## Abstract and Applied Analysis

### Homoclinic Solutions for a Class of the Second-Order Impulsive Hamiltonian Systems

#### Abstract

This paper is concerned with the existence of homoclinic solutions for a class of the second order impulsive Hamiltonian systems. By employing the Mountain Pass Theorem, we demonstrate that the limit of a $2kT$-periodic approximation solution is a homoclinic solution of our problem.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 583107, 8 pages.

Dates
First available in Project Euclid: 26 February 2014

https://projecteuclid.org/euclid.aaa/1393450110

Digital Object Identifier
doi:10.1155/2013/583107

Mathematical Reviews number (MathSciNet)
MR3093764

Zentralblatt MATH identifier
1296.34121

#### Citation

Xie, Jingli; Luo, Zhiguo; Chen, Guoping. Homoclinic Solutions for a Class of the Second-Order Impulsive Hamiltonian Systems. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 583107, 8 pages. doi:10.1155/2013/583107. https://projecteuclid.org/euclid.aaa/1393450110

#### References

• P. H. Rabinowitz, “Homoclinic orbits for a class of Hamiltonian systems,” Proceedings of the Royal Society of Edinburgh. Section A, vol. 114, no. 1-2, pp. 33–38, 1990.
• 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.
• R. P. Agarwal and D. O'Regan, “A multiplicity result for second order impulsive differential equations via the Leggett Williams fixed point theorem,” Applied Mathematics and Computation, vol. 161, no. 2, pp. 433–439, 2005.
• R. P. Agarwal, D. Franco, and D. O'Regan, “Singular boundary value problems for first and second order impulsive differential equations,” Aequationes Mathematicae, vol. 69, no. 1-2, pp. 83–96, 2005.
• J. Chu and J. J. Nieto, “Impulsive periodic solutions of first-order singular differential equations,” Bulletin of the London Mathematical Society, vol. 40, no. 1, pp. 143–150, 2008.
• Z. He and X. He, “Monotone iterative technique for impulsive integro-differential equations with periodic boundary conditions,” Computers & Mathematics with Applications, vol. 48, no. 1-2, pp. 73–84, 2004.
• J. J. Nieto and D. O'Regan, “Variational approach to impulsive differential equations,” Nonlinear Analysis: Real World Applications, vol. 10, no. 2, pp. 680–690, 2009.
• D. Qian and X. Li, “Periodic solutions for ordinary differential equations with sublinear impulsive effects,” Journal of Mathematical Analysis and Applications, vol. 303, no. 1, pp. 288–303, 2005.
• P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, vol. 65 of CBMS Regional Conference Series in Mathematics, American Mathematical Society, Washington, DC, USA, 1986.
• I. Rach\accent23unková and M. Tvrdý, “Non-ordered lower and upper functions in second order impulsive periodic problems,” Dynamics of Continuous, Discrete & Impulsive Systems. Series A, vol. 12, no. 3-4, pp. 397–415, 2005.
• J. Sun, H. Chen, and L. Yang, “Variational methods to fourth-order impulsive differential equations,” Journal of Applied Mathematics and Computing, vol. 35, no. 1-2, pp. 323–340, 2011.
• J. Shen and W. Wang, “Impulsive boundary value problems with nonlinear boundary conditions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 11, pp. 4055–4062, 2008.
• Y. Tian and W. Ge, “Applications of variational methods to boundary-value problem for impulsive differential equations,” Proceedings of the Edinburgh Mathematical Society, vol. 51, no. 2, pp. 509–527, 2008.
• Y. Tian, J. Wang, and W. Ge, “Variational methods to mixed boundary value problem for impulsive differential equations with a parameter,” Taiwanese Journal of Mathematics, vol. 13, no. 4, pp. 1353–1370, 2009.
• J. Xie and Z. Luo, “Multiple solutions for a second-order impulsive Sturm-Liouville equation,” Abstract and Applied Analysis, vol. 2013, Article ID 527082, 6 pages, 2013.
• J. Xie and Z. Luo, “Solutions to a boundary value problem of a fourth-order impulsive differential equation,” Boundary Value Problems, vol. 2013, 154 pages, 2013.
• W. Zuo, D. Jiang, D. O'Regan, and R. P. Agarwal, “Optimal existence conditions for the periodic delay $\phi$-Laplace equation with upper and lower solutions in the reverse order,” Results in Mathematics, vol. 44, no. 3-4, pp. 375–385, 2003.
• H. Zhang and Z. Li, “Periodic and homoclinic solutions generated by impulses,” Nonlinear Analysis: Real World Applications, vol. 12, no. 1, pp. 39–51, 2011.
• H. Zhang and Z. Li, “Variational approach to impulsive differential equations with periodic boundary conditions,” Nonlinear Analysis: Real World Applications, vol. 11, no. 1, pp. 67–78, 2010.
• Z. Zhang and R. Yuan, “An application of variational methods to Dirichlet boundary value problem with impulses,” Nonlinear Analysis: Real World Applications, vol. 11, no. 1, pp. 155–162, 2010.
• X. Lv and S. Lu, “Homoclinic orbits for a class of second-order Hamiltonian systems without a coercive potential,” Journal of Applied Mathematics and Computing, vol. 39, no. 1-2, pp. 121–130, 2012.
• P. D. Makita, “Homoclinic orbits for second order Hamiltonian equations in $\mathbb{R}$,” Journal of Dynamics and Differential Equations, vol. 24, no. 4, pp. 857–871, 2012.
• X. H. Tang and L. Xiao, “Homoclinic solutions for a class ofsecond-order Hamiltonian systems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 3-4, pp. 1140–1152, 2009.
• M. Winkler, “Spatially monotone homoclinic orbits in nonlinear parabolic equations of super-fast diffusion type,” Mathematische Annalen, vol. 355, no. 2, pp. 519–549, 2013.
• X. Zhang and X. Tang, “Subharmonic solutions for a class of non-quadratic second order Hamiltonian systems,” Nonlinear Analysis: Real World Applications, vol. 13, no. 1, pp. 113–130, 2012.
• Z. Zhou, J. Yu, and Y. Chen, “Homoclinic solutions in periodic difference equations with saturable nonlinearity,” Science China Mathematics, vol. 54, no. 1, pp. 83–93, 2011.
• V. Coti-Zelati, I. Ekeland, and E. Séré, “A variational approach to homoclinic orbits in Hamiltonian systems,” Mathematische Annalen, vol. 288, no. 1, pp. 133–160, 1990.
• P.-L. Lions, “The concentration-compactness principle in the calculus of variations. The locally compact case. I,” Annales de l'Institut Henri Poincaré, vol. 1, no. 2, pp. 109–145, 1984.
• H. Hofer and K. Wysocki, “First order elliptic systems and the existence of homoclinic orbits in Hamiltonian systems,” Mathematische Annalen, vol. 288, no. 3, pp. 483–503, 1990.
• A. Ambrosetti and P. H. Rabinowitz, “Dual variational methods in critical point theory and applications,” Journal of Functional Analysis, vol. 14, no. 4, pp. 349–381, 1973.