The asymptotic behavior and oscillation for a class of third-order nonlinear delay dynamic equations

In this paper, we consider a class of third-order nonlinear delay dynamic equations. First, we establish a Kiguradze-type lemma and some useful estimates. Second, we give a sufficient and necessary condition for the existence of eventually positive solutions having upper bounds and tending to zero. Third, we obtain new oscillation criteria by employing the Pötzsche chain rule. Then, using the generalized Riccati transformation technique and averaging method, we establish the Philos-type oscillation criteria. Surprisingly, the integral value of the Philos-type oscillation criteria, which guarantees that all unbounded solutions oscillate, is greater than θ₄(t₁, T). The results of Theorem 3.5 and Remark 3.6 are novel. Finally, we offer four examples to illustrate our results.