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Discrete retract principle for systems of discrete equations. (English) Zbl 0999.39005

The author investigates asymptotic properties of solutions of the difference system \[ \Delta u_k=F(k,u_k), \tag{*} \] where \(u_k\in \mathbf R^n\) and \(F:\mathbf N\times \mathbf R\to \mathbf R\), using the discrete version of the Wazewski retraction principle. In particular, given two sequences \(b_k=(b_k^{[1]},\dots,b_k^{[n]}) \in \mathbf R^n\), \(c_k=(c_k^{[1]},\dots,c_k^{[n]})\in \mathbf R^n\), \(k\in \mathbf N\) with \(b_k^{[i]}<c_k^{[i]}\), \(k\in \mathbf N\), \(i=1,\dots,n\), conditions on the function \(F\) are given which guarantee that there exists a solution \(u_k=(u_k^{[1]},\dots,u_k^{[n]})\) of (*) such that \(b_k^{[i]}<u_k^{[i]}<c_k^{[i]}\), \(k\in \mathbf N\), \(i=1,\dots,n\). General results concerning system (*) are applied to the nonhomogeneous linear system \(\Delta u_k = A_ku_k+g_k\), \(A_k\in \mathbf R^{n\times n}, g_k\in \mathbf R^n\), \(k\in \mathbf N\), and to the scalar Bernoulli type difference equation \(\Delta u_k=u_k(\gamma_k u_k-\beta_k)\) with positive sequences \(\beta_k,\gamma_k\). The discrepancies between discrete and continuous case are pointed out and a number of illustrating examples is given as well.

MSC:

39A11 Stability of difference equations (MSC2000)
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