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

#### 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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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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