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Sufficient conditions for the exponential stability of nonautonomous difference equations. (English) Zbl 1171.39002

The author considers the nonautonomous difference equation
\[ x_n=f_n(x_{n-1}, \ldots, x_{n-m}), \qquad n=1,2, \dots \tag{1} \]
of order \(m,\) where \(f_n:D\rightarrow {\mathbb R},\) \(D\subseteq {\mathbb R}^m\) and \(x_{1-m}, \dots, x_0\) are the initial values.
A point \(\overline{x}\in {\mathbb R}\) is called an equilibrium point of (1) if \(f_n(\overline{x}, \ldots,\overline{x})=\overline{x}\) for all \(n\in {\mathbb N}.\) Equation (1) can be transformed into the first-order nonautonomous vector equation
\[ X_n=V_{f_n}(X_{n-1}),\tag{2} \]
where \(V_{f}(u_1, \ldots, u_m)=(f(u_1,\ldots, u_m),u_1,\ldots, u_{m-1}))\), \(X_{n-1}=(x_{n-1}, \dots, x_{n-m}).\)
If \(\overline{x}\) is an equilibrium point of (1), then \(\overline{X}=(\overline{x}, \ldots, \overline{x})\) is an equilibrium point of (2), i.e., \(V_{f_n}(\overline{X})=\overline{X}\) for all \(n\in {\mathbb N}.\) Let \(\| X\|=\max\{|x_1|, \dots, |x_m|\}\) denote the sup-norm of \(X=(x_1, \ldots, x_m).\) An equilibrium point of (1) is said to be exponentially stable if there is \(\alpha\in (0,1)\) such that for every solution with initial values \(x_0, \dots, x_{1-m}\) in some nontrivial interval containing \(\overline{x},\) one has for all \(n\geq 1,\) \(|x_n-\overline{x}|\leq c\alpha^n,\) where \(c=c(x_0, \ldots, x_{1-m})>0\) is independent of \(n.\) A set \(S\) is called invariant, if it is invariant under all \(V_{f_n}\)’s, i.e., \(V_{f_n}(S)\subseteq S.\)
The main result of the paper under review is as follows. Let \(\overline{x}\) be an equilibrium point of (1) and for fixed \(\alpha\in (0,1),\) let \(A_{\alpha}=\{X\in {\mathbb R}^m: |f_n(X)-\overline{x}|\leq\alpha\| X-\overline{X}\|, \forall n\in {\mathbb N}\}.\) Then \(\overline{X}\) is exponentially stable relative to the largest invariant subset of \(A_{\alpha}.\)

MSC:

39A11 Stability of difference equations (MSC2000)
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References:

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