- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 847184, 7 pages
Lower and Upper Solutions Method for Positive Solutions of Fractional Boundary Value Problems
1Department of Mathematics, Neka Branch, Islamic Azad University, P.O. Box 48411-86114, Neka, Iran
2Department of Mathematics, Sari Branch, Islamic Azad University, P.O. Box 48161-19318, Sari, Iran
3Department of Mathematics, University of Mazandaran, P.O. Box 47416-95447, Babolsar, Iran
4Department of Mathematics and Computer Science, Faculty of Art and Sciences, Cankaya University, Yenimahalle, 06810 Ankara, Turkey
5Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Saudi Arabia
6Institute of Space Sciences, P.O. Box, MG-23, R 76900 Magurele, Bucharest, Romania
Received 10 May 2013; Revised 11 July 2013; Accepted 1 August 2013
Academic Editor: Juan J. Trujillo
Copyright © 2013 R. Darzi 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. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974.
- B. Ross, Ed., The Fractional Calculus and Its Applications, vol. 475 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1975.
- F. B. Tatom, “The relationship between fractional calculus and fractals,” Fractals, vol. 3, no. 1, pp. 217–229, 1995.
- T. F. Nonnenmacher and R. Metzler, “On the Riemann-Liouville fractional calculus and some recent applications,” Fractals, vol. 3, no. 3, pp. 557–566, 1995.
- S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives (Theory and Application), Gordon and Breach Science, Yverdon, Switzerland, 1993.
- 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.
- K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993.
- I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
- V. Lakshmikantham, S. Leela, and J. Vasundhara, Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge, UK, 2009.
- R. P. Agarwal, M. Benchohra, and B. A. Slimani, “Existence results for differential equations with fractional order and impulses,” Memoirs on Differential Equations and Mathematical Physics, vol. 44, pp. 1–21, 2008.
- R. P. Agarwal, M. Benchohra, and S. Hamani, “Boundary value problems for fractional differential equations,” Georgian Mathematical Journal, vol. 16, no. 3, pp. 401–411, 2009.
- C. Yu and G. Z. Gao, “On the solution of nonlinear fractional order differential equation,” Nonlinear Analysis: Theory, Methods & Applications, vol. 63, no. 5-7, pp. e971–e976, 1998.
- B. Ahmad and J. J. Nieto, “Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions,” Computers & Mathematics with Applications, vol. 58, no. 9, pp. 1838–1843, 2009.
- A. M. Nahušev, “The Sturm-Liouville problem for a second order ordinary differential equation with fractional derivatives in the lower terms,” Doklady Akademii Nauk SSSR, vol. 234, no. 2, pp. 308–311, 1977.
- S. Zhang, “Positive solutions for boundary-value problems of nonlinear fractional differential equations,” Electronic Journal of Differential Equations, vol. 36, pp. 1–12, 2006.
- 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.
- D. Băleanu, 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, River Edge, NJ, USA, 2012.
- D. Băleanu and J. J. Trujillo, “On exact solutions of a class of fractional Euler-Lagrange equations,” Nonlinear Dynamics, vol. 52, no. 4, pp. 331–335, 2008.
- D. Băleanu and O. G. Mustafa, “On the global existence of solutions to a class of fractional differential equations,” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1835–1841, 2010.
- M. El-Shahed and J. J. Nieto, “Nontrivial solutions for a nonlinear multi-point boundary value problem of fractional order,” Computers & Mathematics with Applications, vol. 59, no. 11, pp. 3438–3443, 2010.
- M. El-Shahed, “Positive solutions for boundary value problem of nonlinear fractional differential equation,” Abstract and Applied Analysis, vol. 2007, Article ID 10368, 8 pages, 2007.
- S. Liang and J. Zhang, “Positive solutions for boundary value problems of nonlinear fractional differential equation,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 11, pp. 5545–5550, 2009.
- D. Guo and J. Zhang, Nonlinear Fractional Analysis, Science and Technology Press, Jinan, China, 1985.