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\begin{title}
\title{\bf\Large{{A Coupled System of Fractional Difference Equations with Anti-periodic Boundary Conditions}}}
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\begin{author}
\author {Jagan Mohan Jonnalagadda\footnote[1]{Department of Mathematics, Birla Institute of Technology and Science Pilani, Hyderabad - 500078, Telangana, India. email: {j.jaganmohan@hotmail.com}}}
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\noindent{\bf Abstract:} In this article, we give sufficient conditions for the existence, uniqueness and Ulam--Hyers stability of solutions for a coupled system of two-point nabla fractional difference boundary value problems subject to anti-periodic boundary conditions, using the vector approach of Precup \cite{Bo, Je, Ni, Pr}. Some examples are included to illustrate the theory.
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\noindent{\bf Key Words:} Nabla fractional difference, boundary value problem, anti-periodic boundary conditions, existence, uniqueness, Ulam--Hyers stability

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\noindent{\bf AMS Classification:} 39A12, 39A70.

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\section{Introduction}
In \cite{Pr}, Precup described the advantage of vector-valued norms in the study of the semilinear operator system
\begin{equation} \label{System N}
\begin{cases}
N_{1}(u_{1}, u_{2}) = u_{1}, \\
N_{2}(u_{1}, u_{2}) = u_{2}, \\
\end{cases}
\end{equation}
in a Banach space $X$ with norm $|\boldsymbol{\cdot}|$, by some methods of nonlinear analysis. Here $N_{1}$, $N_{2} : X^2 \rightarrow X$ are given nonlinear operators. Obviously, this system can be viewed as a fixed point problem:
\begin{equation}
Nu = u,
\end{equation}
in the space $X^2$, where $u = (u_{1}, u_{2})$ and $N = (N_{1}, N_{2})$. Precup \cite{Pr} proposed the applications of a few fixed point theorems to the system \eqref{System N} in $X^2$, by using the vector-valued norm $$\|u\| = \begin{pmatrix}
|u_{1}| \\ 
|u_{2}|
\end{pmatrix},$$ for $u = (u_{1}, u_{2}) \in X^2$. Also, Precup \cite{Pr} demonstrated that the results obtained by using the vector-valued norm are better than those established by means of any scalar norm in $X^2$.
\begin{theorem} \cite{Pr} \label{Perov}
Assume that
\begin{enumerate}
\item[(H1)] for each $i \in \{1, 2\}$, there exist nonnegative numbers $a_{i}$ and $b_{i}$ such that 
\begin{equation}
|N_{i}(u_{1}, u_{2}) - N_{i}(v_{1}, v_{2})| \leq a_{i}|u_{1} - v_{1}| + b_{i}|u_{2} - v_{2}|,
\end{equation}
for all $(u_{1}, u_{2})$, $(v_{1}, v_{2}) \in X^{2}$;
\item[(H2)] The spectral radius of $M = \begin{pmatrix}
a_1 & b_1 \\ 
a_2 & b_2
\end{pmatrix}$ is less than one.
\end{enumerate}
Then, \eqref{System N} has a unique solution $(u_{1}, u_{2}) \in X^2$.
\end{theorem}
\begin{theorem} \cite{Pr} \label{Schauder}
Assume that
\begin{enumerate}
\item[(H3)] for each $i \in \{1, 2\}$, the operator $N_i$ is completely continuous and, there exist nonnegative numbers $a_{i}$, $b_{i}$ and $c_{i}$ such that 
\begin{equation}
|N_{i}(u_{1}, u_{2})| \leq a_{i}|u_{1}| + b_{i}|u_{2}| + c_{i},
\end{equation}
for all $(u_{1}, u_{2}) \in X^{2}$.
\end{enumerate}
In addition, assume that condition (H2) is satisfied. Then, \eqref{System N} has at least one solution $(u_{1}, u_{2}) \in X^{2}$ satisfying
\begin{equation} \label{Con}
\begin{pmatrix} 
|u_1| \\ 
|u_2|
\end{pmatrix} \leq (I - M)^{-1} \begin{pmatrix}
c_1 \\ 
c_2
\end{pmatrix}.
\end{equation}
\end{theorem}
Further, in \cite{Urs}, the author used the following theorem to establish Ulam--Hyers stability of solutions of \eqref{System N}:
\begin{theorem} \cite{Urs} \label{Urs}
Assume that the hypothesis of Theorem \ref{Perov} holds. Then, the system \eqref{System N} is Ulam--Hyers stable.
\end{theorem} 
Motivated by these results, in this article, we consider the following coupled system of nabla fractional difference equations with anti-periodic boundary conditions
\begin{equation} \label{Coupled System}
\begin{cases}
\Big{(}\nabla^{\alpha_{1} - 1}_{0}\big{(}\nabla u_{1}\big{)}\Big{)}(t) + f_{1}(u_{1}(t), u_{2}(t)) = 0, \quad t \in \mathbb{N}^{T}_{2}, \\ \Big{(}\nabla^{\alpha_{2} - 1}_{0}\big{(}\nabla u_{2}\big{)}\Big{)}(t) + f_{2}(u_{1}(t), u_{2}(t)) = 0, \quad t \in \mathbb{N}^{T}_{2}, \\ u_{1}(0) + u_{1}(T) = 0, \quad \big{(}\nabla u_{1}\big{)}(1) + \big{(}\nabla u_{1}\big{)}(T) = 0, \\
u_{2}(0) + u_{2}(T) = 0, \quad \big{(}\nabla u_{2}\big{)}(1) + \big{(}\nabla u_{2}\big{)}(T) = 0,
\end{cases}
\end{equation}
and apply Theorems \ref{Perov} - \ref{Urs} to establish sufficient conditions on existence, uniqueness, and Ulam--Hyers stability \cite{Ch 1, Ch 2, Du 1, Kh 1, Hy, Ja, Ju F, Ra, Ru 2, Ul} of its solutions. For this purpose, we convert the system \eqref{Coupled System} in the form of \eqref{System N}. But the results may not be straightforward because the computation of nonnegative numbers in each theorem for the system \eqref{Coupled System} is complicated due to the presence of nabla fractional difference operators in it. 

Here $T \in \mathbb{N}_{2}$; $1 < \alpha_{1}, \alpha_{2} < 2$; $f_{1}$, $f_{2} : \mathbb{R}^{2} \rightarrow \mathbb{R}$ are continuous, $\nabla^{\nu}_{0}$ denotes the $\nu^{\text{th}}$-th order Riemann--Liouville type backward (nabla) difference operator where $\nu \in \{\alpha_{1} - 1, \alpha_{2} - 1\}$ and $\nabla$ denotes the first order nabla difference operator.

The present article is organized as follows: Section 2 contains preliminaries. In Section 3, we establish sufficient conditions on existence, uniqueness, and Ulam--Hyers stability of solutions of the system \eqref{Coupled System}. We provide two examples in Section 4 to illustrate the applicability of established results.

\section{Preliminaries}
For our convenience, in this section, we present a few useful definitions and fundamental facts of nabla fractional calculus, which can be found in \cite{Ab 2, Ah, At, Gh, Go, He, Ke, Ki, Po}.

Denote by $\mathbb{N}_{a} = \{a, a + 1, a + 2, \ldots\}$ and $\mathbb{N}^{b}_{a} = \{a, a + 1, a + 2, \ldots, b\}$ for any $a$, $b \in \mathbb{R}$ such that $b - a \in \mathbb{N}_{1}$. The backward jump operator $\rho : \mathbb{N}_{a} \rightarrow \mathbb{N}_{a}$ is defined by $\rho(t) = \max\{a, t - 1\}$, for all $t \in \mathbb{N}_{a}$. Define the $\mu^{th}$-order nabla fractional Taylor monomial by $$H_{\mu}(t, a) = \frac{(t - a)^{\overline{\mu}}}{\Gamma(\mu + 1)} = \frac{\Gamma(t - a + \mu)}{\Gamma(t - a)\Gamma(\mu + 1)}, \quad t \in \mathbb{N}_{a}, \quad \mu \in \mathbb{R} \setminus \{\ldots, -2, -1\}.$$ Here $\Gamma(\cdot)$ denotes the Euler gamma function. Observe that $H_{\mu}(a, a) = 0$ and $H_{\mu}(t, a) = 0$ for all $\mu \in \{\ldots, -2, -1\}$ and $t \in \mathbb{N}_{a}$. The first order backward (nabla) difference of $u: \mathbb{N}_{a} \rightarrow \mathbb{R}$ is defined by $\big{(}\nabla u\big{)}(t) = u(t) - u(t - 1)$, for $t \in \mathbb{N}_{a + 1}$.

\begin{definition}[See \cite{Go}]
Let $u: \mathbb{N}_{a + 1} \rightarrow \mathbb{R}$ and $\nu > 0$. The $\nu^{\text{th}}$-order nabla sum of $u$ based at $a$ is given by
\begin{equation}
\nonumber \big{(}\nabla ^{-\nu}_{a}u\big{)}(t) = \sum^{t}_{s = a + 1}H_{\nu - 1}(t, \rho(s))u(s), \quad t \in \mathbb{N}_{a},
\end{equation}
where by convention $\big{(}\nabla ^{-\nu}_{a}u\big{)}(a) = 0$.
\end{definition}

\begin{definition}[See \cite{Go}]
Let $u: \mathbb{N}_{a + 1} \rightarrow \mathbb{R}$ and $0 < \nu \leq 1$. The $\nu^{\text{th}}$-order nabla difference of $u$ is given by
\begin{equation}
\nonumber \big{(}\nabla ^{\nu}_{a}u\big{)}(t) = \Big{(}\nabla \big{(}\nabla_{a}^{-(1 - \nu)}u\big{)}\Big{)}(t), \quad t\in\mathbb{N}_{a + 1}.
\end{equation}
\end{definition}

\begin{lemma} [See \cite{Go}] \label{Prop}
We have the following properties of nabla fractional Taylor monomials.
\begin{enumerate}
\item $\nabla H_{\mu}(t, a) = H_{\mu - 1}(t, a)$, $t \in \mathbb{N}_{a}$.
\item $\sum^{t}_{s = a + 1} H_{\mu}(s, a) = H_{\mu + 1}(t, a)$, $t \in \mathbb{N}_{a}$.
\item $\sum^{t}_{s = a + 1} H_{\mu}(t, \rho(s)) = H_{\mu + 1}(t, a)$, $t \in \mathbb{N}_{a}$.
\end{enumerate}
\end{lemma}

\begin{proposition} [See \cite{Ik 2}] \label{H}
Let $s \in \mathbb{N}_{a}$ and $-1 < \mu$. The following properties hold:
\begin{enumerate}
\item[(a)] $H_{\mu}(t, \rho(s)) \geq 0$ for $t \in \mathbb{N}_{\rho(s)}$ and, $H_{\mu}(t, \rho(s)) > 0$ for $t \in \mathbb{N}_{s}$.
\item[(b)] $H_{\mu}(t, \rho(s))$ is a decreasing function with respect to $s$ for $t \in \mathbb{N}_{\rho(s)}$ and $\mu \in (0, \infty)$. 
\item[(c)] If $t \in \mathbb{N}_{s}$ and $\mu \in (-1, 0)$, then $H_{\mu}(t, \rho(s))$ is an increasing function of $s$.
\item[(d)] $H_{\mu}(t, \rho(s))$ is a non-decreasing function with respect to $t$ for $t \in \mathbb{N}_{\rho(s)}$ and $\mu \in [0, \infty)$.
\item[(e)] If $t \in \mathbb{N}_{s}$ and $\mu \in (0, \infty)$, then $H_{\mu}(t, \rho(s))$ is an increasing function of $t$.
\item[(f)] $H_{\mu}(t, \rho(s))$ is a decreasing function with respect to $t$ for $t \in \mathbb{N}_{s + 1}$ and $\mu \in (-1, 0)$.
\end{enumerate}
\end{proposition}

\begin{proposition} [See \cite{Ik 2}] \label{Mod}
Let $u$ and $v$ be two nonnegative real-valued functions defined on a set $S$. Further, assume $u$ and $v$ achieve their maximum values in $S$. Then, $$\left|u(t) - v(t)\right| \leq \max \{u(t), v(t)\} \leq \max \big{\{}\max_{t \in S} u(t), \max_{t \in S} v(t)\big{\}},$$ for every fixed $t$ in $S$.
\end{proposition}

\section{Green's Function and Its Property}
Assume $T \in \mathbb{N}_{2}$, $1 < \alpha < 2$, and $h : \mathbb{N}^{T}_{2} \rightarrow \mathbb{R}$. Consider the boundary value problem
\begin{equation} \label{BVP 1}
\begin{cases}
\Big{(}\nabla^{\alpha - 1}_{0}\big{(}\nabla u\big{)}\Big{)}(t) + h(t) = 0, \quad t \in \mathbb{N}^{T}_{2}, \\ u(0) + u(T) = 0, \quad \big{(}\nabla u\big{)}(1) + \big{(}\nabla u\big{)}(T) = 0.
\end{cases}
\end{equation}
First, we construct the Green's function, $G(t, s)$ corresponding to \eqref{BVP 1}, and obtain an expression for its unique solution. Denote by $$D_{1} = \{(t, s) \in \mathbb{N}^{T}_{0} \times \mathbb{N}^{T}_{2} : t \geq s\}, \quad D_{2} = \{(t, s) \in \mathbb{N}^{T}_{0} \times \mathbb{N}^{T}_{2} : t \leq \rho(s)\},$$ and 
\begin{equation}
\xi_{\alpha} = 2\left[1 + H_{\alpha - 2}(T, 0)\right].
\end{equation}

\begin{theorem}
The unique solution of the nabla fractional boundary value problem \eqref{BVP 1} is given by
\begin{equation} \label{Sol}
u(t) = \sum^{T}_{s = 2}G_{\alpha}(t, s)h(s), \quad t \in \mathbb{N}^{T}_{0},
\end{equation}
where
\begin{equation} \label{Jo Green}
G_{\alpha}(t, s) = 
\begin{cases}
K_{\alpha}(t, s) - H_{\alpha - 1}(t, \rho(s)), \quad (t, s) \in D_{1},\\
K_{\alpha}(t, s), \hspace{1.27 in} (t, s) \in D_{2}.
\end{cases}
\end{equation}
Here
\begin{multline}
\nonumber K_{\alpha}(t, s) = \frac{1}{\xi_{\alpha}}\Big{[}H_{\alpha - 1}(T, \rho(s)) + 2H_{\alpha - 1}(t, 0)H_{\alpha - 2}(T, \rho(s)) \\ + H_{\alpha - 1}(T, \rho(s))H_{\alpha - 2}(T, 0) - H_{\alpha - 1}(T, 0)H_{\alpha - 2}(T, \rho(s))\Big{]}.
\end{multline}
\end{theorem}

\begin{proof}
Denote by $$\big{(}\nabla u\big{)}(t) = v(t), \quad t \in \mathbb{N}^{T}_{1}.$$ Subsequently, the difference equation in \eqref{BVP 1} takes the form
\begin{equation} \label{Diff} 
\big{(}\nabla^{\alpha - 1}_{0}v\big{)}(t) + h(t) = 0, \quad t \in \mathbb{N}^{T}_{2}.
\end{equation}
Let $v(1) = c_{2}$. Then, by Lemma 5.1 of \cite{Ab 2}, the unique solution of \eqref{Diff} is given by $$v(t) = H_{\alpha - 2}(t, 0)c_{2} - \big{(}\nabla^{-(\alpha - 1)}_{1}h\big{)}(t), \quad t \in \mathbb{N}^{T}_{1}.$$ That is,
\begin{equation} \label{Diff A} 
\big{(}\nabla u\big{)}(t) = H_{\alpha - 2}(t, 0)c_{2} - \big{(}\nabla^{-(\alpha - 1)}_{1}h\big{)}(t), \quad t \in \mathbb{N}^{T}_{1}.
\end{equation}
Applying the first order nabla sum operator, $\nabla^{-1}$ on both sides of \eqref{Diff A}, we obtain
\begin{equation} \label{Diff B} 
u(t) = c_{1} + H_{\alpha - 1}(t, 0)c_{2} - \big{(}\nabla^{-\alpha}_{1}h\big{)}(t), \quad t \in \mathbb{N}^{T}_{0},
\end{equation}
where $c_{1} = u(0)$. We use the pair of anti-periodic boundary conditions considered in \eqref{BVP 1} to eliminate the constants $c_{1}$ and $c_{2}$ in \eqref{Diff B}. It follows from the first boundary condition $u(0) + u(T) = 0$ that
\begin{equation} \label{Eq 1}
2c_{1} + H_{\alpha - 1}(T, 0)c_{2} = \big{(}\nabla^{-\alpha}_{1}h\big{)}(T).
\end{equation}
The second boundary condition $\big{(}\nabla u\big{)}(1) + \big{(}\nabla u\big{)}(T) = 0$ yields
\begin{equation} \label{Eq 2}
\left[1 + H_{\alpha - 2}(T, 0)\right]c_{2} = \big{(}\nabla^{-(\alpha - 1)}_{1}h\big{)}(T).
\end{equation}
Solving \eqref{Eq 1} and \eqref{Eq 2} for $c_{1}$ and $c_{2}$, we obtain
\begin{align} \label{C}
& c_{1} = \frac{1}{2}\left[\sum^{T}_{s = 2}H_{\alpha - 1}(T, \rho(s))h(s)- \frac{2 H_{\alpha - 1}(T, 0)}{\xi_{\alpha}} \sum^{T}_{s = 2}H_{\alpha - 2}(T, \rho(s))h(s)\right], \\
& c_{2} = \frac{2}{\xi_{\alpha}} \sum^{T}_{s = 2}H_{\alpha - 2}(T, \rho(s))h(s).
\end{align}
Substituting these expressions in \eqref{Diff B}, we achieve \eqref{Jo Green}.
\end{proof}

\begin{lemma}
Observe that
\begin{equation}
\left|K_{\alpha}(t, s)\right| \leq \frac{1}{\xi_{\alpha}} \Big{[}H_{\alpha - 1}(T, 1) + 2H_{\alpha - 1}(T, 0) + H_{\alpha - 2}(T, 0)H_{\alpha - 1}(T, 1)\Big{]},
\end{equation}
for all $(t, s) \in \mathbb{N}^{T}_{0} \times \mathbb{N}^{T}_{2}$.
\end{lemma}

\begin{proof}
Denote by
\begin{multline} \label{K1}
K'_{\alpha}(t, s) = \frac{1}{\xi_{\alpha}}\Big{[}H_{\alpha - 1}(T, \rho(s)) + 2H_{\alpha - 1}(t, 0)H_{\alpha - 2}(T, \rho(s)) \\ + H_{\alpha - 1}(T, \rho(s))H_{\alpha - 2}(T, 0)\Big{]},
\end{multline}
and
\begin{equation} \label{K2}
K''_{\alpha}(t, s) = \frac{1}{\xi_{\alpha}}\Big{[}H_{\alpha - 1}(T, 0)H_{\alpha - 2}(T, \rho(s))\Big{]},
\end{equation}
so that $$K_{\alpha}(t, s) = K'_{\alpha}(t, s) - K''_{\alpha}(t, s), \quad (t, s) \in \mathbb{N}^{T}_{0} \times \mathbb{N}^{T}_{2}.$$ Clearly, from Proposition \ref{H}, $$K'_{\alpha}(t, s) \geq 0, \quad K''_{\alpha}(t, s) > 0, ~ \text{for all} ~ (t, s) \in \mathbb{N}^{T}_{0} \times \mathbb{N}^{T}_{2}.$$ From Proposition \ref{Mod}, it is obvious that
\begin{equation} \label{K3}
\left|K_{\alpha}(t, s)\right| \leq \left\{\max_{(t, s) \in \mathbb{N}^{T}_{0} \times \mathbb{N}^{T}_{2}}K'_{\alpha}(t, s), \max_{(t, s) \in \mathbb{N}^{T}_{0} \times \mathbb{N}^{T}_{2}}K''_{\alpha}(t, s)\right\}.
\end{equation}
First, we evaluate the first backward difference of $K'_{\alpha}(t, s)$ with respect to $t$ for a fixed $s$. Consider $$\nabla K'_{\alpha}(t, s) = \frac{1}{\xi_{\alpha}}\Big{[}2H_{\alpha - 2}(t, 0)H_{\alpha - 2}(T, \rho(s))\Big{]} > 0,$$ for all $(t, s) \in \mathbb{N}^{T}_{0} \times \mathbb{N}^{T}_{2}$, implying that $K'_{\alpha}(t, s)$ is an increasing function of $t$ for a fixed $s$. Thus, we have
\begin{equation} \label{K4}
K'_{\alpha}(t, s) \leq K'_{\alpha}(T, s), \quad (t, s) \in \mathbb{N}^{T}_{0} \times \mathbb{N}^{T}_{2}.
\end{equation}
It follows from \eqref{K1} - \eqref{K4} that
\begin{align*}
& \left|K_{\alpha}(t, s)\right| \\ & \leq \left\{\max_{(t, s) \in \mathbb{N}^{T}_{0} \times \mathbb{N}^{T}_{2}}K'_{\alpha}(t, s), \max_{(t, s) \in \mathbb{N}^{T}_{0} \times \mathbb{N}^{T}_{2}}K''_{\alpha}(t, s)\right\} \\ & \leq \left\{\max_{s \in \mathbb{N}^{T}_{2}}K'_{\alpha}(T, s), \max_{s \in \mathbb{N}^{T}_{2}}K''_{\alpha}(t, s)\right\} \\ & = \max_{s \in \mathbb{N}^{T}_{2}}K'_{\alpha}(T, s) \\ & = \frac{1}{\xi_{\alpha}} \max_{s \in \mathbb{N}^{T}_{2}} \Big{[}H_{\alpha - 1}(T, \rho(s)) + 2H_{\alpha - 1}(T, 0)H_{\alpha - 2}(T, \rho(s)) \\ & \quad + H_{\alpha - 1}(T, \rho(s))H_{\alpha - 2}(T, 0)\Big{]} \\ & \leq \frac{1}{\xi_{\alpha}} \Big{[}\max_{s \in \mathbb{N}^{T}_{2}} H_{\alpha - 1}(T, \rho(s)) + 2H_{\alpha - 1}(T, 0)\max_{s \in \mathbb{N}^{T}_{2}} H_{\alpha - 2}(T, \rho(s)) \\ & \quad + H_{\alpha - 2}(T, 0)\max_{s \in \mathbb{N}^{T}_{2}} H_{\alpha - 1}(T, \rho(s))\Big{]} \\ & = \frac{1}{\xi_{\alpha}} \Big{[}H_{\alpha - 1}(T, \rho(2)) + 2H_{\alpha - 1}(T, 0) H_{\alpha - 2}(T, \rho(T)) + H_{\alpha - 2}(T, 0)H_{\alpha - 1}(T, \rho(2))\Big{]} \\ & = \frac{1}{\xi_{\alpha}} \Big{[}H_{\alpha - 1}(T, 1) + 2H_{\alpha - 1}(T, 0) + H_{\alpha - 2}(T, 0)H_{\alpha - 1}(T, 1)\Big{]}.
\end{align*}
The proof is complete.
\end{proof}

\section{Main Results}
Let $X = \mathbb{R}^{T + 1}$ be the Banach space of all real $(T + 1)$-tuples equipped with the maximum norm $$|u| = \max_{t \in \mathbb{N}^{T}_{0}}|u(t)|.$$ Obviously, the product space $X^2$ is also a Banach space with the vector-norm $$\|u\| = \begin{pmatrix}
|u_{1}| \\ 
|u_{2}|
\end{pmatrix},$$ for $u = (u_{1}, u_{2}) \in X^2$.

For our convenience, denote by
\begin{equation}
\Lambda_i = \frac{1}{\xi_{\alpha_i}} \Big{[}H_{\alpha_i - 1}(T, 1) + 2H_{\alpha_i - 1}(T, 0) + H_{\alpha_i - 2}(T, 0)H_{\alpha_i - 1}(T, 1)\Big{]},
\end{equation}
\begin{align}
& a_{i} = l_{i}\left[\Lambda_i (T - 1) + H_{\alpha_i}(T, 1)\right], \\
& b_{i} = m_{i}\left[\Lambda_i (T - 1) + H_{\alpha_i}(T, 1)\right], \\
& c_{i} = n_{i}\left[\Lambda_i (T - 1) + H_{\alpha_i}(T, 1)\right], 
\end{align}
for $i = 1, 2$.

Define the operator $T : X^2 \rightarrow X^2$ by
\begin{equation}
T(u_{1}, u_{2})(t) = \begin{pmatrix}
T_{1}(u_{1}, u_{2})(t) \\
T_{2}(u_{1}, u_{2})(t)
\end{pmatrix}, \quad t \in \mathbb{N}^{T}_{0},
\end{equation}
where
\begin{align} \label{A 1}
\nonumber & T_{1}(u_{1}, u_{2})(t) \\ \nonumber & = \sum^{T}_{s = 2}G_{\alpha_1}(t, s)f_{1}(u_{1}(s), u_{2}(s)) \\ & = \sum^{T}_{s = 2}K_{\alpha_1}(t, s)f_{1}(u_{1}(s), u_{2}(s)) - \sum^{t}_{s = 2}H_{\alpha_1 - 1}(t, s)f_{1}(u_{1}(s), u_{2}(s)),
\end{align}
and
\begin{align} \label{A 2}
\nonumber & T_{2}(u_{1}, u_{2})(t) \\ \nonumber & = \sum^{T}_{s = 2}G_{\alpha_2}(t, s)f_{2}(u_{1}(s), u_{2}(s)) \\ & = \sum^{T}_{s = 2}K_{\alpha_2}(t, s)f_{2}(u_{1}(s), u_{2}(s)) - \sum^{t}_{s = 2}H_{\alpha_2 - 1}(t, s)f_{2}(u_{1}(s), u_{2}(s)).
\end{align}

\begin{theorem} \label{Equiv}
A couple $(u_{1}, u_{2}) \in X^2$ is a solution of \eqref{Coupled System} if, and only if, 
\begin{equation} \label{System T}
\begin{cases}
T_{1}(u_{1}, u_{2}) = u_{1}, \\
T_{2}(u_{1}, u_{2}) = u_{2}. \\
\end{cases}
\end{equation}
\end{theorem}

In view of Theorem \ref{Equiv} it is enough to apply Theorems \ref{Perov} - \ref{Urs} to the system \eqref{System T}.

\begin{theorem} \label{One}
Assume that
\begin{enumerate}
\item[(I)] for each $i \in \{1, 2\}$, there exist nonnegative numbers $l_{i}$ and $m_{i}$ such that 
\begin{equation}
|f_{i}(u_{1}, u_{2}) - f_{i}(v_{1}, v_{2})| \leq l_{i}|u_{1} - v_{1}| + m_{i}|u_{2} - v_{2}|,
\end{equation}
for all $(u_{1}, u_{2})$, $(v_{1}, v_{2}) \in X^{2}$;
\end{enumerate}
In addition, assume that condition (H2) is satisfied. Then, \eqref{System T} has a unique solution $(u_{1}, u_{2}) \in X^2$.
\end{theorem}

\begin{proof}
For each $i \in \{1, 2\}$ and for all $(u_{1}, u_{2})$, $(v_{1}, v_{2}) \in X^{2}$, consider
\begin{align*}
& |T_{i}(u_{1}, u_{2}) - T_{i}(v_{1}, v_{2})| \\ & \leq \sum^{T}_{s = 2}\left|K_{\alpha_i}(t, s)\right| \left|f_{i}(u_{1}(s), u_{2}(s)) - f_{i}(v_{1}(s), v_{2}(s))\right| \\ & \quad + \sum^{t}_{s = 2}H_{\alpha_i - 1}(t, s)\left|f_{i}(u_{1}(s), u_{2}(s)) - f_{i}(v_{1}(s), v_{2}(s))\right| \\ & \leq \left[l_{i}|u_{1} - v_{1}| + m_{i}|u_{2} - v_{2}|\right] \left[\sum^{T}_{s = 2}\left|K_{\alpha_i}(t, s)\right| + \sum^{t}_{s = 2}H_{\alpha_i - 1}(t, s)\right] \\ & \leq \left[l_{i}|u_{1} - v_{1}| + m_{i}|u_{2} - v_{2}|\right] \left[\Lambda_i (T - 1) + H_{\alpha_i}(t, 1)\right] \\ & \leq \left[l_{i}|u_{1} - v_{1}| + m_{i}|u_{2} - v_{2}|\right] \left[\Lambda_i (T - 1) + H_{\alpha_i}(T, 1)\right] \\ & \leq a_{i}|u_{1} - v_{1}| + b_{i}|u_{2} - v_{2}|,
\end{align*}
implying that (H1) holds. Thus, by Theorem \ref{Perov}, the system \eqref{System T} has a unique solution $(u_{1}, u_{2}) \in X^2$.
\end{proof}

\begin{theorem} \label{Two}
Assume that
\begin{enumerate}
\item[(II)] for each $i \in \{1, 2\}$, there exist nonnegative numbers $a_{i}$, $b_{i}$ and $c_{i}$ such that 
\begin{equation}
|f_{i}(u_{1}, u_{2})| \leq l_{i}|u_{1}| + m_{i}|u_{2}| + n_{i},
\end{equation}
for all $(u_{1}, u_{2}) \in X^{2}$.
\end{enumerate}
In addition, assume that condition (H2) is satisfied. Then, \eqref{System T} has at least one solution $(u_{1}, u_{2}) \in X^2$ satisfying \eqref{Con}.
\end{theorem}

\begin{proof}
Since $T_{i}$, $i = 1, 2$, is a summation operator on a discrete finite set, it is trivially completely continuous on $X^2$. For each $i \in \{1, 2\}$ and for all $(u_{1}, u_{2}) \in X^{2}$, consider
\begin{align*}
& |T_{i}(u_{1}, u_{2})| \\ & \leq \sum^{T}_{s = 2}\left|K_{\alpha_i}(t, s)\right| \left|f_{i}(u_{1}(s), u_{2}(s))\right| + \sum^{t}_{s = 2}H_{\alpha_i - 1}(t, s)\left|f_{i}(u_{1}(s), u_{2}(s))\right| \\ & \leq \left[l_{i}|u_{1}| + m_{i}|u_{2}| + n_i\right] \left[\sum^{T}_{s = 2}\left|K_{\alpha_i}(t, s)\right| + \sum^{t}_{s = 2}H_{\alpha_i - 1}(t, s)\right] \\ & \leq \left[l_{i}|u_{1}| + m_{i}|u_{2}| + n_i\right] \left[\Lambda_i (T - 1) + H_{\alpha_i}(t, 1)\right] \\ & \leq \left[l_{i}|u_{1}| + m_{i}|u_{2}| + n_i\right] \left[\Lambda_i (T - 1) + H_{\alpha_i}(T, 1)\right] \\ & \leq a_{i}|u_{1}| + b_{i}|u_{2}| + c_i,
\end{align*}
implying that (H3) holds. Thus, by Theorem \ref{Schauder}, the system \eqref{System T} has at least one solution $(u_{1}, u_{2}) \in X^2$ satisfying \eqref{Con}.
\end{proof}

\begin{definition} \cite{Urs}
Let $X$ be a Banach space and $T_{1}$, $T_{2} : X \times X \rightarrow X$ be two operators. Then, the system \eqref{System T} is said to be Ulam--Hyers stable if there exist $C_{1}$, $C_{2}$, $C_{3}$, $C_{4} > 0$ such that for each $\varepsilon_{1}$, $\varepsilon_{2} > 0$ and each solution-pair $(u_{1}^{*}, u_{2}^{*}) \in X \times X$ of the in-equations:
\begin{equation}
\begin{cases}
\|u_{1} - T_{1}(u_{1}, u_{2})\|_{X} \leq \varepsilon_{1}, \\
\|u_{2} - T_{2}(u_{1}, u_{2})\|_{X} \leq \varepsilon_{2},
\end{cases}
\end{equation}
there exists a solution $(v_{1}^{*}, v_{2}^{*}) \in X \times X$ of \eqref{System T} such that
\begin{equation}
\begin{cases}
\|u_{1}^{*} - v_{1}^{*}\|_{X} \leq C_{1} \varepsilon_{1} + C_{2} \varepsilon_{2}, \\
\|u_{2}^{*} - v_{2}^{*}\|_{X} \leq C_{3} \varepsilon_{1} + C_{4} \varepsilon_{2}.
\end{cases}
\end{equation}
\end{definition}

\begin{theorem} \label{Three}
Assume that the hypothesis of Theorem \ref{One} holds. Then, the system \eqref{System T} is Ulam--Hyers stable.
\end{theorem} 

\section{Examples}
In this section, we provide two examples to illustrate the applicability of Theorem \ref{One}, Theorem \ref{Two}, and Theorem \ref{Three}.

\begin{exe}
Consider the following boundary value problem for a coupled system of fractional difference equations
\begin{equation} \label{Coupled System E}
\begin{cases}
\Big{(}\nabla^{0.5}_{0}\big{(}\nabla u_{1}\big{)}\Big{)}(t) + (0.001)\left[1 + \tan^{-1}u_{1}(t) + \tan^{-1}u_{2}(t)\right] = 0, \quad t \in \mathbb{N}^{9}_{2}, \\ \Big{(}\nabla^{0.5}_{0}\big{(}\nabla u_{2}\big{)}\Big{)}(t) + (0.002)\left[1 + \sin u_{1}(t) + \sin u_{2}(t)\right] = 0, \quad t \in \mathbb{N}^{9}_{2}, \\ u_{1}(0) + u_{1}(9) = 0, \quad \big{(}\nabla u_{1}\big{)}(1) + \big{(}\nabla u_{1}\big{)}(9) = 0, \\
u_{2}(0) + u_{2}(9) = 0, \quad \big{(}\nabla u_{2}\big{)}(1) + \big{(}\nabla u_{2}\big{)}(9) = 0.
\end{cases}
\end{equation}

Comparing \eqref{Coupled System} and \eqref{Coupled System E}, we have $T = 9$, $\alpha_{1} = \alpha_{2} = 1.5$, $$f_{1}(u_{1}, u_{2}) = (0.001)\left[1 + \tan^{-1}u_{1} + \tan^{-1}u_{2}\right],$$ and $$f_{2}(u_{1}, u_{2}) = (0.002)\left[1 + \sin u_{1} + \sin u_{2}\right],$$ for all $(u_{1}, u_{2}) \in \mathbb{R}^2$. Clearly, $f_{1}$ and $f_{2}$ are continuous on $\mathbb{R}^2$. Next, $f_{1}$ and $f_{2}$ satisfy assumption (I) with $l_{1} = 0.001$, $m_{1} = 0.001$, $l_{2} = 0.002$ and $m_{2} = 0.002$. We have, 
\begin{align*}
& a_{1} = l_{1}\left[\Lambda_1 (T - 1) + H_{\alpha_1}(T, 1)\right] = 0.0527,\\
& a_{2} = l_{2}\left[\Lambda_2 (T - 1) + H_{\alpha_2}(T, 1)\right] = 0.1053, \\
& b_{1} = m_{1}\left[\Lambda_1 (T - 1) + H_{\alpha_1}(T, 1)\right] = 0.0527,\\
& b_{2} = m_{2}\left[\Lambda_2 (T - 1) + H_{\alpha_2}(T, 1)\right] = 0.1053.
\end{align*}
Further, $$M = \begin{pmatrix}
a_1 & b_1 \\ 
a_2 & b_2
\end{pmatrix} = \begin{pmatrix}
0.0527 & 0.0527 \\ 
0.1053 & 0.1053
\end{pmatrix}.$$ The spectral radius of $M$ is $0.158$, which is less than one, implying that $M$ converges to zero. Hence, by Theorem \ref{One}, the system \eqref{Coupled System E} has a unique solution $(u_{1}, u_{2}) \in X^2$. Also, by Theorem \ref{Three}, the unique solution of \eqref{Coupled System E} is Ulam--Hyers stable.
\end{exe}

\begin{exe}
Consider the following boundary value problem for a coupled system of fractional difference equations
\begin{equation} \label{Coupled System F}
\begin{cases}
\Big{(}\nabla^{0.5}_{0}\big{(}\nabla u_{1}\big{)}\Big{)}(t) + (0.01)\left[1 + \frac{1}{\sqrt{1 + u^2_{1}(t)}} + u_{2}(t)\right] = 0, \quad t \in \mathbb{N}^{4}_{2}, \\ \Big{(}\nabla^{0.5}_{0}\big{(}\nabla u_{2}\big{)}\Big{)}(t) + (0.02)\left[1 + u_{1}(t) + \frac{1}{\sqrt{1 + u^2_{2}(t)}}\right] = 0, \quad t \in \mathbb{N}^{4}_{2}, \\ u_{1}(0) + u_{1}(4) = 0, \quad \big{(}\nabla u_{1}\big{)}(1) + \big{(}\nabla u_{1}\big{)}(4) = 0, \\
u_{2}(0) + u_{2}(4) = 0, \quad \big{(}\nabla u_{2}\big{)}(1) + \big{(}\nabla u_{2}\big{)}(4) = 0.
\end{cases}
\end{equation}

Comparing \eqref{Coupled System} and \eqref{Coupled System F}, we have $T = 4$, $\alpha_{1} = \alpha_{2} = 1.5$, $$f_{1}(u_{1}, u_{2}) = (0.01)\left[1 + \frac{1}{\sqrt{1 + u^2_{1}(t)}} + u_{2}(t)\right],$$ and $$f_{2}(u_{1}, u_{2}) = (0.02)\left[1 + u_{1}(t) + \frac{1}{\sqrt{1 + u^2_{2}(t)}}\right],$$ for all $(u_{1}, u_{2}) \in \mathbb{R}^2$. Clearly, $f_{1}$ and $f_{2}$ are continuous on $\mathbb{R}^2$. Next, $f_{1}$ and $f_{2}$ satisfy assumption (II) with $l_{1} = 0.01$, $m_{1} = 0.01$, $l_{2} = 0.02$, $m_{2} = 0.02$, $n_1 = 0.01$ and $n_2 = 0.02$. We have, 
\begin{align*}
& a_{1} = l_{1}\left[\Lambda_1 (T - 1) + H_{\alpha_1}(T, 1)\right] = 0.1219,\\
& a_{2} = l_{2}\left[\Lambda_2 (T - 1) + H_{\alpha_2}(T, 1)\right] = 0.2438, \\
& b_{1} = m_{1}\left[\Lambda_1 (T - 1) + H_{\alpha_1}(T, 1)\right] = 0.1219,\\
& b_{2} = m_{2}\left[\Lambda_2 (T - 1) + H_{\alpha_2}(T, 1)\right] = 0.2438,\\
& c_{1} = n_{1}\left[\Lambda_1 (T - 1) + H_{\alpha_1}(T, 1)\right] = 0.1219,\\
& c_{2} = n_{2}\left[\Lambda_2 (T - 1) + H_{\alpha_2}(T, 1)\right] = 0.2438.
\end{align*}
Further, $$M = \begin{pmatrix}
a_1 & b_1 \\ 
a_2 & b_2
\end{pmatrix} = \begin{pmatrix}
0.1219 & 0.1219 \\ 
0.2438 & 0.2438
\end{pmatrix}.$$ The spectral radius of $M$ is $0.3657$, which is less than one, implying that $M$ converges to zero. Hence, by Theorem \ref{Two}, the system \eqref{Coupled System F} has at least one solution $(u_{1}, u_{2}) \in X^2$ satisfying
\begin{equation*}
\begin{pmatrix} 
|u_1| \\ 
|u_2|
\end{pmatrix} \leq (I - M)^{-1} \begin{pmatrix}
c_1 \\ 
c_2
\end{pmatrix} = \begin{pmatrix}
0.1757 \\ 
0.2658
\end{pmatrix}.
\end{equation*}
\end{exe}

\section*{Conclusion}
In this article, we obtained sufficient conditions on existence, uniqueness and Ulam--Hyers stability of solutions of the system \eqref{Coupled System} using the approaches of Precup and Urs. We also provided two examples to demonstrate the applicability of established results. Observe that Theorem \ref{One} is not applicable to the system \eqref{Coupled System E}.

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\end{document}