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(Delta))(alpha))(Delta) (t) + q(t)x(beta) (t) = 0 on an arbitrary time scale T, where alpha and beta are ratios of positive odd integers, a and q are positive rd-continuous functions on T. The results are obtained when integral(infinity) a(-1/alpha)(s) Delta s < infinity or integral(infinity) a(-1/alpha)(s) Delta s = infinity. 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. (C) 2009 Elsevier Ltd. All rights reserved.