Study of the Atangana-Baleanu-Caputo type fractional system with a generalized Mittag-Leffler kernel

1.   Introduction

Fractional differential equations have a profound physical background and rich theoretical connotations and have been particularly eye-catching in recent years. Several-order differential equations refer to equations that contain fractional derivatives or fractional integrals. Fractional order derivatives and integrals have a wide range of applications in many disciplines such as physics, biology, chemistry, etc., such as power with chaotic dynamic behavior systems, dynamics of quasi-chaotic systems, and complex materials or porous media, random walks with memory, etc. For more information see [1-3]. The approximate controllability of the fractional system can be found in [4-10]. Recently, some researchers have realized the importance of finding new fractional derivatives (FDs) with different singular or nonsingular kernels to meet the need to modeling more real-world problems in different fields of science and engineering. For instance, Caputo and Fabrizio [11] studied a new kind of FDs in the exponential kernel. Atangana and Baleanu (AB) [12] investigated a new type and interesting FD with Mittag-Leffler kernels. Abdeljawad in [13] extended this type for higher arbitrary order and formulated their associated integral operators. But the corresponding integral operators of AB derivative do not have a semigroup property, which makes dealing with them theoretically or mathematically somewhat complicated. Very recently, Abdeljawad in [14,15], introduced a fractional derivative with nonsingular kernel in Atangana-Baleanu-Caputo (ABC) settings with multi-parametered Mittag-Leffler (ML) function and study their semigroup properties, its discrete version in [16]. This diversity of FDs has made the topic of fractional calculus attractive and allows researchers to choose the appropriate operator to obtain better results. For some theoretical works on ABC type FDEs, we refer the reader to the series papers [17-20]. On the other hand side, the study of systems involving FDEs is also important as such systems occur in various problems of applied nature. For some theoretical works on systems of FDEs, we refer to series of papers [21-23].

The topic of stability of systems is one of the most important qualitative characteristics of a solution, for more details about the stability of systems see [24-27].

Abdeljawad et al. [28] studied qualitative analyses of some logistic models in the settings of ABC fractional operators with multi-parameter ML kernels, described as follows:

and

where $_{\theta _{0}}^{ABC}\mathbf{D}^{p, q, v}$ is the generalized left ABC FD of order $p\in \left(0, 1\right], q, v > 0$ and $m, n, l > 0.$

Motivated by the recent advancements of ABC operator, its applications, and by the above works, the aim of the current work is to investigate the existence, uniqueness, stability, and continuous dependence results, and discuss the $\delta$-approximate solutions for a new model in the frame of generalized ABC fractional operators with multi-parameters ML kernels described as follows:

where $_{0}^{ABC}\mathbf{D}^{p, q, v}$ is the generalized ABC FD of order $p\in \left(0, 1\right],$ $q, v > 0$. $\mathcal{F}_{k}\in C\left(\left[0, T\right], \mathbb{R}^{+}\right)$ and satisfies some conditions described later in our analysis. Many researchers in different fields of science and engineering used ABC FD with one parameter ML kernel, but their corresponding AB integral operators do not have a semigroup property, which makes dealing with them theoretically or mathematically somewhat complicated. Nevertheless, in this work, we use a new operator containing interesting kernels, we believe that the qualitative properties of solutions for FDEs should be studied via this operator. This work aims to investigate some properties of solutions for the proposed model via a nonsingular FD in ABC settings with multi-parameter ML kernel introduced lately by [14,15]. Due to the fractional derivative used in this work have semigroup property and recently proposed, the results obtained in this work are new and open the door for the researchers to study more real-world problems in different fields.

Notice that, the considered system is investigated under the generalized ML law. In the case of the ABC fractional operator, the requirement of the vanishing condition of the right hand side of the dynamic system to fulfill the initial data needs recuperation on the modeled population. However, the nature of the generalized ML kernel will enable the emancipation of any restrictions on the initial data.

The structure of our paper is as follows. In Section 2, we present notations, auxiliary lemmas and some basic definitions that are needed for our analysis. In Section 3, we discuss the existence and uniqueness results for the model (1.1). Ulam-Hyers stability results for the model (1.1) are discussed in Section 4. In Section 5, we study the continuous dependence of solution and $\delta$-approximate solutions for the model (1.1). In Section 6, we provide an example to illustrate the validity of our results. The last section is devoted to concluding remarks about our results.

2.   Preliminary results

In order to achieve our main purposes, we present here some definitions and basic auxiliary results that are required throughout our paper. Let $\mathcal{J = }\left[0, T\right] \subset\mathbb{R}^{+}$ and $\mathcal{X} = \left\{ z(\theta):z(\theta)\in C\left(\mathcal{J}, \mathbb{R}^{+}\right) \right\}$ be a space with the norm $\left\Vert z\right\Vert = \sup \{\left\vert z(\theta)\right\vert :\theta \in \mathcal{J}\}.$ Clearly, $\left(\mathcal{X}, \left\Vert \cdot \right\Vert \right)$ is a Banach space. For our analysis, we need defined the product space $\mathcal{G }: = \underbrace{\mathcal{X}\times \mathcal{X}\times \mathcal{X\times }....... \mathcal{X}}_{n-time}$. Undoubtedly that $\mathcal{G}$ is also a Banach space with the following norm

For $0\leq p < 1, q, v > 0,$ we defined the space

Definition 2.1. [3] Let $\lambda \in\mathbb{R}$ and $p, \beta, v, z\in\mathbb{C}$ with $Re\left(p\right) > 0,$ the generalized ML functions $E_{p, \beta }^{v}(\lambda, z)$ are defined by

In the case of $\beta = v = 1,$ Eq (2.1) reduced to

where $\left(v\right) _{k} = v\left(v+1\right).......\left(v+k-1\right).$ Since $\left(1\right) _{k} = k!$, then $E_{p, \beta }^{1}(\lambda, z) = E_{p, \beta }(\lambda, z).$

Definition 2.2. [14,15] Let $p\in \left(0, 1\right),$ $Re\left(q\right) > 0, v\in\mathbb{R}$ and $\lambda = -\frac{p}{1-p}.$ The generalized left ABC-FD and ABR-FD of a function $z$ are defined by

and

repspectively, where $B(p) > 0$ is a normalizing function with $B(0) = B(1) = 1$ and $E_{p, q}^{v}(\lambda, \theta -s)$ is generalized ML functions.

Definition 2.3. [15] Let $p\in \left(0, 1\right], q, v > 0$ and $z(\theta)$ be a function defined on $\left[0, T\right]$. Then, the left generalized AB fractional integral $_{0}^{AB}\mathbf{I}^{p, q, v}z(\theta)$ is given by

If $z(\theta)$ is continuous function at $0\ $and $_{0}^{ABR}\mathbf{D} ^{p, q, v}z(\theta)$ exists, then, we have from [14] that

and

Lemma 2.4. [14,15] For $p\in \left(0, 1\right), q > 0,$ $v\in\mathbb{C}$ and $\lambda = -\frac{p}{1-p},$ we have

Lemma 2.5. [14,15] For any $p\in \left(0, 1\right), q > 0,$ $v\in\mathbb{R}$, and $\lambda = -\frac{p}{1-p},$ we have

Theorem 2.6. [29] Let $\mathcal{X}$ be a Banach space. The operator $\Phi :\mathcal{G}\rightarrow \mathcal{G}$ is Lipschitzian if there exists a constant $0 < L < 1$ such that i.e., $\left\Vert \Phi (z)-\Phi (z^{\ast})\right\Vert \leq L\left\Vert z-z^{\ast }\right\Vert$ for all $z, z^{\ast}\in \mathcal{G}.$ Then $\Phi$ is a contraction.

Theorem 2.7. [29] Let $\Phi :\mathcal{G}\rightarrow \mathcal{G}$ be an operator satisfies

(1) $\Phi$ is completely continuous operator.

(2) The set $\xi (\Phi) = \left\{ z\in \Phi :z = \delta \Phi (z), \delta \in\left[0, 1\right] \right\}$ is bounded.

Then $\Phi$ has a fixed point in $\mathcal{G}$.

Lemma 2.8. [15] For $p\in \left(0, 1\right), q > 0,$ $v\in\mathbb{R}$, $\lambda = -\frac{p}{1-p}$ and let $h(\theta)$ be a continuous functions such that $h(0) = 0$ for $q = 1.$ Then, the following problem

is equivalent to the following fractional integral

Definition 2.9. If $\left(z_{1}, z_{2}, ......, z_{n}\right) \in \mathcal{G},$ then $\left(z_{1}, z_{2}, ......, z_{n}\right)$ is said to be a solution of (1.1), if

(1) $z_{k}(0) = z_{k}^{0}\in\mathbb{R}$ for $k = 1, 2, ......, n.$

(2) $\left(z_{1}, z_{2}, ......, z_{n}\right)$ satisfied the following integral equation

3.   Existence and uniqueness of solutions

We devoted this section to derive the equivalent fractional integral equations for the model (1.1). First of all, by using fixed point technique and mathematical techniques, we prove the existence and uniqueness of solution for model (1.1).

In view of Lemma 2.8, the equivalent fractional integral of model (1.1) is given as follows

Let us consider the continuous operator $\Phi :\mathcal{G} \rightarrow \mathcal{G}$ defined by

where

Notice that the model (1.1) has a solution $\left(z_{1}, z_{2}, ..., z_{n}\right)$ if $\Phi$ has a fixed point. To achieve our results, the following hypothesis must be hold.

$\left(H_{1}\right) :$ Let $\mathcal{F}_{k}:$ $\mathcal{J}\times \mathcal{G} \rightarrow\mathbb{R}, \left(k = 1, 2, ....., n\right)$ be a continuous functions and there exist constants numbers $\lambda _{k}, \varepsilon _{k}^{1}, \varepsilon _{k}^{2}, ..., \varepsilon _{k}^{n} > 0,$ such that

for all $\left(\theta, z_{1}, z_{2}, ..., z_{n}\right) \in \mathcal{J}\times \mathcal{G}.$

$\left(H_{2}\right) :$ The kernels $\mathcal{F}_{k}(\theta, z_{1}(\theta), z_{2}(\theta), ......, z_{n}(\theta))$ satisfies the following Lipschitz condition

such that $0\leq L_{k} < 1, (z_{1}, z_{2}, ...., z_{n}), (z_{1}^{\ast }, z_{2}^{\ast }, ....., z_{n}^{\ast })\in \mathcal{G}.$

To simplify our analysis, we set

and

Theorem 3.1. Assume that $\left(H_{1}\right)$ holds. If $\Lambda_{1} < 1,$ then the operator $\Phi$ is completely continuous.

Proof. First, in view of the continuity of the functions $\mathcal{F}_{k},$ we notice that the operator $\Phi$ is continuous. Define a closed ball

with $r\geq \frac{\Lambda _{2}}{1-\Lambda _{1}},$ where

Now, for $\left(z_{1}, z_{2}, ..., z_{n}\right) \in \mathcal{B}_{r}, \theta \in \mathcal{J},$ then, by (3.1) and $k = 1, 2, ......, n,$ we have

Thus

Hence, $\Phi \left(\mathcal{B}_{r}\right)$ is uniformly bounded. Next, for the equicontinuity of the operator $\Phi$, for any $\theta _{1}, \theta _{2}\in \mathcal{J}, \theta _{1} < \theta _{2}$ and $\left(z_{1}, z_{2}, ....., z_{n}\right) \in \mathcal{B}_{r}$, for $k = 1, 2, ..., n,$ we have

Hence

Thus, $\Phi$ is equicontinuous. According to the above analysis together with Arzela'-Ascoli Theorem, we deduce that $\Phi$ is relatively compact and so completely continuous.

Theorem 3.2. Let $\mathcal{F}_{k}$ be a functions satisfies $\left(H_{1}\right)$ such that $\mathcal{F}_{k}(0, z_{1}(0), z_{2}(0), ..., z_{n}(0))) = 0$ in the case $q = 1$. If $\Lambda_{1} < 1,$ then the model (1.1) has at least one solution $\left(z_{1}, z_{2}, ...., z_{n}\right) \in \mathcal{H}: = \underbrace{C^{p, q, v}\left(\mathcal{J}, \mathbb{R}^{+}\right) \times.......\times C^{p, q, v}\left(\mathcal{J}, \mathbb{R}^{+}\right) }_{n-time}.$

Proof. From Theorem (3.1), we have $\Phi$ is completely continuous. Now, by means of Schaefer's fixed point approaches, we need only prove that the set

is bounded. Let $\left(z_{1}, z_{2}, ...., z_{n}\right) \in \xi (\Phi).$ Then, $z_{k} = \delta \Phi _{k}\left(z_{1}, z_{2}, .., z_{n}\right), k = 1, 2, ..., n.$ For $\theta \in \mathcal{J},$ by (3.1), we get

Thus

Hence $\xi (\Phi)$ is bounded. So, by Theorem (2.7), we deduce that $\Phi$ has one fixed point in $\mathcal{X}$. Consequently, the model (1.1) has at least one solution $\left(z_{1}, z_{2}, ...., z_{n}\right) \in \mathcal{G}$. In addition, by the definition of $\Phi$ and $\left(z_{1}, z_{2}, ...., z_{n}\right)$ possesses the form $z_{k}(\theta) = z_{k}^{0}+_{0}^{AB}\mathbf{I}^{p, q, v}\mathcal{F}_{k}(\theta, z_{1}(\theta), z_{2}(\theta), ....., z_{n}(\theta))$,

we get

By Lemmas 2.4 and 2.5, the identity $_{0}^{ABR}\mathbf{D}^{p, q, v}\mbox{ }_{0}^{AB}\mathbf{I}^{p, q, v}\mbox{ } z_{k}(\theta) = z_{k}(\theta)$, and taking into account that $\mathcal{F} _{k}(0, z_{1}(0), z_{2}(0), ....., z_{n}(0))) = 0$ for $q = 1.$ So, we can shown that $\left(z_{1}, z_{2}, ...., z_{n}\right) (\theta)$ satisfies the model (1.1) if and only if it satisfies (3.1). Finally, we have the estimate

From (3.7), we conclude that

That is $_{0}^{ABC}\mathbf{D}^{p, q, v}z_{k}\in C^{p, q, v}\left(\mathcal{J}, \mathbb{R}^{+}\right)$ and hence

Theorem 3.3. Let $\mathcal{F}_{k}$ be continuous functions satisfies $\left(H_{2}\right).$ Then, the model (1.1) has a unique solution in the space $\mathcal{H}: = \underbrace{C^{p, q, v}\left(\mathcal{J}, \mathbb{R}^{+}\right) \times.......\times C^{p, q, v}\left(\mathcal{J}, \mathbb{R}^{+}\right) }_{n-time},$ provided that $\Delta _{1} < 1.\ $Moreover, the case $q = 1$ requires that $\mathcal{F}_{k}(0, z_{1}(0), z_{2}(0), ....., z_{n}(0))) = 0$.

Proof. Define a closed ball set $\mathcal{A}_{r} = \left\{ (z_{1}, z_{2}, ...., z_{n})\in \mathcal{G}:\left\Vert (z_{1}, z_{2}, ...., z_{n})\right\Vert \leq r\right\}$ with $r\geq \frac{ \Delta _{2}}{1-\Delta _{1}},$ where

and $\mathcal{K}_{k} = \max_{\theta \in \mathcal{J}}\left\vert \mathcal{F} _{k}(\theta, 0, 0, ....., 0)\right\vert.$ In order to prove $\Phi \left(\mathcal{A}_{r}\right) \subset \mathcal{A}_{r},$ let $\left(z_{1}, z_{2}, ...., z_{n}\right) \in \mathcal{A}_{r}$. Then, for $\theta \in \mathcal{J}$, by (3.4), we have

Thus

Hence $\Phi \left(\mathcal{A}_{r}\right) \subset \mathcal{A}_{r}.$ For any $\left(z_{1}, z_{2}, ...., z_{n}\right), \left(z_{1}^{\ast }, z_{2}^{\ast }, ....., z_{n}^{\ast }\right) \in \mathcal{A}_{r}\ $and $\theta \in \mathcal{J }$, we have

Thus

Due to $\Delta _{1} < 1$, we conclude that $\Phi$ a contraction on $C\left(\mathcal{J}, \mathbb{R}^{+}\right)$. Therefore, due to Banach fixed point Theorem, the model (1.1) has a unique fixed point $\left(z_{1}, z_{2}, ...., z_{n}\right) \in \mathcal{G}.$ By the same way in Theorem 3.2, one can prove that $\left(z_{1}, z_{2}, ...., z_{n}\right) \in \mathcal{H}.$ In case $q = 1,$ the condition $\mathcal{F}_{k}(0, z_{1}(0), ..., z_{n}(0)) = 0$ is needed in order to guarantee that solution given by (2.3) will satisfy $z_{k}(0) = z_{k}^{0}, k = 1, 2, ....., n$. However, in case $q\neq 1$, one may note that $z_{k}(0) = z_{k}^{0}$ without any restrictions.

4.   Ulam-Hyers stability

Definition 4.1. [30] The model (1.1) is UH stable if there exists a real number $\mathfrak{N} = \max \left\{ \mathfrak{N}_{1}, \mathfrak{N} _{2}, ......, \mathfrak{N}_{n}\right\} > 0$ such that for each $\epsilon = \max \left\{ \epsilon _{1}, \epsilon _{2}, ....., \epsilon _{n}\right\} > 0$ there exists a solution $\left(\widetilde{z}_{1}, \widetilde{z}_{2}, ....., \widetilde{z}_{n}\right) \in \mathcal{G}$ of the inequality

corresponding to a solution $\left(z_{1}, z_{2}, ...., z_{n}\right) \in \mathcal{G}$ of model (1.1) such that

Remark 4.2. A function $\left(\widetilde{z}_{1}, \widetilde{z}_{2}, ....., \widetilde{z}_{n}\right) \in \mathcal{G}$ satisfies the inequality (4.1) if and only if there exist a small perturbation $\left(z_{1}, z_{2}, ..., z_{n}\right) \in \mathcal{G}$ such that for $k = 1, 2, ...., n,$ we have

(i)$\left\{ \left\vert z_{k}(\theta)\right\vert \leq \epsilon _{k}, \ \theta \in \mathcal{J}.\right.$

(ii)$\mathbf{\ }_{0}^{ABC}\mathbf{D}^{p, q, v}\widetilde{z} _{k}(\theta) = \mathcal{F}_{k}(\theta, \widetilde{z}_{1}(\theta), .., \widetilde{z}_{n}\left(\theta \right))+z_{k}(\theta), \theta \in \mathcal{J }.$

Lemma 4.3. Let $0\leq p < 1, q, v > 0.$ If a function $\left(\widetilde{z}_{1}, \widetilde{z}_{2}, ....., \widetilde{z}_{n}\right) \in \mathcal{G}$ satisfies the inequality (4.1), then $\left(\widetilde{z}_{1}, \widetilde{z}_{2}, ....., \widetilde{z}_{n}\right)$ satisfies the following integral inequalities

where

Proof. Indeed by Remark 4.2, we have the following model

Then, the solution of the above model is given as

It follows that

Hence

Theorem 4.4. Assume that the preconditions of Theorem 3.3 are satisfied. Then the model (1.1) is UH stable in $C\left(\mathcal{J}, \mathbb{R}^{+}\right)$.

Proof. Let $\epsilon = \max \{\epsilon _{1}, \epsilon _{2}, ....., \epsilon _{n}\} > 0$ and

Let $\left(\widetilde{z}_{1}, \widetilde{z}_{2}, ....., \widetilde{z} _{n}\right) \in \mathcal{G}$ be a functions satisfying the inequalities

and let $\left(z_{1}, z_{2}, ...., z_{n}\right) \in \mathcal{G}$ be the unique solution of the following model

Now, in the light of Lemma 2.8$,$ we have

Since $z_{k}(0) = \widetilde{z}_{k}(0)$, $\left(k = 1, 2, ....., n\right),$ we get

Hence, from (3.4) and Lemma 4.3, then for each $\theta \in \mathcal{J},$ we have

which implies

For $\epsilon = \max \left\{ \epsilon _{1}, \epsilon _{2}, ....., \epsilon _{n}\right\} \ $and

we get

Hence the model (1.1) is U-H stable.

5.   Continuous dependence and $\delta$-approximate solutions

In this section, we shall discuss the results of continuous dependence of solutions of the proposed model (1.1) on initial conditions.

Definition 5.1. [29] A function $\left(z_{1}, z_{2}, ...., z_{n}\right) \in \mathcal{G}$ satisfying the following fractional differential inequality

and

is called $\delta$-approximate solutions of model (1.1).

Theorem 5.2. For $p\in \left(0, 1\right), q > 0,$ $v\in\mathbb{R}$ and $\lambda = -\frac{p}{1-p}.$ Let $\mathcal{F}_{k}:\mathcal{J}\times\mathcal{G}\rightarrow\mathbb{R}$ be a continuous function and satisfies Lipschitz condition 3.4, let $\left(z_{1}, z_{2}, ...., z_{n}\right) \ $and $\left(z_{1}^{\ast}, z_{2}^{\ast }, ....., z_{n}^{\ast }\right)$ be a $\delta$-approximation solutions of the model (1.1). If $\bigtriangleup \neq 0$ and

where $l_{ij}\in\mathbb{R}, i, j = 1, 2, ....n$ and

where

then

Proof. Let $\left(z_{1}, z_{2}, ...., z_{n}\right), \left(z_{1}^{\ast }, z_{2}^{\ast }, ....., z_{n}^{\ast }\right) \in \mathcal{G}$ be an $\delta$-approximation solutions of the model (1.1). Then, we have

and

Applying $_{0}^{AB}\mathbf{I}^{p, q, v}$ on both sides of the above inequalities, and using Lemma 2.8, we get

Using the fact $\left\vert z\right\vert -\left\vert y\right\vert \leq \left\vert z-y\right\vert \leq \left\vert z\right\vert +\left\vert y\right\vert$, we get

Then, we have

Consequently, we have the following inequalities

Inequalities (5.3) can be writting as matrices as followes

By simple computations, the above inequality becomes

Since $\Delta \neq 0.$ This leads to

From the fact

It follows that

Remark 5.3. If $\delta _{1} = \delta _{2} = 0$ in the inequality (5.4), then $\left(z_{1}, z_{2}, ...., z_{n}\right), \left(z_{1}^{\ast }, z_{2}^{\ast }, ....., z_{n}^{\ast }\right)$ are solutions of the model (1.1) and the inequality (5.4) reduces to

which provides the continuous dependence of the model (1.1). Also if $z_{0}^{r} = z_{0}^{\ast r}$ for all $r = 1, 2, ..., n,$ then

which provides the uniqueness of a solution of model (1.1).

6.   An example

Consider the following model

Here $p = \frac{1}{2}, q = \frac{1}{2},$ $v = 1,$ $a = 0,$ $\theta = 1,$

Clearly, $\mathcal{F}_{k}\left(\theta, z_{1}(\theta), z_{2}(\theta), ..., z_{n}(\theta)\right)$ are continuous, $\mathcal{F}_{k}\left(0, z_{1}(0), z_{2}(0), ....., z_{n}(0)\right) = 0$ and

for all $\left(z_{1}, z_{2}, ...., z_{n}\right), \left(z_{1}^{\ast }, z_{2}^{\ast }, ...., z_{n}^{\ast }\right) \in \mathcal{G}.$ Also

Here $L_{k} = \lambda _{k} = \varepsilon _{k}^{j} = \frac{1}{3}$ for all $k = 1, 2, ..., n$ and $j = 1, 2, ..., n$. Now, by simple calculation, we get

Thus all conditions of Theorem 3.3 are satisfied. Hence, model (1.1) has a unique solution on $\left(0, 1\right].$ Finally, for $\epsilon = \max \left\{ \epsilon _{1}, \epsilon _{2}, ....., \epsilon _{n}\right\},$ we find that

is satisfied. Hence the model (1.1) is U-H stable with

where

7.   Conclusions

ABC fractional operators with multi-parameters ML kernels on certain time scales and the integral equations expressed by them are some of the keys in developing fractional calculus. In this work, we have obtained some existence, uniqueness, UH stability results for the fractional system (1.1) in the frame of generalized FD in AB settings containing a multi-parameter ML kernel. As well, the data dependence analysis and $\delta$-approximate solutions of the proposed system are discussed. Our approach is based on some fixed point theorems and mathematical techniques. As an application, one example has been provided in order to illustrate the validity of our results. We realized that, if $q\neq 1,$ then the condition $\mathcal{F}_{k}(0\mathfrak{, }z_{1}(0), ......, z_{n}(0)) = 0, (k = 1, 2, ..., n)$ not necessary to guarantee a unique solution. The considered system has been investigated under the generalized ML law. Observed that in the case of the classical ABC fractional operator, the requirement of the vanishing status condition of the right-hand side of the dynamic system to full the initial data needs recuperation on the modeled population. However, the nature of the generalized ML kernel managed to get rid of any restrictions on the initial data. Due to the fractional operators used in this work have semigroup property and recently proposed, the results obtained here are new and open the door for the researchers to study more real-world problems in different fields. Besides, the results obtained in this work are very significant in developing the theory of fractional analytical dynamics of different biological models.

So in the future, the same analysis can be extended to the system of delay equations under the generalized fractional operator.

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University for funding this work through Research Group no. RG-21-09-07.

Conflicts of interest

The authors declare that they have no competing interests.