ1+⋯+θμn,μmθμn+1,μm+⋯+θμm−1,μmφm−1rdθμ0,μ1.
Since the series∑i=0∞θμj,μmφjrdθμ0,μ1 converges, therefore μnn=0∞ is a Cauchy sequence. As U is complete, so there exists μ∈U such that limn⟶∞μn=μ. Next, we will show that μ is an α-fuzzy fixed point. From the triangle inequality, we have(45)dθμ,ℜμα≤θμ,ℜμαdθμ,μn+1+dθμn+1,ℜμα≤θμ,ℜμαdθμ,μn+1+Hθℜμnα,ℜμα≤θμ,ℜμαdθμ,μn+1+φdθμn,μ.
By lettingn⟶∞ and φ0=0, we get dθμ,ℜμα=0. Since ℜμα is closed, we get μ∈ℜμα. Hence, μ is an α-fuzzy fixed point of ℜ.By puttingφℜ=κℜ, where κ∈0,1 in Theorem 8, we get the following corollary.Corollary 3.
LetU,dθ be a complete extended b-metric space with θ:U×U⟶1,∞. Let ℜ:U⟶FU be a fuzzy mapping and α:U⟶0,1 such that, for each μ∈U, ℜμα is a nonempty closed subset of U and φ∈Ψθ such that(46)Hθℜμα,ℜνα≤kdθμ,ν,for all μ,ν∈U, where k∈0,1. Assume k<1/r, where r≥1 is a real constant. Then, ℜ has an α-fuzzy fixed point.Remark 4.
Theorem8 generalizes Theorem 3.1 of [12]. Also, Corollary 3 generalizes Corollary 3.2 and 3.4 of [12].Example 2.
LetU=0,1. Define dθ:U×U⟶0,∞ by dθμ,ν=μ−ν2 and θ:U×U⟶1,∞ by θμ,ν=μ+ν+2, for all μ,ν∈U. Then, U,dθ is a complete extended b-metric space, which is not a b-metric space. Define ℜ:U⟶FU by(47)ℜμt=0,if0≤t<45;45,if45≤t≤4μ+15;47,if4μ+15<t≤1.
Defineα:U⟶0,1 by αμ=4/5, for all μ∈U. Clearly, we can see that, for all μ∈U,(48)ℜμα=45,4μ+15.
Hence,ℜμα is a nonempty closed subset of U. Also, Hθℜμα,ℜνα=16/25μ−ν2=16/25dθμ,ν, where φt=16/25t. Therefore, all the conditions of Theorem 8 are satisfied, and hence, 4/5∈U is an α-fuzzy fixed point of ℜ.Next, we will prove the existence ofα-fuzzy fixed point for multivalued fuzzy contraction mapping under Ćirić type contractive condition in the setting of complete extended b-metric spaces.Theorem 9.
LetU,dθ be a complete extended b-metric space with θ:U×U⟶1,∞. Let ℜ:U⟶FU be a fuzzy mapping and α:U⟶0,1 such that, for each μ∈U, ℜμα is a nonempty closed subset of U and φ∈Ψθ such that(49)Hθℜμα,ℜνα≤φMμ,ν,where(50)Mμ,ν=maxdθμ,ν,dθμ,ℜμα,dθν,ℜνα,dθμ,ℜνα+dθν,ℜμα2θμ,ℜνα,for all μ,ν∈U, where(51)θμ,ℜνα=infν∈ℜναθμ,ν.
Then,ℜ has an α-fuzzy fixed point.Proof.
Let us take an arbitrary pointμ0∈U. Suppose there exists μ1∈ℜμ0α. Recall that ℜμ0α is a nonempty closed subset of U. Thus, clearly if μ0=μ1 and μ1∈ℜμ1α, we get μ1 as an α-fuzzy fixed point of T and the proof is complete. Hence, throughout our proof, we will assume μ0≠μ1 and μ1∉ℜμ1α. Thus, dθμ1,ℜμ1α>0. From the definition of Hausdorff metric, equation (49), and φ∈Ψθ, we have(52)0<dθμ1,ℜμ1α≤Hθℜμ0α,ℜμ1α≤φMdθμ0,μ1,=φmaxdθμ0,μ1,dθμ0,ℜμ0α,dθμ1,ℜμ1α,dθμ0,ℜμ1α+dθμ1,ℜμ0α2θμ0,ℜμ1α≤φmaxdθμ0,μ1,dθμ0,ℜμ0α,dθμ1,ℜμ1α,θμ0,ℜμ1αdθμ0,μ1+dθμ1,ℜμ1α+dθμ1,ℜμ0α2θμ0,ℜμ1α≤φmaxdθμ0,μ1,dθμ0,μ1,dθμ1,ℜμ1α,dθμ0,μ1+dθμ1,ℜμ1α+dθμ1,μ12,=φmaxdθμ0,μ1,dθμ0,μ1,dθμ1,ℜμ1α,dθμ0,μ1+dθμ1,ℜμ1α2,=φmaxdθμ0,μ1,dθμ1,ℜμ1α.
This implies that(53)dθμ1,ℜμ1α≤φmaxdθμ0,μ1,dθμ1,ℜμ1α.
Now, we will take the following two cases:Case 1: ifmaxdθμ0,μ1,dθμ1,ℜμ1α=dθμ1,ℜμ1α, then we have(54)0<dθμ1,ℜμ1α≤φdθμ1,ℜμ1α<dθμ1,ℜμ1α,which is a contradiction.Case 2: ifmaxdθμ0,μ1,dθμ1,ℜμ1α=dθμ0,μ1, then we have(55)0<dθμ1,ℜμ1α≤φdθμ0,μ1<φrdθμ0,μ1,wherer>1 is a real number.
This ensures that there existsμ2∈ℜμ1α with μ1≠μ2 such that(56)0<dθμ1,μ2≤φdθμ0,μ1<φrdθμ0,μ1.
Asℜμ2α is a nonempty closed subset of U, therefore, we assume μ2∉ℜμ2α, so dθμ2,ℜμ2α>0. From the definition of Hausdorff metric, equation (49), and φ∈Ψθ, we have(57)0<dθμ2,ℜμ2α≤Hθℜμ1α,ℜμ2α≤φMdθμ1,μ2=φmaxdθμ1,μ2,dθμ1,ℜμ1α,dθμ2,ℜμ2α,dθμ1,ℜμ2α+dθμ2,ℜμ1α2θμ1,ℜμ2α≤φmaxdθμ1,μ2,dθμ1,ℜμ1α,dθμ2,ℜμ2α,θμ1,ℜμ2αdθμ1,μ2+dθμ2,ℜμ2α+dθμ2,ℜμ1α2θμ1,ℜ2α≤φmaxdθμ1,μ2,dθμ1,μ2,dθμ2,ℜμ2α,dθμ1,μ2+dθμ2,ℜμ2α+dθμ2,μ22,=φmaxdθμ1,μ2,dθμ1,μ2,dθμ2,ℜμ2α,dθμ1,μ2+dθμ2,ℜμ2α2=φmaxdθμ1,μ2,dθμ2,ℜμ2α.
Ifmaxdθμ1,μ2,dθμ2,ℜμ2α=dθμ2,ℜμ2α, then we have(58)0<dθμ2,ℜμ2α≤φdθμ2,ℜμ2α<dθμ2,ℜμ2α,which is a contradiction. Thus, maxdθμ1,μ2,dθμ2,ℜμ2α=dθμ1,μ2. As φ is increasing, we have(59)0<dθμ2,ℜμ2α≤φdθμ1,μ2<φrdθμ1,μ2,where r>1 is a real number. By induction, we can construct a sequence μnn=0∞ in μ such that μn∉ℜμnα, μn+1∈ℜμnα, and(60)0<dθμn,ℜμnα≤Hθℜμn−1α,ℜμnα≤φdθμn−1,μn<φnrdθμ0,μ1,for all n∈ℕ. By applying the same procedure as Theorem 8, we prove that μ is an α-fuzzy fixed point of ℜ.Remark 5.
If we putMμ,ν=dθμ,ν in Theorem 9, we get Theorem 8. Hence, Theorem 9 is an extension of Theorem 8.Example 3.
LetU=0,1,2. Define dθ:U×U⟶0,∞ by(61)dθμ,ν=0,ifμ=ν;16,ifμ≠ν,μ,ν∈0,1;14,ifμ≠ν,μ,ν∈0,2;1,ifμ≠ν,μ,ν∈1,2.
Hence,U,dθ is a complete extended b-metric space, where θ:U×U⟶1,∞ is defined by θμ,ν=μ+ν+1. Define ℜ:U⟶FU by(62)ℜ0t=ℜ1t=14,ift=0;0,ift=1,2,ℜ2t=0,ift=0,2;14,ift=1.
Defineα:U⟶0,1 by αμ=1/2, for all μ∈U. Clearly, we can see that, for all μ∈U,(63)ℜμα=0,ifμ=0,1;1,ifx=2.
Now, for allμ,ν∈U, we get Hθℜ01/2,ℜ11/2=Hθ0,0=0 and Hθℜ01/2,ℜ21/2=Hθ0,1=1/6=Hθℜ11/2,ℜ21/2.
Also since for allμ,ν∈U(64)Mμ,ν=maxdθμ,ν,dθμ,ℜμα,dθν,ℜνα,dθμ,ℜνα+dθν,ℜμα2θμ,ℜνα.
Forμ=0 any ν=1, we have(65)M0,1=maxdθ0,1,dθ0,ℜ01/2,dθ1,ℜ11/2,dθ0,ℜ11/2+dθ1,ℜ01/22θ0,ℜ11/2,M0,1=max16,0,16,0+1/62.
This implies thatM0,1=1/6. Similarly, M0,2=M1,2=1. Define φ:0,∞⟶0,∞ by φt=1/2t for all t>0. Hence, for all μ,ν∈U, we have(66)Hθℜ01/2,ℜ11/2=0<12M0,1,Hθℜ01/2,ℜ21/2=Hθ0,1=16<12M0,2,Hθℜ11/2,ℜ21/2=Hθ0,1=16<12M1,2.
Hence, all the conditions of Theorem9 hold, and therefore, there exists 0∈U such that 0∈ℜ01/2 is an α-fuzzy fixed point of ℜ.Next, we will show that, by utilizing Corollary3, we can prove fixed point results for multivalued mappings.Corollary 4.
LetU,dθ be a complete extended b-metric space with θ:U×U⟶1,∞. Let ℑ:U⟶CLDU be a multivalued mapping such that(67)Hθℑμ,ℑν≤kdθμ,ν,for all μ,ν∈U, where k∈0,1. Assume k<1/r, where r≥1 is a real constant. Then, ℑ has a fixed point.Proof.
Letα:U⟶0,1 be any arbitrary mapping and define ℜ:U⟶FU by(68)ℜμt=α,ift∈ℑμ;0,ift≠ℑμ.
Hence, we obtain(69)ℜμα=t:ℜμt≥α=ℑμ.
Thus, condition (37) becomes (49). Therefore, Corollary 3 can be applied to get an α-fuzzy fixed point μ∈ℜμα=ℑμ, where μ∈U. Hence, the multivalued mapping ℑ has a fixed point.



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*Source: 5579991-2021-03-17.xml*