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 −τ) + m i=1 Qi (t)fi x(t −σi ) = 0, t t0, where m 1 is an integer, τ > 0, σi 0, r,P,Qi ∈ C([t0,∞),R), fi ∈ 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 Qi (t) (i = 1, 2, . . . , m) which means that we allow oscillatory P(t) and Qi (t) (i = 1, 2, . . . , m). In particular, our results improve essentially and extend some known results in the recent references. © 2006 Elsevier Inc. All rights reserved