A 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/(dt)(x(t) -px(t - tau)) + Q(t)G(x(t - sigma)) = f(t), where f,Q is an element of C([T,infinity), (0, infinity)), sigma, tau is an element of(0, infinity), 0 less than or equal to p < 1,G: R --> R such that xG(x) > 0 for x not equal 0, G is nondecreasing, Lipschitzian, and satisfy a sublinear condition integral(0)(+/-k) dx/G(x) < infinity and integral(infinity) f(s) ds < infinity, is either oscillatory or tends to zero asymptotically if and only if integral(T)(infinity) Q(s) ds = infinity. (C) 1997 Academic Press