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

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Existence and uniqueness theorem for state-dependent delay-differential equations of neutral type is given. This theorem generalizes previous results by Grimm and the author.

A neutral impulsive system with a small delay of the argument of the derivative and another delay which differs from a constant by a periodic perturbation of a small amplitude is considered. If the corresponding system with constant delay has an isolated ω-periodic solution and the period of the delay is not rationally dependent on ω, then under a nondegeneracy assumption it is proved that in any sufficiently small neighbourhood of this orbit the perturbed system has a unique almost periodic solution....

The paper is the extension of the author's previous papers and solves more complicated problems. Almost periodic solutions of a certain type of almost periodic linear or quasilinear systems of neutral differential equations are dealt with.

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) $$\underset{t\to \infty}{lim}\frac{x\left(t\right)}{{t}^{k}}\phantom{\rule{4pt}{0ex}}\text{exists}\phantom{\rule{4.0pt}{0ex}}\text{and}\phantom{\rule{4.0pt}{0ex}}\text{is}\phantom{\rule{4.0pt}{0ex}}\text{a}\phantom{\rule{4.0pt}{0ex}}\text{positive}\phantom{\rule{4.0pt}{0ex}}\text{finite}\phantom{\rule{4.0pt}{0ex}}\text{value}\phantom{\rule{4.0pt}{0ex}}\text{if}\phantom{\rule{4.0pt}{0ex}}\text{and}\phantom{\rule{4.0pt}{0ex}}\text{only}\phantom{\rule{4.0pt}{0ex}}\text{if}{\int}_{{t}_{0}}^{\infty}{t}^{n-k-1}F(t,c{\left[g\left(t\right)\right]}^{k})\mathrm{d}t<\infty \phantom{\rule{4.0pt}{0ex}}\text{for}\phantom{\rule{4.0pt}{0ex}}\text{some}\phantom{\rule{4.0pt}{0ex}}c>0.$$ 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.

The main objective of the present paper is to study the approximate solutions for integrodifferential equations of the neutral type with given initial condition. A variant of a certain fundamental integral inequality with explicit estimate is used to establish the results. The discrete analogues of the main results are also given.

In the present paper we study the approximate solutions of a certain difference-differential equation under the given initial conditions. The well known Gronwall-Bellman integral inequality is used to establish the results. Applications to a Volterra type difference-integral equation are also given.