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∈Rc∈R, gg is a Caratheodory function, f∈L1(0,T)f∈L1(0,T), a quotient g(t,s)s lies between 00 and c24+(2πT)2 and a nonlinearity gg 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.