Advances in Difference Equations
Volume 2007 (2007), Article ID 67492, 12 pages
doi:10.1155/2007/67492

Asymptotic expansions for higher-order scalar difference equations

Ravi P. Agarwal1 and Mihály Pituk2

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

Abstract

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 z -transform and the residue theorem.