### A necessary and sufficient condition for the oscillation in a class of even order neutral differential equations.

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Neutral differential equations are studied. Sufficient conditions are obtained to have oscillatory solutions or nonoscillatory solutions. For the existence of solutions, the Schauder-Tikhonov fixed point theorem is used.

The higher order neutral functional differential equation $$\frac{{\mathrm{d}}^{n}}{\mathrm{d}{t}^{n}}\left[x\left(t\right)+h\left(t\right)x\left(\tau \right(t\left)\right)\right]+\sigma f\left(t,x\left(g\right(t\left)\right)\right)=0\phantom{\rule{2.0em}{0ex}}\left(1\right)$$ 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).

The two-point boundary value problem $${u}^{\text{'}\text{'}}+h\left(x\right){u}^{p}=0,\phantom{\rule{1.0em}{0ex}}a<x<b,\phantom{\rule{2.0em}{0ex}}u\left(a\right)=u\left(b\right)=0$$ is considered, where $p>1$, $h\in {C}^{1}[0,1]$ and $h\left(x\right)>0$ for $a\le x\le b$. 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.

Certain hyperbolic equations with continuous distributed deviating arguments are studied, and sufficient conditions are obtained for every solution of some boundary value problems to be oscillatory in a cylindrical domain. Our approach is to reduce the multi-dimensional oscillation problems to one-dimensional oscillation problems for functional differential inequalities by using some integral means of solutions.

The second order linear differential equation $${\left(p\left(x\right){y}^{\text{'}}\right)}^{\text{'}}+q\left(x\right)y=0\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{1.0em}{0ex}}x\in (0,{x}_{0}]$$ is considered, where $p$, $q\in {C}^{1}(0,{x}_{0}]$, $p\left(x\right)>0$, $q\left(x\right)>0$ for $x\in (0,{x}_{0}]$. Sufficient conditions are established for every nontrivial solutions to be nonrectifiable oscillatory near $x=0$ without the Hartman–Wintner condition.

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