Communications in Mathematical Analysis

Heteroclinic Orbits for Discrete Hamiltonian Systems

Yuhua Long, Haiping Shi, and Huafeng Xiao

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Abstract

Recently, the existence and multiplicity results of heteroclinic orbits for a discrete pendulum equation have been investigated. In present paper, we generalize those results to a class of discrete Hamiltonian systems. Since the variational functional is identically infinite, some effective methods, provided by Rabinowitz, have to be adopted to detect critical points corresponding to heteroclinic solutions. .

Article information

Source
Commun. Math. Anal., Volume 9, Number 1 (2010), 1-14.

Dates
First available in Project Euclid: 21 April 2010

Permanent link to this document
https://projecteuclid.org/euclid.cma/1271890714

Mathematical Reviews number (MathSciNet)
MR2569957

Zentralblatt MATH identifier
1192.39002

Subjects
Primary: 39A11

Keywords
heteroclinic solution critical point discrete Hamiltonian systems minimization argument non-degeneracy

Citation

Xiao, Huafeng; Long, Yuhua; Shi, Haiping. Heteroclinic Orbits for Discrete Hamiltonian Systems. Commun. Math. Anal. 9 (2010), no. 1, 1--14. https://projecteuclid.org/euclid.cma/1271890714


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References

  • C. D. Ahlbrandt, Equivalence of discrete Euler equations and discrete Hamiltonian systems. J. Math. Anal. Appl. 180 (1993), pp 498-517.
  • M. Bohner, Linear Hamiltonian difference systems: disconjugacy and Jacobi-type conditions. J. Math. Anal. Appl. 199 (1996), pp 804-826.
  • P. Chen and H. Fang, Existence of periodic and subharmonic solution for second-order p-Laplacian difference equations. Adva. Diff. Equa. 2007 (2007), pp 1-9.
  • S. Z. Chen, Disconjugacy, disfocality, and oscillation of second order difference equations. J. Diff. Equat. 107 (1994), pp 383-394.
  • X. Q. Deng, Periodic Solutions for Subquadratic Discrete Hamiltonian Systems. Adva. Diff. Equa. 2007 (2007), pp 1-16.
  • X. Q. Deng and G. Cheng, Homoclinic Orbits for Second Order Discrete Hamiltonian Systems with Potential Changing Sign. Acta Appl. Math. 103 (2008), pp 301-314.
  • L. H. Erbe and P. X. Yan, Disconjugacy for linear Hamiltonian difference systems. J. Math. Anal. Appl. 167 (1992), pp 355-367.
  • Z. M. Guo and J. S. Yu, Existence of periodic and subharmonic solutions for second-order superlinear difference equations. Sci.China (Ser A) 46 (2003), pp 506-515.
  • Z. M. Guo and J. S. Yu, Periodic and subharmonic solutions for superquadratic discrete Hamiltonian systems.Nonlinear Anal. 55 (2003), pp 969-983.
  • Z. M. Guo and J. S. Yu, The existence of periodic and subharmonic solutions for solutions of subquadratic second order difference equations. J. London Math. Soc. 68 (2003), pp 419-430.
  • Z. M. Guo and J. S. Yu, Multiplicity results for periodic solutions to second-order difference euqations. J. Dyna. Diff. Equa. 18 (2006), pp 943-960.
  • P. Hartman, Difference equations: disconjugacy, principal solutions, Green!-s functions complete monotonicity. Trans. Am. Math. Soc. 246 (1978), pp 1-30.
  • Y. H. Long, Applications of Clark duality to periodic solutions with minimal period for discrete Hamiltonian systems. J. Math. Anal. Appl. 342 (2008), pp 726-741.
  • M. J. Ma and Z. M. Guo, Homoclinic orbits for second order self-adjoint difference equations. J. Math. Anal. Appl. 323 (2006), pp 513-521.
  • M. J. Ma and Z. M. Guo, Homoclinic orbits and subharmonics for nonlinear second order difference equations. Nonlinear Anal. 67 (2007), pp 1737-1745.
  • R. H. Rabinowitz, Periodic and heteroclinic orbits for a periodic Hamiltonian system. Nonlinear Anal. 6 (1989), pp 331-346.
  • H. P. Shi, Homoclinic orbits of nonlinear functional difference equations. Acta. Appl. Math. 106 (2009), pp 135-147.
  • H. F. Xiao and J. S. Yu, Existence of Heteroclinic Orbits for Discrete Pendulum Equation. J. Diff. Equat. Appl. Accept.
  • Y. F. Xue and C. L. Tang, Existence of a periodic solution for subquadratic second-order discrete Hamiltonian system. Nonlinear Anal. 67 (2007), pp 2072-2080.
  • Y. F. Xue and C. L. Tang, Multiple periodic solutions for superquadratic second-order discrete Hamiltonian systems. Appl. Math. Comp. 196 (2008), pp 494-500.
  • J. S. Yu, H. H. Bin and Z. M. Guo,Multiple periodic solutions for discrete Hamiltonian systems. Nonlinear Anal. 66 (2007), pp 1498-1512.
  • J. S. Yu, X. Q. Deng and Z. M. Guo, Periodic solutions of a discrete Hamiltonian system with a change of sign in the potential. J. Math. Anal. Appl. 324 (2006), pp 1140-1151.
  • J. S. Yu and Z. M. Guo, Boundary value problems of discrete generalized Emden-Foeler equation. Sci. China Ser. A 49 (2006), pp 1303-1314.
  • J. S. Yu and Z. M. Guo, On boundary value problems for a discrete generalized Emden-Fowler equation. SJ. Diff. Equat. 231 (2006), pp 18-31.
  • J. S. Yu, Y. H. Long and Z. M. Guo, Subharmonic solutions with prescribed minimal period of a class of a discrete forced pendulum equation. J. Dyna. diff. Equat. 16 (2004), pp 575-586.
  • J. S. Yu, H. P. Shi and Z. M. Guo, Homoclinic orbits for nonlinear difference equations containing both advance and retardation. J. Math. Anal. Appl. 352 (2009), pp 799-806.
  • J. S. Yu, B. S. Zhu and Z. M. Guo, Positive solutions for multiparameter semipositone discrete boundary value problems via variational method. Adv. Difference Equ. 2008 (2008), pp 1-15.
  • B. G. Zhang and G. D. Chen, Oscillation of certain second order nonlinear difference equations. J. Math. Anal. Appl. 199 (1996), pp 827-841.
  • B. Zheng, Multiple periodic solutions to nonlinear discrete Hamiltonian systems, Adva. Diff. Equa. 2007 (2007), pp 1-13.
  • Z. Zhou, J. S. Yu and Z. M. Guo, The existence of periodic and subharmonic solutions to subquadratic discrete Hamiltonian systems. The Australian & Zealand Industrial and Applied Mathematics Journal 47 (2005), pp 89-102.