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Existence and uniqueness of positive solutions for a singular fractional three-point boundary value problem. (English) Zbl 1246.34006
Summary: We investigate the existence and uniqueness of positive solutions for the following singular fractional three-point boundary value problem $D^\alpha_{0^+}u(t) + f(t, u(t)) = 0, 0 < t < 1, u(0) = u'(0) = u''(0) = 0, u''(1) = \beta u''(\eta)$, where $3 < \alpha \leq 4, D^\alpha_{0^+}$ is the standard Riemann-Liouville derivative and $f : (0, 1] \times [0, \infty) \rightarrow [0, \infty)$ with $\lim_{t \rightarrow 0^+} f(t, \cdot) = \infty$ (i.e., $f$ is singular at $t = 0$). Our analysis relies on a fixed point theorem in partially ordered metric spaces.

MSC:
34A08Fractional differential equations
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
34B10Nonlocal and multipoint boundary value problems for ODE
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References:
[1] A. A. Kilbas and J. J. Trujillo, “Differential equations of fractional order: methods, results and problems. I,” Applicable Analysis, vol. 78, no. 1-2, pp. 153-192, 2001. · Zbl 1031.34002 · doi:10.1080/00036810108840931
[2] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science, Amsterdam, The Netherlands, 2006. · Zbl 1092.45003
[3] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993. · Zbl 0789.26002
[4] K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974. · Zbl 0292.26011
[5] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, London, UK, 1999. · Zbl 0924.34008
[6] Z. Bai and H. Lü, “Positive solutions for boundary value problem of nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 311, no. 2, pp. 495-505, 2005. · Zbl 1079.34048 · doi:10.1016/j.jmaa.2005.02.052 · eudml:130751
[7] S. Liang and J. Zhang, “Positive solutions for boundary value problems of nonlinear fractional differential equation,” Nonlinear Analysis A, vol. 71, no. 11, pp. 5545-5550, 2009. · Zbl 1185.26011 · doi:10.1016/j.na.2009.04.045 · eudml:231936
[8] J. Caballero Mena, J. Harjani, and K. Sadarangani, “Existence and uniqueness of positive and nondecreasing solutions for a class of singular fractional boundary value problems,” Boundary Value Problems, vol. 2009, Article ID 421310, 10 pages, 2009. · Zbl 1182.34005 · doi:10.1155/2009/421310 · eudml:45610
[9] S. Liang and J. Zhang, “Existence and uniqueness of strictly nondecreasing and positive solution for a fractional three-point boundary value problem,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1333-1340, 2011. · Zbl 1235.34024 · doi:10.1016/j.camwa.2011.03.073
[10] A. Amini-Harandi and H. Emami, “A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations,” Nonlinear Analysis A, vol. 72, no. 5, pp. 2238-2242, 2010. · Zbl 1197.54054 · doi:10.1016/j.na.2009.10.023
[11] J. J. Nieto and R. Rodríguez-López, “Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations,” Order, vol. 22, no. 3, pp. 223-239, 2005. · Zbl 1095.47013 · doi:10.1007/s11083-005-9018-5