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Nonexistence of homoclinic solutions for a class of discrete Hamiltonian systems. (English) Zbl 1275.39001

Summary: We give several sufficient conditions under which the first-order nonlinear discrete Hamiltonian system \(\Delta x(n) = \alpha(n)x(n + 1) + \beta(n)|y(n)|^{\mu - 2}y(n)\), \(\Delta y(n) = -\gamma(n)|x(n + 1)|^{\nu - 2}x(n + 1) - \alpha(n)y(n)\) has no solution \((x(n), y(n))\) satisfying the condition \(0 < \sum^{+\infty}_{n=-\infty}[|x(n)|^\nu + (1 + \beta(n))|y(n)|^\mu] < +\infty\), where \(\mu, \nu > 1\) and \(1/\mu + 1/\nu = 1\) and \(\alpha(n)\), \(\beta(n)\) and \(\gamma(n)\) are real-valued functions defined on \(\mathbb Z\).

MSC:

39A12 Discrete version of topics in analysis
37C29 Homoclinic and heteroclinic orbits for dynamical systems
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
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[1] Lyapunov, A. M., Probléme général de la stabilité du mouvement, Annde La Faculté, 2, 9, 203-474 (1907) · JFM 38.0738.07
[2] Bohner, M.; Clark, S.; Ridenhour, J., Lyapunov inequalities for time scales, Journal of Inequalities and Applications, 7, 1, 61-77 (2002) · Zbl 1088.34503 · doi:10.1155/S102558340200005X
[3] Cheng, S. S., A discrete analogue of the inequality of Lyapunov, Hokkaido Mathematical Journal, 12, 1, 105-112 (1983) · Zbl 0535.39002
[4] Cheng, S.-S., Lyapunov inequalities for differential and difference equations, Polytechnica Posnaniensis, 23, 25-41 (1991) · Zbl 0753.34017
[5] Clark, S.; Hinton, D., A Liapunov inequality for linear Hamiltonian systems, Mathematical Inequalities & Applications, 1, 2, 201-209 (1998) · Zbl 0909.34033 · doi:10.7153/mia-01-18
[6] Clark, S.; Hinton, D., Discrete Lyapunov inequalities, Dynamic Systems and Applications, 8, 3-4, 369-380 (1999) · Zbl 0940.39013
[7] Guseinov, G. Sh.; Kaymakçalan, B., Lyapunov inequalities for discrete linear Hamiltonian systems, Computers & Mathematics with Applications, 45, 6-9, 1399-1416 (2003) · Zbl 1055.39029 · doi:10.1016/S0898-1221(03)00095-6
[8] Guseinov, G. Sh.; Zafer, A., Stability criteria for linear periodic impulsive Hamiltonian systems, Journal of Mathematical Analysis and Applications, 335, 2, 1195-1206 (2007) · Zbl 1128.34005 · doi:10.1016/j.jmaa.2007.01.095
[9] Hartman, P., Difference equations: disconjugacy, principal solutions, Green’s functions, complete monotonicity, Transactions of the American Mathematical Society, 246, 1-30 (1978) · Zbl 0409.39001 · doi:10.2307/1997963
[10] He, X.; Zhang, Q.-M., A discrete analogue of Lyapunov-type inequalities for nonlinear difference systems, Computers & Mathematics with Applications, 62, 2, 677-684 (2011) · Zbl 1298.39005 · doi:10.1016/j.camwa.2011.05.049
[11] Jiang, L.; Zhou, Z., Lyapunov inequality for linear Hamiltonian systems on time scales, Journal of Mathematical Analysis and Applications, 310, 2, 579-593 (2005) · Zbl 1076.37053 · doi:10.1016/j.jmaa.2005.02.026
[12] Lin, S. H.; Yang, G. S., On discrete analogue of Lyapunov inequality, Tamkang Journal of Mathematics, 20, 2, 169-186 (1989) · Zbl 0697.39004
[13] Wang, X., Stability criteria for linear periodic Hamiltonian systems, Journal of Mathematical Analysis and Applications, 367, 1, 329-336 (2010) · Zbl 1195.34079 · doi:10.1016/j.jmaa.2010.01.027
[14] Tang, X.-H.; Zhang, M., Lyapunov inequalities and stability for linear Hamiltonian systems, Journal of Differential Equations, 252, 1, 358-381 (2012) · Zbl 1242.37039 · doi:10.1016/j.jde.2011.08.002
[15] Tang, X. H.; Zhang, Q.-M.; Zhang, M., Lyapunov-type inequalities for the first-order nonlinear Hamiltonian systems, Computers & Mathematics with Applications, 62, 9, 3603-3613 (2011) · Zbl 1236.34040 · doi:10.1016/j.camwa.2011.09.011
[16] Tiryaki, A.; Ünal, M.; Çakmak, D., Lyapunov-type inequalities for nonlinear systems, Journal of Mathematical Analysis and Applications, 332, 1, 497-511 (2007) · Zbl 1123.34037 · doi:10.1016/j.jmaa.2006.10.010
[17] Ünal, M.; Çakmak, D.; Tiryaki, A., A discrete analogue of Lyapunov-type inequalities for nonlinear systems, Computers & Mathematics with Applications, 55, 11, 2631-2642 (2008) · Zbl 1142.39309 · doi:10.1016/j.camwa.2007.10.014
[18] Ünal, M.; Çakmak, D., Lyapunov-type inequalities for certain nonlinear systems on time scales, Turkish Journal of Mathematics, 32, 3, 255-275 (2008) · Zbl 1166.34005
[19] Zhang, Q.-M.; Tang, X. H., Lyapunov inequalities and stability for discrete linear Hamiltonian systems, Applied Mathematics and Computation, 218, 2, 574-582 (2011) · Zbl 1229.39026 · doi:10.1016/j.amc.2011.05.101
[20] Zhang, Q.-M.; Tang, X. H., Lyapunov inequalities and stability for discrete linear Hamiltonian systems, Journal of Difference Equations and Applications, 18, 9, 1467-1484 (2012) · Zbl 1260.39025 · doi:10.1080/10236198.2011.572071
[21] Agarwal, R.; Ahlbrandt, C.; Bohner, M.; Peterson, A., Discrete linear Hamiltonian systems: a survey, Dynamic Systems and Applications, 8, 3-4, 307-333 (1999) · Zbl 0942.39009
[22] Ahlbrandt, C. D.; Peterson, A. C., Discrete Hamiltonian Systems. Discrete Hamiltonian Systems, Kluwer Texts in the Mathematical Sciences, 16, xiv+374 (1996), Dordrecht, The Netherlands: Kluwer Academic, Dordrecht, The Netherlands · Zbl 0860.39001
[23] Elaydi, S. N., An Introduction to Difference Equations (2004), New York, NY, USA: Springer, New York, NY, USA
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