Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 
 
 

 

Homoclinic orbits for a class of discrete periodic Hamiltonian systems


Author: Qinqin Zhang
Journal: Proc. Amer. Math. Soc. 143 (2015), 3155-3163
MSC (2010): Primary 58E05; Secondary 70H05
DOI: https://doi.org/10.1090/S0002-9939-2015-12107-7
Published electronically: March 18, 2015
MathSciNet review: 3336639
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we establish new criteria for the existence of nontrivial homoclinic orbits to a class of discrete Hamiltonian systems. Our results do not need to suppose that the system satisfies the well-known global Ambrosetti-Rabinowitz superquadratic assumption.


References [Enhancements On Off] (What's this?)

  • [1] Ravi P. Agarwal, Difference equations and inequalities: Theory, methods, and applications, 2nd ed., Monographs and Textbooks in Pure and Applied Mathematics, vol. 228, Marcel Dekker Inc., New York, 2000. MR 1740241 (2001f:39001)
  • [2] Calvin D. Ahlbrandt and Allan C. Peterson, Discrete Hamiltonian systems: Difference equations, continued fractions, and Riccati equations, Kluwer Texts in the Mathematical Sciences, vol. 16, Kluwer Academic Publishers Group, Dordrecht, 1996. MR 1423802 (98m:39043)
  • [3] Giovanna Cerami, An existence criterion for the critical points on unbounded manifolds, Istit. Lombardo Accad. Sci. Lett. Rend. A 112 (1978), no. 2, 332-336 (1979) (Italian). MR 581298 (81k:58021)
  • [4] Xiaoqing Deng and Gong Cheng, Homoclinic orbits for second order discrete Hamiltonian systems with potential changing sign, Acta Appl. Math. 103 (2008), no. 3, 301-314. MR 2430446 (2009f:39012), https://doi.org/10.1007/s10440-008-9237-z
  • [5] Xiaoqing Deng, Gong Cheng, and Haiping Shi, Subharmonic solutions and homoclinic orbits of second order discrete Hamiltonian systems with potential changing sign, Comput. Math. Appl. 58 (2009), no. 6, 1198-1206. MR 2554352 (2010i:37145), https://doi.org/10.1016/j.camwa.2009.06.045
  • [6] Yan Heng Ding, Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems, Nonlinear Anal. 25 (1995), no. 11, 1095-1113. MR 1350732 (96g:34070), https://doi.org/10.1016/0362-546X(94)00229-B
  • [7] Manjun Ma and Zhiming Guo, Homoclinic orbits for second order self-adjoint difference equations, J. Math. Anal. Appl. 323 (2006), no. 1, 513-521. MR 2262222 (2007f:37093), https://doi.org/10.1016/j.jmaa.2005.10.049
  • [8] Manjun Ma and Zhiming Guo, Homoclinic orbits and subharmonics for nonlinear second order difference equations, Nonlinear Anal. 67 (2007), no. 6, 1737-1745. MR 2326026 (2008f:37133), https://doi.org/10.1016/j.na.2006.08.014
  • [9] Xiaoyan Lin and X. H. Tang, Existence of infinitely many homoclinic orbits in discrete Hamiltonian systems, J. Math. Anal. Appl. 373 (2011), no. 1, 59-72. MR 2684457 (2011i:39019), https://doi.org/10.1016/j.jmaa.2010.06.008
  • [10] W. Omana and M. Willem, Homoclinic orbits for a class of Hamiltonian systems, Differential Integral Equations 5 (1992), no. 5, 1115-1120. MR 1171983 (93f:58031)
  • [11] Z. Q. Ou and C. L. Tang, Existence of homoclinic orbits for the second order Hamiltonian systems, J. Math. Anal. Appl. 291(1) (2004), 203-213.
  • [12] X. H. Tang, Xiaoyan Lin, and Li Xiao, Homoclinic solutions for a class of second order discrete Hamiltonian systems, J. Difference Equ. Appl. 16 (2010), no. 11, 1257-1273. MR 2738948 (2012f:39009), https://doi.org/10.1080/10236190902791635
  • [13] X. H. Tang and Xiaoyan Lin, Existence and multiplicity of homoclinic solutions for second-order discrete Hamiltonian systems with subquadratic potential, J. Difference Equ. Appl. 17 (2011), no. 11, 1617-1634. MR 2846503 (2012h:70032), https://doi.org/10.1080/10236191003730514
  • [14] Xian Hua Tang and Xiao Yan Lin, Homoclinic solutions for a class of second order discrete Hamiltonian systems, Acta Math. Sin. (Engl. Ser.) 28 (2012), no. 3, 609-622. MR 2891289, https://doi.org/10.1007/s10114-012-9233-0
  • [15] X. H. Tang and X. Y. Lin, Infinitely many homoclinic orbits for discrete Hamiltonian systems with subquadratic potential, J. Difference. Equ. Appl. 19 (2013), no. 5, 796-813. DOI:10.1080/10236198.2012.691168. MR 349055
  • [16] X. H. Tang and Li Xiao, Homoclinic solutions for ordinary $ p$-Laplacian systems with a coercive potential, Nonlinear Anal. 71 (2009), no. 3-4, 1124-1132. MR 2527532 (2010h:37144), https://doi.org/10.1016/j.na.2008.11.027
  • [17] Jianshe Yu, Haiping Shi, and Zhiming Guo, Homoclinic orbits for nonlinear difference equations containing both advance and retardation, J. Math. Anal. Appl. 352 (2009), no. 2, 799-806. MR 2501925 (2009m:39025), https://doi.org/10.1016/j.jmaa.2008.11.043
  • [18] Zhan Zhou, JianShe Yu, and YuMing Chen, Homoclinic solutions in periodic difference equations with saturable nonlinearity, Sci. China Math. 54 (2011), no. 1, 83-93. MR 2764787 (2011m:39013), https://doi.org/10.1007/s11425-010-4101-9

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 58E05, 70H05

Retrieve articles in all journals with MSC (2010): 58E05, 70H05


Additional Information

Qinqin Zhang
Affiliation: College of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510006, People’s Republic of China
Email: qinqin.zhang0413@gmail.com

DOI: https://doi.org/10.1090/S0002-9939-2015-12107-7
Received by editor(s): November 18, 2012
Received by editor(s) in revised form: December 4, 2012
Published electronically: March 18, 2015
Additional Notes: This project was supported by the Doctoral Program Foundation of the Ministry of Education of China (20104410110001).
Communicated by: Yingfei Yi
Article copyright: © Copyright 2015 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society