Oscillation and asymptotic behavior of third-order nonlinear delay differential equations with positive and negative terms

Abstract

We study the oscillation and nonoscillation of third order delay differential equations with positive and negative terms. We establish Kiguradze Lemma, and offer the novel and useful estimate x(τ1(t))x(t)≥τ1(t)t which plays an important role in our main results. First of all, we give Leighton–Wintner type criteria. Then, by using Riccati transformation, we establish new oscillation criteria including Kamenev-type oscillation criteria. Finally, we present a sufficient and necessary condition which guarantees that the nonoscillatory solution x(t) has an upper bound and tends to zero. Examples illustrate the validity and practicability of our results.