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Mathematical Problems in Engineering
Volume 2013 (2013), Article ID 498781, 21 pages
http://dx.doi.org/10.1155/2013/498781
Research Article

Existence Results for a Coupled System of Nonlinear Singular Fractional Differential Equations with Impulse Effects

1Department of Mathematics, Guangdong University of Business Studies, Guangzhou 510320, China
2Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela, 15782 Santiago de Compostela, Spain
3Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 2 October 2012; Accepted 15 February 2013

Academic Editor: Jocelyn Sabatier

Copyright © 2013 Yuji Liu 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.

Linked References

  1. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, NY, USA, 1993. View at Zentralblatt MATH · View at MathSciNet
  2. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integral and Derivative. Theory and Applications, Gordon and Breach, 1993. View at Zentralblatt MATH · View at MathSciNet
  3. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. B. Ahmad and J. J. Nieto, “Anti-periodic fractional boundary value problems with nonlinear term depending on lower order derivative,” Fractional Calculus and Applied Analysis, vol. 15, no. 3, pp. 451–462, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  5. B. Ahmad, J. J. Nieto, A. Alsaedi, and M. El-Shahed, “A study of nonlinear Langevin equation involving two fractional orders in different intervals,” Nonlinear Analysis: Real World Applications, vol. 13, no. 2, pp. 599–606, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. I. Podlubny and N. Heymans, “Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives,” Rheologica Acta, vol. 45, no. 5, pp. 765–771, 2006. View at Publisher · View at Google Scholar · View at Scopus
  7. 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 Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. A. Carpinteri and F. Mainardi, Eds., Fractals and Fractional Calculus in Continuum Mechanics, CISM Courses and Lectures, International Center for Mechanical Sciences, no. 378, Springer, New York, NY, USA, 1997. View at MathSciNet
  9. R. L. Magin, “Fractional calculus in bioengineering, part 1,” Critical Reviews in Biomedical Engineering, vol. 32, no. 1, pp. 1–104, 2004. View at Publisher · View at Google Scholar
  10. R. L. Magin, “Fractional calculus in bioengineering, part 2,” Critical Reviews in Biomedical Engineering, vol. 32, no. 1, pp. 105–193, 2004. View at Publisher · View at Google Scholar
  11. R. L. Magin, “Fractional calculus in bioengineering, part 3,” Critical Reviews in Biomedical Engineering, vol. 32, no. 3-4, pp. 195–377, 2004. View at Publisher · View at Google Scholar · View at Scopus
  12. X. Chen, L. Wei, J. Sui, and L. Zheng, “Solving the linear time-fractional wave equation by generalized differential transform method,” Applied Mechanics and Materials, vol. 204–208, pp. 4476–4480, 2012. View at Publisher · View at Google Scholar
  13. S. Li, B. Fang, T. Yang, Y. Zhang, L. Tan, and W. Huang, “Dynamics of vibration isolation system obeying fractional differentiation,” Aircraft Engineering and Aerospace Technology, vol. 84, no. 2, pp. 103–108, 2012. View at Publisher · View at Google Scholar
  14. G. S. Priya, P. Prakash, J. J. Nieto, and Z. Kayar, “Higher order numerical scheme for fractional heat equation with Dirichlet and Neumann boundary conditions,” Numerical Heat Transfer B. In press. View at Publisher · View at Google Scholar
  15. J. Zhao, B. Tang, S. Kumar, and Y. Hou, “The extended fractional subequation method for nonlinear fractional differential equations,” Mathematical Problems in Engineering, vol. 2012, Article ID 924956, 11 pages, 2012. View at Publisher · View at Google Scholar
  16. V. V. Lakshmikantham, D. D. Baĭnov, and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. View at MathSciNet
  17. V. Kavitha and M. Mallika Arjunan, “Controllability of impulsive quasi-linear fractional mixed Volterra-Fredholm-type integrodifferential equations in Banach spaces,” The Journal of Nonlinear Science and Its Applications, vol. 4, no. 2, pp. 152–169, 2011. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. B. Ahmad and S. Sivasundaram, “Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations,” Nonlinear Analysis: Hybrid Systems, vol. 3, no. 3, pp. 251–258, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. B. Ahmad and S. Sivasundaram, “Existence of solutions for impulsive integral boundary value problems of fractional order,” Nonlinear Analysis: Hybrid Systems, vol. 4, no. 1, pp. 134–141, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. B. Ahmad and J. J. Nieto, “Existence of solutions for impulsive anti-periodic boundary value problems of fractional order,” Taiwanese Journal of Mathematics, vol. 15, no. 3, pp. 981–993, 2011. View at Google Scholar · View at MathSciNet
  21. Y. Tian and Z. Bai, “Existence results for the three-point impulsive boundary value problem involving fractional differential equations,” Computers & Mathematics with Applications, vol. 59, no. 8, pp. 2601–2609, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. R. P. Agarwal, M. Benchohra, and S. Hamani, “A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions,” Acta Applicandae Mathematicae, vol. 109, no. 3, pp. 973–1033, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. J. Mawhin, Topological Degree Methods in Nonlinear Boundary Value Problems, vol. 40 of NSFCBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, USA, 1979. View at MathSciNet
  24. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. X. Su, “Boundary value problem for a coupled system of nonlinear fractional differential equations,” Applied Mathematics Letters, vol. 22, no. 1, pp. 64–69, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. G. L. Karakostas, “Positive solutions for the Φ-Laplacian when Φ is a sup-multiplicative-like function,” Electronic Journal of Differential Equations, vol. 2004, no. 68, pp. 1–12, 2004. View at Google Scholar · View at MathSciNet