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In this paper, we deal with a problem of the type$$\cases{(\phi(u'))'=\nabla_xF(t,u) & in $[0,T]$\cr & \cr u(0)=u(T)\ , \hskip3pt u'(0)=u'(T)\ ,\cr}$$ where, in particular, $\phi$ is a homeomorphism froman open ball of ${\bf R}^n$ onto ${\bf R}^n$. Using the theory developed byBrezis and Mawhin in [1] jointly with our minimax theorem proved in [3], weobtain a general multiplicity result, under assumptions of qualitative natureonly. Three remarkable corollaries are also presented.

Another multiplicity result for the periodic solutions of certain systems