# On the Existence and Uniqueness Results for Fuzzy Linear and Semilinear Fractional Evolution Equations Involving Caputo Fractional Derivative

Authors: Ali El Mfadel; Said Melliani; M’hamed Elomari
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this manuscript, we establish new existence and uniqueness results for fuzzy linear and semilinear fractional evolution equations involving Caputo fractional derivative. The existence theorems are proved by using fuzzy fractional calculus, Picard’s iteration method, and Banach contraction principle. As application, we conclude this paper by giving an illustrative example to demonstrate the applicability of the obtained results.


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## Body

## 1. Introduction

Fuzzy fractional calculus and fuzzy fractional differential equations are a natural way to model dynamical systems subject to uncertainties. In the past few years, many works have been done by several authors in the theory of fuzzy fractional differential equations (see [1–3]). This theory has been proposed to handle uncertainty due to incomplete information that appears in many mathematical or computer models of some deterministic real-world phenomena. Recently, fractional differential equations have attracted a considerable interest both in mathematics and in applications such as material theory, transport processes, fluid flow phenomena, earthquakes, solute transport, chemistry, wave propagation, signal theory, biology, electromagnetic theory, thermodynamics, mechanics, geology, astrophysics, economics heat conduction in materials with memory, and control theory (see basic books and interesting papers in [4–8]).In many cases, when a real physical phenomenon is modelled by a fractional initial value problem, we cannot usually be sure that the model is perfect. For example, the initial value of this problem may not be known precisely. In order to get a perfect model with a precise initial condition, Agarwal et al. in [9] proposed the concept of solutions for fuzzy fractional differential equations. Arshad and Lupulescu in [10] proved some results on the existence and uniqueness of solution for the fuzzy fractional differential equations under Hukuhara fractional Riemann-Liouville differentiability. Later, Alikhani and Bahrami in [11] have proved the existence and uniqueness results for nonlinear fuzzy fractional integral and integrodifferential equations by using the method of upper and lower solutions. The authors in [12, 13] discussed the concepts about generalized Hukuhara fractional Riemann-Liouville and Caputo differentiability of fuzzy-valued functions, and the equivalence between fuzzy fractional differential equation and fuzzy fractional integral equation is discussed in [14]. For many basic works related to the theory of fractional differential equations and fuzzy fractional differential equations, we refer the readers to the articles [15–20] and references therein.Motivated by the above works, in the present paper, we study the existence result of solution for the following fuzzy linear fractional evolution equation:(1)Dct0+qxt=Atxt,t∈J=t0,T,xt0=x0,and for the following fuzzy semilinear fractional evolution equation:(2)Dct0+qxt=Atxt+ft,xt,t∈J=t0,T,xt0=x0,where cDt0+q is the fuzzy Caputo derivative of xt at order 0<q<1, t0≥0, T>0, At is a bounded linear operator, and f is a fuzzy continuous function.The paper is organized as follows. In Section2, we give some basic properties of fuzzy sets, operations of fuzzy numbers, and some detailed definitions of fuzzy fractional integral and fuzzy fractional derivative which will be used in the rest of this paper. In Section 3, we introduce the existence and uniqueness results of solution for the fuzzy linear fractional evolution equation (1). In Section 4, we discussed the existence and uniqueness results for the fuzzy semilinear fractional evolution equation (2). An illustrative example is presented in Section 5 followed by conclusion and future work in Section 6.


## 2. Preliminaries

In this section, we will briefly give some of notations, definitions, and results from the literature of fuzzy set theory and fuzzy fractional calculus which will be used in the rest of this paper.Definition 1 (see [21]).
A fuzzy number is mappingu:R⟶0,1 such that(1)
u is upper semicontinuous(2)
u is normal; that is, there exist x0∈R such that ux0=1(3)
u is fuzzy convex, that is, uλx+1−λy≥minux,uy for all x,y∈R and λ∈0,1(4)
x∈R,ux>0¯ is compactTheα−cut of a fuzzy number u is defined as follows:(3)uα=x∈R∣ux≥α.Moreover, we also can present theα−cut of fuzzy number u by uα=ulα,urα.Example 1.
Letu be a fuzzy number defined by the following function:(4)μux=x−1,x∈1,2,−x+3,x∈2,3,0,elswhere.
Ifα=1, then the α−cut of the fuzzy number u is given by uα=u1=2.Notations.
(i)
We denote byE1 the collection of all fuzzy numbers(ii)
We denote byCJ,E1 the space of all fuzzy-valued functions which are continuous on J(iii)
We denote byPcR the set of all bounded and closed intervals of R(iv)
We denote also by0E1 the fuzzy zero defined by(5)0E1x=1,ifx=0,0,elswhere.Definition 2 (see [14]).
Letα∈0,1 and u∈E1 such that uα=ulα,urα.
We define the diameter ofα−level set uα of the fuzzy set u as follows:(6)duα=urα−ulα.Definition 3 (see [14]).
The generalized Hukuhara difference of two fuzzy numbersu,v∈E1 is defined as follows:(7)u⊖gHv=w⟺iu=v+w,oriiv=u+−w.Property 4 (see [22]).
Ifu∈E1 and v∈E1, then the following properties hold:(1)
Ifu⊖gHv exists then it is unique(2)
u⊖gHu=0E1(3)
u+v⊖gHv=u(4)
u⊖gHv=0E1⇔u=vDefinition 5 (see [21]).
According to Zadeh’s extension principle, the addition onE1 is defined by(8)u⊕vz=supz=x+yminux,vy.
And scalar multiplication of a fuzzy number is given by(9)keux=ux/k,k∈R−0,0E1,k=0.Remark 6 (see [23]).
Letu,v∈E1 and α∈0,1; then, we have(10)u+vα=uα+vα,u−vα=u1α−v2α,u2α−v1α,kuα=kuα=ku1α,ku2α,ifk≥0,ku2α,ku1α,ifk≤0,uvα=minu1αv1α,u1αv2α,u2αv1α,u2αv2α,maxu1αv1α,u1αv2α,u2αv1α,u2αv2α.Definition 7 (see [24]).
Letu,v∈E1 and α∈0,1; then, the Hausdorf distance between u and v is given by(11)Du,v=supα∈0,1max∣ulα−vlα∣,∣urα−vrα∣.Proposition 8 (see [25]).
D is a metric on E1 and has the following properties:(1)
E1;D is a complete metric space(2)
Du+w,v+w=Du,v, ∀u,v,w∈E1(3)
Dku,kv=∣k∣Du,v, ∀u,v∈E1 and k∈R(4)
Du+w,v+z≤Du,v+Dw,z, ∀u,v,w,z∈E1Remark 9.
Letx,y∈CJ,E1. It is easy to see that the space CJ,E1,Ds is a Banach space where(12)Dsx,y=sups∈JDxs,ys.Definition 10 (see [26]).
Letf:J⟶E1 and t0∈J. We say that f is Hukuhara differentiable at t0 if there exists f′t0∈E1 such that(13)f′t0=limh⟶0+ft0+h⊖gHft0h=limh⟶0−ft0⊖gHft0−hh.Definition 11 (see [25]).
(1)
A functionF:J⟶E1 is strongly measurable if ∀α∈0,1, the set-valued mapping Fα:J⟶PcR defined by Fαt=Ftα is Lebesgue measurable(2)
A functionF:J⟶E1 is called integrably bounded, if there exists an integrable function h such that ∣x∣<ht∀x∈F0tDefinition 12.
LetF:J⟶E1. The integral of F on J denoted by ∫JFtdt is defined by(14)∫JFtdtα=∫JFαtdt=∫Jftdtf:J⟶RisameasurableselectionforFαfor all α∈0,1.Proposition 13 (see [25]).
LetF:J⟶E1 be a fuzzy function. If F is strongly measurable and integrably bounded, then it is integrable.


### 2.1. Fractional Integral and Fractional Derivative of Fuzzy Function

Proposition 14.
Ifu∈E1, then the following properties hold:(1)
uβ⊂uα if 0≤α≤β≤1.(2)
Ifαn⊂0,1 is a nondecreasing sequence which converges to α, then(15)uα=⋂n≥1uαn.Conversely, ifAα=u1α,u2α;α∈0,1 is a family of closed real intervals verifying (1) and (2), then Aα defined a fuzzy number u∈E1 such that uα=Aα.Let0<q<1; the fractional integral of order q of a real function g:J⟶R is given by(16)It0+qgt=1Γq∫0tt−sq−1gsds.Letft∈LJ,E1 such that ft=f1αt,f2αt. Suppose that f1α,f2α∈LJ,R for all α∈0,1 and let(17)Aα=1Γq∫0tt−sq−1f1αsds,1Γq∫0tt−sq−1f2αsds,where Γ. is the Euler gamma function.We have the following lemma.Lemma 15 (see [10]).
The familyAα;α∈0,1 given by (17) defined a fuzzy number u∈E1 such that uα=Aα.Definition 16 (see [10]).
Letft∈LJ,E1.
The fuzzy fractional integral at orderq∈0,1 of f denoted by(18)It0+qft=1Γq∫t0tt−sq−1fsdsis defined by(19)It0+qftα=It0+qflt;α,It0+qfrt;α.Proposition 17 (see [10]).
Letf,g∈LJ,E1 and b∈E1; then, we have(1)
It0+qbft=bIt0+qft(2)
It0+qf+gt=It0+qft+It0+qgt(3)
Iq1Iq2ft=Iq1+q2ft, where q1,q2∈0,12Example 2.
Letx:J⟶E1 be a constant fuzzy function such that xt=u∈E1.
Ifuα=uα1,uα2, then(20)It0+qxtα=1Γq∫t0tt−sq−1uα1sds,1Γq∫t0tt−sq−1uα2sds,It0+qxtα=tqΓq+1uα1,uα2,It0+qxtα=tqΓq+1uα.Definition 18 (see [13]).
Letf∈CJ,E1∩LJ,E1. The function f is called fuzzy Caputo fractional differentiable of order 0<q<1 at t if there exists an element cDt0+qft∈E1 such that(21)cDt0+qft=1Γq∫t0tt−sq−1f′sds.Remark 19 [13].
Sinceftα=flt;α,frt;α for each α∈0,1, then(22)cDt0+qftα=cDt0+qflt;α,cDt0+qfrt;α,where(23)cDt0+qflt;α≔1Γα∫0tt−sα−1fl′s,αds,cDt0+qfrt;α≔1Γα∫t0tt−sα−1fr′s,αds.Example 3.
Letxt=u∈E1. If uα=uα1,uα2, then(24)cDt0+qxtα=1Γq∫t0tt−sq−1uα1′ds,1Γq∫t0tt−sq−1uα2′ds,cDt0+qxtα=0,cDt0+qxt=0E1.


## 2.1. Fractional Integral and Fractional Derivative of Fuzzy Function

Proposition 14.
Ifu∈E1, then the following properties hold:(1)
uβ⊂uα if 0≤α≤β≤1.(2)
Ifαn⊂0,1 is a nondecreasing sequence which converges to α, then(15)uα=⋂n≥1uαn.Conversely, ifAα=u1α,u2α;α∈0,1 is a family of closed real intervals verifying (1) and (2), then Aα defined a fuzzy number u∈E1 such that uα=Aα.Let0<q<1; the fractional integral of order q of a real function g:J⟶R is given by(16)It0+qgt=1Γq∫0tt−sq−1gsds.Letft∈LJ,E1 such that ft=f1αt,f2αt. Suppose that f1α,f2α∈LJ,R for all α∈0,1 and let(17)Aα=1Γq∫0tt−sq−1f1αsds,1Γq∫0tt−sq−1f2αsds,where Γ. is the Euler gamma function.We have the following lemma.Lemma 15 (see [10]).
The familyAα;α∈0,1 given by (17) defined a fuzzy number u∈E1 such that uα=Aα.Definition 16 (see [10]).
Letft∈LJ,E1.
The fuzzy fractional integral at orderq∈0,1 of f denoted by(18)It0+qft=1Γq∫t0tt−sq−1fsdsis defined by(19)It0+qftα=It0+qflt;α,It0+qfrt;α.Proposition 17 (see [10]).
Letf,g∈LJ,E1 and b∈E1; then, we have(1)
It0+qbft=bIt0+qft(2)
It0+qf+gt=It0+qft+It0+qgt(3)
Iq1Iq2ft=Iq1+q2ft, where q1,q2∈0,12Example 2.
Letx:J⟶E1 be a constant fuzzy function such that xt=u∈E1.
Ifuα=uα1,uα2, then(20)It0+qxtα=1Γq∫t0tt−sq−1uα1sds,1Γq∫t0tt−sq−1uα2sds,It0+qxtα=tqΓq+1uα1,uα2,It0+qxtα=tqΓq+1uα.Definition 18 (see [13]).
Letf∈CJ,E1∩LJ,E1. The function f is called fuzzy Caputo fractional differentiable of order 0<q<1 at t if there exists an element cDt0+qft∈E1 such that(21)cDt0+qft=1Γq∫t0tt−sq−1f′sds.Remark 19 [13].
Sinceftα=flt;α,frt;α for each α∈0,1, then(22)cDt0+qftα=cDt0+qflt;α,cDt0+qfrt;α,where(23)cDt0+qflt;α≔1Γα∫0tt−sα−1fl′s,αds,cDt0+qfrt;α≔1Γα∫t0tt−sα−1fr′s,αds.Example 3.
Letxt=u∈E1. If uα=uα1,uα2, then(24)cDt0+qxtα=1Γq∫t0tt−sq−1uα1′ds,1Γq∫t0tt−sq−1uα2′ds,cDt0+qxtα=0,cDt0+qxt=0E1.


## 3. Fuzzy Linear Fractional Evolution Equation

Definition 20 (see [14]).
A fuzzy functionx:J⟶E1 is calledd-increasing(d-decreasing) on J if for every α∈0,1, the real function t⟶dxtα is nondecreasing (nonincreasing), respectively.Remark 21.
Ifx:J⟶E1 is d-increasing or d-decreasing on J, then we say that xt is d-monotone on J.Definition 22.
A fuzzy functionxt is a solution of problem (1) if and only if(1)
xt is continuous and xt∈DAt for all t∈J(2)
Dct0+qxt exists and continuous on J, where 0<q<1(3)
xt satisfies (1)Lemma 23.
Ad-monotone fuzzy function xt is a solution of problem (1) if and only if(1)
xt is continuous and xt∈DAt for all t∈J(2)
xt satisfies the following integral equation(25)xt⊖gHx0=1Γq∫t0tt−sq−1Asxsds(3)
The functiont⟶It0+qAtxt is d-increasing on JProof.
See the proof of Theorem 3 in [14].Theorem 24.
Assume that
H1At is a bounded linear operator on E1 for each t∈J.
H2 The function t⟶At is continuous.
If the assumptionsH1−H2 hold, then problem (1) has a unique solution on J.Proof.
To show that problem (1) has a unique solution defined on J, we use Picard’s iteration method (see [27]).
Let(26)S=supt∈JDAt,0E1,and let T:CJ,E1⟶CJ,E1 be the operator defined as follows:(27)Txt⊖gHx0=1Γα∫t0tt−sq−1Asxsds..
Letx,y∈CJ,E1: then, we have(28)DsTxt,Tyt≤ST−t0qΓq+1DsTx,Ty,and by induction, we can write(29)DTnxt,Tnyt≤ST−t0q/Γq+1n!nDsTnx,Tny.
It follows that(30)DsTnx,Tny≤ST−t0q/Γq+1n!nDsTnx,Tny.
Finally, sinceST−t0q/Γq+1n/n!<1 for large n, then by the well-known generalization of the Banach contraction principle, the operator T has a unique fixed point x∈CJ,E1 which is the solution of problem (1).


## 4. Fuzzy Semilinear Fractional Evolution Equation

Definition 25.
A fuzzy functionxt is a solution of problem (2) if and only if(1)
xt is continuous and xt∈DAt for all t∈J(2)
xt satisfies (2)Lemma 26.
Ad-monotone fuzzy function xt is a solution of problem (2) if and only if(1)
xt is continuous and xt∈DAt for all t∈J(2)
xt satisfies the following integral equation:(31)xt⊖gHx0=1Γq∫t0tt−sq−1Asxsds+1Γq∫t0tt−sq−1fs,xsds(3)
The functiont⟶It0+qAtxt+ft,xt is d-increasing on JProof.
See the proof of Theorem 3 in [14].Theorem 27.
Assume that
H3 The function f is continuous and there exists a positive constant K such that
Dft,x,ft,y≤KDx,y for all x,y∈CJ,E1.
If the assumptionsH1–H3 hold and(32)S+KT−t0qΓq+1<12,
then problem (2) has a unique solution on J.Proof.
LetP:CJ,E1⟶CJ,E1 be the operator defined as follows:(33)Pxt⊖gHx0=1Γq∫t0tt−sq−1Asxsds+1Γq∫t0tt−sq−1fs,xsds.
Letρ be a positive real number such that p≥2Dx0,0E1+f0T−t0q/Γq+1, where f0=Dsf,0E1.
We can prove that the operatorP transforms the ball Bρ0E1,ρ into itself, where(34)Bρ0E1,ρ=y∈CJ,E1:D0E1,y≤ρ.
For this purpose, letx∈Bρ; then, we have(35)DPxt,0E1≤Dx0,0E1+SΓq∫t0tt−sq−1Dxs,0E1ds+1Γq∫t0tt−sq−1Dfs,xs,0E1ds,DPxt,0E1≤Dx0,0E1+ρST−t0qΓq+1+ρK+f0T−t0qΓq+1,DPxt,0E1≤ρ.
It follows thatPBρ⊂Bρ.
Fory,x∈CJ,E1, we have(36)DPyt,Pxt≤1Γq∫t0tt−sq−1DAsys,Asxsds+KΓq∫t0tt−sq−1Dys,xsds.
It follows that(37)DPyt,Pxt≤S+KT−t0qΓq+1Dsy,x,which implies that(38)DsPy,Px≤S+KT−t0qΓq+1Dsy,x.
Thus,(39)DsPy,Px≤12Dsy,x.
Finally,P is a contraction mapping and therefore, there exists a unique fixed point x∈Bρ0E1,ρ such that Pxt=xt which is the solution of problem (2).


## 5. Illustrative Example

In this section, we give an example to illustrate the practical usefulness of the results that we establish in the paper. By using Theorem27, we can solve the following fuzzy semilinear fractional evolution equation:(40)Dc0+2/3ut=120e−tut+e−t9+etut,t∈0,1,u0=u0..We define the operatorA by.(41)At=120e−tI,where I is the identity mapping defined on E1.And the functionf is given by ft,u=e−t/9+etut.It is clear thatAt is a bounded linear operator and the function t⟶At is continuous. Hence, the assumptions H1 and H2 are verified.On the other hand, we have(42)Dft,ut,ft,vt=De−t9+etut,e−t9+etvt,Dft,ut,ft,vt≤e−t9+etDut,vt,Dft,u,ft,v≤110Du,v.Hence, the assumptionH3 in Theorem 27 holds with K=1/10.It remains to be checked that condition (32) in Theorem 27 is also satisfied with t0=0, T=1, K=1/10, S=1/20, and α=2/3.For this purpose, we have(43)K+ST−t0αΓα+1=1/10+1/20×1−02/3Γ2/3+1=320×0.89=0.16<12.Finally, all the conditions of Theorem27 are satisfied for problem (40); then, it has a unique solution on [0,1].Remark 28.
We can show that the fuzzy semilinear fractional evolution equation (40) has a unique solution on [0,1] for some fractional-order 0<β<1. Indeed, we shall check that condition (32) in Theorem 27 is satisfied for some β∈0,1. We have(44)K+ST−t0βΓβ+1<12⟺Γβ+1>310.
Then, by using Theorem27, the fuzzy semilinear fractional evolution equation (40) has a unique solution on [0,1] for the values of β satisfying (44).


## 6. Conclusion and Future Work

In this manuscript, we studied the existence and uniqueness results of fuzzy linear and semilinear fractional evolution equations involving Caputo fractional derivative. The existence theorems are proved by using fuzzy fractional calculus, Picard’s iteration method, and Banach fixed-point theorem. As application, we present an illustrative example to show the applicability of our main results.Our future work is to extend the obtained results in this paper to the infinite dimensional case by using the theory of fuzzy operator semigroups.



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