In this paper we prove two results. The first is an extension of the result of G. D. Jones [4]: (A) Every nontrivial solution for $$\left\}\begin{array}{c}{(-1)}^n{u}^{\left(2n\right)}+f(t,u)=0,\text{in}(\alpha ,\infty ),u^{\left(i\right)}\left(\xi \right)=0,i=0,1,\cdots ,n-1,\text{and}\xi \in (\alpha ,\infty ),\hfill \end{array}\right.$$ must be unbounded, provided $f(t,z)z\ge 0$, in $E\times ℝ$ and for every bounded subset $I$, $f(t,z)$ is bounded in $E\times I$. (B) Every bounded solution for ${(-1)}^n{u}^{\left(2n\right)}+f(t,u)=0$, in $ℝ$, must be constant, provided $f(t,z)z\ge 0$ in $ℝ\times ℝ$ and for every bounded subset $I$, $f(t,z)$ is bounded in $ℝ\times I$.

The neutral differential equation (1.1) $$\frac{{\mathrm{d}}^n}{\mathrm{d}{t}^n}[x\left(t\right)+x(t-\tau )]+\sigma F(t,x\left(g\left(t\right)\right))=0,$$ is considered under the following conditions: $n\ge 2$, $\tau >0$, $\sigma =\pm 1$, $F(t,u)$ is nonnegative on $[t_0,\infty )\times (0,\infty )$ and is nondecreasing in $u\in (0,\infty )$, and $limg\left(t\right)=\infty$ as $t\to \infty$. It is shown that equation (1.1) has a solution $x\left(t\right)$ such that (1.2) $$\underset{t\to \infty }{lim}\frac{x\left(t\right)}{t^k}\text{exists}\text{and}\text{is}\text{a}\text{positive}\text{finite}\text{value}\text{if}\text{and}\text{only}\text{if}{\int }_{t_0}^{\infty }{t}^{n-k-1}F(t,c{\left[g\left(t\right)\right]}^k)\mathrm{d}t<\infty \text{for}\text{some}c>0.$$ Here, $k$ is an integer with $0\le k\le n-1$. To prove the existence of a solution $x\left(t\right)$ satisfying (1.2), the Schauder-Tychonoff fixed point theorem is used.