Oscillatory solutions for second-order difference equations in Hilbert spaces

We consider the difference equation Delta(2)x(n) + f(n,x(n+tau)) = 0, tau = 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. Copyright (C) 2007 C. Gonzalez and A. Jimenez-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.