Series solutions of some anharmonic motion equations

Consider the class of nonlinear oscillators of the form    d2u dt2 + f (u) = g(t ), u(0) = a0, u (0) = 0, (1) where g(t) is a 2T -periodic function, f is a function only dependent on u and is a small parameter.We are interested in the periodic solutions with minimal period 2T , when the restoring term f and g satisfy suitable conditions. By using methods of trigonometric series for solving differential equations we prove the existence of a periodic solution of this perturbed Duffing equation. These results develop and extend a previous ones (Appl. Math. Lett. 14 (8) (2001) 363–368).  2002 Elsevier Science (USA). All rights reserved.