Abstract and Applied Analysis

Nonexistence of Homoclinic Solutions for a Class of Discrete Hamiltonian Systems

Xiaoping Wang

Full-text: Open access

Abstract

We give several sufficient conditions under which the first-order nonlinear discrete Hamiltonian system Δ x ( n ) = α ( n ) x ( n + 1 ) + β ( n ) | y ( n ) | μ - 2 y ( n ) , Δ y ( n ) = - γ ( n ) | x ( n + 1 ) | ν - 2 x ( n + 1 ) - α ( n ) y ( n ) has no solution ( x ( n ) , y ( n ) ) satisfying condition 0 < n = - + [ | x ( n ) | ν + ( 1 + β ( n ) ) | y ( n ) | μ ] < + , where μ , ν > 1 and 1 / μ + 1 / ν = 1 and α ( n ) , β ( n ), and γ ( n ) are real-valued functions defined on .

Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 398681, 6 pages.

Dates
First available in Project Euclid: 27 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393511795

Digital Object Identifier
doi:10.1155/2013/398681

Mathematical Reviews number (MathSciNet)
MR3035382

Zentralblatt MATH identifier
1275.39001

Citation

Wang, Xiaoping. Nonexistence of Homoclinic Solutions for a Class of Discrete Hamiltonian Systems. Abstr. Appl. Anal. 2013 (2013), Article ID 398681, 6 pages. doi:10.1155/2013/398681. https://projecteuclid.org/euclid.aaa/1393511795


Export citation

References

  • A. M. Lyapunov, “Probléme général de la stabilité du mouvement,” Annde La Faculté, vol. 2, no. 9, pp. 203–474, 1907.
  • M. Bohner, S. Clark, and J. Ridenhour, “Lyapunov inequalities for time scales,” Journal of Inequalities and Applications, vol. 7, no. 1, pp. 61–77, 2002.
  • S. S. Cheng, “A discrete analogue of the inequality of Lyapunov,” Hokkaido Mathematical Journal, vol. 12, no. 1, pp. 105–112, 1983.
  • S.-S. Cheng, “Lyapunov inequalities for differential and difference equations,” Polytechnica Posnaniensis, no. 23, pp. 25–-41, 1991.
  • S. Clark and D. Hinton, “A Liapunov inequality for linear Hamiltonian systems,” Mathematical Inequalities & Applications, vol. 1, no. 2, pp. 201–209, 1998.
  • S. Clark and D. Hinton, “Discrete Lyapunov inequalities,” Dynamic Systems and Applications, vol. 8, no. 3-4, pp. 369–380, 1999.
  • G. Sh. Guseinov and B. Kaymakçalan, “Lyapunov inequalities for discrete linear Hamiltonian systems,” Computers & Mathematics with Applications, vol. 45, no. 6–9, pp. 1399–1416, 2003.
  • G. Sh. Guseinov and A. Zafer, “Stability criteria for linear periodic impulsive Hamiltonian systems,” Journal of Mathematical Analysis and Applications, vol. 335, no. 2, pp. 1195–1206, 2007.
  • P. Hartman, “Difference equations: disconjugacy, principal solutions, Green's functions, complete monotonicity,” Transactions of the American Mathematical Society, vol. 246, pp. 1–30, 1978.
  • X. He and Q.-M. Zhang, “A discrete analogue of Lyapunov-type inequalities for nonlinear difference systems,” Computers & Mathematics with Applications, vol. 62, no. 2, pp. 677–684, 2011.
  • L. Jiang and Z. Zhou, “Lyapunov inequality for linear Hamiltonian systems on time scales,” Journal of Mathematical Analysis and Applications, vol. 310, no. 2, pp. 579–593, 2005.
  • S. H. Lin and G. S. Yang, “On discrete analogue of Lyapunov inequality,” Tamkang Journal of Mathematics, vol. 20, no. 2, pp. 169–186, 1989.
  • X. Wang, “Stability criteria for linear periodic Hamiltonian systems,” Journal of Mathematical Analysis and Applications, vol. 367, no. 1, pp. 329–336, 2010.
  • X.-H. Tang and M. Zhang, “Lyapunov inequalities and stability for linear Hamiltonian systems,” Journal of Differential Equations, vol. 252, no. 1, pp. 358–381, 2012.
  • X. H. Tang, Q.-M. Zhang, and M. Zhang, “Lyapunov-type inequalities for the first-order nonlinear Hamiltonian systems,” Computers & Mathematics with Applications, vol. 62, no. 9, pp. 3603–3613, 2011.
  • A. Tiryaki, M. Ünal, and D. Çakmak, “Lyapunov-type inequalities for nonlinear systems,” Journal of Mathematical Analysis and Applications, vol. 332, no. 1, pp. 497–511, 2007.
  • M. Ünal, D. Çakmak, and A. Tiryaki, “A discrete analogue of Lyapunov-type inequalities for nonlinear systems,” Computers & Mathematics with Applications, vol. 55, no. 11, pp. 2631–2642, 2008.
  • M. Ünal and D. Çakmak, “Lyapunov-type inequalities for certain nonlinear systems on time scales,” Turkish Journal of Mathematics, vol. 32, no. 3, pp. 255–275, 2008.
  • Q.-M. Zhang and X. H. Tang, “Lyapunov inequalities and stability for discrete linear Hamiltonian systems,” Applied Mathematics and Computation, vol. 218, no. 2, pp. 574–582, 2011.
  • Q.-M. Zhang and X. H. Tang, “Lyapunov inequalities and stability for discrete linear Hamiltonian systems,” Journal of Difference Equations and Applications, vol. 18, no. 9, pp. 1467–1484, 2012.
  • R. Agarwal, C. Ahlbrandt, M. Bohner, and A. Peterson, “Discrete linear Hamiltonian systems: a survey,” Dynamic Systems and Applications, vol. 8, no. 3-4, pp. 307–333, 1999.
  • C. D. Ahlbrandt and A. C. Peterson, Discrete Hamiltonian Systems, vol. 16 of Kluwer Texts in the Mathematical Sciences, Kluwer Academic, Dordrecht, The Netherlands, 1996.
  • S. N. Elaydi, An Introduction to Difference Equations, Springer, New York, NY, USA, 3rd edition, 2004.