Abstract
In this paper, we discuss by means of a fixed point theorem, the existence of positive solutions of a system of nonlinear Caputo fractional differential equations with integral boundary conditions. An example is given to illustrate the main results.
Similar content being viewed by others
References
Podlubny, I.: Fractional Differential Equations Mathematics in Sciences and Engineering. Academic Press, New York (1999)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
Lakshmikantham, V., Vatsala, A.S.: Basic theory of fractional differential equations. Nonlinear Anal. 69, 2677–2682 (2008)
Ashyralyev, A.: A note on fractional derivatives and fractional powers of operators. J. Math. Anal. Appl. 357, 232–236 (2009)
Tarasov, V.E.: Fractional derivative as fractional power of derivative. Int. J. Math. 18, 281–299 (2007)
Ashyralyev, A., Dal, F., Pınar, Z.: A note on the fractional hyperbolic differential and difference equations. Appl. Math. Comput. 217(9), 4654–4664 (2011)
Agarwal, R.P., O’Regan, D., Stanek, S.: Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations. J. Math. Anal. Appl. 371, 57–68 (2010)
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, 1838–1843 (2009)
Ahmad, B., Nieto, J.J.: Class of differential equations of fractional order with multi-point boundary conditions. Georgian Math. J. 21(3), 243–248 (2014)
Ahmad, B., Alsaedi, A.: Existence and uniqueness of solutions for coupled systems of higher-order nonlinear fractional differential equations. Fixed Point Theory Appl. Article ID 364560 (2010). doi:10.1155/2010/364560
Ahmad, B., Nieto, J.J.: Riemann–Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions. Bound. Value Probl. 36, 1–9 (2011)
Ahmad, B.: Nonlinear fractional differential equations with anti-periodic type fractional boundary conditions. Differ. Equ. Dyn. Syst. 21(4), 387–401 (2013)
Feng, M., Zhang, X., Ge, W.: New existence results for higher-order nonlinear fractional differential equation with integral boundary conditions. Bound. Value Probl. 2011, Art ID 720702 (2011)
Chai, Y., Chen, L., Wu, R.: Inverse projective synchronization between two different hyperchaotic systems with fractional order. J. Appl. Math. 2012, Article ID 762807 (2012)
Bai, C., Fang, J.: The existence of a positive solution for a singular coupled system of nonlinear fractional differential equations. Appl. Math. Comput. 150, 611–621 (2004)
Ntouyas, S.K., Obaid, M.: A coupled system of fractional differential equations with nonlocal integral boundary conditions. Adv. Differ. Equ. Article ID 130 (2012). doi:10.1186/1687-1847-2012-130
Rehman, M., Khan, R.: A note on boundary value problems for a coupled system of fractional differential equations. Comput. Math. Appl. 61, 2630–2637 (2011)
Salem, H.: On the existence of continuous solutions for a singular system of nonlinear fractional differential equations. Appl. Math. Comput. 198, 445–452 (2008)
Su, X.: Existence of solution of boundary value problem for coupled system of fractional differential equations. Eng. Math. 26, 134–137 (2009)
Su, X.: Boundary value problem for a coupled system of nonlinear fractional differential equations. Appl. Math. Lett. 22, 64–69 (2009)
Ashyralyev, A, Sharifov, Y.A.: Existence and uniqueness of solutions for the system of nonlinear fractional differential equations with nonlocal and integral boundary conditions. Abstr. Appl. Anal. Article ID 594802 (2012)
Guezane-Lakoud, A., Khaldi, R.: Solvability of a fractional boundary value problem with fractional integral condition. Nonlinear Anal. 75, 2692–2700 (2012)
Guezane-Lakoud, A., Khaldi, R.: Positive solution to a higher order fractional boundary value problem with fractional integral condition. Rom. J. Math. Comput. Sci. 2(1), 41–54 (2012)
Webb, J.R.L., Infante, G.: Positive solutions of nonlocal boundary value problems involving integral conditions. NoDEA Nonlinear Differ. Equ. Appl. 15(1–2), 45–67 (2008)
Zhao, J., Wang, P., Ge, W.: Existence and nonexistence of positive solutions for a class of third order BVP with integral boundary conditions in Banach spaces. Commun. Nonlinear Sci. Numer. Simulat. 16, 402–413 (2011)
Infante, G., Pietramala, P.: Existence and multiplicity of non-negative solutions for systems of perturbed Hammerstein integral equations. Nonlinear Anal. 71, 1301–1310 (2009)
Guezane-Lakoud, A., Khaldi, R.: Solvability of a two-point fractional boundary value problem. J. Nonlinear Sci. Appl. 5, 64–73 (2012)
Guezane-Lakoud, A., Khaldi, R.: Solvability of a three-point fractional nonlinear boundary value problem. Differ. Equ. Dyn. Syst. 20, 395–403 (2012)
Graef, J.R., Henderson, J.Y., Yang, B.O.: Positive solutions of a nonlinear higher order boundary value problem. Electron J. Differ. Equ. 45, 1–10 (2007)
Henderson, J., Ntouyas, S.K., Purnaras, I.K.: Positive solutions for systems of generalized three-point nonlinear boundary value problems. Comment. Math. Univ. Carolin. 49, 79–91 (2008)
Wang, J., Xiang, H., Liu, Z.: Positive solution to nonzero boundary values problem for a coupled system of nonlinear fractional differential equations. Int. J. Differ. Equ. Article ID 186928 (2010). doi:10.1155/2010/186928
Guo, D.J., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones Vol 5 of Notes and Reports in Mathematics in Science and Engineering. Academic Press, Boston (1988)
Kwong, M.K.: On Krasnoselskii’s cone fixed point theorem. Fixed Theory Appl. Article ID164537 2008)
Acknowledgments
This work was supported by TUBITAK under the Project Number B.14.2.TBT.0.06.01.03.220.01-106923.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Guezane-Lakoud, A., Ashyralyev, A. Positive Solutions for a System of Fractional Differential Equations with Nonlocal Integral Boundary Conditions. Differ Equ Dyn Syst 25, 519–526 (2017). https://doi.org/10.1007/s12591-015-0255-9
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12591-015-0255-9