Advances in Difference Equations
Volume 2006 (2006), Article ID 64534, 15 pages

How the constants in Hille-Nehari theorems depend on time scales

Pavel Řehák

Mathematical Institute, Academy of Sciences of the Czech Republic, Žižkova 22, Brno CZ-61662, Czech Republic

Received 10 January 2006; Revised 7 March 2006; Accepted 17 March 2006

Copyright © 2006 Pavel Řehák. 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 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)tp(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.