## 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.

### MSC:

 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
Full Text:

### References:

 [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 · doi:10.1155/S1085337504306081 [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 · doi:10.1006/jdeq.1994.1103 [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 · doi:10.1016/S0022-247X(02)00442-0 [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 · doi:10.2307/2045743 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.