# Estimation for the region of attraction of nonlinear difference equations

Original article

## Summary.

We consider nonlinear systems of difference equations \(x(n+1)=f(x(n)), \quad n\ge 0\) with \(f(x(n))=Ax(n)+\varphi(x(n))\) and \(f(x(n))=Ax(n)+\mu\varphi(x(n))\), where *A* is any \(N\times N\) matrix, \(\varphi(x)\) is a continuous vector-function, \(\varphi(0) = 0\) and \(\mu\) is a numeral parameter. The spectrum of *A* belongs to the unit circle \(\{|\lambda| < 1\}\). We give the estimations for the region of attraction and the speed of convergence solutions to the zero solution of the systems. We indicate a set *M* such that for solutions of the system with parameter \(\mu \in M\) the limit \(\|x(n)\| \to 0, n \to \infty\) is true.

Mathematics Subject Classification (1991): 39A11

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© Springer-Verlag Berlin Heidelberg 2002