On the oscillation of certain second order nonlinear dynamic equations

We establish some new oscillation criteria for solutions to the second order nonlinear dynamic equation (a (x^Δ)^α)^Δ (t) + q (t) x^β (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.