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
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.