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Homoclinic orbits for discrete Hamiltonian systems with indefinite linear part
1. | College of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510006, China |
References:
[1] |
R. P. Agarwal, Difference Equations and Inequalities: Theory, Method and Applications, second edition, Monographs and Textbooks in Pure and Applied Mathematics, 228, Marcel Dekker, Inc., 2000. |
[2] |
C. D. Ahlbrandt and A. C. Peterson, Discrete Hamiltonian Systems: Difference Equations, Continued Fraction and Riccati Equations, Kluwer Texts in the Mathematical Sciences, 16, Kluwer Academic, Dordrecht, 1996.
doi: 10.1007/978-1-4757-2467-7. |
[3] |
C. J. Batkam, Homoclinic orbits of first-order superquadratic Hamiltonian systems, Discrete Contin. Dyn. Syst., 34 (2014), 3353-3369.
doi: 10.3934/dcds.2014.34.3353. |
[4] |
J. Cruz-Sampedro, Boundary values of the resolvent of Schrödinger Hamiltonian with potentials of order zero, Discrete Contin. Dyn. Syst., 33 (2013), 1061-1076.
doi: 10.3934/dcds.2013.33.1061. |
[5] |
X. Q. Deng and G. Cheng, Homoclinic orbits for second order discrete Hamiltonian systems with potential changing sign, Acta Appl. Math., 103 (2008), 301-314.
doi: 10.1007/s10440-008-9237-z. |
[6] |
D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford, 1987. |
[7] |
M. Izydorek and J. Janczewska, Homoclinic solutions for a class of second order Hamiltonian systems, J. Differential Equations, 219 (2005), 375-389.
doi: 10.1016/j.jde.2005.06.029. |
[8] |
G. B. Li and A. Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763-776.
doi: 10.1142/S0219199702000853. |
[9] |
X. Y. Lin and X. H. Tang, Existence of infinitely many homoclinic orbits in discrete Hamiltonian systems, J. Math. Anal. Appl., 373 (2011), 59-72.
doi: 10.1016/j.jmaa.2010.06.008. |
[10] |
X. Y. Lin and X. H. Tang, Homoclinic orbits for discrete Hamiltonian systems with subquadratic potential, Advances in Difference Equations, 154 (2013), http://www. advancesindifferenceequations.com/content/2013/1/154. |
[11] |
M. Ma and Z. M. Guo, Homoclinic orbits and subharmonics for nonlinear second order difference equations, Nonlinear Anal., 67 (2007), 1737-1745.
doi: 10.1016/j.na.2006.08.014. |
[12] |
W. Omana and M. Willem, Homoclinic orbits for a class of Hamiltonian systems, Differential Integral Equations, 5 (1992), 1115-1120. |
[13] |
P. H. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems, Proc, Roy. Soc. Edinburgh Sect. A, 114 (1990), 33-38.
doi: 10.1017/S0308210500024240. |
[14] |
P. H. Rabinowitz and K. Tanaka, Some results on connecting orbits for a class of Hamiltonian systems, Math. Z., 206 (1991), 473-499.
doi: 10.1007/BF02571356. |
[15] |
J. Sun, J. Chu and Z. Feng, Homoclinic orbits for first order periodic Hamiltonian systems with spectrum zero, Discrete Contin. Dyn. Syst., 33 (2013), 3807-3824.
doi: 10.3934/dcds.2013.33.3807. |
[16] |
X. H. Tang and J. Chen, Infinitely many homoclinic orbits for a class of discrete Hamiltonian systems, Advances in Difference Equations, 242 (2013), http://www. advancesindifferenceequations.com/content/2013/1/242.
doi: 10.1186/1687-1847-2013-242. |
[17] |
X. H. Tang and X. Y. Lin, Existence and multiplicity of homoclinic solutions for second-order discrete Hamiltonian systems with subquadratic potential, J. Differ. Equ. Appl., 17 (2011), 1617-1634.
doi: 10.1080/10236191003730514. |
[18] |
X. H. Tang and X. Y. Lin, Infinitely many homoclinic orbits for discrete Hamiltonian systems with subquadratic potential, J. Differ. Equ. Appl., 19 (2013), 796-813.
doi: 10.1080/10236198.2012.691168. |
[19] |
X. H. Tang, Non-Nehari manifold method for asymptotically linear Schrödinger equation, J. Aust. Math. Soc., 98 (2015), 104-116.
doi: 10.1017/S144678871400041X. |
[20] |
J. S. Yu, H. Shi and Z. M. Guo, Homoclinic orbits for nonlinear difference equations containing both advance and retardation, J. Math. Anal. Appl., 352 (2009), 799-806.
doi: 10.1016/j.jmaa.2008.11.043. |
[21] |
Q. Q. Zhang, Homoclinic orbits for a class of discrete periodic Hamiltonian systems, Proc. Amer. Math. Soc., acceped for publication. |
[22] |
V. Coti Zelati, I. Ekeland and E. Sere, A variational approach to homoclinic orbits in Hamiltonian systems, Math. Ann., 288 (1990), 133-160.
doi: 10.1007/BF01444526. |
[23] |
V. Coti Zelati and P. H. Rabinowitz, Homoclinic orbits for second second order Hamiltonian systems possessing superquadratic potentials, J. Amer. Math. Soc., 4 (1991), 693-727.
doi: 10.2307/2939286. |
[24] |
Z. Zhou and J. S. Yu, On the existence of homoclinic solutions of a class of discrete nonlinear periodic systems, J. Differential Equations, 249 (2010), 1199-1212.
doi: 10.1016/j.jde.2010.03.010. |
[25] |
Z. Zhou and J. S. Yu, Homoclinic solutions in periodic nonlinear difference equations with superlinear nonlinearity, Acta Mathematica Sinica, 29 (2013), 1809-1822.
doi: 10.1007/s10114-013-0736-0. |
[26] |
Z. Zhou, J. S. Yu and Y. Chen, Homoclinic solutions in periodic difference equations with saturable nonlinearity, Science China, Mathematics, 54 (2011), 83-93.
doi: 10.1007/s11425-010-4101-9. |
show all references
References:
[1] |
R. P. Agarwal, Difference Equations and Inequalities: Theory, Method and Applications, second edition, Monographs and Textbooks in Pure and Applied Mathematics, 228, Marcel Dekker, Inc., 2000. |
[2] |
C. D. Ahlbrandt and A. C. Peterson, Discrete Hamiltonian Systems: Difference Equations, Continued Fraction and Riccati Equations, Kluwer Texts in the Mathematical Sciences, 16, Kluwer Academic, Dordrecht, 1996.
doi: 10.1007/978-1-4757-2467-7. |
[3] |
C. J. Batkam, Homoclinic orbits of first-order superquadratic Hamiltonian systems, Discrete Contin. Dyn. Syst., 34 (2014), 3353-3369.
doi: 10.3934/dcds.2014.34.3353. |
[4] |
J. Cruz-Sampedro, Boundary values of the resolvent of Schrödinger Hamiltonian with potentials of order zero, Discrete Contin. Dyn. Syst., 33 (2013), 1061-1076.
doi: 10.3934/dcds.2013.33.1061. |
[5] |
X. Q. Deng and G. Cheng, Homoclinic orbits for second order discrete Hamiltonian systems with potential changing sign, Acta Appl. Math., 103 (2008), 301-314.
doi: 10.1007/s10440-008-9237-z. |
[6] |
D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford, 1987. |
[7] |
M. Izydorek and J. Janczewska, Homoclinic solutions for a class of second order Hamiltonian systems, J. Differential Equations, 219 (2005), 375-389.
doi: 10.1016/j.jde.2005.06.029. |
[8] |
G. B. Li and A. Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763-776.
doi: 10.1142/S0219199702000853. |
[9] |
X. Y. Lin and X. H. Tang, Existence of infinitely many homoclinic orbits in discrete Hamiltonian systems, J. Math. Anal. Appl., 373 (2011), 59-72.
doi: 10.1016/j.jmaa.2010.06.008. |
[10] |
X. Y. Lin and X. H. Tang, Homoclinic orbits for discrete Hamiltonian systems with subquadratic potential, Advances in Difference Equations, 154 (2013), http://www. advancesindifferenceequations.com/content/2013/1/154. |
[11] |
M. Ma and Z. M. Guo, Homoclinic orbits and subharmonics for nonlinear second order difference equations, Nonlinear Anal., 67 (2007), 1737-1745.
doi: 10.1016/j.na.2006.08.014. |
[12] |
W. Omana and M. Willem, Homoclinic orbits for a class of Hamiltonian systems, Differential Integral Equations, 5 (1992), 1115-1120. |
[13] |
P. H. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems, Proc, Roy. Soc. Edinburgh Sect. A, 114 (1990), 33-38.
doi: 10.1017/S0308210500024240. |
[14] |
P. H. Rabinowitz and K. Tanaka, Some results on connecting orbits for a class of Hamiltonian systems, Math. Z., 206 (1991), 473-499.
doi: 10.1007/BF02571356. |
[15] |
J. Sun, J. Chu and Z. Feng, Homoclinic orbits for first order periodic Hamiltonian systems with spectrum zero, Discrete Contin. Dyn. Syst., 33 (2013), 3807-3824.
doi: 10.3934/dcds.2013.33.3807. |
[16] |
X. H. Tang and J. Chen, Infinitely many homoclinic orbits for a class of discrete Hamiltonian systems, Advances in Difference Equations, 242 (2013), http://www. advancesindifferenceequations.com/content/2013/1/242.
doi: 10.1186/1687-1847-2013-242. |
[17] |
X. H. Tang and X. Y. Lin, Existence and multiplicity of homoclinic solutions for second-order discrete Hamiltonian systems with subquadratic potential, J. Differ. Equ. Appl., 17 (2011), 1617-1634.
doi: 10.1080/10236191003730514. |
[18] |
X. H. Tang and X. Y. Lin, Infinitely many homoclinic orbits for discrete Hamiltonian systems with subquadratic potential, J. Differ. Equ. Appl., 19 (2013), 796-813.
doi: 10.1080/10236198.2012.691168. |
[19] |
X. H. Tang, Non-Nehari manifold method for asymptotically linear Schrödinger equation, J. Aust. Math. Soc., 98 (2015), 104-116.
doi: 10.1017/S144678871400041X. |
[20] |
J. S. Yu, H. Shi and Z. M. Guo, Homoclinic orbits for nonlinear difference equations containing both advance and retardation, J. Math. Anal. Appl., 352 (2009), 799-806.
doi: 10.1016/j.jmaa.2008.11.043. |
[21] |
Q. Q. Zhang, Homoclinic orbits for a class of discrete periodic Hamiltonian systems, Proc. Amer. Math. Soc., acceped for publication. |
[22] |
V. Coti Zelati, I. Ekeland and E. Sere, A variational approach to homoclinic orbits in Hamiltonian systems, Math. Ann., 288 (1990), 133-160.
doi: 10.1007/BF01444526. |
[23] |
V. Coti Zelati and P. H. Rabinowitz, Homoclinic orbits for second second order Hamiltonian systems possessing superquadratic potentials, J. Amer. Math. Soc., 4 (1991), 693-727.
doi: 10.2307/2939286. |
[24] |
Z. Zhou and J. S. Yu, On the existence of homoclinic solutions of a class of discrete nonlinear periodic systems, J. Differential Equations, 249 (2010), 1199-1212.
doi: 10.1016/j.jde.2010.03.010. |
[25] |
Z. Zhou and J. S. Yu, Homoclinic solutions in periodic nonlinear difference equations with superlinear nonlinearity, Acta Mathematica Sinica, 29 (2013), 1809-1822.
doi: 10.1007/s10114-013-0736-0. |
[26] |
Z. Zhou, J. S. Yu and Y. Chen, Homoclinic solutions in periodic difference equations with saturable nonlinearity, Science China, Mathematics, 54 (2011), 83-93.
doi: 10.1007/s11425-010-4101-9. |
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