## 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)
Full Text:

### References:

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