Abstract and Applied Analysis
Volume 2004 (2004), Issue 7, Pages 551-565
doi:10.1155/S1085337504306251
On certain comparison theorems for half-linear
dynamic equations on time scales
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.
Abstract
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|α−1sgn x 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.