Existence of periodic solutions of linear Hamiltonian systems with sublinear perturbation
The author considers a Hamiltonian system \(\dot u = JA(t)u+J\nabla G(u,t)\), where \(J\) is the standard symplectic matrix, \(A\) is a symmetric \(2N\times 2N\)-matrix with \(2\pi\)-periodic entries, \(G\) is \(2\pi\)-periodic in \(t\) and \(\nabla G\) has sublinear growth as \(|u|\to\infty\). The main result asserts that if \(G\) satisfies a kind of Ahmad-Lazer-Paul condition on the kernel of the linear part, then this system has a \(2\pi\)-periodic solution. Two more related results are also given. The arguments are variational and use a saddle point theorem of Benci and Rabinowitz.
scientific article; zbMATH DE number 5725777