Abstract and Applied Analysis

Nontrivial Periodic Solutions to Some Semilinear Sixth-Order Difference Equations

Yuhua Long

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We establish some new criteria to guarantee nonexistence, existence, and multiplicity of nontrivial periodic solutions of some semilinear sixth-order difference equations by using minmax method, Z 2 index theory, and variational technique. Our results only make some assumptions on the period T , which are very easy to verify and rather relaxed.

Article information

Abstr. Appl. Anal., Volume 2014 (2014), Article ID 359105, 8 pages.

First available in Project Euclid: 2 October 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Long, Yuhua. Nontrivial Periodic Solutions to Some Semilinear Sixth-Order Difference Equations. Abstr. Appl. Anal. 2014 (2014), Article ID 359105, 8 pages. doi:10.1155/2014/359105. https://projecteuclid.org/euclid.aaa/1412273237

Export citation


  • V. Lakshmikantham and D. Trigiante, Theory of Difference Equations: Numerical Methods and Applications, Academic Press, Boston, Mass, USA, 1988.
  • R. P. Agarwal, Difference Equations and Inequalities: Theory, Methods, and Applications, Marcel Dekker, New York, NY, USA, 1992.
  • X. C. Cai and J. S. Yu, “Existence of periodic solutions for a $2n$th-order nonlinear difference equation,” Journal of Mathematical Analysis and Applications, vol. 329, no. 2, pp. 870–878, 2007.
  • Z. Guo and J. Yu, “Periodic and subharmonic solutions for superquadratic discrete Hamiltonian systems,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 55, no. 7-8, pp. 969–983, 2003.
  • X. Deng, X. Liu, Y. Zhang, and H. Shi, “Periodic and subharmonic solutions for a $2n$th-order difference equation involving $p$-Laplacian,” Indagationes Mathematicae, vol. 24, no. 3, pp. 613–625, 2013.
  • P. W. Bates, P. C. Fife, R. A. Gardner, and C. K. R. T. Jones, “The existence of travelling wave solutions of a generalized phase-field model,” SIAM Journal on Mathematical Analysis, vol. 28, no. 1, pp. 60–93, 1997.
  • G. Caginalp and P. Fife, “Higher-order phase field models and detailed anisotropy,” Physical Review B, vol. 34, no. 7, pp. 4940–4943, 1986.
  • L. A. Peletier, W. C. Vorst, and Van der Vorst, “Stationary solutions of a fourthorder nonlinear diffusion equation,” Differential Equations, vol. 31, pp. 301–314, 1995.
  • R. E. Mickens, Difference Equations: Theory and Application, Van Nostrand Reinhold, New York, NY, USA, 1990.
  • W. G. Kelley and A. C. Peterson, Difference Equations: An Introduction with Applications, Academic Press, Boston, Mass, USA, 1991.
  • A. N. Sharkovsky, Yu. L. Maĭstrenko, and E. Yu. Romanenko, Difference Equations and Their Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993.
  • V. L. Kocić and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Application, Kluwer Academic Publishers, Boston, Mass, USA, 1993.
  • S. N. Elaydi, An Introduction to Difference Equations, Springer, New York, NY, USA, 1996.
  • Z. Zhou, J. Yu, and Z. Guo, “The existence of periodic and subharmonic solutions to subquadratic discrete Hamiltonian systems,” The ANZIAM Journal, vol. 47, no. 1, pp. 89–102, 2005.
  • 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.
  • C. D. Ahlbrandt and A. Peterson, “The $(n,n)$-disconjugacy of a $2n$th order linear difference equation,” Computers & Mathematics with Applications, vol. 28, no. 1–3, pp. 1–9, 1994.
  • T. Peil and A. Peterson, “Asymptotic behavior of solutions of a two term difference equation,” The Rocky Mountain Journal of Mathematics, vol. 24, no. 1, pp. 233–252, 1994.
  • K.-C. Chang, Infinite-Dimensional Morse Theory and Multiple Solution Problems, Birkhäuser, Boston, Mass, USA, 1993.
  • P. H. Rabinowitz, Minmax Methods in Critical Point Theory with applications to Differential Equations, vol. 65 of CBMS Regional Conference Series in Mathematics, American mathematical Society, Providence, RI, USA, 1986.
  • J. X. Sun, Nonlinear Functional Analysis and Applications, Acedemic Press, 2008. \endinput