# Fuzzy Fractional Evolution Equations and Fuzzy Solution Operators

Authors: Atimad Harir; Said Melliani; Lalla Saadia Chadli
Journal: Advances in Fuzzy Systems
 (2019)

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## Abstract

In this paper, the fuzzy fractional evolution equations of orderq (FFEE) with fuzzy Caputo fractional derivative are introduced. We study the existence and uniqueness of mild solutions for FFEE under some conditions. Also, we generalize the definition of the fuzzy fractional integral and derivative order q. The fuzzy Laplace transform is presented and proved. The solvability of the problem (FFEE) and the properties of the fuzzy solution operator and its generator are investigated and developed.


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## Body

## 1. Introduction

The fuzzy fractional differential equations (FFDEs) can also offer a more comprehensive account of the process or phenomenon. This has recently captured much attention in FFDEs. As the derivative of a function is defined in the sense of Riemann–Liouville, Grünwald–Letnikov, or Caputo in fractional calculus, the used derivative is to be specified and defined in FFDEs as well. FFDEs are examined in [1–5]. We adopted the fuzzy Laplace transform method to solve FFDEs because it has the advantage that it solves problems directly without determining a general solution and obtaining nonhomogeneous differential equations [5].C. G. Gal and S. G. Gal studied [6], with more details, fuzzy linear and semilinear (additive and positive homogeneous) operator theory on the complete metric space.Letq∈ℝ+,a>0, and ℝℱ be the set of fuzzy real numbers. Our aim in this paper is to investigate the existence and uniqueness of the fuzzy mild solution for the fuzzy fractional evolution equation:(1)Dqut=Aut+ft,ut,t∈T=0,a,Djutt=0=uj0,j=0,1,2,…,k=q,where A:DA⊂ℝℱ⟶ℝℱ is the infinitesimal generator of a q-resolvent family Sqtt≥0, defined as ℝℱ, f∈T×ℝℱ⟶ℝℱ satisfies some conditions that will be specified later, and the fuzzy fractional derivative Dq is understood here in the caputo sense.


## 2. Preliminaries

In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper.Let us denote byℝℱ=u:ℝ⟶0,1 the class of fuzzy subsets of the real axis satisfying the following properties [7]:(i)
u is normal, i.e., there exists an x0∈ℝ, such that ux0=1.(ii)
u is fuzzy convex, i.e., for x,y∈ℝ and 0<λ≤1,(2)uλx+1−λy≥minux,uy.(iii)
u is upper semicontinuous.(iv)
u0=clx∈ℝux>0 is compact.Then,ℝℱ is called the space of fuzzy numbers. Obviously, ℝ⊂E. For 0<α≤1, denote uα=x∈ℝux≥α, and then from (i) to (iv), it follows that the α-level set uα∈PKℝ for all 0≤α≤1 is a closed bounded interval which we denote by uα=u1α,u2α.WherePKℝ denotes the family of all nonempty compact convex subsets of ℝ and defines the addition and scalar multiplication in PKℝ as usual. For later purposes, we define 0^∈ℝℱ as(3)0^t=1,t=0,0,t≠0.Theorem 1.
(see [8]). If u∈ℝℱ, then(i)
uα∈PKℝfor all0≤α≤1.(ii)
uα2⊂uα1for all0≤α1≤α2≤1.(iii)
αk⊂0,1is a nondecreasing sequence which converges toαand then,(4)uα=∩k≥1uαk.Conversely, if
A
α
=
u
1
α
,
u
2
α
;
α
∈
0,1
is a family of closed real intervals verifying (i) and (ii), then
A
α
defined a fuzzy number
u
∈
ℝ
ℱ
such that
u
α
=
A
α
.Lemma 1.
(see [9]). Let u,v:X⟶0,1 be the fuzzy sets.Then,u=v if and only if uα=vα for all α∈0,1The following arithmetic operations on fuzzy numbers are well known and frequently used below. Ifu,v∈ℝℱ, then(5)u+vα=u1α+v1α,u2α+v2α,u−vα=u1α−v2α,u2α−v1α,λuα=λuα=λu1α,λu2α,ifλ≥0,λu2α,λu1α,ifλ<0.Foru,v∈ℝℱ, if there exists w∈ℝℱ such that u=v+w, then w is the Hukuhara difference of u and v denoted by u−hv.Defined:ℝℱ×ℝℱ⟶ℝ+∪0 by the following equation:(6)du,v=supα∈0,1dHuα,vα,for allu,v∈ℝℱ,where dH is the Hausdorff metric defined in PKℝ.(7)dHuα,vα=maxu1α−v1α,u2α−v2α.It is well known thatℝℱ,d is a complete metric space [10]. We list the following properties of du,v:(8)du+w,v+w=du,v,du,v=dv,u,dku,kv=kdu,v,du,v≤du,w+dw,v,for all u,v,w∈ℝℱ and λ∈ℝ.LetT=0,a⊂ℝ be a compact interval. We denote by CT,ℝℱ the space of all continuous fuzzy functions on T and is a complete metric space with respect to the metric(9)hu,v=supt∈Tdut,vt.Also, we denote byL1T,ℝℱ the space of all fuzzy functions f:T⟶ℝℱ which are Lebesgue integrable on the bounded interval T of ℝ.Letu:T⟶ℝℱ be a fuzzy function. We denote(10)utα=u1αt,u2αt,t∈T,α∈0,1.The derivativeu′t of a fuzzy function u is defined by [11](11)u′tα=u1α′t,u2α′t,α∈0,1,provided this equation defines a fuzzy number u′t∈ℝℱ. The fuzzy integral ∫abutdt, a,b∈T is defined by [12](12)∫abutdtα=∫abu1αtdt,∫abu2αtdt,provided that the Lebesgue integrals on the right exist.From [13], we have the following theorems:Theorem 2.
There exists a real Banach spaceX such that ℝℱ can be the embedding as a convex cone C with vertex 0 into X. Furthermore, the following conditions hold:(i)
The embeddingj is isometric(ii)
Addition inX induces addition in ℝℱ, i.e., for any u,v∈ℝℱ(iii)
Multiplication by a non-negative real number inX induces the corresponding operation in ℝℱ, i.e., for any u∈ℝℱ(iv)
C-C is dense in X(v)
C is closedRemark 1.
Letj˜:ℝℱ⟶X as j˜u=j−1u,u∈ℝℱ. It verifies the following properties:(13)j˜u−j˜v=du,v,j˜su+tv=sj˜u+tj˜v,for allu,v∈ℝℱ,t,s≥0,j˜ℝℱ−jℝℱ=C,since−1ℝℱ=ℝℱ.Theorem 3.
LetX be a Banach space and j an embedding as in Theorem 2, u:T⟶ℝℱ, and assume j∘u is Bochner integrable over T. Then, ∫u∈ℝℱ, and(14)j∫utdt=∫jutdt.Remark 2.
By the definition of fuzzy integral [14], the above equality yields(15)j−1∫jutdt=∫utdt.


## 3. Fuzzy Fractional Integral and Fuzzy Fractional Derivative

Letut∈ℝℱ be such that utα=u1αt,u2αt for all t∈T and q∈ℝ+. Suppose that u1α,u2α∈CT,ℝ∩L1T,ℝ for all α∈0,1, and let(16)Aα≔1Γq∫0tt−sq−1u1αsds,∫0tt−sq−1u2αsds≔ϕqt∗u1αt,ϕqt∗u2αt.Lemma 2.
(see [1]). The family Aα;α∈0,1, given by (16), defined a fuzzy number u∈ℝℱ such that uα=Aα.
We define(17)ϕqt=tq−1Γq,t>0,0,t≤0,ϕ−qt=ϕ1+k−qt∗δ1+kt,k=q,ϕ−nt=δnt,n=0,1,2,…,where δnt is the nth derivative of the delta function and Γ⋅ is the gamma function (for the properties of ϕqt, see [15, 16]). These functions satisfy the semigroup property:(18)ϕqt∗ϕpt=ϕq+pt,forp>0.
The Sobolev spaces can be defined in the following way [17]:(19)Wk,pT,ℝ=u∃φ∈LpT,ℝ:ut=∑j=0kCjtjj!+tkk!∗φt,t∈T.
Note thatφt=uk+1t,Cj=uj0. Let(20)W0k,p=u∈Wk,pT,ℝuj0=0,j=0,1,2,…,k.
So,u∈W0k,p, if ut=tk−1/k−1!∗φt for some φ∈L1T,ℝ.Remark 3.
Letu:T⟶ℝℱ be such that utα=u1αt,u2αt for all t∈T and α∈0,1. Suppose that u1α,u2α∈Wk,1T,ℝ for all α∈0,1, and let(21)Bα≕∑j=0kC1αjtjj!+tkk!∗φ1αt,∑j=0kC2αjtjj!+tkk!∗φ2αt.
Note thatφiαt=uiαjt,Ciαj=uiαj0,fori=1,2.
According to lemma (2) and by the notation of φiα and Ciαj,i=1,2, the family Bα,α∈0,1, given by (21), defined a fuzzy number u∈ℝℱ such that uα=Bα.


### 3.1. Fuzzy Fractional Integral and Derivative

Letu∈CT,ℝℱ∩L1T,ℝℱ. Define the fuzzy fractional primitive of order q>0 of u:(22)Iqut=1Γq∫att−sq−1usds,t∈T,by(23)Iqutα=1Γq∫0tt−sq−1u1αsds,∫0tt−sq−1u2αsds,t∈T=u1αt∗ϕqt,u2αt∗ϕqt,t∈T.Forq=1, we obtain I1ut=∫atusds,t∈T, that is, the integral operator. Also, the following properties are obvious:(i)
Iqcut=cIqut for each constant c∈ℝℱ(ii)
Iqu+vt=Iqut+IqvtProposition 1.
[1]. If u∈CT,ℝℱ∩L1T,ℝℱ, then we have(24)IpIqu=Ip+qu,p,q>0.Definition 1.
Letu∈C1+kT,ℝℱ∩L1T,ℝℱ be a given function such that uα=u1α,u2α for all t∈T, and α∈0,1, the fuzzy fractional differential operator in the Riemann–Liouville sense, is defined for all u satisfying(25)ϕk+1−q∗u∈Wk,1T,(26)Dqut=D1+kϕ1+k−q∗ut=D1+kIk+1−qut=1Γ1+k−qD1+k∫0tust−sk−qds,by(27)Dqutα=D1+kϕ1+k−q∗u1αt,D1+kϕ1+k−q∗u2αt=1Γ1+k−qD1+k∫0tu1αst−sk−qds,D1+k∫01u2αst−sk−qds.
In fact,(28)Dqutα=Dqu1αt,Dqu2αt,for allt∈Tandα∈0,1,and the fuzzy fractional differential operator in the Caputo sense is defined:(29)Dcqut=D1+kϕ1+k−qt∗ut=Ik+1−qD1+kut=1Γ1+k−q∫0tD1+kust−sk−qds,by(30)Dcqutα=D1+ku1αt∗ϕ1+k−qt,D1+ku2αt∗ϕ1+k−qt=1Γ1+k−q∫0tD1+ku1αst−sk−qds,∫0tD1+ku2αst−sk−qds.
Fork=0, we obtain Dcqutα=1/Γ1−q∫0td/dsu1αtt−s−qds,∫0td/dsu2αtt−s−qds, provided that the equation defines a fuzzy number Dcqut∈ℝℱ. In fact,(31)Dcqutα=Dcqu1αt,Dcqu2αt,for allt∈Tandα∈0,1.
Some simple but relevant results valid forq,p,t>0 are(32)Iqϕp=ϕq+p,Dqϕp=ϕp−q,p≥q.Proposition 2.
Letq>0 and k=q. Then, for any u∈L1T,ℝℱ,(33)DqIqut=ut.If moreover (25)
holds, then
(34)
I
q
D
q
u
t
=
u
t
−
h
∑
j
=
0
k
ϕ
k
+
1
−
q
∗
u
j
0
ϕ
q
+
j
−
k
t
.Proof.
Letq>0 and f:T⟶ℝℱ be such that utα=u1αt,u2αt for all t∈T and α∈0,1. If u1α,u2α∈CT,ℝ∩L1T,ℝ for all α∈0,1, then Iquα satisfies (25): Iquα=ϕq∗uα∈L1T,ℝℱ (i.e., ϕq∗u1α,ϕq∗u2α∈L1T,ℝ) and(35)ϕk+1−q∗Iquα=ϕk+1−q∗ϕq∗uα=ϕk+1∗uα∈W0k,1T.
That is,ϕk+1∗u1α,ϕk+1∗u2α∈W0k,1. So, we can apply Dquα to Iquα and thanks to the semigroup property (24)(36)DqIquα=Dk+1Ik+1−qIquα=Dk+1Ik+1uα=uα.
Ifu satisfies (25), then according to (19),(37)ϕk+1−q∗uα=∑j=0kCjϕj+1t+ϕk+1∗φα,where φ1,φ2∈L1T and Cjα=ϕk+1−q∗uj0α. Therefore,(38)IqDquα=IqDk+1Ik+1−quα=IqDk+1ϕk+1−q∗uα=Iqφα.
Convolving both sides of (37) with ϕq and applying the semigroup property (18), we obtain(39)ϕk+1∗fα=∑j=0kCjϕq+j+1t+ϕq+k+1∗φα.
An application ofDk+1 to both sides gives(40)uα=∑j=0kCjϕq+j−kt+ϕq∗φα=∑j=0kCjϕq+j−ktα+ϕq∗φα,and then,(41)uα−h∑j=0kCjϕq+j+1−ktα=ϕq∗φα,which together with (38) implies (34). If ϕk+1−q∗u1α,ϕk+1−q∗u2α∈W0k,1T, that is, Cj=0, j=0,1,…,k, we have IqDqu=u.Remark 4.
(42)
D
q
u
t
=
∑
j
=
0
k
u
j
0
ϕ
j
−
q
t
+
I
k
+
1
−
q
D
k
+
1
u
t
,and then,(43)Dcqut=Dqut−h∑j=0kuj0ϕj−qt=Dqut−h∑j=0kuj0ϕjt.
Therefore,(44)DcqIqut=ut,IqDcqut=ut−h∑j=0kuj0ϕj,and the first identity is valid for all u1α,u2α∈L1T, the second for u1α,u2α∈CkT.


### 3.2. Fuzzy Laplace Transform

Letut∈ℝℱ be such that utα=u1αt,u2αt for all t∈T. Suppose that u1α,u2α∈CT,ℝ∩L1T,ℝ for all α∈0,1. We define the fuzzy Laplace transform [18] by(45)ℒuλα=∫0∞exp−λtu1αtdt,∫0∞exp−λtu2αtdt=Lu1αλ,Lu2αλ,where λ>0 and real.Theorem 4.
[18]. Let u and v are continuous fuzzy-valued functions. Suppose that c1 and c2 are constants. Then,(46)ℒc1uλ+c2vλα=c1ℒuλα+c2ℒvλα,Lemma 3.
Letu is continuous fuzzy-valued function and c∈ℝ:(47)ℒcuλα=cℒuλα.As in [5], we can introduce laplace transforms of derivatives by:Proposition 3.
Letu∈CT,ℝℱ∩L1T,ℝℱ, and then(48)ℒDquλ=λqℒuλ−h∑j=0kϕk+1−q∗uj0λk−j,(49)ℒDcquλ=λqℒuλ−h∑j=0kuj0λq−j.Proof.
We prove (49) because the proof of (48) is similar.
For arbitrary fixedα∈0,1, we have(50)λqℒuλα−h∑j=0kuj0αλq−j=λqLu1αλ−∑j=0ku1αj0λq−j,λqLu2αλ−∑j=0ku2αj0λq−j.
We apply the properties of the Laplace transform, and sinceLϕqλ=λ−q, we obtain(51)LDcqu1αλ=λqLu1αλ−∑j=0ku1αj0λq−j,LDcqu2αλ=λqLu2αλ−∑j=0ku2αj0λq−j.
Then, we conclude that(52)λqℒuλα−h∑j=0kuj0αλq−j=LDcqu1αλ,LDcqu2αλ,by linearity of L,(53)λqℒuλα−h∑j=0kuj0αλq−j=LDcqu1αλ,Dcqu2αλ.
Using (31) leads to obtain(54)λqℒfλα−h∑j=0kfj0αλq−j=ℒDcqfλα.


### 3.3. Fuzzy Solution Operators

We adopt the general definition and theorem of operator theory onℝℱ in [6]. Let A:ℝℱ⟶ℝℱ is linear if(55)Ax+y=Ax+Ay,Ak⋅x=k⋅Ax,for all x,y∈Lℝℱ, k∈ℝ, where Lℝℱ is A∈L+ℝℱ and is continuous at each x∈ℝℱ and L+ℝℱ is semilinear and continuous at 0˜.Let us consider the metricΦ:L+ℝℱ×L+ℝℱ⟶ℝ+(56)ΦA,B=supdAx,Bx;xℱ≤1,where xℱ=d0˜,x and we have ΦA,O˜=Aℱ, where O˜:ℝℱ⟶ℝℱ is given by O˜x=0˜,∀x∈ℝℱ.Theorem 5.
LetA be a bounded linear operator on ℝℱ [6]. If A<1, then i−A is invertible; moreover,(57)limn=∞Φi−A−1,∑k=0nAk=0.
Here,i:ℝℱ⟶ℝℱ denotes the identity function of ℝℱ.Proof.
Recall thatAℱ=supdAx,0˜,x∈ℝℱ,xℱ≤1∀x∈ℝℱ Setting Pnx=∑n=0∞Anx by (Theorem 3.5 and Corollary 3.6 in [6]), it suffices to show that Pnx is a Cauchy sequence in the complete metric space Lℝℱ,Φ. First A∈Lℝℱ, it follows An∈Lℝℱ for n=0,1,2,…. Then for m<n we have(58)dPnx,Pmx=d∑j=0nAjx,∑j=0mAjx≤∑j=mnd0˜,Ajx.
However,(59)d0˜,AAx=A2xℱ≤AℱAxℱ≤Aℱ2xℱ,and, so by induction, one can obtain(60)Ajℱ≤Aℱj,for allj=2,3,….
We obtain(61)dPnx,Pmx≤∑j=mnAℱxℱj,and passing to supremum with xℱ≤1 we get(62)ΦPn,Pm≤∑j=mnAℱj≤∑j=m+∞Aℱj=Aℱm1−Aℱ⟶0asm⟶∞.
Therefore,Pnn≥0 is a uniform Cauchy sequence on −a,a,a>0.(63)i−A∑n=0∞An=∑n=0∞An−∑n=0∞An+1=i+∑n=1∞An−∑n=1∞An=i,and ∑n=0∞Ani−A=i. It now follows that i−A−1=∑n=0∞An; therefore, its limit i−A−1 exists in Lℝℱ. So, limn=∞Φi−A−1,Pn=0 holds uniformly on each compact interval −a,a,a>0.Motivated by the above definitions in [19, 20], we can give the following definition.Definition 2.
The resolvent set of A, denoted byρA, is the set of all real numbers λq such that λqi−A is bijective, i.e.,(64)ρA=λq∈ℝ∗+:λqi−A−1exists.Remark 5.
Letλq>A. Then, A/λq<1. It follows from Theorem 5 that i−A/λq−1 exists, and(65)Rλq,A=λqi−A−1=1λqi−Aλq−1.Lemma 4.
LetA:ℝℱ⟶ℝℱ and A1=jAj−1:C⟶C tow operator.
A is the operator of the resolvent Rλq,A on ℝℱ if and only if A1 is the operator of the resolvent R1λq,A1 defining on the convex closed set C and R1λq,A1=jRλq,Aj−1 and ΦRλq,A,O˜≤M/λq−ω.Proof.
We assume thatΦRλq,A,O˜≤M/λq−ω on ℝℱ. By the properties of j we have, for all u∈C,(66)R1λq,A1u=jRλq,Aj−1u=dRλq,Aj−1u,j−10,dj−1u,j−10=u.
It follows that(67)supu≤1R1λq,A1u=supxℱ≤1dRλq,Ax,0˜,R1λq,A1=ΦdRλq,A,O˜≤Mλq−ω.
Conversely, ifR1λq,A1≤M/λq−ω on C, then for all x∈ℝℱ(68)dRλq,Ax,0˜=dj−1R1λq,A1jx,j−1j0˜=R1λq,A1jx,jx=dx,0˜.
It follows that(69)supxℱ≤1dRλq,Ax,0˜=supu≤1R1λq,A1u,ΦRλq,A,O˜=R1λq,A1≤Mλq−ω,which completes the proof.□
Next, we need to define the fuzzy solution operator (or fuzzyq-resolvent family), which is similar to that given in [19].
Consider the following particular case of (82) for the Caputo fractional derivative evolution equation of order qk<q<k+1 is an integer:(70)Dqut=Aut,t∈T,q∈ℝ+,u0=uj,Djutt=0=uj0=0,j=1,2,3,…,k,where A:ℝℱ⟶ℝℱ.
Applying (44), we obtain that the Cauchy problem (26)(71)ut=u0+1Γq∫0tt−sqAusds,and then we define the solution operator of (70) in terms of the corresponding integral equation (71).Definition 3.
A family of bounded linear operatorsSqtt≥0 on ℝℱ is called a solution operator for (71) (or q-resolvent family), if the following conditions are satisfied:(1)
Sqt is strongly continuous for t≥0 and Sq0=i, the identity mapping on ℝℱ(2)
SqtDA⊂DA and ASqtu0=SqtAu0 for all u0∈DA, t≥0(3)
Sqtu0 is a solution of (71) for all u0∈DA, t≥0Definition 4.
The solution operatorSqt is called exponentially bounded if there are constants M≥1 and w≥0 such that(72)ΦSqt,O˜≤Mewt,t≥0.Proposition 4.
IfSqt is the solution operator of (70) and u0∈ℝℱ, then if for t>0, the Hukuhara difference Sqtu0−hu0 exists, we define(73)Au0=Γ1+qlimt⟶0Sqtu0−hu0tq,which this limit exists in the metric space ℝℱ,d.Proof.
Letv⋅∈CT,ℝℱ, we have(74)vt=ϕ−1∗vt=ϕqϕ−1−q∗vt=ϕq∗vt∗ϕ−1−q=Γ1+qIqvttq.
Takingvt=DqSqtu0 and using (70) and (44), we obtain (73)Lemma 5.
LetA:ℝℱ⟶ℝℱ and A1=jAj−1:C⟶C tow operator.
A is the generator of a fuzzy q-resolvent familySqtt≥0 on ℝℱ if and only if A1 is the generator of an q-resolvent family Sq1tt≥0 defining on the convex closed set C by Sq1=jSqtj−1.Proof.
Follows from the definition ofSqtt≥0 and Sq1tt≥0 (see [19]), then Sqtt≥0 is the fuzzy solution operator on ℝℱ if and only if Sq1tt≥0 is the solution operator on C.
We assume thatA is the generator of a fuzzy q-resolvent family Sqtt≥0 on ℝℱ. By the properties of j we have, for all u0∈j−1DA,(75)Γq+1limt⟶0Sq1tu0−u0tq=Γq+1limt⟶0jSqtj−1u0−jj−1u0tq=jΓq+1limt⟶0Sqtj−1u0−hj−1u0tq=jAj−1u0=A1u0.
Conversely, ifA1 is the generator of a q-resolvent family Sq1tt≥0 on C, then for all u0∈DA(76)Γq+1limt⟶0Sqtu0−hu0tq=Γq+1limt⟶0j−1Sq1tju0−hj−1ju0tq=j−1Γq+1limt⟶0Sq1ju0−hju0tq=j−1A1ju0=Au0.Lemma 6.
LetA is a operator in (70) and j an embedding as in Theorem 2, the solution operators Sqt of (70) is defined by(77)Sqt=ℒ−1λq−1Rλq,At.Proof.
Taking the fuzzy Laplace transform of (70) on both sides, we obtain(78)λqℒutλ−hλq−1u0=Aℒutλ,and using the j,(79)λqjℒutλ−jAℒutλ=λq−1ju0,whereλq>0,λqjℒutλ−jj−1A1jℒutλ=λq−1ju0,λqI−A1jℒutλ=λq−1ju0,λqI−A1Ljutλ=λq−1ju0.
SinceλqI−A1−1 exist, i.e, λq∈ρA1 (see [20]), from the above equation, we obtain(80)Ljutλ=λqI−A1−1λq−1ju0.
Now (87) follows easily by taking the inverse of Laplace transform and applying j−1(81)jut=L−1λq−1R1λq,A1ju0,ut=ℒ−1λq−1j−1R1λq,A1ju0.
This completes the proof.


## 3.1. Fuzzy Fractional Integral and Derivative

Letu∈CT,ℝℱ∩L1T,ℝℱ. Define the fuzzy fractional primitive of order q>0 of u:(22)Iqut=1Γq∫att−sq−1usds,t∈T,by(23)Iqutα=1Γq∫0tt−sq−1u1αsds,∫0tt−sq−1u2αsds,t∈T=u1αt∗ϕqt,u2αt∗ϕqt,t∈T.Forq=1, we obtain I1ut=∫atusds,t∈T, that is, the integral operator. Also, the following properties are obvious:(i)
Iqcut=cIqut for each constant c∈ℝℱ(ii)
Iqu+vt=Iqut+IqvtProposition 1.
[1]. If u∈CT,ℝℱ∩L1T,ℝℱ, then we have(24)IpIqu=Ip+qu,p,q>0.Definition 1.
Letu∈C1+kT,ℝℱ∩L1T,ℝℱ be a given function such that uα=u1α,u2α for all t∈T, and α∈0,1, the fuzzy fractional differential operator in the Riemann–Liouville sense, is defined for all u satisfying(25)ϕk+1−q∗u∈Wk,1T,(26)Dqut=D1+kϕ1+k−q∗ut=D1+kIk+1−qut=1Γ1+k−qD1+k∫0tust−sk−qds,by(27)Dqutα=D1+kϕ1+k−q∗u1αt,D1+kϕ1+k−q∗u2αt=1Γ1+k−qD1+k∫0tu1αst−sk−qds,D1+k∫01u2αst−sk−qds.
In fact,(28)Dqutα=Dqu1αt,Dqu2αt,for allt∈Tandα∈0,1,and the fuzzy fractional differential operator in the Caputo sense is defined:(29)Dcqut=D1+kϕ1+k−qt∗ut=Ik+1−qD1+kut=1Γ1+k−q∫0tD1+kust−sk−qds,by(30)Dcqutα=D1+ku1αt∗ϕ1+k−qt,D1+ku2αt∗ϕ1+k−qt=1Γ1+k−q∫0tD1+ku1αst−sk−qds,∫0tD1+ku2αst−sk−qds.
Fork=0, we obtain Dcqutα=1/Γ1−q∫0td/dsu1αtt−s−qds,∫0td/dsu2αtt−s−qds, provided that the equation defines a fuzzy number Dcqut∈ℝℱ. In fact,(31)Dcqutα=Dcqu1αt,Dcqu2αt,for allt∈Tandα∈0,1.
Some simple but relevant results valid forq,p,t>0 are(32)Iqϕp=ϕq+p,Dqϕp=ϕp−q,p≥q.Proposition 2.
Letq>0 and k=q. Then, for any u∈L1T,ℝℱ,(33)DqIqut=ut.If moreover (25)
holds, then
(34)
I
q
D
q
u
t
=
u
t
−
h
∑
j
=
0
k
ϕ
k
+
1
−
q
∗
u
j
0
ϕ
q
+
j
−
k
t
.Proof.
Letq>0 and f:T⟶ℝℱ be such that utα=u1αt,u2αt for all t∈T and α∈0,1. If u1α,u2α∈CT,ℝ∩L1T,ℝ for all α∈0,1, then Iquα satisfies (25): Iquα=ϕq∗uα∈L1T,ℝℱ (i.e., ϕq∗u1α,ϕq∗u2α∈L1T,ℝ) and(35)ϕk+1−q∗Iquα=ϕk+1−q∗ϕq∗uα=ϕk+1∗uα∈W0k,1T.
That is,ϕk+1∗u1α,ϕk+1∗u2α∈W0k,1. So, we can apply Dquα to Iquα and thanks to the semigroup property (24)(36)DqIquα=Dk+1Ik+1−qIquα=Dk+1Ik+1uα=uα.
Ifu satisfies (25), then according to (19),(37)ϕk+1−q∗uα=∑j=0kCjϕj+1t+ϕk+1∗φα,where φ1,φ2∈L1T and Cjα=ϕk+1−q∗uj0α. Therefore,(38)IqDquα=IqDk+1Ik+1−quα=IqDk+1ϕk+1−q∗uα=Iqφα.
Convolving both sides of (37) with ϕq and applying the semigroup property (18), we obtain(39)ϕk+1∗fα=∑j=0kCjϕq+j+1t+ϕq+k+1∗φα.
An application ofDk+1 to both sides gives(40)uα=∑j=0kCjϕq+j−kt+ϕq∗φα=∑j=0kCjϕq+j−ktα+ϕq∗φα,and then,(41)uα−h∑j=0kCjϕq+j+1−ktα=ϕq∗φα,which together with (38) implies (34). If ϕk+1−q∗u1α,ϕk+1−q∗u2α∈W0k,1T, that is, Cj=0, j=0,1,…,k, we have IqDqu=u.Remark 4.
(42)
D
q
u
t
=
∑
j
=
0
k
u
j
0
ϕ
j
−
q
t
+
I
k
+
1
−
q
D
k
+
1
u
t
,and then,(43)Dcqut=Dqut−h∑j=0kuj0ϕj−qt=Dqut−h∑j=0kuj0ϕjt.
Therefore,(44)DcqIqut=ut,IqDcqut=ut−h∑j=0kuj0ϕj,and the first identity is valid for all u1α,u2α∈L1T, the second for u1α,u2α∈CkT.


## 3.2. Fuzzy Laplace Transform

Letut∈ℝℱ be such that utα=u1αt,u2αt for all t∈T. Suppose that u1α,u2α∈CT,ℝ∩L1T,ℝ for all α∈0,1. We define the fuzzy Laplace transform [18] by(45)ℒuλα=∫0∞exp−λtu1αtdt,∫0∞exp−λtu2αtdt=Lu1αλ,Lu2αλ,where λ>0 and real.Theorem 4.
[18]. Let u and v are continuous fuzzy-valued functions. Suppose that c1 and c2 are constants. Then,(46)ℒc1uλ+c2vλα=c1ℒuλα+c2ℒvλα,Lemma 3.
Letu is continuous fuzzy-valued function and c∈ℝ:(47)ℒcuλα=cℒuλα.As in [5], we can introduce laplace transforms of derivatives by:Proposition 3.
Letu∈CT,ℝℱ∩L1T,ℝℱ, and then(48)ℒDquλ=λqℒuλ−h∑j=0kϕk+1−q∗uj0λk−j,(49)ℒDcquλ=λqℒuλ−h∑j=0kuj0λq−j.Proof.
We prove (49) because the proof of (48) is similar.
For arbitrary fixedα∈0,1, we have(50)λqℒuλα−h∑j=0kuj0αλq−j=λqLu1αλ−∑j=0ku1αj0λq−j,λqLu2αλ−∑j=0ku2αj0λq−j.
We apply the properties of the Laplace transform, and sinceLϕqλ=λ−q, we obtain(51)LDcqu1αλ=λqLu1αλ−∑j=0ku1αj0λq−j,LDcqu2αλ=λqLu2αλ−∑j=0ku2αj0λq−j.
Then, we conclude that(52)λqℒuλα−h∑j=0kuj0αλq−j=LDcqu1αλ,LDcqu2αλ,by linearity of L,(53)λqℒuλα−h∑j=0kuj0αλq−j=LDcqu1αλ,Dcqu2αλ.
Using (31) leads to obtain(54)λqℒfλα−h∑j=0kfj0αλq−j=ℒDcqfλα.


## 3.3. Fuzzy Solution Operators

We adopt the general definition and theorem of operator theory onℝℱ in [6]. Let A:ℝℱ⟶ℝℱ is linear if(55)Ax+y=Ax+Ay,Ak⋅x=k⋅Ax,for all x,y∈Lℝℱ, k∈ℝ, where Lℝℱ is A∈L+ℝℱ and is continuous at each x∈ℝℱ and L+ℝℱ is semilinear and continuous at 0˜.Let us consider the metricΦ:L+ℝℱ×L+ℝℱ⟶ℝ+(56)ΦA,B=supdAx,Bx;xℱ≤1,where xℱ=d0˜,x and we have ΦA,O˜=Aℱ, where O˜:ℝℱ⟶ℝℱ is given by O˜x=0˜,∀x∈ℝℱ.Theorem 5.
LetA be a bounded linear operator on ℝℱ [6]. If A<1, then i−A is invertible; moreover,(57)limn=∞Φi−A−1,∑k=0nAk=0.
Here,i:ℝℱ⟶ℝℱ denotes the identity function of ℝℱ.Proof.
Recall thatAℱ=supdAx,0˜,x∈ℝℱ,xℱ≤1∀x∈ℝℱ Setting Pnx=∑n=0∞Anx by (Theorem 3.5 and Corollary 3.6 in [6]), it suffices to show that Pnx is a Cauchy sequence in the complete metric space Lℝℱ,Φ. First A∈Lℝℱ, it follows An∈Lℝℱ for n=0,1,2,…. Then for m<n we have(58)dPnx,Pmx=d∑j=0nAjx,∑j=0mAjx≤∑j=mnd0˜,Ajx.
However,(59)d0˜,AAx=A2xℱ≤AℱAxℱ≤Aℱ2xℱ,and, so by induction, one can obtain(60)Ajℱ≤Aℱj,for allj=2,3,….
We obtain(61)dPnx,Pmx≤∑j=mnAℱxℱj,and passing to supremum with xℱ≤1 we get(62)ΦPn,Pm≤∑j=mnAℱj≤∑j=m+∞Aℱj=Aℱm1−Aℱ⟶0asm⟶∞.
Therefore,Pnn≥0 is a uniform Cauchy sequence on −a,a,a>0.(63)i−A∑n=0∞An=∑n=0∞An−∑n=0∞An+1=i+∑n=1∞An−∑n=1∞An=i,and ∑n=0∞Ani−A=i. It now follows that i−A−1=∑n=0∞An; therefore, its limit i−A−1 exists in Lℝℱ. So, limn=∞Φi−A−1,Pn=0 holds uniformly on each compact interval −a,a,a>0.Motivated by the above definitions in [19, 20], we can give the following definition.Definition 2.
The resolvent set of A, denoted byρA, is the set of all real numbers λq such that λqi−A is bijective, i.e.,(64)ρA=λq∈ℝ∗+:λqi−A−1exists.Remark 5.
Letλq>A. Then, A/λq<1. It follows from Theorem 5 that i−A/λq−1 exists, and(65)Rλq,A=λqi−A−1=1λqi−Aλq−1.Lemma 4.
LetA:ℝℱ⟶ℝℱ and A1=jAj−1:C⟶C tow operator.
A is the operator of the resolvent Rλq,A on ℝℱ if and only if A1 is the operator of the resolvent R1λq,A1 defining on the convex closed set C and R1λq,A1=jRλq,Aj−1 and ΦRλq,A,O˜≤M/λq−ω.Proof.
We assume thatΦRλq,A,O˜≤M/λq−ω on ℝℱ. By the properties of j we have, for all u∈C,(66)R1λq,A1u=jRλq,Aj−1u=dRλq,Aj−1u,j−10,dj−1u,j−10=u.
It follows that(67)supu≤1R1λq,A1u=supxℱ≤1dRλq,Ax,0˜,R1λq,A1=ΦdRλq,A,O˜≤Mλq−ω.
Conversely, ifR1λq,A1≤M/λq−ω on C, then for all x∈ℝℱ(68)dRλq,Ax,0˜=dj−1R1λq,A1jx,j−1j0˜=R1λq,A1jx,jx=dx,0˜.
It follows that(69)supxℱ≤1dRλq,Ax,0˜=supu≤1R1λq,A1u,ΦRλq,A,O˜=R1λq,A1≤Mλq−ω,which completes the proof.□
Next, we need to define the fuzzy solution operator (or fuzzyq-resolvent family), which is similar to that given in [19].
Consider the following particular case of (82) for the Caputo fractional derivative evolution equation of order qk<q<k+1 is an integer:(70)Dqut=Aut,t∈T,q∈ℝ+,u0=uj,Djutt=0=uj0=0,j=1,2,3,…,k,where A:ℝℱ⟶ℝℱ.
Applying (44), we obtain that the Cauchy problem (26)(71)ut=u0+1Γq∫0tt−sqAusds,and then we define the solution operator of (70) in terms of the corresponding integral equation (71).Definition 3.
A family of bounded linear operatorsSqtt≥0 on ℝℱ is called a solution operator for (71) (or q-resolvent family), if the following conditions are satisfied:(1)
Sqt is strongly continuous for t≥0 and Sq0=i, the identity mapping on ℝℱ(2)
SqtDA⊂DA and ASqtu0=SqtAu0 for all u0∈DA, t≥0(3)
Sqtu0 is a solution of (71) for all u0∈DA, t≥0Definition 4.
The solution operatorSqt is called exponentially bounded if there are constants M≥1 and w≥0 such that(72)ΦSqt,O˜≤Mewt,t≥0.Proposition 4.
IfSqt is the solution operator of (70) and u0∈ℝℱ, then if for t>0, the Hukuhara difference Sqtu0−hu0 exists, we define(73)Au0=Γ1+qlimt⟶0Sqtu0−hu0tq,which this limit exists in the metric space ℝℱ,d.Proof.
Letv⋅∈CT,ℝℱ, we have(74)vt=ϕ−1∗vt=ϕqϕ−1−q∗vt=ϕq∗vt∗ϕ−1−q=Γ1+qIqvttq.
Takingvt=DqSqtu0 and using (70) and (44), we obtain (73)Lemma 5.
LetA:ℝℱ⟶ℝℱ and A1=jAj−1:C⟶C tow operator.
A is the generator of a fuzzy q-resolvent familySqtt≥0 on ℝℱ if and only if A1 is the generator of an q-resolvent family Sq1tt≥0 defining on the convex closed set C by Sq1=jSqtj−1.Proof.
Follows from the definition ofSqtt≥0 and Sq1tt≥0 (see [19]), then Sqtt≥0 is the fuzzy solution operator on ℝℱ if and only if Sq1tt≥0 is the solution operator on C.
We assume thatA is the generator of a fuzzy q-resolvent family Sqtt≥0 on ℝℱ. By the properties of j we have, for all u0∈j−1DA,(75)Γq+1limt⟶0Sq1tu0−u0tq=Γq+1limt⟶0jSqtj−1u0−jj−1u0tq=jΓq+1limt⟶0Sqtj−1u0−hj−1u0tq=jAj−1u0=A1u0.
Conversely, ifA1 is the generator of a q-resolvent family Sq1tt≥0 on C, then for all u0∈DA(76)Γq+1limt⟶0Sqtu0−hu0tq=Γq+1limt⟶0j−1Sq1tju0−hj−1ju0tq=j−1Γq+1limt⟶0Sq1ju0−hju0tq=j−1A1ju0=Au0.Lemma 6.
LetA is a operator in (70) and j an embedding as in Theorem 2, the solution operators Sqt of (70) is defined by(77)Sqt=ℒ−1λq−1Rλq,At.Proof.
Taking the fuzzy Laplace transform of (70) on both sides, we obtain(78)λqℒutλ−hλq−1u0=Aℒutλ,and using the j,(79)λqjℒutλ−jAℒutλ=λq−1ju0,whereλq>0,λqjℒutλ−jj−1A1jℒutλ=λq−1ju0,λqI−A1jℒutλ=λq−1ju0,λqI−A1Ljutλ=λq−1ju0.
SinceλqI−A1−1 exist, i.e, λq∈ρA1 (see [20]), from the above equation, we obtain(80)Ljutλ=λqI−A1−1λq−1ju0.
Now (87) follows easily by taking the inverse of Laplace transform and applying j−1(81)jut=L−1λq−1R1λq,A1ju0,ut=ℒ−1λq−1j−1R1λq,A1ju0.
This completes the proof.


## 4. Fuzzy Fractional Differential Equations

Consider the following fuzzy fractional differential equation:(82)Dqut=Aut+ft,ut,t∈T,q∈ℝ+,Djutt=0=uj0,j=0,1,2,…,k=q,where A generator of q-resolvent family Sqtt≥0 on ℝℱ, Dq is the fuzzy caputo fractional differential operator define by (29) and f∈T×ℝℱ⟶ℝℱ is continuous.Firstly, we consider the following Cauchy problem(83)Dqut=Aut,t∈T,q∈ℝ+,Djutt=0=uj0,j=0,1,2,…,k=q.The roblem (70) is particular case of (83), and if (70) has a solution operator Sqt, then the corresponding problem (83) is uniquely solvable with the solution(84)ut=∑j=0kIjSqtuj,provided uj∈DA, j=0,1,2,…,k. For this reason, we restrict ourselves to problem (71) (in crisp, see [19]).Next, we consider the particular case of (82).(85)Dqut=Aut+ft,ut,t∈T,q∈ℝ+,Djutt=0=u0,Djutt=0=0,j=1,2,…,k=q,where A is a operator and f is an abstract function defined on T×ℝℱ and with values in ℝℱ.Theorem 6.
Let A is an operator,f≔T×ℝℱ⟶ℝℱ be continuous on T and if f satisfies a Hölder condition with an exponent of β∈0,1(86)dfu,fv≤Cdu,vβ.
The functionut∈CT,ℝℱ is a solution of (85) if and only if(87)ut=Sqtu0+∫0tTqt−sfs,usds,where(88)Sqt=ℒ−1λq−1Rλq,At,Tqt=ℒ−1Rλq,At.Proof.
Now applying the fuzzy Riemann–Liouville fractional integral operator (23) in (85) on both sides, we get(89)ut−hu0=ϕq∗Aut+ϕq∗ft,ut,ut=u0+ϕq∗Aut+ϕq∗ft,ut,and taking the fuzzy Laplace transform of (89) on both sides, we obtain(90)ℒutλ=ℒu0λ+ℒϕq∗Autλ+ℒϕq∗ft,utλ,λqℒutλ−hAℒutλ=λq−1u0+ℒft,utλ.
By usingj and if λqI−A1−1 exists, i.e., λq∈ρA1, from the above equation, we obtain(91)jℒutλ=λqI−A1−1λq−1ju0+λqI−A1−1jℒft,utλ,Ljutλ=λqI−A1−1λq−1ju0+λqI−A1−1Ljft,utλ.
Now, (87) follows easily by taking the inverse of Laplace transform:(92)jut=L−1λqI−A1−1λq−1ju0+∫0tL−1λqI−A1−1jfs,usds,and applying j−1 and Lemma 4,(93)ut=ℒ−1λqI−A−1λq−1u0+∫0tℒ−1λqI−A1−1fs,usds.
This completes the proof.Theorem 7.
LetA be a operator. If satisfies (86), then the solutions of the Cauchy problem (82) are fixed points of the operator equation:(94)φut=∑j=0kIjSqtuj+∫0tTqt−sfs,usds.
Theorem 7 leads to the following appropriate definition of a mild solution to (82).Definition 5.
A functionut∈CT,ℝℱ is called a mild solution of (82), if it satisfies the operator equation:(95)ut=∑j=0kIjSqtuj+∫0tTqt−sfs,usds.Theorem 8.
LetA is a operator and f:T×ℝℱ⟶ℝℱ be continuous on T. Assume thatH0there exists M≥1 and ω>0 such that(96)dTqvt,Tqwt≤Meωtdvt,wt,t∈T.H1there exists a constantL>0, such that(97)dft,vt,ft,wt≤Ldvt,wt,for everyvt,wt∈ℝℱ, t∈T. Then, problem (82)has a unique mild solutionut∈CT,ℝℱ.Proof.
Letψ:CT,ℝℱ⟶CT,ℝℱ be the operator defined by(98)ψut=∑j=0kIjSqtuj+∫0tTqt−sfs,usds.
We have to show thatu is mild solution of (82) if and only if ψu=u and vt,wt∈CT,ℝℱ, we find that(99)dψvt,ψwt=d∫0tTqt−sfs,vsds,∫0tTqt−sfs,wsds≤∫0tdTqt−sfs,vs,Tqt−sfs,wsds≤M∫0teωt−sdfs,vs,fs,wsds≤Leωa∫0tdvs,wsds≤Lteωahv,w.
We can deduce that(100)dψ2vt,ψ2wt=Leωa∫0tdψvt,ψwtds≤Leωa∫0tsLeωahv,wds≤Leωat22!hv,w⋮⋮dψnvt,ψnwt=Leωatnn!hv,w≤Leωaann!hv,w,and it follows that(101)hψnv,ψnw=Leωatnn!hv,w,∀v,w∈CT,ℝℱ.
SinceLeωaan/n!⟶0, ∃p∈ℕ such that Leωaap/p!<1.
It follows thatψp is a contraction, and then ∃!u∈CT,ℝℱ such that ψpu=u.
We proveψu=u.
ψ
p
u
=
u implies that ψp+1u=ψu, one can write ψpψu=ψu it follows that ψu is a fixed point of ψp, and since the fixed point is unique, we get ψu=u.
Hence, (85) has a unique solution.



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*Source: 5734190-2019-11-14.xml*