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
2007-01-01

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.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11697/21463
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