OSCILLATIONS OF HIGHER-ORDER NEUTRAL DIFFERENTIAL-EQUATIONS

Consider the nth-order neutral differential equation(~)[GRAPHICS](~)where n greater-than-or-equal-to 1, delta = +/- 1, J, K are initial segments of natural numbers, p(i), tau(i), sigma(k) is-an-element-of R and q(k) greater-than-or-equal-to 0 for i is-an-element-of J and k is-an-element-of K. Then a necessary and sufficient condition for the oscillation of all solutions of (E) is that its characteristic equation [GRAPHICS] has no real roots. The method of proof has the advantage that it results in easily verifiable sufficient conditions (in terms of the coefficients and the arguments only) for the oscillation of all solutions of Equation (E).