Existence of Mild Solution for Mixed Volterra–Fredholm Integro Fractional Differential Equation with Non-instantaneous Impulses

Original Research


We establish a set of sufficient conditions for the existence of mild solution of a class of fractional mixed integro differential equation with not instantaneous impulses. The results are obtained by establishing two theorems by using semigroup theory, Banach fixed point theorem and Krasnoselskii’s fixed point theorem. Two examples are presented to validate the results of the theorems.


Mild solution Fractional differential equation Impulse Banach fixed point theorem Krasnoselskii’s fixed point theorem 



The first author is grateful to North Eastern Regional Institute of Science and Technology, Nirjuli, Arunachal Pradesh, India for granting leave for three years to pursue PhD and to Indian Institute of Technology Guwahati, Guwahati, India for providing opportunity to carry out research. Both authors express their gratitude to the esteemed reviewers for their careful reading of the manuscript and the insightful comments, and also to the Editor for allowing revision of the manuscript which has now definitely reached a much better form.


  1. 1.
    Agarwal, R., Hristova, S., O’Regan, D.: Non-instantaneous impulses in caputo fractional differential equations. Fract. Calc. Appl. Anal. 20, 595–622 (2017)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Agarwal, R., O’Regan, D., Hristova, S.: Monotone iterative technique for the initial value problem for differential equations with non-instantaneous impulses. Appl. Math. Compt. 298, 45–56 (2017)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Anguraj, A., Latha Maheswari, M.: Existence of solutions for fractional impulsive neutral functional infinite delay integro-differential equations with nonlocal conditions. J. Nonlinear Sci. Appl. 20, 271–280 (2017)MATHGoogle Scholar
  4. 4.
    Bai, L., Nieto, J.J.: Variational approach to differential equations with not instantaneous impulses. Appl. Math. Lett. 73, 44–48 (2017)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bai, L., Nieto, J.J., Wang, X.: Variational approach to non-instantaneous impulsive nonlinear differential equations. J. Nonlinear Sci. Appl. 10, 2440–2448 (2017)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bragdi, M., Hazi, M.: Existence and uniqueness of solutions of fractional quasilinear mixed integro-differential equations with nonlocal condition in Banach spaces. Electron. J. Qual. Theory Differ. Equ. 2012(51), 1–16 (2012)CrossRefMATHGoogle Scholar
  7. 7.
    Chang, Y.K., Kavitha, V., Arjunan, M.M.: Existence and uniqueness of mild solutions to a semilinear integrodifferential equation of fractional order. Nonlinear Anal. 71(11), 5551–5559 (2009)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    El-Sayed, A.M.A.: Fractional order evolution equations. J. Fract. Calc. 7(1), 995 (1995)MathSciNetMATHGoogle Scholar
  9. 9.
    Fec, M., Zhou, Y., Wang, J.R.: On the concept and existence of solution for impulsive fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 17(7), 3050–3060 (2012)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Gautam, G.R., Dabas, J.: Mild solution for fractional functional integro-differential equation with not instantaneous impulse. Malaya J. Math. 2(3), 428–437 (2012)MATHGoogle Scholar
  11. 11.
    Hernández, E., O’Regan, D.: On a new class of abstract impulsive differential equations. Proc. Am. Math. Soc. 141(5), 1641–1649 (2013)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)MATHGoogle Scholar
  13. 13.
    Kumar, P., Pandey, D.N., Bahuguna, D.: On a new class of abstract impulsive functional differential equations of fractional order. J. Nonlinear Sci. Appl. 7(2), 102–114 (2014)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, Amsterdam (1993)MATHGoogle Scholar
  15. 15.
    Mophou, Gisèle M.: Existence and uniqueness of mild solutions to impulsive fractional differential equations. Nonlinear Anal. 72(3), 1604–1615 (2010)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    N’Guérékata, G.M.: A Cauchy problem for some fractional abstract differential equation with non local conditions. Nonlinear Anal. 70(5), 1873–1876 (2009)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Pierri, M., O’Regan, D., Rolnik, V.: Existence of solutions for semi-linear abstract differential equations with not instantaneous impulses. Appl. Math. Comput. 219(12), 6743–6749 (2013)MathSciNetMATHGoogle Scholar
  18. 18.
    Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, vol. 198. Academic Press, New York (1998)MATHGoogle Scholar
  19. 19.
    Ravichandran, C., Arjunan, M.M.: Existence results for abstract mixed type impulsive fractional semilinear evolution equations. Int. J. Math. Sci. Comput. 2(1), 14–21 (2012)MathSciNetGoogle Scholar
  20. 20.
    Samoilenko, A.M., Perestyuk, N.A.: Impulsive Differential Equations, vol. 14. World Scientific, Singapore (1995)MATHGoogle Scholar
  21. 21.
    Suganya, S., Baleanu, D., Kalamani, P., Arjunan, M.M.: On fractional neutral integro-differential systems with state-dependent delay and non-instantaneous impulses. Adv. Differ. Equ. 2015(1), 372 (2015)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Wang, J.R., Fečkan, M., Zhou, Y.: On the new concept of solutions and existence results for impulsive fractional evolution equations. Dyn. Partial Differ. Equ. 8(4), 345–361 (2011)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Wang, J.R., Zhou, Y., Lin, Z.: On a new class of impulsive fractional differential equations. Appl. Math. Comput. 242, 649–657 (2014)MathSciNetMATHGoogle Scholar
  24. 24.
    Wang, J.R., Zhou, Y., Wei, W., Xu, H.: Nonlocal problems for fractional integrodifferential equations via fractional operators and optimal controls. Comput. Math. Appl. 62(3), 1427–1441 (2011)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Zhou, Y., Jiao, F.: Existence of mild solutions for fractional neutral evolution equations. Comput. Math. Appl. 59(3), 1063–1077 (2010)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Zhou, Y., Wang, J.R., Zhang, L.: Basic Theory of Fractional Differential Equations. World Scientific, Singapore (2016)CrossRefGoogle Scholar

Copyright information

© Foundation for Scientific Research and Technological Innovation 2018

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology GuwahatiGuwahatiIndia

Personalised recommendations