# Existence ofαL-Fuzzy Fixed Points of L-Fuzzy Mappings

Authors: Shazia Kanwal; Umair Hanif; Maha Eshaq Noorwali; Md. Ashraful Alam
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this research article, fixed point theory is beautifully combined with fuzzy set theory. Two fuzzy fixed point theorems ofL-fuzzy mappings are established and proved for two different contractive type conditions in the scenario of complete b-metric space. In order to give the strength of these results, nontrivial supportive examples for both results are also provided. The notion of L-fuzzy mappings is a generalized form of fuzzy mappings as well as multivalued mappings. In this approach, our results provide uniqueness, extension, and successive generalizations of many valuable recent and conventional results existing in the literature.


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## Body

## 1. Introduction and Preliminaries

Responding to physical problems becomes naiver with the beginning of FS theory which was introduced in 1965 by Zadeh [1], as it benefits in manufacturing the version of fuzziness and flaws stronger and more definite. Now, it is a well-accepted system to grasp confusions originating in different materialistic situations. In 1967, Goguen [2] expanded this idea into the L-FS theory by replacing the interval with a complete distributive lattice L. The concept of a FS is a special case of an L-FS when L = [0, 1]. Then, many results were accomplished by various authors for L-FM. Because the notion of distance function plays an energetic part in approximation theory; therefore, FSs and L-FSs have further been practiced in the classical idea of MSs. The Hausdorff distance for α-cut sets of L-FMs was made known by Rashid et al. to study FP theorems for L-FMs. In 1989, Bakhtin [3] introduced the concept of b-MS. In 1993, Czerwik [4] obtained the results of b-MS. By accepting this idea, many researchers gave generalizations of the Banach contractive principle in b-MS. Boriceanu [5], Bota et al. [6] Czerwik [4], Kir and Kiziltunc [7], Kumam et al. [8], and Pacurar [9] obtained the FP theorems in b-MSs. Afterward, many authors derived and calculated the existence of FP of mappings, satisfying a contractive type condition, for example, Abbas et at. [10] obtained fuzzy common FPs for generalized mappings, Ahmad et al. [11] achieved FPs for locally contractive mappings, Azam et al. [12, 13] established FPs and common FPs for FPs, Estruch and Vidal [14]and Frigon and O’Regan [15] constructed FFPs for FPs, Kanwal and Azam [16] obtained common coincidence points for L-FMs and obtained many useful results for FMs and set-valued mappings as the direct consequences of the main result, Lee and Jin Cho [17] did work in proving the FP theorem for fuzzy contractive-type mappings, Phiangsungnoen and Kumam [18, 19] proved FFP theorems for multivalued fuzzy contractions in b-MSs, and Rashid et al. [20, 21] established L-fuzzy fixed points via beta-L admissible pair and coincidence theorem via α-cuts of L-FMs with applications. Gulzar et al. [22, 23] did work on fuzzy algebra and obtained results in this area.Definition 1 (see [5]).
LetΩ be any non-empty-set and w≥1 be a real-number. A function d:Ω×Ω⟶R+ is called b-metric, if axioms given below are fulfilled for all μ,ν,ξ∈Ω:(1)
dμ,ν≥0anddν,μ=0iffμ=ν(2)
dμ,ν=dν,μ(3)
dμ,ξ≤wdμ,ν+dν,ξ
Then,Ω,d is called b-MS.
If we takew=1, then b-MS becomes ordinary MS. Hence, set of all MSs is a subset of set of all b-MSs.Example 1 (see [5]).
The setlp with 0<p<1, where lp=xn⊂ℝ:∑ι=1∞μιp<∞, together with the function d:lp×lp⟶[0,∞),(1)dμ,ν=∑ι=1∞μι−νιp1/p,where μ=μι,ν=νι∈lp is b metric space with w=21/p>1.) Notice that the abovementioned result holds with 0<p<1 .Definition 2 (see [5]).
LetΩ,d be b-MS and zn be a sequence in Ω. Then,(1)
zn is called a convergent sequence if there exist z∈Ω, such that for all ε>0∃n0ε∈ℕ such that for all n≥n0ε, we have dzn,z<ε. Then, we write limn⟶∞zn=z.(2)
zn is said to be a Cauchy sequence if for all ε>0∃n0ε∈ℕ such that for each n≥n0ε, we have dzn,z<ε.(3)
Ω is called complete if every Cauchy sequence in Ω is convergent in Ω.Definition 3 (see [21]).
SupposeΩ,d be a b-MS, CBΩ be the set of non-empty closed and bounded subsets of Ω, and CLΩ be the set of all non-empty closed subsets of Ω. For z∈Ω and A,B∈CLΩ, we define,(2)dz,A=infa∈Adz,a,dA,B=infda,B:a∈A.
LetΩ,d be a b-MS. Hausdorff b-metric can be defined on CBΩ induced by d as(3)HA,B=maxsupu∈Adu,B,supv∈BdA,v,for all A,B∈CBΩ.Lemma 1 (see [16]).
LetΩ,d be a b-MS and A,B∈CBΩ,(i)
Ifa∈A, then da,B≤HA,B(ii)
ForA,B∈CBΩ and 0<δ∈R. Then, for a∈A there exists b∈B such that(4)da,b≤HA,B+δDefinition 4 (see [16]).
Apartially ordered set (poset) is a set X with binary relation ≺¯ such that for all a,b,c∈X;(1)
a≺a (reflexive)(2)
a≺b and b≺a implies a=b (antisymmetric)(3)
a≺b and b≺c implies a≺c (transitivity)Definition 5 (see [16]).
A posetL,≺L is said to be a(1)
Lattice; if r∧s∈L,r∨s∈L for any r,s∈L.(2)
Complete lattice; if ∨B∈L,∧B∈L for any B⊆L.(3)
Distributive lattice; if r∨s∧t=r∨s∧r∨t,r∧s∨t=r∧s∨r∧t for any r,s,t∈L.(4)
Complete distributive lattice; if r∨∧isi=∧ir∨sir∧∨isi=∨ir∧si for any ri,si∈L.(5)
Bounded lattice; if it is a lattice along with a maximal element 1L and a minimal element 0L, which satisfy 0L≺Lx≺L1L for every x∈L.Definition 6 (see [1]).
A functionW:Ω⟶0,1 is known as FS on a nonempty set ΩDefinition 7 (see [2]).
AnL-FS W on a nonempty set Ω is a function W:Ω⟶L, where L is a bounded complete distributive lattice along with 1L and 0L.Remark 1.
The class ofL-FSs is larger than the class of FSs.Definition 8 (see [16]).
TheγL-level set of an L-FS W is defined as (5)WγL=m∈Ω:γL≺LWm,forγL∈L0L,W0L=m∈Ω:0L≺LWm¯,where D¯ is the closure of the set D (crisp set).
LetFLΩ be the collection of all L-FSs in Ω.Definition 9 (see [21]).
LetΩ1 be any set, Ω2 be a MS. A mapping T is called L-FM, if
T:Ω1⟶FLΩ2. An L-FM T is an L-FS on Ω1×Ω2 with membership function Txy The image Txy is the grade of membership of y in Tx.Definition 10 (see [16]).
LetΩ,d be b-MS and T:Ω⟶FLΩ be an L-FM. A point z∈Ω is the αL-FFP of T if z∈TzαL for some αL∈L\0L.
Now, forx∈Ω,A,B∈FLΩ,αL∈L\0L and AαL,BαL∈CBΩ we define dx,S=infdx,a;a∈S; here, S is a subset of Ω:(6)pαLx,A=infdx,a.;a.∈AαL,pαLA,B=infda,b.;a∈AαL,b.∈BαL,pA,B=supαLpαLA,B,HAαL,BαL=maxsupa∈AαLda,BαL,supb∈BαLdb,AαL.Remark 2.
The functionH:CBΩ×CBΩ⟶ℝ is a Hausdorff b-metric, where Ω is a b-MS and CBΩ is the set of all closed and bounded subsets of Ω.Definition 11 (see [16]).
LetΨb be the class of strictly increasing functions, ψ:[0,∞)⟶[0,∞) such that ∑n=0∞snψnt<+∞ for each t>0, where ψn is the nth iterate of ψ. It is known that for each ψ∈Ψb, we have ψt<t for all t>0 and ψ0=0 for t=0.


## 2.αL-Fuzzy Fixed Points

In this section, we have obtained two different results to findαL-fuzzy fixed points (FFP) in complete b-metric spaces (MS) and established significant examples to validate our results.Theorem 1.
Let (Ω, d) be a complete b-MS with constantb≥1. Let T:Ω⟶ FLΩ be an L-FM and for x,y∈Ω,∃αLx,αLy,ϵL\0L such that TxαLxandTyαLynon−empty and belong to CBΩ satisfying the following condition:(7)HTxαLx,TyαLy≤a1dx,TxαLx+a2dy,TyαLy+a3dx,TyαLy+a4dy,TxαLx+a5dx,y+a6dx,TxαLx1+dx,TxαLx1+dx,y.
Also,ai≥0, where i=1,2,3,…,6 with a1+a2+2ba3+a5+a6<1, and ∑i=16ai<1. Then, T has an αL-FFP.Proof.
Letxo be an arbitrary point in Ω, since TxoαLx0 is nonempty, so there exists x1∈TxoαLx0 and x2∈Tx1αLx1 and so on.
BecauseTxoαLx0 and Tx1αLx1 are closed and bounded subsets of Ω.
By Lemma1,(8)dx1,x2≤HTxoαLx0,Tx1αLx1+a1+ba3+a5+a6,dx1,x2≤a1dxo,TxoαLx0+a2dx1,Tx1αLx1+a3dxo,Tx1αLx1+a4dx1,TxoαLx0+a5dxo,x1+a6dxo,TxoαLx01+dxo,TxoαLx01+dxo,x1+a1+ba3+a5+a6,dx1,x2≤a1dxo,x1+a2dx1,x2+a3dxo,x2+a4dx1,x1+a5dxo,x1+a6dxo,x11+dxo,x11+dxo,x1+a1+ba3+a5+a6,dx1,x2≤a1dxo,x1+a2dx1,x2+a3dxo,x2+a5dxo,x1+a6dxo,x1+a1+ba3+a5+a6,dx1,x2≤a1dxo,x1+a2dx1,x2+ba3dxo,x1+ba3dx1,x2+a5dxo,x1+a6dxo,x1+a1+ba3+a5+a6,dx1,x2≤a1+ba3+a5+a6dxo,x1+a2+ba3dx1,x2+a1+ba3+a5+a6,1−a2+ba3dx1,x2≤a1+ba3+a5+a6dxo,x1+a1+ba3+a5+a6,dx1,x2≤a1+ba3+a5+a61−a2+ba3dxo,x1+a1+ba3+a5+a61−a2+ba3.
Leta1+ba3+a5+a6/1−a2+ba3=γ So,(9)dx1,x2≤γdxo,x1+γ.
Again, sincex2∈Tx1αLx1 and x3∈Tx2αLx2 are bounded and closed subsets of Ω. So,
By Lemma1,(10)dx2,x3≤HTx1αLx1,Tx2αLx2+a1+ba3+a5+a621−a2+ba3,dx2,x3≤a1dx1,Tx1αLx1+a2dx2,Tx2αLx2+a3dx1,Tx2αLx2+a4dx2,Tx1αLx1+a5dx1,x2+a6dx1,Tx1αLx11+dx1,Tx1αLx11+dx1,x2+a1+ba3+a5+a621−a2+ba3,dx2,x3≤a1dx1,x2+a2dx2,x3+a3dx1,x3+a4dx2,x2+a5dx1,x2+a6dx1,x21+dx1,x21+dx1,x2+a1+ba3+a5+a621−a2+ba3,dx2,x3≤a1dx1,x2+a2dx2,x3+a3dx1,x3+a5dx1,x2+a6dx1,x2+a1+ba3+a5+a621−a2+ba3dx2,x3≤a1+ba3+a5+a6dx1,x2+a2+ba3dx2,x3+a1+ba3+a5+a621−a2+ba3,1−a2+ba3dx2,x3≤a1+ba3+a5+a6dx1,x2+a1+ba3+a5+a621−a2+ba3,dx2,x3≤a1+ba3+a5+a61−a2+ba3dx1,x2+a1+ba3+a5+a621−a2+ba32,dx2,x3≤γdx1,x2+γ2.
By using inequality (9), we get(11)dx2,x3≤γγdxo,x1+γ+γ2,dx2,x3≤γ2dxo,x1+γ2+γ2,dx2,x3≤γ2dxo,x1+2γ2.
Continuing in this way by induction, we obtain a sequencexn, such that xn−1∈TxnαL and xn∈Txn+1αL, we have(12)idxn,xn+1≤γndxo,x1+nγn.
Now, for positive integersm,n, and n>m, we have(13)dxm,xn≤bdxm,xm+1+dxm+1,xm+2+…+dxn−1,xn,dxm,xn≤bγmdxo,x1+mγm+γm+1dxo,x1+m+1γ+…+γn−1dxo,x1+n−1γn−1,dxm,xn≤bγm+γm+1+…+γn−1dxo,x1+∑i=mn−1iγi.
Becauseγm+γm+1+…+γn−1 is a geometric series with rcommonratio=γ<1, it is hence convergent. So, we can write above inequality as follows:(14)dxm,xn≤bγm1−γdxo,x1+∑i=mn−1iγi.
Asγ<1, and for m,n⟶∞, the series ∑i=mn−1iγi converges by the Cauchy root test so,(15)dxm,xn⟶0.
Hence,xn is the Cauchy sequence in Ω. As Ω is complete. So, ∃z∈Ω such that xn⟶z as n⟶∞.
Now, we show thatz is a L-fuzzy fixed point. For this, consider(16)dz,TzαL≤bdz,xn+1+dxn+1,TzαL,dz,TzαL≤bdz,xn+1+HTxnαL,TzαL,dz,TzαLz≤bdz,xn+1+a1dxn,TxnαLxn+a2dz,TzαLz+a3dxn,TzαLz+a4dz,TxnαLxn+a5dxn,z+a6dxn,TxnαLxn1+dxn,TxnαLxn1+dxn,z,dz,TzαLz≤bdz,xn+1+a1dxn,xn+1+a2dz,TzαLz+a3dxn,TzαLz+a4dz,xn+1+a5dxn,z+a6dxn,xn+11+dxn,xn+11+dxn,z,dz,TzαLz≤bdz,xn+1+a1dxn,xn+1+a2dz,TzαLz+ba3dxn,z+ba3dz,TzαLz+a4dz,xn+1+a5dxn,z+a6dxn,xn+11+dxn,xn+11+dxn,z.
Alson⟶∞,(17)dz,TzαLz≤ba2dz,TzαLz+ba3dz,TzαLz,dz,TzαLz≤ba2+ba3dz,TzαLz,1−ba2+ba3dz,TzαLz≤0.
As1−ba2+ba3≠0. So, only possibility is dz,TzαLz=0.
Hence,
z∈TzαLz.
So,z is αL-FFP of T.Example 2.
LetΩ=a,a+1 where a∈R. Define d:Ω×Ω⟶R+ by dω,μ=ω−μ. Define L-fuzzy mapping T:Ω⟶FLΩ as(18)Tωt=a+1,a<t≤ω4,a+12,ω4<t≤ω3,a+14,ω3<t≤ω2,a,ω2<t≤a+1.
For allω,μ∈Ω. Here, L=a,a+1 where a∈R. Moreover, for all ω∈Ω,∃αLω=αLμ=a+1 such that TωαL=a,ω/4. Hence, for μ∈Ω,TμαL=a,μ/4.
As(19)HTωαL,TμαL=μ4−ω4,(20)dω,TωαL=ω−ω4,(21)dμ,TμαL=μ−μ4,(22)dω,μ=ω−μ,(23)dω,TμαL=ω−μ4,(24)dμ,TωαL=μ−ω4.
From (19)–(24), we have(25)HTωαL,TμαL≤15ω−ω4+110μ−μ4+115ω−μ4+120μ−ω4+125ω−μ+130⋅ω−ω/41+ω−ω/41+ω−μ.
Hence, all the conditions of the above theorem are satisfied. There exists abL-fuzzy fixed point in complete b-metric. So, a∈Ω is an αL-FFP of T.Corollary 1.
Let (Ω, d) be a complete b-MS with constantb≥1. Let T: Ω ⟶ FΩ be a FM and for x,y∈Ω,∃αx,αy)ε(0,1] such that TxαxandTyαybenon−empty and belongs to CBΩ satisfying the following condition:(26)HTxαx,Tyα(y)≤a1dx,Txαx+a2dy,Tyα(y)+a3dx,Tyα(y)+a4dy,Txαx+a5dx,y+a6dx,Txαx1+dx,Txαx1+dx,y.
Alsoai≥0, where i=1,2,3,…,6 with a1+a2+2ba3+a5+a6<1 and ∑i=16ai<1. Then, T has a FFP.Corollary 2.
Let (Ω,d) be a complete MS. Let T: Ω⟶ FLΩ be an L-FM and for x,y∈Ω,∃αLx,αLy,ϵL/0L such that TxαLandTyαLbenon−empty and belong to CBΩ satisfying the following condition:(27)HTxαLx,TyαLy≤a1dx,TxαLx+a2dy,TyαLy+a3dx,TyαLy+a4dy,TxαLx+a5dx,y+a6dx,TxαLx1+dx,TxαLx1+dx,y.
Alsoai≥0, where i=1,2,3,…,6 with a1+a2+2a3+a5+a6<1 and ∑i=16ai<1. Then, T has an αL-FFP.Corollary 3.
Let (Ω, d) be a complete MS. Let T: Ω⟶FΩ be a FM and for x,y∈Ω,∃αxαy∈(0,1] such that TxαxandTyαybenon−empty and belongs to CBΩ satisfying the following condition:(28)HTxαx,Tyαy≤a1dx,Txαx+a2dy,Tyαy+a3dx,Tyαy+a4dy,Txαx+a5dx,y+a6dx,Txαx1+dx,Txαx1+dx,y.
Alsoai≥0, where i=1,2,3,…,6 with a1+a2+2a3+a5+a6<1 and ∑i=16ai<1. Then, T has a FFP.Theorem 2.
LetΩ,d be a complete b-MS with coefficient s≥1. Let T:Ω⟶FLΩ be L-fuzzy mapping and for all x,y∈Ω, αLx,αLyϵL/0L,TxαLxandTyαy be nonempty closed and bounded subsets of Ω. Suppose T satisfies the following multivalued contraction(29)HTxαLx,TyαLy≤ψdx,ywhere ψ∈Ψb. Then, T has an αL-FFP.Proof.
Letω0 be an arbitrary point in Ω. Suppose that there exists ω1∈Tω0αLω0. Because Tω1αLω1 is a nonempty closed and bounded subset of Ω.Case 1.
Ifω0=ω1, then ω1=ω0∈Tω0αLω0. Hence, ω0 is the required αL-FFP of T.Case 2.
Ifω1∈Tω1αLω1. Then, ω1αL-FFP of T.Case 3.
Now, we assume thatω0≠ω1 and ω1∈Tω0αLω0. so,(30)0<dω1,Tω1αLω1≤HTω0αLω0,Tω1αLω1≤ψdω0,ω1,0<dω1,Tω1αLω1<ψrdω0,ω1,∀r>1.
Because,Tω1αLω1 is a nonempty closed and bounded subset of Ω. Suppose there exists
ω2∈Tω1αLω1 and ω1≠ω2 such that(31)0<dω1,ω2≤ψdω0,ω1≤ψrdω0,ω1.
BecauseTω2αLω2 is a nonempty closed and bounded subset of Ω. We assume that ω2 does not belong Tω2αLω2. Then,(32)0<dω2,Tω2αLω2≤HTω1αLω1,Tω2αLω2≤ψdω1,ω2,0<dω2,Tω2αLω2≤ψdω1,ω2.
From (31),(33)0<dω2,Tω2αLω2≤ψdω1,ω2≤ψ2dω0,ω1,0<dω2,Tω2αLω2≤ψdω1,ω2≤ψ2dω0,ω1<ψ2rdω0,ω1,0<dω2,Tω2αLω2<ψ2rdω0,ω1.
Suppose that there existsω3∈Tω2αLω2 and ω2≠ω3 such that(34)0<dω2,ω3≤ψdω1,ω2<ψ2rdω0,ω1.
By induction, we can construct a sequenceωn in Ω, such that ωn does not belongs to TωnαLωn and ωn+1∈TωnαLωn and(35)0<dωn,TωnαLωn≤HTωn−1αLωn−1,TωnαLωn≤ψdωn−1,ωn≤ψndω0,ω1,0<dωn,TωnαLωn≤ψndω0,ω1<ψnrdω0,ω1,∀n∈N.
Now, form,n∈N with m>n, we have(36)dωn,ωm≤sdωn,ωn+1+dωn+1,ωm,dωn,ωm≤sdωn,ωn+1+sdωn+1,ωm,dωn,ωm≤sdωn,ωn+1+ssdωn+1,ωn+2+dωn+2,ωm,dωn,ωm≤sdωn,ωn+1+s2dωn+1,ωn+2+s2dωn+2,ωm.
Similarly, in this way(37)dωn,ωm≤sdωn,ωn+1+s2dωn+1,ωn+2+s3dωn+2,ωn+3+…+sm−ndωm−1,ωm,dωn,ωm≤sψnrdω0,ω1+s2ψn+1rdω0,ω1+s3ψn+2rdω0,ω1+…+sm−nψm−1rdω0,ω1,dωn,ωm≤1sn−1snψnrdω0,ω1+sn+1ψn+1rdω0,ω1+sn+2ψn+2rdω0,ω1+…+sm−1ψm−1rdω0,ω1,dωn,ωm≤1sn−1∑i=0m−1siψirdω0,ω1.
Becauseiψ∈Ψ, we know that the series ∑i=0m−1siψirdω0,ω1 converges.
Forn⟶∞,dωn,ωm⟶0.
Hence,ωn is a Cauchy sequence in Ω. By the completeness of Ω, there exists ω∗∈Ω such that liml⟶∞ωn=ω∗. We claim that ω∗∈Tω∗αLω∗.
Because(38)dω∗,Tω∗αLω∗≤sdω∗,ωn+1+dωn+1,Tω∗αLω∗,dω∗,Tω∗αLω∗≤sdω∗,ωn+1+HTωnαLωn,Tω∗αLω∗,dω∗,Tω∗αLω∗≤sdω∗,ωn+1+ψdωn,ω∗.
Asn⟶∞,andψ0=0, which implies that(39)dω∗,Tω∗αLω∗=0.
AsTω∗αLω∗ is a closed and bounded subset of Ω. So, ω∗∈Tω∗αLω∗.
Hence,ω∗ is αL-FFP of T.Example 3.
LetΩ=0,1,2. Define metric d on Ω as follows:(40)dω,μ=0,ifω=μ,16,ifω≠μandω,μ∈0,1,12,ifω≠μandω,μ∈0,2,1,ifω≠μandω,μ∈1,2.
It is a complete b-MS with co-efficient3/2.
Define an L-FM as
T:X⟶FLΩ where L=a,b.(41)T0t=T1t=b2,t=0,a,t=1,2,T2t=a,t=0,b2,t=1,2.
DefineαL:Ω⟶L/a by αLω=b/2,∀ω∈Ω.
Now, we obtain(42)Tωb/2=0ω=0,11ω=2
Forω,y∈Ω, we get(43)HT0b/2,T1b/2=maxSupω∈Tωb/2dω,Tμb/2,Supμ∈Tμb/2dμ,Tωb/2.
Defineψ:[0,∞)⟶[0,∞) by ψt=1/3t∀t>0. Thus, we have(44)HT0b/2,T1b/2=0<118=1316=13d0,1,HT0b/2,T2b/2=16=1312=13d0,2,HT1b/2,T2b/2=16<13⋅1=13d1,2.
Therefore, all the conditions of theorem 2 are satisfied. Hence,∃ a point 0∈Ω such that
0∈T0b/2 is an αL-FFP of T.Corollary 4.
LetΩ,d be a complete b-MS with coefficient s≥1. Let T:Ω⟶FΩ be fuzzy mapping and for all x,y∈Ω and for each αx,αyε(0,1],TxαxandTyαy be nonempty closed and bounded subsets of Ω. Suppose T satisfies the following multivalued contraction:(45)HTxαx,Tyαy≤ψdx,y,where ψ∈Ψb. Then, T has an α-FFP.Corollary 5.
LetΩ,d be a complete MS. Let T:Ω⟶FL, for all x,y∈Ω and for each αLx,αLyεL/0L,TxαLx and TyαLy be a nonempty closed and bounded subset of Ω. Suppose T satisfies the following multivalued contraction:(46)HTxαLx,TyαLy≤ψdx,y,where ψ∈Ψ. Then, T has an αL-FFP.Corollary 6.
LetΩ,d be a complete MS. Let T:Ω⟶FΩ be fuzzy mapping for all x,y∈Ω and for each αx,αyε(0,1],Txαx and Tyαy be a nonempty closed and bounded subset of Ω. Suppose T satisfies the following multivalued contraction:(47)HTxαx,Tyαy≤ψdx,y,where ψ∈Ψ. Then, T has an α -FFP.Remark 3.
We have extended fixed point theorems having different contractive conditions toL-fuzzy mappings in complete b-metric spaces and obtained some corollaries as direct consequences of our main results. Fixed point theorems are widely used to obtain solutions of some initial value problems (Fredholm integral equations of 1st and 2nd kinds, Volterra integral equations) to find explicit form of implicit functions, etc. Our work will help to solve problems involving situations mentioned above.


## 3. Conclusion

In the case of the complete b-metric space, two fuzzy fixed point theorems forL-fuzzy mappings are established and proved for two diverse contractive type conditions. Nontrivial supporting examples for both results are also supplied to demonstrate the strength of these findings. Our results give uniqueness, extension, and sequential generalizations of many valuable current and conventional results in the literature using this approach. Some directions for more examinations and work are given in the form of open questions.(1)
Whether these results can be extended to more than one mapping?(2)
In case of complex-valued metric spaces, what type of contractive conditions will be feasible to find fixed points?



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