# On Fuzzy Fixed Points and an Application to Ordinary Fuzzy Differential Equations

Authors: Eskandar Ameer; Hassen Aydi; Muhammad Arshad
Journal: Journal of Function Spaces
 (2020)

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## Abstract

The aim of this paper is to obtain the common fuzz fixed points ofα-fuzzy mappings satisfying generalized almost Y,Λ-contraction in complete metric spaces. Our results are extensions and improvements of the several well-known recent and classical results in literature. We give an example for supporting these results. As an application, we apply our obtained results to study the existence of a solution for a second order nonlinear boundary value problem.


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## Body

## 1. Introduction and Preliminaries

The fixed point result of Banach [1] is an interesting tool that ensures the existence and uniqueness of a fixed point of self-mappings defined on metric spaces. Later, various extensions and generalizations of Banach’s theorem appeared by defining a variety of contractive type conditions for self and non-self-mappings on different spaces (see [2–27]).In [17, 18], Berinde discussed many contraction type mappings. He initiated the notion of almost contractions.Definition 1.
A self-mappingS^ on a metric space (in short MS) ω,∂ is named as an almost contraction, if there are λ∈0,1 and L≥0 so that for all r˘,J^∈ω,
(1)∂S^r˘,S^J^<λ∂r˘,j^+L∂j^,S^j^.
This almost contraction has been generalized as follows.Definition 2.
A self-mappingS^ on a MS ω,∂ is called a generalized almost contraction, if there are λ∈0,1and L≥0 so that for all r˘,J^∈ω,
(2)∂S^r˘,S^J^<λ∂r˘,J^+Lmin∂υ,S^υ,∂ι,S^ι,∂υ,S^ι,∂ι,S^υ.
Wardowski [28] presented the notion of F-contractions and established a related fixed point theorem.Definition 3 (see [28]).
A self-mappingS^ on a MS ω,∂ said to be an F-contraction if there are F∈F and τ>0 such that
(3)∀r˘,J^∈ω,∂S^r˘,S^J^>0⇒τ+F∂S^r˘,S^J^≤F∂r˘,J^,where F is the set of functions F:0,∞⟶−∞,∞ such that
(F1) For all0<r˘<J^, Fr˘<FJ^,
(F2) For each sequencetnn=1∞ of positive numbers,
(4)limn→∞Ft=−∞ifandonlyiflimn→∞tn=0
(F3) There isk∈0,1 so that limt→0+tkFt=0,
Later on, Altun et al. [22] modified Definition 3 by adding the condition (F4):
(F4)FinfP=infFP, for all P⊂0,∞ with infP>0.
Denote byF∗ the set of functions F:0,∞⟶−∞,∞ verifying the conditions F1-F4.Theorem 4 (see [28]).
LetS^:ω⟶ω be an F-contraction on a complete MS. Then, T admits a unique fixed point s∗∈ω, and for each r˘∈ω, the sequence Tjr˘j∈ℕ is convergent to r˘∗.Definition 5 (see [4]).
A self-mappingS^on a MS ω,∂ is said to be a θ-contraction, if there are ℓ∈0,1 and θ∈Θ so that
(5)r˘,J^∈ω,∂S^r˘,S^J^≠0⇒θ∂S^r˘,S^J^≤θ∂r˘,J∧ℓ,where Θ is the set of functions θ:0,∞⟶1,∞ so that
(Θ1) θ is nondecreasing,
(Θ2) For each sequence tn⊂0,∞,
(6)limn→∞θtn=1ifandonlyiflimn→∞tn=0+,
(Θ3) There are r∈0,1 and ℏ∈0,∞ such that limt→0+θt−1/tr=ℏ,
(Θ4) θ is continuous.
The following result was established by Jleli and Samet [4].Theorem 6 (see [4]).
Eachθ-contraction mapping on a complete MS ω,∂ possesses a unique fixed point.
As in [22], Hancer et al. [12] took the following:
(Θ5) θinfQ=infθQ, for all Q⊂0,∞ with infQ>0.
Denote byΘ∗ the set of functions Q:0,∞⟶1,∞ so that Θ1,Θ2,Θ3, and Θ5 hold.
As in [9], we denote by Ξ the set of functions θ:0,∞⟶1,∞ so that
(Θ1)′θ is nondecreasing,
(Θ2)′inft∈0,∞θt=1,
(Θ3)′θ is continuous.Theorem 7 (see [9]).
LetS^ be a self-mapping on a MS ω,∂. We have the equivalence of the following:
(i)
S^ is a θ-contraction with θ∈Ξ;,(ii)
S^ is a F-contraction with F∈F.
Very recently, Liu et al. [9] introduced the notion of (Y,Λ)-type contractions and established new fixed point theorems for such kind of mappings in complete MS.Definition 8.
A self-mappingS^ on a MS ω,∂ is said to be a Y,Λ-type contraction, if there are a comparison function Yandϕ∈Φ such that for all r˘,J^∈ω,
(7)∂S^r˘,S^J^>0⇒Λ∂S^r˘,S^J^≤YΛMr˘,J^,where
(8)Mr˘,J^=max∂r˘,J^,∂r˘,S^r˘,∂J^,S^J^,∂r˘,S^J^+∂J^,S^r˘2,and Φ is the set of functions Λ:0,∞⟶0,∞ so that
(Φ1) Λ is nondecreasing,
(Φ2) For each sequence tn⊂0,∞,
(9)limn→∞Λtn=0ifflimn→∞tn=0,
(Φ3) Λ is continuous.
As in [2], a function Y:0,∞⟶0,∞ is named as a comparison function if
(1)
Y is increasing, that is, t1<t2⇒Yt1≤Yt2,(2)
limn→∞nYt=0 for all t>0, where Yn stands for the nth iterate of Y.
Denote byΨ the set of comparison functions.Lemma 9 (see [9]).
LetΛ:0,∞⟶0,∞ be a nondecreasing and continuous function with inft∈0,∞Λt=0, and tkk be a sequence in 0,∞. Then,
(10)limk→∞Λtk=0ifflimk→∞tk=0.Example 10 (see [2]).
Consider the following comparison functions:(1)
Yt=λt with 0<λ<1,
(11)Yt=t2,0<t<1,t3,1≤t,Yt=tt+1.Example 11 (see [9]).
For allt∈0,∞, consider
(12)Λ1η=t,Λ2η=tt,Λ3η=tet.
Here,Λ1,Λ2,Λ3∈Φ.On the other hand, using the notion of a fuzzy set, Heilpern [29] initiated the family of fuzzy mappings, which generalized the concept of set-valued mappings, and presented a fixed point result for fuzzy contraction mappings in the context of metric linear spaces. Mention that the theorem of Heilpern [29] is considered as a fuzzy extension of the BCP. Later on, several researchers worked on fixed point results involving fuzzy mappings, see ([30–46]).LetCBω be the set family of bounded and closed subsets of a MS ω,∂. For r˘∈ω and A,B∈CBω, set
(13)∂r˘,B=infJ^∈B∂r˘,J^.GivenH:CBω×CBω⟶0,1 as
(14)HA,B=maxsupr˘∈A∂r˘,B,supJ^∈B∂J^,A.H is a metric on CBω. It is named as the Pompeiu-Hausdorff metric induced by ∂. A fuzzy set in ω is a function with domain ω and values in 0,1. Denote by Iω the set fuzzy sets in ω. Let A be a fuzzy set and r˘∈ω, then the function values Ar˘ is named as the grade of membership of r˘ in A. The α-level set of A is denoted by Aα and is defined by
(15)Aα=ω:Λr˘≥αifα∈0,1andA0=ω:Λr˘>0¯.Here,B¯ denotes the closure of the set B. Let Iω be the collection of all fuzzy sets in ω. For A,B∈Iω, A⊂B means Ar˘≤Br˘ for each r˘∈ω. The fuzzy set χr˘ is denoted by r˘ unless and until it is stated, where χr˘ is the characteristic function of the crisp set A. If there is α∈0,1 so that Aα,Bα∈CBω, then consider
(16)PαA,B=infr˘∈Aα,J^∈Bα∂r˘,J^,DαA,B=HAα,Bα.IfAα,Bα∈CBω for each α∈0,1, then take
(17)PA,B=supαPαA,B,∂∞A,B=supαDαA,B.We writepr˘,B instead of pr˘,B. A fuzzy set A in a metric linear space V is said to be an approximate quantity if and only if Aα is compact and convex in V for each α∈0,1 and supr˘∈VAr˘=1. The set of approximate quantities in V is denoted by WV. Let ω be an arbitrary set and Y be a MS. A mapping S^ is called fuzzy mapping (in short, FM) if S^ is a mapping from ω into IY. A fuzzy mapping S^ is a fuzzy subset on ω×Y with the membership function S^r˘J^. The function S^r˘J^ is the grade of membership of J^ in S^r˘.Definition 12.
LetS^,Γ be FM from ω into IY. An element σ in ω is said to be an α-fuzzy fixed point of Γ if there is α∈0,1 so that σ∈Γuα. σ∈ω is named as a common α-fuzzy fixed point of S^ and Γ if there is α∈0,1 so that σ∈S^uα∩Γσα. α=1 is called a common fixed point of fuzzy mappings.In this manuscript, we ensure the existence of some commonα-fuzzy fixed points for fuzzy mappings for almost Y,Λ-contractions in the class of complete MS. Our theorems generalize some known results in literature.Now, we need the following.Lemma 13 (see [16]).
Letω,∂ be a MS and A,B∈CBω, then for every η∈A,
(18)∂η,B≤HA,B.Lemma 14 (see [30]).
LetV be a metric linear space and S^:V⟶WV be a fuzzy mapping and ρ0∈V. Then, there is ρ1∈V so that ρ1⊂S^ρ0.Lemma 15 (see [47]).
Letω,∂ be a metric space, r˘∗∈ω and S^,Γ:ω⟶Iω be FM such that S⌢r˘ is a nonempty compact set for each r˘∈ω. Then, r˘∗∈S^r˘∗ if and only if S^r˘∗r˘∗≥S^r˘∗r˘ for each r˘∈ω.


## 2. Main Results

From now on,ω,∂ is assumed to be a complete MS.Theorem 16.
LetS^,Γ:ω⟶Iω be FM and for each r˘∈ω, and there exist αS^r˘, αΓr˘∈0,1 such that S^r˘αS^r˘ and ΓJ^αΓr˘ are nonempty, closed, and bounded subsets of ω. Assume that there exist Y∈Ψ, Λ∈Φ, and L≥0 such that for all r˘,J^∈ω,HS^r˘αS^r˘,ΓJ^αΓJ^>0 implies
(19)ΛHS^r˘αS^r˘,ΓJ^αΓJ^≤YΛMr˘,J^+LEr˘,J^,where
(20)Mr˘,J^=max∂r˘,J^,∂r˘,S^r˘αS^r˘,∂J^,ΓJ^αΓJ^,∂r˘,ΓJ^αΓJ^+∂J^,S^r˘αS^r˘2,(21)Er˘,J^=min∂r˘,S^r˘αS^r˘,∂J^,ΓJ^αΓJ^,∂r˘,ΓJ^αΓJ^,∂J^,S^r˘αS^r˘.
IfY is continuous, then there exists some u∈ω such that u∈S^uαS^u∩ΓuαΓu.Proof.
Letr˘0∈ω. By hypotheses, there exists αS^r˘0∈0,1 such that S^r˘0αS^r˘0 is a nonempty, closed, and bounded subset of ω. For convenience, we denote αS^r˘0 by α1. Let r˘1∈S^r˘0αS^x0, then there exists αΓr˘1∈0,1 such that Γr˘1αΓr˘1 is a nonempty, closed, and bounded subset of ω. Since Λ is nondecreasing, we have used (19) and Lemma 13(22)Λ∂r˘1,Γr˘1αΓr˘1≤ΛHS^r˘0αS^r˘0,Γr˘1αΓr˘1≤YΛMr˘0,r˘1+LEr˘0,r˘1,where
(23)Mr˘0,r˘1=max∂r˘0,r˘1,∂r˘0,S^r˘0αS^r˘0,∂r˘1,Γr˘1αΓr˘1,∂r˘0,Γr˘1αΓr˘1+∂r˘1,S^r˘0αS^r˘02,(24)Er˘0,r˘1=min∂r˘0,S^r˘0αS^r˘0,∂r˘1,Γr˘1αΓr˘1,∂r˘0,Γr˘1αΓr˘1,∂r˘1,S^r˘0αS^r˘0.
ByΦ3, we have
(25)Λ∂r˘1,Γr˘1αΓr˘1=infJ^∈Γr˘1αΓr˘1Λ∂r˘1,J^.
Thus, there existsr˘1∈Γr˘1αΓr˘1 such that
(26)Λ∂r˘1,Γr˘1αΓr˘1=Λ∂r˘1,r˘2.
Then from (22), we have
(27)Λ∂r˘1,r˘2≤YΛMr˘0,r˘1+LEr˘0,r˘1,where
(28)Mr˘0,r˘1=max∂r˘0,r˘1,∂r˘0,r˘1,∂r˘1,r˘2,∂r˘0,r˘2+∂r˘1,r˘12=max∂r˘0,r˘1,∂r˘1,r˘2,(29)Er˘0,r˘1=min∂r˘0,r˘1,∂r˘1,r˘2,∂x0,x2,∂r˘1,r˘1=0.
Ifmax∂r˘0,r˘1,r˘1,r˘2=∂r˘1,r˘2, then from (27), we have
(30)Λ∂r˘1,r˘2≤YΛ∂r˘1,r˘2<Λ∂r˘1,r˘2,which is a contradiction. Thus, max∂r˘0,r˘1,r˘1,r˘2=∂r˘0,r˘1. By (27), we get that
(31)Λ∂r˘1,r˘2≤YΛ∂r˘0,r˘1.
Next, there existsαS^r˘2∈0,1 such that S^r˘2αS^r˘2 is a nonempty, closed, and bounded subset of ω. By Lemma 13, using (19) and the fact that Λ is nondecreasing, we have
(32)Λ∂r˘2,S^r˘2αS^r˘2≤ΛHS^r˘2αS^r˘2,Γr˘1αΓr˘1≤YΛMr˘1,r˘2+LEr˘1,r˘2,where
(33)Mr˘1,r˘2=max∂r˘1,r˘2,∂r˘2,S^r˘2αS^r˘2,∂r˘1,Γr˘1αΓr˘1,∂r˘2,Γr˘1αΓr˘1+∂r˘1,S^r˘2αS^r˘22,(34)Er˘1,r˘2=min∂r˘2,S^r˘2αS^r˘2,∂r˘1,Γr˘1αΓr˘1,∂r˘2,Γr˘1αΓr˘1,∂r˘1,S^r˘2αS^r˘2.
FromΦ4, we have
(35)Λ∂r˘2,S^r˘2αS^r˘2=infJ^1∈S^r˘2αS^r˘2Λ∂r˘2,J^1.
Thus, there existsr˘3∈S^r˘2αS^r˘2 such that
(36)Λ∂r˘2,S^r˘2αS^r˘2=Λ∂r˘2,r˘3.
Then from (32), we have
(37)Λ∂r˘2,r˘3≤YΛMr˘1,r˘2+LEr˘1,r˘2,where
(38)Mr˘1,r˘2=max∂r˘1,r˘2,∂r˘2,r˘3,dr˘1,r˘2,∂r˘1,r˘3+∂r˘2,r˘22=max∂r˘1,r˘2,∂r˘2,r˘3,(39)Er˘1,r˘2=min∂r˘2,r˘3,∂r˘1,r˘2,∂r˘2,r˘2,∂r˘1,r˘3=0.
Ifmax∂r˘1,r˘2,r˘2,r˘3=∂r˘2,r˘3, then from (37), we have
(40)Λ∂r˘2,r˘3≤YΛ∂r˘2,r˘3<Λ∂r˘2,r˘3,which is a contradiction. Thus, maxr˘1,r˘2,r˘2,r˘3=∂r˘1,r˘2. By (37), we get that
(41)Λ∂r˘2,r˘3≤YΛ∂r˘1,r˘2.
By continuing this process, we construct a sequencer˘n in ω such that r˘2n+1∈S^r˘2nαS^r˘2n and r˘2n+2∈Γr˘2n+1αΓr˘2n+1(42)Λ∂r˘2n+1,r˘2n+2≤YΛ∂r˘2n,r˘2n+1,foralln∈ℕ.(43)Λ∂r˘2n+2,r˘2n+3≤YΛ∂r˘2n+1,r˘2n+2,foralln∈ℕ.
Thus, from (42) and (43), we have
(44)Λ∂r˘n,r˘n+1≤YΛ∂r˘n−1,r˘n,foralln∈ℕ.
This implies that(45)Λ∂r˘n,r˘n+1≤YΛ∂r˘n−1,r˘n≤Y2Λ∂r˘n−2,r˘n−1≤⋯≤YnΛ∂r˘0,r˘1.
Lettingn⟶∞, we get
(46)0≤limn→∞Λ∂r˘n,r˘n+1≤limn→∞YnΛ∂r˘0,r˘1=0.
It yields that(47)limn→∞Λ∂r˘n,r˘n+1=0.
This together withΦ2 and Lemma 9 gives that
(48)limn→∞∂r˘n,r˘n+1=0.
Now, we will prove thatr˘n is a Cauchy sequence. Arguing by contradiction, we assume that there are ε>0 and sequences pnn=1∞ and qnn=1∞ of integers so that for all n∈ℕ,pn>qn>n with dr˘pn,r˘qn≥ε,(49)∂r˘pn−1,r˘qn<ε.
Thus,(50)ε≤∂r˘pn,r˘qn≤∂r˘pn,r˘pn−1+∂r˘pn−1,r˘qn≤∂r˘pn,r˘pn−1+ε.
Lettingn⟶∞ in 2.13, we get
(51)limn→∞∂r˘pn,r˘qn=ε.
Again,(52)∂r˘pn,r˘qn≤∂r˘pn,r˘pn+1+∂r˘pn+1,r˘qn+1+∂r˘qn+1,r˘qn,(53)∂r˘pn+1,r˘qn+1≤∂r˘pn+1,r˘pn+∂r˘pn,r˘qn+∂r˘qn,r˘qn+1.
Takingn⟶∞ in (51) and (52), we get
(54)limn→∞∂r˘pn+1,r˘qn+1=ε.
From (48) and (51), we can choose an integer n0≥1 so that by (19), we get
(55)Λ∂r˘pn+1,r˘qn+1≤ΛHS^r˘pnαS^r˘pn,Γr˘qnαΓr˘qn≤YΛMr˘pn,r˘qn+LEr˘pn,r˘qn,where
(56)Mr˘pn,r˘qn=max∂r˘pn,r˘qn,∂r˘pn,S^r˘pnαS^r˘pn,∂r˘qn,Γr˘qnαΓr˘qn,∂r˘pn,Γr˘qnαΓr˘qn+∂r˘qn,S^r˘pnαS^r˘pn2≤max∂r˘pn,r˘qn,∂r˘pn,r˘pn+1,∂r˘qn,r˘qn+1,∂r˘pn,r˘qn+1+∂r˘qn,r˘pn+12,(57)Er˘pn,r˘qn=min∂r˘pn,S^r˘pnαS^r˘pn,∂r˘qn,Γr˘qnαΓr˘qn,∂r˘pn,Γr˘qnαΓr˘qn,∂r˘qn,S^r˘pnαS^r˘pn≤min∂r˘pn,r˘pn+1,∂r˘qn,r˘qn+1,∂r˘pn,r˘qn+1,∂r˘qn,r˘pn+1=0.
Lettingn⟶∞ in the above inequality, since Y and Λ are continuous and by using (48), (49), (51), and (54), we get
(58)Λε=limn→∞Λ∂r˘pn+1,r˘qn+1≤limn→∞YΛMr˘pn,r˘qn=YΛε<Λε.
It is a contradiction, sor˘n is Cauchy. Since ω is complete, r˘n converges to r˘∗∈ω, i.e., limn→∞dr˘n,r˘∗=0. We claim that r˘∗∈Γr˘∗αΓx∗. Assume that r˘∗∉Γr˘∗αΓx∗ (that is, ∂r˘∗,Γr˘∗αΓr˘∗>0), then there are n0∈ℕ and a subsequence r˘nk of r˘n so that ∂r˘2nk+1,Γr˘∗αΓr˘∗>0, for all nk≥n0. Since ∂r˘2nk+1,Γr˘∗αΓr˘∗>0, for all nk≥n0, so by Φ1, we have
(59)Λ∂r˘2nk+1,Γr˘∗αΓr˘∗≤ΛHS^r˘2nkαS^r˘2nk,Γr˘∗αΓr˘∗≤YΛMr˘2nk,r˘∗+LEr˘2nk,r˘∗,where
(60)Mr˘2nk,r˘∗=max∂r˘2nk,r˘∗,∂r˘2nk,S^r˘2nkαS^r˘2nk,∂r˘∗,Γr˘∗αΓr˘∗,∂r˘2nk,Γr˘∗αΓr˘∗+∂r˘∗,S^r˘2nkαS^r˘2nk2≤max∂r˘2nk,r˘∗,∂r˘2nk,r˘2nk+1,∂r˘∗,Γr˘∗αΓr˘∗,∂r˘2nk,Γr˘∗αΓr˘∗+∂r˘∗,r˘2nk+12,(61)Er˘2nk,r˘∗=min∂r˘2nk,S^r˘2nkαS^r˘2nk,∂r˘∗,Γr˘∗αΓr˘∗,∂r˘2nk,Γr˘∗αΓr˘∗,∂r˘∗,S^r˘2nkαS^r˘2nk≤min∂r˘2nk,r˘2nk+1,∂r˘∗,Γr˘∗αΓr˘∗,∂r˘2nk,Γr˘∗αΓr˘∗,∂r˘∗,r˘2nk+1.
Lettingk⟶∞ and using the continuity of Λ and Y, we have
(62)Λ∂r˘∗,Γr˘∗αΓr˘∗≤YΛ∂r˘∗,Γr˘∗αΓr˘∗+0<Λ∂r˘∗,Γr˘∗αΓr˘∗,which is a contradiction. Hence, ∂r˘∗,Γr˘∗αΓr˘∗=0, and r˘∗∈Γr˘∗αΓr˘∗. Similarly, one can easily prove that r˘∗∈S^r˘∗αS^r˘∗. Therefore, r˘∗∈S^r˘∗αS^r˘∗∩Γr˘∗Γr˘∗.Example 17.
Letω=0,1 endowed with the metric ∂r˘,J^=r˘−J^. Consider Λ,Y:0,∞⟶0,∞ as Λt=t and Yt=99t/100. Here, Λ,Y∈Φ∗ and Y are continuous. For α∈0,1, given S^,Γ:ω⟶Iωby(63)S^r˘t=α,0≤t≤r˘60α2,r˘60≤t≤r˘40α3,r˘40≤t≤r˘20α5,r˘20≤t≤1,(64)Γr˘t=α,0≤t≤r˘15α3,r˘15≤t≤r˘10α4,r˘10≤t≤r˘5α7,r˘5≤t≤1,such that
(65)S^r˘α=0,r˘60,Γr˘α=0,r˘15.
Forr˘,J^∈ω with HS^r˘αS^r˘,ΓJ^αΓJ^>0, we have
(66)ΛHS^r˘αS^r˘,ΓJ^αΓJ^≤YΛMr˘,J^+LEr˘,J^.where
(67)Mr˘,J^=max∂r˘,J^,∂r˘,S^r˘αS^r˘,∂J^,ΓJ^αΓJ^,∂r˘,ΓJ^αΓJ^+∂J^,S^r˘αS^r˘2,(68)Er˘,J^=min∂r˘,S^r˘αS^r˘,∂J^,ΓJ^αΓJ^,∂r˘,ΓJ^αΓJ^,∂J^,S^r˘αS^r˘.
Hence, all the conditions of Theorem16withL=1 are satisfied, and 0∈S0α∩T0α.Corollary 18.
LetS^,Γ be FM from ω into Iω, and for each r˘∈ω, there exist αS^r˘, αΓr˘∈0,1 such that S^r˘αS^r˘ and ΓJ^αΓJ^ are nonempty, closed, and bounded subsets of ω. Assume that there are Y∈Ψ and Λ∈Φ such that for all r˘,J^∈ω,(69)HS^r˘αS^r˘,ΓJ^αΓJ^>0⇒ΛHS^r˘αS^r˘,ΓJ^αΓJ^≤YΛMr˘,J^,where Mr˘,J^ is defined by (20). If Y is continuous, then there is u∈ω such that u∈S^uαS^u∩ΓuαΓu.Proof.
SetL=0 in Theorem 16.Corollary 19.
LetS^ be a FM from ω into Iω, and for each r˘∈ω, there exist αS^r˘∈0,1 such that S^r˘αS^r˘ is nonempty, closed, and bounded subsets of ω. Assume that there are Y∈Ψ, Λ∈Φ, and L≥0 such that for all r˘,J^∈ω,(70)HS^r˘αS^r˘,S^J^αS^J^>0⇒ΛHS^r˘αS^r˘,S^J^αS^J^≤YΛMr˘,J^+LEr˘,J^,where
(71)Mr˘,J^=max∂r˘,J^,Dr˘,S^r˘αS^r˘,DJ^,S^J^αS^J^,Dr˘,S^J^αS^J^+DJ^,S^r˘αS^r˘2,(72)Er˘,J^=minDr˘,S^r˘αS^r˘,DJ^,S^J^αS^J^,Dr˘,S^J^αS^J^,DJ^,S^r˘αS^r˘.
IfY is continuous, then there is u∈ω so that u∈S^uαS^u.Proof.
TakeS^=Γ in Theorem 16.
Now, we consider multivalued mappings.Theorem 20.
LetF,G:ω⟶CBω be multivalued mappings. Assume that there are Y∈Ψ, Λ∈Φ, and L≥0 such that for all r˘,J^∈ω,(73)HFr˘,GJ^>0⇒ΛHFr˘,GJ^≤YΛMr˘,J^+LEr˘,J^,where
(74)Mr˘,J^=max∂r˘,J^,∂r˘,Fr˘,∂J^,GJ^,∂r˘,GJ^+∂J^,Fr˘2,(75)Er˘,J^=min∂r˘,Fr˘,∂J^,GJ^,∂r˘,GJ^,∂J^,Fr˘.
IfY is continuous, then there is u∈ω such that u∈Fu∩Gu.Proof.
Consider a mappingα:ω⟶0,1 and a pair of fuzzy mappings S^,Γ:ω⟶⊑ω defined by
(76)S^r˘t=αr˘,ift∈Fr˘0,ift∉Fr˘,andΓr˘t=αr˘,ift∈Gr˘0,ift∉Gr˘.
Then,(77)S^r˘αr˘=t:S^r˘t≥αr˘=Fr˘,(78)Γr˘αr˘=t:Γr˘t≥αr˘=Gr˘.
Thus, by Theorem16, there is u∈ω so that u∈S^uαS^u∩ΓuαΓu=Fu∩Gu.Theorem 21.
LetS^,Γ:ω⟶Wω be FM. Assume that there are Y∈Ψ, Λ∈Φ, and L≥0 such that for all r˘,J^∈ω,(79)∂∞S^r˘,ΓJ^>0⇒Λ∂∞S^r˘,ΓJ^≤YΛMr˘,J^+LEr˘,J^,where
(80)Mr˘,J^=maxpr˘,J^,pr˘,S^r˘,pJ^,ΓJ^,pr˘,ΓJ^+pJ^,S^r˘2,(81)Er˘,J^=minpr˘,S^r˘,pJ^,ΓJ^,pr˘,ΓJ^,pJ^,S^r˘.
IfY is continuous, then there is u∈ω so that u⊂S^u and u⊂Γu.Proof.
Letr˘∈ω; then by Lemma 14, there is J^∈ω so that J^∈S^r˘1. Similarly, we can find e∈ω such that e∈Γr˘1. It follows that for each r˘∈ω, S^r˘αr˘ and Γr˘αr˘ are nonempty, closed, and bounded subsets of ω. As αr˘=αJ^=1, by the definition of a ∂∞-metric for fuzzy sets, we have
(82)HS^r˘αr˘,ΓJ^αJ^≤∂∞S^r˘,ΓJ^,forallr˘,J^∈ω.
Sinceϕ is nondecreasing, we have
(83)ΛHS^r˘αr˘,ΓJ^αJ^≤Λ∂∞S^r˘,ΓJ^≤YΛMr˘,J^+LEr˘,J^,where
(84)Mr˘,J^=maxpr˘,J^,pr˘,S^r˘,pJ^,ΓJ^,pr˘,ΓJ^+pJ^,S^r˘2,(85)Er˘,J^=minpr˘,S^r˘,pJ^,ΓJ^,pr˘,ΓJ^,pJ^,S^r˘,for all r˘,J^∈ω. Since S^r˘1⊆S^r˘αr˘, we have
(86)∂∞r˘,S^r˘αr˘≤∂r˘,S^r˘1foreachα∈0,1.
It yields that(87)Pr˘,S^r˘≤∂r˘,S^r˘1.
Similarly,(88)pr˘,Γr˘≤∂x,Γr˘1.
It yields that for allr˘,J^∈ω,
(89)ΛHS^r˘1,Γr˘1≤YΛMr˘,J^+LEr˘,J^,where
(90)Mr˘,J^=max∂r˘,J^,∂r˘,S^r˘1,∂J^,Γr˘1,∂r˘,ΓJ^1+∂J^,S^r˘12,(91)Er˘,J^=min∂r˘,S^r˘1,∂J^,ΓJ^1,∂r˘,ΓJ^1,∂J^,S^r˘1.
By Theorem16, we obtain u∈ω such that S^u1∩Γu1, that is, u⊂S^u and u⊂Γu.Corollary 22.
Letϕ,∂ be a complete MS, and S^,Γ:ω⟶Wω be FM. Assume that there exist Y∈Ψ and Λ∈Φ such that for all r˘,J^∈ω,(92)∂∞S^r˘,ΓJ^>0⇒Λ∂∞S^r˘,ΓJ^≤YΛMr˘,J^,where Mr˘,J^ is defined by (80). If Y is continuous, then there is u∈ω so that u⊂S^u and u⊂Γu.Proof.
TakeL=0 in Theorem 21.
We denote byS^ (for details, see [43, 44]) the setvalued mapping induced by a FM S^:ω⟶Fω, i.e.,
(93)S^r˘=J^:S^J^t=maxt∈ωS^r˘t.Corollary 23.
LetS^,Γ:ω⟶Iω be FM such that for all r˘∈ω, S^r˘ and Γ^r˘ are nonempty, closed, and bounded subsets of ω. Assume that there exist Y∈Ψ, Λ∈Φ, and L≥0 such that for all r˘,J^∈ω,(94)HS^r˘,Γ^J^>0⇒ΛHS^r˘,Γ^J^≤YΛMr˘,J^+LEr˘,J^,where
(95)Mr˘,J^=max∂r˘,J^,∂r˘,S^r˘,∂J^,Γ^J^,∂r˘,Γ^J^+∂J^,S^r˘2,(96)Er˘,J^=min∂r˘,S^r˘,∂J^,Γ^J^,∂r˘,Γ^J^,∂J^,S^r˘.
IfY is continuous, then there is r˘∗∈ω so that S^r˘∗r˘∗≥S^r˘∗r˘ and Γr˘∗r˘∗≥Γr˘∗r˘, for all r˘∈ω.Proof.
By Theorem16, there is r˘∗∈ω so that r˘∗∈S^r˘∗∩Γ^r˘∗. Then by Lemma 15, we have
(97)S^r˘∗r˘∗≥S^r˘∗r˘,Γr˘∗r˘∗≥Γr˘∗r˘forallr˘∈ω.Corollary 24.
LetS^,Γ:ω⟶Iω be FM such that for all r˘∈ω, S^r˘ and Γ^r˘ are nonempty, closed, and bounded subsets of ω. Assume that there are Y∈Ψ, Λ∈Φ, and L≥0 so that for all r˘,J^∈ω,(98)HS^r˘,Γ^J^>0⇒ΛHS^r˘,Γ^J^≤YΛMr˘,J^,where
(99)Mr˘,J^=max∂r˘,J^,∂r˘,S^r˘,∂J^,Γ^J^,∂r˘,Γ^J^+∂J^,S^r˘2.
IfY is continuous, then there is r˘∗∈ω such that r˘∗r˘∗≥S^r˘∗r˘ and Γr˘∗r˘∗≥Γr˘∗r˘, for all r˘∈ω.Proof.
TakeL=0 in Corollary 23.


## 3. Some Consequences

Corollary 25.
LetS^,Γ:ω⟶Iω be FM, and for each r˘∈ω, there exist αS^r˘, αΓr˘∈0,1 such that S^r˘αS^r˘ and ΓJ^αΓr˘ are nonempty, closed, and bounded subsets of ω. Assume that there are λ∈0,1 and L≥0 such that for all r˘,J^∈ω,(100)HS^r˘αS^r˘,ΓJ^αΓr˘≤λMr˘,J^+LEr˘,J^,where Mr˘,J^ and Er˘,J^ are defined by (20) and (21), respectively. Then, there is u∈ω such that u∈S^uαS^u∩ΓuαΓu.Proof.
TakeYt=λt and Λt=t in Theorem 16.Corollary 26.
LetS^,Γ:ω⟶Iω be FM, and for each r˘∈ω, there are αS^r˘, αΓr˘∈0,1, such that S^r˘αS^r˘ and ΓJ^αΓr˘ are nonempty, closed, and bounded subsets of ω. Assume that there are θ∈Θ∗ and k∈0,1 such that for all r˘,J^∈ω,HS^r˘αS^r˘,ΓJ^αΓr˘>0 implies
(101)θHS^r˘αS^r˘,ΓJ^αΓr˘≤θMr˘,J^k,where Mr˘,J^ is defined by (20). Then, there is u∈ω such that u∈S^uαS^u∩ΓuαΓu.Proof.
TakeYt≔lnkt, Λt=lnt, and L=0 in Theorem 16.Corollary 27.
LetS^,Γ:ω⟶Iω be FM, and for each r˘∈ω, there exist αS^r˘, αΓr˘∈0,1, such that S^r˘αS^r˘ and ΓJ^αΓr˘ are nonempty, closed, and bounded subsets of ω. Assume that there are F∈⊑∗ and τ>0 such that for all r˘,J^∈ω,HS^r˘αS^r˘,ΓJ^αΓr˘>0 implies
(102)τ+FHS^r˘αS^r˘,ΓJ^αΓr˘≤Fr˘,J^,where Mr˘,J^ is defined by (20). Then, there is u∈X so that u∈S^uαS^u∩ΓuαΓu.Proof.
TakeYt=e−τt, Λt=et, and L=0 in Theorem 16.Corollary 28.
LetS^,Γ:ω⟶Iω be FM, and for each r˘∈ω, there exist αS^r˘, αΓr˘∈0,1, such that S^r˘αS^r˘ and ΓJ^αΓr˘ nonempty, closed, and bounded subsets of ω. Assume that there is L≥0 such that for all r˘,J^∈ω implies
(103)HS^r˘αS^r˘,ΓJ^αΓr˘≤βMr˘,J^Mr˘,J^+LEr˘,J^,where Mr˘,J^ and Er˘,J^ are defined by (20) and (21), respectively, and β:0,′∞⟶0,∞ is such that limr→t+βr<1 for each t∈0,∞. Then, there is u∈ω such that u∈S^uαS^u∩ΓuαΓu.Proof.
It follows from Theorem16 by taking Yt≔βtt and Yt=t.


## 4. Application to an Ordinary Fuzzy Differential Equation

In this section, we apply our obtained results to study the existence of a solution for the second order nonlinear boundary value problem:(104)r˘′′t=Kt,r˘t,r˘′t,t∈0,a,a>0r˘t1=r˘1,t1∈0,ar˘t2=r˘2,t2∈0,a,where K:0,a×Wω×Wω⟶Wω is a continuous function. This problem is equivalent to the integral equation ([45, 46, 48]):
(105)r˘t=∫t1t2Gt,sKs,r˘s,r˘′sds+βt,t∈0,a,where Green’s function G is given by
(106)Gt,s=t2−ts−tt2−t1,ift1≤s≤t≤t2,t2−st−t1t2−t1,ift1≤t≤s≤t2,,and βt satisfies β′′=0, βt1=r˘1, and βt2=r˘2. Let us recall some properties of Gt,s. Particularly,
(107)∫t1t2Gt,sds≤t2−t128,∫t1t2Gtt,sds≤t2−t12.Our investigation is based on the existence of a common fixed point for a pair of integral operators given as follows:(108)S^r˘t=∫t1t2Gt,sK1s,r˘s,r˘′sds+βt,t∈0,a,(109)Γr˘t=∫t1t2Gt,sK2s,r˘s,r˘′sds+βt,t∈0,a,where K1,K2∈C0,a×Wω×Wω,Wω, x∈C10,a,Wω, and β∈C0,a,Wω.Theorem 29.
Suppose that(a)
K1,K2:0,a×Wω×Wω⟶Wω are increasing in their second and third variables,(b)
There isr˘0∈C10,a,Wω such that for all t∈0,a, we have
(110)r˘0t=∫t1t2Gt,sK1s,r˘0s,r˘0′sds+βt,t1,t2∈0,a,(c)
There areγ,δ>0 such that for all t∈0,a, we have
(111)K1t,r˘t,r˘′t−K2t,J^t,J^′t≤γr˘t−J^t+δr˘′t−J^′t,for all r,J^∈C10,a,Wω, with K1.,.,.≠K2.,.,.,(d)
Forγ,δ>0 and t1,t2∈0,a, we have
(112)γt2−t128+δt2−t12<910,(e)
ifr˘,J^∈C10,a,Wω is comparable, then every u∈Γ1r˘1 and every v∈Γ1J^1 are comparable.
Then, the pair of nonlinear integral equations(113)r˘t=∫t1t2Gt,sK1s,r˘s,r˘′sds+βt,t∈0,a,(114)r˘t=∫t1t2Gt,sK2s,r˘s,r˘′sds+βt,t∈0,a,has a common solution in C1t1,t2,Wω.Proof.
ConsiderC=C1t1,t2,Wω with the metric
(115)∂∞r˘,J^=maxt1≤t≤t2γr˘t−J^t+δr˘′t−J^′t.Note thatC,∂∞ is a complete linear MS. Let S^,Γ:C⟶C be two integral operators defined by (108) and (109). Clearly, S^ and Γ are well defined since K1,K2 and β are continuous functions. Now, r˘∗ is a solution of (113) and (114) if and only if r˘∗ is a common fixed point of S^ and Γ. By hypothesis (a), S^,Γ is increasing. Next, for all r˘,J^∈C with K1.,.,.≠K2.,.,., by hypothesis (c), we have successively
(116)S^r˘t−ΓJ^t≤∫t1t2Gt,sK1s,r˘s,r˘′s−K2s,J^s,J∧′sds≤∂∞r˘,J^∫t1t2Gt,sds≤t2−t128∂∞r˘,J^,(117)S^r˘′t−ΓJ^′t≤∫t1t2Gtt,sK1s,r˘s,r˘′s−K2s,J^s,y′sds≤∂∞r˘,J^∫t1t2Gtt,sds≤t2−t12∂∞r˘,J^.From (116) and (117), we easily obtain
(118)∂∞S^r˘,ΓJ^≤γt2−t128+δt2−t12∂∞r˘,J^<910∂∞r˘,J^.It implies that(119)e∂∞S∧r˘,ΓJ^<e9/10∂∞r˘,J^<e∂∞r˘,J^.Therefore,(120)∂∞S^r˘,ΓJ^e∂∞S^r˘,ΓJ^<910∂∞r˘,J^e∂∞r˘,J^.LetΛ,Y:0,∞⟶0,∞ be defined by
(121)Λt=tet,t>0,Yt=9t10,t>0,respectively.Thus, we have(122)Λ∂∞S^r˘,ΓJ^≤YΛ∂∞r˘,J^≤YΛMr˘,J^.Therefore, by Corollary28, S^ and Γ have a common fixed point r˘∗∈C, that is, r˘∗ is a common solution of (113) and (114). As an immediate consequence of Theorem 29, in the case of S^=Γ, we find that the integral equation (105) has a solution in C, and hence, the second order nonlinear boundary value problem (104) has a solution.


## 5. Conclusion

In the present work, we introduced a new concept of fuzzy mappings in complete metric spaces. Also, we derived the existence ofα-fuzzy common fixed points for two fuzzy mappings under generalized almost Y,Λ-contractions in complete metric spaces. We also gave an illustrative example to support our main results. We further showed some relations between multivalued mappings and fuzzy mappings, which can be utilized to ensure the existence of a common fixed point for multivalued mappings. Finally, we applied our main results to provide a solution for a second order nonlinear boundary value problem.



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