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.


39A11 Stability of difference equations (MSC2000)
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[1] Diblík, J., Asymptotic behaviour of solutions of linear differential equations with delay, Ann. Polon. Math., LVIII.2, 131-137 (1993) · Zbl 0784.34053
[2] Diblík, J., Asymptotic representation of solutions of equation \(y(t) = β(t)[y(t)\) − \(y(t\) − τ \((t))]\), J. Math. Anal. Appl., 217, 200-215 (1998) · Zbl 0892.34067
[3] Vrdoljak, B., On behaviour of solutions of system of linear differential equations, Math. Communications, 2, 47-57 (1997) · Zbl 0882.34012
[4] Diblík, J., A multidimensional singular boundary-value problem of the Cauchy-Nicoletti type, Georg. Math. J., 4, 303-312 (1997) · Zbl 0886.34017
[5] Diblík, J., The singular Cauchy-Nicoletti problem for the system of two ordinary differential equations, Math. Bohem., 117, 55-67 (1992) · Zbl 0817.34013
[6] Hartman, Ph., Ordinary Differential Equations (1964), John Wiley & Sons: John Wiley & Sons New York · Zbl 0125.32102
[7] Lakshmikantham, V.; Leela, S., Differential and Integral Inequalities—Ordinary Differential Equations, I (1969), Academic Press: Academic Press New York · Zbl 0177.12403
[8] Ważewski, T., Sur un principe topologique de l’examen de l’allure asymptotique des integrales des equations differentielles ordinaires, Ann. Soc. Polon. Math., 20, 279-313 (1947) · Zbl 0032.35001
[9] Golda, W.; Werbowski, J., Oscillation of linear functional equations of the second order, Funkc. Ekvac., 37, 221-227 (1994) · Zbl 0807.39007
[10] Győri, I.; Pituk, M., Asymptotic formulae for the solutions of a linear delay difference equation, J. Math. Anal. Appl., 195, 376-392 (1995) · Zbl 0846.39003
[11] Győri, I.; Pituk, M., Comparison theorems and asymptotic equilibrium for delay differential and difference equations, Dyn. Systems and Appl., 5, 277-302 (1996) · Zbl 0859.34053
[12] Zhang, S., Stability of infinite delay difference systems, Nonl. Anal. T.M.A., 22, 1121-1129 (1994) · Zbl 0822.39002
[13] Zhang, S., Boundednes of infinite delay difference systems, Nonl. Anal. T.M.A., 22, 1209-1219 (1994) · Zbl 0805.39003
[14] Agarwal, R. P.; Popenda, J., Periodic solutions of first order linear difference equations, Mathl. Comput. Modelling, 22, 1, 11-19 (1995) · Zbl 0871.39002
[15] Agarwal, R. P.; Magnucka-Blandzi, E.; Popenda, J., Best possible Gronwall inequalities, Mathl. Comput. Modelling, 26, 3, 1-8 (1997) · Zbl 0902.26008
[16] Magnucka-Blandzi, E.; Popenda, J., On the asymptotic behavior of a rational system of difference equations, J. Difference Eq. Appl., 5, 271-286 (1999) · Zbl 0933.39016
[17] Popenda, J., On the asymptotic behavior of the solutions of an \(n^{th}\) order difference equation, Annal. Polon. Math., XLIV, 95-111 (1984) · Zbl 0553.39002
[18] Popenda, J.; Werbowski, J., On the discrete analogy of Gronwall lemma, Fasciculi Math., 11, 143-154 (1979) · Zbl 0458.26008
[19] Agarwal, R. P., Differential Equations and Inequalities, Theory, Methods, and Applications (1992), Marcel Dekker: Marcel Dekker New York · Zbl 0784.33008
[20] Borsuk, K., Theory of Retracts (1967), PWN: PWN Warsaw · Zbl 0153.52905
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