About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 937128, 7 pages
http://dx.doi.org/10.1155/2013/937128
Research Article

Nonperiodic Damped Vibration Systems with Asymptotically Quadratic Terms at Infinity: Infinitely Many Homoclinic Orbits

School of Mathematics and Statistics, Anyang Normal University, Anyang, Henan 455000, China

Received 10 September 2013; Accepted 10 October 2013

Academic Editor: Dumitru Motreanu

Copyright © 2013 Guanwei Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. A. Ambrosetti and V. C. Zelati, “Multiple homoclinic orbits for a class of conservative systems,” Rendiconti del Seminario Matematico della Università di Padova, vol. 89, pp. 177–194, 1993. View at MathSciNet
  2. G. Chen and S. Ma, “Periodic solutions for Hamiltonian systems without Ambrosetti-Rabinowitz condition and spectrum 0,” Journal of Mathematical Analysis and Applications, vol. 379, no. 2, pp. 842–851, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  3. Y. H. Ding, “Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 25, no. 11, pp. 1095–1113, 1995. View at Publisher · View at Google Scholar · View at MathSciNet
  4. M. Izydorek and J. Janczewska, “Homoclinic solutions for a class of the second order Hamiltonian systems,” Journal of Differential Equations, vol. 219, no. 2, pp. 375–389, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  5. Y. I. Kim, “Existence of periodic solutions for planar Hamiltonian systems at resonance,” Journal of the Korean Mathematical Society, vol. 48, no. 6, pp. 1143–1152, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  6. J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, vol. 74 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1989. View at MathSciNet
  7. W. Omana and M. Willem, “Homoclinic orbits for a class of Hamiltonian systems,” Differential and Integral Equations, vol. 5, no. 5, pp. 1115–1120, 1992. View at MathSciNet
  8. E. Paturel, “Multiple homoclinic orbits for a class of Hamiltonian systems,” Calculus of Variations and Partial Differential Equations, vol. 12, no. 2, pp. 117–143, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  9. P. H. Rabinowitz, “Homoclinic orbits for a class of Hamiltonian systems,” Proceedings of the Royal Society of Edinburgh A, vol. 114, no. 1-2, pp. 33–38, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  10. P. H. Rabinowitz and K. Tanaka, “Some results on connecting orbits for a class of Hamiltonian systems,” Mathematische Zeitschrift, vol. 206, no. 3, pp. 473–499, 1991. View at Publisher · View at Google Scholar · View at MathSciNet
  11. E. Séré, “Existence of infinitely many homoclinic orbits in Hamiltonian systems,” Mathematische Zeitschrift, vol. 209, no. 1, pp. 27–42, 1992. View at Publisher · View at Google Scholar · View at MathSciNet
  12. J. Sun, H. Chen, and J. J. Nieto, “Homoclinic solutions for a class of subquadratic second-order Hamiltonian systems,” Journal of Mathematical Analysis and Applications, vol. 373, no. 1, pp. 20–29, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  13. X. H. Tang and L. Xiao, “Homoclinic solutions for nonautonomous second-order Hamiltonian systems with a coercive potential,” Journal of Mathematical Analysis and Applications, vol. 351, no. 2, pp. 586–594, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  14. L. L. Wan and C. L. Tang, “Existence and multiplicity of homoclinic orbits for second order Hamiltonian systems without (AR) condition,” Discrete and Continuous Dynamical Systems B, vol. 15, no. 1, pp. 255–271, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  15. J. Xiao and J. J. Nieto, “Variational approach to some damped Dirichlet nonlinear impulsive differential equations,” Journal of the Franklin Institute, vol. 348, no. 2, pp. 369–377, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  16. P. Zhang and C. L. Tang, “Infinitely many periodic solutions for nonautonomous sublinear second-order Hamiltonian systems,” Abstract and Applied Analysis, vol. 2010, Article ID 620438, 10 pages, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  17. Q. Zhang and C. Liu, “Infinitely many homoclinic solutions for second order Hamiltonian systems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 2, pp. 894–903, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  18. W. Zhu, “Existence of homoclinic solutions for a class of second order systems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 75, no. 4, pp. 2455–2463, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  19. Z. Zhang and R. Yuan, “Homoclinic solutions of some second order non-autonomous systems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 11, pp. 5790–5798, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  20. X. Wu and W. Zhang, “Existence and multiplicity of homoclinic solutions for a class of damped vibration problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 13, pp. 4392–4398, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  21. J. Sun, J. J. Nieto, and M. Otero-Novoa, “On homoclinic orbits for a class of damped vibration systems,” Advances in Difference Equations, vol. 2012, article 102, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  22. G. Chen, “Non-periodic damped vibration systems with sublinear terms at infinity: infinitely many homoclinic orbits,” Nonlinear Analysis: Theory, Methods & Applications, vol. 92, pp. 168–176, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  23. G. Chen, “Non-periodic damped 3 vibration systems: infinitelymany homoclinic orbits,” Calculus of Variations and Partial Differential Equations. In press.
  24. D. G. Costa and C. A. Magalhães, “A unified approach to a class of strongly indefinite functionals,” Journal of Differential Equations, vol. 125, no. 2, pp. 521–547, 1996. View at Publisher · View at Google Scholar · View at MathSciNet
  25. D. G. Costa and C. A. Magalhães, “A variational approach to subquadratic perturbations of elliptic systems,” Journal of Differential Equations, vol. 111, no. 1, pp. 103–122, 1994. View at Publisher · View at Google Scholar · View at MathSciNet
  26. J. Wnag, J. Xu, and F. Zhang, “Homoclinic orbits for a class of Hamiltonian systems with superquadratic or asymptotically quadratic potentials,” Communications on Pure and Applied Analysis, vol. 10, no. 1, pp. 269–286, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  27. W. Zou, “Variant fountain theorems and their applications,” Manuscripta Mathematica, vol. 104, no. 3, pp. 343–358, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  28. M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications, Birkhäauser, Boston, Mass, USA, 1996. View at Publisher · View at Google Scholar · View at MathSciNet
  29. V. Benci and P. H. Rabinowitz, “Critical point theorems for indefinite functionals,” Inventiones Mathematicae, vol. 52, no. 3, pp. 241–273, 1979. View at Publisher · View at Google Scholar · View at MathSciNet
  30. P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, vol. 65 of Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, USA, 1986.