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

1.   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 $\psi$-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 $\psi$-Caputo derivative.

Nowadays, Abdo et al. ^[29] proposed an implicit nonlinear FDE including $\psi$-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

where $u, v \in C(I, \mathbb{R})$, $f, g:I\times \mathbb{R}^2 \rightarrow \mathbb{R}$, $h_1, h_2:C(I, \mathbb{R}) \rightarrow \mathbb{R},$ $u_0, v_0 \in \mathbb{R}$ and $^{*}D^{\alpha, \psi}$ is $\psi$-Caputo derivative operator. 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 $\psi$-fractional derivatives unify a wide class of fractional derivatives such as the aforementioned one. On the other hand, the $\psi$-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]}.

2.   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 $C(I, \mathbb{R})$ be the space of all continuous functions $u:I \rightarrow \mathbb{R}$. It is easy to prove $C(I, \mathbb{R})$ is a Banach space with the norm, $\|u\|_{C} = \max \limits_{t \in I} |u(t)|.$ Therefore, $C^n(I, \mathbb{R}), \; n \in \mathbb{N}$, is the space of all $n-$times continuous and differentiable functions from $I$ to $\mathbb{R}$.

The following definition was given in ^[26,27].

Definition 2. Let $\psi \in C^n (I, \mathbb{R})$ be an increasing function such that $\psi' (t) \neq 0$ for all $t \in I$. Consider $u:I \rightarrow \mathbb{R}$ be an integrable function. The $\psi-$Riemann-Liouville fractional integral of order $\alpha > 0, \; \alpha \in (n-1, n)$ for some $n \in \mathbb{N}$, of the function $u$ is defined as

and the $\psi$-Riemann-Liouville fractional derivative of order $\alpha > 0, \; \alpha \in (n-1, n)$ for some $n \in \mathbb{N}$, of the function $u$ is defined as

The following definition was given in ^[27,28].

Definition 3. Let $\psi \in C^n (I, \mathbb{R})$ be an increasing function such that $\psi' (t) \neq 0$ for all $t \in I$. Consider $u:I \rightarrow \mathbb{R}$ be an integrable function. The $\psi$-Caputo fractional derivative of order $\alpha > 0, \; \alpha \in (n-1, n)$ for some $n \in \mathbb{N},$ is defined by $\psi$-Riemann-Liouville fractional derivative as

where, $u^{[k]}_{\psi }(t) = (\frac{1}{\psi ' (t)} \frac{d}{dt})^k\; u(t).$ We recall the following Lemma which was given in ^[28].

Lemma 4. Suppose that $u:I \rightarrow \mathbb{R}$, then

(1) if $u \in C ^n(I, \mathbb{R})$, then $^*D^{\alpha, \psi}_{0^+}\; J^{\alpha, \psi}_{0^+}u(t) = u(t)$,

(2) if $u \in C ^n(I, \mathbb{R})$, then

therefore, if $\alpha \in (0, 1)$, 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 $T:X\rightarrow X$ be a continuous and compact map. Furthermore, let

be a bounded set. Then $T$ has at least one fixed-point in $X.$

Let $f, g:I \times \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ and $h_1, \; h_2: C (I, \mathbb{R})\rightarrow \mathbb{R}$. 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 $h \in C (I, \mathbb{R})$, 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

3.   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,

such that

where $t \in I$ and $u, v \in C(I, \mathbb{R})$.

We will study the proposed system under the following conditions:

(C1) $f, g:I\times \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ are continuous;

(C2) for all $t \in I, \;$ there exists $L_i \in (0, 1), \; i = 1, 2, 3, 4,$ such that

$\forall w_1, w_2, z_1, z_2 \in \mathbb{R};$

(C3) for all $t \in I, \;$ there exists $A_f, B_f, C_f, A_g, B_g, C_g \in C(I, \mathbb{R})$ such that

$\forall u, w \in C(I, \mathbb{R}),$ with

(C4) $h_1, h_2 : C(I, \mathbb{R}) \rightarrow \mathbb{R}$, are continuous and there exists $L_5, L_6 \in (0, 1)$ such that

for all $u, v \in C(I, \mathbb{R}).$

Let $X$ be the product space $X = C(I, \mathbb{R}) \times C(I, \mathbb{R})$ with the norm

we can prove that $X$ is Banach space. Define $F:X \rightarrow X$ 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.

Define $D_r = \{(u, v) \in X:\|(u, v)\|_X \leq r\}.$ Consider $\{(u_n, v_n)\}$ be a sequence in $X$ such that $\{(u_n, v_n)\}$ converges to $(u, v)$ in $D_r$. Hence, $u_n \rightarrow u$ and $v_n \rightarrow v.$ Thus, we have that

where $y_n, y, x_n, x \in C(I, \mathbb{R})$ and

From condition (C2), we get that

which implies that

therefore,

and

Thus, we have that

Since $(u_n, v_n) \rightarrow (u, v)$ as $n \rightarrow \infty$, for all $t \in I$ and from Lebesgue Dominated Converges theorem ^[36], we have

as $n \rightarrow \infty.$ Also,

Thus,

as $n \rightarrow \infty.$ Thus $F$ is continuous operator.

Step 2. $F$ is bounded operator in $X$.

By using (C3), we get

similarly, we find that

therefore, we have

It follows that

Similarly, we have

Thus, we have

and

Therefore, we have

Hence, we have

Thus, $F$ is bounded in $X$.

Step 3. $F$ is equicontinuous operator in $X$.

Let $t_1, t_2 \in I$ such that $t_1 > t_2$. Then, we get

Therefore, we obtain that

When $t_1 \rightarrow t_2$, we get

and

thus, we find that

Hence, $F$ is equicontinuous. Thus, $F$ is completely continuous. From Arzela-Ascoli's theorem ^[37], $F$ is compact operator.

Theorem 9. Let the conditions (C3) and (C4) hold. The proposed system (1.1) has at least one solution if $\; \mathbb{L} < 1,$ where

and

Proof. Let $\mathbb{B}$ defined as

We will show that $\mathbb{B}$ is bounded. Let $(u, v) \in \mathbb{B}$ such that

Then for each $t \in I,$ we have that

From (C3), we get

Thus, we have

Then, we have that

and

Using the last two inequalities, we obtain that

Hence, we have

where

and

Since $\mathbb{L} < 1$, than $\mathbb{B}$ is bounded. Thus, $F$ possess at least a fixed-point.

Theorem 10. Suppose that the conditions (C1) and (C2) hold. If $\mathbb{L}^{*} < \frac{1}{2}$, where

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 $(u, v), (\overline{u}, \overline{v}) \in X$. Then, we have

where $x, \overline{x}, y, \overline{y} \in C(I, \mathbb{R}),$ such that

Therefore, we get

From substitution the last two equations in (3.27), we get

where

By adding the inequalities (3.28) and (3.29) and taking the maximum over $I$, we get

If $\mathbb{L}^* < \frac{1}{2},$ then $F$ has a unique fixed-point in $X.$

4.   Stability analysis

In this section, some types of Ulam stability will be considered. First we recall the definitions of those types of Ulam stability. For more details, see ^[38].

Definition 11. The system (1.1) is called Ulam-Hyers stable(UHS, for short) if there exists $\lambda _1, \lambda_2 \in \mathbb{R}^+$ such that for every $\varepsilon _1 \;, \; \varepsilon _2 > 0$ and each $(u, v) \in X$ of the inequalities

where $t \in I$, there exists a unique solution $(\overline{u}, \overline{v}) \in X$ of the system (1.1) such that

Definition 12. The system (1.1) is called generalized Ulam-Hyers stable (GUHS, for short) if there exists $\varphi _1, \; \varphi_2\; \in C(\mathbb{R}^+, \mathbb{R}^+), \; \varphi _1(0) = \varphi _2(0) = 0, \;$ such that for every $\varepsilon _1 \;, \; \varepsilon _2 > 0$ and each $(u, v) \in X$ solution of the inequalities

where $t \in I$, there exists a unique solution $(\overline{u}, \overline{v}) \in X$ of the system (1.1) such that

Definition 13. The system (1.1) is called Ulam-Hyers-Rassias stable (UHRS, for short) with respect to $\varphi _1, \varphi_2 \in C(I, \mathbb{R}^+)$, if there exists $\kappa _{\varphi_1}, \; \kappa _{\phi_2} \in \mathbb{R}^+$ such that for every $\varepsilon _1 \;, \; \varepsilon _2 > 0$ and each $(u, v) \in X$ solution of the inequalities

where $t \in I$, there exists a unique solution $(\overline{u}, \overline{v}) \in X$ of the system (1.1) such that

Definition 14. The system (1.1) is called generalized Ulam-Hyers-Rassias stable (GUHRS, for short) with respect to $\varphi _1, \varphi_2 \in C(I, \mathbb{R})$, if $\; \exists \; \kappa _{\varphi_1}, \; \kappa _{\phi_2} \in \mathbb{R}^+$ such that for each $(u, v) \in X$ solution of the inequalities

where $t \in I$, there exists a unique solution $(\overline{u}, \overline{v}) \in X$ of the system (1.1) such that

Lemma 15. Let $\alpha \in (0, 1),$ if $(z_1, z_2) \in X$ is the solution of the inequalities (4.1), then $(z_1, z_2)$ is the solution of the following system

Proof. Let $(z_1, z_2) \in X$ be any solution of the inequalities (4.1), then there exists $(\Theta_1, \Theta _2) \in X$ ($\Theta_1$ dependent of $z_1$ and $\Theta_2$ dependent of $z_2$) such that

and

Thus, the system (4.10) is equivalent to the following system

Let

Thus, we have

Theorem 16. Suppose that (C1) and (C2) hold. Then the system (1.1) will be UHS and consequently GUHS under the following condition

Proof. Let $(z_1, z_2) \in X$ be solution of the system (4.1) and $(u, v)$ be the unique solution of the proposed system (1.1). Hence, we get $|N_1(t)| \leq \varepsilon_1$ and $| N_2 (t)| \leq \varepsilon_2 \;, \forall t \in I.$ Therefore

From Lemma 15, we get

where

From (C1), we have

Thus, we get

Therefore, we obtain

and

By Taking the maximum over $t \in I$ for (4.20) and (4.21), we get that

and

where

Then by adding (4.22) and (4.23), we get

Let $\varepsilon = \varepsilon_1 +\varepsilon_2,$ and $E = (\frac{(\psi(T) -\psi (0))^{\alpha}}{\Gamma(\alpha +1)}).$ Hence, if $\; \gamma < \frac{1}{2}\;$, we get that

where

Thus, we have

Thus the system (1.1) is UHS. Therefore, if we put $\varphi_1(\varepsilon) = \varphi_2(\varepsilon) = \hat{ q}\; \varepsilon$, we have that $\varphi_1 (0) = \varphi_2 (0) = 0$ and

Thus, the system (1.1) is GUHS.

Lemma 17. Suppose the following condition holds:

(C5) If $\phi _1, \phi _2 \in C(I, \mathbb{R})$ are increasing. Then, there exists $\mu _{\phi _1}, \; \mu _{\phi _2} \in \mathbb{R}^+$ such that for every $t \in I,$ the following inequalities hold

If $(z_1, z_2) \in X$ 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

From (C5), we have that

Theorem 18. Consider the Conditions (C1), (C2) and (C5) hold. Then the system (1.1) is UHRS and GUHRS under the following condition

Proof. Let $(z_1, z_2) \in X$ 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 $\gamma < \frac{1}{2}$, we get

Thus, we have

Thus, the system (1.1) is UHRS. Therefore, if we put $\varepsilon = 1$, then the system (1) is GUHRS.

5.   An application

The next example is an application to the previous theoretical results. Consider the following $\psi$-Caputo system of FDEs

where $\psi(t) = \sqrt{1+t}\;$, for all $t \in [0, 1].$ It it clear that $\psi$ is increasing and differentiable function on $[0, 1]$. Therefore,

also

It is clear that, $f$ and $g$ are continuous. Then the condition (C1) holds. Therefore, we have

for all $x, y, x_1, y_1, x_2, y_2 \in \mathbb{R}$ and $t \in [0, 1]$. Thus, the conditions (C2) and (C3) hold with

therefore

From the nonlocal condition, we have $u_0 = v_0 = 0$ and $h_1(u) = \frac{1}{10}u(\frac{3}{5})\;, \; h_2(v) = \frac{1}{10}v(\frac{3}{5})$. Hence, we have that

for all $u_1, u_2 \in C([0, 1], \mathbb{R}).$ Hence the condition (C4) holds with $L_5 = L_6 = 0.1$. By using the above conditions, we get that $\mathbb{L}^{*} \cong 0.126191 < \frac{1}{2}$. As a result, the system (5.1) has a unique solution. Therefore

From Theorem 18, the system (5.1) is UHS, GUHS, UHRS and GUHRS.

6.   Conclusions

In this article, we investigated the solvability of a nonlinear coupled system of implicit fractional differential equations including $\psi$-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.

Conflict of interest

The author declares that he has no conflicts of interest.