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) $$ \frac{{\mathrm{d}}^n}{{\mathrm{d}} t^n} [x(t)+x(t-\tau)] + \sigma F(t,x(g(t))) = 0, $$ 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 $\lim g(t) = \infty$ as $t\rightarrow \infty$. It is shown that equation (1.1) has a solution $x(t)$ such that (1.2) $$ \lim_{t\rightarrow \infty} \frac{x(t)}{t^k}\ \text{exists and is a positive finite value if and only if} \int^{\infty}_{t_0} t^{n-k-1} F(t,c[g(t)]^k){\mathrm{d}} t < \infty\text{ for some }c > 0. $$ Here, $k$ is an integer with $0\le k \le n-1$. To prove the existence of a solution $x(t)$ satisfying (1.2), the Schauder-Tychonoff fixed point theorem is used.
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).
We obtain sufficient conditions for every solution of the differential equation [y(t) − p(t)y(r(t))](n) + v(t)G(y(g(t))) − u(t)H(y(h(t))) = f(t) to oscillate or to tend to zero as t approaches infinity. In particular, we extend the results of Karpuz, Rath and Padhy (2008) to the case when G has sub-linear growth at infinity. Our results also apply to the neutral equation [y(t) − p(t)y(r(t))](n) + q(t)G(y(g(t))) = f(t) when q(t) has sign changes. Both bounded and unbounded solutions are consideted here; thus some known results are expanded.