Oscillation behavior of third-order quasilinear neutral delay dynamic equations on time scales

The aim of this paper is to give oscillation criteria for the third-order quasilinear neutral delay dynamic equation [r(t)([x(t)+ p(t)x(?0(t))]??)?]? + q1(t)x?(?1(t)) + q2(t)x?(?2(t)) = 0; on a time scale T, where 0 < ? < ? < ?. By using a generalized Riccati transformation and integral averaging technique, we establish some new sufficient conditions which ensure that every solution of this equation oscillates or converges to zero.

On the Oscillation of Second Order Nonlinear Neutral Delay Dynamic Equations

In this paper we obtain some new oscillation criteria for the second order nonlinear neutral delay dynamic equation (𝑥(𝑡) – 𝑝(𝑡)𝑥(𝑡 – τ 1))ΔΔ + 𝑞(𝑡)𝑓(𝑥(𝑡 – τ 2)) = 0, on a time scale 𝕋. Moreover, a new sufficient condition for the oscillation sublinear equation (𝑥(𝑡) – 𝑝(𝑡)𝑥(𝑡 – τ 1))″ + 𝑞(𝑡)𝑓(𝑥(𝑡 – τ 2)) = 0, is presented, which improves other conditions and an example is given to illustrate our result.