Oscillation criteria for half-linear dynamic equations on time scales

This paper is concerned with oscillation of the second-order half-linear dynamic equation(r(t)(xΔ)γ)Δ+p(t)xγ(t)=0, on a time scale T where γ is the quotient of odd positive integers, r(t) and p(t) are positive rd-continuous functions on T. Our results solve a problem posed by [R.P. Agarwal, D. O'Regan, S.H. Saker, Philos-type oscillation criteria for second-order half linear dynamic equations, Rocky Mountain J. Math. 37 (2007) 1085-1104; S.H. Saker, Oscillation criteria of second order half-linear dynamic equations on time scales, J. Comput. Appl. Math. 177 (2005) 375-387] and our results in the special cases when T=R and T=Z involve and improve some oscillation results for second-order differential and difference equations; and when T=hZ, T=qN0 and T=N02, etc., our oscillation results are essentially new. Some examples illustrating the importance of our results are also included.