Skip Navigation Links
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


Existence of Mild Solutions of Abstract Fractional Differential Equations with Fractional Non-Instantaneous Impulsive Conditions

Discontinuity, Nonlinearity, and Complexity 6(2) (2017) 173--183 | DOI:10.5890/DNC.2017.06.005

A. Anguraj$^{1}$, S. Kanjanadevi$^{1}$ , Juan J. Trujillo$^{2}$

$^{1}$ PSG College of Arts and Science, Coimbatore- 641014, Tamil Nadu, India

$^{2}$ Departamento de Matemáticas, Universidad de La Laguna, Estadística, e I.O., 38271 La Laguna, Tenerife, Spain

Download Full Text PDF



We study the existence and uniqueness of a mild solution of fractional impulsive differential equations with nonlocal conditions. Here we consider fractional derivative in the non-instantaneous impulsive conditions. We use fixed point techniques and resolvent operators to prove our existence results.


This work was partially funded by FEDER and project MTM2013-41704-P from the government of Spain.


  1. [1]  Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J. (2006), Theory and application of fractional differential equations, North-Holland, Amsterdam.
  2. [2]  Miller, K.S. and Ross, B. (1993), An introduction to the fractional calculus and fractional differential equations,Wiley, New York.
  3. [3]  Podlubny, I., (1999), Fractional differential equations,Mathematics in Sciences and Engineering, Academic press, San Diego.
  4. [4]  Delbosco, D. and Rodino, L. (1996), Existence and uniqueness for a nonlinear fractional differential equations, Journal of Mathematical Analysis and Applications, 204, 609-625.
  5. [5]  Delbosco, D. (1994), Fractional calculus and function spaces, Journal of Fractional Calculus, 6, 45-53.
  6. [6]  Lakshmikantham, V. (2008), Theory of fractional functional differential equations, Nonlinear Analysis, 69, 3337-3343.
  7. [7]  Anguraj, A. and Mallika Arjunan, M. (2005), Existence and uniqueness of mild and classical solutions of impulsive evolution equations, Electronic Journal of Differential Equations, 111, 1-8.
  8. [8]  Anguraj, A. and Karthikeyan, K. (2009), Existence of solutions for impulsive neutral functional differential equations with nonlocal conditions, Nonlinear Analysis, 70, 2717-2721.
  9. [9]  Fan, Z. and Gang, L. (2010), Existence results for semilinear differential equations with nonlocal conditions, Journal of Functional Analysis, 258, 1709-1727.
  10. [10]  Lakshmikantham, V., Bainov, D.D., and Simeonov, P.S. (1989), Theory of impulsive differential equations, World Scientific, Singapore.
  11. [11]  Liu, J.H. (1999), Nonlinear impulsive evolution equations, Dynamics ContinuousDiscrete Impulsive Systems, 6, 77-85.
  12. [12]  Nieto, J.J. and O’Regan, D. (2009), Variational approach to impulsive differential equations, Nonlinear Analysis, Real World Applications, 10, 680-690.
  13. [13]  Hernández, 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]  Belmekki, M. and Benchohra, M. (2010), Existence results for fractional order semilinear functional differential equations with nondense domain, Nonlinear Analysis, 72, 925-932.
  15. [15]  Jaradat, O.K., Al-Omari, A., and Momani, S. (2008), Existence of mild solutions for fractional semilinear initial value problems, Nonlinear Analysis, 69(9), 3153-3159.
  16. [16]  Mophou, G.M. (2010), Existence and uniqueness of mild solutions to impulsive fractional differential equations, Nonlinear Analysis, 72, 1604-1615.
  17. [17]  Pandey, D.N., Ujlayan, A., and Bahuguna, D. (2009), On a solution to fractional order integrodifferential equations with analytic semigroups, Nonlinear Analysis, 71, 3690-3698.
  18. [18]  Prüss, J. (1993), Evolutionary integral equations and Applications, Monographs in Mathematics, 87, Birkhäuser Verlag, Basel.
  19. [19]  Hernández, E. and O’Regan, D. (2013), On a new class of abstract impulsive differential equations, Proceedings of the American Mathematical Society, 141, 1641-1649.
  20. [20]  Anguraj, A., Karthikeyan, P., and N’uerekata, G.M. (2009), Nonlocal cauchy problem for some fractional abstract integrodifferential equations in Banach spaces, Communications in Mathematical Analysis, 1, 31-35.
  21. [21]  Chandha, A. and Pandey, D.N. (2014), Existence of the mild solution for impulsive semilinear differential equation, International Journal of Partial Differential Equations, Article ID 640931, 8 pages.
  22. [22]  Gautam, G.R. and Dabas, J. (2014), Existence result of fractional functional integro-differential equation with not instantaneous impulse, International Journal of Advances in Applied Mathematics and Mechanics, 1(3), 11-21.
  23. [23]  Pierri, M., O’Regan, D., and Rolink, V. (2013), Existence of solutions for semi-linear absract differential equations with not instantaneous impulses, Applied mathematics and computation, 219, 6743-6749.
  24. [24]  Wang, J. and Li, X. (2014), Periodic BVP for integer/fractional order nonlinear differential equations with noninstantaneous impulses, Journal of Applied Mathematics and Computation, 46, 321-334.
  25. [25]  Wang, J., Zhou, Y., and Lin, Z. (2014), On a new class of impulsive fractional differential equations, Applied Mathematics and Computation, 242, 649-657.
  26. [26]  Balachandran, K. and Trujillo, J.J. (2010), The nonlocal cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces, Nonlinear Analysis, 72, 4587-4593.
  27. [27]  Byszewski, L. (1991), Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal cauchy problem, Journal of Mathematical Analysis and Applications, 162, 494-505.
  28. [28]  Chandrasekaran, M. (2007), Nonlocal cauchy problem for quasilinear integrodifferential equations in Banach spaces, Electronic Journal of Differential Equations, 2007, 1-6.
  29. [29]  Hernández, E., O’Regan, D. and Balachandran,K. (2013), Existence results for abstract fractional differential equations with nonlocal conditions via resolvent operators, Indagationes mathematicae, 24, 68-82.
  30. [30]  Lin, Y. and Liu, J.H. (1996), Semilinear integrodifferential equations with nonlocal cauchy problem, Nonlinear Analysis, 26, 1023-1033.
  31. [31]  Kosmatov, N. (2013), Initial value problems of fractional order with fractional impulsive conditions, Results in Mathematics, 63, 1289-1310.