About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 143914, 16 pages
http://dx.doi.org/10.1155/2012/143914
Research Article

Multiple Solutions for a Fractional Difference Boundary Value Problem via Variational Approach

1School of Economics and International Trade, Zhejiang University of Finance and Economics, Hangzhou, Zhejiang 310018, China
2Department of Mathematics, Yanbian University, Yanji 133002, China

Received 28 April 2012; Revised 5 November 2012; Accepted 8 November 2012

Academic Editor: Lishan Liu

Copyright © 2012 Zuoshi Xie et al. 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. S. S. Cheng, “Sturmian comparison theorems for three-term recurrence equations,” Journal of Mathematical Analysis and Applications, vol. 111, no. 2, pp. 465–474, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. O. P. Agrawal, J. Gregory, and K. Pericak-Spector, “A Bliss-type multiplier rule for constrained variational problems with time delay,” Journal of Mathematical Analysis and Applications, vol. 210, no. 2, pp. 702–711, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. J. N. Reddy, Energy and Variational Methods in Applied Mechanics: With an Introduction to the Finite Element Method, John Wiley and Sons, New York, NY, USA, 1984.
  4. O. P. Agrawal, “Formulation of Euler-Lagrange equations for fractional variational problems,” Journal of Mathematical Analysis and Applications, vol. 272, no. 1, pp. 368–379, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. B. Gompertz, “On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies,” Philosophical Transactions of the Royal Society of London, vol. 115, pp. 513–585, 1825.
  6. D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, vol. 5, Academic Press, Orlando, Fla, USA, 1988.
  7. H. Brezis and L. Nirenberg, “Remarks on finding critical points,” Communications on Pure and Applied Mathematics, vol. 44, no. 8-9, pp. 939–963, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. P. H. Rabinowitz, Minimax 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.
  9. D. Goeleven, D. Motreanu, and P. D. Panagiotopoulos, “Multiple solutions for a class of eigenvalue problems in hemivariational inequalities,” Nonlinear Analysis: Theory, Methods & Applications, vol. 29, no. 1, pp. 9–26, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. G. Q. Wang and S. S. Cheng, “Steady state solutions of a toroidal neural network via a mountain pass theorem,” Annals of Differential Equations, vol. 22, no. 3, pp. 360–363, 2006. View at Zentralblatt MATH
  11. H. H. Liang and P. X. Weng, “Existence and multiple solutions for a second-order difference boundary value problem via critical point theory,” Journal of Mathematical Analysis and Applications, vol. 326, no. 1, pp. 511–520, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. 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.
  13. C. S. Goodrich, “Solutions to a discrete right-focal fractional boundary value problem,” International Journal of Difference Equations, vol. 5, no. 2, pp. 195–216, 2010.
  14. F. M. Atici and P. W. Eloe, “Two-point boundary value problems for finite fractional difference equations,” Journal of Difference Equations and Applications, vol. 17, no. 4, pp. 445–456, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. F. M. Atici and P. W. Eloe, “Discrete fractional calculus with the nabla operator,” Electronic Journal of Qualitative Theory of Differential Equations Edition I, no. 3, pp. 1–12, 2009. View at Zentralblatt MATH
  16. F. M. Atici and S. Şengül, “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
  17. S. S. Cheng, Partial Difference Equations, vol. 3 of Advances in Discrete Mathematics and Applications, Taylor & Francis, London, UK, 2003. View at Publisher · View at Google Scholar
  18. F. Ayres Jr, Schaum's Outline of Theory and problems of Matrices, Schaum, New York, NY, USA, 1962.