Skip to main content

Advertisement

We’d like to understand how you use our websites in order to improve them. Register your interest.

Periodic solutions of second-order nonautonomous dynamical systems

Abstract

We study the existence of periodic solutions for second-order nonautonomous dynamical systems. We give four sets of hypotheses which guarantee the existence of solutions. We were able to weaken the hypotheses considerably from those used previously for such systems. We employ a new saddle point theorem using linking methods.

[1234567891011121314151617]

References

  1. 1.

    Ambrosetti A, Coti Zelati V: Periodic Solutions of Singular Lagrangian Systems, Progress in Nonlinear Differential Equations and Their Applications. Volume 10. Birkhäuser Boston, Massachusetts; 1993:xii+157.

  2. 2.

    Ben-Naoum AK, Troestler C, Willem M: Existence and multiplicity results for homogeneous second order differential equations. Journal of Differential Equations 1994,112(1):239-249. 10.1006/jdeq.1994.1103

  3. 3.

    Berger MS, Schechter M: On the solvability of semilinear gradient operator equations. Advances in Mathematics 1977,25(2):97-132. 10.1016/0001-8708(77)90001-9

  4. 4.

    Ekeland I, Ghoussoub N: Selected new aspects of the calculus of variations in the large. Bulletin of the American Mathematical Society 2002,39(2):207-265. 10.1090/S0273-0979-02-00929-1

  5. 5.

    Long YM: Nonlinear oscillations for classical Hamiltonian systems with bi-even subquadratic potentials. Nonlinear Analysis 1995,24(12):1665-1671. 10.1016/0362-546X(94)00227-9

  6. 6.

    Mawhin J: Semicoercive monotone variational problems. Académie Royale de Belgique. Bulletin de la Classe des Sciences 1987,73(3-4):118-130.

  7. 7.

    Mawhin J, Willem M: Critical points of convex perturbations of some indefinite quadratic forms and semilinear boundary value problems at resonance. Annales de l'Institut Henri Poincaré. Analyse Non Linéaire 1986,3(6):431-453.

  8. 8.

    Mawhin J, Willem M: Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences. Volume 74. Springer, New York; 1989:xiv+277.

  9. 9.

    Schechter M: New linking theorems. Rendiconti del Seminario Matematico della Università di Padova 1998, 99: 255-269.

  10. 10.

    Schechter M: Linking Methods in Critical Point Theory. Birkhäuser Boston, Massachusetts; 1999:xviii+294.

  11. 11.

    Tang C-L:Periodic solutions of non-autonomous second order systems with-quasisubadditive potential. Journal of Mathematical Analysis and Applications 1995,189(3):671-675. 10.1006/jmaa.1995.1044

  12. 12.

    Tang C-L: Periodic solutions for nonautonomous second order systems with sublinear nonlinearity. Proceedings of the American Mathematical Society 1998,126(11):3263-3270. 10.1090/S0002-9939-98-04706-6

  13. 13.

    Tang C-L, Wu X-P: Periodic solutions for second order systems with not uniformly coercive potential. Journal of Mathematical Analysis and Applications 2001,259(2):386-397. 10.1006/jmaa.2000.7401

  14. 14.

    Tang C-L, Wu X-P: Periodic solutions for a class of nonautonomous subquadratic second order Hamiltonian systems. Journal of Mathematical Analysis and Applications 2002,275(2):870-882. 10.1016/S0022-247X(02)00442-0

  15. 15.

    Tang C-L, Wu X-P: Notes on periodic solutions of subquadratic second order systems. Journal of Mathematical Analysis and Applications 2003,285(1):8-16. 10.1016/S0022-247X(02)00417-1

  16. 16.

    Willem W: Oscillations forcées systèmes hamiltoniens. In Public. Sémin. Analyse Non Linéarie. Université de Franche-Comté, Besancon; 1981.

  17. 17.

    Wu X-P, Tang C-L: Periodic solutions of a class of non-autonomous second-order systems. Journal of Mathematical Analysis and Applications 1999,236(2):227-235. 10.1006/jmaa.1999.6408

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Martin Schechter.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Schechter, M. Periodic solutions of second-order nonautonomous dynamical systems. Bound Value Probl 2006, 25104 (2006). https://doi.org/10.1155/BVP/2006/25104

Download citation

Keywords

  • Differential Equation
  • Dynamical System
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Periodic Solution