The stability of partial difference systems with retarded arguments (Q1827182)

The authors consider the system of partial difference equations \[ Z(x,y)=\begin{cases} \sum_{k=1}^NA_k(x,y)Z(x-p_k(x),y-q_k(y)),&x,y\in \Omega_0,\\ \varphi (x,y), &x,y\in \Omega_2,\end{cases}\tag{*} \] where \(p_k:[0,\infty )\to \mathbb R_{+}, q_k:[0,\infty )\to \mathbb R_{+},\) and \(p_k(.),q_k(.)\) are both continuous functions. \(Z,\varphi \in \mathbb R^n, A_k:\Omega _0\to \mathbb R^{n\times n}\), \(k=1,2,\dots,N,\) are real continuous functions, and \[ \Omega_0=\{(x,y)\mid x\geq 0,y\geq 0\} ,\Omega_1=\{(x,y)\mid x\geq -p,y\geq -q\} ,\Omega_2=\Omega_1/\Omega_0, \] where \(p>0,q>0\). Sufficient conditions are obtained so that the solution \(Z(x,y)\) satisfies \(\| Z(x,y)\| \to 0\). The stability of (*) has been discussed as well.