Electron. J. Diff. Equ., Vol. 2014 (2014), No. 130, pp. 1-13.

Uniqueness and existence of positive solutions for singular fractional differential equations

Nemat Nyamoradi, Tahereh Bashiri, S. Mansour Vaezpour, Dumitru Baleanu

Abstract:
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.

Submitted August 21, 2013. Published June 6, 2014.
Math Subject Classifications: 34A08, 74H20,30E25.
Key Words: Existence of solutions; Banach’s fixed point theorem; Leray-Schauder’s alternative.

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Nemat Nyamoradi
Department of Mathematics, Faculty of Sciences
Razi University, 67149 Kermanshah, Iran
email: nyamoradi@razi.ac.ir
Tahereh Bashiri
Department of Mathematics and Computer Sciences
Amirkabir University of Technology, Tehran, Iran
email: t.bashiri@aut.ac.ir
S. Mansour Vaezpour
Department of Mathematics and Computer Sciences
Amirkabir University of Technology, Tehran, Iran
email: vaez@aut.ac.ir
Dumitru Baleanu
Department of Mathematics and Computer Sciences
Faculty of Art and Sciences, Cankaya University
06530 Ankara, Turkey
email: dumitru@cankaya.edu.tr

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