Non-oscillatory behaviour of higher order functional differential equations of neutral type

Abstract

In this paper, we obtain sufficient conditions so that the neutral functional differential equation $$displaylines{ ig[r(t) [y(t)-p(t)y(au (t))]'ig]^{(n-1)} + q(t) G(y(h(t))) = f(t) }$$ has a bounded and positive solution. Here $ngeq 2$; $q,au, h$ are continuous functions with $q(t) geq 0$; $h(t)$ and $au(t)$ are increasing functions which are less than $t$, and approach infinity as $t o infty$. In our work, $r(t) equiv 1$ is admissible, and neither we assume that $G$ is non-decreasing, that $xG(x) > 0$ for $x eq 0$, nor that $G$ is Lipschitzian. Hence the results of this paper generalize many results in [1] and [4]-[8]