Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations. (English) Zbl 1206.34009

Consider the existence of a positive solution for the singular fractional boundary value problem
\[ D^\alpha u(t)+ f(t,u(t),D^\mu u(t))=0,\,u(0)=u(1)=0, \]
where \(1<\alpha<2\), \(\mu>0\) with \(\alpha-\mu\geq 1,\) \(D^\alpha\) is the standard Riemann-Liouville fractional derivative, the function \(f\) is positive, satisfies the Carathéodory conditions on \( [0,1]\times (0,\infty)\times {\mathbb R}\) and \(f(t,x,y)\) is singular at \(x=0\).
The proofs are based on regularization and sequential techniques and the results are obtained by means of fixed point theorem of cone compression type due to [M. A. Krasnosel’skij, Positive solutions of operator equations. Groningen: The Netherlands: P.Noordhoff Ltd. (1964; Zbl 0121.10604)].


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
47N20 Applications of operator theory to differential and integral equations


Zbl 0121.10604
Full Text: DOI


[1] Agarwal, R. P.; Benchohra, M.; Hamani, S., A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl. Math. (2008)
[2] Agarwal, R. P.; Belmekki, M.; Benchohra, M., A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative, Adv. Difference Equ. (2009), Article ID 981728, 47 pp · Zbl 1182.34103
[3] Bai, Z.; Lü, H., Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl., 311, 495-505 (2005) · Zbl 1079.34048
[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] Bonilla, B.; Rivero, M.; Rodriguez-Germá, L.; Trujillo, J. J., Fractional differential equations as alternative models to nonlinear differential equations, Appl. Math. Comput., 187, 79-88 (2007) · Zbl 1120.34323
[6] 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
[7] Guo, D. J.; Lakshmikantham, V., Nonlinear Problems in Abstract Cones, Notes and Reports Math. Sci. Eng., vol. 5 (1988), Academic Press: Academic Press Boston, MA · Zbl 0661.47045
[8] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations (2006), Elsevier B.V.: Elsevier B.V. Amsterdam, The Netherlands · Zbl 1092.45003
[9] Krasnosel’skii, M. A., Positive Solutions of Operator Equations (1964), P. Noordhoff: P. Noordhoff Groningen, The Netherlands · Zbl 0121.10604
[10] Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations (1993), John Wiley & Sons: John Wiley & Sons New York · Zbl 0789.26002
[11] Podlubny, I., Fractional Differential Equations, Math. Sci. Eng., vol. 198 (1999), Academic Press: Academic Press San Diego · Zbl 0918.34010
[12] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives: Theory and Applications (1993), Gordon and Breach Science: Gordon and Breach Science Switzerland · Zbl 0818.26003
[13] Shi, A.; Zhang, S., Upper and lower solutions method and a fractional differential boundary value problem, Electron. J. Qual. Theory Differ. Equ. (2009), Paper No. 30, 13 pp
[14] Su, X., Boundary value problem for a coupled system of nonlinear fractional differential equations, Appl. Math. Lett., 22, 64-69 (2009) · Zbl 1163.34321
[15] Su, X.; Liu, L., Existence of solution for boundary value problem of nonlinear fractional differential equation, Appl. Math. J. Chinese Univ. Ser. B, 22, 3, 291-298 (2007) · Zbl 1150.34005
[16] Yang, A.; Ge, W., Positive solutions for boundary value problems of \(n\)-dimension nonlinear fractional differential system, Bound. Value Probl. (2008), Article ID 437453, 15 pp
[17] Zhang, S., Existence of solutions for a boundary value problem of fractional order, Acta Math. Sci., 26, 220-228 (2006) · Zbl 1106.34010
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.