×

Homoclinic orbits for a class of noncoercive discrete Hamiltonian systems. (English) Zbl 1283.37038

Summary: A class of first-order noncoercive discrete Hamiltonian systems are considered. Based on a generalized mountain pass theorem, some existence results of homoclinic orbits are obtained when the discrete Hamiltonian system is not periodic and do not necessarily satisfy the global Ambrosetti-Rabinowitz condition.

MSC:

37C29 Homoclinic and heteroclinic orbits for dynamical systems
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] H. Poincar, Les Mthodes Nouvelles De La Mcanique Cleste, Gauthier-Villars, Paris, France, 1899.
[2] Y. Ding and M. Girardi, “Infinitely many homoclinic orbits of a Hamiltonian system with symmetry,” Nonlinear Analysis A, vol. 38, no. 3, pp. 391-415, 1999. · Zbl 0938.37034
[3] 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. · Zbl 0702.34039
[4] M. Ma and Z. Guo, “Homoclinic orbits for second order self-adjoint difference equations,” Journal of Mathematical Analysis and Applications, vol. 323, no. 1, pp. 513-521, 2006. · Zbl 1107.39022
[5] J. Moser, Stable and Random Motions in Dynamical Systems, Princeton University Press, Princeton, NJ, USA, 1973. · Zbl 0271.70009
[6] 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. · Zbl 0759.58018
[7] A. Szulkin and W. Zou, “Homoclinic orbits for asymptotically linear Hamiltonian systems,” Journal of Functional Analysis, vol. 187, no. 1, pp. 25-41, 2001. · Zbl 0984.37072
[8] R. P. Agarwal, Difference Equations and Inequalities: Theory, Methods and Applications, Marcel Dekker, New York, NY, USA, 1992. · Zbl 0925.39001
[9] S. N. Elaydi, An Introduction to Difference Equations, Springer, New York, NY, USA, 1996. · Zbl 0840.39002
[10] W. G. Kelley and A. C. Peterson, Difference Equations: An Introduction with Applications, Academic Press, New York, NY, USA, 1991. · Zbl 0733.39001
[11] V. L. Kocić and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic, Dodrecht, The Netherlands, 1993. · Zbl 0787.39001
[12] R. E. Mickens, Difference Equations: Theory and Application, Van Nostrand Reinhold, New York, NY, USA, 1990. · Zbl 0949.39500
[13] A. N. Sharkovsky, Y. L. Maĭstrenko, and E. Y. Romanenko, Difference Equations and Their Applications, Kluwer Academic, Dodrecht, The Netherlands, 1993.
[14] Y. H. Long, “Multiplicity results for periodic solutions with prescribed minimal periods to discrete Hamiltonian systems,” Journal of Difference Equations and Applications, pp. 1-20, 2010.
[15] Z. Guo and J. Yu, “The existence of periodic and subharmonic solutions of subquadratic second order difference equations,” Journal of the London Mathematical Society, vol. 68, no. 2, pp. 419-430, 2003. · Zbl 1046.39005
[16] Z. Zhou, J. Yu, and Z. Guo, “The existence of periodic and subharmonic solutions to subquadratic discrete Hamiltonian systems,” The ANZIAM Journal, vol. 47, no. 1, pp. 89-102, 2005. · Zbl 1081.39019
[17] J. S. Yu, H. P. Shi, and Z. M. Guo, “Homoclinic orbits for nonlinear difference equations containing both advance and retardation,” Journal of Mathematical Analysis and Applications, vol. 352, no. 2, pp. 799-806, 2009. · Zbl 1160.39311
[18] X. Deng and G. Cheng, “Homoclinic orbits for second order discrete Hamiltonian systems with potential changing sign,” Acta Applicandae Mathematicae, vol. 103, no. 3, pp. 301-314, 2008. · Zbl 1153.39012
[19] Y. H. Long, “Homoclinic solutions of some second-order nonperiodic discrete systems,” Advances in Difference Equations, vol. 64, 12 pages, 2011. · Zbl 1273.39008
[20] P. Rabinowitz, “Minimax methods in critical point theory with applications to differential equations,” in Conference Board of the Mathematical Sciences (CBMS ’89), vol. 65 of Regional Conference Series in Mathematics, 1986. · Zbl 0609.58002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.