- About this Journal ·
- Abstracting and Indexing ·
- 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 797398, 16 pages
Solutions of Sign-Changing Fractional Differential Equation with the Fractional Derivatives
1School of Information and Engineering, Wenzhou Medical College, Zhejiang, Wenzhou 325035, China
2School of Mathematical and Informational Sciences, Yantai University, Shandong, Yantai 264005, China
3Information Engineering Department, Anhui Xinhua University, Anhui, Hefei 230031, China
Received 24 April 2012; Accepted 29 May 2012
Academic Editor: Yonghong Wu
Copyright © 2012 Tunhua Wu 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. Diethelm and A. D. Freed, “On the solutions of nonlinear fractional order differential equations used in the modelling of viscoplasticity,” in Scientific Computing in Chemical Engineering II-Computational Fluid Dynamics, Reaction Engineering and Molecular Properties, F. Keil, W. Mackens, H. Voss, and J. Werthers, Eds., Springer, Heidelberg, Germany, 1999.
- L. Gaul, P. Klein, and S. Kemple, “Damping description involving fractional operators,” Mechanical Systems and Signal Processing, vol. 5, no. 2, pp. 81–88, 1991.
- 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.
- F. Mainardi, “Fractional calculus: some basic problems in continuum and statistical mechanics,” in Fractals and Fractional Calculus in Continuum Mechanics, C. A. Carpinteri and F. Mainardi, Eds., Springer, Vienna, Austria, 1997.
- R. Metzler, W. Schick, H. G. Kilian, and T. F. Nonnenmacher, “Relaxation in filled polymers: a fractional calculus approach,” The Journal of Chemical Physics, vol. 103, no. 16, pp. 7180–7186, 1995.
- K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974.
- X. Zhang, L. Liu, and Y. Wu, “Multiple positive solutions of a singular fractional differential equationwith negatively perturbed term,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 1263–1274, 2012.
- B. Ahmad and J. J. Nieto, “Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions,” Boundary Value Problems, vol. 2011, article 36, 2011.
- C. S. Goodrich, “Existence of a positive solution to systems of differential equations of fractional order,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1251–1268, 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, “Positive solutions to boundary value problems with nonlinear boundary conditions,” Nonlinear Analysis, vol. 75, no. 1, pp. 417–432, 2012.
- X. Zhang and Y. Han, “Existence and uniqueness of positive solutions for higher order nonlocal fractional differential equations,” Applied Mathematics Letters, vol. 25, no. 3, pp. 555–560, 2012.
- Y. Wang, L. Liu, and Y. Wu, “Positive solutions for a nonlocal fractional differential equation,” Nonlinear Analysis, vol. 74, no. 11, pp. 3599–3605, 2011.
- X. Zhang, L. Liu, and Y. Wu, “The eigenvalue problem for a singular higher order fractional differential equation involving fractional derivatives,” Applied Mathematics and Computation, vol. 218, no. 17, pp. 8526–8536, 2012.
- X. Zhang, L. Liu, B. Wiwatanapataphee, and Y. 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.
- J. Wu, X. Zhang, L. Liu, and Y. Wu, “Positive solutions of higher-order nonlinear fractional differential equations with changing-sign measure,” Advances in Difference Equations, vol. 2012, article 71, 2012.
- R. Aris, Introduction to the Analysis of Chemical Reactors, Prentice Hall, Englewood Cliffs, NJ, USA, 1965.
- A. Castro, C. Maya, and R. Shivaji, “Nonlinear eigenvalue problems with semipositone,” Electronic Journal of Differential Equations, no. 5, pp. 33–49, 2000.
- V. Anuradha, D. D. Hai, and R. Shivaji, “Existence results for superlinear semipositone BVP's,” Proceedings of the American Mathematical Society, vol. 124, no. 3, pp. 757–763, 1996.
- J. X. Sun, “Non-zero solutions to superlinear Hammerstein integral equations and applications,” Chinese Annals of Mathematics A, vol. 7, no. 5, pp. 528–535, 1986.
- F. Li and G. Han, “Existence of non-zero solutions to nonlinear Hammerstein integral equation,” Journal of Shanxi University (Natural Science Edition), vol. 26, pp. 283–286, 2003.
- G. Han and Y. Wu, “Nontrivial solutions of singular two-point boundary value problems with sign-changing nonlinear terms,” Journal of Mathematical Analysis and Applications, vol. 325, no. 2, pp. 1327–1338, 2007.
- J. Sun and G. Zhang, “Nontrivial solutions of singular superlinear Sturm-Liouville problems,” Journal of Mathematical Analysis and Applications, vol. 313, no. 2, pp. 518–536, 2006.
- L. Liu, B. Liu, and Y. Wu, “Nontrivial solutions of -point boundary value problems for singular second-order differential equations with a sign-changing nonlinear term,” Journal of Computational and Applied Mathematics, vol. 224, no. 1, pp. 373–382, 2009.
- K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, Germany, 1985.
- I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1999.
- A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006.
- C. Yuan, “Multiple positive solutions for -type semipositone conjugate boundary value problems of nonlinear fractional differential equations,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 36, pp. 1–12, 2010.
- M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Prentice Hall, New York, NY, USA, 1967.