Fractional boundary value problems with Riemann-Liouville fractional derivatives.

In this paper, by employing two fixed point theorems of a sum operators, we investigate the existence and uniqueness of positive solutions for the following fractional boundary value problems: $-D_{0+}^{\alpha}x(t)=f(t, x(t), x(t))+g(t, x(t))$, $0< t <1$, $1< \alpha<2$, where $D_{0+}^{\alpha}$ is the standard Riemann-Liouville fractional derivative, subject to either the boundary conditions $x(0)=x(1)=0$ or $x(0)=0$, $x(1)=\beta x(\eta)$ with $\eta, \beta\eta^{\alpha-1} \in(0,1)$. We also construct an iterative scheme to approximate the solution. As applications of the main results, two examples are given. [ABSTRACT FROM AUTHOR]