Topological Methods in Nonlinear Analysis

Multiple periodic solutions of Hamiltonian systems in the plane

Alessandro Fonda and Luca Ghirardelli

Full-text: Open access

Abstract

Our aim is to prove a multiplicity result for periodic solutions of Hamiltonian systems in the plane, by the use of the Poincaré-Birkhoff Fixed Point Theorem. Our main theorem generalizes previous results obtained for scalar second order equations by Lazer and McKenna [Large scale oscillatory behaviour in loaded asymmetric systems, Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (1987), 243–274] and Del Pino, Manasevich and Murua [On the number of $2\pi$-periodic solutions for $u''+g(u) =s(1+h(t))$ using the Poincaré–Birkhoff Theorem, J. Differential Equations 95 (1992), 240–258].

Article information

Source
Topol. Methods Nonlinear Anal., Volume 36, Number 1 (2010), 27-38.

Dates
First available in Project Euclid: 21 April 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1461251062

Mathematical Reviews number (MathSciNet)
MR2744830

Zentralblatt MATH identifier
1221.34111

Citation

Fonda, Alessandro; Ghirardelli, Luca. Multiple periodic solutions of Hamiltonian systems in the plane. Topol. Methods Nonlinear Anal. 36 (2010), no. 1, 27--38. https://projecteuclid.org/euclid.tmna/1461251062


Export citation

References

  • P. A. Binding and B. P. Rynne, Half-eigenvalues of periodic Sturm–Liouville problems , J. Differential Equations, 206(2004), 280–305 \ref\key 2
  • M. A. Del Pino, R. F. Manásevich and A. Murua, On the number of $2\pi$-periodic solutions for $u''+g(u)=s(1+h(t))$ using the Poincaré–Birkhoff Theorem , J. Differential Equations, 95(1992), 240–258 \ref\key 3
  • W.-Y. Ding, A generalization of the Poincaré–Birkhoff theorem , Proc. Amer. Math. Soc., 88(1983), 341–346 \ref\key 4
  • A. Fonda, Positively homogeneous Hamiltonian systems in the plane , J. Differential Equations, 200(2004), 162–184 \ref\key 5
  • A. Fonda and L. Ghirardelli, Multiple periodic solutions of scalar second order differential equations , Nonlinear Anal., 72 (2010), 4005–4015 \ref\key 6
  • A. C. Lazer and P. J. McKenna, Large scale oscillatory behaviour in loaded asymmetric systems , Ann. Inst. H. Poincaré Anal. Non Linéaire, 4(1987), 243–274 \ref\key 7
  • C. Rebelo, Multiple periodic solutions of second order equations with asymmetric nonlinearities, Discr. Contin. Dynam. Systems, 3 (1997), 25–34 \ref\key 8
  • C. Rebelo and F. Zanolin, Multiplicity results for periodic solutions of second order ODEs with asymmetric nonlinearities , Trans. Amer. Math. Soc., 348(1996), 2349–2389 \ref\key 9
  • C. Zanini and F. Zanolin, A multiplicity result of periodic solutions for parameter dependent asymmetric non-autonomous equations , Dynam. Contin. Discrete Impuls. Systems Ser. A Math. Anal., 12(2005), 343–361