Skip to main content
SpringerLink
Log in

Sufficient close-to-necessary condition for the existence of homoclinic orbits, and applications

  • Original Article
  • Published:
São Paulo Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

In this paper, we give a general and weak sufficient condition to guaranteeing the existence of infinitely many homoclinic solutions for some classes of differential equations, in particular for a class of Hamiltonian systems. We illustrate our main theoretical results by an application.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data Availability

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

References

  1. Ambrosetti, A., Bertotti, M.L.: Homoclinic for second order conservation systems. In: Partial Differential Equations and Related Subjects (Trento, 1990), (Longman Science Technology, Harlow, 1992), pp. 21–37

  2. Ambrosetti, A., Coti Zelati, V.: Multiple homoclinic orbits for a class of conservative systems. Rend. Sem. Mat. Univ. Padova 89, 177–194 (1993)

    MathSciNet  Google Scholar 

  3. Antonacci, F.: Periodic and homoclinic solutions to a class of Hamiltonian systems with indefinite potential in sign. Boll. Un. Mat. Ital. B 10(7), 303–324 (1996)

    MathSciNet  Google Scholar 

  4. Benhassine, A.: Multiple of homoclinic solutions for a perturbed dynamical systems with combined nonlinearities. Mediter. J. Math. 14, 132 (2017). https://doi.org/10.1007/s00009-017-0930-x

    Article  MathSciNet  Google Scholar 

  5. Benhassine, A.: Multiplicity of solutions for nonperiodic perturbed fractional Hamiltonian systems. Electron. J. Differ. Equ. 2017(93), 1–15 (2017)

    MathSciNet  Google Scholar 

  6. Benhassine, A.: Fractional Hamiltonian systems with locally defined potentials. Theor. Math. Phys. 195(1), 560–568 (2018)

    Article  MathSciNet  Google Scholar 

  7. Benhassine, A.: Existence of infinitely many solutions for a class of fractional Hamiltonian systems. J. Elliptic Parabol. Equ. 5 (2019)

  8. Benhassine, A.: Ground state solutions for a class of fractional Hamiltonian systems. Ricerche di Matematica 63, 727–743 (2019)

    Article  MathSciNet  Google Scholar 

  9. Benhassine, A.: Existence and infinitely of many solutions for nonperiodic fractional Hamiltonian systems. Differ. Integral Equ. (to appear)

  10. Benhassine, A.: On a weak condition for a class of \(p-\)Laplacian Hamiltonian systems. Theor. Math. Phys. 208(1), 855–864 (2021). https://doi.org/10.1134/S0040577921070011

    Article  MathSciNet  Google Scholar 

  11. Benhassine, A.: General and weak sufficient condition for Hamiltonian systems. J. Elliptic Parabol. Equ. (2021). https://doi.org/10.1007/s41808-021-00114-z

    Article  MathSciNet  Google Scholar 

  12. Carrião, P.C., Miyagaki, O.H.: Existence of homoclinic solutions for a class of time-dependent Hamiltonian systems. J. Math. Anal. Appl. 230, 157–172 (1999)

    Article  MathSciNet  Google Scholar 

  13. Chen, C.N., Tzeng, S.Y.: Existence and multiplicity results for homoclinic orbits of Hamiltonian systems. Electr. J. Differ. Equ. 1997, 1–19 (1997)

    MathSciNet  Google Scholar 

  14. Coti Zelati, V., Rabinowitz, P.H.: Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials. J. Am. Math. Soc. 4, 693–727 (1991)

    Article  MathSciNet  Google Scholar 

  15. Ding, Y.: Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems. Nonlinear Anal. 25, 1095–1113 (1995)

    Article  MathSciNet  Google Scholar 

  16. Ding, Y., Girardi, M.: Periodic and homoclinic solutions to a class of Hamiltonian systems with the potentials changing sign. Dyn. Syst. Appl. 2, 131–145 (1993)

    MathSciNet  Google Scholar 

  17. Fei, G.: The existence of homoclinic orbits for Hamiltonian systems with the potential changing sign. Chin. Ann. Math. Ser. B 17, 403–410 (1996)

    MathSciNet  Google Scholar 

  18. Felmer, P.L., De, E.A., Silva, B.E.: Homoclinic and periodic orbits for Hamiltonian systems. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 26, 285–301 (1998)

    MathSciNet  Google Scholar 

  19. Kajikiya, R.: A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations. J. Funct. Anal. 225, 352–370 (2005)

    Article  MathSciNet  Google Scholar 

  20. Korman, P., Lazer, A.C.: Homoclinic orbits for a class of symmetric Hamiltonian systems. Electr. J. Differ. Equ. 1994, 1–10 (1994)

    MathSciNet  Google Scholar 

  21. Liu, Y., Chen, H., Sun, J.: Infinitely many homoclinic solutions for some second order Hamiltonian systems. Nonlinear Anal. 74, 6459–6468 (2011)

    Article  MathSciNet  Google Scholar 

  22. Omana, W., Willem, M.: Homoclinic orbits for a class of Hamiltonian systems. Differ. Int. Equ. 5, 1115–1120 (1992)

    MathSciNet  Google Scholar 

  23. Ou, Z., Tang, C.: Existence of homoclinic solutions for the second order Hamiltonian systems. J. Math. Anal. Appl. 291, 203–213 (2004)

    Article  MathSciNet  Google Scholar 

  24. Paturel, E.: Multiple homoclinic orbits for a class of Hamiltonian systems. Calc. Var. Partial Differ. Equ. 12, 117–143 (2001)

    Article  MathSciNet  Google Scholar 

  25. Rabinowitz, P.H.: Minimax methods in critical point theory with applications to differential equations. In: CBMS Regional Conference Series in Mathematics, vol. 65. American Mathematical Society, Providence (1986)

  26. Rabinowitz, P.H.: Homoclinic orbits for a class of Hamiltonian systems. Proc. R. Soc. Edinb. Sect. A 114, 33–38 (1990)

    Article  MathSciNet  Google Scholar 

  27. Rabinowitz, P.H., Tanaka, K.: Some results on connecting orbits for a class of Hamiltonian systems. Math. Z. 206, 473–499 (1990)

    Article  MathSciNet  Google Scholar 

  28. Sun, J., Chen, H., Nieto, J.J.: Homoclinic solutions for a class of subquadratic second-order Hamiltonian systems. J. Math. Anal. Appl. 373, 20–29 (2011)

    Article  MathSciNet  Google Scholar 

  29. Tang, X.H., Lin, X.: Infinitely many homoclinic orbits for Hamiltonian systems with indefinite sign subquadratic potentials. Nonlinear Anal. 74, 6314–6325 (2011)

    Article  MathSciNet  Google Scholar 

  30. Wang, J., Zhang, F., Xu, J.: Existence and multiplicity of homoclinic orbits for the second order Hamiltonian systems. J. Math. Anal. Appl. 366, 569–581 (2010)

    Article  MathSciNet  Google Scholar 

  31. Wei, J., Wang, J.: Infinitely many homoclinic orbits for the second order Hamiltonian systems with general potentials. J. Math. Anal. Appl. 366, 694–699 (2010)

    Article  MathSciNet  Google Scholar 

  32. Willem, M.: Minimax Theorems. Birkhäser, Boston (1996)

    Book  Google Scholar 

  33. Wu, S., Liu, J.: Homoclinic orbits for second order Hamiltonian system with quadratic growth. Appl. Math. J. Chin. Univ. Ser. B 10, 399–410 (1995)

    Article  MathSciNet  Google Scholar 

  34. Yang, J., Zhang, F.: Infinitely many homoclinic orbits for the second order Hamiltonian systems with super-quadratic potentials. Nonlinear Anal. Real World Appl. 10, 1417–1423 (2009)

    Article  MathSciNet  Google Scholar 

  35. Yang, M., Han, Z.: Infinitely many homoclinic solutions for second-order Hamiltonian systems with odd nonlinearities. Nonlinear Anal. 74, 2635–2646 (2011)

    Article  MathSciNet  Google Scholar 

  36. Zhang, Q.: Homoclinic solutions for a class of second order Hamiltonian systems. Math. Nachr. 288(8–9), 1073–1081 (2015). https://doi.org/10.1002/mana.201200293

    Article  MathSciNet  Google Scholar 

  37. Zhang, Q., Chu, L.: Homoclinic solutions for a class of second order Hamiltonian systems with locally defined potentials. Nonlinear Anal. 75, 3188–3197 (2012)

    Article  MathSciNet  Google Scholar 

  38. Zhang, Q., Liu, C.: Infinitely many homoclinic solutions for second order Hamiltonian systems. Nonlinear Anal. 72, 894–903 (2010)

    Article  MathSciNet  Google Scholar 

  39. Zhang, Q., Tang, X.H.: Existence of homoclinic solutions for a class of asymptotically quadratic non-autonomous Hamiltonian systems. Math. Nachr. 285, 778–789 (2012)

    Article  MathSciNet  Google Scholar 

  40. Zhang, Z., Yuan, R.: Homoclinic solutions for a class of non-autonomous subquadratic second-order Hamiltonian systems. Nonlinear Anal. 71, 4125–4130 (2009)

    Article  MathSciNet  Google Scholar 

  41. Zou, W.: Infinitely many homoclinic orbits for the second-order Hamiltonian systems. Appl. Math. Lett. 16, 1283–1287 (2003)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the referees for their careful reading, critical comments and helpful suggestions, which helped to improve the quality of the paper.

Author information

Authors and Affiliations

  1. Higher Institute of Science Computer and Mathematics Monastir, Monastir, Tunisia

    A. Benhassine

  2. Faculty of Sciences of Sfax, Sfax, Tunisia

    S. Farhani & T. Talbi

Authors
  1. A. Benhassine
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Farhani
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. T. Talbi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Benhassine.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Communicated by Liliane Maia.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Benhassine, A., Farhani, S. & Talbi, T. Sufficient close-to-necessary condition for the existence of homoclinic orbits, and applications. São Paulo J. Math. Sci. (2024). https://doi.org/10.1007/s40863-023-00397-1

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40863-023-00397-1

Keywords

Mathematics Subject Classification

Search

Navigation