Open Access

Hille-Kneser-type criteria for second-order dynamic equations on time scales

Advances in Difference Equations20062006:051401

DOI: 10.1155/ADE/2006/51401

Received: 31 January 2006

Accepted: 16 May 2006

Published: 27 July 2006

Abstract

We consider the pair of second-order dynamic equations, (r(t)(xΔ) γ )Δ + p(t)x γ (t) = 0 and (r(t)(xΔ) γ )Δ + p(t)x γσ (t) = 0, on a time scale , where γ > 0 is a quotient of odd positive integers. We establish some necessary and sufficient conditions for nonoscillation of Hille-Kneser type. Our results in the special case when involve the well-known Hille-Kneser-type criteria of second-order linear differential equations established by Hille. For the case of the second-order half-linear differential equation, our results extend and improve some earlier results of Li and Yeh and are related to some work of Došlý and Řehák and some results of Řehák for half-linear equations on time scales. Several examples are considered to illustrate the main results.

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Authors’ Affiliations

(1)
Department of Mathematics, University of Nebraska-Lincoln
(2)
Department of Mathematics, Faculty of Science, Mansoura University

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Copyright

© L.Erbe et al. 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.