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Exponential dichotomy and trichotomy for difference equations. (English) Zbl 0939.39003

The authors focus their attention on some properties of exponential dichotomy and trichotomy of the linear difference equation \[ x(n+1) = A(n)x(n),\;n\in \mathbb{Z}, \] where \(A(n)\) is a \(d\times d\) invertible matrix for each \(n\in \mathbb{Z}\). Under the assumptions that this equation has an exponential dichotomy or trichotomy and the \(d\times d\) invertible matrix \(B(n)\) is such that \(A(n)+B(n)\) is invertible they prove that the perturbed equation \[ x(n+1) = (A(n)+B(n))x(n),\;n\in \mathbb{Z} \] has an exponential dichotomy or trichotomy too, if the norm of \(B(n)\) is sufficiently small. This result improves some known result on the invariance of exponential dichotomy and trichotomy under some perturbations because the radius of the perturbation considered in the paper is larger than those known. Besides the equivalence between the exponential dichotomy for linear difference equations with almost periodic coefficients in an infinite integer interval and in a finite sufficiently long integer interval is proved. This statement is a discrete version of the corresponding equivalence for an almost periodic differential equation \(\frac{dx}{dt}=A(t)x\).

MSC:

39A10 Additive difference equations
34D09 Dichotomy, trichotomy of solutions to ordinary differential equations
34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
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References:

[1] Agarwal, R.P., Difference equations and inequalities, (1992), Marcel Dekker New York · Zbl 0784.33008
[2] Agarwal, R.P.; Wong, P.J.Y., Advanced topics in difference equations, (1997), Kluwer Dordrecht · Zbl 0914.39005
[3] Papaschinopoulos, G.; Schinas, J., Criteria for an exponential dichotomy of difference equations, Czechoslovak math. J., 35, 110, 295-299, (1985) · Zbl 0693.39001
[4] Murty, K.N.; Anand, D.V.S.; Lakshimiprasannam, V., First order difference system-existence and uniqueness, (), 3533-3539 · Zbl 0890.39004
[5] J. Hong and C. Núñez, On the almost periodic type difference equations, Math. Comp. Mod. (to appear).
[6] Palmer, K.J., Exponential dichotomies, the shadowing-lemma and transversal homoclinic points, Dynamics reported, Vol. 1, 265-306, (1988) · Zbl 0676.58025
[7] Papaschinopoulos, G., On exponential trichotomy of linear difference equations, Applicable analysis, 40, 89-100, (1991) · Zbl 0687.39003
[8] Papaschinopoulos, G., Dichotomies in terms of Lyapunov functions for linear difference equations, J.math. anal. appl., 152, 524-535, (1990) · Zbl 0712.39010
[9] Yuan, R.; Hong, J., The existence of almost periodic solutions for a class of differential equations with piecewise constant argument, Nonlinear analysis TMA, 28, 8, 1439-1450, (1997) · Zbl 0869.34038
[10] Palmer, K.J., Exponential dichotomies for almost periodic equations, (), 293-298 · Zbl 0664.34016
[11] Coppel, W.A., ()
[12] Fink, A.M., ()
[13] Vinograd, R.E., Exact bound for exponential dichotomy roughness, I strong dichotomy, J. differential equations, 71, 63-71, (1988) · Zbl 0651.34051
[14] Vinograd, R.E., Exact bound for exponential dichotomy roughness, II an example of attainability, J. differential equations, 90, 203-210, (1990) · Zbl 0743.34050
[15] Vinograd, R.E., Exact bound for exponential dichotomy roughness, III semistrong dichotomy, J. differential equations, 91, 245-267, (1991) · Zbl 0743.34051
[16] Cong, N.D., Topological dynamics of random dynamical systems, (1997), Oxford Mathematical Monographs, Oxford Science Publications
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