Existence of periodic solutions for second order Hamiltonian system (Q432541)

The author considers the following Josephson-type system with unbounded nonlinearities \[ \begin{aligned} \ddot{u}(t)+Au(t)-\nabla F(t,u(t))=h(t),&\quad\text{a.e. }t\in [0,T],\\ u(0)-u(T)=\dot{u}(0)-\dot{u}(T)=0,& \end{aligned}\tag{1} \] where \(A\) is a symmetric \((N\times N)\)-matrix, \(h\in L^{1}(0,T;\mathbb R^{N})\), \(T>0\), and \(F\in C^{1}([0,T]\times\mathbb R^{N};\mathbb R)\). Under a generalized sublinear growth condition and using the least action principle and the saddle-point theorem, he proves that the system (1) has at least one solution.