Advances in Difference Equations 
Volume 2008 (2008), Article ID 505324, 15 pages
doi:10.1155/2008/505324
Research Article

On the Growth of Nonoscillatory Solutions for Difference Equations with Deviating Argument

M. Cecchi,1 Z. Došlá,2 and M. Marini1

 1Department of Electronics and Telecommunications, University of Florence, Via S. Marta 3, 50139 Firenze, Italy
2Department of Mathematics and Statistics, Masaryk University, Janáčkovo nám. 2a, 60200 Brno, Czech Republic
Received 28 March 2008; Accepted 27 May 2008

Recommended by Martin Bohner

Abstract

The half-linear difference equations with the deviating argument Δ(an|Δxn|αsgn Δxn)+bn|xn+q|αsgn xn+q=0 , q ∈ ℤ are considered. We study the role of the deviating argument q, especially as regards the growth of the nonoscillatory solutions and the oscillation. Moreover, the problem of the existence of the intermediate solutions is completely resolved for the classical half-linear equation (q = 1). Some analogies or discrepancies on the growth of the nonoscillatory solutions for the delayed and advanced equations are presented; and the coexistence with different types of nonoscillatory solutions is studied.