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New discrete type inequalities and global stability of nonlinear difference equations. (English) Zbl 1177.39026
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 $$|f(n,x_{n-h_0},x_{n-h_1},\dots,x_{n-h_r})|\leq \sum_{i=0}^r q_i|x_{n-h_i}|\tag1$$ 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 $$|f(n,x_{n-h_0},x_{n-h_1},\dots,x_{n-h_r})|\leq \prod_{i=0}^r \beta_i| x_{n-h_i}|^{\alpha_i}\tag2$$ 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 $0$.

39A30Stability theory (difference equations)
Full Text: DOI
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