## Abstract and Applied Analysis

### Bifurcation for Second-Order Hamiltonian Systems with Periodic Boundary Conditions

#### Abstract

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.

#### Article information

Source
Abstr. Appl. Anal., Volume 2008 (2008), Article ID 756934, 13 pages.

Dates
First available in Project Euclid: 9 September 2008

https://projecteuclid.org/euclid.aaa/1220969147

Digital Object Identifier
doi:10.1155/2008/756934

Mathematical Reviews number (MathSciNet)
MR2393112

Zentralblatt MATH identifier
1144.37025

#### Citation

Faraci, Francesca; Iannizzotto, Antonio. Bifurcation for Second-Order Hamiltonian Systems with Periodic Boundary Conditions. Abstr. Appl. Anal. 2008 (2008), Article ID 756934, 13 pages. doi:10.1155/2008/756934. https://projecteuclid.org/euclid.aaa/1220969147

#### References

• S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, vol. 251 of Grundlehren der Mathematischen Wissenschaften, Springer, New York, NY, USA, 1982.
• 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.
• V. Ďurikovičand M. Ďurikovičová, “On the solutions of nonlinear initial-boundary value problems,” Abstract and Applied Analysis, vol. 2004, no. 5, pp. 407–424, 2004.
• 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.
• 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.
• J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, vol. 74 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1989.
• 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.
• E. Zeidler, Nonlinear Functional Analysis and Its Applications. I, Springer, New York, NY, USA, 1986.
• A. Ambrosetti and G. Prodi, A Primer of Nonlinear Analysis, vol. 34 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1995.
• J. Dieudonné, “Sur les homomorphismes d'espaces normés,” Bulletin des Sciences Mathématiques, vol. 67, pp. 72–84, 1943.