Advances in Difference Equations
Volume 2006 (2006), Article ID 51401, 18 pages
doi:10.1155/ADE/2006/51401

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

L. Erbe,1 A. Peterson,1 and S. H. Saker2

1Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588-0130, USA
2Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Received 31 January 2006; Revised 16 May 2006; Accepted 16 May 2006

Copyright © 2006 L. Erbe et al. 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.

Abstract

We consider the pair of second-order dynamic equations, (r(t)(xΔ)γ)Δ+p(t)xγ(t)=0 and (r(t)(xΔ)γ)Δ+p(t)xγσ(t)=0, on a time scale T, 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 T=ℝ 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.