On Hilfer–Hadamard Tripled System with Symmetric Nonlocal Riemann–Liouville Integral Boundary Conditions

Abstract

The objective of this manuscript is to investigate the existence, uniqueness criteria and Ulam–Hyers stability of solutions to tripled systems of the Hilfer–Hadamard type supplemented with symmetric nonlocal multi-point Riemann–Liouville integral boundary conditions. By converting the considered problem into an equivalent fixed-point problem, the existence and uniqueness are proven by application of the Leray–Schauder nonlinear alternative and Banach’s contraction principle, respectively. In addition, we discuss the Ulam–Hyers stability and generalized Ulam–Hyers stability of the results, and illustrative examples are also presented to demonstrate their correctness and effectiveness.

1. Introduction

In recent decades, fractional differential equations have been considered as flexible tools for modeling complex systems that exhibit memory and inheritance properties due to the fact that these equations involve several types of fractional derivatives. These derivatives differ in type and complement each other regarding performance, which makes differential equations more flexible with regard to scientific interpretability and the accuracy of the mathematics [1]. Riemann–Liouville, Caputo, and Hadamard derivatives are considered the most famous fractional derivatives [2]. However, the need for generalizations of these derivatives emerged when Hilfer studied fractional time evaluation in [3,4]. Consequently, in [5], Hilfer introduced a new generalized derivative known as the Hilfer fractional derivative of type $\tau \in [0,1]$. This derivative interpolates between the Riemann–Liouville and Caputo fractional derivatives when $\gamma$ differs from its smallest to the largest value, respectively. Many researchers are currently focused on utilizing this new derivative in various applications, as seen in [6,7].

The generalization of fractional derivatives is not limited to traditional derivatives, but can also be extended to include an important type of fractional derivative which is unique according to its logarithmic kernel, that is, the Hadamard derivative. Therefore, in 2012, a new combination of Hilfer and Hadamard fractional derivatives was presented by Qasim [8], known as the Hilfer–Hadamard fractional derivative of $\tau$ type. When the value of the parameter $\tau$ differs from 0 to 1, the Hilfer derivative and Caputo–Hadamard derivative, respectively, are special cases of the Hilfer–Hadamard fractional derivative [9]. The authors of [10] applied this new type of derivative to RLC circuit models, while the authors in [11] studied the Hilfer–Hadamard Langevin system.

The study of value problems involving the Hilfer–Hadamard fractional derivative has garnered significant attention. Researchers have explored theoretical studies of Hilfer–Hadamard fractional differential equations by utilizing various fixed-point theorems and examining their stability. For instance, Tudorache et al. [12] investigated a sequential coupled system of Hilfer–Hadamard fractional differential equations with symmetric Hadamard integral boundary conditions. For some studies on Hilfer–Hadamard fractional differential equations, we refer the reader to [13,14] and the references therein. In a variant way, the authors of [15] combined a sequential coupled system of Hilfer–Hadamard integro differential equations with Riemann–Liouville integral boundary conditions, expanding the existing literature on Hilfer–Hadamard derivatives by combining operators that differ in their kernels.

Recently, tripled systems of fractional differential equations, a generalization of coupled fractional systems, have gained more attention due to their applications. For instance, in bio-mathematics, many epidemic models consist of tripled systems, exemplified in [16,17], which can be improved by fractional derivatives, as summarized in the review article [18] and discussed in details in the works [19,20]. The authors of [21,22] established theoretical concepts of the tripled fixed point. As a result, many researchers have been interested in the existence criteria of such systems, especially in the Riemann–Liouville or Caputo sense, and have applied different fixed-point theorems. For instance, the tripled systems presented in [23,24] have fractional orders in the same domain, which allowed the authors to use Krasnoselskii’s theorem to establish the existence results for the considered problems. Alternatively, in [25], Darbo’s fixed point was used to confirm the existence results, while the authors of [26] used the Mawhin’s topological degree theory method to derive sufficient conditions for existence and the author of [27,28,29] applied the Leray–Schauder nonlinear alternative.

Inspired by the above discussion, in this study, we extend the work in [15] and the tripled system in [27] to present the following tripled system of Hilfer–Hadamard differential equations of different fractional orders ${\alpha }_i$ and of type ${\tau }_i\in [0,1];i\in \{1,2,3\}$:

$$\left\{\begin{array}{c}{}^{HH}\mathcal{D}_{1^{+}}^{{\alpha }_1,{\tau }_1}{x}_1\left(u\right)=f(u,x_1\left(u\right),x_2\left(u\right),x_3\left(u\right)),1<{\alpha }_1\le 2,\hfill \\ {}^{HH}\mathcal{D}_{1^{+}}^{{\alpha }_2,{\tau }_2}{x}_2\left(u\right)=g(u,x_1\left(u\right),x_2\left(u\right),x_3\left(u\right)),1<{\alpha }_2\le 2,u\in \mathcal{E}:=[1,\mathcal{T}],\hfill \\ {}^{HH}\mathcal{D}_{1^{+}}^{{\alpha }_3,{\tau }_3}{x}_3\left(u\right)=h(u,x_1\left(u\right),x_2\left(u\right),x_3\left(u\right)),2<{\alpha }_3\le 3,\hfill \end{array}\right.$$

(1)

supplemented with the following symmetric nonlocal multi-point Riemann–Liouville integral boundary conditions:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{x}_1\left(1\right)=0,x_1\left(\mathcal{T}\right)={\sum }_{i=1}^n{a}_i{\mathcal{I}}_{1^{+}}^{{\eta }_i}{x}_2\left(p_i\right),\hfill \\ x_2\left(1\right)=0,x_2\left(\mathcal{T}\right)={\sum }_{j=1}^m{b}_j{\mathcal{I}}_{1^{+}}^{{\xi }_j}{x}_3\left(q_j\right),\hfill \\ x_3\left(1\right)=0,x_3^{\prime }\left(1\right)=0,x_3\left(\mathcal{T}\right)={\sum }_{k=1}^l{c}_k{\mathcal{I}}_{1^{+}}^{{\zeta }_k}{x}_1\left({\rho }_k\right),\hfill \end{array}\right.\end{array}$$

(2)

where $1<p_1<\cdots <p_n<q_1<\cdots <q_m<{\rho }_1<\cdots <{\rho }_l<\mathcal{T}.$ In the problem (1) and (2), we consider ${\mathcal{I}}_{1^{+}}^{\theta }$ to be the Riemann–Liouville fractional integral operator of positive fractional order $\theta \in \{{\eta }_i,{\xi }_j,{\zeta }_k\}$, while $f,g,h:\mathcal{E}\times \mathbb{R}\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ are continuous functions, and $a_i,b_j,c_k$ are nonzero real constants. We establish the criteria for the existence and uniqueness of the solutions of the presented problem by applying standard fixed-point theorems. In addition, we also study the stability issue for the problem (1) and (2) in the Ulam–Hyers sense.

In our manuscript, we aim to enrich the research area on tripled systems by considering Hilfer–Hadamard derivatives with different orders, which, to the best of our knowledge, have not been previously discussed. Moreover, we combine the considered fractional equations with symmetric boundary conditions involving multi-point Riemann–Liouville integrals. It is worth mentioning the importance of the symmetric integral boundary conditions, which are widely used for studying models with long-term memory or nonlocal interactions; see [30,31], for instance.

Here, we note that the most common methodology for studying Hilfer–Hadamard fractional differential equations is to combine them using boundary conditions involving Hadamard operators, see, for example, the works [32,33]. However, the problem (1) and (2) is entirely different and, to the best of our knowledge, it is the first work of its kind in the literature.

Our manuscript is organized as follows: In Section 2, we introduce the necessary mathematical background and basic tools required for our analysis. The main results on the existence and uniqueness of solutions for the considered problem are presented in Section 3, and Section 4 deals with Hyers–Ulam stability. In the last section, we discuss the validity of our results using illustrative examples.

2. Auxiliary Results

Firstly, we introduce some important definitions.

Definition 1

([34]). The left-sided Riemann–Liouville fractional integral of order $\rho >0$ is defined by

$$\begin{array}{c}\hfill {\mathcal{I}}_{a^{+}}^{\rho }h\left(y\right):={\displaystyle \frac{1}{\Gamma \left(\rho \right)}}{\int }_a^y{\left(y-z\right)}^{\rho -1}h\left(z\right)dz,\end{array}$$

(3)

where $h\in C\left(\right[a,\infty ),\mathbb{R})$ and $\Gamma (.)$ denotes the Gamma function.

Definition 2

([34]). The left-sided Hadamard fractional integral of positive order ρ is given by

$$\begin{array}{c}\hfill {}^H\mathcal{I}_{a^{+}}^{\rho }h\left(y\right):={\displaystyle \frac{1}{\Gamma \left(\rho \right)}}{\int }_a^y{\left(log{\displaystyle \frac{y}{z}}\right)}^{\rho -1}{\displaystyle \frac{h\left(z\right)}{z}}dz,\end{array}$$

(4)

where $h:[a,\infty )\to \mathbb{R}$ is a continuous function and $log(.)={log}_e($.$).$

Definition 3

([34]). Let $h:[a,\infty )\to \mathbb{R}$ be a continuous function. The left-sided Hadamard fractional derivative of order $\rho >0$ is defined by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}^H\mathcal{D}_{a^{+}}^{\rho }h\left(y\right)& ={\delta }^m\left({}^H\mathcal{I}_{a^{+}}^{m-\rho }h\left(y\right)\right)\hfill \\ & ={\left(y{\displaystyle \frac{d}{dy}}\right)}^m{\displaystyle \frac{1}{\Gamma (m-\rho )}}{\int }_a^y{\left(log{\displaystyle \frac{y}{z}}\right)}^{m-\rho -1}{\displaystyle \frac{h\left(z\right)}{z}}dz,m=\left[\rho \right]+1,\hfill \end{array}\end{array}$$

(5)

where $\left[\rho \right]$ is the integer part of the real number ρ.

Definition 4

([35]). Assume that $h\left(y\right)\in A{C}_{\delta }^m[a,b]$ and $\rho >0$. Then, the left-sided Caputo–Hadamard fractional derivative of order ρ of h is defined by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}^{CH}\mathcal{D}_{a^{+}}^{\alpha }h\left(y\right)& ={}^H\mathcal{I}_{a^{+}}^{m-\rho }\left({\delta }^{m}h\left(y\right)\right)\hfill \\ & ={\displaystyle \frac{1}{\Gamma (m-\rho )}}{\int }_a^y{\left(log{\displaystyle \frac{y}{z}}\right)}^{m-\rho -1}{\left(z{\displaystyle \frac{d}{dz}}\right)}^m{\displaystyle \frac{h\left(z\right)}{z}}dz,y>a,\hfill \end{array}\end{array}$$

(6)

where the symbol ${}^H\mathcal{I}_{a^{+}}^{(.)}$ denotes the Hadamard integral defined by Definition 2.

Lemma 1

([34]). Assume that $\rho ,\nu >0$. We have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \left(i\right)\left({}^H\mathcal{I}_{a^{+}}^{\rho }{\left(log{\displaystyle \frac{y}{a}}\right)}^{\nu -1}\right)\left(z\right)={\displaystyle \frac{\Gamma \left(\nu \right)}{\Gamma (\nu +\rho )}}{\left(log{\displaystyle \frac{z}{a}}\right)}^{\nu +\rho -1};\hfill \\ & \left(ii\right)\left({}^H\mathcal{D}_{a^{+}}^{\rho }{\left(log{\displaystyle \frac{y}{a}}\right)}^{\nu -1}\right)\left(z\right)={\displaystyle \frac{\Gamma \left(\nu \right)}{\Gamma (\nu -\rho )}}{\left(log{\displaystyle \frac{z}{a}}\right)}^{\nu -\rho -1}.\hfill \end{array}\end{array}$$

(7)

As a special case, if $\nu =1$, we have

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left({}^H\mathcal{D}_{a^{+}}^{\rho }1\right)\left(z\right)={\displaystyle \frac{1}{\Gamma (1-\rho )}}{\left(log{\displaystyle \frac{z}{a}}\right)}^{-\rho }\ne 0,0<\rho <1.\end{array}\end{array}$$

(8)

Definition 5

([8]). Assume that $h\in L^1(a,b)$, $0<a<b<\infty$ and $m-1<\rho \le m$. Then, the left-sided Hilfer–Hadamard fractional derivative of order ρ and type $\tau \in [0,1]$ is defined by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}^{HH}\mathcal{D}_{a^{+}}^{\rho ,\tau }h\left(y\right)=& {(}^H{\mathcal{I}}_{a^{+}}^{\tau (m-\rho )}{\delta }^m{}^H\mathcal{I}_{a^{+}}^{(m-\rho )(1-\tau )}h)\left(y\right)\hfill \\ \hfill =& {(}^H{\mathcal{I}}_{a^{+}}^{\tau (m-\rho )}{\delta }^m{}^H\mathcal{I}_{a^{+}}^{m-\gamma }h)\left(y\right)\hfill \\ \hfill =& {(}^H{\mathcal{I}}_{a^{+}}^{\tau (m-\rho )}{}^H\mathcal{D}_{a^{+}}^{\gamma }h)\left(y\right),\hfill \end{array}\end{array}$$

(9)

where $\gamma =\rho +m\tau -\rho \tau$, ${}^H\mathcal{I}_{a^{+}}^{(.)}$, and ${}^H\mathcal{D}_{a^{+}}^{(.)}$ are defined in Definitions 2 and 3, respectively.

Remark 1

([8]). The Hilfer–Hadamard derivative is the interface between the Hadamard and Caputo–Hadamard derivatives. Indeed, when $\tau =0$, it becomes the Hadamard derivative (5) and when $\tau =1$, it becomes the Caputo–Hadamard fractional derivative (6).

Theorem 1

([36]). Assume that $h\in L^1[a,b]$, ${(}^H{\mathcal{I}}_{a^{+}}^{m-\gamma }h)\left(y\right)\in A{C}_{\delta }^m[a,b]$, and $\rho ,\gamma \in \mathbb{C}$ such that $\tau \in [0,1]$, $\rho >0$, $m=\left[\rho \right]+1$, $\gamma =\rho +m\tau -\rho \tau$. Then, the following is satisfied

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}^H\mathcal{I}_{a^{+}}^{\rho }\left({}^{HH}\mathcal{D}_{a+}^{\rho ,\tau }h\left(y\right)\right)& ={}^H\mathcal{I}_{a^{+}}^{\gamma }\left({}^{HH}\mathcal{D}_{a^{+}}^{\gamma }h\left(y\right)\right)\hfill \\ & =h\left(y\right)-\sum _{k=0}^{m-1}{\displaystyle \frac{\left({\delta }^{(m-k-1)}{(}^H{\mathcal{I}}_{a+}^{m-\gamma }h)\right)\left(a\right)}{\Gamma (\gamma -k)}}{\left(log{\displaystyle \frac{y}{a}}\right)}^{\gamma -k-1},\hfill \end{array}\end{array}$$

(10)

such that for all $k=1,\cdots ,m-1$, and $m-1<\gamma \le m$, the value of $\Gamma (\gamma -k)$ exists.

Definition 6

([21]). An element $\left(u,v,w\right)\in \mathcal{Y}\times \mathcal{Y}\times \mathcal{Y}$ is called a tripled fixed point of a mapping $T:\mathcal{Y}\times \mathcal{Y}\times \mathcal{Y}\to \mathcal{Y}$ if $T\left(u,v,w\right)=u,T\left(v,u,w\right)=v$, and $T\left(w,v,u\right)=w$.

Define an operator $\mathcal{T}$: $\mathcal{Y}\times \mathcal{Y}\times \mathcal{Y}\to \mathcal{Y}\times \mathcal{Y}\times \mathcal{Y}$ such that

$$\mathcal{T}\left(u,v,w\right)=\left(T\left(u,v,w\right),T\left(v,u,w\right),T\left(w,v,u\right)\right).$$

Then, ($u,v,w$) is a tripled fixed point of T iff ($u,v,w$) is a fixed point of $\mathcal{T}$; that is, $\mathcal{T}\left(u,v,w\right)=\left(u,v,w\right)$.

In the following lemma, we solve the linear form of the problem (1) and (2), which transforms it to a fixed-point problem.

Lemma 2.

Let $F,G,H\in C(\mathcal{E},\mathbb{R})$ and $\Delta =1-ABC\ne 0$, where

$$\left\{\begin{array}{c}A={(log\mathcal{T})}^{1-{\gamma }_1}{\sum }_{i=1}^n\frac{a_i}{\Gamma \left({\eta }_i\right)}{\int }_1^{p_i}{\left(p_i-v\right)}^{{\eta }_i-1}{(logv)}^{{\gamma }_2-1}dv,\hfill \\ B={(log\mathcal{T})}^{1-{\gamma }_2}{\sum }_{j=1}^m\frac{b_j}{\Gamma \left({\xi }_j\right)}{\int }_1^{q_j}{\left(q_j-v\right)}^{{\xi }_j-1}{(logv)}^{{\gamma }_3-1}dv,\hfill \\ C={(log\mathcal{T})}^{1-{\gamma }_3}{\sum }_{k=1}^l\frac{c_k}{\Gamma \left({\zeta }_k\right)}{\int }_1^{{\rho }_k}{\left({\rho }_k-v\right)}^{{\zeta }_k-1}{(logv)}^{{\gamma }_1-1}dv.\hfill \end{array}\right.$$

(11)

Then, the solution of the system

$$\left\{\begin{array}{c}{}^{HH}\mathcal{D}_{1^{+}}^{{\alpha }_1,{\tau }_1}{x}_1\left(u\right)=F\left(u\right),1<{\alpha }_1\le 2,0\le {\tau }_i\le 1,u\in \mathcal{E},\hfill \\ {}^{HH}\mathcal{D}_{1^{+}}^{{\alpha }_2,{\tau }_2}{x}_2\left(u\right)=G\left(u\right),1<{\alpha }_2\le 2,\hfill \\ {}^{HH}\mathcal{D}_{1^{+}}^{{\alpha }_3,{\tau }_3}{x}_3\left(u\right)=H\left(u\right),2<{\alpha }_3\le 3,\hfill \end{array}\right.$$

(12)

subject to the boundary conditions (2) is given by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill x_1\left(u\right)={\displaystyle \frac{{(logu)}^{{\gamma }_1-1}}{\Delta }}& \left\{{(log\mathcal{T})}^{1-{\gamma }_1}\right[\sum _{i=1}^n{\displaystyle \frac{a_i}{\Gamma \left({\eta }_i\right)}}{\int }_1^{p_i}{\left(p_i-v\right)}^{{\eta }_i-1}({\displaystyle \frac{1}{\Gamma \left({\alpha }_2\right)}}\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \times {\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_2-1}{\displaystyle \frac{G\left(w\right)}{w}}dw)dv-{\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)}}{\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_1-1}{\displaystyle \frac{F\left(v\right)}{v}}dv]\hfill \\ & +A\left[{(log\mathcal{T})}^{1-{\gamma }_2}\right(\sum _{j=1}^m{\displaystyle \frac{b_j}{\Gamma \left({\xi }_j\right)}}{\int }_1^{q_j}{\left(q_j-v\right)}^{{\xi }_j-1}({\displaystyle \frac{1}{\Gamma \left({\alpha }_3\right)}}\hfill \\ & \times {\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_3-1}{\displaystyle \frac{H\left(w\right)}{w}}dw)dv-{\displaystyle \frac{1}{\Gamma \left({\alpha }_2\right)}}{\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_2-1}{\displaystyle \frac{G\left(v\right)}{v}}dv)]\hfill \\ \hfill & +AB\left[{(log\mathcal{T})}^{1-{\gamma }_3}\right(\sum _{k=1}^l{\displaystyle \frac{c_k}{\Gamma \left({\zeta }_k\right)}}{\int }_1^{{\rho }_k}{\left({\rho }_k-v\right)}^{{\zeta }_k-1}({\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)}}\hfill \\ & \times {\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_1-1}{\displaystyle \frac{F\left(w\right)}{w}}dw)dv-{\displaystyle \frac{1}{\Gamma \left({\alpha }_3\right)}}{\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_3-1}{\displaystyle \frac{H\left(v\right)}{v}}dv)\left]\right\}\hfill \\ & +{\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)}}{\int }_1^u{\left(log{\displaystyle \frac{u}{v}}\right)}^{{\alpha }_1-1}{\displaystyle \frac{F\left(v\right)}{v}}dv,\hfill \end{array}\end{array}$$

(13)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill x_2\left(u\right)={\displaystyle \frac{{(logu)}^{{\gamma }_2-1}}{\Delta }}& \left\{BC\right[{(log\mathcal{T})}^{1-{\gamma }_1}\left(\sum _{i=1}^n{\displaystyle \frac{a_i}{\Gamma \left({\eta }_i\right)}}{\int }_1^{p_i}{\left(p_i-v\right)}^{{\eta }_i-1}\right({\displaystyle \frac{1}{\Gamma \left({\alpha }_2\right)}}\hfill \\ \hfill & \times {\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_2-1}{\displaystyle \frac{G\left(w\right)}{w}}dw)dv-{\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)}}{\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_1-1}{\displaystyle \frac{F\left(v\right)}{v}}dv)]\hfill \\ & +{(log\mathcal{T})}^{1-{\gamma }_2}\left[\sum _{j=1}^m{\displaystyle \frac{b_j}{\Gamma \left({\xi }_j\right)}}{\int }_1^{q_j}{\left(q_j-v\right)}^{{\xi }_j-1}\right({\displaystyle \frac{1}{\Gamma \left({\alpha }_3\right)}}\hfill \\ & \times {\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_3-1}{\displaystyle \frac{H\left(w\right)}{w}}dw)dv-{\displaystyle \frac{1}{\Gamma \left({\alpha }_2\right)}}{\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_2-1}{\displaystyle \frac{G\left(v\right)}{v}}dv]\hfill \\ & +B\left[{(log\mathcal{T})}^{1-{\gamma }_3}\right(\sum _{k=1}^l{\displaystyle \frac{c_k}{\Gamma \left({\zeta }_k\right)}}{\int }_1^{{\rho }_k}{\left({\rho }_k-v\right)}^{{\zeta }_k-1}({\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)}}\hfill \\ \hfill & \times {\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_1-1}{\displaystyle \frac{F\left(w\right)}{w}}dw)dv-{\displaystyle \frac{1}{\Gamma \left({\alpha }_3\right)}}{\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_3-1}{\displaystyle \frac{H\left(v\right)}{v}}dv)\left]\right\}\hfill \\ & +{\displaystyle \frac{1}{\Gamma \left({\alpha }_2\right)}}{\int }_1^u{\left(log{\displaystyle \frac{u}{v}}\right)}^{{\alpha }_2-1}{\displaystyle \frac{G\left(v\right)}{v}}dv,\hfill \end{array}\end{array}$$

(14)

and

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill x_3\left(u\right)={\displaystyle \frac{{(logu)}^{{\gamma }_3-1}}{\Delta }}& \left\{C\right[{(log\mathcal{T})}^{1-{\gamma }_1}\left(\sum _{i=1}^n{\displaystyle \frac{a_i}{\Gamma \left({\eta }_i\right)}}{\int }_1^{p_i}{\left(p_i-v\right)}^{{\eta }_i-1}\right({\displaystyle \frac{1}{\Gamma \left({\alpha }_2\right)}}\hfill \\ \hfill & \times {\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_2-1}{\displaystyle \frac{G\left(w\right)}{w}}dw)dv-{\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)}}{\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_1-1}{\displaystyle \frac{F\left(v\right)}{v}}dv)]\hfill \\ & +AC\left[{(log\mathcal{T})}^{1-{\gamma }_2}\right(\sum _{j=1}^m{\displaystyle \frac{b_j}{\Gamma \left({\xi }_j\right)}}{\int }_1^{q_j}{\left(q_j-v\right)}^{{\xi }_j-1}({\displaystyle \frac{1}{\Gamma \left({\alpha }_3\right)}}\hfill \\ & \times {\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_3-1}{\displaystyle \frac{H\left(w\right)}{w}}dw)dv-{\displaystyle \frac{1}{\Gamma \left({\alpha }_2\right)}}{\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_2-1}{\displaystyle \frac{G\left(v\right)}{v}}dv)]\hfill \\ & +{(log\mathcal{T})}^{1-{\gamma }_3}\left[\sum _{k=1}^l{\displaystyle \frac{c_k}{\Gamma \left({\zeta }_k\right)}}{\int }_1^{{\rho }_k}{\left({\rho }_k-v\right)}^{{\zeta }_k-1}\right({\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)}}\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \hspace{1em}\hspace{1em}\times {\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_1-1}{\displaystyle \frac{F\left(w\right)}{w}}dw)dv-{\displaystyle \frac{1}{\Gamma \left({\alpha }_3\right)}}{\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_3-1}{\displaystyle \frac{H\left(v\right)}{v}}dv]\}\hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma \left({\alpha }_3\right)}}{\int }_1^u{\left(log{\displaystyle \frac{u}{v}}\right)}^{{\alpha }_3-1}{\displaystyle \frac{H\left(v\right)}{v}}dv,\hfill \end{array}\end{array}$$

(15)

where ${\gamma }_1={\alpha }_1+2{\tau }_1-{\alpha }_1{\tau }_1,{\gamma }_2={\alpha }_2+2{\tau }_2-{\alpha }_2{\tau }_2,$ and ${\gamma }_3={\alpha }_3+3{\tau }_3-{\alpha }_3{\tau }_3.$

Proof. 

Using Theorem 1, the solution of system (12) can be written as

$$\begin{array}{cc}\hfill x_1\left(u\right)& ={\lambda }_0{(logu)}^{{\gamma }_1-1}+{\lambda }_1{(logu)}^{{\gamma }_1-2}+{\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)}}{\int }_1^u{\left(log{\displaystyle \frac{u}{v}}\right)}^{{\alpha }_1-1}{\displaystyle \frac{F\left(v\right)}{v}}dv,\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill x_2\left(u\right)& ={\mu }_0{(logu)}^{{\gamma }_2-1}+{\mu }_1{(logu)}^{{\gamma }_2-2}+{\displaystyle \frac{1}{\Gamma \left({\alpha }_2\right)}}{\int }_1^u{\left(log{\displaystyle \frac{u}{v}}\right)}^{{\alpha }_2-1}{\displaystyle \frac{G\left(v\right)}{v}}dv,\hfill \\ \hfill x_3\left(u\right)& ={\nu }_0{(logu)}^{{\gamma }_3-1}+{\nu }_1{(logu)}^{{\gamma }_3-2}+{\nu }_2{(logu)}^{{\gamma }_3-3}\hfill \end{array}$$

(17)

$$\begin{array}{c}\hfill +{\displaystyle \frac{1}{\Gamma \left({\alpha }_3\right)}}{\int }_1^u{\left(log{\displaystyle \frac{u}{v}}\right)}^{{\alpha }_3-1}{\displaystyle \frac{H\left(v\right)}{v}}dv,\end{array}$$

(18)

where ${\lambda }_0,{\lambda }_1,{\mu }_0,{\mu }_1,{\nu }_0,{\nu }_1,$ and ${\nu }_2$ are unknown constants. Applying the conditions $x_1\left(1\right)=x_2\left(1\right)=x_3\left(1\right)=x_3^{\prime }\left(1\right)=0$ in (16)–(18), we find that ${\lambda }_1={\mu }_1={\nu }_1={\nu }_2=0$ as ${\gamma }_1\in [{\alpha }_1,2],{\gamma }_2\in [{\alpha }_2,2]$ and ${\gamma }_3\in [{\alpha }_3,3]$. Thus, we have

$$\begin{array}{c}\hfill x_1\left(u\right)={\lambda }_0{(logu)}^{{\gamma }_1-1}+{\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)}}{\int }_1^u{\left(log{\displaystyle \frac{u}{v}}\right)}^{{\alpha }_1-1}{\displaystyle \frac{F\left(v\right)}{v}}dv,\end{array}$$

(19)

$$\begin{array}{c}\hfill x_2\left(u\right)={\mu }_0{(logu)}^{{\gamma }_2-1}+{\displaystyle \frac{1}{\Gamma \left({\alpha }_2\right)}}{\int }_1^u{\left(log{\displaystyle \frac{u}{v}}\right)}^{{\alpha }_2-1}{\displaystyle \frac{G\left(v\right)}{v}}dv,\end{array}$$

(20)

$$\begin{array}{c}\hfill x_3\left(u\right)={\nu }_0{(logu)}^{{\gamma }_3-1}+{\displaystyle \frac{1}{\Gamma \left({\alpha }_3\right)}}{\int }_1^u{\left(log{\displaystyle \frac{u}{v}}\right)}^{{\alpha }_3-1}{\displaystyle \frac{H\left(v\right)}{v}}dv.\end{array}$$

(21)

Using the conditions $x_1\left(\mathcal{T}\right)={\sum }_{i=1}^n{a}_i{\mathcal{I}}_{1^{+}}^{{\eta }_i}{x}_2\left(p_i\right),x_2\left(\mathcal{T}\right)={\sum }_{j=1}^m{b}_j{\mathcal{I}}_{1^{+}}^{{\xi }_j}{x}_3\left(q_j\right)$ and $x_3\left(\mathcal{T}\right)={\sum }_{k=1}^l{c}_k{\mathcal{I}}_{1^{+}}^{{\zeta }_k}{x}_1\left({\rho }_k\right)$ in (19)–(21), we get

$$\begin{array}{cc}\hfill & {\lambda }_0{(log\mathcal{T})}^{{\gamma }_1-1}+{\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)}}{\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_1-1}{\displaystyle \frac{F\left(v\right)}{v}}dv={\mu }_0\sum _{i=1}^n{\displaystyle \frac{a_i}{\Gamma \left({\eta }_i\right)}}{\int }_1^{p_i}{\left(p_i-v\right)}^{{\eta }_i-1}\times \hfill \\ & {(logv)}^{{\gamma }_2-1}dv+{\displaystyle \frac{1}{\Gamma \left({\alpha }_2\right)}}\sum _{i=1}^n{\displaystyle \frac{a_i}{\Gamma \left({\eta }_i\right)}}{\int }_1^{p_i}{\left(p_i-v\right)}^{{\eta }_i-1}\left({\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_2-1}{\displaystyle \frac{G\left(w\right)}{w}}dw\right)dv,\hfill \\ \hfill \\ \hfill & {\mu }_0{(log\mathcal{T})}^{{\gamma }_2-1}+{\displaystyle \frac{1}{\Gamma \left({\alpha }_2\right)}}{\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_2-1}{\displaystyle \frac{G\left(v\right)}{v}}dv={\nu }_0\sum _{j=1}^m{\displaystyle \frac{b_j}{\Gamma \left({\xi }_j\right)}}{\int }_1^{q_j}{\left(q_j-v\right)}^{{\xi }_j-1}\times \hfill \end{array}$$

(22)

$$\begin{array}{c}\hfill {(logv)}^{{\gamma }_3-1}dv+{\displaystyle \frac{1}{\Gamma \left({\alpha }_3\right)}}\sum _{j=1}^m{\displaystyle \frac{b_j}{\Gamma \left({\xi }_j\right)}}{\int }_1^{q_j}{\left(q_j-v\right)}^{{\xi }_j-1}\left({\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_3-1}{\displaystyle \frac{H\left(w\right)}{w}}dw\right)dv,\end{array}$$

(23)

$$\begin{array}{cc}\hfill & {\nu }_0{(log\mathcal{T})}^{{\gamma }_3-1}+{\displaystyle \frac{1}{\Gamma \left({\alpha }_3\right)}}{\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_3-1}{\displaystyle \frac{H\left(v\right)}{v}}dv={\lambda }_0\sum _{k=1}^l{\displaystyle \frac{c_k}{\Gamma \left({\zeta }_k\right)}}{\int }_1^{{\rho }_k}{\left({\rho }_k-v\right)}^{{\zeta }_k-1}\times \hfill \end{array}$$

$$\begin{array}{c}\hfill {(logv)}^{{\gamma }_1-1}dv+{\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)}}\sum _{k=1}^l{\displaystyle \frac{c_k}{\Gamma \left({\zeta }_k\right)}}{\int }_1^{{\rho }_k}{\left({\rho }_k-v\right)}^{{\zeta }_k-1}\left({\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_1-1}{\displaystyle \frac{F\left(w\right)}{w}}dw\right)dv.\end{array}$$

(24)

From Equations (22)–(24), we arrive at the following system:

$$\left\{\begin{array}{c}{\lambda }_0-{\mu }_{0}A=J_1,\hfill \\ {\mu }_0-{\nu }_{0}B=J_2,\hfill \\ {\nu }_0-{\lambda }_{0}C=J_3,\hfill \end{array}\right.$$

(25)

where the constants $A,B,C$ are defined in (11) and

$$\begin{array}{cc}\hfill J_1& ={(log\mathcal{T})}^{1-{\gamma }_1}[\sum _{i=1}^n{\displaystyle \frac{a_i}{\Gamma \left({\eta }_i\right)}}{\int }_1^{p_i}{\left(p_i-v\right)}^{{\eta }_i-1}\left({\displaystyle \frac{1}{\Gamma \left({\alpha }_2\right)}}{\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_2-1}{\displaystyle \frac{G\left(w\right)}{w}}dw\right)dv\hfill \\ & -{\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)}}{\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_1-1}{\displaystyle \frac{F\left(v\right)}{v}}dv],\hfill \\ \hfill J_2& ={(log\mathcal{T})}^{1-{\gamma }_2}[\sum _{j=1}^m{\displaystyle \frac{b_j}{\Gamma \left({\xi }_j\right)}}{\int }_1^{q_j}{\left(q_j-v\right)}^{{\xi }_j-1}\left({\displaystyle \frac{1}{\Gamma \left({\alpha }_3\right)}}{\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_3-1}{\displaystyle \frac{H\left(w\right)}{w}}dw\right)dv\hfill \\ \hfill & -{\displaystyle \frac{1}{\Gamma \left({\alpha }_2\right)}}{\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_2-1}{\displaystyle \frac{G\left(v\right)}{v}}dv],\hfill \\ \hfill J_3& ={(log\mathcal{T})}^{1-{\gamma }_3}[\sum _{k=1}^l{\displaystyle \frac{c_k}{\Gamma \left({\zeta }_k\right)}}{\int }_1^{{\rho }_k}{\left({\rho }_k-v\right)}^{{\zeta }_k-1}\left({\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)}}{\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_1-1}{\displaystyle \frac{F\left(w\right)}{w}}dw\right)dv\hfill \\ & -{\displaystyle \frac{1}{\Gamma \left({\alpha }_3\right)}}{\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_3-1}{\displaystyle \frac{H\left(v\right)}{v}}dv].\hfill \end{array}$$

By solving system (25), we get the following results

$$\begin{array}{cc}\hfill & {\lambda }_0={\displaystyle \frac{J_1+A{J}_2+AB{J}_3}{\Delta }},\hfill \\ & {\mu }_0={\displaystyle \frac{BC{J}_1+J_2+B{J}_3}{\Delta }},\hfill \\ & {\nu }_0={\displaystyle \frac{C{J}_1+AC{J}_2+J_3}{\Delta }},\hfill \end{array}$$

(26)

where

$$\Delta =1-ABC\ne 0.$$

(27)

Consequently, we obtain the integral Equations (13)–(15) by substituting (26) into (19)–(21). This completes the proof. □

3. Existence and Uniqueness Results

Let $X=C(\mathcal{E},\mathbb{R})$ be a Banach space equipped with the norm defined by $\parallel x\parallel =sup\left\{\right|x\left(u\right)|:u\in \mathcal{E}\}$. Then, $X\times X\times X$ is also a Banach space endowed with the norm $\parallel (x_1,x_2,x_3)\parallel =\parallel x_1\parallel +\parallel x_2\parallel +\parallel x_3\parallel$.

In light of Lemma 2, we define the operator $\mathcal{F}:X\times X\times X\to X\times X\times X$ by

$$\begin{array}{cc}\hfill & \mathcal{F}(x_1\left(u\right),x_2\left(u\right),x_3\left(u\right))\hfill \\ \hfill & =({\mathcal{F}}_1(x_1\left(u\right),x_2\left(u\right),x_3\left(u\right)),{\mathcal{F}}_2(x_1\left(u\right),x_2\left(u\right),x_3\left(u\right)),{\mathcal{F}}_3(x_1\left(u\right),x_2\left(u\right),x_3\left(u\right))),\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill {\mathcal{F}}_1(x_1\left(u\right)& ,x_2\left(u\right),x_3\left(u\right))={\displaystyle \frac{{(logu)}^{{\gamma }_1-1}}{\Delta }}\left\{{(log\mathcal{T})}^{1-{\gamma }_1}\right[\sum _{i=1}^n{\displaystyle \frac{a_i}{\Gamma \left({\eta }_i\right)}}{\int }_1^{p_i}{\left(p_i-v\right)}^{{\eta }_i-1}({\displaystyle \frac{1}{\Gamma \left({\alpha }_2\right)}}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \times {\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_2-1}{\displaystyle \frac{g(w,x_1\left(w\right),x_2\left(w\right),x_3\left(w\right))}{w}}dw)dv-{\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)}}{\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_1-1}\hfill \\ \hfill & \times {\displaystyle \frac{f(v,x_1\left(v\right),x_2\left(v\right),x_3\left(v\right))}{v}}dv]+A[{(log\mathcal{T})}^{1-{\gamma }_2}(\sum _{j=1}^m{\displaystyle \frac{b_j}{\Gamma \left({\xi }_j\right)}}{\int }_1^{q_j}{\left(q_j-v\right)}^{{\xi }_j-1}\hfill \\ & \times \left({\displaystyle \frac{1}{\Gamma \left({\alpha }_3\right)}}{\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_3-1}{\displaystyle \frac{h(w,x_1\left(w\right),x_2\left(w\right),x_3\left(w\right))}{w}}dw\right)dv-{\displaystyle \frac{1}{\Gamma \left({\alpha }_2\right)}}\hfill \\ \hfill & \times {\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_2-1}{\displaystyle \frac{g(v,x_1\left(v\right),x_2\left(v\right),x_3\left(v\right))}{v}}dv\left)\right]+AB\left[{(log\mathcal{T})}^{1-{\gamma }_3}\right(\sum _{k=1}^l{\displaystyle \frac{c_k}{\Gamma \left({\zeta }_k\right)}}\hfill \\ \hfill & \times {\int }_1^{{\rho }_k}{\left({\rho }_k-v\right)}^{{\zeta }_k-1}\left({\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)}}{\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_1-1}{\displaystyle \frac{f(w,x_1\left(w\right),x_2\left(w\right),x_3\left(w\right))}{w}}dw\right)dv\hfill \\ \hfill & -{\displaystyle \frac{1}{\Gamma \left({\alpha }_3\right)}}{\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_3-1}{\displaystyle \frac{h(v,x_1\left(v\right),x_2\left(v\right),x_3\left(v\right))}{v}}dv\left)\right]\}\hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)}}{\int }_1^u{\left(log{\displaystyle \frac{u}{v}}\right)}^{{\alpha }_1-1}{\displaystyle \frac{f(v,x_1\left(v\right),x_2\left(v\right),x_3\left(v\right))}{v}}dv,\hfill \end{array}$$

(28)

$$\begin{array}{c}\\ \hfill {\mathcal{F}}_2(x_1\left(u\right)& ,x_2\left(u\right),x_3\left(u\right))={\displaystyle \frac{{(logu)}^{{\gamma }_2-1}}{\Delta }}\left\{BC\right[{(log\mathcal{T})}^{1-{\gamma }_1}(\sum _{i=1}^n{\displaystyle \frac{a_i}{\Gamma \left({\eta }_i\right)}}{\int }_1^{p_i}{\left(p_i-v\right)}^{{\eta }_i-1}\hfill \\ & \times \left({\displaystyle \frac{1}{\Gamma \left({\alpha }_2\right)}}{\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_2-1}{\displaystyle \frac{g(w,x_1\left(w\right),x_2\left(w\right),x_3\left(w\right))}{w}}dw\right)dv-{\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)}}\hfill \\ & \times {\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_1-1}{\displaystyle \frac{f(v,x_1\left(v\right),x_2\left(v\right),x_3\left(v\right))}{v}}dv\left)\right]+{(log\mathcal{T})}^{1-{\gamma }_2}[\sum _{j=1}^m{\displaystyle \frac{b_j}{\Gamma \left({\xi }_j\right)}}\hfill \\ & \times {\int }_1^{q_j}{\left(q_j-v\right)}^{{\xi }_j-1}\left({\displaystyle \frac{1}{\Gamma \left({\alpha }_3\right)}}{\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_3-1}{\displaystyle \frac{h(w,x_1\left(w\right),x_2\left(w\right),x_3\left(w\right))}{w}}dw\right)dv\hfill \\ & -{\displaystyle \frac{1}{\Gamma \left({\alpha }_2\right)}}{\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_2-1}{\displaystyle \frac{g(v,x_1\left(v\right),x_2\left(v\right),x_3\left(v\right))}{v}}dv]+B[{(log\mathcal{T})}^{1-{\gamma }_3}(\sum _{k=1}^l{\displaystyle \frac{c_k}{\Gamma \left({\zeta }_k\right)}}\hfill \\ & \times {\int }_1^{{\rho }_k}{\left({\rho }_k-v\right)}^{{\zeta }_k-1}\left({\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)}}{\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_1-1}{\displaystyle \frac{f(w,x_1\left(w\right),x_2\left(w\right),x_3\left(w\right))}{w}}dw\right)dv\hfill \\ & -{\displaystyle \frac{1}{\Gamma \left({\alpha }_3\right)}}{\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_3-1}{\displaystyle \frac{h(v,x_1\left(v\right),x_2\left(v\right),x_3\left(v\right))}{v}}dv\left)\right]\}\hfill \\ & +{\displaystyle \frac{1}{\Gamma \left({\alpha }_2\right)}}{\int }_1^u{\left(log{\displaystyle \frac{u}{v}}\right)}^{{\alpha }_2-1}{\displaystyle \frac{g(v,x_1\left(v\right),x_2\left(v\right),x_3\left(v\right))}{v}}dv,\hfill \end{array}$$

(29)

and

$$\begin{array}{cc}\hfill {\mathcal{F}}_3(x_1\left(u\right)& ,x_2\left(u\right),x_3\left(u\right))={\displaystyle \frac{{(logu)}^{{\gamma }_3-1}}{\Delta }}\left\{C\right[{(log\mathcal{T})}^{1-{\gamma }_1}(\sum _{i=1}^n{\displaystyle \frac{a_i}{\Gamma \left({\eta }_i\right)}}{\int }_1^{p_i}{\left(p_i-v\right)}^{{\eta }_i-1}\hfill \\ & \times \left({\displaystyle \frac{1}{\Gamma \left({\alpha }_2\right)}}{\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_2-1}{\displaystyle \frac{g(w,x_1\left(w\right),x_2\left(w\right),x_3\left(w\right))}{w}}dw\right)dv-{\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)}}\hfill \\ & \times {\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_1-1}{\displaystyle \frac{f(v,x_1\left(v\right),x_2\left(v\right),x_3\left(v\right))}{v}}dv\left)\right]+AC\left[{(logT)}^{1-{\gamma }_2}\right(\sum _{j=1}^m{\displaystyle \frac{b_j}{\Gamma \left({\xi }_j\right)}}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \times {\int }_1^{q_j}{\left(q_j-v\right)}^{{\xi }_j-1}\left({\displaystyle \frac{1}{\Gamma \left({\alpha }_3\right)}}{\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_3-1}{\displaystyle \frac{h(w,x_1\left(w\right),x_2\left(w\right),x_3\left(w\right))}{w}}dw\right)dv\hfill \\ \hfill & -{\displaystyle \frac{1}{\Gamma \left({\alpha }_2\right)}}{\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_2-1}{\displaystyle \frac{g(v,x_1\left(v\right),x_2\left(v\right),x_3\left(v\right))}{v}}dv\left)\right]+{(log\mathcal{T})}^{1-{\gamma }_3}[\sum _{k=1}^l{\displaystyle \frac{c_k}{\Gamma \left({\zeta }_k\right)}}\hfill \\ & \times {\int }_1^{{\rho }_k}{\left({\rho }_k-v\right)}^{{\zeta }_k-1}\left({\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)}}{\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_1-1}{\displaystyle \frac{f(w,x_1\left(w\right),x_2\left(w\right),x_3\left(w\right))}{w}}dw\right)dv\hfill \\ \hfill & -{\displaystyle \frac{1}{\Gamma \left({\alpha }_3\right)}}{\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_3-1}{\displaystyle \frac{h(v,x_1\left(v\right),x_2\left(v\right),x_3\left(v\right))}{v}}dv\left]\right\}\hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma \left({\alpha }_3\right)}}{\int }_1^u{\left(log{\displaystyle \frac{u}{v}}\right)}^{{\alpha }_3-1}{\displaystyle \frac{h(v,x_1\left(v\right),x_2\left(v\right),x_3\left(v\right))}{v}}dv.\hfill \end{array}$$

(30)

We set the following notation:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & G_1={\mathbb{L}}_1+{\mathbb{L}}_2+{\mathbb{L}}_3,G_2={\mathbb{M}}_1+{\mathbb{M}}_2+{\mathbb{M}}_3,G_3={\mathbb{N}}_1+{\mathbb{N}}_2+{\mathbb{N}}_3,\hfill \\ \hfill & {\mathbb{L}}_1={\displaystyle \frac{1}{\Gamma ({\alpha }_1+1)}}\left\{{(log\mathcal{T})}^{{\alpha }_1}+{\displaystyle \frac{{(log\mathcal{T})}^{{\alpha }_1}}{|\Delta |}}+A^{✲}{B}^{✲}\sum _{k=1}^l{\displaystyle \frac{|c_k|{({\rho }_k-1)}^{{\zeta }_k}}{|\Delta |\Gamma ({\zeta }_k+1)}}{(log\mathcal{T})}^{{\gamma }_1-{\gamma }_3+{\alpha }_1}\right\},\hfill \\ & {\mathbb{M}}_1={\displaystyle \frac{1}{|\Delta |\Gamma ({\alpha }_2+1)}}\left\{A^{✲}{(log\mathcal{T})}^{{\gamma }_1-{\gamma }_2+{\alpha }_2}+\sum _{i=1}^n{\displaystyle \frac{|a_i|{(p_i-1)}^{{\eta }_i}}{\Gamma ({\eta }_i+1)}}{(log\mathcal{T})}^{{\alpha }_2}\right\},\hfill \\ & {\mathbb{N}}_1={\displaystyle \frac{1}{|\Delta |\Gamma ({\alpha }_3+1)}}\left\{A^{✲}{B}^{✲}{(log\mathcal{T})}^{{\gamma }_1-{\gamma }_3+{\alpha }_3}+A^{✲}\sum _{j=1}^m{\displaystyle \frac{|b_j|{(q_j-1)}^{{\xi }_j}}{\Gamma ({\xi }_j+1)}}{(log\mathcal{T})}^{{\gamma }_1-{\gamma }_2+{\alpha }_3}\right\},\hfill \\ & {\mathbb{L}}_2={\displaystyle \frac{1}{|\Delta |\Gamma ({\alpha }_1+1)}}\left\{B^{✲}{C}^{✲}{(log\mathcal{T})}^{{\gamma }_2-{\gamma }_1+{\alpha }_1}+B^{✲}\sum _{k=1}^l{\displaystyle \frac{|c_k|{({\rho }_k-1)}^{{\zeta }_k}}{\Gamma ({\zeta }_k+1)}}{(log\mathcal{T})}^{{\gamma }_2-{\gamma }_3+{\alpha }_1}\right\},\hfill \\ & {\mathbb{M}}_2={\displaystyle \frac{1}{\Gamma ({\alpha }_2+1)}}\left\{{(log\mathcal{T})}^{{\alpha }_2}+{\displaystyle \frac{{(log\mathcal{T})}^{{\alpha }_2}}{|\Delta |}}+B^{✲}{C}^{✲}\sum _{i=1}^n{\displaystyle \frac{|a_i|{(p_i-1)}^{{\eta }_i}}{|\Delta |\Gamma ({\eta }_i+1)}}{(log\mathcal{T})}^{{\gamma }_2-{\gamma }_1+{\alpha }_2}\right\},\hfill \\ & {\mathbb{N}}_2={\displaystyle \frac{1}{|\Delta |\Gamma ({\alpha }_3+1)}}\left\{B^{✲}{(log\mathcal{T})}^{{\gamma }_2-{\gamma }_3+{\alpha }_3}+\sum _{j=1}^m{\displaystyle \frac{|b_j|{(q_j-1)}^{{\xi }_j}}{\Gamma ({\xi }_j+1)}}{(log\mathcal{T})}^{{\alpha }_3}\right\},\hfill \\ & {\mathbb{L}}_3={\displaystyle \frac{1}{|\Delta |\Gamma ({\alpha }_1+1)}}\left\{C^{✲}{(log\mathcal{T})}^{{\gamma }_3-{\gamma }_1+{\alpha }_1}+\sum _{k=1}^l{\displaystyle \frac{|c_k|{({\rho }_k-1)}^{{\zeta }_k}}{\Gamma ({\zeta }_k+1)}}{(log\mathcal{T})}^{{\alpha }_1}\right\},\hfill \\ & {\mathbb{M}}_3={\displaystyle \frac{1}{|\Delta |\Gamma ({\alpha }_2+1)}}\left\{A^{✲}{C}^{✲}{(log\mathcal{T})}^{{\gamma }_3-{\gamma }_2+{\alpha }_2}+C^{✲}\sum _{i=1}^n{\displaystyle \frac{|a_i|{(p_i-1)}^{{\eta }_i}}{\Gamma ({\eta }_i+1)}}{(log\mathcal{T})}^{{\gamma }_3-{\gamma }_1+{\alpha }_2}\right\},\hfill \\ & {\mathbb{N}}_3={\displaystyle \frac{1}{\Gamma ({\alpha }_3+1)}}\left\{{(log\mathcal{T})}^{{\alpha }_3}+{\displaystyle \frac{{(log\mathcal{T})}^{{\alpha }_3}}{|\Delta |}}+A^{✲}{C}^{✲}\sum _{j=1}^m{\displaystyle \frac{|b_j|{(q_j-1)}^{{\xi }_j}}{|\Delta |\Gamma ({\xi }_j+1)}}{(log\mathcal{T})}^{{\gamma }_3-{\gamma }_2+{\alpha }_3}\right\},\hfill \end{array}\end{array}$$

(31)

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & A^{✲}={(log\mathcal{T})}^{{\gamma }_2-{\gamma }_1}\sum _{i=1}^n{\displaystyle \frac{|a_i|{(p_i-1)}^{{\eta }_i}}{\Gamma ({\eta }_i+1)}},B^{✲}={(log\mathcal{T})}^{{\gamma }_3-{\gamma }_2}\sum _{j=1}^m{\displaystyle \frac{|b_j|{(q_j-1)}^{{\xi }_j}}{\Gamma ({\xi }_j+1)}},\hfill \\ & C^{✲}={(log\mathcal{T})}^{{\gamma }_1-{\gamma }_3}\sum _{k=1}^l{\displaystyle \frac{|c_k|{({\rho }_k-1)}^{{\zeta }_k}}{\Gamma ({\zeta }_k+1)}},\hfill \end{array}\end{array}$$

(32)

with $A\le A^{✲}$, $B\le B^{✲}$, and $C\le C^{✲}$.

In the following result, we apply the Leray–Schauder alternative [37] to establish the existence of solutions to the problem (1) and (2) on $\mathcal{E}$.

Theorem 2.

Let $\Delta \ne 0$, where Δ is defined in (27). We assume that:

$\left(H_1\right)f,g,h:\mathcal{E}\times \mathbb{R}\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ are continuous functions and there exist real constants ${\phi }_i,{\omega }_i,{\varrho }_i⩾0,i=1,2,3,$ and ${\phi }_0,{\omega }_0,{\varrho }_0>0$ such that for all $u\in \mathcal{E}$ and $x_i\in \mathbb{R};i=1,2,3$,

$$\begin{array}{cc}\hfill & |f(u,x_1,x_2,x_3)|\le {\phi }_0+{\phi }_1|x_1|+{\phi }_2|x_2|+{\phi }_3\left|x_3\right|,\hfill \\ & |g(u,x_1,x_2,x_3)|\le {\omega }_0+{\omega }_1|x_1|+{\omega }_2|x_2|+{\omega }_3\left|x_3\right|,\hfill \\ & |h(u,x_1,x_2,x_3)|\le {\varrho }_0+{\varrho }_1|x_1|+{\varrho }_2|x_2|+{\varrho }_3\left|x_3\right|.\hfill \end{array}$$

(33)

Then, the system (1) and (2) has at least one solution on $\mathcal{E}$ provided that each ${\mathbb{S}}_i$ satisfies $0<{\mathbb{S}}_i<1;i=1,2,3$, where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & {\mathbb{S}}_i=G_1{\phi }_i+G_2{\omega }_i+G_3{\varrho }_i,\hfill \end{array}\end{array}$$

(34)

and $G_i;i=1,2,3$ are given in (31).

Proof. 

It is clear that the continuity of the operator $\mathcal{F}:X\times X\times X\to X\times X\times X$ follows from that of the functions $f,g$ and h. Thus, let $\mathsf{\Omega }\subset X\times X\times X$ be bounded so that for all $(x_1,x_2,x_3)\in \mathsf{\Omega }$, we have

$$\begin{array}{cc}\hfill & |f(u,x_1\left(u\right),x_2\left(u\right),x_3\left(u\right))|\le {\sigma }_1,\hfill \\ & |g(u,x_1\left(u\right),x_2\left(u\right),x_3\left(u\right))|\le {\sigma }_2,\hfill \\ & |h(u,x_1\left(u\right),x_2\left(u\right),x_3\left(u\right))|\le {\sigma }_3,\hfill \end{array}$$

(35)

where ${\sigma }_1,{\sigma }_2$ and ${\sigma }_3$ are positive constants. Then, for any $(x_1,x_2,x_3)\in \mathsf{\Omega }$, we get

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & |{\mathcal{F}}_1(x_1\left(u\right),x_2\left(u\right),x_3\left(u\right))|\hfill \\ \hfill \le & {\displaystyle \frac{1}{|\Delta |}}\{\sum _{i=1}^n{\displaystyle \frac{|a_i|}{\Gamma \left({\eta }_i\right)}}{\int }_1^{p_i}{(p_i-v)}^{{\eta }_i-1}\times \hfill \\ & \left({\displaystyle \frac{1}{\Gamma \left({\alpha }_2\right)}}{\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_2-1}{\displaystyle \frac{\left|g\right(w,x_1\left(w\right),x_2\left(w\right),x_3\left(w\right)\left)\right|}{w}}dw\right)dv+{\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)}}{\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_1-1}\hfill \\ & \times {\displaystyle \frac{\left|f\right(v,x_1\left(v\right),x_2\left(v\right),x_3\left(v\right)\left)\right|}{v}}dv+\left|A\right|{(log\mathcal{T})}^{{\gamma }_1-{\gamma }_2}\sum _{j=1}^m{\displaystyle \frac{|b_j|}{\Gamma \left({\xi }_j\right)}}{\int }_1^{q_j}{\left(q_j-v\right)}^{{\xi }_j-1}({\displaystyle \frac{1}{\Gamma \left({\alpha }_3\right)}}\hfill \\ & \times {\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_3-1}{\displaystyle \frac{\left|h\right(w,x_1\left(w\right),x_2\left(w\right),x_3\left(w\right)\left)\right|}{w}}dw)dv+{\displaystyle \frac{{\left|A\right|(log\mathcal{T})}^{{\gamma }_1-{\gamma }_2}}{\Gamma \left({\alpha }_2\right)}}{\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_2-1}\hfill \\ & \times {\displaystyle \frac{\left|g\right(v,x_1\left(v\right),x_2\left(v\right),x_3\left(v\right)\left)\right|}{v}}dv+\left|AB\right|{(log\mathcal{T})}^{{\gamma }_1-{\gamma }_3}\sum _{k=1}^l{\displaystyle \frac{|c_k|}{\Gamma \left({\zeta }_k\right)}}{\int }_1^{{\rho }_k}{\left({\rho }_k-v\right)}^{{\zeta }_k-1}({\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)}}\hfill \\ \hfill & \times {\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_1-1}{\displaystyle \frac{\left|f\right(w,x_1\left(w\right),x_2\left(w\right),x_3\left(w\right)\left)\right|}{w}}dw)dv+{\displaystyle \frac{{\left|AB\right|(log\mathcal{T})}^{{\gamma }_1-{\gamma }_3}}{\Gamma \left({\alpha }_3\right)}}{\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_3-1}\hfill \\ \hfill & \times {\displaystyle \frac{\left|h\right(v,x_1\left(v\right),x_2\left(v\right),x_3\left(v\right)\left)\right|}{v}}dv\}+{\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)}}{\int }_1^u{\left(log{\displaystyle \frac{u}{v}}\right)}^{{\alpha }_1-1}{\displaystyle \frac{\left|f\right(v,x_1\left(v\right),x_2\left(v\right),x_3\left(v\right)\left)\right|}{v}}dv\hfill \\ & \le {\displaystyle \frac{1}{\Gamma ({\alpha }_1+1)}}\left\{{(log\mathcal{T})}^{{\alpha }_1}+{\displaystyle \frac{{(log\mathcal{T})}^{{\alpha }_1}}{|\Delta |}}+A^{✲}{B}^{✲}\sum _{k=1}^l{\displaystyle \frac{|c_k|{({\rho }_k-1)}^{{\zeta }_k}}{|\Delta |\Gamma ({\zeta }_k+1)}}{(log\mathcal{T})}^{{\gamma }_1-{\gamma }_3+{\alpha }_1}\right\}\parallel f\parallel \hfill \\ & +{\displaystyle \frac{1}{|\Delta |\Gamma ({\alpha }_2+1)}}\left\{A^{✲}{(log\mathcal{T})}^{{\gamma }_1-{\gamma }_2+{\alpha }_2}+\sum _{i=1}^n{\displaystyle \frac{|a_i|{(p_i-1)}^{{\eta }_i}}{\Gamma ({\eta }_i+1)}}{(log\mathcal{T})}^{{\alpha }_2}\right\}\parallel g\parallel \hfill \\ \hfill & +{\displaystyle \frac{1}{|\Delta |\Gamma ({\alpha }_3+1)}}\left\{A^{✲}{B}^{✲}{(log\mathcal{T})}^{{\gamma }_1-{\gamma }_3+{\alpha }_3}+A^{✲}\sum _{j=1}^m{\displaystyle \frac{|b_j|{(q_j-1)}^{{\xi }_j}}{\Gamma ({\xi }_j+1)}}{(log\mathcal{T})}^{{\gamma }_1-{\gamma }_2+{\alpha }_3}\right\}\parallel h\parallel .\hfill \end{array}\end{array}$$

Using the notation in (31), we find that

$$\parallel {\mathcal{F}}_1(x_1,x_2,x_3)\parallel \le {\mathbb{L}}_1{\sigma }_1+{\mathbb{M}}_1{\sigma }_2+{\mathbb{N}}_1{\sigma }_3.$$

(36)

Similarly, we can find that

$$\parallel {\mathcal{F}}_2(x_1,x_2,x_3)\parallel \le {\mathbb{L}}_2{\sigma }_1+{\mathbb{M}}_2{\sigma }_2+{\mathbb{N}}_2{\sigma }_3,$$

(37)

and

$$\parallel {\mathcal{F}}_3(x_1,x_2,x_3)\parallel \le {\mathbb{L}}_3{\sigma }_1+{\mathbb{M}}_3{\sigma }_2+{\mathbb{N}}_3{\sigma }_3.$$

(38)

Based on inequalities (36)–(38), we must obtain

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \parallel \mathcal{F}(x_1,x_2,x_3)\parallel \le G_1{\sigma }_1+G_2{\sigma }_2+G_3{\sigma }_3,\end{array}\end{array}$$

(39)

where $G_i;i=1,2,3$ are defined in (31). This proves the uniform boundedness of $\mathcal{F}$.

Next, we prove that $\mathcal{F}$ is equicontinuous. Let $u_1,u_2\in \mathcal{E}$ such that $u_1<u_2$. Then, we get

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & |{\mathcal{F}}_1(x_1\left(u_2\right),x_2\left(u_2\right),x_3\left(u_2\right))-{\mathcal{F}}_1(x_1\left(u_1\right),x_2\left(u_1\right),x_3\left(u_1\right))|\hfill \\ \hfill & \le {\displaystyle \frac{{(log{u}_2)}^{{\gamma }_1-1}-{(log{u}_1)}^{{\gamma }_1-1}}{|\Delta |}}\{{(log\mathcal{T})}^{1-{\gamma }_1}\sum _{i=1}^n{\displaystyle \frac{|a_i|}{\Gamma \left({\eta }_i\right)}}{\int }_1^{p_i}{(p_i-v)}^{{\eta }_i-1}\times \hfill \\ & \left({\displaystyle \frac{1}{\Gamma \left({\alpha }_2\right)}}{\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_2-1}{\displaystyle \frac{\left|g\right(w,x_1\left(w\right),x_2\left(w\right),x_3\left(w\right)\left)\right|}{w}}dw\right)dv+{\displaystyle \frac{{(log\mathcal{T})}^{1-{\gamma }_1}}{\Gamma \left({\alpha }_1\right)}}{\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_1-1}\hfill \\ & \times {\displaystyle \frac{\left|f\right(v,x_1\left(v\right),x_2\left(v\right),x_3\left(v\right)\left)\right|}{v}}dv+\left|A\right|{(log\mathcal{T})}^{1-{\gamma }_2}\sum _{j=1}^m{\displaystyle \frac{|b_j|}{\Gamma \left({\xi }_j\right)}}{\int }_1^{q_j}{\left(q_j-v\right)}^{{\xi }_j-1}({\displaystyle \frac{1}{\Gamma \left({\alpha }_3\right)}}\hfill \\ & \times {\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_3-1}{\displaystyle \frac{\left|h\right(w,x_1\left(w\right),x_2\left(w\right),x_3\left(w\right)\left)\right|}{w}}dw)dv+{\displaystyle \frac{{\left|A\right|(log\mathcal{T})}^{1-{\gamma }_2}}{\Gamma \left({\alpha }_2\right)}}{\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_2-1}\hfill \\ & \times {\displaystyle \frac{\left|g\right(v,x_1\left(v\right),x_2\left(v\right),x_3\left(v\right)\left)\right|}{v}}dv+\left|AB\right|{(log\mathcal{T})}^{1-{\gamma }_3}\sum _{k=1}^l{\displaystyle \frac{|c_k|}{\Gamma \left({\zeta }_k\right)}}{\int }_1^{{\rho }_k}{\left({\rho }_k-v\right)}^{{\zeta }_k-1}({\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)}}\hfill \\ \hfill & \times {\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_1-1}{\displaystyle \frac{\left|f\right(w,x_1\left(w\right),x_2\left(w\right),x_3\left(w\right)\left)\right|}{w}}dw)dv+{\displaystyle \frac{{\left|AB\right|(log\mathcal{T})}^{1-{\gamma }_3}}{\Gamma \left({\alpha }_3\right)}}{\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_3-1}\hfill \\ \hfill & \times {\displaystyle \frac{\left|h\right(v,x_1\left(v\right),x_2\left(v\right),x_3\left(v\right)\left)\right|}{v}}dv\}+{\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)}}\{{\int }_1^{u_1}\left[{\left(log{\displaystyle \frac{u_2}{v}}\right)}^{{\alpha }_1-1}-{\left(log{\displaystyle \frac{u_1}{v}}\right)}^{{\alpha }_1-1}\right]\hfill \\ \hfill & \times {\displaystyle \frac{\left|f\right(v,x_1\left(v\right),x_2\left(v\right),x_3\left(v\right)\left)\right|}{v}}dv+{\int }_{u_1}^{u_2}{\left(log{\displaystyle \frac{u_2}{v}}\right)}^{{\alpha }_1-1}{\displaystyle \frac{\left|f\right(v,x_1\left(v\right),x_2\left(v\right),x_3\left(v\right)\left)\right|}{v}}dv\}.\hfill \end{array}\end{array}$$

Applying (35), we get

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & |{\mathcal{F}}_1(x_1\left(u_2\right),x_2\left(u_2\right),x_3\left(u_2\right))-{\mathcal{F}}_1(x_1\left(u_1\right),x_2\left(u_1\right),x_3\left(u_1\right))|\hfill \\ & \le {\displaystyle \frac{{(log{u}_2)}^{{\gamma }_1-1}-{(log{u}_1)}^{{\gamma }_1-1}}{|\Delta |}}\left\{{\displaystyle \frac{{\sigma }_1}{\Gamma ({\alpha }_1+1)}}\right[A^{✲}{B}^{✲}\sum _{k=1}^l{\displaystyle \frac{|c_k|{({\rho }_k-1)}^{{\zeta }_k}}{\Gamma ({\zeta }_k+1)}}{(log\mathcal{T})}^{1-{\gamma }_3+{\alpha }_1}+|\Delta |\hfill \\ & +{(log\mathcal{T})}^{1-{\gamma }_1+{\alpha }_1}]+{\displaystyle \frac{{\sigma }_2}{\Gamma ({\alpha }_2+1)}}\left[A^{✲}{(log\mathcal{T})}^{1-{\gamma }_2+{\alpha }_2}+\sum _{i=1}^n{\displaystyle \frac{|a_i|{(p_i-1)}^{{\eta }_i}}{\Gamma ({\eta }_i+1)}}{(log\mathcal{T})}^{1-{\gamma }_1+{\alpha }_2}\right]\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & +{\displaystyle \frac{{\sigma }_3}{\Gamma ({\alpha }_3+1)}}\left[A^{✲}{B}^{✲}{(log\mathcal{T})}^{1-{\gamma }_3+{\alpha }_3}+A^{✲}\sum _{j=1}^m{\displaystyle \frac{|b_j|{(q_j-1)}^{{\xi }_j}}{\Gamma ({\xi }_j+1)}}{(log\mathcal{T})}^{1-{\gamma }_2+{\alpha }_3}\right]\}.\hfill \end{array}\end{array}$$

(40)

Similarly, we can obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & |{\mathcal{F}}_2(x_1\left(u_2\right),x_2\left(u_2\right),x_3\left(u_2\right))-{\mathcal{F}}_2(x_1\left(u_1\right),x_2\left(u_1\right),x_3\left(u_1\right))|\hfill \\ \hfill \le & {\displaystyle \frac{{(log{u}_2)}^{{\gamma }_2-1}-{(log{u}_1)}^{{\gamma }_2-1}}{|\Delta |}}\left\{{\displaystyle \frac{{\sigma }_1}{\Gamma ({\alpha }_1+1)}}\right[B^{✲}{C}^{✲}{(log\mathcal{T})}^{1-{\gamma }_1+{\alpha }_1}+B^{✲}\sum _{k=1}^l{\displaystyle \frac{|c_k|{({\rho }_k-1)}^{{\zeta }_k}}{\Gamma ({\zeta }_k+1)}}\hfill \\ & {(log\mathcal{T})}^{1-{\gamma }_3+{\alpha }_1}]+{\displaystyle \frac{{\sigma }_2}{\Gamma ({\alpha }_2+1)}}[|\Delta |+B^{✲}{C}^{✲}\sum _{i=1}^n{\displaystyle \frac{|a_i|{(p_i-1)}^{{\eta }_i}}{\Gamma ({\eta }_i+1)}}{(log\mathcal{T})}^{1-{\gamma }_1+{\alpha }_2}\hfill \\ & +{(log\mathcal{T})}^{1-{\gamma }_2+{\alpha }_2}]+{\displaystyle \frac{{\sigma }_3}{\Gamma ({\alpha }_3+1)}}\left[B^{✲}{(log\mathcal{T})}^{1-{\gamma }_3+{\alpha }_3}+\sum _{j=1}^m{\displaystyle \frac{|b_j|{(q_j-1)}^{{\xi }_j}}{\Gamma ({\xi }_j+1)}}{(log\mathcal{T})}^{1-{\gamma }_2+{\alpha }_3}\right]\},\hfill \end{array}\end{array}$$

(41)

and

$$\begin{array}{cc}\hfill & |{\mathcal{F}}_3(x_1\left(u_2\right),x_2\left(u_2\right),x_3\left(u_2\right))-{\mathcal{F}}_3(x_1\left(u_1\right),x_2\left(u_1\right),x_3\left(u_1\right))|\hfill \\ \hfill \le & {\displaystyle \frac{{(log{u}_2)}^{{\gamma }_3-1}-{(log{u}_1)}^{{\gamma }_3-1}}{|\Delta |}}\left\{{\displaystyle \frac{{\sigma }_1}{\Gamma ({\alpha }_1+1)}}\right[\sum _{k=1}^l{\displaystyle \frac{|c_k|{({\rho }_k-1)}^{{\zeta }_k}}{\Gamma ({\zeta }_k+1)}}{(log\mathcal{T})}^{1-{\gamma }_3+{\alpha }_1}\hfill \\ & +C^{✲}{(log\mathcal{T})}^{1-{\gamma }_1+{\alpha }_1}]+{\displaystyle \frac{{\sigma }_2}{\Gamma ({\alpha }_2+1)}}[C^{✲}\sum _{i=1}^n{\displaystyle \frac{|a_i|{(p_i-1)}^{{\eta }_i}}{\Gamma ({\eta }_i+1)}}{(log\mathcal{T})}^{1-{\gamma }_1+{\alpha }_2}\hfill \\ & +A^{✲}{C}^{✲}{(log\mathcal{T})}^{1-{\gamma }_2+{\alpha }_2}]+{\displaystyle \frac{{\sigma }_3}{\Gamma ({\alpha }_3+1)}}[|\Delta |+{(log\mathcal{T})}^{1-{\gamma }_3+{\alpha }_3}\hfill \\ & +A^{✲}{C}^{✲}\sum _{j=1}^m{\displaystyle \frac{|b_j|{(q_j-1)}^{{\xi }_j}}{\Gamma ({\xi }_j+1)}}{(log\mathcal{T})}^{1-{\gamma }_2+{\alpha }_3}\left]\right\}.\hfill \end{array}$$

(42)

Since inequalities (40)–(42) do not depend on $x_1$, $x_2,$ or $x_3$ and tend to zero as $u_1\to u_2$, we conclude that the operator $\mathcal{F}$ is equicontinuous and, consequently, completely continuous.

Finally, we prove that the set $\mathcal{O}=\{(x_1,x_2,x_3)\in X\times X\times X:(x_1,x_2,x_3)=\nu \mathcal{F}(x_1,x_2,x_3),0\le \nu \le 1\}$ is bounded.

Let $(x_1,x_2,x_3)\in \mathcal{O}$ be with $(x_1,x_2,x_3)=\nu \mathcal{F}(x_1,x_2,x_3)$ for any $u\in \mathcal{E}$. Then, by $\left(H_1\right)$, we get

$$\begin{array}{cc}\hfill \parallel x_1\parallel & =\underset{u\in \mathcal{E}}{sup}|x_1\left(u\right)|\le \underset{u\in \mathcal{E}}{sup}\left|{\mathcal{F}}_1(x_1,x_2,x_3)\left(u\right)\right|\hfill \\ & \le {\displaystyle \frac{1}{|\Delta |}}\left\{\sum _{i=1}^n{\displaystyle \frac{|a_i|}{\Gamma \left({\eta }_i\right)}}{\int }_1^{p_i}{(p_i-v)}^{{\eta }_i-1}\right({\displaystyle \frac{1}{\Gamma \left({\alpha }_2\right)}}{\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_2-1}\hfill \\ & \times {\displaystyle \frac{({\omega }_0+{\omega }_1|x_1|+{\omega }_2|x_2|+{\omega }_3|x_3\left|\right)}{w}}dw)dv+{\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)}}{\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_1-1}\hfill \\ & \times {\displaystyle \frac{({\phi }_0+{\phi }_1|x_1|+{\phi }_2|x_2|+{\phi }_3|x_3\left|\right)}{v}}dv+\left|A\right|{(log\mathcal{T})}^{{\gamma }_1-{\gamma }_2}\sum _{j=1}^m{\displaystyle \frac{|b_j|}{\Gamma \left({\xi }_j\right)}}{\int }_1^{q_j}{\left(q_j-v\right)}^{{\xi }_j-1}\hfill \\ & \times \left({\displaystyle \frac{1}{\Gamma \left({\alpha }_3\right)}}{\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_3-1}{\displaystyle \frac{({\varrho }_0+{\varrho }_1|x_1|+{\varrho }_2|x_2|+{\varrho }_3|x_3\left|\right)}{w}}dw\right)dv\hfill \end{array}$$

$$\begin{array}{cc}\hfill & +{\displaystyle \frac{{\left|A\right|(log\mathcal{T})}^{{\gamma }_1-{\gamma }_2}}{\Gamma \left({\alpha }_2\right)}}{\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_2-1}{\displaystyle \frac{({\omega }_0+{\omega }_1|x_1|+{\omega }_2|x_2|+{\omega }_3|x_3\left|\right)}{v}}dv\hfill \end{array}$$

$$\begin{array}{cc}\hfill & +{\left|AB\right|(log\mathcal{T})}^{{\gamma }_1-{\gamma }_3}\sum _{k=1}^l{\displaystyle \frac{|c_k|}{\Gamma \left({\zeta }_k\right)}}{\int }_1^{{\rho }_k}{\left({\rho }_k-v\right)}^{{\zeta }_k-1}({\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)}}{\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_1-1}\hfill \\ & \times {\displaystyle \frac{({\phi }_0+{\phi }_1|x_1|+{\phi }_2|x_2|+{\phi }_3|x_3\left|\right)}{w}}dw)dv+{\displaystyle \frac{{\left|AB\right|(log\mathcal{T})}^{{\gamma }_1-{\gamma }_3}}{\Gamma \left({\alpha }_3\right)}}{\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_3-1}\hfill \\ \hfill & \times {\displaystyle \frac{({\varrho }_0+{\varrho }_1|x_1|+{\varrho }_2|x_2|+{\varrho }_3|x_3\left|\right)}{v}}dv\}+{\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)}}{\int }_1^u{\left(log{\displaystyle \frac{u}{v}}\right)}^{{\alpha }_1-1}\hfill \\ & \times {\displaystyle \frac{({\phi }_0+{\phi }_1|x_1|+{\phi }_2|x_2|+{\phi }_3|x_3\left|\right)}{v}}dv.\hfill \end{array}$$

Applying the notations in (31) and (32), we get

$$\begin{array}{cc}\hfill \parallel x_1\parallel \le & {\mathbb{L}}_1{\phi }_0+{\mathbb{M}}_1{\omega }_0+{\mathbb{N}}_1{\varrho }_0+({\mathbb{L}}_1{\phi }_1+{\mathbb{M}}_1{\omega }_1+{\mathbb{N}}_1{\varrho }_1)\parallel x_1\parallel \hfill \\ & +({\mathbb{L}}_1{\phi }_2+{\mathbb{M}}_1{\omega }_2+{\mathbb{N}}_1{\varrho }_2)\parallel x_2\parallel +({\mathbb{L}}_1{\phi }_3+{\mathbb{M}}_1{\omega }_3+{\mathbb{N}}_1{\varrho }_3)\parallel x_3\parallel ,\hfill \end{array}$$

(43)

In the same manner, we find that

$$\begin{array}{cc}\hfill \parallel x_2\parallel \le & {\mathbb{L}}_2{\phi }_0+{\mathbb{M}}_2{\omega }_0+{\mathbb{N}}_2{\varrho }_0+({\mathbb{L}}_2{\phi }_1+{\mathbb{M}}_2{\omega }_1+{\mathbb{N}}_2{\varrho }_1)\parallel x_1\parallel \hfill \\ & +({\mathbb{L}}_2{\phi }_2+{\mathbb{M}}_2{\omega }_2+{\mathbb{N}}_2{\varrho }_2)\parallel x_2\parallel +({\mathbb{L}}_2{\phi }_3+{\mathbb{M}}_2{\omega }_3+{\mathbb{N}}_2{\varrho }_3)\parallel x_3\parallel ,\hfill \end{array}$$

(44)

and

$$\begin{array}{cc}\hfill \parallel x_3\parallel \le & {\mathbb{L}}_3{\phi }_0+{\mathbb{M}}_3{\omega }_0+{\mathbb{N}}_3{\varrho }_0+({\mathbb{L}}_3{\phi }_1+{\mathbb{M}}_3{\omega }_1+{\mathbb{N}}_3{\varrho }_1)\parallel x_1\parallel \hfill \\ & +({\mathbb{L}}_3{\phi }_2+{\mathbb{M}}_3{\omega }_2+{\mathbb{N}}_3{\varrho }_2)\parallel x_2\parallel +({\mathbb{L}}_3{\phi }_3+{\mathbb{M}}_3{\omega }_3+{\mathbb{N}}_3{\varrho }_3)\parallel x_3\parallel .\hfill \end{array}$$

(45)

By adding inequalities (43)–(45) and using the notations in (31), we can alternatively find that

$$\begin{array}{cc}\hfill & \parallel (x_1,x_2,x_3)\parallel \le {\displaystyle \frac{1}{K}}\left[G_1{\phi }_0+G_2{\omega }_0+G_3{\varrho }_0\right],\hfill \end{array}$$

(46)

where $K=1-max\{{\mathbb{S}}_i,i=1,2,3\}$ and ${\mathbb{S}}_i$ is defined in (34). Thus, the set $\mathcal{O}$ is bounded. Using the Leray–Schauder alternative [37], we conclude that there is at least one fixed point for the operator $\mathcal{F}$. As a consequence, problem (1) and (2) has at least one solution on $\mathcal{E}$. The proof is complete. □

In the following result, we apply the contraction mapping principle [37] to prove the uniqueness of the solution to the considered problem.

Theorem 3.

Let $\Delta \ne 0$, where $\Delta =1-ABC$. Moreover, we suppose that:

$\left(H_2\right)f,g,h:\mathcal{E}\times \mathbb{R}\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ are continuous functions and there exist positive constants $l_1,l_2$ and $l_3$ such that for all $u\in \mathcal{E}$ and $x_i,{\overline{x}}_i\in \mathbb{R}$, $i=1,2,3$, we have

$$\begin{array}{c}\hfill |f(u,{\overline{x}}_1,{\overline{x}}_2,{\overline{x}}_3)-f(u,x_1,x_2,x_3)|\le l_1\left(\right|{\overline{x}}_1-x_1|+|{\overline{x}}_2-x_1|+|{\overline{x}}_3-x_3\left|\right),\\ \hfill |g(u,{\overline{x}}_1,{\overline{x}}_2,{\overline{x}}_3)-g(u,x_1,x_2,x_3)|\le l_2\left(\right|{\overline{x}}_1-x_1|+|{\overline{x}}_2-x_1|+|{\overline{x}}_3-x_3\left|\right),\\ \hfill |h(u,{\overline{x}}_1,{\overline{x}}_2,{\overline{x}}_3)-h(u,x_1,x_2,x_3)|\le l_3\left(\right|{\overline{x}}_1-x_1|+|{\overline{x}}_2-x_1|+|{\overline{x}}_3-x_3\left|\right),\end{array}$$

(47)

if

$$G_1{l}_1+G_2{l}_2+G_3{l}_3<1,$$

(48)

where $G_i;i=1,2,3$ are defined in (31). Then, there exists a unique solution of the problem (1) and (2) on $\mathcal{E}$.

Proof. 

Let us define ${sup}_{u\in \mathcal{E}}f(u,0,0,0)=Q_1<\infty ,{sup}_{u\in \mathcal{E}}g(u,0,0,0)=Q_2<\infty ,{sup}_{u\in \mathcal{E}}h(u,0,0,0)=Q_3<\infty ,$ and $r>0$ such that

$$r>{\displaystyle \frac{G_1{Q}_1+G_2{Q}_2+G_3{Q}_3}{1-[G_1{l}_1+G_2{l}_2+G_3{l}_3]}}.$$

(49)

First, we prove that $\mathcal{F}{B}_r\subset B_r$ such that $B_r=\{(x_1,x_2,x_3)\in X\times X\times X:\parallel (x_1,x_2,x_3)\parallel \le r\}$. For any $(x_1,x_2,x_3)\in B_r$, using assumption $\left(H_2\right)$, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \left|f\right(u,x_1\left(u\right),x_2\left(u\right),x_3\left(u\right)\left)\right|& \le |f(u,x_1\left(u\right),x_2\left(u\right),x_3\left(u\right))-f(u,0,0,0)|+Q_1\hfill \\ & \le l_1\left(\right|x_1\left(u\right)|+|x_2\left(u\right)|+|x_3\left(u\right)\left|\right)+Q_1\hfill \\ & \le l_1(\parallel x_1\parallel +\parallel x_2\parallel +\parallel x_3\parallel )+Q_1\hfill \\ & \le l_{1}r+Q_1.\hfill \end{array}\end{array}$$

(50)

Similarly, we can obtain

$$\begin{array}{c}\hfill \begin{array}{c}\hfill |g(u,x_1\left(u\right),x_2\left(u\right),x_3\left(u\right))|\le l_2(\parallel x_1\parallel +\parallel x_2\parallel +\parallel x_3\parallel )+Q_2\le l_{2}r+Q_2,\end{array}\end{array}$$

(51)

and

$$\begin{array}{c}\hfill \begin{array}{c}\hfill |h(u,x_1\left(u\right),x_2\left(u\right),x_3\left(u\right))|\le l_3(\parallel x_1\parallel +\parallel x_2\parallel +\parallel x_3\parallel )+Q_3\le l_{3}r+Q_3.\end{array}\end{array}$$

(52)

Using (50)–(52), we get

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |{\mathcal{F}}_1(x_1\left(u\right),x_2\left(u\right),x_3\left(u\right))|\le & {\displaystyle \frac{l_{1}r+Q_1}{\Gamma ({\alpha }_1+1)}}\{A^{✲}{B}^{✲}\sum _{k=1}^l{\displaystyle \frac{|c_k|{({\rho }_k-1)}^{{\zeta }_k}}{|\Delta |\Gamma ({\zeta }_k+1)}}{(log\mathcal{T})}^{{\gamma }_1-{\gamma }_3+{\alpha }_1}\hfill \\ & +{(log\mathcal{T})}^{{\alpha }_1}+{\displaystyle \frac{{(log\mathcal{T})}^{{\alpha }_1}}{|\Delta |}}\}+{\displaystyle \frac{l_{2}r+Q_2}{|\Delta |\Gamma ({\alpha }_2+1)}}\{A^{✲}{(log\mathcal{T})}^{{\gamma }_1-{\gamma }_2+{\alpha }_2}\hfill \\ & +\sum _{i=1}^n{\displaystyle \frac{|a_i|{(p_i-1)}^{{\eta }_i}}{\Gamma ({\eta }_i+1)}}{(log\mathcal{T})}^{{\alpha }_2}\}+{\displaystyle \frac{l_{3}r+Q_3}{|\Delta |\Gamma ({\alpha }_3+1)}}\{\hfill \\ & A^{✲}{B}^{✲}{(log\mathcal{T})}^{{\gamma }_1-{\gamma }_3+{\alpha }_3}+A^{✲}\sum _{j=1}^m{\displaystyle \frac{|b_j|{(q_j-1)}^{{\xi }_j}}{\Gamma ({\xi }_j+1)}}{(log\mathcal{T})}^{{\gamma }_1-{\gamma }_2+{\alpha }_3}\},\hfill \end{array}\end{array}$$

(53)

which, combined with (31), leads to

$$\begin{array}{c}\hfill \parallel {\mathcal{F}}_1(x_1,x_2,x_3)\parallel \le ({\mathbb{L}}_1{l}_1+{\mathbb{M}}_1{l}_2+{\mathbb{N}}_1{l}_3)r+{\mathbb{L}}_1{Q}_1+{\mathbb{M}}_1{Q}_2+{\mathbb{N}}_1{Q}_3.\end{array}$$

(54)

In the same way, we get

$$\begin{array}{c}\hfill \parallel {\mathcal{F}}_2(x_1,x_2,x_3)\parallel \le ({\mathbb{L}}_2{l}_1+{\mathbb{M}}_2{l}_2+{\mathbb{N}}_2{l}_3)r+{\mathbb{L}}_2{Q}_1+{\mathbb{M}}_2{Q}_2+{\mathbb{N}}_2{Q}_3,\end{array}$$

(55)

and

$$\begin{array}{c}\hfill \parallel {\mathcal{F}}_3(x_1,x_2,x_3)\parallel \le ({\mathbb{L}}_3{l}_1+{\mathbb{M}}_3{l}_2+{\mathbb{N}}_3{l}_3)r+{\mathbb{L}}_3{Q}_1+{\mathbb{M}}_3{Q}_2+{\mathbb{N}}_3{Q}_3.\end{array}$$

(56)

Based on inequalities (49) and (54)–(56),we find that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \parallel \mathcal{F}(x_1,x_2,x_3)\parallel \le & [({\mathbb{L}}_1+{\mathbb{L}}_2+{\mathbb{L}}_3)l_1+({\mathbb{M}}_1+{\mathbb{M}}_2+{\mathbb{M}}_3)l_2+({\mathbb{N}}_1+{\mathbb{N}}_2+{\mathbb{N}}_3)l_3]r\hfill \\ & +({\mathbb{L}}_1+{\mathbb{L}}_2+{\mathbb{L}}_3)Q_1+({\mathbb{M}}_1+{\mathbb{M}}_2+{\mathbb{M}}_3)Q_2+({\mathbb{N}}_1+{\mathbb{N}}_2+{\mathbb{N}}_3)Q_3\le r,\hfill \end{array}\end{array}$$

(57)

which guarantees that $\mathcal{F}(x_1,x_2,x_3)\in B_r$. Now, for any $({\overline{x}}_1,{\overline{x}}_2,{\overline{x}}_2),(x_1,x_2,x_3)\in X\times X\times X$ and $u\in \mathcal{E}$, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |{\mathcal{F}}_1& ({\overline{x}}_1,{\overline{x}}_2,{\overline{x}}_3)\left(u\right)-{\mathcal{F}}_1(x_1,x_2,x_3)\left(u\right)|\hfill \\ & \le {\displaystyle \frac{1}{|\Delta |}}\{\sum _{i=1}^n{\displaystyle \frac{|a_i|}{\Gamma \left({\eta }_i\right)}}{\int }_1^{p_i}{(p_i-s)}^{{\eta }_i-1}\times \hfill \\ & \left({\displaystyle \frac{1}{\Gamma \left({\alpha }_2\right)}}{\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_2-1}{\displaystyle \frac{|g(w,{\overline{x}}_1\left(w\right),{\overline{x}}_2\left(w\right),{\overline{x}}_2\left(w\right))-g(w,x_1\left(w\right),x_2\left(w\right),x_3\left(w\right))|}{w}}dw\right)dv\hfill \\ & +{\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)}}{\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_1-1}{\displaystyle \frac{|f(v,{\overline{x}}_1\left(v\right),{\overline{x}}_2\left(v\right),{\overline{x}}_2\left(v\right))-f(v,x_1\left(v\right),x_2\left(v\right),x_3\left(v\right))|}{v}}dv\hfill \\ & +{\left|A\right|(log\mathcal{T})}^{{\gamma }_1-{\gamma }_2}\sum _{j=1}^m{\displaystyle \frac{|b_j|}{\Gamma \left({\xi }_j\right)}}{\int }_1^{q_j}{(q_j-v)}^{{\xi }_j-1}({\displaystyle \frac{1}{\Gamma \left({\alpha }_3\right)}}{\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_3-1}\times \hfill \\ & {\displaystyle \frac{|h(w,{\overline{x}}_1\left(w\right),{\overline{x}}_2\left(w\right),{\overline{x}}_2\left(w\right))-h(w,x_1\left(w\right),x_2\left(w\right),x_3\left(w\right))|}{w}}dw)dv+{\displaystyle \frac{{\left|A\right|(log\mathcal{T})}^{{\gamma }_1-{\gamma }_2}}{\Gamma \left({\alpha }_2\right)}}\hfill \\ & \times {\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_2-1}{\displaystyle \frac{|g(v,{\overline{x}}_1\left(v\right),{\overline{x}}_2\left(v\right),{\overline{x}}_2\left(v\right))-g(v,x_1\left(v\right),x_2\left(v\right),x_3\left(v\right))|}{v}}dv\hfill \\ & +{\left|AB\right|(logT)}^{{\gamma }_1-{\gamma }_3}\sum _{k=1}^l{\displaystyle \frac{|c_k|}{\Gamma \left({\zeta }_k\right)}}{\int }_1^{{\rho }_k}{({\rho }_k-v)}^{{\zeta }_k-1}({\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)}}{\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_1-1}\hfill \\ & \times {\displaystyle \frac{|f(w,{\overline{x}}_1\left(w\right),{\overline{x}}_2\left(w\right),{\overline{x}}_2\left(w\right))-f(w,x_1\left(w\right),x_2\left(w\right),x_3\left(w\right))|}{w}}dw)dv+{\displaystyle \frac{{\left|AB\right|(log\mathcal{T})}^{{\gamma }_1-{\gamma }_3}}{\Gamma \left({\alpha }_3\right)}}\hfill \\ & \times {\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_3-1}{\displaystyle \frac{|h(v,{\overline{x}}_1\left(v\right),{\overline{x}}_2\left(v\right),{\overline{x}}_2\left(v\right))-h(v,x_1\left(v\right),x_2\left(v\right),x_3\left(v\right))|}{v}}dv\}\hfill \\ & +{\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)}}{\int }_1^u{\left(log{\displaystyle \frac{u}{v}}\right)}^{{\alpha }_1-1}{\displaystyle \frac{|f(v,{\overline{x}}_1\left(v\right),{\overline{x}}_2\left(v\right),{\overline{x}}_2\left(v\right))-f(v,x_1\left(v\right),x_2\left(v\right),x_3\left(v\right))|}{v}}dv,\hfill \end{array}\end{array}$$

Using assumption $\left(H_2\right)$, we get

$$\begin{array}{cc}\hfill |{\mathcal{F}}_1& ({\overline{x}}_1,{\overline{x}}_2,{\overline{x}}_3)\left(u\right)-{\mathcal{F}}_1(x_1,x_2,x_3)\left(u\right)|\hfill \\ & \le {\displaystyle \frac{1}{\Gamma ({\alpha }_1+1)}}\left\{{(log\mathcal{T})}^{{\alpha }_1}+{\displaystyle \frac{{(log\mathcal{T})}^{{\alpha }_1}}{|\Delta |}}+A^{✲}{B}^{✲}\sum _{k=1}^l{\displaystyle \frac{|c_k|{({\rho }_k-1)}^{{\zeta }_k}}{|\Delta |\Gamma ({\zeta }_k+1)}}{(log\mathcal{T})}^{{\gamma }_1-{\gamma }_3+{\alpha }_1}\right\}\hfill \\ & \times l_1(\parallel {\overline{x}}_1-x_1\parallel +\parallel {\overline{x}}_2-x_2\parallel +\parallel {\overline{x}}_3-x_3\parallel )\hfill \\ & +{\displaystyle \frac{1}{|\Delta |\Gamma ({\alpha }_2+1)}}\left\{A^{✲}{(log\mathcal{T})}^{{\gamma }_1-{\gamma }_2+{\alpha }_2}+\sum _{i=1}^n{\displaystyle \frac{|a_i|{(p_i-1)}^{{\eta }_i}}{\Gamma ({\eta }_i+1)}}{(log\mathcal{T})}^{{\alpha }_2}\right\}\hfill \\ & \times l_2(\parallel {\overline{x}}_1-x_1\parallel +\parallel {\overline{x}}_2-x_2\parallel +\parallel {\overline{x}}_3-x_3\parallel )\hfill \\ & +{\displaystyle \frac{1}{|\Delta |\Gamma ({\alpha }_3+1)}}\left\{A^{✲}{B}^{✲}{(log\mathcal{T})}^{{\gamma }_1-{\gamma }_3+{\alpha }_3}+A^{✲}\sum _{j=1}^m{\displaystyle \frac{|b_j|{(q_j-1)}^{{\xi }_j}}{\Gamma ({\xi }_j+1)}}{(log\mathcal{T})}^{{\gamma }_1-{\gamma }_2+{\alpha }_3}\right\}\hfill \\ & \times l_3(\parallel {\overline{x}}_1-x_1\parallel +\parallel {\overline{x}}_2-x_2\parallel +\parallel {\overline{x}}_3-x_3\parallel )\hfill \\ & \le ({\mathbb{L}}_1{l}_1+{\mathbb{M}}_1{l}_2+{\mathbb{N}}_1{l}_3)(\parallel {\overline{x}}_1-x_1\parallel +\parallel {\overline{x}}_2-x_2\parallel +\parallel {\overline{x}}_3-x_3\parallel ),\hfill \end{array}$$

which implies that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \parallel {\mathcal{F}}_1& ({\overline{x}}_1,{\overline{x}}_2,{\overline{x}}_3)-{\mathcal{F}}_1(x_1,x_2,x_3))\parallel \hfill \\ & \le ({\mathbb{L}}_1{l}_1+{\mathbb{M}}_1{l}_2+{\mathbb{N}}_1{l}_3)(\parallel {\overline{x}}_1-x_1\parallel +\parallel {\overline{x}}_2-x_2\parallel +\parallel {\overline{x}}_3-x_3\parallel ).\hfill \end{array}\end{array}$$

(58)

Similarly, we can get

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \parallel {\mathcal{F}}_2& ({\overline{x}}_1,{\overline{x}}_2,{\overline{x}}_3)-{\mathcal{F}}_2(x_1,x_2,x_3))\parallel \hfill \\ & \le ({\mathbb{L}}_2{l}_1+{\mathbb{M}}_2{l}_2+{\mathbb{N}}_2{l}_3)(\parallel {\overline{x}}_1-x_1\parallel +\parallel {\overline{x}}_2-x_2\parallel +\parallel {\overline{x}}_3-x_3\parallel ),\hfill \end{array}\end{array}$$

(59)

and

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \parallel {\mathcal{F}}_3& ({\overline{x}}_1,{\overline{x}}_2,{\overline{x}}_3)-{\mathcal{F}}_3(x_1,x_2,x_3))\parallel \hfill \\ & \le ({\mathbb{L}}_3{l}_1+{\mathbb{M}}_3{l}_2+{\mathbb{N}}_3{l}_3)(\parallel {\overline{x}}_1-x_1\parallel +\parallel {\overline{x}}_2-x_2\parallel +\parallel {\overline{x}}_3-x_3\parallel ).\hfill \end{array}\end{array}$$

(60)

It follows directly from (58)–(60) that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \parallel \mathcal{F}& ({\overline{x}}_1,{\overline{x}}_2,{\overline{x}}_3)-\mathcal{F}(x_1,x_2,x_3))\parallel \hfill \\ & \le G_1{l}_1+G_2{l}_2+G_3{l}_3\left]\right(\parallel {\overline{x}}_1-x_1\parallel +\parallel {\overline{x}}_2-x_2\parallel +\parallel {\overline{x}}_3-x_3\parallel ).\hfill \end{array}\end{array}$$

(61)

Inequality (61), together with condition (48), implies that $\mathcal{F}$ is a contraction. Therefore, by the contraction mapping principle, we conclude that the problem (1) and (2) has a unique solution on $\mathcal{E}$. This completes the proof. □

4. Stability Analysis

In this section, we discuss the stability of the solution to system (1) with the conditions in (2). We consider two types of stability: Ulam–Hyers stability and generalized Ulam–Hyers stability.

Consider the following system of inequalities:

$$\left\{\begin{array}{c}{|}^{HH}{\mathcal{D}}_{1^{+}}^{{\alpha }_1,{\tau }_1}{y}_1\left(u\right)-f(u,y_1\left(u\right),y_2\left(u\right),y_3\left(u\right))|\le {ϵ}_1,u\in \mathcal{E},\hfill \\ {|}^{HH}{\mathcal{D}}_{1^{+}}^{{\alpha }_2,{\tau }_2}{y}_2\left(u\right)-g(u,y_1\left(u\right),y_2\left(u\right),y_3\left(u\right))|\le {ϵ}_2,\hfill \\ {|}^{HH}{\mathcal{D}}_{1^{+}}^{{\alpha }_3,{\tau }_3}{y}_3\left(u\right)-h(u,y_1\left(u\right),y_2\left(u\right),y_3\left(u\right))|\le {ϵ}_3,\hfill \end{array}\right.$$

(62)

where ${ϵ}_i;i=1,2,3$ are given positive real numbers.

Remark 2.

The solution of this system of inequalities (62) with the boundary conditions (2) is a function vector $(y_1,y_2,y_3)\in X\times X\times X$ if and only if there exists a function ${\lambda }_i\in C(\mathcal{E},\mathbb{R})$ such that $|{\lambda }_i\left(u\right)|\le {ϵ}_i$ for $i=\{1,2,3\}$ and $(t,s,r)\in X\times X\times X$ satisfies the system

$$\left\{\begin{array}{c}{}^{HH}\mathcal{D}_{1^{+}}^{{\alpha }_1,{\tau }_1}{y}_1\left(u\right)=f(u,y_1\left(u\right),y_2\left(u\right),y_3\left(u\right))+{\lambda }_1\left(u\right),u\in \mathcal{E},\hfill \\ {}^{HH}\mathcal{D}_{1^{+}}^{{\alpha }_2,{\tau }_2}{y}_2\left(u\right)=g(u,y_1\left(u\right),y_2\left(u\right),y_3\left(u\right))+{\lambda }_2\left(u\right),\hfill \\ {}^{HH}\mathcal{D}_{1^{+}}^{{\alpha }_3,{\tau }_3}{y}_3\left(u\right)=h(u,y_1\left(u\right),y_2\left(u\right),y_3\left(u\right))+{\lambda }_3\left(u\right),\hfill \\ y_1\left(1\right)=0,y_1\left(\mathcal{T}\right)={\sum }_{i=1}^n{a}_i{\mathcal{I}}_{1^{+}}^{{\eta }_i}{y}_2\left(p_i\right),\hfill \\ y_2\left(1\right)=0,y_2\left(\mathcal{T}\right)={\sum }_{j=1}^m{b}_j{\mathcal{I}}_{1^{+}}^{{\xi }_j}{y}_3\left(q_j\right),\hfill \\ y_3\left(1\right)=0,y_3^{\prime }\left(1\right)=0,y_3\left(\mathcal{T}\right)={\sum }_{k=1}^l{c}_k{\mathcal{I}}_{1^{+}}^{{\zeta }_k}{y}_1\left({\rho }_k\right).\hfill \end{array}\right.$$

(63)

For the stability analysis, we first state the following definitions for the Ulam–Hyers stability of the considered problem (1) and (2) as follows:

Definition 7

([38,39]). The problem (1) and (2) is said to be Ulam–Hyers stable if, for each $ϵ=({ϵ}_1,{ϵ}_2,{ϵ}_3)>0$ and for each solution $(y_1,y_2,y_3)\in X\times X\times X$ of the system (63), there exists a unique solution $(x_1,x_2,x_3)\in X\times X\times X$ of the problem (1) and (2) satisfying

$$\parallel (y_1,y_2,y_3)-(x_1,x_2,x_3)\parallel \le c_{f,g,h}{ϵ}^T,$$

(64)

where $c_{f,g,h}=(c_f,c_g,c_h)>0$ is a constant vector.

Definition 8

([38,39]). The problem (1) and (2) is called generalized Ulam–Hyers stable if there exists a continuous vector function $\mathsf{\Psi }\left(ϵ\right):{\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ with $\mathsf{\Psi }\left(0\right)=0$ such that for each solution $(y_1,y_2,y_3)\in X\times X\times X$ of (63), there exists a unique solution $(x_1,x_2,x_3)\in X\times X\times X$ of the problem (1) and (2) satisfying

$$\parallel (y_1,y_2,y_3)-(x_1,x_2,x_3)\parallel \le \mathsf{\Psi }\left(ϵ\right).$$

(65)

The following theorem introduces the Ulam–Hyers stability of the considered problem (1) and (2).

Theorem 4.

Let $f,g,h\in C(\mathcal{E}\times \mathbb{R}\times \mathbb{R}\times \mathbb{R},\mathbb{R}).$ If assumption $\left(H_2\right)$ and the conditions (48) are satisfied, then the problem defined in (1) and (2) is Ulam–Hyers stable and hence generalized Ulam–Hyers stable in $X\times X\times X$.

Proof. 

Let $ϵ=({ϵ}_1,{ϵ}_2,{ϵ}_3)>0$ be given. From Lemma 2 and Remark 2, the solution of the system (63) can be written as

$$\begin{array}{cc}\hfill y_1\left(u\right)=& {\displaystyle \frac{{(logu)}^{{\gamma }_1-1}}{\Delta }}\left\{{(log\mathcal{T})}^{1-{\gamma }_1}\right[\sum _{i=1}^n{\displaystyle \frac{a_i}{\Gamma \left({\eta }_i\right)}}{\int }_1^{p_i}{\left(p_i-v\right)}^{{\eta }_i-1}\hfill \\ & \times \left({\displaystyle \frac{1}{\Gamma \left({\alpha }_2\right)}}{\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_2-1}{\displaystyle \frac{g(w,y_1\left(w\right),y_2\left(w\right),y_3\left(w\right))+{\lambda }_2\left(w\right)}{w}}dw\right)dv-{\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)}}\hfill \\ & \times {\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{s}}\right)}^{{\alpha }_1-1}{\displaystyle \frac{f(v,y_1\left(v\right),y_2\left(v\right),y_3\left(v\right))+{\lambda }_1\left(v\right)}{v}}dv]+A[{(log\mathcal{T})}^{1-{\gamma }_2}\hfill \\ & \times \left(\sum _{j=1}^m{\displaystyle \frac{b_j}{\Gamma \left({\xi }_j\right)}}{\int }_1^{q_j}{\left(q_j-v\right)}^{{\xi }_j-1}\right({\displaystyle \frac{1}{\Gamma \left({\alpha }_3\right)}}{\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_3-1}\hfill \\ & \times {\displaystyle \frac{h(w,y_1\left(w\right),y_2\left(w\right),y_3\left(w\right))+{\lambda }_3\left(w\right)}{w}}dw)dv-{\displaystyle \frac{1}{\Gamma \left({\alpha }_2\right)}}{\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_2-1}\hfill \\ \hfill & \times {\displaystyle \frac{g(v,y_1\left(v\right),y_2\left(v\right),y_3\left(v\right))+{\lambda }_2\left(v\right)}{v}}dv\left)\right]+AB\left[{(log\mathcal{T})}^{1-{\gamma }_3}\right(\sum _{k=1}^l{\displaystyle \frac{c_k}{\Gamma \left({\zeta }_k\right)}}{\int }_1^{{\rho }_k}{\left({\rho }_k-v\right)}^{{\zeta }_k-1}\hfill \\ & \times \left({\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)}}{\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_1-1}{\displaystyle \frac{f(w,y_1\left(w\right),y_2\left(w\right),y_3\left(w\right))+{\lambda }_1\left(w\right)}{w}}dw\right)dv\hfill \\ & -{\displaystyle \frac{1}{\Gamma \left({\alpha }_3\right)}}{\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_3-1}{\displaystyle \frac{h(v,y_1\left(v\right),y_2\left(v\right),y_3\left(v\right))+{\lambda }_3\left(v\right)}{v}}dv\left)\right]\}\hfill \\ & +{\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)}}{\int }_1^u{\left(log{\displaystyle \frac{u}{v}}\right)}^{{\alpha }_1-1}{\displaystyle \frac{f(v,y_1\left(v\right),y_2\left(v\right),y_3\left(v\right))+{\lambda }_1\left(v\right)}{v}}dv,\hfill \end{array}$$

(66)

$$\begin{array}{cc}\hfill y_2\left(u\right)=& {\displaystyle \frac{{(logu)}^{{\gamma }_2-1}}{\Delta }}\left\{BC\right[{(log\mathcal{T})}^{1-{\gamma }_1}(\sum _{i=1}^n{\displaystyle \frac{a_i}{\Gamma \left({\eta }_i\right)}}{\int }_1^{p_i}{\left(p_i-v\right)}^{{\eta }_i-1}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \times \left({\displaystyle \frac{1}{\Gamma \left({\alpha }_2\right)}}{\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_2-1}{\displaystyle \frac{g(w,y_1\left(w\right),y_2\left(w\right),y_3\left(w\right))+{\lambda }_2\left(w\right)}{w}}dw\right)dv\hfill \\ & -{\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)}}{\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_1-1}{\displaystyle \frac{f(v,y_1\left(v\right),y_2\left(v\right),y_3\left(v\right))+{\lambda }_1\left(v\right)}{v}}dv\left)\right]\hfill \\ \hfill & +{(log\mathcal{T})}^{1-{\gamma }_2}\left[\sum _{j=1}^m{\displaystyle \frac{b_j}{\Gamma \left({\xi }_j\right)}}{\int }_1^{q_j}{\left(q_j-v\right)}^{{\xi }_j-1}\right({\displaystyle \frac{1}{\Gamma \left({\alpha }_3\right)}}{\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_3-1}\hfill \\ & \times {\displaystyle \frac{h(w,y_1\left(w\right),y_2\left(w\right),y_3\left(w\right))+{\lambda }_3\left(w\right)}{w}}dw)dv-{\displaystyle \frac{1}{\Gamma \left({\alpha }_2\right)}}{\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_2-1}\hfill \\ & \times {\displaystyle \frac{g(v,y_1\left(v\right),y_2\left(v\right),y_3\left(v\right))+{\lambda }_2\left(v\right)}{v}}dv]+B[{(logT)}^{1-{\gamma }_3}(\sum _{k=1}^l{\displaystyle \frac{c_k}{\Gamma \left({\zeta }_k\right)}}{\int }_1^{{\rho }_k}{\left({\rho }_k-v\right)}^{{\zeta }_k-1}\hfill \\ & \times \left({\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)}}{\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_1-1}{\displaystyle \frac{f(w,y_1\left(w\right),y_2\left(w\right),y_3\left(w\right))+{\lambda }_1\left(w\right)}{w}}dw\right)dv\hfill \\ & -{\displaystyle \frac{1}{\Gamma \left({\alpha }_3\right)}}{\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_3-1}{\displaystyle \frac{h(v,y_1\left(v\right),y_2\left(v\right),y_3\left(v\right))+{\lambda }_3\left(v\right)}{v}}dv\left)\right]\}\hfill \\ & +{\displaystyle \frac{1}{\Gamma \left({\alpha }_2\right)}}{\int }_1^u{\left(log{\displaystyle \frac{u}{v}}\right)}^{{\alpha }_2-1}{\displaystyle \frac{g(v,y_1\left(v\right),y_2\left(v\right),y_3\left(v\right))+{\lambda }_2\left(v\right)}{v}}dv,\hfill \end{array}$$

(67)

and

$$\begin{array}{cc}\hfill y_3\left(u\right)=& {\displaystyle \frac{{(logu)}^{{\gamma }_3-1}}{\Delta }}\left\{C\right[{(log\mathcal{T})}^{1-{\gamma }_1}(\sum _{i=1}^n{\displaystyle \frac{a_i}{\Gamma \left({\eta }_i\right)}}{\int }_1^{p_i}{\left(p_i-v\right)}^{{\eta }_i-1}\hfill \\ & \times \left({\displaystyle \frac{1}{\Gamma \left({\alpha }_2\right)}}{\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_2-1}{\displaystyle \frac{g(w,y_1\left(w\right),y_2\left(w\right),y_3\left(w\right))+{\lambda }_2\left(w\right)}{w}}dw\right)dv\hfill \\ & -{\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)}}{\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_1-1}{\displaystyle \frac{f(v,y_1\left(v\right),y_2\left(v\right),y_3\left(v\right))+{\lambda }_1\left(v\right)}{v}}dv\left)\right]\hfill \\ & +AC\left[{(log\mathcal{T})}^{1-{\gamma }_2}\right(\sum _{j=1}^m{\displaystyle \frac{b_j}{\Gamma \left({\xi }_j\right)}}{\int }_1^{q_j}{\left(q_j-v\right)}^{{\xi }_j-1}({\displaystyle \frac{1}{\Gamma \left({\alpha }_3\right)}}{\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_3-1}\hfill \\ & \times {\displaystyle \frac{h(w,y_1\left(w\right),y_2\left(w\right),y_3\left(w\right))+{\lambda }_3\left(w\right)}{w}}dw)dv-{\displaystyle \frac{1}{\Gamma \left({\alpha }_2\right)}}{\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_2-1}\hfill \\ & \times {\displaystyle \frac{g(v,y_1\left(v\right),y_2\left(v\right),y_3\left(v\right))+{\lambda }_2\left(v\right)}{v}}dv\left)\right]+{(log\mathcal{T})}^{1-{\gamma }_3}[\sum _{k=1}^l{\displaystyle \frac{c_k}{\Gamma \left({\zeta }_k\right)}}\times \hfill \\ \hfill & {\int }_1^{{\rho }_k}{\left({\rho }_k-v\right)}^{{\zeta }_k-1}\left({\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)}}{\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_1-1}{\displaystyle \frac{f(w,y_1\left(w\right),y_2\left(w\right),y_3\left(w\right))+{\lambda }_1\left(w\right)}{w}}dw\right)dv\hfill \\ & -{\displaystyle \frac{1}{\Gamma \left({\alpha }_3\right)}}{\int }_1^T{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_3-1}{\displaystyle \frac{h(v,t\left(v\right),s\left(v\right),r\left(v\right))+{\lambda }_3\left(v\right)}{v}}dv\left]\right\}\hfill \\ & +{\displaystyle \frac{1}{\Gamma \left({\alpha }_3\right)}}{\int }_1^u{\left(log{\displaystyle \frac{u}{v}}\right)}^{{\alpha }_3-1}{\displaystyle \frac{h(v,y_1\left(v\right),y_2\left(v\right),y_3\left(v\right))+{\lambda }_3\left(v\right)}{v}}dv.\hfill \end{array}$$

(68)

Applying the notations in (31) and (32) with the aid of $|{\lambda }_i\left(u\right)|\le {ϵ}_i,i=1,2,3$ for any $u\in \mathcal{E}$, we get the following inequalities:

$$\begin{array}{cc}\hfill & \underset{u\in \mathcal{E}}{sup}|y_1\left(u\right)-{\displaystyle \frac{{(logu)}^{{\gamma }_1-1}}{\Delta }}\{{(log\mathcal{T})}^{1-{\gamma }_1}[\sum _{i=1}^n{\displaystyle \frac{a_i}{\Gamma \left({\eta }_i\right)}}{\int }_1^{p_i}{\left(p_i-v\right)}^{{\eta }_i-1}\hfill \\ & \times \left({\displaystyle \frac{1}{\Gamma \left({\alpha }_2\right)}}{\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_2-1}{\displaystyle \frac{g(w,y_1\left(w\right),y_2\left(w\right),y_3\left(w\right))}{w}}dw\right)dv-{\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)}}{\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_1-1}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \times {\displaystyle \frac{f(v,y_1\left(v\right),y_2\left(v\right),y_3\left(v\right))}{v}}dv]+A[{(log\mathcal{T})}^{1-{\gamma }_2}(\sum _{j=1}^m{\displaystyle \frac{b_j}{\Gamma \left({\xi }_j\right)}}{\int }_1^{q_j}{\left(q_j-s\right)}^{{\xi }_j-1}\hfill \\ & \times \left({\displaystyle \frac{1}{\Gamma \left({\alpha }_3\right)}}{\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_3-1}{\displaystyle \frac{h(w,y_1\left(w\right),y_2\left(w\right),y_3\left(w\right))}{w}}dw\right)dv-{\displaystyle \frac{1}{\Gamma \left({\alpha }_2\right)}}{\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_2-1}\hfill \\ & \times {\displaystyle \frac{g(v,y_1\left(v\right),y_2\left(v\right),y_3\left(v\right))}{v}}dv\left)\right]+AB\left[{(log\mathcal{T})}^{1-{\gamma }_3}\right(\sum _{k=1}^l{\displaystyle \frac{c_k}{\Gamma \left({\zeta }_k\right)}}{\int }_1^{{\rho }_k}{\left({\rho }_k-v\right)}^{{\zeta }_k-1}\hfill \\ & \times \left({\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)}}{\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_1-1}{\displaystyle \frac{f(w,y_1\left(w\right),y_2\left(w\right),y_3\left(w\right))}{w}}dw\right)dv-{\displaystyle \frac{1}{\Gamma \left({\alpha }_3\right)}}{\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_3-1}\hfill \\ & \times {\displaystyle \frac{h(v,y_1\left(v\right),y_2\left(v\right),y_3\left(v\right))}{v}}dv\left)\right]\}-{\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)}}{\int }_1^u{\left(log{\displaystyle \frac{u}{v}}\right)}^{{\alpha }_1-1}{\displaystyle \frac{f(v,y_1\left(v\right),y_2\left(v\right),y_3\left(v\right))}{v}}dv|\hfill \\ & \le {ϵ}_1{\mathbb{L}}_1+{ϵ}_2{\mathbb{M}}_1+{ϵ}_3{\mathbb{N}}_1,\hfill \end{array}$$

(69)

$$\begin{array}{cc}\hfill & \underset{u\in \mathcal{E}}{sup}|y_2\left(u\right)-{\displaystyle \frac{{(logu)}^{{\gamma }_2-1}}{\Delta }}\{BC\left[{(log\mathcal{T})}^{1-{\gamma }_1}\right(\sum _{i=1}^n{\displaystyle \frac{a_i}{\Gamma \left({\eta }_i\right)}}{\int }_1^{p_i}{\left(p_i-v\right)}^{{\eta }_i-1}\hfill \\ & \times \left({\displaystyle \frac{1}{\Gamma \left({\alpha }_2\right)}}{\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_2-1}{\displaystyle \frac{g(w,y_1\left(w\right),y_2\left(w\right),y_3\left(w\right))}{w}}dw\right)dv-{\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)}}{\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_1-1}\hfill \\ & \times {\displaystyle \frac{f(v,y_1\left(v\right),y_2\left(v\right),y_3\left(v\right))}{v}}dv\left)\right]+{(log\mathcal{T})}^{1-{\gamma }_2}[\sum _{j=1}^m{\displaystyle \frac{b_j}{\Gamma \left({\xi }_j\right)}}{\int }_1^{q_j}{\left(q_j-v\right)}^{{\xi }_j-1}\hfill \\ & \times \left({\displaystyle \frac{1}{\Gamma \left({\alpha }_3\right)}}{\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_3-1}{\displaystyle \frac{h(w,y_1\left(w\right),y_2\left(w\right),y_3\left(w\right))}{w}}dw\right)dv-{\displaystyle \frac{1}{\Gamma \left({\alpha }_2\right)}}{\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_2-1}\hfill \\ & \times {\displaystyle \frac{g(v,y_1\left(v\right),y_2\left(v\right),y_3\left(v\right))}{v}}dv]+B[{(log\mathcal{T})}^{1-{\gamma }_3}(\sum _{k=1}^l{\displaystyle \frac{c_k}{\Gamma \left({\zeta }_k\right)}}{\int }_1^{{\rho }_k}{\left({\rho }_k-v\right)}^{{\zeta }_k-1}\hfill \\ & \times \left({\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)}}{\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_1-1}{\displaystyle \frac{f(w,y_1\left(w\right),y_2\left(w\right),y_3\left(w\right))}{w}}dw\right)dv-{\displaystyle \frac{1}{\Gamma \left({\alpha }_3\right)}}{\int }_1^T{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_3-1}\hfill \\ & \times {\displaystyle \frac{h(v,y_1\left(v\right),y_2\left(v\right),y_3\left(v\right))}{v}}dv\left)\right]\}+{\displaystyle \frac{1}{\Gamma \left({\alpha }_2\right)}}{\int }_1^u{\left(log{\displaystyle \frac{u}{v}}\right)}^{{\alpha }_2-1}{\displaystyle \frac{g(v,y_1\left(v\right),y_2\left(v\right),y_3\left(v\right))}{v}}dv|\hfill \\ & \le {ϵ}_1{\mathbb{L}}_2+{ϵ}_2{\mathbb{M}}_2+{ϵ}_3{\mathbb{N}}_2,\hfill \end{array}$$

(70)

and

$$\begin{array}{cc}\hfill & \underset{u\in \mathcal{E}}{sup}|y_3\left(u\right)-{\displaystyle \frac{{(logu)}^{{\gamma }_3-1}}{\Delta }}\{C\left[{(log\mathcal{T})}^{1-{\gamma }_1}\right(\sum _{i=1}^n{\displaystyle \frac{a_i}{\Gamma \left({\eta }_i\right)}}{\int }_1^{p_i}{\left(p_i-v\right)}^{{\eta }_i-1}\hfill \\ & \times \left({\displaystyle \frac{1}{\Gamma \left({\alpha }_2\right)}}{\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_2-1}{\displaystyle \frac{g(w,y_1\left(w\right),y_2\left(w\right),y_3\left(w\right))}{w}}dw\right)dv-{\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)}}{\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{s}}\right)}^{{\alpha }_1-1}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \times {\displaystyle \frac{f(v,y_1\left(v\right),y_2\left(v\right),y_3\left(v\right))}{v}}dv\left)\right]+AC\left[{(logT)}^{1-{\gamma }_2}\right(\sum _{j=1}^m{\displaystyle \frac{b_j}{\Gamma \left({\xi }_j\right)}}{\int }_1^{q_j}{\left(q_j-v\right)}^{{\xi }_j-1}\hfill \\ \hfill & \times \left({\displaystyle \frac{1}{\Gamma \left({\alpha }_3\right)}}{\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_3-1}{\displaystyle \frac{h(w,y_1\left(w\right),y_2\left(w\right),y_3\left(w\right))}{w}}dw\right)dv-{\displaystyle \frac{1}{\Gamma \left({\alpha }_2\right)}}{\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_2-1}\hfill \\ \hfill & \times {\displaystyle \frac{g(v,y_1\left(v\right),y_2\left(v\right),y_3\left(v\right))}{v}}dv\left)\right]+{(log\mathcal{T})}^{1-{\gamma }_3}[\sum _{k=1}^l{\displaystyle \frac{c_k}{\Gamma \left({\zeta }_k\right)}}{\int }_1^{{\rho }_k}{\left({\rho }_k-v\right)}^{{\zeta }_k-1}\hfill \\ & \times \left({\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)}}{\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_1-1}{\displaystyle \frac{f(w,y_1\left(w\right),y_2\left(w\right),y_3\left(w\right))}{w}}dw\right)dv-{\displaystyle \frac{1}{\Gamma \left({\alpha }_3\right)}}{\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_3-1}\hfill \\ \hfill & \times {\displaystyle \frac{h(v,y_1\left(v\right),y_2\left(v\right),y_3\left(v\right))}{v}}dv\left]\right\}+{\displaystyle \frac{1}{\Gamma \left({\alpha }_3\right)}}{\int }_1^u{\left(log{\displaystyle \frac{u}{v}}\right)}^{{\alpha }_3-1}{\displaystyle \frac{h(v,y_1\left(v\right),y_2\left(v\right),y_3\left(v\right))}{v}}dv|\hfill \\ & \le {ϵ}_1{\mathbb{L}}_3+{ϵ}_2{\mathbb{M}}_3+{ϵ}_3{\mathbb{N}}_3.\hfill \end{array}$$

(71)

By assumption $\left(H_2\right)$, we can find that

$$\begin{array}{cc}\parallel \hfill & y_1-x_1\parallel =\underset{u\in \mathcal{E}}{sup}|y_1\left(u\right)-x_1\left(u\right)|\hfill \\ & \le {ϵ}_1{\mathbb{L}}_1+{ϵ}_2{\mathbb{M}}_1+{ϵ}_3{\mathbb{N}}_1+{\displaystyle \frac{1}{|\Delta |}}\left\{\sum _{i=1}^n{\displaystyle \frac{|a_i|}{\Gamma \left({\eta }_i\right)}}{\int }_1^{p_i}{(p_i-v)}^{{\eta }_i-1}\right({\displaystyle \frac{1}{\Gamma \left({\alpha }_2\right)}}{\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_2-1}\hfill \\ & \times {\displaystyle \frac{|g(w,y_1\left(w\right),y_2\left(w\right),y_3\left(w\right))-g(w,x_1\left(w\right),x_2\left(w\right),x_3\left(w\right))|}{w}}dw)dv+{\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)}}\hfill \\ & \times {\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_1-1}{\displaystyle \frac{|f(v,y_1\left(v\right),y_2\left(v\right),y_3\left(v\right))-f(v,x_1\left(v\right),x_2\left(v\right),x_3\left(v\right))|}{v}}dv\hfill \\ & +{\left|A\right|(log\mathcal{T})}^{{\gamma }_1-{\gamma }_2}\sum _{j=1}^m{\displaystyle \frac{|b_j|}{\Gamma \left({\xi }_j\right)}}{\int }_1^{q_j}{(q_j-v)}^{{\xi }_j-1}({\displaystyle \frac{1}{\Gamma \left({\alpha }_3\right)}}{\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_3-1}\hfill \\ & \times {\displaystyle \frac{|h(w,y_1\left(w\right),y_2\left(w\right),y_3\left(w\right))-h(w,x_1\left(w\right),x_2\left(w\right),x_3\left(w\right))|}{w}}dw)dv+{\displaystyle \frac{{\left|A\right|(log\mathcal{T})}^{{\gamma }_1-{\gamma }_2}}{\Gamma \left({\alpha }_2\right)}}\hfill \\ & \times {\int }_1^v{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_2-1}{\displaystyle \frac{|g(v,y_1\left(v\right),y_2\left(v\right),y_3\left(v\right))-g(v,x_1\left(v\right),x_2\left(v\right),x_3\left(v\right))|}{v}}dv\hfill \\ & +{\left|AB\right|(logT)}^{{\gamma }_1-{\gamma }_3}\sum _{k=1}^l{\displaystyle \frac{|c_k|}{\Gamma \left({\zeta }_k\right)}}{\int }_1^{{\rho }_k}{({\rho }_k-v)}^{{\zeta }_k-1}({\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)}}{\int }_1^v{\left(log{\displaystyle \frac{v}{w}}\right)}^{{\alpha }_1-1}\hfill \\ & \times {\displaystyle \frac{|f(w,y_1\left(w\right),y_2\left(w\right),y_3\left(w\right))-f(w,x_1\left(w\right),x_2\left(w\right),x_3\left(w\right))|}{w}}dw)dv+{\displaystyle \frac{{\left|AB\right|(log\mathcal{T})}^{{\gamma }_1-{\gamma }_3}}{\Gamma \left({\alpha }_3\right)}}\hfill \\ & \times {\int }_1^{\mathcal{T}}{\left(log{\displaystyle \frac{\mathcal{T}}{v}}\right)}^{{\alpha }_3-1}{\displaystyle \frac{|h(v,y_1\left(v\right),y_2\left(v\right),y_3\left(v\right))-h(v,x_1\left(v\right),x_2\left(v\right),x_3\left(v\right))|}{v}}dv\}\hfill \\ & +{\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)}}{\int }_1^u{\left(log{\displaystyle \frac{u}{v}}\right)}^{{\alpha }_1-1}{\displaystyle \frac{|f(v,y_1\left(v\right),y_2\left(v\right),y_3\left(v\right))-f(v,x_1\left(v\right),x_2\left(v\right),x_3\left(v\right))|}{v}}dv\hfill \\ \hfill \le & {ϵ}_1{\mathbb{L}}_1+{ϵ}_2{\mathbb{M}}_1+{ϵ}_3{\mathbb{N}}_1+({\mathbb{L}}_1{l}_1+{\mathbb{M}}_1{l}_2+{\mathbb{N}}_1{l}_3)(\parallel y_1-x_1\parallel +\parallel y_2-x_2\parallel +\parallel y_3-x_3\parallel ).\hfill \end{array}$$

(72)

In the same manner, we can obtain

$$\begin{array}{cc}\hfill \parallel y_2-x_2\parallel \le & {ϵ}_1{\mathbb{L}}_2+{ϵ}_2{\mathbb{M}}_2+{ϵ}_3{\mathbb{N}}_2\hfill \\ & +({\mathbb{L}}_2{l}_1+{\mathbb{M}}_2{l}_2+{\mathbb{N}}_2{l}_3)(\parallel y_1-x_1\parallel +\parallel y_2-x_2\parallel +\parallel y_3-x_3\parallel ),\hfill \end{array}$$

(73)

and

$$\begin{array}{cc}\hfill \parallel y_3-x_3\parallel \le & {ϵ}_1{\mathbb{L}}_3+{ϵ}_2{\mathbb{M}}_3+{ϵ}_3{\mathbb{N}}_3\hfill \\ & +({\mathbb{L}}_3{l}_1+{\mathbb{M}}_3{l}_2+{\mathbb{N}}_3{l}_3)(\parallel y_1-x_1\parallel +\parallel y_2-x_2\parallel +\parallel y_3-x_3\parallel ).\hfill \end{array}$$

(74)

Therefore, by adding inequalities (72)–(74), we get

$$\begin{array}{cc}\hfill \parallel & (y_1,y_2,y_3)-(x_1,x_2,x_3)\parallel \le {ϵ}_1{G}_1+{ϵ}_2{G}_2+{ϵ}_3{G}_3\hfill \\ & +[G_1{l}_1+G_2{l}_2+G_3{l}_3]\times (\parallel y_1-x_1\parallel +\parallel y_2-x_2\parallel +\parallel y_3-x_3\parallel ),\hfill \end{array}$$

(75)

where $G_i;i=1,2,3$ are defined in (31). Consequently, we have

$$\begin{array}{cc}\hfill \parallel & (y_1,y_2,y_3)-(x_1,x_2,x_3)\parallel \le {\displaystyle \frac{G_1{ϵ}_1+G_2{ϵ}_2+G_3{ϵ}_3}{\mathsf{\Theta }}},\hfill \end{array}$$

(76)

where

$$\mathsf{\Theta }=1-\left(G_1{l}_1+G_2{l}_2+G_3{l}_3\right).$$

(77)

The condition (48) guarantees that all ${\displaystyle \frac{G_i}{\mathsf{\Theta }}}>0;i=1,2,3$. Now, for each $ϵ=({ϵ}_1,{ϵ}_2,{ϵ}_3)$, we can define

$$c_{f,g,h}=(c_f,c_g,c_h):=({\displaystyle \frac{G_1}{\mathsf{\Theta }}},{\displaystyle \frac{G_2}{\mathsf{\Theta }}},{\displaystyle \frac{G_3}{\mathsf{\Theta }}})>0,$$

(78)

to get

$$\begin{array}{cc}\hfill \parallel (y_1,y_2,y_3)-(x_1,x_2,x_3)\parallel & =\parallel y_1-x_1\parallel +\parallel y_2-x_2\parallel +\parallel y_3-x_3\parallel \hfill \\ & \le c_f{ϵ}_1+c_g{ϵ}_2+c_h{ϵ}_3=c_{f,g,h}{ϵ}^T.\hfill \end{array}$$

Hence, the problem (1) and (2) is Ulam–Hyers stable. Furthermore, we can define a function

$$\mathsf{\Psi }\left(ϵ\right)=c_{f,g,h}{ϵ}^T,$$

(79)

where the constant vector $c_{f,g,h}$ is defined in (78). Since the function $\mathsf{\Psi }\left(ϵ\right)$ vanishes only when $ϵ=(0,0,0)$, then the problem (1) and (2) is generalized Ulam–Hyers stable, since

$$\parallel (y_1,y_2,y_3)-(x_1,x_2,x_3)\parallel \le \mathsf{\Psi }\left(ϵ\right).$$

(80)

This completes the proof. □

5. Applications

This section supports our results with illustrative examples.

Example 1.

Consider the following tripled fractional differential system:

$$\left\{\begin{array}{c}{}^{HH}\mathcal{D}_{1^{+}}^{\frac{3}{2},\frac{1}{2}}{x}_1\left(u\right)=\frac{1}{2+u}+e^{-4u}{x}_{1}sin{x}_2+\frac{1}{30}{x}_{2}cos{x}_3+\frac{x_3}{45+u},u\in [1,5],\hfill \\ {}^{HH}\mathcal{D}_{1^{+}}^{\frac{5}{4},\frac{3}{4}}{x}_2\left(u\right)=\frac{1}{6+u}+\frac{e^{-6u}}{5}{x}_1+\frac{e^{-8u}}{5}{x}_2{tan}^{-1}{x}_3+\frac{x_{3}sin{x}_1}{63},\hfill \\ {}^{HH}\mathcal{D}_{1^{+}}^{\frac{5}{2},1}{x}_3\left(u\right)=\sqrt{u+2}+\frac{1}{9}{x}_1+e^{-10u}{x}_{2}cos{x}_1+\frac{x_{3}sin{x}_2}{10+\sqrt{u}},\hfill \\ x_1\left(1\right)=0,x_1\left(5\right)=15{\mathcal{I}}_{1^{+}}^{\frac{7}{4}}{x}_2\left(\frac{11}{10}\right)+7{\mathcal{I}}_{1^{+}}^{\frac{5}{2}}{x}_2\left(\frac{13}{10}\right),\hfill \\ x_2\left(1\right)=0,x_2\left(5\right)=7{\mathcal{I}}_{1^{+}}^{\frac{7}{4}}{x}_3\left(\frac{9}{4}\right)+3{\mathcal{I}}_{1^{+}}^{\frac{13}{8}}{x}_3\left(\frac{14}{5}\right),\hfill \\ x_3\left(1\right)=0,x_3^{\prime }\left(1\right)=0,x_3\left(5\right)=\frac{1}{10}{\mathcal{I}}_{1^{+}}^2{x}_1\left(3\right)+9{\mathcal{I}}_{1^{+}}^{\frac{4}{3}}{x}_1\left(\frac{17}{4}\right).\hfill \end{array}\right.$$

(81)

Here, ${\alpha }_1={\displaystyle \frac{3}{2}},{\alpha }_2={\displaystyle \frac{5}{4}},{\alpha }_3={\displaystyle \frac{5}{2}},{\tau }_1={\displaystyle \frac{1}{2}},{\tau }_2={\displaystyle \frac{3}{4}},{\tau }_3=1,{\gamma }_1={\displaystyle \frac{7}{4}},{\gamma }_2={\displaystyle \frac{29}{16}},{\gamma }_3=3,$$\mathcal{T}=5,a_1=15,a_2=7,{\eta }_1={\displaystyle \frac{7}{4}},{\eta }_2={\displaystyle \frac{5}{2}},p_1={\displaystyle \frac{11}{10}},p_2={\displaystyle \frac{13}{10}},b_1=7,b_2=3,{\xi }_1={\displaystyle \frac{7}{4}},$${\xi }_2={\displaystyle \frac{13}{8}},q_1={\displaystyle \frac{9}{4}},q_2={\displaystyle \frac{14}{5}},c_1={\displaystyle \frac{1}{10}},c_2=9,{\zeta }_1=2,{\zeta }_2={\displaystyle \frac{4}{3}},{\rho }_1=3,$ and ${\rho }_2={\displaystyle \frac{17}{4}}$.

Using the above data, we can calculate all upper bounds defined in (32) as follows:

${\mathbb{L}}_1\backsimeq 3.0985,{\mathbb{L}}_2\backsimeq 11.4898,{\mathbb{L}}_3\backsimeq 0.9751,{\mathbb{M}}_1\backsimeq 0.0075,{\mathbb{M}}_2\backsimeq 3.2277,{\mathbb{M}}_3\backsimeq 0.2739,{\mathbb{N}}_1\backsimeq 0.0545,{\mathbb{N}}_2\backsimeq 0.2022,$ and $N_3\backsimeq 1.9948$.

In addition, the defined functions $f,g$ and h in (81) satisfy condition $\left(H_1\right)$, with ${\phi }_0={\displaystyle \frac{1}{3}},{\phi }_1={\displaystyle \frac{1}{e^4}},$${\phi }_2={\displaystyle \frac{1}{30}},{\phi }_3={\displaystyle \frac{1}{46}},{\omega }_0={\displaystyle \frac{1}{7}},{\omega }_1={\displaystyle \frac{1}{5e^6}},{\omega }_2={\displaystyle \frac{1}{5e^8}},{\omega }_3={\displaystyle \frac{1}{63}},{\varrho }_0=\sqrt{7},{\varrho }_1={\displaystyle \frac{1}{9}},{\varrho }_2={\displaystyle \frac{1}{e^{10}}},$ and ${\varrho }_3={\displaystyle \frac{1}{11}}$.

Consequently, we find that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & {\mathbb{S}}_1\simeq 0.5370,{\mathbb{S}}_2\simeq 0.5191,{\mathbb{S}}_3\simeq 0.5987,\hfill \end{array}\end{array}$$

(82)

which shows that all values ${\mathbb{S}}_i,i=1,2,3$ are still less than 1, which proves the existence of a solution to the problem (81) according to Theorem 2.

Example 2.

Consider the following Hilfer–Hadamard fractional differential system:

$$\left\{\begin{array}{c}{}^{HH}\mathcal{D}_{1^{+}}^{{\alpha }_1,{\tau }_1}{x}_1\left(u\right)=cosu+\frac{e^{-3u}}{\sqrt{u^2+24}}\left(sin{x}_1+\frac{|x_2|}{1+|x_2|}\right),u\in [1,5],\hfill \\ {}^{HH}\mathcal{D}_{1^{+}}^{{\alpha }_2,{\tau }_2}{x}_2\left(u\right)=e^{-u}+\frac{1}{{(u+1)}^3}cos{x}_1+\frac{1}{u+7}{tan}^{-1}{x}_2+\frac{1}{8}\left|x_3\right|,\hfill \\ {}^{HH}\mathcal{D}_{1^{+}}^{{\alpha }_3,{\tau }_3}{x}_3\left(u\right)={tan}^{-1}u+\frac{e^{-2u}}{3}\left(x_1+sin{x}_3\right),\hfill \\ x_1\left(1\right)=0,x_1\left(5\right)=15{\mathcal{I}}_{1^{+}}^{\frac{7}{4}}{x}_2\left(\frac{11}{10}\right)+7{\mathcal{I}}_{1^{+}}^{\frac{5}{2}}{x}_2\left(\frac{13}{10}\right),\hfill \\ x_2\left(1\right)=0,x_2\left(5\right)=7{\mathcal{I}}_{1^{+}}^{\frac{7}{4}}{x}_3\left(\frac{9}{4}\right)+3{\mathcal{I}}_{1^{+}}^{\frac{13}{8}}{x}_3\left(\frac{14}{5}\right),\hfill \\ x_3\left(1\right)=0,x_3^{\prime }\left(1\right)=0,x_3\left(5\right)=\frac{1}{10}{\mathcal{I}}_{1^{+}}^2{x}_1\left(3\right)+9{\mathcal{I}}_{1^{+}}^{\frac{4}{3}}{x}_1\left(\frac{17}{4}\right).\hfill \end{array}\right.$$

(83)

Comparing this problem with the problem (1) and (2), we have $\mathcal{T}=5,a_1=15,a_2=7,$${\eta }_1={\displaystyle \frac{7}{4}},{\eta }_2={\displaystyle \frac{5}{2}},p_1={\displaystyle \frac{11}{10}},p_2={\displaystyle \frac{13}{10}},b_1=7,b_2=3,{\xi }_1={\displaystyle \frac{7}{4}},{\xi }_2={\displaystyle \frac{13}{8}},q_1={\displaystyle \frac{9}{4}},q_2={\displaystyle \frac{14}{5}},$$c_1={\displaystyle \frac{1}{10}},c_2=9,{\zeta }_1=2,{\zeta }_2={\displaystyle \frac{4}{3}},{\rho }_1=3,$ and ${\rho }_2={\displaystyle \frac{17}{4}}$.

In addition, all defined functions $f,g$ and h in (83) satisfy condition ($H_2$) with $l_1={\displaystyle \frac{1}{5e^3}},l_2={\displaystyle \frac{1}{8}},$$l_3={\displaystyle \frac{1}{3e^2}}$.

Now, for $i=1,2,3$, we will fix all values of the parameters ${\tau }_i$ and examine the results when ${\alpha }_i$ changes, as shown in Table 1:

Table 1. Numerical illustration of problem (83).

${\mathit{\tau }}_1=\frac{1}{2},{\mathit{\tau }}_2=\frac{3}{4},{\mathit{\tau }}_3=1$	${\mathit{\alpha }}_1=\frac{9}{8},{\mathit{\alpha }}_2=\frac{21}{20},{\mathit{\alpha }}_3=\frac{11}{4}$	${\mathit{\alpha }}_1=\frac{3}{2},{\mathit{\alpha }}_2=\frac{19}{10},{\mathit{\alpha }}_3=\frac{17}{8}$	${\mathit{\alpha }}_1=\frac{7}{4},{\mathit{\alpha }}_2=\frac{11}{10},{\mathit{\alpha }}_3=\frac{11}{5}$
${\mathbb{L}}_1$	3.2524088700	3.0985360774	2.8845508961
${\mathbb{M}}_1$	0.0091949174	0.0081914168	0.0092299795
${\mathbb{N}}_1$	0.0828257723	0.1208554345	0.1163279406
${\mathbb{L}}_2$	13.9112322005	13.9538292053	13.4585604732
${\mathbb{M}}_2$	3.2528362165	2.7267134656	3.2537913988
${\mathbb{N}}_2$	0.3071287307	0.4481476618	0.4313591259
${\mathbb{L}}_3$	1.1805965109	1.1842115663	1.1421798808
${\mathbb{M}}_3$	0.3364353800	0.2997180174	0.3377182748
${\mathbb{N}}_3$	1.6881838446	2.4633177139	2.3710367504
$G_1{l}_1+G_2{l}_2+G_3{l}_3$	0.7061565695	0.7138407966	0.6847623796
$\mathsf{\Theta }$	0.2938434305	0.2861592034	0.3152376204

Table 1 shows the following:

1. 

Theorem 3 guarantees the uniqueness of the solution of problem (83), since hypothesis $\left(H_2\right)$ and condition (48) are satisfied.

2. 

Since the value of Θ remains positive for different values of the fractional order ${\alpha }_i$, the problem (83) is Ulam–Hyers stable.

3. 

For nonzero ϵ, the problem (83) satisfies generalized Ulam–Hyers stability since the positivity of Θ implies the positivity of function $\mathsf{\Psi }\left(ϵ\right)$.

Therefore, the problem (83) has a unique solution.

6. Conclusions

We have ensured the existence and uniqueness of solutions to a tripled system of Hilfer–Hadamard type by applying Banach contraction mapping and the Leray–Schauder nonlinear alternative. In addition, we have proven the Ulam–Hyers stability of the studied system. We must emphasize that the results of the present work are new and interesting. Furthermore, the study of the Hilfer–Hadamard derivative is expansive and can be extended to more generalized operators. Indeed, for the continuous-time case, the manuscript’s problem can be formulated in the $\psi$-Hilfer derivative since the Hilfer-Hadamard derivative is given as a special case when $\psi \left(t\right)=lnt$ [40]. For the discrete-time, our manuscript can be formulated in the sense of Hilfer fractional difference equations [41].