Abstract
In this chapter we will consider linear difference systems of the form \(x(n + 1) = A(n)x(n)\), where det A(n) ≠ 0 for all n ≥ n 0. Various procedures will be discussed (similar to those in the preceding chapter) for bringing such a system (if possible) into what we have called an L-diagonal form, so that the results of Chap. 3 may be used.
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References
P.M. Batchelder, An Introduction to Linear Difference Equations (Dover Publications Inc., New York, 1967)
Z. Benzaid, D.A. Lutz, Asymptotic representation of solutions of perturbed systems of linear difference equations. Stud. Appl. Math. 77, 195–221 (1987)
S. Bodine, D.A. Lutz, Asymptotic solutions and error estimates for linear systems of difference and differential equations. J. Math. Anal. Appl. 290, 343–362 (2004)
S. Bodine, D.A. Lutz, On asymptotic equivalence of perturbed linear systems of differential and difference equations. J. Math. Anal. Appl. 326, 1174–1189 (2007)
S. Elaydi, Asymptotics for linear difference equations I: basic theory. J. Differ. Equ. Appl. 5, 563–589 (1999)
W.A. Harris, D.A. Lutz, On the asymptotic integration of linear differential systems. J. Math. Anal. Appl. 48, 1–16 (1974)
P.F. Hsieh, F. Xie, Asymptotic diagonalization of a linear ordinary differential system. Kumamoto J. Math. 7, 27–50 (1994)
P.F. Hsieh, F. Xie, Asymptotic diagonalization of a system of linear ordinary differential equations. Dyn. Continuous Discrete Impuls. Syst. 2, 51–74 (1996)
G.K. Immink, Asymptotics of Analytic Difference Equations. Lecture Notes in Mathematics, vol. 1085 (Springer, Berlin, 1984)
K. Knopp, Theorie und Anwendung der unendlichen Reihen (Springer, Berlin/New York, 1964)
R.J. Kooman, Asymptotic diagonalization of matrix systems. J. Approx. Theory 171, 33–64 (2013)
M. Pituk, Asymptotic behavior of a Poincaré recurrence system. J. Approx. Theory 91, 226–243 (1997)
I.M. Rapoport, On Some Asymptotic Methods in the Theory of Differential Equations (Russian). Izdat. Akad. Nauk Ukrain (SSR, Kiev, 1954)
G. Ren, Y. Shi, Y. Wang, Asymptotic behavior of solutions of perturbed linear difference systems. Linear Algebra Appl. 395, 283–302 (2005)
W. Trench, Asymptotic behavior of solutions of Poincaré recurrence systems. Comput. Math. Appl. 28, 317–324 (1994)
W. Trench, Linear asymptotic equilibrium and uniform, exponential, and strict stability of linear difference systems. Comput. Math. Appl. 36, 261–267 (1998)
W. Trench, Linear asymptotic equilibrium of nilpotent systems of linear difference equations. J. Differ. Equ. Appl. 5, 549–556 (1999)
A. Wintner, On a theorem of Bôcher in the theory of ordinary linear differential equations. Am. J. Math. 76, 183–190 (1954)
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Bodine, S., Lutz, D.A. (2015). Conditioning Transformations for Difference Systems. In: Asymptotic Integration of Differential and Difference Equations. Lecture Notes in Mathematics, vol 2129. Springer, Cham. https://doi.org/10.1007/978-3-319-18248-3_5
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DOI: https://doi.org/10.1007/978-3-319-18248-3_5
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