Graduate School of Science and Engineering, Doctor Course, Ehime University, Matsuyama 790-8577, Japan; Present address of the author: Department of Liberal Arts and Engineering Science, Hachinohe National College of Technology, Hachinohe 039-1192, Japan, e-mail: tanaka-g@hachinohe-ct.ac.jp
Abstract: The higher order neutral functional differential equation
\frac{\dd^n}{\dd t^n} \bigl[x(t) + h(t) x(\tau(t))\bigr] + \sigma f\bigl(t,x(g(t))\bigr) = 0 \tag1
is considered under the following conditions: $n\ge2$, $\sigma=\pm1$, $\tau(t)$ is strictly increasing in $t\in[t_0,\infty)$, $\tau(t)<t$ for $t\ge t_0$, $\lim_{t\to\infty} \tau(t)= \infty$, $\lim_{t\to\infty} g(t) = \infty$, 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).
Keywords: neutral differential equation, positive solution
Classification (MSC 2000): 34K11
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