# Positive solutions for some Riemann-Liouville fractional boundary value problems

Volume 9, Issue 7, pp 5093--5106 Publication Date: July 30, 2016
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### Authors

Imed Bachar - Mathematics Department, College of Science, King Saud University, P. O. Box 2455, Riyadh 11451, Saudi Arabia. Habib Mâagli - Department of Mathematics, College of Sciences and Arts, Rabigh Campus, King Abdulaziz University, P. O. Box 344, Rabigh 21911, Saudi Arabia. - Department of Mathematics, Faculte des Sciences de Tunis, Campus Universitaire, 2092 Tunis, Tunisia.

### Abstract

We study the existence and global asymptotic behavior of positive continuous solutions to the following nonlinear fractional boundary value problem $(p_\lambda) \begin{cases} D^\alpha u(t)=\lambda f(t,u(t)),\,\,\,\,\, t\in (0,1),\\ \lim_{t\rightarrow 0^+}t^{2-\alpha} u(t)=\mu, \quad u(1)=\nu, \end{cases}$ where $1 < \alpha\leq 2; D^\alpha$ is the Riemann-Liouville fractional derivative, and $\lambda,\mu$ and $\nu$ are nonnegative constants such that $\mu + \nu > 0$. Our purpose is to give two existence results for the above problem, where $f(t; s)$ is a nonnegative continuous function on $(0; 1)\times[0;\infty)$; nondecreasing with respect to the second variable and satisfying some appropriate integrability condition. Some examples are given to illustrate our existence results.

### Keywords

• Fractional differential equation
• positive solutions
• Green's function
• perturbation arguments
• Schäuder fixed point theorem.

•  34B27
•  34A08
•  34B18
•  34B15
•  47N20

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