About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2014 (2014), Article ID 158436, 7 pages
http://dx.doi.org/10.1155/2014/158436
Research Article

Existence of Positive Solutions to Nonlinear Fractional Boundary Value Problem with Changing Sign Nonlinearity and Advanced Arguments

School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, China

Received 24 March 2014; Accepted 19 June 2014; Published 9 July 2014

Academic Editor: Xinguang Zhang

Copyright © 2014 Zhaocai Hao and Yubo Huang. 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.

Linked References

  1. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, 2000.
  2. F. 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. View at Publisher · View at Google Scholar
  3. R. P. Agarwal, M. Benchohra, and B. Slimani, “Existence results for differential equations with fractional order and impulses,” Georgian Academy of Sciences A: Razmadze Mathematical Institute: Memoirs on Differential Equations and Mathematical Physics, vol. 44, pp. 1–21, 2008. View at MathSciNet
  4. E. Ahmed and H. A. El-Saka, “On fractional order models for Hepatitis C,” Nonlinear Biomedical Physics, vol. 4, article 1, 2010. View at Publisher · View at Google Scholar · View at Scopus
  5. C. Bai, “Positive solutions for nonlinear fractional differential equations with coefficient that changes sign,” Nonlinear Analysis: Theory, Methods and Applications, vol. 64, no. 4, pp. 677–685, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. 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. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. K. Diethelm and A. D. Freed, “On the solution of nonlinear fractional order differential equations used in the modeling 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. Werther, Eds., pp. 217–307, Springer, Heidelberg, Germany, 1999.
  8. X. Ding and Y. Jiang, “Waveform relaxation methods for fractional functional differential equations,” Fractional Calculus and Applied Analysis, vol. 16, no. 3, pp. 573–594, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. 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. View at Publisher · View at Google Scholar · View at Scopus
  10. T. Jankowski, “Existence of positive solutions to third order differential equations with advanced arguments and nonlocal boundary conditions,” Nonlinear Analysis, vol. 75, no. 2, pp. 913–923, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. T. Jankowski, “Positive solutions for second order impulsive differential equations involving Stieltjes integral conditions,” Nonlinear Analysis: Theory, Methods and Applications, vol. 74, no. 11, pp. 3775–3785, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. 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, Amsterdam, The Netherlands, 2006. View at MathSciNet
  13. A. A. Kilbas and J. J. Trujillo, “Differential equations of fractional order: methods, results and problems. I,” Applicable Analysis, vol. 78, no. 1-2, pp. 153–192, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  14. A. A. Kilbas and J. J. Trujillo, “Differential equations of fractional order: methods, results and problems. II,” Applicable Analysis, vol. 81, no. 2, pp. 435–493, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  15. R. C. Koeller, “Applications of fractional calculus to the theory of viscoelasticity,” Journal of Applied Mechanics, vol. 51, no. 2, pp. 299–307, 1984. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. C. Kou, H. Zhou, and Y. Yan, “Existence of solutions of initial value problems for nonlinear fractional differential equations on the half-axis,” Nonlinear Analysis, vol. 74, no. 17, pp. 5975–5986, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  17. P. Kumar and O. P. Agrawal, “An approximate method for numerical solution of fractional differential equations,” Signal Processing, vol. 86, no. 10, pp. 2602–2610, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  18. V. Lakshmikantham, S. Leela, and J. Vasundhara, Theory of Fractional Dynamic Systems, Cambridge Academic, Cambridge, UK, 2009.
  19. 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. View at Publisher · View at Google Scholar · View at MathSciNet
  20. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley and Sons, New York, NY, USA, 1993. View at MathSciNet
  21. I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and Some of their Applications, Academic Press, San Diego, Calif, USA, 1999. View at MathSciNet
  22. I. Podlubny, “Geometric and physical interpretation of fractional integration and fractional differentiation,” Fractional Calculus and Applied Analysis, vol. 5, no. 4, pp. 367–386, 2002. View at MathSciNet
  23. H. M. Srivastava and R. K. Saxena, “Operators of fractional integration and their applications,” Applied Mathematics and Computation, vol. 118, no. 1, pp. 1–52, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  24. X. Zhang, L. Liu, and Y. Wu, “Multiple positive solutions of a singular fractional differential equation with negatively perturbed term,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 1263–1274, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  25. X. Zhang, L. Liu, and Y. Wu, “Existence results for multiple positive solutions of nonlinear higher order perturbed fractional differential equations with derivatives,” Applied Mathematics and Computation, vol. 219, no. 4, pp. 1420–1433, 2012. View at Publisher · View at Google Scholar · View at Scopus
  26. Y. Wang, L. Liu, and Y. Wu, “Positive solutions for a class of fractional boundary value problem with changing sign nonlinearity,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 17, pp. 6434–6441, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  27. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  28. X. Zhang, L. Liu, and Y. Wu, “The uniqueness of positive solution for a singular fractional differential system involving derivatives,” Communications in Nonlinear Science and Numerical Simulation, vol. 18, no. 6, pp. 1400–1409, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  29. 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. View at Publisher · View at Google Scholar · View at Scopus
  30. X. Zhang, L. Liu, Y. Wu, and Y. Lu, “The iterative solutions of nonlinear fractional differential equations,” Applied Mathematics and Computation, vol. 219, no. 9, pp. 4680–4691, 2013. View at Publisher · View at Google Scholar · View at Scopus