Let g:R→R be a continuous function with periodic primitive G

e∈ C[0

2π]. Consider thenonlinear Duffing equation x″(t)+m~2x(t)+g(x(t))=e(t) subjected to the 2π-periodic boundarycondition x(0)--x(2π)=x′(0)--x′(2π)=0. In this paper

by means of min-max theory we givea results concerning the existence of solutions. Our functional does not satisfy the [P.S.] condition.In order to overcome this problem we establish a relationship between two different classes of sets useda min-max characterization of possible critial points.