Advances in Difference Equations
Volume 2007 (2007), Article ID 86925, 9 pages
doi:10.1155/2007/86925

Oscillatory solutions for second-order difference equations in Hilbert spaces

Abstract

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.