duffing equation and action functional

in this paper, we investigate the periodic nonlinear second order ordinary differential equation with friction

x′′(t) + cx′(t) + g(t, x(t)) = f (t), t ∈ [0, t ],

x(0) = x(t ), x′(0) = x′(t ),

where c ∈ r, g is a caratheodory function, f ∈ l1(0, t ), a quotient g(t,s) s lies between 0 and c²/4+( 2π/t )² and a nonlinearity g satisfies the potential landesman–lazer type condition.we introduce a corresponding energy functional and prove that its critical point is a classical solution to this problem.