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Bifurcation for second-order Hamiltonian systems with periodic boundary conditions. (English) Zbl 1144.37025
Summary: Through variational methods, we study nonautonomous systems of second-order ordinary differential equations with periodic boundary conditions. First, we deal with a nonlinear system, depending on a function $u$, and prove that the set of bifurcation points for the solutions of the system is not $\sigma$-compact. Then, we deal with a linear system depending on a real parameter $\lambda >0$ and on a function $u$, and prove that there exists ${\lambda }^{*}$ such that the set of the functions $u$, such that the system admits nontrivial solutions, contains an accumulation point.
##### MSC:
 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods 37J20 Bifurcation problems (finite-dimensional Hamiltonian etc. systems) 47J30 Variational methods (nonlinear operator equations) 37C60 Nonautonomous smooth dynamical systems 34C25 Periodic solutions of ODE