# Stability Analysis for Differential Equations of the General Conformable Type

Authors: Abdellatif Ben Makhlouf; El-Sayed El-Hady; Salah Boulaaras; Mohamed Ali Hammami
Journal: Complexity
 (2022)

---

## Abstract

Fractional calculus is nowadays an efficient tool in modelling many interesting nonlinear phenomena. This study investigates, in a novel way, the Ulam–Hyers (HU) and Ulam–Hyers–Rassias (HUR) stability of differential equations with general conformable derivative (GCD). In our analysis, we employ some version of Banach fixed-point theory (FPT). In this way, we generalize several earlier interesting results. Two examples are given at the end to illustrate our results.


---

## Body

## 1. Introduction

The stability issue gained a considerable attention in various research fields through applications. There are many kinds of stability, among them is the stability introduced by S. M. Ulam, in his famous talk at a conference held in Wisconsin University in 1940. Since then, it is known as HU stability or simply Ulam stability. Its applications for various types of differential equations have been investigated by many researchers. The readers can see the interesting results in1–7, for more details. The stability problem of Ulam can be rewritten in the following form.Consider a groupG∗ and a metric group G∗∗,χ1. Is it true that given ε>0, there exist δ>0 such that if Λ:G∗⟶G∗∗ satisfies(1)χ1Λx1x2,Λx1Λx2<δ.For allx1,x2∈G∗, then a homomorphism Ξ:G∗⟶G∗∗ exists such that(2)χ1Λx1,Ξx1<ε,for every x1∈G∗?The problem of Ulam has been extended in many directions for various interesting settings. In particular, Rassias (see [8]) generalized Ulam’s result for Banach spaces.Initial and boundary value problems with fractional-order derivatives are natural generalization of the classical initial and boundary value problems. It is much more complicated to investigate stability issues of fractional-order problems than their classical analogues; this is because of the singularity and nonlocality in the kernel of fractional differential operators. Fractional derivatives, in general, play negligible roles in a number of fields of science and engineering (see, e.g., [9–13] and the references there in).In particular, during the last few decades, the area of fractional calculus has been investigated qualitatively by using different tools of functional analysis. These tools include but are not limited toGronwall Lemma, see, e.g., [14], Pachpatte’s inequality, see, e.g., [15], Schaefer’s FPT, see, e.g., [16], Schauder’s FPT, see, e.g., [17], Banach FPT, see, e.g., [18], and Picard operator, see, e.g., [15]. Various approaches have been used to define fractional derivatives (see, e.g., 19–28, for more details).It should be remarked that generalized conformable derivative plays an essential role in many applications. For instance, the authors in [29] utilized it to examine some nonlinear evolution equations. Generalized conformable derivative also has been used in [30] to investigate some nonlinear evolution equations. A new generalized version of conformable derivative is given and in [31] with some applications in biological population. In the present study, we generalize several recent interesting works as follows. We use Theorem 2 to generalize the interesting results in [32, 33] by dropping some of the basic assumptions that have been used there. We also use Theorem 3 to generalize the work in [34].The organization of the study is as follows. In Section2, we present some preliminaries and some basic definitions. In Section 3, we introduce our stability results in the sense of HU and HUR. In Section 4, two examples are written to show the validity of our results, and in Section 5, we conclude our work.


## 2. Preliminaries

In this section, some definitions, lemmas, and theorems are given [35–39].Definition 1.
Let us consider a functionϕ defined on c,d; then, the GCD starting from the real c of a function ϕ is defined by(3)Tcυ,ψcϕz=limσ⟶0ϕz+σψcz,υ−ϕzσ.
For allz>c, υ∈0,1 and ψcz,υ is a nonnegative continuous function that satisfies(4)ψcz,1=1,ψc.,υ1≠ψc.,υ2,whereυ1≠υ2andυ1,υ2∈0,1.
IfTcυ,ψcϕz exists, for every z∈c,a; for some a>c, limt⟶c+Tcυ,ψcϕz exists; then, by definition,(5)Tcυ,ψcϕc=limt⟶c+Tcυ,ψcϕz.Remark 1.
To further study the properties of GCD, we suppose thatψcz,υ>0, for all z>c, and 1/ψc.,υ is locally integrable.Definition 2.
Let0<υ<1. The conformable fractional integral starting from c of a function ϕ is defined by(6)Icυ,ψcϕz=∫czϕxψcx,υdx.Lemma 1.
Suppose thatϕ∈Cc,d. Thus,(7)Tcυ,ψIcυ,ψcϕz=ϕz,∀z≥c.Lemma 2.
Suppose thatϕ∈AC1c,d. Thus,(8)Icυ,ψcTcυ,ψϕz=ϕz−ϕc,∀z≥c.Remark 2.
Assume thatϑ∈ℝ∗. If gz:=Eυψcϑ,z,c=eϑ∫cz1/ψcx,υdx, then Tcυ,ψcgz=ϑgz and Icυ,ψcgz=1/ϑgz−1.
The following is the notion of a generalized metric on some setS1.Definition 3.
(see40). Consider a mapping ϱ:S1×S1⟶0,∞. The mapping ϱ is called a generalized metric on set S1 iff ϱ satisfies:M1ϱo1,o2=0 if and only if o1=o2M2ϱo1,o2=ϱo2,o1, for all o1,o2∈S1M3ϱo1,o3≤ϱo1,o2+ϱo2,o3, for all oi∈S1,i=1,2,3
The following theorem (see [40]) represents one of the interesting results of FPT. This theorem plays a fundamental role in our study.Theorem 1.
Suppose thatQ,F is a metric space that is generalized complete. Let B:Q⟶Q be a strictly contractive operator. If there is an integer t≥0 with FΓt+1c,Γtc<∞ for some c∈Q, thus(a)
lims⟶+∞Bsc=c∗, where c∗ is the unique fixed point of Γ in Q∗:=c1∈Q:FBtc,c1<∞(b)
Ifc1∈Q∗, then Fc1,c∗≤1/1−LFBc1,c1
Define the spaceX as X:=CI,ℝ, with I=a,a+T (a is some real number).Lemma 3.
Define a metricη:X×X⟶0,∞ in such a way that(9)ηβ1,β2=infA∈0,∞:β1z−β2zEθψaϑ,z,a≤Aλz,z∈I,where ϑ>0, θ∈0,1, and λ is positive and continuous. Thus, X,η is a generalized complete metric space.The goal of this study is to investigate the stability of the following initial value problem:(10)Taθ,ψayz=ξz,yz,ya=ya,in the sense of HU and HUR. Notice that the solution of the initial value problem (10) is the solution of(11)yz=∫azξp,ypψap,θdp+ya,z∈I.


## 3. Ulam–Hyers–Rassias Stability Results

We use this section to present our main results. The theorem below represents the stability of (10) in the sense of HU.Theorem 2.
Supposeξ is continuous and satisfies(12)|ξz,γ1−ξz,γ2|≤P|γ1−γ2|,∀z∈I,γi∈ℝ,i=1,2.
If an absolutely continuous functionx:I⟶ℝ satisfies(13)|Taθ,ψaxz−ξz,xz|≤ϵ,for some ϵ>0, therefore, there is a unique solution x∗ of (10) with(14)|xz−x∗z|≤ϵP+ϱϱMEθψaP+ϱ,a+T,a,for every z∈I, where ϱ is any positive constant and M=sups∈a,a+TIaθ,ψa1s/EθψaP+ϱ,s,a.Proof.
For anyB1,B2∈X, we define the metric d in this way:(15)dB1,B2=infV∈0,∞:B1z−B2zEθψaP+ϱ,z,a≤V,z∈I.
Define the operatorG:X⟶X such that(16)Gyz:=xa+∫azξs,ysψas,θds,∀y∈X.
Since(17)Gy0z−y0zEθψaP+ϱ,z,a<∞,∀y0∈X,z∈I,so that it is clear that dGy0,y0<∞, in addition, we get y∈X:dy0,y<∞=X.
Now, we prove thatG is strictly contractive:(18)Gy1z−Gy2z≤∫azξs,y1s−ξs,y2sψas,θds≤∫azξs,y1s−ξs,y2sψas,θds≤P∫azy1s−y2sψas,θds≤P∫az1ψas,θy1s−y2sEθψaP+ϱ,s,aEθψaP+ϱ,s,ads≤Pdy1,y2P+ϱEθψaP+ϱ,z,a−1≤Pdy1,y2P+ϱEθψaP+ϱ,z,afor allz∈I.
So, it is clear that(19)Gy1z−Gy2zEθψaP+ϱ,z,a≤PP+ϱdy1,y2,which implies that(20)dGy1,Gy2≤PP+ϱdy1,y2,which prove that the operator G is a strictly contractive one.
We get, from (27),(21)|xz−Gxz|≤ϵ∫at1ψas,θds≤ϵIaθ,ψa1z.
Therefore,(22)dx,Gx≤ϵM.
Now, according to Theorem1, there is some solution x∗ of (10) satisfying(23)dx,x∗≤ϵP+ϱϱM,so that(24)xz−x∗zEθψaP+ϱ,z,a≤ϵP+ϱϱM,which implies that(25)|xz−x∗z|≤ϵP+ϱϱMEθψaP+ϱ,a+T,a.Remark 3.
It is clear that our findings in the sense of HU are some generalized version of the results obtained in [32, 33] as follows. In our analysis, we do not impose any constrains on P unlike equation 5 in Theorem 2 in [32]. In [33], the authors assumed conditions on the function φ which is not the case in our study.
The following theorem represents the stability of (10) in HUR sense.Theorem 3.
Supposeξ is continuous and satisfies(26)|ξz,γ1−ξz,γ2|≤P|γ1−γ2|,∀z∈I,γi∈ℝ,i=1,2.
If an absolutely continuous functionx:I⟶ℝ satisfies(27)|Taθ,ψaxz−ξz,xz|≤κz,where κz is a nondecreasing, continuous function, therefore, there is a unique solution x∗ of (10) with(28)|xz−x∗z|≤P+ϱϱMEθψaP+ϱ,a+T,aκz,for every z∈I, where ϱ is any positive constant and M=sups∈a,a+TIaθ,ψa1s/EθψaP+ϱ,s,a.Proof.
For anyB1,B2∈X, we define the metric d as follows:(29)dB1,B2=infV∈0,∞:B1z−B2zEθψaP+ϱ,z,a≤Vκz,z∈I.
Define the operatorG:X⟶X such that(30)Gyz:=xa+∫azξs,ysψas,θds,∀y∈X.
We havedGy0,y0<∞, for all y0. In addition, we get y∈X:dy0,y<∞=X.
Now, we prove thatG is strictly contractive:(31)Gy1z−Gy2z≤∫azξs,y1s−ξs,y2sψas,θds,≤P∫azy1s−y2sψas,θds≤Pdy1,y2κz∫azEθψaP+ϱ,s,aψas,θds≤Pdy1,y2P+ϱEθψaP+ϱ,z,aκzfor allz∈I.
So,(32)dGy1,Gy2≤PP+ϱdy1,y2,which prove that the operator G is a strictly contractive one.
We get, from (13),(33)|xz−Gxz|≤∫atκsψas,θds≤κzIaθ,ψa1z.
Hence,(34)dx,Gx≤M.
Using Theorem1, there is a solution x∗ of (10) with(35)dx,x∗≤P+ϱϱM.
Thus,(36)|xz−x∗z|≤P+ϱϱMEθψaP+ϱ,a+T,aκz.Remark 4.
Notice that the authors in [34] used conformable fractional Laplace transform to study the HUR stability of several kinds of differential equations. They had to assume some specific conditions, see, e.g., condition 12 in Theorem 3.6 is given in [34].Remark 5.
The authors in [41] obtained stability results for differential equations with integer-order derivatives ψa=1, while in our study it is for GCD. In this sense, we introduce a generalized version of the interesting results [41].


## 4. Examples

This section uses two examples to show the validity of results.Example 1.
Consider equation (10) for ψaz,θ=z−a1−θ, a=0, θ=0.6, T=2, and ξz,ν=z4sinν.
We have(37)|z4sinν1−z4sinν2|≤16|ν1−ν2|,∀z∈0,9,ν1,ν2∈ℝ.
Then,L=16.
Suppose thatν satisfies(38)|T00.6,ψ0νz−ξz,νz|≤0.01,for all z∈0,2.
Here,ϵ=0.01. Using Theorem 2, there is ν∗ and M>0 such that(39)|νz−ν∗z|≤0.01M,∀z∈0,2.Example 2.
Consider equation (10) for ψaz,θ=z−a1−θ, a=3, θ=0.3, T=3, and ξz,ν=zcosν.
We have(40)|zcosν1−zcosν2|≤6|ν1−ν2|,∀z∈3,6,ν1,ν2∈ℝ.
Then,L=6.
Suppose thatν satisfies(41)|T30.3,ψ3νz−ξz,νz|≤z+2,for all z∈3,6.
Here,κz=z+2. Using Theorem 3, there is ν∗ and M>0 such that(42)|νz−ν∗z|≤Mz+2,∀z∈0,2.


## 5. Conclusion

We managed to utilize a version of Banach FPT to present stability results in the sense of HU and HUR for some differential equations involving GCDs. In our analysis, we generalized some interesting results by dropping some of the basic assumptions that have been used in recent known investigations. We used two examples to show the validity of our main findings. We believe that the methodology used in this study can be further applied to many other fractional differential equations.



---

*Source: 7283252-2022-04-14.xml*