HILLE-KNESER-TYPE CRITERIA FOR SECOND-ORDER DYNAMIC EQUATIONS ON TIME SCALES

Authors

Lynn Erbe, University of Nebraska-LincolnFollow
Allan C. Peterson, University of Nebraska-LincolnFollow
S. H. Shaker, Mansoura UniversityFollow

Document Type

Article

Date of this Version

2006

Citation

Volume 2006, Article ID 51401, Pages 1–18

Comments

Copyright © 2006 L. Erbe et al.

This is an open access article

DOI 10.1155/ADE/2006/51401

Abstract

We consider the pair of second-order dynamic equations, (r(t)(x^Δ)^γ)^Δ + p(t)x^γ(t) = 0 and (r(t)(x^Δ)^γ)^Δ + p(t)x^γσ (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 Dosly and Rehak and some results of Rehak for half-linear equations on time scales. Several examples are considered to illustrate the main results.