- 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 2012 (2012), Article ID 468980, 16 pages
Multiple Solutions for a Class of Fractional Boundary Value Problems
Department of Mathematics, Harbin Engineering University, Harbin 150001, China
Received 6 March 2012; Accepted 19 September 2012
Academic Editor: Yong H. Wu
Copyright © 2012 Ge Bin. 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.
- R. Goreno and F. Mainardi, Fractional Calculus: Integral and Differential Equations of Fractional Orders, Fractals and Fractional Calculus in Continuum Mechanics, Springer, New York, NY, USA, 1997.
- B. N. Lundstrom, M. H. Higgs, W. J. Spain, and A. L. Fairhall, “Fractional differentiation by neocortical pyramidal neurons,” Nature Neuroscience, vol. 11, no. 11, pp. 1335–1342, 2008.
- W. G. Glockle and T. F. Nonnenmacher, “A fractional calculus approach to self-similar protein dynamics,” Biophysical Journal, vol. 68, no. 1, pp. 46–53, 1995.
- D. A. Benson, S. W. Wheatcraft, and M. M. Meerschaert, “Application of a fractional advection-dispersion equation,” Water Resources Research, vol. 36, no. 6, pp. 1403–1412, 2000.
- R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing, River Edge, NJ, USA, 2000.
- F. Mainardi, “Fractional calculus: some basic problems in continuum and statistical mechanics,” in Fractals and Fractional Calculus in Continuum Mechanics, A. Carpinteri and F. Mainardi, Eds., pp. 291–348, Springer, Wien, Austria, 1997.
- J. W. Kirchner, X. Feng, and C. Neal, “Frail chemistry and its implications for contaminant transport in catchments,” Nature, vol. 403, no. 6769, pp. 524–527, 2000.
- A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, “Theory and applications of fractional differential equations,” in North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V., Amsterdam, The Netherlands, 2006.
- K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, NY, USA, 1993.
- I. Podlubny, Fractional Differential Equations, vol. 198, Academic Press, San Diego, Calif, USA, 1999.
- S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach, Yverdon, Switzerland, 1993.
- S. Aizicovici, N. S. Papageorgiou, and V. Staicu, “Multiple nontrivial solutions for nonlinear periodic problems with the -Laplacian,” Journal of Differential Equations, vol. 243, no. 2, pp. 504–535, 2007.
- Z. B. Bai and H. S. Lü, “Positive solutions for boundary value problem of nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 311, no. 2, pp. 495–505, 2005.
- S. Q. Zhang, “Existence of solution for a boundary value problem of fractional order,” Acta Mathematica Scientia. Series B, vol. 26, no. 2, pp. 220–228, 2006.
- Z. B. Bai and Y. H. Zhang, “The existence of solutions for a fractional multi-point boundary value problem,” Computers & Mathematics with Applications, vol. 60, no. 8, pp. 2364–2372, 2010.
- W. H. Jiang, “The existence of solutions to boundary value problems of fractional differential equations at resonance,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 5, pp. 1987–1994, 2011.
- S. Q. Zhang, “Existence of a solution for the fractional differential equation with nonlinear boundary conditions,” Computers & Mathematics with Applications, vol. 61, no. 4, pp. 1202–1208, 2011.
- S. H. Liang and J. H. 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.
- X. Zhang, L. Liu, and Y. H. Wu, “The eigenvalue problem for a singular higher order fractional differential equation involving fractional derivatives,” Applied Mathematics and Computation, vol. 218, pp. 8526–8536, 2012.
- X. Zhang, L. Liu, and Y. H. Wu, “Multiple positive solutions of a singular fractional differential equation with negatively perturbed term,” Mathematical and Computer Modelling, vol. 55, pp. 1263–1274, 2012.
- X. Zhang, L. Liu, B. Wiwatanapataphee, and Y. H. Wu, “Positive solutions of Eigenvalue problems for a class of fractional differential equations with derivatives,” Abstract and Applied Analysis, vol. 2012, Article ID 512127, 16 pages, 2012.
- Y. Wang, L. Liu, and Y. Wu, “Positive solutions of a fractional boundary value problem with changing sign nonlinearity,” Abstract and Applied Analysis, vol. 2012, Article ID 149849, 12 pages, 2012.
- Y. Wang, L. Liu, and Y. Wu, “Positive solutions for a nonlocal fractional differential equation,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 11, pp. 3599–3605, 2011.
- J. Jiang, L. Liu, and Y. H. Wu, “Multiple positive solutions of singular fractional differential system involving Stieltjes integral conditions,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 43, pp. 1–18, 2012.
- F. Jiao and Y. Zhou, “Existence of solutions for a class of fractional boundary value problems via critical point theory,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1181–1199, 2011.
- J. Chen and X. H. Tang, “Existence and multiplicity of solutions for some fractional boundary value problem via critical point theory,” Abstract and Applied Analysis, vol. 2012, Article ID 648635, 21 pages, 2012.
- A. Ambrosetti and P. H. Rabinowitz, “Dual variational methods in critical point theory and applications,” Journal of Functional Analysis, vol. 14, pp. 349–381, 1973.
- P. H. Rabinowitz, “Minimax methods in critical point theory with applications to differential equations,” in Proceedings of the CBMS, vol. 65, American Mathematical Society, 1986.
- F. Y. Li, Z. P. Liang, and Q. Zhang, “Existence of solutions to a class of nonlinear second order two-point boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 312, pp. 357–373, 2005.