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.