Consider the following nonlinear difference equation with variable coefficients: \[ \left\{ \begin{array}{l} x(n+1)=x(n)-\dis \sum_{j=0}^m a_j(n) f(x(n-j)), \ \ \ \ n=0,1,2,\cdots, \\ x(j)=x_{j}, \ \ \ -m \leq j \leq 0, \\ \end{array} \right. \] where we assume that $ f(x) $ is a strictly monotone increasing function on $ (-\infty,+\infty) $ such that $ f(0)=0, $ and if $ f(x) \neq x, $ then $ \dis \lim_{x \to -\infty}f(x) $ is finite, and $ \{a_j(n)\}_{n=0}^{\infty}, \ 0 \leq j \leq m $ are nonnegative sequences such that $ \sum_{n=0}^{\infty} \sum_{j=0}^m a_j(n)=+\infty $. In this paper, we establish sufficient conditions for the zero solution of the above equation to be globally asymptotically stable. Applying these conditions to the cases $ f(x)=x $ and $ f(x) = e^x-1 $, we improve the ``$3/2$ criteria" type stability conditions for linear and nonlinear difference equations.

Global stability for nonlinear difference equations with variable coefficients

GUGLIELMI, NICOLA
2007

Abstract

Consider the following nonlinear difference equation with variable coefficients: \[ \left\{ \begin{array}{l} x(n+1)=x(n)-\dis \sum_{j=0}^m a_j(n) f(x(n-j)), \ \ \ \ n=0,1,2,\cdots, \\ x(j)=x_{j}, \ \ \ -m \leq j \leq 0, \\ \end{array} \right. \] where we assume that $ f(x) $ is a strictly monotone increasing function on $ (-\infty,+\infty) $ such that $ f(0)=0, $ and if $ f(x) \neq x, $ then $ \dis \lim_{x \to -\infty}f(x) $ is finite, and $ \{a_j(n)\}_{n=0}^{\infty}, \ 0 \leq j \leq m $ are nonnegative sequences such that $ \sum_{n=0}^{\infty} \sum_{j=0}^m a_j(n)=+\infty $. In this paper, we establish sufficient conditions for the zero solution of the above equation to be globally asymptotically stable. Applying these conditions to the cases $ f(x)=x $ and $ f(x) = e^x-1 $, we improve the ``$3/2$ criteria" type stability conditions for linear and nonlinear difference equations.
Nonlinear difference equations; Global stability
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.12571/3141
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