# Existence and Continuous Dependence on Initial Data of Solution for Initial Value Problem of Fuzzy Multiterm Fractional Differential Equation

Authors: Huichol Choi; Kinam Sin; Sunae Pak; Sungryol So
Journal: Advances in Fuzzy Systems
 (2019)

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## Abstract

In this paper, the fuzzy multiterm fractional differential equation involving Caputo-type fuzzy fractional derivative of order0<α<1 is considered. The uniqueness of solution is established by using the contraction mapping principle and the existence of solution is obtained by Schauder fixed point theorem.


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## Body

## 1. Introduction

Nowadays the fractional differential equations (FDEs) are powerful tools representing many problems in various areas such as control engineering, diffusion processes, signal processing, and electromagnetism.Recently the fuzzy fractional differential equations (FFDEs) have been studied by many researchers in order to analyze some systems with fuzzy initial conditions. But, it is too difficult to find the exact solutions of most FFDEs representing real-world phenomena. Therefore, research about FFDEs can be classified into two classes, namely, existence of solution and numerical methods. Many theoretical researches have been advanced on the existence, uniqueness, and stability of solution of FFDEs [1–12].Also the analytical method and the numerical method are typical methods for solving FFDEs. The analytical method includes the Laplace transform method, monotone iterative method, variation of constant formula, and so on [13–15]. Typical numerical methods are the operational matrix method, fractional Euler method, predictor-corrector method, and so on [16–21].In [22], the existence and uniqueness of the solutions of fuzzy initial value problems of fractional differential equations with the Caputo-type fuzzy fractional derivative have been proved. Under the conditions which the right sides of equations satisfy Hölder continuity or Lipschitz continuity in its all variables, the existence of a solution to the Cauchy problem for fuzzy fractional differential equations was discussed in [23]. In [24], by employing the contraction mapping principle on the complete metric space, the existence and uniqueness result for fuzzy fractional functional integral equation has been proved. The existence results of solutions for fuzzy fractional initial value problem under generalized differentiability conditions are obtained by Banach fixed point theorem in [25]. In [26], researchers discussed the uniqueness and existence of the solutions for FFDEs with Riemann-Liouville H-differentiability of arbitrary order by using Krasnoselskii-Krein type conditions, Kooi type conditions, and Rogers conditions. But, the considerations of researchers in [22–26] were restricted to the case of FFDEs with single derivative term. Ngo et al. [27] presented that the existence and uniqueness results of the solution for fuzzy Caputo-Katugampola (CK) fractional differential equations with initial value and in [28] proved that the fractional fuzzy differential equation is not equal to the fractional fuzzy integral equation in general.Based on the above facts, in this paper, we study the existence and uniqueness of solutions for fuzzy multiterm fractional differential equations of order0<α<1 with fuzzy initial value under Caputo-type H-differentiability.The paper is organized as follows. In Section2, we introduced some definitions and properties of fuzzy fractional calculus. The existence result of solution for proposed problem is described in Section 3. Section 4 presented the continuous dependence on initial data of solution. Finally, the conclusion is summarized in Section 5.


## 2. Preliminaries and Basic Results

We introduce some definitions and notations which will be used throughout our paper.Definition 1 (see [29]).
Let us denote byRF the class of fuzzy subsets u:R→[0,1] satisfying the following properties: (i)
u is normal,i.e., ∃x0∈R for which u(x0)=1,(ii)
u is fuzzy convex,i.e., u(λx+(1-λ)y)≥min{u(x),u(y)} for all ∀x,y∈R, ∀λ∈[0,1],(iii)
u is upper semicontinuous on R,(iv)
suppu=x∈R∣ux>0 is the support of the u, and its closure cl(suppu) is compact.
ThenRF is called the space of fuzzy number and any u∈RF is called fuzzy number.
We denote ther-cut form of fuzzy number u∈RF, 0≤r≤1, by [u]r≔[u1(r),u2(r)].
Also let usu,v∈RF. The metric d:RF×RF→[0,+∞) on RF is defined as follows:(1)du,v≔supr∈0,1⁡max⁡u1r-v1r,u2r-v2r.Definition 2 (see [19]).
Letu,v∈RF. If there exists w∈RF such that u=v⊕w, then w is called the H-difference of uu and v, and it is denoted by w=u⊝Hv. Note that u⊝Hv≠u+(-1)v.Definition 3 (see [19]).
Letf:(a,b)→RF and x0∈(a,b). We say that f is H-differentiable at x0, if for h>0 sufficiently near to 0, there exist the H-differences f(x0+h)⊝Hf(x0), f(x0)⊝Hf(x0-h), and the limits(2)limh→0+⁡fx0+h⊝Hfx0h=limh→0+⁡fx0⊝Hfx0-hh.
Then the limit is denoted byD(1)f(x0).Theorem 4 (see [14]).
Letf:[a,b]→RF be H-differentiable and [f(x)]r≔[f1(x,r),f2(x,r)]. Then f1(x,r),f2(x,r) are all differentiable and (3)D1fxr=f1′x,r,f2′x,r.Definition 5 (see [14]).
A functionf:[a,b]→RF is said to be Riemann integrable on [a,b], if ∃IR∈RF, ∀ε>0, ∃δ>0, for any division of [a,b], Δ:a=t0<t1<⋯<tn=b with norm λ(Δ)<δ, and for any points ξi∈[ti,ti+1], i=0,1,…,n-1,(4)d∑i=0n-1fξi·ti+1-ti,IR<ε.We denote the fuzzy Riemann integral off from a to b by IR=∫abf(t)dt.Lemma 6 (see [14]).
Suppose thatf,g:[a,b]→RF are continuous, then
(i)f is the fuzzy Riemann integrable on [a,b] and F(t)=∫atf(s)ds is differentiable as in Definition 3, namely, D(1)F(t)=f(t).
(ii)d∫abf(s)ds,∫abg(s)ds≤∫abdf(s),g(s)ds.
We introduce following notations:
C F ( I ) is the set of all continuous fuzzy-valued functions on I.
A C F ( I ) is the set of all absolutely continuous fuzzy-valued functions on I.
L F ( I ) is the space of all Lebesque integrable fuzzy-valued functions on I, where I=[0,L] and without losing generality, we promise that L=1.Definition 7 (see [19]).
Letf∈CF(I)∩LF(I). The fuzzy Riemann-Liouville fractional integral of the fuzzy-valued function f is defined as follows:(5)I0+βfx=1Γβ∫0xx-sβ-1fsds,x>0.where I0+β is the Riemann-Liouville integral operator of β and Γ(β) is the Gamma function.Lemma 8 (see [19]).
Letf∈CF(I)∩LF(I). Then Riemann-Liouville integral of the fuzzy-valued function f, based on its r-cut form, can be expressed as follows:(6)I0+βfxr=I0+βf1x,r,I0+βf2x,r,0≤r≤1,where (7)I0+βfix,r≔1Γβ∫0xfis,rx-s1-βds,i=1,2.Lemma 9.
Letf∈CF(I), α,β>0. Then the following relations are satisfied:(8)I0+αI0+βfx=I0+βI0+αfx=I0+α+βfx.Proof.
Let us denote ther-cut form of f by [f(x)]r≔[f1(x,r),f2(x,r)].
Then we have(9)I0+αI0+βfxr=I0+αI0+βf1x,r,I0+αI0+βf2x,r=I0+βI0+αf1x,r,I0+βI0+αf2x,r=I0+βI0+αfxr.
Moreover, sincef∈CF(I), we get(10)I0+αI0+βfxr=I0+α+βf1x,r,I0+α+βf2x,r=I0+α+βfxr.Definition 10 (see [19]).
Let0<β<1. We say that f is fuzzy Riemann-Liouville H-differentiable of order β if (11)I0+1-βfx=1Γ1-β∫0xfsx-sβds,x>0is H-differentiable. Then fuzzy Riemann-Liouville H-derivative of order β of function f is denoted by D0+βRLf(x)≔D(1)I0+1-βf(x).Definition 11 (see [19]).
Let0<β<1. We say that f is a fuzzy Caputo-type differentiable function if H-difference f(x)⊝Hf(0) exists and I0+1-β(f(x)⊝Hf(0))∈ACF(I) satisfies. Then fuzzy Caputo-type derivative of order β of function f is denoted by(12)D0+βcfx≔D0+βRLfx⊝Hf0,x>0.Lemma 12 (see [19]).
Letf∈CF(I)∩LF(I) and [f(x)]r≔[f1(x,r),f2(x,r)] for ∀r∈[0,1]. If f is a fuzzy Caputo-type fractional differentiable function, then(13)D0+βcfxr=I0+1-βDfxr=I0+1-βDf1x,r,I0+1-βDf2x,r.Lemma 13.
Letf:[a,b]→RF be H-differentiable. Then the following relations hold:(i)
f∈CF(0,1]⇒D0+βcI0+βf(x)=f(x).(ii)
f∈CF(0,1]⇒D0+βRLI0+βf(x)=f(x).(iii)
0<λ<β, f∈CF(0,1]⇒D0+λcI0+βf(x)=I0+β-λf(x).Proof.
First we prove (i). From the assumption of Lemma13, we have (14)D0+βcI0+βfx=D0+βRLI0+βfx⊝HI0+βf0=D1I0+1-βI0+βfx⊝HI0+βf0,=D1I0+1-βI0+βfx=D1I0+1fx.
By Lemma6 (i), we get (15)D1I0+1fx=fx.
From the result (i) of lemma, it is obvious that (ii) holds.
Next let prove (iii).(16)D0+λcI0+βfx=D0+λRLI0+βfx⊝HI0+βf0=D1I0+1-λI0+βfx⊝HI0+βf0=D1I0+1-λI0+βfx=D1I0+1+β-λfx.
By Lemma6 (i), we obtain (17)D1I0+1+β-λfx=I0+β-λfx.Lemma 14.
The following facts are true:
(i) Letz∈CF(I). For any positive number σ, the fractional integral I0+σz(x) is continuous in x.
(ii) For any positive numbersσ and k, it holds that I0+σekx≤ekx/kα.
(iii) Letu,v∈CF(I). For any positive number σ, it holds that(18)dI0+σux,I0+σvx≤I0+σdux,vx.Proof.
Let us consider the assertion (i). For anyx0∈(0,1], it is enough to prove that (19)limx→x0⁡dI0+σzx,I0+σzx0=0.
We use the notation[z(x)]r≔[z1(x,r),z2(x,r)] for r-cut representation of z(x).
Sincez∈CF(I), z1(r,x), and z2(r,x) are continuous in x. And by Lemma 8, the following expression holds:(20)dI0+σzx,I0+σzx0=supr∈0,1⁡max⁡I0+σz1x,r-I0+σz1x0,r,I0+σz2x,r-I0+σz2x0,r.
It is well known thatI0+σzi(x,r), i=1,2, are continuous with respective to x. Thus we can see that limx→x0⁡d(I0+σz(x),I0+σz(x0))=0.
Now we prove (ii).
From the definition of fractional integral, the following is true:(21)I0+αekt=1Γα∫0tt-sα-1eksds=1Γα∫0tzα-1ekt-zdz=ektΓα∫0tzα-1e-ktdz=ektΓα∫0ktskα-1e-s1kds=ektΓα·kα·∫0ktsα-1e-sds
By the definition of Gamma function, we have(22)≤ektΓα·kα·∫0∞sα-1e-sds=ektΓα·kαΓα=ektkα.
Let us consider assertion (iii).
We use the notations[u(x)]r≔[u1(x,r),u2(x,r)],[v(x)]r≔[v1(x,r),v2(x,r)] for r-cut representations of u(x),v(x), respectively.
Sinceu,v∈CF(I), u1(x,r),u2(x,r),v1(x,r),v2(x,r) are continuous in x. Therefore the following evaluations are true:(23)dI0+σux,I0+σvx=supr∈0,1⁡max⁡I0+σu1x,r-I0+σv1x,r,I0+σu2x,r-I0+σv2x,r=supr∈0,1⁡max⁡I0+σu1x,r-v1x,r,I0+σu2x,r-v2x,r≤supr∈0,1⁡max⁡I0+σu1x,r-v1x,r,I0+σu2x,r-v2x,r≤I0+σsupr∈0,1⁡max⁡u1x,r-v1x,r,u2x,r-v2x,r=I0+σdz1x,z2x.
The proof is completed.


## 3. Existence of Solution for Fuzzy Multiterm Fractional Differential Equation

Let us consider the existence of solution for initial value problem of following fuzzy multiterm fractional differential equation:(24)D0+αcyx=fx,yx,D0+βcyx,x∈0,1,y0=y0,0<β<α<1,where f:I×RF×RF→RF, y0∈RF, and D0+αc is fuzzy Caputo-type derivative.Definition 15.
Lety:I→RF. We say that y is the solution of initial value problem (24) if D0+βcy(x)∈CF(I) holds and y satisfies (24).
Now let us consider the following:(25)D0+αcyx=zx,x∈0,1,y0=y0,y0∈RF,where z(x)∈CF(I).Lemma 16.
The solution of initial value problem of fuzzy fractional differential equation (25) is represented as (26)yx=y0⊕1Γα∫0xzsx-s1-αds.Proof.
Lety be the solution of initial value problem (25). Then we have (27)D0+αcyx≡zx,x∈I.
Also sincez is the fuzzy continuous, it is the fuzzy integrable and I0+αz(x) exists for x∈I.
Therefore the following relations hold:(28)I0+αD0+αcyx≡I0+αzx,x∈I,I0+αD0+αRLyx⊝Hy0≡I0+αzx,I0+αD1I0+1-αyx⊝Hy0≡I0+αzx.
From the Caputo-type differentiability of fuzzy-valued functiony, we get (29)I0+1-αyx⊝Hy0∈ACFIand since the space of the absolutely continuous functions coincides with the space of primitive functions of Lebesque integrable functions, the following relation is satisfied: (30)∃φ∈LFI;I0+1-αyx⊝Hy0=I0+1φx.
Therefore we obtain(31)yx⊝Hy0=I0+αφx.
By (30) and Lemma 6 (i), the left side of (28) exchanges as (32)I0+αD1I0+1-αyx⊝Hy0=I0+αD1I0+1φx=I0+αφx.
From (28) and (31), y(x)⊝Hy0=I0+αz(x) is satisfied. Namely, we have (33)yx=y0⊕I0+αzx.
Conversely, we prove thaty denoted by (26) is the solution of fuzzy initial value problem (25). Since (34)yx=y0⊕1Γα∫0xzsx-s1-αds=y0⊕I0+αzx,y(x)⊝Hy0=I0+αz(x) holds. Also as z∈CF(I), we get(35)I0+1-αyx⊝Hy0=I0+1zx.
On the other hand, when[z(x)]r≔[z1(x,r),z2(x,r)], by Lemma 8(36)I0+1zxr=∫0xz1s,rds,∫0xz2s,rdsholds ∀r∈[0,1]. Therefore (37)I0+1zx+hr=∫0x+hz1s,rds,∫0x+hz2s,rds=∫0xz1s,rds+∫xx+hz1s,rds,∫0xz2s,rds+∫xx+hz2s,rds=∫0xz1s,rds,∫0xz2s,rds+∫xx+hz1s,rds,∫xx+hz2s,rds.
Since the interval family∫xx+hz1(s,r)ds,∫xx+hz2(s,r)dsr generates obviously a fuzzy number, we can see that there exist the H-differences as (38)I0+1zx+h⊝HI0+1zx,I0+1zx⊝HI0+1zx-h.
Consequently the H-differentiability ofy is leaded.
From (35), we obtain (39)D1I0+1-αyx⊝Hy0=D1I0+1zx=zx.
Also it is obvious thaty satisfies the initial condition.Theorem 17.
Letf of (24) be a fuzzy continuous with respect to every variable. If y(x) is the solution of initial value problem (24), the fuzzy-valued function z(x) which is constructed by z(x)≔D0+αcy(x) is the solution in CF(I) of fuzzy integral equation as(40)zx=fx,y0⊕1Γα∫0xzsx-s1-αds,I0+α-βzx,x∈0,1.Conversely if z(x) is the solution in CF(I) of fuzzy integral equation (40), y(x) which is constructed by (26) is the solution of initial value problem (24).Proof.
Lety(x) be the solution of initial value problem (24). Namely, let assume that y(x) satisfies (41)D0+αcyx=fx,yx,D0+βcyx,x∈0,1,y0=y0,y0∈RF.
Then ifz(x) is denoted by z(x)≔D1αcy(x), by Lemma 16, the following relation is leaded:(42)yx=y0⊕1Γα∫0xzsx-s1-αds=y0⊕I0+αzx.
Applying the operatorD0+βc to the both side of above equation, by Lemma 13 (iii), we get (43)D0+βcyx=I0+α-βzx.
Therefore substituting the above results toD0+αcy(x), D0+βcy(x),y(x) of (41), we obtain (40).
Next letz(x) be the solution of fuzzy integral equation (40). Then it is obvious that (44)yx=y0⊕I0+αzxsatisfies the initial condition of problem (24) from the continuousness of z(x).
Namely(45)y0=y0.
Consequently from (44)(46)yx⊝Hy0=yx⊝Hy0=I0+αzxis leaded and regarding the above equation, we get(47)I0+1-αyx⊝Hy0=I0+1-αI0+αzx=I0+1zx,D1I0+1-αyx⊝Hy0=D1I0+1zx=zx,D0+αcyx=zx=fx,ys,I0+α-βzx.
Moreover since(48)D0+βcyx=D0+βcI0+αzx⊕y0=D0+βcI0+αzx=I0+α-βzx,we obtain (49)D0+αcyx=fx,ys,D0+βcyx.Now we employ the following metric structure inCF(I):(50)∀u,v∈CFI,d∗u,v≔maxt∈I⁡dut,vt.Obviously we can see that(CF(I),d∗) is a complete metric space (see [24]).For any positive numberk, we can consider the metric structure as(51)∀u,v∈CFI,dk∗u,v≔maxt∈I⁡e-ktdut,vt.Then the metricdk∗ is equivalent to the metric d∗. Namely, (52)∃M,m>0;∀u,v∈CFI,mdk∗u,v≤d∗u,v≤Mdk∗u,v.Theorem 18.
Assume that the functionf in (40) is continuous in its all variables and especially, for any y1,y2,z1,z2∈RF, f satisfies the following condition:(53)dfx,y1,z1,fx,y2,z2≤L1·dy1,y2+L2·dz1,z2.Then the fuzzy integral equation (40) has a unique solution.Proof.
Sinceα,α-β>0, there exists k∗>0 that inequality L11/k∗α+L21/k∗α-β<1 is true.
Therefore for anyk∗ satisfying this inequality, we put as follows:(54)q≔L11k∗α+L21k∗α-β.
Also we define the operatorT by (55)Tzx≔fx,y0⊕1Γα∫0xzsx-s1-αds,I0+α-βzx.
For anyz∈CF(I), by (i) of Lemma 14, I0+αz(x),I0+α-βz(x) are continuous and f is continuous by assumptions of theorem. Thus the operator T is a map from CF(I) to CF(I).
Thus the fuzzy integral equation (40) is represented as (56)z=Tz,z∈CFI.
The existence of solution for the fuzzy integral equation (40) is equivalent to the existence of the fixed point of the operator T in CF(I).
For anyz1,z2∈CF(0,1], let us evaluate d(Tz1(x),Tz2(x)).(57)dTz1x,Tz2x=dfx,y0⊕1Γα∫0xz1sx-s1-αds,I0+α-βz1x,fx,y0⊕1Γα∫0xz2sx-s1-αds,I0+α-βz2x≤L1dy0⊕1Γα∫0xz1sx-s1-αds,y0⊕1Γα∫0xz2sx-s1-αds+L2·dI0+α-βz1x,I0+α-βz2x≤L1d1Γα∫0xz1sx-s1-αds,1Γα∫0xz2sx-s1-αds+L2·dI0+α-βz1x,I0+α-βz2xBy (iii) of Lemma 14, we have(58)≤L11Γα∫0xdz1s,z2sx-s1-αds+L2·I0+α-βdz1x,z2x=L11Γα∫0xe-k∗sek∗sdz1s,z2sx-s1-αds+L2·I0+α-βe-k∗xek∗xdz1x,z2x≤L11Γα∫0xek∗sx-s1-αds·dk∗∗z1,z2+L2·I0+α-βek∗s·dk∗∗z1,z2By (ii) of Lemma 14, we have(59)≤L1ek∗xk∗α+L2·ek∗xk∗α-β·dk∗∗z1,z2.
Consequently the following inequality is obtained:(60)dTz1x,Tz2x≤L1ek∗xk∗α+L2·ek∗xk∗α-β·dk∗∗z1,z2.
Multiplying the both sides of the above equation bye-k∗x, we obtain that(61)e-k∗xdTz1x,Tz2x≤L11k∗α+L2·1k∗α-β·dk∗∗z1,z2and(62)dk∗∗Tz1,Tz2≤L11k∗α+L2·1k∗α-β·dk∗∗z1,z2=q·dk∗∗z1,z2.
Thus the operatorT is contractive on CF(I) with respect to the distance dk∗∗ and we obtain the unique fixed point CF(I) of the operator T by contraction mapping principle. By the way, since the distance dk∗∗ is equivalent to d∗ in CF(I), z∗ is also the unique fixed point in sense of the distance d∗. This completes the proof of theorem.Next let us consider the existence of solution in case which have not satisfied the Lipschitz condition.Lemma 19 (Schauder fixed point theorem).
Assume that(E,d) is the complete metric space, U is a nonempty convex closed subset of E, and A is a continuous mapping of U into itself such that A(U) is contained in a compact subset of U, then A has a fixed point in U.Theorem 20.
Suppose that the followings conditions are satisfied:
(i)∃r>d∗(f(·,y0,0^),y0),L1,L2,L3>0,∀x1,x2∈(0,1],∀y1,y2∈Ur(y0),∀z1,z2∈Ur(0^), (63)dfx1,y1,z1,fx2,y2,z2≤L1·dy1,y2+L2·dz1,z2+L3·x1-x2,where Ur(y0)≔u∣d∗(u,y0)≤r,u∈CF(I) and 0^ is zero fuzzy number.
(ii)q∗≔L11/Γ(α+1)+L2·1/Γ(α-β+1)<1.
Let(64)x∗≔max⁡x∣L1·xα+L2·xα-β≤minΓα+1,Γα-β+1r-d∗fx,y0,0^,y0r,xαΓα+1≤1,xαΓα-β+1≤1and(65)U-ry0≔u∣d∗u,y0≤r,u∈CF0,x∗,U-r0^≔u∣d∗u,0^≤r,u∈CF0,x∗.Then the following results hold:(i)
T:U-r(0^)→U-r(0^),(ii)
Tu∣u∈U-r0^ is relatively compact.Proof.
Firstly, let us prove thatT:U-r(0^)→U-r(0^).
We have that, for anyu∈U-r(0^),(66)dy0⊕1Γα∫0xusx-s1-αds,y0≤d1Γα∫0xusx-s1-αds,0^≤1Γα∫0xdus,0^x-s1-αds≤xαd∗u,0^Γα+1≤r.
Thus we can get thaty0⊕1/Γ(α)∫0xu(s)/(x-s)1-αds∈U-r(y0).
Also by (iii) of Lemma14, we have that, for any u∈U-r(0^),(67)dI0+α-βux,0^≤I0+α-βdux,0^≤xα-βΓα-β+1d∗ux,0^≤r.
Thus we can see thatI0+α-βu(x)∈U-r(0^).
Therefore we can get that for anyu∈U-r(0^),(68)dTux,y0=dfx,y0⊕1Γα∫0xusx-s1-αds,I0+α-βux,y0≤dfx,y0⊕1Γα∫0xusx-s1-αds,I0+α-βux,fx,y0,0^+dfx,y0,0^,y0≤L1·dy0⊕1Γα∫0xusx-s1-αds,y0+L2·dI0+α-βux,0^+dfx,y0,0^,y0≤L1·d1Γα∫0xusx-s1-αds,0^+L2·dI0+α-βux,0^+dfx,y0,0^,y0 By (iii) of Lemma 14, we have(69)≤L11Γα∫0xdus,0^x-s1-αds+L2·I0+α-βdux,0^+dfx,y0,0^,y0≤L1·xα·rΓα+1+L2·xα-β·rΓα-β+1+dfx,y0,0^,y0≤L1·x∗α·rmin⁡Γα+1,Γα-β+1+L2·x∗α-β·rmin⁡Γα+1,Γα-β+1+dfx,y0,0^,y0≤r.
Therefore the operatorT is a continuous mapping from convex closed subset U-r(0^) into itself; namely,(70)T:U-r0^→U-r0^.
Next let us prove (ii). The uniformly boundness ofTu∣u∈U-r0^ is obvious from (70).
Let us consider the equicontinuity ofTu∣u∈U-r0^. For any y∈Tu∣u∈U-r0^, there exists z∈U-r(0^) which satisfies y=Tz.
Forx1,x2∈(0,x∗], we estimate d(y(x1),y(x2))=d(Tz(x1),Tz(x2)).
Without losing generality, letx1<x2≤x∗.
Then we have(71)
d
T
z
x
1
,
T
z
x
2
=
d
f
x
1
,
y
0
⊕
1
Γ
α
∫
0
x
1
z
s
x
1
-
s
1
-
α
d
s
,
I
0
+
α
-
β
z
x
x
=
x
1
,
(72)
f
x
2
,
y
0
⊕
1
Γ
α
∫
0
x
2
z
s
x
2
-
s
1
-
α
d
s
,
I
0
+
α
-
β
z
x
x
=
x
2
=
d
f
x
1
,
y
0
⊕
1
Γ
α
∫
0
x
1
z
s
x
1
-
s
1
-
α
d
s
,
1
Γ
α
-
β
∫
0
x
1
z
s
x
1
-
s
1
-
α
-
β
d
s
,
f
x
2
,
y
0
⊕
1
Γ
α
∫
0
x
2
z
s
x
2
-
s
1
-
α
d
s
,
1
Γ
α
-
β
∫
0
x
2
z
s
x
2
-
s
1
-
α
-
β
d
s
≤
d
f
x
1
,
y
0
⊕
1
Γ
α
∫
0
x
1
z
s
x
1
-
s
1
-
α
d
s
,
1
Γ
α
-
β
∫
0
x
1
z
s
x
1
-
s
1
-
α
-
β
d
s
,
f
x
2
,
y
0
⊕
1
Γ
α
∫
0
x
1
z
s
x
1
-
s
1
-
α
d
s
,
1
Γ
α
-
β
∫
0
x
1
z
s
x
1
-
s
1
-
α
-
β
d
s
+
d
f
x
2
,
y
0
⊕
1
Γ
α
∫
0
x
1
z
s
x
1
-
s
1
-
α
d
s
,
1
Γ
α
-
β
∫
0
x
1
z
s
x
1
-
s
1
-
α
-
β
d
s
,
f
x
2
,
y
0
⊕
1
Γ
α
∫
0
x
2
z
s
x
2
-
s
1
-
α
d
s
,
1
Γ
α
-
β
∫
0
x
2
z
s
x
2
-
s
1
-
α
-
β
d
s
≤
L
3
x
1
-
x
2
+
L
1
d
y
0
⊕
1
Γ
α
∫
0
x
1
z
s
x
1
-
s
1
-
α
d
s
,
y
0
⊕
1
Γ
α
∫
0
x
2
z
s
x
2
-
s
1
-
α
d
s
+
L
2
d
1
Γ
α
-
β
∫
0
x
1
z
s
x
1
-
s
1
-
α
-
β
d
s
,
1
Γ
α
-
β
∫
0
x
2
z
s
x
2
-
s
1
-
α
-
β
d
s
=
L
3
x
1
-
x
2
+
L
1
d
1
Γ
α
∫
0
x
1
z
s
x
1
-
s
1
-
α
d
s
,
1
Γ
α
∫
0
x
2
z
s
x
2
-
s
1
-
α
d
s
+
L
2
d
1
Γ
α
-
β
∫
0
x
1
z
s
x
1
-
s
1
-
α
-
β
d
s
,
1
Γ
α
-
β
∫
0
x
2
z
s
x
2
-
s
1
-
α
-
β
d
s
.
Now we estimate the second term of the above expression.(73)d1Γα∫0x1zsx1-s1-αds,1Γα∫0x2zsx2-s1-αds=d1Γα∫0x1zsx1-s1-αds,1Γα∫0x1zsx2-s1-αds⊕1Γα∫x1x2zsx2-s1-αds≤d1Γα∫0x1zsx1-s1-αds,1Γα∫0x1zsx2-s1-αds+d1Γα∫x1x2zsx2-s1-αds,0^≤rΓα∫0x11x1-s1-α-1x2-s1-αds+rΓα∫x1x21x2-s1-αds≤2rΓ1+αx2-x1α+rΓ1+αx2α-x1α.
Estimating the third term of above equation similarly to above, we have(74)d1Γα-β∫0x1zsx1-s1-α-βds,1Γα-β∫0x2zsx2-s1-α-βds≤2rΓ1+α-βx2-x1α-β+rΓ1+α-βx2α-β-x1α-β.
So we obtain as follows:(75)dTzx1,Tzx2≤L3x1-x2+2rL1Γ1+αx2-x1α+rL1Γ1+αx2α-x1α+2rL2Γ1+α-βx2-x1α-β+rL2Γ1+α-βx2α-β-x1α-β.
From right side of the above expression, we can see thatTu∣u∈U-r0^ is equicontinuous.
By Arzelà-Ascoli theorem,Tu∣u∈U-r0^ is relatively compact.By Theorem20 and Schauder fixed point theorem, we can see the operator T has a fixed point in U-r(0^).


## 4. Continuous Dependence on Initial Condition of Solution

We consider the continuous dependence on initial condition of solution for proposed problem (24). Let y1(t,y1,0), y2(t,y2,0) are solutions with initial values y1,0,y2,0∈RF and z1(t,y1,0), z2(t,y2,0) are solutions of corresponding integral equation.The following theorem shows the continuous dependence on initial condition of solution.Theorem 21.
Suppose that the following conditions are satisfied:(i)
∃L1,L2>0;∀x∈[0,1],∀y1,y2,z1,z2∈CF(I),(76)dfx,y1,z1,fx,y2,z2≤L1·dy1,y2+L2·dz1,z2.(ii)
q∗≔L11/Γ(α+1)+L2·1/Γ(α-β+1)<1.
Then following relation holds:(77)dy1t,y1,0,y2t,y2,0≤1+L11-q∗Γα+1dy1,0,y2,0.Proof.
Sincez1(t,y1,0),z2(t,y2,0) are solutions of the integral equation (40), we have (78)z1x,y1,0≡fx,y1,0⊕I0+αz1x,y1,0,I0+α-βz1x,y1,0,z2x,y2,0≡fx,y2,0⊕I0+αz2x,y2,0,I0+α-βz2x,y2,0.
Now we estimated(z1(x,y1,0),z2(x,y2,0)).(79)dz1x,y1,0,z2x,y2,0=dfx,y1,0⊕I0+αz1x,y1,0,I0+α-βz1x,y1,0,fx,y2,0⊕I0+αz2x,y2,0,I0+α-βz2x,y2,0≤L1dy1,0⊕I0+αz1x,y1,0,y2,0⊕I0+αz2x,y2,0+L2dI0+α-βz1x,y1,0,I0+α-βz2x,y2,0≤L1dy1,0,y2,0+dI0+αz1x,y1,0,I0+αz2x,y2,0+L2dI0+α-βz1x,y1,0,I0+α-βz2x,y2,0 By (iii) of Lemma 14, we have(80)≤L1dy1,0,y2,0+I0+αdz1x,y1,0,z2x,y2,0+L2I0+α-βdz1x,y1,0,z2x,y2,0≤L1dy1,0,y2,0+L1Γα+1d∗z1,z2+L2Γα-β+1d∗z1,z2=L1dy1,0,y2,0+q∗d∗z1,z2.
Consequently, we get(81)maxx∈0,1dz1x,y1,0,z2x,y2,0=d∗z1,z2≤L1dy1,0,y2,0+q∗d∗z1,z2,d∗z1,z2≤L11-q∗dy1,0,y2,0.
Next we estimated(y1(t,y1,0),y2(t,y2,0)).(82)dy1t,y1,0,y2t,y2,0=dy1,0⊕1Γα∫0xz1s,y1,0x-s1-αds,y2,0⊕1Γα∫0xz2s,y1,0x-s1-αds≤dy1,0,y2,0+d1Γα∫0xz1s,y1,0x-s1-αds,1Γα∫0xz2s,y1,0x-s1-αdsBy (iii) of Lemma 14, we have(83)≤dy1,0,y2,0+1Γα∫0xdz1s,y1,0,z2s,y1,0x-s1-αds≤dy1,0,y2,0+1Γα+1d∗z1,z2.
By (81), we obtain(84)dy1t,y1,0,y2t,y2,0≤1+L11-q∗Γα+1dy1,0,y2,0.


## 5. Conclusions

We obtained the uniqueness results by employing contraction mapping principle and the existence results by using Schauder fixed point theorem for the solution of the fuzzy multiterm fractional differential equation involving Caputo-type fuzzy fractional derivative of order0<α<1 on complete metric space of continuous fuzzy number value functions. We also established the continuous dependence of solution on its initial condition. Our results can be expanded to the case in which the right side of equation involves more than two derivative terms.



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*Source: 6457548-2019-06-19.xml*