Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2016, Article ID 8101802, 20 pages
http://dx.doi.org/10.1155/2016/8101802
Research Article

The General Solution of Impulsive Systems with Caputo-Hadamard Fractional Derivative of Order

1School of Electronic Engineering, Jiujiang University, Jiujiang, Jiangxi 332005, China
2School of Chemical and Environmental Engineering, Jiujiang University, Jiujiang, Jiangxi 332005, China
3School of Chemistry and Chemical Engineering, Chongqing University, Chongqing 400044, China

Received 21 December 2015; Accepted 19 January 2016

Academic Editor: José A. T. Machado

Copyright © 2016 Xianmin Zhang 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. A. A. Kilbas, “Hadamard-type fractional calculus,” Journal of the Korean Mathematical Society, vol. 38, no. 6, pp. 1191–1204, 2001. View at Google Scholar · View at MathSciNet · View at Scopus
  2. P. L. Butzer, A. A. Kilbas, and J. J. Trujillo, “Compositions of Hadamard-type fractional integration operators and the semigroup property,” Journal of Mathematical Analysis and Applications, vol. 269, no. 2, pp. 387–400, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. P. L. Butzer, A. A. Kilbas, and J. J. Trujillo, “Mellin transform analysis and integration by parts for Hadamard-type fractional integrals,” Journal of Mathematical Analysis and Applications, vol. 270, no. 1, pp. 1–15, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. M. Klimek, “Sequential fractional differential equations with Hadamard derivative,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 12, pp. 4689–4697, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. B. Ahmad and S. K. Ntouyas, “A fully Hadamard type integral boundary value problem of a coupled system of fractional differential equations,” Fractional Calculus and Applied Analysis, vol. 17, no. 2, pp. 348–360, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. P. Thiramanus, S. K. Ntouyas, and J. Tariboon, “Existence and uniqueness results for Hadamard-type fractional differential equations with nonlocal fractional integral boundary conditions,” Abstract and Applied Analysis, vol. 2014, Article ID 902054, 9 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. A. A. Kilbas, H. H. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006.
  8. F. Jarad, T. Abdeljawad, and D. Baleanu, “Caputo-type modification of the Hadamard fractional derivatives,” Advances in Difference Equations, vol. 2012, article 142, 8 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. Y. Y. Gambo, F. Jarad, D. Baleanu, and T. Abdeljawad, “On Caputo modification of the Hadamard fractional derivatives,” Advances in Difference Equations, vol. 2014, article 10, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. 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 MathSciNet · View at Scopus
  11. 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 MathSciNet · View at Scopus
  12. B. Ahmad and G. Wang, “A study of an impulsive four-point nonlocal boundary value problem of nonlinear fractional differential equations,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1341–1349, 2011. View at Google Scholar
  13. 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 MathSciNet · View at Scopus
  14. J. Cao and H. Chen, “Some results on impulsive boundary value problem for fractional differential inclusions,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 2010, no. 11, pp. 1–24, 2010. View at Publisher · View at Google Scholar
  15. G. Wang, B. Ahmad, and L. Zhang, “Impulsive anti-periodic boundary value problem for nonlinear differential equations of fractional order,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 3, pp. 792–804, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. G. Wang, L. Zhang, and G. Song, “Systems of first order impulsive functional differential equations with deviating arguments and nonlinear boundary conditions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 3, pp. 974–982, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. G. Wang, B. Ahmad, and L. Zhang, “Some existence results for impulsive nonlinear fractional differential equations with mixed boundary conditions,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1389–1397, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. X. Wang, “Impulsive boundary value problem for nonlinear differential equations of fractional order,” Computers & Mathematics with Applications, vol. 62, no. 5, pp. 2383–2391, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. M. Feckan, Y. Zhou, and J. R. Wang, “On the concept and existence of solution for impulsive fractional differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 7, pp. 3050–3060, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. I. Stamova and G. Stamov, “Stability analysis of impulsive functional systems of fractional order,” Communications in Nonlinear Science and Numerical Simulation, vol. 19, no. 3, pp. 702–709, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. S. Abbas and M. Benchohra, “Impulsive hyperbolic functional differential equations of fractional order with state-dependent delay,” Fractional Calculus and Applied Analysis, vol. 13, pp. 225–242, 2010. View at Google Scholar
  22. S. Abbas and M. Benchohra, “Upper and lower solutions method for impulsive partial hyperbolic differential equations with fractional order,” Nonlinear Analysis: Hybrid Systems, vol. 4, no. 3, pp. 406–413, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. S. Abbas, R. P. Agarwal, and M. Benchohra, “Darboux problem for impulsive partial hyperbolic differential equations of fractional order with variable times and infinite delay,” Nonlinear Analysis: Hybrid Systems, vol. 4, no. 4, pp. 818–829, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  24. S. Abbas, M. Benchohra, and L. G{\`o}rniewicz, “Existence theory for impulsive partial hyperbolic functional differential equations involving the Caputo fractional derivative,” Scientiae Mathematicae Japonicae, vol. 72, no. 1, pp. 49–60, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. M. Benchohra and D. Seba, “Impulsive partial hyperbolic fractional order differential equations in banach spaces,” Journal of Fractional Calculus and Applications, vol. 1, no. 4, pp. 1–12, 2011. View at Google Scholar
  26. T. L. Guo and K. Zhang, “Impulsive fractional partial differential equations,” Applied Mathematics and Computation, vol. 257, pp. 581–590, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  27. X. Zhang, X. Zhang, and M. Zhang, “On the concept of general solution for impulsive differential equations of fractional order q(0,1),” Applied Mathematics and Computation, vol. 247, pp. 72–89, 2014. View at Publisher · View at Google Scholar
  28. X. Zhang, “On the concept of general solution for impulsive differential equations of fractional-order q(1, 2),” Applied Mathematics and Computation, vol. 268, pp. 103–120, 2015. View at Publisher · View at Google Scholar
  29. X. Zhang, “The general solution of differential equations with Caputo-Hadamard fractional derivatives and impulsive effect,” Advances in Difference Equations, vol. 2015, article 215, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  30. X. Zhang, P. Agarwal, Z. Liu, and H. Peng, “The general solution for impulsive differential equations with Riemann-Liouville fractional-order q ϵ (1,2),” Open Mathematics, vol. 13, pp. 908–923, 2015. View at Publisher · View at Google Scholar · View at MathSciNet