Skip Navigation Links
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain


Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email:

On Representation of Solutions of Abstract Fractional Differential Equations

Journal of Applied Nonlinear Dynamics 8(4) (2019) 677--687 | DOI:10.5890/JAND.2019.12.012

K. Balachandran$^{1}$, R. Mabel Lizzy$^{2}$, J. J. Trujillo$^{3}$

$^{1}$ Department of Mathematics, Bharathiar University, Coimbatore 641 046, India

$^{2}$ Chair of Numerical Analysis, Technical University of Munich, 85748 Garching, Germany

$^{3}$ Department de An´alisis Mathem´atico, Universidad de La Laquna, 38271 La Laguna, Tenerife, Spain

Download Full Text PDF



In this paper we discuss about the solution representation of abstract fractional differential equations with different conditions on the operator A. The solution representation obtained by using Mittag-Leffler function is correct when the operator A is bounded. However when the operator A is unbounded the representation derived in [1] seems to be incorrect. We also obtain a suitable form of solution to stochastic fractional differential equation with unbounded operators.


The first author is thankful to the University Grant Commission for providing a UGC-BSR Faculty Fellowship to carry out this work. The second author is supported by the University Grant Commission under the grant no. MANF-2015-17-TAM-50645 from the Government of India.


  1. [1]  El-Bhori, M. (2002), Some probability densities and fundamental solutions of fractional evolution equations, Chaos, Solitons and Fractals, 14, 433-440.
  2. [2]  Podlubny, I. (1999), Fractional Differential Equations, Academic Press, New York.
  3. [3]  Mainardi, F., Mura, A., and Pagnini, G. (2010), The M-Wright function in time-fractional diffusion processes: a tutorial survey, International Journal of Differential Equations, 2010, 104505 (29 pages)
  4. [4]  Bazhlekova, E. (1998), The abstract Cauchy problem for the fractional evolution equation, Fractional Calculus and Applied Analysis, 1, 255-270.
  5. [5]  Kilbas, A., Srivastava, H.M., and Trujillo, J.J. (2006), Theory and Application of Fractional Differential Equations, Elsevier, New York.
  6. [6]  K. Balachandran, Matar, M., and Trujillo, J.J. (2016), Note on controllability of linear fractional dynamical systems, Journal of Control and Decision, 3, 267-279.
  7. [7]  Kreyszig, E. (1978), Introductory Functional Analysis with Applications, JohnWiley and Sons Inc, New York.
  8. [8]  Zhou, Y., Shen, X.H., and Zhang, L. (2013), Cauchy problem for fractional evolution equations with Caputo derivative, The European Physical Journal Special Topics, 222, 1747-1764.
  9. [9]  Pazy, A. (1983), Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer- Verlag, New York Inc..
  10. [10]  Chow, P.L. (2007), Stochastic Partial Differential Equations, Chapman & Hall/CRC, New York.
  11. [11]  Da Prato, G. and Zabczyk, J. (1992), Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge.
  12. [12]  K. Balachandran and S. Kiruthika, (2011), Existence results for fractional integrodifferential equations with nonlocal conditions via resolvent operators, Computers and Mathematics with Applications, 62, 1350-1358.
  13. [13]  Hernandez, E., O’Regan, D., and Balachandran, K. (2010), On recent developments in the theory of abstract differential equations with fractional derivatives, Nonlinear Analysis, 73, 3462-3471.
  14. [14]  Prüss, J. (1993), Evolutionary Integral Equations and Applications, Birkh¨auser Verlag, Basel.
  15. [15]  Karczewska, A. and Lizama, C. (2007), On stochastic fractional Volterra equations in Hilbert space, Discrete and Continuous Dynamical Systems, 2007, 541-550.
  16. [16]  Miller, K.S. and Ross, B. (1993), An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley and Sons, New York.