Abstract and Applied Analysis
Volume 2004 (2004), Issue 7, Pages 551-565

On certain comparison theorems for half-linear dynamic equations on time scales

Pavel Řehák

Mathematical Institute, Academy of Sciences of the Czech Republic, Žižkova 22, Brno 61662, Czech Republic

Received 9 October 2002

Copyright © 2004 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 obtain comparison theorems for the second-order half-linear dynamic equation [r(t)Φ(yΔ)]Δ+p(t)Φ(yσ)=0, where Φ(x)=|x|α1sgnx with α>1. In particular, it is shown that the nonoscillation of the previous dynamic equation is preserved if we multiply the coefficient p(t) by a suitable function q(t) and lower the exponent α in the nonlinearity Φ, under certain assumptions. Moreover, we give a generalization of Hille-Wintner comparison theorem. In addition to the aspect of unification and extension, our theorems provide some new results even in the continuous and the discrete case.