Abstract and Applied Analysis

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

Guanwei Chen

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Abstract

We study a class of nonperiodic damped vibration systems with asymptotically quadratic terms at infinity. We obtain infinitely many nontrivial homoclinic orbits by a variant fountain theorem developed recently by Zou. To the best of our knowledge, there is no result published concerning the existence (or multiplicity) of nontrivial homoclinic orbits for this class of non-periodic damped vibration systems with asymptotically quadratic terms at infinity.

Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 937128, 7 pages.

Dates
First available in Project Euclid: 27 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393512125

Digital Object Identifier
doi:10.1155/2013/937128

Mathematical Reviews number (MathSciNet)
MR3126799

Zentralblatt MATH identifier
07095512

Citation

Chen, Guanwei. Nonperiodic Damped Vibration Systems with Asymptotically Quadratic Terms at Infinity: Infinitely Many Homoclinic Orbits. Abstr. Appl. Anal. 2013 (2013), Article ID 937128, 7 pages. doi:10.1155/2013/937128. https://projecteuclid.org/euclid.aaa/1393512125


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References

  • A. Ambrosetti and V. C. Zelati, “Multiple homoclinic orbits for aclass of conservative systems,” Rendiconti del Seminario Matematico della Università di Padova, vol. 89, pp. 177–194, 1993.
  • 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.
  • 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.
  • M. Izydorek and J. Janczewska, “Homoclinic solutions for a classof the second order Hamiltonian systems,” Journal of Differential Equations, vol. 219, no. 2, pp. 375–389, 2005.
  • 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.
  • J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, vol. 74 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1989.
  • 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.
  • 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.
  • 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.
  • 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.
  • E. Séré, “Existence of infinitely many homoclinic orbits in Hamiltonian systems,” Mathematische Zeitschrift, vol. 209, no. 1, pp. 27–42, 1992.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • W. Zhu, “Existence of homoclinic solutions for a class of secondorder systems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 75, no. 4, pp. 2455–2463, 2012.
  • 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.
  • 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.
  • 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.
  • 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.
  • G. Chen, “Non-periodic damped 3 vibration systems: infinitelymany homoclinic orbits,” Calculus of Variations and Partial Dif-ferential Equations. In press.
  • D. G. Costa and C. A. Magalhães, “A unified approach to a classof strongly indefinite functionals,” Journal of Differential Equations, vol. 125, no. 2, pp. 521–547, 1996.
  • D. G. Costa and C. A. Magalhães, “A variational approach tosubquadratic perturbations of elliptic systems,” Journal of Differential Equations, vol. 111, no. 1, pp. 103–122, 1994.
  • 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.
  • W. Zou, “Variant fountain theorems and their applications,” Manuscripta Mathematica, vol. 104, no. 3, pp. 343–358, 2001.
  • M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications, Birkhäauser, Boston, Mass, USA, 1996.
  • V. Benci and P. H. Rabinowitz, “Critical point theorems for indefinite functionals,” Inventiones Mathematicae, vol. 52, no. 3, pp. 241–273, 1979.
  • 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. \endinput