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 \(\lambda ^{\ast }\) such that the set of the functions \(u\), such that the system admits nontrivial solutions, contains an accumulation point.


37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
37J20 Bifurcation problems for finite-dimensional Hamiltonian and Lagrangian systems
47J30 Variational methods involving nonlinear operators
37C60 Nonautonomous smooth dynamical systems
34C25 Periodic solutions to ordinary differential equations
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[1] S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, vol. 251 of Grundlehren der Mathematischen Wissenschaften, Springer, New York, NY, USA, 1982. · Zbl 0487.47039
[2] B. Ricceri, “On the singular set of certain potential operators in Hilbert spaces,” in Differential Equations, Chaos and Variational Problems, V. Staicu, Ed., Progress in Nonlinear Differential Equations and Applications, pp. 377-391, Birkhäuser, Boston, Mass, USA, 2007. · Zbl 1223.47055
[3] V. \vDurikovi\vc and M. \vDurikovi, “On the solutions of nonlinear initial-boundary value problems,” Abstract and Applied Analysis, vol. 2004, no. 5, pp. 407-424, 2004. · Zbl 1063.35017
[4] A. K. Ben-Naoum, C. Troestler, and M. Willem, “Existence and multiplicity results for homogeneous second order differential equations,” Journal of Differential Equations, vol. 112, no. 1, pp. 239-249, 1994. · Zbl 0808.58013
[5] C.-L. Tang and X.-P. Wu, “Periodic solutions for a class of nonautonomous subquadratic second order Hamiltonian systems,” Journal of Mathematical Analysis and Applications, vol. 275, no. 2, pp. 870-882, 2002. · Zbl 1043.34045
[6] J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, vol. 74 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1989. · Zbl 0676.58017
[7] R. S. Sadyrkhanov, “On infinite-dimensional features of proper and closed mappings,” Proceedings of the American Mathematical Society, vol. 98, no. 4, pp. 643-648, 1986. · Zbl 0609.58006
[8] E. Zeidler, Nonlinear Functional Analysis and Its Applications. I, Springer, New York, NY, USA, 1986. · Zbl 0583.47050
[9] A. Ambrosetti and G. Prodi, A Primer of Nonlinear Analysis, vol. 34 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1995. · Zbl 0818.47059
[10] J. Dieudonné, “Sur les homomorphismes d/espaces normés,” Bulletin des Sciences Mathématiques, vol. 67, pp. 72-84, 1943. · Zbl 0028.23301
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