Existence of nonoscillatory and oscillatory solutions of neutral differential equations with positive and negative coefficients

summary:In this paper, we study the existence of oscillatory and nonoscillatory solutions of neutral differential equations of the form \(x(t)-cx(t-r)\)'\pm\(P(t)x(t-\theta)-Q(t)x(t-\delta)\)=0 where $c>0$, $r>0$, $\theta>\delta\geq0$ are constants, and $P$, $Q\in C(\bb R^+\!,\bb R^+)$. We obtain some sufficient and some necessary conditions for the existence of bounded and unbounded positive solutions, as well as some sufficient conditions for the existence of bounded and unbounded oscillatory solutions