Existence and multiplicity of positive solutions for singular fractional boundary value problems. (English) Zbl 1247.34006

Summary: We discuss the existence and multiplicity of positive solutions for the singular fractional boundary value problem \[ D^{\alpha}_{0+}u(t)+f((t,u(t),D^{\nu}_{0+}u(t),D^{\mu}_{0+}u(t))=0, \]
\[ u(0)=u'(0)=u''(0)=u''(1)=0, \] where \(3<\alpha\leq 4\), \(0<\nu\leq 1\), \(1<\mu \leq 2\), \(D^{\alpha}_{0+}\) is the standard Riemann-Liouville fractional derivative, \(f\) is a Carathédory function and \(f(t,x,y,z)\) is singular at the value \(0\) of its arguments \(x,y,z\). By means of a fixed point theorem, the existence and multiplicity of positive solutions are obtained.


34A08 Fractional ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
45J05 Integro-ordinary differential equations
Full Text: DOI


[1] Agarwal, R.P.; O’Regan, D.; Wong, P.J.Y., Positive solutions of differential, difference and integral equations, (1999), Kluwer Academic Publishers Netherlands · Zbl 0923.39002
[2] Agarwal, R.P.; Zhou, Y.; He, Y., Existence of fractional neutral functional differential equations, Comput. math. appl., 59, 3, 1095-1100, (2010) · Zbl 1189.34152
[3] Bai, Z., On positive solutions of a nonlocal fractional boundary value problem, Nonlinear anal., 72, 916-924, (2010) · Zbl 1187.34026
[4] Benchohra, M.; Hamani, S.; Ntouyas, S.K., Boundary value problems for differential equations with fractional order and nonlocal conditions, Nonlinear anal., 71, 2391-2396, (2009) · Zbl 1198.26007
[5] Chang, Y.; Nieto, J.J., Some new existence results for fractional differential inclusions with boundary conditions, Math. comput. modelling, 49, 605-609, (2009) · Zbl 1165.34313
[6] Chen, F.; Zhou, Y., Attractivity of fractional functional differential equations, Comput. math. appl., 62, 3, 1359-1369, (2011) · Zbl 1228.34017
[7] Chen, F.; Nieto, J.J.; Zhou, Y., Global attractivity for nonlinear fractional differential equations, Nonlinear anal., 13, 287-298, (2012) · Zbl 1238.34011
[8] Daftardar-Gejji, V.; Bhalekar, S., Boundary value problems for multi-term fractional differential equations, J. math. anal. appl., 345, 754-765, (2008) · Zbl 1151.26004
[9] Diethelm, K., ()
[10] Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J., Theory and applications of fractional differential equations, (2006), Elsevier B.V. Amsterdam, The Netherlands · Zbl 1092.45003
[11] Krasnosel’skii, M.A., Positive solutions of operator equations, (1964), P. Noordhoff Groningen, The Netherlands · Zbl 0121.10604
[12] Lakshmikantham, V.; Vatsala, A.S., Basic theory of fractional differential equations, Nonlinear anal., 69, 2677-2682, (2008) · Zbl 1161.34001
[13] Liu, X.; Jia, M., Multiple solutions for fractional differential equations with nonlinear boundary conditions, Comput. math. appl., 59, 2880-2886, (2010) · Zbl 1193.34037
[14] Podlubny, I., ()
[15] Wang, J.; Zhou, Y.; Wei, W., A class of fractional delay nonlinear integrodifferential controlled systems in Banach spaces, Commun. nonlinear sci. numer. simul., 16, 4049-4059, (2011) · Zbl 1223.45007
[16] Wang, J.; Zhou, Y., Analysis of nonlinear fractional control systems in Banach spaces, Nonlinear anal. TMA, 74, 5929-5942, (2011) · Zbl 1223.93059
[17] Wang, J.; Zhou, Y., Existence and controllability results for fractional semilinear differential inclusions, Nonlinear anal. RWA, 12, 3642-3653, (2011) · Zbl 1231.34108
[18] Wang, J.; Wei, W.; Zhou, Y., Fractional finite time delay evolution systems and optimal controls in infinite dimensional spaces, J. dyn. control syst., 17, 515-535, (2011) · Zbl 1241.26005
[19] Wang, J.; Zhou, Y.; Wei, W.; Xu, H., Nonlocal problems for fractional integrodifferential equations via fractional operators and optimal controls, Comput. math. appl., 62, 1427-1441, (2011) · Zbl 1228.45015
[20] Wang, J.; Zhou, Y., Study of an approximation process of time optimal control for fractional evolution systems in Banach spaces, Adv. difference equ., 2011, 1-16, (2011), Article ID 385324 · Zbl 1222.49006
[21] Wei, Z.; Dong, W.; Che, J., Periodic boundary value problems for fractional differential equations involving a riemann – liouville fractional derivative, Nonlinear anal. TMA, 73, 3232-3238, (2010) · Zbl 1202.26017
[22] Zhou, Y.; Jiao, F.; Li, J., Existence and uniqueness for fractional neutral differential equations with infinite delay, Nonlinear anal. TMA, 71, 3249-3256, (2009) · Zbl 1177.34084
[23] Bai, Z.; Lü, H., Positive solutions of boundary value problems of nonlinear fractional differential equation, J. math. anal. appl., 311, 495-505, (2005) · Zbl 1079.34048
[24] Bai, Z.; Qiu, T., Existence of positive solution for singular fractional differential equation, Appl. math. comput., 215, 2761-2767, (2009) · Zbl 1185.34004
[25] Staněk, S., The existence of positive solutions of singular fractional boundary value problems, Comput. math. appl., 62, 3, 1379-1388, (2011) · Zbl 1228.34020
[26] Xu, X.; Jiang, D.; Yuan, C., Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation, Nonlinear anal., 71, 4676-4688, (2009) · Zbl 1178.34006
[27] Zhang, S., Positive solutions to singular boundary value problem for nonlinear fractional differential equation, Comput. math. appl., 59, 1300-1309, (2010) · Zbl 1189.34050
[28] Y. Zhao, S. Sun, Z. Han, M. Zhang, Existence on positive solutions for boundary value problems of singular nonlinear fractional differential equations, in: 2010 Sixth IEEE/ASME International Conference on Mechatronic and Embedded Systems and Applications, Qingdao, 2010, pp. 480-485.
[29] Agarwal, R.P.; O’Regan, D.; Staněk, S., Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations, J. math. anal. appl., 371, 57-68, (2010) · Zbl 1206.34009
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