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

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.


Introduction
e 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][2][3][4][5][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 Section 2, 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: We investigate the existence of a unique solution and the Gauss hypergeometric stability of this fractional-order differential equation.
In the above equation, c D α J 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 ∈ C(J × R 2 , R), h ∈ C(J, J) (J is the colure of J ) with h(J) ≤ 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.

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][12][13][14]. In the continuation, by introducing the Gauss hypergeometric series, we define the Gauss hypergeometric stability of equation (1) [15][16][17]. Also, we consider the Picard operator and Henry-Gronwall inequality, which we use in the next section [18,19]. (2) e Riemann-Liouville derivative of order τ and the Caputo derivative of order τ, respectively, are defined by Now, we consider the Gauss hypergeometric series (see [17,20]) F by 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 for k ∈ C(J × R 2 , R) and ε > 0, then w also satisfies the following integral inequality: Remark 2. Assume that w is a function such that w ∈ C(J, R). en, w is a solution of inequality (5) if and only if we can find a function f w ∈ C(J, R) such that Mathematical Problems in Engineering Definition 3. We say that equation (1) is Gauss hypergeometric stable w.r.t F(ℵ, β; ℘; J α ) if, for each ε > 0, there exists a constant value c F > 0, such that, for a differentiable function w satisfying (5), we can find ρ such that the following holds: en, we have

Lemma 1. If we consider an increasing Picard operator
In the sequel, we express the Gronwall lemma (see Lemma 7.1.1 in [21]). erefore, Remark 3. For continuous and nondecreasing functions ](j) and μ(J) on J, we have 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]). ere are some slight mistakes and typos in [13]. In Definition 2.4 and Remark 2.6 in [13], the For the precise definition needed in [13], we note that the statement (some typos) and proof of eorem 1 in [13] is fine once with the norm (there was accidently a typo in relation to the norm in [13]

Main Results
As mentioned in Section 1, 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 fractionalorder differential equation. We prove the first theorem utilizing Chebyshev norm ‖.‖ C (‖ρ‖ C � max J∈J |ρ(J)|).

Theorem 1.
We assume that the functions k ∈ C(J × R 2 , R) and h ∈ C(J, J) exist, such that h(J) ≤ J. Assume the following are satisfied: erefore, (1) has a unique solution in C(J, R) and is Gauss hypergeometric stable.
Proof. According to equation (1), we have 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 Λ k in Y: � C(J, R) as Now, we need to demonstrate that Λ k determined in (13) is a contraction mapping on Y: � C(J, R).

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According (i), we have for all J ∈ J, which implies that us, Λ k 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 that ρ is a unique solution to equation (1). en, for w ∈ C(J, R) that is a given differentiable function, satisfying (5), we have Obviously, Remark 1 implies that for J ∈ J. Also according to (i), we have for all J ∈ J. Now, we consider the operator T: C(J, R + ) ⟶ C(J, R + ), for each σ ∈ C(J, R + ), as follows: In the continuation, we show that T is a Picard operator. By (ii), we have for all J ∈ J and for every σ, ϑ ∈ C(J, R + ), which implies that Consequently, T w.r.t. the Chebyshev norm ‖.‖ C is a contraction mapping on C(J, R + ).

□
In the following, we give an example to illustrate the previous theorem.

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Example 1. Consider a differential equation for α � 1/2 and for any J ∈ (0, 1] (q � 1) as follows: Based on what has been said in the above theorem, we have According to equation (5) and using the fractional-order differential equation, the following inequality is established for the differentiable function w: e functions k and h used in the above inequality are considered as follows: To be Ulam-Hyers-hypergeometric stable, we need a positive constant L k and a coefficient of Lipschitz 2L k q α /Γ(α + 1). en, we consider L k � 2/5 and as a result 2L k q α /Γ(α + 1) � 8/5 � � π √ ≈ 0.9 < 1. According to eorem 1, all the conditions for the existence of the solution and the Gauss hypergeometric stability are established.
Proof. Here, we prove that Λ k : Y ⟶ Y is defined in eorem 1 with Bielecki's norm ‖.‖ B , which is a contraction operator. Also, we ignore from saying the similar arguments that are expressed in eorem 1. en, for all J ∈ J, we have Now, we utilize Holder's inequality, for α ∈ (1/2, 1). en, erefore, Consequently, we have erefore, Λ k 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 eorem 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 ∈ C(J, R + ), we have en, Consequently, T is a contraction mapping on C(J, R + ) with Bielecki's norm ‖.‖ B . e Gauss hypergeometric stability proof is exactly the same as eorem 1. □ Example 2. Consider a fractional-order differential equation as for α � 2/3 and for any J ∈ (0, 1/4]. Based on what has been said in eorem 2, we have Let L k � 1/2 and η � 1/2. Consequently, According to equation (5) and using the fractional-order differential equation, for the differentiable function w, we have All the conditions for the existence of the solution and the Gauss hypergeometric stability are established by eorem 2. erefore, the coefficient c F � F(ℵ, β; ℘; (1/4) 2/3 ) exists, and we have According to equation (39) for the function ρ, we have Figure 2 for every J ∈ (0, 1/4).
Let ; therefore, we have

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

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.

Authors' Contributions
All authors participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.  Mathematical Problems in Engineering