For the differential equation where g is - periodic in t, L. Amaral and M. P. Pera [Boll. Unione Mat. Ital., V. Ser., C, Anal. Funz. Appl. 18, 107-117 (1981; Zbl 0472.34028)] showed that there is a -periodic solution if there are constants and , with , such that for large x,
In the paper under review it is shown that if (sgn x) g(t,x) is bounded below, then the integral condition just stated can be replaced by
This theorem and the theorem of Amaral and Pera are derived using the variational methods introduced by P. Rabinowitz [Nonlinear analysis, Collect. Pap. Honor H. Rothe, 161-177 (1978; Zbl 0466.58015)].