The European Physical Journal Special Topics

, Volume 222, Issue 8, pp 1749–1765 | Cite as

Cauchy problem for fractional evolution equations with Caputo derivative

  • Y. ZhouEmail author
  • X. H. Shen
  • L. Zhang
Regular Article


This paper concerns the abstract nonlocal Cauchy problem of a class of fractional evolution equations with Caputo derivative. A suitable mild solution of evolution equations with Caputo derivative is introduced. In the cases C 0 semigroup is compact or noncompact, the existence theorems of mild solutions for the nonlocal Cauchy problem are established by means of fractional calculus, theory of Hausdorff measure of noncompactness and fixed point theorems.


Banach Space Point Theorem European Physical Journal Special Topic Fractional Derivative Fractional Calculus 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    L. Byszewski, J. Math. Anal. Appl. 162, 494 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    L. Byszewski, V. Lakshmikantham, Appl. Anal. 40, 11 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    M. Belmekki, M. Benchohra Nonlinear Analysis: Theory, Methods and Applications 72, 925 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    M.M. Meerschaert, D.A. Benson, H. Scheffler, B. Baeumer, Phys. Rev. E 65, 1103 (2002)MathSciNetADSCrossRefGoogle Scholar
  5. 5.
    S.D. Eidelman, A.N. Kochubei, J. Diff. Equa. 199, 211 (2004)MathSciNetCrossRefzbMATHADSGoogle Scholar
  6. 6.
    M.A. Darwish, J. Henderson, S.K. Ntouyas, Nonlinear Stud. 16, 197 (2009)MathSciNetzbMATHGoogle Scholar
  7. 7.
    W.R. Schneider, W. Wayes, J. Math. Phys. 30, 134 (1989)MathSciNetADSCrossRefzbMATHGoogle Scholar
  8. 8.
    A. Hanyga, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 458, 933 (2002)MathSciNetADSCrossRefzbMATHGoogle Scholar
  9. 9.
    L. Hu, Y. Ren, R. Sakthivel, Semigroup Forum 79, 507 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    G. Zaslavsky, chaotic advection, tracer dynamics and turbulent dispersion, Physica D 76, 110 (1994)MathSciNetzbMATHGoogle Scholar
  11. 11.
    N. Hayashi, E. I. Kaikina, P. I. Naumkin, J. London Math. Soc. 72, 663 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    E. Hernandez, D. O’Regan, Krishnan Balachandran, Nonlinear Anal. 73, 3462 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Y. Zhou, F. Jiao, Nonlinear Anal.: RWA 11, 4465 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    J. Wang, Y. Zhou, Nonlinear Anal.: RWA 12, 262 (2011)CrossRefzbMATHGoogle Scholar
  15. 15.
    J. Wang, M. Feckan, Y. Zhou, Dyn. Partial Diff. Equs. 8, 345 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    R.P. Agarwal, B. Ahmad, Dynamics Contin. Discr. Impulsive Sys., Series A: Math. Anal. 18, 457 (2011)MathSciNetzbMATHGoogle Scholar
  17. 17.
    S. Kumar, N. Sukavanam, J. Diff. Equs. 252, 6163 (2012)MathSciNetCrossRefzbMATHADSGoogle Scholar
  18. 18.
    R. Wang, D. Chen, T. J. Xiao, J. Diff. Equs. 252, 202 (2012)MathSciNetCrossRefzbMATHADSGoogle Scholar
  19. 19.
    X.B. Shu, Y.Z. Lai, Y.M. Chen, Nonlinear Anal.: TMA 74, 2003 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    K. Li, J. Peng, J. Jia, J. Functional Anal. 263, 476 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    M.M. Fall, T. Weth, J. Functional Anal. 263, 2205 (2012)Google Scholar
  22. 22.
    A.A. Kilbas, H.M. Srivastava, J. Juan Trujillo, Theory and Applications of Fractional Differential Equations (Elsevier Science B.V., Amsterdam, 2006)Google Scholar
  23. 23.
    J. Banas, K. Goebel, Measure of Noncompactness in Banach Spaces (Marcel Dekker Inc., New York, 1980)Google Scholar
  24. 24.
    K. Deimling, Nonlinear Functional Analysis (Springer-Verlag, 1985)Google Scholar
  25. 25.
    H.-P. Heinz, Nonlinear Anal.: TMA 7, 1357 (1983)MathSciNetCrossRefGoogle Scholar
  26. 26.
    V. Lakshmikantham, S. Leela, Nonlinear Differential Equations in Abstract Spaces (Pergamon Press, New York, 1969)Google Scholar
  27. 27.
    D.J. Guo, V. Lakshmikantham, X.Z. Liu, Nonlinear Integral Equations in Abstract Spaces (Kluwer Academic, Dordrecht, 1996)Google Scholar
  28. 28.
    H. Mönch, Nonlinear Anal.: TMA 4, 985 (1980)CrossRefzbMATHGoogle Scholar
  29. 29.
    D. Bothe, lsrael. J. Math. 108, 109 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    J. Wang, Y. Zhou, M. Feckan, Nonlinear Dyn., doi: 10.1007/s11071-012-0452-9 (in press)
  31. 31.
    L. Liu, F. Guo, C. Wu, Y. Wu, J. Math. Anal. Appl. 309, 638 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    F. Mainardi, P. Paraddisi, R. Gorenflo, Probability distributions generated by fractional diffusion equations, edited by J. Kertesz, I. Kondor, Econophysics: An Emerging Science (Kluwer, Dordrecht, 2000)Google Scholar
  33. 33.
    A.Z.-A.M. Tazali, Ordinary and Partial Differential Equations, Lecture Notes in Math., Springer, Dundee 964, 652 (1982)Google Scholar
  34. 34.
    A. Pazy, Applied Mathematical Sciences (Springer-Verlag, Berlin-New York, 1983), p. 44Google Scholar

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© EDP Sciences and Springer 2013

Authors and Affiliations

  1. 1.Department of MathematicsXiangtan UniversityXiangtanPR China

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