Advances in Difference Equations
Volume 2007 (2007), Article ID 67492, 12 pages
Asymptotic Expansions for Higher-Order Scalar Difference Equations
1Department of Mathematical Sciences, Florida Institute of Technology, Melbourne 32901-6975, FL, USA
2Department of Mathematics and Computing, University of Veszprém, P.O. Box 158, Veszprém 8201, Hungary
Received 26 November 2006; Accepted 23 February 2007
Academic Editor: Mariella Cecchi
Copyright © 2007 Ravi P. Agarwal and Mihály Pituk. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We give an asymptotic expansion of the solutions of higher-order Poincaré difference equation in terms of the characteristic solutions of the limiting equation. As a consequence, we obtain an asymptotic description of the solutions approaching a hyperbolic equilibrium of a higher-order nonlinear difference equation with sufficiently smooth nonlinearity. The proof is based on the inversion formula for the
-transform and the residue theorem.