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Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 104276, 9 pages
Existence Results for Fractional Difference Equations with Three-Point Fractional Sum Boundary Conditions
1Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology, North Bangkok, Bangkok, Thailand
2Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece
Received 3 June 2013; Accepted 7 August 2013
Academic Editor: Eric R. Kaufmann
Copyright © 2013 Thanin Sitthiwirattham 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.
- K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993.
- S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Yverdon, Switzerland, 1993.
- I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999.
- A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherlands, 2006.
- V. Lakshmikantham, S. Leela, and J. Vasundhara Devi, Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge, UK, 2009.
- D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus Models and Numerical Methods, vol. 3 of Series on Complexity, Nonlinearity and Chaos, World Scientific, Boston, Mass, USA, 2012.
- K. Diethelm, The Analysis of Fractional Differential Equations. An Application-oriented Exposition Using Differential Operators of Caputo Type, vol. 2004 of Lecture Notes in Mathematics, Springer, 2010.
- A. Guezane-Lakoud and R. Khaldi, “Solvability of a three-point fractional nonlinear boundary value problem,” Differential Equations and Dynamical Systems, vol. 20, no. 4, pp. 395–403, 2012.
- A. Guezane-Lakoud and R. Khaldi, “Positive solution to a higher order fractional boundary value problem with fractional integral condition,” Romanian Journal of Mathematics and Computer Science, vol. 2, no. 1, pp. 41–54, 2012.
- E. R. Kaufmann, “Existence and nonexistence of positive solutions for a nonlinear fractional boundary value problem,” Discrete and Continuous Dynamical Systems A, vol. 2009, pp. 416–423, 2009.
- J. Wang, H. Xiang, and Z. Liu, “Positive solution to nonzero boundary values problem for a coupled system of nonlinear fractional differential equations,” International Journal of Differential Equations, vol. 2010, Article ID 186928, 12 pages, 2010.
- Z. Bai, “On positive solutions of a nonlocal fractional boundary value problem,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 2, pp. 916–924, 2010.
- W. Sudsutad and J. Tariboon, “Boundary value problems for fractional differential equations with three-point fractional integral boundary conditions,” Advances in Difference Equations, vol. 2012, article 93, 2012.
- S. K. Ntouyas, “Boundary value problems for nonlinear fractional differential equations and inclusions with nonlocal and fractional integral boundary conditions,” Opuscula Mathematica, vol. 33, no. 1, pp. 117–138, 2013.
- A. Guezane-Lakoud and R. Khaldi, “Solvability of a fractional boundary value problem with fractional integral condition,” Nonlinear Analysis: Theory, Methods & Applications, vol. 75, no. 4, pp. 2692–2700, 2012.
- B. Ahmad, S. K. Ntouyas, and A. Assolami, “Caputo type fractional differential equations with nonlocal Riemann-Liouville integral boundary conditions,” Journal of Applied Mathematics and Computing, vol. 41, no. 1-2, pp. 339–350, 2013.
- 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.
- C. S. Goodrich, “Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions,” Computers & Mathematics with Applications, vol. 61, no. 2, pp. 191–202, 2011.
- C. S. Goodrich, “Continuity of solutions to discrete fractional initial value problems,” Computers & Mathematics with Applications, vol. 59, no. 11, pp. 3489–3499, 2010.
- 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.
- C. S. Goodrich, “Existence of a positive solution to a system of discrete fractional boundary value problems,” Applied Mathematics and Computation, vol. 217, no. 9, pp. 4740–4753, 2011.
- C. S. Goodrich, “On a discrete fractional three-point boundary value problem,” Journal of Difference Equations and Applications, vol. 18, no. 3, pp. 397–415, 2012.
- F. Chen, X. Luo, and Y. Zhou, “Existence results for nonlinear fractional difference equation,” Advances in Difference Equations, vol. 2011, Article ID 713201, 12 pages, 2011.
- Y. Pan, Z. Han, S. Sun, and Z. Huang, “The existence and uniqueness of solutions to boundary value problems of fractional difference equations,” Mathematical Sciences, vol. 6, article 7, 7 pages, 2012.
- T. Abdeljawad, “On Riemann and Caputo fractional differences,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1602–1611, 2011.
- T. Sitthiwirattham, J. Tariboon, and S. K. Ntouyas, “Boundary value problems for fractional difference equations with three-point fractional sum boundary conditions,” Advances in Difference Equations. In press.
- V. Lakshmikantham and S. Leela, Nonlinear Differential Equations in Abstract Spaces, vol. 2 of International Series in Nonlinear Mathematics: Theory, Methods and Applications, Pergamon Press, Oxford, UK, 1981.
- W. Rudin, Functional Analysis, McGraw-Hill, New York, NY, USA, 1973.
- M. A. Krasnoselskii, “Two remarks on the method of successive approximations,” Uspekhi Matematicheskikh Nauk, vol. 10, pp. 123–127, 1955.
- A. Granas and J. Dugundji, Fixed Point Theory, Springer, New York, NY, USA, 2003.