Nontrivial periodic solutions of asymptotically linear Hamiltonian systems (Q2761576)

This paper is devoted to study the Hamiltonian system NEWLINE\[NEWLINE\dot z=J H'(t,z), \tag{1}NEWLINE\]NEWLINE where \(H\in C^2([0,1] \times\mathbb{R}^{2N}, \mathbb{R})\) is a 1-periodic function in \(t\), and \(H'(t,z)\) denotes the gradient of \(H(t,z)\) with respect to the \(z\) variable. Here \(N\) is a positive integer and \(J=\left( \begin{smallmatrix} 0 & -I_N\\ I_N & 0\end{smallmatrix} \right)\) is the standard \(2N\times 2N\) symplectic matrix. Under some suitable assumptions on \(H\) and using the saddle point theorem, and Conley index theory, the author presents new results on the existence of periodic solutions for the asymptotically linear Hamiltonian system (1).