Global stability of some classes of higher-order nonlinear difference equations. (English) Zbl 1194.39012
The author gives proofs of some known results on global stability of higher-order nonlinear difference equations
\[ x_{n+1} - x_n = -p x_n + f(n,x_{n - h_1},\dots,x_{n - h_r}),\quad n = 0,1,2,\dots, \]
where \(h_i \in N\), \(i =1,\dots,r\), \(r \in N\), \(p \in [0,1]\) and the function \(f\) satisfies inequalities
\[ |f(n,y_0,\dots,y_r)| \leq \sum_{i=0}^r q_i |y_i| \] or
\[ |f(n,y_0,\dots,y_r)| \leq \beta\prod_{i=0}^r |y_{i}|^{\alpha_i}, \] \(\sum_{i=0}^r \alpha_i= 1, \alpha_i > 0, \beta <1\).
\[ x_{n+1} - x_n = -p x_n + f(n,x_{n - h_1},\dots,x_{n - h_r}),\quad n = 0,1,2,\dots, \]
where \(h_i \in N\), \(i =1,\dots,r\), \(r \in N\), \(p \in [0,1]\) and the function \(f\) satisfies inequalities
\[ |f(n,y_0,\dots,y_r)| \leq \sum_{i=0}^r q_i |y_i| \] or
\[ |f(n,y_0,\dots,y_r)| \leq \beta\prod_{i=0}^r |y_{i}|^{\alpha_i}, \] \(\sum_{i=0}^r \alpha_i= 1, \alpha_i > 0, \beta <1\).
Reviewer: Victor I. Tkachenko (Kyïv)
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