Abstract
In this paper, we study the existence of periodic or nonnegative periodic solutions of the nonlinear neutral differential equation
d/dt [x(t) - Q(t, x(t - tau(t)))] = -a(t)h(x(t - tau(t))) + G(t, x(t), x(t - tau(t))).
We invert this equation to construct a sum of a compact map and a large contraction which is suitable for applying the modification of Krasnoselskii's theorem. The Caratheodory condition is used for the functions Q and G.