Discrete Dynamics in Nature and Society
Volume 2007 (2007), Article ID 56813, 20 pages
doi:10.1155/2007/56813
Research Article

Asymptotics of Some Classes of Higher-Order Difference Equations

Stevo Stević

Mathematical Institute of the Serbian Academy of Science, Knez Mihailova 35/I, Beograd 11000, Serbia

Received 9 February 2007; Accepted 23 April 2007

Copyright © 2007 Stevo Stević. 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.

Abstract

We present some methods for finding asymptotics of some classes of nonlinear higher-order difference equations. Among others, we confirm a conjecture posed by S. Stević (2005). Monotonous solutions of the equation yn=A+(yn−k/∑j=1mβjyn−qj)p, n∈ℕ0, where p,A∈(0,∞), k,m∈ℕ, qj,j∈{1,…,m}, are natural numbers such that q1<q2<⋯<qm,βj∈(0,+∞), j∈{1,…,m}, ∑j=1mβj=1, and y−s,y−s+1,…,y−1∈(0,∞), where s=max{k,qm}, are found. A new inclusion theorem is proved. Also, some open problems and conjectures are posed.