Oscillations of higher order neutral differential equations of mixed type

In this paper, we establish some new oscillation theorems for neutral highe r order functional differential equations of the form (E) d(n)/dt(n) (x(t) + cx(t - h) + Cx(t + H)) + qx(t - g) + Qx(t + G) = 0, where c, C, g, G, h and H are real constants, and q and Q are nonnegative r eal constants. The results of this paper improve noticeably the known oscil lation theorems. By a new analysis technique we give weaker sufficient cond itions for all solutions of equation (E) to be oscillatory.