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International Journal of Differential Equations
Volume 2013 (2013), Article ID 490673, 9 pages
http://dx.doi.org/10.1155/2013/490673
Research Article

Nonlocal Problems for Fractional Differential Equations via Resolvent Operators

1Department of Mathematics, Changshu Institute of Technology, Suzhou, Jiangsu 215500, China
2Laboratoire CEREGMIA, Université des Antilles et de la Guyane, Campus Fouillole, Guadeloupe (FWI) 97159 Pointe-à-Pitre, France

Received 7 April 2013; Accepted 15 May 2013

Academic Editor: Sotiris Ntouyas

Copyright © 2013 Zhenbin Fan and Gisèle Mophou. 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. I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. View at MathSciNet
  2. J. Prüss, Evolutionary Integral Equations and Applications, vol. 87 of Monographs in Mathematics, Birkhäuser, Basel, Switzerland, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  3. R. P. Agarwal, Y. Zhou, and Y. He, “Existence of fractional neutral functional differential equations,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1095–1100, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. R. P. Agarwal, M. Belmekki, and M. Benchohra, “Existence results for semilinear functional differential inclusions involving Riemann-Liouville fractional derivative,” Dynamics of Continuous, Discrete & Impulsive Systems, vol. 17, no. 3, pp. 347–361, 2010. View at Zentralblatt MATH · View at MathSciNet
  5. R. P. Agarwal, V. Lakshmikantham, and J. J. Nieto, “On the concept of solution for fractional differential equations with uncertainty,” Nonlinear Analysis, vol. 72, no. 6, pp. 2859–2862, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. E. Bajlekova, Fractional evolution equations in Banach spaces [Ph.D. thesis], University Press Facilities, Eindhoven University of Technology, 2001.
  7. M. Benchohra, J. Henderson, S. K. Ntouyas, and A. Ouahab, “Existence results for fractional order functional differential equations with infinite delay,” Journal of Mathematical Analysis and Applications, vol. 338, no. 2, pp. 1340–1350, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. M. Benchohra and B. A. Slimani, “Existence and uniqueness of solutions to impulsive fractional differential equations,” Electronic Journal of Differential Equations, no. 10, pp. 1–11, 2009. View at Zentralblatt MATH · View at MathSciNet
  9. M. Fečkan, J. Wang, and Y. Zhou, “Controllability of fractional functional evolution equations of Sobolev type via characteristic solution operators,” Journal of Optimization Theory and Applications, vol. 156, no. 1, pp. 79–95, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  10. E. Hernández, D. O'Regan, and K. Balachandran, “On recent developments in the theory of abstract differential equations with fractional derivatives,” Nonlinear Analysis, vol. 73, no. 10, pp. 3462–3471, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. E. Hernández, D. O'Regan, and K. Balachandran, “Existence results for abstract fractional differential equations with nonlocal conditions via resolvent operators,” Indagationes Mathematicae, vol. 24, no. 1, pp. 68–82, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. C. Lizama, “Regularized solutions for abstract Volterra equations,” Journal of Mathematical Analysis and Applications, vol. 243, no. 2, pp. 278–292, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. C. Lizama, “An operator theoretical approach to a class of fractional order differential equations,” Applied Mathematics Letters, vol. 24, no. 2, pp. 184–190, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. C. Lizama and G. M. N'Guérékata, “Bounded mild solutions for semilinear integro differential equations in Banach spaces,” Integral Equations and Operator Theory, vol. 68, no. 2, pp. 207–227, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. M. Li, C. Chen, and F.-B. Li, “On fractional powers of generators of fractional resolvent families,” Journal of Functional Analysis, vol. 259, no. 10, pp. 2702–2726, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. K. Li, J. Peng, and J. Jia, “Cauchy problems for fractional differential equations with Riemann-Liouville fractional derivatives,” Journal of Functional Analysis, vol. 263, no. 2, pp. 476–510, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  17. G. M. Mophou and G. M. N'Guérékata, “Existence of mild solutions of some semilinear neutral fractional functional evolution equations with infinite delay,” Applied Mathematics and Computation, vol. 216, no. 1, pp. 61–69, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. G. M. Mophou and G. M. N'Guérékata, “A note on a semilinear fractional differential equation of neutral type with infinite delay,” Advances in Difference Equations, vol. 2010, Article ID 674630, 8 pages, 2010. View at Zentralblatt MATH · View at MathSciNet
  19. J. Wang, Z. Fan, and Y. Zhou, “Nonlocal controllability of semilinear dynamic systems with fractional derivative in Banach spaces,” Journal of Optimization Theory and Applications, vol. 154, no. 1, pp. 292–302, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. J. Wang, Y. Zhou, and M. Fečkan, “Abstract Cauchy problem for fractional differential equations,” Nonlinear Dynamics, vol. 71, no. 4, pp. 685–700, 2013.
  21. 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
  22. L. Chen and Z. Fan, “On mild solutions to fractional differential equations with nonlocal conditions,” Electronic Journal of Qualitative Theory of Differential Equations, no. 53, pp. 1–13, 2011. View at MathSciNet
  23. Z. Fan, “Existence of nondensely defined evolution equations with nonlocal conditions,” Nonlinear Analysis, vol. 70, no. 11, pp. 3829–3836, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. Z. Fan, “Impulsive problems for semilinear differential equations with nonlocal conditions,” Nonlinear Analysis, vol. 72, no. 2, pp. 1104–1109, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. Z. Fan and G. Li, “Existence results for semilinear differential equations with nonlocal and impulsive conditions,” Journal of Functional Analysis, vol. 258, no. 5, pp. 1709–1727, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. E. Hernández, J. S. dos Santos, and K. A. G. Azevedo, “Existence of solutions for a class of abstract differential equations with nonlocal conditions,” Nonlinear Analysis, vol. 74, no. 7, pp. 2624–2634, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. J. Liang, J. H. Liu, and T.-J. Xiao, “Nonlocal problems for integrodifferential equations,” Dynamics of Continuous, Discrete & Impulsive Systems, vol. 15, no. 6, pp. 815–824, 2008. View at Zentralblatt MATH · View at MathSciNet
  28. J. Liang, J. H. Liu, and T.-J. Xiao, “Nonlocal impulsive problems for nonlinear differential equations in Banach spaces,” Mathematical and Computer Modelling, vol. 49, no. 3-4, pp. 798–804, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. J. Liang and Z. Fan, “Nonlocal impulsive Cauchy problems for evolution equations,” Advances in Difference Equations, vol. 2011, Article ID 784161, 17 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. X. M. Xue, “Existence of semilinear differential equations with nonlocal initial conditions,” Acta Mathematica Sinica, vol. 23, no. 6, pp. 983–988, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. Y. Zhou and F. Jiao, “Nonlocal Cauchy problem for fractional evolution equations,” Nonlinear Analysis, vol. 11, no. 5, pp. 4465–4475, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. L. Zhu and G. Li, “Existence results of semilinear differential equations with nonlocal initial conditions in Banach spaces,” Nonlinear Analysis, vol. 74, no. 15, pp. 5133–5140, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. L. Zhu, Q. Huang, and G. Li, “Existence and asymptotic properties of solutions of nonlinear multivalued differential inclusions with nonlocal conditions,” Journal of Mathematical Analysis and Applications, vol. 390, no. 2, pp. 523–534, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  34. J. Banaś and K. Goebel, Measures of Noncompactness in Banach Spaces, vol. 60 of Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1980. View at MathSciNet