Nonlinear functional differential equations with Properties A and B

In this paper, an nth order functional differential equation is considered for which the generalized Emden-Fowler-type equation (0.1)u(n)(t)+p(t)|u(t)|μ(t)signu(t)=0,t⩾0, can be considered as a nonlinear model. Here, we assume that n⩾2, p∈Lloc(R+;R), and μ∈C(R+;(0,1]) is a nondecreasing function. In case μ(t)≡const>0, oscillatory properties of Eq. (0.1) have been extensively studied, where as if μ(t)≢const, to the extent of authors' knowledge, the analogous questions have not been examined. It turns out that the oscillatory properties of Eq. (0.1) substantially depend on the rate at which the function μ+−μ(t) tends to zero as t→+∞, where μ+=limt→+∞μ(t). In this paper, new sufficient conditions for a general class of nonlinear functional differential equations to have Properties A and B are established, and these results apply to the special case of Eq. (0.1) as well.