  In this paper, we deal with a problem of the type
$$\cases{(\phi(u'))'=\nabla_xF(t,u) & in $[0,T]$\cr & \cr u(0)=u(T)\ , \hskip
3pt u'(0)=u'(T)\ ,\cr}$$ where, in particular, $\phi$ is a homeomorphism from
an open ball of ${\bf R}^n$ onto ${\bf R}^n$. Using the theory developed by
Brezis and Mawhin in [1] jointly with our minimax theorem proved in [3], we
obtain a general multiplicity result, under assumptions of qualitative nature
only. Three remarkable corollaries are also presented.
