On coupled impulsive fractional integro-differential equations with Riemann-Liouville derivatives

1.   Introduction

Fractional order differential equations are the generalizations of the classical integer order differential equations. The idea about the fractional order derivative was introduced at the end of the sixteenth century $(1695)$ when Leibniz used the notation $\frac{d^{n}}{d\sigma^{n}}$ for $n^{th}$ order derivative. By writing a letter to him, L'Hospital asked the question: what would be the result if $n = \frac{1}{2}?$ Leibniz answered in such words, "An apparent Paradox, from which one day useful consequences will be drawn", and this question became the foundation of fractional calculus. Fractional calculus has become a speedily developing area and its applications can be found in diverse fields ranging from physical sciences, porous media, electrochemistry, economics, electromagnetics, medicine and engineering to biological sciences. Progressively, fractional differential equations play a very important role in fields such as thermodynamics, statistical physics, nonlinear oscillation of earthquakes, viscoelasticity, defence, optics, control, signal processing, electrical circuits, astronomy etc. There are some outstanding articles which provide the main theoretical tools for the qualitative analysis of this research field, and at the same time, shows the interconnection as well as the distinction between integral models, classical and fractional differential equations, see ^{[14,16,18,19,22,25,26,28,30]}

Impulsive fractional differential equations are used to describe both physical, social sciences and many dynamical systems such as evolution processes pharmacotherapy. There are two types of impulsive fractional differential equations the first one is instantaneous impulsive fractional differential equations while the other one is non-instantaneous impulsive fractional differential equations. In last few decades, the theory of impulsive fractional differential equations are well utilized in medicine, mechanical engineering, ecology, biology and astronomy etc. There are some remarkable monographs ^{[3,6,8,15,20,23,33,34]}, considering fractional differential equations with impulses.

The most preferable research area in the field of fractional differential equations $(\mathcal{FDE}'s)$, which received great attention from the researchers is the theory regarding the existence of solutions. Many researchers developed some interesting results about the existence of solutions of different boundary value problems (BVPs) using different fixed point theorems. For details we refer the reader to ^{[2,7,9,10,11,13,27]}. Most of the time, it is quite intricate to find the exact solutions of nonlinear differential equations, in such a situation different approximation techniques are introduced. The difference between exact and approximate solutions is nowadays dealt with using Hyers-Ulam $(\mathcal{HU})$ type stabilities, which were first introduced in 1940 by Ulam ^[29] and then answered by Hyers in the following year in the context of Banach spaces. Many researchers investigated $\mathcal{HU}$ type stabilities for different problems with different approaches ^{[12,17,31,35,36,37,39,40]}.

Zada and Dayyan ^[38], investigated the existence, uniqueness and Ulam's type stability for the implicit fractional differential equation with instantaneous impulses and Riemann-Liouville fractional integral boundary conditions having the following form

where $I = [0, \mathbb{T}]$, and ${}^{c}\mathcal{D}_{0, \sigma}^{\alpha}$ is a generalization of classical Caputo derivative of order $\alpha$ with lower bound at $0$, $\phi_{1}:I\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$ is a continuous function. Furthermore, $\texttt{u}(\sigma^{+}_{j})$ and $\texttt{u}(\sigma^{+}_{j})$ represent the right-sided and left-sided limits respectively at $\sigma = \sigma_{j}$ for $j = 1, 2, \dots, q-1$.

Ali et al. ^[4], studied a coupled system for the existence and uniqueness of solution using Riemann-Liouville derivative

where $\sigma\in\mathcal{J} = [0, \mathbb{T}]$, $\mathbb{T} > 0$, $\alpha, \beta\in(1, 2]$, and $\beta_{1}, \beta_{2}, \gamma_{1}, \gamma_{2}\ne1$. $\mathcal{D}^{\alpha}$, $\mathcal{D}^{\beta}$ are the Riemann-Liouville fractional derivatives and $\phi_{1}, \phi_{2}:[0, 1]\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$ are continuous functions.

Wang et al. ^[32], presented stability of the following coupled system of implicit fractional integro-differential equations having anti-periodic boundary conditions:

where $1 < \alpha, \beta\leq2$, $0\leq\texttt{r}_{1}, \texttt{r}_{2}\leq2$, $\gamma_{1}, \gamma_{2} > 0$, and $\mathcal{J} = [0, \mathbb{T}]$, $\mathbb{T} > 0$. $\phi_{1}, \phi_{2}, f, g:\mathcal{J}\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$ are continuous functions.

Motivated by the above work, we focus our attention on the following coupled impulsive fractional integro-differential equations with Riemann-Liouville derivatives of the form:

where $1 < \alpha, \beta\le2$, $\phi_1, \phi_2:[0, \mathbb{T}]\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$ being continuous functions and

where $\texttt{u}(\sigma_{j}^+), \texttt{v}(\sigma_{k}^+)$ and $\texttt{u}(\sigma_{j}^-), \texttt{v}(\sigma_{k}^-)$ are the right limits and left limits respectively, $\mathcal{E}_{j}, \mathcal{E}_{j}^*, \mathcal{E}_{k}, \mathcal{E}_{k}^*:\mathbb{R}\rightarrow\mathbb{R}$ are continuous functions, and $\mathcal{D}^{\alpha}, \mathcal{I}^{\alpha}$ are the $\alpha$-order Riemann-Liouville fractional derivative and integral operators respectively.

The remaining article is arranged as follows: In Section 2, we present some basic definitions, theorems, and lemmas that will be used in our main results. In Section 3, we use suitable cases for the existence and uniqueness of solution for the proposed system (1.1) using Kransnoselskii's type fixed point theorem. In Section 4, we discuss different kinds of stabilities in the sense of Ulam under certain conditions. In Section 5, an example is given to support the main results.

2.   Auxiliary results

In this section, we present some basics notations, definitions, and results that are used in the whole article.

Let $\mathbb{T} > 0$, $\omega = [0, \mathbb{T}]$. The Banach space of all continuous functions from $\omega$ into $\mathbb{R}$ is denoted by $\mathcal{C}(\omega, \mathbb{R})$ with the norm

and the product of these spaces is also a Banach space with the norm

The piecewise continuous functions with $1 < \alpha, \beta\leq2$ are denoted as follows:

with the norms

respectively. Their product $\vartheta = \vartheta_{1}\times\vartheta_{2}$ is also a Banach space with the norm $\|(\texttt{u}, \texttt{v})\|_{\vartheta} = \|\texttt{u}\|_{\vartheta_{1}}+\|\texttt{v}\|_{\vartheta_{2}}$.

Definition 2.1. ^[1] The Riemann-Liouville fractional integral of order $\alpha > 0$ for a function ${\mathit{\mathtt{u}}}:\mathbb{R}^{+}\rightarrow\mathbb{R}$ is defined as

where $\Gamma(\cdot)$ is the Euler gamma function defined by $\Gamma(\alpha) = \int_{0}^{\infty}e^{-\sigma}\sigma^{\alpha-1}d\sigma, \; \; \alpha > 0.$

Definition 2.2. For a function ${\mathit{\mathtt{u}}}:\mathbb{R}^{+}\rightarrow\mathbb{R}$, the Riemann-Liouville derivative of fractional order $\alpha > 0$, $p = [\alpha]+1$, is defined as

provided that integral on the right side exists. $[\alpha]$ denotes the integer part of the real number $\alpha$. For more properties, the reader may refer to ^[1].

Lemma 2.1. ^[1] Let ${\mathit{\mathtt{u}}}$ be any function, and let $\alpha > 0$, then the Riemann-Liouville fractional derivative for the Homogeneous differential equation

has a solution

and for non-homogeneous differential equation

has a solution

where $p = [\alpha]+1$ and $c_{i}, i = 1, 2, \dots, p,$ are real constants.

Theorem 2.1. (Altman ^[5]) Let $\Lambda\neq0$ be a convex and closed subset of Banach space $\vartheta$. Consider two operators $\Im_{1}, \Im_{2}$ such that

(1) $\Im_{1}({\mathit{\mathtt{u}}}, {\mathit{\mathtt{v}}})+\Im_{2}({\mathit{\mathtt{u}}}, {\mathit{\mathtt{v}}})\in\Lambda$;

(2) $\Im_{1}$ is a contractive operator;

(3) $\Im_{2}$ is a compact and continuous operator.

Then there exists $({\mathit{\mathtt{u}}}, {\mathit{\mathtt{v}}})\in\Lambda$ such that $\Im_{1}({\mathit{\mathtt{u}}}, {\mathit{\mathtt{v}}})+\Im_{2}({\mathit{\mathtt{u}}}, {\mathit{\mathtt{v}}}) = ({\mathit{\mathtt{u}}}, {\mathit{\mathtt{v}}})\in\vartheta.$

2.1.   Ulam's Stabilities and Remarks

The following definitions and remarks are taken from ^[21,24].

Definition 2.3. The given system (1.1) is $\mathcal{HU}$ stable if there exists $\mathcal{N}_{\alpha, \beta} = \max\{\mathcal{N}_{\alpha}, \mathcal{N}_{\beta}\} > 0$ such that, for $\kappa = \max\{\kappa_{\alpha, }, \kappa_{\beta}\} > 0$ and for every solution $(\xi, \zeta)\in \vartheta$ of the inequality

there exists a solution $({\mathit{\mathtt{u}}}, {\mathit{\mathtt{v}}})\in\vartheta$ with

Definition 2.4. The given system (1.1) is generalized $\mathcal{HU}$ stable if there exists $\mathcal{N}'\in \mathcal{C}(\mathbb{R}^+, \mathbb{R}^+)$ with $\mathcal{N}'(0) = 0$ such that, for any approximate solution $(\xi, \zeta)\in \vartheta$ of inequality (2.1), there exists a solution $({\mathit{\mathtt{u}}}, {\mathit{\mathtt{v}}})\in\vartheta$ of (1.1) satisfying

Definition 2.5. The given system (1.1) is $\mathcal{HUR}$ stable with respect to $\psi_{\alpha, \beta} = \max\{\psi_{\alpha}, \psi_{\beta}\}$ with $\psi_{\alpha, \beta}\in\mathcal{C}(\omega, \mathbb{R})$ if there exists a constant $\mathcal{N}_{\psi_{\alpha}, \psi_{\beta}} = \max\{\mathcal{N}_{\psi_{\alpha}}, \mathcal{N}_{\psi_{\beta}}\} > 0$ such that, for any $\kappa = \max\{\kappa_{\alpha, }, \kappa_{\beta}\} > 0$ and for any approximate solution $(\xi, \zeta)\in\vartheta$ of the inequality

there exists a solution $({\mathit{\mathtt{u}}}, {\mathit{\mathtt{v}}})\in \vartheta$ with

Definition 2.6. The given system (1.1) is generalized $\mathcal{HUR}$ stable with respect to $\psi_{\alpha, \beta} = \max\{\psi_{\alpha}, \psi_{\beta}\}$ with $\psi_{\alpha, \beta}\in\mathcal{C}(\omega, \mathbb{R})$ if there exists a constant $\mathcal{N}_{\psi_{\alpha}, \psi_{\beta}} = \max\{\mathcal{N}_{\psi_{\alpha}}, \mathcal{N}_{\psi_{\beta}}\} > 0$ such that, for any approximate solution $(\xi, \zeta)\in\vartheta$ of inequality (2.2), there exists a solution $({\mathit{\mathtt{u}}}, {\mathit{\mathtt{v}}})\in\vartheta$ of (1.1) satisfying

Remark 2.1. Let $(\xi, \zeta)\in\vartheta$ be a solution of inequalities (2.1) if there exist functions $\mathfrak{K}_{\phi_1}, \mathfrak{L}_{\phi_2}\in\mathcal{C}(\omega, \mathbb{R})$ depending on $\xi, \zeta$ respectively such that

(1) $|\mathfrak{K}_{\phi_1}(\sigma)|\le\kappa_{\alpha}, \; \; |\mathfrak{L}_{\phi_2}(\sigma)|\le\kappa_{\beta}, \; \; \sigma\in\omega$;

(2)

3.   Existence and uniqueness

In this section, we discuss the existence and uniqueness of solution of the proposed system (1.1).

Theorem 3.1. Let $\alpha, \beta\in(1, 2]$ and $\phi_1$ be any linear and continuous function. The fractional impulsive differential equation

has a solution

Proof. Consider

For $\sigma\in[0, \sigma_{1}]$, Lemma 2.1 gives

Again, for $\sigma\in(\sigma_{1}, \sigma_{2}]$, Lemma 2.1 gives

Hence it follows that

Using

we obtain

Substituting the values of $b_{1}$, $b_{2}$ in (3.5), we get

Similarly, for $\sigma\in(\sigma_{j}, \sigma_{j+1}]$,

Finally, after applying conditions $\nu_{1}\mathcal{D}^{\alpha-2}\texttt{u}(\sigma)|_{\sigma = 0} = \texttt{u}_{1},$ and $\mu_{1}\texttt{u}(\sigma)|_{\sigma = \mathbb{T}}+\nu_{2}\mathcal{I}^{\alpha-1}\texttt{u}(\sigma)|_{\sigma = \mathbb{T}} = \texttt{u}_{2}$ to (3.6) and finding the values of $a_{1}$ and $a_{2}$, we obtain Eq (2.2).

Corollary 1. In view of Theorem 3.1, our coupled system (1.1) has the following solution:

and

Now, for transformation of the given system (1.1) into a fixed point problem, let the operators $\Im_1, \Im_2:\vartheta\rightarrow\vartheta$ be define as follows:

and

For additional analysis, the following hypothesis needs to hold:

(H₁) ● For $\sigma\in \omega$ there exist bounded functions $o, \tau, \upsilon\in \vartheta$ such that

with $o_1 = \sup_{\sigma\in\omega} o(\sigma)$, $\tau_{1} = \sup_{\sigma\in\omega}\tau(\sigma)$, and $\upsilon_{1} = \sup_{\sigma\in\omega}\upsilon(\sigma) < 1$.

● Similarly, for $\sigma\in\omega$ there exist bounded functions $o^{*}, \tau^{*}, \upsilon^{*}\in\vartheta$ such that

with $o_{2} = \sup_{\sigma\in\omega} o^*(\sigma)$, $\tau_{2} = \sup_{\sigma\in\omega}\tau^*(\sigma)$, and $\upsilon_{2} = \sup_{\sigma\in\omega}\upsilon^*(\sigma) < 1.$

(H₂) $\mathcal{E}_{j}, \mathcal{E}_{j}^*:\mathbb{R}\rightarrow\mathbb{R}$ are continuous and there exist constants $\mathcal{G}_{\mathcal{E}}, \mathcal{G}_{\mathcal{E}^*}, \mathcal{G}'_{\mathcal{E}}, \mathcal{G}'_{\mathcal{E}^*}, \mathcal{\widehat{G}}_{\mathcal{E}}, \mathcal{\widehat{G}}_{\mathcal{E}^*}, \mathcal{\widehat{G}}'_{\mathcal{E}}, \mathcal{\widehat{G}}'_{\mathcal{E}^*} > 0$ such that, for any $(\texttt{u}, \texttt{v})\in\vartheta$,

where $z = 1, 2, \dots, p.$

(H₃) ● For all $x_{1}, x_{2}, x_{1}^*, x_{2}^*\in\mathbb{R}$ and for each $\sigma\in\omega$, there exist constants $\mathcal{L}_{\phi_1} > 0$, $0 < \mathcal{L}_{\phi_1}^* < 1$ such that

● Similarly, for all $x_{1}, x_{2}, x_{1}^*, x_{2}^*\in\mathbb{R}$ and for each $\sigma\in\omega$, there exist constants $\mathcal{L}_{\phi_2} > 0$, $0 < \mathcal{L}_{\phi_2}^* < 1$ such that

(H₄) $\mathcal{E}_{z}, \mathcal{E}_{z}^*:\mathbb{R}\rightarrow\mathbb{R}$ are continuous and there exist constants $\mathcal{L}_{\mathcal{E}}, \mathcal{L}_{\mathcal{E}^*}, \mathcal{L}_{\mathcal{E}}^*, \mathcal{L}_{\mathcal{E^*}}^*$ such that, for any $(\texttt{u}, \texttt{v}), (\texttt{u}^*, \texttt{v}^*)\in\vartheta$,

Here we use Kransnoselskii's fixed point theorem to show that the operator $\Im_1+\Im_2$ has at least one fixed point. Therefore, we choose a closed ball

where

Theorem 3.2. If hypotheses $(\boldsymbol{H}_{1})$–$(\boldsymbol{H}_{4})$ are hold, then the given system (1.1) has at least one solution.

Proof. 1) For any $(\texttt{u}, \texttt{v})\in\vartheta_{r}$, we have

From (3.9), we get

This implies that

Similarly, we can obtain

where

Also, we have

Now by $(\textbf{H}_{1}$)

Now, taking $\sup_{\sigma\in\omega}$ on both sides, we get

Now taking $\sup_{\sigma\in\omega}$ of (3.15) and using (3.16) in (3.15), we get

Similarly,

where

Putting (3.13), (3.14), (3.17) and (3.18) in (3.11), we get

Hence, $\|\Im_{1}(\texttt{u}, \texttt{v})+\Im_{2}(\texttt{u}, \texttt{v})\|_{\vartheta}\in \vartheta_{r}.$

2) Next, for any $\sigma\in\omega$, $(\texttt{u}, \texttt{v}), (\xi, \zeta)\in\vartheta$

Here $\mathcal{L} = \max\{\mathcal{L}_{\mathcal{E}}, \mathcal{L}_{\mathcal{E}^*}, \mathcal{L}_{\mathcal{E}}^*, \mathcal{L}_{\mathcal{E}^*}^*\},$

and

Therefore, $\Im_{1}$ is a contractive operator.

3) Now, for the continuity and compactness of $\Im_{2}$, we make a sequence $T_{s} = (\texttt{u}_{s}, \texttt{v}_{s})$ in $\vartheta_{r}$ such that $(\texttt{u}_{s}, \texttt{v}_{s})\rightarrow(\texttt{u}, \texttt{v})$ for $s\rightarrow\infty$ in $\vartheta_{r}$. Thus, we have

This implies $\|\Im_2(\texttt{u}_{s}, \texttt{v}_{s})-\Im_2(\texttt{u}, \texttt{v})\|_{\vartheta}\rightarrow0$ as $s\rightarrow\infty$, therefore $\Im_{2}$ is continuous.

Next, we show that $\Im_{2}$ is uniformly bounded on $\vartheta_{r}$. From (3.17) and (3.18), we have

Thus, $\Im_{2}$ is uniformly bounded on $\vartheta_{r}$.

For equicontinuity, suppose $\eta_{1}, \eta_{2}\in\omega$ with $\eta_{1} < \eta_{2}$, and for any $(\texttt{u}, \texttt{v})\in\vartheta_{r}\subset\vartheta$ where $\vartheta_{r}$ is clearly bounded, we have

This implies that

In the same way, we have

Hence

Thus, $\Im_{2}$ is equicontinuous. So $\Im_{2}$ is relatively compact on $\vartheta_{r}$. Hence, by the Arzel$\grave{a}$–Ascoli Theorem, $\Im_{2}$ is compact on $\vartheta_{r}.$ Thus all the condition of Theorem 2.1 are satisfied. So the given system (1.1) has at least one solution.

Theorem 3.3. Let hypotheses $(\boldsymbol{H}_{3})$, $(\boldsymbol{H}_{4})$ be satisfied with

then the given system (1.1) has unique solution.

Proof. First we define an operator $\varphi = (\varphi_{1}, \varphi_{2}):\vartheta\rightarrow\vartheta$, i.e., $\varphi(\texttt{u}, \texttt{v})(\sigma) = (\varphi_{1}(\texttt{u}, \texttt{v}), \varphi_{2}(\texttt{u}, \texttt{v}))(\sigma)$, where

and

In view of Theorem 3.2, we have

Taking $\sup_{\sigma\in\omega}$, we get

where

Similarly,

where

Hence

This implies that the operator $\varphi$ is a contraction. Therefore, (1.1) has a unique solution.

4.   Ulam's stability analysis

In this section, we study different kinds of stabilities, like $\mathcal{HU}$, generalized $\mathcal{HU}$, $\mathcal{HUR}$, and generalized $\mathcal{HUR}$ stability of the proposed system.

Theorem 4.1. If assumptions $(\boldsymbol{H}_{3})$, $(\boldsymbol{H}_{4})$ and inequality (3.19) are satisfied and

then the unique solution of the coupled system (1.1) is $\mathcal{HU}$ stable and consequently generalized $\mathcal{HU}$ stable.

Proof. Let $(\xi, \zeta)\in\vartheta$ is a solution of inequality (2.1), and let $(\texttt{u}, \texttt{v})\in\vartheta$ be the unique solution of the coupled system given by

By Remark 2.1 we have

By Corollary 1, the solution of problem (4.2) is

and

We consider

As in Theorem 3.3, we get

and

From (4.5) and (4.6), we have

and

respectively. Let

Then the last two inequalities can be written in a matrix form as follows:

where

From system (4.7) we have

which implies that

If $\kappa = \max\{\kappa_{\alpha}, \kappa_{\beta}\}$ and $\mathcal{N}_{\alpha, \beta} = \frac{\mathcal{P}_{2}}{\mathcal{F}}+\frac{\mathcal{P}_{1}\mathcal{P}_{4}}{\mathcal{F}}+\frac{\mathcal{P}_{2}\mathcal{P}_{3}}{\mathcal{F}}+\frac{\mathcal{P}_{4}}{\mathcal{F}},$ then

Thus system (1.1) is $\mathcal{HU}$ stable. Also, if

with $\mathcal{N'}(0) = 0,$ then the given system (1.1) is generalized $\mathcal{HU}$ stable.

For the next result, we assume the following:

(H₅) Let there exists two nondecreasing functions $w_{\alpha}, w_{\beta}\in\mathcal{C}(\omega, \mathbb{R^+})$ such that

Theorem 4.2. If assumptions $(boldsymbol)$–$(\boldsymbol{H}_{5})$ and inequality (3.19) are satisfied and

then the unique solution of the given system (1.1) is $\mathcal{HUR}$ stable and accordingly generalized $\mathcal{HUR}$ stable.

Proof. With the help of Definitions 2.5 and 2.6, we can achieve our result doing the same steps as in Theorem 4.1.

5.   Example

Here we present a specific example, as follows.

Example 5.1. Let

From system (5.1), we see that $\alpha = \frac{6}{5}$, $\beta = \frac{5}{4}$, $\mu_{1} = \mu_{2} = -50$, $\nu_{1} = \nu_{3} = 1$, $\nu_{2} = \nu_{4} = \frac{1}{85}$, $\mathbb{T} = e$, $\sigma_{1} = \frac{3}{2}$, and ${\mathit{\mathtt{u}}}_{1}, {\mathit{\mathtt{u}}}_{2}, {\mathit{\mathtt{v}}}_{1}, {\mathit{\mathtt{v}}}_{2}\in\mathbb{R}$.

Set

Now, for all ${\mathit{\mathtt{u}}}, {\mathit{\mathtt{u}}}^{*}, {\mathit{\mathtt{v}}}, {\mathit{\mathtt{v}}}^{*}\in\mathbb{R}$, and $\sigma\in[0, e]$, we obtain

and

These satisfy condition $(\boldsymbol{H}_{3})$ with $\mathcal{L}_{\phi_{1}} = \mathcal{L}_{\phi_{1}}^* = \frac{1}{80e^{90}}$, $\mathcal{L}_{\phi_{2}} = \mathcal{L}_{\phi_{2}}^* = \frac{1}{95}.$

Set

Then we have

These satisfy condition $(\boldsymbol{H}_{4})$ with $\mathcal{L}_{\mathcal{E}} = \mathcal{L}_{\mathcal{E}}^* = \mathcal{L}_{\mathcal{E}^*} = \mathcal{L}_{\mathcal{E}^*}^* = \frac{1}{70}.$

From Theorem 3.3, we use the inequality and get

hence (5.1) has a unique solution, so (5.1) has a solution $({\mathit{\mathtt{u}}}, {\mathit{\mathtt{v}}})\in\vartheta$. The solution of (5.1) is given by

and

(i) If we take $\phi_{1}(\sigma, \mathcal{I}^{\alpha}{\mathit{\mathtt{u}}}(\sigma), \mathcal{I}^{\beta}{\mathit{\mathtt{v}}}(\sigma)) = \frac{1}{80e^{\sigma+90}},$ $\phi_{2}(\sigma, \mathcal{I}^{\alpha}{\mathit{\mathtt{u}}}(\sigma), \mathcal{I}^{\beta}{\mathit{\mathtt{v}}}(\sigma)) = \frac{\sigma\cos(\sigma)-\sin(\sigma)}{95}-\frac{1}{95},$ $\mathcal{E}_{1}\left({\mathit{\mathtt{u}}}\left(\frac{3}{2}\right)\right) = \mathcal{E}_{1}^{*}\left({\mathit{\mathtt{u}}}\left(\frac{3}{2}\right)\right) = \mathcal{E}_{1}\left({\mathit{\mathtt{v}}}\left(\frac{3}{2}\right)\right) = \mathcal{E}_{1}^{*}\left({\mathit{\mathtt{v}}}\left(\frac{3}{2}\right)\right) = \frac{1}{70},$ and ${\mathit{\mathtt{u}}}(\sigma) = {\mathit{\mathtt{v}}}(\sigma) = \sigma$ then with the constant values ${\mathit{\mathtt{u}}}_{1} = {\mathit{\mathtt{v}}}_{1} = \frac{1}{15}$, ${\mathit{\mathtt{u}}}_{2} = {\mathit{\mathtt{v}}}_{2} = 2$, the graph of the solution is shown in Figure 1.

(ii) If we take $\phi_{1}(\sigma, \mathcal{I}^{\alpha}{\mathit{\mathtt{u}}}(\sigma), \mathcal{I}^{\beta}{\mathit{\mathtt{v}}}(\sigma)) = \frac{\sigma+1}{80e^{\sigma+90}},$ $\phi_{2}(\sigma, \mathcal{I}^{\alpha}{\mathit{\mathtt{u}}}(\sigma), \mathcal{I}^{\beta}{\mathit{\mathtt{v}}}(\sigma)) = \frac{\sigma^{2}+1}{95}-\frac{\sigma}{95},$ $\mathcal{E}_{1}\left({\mathit{\mathtt{u}}}\left(\frac{3}{2}\right)\right) = \mathcal{E}_{1}^{*}\left({\mathit{\mathtt{u}}}\left(\frac{3}{2}\right)\right) = \mathcal{E}_{1}\left({\mathit{\mathtt{v}}}\left(\frac{3}{2}\right)\right) = \mathcal{E}_{1}^{*}\left({\mathit{\mathtt{v}}}\left(\frac{3}{2}\right)\right) = \frac{1}{70},$ and ${\mathit{\mathtt{u}}}(\sigma) = {\mathit{\mathtt{v}}}(\sigma) = \sigma$ then with the constant values ${\mathit{\mathtt{u}}}_{1} = {\mathit{\mathtt{v}}}_{1} = -\frac{1}{15}$, ${\mathit{\mathtt{u}}}_{2} = {\mathit{\mathtt{v}}}_{2} = -2$, the graph of the solution is shown in Figure 2.

From Theorem 4.1, we use the inequality and get

thus, the given system (5.1) is $\mathcal{HU}$ stable and also generalized $\mathcal{HU}$ stable. Likewise, we can justify the condition of Theorems 3.2 and 4.2.

6.   Conclusion

In this article, we used the Kransnoselskii's fixed point theorem and acquired the necessary cases for the existence and uniqueness of solution for the given fractional integro-differential Eqs (1.1). Furthermore, under specific assumptions and conditions, we proved different kinds of Ulam's stability of system (1.1). The concept of Ulam's stability is very important because it gives a relationship between approximate and exact solutions, so our results may be very helpful in approximation theory and numerical analysis. The mentioned stability is rarely investigated for impulsive fractional integro-differential equations. Finally, we illustrated the main results by giving a suitable example.

Acknowledgments

This research was supported by the Natural Science Foundation of Jiangxi Province (Grant Nos. 20192BAB201011, 20192BCBL23030 and 20192ACBL21053) and the National Natural Science Foundation of China (Grant No. 11861053).

Conflict of interest

The authors declare that they have no conflict of interest.