## Positive solutions for a class of singular fractional boundary value problems.(English)Zbl 1235.34010

Summary: We investigate the existence and uniqueness of positive solutions for the following singular fractional boundary value problem $D^\alpha_{0^+}u(t)+f(t,u(t))=0,\;0<t<1,\quad u(0)=u(1)=0,$ where $$1<\alpha\leq 2, D^\alpha_{0^+}$$ is the standard Riemann-Liouville differentiation and $$f:(0,1 \times[0,\infty)\to[0,\infty)$$ with $$\lim_{t\to 0^+}f(t,-)= \infty$$ (i.e., $$f$$ is singular at $$t=0)$$. Our analysis relies on a fixed point theorem in partially ordered sets.

### MSC:

 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
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### References:

 [1] 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 [2] Caballero, J.; Harjani, J.; Sadarangani, K., Existence and uniqueness of positive and nondecreasing solutions for a class of singular fractional boundary value problem, Bound. value probl., 2009, (2009), Article ID 421310, 10 pages · Zbl 1182.34005 [3] Campos, L.M.B.C., On the solutions of some simple fractional differential equations, Int. J. math. sci., 13, 481-496, (1990) · Zbl 0711.34019 [4] Delbosco, D.; Rodino, L., Existence and uniqueness of a nonlinear fractional differential equation, J. math. anal. appl., 204, 609-625, (1996) · Zbl 0881.34005 [5] Kilbas, A.A.; Trujillo, J.J., Differential equations of fractional order: methods, results and problems-I, Appl. anal., 78, 153-192, (2001) · Zbl 1031.34002 [6] Li, C.F.; Luo, X.N.; Zhou, Y., Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations, Comput. math. appl., 59, 1363-1375, (2010) · Zbl 1189.34014 [7] Ling, Y.; Ding, S., A class of analytic functions defined by fractional derivation, J. math. anal. appl., 186, 504-513, (1994) · Zbl 0813.30016 [8] Miller, K.S.; Ross, B., An introduction to the fractional calculus and fractional differential equation, (1993), Wiley New York · Zbl 0789.26002 [9] Qui, T.; Bai, Z., Existence of positive solutions for singular fractional differential equations, Electron. J. differential equations, 146, 1-9, (2008) [10] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integral and Derivative. Theory and Applications, Gordon and Breach Science, Yverdon, Switzerland, 1993. · Zbl 0818.26003 [11] Zhang, S., The existence of a positive solution for a nonlinear fractional differential equation, J. math. anal. appl., 252, 804-812, (2000) · Zbl 0972.34004 [12] Zhou, Y.; Jiao, F., Nonlocal Cauchy problem for fractional evolution equations, Nonlinear anal. real world appl., 11, 4465-4475, (2010) · Zbl 1260.34017 [13] Amini-Harandi, A.; Emani, H., A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations, Nonlinear anal., 72, 2238-2242, (2010) · Zbl 1197.54054 [14] Harjani, J.; Sadarangani, K., Fixed point theorems for weakly contractive mappings in partially ordered sets, Nonlinear anal., 71, 3403-3410, (2009) · Zbl 1221.54058 [15] Nieto, J.J.; Rodríguez-López, R., Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order, 22, 223-239, (2005) · Zbl 1095.47013 [16] Nieto, J.J.; Rodríguez-López, R., Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta math. sinica, 23, 2205-2212, (2007) · Zbl 1140.47045 [17] O’Regan, D.; Petrusel, A., Fixed point theorems for generalized contractions in ordered metric spaces, J. math. anal. appl., 341, 1241-1252, (2008) · Zbl 1142.47033
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