Oscillations of solutions of neutral differential equations with positive and negative coefficients

We obtain some new results for oscillation of all solutions of tile neutral differential equation with positive and negative coefficients [(d)/(dt)[y( t) - R(t)y(t - r)] + P(t)y(t - tau) - Q(t)y(t - sigma) = 0, where P,Q,R is an element of C([t(0),infinity),R), r is an element of (0,infinity), and ta u,sigma is an element of [0,infinity). These new results are obtained by es tablishing and using some new lemmas which are interesting in their own rig ht and which may have further applications in analysis. (C) 2001 Elsevier S cience Ltd. All rights reserved.