Discrete Dynamics in Nature and Society
Volume 2011 (2011), Article ID 570139, 12 pages
http://dx.doi.org/10.1155/2011/570139
Research Article

On the Characterization of a Class of Difference Equations

Muhammed Altun

Faculty of Arts and Sciences, Adıyaman University, 02040 Adıyaman, Turkey

Received 7 April 2011; Accepted 17 May 2011

Academic Editor: Jianshe Yu

Copyright © 2011 Muhammed Altun. 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 focus on the behavior of solutions of the difference equation 𝑥 𝑛 = 𝑏 1 𝑥 𝑛 − 1 + 𝑏 2 𝑥 𝑛 − 2 + ⋯ + 𝑏 𝑛 𝑥 0 + 𝑦 𝑛 , 𝑛 = 1 , 2 , … , where ( 𝑏 𝑘 ) is a fixed sequence of complex numbers, and ( 𝑦 𝑘 ) is a fixed sequence in a complex Banach space. We give the general solution of this difference equation. To examine the asymptotic behavior of solutions, we compute the spectra of operators which correspond to such type of difference equations. These operators are represented by upper triangular or lower triangular infinite banded Toeplitz matrices.