Quasi-linear functional differential equations with Property A

We study oscillatory properties of solutions of a functional differential equation of the form u((n)) (t) + F (u) (t) = 0, (0.1) where n >= 2 and F: C(R+; R) -> L-loc(R+; R) is a continuous mapping. Sufficient conditions are established for this equation to have the so-called Property A. The obtained results are also new for the generalized Emden-Fowler type ordinary differential equation. The method by which the oscillatory properties of Eq. (0.1) are established enables one to obtain optimal conditions for (0.1) to have Property A for sufficiently general equations (for some classes of functions the obtained sufficient conditions are necessary as well). (c) 2006 Elsevier Inc. All rights reserved.