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).

We investigate an asymptotic behaviour of damped non-oscillatory solutions of the initial value problem with a time singularity ${\left(p\left(t\right)u^{\text{'}}\left(t\right)\right)}^{\text{'}}+p\left(t\right)f\left(u\left(t\right)\right)=0$, $u\left(0\right)=u_0$, $u^{\text{'}}\left(0\right)=0$ on the unbounded domain $[0,\infty )$. Function $f$ is locally Lipschitz continuous on $ℝ$ and has at least three zeros $L_0<0$, $0$ and $L>0$. The initial value $u_0\in (L_0,L)\setminus \left\{0\right\}$. Function $p$ is continuous on $[0,\infty ),$ has a positive continuous derivative on $(0,\infty )$ and $p\left(0\right)=0$. Asymptotic formulas for damped non-oscillatory solutions and their first derivatives are derived under some additional assumptions. Further,...

The paper investigates the singular initial problem[4pt] ${\left(p\left(t\right)u^{\text{'}}\left(t\right)\right)}^{\text{'}}+q\left(t\right)f\left(u\left(t\right)\right)=0,u\left(0\right)=u_0,u^{\text{'}}\left(0\right)=0$[4pt] on the half-line $[0,\infty )$. Here $u_0\in [L_0,L]$, where $L_0$, $0$ and $L$ are zeros of $f$, which is locally Lipschitz continuous on $ℝ$. Function $p$ is continuous on $[0,\infty )$, has a positive continuous derivative on $(0,\infty )$ and $p\left(0\right)=0$. Function $q$ is continuous on $[0,\infty )$ and positive on $(0,\infty )$. For specific values $u_0$ we prove the existence and uniqueness of damped solutions of this problem. With additional conditions for $f$, $p$ and $q$ it is shown that the problem has for each specified...