Online First version
https://doi.org/10.7494/OpMath.202601081

 
Opuscula Mathematica

Applying the Poincaré-Birkhoff theorem to antiperiodic problems

Alessandro Fonda
Natnael Gezahegn Mamo
Andrea Sfecci
Wahid Ullah

Abstract. We show how the Poincaré-Birkhoff theorem for Hamiltonian systems can be used to find multiple solutions of the antiperiodic problem. Applications are given to scalar second order differential equations whose nonlinearities provide a twist in the phase plane, among which those with a superlinear or sublinear behaviour at infinity.

Keywords: antiperiodic solutions, Poincaré-Birkhoff theorem, Hamiltonian systems.

Mathematics Subject Classification: 34B15.

Full text (pdf)

  1. G.D. Birkhoff, Dynamical Systems, Amer. Math. Soc., New York, 1927.
  2. A. Boscaggin, Subharmonic solutions of planar Hamiltonian systems: a rotation number approach, Adv. Nonlin. Stud. 11 (2011), 77-103. https://doi.org/10.1515/ans-2011-0104
  3. A. Boscaggin, Periodic solutions to superlinear planar Hamiltonian systems, Port. Math. 69 (2012), 127-140. https://doi.org/10.4171/pm/1909
  4. A. Boscaggin, M. Garrione, Resonance and rotation numbers for planar Hamiltonian systems: multiplicity results via the Poincaré-Birkhoff theorem, Nonlinear Anal. 74 (2011), 4166-4185. https://doi.org/10.1016/j.na.2011.03.051
  5. M. Brown, W.D. Neumann, Proof of the Poincaré-Birkhoff fixed point theorem, Michigan Math. J. 24 (1977), 21-31. https://doi.org/10.1307/mmj/1029001816
  6. P. Buttazzoni, A. Fonda, Periodic perturbations of scalar second order differential equations, Discrete Contin. Dyn. Syst. 3 (1997), 451-455. https://doi.org/10.3934/dcds.1997.3.451
  7. F. Dalbono, C. Rebelo, Poincaré-Birkhoff fixed point theorem and periodic solutions of asymptotically linear planar Hamiltonian systems, Turin Fortnight Lectures on Nonlinear Analysis (2001), Rend. Sem. Mat. Univ. Politec. Torino 60 (2002), 233-263 (2003).
  8. T. Ding, F. Zanolin, Periodic solutions of Duffing's equations with superquadratic potential, J. Differential Equations 97 (1992), 328-378. https://doi.org/10.1016/0022-0396(92)90076-y
  9. T. Ding, F. Zanolin, Subharmonic solutions of second order nonlinear equations: a time-map approach, Nonlinear Anal. 20 (1993), 509-532. https://doi.org/10.1016/0362-546x(93)90036-r
  10. A. Fonda, Periodic solutions of Hamiltonian systems with symmetries, [in:] Topological Methods for Delay and Ordinary Differential Equations with Applications to Continuum Mechanics, Springer, 2024, 1-19. https://doi.org/10.1007/978-3-031-61337-1_1
  11. A. Fonda, A. Sfecci, Periodic solutions of weakly coupled superlinear systems, J. Differential Equations 260 (2016), 2150-2162. https://doi.org/10.1016/j.jde.2015.09.056
  12. A. Fonda, A. Sfecci, Multiple periodic solutions of Hamiltonian systems confined in a box, Discrete Contin. Dyn. Syst. 37 (2017), 1425-1436. https://doi.org/10.3934/dcds.2017059
  13. A. Fonda, R. Toader, Subharmonic solutions of Hamiltonian systems displaying some kind of sublinear growth, Adv. Nonlinear Anal. 8 (2019), 583-602. https://doi.org/10.1515/anona-2017-0040
  14. A. Fonda, A.J. Ureña, A higher dimensional Poincaré-Birkhoff theorem for Hamiltonian flows, Ann. Inst. H. Poincaré Anal. Non Linéaire 34 (2017), 679-698. https://doi.org/10.1016/j.anihpc.2016.04.002
  15. A. Fonda, F. Zanolin, Periodic oscillations of forced pendulums with a very small length, Proc. Royal Soc. Edinburgh 127 (1997), 67-76. https://doi.org/10.1017/s0308210500023519
  16. A. Fonda, R.F. Manasevich, F. Zanolin, Subharmonic solutions for some second order differential equations with singularities, SIAM J. Math. Anal. 24 (1993), 1294-1311. https://doi.org/10.1137/0524074
  17. A. Fonda, M. Sabatini, F. Zanolin, Periodic solutions of perturbed Hamiltonian systems in the plane by the use of the Poincaré-Birkhoff theorem, Topol. Methods Nonlinear Anal. 40 (2012), 29-52.
  18. A. Fonda, Z. Schneider, F. Zanolin, Periodic oscillations for a nonlinear suspension bridge model, J. Comp. Appl. Math. 52 (1994), 113-140. https://doi.org/10.1016/0377-0427(94)90352-2
  19. P. Gidoni, A. Margheri, Lower bounds on the number of periodic solutions for asymptotically linear planar Hamiltonian systems, Discrete Contin. Dyn. Syst. 39 (2019), 585-606. https://doi.org/10.3934/dcds.2019024
  20. P. Hartman, On boundary value problems for superlinear second order differential equations, J. Differential Equations 26 (1977), 37-53. https://doi.org/10.1016/0022-0396(77)90097-3
  21. H. Jacobowitz, Periodic solutions of \(x''+f(t,x)=0\) via the Poincaré-Birkhoff theorem, J. Differential Equations 20 (1976), 37-52. https://doi.org/10.1016/0022-0396(76)90094-2
  22. P. Le Calvez, About Poincaré-Birkhoff theorem, Publ. Mat. Urug. 13 (2011), 61-98.
  23. A. Margheri, C. Rebelo, F. Zanolin, Maslov index, Poincaré-Birkhoff theorem and periodic solutions of asymptotically linear planar Hamiltonian systems, J. Differential Equations 183 (2002), 342-367. https://doi.org/10.1006/jdeq.2001.4122
  24. H. Poincaré, Sur un théorème de géométrie, Rend. Circ. Mat. Palermo 33 (1912), 375-407. https://doi.org/10.1007/bf03015314
  25. D. Qian, P.J. Torres, P. Wang, Periodic solutions of second order equations via rotation numbers, J. Differential Equations 266 (2019), 4746-4768. https://doi.org/10.1016/j.jde.2018.10.010
  26. C. Rebelo, A note on the Poincaré-Birkhoff fixed point theorem and periodic solutions of planar systems, Nonlinear Anal. 29 (1997), 291-311. https://doi.org/10.1016/s0362-546x(96)00065-x
  27. C. Zanini, Rotation numbers, eigenvalues, and the Poincaré–Birkhoff theorem, J. Math. Anal. Appl. 279 (2003), 290-307. https://doi.org/10.1016/s0022-247x(03)00012-x
  • Alessandro Fonda (corresponding author)
  • ORCID iD https://orcid.org/0000-0001-6230-3101
  • Dipartimento di Matematica, Informatica e Geoscienze, Università degli Studi di Trieste, P.le Europa 1, 34127 Trieste, Italy
  • Natnael Gezahegn Mamo
  • ORCID iD https://orcid.org/0000-0002-3765-103X
  • Dipartimento di Matematica, Informatica e Geoscienze, Università degli Studi di Trieste, P.le Europa 1, 34127 Trieste, Italy
  • Communicated by Marek Galewski.
  • Received: 2025-11-18.
  • Revised: 2026-01-08.
  • Accepted: 2026-01-08.
  • Published online: 2026-03-18.
Opuscula Mathematica - cover

We advise that this website uses cookies to help us understand how the site is used. All data is anonymized. Recent versions of popular browsers provide users with control over cookies, allowing them to set their preferences to accept or reject all cookies or specific ones.