  A topological degree based averaging principle has been proposed by J. Mawhin
in his PhD thesis [J. Mawhin, Le Probleme des Solutions Periodiques en
Mecanique non Lineaire, These de doctorat en sciences, Universite de Liege,
1969]. In the present paper we extend this result for the case when the
unperturbed system is nonlinear but possesses a set of T-periodic solutions
whose initial conditions constitute a boundary of some open set.
