Let \(\mathbb{Z}^+\) be the set of positive integers and consider the difference equation
\[ \Delta x_n = -px_n + f(n,x_{n-h_0},x_{n-h_1},\dots,x_{n-h_r}),\quad n=0,1,2,\dots \]
where \(r\) is a positive integer, \(0=h_0<h_1<\cdots <h_r\), and \(h_i\in \mathbb{Z}^+\) for \(i=1,\dots,r\).
The authors’ main results can be stated as follows: The zero solution is globally asymptotically stable if
\[ |f(n,x_{n-h_0},x_{n-h_1},\dots,x_{n-h_r})|\leq \sum_{i=0}^r q_i|x_{n-h_i}|\tag{1} \]
where \(q_i\geq 0,~i=1,\dots,r-1\), \(q_r>0\), and \(\sum_{i=0}^r q_i<p\leq 1\) for some positive real number \(p\) or
\[ |f(n,x_{n-h_0},x_{n-h_1},\dots,x_{n-h_r})|\leq \prod_{i=0}^r \beta_i| x_{n-h_i}|^{\alpha_i}\tag{2} \]
where \(\alpha_i>0\) for \(i=1,\dots,r\) with \(\sum_{i=0}^r \alpha_i=1\), \(\beta_i>0\) for \(i=1,\dots,r\), and \(\prod_{i=0}^r \beta_i<p\leq 1\) for some positive real number \(p\).
It appears to me that the authors’ results are valid provided that the index \(i\) (for the parameters \(q_i,~\alpha_i\), and \(\beta_i\)) starts from \(0\).