- About this Journal
- Abstracting and Indexing
- Aims and Scope
- 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
Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 425408, 16 pages
Positive Solution of a Nonlinear Fractional Differential Equation Involving Caputo Derivative
1College of Mathematics and Physics, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
2Key Laboratory of Industrial Internet of Things & Networked Control of Ministry of Education, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
3College of Applied Sciences, Beijing University of Technology, Beijing 100124, China
Received 13 August 2012; Accepted 5 September 2012
Academic Editor: Seenith Sivasundaram
Copyright © 2012 Changyou Wang 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.
- F. Mainardi, “The fundamental solutions for the fractional diffusion-wave equation,” Applied Mathematics Letters, vol. 9, no. 6, pp. 23–28, 1996.
- E. Buckwar and Y. Luchko, “Invariance of a partial differential equation of fractional order under the Lie group of scaling transformations,” Journal of Mathematical Analysis and Applications, vol. 227, no. 1, pp. 81–97, 1998.
- Z. Zheng-you, L. Gen-guo, and C. Chang-jun, “Quasi-static and dynamical analysis for viscoelastic Timoshenko beam with fractional derivative constitutive relation,” Applied Mathematics and Mechanics, vol. 23, no. 1, pp. 1–12, 2002.
- R. Gorenflo and S. Vessella, Abel Integral Equations, vol. 1461 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1991.
- K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley & Sons, New York, NY, USA, 1993.
- B. Ross and B. K. Sachdeva, “The solution of certain integral equations by means of operators of arbitrary order,” The American Mathematical Monthly, vol. 97, no. 6, pp. 498–503, 1990.
- S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science, Yverdon, Switzerland, 1993.
- H. M. Srivastava and R. G. Buschman, Theory and Applications of Convolution Integral Equations, vol. 79 of Mathematics and Its Applications, Kluwer Academic Publishers Group, Dordrecht, The Netherlands, 1992.
- K. Diethelm and N. J. Ford, “Analysis of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 265, no. 2, pp. 229–248, 2002.
- V. Daftardar-Gejji and H. Jafari, “Analysis of a system of nonautonomous fractional differential equations involving Caputo derivatives,” Journal of Mathematical Analysis and Applications, vol. 328, no. 2, pp. 1026–1033, 2007.
- D. Delbosco and L. Rodino, “Existence and uniqueness for a nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 204, no. 2, pp. 609–625, 1996.
- S. Zhang, “The existence of a positive solution for a nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 252, no. 2, pp. 804–812, 2000.
- Q. L. Yao, “Existence of positive solution to a class of sublinear fractional differential equations,” Acta Mathematicae Applicatae Sinica, vol. 28, no. 3, pp. 429–434, 2005 (Chinese).
- V. Lakshmikantham and A. S. Vatsala, “Basic theory of fractional differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 8, pp. 2677–2682, 2008.
- S. Zhang, “Positive solutions to singular boundary value problem for nonlinear fractional differential equation,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1300–1309, 2010.
- C. Wang, “Existence and stability of periodic solutions for parabolic systems with time delays,” Journal of Mathematical Analysis and Applications, vol. 339, no. 2, pp. 1354–1361, 2008.
- C. Wang, R. Wang, S. Wang, and C. Yang, “Positive solution of singular boundary value problem for a nonlinear fractional differential equation,” Boundary Value Problems, vol. 2011, Article ID 297026, 12 pages, 2011.
- V. Lakshmikantham, S. Leela, and J. Vasundhara Devi, Theory of Fractional Dynamic Systems, Cambridge Academic, Cambridge, UK, 2009.
- Y.-K. Chang and J. J. Nieto, “Some new existence results for fractional differential inclusions with boundary conditions,” Mathematical and Computer Modelling, vol. 49, no. 3-4, pp. 605–609, 2009.
- N. Kosmatov, “Integral equations and initial value problems for nonlinear differential equations of fractional order,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 7, pp. 2521–2529, 2009.
- G. Jumarie, “An approach via fractional analysis to non-linearity induced by coarse-graining in space,” Nonlinear Analysis: Real World Applications, vol. 11, no. 1, pp. 535–546, 2010.
- R. P. Agarwal, V. Lakshmikantham, and J. J. Nieto, “On the concept of solution for fractional differential equations with uncertainty,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 6, pp. 2859–2862, 2010.
- Z. Wei, Q. Li, and J. Che, “Initial value problems for fractional differential equations involving Riemann-Liouville sequential fractional derivative,” Journal of Mathematical Analysis and Applications, vol. 367, no. 1, pp. 260–272, 2010.
- K. Balachandran and J. J. Trujillo, “The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 12, pp. 4587–4593, 2010.
- Y. Luchko and R. Gorenflo, “An operational method for solving fractional differential equations with the Caputo derivatives,” Acta Mathematica Vietnamica, vol. 24, no. 2, pp. 207–233, 1999.
- I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.