Ulam stabilities of nonlinear coupled system of fractional differential equations including generalized Caputo fractional derivative

Abstract: In this paper, we establish the existence and uniqueness of solution for a nonlinear coupled system of implicit fractional differential equations including ψ-Caputo fractional operator under nonlocal conditions. Schaefer’s and Banach fixed–point theorems are applied to obtain the solvability results for the proposed system. Furthermore, we extend the results to investigate several types of Ulam stability for the proposed system by using classical tool of nonlinear analysis. Finally, an example is provided to illustrate the abstract results.


Introduction
Fractional calculus is very significant and important part of mathematical analysis. As is well known, fractional calculus plays a very vital role for the study of fractional order systems. Fractional differential equations (FDEs, for short) are utilized in many applications of engineering and natural science topics, such as dynamics of biology, finance, optimal control, automation system, image analysis, fluid and elasticity [1][2][3][4][5][6][7].
Different natural problems can be modeled as mathematical problems. The solvability of these mathematical problems is an important aspect of modern mathematics. The fixed-point theory can be applied to investigate the solvability for those natural problems. A large number of works dealing with refinements, applications of fixed-point theory in different types of FDEs have appeared [8][9][10][11]. In particular, Schaefer's and Banach fixed-point theorems are effectively used to establish the solvability of a lot of nonlinear natural problems [12].
With the development of powerful mainframe computers and parallel computing methods, the investigation of stability of solutions of FDEs has become important tool to achieve and to confirm the study, which led that stability study to be one of the most important research subjects in functional analysis. Additionally, sometimes it is too difficult to find an exact solution for nonlinear FDEs, in such case stability analysis plays a very important role in the study. The theory of Ulam stability was introduced in 1940 by Ulam [13]. The following year, Hyers [14] established a type of stability in the Banach space which was more generalized than the concept of Ulam stability, and applied this type to achieve the stability conditions of functional equations. After that, Rassias [15] considered another definition of stability condition, that definition was more improved than Ulam and Hyers stabilities, and used this definition to study the stability of FDEs [16,17].
Recently, more and more researchers are interested in applying the Hyers-Ulam stability [18][19][20][21][22]. In 2017, Ali et al. [23] established the existence of solutions for the coupled of FDEs involving Riemann-Liouville fractional operators and achieved Hyers-Ulam stability (HUS). In the same year, Khan et al. [24] considered the existence of solution, therefore HUS for the coupled system of FDEs in Caputo's sense. Recently, in 2019, Ali et al. [25] greatly given Ulam Stability conditions of solution for nonlinear coupled implicit FDEs.
There are several approaches to define the fractional derivative such as Riemann-Liouville, Hadamard, Erdely Kober and Caputo [26]. Recently, in 2017, Almeida [27] proposed new definition of the fractional derivative and called this operator ψ-Caputo derivative. This new definition is more generalized than Riemann-Liouville, Hadamard, Erdely Kober and Caputo operators types. He given a lot of properties of these new fractional derivative operator. After one year, Almeida et al. [28] studied the uniqueness of solution for the system of FDEs including ψ-Caputo derivative.
Nowadays, Abdo et al. [29] proposed an implicit nonlinear FDE including ψ-Caputo derivative under nonlocal condition. They obtained the existence and uniqueness of solution for this problem under some hypotheses.
In this paper, we investigate the existence and uniqueness of solution for the following couples system of FDEs We apply Schaefer's and Banach-fixed-point theorems to obtain the existence and the uniqueness of solution for the proposed system (1.1) under certain hypotheses. Furthermore, we study several kinds of Ulam stability for the proposed system (1.1). The system (1.1) is very general since it is involved fractional derivatives with unlike kernels. Further, it has been shown that these ψ-fractional derivatives unify a wide class of fractional derivatives such as the aforementioned one. On the other hand, the ψ-Caputo operators are found to be more effective and practical than the classical Caputo operators in the mathematical modeling of numerous phenomena and process in dynamics systems [30][31][32][33][34][35].

Preliminaries
In this section, we give some basic concepts and auxiliary results. The following definition was given in [25]. Definition 1. Let I = [0, T ], T > 0, and C(I, R) be the space of all continuous functions u : I → R. It is easy to prove C(I, R) is a Banach space with the norm, u C = max t∈I |u(t)|. Therefore, C n (I, R), n ∈ N, is the space of all n−times continuous and differentiable functions from I to R.
The following definition was given in [26,27]. Definition 2. Let ψ ∈ C n (I, R) be an increasing function such that ψ (t) 0 for all t ∈ I. Consider u : I → R be an integrable function. The ψ−Riemann-Liouville fractional integral of order α > 0, α ∈ (n − 1, n) for some n ∈ N, of the function u is defined as and the ψ-Riemann-Liouville fractional derivative of order α > 0, α ∈ (n − 1, n) for some n ∈ N, of the function u is defined as The following definition was given in [27,28]. Definition 3. Let ψ ∈ C n (I, R) be an increasing function such that ψ (t) 0 for all t ∈ I. Consider u : I → R be an integrable function. The ψ-Caputo fractional derivative of order α > 0, α ∈ (n − 1, n) for some n ∈ N, is defined by ψ-Riemann-Liouville fractional derivative as * D α,ψ We recall the following Lemma which was given in [28]. Lemma 4. Suppose that u : I → R, then (1) if u ∈ C n (I, R), then * D α,ψ 0 + J α,ψ 0 + u(t) = u(t), (2) if u ∈ C n (I, R), then . Now we recall Schaefer's fixed-point theorem which was given in [36]. Theorem 5. (Schaefer's fixed-point theorem) Let X be a Banach space and T : X → X be a continuous and compact map. Furthermore, let Ω = {x ∈ X : x = λT x, λ ∈ (0, 1)}, be a bounded set. Then T has at least one fixed-point in X.
Let f, g : I × R × R → R and h 1 , h 2 : C(I, R) → R. Then we study the Ulam stabilities of the following proposed problem.
The following two Lemmas will be utilized throughout in the paper. Lemma 6. [29] Let h ∈ C(I, R), then the FDE can be written as the following integral equation From Lemma 6, we can show the following interesting Lemma. Lemma 7. The solution of the system (1.1) is equivalent to the following coupled system of nonlinear integral equations

Existence and uniqueness
In this partition we will investigate the solvability of the proposed system (1.1). The system (1.1) can be written as Now, define the following two operators, where t ∈ I and u, v ∈ C(I, R). We will study the proposed system under the following conditions: (C4) h 1 , h 2 : C(I, R) → R, are continuous and there exists L 5 , L 6 ∈ (0, 1) such that The existence of solution for the proposed system (1.1) is equivalent the existence of the fixed-point for the operator equation Theorem 8. Let the conditions (C1)-(C4) hold. Then F is compact.
Proof. Arzela-Ascoli's theorem will be applied by doing the following steps.
Step 1. F is continuous.
where y n , y, x n , x ∈ C(I, R) and x n = f (t, u n (t), y n (t)) , y n = g(t, v n (t), x n (t)).
From condition (C2), we get that and Thus, we have that Since (u n , v n ) → (u, v) as n → ∞, for all t ∈ I and from Lebesgue Dominated Converges theorem [36], we have as n → ∞. Thus F is continuous operator.
Step 2. F is bounded operator in X. By using (C3), we get similarly, we find that Similarly, we have and (3.11) Therefore, we have Hence, we have Thus, F is bounded in X .
Step 3. F is equicontinuous operator in X. Let t 1 , t 2 ∈ I such that t 1 > t 2 . Then, we get Therefore, we obtain that and thus, we find that Hence, F is equicontinuous. Thus, F is completely continuous. From Arzela-Ascoli's theorem [37], F is compact operator.
Proof. Let B defined as We will show that B is bounded. Let (u, v) ∈ B such that Then for each t ∈ I, we have that Thus, we have 20) Then, we have that

22) and
Using the last two inequalities, we obtain that

26)
then the system (1.1) has a unique solution in X.
Proof. We apply Banach contraction approach to prove F has a unique fixed-point. Let (u, v), (u, v) ∈ X. Then, we have where x, x, y, y ∈ C(I, R), such that x = f (t, u, y) , y = g(t, v, x).
Therefore, we get From substitution the last two equations in (3.27), we get (3.30) By adding the inequalities (3.28) and (3.29) and taking the maximum over I , we get If L * < 1 2 , then F has a unique fixed-point in X.
Lemma 17. Suppose the following condition holds: (C5) If φ 1 , φ 2 ∈ C(I, R) are increasing. Then, there exists µ φ 1 , µ φ 2 ∈ R + such that for every t ∈ I, the following inequalities hold If (z 1 , z 2 ) ∈ X is the solution of the inequalities (4.5), then (z 1 , z 2 ) are the solution of the following inequalities Proof. From Lemma 15, we get Proof. Let (z 1 , z 2 ) ∈ X be any solution of the system of inequalities (4.5) and (u, v) be the unique solution of the system (1.1). By using Lemma 17 and by doing the same steps as in the proof of Theorem 16, we get and Thus, we have where and If γ < 1 2 , we get Thus, we have Thus, the system (1.1) is UHRS. Therefore, if we put ε = 1, then the system (1) is GUHRS.

Conclusions
In this article, we investigated the solvability of a nonlinear coupled system of implicit fractional differential equations including ψ-Caputo fractional operator under nonlocal conditions. The proposed system is more general than many of the systems proposed in pervious literature. We applied Schaefers and Banach fixedpoint theorems to study the solvability of the proposed system. Furthermore, we extended the results to investigate several of Ulam stability for the proposed system. Finally, we provided an example to illustrate the abstract results. The results given in this article extended and developed some previous works.