DOI QR코드

DOI QR Code

EXISTENCE OF PERIODIC SOLUTIONS OF A HIGHER ORDER DIFFERENCE SYSTEM

  • Hu, Ronghui (COLLEGE OF MATHEMATICS AND ECONOMETRICS HUNAN UNIVERSITY) ;
  • Huang, Lihong (COLLEGE OF MATHEMATICS AND ECONOMETRICS HUNAN UNIVERSITY)
  • Published : 2008.03.31

Abstract

By using critical point theorem, we study a higher order difference system, and obtain some new sufficient conditions ensuring the existence of periodic solutions for such a system.

Keywords

References

  1. R. P. Agarwal, Difference Equations and Inequalities, Theory, methods, and applications. Monographs and Textbooks in Pure and Applied Mathematics, 155. Marcel Dekker, Inc., New York, 1992
  2. R. P. Agarwal and W. Zhang, Periodic solutions of difference equations with general periodicity, Comput. Math. Appl. 42 (2001), no. 3-5, 719-727 https://doi.org/10.1016/S0898-1221(01)00191-2
  3. Jingliang Chen and Xianghui Chen, The Particular Matrix, Tsinghua University Press, 2000
  4. S. N. Eladi, An Introduction to Difference Equations, Second edition. Undergraduate Texts in Mathematics. Springer-Verlag, New York, 1999
  5. S. N. Elaydi and S. Zhang, Stability and periodicity of difference equations with finite delay, Funkcial. Ekvac. 37 (1994), no. 3, 401-413
  6. L. H. Erbe, H. Xia, and J. S. Yu, Global stability of a linear nonautonomous delay difference equation, J. Differ. Equations Appl. 1 (1995), no. 2, 151-161 https://doi.org/10.1080/10236199508808016
  7. Z. M. Guo and J. S. Yu, Existence of periodic and subharmonic solutions for second-order superlinear difference equations, Sci. China Ser. A 46 (2003), no. 4, 506-515 https://doi.org/10.1007/BF02884022
  8. Z. M. Guo and J. S. Yu, The existence of periodic and subharmonic solutions of subquadratic second order difference equations, J. London Math. Soc. (2) 68 (2003), no. 2, 419-430 https://doi.org/10.1112/S0024610703004563
  9. J. K. Hale and J. Mawhin, Coincidence degree and periodic solutions of neutral equations, J. Differential Equations 15 (1974), 295-307 https://doi.org/10.1016/0022-0396(74)90081-3
  10. J. L. Kaplan and J. A. Yorke, Ordinary differential equations which yield periodic solutions of differential delay equations, J. Math. Anal. Appl. 48 (1974), 317-324 https://doi.org/10.1016/0022-247X(74)90162-0
  11. V. L. Kocic and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Mathematics and its Applications, 256. Kluwer Academic Publishers Group, Dordrecht, 1993
  12. H. Matsunaga, T. Hara, and S. Sakata, Global attractivity for a nonlinear difference equation with variable delay, Comput. Math. Appl. 41 (2001), no. 5-6, 543-551 https://doi.org/10.1016/S0898-1221(00)00297-2
  13. J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences, 74. Springer-Verlag, New York, 1989
  14. P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics, 65. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986
  15. X. H. Tang and J. S. Yu, Oscillation of nonlinear delay difference equations, J. Math. Anal. Appl. 249 (2000), no. 2, 476-490 https://doi.org/10.1006/jmaa.2000.6902
  16. K. Yosida, Functional Analysis, Springer-Verlag, Berlin, 1995
  17. J. S. Yu, Asymptotic stability for a linear difference equation with variable delay, Comput. Math. Appl. 36 (1998), no. 10-12, 203-210 https://doi.org/10.1016/S0898-1221(98)80021-7
  18. G. Zhang, Critical point Theory and Application, Shanghai Press, 1986
  19. Z. Zhou and Q. Zhang, Uniform stability of nonlinear difference systems, J. Math. Anal. Appl. 225 (1998), no. 2, 486-500 https://doi.org/10.1006/jmaa.1998.6039
  20. Z. Zhou, Periodic orbits on discrete dynamical systems, Comput. Math. Appl. 45 (2003), no. 6-9, 1155-1161 https://doi.org/10.1016/S0898-1221(03)00075-0
  21. Z. Zhou, J. Yu, and Z. Guo, Periodic solutions of higher-dimensional discrete systems, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004), no. 5, 1013-1022 https://doi.org/10.1017/S0308210500003607
  22. Z. Zhou, J. S. Yu and Z. M. Guo, The existence of periodic and subharmonic solutions to subquadratic discrete Hamiltonian systems, ANZIAM J. 47 (2005), no. 1, 89-102 https://doi.org/10.1017/S1446181100009792

Cited by

  1. Multiplicity of Periodic Solutions for a Higher Order Difference Equation vol.2014, 2014, https://doi.org/10.1155/2014/925290
  2. Periodic solutions for second-order difference equations with resonance at infinity vol.18, pp.1, 2012, https://doi.org/10.1080/10236191003730498
  3. Existence and Multiple Solutions for Higher Order Difference Dirichlet Boundary Value Problems vol.19, pp.5, 2018, https://doi.org/10.1515/ijnsns-2017-0176