Necessary and Sufficient Condition for the Solutions of a Functional Differential Equation to Be Oscillatory or Tend to Zero

It is shown that every solution of the nonhomogeneous functional differential equation d x t.ypx tyt ..qQ t.G x tys ..sf t., dt where f, QgC wT,`., 0,`..,s,tg 0, `., 0Fp-1,G: RªR such that xG x. )0 for x/0, G is nondecreasing, Lipschitzian, and satisfy a sublinear condition "k dx ` H -` and Hf s. ds-`, 0 G x. is either oscillatory or tends to zero asymptotically if and only if ` Q s dss`.