Oscillation and asymptotic behaviour of a higher order neutral differential equation with positive and negative coefficients

In this paper, we obtain necessary and sufficient conditions so that every solution of $$ ig(y(t)- p(t) y(r(t))ig)^{(n)}+ q(t)G( y(g(t)))-u(t)H(y(h(t))) = f(t) $$ oscillates or tends to zero as $t o infty$, where $n$ is an integer $n geq 2$, $q>0$, $ugeq 0$. Both bounded and unbounded solutions are considered in this paper. The results hold also when $uequiv 0$, $f(t)equiv 0$, and $G(u)equiv u$. This paper extends and generalizes some recent results.