Advances in Difference Equations
Volume 2006 (2006), Article ID 64534, 15 pages
doi:10.1155/ADE/2006/64534
Abstract
We present criteria of Hille-Nehari-type for the linear dynamic equation (r(t)yΔ)Δ+p(t)yσ=0, that is, the criteria in terms of the limit behavior of (∫at1/r(s)Δs)∫t∞p(s)Δs as t→∞. As a particular important case, we get that there is a (sharp) critical constant in those criteria which belongs to the interval [0,1/4], and its value depends on the graininess μ and the coefficient r. Also we offer some applications, for example, criteria for strong (non-) oscillation and Kneser-type criteria, comparison with existing results (our theorems turn out to be new even in the discrete case as well as in many other situations), and comments with examples.