Existence of Positive Solutions for a Class of Higher Order Neutral Functional Differential Equations

Abstract

The higher order neutral functional differential equation

$$(1) \frac {d^n}{dt^n} [x(t)+ h(t)x ( \tau(t))]+ \sigma f (t,x (g(t)))=0$$

is considered under the following conditions: $n \geqslant 2, \sigma = \pm 1, \tau (t)$ is strictly increasing in $t \in \left[ {t_0 ,\infty } \right), \tau (t) < t {\text{for}} t \geqslant t_0 ,\mathop { \lim }\limits_{t \to \infty } \tau (t) = \infty ,\mathop { \lim }\limits_{t \to \infty } g(t) = \infty , {\text{and}} f(t,u)$ is nonnegative on $\left[ {t_0 ,\infty } \right) \times \left( {0,\infty } \right)$ and nondecreasing in $u \in (0,\infty )$. A necessary and sufficient condition is derived for the existence of certain positive solutions of (1).