# Ulam-Hyers-Rassias Stability of Nonlinear Differential Equations with Riemann-Liouville Fractional Derivative

Authors: El-sayed El-hady; Abdellatif Ben Makhlouf; Salah Boulaaras; Lassaad Mchiri
Journal: Journal of Function Spaces
 (2022)

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## Abstract

Fractional derivatives are used to model the transmission of many real world problems like COVID-19. It is always hard to find analytical solutions for such models. Thus, approximate solutions are of interest in many interesting applications. Stability theory introduces such approximate solutions using some conditions. This article is devoted to the investigation of the stability of nonlinear differential equations with Riemann-Liouville fractional derivative. We employed a version of Banach fixed point theory to study the stability in the sense of Ulam-Hyers-Rassias (UHR). In the end, we provide a couple of examples to illustrate our results. In this way, we extend several earlier outcomes.


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## Body

## 1. Introduction

Fractional calculus (FC) has been appearing in a wide range of fields, such as chemistry, economics, polymer rheology, and aerodynamics. This is due to the existence of many nice tools (see e.g., [1, 2]) that are not available in the classical calculus. In particular, FC enables researches to model in an efficient way many complicated real world problems, e.g., COVID-19 (see [3]), Ebola virus (see [4]), and HIV (see [5]). Moreover, it has recent interesting applications in image processing (see [6]) and in diabetes (see [7]).The stability problem named after Ulam is currently a research trend in many applications (see e.g., [8] for more references and details). It pupped up as a consequence of the famous question asked by Ulam at a conference held in Wisconsin University in the fall term of 1940 (see [9]). The mentioned Ulam’s stability problem can be rewritten as follows:LetG∗∗,ρ be a metric group and G∗ be a group. Is it true that for some ε1>0, there is a δ1>0 that satisfies if T:G∗⟶G∗∗ verifies
(1)ρTt1t2,Tt1Tt2<δ1,for every t1,t2∈G∗; thus, there exists a homomorphism g:G∗⟶G∗∗ fulfilling
(2)ρTt1,gt1<ε1,for all t1∈G∗.Answers have been introduced for the question of Ulam by many mathematicians. For instance, in 1941, Hyers gave an exact answer to Ulam’s question. Afterwards, Rassias in 1978 (see [10]) introduced a general form of the result of Hyers. The famous result obtained by Rassias can be rewritten:Theorem 1. [10].
Assume thatB∗,B∗∗ are Banach spaces and assume some continuous mapping from ℝ into B∗∗. Suppose that there exists ω≥0 and ϑ∈0,1 such that
(3)hb1+b2−hb1−hb2≤ωb1ϑ+b2ϑ,b1,b2∈B∗\0.
Then, there is a unique solutionΨ:B∗⟶B∗∗ of the Cauchy equation hb1+b2=hb1+hb2 with
(4)hb1−Ψb1≤2ωb1ϑ2−2ϑ,b1∈B∗\0.Through the past six decades, the stability subject has been a common issue of investigations in many places (see, e.g., [12, 15, 22, 23, 9, 20, 21, 25, 27, 26, 28]). As a consequence of the interesting results presented in this direction, many articles devoted to this subject have been introduced ([24, 16, 29] and the references therein). In 2010, Jung employed a fixed point technique (FPT) to study the stability of the equation ρ′=ρθ,λ (see [11]). It should be remarked that Jung in [11] generalized the work of Alsina and Ger to the nonlinear case. In 2012, Bojor (see [12]) used different assumptions to study the stability of
(5)h′x+mxhx=rx,and improved the result of Jung in [11].In 2015, Tunç and Biçer in [13] improved the approach of Jung in [11] for the functional differential equation:
(6)z′x1=Fx1,zx1,zx1−τ.In [14], Huang et al. investigated the stability of the following equation:
(7)Tnx1=Fx1,Tx1,T′x1,⋯,Tn−1x1.In [15], Popa and Pugna studied the UH stability (UHS) of Euler’s equation. In [16], Shen introduced Ulam stability for equations on time scales. In [17], the authors employed weakly Picard operator theory to investigate the UHS of some kind of equations in Banach Spaces. Furthermore, they obtained the UHR stability for such kind of equations via Pachpatte’s integral inequalities. FPT has been employed in [18] to study the stability of a nonlinear Volterra integrodifferential equation with delay and in [19] to study the stability of impulsive Volterra integral equation.The framework of the paper is as follows. In Section2, we introduce some preliminaries; in Section 3, we present the stability results in UHR sense; in Section 4, we illustrate our results with two examples, and Section 5 is devoted to the conclusion.


## 2. Preliminaries

From now on, we useℝ to denote real numbers set and ℂ to denote the complex numbers set. We define the generalized metric on a nonempty set S as follows.Definition 2. [20].
The mappingσ:S×S⟶0,∞ is said to be a generalized metric on S if and only if σ fulfills the assertions:
G1σr1,r2=0 if and only if r1=r2;
G2σr1,r2=σr2,r1 for all r1,r2∈S;
G3σr1,r3≤σr1,r2+σr2,r3 for all r1,r2,r3∈S.Now, we present the notion of UHR stability.Definition 3.
The following fractional differential equation(8)Fx,ν,DTλxν=0is UHR stable if for given ϵ>0 and a function xν which satisfies
(9)Fx,ν,DTλxν≤ϵΠν.There is a solution x0ν of (8) with xν−x0ν≤∈ΠνΦν, where Π· and Φ· are some functions that do not depends on x and x0.The following theorem represents one of the central results of FPT (see [20]).Theorem 4.
For a generalized complete metric spaceZ,γ. Suppose an operator Γ:Z⟶Z that is strictly contractive with some Lipschitz constant L<1, if there exists an integer that is nonnegative k such that γΓk+1y,Γky<∞ for some y∈Z, then the following are true:
(a)
The sequenceΓny converges to a fixed point y∗ of Γy∗ is the unique fixed point of Γ in Z∗≔y1∈Z:γΓky,y1<∞(c)
Ify1∈Z∗, then γy1,y∗≤1/1−LγΓy1,y1The current article is written to study the stability of the following differential equation with right-sided Riemann-Liouville fractional derivative(10)DTλxν=fν,xν,forallν∈0,T,with initial conditions
(11)DTλ−ixT=ai∈ℝ,i=1,2⋯,n−1,an=ITn−λxT=0,where f:0,T×ℝ⟶ℝ is some continuous nonlinear function and n=−−λ, where · is the well-known greatest integer function.


## 3. Stability Results

This section is used to present the main findings of this article. In other words, we use it to prove the UHR stability of (10).Let us first useE=C0,T,ℝ to denote the space of all continuous functions from the interval 0,T into the set of reals ℝ. In the next subsections, we investigate the stability of (10) when 0<λ<1 and when λ>1. We start with the case 0<λ<1 as follows.


### 3.1. The Case0<λ<1

The following theorem represents the stability of (10) in the sense of UHR.Theorem 5.
Assume thatf:0,T×ℝ⟶ℝ satisfies
(12)fν,ρ1−fν,ρ2≤Kρ1−ρ2,∀ν∈0,T,ρs∈ℝ,s=1,2.
If a continuous functionx∈E satisfies IT1−λxT=0, then
(13)DTλxν−fν,xν≤ϵψν,for all ν∈0,T, where ψν is a nonincreasing function. Then, there is a unique function x0 such that
(14)x0ν=1Γλ∫νTs−νλ−1fs,x0sds,xν−x0ν≤δ+KKTλΓλ+1EλK+δTλϵψν,∀ν∈0,T,for any positive constants δ.Proof.
We start the proof by defining the metric onE in this manner
(15)dx1,x2≔infM>0:x1ν−x2νEλK+δT−νλ≤Mψν.
We can prove that the spaceE,d is a complete generalized metric space (see Lemma 1 in [21]).
Define the operatorA:E⟶E with
(16)Auν≔1Γλ∫νTs−νλ−1fs,usds.
Since we haveAu∈E for all u∈E and
(17)Au0ν−u0νEλK+δT−νλ<+∞,∀u0∈E,ν∈0,T,Therefore, dAu0,u0<∞. Note also that we have du0,u<∞, ∀u∈E, and then u∈E:du0,u<∞=E.
In addition, for anyu1,u2∈E we get
(18)Au1ν−Au2ν=1Γλ∫νTs−νλ−1fs,u1s−fs,u2sds≤KΓλ∫νTs−νλ−1u1s−u2sds≤KΓλ∫νTs−νλ−1u1s−u2sEλK+δT−sλEλK+δT−sλds≤Kdu1,u2Γλ∫νTψss−νλ−1EλK+δT−sλds≤Kdu1,u2Γλψν∫νTs−νλ−1EλK+δT−sλds,for all ν∈0,T.
Now, using the fact that(19)∫νTs−νλ−1EλK+δT−sλds≤ΓλK+δEλK+δT−νλ,for all ν∈0,T.
Then,(20)Au1ν−Au2ν≤KK+δdu1,u2EλK+δT−νλψν,which implies that
(21)dAu1,Au2≤KK+δdu1,u2,which proves that A is a strictly contractive. Following the same way as in the proof of Theorem 7.1 in [22], we get
(22)xν−1Γλ∫νTs−νλ−1fs,xsds≤1Γλ∫νTs−νλ−1ϵψsds≤ϵΓλψν∫νTs−νλ−1ds≤ϵΓλ+1ψνT−νλ,which means that
(23)dx,Ax≤ϵΓ1+λTλ.
Now, in view of Theorem4 there is a solution x∗ with
(24)dx∗,x≤δ+KδϵTλΓ1+λ,and then
(25)x∗ν−xν≤δ+KδTλΓλ+1EλK+δTλϵψν,for all ν∈0,T. This results prove that in view of Definition 3, (10) is UHR stable.Remark 6.
In the current work, we do not assume any constrains onK unlike the case of results in [22] where the assumption 0<KLM<1 is a basic condition.Now, we investigate the stability of (10) in the case where λ>1 as follows.


### 3.2. The Caseλ>1

Theorem 7.
Assume thatf:0,T×ℝ⟶ℝ satisfies
(26)fν,ρ1−fν,ρ2≤Kρ1−ρ2,∀ν∈0,T,ρi∈ℝ,i=1,2.
If a continuous functionx∈E satisfies DTλ−ixT=ai,i=1,2⋯,n−1,an=ITn−λxT=0,n=−−λ, then
(27)DTλxν−fν,xν≤ϵψν,for all ν∈0,T, where ψν is a nonincreasing function. Then, there is a unique function x0 with
(28)x0ν=∑j=1n−1−1n−jajΓλ−j+1T−νλ−j+1Γλ∫νTs−νλ−1fs,x0sds,xν−x0ν≤TλeK+δT1−cΓλ+1ϵψν,∀ν∈0,T,where c=KTλ−1/ΓλK+δ and some positive constant δ such that 0<c<1.Proof.
We start by defining the metric onE by the form
(29)dx1,x2≔infM>0:x1ν−x2νeK+δT−ν≤Mψν,and we define the operator A:E⟶E such that
(30)Auν≔∑j=1n−1−1n−jajΓλ−j+1T−νλ−j+1Γλ∫νTs−νλ−1fs,usds.
Since we haveAu∈E for all u∈E and
(31)Au0ν−u0νeK+δT−ν<+∞,∀u0∈E,ν∈0,T,so that it is clear that dAu0,u0<∞. Note also that we have du0,u<∞, ∀u∈E, and then u∈E:du0,u<∞=E.
In addition, for anyu1,u2∈E, we get
(32)Au1ν−Au2ν=1Γλ∫νTs−νλ−1fs,u1s−fs,u2sds≤KΓλ∫νTs−νλ−1u1s−u2sds≤KTλ−1du1,u2ΓλψνeK+δT−νK+δ.
Then, (using0<c<1)
(33)dAu1,Au2≤KTλ−1ΓλK+δdu1,u2≤cdu1,u2,which proves that the operator A is strictly contractive. Following the same way as in the proof of Theorem 5 in [22], we have
(34)xν−Axν≤1Γλ∫νTs−νλ−1ϵψsds≤ϵΓλ+1ψνT−νλ.Then,
(35)dx,Ax≤ϵΓλ+1Tλ.
Now, there is a solutionx∗ (due to Theorem 4) with
(36)dx∗,x≤11−cϵTλΓλ+1,and then
(37)x∗ν−xν≤TλeK+δT1−cΓλ+1ϵψν,for all ν∈0,T.Remark 8.
Notice that in the current work, we do not assume any condition onK unlike the case of Theorem 7in [22] where the condition 0<KLM<1 is a basic condition.


## 3.1. The Case0<λ<1

The following theorem represents the stability of (10) in the sense of UHR.Theorem 5.
Assume thatf:0,T×ℝ⟶ℝ satisfies
(12)fν,ρ1−fν,ρ2≤Kρ1−ρ2,∀ν∈0,T,ρs∈ℝ,s=1,2.
If a continuous functionx∈E satisfies IT1−λxT=0, then
(13)DTλxν−fν,xν≤ϵψν,for all ν∈0,T, where ψν is a nonincreasing function. Then, there is a unique function x0 such that
(14)x0ν=1Γλ∫νTs−νλ−1fs,x0sds,xν−x0ν≤δ+KKTλΓλ+1EλK+δTλϵψν,∀ν∈0,T,for any positive constants δ.Proof.
We start the proof by defining the metric onE in this manner
(15)dx1,x2≔infM>0:x1ν−x2νEλK+δT−νλ≤Mψν.
We can prove that the spaceE,d is a complete generalized metric space (see Lemma 1 in [21]).
Define the operatorA:E⟶E with
(16)Auν≔1Γλ∫νTs−νλ−1fs,usds.
Since we haveAu∈E for all u∈E and
(17)Au0ν−u0νEλK+δT−νλ<+∞,∀u0∈E,ν∈0,T,Therefore, dAu0,u0<∞. Note also that we have du0,u<∞, ∀u∈E, and then u∈E:du0,u<∞=E.
In addition, for anyu1,u2∈E we get
(18)Au1ν−Au2ν=1Γλ∫νTs−νλ−1fs,u1s−fs,u2sds≤KΓλ∫νTs−νλ−1u1s−u2sds≤KΓλ∫νTs−νλ−1u1s−u2sEλK+δT−sλEλK+δT−sλds≤Kdu1,u2Γλ∫νTψss−νλ−1EλK+δT−sλds≤Kdu1,u2Γλψν∫νTs−νλ−1EλK+δT−sλds,for all ν∈0,T.
Now, using the fact that(19)∫νTs−νλ−1EλK+δT−sλds≤ΓλK+δEλK+δT−νλ,for all ν∈0,T.
Then,(20)Au1ν−Au2ν≤KK+δdu1,u2EλK+δT−νλψν,which implies that
(21)dAu1,Au2≤KK+δdu1,u2,which proves that A is a strictly contractive. Following the same way as in the proof of Theorem 7.1 in [22], we get
(22)xν−1Γλ∫νTs−νλ−1fs,xsds≤1Γλ∫νTs−νλ−1ϵψsds≤ϵΓλψν∫νTs−νλ−1ds≤ϵΓλ+1ψνT−νλ,which means that
(23)dx,Ax≤ϵΓ1+λTλ.
Now, in view of Theorem4 there is a solution x∗ with
(24)dx∗,x≤δ+KδϵTλΓ1+λ,and then
(25)x∗ν−xν≤δ+KδTλΓλ+1EλK+δTλϵψν,for all ν∈0,T. This results prove that in view of Definition 3, (10) is UHR stable.Remark 6.
In the current work, we do not assume any constrains onK unlike the case of results in [22] where the assumption 0<KLM<1 is a basic condition.Now, we investigate the stability of (10) in the case where λ>1 as follows.


## 3.2. The Caseλ>1

Theorem 7.
Assume thatf:0,T×ℝ⟶ℝ satisfies
(26)fν,ρ1−fν,ρ2≤Kρ1−ρ2,∀ν∈0,T,ρi∈ℝ,i=1,2.
If a continuous functionx∈E satisfies DTλ−ixT=ai,i=1,2⋯,n−1,an=ITn−λxT=0,n=−−λ, then
(27)DTλxν−fν,xν≤ϵψν,for all ν∈0,T, where ψν is a nonincreasing function. Then, there is a unique function x0 with
(28)x0ν=∑j=1n−1−1n−jajΓλ−j+1T−νλ−j+1Γλ∫νTs−νλ−1fs,x0sds,xν−x0ν≤TλeK+δT1−cΓλ+1ϵψν,∀ν∈0,T,where c=KTλ−1/ΓλK+δ and some positive constant δ such that 0<c<1.Proof.
We start by defining the metric onE by the form
(29)dx1,x2≔infM>0:x1ν−x2νeK+δT−ν≤Mψν,and we define the operator A:E⟶E such that
(30)Auν≔∑j=1n−1−1n−jajΓλ−j+1T−νλ−j+1Γλ∫νTs−νλ−1fs,usds.
Since we haveAu∈E for all u∈E and
(31)Au0ν−u0νeK+δT−ν<+∞,∀u0∈E,ν∈0,T,so that it is clear that dAu0,u0<∞. Note also that we have du0,u<∞, ∀u∈E, and then u∈E:du0,u<∞=E.
In addition, for anyu1,u2∈E, we get
(32)Au1ν−Au2ν=1Γλ∫νTs−νλ−1fs,u1s−fs,u2sds≤KΓλ∫νTs−νλ−1u1s−u2sds≤KTλ−1du1,u2ΓλψνeK+δT−νK+δ.
Then, (using0<c<1)
(33)dAu1,Au2≤KTλ−1ΓλK+δdu1,u2≤cdu1,u2,which proves that the operator A is strictly contractive. Following the same way as in the proof of Theorem 5 in [22], we have
(34)xν−Axν≤1Γλ∫νTs−νλ−1ϵψsds≤ϵΓλ+1ψνT−νλ.Then,
(35)dx,Ax≤ϵΓλ+1Tλ.
Now, there is a solutionx∗ (due to Theorem 4) with
(36)dx∗,x≤11−cϵTλΓλ+1,and then
(37)x∗ν−xν≤TλeK+δT1−cΓλ+1ϵψν,for all ν∈0,T.Remark 8.
Notice that in the current work, we do not assume any condition onK unlike the case of Theorem 7in [22] where the condition 0<KLM<1 is a basic condition.


## 4. Examples

The following examples are used to illustrate our findings.Example 9.
Consider equation (10) for λ=0.5, T=10, and fν,x=ν2cosx.
We have(38)ν2cosx1−ν2cosx2≤100x1−x2,∀ν∈0,10,x1,x2∈ℝ.
Then,K=100.
Suppose thatx∈C0,10,ℝ satisfies I100.5x10=0 and
(39)D100.5xν−ν2cosxν≤1,for all ν∈0,10.
Here,ϵ=1 and ψν=1. Using Theorem 5, there is a continuous function x0 such that
(40)x0ν=1Γ0.5∫ν10s−ν−0.5s2cosx0sds,xν−x0ν≤11E0.51110Γ1.5,∀ν∈0,10.Example 10.
Consider equation (10) for λ=1.5, T=5, and fν,x=ν3sinx.
We have(41)ν3x1−ν3x2≤125x1−x2,∀ν∈0,5,x1,x2∈ℝ.
Then,K=125.
Suppose thatx∈C0,5,ℝ satisfies D50.5x5=a∈ℝ, I50.5x5=0 and
(42)D51.5xν−ν3sinxν≤1,for all ν∈0,5.
Here,ϵ=1 and ψν=1. Using Theorem 7, there is a continuous function x0 such that
(43)x0ν=−aΓ1.55−ν0.5+1Γ1.5∫ν5s−ν0.5s3sinx0sds,xν−x0ν≤51.5e1600Γ2.51−1255/320Γ1.5,∀ν∈0,5.


## 5. Conclusion

A version of Banach’s contraction principle has been successfully utilized in this work to study the UHR stability of nonlinear differential equations with Riemann-Liouville fractional derivatives. In this way, under specific assumptions and conditions, the stability results have been obtained. In our analysis, we get rid of some constrains that have been posed on the lipschitz constants in some interesting recent related works. Two illustrative examples are given at the end to apply our theoretical results and show its validity. Potential future directions of our work can be dedicated to applying our obtained results to some practical applications. Some possible extensions and generalizations of our obtained results can also be our future investigations.



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