Multiple Periodic Solutions for Some Classes of First-Order Hamiltonian Systems
Mohsen Timoumi
DOI: 10.4236/am.2011.27114   PDF    HTML     4,711 Downloads   8,759 Views   Citations


Considering a decomposition R2N=A⊕B of R2N , we prove in this work, the existence of at least (1+dimA) geometrically distinct periodic solutions for the first-order Hamiltonian system Jx'(t)+H'(t,x(t))+e(t)=0 when the Hamiltonian H(t,u+v) is periodic in (t,u) and its growth at infinity in v is at most like or faster than |v|a, 0≤a<1 , and e is a forcing term. For the proof, we use the Least Action Principle and a Generalized Saddle Point Theorem.

Share and Cite:

Timoumi, M. (2011) Multiple Periodic Solutions for Some Classes of First-Order Hamiltonian Systems. Applied Mathematics, 2, 846-853. doi: 10.4236/am.2011.27114.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] T. An, “Periodic Solutions of Superlinear Autonomous Hamiltonian Systems with Prescribed Period,” Journal of Mathematical Analysis Application, Vol. 323, No. 15, 2006, pp. 854-863. doi:10.1016/j.jmaa.2005.11.004
[2] P. L. Felmer, “Periodic Solutions of Superquadratic Hamiltonian Systems,” Journal of Differential Equation, Vol. 102, No. 1, 1993, pp. 188-207. doi:10.1006/jdeq.1993.1027
[3] Z. Q. Ou and C. L. Tang, “Periodic and Subharmonic Solutions for a Class of Superquadratic Hamiltonian Systems,” Nonlinear Analysis, Vol. 58, No. 3-4, 2004, pp. 245-258. doi:10.1016/
[4] C. L. Tang and X. P. Wu, “Periodic Solutions for Second Order Systems with not Uniformly Coercive Potential,” Journal of Mathematical Analysis Application, Vol. 259, No. 2, 2001, pp. 386-397. doi:10.1006/jmaa.2000.7401
[5] M. Timoumi, “Periodic Solutions for Noncoercive Hamiltonian Systems,” Demonstratio Mathematica, Vol. 35, No. 4, 2002, pp. 899-913.
[6] K. C. Chang, “On the Periodic Nonlinearity and the Multiplicity of Solutions,” Nonlinear Analysis, Vol. 13, No. 5 1989, pp. 527-537. doi:10.1016/0362-546X(89)90062-X
[7] S. X. Chen, X. Wu and F. Zhao, “New Existence and Multiplicity Theorems of Periodic Solutions for Non-Autonomous Second Order Hamiltonian Systems,” Mathematical and Computer Modeling, Vol. 46, No. 3-4, 2007, pp. 550-556. doi:10.1016/j.mcm.2006.11.019
[8] I. Ekeland and J. M. Lasry, “On the Number of Periodic Trajectories for a Hamiltonian Flow on a Convex Energy Surface,” Annals of Mathematics, Vol. 112, No. 2, 1980, pp. 283-319. doi:10.2307/1971148
[9] M. Timoumi, “On the Multiplicity of Periodic Solutions of a Hamiltonian System,” Demonstratio Mathematica, Vol. 35, No. 4, 2002, pp. 899-913.
[10] G. Fournier, D. Lupo, M. Ramos and M. Willem, “Limit Relative Category and Critical Point Theory,” Dynamics Reported, Vol. 3, 1994, pp. 1-24.

Copyright © 2024 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.