Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Asymptotic Expansions for Higher-Order Scalar Difference Equations | Hindawi Publishing Corporation Advances in Difference Equations Volume 2007 Article ID 67492 12 pages doi 2007 67492 Research Article Asymptotic Expansions for Higher-Order Scalar Difference Equations Ravi P. Agarwal and Mihaly Pituk Received 26 November 2006 Accepted 23 February 2007 Recommended by Mariella Cecchi We give an asymptotic expansion of the solutions of higher-order Poincare 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. Copyright 2007 R. P. Agarwal and M. 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. 1. Introduction Our principal interest in this paper is the asymptotic behavior of the solutions of higher-order nonlinear difference equations in a neighborhood of an equilibrium. In the specific case of rational difference equations this problem has been studied in several recent papers see . 1-3 and the references therein . In this paper we will deal with general nonlinear difference equations. Our main result Theorem applies in the case when the equilibrium is hyperbolic and the nonlinearity is sufficiently smooth. Roughly speaking it says that in this case the solutions of the nonlinear equation approach the equilibrium along the solutions of the corresponding linearized equation. This result is in fact a consequence of the asymptotic expansion of the solutions of Poincare difference equation established in Theorem . Our results maybe viewed as the discrete analogs of similar qualitative results known for ordinary and functional differential .
