On the oscillation of certain second order nonlinear dynamic equations

Abstract

We establish some new oscillation criteria for solutions to the second order nonlinear dynamic equation (a (xΔ)α)Δ (t) + q (t) xβ (t) = 0 on an arbitrary time scale T, where α and β are ratios of positive odd integers, a and q are positive rd-continuous functions on T. The results are obtained when ∫∞ a- 1 / α (s) Δ s < ∞ or ∫∞ a- 1 / α (s) Δ s = ∞. These criteria unify and extend known results for the corresponding half-linear and nonlinear differential and difference equations. Some of our results are new even in the continuous and discrete cases. © 2009 Elsevier Ltd. All rights reserved.