Journal of Applied Mathematics
Volume 2011 (2011), Article ID 262316, 18 pages
http://dx.doi.org/10.1155/2011/262316
Research Article

Convergence and Divergence of the Solutions of a Neutral Difference Equation

G. E. Chatzarakis and G. N. Miliaras

Department of Electrical Engineering Educators, School of Pedagogical and Technological Education (ASPETE), N. Heraklion, 14121 Athens, Greece

Received 9 June 2011; Accepted 7 August 2011

Academic Editor: Alexandre Carvalho

Copyright © 2011 G. E. Chatzarakis and G. N. Miliaras. 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 investigate the asymptotic behavior of the solutions of a neutral type difference equation of the form Δ [ 𝑥 ( 𝑛 ) + 𝑐 𝑥 ( 𝜏 ( 𝑛 ) ) ] + 𝑝 ( 𝑛 ) 𝑥 ( 𝜎 ( 𝑛 ) ) = 0 , where 𝜏 ( 𝑛 ) is a general retarded argument, 𝜎 ( 𝑛 ) is a general deviated argument (retarded or advanced), 𝑐 ∈ ℝ , ( 𝑝 ( 𝑛 ) ) 𝑛 ≥ 0 is a sequence of positive real numbers such that 𝑝 ( 𝑛 ) ≥ 𝑝 , 𝑝 ∈ ℝ + , and Δ denotes the forward difference operator Δ 𝑥 ( 𝑛 ) = 𝑥 ( 𝑛 + 1 ) − 𝑥 ( 𝑛 ) . Also, we examine the asymptotic behavior of the solutions in case they are continuous and differentiable with respect to 𝑐 .