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Abstract and Applied Analysis
Volume 2014, Article ID 730695, 8 pages
http://dx.doi.org/10.1155/2014/730695
Research Article

Complete Controllability for Fractional Evolution Equations

Xia Yang1 and Haibo Gu2,3

1School of Science, Shihezi University, Shihezi, Xinjiang 832003, China
2School of Mathematics Sciences, Xinjiang Normal University, Urumqi, Xinjiang 830054, China
3School of Mathematics and Computational Science, Xiangtan University, Xiangtan, Hunan 411105, China

Received 3 December 2013; Revised 13 March 2014; Accepted 20 March 2014; Published 10 April 2014

Academic Editor: Simeon Reich

Copyright © 2014 Xia Yang and Haibo Gu. 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, H. M. Srivastava, and J. J. Trujillo, “Theory an application of fractional differential equations,” in North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V., Amsterdam, The Netherlands, 2006. View at Google Scholar
  2. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley & Sons, New York, NY, USA, 1993. View at MathSciNet
  3. I. Podlubny, Fractional Differential Equations, vol. 198, Academic Press, San Diego, Calif, USA, 1999. View at MathSciNet
  4. V. Lakshmikantham, S. Leela, and J. V. Devi, Theory of Fractional Dynamic Systems, Cambridge Scientific, Cambridge, UK, 2009.
  5. 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 A: Theory and Methods, vol. 72, no. 6, pp. 2859–2862, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. Z. B. Bai and Y. H. Zhang, “Solvability of fractional three-point boundary value problems with nonlinear growth,” Applied Mathematics and Computation, vol. 218, no. 5, pp. 1719–1725, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. E. Bazhlekova, Fractional evolution equations in Banach space [Ph.D. thesis], Eindhoven University of Technology, Eindhoven, The Netherlands, 2001.
  8. M. Benchohra and A. Ouahab, “Controllability results for functional semilinear differential inclusions in Fréchet spaces,” Nonlinear Analysis. Theory, Methods & Applications A: Theory and Methods, vol. 61, no. 3, pp. 405–423, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. F. L. Chen, J. J. Nieto, and Y. Zhou, “Global attractivity for nonlinear fractional differential equations,” Nonlinear Analysis. Real World Applications, vol. 13, no. 1, pp. 287–298, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. S. D. Eidelman and A. N. Kochubei, “Cauchy problem for fractional diffusion equations,” Journal of Differential Equations, vol. 199, no. 2, pp. 211–255, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. V. Lakshmikantham and A. S. Vatsala, “Basic theory of fractional differential equations,” Nonlinear Analysis. Theory, Methods & Applications A: Theory and Methods, vol. 69, no. 8, pp. 2677–2682, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. J. P. Yan and C. P. Li, “On chaos synchronization of fractional differential equations,” Chaos, Solitons & Fractals, vol. 32, no. 2, pp. 725–735, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. Y. Li, Y. Chen, and I. Podlubny, “Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability,” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1810–1821, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. H. P. Ye, J. M. Gao, and Y. S. Ding, “A generalized Gronwall inequality and its application to a fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 328, no. 2, pp. 1075–1081, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. R. P. Agarwal, M. Belmekki, and M. Benchohra, “A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative,” Advances in Difference Equations, vol. 2009, Article ID 981728, 47 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. M. Belmekki and M. Benchohra, “Existence results for fractional order semilinear functional differential equations with nondense domain,” Nonlinear Analysis. Theory, Methods & Applications A: Theory and Methods, vol. 72, no. 2, pp. 925–932, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. E. Hernández, D. O'Regan, and K. Balachandran, “On recent developments in the theory of abstract differential equations with fractional derivatives,” Nonlinear Analysis. Theory, Methods & Applications A: Theory and Methods, vol. 73, no. 10, pp. 3462–3471, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. L. Hu, Y. Ren, and R. Sakthivel, “Existence and uniqueness of mild solutions for semilinear integro-differential equations of fractional order with nonlocal initial conditions and delays,” Semigroup Forum, vol. 79, no. 3, pp. 507–514, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. G. M. Mophou, “Existence and uniqueness of mild solutions to impulsive fractional differential equations,” Nonlinear Analysis. Theory, Methods & Applications A: Theory and Methods, vol. 72, no. 3-4, pp. 1604–1615, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. X.-B. Shu, Y. Lai, and Y. Chen, “The existence of mild solutions for impulsive fractional partial differential equations,” Nonlinear Analysis. Theory, Methods & Applications A: Theory and Methods, vol. 74, no. 5, pp. 2003–2011, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. J. R. Wang and Y. Zhou, “Complete controllability of fractional evolution systems,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 11, pp. 4346–4355, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. Y. Zhou and F. Jiao, “Nonlocal Cauchy problem for fractional evolution equations,” Nonlinear Analysis. Real World Applications, vol. 11, no. 5, pp. 4465–4475, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. Y. Zhou and F. Jiao, “Existence of mild solutions for fractional neutral evolution equations,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1063–1077, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. M. Fečkan, Y. Zhou, and J. 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 Zentralblatt MATH · View at MathSciNet
  25. X.-B. Shu and Q. Wang, “The existence and uniqueness of mild solutions for fractional differential equations with nonlocal conditions of order 1<α<2,” Computers & Mathematics with Applications, vol. 64, no. 6, pp. 2100–2110, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. R. P. Agarwal, B. Ahmad, A. Alsaedi, and N. Shahzad, “Existence and dimension of the set of mild solutions to semilinear fractional differential inclusions,” Advances in Difference Equations, vol. 2012, p. 74, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  27. N. U. Ahmed, Dynamic Systems and Control with Applications, World Scientific, Hackensack, NJ, USA, 2006. View at MathSciNet
  28. K. Balachandran and R. Sakthivel, “Controllability of functional semilinear integrodifferential systems in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 255, no. 2, pp. 447–457, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. Y. K. Chang, J. J. Nieto, and W. S. Li, “Controllability of semilinear differential systems with nonlocal initial conditions in Banach spaces,” Journal of Optimization Theory and Applications, vol. 142, no. 2, pp. 267–273, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. Z. Tai and X. Wang, “Controllability of fractional-order impulsive neutral functional infinite delay integrodifferential systems in Banach spaces,” Applied Mathematics Letters, vol. 22, no. 11, pp. 1760–1765, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. Z. Yan, “Controllability of fractional-order partial neutral functional integrodifferential inclusions with infinite delay,” Journal of the Franklin Institute. Engineering and Applied Mathematics, vol. 348, no. 8, pp. 2156–2173, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. L. Byszewski, “Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem,” Journal of Mathematical Analysis and Applications, vol. 162, no. 2, pp. 494–505, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. L. Byszewski and V. Lakshmikantham, “Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space,” Applicable Analysis, vol. 40, no. 1, pp. 11–19, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  34. S. Reich, “Fixed points of condensing functions,” Journal of Mathematical Analysis and Applications, vol. 41, pp. 460–467, 1973. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet