Positive solutions for some RiemannLiouville fractional boundary value problems
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 RiemannLiouville 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.
MSC
 34B27
 34A08
 34B18
 34B15
 47N20
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