Quasi-linear functional differential equations with Property A

We study oscillatory properties of solutions of a functional differential equation of the formequation(0.1)u(n)(t)+F(u)(t)=0,u(n)(t)+F(u)(t)=0, where n⩾2n⩾2 and F:C(R+;R)→Lloc(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).