Asymptotic behavior of solutions of nonautonomous difference equations
Let \(B_ r\) denote the ball of radius \(r\) in \(\mathbb{R}^ d\). The author proves that if there exist functions \(V(n,x) : \mathbb{N}_ 0 \times B_ r \to \mathbb{R}\), \(W(n,x) : \mathbb{N}_ 0 \times B_ r \to \mathbb{R}_ +\), \(\alpha : \mathbb{R}_ + \to \mathbb{R}_ +\), \(p,q : \mathbb{N}_ 0 \to \mathbb{R}_ +\) \((W(n,x), V(n,x), \alpha (x)\) are continuous in \(x, \alpha\)-nondecreasing, \(V(n,x)\) bounded from below by some real constant, \(\sum^ \infty_{k = 1} q(k) < \infty\), \(\sum^ \infty_{k = 1} p(m_ k) = \infty\) for any sequence \(\{m_ k\}\) of positive integers \(m_ k \to \infty)\) and such that \(V(n + 1, f(n,x)) - V(n,x) \leq - p(n) \alpha (W(n,x)) + q(n)\), \((n,x) \in \mathbb{N}_ 0 \times B_ r\), then for every solution \(\{x(n)\}\) of the difference equation \(x(n + 1) = f(n,x(n))\) there is \(W(n,x(n)) \to 0\) as \(n \to \infty\). The author gives examples of difference equations for which applying his result he deduces that the zero solution is asymptotically stable, and for which earlier results like discrete LaSalle's invariance principle or discrete Salvadori's theorem cannot be applied.
scientific article; zbMATH DE number 635928