Homoclinic orbits for a class of discrete periodic Hamiltonian systems

Zhang, Qinqin

doi:10.1090/S0002-9939-2015-12107-7

Homoclinic orbits for a class of discrete periodic Hamiltonian systems
HTML articles powered by AMS MathViewer

by Qinqin Zhang

Proc. Amer. Math. Soc. 143 (2015), 3155-3163

DOI: https://doi.org/10.1090/S0002-9939-2015-12107-7

Published electronically: March 18, 2015

PDF | Request permission

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

Ravi P. Agarwal, Difference equations and inequalities, 2nd ed., Monographs and Textbooks in Pure and Applied Mathematics, vol. 228, Marcel Dekker, Inc., New York, 2000. Theory, methods, and applications. MR 1740241
Calvin D. Ahlbrandt and Allan C. Peterson, Discrete Hamiltonian systems, Kluwer Texts in the Mathematical Sciences, vol. 16, Kluwer Academic Publishers Group, Dordrecht, 1996. Difference equations, continued fractions, and Riccati equations. MR 1423802, DOI 10.1007/978-1-4757-2467-7
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
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, DOI 10.1007/s10440-008-9237-z
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, DOI 10.1016/j.camwa.2009.06.045
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, DOI 10.1016/0362-546X(94)00229-B
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, DOI 10.1016/j.jmaa.2005.10.049
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, DOI 10.1016/j.na.2006.08.014
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, DOI 10.1016/j.jmaa.2010.06.008
W. Omana and M. Willem, Homoclinic orbits for a class of Hamiltonian systems, Differential Integral Equations 5 (1992), no. 5, 1115–1120. MR 1171983
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.
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, DOI 10.1080/10236190902791635
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, DOI 10.1080/10236191003730514
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, DOI 10.1007/s10114-012-9233-0
František Havelka, Über Beziehungen der darstellenden Geometrie und Nomographie zu einander, Sb. Prací Přírodověd. Fak. Univ. Palackého v Olomouci 37 (1972), 51–56 (German, with Czech summary). MR 0349055
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, DOI 10.1016/j.na.2008.11.027
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, DOI 10.1016/j.jmaa.2008.11.043
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, DOI 10.1007/s11425-010-4101-9