Results in Mathematics

, Volume 63, Issue 3–4, pp 1289–1310 | Cite as

Initial Value Problems of Fractional Order with Fractional Impulsive Conditions

  • Nickolai KosmatovEmail author


In this paper we intend to accomplish two tasks firstly, we address some basic errors in several recent results involving impulsive fractional equations with the Caputo derivative, and, secondly, we study initial value problems for nonlinear differential equations with the Riemann–Liouville derivative of order 0 < α ≤ 1 and the Caputo derivatives of order 1 < δ < 2. In both cases, the corresponding fractional derivative of lower order is involved in the formulation of impulsive conditions.

Mathematics Subject Classification (2000)

34A08 34A37 


Caputo derivative fixed point impulsive condition Riemann–Liouville derivative 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Agarwal R.P., Benchohra M., Slimani B.A.: Existence results for differential equations with fractional order and impulses. Mem. Differ. Equ. Math. Phys. 44, 1–21 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Agarwal R.P., Karakoç F.: A survey on oscillation of impulsive delay differential equations. Comp. Math. Appl. 60, 1648–1685 (2010)zbMATHCrossRefGoogle Scholar
  3. 3.
    Ahmad, B., Nieto, J.J.: Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions. Bound. Value Probl. 2009, 1–11 (2009), ID 708576Google Scholar
  4. 4.
    Ahmad B., Sivasundaram S.: Existence of solutions for impulsive integral boundary value problems of fractional order. Nonlinear Anal. Hybrid Syst. 4, 134–141 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bai, C.: Existence of positive solutions for a functional fractional boundary value problem. Abstr. Appl. Anal. 2010, 1–13 (2010), Art. ID 127363Google Scholar
  6. 6.
    Bainov D.D., Simeonov P.S.: Systems with Impulse Effect: Stability, Theory and Applications. In: Ellis Horwood Series: Mathematics and its Applications. Ellis Horwood, Chichester (1989)Google Scholar
  7. 7.
    Benchohra M., Henderson J., Ntouyas S.K.: Impulsive Differential Equations and Inclusions. Hindawi Publishing Corporation, New York (2006)zbMATHCrossRefGoogle Scholar
  8. 8.
    Benchohra, M., Slimani, B.A.: Existence and uniqueness of solutions to impulsive fractional differential equations. Electron. J. Differ. Equ. (no. 10), 1–11 (2009)Google Scholar
  9. 9.
    Benchohra, M., Hamani, S., Nieto, J.J., Slimani, B.A.: Existence of solutions to differential inclusions with fractional order and impusses. Electron. J. Differ. Equ. (no. 80), 1–18 (2010)Google Scholar
  10. 10.
    Benchohra, M., Seba, D.: Impulsive fractional differential equations in Banach spaces. Electron. J. Qual. Theory Differ. Equ. Special Edn. I. (no. 8), 1–14 (2009)Google Scholar
  11. 11.
    Boichuk A.A., Samoilenko A.M.: Generalised inverse operators and Fredholm boundary-value problems. VSP, Utrecht (2004)CrossRefGoogle Scholar
  12. 12.
    Caputo M.: Linear models of dissipation whose Q is almost frequency independent (Part II). Geophys. J. R. Astron. Soc. 13, 529–539 (1967)CrossRefGoogle Scholar
  13. 13.
    Chen J., Tisdell C.C., Yuan R.: On the solvability of periodic boundary value problems with impulse. J. Math. Anal. Appl. 331, 902–912 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Goodrich C.: Continuity of solutions to discrete fractional initial value problems. Comp. Math. Appl. 59, 3489–3499 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Guo D.: Existence of positive solutions for nth-order nonlinear impulsive singular integro-differential equations in Banach spaces. Nonlinear Anal. 68, 2727–2740 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Guo D.: Positive solutions of an infinite boundary value problem for nth-order nonlinear impulsive singular integro-differential equations in Banach spaces. Nonlinear Anal. 70, 2078–2090 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Herzallah, M.A.E., Baleanu, D.: Fractional-order variational calculus with generalized boundary conditions. Adv. Differ. Equ. 2011, 1–9 (2011), ID 357580Google Scholar
  18. 18.
    Kaufmann, E.R.: Impulsive periodic boundary value problems for dynamic equations on time scale. Adv. Differ. Equ. 2009, 1–10 (2009), ID 603271Google Scholar
  19. 19.
    Kilbas A.A., Srivastava H.M., Trujillo J.J.: Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies. Elsevier Science, Amsterdam (2006)Google Scholar
  20. 20.
    Krasnosel’skiĭ M.A.: Some problems in nonlinear analysis. Amer. Math. Soc. Transl. Ser. 2(10), 345–409 (1958)Google Scholar
  21. 21.
    Labidi S., Tatar N.: Unboundedness for the Euler–Bernoulli beam equation with a fractional boundary dissipation. Appl. Math. Comput. 161, 697–706 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Lakshmikantham V.: Theory of fractional functional differential equations. Nonlinear Anal. 69, 3337–3343 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Lakshmikantham V., Bainov D.D., Simeonov P.S.: Theory of impulsive differential equations. Series in modern applied mathematics. World Scientific, New Jersey (1994)Google Scholar
  24. 24.
    Lakshmikantham V., Vatsala A.S.: Basic theory of fractional differential equations. Nonlinear Anal. 69, 2677–2682 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Luo Z., Nieto J.J.: New results for the periodic boundary value problem for impulsive integro-differential equations. Nonlinear Anal. 70, 2248–2260 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Luo Z., Jing Z.: Periodic boundary value problem for first-order impulsive functional differential equations. Comp. Math. Appl. 55, 2094–2107 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Podlubny I.: Fractional Differential Equations, Mathematics in Sciences and Applications. Academic Press, New York (1999)Google Scholar
  28. 28.
    Sabatier J., Agrawal O.P., Tenreiro-Machado J.A.: Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, The Netherlands (2007)zbMATHCrossRefGoogle Scholar
  29. 29.
    Samko S.G., Kilbas A.A., Mirichev O.I.: Fractional Integral and Derivatives (Theory and Applications). Gordon and Breach, Switzerland (1993)Google Scholar
  30. 30.
    Samoĭlenko A.M., Perestyuk N.A.: Impulsive Differential Equations, World Scientific Series on Nonlinear Science, Series A: Monographs and Treatises. World Scientific, New Jersey (1995)Google Scholar
  31. 31.
    Shu, X.B., Lai, Y., Chen, Y.: The existence of mild solutions for impulsive fractional partial differential equations. Nonlinear Anal. (2012) (in press)Google Scholar
  32. 32.
    Tian Y., Bai Z.: Existence results for the three-point impulsive boundary value problem involving fractional differential equations. Comp. Math. Appl. 59, 2601–2609 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Tian Y., Ge W.: Variational methods to Sturm–Liouville boundary value problem for impulsive differential equations. Nonlinear Anal. 72, 277–287 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Wang, J., Zhou, Y., Fecčkan, M.: On recent developments in the theory of boundary value problem for impulsive fractional differential equations. Comp. Math. Appl. doi: 10.1016/j.camwa.2011.12.064
  35. 35.
    Wei Z., Li Q., Che J.: Initial value problems for fractional differential equations involving Riemann-Liouville sequential fractional derivative. J. Math. Anal. Appl. 367, 260–272 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Zhang X., Huang X., Lou Z.: The existence and uniqueness of mild solutions for impulsive fractional equations with nonlocal conditions and infinite delay. Nonlinear Anal. Hybrid Syst. 4, 775–781 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Zeidler E.: Nonlinear Functional Analysis and Applications, I: Fixed Point Theorems. Springer, New York (1986)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of Arkansas at Little RockLittle RockUSA

Personalised recommendations