Oscillation criteria for first order neutral differential equations with positive and negative coefficients

Sufficient conditions for the oscillation of the equation d/dt[x(t) - R(t)x (t - r)] + p(t)x(t - tau) - Q(t)x(t - delta) = 0, where P, Q, R E C([t(0),i nfinity), R+), r is an element of (0,infinity), tau, delta is an element of R+, tau greater than or equal to delta, are obtained where R(t) + integral (t)(t-tau+delta)Q(u) du - 1 is allowed to oscillate and the condition inte gral (infinity)(t0)[P(s)- Q(s - tau + delta)]integral (infinity)(S)[P(u)- Q (u - tau + delta)]duds = infinity is not necessary. Some examples are given , which show that the results here are almost sharp. (C) 2001 Academic Pres s.