# Fuzzy Fixed Point Results inF-Metric Spaces with Applications

Authors: Monairah Alansari; Shehu Shagari Mohammed; Akbar Azam
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this paper, some concepts ofF-metric spaces are used to study a few fuzzy fixed point theorems. Consequently, corresponding fixed point theorems of multivalued and single-valued mappings are discussed. Moreover, one of our obtained results is applied to establish some conditions for existence of solutions of fuzzy Cauchy problems. It is hoped that the established ideas in this work will awake new research directions in fuzzy fixed point theory and related hybrid models in the framework of F-metric spaces.


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## Body

## 1. Introduction

One of the challenges in mathematical modeling of practical phenomena relates to the indeterminacy induced by our inability to categorize events with adequate precision. It has been understood that classical mathematics cannot cope effectively with imprecisions. As a result, the concept of fuzzy set was initiated by Zadeh [1] in 1965 as one of the uncertainty approaches to construct mathematical models compatible with real world problems in engineering, life science, economics, medicine, language theory, and so on. The basic ideas of fuzzy set have been extended in different directions. In particular, the notion of fixed point results for fuzzy set-valued mappings and fuzzy contractions was initiated by Heilpern [2] who proved a fixed point theorem parallel to the Banach fixed point theorem (see [3]) in the frame of fuzzy set. Thereafter, several authors have studied and applied fuzzy fixed point results in different settings [4, 5], see, for example [6–14] and the references therein.Not long ago, Jleli and Samet [9] initiated the concepts of F-metric spaces and obtained a generalization of the Banach fixed point theorem. Meanwhile, researchers have picked keen interests in establishing and improving different results in F-metric spaces, see, for instance, [15–17].The aim of this paper is twofold. First, we study some common fuzzy fixed point results in the setting ofF-complete F-metric spaces. Consequently, corresponding fixed point theorems of multivalued and single-valued mappings are derived. Thereafter, one of our obtained results is applied to discuss some solvability conditions of fuzzy initial value problems. As far as we know, in the setting of F-metric spaces and fuzzy mappings, the results presented herein are new and fundamental. On this note, it can be improved upon when discussed in other generalized hybrid models within the scope of fuzzy mathematics.


## 2. Preliminaries

In this section, we record some specific definitions and results that will be useful in what follows hereafter.Definition 1 (see [9]).
LetF be the set of functions f:0,+∞⟶ℝ satisfying the following conditions:
(F1) f is nondecreasing, i.e., 0<s<t implies fs≤ft
(F2) for every sequence tn⊂0,+∞, we have
(1)limn−→+∞tn=0implieslimn−→+∞ftn=−∞.Definition 2 (see [9]).
LetX be a nonempty set and D:X×X⟶0,+∞ be a given mapping. Suppose that there exists f,ρ∈F×0,+∞ such that
(D1) x,y∈X×X,Dx,y=0⇐⇒x=y
(D2) Dx,y=Dy,x,forallx,y∈X×X
(D3) for every x,y∈X×X, for every k∈ℕ,k≥2, and for every sequence uii=1n⊂X with u1,uk=x,y, we have
(2)Dx,y>0⇒fDx,y≤f∑i=1k−1Dui,ui−1+ρ.
Then,D is said to be an F-metric and the pair X,D is called an F-metric space.Example 3 (see [9]).
LetX=ℝ. If we define D:X×X⟶0,+∞ by
(3)Dx,y=x−y2,ifx,y∈0,3×0,3,x−y,ifx,y∉0,3×0,3,with ft=lnt and ρ=ln3, then X,Dis an F-metric space.Definition 4 (see [9]).
LetX,D be an F-metric space and xnn∈ℕ be a sequence in X.(i)
We say thatxnn∈ℕ is F-convergent to x if xnn∈ℕ is convergent to x with respect to the F-metric D(ii)
xnn∈ℕ is said to be F-Cauchy, if(4)limk,n−→∞Dxk,xn=0.(iii)
X,D is F-complete, if every F-Cauchy sequence in X is F-convergent to an element in X
In the sequel, we shall adopt the following notations and definitions in the setting ofF-metric space. We shall denote an F-metric by DF so that X,DF represents an F-metric space. Let C2X be the set of all nonempty compact subsets of X and A,B∈C2X. Then
(5)DFx,A=infDFx,a:a∈A,DFA,B=infDFa,b:a∈A,b∈B.
Then, the Hausdorff metricHF on C2X induced by the metric DF is defined as
(6)HFA,B=maxsupa∈ADFa,B,supb∈BDFA,b,∞,ifitexists,otherwise.Definition 5 (see [1, 2]).
LetX be an arbitrary nonempty set. Then, a fuzzy set in X is a function with domain X and values in 0,1=I. If B is a fuzzy set in X and x∈X, then the function values Bx is called the grade of membership of x∈X. The α-level set of B, denoted by Bα, is defined as
(7)Bα=x∈X:Bx≥αifx∈0,1,B0=Ax>0¯.
Here,M¯ denotes the closure of the crisp set M. Also, the family of fuzzy sets in a metric space X shall be denoted by IX.Definition 6.
LetX be an F-metric space. A subset A of X is called proximal, if for each x∈X, there exists a∈A such that DFx,a=DFx,A.
Let the set of all nonempty bounded proximal sets inX be denoted by PrX and the set of all nonempty closed and bounded subsets of X be represented by CBX. Since every compact set is proximal and any proximal set is closed, we have the inclusions:
(8)C2X⊆PrX⊆CBX.Definition 7 (see [2]).
LetX be an arbitrary set and Y a metric space. A mapping B:X⟶IX is called a fuzzy mapping. A fuzzy mapping B is a fuzzy subset of X×Y with membership value Bxy.Definition 8 (see [2, 7]).
LetA and B be fuzzy mappings from X into IX. A pointu∈X is called fuzzy fixed point of A if u∈Auα. The point u is known as a common fuzzy fixed point of A and B if u∈Auα∩Buα.Definition 9 (see [18, 19]).
A nondecreasing functionφ:0,∞⟶0,∞ is said to be a comparison function, if φnt⟶0 as n⟶∞ for every t∈0,∞, where φntdenotes the nth iterate of φ.Denote byΩ the set of all comparison functions.Lemma 10 (see [18, 19]).
Letφ∈Ω. Then, the following properties hold:
(i)
Each iterateφi of φ, for i≥1 is a comparison function(ii)
φ is continuous at 0(iii)
φt<t for all t>0


## 3. Main Results

First, we present the following auxiliary result.Lemma 11.
LetA and B be nonempty closed and compact subsets of an F-metric space X,DF. If a∈A, then DFa,B≤HFA,B.Proof.
The proof is a direct consequence of the definition ofHFA,B.Theorem 12.
LetX,DF be an F-complete F-metric space and A,B:X⟶IX be fuzzy mappings. Assume that for every x∈X, there exist αAx,αBx∈0,1 such that AxαAx,BxαBx∈C2X. Suppose also that the following condition holds:
(9)HFAxαAx,ByαBx≤φmaxDFx,y,DFx,AxαAx,DFy,ByαBy,DFy,ByαBy+DFx,AxαAx2,for all x,y∈X, where ϕ∈Ω. Then, there exists u∈X such that u∈AuαAu∩BuαBu.Proof.
Letx0∈X be arbitrary. By hypothesis, there exists αAx0∈0,1 such that Ax0αAx0∈C2X. Since Ax0αAx0 is a nonempty compact subset of X, there exists x1∈Ax0αAx0 such that DFx0,x1=DFx0,Ax0αAx0. Similarly, we can find αBx1∈0,1 such that Bx1αBx1∈C2X and by compactness of Bx1αBx1, we can choose x2∈Bx1αBx1 such that DFx1,x2=DFx1,Bx1αBx1. For convenience, denote αAxi and αBxi byαi+1, where i=0,1,2,⋯.
By Lemma11, we have
(10)DFx1,x2=DFx1,Bx1α2≤HFAx0α1,Bx1α2.
Therefore, using (10) together with (9), we have
(11)DFx1,x2≤φmaxDFx0,x1,DFx0,Ax1α1,DFx1,Bx1α2,DFx0,Bx1α2+DFx1,Ax0α12≤φmaxDFx0,x1,DFx0,x1,DFx1,x2,DFx0,x2+DFx1,x12≤φmaxDFx0,x1,DFx1,x2,DFx0,x1+DFx1,x22≤φmaxDFx0,x1,DFx1,x2.
IfmaxDFx0,x1,DFx1,x2=DFx1,x2, then (11) becomes
(12)DFx1,x2≤φDFx0,x1<DFx1,x2,which is a contradiction. It follows that maxDx0,x1,DFx1,x2=Dx0,x1. Therefore, we have
(13)DFx1,x2≤φDFx0,x1.
By continuous repetition of the above steps, we generate a sequencexnn∈ℕ of elements of X with
(14)x2n−1∈Ax2n−2αAx2n−1,x2n∈Bx2n−1αBx2n−1,such that
(15)DFx2n,x2n+1≤φDFx2n−1,x2n,DFx2n−1,x2n≤φDFx2n−2,x2n−1.
Consequently, by induction, for alln∈ℕ, we have
(16)DFxn,xn+1≤φDFxn−1,xn≤⋯≤φnDFx0,x1.
Letη>0 be a given positive number and f,ρ∈F×0,∞ such that condition D3 is satisfied. By F2, there exists λ>0 such that
(17)0<t<λimpliesft≤fη–ρ.
Letnη∈ℕ such that 0<∑n≥nηφnDFx0,x1<λ. Hence, by (17) and (F1), we get
(18)f∑i=nk−1φiDFx0,x1≤f∑n≥nηφnDFx0,x1≤fη−ρ.
Now, forDFxk,xn>0, by (D3), (16), and (18), we obtain
(19)fDFxk,xn≤f∑i=nk−1DFxi,xi+1+ρ≤f∑i=nk−1φiDFx0,x1+ρ≤f∑n≥nηφnDFx0,x1≤fη.
It follows from (F1) that
(20)DFxk,xn<η,k>n≥nη.This shows that xnn∈ℕ is F-Cauchy. Hence, F-completeness of X,DF implies that there exists u∈X such that xn⟶u as n⟶∞. Now, to prove that u∈AuαAu, assume that DFu,AuαAu>0. Then by (D3), we get
(21)fDFu,AuαAu≤fDFu,x2n+HFBx2n−1αBx2n−1,AuαAu+ρ≤fDFu,x2n+φmaxDFu,x2n−1,DFu,AuαAu,DFx2n−1,Bx2n−1αBx2n−1,DFu,Bx2n−1αBx2n−1+DFx2n−1,AuαAu2+ρ≤fDFu,x2n+φmaxDFu,x2n−1,DFu,AuαAu,DFx2n−1,x2n,DFu,x2n+DFx2n−1,AuαAu2+ρ.Now, we analyze (21) under the following cases:Case (i).
If(22)maxDFu,x2n−1,DFu,AuαAu,DFx2n−1,x2n,DFu,x2n+DFx2n−1,AuαAu2=DFu,x2n−1,then (21) becomes
(23)fDFu,AuαAu≤fDFu,x2n+φDFu,x2n−1+ρ.
Sincexn⟶u as n⟶∞, then by (F2) and the properties of ϕ∈Ω,
(24)limn→∞fDFu,x2n+DFu,x2n−1+ρ=−∞, which is a contradiction.Case (ii).
If(25)maxDFu,x2n−1,DFu,AuαAu,DFx2n−1,x2n,DFu,x2n+DFx2n−1,AuαAu2=DFx2n−1,x2n,then,
(26)fDFu,AuαAu≤fDFu,x2n+φDFx2n−1,x2n+ρ.
Hence, by (F2) and the properties of ϕ∈Ω, limn→∞fDFu,x2n+DFx2n−1,x2n+ρ=−∞, a contradiction.Case (iii).
If(27)maxDFu,x2n−1,DFu,AuαAu,DFx2n−1,x2n,DFu,x2n+DFx2n−1,AuαAu2=DFu,x2n+DFx2n−1,AuαAu2,then,
(28)fDFu,AuαAu≤fDFu,x2n+DFu,x2n+DFx2n−1,AuαAu2+ρ.
By condition (F1), from (28), for ρ=0, we have
(29)DFu,AuαAu<DFu,x2n+DFu,x2n+DFx2n−1,AuαAu2.
Asn⟶∞ in (29), we obtain
(30)DFu,AuαAu<DFu,AuαAu2<DFu,AuαAu, which is a contradiction.Case (iv).
If(31)maxDFu,x2n−1,DFu,AuαAu,DFx2n−1,x2n,DFu,x2n+DFx2n−1,AuαAu2=DFu,AuαAu,then,
(32)fDFu,AuαAu≤fDFu,x2n+φDFu,AuαAu+ρ.
By (F1), from (32), for ρ=0, and applying ϕt<t for all t>0, we get
(33)DFu,AuαAu<DFu,x2n+DFu,AuαAu<DFu,AuαAu,asn⟶∞, which is a contradiction. It follows that DFu,AuαAu=0. On the same steps, one can show that DFu,BuαBu=0. Consequently, u∈AuαAu∩BuαBu.Next, we give an example to support the validity of the hypotheses of Theorem12.Example 17.
LetX=ℝ+=0,∞ and DF:X×X→0,∞ be defined by
(34)DFx,y=x–y2,ifx,y∈0,5×0,5,x–y,ifx,y∉0,5×0,5,for all x,y∈X. It can be seen that DF satisfies (D1), (D2), and (D3); hence, X,DF is an F-metric space with ft=lnt, t>0, and ρ=ln5. Notice that DF does not satisfy the triangle inequality, since
(35)d1,5=16>d1,4+d4,5=10.
Moreover, letα∈0,1 and consider two fuzzy mappings A,B:X⟶IX defined as follows:
(i)
Ifx=0(36)Axt=Bxt=1,ift=0,0,ift≠0,(ii)
If0<x<∞(37)Axt=α,if0≤t<x260α4,ifx260≤t<x240,α8,ifx240≤t<x2,0,ifx2≤t<∞,Bxt=β,if0≤t<x250,β4,ifx250≤t<x240,β32,ifx240≤t<x2,0,ifx2≤t<∞.
Now, defineφ:0,∞⟶0,∞byφt=t/16,t≥0. Clearly, φ∈Ω.
Now, forx∈X, there exist αAx=α/4∈0,1 and αBx=β/8∈0,1 such that Axα/4,Bxβ/8∈C2X. Specifically, if x=y=0, then Axα/4,Bxβ/8=0, and hence
(38)HFAxα/4,Byβ/8=0≤φmaxDFx,y,DFx,Axα/4,DFy,Byβ/8,DFx,Byβ/8+DFy,Axα/42.
Ifx,y∈0,∞, then
(39)Axα/4=t∈X:Axt≥α4=0,x240.
Similarly,Byβ/8=0,y2/40. Therefore, for x≠y, by the definition of DF, we get
(40)HFAxα/4,Byβ/8=x240−y2402≤x+y40x−y2≤116x−y2≤116DFx,y≤φmaxDFx,y,DFx,Axα/4,DFy,Byβ/8,DFx,Byβ/8+DFy,Axα/42.Consequently, all the conditions of Theorem12 are satisfied to find u=0∈A0α/4∩B0β/8.By imposing continuity condition on the functionf∈F, we have the following modification of Theorem 12.Theorem 18.
LetX,DF be an F-complete F-metric space and A,B:X⟶IX be fuzzy mappings. Assume that for every x∈X, there exist αAx,αBx∈0,1 such that AxαAx,BxαB∈xC2X. Suppose also that the following conditions hold:
(i)
The functionf∈F is assumed to be continuous. In addition, suppose φ∈Ω satisfies ft>fφt+ρ for all t∈0,∞(ii)
And for allx,y∈X, we have(41)HFAxαAx,ByαBy≤φmaxDFx,y,DFx,AxαAx,DFy,ByαBy,DFx,ByαBy+DFy,AxαAx2.
Then, there existsu∈X such that u∈AuαAu∩BuαBu.Proof.
Following the proof of Theorem12, we obtain that xnn∈ℕ is an F-Cauchy sequence in the F-complete metric space X,DF. Therefore, there exists u∈X such that
(42)limn→∞DFxn,u=0.
Now, to prove thatu∈AuαAu, we argue by contradiction. So assume DFu,AuαAu>0. Then by (D3), and inequation (41), we get
(43)ƒDFu,AuαAu≤ƒDFu,x2n+HFBx2n−1αBx2n−1,AuαAu+ρ≤ƒDFu,x2n+φmaxDFu,x2n−1,DFu,AuαAu,DFx2n−1,Bx2n−1αBx2n−1,DFu,Bx2n−1αBx2n−1+DFx2n−1,AuαAu2+ρ≤ƒDFu,x2n+φmaxDFu,x2n−1,DFu,AuαAu,DFx2n−1,x2n,DFu,x2n+DFx2n−1,AuαAu2+ρ≤ƒDFu,x2n+φmaxDFu,x2n−1,DFu,AuαAu,DFx2n−1,x2n,DFu,x2n+DFx2n−1,AuαAu2+ρ≤ƒDFu,x2n+φmaxDFu,x2n−1,DFu,AuαAu,DFx2n−1,x2n,DFu,x2n+DFx2n−1,AuαAu+ρ.
Taking the limit in (43) as n→∞ and using (42) together with the continuity of f and φ, we have
(44)ƒDFu,AuαAu≤ƒφDFu,AuαAu+ρ,which is a contradiction to the condition on f. It follows that DFu,AuαAu=0. On similar steps, we can show that DFu,BuαBu=0. Consequently, we have
(45)u∈AuαAu∩BuαBu.Example 19.
In line with Example17, take ƒt=lnt,t>0,φt=t/16 and ρ=ln5; then, f is continuous on 0,∞ and φ is continuous for all t. Notice that the condition ft>fφt+ρ for t>0 becomes lnt>lnt/16+ln5. Therefore, following the remaining constructions of Example 17, one can easily verify that all the hypotheses of Theorem 18 are satisfied to find some u∈X such that u∈AuαAu∩BuαBu.


## 4. Consequences

In this section, we apply Theorems12 and 18 to deduce some fixed point results of multivalued and single-valued mappings in the context of F-metric spaces. To this end, recall that a point u∈X is called a fixed point of a multivalued (single-valued) mapping T on X, if u∈Tuu=Tu.Corollary 20.
LetX,DF be an F-complete F-metric space and S,T:X⟶C2X be multivalued mappings. Suppose that the following condition holds:
(46)HFSx,Ty≤φmaxDFx,y,DFx,Sx,DFy,Ty,DFx,Ty+DFy,Sx2,for all x,y∈X, where φ∈Ω. Then, there exists u∈X such that u∈Su∩Tu.Proof.
LetαA,αB:X⟶0,1 be any two arbitrary mappings, and consider two fuzzy set-valued maps A,B:X⟶IX defined as follows:
(47)Axt=αAx,ift∈Sx,0,ift∉Sx,Bxt=αBx,ift∈Tx,0,ift∉Tx.
Then, for allx∈X, we have
(48)AxαAx=t∈X:Axt≥αAx=Sx.
Similarly,BxαBx=Tx. Consequently, Theorem 12 can be applied to find u∈X such that u∈AuαAu∩BuαBu=Su∩Tu.Following the proof of Corollary20, we can also apply Theorem 18 to establish the following result.Corollary 21.
LetX,DF be an F-complete F-metric space and S,T:X⟶C2X be multivalued mappings. Suppose that the following conditions hold:
(i)
The functionf∈F is assumed to be continuous. In addition, suppose φ∈Ω satisfiesft>fφt+ρ for allt∈0,∞(ii)
And for allx,y∈X, we have(49)HFSx,Ty≤φmaxDFx,y,DFx,Sx,DFy,Ty,DFx,Ty+DFy,Sx2
Then, there existsu∈X such that u∈Su∩Tu.Corollary 22.
LetX,DF be an F-complete F-metric space and g,h:X⟶X be single-valued mappings. Suppose that the following condition holds:
(50)DFgx,hy≤φmax,DFx,y,DFx,gx,DFy,hy,DFx,hy+DFy,gx2,for all x,y∈X, where φ∈Ω. Then, there exists u∈X such that u=gu=hu.Proof.
LetαAx,αBx∈0,1 for all x∈X. Then, define two fuzzy set-valued maps A,B:X⟶IX as follows:
(51)Axt=αAx,ift=gx,0,ift≠gx,Bxt=αBx,ift=hx,0,ift≠hx.
Then,(52)AxαAx=t∈X:Axt≥αAx=gx.
Similarly,BxαBx=hx. Obviously, gx,hx∈C2X, for all x∈X. Notice that in this case, HFAxαAx,ByαBy=DFgx,hy. Therefore, Theorem 12 can be applied to find u∈X such that u∈AuαAu=gu and BuαBu=hu, which further implies that u=gu=hu.
Following the proof of Corollary22, one can also employ Theorem 18 to establish the following result.Corollary 23.
LetX,DF be an F-complete F-metric space and g,h:X⟶X be single-valued mappings. Suppose that the following conditions hold:
(i)
The functionf∈F is assumed to be continuous. In addition, suppose that φ∈Ω satisfies ft>fφtt+ρ for all t∈0,∞ and for all x,y∈X(ii)
And for allx,y∈X, we have(53)DFgx,hy≤φmaxDFx,y,DFx,gx,DFy,hy,DFx,hy+DFy,gx2
Then, there existsu∈X such that u=gu=hu.
In the following, we apply Corollary22 to deduce the main result of Jlei and Samet [9].Corollary 24 (see [9]).
LetX,DF be an F-complete F-metric space and h:X⟶X be a single-valued mapping. If for all x,y∈X, there exists λ∈0,1 such that
(54)DFhx,hy≤λDFx,y,then there exists u∈X such that hu=u.Proof.
Consider Corollary22. Define the function φ:0,∞⟶0,∞ by φt=λt, for all t≥0 and λ∈0,1. Then, by taking g=h, all the hypotheses of Corollary 22 coincide with that of Corollary 24; and so, we can find u∈X such that hu=u.Remark 25.
It is obvious that more consequences of Theorems12 and 18 can be obtained, but we skip them due to the length of the paper.


## 5. Application to Fuzzy Initial Value Problems

Fuzzy differential equations (FDEs) and fuzzy integral equations (FIEs) play significant roles in modeling dynamic systems in which uncertainties or vague notions flourish. These concepts have been established in different theoretical directions, and a large number of applications in practical problems have been studied (see, for example, [20–22]). Several techniques for studying FDEs have been presented. The first most popular is using the Hukuhara differentiability (H-differentiability) for fuzzy valued functions (see [20, 23, 24]). On the other hand, the concept of FIEs was initiated by Kaleva [22] and Seikkala [25]. In the study of existence and uniqueness conditions for solutions of FDEs and FIEs, many authors have applied different fixed point theorems. By using the classical Banach fixed point theorem, Subrahmanyam and Sudarsanam [26] proved an existence and uniqueness result for some Volterra integral equations involving fuzzy set-valued mappings. With the help of Shaulder’s fixed point theorem and Arzela-Ascoli’s theorem, Allahviranloo et al. [27] studied the existence and uniqueness conditions of solutions of some nonlinear fuzzy Volterra integral equations. In [28], the authors discussed some existence results for a fuzzy initial value problem (FIVP) by employing some contractive-like mapping techniques. Congxin and Shiji [29] studied a Cauchy problem of fuzzy differential equation on the basis of the definition of H-differentiability for fuzzy set-valued mappings. They obtained the existence and uniqueness theorem for the Cauchy problem under some generalized Lipschitz condition. Similarly, Villamizar-Roa et al. [30] studied the existence and uniqueness of solution of FIVP in the setting of generalized Hukuhara derivatives. For some intricacies involved in the theory of fuzzy differential equations, the interested reader may consult [22, 24, 31].In this section, using the ideas of fuzzy mappings in anF-complete F-metric space, we provide some conditions for the existence of solutions of a FIVP. In line with the existence methods, our technique is connected with studying the existence of solutions of the equivalent Volterra integral reformulation of the FIVP.First, in what follows, we recall a few known results that are needed in the sequel. For most of these basic concepts, we follow [30, 32]. Let Pκℝ denote the family of nonempty compact subsets of ℝ. Define addition and multiplication in Pκℝ as usual, that is, for A,B∈Pκℝ and η∈ℝ, we have
(55)A+B=a+b:a∈A,b∈B,ηA=ηa:a∈A.The Hausdorff metricH in Pκℝ is defined as
(56)HA,B=maxsupa∈A,b∈Binfa−bℝ,supb∈B,a∈Ainfa−bℝ.It is well known that the couplePκℝ,H is a complete metric space. Moreover, the metric H satisfies the following properties for all A,B,C,D∈Pκℝ:
(57)HηA,ηB=ηHA,B,HA+B,C+D≤HA,C+HB,D,HA+C,B+C=HA,B.In general,A+−A≠0, where −1A=−a:a∈A, and hence Pκℝ is not a linear space (cf. [30]).Definition 26.
A fuzzy number inℝ is a function x:ℝ⟶0,1 having the following properties:
(i)
x is normal, that is, there exists t0∈ℝ such that xt0=1(ii)
x is fuzzy convex, that is(58)xηt1+1−ηt2≥minxt1,xt2,∀t1,t2∈ℝ,η∈0,1(iii)
x is upper semicontinuous, that is, xα is closed for all α∈0,1(iv)
x0=t∈R:xt>0¯ is compact
Throughout this section, we shall denote the set of all fuzzy numbers inℝ by I1. The set xα=t∈ℝ:xt≥α=xαl,xαr denotes the α-level set of x∈I1. It follows from (i) to (iv) that xα∈Pκℝ.
The supremum onI1 is defined as
(59)D∞x,y=sup0≤α≤1maxxα1,l−xα2,l,xα1,r−xα2,r,for every x, y∈I1, where xαr−xαl=diamxα is called the diameter of xα.
We shall callCa,b,I1 the set of all continuous fuzzy functions defined on a,b. It is verifiable that Ca,b,I1 is an F-complete F-metric space with respect to the F-metric:
(60)DFx,y=supt∈JD∞xt,yt,x,y∈Ca,b,I1.
The following lemma summarizes some basic properties of the integral of fuzzy functions.Lemma 27 (see [22]).
Letx,y:a,b⟶I1 be fuzzy functions and η∈ℝ. Then,
(61)∫abx+ytdt=∫abxtdt+∫abytdt,∫abηxtdt=η∫abxtdt,D∞xt,ytisintegrable,D∞∫abxtdt,∫abytdt≤∫abD∞xt,ytdt.Definition 28 (see [30]).
LetIn denote the set of all fuzzy numbers in ℝn and x,y,z∈In. An element z is called the Hukuhara difference (or H-difference) of x and y, if it satisfies the equation x=y+z. If the H-difference of x and y exists, it is denoted by x⊖Hy (or x−y). It is easy to see that x⊖Hx=0, and if x⊖Hy exists, it is unique.Definition 29 (see [30]).
Letg:a,b⟶In. The function g is said to be strongly generalized differentiable (or GH-differentiable) at t0∈a,b, if there exists an element gG′t0∈In such that there exists the Hukuhara differences:
(62)gt0+δ⊖Hgt0,gt0⊖Hgt0−δ,limδ→0+gt0+δ⊖Hgt0δ=limδ→0+gt0⊖Hgt0−δδ=gG′t0.
Here, the limit is taken in the metric spaceIn,D, and at the end points of a,b, only one-sided derivatives are considered.Consider the following fuzzy initial value (FIVP):(63)x′t=gt,xt,t∈J=a,ρ,xa=ρ,where the derivative is considered in the sense of GH-differentiability, the fuzzy function g:J×I1⟶I1 is continuous, and the initial condition xa is a fuzzy number. We denote by C′J,I1 the set of all continuous functions g:J×I1⟶I1 with a continuous derivative.Lemma 30 (see [20]).
A functionx∈C′J,I1 is a solution of the FIPV (63) if and only if it satisfies the fuzzy Volterra integral equation:
(64)xt=ρ⊖H−1∫atgu,xudu,t∈J=a,ρ.
With the above preliminaries, we apply Theorem12 together with the following result to study the existence conditions of solutions of the FIVP (63)Theorem 31.
Letg:J×I1⟶I1 be continuous and assume that the following conditions hold:
(i)
The functiong is strictly increasing in the second variable, that is, ifx<y, then gt,x<gt,y(ii)
There exist some constantsτ>0 and λ∈0,1/2ρ−a such that(65)gt,xt−gt,ytℝ≤λmaxt∈JD∞x,ye−τt−aif x<y for each t∈J and x,y∈I1, where D∞x,y is the supremum on I1.
Then, the FIVP (63) has a fuzzy solution in CJ,I1.Proof.
Letτ>0 be a constant. We consider the space CJ,I1 endowed with the weighted F-metric:
(66)Dτx,y=supt∈JD∞xt,yte−τt−a,x,y∈CJ,I1.
Notice thatDτx,y is equivalent to the F-metric DFx,y, because
(67)eϱ−aτDFx,y≤Dτx,y≤DFx,y,for all x,y∈CJ,I1.
LetM,Q:X⟶0,1 be any two mappings. For x∈X, take
(68)Fτt=ρ⊖H−1∫atgu,xudu.
Assumex<y. Then, by hypothesis (i), we have
(69)Fxt=ρ⊖H−1∫atgu,xudu<ρ⊖H−1∫atgu,yudu=θyt.
Hence,Fxt≠θyt. Consider two fuzzy mappings A,B:X⟶IX defined by
(70)μAxr=Mx,ifrt=Fxt,0,otherwise,μByr=Qy,ifrt=θyt,0,otherwise.
By takingαAx=Mx and αBy=Qy, we have
(71)AxαAx=r∈X:Axt≥Mx=Fxt,and similarly, ByαBy=θyt. Therefore, we have
(72)HFAxαAx,ByαBy=maxsupx∈AxαAx,y∈ByαByinfx−yℝ,supy∈ByαBy,x∈Axαxyinfx−yℝ,≤maxsupt∈JFxt−θytℝ=supt∈JFxt−θytℝ=supt∈J∫atgu,xudu−∫atgu,yuduℝ≤supt∈J∫atgu,xu−gu,yuℝdu≤supt∈J∫atduλmaxD∞x,ye−τt−a≤λsupt∈Jt−amaxD∞x,ye−τt−a≤λϱ−aDτx,y12DFx,yφAx,y,where
(73)Ax,y=maxDFx,y,DFx,AxαAx,DFy,ByαBy,DFx,ByαBy+DFy,AxαAx2.
Hence, all the conditions of Theorem12 are satisfied with φt=t/2, for all t≥0. It follows that there exists x∗∈X such that x∗∈Ax∗αAx∗∩Bx∗αBx∗. Consequently, x∗ is a solution of the FIVP (63).


## 6. Conclusion

Two problems are addressed in this article. First, two fuzzy fixed point theorems in the context ofF-metric spaces are established. Consequently, corresponding fixed point theorems of multivalued and single-valued mappings are deduced. From an application perspective, in the second direction, one of our results is employed to provide some existence conditions for solutions of fuzzy initial value problems. As far as we know, in the setting of F-metric spaces and fuzzy set, the results presented herein are new and fundamental. Consequently, it can be improved upon when discussed in the setting of L-fuzzy mappings, intuitionistic fuzzy mappings, soft set-valued maps, and other generalized hybrid models within the scope of fuzzy mathematics.



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*Source: 5142815-2020-07-06.xml*