A Coupled System of Fractional Difference Equations with Nonlocal Fractional Sum Boundary Conditions on the Discrete Half-Line

Abstract

In this article, we propose a coupled system of fractional difference equations with nonlocal fractional sum boundary conditions on the discrete half-line and study its existence result by using Schauder’s fixed point theorem. An example is provided to illustrate the results.

1. Introduction

Recently, many mathematicians and researchers have extensively studied fractional difference calculus since this subject can be used for describing many problems of real-world phenomena such as mechanical, control systems, flow in porous media, and electrical networks (see [1,2] and the references therein). The basic definitions and properties of fractional difference calculus are given in the book [3]. The applications and developments of the theory can be found in [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47] and the references cited therein. For example, Ferreira [20] studied the fractional difference equation of order less than one. Goodrich [22] presented the fractional difference equation of order $1<\alpha \le 2$ with a constant boundary condition. Chen et al. [28] proposed the initial value problem of order less than one. Chen and Zhou [29] studied the antiperiodic boundary value problem of order $1<\alpha \le 2$. Sitthiwirattham et al. [38] initiated the study of the fractional sum boundary value problem of order $1<\alpha \le 2$. Sitthiwirattham [40] proposed the sequential fractional difference equation with the fractional sum boundary condition. We observe that these research works are fractional problems containing only one equation.

The study of coupled systems of fractional differential equations is an important topic in this area (see [48,49,50,51,52,53] and the references cited therein), and a recent example of the application of systems of fractional difference equations is [54].

For the boundary value problems for systems of discrete fractional equations, there are some studies in this area (see [55,56,57,58,59,60] and the references cited therein).

Pan et al. [55] proposed the system of discrete fractional difference equations as given by:

$$\begin{array}{ccc}\hfill -{\Delta }^{\nu }{y}_1\left(t\right)& =& f\left(y_1(t+\nu ),y_2(t+\mu -1)\right),\hfill \\ \hfill -{\Delta }^{\mu }{y}_2\left(t\right)& =& g\left(y_1(t+\nu ),y_2(t+\mu -1)\right),\hfill \end{array}$$

(1)

for $t\in {\mathbb{N}}_{0,b+1}:=\{0,1,2,\dots ,b+1\}$, with the difference boundary conditions:

$$\begin{array}{c}{y}_1(\nu -2)=\Delta y_1(\nu +b)=0,\\ y_2(\mu -2)=\Delta y_2(\mu +b)=0,\end{array}$$

(2)

where $b\in {\mathbb{N}}_0:=\mathbb{N}\cup \{0\}$; $1<\mu ,\nu \le 2$; $0<\beta \le 1$; and $f,g:\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ are continuous functions. ${\Delta }^{\nu }$ and ${\Delta }^{\mu }$ are fractional difference operator of order $\nu$ and $\mu$, respectively.

In 2015, Goodrich [58] discussed the coupled system of discrete fractional difference equations:

$$\begin{array}{ccc}\hfill -{\Delta }^{-\nu }x\left(t\right)& =& {\lambda }_{1}f\left(t+\nu -1,y(t+\mu -1)\right),t\in {\mathbb{N}}_{0,b+1},\hfill \\ \hfill -{\Delta }^{-\mu }y\left(t\right)& =& {\lambda }_{2}g\left(t+\mu -1,y(t+\nu -1)\right),\hfill \end{array}$$

(3)

with the nonlinearities satisfying no growth conditions:

$$\begin{array}{cc}x(\nu -2)=H_1\left({\sum }_{i=1}^n{a}_{i}y\left({\xi }_i\right)\right),\hfill & x(\nu +b+1)=0,\hfill \\ y(\mu -2)=H_2\left({\sum }_{j=1}^m{b}_{j}x\left({\zeta }_i\right)\right),\hfill & y(\mu +b+1)=0,\hfill \end{array}$$

(4)

where $1<\nu \le 2$; $1<\mu \le 2$; ${\lambda }_1,{\lambda }_2>0$; ${\left\{a_i\right\}}_{i=1}^n,{\left\{b_j\right\}}_{j=1}^m\subseteq (0,\infty )$; and $H_1,H_2:[0,\infty )\to [0,\infty )$ are continuous functions.

In this paper, we considered the coupled system of fractional difference equations:

$$\left\{\begin{array}{c}{\Delta }^{{\alpha }_1}{u}_1\left(t\right)=F_1\left(t+{\alpha }_1-1,t+{\alpha }_2-1,{\Delta }^{{\beta }_1}{u}_1(t+{\alpha }_1-{\beta }_1),u_2(t+{\alpha }_2-1)\right),\hfill \\ {\Delta }^{{\alpha }_2}{u}_2\left(t\right)=F_2\left(t+{\alpha }_1-1,t+{\alpha }_2-1,{\Delta }^{{\beta }_2}{u}_2(t+{\alpha }_2-{\beta }_1),u_1(t+{\alpha }_1-1)\right),\hfill \end{array}\right.$$

(5)

for $t\in {\mathbb{N}}_0$, subject to the nonlocal fractional sum boundary conditions on the discrete half-line ${\mathbb{N}}_0$:

$$\left\{\begin{array}{c}{u}_1({\alpha }_1-2)={\varphi }_1(u_1,u_2),\hfill \\ u_2({\alpha }_2-2)={\varphi }_2(u_1,u_2),\hfill \\ \underset{t\to \infty }{lim}{u}_1(t+{\alpha }_1-2)={\lambda }_2{\Delta }^{-{\theta }_2}{g}_2({\eta }_2+{\theta }_2)u_2({\eta }_2+{\theta }_2),\hfill \\ \underset{t\to \infty }{lim}{u}_2(t+{\alpha }_2-2)={\lambda }_1{\Delta }^{-{\theta }_1}{g}_1({\eta }_1+{\theta }_1)u_1({\eta }_1+{\theta }_1).\hfill \end{array}\right.$$

(6)

For $i=1,2$, ${\alpha }_i\in (1,2];{\nu }_i,{\gamma }_i,{\theta }_i\in (0,1];{\beta }_i\in ({\alpha }_i-1,{\alpha }_i);{\lambda }_1,{\lambda }_2>0$, and ${\eta }_i\in {\mathbb{N}}_{{\alpha }_i-1,T+{\alpha }_i-1}$ are given constants; $F_i\in C\left({\mathbb{N}}_{{\alpha }_1-2}\times {\mathbb{N}}_{{\alpha }_2-2}\times {\mathbb{R}}^2,\mathbb{R}\right)$ and $g_i\in C\left({\mathbb{N}}_{{\alpha }_i-2,T+{\alpha }_i},{\mathbb{R}}^{+}\right)$ are given functions; ${\varphi }_i(u_1,u_2)$ are given functionals; and ${\Delta }^{-{\theta }_i}$ are fractional sums of order ${\theta }_i$.

The goal of this study is to show the existence of solutions of the governing problem (5) and (6). The paper is structured as follows. Some definitions and basic lemmas are recalled in Section 2. In Section 3, we prove the existence of solutions of the boundary value problem (5) by employing Schauder’s fixed point theorem. Finally, we present an example to illustrate our result in the last section.

2. Preliminaries

In what follows, the notation, definitions, and lemmas used in the main results are given.

Definition 1.

The generalized falling function is defined by ${\displaystyle t^{\underline{\alpha }}:=\frac{\mathsf{\Gamma }(t+1)}{\mathsf{\Gamma }(t+1-\alpha )}}$, for any t and α for which the right-hand side is defined. If $t+1-\alpha$ is a pole of the Gamma function and $t+1$ is not a pole, then $t^{\underline{\alpha }}=0$.

Lemma 1.

[4] Assume the falling factorial functions are well defined. If $t\le r$, then $t^{\underline{\alpha }}\le r^{\underline{\alpha }}$ for any $\alpha >0$.

Definition 2.

For $\alpha >0$ and f defined on ${\mathbb{N}}_a$, the α-order fractional sum of f is defined by:

$${\Delta }^{-\alpha }f\left(t\right):=\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}\sum _{s=a}^{t-\alpha }{(t-\sigma \left(s\right))}^{\underline{\alpha -1}}f\left(s\right),$$

where $t\in {\mathbb{N}}_{a+\alpha }$ and $\sigma \left(s\right)=s+1$.

Definition 3.

For $\alpha >0$ and f defined on ${\mathbb{N}}_a$, the α-order Riemann–Liouville fractional difference of f is defined by:

$${\Delta }^{\alpha }f\left(t\right):={\Delta }^N{\Delta }^{-(N-\alpha )}f\left(t\right)=\frac{1}{\mathsf{\Gamma }(-\alpha )}\sum _{s=a}^{t+\alpha }{(t-\sigma \left(s\right))}^{\underline{-\alpha -1}}f\left(s\right),$$

where $t\in {\mathbb{N}}_{a+N-\alpha }$ and $N\in \mathbb{N}$ are chosen so that $0\le N-1<\alpha \le N$.

Lemma 2.

[4] Let $0\le N-1<\alpha \le N.$ Then,

$${\Delta }^{-\alpha }{\Delta }^{\alpha }y\left(t\right)=y\left(t\right)+C_1{t}^{\underline{\alpha -1}}+C_2{t}^{\underline{\alpha -2}}+\dots +C_N{t}^{\underline{\alpha -N}},$$

for some $C_i\in \mathbb{R}$, with $1\le i\le N.$

The following lemma deals with the linear variant of the boundary value problems (5) and (6) and gives a representation of the solution.

Lemma 3.

Let ${\alpha }_i\in (1,2],{\theta }_i\in (0,1],{\lambda }_1,{\lambda }_2>0$ and ${\eta }_i\in {\mathbb{N}}_{{\alpha }_i-1,T+{\alpha }_i-1}$ be given constants, $k_i\in C\left({\mathbb{N}}_{{\alpha }_i-2},\mathbb{R}\right)$ and $g_i\in C\left({\mathbb{N}}_{{\alpha }_i-2,T+{\alpha }_i},{\mathbb{R}}^{+}\right)$ given functions, and ${\varphi }_i(u_1,u_2)$ given functionals. For each $i,j\in \{1,2\}$ and $i\ne j$, then the problems:

$$\begin{array}{c}{\Delta }^{{\alpha }_i}{u}_i\left(t\right)=k_i(t+{\alpha }_i-1),t\in {\mathbb{N}}_0,\hfill \end{array}$$

(7)

$$\begin{array}{c}{u}_i({\alpha }_i-2)={\varphi }_i(u_1,u_2),\hfill \end{array}$$

(8)

$$\begin{array}{c}\underset{t\to \infty }{lim}{u}_i(t+{\alpha }_i)={\lambda }_j{\Delta }^{-{\theta }_j}{g}_j({\eta }_j+{\theta }_j)u_j({\eta }_j+{\theta }_j).\hfill \end{array}$$

(9)

have the unique solutions:

$$\begin{array}{cc}\hfill u_1\left(t_1\right)=& t_1^{\underline{{\alpha }_1-1}}\{\frac{{\lambda }_1}{\mathsf{\Lambda }\mathsf{\Gamma }\left({\theta }_1\right)}\sum _{s={\alpha }_1-2}^{{\eta }_1}{({\eta }_1+{\theta }_1-\sigma \left(s\right))}^{\underline{{\theta }_1-1}}{g}_1\left(s\right)s^{\underline{{\alpha }_1-1}}\mathcal{P}(k_1,k_2)\hfill \\ & -\frac{{\lambda }_2}{\mathsf{\Lambda }\mathsf{\Gamma }\left({\theta }_2\right)}\sum _{s={\alpha }_2-2}^{{\eta }_2}{({\eta }_2+{\theta }_2-\sigma \left(s\right))}^{\underline{{\theta }_2-1}}{g}_2\left(s\right)s^{\underline{{\alpha }_2-1}}\mathcal{Q}(k_1,k_2)\}\hfill \\ & +\frac{t_1^{\underline{{\alpha }_1-2}}{\varphi }_1(u_1,u_2)}{\mathsf{\Gamma }\left({\alpha }_1\right)}+\frac{1}{\mathsf{\Gamma }\left({\alpha }_1\right)}\sum _{s=0}^{t_1-{\alpha }_1}{(t_1-\sigma \left(s\right))}^{\underline{{\alpha }_1-1}}{k}_1(s+{\alpha }_1-1),t_1\in {\mathbb{N}}_{{\alpha }_1-2},\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill u_2\left(t_2\right)=& t_2^{\underline{{\alpha }_2-1}}\left\{\frac{{\displaystyle \underset{t_2\to \infty }{lim}{t}_2^{\underline{{\alpha }_2-1}}}}{\mathsf{\Lambda }}\mathcal{P}(k_1,k_2)-\frac{{\displaystyle \underset{t_1\to \infty }{lim}{t}_1^{\underline{{\alpha }_1-1}}}}{\mathsf{\Lambda }}\mathcal{Q}(k_1,k_2)\right\}\hfill \\ & +\frac{t_2^{\underline{{\alpha }_2-2}}{\varphi }_2(u_1,u_2)}{\mathsf{\Gamma }\left({\alpha }_2\right)}+\frac{1}{\mathsf{\Gamma }\left({\alpha }_2\right)}\sum _{s=0}^{t_2-{\alpha }_2}{(t_2-\sigma \left(s\right))}^{\underline{{\alpha }_2-1}}{k}_2(s+{\alpha }_2-1),t_2\in {\mathbb{N}}_{{\alpha }_2-2},\hfill \end{array}$$

(11)

provided that both $u_1\left(t_1\right),u_2\left(t_2\right)$ are uniformly bounded on ${\mathbb{N}}_{{\alpha }_1-2}$ and ${\mathbb{N}}_{{\alpha }_2-2}$, respectively, and:

$$\begin{array}{cc}\hfill \mathsf{\Lambda }=& {\displaystyle \frac{{\displaystyle {\lambda }_2\underset{t_2\to \infty }{lim}{t}_2^{\underline{{\alpha }_2-1}}}}{\mathsf{\Gamma }\left({\alpha }_2\right)}}\sum _{s={\alpha }_2-1}^{{\eta }_2}{({\eta }_2+{\theta }_2-\sigma \left(s\right))}^{\underline{{\theta }_2-1}}{g}_2\left(s\right)s^{\underline{{\alpha }_2-1}}\hfill \\ & -{\displaystyle \frac{{\displaystyle {\lambda }_1\underset{t_1\to \infty }{lim}{t}_1^{\underline{{\alpha }_1-1}}}}{\mathsf{\Gamma }\left({\alpha }_1\right)}}\sum _{s={\alpha }_1-1}^{{\eta }_1}{({\eta }_1+{\theta }_1-\sigma \left(s\right))}^{\underline{{\theta }_1-1}}{g}_1\left(s\right)s^{\underline{{\alpha }_1-1}},t_i\in {\mathbb{N}}_{{\alpha }_i-2},\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill \mathcal{P}(k_1,k_2)=& \frac{{\displaystyle \underset{t_1\to \infty }{lim}{t}_1^{\underline{{\alpha }_1-2}}{\varphi }_1(u_1,u_2)}}{\mathsf{\Gamma }\left({\alpha }_1\right)}-\frac{{\lambda }_2{\varphi }_2(u_1,u_2)}{\mathsf{\Gamma }\left({\alpha }_2\right)\mathsf{\Gamma }\left({\theta }_2\right)}\sum _{s={\alpha }_2-2}^{{\eta }_2}{({\eta }_2+{\theta }_2-\sigma \left(s\right))}^{\underline{{\theta }_2-1}}{g}_2\left(s\right)s^{\underline{{\alpha }_2-2}}\hfill \\ & {\displaystyle +\frac{1}{\mathsf{\Gamma }\left({\alpha }_1\right)}\underset{t_1\to \infty }{lim}\sum _{s=0}^{t_1-{\alpha }_1}{(t_1-\sigma \left(s\right))}^{\underline{{\alpha }_1-1}}{k}_1(s+{\alpha }_1-1)-\frac{{\lambda }_2}{\mathsf{\Gamma }\left({\alpha }_2\right)\mathsf{\Gamma }\left({\theta }_2\right)}\times }\hfill \\ & \sum _{\xi ={\alpha }_2}^{{\eta }_2}\sum _{s=0}^{\xi -{\alpha }_2}{({\eta }_2+{\theta }_2-\sigma \left(\xi \right))}^{\underline{{\theta }_2-1}}{(\xi -\sigma \left(s\right))}^{\underline{{\alpha }_2-1}}{g}_2(s+{\alpha }_2-1)k_2(s+{\alpha }_2-1),\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill \mathcal{Q}(k_1,k_2)=& \frac{{\displaystyle \underset{t_2\to \infty }{lim}{t}_2^{\underline{{\alpha }_2-2}}{\varphi }_2(u_1,u_2)}}{\mathsf{\Gamma }\left({\alpha }_2\right)}-\frac{{\lambda }_1{\varphi }_1(u_1,u_2)}{\mathsf{\Gamma }\left({\alpha }_1\right)\mathsf{\Gamma }\left({\theta }_1\right)}\sum _{s={\alpha }_1-2}^{{\eta }_1}{({\eta }_1+{\theta }_1-\sigma \left(s\right))}^{\underline{{\theta }_1-1}}{g}_1\left(s\right)s^{\underline{{\alpha }_1-2}}\hfill \\ & {\displaystyle +\frac{1}{\mathsf{\Gamma }\left({\alpha }_2\right)}\underset{t_2\to \infty }{lim}\sum _{s=0}^{t_2-{\alpha }_2}{(t_2-\sigma \left(s\right))}^{\underline{{\alpha }_2-1}}{k}_2(s+{\alpha }_2-1)-\frac{{\lambda }_1}{\mathsf{\Gamma }\left({\alpha }_1\right)\mathsf{\Gamma }\left({\theta }_1\right)}\times }\hfill \\ & \sum _{\xi ={\alpha }_1}^{{\eta }_1}\sum _{s=0}^{\xi -{\alpha }_1}{({\eta }_1+{\theta }_1-\sigma \left(\xi \right))}^{\underline{{\theta }_1-1}}{(\xi -\sigma \left(s\right))}^{\underline{{\alpha }_1-1}}{g}_1(s+{\alpha }_1-1)k_1(s+{\alpha }_1-1).\hfill \end{array}$$

(14)

Proof. 

For each $i,j\in \{1,2\}$ and $i\ne j$, using Lemma 2 and the fractional sum of order $\alpha \in (1,2]$ for (7), we obtain:

$$\begin{array}{c}\hfill u_i\left(t_i\right)=C_{1i}{t}_i^{\underline{{\alpha }_i-1}}+C_{2i}{t}_i^{\underline{{\alpha }_i-2}}+\frac{1}{\mathsf{\Gamma }\left({\alpha }_i\right)}\sum _{s=0}^{t_i-{\alpha }_i}{(t_i-\sigma \left(s\right))}^{\underline{{\alpha }_i-1}}{k}_i(s+{\alpha }_i-1),\end{array}$$

(15)

for $t_i\in {\mathbb{N}}_{{\alpha }_i-2}$.

By using the boundary condition (8), we find that:

$$\begin{array}{c}\hfill C_{2i}=\frac{{\varphi }_i(u_1,u_2)}{\mathsf{\Gamma }\left({\alpha }_i\right)}.\end{array}$$

(16)

Then, for $t_i\in {\mathbb{N}}_{{\alpha }_i-2}$, we have:

$$\begin{array}{ccc}\hfill u_i\left(t_i\right)& =& C_{1i}{t}_i^{\underline{{\alpha }_i-1}}+\frac{{\varphi }_i(u_1,u_2)}{\mathsf{\Gamma }\left({\alpha }_i\right)}{t}_i^{\underline{{\alpha }_i-2}}\hfill \\ & & +\frac{1}{\mathsf{\Gamma }\left({\alpha }_i\right)}\sum _{s=0}^{t_i-{\alpha }_i}{(t_i-\sigma \left(s\right))}^{\underline{{\alpha }_i-1}}{k}_i(s+{\alpha }_i-1).\hfill \end{array}$$

(17)

Taking the fractional sum of order $0<{\theta }_i\le 1$ for (17), we obtain:

$$\begin{array}{cc}& {\Delta }^{-{\theta }_i}{u}_i\left(t_i\right)\hfill \\ =& \frac{C_{1i}}{\mathsf{\Gamma }\left({\theta }_i\right)}\sum _{s={\alpha }_i-2}^{t_i}{(t_i+\theta -\sigma \left(s\right))}^{\underline{{\theta }_i-1}}{g}_i\left(s\right)s^{\underline{{\alpha }_i-1}}+\frac{{\varphi }_i(u_1,u_2)}{\mathsf{\Gamma }\left({\alpha }_i\right)}\times \hfill \\ & \sum _{s={\alpha }_i-2}^{t_i}{(t_i+{\theta }_i-\sigma \left(s\right))}^{\underline{{\theta }_i-1}}{g}_i\left(s\right)s^{\underline{{\alpha }_i-2}}+\frac{1}{\mathsf{\Gamma }\left({\theta }_i\right)\mathsf{\Gamma }\left({\alpha }_i\right)}\sum _{\xi ={\alpha }_i}^{t_i}\sum _{s=0}^{\xi -{\alpha }_i}{(t_i+{\theta }_i-\sigma \left(\xi \right))}^{\underline{{\theta }_i-1}}\times \hfill \\ & {(\xi -\sigma \left(s\right))}^{\underline{{\alpha }_i-1}}{g}_i(s+{\alpha }_i-1)k_i(s+{\alpha }_i-1),\hfill \end{array}$$

(18)

for $t_i\in {\mathbb{N}}_{{\alpha }_i-2}$.

Employing the boundary condition (9), this implies that:

$$\begin{array}{cc}& C_{11}\underset{t_1\to \infty }{lim}{t}_1^{\underline{{\alpha }_1-1}}+\frac{{\varphi }_1(u_1,u_2)}{\mathsf{\Gamma }\left({\alpha }_1\right)}\underset{t_1\to \infty }{lim}{t}_1^{\underline{{\alpha }_1-2}}+\frac{{lim}_{t_1\to \infty }}{\mathsf{\Gamma }\left({\alpha }_1\right)}\sum _{s=0}^{t_1-{\alpha }_1}{(t_i-\sigma \left(s\right))}^{\underline{{\alpha }_1-1}}{k}_1(s+{\alpha }_1-1)\hfill \\ \hfill =& \frac{{\lambda }_2{C}_{12}}{\mathsf{\Gamma }\left({\theta }_2\right)}\sum _{s={\alpha }_2-1}^{{\eta }_2}{({\eta }_2+{\theta }_2-\sigma \left(s\right))}^{\underline{{\theta }_2-1}}{g}_2\left(s\right)s^{\underline{{\alpha }_2-1}}\hfill \\ & +\frac{{\lambda }_2{\varphi }_2(u_1,u_2)}{\mathsf{\Gamma }\left({\alpha }_2\right)\mathsf{\Gamma }\left({\theta }_2\right)}\sum _{s={\alpha }_2-2}^{{\eta }_2}{({\eta }_2+{\theta }_2-\sigma \left(s\right))}^{\underline{{\theta }_2-1}}{g}_2\left(s\right)s^{\underline{{\alpha }_2-2}}\hfill \\ & +\frac{{\lambda }_2}{\mathsf{\Gamma }\left({\alpha }_2\right)\mathsf{\Gamma }\left({\theta }_2\right)}\sum _{\xi ={\alpha }_2}^{{\eta }_2}\sum _{s=0}^{\xi -{\alpha }_2}{({\eta }_2+{\theta }_2-\sigma \left(\xi \right))}^{\underline{{\theta }_2-1}}{(\xi -\sigma \left(s\right))}^{\underline{{\alpha }_2-1}}{g}_2(s+{\alpha }_2-1)k_2(s+{\alpha }_2-1),\hfill \end{array}$$

(19)

and:

$$\begin{array}{cc}& C_{12}\underset{t_2\to \infty }{lim}{t}_2^{\underline{{\alpha }_2-1}}+\frac{{\varphi }_2(u_1,u_2)}{\mathsf{\Gamma }\left({\alpha }_2\right)}\underset{t_2\to \infty }{lim}{t}_2^{\underline{{\alpha }_2-2}}+\frac{1}{\mathsf{\Gamma }\left({\alpha }_2\right)}\underset{t_2\to \infty }{lim}\sum _{s=0}^{t_2-{\alpha }_2}{(t_2-\sigma \left(s\right))}^{\underline{{\alpha }_2-1}}{k}_2(s+{\alpha }_2-1)\hfill \\ \hfill =& \frac{{\lambda }_1{C}_{11}}{\mathsf{\Gamma }\left({\theta }_1\right)}\sum _{s={\alpha }_1-1}^{{\eta }_1}{({\eta }_1+{\theta }_1-\sigma \left(s\right))}^{\underline{{\theta }_1-1}}{g}_1\left(s\right)s^{\underline{{\alpha }_1-1}}\hfill \\ & +\frac{{\lambda }_1{\varphi }_1(u_1,u_2)}{\mathsf{\Gamma }\left({\alpha }_1\right)\mathsf{\Gamma }\left({\theta }_1\right)}\sum _{s={\alpha }_1-2}^{{\eta }_1}{({\eta }_1+{\theta }_1-\sigma \left(s\right))}^{\underline{{\theta }_1-1}}{g}_1\left(s\right)s^{\underline{{\alpha }_1-2}}\hfill \\ & +\frac{{\lambda }_1}{\mathsf{\Gamma }\left({\alpha }_1\right)\mathsf{\Gamma }\left({\theta }_1\right)}\sum _{\xi ={\alpha }_1}^{{\eta }_1}\sum _{s=0}^{\xi -{\alpha }_1}{({\eta }_1+{\theta }_1-\sigma \left(\xi \right))}^{\underline{{\theta }_1-1}}{(\xi -\sigma \left(s\right))}^{\underline{{\alpha }_1-1}}{g}_1(s+{\alpha }_1-1)k_1(s+{\alpha }_1-1).\hfill \end{array}$$

(20)

After solving the system of Equations (19) and (20), we obtain:

$$\begin{array}{ccc}\hfill C_{11}& =& \frac{{\lambda }_1}{\mathsf{\Lambda }\mathsf{\Gamma }\left({\theta }_1\right)}\sum _{s={\alpha }_1-2}^{{\eta }_1}{({\eta }_1+{\theta }_1-\sigma \left(s\right))}^{\underline{{\theta }_1-1}}{g}_1\left(s\right)s^{\underline{{\alpha }_1-1}}\mathcal{P}(k_1,k_2)\hfill \\ & & -\frac{{\lambda }_2}{\mathsf{\Lambda }\mathsf{\Gamma }\left({\theta }_2\right)}\sum _{s={\alpha }_2-2}^{{\eta }_2}{({\eta }_2+{\theta }_2-\sigma \left(s\right))}^{\underline{{\theta }_2-1}}{g}_2\left(s\right)s^{\underline{{\alpha }_2-1}}\mathcal{Q}(k_1,k_2),\hfill \end{array}$$

(21)

and:

$$\begin{array}{ccc}\hfill C_{12}& =& \frac{{\displaystyle \underset{t_2\to \infty }{lim}{t}_2^{\underline{{\alpha }_2-1}}}}{\mathsf{\Lambda }}\mathcal{P}(k_1,k_2)-\frac{{\displaystyle \underset{t_1\to \infty }{lim}{t}_1^{\underline{{\alpha }_1-1}}}}{\mathsf{\Lambda }}\mathcal{Q}(k_1,k_2),\hfill \end{array}$$

(22)

where $\mathsf{\Lambda },\mathcal{P}(k_1,k_2)$ and $\mathcal{Q}(k_1,k_2)$ are defined as (12)–(14), respectively.  □

The following lemma deals with the solutions $u_i\left(t_i\right),i=1,2$ of the problems (7)–(9), and ${\Delta }^{{\beta }_i}{u}_i(t_i-{\beta }_i+1)$ are uniformly bounded on ${\mathbb{N}}_{{\alpha }_i-2},{\beta }_i\in ({\alpha }_i-1,{\alpha }_i)$.

Lemma 4.

For each $i,j\in \{1,2\}$ and $i\ne j$, let $k_i\in C\left({\mathbb{N}}_{{\alpha }_i-2},\mathbb{R}\right)$ and $g_i\in C\left({\mathbb{N}}_{{\alpha }_i-2},{\mathbb{R}}^{+}\right)$ be given functions, ${\varphi }_i(u_1,u_2)$ be given functionals, ${\rho }_i>max\{{\beta }_i-{\alpha }_i\},{\beta }_i\in ({\alpha }_i-1,{\alpha }_i)$, and $0<g_i\le g_i\left(s_i\right)\le G_i,$ for each $s_i\in {\mathbb{N}}_{{\alpha }_i-2,T+{\alpha }_i}$.

The solution $u_i\left(t_i\right)$ of the problems (7)–(9) and ${\Delta }^{{\beta }_i}{u}_i(t_i-{\beta }_i+1)$ are uniformly bounded on ${\mathbb{N}}_{{\alpha }_i-2}$, if and only if $u_i\left(t_i\right)$ and ${\Delta }^{{\beta }_i}{u}_i(t_i-{\beta }_i+1)$ satisfy the following properties:

$\left(A_1\right)$

There exist constants $M_1,N_1,m_1,n_1>0$ such that, for $u_1$ and ${\Delta }^{{\beta }_1}{u}_1$,

$$\begin{array}{cc}\hfill |k_i\left(t_i\right)|\le & M_1{e}^{-m_1(2t_1+t_2)},\hfill \\ \hfill |{\varphi }_i(u_1,u_2)|\le & N_1{(t_1+{\rho }_1)}^{\underline{{\rho }_1}}{e}^{-n_1(t_1+1)}.\hfill \end{array}$$

$\left(A_2\right)$

There exist constants $M_2,N_2,m_2,n_2>0$ such that, for $u_2$ and ${\Delta }^{{\beta }_2}{u}_2$,

$$\begin{array}{cc}\hfill |k_i\left(t_i\right)|\le & M_2{(t_2+{\rho }_2)}^{\underline{{\rho }_2}}{e}^{-m_2(t_1+2t_2)},\hfill \\ \hfill |{\varphi }_i(u_1,u_2)|\le & N_2{(t_1+{\rho }_1)}^{\underline{{\rho }_1}}{\left[{(t_2+{\rho }_2)}^{\underline{{\rho }_2}}\right]}^2{e}^{-n_2(t_2+1)}.\hfill \end{array}$$

$\left(A_3\right)$

There exist constants ${\mathsf{\Omega }}_i>0,i=1,2$ such that,

$$\begin{array}{c}\left\{\frac{{(t_1-{\alpha }_1+1)}^{\underline{2-{\alpha }_1}}{(t_2-{\alpha }_2+1)}^{\underline{2-{\alpha }_2}}}{1+{(t_1+{\rho }_1)}^{\underline{{\rho }_1}}{(t_2+{\rho }_2)}^{\underline{{\rho }_2}}}\right\}|u_i\left(t_i\right)|<{\mathsf{\Omega }}_i,\hfill \\ \left\{\frac{{(t_1-{\alpha }_1+1)}^{\underline{2-{\alpha }_1}}{(t_2-{\alpha }_2+1)}^{\underline{2-{\alpha }_2}}}{1+{(t_1+{\rho }_1)}^{\underline{{\rho }_1}}{(t_2+{\rho }_2)}^{\underline{{\rho }_2}}}\right\}|{\Delta }^{{\beta }_i}{u}_i\left(t_i\right)|<{\mathsf{\Omega }}_i.\hfill \end{array}$$

Proof. 

Firstly, taking the fractional difference of order ${\alpha }_i-1<{\beta }_i<{\alpha }_i,i=1,2$ for (10) and (11), we obtain:

$$\begin{array}{cc}& {\Delta }^{{\beta }_1}{u}_1\left(t_1\right)\hfill \\ \hfill =& \frac{1}{\mathsf{\Gamma }(-{\beta }_1)}\sum _{s={\alpha }_1-1}^{t_1+1}{(t_1-{\beta }_1+1-\sigma \left(s\right))}^{\underline{-{\beta }_1-1}}{s}^{\underline{{\alpha }_1-1}}\times \{\frac{{\lambda }_1}{\mathsf{\Lambda }\mathsf{\Gamma }\left({\theta }_1\right)}\times \hfill \\ & \sum _{s={\alpha }_1-2}^{{\eta }_1}{({\eta }_1+{\theta }_1-\sigma \left(s\right))}^{\underline{{\theta }_1-1}}{g}_1\left(s\right)s^{\underline{{\alpha }_1-1}}\mathcal{P}(k_1,k_2)-\frac{{\lambda }_2}{\mathsf{\Lambda }\mathsf{\Gamma }\left({\theta }_2\right)}\times \hfill \\ & \sum _{s={\alpha }_2-2}^{{\eta }_2}{({\eta }_2+{\theta }_2-\sigma \left(s\right))}^{\underline{{\theta }_2-1}}{g}_2\left(s\right)s^{\underline{{\alpha }_2-1}}\mathcal{Q}(k_1,k_2)\}\hfill \\ & +\frac{{\varphi }_1(u_1,u_2)}{\mathsf{\Gamma }(-{\beta }_1)\mathsf{\Gamma }\left({\alpha }_1\right)}\sum _{s={\alpha }_1-2}^{t_1+1}{(t_1-{\beta }_1+1-\sigma \left(s\right))}^{\underline{-{\beta }_1-1}}{s}^{\underline{{\alpha }_1-2}}\hfill \\ & +\frac{1}{\mathsf{\Gamma }(-{\beta }_1)\mathsf{\Gamma }\left({\alpha }_1\right)}\sum _{\xi ={\alpha }_1}^{t_1+1}\sum _{s=0}^{\xi -{\alpha }_1}{(t_1-{\beta }_1+1-\sigma \left(s\right))}^{\underline{-{\beta }_1-1}}{(\xi -\sigma \left(s\right))}^{\underline{{\alpha }_1-1}}{k}_1(s+{\alpha }_1-1),\hfill \end{array}$$

(23)

and

$$\begin{array}{cc}& {\Delta }^{{\beta }_2}{u}_2\left(t_2\right)\hfill \\ \hfill =& \frac{1}{\mathsf{\Gamma }(-{\beta }_2)}\sum _{s={\alpha }_2-1}^{t_2+1}{(t_2-{\beta }_2+1-\sigma \left(s\right))}^{\underline{-{\beta }_2-1}}{s}^{\underline{{\alpha }_2-1}}\times \{\frac{{\displaystyle \underset{t_2\to \infty }{lim}{t}_2^{\underline{{\alpha }_2-1}}}}{\mathsf{\Lambda }}\mathcal{P}(k_1,k_2)\hfill \\ & -\frac{{\displaystyle \underset{t_1\to \infty }{lim}{t}_1^{\underline{{\alpha }_1-1}}}}{\mathsf{\Lambda }}\mathcal{Q}(k_1,k_2)\}+\frac{{\varphi }_2(u_1,u_2)}{\mathsf{\Gamma }(-{\beta }_2)\mathsf{\Gamma }\left({\alpha }_2\right)}\sum _{s={\alpha }_2-2}^{t_2+1}{(t_2-{\beta }_1+1-\sigma \left(s\right))}^{\underline{-{\beta }_2-1}}{s}^{\underline{{\alpha }_2-2}}\hfill \\ & +\frac{1}{\mathsf{\Gamma }(-{\beta }_2)\mathsf{\Gamma }\left({\alpha }_2\right)}\sum _{\xi ={\alpha }_2}^{t_2+1}\sum _{s=0}^{\xi -{\alpha }_2}{(t_2-{\beta }_2+1-\sigma \left(s\right))}^{\underline{-{\beta }_2-1}}{(\xi -\sigma \left(s\right))}^{\underline{{\alpha }_2-1}}{k}_2(s+{\alpha }_2-1).\hfill \end{array}$$

(24)

If $u_i\left(t_i\right)$ and ${\Delta }^{{\beta }_i}{u}_i(t_i-{\beta }_i+1)$ are uniformly bounded on ${\mathbb{N}}_{{\alpha }_i-2}$, we have:

$$\begin{array}{ccc}\hfill \left|\mathsf{\Lambda }\right|& \le & |{\displaystyle \frac{{\displaystyle {\lambda }_2\underset{t_2\to \infty }{lim}{t}_2^{\underline{{\alpha }_2-1}}}}{\mathsf{\Gamma }\left({\alpha }_2\right)}}\sum _{s={\alpha }_2-1}^{{\eta }_2}{({\eta }_2+{\theta }_2-\sigma \left(s\right))}^{\underline{{\theta }_2-1}}{g}_2\left(s\right)s^{\underline{{\alpha }_2-1}}\hfill \\ & & -{\displaystyle \frac{{\displaystyle {\lambda }_1\underset{t_1\to \infty }{lim}{t}_1^{\underline{{\alpha }_1-1}}}}{\mathsf{\Gamma }\left({\alpha }_1\right)}}\sum _{s={\alpha }_1-1}^{{\eta }_1}{({\eta }_1+{\theta }_1-\sigma \left(s\right))}^{\underline{{\theta }_1-1}}{g}_1\left(s\right)s^{\underline{{\alpha }_1-1}}|\hfill \\ & \le & {\displaystyle max\left\{\right|\underset{t_2\to \infty }{lim}{t}_2^{\underline{{\alpha }_2-1}}{({\eta }_2+{\theta }_2-{\alpha }_2)}^{\underline{{\theta }_2-1}}{G}_2{\lambda }_2{\mathcal{A}}_2|,}\hfill \\ & & {\displaystyle \left|\underset{t_1\to \infty }{lim}{t}_1^{\underline{{\alpha }_1-1}}{({\eta }_1+{\theta }_1-{\alpha }_1)}^{\underline{{\theta }_1-1}}{G}_1{\lambda }_1{\mathcal{A}}_1\right|\}.}\hfill \end{array}$$

(25)

Furthermore, considering $u_1\left(t_i\right)$ and ${\Delta }^{{\beta }_1}{u}_i\left(t_i\right)$, we obtain:

$$|k_i\left(t_i\right)|<\left\{\begin{array}{cc}{M}_1{e}^{1-(t_1+t_2)}\hfill & ,\mathrm{for}{u}_1\left(t_1\right)\hfill \\ M_1{e}^{-(2t_1+t_2)}\hfill & ,\mathrm{for}{\Delta }^{{\beta }_1}{u}_1(t_1-{\beta }_1+1)\hfill \\ M_2{(t_2+{\rho }_2)}^{\underline{{\rho }_2}}{e}^{-(t_1+t_2)}\hfill & ,\mathrm{for}{u}_2\left(t_2\right)\hfill \\ M_2{(t_2+{\rho }_2)}^{\underline{{\rho }_2}}{e}^{-(t_1+2t_2)}\hfill & ,\mathrm{for}{\Delta }^{{\beta }_2}{u}_2(t_2-{\beta }_2+1)\hfill \end{array}\right.$$

(26)

and:

$$|{\varphi }_i(t_1,t_2)|<\left\{\begin{array}{cc}{N}_1{(t_1+{\rho }_1+1)}^{\underline{{\rho }_1}}\hfill & ,\mathrm{for}{u}_1\left(t_1\right)\hfill \\ N_1{(t_1+{\rho }_1)}^{\underline{{\rho }_1}}{e}^{-(t_1+1)}\hfill & ,\mathrm{for}{\Delta }^{{\beta }_1}{u}_1(t_1-{\beta }_1+1)\hfill \\ N_2{(t_1+{\rho }_1)}^{\underline{{\rho }_1}}{\left[{(t_2+{\rho }_2+1)}^{\underline{{\rho }_2}}\right]}^2\hfill & ,\mathrm{for}{u}_2\left(t_2\right)\hfill \\ N_2{(t_1+{\rho }_1)}^{\underline{{\rho }_1}}{\left[{(t_2+{\rho }_2)}^{\underline{{\rho }_2}}\right]}^2{e}^{-(t_2+1)}\hfill & ,\mathrm{for}{\Delta }^{{\beta }_2}{u}_2(t_2-{\beta }_2+1)\hfill \end{array}\right.$$

(27)

where:

$$\begin{array}{cc}\hfill M_1=& min\{\frac{{\lambda }_1{G}_1{({\eta }_1-{\alpha }_1+{\theta }_1)}^{\underline{{\theta }_1-1}}{\mathcal{A}}_1}{{\lambda }_2{g}_2\mathsf{\Gamma }\left({\alpha }_2\right){\mathcal{A}}_2},\frac{G_1{({\eta }_1-{\alpha }_1+{\theta }_1)}^{\underline{{\theta }_1-1}}}{g_1},\mathsf{\Gamma }\left({\alpha }_1\right),\hfill \\ & \frac{G_1{({\eta }_1-{\alpha }_1+{\theta }_1)}^{\underline{{\theta }_1-1}}}{{\lambda }_2{g}_1{g}_2\mathsf{\Gamma }\left({\alpha }_1\right){\mathcal{C}}_2},\frac{G_1{({\eta }_1-{\alpha }_1+{\theta }_1)}^{\underline{{\theta }_1-1}}{\mathcal{A}}_1}{{\lambda }_2{g}_1{g}_2\mathsf{\Gamma }\left({\alpha }_2\right){\mathcal{A}}_2{\mathcal{C}}_1}\},\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill M_2=& min\{\mathsf{\Gamma }\left({\alpha }_2\right),{\lambda }_2{G}_2{({\eta }_2-{\alpha }_2+{\theta }_2)}^{\underline{{\theta }_2-1}}{\mathcal{A}}_2,G_1{({\eta }_1-{\alpha }_1+{\theta }_1)}^{\underline{{\theta }_1-1}}{\mathcal{A}}_1,\hfill \\ & \frac{G_1{({\eta }_1-{\alpha }_1+{\theta }_1)}^{\underline{{\theta }_1-1}}{\mathcal{A}}_1}{g_1{\mathcal{C}}_1},\frac{G_2{({\eta }_2-{\alpha }_2+{\theta }_2)}^{\underline{{\theta }_2-1}}{\mathcal{A}}_2}{g_2{\mathcal{C}}_2}\},\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill N_1=& min\left\{\mathsf{\Gamma }\left({\theta }_1\right),\mathsf{\Gamma }\left({\theta }_2\right),\mathsf{\Gamma }\left({\alpha }_1\right),\frac{\mathsf{\Gamma }\left({\theta }_1\right)}{{\lambda }_2{g}_2\mathsf{\Gamma }\left({\alpha }_1\right){\mathcal{B}}_2},\frac{G_1{({\eta }_1-{\alpha }_1+{\theta }_1)}^{\underline{{\theta }_1-1}}{\mathcal{A}}_1}{{\lambda }_2{g}_1{g}_2\mathsf{\Gamma }({\alpha }_1-1){\mathcal{B}}_1{\mathcal{A}}_2}\right\},\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill N_2=& min\{\mathsf{\Gamma }\left({\alpha }_2\right),{\lambda }_2{G}_2\mathsf{\Gamma }\left({\alpha }_1\right){({\eta }_2-{\alpha }_2+{\theta }_2)}^{\underline{{\theta }_2-1}}{\mathcal{A}}_2,{\lambda }_1{G}_1\mathsf{\Gamma }\left({\alpha }_2\right){({\eta }_1-{\alpha }_1+{\theta }_1)}^{\underline{{\theta }_1-1}}{\mathcal{A}}_1,\hfill \\ & \frac{G_2{({\eta }_2-{\alpha }_2+{\theta }_2)}^{\underline{{\theta }_2-1}}{\mathcal{A}}_2}{g_2{\mathcal{B}}_2},\frac{G_1{({\eta }_1-{\alpha }_1+{\theta }_1)}^{\underline{{\theta }_1-1}}{\mathcal{A}}_1}{g_1{\mathcal{B}}_1}\}.\hfill \end{array}$$

(31)

Consequently, the conditions $\left(A1\right)$ and $\left(A2\right)$ hold.

We next show that the condition $\left(A3\right)$ holds. By using the conditions $\left(A1\right)$ and $\left(A2\right)$, we obtain:

$$\begin{array}{cc}\hfill \left|u_i\left(t_i\right)\right|\le t_1^{\underline{{\alpha }_1}}{t}_2^{\underline{{\alpha }_2}}{\mathsf{\Omega }}_i<& t_1^{\underline{{\alpha }_1+{\rho }_1-1}}{t}_2^{\underline{{\alpha }_2+{\rho }_2-1}}{\mathsf{\Omega }}_i\hfill \\ \hfill <& \left[\frac{1+t_1^{\underline{{\rho }_1+2}}{t}_2^{\underline{{\rho }_2+2}}}{{(t_1-{\alpha }_1+1)}^{\underline{2-{\alpha }_1}}{(t_2-{\alpha }_2+1)}^{\underline{2-{\alpha }_2}}}\right]{\mathsf{\Omega }}_i\hfill \end{array}$$

and:

$$\begin{array}{cc}\hfill \left|{\Delta }^{{\beta }_i}{u}_i(t_i-{\beta }_i+1)\right|\le & {(t_i-{\alpha }_i-{\beta }_i+1)}^{\underline{-{\beta }_i}}{t}_j^{\underline{{\alpha }_j}}{\mathsf{\Omega }}_i<t_i^{\underline{{\alpha }_i}}{t}_j^{\underline{{\alpha }_j}}{\mathsf{\Omega }}_i\hfill \\ \hfill <& \left[\frac{1+t_1^{\underline{{\rho }_1+2}}{t}_2^{\underline{{\rho }_2+2}}}{{(t_1-{\alpha }_1+1)}^{\underline{2-{\alpha }_1}}{(t_2-{\alpha }_2+1)}^{\underline{2-{\alpha }_2}}}\right]{\mathsf{\Omega }}_i,i\ne j=1,2\hfill \end{array}$$

where:

$$\begin{array}{cc}\hfill {\mathsf{\Omega }}_1=& max\{\frac{N_1}{\mathsf{\Gamma }\left({\theta }_1\right)}+M_1+\frac{{\lambda }_2{G}_2\mathsf{\Gamma }\left({\alpha }_1\right){({\eta }_2-{\alpha }_2+{\theta }_2)}^{\underline{{\theta }_2-1}}}{\mathsf{\Gamma }\left({\theta }_2\right)}\times \hfill \\ & \left(N_2{\mathcal{B}}_2+\frac{M_2{\mathcal{C}}_2}{{\lambda }_1{G}_1{({\eta }_1-{\alpha }_1+{\theta }_1)}^{\underline{{\theta }_1-1}}{\mathcal{A}}_1}\right),\hfill \\ & \frac{N_2}{\mathsf{\Gamma }\left({\theta }_2\right)}+M_2+\frac{{\lambda }_1{G}_1\mathsf{\Gamma }\left({\alpha }_2\right){({\eta }_1-{\alpha }_1+{\theta }_1)}^{\underline{{\theta }_1-1}}}{\mathsf{\Gamma }\left({\theta }_1\right)}\times \hfill \\ & \left(N_1{\mathcal{B}}_1+\frac{M_1{\mathcal{C}}_1}{{\lambda }_2{G}_2{({\eta }_2-{\alpha }_2+{\theta }_2)}^{\underline{{\theta }_2-1}}{\mathcal{A}}_2}\right)\}\hfill \\ & +\frac{1}{\mathsf{\Gamma }\left({\alpha }_1\right)}(M_1+N_1),\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill {\mathsf{\Omega }}_2=& max\{\frac{1}{G_2{({\eta }_2-{\alpha }_2+{\theta }_2)}^{\underline{{\theta }_2-1}}{\mathcal{A}}_2}\left[\frac{M_1+N_1}{\mathsf{\Gamma }\left({\alpha }_1\right){\lambda }_2}+\frac{1}{\mathsf{\Gamma }\left({\theta }_2\right){\mathcal{A}}_2}\left(N_2{\mathcal{B}}_2+M_2{\mathcal{C}}_2\right)\right],\hfill \\ & \frac{1}{G_1{({\eta }_1-{\alpha }_1+{\theta }_1)}^{\underline{{\theta }_1-1}}{\mathcal{A}}_1}\left[\frac{M_2+N_2}{\mathsf{\Gamma }\left({\alpha }_2\right){\lambda }_1}+\frac{1}{\mathsf{\Gamma }\left({\theta }_1\right){\mathcal{A}}_1}\left(N_1{\mathcal{B}}_1+M_1{\mathcal{C}}_1\right)\right]\}\hfill \\ & +\frac{1}{\mathsf{\Gamma }\left({\alpha }_2\right)}(M_2+N_2),\hfill \end{array}$$

(33)

with

$$\begin{array}{c}{\mathcal{A}}_i={}_2{F}_1\left({\alpha }_i,{\alpha }_i-{\eta }_i-1;{\alpha }_i-{\eta }_i-{\theta }_i;1\right)\hfill \end{array}$$

(34)

$$\begin{array}{c}{\mathcal{B}}_i={}_2{F}_1\left({\alpha }_i-1,{\alpha }_i-{\eta }_i-1;{\alpha }_i-{\eta }_i-{\theta }_i-1;1\right)\hfill \end{array}$$

(35)

$$\begin{array}{c}{\mathcal{C}}_i={}_2{F}_1\left({\alpha }_i+1,{\alpha }_i-{\eta }_i;{\alpha }_i-{\eta }_i-{\theta }_i+1;1\right).\hfill \end{array}$$

(36)

Therefore, the condition $\left(A3\right)$ holds.

Finally, if the conditions $\left(A1\right)$–$\left(A3\right)$ hold, it is clear that $u_i\left(t_i\right)$ and ${\Delta }^{{\beta }_i}{u}_i(t_i-{\beta }_i+1)$ are uniformly bounded on ${\mathbb{N}}_{{\alpha }_i-2}$. Our proof is complete.  □

We next provide the following theorems used for proving the existence result for the problems (5) and (6).

Theorem 1.

(Arzelá–Ascoli theorem [61])

A set of functions in $C[a,b]$ with the sup norm is relatively compact if and only if it is uniformly bounded and equicontinuous on $[a,b]$.

Theorem 2.

[61] If a set is closed and relatively compact, then it is compact.

Theorem 3.

(Schauder’s fixed point theorem [61])

If S is a convex compact subset of a normed space, every continuous mapping of S into itself has a fixed point.

3. Main Result

In this section, we aim to establish the existence result for the problems (5) and (6). To accomplish this, we let ${\mathcal{C}}_i=C({\mathbb{N}}_{{\alpha }_i-2},\mathbb{R})$ be a Banach space of all functions on ${\mathbb{N}}_{{\alpha }_i-2}$, for each $i,j\in \{1,2\}$ and $i\ne j$. Obviously, the product spaces:

$$\begin{array}{cc}\hfill {\mathcal{U}}_i=& \left\{\right(u_1,u_2)\in {\mathcal{C}}_1\times {\mathcal{C}}_2:{\Delta }^{{\beta }_i}{u}_i(t_i-{\beta }_i+1)\in {\mathcal{C}}_i\mathrm{and}\chi \left|u_j\left(t_j\right)\right|,\hfill \\ & \chi |{\Delta }^{{\beta }_i}{u}_i(t_i-{\beta }_i+1)|\mathrm{are}\mathrm{bounded}\mathrm{on}{\mathbb{N}}_{{\alpha }_j-2},{\mathbb{N}}_{{\alpha }_i-2},\mathrm{respectively},\}\hfill \end{array}$$

is also the Banach space endowed with the norm defined by:

$$\parallel (u_1,u_2){\parallel }_{{\mathcal{U}}_i}=\parallel {\Delta }^{{\beta }_i}{u}_i{\parallel }_{{\mathcal{C}}_i}+{\parallel u_j\parallel }_{{\mathcal{C}}_j},$$

where:

$${\displaystyle \parallel {\Delta }^{{\beta }_i}{u}_i{\parallel }_{{\mathcal{C}}_i}=\underset{t_i\in {\mathbb{N}}_{{\alpha }_i-2}}{max}\chi |{\Delta }^{{\beta }_i}{u}_i(t_i-{\beta }_i+1,t_j)|\mathrm{and}\parallel u_j{\parallel }_{{\mathcal{C}}_j}=\underset{t_j\in {\mathbb{N}}_{{\alpha }_j-2}}{max}\chi |u_j(t_i,t_j)|,}$$

with for ${\rho }_i>max\{{\beta }_i-{\alpha }_i\}$ and ${\beta }_i\in ({\alpha }_i-1,{\alpha }_i)$,

$$\begin{array}{cc}\hfill \chi =& \frac{{(t_1-{\alpha }_1+1)}^{\underline{2-{\alpha }_1}}{(t_2-{\alpha }_2+1)}^{\underline{2-{\alpha }_2}}}{1+t_1^{\underline{{\rho }_1+2}}{t}_2^{\underline{{\rho }_2+2}}}.\hfill \end{array}$$

(37)

Let $\mathcal{U}={\mathcal{U}}_1\cap {\mathcal{U}}_2$; clearly, the space $\left(\mathcal{U},\parallel (u_1,u_2){\parallel }_{\mathcal{U}}\right)$ is the Banach space with the norm:

$$\parallel (u_1,u_2){\parallel }_{\mathcal{U}}=max\left\{\parallel (u_1,u_2){\parallel }_{{\mathcal{U}}_1},{\parallel (u_1,u_2)\parallel }_{{\mathcal{U}}_2}\right\}.$$

Next, we define the operator $\mathcal{F}:\mathcal{U}\to \mathcal{U}$ by:

$$\begin{array}{ccc}\hfill \left(\mathcal{F}(u_1,u_2)\right)(t_1,t_2)& =& \left(\left({\mathcal{F}}_1(u_1,u_2)\right)(t_1,t_2),\left({\mathcal{F}}_2(u_1,u_2)\right)(t_1,t_2)\right),\hfill \end{array}$$

(38)

and:

$$\begin{array}{cc}\hfill \left({\mathcal{F}}_1(u_1,u_2)\right)(t_1,t_2)=& \frac{t_1^{\underline{{\alpha }_1-1}}}{\mathsf{\Lambda }}\{\frac{{\lambda }_1}{\mathsf{\Gamma }\left({\theta }_1\right)}\sum _{s={\alpha }_1-2}^{{\eta }_1}{({\eta }_1+{\theta }_1-\sigma \left(s\right))}^{\underline{{\theta }_1-1}}{g}_1\left(s\right)s^{\underline{{\alpha }_1-1}}\mathcal{P}(F_1,F_2)\hfill \\ & -\frac{{\lambda }_2}{\mathsf{\Gamma }\left({\theta }_2\right)}\sum _{s={\alpha }_2-2}^{{\eta }_2}{({\eta }_2+{\theta }_2-\sigma \left(s\right))}^{\underline{{\theta }_2-1}}{g}_2\left(s\right)s^{\underline{{\alpha }_2-1}}\mathcal{Q}(F_1,F_2)\}\hfill \\ & +\frac{t_1^{\underline{{\alpha }_1-2}}{\varphi }_1(u_1,u_2)}{\mathsf{\Gamma }\left({\alpha }_1\right)}+\frac{1}{\mathsf{\Gamma }\left({\alpha }_1\right)}\sum _{s={\alpha }_1-1}^{t_1-1}{(t_1+{\alpha }_1-1-\sigma \left(s\right))}^{\underline{{\alpha }_1-1}}\times \hfill \\ & F_1\left(s,t_2,{\Delta }^{{\beta }_1}{u}_1(s-{\beta }_1+1),u_2\left(t_2\right)\right),t_i\in {\mathbb{N}}_{{\alpha }_i-2},\hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill \left({\mathcal{F}}_2(u_1,u_2)\right)(t_1,t_2)=& {\displaystyle \frac{t_2^{\underline{{\alpha }_2-1}}}{\mathsf{\Lambda }}\left\{\underset{t_2\to \infty }{lim}{t}_2^{\underline{{\alpha }_2-1}}\mathcal{P}(F_1,F_2)-\underset{t_1\to \infty }{lim}{t}_1^{\underline{{\alpha }_1-1}}\mathcal{Q}(F_1,F_2)\right\}}\hfill \\ & +\frac{t_2^{\underline{{\alpha }_2-2}}{\varphi }_2(u_1,u_2)}{\mathsf{\Gamma }\left({\alpha }_2\right)}+\frac{1}{\mathsf{\Gamma }\left({\alpha }_2\right)}\sum _{s={\alpha }_2-1}^{t_2-1}{(t_2+{\alpha }_2-1-\sigma \left(s\right))}^{\underline{{\alpha }_2-1}}\times \hfill \\ & F_2\left(t_1,s,u_1\left(t_1\right),{\Delta }^{{\beta }_2}{u}_2(s-{\beta }_2+1)\right),t_i\in {\mathbb{N}}_{{\alpha }_i-2},\hfill \end{array}$$

(40)

where $\mathsf{\Lambda }$ is defined as (12), and:

$$\begin{array}{cc}\hfill \mathcal{P}(F_1,F_2)=& \frac{{\displaystyle \underset{t_1\to \infty }{lim}{t}_1^{\underline{{\alpha }_1-2}}{\varphi }_1(u_1,u_2)}}{\mathsf{\Gamma }\left({\alpha }_1\right)}-\frac{{\lambda }_2{\varphi }_2(u_1,u_2)}{\mathsf{\Gamma }\left({\alpha }_2\right)\mathsf{\Gamma }\left({\theta }_2\right)}\sum _{s={\alpha }_2-2}^{{\eta }_2}{({\eta }_2+{\theta }_2-\sigma \left(s\right))}^{\underline{{\theta }_2-1}}{g}_2\left(s\right)s^{\underline{{\alpha }_2-2}}\hfill \\ & {\displaystyle +\frac{1}{\mathsf{\Gamma }\left({\alpha }_1\right)}\underset{t_1\to \infty }{lim}\sum _{s={\alpha }_1-1}^{t_1-1}{(t_1+{\alpha }_1-1-\sigma \left(s\right))}^{\underline{{\alpha }_1-1}}{F}_1\left(s,t_2,{\Delta }^{{\beta }_1}{u}_1(s-{\beta }_1+1),u_2\left(t_2\right)\right)}\hfill \\ & -\frac{{\lambda }_2}{\mathsf{\Gamma }\left({\alpha }_2\right)\mathsf{\Gamma }\left({\theta }_2\right)}\sum _{\xi ={\alpha }_2}^{{\eta }_2}\sum _{s={\alpha }_2-1}^{\xi -1}{({\eta }_2+{\theta }_2-\sigma \left(\xi \right))}^{\underline{{\theta }_2-1}}{(\xi +{\alpha }_2-1-\sigma \left(s\right))}^{\underline{{\alpha }_2-1}}\times \hfill \\ & g_2\left(s\right)F_2\left(t_1,s,u_1\left(t_1\right),{\Delta }^{{\beta }_2}{u}_2(s-{\beta }_2+1)\right),\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill \mathcal{Q}(F_1,F_2)=& \frac{{\displaystyle \underset{t_2\to \infty }{lim}{t}_2^{\underline{{\alpha }_2-2}}{\varphi }_2(u_1,u_2)}}{\mathsf{\Gamma }\left({\alpha }_2\right)}-\frac{{\lambda }_1{\varphi }_1(u_1,u_2)}{\mathsf{\Gamma }\left({\alpha }_1\right)\mathsf{\Gamma }\left({\theta }_1\right)}\sum _{s={\alpha }_1-2}^{{\eta }_1}{({\eta }_1+{\theta }_1-\sigma \left(s\right))}^{\underline{{\theta }_1-1}}{g}_1\left(s\right)s^{\underline{{\alpha }_1-2}}\hfill \\ & {\displaystyle +\frac{1}{\mathsf{\Gamma }\left({\alpha }_2\right)}\underset{t_2\to \infty }{lim}\sum _{s={\alpha }_2-1}^{t_2-1}{(t_2+{\alpha }_2-1-\sigma \left(s\right))}^{\underline{{\alpha }_2-1}}{F}_2\left(t_1,s,u_1\left(t_1\right),{\Delta }^{{\beta }_2}{u}_2(s-{\beta }_2+1)\right)}\hfill \\ & -\frac{{\lambda }_1}{\mathsf{\Gamma }\left({\alpha }_1\right)\mathsf{\Gamma }\left({\theta }_1\right)}\sum _{\xi ={\alpha }_1}^{{\eta }_1}\sum _{s={\alpha }_1-1}^{\xi -1}{({\eta }_1+{\theta }_1-\sigma \left(\xi \right))}^{\underline{{\theta }_1-1}}{(\xi +{\alpha }_1-1-\sigma \left(s\right))}^{\underline{{\alpha }_1-1}}\times \hfill \\ & g_1\left(s\right)F_1\left(s,t_2,{\Delta }^{{\beta }_1}{u}_1(s-{\beta }_1+1),u_2\left(t_2\right)\right).\hfill \end{array}$$

(42)

We next make the following assumptions:

$\left(H_1\right)$

There exist positive numbers ${}_{ip}{\rho }_2\in (-1,{\rho }_2)$ and $M_{ip},m_{ip}>0(i=1,2\mathrm{and}p=1,2,3)$ such that, for each $t_i\in {\mathbb{N}}_{{\alpha }_i-2}$ and $v_i\in \mathbb{R}$,

$$\begin{array}{cc}\hfill |F_i\left(t_1,t_2,\frac{1}{\chi }{v}_1,\frac{1}{\chi }{v}_2\right)-& M_{i1}{\left(t_2+{}_{i1}{\rho }_2\right)}^{{}_{i1}{\rho }_2}{e}^{-m_{i1}(t_1+t_2)}|\hfill \\ \hfill \le & M_{i2}{\left(t_2+{}_{i2}{\rho }_2\right)}^{{}_{i2}{\rho }_2}{e}^{-m_{i2}(t_1+t_2)}\left|v_1\right|\hfill \\ \hfill +& M_{i3}{\left(t_2+{}_{i3}{\rho }_2\right)}^{{}_{i3}{\rho }_2}{e}^{-m_{i3}(t_1+t_2)}\left|v_2\right|.\hfill \end{array}$$

$\left(H_2\right)$

There exist positive numbers ${}_{ip}{\tilde{\rho }}_i\in (-1,{\rho }_i)$ and $N_{ip},n_{ip}>0$$(i=1,2\mathrm{and}p=1,2,3)$ such that, for $v_i\in {\mathcal{C}}_i$,

$$\begin{array}{cc}\hfill |{\varphi }_i\left(\frac{1}{\chi }{v}_1,\frac{1}{\chi }{v}_2\right)-& N_{i1}{\left(t_1+{}_{i1}{\tilde{\rho }}_1\right)}^{{}_{i1}{\tilde{\rho }}_1}{\left[{\left(t_2+{}_{i1}{\tilde{\rho }}_2\right)}^{{}_{i1}{\tilde{\rho }}_2}\right]}^2{e}^{-n_{i1}(t_1+t_2)}|\hfill \\ \hfill \le & N_{i2}{\left(t_1+{}_{i2}{\tilde{\rho }}_1\right)}^{{}_{i2}{\tilde{\rho }}_1}{\left[{\left(t_2+{}_{i2}{\tilde{\rho }}_2\right)}^{{}_{i2}{\tilde{\rho }}_2}\right]}^2{e}^{-n_{i2}(t_1+t_2)}\parallel v_1\parallel \hfill \\ \hfill +& N_{i3}{\left(t_1+{}_{i3}{\tilde{\rho }}_1\right)}^{{}_{i3}{\tilde{\rho }}_1}{\left[{\left(t_2+{}_{i3}{\tilde{\rho }}_2\right)}^{{}_{i3}{\tilde{\rho }}_2}\right]}^2{e}^{-n_{i3}(t_1+t_2)}\parallel v_2\parallel .\hfill \end{array}$$

$\left(H_3\right)$

$g_i\le g_i\left({\eta }_i\right)$ for all ${\eta }_i\in {\mathbb{N}}_{{\alpha }_i-1,T+{\alpha }_i-1}.$

Lemma 5.

Suppose that $\left(H_1\right)$–$\left(H_3\right)$ hold. Then, the fixed point of $\mathcal{F}$ coincides with the solution of the problems (5) and (6), and $\mathcal{F}:\mathcal{U}\to \mathcal{U}$ is completely continuous.

Proof. 

Let $(u_1,u_2)\in \mathcal{U}$, for each $i,j\in \{1,2\}$ and $i\ne j$. By the above assumptions $\left(H_1\right)$ and $\left(H_2\right)$, it follows that:

$$\begin{array}{cc}& \left|F_i\left(t_1,t_2,{\Delta }^{{\beta }_i}{u}_i(t_i-{\beta }_i+1),u_j\left(t_j\right)\right)\right|\hfill \\ \hfill =& \left|F_i\left(t_1,t_2,\frac{1}{\chi }\left[\chi {\Delta }^{{\beta }_i}{u}_i(t_i-{\beta }_i+1)\right],\frac{1}{\chi }\left[\chi u_j\left(t_j\right)\right]\right)\right|\hfill \\ \hfill \le & M_{i1}{\left(t_2+{}_{i1}{\rho }_2\right)}^{{}_{i1}{\rho }_2}{e}^{-m_{i1}(t_1+t_2)}+M_{i2}{\left(t_2+{}_{i2}{\rho }_2\right)}^{{}_{i2}{\rho }_2}{e}^{-m_{i2}(t_1+t_2)}{\parallel {\Delta }^{{\beta }_i}{u}_i\parallel }_{{\mathcal{C}}_i}\hfill \\ & +M_{i3}{\left(t_2+{}_{i3}{\rho }_2\right)}^{{}_{i3}{\rho }_2}{e}^{-m_{i3}(t_1+t_2)}{\parallel u_j\parallel }_{{\mathcal{C}}_j},\hfill \end{array}$$

(43)

and

$$\begin{array}{cc}\hfill \left|{\varphi }_i\left(u_1,u_2\right)\right|=& \left|{\varphi }_i\left(\frac{1}{\chi }\left[\chi u_1\right],\frac{1}{\chi }\left[\chi u_2\right]\right)\right|\hfill \\ \hfill \le & N_{i1}{\left(t_1+{}_{i1}{\tilde{\rho }}_1\right)}^{{}_{i1}{\tilde{\rho }}_1}{\left[{\left(t_2+{}_{i1}{\tilde{\rho }}_2\right)}^{{}_{i1}{\tilde{\rho }}_2}\right]}^2{e}^{-n_{i1}(t_1+t_2)}\hfill \\ & +N_{i2}{\left(t_1+{}_{i2}{\tilde{\rho }}_1\right)}^{{}_{i2}{\tilde{\rho }}_1}{\left[{\left(t_2+{}_{i2}{\tilde{\rho }}_2\right)}^{{}_{i2}{\tilde{\rho }}_2}\right]}^2{e}^{-n_{i2}(t_1+t_2)}{\parallel u_1\parallel }_{{\mathcal{C}}_1}\hfill \\ & +N_{i3}{\left(t_1+{}_{i3}{\tilde{\rho }}_1\right)}^{{}_{i3}{\tilde{\rho }}_1}{\left[{\left(t_2+{}_{i3}{\tilde{\rho }}_2\right)}^{{}_{i3}{\tilde{\rho }}_2}\right]}^2{e}^{-n_{i3}(t_1+t_2)}{\parallel u_2\parallel }_{{\mathcal{C}}_2}.\hfill \end{array}$$

(44)

The rest of the proof follows from Lemmas 3 and 4. This implies that the fixed point of $\mathcal{F}$ coincides with the solution of the problems (5) and (6).

To show that $\mathcal{F}$ is completely continuous, we organize the proof as the following four steps.

Step I.$\mathcal{F}$ is well defined and maps bounded sets into bounded sets.

Let $B_R=\left\{(u_1,u_2)\in \mathcal{U}:{\parallel (u_1,u_2)\parallel }_{\mathcal{U}}\le R\right\}$, then for $(u_1,u_2)\in \mathcal{U}$:

$$\begin{array}{ccc}\hfill R& \ge & max\left\{\parallel (u_1,u_2){\parallel }_{{\mathcal{U}}_1},{\parallel (u_1,u_2)\parallel }_{{\mathcal{U}}_2}\right\}\hfill \\ & =& max\{\chi \left[\left|{\Delta }^{{\beta }_1}{u}_1(t_1-{\beta }_1+1)\right|+\left|u_2\left(t_2\right)\right|\right],\hfill \\ & & \chi \left[|u_1\left(t_1\right)|+|{\Delta }^{{\beta }_2}{u}_2(t_2-{\beta }_2+1)|\right]\}.\hfill \end{array}$$

(45)

By the definition of $\mathcal{F}$, we get ${\mathcal{F}}_i(u_1,u_2),{\Delta }^{{\beta }_i}{\mathcal{F}}_i(u_1,u_2)\in \mathcal{U}$. Therefore, (43) and (44) imply that:

$$\begin{array}{cc}\hfill \left|F_i\right(t_1,t_2,{\Delta }^{{\beta }_i}& u_i(t_i-{\beta }_i+1),u_j\left(t_j\right)\left)\right|\hfill \\ \hfill =& \left|F_i\left(t_1,t_2,\frac{1}{\chi }\left[\chi {\Delta }^{{\beta }_i}{u}_i(t_i-{\beta }_i+1)\right],\frac{1}{\chi }\left[\chi u_j\left(t_j\right)\right]\right)\right|\hfill \\ \hfill \le & M_{i1}{\left(t_2+{}_{i1}{\rho }_2\right)}^{{}_{i1}{\rho }_2}{e}^{-m_{i1}(t_1+t_2)}+R{M}_{i2}{\left(t_2+{}_{i2}{\rho }_2\right)}^{{}_{i2}{\rho }_2}\times \hfill \\ & e^{-m_{i2}(t_1+t_2)}+R{M}_{i3}{\left(t_2+{}_{i3}{\rho }_2\right)}^{{}_{i3}{\rho }_2}{e}^{-m_{i3}(t_1+t_2)},\hfill \end{array}$$

(46)

and

$$\begin{array}{cc}\hfill \left|{\varphi }_i\left(u_1,u_2\right)\right|=& \left|{\varphi }_i\left(\frac{1}{\chi }\left[\chi u_1\right],\frac{1}{\chi }\left[\chi u_2\right]\right)\right|\hfill \\ \hfill \le & N_{i1}{\left(t_1+{}_{i1}{\tilde{\rho }}_1\right)}^{{}_{i1}{\tilde{\rho }}_1}{\left[{\left(t_2+{}_{i1}{\tilde{\rho }}_2\right)}^{{}_{i1}{\tilde{\rho }}_2}\right]}^2{e}^{-n_{i1}(t_1+t_2)}\hfill \\ & +R{N}_{i2}{\left(t_1+{}_{i2}{\tilde{\rho }}_1\right)}^{{}_{i2}{\tilde{\rho }}_1}{\left[{\left(t_2+{}_{i2}{\tilde{\rho }}_2\right)}^{{}_{i2}{\tilde{\rho }}_2}\right]}^2{e}^{-n_{i2}(t_1+t_2)}\hfill \\ & +R{N}_{i3}{\left(t_1+{}_{i3}{\tilde{\rho }}_1\right)}^{{}_{i3}{\tilde{\rho }}_1}{\left[{\left(t_2+{}_{i3}{\tilde{\rho }}_2\right)}^{{}_{i3}{\tilde{\rho }}_2}\right]}^2{e}^{-n_{i3}(t_1+t_2)}.\hfill \end{array}$$

(47)

Let

$$\begin{array}{cc}\hfill {\tilde{\mathsf{\Omega }}}_1& =:\left(N_{11}+N_{12}R+N_{13}R\right){\tilde{\mathsf{\Omega }}}_{11}+\left(N_{21}+N_{22}R+N_{23}R\right){\tilde{\mathsf{\Omega }}}_{12}\hfill \\ & +\left(M_{11}+M_{12}R+M_{13}R\right){\tilde{\mathsf{\Omega }}}_{13}+\left(M_{21}+M_{22}R+M_{23}R\right){\tilde{\mathsf{\Omega }}}_{14}\hfill \\ \hfill {\tilde{\mathsf{\Omega }}}_2& =:\left(N_{11}+N_{12}R+N_{13}R\right){\tilde{\mathsf{\Omega }}}_{21}+\left(N_{21}+N_{22}R+N_{23}R\right){\tilde{\mathsf{\Omega }}}_{22}\hfill \\ & +\left(M_{11}+M_{12}R+M_{13}R\right){\tilde{\mathsf{\Omega }}}_{23}+\left(M_{21}+M_{22}R+M_{23}R\right){\tilde{\mathsf{\Omega }}}_{24}\hfill \end{array}$$

where:

$$\begin{array}{cc}\hfill {\tilde{\mathsf{\Omega }}}_{11}& =\left[max\left\{\frac{1}{\mathsf{\Gamma }\left({\theta }_1\right)},\frac{{\lambda }_1{G}_1\mathsf{\Gamma }\left({\alpha }_2\right)}{\mathsf{\Gamma }\left({\theta }_1\right)}{({\eta }_1-{\alpha }_1+{\theta }_1)}^{\underline{{\theta }_1-1}}{\mathcal{B}}_1\right\}+\frac{1}{\mathsf{\Gamma }\left({\alpha }_1\right)}\right],\hfill \\ \hfill {\tilde{\mathsf{\Omega }}}_{12}& =max\left\{\frac{1}{\mathsf{\Gamma }\left({\theta }_2\right)},\frac{{\lambda }_2{G}_2\mathsf{\Gamma }\left({\alpha }_1\right)}{\mathsf{\Gamma }\left({\theta }_2\right)}{({\eta }_2-{\alpha }_2+{\theta }_2)}^{\underline{{\theta }_2-1}}{\mathcal{B}}_2\right\},\hfill \\ \hfill {\tilde{\mathsf{\Omega }}}_{13}& =\left[max\left\{1,\frac{{\lambda }_1{G}_1\mathsf{\Gamma }\left({\alpha }_2\right)}{\mathsf{\Gamma }\left({\theta }_1\right)}{({\eta }_1-{\alpha }_1+{\theta }_1-1)}^{\underline{{\theta }_1-1}}{\mathcal{C}}_1\right\}+\frac{1}{\mathsf{\Gamma }\left({\alpha }_1\right)}\right],\hfill \\ \hfill {\tilde{\mathsf{\Omega }}}_{14}& =max\left\{1,\frac{{\lambda }_2{G}_2\mathsf{\Gamma }\left({\alpha }_1\right)}{\mathsf{\Gamma }\left({\theta }_2\right)}{({\eta }_2-{\alpha }_2+{\theta }_2-1)}^{\underline{{\theta }_2-1}}{\mathcal{C}}_2\right\},\hfill \\ \hfill {\tilde{\mathsf{\Omega }}}_{21}& =max\left\{\frac{1}{G_2{\lambda }_2\mathsf{\Gamma }\left({\alpha }_1\right){({\eta }_2-{\alpha }_2+{\theta }_2)}^{\underline{{\theta }_2-1}}{\mathcal{A}}_2},\frac{{\mathcal{B}}_1}{\mathsf{\Gamma }\left({\theta }_1\right){\mathcal{A}}_1}\right\},\hfill \\ \hfill {\tilde{\mathsf{\Omega }}}_{22}& =\left[max\left\{\frac{1}{G_1{\lambda }_1\mathsf{\Gamma }\left({\alpha }_2\right){({\eta }_1-{\alpha }_1+{\theta }_1)}^{\underline{{\theta }_1-1}}{\mathcal{A}}_1},\frac{{\mathcal{B}}_2}{\mathsf{\Gamma }\left({\theta }_2\right){\mathcal{A}}_2}\right\}+\frac{1}{\mathsf{\Gamma }\left({\alpha }_2\right)}\right],\hfill \\ \hfill {\tilde{\mathsf{\Omega }}}_{23}& =max\left\{\frac{1}{G_2{\lambda }_2\mathsf{\Gamma }\left({\alpha }_1\right){({\eta }_2-{\alpha }_2+{\theta }_2)}^{\underline{{\theta }_2-1}}{\mathcal{A}}_2},\frac{{\mathcal{C}}_1}{\mathsf{\Gamma }\left({\theta }_1\right){\mathcal{A}}_1}\right\},\hfill \\ \hfill {\tilde{\mathsf{\Omega }}}_{24}& =\left[max\left\{\frac{1}{G_1{\lambda }_1\mathsf{\Gamma }\left({\alpha }_2\right){({\eta }_1-{\alpha }_1+{\theta }_1)}^{\underline{{\theta }_1-1}}{\mathcal{A}}_1},\frac{{\mathcal{C}}_2}{\mathsf{\Gamma }\left({\theta }_2\right){\mathcal{A}}_2}\right\}+\frac{1}{\mathsf{\Gamma }\left({\alpha }_2\right)}\right].\hfill \end{array}$$

Hence, we obtain:

$$\begin{array}{c}\chi \left|\left({\mathcal{F}}_1(u_1,u_2)\right)(t_1,t_2)\right|\hfill \\ \le \left(N_{11}+N_{12}R+N_{13}R\right)\left[max\left\{\frac{1}{\mathsf{\Gamma }\left({\theta }_1\right)},\frac{{\lambda }_1{G}_1\mathsf{\Gamma }\left({\alpha }_2\right)}{\mathsf{\Gamma }\left({\theta }_1\right)}{({\eta }_1-{\alpha }_1+{\theta }_1)}^{\underline{{\theta }_1-1}}{\mathcal{B}}_1\right\}+\frac{1}{\mathsf{\Gamma }\left({\alpha }_1\right)}\right]\hfill \\ +\left(N_{21}+N_{22}R+N_{23}R\right)max\left\{\frac{1}{\mathsf{\Gamma }\left({\theta }_2\right)},\frac{{\lambda }_2{G}_2\mathsf{\Gamma }\left({\alpha }_1\right)}{\mathsf{\Gamma }\left({\theta }_2\right)}{({\eta }_2-{\alpha }_2+{\theta }_2)}^{\underline{{\theta }_2-1}}{\mathcal{B}}_2\right\}\hfill \\ +\left(M_{11}+M_{12}R+M_{13}R\right)\left[max\left\{1,\frac{{\lambda }_1{G}_1\mathsf{\Gamma }\left({\alpha }_2\right)}{\mathsf{\Gamma }\left({\theta }_1\right)}{({\eta }_1-{\alpha }_1+{\theta }_1-1)}^{\underline{{\theta }_1-1}}{\mathcal{C}}_1\right\}+\frac{1}{\mathsf{\Gamma }\left({\alpha }_1\right)}\right]\hfill \\ +\left(M_{21}+M_{22}R+M_{23}R\right)max\left\{1,\frac{{\lambda }_2{G}_2\mathsf{\Gamma }\left({\alpha }_1\right)}{\mathsf{\Gamma }\left({\theta }_2\right)}{({\eta }_2-{\alpha }_2+{\theta }_2-1)}^{\underline{{\theta }_2-1}}{\mathcal{C}}_2\right\}\hfill \\ ={\tilde{\mathsf{\Omega }}}_1,\hfill \end{array}$$

(48)

and:

$$\begin{array}{c}\chi \left|\left({\mathcal{F}}_2(u_1,u_2)\right)(t_1,t_2)\right|\hfill \\ \le \left(N_{11}+N_{12}R+N_{13}R\right)max\left\{\frac{1}{G_2{\lambda }_2\mathsf{\Gamma }\left({\alpha }_1\right){({\eta }_2-{\alpha }_2+{\theta }_2)}^{\underline{{\theta }_2-1}}{\mathcal{A}}_2},\frac{{\mathcal{B}}_1}{\mathsf{\Gamma }\left({\theta }_1\right){\mathcal{A}}_1}\right\}\hfill \\ +\left(N_{21}+N_{22}R+N_{23}R\right)\times \hfill \\ \left[max\left\{\frac{1}{G_1{\lambda }_1\mathsf{\Gamma }\left({\alpha }_2\right){({\eta }_1-{\alpha }_1+{\theta }_1)}^{\underline{{\theta }_1-1}}{\mathcal{A}}_1},\frac{{\mathcal{B}}_2}{\mathsf{\Gamma }\left({\theta }_2\right){\mathcal{A}}_2}\right\}+\frac{1}{\mathsf{\Gamma }\left({\alpha }_2\right)}\right]\hfill \\ +\left(M_{11}+M_{12}R+M_{13}R\right)max\left\{\frac{1}{G_2{\lambda }_2\mathsf{\Gamma }\left({\alpha }_1\right){({\eta }_2-{\alpha }_2+{\theta }_2)}^{\underline{{\theta }_2-1}}{\mathcal{A}}_2},\frac{{\mathcal{C}}_1}{\mathsf{\Gamma }\left({\theta }_1\right){\mathcal{A}}_1}\right\}\hfill \\ +\left(M_{21}+M_{22}R+M_{23}R\right)\times \hfill \\ \left[max\left\{\frac{1}{G_1{\lambda }_1\mathsf{\Gamma }\left({\alpha }_2\right){({\eta }_1-{\alpha }_1+{\theta }_1)}^{\underline{{\theta }_1-1}}{\mathcal{A}}_1},\frac{{\mathcal{C}}_2}{\mathsf{\Gamma }\left({\theta }_2\right){\mathcal{A}}_2}\right\}+\frac{1}{\mathsf{\Gamma }\left({\alpha }_2\right)}\right]\hfill \\ ={\tilde{\mathsf{\Omega }}}_2.\hfill \end{array}$$

(49)

Similarly, we have:

$$\chi \left|{\Delta }^{{\beta }_1}\left({\mathcal{F}}_1(u_1,u_2)\right)(t_1-{\beta }_1+1,t_2)\right|<{\tilde{\mathsf{\Omega }}}_1,$$

(50)

$$\chi \left|{\Delta }^{{\beta }_2}\left({\mathcal{F}}_2(u_1,u_2)\right)(t_1,t_2-{\beta }_2+1)\right|<{\tilde{\mathsf{\Omega }}}_2.$$

(51)

Therefore, ${\mathcal{F}}_i(u_1,u_2)\in \mathcal{U}$. This implies that $\mathcal{F}:\mathcal{U}⟶\mathcal{U}$ is well defined.

Furthermore, we obtain:

$$\begin{array}{cc}\hfill \parallel \mathcal{F}(u_1,u_2){\parallel }_{{\mathcal{U}}_i}& =max\left\{\chi \right|{\Delta }^{{\beta }_i}\left({\mathcal{F}}_i(u_1,u_2)\right)(t_i-{\beta }_i+1,t_j)|\hfill \\ & +\chi \left|\left({\mathcal{F}}_j(u_1,u_2)\right)(t_i,t_j)\right|\mathrm{for}i,j\in \{1,2\},i\ne j\}.\hfill \end{array}$$

(52)

Hence,

$$\begin{array}{ccc}\hfill \parallel \mathcal{F}(u_1,u_2){\parallel }_{\mathcal{U}}& =& max\left\{\parallel \mathcal{F}(u_1,u_2){\parallel }_{{\mathcal{U}}_1},{\parallel \mathcal{F}(u_1,u_2)\parallel }_{{\mathcal{U}}_2}\right\}<{\tilde{\mathsf{\Omega }}}_1+{\tilde{\mathsf{\Omega }}}_2.\hfill \end{array}$$

(53)

Thus, $\mathcal{F}$ maps bounded sets into bounded sets.

Step II.$\mathcal{F}$ is continuous.

Let $ϵ>0$ be given. Since $F_i$ and ${\varphi }_i$ are continuous, then $F_i$ and ${\varphi }_i$ are uniformly continuous. Therefore, there exists $\delta =min\left\{{\delta }_i,{\widehat{\delta }}_i\right\}>0$ such that, for each $t_i\in {\mathcal{N}}_{{\alpha }_i-2}$, $u_i,v_i\in {\mathcal{C}}_i$ with $max\left\{\chi \right|{\Delta }^{{\beta }_i}{u}_i(t_i-{\beta }_i+1)-{\Delta }^{{\beta }_i}{v}_i\left(t_i\right)|+\chi |u_j\left(t_i\right)-v_j\left(t_j\right)|\}<{\delta }_i$,

$$\begin{array}{cc}& |F_i\left(t_1,t_2,{\Delta }^{{\beta }_i}{u}_i(t_i-{\beta }_i+1),u_j\left(t_j\right)\right)-F_i\left(t_1,t_2,{\Delta }^{{\beta }_i}{v}_i(t_i-{\beta }_i+1),v_j\left(t_j\right)\right)|\hfill \\ \hfill =& |F_i\left(t_1,t_2,\frac{1}{\chi }\left[\chi {\Delta }^{{\beta }_i}{u}_i(t_i-{\beta }_i+1)\right],\frac{1}{\chi }\left[\chi u_j\left(t_j\right)\right]\right)\hfill \\ \hfill -& F_i\left(t_1,t_2,\frac{1}{\chi }\left[\chi {\Delta }^{{\beta }_i}{v}_i(t_i-{\beta }_i+1)\right],\frac{1}{\chi }\left[\chi v_j\left(t_j\right)\right]\right)|\hfill \\ \hfill <& 2M_{i1}+2M_{i2}R+2M_{i3}R<\frac{ϵ}{4{\mathsf{\Omega }}_i}.\hfill \end{array}$$

(54)

For each $u_i,v_i\in {\mathcal{C}}_i$ with $|u_i-v_i|<{\widehat{\delta }}_i$,

$$\begin{array}{cc}\hfill |{\varphi }_i\left(u_1,u_2\right)-{\varphi }_i\left(v_1,v_2\right)|=& |{\varphi }_i\left(\frac{1}{\chi }\left[\chi u_1\right],\frac{1}{\chi }\left[\chi u_2\right]\right)-{\varphi }_i\left(\frac{1}{\chi }\left[\chi v_1\right],\frac{1}{\chi }\left[\chi v_2\right]\right)|\hfill \\ \hfill <& 2N_{i1}+2N_{i2}R+2N_{i3}R<\frac{ϵ}{4{\mathsf{\Omega }}_i}.\hfill \end{array}$$

(55)

Similar to Step I, we obtain:

$$\chi |\left({\mathcal{F}}_i(u_1,u_2)\right)-\left({\mathcal{F}}_i(v_1,v_2)\right)|<2{\tilde{\mathsf{\Omega }}}_i<\frac{ϵ}{2}$$

and $\chi |{\Delta }^{{\beta }_i}\left({\mathcal{F}}_i(u_1,u_2)\right)-{\Delta }^{{\beta }_i}\left({\mathcal{F}}_i(v_1,v_2)\right)|<2{\tilde{\mathsf{\Omega }}}_i<\frac{ϵ}{2}$.

Thus, we have:

$$\begin{array}{cc}& \parallel {\mathcal{F}}_i(u_1,u_2)-{\mathcal{F}}_i(v_1,v_2){\parallel }_{{\mathcal{U}}_i}\hfill \\ =& \parallel {\Delta }^{{\beta }_i}{\mathcal{F}}_i(u_1,u_2)-{\Delta }^{{\beta }_i}{\mathcal{F}}_i(v_1,v_2){\parallel }_{{\mathcal{C}}_i}+{\parallel {\mathcal{F}}_j(u_1,u_2)-{\mathcal{F}}_j(v_1,v_2)\parallel }_{{\mathcal{C}}_j}\hfill \\ <& 2({\tilde{\mathsf{\Omega }}}_1+{\tilde{\mathsf{\Omega }}}_2)<ϵ.\hfill \end{array}$$

(56)

This means that each ${\mathcal{F}}_i,i=1,2$ is continuous. This shows $\mathcal{F}$ is continuous.

In order to prove that $\mathcal{F}$ maps bounded sets of $\mathcal{U}\subset {\mathcal{C}}_1\times {\mathcal{C}}_2$ to relatively compact sets of $\mathcal{U}\subset {\mathcal{C}}_1\times {\mathcal{C}}_2$, it suffices to show that both ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$ map bounded sets to relatively compact sets. Let ${\mathsf{\Theta }}_i\subset {\mathcal{C}}_i$, $i=1,2$ be bounded sets and ${\mathsf{\Theta }}_1\times {\mathsf{\Theta }}_2\subset \mathcal{U}$. Recall that ${\mathsf{\Theta }}_i$ are relatively compact if:

• both ${\mathsf{\Theta }}_i$ are bounded,
• both $\chi {\mathsf{\Theta }}_i$ are equicontinuous on any closed subintervals of ${\mathbb{N}}_{{\alpha }_i-2}$,
• both $\chi {\mathsf{\Theta }}_i$ are equiconvergent as $t_i\to \infty$.

It has been shown from in Step I that both ${\mathcal{F}}_i$ are uniformly bounded. Now, we show that ${\mathcal{F}}_i$ maps bounded sets into equicontinuous sets of $\mathcal{U}$.

Step III. Both ${\mathcal{F}}_i:{\mathsf{\Theta }}_1\times {\mathsf{\Theta }}_2\to \mathcal{U}$ are equicontinuous on $\left([a_1,b_1]\cap {\mathbb{N}}_{{\alpha }_1-2}\right)\times \left([a_2,b_2]\cap {\mathbb{N}}_{{\alpha }_2-2}\right):=\mathcal{D}$.

For any $ϵ>0$, there exists $\delta >0$ such that, for each $t_{i1},t_{i2}\in {\mathbb{N}}_{{\alpha }_i-2}\cap [a_i,b_i]$,

$$\begin{array}{c}\hfill \left|{(t_{11}+{\rho }_1)}^{{\underline{\rho }}_1}{(t_{21}+{\rho }_2)}^{{\underline{\rho }}_2}-{(t_{12}+{\rho }_1)}^{{\underline{\rho }}_1}{(t_{22}+{\rho }_2)}^{{\underline{\rho }}_2}\right|\le \frac{ϵ}{2max\{{\tilde{\mathsf{\Omega }}}_1^{*}+{\tilde{\mathsf{\Omega }}}_2^{*}\}}=\delta ,\end{array}$$

(57)

where:

$$\begin{array}{ccc}\hfill {\tilde{\mathsf{\Omega }}}_1^{*}& =& \left([N_{11}+N_{12}R+N_{13}R]+[M_{11}+M_{12}R+M_{13}R]\right)\left[\frac{1}{\mathsf{\Gamma }\left({\theta }_1\right)}+\frac{1}{\mathsf{\Gamma }\left({\alpha }_1\right)}\right]\hfill \\ & & +\left([N_{11}+N_{12}R+N_{13}R]{\mathcal{B}}_1+[M_{11}+M_{12}R+M_{13}R]{\mathcal{C}}_1\right)\times \hfill \\ & & \frac{{\lambda }_1{G}_1\mathsf{\Gamma }\left({\alpha }_2\right){({\eta }_1-{\alpha }_1+{\theta }_1)}^{\underline{{\theta }_1-1}}}{\mathsf{\Gamma }\left({\theta }_1\right)\mathsf{\Gamma }\left({\theta }_2\right)}\hfill \\ & & +\left([N_{21}+N_{22}R+N_{23}R]+[M_{21}+M_{22}R+M_{23}R]\right)\frac{1}{\mathsf{\Gamma }\left({\theta }_2\right)}\hfill \\ & & +\left([N_{21}+N_{22}R+N_{23}R]{\mathcal{B}}_2+[M_{21}+M_{22}R+M_{23}R]{\mathcal{C}}_2\right)\times \hfill \\ & & \frac{{\lambda }_2{G}_2\mathsf{\Gamma }\left({\alpha }_1\right){({\eta }_2-{\alpha }_2+{\theta }_2)}^{\underline{{\theta }_2-1}}}{\mathsf{\Gamma }\left({\theta }_1\right)\mathsf{\Gamma }\left({\theta }_2\right)},\hfill \end{array}$$

(58)

$$\begin{array}{ccc}\hfill {\tilde{\mathsf{\Omega }}}_2^{*}& =& [N_{11}+N_{12}R+N_{13}R]\left[\frac{1}{{\lambda }_2{G}_2{({\eta }_2-{\alpha }_2+{\theta }_2)}^{\underline{{\theta }_2-1}}{\mathcal{A}}_2}+\frac{1}{\mathsf{\Gamma }\left({\alpha }_2\right)}\right]\hfill \\ & & +[N_{21}+N_{22}R+N_{23}R]\frac{1}{{\lambda }_1{G}_1{({\eta }_1-{\alpha }_1+{\theta }_1)}^{\underline{{\theta }_1-1}}{\mathcal{A}}_1}\hfill \\ & & +[M_{11}+M_{12}R+M_{13}R]\frac{{\mathcal{B}}_1}{\mathsf{\Gamma }\left({\theta }_1\right){\mathcal{A}}_1}+[M_{21}+M_{22}R+M_{23}R]\frac{{\mathcal{B}}_2}{\mathsf{\Gamma }\left({\theta }_2\right){\mathcal{A}}_2}.\hfill \end{array}$$

(59)

Hence, for each $t_{i1},t_{i2}\in {\mathbb{N}}_{{\alpha }_i-2}\cap [a_i,b_i]$, and $u_i\in {\mathsf{\Theta }}_i$, we have:

$$\begin{array}{cc}& |\chi \left({\mathcal{F}}_1{u}_1\right)(t_{11},t_{21})-\chi \left({\mathcal{F}}_1{u}_1\right)(t_{12},t_{22})|\hfill \\ \le & \chi \left|t_{11}^{\underline{{\alpha }_1}}\right\{\left[\frac{|{\varphi }_1(u_1,u_2)|}{\mathsf{\Gamma }\left({\theta }_1\right)}-\frac{|{\varphi }_2(u_1,u_2)|}{\mathsf{\Gamma }\left({\theta }_2\right)}+\frac{|{\varphi }_1(u_1,u_2)|}{\mathsf{\Gamma }\left({\alpha }_1\right)}\right]+\frac{1}{\mathsf{\Gamma }\left({\theta }_1\right)\mathsf{\Gamma }\left({\theta }_2\right)}\times \hfill \\ & \left[\frac{{\lambda }_1\mathsf{\Gamma }\left({\alpha }_2\right){\mathcal{B}}_1\left|{\varphi }_1(u_1,u_2)\right|}{t_{21}^{\underline{{\alpha }_2-1}}\mathsf{\Gamma }\left({\alpha }_1\right)}+\frac{{\lambda }_2\mathsf{\Gamma }\left({\alpha }_1\right){\mathcal{B}}_2\left|{\varphi }_2(u_1,u_2)\right|}{t_{11}^{\underline{{\alpha }_1-1}}\mathsf{\Gamma }\left({\alpha }_2\right)}\right]+\frac{1}{\mathsf{\Gamma }\left({\theta }_1\right)\mathsf{\Gamma }\left({\theta }_2\right)}\times \hfill \\ & +\left[\frac{{\lambda }_1\mathsf{\Gamma }\left({\alpha }_2\right){\mathcal{B}}_1\left|F_1(t_1,t_2,{\Delta }^{{\beta }_1}{u}_1,u_2)\right|}{t_{22}^{\underline{{\alpha }_2-1}}\mathsf{\Gamma }\left({\alpha }_1\right)}+\frac{{\lambda }_2\mathsf{\Gamma }\left({\alpha }_1\right){\mathcal{B}}_2\left|F_2(t_1,t_2,u_1,{\Delta }^{{\beta }_2}{u}_2)\right|}{t_{12}^{\underline{{\alpha }_1-1}}\mathsf{\Gamma }\left({\alpha }_2\right)}\right]\hfill \\ & \left[\frac{|F_1(t_1,t_2,{\Delta }^{{\beta }_1}{u}_1,u_2)|}{\mathsf{\Gamma }\left({\theta }_1\right)}-\frac{|F_2(t_1,t_2,u_1,{\Delta }^{{\beta }_2}{u}_2)|}{\mathsf{\Gamma }\left({\theta }_2\right)}+\frac{|F_1(t_1,t_2,{\Delta }^{{\beta }_1}{u}_1,u_2)|}{\mathsf{\Gamma }\left({\alpha }_1\right)}\right]\}\hfill \\ & -t_{12}^{\underline{{\alpha }_1}}\{\left[\frac{|{\varphi }_1(u_1,u_2)|}{\mathsf{\Gamma }\left({\theta }_1\right)}-\frac{|{\varphi }_2(u_1,u_2)|}{\mathsf{\Gamma }\left({\theta }_2\right)}+\frac{|{\varphi }_1(u_1,u_2)|}{\mathsf{\Gamma }\left({\alpha }_1\right)}\right]+\frac{1}{\mathsf{\Gamma }\left({\theta }_1\right)\mathsf{\Gamma }\left({\theta }_2\right)}\times \hfill \\ & \left[\frac{{\lambda }_1\mathsf{\Gamma }\left({\alpha }_2\right){\mathcal{B}}_1\left|{\varphi }_1(u_1,u_2)\right|}{t_{21}^{\underline{{\alpha }_2-1}}\mathsf{\Gamma }\left({\alpha }_1\right)}+\frac{{\lambda }_2\mathsf{\Gamma }\left({\alpha }_1\right){\mathcal{B}}_2\left|{\varphi }_2(u_1,u_2)\right|}{t_{11}^{\underline{{\alpha }_1-1}}\mathsf{\Gamma }\left({\alpha }_2\right)}\right]+\frac{1}{\mathsf{\Gamma }\left({\theta }_1\right)\mathsf{\Gamma }\left({\theta }_2\right)}\times \hfill \\ & +\left[\frac{{\lambda }_1\mathsf{\Gamma }\left({\alpha }_2\right){\mathcal{B}}_1\left|F_1(t_1,t_2,{\Delta }^{{\beta }_1}{u}_1,u_2)\right|}{t_{22}^{\underline{{\alpha }_2-1}}\mathsf{\Gamma }\left({\alpha }_1\right)}+\frac{{\lambda }_2\mathsf{\Gamma }\left({\alpha }_1\right){\mathcal{B}}_2\left|F_2(t_1,t_2,u_1,{\Delta }^{{\beta }_2}{u}_2)\right|}{t_{12}^{\underline{{\alpha }_1-1}}\mathsf{\Gamma }\left({\alpha }_2\right)}\right]\hfill \\ & \left[\frac{|F_1(t_1,t_2,{\Delta }^{{\beta }_1}{u}_1,u_2)|}{\mathsf{\Gamma }\left({\theta }_1\right)}-\frac{|F_2(t_1,t_2,u_1,{\Delta }^{{\beta }_2}{u}_2)|}{\mathsf{\Gamma }\left({\theta }_2\right)}+\frac{|F_1(t_1,t_2,{\Delta }^{{\beta }_1}{u}_1,u_2)|}{\mathsf{\Gamma }\left({\alpha }_1\right)}\right]\left\}\right|\hfill \\ <& |{(t_{11}+{\rho }_1)}^{\underline{{\rho }_1}}{(t_{21}+{\rho }_2)}^{\underline{{\rho }_2}}-{(t_{12}+{\rho }_1)}^{\underline{{\rho }_1}}{(t_{22}+{\rho }_2)}^{\underline{{\rho }_2}}|{\tilde{\mathsf{\Omega }}}_1^{*}\hfill \\ <& \frac{ϵ}{2},\hfill \end{array}$$

(60)

and:

$$\begin{array}{cc}& |\chi \left({\mathcal{F}}_2{u}_2\right)(t_{11},t_{21})-\chi \left({\mathcal{F}}_2{u}_2\right)(t_{12},t_{22})|\hfill \\ \le & \chi \left|t_{21}^{\underline{{\alpha }_2}}\right\{\left[\frac{t_{11}^{\underline{{\alpha }_1-1}}\left|{\varphi }_1(u_1,u_2)\right|}{{\lambda }_2{G}_2{({\eta }_2-{\alpha }_2+{\theta }_2)}^{\underline{{\theta }_2-1}}{\mathcal{A}}_2}-\frac{t_{21}^{\underline{{\alpha }_2-1}}\left|{\varphi }_2(u_1,u_2)\right|}{{\lambda }_1{G}_1{({\eta }_1-{\alpha }_1+{\theta }_1)}^{\underline{{\theta }_1-1}}{\mathcal{A}}_1}+\frac{|{\varphi }_2(u_1,u_2)|}{\mathsf{\Gamma }\left({\alpha }_2\right)}\right]\hfill \\ & +t_{21}^{\underline{{\alpha }_2-1}}\left[\frac{{\mathcal{B}}_1\left|{\varphi }_1(u_1,u_2)\right|}{\mathsf{\Gamma }\left({\theta }_1\right){\mathcal{A}}_1}-\frac{{\mathcal{B}}_2\left|{\varphi }_2(u_1,u_2)\right|}{\mathsf{\Gamma }\left({\theta }_2\right){\mathcal{A}}_2}\right]+[\frac{t_{11}^{\underline{{\alpha }_1-1}}\left|F_1(t_1,t_2,{\Delta }^{{\beta }_1}{u}_1,u_2)\right|}{{\lambda }_2{G}_2{({\eta }_2-{\alpha }_2+{\theta }_2)}^{\underline{{\theta }_2-1}}{\mathcal{A}}_2}\hfill \\ & -\frac{t_{21}^{\underline{{\alpha }_2-1}}\left|F_2(t_1,t_2,u_1,{\Delta }^{{\beta }_2}{u}_2)\right|}{{\lambda }_1{G}_1{({\eta }_1-{\alpha }_1+{\theta }_1)}^{\underline{{\theta }_1-1}}{\mathcal{A}}_1}+\frac{|F_2(t_1,t_2,u_1,{\Delta }^{{\beta }_2}{u}_2)|}{\mathsf{\Gamma }\left({\alpha }_2\right)}]\hfill \\ & +t_{21}^{\underline{{\alpha }_2-1}}\left[\frac{t_{11}^{\underline{{\alpha }_1}}{\mathcal{C}}_1\left|F_1(t_1,t_2,{\Delta }^{{\beta }_1}{u}_1,u_2)\right|}{\mathsf{\Gamma }\left({\theta }_1\right){\mathcal{A}}_1}-\frac{t_{21}^{\underline{{\alpha }_2}}{\mathcal{C}}_2\left|F_2(t_1,t_2,u_1,{\Delta }^{{\beta }_2}{u}_2)\right|}{\mathsf{\Gamma }\left({\theta }_2\right){\mathcal{A}}_2}\right]\}\hfill \\ & -t_{22}^{\underline{{\alpha }_2}}\{\left[\frac{t_{11}^{\underline{{\alpha }_1-1}}\left|{\varphi }_1(u_1,u_2)\right|}{{\lambda }_2{G}_2{({\eta }_2-{\alpha }_2+{\theta }_2)}^{\underline{{\theta }_2-1}}{\mathcal{A}}_2}-\frac{t_{21}^{\underline{{\alpha }_2-1}}\left|{\varphi }_2(u_1,u_2)\right|}{{\lambda }_1{G}_1{({\eta }_1-{\alpha }_1+{\theta }_1)}^{\underline{{\theta }_1-1}}{\mathcal{A}}_1}+\frac{|{\varphi }_2(u_1,u_2)|}{\mathsf{\Gamma }\left({\alpha }_2\right)}\right]\hfill \\ & +t_{21}^{\underline{{\alpha }_2-1}}\left[\frac{{\mathcal{B}}_1\left|{\varphi }_1(u_1,u_2)\right|}{\mathsf{\Gamma }\left({\theta }_1\right){\mathcal{A}}_1}-\frac{{\mathcal{B}}_2\left|{\varphi }_2(u_1,u_2)\right|}{\mathsf{\Gamma }\left({\theta }_2\right){\mathcal{A}}_2}\right]+[\frac{t_{11}^{\underline{{\alpha }_1-1}}\left|F_1(t_1,t_2,{\Delta }^{{\beta }_1}{u}_1,u_2)\right|}{{\lambda }_2{G}_2{({\eta }_2-{\alpha }_2+{\theta }_2)}^{\underline{{\theta }_2-1}}{\mathcal{A}}_2}\hfill \\ & -\frac{t_{21}^{\underline{{\alpha }_2-1}}\left|F_2(t_1,t_2,u_1,{\Delta }^{{\beta }_2}{u}_2)\right|}{{\lambda }_1{G}_1{({\eta }_1-{\alpha }_1+{\theta }_1)}^{\underline{{\theta }_1-1}}{\mathcal{A}}_1}+\frac{|F_2(t_1,t_2,u_1,{\Delta }^{{\beta }_2}{u}_2)|}{\mathsf{\Gamma }\left({\alpha }_2\right)}]\hfill \\ & +t_{21}^{\underline{{\alpha }_2-1}}\left[\frac{t_{11}^{\underline{{\alpha }_1}}{\mathcal{C}}_1\left|F_1(t_1,t_2,{\Delta }^{{\beta }_1}{u}_1,u_2)\right|}{\mathsf{\Gamma }\left({\theta }_1\right){\mathcal{A}}_1}-\frac{t_{21}^{\underline{{\alpha }_2}}{\mathcal{C}}_2\left|F_2(t_1,t_2,u_1,{\Delta }^{{\beta }_2}{u}_2)\right|}{\mathsf{\Gamma }\left({\theta }_2\right){\mathcal{A}}_2}\right]\}\hfill \\ <& |{(t_{11}+{\rho }_1)}^{\underline{{\rho }_1}}{(t_{21}+{\rho }_2)}^{\underline{{\rho }_2}}-{(t_{12}+{\rho }_1)}^{\underline{{\rho }_1}}{(t_{22}+{\rho }_2)}^{\underline{{\rho }_2}}|{\tilde{\mathsf{\Omega }}}_2^{*}\hfill \\ <& \frac{ϵ}{2}.\hfill \end{array}$$

(61)

Similarly, for each $i,j\in \{1,2\}$ and $j\ne i$, we obtain:

$$\begin{array}{c}\hfill |{\Delta }^{{\beta }_i}\left({\mathcal{F}}_i(u_1,u_2)\right)(t_{i1}-{\beta }_i+1,t_{j1})-{\Delta }^{{\beta }_i}\left({\mathcal{F}}_i(u_1,u_2)\right)(t_{i2}-{\beta }_i+1,t_{j2})|<\frac{ϵ}{2}.\end{array}$$

(62)

Hence:

$$\begin{array}{cc}& \parallel {\mathcal{F}}_i(u_1,u_2)(t_{11},t_{21})-{\mathcal{F}}_i(u_1,u_2)(t_{12},t_{22}){\parallel }_{{\mathcal{U}}_i}\hfill \\ \hfill =& \parallel {\Delta }^{{\beta }_i}{\mathcal{F}}_i(u_1,u_2)(t_{i1}-{\beta }_i+1,t_{j1})-{\Delta }^{{\beta }_i}{\mathcal{F}}_i(u_1,u_2)(t_{i2}-{\beta }_i+1,t_{j2}){\parallel }_{{\mathcal{C}}_i}\hfill \\ \hfill +& \parallel {\mathcal{F}}_j(u_1,u_2)(t_{11},t_{21})-{\mathcal{F}}_j(u_1,u_2)(t_{12},t_{22}){\parallel }_{{\mathcal{C}}_j}\hfill \\ \hfill <& \frac{ϵ}{2}+\frac{ϵ}{2}=ϵ.\hfill \end{array}$$

(63)

This implies that both ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$ are equicontinuous on $\mathcal{D}$, which shows that $\mathcal{F}$ is equicontinuous on $\mathcal{D}$. Therefore, by the Arzelá–Ascoli theorem and Theorem 2, we can conclude that $\mathcal{F}$ is completely continuous.

Step IV. Both ${\mathcal{F}}_i:{\mathsf{\Theta }}_1\times {\mathsf{\Theta }}_2\to \mathcal{U}$ are equiconvergent as $t_1,t_2\to \infty$.

By the assumption $\left(H1\right)-\left(H2\right)$, we obtain:

$$\begin{array}{c}\hfill \chi |\left({\mathcal{F}}_1(u_1,u_2)\right)(t_1,t_2)|<\frac{{\tilde{\mathsf{\Omega }}}_1^{*}}{t_1^{\underline{{\alpha }_1}}{t}_2^{\underline{{\alpha }_2}}}\to 0\mathrm{uniformly}\mathrm{in}{\mathsf{\Theta }}_1\times {\mathsf{\Theta }}_2\mathrm{as}{t}_1,t_2\to \infty ,\\ \hfill \chi |\left({\mathcal{F}}_2(u_1,u_2)\right)(t_1,t_2)|<\frac{{\tilde{\mathsf{\Omega }}}_2^{*}}{t_1^{\underline{{\alpha }_1}}{t}_2^{\underline{{\alpha }_2}}}\to 0\mathrm{uniformly}\mathrm{in}{\mathsf{\Theta }}_1\times {\mathsf{\Theta }}_2\mathrm{as}{t}_1,t_2\to \infty ,\end{array}$$

where ${\tilde{\mathsf{\Omega }}}_1^{*},{\tilde{\mathsf{\Omega }}}_2^{*}$ are defined as (58) and (59).

Furthermore, we have:

$$\begin{array}{ccc}\hfill \chi \left|{\Delta }^{{\beta }_1}\left({\mathcal{F}}_1(u_1,u_2)\right)(t_1-{\beta }_1+1,t_2)\right|& <& \frac{{\tilde{\mathsf{\Omega }}}_1^{*}}{t_1^{\underline{{\alpha }_1}}{t}_2^{\underline{{\alpha }_2}}}\to 0\hfill \\ & & \mathrm{uniformly}\mathrm{in}{\mathsf{\Theta }}_1\times {\mathsf{\Theta }}_2\mathrm{as}{t}_1,t_2\to \infty ,\hfill \\ \hfill \chi \left|{\Delta }^{{\beta }_2}\left({\mathcal{F}}_2(u_1,u_2)\right)(t_1,t_2-{\beta }_2+1)\right|& <& \frac{{\tilde{\mathsf{\Omega }}}_2^{*}}{t_1^{\underline{{\alpha }_1}}{t}_2^{\underline{{\alpha }_2}}}\to 0\hfill \\ & & \mathrm{uniformly}\mathrm{in}{\mathsf{\Theta }}_1\times {\mathsf{\Theta }}_2\mathrm{as}{t}_1,t_2\to \infty .\hfill \end{array}$$

Hence, both ${\mathcal{F}}_i$ are equiconvergent as $t_1,t_2\to \infty$.

Consequently, from Step I–Step IV, we conclude that $\mathcal{F}$ is completely continuous.

This complete the proof. □

Finally, we present the main result of the article. For the sake of convenience, we set:

$$\begin{array}{cc}\hfill {\mathsf{\Psi }}_1=& \left(N_{12}+N_{13}\right){\tilde{\mathsf{\Omega }}}_{11}+\left(N_{22}+N_{23}\right){\tilde{\mathsf{\Omega }}}_{12}+\left(M_{12}+M_{13}\right){\tilde{\mathsf{\Omega }}}_{13}\hfill \\ & +\left(M_{22}+M_{23}\right){\tilde{\mathsf{\Omega }}}_{14},\hfill \end{array}$$

(64)

and

$$\begin{array}{cc}\hfill {\mathsf{\Psi }}_2=& \left(N_{12}+N_{13}\right){\tilde{\mathsf{\Omega }}}_{21}+\left(N_{22}+N_{23}\right){\tilde{\mathsf{\Omega }}}_{22}+\left(M_{12}+M_{13}\right){\tilde{\mathsf{\Omega }}}_{23}\hfill \\ & +\left(M_{22}+M_{23}\right){\tilde{\mathsf{\Omega }}}_{24},\hfill \end{array}$$

(65)

where ${\tilde{\mathsf{\Omega }}}_{1p},{\tilde{\mathsf{\Omega }}}_{2p},p=1,2,3,4$ are defined as (48) and (49).

Theorem 4.

Suppose that $\left(H_1\right)$–$\left(H_2\right)$ hold. Then, the problems (5) and (6) has at least one solution if:

$$\begin{array}{c}\hfill {\mathsf{\Psi }}_1+{\mathsf{\Psi }}_2<1.\end{array}$$

(66)

Proof. 

Under the Banach space $\mathcal{U}$ equipped with the norm ${\parallel ·\parallel }_{\mathcal{U}}$, we let:

$$\begin{array}{cc}& {\omega }_1(t_1,t_2)\hfill \\ \hfill =& \frac{t_1^{\underline{{\alpha }_1-1}}}{\mathsf{\Lambda }}\left\{\frac{{\lambda }_1{G}_1\mathsf{\Gamma }\left({\alpha }_1\right){({\eta }_1-{\alpha }_1+{\theta }_1)}^{\underline{{\theta }_1-1}}{\mathcal{A}}_1}{\mathsf{\Gamma }\left({\theta }_1\right)}\right[{(t_1+{\rho }_1)}^{\underline{{\rho }_1}}{e}^{-n_1(t_1+1)}\times \hfill \\ & \left(\frac{N_{11}{t}_1^{\underline{{\alpha }_1-2}}}{\mathsf{\Gamma }\left({\alpha }_1\right)}-\frac{N_{21}{\lambda }_2{G}_2{({\eta }_2-{\alpha }_2+{\theta }_2)}^{\underline{{\theta }_2-1}}{\mathcal{B}}_2}{\mathsf{\Gamma }\left({\theta }_2\right)}\right)+\frac{M_{11}{e}^{-m_1{t}_2}}{\mathsf{\Gamma }\left({\alpha }_1\right)}\times \hfill \\ & \sum _{s=0}^{t_1-{\alpha }_1}{(t_1-\sigma \left(s\right))}^{\underline{{\alpha }_1-1}}{e}^{-2m_{1}s}-\frac{M_{21}{G}_2{({\eta }_2-{\alpha }_2+{\theta }_2-1)}^{\underline{{\theta }_2-1}}{e}^{-m_1(2t_1+t_2)}}{\mathsf{\Gamma }\left({\theta }_2\right)}]\hfill \\ & +\frac{{\lambda }_2{G}_2\mathsf{\Gamma }\left({\alpha }_2\right){({\eta }_2-{\alpha }_2+{\theta }_2)}^{\underline{{\theta }_2-1}}{\mathcal{A}}_2}{\mathsf{\Gamma }\left({\theta }_2\right)}[{(t_1+{\rho }_1)}^{\underline{{\rho }_1}}{e}^{-n_1(t_1+1)}\times \hfill \\ & \left(\frac{N_{21}{t}_2^{\underline{{\alpha }_2-2}}}{\mathsf{\Gamma }\left({\alpha }_2\right)}-\frac{N_{11}{\lambda }_1{G}_1{({\eta }_1-{\alpha }_1+{\theta }_1)}^{\underline{{\theta }_1-1}}{\mathcal{B}}_1}{\mathsf{\Gamma }\left({\theta }_1\right)}\right)+\frac{M_{21}{e}^{-2m_1{t}_1}}{\mathsf{\Gamma }\left({\alpha }_2\right)}\times \hfill \\ & \sum _{s=0}^{t_2-{\alpha }_2}{(t_2-\sigma \left(s\right))}^{\underline{{\alpha }_2-1}}{e}^{-m_{1}s}-\frac{M_{11}{G}_1{({\eta }_1-{\alpha }_1+{\theta }_1-1)}^{\underline{{\theta }_1-1}}{e}^{-m_1(2t_1+t_2)}}{\mathsf{\Gamma }\left({\theta }_1\right)}\left]\right\}\hfill \\ & +\frac{N_{11}}{\mathsf{\Gamma }\left({\alpha }_1\right)}{(t_1+{\rho }_1)}^{\underline{{\rho }_1}}{e}^{-n_1(t_1+1)}+\frac{M_{11}{e}^{-m_1{t}_2}}{\mathsf{\Gamma }\left({\alpha }_1\right)}\sum _{s=0}^{t_1-{\alpha }_1}{(t_1-\sigma \left(s\right))}^{\underline{{\alpha }_1-1}}{e}^{-2m_{1}s},\hfill \end{array}$$

(67)

and

$$\begin{array}{cc}& {\omega }_2(t_1,t_2)\hfill \\ \hfill =& \frac{t_2^{\underline{{\alpha }_2-1}}}{\mathsf{\Lambda }}\left\{t_2^{\underline{{\alpha }_2-1}}\right[{(t_1+{\rho }_1)}^{\underline{{\rho }_1}}{\left[{(t_2+{\rho }_2)}^{\underline{{\rho }_2}}\right]}^2{e}^{-n_2(t_2+1)}(\frac{N_{12}{t}_1^{\underline{{\alpha }_1-2}}}{\mathsf{\Gamma }\left({\alpha }_1\right)}\hfill \\ & -\frac{N_{22}{\lambda }_2{G}_2{({\eta }_2-{\alpha }_2+{\theta }_2)}^{\underline{{\theta }_2-1}}{\mathcal{B}}_2}{\mathsf{\Gamma }\left({\theta }_2\right)})+\frac{M_{12}{e}^{-m_2{t}_2}}{\mathsf{\Gamma }\left({\alpha }_1\right)}{(t_2+{\rho }_2)}^{\underline{{\rho }_2}}\sum _{s=0}^{t_1-{\alpha }_1}{(t_1-\sigma \left(s\right))}^{\underline{{\alpha }_1-1}}\times \hfill \\ & e^{-2m_{2}s}-\frac{M_{22}{G}_2{({\eta }_2-{\alpha }_2+{\theta }_2-1)}^{\underline{{\theta }_2-1}}{e}^{-m_2(t_1+2t_2)}{(t_2+{\rho }_2)}^{\underline{{\rho }_2}}}{\mathsf{\Gamma }\left({\theta }_2\right)}]+t_1^{\underline{{\alpha }_1-1}}\times \hfill \\ & [{(t_1+{\rho }_1)}^{\underline{{\rho }_1}}{\left[{(t_2+{\rho }_2)}^{\underline{{\rho }_2}}\right]}^2{e}^{-n_2(t_2+1)}\left(\frac{N_{22}{t}_2^{\underline{{\alpha }_2-2}}}{\mathsf{\Gamma }\left({\alpha }_2\right)}-\frac{N_{12}{\lambda }_1{G}_1{({\eta }_1-{\alpha }_1+{\theta }_1)}^{\underline{{\theta }_1-1}}{\mathcal{B}}_1}{\mathsf{\Gamma }\left({\theta }_1\right)}\right)\hfill \\ & +\frac{M_{22}{e}^{-m_2{t}_1}}{\mathsf{\Gamma }\left({\alpha }_2\right)}\sum _{s=0}^{t_2-{\alpha }_2}{(t_2-\sigma \left(s\right))}^{\underline{{\alpha }_2-1}}{e}^{-2m_{2}s}{(t_2+{\rho }_2)}^{\underline{{\rho }_2}}\hfill \\ & -\frac{M_{22}{G}_1{({\eta }_1-{\alpha }_1+{\theta }_1-1)}^{\underline{{\theta }_1-1}}{e}^{-m_2(t_2+2t_2)}}{\mathsf{\Gamma }\left({\theta }_1\right)}]+\frac{N_{22}}{\mathsf{\Gamma }\left({\alpha }_2\right)}{\left[{(t_2+{\rho }_2)}^{\underline{{\rho }_2}}\right]}^2{e}^{-n_2(t_2+1)}{t}_2^{\underline{{\alpha }_2-2}}\}\hfill \\ & +\frac{M_{22}{e}^{-m_2{t}_1}}{\mathsf{\Gamma }\left({\alpha }_2\right)}\sum _{s=0}^{t_2-{\alpha }_2}{(t_2-\sigma \left(s\right))}^{\underline{{\alpha }_2-1}}{e}^{-2m_{2}s}{(t_2+{\rho }_2)}^{\underline{{\rho }_2}}.\hfill \end{array}$$

(68)

It is clear that $({\omega }_1,{\omega }_2)\in \mathcal{U}$. For $\ell >0$, we define:

$$\begin{array}{c}\hfill {\mathsf{\Xi }}_{\ell }=\left\{(u_1,u_2)\in \mathcal{U}:{\parallel (u_1,u_2)-({\omega }_1,{\omega }_2)\parallel }_{\mathcal{U}}\le \ell \right\}.\end{array}$$

(69)

For $(u_1,u_2)\in {\mathsf{\Xi }}_{\ell }$, we have:

$$\begin{array}{cc}\hfill \parallel (u_1,u_2){\parallel }_{\mathcal{U}}\le & \parallel (u_1,u_2)-({\omega }_1,{\omega }_2){\parallel }_{\mathcal{U}}+\parallel ({\omega }_1,{\omega }_2){\parallel }_{\mathcal{U}}\le \ell +{\parallel ({\omega }_1,{\omega }_2)\parallel }_{\mathcal{U}},\hfill \end{array}$$

(70)

$$\begin{array}{cc}\hfill \parallel (u_1,u_2){\parallel }_{\mathcal{U}}=& max\left\{\parallel (u_1,u_2){\parallel }_{{\mathcal{U}}_1},{\parallel (u_1,u_2)\parallel }_{{\mathcal{U}}_2}\right\}\le \ell +{\parallel ({\omega }_1,{\omega }_2)\parallel }_{\mathcal{U}}.\hfill \end{array}$$

(71)

Using the conditions $\left(H_1\right)$–$\left(H_2\right)$, together with the procedure employed in Lemma 5, we have:

$$\begin{array}{cc}& |F_i\left(t_1,t_2,{\Delta }^{{\beta }_i}{u}_i,u_j\right)-M_{i1}{\left(t_2+{}_{i1}{\rho }_2\right)}^{\underline{{}_{i1}{\rho }_2}}{e}^{-m_{i1}(t_1+t_2)}|\hfill \\ \hfill \le & \{M_{i2}{\left(t_2+{}_{i2}{\rho }_2\right)}^{\underline{{}_{i2}{\rho }_2}}{e}^{-m_{i2}(t_1+t_2)}\hfill \\ & +M_{i3}{\left(t_2+{}_{i3}{\rho }_2\right)}^{\underline{{}_{i3}{\rho }_2}}{e}^{-m_{i3}(t_1+t_2)}\}{\parallel (u_1,u_2)\parallel }_{\mathcal{U}},\hfill \end{array}$$

(72)

and

$$\begin{array}{cc}& |{\varphi }_i\left(u_1,u_2\right)-N_{i1}{\left(t_1+{}_{i1}{\tilde{\rho }}_1\right)}^{\underline{{}_{i1}{\tilde{\rho }}_1}}{\left[{\left(t_2+{}_{i1}{\tilde{\rho }}_2\right)}^{\underline{{}_{i1}{\tilde{\rho }}_2}}\right]}^2{e}^{-n_{i1}(t_1+t_2)}|\hfill \\ \hfill =& \{N_{i2}{\left(t_1+{}_{i2}{\tilde{\rho }}_1\right)}^{\underline{{}_{i2}{\tilde{\rho }}_1}}{\left[{\left(t_2+{}_{i2}{\tilde{\rho }}_2\right)}^{\underline{{}_{i2}{\tilde{\rho }}_2}}\right]}^2{e}^{-n_{i2}(t_1+t_2)}\hfill \\ & +N_{i3}{\left(t_1+{}_{i3}{\tilde{\rho }}_1\right)}^{\underline{{}_{i3}{\tilde{\rho }}_1}}{\left[{\left(t_2+{}_{i3}{\tilde{\rho }}_2\right)}^{\underline{{}_{i3}{\tilde{\rho }}_2}}\right]}^2{e}^{-n_{i3}(t_1+t_2)}\}{\parallel (u_1,u_2)\parallel }_{\mathcal{U}}.\hfill \end{array}$$

(73)

Therefore, we obtain:

$$\begin{array}{c}\hfill \chi |\left({\mathcal{F}}_i(u_i,u_j)\right)(t_1,t_2)-{\omega }_i(t_1,t_2)|\le \left(\ell +\parallel ({\omega }_1,{\omega }_2){\parallel }_{\mathcal{U}}\right){\mathsf{\Psi }}_i.\end{array}$$

(74)

Furthermore, we have:

$$\begin{array}{cc}& \chi |{\Delta }^{{\beta }_i}\left({\mathcal{F}}_i(u_i,u_j)\right)(t_i-{\beta }_i+1,t_j)-{\Delta }^{{\beta }_i}{\omega }_i(t_i-{\beta }_i+1,t_j)|\hfill \\ \hfill \le & \left(\ell +\parallel ({\omega }_1,{\omega }_2){\parallel }_{\mathcal{U}}\right){\mathsf{\Psi }}_i.\hfill \end{array}$$

(75)

Hence, it follows that:

$$\begin{array}{c}\hfill \parallel \left({\mathcal{F}}_i(u_1,u_2)\right)-{\omega }_i{\parallel }_{{\mathcal{U}}_i}\le \left(\ell +\parallel ({\omega }_1,{\omega }_2){\parallel }_{\mathcal{U}}\right)2{\mathsf{\Psi }}_i.\end{array}$$

(76)

Therefore,

$$\begin{array}{c}\hfill \parallel \left(\mathcal{F}(u_1,u_2)\right)-({\omega }_1,{\omega }_2){\parallel }_{\mathcal{U}}\le \left(\ell +\parallel ({\omega }_1,{\omega }_2){\parallel }_{\mathcal{U}}\right)2max\left\{{\mathsf{\Psi }}_1,{\mathsf{\Psi }}_2\right\}.\end{array}$$

(77)

Choosing:

$$\begin{array}{c}\hfill \ell \ge \frac{\parallel ({\omega }_1,{\omega }_2){\parallel }_{\mathcal{U}}2max\left\{{\mathsf{\Psi }}_1,{\mathsf{\Psi }}_2\right\}}{1-2max\left\{{\mathsf{\Psi }}_1,{\mathsf{\Psi }}_2\right\}},\end{array}$$

(78)

and for $(u_1,u_2)\in {\mathsf{\Xi }}_{\ell }$, we consequently obtain:

$$\begin{array}{c}\hfill \parallel \left(\mathcal{F}(u_1,u_2)\right)-({\omega }_1,{\omega }_2){\parallel }_{\mathcal{U}}\le \ell .\end{array}$$

(79)

From the Schauder fixed point theorem, this implies that $\mathcal{F}$ has a fixed point $(u_1,u_2)\in {\mathsf{\Xi }}_{\ell }$, which is a bounded solution of the problems (5) and (6). The proof is complete.  □

4. Example

In order to illustrate our result, we consider the following fractional sum boundary value problem:

$$\begin{array}{ccc}\hfill {\Delta }^{\frac{3}{2}}{u}_1\left(t\right)& =& \frac{2}{3}\left(t+\frac{4}{3}\right)e^{-(12t+5)}+\frac{{\left(t+\frac{5}{3}\right)}^{\frac{4}{3}}{e}^{-\left(25t+\frac{31}{3}\right)}{u}_2\left(t+\frac{1}{3}\right)}{4000{\left(t+\frac{301}{3}\right)}^2\left(1+{\cos }^2{u}_2\pi \right)}\hfill \\ & & +\frac{{\left(t+\frac{11}{6}\right)}^{\frac{3}{2}}{e}^{-\left[\left(12t+5\right)+\left(t+\frac{1}{2}\right)\pi \right]}{\Delta }^{\frac{1}{3}}{u}_1\left(t+\frac{4}{3}\right)}{5000e+10{\cos }^2\left(t+\frac{1}{2}\right)\pi },t\in {\mathbb{N}}_0\hfill \\ \hfill {\Delta }^{\frac{4}{3}}{u}_2\left(t\right)& =& \frac{1}{2}\left(t+\frac{4}{3}\right)e^{-(12t+5)}+\frac{{\left(t+\frac{5}{3}\right)}^{\frac{4}{3}}{e}^{-\left(25t+\frac{21}{3}\right)}{u}_1\left(t+\frac{1}{2}\right)}{1000{\left(e^{\left(t+\frac{1}{2}\right)}+10\right)}^2}\hfill \\ & & +\frac{{\left(t+\frac{11}{6}\right)}^{\frac{3}{2}}{e}^{-\left(12t+5\right)}\arctan \left({\cos }^2\left(t+\frac{1}{3}\right)\pi \right){\Delta }^{\frac{3}{4}}{u}_2\left(t_2+\frac{7}{12}\right)}{1000\pi {\left(t+\frac{10}{3}\right)}^2},t\in {\mathbb{N}}_0\hfill \\ \hfill u_1\left(-\frac{1}{2}\right)& =& {\varphi }_1(u_1,u_2)=\frac{|u_1|}{2000e^3}{\cos }^2\left|\pi u_1\right|+\frac{|u_2\left|\right|u_2^2{+2|}^{\underline{1-|u_2^2+2|}}}{4000{\pi }^2(u_2^2+e)},\hfill \\ \hfill u_2\left(-\frac{2}{3}\right)& =& {\varphi }_2(u_1,u_2)=\frac{|u_2|}{5000e^2}{\sin }^2\left|\pi u_2\right|+\frac{|u_1\left|\right|u_1^2{+3|}^{\underline{1-|u_1^2+3|}}}{2000\pi (u_1^2+\pi )},\hfill \\ \hfill \underset{t⟶\infty }{lim}{u}_1\left(t-\frac{1}{2}\right)& =& \frac{1}{2}{\Delta }^{-\frac{1}{4}}{\left(12e+\cos \left(4\right)\right)}^2{u}_2\left(4\right)\hfill \\ \hfill \underset{t⟶\infty }{lim}{u}_2\left(t-\frac{2}{3}\right)& =& \frac{3}{4}{\Delta }^{-\frac{2}{3}}{\left(10e-\sin \left(\frac{15}{4}\right)\right)}^3{u}_1\left(\frac{15}{4}\right).\hfill \end{array}$$

(80)

Here, ${\alpha }_1=\frac{3}{2},{\alpha }_2=\frac{4}{3},{\beta }_1=\frac{1}{3},{\beta }_2=\frac{3}{4},{\gamma }_1=\frac{3}{4},{\gamma }_2=\frac{5}{6},{\theta }_1=\frac{1}{4},{\theta }_2=\frac{2}{3},{\eta }_1=\frac{7}{2},{\eta }_2=\frac{10}{3},{\lambda }_1=\frac{1}{2},{\lambda }_2=\frac{3}{4},T=4,g_1\left(t_1\right)={(10e-\sin t_1)}^3,g_2\left(t_2\right)={(12e+\cos t_2)}^2,$ and:

$$\begin{array}{cc}\hfill F_1\left(t_1,t_2,{\Delta }^{\frac{1}{3}}{u}_1\left(t_1+\frac{2}{3}\right),u_2\left(t_2\right)\right)& =\frac{2}{3}\left(t_2+1\right)e^{-6(t_1+t_2)}+\frac{{\left(t_2+\frac{4}{3}\right)}^{\frac{4}{3}}{e}^{-\left[12\left(t_1+t_2\right)+t_2\right]}{u}_2\left(t_2\right)}{4000{\left(t_2+10\right)}^2\left(1+{\cos }^2{u}_2\pi \right)}\hfill \\ & +\frac{{\left(t_2+\frac{3}{2}\right)}^{\frac{3}{2}}{e}^{-\left[6\left(t_1+t_2\right)+t_1\pi \right]}{\Delta }^{\frac{1}{3}}{u}_1\left(t_1+\frac{2}{3}\right)}{5000e+10{\cos }^2{t}_1\pi }\hfill \\ \hfill F_2\left(t_1,t_2,{\Delta }^{\frac{3}{4}}{u}_2\left(t_2+\frac{1}{4}\right),u_1\left(t_1\right)\right)& =\frac{1}{2}\left(t_2+1\right)e^{-6(t_1+t_2)}+\frac{{\left(t_2+\frac{4}{3}\right)}^{\frac{4}{3}}{e}^{-\left[12\left(t_1+t_2\right)+t_1\right]}{u}_1\left(t_1\right)}{1000{(e^{t_1}+10)}^2}\hfill \\ & +\frac{{\left(t_2+\frac{3}{2}\right)}^{\frac{3}{2}}{e}^{-6\left(t_1+t_2\right)}\arctan \left({\cos }^2{t}_2\pi \right){\Delta }^{\frac{3}{4}}{u}_2\left(t_2+\frac{1}{4}\right)}{1000\pi {\left(t_2+3\right)}^2}.\hfill \end{array}$$

Choose ${\rho }_1=1,{\rho }_2=2,{}_{i1}{\rho }_1={}_{i1}{\tilde{\rho }}_1=\frac{1}{2},{}_{i2}{\rho }_1={}_{i2}{\tilde{\rho }}_1=\frac{2}{3},{}_{i3}{\rho }_1={}_{i3}{\tilde{\rho }}_1=\frac{3}{4}{}_{i1}{\rho }_2=1,{}_{i2}{\rho }_2=\frac{3}{2},{}_{i3}{\rho }_2=\frac{4}{3}$, $m_{i1}=n_{i1}=6,m_{i2}=n_{i2}=6,m_{i3}=n_{i3}=12,$ where ${\rho }_i>max\{{\beta }_1-{\alpha }_1,{\beta }_2-{\alpha }_2\},{}_{ip}{\rho }_1\in (-1,1),{}_{ip}{\rho }_2\in (-1,2)$ for $i=1,2$ and $p=1,2,3$.

Let $t_1\in {\mathbb{N}}_{-\frac{1}{2},\frac{11}{2}},t_2\in {\mathbb{N}}_{-\frac{2}{3},\frac{16}{3}}$ and $\chi =\frac{{\left(t_1-\frac{1}{2}\right)}^{\underline{1/2}}{\left(t_2-\frac{1}{3}\right)}^{\underline{2/3}}}{1+t_1^{\underline{3}}{t}_2^{\underline{4}}}.$ Since:

$$\begin{array}{cc}& |F_1\left(t_1,t_2,\frac{1}{\chi }{\Delta }^{\frac{1}{3}}{u}_1,\frac{1}{\chi }{u}_2\right)-\frac{2}{3}\left(t_2+1\right)e^{-6(t_1+t_2)}|\hfill \\ \hfill \le & \frac{1}{361000}{\left(t_2+\frac{4}{3}\right)}^{\frac{4}{3}}{e}^{-12\left(t_1+t_2\right)}\left|u_2\right|+\frac{1}{5000e^{10}+10}{\left(t_2+\frac{3}{2}\right)}^{\frac{3}{2}}{e}^{-6\left(t_1+t_2\right)}\left|{\Delta }^{\frac{1}{3}}{u}_1\right|,\hfill \\ & |F_2\left(t_1,t_2,\frac{1}{\chi }{\Delta }^{\frac{3}{4}}{u}_2,\frac{1}{\chi }{u}_1\right)-\frac{1}{2}\left(t_2+1\right)e^{-6(t_1+t_2)}|\hfill \\ \hfill \le & \frac{1}{121000}{\left(t_2+\frac{4}{3}\right)}^{\frac{4}{3}}{e}^{-12\left(t_1+t_2\right)}\left|u_2\right|+\frac{9}{196000}{\left(t_2+\frac{3}{2}\right)}^{\frac{3}{2}}{e}^{-6\left(t_1+t_2\right)}\left|{\Delta }^{\frac{3}{4}}{u}_2\right|,\hfill \end{array}$$

we find that $\left(H_1\right)$ holds with $M_{11}=0.666,M_{12}=9.080,M_{13}=2.770$ and $M_{21}=0.500,M_{22}=0.000046,M_{23}=0.0000083$.

Furthermore, we obtain:

$$\begin{array}{cc}& |{\varphi }_1\left(\frac{1}{\chi }{u}_1,\frac{1}{\chi }{u}_2\right)-\frac{2}{5}{\left(t_1+\frac{1}{2}\right)}^{\frac{1}{2}}\left(t_2+\frac{1}{2}\right)e^{-6(t_1+t_2)}|\hfill \\ \hfill \le & \frac{1}{2000e^3}{\left(t_1+\frac{2}{3}\right)}^{\frac{2}{3}}{\left(t_2+\frac{2}{3}\right)}^{\frac{4}{3}}{e}^{-6(t_1+t_2)}\parallel u_1\parallel +\frac{1}{4000{\pi }^2}{\left(t_1+\frac{3}{4}\right)}^{\frac{3}{4}}{\left(t_2+\frac{3}{4}\right)}^{\frac{3}{2}}\times \hfill \\ & e^{-3(t_1+t_2)}\parallel u_2\parallel ,\hfill \\ & |{\varphi }_2\left(\frac{1}{\chi }{u}_1,\frac{1}{\chi }{u}_2\right)-\frac{5}{6}{\left(t_1+\frac{1}{2}\right)}^{\frac{1}{2}}\left(t_2+\frac{1}{2}\right)e^{-6(t_1+t_2)}|\hfill \\ \hfill \le & \frac{1}{2000\pi }{\left(t_1+\frac{2}{3}\right)}^{\frac{2}{3}}{\left(t_2+\frac{2}{3}\right)}^{\frac{4}{3}}{e}^{-6(t_1+t_2)}\parallel u_1\parallel +\frac{1}{500e^2}{\left(t_1+\frac{3}{4}\right)}^{\frac{3}{4}}{\left(t_2+\frac{3}{4}\right)}^{\frac{3}{2}}\times \hfill \\ & e^{-3(t_1+t_2)}\parallel u_2\parallel ,\hfill \end{array}$$

and ${(10e-1)}^3<g_1\left(t_1\right)<{(10e+1)}^3$ and ${(12e-1)}^2<g_2\left(t_2\right)<{(12e+1)}^2.$

Thus, $\left(H_2\right),\left(H3\right)$ hold with $N_{11}=0.4,N_{12}=0.0000249,N_{13}=0.0000253,N_{21}=0.833,N_{22}=0.000159,N_{23}=0.000271,g_1=17949.37,g_2=999.79,G_1=22384.80$, and $G_2=1130.26.$

Finally, we find that:

$${\tilde{\mathsf{\Omega }}}_{11}=2313.238,{\tilde{\mathsf{\Omega }}}_{12}=319.647,{\tilde{\mathsf{\Omega }}}_{13}=27053.522,{\tilde{\mathsf{\Omega }}}_{14}=2597.063,$$

$${\tilde{\mathsf{\Omega }}}_{21}=0.0212,{\tilde{\mathsf{\Omega }}}_{22}=1.1403,{\tilde{\mathsf{\Omega }}}_{23}=0.0198,{\tilde{\mathsf{\Omega }}}_{24}=1.5093.$$

Therefore, we have:

$$\begin{array}{c}\hfill {\mathsf{\Psi }}_1+{\mathsf{\Psi }}_2=0.00057+0.4692=0.4698<1.\end{array}$$

Hence, by Theorem 4, this boundary value problem has at least one solution.  □