Nonlinear Neutral Differential Equations with Multiple Delays: An Oscillation Analysis

This paper investigates the oscillatory behavior of the second-order nonlinear neutral differential equation ${\left(r\left(s\right){\left({\omega }^{\prime }\left(s\right)\right)}^{\alpha }\right)}^{\prime }+{\displaystyle \sum _{i=1}^n}{\phi }_i\left(s\right)x^{\gamma }\left({\sigma }_i\left(s\right)\right)=0$, $\omega \left(s\right):=x\left(s\right)+\vartheta \left(s\right)x\left(\tau \left(s\right)\right)$, in the noncanonical setting ${\int }^{\infty }{r}^{-1/\alpha }\left(\xi \right)\mathrm{d}\xi <\infty$, under the large neutral coefficient assumption $\vartheta \left(s\right)\ge \ell >1$. The aim is to establish new sufficient conditions for the oscillation of all solutions within a unified analytical framework that covers several delayed arguments simultaneously. The analysis combines the eventual monotonic properties of the neutral function with linearization and comparison techniques and treats separately the two parameter regimes $\alpha \ge \gamma$ and $\alpha <\gamma$, which exhibit different asymptotic behavior. Six oscillation criteria are obtained, each expressed as a single integral condition, and these criteria are shown to form a hierarchy in which one criterion in each regime is the least restrictive of the six; this is a comparison among sufficient conditions and not a sharpness (necessity) statement. Two refinements are also established: a delay-selection version of the criteria, which treats the delays separately and removes the need for the ordering ${\sigma }_i\le \tau$ for all but one delay, and a sharper lower bound for the constant produced by the neutral relation. Five examples are presented covering both parameter regimes, including one with a non-constant neutral coefficient and distinct forcing coefficients, and a linear Euler-type example for which the oscillation threshold can be bracketed rigorously so that the distance of the criteria from optimality can be measured, together with a quantitative comparison of the resulting oscillation thresholds and their dependence on the delay ratio and on the neutral bound. They illustrate the applicability of the criteria in a parameter range not covered by several closely related studies, which assume $0\le \vartheta \left(s\right)<1$.