Advances in Difference Equations
Volume 2006 (2006), Article ID 64534, 15 pages
How the constants in Hille-Nehari theorems depend on time scales
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 , that is, the criteria in terms of the limit behavior of as . As a particular important case, we get that there is a (sharp)
critical constant in those criteria which belongs to the interval , and its value depends on the graininess and the coefficient . 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.