The higher order neutral functional differential equation \[ \frac{\mathrm{d}^n}{\mathrm{d}t^n} \bigl [x(t) + h(t) x(\tau (t))\bigr ] + \sigma f\bigl (t,x(g(t))\bigr ) = 0 \qquad \mathrm{(1)}\] is considered under the following conditions: $n\ge 2$, $\sigma =\pm 1$, $\tau (t)$ is strictly increasing in $t\in [t_0,\infty )$, $\tau (t)<t$ for $t\ge t_0$, $\lim _{t\rightarrow \infty } \tau (t)= \infty $, $\lim _{t\rightarrow \infty } g(t) = \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).
The two-point boundary value problem u ′′ + h(x)u p = 0, a < x < b, u(a) = u(b) = 0 is considered, where p > 1, h ∈ C 1 [0, 1] and h(x) > 0 for a ≤ x ≤ b. The existence of positive solutions is well-known. Several sufficient conditions have been obtained for the uniqueness of positive solutions. On the other hand, a non-uniqueness example was given by Moore and Nehari in 1959. In this paper, new uniqueness results are presented.