This paper establishes new monotonic properties of nonoscillatory solutions for second-order half-linear functional differential equations with delayed argument

$$(a(r)(u^{\prime }(r){)}^m{)}^{\prime }=b(r)u^m(\phi (r))$$

where $m\in (0,1)$. We develop several key monotonicity results for Kneser solutions and use these properties to derive criteria for the elimination of bounded nonoscillatory solutions. Our approach extends known techniques from linear differential equations to the half-linear case, providing new insights into the qualitative behavior of solutions.

In this paper, we investigate the oscillatory behavior of second-order neutral multi-delay differential equations

$${\left(r\left(t\right){\left[{\left(u\left(t\right)+\sum _{i=1}^n{b}_i(t)u({\psi }_i(t))\right)}^{\prime }\right]}^{\alpha }\right)}^{\mathrm{\prime }}+Q\left(t,u\left(\tau \left(t\right)\right)\right)=0.$$

We introduce new monotonic properties of the non-oscillatory solutions of the equation, which are then used to linearize the equation and derive new oscillatory criteria. Our results are further supported by a numerical simulation example.