New results for a coupled system of ABR fractional differential equations with sub-strip boundary conditions

1.   Introduction

For the last three decades, fractional calculus has caught importance and popularity among researchers due to its applicability in modeling many phenomena of the real-world such as propagation in complex mediums, polymers, biological tissues, earth sediments, etc. For more details about applications of fractional calculus, we refer the reader to monographs of Podlubny ^[1], Samko ^[2], Kilbas ^[3], Hilfer ^[4], and references therein. One of the features of fractional calculus is the fact there are many types of derivatives and thus the researchers can use the most suitable fractional derivative for the model they work on. Some of these researchers realized the need for fractional operators with non-singular kernels in modeling some phenomena. Caputo and Fabrizio in ^[5] studied a new kind of fractional derivative with an exponential kernel. A new type and interesting fractional derivative with Mittag-Leffler kernels were developed by Atangana and Baleanu in ^[6]. Abdeljawad in ^[7] extended this fractional derivative from order between zero and one to higher arbitrary order and formulated their associated integral operators. Atangana ^[8,9] introduced some new types of fractional derivatives in the form of power-law and generalized Mittag-Leffler. Many researchers have realized the importance of these new fractional derivatives and applied them to study some properties of solutions for some problems in different fields of science and engineering (see ^{[10,11,12,13,14]}). The famous kinds of stability of fractional differential equations are Ulam, Ulam-Hyers, and Ulam-Hyers-Rassias stability. For more details on kinds of stability, we refer the reader to monographs of Ulam ^[15], Hyers ^[16] and Rassias ^[17].

Coupled systems of fractional differential equations appear in modeling many phenomena of real-world problems. Ahmad et al. ^[18,19] studied existence results for a coupled system of Caputo type sequential fractional differential equations with nonlocal integral boundary conditions. Recently, Almalahi et al. ^[20] studied the existence, uniqueness, and Ulam-Hyers stability results for a coupled system of generalized Hilfer sequential fractional differential equations with two-point boundary conditions by means of Leray-Schauder alternative and Banach fixed point theorem. Almalahi et al. ^[21] studied stability results of positive solutions for a system of generalized Hilfer fractional differential equations building upper and lower control functions and using some techniques of nonlinear functional analysis. Utilizing the Banach and Krasnoselskii fixed point theorems. Alsaedi et al. ^[22] studied the existence and uniqueness results for a nonlinear Caputo-Riemann-Liouville type fractional integro-differential boundary value problem with multi-point sub-strip boundary conditions in the form

where $^{C}D^{\varrho }$ represents the Caputo fractional derivative operator of order $\varrho \in \left(m-1, m\right], m\in \mathbb{N}, m\geq 2, p_{i} > 0, 0 < \tau, \eta _{1}, \eta _{2}, ..., \eta _{p} < 1,$ $\alpha, \beta, \varpi _{1}, \varpi _{2}\in \mathbb{R},$ $\mu _{i}\in \mathbb{R}, i = 1, 2, ...p$ and $f, g_{i}:\left[0, 1\right] \times \mathbb{R} \rightarrow \mathbb{R}, i = 1, 2, ...k$ are continuous functions.

In ^[23], Alsaedi et al. discussed the existence and uniqueness of solutions for the following coupled system

subjected to the conditions

where $^{C}D^{\varrho _{1}}, ^{C}D^{\varrho _{2}}$ represents the Caputo fractional derivative of order $\varrho _{1}, \varrho _{2}\in \left(1, 2 \right].$

Motivated by the novel advancements of Atangana-Baleanu and its applications and the above argumentations, the intent of this work is to investigate the existence, uniqueness, and stability results of a new coupled system under a new fractional derivative so-called ABR fractional derivative of order $1 < \varrho _{1}, \varrho _{2}\leq 2$ with multi-point sub-strip boundary conditions described by

where

$\bullet$ $^{ABR}D_{a^{+}}^{p}$ represents the Atangana-Baleanu-Riemann fractional derivative of order $p = \left\{ \varrho _{1}, \varrho _{2}\right\} \subset \left(1, 2\right]$.

$\bullet$ $\alpha _{1}, \alpha _{2}, \varpi _{1}, \varpi _{2}, \mu _{i}, \xi _{i}\in \mathbb{R},$ and $\eta _{i}, \tau \in \left(0, 1\right)$, $i = 1, 2, ,..., m$.

$\bullet$ $f_{j}:\left[0, 1\right] \times \mathbb{R} ^{2}\longrightarrow \mathbb{R}$ are continuous function, $j = 1, 2.$

In this work, we consider a new type of coupled system involving new fractional operators which extended lately to higher-order by Abdeljawad ^[7]. We considered the system (1.1) with multipoint sub strip conditions, which means our results yield some new results related to choosing the parameters, if $\varpi _{1} = \varpi _{2} = 0,$ then the system (1.1) reduce to the system with coupled multi-point boundary conditions and if $\mu _{i} = \xi _{i} = 0,$ then the system (1.1) reduce to the system with coupled sub-strip boundary conditions.

We investigated the existence and uniqueness of the solution as well as Ulam-Hyers and generalized Ulam-Hyers stability of the proposed coupled system by using minimal conditions.

The main contribution of this work is to find an equivalent fractional integral equation for the suggested system and to prove its existence, uniqueness, and Ulam-Hyers (UH) stability results for a new system under a new fractional derivative. The fixed point theorems of Banach and Leray-Schauder are used in our analysis. Despite the fact that we employ common methods to get our conclusions, the application of it to the suggested system is novel. Furthermore, the results acquired in this study may be extended to an n-tuple fractional system. Our results obtained include the results of Alsaedi et al. in ^[22,23]. With regard to the boundary condition at the terminal position $\sigma = 1$ used in this work, the linear combination of the unknown function and its derivative is associated with the contribution due to sub-strip $\left(0, \tau \right)$ and finitely many nonlocal positions between them within the domain $\left[0.1\right]$. This boundary condition covers many interesting situations, for example, it corresponds to the two-strip aperture condition for all $\mu _{i} = \xi _{i} = 0, i = 1, 2, ...., m$. By choosing $\varpi _{1} = \varpi _{2} = 0$, this condition reduces to a multi-point nonlocal boundary condition. It's worth noting that integral boundary conditions play a critical role in the research of practical problems like blood flow problems ^[24] and bacterial self-regularization ^[25], among others. For more applications about strip conditions in engineering and real-world problems (see ^[26,27]). To the best of our knowledge, this is the first work dealing with the ABR fractional derivative of order $\varrho _{1}, \varrho _{2}$ $\in \left(1, 2\right]$ with multi-point sub-strip boundary conditions. In consequence, the results of this work will be a useful contribution to the existing literature on this topic.

The paper is organized as follows: In Section 2, we present notations and some preliminary facts used throughout the paper. Section 3 discusses the existence and uniqueness results for ABR-System (1.1). The stability analysis in the frame of Ulam-Hyers has been discussed in Section 4. Section 5 provides an example to illustrate the validity of our results. Concluding remarks about our results in the last Section.

2.   Preliminaries

To achieve our main objectives, we present here some definitions and basic auxiliary results that are required throughout the paper. Let $\mathcal{J} = \left[0, 1\right] \subset \mathbb{R}$ and $\mathcal{X} = C\left(\mathcal{J}, \mathbb{R} \right)$ be the space of continuous functions $\vartheta :\mathcal{J} \rightarrow \mathbb{R}$ equipped with the norm $\left\Vert \vartheta \right\Vert = \sup_{\sigma \in \mathcal{J}}\left\vert \vartheta \left(\sigma \right) \right\vert$. Evidently, ($\mathcal{X}$, $\left\Vert \cdot \right\Vert$) is a Banach space and hence the product space $\mathcal{H}: = \mathcal{X}\times \mathcal{X}$ is also a Banach space with the following norm $\left\Vert \left(\vartheta _{1}, \vartheta _{2}\right) \right\Vert = \left\Vert \vartheta _{1}\right\Vert +\left\Vert \vartheta _{2}\right\Vert$.

Definition 2.1. ^[6] Let $0 < \varrho \leq 1\mathfrak{\ }$and $\vartheta \in H^{1}\left(\mathcal{J}\right).$ Then the left-sided $ABR$ fractional derivative of order $\varrho$ for a function $\vartheta$ with the lower limit zero is defined by

where $B\mathcal{(}\varrho \mathcal{)} = \frac{\varrho }{2-\varrho } > 0$ is the normalization function such that $B\mathcal{(}0\mathcal{)} = B\mathcal{(}1 \mathcal{)} = 1$ and $E_{\varrho }$ is the Mittag-Leffler function defined by

The associated Atangana-Baleanu ($AB$) fractional integral is given by

Definition 2.2. ^[7] The relation between the ABR and ABC fractional differential equations is given by

Lemma 2.3. ^[6] Let $\vartheta > 0$. Then $^{AB}I_{0^{+}}^{\varrho }$ isbounded from $\mathcal{X}$ into $\mathcal{X}$.

Definition 2.4. (^[7] Definition 3.1) Let $n < \varrho \leq n+1\mathfrak{\ }$ and $\vartheta ^{\left(n\right) }\in H^{1}\left(0, 1\right).$ Let $\beta = \varrho -n.$ Then, $0 < \beta \leq 1$ and the left-sided $ABR$ fractional derivative of order $\varrho$ for a function $\vartheta$ with the lower limit zero is defined by

The correspondent fractional integral is given by

Lemma 2.5. (^[7] Proposition 3.1) Let $\vartheta (\sigma)$ be afunction defined on $\left[0, b\right]$ and $n < \varrho \leq n+1$. Then, forsome $n\in \mathbb{N} _{0}$, we have

Lemma 2.6. (^[7] Theorem 4.2) Let $\varrho \in \left(1, 2\right]$and $\hbar _{i}\in \mathcal{X}, i = 1, 2.$ Then the solution of the followingproblem

is given by

Theorem 2.7. ^[28] Let $K$ be closed subset from a Banach space $\mathcal{X}$, and $\mathcal{G}:\mathcal{K}\rightarrow \mathcal{K}$, be a strictcontraction i.e., $\left\Vert \mathcal{G}(x)-\mathcal{G}(y)\right\Vert \leq L\left\Vert x-y\right\Vert$ for some $0 < L < 1$ and all $x, y\in \mathcal{K}.$Then $\mathcal{G}$ has a fixed point in $\mathcal{K}$.

Lemma 2.8. ^[29] Let $\mathcal{G}:\mathcal{X}\rightarrow \mathcal{X}$ bean operator satisfies

$\bullet$ The operator $\mathcal{G}$ is completely continuous,

$\bullet$ The set $\xi (\mathcal{G}) = \left\{ \vartheta \in \mathcal{X}:\vartheta = \delta \mathcal{G}(\vartheta), \delta \in \left[0, 1\right]\right\} \ $is bounded.

Then, $\mathcal{G}$ has at least one fixed point.

Theorem 2.9. Let $\varrho _{1}, \varrho _{2}\in \left(1, 2\right], \Theta = \alpha _{1}\alpha _{2}-\left(\varpi _{1}\tau +\sum_{i = 1}^{m}\mu _{i}\eta_{i}\right) \left(\varpi _{2}\tau +\sum_{i = 1}^{m}\xi _{i}\eta _{i}\right)\neq 0, \alpha _{1}, \alpha _{2}, \varpi _{1}, \varpi _{2}, \mu _{i}, \xi _{i}\in \mathbb{R},$ and $\eta _{i}, \tau \in \left(0, 1\right)$, $i = 1, 2, ,..., m$ and$\ \hbar_{1}, \hbar _{2}\in \mathcal{X}.$ The unique solution $\left(\vartheta_{1}, \vartheta _{2}\right) \in \mathcal{H}$ of the following problem

is given by

and

where

and

Proof. Assume that $\left(\vartheta _{1}, \vartheta _{2}\right) \in \mathcal{H}$ is a solution of the following equations

Then, by Lemma 2.6, we get

where, $c_{1}, c_{2}$ are arbitrary constants. Applying the conditions ($\alpha _{1}\vartheta _{1}(1)-\varpi _{1}\int_{0}^{\tau }\vartheta _{2}(s)ds = \sum_{i = 1}^{m}\mu _{i}\vartheta _{2}(\eta _{i})$ and$\ \alpha _{2}\vartheta _{2}(1)-\varpi _{2}\int_{0}^{\tau }\vartheta _{1}(s)ds = \sum_{i = 1}^{m}\xi _{i}\vartheta _{1}(\eta _{i})$), we obtain

and

Equations (2.6) and (2.7) can be written as the following system

where

and

Solving system (2.8) for $c_{1}$ and $c_{2},$ we obtain

Substituting the values of $c_{1}$ and $c_{2}$ in (2.4) and (2.5) respectively, we get (2.2) and 2.3. Conversely, apply the operators $^{ABR}{\bf{D}}_{0^{+}}^{\varrho _{1}}, ^{ABR}{\bf{D}} _{0^{+}}^{\varrho _{2}}$ on (2.2) and (2.3) respectively and making use the Lemma 2.5 and note that $^{ABR}{\bf{D}} _{0^{+}}^{\varrho _{i}}c = ^{ABR}D_{0^{+}}^{\beta _{i}}\frac{d}{d\sigma } c = 0, (\beta _{i} = \varrho _{i}-n), i = 1, 2,$ we obtain (2.1). Hence, $\left(\vartheta _{1}, \vartheta _{2}\right)$ satisfies (2.1) if and only if it satisfies (2.2) and (2.3). The proof is completed.

3.   Existence and uniqueness of solutions for ABR-System (1.1)

In view of Lemma 2.9, we define an operator $\Upsilon :\mathcal{H} \rightarrow \mathcal{H}$ by

where

and

such that, $F_{i, \vartheta }(s) = f_{i}(s, \vartheta _{1}(s), \vartheta _{2}(s)), i = 1, 2.$ In the sequel, to simplify our analysis, we take the following notations

and

In the forthcoming theorems, we will prove the existence and uniqueness of solutions for the ABR-System (1.1) utilizing Leray-Schauder alternative and Banach contraction mapping principle.

Theorem 3.1. Let $f_{1}, f_{2}:\left[0, 1\right] \times \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ be are continuous functions. In addition, we assume that:

$\left(H_{1}\right) :\left\vert f_{i}(\sigma, \vartheta, v)\right\vert \leq\varepsilon _{i}+\theta _{i}\left\vert \vartheta \right\vert +\lambda_{i}\left\vert v\right\vert, \varepsilon _{i}, \theta _{i}, \lambda_{i} > 0, i = 1, 2.$

Then, the ABR-System (1.1) has at least one solution, provided that $\Lambda _{1} < 1,$ where

Proof. Notice that the ABR-System (1.1) has at least one solution $\left(\vartheta _{1}, \vartheta _{2}\right)$ if the operator $\Upsilon$ defined by (3.1) has a fixed point. For that, we shall divide the proof into the next steps:

Step 1: $\Upsilon$ is continuous. Since the functions $f_{1}$ and $f_{2}$ are continuous$,$ we conclude that the operator $\Upsilon$ is continuous too.

Step 2: $\Upsilon$ is compact.

Define a closed ball $\mathbb{B}_{R} = \left\{ \left(\vartheta _{1}, \vartheta _{2}\right) \in \mathcal{H}:\left\Vert \left(\vartheta _{1}, \vartheta _{2}\right) \right\Vert \leq R\right\}$ with

First, we show that $\Upsilon$ is uniformly bounded on $\mathbb{B}_{R}$. For each $\left(\vartheta _{1}, \vartheta _{2}\right) \in \mathbb{B}_{R},$ we have

Similarly, we can find that

Consequently, we get

Hence $\Upsilon$ is uniformly bounded.

Next, we show that $\Upsilon$ is equicontinuous. Since $\mathbb{B} _{R}\subset \mathcal{H}$ is bounded. Then, for all $\left(\vartheta _{1}, \vartheta _{2}\right) \in \mathbb{B}_{R}$, there exist constants $\varphi _{1}, \varphi _{2} > 0$ such that $\left\vert F_{1, \vartheta }(\sigma)\right\vert \leq \varphi _{1}$ and $\left\vert F_{2, \vartheta }(\sigma)\right\vert \leq \varphi _{2}.$ Let $\sigma _{1}, \sigma _{2}\in \mathcal{J}$ such that $\sigma _{1} < \sigma _{2}.$ Then, we have

Take $\sigma _{2}\rightarrow \sigma _{1}$, we get

In the same technique, we get

It follows that

Hence $\Upsilon$ is equicontinuous. Due to the Arzelá-Ascoli theorem, we conclude that the operator $\Upsilon$ is compact in $\mathcal{H}$. Therefore, from the above steps, we deduce that $\Upsilon$ is completely continuous.

Step 3: In the last step, we show tht the set $\xi (\Upsilon) = \left\{ \left(\vartheta _{1}, \vartheta _{2}\right) \in \mathcal{H}:\left(\vartheta _{1}, \vartheta _{2}\right) = \mathfrak{\beta }\Upsilon \left(\vartheta _{1}, \vartheta _{2}\right), \; \; \mathfrak{\beta }\in \left(0, 1\right) \right\}$ is bounded.

Let $\left(\vartheta _{1}, \vartheta _{2}\right) \in \xi (\Upsilon).$ Then $\left(\vartheta _{1}, \vartheta _{2}\right) = \mathfrak{\beta }\Upsilon \left(\vartheta _{1}, \vartheta _{2}\right).$ Now, for $\sigma \in \mathcal{ J},$ we have $\vartheta _{1}(\sigma) = \mathfrak{\beta }\Upsilon _{1}\left(\vartheta _{1}, \vartheta _{2}\right)$ and $\vartheta _{2}(\sigma) = \mathfrak{\beta }\Upsilon _{2}\left(\vartheta _{1}, \vartheta _{2}\right).$ According to $\left(H_{1}\right),$ we obtain

By the same technique, we get

Then, we have

Since $\Lambda _{1} < 1$, therefore

Hence, the set $\xi (\Upsilon)$ is bounded. Due to the above steps with Theorem 2.8, we deduce that $\Upsilon$ has at least one fixed point. Consequently, the ABR-System (1.1) has at least one solution$.$

Theorem 3.2. Let $f_{1}, f_{2}:\left[0, 1\right] \times \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ be are continuous functions. In addition, we assume that:

$\left(H_{2}\right) :\left\vert f_{i}(\sigma, \vartheta_{1}, v_{1})-f_{i}(\sigma, \vartheta _{2}, v_{2})\right\vert \leq L_{i}\left(\left\vert \vartheta _{1}-\vartheta _{2}\right\vert +\left\vert v_{1}-v_{2}\right\vert \right), L_{i} > 0, i = 1, 2.$

Then, the system (2.1) has a unique solution, provided that $\sigma < 1,$where

Proof. Let us consider a closed ball set $\mathbb{B}_{R}$ defined in Theorem 3.1. In order to apply Theorem 2.7, we will divide the proof into the following steps:

Step (1): We show that $\Upsilon (\mathbb{B}_{R})\subset \mathbb{B}_{R}.$ By the second step in Theorem 3.1, we have $\Upsilon (\mathbb{B} _{R})\subset \mathbb{B}_{R}.$

Step (1): We need to prove that $\Upsilon$ is a contraction map. Let $(\vartheta _{1}, \vartheta _{2}), (x_{1}, x_{2})\in \mathcal{H}$ and $\sigma \in \mathcal{J}$. Then, we obtain

and consequently, we obtain

By the same way, one can obtain

It follows from (3.3) and (3.4) that

Due to $\sigma < 1$, we conclude that the operator $\Upsilon$ is a contraction. Hence, by Theorem 2.7, the ABR-System (1.1) has a unique solution.

4.   Ulam-Hyers stability

In this section, we shall discuss the Ulam-Hyers (UH) stability of the ABR-System (1.1).

Remark 4.1. ^[20] A function $\left(\widehat{\vartheta }_{1}, \widehat{ \vartheta }_{2}\right) \in \mathcal{H}$ satisfies the following inequalities

if and only if there exists a functions $\kappa _{1}, \kappa _{2}\in D$ such that

${\bf{(i)}}$ $\left\{ \begin{array}{c} \left\vert \kappa _{1}(\sigma)\right\vert \leq \varepsilon _{1}, \\ \left\vert \kappa _{2}(\sigma)\right\vert \leq \varepsilon _{2}. \end{array} \right.$

${\bf{(ii)}}$ $\left\{ \begin{array}{c} ^{ABR}{\bf{D}}_{0^{+}}^{\varrho _{1}}\widehat{\vartheta }_{1}(\sigma) = F_{1, \widehat{\vartheta }}(\sigma)+\kappa _{1}(\sigma), \\ ^{ABR}{\bf{D}}_{0^{+}}^{\varrho _{2}}\widehat{\vartheta }_{2}(\sigma) = F_{2, \widehat{\vartheta }}(\sigma)+\kappa _{2}(\sigma). \end{array} \right..$

Definition 4.2. ^[20] The ABR-System (1.1) is UH stable if there exists $\mathcal{M} > 0$ such that, for each $\varepsilon = \max \{\varepsilon _{1}, \varepsilon _{2}\} > 0$ and each solution $\left(\widehat{ \vartheta }_{1}, \widehat{\vartheta }_{2}\right) \in \mathcal{H}$ of the inequalities (4.1), there exists a solution $\left(\vartheta _{1}, \vartheta _{2}\right) \in \mathcal{H}$ of the ABR-System (1.1) with

Lemma 4.3. Let $\varrho _{1}, \varrho _{2}\in \left(1, 2\right).$ If afunction $\left(\widehat{\vartheta }_{1}, \widehat{\vartheta }_{2}\right)\in \mathcal{H}$ satisfies the inequalities (4.1), then $\left(\widehat{\vartheta }_{1}, \widehat{\vartheta }_{2}\right)$ satisfies thefollowing integral inequalities

where

and

Proof. By Remark 4.1, we have

Then, in view of Theorem 2.9 and Lemma 4.3, we get

In the same way, one can obtain

Theorem 4.4. Assume that $(H_{2})$ hold$.$ If

Then

are Ulam-Hyers stable.

Proof. Let $\varepsilon = \max \{\varepsilon _{1}, \varepsilon _{2}\} > 0\ $ and $\left(\widehat{\vartheta }_{1}, \widehat{\vartheta }_{2}\right) \in \mathcal{ H}$ be a function that satisfying the inequalities 4.1 and let $\left(\vartheta _{1}, \vartheta _{2}\right) \in \mathcal{H}$ be the unique solution of the following system

Now, by Theorem 2.9, we have

Since $\alpha _{1}\vartheta _{1}(1) = \alpha _{1}\widehat{\vartheta }_{1}(1)$ and $\alpha _{2}\vartheta _{2}(1) = \alpha _{2}\widehat{\vartheta }_{2}(1),$ we can proof that $\mathfrak{R}_{\vartheta _{1}} = \mathfrak{R}_{\widehat{ \vartheta }_{1}}$ and $\mathfrak{R}_{\vartheta _{2}} = \mathfrak{R}_{\widehat{ \vartheta }_{2}}$. Hence, from (H$_{2}$) with Lemma 4.3, and for each $\sigma \in \left[0, 1\right],$ we have $\ $

Hence

By the same technique, we get

Thus

where $\varepsilon = \max \{\varepsilon _{1}, \varepsilon _{2}\}$ and

Hence, from (4.6) and Definition 4.2, we deduce that the coupled system (4.2) is Ulam-Hyers (UH) stable.

5.   An example

In this section, we will demonstrate the applicability of our main results through the following example.

Example 5.1. Consider the following system

Here $\varrho _{1} = \frac{3}{2},$ $\varrho _{2} = \frac{5}{4}, \alpha _{1} = \frac{ 1}{4}, \alpha _{2} = \frac{1}{2}, \tau = \frac{1}{7}, \varpi _{1} = \varpi _{2} = 1, m = 3, \mu _{1} = \frac{1}{2}, \mu _{2} = \frac{3}{4}, \mu _{3} = \xi _{3} = 1,$ $\xi _{i} = \frac{i}{3}, (i = 1, 2), \eta _{i} = \frac{i}{5}, (i = 1, 2, 3)$ and

Clearly, for each $\vartheta _{i}, v_{i}\in \mathbb{R}, i = 1, 2,$ we have

and

with $L_{1} = \frac{1}{234}$ and $L_{2} = \frac{1}{240}.$ By the given data, we get $Q_{\psi }\simeq 5.7, Q_{\pi }\simeq 1.8, M_{\pi }\simeq 1.7, M_{\psi }\simeq 0.3$ and $\sigma \simeq 0.04 < 1$. Then, all conditions in Theorem 3.2 are hold. Consequently, the coupled system (5.1) has a unique solution. On the other hand, by simple calculation, we get $\Lambda _{1}\simeq 0.02 < 1$ and hence all hypothesis in Theorem 3.1 are satisfied $.$ Thus, the coupled system (5.1) has at least one solution in $\left[0, 1\right].$ For every $\varepsilon = \max \{\varepsilon _{1}, \varepsilon _{2}\} > 0$ and each $\left(\widehat{\vartheta }_{1}, \widehat{\vartheta }_{2}\right) \in \mathcal{H}$ satisfies

there exists a solution $\left(\vartheta _{1}, \vartheta _{2}\right) \in \mathcal{H}$ of the coupled system (5.1) with

where

and

Therefore, all conditions in Theorem 4.4 are satisfied and hence the coupled system 4.2 is UH stable and GUH.

6.   Conclusions

In recent years, the subject of fractional operators involving nonsingular kernels is novel and has very important significance in modeling many phenomena in the real world, thus there is interest from some researchers to study some qualitative properties of FDEs. In this paper, we have discussed a new system of the nonlinear fractional operators with nonsingular Mittag-Leffler function kernels from order $1 < \varrho _{1}, \varrho _{2}\leq 2$ with multipoint sub-strip boundary conditions. The results obtained in this work are new and cover some new results by choices of the parameters. We proved the existence, uniqueness, and UH, GUH results by means of the Banach fixed point theorem for initial value problems in the frame of ABR derivatives.

Conflict of interest

The authors declare no conflict of interest.