The aim of this paper is to deduce oscillatory and asymptotic behavior of the solutions of the $n-$th order neutral differential equation $${(x\left(t\right)-px(t-\tau ))}^{\left(n\right)}-q\left(t\right)x\left(\sigma \left(t\right)\right)=0,$$ where $\sigma \left(t\right)$ is a delayed or advanced argument.

The a priori boundedness principle is proved for the Dirichlet boundary value problems for strongly singular higher-order nonlinear functional-differential equations. Several sufficient conditions of solvability of the Dirichlet problem under consideration are derived from the a priori boundedness principle. The proof of the a priori boundedness principle is based on the Agarwal-Kiguradze type theorems, which guarantee the existence of the Fredholm property for strongly singular higher-order linear...