Advances in Difference Equations
Volume 2007 (2007), Article ID 86925, 9 pages
Research Article

Oscillatory Solutions for Second-Order Difference Equations in Hilbert Spaces

Cristóbal González and Antonio Jiménez-Melado

Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, Málaga 29071, Spain

Received 16 March 2007; Revised 24 July 2007; Accepted 27 July 2007

Academic Editor: Donal O'Regan

Copyright © 2007 Cristóbal González and Antonio Jiménez-Melado. 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.


We consider the difference equation Δ2xn+f(n,xn+τ)=0, τ=0,1,, in the context of a Hilbert space. In this setting, we propose a concept of oscillation with respect to a direction and give sufficient conditions so that all its solutions be directionally oscillatory, as well as conditions which guarantee the existence of directionally positive monotone increasing solutions.