Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions. (English) Zbl 1205.34003

The paper mainly concerns the existence of a solution to a class of boundary value problems for coupled systems of nonlinear fractional differential equations. The article is very similar to the reference [X. Su, Appl. Math. Lett. 22, No. 1, 64–69 (2009; Zbl 1163.34321)]. The novelty is the condition on the boundary which introduces more difficulty in the analysis of the existence of solutions. In particular, the result is obtained by the Schauder fixed point theorem to an operator defined on a suitable function space.


34A08 Fractional ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B27 Green’s functions for ordinary differential equations
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.


Zbl 1163.34321


Full Text: DOI


[1] Bai, Z.; Lü, H., Positive solutions for boundary value problems of nonlinear fractional differential equations, J. math. anal. appl., 311, 495-505, (2005) · Zbl 1079.34048
[2] Chang, Y.K.; Nieto, J.J., Some new existence results for fractional differential inclusions with boundary conditions, Math. comput. modelling, 49, 605-609, (2009) · Zbl 1165.34313
[3] Deng, W., Numerical algorithm for the time fractional fokker – planck equation, J. comput. phys., 227, 1510-1522, (2007) · Zbl 1388.35095
[4] Ibrahim, R.W.; Darus, M., Subordination and superordination for univalent solutions for fractional differential equations, J. math. anal. appl., 345, 871-879, (2008) · Zbl 1147.30009
[5] Ladaci, S.; Loiseau, J.L.; Charef, A., Fractional order adaptive high-gain controllers for a class of linear systems, Commun. nonlinear sci. numer. simul., 13, 707-714, (2008) · Zbl 1221.93128
[6] Rida, S.Z.; El-Sherbiny, H.M.; Arafa, A.A.M., On the solution of the fractional nonlinear Schrödinger equation, Phys. lett. A, 372, 553-558, (2008) · Zbl 1217.81068
[7] Yang, A.; Ge, W., Positive solutions for boundary value problems of N-dimension nonlinear fractional differential system, Bound. value probl., 2008, (2008), 15 pages. Article ID 437453 · Zbl 1167.34314
[8] Su, X.; Zhang, S., Solutions to boundary-value problems for nonlinear differential equations of fractional order, Electron. J. differential equations, 2009, 26, 115, (2009)
[9] Ahmad, B.; Nieto, J.J., Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions, Bound. value probl., 2009, (2009), 11 pages. Article ID 708576 · Zbl 1167.45003
[10] Ahmad, B.; Nieto, J.J., Existence of solutions for nonlocal boundary value problems of higher-order nonlinear fractional differential equations, Abstr. appl. anal., 2009, (2009), 9 pages. Article ID 494720 · Zbl 1186.34009
[11] 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) · Zbl 1061.34001
[12] Chen, Y.; An, H., Numerical solutions of coupled Burgers equations with time and space fractional derivatives, Appl. math. comput., 200, 87-95, (2008) · Zbl 1143.65102
[13] Gafiychuk, V.; Datsko, B.; Meleshko, V., Mathematical modeling of time fractional reaction – diffusion systems, J. comput. appl. math., 220, 215-225, (2008) · Zbl 1152.45008
[14] Gejji, V.D., Positive solutions of a system of non-autonomous fractional differential equations, J. math. anal. appl., 302, 56-64, (2005) · Zbl 1064.34004
[15] Lazarević, M.P., Finite time stability analysis of PD^{α} fractional control of robotic time-delay systems, Mech. res. comm., 33, 269-279, (2006) · Zbl 1192.70008
[16] Su, X., Boundary value problem for a coupled system of nonlinear fractional differential equations, Appl. math. lett., 22, 64-69, (2009) · Zbl 1163.34321
[17] Gafiychuk, V.; Datsko, B.; Meleshko, V.; Blackmore, D., Analysis of the solutions of coupled nonlinear fractional reaction – diffusion equations, Chaos solitons fractals, 41, 1095-1104, (2009) · Zbl 1198.35123
[18] Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J., ()
[19] Podlubny, I., Fractional differential equations, (1999), Academic Press San Diego · Zbl 0918.34010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.