Advances in Difference Equations

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

Advances in Difference Equations20062006:064534

https://doi.org/10.1155/ADE/2006/64534

©  Pavel Řehák 2006

Received: 10 January 2006

Accepted: 17 March 2006

Published: 7 June 2006

Abstract

We present criteria of Hille-Nehari-type for the linear dynamic equation (r(t)yΔ)Δ + p(t)y σ = 0, that is, the criteria in terms of the limit behavior of 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.

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Authors’ Affiliations

(1)
Mathematical Institute, Academy of Sciences of the Czech Republic

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Copyright

© Pavel Řehák 2006

This article is published under license to BioMed Central Ltd. 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.