# Picard Method for Existence, Uniqueness, and Gauss Hypergeomatric Stability of the Fractional-Order Differential Equations

Authors: Zahra Eidinejad; Reza Saadati; Manuel De La Sen
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, we consider a class of fractional-order differential equations and investigate two aspects of these equations. First, we consider the existence of a unique solution, and then, using a new class of control functions, we investigate the Gauss hypergeometric stability. We use Chebyshev and Bielecki norms in order to prove these aspects by the Picard method. Finally, we give some examples to illustrate our results.


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## Body

## 1. Introduction

The topic of fractional calculus and its significant applications have appeared to be an appropriate tool in study widespread fields of engineering and science. In fact, there are a lot of phenomena in which highly accurate modeling is very important to study them. Fluid mechanics, biological models, electochemistry, viscoelasticity, and electromagnetics are some examples of these fields. Also, research on fractional calculus (FC) and its applications is an important part of mathematical analysis (see [1–6] and references therein). Its importance can also be explored in other fields such as fluid dynamics traffic models, oscillation due to earthquakes, and flow in porous media due to seepage.Researchers have always tried to create the conditions for the existence and uniqueness of the solution for fractional differential equations. Since most problems cannot be solved for exact solutions, we need powerful analytical techniques. For good results, one needs stable algorithms and methods. For such needs, the stability theory was founded [7, 8]. It has been found that the notion of fractional-order differential equations can well describe these models in science and engendering. Recently, the study of fractional differential equations has been increased among researchers.Our article is organized as follows. In Section2, we explain the definitions and some important results that we use in the proofs of this article. In Section 3, we consider the following fractional-order differential equation:(1)DcJαρJ=kJ,ρJ,ρhJ,j∈J⊂ℝ,α∈0,1.We investigate the existence of a unique solution and the Gauss hypergeometric stability of this fractional-order differential equation.In the above equation,DcJα for a function ρ given on the interval J=0,q,q∈ℝ+ is the Caputo fractional derivative of order α and the functions k∈CJ×ℝ2,ℝ,h∈CJ¯,J¯ (J¯ is the colure of J) with hJ≤J. In the first theorem, we investigate the Gauss hypergeometric stability of the fractional-order differential equation using the Chebyshev norm, and in the second theorem, we have proved the Gauss hypergeometric stability of the equation by using Bielecki norm. At the end of each theorem, we provide examples that demonstrate our results well.


## 2. Preliminaries

Discontinuous control strategy is a method that is used in various issues. Among its applications, we can mention the study of the dynamic behavior of the computer worm system (see [9, 10], for more information). In this paper, we consider the Gauss hypergeometric function as a control function and use this function to investigate the stability of Gauss hypergeometric. In this section, we present the definitions of the fractional integral, the Riemann–Liouville fractional derivative, and the Caputo fractional derivative of order τ, which we utilize in this paper, for more detail, we refer to [11–14]. In the continuation, by introducing the Gauss hypergeometric series, we define the Gauss hypergeometric stability of equation (1) [15–17]. Also, we consider the Picard operator and Henry–Gronwall inequality, which we use in the next section [18, 19].Definition 1.
Letτ>0 and k∈CJ×ℝ2,ℝ, and Γτ is the Gamma function. On an interval 0,J, the Riemann–Liouville fractional integral of order τ is defined by(2)IJτkJ=1Γτ∫0JkℓJ−ℓ1−τdℓ,J>0,τ>0.Definition 2.
Assume thatn−1<τ<n, k:0,∞⟶ℝ. The Riemann–Liouville derivative of order τ and the Caputo derivative of order τ, respectively, are defined by(3)DLJτkJ=1Γn−τdndJn∫0JkℓJ−ℓτ+1−ndℓ,J>0,DcJτkJ=DLJτkJ−∑k=0n−1Jkk!kk0,J>0.Now, we consider the Gauss hypergeometric series (see [17, 20]) F by(4)Fℵ,β;℘;z=∑k=0∞ℵkβk℘kzkk!=Γ℘ΓℵΓβ∑k=0∞Γℵ+kΓβ+kΓ℘+kzkk!,where in ℵ,β,℘∈ℝ+ and z∈ℝ.Remark 1.
We consider fractional-order differential equation (1) that is controlled by Gauss hypergeometric F. If w is a differential function, satisfying(5)DcJαwt−kJ,wJ,whJ≤εFℵ,β;℘;Jα,for k∈CJ×ℝ2,ℝ and ε>0, then w also satisfies the following integral inequality:(6)wJ−w0−1Γα∫0JJ−ℓα−1kℓ,wℓ,whℓdℓ≤εΓα∫0JJ−ℓα−1Fℵ,β;℘,ℓαdℓ≤εΓα∫0JJ−ℓα−1ΓcΓℵΓβ∑k=0∞Γℵ+kΓβ+kΓ℘+kℓαkk!dℓ=εΓαΓ℘ΓℵΓβ∑k=0∞Γℵ+kΓβ+kΓ℘+k1k!∫0JJ−ℓα−1ℓαkdℓ=εΓαΓcΓaΓb∑k=0∞Γℵ+kΓβ+kΓ℘+kJk+1αΓkα+1ΓαΓk+1α+1k!≤εΓ℘ΓℵΓβ∑n=0∞Γℵ+kΓβ+kΓ℘+kJnαΓnα+1Γn+1α+1n!≤εΓ℘ΓℵΓβ∑n=0∞Γℵ+kΓβ+kΓ℘+k1n!JnαΓnα+1Γnα+1=εFℵ,β;℘,Jα.Remark 2.
Assume thatw is a function such that w∈CJ,ℝ. Then, w is a solution of inequality (5) if and only if we can find a function f˜w∈CJ,ℝ such that(i)
f˜J≤εFℵ,β;℘;Jα, for all J∈J(ii)
DcJαwJ=kJ,wJ,whJ+f˜J, for all J∈JDefinition 3.
We say that equation (1) is Gauss hypergeometric stable w.r.t Fℵ,β;℘;Jα if, for each ε>0, there exists a constant value cF>0, such that, for a differentiable function w satisfying (5), we can find ρ such that the following holds:(7)DcJαρJ=kJ,ρJ,ρhJ,J∈J⊂ℝ,α∈0,1.
Then, we have(8)wJ−ρJ≤cFεFℵ,β;℘;Jα,J∈J.Definition 4.
Consider the metric spaceY,δ and the operator T on Y. Now, T is a Picard operator if there exists y∗∈Y such that(i)
ET=y∗ wherever ET=y∈Y:Ty=y is the fixed point set of T(ii)
The sequenceTny0n∈ℕ converges to y∗, for all y0∈Y.Lemma 1.
If we consider an increasing Picard operatorT:Y⟶Y, then Y,δ,≤ is an ordered metric space. Therefore, for each y∈Y, y≤Ty, we have y≤yT∗.In the sequel, we express the Gronwall lemma (see Lemma 7.1.1 in [21]).Lemma 2.
We consider the continuous functionsμ,ν:J⟶0,∞. For the nondecreasing function ν, there are positive constants κ and 0<α<1 such that(9)μJ≤νJ+κ∫0JJ−ℓα−1μℓdℓ,J∈J.Therefore,(10)μJ≤νJ+∫0J∑n=1∞κΓαnΓnαJ−ℓnα−1νℓdℓ,J∈J.Remark 3.
For continuous and nondecreasing functionsνj and μJ on J, we have(11)μJ≤νJFℵ,β;℘;κΓαJα.Remark 4.
We propose the Picard method for the uniqueness and stability of this problem, in which case both its conditions will be weaker and it will be easier to prove it.Remark 5 (see [22]).
There are some slight mistakes and typos in [13]. In Definition 2.4 and Remark 2.6 in [13], the space C−h,d should be C−h,0∩C1−α;ψ0,d. For the precise definition needed in [13], we note that the statement (some typos) and proof of Theorem 1 in [13] is fine once X=C−h,d in [13] is replaced by C−h,0∩C1−α;ψ0,d with the norm (there was accidently a typo in relation to the norm in [13]) max.C−h,0,.C1−α;ψ0,d.


## 3. Main Results

As mentioned in Section1, we use stability algorithms and methods to get good results. As we know, there are different types of stability such as exponential, Mittag-Leffler and Lyapunov type (see [9, 23], for more details). Here, we prove the Gauss hypergeometric stability of equation (1). Before proving the Gauss hypergeometric stability, we first investigate the existence of a unique solution to the fractional-order differential equation. We prove the first theorem utilizing Chebyshev norm .CρC=maxJ∈JρJ.Theorem 1.
We assume that the functionsk∈CJ×ℝ2,ℝ and h∈CJ¯,J¯ exist, such that hJ≤J. Assume the following are satisfied:(i)
There existsLk>0 such that kJ,u1,u2−kJ,v1,v2≤Lk∑i=12ui−vi, for all J∈J,ui,vi∈ℝ,i=1,2(ii)
2Lkqα/Γα+1<1
Therefore, (1) has a unique solution in CJ,ℝ and is Gauss hypergeometric stable.Proof.
According to equation (1), we have(12)ρJ=φ0+1Γα∫0JJ−ℓα−1kℓ,ρℓ,ρhℓdℓ,J∈J.
It is easy to see that (1) is equivalent to (12).
We first prove the existence of solution for equation (12). We define the mapping Λk in Y:=CJ,ℝ as(13)ΛkρJ=φ0+1Γα∫0JJ−ℓα−1kℓ,ρℓ,ρhℓdℓ,J∈J.
Now, we need to demonstrate thatΛk determined in (13) is a contraction mapping on Y:=CJ,ℝ.
According (i), we have(14)ΛkρJ−ΛkwJ≤1Γα∫0JJ−ℓα−1kℓ,ρℓ,ρhℓ−kℓ,wℓ,whℓdℓ≤LkΓα∫0JJ−ℓα−1max0≤ℓ≤qρℓ−wℓ+max0≤ℓ≤qρhℓ−whℓdℓ≤2LkΓαρ−wc∫0JJ−ℓα−1dℓ≤2LkqαΓα+1ρ−wc,for all J∈J, which implies that(15)Λkρ−ΛkwC≤2LkqαΓα+1ρ−wC,ρ,w∈CJ,ℝ.
Thus,Λk is a contraction mapping w.r.t. the Chebyshev norm .C on Y. Now, to continue the proof, we utilize the Banach contraction principle.
Now, we prove Gauss hypergeometric stability. We assume thatρ is a unique solution to equation (1). Then, for w∈CJ,ℝ that is a given differentiable function, satisfying (5), we have(16)ρJ=w0+1Γα∫0JJ−ℓα−1kℓ,ρℓ,ρhℓdℓ,J∈J.
Obviously, Remark1 implies that(17)wJ−w0−1Γα∫0JJ−ℓα−1kℓ,wℓ,whℓdℓ≤εFℵ;β;℘;Jα,for J∈J.
Also according to (i), we have(18)wJ−ρJ≤wJ−w0−1Γα∫0JJ−ℓα−1kℓ,wℓ,whℓdℓ+1Γα∫0JJ−ℓα−1kℓ,wℓ,whℓdℓ−1Γα∫0JJ−ℓα−1kℓ,ρℓ,ρhℓdℓ≤εFℵ,β;℘;Jα+LkΓα∫0JJ−ℓα−1wℓ−ρℓdℓ+whℓ−ρhℓdℓ,for all J∈J. Now, we consider the operator T:CJ,ℝ+⟶CJ,ℝ+, for each σ∈CJ,ℝ+, as follows:(19)TσJ=εFℵ,β;℘;Jα+LkΓα∫0JJ−ℓα−1σℓdℓ+∫0JJ−ℓα−1σhℓdℓ,J∈J.
In the continuation, we show thatT is a Picard operator. By (ii), we have(20)TσJ−TϑJ≤LkΓα∫0JJ−ℓα−1σℓ−ϑℓdℓ+σhℓ−ϑhℓdℓ≤2LkqαΓα+1σ−ϑC,for all J∈J and for every σ,ϑ∈CJ,ℝ+, which implies that(21)Tσ−TϑC≤2LkqαΓα+1σ−ϑC.
Consequently,T w.r.t. the Chebyshev norm .C is a contraction mapping on CJ,ℝ+. Thus, utilizing the Banach contraction principle, we obtain that T is a Picard operator and ET=σ∗. Now, for,(22)σ∗J=εFℵ,β;℘;Jα+LkΓα∫0JJ−ℓα−1σ∗ℓdℓ+∫0JJ−ℓα−1σ∗hℓdℓ.
In the sequel, we show that the solutionσ∗ is increasing. Suppose that and denote m:=minℓ∈Jσ∗ℓ+σ∗hℓ∈ℝ+, and we have(23)σ∗J2−σ∗J1=εFℵ,β;℘;J2α−Fℵ,β,℘;J1αLkΓα∫0J1J2−ℓα−1−J1−ℓα−1σ∗ℓ+σ∗hℓdℓ+LkΓα∫J1J2J2−ℓα−1σ∗ℓ+σ∗hℓdℓ≥εFℵ,β;℘;J2α−Fℵ,β;℘;J1α+mLkΓα∫0J1J2−ℓα−1−J1−ℓα−1dℓ+mLkΓα∫J1J2J2−ℓα−1dℓ=εFℵ,β;℘;J2α−Fℵ,β;℘;J1α+mLkΓα+1J2α−J1α>0.
Then,σ∗ is increasing. From hJ≤J, we obtain(24)σ∗hJ≤σ∗J,σ∗J≤εFℵ,β;℘;Jα+2LkΓα∫0JJ−ℓα−1σ∗ℓdℓ.
Now, utilizing Lemma2, Remark 2, and cF:=Fℵ,β;℘;2Lkqα, we obtain(25)σ∗J≤cFεFℵ,β;℘;Jα,J∈J.
SinceT is a Picard operator and increasing, by Lemma 1 and from (18), we conclude that σ≤σ∗. If we consider σ=w−ρ, then(26)wJ−ρJ≤cFεFℵ,β;wp;Jα,J∈J.
Therefore, equation (1) is Gauss hypergeometric stable.In the following, we give an example to illustrate the previous theorem.Example 1.
Consider a differential equation forα=1/2 and for any J∈0,1 (q=1) as follows:(27)DcJ1/2ρJ=0.01ρ5/2π1+J−3/2,J∈0,1.
Based on what has been said in the above theorem, we have(28)ρJ=w0+0.01Γ1/2∫0JJ−ℓ−1/2ρJ5/2π1+ℓ−3/2dℓ.
According to equation (5) and using the fractional-order differential equation, the following inequality is established for the differentiable function w:(29)DcJ1/2wJ−kJ,wJ,w1+J−3/2≤εFℵ,β;℘;J1/2.
The functionsk and h used in the above inequality are considered as follows:(30)hℑ=1+ℑ−3/2,kℑ,ρℑ,ρhℑ=0.01ρ5/2π1+ℑ−3/2.
To be Ulam–Hyers-hypergeometric stable, we need a positive constantLk and a coefficient of Lipschitz 2Lkqα/Γα+1. Then, we consider Lk=2/5 and as a result 2Lkqα/Γα+1=8/5π≈0.9<1.
According to Theorem1, all the conditions for the existence of the solution and the Gauss hypergeometric stability are established. Thus, the coefficient cF=Fℵ,β;℘;4/5 exists to us so that(31)wJ−ρJ≤cFεFℵ,β;℘;J1/2,J∈0,1.
Based on equation (28), we have a nonlinear equation whose diagram for the arbitrary values in the interval 0,1 is shown in Figure 1.
In the next theorem, we prove the Gauss hypergeometric stability utilizing Bielecki’s norm.BρB:=maxJ∈IρJexp−ηJ,η>0,I⊂ℝ+.Figure 1
Fraction integral equation diagram.F is the nonlinear equation obtained from equation (28).Theorem 2.
Assume that the functionsk∈CJ×ℝ2,ℝ,h∈CJ¯,J¯ exist, such that hJ≤J. Assume the following are satisfied:(i)
There existsLk>0 such that kJ,u1,u2−kJ,v1,v2≤Lk∑i=12ui−vi for all J∈J,ui,vi∈ℝ,i=1,2(ii)
2Lkqαexpηq/Γα22α−1η<1, for some α∈1/2,1 and η>0
Therefore, equation (1) has a unique solution in CC,ℝ and is Gauss hypergeometric stable.Proof.
Here, we prove thatΛk:Y⟶Y is defined in Theorem 1 with Bielecki’s norm .B, which is a contraction operator. Also, we ignore from saying the similar arguments that are expressed in Theorem 1. Then, for all J∈J, we have(32)ΛkρJ−ΛkwJ≤1Γα∫0JJ−ℓα−1kℓ,ρℓ,ρhℓ−kℓ,wℓ,whℓdℓ≤LkΓα∫0JJ−ℓα−1expηℓmax0≤ℓ≤qρℓ−wℓexp−ηℓ+max0≤ℓ≤qρhℓ−whℓexp−ηℓdℓ≤2LkΓαρ−wB∫0JJ−ℓα−1expηℓdℓ.
Now, we utilize Holder’s inequality, forα∈1/2,1. Then,(33)∫0JJ−ℓα−1expηℓ≤∫0JJ−ℓ2α−1dℓ1/2∫0Jexp2ηℓdl1/2≤12ηqα2α−1expηq.
Therefore,(34)Λkρ−ΛkwB≤2LkqαexpηqΓα22α−1ηρ−wB.
Consequently, we have(35)Λkρ−ΛkwB≤2LkqαexpηqΓα22α−1ηρ−wB,ρ,w∈CJ,ℝ.
Therefore,Λk on Y is a contraction mapping w.r.t. Bielecki’s norm .B. Now, we apply the Banach contraction principle to prove (ii). To prove that T in Theorem 1 is a Picard operator, we show that T is a contraction. For all J∈J and α∈1/2,1 and for every σ,ϑ∈CJ,ℝ+, we have(36)TσJ−TϑJ≤2LkqαexpηqΓα22α−1ηρ−wB.
Then,(37)Tσ−TϑB≤2LkqαexpηqΓα22α−1ηρ−wB.
Consequently,T is a contraction mapping on CJ,ℝ+ with Bielecki’s norm .B. The Gauss hypergeometric stability proof is exactly the same as Theorem 1.Example 2.
Consider a fractional-order differential equation as(38)DcJ2/3ρJ=127Γ23J−h,J∈0,14,for α=2/3 and for any J∈0,1/4. Based on what has been said in Theorem 2, we have(39)ρJ=w0+127Γ2/3∫0JJ−ℓ−9/10Γ23J−0.01dℓ.
LetLk=1/2 and η=1/2. Consequently,(40)2LkqαexpηqΓα22α−1η=1/42/3exp1/8Γ2/31/3<0.58<1.
According to equation (5) and using the fractional-order differential equation, for the differentiable function w, we have(41)DcJ2/3wJ−kJ,wJ,wJ−h≤εFℵ,β;℘;J2/3.
All the conditions for the existence of the solution and the Gauss hypergeometric stability are established by Theorem2. Therefore, the coefficient cF=Fℵ,β;℘;1/42/3 exists, and we have(42)wJ−ρJ≤cFεFℵ,β;℘;J2/3,J∈0,14.
According to equation (39) for the function ρ, we have Figure 2 for every J∈0,1/4.Figure 2
Fractional integral equation diagram.Remark 6.
In Theorem2, we examined our problem for α∈1/2,1 by utilizing Holder’s inequality. At present, we re-examine the problem for α∈0,1.
Let ; therefore, we have(43)ΛkρJ−ΛkwJ≤2LkΓαρ−wB∫0JJ−ℓα−1expηℓdℓ≤2LkqαexpηqΓα+1ρ−wB.
For someα∈0,1 and η>0, we have 2Lkqαexpηq/Γα+1<1. Now, we be able demonstrate that Λk is a contraction via Bielecki’s norm .B on Y as well.


## 4. Conclusion

In this paper, we applied the Picard method to investigate existence, uniqueness, and Gauss hypergeometric stability of fractional-order differential equations.



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