The existence of nontrivial periodic solutions of the scalar equation x^^・(t) = - f(x(t))(g(x(t)) + h(x(t - 1))) is mainly discussed by using a fixed point theorem for a closed convex set. As an application of the main results, we show that a conjecture by G. Seifert is right. Moreover we give a negative answer to a question by G. Seifert.