Open Access

Periodic solutions of second-order nonautonomous dynamical systems

Boundary Value Problems20062006:25104

DOI: 10.1155/BVP/2006/25104

Received: 13 March 2006

Accepted: 15 May 2006

Published: 31 August 2006


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.


Authors’ Affiliations

Department of Mathematics, University of California


  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.MATHGoogle Scholar
  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.1994.1103MathSciNetView ArticleMATHGoogle Scholar
  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)90001-9MathSciNetView ArticleMATHGoogle Scholar
  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/S0273-0979-02-00929-1MathSciNetView ArticleMATHGoogle Scholar
  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)00227-9MathSciNetView ArticleMATHGoogle Scholar
  6. Mawhin J: Semicoercive monotone variational problems. Académie Royale de Belgique. Bulletin de la Classe des Sciences 1987,73(3-4):118-130.MathSciNetMATHGoogle Scholar
  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.MathSciNetMATHGoogle Scholar
  8. Mawhin J, Willem M: Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences. Volume 74. Springer, New York; 1989:xiv+277.View ArticleMATHGoogle Scholar
  9. Schechter M: New linking theorems. Rendiconti del Seminario Matematico della Università di Padova 1998, 99: 255-269.MathSciNetMATHGoogle Scholar
  10. Schechter M: Linking Methods in Critical Point Theory. Birkhäuser Boston, Massachusetts; 1999:xviii+294.View ArticleMATHGoogle Scholar
  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.1995.1044MathSciNetView ArticleMATHGoogle Scholar
  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/S0002-9939-98-04706-6MathSciNetView ArticleMATHGoogle Scholar
  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.2000.7401MathSciNetView ArticleMATHGoogle Scholar
  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)00442-0MathSciNetView ArticleMATHGoogle Scholar
  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)00417-1MathSciNetView ArticleMATHGoogle Scholar
  16. Willem W: Oscillations forcées systèmes hamiltoniens. In Public. Sémin. Analyse Non Linéarie. Université de Franche-Comté, Besancon; 1981.Google Scholar
  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.1999.6408MathSciNetView ArticleMATHGoogle Scholar


© Schechter 2006

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