Abstract: In this paper,we consider the following systems x″+ Bx′+ gradG（x,t） = 0.Weak sufficient condition for the existence of a unique 2π-periodic solution of the systems is given and the results in [1]~ [3],[7]~ [8]are consequences of Theorem 2 in this paper if ‖B‖2<1. 

Key words:  , nonconservative system, periodic solution, homeomorphism