APPLICATION OF PSEUDO $Z_p$ INDEX THEORY TO PERIODIC SOLUTIONS WITH MINIMAL PERIOD FOR DISCRETE HAMILTONIAN SYSTEMS

  • Yuhua, Long (School of Mathematics and Information Science, Guangzhou University)
  • Received : 2009.10.24
  • Accepted : 2009.12.05
  • Published : 2010.05.30

Abstract

By making use of minimax theory and pseudo $Z_p$ index theory, some results on the existence and multiplicity of periodic solutions with minimal period to nonconvex superquadratic discrete Hamiltonian systems are obtained.

Keywords

References

  1. Birkhoff G D, Lewis D C, On the periodic motion near a given periodic motion of a dynamical system, Annals of Mathematics on Pure Applications, 12 (1933), 117-133.
  2. Morse J, Proof of generalized fixed point theorem due to G. D. Birkhoff, Lecture Notes in Mathematics, Springer-verlag, New York, 1977.
  3. Meyer K R, Hall G R, Intoduction to Hamiltonian Dynamical Systems and the N-boby Proble, Springer-verlag, New YOrk, 1992.
  4. Arnold V A, Mathematical Methods of Classical Mechanics, Springer-verlag, New YOrk, 1983.
  5. Rabinowitz P. H., Varianttional methods for nonlinear eigenvalue problems, in Eigenvalues of Nonlinear Problems (G. Prodi, Ed.), Edizioni Cremonese, Rome, 1974.
  6. Long Y, Index Theory for Symplectic Paths with Applications, Birkhauser, Boston, 2002.
  7. Benci V, Rabinowitz P H, Critical point theorems for indefinite functionals, Inventions Mathematica, 53 (1979), 241-273.
  8. Ekeland I, Hofer H, Period solution with prescribed period for convex Hamiltonian systems, Inventions Mathematicae, 81 (1985), 155-188. https://doi.org/10.1007/BF01388776
  9. Ambrosetti A, Mancini G, Solutions of minimal period for a class of convex Hamiltonian systems, Mathematica Annalen, 255 (1981), 405-421. https://doi.org/10.1007/BF01450713
  10. Chang K C, Infinite Dimensional Morse Theory and Multiple Solution Problems, Birkhauser, Boston, 1993.
  11. Rabinowitz P H, Periodic solution of Hamiltonian systems, Communication on Pure and Applied Mathematics, 40 (1978), 157-184.
  12. Rabinowitz P H, On subharmonic solution of Hamiltonian systems, Communication on Pure and Applied Mathematics, 33 1980, 609-633. https://doi.org/10.1002/cpa.3160330504
  13. Agarwal R. P, Difference Equations and Inequalities: Theory, Methods and Applications, Marcel Dekker, New York, 2000.
  14. Kelley W. G, Peterson A. C, Difference Equations: An Introduction with Applications, Academic Press, New York, 1991.
  15. Chang K. C., Critical point theory and its applications, Science and technical Press, Shanghai, China, 1980.
  16. Michalek R. and Tarantello G., Subharmonics solutions with prescriebed minimal period for nonautonomous Hamiltonian systems, J. Diff. Eqns., 72 (1988), 28-55. https://doi.org/10.1016/0022-0396(88)90148-9
  17. Liu J. Q. and Wang Z. Q., Remarks on subharmonics with minimal periods of Hamiltonian systems, Nonlinear Anal. T. M. A., 20 (1993), 803-821. https://doi.org/10.1016/0362-546X(93)90070-9
  18. Guo Z. M. and Yu J. S., The existence of the periodic and subharmonic solutions for superquadratic discrete Hamiltonian systems, Nonlinear Anal. T. M. Appl., 55 (2003), 969-983. https://doi.org/10.1016/j.na.2003.07.019
  19. Zhou Z., Yu J. S. and Guo Z. M., The existence of the periodic and subharmonics solutions to subquadratic discrete Hamiltonian systems, ANZIAM J. Australian Math. Soci., 47 (2005), 89-102. https://doi.org/10.1017/S1446181100009792
  20. Yu J. S., Bin H. H. and Guo Z. M., Periodic solutions for discrete convex Hamiltonian systems via Clark duality, Discrete and Continuous Dyna. Syst., 15 (2006), 939-950.