# The Ulam Stability of Fractional Differential Equation with the Caputo-Fabrizio Derivative

Authors: Shuyi Wang
Journal: Journal of Function Spaces
 (2022)

---

## Abstract

The aim of this paper is to establish the Ulam stability of the Caputo-Fabrizio fractional differential equation with integral boundary condition. We also present the existence and uniqueness results of the solution for the Caputo-Fabrizio fractional differential equation by Krasnoselskii’s fixed point theorem and Banach fixed point theorem. Some examples are provided to illustrate our theorems.


---

## Body

## 1. Introduction

Ulam [1] proposed to study the approximation degree of the approximate solution and the exact solution of the equation in 1940. Hyers [2] responded to Ulam’s proposal and defined the Hyers-Ulam stability of equation in 1941. Later on, Rassias [3] extended Hyers’s work and defined the Hyers-Ulam-Rassias stability of equation in 1978. The Hyers-Ulam stability and Hyers-Ulam-Rassias stability are collectively referred to as the Ulam stability. Subsequently, researchers initiated a research on the Ulam stability of integer-order differential equations (see [4–10]). Obloza [4], Cemil and Emel [5] proved the Hyers-Ulam stability and Hyers-Ulam-Rassias stability of the first-order differential equation, respectively. Wang et al. [6] studied the Ulam stability of the first-order differential equation with a boundary value condition. Otrocol and Ilea [7] obtained the Ulam stability of the first-order delay differential equation. Huang and Li [8] also obtained the Hyers-Ulam stability of another class of the first-order delay differential equation. Zada et al. [9] studied the Hyers-Ulam-Rassias stability of the higher order delay differential equation. However, the study on the Ulam stability of fractional differential equations is in its infancy.Fractional differential equations are widely applied in physics [11, 12], control systems [13], chemical technology [14], and biosciences [15]. Fractional integral boundary value problems have been explored by many researchers. In particular, the integral boundary value problem provides a feasible method for the modeling of population dynamics and chemical engineering problems (see [16–18]). Although fractional integral boundary value problems are widely used, it is not easy to solve the equation, and the exact solution is often not obtained. Therefore, it is necessary to study the Ulam stability of fractional differential equations and use the approximate solution to replace the exact solution. So far, researchers have studied the Ulam stability and the existence and uniqueness of a solution for fractional differential equations with Hilfer-Hadamard, Caputo, and Caputo-Fabrizio fractional derivatives (see [19–22]). Abbas et al. [19] proved the existence and the Ulam stability of a fractional differential equation with the Hilfer-Hadamard derivative.In [20], Wang et al. established the Ulam stability and data dependence for the Caputo fractional differential equation
(1)Dcβxt=kt,xt,t∈a,+∞.In [21], Dai et al. studied the Ulam stability of the Caputo fractional differential equation with an integral boundary condition
(2)x′t+cD0+βxt=kt,xt,t∈0,1,x1=I0+γxη,where I0+γ· is the Riemann-Liouville fractional integral, γ>0.In [22], Liu et al. obtained the Hyers-Ulam stability and the existence of solutions for the Caputo-Fabrizio fractional differential equation
(3)DCFβxt=kt,xt,t∈0,T,where CFDβ· is the Caputo-Fabrizio fractional derivative, β∈0,1.Motivated by [20–22], in this paper, our purpose is to study the existence and uniqueness of a solution and the Ulam stability of the following Caputo-Fabrizio fractional differential equation with boundary value condition:
(4)x′t+DCFβxt=kt,xt,t∈0,1,x1=I0+γxξ,where xt is a continuous differentiable function on 0,1; k:0,1×ℝ⟶ℝ is continuous; CFDβ· is the Caputo-Fabrizio fractional derivative, β∈0,1; and I0+γ· is the Riemann-Liouville fractional integral, γ>0, ξ∈0,1.Equation (4) is a new kind of the Korteweg-de Vries-Bergers (KDVB) equation model. In [23], Equation (4) is used to describe unusual irregularities and nonlinearities in wave dynamics and liquids motions.The main contributions are as follows: Firstly, we give the definitions of the Hyers-Ulam stability and Hyers-Ulam-Rassias stability for Equation (4). Then, we obtain a sufficient condition to derive the uniqueness of the solution for Equation (4) by the Banach contraction principle. Next, we give a sufficient condition to prove the existence of the solution for Equation (4) by Krasnoselskii’s fixed point theorem. On this basis, we give the Ulam stability results for Equation (4) by the Laplace transform and inequality results.The rest of our article is arranged as follows. Some basic definitions and necessary theorems are presented in Section2. We establish sufficient conditions to show existence and uniqueness of solution for the Caputo-Fabrizio fractional differential equation in Section 3. In Section 4, we prove the Ulam stability of the Caputo-Fabrizio fractional differential equation. Two examples are provided in Section 5 to illustrate our theorems.


## 2. Preliminaries

We will denote byC10,1 the space of continuous differentiable functions on 0,1 with norm
(5)x=supxt,t∈0,1.Definition 1 [24].
The Caputo-Fabrizio fractional derivative of orderβ of a continuous differentiable function x is given by
(6)DCFβxt=2−βMβ21−β∫0texp−β1−βt−τx′τdτ,t≥0,
the normalization functionMβ depends on β.Definition 2 [25].
The Riemann-Liouville fractional integral of orderγ of a function x is given by
(7)I0+γxt=1Γγ∫0tt−τγ−1xτdτ,t≥0.Based on Definition 2 in [5] and Definition 2.1 in [9], we give the definitions of the Hyers-Ulam stability and the Hyers-Ulam-Rassias stability for Equation (4).Definition 3.
Equation (4) has the Hyers-Ulam stability if and only if for any solution xt of
(8)x′t+DCFβxt−kt,xt≤ε,t∈0,1,where ε>0, there is a constant C>0 and a solution yt of Equation (4) satisfying
(9)xt−yt≤C×ε,t∈0,1.Definition 4.
Equation (4) has the Hyers-Ulam-Rassias stability if and only if for any solution xt of
(10)x′t+CFDβxt−kt,xt≤δt,t∈0,1,where δt∈C0,1,R+, there is a constant Kk,δ>0 and a solution yt of Equation (4) satisfying
(11)xt−yt≤Kk,δ×δt,t∈0,1.Theorem 5 [26].
Ifx is a piecewise continuous function and there exist K>0 and μ such that
(12)xt≤Keμt,t≥t0,then the Laplace transform Lxts exists.Theorem 6 [27].
Letβ∈0,1. The Laplace transform of DCFβxt is
(13)LDCFβxts=2−βMβ2s+β1−ssLxts−x0,s>0,where Lxts is the Laplace transform of xt.Theorem 7.
The solution of the following fractional problem(14)x′t+DCFβxt=kt,xt,t∈0,1,x1=I0+γxξ.is given by
(15)xt=I0+γxξ+∫01Gt,sks,xsds,where
(16)Gt,s=1−bβaβexp−aβt−s−1−bβaβexp−aβ1−s,0≤s≤t,−1−bβaβexp−aβ1−s−bβaβ,t≤s≤1,(17)aβ=2β+2−βMβ21−β,bβ=2β21−β.Proof.
Sincext is continuous differentiable function on 0,1, x′t is bounded function on 0,1. By Definition 1, DCFβxt is also a bounded function. Then, there exist constants k1,k2>0 and μ1,μ2 such that
(18)x′t≤k1eμ1t,t≥t0>0,DCFβxt≤k2eμ2t,t≥t0>0.
From Theorem5, the Laplace transform of x′t and CFDβxt exists.
Taking the Laplace transform for the first formula of Equation (14), we conclude
(19)sx~s−x0+2−βMβ2s+β1−ssx~s−x0=k~s,xs,or
(20)x~s=1sx0+1s+2β+2−βMβ/21−βk~s,xs+2β/21−βss+2β+2−βMβ/21−βk~s,xs.
Taking the Laplace inverse transform for the above equation, we conclude(21)xt=x0+1−bβaβ∫0texp−aβt−sks,xsds+bβaβ∫0tks,xsds.
Then(22)x1=x0+1−bβaβ∫01exp−aβ1−sks,xsds+bβaβ∫01ks,xsds.
Sincex1=I0+γxξ, thus
(23)x0=I0+γxξ−1−bβaβ∫01exp−aβ1−sks,xsds−bβaβ∫01ks,xsds.
Then(24)xt=I0+γxξ−1−bβaβ∫01exp−aβ1−sks,xsds−bβaβ∫01ks,xsds+1−bβaβ∫0texp−aβt−sks,xsds+bβaβ∫0tks,xsds.
By the definition ofGt,s, we conclude
(25)xt=I0+γxξ+∫01Gt,sks,xsds.Remark 8.
(26)∫0tGt,sds=∫0t1−bβaβexp−aβt−s−1−bβaβexp−aβ1−sds≤∫0t1−bβaβexp−aβt−s+1−bβaβexp−aβ1−sds≤∫0texp−aβt−s+exp−aβ1−sds=exp−aβt+exp−aβ·∫0texpaβsds=1aβ·exp−aβt+exp−aβ·expaβt−1≤2aβ·1−exp−aβ=E.
Thus, there exists a constantE>0 such that
(27)∫0tGt,sds≤E,t∈0,1.Theorem 9 (Krasnoselskii’s fixed point theorem).
LetS be a bounded convex closed subset of a Banach space W, and P,Q:S⟶W satisfy the following:
(i)
Px+Qy∈S, for all x,y∈S(ii)
P is completely continuous(iii)
Q is a contraction mapping
Then,P+Q has at least one fixed point.


## 3. Existence and Uniqueness Theorems for Fractional Differential Equation

The following assumption will be needed throughout the paper:(S1): k:0,1×ℝ⟶ℝ is a continuous function.(S2): kt,x satisfies the following Lipschitz condition for the second variable:
(28)kt,x1−kt,x2≤ckx1−x2,x1,x2∈ℝ,t∈0,1.(S3): Let δt:0,1⟶ℝ+ satisfy
(29)∫0tδsds≤Lδ·δt,Lδ>0,t∈0,1..Theorem 10.
Suppose that (S1) and (S2) are satisfied; then Equation (4) has a unique solution provided that ξγ/Γγ+1+Eck<1.Proof.
Sincek∈C0,1×ℝ,ℝ, there exists T>0 such that
(30)T=maxt∈0,1,s∈ℝkt,s.
Similar to the proof of Theorem 3 in [22]. Let operator F be given by
(31)Fxt=I0+γxξ+∫01Gt,sks,xsds.
Firstly, we prove thatF maps a closed set into a closed set.
LetUb=x∈C10,1,ℝ∣x≤b,b≥ET/1−ξγ/Γγ+1>0. For x∈Ub, it follows that
(32)Fxt≤1Γγ∫0ξξ−sγ−1xsds+∫01Gt,sks,xsds≤ξγΓγ+1b+ET≤b.
This impliesFUb⊆Ub.
Then, we prove thatF is a strict contraction.
Letx1,x2∈C10,1,ℝ, for any t∈0,1; it follows that
(33)∣Fx1t−Fx2t∣≤1Γγ∫0ξξ−sγ−1x1s−x2sds+∫01Gt,sks,x1s−ks,x2sds≤ξγΓγ+1+Eckx1−x2.
Asξγ/Γγ+1+Eck<1, for x1,x2∈C10,1,ℝ, F is a strict contraction. From the Banach fixed point theorem, F has a unique fixed point x∗t∈C10,1,ℝ; accordingly, Equation (4) has a unique solution.Theorem 11.
Suppose that (S1) and (S2) are satisfied; then Equation (4) has at least one solution provided that ξγ/Γγ+1+Eck<1.Proof.
Sincek∈C0,1×ℝ,ℝ, there exists T>0 such that
(34)T=maxt∈0,1,s∈ℝkt,s.
LetUc=x∈C10,1∣x≤c,c≥ET/1−ξγ/Γγ+1>0.
Let operatorsP and Q be given by
(35)Pxt=−1−bβaβ∫t1exp−aβ1−sks,xsds−bβaβ∫t1ks,xsds,Qxt=I0+γxξ+1−bβaβ∫0texp−aβt−sks,xs−exp−aβ1−sks,xsds.
Firstly, for allx1,x2∈Uc, using Remark 8, it follows that
(36)Px1+Qx2=sup−1−bβaβ∫t1exp−aβ1−sks,x1sds−bβaβ∫t1ks,x1sds+I0+γx2ξ+1−bβaβ∫0texp−aβt−sks,x2s−exp−aβ1−sks,x2sds≤sup∫0tGt,sks,x2sds+∫t1Gt,sks,x1sds+sup1Γγ∫0ξξ−sγ−1x2sds≤sup∫0tGt,sks,x2sds+∫t1Gt,sks,x1sds+sup1Γγ∫0ξξ−sγ−1ds·c≤sup∫01Gt,sds·T+ξγΓγ+1c≤ξγΓγ+1c+ET≤c.
Hence, we havePx1+Qx2∈Uc.
Then, for allx1,x2∈C10,1,
(37)Qx1−Qx2=supI0+γx1ξ−I0+γx2ξ+1−bβaβ∫0texp−aβt−s−exp−aβ1−s·ks,x1s−ks,x2sds≤sup1Γγ∫0ξξ−sγ−1x1s−x2sds+sup∫0tGt,s·ks,x1s−ks,x2sds≤ξγΓγ+1+Eckx1−x2.
Asξγ/Γγ+1+Eck<1, Q is a contraction mapping.
Finally, we prove operatorP is completely continuous.
Step 1. OperatorP is continuous.
Letxn be a convergent sequence, xn⟶x∈C10,1,ℝ, by Remark 8 and S2; it follows that
(38)Pxnt−Pxt=1−bβaβ∫t1exp−aβ1−sks,xns−ks,xsds+bβaβ∫t1ks,xns−ks,xsds≤∫t1Gt,sks,xns−ks,xsds≤Eckxn−x.
Sincexn⟶x, we have Pxn⟶Px; then operator P is continuous.
Step 2. OperatorP is bounded on Uc.
(39)Pxt=−1−bβaβ∫t1exp−aβ1−sks,xsds−bβaβ∫t1ks,xsds=∫t1Gt,sks,xsds≤∫t1Gt,sks,xsds≤ET.
Step 3. OperatorP is equicontinuous in C10,1,ℝ.
Lett1,t2∈0,1 and t2<t1, x∈Uc; it follows that
(40)Pxt1−Pxt2=1−bβaβ∫t2t1exp−aβ1−sks,xsds+bβaβ∫t2t1ks,xsds≤1−bβaβ∫t2t1exp−aβ1−sds+bβaβ∫t2t1ds·T≤T·t1−t2.
Then, operatorP is equicontinuous.
From Step 1-Step 3 and the Arzela-Ascoli theorem,P is completely continuous. By Theorem 9, P+Q has at least one fixed point, since
(41)Px+Qxt=I0+γxξ+∫01Gt,sks,xsds.
From Theorem7, Equation (4) has at least one solution.


## 4. Stability Results

Theorem 12.
Suppose that (S1) and (S2) are satisfied; then Equation (4) has the Hyers-Ulam stability on 0,1.Proof.
SinceS1 and S2 hold, by Theorems 10 and 11, Equation (4) has a unique solution. From Theorem 7, Equation (4) has the unique solution
(42)xt=x0+1−bβaβ∫0texp−aβt−sks,xsds+bβaβ∫0tks,xsds.
Letyt satisfy y0=x0 and be a solution of the inequality
(43)y′t+DCFβyt−kt,yt≤ε,t∈0,1.
Set(44)Gt=y′t+DCFβyt−kt,yt,t∈0,1.
Then(45)y′t+DCFβyt=Gt+kt,yt,t∈0,1,Gt≤ε,t∈0,1.
From the proof of Theorem7, we conclude
(46)yt=y0+1−bβaβ∫0texp−aβt−sGs+ks,ysds+bβaβ∫0tGs+ks,ysds.
Then(47)yt−y0−1−bβaβ∫0texp−aβt−sks,ysds−bβaβ∫0tks,ysds=1−bβaβ∫0texp−aβt−sGsds+bβaβ∫0tGsds≤1−bβaβ∫0tGsds+bβaβ∫0tGsds≤∫0tGsds≤ε.
Thus(48)yt−xt=yt−x0−1−bβaβ∫0texp−aβt−sks,xsds−bβaβ∫0tks,xsds≤yt−y0−1−bβaβ∫0texp−aβt−sks,ysds−bβaβ∫0tks,ysds+1−bβaβ∫0texp−aβt−sks,ys−ks,xsds+bβaβ∫0tks,ys−ks,xsds≤ε+ck∫0tys−xsds.
From the Gronwall-Bellman inequality, we conclude(49)yt−xt≤exp∫0tckds·ε≤expck·ε.
From Definition3, Equation (4) has the Hyers-Ulam stability.Theorem 13.
Suppose that (S1), (S2), and (S3) are satisfied; then Equation (4) has the Hyers-Ulam-Rassias stability on 0,1.Proof.
Since (S1) and (S2) hold, by Theorems 10 and 11, Equation (4) has a unique solution. From Theorem 7, Equation (4) has the unique solution
(50)xt=x0+1−bβaβ∫0texp−aβt−sks,xsds+bβaβ∫0tks,xsds.
Letyt satisfy y0=x0 and be a solution of the inequality
(51)y′t+DCFβyt−kt,yt≤δt,t∈0,1.
Set(52)Gt=y′t+DCFβyt−kt,yt,t∈0,1.
Then(53)y′t+DCFβyt=Gt+kt,yt,t∈0,1,Gt≤δt,t∈0,1.
From the proof of Theorem7, we conclude
(54)yt=y0+1−bβaβ∫0texp−aβt−sGs+ks,ysds+bβaβ∫0tGs+ks,ysds.
Then byS3, it follows that
(55)yt−y0−1−bβaβ∫0texp−aβt−sks,ysds−bβaβ∫0tks,ysds=1−bβaβ∫0texp−aβt−sGsds+bβaβ∫0tGsds≤1−bβaβ∫0tGsds+bβaβ∫0tGsds≤∫0tGsds≤∫0tδsds≤Lδ·δt.
Thus(56)yt−xt=yt−x0−1−bβaβ∫0texp−aβt−sks,xsds−bβaβ∫0tks,xsds≤yt−y0−1−bβaβ∫0texp−aβt−sks,ysds−bβaβ∫0tks,ysds+1−bβaβ∫0texp−aβt−sks,ys−ks,xsds+bβaβ∫0tks,ys−ks,xsds≤Lδ·δt+ck∫0tys−xsds.
From the Gronwall-Bellman inequality, we conclude(57)yt−xt≤Lδ×δt+∫0tLδ×δs×ckexp∫stckdtds≤Lδ+Lδ2ckexpck×δt.
From Definition4, Equation (4) has the Hyers-Ulam-Rassias stability on 0,1.


## 5. Example

In this section, we give two examples to illustrate our main results.Example 1.
Consider the following problem of the Caputo-Fabrizio fractional differential equation of form(58)x′t+CFD13xt=e−tx+8,t∈0,1,x1=I0+12x14,and the following inequality
(59)y′t+CFD13yt−e−ty+8≤δt,t∈0,1.
Let(60)β=13,γ=12,ξ=14.
Then(61)M13=65,a13=2,b13=12,since
(62)kt,x=e−tx+8,t,x∈0,1×ℝ.
Then, it follows that(63)kt,x1−kt,x2=e−t1∣x1∣+8−1∣x2∣+8≤e−t1x1+8x2+8∣x1−x2∣≤e−tx1−x264≤164x1−x2.
Hence,ck=1/64.Therefore,S1 and S2 are satisfied, ξγ/Γγ+1+Eck=1/41/2/Γ1/2+1+3×1/64<1. By Theorems 10 and 11, Equation (4) has a unique solution
(64)xt=x0+34∫0texp−2t−se−sx+8ds+14∫0te−sx+8ds.Setδt=et∈C0,1,0,+∞, ∫0tδsds=∫0tesds=et−1≤et; we conclude Lδ=1>0.Becauseyt satisfies the following inequality:
(65)y′t+CFD13yt−e−t∣y∣+8≤δt,t∈0,1,it follows that
(66)yt−x0−1−b1/3a1/3∫0texp−a13t−sks,ysds−b1/3a1/3∫0tks,ysds≤et.Because (S1), (S2), and (S3) are satisfied, by Theorem 13, it follows that
(67)yt−xt≤Lδ+Lδ2ckexpck·et≤1+164e164·et.Consequently, the equation has the Hyers-Ulam-Rassias stability.Example 2.
Consider the following problem of the Caputo-Fabrizio fractional differential equation of form(68)x′t+CFD12xt=tx+8,t∈0,1,x1=I0+13x12,and the following inequality
(69)y′t+CFD12yt−ty+8≤ε,t∈0,1.
Let(70)β=12,γ=13,ξ=12.
Then(71)M12=43,a12=3,b12=1,since
(72)kt,x=t∣x∣+8,t,x∈0,1×ℝ.
Then, it follows that(73)kt,x1−kt,x2=t1∣x1∣+8−1∣x2∣+8≤t1x1+8x2+8x1−x2≤tx1−x264≤164x1−x2.
Hence,ck=1/64.Therefore, (S1) and (S2) are satisfied, ξγ/Γγ+1+Eck=1/21/3/Γ1/3+1+3×1/64<1. By Theorems 10 and 11, Equation (4) has a unique solution
(74)xt=x0+23∫0texp−3t−ssx+8ds+13∫0tsx+8ds.Setyt=et∈C0,1,0,+∞, and fix ε=9/32; it follows that
(75)y′t+CFD12yt−t∣y∣+8≤932=ε,t∈0,1.Because (S1) and (S2) are satisfied, by Theorem 12, we conclude
(76)yt−xt≤exp164·932=exp164·ε.Consequently, the equation has the Hyers-Ulam stability.


## 6. Conclusions

In this article, we established the Ulam stability of the Caputo-Fabrizio fractional differential equation with an integral boundary condition by the Laplace transform method. Krasnoselskii’s fixed point theorem and Banach fixed point theorem are employed to prove the existence and uniqueness results of the solution for the Caputo-Fabrizio fractional differential equation. Besides, we constructed a solution for the equation via new Green’s functionGt,s. The Ulam stability of the Caputo-Fabrizio fractional differential equation is used to study unusual irregularities and nonlinearities in wave dynamics and liquids motions. Because the Ulam stability is widely used, we will study the Ulam stability of the ABC fractional differential equation in the future study.



---

*Source: 7268518-2022-01-07.xml*