On a Riemann–Liouville Type Implicit Coupled System via Generalized Boundary Conditions ^†

Abstract

We study a coupled system of implicit differential equations with fractional-order differential boundary conditions and the Riemann–Liouville derivative. The existence, uniqueness, and at least one solution are established by applying the Banach contraction and Leray–Schauder fixed point theorem. Furthermore, Hyers–Ulam type stabilities are discussed. An example is presented to illustrate our main result. The suggested system is the generalization of fourth-order ordinary differential equations with anti-periodic, classical, and initial boundary conditions.

1. Introduction

The generalization of ordinary derivatives leads us to the theory of fractional derivatives. The concept of fractional derivatives was established in 1695, after the well-known conversation of Leibniz and L’Hospital [1]. Mathematicians like Riemann, Liouville, Caputo, Hadamard, Fourier, and Laplace contributed a lot and made the area more interesting for researchers. A fractional-order derivative is a global operator, which may act as a tool to modify or modernize different physical phenomena like control theory [2], dynamical process [3], electro-chemistry [4], mathematical biology [5], image and signal processing [6], etc. For more applications of the fractional differential equations ($\mathrm{F}\mathrm{D}\mathrm{E}\mathrm{s}$), we refer the reader to the works in [7,8,9,10,11]. Furthermore, the theory of coupled systems of differential equations is referred to as an important theory in the applied sciences envisaging different areas of biochemistry, ecology, biology, and classical fields of physical sciences and engineering. For details see in [12,13,14].

The theory regarding the existence of solutions of $\mathrm{F}\mathrm{D}\mathrm{E}\mathrm{s}$, drew significant attention of the researchers working on different boundary conditions, e.g., classical, integral, multi-point, non-local, periodic, and anti-periodic [15,16,17,18]. Among the qualitative properties of $\mathrm{F}\mathrm{D}\mathrm{E}\mathrm{s}$, the stability property of the solution is the central one, particularly the Hyers–Ulam ($\mathrm{H}\mathrm{U}$) stability [19,20,21,22,23,24,25,26]. Stability theory in the sense of $\mathrm{H}\mathrm{U}$ was first discussed by Ulam [27] in the form of a question in 1940 and the following year, Hyers [28] answered his question in the context of Banach spaces. Recently, generalized $\mathrm{H}\mathrm{U}$ stability was discussed by Alqifiary et al. [29] for linear differential equations. Razaei et al. [30] presented Laplace transform and $\mathrm{H}\mathrm{U}$ stability of linear differential equations. Wang et al. [31] studied $\mathrm{H}\mathrm{U}$ stability for two types of linear $\mathrm{F}\mathrm{D}\mathrm{E}\mathrm{s}$. Shen et al. [32] worked on the $\mathrm{H}\mathrm{U}$ stability of linear $\mathrm{F}\mathrm{D}\mathrm{E}\mathrm{s}$ with constant coefficients using Laplace transform method. Liu et al. [33] proved the $\mathrm{H}\mathrm{U}$ stability of linear Caputo–Fabrizio $\mathrm{F}\mathrm{D}\mathrm{E}\mathrm{s}$. Liu et al. [34] studied the $\mathrm{H}\mathrm{U}$ stability of linear Caputo–Fabrizio $\mathrm{F}\mathrm{D}\mathrm{E}\mathrm{s}$ with the Mittag–Leffler kernel by Laplace transform method.

The above work motivate us to study the coupled implicit $\mathrm{F}\mathrm{D}\mathrm{E}\mathrm{s}$ with fractional-order differential boundary conditions:

$$\left\{\begin{array}{cc}\hfill & {\mathfrak{D}}^{\alpha }\mathsf{v}\left(\mathsf{t}\right)-{\chi }_1(\mathsf{t},\mathsf{u}\left(\mathsf{t}\right),{\mathfrak{D}}^{\alpha }\mathsf{v}\left(\mathsf{t}\right))=0;\mathsf{t}\in \mathfrak{J},\hfill \\ & {\mathfrak{D}}^{\kappa }\mathsf{u}\left(\mathsf{t}\right)-{\chi }_2(\mathsf{t},\mathsf{v}\left(\mathsf{t}\right),{\mathfrak{D}}^{\kappa }\mathsf{u}\left(\mathsf{t}\right))=0;\mathsf{t}\in \mathfrak{J},\hfill \\ & {\mathfrak{D}}^{\alpha -4}\mathsf{v}\left(0\right)={\eta }_1{\mathfrak{D}}^{\alpha -4}\mathsf{v}\left(\sigma \right),{\mathfrak{D}}^{\alpha -3}\mathsf{v}\left(0\right)={\eta }_2{\mathfrak{D}}^{\alpha -3}\mathsf{v}\left(\sigma \right),\hfill \\ & {\mathfrak{D}}^{\alpha -2}\mathsf{v}\left(0\right)={\eta }_3{\mathfrak{D}}^{\alpha -2}\mathsf{v}\left(\sigma \right),{\mathfrak{D}}^{\alpha -1}\mathsf{v}\left(0\right)={\eta }_4{\mathfrak{D}}^{\alpha -1}\mathsf{v}\left(\sigma \right),\hfill \\ & {\mathfrak{D}}^{\kappa -4}\mathsf{u}\left(0\right)={\eta }_5{\mathfrak{D}}^{\kappa -4}\mathsf{u}\left(\sigma \right),{\mathfrak{D}}^{\kappa -3}\mathsf{u}\left(0\right)={\eta }_6{\mathfrak{D}}^{\kappa -3}\mathsf{u}\left(\sigma \right),\hfill \\ & {\mathfrak{D}}^{\kappa -2}\mathsf{u}\left(0\right)={\eta }_7{\mathfrak{D}}^{\kappa -2}\mathsf{u}\left(\sigma \right),{\mathfrak{D}}^{\kappa -1}\mathsf{u}\left(0\right)={\eta }_8{\mathfrak{D}}^{\kappa -1}\mathsf{u}\left(\sigma \right),\hfill \end{array}\right.$$

(1)

where $3<\alpha ,\kappa \le 4,$ $\mathfrak{J}=[0,\sigma ],\sigma >0$ and ${\eta }_{\mathcal{i}}\ne 1$ for $\mathcal{i}=1,2,\dots ,8$. ${\mathfrak{D}}^{\alpha },{\mathfrak{D}}^{\kappa }$ be the $\alpha ,\kappa$ order denotes Riemann–Liouville fractional derivatives and ${\chi }_1,{\chi }_2:\mathfrak{J}\times \mathcal{R}\times \mathcal{R}\to \mathcal{R}$ be continuous functions.

Higher-order ordinary differential equations $\left(\mathrm{O}\mathrm{D}\mathrm{E}\mathrm{s}\right)$ can be used to model problems arising from the field of applied sciences and engineering [35,36]. The generalization of fourth-order $\mathrm{O}\mathrm{D}\mathrm{E}\mathrm{s}$ are $\mathrm{F}\mathrm{D}\mathrm{E}\mathrm{s}$ (1) if $\alpha =\kappa =4$. Fourth-order differential equations have important applications in mechanics, thus have attracted considerable attention over the last three decades. The problem of static deflection of a uniform beam, which can be modeled as a fourth-order initial value problem is a good example of a real problem in engineering [37,38].

This problem has been extensively analyzed, some new techniques were developed and numerous general and impressive results regarding the existence of solutions were established in [39,40,41,42]. Sometimes, mathematical modeling of the various physical phenomena may arise as a coupled system of the forgoing $\mathrm{O}\mathrm{D}\mathrm{E}\mathrm{s}$. Furthermore, for ${\eta }_{\mathcal{i}}=-1(\mathcal{i}=1,2,\dots ,8),$ we can obtain anti-periodic boundary conditions which are applicable in several mathematical models, some are given in [43,44].

The manuscript is categorized as follows. For our main results, we establish some basic notations, definitions, and lemma in Section 2. In Section 3, we present existence, uniqueness, and at least one solution of system (1) by applying the Banach contraction fixed point theorem and Leray–Schauder fixed point theorem. In Section 4, we discuss definitions of $\mathrm{H}\mathrm{U}$ type stabilities, which help us to show that system (1) has $\mathrm{H}\mathrm{U}$ type stabilities by two different approaches. In Section 5, by a particular example of the system (1), we show that our results are applicable.

2. Background Materials

In this fragment, we present basic notations with Banach spaces, definitions of the considered derivative and integral, and lemma, which will be utilized in the next sections.

Suppose $\mathbb{C}\left(\mathfrak{J}\right)$ is a Banach space with a norm defined as $\parallel \mathsf{v}\parallel ={sup}_{\mathsf{t}\in \mathfrak{J}}\left|\mathsf{v}\left(\mathsf{t}\right)\right|.$ For $\mathsf{t}\in \mathfrak{J},$ we define ${\mathsf{v}}_r\left(\mathsf{t}\right)={\mathsf{t}}^r\mathsf{v}\left(\mathsf{t}\right),r\ge 0.$ Suppose that ${\mathsf{S}}_1={\mathbb{C}}_r\left(\mathfrak{J}\right)\subset \mathbb{C}\left(\mathfrak{J}\right)$ be the space of all functions $\mathsf{v}$ such that ${\mathsf{v}}_r\in {\mathsf{S}}_1$ which yields to be a Banach space when endowed with the norm

$${\parallel \mathsf{v}\parallel }_{{\mathsf{S}}_1}=max\left\{\underset{\mathsf{t}\in \mathfrak{J}}{sup}{\mathsf{t}}^r\right|\mathsf{v}\left(\mathsf{t}\right)|,\underset{\mathsf{t}\in \mathfrak{J}}{sup}{\mathsf{t}}^r|{\mathfrak{D}}^{\alpha }\mathsf{v}\left(\mathsf{t}\right)\left|\right\}.$$

Similarly, ${\parallel (\mathsf{v},\mathsf{u})\parallel }_{\mathsf{S}}={\parallel \mathsf{v}\parallel }_{{\mathsf{S}}_1}+{\parallel \mathsf{u}\parallel }_{{\mathsf{S}}_2}$ is the norm defined on the product space, where $\mathsf{S}={\mathsf{S}}_1\times {\mathsf{S}}_2$. Obviously $\left({\mathsf{S},\parallel (\mathsf{v},\mathsf{u})\parallel }_{\mathsf{S}}\right)$ is a Banach space.

Definition 1.

[45] For a continuous function $\mathsf{v}:{\mathcal{R}}^{+}\to \mathcal{R}$, the Riemann–Liouville integral of order $\alpha >0$ is defined as

$${\mathfrak{I}}^{\alpha }\mathsf{v}\left(\mathsf{t}\right)=\frac{1}{\Gamma \left(\alpha \right)}{\int }_0^{\mathsf{t}}\frac{\mathsf{v}\left(\tau \right)}{{(\mathsf{t}-\tau )}^{1-\alpha }}\mathsf{d}\tau ,$$

such that the integral is pointwise defined on ${\mathcal{R}}^{+}$.

Definition 2.

[45] For a continuous function $\mathsf{v}:{\mathcal{R}}^{+}\to \mathcal{R}$, the Riemann–Liouville derivative of order $\alpha >0$ is defined as

$${\mathfrak{D}}^{\alpha }\mathsf{v}\left(\mathsf{t}\right)=\frac{1}{\Gamma (n-\alpha )}{\left(\frac{\mathsf{d}}{\mathsf{d}\mathsf{t}}\right)}^n{\int }_0^{\mathsf{t}}\frac{\mathsf{v}\left(\tau \right)}{{(\mathsf{t}-\tau )}^{\alpha -n+1}}\mathsf{d}\tau ,$$

where $\left[\alpha \right]$ represents the integer part of α and $n=\left[\alpha \right]+1$. We note that for $\varrho >-1,\varrho \ne \alpha -1,\alpha -2,\dots ,\alpha -n,$ we have

$${\mathfrak{D}}^{\alpha }{\mathsf{t}}^{\varrho }=\frac{\Gamma (\varrho +1)}{\Gamma (\varrho -\alpha +1)}{\mathsf{t}}^{\varrho -\alpha }$$

and

$${\mathfrak{D}}^{\alpha }{\mathsf{t}}^{\alpha -i}=0,i=1,2,3,\dots ,n.$$

Lemma 1.

[45] Solution of the following Riemann–Liouville $\mathrm{F}\mathrm{D}\mathrm{E}$ of order $n-1<\alpha \le n$

$${\mathfrak{D}}^{\alpha }\mathsf{v}\left(\mathsf{t}\right)=\vartheta \left(\mathsf{t}\right),$$

is

$${\mathfrak{I}}^{\alpha }{\mathfrak{D}}^{\alpha }\mathsf{v}\left(\mathsf{t}\right)={\mathfrak{I}}^{\alpha }\vartheta \left(\mathsf{t}\right)+k_0{\mathsf{t}}^{\alpha -n}+k_1{\mathsf{t}}^{\alpha -n-1}+\dots +k_{n-2}{\mathsf{t}}^{\alpha -2}+k_{n-1}{\mathsf{t}}^{\alpha -1},$$

where $k_{\mathcal{i}}(i=1,2,3,\dots ,n)$ are unknowns.

3. Existence Theory

This section is devoted to the equivalent integral form of the proposed problem.

Lemma 2.

Let $\vartheta \in \mathbb{C}\left(\mathfrak{J}\right)$, the following $\alpha \in (3,4]$ order $\mathrm{F}\mathrm{D}\mathrm{E}$ with boundary conditions

$$\left\{\begin{array}{cc}& {\mathfrak{D}}^{\alpha }\mathsf{v}\left(\mathsf{t}\right)=\vartheta \left(\mathsf{t}\right);\mathsf{t}\in \mathfrak{J},\hfill \\ & {\mathfrak{D}}^{\alpha -4}\mathsf{v}\left(0\right)={\eta }_1{\mathfrak{D}}^{\alpha -4}\mathsf{v}\left(\sigma \right),{\mathfrak{D}}^{\alpha -3}\mathsf{v}\left(0\right)={\eta }_2{\mathfrak{D}}^{\alpha -3}\mathsf{v}\left(\sigma \right),\hfill \\ & {\mathfrak{D}}^{\alpha -2}\mathsf{v}\left(0\right)={\eta }_3{\mathfrak{D}}^{\alpha -2}\mathsf{v}\left(\sigma \right),{\mathfrak{D}}^{\alpha -1}\mathsf{v}\left(0\right)={\eta }_4{\mathfrak{D}}^{\alpha -1}\mathsf{v}\left(\sigma \right),\hfill \end{array}\right.$$

(2)

have the solution

$$\mathsf{v}\left(\mathsf{t}\right)={\int }_0^{\sigma }{\mathbf{G}}_{\alpha }(\mathsf{t},\tau )\vartheta \left(\tau \right)\mathsf{d}\tau ,$$

where

$${\mathbf{G}}_{\alpha }(\mathsf{t},\tau )=\left\{\begin{array}{c}\frac{{\left(\mathsf{t}-\tau \right)}^{\alpha -1}}{\Gamma \left(\alpha \right)}+\frac{{\eta }_1{\mathsf{t}}^{\alpha -4}{\left(\sigma -\tau \right)}^3}{6(1-{\eta }_1)\Gamma (\alpha -3)}+\frac{\left[(1-{\eta }_1){\eta }_2{\mathsf{t}}^{\alpha -3}+{\eta }_1{\eta }_2\sigma {\mathsf{t}}^{\alpha -4}(\alpha -3)\right]{\left(\sigma -\tau \right)}^2}{2(1-{\eta }_1)(1-{\eta }_2)\Gamma (\alpha -2)}\hfill \\ +\frac{{\eta }_3{\mathsf{t}}^{\alpha -2}\left(\sigma -\tau \right)}{(1-{\eta }_3)\Gamma (\alpha -1)}+\frac{{\eta }_2{\eta }_3\sigma {\mathsf{t}}^{\alpha -3}\left(\sigma -\tau \right)}{(1-{\eta }_2)(1-{\eta }_3)\Gamma (\alpha -2)}+\frac{{\eta }_1(1+{\eta }_2){\eta }_3{\sigma }^2{\mathsf{t}}^{\alpha -4}\left(\sigma -\tau \right)}{2(1-{\eta }_1)(1-{\eta }_2)(1-{\eta }_3)\Gamma (\alpha -3)}\hfill \\ +\frac{{\eta }_4{\mathsf{t}}^{\alpha -1}}{(1-{\eta }_4)\Gamma \left(\alpha \right)}+\frac{{\eta }_3{\eta }_4\sigma {\mathsf{t}}^{\alpha -2}}{(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -1)}+\frac{{\eta }_2(1+{\eta }_3){\eta }_4{\sigma }^2{\mathsf{t}}^{\alpha -3}}{2(1-{\eta }_2)(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -2)}\hfill \\ +\frac{{\eta }_1\left((1+{\eta }_2)(1+{\eta }_3)+{\eta }_2+{\eta }_3\right){\eta }_4{\sigma }^3{\mathsf{t}}^{\alpha -4}}{6(1-{\eta }_1)(1-{\eta }_2)(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -3)},\hfill & & 0\le \tau <\mathsf{t}\le \sigma ,\hfill \\ \\ \frac{{\eta }_1{\mathsf{t}}^{\alpha -4}{\left(\sigma -\tau \right)}^3}{6(1-{\eta }_1)\Gamma (\alpha -3)}+\frac{\left[(1-{\eta }_1){\eta }_2{\mathsf{t}}^{\alpha -3}+{\eta }_1{\eta }_2\sigma {\mathsf{t}}^{\alpha -4}(\alpha -3)\right]{\left(\sigma -\tau \right)}^2}{2(1-{\eta }_1)(1-{\eta }_2)\Gamma (\alpha -2)}\hfill \\ +\frac{{\eta }_3{\mathsf{t}}^{\alpha -2}\left(\sigma -\tau \right)}{(1-{\eta }_3)\Gamma (\alpha -1)}+\frac{{\eta }_2{\eta }_3\sigma {\mathsf{t}}^{\alpha -3}\left(\sigma -\tau \right)}{(1-{\eta }_2)(1-{\eta }_3)\Gamma (\alpha -2)}+\frac{{\eta }_1(1+{\eta }_2){\eta }_3{\sigma }^2{\mathsf{t}}^{\alpha -4}\left(\sigma -\tau \right)}{2(1-{\eta }_1)(1-{\eta }_2)(1-{\eta }_3)\Gamma (\alpha -3)}\hfill \\ +\frac{{\eta }_4{\mathsf{t}}^{\alpha -1}}{(1-{\eta }_4)\Gamma \left(\alpha \right)}+\frac{{\eta }_3{\eta }_4\sigma {\mathsf{t}}^{\alpha -2}}{(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -1)}+\frac{{\eta }_2(1+{\eta }_3){\eta }_4{\sigma }^2{\mathsf{t}}^{\alpha -3}}{2(1-{\eta }_2)(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -2)}\hfill \\ +\frac{{\eta }_1\left((1+{\eta }_2)(1+{\eta }_3)+{\eta }_2+{\eta }_3\right){\eta }_4{\sigma }^3{\mathsf{t}}^{\alpha -4}}{6(1-{\eta }_1)(1-{\eta }_2)(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -3)},\hfill & & 0\le \mathsf{t}<\tau \le \sigma .\hfill \end{array}\right.$$

(3)

Proof. 

Using Lemma 1 on $\mathrm{F}\mathrm{D}\mathrm{E}$ (2), we have

$$\mathsf{v}\left(\mathsf{t}\right)=\frac{1}{\Gamma \left(\alpha \right)}{\int }_0^{\mathsf{t}}{(\mathsf{t}-\tau )}^{\alpha -1}\vartheta \left(\tau \right)\mathsf{d}\tau +k_3{\mathsf{t}}^{\alpha -1}+k_2{\mathsf{t}}^{\alpha -2}+k_1{\mathsf{t}}^{\alpha -3}+k_0{\mathsf{t}}^{\alpha -4}.$$

(4)

Applying boundary conditions of (2) on (4), we get unknowns

$$\begin{array}{cc}\hfill k_0=& \frac{{\eta }_1}{(1-{\eta }_1)\Gamma (\alpha -3)}[\frac{1}{6}{\int }_0^{\sigma }{(\sigma -\tau )}^3\vartheta \left(\tau \right)\mathsf{d}\tau +\frac{{\eta }_2\sigma }{2(1-{\eta }_2)}{\int }_0^{\sigma }{\left(\sigma -\tau \right)}^2\vartheta \left(\tau \right)\mathsf{d}\tau \hfill \\ \hfill & +\frac{(1+{\eta }_2){\eta }_3{\sigma }^2}{2(1-{\eta }_2)(1-{\eta }_3)}{\int }_0^{\sigma }\left(\sigma -\tau \right)\vartheta \left(\tau \right)\mathsf{d}\tau +\frac{\left((1+{\eta }_2)(1+{\eta }_3)+{\eta }_2+{\eta }_3\right){\eta }_4{\sigma }^3}{6(1-{\eta }_2)(1-{\eta }_3)(1-{\eta }_4)}{\int }_0^{\sigma }\vartheta \left(\tau \right)\mathsf{d}\tau ],\hfill \\ \hfill k_1=& \frac{{\eta }_2}{(1-{\eta }_2)\Gamma (\alpha -2)}[\frac{1}{2}{\int }_0^{\sigma }{\left(\sigma -\tau \right)}^2\vartheta \left(\tau \right)\mathsf{d}\tau +\frac{{\eta }_3\sigma }{(1-{\eta }_3)}{\int }_0^{\sigma }(\sigma -\tau )\vartheta \left(\tau \right)\mathsf{d}\tau \hfill \\ \hfill & +\frac{(1+{\eta }_3){\eta }_4{\sigma }^2}{2(1-{\eta }_3)(1-{\eta }_4)}{\int }_0^{\sigma }\vartheta \left(\tau \right)\mathsf{d}\tau ],\hfill \\ \hfill k_2=& \frac{{\eta }_3}{(1-{\eta }_3)\Gamma (\alpha -1)}\left[{\int }_0^{\sigma }(\sigma -\tau )\vartheta \left(\tau \right)\mathsf{d}\tau +\frac{{\eta }_4\sigma }{(1-{\eta }_4)}{\int }_0^{\sigma }\vartheta \left(\tau \right)\mathsf{d}\tau \right],\hfill \\ \hfill k_3=& \frac{{\eta }_4}{(1-{\eta }_4)\Gamma \left(\alpha \right)}{\int }_0^{\sigma }\vartheta \left(\tau \right)\mathsf{d}\tau .\hfill \end{array}$$

Put the values of $k_0,k_1,k_2$ and $k_3$ in Equation (4), we obtain

$$\begin{array}{cc}\hfill \mathsf{v}\left(\mathsf{t}\right)=& \frac{1}{\Gamma \left(\alpha \right)}{\int }_0^{\mathsf{t}}{\left(\mathsf{t}-\tau \right)}^{\alpha -1}\vartheta \left(\tau \right)\mathsf{d}\tau +\frac{{\eta }_1{\mathsf{t}}^{\alpha -4}}{6(1-{\eta }_1)\Gamma (\alpha -3)}{\int }_0^{\sigma }{\left(\sigma -\tau \right)}^3\vartheta \left(\tau \right)\mathsf{d}\tau \hfill \\ \hfill & +\left[\frac{{\eta }_2{\mathsf{t}}^{\alpha -3}}{2(1-{\eta }_2)\Gamma (\alpha -2)}+\frac{{\eta }_1{\eta }_2\sigma {\mathsf{t}}^{\alpha -4}}{2(1-{\eta }_1)(1-{\eta }_2)\Gamma (\alpha -3)}\right]{\int }_0^{\sigma }{\left(\sigma -\tau \right)}^2\vartheta \left(\tau \right)\mathsf{d}\tau \hfill \\ \hfill & +\left[\frac{{\eta }_3{\mathsf{t}}^{\alpha -2}}{(1-{\eta }_3)\Gamma (\alpha -1)}+\frac{{\eta }_2{\eta }_3\sigma {\mathsf{t}}^{\alpha -3}}{(1-{\eta }_2)(1-{\eta }_3)\Gamma (\alpha -2)}+\frac{{\eta }_1(1+{\eta }_2){\eta }_3{\sigma }^2{\mathsf{t}}^{\alpha -4}}{2(1-{\eta }_1)(1-{\eta }_2)(1-{\eta }_3)\Gamma (\alpha -3)}\right]\hfill \\ \hfill & \times {\int }_0^{\sigma }\left(\sigma -\tau \right)\vartheta \left(\tau \right)\mathsf{d}\tau +[\frac{{\eta }_4{\mathsf{t}}^{\alpha -1}}{(1-{\eta }_4)\Gamma \left(\alpha \right)}+\frac{{\eta }_3{\eta }_4\sigma {\mathsf{t}}^{\alpha -2}}{(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -1)}\hfill \\ \hfill & +\frac{{\eta }_2(1+{\eta }_3){\eta }_4{\sigma }^2{\mathsf{t}}^{\alpha -3}}{2(1-{\eta }_2)(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -2)}+\hfill \\ \hfill & +\frac{{\eta }_1\left((1+{\eta }_2)(1+{\eta }_3)+{\eta }_2+{\eta }_3\right){\eta }_4{\sigma }^3{\mathsf{t}}^{\alpha -4}}{6(1-{\eta }_1)(1-{\eta }_2)(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -3)}]{\int }_0^{\sigma }\vartheta \left(\tau \right)\mathsf{d}\tau \hfill \\ \hfill =& {\int }_0^{\sigma }{\mathbf{G}}_{\alpha }(\mathsf{t},\tau )\vartheta \left(\tau \right)\mathsf{d}\tau ,\hfill \end{array}$$

(5)

where ${\mathbf{G}}_{\alpha }(\mathsf{t},\tau )$ is given by (3). □

Remark 1.

Let $\mu \in \mathbb{C}\left(\mathfrak{J}\right)$, the following $\kappa \in (3,4]$ order $\mathrm{F}\mathrm{D}\mathrm{E}$ with boundary conditions

$$\left\{\begin{array}{cc}& {\mathfrak{D}}^{\kappa }\mathsf{u}\left(\mathsf{t}\right)=\mu \left(\mathsf{t}\right);\mathsf{t}\in \mathfrak{J},\hfill \\ & {\mathfrak{D}}^{\kappa -4}\mathsf{u}\left(0\right)={\eta }_5{\mathfrak{D}}^{\kappa -4}\mathsf{u}\left(\sigma \right),{\mathfrak{D}}^{\kappa -3}\mathsf{u}\left(0\right)={\eta }_6{\mathfrak{D}}^{\kappa -3}\mathsf{u}\left(\sigma \right),\hfill \\ & {\mathfrak{D}}^{\kappa -2}\mathsf{u}\left(0\right)={\eta }_7{\mathfrak{D}}^{\kappa -2}\mathsf{u}\left(\sigma \right),{\mathfrak{D}}^{\kappa -1}\mathsf{u}\left(0\right)={\eta }_8{\mathfrak{D}}^{\kappa -1}\mathsf{u}\left(\sigma \right)\hfill \end{array}\right.$$

has the solution

$$\mathsf{u}\left(\mathsf{t}\right)={\int }_0^{\sigma }{\mathbf{G}}_{\kappa }(\mathsf{t},\tau )\mu \left(\tau \right)\mathrm{D}\tau ,$$

where ${\mathbf{G}}_{\kappa }(\mathsf{t},\tau )$ is given by

$${\mathbf{G}}_{\kappa }(\mathsf{t},\tau )=\left\{\begin{array}{c}\frac{{\left(\mathsf{t}-\tau \right)}^{\kappa -1}}{\Gamma \left(\kappa \right)}+\frac{{\eta }_5{\mathsf{t}}^{\kappa -4}{\left(\sigma -\tau \right)}^3}{6(1-{\eta }_5)\Gamma (\kappa -3)}+\frac{\left[(1-{\eta }_5){\eta }_6{\mathsf{t}}^{\kappa -3}+{\eta }_5{\eta }_6\sigma {\mathsf{t}}^{\kappa -4}(\kappa -3)\right]{\left(\sigma -\tau \right)}^2}{2(1-{\eta }_5)(1-{\eta }_6)\Gamma (\kappa -2)}\hfill \\ +\frac{{\eta }_7{\mathsf{t}}^{\kappa -2}\left(\sigma -\tau \right)}{(1-{\eta }_7)\Gamma (\kappa -1)}+\frac{{\eta }_6{\eta }_7\sigma {\mathsf{t}}^{\kappa -3}\left(\sigma -\tau \right)}{(1-{\eta }_6)(1-{\eta }_7)\Gamma (\kappa -2)}+\frac{{\eta }_5(1+{\eta }_6){\eta }_7{\sigma }^2{\mathsf{t}}^{\kappa -4}\left(\sigma -\tau \right)}{2(1-{\eta }_5)(1-{\eta }_6)(1-{\eta }_7)\Gamma (\kappa -3)}\hfill \\ +\frac{{\eta }_8{\mathsf{t}}^{\kappa -1}}{(1-{\eta }_8)\Gamma \left(\kappa \right)}+\frac{{\eta }_7{\eta }_8\sigma {\mathsf{t}}^{\kappa -2}}{(1-{\eta }_7)(1-{\eta }_8)\Gamma (\kappa -1)}+\frac{{\eta }_6(1+{\eta }_7){\eta }_8{\sigma }^2{\mathsf{t}}^{\kappa -3}}{2(1-{\eta }_6)(1-{\eta }_7)(1-{\eta }_8)\Gamma (\kappa -2)}\hfill \\ +\frac{{\eta }_5\left((1+{\eta }_6)(1+{\eta }_7)+{\eta }_6+{\eta }_7\right){\eta }_8{\sigma }^3{\mathsf{t}}^{\kappa -4}}{6(1-{\eta }_5)(1-{\eta }_6)(1-{\eta }_7)(1-{\eta }_8)\Gamma (\kappa -3)},\hfill & & 0\le \tau <\mathsf{t}\le \sigma ,\hfill \\ \\ \frac{{\eta }_5{\mathsf{t}}^{\kappa -4}{\left(\sigma -\tau \right)}^3}{6(1-{\eta }_5)\Gamma (\kappa -3)}+\frac{\left[(1-{\eta }_5){\eta }_6{\mathsf{t}}^{\kappa -3}+{\eta }_5{\eta }_6\sigma {\mathsf{t}}^{\kappa -4}(\kappa -3)\right]{\left(\sigma -\tau \right)}^2}{2(1-{\eta }_5)(1-{\eta }_6)\Gamma (\kappa -2)}\hfill \\ +\frac{{\eta }_7{\mathsf{t}}^{\kappa -2}\left(\sigma -\tau \right)}{(1-{\eta }_7)\Gamma (\kappa -1)}+\frac{{\eta }_6{\eta }_7\sigma {\mathsf{t}}^{\kappa -3}\left(\sigma -\tau \right)}{(1-{\eta }_6)(1-{\eta }_7)\Gamma (\kappa -2)}+\frac{{\eta }_5(1+{\eta }_6){\eta }_7{\sigma }^2{\mathsf{t}}^{\kappa -4}\left(\sigma -\tau \right)}{2(1-{\eta }_5)(1-{\eta }_6)(1-{\eta }_7)\Gamma (\kappa -3)}\hfill \\ +\frac{{\eta }_8{\mathsf{t}}^{\kappa -1}}{(1-{\eta }_8)\Gamma \left(\kappa \right)}+\frac{{\eta }_7{\eta }_8\sigma {\mathsf{t}}^{\kappa -2}}{(1-{\eta }_7)(1-{\eta }_8)\Gamma (\kappa -1)}+\frac{{\eta }_6(1+{\eta }_7){\eta }_8{\sigma }^2{\mathsf{t}}^{\kappa -3}}{2(1-{\eta }_6)(1-{\eta }_7)(1-{\eta }_8)\Gamma (\kappa -2)}\hfill \\ +\frac{{\eta }_5\left((1+{\eta }_6)(1+{\eta }_7)+{\eta }_6+{\eta }_7\right){\eta }_8{\sigma }^3{\mathsf{t}}^{\kappa -4}}{6(1-{\eta }_5)(1-{\eta }_6)(1-{\eta }_7)(1-{\eta }_8)\Gamma (\kappa -3)},\hfill & & 0\le \mathsf{t}<\tau \le \sigma .\hfill \end{array}\right.$$

Remark 2.

Putting $\alpha =4$ and ${\eta }_1={\eta }_2={\eta }_3={\eta }_4=-1$ in (3), gives Green’s function ${\mathbf{G}}_{\alpha }(\mathsf{t},\tau )$ of fourth-order $\mathrm{O}\mathrm{D}\mathrm{E}$ with anti-periodic boundary conditions.

Remark 3.

Putting $\alpha =4$ and ${\eta }_1={\eta }_2={\eta }_3={\eta }_4=0$ in (5), gives the solution of fourth-order $\mathrm{O}\mathrm{D}\mathrm{E}$ having initial conditions.

For the reason of advantage, we set the following notations:

$$\begin{array}{cc}\hfill {\mathcal{Q}}_{\alpha }=& max\{\frac{{\sigma }^4}{\Gamma (\alpha +1)}+|\frac{{\eta }_1{\sigma }^4}{24(1-{\eta }_1)\Gamma (\alpha -3)}|+\frac{{\eta }_2(1-{\eta }_1){\sigma }^4+{\eta }_1{\eta }_2{\sigma }^4(\alpha -3)}{6(1-{\eta }_1)(1-{\eta }_2)\Gamma (\alpha -2)}|\hfill \\ \hfill & +\left|\frac{{\eta }_3(1-{\eta }_2){\sigma }^4+{\eta }_2{\eta }_3{\sigma }^4(\alpha -2)}{2(1-{\eta }_2)(1-{\eta }_3)\Gamma (\alpha -1)}\right|+\left|\frac{{\eta }_1(1+{\eta }_2){\eta }_3{\sigma }^4}{2(1-{\eta }_1)(1-{\eta }_2)(1-{\eta }_3)\Gamma (\alpha -3)}\right|\hfill \\ \hfill & +\left|\frac{(1-{\eta }_3){\eta }_4{\sigma }^4+{\eta }_3{\eta }_4{\sigma }^4(\alpha -1)}{(1-{\eta }_3)(1-{\eta }_4)\Gamma \left(\alpha \right)}\right|+\left|\frac{{\eta }_2(1+{\eta }_3){\eta }_4{\sigma }^4}{2(1-{\eta }_2)(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -2)}\right|\hfill \\ \hfill & +\left|\frac{{\eta }_1\left((1+{\eta }_2)(1+{\eta }_3)+{\eta }_2+{\eta }_3\right){\eta }_4{\sigma }^4}{6(1-{\eta }_1)(1-{\eta }_2)(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -3)}\right|\}\hfill \end{array}$$

(6)

and

$$\begin{array}{cc}\hfill {\mathcal{Q}}_{\kappa }=& max\{\frac{{\sigma }^4}{\Gamma (\kappa +1)}+|\frac{{\eta }_5{\sigma }^4}{24(1-{\eta }_5)\Gamma (\kappa -3)}|+\frac{{\eta }_6(1-{\eta }_5){\sigma }^4+{\eta }_5{\eta }_6{\sigma }^4(\kappa -3)}{6(1-{\eta }_5)(1-{\eta }_6)\Gamma (\kappa -2)}|\hfill \\ \hfill & +\left|\frac{{\eta }_7(1-{\eta }_6){\sigma }^4+{\eta }_6{\eta }_7{\sigma }^4(\kappa -2)}{2(1-{\eta }_6)(1-{\eta }_7)\Gamma (\kappa -1)}\right|+\left|\frac{{\eta }_5(1+{\eta }_6){\eta }_7{\sigma }^4}{2(1-{\eta }_5)(1-{\eta }_6)(1-{\eta }_7)\Gamma (\kappa -3)}\right|\hfill \\ \hfill & +\left|\frac{(1-{\eta }_7){\eta }_8{\sigma }^4+{\eta }_7{\eta }_8{\sigma }^4(\kappa -1)}{(1-{\eta }_7)(1-{\eta }_8)\Gamma \left(\kappa \right)}\right|+\left|\frac{{\eta }_6(1+{\eta }_7){\eta }_8{\sigma }^4}{2(1-{\eta }_6)(1-{\eta }_7)(1-{\eta }_8)\Gamma (\kappa -2)}\right|\hfill \\ \hfill & +\left|\frac{{\eta }_5\left((1+{\eta }_6)(1+{\eta }_7)+{\eta }_6+{\eta }_7\right){\eta }_8{\sigma }^4}{6(1-{\eta }_5)(1-{\eta }_6)(1-{\eta }_7)(1-{\eta }_8)\Gamma (\kappa -3)}\right|\}\hfill \end{array}$$

(7)

If solution of system (1) is $(\mathsf{v},\mathsf{u})$ and $\mathsf{t}\in \mathfrak{J},$ then

$$\begin{array}{cc}& \mathsf{v}\left(\mathsf{t}\right)=\frac{1}{\Gamma \left(\alpha \right)}{\int }_0^{\mathsf{t}}{\left(\mathsf{t}-\tau \right)}^{\alpha -1}{\chi }_1(\tau ,\mathsf{u}\left(\tau \right),{\mathfrak{D}}^{\alpha }\mathsf{v}\left(\tau \right))\mathsf{d}\tau \hfill \\ \hfill & +\frac{{\eta }_1{\mathsf{t}}^{\alpha -4}}{6(1-{\eta }_1)\Gamma (\alpha -3)}{\int }_0^{\sigma }{\left(\sigma -\tau \right)}^3{\chi }_1(\tau ,\mathsf{u}\left(\tau \right),{\mathfrak{D}}^{\alpha }\mathsf{v}\left(\tau \right))\mathsf{d}\tau \hfill \\ \hfill & +\left[\frac{{\eta }_2{\mathsf{t}}^{\alpha -3}}{2(1-{\eta }_2)\Gamma (\alpha -2)}+\frac{{\eta }_1{\eta }_2\sigma {\mathsf{t}}^{\alpha -4}}{2(1-{\eta }_1)(1-{\eta }_2)\Gamma (\alpha -3)}\right]{\int }_0^{\sigma }{\left(\sigma -\tau \right)}^2{\chi }_1(\tau ,\mathsf{u}\left(\tau \right),{\mathfrak{D}}^{\alpha }\mathsf{v}\left(\tau \right))\mathsf{d}\tau \hfill \\ \hfill & +\left[\frac{{\eta }_3{\mathsf{t}}^{\alpha -2}}{(1-{\eta }_3)\Gamma (\alpha -1)}+\frac{{\eta }_2{\eta }_3\sigma {\mathsf{t}}^{\alpha -3}}{(1-{\eta }_2)(1-{\eta }_3)\Gamma (\alpha -2)}+\frac{{\eta }_1(1+{\eta }_2){\eta }_3{\sigma }^2{\mathsf{t}}^{\alpha -4}}{2(1-{\eta }_1)(1-{\eta }_2)(1-{\eta }_3)\Gamma (\alpha -3)}\right]\hfill \\ \hfill & \times {\int }_0^{\sigma }\left(\sigma -\tau \right){\chi }_1(\tau ,\mathsf{u}\left(\tau \right),{\mathfrak{D}}^{\alpha }\mathsf{v}\left(\tau \right))\mathsf{d}\tau +[\frac{{\eta }_4{\mathsf{t}}^{\alpha -1}}{(1-{\eta }_4)\Gamma \left(\alpha \right)}+\frac{{\eta }_3{\eta }_4\sigma {\mathsf{t}}^{\alpha -2}}{(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -1)}\hfill \\ \hfill & +\frac{{\eta }_2(1+{\eta }_3){\eta }_4{\sigma }^2{\mathsf{t}}^{\alpha -3}}{2(1-{\eta }_2)(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -2)}+\frac{{\eta }_1\left((1+{\eta }_2)(1+{\eta }_3)+{\eta }_2+{\eta }_3\right){\eta }_4{\sigma }^3{\mathsf{t}}^{\alpha -4}}{6(1-{\eta }_1)(1-{\eta }_2)(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -3)}]\hfill \\ \hfill & \times {\int }_0^{\sigma }{\chi }_1(\tau ,\mathsf{u}\left(\tau \right),{\mathfrak{D}}^{\alpha }\mathsf{v}\left(\tau \right))\mathsf{d}\tau \hfill \\ \hfill =& {\int }_0^{\sigma }{\mathbf{G}}_{\alpha }(\mathsf{t},\tau ){\chi }_1(\tau ,\mathsf{u}\left(\tau \right),{\mathfrak{D}}^{\alpha }\mathsf{v}\left(\tau \right))\mathsf{d}\tau ,\hfill \end{array}$$

and

$$\begin{array}{cc}& \mathsf{u}\left(\mathsf{t}\right)=\frac{1}{\Gamma \left(\kappa \right)}{\int }_0^{\mathsf{t}}{\left(\mathsf{t}-\tau \right)}^{\kappa -1}{\chi }_2(\tau ,\mathsf{v}\left(\tau \right),{\mathfrak{D}}^{\kappa }\mathsf{u}\left(\tau \right))\mathsf{d}\tau \hfill \\ \hfill & +\frac{{\eta }_5{\mathsf{t}}^{\kappa -4}}{6(1-{\eta }_5)\Gamma (\kappa -3)}{\int }_0^{\sigma }{\left(\sigma -\tau \right)}^3{\chi }_2(\tau ,\mathsf{v}\left(\tau \right),{\mathfrak{D}}^{\kappa }\mathsf{u}\left(\tau \right))\mathsf{d}\tau \hfill \\ \hfill & +\left[\frac{{\eta }_6{\mathsf{t}}^{\kappa -3}}{2(1-{\eta }_6)\Gamma (\kappa -2)}+\frac{{\eta }_5{\eta }_6\sigma {\mathsf{t}}^{\kappa -4}}{2(1-{\eta }_5)(1-{\eta }_6)\Gamma (\kappa -3)}\right]{\int }_0^{\sigma }{\left(\sigma -\tau \right)}^2{\chi }_2(\tau ,\mathsf{v}\left(\tau \right),{\mathfrak{D}}^{\kappa }\mathsf{u}\left(\tau \right))\mathsf{d}\tau \hfill \\ \hfill & +\left[\frac{{\eta }_7{\mathsf{t}}^{\kappa -2}}{(1-{\eta }_7)\Gamma (\kappa -1)}+\frac{{\eta }_6{\eta }_7\sigma {\mathsf{t}}^{\kappa -3}}{(1-{\eta }_6)(1-{\eta }_7)\Gamma (\kappa -2)}+\frac{{\eta }_5(1+{\eta }_6){\eta }_7{\sigma }^2{\mathsf{t}}^{\kappa -4}}{2(1-{\eta }_5)(1-{\eta }_6)(1-{\eta }_7)\Gamma (\kappa -3)}\right]\hfill \\ \hfill & \times {\int }_0^{\sigma }\left(\sigma -\tau \right){\chi }_2(\tau ,\mathsf{v}\left(\tau \right),{\mathfrak{D}}^{\kappa }\mathsf{u}\left(\tau \right))\mathsf{d}\tau +[\frac{{\eta }_8{\mathsf{t}}^{\kappa -1}}{(1-{\eta }_8)\Gamma \left(\kappa \right)}+\frac{{\eta }_7{\eta }_8\sigma {\mathsf{t}}^{\kappa -2}}{(1-{\eta }_7)(1-{\eta }_8)\Gamma (\kappa -1)}\hfill \\ \hfill & +\frac{{\eta }_6(1+{\eta }_7){\eta }_8{\sigma }^2{\mathsf{t}}^{\kappa -3}}{2(1-{\eta }_6)(1-{\eta }_7)(1-{\eta }_8)\Gamma (\kappa -2)}+\frac{{\eta }_5\left((1+{\eta }_6)(1+{\eta }_7)+{\eta }_6+{\eta }_7\right){\eta }_8{\sigma }^3{\mathsf{t}}^{\kappa -4}}{6(1-{\eta }_5)(1-{\eta }_6)(1-{\eta }_7)(1-{\eta }_8)\Gamma (\kappa -3)}]\hfill \\ \hfill & \times {\int }_0^{\sigma }{\chi }_2(\tau ,\mathsf{v}\left(\tau \right),{\mathfrak{D}}^{\kappa }\mathsf{u}\left(\tau \right))\mathsf{d}\tau \hfill \\ \hfill =& {\int }_0^{\sigma }{\mathbf{G}}_{\kappa }(\mathsf{t},\tau ){\chi }_2(\tau ,\mathsf{v}\left(\tau \right),{\mathfrak{D}}^{\kappa }\mathsf{u}\left(\tau \right))\mathsf{d}\tau .\hfill \end{array}$$

We use the following notations for convenience:

$$\begin{array}{cc}\hfill & y\left(\mathsf{t}\right)={\chi }_1(\mathsf{t},\mathsf{u}\left(\mathsf{t}\right),{\mathfrak{D}}^{\alpha }\mathsf{v}\left(\mathsf{t}\right))={\chi }_1(\mathsf{t},\mathsf{u}\left(\mathsf{t}\right),y\left(\mathsf{t}\right))\hfill \\ \hfill & x\left(\mathsf{t}\right)={\chi }_2(\mathsf{t},\mathsf{v}\left(\mathsf{t}\right),{\mathfrak{D}}^{\kappa }\mathsf{u}\left(\mathsf{t}\right))={\chi }_2(\mathsf{t},\mathsf{v}\left(\mathsf{t}\right),x\left(\mathsf{t}\right)).\hfill \end{array}$$

Now, transform system (1) to the fixed point problem, let $\mathsf{F}:\mathsf{S}\to \mathsf{S}$ is an operator defined by

$$\mathsf{F}(\mathsf{v},\mathsf{u})\left(\mathsf{t}\right)=\left(\begin{array}{c}\underset{0}{\overset{\mathsf{t}}{\int }}{\mathbf{G}}_{\alpha }(\mathsf{t},\tau ){\chi }_1(\tau ,\mathsf{u}\left(\tau \right),y\left(\tau \right))\mathsf{d}\tau \\ \underset{0}{\overset{\mathsf{t}}{\int }}{\mathbf{G}}_{\kappa }(\mathsf{t},\tau ){\chi }_2(\tau ,\mathsf{v}\left(\tau \right),x\left(\tau \right))\mathsf{d}\tau \end{array}\right)=\left(\begin{array}{c}{\mathsf{F}}_{\alpha }(\mathsf{u},y)\left(\mathsf{t}\right)\\ {\mathsf{F}}_{\kappa }(\mathsf{v},x)\left(\mathsf{t}\right)\end{array}\right)=\left(\begin{array}{c}{\mathsf{F}}_{\alpha }\left(\mathsf{v}\right)\left(\mathsf{t}\right)\\ {\mathsf{F}}_{\kappa }\left(\mathsf{u}\right)\left(\mathsf{t}\right)\end{array}\right).$$

(8)

Then, the fixed point of $\mathsf{F}$ and the solution of system (1) coincided, i.e.,

$$\begin{array}{cc}& {\mathsf{F}}_{\alpha }\left(\mathsf{v}\right)\left(\mathsf{t}\right)=\hfill \\ \hfill & \frac{1}{\Gamma \left(\alpha \right)}{\int }_0^{\mathsf{t}}{\left(\mathsf{t}-\tau \right)}^{\alpha -1}y\left(\tau \right)\mathsf{d}\tau +\frac{{\eta }_1{\mathsf{t}}^{\alpha -4}}{6(1-{\eta }_1)\Gamma (\alpha -3)}{\int }_0^{\sigma }{\left(\sigma -\tau \right)}^{3}v\left(\tau \right)\mathsf{d}\tau \hfill \\ \hfill & +\left[\frac{{\eta }_2{\mathsf{t}}^{\alpha -3}}{2(1-{\eta }_2)\Gamma (\alpha -2)}+\frac{{\eta }_1{\eta }_2\sigma {\mathsf{t}}^{\alpha -4}}{2(1-{\eta }_1)(1-{\eta }_2)\Gamma (\alpha -3)}\right]{\int }_0^{\sigma }{\left(\sigma -\tau \right)}^{2}y\left(\tau \right)\mathsf{d}\tau \hfill \\ \hfill & +\left[\frac{{\eta }_3{\mathsf{t}}^{\alpha -2}}{(1-{\eta }_3)\Gamma (\alpha -1)}+\frac{{\eta }_2{\eta }_3\sigma {\mathsf{t}}^{\alpha -3}}{(1-{\eta }_2)(1-{\eta }_3)\Gamma (\alpha -2)}+\frac{{\eta }_1(1+{\eta }_2){\eta }_3{\sigma }^2{\mathsf{t}}^{\alpha -4}}{2(1-{\eta }_1)(1-{\eta }_2)(1-{\eta }_3)\Gamma (\alpha -3)}\right]\hfill \\ \hfill & \times {\int }_0^{\sigma }\left(\sigma -\tau \right)y\left(\tau \right)\mathsf{d}\tau +[\frac{{\eta }_4{\mathsf{t}}^{\alpha -1}}{(1-{\eta }_4)\Gamma \left(\alpha \right)}+\frac{{\eta }_3{\eta }_4\sigma {\mathsf{t}}^{\alpha -2}}{(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -1)}\hfill \\ \hfill & +\frac{{\eta }_2(1+{\eta }_3){\eta }_4{\sigma }^2{\mathsf{t}}^{\alpha -3}}{2(1-{\eta }_2)(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -2)}+\frac{{\eta }_1\left((1+{\eta }_2)(1+{\eta }_3)+{\eta }_2+{\eta }_3\right){\eta }_4{\sigma }^3{\mathsf{t}}^{\alpha -4}}{6(1-{\eta }_1)(1-{\eta }_2)(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -3)}]{\int }_0^{\sigma }y\left(\tau \right)\mathsf{d}\tau \hfill \end{array}$$

and

$$\begin{array}{cc}& {\mathsf{F}}_{\kappa }\left(\mathsf{u}\right)\left(\mathsf{t}\right)=\hfill \\ \hfill & \frac{1}{\Gamma \left(\kappa \right)}{\int }_0^{\mathsf{t}}{\left(\mathsf{t}-\tau \right)}^{\kappa -1}x\left(\tau \right)\mathsf{d}\tau +\frac{{\eta }_5{\mathsf{t}}^{\kappa -4}}{6(1-{\eta }_5)\Gamma (\kappa -3)}{\int }_0^{\sigma }{\left(\sigma -\tau \right)}^{3}x\left(\tau \right)\mathsf{d}\tau \hfill \\ \hfill & +\left[\frac{{\eta }_6{\mathsf{t}}^{\kappa -3}}{2(1-{\eta }_6)\Gamma (\kappa -2)}+\frac{{\eta }_5{\eta }_6\sigma {\mathsf{t}}^{\kappa -4}}{2(1-{\eta }_5)(1-{\eta }_6)\Gamma (\kappa -3)}\right]{\int }_0^{\sigma }{\left(\sigma -\tau \right)}^{2}x\left(\tau \right)\mathsf{d}\tau \hfill \\ \hfill & +\left[\frac{{\eta }_7{\mathsf{t}}^{\kappa -2}}{(1-{\eta }_7)\Gamma (\kappa -1)}+\frac{{\eta }_6{\eta }_7\sigma {\mathsf{t}}^{\kappa -3}}{(1-{\eta }_6)(1-{\eta }_7)\Gamma (\kappa -2)}+\frac{{\eta }_5(1+{\eta }_6){\eta }_7{\sigma }^2{\mathsf{t}}^{\kappa -4}}{2(1-{\eta }_5)(1-{\eta }_6)(1-{\eta }_7)\Gamma (\kappa -3)}\right]\hfill \\ \hfill & \times {\int }_0^{\sigma }\left(\sigma -\tau \right)x\left(\tau \right)\mathsf{d}\tau +[\frac{{\eta }_8{\mathsf{t}}^{\kappa -1}}{(1-{\eta }_8)\Gamma \left(\kappa \right)}+\frac{{\eta }_7{\eta }_8\sigma {\mathsf{t}}^{\kappa -2}}{(1-{\eta }_7)(1-{\eta }_8)\Gamma (\kappa -1)}\hfill \\ \hfill & +\frac{{\eta }_6(1+{\eta }_7){\eta }_8{\sigma }^2{\mathsf{t}}^{\kappa -3}}{2(1-{\eta }_6)(1-{\eta }_7)(1-{\eta }_8)\Gamma (\kappa -2)}+\frac{{\eta }_5\left((1+{\eta }_6)(1+{\eta }_7)+{\eta }_6+{\eta }_7\right){\eta }_8{\sigma }^3{\mathsf{t}}^{\kappa -4}}{6(1-{\eta }_5)(1-{\eta }_6)(1-{\eta }_7)(1-{\eta }_8)\Gamma (\kappa -3)}]{\int }_0^{\sigma }x\left(\tau \right)\mathsf{d}\tau .\hfill \end{array}$$

Using Banach contraction theorem in the following, we prove the uniqueness of solution of system (1).

Theorem 1.

Let the functions ${\chi }_1,{\chi }_2:\mathfrak{J}\times \mathcal{R}\times \mathcal{R}\to \mathcal{R}$ are continuous and satisfy the hypothesis:

${\mathbf{H}}_1$: For every $\mathsf{t}\in \mathfrak{J}$ and $\mathsf{u},\mathsf{v},y,x,\overline{\mathsf{u}},\overline{\mathsf{v}},\overline{y},\overline{x}:\mathfrak{J}\to \mathcal{R}$, there are ${\mathcal{L}}_{{\chi }_1},{\mathcal{L}}_{{\chi }_2},{\overline{\mathcal{L}}}_{{\chi }_1},{\overline{\mathcal{L}}}_{{\chi }_2}$, such that

$$|{\chi }_1(\mathsf{t},\mathsf{u}\left(\mathsf{t}\right),y\left(\mathsf{t}\right))-{\chi }_1(\mathsf{t},\overline{\mathsf{u}}\left(\mathsf{t}\right),\overline{y}\left(\mathsf{t}\right))|\le {\mathcal{L}}_{{\chi }_1}|\mathsf{u}\left(\mathsf{t}\right)-\overline{\mathsf{u}}\left(\mathsf{t}\right)|+{\overline{\mathcal{L}}}_{{\chi }_1}|y\left(\mathsf{t}\right)-\overline{y}\left(\mathsf{t}\right)|,$$

$$|{\chi }_2(\mathsf{t},\mathsf{v}\left(\mathsf{t}\right),x\left(\mathsf{t}\right))-{\chi }_2(\mathsf{t},\overline{v}\left(\mathsf{t}\right),\overline{x}\left(\mathsf{t}\right))|\le {\mathcal{L}}_{{\chi }_2}|\mathsf{v}\left(\mathsf{t}\right)-\overline{v}\left(\mathsf{t}\right)|+{\overline{\mathcal{L}}}_{{\chi }_2}|x\left(\mathsf{t}\right)-\overline{x}\left(\mathsf{t}\right)|.$$

In addition, suppose that

$$\begin{array}{c}\hfill \frac{{\mathcal{Q}}_{\alpha }{\mathcal{L}}_{{\chi }_1}(1-{\overline{\mathcal{L}}}_{{\chi }_2})+{\mathcal{Q}}_{\kappa }{\mathcal{L}}_{{\chi }_2}(1-{\overline{\mathcal{L}}}_{{\chi }_1})}{(1-{\overline{\mathcal{L}}}_{{\chi }_2})(1-{\overline{\mathcal{L}}}_{{\chi }_1})}<1,\end{array}$$

where ${\mathcal{Q}}_{\alpha }$ and ${\mathcal{Q}}_{\kappa }$ are defined by Equations (6) and (7), respectively. Furthermore, $0\le {\overline{\mathcal{L}}}_{{\chi }_1},{\overline{\mathcal{L}}}_{{\chi }_2}<1$ (through out the paper). Then, the solution of system (1) is unique.

Proof. 

Consider ${sup}_{\mathsf{t}\in \mathfrak{J}}{\chi }_1(\mathsf{t},0,0)=\Phi *<\infty$ and ${sup}_{\mathsf{t}\in \mathfrak{J}}{\chi }_2(\mathsf{t},0,0)={\Psi }^{*}<\infty ,$ such that

$$\begin{array}{c}\hfill r\ge \frac{2{\mathcal{Q}}_{\alpha }{\Phi }^{*}(1-{\overline{\mathcal{L}}}_{{\chi }_2})+2{\mathcal{Q}}_{\kappa }{\Psi }^{*}(1-{\overline{\mathcal{L}}}_{{\chi }_1})}{2(1-{\overline{\mathcal{L}}}_{{\chi }_1})(1-{\overline{\mathcal{L}}}_{{\chi }_2})-{\mathcal{Q}}_{\alpha }{\mathcal{L}}_{{\chi }_1}-{\mathcal{Q}}_{\kappa }{\mathcal{L}}_{{\chi }_2}}.\end{array}$$

We show that $\mathsf{F}\left({\mathcal{B}}_r\right)\subset {\mathcal{B}}_r,$ where

$$\begin{array}{cc}\hfill {\mathcal{B}}_r& =\left\{(\mathsf{v},\mathsf{u})\in {\mathsf{S}:\parallel (\mathsf{v},\mathsf{u})\parallel }_{\mathsf{S}}\le r,\parallel \mathsf{v}\parallel \le \frac{r}{2},\parallel \mathsf{u}\parallel \le \frac{r}{2}\right\}.\hfill \end{array}$$

For $(\mathsf{v},\mathsf{u})\in {\mathcal{B}}_r,$ we have

$$\begin{array}{cc}\hfill & {\mathsf{t}}^{4-\alpha }\left|{\mathsf{F}}_{\alpha }\left(\mathsf{v}\right)\left(\mathsf{t}\right)\right|\hfill \\ \hfill \le & \frac{{\mathsf{t}}^{4-\alpha }}{\Gamma \left(\alpha \right)}{\int }_0^{\mathsf{t}}{\left(\mathsf{t}-\tau \right)}^{\alpha -1}\left(\right|{\chi }_1(\tau ,\mathsf{u}\left(\tau \right),y\left(\tau \right))-{\chi }_1(\tau ,0,0)|+|{\chi }_1(\tau ,0,0)\left|\right)\mathsf{d}\tau +\left|\frac{{\eta }_1}{6(1-{\eta }_1)\Gamma (\alpha -3)}\right|\hfill \\ \hfill & \times {\int }_0^{\sigma }{\left(\sigma -\tau \right)}^3\left(\right|{\chi }_1(\tau ,\mathsf{u}\left(\tau \right),y\left(\tau \right))-{\chi }_1(\tau ,0,0)|+|{\chi }_1(\tau ,0,0)\left|\right)\mathsf{d}\tau +|\frac{{\eta }_2\mathsf{t}}{2(1-{\eta }_2)\Gamma (\alpha -2)}\hfill \\ \hfill & +\frac{{\eta }_1{\eta }_2\sigma }{2(1-{\eta }_1)(1-{\eta }_2)\Gamma (\alpha -3)}|{\int }_0^T{\left(\sigma -\tau \right)}^2\left(\right|{\chi }_1(\tau ,\mathsf{u}\left(\tau \right),v\left(\tau \right))-{\chi }_1(\tau ,0,0)|+\left|{\chi }_1(\tau ,0,0)\right|)\mathsf{d}\tau \hfill \\ \hfill & +|\frac{{\eta }_3{\mathsf{t}}^2}{(1-{\eta }_3)\Gamma (\alpha -1)}+\frac{{\eta }_2{\eta }_3\mathsf{t}\sigma }{(1-{\eta }_2)(1-{\eta }_3)\Gamma (\alpha -2)}+\frac{{\eta }_1(1+{\eta }_2){\eta }_3{\sigma }^2}{2(1-{\eta }_1)(1-{\eta }_2)(1-{\eta }_3)\Gamma (\alpha -3)}|\hfill \\ \hfill & \times {\int }_0^{\sigma }\left(\sigma -\tau \right)\left(|{\chi }_1(\tau ,\mathsf{u}\left(\tau \right),v\left(\tau \right))-{\chi }_1(\tau ,0,0)|+\left|{\chi }_1(\tau ,0,0)\right|\right)\mathsf{d}\tau +|\frac{{\eta }_3{\eta }_4\sigma {\mathsf{t}}^2}{(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -1)}\hfill \\ \hfill & +\frac{{\eta }_4{\mathsf{t}}^3}{(1-{\eta }_4)\Gamma \left(\alpha \right)}+\frac{{\eta }_1\left((1+{\eta }_2)(1+{\eta }_3)+{\eta }_2+{\eta }_3\right){\eta }_4{\sigma }^3}{6(1-{\eta }_1)(1-{\eta }_2)(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -3)}\hfill \\ \hfill & +\frac{{\eta }_2(1+{\eta }_3){\eta }_4{\sigma }^2\mathsf{t}}{2(1-{\eta }_2)(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -2)}|{\int }_0^{\sigma }\left(|{\chi }_1(\tau ,\mathsf{u}\left(\tau \right),v\left(\tau \right))-{\chi }_1(\tau ,0,0)|+\left|{\chi }_1(\tau ,0,0)\right|\right)\mathsf{d}\tau .\hfill \end{array}$$

(9)

Consider

$$\begin{array}{ccc}\hfill \left|v\right(\mathsf{t}\left)\right|& \le & |{\chi }_1(\mathsf{t},\mathsf{u}\left(\mathsf{t}\right),v\left(\mathsf{t}\right))-{\chi }_1(\mathsf{t},0,0)|+|{\chi }_1(\mathsf{t},0,0)|\hfill \\ \hfill & \le & |{\chi }_1(\mathsf{t},0,0)|+{\mathcal{L}}_{{\chi }_1}\left|\mathsf{u}\left(\mathsf{t}\right)\right|+{\overline{\mathcal{L}}}_{{\chi }_1}\left|v\left(\mathsf{t}\right)\right|\hfill \\ & \le & \frac{|{\chi }_1(\mathsf{t},0,0)|+{\mathcal{L}}_{{\chi }_1}\left|\mathsf{u}\left(\mathsf{t}\right)\right|}{1-{\overline{\mathcal{L}}}_{{\chi }_1}}.\hfill \end{array}$$

(10)

Substituting (10) in (9), we get

$$\begin{array}{cc}\hfill & \parallel {\mathsf{F}}_{\alpha }\left(\mathsf{v}\right)\parallel \hfill \\ \hfill \le & [\frac{{\sigma }^4}{\Gamma (\alpha +1)}+|\frac{{\eta }_1{\sigma }^4}{24(1-{\eta }_1)\Gamma (\alpha -3)}|+\frac{{\eta }_2(1-{\eta }_1){\sigma }^4+{\eta }_1{\eta }_2{\sigma }^4(\alpha -3)}{6(1-{\eta }_1)(1-{\eta }_2)\Gamma (\alpha -2)}|\hfill \\ \hfill & +\left|\frac{{\eta }_3(1-{\eta }_2){\sigma }^4+{\eta }_2{\eta }_3{\sigma }^4(\alpha -2)}{2(1-{\eta }_2)(1-{\eta }_3)\Gamma (\alpha -1)}\right|+\left|\frac{{\eta }_1(1+{\eta }_2){\eta }_3{\sigma }^4}{2(1-{\eta }_1)(1-{\eta }_2)(1-{\eta }_3)\Gamma (\alpha -3)}\right|\hfill \\ \hfill & +\left|\frac{(1-{\eta }_3){\eta }_4{\sigma }^4+{\eta }_3{\eta }_4{\sigma }^4(\alpha -1)}{(1-{\eta }_3)(1-{\eta }_4)\Gamma \left(\alpha \right)}\right|+\left|\frac{{\eta }_2(1+{\eta }_3){\eta }_4{\sigma }^4}{2(1-{\eta }_2)(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -2)}\right|\hfill \\ \hfill & +\left|\frac{{\eta }_1\left((1+{\eta }_2)(1+{\eta }_3)+{\eta }_2+{\eta }_3\right){\eta }_4{\sigma }^4}{6(1-{\eta }_1)(1-{\eta }_2)(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -3)}\right|]\frac{2{\Phi }^{*}+{\mathcal{L}}_{{\chi }_1}r}{2(1-{\overline{\mathcal{L}}}_{{\chi }_1})}.\hfill \end{array}$$

Therefore,

$$\begin{array}{cc}\hfill \parallel {\mathsf{F}}_{\alpha }\left(\mathsf{v}\right)\parallel & \le {\mathcal{Q}}_{\alpha }\frac{2{\Phi }^{*}+{\mathcal{L}}_{{\chi }_1}r}{2(1-{\overline{\mathcal{L}}}_{{\chi }_1})}.\hfill \end{array}$$

(11)

On the same way, we can write

$$\begin{array}{cc}\hfill \parallel {\mathsf{F}}_{\kappa }\left(\mathsf{u}\right)\parallel & \le {\mathcal{Q}}_{\kappa }\frac{2{\Psi }^{*}+{\mathcal{L}}_{{\chi }_2}r}{2(1-{\overline{\mathcal{L}}}_{{\chi }_2})}.\hfill \end{array}$$

(12)

Inequalities (11) and (12) combined give

$$\begin{array}{c}\hfill {\parallel \mathsf{F}(\mathsf{v},\mathsf{u})\parallel }_{\mathsf{S}}\le r.\end{array}$$

For any $\mathsf{t}\in \mathfrak{J},$ and $({\mathsf{v}}_1,{\mathsf{u}}_1),({\mathsf{v}}_2,{\mathsf{u}}_2)\in \mathsf{S}$, we get

$$\begin{array}{cc}\hfill & {\mathsf{t}}^{4-\alpha }|{\mathsf{F}}_{\alpha }\left({\mathsf{v}}_1\right)\left(\mathsf{t}\right)-{\mathsf{F}}_{\alpha }\left({\mathsf{v}}_2\right)\left(\mathsf{t}\right)|\hfill \\ \hfill \le & \frac{{\mathsf{t}}^{4-\alpha }}{\Gamma \left(\alpha \right)}{\int }_0^{\mathsf{t}}{\left(\mathsf{t}-\tau \right)}^{\alpha -1}|{\chi }_1(\tau ,{\mathsf{u}}_1\left(\tau \right),y_1\left(\tau \right))-{\chi }_1(\tau ,{\mathsf{u}}_2\left(\tau \right),y_2\left(\tau \right))|\mathsf{d}\tau +\left|\frac{{\eta }_1}{6(1-{\eta }_1)\Gamma (\alpha -3)}\right|\hfill \\ \hfill & \times {\int }_0^{\sigma }{\left(\sigma -\tau \right)}^3|{\chi }_1(\tau ,{\mathsf{u}}_1\left(\tau \right),y_1\left(\tau \right))-{\chi }_1(\tau ,{\mathsf{u}}_2\left(\tau \right),y_2\left(\tau \right))|\mathsf{d}\tau +|\frac{{\eta }_2\mathsf{t}}{2(1-{\eta }_2)\Gamma (\alpha -2)}\hfill \\ \hfill & +\frac{{\eta }_1{\eta }_2\sigma }{2(1-{\eta }_1)(1-{\eta }_2)\Gamma (\alpha -3)}\left|{\int }_0^{\sigma }{\left(\sigma -\tau \right)}^2\right|{\chi }_1(\tau ,{\mathsf{u}}_1\left(\tau \right),y_1\left(\tau \right))-{\chi }_1(\tau ,{\mathsf{u}}_2\left(\tau \right),y_2\left(\tau \right))|\mathsf{d}\tau \hfill \\ \hfill & +|\frac{{\eta }_3{\mathsf{t}}^2}{(1-{\eta }_3)\Gamma (\alpha -1)}+\frac{{\eta }_2{\eta }_3\mathsf{t}\sigma }{(1-{\eta }_2)(1-{\eta }_3)\Gamma (\alpha -2)}+\frac{{\eta }_1(1+{\eta }_2){\eta }_3{\sigma }^2}{2(1-{\eta }_1)(1-{\eta }_2)(1-{\mathfrak{c}}_0)\Gamma (\alpha -3)}|\hfill \\ \hfill & \times {\int }_0^{\sigma }\left(\sigma -\tau \right)|{\chi }_1(\tau ,{\mathsf{u}}_1\left(\tau \right),y_1\left(\tau \right))-{\chi }_1(\tau ,{\mathsf{u}}_2\left(\tau \right),y_2\left(\tau \right))|\mathsf{d}\tau +|\frac{{\eta }_3{\eta }_4\sigma {\mathsf{t}}^2}{(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -1)}\hfill \\ \hfill & +\frac{{\eta }_4{\mathsf{t}}^3}{(1-{\eta }_4)\Gamma \left(\alpha \right)}+\frac{{\eta }_1\left((1+{\eta }_2)(1+{\eta }_3)+{\eta }_2+{\eta }_3\right){\eta }_4{\sigma }^3}{6(1-{\eta }_1)(1-{\eta }_2)(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -3)}\hfill \\ \hfill & +\frac{{\eta }_2(1+{\eta }_3){\eta }_4{\sigma }^2\mathsf{t}}{2(1-{\eta }_2)(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -2)}\left|{\int }_0^{\sigma }\right|{\chi }_1(\tau ,{\mathsf{u}}_1\left(\tau \right),y_1\left(\tau \right))-{\chi }_1(\tau ,{\mathsf{u}}_2\left(\tau \right),y_2\left(\tau \right))|\mathsf{d}\tau \hfill \\ \hfill \le & [\frac{{\sigma }^4}{\Gamma (\alpha +1)}+|\frac{{\eta }_1{\sigma }^4}{24(1-{\eta }_1)\Gamma (\alpha -3)}|+\frac{{\eta }_2(1-{\eta }_1){\sigma }^4+{\eta }_1{\eta }_2{\sigma }^4(\alpha -3)}{6(1-{\eta }_1)(1-{\eta }_2)\Gamma (\alpha -2)}|\hfill \\ \hfill & +\left|\frac{{\eta }_3(1-{\eta }_2){\sigma }^4+{\eta }_2{\eta }_3{\sigma }^4(\alpha -2)}{2(1-{\eta }_2)(1-{\eta }_3)\Gamma (\alpha -1)}\right|+\left|\frac{{\eta }_1(1+{\eta }_2){\eta }_3{\sigma }^4}{2(1-{\eta }_1)(1-{\eta }_2)(1-{\eta }_3)\Gamma (\alpha -3)}\right|\hfill \\ \hfill & +\left|\frac{(1-{\eta }_3){\eta }_4{\sigma }^4+{\eta }_3{\eta }_4{\sigma }^4(\alpha -1)}{(1-{\eta }_3)(1-{\eta }_4)\Gamma \left(\alpha \right)}\right|+\left|\frac{{\eta }_2(1+{\eta }_3){\eta }_4{\sigma }^4}{2(1-{\eta }_2)(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -2)}\right|\hfill \\ \hfill & +\left|\frac{{\eta }_1\left((1+{\eta }_2)(1+{\eta }_3)+{\eta }_2+{\eta }_3\right){\eta }_4{\sigma }^4}{6(1-{\eta }_1)(1-{\eta }_2)(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -3)}\right|]\frac{{\mathcal{L}}_{{\chi }_1}}{1-{\overline{\mathcal{L}}}_{{\chi }_1}}\parallel {\mathsf{u}}_1-{\mathsf{u}}_2\parallel \hfill \end{array}$$

and thus we get

$$\begin{array}{cc}\hfill \parallel {\mathsf{F}}_{\alpha }\left({\mathsf{v}}_1\right)-{\mathsf{F}}_{\alpha }\left({\mathsf{v}}_2\right)\parallel & \le \frac{{\mathcal{Q}}_{\alpha }{\mathcal{L}}_{{\chi }_1}}{1-{\overline{\mathcal{L}}}_{{\chi }_1}}\parallel {\mathsf{u}}_1-{\mathsf{u}}_2\parallel .\hfill \end{array}$$

(13)

Similarly,

$$\begin{array}{cc}\hfill \parallel {\mathsf{F}}_{\kappa }\left({\mathsf{u}}_1\right)-{\mathsf{F}}_{\kappa }\left({\mathsf{u}}_2\right)\parallel & \le \frac{{\mathcal{Q}}_{\kappa }{\mathcal{L}}_{{\chi }_2}}{1-{\overline{\mathcal{L}}}_{{\chi }_2}}\parallel {\mathsf{v}}_1-{\mathsf{v}}_2\parallel .\hfill \end{array}$$

(14)

From the inequalities (13) and (14), we get that

$$\begin{array}{cc}\hfill \parallel \mathsf{F}({\mathsf{v}}_1,{\mathsf{u}}_1)-\mathsf{F}({\mathsf{v}}_2,{\mathsf{u}}_2){\parallel }_{\mathsf{S}}& \le \frac{{\mathcal{Q}}_{\alpha }{\mathcal{L}}_{{\chi }_1}(1-{\overline{\mathcal{L}}}_{{\chi }_2})+{\mathcal{Q}}_{\kappa }{\mathcal{L}}_{{\chi }_2}(1-{\overline{\mathcal{L}}}_{{\chi }_1})}{(1-{\overline{\mathcal{L}}}_{{\chi }_2})(1-{\overline{\mathcal{L}}}_{{\chi }_1})}{\parallel ({\mathsf{v}}_1,{\mathsf{u}}_1)-({\mathsf{v}}_2,{\mathsf{u}}_2)\parallel }_{\mathsf{S}}.\hfill \end{array}$$

Therefore, $\mathsf{F}$ is a contraction operator. Therefore, by Banach’s fixed point theorem, $\mathsf{F}$ has a unique fixed point, so the solution of the problem (1) is unique. □

The next result is based on the following Leray–Schauder alternative theorem.

Theorem 2.

[46] Let $\mathsf{F}:\mathsf{S}\to \mathsf{S}$ be an operator which is completely continuous (i.e., a map that restricted to any bounded set in $\mathsf{S}$ is compact). Suppose

$$\begin{array}{cc}\hfill \mathcal{B}\left(\mathsf{F}\right)& =\left\{\mathsf{v}\in \mathsf{S}:\mathsf{v}=\lambda \mathsf{F}\left(\mathsf{v}\right),\lambda \in [0,1]\right\}.\hfill \end{array}$$

Then, either the operator $\mathsf{F}$ has at least one fixed point or the set $\mathcal{B}\left(\mathsf{F}\right)$ is unbounded.

Theorem 3.

Suppose the functions ${\chi }_1,{\chi }_2:\mathfrak{J}\times \mathcal{R}\times \mathcal{R}\to \mathcal{R}$ are continuous and satisfy the following hypothesis:

H₂:

For every $\mathsf{t}\in \mathfrak{J}$ and $\mathsf{u},v:\mathfrak{J}\to \mathcal{R}$, there are ${\varphi }_{\mathcal{i}}(\mathcal{i}=1,2,3):\mathfrak{J}\to {\mathcal{R}}^{+}$, such that

$$\left|{\chi }_1(\mathsf{t},\mathsf{u}\left(\mathsf{t}\right),y\left(\mathsf{t}\right))\right|\le {\varphi }_1\left(\mathsf{t}\right)+{\varphi }_2\left(\mathsf{t}\right)\left|\mathsf{u}\left(\mathsf{t}\right)\right|+{\varphi }_3\left(\mathsf{t}\right)\left|y\left(\mathsf{t}\right)\right|.$$

Similarly, for every $\mathsf{t}\in \mathfrak{J}$ and $\mathsf{v},x:\mathfrak{J}\to \mathcal{R}$, there are ${\phi }_{\mathcal{i}}(\mathcal{i}=1,2,3):\mathfrak{J}\to {\mathcal{R}}^{+}$, such that

$$\left|{\chi }_2(\mathsf{t},\mathsf{v}\left(\mathsf{t}\right),x\left(\mathsf{t}\right))\right|\le {\phi }_1\left(\mathsf{t}\right)+{\phi }_2\left(\mathsf{t}\right)\left|\mathsf{u}\left(\mathsf{t}\right)\right|+{\phi }_3\left(\mathsf{t}\right)\left|x\left(\mathsf{t}\right)\right|,$$

with ${sup}_{\mathsf{t}\in \mathfrak{J}}{\varphi }_{\mathcal{i}}\left(\mathsf{t}\right)={\varphi }_{\mathcal{i}}^{*},{sup}_{\mathsf{t}\in \mathfrak{J}}{\phi }_{\mathcal{i}}\left(\mathsf{t}\right)={\phi }_{\mathcal{i}}^{*}(\mathcal{i}=1,2,3)$.

In addition, it is assumed that

$$\begin{array}{cc}\hfill {\mathcal{Q}}_0& =max\left\{\frac{{\mathcal{Q}}_{\kappa }{\phi }_2^{*}}{1-{\phi }_3^{*}},\frac{{\mathcal{Q}}_{\alpha }{\varphi }_2^{*}}{1-{\varphi }_3^{*}}\right\}<1\mathit{and}0\le {\varphi }_3^{*},{\phi }_3^{*}<1.\hfill \end{array}$$

(15)

Then, the system (1) has at least one solution.

Proof. 

First, we prove that $\mathsf{F}$ is completely continuous. In view of continuity of ${\chi }_1,{\chi }_2,$ the operator $\mathsf{F}$ is also continuous. For any $(\mathsf{v},\mathsf{u})\in {\mathcal{B}}_r$, we have

$$\begin{array}{cc}\hfill & {\mathsf{t}}^{4-\alpha }\left|{\mathsf{F}}_{\alpha }\left(\mathsf{v}\right)\left(\mathsf{t}\right)\right|\hfill \\ \hfill \le & \frac{{\mathsf{t}}^{4-\alpha }}{\Gamma \left(\alpha \right)}{\int }_0^{\mathsf{t}}{\left(\mathsf{t}-\tau \right)}^{\alpha -1}\left|y\left(\tau \right)\right|\mathsf{d}\tau +\left|\frac{{\eta }_1}{6(1-{\eta }_1)\Gamma (\alpha -3)}\right|{\int }_0^{\sigma }{\left(\sigma -\tau \right)}^3\left|y\left(\tau \right)\right|\mathsf{d}\tau +|\frac{{\eta }_2\mathsf{t}}{2(1-{\eta }_2)\Gamma (\alpha -2)}\hfill \\ \hfill & +\frac{{\eta }_1{\eta }_2\sigma }{2(1-{\eta }_1)(1-{\eta }_2)\Gamma (\alpha -3)}|{\int }_0^{\sigma }{\left(\sigma -\tau \right)}^2\left|y\left(\tau \right)\right|\mathsf{d}\tau +|\frac{{\eta }_3{\mathsf{t}}^2}{(1-{\eta }_3)\Gamma (\alpha -1)}\hfill \\ \hfill & +\frac{{\eta }_2{\eta }_3\mathsf{t}\sigma }{(1-{\eta }_2)(1-{\eta }_3)\Gamma (\alpha -2)}+\frac{{\eta }_1(1+{\eta }_2){\eta }_3{\sigma }^2}{2(1-{\eta }_1)(1-{\eta }_2)(1-{\eta }_3)\Gamma (\alpha -3)}|{\int }_0^{\sigma }\left(\sigma -\tau \right)\left|y\left(\tau \right)\right|\mathsf{d}\tau \hfill \\ \hfill & +|\frac{{\eta }_4{\mathsf{t}}^3}{(1-{\eta }_4)\Gamma \left(\alpha \right)}+\frac{{\eta }_3{\eta }_4\sigma {\mathsf{t}}^2}{(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -1)}+\frac{{\eta }_2(1+{\eta }_3){\eta }_4{\sigma }^2\mathsf{t}}{2(1-{\eta }_2)(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -2)}\hfill \\ \hfill & +\frac{{\eta }_1\left((1+{\eta }_2)(1+{\eta }_3)+{\eta }_2+{\eta }_3\right){\eta }_4{\sigma }^3}{6(1-{\eta }_1)(1-{\eta }_2)(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -3)}|{\int }_0^{\sigma }\left|y\left(\tau \right)\right|\mathsf{d}\tau .\hfill \end{array}$$

(16)

Now by ${\mathbf{H}}_2$, we have

$$\begin{array}{ccc}\hfill \left|y\right(\mathsf{t}\left)\right|& =& |{\chi }_1(\mathsf{t},\mathsf{u}\left(\mathsf{t}\right),y\left(\mathsf{t}\right))|\hfill \\ & \le & {\varphi }_1\left(\mathsf{t}\right)+{\varphi }_2\left(\mathsf{t}\right)\left|\mathsf{u}\left(\mathsf{t}\right)\right|+{\varphi }_3\left(\mathsf{t}\right)\left|y\left(\mathsf{t}\right)\right|\hfill \\ & \le & \frac{{\varphi }_1\left(\mathsf{t}\right)+{\varphi }_2\left(\mathsf{t}\right)\left|\mathsf{u}\left(\mathsf{t}\right)\right|}{1-{\varphi }_3\left(\mathsf{t}\right)}.\hfill \end{array}$$

Therefore, (16) implies

$$\begin{array}{cc}\hfill \parallel {\mathsf{F}}_{\alpha }\left(\mathsf{v}\right)\parallel \le & [\frac{{\sigma }^4}{\Gamma (\alpha +1)}+|\frac{{\eta }_1{\sigma }^4}{24(1-{\eta }_1)\Gamma (\alpha -3)}|+\frac{{\eta }_2(1-{\eta }_1){\sigma }^4+{\eta }_1{\eta }_2{\sigma }^4(\alpha -3)}{6(1-{\eta }_1)(1-{\eta }_2)\Gamma (\alpha -2)}|\hfill \\ \hfill & +\left|\frac{{\eta }_3(1-{\eta }_2){\sigma }^4+{\eta }_2{\eta }_3{\sigma }^4(\alpha -2)}{2(1-{\eta }_2)(1-{\eta }_3)\Gamma (\alpha -1)}\right|+\left|\frac{{\eta }_1(1+{\eta }_2){\eta }_3{\sigma }^4}{2(1-{\eta }_1)(1-{\eta }_2)(1-{\eta }_3)\Gamma (\alpha -3)}\right|\hfill \\ \hfill & +\left|\frac{(1-{\eta }_3){\eta }_4{\sigma }^4+{\eta }_3{\eta }_4{\sigma }^4(\alpha -1)}{(1-{\eta }_3)(1-{\eta }_4)\Gamma \left(\alpha \right)}\right|+\left|\frac{{\eta }_2(1+{\eta }_3){\eta }_4{\sigma }^4}{2(1-{\eta }_2)(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -2)}\right|\hfill \\ \hfill & +\left|\frac{{\eta }_1\left((1+{\eta }_2)(1+{\eta }_3)+{\eta }_2+{\eta }_3\right){\eta }_4{\sigma }^4}{6(1-{\eta }_1)(1-{\eta }_2)(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -3)}\right|]\frac{2{\varphi }_1^{*}+{\varphi }_2^{*}r}{2(1-{\varphi }_3^{*})},\hfill \end{array}$$

(17)

which implies that

$$\begin{array}{cc}\hfill \parallel {\mathsf{F}}_{\alpha }\left(\mathsf{v}\right)\parallel & \le {\mathcal{Q}}_{\alpha }\frac{2{\varphi }_1^{*}+{\varphi }_2^{*}r}{2(1-{\varphi }_3^{*})}.\hfill \end{array}$$

(18)

Similarly, we get

$$\begin{array}{cc}\hfill \parallel {\mathsf{F}}_{\kappa }\left(\mathsf{u}\right)\parallel & \le {\mathcal{Q}}_{\kappa }\frac{2{\phi }_1^{*}+{\phi }_2^{*}r}{2(1-{\phi }_3^{*})}.\hfill \end{array}$$

(19)

Thus, it follows from the inequalities (18) and (19) that $\mathsf{F}$ is uniformly bounded.

Now, we prove that $\mathsf{F}$ is equicontinuous. Let $0\le {\mathsf{t}}_2\le {\mathsf{t}}_1\le \mathsf{t}$. Then, we have

$$\begin{array}{cc}\hfill & |{\mathsf{t}}_1^{4-\alpha }{\mathsf{F}}_{\alpha }\left(\mathsf{v}\right)\left({\mathsf{t}}_1\right)-{\mathsf{t}}_2^{4-\alpha }{\mathsf{F}}_{\alpha }\left(\mathsf{v}\right)\left({\mathsf{t}}_2\right)|=\hfill \\ \hfill =& |\frac{1}{\Gamma \left(\alpha \right)}{\int }_0^{{\mathsf{t}}_1}\left[{\mathsf{t}}_1^{4-\alpha }{({\mathsf{t}}_1-\tau )}^{\alpha -1}-{\mathsf{t}}_2^{4-\alpha }{({\mathsf{t}}_2-\tau )}^{\alpha -1}\right]v\left(\tau \right)\mathsf{d}\tau -\frac{1}{\Gamma \left(\alpha \right)}{\int }_{{\mathsf{t}}_1}^{{\mathsf{t}}_2}{\mathsf{t}}_2^{4-\alpha }{({\mathsf{t}}_2-\tau )}^{\alpha -1}y\left(\tau \right)\mathsf{d}\tau \hfill \\ \hfill & +\left[\frac{{\eta }_4({\mathsf{t}}_1^3-{\mathsf{t}}_2^3)}{(1-{\eta }_4)\Gamma \left(\alpha \right)}+\frac{{\eta }_3{\eta }_4\sigma ({\mathsf{t}}_1^2-{\mathsf{t}}_2^2)}{(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -1)}+\frac{{\eta }_2(1+{\eta }_3){\eta }_4{\sigma }^2({\mathsf{t}}_1-{\mathsf{t}}_2)}{2(1-{\eta }_2)(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -2)}\right]{\int }_0^{\sigma }y\left(\tau \right)\mathsf{d}\tau \hfill \\ \hfill & +\left[\frac{{\eta }_3({\mathsf{t}}_1^2-{\mathsf{t}}_2^2)}{(1-{\eta }_3)\Gamma (\alpha -1)}+\frac{{\eta }_2{\eta }_3\sigma ({\mathsf{t}}_1-{\mathsf{t}}_2)}{(1-{\eta }_2)(1-{\eta }_3)\Gamma (\alpha -2)}\right]{\int }_0^{\sigma }\left(\sigma -\tau \right)y\left(\tau \right)\mathsf{d}\tau \hfill \\ \hfill & +\frac{{\eta }_2({\mathsf{t}}_1-{\mathsf{t}}_2)}{2(1-{\eta }_2)\Gamma (\alpha -2)}{\int }_0^{\sigma }{\left(\sigma -\tau \right)}^{2}y\left(\tau \right)\mathsf{d}\tau |.\hfill \end{array}$$

Therefore, we get

$$\begin{array}{cc}\hfill & |{\mathsf{t}}_1^{4-\alpha }{\mathsf{F}}_{\alpha }\left(\mathsf{v}\right)\left({\mathsf{t}}_1\right)-{\mathsf{t}}_2^{4-\alpha }{\mathsf{F}}_{\alpha }\left(\mathsf{v}\right)\left({\mathsf{t}}_2\right)|\hfill \\ \hfill \le & \left[\right|\frac{1}{\Gamma \left(\alpha \right)}{\int }_0^{{\mathsf{t}}_1}\left[{\mathsf{t}}_1^{4-\alpha }{({\mathsf{t}}_1-\tau )}^{\alpha -1}-{\mathsf{t}}_2^{4-\alpha }{({\mathsf{t}}_2-\tau )}^{\alpha -1}\right]\mathsf{d}\tau -\frac{1}{\Gamma \left(\alpha \right)}{\int }_{{\mathsf{t}}_1}^{{\mathsf{t}}_2}{\mathsf{t}}_2^{4-\alpha }{({\mathsf{t}}_2-\tau )}^{\alpha -1}\mathsf{d}\tau |\hfill \\ \hfill & +|\frac{{\eta }_4\sigma ({\mathsf{t}}_1^3-{\mathsf{t}}_2^3)}{(1-{\eta }_4)\Gamma \left(\alpha \right)}+\frac{{\eta }_3{\eta }_4{\sigma }^2({\mathsf{t}}_1^2-{\mathsf{t}}_2^2)}{(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -1)}+\frac{{\eta }_2(1+{\eta }_3){\eta }_4{\sigma }^3({\mathsf{t}}_1-{\mathsf{t}}_2)}{2(1-{\eta }_2)(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -2)}|\hfill \\ \hfill & +|\frac{{\eta }_3{\sigma }^2({\mathsf{t}}_1^2-{\mathsf{t}}_2^2)}{2(1-{\eta }_3)\Gamma (\alpha -1)}+\frac{{\eta }_2{\eta }_3{\sigma }^3({\mathsf{t}}_1-{\mathsf{t}}_2)}{2(1-{\eta }_2)(1-{\eta }_3)\Gamma (\alpha -2)}|\hfill \\ \hfill & +\frac{{\eta }_2{\sigma }^3({\mathsf{t}}_1-{\mathsf{t}}_2)}{6(1-{\eta }_2)\Gamma (\alpha -2)}\left|\right]\frac{{\varphi }_1^{*}+{\varphi }_2^{*}\left|\mathsf{u}\right|}{1-{\varphi }_3^{*}}⟶0\mathrm{as}{\mathsf{t}}_1⟶{\mathsf{t}}_2.\hfill \end{array}$$

Similarly

$$\begin{array}{cc}\hfill & |{\mathsf{t}}_1^{4-\kappa }{\mathsf{F}}_{\kappa }\left(\mathsf{u}\right)\left({\mathsf{t}}_1\right)-{\mathsf{t}}_2^{4-\kappa }{\mathsf{F}}_{\kappa }\left(\mathsf{u}\right)\left({\mathsf{t}}_2\right)|\hfill \\ \hfill \le & \left[\right|\frac{1}{\Gamma \left(\kappa \right)}{\int }_0^{{\mathsf{t}}_1}\left[{\mathsf{t}}_1^{4-\kappa }{({\mathsf{t}}_1-\tau )}^{\kappa -1}-{\mathsf{t}}_2^{4-\kappa }{({\mathsf{t}}_2-\tau )}^{\kappa -1}\right]\mathsf{d}\tau -\frac{1}{\Gamma \left(\kappa \right)}{\int }_{{\mathsf{t}}_1}^{{\mathsf{t}}_2}{\mathsf{t}}_2^{4-\kappa }{({\mathsf{t}}_2-\tau )}^{\kappa -1}\mathsf{d}\tau |\hfill \\ \hfill & +|\frac{{\eta }_4\sigma ({\mathsf{t}}_1^3-{\mathsf{t}}_2^3)}{(1-{\eta }_4)\Gamma \left(\kappa \right)}+\frac{{\eta }_3{\eta }_4{\sigma }^2({\mathsf{t}}_1^2-{\mathsf{t}}_2^2)}{(1-{\eta }_3)(1-{\eta }_4)\Gamma (\kappa -1)}+\frac{{\eta }_2(1+{\eta }_3){\eta }_4{\sigma }^3({\mathsf{t}}_1-{\mathsf{t}}_2)}{2(1-{\eta }_2)(1-{\eta }_3)(1-{\eta }_4)\Gamma (\kappa -2)}|\hfill \\ \hfill & +|\frac{{\eta }_3{\sigma }^2({\mathsf{t}}_1^2-{\mathsf{t}}_2^2)}{2(1-{\eta }_3)\Gamma (\kappa -1)}+\frac{{\eta }_2{\eta }_3{\sigma }^3({\mathsf{t}}_1-{\mathsf{t}}_2)}{2(1-{\eta }_2)(1-{\eta }_3)\Gamma (\kappa -2)}|\hfill \\ \hfill & +\frac{{\eta }_2{\sigma }^3({\mathsf{t}}_1-{\mathsf{t}}_2)}{6(1-{\eta }_2)\Gamma (\kappa -2)}\left|\right]\frac{{\phi }_1^{*}+{\phi }_2^{*}\left|\mathsf{v}\right|}{1-{\phi }_3^{*}}⟶0\mathrm{as}{\mathsf{t}}_1⟶{\mathsf{t}}_2.\hfill \end{array}$$

Therefore, $\mathsf{F}(\mathsf{v},\mathsf{u})$ is equicontinuous. Thus, we proved that the operator $\mathsf{F}(\mathsf{v},\mathsf{u})$ is continuous, uniformly bounded, and equicontinuous, concluding that $\mathsf{F}(\mathsf{v},\mathsf{u})$ is completely continuous. Now, by using Arzela–Ascoli theorem, the operator $\mathsf{F}(\mathsf{v},\mathsf{u})$ is compact.

Finally, we are going to check that $\mathcal{B}=\left\{(\mathsf{v},\mathsf{u})\in \mathsf{S}\left|\right(\mathsf{v},\mathsf{u})=\lambda \mathsf{F}(\mathsf{v},\mathsf{u}),\lambda \in [0,1]\right\}$ is bounded. Suppose $(\mathsf{v},\mathsf{u})\in \mathcal{B},$ then $(\mathsf{v},\mathsf{u})=\lambda \mathsf{F}(\mathsf{v},\mathsf{u}).$ For $\mathsf{t}\in \mathfrak{J},$ we have

$$\mathsf{v}\left(\mathsf{t}\right)=\lambda {\mathsf{F}}_{\alpha }\left(\mathsf{v}\right)\left(\mathsf{t}\right),\mathsf{u}\left(\mathsf{t}\right)=\lambda {\mathsf{F}}_{\kappa }\left(\mathsf{u}\right)\left(\mathsf{t}\right).$$

Then,

$$\begin{array}{cc}\hfill & {\mathsf{t}}^{4-\alpha }\left|\mathsf{v}\left(\mathsf{t}\right)\right|\hfill \\ \hfill \le & [\frac{{\sigma }^4}{\Gamma (\alpha +1)}+|\frac{{\eta }_1{\sigma }^4}{24(1-{\eta }_1)\Gamma (\alpha -3)}|+\frac{{\eta }_2(1-{\eta }_1){\sigma }^4+{\eta }_1{\eta }_2{\sigma }^4(\alpha -3)}{6(1-{\eta }_1)(1-{\eta }_2)\Gamma (\alpha -2)}|\hfill \\ \hfill & +\left|\frac{{\eta }_3(1-{\eta }_2){\sigma }^4+{\eta }_2{\eta }_3{\sigma }^4(\alpha -2)}{2(1-{\eta }_2)(1-{\eta }_3)\Gamma (\alpha -1)}\right|+\left|\frac{{\eta }_1(1+{\eta }_2){\eta }_3{\sigma }^4}{2(1-{\eta }_1)(1-{\eta }_2)(1-{\eta }_3)\Gamma (\alpha -3)}\right|\hfill \\ \hfill & +\left|\frac{(1-{\eta }_3){\eta }_4{\sigma }^4+{\eta }_3{\eta }_4{\sigma }^4(\alpha -1)}{(1-{\eta }_3)(1-{\eta }_4)\Gamma \left(\alpha \right)}\right|+\left|\frac{{\eta }_2(1+{\eta }_3){\eta }_4{\sigma }^4}{2(1-{\eta }_2)(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -2)}\right|\hfill \\ \hfill & +\left|\frac{{\eta }_1\left((1+{\eta }_2)(1+{\eta }_3)+{\eta }_2+{\eta }_3\right){\eta }_4{\sigma }^4}{6(1-{\eta }_1)(1-{\eta }_2)(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -3)}\right|]\frac{{\varphi }_1\left(\mathsf{t}\right)+{\varphi }_2\left(\mathsf{t}\right)\left|\mathsf{u}\left(\mathsf{t}\right)\right|}{1-{\varphi }_3\left(\mathsf{t}\right)}\hfill \end{array}$$

(20)

and

$$\begin{array}{cc}\hfill & {\mathsf{t}}^{4-\kappa }\left|\mathsf{u}\left(\mathsf{t}\right)\right|\hfill \\ \hfill \le & [\frac{{\sigma }^4}{\Gamma (\kappa +1)}+|\frac{{\eta }_5{\sigma }^4}{24(1-{\eta }_5)\Gamma (\kappa -3)}|+\frac{{\eta }_6(1-{\eta }_5){\sigma }^4+{\eta }_5{\eta }_6{\sigma }^4(\kappa -3)}{6(1-{\eta }_5)(1-{\eta }_6)\Gamma (\kappa -2)}|\hfill \\ \hfill & +\left|\frac{{\eta }_7(1-{\eta }_6){\sigma }^4+{\eta }_6{\eta }_7{\sigma }^4(\kappa -2)}{2(1-{\eta }_6)(1-{\eta }_7)\Gamma (\kappa -1)}\right|+\left|\frac{{\eta }_5(1+{\eta }_6){\eta }_7{\sigma }^4}{2(1-{\eta }_5)(1-{\eta }_6)(1-{\eta }_7)\Gamma (\kappa -3)}\right|\hfill \\ \hfill & +\left|\frac{(1-{\eta }_7){\eta }_8{\sigma }^4+{\eta }_7{\eta }_8{\sigma }^4(\kappa -1)}{(1-{\eta }_7)(1-{\eta }_8)\Gamma \left(\kappa \right)}\right|+\left|\frac{{\eta }_6(1+{\eta }_7){\eta }_8{\sigma }^4}{2(1-{\eta }_6)(1-{\eta }_7)(1-{\eta }_8)\Gamma (\kappa -2)}\right|\hfill \\ \hfill & +\left|\frac{{\eta }_5\left((1+{\eta }_6)(1+{\eta }_7)+{\eta }_6+{\eta }_7\right){\eta }_8{\sigma }^4}{6(1-{\eta }_5)(1-{\eta }_6)(1-{\eta }_7)(1-{\eta }_8)\Gamma (\kappa -3)}\right|]\frac{{\phi }_1\left(\mathsf{t}\right)+{\phi }_2\left(\mathsf{t}\right)\left|\mathsf{v}\left(\mathsf{t}\right)\right|}{1-{\phi }_3\left(\mathsf{t}\right)}.\hfill \end{array}$$

(21)

Therefore, from (20) and (21), we have

$$\begin{array}{cc}\hfill \parallel \mathsf{v}\parallel & \le {\mathcal{Q}}_{\alpha }\frac{{\varphi }_1^{*}+{\varphi }_2^{*}\parallel \mathsf{u}\parallel }{1-{\varphi }_3^{*}}\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \parallel \mathsf{u}\parallel & \le {\mathcal{Q}}_{\kappa }\frac{{\phi }_1^{*}+{\phi }_2^{*}\parallel \mathsf{v}\parallel }{1-{\phi }_3^{*}},\hfill \end{array}$$

which imply that

$$\begin{array}{cc}\hfill \parallel \mathsf{v}\parallel +\parallel \mathsf{u}\parallel & =\frac{{\mathcal{Q}}_{\alpha }{\varphi }_1^{*}}{1-{\varphi }_3^{*}}+\frac{{\mathcal{Q}}_{\kappa }{\phi }_1^{*}}{1-{\phi }_3^{*}}+\frac{{\mathcal{Q}}_{\kappa }{\phi }_2^{*}\parallel \mathsf{v}\parallel }{1-{\phi }_3^{*}}+\frac{{\mathcal{Q}}_{\alpha }{\varphi }_2^{*}\parallel \mathsf{u}\parallel }{1-{\varphi }_3^{*}}.\hfill \end{array}$$

Consequently, we get

$$\begin{array}{cc}\hfill {\parallel (\mathsf{v},\mathsf{u})\parallel }_{\mathsf{S}}& \le \frac{{\mathcal{Q}}_{\alpha }{\varphi }_1^{*}+{\mathcal{Q}}_{\kappa }{\phi }_1^{*}}{(1-{\varphi }_3^{*})(1-{\phi }_3^{*})(1-{\mathcal{Q}}_0)},\hfill \end{array}$$

for any $\mathsf{t}\in \mathfrak{J}$, where ${\mathcal{Q}}_0$ is defined by (15), which infer that $\mathcal{B}$ is bounded. Therefore, by Theorem 2, $\mathsf{F}$ has at least one fixed point. Thus, the system (1) has at least one solution. □

4. Stability Results

Let us recall some definitions related to $\mathrm{H}\mathrm{U}$ stabilities:

Suppose the functions ${\Theta }_{\alpha },{\Theta }_{\kappa }:\mathfrak{J}\to {\mathcal{R}}^{+}$ are nondecreasing and ${ϵ}_{\alpha },{ϵ}_{\kappa }>0.$ Consider the inequalities given below.

$$\left\{\begin{array}{cc}\hfill & |{\mathfrak{D}}^{\alpha }\mathsf{v}\left(\mathsf{t}\right)-{\chi }_1(\mathsf{t},\mathsf{u}\left(\mathsf{t}\right),{\mathfrak{D}}^{\alpha }\mathsf{v}\left(\mathsf{t}\right))|\le {ϵ}_{\alpha },\mathsf{t}\in \mathfrak{J},\hfill \\ & |{\mathfrak{D}}^{\kappa }\mathsf{u}\left(\mathsf{t}\right)-{\chi }_2(\mathsf{t},\mathsf{v}\left(\mathsf{t}\right),{\mathfrak{D}}^{\kappa }\mathsf{u}\left(\mathsf{t}\right))|\le {ϵ}_{\kappa },\mathsf{t}\in \mathfrak{J},\hfill \end{array}\right.$$

(22)

$$\left\{\begin{array}{cc}\hfill & |{\mathfrak{D}}^{\alpha }\mathsf{v}\left(\mathsf{t}\right)-{\chi }_1(\mathsf{t},\mathsf{u}\left(\mathsf{t}\right),{\mathfrak{D}}^{\alpha }\mathsf{v}\left(\mathsf{t}\right))|\le {\Theta }_{\alpha }\left(\mathsf{t}\right){ϵ}_{\alpha },\mathsf{t}\in \mathfrak{J},\hfill \\ & |{\mathfrak{D}}^{\kappa }\mathsf{u}\left(\mathsf{t}\right)-{\chi }_2(\mathsf{t},\mathsf{v}\left(\mathsf{t}\right),{\mathfrak{D}}^{\kappa }\mathsf{u}\left(\mathsf{t}\right))|\le {\Theta }_{\kappa }\left(\mathsf{t}\right){ϵ}_{\kappa },\mathsf{t}\in \mathfrak{J},\hfill \end{array}\right.$$

(23)

$$\left\{\begin{array}{cc}\hfill & |{\mathfrak{D}}^{\alpha }\mathsf{v}\left(\mathsf{t}\right)-{\chi }_1(\mathsf{t},\mathsf{u}\left(\mathsf{t}\right),{\mathfrak{D}}^{\alpha }\mathsf{v}\left(\mathsf{t}\right))|\le {\Theta }_{\alpha }\left(\mathsf{t}\right),\mathsf{t}\in \mathfrak{J},\hfill \\ & |{\mathfrak{D}}^{\kappa }\mathsf{u}\left(\mathsf{t}\right)-{\chi }_2(\mathsf{t},\mathsf{v}\left(\mathsf{t}\right),{\mathfrak{D}}^{\kappa }\mathsf{u}\left(\mathsf{t}\right))|\le {\Theta }_{\kappa }\left(\mathsf{t}\right),\mathsf{t}\in \mathfrak{J}.\hfill \end{array}\right.$$

(24)

Definition 3.

[47] System (1) is $\mathrm{H}\mathrm{U}$ stable, if there are ${\mathbf{C}}_{\alpha ,\kappa }=max({\mathbf{C}}_{\alpha },{\mathbf{C}}_{\kappa })>0$ such that for some $ϵ=max({ϵ}_{\alpha },{ϵ}_{\kappa })>0$ and for each solution $(\mathsf{v},\mathsf{u})\in \mathsf{S}$ of the inequality (22). There is a solution $(\mathsf{w},\zeta )\in \mathsf{S}$ with

$$\parallel (\mathsf{v},\mathsf{u})\left(\mathsf{t}\right)-(\mathsf{w},\zeta )\left(\mathsf{t}\right)\parallel \le {\mathbf{C}}_{\alpha ,\kappa }ϵ,\mathsf{t}\in \mathfrak{J}.$$

(25)

Definition 4.

[47] System (1) is generalized $\mathrm{H}\mathrm{U}$ stable, if there is ${\Phi }_{\alpha ,\kappa }\in \mathbb{C}({\mathcal{R}}^{+},{\mathcal{R}}^{+})$ with ${\Phi }_{\alpha ,\kappa }\left(0\right)=0$, such that for each solution $(\mathsf{v},\mathsf{u})\in \mathsf{S}$ of (22), there is a solution $(\mathsf{w},\zeta )\in \mathsf{S}$ of problem (1), which satisfies

$$\parallel (\mathsf{v},\mathsf{u})\left(\mathsf{t}\right)-(\mathsf{w},\zeta )\left(\mathsf{t}\right)\parallel \le {\Phi }_{\alpha ,\kappa }\left(ϵ\right),\mathsf{t}\in \mathfrak{J}.$$

(26)

Definition 5.

[47] System (1) is $\mathrm{H}\mathrm{U}$–Rassias stable with respect to ${\Theta }_{\alpha ,\kappa }=max({\Theta }_{\alpha },{\Theta }_{\kappa })\in \mathbb{C}(\mathfrak{J},\mathcal{R}),$ if there are constants ${\mathbf{C}}_{{\Theta }_{\alpha },{\Theta }_{\kappa }}=max({\mathbf{C}}_{{\Theta }_{\alpha }},{\mathbf{C}}_{{\Theta }_{\kappa }})>0$ such that for some $ϵ=({ϵ}_{\alpha },{ϵ}_{\kappa })>0$ and for each solution $(\mathsf{v},\mathsf{u})\in \mathsf{S}$ of the inequality (23). There is a solution $(\mathsf{w},\zeta )\in \mathsf{S}$ with

$$\parallel (\mathsf{v},\mathsf{u})\left(\mathsf{t}\right)-(\mathsf{w},\zeta )\left(\mathsf{t}\right)\parallel \le {\mathbf{C}}_{{\Theta }_{\alpha },{\Theta }_{\kappa }}{\Theta }_{\alpha ,\kappa }\left(\mathsf{t}\right)ϵ,\mathsf{t}\in \mathfrak{J}.$$

(27)

Definition 6.

[47] System (1) is generalized $\mathrm{H}\mathrm{U}$–Rassias stable with respect to ${\Theta }_{\alpha ,\kappa }=max({\Theta }_{\alpha },{\Theta }_{\kappa })\in \mathbb{C}(\mathfrak{J},\mathcal{R})$, if there is constant ${\mathbf{C}}_{{\Theta }_{\alpha },{\Theta }_{\kappa }}=max({\mathbf{C}}_{{\Theta }_{\alpha }},{\mathbf{C}}_{{\Theta }_{\kappa }})>0$, such that for each solution $(\mathsf{v},\mathsf{u})\in \mathsf{S}$ of the inequality (24). There is a solution $(\mathsf{w},\zeta )\in \mathsf{S}$ of (1), which satisfies

$$\parallel (\mathsf{v},\mathsf{u})\left(\mathsf{t}\right)-(\mathsf{w},\zeta )\left(\mathsf{t}\right)\parallel \le {\mathbf{C}}_{{\Theta }_{\alpha },{\Theta }_{\kappa }}{\Theta }_{\alpha ,\kappa }\left(\mathsf{t}\right),\mathsf{t}\in \mathfrak{J}.$$

(28)

Remark 4.

We say that $(\mathsf{v},\mathsf{u})\in \mathsf{S}$ is a solution of the inequality (22), if there are ${\Psi }_{{\chi }_1},{\Psi }_{{\chi }_2}\in \mathbb{C}(\mathfrak{J},\mathcal{R})$, which depends on $\mathsf{v},\mathsf{u}$, respectively, such that$\left(A_1\right)$$\left|{\Psi }_{{\chi }_1}\left(\mathsf{t}\right)\right|\le {ϵ}_{\alpha },\left|{\Psi }_{{\chi }_2}\left(\mathsf{t}\right)\right|\le {ϵ}_{\kappa },\mathsf{t}\in \mathfrak{J}$;

$\left(A_2\right)$

$$\left\{\begin{array}{cc}& {\mathfrak{D}}^{\alpha }\mathsf{v}\left(\mathsf{t}\right)={\chi }_1(\mathsf{t},\mathsf{u}\left(\mathsf{t}\right),{\mathfrak{D}}^{\alpha }\mathsf{v}\left(\mathsf{t}\right))+{\Psi }_{{\chi }_1}\left(\mathsf{t}\right),\mathsf{t}\in \mathfrak{J},\hfill \\ & {\mathfrak{D}}^{\kappa }\mathsf{u}\left(\mathsf{t}\right)={\chi }_2(\mathsf{t},\mathsf{v}\left(\mathsf{t}\right),{\mathfrak{D}}^{\kappa }\mathsf{u}\left(\mathsf{t}\right))+{\Psi }_{{\chi }_2}\left(\mathsf{t}\right),\mathsf{t}\in \mathfrak{J}.\hfill \end{array}\right.$$

Lemma 3.

Let $(\mathsf{v},\mathsf{u})\in \mathsf{S}$ be the solution of inequality (22), then we have

$$\left\{\begin{array}{cc}& \parallel \mathsf{v}-m_1\parallel \le {\mathcal{Q}}_{\alpha }{ϵ}_{\alpha },\mathsf{t}\in \mathfrak{J},\hfill \\ & \parallel \mathsf{u}-m_2\parallel \le {\mathcal{Q}}_{\kappa }{ϵ}_{\kappa },\mathsf{t}\in \mathfrak{J}.\hfill \end{array}\right.$$

Proof. 

By $\left(A_2\right)$ of Remark 4 and for $\mathsf{t}\in \mathfrak{J}$, we have

$$\left\{\begin{array}{cc}\hfill & {\mathfrak{D}}^{\alpha }\mathsf{v}\left(\mathsf{t}\right)={\chi }_1(\mathsf{t},\mathsf{u}\left(\mathsf{t}\right),{\mathfrak{D}}^{\alpha }\mathsf{v}\left(\mathsf{t}\right))+{\Psi }_{{\chi }_1}\left(\mathsf{t}\right),\hfill \\ & {\mathfrak{D}}^{\kappa }\mathsf{u}\left(\mathsf{t}\right)={\chi }_2(\mathsf{t},\mathsf{v}\left(\mathsf{t}\right),{\mathfrak{D}}^{\kappa }\mathsf{u}\left(\mathsf{t}\right))+{\Psi }_{{\chi }_2}\left(\mathsf{t}\right),\hfill \\ & {\mathfrak{D}}^{\alpha -4}\mathsf{v}\left(0\right)={\eta }_1{\mathfrak{D}}^{\alpha -4}\mathsf{v}\left(\sigma \right),{\mathfrak{D}}^{\alpha -3}\mathsf{v}\left(0\right)={\eta }_2{\mathfrak{D}}^{\alpha -3}\mathsf{v}\left(\sigma \right),\hfill \\ & {\mathfrak{D}}^{\alpha -2}\mathsf{v}\left(0\right)={\eta }_3{\mathfrak{D}}^{\alpha -2}\mathsf{v}\left(\sigma \right),{\mathfrak{D}}^{\alpha -1}\mathsf{v}\left(0\right)={\eta }_4{\mathfrak{D}}^{\alpha -1}\mathsf{v}\left(\sigma \right),\hfill \\ & {\mathfrak{D}}^{\kappa -4}\mathsf{u}\left(0\right)={\eta }_5{\mathfrak{D}}^{\kappa -4}\mathsf{u}\left(\sigma \right),{\mathfrak{D}}^{\kappa -3}\mathsf{u}\left(0\right)={\eta }_6{\mathfrak{D}}^{\kappa -3}\mathsf{u}\left(\sigma \right)\hfill \\ & {\mathfrak{D}}^{\kappa -2}\mathsf{u}\left(0\right)={\eta }_7{\mathfrak{D}}^{\kappa -2}\mathsf{u}\left(\sigma \right),{\mathfrak{D}}^{\kappa -1}\mathsf{u}\left(0\right)={\eta }_8{\mathfrak{D}}^{\kappa -1}\mathsf{u}\left(\sigma \right).\hfill \end{array}\right.$$

(29)

By Lemma 1, the solution of (29) can be written as

$$\left\{\begin{array}{cc}\hfill & \mathsf{v}\left(\mathsf{t}\right)=\frac{1}{\Gamma \left(\alpha \right)}{\int }_0^{\mathsf{t}}{\left(\mathsf{t}-\tau \right)}^{\alpha -1}\left[{\chi }_1(\tau ,\mathsf{u}\left(\tau \right),{\mathfrak{D}}^{\alpha }\mathsf{v}\left(\tau \right))+{\Psi }_{{\chi }_1}\left(\tau \right)\right]\mathsf{d}\tau +\frac{{\eta }_1{\mathsf{t}}^{\alpha -4}}{6(1-{\eta }_1)\Gamma (\alpha -3)}\hfill \\ \hfill & \times {\int }_0^{\sigma }{\left(\sigma -\tau \right)}^3\left[{\chi }_1(\tau ,\mathsf{u}\left(\tau \right),{\mathfrak{D}}^{\alpha }\mathsf{v}\left(\tau \right))+{\Psi }_{{\chi }_1}\left(\tau \right)\right]\mathsf{d}\tau +\left[\frac{{\eta }_2{\mathsf{t}}^{\alpha -3}}{2(1-{\eta }_2)\Gamma (\alpha -2)}+\frac{{\eta }_1{\eta }_2\sigma {\mathsf{t}}^{\alpha -4}}{2(1-{\eta }_1)(1-{\eta }_2)\Gamma (\alpha -3)}\right]\hfill \\ \hfill & \times {\int }_0^{\sigma }{\left(\sigma -\tau \right)}^2\left[{\chi }_1(\tau ,\mathsf{u}\left(\tau \right),{\mathfrak{D}}^{\alpha }\mathsf{v}\left(\tau \right))+{\Psi }_{{\chi }_1}\left(\tau \right)\right]\mathsf{d}\tau +[\frac{{\eta }_3{\mathsf{t}}^{\alpha -2}}{(1-{\eta }_3)\Gamma (\alpha -1)}+\frac{{\eta }_2{\eta }_3\sigma {\mathsf{t}}^{\alpha -3}}{(1-{\eta }_2)(1-{\eta }_3)\Gamma (\alpha -2)}\hfill \\ \hfill & +\frac{{\eta }_1(1+{\eta }_2){\eta }_3{\sigma }^2{\mathsf{t}}^{\alpha -4}}{2(1-{\eta }_1)(1-{\eta }_2)(1-{\eta }_3)\Gamma (\alpha -3)}]{\int }_0^{\sigma }\left(\sigma -\tau \right)\left[{\chi }_1(\tau ,\mathsf{u}\left(\tau \right),{\mathfrak{D}}^{\alpha }\mathsf{v}\left(\tau \right))+{\Psi }_{{\chi }_1}\left(\tau \right)\right]\mathsf{d}\tau \hfill \\ \hfill & +[\frac{{\eta }_4{\mathsf{t}}^{\alpha -1}}{(1-{\eta }_4)\Gamma \left(\alpha \right)}+\frac{{\eta }_3{\eta }_4\sigma {\mathsf{t}}^{\alpha -2}}{(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -1)}+\frac{{\eta }_2(1+{\eta }_3){\eta }_4{\sigma }^2{\mathsf{t}}^{\alpha -3}}{2(1-{\eta }_2)(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -2)}\hfill \\ \hfill & +\frac{{\eta }_1\left((1+{\eta }_2)(1+{\eta }_3)+{\eta }_2+{\eta }_3\right){\eta }_4{\sigma }^3{\mathsf{t}}^{\alpha -4}}{6(1-{\eta }_1)(1-{\eta }_2)(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -3)}]{\int }_0^{\sigma }\left[{\chi }_1(\tau ,\mathsf{u}\left(\tau \right),{\mathfrak{D}}^{\alpha }\mathsf{v}\left(\tau \right))+{\Psi }_{{\chi }_1}\left(\tau \right)\right]\mathsf{d}\tau ,\hfill \\ \hfill & \mathsf{u}\left(\mathsf{t}\right)=\frac{1}{\Gamma \left(\kappa \right)}{\int }_0^{\mathsf{t}}{\left(\mathsf{t}-\tau \right)}^{\kappa -1}\left[{\chi }_2(\tau ,\mathsf{v}\left(\tau \right),{\mathfrak{D}}^{\kappa }\mathsf{u}\left(\tau \right))+{\Psi }_{{\chi }_2}\left(\tau \right)\right]\mathsf{d}\tau +\frac{{\eta }_5{\mathsf{t}}^{\kappa -4}}{6(1-{\eta }_5)\Gamma (\kappa -3)}\hfill \\ \hfill & \times {\int }_0^{\sigma }{\left(\sigma -\tau \right)}^3\left[{\chi }_2(\tau ,\mathsf{v}\left(\tau \right),{\mathfrak{D}}^{\kappa }\mathsf{u}\left(\tau \right))+{\Psi }_{{\chi }_2}\left(\tau \right)\right]\mathsf{d}\tau +\left[\frac{{\eta }_6{\mathsf{t}}^{\kappa -3}}{2(1-{\eta }_6)\Gamma (\kappa -2)}+\frac{{\eta }_5{\eta }_6\sigma {\mathsf{t}}^{\kappa -4}}{2(1-{\eta }_5)(1-{\eta }_6)\Gamma (\kappa -3)}\right]\hfill \\ \hfill & \times {\int }_0^{\sigma }{\left(\sigma -\tau \right)}^2\left[{\chi }_2(\tau ,\mathsf{v}\left(\tau \right),{\mathfrak{D}}^{\kappa }\mathsf{u}\left(\tau \right))+{\Psi }_{{\chi }_2}\left(\tau \right)\right]\mathsf{d}\tau +[\frac{{\eta }_7{\mathsf{t}}^{\kappa -2}}{(1-{\eta }_7)\Gamma (\kappa -1)}+\frac{{\eta }_6{\eta }_7\sigma {\mathsf{t}}^{\kappa -3}}{(1-{\eta }_6)(1-{\eta }_7)\Gamma (\kappa -2)}\hfill \\ \hfill & +\frac{{\eta }_5(1+{\eta }_6){\eta }_7{\sigma }^2{\mathsf{t}}^{\kappa -4}}{2(1-{\eta }_5)(1-{\eta }_6)(1-{\eta }_7)\Gamma (\kappa -3)}]{\int }_0^{\sigma }\left(\sigma -\tau \right)\left[{\chi }_2(\tau ,\mathsf{v}\left(\tau \right),{\mathfrak{D}}^{\kappa }\mathsf{u}\left(\tau \right))+{\Psi }_{{\chi }_2}\left(\tau \right)\right]\mathsf{d}\tau \hfill \\ \hfill & +[\frac{{\eta }_8{\mathsf{t}}^{\kappa -1}}{(1-{\eta }_8)\Gamma \left(\kappa \right)}+\frac{{\eta }_7{\eta }_8\sigma {\mathsf{t}}^{\kappa -2}}{(1-{\eta }_7)(1-{\eta }_8)\Gamma (\kappa -1)}+\frac{{\eta }_6(1+{\eta }_7){\eta }_8{\sigma }^2{\mathsf{t}}^{\kappa -3}}{2(1-{\eta }_6)(1-{\eta }_7)(1-{\eta }_8)\Gamma (\kappa -2)}\hfill \\ \hfill & +\frac{{\eta }_5\left((1+{\eta }_6)(1+{\eta }_7)+{\eta }_6+{\eta }_7\right){\eta }_8{\sigma }^3{\mathsf{t}}^{\kappa -4}}{6(1-{\eta }_5)(1-{\eta }_6)(1-{\eta }_7)(1-{\eta }_8)\Gamma (\kappa -3)}]{\int }_0^{\sigma }\left[{\chi }_2(\tau ,\mathsf{v}\left(\tau \right),{\mathfrak{D}}^{\kappa }\mathsf{u}\left(\tau \right))+{\Psi }_{{\chi }_2}\left(\tau \right)\right]\mathsf{d}\tau .\hfill \end{array}\right.$$

(30)

From first equation of (30), we have

$$\begin{array}{cc}\hfill & {\mathsf{t}}^{4-\alpha }|\mathsf{v}\left(\mathsf{t}\right)-m_1\left(\mathsf{t}\right)|\le \frac{{\mathsf{t}}^{4-\alpha }}{\Gamma \left(\alpha \right)}{\int }_0^{\mathsf{t}}{\left(\mathsf{t}-\tau \right)}^{\alpha -1}\left|{\Psi }_{{\chi }_1}\left(\tau \right)\right|\mathsf{d}\tau \hfill \\ \hfill & +\left|\frac{{\eta }_1}{6(1-{\eta }_1)\Gamma (\alpha -3)}\right|{\int }_0^{\sigma }{\left(\sigma -\tau \right)}^3\left|{\Psi }_{{\chi }_1}\left(\tau \right)\right|\mathsf{d}\tau \hfill \\ \hfill & +|\frac{{\eta }_2\mathsf{t}}{2(1-{\eta }_2)\Gamma (\alpha -2)}+\frac{{\eta }_1{\eta }_2\sigma }{2(1-{\eta }_1)(1-{\eta }_2)\Gamma (\alpha -3)}|{\int }_0^{\sigma }{\left(\sigma -\tau \right)}^2\left|{\Psi }_{{\chi }_1}\left(\tau \right)\right|\mathsf{d}\tau \hfill \\ \hfill & +|\frac{{\eta }_3{\mathsf{t}}^2}{(1-{\eta }_3)\Gamma (\alpha -1)}+\frac{{\eta }_2{\eta }_3\mathsf{t}\sigma }{(1-{\eta }_2)(1-{\eta }_3)\Gamma (\alpha -2)}\hfill \\ \hfill & +\frac{{\eta }_1(1+{\eta }_2){\eta }_3{\sigma }^2}{2(1-{\eta }_1)(1-{\eta }_2)(1-{\eta }_3)\Gamma (\alpha -3)}\left|{\int }_0^{\sigma }\left(\sigma -\tau \right)\right|{\Psi }_{{\chi }_1}\left(\tau \right)|\mathsf{d}\tau +|\frac{{\eta }_4{\mathsf{t}}^3}{(1-{\eta }_4)\Gamma \left(\alpha \right)}\hfill \\ \hfill & +\frac{{\eta }_3{\eta }_4\sigma {\mathsf{t}}^2}{(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -1)}+\frac{{\eta }_2(1+{\eta }_3){\eta }_4{\sigma }^2\mathsf{t}}{2(1-{\eta }_2)(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -2)}\hfill \\ \hfill & +\frac{{\eta }_1\left((1+{\eta }_2)(1+{\eta }_3)+{\eta }_2+{\eta }_3\right){\eta }_4{\sigma }^3}{6(1-{\eta }_1)(1-{\eta }_2)(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -3)}\left|{\int }_0^{\sigma }\right|{\Psi }_{{\chi }_1}\left(\tau \right)|\mathsf{d}\tau .\hfill \end{array}$$

(31)

where $m_1\left(\mathsf{t}\right)$ are those terms which are free of ${\Psi }_{{\chi }_1}.$ Using (6) and $\left(A_1\right)$ of Remark 4, (31) becomes

$$\begin{array}{cc}\hfill \parallel \mathsf{v}-m_1\parallel & \le {\mathcal{Q}}_{\alpha }{ϵ}_{\alpha }.\hfill \end{array}$$

Similarly for second equation of (30), we obtain

$$\begin{array}{cc}\hfill \parallel \mathsf{u}-m_2\parallel & \le {\mathcal{Q}}_{\kappa }{ϵ}_{\kappa }.\hfill \end{array}$$

□

4.1. Method (I)

Theorem 4.

If hypothesis ${\mathbf{H}}_1$ and

$$\begin{array}{c}\hfill \Lambda =1-\frac{{\mathcal{Q}}_{\alpha }{\mathcal{Q}}_{\kappa }{\mathcal{L}}_{{\chi }_1}{\mathcal{L}}_{{\chi }_2}}{(1-{\mathcal{Q}}_{\alpha }{\overline{\mathcal{L}}}_{{\chi }_1})(1-{\mathcal{Q}}_{\kappa }{\overline{\mathcal{L}}}_{{\chi }_2})}>0\end{array}$$

(32)

hold, with $0\le {\mathcal{Q}}_{\alpha }{\overline{\mathcal{L}}}_{{\chi }_1},{\mathcal{Q}}_{\kappa }{\overline{\mathcal{L}}}_{{\chi }_2}<1$. Then system (1) is $\mathrm{H}\mathrm{U}$ stable.

Proof. 

Let $(\mathsf{v},\mathsf{u})\in \mathsf{S}$ be the solution of (22) and $(\mathsf{w},\zeta )\in \mathsf{S}$ be the solution of following system:

$$\left\{\begin{array}{cc}\hfill & {\mathfrak{D}}^{\alpha }\mathsf{w}\left(\mathsf{t}\right)-{\chi }_1(\mathsf{t},\zeta \left(\mathsf{t}\right),{\mathfrak{D}}^{\alpha }\mathsf{w}\left(\mathsf{t}\right))=0,\mathsf{t}\in \mathfrak{J},\hfill \\ & {\mathfrak{D}}^{\kappa }\zeta \left(\mathsf{t}\right)-{\chi }_2(\mathsf{t},\mathsf{w}\left(\mathsf{t}\right),{\mathfrak{D}}^{\kappa }\zeta \left(\mathsf{t}\right))=0,\mathsf{t}\in \mathfrak{J},\hfill \\ & {\mathfrak{D}}^{\alpha -4}\mathsf{w}\left(0\right)={\eta }_1{\mathfrak{D}}^{\alpha -4}\mathsf{w}\left(\sigma \right),{\mathfrak{D}}^{\alpha -3}\mathsf{w}\left(0\right)={\eta }_2{\mathfrak{D}}^{\alpha -3}\mathsf{w}\left(\sigma \right),\hfill \\ & {\mathfrak{D}}^{\alpha -2}\mathsf{w}\left(0\right)={\eta }_3{\mathfrak{D}}^{\alpha -2}\mathsf{w}\left(\sigma \right),{\mathfrak{D}}^{\alpha -1}\mathsf{w}\left(0\right)={\eta }_4{\mathfrak{D}}^{\alpha -1}\mathsf{w}\left(\sigma \right),\hfill \\ & {\mathfrak{D}}^{\kappa -4}\zeta \left(0\right)={\eta }_5{\mathfrak{D}}^{\kappa -4}\zeta \left(\sigma \right),{\mathfrak{D}}^{\kappa -3}\zeta \left(0\right)={\eta }_6{\mathfrak{D}}^{\kappa -3}\zeta \left(\sigma \right),\hfill \\ & {\mathfrak{D}}^{\kappa -2}\zeta \left(0\right)={\eta }_7{\mathfrak{D}}^{\kappa -2}\zeta \left(\sigma \right),{\mathfrak{D}}^{\kappa -1}\mathsf{w}\left(0\right)={\eta }_8{\mathfrak{D}}^{\kappa -1}\zeta \left(\sigma \right).\hfill \end{array}\right.$$

(33)

Then in view of Lemma 1, for $\mathsf{t}\in \mathfrak{J}$ the solution of (33) is given by:

$$\left\{\begin{array}{cc}\hfill & \mathsf{w}\left(\mathsf{t}\right)=\frac{1}{\Gamma \left(\alpha \right)}{\int }_0^{\mathsf{t}}{\left(\mathsf{t}-\tau \right)}^{\alpha -1}{\chi }_1(\tau ,\zeta \left(\tau \right),{\mathfrak{D}}^{\alpha }\mathsf{w}\left(\tau \right))\mathsf{d}\tau +\frac{{\eta }_1{\mathsf{t}}^{\alpha -4}}{6(1-{\eta }_1)\Gamma (\alpha -3)}\hfill \\ \hfill & \times {\int }_0^{\sigma }{\left(\sigma -\tau \right)}^3{\chi }_1(\tau ,\zeta \left(\tau \right),{\mathfrak{D}}^{\alpha }\mathsf{w}\left(\tau \right))\mathsf{d}\tau +\left[\frac{{\eta }_2{\mathsf{t}}^{\alpha -3}}{2(1-{\eta }_2)\Gamma (\alpha -2)}+\frac{{\eta }_1{\eta }_2\sigma {\mathsf{t}}^{\alpha -4}}{2(1-{\eta }_1)(1-{\eta }_2)\Gamma (\alpha -3)}\right]\hfill \\ \hfill & \times {\int }_0^{\sigma }{\left(\sigma -\tau \right)}^2{\chi }_1(\tau ,\zeta \left(\tau \right),{\mathfrak{D}}^{\alpha }\mathsf{w}\left(\tau \right))\mathsf{d}\tau +[\frac{{\eta }_3{\mathsf{t}}^{\alpha -2}}{(1-{\eta }_3)\Gamma (\alpha -1)}+\frac{{\eta }_2{\eta }_3\sigma {\mathsf{t}}^{\alpha -3}}{(1-{\eta }_2)(1-{\eta }_3)\Gamma (\alpha -2)}\hfill \\ \hfill & +\frac{{\eta }_1(1+{\eta }_2){\eta }_3{\sigma }^2{\mathsf{t}}^{\alpha -4}}{2(1-{\eta }_1)(1-{\eta }_2)(1-{\eta }_3)\Gamma (\alpha -3)}]{\int }_0^{\sigma }\left(\sigma -\tau \right){\chi }_1(\tau ,\zeta \left(\tau \right),{\mathfrak{D}}^{\alpha }\mathsf{w}\left(\tau \right))\mathsf{d}\tau \hfill \\ \hfill & +[\frac{{\eta }_4{\mathsf{t}}^{\alpha -1}}{(1-{\eta }_4)\Gamma \left(\alpha \right)}+\frac{{\eta }_3{\eta }_4\sigma {\mathsf{t}}^{\alpha -2}}{(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -1)}+\frac{{\eta }_2(1+{\eta }_3){\eta }_4{\sigma }^2{\mathsf{t}}^{\alpha -3}}{2(1-{\eta }_2)(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -2)}\hfill \\ \hfill & +\frac{{\eta }_1\left((1+{\eta }_2)(1+{\eta }_3)+{\eta }_2+{\eta }_3\right){\eta }_4{\sigma }^3{\mathsf{t}}^{\alpha -4}}{6(1-{\eta }_1)(1-{\eta }_2)(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -3)}]{\int }_0^{\sigma }{\chi }_1(\tau ,\zeta \left(\tau \right),{\mathfrak{D}}^{\alpha }\mathsf{w}\left(\tau \right))\mathsf{d}\tau ,\hfill \\ \hfill & \zeta \left(\mathsf{t}\right)=\frac{1}{\Gamma \left(\kappa \right)}{\int }_0^{\mathsf{t}}{\left(\mathsf{t}-\tau \right)}^{\kappa -1}{\chi }_2(\tau ,\mathsf{w}\left(\tau \right),{\mathfrak{D}}^{\kappa }\zeta \left(\tau \right))\mathsf{d}\tau +\frac{{\eta }_5{\mathsf{t}}^{\kappa -4}}{6(1-{\eta }_5)\Gamma (\kappa -3)}\hfill \\ \hfill & \times {\int }_0^{\sigma }{\left(\sigma -\tau \right)}^3{\chi }_2(\tau ,\mathsf{w}\left(\tau \right),{\mathfrak{D}}^{\kappa }\zeta \left(\tau \right))\mathsf{d}\tau +\left[\frac{{\eta }_6{\mathsf{t}}^{\kappa -3}}{2(1-{\eta }_6)\Gamma (\kappa -2)}+\frac{{\eta }_5{\eta }_6\sigma {\mathsf{t}}^{\kappa -4}}{2(1-{\eta }_5)(1-{\eta }_6)\Gamma (\kappa -3)}\right]\hfill \\ \hfill & \times {\int }_0^{\sigma }{\left(\sigma -\tau \right)}^2{\chi }_2(\tau ,\mathsf{w}\left(\tau \right),{\mathfrak{D}}^{\kappa }\zeta \left(\tau \right))\mathsf{d}\tau +[\frac{{\eta }_7{\mathsf{t}}^{\kappa -2}}{(1-{\eta }_7)\Gamma (\kappa -1)}+\frac{{\eta }_6{\eta }_7\sigma {\mathsf{t}}^{\kappa -3}}{(1-{\eta }_6)(1-{\eta }_7)\Gamma (\kappa -2)}\hfill \\ \hfill & +\frac{{\eta }_5(1+{\eta }_6){\eta }_7{\sigma }^2{\mathsf{t}}^{\kappa -4}}{2(1-{\eta }_5)(1-{\eta }_6)(1-{\eta }_7)\Gamma (\kappa -3)}]{\int }_0^{\sigma }\left(\sigma -\tau \right){\chi }_2(\tau ,\mathsf{w}\left(\tau \right),{\mathfrak{D}}^{\kappa }\zeta \left(\tau \right))\mathsf{d}\tau \hfill \\ \hfill & +[\frac{{\eta }_8{\mathsf{t}}^{\kappa -1}}{(1-{\eta }_8)\Gamma \left(\kappa \right)}+\frac{{\eta }_7{\eta }_8\sigma {\mathsf{t}}^{\kappa -2}}{(1-{\eta }_7)(1-{\eta }_8)\Gamma (\kappa -1)}+\frac{{\eta }_6(1+{\eta }_7){\eta }_8{\sigma }^2{\mathsf{t}}^{\kappa -3}}{2(1-{\eta }_6)(1-{\eta }_7)(1-{\eta }_8)\Gamma (\kappa -2)}\hfill \\ \hfill & +\frac{{\eta }_5\left((1+{\eta }_6)(1+{\eta }_7)+{\eta }_6+{\eta }_7\right){\eta }_8{\sigma }^3{\mathsf{t}}^{\kappa -4}}{6(1-{\eta }_5)(1-{\eta }_6)(1-{\eta }_7)(1-{\eta }_8)\Gamma (\kappa -3)}]{\int }_0^{\sigma }{\chi }_2(\tau ,\mathsf{w}\left(\tau \right),{\mathfrak{D}}^{\kappa }\zeta \left(\tau \right))\mathsf{d}\tau .\hfill \end{array}\right.$$

(34)

Consider

$$\begin{array}{cc}\hfill {\mathsf{t}}^{4-\alpha }|\mathsf{v}\left(\mathsf{t}\right)-\mathsf{w}\left(\mathsf{t}\right)|& \le {\mathsf{t}}^{4-\alpha }|\mathsf{v}\left(\mathsf{t}\right)-m_1\left(\mathsf{t}\right)|+{\mathsf{t}}^{4-\alpha }|m_1\left(\mathsf{t}\right)-\mathsf{w}\left(\mathsf{t}\right)|.\hfill \end{array}$$

(35)

Applying Lemma 3 in (35), we get

$$\begin{array}{cc}\hfill & {\mathsf{t}}^{4-\alpha }|\mathsf{v}\left(\mathsf{t}\right)-\mathsf{w}\left(\mathsf{t}\right)|\le {\mathcal{Q}}_{\alpha }{ϵ}_{\alpha }\hfill \\ \hfill & +\frac{{\mathsf{t}}^{4-\alpha }}{\Gamma \left(\alpha \right)}{\int }_0^{\mathsf{t}}{\left(\mathsf{t}-\tau \right)}^{\alpha -1}|{\chi }_1(\tau ,\mathsf{u}\left(\tau \right),{\mathfrak{D}}^{\alpha }\mathsf{v}\left(\tau \right))-{\chi }_1(\tau ,\zeta \left(\tau \right),{\mathfrak{D}}^{\alpha }\mathsf{w}\left(\tau \right))|\mathsf{d}\tau +\left|\frac{{\eta }_1}{6(1-{\eta }_1)\Gamma (\alpha -3)}\right|\hfill \\ \hfill & \times {\int }_0^{\sigma }{\left(\sigma -\tau \right)}^3|{\chi }_1(\tau ,\mathsf{u}\left(\tau \right),{\mathfrak{D}}^{\alpha }\mathsf{v}\left(\tau \right))-{\chi }_1(\tau ,\zeta \left(\tau \right),{\mathfrak{D}}^{\alpha }\mathsf{w}\left(\tau \right))|\mathsf{d}\tau +|\frac{{\eta }_2\mathsf{t}}{2(1-{\eta }_2)\Gamma (\alpha -2)}\hfill \\ \hfill & +\frac{{\eta }_1{\eta }_2\sigma }{2(1-{\eta }_1)(1-{\eta }_2)\Gamma (\alpha -3)}\left|{\int }_0^{\sigma }{\left(\sigma -\tau \right)}^2\right|{\chi }_1(\tau ,\mathsf{u}\left(\tau \right),{\mathfrak{D}}^{\alpha }\mathsf{v}\left(\tau \right))-{\chi }_1(\tau ,\zeta \left(\tau \right),{\mathfrak{D}}^{\alpha }\mathsf{w}\left(\tau \right))|\mathsf{d}\tau \hfill \\ \hfill & +|\frac{{\eta }_3{\mathsf{t}}^2}{(1-{\eta }_3)\Gamma (\alpha -1)}+\frac{{\eta }_2{\eta }_3\mathsf{t}\sigma }{(1-{\eta }_2)(1-{\eta }_3)\Gamma (\alpha -2)}+\frac{{\eta }_1(1+{\eta }_2){\eta }_3{\sigma }^2}{2(1-{\eta }_1)(1-{\eta }_2)(1-{\eta }_3)\Gamma (\alpha -3)}|\hfill \\ \hfill & \times {\int }_0^{\sigma }\left(\sigma -\tau \right)|{\chi }_1(\tau ,\mathsf{u}\left(\tau \right),{\mathfrak{D}}^{\alpha }\mathsf{v}\left(\tau \right))-{\chi }_1(\tau ,\zeta \left(\tau \right),{\mathfrak{D}}^{\alpha }\mathsf{w}\left(\tau \right))|\mathsf{d}\tau +|\frac{{\eta }_4{\mathsf{t}}^3}{(1-{\eta }_4)\Gamma \left(\alpha \right)}\hfill \\ \hfill & +\frac{{\eta }_3{\eta }_4\sigma {\mathsf{t}}^2}{(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -1)}+\frac{{\eta }_2(1+{\eta }_3){\eta }_4{\sigma }^2\mathsf{t}}{2(1-{\eta }_2)(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -2)}\hfill \\ \hfill & +\frac{{\eta }_1\left((1+{\eta }_2)(1+{\eta }_3)+{\eta }_2+{\eta }_3\right){\eta }_4{\sigma }^3}{6(1-{\eta }_1)(1-{\eta }_2)(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -3)}|\hfill \\ \hfill & \times {\int }_0^{\sigma }|{\chi }_1(\tau ,\mathsf{u}\left(\tau \right),{\mathfrak{D}}^{\alpha }\mathsf{v}\left(\tau \right))-{\chi }_1(\tau ,\zeta \left(\tau \right),{\mathfrak{D}}^{\alpha }\mathsf{w}\left(\tau \right))|\mathsf{d}\tau \hfill \\ \hfill \le & {\mathcal{Q}}_{\alpha }{ϵ}_{\alpha }+[\frac{{\sigma }^4}{\Gamma (\alpha +1)}+|\frac{{\eta }_1{\sigma }^4}{24(1-{\eta }_1)\Gamma (\alpha -3)}|+\frac{{\eta }_2(1-{\eta }_1){\sigma }^4+{\eta }_1{\eta }_2{\sigma }^4(\alpha -3)}{6(1-{\eta }_1)(1-{\eta }_2)\Gamma (\alpha -2)}|\hfill \\ \hfill & +\left|\frac{{\eta }_3(1-{\eta }_2){\sigma }^4+{\eta }_2{\eta }_3{\sigma }^4(\alpha -2)}{2(1-{\eta }_2)(1-{\eta }_3)\Gamma (\alpha -1)}\right|+\left|\frac{{\eta }_1(1+{\eta }_2){\eta }_3{\sigma }^4}{2(1-{\eta }_1)(1-{\eta }_2)(1-{\eta }_3)\Gamma (\alpha -3)}\right|\hfill \\ \hfill & +\left|\frac{(1-{\eta }_3){\eta }_4{\sigma }^4+{\eta }_3{\eta }_4{\sigma }^4(\alpha -1)}{(1-{\eta }_3)(1-{\eta }_4)\Gamma \left(\alpha \right)}\right|+\left|\frac{{\eta }_2(1+{\eta }_3){\eta }_4{\sigma }^4}{2(1-{\eta }_2)(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -2)}\right|\hfill \\ \hfill & +\left|\frac{{\eta }_1\left((1+{\eta }_2)(1+{\eta }_3)+{\eta }_2+{\eta }_3\right){\eta }_4{\sigma }^4}{6(1-{\eta }_1)(1-{\eta }_2)(1-{\eta }_3)(1-{\eta }_4)\Gamma (\alpha -3)}\right|]\left({\mathcal{L}}_{{\chi }_1}\parallel \mathsf{u}-\zeta \parallel +{\overline{\mathcal{L}}}_{{\chi }_1}\parallel {\mathfrak{D}}^{\alpha }\mathsf{v}-{\mathfrak{D}}^{\alpha }\mathsf{w}\parallel \right).\hfill \end{array}$$

(36)

Using ${\mathbf{H}}_1$ of Theorem 1 and (6) in (36), we have

$$\parallel \mathsf{v}-\mathsf{w}\parallel \le \frac{{\mathcal{Q}}_{\alpha }{ϵ}_{\alpha }}{1-{\mathcal{Q}}_{\alpha }{\overline{\mathcal{L}}}_{{\chi }_1}}+\frac{{\mathcal{Q}}_{\alpha }{\mathcal{L}}_{{\chi }_1}}{1-{\mathcal{Q}}_{\alpha }{\overline{\mathcal{L}}}_{{\chi }_1}}\parallel \mathsf{u}-\zeta \parallel .$$

(37)

Similarly, we can get

$$\parallel \mathsf{u}-\zeta \parallel \le \frac{{\mathcal{Q}}_{\kappa }{ϵ}_{\kappa }}{1-{\mathcal{Q}}_{\kappa }{\overline{\mathcal{L}}}_{{\chi }_2}}+\frac{{\mathcal{Q}}_{\kappa }{\mathcal{L}}_{{\chi }_2}}{1-{\mathcal{Q}}_{\kappa }{\overline{\mathcal{L}}}_{{\chi }_2}}\parallel \mathsf{v}-\mathsf{w}\parallel .$$

(38)

We write (37) and (38) as

$$\begin{array}{cc}& \parallel \mathsf{v}-\mathsf{w}\parallel -\frac{{\mathcal{Q}}_{\alpha }{\mathcal{L}}_{{\chi }_1}}{1-{\mathcal{Q}}_{\alpha }{\overline{\mathcal{L}}}_{{\chi }_1}}\parallel \mathsf{u}-\zeta \parallel \le \frac{{\mathcal{Q}}_{\alpha }{ϵ}_{\alpha }}{1-{\mathcal{Q}}_{\alpha }{\overline{\mathcal{L}}}_{{\chi }_1}},\hfill \\ & \parallel \mathsf{u}-\zeta \parallel -\frac{{\mathcal{Q}}_{\kappa }{\mathcal{L}}_{{\chi }_2}}{1-{\mathcal{Q}}_{\kappa }{\overline{\mathcal{L}}}_{{\chi }_2}}\parallel \mathsf{v}-\mathsf{w}\parallel \le \frac{{\mathcal{Q}}_{\kappa }{ϵ}_{\kappa }}{1-{\mathcal{Q}}_{\kappa }{\overline{\mathcal{L}}}_{{\chi }_2}},\hfill \\ \\ & \left[\begin{array}{cc}1& -\frac{{\mathcal{Q}}_{\alpha }{\mathcal{L}}_{{\chi }_1}}{1-{\mathcal{Q}}_{\alpha }{\overline{\mathcal{L}}}_{{\chi }_1}}\\ \\ -\frac{{\mathcal{Q}}_{\kappa }{\mathcal{L}}_{{\chi }_2}}{1-{\mathcal{Q}}_{\kappa }{\overline{\mathcal{L}}}_{{\chi }_2}}& 1\end{array}\right]\left[\begin{array}{c}\parallel \mathsf{v}-\mathsf{w}\parallel \\ \\ \parallel \mathsf{u}-\zeta \parallel \end{array}\right]\le \left[\begin{array}{c}\frac{{\mathcal{Q}}_{\alpha }{ϵ}_{\alpha }}{1-{\mathcal{Q}}_{\alpha }{\overline{\mathcal{L}}}_{{\chi }_1}}\\ \\ \frac{{\mathcal{Q}}_{\kappa }{ϵ}_{\kappa }}{1-{\mathcal{Q}}_{\kappa }{\overline{\mathcal{L}}}_{{\chi }_2}}\end{array}\right].\hfill \end{array}$$

From the above, we get

$$\begin{array}{cc}& \left[\begin{array}{c}\parallel \mathsf{v}-\mathsf{w}\parallel \\ \\ \parallel \mathsf{u}-\zeta \parallel \end{array}\right]\le \left[\begin{array}{cc}\frac{1}{\Lambda }& \frac{{\mathcal{Q}}_{\alpha }{\mathcal{L}}_{{\chi }_1}}{\Lambda (1-{\mathcal{Q}}_{\alpha }{\overline{\mathcal{L}}}_{{\chi }_1})}\\ \\ \frac{{\mathcal{Q}}_{\kappa }{\mathcal{L}}_{{\chi }_2}}{\Lambda (1-{\mathcal{Q}}_{\kappa }{\overline{\mathcal{L}}}_{{\chi }_2})}& \frac{1}{\Lambda }\end{array}\right]\left[\begin{array}{c}\frac{{\mathcal{Q}}_{\alpha }{ϵ}_{\alpha }}{1-{\mathcal{Q}}_{\alpha }{\overline{\mathcal{L}}}_{{\chi }_1}}\\ \\ \frac{{\mathcal{Q}}_{\kappa }{ϵ}_{\kappa }}{1-{\mathcal{Q}}_{\kappa }{\overline{\mathcal{L}}}_{{\chi }_2}}\end{array}\right],\hfill \end{array}$$

where

$$\Lambda =1-\frac{{\mathcal{Q}}_{\alpha }{\mathcal{Q}}_{\kappa }{\mathcal{L}}_{{\chi }_1}{\mathcal{L}}_{{\chi }_2}}{(1-{\mathcal{Q}}_{\alpha }{\overline{\mathcal{L}}}_{{\chi }_1})(1-{\mathcal{Q}}_{\kappa }{\overline{\mathcal{L}}}_{{\chi }_2})}>0.$$

Further simplification gives

$$\begin{array}{cc}\hfill & \parallel \mathsf{v}-\mathsf{w}\parallel \le \frac{{\mathcal{Q}}_{\alpha }{ϵ}_{\alpha }}{\Lambda (1-{\mathcal{Q}}_{\alpha }{\overline{\mathcal{L}}}_{{\chi }_1})}+\frac{{\mathcal{Q}}_{\alpha }{\mathcal{Q}}_{\kappa }{\mathcal{L}}_{{\chi }_1}{ϵ}_{\kappa }}{\Lambda (1-{\mathcal{Q}}_{\alpha }{\overline{\mathcal{L}}}_{{\chi }_1})(1-{\mathcal{Q}}_{\kappa }{\overline{\mathcal{L}}}_{{\chi }_2})},\hfill \\ & \parallel \mathsf{u}-\zeta \parallel \le \frac{{\mathcal{Q}}_{\kappa }{ϵ}_{\kappa }}{\Lambda (1-{\mathcal{Q}}_{\kappa }{\overline{\mathcal{L}}}_{{\chi }_2})}+\frac{{\mathcal{Q}}_{\alpha }{\mathcal{Q}}_{\kappa }{\mathcal{L}}_{{\chi }_2}{ϵ}_{\alpha }}{\Lambda (1-{\mathcal{Q}}_{\alpha }{\overline{\mathcal{L}}}_{{\chi }_1})(1-{\mathcal{Q}}_{\kappa }{\overline{\mathcal{L}}}_{{\chi }_2})},\hfill \end{array}$$

from which we have

$$\begin{array}{cc}\hfill \parallel \mathsf{v}-\mathsf{w}\parallel +\parallel \mathsf{u}-\zeta \parallel \le & \frac{{\mathcal{Q}}_{\alpha }{ϵ}_{\alpha }}{\Lambda (1-{\mathcal{Q}}_{\alpha }{\overline{\mathcal{L}}}_{{\chi }_1})}+\frac{{\mathcal{Q}}_{\kappa }{ϵ}_{\kappa }}{\Lambda (1-{\mathcal{Q}}_{\kappa }{\overline{\mathcal{L}}}_{{\chi }_2})}+\frac{{\mathcal{Q}}_{\alpha }{\mathcal{Q}}_{\kappa }{\mathcal{L}}_{{\chi }_1}{ϵ}_{\kappa }}{\Lambda (1-{\mathcal{Q}}_{\alpha }{\overline{\mathcal{L}}}_{{\chi }_1})(1-{\mathcal{Q}}_{\kappa }{\overline{\mathcal{L}}}_{{\chi }_2})}\hfill \\ \hfill & +\frac{{\mathcal{Q}}_{\alpha }{\mathcal{Q}}_{\kappa }{\mathcal{L}}_{{\chi }_2}{ϵ}_{\alpha }}{\Lambda (1-{\mathcal{Q}}_{\alpha }{\overline{\mathcal{L}}}_{{\chi }_1})(1-{\mathcal{Q}}_{\kappa }{\overline{\mathcal{L}}}_{{\chi }_2})}.\hfill \end{array}$$

(39)

Let $ϵ=max\left\{{ϵ}_{\alpha },{ϵ}_{\kappa }\right\},$ then from (39) we have

$${\parallel (\mathsf{v},\mathsf{u})-(\mathsf{w},\zeta )\parallel }_{\mathsf{S}}\le {\mathbf{C}}_{\alpha ,\kappa }ϵ,$$

(40)

where

$$\begin{array}{cc}\hfill {\mathbf{C}}_{\alpha ,\kappa }=& \frac{{\mathcal{Q}}_{\alpha }}{\Lambda (1-{\mathcal{Q}}_{\alpha }{\overline{\mathcal{L}}}_{{\chi }_1})}+\frac{{\mathcal{Q}}_{\kappa }}{\Lambda (1-{\mathcal{Q}}_{\kappa }{\overline{\mathcal{L}}}_{{\chi }_2})}+\frac{{\mathcal{Q}}_{\alpha }{\mathcal{Q}}_{\kappa }{\mathcal{L}}_{{\chi }_1}}{\Lambda (1-{\mathcal{Q}}_{\alpha }{\overline{\mathcal{L}}}_{{\chi }_1})(1-{\mathcal{Q}}_{\kappa }{\overline{\mathcal{L}}}_{{\chi }_2})}\hfill \\ \hfill & +\frac{{\mathcal{Q}}_{\alpha }{\mathcal{Q}}_{\kappa }{\mathcal{L}}_{{\chi }_2}}{\Lambda (1-{\mathcal{Q}}_{\alpha }{\overline{\mathcal{L}}}_{{\chi }_1})(1-{\mathcal{Q}}_{\kappa }{\overline{\mathcal{L}}}_{{\chi }_2})}.\hfill \end{array}$$

□

Remark 5.

By setting ${\Phi }_{\alpha ,\kappa }\left(ϵ\right)={\mathbf{C}}_{\alpha ,\kappa }ϵ,{\Phi }_{\alpha ,\kappa }\left(0\right)=0$ in (40), then by Definition 4 the problem (1) is generalized $\mathrm{H}\mathrm{U}$ stable.

H₃:

Let functions ${\Theta }_{\alpha },{\Theta }_{\kappa }:\mathfrak{J}\to {\mathcal{R}}^{+}$ be nondecreasing. Then, there are ${\zeta }_{{\Theta }_{\alpha }},{\zeta }_{{\Theta }_{\kappa }}>0$, such that for every $\mathsf{t}\in \mathfrak{J},$ the inequalities

$${\mathfrak{I}}^{\alpha }{\Theta }_{\alpha }\left(\mathsf{t}\right)\le {\zeta }_{{\Theta }_{\alpha }}{\Theta }_{\alpha }\left(\mathsf{t}\right)\mathrm{and}{\mathfrak{I}}^{\kappa }{\Theta }_{\kappa }\left(\mathsf{t}\right)\le {\zeta }_{{\Theta }_{\kappa }}{\Theta }_{\kappa }\left(\mathsf{t}\right)$$

holds.

Remark 6.

Lemma 3 and Theorem 4 gives that the system (1) is $\mathrm{H}\mathrm{U}$–Rassias and generalized $\mathrm{H}\mathrm{U}$–Rassias stable, if ${ϵ}_{\alpha }={\Theta }_{\alpha }\left(\mathsf{t}\right){ϵ}_{\alpha }$ and ${ϵ}_{\kappa }={\Theta }_{\kappa }\left(\mathsf{t}\right){ϵ}_{\kappa }$ with ${\mathbf{H}}_3$ and $\Lambda >0$.

4.2. Method (II)

Theorem 5.

Under the hypothesis ${\mathbf{H}}_1$ and if ${\Lambda }^{*}=1-\left[\frac{{\mathcal{Q}}_{\kappa }{\mathcal{L}}_{{\chi }_2}}{1-{\mathcal{Q}}_{\kappa }{\overline{\mathcal{L}}}_{{\chi }_2}}+\frac{{\mathcal{Q}}_{\alpha }{\mathcal{L}}_{{\chi }_1}}{1-{\mathcal{Q}}_{\alpha }{\overline{\mathcal{L}}}_{{\chi }_1}}\right]>0.$ Then system (1) is $\mathrm{H}\mathrm{U}$ stable.

Proof. 

From inequality (37) and (38), we have

$$\begin{array}{cc}\hfill & \parallel \mathsf{v}-\mathsf{w}\parallel +\parallel \mathsf{u}-\zeta \parallel \hfill \\ \hfill \le & \frac{{\mathcal{Q}}_{\alpha }{ϵ}_{\alpha }}{1-{\mathcal{Q}}_{\alpha }{\overline{\mathcal{L}}}_{{\chi }_1}}+\frac{{\mathcal{Q}}_{\kappa }{ϵ}_{\kappa }}{1-{\mathcal{Q}}_{\kappa }{\overline{\mathcal{L}}}_{{\chi }_2}}+\frac{{\mathcal{Q}}_{\kappa }{\mathcal{L}}_{{\chi }_2}}{1-{\mathcal{Q}}_{\kappa }{\overline{\mathcal{L}}}_{{\chi }_2}}\parallel \mathsf{v}-\mathsf{w}\parallel +\frac{{\mathcal{Q}}_{\alpha }{\mathcal{L}}_{{\chi }_1}}{1-{\mathcal{Q}}_{\alpha }{\overline{\mathcal{L}}}_{{\chi }_1}}\parallel \mathsf{u}-\zeta \parallel .\hfill \end{array}$$

(41)

Let $max\left\{{ϵ}_{\alpha },{ϵ}_{\kappa }\right\}=ϵ,$ then from (41) we obtain

$$\begin{array}{cc}\hfill {\parallel (\mathsf{v},\mathsf{u})-(\mathsf{w},\zeta )\parallel }_{\mathsf{S}}& \le {\mathbf{C}}_{\alpha ,\kappa }ϵ,\hfill \end{array}$$

(42)

where

$$\begin{array}{c}\hfill {\mathbf{C}}_{\alpha ,\kappa }=\left[\frac{{\mathcal{Q}}_{\alpha }}{{\Lambda }^{*}(1-{\mathcal{Q}}_{\alpha }{\overline{\mathcal{L}}}_{{\chi }_1})}+\frac{{\mathcal{Q}}_{\kappa }}{{\Lambda }^{*}(1-{\mathcal{Q}}_{\kappa }{\overline{\mathcal{L}}}_{{\chi }_2})}\right].\end{array}$$

□

Remark 7.

With the help of Remark 5, we can obtain the generalized $\mathrm{H}\mathrm{U}$ stability of system (1).

Remark 8.

Lemma 3 and Theorem 5 gives that the system (1) is $\mathrm{H}\mathrm{U}$–Rassias and generalized $\mathrm{H}\mathrm{U}$–Rassias stable, if ${ϵ}_{\alpha }={\Theta }_{\alpha }\left(\mathsf{t}\right){ϵ}_{\alpha }$ and ${ϵ}_{\kappa }={\Theta }_{\kappa }\left(\mathsf{t}\right){ϵ}_{\kappa }$ with ${\mathbf{H}}_3$ and ${\Lambda }^{*}>0$.

Remark 9.

The results of coupled systems of fourth-order nonlinear $\mathrm{F}\mathrm{D}\mathrm{E}\mathrm{s}$ gives the results of fourth-order nonlinear system of $\mathrm{O}\mathrm{D}\mathrm{E}\mathrm{s}$ (If $\alpha ,\kappa =4$) with anti-periodic and initial conditions, if ${\eta }_{\mathcal{i}}=-1(\mathcal{i}=1,2,\dots ,8)$ and ${\eta }_{\mathcal{i}}=0(\mathcal{i}=1,2,\dots ,8)$ respectively.

5. Example

Example 1.

Consider the following coupled system of $\mathrm{F}\mathrm{D}\mathrm{E}\mathrm{s}$:

$$\left\{\begin{array}{cc}\hfill & {\mathfrak{D}}^{\alpha }\mathsf{v}\left(\mathsf{t}\right)-\left[\frac{1}{4{(\mathsf{t}+2)}^2}\frac{\left|{\mathfrak{D}}^{\alpha }\mathsf{v}\left(\mathsf{t}\right)\right|}{1+\left|{\mathfrak{D}}^{\alpha }\mathsf{v}\left(\mathsf{t}\right)\right|}+\frac{1}{16}{sin}^2\mathsf{u}\left(\mathsf{t}\right)\right]=0,\mathsf{t}\in [0,1],\hfill \\ & {\mathfrak{D}}^{\kappa }\mathsf{u}\left(\mathsf{t}\right)-\left[\frac{1}{32\pi }sin\left(2\pi \mathsf{v}\left(\mathsf{t}\right)\right)+\frac{\left|{\mathfrak{D}}^{\kappa }\mathsf{u}\left(\mathsf{t}\right)\right|}{16\left(1+\left|{\mathfrak{D}}^{\kappa }\mathsf{u}\left(\mathsf{t}\right)\right|\right)}+\frac{1}{2}\right]=0,\mathsf{t}\in [0,1],\hfill \\ & {\mathfrak{D}}^{\alpha -4}\mathsf{v}\left(0\right)={\eta }_1{\mathfrak{D}}^{\alpha -4}\mathsf{v}\left(\sigma \right),{\mathfrak{D}}^{\alpha -3}\mathsf{v}\left(0\right)={\eta }_2{\mathfrak{D}}^{\alpha -3}\mathsf{v}\left(\sigma \right),\hfill \\ & {\mathfrak{D}}^{\alpha -2}\mathsf{v}\left(0\right)={\eta }_3{\mathfrak{D}}^{\alpha -2}\mathsf{v}\left(\sigma \right),{\mathfrak{D}}^{\alpha -1}\mathsf{v}\left(0\right)={\eta }_4{\mathfrak{D}}^{\alpha -1}\mathsf{v}\left(\sigma \right),\hfill \\ & {\mathfrak{D}}^{\kappa -4}\mathsf{u}\left(0\right)={\eta }_5{\mathfrak{D}}^{\kappa -4}\mathsf{u}\left(\sigma \right),{\mathfrak{D}}^{\kappa -3}\mathsf{u}\left(0\right)={\eta }_6{\mathfrak{D}}^{\kappa -3}\mathsf{u}\left(\sigma \right),\hfill \\ & {\mathfrak{D}}^{\kappa -2}\mathsf{u}\left(0\right)={\eta }_7{\mathfrak{D}}^{\kappa -2}\mathsf{u}\left(\sigma \right),{\mathfrak{D}}^{\kappa -1}\mathsf{u}\left(0\right)={\eta }_8{\mathfrak{D}}^{\kappa -1}\mathsf{u}\left(\sigma \right).\hfill \end{array}\right.$$

(43)

From system (43), we can see $\alpha =\kappa =\frac{10}{3},\sigma =1,{\eta }_1={\eta }_5=\frac{1}{2},{\eta }_2={\eta }_6=\frac{1}{3}$, ${\eta }_3={\eta }_7=-1$ and ${\eta }_4={\eta }_8=-1$. Moreover, we have

$$\begin{array}{cc}\hfill |{\chi }_1(\mathsf{t},{\mathsf{u}}_1\left(\mathsf{t}\right),{\mathfrak{D}}^{\alpha }{\mathsf{v}}_1\left(\mathsf{t}\right))-{\chi }_1(\mathsf{t},{\mathsf{u}}_2\left(\mathsf{t}\right),{\mathfrak{D}}^{\alpha }{\mathsf{v}}_2\left(\mathsf{t}\right))|& \le \frac{1}{16}|{\mathsf{u}}_1\left(\mathsf{t}\right)-{\mathsf{u}}_2\left(\mathsf{t}\right)|+\frac{1}{16}|{\mathfrak{D}}^{\alpha }{\mathsf{v}}_1\left(\mathsf{t}\right)-{\mathfrak{D}}^{\alpha }{\mathsf{v}}_2\left(\mathsf{t}\right)|,\hfill \\ \hfill |{\chi }_2(\mathsf{t},{\mathsf{v}}_1\left(\mathsf{t}\right),{\mathfrak{D}}^{\kappa }{\mathsf{u}}_1\left(\mathsf{t}\right))-{\chi }_2(\mathsf{t},{\mathsf{v}}_2\left(\mathsf{t}\right),{\mathfrak{D}}^{\kappa }{\mathsf{u}}_2\left(\mathsf{t}\right))|& \le \frac{1}{16}|{\mathsf{v}}_1\left(\mathsf{t}\right)-{\mathsf{v}}_2\left(\mathsf{t}\right)|+\frac{1}{16}|{\mathfrak{D}}^{\kappa }{\mathsf{u}}_1\left(\mathsf{t}\right)-{\mathfrak{D}}^{\kappa }{\mathsf{u}}_2\left(\mathsf{t}\right)|.\hfill \end{array}$$

Therefore, we get ${\mathcal{L}}_{{\chi }_1}={\overline{\mathcal{L}}}_{{\chi }_1}={\mathcal{L}}_{{\chi }_2}={\overline{\mathcal{L}}}_{{\chi }_2}=\frac{1}{16}$. Therefore,

$$\begin{array}{c}\hfill \frac{{\mathcal{Q}}_{\alpha }{\mathcal{L}}_{{\chi }_1}(1-{\overline{\mathcal{L}}}_{{\chi }_2})+{\mathcal{Q}}_{\kappa }{\mathcal{L}}_{{\chi }_2}(1-{\overline{\mathcal{L}}}_{{\chi }_1})}{(1-{\overline{\mathcal{L}}}_{{\chi }_2})(1-{\overline{\mathcal{L}}}_{{\chi }_1})}\approx 0.75141<1,\end{array}$$

Thus, solution of (43) is unique. Moreover, system (43) is $\mathrm{H}\mathrm{U}$, generalized $\mathrm{H}\mathrm{U}$, $\mathrm{H}\mathrm{U}$–Rassias and generalized $\mathrm{H}\mathrm{U}$–Rassias stable by two different approaches under the conditions of Theorem 4 and Theorem 5, i.e., $\Lambda >0$ and ${\Lambda }^{*}>0$.

6. Conclusions

This paper concluded that the solution of coupled implicit $\mathrm{F}\mathrm{D}\mathrm{E}\mathrm{s}\mathrm{s}$ (1) is unique and exists by using the Banach contraction theorem and Leray–Schauder fixed point theorem. Under some assumptions, the aforesaid coupled system has at least one solution. Besides this, the considered coupled system is $\mathrm{H}\mathrm{U}$, generalized $\mathrm{H}\mathrm{U}$, $\mathrm{H}\mathrm{U}$–Rassias and generalized $\mathrm{H}\mathrm{U}$–Rassias stable. An example is presented to illustrate our obtained results. The proposed system (1) gives the following well-known system of $\mathrm{O}\mathrm{D}\mathrm{E}\mathrm{s}$, which has wide applications in applied sciences [5]

• ${\eta }_{\mathcal{i}}=-1(\mathcal{i}=1,2,\dots ,8)$ and $\alpha ,\kappa =4$, then we get fourth-order $\mathrm{O}\mathrm{D}\mathrm{E}\mathrm{s}$ system with anti-periodic boundary conditions.
• ${\eta }_{\mathcal{i}}=0(\mathcal{i}=1,2,\dots ,8)$ and $\alpha ,\kappa =4$, then we get fourth-order $\mathrm{O}\mathrm{D}\mathrm{E}\mathrm{s}$ system with initial conditions.