Existence for nonoscillatory solutions of second-order nonlinear differential equations

In this paper, the existence of nonoscillatory solutions of the second-order nonlinear neutral differential equation [r(t) (x(t) + P(t)x(t - tau))']' + Sigma(m)(1=l) Q(i)(t) f(i)(x(t - sigma(i))) = 0, t >= t(0), where m >= 1 is an integer, tau > 0, sigma(i) >= 0, r, P, Qi is an element of C ([t(0), infinity), R), f(i) E C(R, R) (i = 1, 2,..., m), are studied. Some new sufficient conditions for the existence of a nonoscillatory solution of above equation are obtained for general P(t) and Q(i)(t) (i = 1, 2,..., in) which means that we allow oscillatory P(t) pd Q(i) (t) (i = 1, 2,..., in). In particular, our results improve essentially and extend some known results in the recent references. (c) 2006 Elsevier Inc. All rights reserved.