The paper is devoted to the study of oscillation of solutions to a class of second-order half-linear neutral differential equations with delayed arguments. New oscillation criteria are established, and they essentially improve the well-known results reported in the literature, including those for non-neutral differential equations. The adopted approach refines the classical Riccati transformation technique by taking into account such part of the overall impact of the delay that has been neglected in the earlier results. The effectiveness of the obtained criteria is illustrated via examples.

In this paper, effective oscillation criteria for third-order delay differential equations of the form, ${\left(r_2{\left(r_1{y}^{\prime }\right)}^{\prime }\right)}^{\prime }\left(t\right)+q\left(t\right)y\left(\tau \left(t\right)\right)=0$ ensuring that any nonoscillatory solution tends to zero asymptotically, are established. The results become sharp when applied to a Euler-type delay differential equation and, to the best of our knowledge, improve all existing results from the literature. Examples are provided to illustrate the importance of the main results.

In this paper, we are concerned with the oscillation properties of the third order differential equation $$ \left( {b(t) \left( {[a(t)x'(t)'} \right)^\gamma } \right)^\prime + q(t)x^\gamma (t) = 0, \gamma > 0 $$. Some new sufficient conditions which insure that every solution oscillates or converges to zero are established. The obtained results extend the results known in the literature for γ = 1. Some examples are considered to illustrate our main results.