Existence of solutions of boundary value problems for nonlinear fractional differential equations with integral conditions


  • Ahcene Boukehila University of Laghouat.




Fractional differential equation, Green's function, Banach contraction principle, Schauder fixed point theorem


In this work we investigate the existence and uniqueness of solutions of boundary value problems for fractional differential equations involving the Caputo fractional derivative with integral conditions and the nonlinear term depends on the fractional derivative of an unknown function. Our existence results are based on Banach contraction principle and Schauder fixed point theorem. Two examples are provided to illustrate our results.

Author Biography

Ahcene Boukehila, University of Laghouat.

Laboratory of Pure and Applied Mathematics.


B. Ahmad and S. K. Ntouyas, ”Existence results for a coupled system of Caputo type sequential fractional differential equations with nonlocal integral boundary conditions”, Applied mathematics and computation, vol. 266, no. 1, pp. 615-622, 2015, https://doi.org/10.1016/j.amc.2015.05.116.

R. L. Bagley and P. L. Torvik, ”A theoretical basis for the application of fractional calculus to viscoelasticity”, Journal of Rheology, vol. 27, no. 3, pp. 201-210, 1983, https://doi.org/10.1122/1.549724

M. Benchohra, J. R. Graef and S. Hamani, ”Existence results for boundary value problems with non—linear fractional differential equations”, Applicable analysis , vol. 87, no. 7, pp. 851-863, 2008, https://doi.org/10.1080/00036810802307579

F. Chen, J. J. Nieto and Y. Zhou, ”Global attractivity for non-linear fractional differential equations”, Nonlinear Analysis Real World Applications, Vol. 13, no. 1, pp. 287-298, 2012, https://doi.org/10.1016/j.nonrwa.2011.07.034

D. Delbosco and L. Rodino, ”Existence and uniqueness for a nonlinear fractional differential equation”, Journal of Mathematical Analysis and Applications, vol. 204, no. 2, pp. 609-625, 1996, https://doi.org/10.1006/jmaa.1996.0456

A. Guezane-Lakoud and R. Khaldi, ”Solvability of a fractional boundary value problem with fractional integral condition”, Nonlinear Analysis: Theory, Methods and Applications, Vol 75, no. 4, pp. 2692-2700, 2012, https://doi.org/10.1016/j.na.2011.11.014.

R. Hilfer, Applications of Fractional Calculus in Physics. Singapore: World Scientific, 2000.

J. Henderson, R. Luca and A. Tudorache, ”On a system of fractional differential equations with coupled integral boundary conditions”, Fractional Calculus and Applied Analysis, vol. 18, no. 2, pp. 361-386, 2015, https://doi.org/10.1515/fca-2015-0024

A. A. Kilbas, H. M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Amsterdam: Elsevier B.V, 2006.

V. Lakshmikantham and A. S. Vatsala, ”Basic theory of fractional differential equations”, Nonlinear Analysis: Theory, Methods and Applications, vol. 69, no. 8, pp. 2677-2681, 2008, https://doi.org/10.1016/j.na.2007.08.042

C. Li and W. Deng, ”Remarks on fractional derivatives”, Applied Mathematics and Computation, vol. 187, no. 2, pp. 777-784, 2007, https://doi.org/10.1016/j.amc.2006.08.163

F. Meral, T. Royston and R. Magin, ”Fractional calculus in viscoelasticity: an experimental study”, Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 4, pp. 939-945, 2010, https://doi.org/10.1016/j.cnsns.2009.05.004

I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering. New York: Academic Press, 1999.

M. Rehman and R. Khan, ”Existence and uniqueness of solutions for multi-point boundary value problems for fractional differential equations”, Applied Mathematics Letters, vol. 23, no. 9, pp. 1038-1044, 2010, https://doi.org/10.1016/j.aml.2010.04.033

X. Su, ”Boundary value problem for a coupled system of nonlinear fractional differential equations”, Applied Mathematics Letters, vol. 22, no. 1, pp. 64-69, 2009, https://doi.org/10.1016/j.aml.2008.03.001

Y. Zhou, Basic Theory of Fractional Differential Equations. Singapore: World Scientific, 2014.



How to Cite

A. Boukehila, “Existence of solutions of boundary value problems for nonlinear fractional differential equations with integral conditions”, Proyecciones (Antofagasta, On line), vol. 40, no. 5, pp. 1117-1135, Sep. 2021.