Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2012, Article ID 879657, 14 pages
http://dx.doi.org/10.1155/2012/879657
Research Article

Asymptotic Stability Results for Nonlinear Fractional Difference Equations

Department of Mathematics, Xiangnan University, Chenzhou 423000, China

Received 1 August 2011; Revised 27 December 2011; Accepted 2 January 2012

Academic Editor: Michela Redivo-Zaglia

Copyright © 2012 Fulai Chen and Zhigang Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. R. P. Agarwal, V. Lakshmikantham, and J. J. Nieto, “On the concept of solution for fractional differential equations with uncertainty,” Nonlinear Analysis. Theory, Methods & Applications, vol. 72, no. 6, pp. 2859–2862, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. M. Benchohra, J. Henderson, S. K. Ntouyas, and A. Ouahab, “Existence results for fractional order functional differential equations with infinite delay,” Journal of Mathematical Analysis and Applications, vol. 338, no. 2, pp. 1340–1350, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. F. Chen, A. Chen, and X. Wang, “On the solutions for impulsive fractional functional differential equations,” Differential Equations and Dynamical Systems, vol. 17, no. 4, pp. 379–391, 2009. View at Publisher · View at Google Scholar
  4. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, the Netherlands, 2006.
  5. V. Lakshmikantham, “Theory of fractional functional differential equations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 69, no. 10, pp. 3337–3343, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. V. Lakshmikantham, S. Leela, and J. V. Devi, Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers, Cambridge, UK, 2009.
  7. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993.
  8. J. J. Nieto, “Maximum principles for fractional differential equations derived from Mittag-Leffler functions,” Applied Mathematics Letters, vol. 23, no. 10, pp. 1248–1251, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999.
  10. J. Wang and Y. Zhou, “A class of fractional evolution equations and optimal controls,” Nonlinear Analysis. Real World Applications, vol. 12, no. 1, pp. 262–272, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. Y. Zhou, F. Jiao, and J. Li, “Existence and uniqueness for P-type fractional neutral differential equations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 7-8, pp. 2724–2733, 2009. View at Publisher · View at Google Scholar
  12. Y. Zhou, F. Jiao, and J. Li, “Existence and uniqueness for fractional neutral differential equations with infinite delay,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 7-8, pp. 3249–3256, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. G. A. Anastassiou, “Discrete fractional calculus and inequalities,” Classical Analysis and ODEs. In press.
  14. F. M. Atici and P. W. Eloe, “Initial value problems in discrete fractional calculus,” Proceedings of the American Mathematical Society, vol. 137, no. 3, pp. 981–989, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. F. M. Atici and P. W. Eloe, “A transform method in discrete fractional calculus,” International Journal of Difference Equations, vol. 2, no. 2, pp. 165–176, 2007. View at Google Scholar
  16. F. M. Atıcı and P. W. Eloe, “Discrete fractional calculus with the nabla operator,” Electronic Journal of Qualitative Theory of Differential Equations I, no. 3, pp. 1–12, 2009. View at Google Scholar · View at Zentralblatt MATH
  17. F. M. Atici and S. Sengül, “Modeling with fractional difference equations,” Journal of Mathematical Analysis and Applications, vol. 369, no. 1, pp. 1–9, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. F. Chen, “Fixed points and asymptotic stability of nonlinear fractional difference equations,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 39, pp. 1–18, 2011. View at Google Scholar
  19. F. Chen and Y. Zhou, “Attractivity of fractional functional differential equations,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1359–1369, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. T. A. Burton and T. Furumochi, “Krasnoselskii's fixed point theorem and stability,” Nonlinear Analysis. Theory, Methods & Applications, vol. 49, no. 4, pp. 445–454, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. T. A. Burton, “Fixed points, stability, and exact linearization,” Nonlinear Analysis. Theory, Methods & Applications, vol. 61, no. 5, pp. 857–870, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. T. A. Burton, Stability by Fixed Point Theory for Functional Differential Equations, Dover Publications, Mineola, NY, USA, 2006.
  23. B. C. Dhage, “Global attractivity results for nonlinear functional integral equations via a Krasnoselskii type fixed point theorem,” Nonlinear Analysis. Theory, Methods & Applications, vol. 70, no. 7, pp. 2485–2493, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. C. Jin and J. Luo, “Stability in functional differential equations established using fixed point theory,” Nonlinear Analysis. Theory, Methods & Applications, vol. 68, no. 11, pp. 3307–3315, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. Y. N. Raffoul, “Stability in neutral nonlinear differential equations with functional delays using fixed-point theory,” Mathematical and Computer Modelling, vol. 40, no. 7-8, pp. 691–700, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  26. Y. Zhou, Oscillatory Behavior of Delay Differential Equations, Science Press, Beijing, China, 2007.
  27. S. S. Cheng and W. T. Patula, “An existence theorem for a nonlinear difference equation,” Nonlinear Analysis. Theory, Methods & Applications, vol. 20, no. 3, pp. 193–203, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH