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.
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References ▼
  1. [1] G.D. Birkhoff, Dynamical Systems, Amer. Math. Soc., New York, 1927.
  2. [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. [3] A. Boscaggin, Periodic solutions to superlinear planar Hamiltonian systems, Port. Math. 69 (2012), 127-140. https://doi.org/10.4171/pm/1909
  4. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [22] P. Le Calvez, About Poincaré-Birkhoff theorem, Publ. Mat. Urug. 13 (2011), 61-98.
  23. [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. [24] H. Poincaré, Sur un théorème de géométrie, Rend. Circ. Mat. Palermo 33 (1912), 375-407. https://doi.org/10.1007/bf03015314
  25. [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. [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. [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
About the authors ▼
Alessandro Fonda (corresponding author)
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
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
Andrea Sfecci
https://orcid.org/0000-0002-8580-3026
Dipartimento di Matematica, Informatica e Geoscienze, Università degli Studi di Trieste, P.le Europa 1, 34127 Trieste, Italy
Wahid Ullah
https://orcid.org/0000-0002-0593-617X
Dipartimento di Matematica, Informatica e Geoscienze, Università degli Studi di Trieste, P.le Europa 1, 34127 Trieste, Italy
About the article ▼
Communicated by Marek Galewski.
Received: 2025-11-18.
Revised: 2026-01-08.
Accepted: 2026-01-08.
Published online: 2026-03-18.