Geodesic
Uniqueness and existence of positive solutions for singular fractional differential equations
Electronic journal of differential equations, Tome 2014 (2014)
In this article, we study the existence of positive solutions for the singular fractional boundary value problem

$\displaylines{ - D^\alpha u(t) = A f (t, u (t))+\sum_{i=1}^k B_i I^{\beta_i} g_i (t, u(t)) , \quad t \in (0, 1),\cr D^\delta u (0) = 0,\quad D^\delta u (1) = a D^{\frac{\alpha-\delta-1}{2}}(D^\delta u (t))\big|_{t=\xi} }$

where $1\alpha\leq 2, 0\xi \leq 1/2, a \in [0,\infty), 1\alpha-\delta 2, 0\beta_i 1, A,B_i, 1\leq i \leq k$, are real constant, $D^\alpha$ is the Riemann-Liouville fractional derivative of order $\alpha$. By using the Banach's fixed point theorem and Leray-Schauder's alternative, the existence of positive solutions is obtained. At last, an example is given for illustration.
Classification : 34A08, 74H20, 30E25
Keywords: existence of solutions, Banach's fixed point theorem, Leray-Schauder's alternative 
@article{EJDE_2014__2014__a30,
     author = {Nyamoradi,  Nemat and Bashiri,  Tahereh and Vaezpour,  S.Mansour and Baleanu,  Dumitru},
     title = {Uniqueness and existence of positive solutions for singular fractional differential equations},
     journal = {Electronic journal of differential equations},
     year = {2014},
     volume = {2014},
     zbl = {1300.34022},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2014__2014__a30/}
}
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JO  - Electronic journal of differential equations
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%A Baleanu,  Dumitru
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%J Electronic journal of differential equations
%D 2014
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%F EJDE_2014__2014__a30
Nyamoradi,  Nemat; Bashiri,  Tahereh; Vaezpour,  S.Mansour; Baleanu,  Dumitru. Uniqueness and existence of positive solutions for singular fractional differential equations. Electronic journal of differential equations, Tome 2014 (2014). http://geodesic.mathdoc.fr/item/EJDE_2014__2014__a30/