We study the problem of the existence of periodic solutions for ordinary differential equations of the form \(x^{(m)}=g(t,x,...,x^{(m)})\), where g is a nonlinear vectorial function and m is arbitrary. For it, we transform our problem in the abstract form \(Lx=Nx\), where L and N are operators, linear and nonlinear respectively, defined from a normed space X into a normed space Z and then, to study this equation by using the coincidence degree theory developed by G. Hetzer [Ann. Soc. Sci. Bruxelles, Sér. I 89, 497-508 (1975; Zbl 0316.47041)]. Our results include the case where g is quasibounded or it is of exponential type.