Existence, nonexistence and multiplicity of positive solutions for nonlinear fractional differential equations

In this paper, we consider the existence, nonexistence and multiplicity of positive solutions for nonlinear fractional differential equation boundary-value problem $$\left\{ \begin{array}{@{}l}-D^{\alpha}_{0+}u(t)=f(t,u(t)), \quad t\in0,1]\\3pt]u(0)=u(1)=u''(0)=0\end{array} \right.$$ where 2<????3 is a real number, and $D^{\alpha}_{0+}$ is the Caputo??s fractional derivative, and f:0,1]×0,+??)??0,+??) is continuous. By means of a fixed-point theorem on cones, some existence, nonexistence and multiplicity of positive solutions are obtained.