Necessary and sufficient conditions for oscillation of first order nonlinear neutral differential equations

In this paper, we prove that every solution of the first order nonlinear neutral differential equation [x(t) - px(t - tau)]' + q(t) (j=1)Pi(m) vertical bar x(t - sigma(j))vertical bar(beta j) sign[x(t - sigma(1))] = 0. t >= t(0), oscillates if and only if (t0)integral(infinity) q(s) exp[tau(-1) ln p((j=1)Sigma(m) beta(j) - 1)s]ds = infinity, when (Sigma(m)(j=1) beta(j) - 1) ln p < 0, and (t0)integral(infinity) q(s)ds = infinity, when (Sigma(m)(j=1) beta(j) - 1) ln p > 0, where p, tau > 0, beta(j) > 0, sigma(j) >= 0, j = 1, 2, . . ., m, q is an element of C ([t(0), infinity), [0, infinity)). (c) 2005 Published by Elsevier Inc.