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This paper is concerned with oscillation of second-order nonlinear neutral dynamic equations on time scales with a variable delay. By using the generalized Riccati technique and integral averaging technique, new oscillation criteria are obtained for all solutions of the equation. Some results extend known results for difference equations when the time scale is the set Z + of positive integers and for differential equations when the time scale is R . Several examples are given to illustrate the results of the paper.

In this paper, we first establish some differential inequalities and then some Sturm comparison theorems are derived for the second order nonlinear differential equations. Also, our results generalize some classical Sturm comparison theorems.

Some oscillation criteria for the second order neutral nonlinear differential equation[y(t)+p(t)y(@s(t))]^''[email protected]?"i"="1^nq"i(t)f"i(y(@t"i(t)))=0,t>=t"0are established. New oscillation criteria are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of [t"0,~), rather than on the whole half-line. Our results are more natural according to the Sturm Separation Theorem.