Existence of Solutions for a Class of Fractional Boundary Value Equations with Impulsive Effects via Critical Point Theory

  • Nemat Nyamoradi
  • Elham Tayyebi


In this paper, we study the existence of weak solutions for a class of fractional boundary value equations with impulsive effects. Existence results are obtained using the variational methods and the critical point theory. Our theorems mainly extend the recent results of Wang et al. (Mediterr J Math 13(6): 4845–4866, 2016). Finally, some examples are presented to illustrate our results.


Fractional differential equations Impulsive problems Variational methods Critical points 

Mathematics Subject Classification

34A08 34B15 34B37 


  1. 1.
    Diethelm, K., Freed, A.: On the solution of nonlinear fractional order differential equations used in the modeling of viscoelasticity. In: Keil, F., Mackens, W., Voss, H., Werther, J. (eds.) Scientific Computing in Chemical Engineering II-Computational Fluid Dynamics, Reaction Engineering and Molecular Properties, pp. 217–224. Springer, Heidelberg (1999)Google Scholar
  2. 2.
    Lundstrom, B., Higgs, M., Spain, W., Fairhall, A.: Fractional differentiation by neocortical pyramidal neurons. Nat. Neurosci. 11, 1335–1342 (2008)CrossRefGoogle Scholar
  3. 3.
    Glockle, W., Nonnenmacher, T.: A fractional calculus approach of self-similar protein dynamics. Biophys. J. 68, 46–53 (1995)CrossRefGoogle Scholar
  4. 4.
    Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)CrossRefzbMATHGoogle Scholar
  5. 5.
    Mainardi, F.: Fractional calculus, some basic problems in continuum and statistical mechanics. In: Carpinteri, A., Mainardi, F. (eds.) Fractals and Fractional Calculus in Continuum Mechanics, pp. 291–348. Springer, Wien (1997)CrossRefGoogle Scholar
  6. 6.
    Kirchner, J., Feng, X., Neal, C.: Fractal streamchemistry and its implications for contaminant transport in catchments. Nature 403, 524–526 (2000)CrossRefGoogle Scholar
  7. 7.
    Kilbas, A., Srivastava, H., Trujillo, J.: Theory and applications of fractional differential equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)Google Scholar
  8. 8.
    Miller, K., Ross, B.: An Introduction to the Fractional Calculus and Differential Equations. Wiley, New York (1993)zbMATHGoogle Scholar
  9. 9.
    Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)zbMATHGoogle Scholar
  10. 10.
    Samko, S., Kilbas, A., Marichev, O.: Fractional Integral and Derivatives, Theory and Applications. Gordon and Breach, Longhorne (1993)zbMATHGoogle Scholar
  11. 11.
    Benson, D., Wheatcraft, S., Meerschaert, M.: Application of a fractional advection-dispersion equation. Water Resour. Res. 36, 1403–1412 (2000)CrossRefGoogle Scholar
  12. 12.
    Benson, D., Wheatcraft, S., Meerschaert, M.: The fractional-order governing equation of Lévy motion. Water Resour. Res. 36, 1413–1423 (2000)CrossRefGoogle Scholar
  13. 13.
    Fix, G., Roop, J.: Least squares finite-element solution of a fractional order two-point boundary value problem. Comput. Math. Appl. 48, 1017–1033 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Erwin, V., Roop, J.: Variational formulation for the stationary fractional advection dispersion equation. Numer. Methods Partial Differ. Equ. 22, 58–76 (2006)MathSciNetGoogle Scholar
  15. 15.
    Lakshmikantham, V., Vatsala, A.: Basic theory of fractional differential equations. Nonlinear Anal. TMA 69, 2677–2682 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Devi, J., Lakshmikantham, V.: Nonsmooth analysis and fractional differential equations. Nonlinear Anal. TMA 70, 4151–4157 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Zhou, Y., Jiao, F., Li, J.: Existence and uniqueness for p-type fractional neutral differential equations. Nonlinear Anal. TMA 71, 2724–2733 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Zhou, Y., Jiao, F., Li, J.: Existence and uniqueness for fractional neutral differential equations with infinite delay. Nonlinear Anal. TMA 71, 3249–3256 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Zhou, Y., Jiao, F.: Nonlocal Cauchy problem for fractional evolution equations. Nonlinear Anal. RWA 11, 4465–4475 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Wang, J., Zhou, Y.: A class of fractional evolution equations and optimal controls. Nonlinear Anal. RWA 12, 262–272 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Nieto, J.: Variational formulation of a damped Dirichlet impulsive problem. Appl. Math. Lett. 23, 940–942 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Agarwal, R., Benchohra, M., Hamani, S.: A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Appl. Math. 109, 973–1033 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Zhang, S.: Positive solutions to singular boundary value problem for nonlinear fractional differential equation. Comput. Math. Appl. 59, 1300–1309 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Nyamoradi, N.: Existence and multiplicity of solutions for impulsive fractional differential equations. Mediterr. J. Math 14, 85 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Wang, Y., Li, Y., Zhou, J.: Solvability of boundary value problems for impulsive fractional differential equations via critical point theory. Mediterr. J. Math. 13(6), 4845–4866 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Bai, C.: Existence of solutions for a nonlinear fractional boundary value problem via a local minimum theorem. Elect. J. Differ. Equat. 176, 1–9 (2012)MathSciNetGoogle Scholar
  27. 27.
    Bonanno, G., Rodriguez-Löpez, R., Tersian, S.: Existence of solutions to boundary value problem for impulsive fractional differential equations. Fract. Calc. Appl. Anal. 17(3), 717–744 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Rodriguez-Löpez, R., Tersian, S.: Multiple solutions to boundary value problem for impulsive fractional differential equations. Fract. Calc. Appl. Anal. 17(4), 1016–1038 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Jiao, F., Zhou, Y.: Existence of solutions for a class of fractional boundary value problems via critical point theory. Comput. Math. Appl. 62, 1181–1199 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Brézis, H., Nirenberg, L.: Remarks on finding critical points. Commun. Pure Appl. Math. 44, 939–963 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Rabinowitz, P.H.: Minimax methods in critical point theory with applications to differential equations. CBMS Regional Conference Series in Mathematics, vol. 65. Washington, DC: American Mathematical Society (1986)Google Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of SciencesRazi UniversityKermanshahIran

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