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(xΔ)α)Δ(t)+q(t)xβ(t)=0(a(xΔ)α)Δ(t)+q(t)xβ(t)=0 on an arbitrary time scale TT, where αα and ββ are ratios of positive odd integers, aa and qq are positive rd-continuous functions on TT. The results are obtained when ∫∞a−1/α(s)Δs<∞ or ∫∞a−1/α(s)Δs=∞. 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.