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

Yuhua, Long

J. Appl. Math. 2012, Article ID 720139, 21 p. (2012).

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)

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