Behavior of Non-Oscillatory Solutions of Fourth-Order Neutral Differential Equations

In this paper, we deal with the asymptotics and oscillation of the solutions of fourth-order neutral differential equations of the form r (t) (z''' (t))(alpha))' + q (t) x(alpha) (g (t)) = 0, where z (t) : = x (t) + p ( t) x (delta (t)). By using a generalized Riccati transformation, we study asymptotic behavior and derive some new oscillation criteria. Our results extend and improve some well-known results which were published recently in the literature. Symmetry ideas are often invisible in these studies, but they help us decide the right way to study them, and to show us the correct direction for future developments. An example is given to illustrate the importance of our results.