Second order neutral multi-delay differential equations and its oscillatory criteria derived by linearization

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.