Abstract
In this paper, by applying the Schauder fixed point theorem, the Leray–Schauder nonlinear alternative and the Banach contraction principle, we establish some sufficient conditions for the existence and uniqueness of solutions for a coupled system of nonlinear fractional differential equations with fractional integral conditions, involving the Caputo fractional derivative. Some examples are given to illustrate our results.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)
Podlubny, I.: Fractional Differential Equation. Academic Press, San Diego (1999)
Kilbas, A.A., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
Lakshmikantham, V., Leela, S., Vasundhara, D.J.: Theory of Fractional Dynamic Systems. Cambridge Scientific Publ, Cambridge (2009)
Webb, J.R.L., Infante, G.: Nonlocal boundary value problems of arbitrary order. J. Lond. Math. Soc. 79(1), 238–258 (2009)
Wei, Z., Li, Q., Che, J.: Initial value problems for fractional differential equations involving Rieman–Liouville sequential fractional derivative. J. Math. Anal. Appl. 367(1), 260–272 (2010)
Agarwal, R.P., O’Regan, D., Stane, K.S.: Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations. J. Math. Anal. Appl. 371(1), 57–68 (2010)
Bai, Z.: On positive solutions of a nonlocal fractional boundary value problem. Nonlinear Anal. 72(2), 916–924 (2010)
Zhao, Y., Sun, S., Han, Z., et al.: The existence of multiple positive solutions for boundary value problems of nonlinear fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 16(4), 2086–2097 (2011)
Cabada, A., Wang, G.: Positive solutions of nonlinear fractional differential equations with integral boundary value conditions. J. Math. Anal. Appl. 389(1), 403–411 (2012)
Jiang, W.: Eigenvalue interval for multi-point boundary value problems of fractional differential equations. Appl. Math. Comput. 219(9), 4570–4575 (2013)
Ahmad, B., Ntouyas, S.K., Alsaedi, A.: An existence result for fractional differential inclusions with nonlinear integral boundary conditions. J. Inequal. Appl. 296(1), 1–9 (2013)
Dahal, R., Duncan, D., Goodrich, C.S.: Systems of semipositone discrete fractional boundary value problems. J. Differ. Equ. Appl. 20(3), 473–491 (2014)
Graef, J.R., Kong, L., Wang, M.: A Chebyshev spectral method for solving Riemann–Liouville fractional boundary value problems. Appl. Math. Comput. 241(1), 140–150 (2014)
Guezane-Lakoud, A., Khaldi, R.: Solvability of a fractional boundary value problem with fractional integral condition. Nonlinear Anal. 75(4), 2692–2700 (2012)
Su, X.: Boundary value problem for a coupled system of nonlinear fractional differential equations. Appl. Math. Lett. 22(1), 64–69 (2009)
Bai, C., Fang, J.: The existence of a positive solution for a singular coupled system of nonlinear fractional differential equations. Appl. Math. Comput. 150(3), 611–621 (2004)
Li, Y., Wei, Z.: Positive solutions for a coupled systems of mixed higher-order nonlinear singular differential equations. Fixed Point Theory. 15(1), 167–178 (2014)
Zhang, Y., Bai, Z., Feng, T.: Existence results for a coupled system of nonlinear fractional three-point boundary value problems at resonance. Comput. Math. Appl. 61(4), 1032–1047 (2011)
Yuan, C.: Multiple positive solutions for (n-1,1) type semipositone conjugate boundary value problems for coupled systems of nonlinear fractional differential equations. Electron. J. Qual. Theory Differ. Equ. 2011(13), 1–12 (2011)
Zhao, Y., Sun, S., Han, Z., Feng, W.: Positive solutions for for a coupled system of nonlinear differential equations of mixed fractional orders. Adv. Differ. Equ. 2011(1), 64–69 (2011)
Chen, Y., Chen, D., Lv, Z.: The existence results for a coupled system of nonlinear fractional differential equations with multi-point boundary conditions. Bull. Iranian Math. Soc. 38(3), 607–624 (2012)
Zhang, X., Zhu, C., Wu, Z.: Solvability for a coupled system of fractional differential equations with impulses at resonance. Bound. Value Probl. 2013(1), 1–23 (2013)
Ahmad, B., Nieto, J.J.: Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions. Comput. Math. Appl. 58(9), 1838–1843 (2009)
Zhu, C., Zhang, X., Wu, Z.: Solvability for a coupled system of fractional differential equations with integral boundary conditions. Taiwanese J. Math. 17(6), 2039–2054 (2013)
Ntouyas, S., Obaid, M.: A coupled system of fractional differential equations with nonlocal integral boundary conditions. Adv. Differ. Equ. 2012(1), 1–8 (2012)
Deimling, K.: Nonlinear Functional Analysis. Springer, Berlin (1985)
Acknowledgments
The authors are grateful to the referees for their careful reading. This research is supported by the Nature Science Foundation of Anhui Provincial Education Department (Grant No. KJ2014A252), the National Natural Science Foundation of China, Tianyuan Foundation (Grant No. 11226119) and the Excellent Youth Foundation of Suzhou University (Grant No. 2014XQNRL001). Conflict of interest The authors declare that they have no competing interests.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Li, Y., Sang, Y. & Zhang, H. Solvability of a coupled system of nonlinear fractional differential equations with fractional integral conditions. J. Appl. Math. Comput. 50, 73–91 (2016). https://doi.org/10.1007/s12190-014-0859-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12190-014-0859-1