On the existence of positive solutions for a local fractional boundary value problem with an integral boundary condition

Abstract

In this work, we are concerened with the fractional differential equation     
\begin{displaymath}
D^{\alpha}_{0^+} u(t)+f(t,u(s))=0,\quad 1<\alpha\leq 2
\end{displaymath}
where $D^\alpha_{0^+}$ is  the standard Riemann-Liouville fractional  derivative, subject to the local boundary conditions
\begin{displaymath}
u(0)=0,\quad u(1)+\int_0^\eta u(t)dt=0, \quad 0\leq \eta< 1.
\end{displaymath}
We try to obtain the existence of positive solutions by using some fixed point theorems.
\end{abstract}