Note on homoclinic solutions to nonautonomous Hamiltonian systems with sign-changing nonlinear part

In the paper, we utilize the recent variational, abstract theorem to show the existence of homoclinic solutions to the Hamiltonian system $$ \dot{z} = J D_z H(z, t), \quad t \in \mathbb{R}, $$ where the Hamiltonian $H : \mathbb{R}^{2N} \times \mathbb{R} \rightarrow \mathbb{R}$ is of the form $$ H(z, t) = \frac12 Az \cdot z + \Gamma(t) \left( F(z) - \lambda G(z) \right) $$ for some symmetric matrix $A$.