# Fuzzy Fixed Point Results of Fuzzy Mappings onb-Metric Spaces via α∗,F-Contractions

Authors: Samina Batul; Dur-e-Shehwar Sagheer; Muhammad Anwar; Hassen Aydi; Vahid Parvaneh
Journal: Advances in Mathematical Physics
 (2022)

---

## Abstract

In this manuscript, we establish some fixed point results for fuzzy mappings viaα∗,F -contractions. For validation of the proved results, some nontrivial examples are presented. Few interesting consequences are also stated which authenticate that our results generalize many existing ones in the literature.


---

## Body

## 1. Introduction

Fixed point theory has attained an important and significant role in analysis. The literature of the last four decades ornamented with results which discover fixed points of self and non-self nonlinear operators in a metric space. This branch of mathematics provides a strong tool for finding out the solution of integral, differential, and eigenvalue equations. Since 1922, the Banach contraction principle (BCP) has become the center of the attention for researchers working in different areas. The eminent Banach contraction principle (BCP) [1] states that for a complete metric space Ω,d, each self-mapping η on Ω satisfying
(1)dηξ1,ηξ2≤λdξ1,ξ2,∀ξ1,ξ2∈Ω,whereλ∈0,1has a unique fixed point.A lot of literature can be found on generalizations and extensions of the famous BCP by changing either the space under consideration or the condition on the mapping. In recent years, various authors presented interesting generalizations of a metric space, for example, uniform space [2, 3], b-metric space [4], C∗-algebra valued metric space [5, 6], Gpb-metric space [7], and b2-metric space [8].A very interesting generalization of metric space is cone metric space. Gupta and Chauhan [9] presented an analogous to the BCP on cone b-metric spaces in 2021. In contrast, Wardowski [10] extended the Banach contraction to a more generalized form, known as F-contractions, and established a fixed point theorem in complete metric spaces. Furthermore, many mathematicians used F-contractions for the existence of a fixed point, see [11–13]. Nadler’s theorem [14] states that for a complete metric space Ω,d, each non-self-mapping η:Ω⟶CΩ satisfying
(2)Hηξ1,ηξ2≤λdξ1,ξ2,∀ξ1,ξ2∈Ω,whereλ∈0,1has a fixed point. Here, H denotes the Hausdorff metric defined on CΩ, the set of bounded and closed subsets of Ω. Throughout the article, a b−metric space and a fuzzy fixed point are denoted by a bms and a ffp, respectively. Some important definitions are presented before constructing the main results. These definitions are inevitable for next discussion.Definition 1.
LetΩ be a nonempty set. A function Db:Ω×Ω⟶0,∞ is called a b-metric if, for all ξ1,ξ2,ξ3∈Ω and b≥1, the following assertions are satisfied:
(i)
Dbξ1,ξ2=0,ifandonlyifξ1=ξ2(ii)
Dbξ1,ξ2=Dbξ2,ξ1(iii)
Dbξ1,ξ2≤bDbξ1,ξ3+Dbξ3,ξ2The pairΩ,Db is called a bms. if we consider b=1, then above definition coincides with the definition of a metric space.Example 2.
LetΩ=ℝ. Define Db:Ω×Ω⟶ℝ by Dbξ1,ξ2=ξ1−ξ22. Then, it is easy to show that Db,Ω is a bms with b=2.Definition 3.
LetΩ,Db be a bms.
(i)
A sequenceξn∈Ω,Db is said to be a Cauchy sequence if, given ε>0, there exists n0∈ℕ such that, for all n,m≥n0, we have Dbξm,ξn<ε or limn,m⟶∞Dbξm,ξn=0(ii)
A sequenceξn converges to ξ if limn⟶∞Dbξn,ξ=0Among numerous advances of fuzzy sets theory, a significant development is made to find the fuzzy analogues of fixed point results of the classical fixed point theorems. The concept of fuzzy sets along with its related notions is given by Zadeh in [15]. In 1975, the idea of fuzzy metric space is introduced. This interesting concepts is investigated by many researchers with different contractions, see for example [16, 17].Furthermore, Weiss [18] and Butnariu [19] used the notions of fuzzy maps to established various results. Heilpern [20] proved a fixed point theorem for fuzzy maps which is the analogue of Nadler’s multivalued result [14] in metric spaces. Notion of fuzzy maps is investigated by many mathematician in different directions. In this context, in [21], the existence of common fuzzy fixed points under a rational contractive condition has been established which is further generalized by Shoaib et al. [22] in dislocated complete metric spaces. Motivated by these research ideas, we formulate some results for fuzzy contractions using the Hausdorff metric. Namely, the present article provides certain fixed point results for fuzzy mappings via α∗,F contractions. A mapping F:ℝ+⟶ℝ is said to be an F-mapping [10], if the following conditions are satisfied:(F1):F is a strictly increasing function, that is, for ξ1,ξ2∈ℝ+, with ξ1<ξ2 then Fξ1<Fξ2(F2): for each sequenceξn of the positive real numbers ℝ+,
(3)limn⟶∞ξn=0ifandonlyiflimn⟶∞Fξn=−∞.(F3): There is a real numberc∈0,1 such as
(4)limξ⟶0+ξcFξ=0.Afterward, Altun et al. [13] altered the above definition by adding another condition given below:(F4):FinfA=infFA∀A⊂0,∞The set of all functions satisfyingF1 to F4 is denoted by F. Let Ω,d be a metric space and η:Ω⟶Ω be a self mapping on Ω. η is said to be an F-contraction [10] if there exists τ>0 and F satisfying F1-F3 such that
(5)dηξ1,ηξ2>0⇒τ+Fdηξ1,ηξ2≤Fdξ1,ξ2∀ξ1,ξ2∈Ω.Definition 4. (see [22]).
LetM be a nonempty subset of a bms Ω,Db, and let ξ1∈Ω. An element ξ0∈M is called the best approximation in M if Dbξ1,M=Dbξ1,ξ0, where
(6)Dbξ1,M=infξ2∈ΩDbξ1,ξ2.
The setM is called a proximinal set if every ξ∈Ω has at least one best approximation in M. The set of all proximinal subsets of Ω is denoted by PΩ.Definition 5. (see [22]).
The function,HDb:PΩ×PΩ⟶ℝ, defined by
(7)HDbM,N=maxsupξ1∈MDbξ1,N,supξ2∈NDbM,ξ2is called a Hausdorff b-metric on PΩ.A function with domainΩ and values in 0,1 is called a fuzzy set. We will denote the collection of all fuzzy sets in Ω by FΩ. If ξ∈Ω, and M is a fuzzy set, then Mξ is called the grade of membership of ξ in M. The set Mα is called an α-level set of a fuzzy set M, and it is given by
(8)Mα=ξ:Mξ≥α,whereα∈0,1,M0=ξ:Mξ>0.Ifη is a mapping from Ω into FΩ, then η is called a fuzzy mapping. A fuzzy mapping η is a fuzzy subset on Ω×Ω with membership function ηξ1ξ2. The function ηξ1ξ2 is the grade of membership of ξ2 in ηξ1.Definition 6. (see [21]).
Letη:Ω⟶FΩ be a fuzzy mapping. A point ξ1∈Ω is called an α−fuzzy fixed point (ffp) if there exists α∈0,1 such that ξ∈ηξ1α.Lemma 7. (see [22]).
LetM and N be nonempty proximal subsets of a bms Ω,Db.
Ifξ1∈M, then Dbξ1,N≤HDbM,N.Lemma 8. (see [22]).
LetΩ,Db be a bms and suppose that PΩ,HDb is a Hausdorff bms. If, for all M,N∈PΩ and for each ξ1∈M, there exists ξ2∈N satisfying Dbξ1,N=Dbξ1,ξ2, then HDbM,N≥Dbξ1,ξ2.


## 2. Main Results

Definition 9.
Consider a bmsΩ,Db. A mapping η:Ω⟶FΩ is said to be a fuzzy α∗,F-contractive mapping if for a function α∗:Ω×Ω⟶0,∞, there exists F:0,+∞⟶ℝ such that condition (F1) holds, and the following conditions are satisfied:
(Fb2): for each sequenceξn of positive numbers, if limn⟶∞Fξn=−∞, then limn⟶∞ξn=0
(Fb3):k∈0,1/1+log2b such that ξkFξ=0
(Fb4): there existsτ>0 for which
(9)τ+Fα∗ξ1,ξ2Hηξ1αξ1,ηξ2αξ2≤Fa1Dbξ1,ηξ1αξ1+a2Dbξ2,ηξ2αξ2+a3Dbξ1,ηξ2αξ2+a4Dbξ2,ηξ1αξ1+a5Dbξ1,ξ2+a6Dbξ1,ηξ1αξ11+Dbξ1,ηξ1αξ11+Dbξ1,ξ2,for all ξ1,ξ2∈Ω,andai≥0for1≤i≤6,alsoa1+a2+2ba3+a4+a5+a6<1.Theorem 10.
LetΩ,Db be a complete bms with b≥1. Let η:Ω⟶FΩ be a fuzzy α∗,F-contractive mapping. Then, η has an α−ffp if following conditions are satisfied:
(i)
η is an α∗-admissible mapping(ii)
There existsξ0∈Ωandξ1∈ηξ0αξ0suchthatα∗ξ0,ξ1≥1(iii)
For any sequenceξn⊂Ωwhichconvergestoξ∈Ωwithα∗ξn,ξn+1≥1∀n∈ℕ∪0,wehavelimn⟶∞α∗ξn,ξ≥1Proof.
Letξ0∈Ω be an arbitrary point of Ω. Choose ξ1∈ηξ0αξ0. If ξ1∈ηξ1αξ1, then ξ1 is a fixed point, and there is nothing to prove; so, ξ1∉ηξ1αξ1. Therefore, ηξ0αξ0≠ηξ1αξ1. Now, by Lemma 8, there exists ξ1∈ηξ2αξ2 such that
(10)Dbξ1,ηξ1<Hηξ0αξ0,ηξ1αξ1.
Then,(11)Dbξ1,ξ2<Hηξ0αξ0,ηξ1αξ1.
Consider(12)FDbξ2,ξ1<Fα∗ξ0,ξ1Hϕξ1αξ1,ηξ0αξ0τ+FDbξ2,ξ1<τ+Fα∗ξ0,ξ1Hϕξ1αξ1,ηξ0αξ0≤Fa1Dbξ1,ϕξ1αξ1+a2Dbξ0,ηξ0αξ0+a3Dbξ1,ηξ0αξ0+a4Dbξ0ϕξ1αξ1+a5Dbξ0,ξ1≤Fa1Dbξ1,ξ2+a2Dbξ0,ξ1+a3Dbξ1,ξ1+a4Dbξ0,ξ2+a5Dbξ0,ξ1≤Fa1Dbξ1,ξ2+a2Dbξ0,ξ1+a4bDbξ0,ξ1+Dbξ1,ξ2+a5Dbξ0,ξ1≤Fa2+ba4+a5Dbξ0,ξ1+a1+ba4Dbξ1,ξ2.
Thus,(13)FDbξ1,ξ2≤Fa1+a3b+a5+a6Dbξ0,ξ1+a2+a3bDbξ1,ξ2.
SinceF is increasing, one writes
(14)Dbξ1,ξ2≤a1+a3b+a5+a6Dbξ0,ξ1+a2+a3bDbξ1,ξ2.
We have(15)Dbξ1,ξ2≤a1+a3b+a5+a61−a2+a3bDbξ0,ξ1.
That is,(16)Dbξ1,ξ2≤Dbξ0,ξ1because a1+a2+2ba3+a4+a5+a6<1. Consequently, τ+FDbξ1,ξ2≤FDbξ0,ξ1. Continuing in the same manner, we can define a sequence ξn such that ξn∉ηξnαξn,ξn+1∈ηξnαξn and
(17)τ+FDbξn+1,ξn+2≤FDbξn,ξn+1∀n∈ℕ∪0.
Now, letdn=Dbξn,ξn+1>0,∀n∈ℕ∪0. By (2.2),
(18)Fdn≤Fdn−1−τ≤⋯≤Fd0−nτ,n∈ℕand hence limn⟶+∞Fdn=−∞. By property (Fb2), we get that dn⟶0 as n⟶∞. Then, by condition (Fb3), there exists k∈0,1/1+log2b such that
(19)limn⟶∞dnkFdn=0.
Therefore, we have(20)dnkFdn−dnkFd0≤−ndnkτ.
Applying limitn⟶∞, we have limn⟶∞dnkn=0. Therefore, there exists n0∈ℕ such that
(21)dnk≤1n⇒dn≤1n1/k∀,n≥n0.
Using Lemma7 given in [23], ξn is a Cauchy sequence in Ω. Since Ω is complete, there exists ξ∈Ω such that limn⟶∞ξn=ξ. We claim that η has an α−ffp. Consider
(22)Dbξ,ηξαξ≤bDbξ,ξn+1+dξn+1,ηξαξ≤bDbξ,ξn+1+Hηξnαξn,ηξαξ≤bDbξ,ξn+1+Hα∗ξn,ξηξnαξn,ηξαξ≤bDbξ,ξn+1+a1Dbξn,ηξnαξn+a2Dbξ,ηξαξ+a3Dbξn,ηξαξ+a4Dbξ,ηξnαξn+a5Dbξn,ξ+a6Dbξn,ηξnαξn1+Dbξn,ηξnαξn1+Dbξn,ξ≤bDbξ,ξn+1+a1Dbξn,ξn+1+a2Dbξ,ηξαξ+a3Dbξn,ηξαξ+a4Dbξ,ξn+1+a5Dbξn,ξ+a6Dbξn,ξn+11+Dbξn,ηξnαξn1+Dbξn,ξ.
Taking limitn⟶∞, we get
(23)1−ba2−ba3Dbξ,ηξαξ≤0.
So, we getDbξ,ηξαξ=0, which implies that ξ∈ηξαξ. Hence, ξ∈Ω is an α−ffp of η.Lettingb=1 in Theorem 10, we obtain the following:Corollary 11.
LetΩ,D be a complete metric space. Let η:Ω⟶FΩ be a fuzzy α∗,F-contractive mapping, and then η has an α−ffp if following conditions are satisfied:
(i)
η is an α∗-admissible mapping(ii)
There existsξ0∈Ω and ξ1∈ηξ0αξ0 such that α∗ξ0,ξ1≥1(iii)
For any sequenceξn⊂Ω which converges to ξ∈Ω with α∗ξn,ξn+1≥1∀n∈ℕ∪0, we have limn⟶∞α∗ξn,ξ≥1If we considerb=1 and α∗ξ1,ξ2=1 for all ξ1,ξ2∈Ω, then we have the following result proved by Ahmed et al. [11].Corollary 12.
LetΩ,D be a complete metric space, and let η:Ω⟶PΩ. Suppose for each ξ1 and ξ2∈Ω, there exists αξ1 and αξ2∈0,1 such that ηξ1αξ1,ηξ2αξ2∈CBΩ. Assume there exists F∈F such that
(24)τ+Hηξ1αξ1,ηξ2αξ2≤FDξ1,ξ2,then η has an α−ffp.Example 13.
LetΩ=0,1 and define Db=ξ1−ξ22. Then, Ω,Db is a complete bms with b=2. Define a fuzzy mapping η:Ω⟶FΩ by
(25)ηξt=1,for0≤t≤ξ4;12,forξ4≤t≤ξ3;14,forξ3≤t≤ξ2;0,forξ2≤t≤1,and for all ξ∈Ω, there exists αξ=1 such that
(26)ηξαξ=0,ξ4.α∗ξ1,ξ2=1,ifξ1,ξ2∈Ω−1;0,otherwise.
Leta1=1/10, a2=a3=0, a4=1/20, a5=1/30, and a6=1/40, and then we have
(27)Hηξ1αξ1,ηξ2αξ2ηξ2αξ2≤110ξ1−ξ122+0ξ2−ξ222+0ξ1−ξ222+120ξ2−ξ122+130ξ1−ξ22+140ξ1−ξ1/221+ξ1−ξ1/221+ξ1−ξ22,where μ=a1+ba3+a5+a6/1−a2+ba3<1/b. Therefore,
(28)τ+Fα∗ξ1,ξ2Hηξ1αξ1,ηξ2αξ2≤Fa1Dbξ1,ηξ1αξ1+a2Dbξ2,ηξ2αξ2+a3Dbξ1,ηξ2αξ2+a4Dbξ2,ηξ1αξ1+a5Dbξ1,ξ2+a6Dbξ1,ηξ1αξ11+Dbξ1,ηξ1αξ11+Dbξ1,ξ2,where Fx=lnx for all x∈0,∞. It follows that all the conditions of Theorem 10 are satisfied, and 0 is an α−ffp of η.Definition 14.
LetΩ,Db be a bms. Let α∗:Ω×Ω⟶0,∞ and ϕ,η be a pair of fuzzy mappings from Ω into FΩ. The pair ϕ,η is said to be α∗-admissible if it satisfies the following conditions:
(i)
For eachξ1∈Ω, there exists ξ2∈ηξ1αξ1 such that α∗ξ2,ξ3≥1 for all ξ3∈ϕξ2αξ2(ii)
For eachξ1∈Ω, there exists ξ2∈ϕξ1αξ1 such that α∗ξ2,ξ3≥1 for all ξ3∈ηξ2αξ2Definition 15.
Consider a complete bmsΩ,Db. A pair of fuzzy mappings ϕ,η is said to be an α∗,F-contraction if there exists F:0,+∞⟶ℝ so that the condition (F1) holds and the following conditions are satisfied:
(F′b2): for each sequence ξn of positive numbers, if limn⟶∞Fξn=−∞, then limn⟶∞ξn=0;
(F′b3): there exists k∈0,1/1+log2b such that ξkFξ=0;
(F′b4): there exists τ>0, for which
(29)τ+Fα∗ξ1,ξ2Hηξ1αξ1,ϕξ2αξ2≤Fa1Dbξ1,ηξ1αξ1+a2Dbξ2,ϕξ2αξ2+a3Dbξ1,ϕξ2αξ2+a4Dbξ2,ηξ1αξ1+a5Dbξ1,ξ2
∀ξ1,ξ2∈Ω, with ai≥0,for1≤i≤5,a1+a2b+2+ba3+a4b+2+2ba5<2.Theorem 16.
LetΩ,Db be a complete bms with b≥1. Let η,ϕ be a pair of α∗,F fuzzy contraction mappings, and then η and ϕ have a common α− ffp if following conditions are satisfied:
(i)
Bothη and ϕ are α∗-admissible mappings(ii)
There existsξ0∈Ω and ξ1∈ηξ0αξ0 such that α∗ξ0,ξ1≥1(iii)
For any sequenceξn⊂Ω which converges to ξ∈Ω with α∗ξn,ξn+1≥1∀n∈ℕ∪0, we have limn⟶∞α∗ξn,ξ≥1Proof.
Letξ0∈Ω be an arbitrary point of Ω. Choose ξ1∈ηξ0αξ0; then by Lemma 8, there exists ξ2∈ϕξ1αξ1 such that
(30)Dbξ1,ϕξ1αξ1<Hηξ0αξ0,ϕξ1αξ1.
Now, by Lemma8,
(31)Dbξ1,ξ2<Hηξ0αξ0,ϕξ1αξ1.
By using assumption (ii),(32)FDbξ1,ξ2<Fα∗ξ0,ξ1Hηξ0αξ0,ϕξ1αξ1τ+FDbξ1,ξ2<τ+Fα∗ξ0,ξ1Hηξ0αξ0,ϕξ1αξ1≤Fa1Dbξ0,ηξ0αξ0+a2Dbξ1,ϕξ1αξ1+a3Dbξ0,ϕξ1αξ1+a4Dbξ1,ηξ0αξ0+a5Dbξ0,ξ1≤Fa1Dbξ0,ξ1+a2Dbξ1,ξ2+a3Dbξ0,ξ2+a5Dbξ0,ξ1≤Fa1Dbξ0,ξ1+a2Dbξ1,ξ2+a3bDbξ0,ξ1+Dbξ1,ξ2+a5Dbξ0,ξ1≤Fa1+a3b+a5Dbξ0,ξ1+a2+a3bDbξ1,ξ2.⇒FDξ1,ξ2≤Fa1+a3b+a5Dbξ0,ξ1+a2+a3bDbξ1,ξ2.
SinceF is increasing, one writes
(33)Dbξ1,ξ2≤a1+a3b+a5Dbξ0,ξ1+a2+a3bDbξ1,ξ2.
That is,(34)1−a2+a3bDbξ1,ξ2≤a1+a3bDbξ0,ξ1.
So, we have(35)Dbξ1,ξ2≤a1+a3b+a51−a2+a3bDbξ0,ξ1.
By using symmetric property of a bm, we have(36)Dbξ2,ξ1<Hϕξ1αξ1,ηξ0αξ0.
Then,(37)FDbξ2,ξ1<FHϕξ1αξ1,ηξ0αξ0.
Consider(38)FDbξ2,ξ1<Fα∗ξ0,ξ1Hϕξ1αξ1,ηξ0αξ0τ+FDbξ2,ξ1<τ+Fα∗ξ0,ξ1Hϕξ1αξ1,ηξ0αξ0≤Fa1Dbξ1,ϕξ1αξ1+a2Dbξ0,ηξ0αξ0+a3Dbξ1,ηξ0αξ0+a4Dbξ0,ϕξ1αξ1+a5Dbξ0,ξ1≤Fa1Dbξ1,ξ2+a2Dbξ0,ξ1+a3Dbξ1,ξ1+a4Dbξ0,ξ2+a5Dbξ0,ξ1≤Fa1Dbξ1,ξ2+a2Dbξ0,ξ1+a4bDbξ0,ξ1+Dbξ1,ξ2+a5Dbξ0,ξ1≤Fa2+ba4+a5Dbξ0,ξ1+a1+ba4Dbξ1,ξ2.
Thus,(39)FDbξ1,ξ2≤Fa2+a4b+a5Dbξ0,ξ1+a1+a4bDbξ1,ξ2.
SinceF is increasing, we have
(40)Dbξ2,ξ1≤a2+ba4+a5Dbξ0,ξ1+a1+ba4Dbξ1,ξ2.
That is,(41)1−a1+ba4Dbξ1,ξ2≤a2+ba4+a5Dbξ0,ξ1.
Therefore, we obtain(42)Dbξ2,ξ1≤a2+a4b+a51−a1+a4bDbξ0,ξ1.
Add (35) and (42) to conclude
(43)Dbξ1,ξ2≤a1+a2+ba3+ba4+2a52−2a1+a2+ba3+ba4Dbξ0,ξ1.
By using the given condition, we have(44)Dbξ1,ξ2≤Dbξ0,ξ1.
Consequently,τ+FDbξ1,ξ2≤FDbξ0,ξ1. Continuing in the same manner, we can define a sequence ξn such that ξn∉ηξnαξn,ξn+1∈ηξnαξn and
(45)τ+FDbξn+1,ξn+2≤FDbξn,ξn+1∀n∈ℕ∪0.
Now, letdn=Dbξn,ξn+1>0,∀n∈ℕ∪0. By (2.10),
(46)Fdn≤Fdn−1−τ≤⋯≤Fd0−nτ,n∈ℕ,and hence limn⟶+∞Fdn=−∞. By property (F′b2), we obtain dn⟶0 as n⟶∞. Then, by condition (F′b3), there exists k∈0,1/1+log2b such that
(47)limn⟶∞dnkFdn=0.
From (47),
(48)dnkFdn−dnkFd0≤−ndnkτ.
Applying limitn⟶∞, we have limn⟶∞dnkn=0. Therefore, there exists n0∈ℕ such that
(49)dnk≤1n⇒dn≤1n1/k∀n≥n0.
Using Lemma7 given in [23], ξn is a Cauchy sequence in Ω which is complete; so, there exists ξ∈Ω such that limn⟶∞ξn=ξ. We shall prove that Φ has a fixed point. Consider
(50)Dbξ,ϕξαξ≤bDbξ,ξn+1+Dbξn+1,ϕξαξ≤bDbξ,ξn+1+Hηξnαξn,ϕξαξ≤bDbξ,ξn+1+α∗ξn,ξHηξnαξn,ϕξαξ.
SinceF is increasing and
(51)Hηξnαξn,ϕξαξ≤α∗ξn,ξHηξnαξn,ϕξαξ,one writes
(52)Dbξ,ϕξαξ≤Fα∗ξn,ξHηξnαξn,ϕξαξ≤bDbξ,ξn+1+a1Dbξn,ηξnαξn+a2Dbξ,ϕξαξ+a3Dbξn,ϕξαξ+a4Dbξ,ηξnαξn+a5Dbξn,ξ.
Using (52), we obtain
(53)Dbξ,ϕξαξ≤Fα∗ξn,ξHηξnαξn,ϕξαξ≤bDbξ,ξn+1+a1Dbξn,ηξnαξn+a2Dbξ,ϕξαξ+a3Dbξn,ϕξαξ+a4Dbξ,ηξnαξn+a5Dbξn,ξ.
Taking limitn⟶∞, we obtain
(54)Dbξ,ϕξαξ≤0.
Therefore,ξ∈ϕξαξ. This implies that ξ is an α−ffp of ϕ. By the same procedure, one can show that ξ∈ηξαξ. Hence, ξ∈Ω is a common α−ffp of ϕ and η.If we considerb=1 and α∗ξ1,ξ2=1 for all ξ1,ξ2 in Ω, then we have the following result proved by Ahmed et al. [11].Corollary 17.
LetΩ,D be a complete metric space, and let η,ϕ:Ω⟶PΩ. Suppose for each ξ1 and ξ2∈Ω, there exists αξ1 and αξ2∈0,1 such that ηξ1αξ1,ϕξ2αξ2∈CBΩ. Assume there exists F∈F such that
(55)τ+Hηξ1αξ1,ϕξ2αξ2≤FDξ1,ξ2,then η and ϕ have a common α−ffp.Example 18.
LetΩ=0,1,2 and Db be a bm (with b>1) defined by
(56)Dbξ1,ξ2=0,forξ1=ξ2andξ1,ξ2∈0,1,2;12,forξ1≠ξ2andξ1,ξ2∈0,2;14,forξ1≠ξ2andξ1,ξ2∈1,2;1,forξ1≠ξ2andξ1,ξ2∈0,1.
It is clear that it is a bms withb=4/3, but it is not a metric space.
Takeϕ,η:Ω⟶FΩ as fuzzy mappings defined by
(57)η0t=0,fort=1,2;ξ=0;12,fort=0,ξ=0;η1t=0,fort=0,1;12,fort=2;η2t=0,fort=0;12,fort=1;ϕ0t=ϕ1t=ϕ2t=0,fort=0,1,2.
Now, we defineαηξ=αϕξ=α∈0,1, and we have
(58)ηξαηξ=0,forξ=0;1,forξ=1;2,forξ=2;ϕξαϕξ=0,forξ=0,1,2.
Therefore,(59)Hηξαηξ,ϕξαϕξ=H0,0=0,forξ=0;H1,0=1,forξ=1;H2,0=2,forξ=2;α∗ξ1,ξ2=1,ifξ1,ξ2∈Ω−1;0,otherwise.
Leta1=a2=1/4, a3=a5=0, and a4=1/8, and then we have bμ=4/5<1 and μ=a1+a2+ba3+ba4+2a5/2−a1+a2+ba3+ba4=3/5. Further, it can also be seen that ai≥0 for i=1,2,⋯5, where a1+a2b+1+bb+1a3+a4+2ba5=189/128<2 and ∑i=15ai<1. Further, it is easy to verify that all the conditions of Theorem 10 are satisfied with Fx=lnx for all x∈0,∞. Hence, there exists 0∈ηξ1αηξ1∩ϕξ2αϕξ2 which is a common fixed point.


## 3. Conclusion

In this paper, the notion ofα∗,F fuzzy contractive mappings has been introduced, and few results have been established for the existence of α fuzzy fixed points of an α∗,F-contraction and a pair of α∗,F-contractions. To elaborate the main results, examples have also been presented. Few corollaries have been established to show that our results generalize and extend many existing classical results available in the literature.



---

*Source: 4511632-2022-07-29.xml*