Minimal period estimates on \(P\)-symmetric periodic solutions of first-order mild superquadratic Hamiltonian systems
A \(P\)-boundary problem of the autonomous Hamiltonian system \[ -J\dot{z}(t)=\nabla H(z(t)), \qquad z(\tau)=Pz(t), \] is considered. Here \(P\) is a \(2n\times 2n\) symplectic matrix, \(H\in C^2(\mathbb R^{2n},\mathbb R)\) with \(H(Pz)=H(z)\), and \(J\) is the standard symplectic matrix. It is assumed that \(P^k=I_{2n}\) and \(P^m\neq I_{2n}\) for smaller \(m\). A solution of the above system can be extended to a \(k\tau\)-periodic solution. A \(T\)-periodic solution is called \(P\)-symmetric if \(z(T/k)=Pz(0)\). Under assumptions similar to those in [\textit{A. Abbondandolo}, Morse theory for Hamiltonian systems. Boca Raton, FL: Chapman \& Hall/CRC (2001; Zbl 0967.37002)] the authors obtain estimates on Maslov \(P\)-index and minimal period estimates. The conditions are a bit different from the well known Ambrosetti-Rabinowitz conditions.
scientific article; zbMATH DE number 7371550
scientific article