On an even order neutral differential inequality (Q1885412)

The even-order neutral differential inequality \[ [x(t)+c(t)x(t-\tau)]^{(n)}+\int_{a}^{b}p(t,\xi)f(x[g(t,\xi)])d\sigma(\xi)\leq 0, t\geq t_0, \] is considered. The solution \(x(.)\in C^{(n)}([t_0,\infty),\mathbb{R})\) of this inequality is said to be eventually positive if there exists a positive number \(T\geq t_0\) such that \(x(t)\geq 0\) for \( t\geq T.\) Nonexistence criteria of eventually positive solution are established.