where e is a small real parameter, o1, * , oare real positive numbers, and each fj is a real valued function periodic in the real variable t of period 27r/w, w>0. In a previous paper R. A. Gambill and J. K. Hale [6] have given sufficient conditions for the existence of periodic solutions of (1) (and more general systems), whose dominant terms have periods in a rational ratio with 27r/w (harmonics, subharmonics, ultra-subharmonics). Also, a number of examples and applications were given in [6]. The aim of the present paper is to prove a new general statement which contains as particular cases two of the various theorems proved in [6]. We shall use exactly the same method used in [6]. This method has been successively developed by L. Cesari, J. K. Hale and R. A. Gambill, in a series of papers concerning boundedness of solutions of linear differential systems with periodic coefficients [1; 4; 5; 8], cycles of autonomous weakly nonlinear differential systems [9], and harmonics and subharmonics of periodic weakly nonlinear differential systems [6]. The method will be reviewed below so as to make the present paper independent. For bibliographical indications on the vast subject we refer to the papers quoted in the bibliography.