Asymptotic properties of solutions of certain third-order dynamic equations

In this paper, the well known oscillation criteria due to Hille and Nehari for second-order linear differential equations will be generalized and extended to the third-order nonlinear dynamic equation (r2(t)((r1(t)xΔ(t))Δ)γ)Δ+q(t)f(x(t))=0(r2(t)((r1(t)xΔ(t))Δ)γ)Δ+q(t)f(x(t))=0 on time scale TT, where γ≥1γ≥1 is a ratio of odd positive integers. Our results are essentially new even for third-order differential and difference equations, i.e., when T=RT=R and T=NT=N. Two examples of dynamic equations on different time scales are given to show the applications of our main results.