Integral criteria are established for $dim{V}_i\left(p\right)=0$ and $dim{V}_i\left(p\right)=1,i\in \{0,1\}$, where $V_i\left(p\right)$ is the space of solutions $u$ of the equation $$u^{\text{'}\text{'}}+p\left(t\right)u=0$$ satisfying the condition $${\int }^{+\infty }\frac{u^2\left(s\right)}{s^i}ds<+\infty .$$

The higher order neutral functional differential equation $$\frac{{\mathrm{d}}^n}{\mathrm{d}{t}^n}\left[x\left(t\right)+h\left(t\right)x\left(\tau \right(t\left)\right)\right]+\sigma f\left(t,x\left(g\right(t\left)\right)\right)=0\left(1\right)$$ is considered under the following conditions: $n\ge 2$, $\sigma =\pm 1$, $\tau \left(t\right)$ is strictly increasing in $t\in [t_0,\infty )$, $\tau \left(t\right)<t$ for $t\ge t_0$, ${lim}_{t\to \infty }\tau \left(t\right)=\infty$, ${lim}_{t\to \infty }g\left(t\right)=\infty$, and $f(t,u)$ is nonnegative on $[t_0,\infty )\times (0,\infty )$ and nondecreasing in $u\in (0,\infty )$. A necessary and sufficient condition is derived for the existence of certain positive solutions of (1).