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

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.