Abstract and Applied Analysis

Subharmonics with Minimal Periods for Convex Discrete Hamiltonian Systems

Honghua Bin

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Abstract

We consider the subharmonics with minimal periods for convex discrete Hamiltonian systems. By using variational methods and dual functional, we obtain that the system has a p T -periodic solution for each positive integer p , and solution of system has minimal period p T as H subquadratic growth both at 0 and infinity.

Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 508247, 9 pages.

Dates
First available in Project Euclid: 26 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393450360

Digital Object Identifier
doi:10.1155/2013/508247

Mathematical Reviews number (MathSciNet)
MR3039168

Zentralblatt MATH identifier
1383.37051

Citation

Bin, Honghua. Subharmonics with Minimal Periods for Convex Discrete Hamiltonian Systems. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 508247, 9 pages. doi:10.1155/2013/508247. https://projecteuclid.org/euclid.aaa/1393450360


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References

  • Y.-T. Xu, “Subharmonic solutions for convex nonautonomous Hamiltonian systems,” Nonlinear Analysis. Theory, Methods & Applications, vol. 28, no. 8, pp. 1359–1371, 1997.
  • J. Q. Liu and Z. Q. Wang, “Remarks on subharmonics with minimal periods of Hamiltonian systems,” Nonlinear Analysis. Theory, Methods & Applications, vol. 20, no. 7, pp. 803–821, 1993.
  • R. Michalek and G. Tarantello, “Subharmonic solutions with prescribed minimal period for nonautonomous Hamiltonian systems,” Journal of Differential Equations, vol. 72, no. 1, pp. 28–55, 1988.
  • P. H. Rabinowitz, “Minimax methods on critical point theory with applications to differentiable equations,” in Proceedings of the CBMS, vol. 65, American Mathematical Society, Providence, RI, USA, 1986.
  • Y.-T. Xu and Z.-M. Guo, “Applications of a ${Z}_{p}$ index theory to periodic solutions for a class of functional differential equations,” Journal of Mathematical Analysis and Applications, vol. 257, no. 1, pp. 189–205, 2001.
  • Z. Guo and J. Yu, “Existence of periodic and subharmonic solutions for second-order superlinear difference equations,” Science in China A, vol. 46, no. 4, pp. 506–515, 2003.
  • Z. Guo and J. Yu, “The existence of periodic and subharmonic solutions for superquadratic discrete Hamiltonian systems,” Nonlinear Analysis. Theory, Methods & Applications, vol. 55, no. 7-8, pp. 969–983, 2003.
  • J. Yu, H. Bin, and Z. Guo, “Periodic solutions for discrete convex Hamiltonian systems via Clarke duality,” Discrete and Continuous Dynamical Systems A, vol. 15, no. 3, pp. 939–950, 2006.
  • F. H. Clarke, “A classical variational principle for periodic Hamiltonian trajectories,” Proceedings of the American Mathematical Society, vol. 76, no. 1, pp. 186–188, 1979.
  • F. H. Clarke, “Periodic solutions of Hamilton's equations and local minima of the dual action,” Transactions of the American Mathematical Society, vol. 287, no. 1, pp. 239–251, 1985.
  • F. H. Clarke and I. Ekeland, “Nonlinear oscillations and boundary value problems for Hamiltonian systems,” Archive for Rational Mechanics and Analysis, vol. 78, no. 4, pp. 315–333, 1982.
  • J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, vol. 74, Springer, New York, NY, USA, 1989.
  • A. Ambrosetti and G. Mancini, “Solutions of minimal period for a class of convex Hamiltonian systems,” Mathematische Annalen, vol. 255, no. 3, pp. 405–421, 1981.
  • G. Cordaro, “Existence and location of periodic solutions to convex and non coercive Hamiltonian systems,” Discrete and Continuous Dynamical Systems A, vol. 12, no. 5, pp. 983–996, 2005.
  • F. M. Atici and A. Cabada, “Existence and uniqueness results for discrete second-order periodic boundary value problems,” Computers & Mathematics with Applications, vol. 45, no. 6-9, pp. 1417–1427, 2003.
  • F. M. Atici and G. S. Guseinov, “Positive periodic solutions for nonlinear difference equations with periodic coefficients,” Journal of Mathematical Analysis and Applications, vol. 232, no. 1, pp. 166–182, 1999.
  • Z. Huang, C. Feng, and S. Mohamad, “Multistability analysis for a general class of delayed Cohen-Grossberg neural networks,” Information Sciences, vol. 187, pp. 233–244, 2012.
  • Z. Huang and Y. N. Raffoul, “Biperiodicity in neutral-type delayed difference neural networks,” Advances in Difference Equations, vol. 2012, article 5, 2012.
  • D. Jiang and R. P. Agarwal, “Existence of positive periodic solutions for a class of difference equations with several deviating arguments,” Computers & Mathematics with Applications, vol. 45, no. 6-9, pp. 1303–1309, 2003.
  • R. P. Agarwal, Difference Equations and Inequalities: Theory, Methods, and Applications, vol. 228, Marcel Dekker Inc., New York, NY, USA, 2nd edition, 2000.
  • Y. T. Xu and Z. M. Guo, “Applications of a geometrical index theory to functional differential equations,” Acta Mathematica Sinica, vol. 44, no. 6, pp. 1027–1036, 2001.